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

         ↳AFS

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)

The following rules can be oriented:

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)

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(proper(x₁)) = 1 + x₁

POL(match(x₁, x₂)) = x₁ + x₂

POL(X) = 1

POL(check(x₁)) = 1 + x₁

POL(mark(x₁)) = 1 + x₁

POL(ok(x₁)) = 1 + x₁

POL(F(x₁)) = x₁

POL(f(x₁)) = x₁

POL(start(x₁)) = x₁

POL(top(x₁)) = 1 + x₁

POL(active(x₁)) = 1 + x₁

POL(c) = 0

POL(found(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)
found(x₁) -> found(x₁)
active(x₁) -> active(x₁)
f(x₁) -> f(x₁)
top(x₁) -> top(x₁)
check(x₁) -> check(x₁)
start(x₁) -> start(x₁)
match(x₁, x₂) -> match(x₁, x₂)
proper(x₁) -> proper(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

         ↳AFS

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

         ↳AFS

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)

The following rules can be oriented:

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)

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(top(x₁)) = x₁

POL(ACTIVE(x₁)) = x₁

POL(active(x₁)) = x₁

POL(proper(x₁)) = x₁

POL(c) = 0

POL(check(x₁)) = x₁

POL(found(x₁)) = x₁

POL(mark(x₁)) = x₁

POL(ok(x₁)) = x₁

POL(f(x₁)) = 1 + x₁

POL(start(x₁)) = x₁

resulting in one new DP problem.
Used Argument Filtering System:

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

         ↳AFS

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

         ↳AFS

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)

The following rules can be oriented:

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)

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(top(x₁)) = x₁

POL(active(x₁)) = x₁

POL(proper(x₁)) = x₁

POL(c) = 0

POL(PROPER(x₁)) = x₁

POL(check(x₁)) = x₁

POL(found(x₁)) = x₁

POL(mark(x₁)) = x₁

POL(ok(x₁)) = x₁

POL(f(x₁)) = 1 + x₁

POL(start(x₁)) = x₁

resulting in one new DP problem.
Used Argument Filtering System:

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

         ↳AFS

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

         ↳AFS

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)

The following rules can be oriented:

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)

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(top(x₁)) = x₁

POL(active(x₁)) = x₁

POL(proper(x₁)) = x₁

POL(c) = 0

POL(MATCH(x₁, x₂)) = x₁ + x₂

POL(check(x₁)) = x₁

POL(found(x₁)) = x₁

POL(mark(x₁)) = x₁

POL(ok(x₁)) = x₁

POL(f(x₁)) = 1 + x₁

POL(start(x₁)) = x₁

resulting in one new DP problem.
Used Argument Filtering System:

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

         ↳AFS

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

         ↳AFS

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)

The following rules can be oriented:

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)

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(top(x₁)) = x₁

POL(active(x₁)) = x₁

POL(proper(x₁)) = x₁

POL(c) = 0

POL(check(x₁)) = x₁

POL(found(x₁)) = x₁

POL(mark(x₁)) = x₁

POL(CHECK(x₁)) = x₁

POL(ok(x₁)) = x₁

POL(f(x₁)) = 1 + x₁

POL(start(x₁)) = x₁

resulting in one new DP problem.
Used Argument Filtering System:

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

         ↳AFS

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

         ↳Argument Filtering and Ordering

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)

The following dependency pair can be strictly oriented:

TOP(active(c)) -> TOP(mark(c))

The following rules can be oriented:

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

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(top(x₁)) = x₁

POL(active(x₁)) = x₁

POL(proper) = 1

POL(c) = 1

POL(X) = 1

POL(check) = 0

POL(found(x₁)) = x₁

POL(mark) = 0

POL(TOP(x₁)) = x₁

POL(ok(x₁)) = x₁

POL(f) = 0

POL(start(x₁)) = x₁

resulting in one new DP problem.
Used Argument Filtering System:

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

         ↳AFS

           →DP Problem 12

             ↳Argument Filtering and Ordering

Dependency Pairs:

TOP(found(x)) -> TOP(active(x))
TOP(mark(x)) -> TOP(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:

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

The following rules can be oriented:

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

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(top) = 0

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

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳AFS

           →DP Problem 12

             ↳AFS

...

               →DP Problem 13

                 ↳Argument Filtering and Ordering

Dependency Pair:

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)

The following dependency pair can be strictly oriented:

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

The following rules 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))
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))
start(ok(x)) -> found(x)

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(proper(x₁)) = x₁

POL(match(x₁, x₂)) = x₁ + x₂

POL(X) = 0

POL(check(x₁)) = 1 + x₁

POL(mark(x₁)) = x₁

POL(TOP(x₁)) = 1 + x₁

POL(ok(x₁)) = x₁

POL(f(x₁)) = x₁

POL(start(x₁)) = 1 + x₁

POL(top) = 0

POL(active(x₁)) = x₁

POL(c) = 0

POL(found(x₁)) = 1 + x₁

resulting in one new DP problem.
Used Argument Filtering System:

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

         ↳AFS

           →DP Problem 12

             ↳AFS

...

               →DP Problem 14

                 ↳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)

Using the Dependency Graph resulted in no new DP problems.

Termination of R successfully shown.
Duration:
0:36 minutes

POL(proper(x₁))	= 1 + x₁
POL(match(x₁, x₂))	= x₁ + x₂
POL(X)	= 1
POL(check(x₁))	= 1 + x₁
POL(mark(x₁))	= 1 + x₁
POL(ok(x₁))	= 1 + x₁
POL(F(x₁))	= x₁
POL(f(x₁))	= x₁
POL(start(x₁))	= x₁
POL(top(x₁))	= 1 + x₁
POL(active(x₁))	= 1 + x₁
POL(c)	= 0
POL(found(x₁))	= 1 + x₁

POL(top)	= 0
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