Term Rewriting System R:
[x, y]
f(f(0, x), 1) -> f(g(f(x, x)), x)
f(g(x), y) -> g(f(x, y))

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

F(f(0, x), 1) -> F(g(f(x, x)), x)
F(f(0, x), 1) -> F(x, x)
F(g(x), y) -> F(x, y)

Furthermore, R contains one SCC.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Forward Instantiation Transformation`

Dependency Pairs:

F(f(0, x), 1) -> F(x, x)
F(g(x), y) -> F(x, y)
F(f(0, x), 1) -> F(g(f(x, x)), x)

Rules:

f(f(0, x), 1) -> f(g(f(x, x)), x)
f(g(x), y) -> g(f(x, y))

Strategy:

innermost

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

F(f(0, x), 1) -> F(x, x)
one new Dependency Pair is created:

F(f(0, g(x'')), 1) -> F(g(x''), g(x''))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 2`
`             ↳Forward Instantiation Transformation`

Dependency Pairs:

F(f(0, g(x'')), 1) -> F(g(x''), g(x''))
F(f(0, x), 1) -> F(g(f(x, x)), x)
F(g(x), y) -> F(x, y)

Rules:

f(f(0, x), 1) -> f(g(f(x, x)), x)
f(g(x), y) -> g(f(x, y))

Strategy:

innermost

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

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

F(g(g(x'')), y'') -> F(g(x''), y'')
F(g(f(0, x'')), 1) -> F(f(0, x''), 1)
F(g(f(0, g(x''''))), 1) -> F(f(0, g(x'''')), 1)

The transformation is resulting in one new DP problem:

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

Dependency Pairs:

F(g(f(0, g(x''''))), 1) -> F(f(0, g(x'''')), 1)
F(f(0, x), 1) -> F(g(f(x, x)), x)
F(g(f(0, x'')), 1) -> F(f(0, x''), 1)
F(g(g(x'')), y'') -> F(g(x''), y'')
F(f(0, g(x'')), 1) -> F(g(x''), g(x''))

Rules:

f(f(0, x), 1) -> f(g(f(x, x)), x)
f(g(x), y) -> g(f(x, y))

Strategy:

innermost

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

F(f(0, x), 1) -> F(g(f(x, x)), x)
one new Dependency Pair is created:

F(f(0, g(x'')), 1) -> F(g(g(f(x'', g(x'')))), g(x''))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 2`
`             ↳FwdInst`
`             ...`
`               →DP Problem 4`
`                 ↳Polynomial Ordering`

Dependency Pairs:

F(f(0, g(x'')), 1) -> F(g(g(f(x'', g(x'')))), g(x''))
F(g(f(0, x'')), 1) -> F(f(0, x''), 1)
F(g(g(x'')), y'') -> F(g(x''), y'')
F(f(0, g(x'')), 1) -> F(g(x''), g(x''))
F(g(f(0, g(x''''))), 1) -> F(f(0, g(x'''')), 1)

Rules:

f(f(0, x), 1) -> f(g(f(x, x)), x)
f(g(x), y) -> g(f(x, y))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

F(f(0, g(x'')), 1) -> F(g(g(f(x'', g(x'')))), g(x''))
F(f(0, g(x'')), 1) -> F(g(x''), 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(0) =  0 POL(g(x1)) =  0 POL(1) =  1 POL(f(x1, x2)) =  0 POL(F(x1, x2)) =  x2

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 2`
`             ↳FwdInst`
`             ...`
`               →DP Problem 5`
`                 ↳Dependency Graph`

Dependency Pairs:

F(g(f(0, x'')), 1) -> F(f(0, x''), 1)
F(g(g(x'')), y'') -> F(g(x''), y'')
F(g(f(0, g(x''''))), 1) -> F(f(0, g(x'''')), 1)

Rules:

f(f(0, x), 1) -> f(g(f(x, x)), x)
f(g(x), y) -> g(f(x, y))

Strategy:

innermost

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

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 2`
`             ↳FwdInst`
`             ...`
`               →DP Problem 6`
`                 ↳Polynomial Ordering`

Dependency Pair:

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

Rules:

f(f(0, x), 1) -> f(g(f(x, x)), x)
f(g(x), y) -> g(f(x, y))

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

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(F(x1, x2)) =  1 + x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 2`
`             ↳FwdInst`
`             ...`
`               →DP Problem 7`
`                 ↳Dependency Graph`

Dependency Pair:

Rules:

f(f(0, x), 1) -> f(g(f(x, x)), x)
f(g(x), y) -> g(f(x, y))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:00 minutes