Termination Proof using AProVE

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.

     ↳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.

     ↳DPs

       →DP Problem 1

         ↳Narrowing Transformation

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

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:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

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

three new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 3

                 ↳Forward Instantiation Transformation

Dependency Pairs:

F(f(a, a), f(f(y'', z''), z')) -> F(a, f(b, f(f(f(a, y''), z''), z')))
F(f(a, a), f(a, z'')) -> F(a, f(b, f(a, f(b, f(a, z'')))))
F(f(a, a), f(y'', z'')) -> F(a, f(f(b, f(a, y'')), z''))
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(f(b, 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 Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

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

one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 4

                 ↳Forward Instantiation Transformation

Dependency Pairs:

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

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)

five 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''''))
F(f(a, a), f(f(a, z''''), z)) -> F(f(a, a), f(a, z''''))
F(f(a, a), f(f(f(y'''', z''''), z'0'), z)) -> F(f(a, a), f(f(y'''', z''''), z'0'))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 5

                 ↳Polynomial Ordering

Dependency Pairs:

F(f(a, a), f(f(f(y'''', z''''), z'0'), z)) -> F(f(a, a), f(f(y'''', z''''), z'0'))
F(f(a, a), f(f(a, z''''), z)) -> F(f(a, a), f(a, z''''))
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(f(a, a), f(f(y'', z''), z')) -> F(a, f(b, f(f(f(a, y''), z''), z')))
F(f(a, a), f(a, z'')) -> F(a, f(b, f(a, f(b, f(a, z'')))))
F(f(a, a), f(y'', z'')) -> F(a, f(f(b, f(a, y'')), z''))
F(f(a, a), x'') -> F(a, f(f(b, a), x''))
F(x'', f(f(y'', z''), z)) -> F(x'', f(y'', z''))
F(f(a, a), x) -> F(b, f(a, x))
F(x, f(y, z)) -> F(f(x, y), z)
F(f(a, a), f(y'', z'')) -> F(a, f(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

The following dependency pair can be strictly oriented:

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

POL(F(x₁, x₂)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 6

                 ↳Polynomial Ordering

Dependency Pairs:

F(f(a, a), f(f(f(y'''', z''''), z'0'), z)) -> F(f(a, a), f(f(y'''', z''''), z'0'))
F(f(a, a), f(f(a, z''''), z)) -> F(f(a, a), f(a, z''''))
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(f(a, a), f(f(y'', z''), z')) -> F(a, f(b, f(f(f(a, y''), z''), z')))
F(f(a, a), f(a, z'')) -> F(a, f(b, f(a, f(b, f(a, z'')))))
F(f(a, a), f(y'', z'')) -> F(a, f(f(b, f(a, y'')), z''))
F(f(a, a), x'') -> F(a, f(f(b, a), x''))
F(x'', f(f(y'', z''), z)) -> F(x'', f(y'', z''))
F(x, f(y, z)) -> F(f(x, y), z)
F(f(a, a), f(y'', z'')) -> F(a, f(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

The following dependency pair can be strictly oriented:

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

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.

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 7

                 ↳Narrowing Transformation

Dependency Pairs:

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

two new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 8

                 ↳Narrowing Transformation

Dependency Pairs:

F(f(a, a), f(f(y', z'), z'')) -> F(a, f(f(b, f(f(a, y'), z')), z''))
F(f(a, a), f(y''', z'')) -> F(a, f(f(f(b, a), y'''), z''))
F(f(a, a), f(f(a, z''''), z)) -> F(f(a, a), f(a, z''''))
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(f(a, a), f(f(y'', z''), z')) -> F(a, f(b, f(f(f(a, y''), z''), z')))
F(f(a, a), f(a, z'')) -> F(a, f(b, f(a, f(b, f(a, z'')))))
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(f(f(y'''', z''''), z'0'), z)) -> F(f(a, a), f(f(y'''', z''''), z'0'))

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

three new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 9

                 ↳Narrowing Transformation

Dependency Pairs:

F(f(a, a), f(f(f(y'''', z''''), z'0'), z)) -> F(f(a, a), f(f(y'''', z''''), z'0'))
F(f(a, a), f(a, z''')) -> F(a, f(b, f(a, f(f(b, a), z'''))))
F(f(a, a), f(a, z''')) -> F(a, f(b, f(f(a, b), f(a, z'''))))
F(f(a, a), f(a, z''')) -> F(a, f(f(b, a), f(b, f(a, z'''))))
F(f(a, a), f(y''', z'')) -> F(a, f(f(f(b, a), y'''), z''))
F(f(a, a), f(f(a, z''''), z)) -> F(f(a, a), f(a, z''''))
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(f(a, a), f(f(y'', z''), z')) -> F(a, f(b, f(f(f(a, y''), z''), z')))
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(f(y', z'), z'')) -> F(a, f(f(b, f(f(a, y'), z')), 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

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

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

three new Dependency Pairs are created:

F(f(a, a), f(f(y''', z'''), z'0)) -> F(a, f(f(b, f(f(a, y'''), z''')), z'0))
F(f(a, a), f(f(a, z'''), z')) -> F(a, f(b, f(f(a, f(b, f(a, z'''))), z')))
F(f(a, a), f(f(f(y', z0), z''), z')) -> F(a, f(b, f(f(f(f(a, y'), z0), z''), z')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 10

                 ↳Narrowing Transformation

Dependency Pairs:

F(f(a, a), f(f(f(y', z0), z''), z')) -> F(a, f(b, f(f(f(f(a, y'), z0), z''), z')))
F(f(a, a), f(f(a, z'''), z')) -> F(a, f(b, f(f(a, f(b, f(a, z'''))), z')))
F(f(a, a), f(f(y''', z'''), z'0)) -> F(a, f(f(b, f(f(a, y'''), z''')), z'0))
F(f(a, a), f(f(y', z'), z'')) -> F(a, f(f(b, f(f(a, y'), z')), z''))
F(f(a, a), f(a, z''')) -> F(a, f(b, f(a, f(f(b, a), z'''))))
F(f(a, a), f(a, z''')) -> F(a, f(b, f(f(a, b), f(a, z'''))))
F(f(a, a), f(a, z''')) -> F(a, f(f(b, a), f(b, f(a, z'''))))
F(f(a, a), f(y''', z'')) -> F(a, f(f(f(b, a), y'''), z''))
F(f(a, a), f(f(a, z''''), z)) -> F(f(a, a), f(a, z''''))
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(f(f(y'''', z''''), z'0'), z)) -> F(f(a, a), f(f(y'''', z''''), z'0'))

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

three new Dependency Pairs are created:

F(f(a, a), f(f(y'', z'''), z'')) -> F(a, f(f(f(b, f(a, y'')), z'''), z''))
F(f(a, a), f(f(a, z'''), z'')) -> F(a, f(f(b, f(a, f(b, f(a, z''')))), z''))
F(f(a, a), f(f(f(y'', z0), z'), z'')) -> F(a, f(f(b, f(f(f(a, y''), z0), z')), z''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 11

                 ↳Narrowing Transformation

Dependency Pairs:

F(f(a, a), f(f(f(y'', z0), z'), z'')) -> F(a, f(f(b, f(f(f(a, y''), z0), z')), z''))
F(f(a, a), f(f(a, z'''), z'')) -> F(a, f(f(b, f(a, f(b, f(a, z''')))), z''))
F(f(a, a), f(f(y'', z'''), z'')) -> F(a, f(f(f(b, f(a, y'')), z'''), z''))
F(f(a, a), f(f(a, z'''), z')) -> F(a, f(b, f(f(a, f(b, f(a, z'''))), z')))
F(f(a, a), f(f(y''', z'''), z'0)) -> F(a, f(f(b, f(f(a, y'''), z''')), z'0))
F(f(a, a), f(f(f(y'''', z''''), z'0'), z)) -> F(f(a, a), f(f(y'''', z''''), z'0'))
F(f(a, a), f(a, z''')) -> F(a, f(b, f(a, f(f(b, a), z'''))))
F(f(a, a), f(a, z''')) -> F(a, f(b, f(f(a, b), f(a, z'''))))
F(f(a, a), f(a, z''')) -> F(a, f(f(b, a), f(b, f(a, z'''))))
F(f(a, a), f(y''', z'')) -> F(a, f(f(f(b, a), y'''), z''))
F(f(a, a), f(f(a, z''''), z)) -> F(f(a, a), f(a, z''''))
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(f(f(y', z0), z''), z')) -> F(a, f(b, f(f(f(f(a, y'), z0), z''), 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

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

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

two new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 12

                 ↳Narrowing Transformation

Dependency Pairs:

F(f(a, a), f(f(a, z'''), z'')) -> F(a, f(f(b, f(a, f(b, f(a, z''')))), z''))
F(f(a, a), f(f(y'', z'''), z'')) -> F(a, f(f(f(b, f(a, y'')), z'''), z''))
F(f(a, a), f(f(f(y', z0), z''), z')) -> F(a, f(b, f(f(f(f(a, y'), z0), z''), z')))
F(f(a, a), f(f(a, z'''), z')) -> F(a, f(b, f(f(a, f(b, f(a, z'''))), z')))
F(f(a, a), f(f(y''', z'''), z'0)) -> F(a, f(f(b, f(f(a, y'''), z''')), z'0))
F(f(a, a), f(f(f(y'''', z''''), z'0'), z)) -> F(f(a, a), f(f(y'''', z''''), z'0'))
F(f(a, a), f(a, z'''')) -> F(a, f(f(b, a), f(f(b, a), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(f(b, a), b), f(a, z'''')))
F(f(a, a), f(a, z''')) -> F(a, f(b, f(a, f(f(b, a), z'''))))
F(f(a, a), f(a, z''')) -> F(a, f(b, f(f(a, b), f(a, z'''))))
F(f(a, a), f(y''', z'')) -> F(a, f(f(f(b, a), y'''), z''))
F(f(a, a), f(f(a, z''''), z)) -> F(f(a, a), f(a, z''''))
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(f(f(y'', z0), z'), z'')) -> F(a, f(f(b, f(f(f(a, y''), z0), z')), 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

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

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

two new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 13

                 ↳Narrowing Transformation

Dependency Pairs:

F(f(a, a), f(f(f(y'', z0), z'), z'')) -> F(a, f(f(b, f(f(f(a, y''), z0), z')), z''))
F(f(a, a), f(f(y'', z'''), z'')) -> F(a, f(f(f(b, f(a, y'')), z'''), z''))
F(f(a, a), f(f(f(y', z0), z''), z')) -> F(a, f(b, f(f(f(f(a, y'), z0), z''), z')))
F(f(a, a), f(f(a, z'''), z')) -> F(a, f(b, f(f(a, f(b, f(a, z'''))), z')))
F(f(a, a), f(f(y''', z'''), z'0)) -> F(a, f(f(b, f(f(a, y'''), z''')), z'0))
F(f(a, a), f(f(f(y'''', z''''), z'0'), z)) -> F(f(a, a), f(f(y'''', z''''), z'0'))
F(f(a, a), f(a, z'''')) -> F(a, f(b, f(f(f(a, b), a), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(b, f(a, b)), f(a, z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(b, a), f(f(b, a), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(f(b, a), b), f(a, z'''')))
F(f(a, a), f(a, z''')) -> F(a, f(b, f(a, f(f(b, a), z'''))))
F(f(a, a), f(y''', z'')) -> F(a, f(f(f(b, a), y'''), z''))
F(f(a, a), f(f(a, z''''), z)) -> F(f(a, a), f(a, z''''))
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(f(a, z'''), z'')) -> F(a, f(f(b, f(a, f(b, f(a, z''')))), 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

