Term Rewriting System R:
[x, y]
*(x, *(minus(y), y)) -> *(minus(*(y, y)), x)

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC. 


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


Dependency Pair:

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


Rule:


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


Strategy:

innermost




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


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


Dependency Pair:

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


Rule:

none


Strategy:

innermost




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

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

We obtain no new DP problems.

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