TIMEOUT

The TRS could not be proven terminating. The proof attempt took 60000 ms.

The following DP Processors were used


Problem 1 was processed with processor DependencyGraph (484ms).
 | – Problem 2 was processed with processor SubtermCriterion (1ms).
 | – Problem 3 was processed with processor SubtermCriterion (1ms).
 | – Problem 4 was processed with processor SubtermCriterion (1ms).
 | – Problem 5 was processed with processor SubtermCriterion (1ms).
 | – Problem 6 was processed with processor SubtermCriterion (0ms).
 | – Problem 7 was processed with processor SubtermCriterion (2ms).
 | – Problem 8 was processed with processor ForwardNarrowing (5ms).
 |    | – Problem 9 was processed with processor ForwardNarrowing (6ms).
 |    |    | – Problem 10 was processed with processor ForwardNarrowing (2ms).
 |    |    |    | – Problem 11 was processed with processor ForwardNarrowing (6ms).
 |    |    |    |    | – Problem 12 was processed with processor ForwardNarrowing (7ms).
 |    |    |    |    |    | – Problem 13 was processed with processor ForwardNarrowing (7ms).
 |    |    |    |    |    |    | – Problem 14 was processed with processor ForwardNarrowing (5ms).
 |    |    |    |    |    |    |    | – Problem 15 was processed with processor ForwardNarrowing (5ms).
 |    |    |    |    |    |    |    |    | – Problem 16 was processed with processor ForwardNarrowing (5ms).
 |    |    |    |    |    |    |    |    |    | – Problem 17 was processed with processor ForwardNarrowing (4ms).
 |    |    |    |    |    |    |    |    |    |    | – Problem 18 was processed with processor ForwardNarrowing (7ms).
 |    |    |    |    |    |    |    |    |    |    |    | – Problem 19 was processed with processor ForwardNarrowing (40ms).
 |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 20 was processed with processor ForwardNarrowing (6ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 21 was processed with processor ForwardNarrowing (8ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 22 was processed with processor ForwardNarrowing (8ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 23 was processed with processor ForwardNarrowing (8ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 24 was processed with processor ForwardNarrowing (8ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 25 was processed with processor ForwardNarrowing (9ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 26 was processed with processor ForwardNarrowing (10ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 27 was processed with processor ForwardNarrowing (7ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 28 was processed with processor ForwardNarrowing (38ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 29 remains open; application of the following processors failed [ForwardNarrowing (89ms), ForwardNarrowing (45ms), ForwardNarrowing (125ms), ForwardNarrowing (135ms), ForwardNarrowing (78ms), ForwardNarrowing (86ms), ForwardNarrowing (212ms), ForwardNarrowing (73ms), ForwardNarrowing (179ms), ForwardNarrowing (233ms), ForwardNarrowing (222ms), ForwardNarrowing (90ms), ForwardNarrowing (311ms), ForwardNarrowing (294ms), ForwardNarrowing (313ms), ForwardNarrowing (186ms), ForwardNarrowing (367ms), ForwardNarrowing (329ms), ForwardNarrowing (590ms), ForwardNarrowing (227ms), ForwardNarrowing (188ms), ForwardNarrowing (395ms), ForwardNarrowing (629ms), ForwardNarrowing (284ms), ForwardNarrowing (436ms), ForwardNarrowing (1024ms), ForwardNarrowing (300ms), ForwardNarrowing (500ms), ForwardNarrowing (799ms), ForwardNarrowing (366ms), ForwardNarrowing (236ms), ForwardNarrowing (1017ms), ForwardNarrowing (1270ms), ForwardNarrowing (1222ms), ForwardNarrowing (timeout)].

The following open problems remain:

Open Dependency Pair Problem 8

Dependency Pairs

top#(mark(X)) → top#(proper(X))top#(ok(X)) → top#(active(X))

Rewrite Rules

active(f(0)) → mark(cons(0, f(s(0))))active(f(s(0))) → mark(f(p(s(0))))
active(p(s(X))) → mark(X)active(f(X)) → f(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)active(s(X)) → s(active(X))
active(p(X)) → p(active(X))f(mark(X)) → mark(f(X))
cons(mark(X1), X2) → mark(cons(X1, X2))s(mark(X)) → mark(s(X))
p(mark(X)) → mark(p(X))proper(f(X)) → f(proper(X))
proper(0) → ok(0)proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))proper(p(X)) → p(proper(X))
f(ok(X)) → ok(f(X))cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))p(ok(X)) → ok(p(X))
top(mark(X)) → top(proper(X))top(ok(X)) → top(active(X))

Original Signature

Termination of terms over the following signature is verified: f, 0, s, p, active, ok, mark, proper, top, cons

Problem 1: DependencyGraph

Dependency Pair Problem

Dependency Pairs

proper#(cons(X1, X2)) → proper#(X1)top#(ok(X)) → top#(active(X))
cons#(mark(X1), X2) → cons#(X1, X2)active#(f(s(0))) → f#(p(s(0)))
cons#(ok(X1), ok(X2)) → cons#(X1, X2)p#(mark(X)) → p#(X)
top#(ok(X)) → active#(X)proper#(p(X)) → proper#(X)
active#(cons(X1, X2)) → cons#(active(X1), X2)active#(p(X)) → p#(active(X))
proper#(p(X)) → p#(proper(X))f#(mark(X)) → f#(X)
top#(mark(X)) → proper#(X)f#(ok(X)) → f#(X)
active#(f(0)) → s#(0)proper#(f(X)) → f#(proper(X))
top#(mark(X)) → top#(proper(X))proper#(f(X)) → proper#(X)
active#(f(s(0))) → p#(s(0))proper#(cons(X1, X2)) → proper#(X2)
active#(p(X)) → active#(X)active#(s(X)) → s#(active(X))
s#(ok(X)) → s#(X)active#(f(s(0))) → s#(0)
s#(mark(X)) → s#(X)proper#(s(X)) → proper#(X)
active#(f(X)) → f#(active(X))active#(f(0)) → cons#(0, f(s(0)))
proper#(cons(X1, X2)) → cons#(proper(X1), proper(X2))active#(s(X)) → active#(X)
proper#(s(X)) → s#(proper(X))active#(f(0)) → f#(s(0))
p#(ok(X)) → p#(X)active#(f(X)) → active#(X)
active#(cons(X1, X2)) → active#(X1)

Rewrite Rules

active(f(0)) → mark(cons(0, f(s(0))))active(f(s(0))) → mark(f(p(s(0))))
active(p(s(X))) → mark(X)active(f(X)) → f(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)active(s(X)) → s(active(X))
active(p(X)) → p(active(X))f(mark(X)) → mark(f(X))
cons(mark(X1), X2) → mark(cons(X1, X2))s(mark(X)) → mark(s(X))
p(mark(X)) → mark(p(X))proper(f(X)) → f(proper(X))
proper(0) → ok(0)proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))proper(p(X)) → p(proper(X))
f(ok(X)) → ok(f(X))cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))p(ok(X)) → ok(p(X))
top(mark(X)) → top(proper(X))top(ok(X)) → top(active(X))

Original Signature

Termination of terms over the following signature is verified: f, 0, s, p, active, mark, ok, proper, cons, top

Strategy

The following SCCs where found

f#(mark(X)) → f#(X)f#(ok(X)) → f#(X)
cons#(mark(X1), X2) → cons#(X1, X2)cons#(ok(X1), ok(X2)) → cons#(X1, X2)
p#(ok(X)) → p#(X)p#(mark(X)) → p#(X)
active#(s(X)) → active#(X)active#(p(X)) → active#(X)
active#(f(X)) → active#(X)active#(cons(X1, X2)) → active#(X1)
proper#(s(X)) → proper#(X)proper#(cons(X1, X2)) → proper#(X1)
proper#(f(X)) → proper#(X)proper#(cons(X1, X2)) → proper#(X2)
proper#(p(X)) → proper#(X)
top#(mark(X)) → top#(proper(X))top#(ok(X)) → top#(active(X))
s#(mark(X)) → s#(X)s#(ok(X)) → s#(X)

Problem 2: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

s#(mark(X)) → s#(X)s#(ok(X)) → s#(X)

Rewrite Rules

active(f(0)) → mark(cons(0, f(s(0))))active(f(s(0))) → mark(f(p(s(0))))
active(p(s(X))) → mark(X)active(f(X)) → f(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)active(s(X)) → s(active(X))
active(p(X)) → p(active(X))f(mark(X)) → mark(f(X))
cons(mark(X1), X2) → mark(cons(X1, X2))s(mark(X)) → mark(s(X))
p(mark(X)) → mark(p(X))proper(f(X)) → f(proper(X))
proper(0) → ok(0)proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))proper(p(X)) → p(proper(X))
f(ok(X)) → ok(f(X))cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))p(ok(X)) → ok(p(X))
top(mark(X)) → top(proper(X))top(ok(X)) → top(active(X))

Original Signature

Termination of terms over the following signature is verified: f, 0, s, p, active, mark, ok, proper, cons, top

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

s#(mark(X)) → s#(X)s#(ok(X)) → s#(X)

Problem 3: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

proper#(s(X)) → proper#(X)proper#(cons(X1, X2)) → proper#(X1)
proper#(f(X)) → proper#(X)proper#(cons(X1, X2)) → proper#(X2)
proper#(p(X)) → proper#(X)

Rewrite Rules

active(f(0)) → mark(cons(0, f(s(0))))active(f(s(0))) → mark(f(p(s(0))))
active(p(s(X))) → mark(X)active(f(X)) → f(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)active(s(X)) → s(active(X))
active(p(X)) → p(active(X))f(mark(X)) → mark(f(X))
cons(mark(X1), X2) → mark(cons(X1, X2))s(mark(X)) → mark(s(X))
p(mark(X)) → mark(p(X))proper(f(X)) → f(proper(X))
proper(0) → ok(0)proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))proper(p(X)) → p(proper(X))
f(ok(X)) → ok(f(X))cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))p(ok(X)) → ok(p(X))
top(mark(X)) → top(proper(X))top(ok(X)) → top(active(X))

Original Signature

Termination of terms over the following signature is verified: f, 0, s, p, active, mark, ok, proper, cons, top

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

proper#(s(X)) → proper#(X)proper#(cons(X1, X2)) → proper#(X1)
proper#(f(X)) → proper#(X)proper#(cons(X1, X2)) → proper#(X2)
proper#(p(X)) → proper#(X)

Problem 4: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

active#(s(X)) → active#(X)active#(p(X)) → active#(X)
active#(f(X)) → active#(X)active#(cons(X1, X2)) → active#(X1)

Rewrite Rules

active(f(0)) → mark(cons(0, f(s(0))))active(f(s(0))) → mark(f(p(s(0))))
active(p(s(X))) → mark(X)active(f(X)) → f(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)active(s(X)) → s(active(X))
active(p(X)) → p(active(X))f(mark(X)) → mark(f(X))
cons(mark(X1), X2) → mark(cons(X1, X2))s(mark(X)) → mark(s(X))
p(mark(X)) → mark(p(X))proper(f(X)) → f(proper(X))
proper(0) → ok(0)proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))proper(p(X)) → p(proper(X))
f(ok(X)) → ok(f(X))cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))p(ok(X)) → ok(p(X))
top(mark(X)) → top(proper(X))top(ok(X)) → top(active(X))

Original Signature

Termination of terms over the following signature is verified: f, 0, s, p, active, mark, ok, proper, cons, top

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

active#(p(X)) → active#(X)active#(s(X)) → active#(X)
active#(f(X)) → active#(X)active#(cons(X1, X2)) → active#(X1)

Problem 5: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

cons#(mark(X1), X2) → cons#(X1, X2)cons#(ok(X1), ok(X2)) → cons#(X1, X2)

Rewrite Rules

active(f(0)) → mark(cons(0, f(s(0))))active(f(s(0))) → mark(f(p(s(0))))
active(p(s(X))) → mark(X)active(f(X)) → f(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)active(s(X)) → s(active(X))
active(p(X)) → p(active(X))f(mark(X)) → mark(f(X))
cons(mark(X1), X2) → mark(cons(X1, X2))s(mark(X)) → mark(s(X))
p(mark(X)) → mark(p(X))proper(f(X)) → f(proper(X))
proper(0) → ok(0)proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))proper(p(X)) → p(proper(X))
f(ok(X)) → ok(f(X))cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))p(ok(X)) → ok(p(X))
top(mark(X)) → top(proper(X))top(ok(X)) → top(active(X))

Original Signature

Termination of terms over the following signature is verified: f, 0, s, p, active, mark, ok, proper, cons, top

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

cons#(mark(X1), X2) → cons#(X1, X2)cons#(ok(X1), ok(X2)) → cons#(X1, X2)

Problem 6: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

f#(mark(X)) → f#(X)f#(ok(X)) → f#(X)

Rewrite Rules

active(f(0)) → mark(cons(0, f(s(0))))active(f(s(0))) → mark(f(p(s(0))))
active(p(s(X))) → mark(X)active(f(X)) → f(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)active(s(X)) → s(active(X))
active(p(X)) → p(active(X))f(mark(X)) → mark(f(X))
cons(mark(X1), X2) → mark(cons(X1, X2))s(mark(X)) → mark(s(X))
p(mark(X)) → mark(p(X))proper(f(X)) → f(proper(X))
proper(0) → ok(0)proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))proper(p(X)) → p(proper(X))
f(ok(X)) → ok(f(X))cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))p(ok(X)) → ok(p(X))
top(mark(X)) → top(proper(X))top(ok(X)) → top(active(X))

Original Signature

Termination of terms over the following signature is verified: f, 0, s, p, active, mark, ok, proper, cons, top

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

f#(mark(X)) → f#(X)f#(ok(X)) → f#(X)

Problem 7: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

p#(ok(X)) → p#(X)p#(mark(X)) → p#(X)

Rewrite Rules

active(f(0)) → mark(cons(0, f(s(0))))active(f(s(0))) → mark(f(p(s(0))))
active(p(s(X))) → mark(X)active(f(X)) → f(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)active(s(X)) → s(active(X))
active(p(X)) → p(active(X))f(mark(X)) → mark(f(X))
cons(mark(X1), X2) → mark(cons(X1, X2))s(mark(X)) → mark(s(X))
p(mark(X)) → mark(p(X))proper(f(X)) → f(proper(X))
proper(0) → ok(0)proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))proper(p(X)) → p(proper(X))
f(ok(X)) → ok(f(X))cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))p(ok(X)) → ok(p(X))
top(mark(X)) → top(proper(X))top(ok(X)) → top(active(X))

Original Signature

Termination of terms over the following signature is verified: f, 0, s, p, active, mark, ok, proper, cons, top

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

p#(ok(X)) → p#(X)p#(mark(X)) → p#(X)

Problem 8: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

top#(mark(X)) → top#(proper(X))top#(ok(X)) → top#(active(X))

Rewrite Rules

active(f(0)) → mark(cons(0, f(s(0))))active(f(s(0))) → mark(f(p(s(0))))
active(p(s(X))) → mark(X)active(f(X)) → f(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)active(s(X)) → s(active(X))
active(p(X)) → p(active(X))f(mark(X)) → mark(f(X))
cons(mark(X1), X2) → mark(cons(X1, X2))s(mark(X)) → mark(s(X))
p(mark(X)) → mark(p(X))proper(f(X)) → f(proper(X))
proper(0) → ok(0)proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))proper(p(X)) → p(proper(X))
f(ok(X)) → ok(f(X))cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))p(ok(X)) → ok(p(X))
top(mark(X)) → top(proper(X))top(ok(X)) → top(active(X))

Original Signature

Termination of terms over the following signature is verified: f, 0, s, p, active, mark, ok, proper, cons, top

