Term Rewriting System R:
[x, y]
active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
top(active(c)) -> top(mark(c))
top(mark(x)) -> top(check(x))
top(found(x)) -> top(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

ACTIVE(f(x)) -> F(active(x))
ACTIVE(f(x)) -> ACTIVE(x)
TOP(active(c)) -> TOP(mark(c))
TOP(mark(x)) -> TOP(check(x))
TOP(mark(x)) -> CHECK(x)
TOP(found(x)) -> TOP(active(x))
TOP(found(x)) -> ACTIVE(x)
CHECK(f(x)) -> F(check(x))
CHECK(f(x)) -> CHECK(x)
CHECK(x) -> START(match(f(X), x))
CHECK(x) -> MATCH(f(X), x)
CHECK(x) -> F(X)
MATCH(f(x), f(y)) -> F(match(x, y))
MATCH(f(x), f(y)) -> MATCH(x, y)
MATCH(X, x) -> PROPER(x)
PROPER(f(x)) -> F(proper(x))
PROPER(f(x)) -> PROPER(x)
F(ok(x)) -> F(x)
F(found(x)) -> F(x)
F(mark(x)) -> F(x)

Furthermore, R contains six SCCs.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Forward Instantiation Transformation`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳FwdInst`
`       →DP Problem 5`
`         ↳FwdInst`
`       →DP Problem 6`
`         ↳Nar`

Dependency Pairs:

F(mark(x)) -> F(x)
F(found(x)) -> F(x)
F(ok(x)) -> F(x)

Rules:

active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
top(active(c)) -> top(mark(c))
top(mark(x)) -> top(check(x))
top(found(x)) -> top(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)

Strategy:

innermost

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

F(ok(x)) -> F(x)
three new Dependency Pairs are created:

F(ok(ok(x''))) -> F(ok(x''))
F(ok(found(x''))) -> F(found(x''))
F(ok(mark(x''))) -> F(mark(x''))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 7`
`             ↳Forward Instantiation Transformation`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳FwdInst`
`       →DP Problem 5`
`         ↳FwdInst`
`       →DP Problem 6`
`         ↳Nar`

Dependency Pairs:

F(ok(mark(x''))) -> F(mark(x''))
F(ok(found(x''))) -> F(found(x''))
F(ok(ok(x''))) -> F(ok(x''))
F(found(x)) -> F(x)
F(mark(x)) -> F(x)

Rules:

active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
top(active(c)) -> top(mark(c))
top(mark(x)) -> top(check(x))
top(found(x)) -> top(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)

Strategy:

innermost

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

F(found(x)) -> F(x)
five new Dependency Pairs are created:

F(found(found(x''))) -> F(found(x''))
F(found(mark(x''))) -> F(mark(x''))
F(found(ok(ok(x'''')))) -> F(ok(ok(x'''')))
F(found(ok(found(x'''')))) -> F(ok(found(x'''')))
F(found(ok(mark(x'''')))) -> F(ok(mark(x'''')))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 7`
`             ↳FwdInst`
`             ...`
`               →DP Problem 8`
`                 ↳Forward Instantiation Transformation`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳FwdInst`
`       →DP Problem 5`
`         ↳FwdInst`
`       →DP Problem 6`
`         ↳Nar`

Dependency Pairs:

F(found(ok(mark(x'''')))) -> F(ok(mark(x'''')))
F(found(ok(found(x'''')))) -> F(ok(found(x'''')))
F(found(ok(ok(x'''')))) -> F(ok(ok(x'''')))
F(found(mark(x''))) -> F(mark(x''))
F(found(found(x''))) -> F(found(x''))
F(ok(found(x''))) -> F(found(x''))
F(ok(ok(x''))) -> F(ok(x''))
F(mark(x)) -> F(x)
F(ok(mark(x''))) -> F(mark(x''))

Rules:

active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
top(active(c)) -> top(mark(c))
top(mark(x)) -> top(check(x))
top(found(x)) -> top(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)

Strategy:

innermost

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

F(mark(x)) -> F(x)
nine new Dependency Pairs are created:

F(mark(mark(x''))) -> F(mark(x''))
F(mark(ok(ok(x'''')))) -> F(ok(ok(x'''')))
F(mark(ok(found(x'''')))) -> F(ok(found(x'''')))
F(mark(ok(mark(x'''')))) -> F(ok(mark(x'''')))
F(mark(found(found(x'''')))) -> F(found(found(x'''')))
F(mark(found(mark(x'''')))) -> F(found(mark(x'''')))
F(mark(found(ok(ok(x''''''))))) -> F(found(ok(ok(x''''''))))
F(mark(found(ok(found(x''''''))))) -> F(found(ok(found(x''''''))))
F(mark(found(ok(mark(x''''''))))) -> F(found(ok(mark(x''''''))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 7`
`             ↳FwdInst`
`             ...`
`               →DP Problem 9`
`                 ↳Forward Instantiation Transformation`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳FwdInst`
`       →DP Problem 5`
`         ↳FwdInst`
`       →DP Problem 6`
`         ↳Nar`

Dependency Pairs:

F(mark(found(ok(mark(x''''''))))) -> F(found(ok(mark(x''''''))))
F(found(ok(found(x'''')))) -> F(ok(found(x'''')))
F(mark(found(ok(found(x''''''))))) -> F(found(ok(found(x''''''))))
F(found(ok(ok(x'''')))) -> F(ok(ok(x'''')))
F(mark(found(ok(ok(x''''''))))) -> F(found(ok(ok(x''''''))))
F(mark(found(mark(x'''')))) -> F(found(mark(x'''')))
F(mark(found(found(x'''')))) -> F(found(found(x'''')))
F(mark(ok(mark(x'''')))) -> F(ok(mark(x'''')))
F(mark(ok(found(x'''')))) -> F(ok(found(x'''')))
F(found(mark(x''))) -> F(mark(x''))
F(found(found(x''))) -> F(found(x''))
F(ok(found(x''))) -> F(found(x''))
F(ok(ok(x''))) -> F(ok(x''))
F(mark(ok(ok(x'''')))) -> F(ok(ok(x'''')))
F(mark(mark(x''))) -> F(mark(x''))
F(ok(mark(x''))) -> F(mark(x''))
F(found(ok(mark(x'''')))) -> F(ok(mark(x'''')))

Rules:

active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
top(active(c)) -> top(mark(c))
top(mark(x)) -> top(check(x))
top(found(x)) -> top(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)

Strategy:

innermost

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

F(ok(ok(x''))) -> F(ok(x''))
three new Dependency Pairs are created:

F(ok(ok(ok(x'''')))) -> F(ok(ok(x'''')))
F(ok(ok(found(x'''')))) -> F(ok(found(x'''')))
F(ok(ok(mark(x'''')))) -> F(ok(mark(x'''')))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 7`
`             ↳FwdInst`
`             ...`
`               →DP Problem 10`
`                 ↳Forward Instantiation Transformation`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳FwdInst`
`       →DP Problem 5`
`         ↳FwdInst`
`       →DP Problem 6`
`         ↳Nar`

Dependency Pairs:

F(found(ok(found(x'''')))) -> F(ok(found(x'''')))
F(mark(found(ok(found(x''''''))))) -> F(found(ok(found(x''''''))))
F(ok(ok(mark(x'''')))) -> F(ok(mark(x'''')))
F(found(ok(ok(x'''')))) -> F(ok(ok(x'''')))
F(mark(found(ok(ok(x''''''))))) -> F(found(ok(ok(x''''''))))
F(mark(found(mark(x'''')))) -> F(found(mark(x'''')))
F(mark(found(found(x'''')))) -> F(found(found(x'''')))
F(mark(ok(mark(x'''')))) -> F(ok(mark(x'''')))
F(mark(ok(found(x'''')))) -> F(ok(found(x'''')))
F(found(mark(x''))) -> F(mark(x''))
F(found(found(x''))) -> F(found(x''))
F(ok(found(x''))) -> F(found(x''))
F(ok(ok(found(x'''')))) -> F(ok(found(x'''')))
F(ok(ok(ok(x'''')))) -> F(ok(ok(x'''')))
F(mark(ok(ok(x'''')))) -> F(ok(ok(x'''')))
F(mark(mark(x''))) -> F(mark(x''))
F(ok(mark(x''))) -> F(mark(x''))
F(found(ok(mark(x'''')))) -> F(ok(mark(x'''')))
F(mark(found(ok(mark(x''''''))))) -> F(found(ok(mark(x''''''))))

Rules:

active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
top(active(c)) -> top(mark(c))
top(mark(x)) -> top(check(x))
top(found(x)) -> top(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)

Strategy:

innermost

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

F(ok(found(x''))) -> F(found(x''))
five new Dependency Pairs are created:

F(ok(found(found(x'''')))) -> F(found(found(x'''')))
F(ok(found(mark(x'''')))) -> F(found(mark(x'''')))
F(ok(found(ok(ok(x''''''))))) -> F(found(ok(ok(x''''''))))
F(ok(found(ok(found(x''''''))))) -> F(found(ok(found(x''''''))))
F(ok(found(ok(mark(x''''''))))) -> F(found(ok(mark(x''''''))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 7`
`             ↳FwdInst`
`             ...`
`               →DP Problem 11`
`                 ↳Forward Instantiation Transformation`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳FwdInst`
`       →DP Problem 5`
`         ↳FwdInst`
`       →DP Problem 6`
`         ↳Nar`

Dependency Pairs:

F(mark(found(ok(mark(x''''''))))) -> F(found(ok(mark(x''''''))))
F(mark(found(ok(found(x''''''))))) -> F(found(ok(found(x''''''))))
F(mark(found(ok(ok(x''''''))))) -> F(found(ok(ok(x''''''))))
F(mark(found(mark(x'''')))) -> F(found(mark(x'''')))
F(mark(found(found(x'''')))) -> F(found(found(x'''')))
F(mark(ok(mark(x'''')))) -> F(ok(mark(x'''')))
F(found(ok(mark(x'''')))) -> F(ok(mark(x'''')))
F(ok(found(ok(mark(x''''''))))) -> F(found(ok(mark(x''''''))))
F(ok(found(ok(found(x''''''))))) -> F(found(ok(found(x''''''))))
F(mark(ok(found(x'''')))) -> F(ok(found(x'''')))
F(ok(mark(x''))) -> F(mark(x''))
F(ok(ok(mark(x'''')))) -> F(ok(mark(x'''')))
F(found(ok(ok(x'''')))) -> F(ok(ok(x'''')))
F(ok(found(ok(ok(x''''''))))) -> F(found(ok(ok(x''''''))))
F(ok(found(mark(x'''')))) -> F(found(mark(x'''')))
F(ok(ok(found(x'''')))) -> F(ok(found(x'''')))
F(ok(ok(ok(x'''')))) -> F(ok(ok(x'''')))
F(mark(ok(ok(x'''')))) -> F(ok(ok(x'''')))
F(mark(mark(x''))) -> F(mark(x''))
F(found(mark(x''))) -> F(mark(x''))
F(found(found(x''))) -> F(found(x''))
F(ok(found(found(x'''')))) -> F(found(found(x'''')))
F(found(ok(found(x'''')))) -> F(ok(found(x'''')))

Rules:

active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
top(active(c)) -> top(mark(c))
top(mark(x)) -> top(check(x))
top(found(x)) -> top(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)

Strategy:

innermost

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

F(ok(mark(x''))) -> F(mark(x''))
nine new Dependency Pairs are created:

F(ok(mark(mark(x'''')))) -> F(mark(mark(x'''')))
F(ok(mark(ok(ok(x''''''))))) -> F(mark(ok(ok(x''''''))))
F(ok(mark(ok(found(x''''''))))) -> F(mark(ok(found(x''''''))))
F(ok(mark(ok(mark(x''''''))))) -> F(mark(ok(mark(x''''''))))
F(ok(mark(found(found(x''''''))))) -> F(mark(found(found(x''''''))))
F(ok(mark(found(mark(x''''''))))) -> F(mark(found(mark(x''''''))))
F(ok(mark(found(ok(ok(x'''''''')))))) -> F(mark(found(ok(ok(x'''''''')))))
F(ok(mark(found(ok(found(x'''''''')))))) -> F(mark(found(ok(found(x'''''''')))))
F(ok(mark(found(ok(mark(x'''''''')))))) -> F(mark(found(ok(mark(x'''''''')))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 7`
`             ↳FwdInst`
`             ...`
`               →DP Problem 12`
`                 ↳Forward Instantiation Transformation`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳FwdInst`
`       →DP Problem 5`
`         ↳FwdInst`
`       →DP Problem 6`
`         ↳Nar`

Dependency Pairs:

F(ok(mark(found(ok(mark(x'''''''')))))) -> F(mark(found(ok(mark(x'''''''')))))
F(ok(found(ok(mark(x''''''))))) -> F(found(ok(mark(x''''''))))
F(ok(found(ok(found(x''''''))))) -> F(found(ok(found(x''''''))))
F(found(ok(found(x'''')))) -> F(ok(found(x'''')))
F(mark(found(ok(found(x''''''))))) -> F(found(ok(found(x''''''))))
F(ok(mark(found(ok(found(x'''''''')))))) -> F(mark(found(ok(found(x'''''''')))))
F(mark(found(ok(ok(x''''''))))) -> F(found(ok(ok(x''''''))))
F(ok(mark(found(ok(ok(x'''''''')))))) -> F(mark(found(ok(ok(x'''''''')))))
F(mark(found(mark(x'''')))) -> F(found(mark(x'''')))
F(ok(mark(found(mark(x''''''))))) -> F(mark(found(mark(x''''''))))
F(mark(found(found(x'''')))) -> F(found(found(x'''')))
F(ok(mark(found(found(x''''''))))) -> F(mark(found(found(x''''''))))
F(mark(ok(mark(x'''')))) -> F(ok(mark(x'''')))
F(ok(mark(ok(mark(x''''''))))) -> F(mark(ok(mark(x''''''))))
F(ok(mark(ok(found(x''''''))))) -> F(mark(ok(found(x''''''))))
F(ok(mark(ok(ok(x''''''))))) -> F(mark(ok(ok(x''''''))))
F(ok(ok(mark(x'''')))) -> F(ok(mark(x'''')))
F(found(ok(ok(x'''')))) -> F(ok(ok(x'''')))
F(ok(found(ok(ok(x''''''))))) -> F(found(ok(ok(x''''''))))
F(ok(found(mark(x'''')))) -> F(found(mark(x'''')))
F(mark(ok(found(x'''')))) -> F(ok(found(x'''')))
F(found(mark(x''))) -> F(mark(x''))
F(found(found(x''))) -> F(found(x''))
F(ok(found(found(x'''')))) -> F(found(found(x'''')))
F(ok(ok(found(x'''')))) -> F(ok(found(x'''')))
F(ok(ok(ok(x'''')))) -> F(ok(ok(x'''')))
F(mark(ok(ok(x'''')))) -> F(ok(ok(x'''')))
F(mark(mark(x''))) -> F(mark(x''))
F(ok(mark(mark(x'''')))) -> F(mark(mark(x'''')))
F(found(ok(mark(x'''')))) -> F(ok(mark(x'''')))
F(mark(found(ok(mark(x''''''))))) -> F(found(ok(mark(x''''''))))

Rules:

active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
top(active(c)) -> top(mark(c))
top(mark(x)) -> top(check(x))
top(found(x)) -> top(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)

Strategy:

innermost

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

F(found(found(x''))) -> F(found(x''))
five new Dependency Pairs are created:

F(found(found(found(x'''')))) -> F(found(found(x'''')))
F(found(found(mark(x'''')))) -> F(found(mark(x'''')))
F(found(found(ok(ok(x''''''))))) -> F(found(ok(ok(x''''''))))
F(found(found(ok(found(x''''''))))) -> F(found(ok(found(x''''''))))
F(found(found(ok(mark(x''''''))))) -> F(found(ok(mark(x''''''))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 7`
`             ↳FwdInst`
`             ...`
`               →DP Problem 13`
`                 ↳Forward Instantiation Transformation`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳FwdInst`
`       →DP Problem 5`
`         ↳FwdInst`
`       →DP Problem 6`
`         ↳Nar`

Dependency Pairs:

F(mark(found(ok(found(x''''''))))) -> F(found(ok(found(x''''''))))
F(ok(mark(found(ok(found(x'''''''')))))) -> F(mark(found(ok(found(x'''''''')))))
F(mark(found(ok(ok(x''''''))))) -> F(found(ok(ok(x''''''))))
F(ok(mark(found(ok(ok(x'''''''')))))) -> F(mark(found(ok(ok(x'''''''')))))
F(mark(found(mark(x'''')))) -> F(found(mark(x'''')))
F(ok(mark(found(mark(x''''''))))) -> F(mark(found(mark(x''''''))))
F(found(found(ok(mark(x''''''))))) -> F(found(ok(mark(x''''''))))
F(ok(found(ok(mark(x''''''))))) -> F(found(ok(mark(x''''''))))
F(ok(found(ok(found(x''''''))))) -> F(found(ok(found(x''''''))))
F(found(ok(found(x'''')))) -> F(ok(found(x'''')))
F(found(found(ok(found(x''''''))))) -> F(found(ok(found(x''''''))))
F(found(found(ok(ok(x''''''))))) -> F(found(ok(ok(x''''''))))
F(mark(found(found(x'''')))) -> F(found(found(x'''')))
F(ok(mark(found(found(x''''''))))) -> F(mark(found(found(x''''''))))
F(mark(ok(mark(x'''')))) -> F(ok(mark(x'''')))
F(ok(mark(ok(mark(x''''''))))) -> F(mark(ok(mark(x''''''))))
F(ok(mark(ok(found(x''''''))))) -> F(mark(ok(found(x''''''))))
F(ok(mark(ok(ok(x''''''))))) -> F(mark(ok(ok(x''''''))))
F(ok(ok(mark(x'''')))) -> F(ok(mark(x'''')))
F(found(ok(ok(x'''')))) -> F(ok(ok(x'''')))
F(ok(found(ok(ok(x''''''))))) -> F(found(ok(ok(x''''''))))
F(ok(found(mark(x'''')))) -> F(found(mark(x'''')))
F(mark(ok(found(x'''')))) -> F(ok(found(x'''')))
F(found(mark(x''))) -> F(mark(x''))
F(found(found(mark(x'''')))) -> F(found(mark(x'''')))
F(found(found(found(x'''')))) -> F(found(found(x'''')))
F(ok(found(found(x'''')))) -> F(found(found(x'''')))
F(ok(ok(found(x'''')))) -> F(ok(found(x'''')))
F(ok(ok(ok(x'''')))) -> F(ok(ok(x'''')))
F(mark(ok(ok(x'''')))) -> F(ok(ok(x'''')))
F(mark(mark(x''))) -> F(mark(x''))
F(ok(mark(mark(x'''')))) -> F(mark(mark(x'''')))
F(found(ok(mark(x'''')))) -> F(ok(mark(x'''')))
F(mark(found(ok(mark(x''''''))))) -> F(found(ok(mark(x''''''))))
F(ok(mark(found(ok(mark(x'''''''')))))) -> F(mark(found(ok(mark(x'''''''')))))

Rules:

active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
top(active(c)) -> top(mark(c))
top(mark(x)) -> top(check(x))
top(found(x)) -> top(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)

Strategy:

innermost

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

F(found(mark(x''))) -> F(mark(x''))
nine new Dependency Pairs are created:

F(found(mark(mark(x'''')))) -> F(mark(mark(x'''')))
F(found(mark(ok(ok(x''''''))))) -> F(mark(ok(ok(x''''''))))
F(found(mark(ok(found(x''''''))))) -> F(mark(ok(found(x''''''))))
F(found(mark(ok(mark(x''''''))))) -> F(mark(ok(mark(x''''''))))
F(found(mark(found(found(x''''''))))) -> F(mark(found(found(x''''''))))
F(found(mark(found(mark(x''''''))))) -> F(mark(found(mark(x''''''))))
F(found(mark(found(ok(ok(x'''''''')))))) -> F(mark(found(ok(ok(x'''''''')))))
F(found(mark(found(ok(found(x'''''''')))))) -> F(mark(found(ok(found(x'''''''')))))
F(found(mark(found(ok(mark(x'''''''')))))) -> F(mark(found(ok(mark(x'''''''')))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 7`
`             ↳FwdInst`
`             ...`
`               →DP Problem 14`
`                 ↳Forward Instantiation Transformation`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳FwdInst`
`       →DP Problem 5`
`         ↳FwdInst`
`       →DP Problem 6`
`         ↳Nar`

Dependency Pairs:

F(ok(found(ok(mark(x''''''))))) -> F(found(ok(mark(x''''''))))
F(ok(found(ok(found(x''''''))))) -> F(found(ok(found(x''''''))))
F(ok(mark(found(ok(mark(x'''''''')))))) -> F(mark(found(ok(mark(x'''''''')))))
F(ok(mark(found(ok(found(x'''''''')))))) -> F(mark(found(ok(found(x'''''''')))))
F(ok(mark(found(ok(ok(x'''''''')))))) -> F(mark(found(ok(ok(x'''''''')))))
F(mark(found(ok(mark(x''''''))))) -> F(found(ok(mark(x''''''))))
F(found(mark(found(ok(mark(x'''''''')))))) -> F(mark(found(ok(mark(x'''''''')))))
F(found(mark(found(ok(found(x'''''''')))))) -> F(mark(found(ok(found(x'''''''')))))
F(mark(found(ok(ok(x''''''))))) -> F(found(ok(ok(x''''''))))
F(found(mark(found(ok(ok(x'''''''')))))) -> F(mark(found(ok(ok(x'''''''')))))
F(found(mark(found(mark(x''''''))))) -> F(mark(found(mark(x''''''))))
F(found(mark(found(found(x''''''))))) -> F(mark(found(found(x''''''))))
F(found(mark(ok(mark(x''''''))))) -> F(mark(ok(mark(x''''''))))
F(mark(found(mark(x'''')))) -> F(found(mark(x'''')))
F(ok(mark(found(mark(x''''''))))) -> F(mark(found(mark(x''''''))))
F(found(ok(mark(x'''')))) -> F(ok(mark(x'''')))
F(found(found(ok(mark(x''''''))))) -> F(found(ok(mark(x''''''))))
F(found(found(ok(found(x''''''))))) -> F(found(ok(found(x''''''))))
F(found(found(ok(ok(x''''''))))) -> F(found(ok(ok(x''''''))))
F(mark(found(found(x'''')))) -> F(found(found(x'''')))
F(ok(mark(found(found(x''''''))))) -> F(mark(found(found(x''''''))))
F(mark(ok(mark(x'''')))) -> F(ok(mark(x'''')))
F(ok(mark(ok(mark(x''''''))))) -> F(mark(ok(mark(x''''''))))
F(ok(mark(ok(found(x''''''))))) -> F(mark(ok(found(x''''''))))
F(ok(mark(ok(ok(x''''''))))) -> F(mark(ok(ok(x''''''))))
F(ok(mark(mark(x'''')))) -> F(mark(mark(x'''')))
F(ok(ok(mark(x'''')))) -> F(ok(mark(x'''')))
F(found(ok(ok(x'''')))) -> F(ok(ok(x'''')))
F(ok(found(ok(ok(x''''''))))) -> F(found(ok(ok(x''''''))))
F(mark(ok(found(x'''')))) -> F(ok(found(x'''')))
F(found(mark(ok(found(x''''''))))) -> F(mark(ok(found(x''''''))))
F(found(mark(ok(ok(x''''''))))) -> F(mark(ok(ok(x''''''))))
F(ok(found(mark(x'''')))) -> F(found(mark(x'''')))
F(ok(ok(found(x'''')))) -> F(ok(found(x'''')))
F(ok(ok(ok(x'''')))) -> F(ok(ok(x'''')))
F(mark(ok(ok(x'''')))) -> F(ok(ok(x'''')))
F(mark(mark(x''))) -> F(mark(x''))
F(found(mark(mark(x'''')))) -> F(mark(mark(x'''')))
F(found(found(mark(x'''')))) -> F(found(mark(x'''')))
F(found(found(found(x'''')))) -> F(found(found(x'''')))
F(ok(found(found(x'''')))) -> F(found(found(x'''')))
F(found(ok(found(x'''')))) -> F(ok(found(x'''')))
F(mark(found(ok(found(x''''''))))) -> F(found(ok(found(x''''''))))

Rules:

active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
top(active(c)) -> top(mark(c))
top(mark(x)) -> top(check(x))
top(found(x)) -> top(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)

Strategy:

innermost

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

F(found(ok(ok(x'''')))) -> F(ok(ok(x'''')))
three new Dependency Pairs are created:

F(found(ok(ok(ok(x''''''))))) -> F(ok(ok(ok(x''''''))))
F(found(ok(ok(found(x''''''))))) -> F(ok(ok(found(x''''''))))
F(found(ok(ok(mark(x''''''))))) -> F(ok(ok(mark(x''''''))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 7`
`             ↳FwdInst`
`             ...`
`               →DP Problem 15`
`                 ↳Forward Instantiation Transformation`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳FwdInst`
`       →DP Problem 5`
`         ↳FwdInst`
`       →DP Problem 6`
`         ↳Nar`

Dependency Pairs:

F(found(found(ok(mark(x''''''))))) -> F(found(ok(mark(x''''''))))
F(found(found(ok(found(x''''''))))) -> F(found(ok(found(x''''''))))
F(ok(mark(found(ok(mark(x'''''''')))))) -> F(mark(found(ok(mark(x'''''''')))))
F(ok(mark(found(ok(found(x'''''''')))))) -> F(mark(found(ok(found(x'''''''')))))
F(ok(mark(found(ok(ok(x'''''''')))))) -> F(mark(found(ok(ok(x'''''''')))))
F(mark(found(ok(mark(x''''''))))) -> F(found(ok(mark(x''''''))))
F(found(mark(found(ok(mark(x'''''''')))))) -> F(mark(found(ok(mark(x'''''''')))))
F(ok(found(ok(found(x''''''))))) -> F(found(ok(found(x''''''))))
F(ok(found(ok(ok(x''''''))))) -> F(found(ok(ok(x''''''))))
F(found(ok(found(x'''')))) -> F(ok(found(x'''')))
F(mark(found(ok(found(x''''''))))) -> F(found(ok(found(x''''''))))
F(found(mark(found(ok(found(x'''''''')))))) -> F(mark(found(ok(found(x'''''''')))))
F(mark(found(ok(ok(x''''''))))) -> F(found(ok(ok(x''''''))))
F(found(mark(found(ok(ok(x'''''''')))))) -> F(mark(found(ok(ok(x'''''''')))))
F(found(mark(found(mark(x''''''))))) -> F(mark(found(mark(x''''''))))
F(found(mark(found(found(x''''''))))) -> F(mark(found(found(x''''''))))
F(mark(found(mark(x'''')))) -> F(found(mark(x'''')))
F(ok(mark(found(mark(x''''''))))) -> F(mark(found(mark(x''''''))))
F(ok(ok(mark(x'''')))) -> F(ok(mark(x'''')))
F(found(ok(ok(mark(x''''''))))) -> F(ok(ok(mark(x''''''))))
F(found(ok(ok(found(x''''''))))) -> F(ok(ok(found(x''''''))))
F(found(ok(ok(ok(x''''''))))) -> F(ok(ok(ok(x''''''))))
F(found(found(ok(ok(x''''''))))) -> F(found(ok(ok(x''''''))))
F(mark(found(found(x'''')))) -> F(found(found(x'''')))
F(ok(mark(found(found(x''''''))))) -> F(mark(found(found(x''''''))))
F(ok(mark(ok(mark(x''''''))))) -> F(mark(ok(mark(x''''''))))
F(ok(mark(ok(found(x''''''))))) -> F(mark(ok(found(x''''''))))
F(ok(mark(ok(ok(x''''''))))) -> F(mark(ok(ok(x''''''))))
F(mark(ok(mark(x'''')))) -> F(ok(mark(x'''')))
F(found(mark(ok(mark(x''''''))))) -> F(mark(ok(mark(x''''''))))
F(ok(found(mark(x'''')))) -> F(found(mark(x'''')))
F(mark(ok(found(x'''')))) -> F(ok(found(x'''')))
F(found(mark(ok(found(x''''''))))) -> F(mark(ok(found(x''''''))))
F(found(mark(ok(ok(x''''''))))) -> F(mark(ok(ok(x''''''))))
F(found(mark(mark(x'''')))) -> F(mark(mark(x'''')))
F(found(found(mark(x'''')))) -> F(found(mark(x'''')))
F(found(found(found(x'''')))) -> F(found(found(x'''')))
F(ok(found(found(x'''')))) -> F(found(found(x'''')))
F(ok(ok(found(x'''')))) -> F(ok(found(x'''')))
F(ok(ok(ok(x'''')))) -> F(ok(ok(x'''')))
F(mark(ok(ok(x'''')))) -> F(ok(ok(x'''')))
F(mark(mark(x''))) -> F(mark(x''))
F(ok(mark(mark(x'''')))) -> F(mark(mark(x'''')))
F(found(ok(mark(x'''')))) -> F(ok(mark(x'''')))
F(ok(found(ok(mark(x''''''))))) -> F(found(ok(mark(x''''''))))

Rules:

active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
top(active(c)) -> top(mark(c))
top(mark(x)) -> top(check(x))
top(found(x)) -> top(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)

Strategy:

innermost

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

F(found(ok(found(x'''')))) -> F(ok(found(x'''')))
five new Dependency Pairs are created:

F(found(ok(found(found(x''''''))))) -> F(ok(found(found(x''''''))))
F(found(ok(found(mark(x''''''))))) -> F(ok(found(mark(x''''''))))
F(found(ok(found(ok(ok(x'''''''')))))) -> F(ok(found(ok(ok(x'''''''')))))
F(found(ok(found(ok(found(x'''''''')))))) -> F(ok(found(ok(found(x'''''''')))))
F(found(ok(found(ok(mark(x'''''''')))))) -> F(ok(found(ok(mark(x'''''''')))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 7`
`             ↳FwdInst`
`             ...`
`               →DP Problem 16`
`                 ↳Forward Instantiation Transformation`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳FwdInst`
`       →DP Problem 5`
`         ↳FwdInst`
`       →DP Problem 6`
`         ↳Nar`

Dependency Pairs:

F(found(found(ok(found(x''''''))))) -> F(found(ok(found(x''''''))))
F(ok(mark(found(ok(mark(x'''''''')))))) -> F(mark(found(ok(mark(x'''''''')))))
F(ok(mark(found(ok(found(x'''''''')))))) -> F(mark(found(ok(found(x'''''''')))))
F(ok(mark(found(ok(ok(x'''''''')))))) -> F(mark(found(ok(ok(x'''''''')))))
F(mark(found(ok(mark(x''''''))))) -> F(found(ok(mark(x''''''))))
F(found(mark(found(ok(mark(x'''''''')))))) -> F(mark(found(ok(mark(x'''''''')))))
F(ok(found(ok(mark(x''''''))))) -> F(found(ok(mark(x''''''))))
F(found(ok(found(ok(mark(x'''''''')))))) -> F(ok(found(ok(mark(x'''''''')))))
F(ok(found(ok(found(x''''''))))) -> F(found(ok(found(x''''''))))
F(found(ok(found(ok(found(x'''''''')))))) -> F(ok(found(ok(found(x'''''''')))))
F(ok(found(ok(ok(x''''''))))) -> F(found(ok(ok(x''''''))))
F(found(ok(found(ok(ok(x'''''''')))))) -> F(ok(found(ok(ok(x'''''''')))))
F(found(ok(found(mark(x''''''))))) -> F(ok(found(mark(x''''''))))
F(found(ok(found(found(x''''''))))) -> F(ok(found(found(x''''''))))
F(mark(found(ok(found(x''''''))))) -> F(found(ok(found(x''''''))))
F(found(mark(found(ok(found(x'''''''')))))) -> F(mark(found(ok(found(x'''''''')))))
F(mark(found(ok(ok(x''''''))))) -> F(found(ok(ok(x''''''))))
F(found(mark(found(ok(ok(x'''''''')))))) -> F(mark(found(ok(ok(x'''''''')))))
F(found(mark(found(mark(x''''''))))) -> F(mark(found(mark(x''''''))))
F(found(mark(found(found(x''''''))))) -> F(mark(found(found(x''''''))))
F(mark(found(mark(x'''')))) -> F(found(mark(x'''')))
F(ok(mark(found(mark(x''''''))))) -> F(mark(found(mark(x''''''))))
F(ok(ok(mark(x'''')))) -> F(ok(mark(x'''')))
F(found(ok(ok(mark(x''''''))))) -> F(ok(ok(mark(x''''''))))
F(found(ok(ok(found(x''''''))))) -> F(ok(ok(found(x''''''))))
F(found(ok(ok(ok(x''''''))))) -> F(ok(ok(ok(x''''''))))
F(found(found(ok(ok(x''''''))))) -> F(found(ok(ok(x''''''))))
F(mark(found(found(x'''')))) -> F(found(found(x'''')))
F(ok(mark(found(found(x''''''))))) -> F(mark(found(found(x''''''))))
F(ok(mark(ok(mark(x''''''))))) -> F(mark(ok(mark(x''''''))))
F(ok(mark(ok(found(x''''''))))) -> F(mark(ok(found(x''''''))))
F(ok(mark(ok(ok(x''''''))))) -> F(mark(ok(ok(x''''''))))
F(mark(ok(mark(x'''')))) -> F(ok(mark(x'''')))
F(found(mark(ok(mark(x''''''))))) -> F(mark(ok(mark(x''''''))))
F(ok(found(mark(x'''')))) -> F(found(mark(x'''')))
F(mark(ok(found(x'''')))) -> F(ok(found(x'''')))
F(found(mark(ok(found(x''''''))))) -> F(mark(ok(found(x''''''))))
F(found(mark(ok(ok(x''''''))))) -> F(mark(ok(ok(x''''''))))
F(found(mark(mark(x'''')))) -> F(mark(mark(x'''')))
F(found(found(mark(x'''')))) -> F(found(mark(x'''')))
F(found(found(found(x'''')))) -> F(found(found(x'''')))
F(ok(found(found(x'''')))) -> F(found(found(x'''')))
F(ok(ok(found(x'''')))) -> F(ok(found(x'''')))
F(ok(ok(ok(x'''')))) -> F(ok(ok(x'''')))
F(mark(ok(ok(x'''')))) -> F(ok(ok(x'''')))
F(mark(mark(x''))) -> F(mark(x''))
F(ok(mark(mark(x'''')))) -> F(mark(mark(x'''')))
F(found(ok(mark(x'''')))) -> F(ok(mark(x'''')))
F(found(found(ok(mark(x''''''))))) -> F(found(ok(mark(x''''''))))

Rules:

active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
top(active(c)) -> top(mark(c))
top(mark(x)) -> top(check(x))
top(found(x)) -> top(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)

Strategy:

innermost

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

F(found(ok(mark(x'''')))) -> F(ok(mark(x'''')))
nine new Dependency Pairs are created:

F(found(ok(mark(mark(x''''''))))) -> F(ok(mark(mark(x''''''))))
F(found(ok(mark(ok(ok(x'''''''')))))) -> F(ok(mark(ok(ok(x'''''''')))))
F(found(ok(mark(ok(found(x'''''''')))))) -> F(ok(mark(ok(found(x'''''''')))))
F(found(ok(mark(ok(mark(x'''''''')))))) -> F(ok(mark(ok(mark(x'''''''')))))
F(found(ok(mark(found(found(x'''''''')))))) -> F(ok(mark(found(found(x'''''''')))))
F(found(ok(mark(found(mark(x'''''''')))))) -> F(ok(mark(found(mark(x'''''''')))))
F(found(ok(mark(found(ok(ok(x''''''''''))))))) -> F(ok(mark(found(ok(ok(x''''''''''))))))
F(found(ok(mark(found(ok(found(x''''''''''))))))) -> F(ok(mark(found(ok(found(x''''''''''))))))
F(found(ok(mark(found(ok(mark(x''''''''''))))))) -> F(ok(mark(found(ok(mark(x''''''''''))))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 7`
`             ↳FwdInst`
`             ...`
`               →DP Problem 17`
`                 ↳Forward Instantiation Transformation`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳FwdInst`
`       →DP Problem 5`
`         ↳FwdInst`
`       →DP Problem 6`
`         ↳Nar`

Dependency Pairs:

F(found(mark(found(ok(mark(x'''''''')))))) -> F(mark(found(ok(mark(x'''''''')))))
F(mark(found(ok(mark(x''''''))))) -> F(found(ok(mark(x''''''))))
F(ok(mark(found(ok(mark(x'''''''')))))) -> F(mark(found(ok(mark(x'''''''')))))
F(found(ok(mark(found(ok(mark(x''''''''''))))))) -> F(ok(mark(found(ok(mark(x''''''''''))))))
F(ok(mark(found(ok(found(x'''''''')))))) -> F(mark(found(ok(found(x'''''''')))))
F(found(ok(mark(found(ok(found(x''''''''''))))))) -> F(ok(mark(found(ok(found(x''''''''''))))))
F(ok(mark(found(ok(ok(x'''''''')))))) -> F(mark(found(ok(ok(x'''''''')))))
F(found(ok(mark(found(ok(ok(x''''''''''))))))) -> F(ok(mark(found(ok(ok(x''''''''''))))))
F(ok(found(ok(mark(x''''''))))) -> F(found(ok(mark(x''''''))))
F(found(ok(found(ok(mark(x'''''''')))))) -> F(ok(found(ok(mark(x'''''''')))))
F(ok(found(ok(found(x''''''))))) -> F(found(ok(found(x''''''))))
F(found(ok(found(ok(found(x'''''''')))))) -> F(ok(found(ok(found(x'''''''')))))
F(found(ok(found(ok(ok(x'''''''')))))) -> F(ok(found(ok(ok(x'''''''')))))
F(found(ok(found(mark(x''''''))))) -> F(ok(found(mark(x''''''))))
F(mark(found(ok(found(x''''''))))) -> F(found(ok(found(x''''''))))
F(found(mark(found(ok(found(x'''''''')))))) -> F(mark(found(ok(found(x'''''''')))))
F(mark(found(ok(ok(x''''''))))) -> F(found(ok(ok(x''''''))))
F(found(mark(found(ok(ok(x'''''''')))))) -> F(mark(found(ok(ok(x'''''''')))))
F(found(mark(found(mark(x''''''))))) -> F(mark(found(mark(x''''''))))
F(found(mark(found(found(x''''''))))) -> F(mark(found(found(x''''''))))
F(found(mark(ok(mark(x''''''))))) -> F(mark(ok(mark(x''''''))))
F(mark(found(mark(x'''')))) -> F(found(mark(x'''')))
F(ok(mark(found(mark(x''''''))))) -> F(mark(found(mark(x''''''))))
F(found(ok(mark(found(mark(x'''''''')))))) -> F(ok(mark(found(mark(x'''''''')))))
F(found(ok(mark(found(found(x'''''''')))))) -> F(ok(mark(found(found(x'''''''')))))
F(found(ok(mark(ok(mark(x'''''''')))))) -> F(ok(mark(ok(mark(x'''''''')))))
F(found(ok(mark(ok(found(x'''''''')))))) -> F(ok(mark(ok(found(x'''''''')))))
F(found(ok(mark(ok(ok(x'''''''')))))) -> F(ok(mark(ok(ok(x'''''''')))))
F(found(ok(mark(mark(x''''''))))) -> F(ok(mark(mark(x''''''))))
F(found(found(ok(mark(x''''''))))) -> F(found(ok(mark(x''''''))))
F(found(found(ok(ok(x''''''))))) -> F(found(ok(ok(x''''''))))
F(mark(found(found(x'''')))) -> F(found(found(x'''')))
F(ok(mark(found(found(x''''''))))) -> F(mark(found(found(x''''''))))
F(mark(ok(mark(x'''')))) -> F(ok(mark(x'''')))
F(ok(mark(ok(mark(x''''''))))) -> F(mark(ok(mark(x''''''))))
F(ok(mark(ok(found(x''''''))))) -> F(mark(ok(found(x''''''))))
F(ok(mark(ok(ok(x''''''))))) -> F(mark(ok(ok(x''''''))))
F(ok(mark(mark(x'''')))) -> F(mark(mark(x'''')))
F(ok(ok(mark(x'''')))) -> F(ok(mark(x'''')))
F(found(ok(ok(mark(x''''''))))) -> F(ok(ok(mark(x''''''))))
F(found(ok(ok(found(x''''''))))) -> F(ok(ok(found(x''''''))))
F(found(ok(ok(ok(x''''''))))) -> F(ok(ok(ok(x''''''))))
F(ok(found(ok(ok(x''''''))))) -> F(found(ok(ok(x''''''))))
F(mark(ok(found(x'''')))) -> F(ok(found(x'''')))
F(found(mark(ok(found(x''''''))))) -> F(mark(ok(found(x''''''))))
F(found(mark(ok(ok(x''''''))))) -> F(mark(ok(ok(x''''''))))
F(ok(found(mark(x'''')))) -> F(found(mark(x'''')))
F(ok(ok(found(x'''')))) -> F(ok(found(x'''')))
F(ok(ok(ok(x'''')))) -> F(ok(ok(x'''')))
F(mark(ok(ok(x'''')))) -> F(ok(ok(x'''')))
F(mark(mark(x''))) -> F(mark(x''))
F(found(mark(mark(x'''')))) -> F(mark(mark(x'''')))
F(found(found(mark(x'''')))) -> F(found(mark(x'''')))
F(found(found(found(x'''')))) -> F(found(found(x'''')))
F(ok(found(found(x'''')))) -> F(found(found(x'''')))
F(found(ok(found(found(x''''''))))) -> F(ok(found(found(x''''''))))
F(found(found(ok(found(x''''''))))) -> F(found(ok(found(x''''''))))

Rules:

active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
top(active(c)) -> top(mark(c))
top(mark(x)) -> top(check(x))
top(found(x)) -> top(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)

Strategy:

innermost

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

F(mark(mark(x''))) -> F(mark(x''))
nine new Dependency Pairs are created:

F(mark(mark(mark(x'''')))) -> F(mark(mark(x'''')))
F(mark(mark(ok(ok(x''''''))))) -> F(mark(ok(ok(x''''''))))
F(mark(mark(ok(found(x''''''))))) -> F(mark(ok(found(x''''''))))
F(mark(mark(ok(mark(x''''''))))) -> F(mark(ok(mark(x''''''))))
F(mark(mark(found(found(x''''''))))) -> F(mark(found(found(x''''''))))
F(mark(mark(found(mark(x''''''))))) -> F(mark(found(mark(x''''''))))
F(mark(mark(found(ok(ok(x'''''''')))))) -> F(mark(found(ok(ok(x'''''''')))))
F(mark(mark(found(ok(found(x'''''''')))))) -> F(mark(found(ok(found(x'''''''')))))
F(mark(mark(found(ok(mark(x'''''''')))))) -> F(mark(found(ok(mark(x'''''''')))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 7`
`             ↳FwdInst`
`             ...`
`               →DP Problem 18`
`                 ↳Forward Instantiation Transformation`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳FwdInst`
`       →DP Problem 5`
`         ↳FwdInst`
`       →DP Problem 6`
`         ↳Nar`

Dependency Pairs:

F(mark(mark(found(ok(mark(x'''''''')))))) -> F(mark(found(ok(mark(x'''''''')))))
F(mark(mark(found(ok(found(x'''''''')))))) -> F(mark(found(ok(found(x'''''''')))))
F(mark(mark(found(ok(ok(x'''''''')))))) -> F(mark(found(ok(ok(x'''''''')))))
F(mark(mark(found(mark(x''''''))))) -> F(mark(found(mark(x''''''))))
F(mark(mark(found(found(x''''''))))) -> F(mark(found(found(x''''''))))
F(mark(mark(ok(mark(x''''''))))) -> F(mark(ok(mark(x''''''))))
F(found(found(ok(mark(x''''''))))) -> F(found(ok(mark(x''''''))))
F(found(found(ok(found(x''''''))))) -> F(found(ok(found(x''''''))))
F(ok(mark(found(ok(mark(x'''''''')))))) -> F(mark(found(ok(mark(x'''''''')))))
F(found(ok(mark(found(ok(mark(x''''''''''))))))) -> F(ok(mark(found(ok(mark(x''''''''''))))))
F(ok(mark(found(ok(found(x'''''''')))))) -> F(mark(found(ok(found(x'''''''')))))
F(found(ok(mark(found(ok(found(x''''''''''))))))) -> F(ok(mark(found(ok(found(x''''''''''))))))
F(ok(mark(found(ok(ok(x'''''''')))))) -> F(mark(found(ok(ok(x'''''''')))))
F(found(ok(mark(found(ok(ok(x''''''''''))))))) -> F(ok(mark(found(ok(ok(x''''''''''))))))
F(found(ok(mark(found(mark(x'''''''')))))) -> F(ok(mark(found(mark(x'''''''')))))
F(found(ok(mark(found(found(x'''''''')))))) -> F(ok(mark(found(found(x'''''''')))))
F(found(ok(mark(ok(mark(x'''''''')))))) -> F(ok(mark(ok(mark(x'''''''')))))
F(found(ok(mark(ok(found(x'''''''')))))) -> F(ok(mark(ok(found(x'''''''')))))
F(found(ok(mark(ok(ok(x'''''''')))))) -> F(ok(mark(ok(ok(x'''''''')))))
F(ok(found(ok(mark(x''''''))))) -> F(found(ok(mark(x''''''))))
F(found(ok(found(ok(mark(x'''''''')))))) -> F(ok(found(ok(mark(x'''''''')))))
F(ok(found(ok(found(x''''''))))) -> F(found(ok(found(x''''''))))
F(found(ok(found(ok(found(x'''''''')))))) -> F(ok(found(ok(found(x'''''''')))))
F(ok(found(ok(ok(x''''''))))) -> F(found(ok(ok(x''''''))))
F(found(ok(found(ok(ok(x'''''''')))))) -> F(ok(found(ok(ok(x'''''''')))))
F(found(ok(found(mark(x''''''))))) -> F(ok(found(mark(x''''''))))
F(found(ok(found(found(x''''''))))) -> F(ok(found(found(x''''''))))
F(mark(found(ok(found(x''''''))))) -> F(found(ok(found(x''''''))))
F(found(mark(found(ok(found(x'''''''')))))) -> F(mark(found(ok(found(x'''''''')))))
F(mark(found(ok(ok(x''''''))))) -> F(found(ok(ok(x''''''))))
F(found(mark(found(ok(ok(x'''''''')))))) -> F(mark(found(ok(ok(x'''''''')))))
F(found(mark(found(mark(x''''''))))) -> F(mark(found(mark(x''''''))))
F(found(mark(found(found(x''''''))))) -> F(mark(found(found(x''''''))))
F(mark(found(mark(x'''')))) -> F(found(mark(x'''')))
F(ok(mark(found(mark(x''''''))))) -> F(mark(found(mark(x''''''))))
F(ok(ok(mark(x'''')))) -> F(ok(mark(x'''')))
F(found(ok(ok(mark(x''''''))))) -> F(ok(ok(mark(x''''''))))
F(found(ok(ok(found(x''''''))))) -> F(ok(ok(found(x''''''))))
F(found(ok(ok(ok(x''''''))))) -> F(ok(ok(ok(x''''''))))
F(found(found(ok(ok(x''''''))))) -> F(found(ok(ok(x''''''))))
F(mark(found(found(x'''')))) -> F(found(found(x'''')))
F(ok(mark(found(found(x''''''))))) -> F(mark(found(found(x''''''))))
F(ok(mark(ok(mark(x''''''))))) -> F(mark(ok(mark(x''''''))))
F(ok(mark(ok(found(x''''''))))) -> F(mark(ok(found(x''''''))))
F(ok(mark(ok(ok(x''''''))))) -> F(mark(ok(ok(x''''''))))
F(mark(ok(mark(x'''')))) -> F(ok(mark(x'''')))
F(found(mark(ok(mark(x''''''))))) -> F(mark(ok(mark(x''''''))))
F(found(mark(ok(found(x''''''))))) -> F(mark(ok(found(x''''''))))
F(found(mark(ok(ok(x''''''))))) -> F(mark(ok(ok(x''''''))))
F(ok(found(mark(x'''')))) -> F(found(mark(x'''')))
F(mark(ok(found(x'''')))) -> F(ok(found(x'''')))
F(mark(mark(ok(found(x''''''))))) -> F(mark(ok(found(x''''''))))
F(found(mark(mark(x'''')))) -> F(mark(mark(x'''')))
F(found(found(mark(x'''')))) -> F(found(mark(x'''')))
F(found(found(found(x'''')))) -> F(found(found(x'''')))
F(ok(found(found(x'''')))) -> F(found(found(x'''')))
F(ok(ok(found(x'''')))) -> F(ok(found(x'''')))
F(ok(ok(ok(x'''')))) -> F(ok(ok(x'''')))
F(mark(ok(ok(x'''')))) -> F(ok(ok(x'''')))
F(mark(mark(ok(ok(x''''''))))) -> F(mark(ok(ok(x''''''))))
F(mark(mark(mark(x'''')))) -> F(mark(mark(x'''')))
F(ok(mark(mark(x'''')))) -> F(mark(mark(x'''')))
F(found(ok(mark(mark(x''''''))))) -> F(ok(mark(mark(x''''''))))
F(mark(found(ok(mark(x''''''))))) -> F(found(ok(mark(x''''''))))
F(found(mark(found(ok(mark(x'''''''')))))) -> F(mark(found(ok(mark(x'''''''')))))

Rules:

active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
top(active(c)) -> top(mark(c))
top(mark(x)) -> top(check(x))
top(found(x)) -> top(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)

Strategy:

innermost

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

F(mark(ok(ok(x'''')))) -> F(ok(ok(x'''')))
three new Dependency Pairs are created:

F(mark(ok(ok(ok(x''''''))))) -> F(ok(ok(ok(x''''''))))
F(mark(ok(ok(found(x''''''))))) -> F(ok(ok(found(x''''''))))
F(mark(ok(ok(mark(x''''''))))) -> F(ok(ok(mark(x''''''))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 7`
`             ↳FwdInst`
`             ...`
`               →DP Problem 19`
`                 ↳Forward Instantiation Transformation`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳FwdInst`
`       →DP Problem 5`
`         ↳FwdInst`
`       →DP Problem 6`
`         ↳Nar`

Dependency Pairs:

F(mark(mark(found(ok(found(x'''''''')))))) -> F(mark(found(ok(found(x'''''''')))))
F(mark(mark(found(ok(ok(x'''''''')))))) -> F(mark(found(ok(ok(x'''''''')))))
F(mark(mark(found(mark(x''''''))))) -> F(mark(found(mark(x''''''))))
F(mark(mark(found(found(x''''''))))) -> F(mark(found(found(x''''''))))
F(mark(mark(ok(mark(x''''''))))) -> F(mark(ok(mark(x''''''))))
F(found(found(ok(mark(x''''''))))) -> F(found(ok(mark(x''''''))))
F(ok(mark(found(ok(mark(x'''''''')))))) -> F(mark(found(ok(mark(x'''''''')))))
F(found(ok(mark(found(ok(mark(x''''''''''))))))) -> F(ok(mark(found(ok(mark(x''''''''''))))))
F(ok(mark(found(ok(found(x'''''''')))))) -> F(mark(found(ok(found(x'''''''')))))
F(found(ok(mark(found(ok(found(x''''''''''))))))) -> F(ok(mark(found(ok(found(x''''''''''))))))
F(ok(mark(found(ok(ok(x'''''''')))))) -> F(mark(found(ok(ok(x'''''''')))))
F(found(ok(mark(found(ok(ok(x''''''''''))))))) -> F(ok(mark(found(ok(ok(x''''''''''))))))
F(found(mark(found(ok(mark(x'''''''')))))) -> F(mark(found(ok(mark(x'''''''')))))
F(mark(found(ok(found(x''''''))))) -> F(found(ok(found(x''''''))))
F(found(mark(found(ok(found(x'''''''')))))) -> F(mark(found(ok(found(x'''''''')))))
F(mark(found(ok(ok(x''''''))))) -> F(found(ok(ok(x''''''))))
F(found(mark(found(ok(ok(x'''''''')))))) -> F(mark(found(ok(ok(x'''''''')))))
F(found(mark(found(mark(x''''''))))) -> F(mark(found(mark(x''''''))))
F(found(mark(found(found(x''''''))))) -> F(mark(found(found(x''''''))))
F(found(mark(ok(mark(x''''''))))) -> F(mark(ok(mark(x''''''))))
F(found(mark(ok(found(x''''''))))) -> F(mark(ok(found(x''''''))))
F(mark(found(mark(x'''')))) -> F(found(mark(x'''')))
F(ok(mark(found(mark(x''''''))))) -> F(mark(found(mark(x''''''))))
F(found(ok(mark(found(mark(x'''''''')))))) -> F(ok(mark(found(mark(x'''''''')))))
F(found(ok(mark(found(found(x'''''''')))))) -> F(ok(mark(found(found(x'''''''')))))
F(found(ok(mark(ok(mark(x'''''''')))))) -> F(ok(mark(ok(mark(x'''''''')))))
F(found(ok(mark(ok(found(x'''''''')))))) -> F(ok(mark(ok(found(x'''''''')))))
F(found(ok(mark(ok(ok(x'''''''')))))) -> F(ok(mark(ok(ok(x'''''''')))))
F(ok(found(ok(mark(x''''''))))) -> F(found(ok(mark(x''''''))))
F(found(ok(found(ok(mark(x'''''''')))))) -> F(ok(found(ok(mark(x'''''''')))))
F(ok(found(ok(found(x''''''))))) -> F(found(ok(found(x''''''))))
F(found(ok(found(ok(found(x'''''''')))))) -> F(ok(found(ok(found(x'''''''')))))
F(ok(found(ok(ok(x''''''))))) -> F(found(ok(ok(x''''''))))
F(found(ok(found(ok(ok(x'''''''')))))) -> F(ok(found(ok(ok(x'''''''')))))
F(found(ok(found(mark(x''''''))))) -> F(ok(found(mark(x''''''))))
F(found(ok(found(found(x''''''))))) -> F(ok(found(found(x''''''))))
F(found(found(ok(found(x''''''))))) -> F(found(ok(found(x''''''))))
F(found(ok(ok(mark(x''''''))))) -> F(ok(ok(mark(x''''''))))
F(found(ok(ok(found(x''''''))))) -> F(ok(ok(found(x''''''))))
F(found(ok(ok(ok(x''''''))))) -> F(ok(ok(ok(x''''''))))
F(found(found(ok(ok(x''''''))))) -> F(found(ok(ok(x''''''))))
F(mark(found(found(x'''')))) -> F(found(found(x'''')))
F(ok(mark(found(found(x''''''))))) -> F(mark(found(found(x''''''))))
F(mark(ok(mark(x'''')))) -> F(ok(mark(x'''')))
F(ok(mark(ok(mark(x''''''))))) -> F(mark(ok(mark(x''''''))))
F(ok(mark(ok(found(x''''''))))) -> F(mark(ok(found(x''''''))))
F(ok(mark(ok(ok(x''''''))))) -> F(mark(ok(ok(x''''''))))
F(ok(ok(mark(x'''')))) -> F(ok(mark(x'''')))
F(mark(ok(ok(mark(x''''''))))) -> F(ok(ok(mark(x''''''))))
F(mark(ok(ok(found(x''''''))))) -> F(ok(ok(found(x''''''))))
F(found(mark(ok(ok(x''''''))))) -> F(mark(ok(ok(x''''''))))
F(ok(found(mark(x'''')))) -> F(found(mark(x'''')))
F(mark(ok(found(x'''')))) -> F(ok(found(x'''')))
F(mark(mark(ok(found(x''''''))))) -> F(mark(ok(found(x''''''))))
F(found(mark(mark(x'''')))) -> F(mark(mark(x'''')))
F(found(found(mark(x'''')))) -> F(found(mark(x'''')))
F(found(found(found(x'''')))) -> F(found(found(x'''')))
F(ok(found(found(x'''')))) -> F(found(found(x'''')))
F(ok(ok(found(x'''')))) -> F(ok(found(x'''')))
F(ok(ok(ok(x'''')))) -> F(ok(ok(x'''')))
F(mark(ok(ok(ok(x''''''))))) -> F(ok(ok(ok(x''''''))))
F(mark(mark(ok(ok(x''''''))))) -> F(mark(ok(ok(x''''''))))
F(mark(mark(mark(x'''')))) -> F(mark(mark(x'''')))
F(ok(mark(mark(x'''')))) -> F(mark(mark(x'''')))
F(found(ok(mark(mark(x''''''))))) -> F(ok(mark(mark(x''''''))))
F(mark(found(ok(mark(x''''''))))) -> F(found(ok(mark(x''''''))))
F(mark(mark(found(ok(mark(x'''''''')))))) -> F(mark(found(ok(mark(x'''''''')))))

Rules:

active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
top(active(c)) -> top(mark(c))
top(mark(x)) -> top(check(x))
top(found(x)) -> top(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)

Strategy:

innermost

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

F(mark(ok(found(x'''')))) -> F(ok(found(x'''')))
five new Dependency Pairs are created:

F(mark(ok(found(found(x''''''))))) -> F(ok(found(found(x''''''))))
F(mark(ok(found(mark(x''''''))))) -> F(ok(found(mark(x''''''))))
F(mark(ok(found(ok(ok(x'''''''')))))) -> F(ok(found(ok(ok(x'''''''')))))
F(mark(ok(found(ok(found(x'''''''')))))) -> F(ok(found(ok(found(x'''''''')))))
F(mark(ok(found(ok(mark(x'''''''')))))) -> F(ok(found(ok(mark(x'''''''')))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 7`
`             ↳FwdInst`
`             ...`
`               →DP Problem 20`
`                 ↳Forward Instantiation Transformation`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳FwdInst`
`       →DP Problem 5`
`         ↳FwdInst`
`       →DP Problem 6`
`         ↳Nar`

Dependency Pairs:

F(mark(mark(found(ok(mark(x'''''''')))))) -> F(mark(found(ok(mark(x'''''''')))))
F(mark(mark(found(ok(ok(x'''''''')))))) -> F(mark(found(ok(ok(x'''''''')))))
F(mark(mark(found(mark(x''''''))))) -> F(mark(found(mark(x''''''))))
F(mark(mark(found(found(x''''''))))) -> F(mark(found(found(x''''''))))
F(mark(mark(ok(mark(x''''''))))) -> F(mark(ok(mark(x''''''))))
F(ok(mark(found(ok(mark(x'''''''')))))) -> F(mark(found(ok(mark(x'''''''')))))
F(found(ok(mark(found(ok(mark(x''''''''''))))))) -> F(ok(mark(found(ok(mark(x''''''''''))))))
F(ok(mark(found(ok(found(x'''''''')))))) -> F(mark(found(ok(found(x'''''''')))))
F(found(ok(mark(found(ok(found(x''''''''''))))))) -> F(ok(mark(found(ok(found(x''''''''''))))))
F(ok(mark(found(ok(ok(x'''''''')))))) -> F(mark(found(ok(ok(x'''''''')))))
F(found(ok(mark(found(ok(ok(x''''''''''))))))) -> F(ok(mark(found(ok(ok(x''''''''''))))))
F(mark(found(ok(mark(x''''''))))) -> F(found(ok(mark(x''''''))))
F(found(mark(found(ok(mark(x'''''''')))))) -> F(mark(found(ok(mark(x'''''''')))))
F(found(mark(found(ok(found(x'''''''')))))) -> F(mark(found(ok(found(x'''''''')))))
F(mark(found(ok(ok(x''''''))))) -> F(found(ok(ok(x''''''))))
F(found(mark(found(ok(ok(x'''''''')))))) -> F(mark(found(ok(ok(x'''''''')))))
F(found(mark(found(mark(x''''''))))) -> F(mark(found(mark(x''''''))))
F(found(mark(found(found(x''''''))))) -> F(mark(found(found(x''''''))))
F(found(mark(ok(mark(x''''''))))) -> F(mark(ok(mark(x''''''))))
F(found(mark(ok(found(x''''''))))) -> F(mark(ok(found(x''''''))))
F(mark(found(mark(x'''')))) -> F(found(mark(x'''')))
F(ok(mark(found(mark(x''''''))))) -> F(mark(found(mark(x''''''))))
F(found(ok(mark(found(mark(x'''''''')))))) -> F(ok(mark(found(mark(x'''''''')))))
F(found(ok(mark(found(found(x'''''''')))))) -> F(ok(mark(found(found(x'''''''')))))
F(found(found(ok(mark(x''''''))))) -> F(found(ok(mark(x''''''))))
F(found(found(ok(found(x''''''))))) -> F(found(ok(found(x''''''))))
F(found(found(ok(ok(x''''''))))) -> F(found(ok(ok(x''''''))))
F(mark(found(found(x'''')))) -> F(found(found(x'''')))
F(ok(mark(found(found(x''''''))))) -> F(mark(found(found(x''''''))))
F(mark(ok(mark(x'''')))) -> F(ok(mark(x'''')))
F(ok(mark(ok(mark(x''''''))))) -> F(mark(ok(mark(x''''''))))
F(found(ok(mark(ok(mark(x'''''''')))))) -> F(ok(mark(ok(mark(x'''''''')))))
F(mark(ok(found(ok(mark(x'''''''')))))) -> F(ok(found(ok(mark(x'''''''')))))
F(ok(mark(ok(found(x''''''))))) -> F(mark(ok(found(x''''''))))
F(found(ok(mark(ok(found(x'''''''')))))) -> F(ok(mark(ok(found(x'''''''')))))
F(ok(mark(ok(ok(x''''''))))) -> F(mark(ok(ok(x''''''))))
F(found(ok(mark(ok(ok(x'''''''')))))) -> F(ok(mark(ok(ok(x'''''''')))))
F(found(ok(mark(mark(x''''''))))) -> F(ok(mark(mark(x''''''))))
F(ok(found(ok(mark(x''''''))))) -> F(found(ok(mark(x''''''))))
F(found(ok(found(ok(mark(x'''''''')))))) -> F(ok(found(ok(mark(x'''''''')))))
F(found(ok(found(ok(found(x'''''''')))))) -> F(ok(found(ok(found(x'''''''')))))
F(found(ok(found(ok(ok(x'''''''')))))) -> F(ok(found(ok(ok(x'''''''')))))
F(found(ok(found(mark(x''''''))))) -> F(ok(found(mark(x''''''))))
F(ok(found(ok(found(x''''''))))) -> F(found(ok(found(x''''''))))
F(mark(ok(found(ok(found(x'''''''')))))) -> F(ok(found(ok(found(x'''''''')))))
F(found(ok(ok(mark(x''''''))))) -> F(ok(ok(mark(x''''''))))
F(found(ok(ok(found(x''''''))))) -> F(ok(ok(found(x''''''))))
F(found(ok(ok(ok(x''''''))))) -> F(ok(ok(ok(x''''''))))
F(ok(found(ok(ok(x''''''))))) -> F(found(ok(ok(x''''''))))
F(mark(ok(found(ok(ok(x'''''''')))))) -> F(ok(found(ok(ok(x'''''''')))))
F(mark(ok(found(mark(x''''''))))) -> F(ok(found(mark(x''''''))))
F(mark(ok(found(found(x''''''))))) -> F(ok(found(found(x''''''))))
F(mark(mark(ok(found(x''''''))))) -> F(mark(ok(found(x''''''))))
F(ok(mark(mark(x'''')))) -> F(mark(mark(x'''')))
F(ok(ok(mark(x'''')))) -> F(ok(mark(x'''')))
F(mark(ok(ok(mark(x''''''))))) -> F(ok(ok(mark(x''''''))))
F(mark(ok(ok(found(x''''''))))) -> F(ok(ok(found(x''''''))))
F(found(mark(ok(ok(x''''''))))) -> F(mark(ok(ok(x''''''))))
F(ok(found(mark(x'''')))) -> F(found(mark(x'''')))
F(ok(ok(found(x'''')))) -> F(ok(found(x'''')))
F(ok(ok(ok(x'''')))) -> F(ok(ok(x'''')))
F(mark(ok(ok(ok(x''''''))))) -> F(ok(ok(ok(x''''''))))
F(mark(mark(ok(ok(x''''''))))) -> F(mark(ok(ok(x''''''))))
F(mark(mark(mark(x'''')))) -> F(mark(mark(x'''')))
F(found(mark(mark(x'''')))) -> F(mark(mark(x'''')))
F(found(found(mark(x'''')))) -> F(found(mark(x'''')))
F(found(found(found(x'''')))) -> F(found(found(x'''')))
F(ok(found(found(x'''')))) -> F(found(found(x'''')))
F(found(ok(found(found(x''''''))))) -> F(ok(found(found(x''''''))))
F(mark(found(ok(found(x''''''))))) -> F(found(ok(found(x''''''))))
F(mark(mark(found(ok(found(x'''''''')))))) -> F(mark(found(ok(found(x'''''''')))))