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

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

two new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 14

                 ↳Narrowing Transformation

Dependency Pairs:

F(f(a, a), f(f(a, z'''), z'')) -> F(a, f(f(b, f(a, f(b, f(a, z''')))), z''))
F(f(a, a), f(f(y'', z'''), z'')) -> F(a, f(f(f(b, f(a, y'')), z'''), z''))
F(f(a, a), f(f(f(y', z0), z''), z')) -> F(a, f(b, f(f(f(f(a, y'), z0), z''), z')))
F(f(a, a), f(f(a, z'''), z')) -> F(a, f(b, f(f(a, f(b, f(a, z'''))), z')))
F(f(a, a), f(f(y''', z'''), z'0)) -> F(a, f(f(b, f(f(a, y'''), z''')), z'0))
F(f(a, a), f(f(f(y'''', z''''), z'0'), z)) -> F(f(a, a), f(f(y'''', z''''), z'0'))
F(f(a, a), f(a, z'''')) -> F(a, f(b, f(f(a, f(b, a)), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(b, a), f(f(b, a), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(b, f(f(f(a, b), a), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(b, f(a, b)), f(a, z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(b, a), f(f(b, a), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(f(b, a), b), f(a, z'''')))
F(f(a, a), f(y''', z'')) -> F(a, f(f(f(b, a), y'''), z''))
F(f(a, a), f(f(a, z''''), z)) -> F(f(a, a), f(a, z''''))
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(f(f(y'', z0), z'), z'')) -> F(a, f(f(b, f(f(f(a, y''), z0), z')), 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

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

F(f(a, a), f(f(y''', z'''), z'0)) -> F(a, f(f(b, f(f(a, y'''), z''')), z'0))

three new Dependency Pairs are created:

F(f(a, a), f(f(y'''', z''''), z'0)) -> F(a, f(f(f(b, f(a, y'''')), z''''), z'0))
F(f(a, a), f(f(a, z''''), z'0)) -> F(a, f(f(b, f(a, f(b, f(a, z'''')))), z'0))
F(f(a, a), f(f(f(y', z'), z'''), z'0)) -> F(a, f(f(b, f(f(f(a, y'), z'), z''')), z'0))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 15

                 ↳Narrowing Transformation

Dependency Pairs:

F(f(a, a), f(f(f(y', z'), z'''), z'0)) -> F(a, f(f(b, f(f(f(a, y'), z'), z''')), z'0))
F(f(a, a), f(f(a, z''''), z'0)) -> F(a, f(f(b, f(a, f(b, f(a, z'''')))), z'0))
F(f(a, a), f(f(y'''', z''''), z'0)) -> F(a, f(f(f(b, f(a, y'''')), z''''), z'0))
F(f(a, a), f(f(f(y'', z0), z'), z'')) -> F(a, f(f(b, f(f(f(a, y''), z0), z')), z''))
F(f(a, a), f(f(y'', z'''), z'')) -> F(a, f(f(f(b, f(a, y'')), z'''), z''))
F(f(a, a), f(f(f(y', z0), z''), z')) -> F(a, f(b, f(f(f(f(a, y'), z0), z''), z')))
F(f(a, a), f(f(a, z'''), z')) -> F(a, f(b, f(f(a, f(b, f(a, z'''))), z')))
F(f(a, a), f(f(f(y'''', z''''), z'0'), z)) -> F(f(a, a), f(f(y'''', z''''), z'0'))
F(f(a, a), f(a, z'''')) -> F(a, f(b, f(f(a, f(b, a)), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(b, a), f(f(b, a), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(b, f(f(f(a, b), a), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(b, f(a, b)), f(a, z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(b, a), f(f(b, a), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(f(b, a), b), f(a, z'''')))
F(f(a, a), f(y''', z'')) -> F(a, f(f(f(b, a), y'''), z''))
F(f(a, a), f(f(a, z''''), z)) -> F(f(a, a), f(a, z''''))
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(f(a, z'''), z'')) -> F(a, f(f(b, f(a, f(b, f(a, z''')))), 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

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

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

three new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 16

                 ↳Narrowing Transformation

Dependency Pairs:

F(f(a, a), f(f(a, z''''), z')) -> F(a, f(b, f(f(a, f(f(b, a), z'''')), z')))
F(f(a, a), f(f(a, z''''), z')) -> F(a, f(b, f(f(f(a, b), f(a, z'''')), z')))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(a, f(b, f(a, z'''')))), z''))
F(f(a, a), f(f(a, z''''), z'0)) -> F(a, f(f(b, f(a, f(b, f(a, z'''')))), z'0))
F(f(a, a), f(f(y'''', z''''), z'0)) -> F(a, f(f(f(b, f(a, y'''')), z''''), z'0))
F(f(a, a), f(f(f(y'', z0), z'), z'')) -> F(a, f(f(b, f(f(f(a, y''), z0), z')), z''))
F(f(a, a), f(f(a, z'''), z'')) -> F(a, f(f(b, f(a, f(b, f(a, z''')))), z''))
F(f(a, a), f(f(y'', z'''), z'')) -> F(a, f(f(f(b, f(a, y'')), z'''), z''))
F(f(a, a), f(f(f(y', z0), z''), z')) -> F(a, f(b, f(f(f(f(a, y'), z0), z''), z')))
F(f(a, a), f(f(f(y'''', z''''), z'0'), z)) -> F(f(a, a), f(f(y'''', z''''), z'0'))
F(f(a, a), f(a, z'''')) -> F(a, f(b, f(f(a, f(b, a)), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(b, a), f(f(b, a), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(b, f(f(f(a, b), a), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(b, f(a, b)), f(a, z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(b, a), f(f(b, a), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(f(b, a), b), f(a, z'''')))
F(f(a, a), f(y''', z'')) -> F(a, f(f(f(b, a), y'''), z''))
F(f(a, a), f(f(a, z''''), z)) -> F(f(a, a), f(a, z''''))
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(f(f(y', z'), z'''), z'0)) -> F(a, f(f(b, f(f(f(a, y'), z'), z''')), z'0))

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), f(f(f(y', z0), z''), z')) -> F(a, f(b, f(f(f(f(a, y'), z0), z''), z')))

three new Dependency Pairs are created:

F(f(a, a), f(f(f(y'', z0'), z'''), z'0)) -> F(a, f(f(b, f(f(f(a, y''), z0'), z''')), z'0))
F(f(a, a), f(f(f(a, z0'), z''), z')) -> F(a, f(b, f(f(f(a, f(b, f(a, z0'))), z''), z')))
F(f(a, a), f(f(f(f(y'', z1), z0), z''), z')) -> F(a, f(b, f(f(f(f(f(a, y''), z1), z0), z''), z')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 17

                 ↳Narrowing Transformation

Dependency Pairs:

F(f(a, a), f(f(f(f(y'', z1), z0), z''), z')) -> F(a, f(b, f(f(f(f(f(a, y''), z1), z0), z''), z')))
F(f(a, a), f(f(f(a, z0'), z''), z')) -> F(a, f(b, f(f(f(a, f(b, f(a, z0'))), z''), z')))
F(f(a, a), f(f(f(y'', z0'), z'''), z'0)) -> F(a, f(f(b, f(f(f(a, y''), z0'), z''')), z'0))
F(f(a, a), f(f(a, z''''), z')) -> F(a, f(b, f(f(f(a, b), f(a, z'''')), z')))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(a, f(b, f(a, z'''')))), z''))
F(f(a, a), f(f(f(y', z'), z'''), z'0)) -> F(a, f(f(b, f(f(f(a, y'), z'), z''')), z'0))
F(f(a, a), f(f(a, z''''), z'0)) -> F(a, f(f(b, f(a, f(b, f(a, z'''')))), z'0))
F(f(a, a), f(f(y'''', z''''), z'0)) -> F(a, f(f(f(b, f(a, y'''')), z''''), z'0))
F(f(a, a), f(f(f(y'', z0), z'), z'')) -> F(a, f(f(b, f(f(f(a, y''), z0), z')), z''))
F(f(a, a), f(f(a, z'''), z'')) -> F(a, f(f(b, f(a, f(b, f(a, z''')))), z''))
F(f(a, a), f(f(y'', z'''), z'')) -> F(a, f(f(f(b, f(a, y'')), z'''), z''))
F(f(a, a), f(f(f(y'''', z''''), z'0'), z)) -> F(f(a, a), f(f(y'''', z''''), z'0'))
F(f(a, a), f(a, z'''')) -> F(a, f(b, f(f(a, f(b, a)), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(b, a), f(f(b, a), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(b, f(f(f(a, b), a), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(b, f(a, b)), f(a, z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(b, a), f(f(b, a), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(f(b, a), b), f(a, z'''')))
F(f(a, a), f(y''', z'')) -> F(a, f(f(f(b, a), y'''), z''))
F(f(a, a), f(f(a, z''''), z)) -> F(f(a, a), f(a, z''''))
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(f(a, z''''), z')) -> F(a, f(b, f(f(a, f(f(b, a), z'''')), 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

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

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

two new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 18

                 ↳Narrowing Transformation

Dependency Pairs:

F(f(a, a), f(f(f(y', z'), z'''), z'')) -> F(a, f(f(f(b, f(f(a, y'), z')), z'''), z''))
F(f(a, a), f(f(y''', z'''), z'')) -> F(a, f(f(f(f(b, a), y'''), z'''), z''))
F(f(a, a), f(f(f(a, z0'), z''), z')) -> F(a, f(b, f(f(f(a, f(b, f(a, z0'))), z''), z')))
F(f(a, a), f(f(f(y'', z0'), z'''), z'0)) -> F(a, f(f(b, f(f(f(a, y''), z0'), z''')), z'0))
F(f(a, a), f(f(a, z''''), z')) -> F(a, f(b, f(f(a, f(f(b, a), z'''')), z')))
F(f(a, a), f(f(a, z''''), z')) -> F(a, f(b, f(f(f(a, b), f(a, z'''')), z')))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(a, f(b, f(a, z'''')))), z''))
F(f(a, a), f(f(f(y', z'), z'''), z'0)) -> F(a, f(f(b, f(f(f(a, y'), z'), z''')), z'0))
F(f(a, a), f(f(a, z''''), z'0)) -> F(a, f(f(b, f(a, f(b, f(a, z'''')))), z'0))
F(f(a, a), f(f(y'''', z''''), z'0)) -> F(a, f(f(f(b, f(a, y'''')), z''''), z'0))
F(f(a, a), f(f(f(y'', z0), z'), z'')) -> F(a, f(f(b, f(f(f(a, y''), z0), z')), z''))
F(f(a, a), f(f(a, z'''), z'')) -> F(a, f(f(b, f(a, f(b, f(a, z''')))), z''))
F(f(a, a), f(f(f(y'''', z''''), z'0'), z)) -> F(f(a, a), f(f(y'''', z''''), z'0'))
F(f(a, a), f(a, z'''')) -> F(a, f(b, f(f(a, f(b, a)), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(b, a), f(f(b, a), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(b, f(f(f(a, b), a), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(b, f(a, b)), f(a, z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(b, a), f(f(b, a), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(f(b, a), b), f(a, z'''')))
F(f(a, a), f(y''', z'')) -> F(a, f(f(f(b, a), y'''), z''))
F(f(a, a), f(f(a, z''''), z)) -> F(f(a, a), f(a, z''''))
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(f(f(f(y'', z1), z0), z''), z')) -> F(a, f(b, f(f(f(f(f(a, y''), z1), z0), z''), 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

