YES

The TRS could be proven terminating. The proof took 1332 ms.

The following DP Processors were used


Problem 1 was processed with processor DependencyGraph (139ms).
 | – Problem 2 was processed with processor SubtermCriterion (9ms).
 | – Problem 3 was processed with processor SubtermCriterion (12ms).
 | – Problem 4 was processed with processor SubtermCriterion (6ms).
 | – Problem 5 was processed with processor PolynomialLinearRange4iUR (530ms).
 |    | – Problem 8 was processed with processor PolynomialLinearRange4iUR (427ms).
 |    |    | – Problem 9 was processed with processor PolynomialLinearRange4iUR (40ms).
 | – Problem 6 was processed with processor SubtermCriterion (1ms).
 | – Problem 7 was processed with processor SubtermCriterion (0ms).

Problem 1: DependencyGraph



Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

active(f(x))mark(x)top(active(c))top(mark(c))
top(mark(x))top(check(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))
start(ok(x))found(x)f(found(x))found(f(x))
top(found(x))top(active(x))active(f(x))f(active(x))
f(mark(x))mark(f(x))

Original Signature

Termination of terms over the following signature is verified: f, check, c, start, active, mark, ok, match, proper, X, top, found

Strategy


The following SCCs where found

match#(f(x), f(y)) → match#(x, y)

check#(f(x)) → check#(x)

f#(mark(x)) → f#(x)f#(found(x)) → f#(x)
f#(ok(x)) → f#(x)

proper#(f(x)) → proper#(x)

top#(found(x)) → top#(active(x))top#(active(c)) → top#(mark(c))
top#(mark(x)) → top#(check(x))

active#(f(x)) → active#(x)

Problem 2: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

match#(f(x), f(y))match#(x, y)

Rewrite Rules

active(f(x))mark(x)top(active(c))top(mark(c))
top(mark(x))top(check(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))
start(ok(x))found(x)f(found(x))found(f(x))
top(found(x))top(active(x))active(f(x))f(active(x))
f(mark(x))mark(f(x))

Original Signature

Termination of terms over the following signature is verified: f, check, c, start, active, mark, ok, match, proper, X, top, found

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

match#(f(x), f(y))match#(x, y)

Problem 3: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

f#(mark(x))f#(x)f#(found(x))f#(x)
f#(ok(x))f#(x)

Rewrite Rules

active(f(x))mark(x)top(active(c))top(mark(c))
top(mark(x))top(check(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))
start(ok(x))found(x)f(found(x))found(f(x))
top(found(x))top(active(x))active(f(x))f(active(x))
f(mark(x))mark(f(x))

Original Signature

Termination of terms over the following signature is verified: f, check, c, start, active, mark, ok, match, proper, X, top, found

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

f#(mark(x))f#(x)f#(found(x))f#(x)
f#(ok(x))f#(x)

Problem 4: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

check#(f(x))check#(x)

Rewrite Rules

active(f(x))mark(x)top(active(c))top(mark(c))
top(mark(x))top(check(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))
start(ok(x))found(x)f(found(x))found(f(x))
top(found(x))top(active(x))active(f(x))f(active(x))
f(mark(x))mark(f(x))

Original Signature

Termination of terms over the following signature is verified: f, check, c, start, active, mark, ok, match, proper, X, top, found

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

check#(f(x))check#(x)

Problem 5: PolynomialLinearRange4iUR



Dependency Pair Problem

Dependency Pairs

top#(found(x))top#(active(x))top#(active(c))top#(mark(c))
top#(mark(x))top#(check(x))

Rewrite Rules

active(f(x))mark(x)top(active(c))top(mark(c))
top(mark(x))top(check(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))
start(ok(x))found(x)f(found(x))found(f(x))
top(found(x))top(active(x))active(f(x))f(active(x))
f(mark(x))mark(f(x))

Original Signature

Termination of terms over the following signature is verified: f, check, c, start, active, mark, ok, match, proper, X, top, found

Strategy


Polynomial Interpretation

Improved Usable rules

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

The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:

top#(active(c))top#(mark(c))

Problem 8: PolynomialLinearRange4iUR



Dependency Pair Problem

Dependency Pairs

top#(found(x))top#(active(x))top#(mark(x))top#(check(x))

Rewrite Rules

active(f(x))mark(x)top(active(c))top(mark(c))
top(mark(x))top(check(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))
start(ok(x))found(x)f(found(x))found(f(x))
top(found(x))top(active(x))active(f(x))f(active(x))
f(mark(x))mark(f(x))

Original Signature

Termination of terms over the following signature is verified: f, check, start, c, active, ok, mark, proper, match, X, found, top

Strategy


Polynomial Interpretation

Improved Usable rules

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

The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:

top#(mark(x))top#(check(x))

Problem 9: PolynomialLinearRange4iUR



Dependency Pair Problem

Dependency Pairs

top#(found(x))top#(active(x))

Rewrite Rules

active(f(x))mark(x)top(active(c))top(mark(c))
top(mark(x))top(check(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))
start(ok(x))found(x)f(found(x))found(f(x))
top(found(x))top(active(x))active(f(x))f(active(x))
f(mark(x))mark(f(x))

Original Signature

Termination of terms over the following signature is verified: f, check, c, start, active, mark, ok, match, proper, X, top, found

Strategy


Polynomial Interpretation

Improved Usable rules

f(mark(x))mark(f(x))f(found(x))found(f(x))
f(ok(x))ok(f(x))active(f(x))f(active(x))
active(f(x))mark(x)

The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:

top#(found(x))top#(active(x))

Problem 6: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

active#(f(x))active#(x)

Rewrite Rules

active(f(x))mark(x)top(active(c))top(mark(c))
top(mark(x))top(check(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))
start(ok(x))found(x)f(found(x))found(f(x))
top(found(x))top(active(x))active(f(x))f(active(x))
f(mark(x))mark(f(x))

Original Signature

Termination of terms over the following signature is verified: f, check, c, start, active, mark, ok, match, proper, X, top, found

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

active#(f(x))active#(x)

Problem 7: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

proper#(f(x))proper#(x)

Rewrite Rules

active(f(x))mark(x)top(active(c))top(mark(c))
top(mark(x))top(check(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))
start(ok(x))found(x)f(found(x))found(f(x))
top(found(x))top(active(x))active(f(x))f(active(x))
f(mark(x))mark(f(x))

Original Signature

Termination of terms over the following signature is verified: f, check, c, start, active, mark, ok, match, proper, X, top, found

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

proper#(f(x))proper#(x)