Term Rewriting System R:
[x, y, z]
f(f(a, a), x) -> f(a, f(b, f(a, x)))
f(x, f(y, z)) -> f(f(x, y), z)

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

F(f(a, a), x) -> F(a, f(b, f(a, x)))
F(f(a, a), x) -> F(b, f(a, x))
F(f(a, a), x) -> F(a, x)
F(x, f(y, z)) -> F(f(x, y), z)
F(x, f(y, z)) -> F(x, y)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Polynomial Ordering


Dependency Pairs:

F(f(a, a), x) -> F(a, x)
F(x, f(y, z)) -> F(x, y)
F(f(a, a), x) -> F(b, f(a, x))
F(x, f(y, z)) -> F(f(x, y), z)
F(f(a, a), x) -> F(a, f(b, f(a, x)))


Rules:


f(f(a, a), x) -> f(a, f(b, f(a, x)))
f(x, f(y, z)) -> f(f(x, y), z)


Strategy:

innermost




The following dependency pairs can be strictly oriented:

F(f(a, a), x) -> F(a, x)
F(f(a, a), x) -> F(b, f(a, x))


Additionally, the following usable rules for innermost can be oriented:

f(f(a, a), x) -> f(a, f(b, f(a, x)))
f(x, f(y, z)) -> f(f(x, y), z)


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(b)=  0  
  POL(a)=  1  
  POL(f(x1, x2))=  x1 + x2  
  POL(F(x1, x2))=  1 + x1 + x2  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
           →DP Problem 2
             ↳Narrowing Transformation


Dependency Pairs:

F(x, f(y, z)) -> F(x, y)
F(x, f(y, z)) -> F(f(x, y), z)
F(f(a, a), x) -> F(a, f(b, f(a, x)))


Rules:


f(f(a, a), x) -> f(a, f(b, f(a, x)))
f(x, f(y, z)) -> f(f(x, y), z)


Strategy:

innermost




On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

F(f(a, a), x) -> F(a, f(b, f(a, x)))
two new Dependency Pairs are created:

F(f(a, a), x'') -> F(a, f(f(b, a), x''))
F(f(a, a), f(y', z')) -> F(a, f(b, f(f(a, y'), z')))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 3
                 ↳Forward Instantiation Transformation


Dependency Pairs:

F(f(a, a), f(y', z')) -> F(a, f(b, f(f(a, y'), z')))
F(f(a, a), x'') -> F(a, f(f(b, a), x''))
F(x, f(y, z)) -> F(f(x, y), z)
F(x, f(y, z)) -> F(x, y)


Rules:


f(f(a, a), x) -> f(a, f(b, f(a, x)))
f(x, f(y, z)) -> f(f(x, y), z)


Strategy:

innermost




On this DP problem, a Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

F(x, f(y, z)) -> F(x, y)
three new Dependency Pairs are created:

F(x'', f(f(y'', z''), z)) -> F(x'', f(y'', z''))
F(f(a, a), f(y', z)) -> F(f(a, a), y')
F(f(a, a), f(f(y''', z'''), z)) -> F(f(a, a), f(y''', z'''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 4
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

F(f(a, a), f(f(y''', z'''), z)) -> F(f(a, a), f(y''', z'''))
F(f(a, a), f(y', z)) -> F(f(a, a), y')
F(x'', f(f(y'', z''), z)) -> F(x'', f(y'', z''))
F(f(a, a), x'') -> F(a, f(f(b, a), x''))
F(x, f(y, z)) -> F(f(x, y), z)
F(f(a, a), f(y', z')) -> F(a, f(b, f(f(a, y'), z')))


Rules:


f(f(a, a), x) -> f(a, f(b, f(a, x)))
f(x, f(y, z)) -> f(f(x, y), z)


Strategy:

innermost



Innermost Termination of R could not be shown.
Duration:
0:00 minutes