Termination Proof using AProVE

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.

     ↳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.

     ↳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:

     ↳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:

     ↳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:

     ↳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:

     ↳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:

     ↳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:

     ↳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:

     ↳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:

     ↳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:

     ↳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:

     ↳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:

     ↳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:

     ↳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:

     ↳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:

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:

     ↳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:

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:

     ↳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:

     ↳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:

     ↳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:

     ↳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:

     ↳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:

     ↳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 w.r.t. to the AFS 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)

     ↳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.

     ↳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 w.r.t. to the AFS 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)

     ↳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:

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.

     ↳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 w.r.t. to the AFS 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)

     ↳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.

     ↳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:

     ↳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:

     ↳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 w.r.t. to the AFS can be oriented:

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

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(ACTIVE(x₁)) = 1 + x₁

POL(found(x₁)) = 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)

     ↳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.

     ↳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:

     ↳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:

     ↳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 w.r.t. to the AFS can be oriented:

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

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(PROPER(x₁)) = 1 + x₁

POL(found(x₁)) = 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)

     ↳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.

     ↳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:

     ↳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:

     ↳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 w.r.t. to the AFS can be oriented:

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

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(MATCH(x₁, x₂)) = 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)

     ↳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.

     ↳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:

     ↳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:

     ↳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 w.r.t. to the AFS can be oriented:

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

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(found(x₁)) = 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)

     ↳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.

     ↳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:

     ↳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:

     ↳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:

     ↳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:

     ↳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:

     ↳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:

     ↳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:

     ↳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 w.r.t. to the AFS can be oriented:

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

     ↳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 w.r.t. to the AFS can be oriented:

match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> 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)
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)
proper(x₁) -> proper(x₁)
ok(x₁) -> ok(x₁)

     ↳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 w.r.t. to the AFS can be oriented:

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

     ↳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 w.r.t. to the AFS can be oriented:

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

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(active(x₁)) = 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)
mark(x₁) -> mark(x₁)
ok(x₁) -> ok(x₁)

     ↳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:
0:42 minutes

POL(found(x₁))	= 1 + x₁
POL(mark(x₁))	= x₁
POL(ok(x₁))	= x₁
POL(F(x₁))	= 1 + x₁

POL(ACTIVE(x₁))	= 1 + x₁
POL(found(x₁))	= x₁
POL(mark(x₁))	= x₁
POL(ok(x₁))	= x₁
POL(f(x₁))	= 1 + x₁

POL(PROPER(x₁))	= 1 + x₁
POL(found(x₁))	= x₁
POL(mark(x₁))	= x₁
POL(ok(x₁))	= x₁
POL(f(x₁))	= 1 + x₁

POL(MATCH(x₁, x₂))	= 1 + x₁ + x₂
POL(found(x₁))	= x₁
POL(mark(x₁))	= x₁
POL(ok(x₁))	= x₁
POL(f(x₁))	= 1 + x₁