Term Rewriting System R:
[X]
active(g(X)) -> mark(h(X))
active(c) -> mark(d)
active(h(d)) -> mark(g(c))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
proper(c) -> ok(c)
proper(d) -> ok(d)
g(ok(X)) -> ok(g(X))
h(ok(X)) -> ok(h(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

ACTIVE(g(X)) -> H(X)
ACTIVE(h(d)) -> G(c)
PROPER(g(X)) -> G(proper(X))
PROPER(g(X)) -> PROPER(X)
PROPER(h(X)) -> H(proper(X))
PROPER(h(X)) -> PROPER(X)
G(ok(X)) -> G(X)
H(ok(X)) -> H(X)
TOP(mark(X)) -> TOP(proper(X))
TOP(mark(X)) -> PROPER(X)
TOP(ok(X)) -> TOP(active(X))
TOP(ok(X)) -> ACTIVE(X)

Furthermore, R contains four SCCs.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polynomial Ordering`
`       →DP Problem 2`
`         ↳Polo`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Nar`

Dependency Pair:

G(ok(X)) -> G(X)

Rules:

active(g(X)) -> mark(h(X))
active(c) -> mark(d)
active(h(d)) -> mark(g(c))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
proper(c) -> ok(c)
proper(d) -> ok(d)
g(ok(X)) -> ok(g(X))
h(ok(X)) -> ok(h(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

The following dependency pair can be strictly oriented:

G(ok(X)) -> 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(G(x1)) =  x1 POL(ok(x1)) =  1 + x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`           →DP Problem 5`
`             ↳Dependency Graph`
`       →DP Problem 2`
`         ↳Polo`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Nar`

Dependency Pair:

Rules:

active(g(X)) -> mark(h(X))
active(c) -> mark(d)
active(h(d)) -> mark(g(c))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
proper(c) -> ok(c)
proper(d) -> ok(d)
g(ok(X)) -> ok(g(X))
h(ok(X)) -> ok(h(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Polynomial Ordering`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Nar`

Dependency Pair:

H(ok(X)) -> H(X)

Rules:

active(g(X)) -> mark(h(X))
active(c) -> mark(d)
active(h(d)) -> mark(g(c))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
proper(c) -> ok(c)
proper(d) -> ok(d)
g(ok(X)) -> ok(g(X))
h(ok(X)) -> ok(h(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

The following dependency pair can be strictly oriented:

H(ok(X)) -> H(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(H(x1)) =  x1 POL(ok(x1)) =  1 + x1

resulting in one new DP problem.

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

Dependency Pair:

Rules:

active(g(X)) -> mark(h(X))
active(c) -> mark(d)
active(h(d)) -> mark(g(c))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
proper(c) -> ok(c)
proper(d) -> ok(d)
g(ok(X)) -> ok(g(X))
h(ok(X)) -> ok(h(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Polo`
`       →DP Problem 3`
`         ↳Polynomial Ordering`
`       →DP Problem 4`
`         ↳Nar`

Dependency Pairs:

PROPER(h(X)) -> PROPER(X)
PROPER(g(X)) -> PROPER(X)

Rules:

active(g(X)) -> mark(h(X))
active(c) -> mark(d)
active(h(d)) -> mark(g(c))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
proper(c) -> ok(c)
proper(d) -> ok(d)
g(ok(X)) -> ok(g(X))
h(ok(X)) -> ok(h(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

The following dependency pair can be strictly oriented:

PROPER(h(X)) -> PROPER(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(g(x1)) =  x1 POL(PROPER(x1)) =  x1 POL(h(x1)) =  1 + x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Polo`
`       →DP Problem 3`
`         ↳Polo`
`           →DP Problem 7`
`             ↳Polynomial Ordering`
`       →DP Problem 4`
`         ↳Nar`

Dependency Pair:

PROPER(g(X)) -> PROPER(X)

Rules:

active(g(X)) -> mark(h(X))
active(c) -> mark(d)
active(h(d)) -> mark(g(c))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
proper(c) -> ok(c)
proper(d) -> ok(d)
g(ok(X)) -> ok(g(X))
h(ok(X)) -> ok(h(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

The following dependency pair can be strictly oriented:

PROPER(g(X)) -> PROPER(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(g(x1)) =  1 + x1 POL(PROPER(x1)) =  x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Polo`
`       →DP Problem 3`
`         ↳Polo`
`           →DP Problem 7`
`             ↳Polo`
`             ...`
`               →DP Problem 8`
`                 ↳Dependency Graph`
`       →DP Problem 4`
`         ↳Nar`

Dependency Pair:

Rules:

active(g(X)) -> mark(h(X))
active(c) -> mark(d)
active(h(d)) -> mark(g(c))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
proper(c) -> ok(c)
proper(d) -> ok(d)
g(ok(X)) -> ok(g(X))
h(ok(X)) -> ok(h(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Polo`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Narrowing Transformation`

Dependency Pairs:

TOP(ok(X)) -> TOP(active(X))
TOP(mark(X)) -> TOP(proper(X))

Rules:

active(g(X)) -> mark(h(X))
active(c) -> mark(d)
active(h(d)) -> mark(g(c))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
proper(c) -> ok(c)
proper(d) -> ok(d)
g(ok(X)) -> ok(g(X))
h(ok(X)) -> ok(h(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

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

TOP(mark(X)) -> TOP(proper(X))
four new Dependency Pairs are created:

TOP(mark(g(X''))) -> TOP(g(proper(X'')))
TOP(mark(h(X''))) -> TOP(h(proper(X'')))
TOP(mark(c)) -> TOP(ok(c))
TOP(mark(d)) -> TOP(ok(d))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Polo`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Nar`
`           →DP Problem 9`
`             ↳Narrowing Transformation`

Dependency Pairs:

TOP(mark(d)) -> TOP(ok(d))
TOP(mark(c)) -> TOP(ok(c))
TOP(mark(h(X''))) -> TOP(h(proper(X'')))
TOP(mark(g(X''))) -> TOP(g(proper(X'')))
TOP(ok(X)) -> TOP(active(X))

Rules:

active(g(X)) -> mark(h(X))
active(c) -> mark(d)
active(h(d)) -> mark(g(c))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
proper(c) -> ok(c)
proper(d) -> ok(d)
g(ok(X)) -> ok(g(X))
h(ok(X)) -> ok(h(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

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

TOP(ok(X)) -> TOP(active(X))
three new Dependency Pairs are created:

TOP(ok(g(X''))) -> TOP(mark(h(X'')))
TOP(ok(c)) -> TOP(mark(d))
TOP(ok(h(d))) -> TOP(mark(g(c)))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Polo`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Nar`
`           →DP Problem 9`
`             ↳Nar`
`             ...`
`               →DP Problem 10`
`                 ↳Polynomial Ordering`

Dependency Pairs:

TOP(ok(h(d))) -> TOP(mark(g(c)))
TOP(ok(g(X''))) -> TOP(mark(h(X'')))
TOP(mark(g(X''))) -> TOP(g(proper(X'')))
TOP(mark(h(X''))) -> TOP(h(proper(X'')))

Rules:

active(g(X)) -> mark(h(X))
active(c) -> mark(d)
active(h(d)) -> mark(g(c))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
proper(c) -> ok(c)
proper(d) -> ok(d)
g(ok(X)) -> ok(g(X))
h(ok(X)) -> ok(h(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

The following dependency pair can be strictly oriented:

TOP(ok(h(d))) -> TOP(mark(g(c)))

Additionally, the following usable rules for innermost w.r.t. to the implicit AFS can be oriented:

g(ok(X)) -> ok(g(X))
h(ok(X)) -> ok(h(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
proper(c) -> ok(c)
proper(d) -> ok(d)

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(proper(x1)) =  x1 POL(c) =  0 POL(g(x1)) =  x1 POL(d) =  1 POL(h(x1)) =  x1 POL(mark(x1)) =  x1 POL(ok(x1)) =  x1 POL(TOP(x1)) =  1 + x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Polo`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Nar`
`           →DP Problem 9`
`             ↳Nar`
`             ...`
`               →DP Problem 11`
`                 ↳Polynomial Ordering`

Dependency Pairs:

TOP(ok(g(X''))) -> TOP(mark(h(X'')))
TOP(mark(g(X''))) -> TOP(g(proper(X'')))
TOP(mark(h(X''))) -> TOP(h(proper(X'')))

Rules:

active(g(X)) -> mark(h(X))
active(c) -> mark(d)
active(h(d)) -> mark(g(c))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
proper(c) -> ok(c)
proper(d) -> ok(d)
g(ok(X)) -> ok(g(X))
h(ok(X)) -> ok(h(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

The following dependency pair can be strictly oriented:

TOP(ok(g(X''))) -> TOP(mark(h(X'')))

Additionally, the following usable rules for innermost w.r.t. to the implicit AFS can be oriented:

h(ok(X)) -> ok(h(X))
g(ok(X)) -> ok(g(X))

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(proper(x1)) =  0 POL(c) =  0 POL(g(x1)) =  1 POL(d) =  0 POL(h(x1)) =  0 POL(mark(x1)) =  x1 POL(ok(x1)) =  x1 POL(TOP(x1)) =  x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Polo`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Nar`
`           →DP Problem 9`
`             ↳Nar`
`             ...`
`               →DP Problem 12`
`                 ↳Forward Instantiation Transformation`

Dependency Pairs:

TOP(mark(g(X''))) -> TOP(g(proper(X'')))
TOP(mark(h(X''))) -> TOP(h(proper(X'')))

Rules:

active(g(X)) -> mark(h(X))
active(c) -> mark(d)
active(h(d)) -> mark(g(c))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
proper(c) -> ok(c)
proper(d) -> ok(d)
g(ok(X)) -> ok(g(X))
h(ok(X)) -> ok(h(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

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

TOP(mark(g(X''))) -> TOP(g(proper(X'')))
no new Dependency Pairs are created.
The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Polo`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Nar`
`           →DP Problem 9`
`             ↳Nar`
`             ...`
`               →DP Problem 13`
`                 ↳Forward Instantiation Transformation`

Dependency Pair:

TOP(mark(h(X''))) -> TOP(h(proper(X'')))

Rules:

active(g(X)) -> mark(h(X))
active(c) -> mark(d)
active(h(d)) -> mark(g(c))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
proper(c) -> ok(c)
proper(d) -> ok(d)
g(ok(X)) -> ok(g(X))
h(ok(X)) -> ok(h(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

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

TOP(mark(h(X''))) -> TOP(h(proper(X'')))
no new Dependency Pairs are created.
The transformation is resulting in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:01 minutes