Strategy

The right-hand side of the rule top#(mark(X)) → top#(proper(X)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
top#(ok(0)) 
top#(f(proper(_x21))) 
top#(p(proper(_x21))) 
top#(cons(proper(_x21), proper(_x22))) 
top#(s(proper(_x21))) 
Thus, the rule top#(mark(X)) → top#(proper(X)) is replaced by the following rules:
top#(mark(0)) → top#(ok(0))top#(mark(f(_x21))) → top#(f(proper(_x21)))
top#(mark(cons(_x21, _x22))) → top#(cons(proper(_x21), proper(_x22)))top#(mark(s(_x21))) → top#(s(proper(_x21)))
top#(mark(p(_x21))) → top#(p(proper(_x21)))

Problem 9: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

top#(mark(0)) → top#(ok(0))top#(ok(X)) → top#(active(X))
top#(mark(f(_x21))) → top#(f(proper(_x21)))top#(mark(cons(_x21, _x22))) → top#(cons(proper(_x21), proper(_x22)))
top#(mark(s(_x21))) → top#(s(proper(_x21)))top#(mark(p(_x21))) → top#(p(proper(_x21)))

Rewrite Rules

active(f(0)) → mark(cons(0, f(s(0))))active(f(s(0))) → mark(f(p(s(0))))
active(p(s(X))) → mark(X)active(f(X)) → f(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)active(s(X)) → s(active(X))
active(p(X)) → p(active(X))f(mark(X)) → mark(f(X))
cons(mark(X1), X2) → mark(cons(X1, X2))s(mark(X)) → mark(s(X))
p(mark(X)) → mark(p(X))proper(f(X)) → f(proper(X))
proper(0) → ok(0)proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))proper(p(X)) → p(proper(X))
f(ok(X)) → ok(f(X))cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))p(ok(X)) → ok(p(X))
top(mark(X)) → top(proper(X))top(ok(X)) → top(active(X))

Original Signature

Termination of terms over the following signature is verified: f, 0, s, p, active, ok, mark, proper, top, cons

Strategy

The right-hand side of the rule top#(ok(X)) → top#(active(X)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
top#(s(active(_x21))) 
top#(mark(_x21)) 
top#(p(active(_x21))) 
top#(cons(active(_x21), _x22)) 
top#(f(active(_x21))) 
top#(mark(cons(0, f(s(0))))) 
top#(mark(f(p(s(0))))) 
Thus, the rule top#(ok(X)) → top#(active(X)) is replaced by the following rules:
top#(ok(f(0))) → top#(mark(cons(0, f(s(0)))))top#(ok(f(s(0)))) → top#(mark(f(p(s(0)))))
top#(ok(s(_x21))) → top#(s(active(_x21)))top#(ok(cons(_x21, _x22))) → top#(cons(active(_x21), _x22))
top#(ok(f(_x21))) → top#(f(active(_x21)))top#(ok(p(_x21))) → top#(p(active(_x21)))
top#(ok(p(s(_x21)))) → top#(mark(_x21))

Problem 10: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

top#(mark(0)) → top#(ok(0))top#(ok(f(0))) → top#(mark(cons(0, f(s(0)))))
top#(ok(f(s(0)))) → top#(mark(f(p(s(0)))))top#(ok(s(_x21))) → top#(s(active(_x21)))
top#(mark(f(_x21))) → top#(f(proper(_x21)))top#(mark(s(_x21))) → top#(s(proper(_x21)))
top#(mark(cons(_x21, _x22))) → top#(cons(proper(_x21), proper(_x22)))top#(ok(cons(_x21, _x22))) → top#(cons(active(_x21), _x22))
top#(mark(p(_x21))) → top#(p(proper(_x21)))top#(ok(f(_x21))) → top#(f(active(_x21)))
top#(ok(p(_x21))) → top#(p(active(_x21)))top#(ok(p(s(_x21)))) → top#(mark(_x21))

Rewrite Rules

active(f(0)) → mark(cons(0, f(s(0))))active(f(s(0))) → mark(f(p(s(0))))
active(p(s(X))) → mark(X)active(f(X)) → f(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)active(s(X)) → s(active(X))
active(p(X)) → p(active(X))f(mark(X)) → mark(f(X))
cons(mark(X1), X2) → mark(cons(X1, X2))s(mark(X)) → mark(s(X))
p(mark(X)) → mark(p(X))proper(f(X)) → f(proper(X))
proper(0) → ok(0)proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))proper(p(X)) → p(proper(X))
f(ok(X)) → ok(f(X))cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))p(ok(X)) → ok(p(X))
top(mark(X)) → top(proper(X))top(ok(X)) → top(active(X))

Original Signature

Termination of terms over the following signature is verified: f, 0, s, p, active, mark, ok, proper, cons, top

Strategy

The right-hand side of the rule top#(mark(0)) → top#(ok(0)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
Thus, the rule top#(mark(0)) → top#(ok(0)) is deleted.

Problem 11: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

top#(ok(f(s(0)))) → top#(mark(f(p(s(0)))))top#(ok(f(0))) → top#(mark(cons(0, f(s(0)))))
top#(ok(s(_x21))) → top#(s(active(_x21)))top#(mark(f(_x21))) → top#(f(proper(_x21)))
top#(ok(cons(_x21, _x22))) → top#(cons(active(_x21), _x22))top#(mark(cons(_x21, _x22))) → top#(cons(proper(_x21), proper(_x22)))
top#(mark(s(_x21))) → top#(s(proper(_x21)))top#(ok(p(s(_x21)))) → top#(mark(_x21))
top#(ok(p(_x21))) → top#(p(active(_x21)))top#(ok(f(_x21))) → top#(f(active(_x21)))
top#(mark(p(_x21))) → top#(p(proper(_x21)))

Rewrite Rules

active(f(0)) → mark(cons(0, f(s(0))))active(f(s(0))) → mark(f(p(s(0))))
active(p(s(X))) → mark(X)active(f(X)) → f(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)active(s(X)) → s(active(X))
active(p(X)) → p(active(X))f(mark(X)) → mark(f(X))
cons(mark(X1), X2) → mark(cons(X1, X2))s(mark(X)) → mark(s(X))
p(mark(X)) → mark(p(X))proper(f(X)) → f(proper(X))
proper(0) → ok(0)proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))proper(p(X)) → p(proper(X))
f(ok(X)) → ok(f(X))cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))p(ok(X)) → ok(p(X))
top(mark(X)) → top(proper(X))top(ok(X)) → top(active(X))

Original Signature

Termination of terms over the following signature is verified: f, 0, s, p, active, ok, mark, proper, top, cons

Strategy

The right-hand side of the rule top#(ok(s(_x21))) → top#(s(active(_x21))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
top#(s(mark(cons(0, f(s(0)))))) 
top#(s(cons(active(_x41), _x42))) 
top#(s(mark(f(p(s(0)))))) 
top#(s(f(active(_x41)))) 
top#(s(mark(_x41))) 
top#(s(p(active(_x41)))) 
top#(s(s(active(_x41)))) 
Thus, the rule top#(ok(s(_x21))) → top#(s(active(_x21))) is replaced by the following rules:
top#(ok(s(p(s(_x41))))) → top#(s(mark(_x41)))top#(ok(s(cons(_x41, _x42)))) → top#(s(cons(active(_x41), _x42)))
top#(ok(s(f(0)))) → top#(s(mark(cons(0, f(s(0))))))top#(ok(s(f(_x41)))) → top#(s(f(active(_x41))))
top#(ok(s(s(_x41)))) → top#(s(s(active(_x41))))top#(ok(s(p(_x41)))) → top#(s(p(active(_x41))))
top#(ok(s(f(s(0))))) → top#(s(mark(f(p(s(0))))))

Problem 12: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

top#(ok(s(f(0)))) → top#(s(mark(cons(0, f(s(0))))))top#(ok(s(f(_x41)))) → top#(s(f(active(_x41))))
top#(ok(s(f(s(0))))) → top#(s(mark(f(p(s(0))))))top#(ok(s(p(_x41)))) → top#(s(p(active(_x41))))
top#(mark(s(_x21))) → top#(s(proper(_x21)))top#(mark(cons(_x21, _x22))) → top#(cons(proper(_x21), proper(_x22)))
top#(ok(f(_x21))) → top#(f(active(_x21)))top#(ok(p(_x21))) → top#(p(active(_x21)))
top#(ok(s(p(s(_x41))))) → top#(s(mark(_x41)))top#(ok(s(cons(_x41, _x42)))) → top#(s(cons(active(_x41), _x42)))
top#(ok(f(0))) → top#(mark(cons(0, f(s(0)))))top#(ok(f(s(0)))) → top#(mark(f(p(s(0)))))
top#(ok(s(s(_x41)))) → top#(s(s(active(_x41))))top#(mark(f(_x21))) → top#(f(proper(_x21)))
top#(ok(cons(_x21, _x22))) → top#(cons(active(_x21), _x22))top#(ok(p(s(_x21)))) → top#(mark(_x21))
top#(mark(p(_x21))) → top#(p(proper(_x21)))

Rewrite Rules

active(f(0)) → mark(cons(0, f(s(0))))active(f(s(0))) → mark(f(p(s(0))))
active(p(s(X))) → mark(X)active(f(X)) → f(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)active(s(X)) → s(active(X))
active(p(X)) → p(active(X))f(mark(X)) → mark(f(X))
cons(mark(X1), X2) → mark(cons(X1, X2))s(mark(X)) → mark(s(X))
p(mark(X)) → mark(p(X))proper(f(X)) → f(proper(X))
proper(0) → ok(0)proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))proper(p(X)) → p(proper(X))
f(ok(X)) → ok(f(X))cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))p(ok(X)) → ok(p(X))
top(mark(X)) → top(proper(X))top(ok(X)) → top(active(X))

Original Signature

Termination of terms over the following signature is verified: f, 0, s, p, active, mark, ok, proper, cons, top

Strategy

The right-hand side of the rule top#(ok(s(f(0)))) → top#(s(mark(cons(0, f(s(0)))))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
top#(mark(s(cons(0, f(s(0)))))) 
Thus, the rule top#(ok(s(f(0)))) → top#(s(mark(cons(0, f(s(0)))))) is replaced by the following rules:
top#(ok(s(f(0)))) → top#(mark(s(cons(0, f(s(0))))))

Problem 13: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

top#(ok(s(f(0)))) → top#(mark(s(cons(0, f(s(0))))))top#(ok(s(f(_x41)))) → top#(s(f(active(_x41))))
top#(ok(s(f(s(0))))) → top#(s(mark(f(p(s(0))))))top#(ok(s(p(_x41)))) → top#(s(p(active(_x41))))
top#(mark(s(_x21))) → top#(s(proper(_x21)))top#(mark(cons(_x21, _x22))) → top#(cons(proper(_x21), proper(_x22)))
top#(ok(f(_x21))) → top#(f(active(_x21)))top#(ok(p(_x21))) → top#(p(active(_x21)))
top#(ok(s(p(s(_x41))))) → top#(s(mark(_x41)))top#(ok(s(cons(_x41, _x42)))) → top#(s(cons(active(_x41), _x42)))
top#(ok(f(0))) → top#(mark(cons(0, f(s(0)))))top#(ok(f(s(0)))) → top#(mark(f(p(s(0)))))
top#(ok(s(s(_x41)))) → top#(s(s(active(_x41))))top#(mark(f(_x21))) → top#(f(proper(_x21)))
top#(ok(cons(_x21, _x22))) → top#(cons(active(_x21), _x22))top#(mark(p(_x21))) → top#(p(proper(_x21)))
top#(ok(p(s(_x21)))) → top#(mark(_x21))

Rewrite Rules

active(f(0)) → mark(cons(0, f(s(0))))active(f(s(0))) → mark(f(p(s(0))))
active(p(s(X))) → mark(X)active(f(X)) → f(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)active(s(X)) → s(active(X))
active(p(X)) → p(active(X))f(mark(X)) → mark(f(X))
cons(mark(X1), X2) → mark(cons(X1, X2))s(mark(X)) → mark(s(X))
p(mark(X)) → mark(p(X))proper(f(X)) → f(proper(X))
proper(0) → ok(0)proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))proper(p(X)) → p(proper(X))
f(ok(X)) → ok(f(X))cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))p(ok(X)) → ok(p(X))
top(mark(X)) → top(proper(X))top(ok(X)) → top(active(X))

Original Signature

Termination of terms over the following signature is verified: f, 0, s, p, active, ok, mark, proper, top, cons

Strategy

The right-hand side of the rule top#(ok(s(f(_x41)))) → top#(s(f(active(_x41)))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
top#(s(f(cons(active(_x51), _x52)))) 
top#(s(f(mark(cons(0, f(s(0))))))) 
top#(s(f(p(active(_x51))))) 
top#(s(f(f(active(_x51))))) 
top#(s(f(s(active(_x51))))) 
top#(s(f(mark(f(p(s(0))))))) 
top#(s(f(mark(_x51)))) 
Thus, the rule top#(ok(s(f(_x41)))) → top#(s(f(active(_x41)))) is replaced by the following rules:
top#(ok(s(f(f(_x51))))) → top#(s(f(f(active(_x51)))))top#(ok(s(f(p(s(_x51)))))) → top#(s(f(mark(_x51))))
top#(ok(s(f(p(_x51))))) → top#(s(f(p(active(_x51)))))top#(ok(s(f(f(0))))) → top#(s(f(mark(cons(0, f(s(0)))))))
top#(ok(s(f(cons(_x51, _x52))))) → top#(s(f(cons(active(_x51), _x52))))top#(ok(s(f(f(s(0)))))) → top#(s(f(mark(f(p(s(0)))))))
top#(ok(s(f(s(_x51))))) → top#(s(f(s(active(_x51)))))

Problem 14: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

top#(ok(s(f(0)))) → top#(mark(s(cons(0, f(s(0))))))top#(ok(s(f(p(s(_x51)))))) → top#(s(f(mark(_x51))))
top#(ok(s(f(s(0))))) → top#(s(mark(f(p(s(0))))))top#(ok(s(p(_x41)))) → top#(s(p(active(_x41))))
top#(mark(s(_x21))) → top#(s(proper(_x21)))top#(mark(cons(_x21, _x22))) → top#(cons(proper(_x21), proper(_x22)))
top#(ok(f(_x21))) → top#(f(active(_x21)))top#(ok(p(_x21))) → top#(p(active(_x21)))
top#(ok(s(f(s(_x51))))) → top#(s(f(s(active(_x51)))))top#(ok(s(p(s(_x41))))) → top#(s(mark(_x41)))
top#(ok(s(cons(_x41, _x42)))) → top#(s(cons(active(_x41), _x42)))top#(ok(f(0))) → top#(mark(cons(0, f(s(0)))))
top#(ok(f(s(0)))) → top#(mark(f(p(s(0)))))top#(ok(s(f(f(_x51))))) → top#(s(f(f(active(_x51)))))
top#(ok(s(s(_x41)))) → top#(s(s(active(_x41))))top#(ok(s(f(p(_x51))))) → top#(s(f(p(active(_x51)))))
top#(ok(s(f(f(0))))) → top#(s(f(mark(cons(0, f(s(0)))))))top#(mark(f(_x21))) → top#(f(proper(_x21)))
top#(ok(cons(_x21, _x22))) → top#(cons(active(_x21), _x22))top#(ok(p(s(_x21)))) → top#(mark(_x21))
top#(mark(p(_x21))) → top#(p(proper(_x21)))top#(ok(s(f(cons(_x51, _x52))))) → top#(s(f(cons(active(_x51), _x52))))
top#(ok(s(f(f(s(0)))))) → top#(s(f(mark(f(p(s(0)))))))

