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

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

*'(x, *(y, z)) -> *'(otimes(x, y), z)
*'(+(x, y), z) -> *'(x, z)
*'(+(x, y), z) -> *'(y, z)
*'(x, oplus(y, z)) -> *'(x, y)
*'(x, oplus(y, z)) -> *'(x, z)

Furthermore, R contains one SCC. 


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


Dependency Pairs:

*'(x, oplus(y, z)) -> *'(x, z)
*'(+(x, y), z) -> *'(y, z)
*'(+(x, y), z) -> *'(x, z)
*'(x, oplus(y, z)) -> *'(x, y)
*'(x, *(y, z)) -> *'(otimes(x, y), z)


Rules:


*(x, *(y, z)) -> *(otimes(x, y), z)
*(1, y) -> y
*(+(x, y), z) -> oplus(*(x, z), *(y, z))
*(x, oplus(y, z)) -> oplus(*(x, y), *(x, z))





We number the DPs as follows:
  1. *'(x, oplus(y, z)) -> *'(x, z)
  2. *'(+(x, y), z) -> *'(y, z)
  3. *'(+(x, y), z) -> *'(x, z)
  4. *'(x, oplus(y, z)) -> *'(x, y)
  5. *'(x, *(y, z)) -> *'(otimes(x, y), z)
and get the following Size-Change Graph(s):
{4, 1} , {4, 1}
1=1
2>2
{3, 2} , {3, 2}
1>1
2=2
{5} , {5}
2>2

which lead(s) to this/these maximal multigraph(s):
{5} , {5}
2>2
{3, 2} , {3, 2}
1>1
2=2
{4, 1} , {4, 1}
1=1
2>2
{4, 1} , {3, 2}
1>1
2>2
{4, 1} , {5}
2>2
{3, 2} , {4, 1}
1>1
2>2
{5} , {4, 1}
2>2
{3, 2} , {4, 1}
2>2
{5} , {3, 2}
2>2
{4, 1} , {4, 1}
2>2
{4, 1} , {4, 1}
1>1
2>2
{3, 2} , {3, 2}
1>1
2>2
{3, 2} , {3, 2}
2>2
{4, 1} , {3, 2}
2>2

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
otimes(x1, x2) -> otimes(x1, x2)
*(x1, x2) -> *(x1, x2)
oplus(x1, x2) -> oplus(x1, x2)
+(x1, x2) -> +(x1, x2)

We obtain no new DP problems.

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