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)

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

         ↳Argument Filtering and Ordering

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →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)

The following dependency pairs can be strictly oriented:

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

There are no usable rules w.r.t. to the AFS that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:

F(x₁) -> F(x₁)
ok(x₁) -> ok(x₁)
mark(x₁) -> mark(x₁)
found(x₁) -> found(x₁)

     ↳DPs

       →DP Problem 1

         ↳AFS

           →DP Problem 7

             ↳Dependency Graph

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →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)

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳Argument Filtering and Ordering

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →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)

The following dependency pair can be strictly oriented:

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

There are no usable rules w.r.t. to the AFS that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:

ACTIVE(x₁) -> ACTIVE(x₁)
f(x₁) -> f(x₁)

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

           →DP Problem 8

             ↳Dependency Graph

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →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)

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳Argument Filtering and Ordering

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →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)

The following dependency pair can be strictly oriented:

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

There are no usable rules w.r.t. to the AFS that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:

PROPER(x₁) -> PROPER(x₁)
f(x₁) -> f(x₁)

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

           →DP Problem 9

             ↳Dependency Graph

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →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)

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳Argument Filtering and Ordering

       →DP Problem 5

         ↳AFS

       →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)

The following dependency pair can be strictly oriented:

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

There are no usable rules w.r.t. to the AFS that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:

MATCH(x₁, x₂) -> MATCH(x₁, x₂)
f(x₁) -> f(x₁)

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

           →DP Problem 10

             ↳Dependency Graph

       →DP Problem 5

         ↳AFS

       →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)

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳Argument Filtering and Ordering

       →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)

The following dependency pair can be strictly oriented:

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

There are no usable rules w.r.t. to the AFS that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:

CHECK(x₁) -> CHECK(x₁)
f(x₁) -> f(x₁)

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

           →DP Problem 11

             ↳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)

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →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)

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

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳Nar

           →DP Problem 12

             ↳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)

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

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳Nar

           →DP Problem 12

             ↳Nar

...

               →DP Problem 13

                 ↳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)

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

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳Nar

           →DP Problem 12

             ↳Nar

...

               →DP Problem 14

                 ↳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)

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

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳Nar

           →DP Problem 12

             ↳Nar

...

               →DP Problem 15

                 ↳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)

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

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳Nar

           →DP Problem 12

             ↳Nar

...

               →DP Problem 16

                 ↳Argument Filtering and Ordering

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)

The following dependency pairs can be strictly oriented:

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

The following usable rules 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: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:

TOP(x₁) -> TOP(x₁)
mark(x₁) -> x₁
f(x₁) -> f(x₁)
start(x₁) -> x₁
match(x₁, x₂) -> x₂
found(x₁) -> x₁
check(x₁) -> x₁
active(x₁) -> x₁
proper(x₁) -> x₁
ok(x₁) -> x₁

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳Nar

           →DP Problem 12

             ↳Nar

...

               →DP Problem 17

                 ↳Narrowing Transformation

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)

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

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

two new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳Nar

           →DP Problem 12

             ↳Nar

...

               →DP Problem 18

                 ↳Narrowing Transformation

Dependency Pairs:

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

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

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

one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳Nar

           →DP Problem 12

             ↳Nar

...

               →DP Problem 19

                 ↳Narrowing Transformation

Dependency Pairs:

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

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

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

one new Dependency Pair is created:

TOP(mark(f(y''))) -> TOP(start(f(proper(y''))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳Nar

           →DP Problem 12

             ↳Nar

...

               →DP Problem 20

                 ↳Narrowing Transformation

Dependency Pairs:

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

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

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

two new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳Nar

           →DP Problem 12

             ↳Nar

...

               →DP Problem 21

                 ↳Narrowing Transformation

Dependency Pairs:

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

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

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

two new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳Nar

           →DP Problem 12

             ↳Nar

...

               →DP Problem 22

                 ↳Narrowing Transformation

Dependency Pairs:

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

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

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

one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳Nar

           →DP Problem 12

             ↳Nar

...

