Term Rewriting System R:
[X]
f(a) -> f(c(a))
f(c(X)) -> X
f(c(a)) -> f(d(b))
f(a) -> f(d(a))
f(d(X)) -> X
f(c(b)) -> f(d(a))
e(g(X)) -> e(X)

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

F(a) -> F(c(a))
F(c(a)) -> F(d(b))
F(a) -> F(d(a))
F(c(b)) -> F(d(a))
E(g(X)) -> E(X)

Furthermore, R contains one SCC.

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

Dependency Pair:

E(g(X)) -> E(X)

Rules:

f(a) -> f(c(a))
f(c(X)) -> X
f(c(a)) -> f(d(b))
f(a) -> f(d(a))
f(d(X)) -> X
f(c(b)) -> f(d(a))
e(g(X)) -> e(X)

Strategy:

innermost

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

E(g(X)) -> E(X)
one new Dependency Pair is created:

E(g(g(X''))) -> E(g(X''))

The transformation is resulting in one new DP problem:

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

Dependency Pair:

E(g(g(X''))) -> E(g(X''))

Rules:

f(a) -> f(c(a))
f(c(X)) -> X
f(c(a)) -> f(d(b))
f(a) -> f(d(a))
f(d(X)) -> X
f(c(b)) -> f(d(a))
e(g(X)) -> e(X)

Strategy:

innermost

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

E(g(g(X''))) -> E(g(X''))
one new Dependency Pair is created:

E(g(g(g(X'''')))) -> E(g(g(X'''')))

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:

E(g(g(g(X'''')))) -> E(g(g(X'''')))

Rules:

f(a) -> f(c(a))
f(c(X)) -> X
f(c(a)) -> f(d(b))
f(a) -> f(d(a))
f(d(X)) -> X
f(c(b)) -> f(d(a))
e(g(X)) -> e(X)

Strategy:

innermost

The following dependency pair can be strictly oriented:

E(g(g(g(X'''')))) -> E(g(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(E(x1)) =  1 + x1 POL(g(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(a) -> f(c(a))
f(c(X)) -> X
f(c(a)) -> f(d(b))
f(a) -> f(d(a))
f(d(X)) -> X
f(c(b)) -> f(d(a))
e(g(X)) -> e(X)

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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