Termination Proof using AProVE

Term Rewriting System R:
[x, y]
active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
top(active(c)) -> top(mark(c))
top(mark(x)) -> top(check(x))
top(found(x)) -> top(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

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

Furthermore, R contains six SCCs.

     ↳DPs

       →DP Problem 1

         ↳Usable Rules (Innermost)

       →DP Problem 2

         ↳UsableRules

       →DP Problem 3

         ↳UsableRules

       →DP Problem 4

         ↳UsableRules

       →DP Problem 5

         ↳UsableRules

       →DP Problem 6

         ↳UsableRules

Dependency Pairs:

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

Rules:

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

Strategy:

innermost

As we are in the innermost case, we can delete all 15 non-usable-rules.

     ↳DPs

       →DP Problem 1

         ↳UsableRules

           →DP Problem 7

             ↳Size-Change Principle

       →DP Problem 2

         ↳UsableRules

       →DP Problem 3

         ↳UsableRules

       →DP Problem 4

         ↳UsableRules

       →DP Problem 5

         ↳UsableRules

       →DP Problem 6

         ↳UsableRules

Dependency Pairs:

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

Rule:

none

Strategy:

innermost

We number the DPs as follows:

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

and get the following Size-Change Graph(s):

{1, 2, 3}	,	{1, 2, 3}
1	>	1

which lead(s) to this/these maximal multigraph(s):

{1, 2, 3}	,	{1, 2, 3}
1	>	1

D_P: 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.

     ↳DPs

       →DP Problem 1

         ↳UsableRules

       →DP Problem 2

         ↳Usable Rules (Innermost)

       →DP Problem 3

         ↳UsableRules

       →DP Problem 4

         ↳UsableRules

       →DP Problem 5

         ↳UsableRules

       →DP Problem 6

         ↳UsableRules

Dependency Pair:

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

Rules:

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

Strategy:

innermost

As we are in the innermost case, we can delete all 15 non-usable-rules.

     ↳DPs

       →DP Problem 1

         ↳UsableRules

       →DP Problem 2

         ↳UsableRules

           →DP Problem 8

             ↳Size-Change Principle

       →DP Problem 3

         ↳UsableRules

       →DP Problem 4

         ↳UsableRules

       →DP Problem 5

         ↳UsableRules

       →DP Problem 6

         ↳UsableRules

Dependency Pair:

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

Rule:

none

Strategy:

innermost

We number the DPs as follows:

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

D_P: 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.

     ↳DPs

       →DP Problem 1

         ↳UsableRules

       →DP Problem 2

         ↳UsableRules

       →DP Problem 3

         ↳Usable Rules (Innermost)

       →DP Problem 4

         ↳UsableRules

       →DP Problem 5

         ↳UsableRules

       →DP Problem 6

         ↳UsableRules

Dependency Pair:

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

Rules:

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

Strategy:

innermost

As we are in the innermost case, we can delete all 15 non-usable-rules.

     ↳DPs

       →DP Problem 1

         ↳UsableRules

       →DP Problem 2

         ↳UsableRules

       →DP Problem 3

         ↳UsableRules

           →DP Problem 9

             ↳Size-Change Principle

       →DP Problem 4

         ↳UsableRules

       →DP Problem 5

         ↳UsableRules

       →DP Problem 6

         ↳UsableRules

Dependency Pair:

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

Rule:

none

Strategy:

innermost

We number the DPs as follows:

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

D_P: 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.

     ↳DPs

       →DP Problem 1

         ↳UsableRules

       →DP Problem 2

         ↳UsableRules

       →DP Problem 3

         ↳UsableRules

       →DP Problem 4

         ↳Usable Rules (Innermost)

       →DP Problem 5

         ↳UsableRules

       →DP Problem 6

         ↳UsableRules

Dependency Pair:

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

Rules:

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

Strategy:

innermost

As we are in the innermost case, we can delete all 15 non-usable-rules.

     ↳DPs

       →DP Problem 1

         ↳UsableRules

       →DP Problem 2

         ↳UsableRules

       →DP Problem 3

         ↳UsableRules

       →DP Problem 4

         ↳UsableRules

           →DP Problem 10

             ↳Size-Change Principle

       →DP Problem 5

         ↳UsableRules

       →DP Problem 6

         ↳UsableRules

Dependency Pair:

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

Rule:

none

Strategy:

innermost

We number the DPs as follows:

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

D_P: 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.

     ↳DPs

       →DP Problem 1

         ↳UsableRules

       →DP Problem 2

         ↳UsableRules

       →DP Problem 3

         ↳UsableRules

       →DP Problem 4

         ↳UsableRules

       →DP Problem 5

         ↳Usable Rules (Innermost)

       →DP Problem 6

         ↳UsableRules

Dependency Pair:

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

Rules:

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

Strategy:

innermost

As we are in the innermost case, we can delete all 15 non-usable-rules.

     ↳DPs

       →DP Problem 1

         ↳UsableRules

       →DP Problem 2

         ↳UsableRules

       →DP Problem 3

         ↳UsableRules

       →DP Problem 4

         ↳UsableRules

       →DP Problem 5

         ↳UsableRules

           →DP Problem 11

             ↳Size-Change Principle

       →DP Problem 6

         ↳UsableRules

Dependency Pair:

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

Rule:

none

Strategy:

innermost

We number the DPs as follows:

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

D_P: 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.

