Term Rewriting System R:
[X, Y, X1, X2]
f(g(X), Y) -> f(X, nf(g(X), activate(Y)))
f(X1, X2) -> nf(X1, X2)
activate(nf(X1, X2)) -> f(X1, X2)
activate(X) -> X

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

F(g(X), Y) -> F(X, nf(g(X), activate(Y)))
F(g(X), Y) -> ACTIVATE(Y)
ACTIVATE(nf(X1, X2)) -> F(X1, X2)

Furthermore, R contains one SCC.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Narrowing Transformation`

Dependency Pairs:

ACTIVATE(nf(X1, X2)) -> F(X1, X2)
F(g(X), Y) -> ACTIVATE(Y)
F(g(X), Y) -> F(X, nf(g(X), activate(Y)))

Rules:

f(g(X), Y) -> f(X, nf(g(X), activate(Y)))
f(X1, X2) -> nf(X1, X2)
activate(nf(X1, X2)) -> f(X1, X2)
activate(X) -> X

Strategy:

innermost

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

F(g(X), Y) -> F(X, nf(g(X), activate(Y)))
two new Dependency Pairs are created:

F(g(X), nf(X1', X2')) -> F(X, nf(g(X), f(X1', X2')))
F(g(X), Y') -> F(X, nf(g(X), Y'))

The transformation is resulting in one new DP problem:

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

Dependency Pairs:

F(g(X), Y') -> F(X, nf(g(X), Y'))
F(g(X), nf(X1', X2')) -> F(X, nf(g(X), f(X1', X2')))
F(g(X), Y) -> ACTIVATE(Y)
ACTIVATE(nf(X1, X2)) -> F(X1, X2)

Rules:

f(g(X), Y) -> f(X, nf(g(X), activate(Y)))
f(X1, X2) -> nf(X1, X2)
activate(nf(X1, X2)) -> f(X1, X2)
activate(X) -> X

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) -> ACTIVATE(Y)
one new Dependency Pair is created:

F(g(X), nf(X1'', X2'')) -> ACTIVATE(nf(X1'', X2''))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳FwdInst`
`             ...`
`               →DP Problem 3`
`                 ↳Forward Instantiation Transformation`

Dependency Pairs:

ACTIVATE(nf(X1, X2)) -> F(X1, X2)
F(g(X), nf(X1'', X2'')) -> ACTIVATE(nf(X1'', X2''))
F(g(X), nf(X1', X2')) -> F(X, nf(g(X), f(X1', X2')))
F(g(X), Y') -> F(X, nf(g(X), Y'))

Rules:

f(g(X), Y) -> f(X, nf(g(X), activate(Y)))
f(X1, X2) -> nf(X1, X2)
activate(nf(X1, X2)) -> f(X1, X2)
activate(X) -> X

Strategy:

innermost

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

ACTIVATE(nf(X1, X2)) -> F(X1, X2)
three new Dependency Pairs are created:

ACTIVATE(nf(g(X''), nf(X1''', X2'''))) -> F(g(X''), nf(X1''', X2'''))
ACTIVATE(nf(g(X''), X2')) -> F(g(X''), X2')
ACTIVATE(nf(g(X''), nf(X1'''', X2''''))) -> F(g(X''), nf(X1'''', X2''''))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳FwdInst`
`             ...`
`               →DP Problem 4`
`                 ↳Forward Instantiation Transformation`

Dependency Pairs:

ACTIVATE(nf(g(X''), nf(X1'''', X2''''))) -> F(g(X''), nf(X1'''', X2''''))
ACTIVATE(nf(g(X''), X2')) -> F(g(X''), X2')
F(g(X), Y') -> F(X, nf(g(X), Y'))
F(g(X), nf(X1', X2')) -> F(X, nf(g(X), f(X1', X2')))
ACTIVATE(nf(g(X''), nf(X1''', X2'''))) -> F(g(X''), nf(X1''', X2'''))
F(g(X), nf(X1'', X2'')) -> ACTIVATE(nf(X1'', X2''))

Rules:

f(g(X), Y) -> f(X, nf(g(X), activate(Y)))
f(X1, X2) -> nf(X1, X2)
activate(nf(X1, X2)) -> f(X1, X2)
activate(X) -> X

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, nf(g(X), Y'))
two new Dependency Pairs are created:

F(g(g(X'')), Y'') -> F(g(X''), nf(g(g(X'')), Y''))
F(g(g(X'')), Y''') -> F(g(X''), nf(g(g(X'')), Y'''))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳FwdInst`
`             ...`
`               →DP Problem 5`
`                 ↳Remaining Obligation(s)`

The following remains to be proven:
Dependency Pairs:

ACTIVATE(nf(g(X''), X2')) -> F(g(X''), X2')
F(g(g(X'')), Y''') -> F(g(X''), nf(g(g(X'')), Y'''))
F(g(g(X'')), Y'') -> F(g(X''), nf(g(g(X'')), Y''))
ACTIVATE(nf(g(X''), nf(X1''', X2'''))) -> F(g(X''), nf(X1''', X2'''))
F(g(X), nf(X1'', X2'')) -> ACTIVATE(nf(X1'', X2''))
F(g(X), nf(X1', X2')) -> F(X, nf(g(X), f(X1', X2')))
ACTIVATE(nf(g(X''), nf(X1'''', X2''''))) -> F(g(X''), nf(X1'''', X2''''))

Rules:

f(g(X), Y) -> f(X, nf(g(X), activate(Y)))
f(X1, X2) -> nf(X1, X2)
activate(nf(X1, X2)) -> f(X1, X2)
activate(X) -> X

Strategy:

innermost

Innermost Termination of R could not be shown.
Duration:
0:00 minutes