Term Rewriting System R:
[X, Z, Y]
h(X, Z) -> f(X, s(X), Z)
f(X, Y, g(X, Y)) -> h(0, g(X, Y))
g(0, Y) -> 0
g(X, s(Y)) -> g(X, Y)

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

H(X, Z) -> F(X, s(X), Z)
F(X, Y, g(X, Y)) -> H(0, g(X, Y))
G(X, s(Y)) -> G(X, Y)

Furthermore, R contains one SCC.

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

Dependency Pair:

G(X, s(Y)) -> G(X, Y)

Rules:

h(X, Z) -> f(X, s(X), Z)
f(X, Y, g(X, Y)) -> h(0, g(X, Y))
g(0, Y) -> 0
g(X, s(Y)) -> g(X, Y)

Strategy:

innermost

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

G(X, s(Y)) -> G(X, Y)
one new Dependency Pair is created:

G(X'', s(s(Y''))) -> G(X'', s(Y''))

The transformation is resulting in one new DP problem:

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

Dependency Pair:

G(X'', s(s(Y''))) -> G(X'', s(Y''))

Rules:

h(X, Z) -> f(X, s(X), Z)
f(X, Y, g(X, Y)) -> h(0, g(X, Y))
g(0, Y) -> 0
g(X, s(Y)) -> g(X, Y)

Strategy:

innermost

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

G(X'', s(s(Y''))) -> G(X'', s(Y''))
one new Dependency Pair is created:

G(X'''', s(s(s(Y'''')))) -> G(X'''', s(s(Y'''')))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 2`
`             ↳FwdInst`
`             ...`
`               →DP Problem 3`
`                 ↳Argument Filtering and Ordering`

Dependency Pair:

G(X'''', s(s(s(Y'''')))) -> G(X'''', s(s(Y'''')))

Rules:

h(X, Z) -> f(X, s(X), Z)
f(X, Y, g(X, Y)) -> h(0, g(X, Y))
g(0, Y) -> 0
g(X, s(Y)) -> g(X, Y)

Strategy:

innermost

The following dependency pair can be strictly oriented:

G(X'''', s(s(s(Y'''')))) -> G(X'''', s(s(Y'''')))

There are no usable rules for innermost that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:
G(x1, x2) -> G(x1, x2)
s(x1) -> s(x1)

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

Dependency Pair:

Rules:

h(X, Z) -> f(X, s(X), Z)
f(X, Y, g(X, Y)) -> h(0, g(X, Y))
g(0, Y) -> 0
g(X, s(Y)) -> g(X, Y)

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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