Termination Proof using AProVE

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

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

F(g(x), a) -> F(x, g(a))
F(g(x), g(y)) -> H(g(y), x, g(y))
H(g(x), y, z) -> F(y, h(x, y, z))
H(g(x), y, z) -> H(x, y, z)

Furthermore, R contains one SCC.

     ↳DPs

       →DP Problem 1

         ↳Narrowing Transformation

Dependency Pairs:

H(g(x), y, z) -> H(x, y, z)
H(g(x), y, z) -> F(y, h(x, y, z))
F(g(x), g(y)) -> H(g(y), x, g(y))
F(g(x), a) -> F(x, g(a))

Rules:

f(a, g(y)) -> g(g(y))
f(g(x), a) -> f(x, g(a))
f(g(x), g(y)) -> h(g(y), x, g(y))
h(g(x), y, z) -> f(y, h(x, y, z))
h(a, y, z) -> z

Strategy:

innermost

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

H(g(x), y, z) -> F(y, h(x, y, z))

two new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Narrowing Transformation

Dependency Pairs:

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

Rules:

f(a, g(y)) -> g(g(y))
f(g(x), a) -> f(x, g(a))
f(g(x), g(y)) -> h(g(y), x, g(y))
h(g(x), y, z) -> f(y, h(x, y, z))
h(a, y, z) -> z

Strategy:

innermost

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

H(g(g(x'')), y'', z'') -> F(y'', f(y'', h(x'', y'', z'')))

two new Dependency Pairs are created:

H(g(g(g(x'))), y''', z''') -> F(y''', f(y''', f(y''', h(x', y''', z'''))))
H(g(g(a)), y''', z''') -> F(y''', f(y''', 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:

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

Rules:

f(a, g(y)) -> g(g(y))
f(g(x), a) -> f(x, g(a))
f(g(x), g(y)) -> h(g(y), x, g(y))
h(g(x), y, z) -> f(y, h(x, y, z))
h(a, y, z) -> z

Strategy:

innermost

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

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

one new Dependency Pair is created:

F(g(g(x'')), a) -> F(g(x''), g(a))

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:

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

Rules:

f(a, g(y)) -> g(g(y))
f(g(x), a) -> f(x, g(a))
f(g(x), g(y)) -> h(g(y), x, g(y))
h(g(x), y, z) -> f(y, h(x, y, z))
h(a, y, z) -> z

Strategy:

innermost

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

H(g(x), y, z) -> H(x, y, z)

four new Dependency Pairs are created:

H(g(g(x'')), y'', z'') -> H(g(x''), y'', z'')
H(g(g(a)), y', z') -> H(g(a), y', z')
H(g(g(g(g(x''')))), y', z') -> H(g(g(g(x'''))), y', z')
H(g(g(g(a))), y', z') -> H(g(g(a)), y', z')

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 5

                 ↳Forward Instantiation Transformation

Dependency Pairs:

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

Rules:

f(a, g(y)) -> g(g(y))
f(g(x), a) -> f(x, g(a))
f(g(x), g(y)) -> h(g(y), x, g(y))
h(g(x), y, z) -> f(y, h(x, y, z))
h(a, y, z) -> z

Strategy:

innermost

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

F(g(x), g(y)) -> H(g(y), x, g(y))

six new Dependency Pairs are created:

F(g(x'), g(a)) -> H(g(a), x', g(a))
F(g(x'), g(g(g(x''')))) -> H(g(g(g(x'''))), x', g(g(g(x'''))))
F(g(x'), g(g(a))) -> H(g(g(a)), x', g(g(a)))
F(g(x'), g(g(x''''))) -> H(g(g(x'''')), x', g(g(x'''')))
F(g(x'), g(g(g(g(x'''''))))) -> H(g(g(g(g(x''''')))), x', g(g(g(g(x''''')))))
F(g(x'), g(g(g(a)))) -> H(g(g(g(a))), x', g(g(g(a))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 6

                 ↳Forward Instantiation Transformation

Dependency Pairs:

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

Rules:

f(a, g(y)) -> g(g(y))
f(g(x), a) -> f(x, g(a))
f(g(x), g(y)) -> h(g(y), x, g(y))
h(g(x), y, z) -> f(y, h(x, y, z))
h(a, y, z) -> z

