Term Rewriting System R:
[x, y, z]
max(L(x)) -> x
max(N(L(0), L(y))) -> y
max(N(L(s(x)), L(s(y)))) -> s(max(N(L(x), L(y))))
max(N(L(x), N(y, z))) -> max(N(L(x), L(max(N(y, z)))))

Termination of R to be shown.

`   R`
`     ↳Removing Redundant Rules`

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

max(N(L(0), L(y))) -> y

where the Polynomial interpretation:
 POL(0) =  0 POL(L(x1)) =  x1 POL(N(x1, x2)) =  1 + x1 + x2 POL(max(x1)) =  x1 POL(s(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`
`         ↳Removing Redundant Rules`

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

max(N(L(s(x)), L(s(y)))) -> s(max(N(L(x), L(y))))

where the Polynomial interpretation:
 POL(L(x1)) =  x1 POL(N(x1, x2)) =  x1 + x2 POL(max(x1)) =  x1 POL(s(x1)) =  1 + x1
was used.

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

`   R`
`     ↳RRRPolo`
`       →TRS2`
`         ↳RRRPolo`
`           →TRS3`
`             ↳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`
`     ↳RRRPolo`
`       →TRS2`
`         ↳RRRPolo`
`           →TRS3`
`             ↳OC`
`             ...`
`               →TRS4`
`                 ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

MAX(N(L(x), N(y, z))) -> MAX(N(L(x), L(max(N(y, z)))))
MAX(N(L(x), N(y, z))) -> MAX(N(y, z))

Furthermore, R contains one SCC.

`   R`
`     ↳RRRPolo`
`       →TRS2`
`         ↳RRRPolo`
`           →TRS3`
`             ↳OC`
`             ...`
`               →DP Problem 1`
`                 ↳Usable Rules (Innermost)`

Dependency Pair:

MAX(N(L(x), N(y, z))) -> MAX(N(y, z))

Rules:

max(N(L(x), N(y, z))) -> max(N(L(x), L(max(N(y, z)))))
max(L(x)) -> x

Strategy:

innermost

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

`   R`
`     ↳RRRPolo`
`       →TRS2`
`         ↳RRRPolo`
`           →TRS3`
`             ↳OC`
`             ...`
`               →DP Problem 2`
`                 ↳Size-Change Principle`

Dependency Pair:

MAX(N(L(x), N(y, z))) -> MAX(N(y, z))

Rule:

none

Strategy:

innermost

We number the DPs as follows:
1. MAX(N(L(x), N(y, z))) -> MAX(N(y, z))
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:
L(x1) -> L(x1)
N(x1, x2) -> N(x1, x2)

We obtain no new DP problems.

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