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)

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC. 


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


Dependency Pairs:

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


Rules:


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)


Strategy:

innermost




As we are in the innermost case, we can delete all 6 non-usable-rules.


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
           →DP Problem 2
             ↳Size-Change Principle


Dependency Pairs:

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


Rules:


minus(0) -> 0
minus(minus(x)) -> x


Strategy:

innermost




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

which lead(s) to this/these maximal multigraph(s):
{1} , {1}
1=1
2>2
{1} , {2}
2>2
{2} , {1}
2>2
{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:
+(x1, x2) -> +(x1, x2)

We obtain no new DP problems.

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