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

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

three new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 19

                 ↳Narrowing Transformation

Dependency Pairs:

F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(a, f(f(b, a), z''''))), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(f(a, b), f(a, z''''))), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(f(b, a), f(b, f(a, z''''))), z''))
F(f(a, a), f(f(y''', z'''), z'')) -> F(a, f(f(f(f(b, a), y'''), z'''), z''))
F(f(a, a), f(f(f(f(y'', z1), z0), z''), z')) -> F(a, f(b, f(f(f(f(f(a, y''), z1), z0), z''), z')))
F(f(a, a), f(f(f(a, z0'), z''), z')) -> F(a, f(b, f(f(f(a, f(b, f(a, z0'))), z''), z')))
F(f(a, a), f(f(f(y'', z0'), z'''), z'0)) -> F(a, f(f(b, f(f(f(a, y''), z0'), z''')), z'0))
F(f(a, a), f(f(a, z''''), z')) -> F(a, f(b, f(f(a, f(f(b, a), z'''')), z')))
F(f(a, a), f(f(a, z''''), z')) -> F(a, f(b, f(f(f(a, b), f(a, z'''')), z')))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(a, f(b, f(a, z'''')))), z''))
F(f(a, a), f(f(f(y', z'), z'''), z'0)) -> F(a, f(f(b, f(f(f(a, y'), z'), z''')), z'0))
F(f(a, a), f(f(a, z''''), z'0)) -> F(a, f(f(b, f(a, f(b, f(a, z'''')))), z'0))
F(f(a, a), f(f(y'''', z''''), z'0)) -> F(a, f(f(f(b, f(a, y'''')), z''''), z'0))
F(f(a, a), f(f(f(y'', z0), z'), z'')) -> F(a, f(f(b, f(f(f(a, y''), z0), z')), z''))
F(f(a, a), f(f(f(y'''', z''''), z'0'), z)) -> F(f(a, a), f(f(y'''', z''''), z'0'))
F(f(a, a), f(a, z'''')) -> F(a, f(b, f(f(a, f(b, a)), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(b, a), f(f(b, a), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(b, f(f(f(a, b), a), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(b, f(a, b)), f(a, z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(b, a), f(f(b, a), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(f(b, a), b), f(a, z'''')))
F(f(a, a), f(y''', z'')) -> F(a, f(f(f(b, a), y'''), z''))
F(f(a, a), f(f(a, z''''), z)) -> F(f(a, a), f(a, z''''))
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(f(f(y', z'), z'''), z'')) -> F(a, f(f(f(b, f(f(a, y'), z')), z'''), 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

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

F(f(a, a), f(f(f(y'', z0), z'), z'')) -> F(a, f(f(b, f(f(f(a, y''), z0), z')), z''))

three new Dependency Pairs are created:

F(f(a, a), f(f(f(y''', z0'), z'''), z'')) -> F(a, f(f(f(b, f(f(a, y'''), z0')), z'''), z''))
F(f(a, a), f(f(f(a, z0'), z'), z'')) -> F(a, f(f(b, f(f(a, f(b, f(a, z0'))), z')), z''))
F(f(a, a), f(f(f(f(y', z1), z0), z'), z'')) -> F(a, f(f(b, f(f(f(f(a, y'), z1), z0), z')), z''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 20

                 ↳Narrowing Transformation

Dependency Pairs:

F(f(a, a), f(f(f(f(y', z1), z0), z'), z'')) -> F(a, f(f(b, f(f(f(f(a, y'), z1), z0), z')), z''))
F(f(a, a), f(f(f(a, z0'), z'), z'')) -> F(a, f(f(b, f(f(a, f(b, f(a, z0'))), z')), z''))
F(f(a, a), f(f(f(y''', z0'), z'''), z'')) -> F(a, f(f(f(b, f(f(a, y'''), z0')), z'''), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(f(a, b), f(a, z''''))), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(f(b, a), f(b, f(a, z''''))), z''))
F(f(a, a), f(f(f(y', z'), z'''), z'')) -> F(a, f(f(f(b, f(f(a, y'), z')), z'''), z''))
F(f(a, a), f(f(y''', z'''), z'')) -> F(a, f(f(f(f(b, a), y'''), z'''), z''))
F(f(a, a), f(f(f(f(y'', z1), z0), z''), z')) -> F(a, f(b, f(f(f(f(f(a, y''), z1), z0), z''), z')))
F(f(a, a), f(f(f(a, z0'), z''), z')) -> F(a, f(b, f(f(f(a, f(b, f(a, z0'))), z''), z')))
F(f(a, a), f(f(f(y'', z0'), z'''), z'0)) -> F(a, f(f(b, f(f(f(a, y''), z0'), z''')), z'0))
F(f(a, a), f(f(a, z''''), z')) -> F(a, f(b, f(f(a, f(f(b, a), z'''')), z')))
F(f(a, a), f(f(a, z''''), z')) -> F(a, f(b, f(f(f(a, b), f(a, z'''')), z')))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(a, f(b, f(a, z'''')))), z''))
F(f(a, a), f(f(f(y', z'), z'''), z'0)) -> F(a, f(f(b, f(f(f(a, y'), z'), z''')), z'0))
F(f(a, a), f(f(a, z''''), z'0)) -> F(a, f(f(b, f(a, f(b, f(a, z'''')))), z'0))
F(f(a, a), f(f(y'''', z''''), z'0)) -> F(a, f(f(f(b, f(a, y'''')), z''''), z'0))
F(f(a, a), f(f(f(y'''', z''''), z'0'), z)) -> F(f(a, a), f(f(y'''', z''''), z'0'))
F(f(a, a), f(a, z'''')) -> F(a, f(b, f(f(a, f(b, a)), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(b, a), f(f(b, a), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(b, f(f(f(a, b), a), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(b, f(a, b)), f(a, z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(b, a), f(f(b, a), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(f(b, a), b), f(a, z'''')))
F(f(a, a), f(y''', z'')) -> F(a, f(f(f(b, a), y'''), z''))
F(f(a, a), f(f(a, z''''), z)) -> F(f(a, a), f(a, z''''))
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(f(a, z''''), z'')) -> F(a, f(f(b, f(a, f(f(b, a), z''''))), 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

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

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

one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 21

                 ↳Narrowing Transformation

Dependency Pairs:

F(f(a, a), f(f(f(a, z0'), z'), z'')) -> F(a, f(f(b, f(f(a, f(b, f(a, z0'))), z')), z''))
F(f(a, a), f(f(f(y''', z0'), z'''), z'')) -> F(a, f(f(f(b, f(f(a, y'''), z0')), z'''), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(a, f(f(b, a), z''''))), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(f(a, b), f(a, z''''))), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(f(b, a), f(b, f(a, z''''))), z''))
F(f(a, a), f(f(f(y', z'), z'''), z'')) -> F(a, f(f(f(b, f(f(a, y'), z')), z'''), z''))
F(f(a, a), f(f(y''', z'''), z'')) -> F(a, f(f(f(f(b, a), y'''), z'''), z''))
F(f(a, a), f(f(f(f(y'', z1), z0), z''), z')) -> F(a, f(b, f(f(f(f(f(a, y''), z1), z0), z''), z')))
F(f(a, a), f(f(f(a, z0'), z''), z')) -> F(a, f(b, f(f(f(a, f(b, f(a, z0'))), z''), z')))
F(f(a, a), f(f(f(y'', z0'), z'''), z'0)) -> F(a, f(f(b, f(f(f(a, y''), z0'), z''')), z'0))
F(f(a, a), f(f(a, z''''), z')) -> F(a, f(b, f(f(a, f(f(b, a), z'''')), z')))
F(f(a, a), f(f(a, z''''), z')) -> F(a, f(b, f(f(f(a, b), f(a, z'''')), z')))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(a, f(b, f(a, z'''')))), z''))
F(f(a, a), f(f(f(y', z'), z'''), z'0)) -> F(a, f(f(b, f(f(f(a, y'), z'), z''')), z'0))
F(f(a, a), f(f(a, z''''), z'0)) -> F(a, f(f(b, f(a, f(b, f(a, z'''')))), z'0))
F(f(a, a), f(f(y'''', z''''), z'0)) -> F(a, f(f(f(b, f(a, y'''')), z''''), z'0))
F(f(a, a), f(f(f(y'''', z''''), z'0'), z)) -> F(f(a, a), f(f(y'''', z''''), z'0'))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(f(b, a), b), a), z'''''))
F(f(a, a), f(a, z'''')) -> F(a, f(b, f(f(a, f(b, a)), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(b, a), f(f(b, a), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(b, f(f(f(a, b), a), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(b, f(a, b)), f(a, z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(b, a), f(f(b, a), z'''')))
F(f(a, a), f(y''', z'')) -> F(a, f(f(f(b, a), y'''), z''))
F(f(a, a), f(f(a, z''''), z)) -> F(f(a, a), f(a, z''''))
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(f(f(f(y', z1), z0), z'), z'')) -> F(a, f(f(b, f(f(f(f(a, y'), z1), z0), z')), 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

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

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

one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 22

                 ↳Narrowing Transformation

Dependency Pairs:

F(f(a, a), f(f(f(f(y', z1), z0), z'), z'')) -> F(a, f(f(b, f(f(f(f(a, y'), z1), z0), z')), z''))
F(f(a, a), f(f(f(y''', z0'), z'''), z'')) -> F(a, f(f(f(b, f(f(a, y'''), z0')), z'''), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(a, f(f(b, a), z''''))), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(f(a, b), f(a, z''''))), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(f(b, a), f(b, f(a, z''''))), z''))
F(f(a, a), f(f(f(y', z'), z'''), z'')) -> F(a, f(f(f(b, f(f(a, y'), z')), z'''), z''))
F(f(a, a), f(f(y''', z'''), z'')) -> F(a, f(f(f(f(b, a), y'''), z'''), z''))
F(f(a, a), f(f(f(f(y'', z1), z0), z''), z')) -> F(a, f(b, f(f(f(f(f(a, y''), z1), z0), z''), z')))
F(f(a, a), f(f(f(a, z0'), z''), z')) -> F(a, f(b, f(f(f(a, f(b, f(a, z0'))), z''), z')))
F(f(a, a), f(f(f(y'', z0'), z'''), z'0)) -> F(a, f(f(b, f(f(f(a, y''), z0'), z''')), z'0))
F(f(a, a), f(f(a, z''''), z')) -> F(a, f(b, f(f(a, f(f(b, a), z'''')), z')))
F(f(a, a), f(f(a, z''''), z')) -> F(a, f(b, f(f(f(a, b), f(a, z'''')), z')))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(a, f(b, f(a, z'''')))), z''))
F(f(a, a), f(f(f(y', z'), z'''), z'0)) -> F(a, f(f(b, f(f(f(a, y'), z'), z''')), z'0))
F(f(a, a), f(f(a, z''''), z'0)) -> F(a, f(f(b, f(a, f(b, f(a, z'''')))), z'0))
F(f(a, a), f(f(y'''', z''''), z'0)) -> F(a, f(f(f(b, f(a, y'''')), z''''), z'0))
F(f(a, a), f(f(f(y'''', z''''), z'0'), z)) -> F(f(a, a), f(f(y'''', z''''), z'0'))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, a), f(b, a)), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(f(b, a), b), a), z'''''))
F(f(a, a), f(a, z'''')) -> F(a, f(b, f(f(a, f(b, a)), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(b, a), f(f(b, a), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(b, f(f(f(a, b), a), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(b, f(a, b)), f(a, z'''')))
F(f(a, a), f(y''', z'')) -> F(a, f(f(f(b, a), y'''), z''))
F(f(a, a), f(f(a, z''''), z)) -> F(f(a, a), f(a, z''''))
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(f(f(a, z0'), z'), z'')) -> F(a, f(f(b, f(f(a, f(b, f(a, z0'))), z')), 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