     ↳DPs

       →DP Problem 1

         ↳UsableRules

       →DP Problem 2

         ↳UsableRules

       →DP Problem 3

         ↳UsableRules

       →DP Problem 4

         ↳UsableRules

       →DP Problem 5

         ↳UsableRules

       →DP Problem 6

         ↳Usable Rules (Innermost)

Dependency Pairs:

TOP(found(x)) -> TOP(active(x))
TOP(mark(x)) -> TOP(check(x))
TOP(active(c)) -> TOP(mark(c))

Rules:

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

Strategy:

innermost

As we are in the innermost case, we can delete all 3 non-usable-rules.

     ↳DPs

       →DP Problem 1

         ↳UsableRules

       →DP Problem 2

         ↳UsableRules

       →DP Problem 3

         ↳UsableRules

       →DP Problem 4

         ↳UsableRules

       →DP Problem 5

         ↳UsableRules

       →DP Problem 6

         ↳UsableRules

           →DP Problem 12

             ↳Negative Polynomial Order

Dependency Pairs:

TOP(found(x)) -> TOP(active(x))
TOP(mark(x)) -> TOP(check(x))
TOP(active(c)) -> TOP(mark(c))

Rules:

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

Strategy:

innermost

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.

f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
f(ok(x)) -> ok(f(x))
active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
match(X, x) -> proper(x)
match(f(x), f(y)) -> f(match(x, y))
proper(f(x)) -> f(proper(x))
proper(c) -> ok(c)
start(ok(x)) -> found(x)
check(x) -> start(match(f(X), x))
check(f(x)) -> f(check(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( ok(x₁) ) = x₁

POL( match(x₁, x₂) ) = x₁

POL( X ) = 1

POL( proper(x₁) ) = 1

POL( start(x₁) ) = x₁

This results in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳UsableRules

       →DP Problem 2

         ↳UsableRules

       →DP Problem 3

         ↳UsableRules

       →DP Problem 4

         ↳UsableRules

       →DP Problem 5

         ↳UsableRules

       →DP Problem 6

         ↳UsableRules

           →DP Problem 12

             ↳Neg POLO

...

               →DP Problem 13

                 ↳Modular Removal of Rules

Dependency Pairs:

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

Rules:

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

Strategy:

innermost

We have the following set of usable rules:

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

To remove rules and DPs from this DP problem we used the following monotonic and C_E-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.

     ↳DPs

       →DP Problem 1

         ↳UsableRules

       →DP Problem 2

         ↳UsableRules

       →DP Problem 3

         ↳UsableRules

       →DP Problem 4

         ↳UsableRules

       →DP Problem 5

         ↳UsableRules

       →DP Problem 6

         ↳UsableRules

           →DP Problem 12

             ↳Neg POLO

...

               →DP Problem 14

                 ↳Modular Removal of Rules

Dependency Pairs:

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

Rules:

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

Strategy:

innermost

We have the following set of usable rules:

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

To remove rules and DPs from this DP problem we used the following monotonic and C_E-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.

     ↳DPs

       →DP Problem 1

         ↳UsableRules

       →DP Problem 2

         ↳UsableRules

       →DP Problem 3

         ↳UsableRules

       →DP Problem 4

         ↳UsableRules

       →DP Problem 5

         ↳UsableRules

       →DP Problem 6

         ↳UsableRules

           →DP Problem 12

             ↳Neg POLO

...

               →DP Problem 15

                 ↳Modular Removal of Rules

Dependency Pairs:

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

Rules:

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

Strategy:

innermost

We have the following set of usable rules:

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

To remove rules and DPs from this DP problem we used the following monotonic and C_E-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.

     ↳DPs

       →DP Problem 1

         ↳UsableRules

       →DP Problem 2

         ↳UsableRules

       →DP Problem 3

         ↳UsableRules

       →DP Problem 4

         ↳UsableRules

       →DP Problem 5

         ↳UsableRules

       →DP Problem 6

         ↳UsableRules

           →DP Problem 12

             ↳Neg POLO

...

               →DP Problem 16

                 ↳Modular Removal of Rules

Dependency Pair:

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

Rules:

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

Strategy:

innermost

We have the following set of usable rules:

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

To remove rules and DPs from this DP problem we used the following monotonic and C_E-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.

     ↳DPs

       →DP Problem 1

         ↳UsableRules

       →DP Problem 2

         ↳UsableRules

       →DP Problem 3

         ↳UsableRules

       →DP Problem 4

         ↳UsableRules

       →DP Problem 5

         ↳UsableRules

       →DP Problem 6

         ↳UsableRules

           →DP Problem 12

             ↳Neg POLO

...

               →DP Problem 17

                 ↳Modular Removal of Rules

Dependency Pair:

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

Rules:

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

Strategy:

innermost

We have the following set of usable rules:

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

To remove rules and DPs from this DP problem we used the following monotonic and C_E-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.

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

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