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

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

F(0, 1, g(x, y), z) -> F(g(x, y), g(x, y), g(x, y), h(x))
F(0, 1, g(x, y), z) -> H(x)
H(g(x, y)) -> H(x)

Furthermore, R contains one SCC.

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

Dependency Pair:

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

Rules:

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

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)) -> H(x)
one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem:

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

Dependency Pair:

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

Rules:

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

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''), y)) -> H(g(x'', y''))
one new Dependency Pair is created:

H(g(g(g(x'''', y''''), y''0), y)) -> H(g(g(x'''', y''''), y''0))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 2`
`             ↳FwdInst`
`             ...`
`               →DP Problem 3`
`                 ↳Polynomial Ordering`

Dependency Pair:

H(g(g(g(x'''', y''''), y''0), y)) -> H(g(g(x'''', y''''), y''0))

Rules:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

H(g(g(g(x'''', y''''), y''0), y)) -> H(g(g(x'''', y''''), y''0))

Additionally, the following usable rules for innermost w.r.t. to the implicit AFS can be oriented:

g(0, 1) -> 0
g(0, 1) -> 1

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

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 2`
`             ↳FwdInst`
`             ...`
`               →DP Problem 4`
`                 ↳Dependency Graph`

Dependency Pair:

Rules:

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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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