Term Rewriting System R:
[x, y, z]
+(*(x, y), *(x, z)) -> *(x, +(y, z))
+(+(x, y), z) -> +(x, +(y, z))
+(*(x, y), +(*(x, z), u)) -> +(*(x, +(y, z)), u)

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

+'(*(x, y), *(x, z)) -> +'(y, z)
+'(+(x, y), z) -> +'(x, +(y, z))
+'(+(x, y), z) -> +'(y, z)
+'(*(x, y), +(*(x, z), u)) -> +'(*(x, +(y, z)), u)
+'(*(x, y), +(*(x, z), u)) -> +'(y, z)

Furthermore, R contains one SCC. 


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


Dependency Pairs:

+'(+(x, y), z) -> +'(y, z)
+'(*(x, y), +(*(x, z), u)) -> +'(y, z)
+'(+(x, y), z) -> +'(x, +(y, z))
+'(*(x, y), *(x, z)) -> +'(y, z)


Rules:


+(*(x, y), *(x, z)) -> *(x, +(y, z))
+(+(x, y), z) -> +(x, +(y, z))
+(*(x, y), +(*(x, z), u)) -> +(*(x, +(y, z)), u)


Strategy:

innermost




We number the DPs as follows:
  1. +'(+(x, y), z) -> +'(y, z)
  2. +'(*(x, y), +(*(x, z), u)) -> +'(y, z)
  3. +'(+(x, y), z) -> +'(x, +(y, z))
  4. +'(*(x, y), *(x, z)) -> +'(y, z)
and get the following Size-Change Graph(s):
{1} , {1}
1>1
2=2
{4, 2} , {4, 2}
1>1
2>2
{3} , {3}
1>1

which lead(s) to this/these maximal multigraph(s):
{1} , {1}
1>1
2=2
{4, 2} , {4, 2}
1>1
2>2
{4, 2} , {3}
1>1
{3} , {4, 2}
1>1
{1} , {4, 2}
1>1
2>2
{4, 2} , {1}
1>1
2>2
{1} , {4, 2}
1>1
{4, 2} , {4, 2}
1>1
{3} , {1}
1>1
{1} , {1}
1>1
2>2
{4, 2} , {1}
1>1
{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:
*(x1, x2) -> *(x1, x2)

We obtain no new DP problems.

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