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`
`         ↳Size-Change Principle`
`       →DP Problem 2`
`         ↳SCP`
`       →DP Problem 3`
`         ↳SCP`
`       →DP Problem 4`
`         ↳SCP`
`       →DP Problem 5`
`         ↳SCP`
`       →DP Problem 6`
`         ↳MRR`

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)

We number the DPs as follows:
1. F(mark(x)) -> F(x)
2. F(found(x)) -> F(x)
3. F(ok(x)) -> F(x)
and get the following Size-Change Graph(s):
{3, 2, 1} , {3, 2, 1}
1>1

which lead(s) to this/these maximal multigraph(s):
{3, 2, 1} , {3, 2, 1}
1>1

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
found(x1) -> found(x1)
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)

We obtain no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳SCP`
`       →DP Problem 2`
`         ↳Size-Change Principle`
`       →DP Problem 3`
`         ↳SCP`
`       →DP Problem 4`
`         ↳SCP`
`       →DP Problem 5`
`         ↳SCP`
`       →DP Problem 6`
`         ↳MRR`

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)

We number the DPs as follows:
1. ACTIVE(f(x)) -> ACTIVE(x)
and get the following Size-Change Graph(s):
{1} , {1}
1>1

which lead(s) to this/these maximal multigraph(s):
{1} , {1}
1>1

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
f(x1) -> f(x1)

We obtain no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳SCP`
`       →DP Problem 2`
`         ↳SCP`
`       →DP Problem 3`
`         ↳Size-Change Principle`
`       →DP Problem 4`
`         ↳SCP`
`       →DP Problem 5`
`         ↳SCP`
`       →DP Problem 6`
`         ↳MRR`

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)

We number the DPs as follows:
1. PROPER(f(x)) -> PROPER(x)
and get the following Size-Change Graph(s):
{1} , {1}
1>1

which lead(s) to this/these maximal multigraph(s):
{1} , {1}
1>1

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
f(x1) -> f(x1)

We obtain no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳SCP`
`       →DP Problem 2`
`         ↳SCP`
`       →DP Problem 3`
`         ↳SCP`
`       →DP Problem 4`
`         ↳Size-Change Principle`
`       →DP Problem 5`
`         ↳SCP`
`       →DP Problem 6`
`         ↳MRR`

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)

We number the DPs as follows:
1. MATCH(f(x), f(y)) -> MATCH(x, y)
and get the following Size-Change Graph(s):
{1} , {1}
1>1
2>2

which lead(s) to this/these maximal multigraph(s):
{1} , {1}
1>1
2>2

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
f(x1) -> f(x1)

We obtain no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳SCP`
`       →DP Problem 2`
`         ↳SCP`
`       →DP Problem 3`
`         ↳SCP`
`       →DP Problem 4`
`         ↳SCP`
`       →DP Problem 5`
`         ↳Size-Change Principle`
`       →DP Problem 6`
`         ↳MRR`

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)

We number the DPs as follows:
1. CHECK(f(x)) -> CHECK(x)
and get the following Size-Change Graph(s):
{1} , {1}
1>1

which lead(s) to this/these maximal multigraph(s):
{1} , {1}
1>1

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
f(x1) -> f(x1)

We obtain no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳SCP`
`       →DP Problem 2`
`         ↳SCP`
`       →DP Problem 3`
`         ↳SCP`
`       →DP Problem 4`
`         ↳SCP`
`       →DP Problem 5`
`         ↳SCP`
`       →DP Problem 6`
`         ↳Modular Removal of Rules`

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)

We have the following set of usable rules:

active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
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))
To remove rules and DPs from this DP problem we used the following monotonic and CE-compatible order: Polynomial ordering.
Polynomial interpretation:
 POL(active(x1)) =  x1 POL(proper(x1)) =  x1 POL(c) =  0 POL(match(x1, x2)) =  x1 + x2 POL(X) =  0 POL(check(x1)) =  x1 POL(found(x1)) =  x1 POL(mark(x1)) =  x1 POL(TOP(x1)) =  x1 POL(ok(x1)) =  x1 POL(f(x1)) =  x1 POL(start(x1)) =  x1

We have the following set D of usable symbols: {proper, active, c, match, X, check, found, mark, TOP, ok, f, start}
No Dependency Pairs can be deleted.
3 non usable rules have been deleted.

The result of this processor delivers one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳SCP`
`       →DP Problem 2`
`         ↳SCP`
`       →DP Problem 3`
`         ↳SCP`
`       →DP Problem 4`
`         ↳SCP`
`       →DP Problem 5`
`         ↳SCP`
`       →DP Problem 6`
`         ↳MRR`
`           →DP Problem 7`
`             ↳Negative Polynomial Order`

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

The following Dependency Pair can be strictly oriented using the given order.

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

Moreover, the following usable rules (regarding the implicit AFS) are oriented.

active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
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 Order with Interpretation:

POL( TOP(x1) ) = x1

POL( active(x1) ) = x1

POL( c ) = 1

POL( mark(x1) ) = 0

POL( found(x1) ) = x1

POL( check(x1) ) = 0

POL( f(x1) ) = 0

POL( start(x1) ) = x1

POL( match(x1, x2) ) = x1

POL( ok(x1) ) = x1

POL( X ) = 1

POL( proper(x1) ) = 1

This results in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳SCP`
`       →DP Problem 2`
`         ↳SCP`
`       →DP Problem 3`
`         ↳SCP`
`       →DP Problem 4`
`         ↳SCP`
`       →DP Problem 5`
`         ↳SCP`
`       →DP Problem 6`
`         ↳MRR`
`           →DP Problem 7`
`             ↳Neg POLO`
`             ...`
`               →DP Problem 8`
`                 ↳Modular Removal of Rules`

