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
Argument Filtering and 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(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(found(mark(ok(mark(x''''''))))) -> F(mark(ok(mark(x''''''))))
F(found(mark(ok(found(x''''''))))) -> F(mark(ok(found(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(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(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(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(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(found(ok(found(found(x''''''))))) -> F(ok(found(found(x''''''))))


There are no usable rules for innermost that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(found(x1))=  1 + x1  
  POL(mark(x1))=  x1  
  POL(ok(x1))=  x1  
  POL(F(x1))=  1 + x1  

resulting in one new DP problem.
Used Argument Filtering System:
F(x1) -> F(x1)
found(x1) -> found(x1)
ok(x1) -> ok(x1)
mark(x1) -> mark(x1)


   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(mark(found(mark(found(ok(mark(x''''''''''))))))) -> F(found(mark(found(ok(mark(x''''''''''))))))
F(mark(found(mark(found(ok(found(x''''''''''))))))) -> F(found(mark(found(ok(found(x''''''''''))))))
F(mark(found(mark(found(ok(ok(x''''''''''))))))) -> F(found(mark(found(ok(ok(x''''''''''))))))
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(ok(mark(found(ok(mark(x'''''''''''')))))))) -> F(found(ok(mark(found(ok(mark(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(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(mark(found(mark(ok(mark(x'''''''')))))) -> F(found(mark(ok(mark(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(mark(found(found(ok(mark(x'''''''')))))) -> F(found(found(ok(mark(x'''''''')))))
F(mark(found(found(ok(found(x'''''''')))))) -> F(found(found(ok(found(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(mark(ok(found(ok(mark(x'''''''')))))) -> F(ok(found(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(mark(ok(found(ok(found(x'''''''')))))) -> F(ok(found(ok(found(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(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(ok(found(found(x'''')))) -> F(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
Argument Filtering and 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(ok(mark(ok(mark(x'''''''')))))) -> F(ok(mark(ok(mark(x'''''''')))))
F(mark(ok(mark(ok(ok(x'''''''')))))) -> F(ok(mark(ok(ok(x'''''''')))))
F(ok(mark(ok(mark(x''''''))))) -> F(mark(ok(mark(x''''''))))
F(mark(ok(ok(mark(x''''''))))) -> F(ok(ok(mark(x''''''))))
F(ok(mark(ok(ok(x''''''))))) -> F(mark(ok(ok(x''''''))))
F(ok(ok(mark(x'''')))) -> F(ok(mark(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(mark(ok(mark(mark(x''''''))))) -> F(ok(mark(mark(x''''''))))
F(mark(mark(ok(mark(x''''''))))) -> F(mark(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




The following dependency pairs can be strictly oriented:

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


There are no usable rules for innermost that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(mark(x1))=  x1  
  POL(ok(x1))=  1 + x1  
  POL(F(x1))=  1 + x1  

resulting in one new DP problem.
Used Argument Filtering System:
F(x1) -> F(x1)
ok(x1) -> ok(x1)
mark(x1) -> mark(x1)


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




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
Argument Filtering and Ordering
       →DP Problem 2
FwdInst
       →DP Problem 3
FwdInst
       →DP Problem 4
FwdInst
       →DP Problem 5
FwdInst
       →DP Problem 6
Nar


Dependency Pair:

F(mark(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




The following dependency pair can be strictly oriented:

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


There are no usable rules for innermost that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(mark(x1))=  1 + x1  
  POL(F(x1))=  1 + x1  

resulting in one new DP problem.
Used Argument Filtering System:
F(x1) -> F(x1)
mark(x1) -> mark(x1)


   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
Argument Filtering and 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'''')))


The following usable rules for innermost 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))=  x1  
  POL(mark(x1))=  x1  
  POL(ok(x1))=  x1  
  POL(f(x1))=  1 + x1  

resulting in one new DP problem.
Used Argument Filtering System:
ACTIVE(x1) -> ACTIVE(x1)
f(x1) -> f(x1)
ok(x1) -> ok(x1)
found(x1) -> found(x1)
mark(x1) -> mark(x1)


   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
Argument Filtering and 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'''')))


The following usable rules for innermost 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))=  x1  
  POL(mark(x1))=  x1  
  POL(ok(x1))=  x1  
  POL(f(x1))=  1 + x1  

resulting in one new DP problem.
Used Argument Filtering System:
PROPER(x1) -> PROPER(x1)
f(x1) -> f(x1)
ok(x1) -> ok(x1)
found(x1) -> found(x1)
mark(x1) -> mark(x1)


