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.

R
Dependency Pair Analysis

R contains the following Dependency Pairs:

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

Furthermore, R contains six SCCs.

R
DPs
→DP Problem 1
Polynomial Ordering
→DP Problem 2
Polo
→DP Problem 3
Polo
→DP Problem 4
Polo
→DP Problem 5
Polo
→DP Problem 6
Polo

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 pair can be strictly oriented:

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

Additionally, 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))
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))

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

resulting in one new DP problem.

R
DPs
→DP Problem 1
Polo
→DP Problem 7
Polynomial Ordering
→DP Problem 2
Polo
→DP Problem 3
Polo
→DP Problem 4
Polo
→DP Problem 5
Polo
→DP Problem 6
Polo

Dependency Pairs:

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 pair can be strictly oriented:

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

Additionally, 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))
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))

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

resulting in one new DP problem.

R
DPs
→DP Problem 1
Polo
→DP Problem 7
Polo
...
→DP Problem 8
Polynomial Ordering
→DP Problem 2
Polo
→DP Problem 3
Polo
→DP Problem 4
Polo
→DP Problem 5
Polo
→DP Problem 6
Polo

Dependency Pair:

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 pair can be strictly oriented:

F(ok(x)) -> F(x)

Additionally, 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))
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))

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

resulting in one new DP problem.

R
DPs
→DP Problem 1
Polo
→DP Problem 7
Polo
...
→DP Problem 9
Dependency Graph
→DP Problem 2
Polo
→DP Problem 3
Polo
→DP Problem 4
Polo
→DP Problem 5
Polo
→DP Problem 6
Polo

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.

R
DPs
→DP Problem 1
Polo
→DP Problem 2
Polynomial Ordering
→DP Problem 3
Polo
→DP Problem 4
Polo
→DP Problem 5
Polo
→DP Problem 6
Polo

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)

Additionally, 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))
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))

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

resulting in one new DP problem.

R
DPs
→DP Problem 1
Polo
→DP Problem 2
Polo
→DP Problem 10
Dependency Graph
→DP Problem 3
Polo
→DP Problem 4
Polo
→DP Problem 5
Polo
→DP Problem 6
Polo

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.

R
DPs
→DP Problem 1
Polo
→DP Problem 2
Polo
→DP Problem 3
Polynomial Ordering
→DP Problem 4
Polo
→DP Problem 5
Polo
→DP Problem 6
Polo

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)

Additionally, 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))
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))

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

resulting in one new DP problem.

R
DPs
→DP Problem 1
Polo
→DP Problem 2
Polo
→DP Problem 3
Polo
→DP Problem 11
Dependency Graph
→DP Problem 4
Polo
→DP Problem 5
Polo
→DP Problem 6
Polo

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.

R
DPs
→DP Problem 1
Polo
→DP Problem 2
Polo
→DP Problem 3
Polo
→DP Problem 4
Polynomial Ordering
→DP Problem 5
Polo
→DP Problem 6
Polo

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)

Additionally, 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))
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))

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

resulting in one new DP problem.

R
DPs
→DP Problem 1
Polo
→DP Problem 2
Polo
→DP Problem 3
Polo
→DP Problem 4
Polo
→DP Problem 12
Dependency Graph
→DP Problem 5
Polo
→DP Problem 6
Polo

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.

R
DPs
→DP Problem 1
Polo
→DP Problem 2
Polo
→DP Problem 3
Polo
→DP Problem 4
Polo
→DP Problem 5
Polynomial Ordering
→DP Problem 6
Polo

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)

Additionally, 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))
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))

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

resulting in one new DP problem.

R
DPs
→DP Problem 1
Polo
→DP Problem 2
Polo
→DP Problem 3
Polo
→DP Problem 4
Polo
→DP Problem 5
Polo
→DP Problem 13
Dependency Graph
→DP Problem 6
Polo

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.

R
DPs
→DP Problem 1
Polo
→DP Problem 2
Polo
→DP Problem 3
Polo
→DP Problem 4
Polo
→DP Problem 5
Polo
→DP Problem 6
Polynomial 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))

Additionally, 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))
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))

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

resulting in one new DP problem.

R
DPs
→DP Problem 1
Polo
→DP Problem 2
Polo
→DP Problem 3
Polo
→DP Problem 4
Polo
→DP Problem 5
Polo
→DP Problem 6
Polo
→DP Problem 14
Polynomial 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(found(x)) -> TOP(active(x))

Additionally, 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))
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))

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

resulting in one new DP problem.

R
DPs
→DP Problem 1
Polo
→DP Problem 2
Polo
→DP Problem 3
Polo
→DP Problem 4
Polo
→DP Problem 5
Polo
→DP Problem 6
Polo
→DP Problem 14
Polo
...
→DP Problem 15
Polynomial Ordering

Dependency Pair:

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))

Additionally, 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))
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))

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

resulting in one new DP problem.

R
DPs
→DP Problem 1
Polo
→DP Problem 2
Polo
→DP Problem 3
Polo
→DP Problem 4
Polo
→DP Problem 5
Polo
→DP Problem 6
Polo
→DP Problem 14
Polo
...
→DP Problem 16
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:
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