Rewrite Rules

active(f(0)) → mark(cons(0, f(s(0))))active(f(s(0))) → mark(f(p(s(0))))
active(p(s(X))) → mark(X)active(f(X)) → f(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)active(s(X)) → s(active(X))
active(p(X)) → p(active(X))f(mark(X)) → mark(f(X))
cons(mark(X1), X2) → mark(cons(X1, X2))s(mark(X)) → mark(s(X))
p(mark(X)) → mark(p(X))proper(f(X)) → f(proper(X))
proper(0) → ok(0)proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))proper(p(X)) → p(proper(X))
f(ok(X)) → ok(f(X))cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))p(ok(X)) → ok(p(X))
top(mark(X)) → top(proper(X))top(ok(X)) → top(active(X))

Original Signature

Termination of terms over the following signature is verified: f, 0, s, p, active, mark, ok, proper, cons, top

Strategy

The right-hand side of the rule top#(ok(s(f(p(s(_x51)))))) → top#(s(f(mark(_x51)))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
top#(s(mark(f(_x31)))) 
Thus, the rule top#(ok(s(f(p(s(_x51)))))) → top#(s(f(mark(_x51)))) is replaced by the following rules:
top#(ok(s(f(p(s(_x31)))))) → top#(s(mark(f(_x31))))

Problem 15: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

top#(ok(s(f(p(s(_x31)))))) → top#(s(mark(f(_x31))))top#(ok(s(f(0)))) → top#(mark(s(cons(0, f(s(0))))))
top#(ok(s(f(s(0))))) → top#(s(mark(f(p(s(0))))))top#(ok(s(p(_x41)))) → top#(s(p(active(_x41))))
top#(mark(s(_x21))) → top#(s(proper(_x21)))top#(mark(cons(_x21, _x22))) → top#(cons(proper(_x21), proper(_x22)))
top#(ok(f(_x21))) → top#(f(active(_x21)))top#(ok(p(_x21))) → top#(p(active(_x21)))
top#(ok(s(f(s(_x51))))) → top#(s(f(s(active(_x51)))))top#(ok(s(p(s(_x41))))) → top#(s(mark(_x41)))
top#(ok(s(cons(_x41, _x42)))) → top#(s(cons(active(_x41), _x42)))top#(ok(f(0))) → top#(mark(cons(0, f(s(0)))))
top#(ok(f(s(0)))) → top#(mark(f(p(s(0)))))top#(ok(s(f(f(_x51))))) → top#(s(f(f(active(_x51)))))
top#(ok(s(f(f(0))))) → top#(s(f(mark(cons(0, f(s(0)))))))top#(ok(s(f(p(_x51))))) → top#(s(f(p(active(_x51)))))
top#(ok(s(s(_x41)))) → top#(s(s(active(_x41))))top#(mark(f(_x21))) → top#(f(proper(_x21)))
top#(ok(cons(_x21, _x22))) → top#(cons(active(_x21), _x22))top#(mark(p(_x21))) → top#(p(proper(_x21)))
top#(ok(p(s(_x21)))) → top#(mark(_x21))top#(ok(s(f(cons(_x51, _x52))))) → top#(s(f(cons(active(_x51), _x52))))
top#(ok(s(f(f(s(0)))))) → top#(s(f(mark(f(p(s(0)))))))

Rewrite Rules

active(f(0)) → mark(cons(0, f(s(0))))active(f(s(0))) → mark(f(p(s(0))))
active(p(s(X))) → mark(X)active(f(X)) → f(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)active(s(X)) → s(active(X))
active(p(X)) → p(active(X))f(mark(X)) → mark(f(X))
cons(mark(X1), X2) → mark(cons(X1, X2))s(mark(X)) → mark(s(X))
p(mark(X)) → mark(p(X))proper(f(X)) → f(proper(X))
proper(0) → ok(0)proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))proper(p(X)) → p(proper(X))
f(ok(X)) → ok(f(X))cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))p(ok(X)) → ok(p(X))
top(mark(X)) → top(proper(X))top(ok(X)) → top(active(X))

Original Signature

Termination of terms over the following signature is verified: f, 0, s, p, active, ok, mark, proper, top, cons

Strategy

The right-hand side of the rule top#(ok(s(f(p(s(_x31)))))) → top#(s(mark(f(_x31)))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
top#(s(mark(ok(f(_x51))))) 
top#(s(mark(mark(f(_x51))))) 
top#(mark(s(f(_x31)))) 
Thus, the rule top#(ok(s(f(p(s(_x31)))))) → top#(s(mark(f(_x31)))) is replaced by the following rules:
top#(ok(s(f(p(s(mark(_x51))))))) → top#(s(mark(mark(f(_x51)))))top#(ok(s(f(p(s(ok(_x51))))))) → top#(s(mark(ok(f(_x51)))))
top#(ok(s(f(p(s(_x31)))))) → top#(mark(s(f(_x31))))

Problem 16: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

top#(ok(s(f(0)))) → top#(mark(s(cons(0, f(s(0))))))top#(ok(s(f(p(s(mark(_x51))))))) → top#(s(mark(mark(f(_x51)))))
top#(ok(s(f(p(s(ok(_x51))))))) → top#(s(mark(ok(f(_x51)))))top#(ok(s(p(_x41)))) → top#(s(p(active(_x41))))
top#(ok(f(0))) → top#(mark(cons(0, f(s(0)))))top#(ok(s(f(f(0))))) → top#(s(f(mark(cons(0, f(s(0)))))))
top#(ok(s(f(p(_x51))))) → top#(s(f(p(active(_x51)))))top#(ok(s(s(_x41)))) → top#(s(s(active(_x41))))
top#(mark(f(_x21))) → top#(f(proper(_x21)))top#(ok(cons(_x21, _x22))) → top#(cons(active(_x21), _x22))
top#(mark(p(_x21))) → top#(p(proper(_x21)))top#(ok(p(s(_x21)))) → top#(mark(_x21))
top#(ok(s(f(cons(_x51, _x52))))) → top#(s(f(cons(active(_x51), _x52))))top#(ok(s(f(f(s(0)))))) → top#(s(f(mark(f(p(s(0)))))))
top#(ok(s(f(s(0))))) → top#(s(mark(f(p(s(0))))))top#(mark(cons(_x21, _x22))) → top#(cons(proper(_x21), proper(_x22)))
top#(mark(s(_x21))) → top#(s(proper(_x21)))top#(ok(p(_x21))) → top#(p(active(_x21)))
top#(ok(f(_x21))) → top#(f(active(_x21)))top#(ok(s(f(p(s(_x31)))))) → top#(mark(s(f(_x31))))
top#(ok(s(f(s(_x51))))) → top#(s(f(s(active(_x51)))))top#(ok(s(p(s(_x41))))) → top#(s(mark(_x41)))
top#(ok(s(cons(_x41, _x42)))) → top#(s(cons(active(_x41), _x42)))top#(ok(f(s(0)))) → top#(mark(f(p(s(0)))))
top#(ok(s(f(f(_x51))))) → top#(s(f(f(active(_x51)))))

Rewrite Rules

active(f(0)) → mark(cons(0, f(s(0))))active(f(s(0))) → mark(f(p(s(0))))
active(p(s(X))) → mark(X)active(f(X)) → f(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)active(s(X)) → s(active(X))
active(p(X)) → p(active(X))f(mark(X)) → mark(f(X))
cons(mark(X1), X2) → mark(cons(X1, X2))s(mark(X)) → mark(s(X))
p(mark(X)) → mark(p(X))proper(f(X)) → f(proper(X))
proper(0) → ok(0)proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))proper(p(X)) → p(proper(X))
f(ok(X)) → ok(f(X))cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))p(ok(X)) → ok(p(X))
top(mark(X)) → top(proper(X))top(ok(X)) → top(active(X))

Original Signature

Termination of terms over the following signature is verified: f, 0, s, p, active, mark, ok, proper, cons, top

Strategy

The right-hand side of the rule top#(ok(s(f(p(s(mark(_x51))))))) → top#(s(mark(mark(f(_x51))))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
top#(s(mark(mark(mark(f(_x61)))))) 
top#(mark(s(mark(f(_x51))))) 
top#(s(mark(mark(ok(f(_x61)))))) 
Thus, the rule top#(ok(s(f(p(s(mark(_x51))))))) → top#(s(mark(mark(f(_x51))))) is replaced by the following rules:
top#(ok(s(f(p(s(mark(mark(_x61)))))))) → top#(s(mark(mark(mark(f(_x61))))))top#(ok(s(f(p(s(mark(_x51))))))) → top#(mark(s(mark(f(_x51)))))
top#(ok(s(f(p(s(mark(ok(_x61)))))))) → top#(s(mark(mark(ok(f(_x61))))))

Problem 17: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

top#(ok(s(f(0)))) → top#(mark(s(cons(0, f(s(0))))))top#(ok(s(p(_x41)))) → top#(s(p(active(_x41))))
top#(ok(s(f(p(s(ok(_x51))))))) → top#(s(mark(ok(f(_x51)))))top#(ok(f(0))) → top#(mark(cons(0, f(s(0)))))
top#(ok(s(s(_x41)))) → top#(s(s(active(_x41))))top#(ok(s(f(p(_x51))))) → top#(s(f(p(active(_x51)))))
top#(ok(s(f(f(0))))) → top#(s(f(mark(cons(0, f(s(0)))))))top#(mark(f(_x21))) → top#(f(proper(_x21)))
top#(ok(cons(_x21, _x22))) → top#(cons(active(_x21), _x22))top#(ok(p(s(_x21)))) → top#(mark(_x21))
top#(mark(p(_x21))) → top#(p(proper(_x21)))top#(ok(s(f(cons(_x51, _x52))))) → top#(s(f(cons(active(_x51), _x52))))
top#(ok(s(f(f(s(0)))))) → top#(s(f(mark(f(p(s(0)))))))top#(ok(s(f(s(0))))) → top#(s(mark(f(p(s(0))))))
top#(mark(cons(_x21, _x22))) → top#(cons(proper(_x21), proper(_x22)))top#(mark(s(_x21))) → top#(s(proper(_x21)))
top#(ok(p(_x21))) → top#(p(active(_x21)))top#(ok(f(_x21))) → top#(f(active(_x21)))
top#(ok(s(f(p(s(mark(ok(_x61)))))))) → top#(s(mark(mark(ok(f(_x61))))))top#(ok(s(f(p(s(mark(_x51))))))) → top#(mark(s(mark(f(_x51)))))
top#(ok(s(f(p(s(_x31)))))) → top#(mark(s(f(_x31))))top#(ok(s(f(s(_x51))))) → top#(s(f(s(active(_x51)))))
top#(ok(s(p(s(_x41))))) → top#(s(mark(_x41)))top#(ok(s(cons(_x41, _x42)))) → top#(s(cons(active(_x41), _x42)))
top#(ok(f(s(0)))) → top#(mark(f(p(s(0)))))top#(ok(s(f(p(s(mark(mark(_x61)))))))) → top#(s(mark(mark(mark(f(_x61))))))
top#(ok(s(f(f(_x51))))) → top#(s(f(f(active(_x51)))))

Rewrite Rules

active(f(0)) → mark(cons(0, f(s(0))))active(f(s(0))) → mark(f(p(s(0))))
active(p(s(X))) → mark(X)active(f(X)) → f(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)active(s(X)) → s(active(X))
active(p(X)) → p(active(X))f(mark(X)) → mark(f(X))
cons(mark(X1), X2) → mark(cons(X1, X2))s(mark(X)) → mark(s(X))
p(mark(X)) → mark(p(X))proper(f(X)) → f(proper(X))
proper(0) → ok(0)proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))proper(p(X)) → p(proper(X))
f(ok(X)) → ok(f(X))cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))p(ok(X)) → ok(p(X))
top(mark(X)) → top(proper(X))top(ok(X)) → top(active(X))

Original Signature

Termination of terms over the following signature is verified: f, 0, s, p, active, ok, mark, proper, top, cons

Strategy

The right-hand side of the rule top#(ok(s(f(p(s(ok(_x51))))))) → top#(s(mark(ok(f(_x51))))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
top#(s(mark(ok(mark(f(_x61)))))) 
top#(mark(s(ok(f(_x51))))) 
top#(s(mark(ok(ok(f(_x61)))))) 
Thus, the rule top#(ok(s(f(p(s(ok(_x51))))))) → top#(s(mark(ok(f(_x51))))) is replaced by the following rules:
top#(ok(s(f(p(s(ok(ok(_x61)))))))) → top#(s(mark(ok(ok(f(_x61))))))top#(ok(s(f(p(s(ok(mark(_x61)))))))) → top#(s(mark(ok(mark(f(_x61))))))
top#(ok(s(f(p(s(ok(_x51))))))) → top#(mark(s(ok(f(_x51)))))

Problem 18: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

top#(ok(s(f(0)))) → top#(mark(s(cons(0, f(s(0))))))top#(ok(s(p(_x41)))) → top#(s(p(active(_x41))))
top#(ok(f(0))) → top#(mark(cons(0, f(s(0)))))top#(ok(s(f(f(0))))) → top#(s(f(mark(cons(0, f(s(0)))))))
top#(ok(s(f(p(_x51))))) → top#(s(f(p(active(_x51)))))top#(ok(s(s(_x41)))) → top#(s(s(active(_x41))))
top#(mark(f(_x21))) → top#(f(proper(_x21)))top#(ok(s(f(p(s(ok(mark(_x61)))))))) → top#(s(mark(ok(mark(f(_x61))))))
top#(ok(cons(_x21, _x22))) → top#(cons(active(_x21), _x22))top#(mark(p(_x21))) → top#(p(proper(_x21)))
top#(ok(p(s(_x21)))) → top#(mark(_x21))top#(ok(s(f(cons(_x51, _x52))))) → top#(s(f(cons(active(_x51), _x52))))
top#(ok(s(f(p(s(ok(_x51))))))) → top#(mark(s(ok(f(_x51)))))top#(ok(s(f(f(s(0)))))) → top#(s(f(mark(f(p(s(0)))))))
top#(ok(s(f(s(0))))) → top#(s(mark(f(p(s(0))))))top#(mark(cons(_x21, _x22))) → top#(cons(proper(_x21), proper(_x22)))
top#(mark(s(_x21))) → top#(s(proper(_x21)))top#(ok(p(_x21))) → top#(p(active(_x21)))
top#(ok(f(_x21))) → top#(f(active(_x21)))top#(ok(s(f(p(s(mark(ok(_x61)))))))) → top#(s(mark(mark(ok(f(_x61))))))
top#(ok(s(f(p(s(mark(_x51))))))) → top#(mark(s(mark(f(_x51)))))top#(ok(s(f(p(s(_x31)))))) → top#(mark(s(f(_x31))))
top#(ok(s(f(s(_x51))))) → top#(s(f(s(active(_x51)))))top#(ok(s(p(s(_x41))))) → top#(s(mark(_x41)))
top#(ok(s(cons(_x41, _x42)))) → top#(s(cons(active(_x41), _x42)))top#(ok(s(f(p(s(ok(ok(_x61)))))))) → top#(s(mark(ok(ok(f(_x61))))))
top#(ok(f(s(0)))) → top#(mark(f(p(s(0)))))top#(ok(s(f(p(s(mark(mark(_x61)))))))) → top#(s(mark(mark(mark(f(_x61))))))
top#(ok(s(f(f(_x51))))) → top#(s(f(f(active(_x51)))))