   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
Argument Filtering and 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'''')))


The following usable rules for innermost 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))=  1 + x1 + x2  
  POL(found(x1))=  x1  
  POL(mark(x1))=  x1  
  POL(ok(x1))=  x1  
  POL(f(x1))=  1 + x1  

resulting in one new DP problem.
Used Argument Filtering System:
MATCH(x1, x2) -> MATCH(x1, x2)
f(x1) -> f(x1)
ok(x1) -> ok(x1)
found(x1) -> found(x1)
mark(x1) -> mark(x1)


   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
Argument Filtering and 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'''')))


The following usable rules for innermost 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))=  x1  
  POL(mark(x1))=  x1  
  POL(CHECK(x1))=  1 + x1  
  POL(ok(x1))=  x1  
  POL(f(x1))=  1 + x1  

resulting in one new DP problem.
Used Argument Filtering System:
CHECK(x1) -> CHECK(x1)
f(x1) -> f(x1)
ok(x1) -> ok(x1)
found(x1) -> found(x1)
mark(x1) -> mark(x1)


   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
Argument Filtering and 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')))


The following usable rules for innermost can be oriented:

f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
active(f(x)) -> mark(x)
active(f(x)) -> f(active(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(check(x1))=  x1  
  POL(found(x1))=  x1  
  POL(mark(x1))=  x1  
  POL(TOP(x1))=  1 + x1  
  POL(ok(x1))=  x1  
  POL(f(x1))=  1 + x1  
  POL(start(x1))=  x1  

resulting in one new DP problem.
Used Argument Filtering System:
TOP(x1) -> TOP(x1)
found(x1) -> found(x1)
mark(x1) -> mark(x1)
f(x1) -> f(x1)
start(x1) -> start(x1)
match(x1, x2) -> x2
check(x1) -> check(x1)
active(x1) -> active(x1)
ok(x1) -> ok(x1)
proper(x1) -> proper(x1)


   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
Argument Filtering and 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 pair can be strictly oriented:

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


The following usable rules for innermost can be oriented:

start(ok(x)) -> found(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)
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
active(f(x)) -> mark(x)
active(f(x)) -> f(active(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))=  x1 + x2  
  POL(X)=  0  
  POL(check(x1))=  1 + x1  
  POL(found(x1))=  x1  
  POL(mark(x1))=  1 + x1  
  POL(TOP(x1))=  1 + x1  
  POL(ok(x1))=  x1  
  POL(f(x1))=  1 + x1  
  POL(start(x1))=  x1  

resulting in one new DP problem.
Used Argument Filtering System:
TOP(x1) -> TOP(x1)
mark(x1) -> mark(x1)
start(x1) -> start(x1)
f(x1) -> f(x1)
match(x1, x2) -> match(x1, x2)
check(x1) -> check(x1)
found(x1) -> found(x1)
active(x1) -> active(x1)
ok(x1) -> ok(x1)
proper(x1) -> proper(x1)


   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
Argument Filtering and Ordering


Dependency Pairs:

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


The following usable rules for innermost can be oriented:

f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
start(ok(x)) -> found(x)
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
active(f(x)) -> mark(x)
active(f(x)) -> f(active(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(check(x1))=  x1  
  POL(found(x1))=  x1  
  POL(mark(x1))=  1 + x1  
  POL(TOP(x1))=  1 + x1  
  POL(ok(x1))=  x1  
  POL(f(x1))=  1 + x1  
  POL(start(x1))=  x1  

resulting in one new DP problem.
Used Argument Filtering System:
TOP(x1) -> TOP(x1)
mark(x1) -> mark(x1)
f(x1) -> f(x1)
check(x1) -> check(x1)
start(x1) -> start(x1)
match(x1, x2) -> x2
found(x1) -> found(x1)
active(x1) -> active(x1)
ok(x1) -> ok(x1)
proper(x1) -> proper(x1)


   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
Argument Filtering and 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'))))


The following usable rules for innermost 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))


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

resulting in one new DP problem.
Used Argument Filtering System:
TOP(x1) -> TOP(x1)
found(x1) -> found(x1)
f(x1) -> f(x1)
active(x1) -> active(x1)
ok(x1) -> ok(x1)
mark(x1) -> mark(x1)


   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 54
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:
0:24 minutes