               →DP Problem 23

                 ↳Narrowing Transformation

Dependency Pairs:

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

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

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

one new Dependency Pair is created:

TOP(mark(f(f(y'')))) -> TOP(f(start(f(proper(y'')))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳Nar

           →DP Problem 12

             ↳Nar

...

               →DP Problem 24

                 ↳Narrowing Transformation

Dependency Pairs:

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

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

TOP(mark(f(y''))) -> TOP(start(f(proper(y''))))

two new Dependency Pairs are created:

TOP(mark(f(c))) -> TOP(start(f(ok(c))))
TOP(mark(f(f(x')))) -> TOP(start(f(f(proper(x')))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳Nar

           →DP Problem 12

             ↳Nar

...

               →DP Problem 25

                 ↳Narrowing Transformation

Dependency Pairs:

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

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

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

one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳Nar

           →DP Problem 12

             ↳Nar

...

               →DP Problem 26

                 ↳Narrowing Transformation

Dependency Pairs:

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

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

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

two new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳Nar

           →DP Problem 12

             ↳Nar

...

               →DP Problem 27

                 ↳Narrowing Transformation

Dependency Pairs:

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

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

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

two new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳Nar

           →DP Problem 12

             ↳Nar

...

               →DP Problem 28

                 ↳Narrowing Transformation

Dependency Pairs:

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

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

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

one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳Nar

           →DP Problem 12

             ↳Nar

...

               →DP Problem 29

                 ↳Narrowing Transformation

Dependency Pairs:

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

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

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

one new Dependency Pair is created:

TOP(mark(f(f(f(y''))))) -> TOP(f(f(start(f(proper(y''))))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳Nar

           →DP Problem 12

             ↳Nar

...

               →DP Problem 30

                 ↳Narrowing Transformation

Dependency Pairs:

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

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

TOP(mark(f(f(y'')))) -> TOP(f(start(f(proper(y'')))))

two new Dependency Pairs are created:

TOP(mark(f(f(c)))) -> TOP(f(start(f(ok(c)))))
TOP(mark(f(f(f(x'))))) -> TOP(f(start(f(f(proper(x'))))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳Nar

           →DP Problem 12

             ↳Nar

...

               →DP Problem 31

                 ↳Narrowing Transformation

Dependency Pairs:

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

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

TOP(mark(f(c))) -> TOP(start(f(ok(c))))

one new Dependency Pair is created:

TOP(mark(f(c))) -> TOP(start(ok(f(c))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳Nar

           →DP Problem 12

             ↳Nar

...

               →DP Problem 32

                 ↳Narrowing Transformation

Dependency Pairs:

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

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

TOP(mark(f(f(x')))) -> TOP(start(f(f(proper(x')))))

two new Dependency Pairs are created:

TOP(mark(f(f(c)))) -> TOP(start(f(f(ok(c)))))
TOP(mark(f(f(f(x''))))) -> TOP(start(f(f(f(proper(x''))))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳Nar

           →DP Problem 12

             ↳Nar

...

               →DP Problem 33

                 ↳Narrowing Transformation

Dependency Pairs:

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

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

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

four new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳Nar

           →DP Problem 12

             ↳Nar

...

               →DP Problem 34

                 ↳Narrowing Transformation

Dependency Pairs:

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

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

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

one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳Nar

           →DP Problem 12

             ↳Nar

...

               →DP Problem 35

                 ↳Narrowing Transformation

Dependency Pairs:

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

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

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

two new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳Nar

           →DP Problem 12

             ↳Nar

...

               →DP Problem 36

                 ↳Argument Filtering and Ordering

Dependency Pairs:

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

The following dependency pairs can be strictly oriented:

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

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

Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:

TOP(x₁) -> TOP(x₁)
found(x₁) -> x₁
f(x₁) -> f(x₁)
mark(x₁) -> x₁
check(x₁) -> x₁
start(x₁) -> x₁
ok(x₁) -> x₁
proper(x₁) -> x₁
match(x₁, x₂) -> x₂
active(x₁) -> x₁

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳Nar

           →DP Problem 12

             ↳Nar

...

               →DP Problem 37

                 ↳Remaining Obligation(s)

The following remains to be proven:
Dependency Pairs:

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

Termination of R could not be shown.
Duration:
0:03 minutes