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.

`   R`
`     ↳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.

`   R`
`     ↳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:

`   R`
`     ↳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:

`   R`
`     ↳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:

`   R`
`     ↳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:

`   R`
`     ↳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:

`   R`
`     ↳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:

`   R`
`     ↳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:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 9`
`                 ↳Polynomial 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:

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

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(g(x1)) =  1 + x1 POL(a) =  0 POL(H(x1, x2, x3)) =  x2 POL(F(x1, x2)) =  x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 13`
`                 ↳Dependency Graph`

Dependency Pair:

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

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳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:

`   R`
`     ↳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:

`   R`
`     ↳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:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 12`
`                 ↳Polynomial 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 implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(g(x1)) =  1 + x1 POL(h(x1, x2, x3)) =  0 POL(a) =  0 POL(H(x1, x2, x3)) =  x2 POL(f(x1, x2)) =  0 POL(F(x1, x2)) =  x1

resulting in one new DP problem.

`   R`
`     ↳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.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 15`
`                 ↳Polynomial 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 implicit 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.

`   R`
`     ↳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:
0:04 minutes