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

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

two new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 23

                 ↳Narrowing Transformation

Dependency Pairs:

F(f(a, a), f(f(f(a, z0'), z'), z'')) -> F(a, f(f(b, f(f(a, f(b, f(a, z0'))), z')), z''))
F(f(a, a), f(f(f(y''', z0'), z'''), z'')) -> F(a, f(f(f(b, f(f(a, y'''), z0')), z'''), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(a, f(f(b, a), z''''))), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(f(a, b), f(a, z''''))), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(f(b, a), f(b, f(a, z''''))), z''))
F(f(a, a), f(f(f(y', z'), z'''), z'')) -> F(a, f(f(f(b, f(f(a, y'), z')), z'''), z''))
F(f(a, a), f(f(y''', z'''), z'')) -> F(a, f(f(f(f(b, a), y'''), z'''), z''))
F(f(a, a), f(f(f(f(y'', z1), z0), z''), z')) -> F(a, f(b, f(f(f(f(f(a, y''), z1), z0), z''), z')))
F(f(a, a), f(f(f(a, z0'), z''), z')) -> F(a, f(b, f(f(f(a, f(b, f(a, z0'))), z''), z')))
F(f(a, a), f(f(f(y'', z0'), z'''), z'0)) -> F(a, f(f(b, f(f(f(a, y''), z0'), z''')), z'0))
F(f(a, a), f(f(a, z''''), z')) -> F(a, f(b, f(f(a, f(f(b, a), z'''')), z')))
F(f(a, a), f(f(a, z''''), z')) -> F(a, f(b, f(f(f(a, b), f(a, z'''')), z')))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(a, f(b, f(a, z'''')))), z''))
F(f(a, a), f(f(f(y', z'), z'''), z'0)) -> F(a, f(f(b, f(f(f(a, y'), z'), z''')), z'0))
F(f(a, a), f(f(a, z''''), z'0)) -> F(a, f(f(b, f(a, f(b, f(a, z'''')))), z'0))
F(f(a, a), f(f(y'''', z''''), z'0)) -> F(a, f(f(f(b, f(a, y'''')), z''''), z'0))
F(f(a, a), f(f(f(y'''', z''''), z'0'), z)) -> F(f(a, a), f(f(y'''', z''''), z'0'))
F(f(a, a), f(a, z'''')) -> F(a, f(f(f(b, a), b), f(a, z'''')))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, f(a, b)), a), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, a), f(b, a)), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(f(b, a), b), a), z'''''))
F(f(a, a), f(a, z'''')) -> F(a, f(b, f(f(a, f(b, a)), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(b, a), f(f(b, a), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(b, f(f(f(a, b), a), z'''')))
F(f(a, a), f(y''', z'')) -> F(a, f(f(f(b, a), y'''), z''))
F(f(a, a), f(f(a, z''''), z)) -> F(f(a, a), f(a, z''''))
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(f(f(f(y', z1), z0), z'), z'')) -> F(a, f(f(b, f(f(f(f(a, y'), z1), z0), z')), 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

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

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

one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 24

                 ↳Narrowing Transformation

Dependency Pairs:

F(f(a, a), f(f(f(f(y', z1), z0), z'), z'')) -> F(a, f(f(b, f(f(f(f(a, y'), z1), z0), z')), z''))
F(f(a, a), f(f(f(y''', z0'), z'''), z'')) -> F(a, f(f(f(b, f(f(a, y'''), z0')), z'''), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(a, f(f(b, a), z''''))), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(f(a, b), f(a, z''''))), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(f(b, a), f(b, f(a, z''''))), z''))
F(f(a, a), f(f(f(y', z'), z'''), z'')) -> F(a, f(f(f(b, f(f(a, y'), z')), z'''), z''))
F(f(a, a), f(f(y''', z'''), z'')) -> F(a, f(f(f(f(b, a), y'''), z'''), z''))
F(f(a, a), f(f(f(f(y'', z1), z0), z''), z')) -> F(a, f(b, f(f(f(f(f(a, y''), z1), z0), z''), z')))
F(f(a, a), f(f(f(a, z0'), z''), z')) -> F(a, f(b, f(f(f(a, f(b, f(a, z0'))), z''), z')))
F(f(a, a), f(f(f(y'', z0'), z'''), z'0)) -> F(a, f(f(b, f(f(f(a, y''), z0'), z''')), z'0))
F(f(a, a), f(f(a, z''''), z')) -> F(a, f(b, f(f(a, f(f(b, a), z'''')), z')))
F(f(a, a), f(f(a, z''''), z')) -> F(a, f(b, f(f(f(a, b), f(a, z'''')), z')))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(a, f(b, f(a, z'''')))), z''))
F(f(a, a), f(f(f(y', z'), z'''), z'0)) -> F(a, f(f(b, f(f(f(a, y'), z'), z''')), z'0))
F(f(a, a), f(f(a, z''''), z'0)) -> F(a, f(f(b, f(a, f(b, f(a, z'''')))), z'0))
F(f(a, a), f(f(y'''', z''''), z'0)) -> F(a, f(f(f(b, f(a, y'''')), z''''), z'0))
F(f(a, a), f(f(f(y'''', z''''), z'0'), z)) -> F(f(a, a), f(f(y'''', z''''), z'0'))
F(f(a, a), f(a, z''''')) -> F(a, f(f(b, f(f(a, b), a)), z'''''))
F(f(a, a), f(a, z'''')) -> F(a, f(f(f(b, a), b), f(a, z'''')))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, f(a, b)), a), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, a), f(b, a)), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(f(b, a), b), a), z'''''))
F(f(a, a), f(a, z'''')) -> F(a, f(b, f(f(a, f(b, a)), z'''')))
F(f(a, a), f(a, z'''')) -> F(a, f(f(b, a), f(f(b, a), z'''')))
F(f(a, a), f(y''', z'')) -> F(a, f(f(f(b, a), y'''), z''))
F(f(a, a), f(f(a, z''''), z)) -> F(f(a, a), f(a, z''''))
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(f(f(a, z0'), z'), z'')) -> F(a, f(f(b, f(f(a, f(b, f(a, z0'))), z')), 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

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

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

one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 25

                 ↳Narrowing Transformation

Dependency Pairs:

F(f(a, a), f(f(f(a, z0'), z'), z'')) -> F(a, f(f(b, f(f(a, f(b, f(a, z0'))), z')), z''))
F(f(a, a), f(f(f(y''', z0'), z'''), z'')) -> F(a, f(f(f(b, f(f(a, y'''), z0')), z'''), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(a, f(f(b, a), z''''))), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(f(a, b), f(a, z''''))), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(f(b, a), f(b, f(a, z''''))), z''))
F(f(a, a), f(f(f(y', z'), z'''), z'')) -> F(a, f(f(f(b, f(f(a, y'), z')), z'''), z''))
F(f(a, a), f(f(y''', z'''), z'')) -> F(a, f(f(f(f(b, a), y'''), z'''), z''))
F(f(a, a), f(f(f(f(y'', z1), z0), z''), z')) -> F(a, f(b, f(f(f(f(f(a, y''), z1), z0), z''), z')))
F(f(a, a), f(f(f(a, z0'), z''), z')) -> F(a, f(b, f(f(f(a, f(b, f(a, z0'))), z''), z')))
F(f(a, a), f(f(f(y'', z0'), z'''), z'0)) -> F(a, f(f(b, f(f(f(a, y''), z0'), z''')), z'0))
F(f(a, a), f(f(a, z''''), z')) -> F(a, f(b, f(f(a, f(f(b, a), z'''')), z')))
F(f(a, a), f(f(a, z''''), z')) -> F(a, f(b, f(f(f(a, b), f(a, z'''')), z')))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(a, f(b, f(a, z'''')))), z''))
F(f(a, a), f(f(f(y', z'), z'''), z'0)) -> F(a, f(f(b, f(f(f(a, y'), z'), z''')), z'0))
F(f(a, a), f(f(a, z''''), z'0)) -> F(a, f(f(b, f(a, f(b, f(a, z'''')))), z'0))
F(f(a, a), f(f(y'''', z''''), z'0)) -> F(a, f(f(f(b, f(a, y'''')), z''''), z'0))
F(f(a, a), f(f(f(y'''', z''''), z'0'), z)) -> F(f(a, a), f(f(y'''', z''''), z'0'))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, a), f(b, a)), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(b, f(f(a, b), a)), z'''''))
F(f(a, a), f(a, z'''')) -> F(a, f(f(f(b, a), b), f(a, z'''')))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, f(a, b)), a), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, a), f(b, a)), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(f(b, a), b), a), z'''''))
F(f(a, a), f(a, z'''')) -> F(a, f(b, f(f(a, f(b, a)), z'''')))
F(f(a, a), f(y''', z'')) -> F(a, f(f(f(b, a), y'''), z''))
F(f(a, a), f(f(a, z''''), z)) -> F(f(a, a), f(a, z''''))
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(f(f(f(y', z1), z0), z'), z'')) -> F(a, f(f(b, f(f(f(f(a, y'), z1), z0), z')), 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

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

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

two new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 26

                 ↳Narrowing Transformation

Dependency Pairs:

F(f(a, a), f(f(f(f(y', z1), z0), z'), z'')) -> F(a, f(f(b, f(f(f(f(a, y'), z1), z0), z')), z''))
F(f(a, a), f(f(f(y''', z0'), z'''), z'')) -> F(a, f(f(f(b, f(f(a, y'''), z0')), z'''), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(a, f(f(b, a), z''''))), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(f(a, b), f(a, z''''))), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(f(b, a), f(b, f(a, z''''))), z''))
F(f(a, a), f(f(f(y', z'), z'''), z'')) -> F(a, f(f(f(b, f(f(a, y'), z')), z'''), z''))
F(f(a, a), f(f(y''', z'''), z'')) -> F(a, f(f(f(f(b, a), y'''), z'''), z''))
F(f(a, a), f(f(f(f(y'', z1), z0), z''), z')) -> F(a, f(b, f(f(f(f(f(a, y''), z1), z0), z''), z')))
F(f(a, a), f(f(f(a, z0'), z''), z')) -> F(a, f(b, f(f(f(a, f(b, f(a, z0'))), z''), z')))
F(f(a, a), f(f(f(y'', z0'), z'''), z'0)) -> F(a, f(f(b, f(f(f(a, y''), z0'), z''')), z'0))
F(f(a, a), f(f(a, z''''), z')) -> F(a, f(b, f(f(a, f(f(b, a), z'''')), z')))
F(f(a, a), f(f(a, z''''), z')) -> F(a, f(b, f(f(f(a, b), f(a, z'''')), z')))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(a, f(b, f(a, z'''')))), z''))
F(f(a, a), f(f(f(y', z'), z'''), z'0)) -> F(a, f(f(b, f(f(f(a, y'), z'), z''')), z'0))
F(f(a, a), f(f(a, z''''), z'0)) -> F(a, f(f(b, f(a, f(b, f(a, z'''')))), z'0))
F(f(a, a), f(f(y'''', z''''), z'0)) -> F(a, f(f(f(b, f(a, y'''')), z''''), z'0))
F(f(a, a), f(f(f(y'''', z''''), z'0'), z)) -> F(f(a, a), f(f(y'''', z''''), z'0'))
F(f(a, a), f(a, z'''')) -> F(a, f(b, f(f(f(a, b), a), z'''')))
F(f(a, a), f(a, z''''')) -> F(a, f(f(b, f(a, f(b, a))), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, a), f(b, a)), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(b, f(f(a, b), a)), z'''''))
F(f(a, a), f(a, z'''')) -> F(a, f(f(f(b, a), b), f(a, z'''')))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, f(a, b)), a), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, a), f(b, a)), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(f(b, a), b), a), z'''''))
F(f(a, a), f(y''', z'')) -> F(a, f(f(f(b, a), y'''), z''))
F(f(a, a), f(f(a, z''''), z)) -> F(f(a, a), f(a, z''''))
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(f(f(a, z0'), z'), z'')) -> F(a, f(f(b, f(f(a, f(b, f(a, z0'))), z')), 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

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

F(f(a, a), f(f(y'''', z''''), z'0)) -> F(a, f(f(f(b, f(a, y'''')), z''''), z'0))

two new Dependency Pairs are created:

F(f(a, a), f(f(y''''', z''''), z'0)) -> F(a, f(f(f(f(b, a), y'''''), z''''), z'0))
F(f(a, a), f(f(f(y', z'), z''''), z'0)) -> F(a, f(f(f(b, f(f(a, y'), z')), z''''), z'0))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 27

                 ↳Narrowing Transformation

Dependency Pairs:

F(f(a, a), f(f(f(y', z'), z''''), z'0)) -> F(a, f(f(f(b, f(f(a, y'), z')), z''''), z'0))
F(f(a, a), f(f(y''''', z''''), z'0)) -> F(a, f(f(f(f(b, a), y'''''), z''''), z'0))
F(f(a, a), f(f(f(a, z0'), z'), z'')) -> F(a, f(f(b, f(f(a, f(b, f(a, z0'))), z')), z''))
F(f(a, a), f(f(f(y''', z0'), z'''), z'')) -> F(a, f(f(f(b, f(f(a, y'''), z0')), z'''), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(a, f(f(b, a), z''''))), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(f(a, b), f(a, z''''))), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(f(b, a), f(b, f(a, z''''))), z''))
F(f(a, a), f(f(f(y', z'), z'''), z'')) -> F(a, f(f(f(b, f(f(a, y'), z')), z'''), z''))
F(f(a, a), f(f(y''', z'''), z'')) -> F(a, f(f(f(f(b, a), y'''), z'''), z''))
F(f(a, a), f(f(f(f(y'', z1), z0), z''), z')) -> F(a, f(b, f(f(f(f(f(a, y''), z1), z0), z''), z')))
F(f(a, a), f(f(f(a, z0'), z''), z')) -> F(a, f(b, f(f(f(a, f(b, f(a, z0'))), z''), z')))
F(f(a, a), f(f(f(y'', z0'), z'''), z'0)) -> F(a, f(f(b, f(f(f(a, y''), z0'), z''')), z'0))
F(f(a, a), f(f(a, z''''), z')) -> F(a, f(b, f(f(a, f(f(b, a), z'''')), z')))
F(f(a, a), f(f(a, z''''), z')) -> F(a, f(b, f(f(f(a, b), f(a, z'''')), z')))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(a, f(b, f(a, z'''')))), z''))
F(f(a, a), f(f(f(y', z'), z'''), z'0)) -> F(a, f(f(b, f(f(f(a, y'), z'), z''')), z'0))
F(f(a, a), f(f(a, z''''), z'0)) -> F(a, f(f(b, f(a, f(b, f(a, z'''')))), z'0))
F(f(a, a), f(f(f(y'''', z''''), z'0'), z)) -> F(f(a, a), f(f(y'''', z''''), z'0'))
F(f(a, a), f(a, z'''')) -> F(a, f(b, f(f(f(a, b), a), z'''')))
F(f(a, a), f(a, z''''')) -> F(a, f(f(b, f(a, f(b, a))), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, a), f(b, a)), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(b, f(f(a, b), a)), z'''''))
F(f(a, a), f(a, z'''')) -> F(a, f(f(f(b, a), b), f(a, z'''')))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, f(a, b)), a), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, a), f(b, a)), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(f(b, a), b), a), z'''''))
F(f(a, a), f(y''', z'')) -> F(a, f(f(f(b, a), y'''), z''))
F(f(a, a), f(f(a, z''''), z)) -> F(f(a, a), f(a, z''''))
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(f(f(f(y', z1), z0), z'), z'')) -> F(a, f(f(b, f(f(f(f(a, y'), z1), z0), z')), 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

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

F(f(a, a), f(f(a, z''''), z'0)) -> F(a, f(f(b, f(a, f(b, f(a, z'''')))), z'0))

three new Dependency Pairs are created:

F(f(a, a), f(f(a, z'''''), z'0)) -> F(a, f(f(f(b, a), f(b, f(a, z'''''))), z'0))
F(f(a, a), f(f(a, z'''''), z'0)) -> F(a, f(f(b, f(f(a, b), f(a, z'''''))), z'0))
F(f(a, a), f(f(a, z'''''), z'0)) -> F(a, f(f(b, f(a, f(f(b, a), z'''''))), z'0))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 28

                 ↳Narrowing Transformation

Dependency Pairs:

F(f(a, a), f(f(a, z'''''), z'0)) -> F(a, f(f(b, f(a, f(f(b, a), z'''''))), z'0))
F(f(a, a), f(f(a, z'''''), z'0)) -> F(a, f(f(b, f(f(a, b), f(a, z'''''))), z'0))
F(f(a, a), f(f(a, z'''''), z'0)) -> F(a, f(f(f(b, a), f(b, f(a, z'''''))), z'0))
F(f(a, a), f(f(y''''', z''''), z'0)) -> F(a, f(f(f(f(b, a), y'''''), z''''), z'0))
F(f(a, a), f(f(f(f(y', z1), z0), z'), z'')) -> F(a, f(f(b, f(f(f(f(a, y'), z1), z0), z')), z''))
F(f(a, a), f(f(f(a, z0'), z'), z'')) -> F(a, f(f(b, f(f(a, f(b, f(a, z0'))), z')), z''))
F(f(a, a), f(f(f(y''', z0'), z'''), z'')) -> F(a, f(f(f(b, f(f(a, y'''), z0')), z'''), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(a, f(f(b, a), z''''))), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(f(a, b), f(a, z''''))), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(f(b, a), f(b, f(a, z''''))), z''))
F(f(a, a), f(f(f(y', z'), z'''), z'')) -> F(a, f(f(f(b, f(f(a, y'), z')), z'''), z''))
F(f(a, a), f(f(y''', z'''), z'')) -> F(a, f(f(f(f(b, a), y'''), z'''), z''))
F(f(a, a), f(f(f(f(y'', z1), z0), z''), z')) -> F(a, f(b, f(f(f(f(f(a, y''), z1), z0), z''), z')))
F(f(a, a), f(f(f(a, z0'), z''), z')) -> F(a, f(b, f(f(f(a, f(b, f(a, z0'))), z''), z')))
F(f(a, a), f(f(f(y'', z0'), z'''), z'0)) -> F(a, f(f(b, f(f(f(a, y''), z0'), z''')), z'0))
F(f(a, a), f(f(a, z''''), z')) -> F(a, f(b, f(f(a, f(f(b, a), z'''')), z')))
F(f(a, a), f(f(a, z''''), z')) -> F(a, f(b, f(f(f(a, b), f(a, z'''')), z')))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(a, f(b, f(a, z'''')))), z''))
F(f(a, a), f(f(f(y', z'), z'''), z'0)) -> F(a, f(f(b, f(f(f(a, y'), z'), z''')), z'0))
F(f(a, a), f(f(f(y'''', z''''), z'0'), z)) -> F(f(a, a), f(f(y'''', z''''), z'0'))
F(f(a, a), f(a, z'''')) -> F(a, f(b, f(f(f(a, b), a), z'''')))
F(f(a, a), f(a, z''''')) -> F(a, f(f(b, f(a, f(b, a))), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, a), f(b, a)), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(b, f(f(a, b), a)), z'''''))
F(f(a, a), f(a, z'''')) -> F(a, f(f(f(b, a), b), f(a, z'''')))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, f(a, b)), a), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, a), f(b, a)), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(f(b, a), b), a), z'''''))
F(f(a, a), f(y''', z'')) -> F(a, f(f(f(b, a), y'''), z''))
F(f(a, a), f(f(a, z''''), z)) -> F(f(a, a), f(a, z''''))
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(f(f(y', z'), z''''), z'0)) -> F(a, f(f(f(b, f(f(a, y'), z')), z''''), z'0))

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), f(f(f(y', z'), z'''), z'0)) -> F(a, f(f(b, f(f(f(a, y'), z'), z''')), z'0))

three new Dependency Pairs are created:

F(f(a, a), f(f(f(y'', z''), z''''), z'0)) -> F(a, f(f(f(b, f(f(a, y''), z'')), z''''), z'0))
F(f(a, a), f(f(f(a, z''), z'''), z'0)) -> F(a, f(f(b, f(f(a, f(b, f(a, z''))), z''')), z'0))
F(f(a, a), f(f(f(f(y'', z''), z'), z'''), z'0)) -> F(a, f(f(b, f(f(f(f(a, y''), z''), z'), z''')), z'0))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 29

                 ↳Narrowing Transformation

Dependency Pairs:

F(f(a, a), f(f(f(f(y'', z''), z'), z'''), z'0)) -> F(a, f(f(b, f(f(f(f(a, y''), z''), z'), z''')), z'0))
F(f(a, a), f(f(f(a, z''), z'''), z'0)) -> F(a, f(f(b, f(f(a, f(b, f(a, z''))), z''')), z'0))
F(f(a, a), f(f(f(y'', z''), z''''), z'0)) -> F(a, f(f(f(b, f(f(a, y''), z'')), z''''), z'0))
F(f(a, a), f(f(a, z'''''), z'0)) -> F(a, f(f(b, f(f(a, b), f(a, z'''''))), z'0))
F(f(a, a), f(f(a, z'''''), z'0)) -> F(a, f(f(f(b, a), f(b, f(a, z'''''))), z'0))
F(f(a, a), f(f(f(y', z'), z''''), z'0)) -> F(a, f(f(f(b, f(f(a, y'), z')), z''''), z'0))
F(f(a, a), f(f(y''''', z''''), z'0)) -> F(a, f(f(f(f(b, a), y'''''), z''''), z'0))
F(f(a, a), f(f(f(f(y', z1), z0), z'), z'')) -> F(a, f(f(b, f(f(f(f(a, y'), z1), z0), z')), z''))
F(f(a, a), f(f(f(a, z0'), z'), z'')) -> F(a, f(f(b, f(f(a, f(b, f(a, z0'))), z')), z''))
F(f(a, a), f(f(f(y''', z0'), z'''), z'')) -> F(a, f(f(f(b, f(f(a, y'''), z0')), z'''), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(a, f(f(b, a), z''''))), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(f(a, b), f(a, z''''))), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(f(b, a), f(b, f(a, z''''))), z''))
F(f(a, a), f(f(f(y', z'), z'''), z'')) -> F(a, f(f(f(b, f(f(a, y'), z')), z'''), z''))
F(f(a, a), f(f(y''', z'''), z'')) -> F(a, f(f(f(f(b, a), y'''), z'''), z''))
F(f(a, a), f(f(f(f(y'', z1), z0), z''), z')) -> F(a, f(b, f(f(f(f(f(a, y''), z1), z0), z''), z')))
F(f(a, a), f(f(f(a, z0'), z''), z')) -> F(a, f(b, f(f(f(a, f(b, f(a, z0'))), z''), z')))
F(f(a, a), f(f(f(y'', z0'), z'''), z'0)) -> F(a, f(f(b, f(f(f(a, y''), z0'), z''')), z'0))
F(f(a, a), f(f(a, z''''), z')) -> F(a, f(b, f(f(a, f(f(b, a), z'''')), z')))
F(f(a, a), f(f(a, z''''), z')) -> F(a, f(b, f(f(f(a, b), f(a, z'''')), z')))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(a, f(b, f(a, z'''')))), z''))
F(f(a, a), f(f(f(y'''', z''''), z'0'), z)) -> F(f(a, a), f(f(y'''', z''''), z'0'))
F(f(a, a), f(a, z'''')) -> F(a, f(b, f(f(f(a, b), a), z'''')))
F(f(a, a), f(a, z''''')) -> F(a, f(f(b, f(a, f(b, a))), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, a), f(b, a)), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(b, f(f(a, b), a)), z'''''))
F(f(a, a), f(a, z'''')) -> F(a, f(f(f(b, a), b), f(a, z'''')))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, f(a, b)), a), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, a), f(b, a)), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(f(b, a), b), a), z'''''))
F(f(a, a), f(y''', z'')) -> F(a, f(f(f(b, a), y'''), z''))
F(f(a, a), f(f(a, z''''), z)) -> F(f(a, a), f(a, z''''))
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(f(a, z'''''), z'0)) -> F(a, f(f(b, f(a, f(f(b, a), z'''''))), z'0))

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

three new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 30

                 ↳Narrowing Transformation

Dependency Pairs:

F(f(a, a), f(f(a, z'''''), z'')) -> F(a, f(f(b, f(a, f(f(b, a), z'''''))), z''))
F(f(a, a), f(f(a, z'''''), z'')) -> F(a, f(f(b, f(f(a, b), f(a, z'''''))), z''))
F(f(a, a), f(f(a, z'''''), z'')) -> F(a, f(f(f(b, a), f(b, f(a, z'''''))), z''))
F(f(a, a), f(f(f(a, z''), z'''), z'0)) -> F(a, f(f(b, f(f(a, f(b, f(a, z''))), z''')), z'0))
F(f(a, a), f(f(f(y'', z''), z''''), z'0)) -> F(a, f(f(f(b, f(f(a, y''), z'')), z''''), z'0))
F(f(a, a), f(f(a, z'''''), z'0)) -> F(a, f(f(b, f(a, f(f(b, a), z'''''))), z'0))
F(f(a, a), f(f(a, z'''''), z'0)) -> F(a, f(f(b, f(f(a, b), f(a, z'''''))), z'0))
F(f(a, a), f(f(a, z'''''), z'0)) -> F(a, f(f(f(b, a), f(b, f(a, z'''''))), z'0))
F(f(a, a), f(f(f(y', z'), z''''), z'0)) -> F(a, f(f(f(b, f(f(a, y'), z')), z''''), z'0))
F(f(a, a), f(f(y''''', z''''), z'0)) -> F(a, f(f(f(f(b, a), y'''''), z''''), z'0))
F(f(a, a), f(f(f(f(y', z1), z0), z'), z'')) -> F(a, f(f(b, f(f(f(f(a, y'), z1), z0), z')), z''))
F(f(a, a), f(f(f(a, z0'), z'), z'')) -> F(a, f(f(b, f(f(a, f(b, f(a, z0'))), z')), z''))
F(f(a, a), f(f(f(y''', z0'), z'''), z'')) -> F(a, f(f(f(b, f(f(a, y'''), z0')), z'''), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(a, f(f(b, a), z''''))), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(f(a, b), f(a, z''''))), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(f(b, a), f(b, f(a, z''''))), z''))
F(f(a, a), f(f(f(y', z'), z'''), z'')) -> F(a, f(f(f(b, f(f(a, y'), z')), z'''), z''))
F(f(a, a), f(f(y''', z'''), z'')) -> F(a, f(f(f(f(b, a), y'''), z'''), z''))
F(f(a, a), f(f(f(f(y'', z1), z0), z''), z')) -> F(a, f(b, f(f(f(f(f(a, y''), z1), z0), z''), z')))
F(f(a, a), f(f(f(a, z0'), z''), z')) -> F(a, f(b, f(f(f(a, f(b, f(a, z0'))), z''), z')))
F(f(a, a), f(f(f(y'', z0'), z'''), z'0)) -> F(a, f(f(b, f(f(f(a, y''), z0'), z''')), z'0))
F(f(a, a), f(f(a, z''''), z')) -> F(a, f(b, f(f(a, f(f(b, a), z'''')), z')))
F(f(a, a), f(f(a, z''''), z')) -> F(a, f(b, f(f(f(a, b), f(a, z'''')), z')))
F(f(a, a), f(f(f(y'''', z''''), z'0'), z)) -> F(f(a, a), f(f(y'''', z''''), z'0'))
F(f(a, a), f(a, z'''')) -> F(a, f(b, f(f(f(a, b), a), z'''')))
F(f(a, a), f(a, z''''')) -> F(a, f(f(b, f(a, f(b, a))), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, a), f(b, a)), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(b, f(f(a, b), a)), z'''''))
F(f(a, a), f(a, z'''')) -> F(a, f(f(f(b, a), b), f(a, z'''')))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, f(a, b)), a), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, a), f(b, a)), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(f(b, a), b), a), z'''''))
F(f(a, a), f(y''', z'')) -> F(a, f(f(f(b, a), y'''), z''))
F(f(a, a), f(f(a, z''''), z)) -> F(f(a, a), f(a, z''''))
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(f(f(f(y'', z''), z'), z'''), z'0)) -> F(a, f(f(b, f(f(f(f(a, y''), z''), z'), z''')), z'0))

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

two new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 31

                 ↳Narrowing Transformation

Dependency Pairs:

F(f(a, a), f(f(a, z'''''), z')) -> F(a, f(b, f(f(f(f(a, b), a), z'''''), z')))
F(f(a, a), f(f(a, z'''''), z'')) -> F(a, f(f(b, f(f(a, b), f(a, z'''''))), z''))
F(f(a, a), f(f(a, z'''''), z'')) -> F(a, f(f(b, f(f(a, b), f(a, z'''''))), z''))
F(f(a, a), f(f(a, z'''''), z'')) -> F(a, f(f(f(b, a), f(b, f(a, z'''''))), z''))
F(f(a, a), f(f(f(f(y'', z''), z'), z'''), z'0)) -> F(a, f(f(b, f(f(f(f(a, y''), z''), z'), z''')), z'0))
F(f(a, a), f(f(f(a, z''), z'''), z'0)) -> F(a, f(f(b, f(f(a, f(b, f(a, z''))), z''')), z'0))
F(f(a, a), f(f(f(y'', z''), z''''), z'0)) -> F(a, f(f(f(b, f(f(a, y''), z'')), z''''), z'0))
F(f(a, a), f(f(a, z'''''), z'0)) -> F(a, f(f(b, f(a, f(f(b, a), z'''''))), z'0))
F(f(a, a), f(f(a, z'''''), z'0)) -> F(a, f(f(b, f(f(a, b), f(a, z'''''))), z'0))
F(f(a, a), f(f(a, z'''''), z'0)) -> F(a, f(f(f(b, a), f(b, f(a, z'''''))), z'0))
F(f(a, a), f(f(f(y', z'), z''''), z'0)) -> F(a, f(f(f(b, f(f(a, y'), z')), z''''), z'0))
F(f(a, a), f(f(y''''', z''''), z'0)) -> F(a, f(f(f(f(b, a), y'''''), z''''), z'0))
F(f(a, a), f(f(f(f(y', z1), z0), z'), z'')) -> F(a, f(f(b, f(f(f(f(a, y'), z1), z0), z')), z''))
F(f(a, a), f(f(f(a, z0'), z'), z'')) -> F(a, f(f(b, f(f(a, f(b, f(a, z0'))), z')), z''))
F(f(a, a), f(f(f(y''', z0'), z'''), z'')) -> F(a, f(f(f(b, f(f(a, y'''), z0')), z'''), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(a, f(f(b, a), z''''))), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(f(a, b), f(a, z''''))), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(f(b, a), f(b, f(a, z''''))), z''))
F(f(a, a), f(f(f(y', z'), z'''), z'')) -> F(a, f(f(f(b, f(f(a, y'), z')), z'''), z''))
F(f(a, a), f(f(y''', z'''), z'')) -> F(a, f(f(f(f(b, a), y'''), z'''), z''))
F(f(a, a), f(f(f(f(y'', z1), z0), z''), z')) -> F(a, f(b, f(f(f(f(f(a, y''), z1), z0), z''), z')))
F(f(a, a), f(f(f(a, z0'), z''), z')) -> F(a, f(b, f(f(f(a, f(b, f(a, z0'))), z''), z')))
F(f(a, a), f(f(f(y'', z0'), z'''), z'0)) -> F(a, f(f(b, f(f(f(a, y''), z0'), z''')), z'0))
F(f(a, a), f(f(a, z''''), z')) -> F(a, f(b, f(f(a, f(f(b, a), z'''')), z')))
F(f(a, a), f(f(f(y'''', z''''), z'0'), z)) -> F(f(a, a), f(f(y'''', z''''), z'0'))
F(f(a, a), f(a, z'''')) -> F(a, f(b, f(f(f(a, b), a), z'''')))
F(f(a, a), f(a, z''''')) -> F(a, f(f(b, f(a, f(b, a))), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, a), f(b, a)), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(b, f(f(a, b), a)), z'''''))
F(f(a, a), f(a, z'''')) -> F(a, f(f(f(b, a), b), f(a, z'''')))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, f(a, b)), a), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, a), f(b, a)), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(f(b, a), b), a), z'''''))
F(f(a, a), f(y''', z'')) -> F(a, f(f(f(b, a), y'''), z''))
F(f(a, a), f(f(a, z''''), z)) -> F(f(a, a), f(a, z''''))
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(f(a, z'''''), z'')) -> F(a, f(f(b, f(a, f(f(b, a), z'''''))), 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