Rewrite Rules

active(f(0)) → mark(cons(0, f(s(0))))active(f(s(0))) → mark(f(p(s(0))))
active(p(s(X))) → mark(X)active(f(X)) → f(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)active(s(X)) → s(active(X))
active(p(X)) → p(active(X))f(mark(X)) → mark(f(X))
cons(mark(X1), X2) → mark(cons(X1, X2))s(mark(X)) → mark(s(X))
p(mark(X)) → mark(p(X))proper(f(X)) → f(proper(X))
proper(0) → ok(0)proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))proper(p(X)) → p(proper(X))
f(ok(X)) → ok(f(X))cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))p(ok(X)) → ok(p(X))
top(mark(X)) → top(proper(X))top(ok(X)) → top(active(X))

Original Signature

Termination of terms over the following signature is verified: f, 0, s, p, active, mark, ok, proper, cons, top

Strategy

The right-hand side of the rule top#(ok(s(p(_x41)))) → top#(s(p(active(_x41)))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
top#(s(p(mark(_x51)))) 
top#(s(p(cons(active(_x51), _x52)))) 
top#(s(p(s(active(_x51))))) 
top#(s(p(mark(cons(0, f(s(0))))))) 
top#(s(p(mark(f(p(s(0))))))) 
top#(s(p(f(active(_x51))))) 
top#(s(p(p(active(_x51))))) 
Thus, the rule top#(ok(s(p(_x41)))) → top#(s(p(active(_x41)))) is replaced by the following rules:
top#(ok(s(p(p(s(_x51)))))) → top#(s(p(mark(_x51))))top#(ok(s(p(cons(_x51, _x52))))) → top#(s(p(cons(active(_x51), _x52))))
top#(ok(s(p(p(_x51))))) → top#(s(p(p(active(_x51)))))top#(ok(s(p(f(s(0)))))) → top#(s(p(mark(f(p(s(0)))))))
top#(ok(s(p(f(_x51))))) → top#(s(p(f(active(_x51)))))top#(ok(s(p(f(0))))) → top#(s(p(mark(cons(0, f(s(0)))))))
top#(ok(s(p(s(_x51))))) → top#(s(p(s(active(_x51)))))

Problem 19: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

top#(ok(s(f(0)))) → top#(mark(s(cons(0, f(s(0))))))top#(ok(f(0))) → top#(mark(cons(0, f(s(0)))))
top#(ok(s(s(_x41)))) → top#(s(s(active(_x41))))top#(ok(s(f(p(_x51))))) → top#(s(f(p(active(_x51)))))
top#(ok(s(f(f(0))))) → top#(s(f(mark(cons(0, f(s(0)))))))top#(mark(f(_x21))) → top#(f(proper(_x21)))
top#(ok(s(p(f(0))))) → top#(s(p(mark(cons(0, f(s(0)))))))top#(ok(s(f(p(s(ok(mark(_x61)))))))) → top#(s(mark(ok(mark(f(_x61))))))
top#(ok(cons(_x21, _x22))) → top#(cons(active(_x21), _x22))top#(ok(p(s(_x21)))) → top#(mark(_x21))
top#(mark(p(_x21))) → top#(p(proper(_x21)))top#(ok(s(f(p(s(ok(_x51))))))) → top#(mark(s(ok(f(_x51)))))
top#(ok(s(f(cons(_x51, _x52))))) → top#(s(f(cons(active(_x51), _x52))))top#(ok(s(f(f(s(0)))))) → top#(s(f(mark(f(p(s(0)))))))
top#(ok(s(p(p(s(_x51)))))) → top#(s(p(mark(_x51))))top#(ok(s(p(p(_x51))))) → top#(s(p(p(active(_x51)))))
top#(ok(s(f(s(0))))) → top#(s(mark(f(p(s(0))))))top#(mark(cons(_x21, _x22))) → top#(cons(proper(_x21), proper(_x22)))
top#(mark(s(_x21))) → top#(s(proper(_x21)))top#(ok(s(p(s(_x51))))) → top#(s(p(s(active(_x51)))))
top#(ok(p(_x21))) → top#(p(active(_x21)))top#(ok(f(_x21))) → top#(f(active(_x21)))
top#(ok(s(f(p(s(mark(ok(_x61)))))))) → top#(s(mark(mark(ok(f(_x61))))))top#(ok(s(f(p(s(mark(_x51))))))) → top#(mark(s(mark(f(_x51)))))
top#(ok(s(f(p(s(_x31)))))) → top#(mark(s(f(_x31))))top#(ok(s(f(s(_x51))))) → top#(s(f(s(active(_x51)))))
top#(ok(s(p(s(_x41))))) → top#(s(mark(_x41)))top#(ok(s(f(p(s(ok(ok(_x61)))))))) → top#(s(mark(ok(ok(f(_x61))))))
top#(ok(s(cons(_x41, _x42)))) → top#(s(cons(active(_x41), _x42)))top#(ok(f(s(0)))) → top#(mark(f(p(s(0)))))
top#(ok(s(p(cons(_x51, _x52))))) → top#(s(p(cons(active(_x51), _x52))))top#(ok(s(f(p(s(mark(mark(_x61)))))))) → top#(s(mark(mark(mark(f(_x61))))))
top#(ok(s(f(f(_x51))))) → top#(s(f(f(active(_x51)))))top#(ok(s(p(f(s(0)))))) → top#(s(p(mark(f(p(s(0)))))))
top#(ok(s(p(f(_x51))))) → top#(s(p(f(active(_x51)))))

Rewrite Rules

active(f(0)) → mark(cons(0, f(s(0))))active(f(s(0))) → mark(f(p(s(0))))
active(p(s(X))) → mark(X)active(f(X)) → f(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)active(s(X)) → s(active(X))
active(p(X)) → p(active(X))f(mark(X)) → mark(f(X))
cons(mark(X1), X2) → mark(cons(X1, X2))s(mark(X)) → mark(s(X))
p(mark(X)) → mark(p(X))proper(f(X)) → f(proper(X))
proper(0) → ok(0)proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))proper(p(X)) → p(proper(X))
f(ok(X)) → ok(f(X))cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))p(ok(X)) → ok(p(X))
top(mark(X)) → top(proper(X))top(ok(X)) → top(active(X))

Original Signature

Termination of terms over the following signature is verified: f, 0, s, p, active, ok, mark, proper, top, cons

Strategy

The right-hand side of the rule top#(ok(s(f(f(0))))) → top#(s(f(mark(cons(0, f(s(0))))))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
top#(s(mark(f(cons(0, f(s(0))))))) 
Thus, the rule top#(ok(s(f(f(0))))) → top#(s(f(mark(cons(0, f(s(0))))))) is replaced by the following rules:
top#(ok(s(f(f(0))))) → top#(s(mark(f(cons(0, f(s(0)))))))

Problem 20: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

top#(ok(s(f(0)))) → top#(mark(s(cons(0, f(s(0))))))top#(ok(s(f(f(0))))) → top#(s(mark(f(cons(0, f(s(0)))))))
top#(ok(f(0))) → top#(mark(cons(0, f(s(0)))))top#(ok(s(f(p(_x51))))) → top#(s(f(p(active(_x51)))))
top#(ok(s(s(_x41)))) → top#(s(s(active(_x41))))top#(mark(f(_x21))) → top#(f(proper(_x21)))
top#(ok(s(p(f(0))))) → top#(s(p(mark(cons(0, f(s(0)))))))top#(ok(s(f(p(s(ok(mark(_x61)))))))) → top#(s(mark(ok(mark(f(_x61))))))
top#(ok(cons(_x21, _x22))) → top#(cons(active(_x21), _x22))top#(mark(p(_x21))) → top#(p(proper(_x21)))
top#(ok(p(s(_x21)))) → top#(mark(_x21))top#(ok(s(f(cons(_x51, _x52))))) → top#(s(f(cons(active(_x51), _x52))))
top#(ok(s(f(p(s(ok(_x51))))))) → top#(mark(s(ok(f(_x51)))))top#(ok(s(f(f(s(0)))))) → top#(s(f(mark(f(p(s(0)))))))
top#(ok(s(p(p(s(_x51)))))) → top#(s(p(mark(_x51))))top#(ok(s(p(p(_x51))))) → top#(s(p(p(active(_x51)))))
top#(ok(s(f(s(0))))) → top#(s(mark(f(p(s(0))))))top#(mark(cons(_x21, _x22))) → top#(cons(proper(_x21), proper(_x22)))
top#(mark(s(_x21))) → top#(s(proper(_x21)))top#(ok(s(p(s(_x51))))) → top#(s(p(s(active(_x51)))))
top#(ok(p(_x21))) → top#(p(active(_x21)))top#(ok(f(_x21))) → top#(f(active(_x21)))
top#(ok(s(f(p(s(mark(ok(_x61)))))))) → top#(s(mark(mark(ok(f(_x61))))))top#(ok(s(f(p(s(mark(_x51))))))) → top#(mark(s(mark(f(_x51)))))
top#(ok(s(f(p(s(_x31)))))) → top#(mark(s(f(_x31))))top#(ok(s(f(s(_x51))))) → top#(s(f(s(active(_x51)))))
top#(ok(s(p(s(_x41))))) → top#(s(mark(_x41)))top#(ok(s(cons(_x41, _x42)))) → top#(s(cons(active(_x41), _x42)))
top#(ok(s(f(p(s(ok(ok(_x61)))))))) → top#(s(mark(ok(ok(f(_x61))))))top#(ok(f(s(0)))) → top#(mark(f(p(s(0)))))
top#(ok(s(p(cons(_x51, _x52))))) → top#(s(p(cons(active(_x51), _x52))))top#(ok(s(f(p(s(mark(mark(_x61)))))))) → top#(s(mark(mark(mark(f(_x61))))))
top#(ok(s(f(f(_x51))))) → top#(s(f(f(active(_x51)))))top#(ok(s(p(f(s(0)))))) → top#(s(p(mark(f(p(s(0)))))))
top#(ok(s(p(f(_x51))))) → top#(s(p(f(active(_x51)))))

Rewrite Rules

active(f(0)) → mark(cons(0, f(s(0))))active(f(s(0))) → mark(f(p(s(0))))
active(p(s(X))) → mark(X)active(f(X)) → f(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)active(s(X)) → s(active(X))
active(p(X)) → p(active(X))f(mark(X)) → mark(f(X))
cons(mark(X1), X2) → mark(cons(X1, X2))s(mark(X)) → mark(s(X))
p(mark(X)) → mark(p(X))proper(f(X)) → f(proper(X))
proper(0) → ok(0)proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))proper(p(X)) → p(proper(X))
f(ok(X)) → ok(f(X))cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))p(ok(X)) → ok(p(X))
top(mark(X)) → top(proper(X))top(ok(X)) → top(active(X))

Original Signature

Termination of terms over the following signature is verified: f, 0, s, p, active, mark, ok, proper, cons, top

Strategy

The right-hand side of the rule top#(ok(s(f(f(0))))) → top#(s(mark(f(cons(0, f(s(0))))))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
top#(mark(s(f(cons(0, f(s(0))))))) 
Thus, the rule top#(ok(s(f(f(0))))) → top#(s(mark(f(cons(0, f(s(0))))))) is replaced by the following rules:
top#(ok(s(f(f(0))))) → top#(mark(s(f(cons(0, f(s(0)))))))

Problem 21: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

top#(ok(s(f(0)))) → top#(mark(s(cons(0, f(s(0))))))top#(ok(f(0))) → top#(mark(cons(0, f(s(0)))))
top#(ok(s(s(_x41)))) → top#(s(s(active(_x41))))top#(ok(s(f(p(_x51))))) → top#(s(f(p(active(_x51)))))
top#(mark(f(_x21))) → top#(f(proper(_x21)))top#(ok(s(p(f(0))))) → top#(s(p(mark(cons(0, f(s(0)))))))
top#(ok(s(f(p(s(ok(mark(_x61)))))))) → top#(s(mark(ok(mark(f(_x61))))))top#(ok(cons(_x21, _x22))) → top#(cons(active(_x21), _x22))
top#(ok(p(s(_x21)))) → top#(mark(_x21))top#(mark(p(_x21))) → top#(p(proper(_x21)))
top#(ok(s(f(p(s(ok(_x51))))))) → top#(mark(s(ok(f(_x51)))))top#(ok(s(f(cons(_x51, _x52))))) → top#(s(f(cons(active(_x51), _x52))))
top#(ok(s(f(f(s(0)))))) → top#(s(f(mark(f(p(s(0)))))))top#(ok(s(p(p(s(_x51)))))) → top#(s(p(mark(_x51))))
top#(ok(s(p(p(_x51))))) → top#(s(p(p(active(_x51)))))top#(ok(s(f(f(0))))) → top#(mark(s(f(cons(0, f(s(0)))))))
top#(ok(s(f(s(0))))) → top#(s(mark(f(p(s(0))))))top#(mark(cons(_x21, _x22))) → top#(cons(proper(_x21), proper(_x22)))
top#(mark(s(_x21))) → top#(s(proper(_x21)))top#(ok(s(p(s(_x51))))) → top#(s(p(s(active(_x51)))))
top#(ok(p(_x21))) → top#(p(active(_x21)))top#(ok(f(_x21))) → top#(f(active(_x21)))
top#(ok(s(f(p(s(mark(ok(_x61)))))))) → top#(s(mark(mark(ok(f(_x61))))))top#(ok(s(f(p(s(mark(_x51))))))) → top#(mark(s(mark(f(_x51)))))
top#(ok(s(f(p(s(_x31)))))) → top#(mark(s(f(_x31))))top#(ok(s(f(s(_x51))))) → top#(s(f(s(active(_x51)))))
top#(ok(s(p(s(_x41))))) → top#(s(mark(_x41)))top#(ok(s(f(p(s(ok(ok(_x61)))))))) → top#(s(mark(ok(ok(f(_x61))))))
top#(ok(s(cons(_x41, _x42)))) → top#(s(cons(active(_x41), _x42)))top#(ok(f(s(0)))) → top#(mark(f(p(s(0)))))
top#(ok(s(p(cons(_x51, _x52))))) → top#(s(p(cons(active(_x51), _x52))))top#(ok(s(f(p(s(mark(mark(_x61)))))))) → top#(s(mark(mark(mark(f(_x61))))))
top#(ok(s(f(f(_x51))))) → top#(s(f(f(active(_x51)))))top#(ok(s(p(f(s(0)))))) → top#(s(p(mark(f(p(s(0)))))))
top#(ok(s(p(f(_x51))))) → top#(s(p(f(active(_x51)))))

Rewrite Rules

active(f(0)) → mark(cons(0, f(s(0))))active(f(s(0))) → mark(f(p(s(0))))
active(p(s(X))) → mark(X)active(f(X)) → f(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)active(s(X)) → s(active(X))
active(p(X)) → p(active(X))f(mark(X)) → mark(f(X))
cons(mark(X1), X2) → mark(cons(X1, X2))s(mark(X)) → mark(s(X))
p(mark(X)) → mark(p(X))proper(f(X)) → f(proper(X))
proper(0) → ok(0)proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))proper(p(X)) → p(proper(X))
f(ok(X)) → ok(f(X))cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))p(ok(X)) → ok(p(X))
top(mark(X)) → top(proper(X))top(ok(X)) → top(active(X))

Original Signature

Termination of terms over the following signature is verified: f, 0, s, p, active, ok, mark, proper, top, cons

