Term Rewriting System R:
[x, y, z]
minus(0) -> 0
minus(minus(x)) -> x
+(x, 0) -> x
+(0, y) -> y
+(minus(1), 1) -> 0
+(x, minus(y)) -> minus(+(minus(x), y))
+(x, +(y, z)) -> +(+(x, y), z)
+(minus(+(x, 1)), 1) -> minus(x)

Termination of R to be shown.

`   R`
`     ↳Removing Redundant Rules`

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

minus(0) -> 0
minus(minus(x)) -> x
+(minus(1), 1) -> 0

where the Polynomial interpretation:
 POL(0) =  0 POL(1) =  0 POL(minus(x1)) =  1 + x1 POL(+(x1, x2)) =  x1 + 2·x2
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:

+(x, 0) -> x
+(0, y) -> y

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

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

+(x, +(y, z)) -> +(+(x, y), z)
+(minus(+(x, 1)), 1) -> minus(x)

where the Polynomial interpretation:
 POL(1) =  0 POL(minus(x1)) =  x1 POL(+(x1, x2)) =  1 + x1 + 2·x2
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`
`             ↳RRRPolo`
`             ...`
`               →TRS4`
`                 ↳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`
`             ↳RRRPolo`
`             ...`
`               →TRS5`
`                 ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

+'(x, minus(y)) -> +'(minus(x), y)

Furthermore, R contains one SCC.

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

Dependency Pair:

+'(x, minus(y)) -> +'(minus(x), y)

Rule:

+(x, minus(y)) -> minus(+(minus(x), y))

Strategy:

innermost

As we are in the innermost case, we can delete all 1 non-usable-rules.

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

Dependency Pair:

+'(x, minus(y)) -> +'(minus(x), y)

Rule:

none

Strategy:

innermost

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

which lead(s) to this/these maximal multigraph(s):
{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:
minus(x1) -> minus(x1)

We obtain no new DP problems.

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