Rules:

active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
top(active(c)) -> top(mark(c))
top(mark(x)) -> top(check(x))
top(found(x)) -> top(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)

Strategy:

innermost

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

F(mark(ok(mark(x'''')))) -> F(ok(mark(x'''')))
nine new Dependency Pairs are created:

F(mark(ok(mark(mark(x''''''))))) -> F(ok(mark(mark(x''''''))))
F(mark(ok(mark(ok(ok(x'''''''')))))) -> F(ok(mark(ok(ok(x'''''''')))))
F(mark(ok(mark(ok(found(x'''''''')))))) -> F(ok(mark(ok(found(x'''''''')))))
F(mark(ok(mark(ok(mark(x'''''''')))))) -> F(ok(mark(ok(mark(x'''''''')))))
F(mark(ok(mark(found(found(x'''''''')))))) -> F(ok(mark(found(found(x'''''''')))))
F(mark(ok(mark(found(mark(x'''''''')))))) -> F(ok(mark(found(mark(x'''''''')))))
F(mark(ok(mark(found(ok(ok(x''''''''''))))))) -> F(ok(mark(found(ok(ok(x''''''''''))))))
F(mark(ok(mark(found(ok(found(x''''''''''))))))) -> F(ok(mark(found(ok(found(x''''''''''))))))
F(mark(ok(mark(found(ok(mark(x''''''''''))))))) -> F(ok(mark(found(ok(mark(x''''''''''))))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 7`
`             ↳FwdInst`
`             ...`
`               →DP Problem 21`
`                 ↳Forward Instantiation Transformation`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳FwdInst`
`       →DP Problem 5`
`         ↳FwdInst`
`       →DP Problem 6`
`         ↳Nar`

Dependency Pairs:

F(mark(mark(found(ok(found(x'''''''')))))) -> F(mark(found(ok(found(x'''''''')))))
F(mark(mark(found(ok(ok(x'''''''')))))) -> F(mark(found(ok(ok(x'''''''')))))
F(mark(mark(found(mark(x''''''))))) -> F(mark(found(mark(x''''''))))
F(mark(mark(found(found(x''''''))))) -> F(mark(found(found(x''''''))))
F(mark(mark(ok(mark(x''''''))))) -> F(mark(ok(mark(x''''''))))
F(found(ok(mark(found(ok(mark(x''''''''''))))))) -> F(ok(mark(found(ok(mark(x''''''''''))))))
F(found(ok(mark(found(ok(found(x''''''''''))))))) -> F(ok(mark(found(ok(found(x''''''''''))))))
F(found(ok(mark(found(ok(ok(x''''''''''))))))) -> F(ok(mark(found(ok(ok(x''''''''''))))))
F(found(mark(found(ok(mark(x'''''''')))))) -> F(mark(found(ok(mark(x'''''''')))))
F(found(mark(found(ok(found(x'''''''')))))) -> F(mark(found(ok(found(x'''''''')))))
F(found(mark(found(ok(ok(x'''''''')))))) -> F(mark(found(ok(ok(x'''''''')))))
F(found(mark(found(mark(x''''''))))) -> F(mark(found(mark(x''''''))))
F(found(mark(found(found(x''''''))))) -> F(mark(found(found(x''''''))))
F(ok(mark(found(ok(mark(x'''''''')))))) -> F(mark(found(ok(mark(x'''''''')))))
F(mark(ok(mark(found(ok(mark(x''''''''''))))))) -> F(ok(mark(found(ok(mark(x''''''''''))))))
F(mark(found(ok(found(x''''''))))) -> F(found(ok(found(x''''''))))
F(ok(mark(found(ok(found(x'''''''')))))) -> F(mark(found(ok(found(x'''''''')))))
F(mark(ok(mark(found(ok(found(x''''''''''))))))) -> F(ok(mark(found(ok(found(x''''''''''))))))
F(mark(found(ok(ok(x''''''))))) -> F(found(ok(ok(x''''''))))
F(ok(mark(found(ok(ok(x'''''''')))))) -> F(mark(found(ok(ok(x'''''''')))))
F(mark(ok(mark(found(ok(ok(x''''''''''))))))) -> F(ok(mark(found(ok(ok(x''''''''''))))))
F(mark(ok(mark(found(mark(x'''''''')))))) -> F(ok(mark(found(mark(x'''''''')))))
F(found(mark(ok(mark(x''''''))))) -> F(mark(ok(mark(x''''''))))
F(mark(ok(found(ok(mark(x'''''''')))))) -> F(ok(found(ok(mark(x'''''''')))))
F(found(mark(ok(found(x''''''))))) -> F(mark(ok(found(x''''''))))
F(mark(found(mark(x'''')))) -> F(found(mark(x'''')))
F(ok(mark(found(mark(x''''''))))) -> F(mark(found(mark(x''''''))))
F(found(ok(mark(found(mark(x'''''''')))))) -> F(ok(mark(found(mark(x'''''''')))))
F(found(ok(mark(found(found(x'''''''')))))) -> F(ok(mark(found(found(x'''''''')))))
F(found(found(ok(mark(x''''''))))) -> F(found(ok(mark(x''''''))))
F(found(found(ok(found(x''''''))))) -> F(found(ok(found(x''''''))))
F(found(found(ok(ok(x''''''))))) -> F(found(ok(ok(x''''''))))
F(mark(found(found(x'''')))) -> F(found(found(x'''')))
F(ok(mark(found(found(x''''''))))) -> F(mark(found(found(x''''''))))
F(mark(ok(mark(found(found(x'''''''')))))) -> F(ok(mark(found(found(x'''''''')))))
F(mark(ok(mark(ok(mark(x'''''''')))))) -> F(ok(mark(ok(mark(x'''''''')))))
F(mark(ok(mark(ok(found(x'''''''')))))) -> F(ok(mark(ok(found(x'''''''')))))
F(mark(ok(mark(ok(ok(x'''''''')))))) -> F(ok(mark(ok(ok(x'''''''')))))
F(mark(ok(mark(mark(x''''''))))) -> F(ok(mark(mark(x''''''))))
F(ok(mark(ok(mark(x''''''))))) -> F(mark(ok(mark(x''''''))))
F(found(ok(mark(ok(mark(x'''''''')))))) -> F(ok(mark(ok(mark(x'''''''')))))
F(found(ok(mark(ok(found(x'''''''')))))) -> F(ok(mark(ok(found(x'''''''')))))
F(found(ok(mark(ok(ok(x'''''''')))))) -> F(ok(mark(ok(ok(x'''''''')))))
F(ok(found(ok(mark(x''''''))))) -> F(found(ok(mark(x''''''))))
F(found(ok(found(ok(mark(x'''''''')))))) -> F(ok(found(ok(mark(x'''''''')))))
F(found(ok(found(ok(found(x'''''''')))))) -> F(ok(found(ok(found(x'''''''')))))
F(found(ok(found(ok(ok(x'''''''')))))) -> F(ok(found(ok(ok(x'''''''')))))
F(found(ok(found(mark(x''''''))))) -> F(ok(found(mark(x''''''))))
F(found(ok(found(found(x''''''))))) -> F(ok(found(found(x''''''))))
F(ok(found(ok(found(x''''''))))) -> F(found(ok(found(x''''''))))
F(mark(ok(found(ok(found(x'''''''')))))) -> F(ok(found(ok(found(x'''''''')))))
F(found(ok(ok(mark(x''''''))))) -> F(ok(ok(mark(x''''''))))
F(found(ok(ok(found(x''''''))))) -> F(ok(ok(found(x''''''))))
F(found(ok(ok(ok(x''''''))))) -> F(ok(ok(ok(x''''''))))
F(ok(found(ok(ok(x''''''))))) -> F(found(ok(ok(x''''''))))
F(mark(ok(found(ok(ok(x'''''''')))))) -> F(ok(found(ok(ok(x'''''''')))))
F(ok(mark(ok(found(x''''''))))) -> F(mark(ok(found(x''''''))))
F(ok(mark(ok(ok(x''''''))))) -> F(mark(ok(ok(x''''''))))
F(ok(ok(mark(x'''')))) -> F(ok(mark(x'''')))
F(mark(ok(ok(mark(x''''''))))) -> F(ok(ok(mark(x''''''))))
F(mark(ok(ok(found(x''''''))))) -> F(ok(ok(found(x''''''))))
F(found(mark(ok(ok(x''''''))))) -> F(mark(ok(ok(x''''''))))
F(ok(found(mark(x'''')))) -> F(found(mark(x'''')))
F(mark(ok(found(mark(x''''''))))) -> F(ok(found(mark(x''''''))))
F(mark(ok(found(found(x''''''))))) -> F(ok(found(found(x''''''))))
F(mark(mark(ok(found(x''''''))))) -> F(mark(ok(found(x''''''))))
F(found(mark(mark(x'''')))) -> F(mark(mark(x'''')))
F(found(found(mark(x'''')))) -> F(found(mark(x'''')))
F(found(found(found(x'''')))) -> F(found(found(x'''')))
F(ok(found(found(x'''')))) -> F(found(found(x'''')))
F(ok(ok(found(x'''')))) -> F(ok(found(x'''')))
F(ok(ok(ok(x'''')))) -> F(ok(ok(x'''')))
F(mark(ok(ok(ok(x''''''))))) -> F(ok(ok(ok(x''''''))))
F(mark(mark(ok(ok(x''''''))))) -> F(mark(ok(ok(x''''''))))
F(mark(mark(mark(x'''')))) -> F(mark(mark(x'''')))
F(ok(mark(mark(x'''')))) -> F(mark(mark(x'''')))
F(found(ok(mark(mark(x''''''))))) -> F(ok(mark(mark(x''''''))))
F(mark(found(ok(mark(x''''''))))) -> F(found(ok(mark(x''''''))))
F(mark(mark(found(ok(mark(x'''''''')))))) -> F(mark(found(ok(mark(x'''''''')))))

Rules:

active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
top(active(c)) -> top(mark(c))
top(mark(x)) -> top(check(x))
top(found(x)) -> top(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)

Strategy:

innermost

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

F(mark(found(found(x'''')))) -> F(found(found(x'''')))
five new Dependency Pairs are created:

F(mark(found(found(found(x''''''))))) -> F(found(found(found(x''''''))))
F(mark(found(found(mark(x''''''))))) -> F(found(found(mark(x''''''))))
F(mark(found(found(ok(ok(x'''''''')))))) -> F(found(found(ok(ok(x'''''''')))))
F(mark(found(found(ok(found(x'''''''')))))) -> F(found(found(ok(found(x'''''''')))))
F(mark(found(found(ok(mark(x'''''''')))))) -> F(found(found(ok(mark(x'''''''')))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 7`
`             ↳FwdInst`
`             ...`
`               →DP Problem 22`
`                 ↳Forward Instantiation Transformation`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳FwdInst`
`       →DP Problem 5`
`         ↳FwdInst`
`       →DP Problem 6`
`         ↳Nar`

Dependency Pairs:

F(mark(mark(found(ok(mark(x'''''''')))))) -> F(mark(found(ok(mark(x'''''''')))))
F(mark(mark(found(ok(ok(x'''''''')))))) -> F(mark(found(ok(ok(x'''''''')))))
F(mark(mark(found(mark(x''''''))))) -> F(mark(found(mark(x''''''))))
F(mark(mark(found(found(x''''''))))) -> F(mark(found(found(x''''''))))
F(mark(mark(ok(mark(x''''''))))) -> F(mark(ok(mark(x''''''))))
F(found(mark(found(ok(mark(x'''''''')))))) -> F(mark(found(ok(mark(x'''''''')))))
F(found(mark(found(ok(found(x'''''''')))))) -> F(mark(found(ok(found(x'''''''')))))
F(found(mark(found(ok(ok(x'''''''')))))) -> F(mark(found(ok(ok(x'''''''')))))
F(found(mark(found(mark(x''''''))))) -> F(mark(found(mark(x''''''))))
F(found(mark(found(found(x''''''))))) -> F(mark(found(found(x''''''))))
F(found(ok(mark(found(ok(mark(x''''''''''))))))) -> F(ok(mark(found(ok(mark(x''''''''''))))))
F(found(ok(mark(found(ok(found(x''''''''''))))))) -> F(ok(mark(found(ok(found(x''''''''''))))))
F(found(ok(mark(found(ok(ok(x''''''''''))))))) -> F(ok(mark(found(ok(ok(x''''''''''))))))
F(mark(found(ok(mark(x''''''))))) -> F(found(ok(mark(x''''''))))
F(ok(mark(found(ok(mark(x'''''''')))))) -> F(mark(found(ok(mark(x'''''''')))))
F(mark(ok(mark(found(ok(mark(x''''''''''))))))) -> F(ok(mark(found(ok(mark(x''''''''''))))))
F(ok(mark(found(ok(found(x'''''''')))))) -> F(mark(found(ok(found(x'''''''')))))
F(mark(ok(mark(found(ok(found(x''''''''''))))))) -> F(ok(mark(found(ok(found(x''''''''''))))))
F(mark(found(ok(ok(x''''''))))) -> F(found(ok(ok(x''''''))))
F(ok(mark(found(ok(ok(x'''''''')))))) -> F(mark(found(ok(ok(x'''''''')))))
F(mark(ok(mark(found(ok(ok(x''''''''''))))))) -> F(ok(mark(found(ok(ok(x''''''''''))))))
F(mark(ok(mark(found(mark(x'''''''')))))) -> F(ok(mark(found(mark(x'''''''')))))
F(found(mark(ok(mark(x''''''))))) -> F(mark(ok(mark(x''''''))))
F(found(mark(ok(found(x''''''))))) -> F(mark(ok(found(x''''''))))
F(mark(found(mark(x'''')))) -> F(found(mark(x'''')))
F(ok(mark(found(mark(x''''''))))) -> F(mark(found(mark(x''''''))))
F(found(ok(mark(found(mark(x'''''''')))))) -> F(ok(mark(found(mark(x'''''''')))))
F(found(ok(mark(found(found(x'''''''')))))) -> F(ok(mark(found(found(x'''''''')))))
F(found(found(ok(mark(x''''''))))) -> F(found(ok(mark(x''''''))))
F(mark(found(found(ok(mark(x'''''''')))))) -> F(found(found(ok(mark(x'''''''')))))
F(found(found(ok(found(x''''''))))) -> F(found(ok(found(x''''''))))
F(mark(found(found(ok(found(x'''''''')))))) -> F(found(found(ok(found(x'''''''')))))
F(found(found(ok(ok(x''''''))))) -> F(found(ok(ok(x''''''))))
F(mark(found(found(ok(ok(x'''''''')))))) -> F(found(found(ok(ok(x'''''''')))))
F(mark(found(found(mark(x''''''))))) -> F(found(found(mark(x''''''))))
F(mark(found(found(found(x''''''))))) -> F(found(found(found(x''''''))))
F(ok(mark(found(found(x''''''))))) -> F(mark(found(found(x''''''))))
F(mark(ok(mark(found(found(x'''''''')))))) -> F(ok(mark(found(found(x'''''''')))))
F(mark(ok(mark(ok(mark(x'''''''')))))) -> F(ok(mark(ok(mark(x'''''''')))))
F(mark(ok(mark(ok(found(x'''''''')))))) -> F(ok(mark(ok(found(x'''''''')))))
F(mark(ok(mark(ok(ok(x'''''''')))))) -> F(ok(mark(ok(ok(x'''''''')))))
F(mark(ok(mark(mark(x''''''))))) -> F(ok(mark(mark(x''''''))))
F(ok(mark(ok(mark(x''''''))))) -> F(mark(ok(mark(x''''''))))
F(found(ok(mark(ok(mark(x'''''''')))))) -> F(ok(mark(ok(mark(x'''''''')))))
F(mark(ok(found(ok(mark(x'''''''')))))) -> F(ok(found(ok(mark(x'''''''')))))
F(ok(mark(ok(found(x''''''))))) -> F(mark(ok(found(x''''''))))
F(found(ok(mark(ok(found(x'''''''')))))) -> F(ok(mark(ok(found(x'''''''')))))
F(ok(mark(ok(ok(x''''''))))) -> F(mark(ok(ok(x''''''))))
F(found(ok(mark(ok(ok(x'''''''')))))) -> F(ok(mark(ok(ok(x'''''''')))))
F(found(ok(mark(mark(x''''''))))) -> F(ok(mark(mark(x''''''))))
F(ok(found(ok(mark(x''''''))))) -> F(found(ok(mark(x''''''))))
F(found(ok(found(ok(mark(x'''''''')))))) -> F(ok(found(ok(mark(x'''''''')))))
F(found(ok(found(ok(found(x'''''''')))))) -> F(ok(found(ok(found(x'''''''')))))
F(found(ok(found(ok(ok(x'''''''')))))) -> F(ok(found(ok(ok(x'''''''')))))
F(found(ok(found(mark(x''''''))))) -> F(ok(found(mark(x''''''))))
F(ok(found(ok(found(x''''''))))) -> F(found(ok(found(x''''''))))
F(mark(ok(found(ok(found(x'''''''')))))) -> F(ok(found(ok(found(x'''''''')))))
F(found(ok(ok(mark(x''''''))))) -> F(ok(ok(mark(x''''''))))
F(found(ok(ok(found(x''''''))))) -> F(ok(ok(found(x''''''))))
F(found(ok(ok(ok(x''''''))))) -> F(ok(ok(ok(x''''''))))
F(ok(found(ok(ok(x''''''))))) -> F(found(ok(ok(x''''''))))
F(mark(ok(found(ok(ok(x'''''''')))))) -> F(ok(found(ok(ok(x'''''''')))))
F(mark(ok(found(mark(x''''''))))) -> F(ok(found(mark(x''''''))))
F(mark(ok(found(found(x''''''))))) -> F(ok(found(found(x''''''))))
F(mark(mark(ok(found(x''''''))))) -> F(mark(ok(found(x''''''))))
F(ok(mark(mark(x'''')))) -> F(mark(mark(x'''')))
F(ok(ok(mark(x'''')))) -> F(ok(mark(x'''')))
F(mark(ok(ok(mark(x''''''))))) -> F(ok(ok(mark(x''''''))))
F(mark(ok(ok(found(x''''''))))) -> F(ok(ok(found(x''''''))))
F(found(mark(ok(ok(x''''''))))) -> F(mark(ok(ok(x''''''))))
F(ok(found(mark(x'''')))) -> F(found(mark(x'''')))
F(ok(ok(found(x'''')))) -> F(ok(found(x'''')))
F(ok(ok(ok(x'''')))) -> F(ok(ok(x'''')))
F(mark(ok(ok(ok(x''''''))))) -> F(ok(ok(ok(x''''''))))
F(mark(mark(ok(ok(x''''''))))) -> F(mark(ok(ok(x''''''))))
F(mark(mark(mark(x'''')))) -> F(mark(mark(x'''')))
F(found(mark(mark(x'''')))) -> F(mark(mark(x'''')))
F(found(found(mark(x'''')))) -> F(found(mark(x'''')))
F(found(found(found(x'''')))) -> F(found(found(x'''')))
F(ok(found(found(x'''')))) -> F(found(found(x'''')))
F(found(ok(found(found(x''''''))))) -> F(ok(found(found(x''''''))))
F(mark(found(ok(found(x''''''))))) -> F(found(ok(found(x''''''))))
F(mark(mark(found(ok(found(x'''''''')))))) -> F(mark(found(ok(found(x'''''''')))))

Rules:

active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
top(active(c)) -> top(mark(c))
top(mark(x)) -> top(check(x))
top(found(x)) -> top(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)

Strategy:

innermost

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

F(mark(found(mark(x'''')))) -> F(found(mark(x'''')))
nine new Dependency Pairs are created:

F(mark(found(mark(mark(x''''''))))) -> F(found(mark(mark(x''''''))))
F(mark(found(mark(ok(ok(x'''''''')))))) -> F(found(mark(ok(ok(x'''''''')))))
F(mark(found(mark(ok(found(x'''''''')))))) -> F(found(mark(ok(found(x'''''''')))))
F(mark(found(mark(ok(mark(x'''''''')))))) -> F(found(mark(ok(mark(x'''''''')))))
F(mark(found(mark(found(found(x'''''''')))))) -> F(found(mark(found(found(x'''''''')))))
F(mark(found(mark(found(mark(x'''''''')))))) -> F(found(mark(found(mark(x'''''''')))))
F(mark(found(mark(found(ok(ok(x''''''''''))))))) -> F(found(mark(found(ok(ok(x''''''''''))))))
F(mark(found(mark(found(ok(found(x''''''''''))))))) -> F(found(mark(found(ok(found(x''''''''''))))))
F(mark(found(mark(found(ok(mark(x''''''''''))))))) -> F(found(mark(found(ok(mark(x''''''''''))))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 7`
`             ↳FwdInst`
`             ...`
`               →DP Problem 23`
`                 ↳Forward Instantiation Transformation`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳FwdInst`
`       →DP Problem 5`
`         ↳FwdInst`
`       →DP Problem 6`
`         ↳Nar`

Dependency Pairs:

F(mark(mark(found(ok(found(x'''''''')))))) -> F(mark(found(ok(found(x'''''''')))))
F(mark(mark(found(ok(ok(x'''''''')))))) -> F(mark(found(ok(ok(x'''''''')))))
F(mark(mark(found(mark(x''''''))))) -> F(mark(found(mark(x''''''))))
F(mark(mark(found(found(x''''''))))) -> F(mark(found(found(x''''''))))
F(mark(mark(ok(mark(x''''''))))) -> F(mark(ok(mark(x''''''))))
F(found(ok(mark(found(ok(mark(x''''''''''))))))) -> F(ok(mark(found(ok(mark(x''''''''''))))))
F(found(ok(mark(found(ok(found(x''''''''''))))))) -> F(ok(mark(found(ok(found(x''''''''''))))))
F(found(ok(mark(found(ok(ok(x''''''''''))))))) -> F(ok(mark(found(ok(ok(x''''''''''))))))
F(found(mark(found(ok(mark(x'''''''')))))) -> F(mark(found(ok(mark(x'''''''')))))
F(mark(found(mark(found(ok(mark(x''''''''''))))))) -> F(found(mark(found(ok(mark(x''''''''''))))))
F(found(mark(found(ok(found(x'''''''')))))) -> F(mark(found(ok(found(x'''''''')))))
F(mark(found(mark(found(ok(found(x''''''''''))))))) -> F(found(mark(found(ok(found(x''''''''''))))))
F(found(mark(found(ok(ok(x'''''''')))))) -> F(mark(found(ok(ok(x'''''''')))))
F(mark(found(mark(found(ok(ok(x''''''''''))))))) -> F(found(mark(found(ok(ok(x''''''''''))))))
F(found(mark(found(mark(x''''''))))) -> F(mark(found(mark(x''''''))))
F(mark(found(mark(found(mark(x'''''''')))))) -> F(found(mark(found(mark(x'''''''')))))
F(found(mark(found(found(x''''''))))) -> F(mark(found(found(x''''''))))
F(mark(found(mark(found(found(x'''''''')))))) -> F(found(mark(found(found(x'''''''')))))
F(ok(mark(found(ok(mark(x'''''''')))))) -> F(mark(found(ok(mark(x'''''''')))))
F(mark(ok(mark(found(ok(mark(x''''''''''))))))) -> F(ok(mark(found(ok(mark(x''''''''''))))))
F(mark(found(ok(found(x''''''))))) -> F(found(ok(found(x''''''))))
F(ok(mark(found(ok(found(x'''''''')))))) -> F(mark(found(ok(found(x'''''''')))))
F(mark(ok(mark(found(ok(found(x''''''''''))))))) -> F(ok(mark(found(ok(found(x''''''''''))))))
F(mark(found(ok(ok(x''''''))))) -> F(found(ok(ok(x''''''))))
F(ok(mark(found(ok(ok(x'''''''')))))) -> F(mark(found(ok(ok(x'''''''')))))
F(mark(ok(mark(found(ok(ok(x''''''''''))))))) -> F(ok(mark(found(ok(ok(x''''''''''))))))
F(mark(ok(mark(found(mark(x'''''''')))))) -> F(ok(mark(found(mark(x'''''''')))))
F(found(mark(ok(mark(x''''''))))) -> F(mark(ok(mark(x''''''))))
F(mark(found(mark(ok(mark(x'''''''')))))) -> F(found(mark(ok(mark(x'''''''')))))
F(mark(ok(found(ok(mark(x'''''''')))))) -> F(ok(found(ok(mark(x'''''''')))))
F(found(mark(ok(found(x''''''))))) -> F(mark(ok(found(x''''''))))
F(mark(found(mark(ok(found(x'''''''')))))) -> F(found(mark(ok(found(x'''''''')))))
F(mark(found(mark(ok(ok(x'''''''')))))) -> F(found(mark(ok(ok(x'''''''')))))
F(mark(found(mark(mark(x''''''))))) -> F(found(mark(mark(x''''''))))
F(ok(mark(found(mark(x''''''))))) -> F(mark(found(mark(x''''''))))
F(found(ok(mark(found(mark(x'''''''')))))) -> F(ok(mark(found(mark(x'''''''')))))
F(found(ok(mark(found(found(x'''''''')))))) -> F(ok(mark(found(found(x'''''''')))))
F(found(found(ok(mark(x''''''))))) -> F(found(ok(mark(x''''''))))
F(mark(found(found(ok(mark(x'''''''')))))) -> F(found(found(ok(mark(x'''''''')))))
F(found(found(ok(found(x''''''))))) -> F(found(ok(found(x''''''))))
F(mark(found(found(ok(found(x'''''''')))))) -> F(found(found(ok(found(x'''''''')))))
F(found(found(ok(ok(x''''''))))) -> F(found(ok(ok(x''''''))))
F(mark(found(found(ok(ok(x'''''''')))))) -> F(found(found(ok(ok(x'''''''')))))
F(mark(found(found(mark(x''''''))))) -> F(found(found(mark(x''''''))))
F(mark(found(found(found(x''''''))))) -> F(found(found(found(x''''''))))
F(ok(mark(found(found(x''''''))))) -> F(mark(found(found(x''''''))))
F(mark(ok(mark(found(found(x'''''''')))))) -> F(ok(mark(found(found(x'''''''')))))
F(mark(ok(mark(ok(mark(x'''''''')))))) -> F(ok(mark(ok(mark(x'''''''')))))
F(mark(ok(mark(ok(found(x'''''''')))))) -> F(ok(mark(ok(found(x'''''''')))))
F(mark(ok(mark(ok(ok(x'''''''')))))) -> F(ok(mark(ok(ok(x'''''''')))))
F(mark(ok(mark(mark(x''''''))))) -> F(ok(mark(mark(x''''''))))
F(ok(mark(ok(mark(x''''''))))) -> F(mark(ok(mark(x''''''))))
F(found(ok(mark(ok(mark(x'''''''')))))) -> F(ok(mark(ok(mark(x'''''''')))))
F(found(ok(mark(ok(found(x'''''''')))))) -> F(ok(mark(ok(found(x'''''''')))))
F(found(ok(mark(ok(ok(x'''''''')))))) -> F(ok(mark(ok(ok(x'''''''')))))
F(ok(found(ok(mark(x''''''))))) -> F(found(ok(mark(x''''''))))
F(found(ok(found(ok(mark(x'''''''')))))) -> F(ok(found(ok(mark(x'''''''')))))
F(found(ok(found(ok(found(x'''''''')))))) -> F(ok(found(ok(found(x'''''''')))))
F(found(ok(found(ok(ok(x'''''''')))))) -> F(ok(found(ok(ok(x'''''''')))))
F(found(ok(found(mark(x''''''))))) -> F(ok(found(mark(x''''''))))
F(found(ok(found(found(x''''''))))) -> F(ok(found(found(x''''''))))
F(ok(found(ok(found(x''''''))))) -> F(found(ok(found(x''''''))))
F(mark(ok(found(ok(found(x'''''''')))))) -> F(ok(found(ok(found(x'''''''')))))
F(found(ok(ok(mark(x''''''))))) -> F(ok(ok(mark(x''''''))))
F(found(ok(ok(found(x''''''))))) -> F(ok(ok(found(x''''''))))
F(found(ok(ok(ok(x''''''))))) -> F(ok(ok(ok(x''''''))))
F(ok(found(ok(ok(x''''''))))) -> F(found(ok(ok(x''''''))))
F(mark(ok(found(ok(ok(x'''''''')))))) -> F(ok(found(ok(ok(x'''''''')))))
F(ok(mark(ok(found(x''''''))))) -> F(mark(ok(found(x''''''))))
F(ok(mark(ok(ok(x''''''))))) -> F(mark(ok(ok(x''''''))))
F(ok(ok(mark(x'''')))) -> F(ok(mark(x'''')))
F(mark(ok(ok(mark(x''''''))))) -> F(ok(ok(mark(x''''''))))
F(mark(ok(ok(found(x''''''))))) -> F(ok(ok(found(x''''''))))
F(found(mark(ok(ok(x''''''))))) -> F(mark(ok(ok(x''''''))))
F(ok(found(mark(x'''')))) -> F(found(mark(x'''')))
F(mark(ok(found(mark(x''''''))))) -> F(ok(found(mark(x''''''))))
F(mark(ok(found(found(x''''''))))) -> F(ok(found(found(x''''''))))
F(mark(mark(ok(found(x''''''))))) -> F(mark(ok(found(x''''''))))
F(found(mark(mark(x'''')))) -> F(mark(mark(x'''')))
F(found(found(mark(x'''')))) -> F(found(mark(x'''')))
F(found(found(found(x'''')))) -> F(found(found(x'''')))
F(ok(found(found(x'''')))) -> F(found(found(x'''')))
F(ok(ok(found(x'''')))) -> F(ok(found(x'''')))
F(ok(ok(ok(x'''')))) -> F(ok(ok(x'''')))
F(mark(ok(ok(ok(x''''''))))) -> F(ok(ok(ok(x''''''))))
F(mark(mark(ok(ok(x''''''))))) -> F(mark(ok(ok(x''''''))))
F(mark(mark(mark(x'''')))) -> F(mark(mark(x'''')))
F(ok(mark(mark(x'''')))) -> F(mark(mark(x'''')))
F(found(ok(mark(mark(x''''''))))) -> F(ok(mark(mark(x''''''))))
F(mark(found(ok(mark(x''''''))))) -> F(found(ok(mark(x''''''))))
F(mark(mark(found(ok(mark(x'''''''')))))) -> F(mark(found(ok(mark(x'''''''')))))

Rules:

active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
top(active(c)) -> top(mark(c))
top(mark(x)) -> top(check(x))
top(found(x)) -> top(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)

Strategy:

innermost

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

F(mark(found(ok(ok(x''''''))))) -> F(found(ok(ok(x''''''))))
three new Dependency Pairs are created:

F(mark(found(ok(ok(ok(x'''''''')))))) -> F(found(ok(ok(ok(x'''''''')))))
F(mark(found(ok(ok(found(x'''''''')))))) -> F(found(ok(ok(found(x'''''''')))))
F(mark(found(ok(ok(mark(x'''''''')))))) -> F(found(ok(ok(mark(x'''''''')))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 7`
`             ↳FwdInst`
`             ...`
`               →DP Problem 24`
`                 ↳Forward Instantiation Transformation`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳FwdInst`
`       →DP Problem 5`
`         ↳FwdInst`
`       →DP Problem 6`
`         ↳Nar`

Dependency Pairs:

F(mark(mark(found(ok(mark(x'''''''')))))) -> F(mark(found(ok(mark(x'''''''')))))
F(mark(mark(found(ok(ok(x'''''''')))))) -> F(mark(found(ok(ok(x'''''''')))))
F(mark(mark(found(mark(x''''''))))) -> F(mark(found(mark(x''''''))))
F(mark(mark(found(found(x''''''))))) -> F(mark(found(found(x''''''))))
F(mark(mark(ok(mark(x''''''))))) -> F(mark(ok(mark(x''''''))))
F(found(mark(found(ok(mark(x'''''''')))))) -> F(mark(found(ok(mark(x'''''''')))))
F(mark(found(mark(found(ok(mark(x''''''''''))))))) -> F(found(mark(found(ok(mark(x''''''''''))))))
F(found(mark(found(ok(found(x'''''''')))))) -> F(mark(found(ok(found(x'''''''')))))
F(mark(found(mark(found(ok(found(x''''''''''))))))) -> F(found(mark(found(ok(found(x''''''''''))))))
F(found(mark(found(ok(ok(x'''''''')))))) -> F(mark(found(ok(ok(x'''''''')))))
F(mark(found(mark(found(ok(ok(x''''''''''))))))) -> F(found(mark(found(ok(ok(x''''''''''))))))
F(found(mark(found(mark(x''''''))))) -> F(mark(found(mark(x''''''))))
F(mark(found(mark(found(mark(x'''''''')))))) -> F(found(mark(found(mark(x'''''''')))))
F(found(mark(found(found(x''''''))))) -> F(mark(found(found(x''''''))))
F(mark(found(mark(found(found(x'''''''')))))) -> F(found(mark(found(found(x'''''''')))))
F(found(ok(mark(found(ok(mark(x''''''''''))))))) -> F(ok(mark(found(ok(mark(x''''''''''))))))
F(found(ok(mark(found(ok(found(x''''''''''))))))) -> F(ok(mark(found(ok(found(x''''''''''))))))
F(found(ok(mark(found(ok(ok(x''''''''''))))))) -> F(ok(mark(found(ok(ok(x''''''''''))))))
F(mark(found(ok(mark(x''''''))))) -> F(found(ok(mark(x''''''))))
F(ok(mark(found(ok(mark(x'''''''')))))) -> F(mark(found(ok(mark(x'''''''')))))
F(mark(ok(mark(found(ok(mark(x''''''''''))))))) -> F(ok(mark(found(ok(mark(x''''''''''))))))
F(ok(mark(found(ok(found(x'''''''')))))) -> F(mark(found(ok(found(x'''''''')))))
F(mark(ok(mark(found(ok(found(x''''''''''))))))) -> F(ok(mark(found(ok(found(x''''''''''))))))
F(mark(found(ok(ok(mark(x'''''''')))))) -> F(found(ok(ok(mark(x'''''''')))))
F(mark(found(ok(ok(found(x'''''''')))))) -> F(found(ok(ok(found(x'''''''')))))
F(mark(found(ok(ok(ok(x'''''''')))))) -> F(found(ok(ok(ok(x'''''''')))))
F(ok(mark(found(ok(ok(x'''''''')))))) -> F(mark(found(ok(ok(x'''''''')))))
F(mark(ok(mark(found(ok(ok(x''''''''''))))))) -> F(ok(mark(found(ok(ok(x''''''''''))))))
F(mark(ok(mark(found(mark(x'''''''')))))) -> F(ok(mark(found(mark(x'''''''')))))
F(found(mark(ok(mark(x''''''))))) -> F(mark(ok(mark(x''''''))))
F(mark(found(mark(ok(mark(x'''''''')))))) -> F(found(mark(ok(mark(x'''''''')))))
F(found(mark(ok(found(x''''''))))) -> F(mark(ok(found(x''''''))))
F(mark(found(mark(ok(found(x'''''''')))))) -> F(found(mark(ok(found(x'''''''')))))
F(mark(found(mark(ok(ok(x'''''''')))))) -> F(found(mark(ok(ok(x'''''''')))))
F(mark(found(mark(mark(x''''''))))) -> F(found(mark(mark(x''''''))))
F(ok(mark(found(mark(x''''''))))) -> F(mark(found(mark(x''''''))))
F(found(ok(mark(found(mark(x'''''''')))))) -> F(ok(mark(found(mark(x'''''''')))))
F(found(ok(mark(found(found(x'''''''')))))) -> F(ok(mark(found(found(x'''''''')))))
F(found(found(ok(mark(x''''''))))) -> F(found(ok(mark(x''''''))))
F(mark(found(found(ok(mark(x'''''''')))))) -> F(found(found(ok(mark(x'''''''')))))
F(found(found(ok(found(x''''''))))) -> F(found(ok(found(x''''''))))
F(mark(found(found(ok(found(x'''''''')))))) -> F(found(found(ok(found(x'''''''')))))
F(found(found(ok(ok(x''''''))))) -> F(found(ok(ok(x''''''))))
F(mark(found(found(ok(ok(x'''''''')))))) -> F(found(found(ok(ok(x'''''''')))))
F(mark(found(found(mark(x''''''))))) -> F(found(found(mark(x''''''))))
F(mark(found(found(found(x''''''))))) -> F(found(found(found(x''''''))))
F(ok(mark(found(found(x''''''))))) -> F(mark(found(found(x''''''))))
F(mark(ok(mark(found(found(x'''''''')))))) -> F(ok(mark(found(found(x'''''''')))))
F(mark(ok(mark(ok(mark(x'''''''')))))) -> F(ok(mark(ok(mark(x'''''''')))))
F(mark(ok(mark(ok(found(x'''''''')))))) -> F(ok(mark(ok(found(x'''''''')))))
F(mark(ok(mark(ok(ok(x'''''''')))))) -> F(ok(mark(ok(ok(x'''''''')))))
F(mark(ok(mark(mark(x''''''))))) -> F(ok(mark(mark(x''''''))))
F(ok(mark(ok(mark(x''''''))))) -> F(mark(ok(mark(x''''''))))
F(found(ok(mark(ok(mark(x'''''''')))))) -> F(ok(mark(ok(mark(x'''''''')))))
F(mark(ok(found(ok(mark(x'''''''')))))) -> F(ok(found(ok(mark(x'''''''')))))
F(ok(mark(ok(found(x''''''))))) -> F(mark(ok(found(x''''''))))
F(found(ok(mark(ok(found(x'''''''')))))) -> F(ok(mark(ok(found(x'''''''')))))
F(ok(mark(ok(ok(x''''''))))) -> F(mark(ok(ok(x''''''))))
F(found(ok(mark(ok(ok(x'''''''')))))) -> F(ok(mark(ok(ok(x'''''''')))))
F(found(ok(mark(mark(x''''''))))) -> F(ok(mark(mark(x''''''))))
F(ok(found(ok(mark(x''''''))))) -> F(found(ok(mark(x''''''))))
F(found(ok(found(ok(mark(x'''''''')))))) -> F(ok(found(ok(mark(x'''''''')))))
F(found(ok(found(ok(found(x'''''''')))))) -> F(ok(found(ok(found(x'''''''')))))
F(found(ok(found(ok(ok(x'''''''')))))) -> F(ok(found(ok(ok(x'''''''')))))
F(found(ok(found(mark(x''''''))))) -> F(ok(found(mark(x''''''))))
F(ok(found(ok(found(x''''''))))) -> F(found(ok(found(x''''''))))
F(mark(ok(found(ok(found(x'''''''')))))) -> F(ok(found(ok(found(x'''''''')))))
F(found(ok(ok(mark(x''''''))))) -> F(ok(ok(mark(x''''''))))
F(found(ok(ok(found(x''''''))))) -> F(ok(ok(found(x''''''))))
F(found(ok(ok(ok(x''''''))))) -> F(ok(ok(ok(x''''''))))
F(ok(found(ok(ok(x''''''))))) -> F(found(ok(ok(x''''''))))
F(mark(ok(found(ok(ok(x'''''''')))))) -> F(ok(found(ok(ok(x'''''''')))))
F(mark(ok(found(mark(x''''''))))) -> F(ok(found(mark(x''''''))))
F(mark(ok(found(found(x''''''))))) -> F(ok(found(found(x''''''))))
F(mark(mark(ok(found(x''''''))))) -> F(mark(ok(found(x''''''))))
F(ok(mark(mark(x'''')))) -> F(mark(mark(x'''')))
F(ok(ok(mark(x'''')))) -> F(ok(mark(x'''')))
F(mark(ok(ok(mark(x''''''))))) -> F(ok(ok(mark(x''''''))))
F(mark(ok(ok(found(x''''''))))) -> F(ok(ok(found(x''''''))))
F(found(mark(ok(ok(x''''''))))) -> F(mark(ok(ok(x''''''))))
F(ok(found(mark(x'''')))) -> F(found(mark(x'''')))
F(ok(ok(found(x'''')))) -> F(ok(found(x'''')))
F(ok(ok(ok(x'''')))) -> F(ok(ok(x'''')))
F(mark(ok(ok(ok(x''''''))))) -> F(ok(ok(ok(x''''''))))
F(mark(mark(ok(ok(x''''''))))) -> F(mark(ok(ok(x''''''))))
F(mark(mark(mark(x'''')))) -> F(mark(mark(x'''')))
F(found(mark(mark(x'''')))) -> F(mark(mark(x'''')))
F(found(found(mark(x'''')))) -> F(found(mark(x'''')))
F(found(found(found(x'''')))) -> F(found(found(x'''')))
F(ok(found(found(x'''')))) -> F(found(found(x'''')))
F(found(ok(found(found(x''''''))))) -> F(ok(found(found(x''''''))))
F(mark(found(ok(found(x''''''))))) -> F(found(ok(found(x''''''))))
F(mark(mark(found(ok(found(x'''''''')))))) -> F(mark(found(ok(found(x'''''''')))))