Strategy

The right-hand side of the rule top#(ok(s(f(p(_x51))))) → top#(s(f(p(active(_x51))))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
top#(s(f(p(mark(_x61))))) 
top#(s(f(p(s(active(_x61)))))) 
top#(s(f(p(mark(cons(0, f(s(0)))))))) 
top#(s(f(p(cons(active(_x61), _x62))))) 
top#(s(f(p(p(active(_x61)))))) 
top#(s(f(p(f(active(_x61)))))) 
top#(s(f(p(mark(f(p(s(0)))))))) 
Thus, the rule top#(ok(s(f(p(_x51))))) → top#(s(f(p(active(_x51))))) is replaced by the following rules:
top#(ok(s(f(p(cons(_x61, _x62)))))) → top#(s(f(p(cons(active(_x61), _x62)))))top#(ok(s(f(p(p(s(_x61))))))) → top#(s(f(p(mark(_x61)))))
top#(ok(s(f(p(f(_x61)))))) → top#(s(f(p(f(active(_x61))))))top#(ok(s(f(p(f(s(0))))))) → top#(s(f(p(mark(f(p(s(0))))))))
top#(ok(s(f(p(p(_x61)))))) → top#(s(f(p(p(active(_x61))))))top#(ok(s(f(p(s(_x61)))))) → top#(s(f(p(s(active(_x61))))))
top#(ok(s(f(p(f(0)))))) → top#(s(f(p(mark(cons(0, f(s(0))))))))

Problem 22: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

top#(ok(s(f(0)))) → top#(mark(s(cons(0, f(s(0))))))top#(ok(s(f(p(p(_x61)))))) → top#(s(f(p(p(active(_x61))))))
top#(ok(f(0))) → top#(mark(cons(0, f(s(0)))))top#(ok(s(s(_x41)))) → top#(s(s(active(_x41))))
top#(mark(f(_x21))) → top#(f(proper(_x21)))top#(ok(s(p(f(0))))) → top#(s(p(mark(cons(0, f(s(0)))))))
top#(ok(s(f(p(s(ok(mark(_x61)))))))) → top#(s(mark(ok(mark(f(_x61))))))top#(ok(cons(_x21, _x22))) → top#(cons(active(_x21), _x22))
top#(mark(p(_x21))) → top#(p(proper(_x21)))top#(ok(p(s(_x21)))) → top#(mark(_x21))
top#(ok(s(f(cons(_x51, _x52))))) → top#(s(f(cons(active(_x51), _x52))))top#(ok(s(f(p(s(ok(_x51))))))) → top#(mark(s(ok(f(_x51)))))
top#(ok(s(f(f(s(0)))))) → top#(s(f(mark(f(p(s(0)))))))top#(ok(s(p(p(s(_x51)))))) → top#(s(p(mark(_x51))))
top#(ok(s(f(p(cons(_x61, _x62)))))) → top#(s(f(p(cons(active(_x61), _x62)))))top#(ok(s(p(p(_x51))))) → top#(s(p(p(active(_x51)))))
top#(ok(s(f(p(f(_x61)))))) → top#(s(f(p(f(active(_x61))))))top#(ok(s(f(f(0))))) → top#(mark(s(f(cons(0, f(s(0)))))))
top#(ok(s(f(s(0))))) → top#(s(mark(f(p(s(0))))))top#(ok(s(f(p(f(s(0))))))) → top#(s(f(p(mark(f(p(s(0))))))))
top#(mark(cons(_x21, _x22))) → top#(cons(proper(_x21), proper(_x22)))top#(mark(s(_x21))) → top#(s(proper(_x21)))
top#(ok(s(p(s(_x51))))) → top#(s(p(s(active(_x51)))))top#(ok(p(_x21))) → top#(p(active(_x21)))
top#(ok(f(_x21))) → top#(f(active(_x21)))top#(ok(s(f(p(s(_x31)))))) → top#(mark(s(f(_x31))))
top#(ok(s(f(p(s(mark(_x51))))))) → top#(mark(s(mark(f(_x51)))))top#(ok(s(f(p(s(mark(ok(_x61)))))))) → top#(s(mark(mark(ok(f(_x61))))))
top#(ok(s(f(s(_x51))))) → top#(s(f(s(active(_x51)))))top#(ok(s(f(p(f(0)))))) → top#(s(f(p(mark(cons(0, f(s(0))))))))
top#(ok(s(p(s(_x41))))) → top#(s(mark(_x41)))top#(ok(s(cons(_x41, _x42)))) → top#(s(cons(active(_x41), _x42)))
top#(ok(s(f(p(s(ok(ok(_x61)))))))) → top#(s(mark(ok(ok(f(_x61))))))top#(ok(f(s(0)))) → top#(mark(f(p(s(0)))))
top#(ok(s(p(cons(_x51, _x52))))) → top#(s(p(cons(active(_x51), _x52))))top#(ok(s(f(p(s(mark(mark(_x61)))))))) → top#(s(mark(mark(mark(f(_x61))))))
top#(ok(s(f(p(p(s(_x61))))))) → top#(s(f(p(mark(_x61)))))top#(ok(s(f(f(_x51))))) → top#(s(f(f(active(_x51)))))
top#(ok(s(p(f(s(0)))))) → top#(s(p(mark(f(p(s(0)))))))top#(ok(s(p(f(_x51))))) → top#(s(p(f(active(_x51)))))
top#(ok(s(f(p(s(_x61)))))) → top#(s(f(p(s(active(_x61))))))

Rewrite Rules

active(f(0)) → mark(cons(0, f(s(0))))active(f(s(0))) → mark(f(p(s(0))))
active(p(s(X))) → mark(X)active(f(X)) → f(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)active(s(X)) → s(active(X))
active(p(X)) → p(active(X))f(mark(X)) → mark(f(X))
cons(mark(X1), X2) → mark(cons(X1, X2))s(mark(X)) → mark(s(X))
p(mark(X)) → mark(p(X))proper(f(X)) → f(proper(X))
proper(0) → ok(0)proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))proper(p(X)) → p(proper(X))
f(ok(X)) → ok(f(X))cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))p(ok(X)) → ok(p(X))
top(mark(X)) → top(proper(X))top(ok(X)) → top(active(X))

Original Signature

Termination of terms over the following signature is verified: f, 0, s, p, active, mark, ok, proper, cons, top

Strategy

The right-hand side of the rule top#(ok(s(f(p(p(_x61)))))) → top#(s(f(p(p(active(_x61)))))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
top#(s(f(p(p(p(active(_x71))))))) 
top#(s(f(p(p(mark(f(p(s(0))))))))) 
top#(s(f(p(p(mark(_x71)))))) 
top#(s(f(p(p(mark(cons(0, f(s(0))))))))) 
top#(s(f(p(p(cons(active(_x71), _x72)))))) 
top#(s(f(p(p(s(active(_x71))))))) 
top#(s(f(p(p(f(active(_x71))))))) 
Thus, the rule top#(ok(s(f(p(p(_x61)))))) → top#(s(f(p(p(active(_x61)))))) is replaced by the following rules:
top#(ok(s(f(p(p(f(0))))))) → top#(s(f(p(p(mark(cons(0, f(s(0)))))))))top#(ok(s(f(p(p(f(s(0)))))))) → top#(s(f(p(p(mark(f(p(s(0)))))))))
top#(ok(s(f(p(p(cons(_x71, _x72))))))) → top#(s(f(p(p(cons(active(_x71), _x72))))))top#(ok(s(f(p(p(s(_x71))))))) → top#(s(f(p(p(s(active(_x71)))))))
top#(ok(s(f(p(p(p(_x71))))))) → top#(s(f(p(p(p(active(_x71)))))))top#(ok(s(f(p(p(p(s(_x71)))))))) → top#(s(f(p(p(mark(_x71))))))
top#(ok(s(f(p(p(f(_x71))))))) → top#(s(f(p(p(f(active(_x71)))))))

Problem 23: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

top#(ok(s(f(p(p(f(s(0)))))))) → top#(s(f(p(p(mark(f(p(s(0)))))))))top#(ok(s(f(0)))) → top#(mark(s(cons(0, f(s(0))))))
top#(ok(s(f(p(p(cons(_x71, _x72))))))) → top#(s(f(p(p(cons(active(_x71), _x72))))))top#(ok(s(f(p(p(p(_x71))))))) → top#(s(f(p(p(p(active(_x71)))))))
top#(ok(s(f(p(p(p(s(_x71)))))))) → top#(s(f(p(p(mark(_x71))))))top#(ok(s(f(p(p(f(0))))))) → top#(s(f(p(p(mark(cons(0, f(s(0)))))))))
top#(ok(f(0))) → top#(mark(cons(0, f(s(0)))))top#(ok(s(s(_x41)))) → top#(s(s(active(_x41))))
top#(mark(f(_x21))) → top#(f(proper(_x21)))top#(ok(s(p(f(0))))) → top#(s(p(mark(cons(0, f(s(0)))))))
top#(ok(s(f(p(s(ok(mark(_x61)))))))) → top#(s(mark(ok(mark(f(_x61))))))top#(ok(cons(_x21, _x22))) → top#(cons(active(_x21), _x22))
top#(mark(p(_x21))) → top#(p(proper(_x21)))top#(ok(p(s(_x21)))) → top#(mark(_x21))
top#(ok(s(f(cons(_x51, _x52))))) → top#(s(f(cons(active(_x51), _x52))))top#(ok(s(f(p(s(ok(_x51))))))) → top#(mark(s(ok(f(_x51)))))
top#(ok(s(f(f(s(0)))))) → top#(s(f(mark(f(p(s(0)))))))top#(ok(s(p(p(s(_x51)))))) → top#(s(p(mark(_x51))))
top#(ok(s(f(p(cons(_x61, _x62)))))) → top#(s(f(p(cons(active(_x61), _x62)))))top#(ok(s(p(p(_x51))))) → top#(s(p(p(active(_x51)))))
top#(ok(s(f(p(f(_x61)))))) → top#(s(f(p(f(active(_x61))))))top#(ok(s(f(f(0))))) → top#(mark(s(f(cons(0, f(s(0)))))))
top#(ok(s(f(s(0))))) → top#(s(mark(f(p(s(0))))))top#(ok(s(f(p(f(s(0))))))) → top#(s(f(p(mark(f(p(s(0))))))))
top#(ok(s(f(p(p(s(_x71))))))) → top#(s(f(p(p(s(active(_x71)))))))top#(mark(s(_x21))) → top#(s(proper(_x21)))
top#(mark(cons(_x21, _x22))) → top#(cons(proper(_x21), proper(_x22)))top#(ok(f(_x21))) → top#(f(active(_x21)))
top#(ok(p(_x21))) → top#(p(active(_x21)))top#(ok(s(p(s(_x51))))) → top#(s(p(s(active(_x51)))))
top#(ok(s(f(p(s(mark(ok(_x61)))))))) → top#(s(mark(mark(ok(f(_x61))))))top#(ok(s(f(p(s(mark(_x51))))))) → top#(mark(s(mark(f(_x51)))))
top#(ok(s(f(p(s(_x31)))))) → top#(mark(s(f(_x31))))top#(ok(s(f(p(f(0)))))) → top#(s(f(p(mark(cons(0, f(s(0))))))))
top#(ok(s(f(s(_x51))))) → top#(s(f(s(active(_x51)))))top#(ok(s(f(p(p(f(_x71))))))) → top#(s(f(p(p(f(active(_x71)))))))
top#(ok(s(p(s(_x41))))) → top#(s(mark(_x41)))top#(ok(s(f(p(s(ok(ok(_x61)))))))) → top#(s(mark(ok(ok(f(_x61))))))
top#(ok(s(cons(_x41, _x42)))) → top#(s(cons(active(_x41), _x42)))top#(ok(f(s(0)))) → top#(mark(f(p(s(0)))))
top#(ok(s(p(cons(_x51, _x52))))) → top#(s(p(cons(active(_x51), _x52))))top#(ok(s(f(p(p(s(_x61))))))) → top#(s(f(p(mark(_x61)))))
top#(ok(s(f(p(s(mark(mark(_x61)))))))) → top#(s(mark(mark(mark(f(_x61))))))top#(ok(s(f(f(_x51))))) → top#(s(f(f(active(_x51)))))
top#(ok(s(p(f(s(0)))))) → top#(s(p(mark(f(p(s(0)))))))top#(ok(s(p(f(_x51))))) → top#(s(p(f(active(_x51)))))
top#(ok(s(f(p(s(_x61)))))) → top#(s(f(p(s(active(_x61))))))

Rewrite Rules

active(f(0)) → mark(cons(0, f(s(0))))active(f(s(0))) → mark(f(p(s(0))))
active(p(s(X))) → mark(X)active(f(X)) → f(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)active(s(X)) → s(active(X))
active(p(X)) → p(active(X))f(mark(X)) → mark(f(X))
cons(mark(X1), X2) → mark(cons(X1, X2))s(mark(X)) → mark(s(X))
p(mark(X)) → mark(p(X))proper(f(X)) → f(proper(X))
proper(0) → ok(0)proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))proper(p(X)) → p(proper(X))
f(ok(X)) → ok(f(X))cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))p(ok(X)) → ok(p(X))
top(mark(X)) → top(proper(X))top(ok(X)) → top(active(X))

Original Signature

Termination of terms over the following signature is verified: f, 0, s, p, active, ok, mark, proper, top, cons

Strategy

The right-hand side of the rule top#(ok(s(f(p(p(f(s(0)))))))) → top#(s(f(p(p(mark(f(p(s(0))))))))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
top#(s(f(p(mark(p(f(p(s(0))))))))) 
Thus, the rule top#(ok(s(f(p(p(f(s(0)))))))) → top#(s(f(p(p(mark(f(p(s(0))))))))) is replaced by the following rules:
top#(ok(s(f(p(p(f(s(0)))))))) → top#(s(f(p(mark(p(f(p(s(0)))))))))

Problem 24: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

top#(ok(s(f(0)))) → top#(mark(s(cons(0, f(s(0))))))top#(ok(s(f(p(p(cons(_x71, _x72))))))) → top#(s(f(p(p(cons(active(_x71), _x72))))))
top#(ok(s(f(p(p(p(s(_x71)))))))) → top#(s(f(p(p(mark(_x71))))))top#(ok(s(f(p(p(p(_x71))))))) → top#(s(f(p(p(p(active(_x71)))))))
top#(ok(s(f(p(p(f(s(0)))))))) → top#(s(f(p(mark(p(f(p(s(0)))))))))top#(ok(s(f(p(p(f(0))))))) → top#(s(f(p(p(mark(cons(0, f(s(0)))))))))
top#(ok(f(0))) → top#(mark(cons(0, f(s(0)))))top#(ok(s(s(_x41)))) → top#(s(s(active(_x41))))
top#(mark(f(_x21))) → top#(f(proper(_x21)))top#(ok(s(p(f(0))))) → top#(s(p(mark(cons(0, f(s(0)))))))
top#(ok(s(f(p(s(ok(mark(_x61)))))))) → top#(s(mark(ok(mark(f(_x61))))))top#(ok(cons(_x21, _x22))) → top#(cons(active(_x21), _x22))
top#(mark(p(_x21))) → top#(p(proper(_x21)))top#(ok(p(s(_x21)))) → top#(mark(_x21))
top#(ok(s(f(cons(_x51, _x52))))) → top#(s(f(cons(active(_x51), _x52))))top#(ok(s(f(p(s(ok(_x51))))))) → top#(mark(s(ok(f(_x51)))))
top#(ok(s(f(f(s(0)))))) → top#(s(f(mark(f(p(s(0)))))))top#(ok(s(p(p(s(_x51)))))) → top#(s(p(mark(_x51))))
top#(ok(s(f(p(cons(_x61, _x62)))))) → top#(s(f(p(cons(active(_x61), _x62)))))top#(ok(s(p(p(_x51))))) → top#(s(p(p(active(_x51)))))
top#(ok(s(f(p(f(_x61)))))) → top#(s(f(p(f(active(_x61))))))top#(ok(s(f(f(0))))) → top#(mark(s(f(cons(0, f(s(0)))))))
top#(ok(s(f(s(0))))) → top#(s(mark(f(p(s(0))))))top#(ok(s(f(p(f(s(0))))))) → top#(s(f(p(mark(f(p(s(0))))))))
top#(ok(s(f(p(p(s(_x71))))))) → top#(s(f(p(p(s(active(_x71)))))))top#(mark(cons(_x21, _x22))) → top#(cons(proper(_x21), proper(_x22)))
top#(mark(s(_x21))) → top#(s(proper(_x21)))top#(ok(s(p(s(_x51))))) → top#(s(p(s(active(_x51)))))
top#(ok(p(_x21))) → top#(p(active(_x21)))top#(ok(f(_x21))) → top#(f(active(_x21)))
top#(ok(s(f(p(s(_x31)))))) → top#(mark(s(f(_x31))))top#(ok(s(f(p(s(mark(_x51))))))) → top#(mark(s(mark(f(_x51)))))
top#(ok(s(f(p(s(mark(ok(_x61)))))))) → top#(s(mark(mark(ok(f(_x61))))))top#(ok(s(f(p(p(f(_x71))))))) → top#(s(f(p(p(f(active(_x71)))))))
top#(ok(s(f(s(_x51))))) → top#(s(f(s(active(_x51)))))top#(ok(s(f(p(f(0)))))) → top#(s(f(p(mark(cons(0, f(s(0))))))))
top#(ok(s(p(s(_x41))))) → top#(s(mark(_x41)))top#(ok(s(cons(_x41, _x42)))) → top#(s(cons(active(_x41), _x42)))
top#(ok(s(f(p(s(ok(ok(_x61)))))))) → top#(s(mark(ok(ok(f(_x61))))))top#(ok(f(s(0)))) → top#(mark(f(p(s(0)))))
top#(ok(s(p(cons(_x51, _x52))))) → top#(s(p(cons(active(_x51), _x52))))top#(ok(s(f(p(s(mark(mark(_x61)))))))) → top#(s(mark(mark(mark(f(_x61))))))
top#(ok(s(f(p(p(s(_x61))))))) → top#(s(f(p(mark(_x61)))))top#(ok(s(f(f(_x51))))) → top#(s(f(f(active(_x51)))))
top#(ok(s(p(f(s(0)))))) → top#(s(p(mark(f(p(s(0)))))))top#(ok(s(p(f(_x51))))) → top#(s(p(f(active(_x51)))))
top#(ok(s(f(p(s(_x61)))))) → top#(s(f(p(s(active(_x61))))))

