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

Furthermore, R contains one SCC.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polynomial Ordering`

Dependency Pairs:

ACTIVATE(ng(X)) -> ACTIVATE(X)
ACTIVATE(nf(X1, X2)) -> ACTIVATE(X1)

Rules:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

ACTIVATE(ng(X)) -> ACTIVATE(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(n__f(x1, x2)) =  x1 POL(n__g(x1)) =  1 + x1 POL(ACTIVATE(x1)) =  x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`           →DP Problem 2`
`             ↳Polynomial Ordering`

Dependency Pair:

ACTIVATE(nf(X1, X2)) -> ACTIVATE(X1)

Rules:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

ACTIVATE(nf(X1, X2)) -> ACTIVATE(X1)

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(n__f(x1, x2)) =  1 + x1 POL(ACTIVATE(x1)) =  x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`           →DP Problem 2`
`             ↳Polo`
`             ...`
`               →DP Problem 3`
`                 ↳Dependency Graph`

Dependency Pair:

Rules:

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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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