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

We have the following set of usable rules:

active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
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))
To remove rules and DPs from this DP problem we used the following monotonic and CE-compatible order: Polynomial ordering.
Polynomial interpretation:
 POL(active(x1)) =  x1 POL(proper(x1)) =  x1 POL(c) =  0 POL(match(x1, x2)) =  x1 + x2 POL(X) =  0 POL(check(x1)) =  1 + x1 POL(found(x1)) =  x1 POL(mark(x1)) =  1 + x1 POL(TOP(x1)) =  1 + x1 POL(ok(x1)) =  x1 POL(f(x1)) =  1 + x1 POL(start(x1)) =  x1

We have the following set D of usable symbols: {proper, active, c, match, X, check, found, mark, TOP, ok, f, start}
No Dependency Pairs can be deleted.
The following rules can be deleted as the lhs is strictly greater than the corresponding rhs:

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

The result of this processor delivers one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳SCP`
`       →DP Problem 2`
`         ↳SCP`
`       →DP Problem 3`
`         ↳SCP`
`       →DP Problem 4`
`         ↳SCP`
`       →DP Problem 5`
`         ↳SCP`
`       →DP Problem 6`
`         ↳MRR`
`           →DP Problem 7`
`             ↳Neg POLO`
`             ...`
`               →DP Problem 9`
`                 ↳Modular Removal of Rules`

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

We have the following set of usable rules:

active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
To remove rules and DPs from this DP problem we used the following monotonic and CE-compatible order: Polynomial ordering.
Polynomial interpretation:
 POL(active(x1)) =  x1 POL(match(x1, x2)) =  x1 + x2 POL(X) =  0 POL(check(x1)) =  x1 POL(found(x1)) =  x1 POL(mark(x1)) =  x1 POL(TOP(x1)) =  x1 POL(ok(x1)) =  x1 POL(f(x1)) =  x1 POL(start(x1)) =  x1

We have the following set D of usable symbols: {active, match, X, check, found, mark, TOP, ok, f, start}
No Dependency Pairs can be deleted.
4 non usable rules have been deleted.

The result of this processor delivers one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳SCP`
`       →DP Problem 2`
`         ↳SCP`
`       →DP Problem 3`
`         ↳SCP`
`       →DP Problem 4`
`         ↳SCP`
`       →DP Problem 5`
`         ↳SCP`
`       →DP Problem 6`
`         ↳MRR`
`           →DP Problem 7`
`             ↳Neg POLO`
`             ...`
`               →DP Problem 10`
`                 ↳Modular Removal of Rules`

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))
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))

We have the following set of usable rules:

active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
To remove rules and DPs from this DP problem we used the following monotonic and CE-compatible order: Polynomial ordering.
Polynomial interpretation:
 POL(active(x1)) =  x1 POL(match(x1, x2)) =  x1 + x2 POL(X) =  0 POL(check(x1)) =  x1 POL(found(x1)) =  1 + x1 POL(mark(x1)) =  x1 POL(TOP(x1)) =  1 + x1 POL(ok(x1)) =  x1 POL(f(x1)) =  x1 POL(start(x1)) =  x1

We have the following set D of usable symbols: {active, match, X, check, found, mark, TOP, ok, f, start}
The following Dependency Pairs can be deleted as the lhs is strictly greater than the corresponding rhs:

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

No Rules can be deleted.

The result of this processor delivers one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳SCP`
`       →DP Problem 2`
`         ↳SCP`
`       →DP Problem 3`
`         ↳SCP`
`       →DP Problem 4`
`         ↳SCP`
`       →DP Problem 5`
`         ↳SCP`
`       →DP Problem 6`
`         ↳MRR`
`           →DP Problem 7`
`             ↳Neg POLO`
`             ...`
`               →DP Problem 11`
`                 ↳Modular Removal of Rules`

Dependency Pair:

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

Rules:

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))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))

We have the following set of usable rules:

check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
To remove rules and DPs from this DP problem we used the following monotonic and CE-compatible order: Polynomial ordering.
Polynomial interpretation:
 POL(match(x1, x2)) =  x1 + x2 POL(X) =  0 POL(check(x1)) =  x1 POL(found(x1)) =  x1 POL(mark(x1)) =  x1 POL(TOP(x1)) =  1 + x1 POL(ok(x1)) =  x1 POL(f(x1)) =  x1 POL(start(x1)) =  x1

We have the following set D of usable symbols: {match, X, check, found, mark, TOP, ok, f, start}
No Dependency Pairs can be deleted.
2 non usable rules have been deleted.

The result of this processor delivers one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳SCP`
`       →DP Problem 2`
`         ↳SCP`
`       →DP Problem 3`
`         ↳SCP`
`       →DP Problem 4`
`         ↳SCP`
`       →DP Problem 5`
`         ↳SCP`
`       →DP Problem 6`
`         ↳MRR`
`           →DP Problem 7`
`             ↳Neg POLO`
`             ...`
`               →DP Problem 12`
`                 ↳Modular Removal of Rules`

Dependency Pair:

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

Rules:

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

We have the following set of usable rules:

check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
To remove rules and DPs from this DP problem we used the following monotonic and CE-compatible order: Polynomial ordering.
Polynomial interpretation:
 POL(match(x1, x2)) =  x1 + x2 POL(X) =  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)) =  x1 POL(start(x1)) =  x1

We have the following set D of usable symbols: {match, X, check, found, mark, TOP, ok, f, start}
The following Dependency Pairs can be deleted as the lhs is strictly greater than the corresponding rhs:

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

No Rules can be deleted.

After the removal, there are no SCCs in the dependency graph which results in no DP problems which have to be solved.

Termination of R successfully shown.
Duration:
0:01 minutes