Rewrite Rules

active(f(0)) → mark(cons(0, f(s(0))))active(f(s(0))) → mark(f(p(s(0))))
active(p(s(X))) → mark(X)active(f(X)) → f(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)active(s(X)) → s(active(X))
active(p(X)) → p(active(X))f(mark(X)) → mark(f(X))
cons(mark(X1), X2) → mark(cons(X1, X2))s(mark(X)) → mark(s(X))
p(mark(X)) → mark(p(X))proper(f(X)) → f(proper(X))
proper(0) → ok(0)proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))proper(p(X)) → p(proper(X))
f(ok(X)) → ok(f(X))cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))p(ok(X)) → ok(p(X))
top(mark(X)) → top(proper(X))top(ok(X)) → top(active(X))

Original Signature

Termination of terms over the following signature is verified: f, 0, s, p, active, mark, ok, proper, cons, top

Strategy

The right-hand side of the rule top#(ok(s(f(p(p(cons(_x71, _x72))))))) → top#(s(f(p(p(cons(active(_x71), _x72)))))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
top#(s(f(p(p(cons(mark(_x81), _x72)))))) 
top#(s(f(p(p(cons(cons(active(_x81), _x82), _x72)))))) 
top#(s(f(p(p(cons(s(active(_x81)), _x72)))))) 
top#(s(f(p(p(cons(f(active(_x81)), _x72)))))) 
top#(s(f(p(p(cons(mark(cons(0, f(s(0)))), _x72)))))) 
top#(s(f(p(p(cons(mark(f(p(s(0)))), _x72)))))) 
top#(s(f(p(p(cons(p(active(_x81)), _x72)))))) 
Thus, the rule top#(ok(s(f(p(p(cons(_x71, _x72))))))) → top#(s(f(p(p(cons(active(_x71), _x72)))))) is replaced by the following rules:
top#(ok(s(f(p(p(cons(p(_x81), _x72))))))) → top#(s(f(p(p(cons(p(active(_x81)), _x72))))))top#(ok(s(f(p(p(cons(f(_x81), _x72))))))) → top#(s(f(p(p(cons(f(active(_x81)), _x72))))))
top#(ok(s(f(p(p(cons(f(s(0)), _x72))))))) → top#(s(f(p(p(cons(mark(f(p(s(0)))), _x72))))))top#(ok(s(f(p(p(cons(s(_x81), _x72))))))) → top#(s(f(p(p(cons(s(active(_x81)), _x72))))))
top#(ok(s(f(p(p(cons(f(0), _x72))))))) → top#(s(f(p(p(cons(mark(cons(0, f(s(0)))), _x72))))))top#(ok(s(f(p(p(cons(p(s(_x81)), _x72))))))) → top#(s(f(p(p(cons(mark(_x81), _x72))))))
top#(ok(s(f(p(p(cons(cons(_x81, _x82), _x72))))))) → top#(s(f(p(p(cons(cons(active(_x81), _x82), _x72))))))

Problem 25: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

top#(ok(s(f(0)))) → top#(mark(s(cons(0, f(s(0))))))top#(ok(s(f(p(p(f(0))))))) → top#(s(f(p(p(mark(cons(0, f(s(0)))))))))
top#(ok(s(f(p(p(cons(f(_x81), _x72))))))) → top#(s(f(p(p(cons(f(active(_x81)), _x72))))))top#(ok(s(f(p(p(cons(f(s(0)), _x72))))))) → top#(s(f(p(p(cons(mark(f(p(s(0)))), _x72))))))
top#(ok(s(s(_x41)))) → top#(s(s(active(_x41))))top#(mark(f(_x21))) → top#(f(proper(_x21)))
top#(ok(s(p(f(0))))) → top#(s(p(mark(cons(0, f(s(0)))))))top#(ok(cons(_x21, _x22))) → top#(cons(active(_x21), _x22))
top#(ok(p(s(_x21)))) → top#(mark(_x21))top#(ok(s(f(p(s(ok(_x51))))))) → top#(mark(s(ok(f(_x51)))))
top#(ok(s(f(cons(_x51, _x52))))) → top#(s(f(cons(active(_x51), _x52))))top#(ok(s(f(f(s(0)))))) → top#(s(f(mark(f(p(s(0)))))))
top#(ok(s(p(p(s(_x51)))))) → top#(s(p(mark(_x51))))top#(ok(s(f(p(cons(_x61, _x62)))))) → top#(s(f(p(cons(active(_x61), _x62)))))
top#(ok(s(p(p(_x51))))) → top#(s(p(p(active(_x51)))))top#(ok(s(f(p(f(_x61)))))) → top#(s(f(p(f(active(_x61))))))
top#(ok(s(f(f(0))))) → top#(mark(s(f(cons(0, f(s(0)))))))top#(ok(s(f(p(p(s(_x71))))))) → top#(s(f(p(p(s(active(_x71)))))))
top#(mark(s(_x21))) → top#(s(proper(_x21)))top#(ok(s(p(s(_x51))))) → top#(s(p(s(active(_x51)))))
top#(ok(s(f(p(s(mark(_x51))))))) → top#(mark(s(mark(f(_x51)))))top#(ok(s(f(p(p(f(_x71))))))) → top#(s(f(p(p(f(active(_x71)))))))
top#(ok(s(f(s(_x51))))) → top#(s(f(s(active(_x51)))))top#(ok(s(f(p(f(0)))))) → top#(s(f(p(mark(cons(0, f(s(0))))))))
top#(ok(s(p(s(_x41))))) → top#(s(mark(_x41)))top#(ok(s(f(p(s(ok(ok(_x61)))))))) → top#(s(mark(ok(ok(f(_x61))))))
top#(ok(s(cons(_x41, _x42)))) → top#(s(cons(active(_x41), _x42)))top#(ok(s(p(cons(_x51, _x52))))) → top#(s(p(cons(active(_x51), _x52))))
top#(ok(s(f(p(p(s(_x61))))))) → top#(s(f(p(mark(_x61)))))top#(ok(s(f(p(s(mark(mark(_x61)))))))) → top#(s(mark(mark(mark(f(_x61))))))
top#(ok(s(p(f(_x51))))) → top#(s(p(f(active(_x51)))))top#(ok(s(f(p(s(_x61)))))) → top#(s(f(p(s(active(_x61))))))
top#(ok(s(f(p(p(cons(p(_x81), _x72))))))) → top#(s(f(p(p(cons(p(active(_x81)), _x72))))))top#(ok(s(f(p(p(cons(s(_x81), _x72))))))) → top#(s(f(p(p(cons(s(active(_x81)), _x72))))))
top#(ok(s(f(p(p(cons(p(s(_x81)), _x72))))))) → top#(s(f(p(p(cons(mark(_x81), _x72))))))top#(ok(s(f(p(p(p(_x71))))))) → top#(s(f(p(p(p(active(_x71)))))))
top#(ok(s(f(p(p(p(s(_x71)))))))) → top#(s(f(p(p(mark(_x71))))))top#(ok(s(f(p(p(f(s(0)))))))) → top#(s(f(p(mark(p(f(p(s(0)))))))))
top#(ok(f(0))) → top#(mark(cons(0, f(s(0)))))top#(ok(s(f(p(s(ok(mark(_x61)))))))) → top#(s(mark(ok(mark(f(_x61))))))
top#(mark(p(_x21))) → top#(p(proper(_x21)))top#(ok(s(f(p(p(cons(cons(_x81, _x82), _x72))))))) → top#(s(f(p(p(cons(cons(active(_x81), _x82), _x72))))))
top#(ok(s(f(s(0))))) → top#(s(mark(f(p(s(0))))))top#(ok(s(f(p(f(s(0))))))) → top#(s(f(p(mark(f(p(s(0))))))))
top#(ok(s(f(p(p(cons(f(0), _x72))))))) → top#(s(f(p(p(cons(mark(cons(0, f(s(0)))), _x72))))))top#(mark(cons(_x21, _x22))) → top#(cons(proper(_x21), proper(_x22)))
top#(ok(f(_x21))) → top#(f(active(_x21)))top#(ok(p(_x21))) → top#(p(active(_x21)))
top#(ok(s(f(p(s(_x31)))))) → top#(mark(s(f(_x31))))top#(ok(s(f(p(s(mark(ok(_x61)))))))) → top#(s(mark(mark(ok(f(_x61))))))
top#(ok(f(s(0)))) → top#(mark(f(p(s(0)))))top#(ok(s(f(f(_x51))))) → top#(s(f(f(active(_x51)))))
top#(ok(s(p(f(s(0)))))) → top#(s(p(mark(f(p(s(0)))))))

Rewrite Rules

active(f(0)) → mark(cons(0, f(s(0))))active(f(s(0))) → mark(f(p(s(0))))
active(p(s(X))) → mark(X)active(f(X)) → f(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)active(s(X)) → s(active(X))
active(p(X)) → p(active(X))f(mark(X)) → mark(f(X))
cons(mark(X1), X2) → mark(cons(X1, X2))s(mark(X)) → mark(s(X))
p(mark(X)) → mark(p(X))proper(f(X)) → f(proper(X))
proper(0) → ok(0)proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))proper(p(X)) → p(proper(X))
f(ok(X)) → ok(f(X))cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))p(ok(X)) → ok(p(X))
top(mark(X)) → top(proper(X))top(ok(X)) → top(active(X))

Original Signature

Termination of terms over the following signature is verified: f, 0, s, p, active, ok, mark, proper, top, cons

Strategy

The right-hand side of the rule top#(ok(s(f(p(p(f(0))))))) → top#(s(f(p(p(mark(cons(0, f(s(0))))))))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
top#(s(f(p(mark(p(cons(0, f(s(0))))))))) 
Thus, the rule top#(ok(s(f(p(p(f(0))))))) → top#(s(f(p(p(mark(cons(0, f(s(0))))))))) is replaced by the following rules:
top#(ok(s(f(p(p(f(0))))))) → top#(s(f(p(mark(p(cons(0, f(s(0)))))))))

Problem 26: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

top#(ok(s(f(0)))) → top#(mark(s(cons(0, f(s(0))))))top#(ok(s(f(p(p(cons(f(_x81), _x72))))))) → top#(s(f(p(p(cons(f(active(_x81)), _x72))))))
top#(ok(s(f(p(p(cons(f(s(0)), _x72))))))) → top#(s(f(p(p(cons(mark(f(p(s(0)))), _x72))))))top#(ok(s(s(_x41)))) → top#(s(s(active(_x41))))
top#(mark(f(_x21))) → top#(f(proper(_x21)))top#(ok(s(p(f(0))))) → top#(s(p(mark(cons(0, f(s(0)))))))
top#(ok(cons(_x21, _x22))) → top#(cons(active(_x21), _x22))top#(ok(p(s(_x21)))) → top#(mark(_x21))
top#(ok(s(f(p(s(ok(_x51))))))) → top#(mark(s(ok(f(_x51)))))top#(ok(s(f(cons(_x51, _x52))))) → top#(s(f(cons(active(_x51), _x52))))
top#(ok(s(f(f(s(0)))))) → top#(s(f(mark(f(p(s(0)))))))top#(ok(s(f(p(p(f(0))))))) → top#(s(f(p(mark(p(cons(0, f(s(0)))))))))
top#(ok(s(p(p(s(_x51)))))) → top#(s(p(mark(_x51))))top#(ok(s(f(p(cons(_x61, _x62)))))) → top#(s(f(p(cons(active(_x61), _x62)))))
top#(ok(s(p(p(_x51))))) → top#(s(p(p(active(_x51)))))top#(ok(s(f(f(0))))) → top#(mark(s(f(cons(0, f(s(0)))))))
top#(ok(s(f(p(f(_x61)))))) → top#(s(f(p(f(active(_x61))))))top#(ok(s(f(p(p(s(_x71))))))) → top#(s(f(p(p(s(active(_x71)))))))
top#(mark(s(_x21))) → top#(s(proper(_x21)))top#(ok(s(p(s(_x51))))) → top#(s(p(s(active(_x51)))))
top#(ok(s(f(p(s(mark(_x51))))))) → top#(mark(s(mark(f(_x51)))))top#(ok(s(f(p(f(0)))))) → top#(s(f(p(mark(cons(0, f(s(0))))))))
top#(ok(s(f(s(_x51))))) → top#(s(f(s(active(_x51)))))top#(ok(s(f(p(p(f(_x71))))))) → top#(s(f(p(p(f(active(_x71)))))))
top#(ok(s(p(s(_x41))))) → top#(s(mark(_x41)))top#(ok(s(f(p(s(ok(ok(_x61)))))))) → top#(s(mark(ok(ok(f(_x61))))))
top#(ok(s(cons(_x41, _x42)))) → top#(s(cons(active(_x41), _x42)))top#(ok(s(p(cons(_x51, _x52))))) → top#(s(p(cons(active(_x51), _x52))))
top#(ok(s(f(p(p(s(_x61))))))) → top#(s(f(p(mark(_x61)))))top#(ok(s(f(p(s(mark(mark(_x61)))))))) → top#(s(mark(mark(mark(f(_x61))))))
top#(ok(s(p(f(_x51))))) → top#(s(p(f(active(_x51)))))top#(ok(s(f(p(s(_x61)))))) → top#(s(f(p(s(active(_x61))))))
top#(ok(s(f(p(p(cons(p(_x81), _x72))))))) → top#(s(f(p(p(cons(p(active(_x81)), _x72))))))top#(ok(s(f(p(p(cons(s(_x81), _x72))))))) → top#(s(f(p(p(cons(s(active(_x81)), _x72))))))
top#(ok(s(f(p(p(cons(p(s(_x81)), _x72))))))) → top#(s(f(p(p(cons(mark(_x81), _x72))))))top#(ok(s(f(p(p(p(_x71))))))) → top#(s(f(p(p(p(active(_x71)))))))
top#(ok(s(f(p(p(p(s(_x71)))))))) → top#(s(f(p(p(mark(_x71))))))top#(ok(s(f(p(p(f(s(0)))))))) → top#(s(f(p(mark(p(f(p(s(0)))))))))
top#(ok(f(0))) → top#(mark(cons(0, f(s(0)))))top#(ok(s(f(p(s(ok(mark(_x61)))))))) → top#(s(mark(ok(mark(f(_x61))))))
top#(mark(p(_x21))) → top#(p(proper(_x21)))top#(ok(s(f(p(p(cons(cons(_x81, _x82), _x72))))))) → top#(s(f(p(p(cons(cons(active(_x81), _x82), _x72))))))
top#(ok(s(f(s(0))))) → top#(s(mark(f(p(s(0))))))top#(ok(s(f(p(f(s(0))))))) → top#(s(f(p(mark(f(p(s(0))))))))
top#(ok(s(f(p(p(cons(f(0), _x72))))))) → top#(s(f(p(p(cons(mark(cons(0, f(s(0)))), _x72))))))top#(mark(cons(_x21, _x22))) → top#(cons(proper(_x21), proper(_x22)))
top#(ok(f(_x21))) → top#(f(active(_x21)))top#(ok(p(_x21))) → top#(p(active(_x21)))
top#(ok(s(f(p(s(mark(ok(_x61)))))))) → top#(s(mark(mark(ok(f(_x61))))))top#(ok(s(f(p(s(_x31)))))) → top#(mark(s(f(_x31))))
top#(ok(f(s(0)))) → top#(mark(f(p(s(0)))))top#(ok(s(f(f(_x51))))) → top#(s(f(f(active(_x51)))))
top#(ok(s(p(f(s(0)))))) → top#(s(p(mark(f(p(s(0)))))))

