Term Rewriting System R:
[x, y, z]
*(x, 1) -> x
*(1, y) -> y
*(i(x), x) -> 1
*(x, i(x)) -> 1
*(x, *(y, z)) -> *(*(x, y), z)
*(*(x, y), i(y)) -> x
*(*(x, i(y)), y) -> x
*(k(x, y), k(y, x)) -> 1
*(*(i(x), k(y, z)), x) -> k(*(*(i(x), y), x), *(*(i(x), z), x))
i(1) -> 1
i(i(x)) -> x
i(*(x, y)) -> *(i(y), i(x))
k(x, 1) -> 1
k(x, x) -> 1
k(*(x, i(y)), *(y, i(x))) -> 1

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

*'(x, *(y, z)) -> *'(*(x, y), z)
*'(x, *(y, z)) -> *'(x, y)
*'(*(i(x), k(y, z)), x) -> K(*(*(i(x), y), x), *(*(i(x), z), x))
*'(*(i(x), k(y, z)), x) -> *'(*(i(x), y), x)
*'(*(i(x), k(y, z)), x) -> *'(i(x), y)
*'(*(i(x), k(y, z)), x) -> *'(*(i(x), z), x)
*'(*(i(x), k(y, z)), x) -> *'(i(x), z)
I(*(x, y)) -> *'(i(y), i(x))
I(*(x, y)) -> I(y)
I(*(x, y)) -> I(x)

Furthermore, R contains two SCCs. 


   R
     ↳DPs
       →DP Problem 1
         ↳Usable Rules (Innermost)
       →DP Problem 2
         ↳Polo


Dependency Pair:

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


Rules:


*(x, 1) -> x
*(1, y) -> y
*(i(x), x) -> 1
*(x, i(x)) -> 1
*(x, *(y, z)) -> *(*(x, y), z)
*(*(x, y), i(y)) -> x
*(*(x, i(y)), y) -> x
*(k(x, y), k(y, x)) -> 1
*(*(i(x), k(y, z)), x) -> k(*(*(i(x), y), x), *(*(i(x), z), x))
i(1) -> 1
i(i(x)) -> x
i(*(x, y)) -> *(i(y), i(x))
k(x, 1) -> 1
k(x, x) -> 1
k(*(x, i(y)), *(y, i(x))) -> 1


Strategy:

innermost




As we are in the innermost case, we can delete all 15 non-usable-rules.


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
           →DP Problem 3
             ↳Size-Change Principle
       →DP Problem 2
         ↳Polo


Dependency Pair:

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


Rule:

none


Strategy:

innermost




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

which lead(s) to this/these maximal multigraph(s):
{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. 


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
       →DP Problem 2
         ↳Polynomial Ordering


Dependency Pairs:

*'(*(i(x), k(y, z)), x) -> *'(i(x), z)
*'(*(i(x), k(y, z)), x) -> *'(*(i(x), z), x)
*'(x, *(y, z)) -> *'(x, y)
*'(*(i(x), k(y, z)), x) -> *'(i(x), y)
*'(*(i(x), k(y, z)), x) -> *'(*(i(x), y), x)
*'(x, *(y, z)) -> *'(*(x, y), z)


Rules:


*(x, 1) -> x
*(1, y) -> y
*(i(x), x) -> 1
*(x, i(x)) -> 1
*(x, *(y, z)) -> *(*(x, y), z)
*(*(x, y), i(y)) -> x
*(*(x, i(y)), y) -> x
*(k(x, y), k(y, x)) -> 1
*(*(i(x), k(y, z)), x) -> k(*(*(i(x), y), x), *(*(i(x), z), x))
i(1) -> 1
i(i(x)) -> x
i(*(x, y)) -> *(i(y), i(x))
k(x, 1) -> 1
k(x, x) -> 1
k(*(x, i(y)), *(y, i(x))) -> 1


Strategy:

innermost




The following dependency pairs can be strictly oriented:

*'(*(i(x), k(y, z)), x) -> *'(i(x), z)
*'(*(i(x), k(y, z)), x) -> *'(*(i(x), z), x)
*'(*(i(x), k(y, z)), x) -> *'(i(x), y)
*'(*(i(x), k(y, z)), x) -> *'(*(i(x), y), x)


Additionally, the following usable rules w.r.t. the implicit AFS can be oriented:

i(i(x)) -> x
i(1) -> 1
i(*(x, y)) -> *(i(y), i(x))
*(*(x, i(y)), y) -> x
*(*(x, y), i(y)) -> x
*(x, *(y, z)) -> *(*(x, y), z)
*(x, 1) -> x
*(x, i(x)) -> 1
*(i(x), x) -> 1
*(*(i(x), k(y, z)), x) -> k(*(*(i(x), y), x), *(*(i(x), z), x))
*(k(x, y), k(y, x)) -> 1
*(1, y) -> y
k(*(x, i(y)), *(y, i(x))) -> 1
k(x, x) -> 1
k(x, 1) -> 1


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(i(x1))=  x1  
  POL(*'(x1, x2))=  1 + x1 + x1·x2 + x2  
  POL(1)=  0  
  POL(*(x1, x2))=  x1 + x1·x2 + x2  
  POL(k(x1, x2))=  1 + x1 + x2  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
       →DP Problem 2
         ↳Polo
           →DP Problem 4
             ↳Dependency Graph


Dependency Pairs:

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


Rules:


*(x, 1) -> x
*(1, y) -> y
*(i(x), x) -> 1
*(x, i(x)) -> 1
*(x, *(y, z)) -> *(*(x, y), z)
*(*(x, y), i(y)) -> x
*(*(x, i(y)), y) -> x
*(k(x, y), k(y, x)) -> 1
*(*(i(x), k(y, z)), x) -> k(*(*(i(x), y), x), *(*(i(x), z), x))
i(1) -> 1
i(i(x)) -> x
i(*(x, y)) -> *(i(y), i(x))
k(x, 1) -> 1
k(x, x) -> 1
k(*(x, i(y)), *(y, i(x))) -> 1


Strategy:

innermost




Using the Dependency Graph the DP problem was split into 1 DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
       →DP Problem 2
         ↳Polo
           →DP Problem 4
             ↳DGraph
             ...
               →DP Problem 5
                 ↳Usable Rules (Innermost)


Dependency Pair:

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


Rules:


*(x, 1) -> x
*(1, y) -> y
*(i(x), x) -> 1
*(x, i(x)) -> 1
*(x, *(y, z)) -> *(*(x, y), z)
*(*(x, y), i(y)) -> x
*(*(x, i(y)), y) -> x
*(k(x, y), k(y, x)) -> 1
*(*(i(x), k(y, z)), x) -> k(*(*(i(x), y), x), *(*(i(x), z), x))
i(1) -> 1
i(i(x)) -> x
i(*(x, y)) -> *(i(y), i(x))
k(x, 1) -> 1
k(x, x) -> 1
k(*(x, i(y)), *(y, i(x))) -> 1


Strategy:

innermost




As we are in the innermost case, we can delete all 15 non-usable-rules.


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
       →DP Problem 2
         ↳Polo
           →DP Problem 4
             ↳DGraph
             ...
               →DP Problem 6
                 ↳Size-Change Principle


Dependency Pair:

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


Rule:

none


Strategy:

innermost




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

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:
*(x1, x2) -> *(x1, x2)

We obtain no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:07 minutes