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

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

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

Furthermore, R contains three SCCs.

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

Dependency Pair:

F(s(X)) -> F(X)

Rules:

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

Strategy:

innermost

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

F(s(X)) -> F(X)
one new Dependency Pair is created:

F(s(s(X''))) -> F(s(X''))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 4`
`             ↳Forward Instantiation Transformation`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Nar`

Dependency Pair:

F(s(s(X''))) -> F(s(X''))

Rules:

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

Strategy:

innermost

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

F(s(s(X''))) -> F(s(X''))
one new Dependency Pair is created:

F(s(s(s(X'''')))) -> F(s(s(X'''')))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 4`
`             ↳FwdInst`
`             ...`
`               →DP Problem 5`
`                 ↳Polynomial Ordering`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Nar`

Dependency Pair:

F(s(s(s(X'''')))) -> F(s(s(X'''')))

Rules:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

F(s(s(s(X'''')))) -> F(s(s(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(s(x1)) =  1 + x1 POL(F(x1)) =  1 + x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 4`
`             ↳FwdInst`
`             ...`
`               →DP Problem 6`
`                 ↳Dependency Graph`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Nar`

Dependency Pair:

Rules:

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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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

Dependency Pair:

G(cons(0, Y)) -> G(Y)

Rules:

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

Strategy:

innermost

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

G(cons(0, Y)) -> G(Y)
one new Dependency Pair is created:

G(cons(0, cons(0, Y''))) -> G(cons(0, Y''))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`           →DP Problem 7`
`             ↳Forward Instantiation Transformation`
`       →DP Problem 3`
`         ↳Nar`

Dependency Pair:

G(cons(0, cons(0, Y''))) -> G(cons(0, Y''))

Rules:

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

Strategy:

innermost

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

G(cons(0, cons(0, Y''))) -> G(cons(0, Y''))
one new Dependency Pair is created:

G(cons(0, cons(0, cons(0, Y'''')))) -> G(cons(0, cons(0, Y'''')))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`           →DP Problem 7`
`             ↳FwdInst`
`             ...`
`               →DP Problem 8`
`                 ↳Polynomial Ordering`
`       →DP Problem 3`
`         ↳Nar`

Dependency Pair:

G(cons(0, cons(0, cons(0, Y'''')))) -> G(cons(0, cons(0, Y'''')))

Rules:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

G(cons(0, cons(0, cons(0, Y'''')))) -> G(cons(0, cons(0, Y'''')))

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(0) =  0 POL(G(x1)) =  1 + x1 POL(cons(x1, x2)) =  1 + x2

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`           →DP Problem 7`
`             ↳FwdInst`
`             ...`
`               →DP Problem 9`
`                 ↳Dependency Graph`
`       →DP Problem 3`
`         ↳Nar`

Dependency Pair:

Rules:

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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Narrowing Transformation`

Dependency Pair:

H(cons(X, Y)) -> H(g(cons(X, Y)))

Rules:

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

Strategy:

innermost

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

H(cons(X, Y)) -> H(g(cons(X, Y)))
two new Dependency Pairs are created:

H(cons(0, Y'')) -> H(g(Y''))
H(cons(s(X''), Y'')) -> H(s(X''))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Nar`
`           →DP Problem 10`
`             ↳Narrowing Transformation`

Dependency Pair:

H(cons(0, Y'')) -> H(g(Y''))

Rules:

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

Strategy:

innermost

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

H(cons(0, Y'')) -> H(g(Y''))
two new Dependency Pairs are created:

H(cons(0, cons(0, Y'))) -> H(g(Y'))
H(cons(0, cons(s(X'), Y'))) -> H(s(X'))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Nar`
`           →DP Problem 10`
`             ↳Nar`
`             ...`
`               →DP Problem 11`
`                 ↳Forward Instantiation Transformation`

Dependency Pair:

H(cons(0, cons(0, Y'))) -> H(g(Y'))

Rules:

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

Strategy:

innermost

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

H(cons(0, cons(0, Y'))) -> H(g(Y'))
no new Dependency Pairs are created.
The transformation is resulting in no new DP problems.

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