Rules:

active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
top(active(c)) -> top(mark(c))
top(mark(x)) -> top(check(x))
top(found(x)) -> top(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)

Strategy:

innermost

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

F(mark(found(ok(found(x''''''))))) -> F(found(ok(found(x''''''))))
five new Dependency Pairs are created:

F(mark(found(ok(found(found(x'''''''')))))) -> F(found(ok(found(found(x'''''''')))))
F(mark(found(ok(found(mark(x'''''''')))))) -> F(found(ok(found(mark(x'''''''')))))
F(mark(found(ok(found(ok(ok(x''''''''''))))))) -> F(found(ok(found(ok(ok(x''''''''''))))))
F(mark(found(ok(found(ok(found(x''''''''''))))))) -> F(found(ok(found(ok(found(x''''''''''))))))
F(mark(found(ok(found(ok(mark(x''''''''''))))))) -> F(found(ok(found(ok(mark(x''''''''''))))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 7`
`             ↳FwdInst`
`             ...`
`               →DP Problem 25`
`                 ↳Forward Instantiation Transformation`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳FwdInst`
`       →DP Problem 5`
`         ↳FwdInst`
`       →DP Problem 6`
`         ↳Nar`

Dependency Pairs:

F(mark(mark(found(ok(found(x'''''''')))))) -> F(mark(found(ok(found(x'''''''')))))
F(mark(mark(found(ok(ok(x'''''''')))))) -> F(mark(found(ok(ok(x'''''''')))))
F(mark(mark(found(mark(x''''''))))) -> F(mark(found(mark(x''''''))))
F(mark(mark(found(found(x''''''))))) -> F(mark(found(found(x''''''))))
F(mark(mark(ok(mark(x''''''))))) -> F(mark(ok(mark(x''''''))))
F(found(ok(mark(found(ok(mark(x''''''''''))))))) -> F(ok(mark(found(ok(mark(x''''''''''))))))
F(found(ok(mark(found(ok(found(x''''''''''))))))) -> F(ok(mark(found(ok(found(x''''''''''))))))
F(found(ok(mark(found(ok(ok(x''''''''''))))))) -> F(ok(mark(found(ok(ok(x''''''''''))))))
F(found(mark(found(ok(mark(x'''''''')))))) -> F(mark(found(ok(mark(x'''''''')))))
F(mark(found(mark(found(ok(mark(x''''''''''))))))) -> F(found(mark(found(ok(mark(x''''''''''))))))
F(found(mark(found(ok(found(x'''''''')))))) -> F(mark(found(ok(found(x'''''''')))))
F(mark(found(mark(found(ok(found(x''''''''''))))))) -> F(found(mark(found(ok(found(x''''''''''))))))
F(found(mark(found(ok(ok(x'''''''')))))) -> F(mark(found(ok(ok(x'''''''')))))
F(mark(found(mark(found(ok(ok(x''''''''''))))))) -> F(found(mark(found(ok(ok(x''''''''''))))))
F(found(mark(found(mark(x''''''))))) -> F(mark(found(mark(x''''''))))
F(mark(found(mark(found(mark(x'''''''')))))) -> F(found(mark(found(mark(x'''''''')))))
F(found(mark(found(found(x''''''))))) -> F(mark(found(found(x''''''))))
F(mark(found(mark(found(found(x'''''''')))))) -> F(found(mark(found(found(x'''''''')))))
F(ok(mark(found(ok(mark(x'''''''')))))) -> F(mark(found(ok(mark(x'''''''')))))
F(mark(ok(mark(found(ok(mark(x''''''''''))))))) -> F(ok(mark(found(ok(mark(x''''''''''))))))
F(mark(found(ok(found(ok(mark(x''''''''''))))))) -> F(found(ok(found(ok(mark(x''''''''''))))))
F(mark(found(ok(found(ok(found(x''''''''''))))))) -> F(found(ok(found(ok(found(x''''''''''))))))
F(mark(found(ok(found(ok(ok(x''''''''''))))))) -> F(found(ok(found(ok(ok(x''''''''''))))))
F(mark(found(ok(found(mark(x'''''''')))))) -> F(found(ok(found(mark(x'''''''')))))
F(mark(found(ok(found(found(x'''''''')))))) -> F(found(ok(found(found(x'''''''')))))
F(ok(mark(found(ok(found(x'''''''')))))) -> F(mark(found(ok(found(x'''''''')))))
F(mark(ok(mark(found(ok(found(x''''''''''))))))) -> F(ok(mark(found(ok(found(x''''''''''))))))
F(mark(found(ok(ok(mark(x'''''''')))))) -> F(found(ok(ok(mark(x'''''''')))))
F(mark(found(ok(ok(found(x'''''''')))))) -> F(found(ok(ok(found(x'''''''')))))
F(mark(found(ok(ok(ok(x'''''''')))))) -> F(found(ok(ok(ok(x'''''''')))))
F(ok(mark(found(ok(ok(x'''''''')))))) -> F(mark(found(ok(ok(x'''''''')))))
F(mark(ok(mark(found(ok(ok(x''''''''''))))))) -> F(ok(mark(found(ok(ok(x''''''''''))))))
F(mark(ok(mark(found(mark(x'''''''')))))) -> F(ok(mark(found(mark(x'''''''')))))
F(found(mark(ok(mark(x''''''))))) -> F(mark(ok(mark(x''''''))))
F(mark(found(mark(ok(mark(x'''''''')))))) -> F(found(mark(ok(mark(x'''''''')))))
F(mark(ok(found(ok(mark(x'''''''')))))) -> F(ok(found(ok(mark(x'''''''')))))
F(found(mark(ok(found(x''''''))))) -> F(mark(ok(found(x''''''))))
F(mark(found(mark(ok(found(x'''''''')))))) -> F(found(mark(ok(found(x'''''''')))))
F(mark(found(mark(ok(ok(x'''''''')))))) -> F(found(mark(ok(ok(x'''''''')))))
F(mark(found(mark(mark(x''''''))))) -> F(found(mark(mark(x''''''))))
F(ok(mark(found(mark(x''''''))))) -> F(mark(found(mark(x''''''))))
F(found(ok(mark(found(mark(x'''''''')))))) -> F(ok(mark(found(mark(x'''''''')))))
F(found(ok(mark(found(found(x'''''''')))))) -> F(ok(mark(found(found(x'''''''')))))
F(found(found(ok(mark(x''''''))))) -> F(found(ok(mark(x''''''))))
F(mark(found(found(ok(mark(x'''''''')))))) -> F(found(found(ok(mark(x'''''''')))))
F(found(found(ok(found(x''''''))))) -> F(found(ok(found(x''''''))))
F(mark(found(found(ok(found(x'''''''')))))) -> F(found(found(ok(found(x'''''''')))))
F(found(found(ok(ok(x''''''))))) -> F(found(ok(ok(x''''''))))
F(mark(found(found(ok(ok(x'''''''')))))) -> F(found(found(ok(ok(x'''''''')))))
F(mark(found(found(mark(x''''''))))) -> F(found(found(mark(x''''''))))
F(mark(found(found(found(x''''''))))) -> F(found(found(found(x''''''))))
F(ok(mark(found(found(x''''''))))) -> F(mark(found(found(x''''''))))
F(mark(ok(mark(found(found(x'''''''')))))) -> F(ok(mark(found(found(x'''''''')))))
F(mark(ok(mark(ok(mark(x'''''''')))))) -> F(ok(mark(ok(mark(x'''''''')))))
F(mark(ok(mark(ok(found(x'''''''')))))) -> F(ok(mark(ok(found(x'''''''')))))
F(mark(ok(mark(ok(ok(x'''''''')))))) -> F(ok(mark(ok(ok(x'''''''')))))
F(mark(ok(mark(mark(x''''''))))) -> F(ok(mark(mark(x''''''))))
F(ok(mark(ok(mark(x''''''))))) -> F(mark(ok(mark(x''''''))))
F(found(ok(mark(ok(mark(x'''''''')))))) -> F(ok(mark(ok(mark(x'''''''')))))
F(found(ok(mark(ok(found(x'''''''')))))) -> F(ok(mark(ok(found(x'''''''')))))
F(found(ok(mark(ok(ok(x'''''''')))))) -> F(ok(mark(ok(ok(x'''''''')))))
F(ok(found(ok(mark(x''''''))))) -> F(found(ok(mark(x''''''))))
F(found(ok(found(ok(mark(x'''''''')))))) -> F(ok(found(ok(mark(x'''''''')))))
F(found(ok(found(ok(found(x'''''''')))))) -> F(ok(found(ok(found(x'''''''')))))
F(found(ok(found(ok(ok(x'''''''')))))) -> F(ok(found(ok(ok(x'''''''')))))
F(found(ok(found(mark(x''''''))))) -> F(ok(found(mark(x''''''))))
F(found(ok(found(found(x''''''))))) -> F(ok(found(found(x''''''))))
F(ok(found(ok(found(x''''''))))) -> F(found(ok(found(x''''''))))
F(mark(ok(found(ok(found(x'''''''')))))) -> F(ok(found(ok(found(x'''''''')))))
F(found(ok(ok(mark(x''''''))))) -> F(ok(ok(mark(x''''''))))
F(found(ok(ok(found(x''''''))))) -> F(ok(ok(found(x''''''))))
F(found(ok(ok(ok(x''''''))))) -> F(ok(ok(ok(x''''''))))
F(ok(found(ok(ok(x''''''))))) -> F(found(ok(ok(x''''''))))
F(mark(ok(found(ok(ok(x'''''''')))))) -> F(ok(found(ok(ok(x'''''''')))))
F(ok(mark(ok(found(x''''''))))) -> F(mark(ok(found(x''''''))))
F(ok(mark(ok(ok(x''''''))))) -> F(mark(ok(ok(x''''''))))
F(ok(ok(mark(x'''')))) -> F(ok(mark(x'''')))
F(mark(ok(ok(mark(x''''''))))) -> F(ok(ok(mark(x''''''))))
F(mark(ok(ok(found(x''''''))))) -> F(ok(ok(found(x''''''))))
F(found(mark(ok(ok(x''''''))))) -> F(mark(ok(ok(x''''''))))
F(ok(found(mark(x'''')))) -> F(found(mark(x'''')))
F(mark(ok(found(mark(x''''''))))) -> F(ok(found(mark(x''''''))))
F(mark(ok(found(found(x''''''))))) -> F(ok(found(found(x''''''))))
F(mark(mark(ok(found(x''''''))))) -> F(mark(ok(found(x''''''))))
F(found(mark(mark(x'''')))) -> F(mark(mark(x'''')))
F(found(found(mark(x'''')))) -> F(found(mark(x'''')))
F(found(found(found(x'''')))) -> F(found(found(x'''')))
F(ok(found(found(x'''')))) -> F(found(found(x'''')))
F(ok(ok(found(x'''')))) -> F(ok(found(x'''')))
F(ok(ok(ok(x'''')))) -> F(ok(ok(x'''')))
F(mark(ok(ok(ok(x''''''))))) -> F(ok(ok(ok(x''''''))))
F(mark(mark(ok(ok(x''''''))))) -> F(mark(ok(ok(x''''''))))
F(mark(mark(mark(x'''')))) -> F(mark(mark(x'''')))
F(ok(mark(mark(x'''')))) -> F(mark(mark(x'''')))
F(found(ok(mark(mark(x''''''))))) -> F(ok(mark(mark(x''''''))))
F(mark(found(ok(mark(x''''''))))) -> F(found(ok(mark(x''''''))))
F(mark(mark(found(ok(mark(x'''''''')))))) -> F(mark(found(ok(mark(x'''''''')))))

Rules:

active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
top(active(c)) -> top(mark(c))
top(mark(x)) -> top(check(x))
top(found(x)) -> top(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)

Strategy:

innermost

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

F(mark(found(ok(mark(x''''''))))) -> F(found(ok(mark(x''''''))))
nine new Dependency Pairs are created:

F(mark(found(ok(mark(mark(x'''''''')))))) -> F(found(ok(mark(mark(x'''''''')))))
F(mark(found(ok(mark(ok(ok(x''''''''''))))))) -> F(found(ok(mark(ok(ok(x''''''''''))))))
F(mark(found(ok(mark(ok(found(x''''''''''))))))) -> F(found(ok(mark(ok(found(x''''''''''))))))
F(mark(found(ok(mark(ok(mark(x''''''''''))))))) -> F(found(ok(mark(ok(mark(x''''''''''))))))
F(mark(found(ok(mark(found(found(x''''''''''))))))) -> F(found(ok(mark(found(found(x''''''''''))))))
F(mark(found(ok(mark(found(mark(x''''''''''))))))) -> F(found(ok(mark(found(mark(x''''''''''))))))
F(mark(found(ok(mark(found(ok(ok(x'''''''''''')))))))) -> F(found(ok(mark(found(ok(ok(x'''''''''''')))))))
F(mark(found(ok(mark(found(ok(found(x'''''''''''')))))))) -> F(found(ok(mark(found(ok(found(x'''''''''''')))))))
F(mark(found(ok(mark(found(ok(mark(x'''''''''''')))))))) -> F(found(ok(mark(found(ok(mark(x'''''''''''')))))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 7`
`             ↳FwdInst`
`             ...`
`               →DP Problem 26`
`                 ↳Polynomial Ordering`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳FwdInst`
`       →DP Problem 5`
`         ↳FwdInst`
`       →DP Problem 6`
`         ↳Nar`

Dependency Pairs:

F(mark(mark(found(ok(mark(x'''''''')))))) -> F(mark(found(ok(mark(x'''''''')))))
F(mark(mark(found(ok(ok(x'''''''')))))) -> F(mark(found(ok(ok(x'''''''')))))
F(mark(mark(found(mark(x''''''))))) -> F(mark(found(mark(x''''''))))
F(mark(mark(found(found(x''''''))))) -> F(mark(found(found(x''''''))))
F(mark(mark(ok(mark(x''''''))))) -> F(mark(ok(mark(x''''''))))
F(found(mark(found(ok(mark(x'''''''')))))) -> F(mark(found(ok(mark(x'''''''')))))
F(mark(found(mark(found(ok(mark(x''''''''''))))))) -> F(found(mark(found(ok(mark(x''''''''''))))))
F(found(mark(found(ok(found(x'''''''')))))) -> F(mark(found(ok(found(x'''''''')))))
F(mark(found(mark(found(ok(found(x''''''''''))))))) -> F(found(mark(found(ok(found(x''''''''''))))))
F(found(mark(found(ok(ok(x'''''''')))))) -> F(mark(found(ok(ok(x'''''''')))))
F(mark(found(mark(found(ok(ok(x''''''''''))))))) -> F(found(mark(found(ok(ok(x''''''''''))))))
F(found(mark(found(mark(x''''''))))) -> F(mark(found(mark(x''''''))))
F(mark(found(mark(found(mark(x'''''''')))))) -> F(found(mark(found(mark(x'''''''')))))
F(found(mark(found(found(x''''''))))) -> F(mark(found(found(x''''''))))
F(mark(found(mark(found(found(x'''''''')))))) -> F(found(mark(found(found(x'''''''')))))
F(found(ok(mark(found(ok(mark(x''''''''''))))))) -> F(ok(mark(found(ok(mark(x''''''''''))))))
F(mark(found(ok(mark(found(ok(mark(x'''''''''''')))))))) -> F(found(ok(mark(found(ok(mark(x'''''''''''')))))))
F(found(ok(mark(found(ok(found(x''''''''''))))))) -> F(ok(mark(found(ok(found(x''''''''''))))))
F(mark(found(ok(mark(found(ok(found(x'''''''''''')))))))) -> F(found(ok(mark(found(ok(found(x'''''''''''')))))))
F(found(ok(mark(found(ok(ok(x''''''''''))))))) -> F(ok(mark(found(ok(ok(x''''''''''))))))
F(mark(found(ok(mark(found(ok(ok(x'''''''''''')))))))) -> F(found(ok(mark(found(ok(ok(x'''''''''''')))))))
F(mark(found(ok(mark(found(mark(x''''''''''))))))) -> F(found(ok(mark(found(mark(x''''''''''))))))
F(mark(found(ok(mark(found(found(x''''''''''))))))) -> F(found(ok(mark(found(found(x''''''''''))))))
F(mark(found(ok(mark(ok(mark(x''''''''''))))))) -> F(found(ok(mark(ok(mark(x''''''''''))))))
F(mark(found(ok(mark(ok(found(x''''''''''))))))) -> F(found(ok(mark(ok(found(x''''''''''))))))
F(mark(found(ok(mark(ok(ok(x''''''''''))))))) -> F(found(ok(mark(ok(ok(x''''''''''))))))
F(mark(found(ok(mark(mark(x'''''''')))))) -> F(found(ok(mark(mark(x'''''''')))))
F(ok(mark(found(ok(mark(x'''''''')))))) -> F(mark(found(ok(mark(x'''''''')))))
F(mark(ok(mark(found(ok(mark(x''''''''''))))))) -> F(ok(mark(found(ok(mark(x''''''''''))))))
F(mark(found(ok(found(ok(mark(x''''''''''))))))) -> F(found(ok(found(ok(mark(x''''''''''))))))
F(mark(found(ok(found(ok(found(x''''''''''))))))) -> F(found(ok(found(ok(found(x''''''''''))))))
F(mark(found(ok(found(ok(ok(x''''''''''))))))) -> F(found(ok(found(ok(ok(x''''''''''))))))
F(mark(found(ok(found(mark(x'''''''')))))) -> F(found(ok(found(mark(x'''''''')))))
F(ok(mark(found(ok(found(x'''''''')))))) -> F(mark(found(ok(found(x'''''''')))))
F(mark(ok(mark(found(ok(found(x''''''''''))))))) -> F(ok(mark(found(ok(found(x''''''''''))))))
F(mark(found(ok(ok(mark(x'''''''')))))) -> F(found(ok(ok(mark(x'''''''')))))
F(mark(found(ok(ok(found(x'''''''')))))) -> F(found(ok(ok(found(x'''''''')))))
F(mark(found(ok(ok(ok(x'''''''')))))) -> F(found(ok(ok(ok(x'''''''')))))
F(ok(mark(found(ok(ok(x'''''''')))))) -> F(mark(found(ok(ok(x'''''''')))))
F(mark(ok(mark(found(ok(ok(x''''''''''))))))) -> F(ok(mark(found(ok(ok(x''''''''''))))))
F(mark(ok(mark(found(mark(x'''''''')))))) -> F(ok(mark(found(mark(x'''''''')))))
F(found(mark(ok(mark(x''''''))))) -> F(mark(ok(mark(x''''''))))
F(mark(found(mark(ok(mark(x'''''''')))))) -> F(found(mark(ok(mark(x'''''''')))))
F(found(mark(ok(found(x''''''))))) -> F(mark(ok(found(x''''''))))
F(mark(found(mark(ok(found(x'''''''')))))) -> F(found(mark(ok(found(x'''''''')))))
F(mark(found(mark(ok(ok(x'''''''')))))) -> F(found(mark(ok(ok(x'''''''')))))
F(mark(found(mark(mark(x''''''))))) -> F(found(mark(mark(x''''''))))
F(ok(mark(found(mark(x''''''))))) -> F(mark(found(mark(x''''''))))
F(found(ok(mark(found(mark(x'''''''')))))) -> F(ok(mark(found(mark(x'''''''')))))
F(found(ok(mark(found(found(x'''''''')))))) -> F(ok(mark(found(found(x'''''''')))))
F(found(found(ok(mark(x''''''))))) -> F(found(ok(mark(x''''''))))
F(mark(found(found(ok(mark(x'''''''')))))) -> F(found(found(ok(mark(x'''''''')))))
F(found(found(ok(found(x''''''))))) -> F(found(ok(found(x''''''))))
F(mark(found(found(ok(found(x'''''''')))))) -> F(found(found(ok(found(x'''''''')))))
F(found(found(ok(ok(x''''''))))) -> F(found(ok(ok(x''''''))))
F(mark(found(found(ok(ok(x'''''''')))))) -> F(found(found(ok(ok(x'''''''')))))
F(mark(found(found(mark(x''''''))))) -> F(found(found(mark(x''''''))))
F(mark(found(found(found(x''''''))))) -> F(found(found(found(x''''''))))
F(ok(mark(found(found(x''''''))))) -> F(mark(found(found(x''''''))))
F(mark(ok(mark(found(found(x'''''''')))))) -> F(ok(mark(found(found(x'''''''')))))
F(mark(ok(mark(ok(mark(x'''''''')))))) -> F(ok(mark(ok(mark(x'''''''')))))
F(mark(ok(mark(ok(found(x'''''''')))))) -> F(ok(mark(ok(found(x'''''''')))))
F(mark(ok(mark(ok(ok(x'''''''')))))) -> F(ok(mark(ok(ok(x'''''''')))))
F(mark(ok(mark(mark(x''''''))))) -> F(ok(mark(mark(x''''''))))
F(ok(mark(ok(mark(x''''''))))) -> F(mark(ok(mark(x''''''))))
F(found(ok(mark(ok(mark(x'''''''')))))) -> F(ok(mark(ok(mark(x'''''''')))))
F(mark(ok(found(ok(mark(x'''''''')))))) -> F(ok(found(ok(mark(x'''''''')))))
F(ok(mark(ok(found(x''''''))))) -> F(mark(ok(found(x''''''))))
F(found(ok(mark(ok(found(x'''''''')))))) -> F(ok(mark(ok(found(x'''''''')))))
F(ok(mark(ok(ok(x''''''))))) -> F(mark(ok(ok(x''''''))))
F(found(ok(mark(ok(ok(x'''''''')))))) -> F(ok(mark(ok(ok(x'''''''')))))
F(found(ok(mark(mark(x''''''))))) -> F(ok(mark(mark(x''''''))))
F(ok(found(ok(mark(x''''''))))) -> F(found(ok(mark(x''''''))))
F(found(ok(found(ok(mark(x'''''''')))))) -> F(ok(found(ok(mark(x'''''''')))))
F(found(ok(found(ok(found(x'''''''')))))) -> F(ok(found(ok(found(x'''''''')))))
F(found(ok(found(ok(ok(x'''''''')))))) -> F(ok(found(ok(ok(x'''''''')))))
F(found(ok(found(mark(x''''''))))) -> F(ok(found(mark(x''''''))))
F(ok(found(ok(found(x''''''))))) -> F(found(ok(found(x''''''))))
F(mark(ok(found(ok(found(x'''''''')))))) -> F(ok(found(ok(found(x'''''''')))))
F(found(ok(ok(mark(x''''''))))) -> F(ok(ok(mark(x''''''))))
F(found(ok(ok(found(x''''''))))) -> F(ok(ok(found(x''''''))))
F(found(ok(ok(ok(x''''''))))) -> F(ok(ok(ok(x''''''))))
F(ok(found(ok(ok(x''''''))))) -> F(found(ok(ok(x''''''))))
F(mark(ok(found(ok(ok(x'''''''')))))) -> F(ok(found(ok(ok(x'''''''')))))
F(mark(ok(found(mark(x''''''))))) -> F(ok(found(mark(x''''''))))
F(mark(ok(found(found(x''''''))))) -> F(ok(found(found(x''''''))))
F(mark(mark(ok(found(x''''''))))) -> F(mark(ok(found(x''''''))))
F(ok(mark(mark(x'''')))) -> F(mark(mark(x'''')))
F(ok(ok(mark(x'''')))) -> F(ok(mark(x'''')))
F(mark(ok(ok(mark(x''''''))))) -> F(ok(ok(mark(x''''''))))
F(mark(ok(ok(found(x''''''))))) -> F(ok(ok(found(x''''''))))
F(found(mark(ok(ok(x''''''))))) -> F(mark(ok(ok(x''''''))))
F(ok(found(mark(x'''')))) -> F(found(mark(x'''')))
F(ok(ok(found(x'''')))) -> F(ok(found(x'''')))
F(ok(ok(ok(x'''')))) -> F(ok(ok(x'''')))
F(mark(ok(ok(ok(x''''''))))) -> F(ok(ok(ok(x''''''))))
F(mark(mark(ok(ok(x''''''))))) -> F(mark(ok(ok(x''''''))))
F(mark(mark(mark(x'''')))) -> F(mark(mark(x'''')))
F(found(mark(mark(x'''')))) -> F(mark(mark(x'''')))
F(found(found(mark(x'''')))) -> F(found(mark(x'''')))
F(found(found(found(x'''')))) -> F(found(found(x'''')))
F(ok(found(found(x'''')))) -> F(found(found(x'''')))
F(found(ok(found(found(x''''''))))) -> F(ok(found(found(x''''''))))
F(mark(found(ok(found(found(x'''''''')))))) -> F(found(ok(found(found(x'''''''')))))
F(mark(mark(found(ok(found(x'''''''')))))) -> F(mark(found(ok(found(x'''''''')))))