Rewrite Rules

active(f(0)) → mark(cons(0, f(s(0))))active(f(s(0))) → mark(f(p(s(0))))
active(p(s(X))) → mark(X)active(f(X)) → f(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)active(s(X)) → s(active(X))
active(p(X)) → p(active(X))f(mark(X)) → mark(f(X))
cons(mark(X1), X2) → mark(cons(X1, X2))s(mark(X)) → mark(s(X))
p(mark(X)) → mark(p(X))proper(f(X)) → f(proper(X))
proper(0) → ok(0)proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))proper(p(X)) → p(proper(X))
f(ok(X)) → ok(f(X))cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))p(ok(X)) → ok(p(X))
top(mark(X)) → top(proper(X))top(ok(X)) → top(active(X))

Original Signature

Termination of terms over the following signature is verified: f, 0, s, p, active, mark, ok, proper, cons, top

Strategy

The right-hand side of the rule top#(ok(s(f(p(p(cons(f(_x81), _x72))))))) → top#(s(f(p(p(cons(f(active(_x81)), _x72)))))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
top#(s(f(p(p(cons(f(mark(f(p(s(0))))), _x72)))))) 
top#(s(f(p(p(cons(f(mark(_x91)), _x72)))))) 
top#(s(f(p(p(cons(f(mark(cons(0, f(s(0))))), _x72)))))) 
top#(s(f(p(p(cons(f(f(active(_x91))), _x72)))))) 
top#(s(f(p(p(cons(f(p(active(_x91))), _x72)))))) 
top#(s(f(p(p(cons(f(s(active(_x91))), _x72)))))) 
top#(s(f(p(p(cons(f(cons(active(_x91), _x92)), _x72)))))) 
Thus, the rule top#(ok(s(f(p(p(cons(f(_x81), _x72))))))) → top#(s(f(p(p(cons(f(active(_x81)), _x72)))))) is replaced by the following rules:
top#(ok(s(f(p(p(cons(f(p(_x91)), _x72))))))) → top#(s(f(p(p(cons(f(p(active(_x91))), _x72))))))top#(ok(s(f(p(p(cons(f(p(s(_x91))), _x72))))))) → top#(s(f(p(p(cons(f(mark(_x91)), _x72))))))
top#(ok(s(f(p(p(cons(f(cons(_x91, _x92)), _x72))))))) → top#(s(f(p(p(cons(f(cons(active(_x91), _x92)), _x72))))))top#(ok(s(f(p(p(cons(f(s(_x91)), _x72))))))) → top#(s(f(p(p(cons(f(s(active(_x91))), _x72))))))
top#(ok(s(f(p(p(cons(f(f(s(0))), _x72))))))) → top#(s(f(p(p(cons(f(mark(f(p(s(0))))), _x72))))))top#(ok(s(f(p(p(cons(f(f(_x91)), _x72))))))) → top#(s(f(p(p(cons(f(f(active(_x91))), _x72))))))
top#(ok(s(f(p(p(cons(f(f(0)), _x72))))))) → top#(s(f(p(p(cons(f(mark(cons(0, f(s(0))))), _x72))))))

Problem 27: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

top#(ok(s(f(p(p(cons(f(p(s(_x91))), _x72))))))) → top#(s(f(p(p(cons(f(mark(_x91)), _x72))))))top#(ok(s(f(0)))) → top#(mark(s(cons(0, f(s(0))))))
top#(ok(s(f(p(p(cons(f(s(0)), _x72))))))) → top#(s(f(p(p(cons(mark(f(p(s(0)))), _x72))))))top#(ok(s(s(_x41)))) → top#(s(s(active(_x41))))
top#(mark(f(_x21))) → top#(f(proper(_x21)))top#(ok(s(p(f(0))))) → top#(s(p(mark(cons(0, f(s(0)))))))
top#(ok(cons(_x21, _x22))) → top#(cons(active(_x21), _x22))top#(ok(p(s(_x21)))) → top#(mark(_x21))
top#(ok(s(f(p(s(ok(_x51))))))) → top#(mark(s(ok(f(_x51)))))top#(ok(s(f(cons(_x51, _x52))))) → top#(s(f(cons(active(_x51), _x52))))
top#(ok(s(f(f(s(0)))))) → top#(s(f(mark(f(p(s(0)))))))top#(ok(s(f(p(p(f(0))))))) → top#(s(f(p(mark(p(cons(0, f(s(0)))))))))
top#(ok(s(p(p(s(_x51)))))) → top#(s(p(mark(_x51))))top#(ok(s(f(p(cons(_x61, _x62)))))) → top#(s(f(p(cons(active(_x61), _x62)))))
top#(ok(s(p(p(_x51))))) → top#(s(p(p(active(_x51)))))top#(ok(s(f(p(f(_x61)))))) → top#(s(f(p(f(active(_x61))))))
top#(ok(s(f(f(0))))) → top#(mark(s(f(cons(0, f(s(0)))))))top#(ok(s(f(p(p(s(_x71))))))) → top#(s(f(p(p(s(active(_x71)))))))
top#(mark(s(_x21))) → top#(s(proper(_x21)))top#(ok(s(p(s(_x51))))) → top#(s(p(s(active(_x51)))))
top#(ok(s(f(p(p(cons(f(f(_x91)), _x72))))))) → top#(s(f(p(p(cons(f(f(active(_x91))), _x72))))))top#(ok(s(f(p(s(mark(_x51))))))) → top#(mark(s(mark(f(_x51)))))
top#(ok(s(f(p(p(f(_x71))))))) → top#(s(f(p(p(f(active(_x71)))))))top#(ok(s(f(s(_x51))))) → top#(s(f(s(active(_x51)))))
top#(ok(s(f(p(f(0)))))) → top#(s(f(p(mark(cons(0, f(s(0))))))))top#(ok(s(p(s(_x41))))) → top#(s(mark(_x41)))
top#(ok(s(f(p(s(ok(ok(_x61)))))))) → top#(s(mark(ok(ok(f(_x61))))))top#(ok(s(cons(_x41, _x42)))) → top#(s(cons(active(_x41), _x42)))
top#(ok(s(f(p(p(cons(f(s(_x91)), _x72))))))) → top#(s(f(p(p(cons(f(s(active(_x91))), _x72))))))top#(ok(s(p(cons(_x51, _x52))))) → top#(s(p(cons(active(_x51), _x52))))
top#(ok(s(f(p(p(s(_x61))))))) → top#(s(f(p(mark(_x61)))))top#(ok(s(f(p(s(mark(mark(_x61)))))))) → top#(s(mark(mark(mark(f(_x61))))))
top#(ok(s(p(f(_x51))))) → top#(s(p(f(active(_x51)))))top#(ok(s(f(p(s(_x61)))))) → top#(s(f(p(s(active(_x61))))))
top#(ok(s(f(p(p(cons(p(_x81), _x72))))))) → top#(s(f(p(p(cons(p(active(_x81)), _x72))))))top#(ok(s(f(p(p(cons(f(cons(_x91, _x92)), _x72))))))) → top#(s(f(p(p(cons(f(cons(active(_x91), _x92)), _x72))))))
top#(ok(s(f(p(p(cons(s(_x81), _x72))))))) → top#(s(f(p(p(cons(s(active(_x81)), _x72))))))top#(ok(s(f(p(p(cons(p(s(_x81)), _x72))))))) → top#(s(f(p(p(cons(mark(_x81), _x72))))))
top#(ok(s(f(p(p(p(_x71))))))) → top#(s(f(p(p(p(active(_x71)))))))top#(ok(s(f(p(p(p(s(_x71)))))))) → top#(s(f(p(p(mark(_x71))))))
top#(ok(s(f(p(p(cons(f(f(0)), _x72))))))) → top#(s(f(p(p(cons(f(mark(cons(0, f(s(0))))), _x72))))))top#(ok(s(f(p(p(f(s(0)))))))) → top#(s(f(p(mark(p(f(p(s(0)))))))))
top#(ok(s(f(p(p(cons(f(p(_x91)), _x72))))))) → top#(s(f(p(p(cons(f(p(active(_x91))), _x72))))))top#(ok(f(0))) → top#(mark(cons(0, f(s(0)))))
top#(ok(s(f(p(s(ok(mark(_x61)))))))) → top#(s(mark(ok(mark(f(_x61))))))top#(ok(s(f(p(p(cons(f(f(s(0))), _x72))))))) → top#(s(f(p(p(cons(f(mark(f(p(s(0))))), _x72))))))
top#(mark(p(_x21))) → top#(p(proper(_x21)))top#(ok(s(f(p(p(cons(cons(_x81, _x82), _x72))))))) → top#(s(f(p(p(cons(cons(active(_x81), _x82), _x72))))))
top#(ok(s(f(s(0))))) → top#(s(mark(f(p(s(0))))))top#(ok(s(f(p(f(s(0))))))) → top#(s(f(p(mark(f(p(s(0))))))))
top#(mark(cons(_x21, _x22))) → top#(cons(proper(_x21), proper(_x22)))top#(ok(s(f(p(p(cons(f(0), _x72))))))) → top#(s(f(p(p(cons(mark(cons(0, f(s(0)))), _x72))))))
top#(ok(p(_x21))) → top#(p(active(_x21)))top#(ok(f(_x21))) → top#(f(active(_x21)))
top#(ok(s(f(p(s(_x31)))))) → top#(mark(s(f(_x31))))top#(ok(s(f(p(s(mark(ok(_x61)))))))) → top#(s(mark(mark(ok(f(_x61))))))
top#(ok(f(s(0)))) → top#(mark(f(p(s(0)))))top#(ok(s(f(f(_x51))))) → top#(s(f(f(active(_x51)))))
top#(ok(s(p(f(s(0)))))) → top#(s(p(mark(f(p(s(0)))))))

Rewrite Rules

active(f(0)) → mark(cons(0, f(s(0))))active(f(s(0))) → mark(f(p(s(0))))
active(p(s(X))) → mark(X)active(f(X)) → f(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)active(s(X)) → s(active(X))
active(p(X)) → p(active(X))f(mark(X)) → mark(f(X))
cons(mark(X1), X2) → mark(cons(X1, X2))s(mark(X)) → mark(s(X))
p(mark(X)) → mark(p(X))proper(f(X)) → f(proper(X))
proper(0) → ok(0)proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))proper(p(X)) → p(proper(X))
f(ok(X)) → ok(f(X))cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))p(ok(X)) → ok(p(X))
top(mark(X)) → top(proper(X))top(ok(X)) → top(active(X))

Original Signature

Termination of terms over the following signature is verified: f, 0, s, p, active, ok, mark, proper, top, cons

Strategy

The right-hand side of the rule top#(ok(s(f(p(p(cons(f(p(s(_x91))), _x72))))))) → top#(s(f(p(p(cons(f(mark(_x91)), _x72)))))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
top#(s(f(p(p(cons(mark(f(_x71)), _x72)))))) 
Thus, the rule top#(ok(s(f(p(p(cons(f(p(s(_x91))), _x72))))))) → top#(s(f(p(p(cons(f(mark(_x91)), _x72)))))) is replaced by the following rules:
top#(ok(s(f(p(p(cons(f(p(s(_x71))), _x72))))))) → top#(s(f(p(p(cons(mark(f(_x71)), _x72))))))