Strategy:

innermost

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

H(g(a), y'', z'') -> F(y'', z'')

seven new Dependency Pairs are created:

H(g(a), g(g(x'''')), a) -> F(g(g(x'''')), a)
H(g(a), g(x'''), g(a)) -> F(g(x'''), g(a))
H(g(a), g(x'''), g(g(g(x''''')))) -> F(g(x'''), g(g(g(x'''''))))
H(g(a), g(x'''), g(g(a))) -> F(g(x'''), g(g(a)))
H(g(a), g(x'''), g(g(x''''''))) -> F(g(x'''), g(g(x'''''')))
H(g(a), g(x'''), g(g(g(g(x'''''''))))) -> F(g(x'''), g(g(g(g(x''''''')))))
H(g(a), g(x'''), g(g(g(a)))) -> F(g(x'''), g(g(g(a))))

The transformation is resulting in two new DP problems:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 7

                 ↳Forward Instantiation Transformation

Dependency Pairs:

H(g(a), g(x'''), g(a)) -> F(g(x'''), g(a))
F(g(x'), g(a)) -> H(g(a), x', g(a))

Rules:

f(a, g(y)) -> g(g(y))
f(g(x), a) -> f(x, g(a))
f(g(x), g(y)) -> h(g(y), x, g(y))
h(g(x), y, z) -> f(y, h(x, y, z))
h(a, y, z) -> z

Strategy:

innermost

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

F(g(x'), g(a)) -> H(g(a), x', g(a))

one new Dependency Pair is created:

F(g(g(x''''')), g(a)) -> H(g(a), g(x'''''), g(a))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 9

                 ↳Argument Filtering and Ordering

Dependency Pairs:

F(g(g(x''''')), g(a)) -> H(g(a), g(x'''''), g(a))
H(g(a), g(x'''), g(a)) -> F(g(x'''), g(a))

Rules:

f(a, g(y)) -> g(g(y))
f(g(x), a) -> f(x, g(a))
f(g(x), g(y)) -> h(g(y), x, g(y))
h(g(x), y, z) -> f(y, h(x, y, z))
h(a, y, z) -> z

Strategy:

innermost

The following dependency pair can be strictly oriented:

H(g(a), g(x'''), g(a)) -> F(g(x'''), g(a))

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:

POL(g(x₁)) = 1 + x₁

POL(H(x₁, x₂, x₃)) = x₁ + x₂ + x₃

POL(a) = 0

POL(F(x₁, x₂)) = x₁ + x₂

resulting in one new DP problem.
Used Argument Filtering System:

H(x₁, x₂, x₃) -> H(x₁, x₂, x₃)
F(x₁, x₂) -> F(x₁, x₂)
g(x₁) -> g(x₁)

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 13

                 ↳Dependency Graph

Dependency Pair:

F(g(g(x''''')), g(a)) -> H(g(a), g(x'''''), g(a))

Rules:

f(a, g(y)) -> g(g(y))
f(g(x), a) -> f(x, g(a))
f(g(x), g(y)) -> h(g(y), x, g(y))
h(g(x), y, z) -> f(y, h(x, y, z))
h(a, y, z) -> z

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 8

                 ↳Forward Instantiation Transformation

Dependency Pairs:

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

Rules:

f(a, g(y)) -> g(g(y))
f(g(x), a) -> f(x, g(a))
f(g(x), g(y)) -> h(g(y), x, g(y))
h(g(x), y, z) -> f(y, h(x, y, z))
h(a, y, z) -> z

Strategy:

innermost

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

H(g(g(x'')), y'', z'') -> H(g(x''), y'', z'')

11 new Dependency Pairs are created:

H(g(g(g(g(x'''')))), y''', z''') -> H(g(g(g(x''''))), y''', z''')
H(g(g(g(a))), y''', z''') -> H(g(g(a)), y''', z''')
H(g(g(g(x''''))), y'''', z'''') -> H(g(g(x'''')), y'''', z'''')
H(g(g(g(a))), y'''', z'''') -> H(g(g(a)), y'''', z'''')
H(g(g(g(g(g(x'''''))))), y'''', z'''') -> H(g(g(g(g(x''''')))), y'''', z'''')
H(g(g(g(g(a)))), y'''', z'''') -> H(g(g(g(a))), y'''', z'''')
H(g(g(a)), g(x'''''), g(g(g(x''''''')))) -> H(g(a), g(x'''''), g(g(g(x'''''''))))
H(g(g(a)), g(x'''''), g(g(a))) -> H(g(a), g(x'''''), g(g(a)))
H(g(g(a)), g(x'''''), g(g(x''''''''))) -> H(g(a), g(x'''''), g(g(x'''''''')))
H(g(g(a)), g(x'''''), g(g(g(g(x'''''''''))))) -> H(g(a), g(x'''''), g(g(g(g(x''''''''')))))
H(g(g(a)), g(x'''''), g(g(g(a)))) -> H(g(a), g(x'''''), g(g(g(a))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 10

                 ↳Forward Instantiation Transformation

Dependency Pairs:

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

Rules:

f(a, g(y)) -> g(g(y))
f(g(x), a) -> f(x, g(a))
f(g(x), g(y)) -> h(g(y), x, g(y))
h(g(x), y, z) -> f(y, h(x, y, z))
h(a, y, z) -> z

Strategy:

innermost

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

H(g(g(a)), y', z') -> H(g(a), y', z')

five new Dependency Pairs are created:

H(g(g(a)), g(x'''''), g(g(g(x''''''')))) -> H(g(a), g(x'''''), g(g(g(x'''''''))))
H(g(g(a)), g(x'''''), g(g(a))) -> H(g(a), g(x'''''), g(g(a)))
H(g(g(a)), g(x'''''), g(g(x''''''''))) -> H(g(a), g(x'''''), g(g(x'''''''')))
H(g(g(a)), g(x'''''), g(g(g(g(x'''''''''))))) -> H(g(a), g(x'''''), g(g(g(g(x''''''''')))))
H(g(g(a)), g(x'''''), g(g(g(a)))) -> H(g(a), g(x'''''), g(g(g(a))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 11

                 ↳Forward Instantiation Transformation

Dependency Pairs:

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

Rules:

f(a, g(y)) -> g(g(y))
f(g(x), a) -> f(x, g(a))
f(g(x), g(y)) -> h(g(y), x, g(y))
h(g(x), y, z) -> f(y, h(x, y, z))
h(a, y, z) -> z

Strategy:

innermost

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

F(g(x'), g(g(x''''))) -> H(g(g(x'''')), x', g(g(x'''')))

eight new Dependency Pairs are created:

F(g(x'0), g(g(g(x''')))) -> H(g(g(g(x'''))), x'0, g(g(g(x'''))))
F(g(x''), g(g(a))) -> H(g(g(a)), x'', g(g(a)))
F(g(x''), g(g(g(g(x''''''))))) -> H(g(g(g(g(x'''''')))), x'', g(g(g(g(x'''''')))))
F(g(x''), g(g(g(a)))) -> H(g(g(g(a))), x'', g(g(g(a))))
F(g(x''), g(g(g(x'''''')))) -> H(g(g(g(x''''''))), x'', g(g(g(x''''''))))
F(g(x''), g(g(g(g(g(x''''''')))))) -> H(g(g(g(g(g(x'''''''))))), x'', g(g(g(g(g(x'''''''))))))
F(g(x''), g(g(g(g(a))))) -> H(g(g(g(g(a)))), x'', g(g(g(g(a)))))
F(g(g(x''''''')), g(g(a))) -> H(g(g(a)), g(x'''''''), g(g(a)))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 12

                 ↳Argument Filtering and Ordering

Dependency Pairs:

F(g(g(x''''''')), g(g(a))) -> H(g(g(a)), g(x'''''''), g(g(a)))
H(g(g(a)), g(x'''''), g(g(g(a)))) -> H(g(a), g(x'''''), g(g(g(a))))
H(g(g(a)), g(x'''''), g(g(g(g(x'''''''''))))) -> H(g(a), g(x'''''), g(g(g(g(x''''''''')))))
H(g(g(a)), g(x'''''), g(g(x''''''''))) -> H(g(a), g(x'''''), g(g(x'''''''')))
H(g(g(a)), g(x'''''), g(g(a))) -> H(g(a), g(x'''''), g(g(a)))
H(g(g(a)), g(x'''''), g(g(g(x''''''')))) -> H(g(a), g(x'''''), g(g(g(x'''''''))))
F(g(x''), g(g(g(a)))) -> H(g(g(g(a))), x'', g(g(g(a))))
H(g(a), g(x'''), g(g(g(a)))) -> F(g(x'''), g(g(g(a))))
H(g(g(a)), g(x'''''), g(g(g(a)))) -> H(g(a), g(x'''''), g(g(g(a))))
F(g(x''), g(g(g(g(a))))) -> H(g(g(g(g(a)))), x'', g(g(g(g(a)))))
F(g(x''), g(g(g(g(g(x''''''')))))) -> H(g(g(g(g(g(x'''''''))))), x'', g(g(g(g(g(x'''''''))))))
F(g(x''), g(g(g(x'''''')))) -> H(g(g(g(x''''''))), x'', g(g(g(x''''''))))
F(g(x''), g(g(g(g(x''''''))))) -> H(g(g(g(g(x'''''')))), x'', g(g(g(g(x'''''')))))
H(g(g(g(g(a)))), y'''', z'''') -> H(g(g(g(a))), y'''', z'''')
F(g(x'0), g(g(g(x''')))) -> H(g(g(g(x'''))), x'0, g(g(g(x'''))))
H(g(a), g(x'''), g(g(g(g(x'''''''))))) -> F(g(x'''), g(g(g(g(x''''''')))))
H(g(g(a)), g(x'''''), g(g(g(g(x'''''''''))))) -> H(g(a), g(x'''''), g(g(g(g(x''''''''')))))
H(g(g(g(a))), y'''', z'''') -> H(g(g(a)), y'''', z'''')
H(g(g(g(x''''))), y'''', z'''') -> H(g(g(x'''')), y'''', z'''')
F(g(x'), g(g(g(a)))) -> H(g(g(g(a))), x', g(g(g(a))))
H(g(a), g(x'''), g(g(x''''''))) -> F(g(x'''), g(g(x'''''')))
H(g(g(a)), g(x'''''), g(g(g(x''''''')))) -> H(g(a), g(x'''''), g(g(g(x'''''''))))
H(g(g(g(a))), y''', z''') -> H(g(g(a)), y''', z''')
H(g(g(g(g(x'''')))), y''', z''') -> H(g(g(g(x''''))), y''', z''')
F(g(x'), g(g(g(g(x'''''))))) -> H(g(g(g(g(x''''')))), x', g(g(g(g(x''''')))))
H(g(a), g(x'''), g(g(g(x''''')))) -> F(g(x'''), g(g(g(x'''''))))
H(g(g(a)), g(x'''''), g(g(x''''''''))) -> H(g(a), g(x'''''), g(g(x'''''''')))
F(g(x''), g(g(a))) -> H(g(g(a)), x'', g(g(a)))
H(g(a), g(x'''), g(g(a))) -> F(g(x'''), g(g(a)))
H(g(g(a)), g(x'''''), g(g(a))) -> H(g(a), g(x'''''), g(g(a)))
F(g(x'), g(g(a))) -> H(g(g(a)), x', g(g(a)))
H(g(g(a)), y''', z''') -> F(y''', f(y''', z'''))
H(g(g(g(a))), y', z') -> H(g(g(a)), y', z')
H(g(g(g(g(x''')))), y', z') -> H(g(g(g(x'''))), y', z')
F(g(x'), g(g(g(x''')))) -> H(g(g(g(x'''))), x', g(g(g(x'''))))
H(g(g(g(x'))), y''', z''') -> F(y''', f(y''', f(y''', h(x', y''', z'''))))
H(g(g(g(g(g(x'''''))))), y'''', z'''') -> H(g(g(g(g(x''''')))), y'''', z'''')

Rules:

f(a, g(y)) -> g(g(y))
f(g(x), a) -> f(x, g(a))
f(g(x), g(y)) -> h(g(y), x, g(y))
h(g(x), y, z) -> f(y, h(x, y, z))
h(a, y, z) -> z

Strategy:

innermost

The following dependency pairs can be strictly oriented:

F(g(g(x''''''')), g(g(a))) -> H(g(g(a)), g(x'''''''), g(g(a)))
F(g(x''), g(g(g(a)))) -> H(g(g(g(a))), x'', g(g(g(a))))
F(g(x''), g(g(g(g(a))))) -> H(g(g(g(g(a)))), x'', g(g(g(g(a)))))
F(g(x''), g(g(g(g(g(x''''''')))))) -> H(g(g(g(g(g(x'''''''))))), x'', g(g(g(g(g(x'''''''))))))
F(g(x''), g(g(g(x'''''')))) -> H(g(g(g(x''''''))), x'', g(g(g(x''''''))))
F(g(x''), g(g(g(g(x''''''))))) -> H(g(g(g(g(x'''''')))), x'', g(g(g(g(x'''''')))))
F(g(x'0), g(g(g(x''')))) -> H(g(g(g(x'''))), x'0, g(g(g(x'''))))
F(g(x'), g(g(g(a)))) -> H(g(g(g(a))), x', g(g(g(a))))
F(g(x'), g(g(g(g(x'''''))))) -> H(g(g(g(g(x''''')))), x', g(g(g(g(x''''')))))
F(g(x''), g(g(a))) -> H(g(g(a)), x'', g(g(a)))
F(g(x'), g(g(a))) -> H(g(g(a)), x', g(g(a)))
F(g(x'), g(g(g(x''')))) -> H(g(g(g(x'''))), x', g(g(g(x'''))))

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:

POL(g(x₁)) = 1 + x₁

resulting in one new DP problem.
Used Argument Filtering System:

H(x₁, x₂, x₃) -> x₂
g(x₁) -> g(x₁)
F(x₁, x₂) -> x₁

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 14

                 ↳Dependency Graph

Dependency Pairs:

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

Rules:

f(a, g(y)) -> g(g(y))
f(g(x), a) -> f(x, g(a))
f(g(x), g(y)) -> h(g(y), x, g(y))
h(g(x), y, z) -> f(y, h(x, y, z))
h(a, y, z) -> z

Strategy:

innermost

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

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 15

                 ↳Argument Filtering and Ordering

Dependency Pairs:

H(g(g(g(g(g(x'''''))))), y'''', z'''') -> H(g(g(g(g(x''''')))), y'''', z'''')
H(g(g(g(g(x'''')))), y''', z''') -> H(g(g(g(x''''))), y''', z''')
H(g(g(g(g(x''')))), y', z') -> H(g(g(g(x'''))), y', z')
H(g(g(g(x''''))), y'''', z'''') -> H(g(g(x'''')), y'''', z'''')
H(g(g(g(g(a)))), y'''', z'''') -> H(g(g(g(a))), y'''', z'''')

Rules:

f(a, g(y)) -> g(g(y))
f(g(x), a) -> f(x, g(a))
f(g(x), g(y)) -> h(g(y), x, g(y))
h(g(x), y, z) -> f(y, h(x, y, z))
h(a, y, z) -> z

Strategy:

innermost

The following dependency pairs can be strictly oriented:

H(g(g(g(g(g(x'''''))))), y'''', z'''') -> H(g(g(g(g(x''''')))), y'''', z'''')
H(g(g(g(g(x'''')))), y''', z''') -> H(g(g(g(x''''))), y''', z''')
H(g(g(g(g(x''')))), y', z') -> H(g(g(g(x'''))), y', z')
H(g(g(g(x''''))), y'''', z'''') -> H(g(g(x'''')), y'''', z'''')
H(g(g(g(g(a)))), y'''', z'''') -> H(g(g(g(a))), y'''', z'''')

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:

POL(g(x₁)) = 1 + x₁

POL(H(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃

POL(a) = 0

resulting in one new DP problem.
Used Argument Filtering System:

H(x₁, x₂, x₃) -> H(x₁, x₂, x₃)
g(x₁) -> g(x₁)

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 16

                 ↳Dependency Graph

Dependency Pair:

Rules:

f(a, g(y)) -> g(g(y))
f(g(x), a) -> f(x, g(a))
f(g(x), g(y)) -> h(g(y), x, g(y))
h(g(x), y, z) -> f(y, h(x, y, z))
h(a, y, z) -> z

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:06 minutes

POL(g(x₁))	= 1 + x₁
POL(H(x₁, x₂, x₃))	= x₁ + x₂ + x₃
POL(a)	= 0
POL(F(x₁, x₂))	= x₁ + x₂