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

Innermost Termination of R to be shown.

`   R`
`     ↳Removing Redundant Rules`

Removing the following rules from R which fullfill a polynomial ordering:

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

where the Polynomial interpretation:
 POL(c(x1, x2)) =  x1 + x2 POL(g(x1)) =  x1 POL(s(x1)) =  1 + x1 POL(f(x1)) =  x1
was used.

Not all Rules of R can be deleted, so we still have to regard a part of R.

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

R contains the following Dependency Pairs:

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

Furthermore, R contains two SCCs.

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

Dependency Pair:

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

Rules:

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

Strategy:

innermost

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

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

Dependency Pair:

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

Rule:

none

Strategy:

innermost

We number the DPs as follows:
1. G(c(x, s(y))) -> G(c(s(x), 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, x2) -> c(x2)
s(x1) -> s(x1)

We obtain no new DP problems.

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

Dependency Pair:

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

Rules:

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

Strategy:

innermost

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

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

Dependency Pair:

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

Rule:

none

Strategy:

innermost

We number the DPs as follows:
1. F(c(s(x), y)) -> F(c(x, 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, x2) -> c(x1)
s(x1) -> s(x1)

We obtain no new DP problems.

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