Term Rewriting System R:
[x, y, z]
f(f(a, b), x) -> f(b, f(a, f(c, 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, b), x) -> F(b, f(a, f(c, f(b, f(a, x)))))
F(f(a, b), x) -> F(a, f(c, f(b, f(a, x))))
F(f(a, b), x) -> F(c, f(b, f(a, x)))
F(f(a, b), x) -> F(b, f(a, x))
F(f(a, b), 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
         ↳Narrowing Transformation


Dependency Pairs:

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


Rules:


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

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

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

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


Rules:


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

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

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 3
                 ↳Narrowing Transformation


Dependency Pairs:

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


Rules:


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

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

The transformation is resulting in one new DP problem: 



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




The following remains to be proven:
Dependency Pairs:

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


Rules:


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