Rules:

active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
top(active(c)) -> top(mark(c))
top(mark(x)) -> top(check(x))
top(found(x)) -> top(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)

Strategy:

innermost

The following dependency pairs can be strictly oriented:

F(ok(mark(found(ok(mark(x'''''''')))))) -> F(mark(found(ok(mark(x'''''''')))))
F(ok(mark(found(ok(found(x'''''''')))))) -> F(mark(found(ok(found(x'''''''')))))
F(ok(mark(found(ok(ok(x'''''''')))))) -> F(mark(found(ok(ok(x'''''''')))))
F(ok(mark(found(mark(x''''''))))) -> F(mark(found(mark(x''''''))))
F(ok(mark(found(found(x''''''))))) -> F(mark(found(found(x''''''))))
F(ok(mark(ok(mark(x''''''))))) -> F(mark(ok(mark(x''''''))))
F(ok(mark(ok(found(x''''''))))) -> F(mark(ok(found(x''''''))))
F(ok(mark(ok(ok(x''''''))))) -> F(mark(ok(ok(x''''''))))
F(ok(found(ok(mark(x''''''))))) -> F(found(ok(mark(x''''''))))
F(ok(found(ok(found(x''''''))))) -> F(found(ok(found(x''''''))))
F(ok(found(ok(ok(x''''''))))) -> F(found(ok(ok(x''''''))))
F(ok(mark(mark(x'''')))) -> F(mark(mark(x'''')))
F(ok(ok(mark(x'''')))) -> F(ok(mark(x'''')))
F(ok(found(mark(x'''')))) -> F(found(mark(x'''')))
F(ok(ok(found(x'''')))) -> F(ok(found(x'''')))
F(ok(ok(ok(x'''')))) -> F(ok(ok(x'''')))
F(ok(found(found(x'''')))) -> F(found(found(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(found(x1)) =  x1 POL(mark(x1)) =  x1 POL(ok(x1)) =  1 + x1 POL(F(x1)) =  1 + x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 7`
`             ↳FwdInst`
`             ...`
`               →DP Problem 27`
`                 ↳Dependency Graph`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳FwdInst`
`       →DP Problem 5`
`         ↳FwdInst`
`       →DP Problem 6`
`         ↳Nar`

Dependency Pairs:

F(mark(mark(found(ok(mark(x'''''''')))))) -> F(mark(found(ok(mark(x'''''''')))))
F(mark(mark(found(ok(ok(x'''''''')))))) -> F(mark(found(ok(ok(x'''''''')))))
F(mark(mark(found(mark(x''''''))))) -> F(mark(found(mark(x''''''))))
F(mark(mark(found(found(x''''''))))) -> F(mark(found(found(x''''''))))
F(mark(mark(ok(mark(x''''''))))) -> F(mark(ok(mark(x''''''))))
F(found(mark(found(ok(mark(x'''''''')))))) -> F(mark(found(ok(mark(x'''''''')))))
F(mark(found(mark(found(ok(mark(x''''''''''))))))) -> F(found(mark(found(ok(mark(x''''''''''))))))
F(found(mark(found(ok(found(x'''''''')))))) -> F(mark(found(ok(found(x'''''''')))))
F(mark(found(mark(found(ok(found(x''''''''''))))))) -> F(found(mark(found(ok(found(x''''''''''))))))
F(found(mark(found(ok(ok(x'''''''')))))) -> F(mark(found(ok(ok(x'''''''')))))
F(mark(found(mark(found(ok(ok(x''''''''''))))))) -> F(found(mark(found(ok(ok(x''''''''''))))))
F(found(mark(found(mark(x''''''))))) -> F(mark(found(mark(x''''''))))
F(mark(found(mark(found(mark(x'''''''')))))) -> F(found(mark(found(mark(x'''''''')))))
F(found(mark(found(found(x''''''))))) -> F(mark(found(found(x''''''))))
F(mark(found(mark(found(found(x'''''''')))))) -> F(found(mark(found(found(x'''''''')))))
F(found(ok(mark(found(ok(mark(x''''''''''))))))) -> F(ok(mark(found(ok(mark(x''''''''''))))))
F(mark(found(ok(mark(found(ok(mark(x'''''''''''')))))))) -> F(found(ok(mark(found(ok(mark(x'''''''''''')))))))
F(found(ok(mark(found(ok(found(x''''''''''))))))) -> F(ok(mark(found(ok(found(x''''''''''))))))
F(mark(found(ok(mark(found(ok(found(x'''''''''''')))))))) -> F(found(ok(mark(found(ok(found(x'''''''''''')))))))
F(found(ok(mark(found(ok(ok(x''''''''''))))))) -> F(ok(mark(found(ok(ok(x''''''''''))))))
F(mark(found(ok(mark(found(ok(ok(x'''''''''''')))))))) -> F(found(ok(mark(found(ok(ok(x'''''''''''')))))))
F(mark(found(ok(mark(found(mark(x''''''''''))))))) -> F(found(ok(mark(found(mark(x''''''''''))))))
F(mark(found(ok(mark(found(found(x''''''''''))))))) -> F(found(ok(mark(found(found(x''''''''''))))))
F(mark(found(ok(mark(ok(mark(x''''''''''))))))) -> F(found(ok(mark(ok(mark(x''''''''''))))))
F(mark(found(ok(mark(ok(found(x''''''''''))))))) -> F(found(ok(mark(ok(found(x''''''''''))))))
F(mark(found(ok(mark(ok(ok(x''''''''''))))))) -> F(found(ok(mark(ok(ok(x''''''''''))))))
F(mark(found(ok(mark(mark(x'''''''')))))) -> F(found(ok(mark(mark(x'''''''')))))
F(mark(ok(mark(found(ok(mark(x''''''''''))))))) -> F(ok(mark(found(ok(mark(x''''''''''))))))
F(mark(found(ok(found(ok(mark(x''''''''''))))))) -> F(found(ok(found(ok(mark(x''''''''''))))))
F(mark(found(ok(found(ok(found(x''''''''''))))))) -> F(found(ok(found(ok(found(x''''''''''))))))
F(mark(found(ok(found(ok(ok(x''''''''''))))))) -> F(found(ok(found(ok(ok(x''''''''''))))))
F(mark(found(ok(found(mark(x'''''''')))))) -> F(found(ok(found(mark(x'''''''')))))
F(mark(ok(mark(found(ok(found(x''''''''''))))))) -> F(ok(mark(found(ok(found(x''''''''''))))))
F(mark(found(ok(ok(mark(x'''''''')))))) -> F(found(ok(ok(mark(x'''''''')))))
F(mark(found(ok(ok(found(x'''''''')))))) -> F(found(ok(ok(found(x'''''''')))))
F(mark(found(ok(ok(ok(x'''''''')))))) -> F(found(ok(ok(ok(x'''''''')))))
F(mark(ok(mark(found(ok(ok(x''''''''''))))))) -> F(ok(mark(found(ok(ok(x''''''''''))))))
F(mark(ok(mark(found(mark(x'''''''')))))) -> F(ok(mark(found(mark(x'''''''')))))
F(found(mark(ok(mark(x''''''))))) -> F(mark(ok(mark(x''''''))))
F(mark(found(mark(ok(mark(x'''''''')))))) -> F(found(mark(ok(mark(x'''''''')))))
F(found(mark(ok(found(x''''''))))) -> F(mark(ok(found(x''''''))))
F(mark(found(mark(ok(found(x'''''''')))))) -> F(found(mark(ok(found(x'''''''')))))
F(mark(found(mark(ok(ok(x'''''''')))))) -> F(found(mark(ok(ok(x'''''''')))))
F(mark(found(mark(mark(x''''''))))) -> F(found(mark(mark(x''''''))))
F(found(ok(mark(found(mark(x'''''''')))))) -> F(ok(mark(found(mark(x'''''''')))))
F(found(ok(mark(found(found(x'''''''')))))) -> F(ok(mark(found(found(x'''''''')))))
F(found(found(ok(mark(x''''''))))) -> F(found(ok(mark(x''''''))))
F(mark(found(found(ok(mark(x'''''''')))))) -> F(found(found(ok(mark(x'''''''')))))
F(found(found(ok(found(x''''''))))) -> F(found(ok(found(x''''''))))
F(mark(found(found(ok(found(x'''''''')))))) -> F(found(found(ok(found(x'''''''')))))
F(found(found(ok(ok(x''''''))))) -> F(found(ok(ok(x''''''))))
F(mark(found(found(ok(ok(x'''''''')))))) -> F(found(found(ok(ok(x'''''''')))))
F(mark(found(found(mark(x''''''))))) -> F(found(found(mark(x''''''))))
F(mark(found(found(found(x''''''))))) -> F(found(found(found(x''''''))))
F(mark(ok(mark(found(found(x'''''''')))))) -> F(ok(mark(found(found(x'''''''')))))
F(mark(ok(mark(ok(mark(x'''''''')))))) -> F(ok(mark(ok(mark(x'''''''')))))
F(mark(ok(mark(ok(found(x'''''''')))))) -> F(ok(mark(ok(found(x'''''''')))))
F(mark(ok(mark(ok(ok(x'''''''')))))) -> F(ok(mark(ok(ok(x'''''''')))))
F(mark(ok(mark(mark(x''''''))))) -> F(ok(mark(mark(x''''''))))
F(found(ok(mark(ok(mark(x'''''''')))))) -> F(ok(mark(ok(mark(x'''''''')))))
F(mark(ok(found(ok(mark(x'''''''')))))) -> F(ok(found(ok(mark(x'''''''')))))
F(found(ok(mark(ok(found(x'''''''')))))) -> F(ok(mark(ok(found(x'''''''')))))
F(found(ok(mark(ok(ok(x'''''''')))))) -> F(ok(mark(ok(ok(x'''''''')))))
F(found(ok(mark(mark(x''''''))))) -> F(ok(mark(mark(x''''''))))
F(found(ok(found(ok(mark(x'''''''')))))) -> F(ok(found(ok(mark(x'''''''')))))
F(found(ok(found(ok(found(x'''''''')))))) -> F(ok(found(ok(found(x'''''''')))))
F(found(ok(found(ok(ok(x'''''''')))))) -> F(ok(found(ok(ok(x'''''''')))))
F(found(ok(found(mark(x''''''))))) -> F(ok(found(mark(x''''''))))
F(mark(ok(found(ok(found(x'''''''')))))) -> F(ok(found(ok(found(x'''''''')))))
F(found(ok(ok(mark(x''''''))))) -> F(ok(ok(mark(x''''''))))
F(found(ok(ok(found(x''''''))))) -> F(ok(ok(found(x''''''))))
F(found(ok(ok(ok(x''''''))))) -> F(ok(ok(ok(x''''''))))
F(mark(ok(found(ok(ok(x'''''''')))))) -> F(ok(found(ok(ok(x'''''''')))))
F(mark(ok(found(mark(x''''''))))) -> F(ok(found(mark(x''''''))))
F(mark(ok(found(found(x''''''))))) -> F(ok(found(found(x''''''))))
F(mark(mark(ok(found(x''''''))))) -> F(mark(ok(found(x''''''))))
F(mark(ok(ok(mark(x''''''))))) -> F(ok(ok(mark(x''''''))))
F(mark(ok(ok(found(x''''''))))) -> F(ok(ok(found(x''''''))))
F(found(mark(ok(ok(x''''''))))) -> F(mark(ok(ok(x''''''))))
F(mark(ok(ok(ok(x''''''))))) -> F(ok(ok(ok(x''''''))))
F(mark(mark(ok(ok(x''''''))))) -> F(mark(ok(ok(x''''''))))
F(mark(mark(mark(x'''')))) -> F(mark(mark(x'''')))
F(found(mark(mark(x'''')))) -> F(mark(mark(x'''')))
F(found(found(mark(x'''')))) -> F(found(mark(x'''')))
F(found(found(found(x'''')))) -> F(found(found(x'''')))
F(found(ok(found(found(x''''''))))) -> F(ok(found(found(x''''''))))
F(mark(found(ok(found(found(x'''''''')))))) -> F(found(ok(found(found(x'''''''')))))
F(mark(mark(found(ok(found(x'''''''')))))) -> F(mark(found(ok(found(x'''''''')))))

Rules:

active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
top(active(c)) -> top(mark(c))
top(mark(x)) -> top(check(x))
top(found(x)) -> top(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)

Strategy:

innermost

Using the Dependency Graph the DP problem was split into 1 DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 7`
`             ↳FwdInst`
`             ...`
`               →DP Problem 28`
`                 ↳Polynomial Ordering`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳FwdInst`
`       →DP Problem 5`
`         ↳FwdInst`
`       →DP Problem 6`
`         ↳Nar`

Dependency Pairs:

F(mark(found(mark(found(mark(x'''''''')))))) -> F(found(mark(found(mark(x'''''''')))))
F(mark(found(mark(found(found(x'''''''')))))) -> F(found(mark(found(found(x'''''''')))))
F(found(mark(found(mark(x''''''))))) -> F(mark(found(mark(x''''''))))
F(mark(found(found(mark(x''''''))))) -> F(found(found(mark(x''''''))))
F(found(mark(found(found(x''''''))))) -> F(mark(found(found(x''''''))))
F(found(found(mark(x'''')))) -> F(found(mark(x'''')))
F(found(found(found(x'''')))) -> F(found(found(x'''')))
F(mark(found(found(found(x''''''))))) -> F(found(found(found(x''''''))))
F(mark(mark(found(found(x''''''))))) -> F(mark(found(found(x''''''))))
F(mark(mark(mark(x'''')))) -> F(mark(mark(x'''')))
F(found(mark(mark(x'''')))) -> F(mark(mark(x'''')))
F(mark(found(mark(mark(x''''''))))) -> F(found(mark(mark(x''''''))))
F(mark(mark(found(mark(x''''''))))) -> F(mark(found(mark(x''''''))))

Rules:

active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
top(active(c)) -> top(mark(c))
top(mark(x)) -> top(check(x))
top(found(x)) -> top(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)

Strategy:

innermost

The following dependency pairs can be strictly oriented:

F(mark(found(mark(found(mark(x'''''''')))))) -> F(found(mark(found(mark(x'''''''')))))
F(mark(found(mark(found(found(x'''''''')))))) -> F(found(mark(found(found(x'''''''')))))
F(mark(found(found(mark(x''''''))))) -> F(found(found(mark(x''''''))))
F(mark(found(found(found(x''''''))))) -> F(found(found(found(x''''''))))
F(mark(mark(found(found(x''''''))))) -> F(mark(found(found(x''''''))))
F(mark(mark(mark(x'''')))) -> F(mark(mark(x'''')))
F(mark(found(mark(mark(x''''''))))) -> F(found(mark(mark(x''''''))))
F(mark(mark(found(mark(x''''''))))) -> F(mark(found(mark(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(found(x1)) =  x1 POL(mark(x1)) =  1 + x1 POL(F(x1)) =  1 + x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 7`
`             ↳FwdInst`
`             ...`
`               →DP Problem 29`
`                 ↳Dependency Graph`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳FwdInst`
`       →DP Problem 5`
`         ↳FwdInst`
`       →DP Problem 6`
`         ↳Nar`

Dependency Pairs:

F(found(mark(found(mark(x''''''))))) -> F(mark(found(mark(x''''''))))
F(found(mark(found(found(x''''''))))) -> F(mark(found(found(x''''''))))
F(found(found(mark(x'''')))) -> F(found(mark(x'''')))
F(found(found(found(x'''')))) -> F(found(found(x'''')))
F(found(mark(mark(x'''')))) -> F(mark(mark(x'''')))

Rules:

active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
top(active(c)) -> top(mark(c))
top(mark(x)) -> top(check(x))
top(found(x)) -> top(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)

Strategy:

innermost

Using the Dependency Graph the DP problem was split into 1 DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 7`
`             ↳FwdInst`
`             ...`
`               →DP Problem 30`
`                 ↳Polynomial Ordering`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳FwdInst`
`       →DP Problem 5`
`         ↳FwdInst`
`       →DP Problem 6`
`         ↳Nar`

Dependency Pair:

F(found(found(found(x'''')))) -> F(found(found(x'''')))

Rules:

active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
top(active(c)) -> top(mark(c))
top(mark(x)) -> top(check(x))
top(found(x)) -> top(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)

Strategy:

innermost

The following dependency pair can be strictly oriented:

F(found(found(found(x'''')))) -> F(found(found(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(found(x1)) =  1 + x1 POL(F(x1)) =  1 + x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 7`
`             ↳FwdInst`
`             ...`
`               →DP Problem 31`
`                 ↳Dependency Graph`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳FwdInst`
`       →DP Problem 5`
`         ↳FwdInst`
`       →DP Problem 6`
`         ↳Nar`

Dependency Pair:

Rules:

active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
top(active(c)) -> top(mark(c))
top(mark(x)) -> top(check(x))
top(found(x)) -> top(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)

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`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳FwdInst`
`       →DP Problem 5`
`         ↳FwdInst`
`       →DP Problem 6`
`         ↳Nar`

Dependency Pair:

ACTIVE(f(x)) -> ACTIVE(x)

Rules:

active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
top(active(c)) -> top(mark(c))
top(mark(x)) -> top(check(x))
top(found(x)) -> top(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)

Strategy:

innermost

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

ACTIVE(f(x)) -> ACTIVE(x)
one new Dependency Pair is created:

ACTIVE(f(f(x''))) -> ACTIVE(f(x''))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`           →DP Problem 32`
`             ↳Forward Instantiation Transformation`
`       →DP Problem 3`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳FwdInst`
`       →DP Problem 5`
`         ↳FwdInst`
`       →DP Problem 6`
`         ↳Nar`

Dependency Pair:

ACTIVE(f(f(x''))) -> ACTIVE(f(x''))

Rules:

active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
top(active(c)) -> top(mark(c))
top(mark(x)) -> top(check(x))
top(found(x)) -> top(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)

Strategy:

innermost

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

ACTIVE(f(f(x''))) -> ACTIVE(f(x''))
one new Dependency Pair is created:

ACTIVE(f(f(f(x'''')))) -> ACTIVE(f(f(x'''')))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`           →DP Problem 32`
`             ↳FwdInst`
`             ...`
`               →DP Problem 33`
`                 ↳Polynomial Ordering`
`       →DP Problem 3`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳FwdInst`
`       →DP Problem 5`
`         ↳FwdInst`
`       →DP Problem 6`
`         ↳Nar`

Dependency Pair:

ACTIVE(f(f(f(x'''')))) -> ACTIVE(f(f(x'''')))

Rules:

active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
top(active(c)) -> top(mark(c))
top(mark(x)) -> top(check(x))
top(found(x)) -> top(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)

Strategy:

innermost

The following dependency pair can be strictly oriented:

ACTIVE(f(f(f(x'''')))) -> ACTIVE(f(f(x'''')))

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

f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(ACTIVE(x1)) =  1 + x1 POL(found(x1)) =  0 POL(mark(x1)) =  0 POL(ok(x1)) =  0 POL(f(x1)) =  1 + x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`           →DP Problem 32`
`             ↳FwdInst`
`             ...`
`               →DP Problem 34`
`                 ↳Dependency Graph`
`       →DP Problem 3`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳FwdInst`
`       →DP Problem 5`
`         ↳FwdInst`
`       →DP Problem 6`
`         ↳Nar`

Dependency Pair:

Rules:

active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
top(active(c)) -> top(mark(c))
top(mark(x)) -> top(check(x))
top(found(x)) -> top(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)

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`
`         ↳Forward Instantiation Transformation`
`       →DP Problem 4`
`         ↳FwdInst`
`       →DP Problem 5`
`         ↳FwdInst`
`       →DP Problem 6`
`         ↳Nar`

Dependency Pair:

PROPER(f(x)) -> PROPER(x)

Rules:

active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
top(active(c)) -> top(mark(c))
top(mark(x)) -> top(check(x))
top(found(x)) -> top(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)

Strategy:

innermost

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

PROPER(f(x)) -> PROPER(x)
one new Dependency Pair is created:

PROPER(f(f(x''))) -> PROPER(f(x''))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳FwdInst`
`           →DP Problem 35`
`             ↳Forward Instantiation Transformation`
`       →DP Problem 4`
`         ↳FwdInst`
`       →DP Problem 5`
`         ↳FwdInst`
`       →DP Problem 6`
`         ↳Nar`

Dependency Pair:

PROPER(f(f(x''))) -> PROPER(f(x''))

Rules:

active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
top(active(c)) -> top(mark(c))
top(mark(x)) -> top(check(x))
top(found(x)) -> top(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)

Strategy:

innermost

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

PROPER(f(f(x''))) -> PROPER(f(x''))
one new Dependency Pair is created:

PROPER(f(f(f(x'''')))) -> PROPER(f(f(x'''')))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳FwdInst`
`           →DP Problem 35`
`             ↳FwdInst`
`             ...`
`               →DP Problem 36`
`                 ↳Polynomial Ordering`
`       →DP Problem 4`
`         ↳FwdInst`
`       →DP Problem 5`
`         ↳FwdInst`
`       →DP Problem 6`
`         ↳Nar`

Dependency Pair:

PROPER(f(f(f(x'''')))) -> PROPER(f(f(x'''')))

Rules:

active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
top(active(c)) -> top(mark(c))
top(mark(x)) -> top(check(x))
top(found(x)) -> top(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)

Strategy:

innermost

The following dependency pair can be strictly oriented:

PROPER(f(f(f(x'''')))) -> PROPER(f(f(x'''')))

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

f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(PROPER(x1)) =  1 + x1 POL(found(x1)) =  0 POL(mark(x1)) =  0 POL(ok(x1)) =  0 POL(f(x1)) =  1 + x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳FwdInst`
`           →DP Problem 35`
`             ↳FwdInst`
`             ...`
`               →DP Problem 37`
`                 ↳Dependency Graph`
`       →DP Problem 4`
`         ↳FwdInst`
`       →DP Problem 5`
`         ↳FwdInst`
`       →DP Problem 6`
`         ↳Nar`

Dependency Pair:

Rules:

active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
top(active(c)) -> top(mark(c))
top(mark(x)) -> top(check(x))
top(found(x)) -> top(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)

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`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳Forward Instantiation Transformation`
`       →DP Problem 5`
`         ↳FwdInst`
`       →DP Problem 6`
`         ↳Nar`

Dependency Pair:

MATCH(f(x), f(y)) -> MATCH(x, y)

Rules:

active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
top(active(c)) -> top(mark(c))
top(mark(x)) -> top(check(x))
top(found(x)) -> top(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)

Strategy:

innermost

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

MATCH(f(x), f(y)) -> MATCH(x, y)
one new Dependency Pair is created:

MATCH(f(f(x'')), f(f(y''))) -> MATCH(f(x''), f(y''))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳FwdInst`
`           →DP Problem 38`
`             ↳Forward Instantiation Transformation`
`       →DP Problem 5`
`         ↳FwdInst`
`       →DP Problem 6`
`         ↳Nar`

Dependency Pair:

MATCH(f(f(x'')), f(f(y''))) -> MATCH(f(x''), f(y''))

Rules:

active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
top(active(c)) -> top(mark(c))
top(mark(x)) -> top(check(x))
top(found(x)) -> top(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)

Strategy:

innermost

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

MATCH(f(f(x'')), f(f(y''))) -> MATCH(f(x''), f(y''))
one new Dependency Pair is created:

MATCH(f(f(f(x''''))), f(f(f(y'''')))) -> MATCH(f(f(x'''')), f(f(y'''')))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳FwdInst`
`           →DP Problem 38`
`             ↳FwdInst`
`             ...`
`               →DP Problem 39`
`                 ↳Polynomial Ordering`
`       →DP Problem 5`
`         ↳FwdInst`
`       →DP Problem 6`
`         ↳Nar`

Dependency Pair:

MATCH(f(f(f(x''''))), f(f(f(y'''')))) -> MATCH(f(f(x'''')), f(f(y'''')))

Rules:

active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
top(active(c)) -> top(mark(c))
top(mark(x)) -> top(check(x))
top(found(x)) -> top(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)

Strategy:

innermost

The following dependency pair can be strictly oriented:

MATCH(f(f(f(x''''))), f(f(f(y'''')))) -> MATCH(f(f(x'''')), f(f(y'''')))

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

f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(MATCH(x1, x2)) =  x2 POL(found(x1)) =  0 POL(mark(x1)) =  0 POL(ok(x1)) =  0 POL(f(x1)) =  1 + x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳FwdInst`
`           →DP Problem 38`
`             ↳FwdInst`
`             ...`
`               →DP Problem 40`
`                 ↳Dependency Graph`
`       →DP Problem 5`
`         ↳FwdInst`
`       →DP Problem 6`
`         ↳Nar`

Dependency Pair:

Rules:

active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
top(active(c)) -> top(mark(c))
top(mark(x)) -> top(check(x))
top(found(x)) -> top(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)

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`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳FwdInst`
`       →DP Problem 5`
`         ↳Forward Instantiation Transformation`
`       →DP Problem 6`
`         ↳Nar`

Dependency Pair:

CHECK(f(x)) -> CHECK(x)

Rules:

active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
top(active(c)) -> top(mark(c))
top(mark(x)) -> top(check(x))
top(found(x)) -> top(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)

Strategy:

innermost

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

CHECK(f(x)) -> CHECK(x)
one new Dependency Pair is created:

CHECK(f(f(x''))) -> CHECK(f(x''))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳FwdInst`
`       →DP Problem 5`
`         ↳FwdInst`
`           →DP Problem 41`
`             ↳Forward Instantiation Transformation`
`       →DP Problem 6`
`         ↳Nar`

Dependency Pair:

CHECK(f(f(x''))) -> CHECK(f(x''))

Rules:

active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
top(active(c)) -> top(mark(c))
top(mark(x)) -> top(check(x))
top(found(x)) -> top(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)

Strategy:

innermost

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

CHECK(f(f(x''))) -> CHECK(f(x''))
one new Dependency Pair is created:

CHECK(f(f(f(x'''')))) -> CHECK(f(f(x'''')))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳FwdInst`
`       →DP Problem 5`
`         ↳FwdInst`
`           →DP Problem 41`
`             ↳FwdInst`
`             ...`
`               →DP Problem 42`
`                 ↳Polynomial Ordering`
`       →DP Problem 6`
`         ↳Nar`

Dependency Pair:

CHECK(f(f(f(x'''')))) -> CHECK(f(f(x'''')))

Rules:

active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
top(active(c)) -> top(mark(c))
top(mark(x)) -> top(check(x))
top(found(x)) -> top(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)

Strategy:

innermost

The following dependency pair can be strictly oriented:

CHECK(f(f(f(x'''')))) -> CHECK(f(f(x'''')))

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

f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(found(x1)) =  0 POL(mark(x1)) =  0 POL(ok(x1)) =  0 POL(CHECK(x1)) =  1 + x1 POL(f(x1)) =  1 + x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳FwdInst`
`       →DP Problem 5`
`         ↳FwdInst`
`           →DP Problem 41`
`             ↳FwdInst`
`             ...`
`               →DP Problem 43`
`                 ↳Dependency Graph`
`       →DP Problem 6`
`         ↳Nar`

Dependency Pair:

Rules:

active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
top(active(c)) -> top(mark(c))
top(mark(x)) -> top(check(x))
top(found(x)) -> top(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)

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`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳FwdInst`
`       →DP Problem 5`
`         ↳FwdInst`
`       →DP Problem 6`
`         ↳Narrowing Transformation`

Dependency Pairs:

TOP(found(x)) -> TOP(active(x))
TOP(mark(x)) -> TOP(check(x))
TOP(active(c)) -> TOP(mark(c))

Rules:

active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
top(active(c)) -> top(mark(c))
top(mark(x)) -> top(check(x))
top(found(x)) -> top(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(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(check(x))
two new Dependency Pairs are created:

TOP(mark(f(x''))) -> TOP(f(check(x'')))
TOP(mark(x'')) -> TOP(start(match(f(X), x'')))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳FwdInst`
`       →DP Problem 5`
`         ↳FwdInst`
`       →DP Problem 6`
`         ↳Nar`
`           →DP Problem 44`
`             ↳Narrowing Transformation`

Dependency Pairs:

TOP(mark(f(x''))) -> TOP(f(check(x'')))
TOP(mark(x'')) -> TOP(start(match(f(X), x'')))
TOP(active(c)) -> TOP(mark(c))
TOP(found(x)) -> TOP(active(x))

Rules:

active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
top(active(c)) -> top(mark(c))
top(mark(x)) -> top(check(x))
top(found(x)) -> top(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)

Strategy:

innermost

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

TOP(mark(f(x''))) -> TOP(f(check(x'')))
two new Dependency Pairs are created:

TOP(mark(f(f(x')))) -> TOP(f(f(check(x'))))
TOP(mark(f(x'''))) -> TOP(f(start(match(f(X), x'''))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳FwdInst`
`       →DP Problem 5`
`         ↳FwdInst`
`       →DP Problem 6`
`         ↳Nar`
`           →DP Problem 44`
`             ↳Nar`
`             ...`
`               →DP Problem 45`
`                 ↳Narrowing Transformation`

Dependency Pairs:

TOP(mark(f(x'''))) -> TOP(f(start(match(f(X), x'''))))
TOP(mark(f(f(x')))) -> TOP(f(f(check(x'))))
TOP(found(x)) -> TOP(active(x))
TOP(active(c)) -> TOP(mark(c))
TOP(mark(x'')) -> TOP(start(match(f(X), x'')))

Rules:

active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
top(active(c)) -> top(mark(c))
top(mark(x)) -> top(check(x))
top(found(x)) -> top(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(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(start(match(f(X), x'')))
one new Dependency Pair is created:

TOP(mark(f(y'))) -> TOP(start(f(match(X, y'))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳FwdInst`
`       →DP Problem 5`
`         ↳FwdInst`
`       →DP Problem 6`
`         ↳Nar`
`           →DP Problem 44`
`             ↳Nar`
`             ...`
`               →DP Problem 46`
`                 ↳Narrowing Transformation`

Dependency Pairs:

TOP(mark(f(y'))) -> TOP(start(f(match(X, y'))))
TOP(mark(f(f(x')))) -> TOP(f(f(check(x'))))
TOP(found(x)) -> TOP(active(x))
TOP(mark(f(x'''))) -> TOP(f(start(match(f(X), x'''))))

Rules:

active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
top(active(c)) -> top(mark(c))
top(mark(x)) -> top(check(x))
top(found(x)) -> top(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)

Strategy:

innermost

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

TOP(found(x)) -> TOP(active(x))
two new Dependency Pairs are created:

TOP(found(f(x''))) -> TOP(mark(x''))
TOP(found(f(x''))) -> TOP(f(active(x'')))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳FwdInst`
`       →DP Problem 5`
`         ↳FwdInst`
`       →DP Problem 6`
`         ↳Nar`
`           →DP Problem 44`
`             ↳Nar`
`             ...`
`               →DP Problem 47`
`                 ↳Narrowing Transformation`

Dependency Pairs:

TOP(found(f(x''))) -> TOP(f(active(x'')))
TOP(found(f(x''))) -> TOP(mark(x''))
TOP(mark(f(x'''))) -> TOP(f(start(match(f(X), x'''))))
TOP(mark(f(f(x')))) -> TOP(f(f(check(x'))))
TOP(mark(f(y'))) -> TOP(start(f(match(X, y'))))

Rules:

active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
top(active(c)) -> top(mark(c))
top(mark(x)) -> top(check(x))
top(found(x)) -> top(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)

Strategy:

innermost

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

TOP(found(f(x''))) -> TOP(f(active(x'')))
two new Dependency Pairs are created:

TOP(found(f(f(x')))) -> TOP(f(mark(x')))
TOP(found(f(f(x')))) -> TOP(f(f(active(x'))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳FwdInst`
`       →DP Problem 5`
`         ↳FwdInst`
`       →DP Problem 6`
`         ↳Nar`
`           →DP Problem 44`
`             ↳Nar`
`             ...`
`               →DP Problem 48`
`                 ↳Rewriting Transformation`

Dependency Pairs:

TOP(found(f(f(x')))) -> TOP(f(f(active(x'))))
TOP(found(f(f(x')))) -> TOP(f(mark(x')))
TOP(mark(f(y'))) -> TOP(start(f(match(X, y'))))
TOP(mark(f(x'''))) -> TOP(f(start(match(f(X), x'''))))
TOP(mark(f(f(x')))) -> TOP(f(f(check(x'))))
TOP(found(f(x''))) -> TOP(mark(x''))

Rules:

active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
top(active(c)) -> top(mark(c))
top(mark(x)) -> top(check(x))
top(found(x)) -> top(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)

Strategy:

innermost

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

TOP(found(f(f(x')))) -> TOP(f(mark(x')))
one new Dependency Pair is created:

TOP(found(f(f(x')))) -> TOP(mark(f(x')))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳FwdInst`
`       →DP Problem 5`
`         ↳FwdInst`
`       →DP Problem 6`
`         ↳Nar`
`           →DP Problem 44`
`             ↳Nar`
`             ...`
`               →DP Problem 49`
`                 ↳Forward Instantiation Transformation`

Dependency Pairs:

TOP(found(f(f(x')))) -> TOP(mark(f(x')))
TOP(found(f(x''))) -> TOP(mark(x''))
TOP(mark(f(y'))) -> TOP(start(f(match(X, y'))))
TOP(mark(f(x'''))) -> TOP(f(start(match(f(X), x'''))))
TOP(mark(f(f(x')))) -> TOP(f(f(check(x'))))
TOP(found(f(f(x')))) -> TOP(f(f(active(x'))))

Rules:

active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
top(active(c)) -> top(mark(c))
top(mark(x)) -> top(check(x))
top(found(x)) -> top(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)

Strategy:

innermost

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

TOP(found(f(x''))) -> TOP(mark(x''))
three new Dependency Pairs are created:

TOP(found(f(f(f(x''''))))) -> TOP(mark(f(f(x''''))))
TOP(found(f(f(x''''')))) -> TOP(mark(f(x''''')))
TOP(found(f(f(y''')))) -> TOP(mark(f(y''')))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳FwdInst`
`       →DP Problem 5`
`         ↳FwdInst`
`       →DP Problem 6`
`         ↳Nar`
`           →DP Problem 44`
`             ↳Nar`
`             ...`
`               →DP Problem 50`
`                 ↳Polynomial Ordering`

Dependency Pairs:

TOP(found(f(f(y''')))) -> TOP(mark(f(y''')))
TOP(found(f(f(x''''')))) -> TOP(mark(f(x''''')))
TOP(found(f(f(f(x''''))))) -> TOP(mark(f(f(x''''))))
TOP(found(f(f(x')))) -> TOP(f(f(active(x'))))
TOP(mark(f(y'))) -> TOP(start(f(match(X, y'))))
TOP(mark(f(x'''))) -> TOP(f(start(match(f(X), x'''))))
TOP(mark(f(f(x')))) -> TOP(f(f(check(x'))))
TOP(found(f(f(x')))) -> TOP(mark(f(x')))

Rules:

active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
top(active(c)) -> top(mark(c))
top(mark(x)) -> top(check(x))
top(found(x)) -> top(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)

Strategy:

innermost

The following dependency pairs can be strictly oriented:

TOP(found(f(f(y''')))) -> TOP(mark(f(y''')))
TOP(found(f(f(x''''')))) -> TOP(mark(f(x''''')))
TOP(found(f(f(f(x''''))))) -> TOP(mark(f(f(x''''))))
TOP(found(f(f(x')))) -> TOP(mark(f(x')))

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

f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
start(ok(x)) -> found(x)
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(active(x1)) =  x1 POL(proper(x1)) =  x1 POL(c) =  0 POL(match(x1, x2)) =  x2 POL(X) =  0 POL(check(x1)) =  x1 POL(found(x1)) =  x1 POL(mark(x1)) =  x1 POL(ok(x1)) =  x1 POL(TOP(x1)) =  1 + x1 POL(f(x1)) =  1 + x1 POL(start(x1)) =  x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳FwdInst`
`       →DP Problem 5`
`         ↳FwdInst`
`       →DP Problem 6`
`         ↳Nar`
`           →DP Problem 44`
`             ↳Nar`
`             ...`
`               →DP Problem 51`
`                 ↳Polynomial Ordering`

Dependency Pairs:

TOP(found(f(f(x')))) -> TOP(f(f(active(x'))))
TOP(mark(f(y'))) -> TOP(start(f(match(X, y'))))
TOP(mark(f(x'''))) -> TOP(f(start(match(f(X), x'''))))
TOP(mark(f(f(x')))) -> TOP(f(f(check(x'))))

Rules:

active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
top(active(c)) -> top(mark(c))
top(mark(x)) -> top(check(x))
top(found(x)) -> top(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)

Strategy:

innermost

The following dependency pairs can be strictly oriented:

TOP(mark(f(y'))) -> TOP(start(f(match(X, y'))))
TOP(mark(f(x'''))) -> TOP(f(start(match(f(X), x'''))))
TOP(mark(f(f(x')))) -> TOP(f(f(check(x'))))

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

active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
start(ok(x)) -> found(x)
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(active(x1)) =  x1 POL(proper(x1)) =  x1 POL(c) =  0 POL(match(x1, x2)) =  x2 POL(X) =  0 POL(check(x1)) =  x1 POL(found(x1)) =  x1 POL(mark(x1)) =  1 + x1 POL(ok(x1)) =  x1 POL(TOP(x1)) =  1 + x1 POL(f(x1)) =  1 + x1 POL(start(x1)) =  x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳FwdInst`
`       →DP Problem 5`
`         ↳FwdInst`
`       →DP Problem 6`
`         ↳Nar`
`           →DP Problem 44`
`             ↳Nar`
`             ...`
`               →DP Problem 52`
`                 ↳Polynomial Ordering`

Dependency Pair:

TOP(found(f(f(x')))) -> TOP(f(f(active(x'))))

Rules:

active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
top(active(c)) -> top(mark(c))
top(mark(x)) -> top(check(x))
top(found(x)) -> top(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)

Strategy:

innermost

The following dependency pair can be strictly oriented:

TOP(found(f(f(x')))) -> TOP(f(f(active(x'))))

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

active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(active(x1)) =  0 POL(found(x1)) =  1 POL(mark(x1)) =  0 POL(ok(x1)) =  0 POL(TOP(x1)) =  1 + x1 POL(f(x1)) =  x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳FwdInst`
`       →DP Problem 5`
`         ↳FwdInst`
`       →DP Problem 6`
`         ↳Nar`
`           →DP Problem 44`
`             ↳Nar`
`             ...`
`               →DP Problem 53`
`                 ↳Dependency Graph`

Dependency Pair:

Rules:

active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
top(active(c)) -> top(mark(c))
top(mark(x)) -> top(check(x))
top(found(x)) -> top(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:22 minutes