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

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

two new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 32

                 ↳Narrowing Transformation

Dependency Pairs:

F(f(a, a), f(f(a, z'''''), z')) -> F(a, f(b, f(f(f(a, f(b, a)), z'''''), z')))
F(f(a, a), f(f(a, z'''''), z'')) -> F(a, f(f(b, f(a, f(f(b, a), z'''''))), z''))
F(f(a, a), f(f(a, z'''''), z'')) -> F(a, f(f(b, f(f(a, b), f(a, z'''''))), z''))
F(f(a, a), f(f(a, z'''''), z'')) -> F(a, f(f(b, f(a, f(f(b, a), z'''''))), z''))
F(f(a, a), f(f(a, z'''''), z'')) -> F(a, f(f(b, f(f(a, b), f(a, z'''''))), z''))
F(f(a, a), f(f(a, z'''''), z'')) -> F(a, f(f(f(b, a), f(b, f(a, z'''''))), z''))
F(f(a, a), f(f(f(f(y'', z''), z'), z'''), z'0)) -> F(a, f(f(b, f(f(f(f(a, y''), z''), z'), z''')), z'0))
F(f(a, a), f(f(f(a, z''), z'''), z'0)) -> F(a, f(f(b, f(f(a, f(b, f(a, z''))), z''')), z'0))
F(f(a, a), f(f(f(y'', z''), z''''), z'0)) -> F(a, f(f(f(b, f(f(a, y''), z'')), z''''), z'0))
F(f(a, a), f(f(a, z'''''), z'0)) -> F(a, f(f(b, f(a, f(f(b, a), z'''''))), z'0))
F(f(a, a), f(f(a, z'''''), z'0)) -> F(a, f(f(b, f(f(a, b), f(a, z'''''))), z'0))
F(f(a, a), f(f(a, z'''''), z'0)) -> F(a, f(f(f(b, a), f(b, f(a, z'''''))), z'0))
F(f(a, a), f(f(f(y', z'), z''''), z'0)) -> F(a, f(f(f(b, f(f(a, y'), z')), z''''), z'0))
F(f(a, a), f(f(y''''', z''''), z'0)) -> F(a, f(f(f(f(b, a), y'''''), z''''), z'0))
F(f(a, a), f(f(f(f(y', z1), z0), z'), z'')) -> F(a, f(f(b, f(f(f(f(a, y'), z1), z0), z')), z''))
F(f(a, a), f(f(f(a, z0'), z'), z'')) -> F(a, f(f(b, f(f(a, f(b, f(a, z0'))), z')), z''))
F(f(a, a), f(f(f(y''', z0'), z'''), z'')) -> F(a, f(f(f(b, f(f(a, y'''), z0')), z'''), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(a, f(f(b, a), z''''))), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(f(a, b), f(a, z''''))), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(f(b, a), f(b, f(a, z''''))), z''))
F(f(a, a), f(f(f(y', z'), z'''), z'')) -> F(a, f(f(f(b, f(f(a, y'), z')), z'''), z''))
F(f(a, a), f(f(y''', z'''), z'')) -> F(a, f(f(f(f(b, a), y'''), z'''), z''))
F(f(a, a), f(f(f(f(y'', z1), z0), z''), z')) -> F(a, f(b, f(f(f(f(f(a, y''), z1), z0), z''), z')))
F(f(a, a), f(f(f(a, z0'), z''), z')) -> F(a, f(b, f(f(f(a, f(b, f(a, z0'))), z''), z')))
F(f(a, a), f(f(f(y'', z0'), z'''), z'0)) -> F(a, f(f(b, f(f(f(a, y''), z0'), z''')), z'0))
F(f(a, a), f(f(f(y'''', z''''), z'0'), z)) -> F(f(a, a), f(f(y'''', z''''), z'0'))
F(f(a, a), f(a, z'''')) -> F(a, f(b, f(f(f(a, b), a), z'''')))
F(f(a, a), f(a, z''''')) -> F(a, f(f(b, f(a, f(b, a))), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, a), f(b, a)), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(b, f(f(a, b), a)), z'''''))
F(f(a, a), f(a, z'''')) -> F(a, f(f(f(b, a), b), f(a, z'''')))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, f(a, b)), a), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, a), f(b, a)), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(f(b, a), b), a), z'''''))
F(f(a, a), f(y''', z'')) -> F(a, f(f(f(b, a), y'''), z''))
F(f(a, a), f(f(a, z''''), z)) -> F(f(a, a), f(a, z''''))
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(f(a, z'''''), z')) -> F(a, f(b, f(f(f(f(a, b), a), z'''''), 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

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

F(f(a, a), f(f(f(y'', z0'), z'''), z'0)) -> F(a, f(f(b, f(f(f(a, y''), z0'), z''')), z'0))

three new Dependency Pairs are created:

F(f(a, a), f(f(f(y''', z0''), z''''), z'0)) -> F(a, f(f(f(b, f(f(a, y'''), z0'')), z''''), z'0))
F(f(a, a), f(f(f(a, z0''), z'''), z'0)) -> F(a, f(f(b, f(f(a, f(b, f(a, z0''))), z''')), z'0))
F(f(a, a), f(f(f(f(y', z'), z0'), z'''), z'0)) -> F(a, f(f(b, f(f(f(f(a, y'), z'), z0'), z''')), z'0))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 33

                 ↳Narrowing Transformation

Dependency Pairs:

F(f(a, a), f(f(f(f(y', z'), z0'), z'''), z'0)) -> F(a, f(f(b, f(f(f(f(a, y'), z'), z0'), z''')), z'0))
F(f(a, a), f(f(f(a, z0''), z'''), z'0)) -> F(a, f(f(b, f(f(a, f(b, f(a, z0''))), z''')), z'0))
F(f(a, a), f(f(f(y''', z0''), z''''), z'0)) -> F(a, f(f(f(b, f(f(a, y'''), z0'')), z''''), z'0))
F(f(a, a), f(f(a, z'''''), z'')) -> F(a, f(f(b, f(a, f(f(b, a), z'''''))), z''))
F(f(a, a), f(f(a, z'''''), z')) -> F(a, f(b, f(f(f(f(a, b), a), z'''''), z')))
F(f(a, a), f(f(a, z'''''), z'')) -> F(a, f(f(b, f(f(a, b), f(a, z'''''))), z''))
F(f(a, a), f(f(a, z'''''), z'')) -> F(a, f(f(b, f(a, f(f(b, a), z'''''))), z''))
F(f(a, a), f(f(a, z'''''), z'')) -> F(a, f(f(b, f(f(a, b), f(a, z'''''))), z''))
F(f(a, a), f(f(a, z'''''), z'')) -> F(a, f(f(f(b, a), f(b, f(a, z'''''))), z''))
F(f(a, a), f(f(f(f(y'', z''), z'), z'''), z'0)) -> F(a, f(f(b, f(f(f(f(a, y''), z''), z'), z''')), z'0))
F(f(a, a), f(f(f(a, z''), z'''), z'0)) -> F(a, f(f(b, f(f(a, f(b, f(a, z''))), z''')), z'0))
F(f(a, a), f(f(f(y'', z''), z''''), z'0)) -> F(a, f(f(f(b, f(f(a, y''), z'')), z''''), z'0))
F(f(a, a), f(f(a, z'''''), z'0)) -> F(a, f(f(b, f(a, f(f(b, a), z'''''))), z'0))
F(f(a, a), f(f(a, z'''''), z'0)) -> F(a, f(f(b, f(f(a, b), f(a, z'''''))), z'0))
F(f(a, a), f(f(a, z'''''), z'0)) -> F(a, f(f(f(b, a), f(b, f(a, z'''''))), z'0))
F(f(a, a), f(f(f(y', z'), z''''), z'0)) -> F(a, f(f(f(b, f(f(a, y'), z')), z''''), z'0))
F(f(a, a), f(f(y''''', z''''), z'0)) -> F(a, f(f(f(f(b, a), y'''''), z''''), z'0))
F(f(a, a), f(f(f(f(y', z1), z0), z'), z'')) -> F(a, f(f(b, f(f(f(f(a, y'), z1), z0), z')), z''))
F(f(a, a), f(f(f(a, z0'), z'), z'')) -> F(a, f(f(b, f(f(a, f(b, f(a, z0'))), z')), z''))
F(f(a, a), f(f(f(y''', z0'), z'''), z'')) -> F(a, f(f(f(b, f(f(a, y'''), z0')), z'''), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(a, f(f(b, a), z''''))), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(f(a, b), f(a, z''''))), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(f(b, a), f(b, f(a, z''''))), z''))
F(f(a, a), f(f(f(y', z'), z'''), z'')) -> F(a, f(f(f(b, f(f(a, y'), z')), z'''), z''))
F(f(a, a), f(f(y''', z'''), z'')) -> F(a, f(f(f(f(b, a), y'''), z'''), z''))
F(f(a, a), f(f(f(f(y'', z1), z0), z''), z')) -> F(a, f(b, f(f(f(f(f(a, y''), z1), z0), z''), z')))
F(f(a, a), f(f(f(a, z0'), z''), z')) -> F(a, f(b, f(f(f(a, f(b, f(a, z0'))), z''), z')))
F(f(a, a), f(f(f(y'''', z''''), z'0'), z)) -> F(f(a, a), f(f(y'''', z''''), z'0'))
F(f(a, a), f(a, z'''')) -> F(a, f(b, f(f(f(a, b), a), z'''')))
F(f(a, a), f(a, z''''')) -> F(a, f(f(b, f(a, f(b, a))), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, a), f(b, a)), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(b, f(f(a, b), a)), z'''''))
F(f(a, a), f(a, z'''')) -> F(a, f(f(f(b, a), b), f(a, z'''')))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, f(a, b)), a), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, a), f(b, a)), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(f(b, a), b), a), z'''''))
F(f(a, a), f(y''', z'')) -> F(a, f(f(f(b, a), y'''), z''))
F(f(a, a), f(f(a, z''''), z)) -> F(f(a, a), f(a, z''''))
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(f(a, z'''''), z')) -> F(a, f(b, f(f(f(a, f(b, a)), z'''''), 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

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

F(f(a, a), f(f(f(a, z0'), z''), z')) -> F(a, f(b, f(f(f(a, f(b, f(a, z0'))), z''), z')))

three new Dependency Pairs are created:

F(f(a, a), f(f(f(a, z0''), z'''), z'0)) -> F(a, f(f(b, f(f(a, f(b, f(a, z0''))), z''')), z'0))
F(f(a, a), f(f(f(a, z0''), z''), z')) -> F(a, f(b, f(f(f(f(a, b), f(a, z0'')), z''), z')))
F(f(a, a), f(f(f(a, z0''), z''), z')) -> F(a, f(b, f(f(f(a, f(f(b, a), z0'')), z''), z')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 34

                 ↳Narrowing Transformation

Dependency Pairs:

F(f(a, a), f(f(f(a, z0''), z''), z')) -> F(a, f(b, f(f(f(a, f(f(b, a), z0'')), z''), z')))
F(f(a, a), f(f(f(a, z0''), z''), z')) -> F(a, f(b, f(f(f(f(a, b), f(a, z0'')), z''), z')))
F(f(a, a), f(f(f(a, z0''), z'''), z'0)) -> F(a, f(f(b, f(f(a, f(b, f(a, z0''))), z''')), z'0))
F(f(a, a), f(f(f(a, z0''), z'''), z'0)) -> F(a, f(f(b, f(f(a, f(b, f(a, z0''))), z''')), z'0))
F(f(a, a), f(f(f(y''', z0''), z''''), z'0)) -> F(a, f(f(f(b, f(f(a, y'''), z0'')), z''''), z'0))
F(f(a, a), f(f(a, z'''''), z')) -> F(a, f(b, f(f(f(a, f(b, a)), z'''''), z')))
F(f(a, a), f(f(a, z'''''), z'')) -> F(a, f(f(b, f(a, f(f(b, a), z'''''))), z''))
F(f(a, a), f(f(a, z'''''), z')) -> F(a, f(b, f(f(f(f(a, b), a), z'''''), z')))
F(f(a, a), f(f(a, z'''''), z'')) -> F(a, f(f(b, f(f(a, b), f(a, z'''''))), z''))
F(f(a, a), f(f(a, z'''''), z'')) -> F(a, f(f(b, f(a, f(f(b, a), z'''''))), z''))
F(f(a, a), f(f(a, z'''''), z'')) -> F(a, f(f(b, f(f(a, b), f(a, z'''''))), z''))
F(f(a, a), f(f(a, z'''''), z'')) -> F(a, f(f(f(b, a), f(b, f(a, z'''''))), z''))
F(f(a, a), f(f(f(f(y'', z''), z'), z'''), z'0)) -> F(a, f(f(b, f(f(f(f(a, y''), z''), z'), z''')), z'0))
F(f(a, a), f(f(f(a, z''), z'''), z'0)) -> F(a, f(f(b, f(f(a, f(b, f(a, z''))), z''')), z'0))
F(f(a, a), f(f(f(y'', z''), z''''), z'0)) -> F(a, f(f(f(b, f(f(a, y''), z'')), z''''), z'0))
F(f(a, a), f(f(a, z'''''), z'0)) -> F(a, f(f(b, f(a, f(f(b, a), z'''''))), z'0))
F(f(a, a), f(f(a, z'''''), z'0)) -> F(a, f(f(b, f(f(a, b), f(a, z'''''))), z'0))
F(f(a, a), f(f(a, z'''''), z'0)) -> F(a, f(f(f(b, a), f(b, f(a, z'''''))), z'0))
F(f(a, a), f(f(f(y', z'), z''''), z'0)) -> F(a, f(f(f(b, f(f(a, y'), z')), z''''), z'0))
F(f(a, a), f(f(y''''', z''''), z'0)) -> F(a, f(f(f(f(b, a), y'''''), z''''), z'0))
F(f(a, a), f(f(f(f(y', z1), z0), z'), z'')) -> F(a, f(f(b, f(f(f(f(a, y'), z1), z0), z')), z''))
F(f(a, a), f(f(f(a, z0'), z'), z'')) -> F(a, f(f(b, f(f(a, f(b, f(a, z0'))), z')), z''))
F(f(a, a), f(f(f(y''', z0'), z'''), z'')) -> F(a, f(f(f(b, f(f(a, y'''), z0')), z'''), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(a, f(f(b, a), z''''))), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(f(a, b), f(a, z''''))), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(f(b, a), f(b, f(a, z''''))), z''))
F(f(a, a), f(f(f(y', z'), z'''), z'')) -> F(a, f(f(f(b, f(f(a, y'), z')), z'''), z''))
F(f(a, a), f(f(y''', z'''), z'')) -> F(a, f(f(f(f(b, a), y'''), z'''), z''))
F(f(a, a), f(f(f(f(y'', z1), z0), z''), z')) -> F(a, f(b, f(f(f(f(f(a, y''), z1), z0), z''), z')))
F(f(a, a), f(f(f(y'''', z''''), z'0'), z)) -> F(f(a, a), f(f(y'''', z''''), z'0'))
F(f(a, a), f(a, z'''')) -> F(a, f(b, f(f(f(a, b), a), z'''')))
F(f(a, a), f(a, z''''')) -> F(a, f(f(b, f(a, f(b, a))), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, a), f(b, a)), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(b, f(f(a, b), a)), z'''''))
F(f(a, a), f(a, z'''')) -> F(a, f(f(f(b, a), b), f(a, z'''')))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, f(a, b)), a), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, a), f(b, a)), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(f(b, a), b), a), z'''''))
F(f(a, a), f(y''', z'')) -> F(a, f(f(f(b, a), y'''), z''))
F(f(a, a), f(f(a, z''''), z)) -> F(f(a, a), f(a, z''''))
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(f(f(f(y', z'), z0'), z'''), z'0)) -> F(a, f(f(b, f(f(f(f(a, y'), z'), z0'), z''')), z'0))

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), f(f(f(f(y'', z1), z0), z''), z')) -> F(a, f(b, f(f(f(f(f(a, y''), z1), z0), z''), z')))

three new Dependency Pairs are created:

F(f(a, a), f(f(f(f(y''', z1'), z0'), z'''), z'0)) -> F(a, f(f(b, f(f(f(f(a, y'''), z1'), z0'), z''')), z'0))
F(f(a, a), f(f(f(f(a, z1'), z0), z''), z')) -> F(a, f(b, f(f(f(f(a, f(b, f(a, z1'))), z0), z''), z')))
F(f(a, a), f(f(f(f(f(y', z2), z1), z0), z''), z')) -> F(a, f(b, f(f(f(f(f(f(a, y'), z2), z1), z0), z''), z')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 35

                 ↳Remaining Obligation(s)

The following remains to be proven:
Dependency Pairs:

F(f(a, a), f(f(f(f(f(y', z2), z1), z0), z''), z')) -> F(a, f(b, f(f(f(f(f(f(a, y'), z2), z1), z0), z''), z')))
F(f(a, a), f(f(f(f(a, z1'), z0), z''), z')) -> F(a, f(b, f(f(f(f(a, f(b, f(a, z1'))), z0), z''), z')))
F(f(a, a), f(f(f(f(y''', z1'), z0'), z'''), z'0)) -> F(a, f(f(b, f(f(f(f(a, y'''), z1'), z0'), z''')), z'0))
F(f(a, a), f(f(f(a, z0''), z''), z')) -> F(a, f(b, f(f(f(f(a, b), f(a, z0'')), z''), z')))
F(f(a, a), f(f(f(a, z0''), z'''), z'0)) -> F(a, f(f(b, f(f(a, f(b, f(a, z0''))), z''')), z'0))
F(f(a, a), f(f(f(f(y', z'), z0'), z'''), z'0)) -> F(a, f(f(b, f(f(f(f(a, y'), z'), z0'), z''')), z'0))
F(f(a, a), f(f(f(a, z0''), z'''), z'0)) -> F(a, f(f(b, f(f(a, f(b, f(a, z0''))), z''')), z'0))
F(f(a, a), f(f(f(y''', z0''), z''''), z'0)) -> F(a, f(f(f(b, f(f(a, y'''), z0'')), z''''), z'0))
F(f(a, a), f(f(a, z'''''), z')) -> F(a, f(b, f(f(f(a, f(b, a)), z'''''), z')))
F(f(a, a), f(f(a, z'''''), z'')) -> F(a, f(f(b, f(a, f(f(b, a), z'''''))), z''))
F(f(a, a), f(f(a, z'''''), z')) -> F(a, f(b, f(f(f(f(a, b), a), z'''''), z')))
F(f(a, a), f(f(a, z'''''), z'')) -> F(a, f(f(b, f(f(a, b), f(a, z'''''))), z''))
F(f(a, a), f(f(a, z'''''), z'')) -> F(a, f(f(b, f(a, f(f(b, a), z'''''))), z''))
F(f(a, a), f(f(a, z'''''), z'')) -> F(a, f(f(b, f(f(a, b), f(a, z'''''))), z''))
F(f(a, a), f(f(a, z'''''), z'')) -> F(a, f(f(f(b, a), f(b, f(a, z'''''))), z''))
F(f(a, a), f(f(f(f(y'', z''), z'), z'''), z'0)) -> F(a, f(f(b, f(f(f(f(a, y''), z''), z'), z''')), z'0))
F(f(a, a), f(f(f(a, z''), z'''), z'0)) -> F(a, f(f(b, f(f(a, f(b, f(a, z''))), z''')), z'0))
F(f(a, a), f(f(f(y'', z''), z''''), z'0)) -> F(a, f(f(f(b, f(f(a, y''), z'')), z''''), z'0))
F(f(a, a), f(f(a, z'''''), z'0)) -> F(a, f(f(b, f(a, f(f(b, a), z'''''))), z'0))
F(f(a, a), f(f(a, z'''''), z'0)) -> F(a, f(f(b, f(f(a, b), f(a, z'''''))), z'0))
F(f(a, a), f(f(a, z'''''), z'0)) -> F(a, f(f(f(b, a), f(b, f(a, z'''''))), z'0))
F(f(a, a), f(f(f(y', z'), z''''), z'0)) -> F(a, f(f(f(b, f(f(a, y'), z')), z''''), z'0))
F(f(a, a), f(f(y''''', z''''), z'0)) -> F(a, f(f(f(f(b, a), y'''''), z''''), z'0))
F(f(a, a), f(f(f(f(y', z1), z0), z'), z'')) -> F(a, f(f(b, f(f(f(f(a, y'), z1), z0), z')), z''))
F(f(a, a), f(f(f(a, z0'), z'), z'')) -> F(a, f(f(b, f(f(a, f(b, f(a, z0'))), z')), z''))
F(f(a, a), f(f(f(y''', z0'), z'''), z'')) -> F(a, f(f(f(b, f(f(a, y'''), z0')), z'''), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(a, f(f(b, a), z''''))), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(b, f(f(a, b), f(a, z''''))), z''))
F(f(a, a), f(f(a, z''''), z'')) -> F(a, f(f(f(b, a), f(b, f(a, z''''))), z''))
F(f(a, a), f(f(f(y', z'), z'''), z'')) -> F(a, f(f(f(b, f(f(a, y'), z')), z'''), z''))
F(f(a, a), f(f(y''', z'''), z'')) -> F(a, f(f(f(f(b, a), y'''), z'''), z''))
F(f(a, a), f(f(f(y'''', z''''), z'0'), z)) -> F(f(a, a), f(f(y'''', z''''), z'0'))
F(f(a, a), f(a, z'''')) -> F(a, f(b, f(f(f(a, b), a), z'''')))
F(f(a, a), f(a, z''''')) -> F(a, f(f(b, f(a, f(b, a))), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, a), f(b, a)), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(b, f(f(a, b), a)), z'''''))
F(f(a, a), f(a, z'''')) -> F(a, f(f(f(b, a), b), f(a, z'''')))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, f(a, b)), a), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(b, a), f(b, a)), z'''''))
F(f(a, a), f(a, z''''')) -> F(a, f(f(f(f(b, a), b), a), z'''''))
F(f(a, a), f(y''', z'')) -> F(a, f(f(f(b, a), y'''), z''))
F(f(a, a), f(f(a, z''''), z)) -> F(f(a, a), f(a, z''''))
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(f(f(a, z0''), z''), z')) -> F(a, f(b, f(f(f(a, f(f(b, a), z0'')), z''), 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:
0:13 minutes

POL(b)	= 0
POL(a)	= 1
POL(f(x₁, x₂))	= x₁
POL(F(x₁, x₂))	= x₁