Problem 28: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

top#(mark(f(cons(s(cons(cons(0, _x72), f(p(cons(_x101, _x102))))), 0)))) → top#(f(cons(s(cons(cons(ok(0), proper(_x72)), f(p(cons(proper(_x101), proper(_x102)))))), ok(0))))top#(mark(f(cons(f(_x51), f(_x51))))) → top#(f(cons(f(proper(_x51)), f(proper(_x51)))))
top#(mark(f(cons(s(_x51), p(_x51))))) → top#(f(cons(s(proper(_x51)), p(proper(_x51)))))top#(mark(f(cons(s(cons(cons(_x71, _x72), f(p(cons(_x101, cons(_x111, _x112)))))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), proper(_x72)), f(p(cons(proper(_x101), cons(proper(_x111), proper(_x112))))))), ok(0))))
top#(mark(f(cons(s(cons(cons(_x71, p(_x81)), f(p(_x91)))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), p(proper(_x81))), f(p(proper(_x91))))), ok(0))))top#(mark(f(cons(f(0), s(s(_x71)))))) → top#(f(cons(f(ok(0)), s(s(proper(_x71))))))
top#(mark(f(cons(s(cons(cons(_x71, s(_x81)), f(p(cons(_x101, _x102))))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), s(proper(_x81))), f(p(cons(proper(_x101), proper(_x102)))))), ok(0))))top#(ok(s(f(p(cons(_x61, _x62)))))) → top#(s(f(p(cons(active(_x61), _x62)))))
top#(mark(f(cons(s(cons(cons(0, _x72), f(p(_x91)))), 0)))) → top#(f(cons(s(cons(cons(ok(0), proper(_x72)), f(p(proper(_x91))))), ok(0))))top#(mark(f(cons(s(cons(cons(_x71, p(0)), f(p(p(_x101))))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), p(ok(0))), f(p(p(proper(_x101)))))), ok(0))))
top#(ok(s(f(f(0))))) → top#(mark(s(f(cons(0, f(s(0)))))))top#(ok(s(f(p(f(_x61)))))) → top#(s(f(p(f(active(_x61))))))
top#(mark(f(cons(f(_x51), s(_x51))))) → top#(f(cons(f(proper(_x51)), s(proper(_x51)))))top#(mark(f(cons(s(cons(cons(s(_x81), _x72), f(p(p(_x101))))), 0)))) → top#(f(cons(s(cons(cons(s(proper(_x81)), proper(_x72)), f(p(p(proper(_x101)))))), ok(0))))
top#(mark(f(cons(s(cons(cons(_x71, _x72), f(p(0)))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), proper(_x72)), f(p(ok(0))))), ok(0))))top#(mark(f(cons(s(cons(_x61, p(s(_x91)))), 0)))) → top#(f(cons(s(cons(proper(_x61), p(s(proper(_x91))))), ok(0))))
top#(ok(s(f(p(p(cons(f(f(_x91)), _x72))))))) → top#(s(f(p(p(cons(f(f(active(_x91))), _x72))))))top#(mark(f(cons(s(cons(cons(_x71, _x72), f(p(p(cons(_x111, _x112)))))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), proper(_x72)), f(p(p(cons(proper(_x111), proper(_x112))))))), ok(0))))
top#(ok(s(cons(_x41, _x42)))) → top#(s(cons(active(_x41), _x42)))top#(ok(s(f(p(s(mark(mark(_x61)))))))) → top#(s(mark(mark(mark(f(_x61))))))
top#(mark(f(cons(s(s(_x61)), 0)))) → top#(f(cons(s(s(proper(_x61))), ok(0))))top#(mark(f(cons(s(cons(_x61, f(p(0)))), 0)))) → top#(f(cons(s(cons(proper(_x61), f(p(ok(0))))), ok(0))))
top#(mark(f(cons(s(cons(cons(_x71, _x72), f(p(cons(_x101, s(_x111)))))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), proper(_x72)), f(p(cons(proper(_x101), s(proper(_x111))))))), ok(0))))top#(mark(f(cons(s(cons(p(_x71), p(0))), 0)))) → top#(f(cons(s(cons(p(proper(_x71)), p(ok(0)))), ok(0))))
top#(mark(f(cons(s(cons(cons(_x71, cons(_x81, _x82)), f(p(p(s(_x111)))))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), cons(proper(_x81), proper(_x82))), f(p(p(s(proper(_x111))))))), ok(0))))top#(mark(f(cons(f(cons(_x61, _x62)), _x42)))) → top#(f(cons(f(cons(proper(_x61), proper(_x62))), proper(_x42))))
top#(ok(s(f(p(s(_x61)))))) → top#(s(f(p(s(active(_x61))))))top#(mark(f(cons(s(cons(cons(_x71, p(_x81)), f(p(cons(s(_x111), _x102))))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), p(proper(_x81))), f(p(cons(s(proper(_x111)), proper(_x102)))))), ok(0))))
top#(mark(f(cons(s(cons(_x61, f(p(f(_x101))))), 0)))) → top#(f(cons(s(cons(proper(_x61), f(p(f(proper(_x101)))))), ok(0))))top#(mark(f(cons(s(cons(cons(_x71, s(_x81)), f(p(_x91)))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), s(proper(_x81))), f(p(proper(_x91))))), ok(0))))
top#(mark(f(cons(s(cons(cons(cons(_x81, _x82), _x72), f(p(p(_x101))))), 0)))) → top#(f(cons(s(cons(cons(cons(proper(_x81), proper(_x82)), proper(_x72)), f(p(p(proper(_x101)))))), ok(0))))top#(ok(s(f(p(p(cons(s(_x81), _x72))))))) → top#(s(f(p(p(cons(s(active(_x81)), _x72))))))
top#(mark(f(cons(s(cons(_x61, p(0))), 0)))) → top#(f(cons(s(cons(proper(_x61), ok(p(0)))), ok(0))))top#(ok(s(f(p(p(f(s(0)))))))) → top#(s(f(p(mark(p(f(p(s(0)))))))))
top#(mark(f(cons(f(0), s(p(_x71)))))) → top#(f(cons(f(ok(0)), s(p(proper(_x71))))))top#(mark(f(cons(s(cons(cons(_x71, s(_x81)), f(p(p(_x101))))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), s(proper(_x81))), f(p(p(proper(_x101)))))), ok(0))))
top#(mark(f(cons(s(cons(_x61, f(s(_x91)))), 0)))) → top#(f(cons(s(cons(proper(_x61), f(s(proper(_x91))))), ok(0))))top#(ok(f(0))) → top#(mark(cons(0, f(s(0)))))
top#(mark(f(cons(s(cons(cons(_x71, p(f(_x101))), f(p(p(_x101))))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), p(f(proper(_x101)))), f(p(p(proper(_x101)))))), ok(0))))top#(mark(f(cons(f(cons(_x61, _x62)), 0)))) → top#(f(cons(f(cons(proper(_x61), proper(_x62))), ok(0))))
top#(mark(f(cons(s(cons(p(_x71), f(_x71))), 0)))) → top#(f(cons(s(cons(p(proper(_x71)), f(proper(_x71)))), ok(0))))top#(mark(f(cons(0, _x42)))) → top#(f(cons(ok(0), proper(_x42))))
top#(mark(f(cons(s(cons(cons(_x71, p(p(_x101))), f(p(cons(_x101, _x102))))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), p(p(proper(_x101)))), f(p(cons(proper(_x101), proper(_x102)))))), ok(0))))top#(mark(f(cons(s(cons(cons(f(_x81), f(_x81)), f(p(_x91)))), 0)))) → top#(f(cons(s(cons(cons(f(proper(_x81)), f(proper(_x81))), f(p(proper(_x91))))), ok(0))))
top#(ok(s(f(p(p(cons(cons(_x81, _x82), _x72))))))) → top#(s(f(p(p(cons(cons(active(_x81), _x82), _x72))))))top#(ok(s(s(f(0))))) → top#(s(s(mark(cons(0, f(s(0)))))))
top#(mark(f(cons(s(cons(cons(_x71, p(s(_x101))), f(p(cons(_x101, _x102))))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), p(s(proper(_x101)))), f(p(cons(proper(_x101), proper(_x102)))))), ok(0))))top#(mark(f(cons(_x41, s(s(_x71)))))) → top#(f(cons(proper(_x41), s(s(proper(_x71))))))
top#(mark(f(cons(s(cons(_x61, p(f(_x91)))), 0)))) → top#(f(cons(s(cons(proper(_x61), p(f(proper(_x91))))), ok(0))))top#(ok(s(s(p(_x51))))) → top#(s(s(p(active(_x51)))))
top#(mark(f(cons(_x41, s(f(_x71)))))) → top#(f(cons(proper(_x41), s(f(proper(_x71))))))top#(ok(f(_x21))) → top#(f(active(_x21)))
top#(ok(p(_x21))) → top#(p(active(_x21)))top#(ok(s(f(p(s(_x31)))))) → top#(mark(s(f(_x31))))
top#(ok(s(f(p(s(mark(ok(_x61)))))))) → top#(s(mark(mark(ok(f(_x61))))))top#(mark(f(cons(s(cons(cons(_x71, f(_x81)), f(p(cons(_x101, _x102))))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), f(proper(_x81))), f(p(cons(proper(_x101), proper(_x102)))))), ok(0))))
top#(mark(f(cons(s(cons(_x61, f(p(cons(_x101, _x102))))), 0)))) → top#(f(cons(s(cons(proper(_x61), f(p(cons(proper(_x101), proper(_x102)))))), ok(0))))top#(mark(f(cons(s(cons(cons(s(_x81), _x72), f(p(cons(_x101, _x102))))), 0)))) → top#(f(cons(s(cons(cons(s(proper(_x81)), proper(_x72)), f(p(cons(proper(_x101), proper(_x102)))))), ok(0))))
top#(ok(s(f(f(_x51))))) → top#(s(f(f(active(_x51)))))top#(mark(f(cons(f(0), p(_x51))))) → top#(f(cons(f(ok(0)), p(proper(_x51)))))
top#(mark(f(cons(s(cons(cons(s(_x81), cons(_x81, _x82)), f(p(p(_x101))))), 0)))) → top#(f(cons(s(cons(cons(s(proper(_x81)), cons(proper(_x81), proper(_x82))), f(p(p(proper(_x101)))))), ok(0))))top#(mark(f(cons(s(cons(cons(_x71, _x72), p(_x71))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), proper(_x72)), p(proper(_x71)))), ok(0))))
top#(mark(f(cons(s(cons(cons(_x71, f(cons(_x101, _x102))), f(p(_x91)))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), f(cons(proper(_x101), proper(_x102)))), f(p(proper(_x91))))), ok(0))))top#(mark(f(cons(s(cons(cons(_x71, _x72), f(p(cons(_x101, p(_x111)))))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), proper(_x72)), f(p(cons(proper(_x101), p(proper(_x111))))))), ok(0))))
top#(mark(f(cons(s(cons(_x61, p(cons(_x91, _x92)))), 0)))) → top#(f(cons(s(cons(proper(_x61), p(cons(proper(_x91), proper(_x92))))), ok(0))))top#(ok(s(f(0)))) → top#(mark(s(cons(0, f(s(0))))))
top#(mark(f(cons(s(cons(cons(0, f(_x81)), f(p(_x91)))), 0)))) → top#(f(cons(s(cons(cons(ok(0), f(proper(_x81))), f(p(proper(_x91))))), ok(0))))top#(mark(f(cons(s(cons(cons(_x71, p(_x81)), f(p(cons(_x101, 0))))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), p(proper(_x81))), f(p(cons(proper(_x101), ok(0)))))), ok(0))))
top#(mark(f(cons(s(cons(0, p(0))), 0)))) → top#(f(cons(s(cons(ok(0), p(ok(0)))), ok(0))))top#(mark(f(cons(s(cons(s(_x71), p(0))), 0)))) → top#(f(cons(s(cons(s(proper(_x71)), p(ok(0)))), ok(0))))
top#(mark(f(0))) → top#(f(ok(0)))top#(mark(f(cons(s(cons(cons(0, _x72), f(p(p(s(_x111)))))), 0)))) → top#(f(cons(s(cons(cons(ok(0), proper(_x72)), f(p(p(s(proper(_x111))))))), ok(0))))
top#(mark(f(cons(s(cons(cons(_x71, cons(s(_x101), _x82)), f(p(p(_x101))))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), cons(s(proper(_x101)), proper(_x82))), f(p(p(proper(_x101)))))), ok(0))))top#(mark(f(cons(s(cons(f(_x71), _x62)), 0)))) → top#(f(cons(s(cons(f(proper(_x71)), proper(_x62))), ok(0))))
top#(ok(s(p(f(0))))) → top#(s(p(mark(cons(0, f(s(0)))))))top#(mark(f(cons(s(cons(cons(_x71, p(f(_x101))), f(p(cons(_x101, _x102))))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), p(f(proper(_x101)))), f(p(cons(proper(_x101), proper(_x102)))))), ok(0))))
top#(ok(s(s(s(_x51))))) → top#(s(s(s(active(_x51)))))top#(ok(p(s(_x21)))) → top#(mark(_x21))
top#(ok(s(f(cons(_x51, _x52))))) → top#(s(f(cons(active(_x51), _x52))))top#(ok(cons(p(_x41), _x22))) → top#(cons(p(active(_x41)), _x22))
top#(mark(f(cons(s(cons(cons(_x71, p(_x81)), f(p(cons(f(_x111), _x102))))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), p(proper(_x81))), f(p(cons(f(proper(_x111)), proper(_x102)))))), ok(0))))top#(mark(f(cons(s(cons(cons(_x71, cons(_x81, p(_x111))), f(p(p(_x101))))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), cons(proper(_x81), p(proper(_x111)))), f(p(p(proper(_x101)))))), ok(0))))
top#(mark(f(cons(s(cons(cons(_x71, p(_x81)), f(p(cons(_x101, s(_x111)))))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), p(proper(_x81))), f(p(cons(proper(_x101), s(proper(_x111))))))), ok(0))))top#(mark(f(cons(s(cons(cons(_x71, _x72), f(p(cons(cons(_x111, _x112), _x102))))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), proper(_x72)), f(p(cons(cons(proper(_x111), proper(_x112)), proper(_x102)))))), ok(0))))
top#(mark(f(cons(s(cons(cons(_x71, _x72), f(p(cons(_x101, 0))))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), proper(_x72)), f(p(cons(proper(_x101), ok(0)))))), ok(0))))top#(mark(f(cons(s(cons(_x61, f(0))), 0)))) → top#(f(cons(s(cons(proper(_x61), f(ok(0)))), ok(0))))
top#(mark(f(cons(f(_x51), p(_x51))))) → top#(f(cons(f(proper(_x51)), p(proper(_x51)))))top#(mark(f(cons(s(p(_x61)), 0)))) → top#(f(cons(s(p(proper(_x61))), ok(0))))
top#(mark(f(cons(s(cons(cons(_x71, _x72), f(_x71))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), proper(_x72)), f(proper(_x71)))), ok(0))))top#(mark(f(cons(s(cons(_x61, p(p(_x91)))), 0)))) → top#(f(cons(s(cons(proper(_x61), p(p(proper(_x91))))), ok(0))))
top#(mark(s(_x21))) → top#(s(proper(_x21)))top#(mark(f(cons(s(cons(cons(p(_x81), f(_x81)), f(p(_x91)))), 0)))) → top#(f(cons(s(cons(cons(p(proper(_x81)), f(proper(_x81))), f(p(proper(_x91))))), ok(0))))
top#(ok(s(p(s(_x51))))) → top#(s(p(s(active(_x51)))))top#(mark(f(cons(s(cons(cons(cons(_x81, _x82), p(_x81)), f(p(cons(_x101, _x102))))), 0)))) → top#(f(cons(s(cons(cons(cons(proper(_x81), proper(_x82)), p(proper(_x81))), f(p(cons(proper(_x101), proper(_x102)))))), ok(0))))
top#(ok(s(f(p(s(mark(_x51))))))) → top#(mark(s(mark(f(_x51)))))top#(mark(f(cons(s(cons(p(_x71), _x62)), 0)))) → top#(f(cons(s(cons(p(proper(_x71)), proper(_x62))), ok(0))))
top#(ok(s(f(p(f(0)))))) → top#(s(f(p(mark(cons(0, f(s(0))))))))top#(mark(f(cons(s(cons(_x61, f(p(s(_x101))))), 0)))) → top#(f(cons(s(cons(proper(_x61), f(p(s(proper(_x101)))))), ok(0))))
top#(mark(f(cons(f(0), s(f(_x71)))))) → top#(f(cons(f(ok(0)), s(f(proper(_x71))))))top#(mark(f(cons(s(cons(f(_x71), f(_x71))), 0)))) → top#(f(cons(s(cons(f(proper(_x71)), f(proper(_x71)))), ok(0))))
top#(ok(s(f(p(p(cons(f(s(0)), _x62))))))) → top#(s(f(p(mark(p(cons(f(p(s(0))), _x62)))))))top#(mark(f(cons(s(_x51), cons(_x51, _x52))))) → top#(f(cons(s(proper(_x51)), cons(proper(_x51), proper(_x52)))))
top#(mark(f(p(_x41)))) → top#(f(p(proper(_x41))))top#(mark(f(cons(f(s(_x61)), _x42)))) → top#(f(cons(f(s(proper(_x61))), proper(_x42))))
top#(mark(f(cons(s(cons(cons(p(_x81), p(_x81)), f(p(cons(_x101, _x102))))), 0)))) → top#(f(cons(s(cons(cons(p(proper(_x81)), p(proper(_x81))), f(p(cons(proper(_x101), proper(_x102)))))), ok(0))))top#(mark(f(cons(s(cons(cons(_x71, p(p(_x101))), f(p(p(_x101))))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), p(p(proper(_x101)))), f(p(p(proper(_x101)))))), ok(0))))
top#(mark(f(cons(s(cons(p(_x71), f(p(_x91)))), 0)))) → top#(f(cons(s(cons(p(proper(_x71)), f(p(proper(_x91))))), ok(0))))top#(mark(f(f(_x41)))) → top#(f(f(proper(_x41))))
top#(mark(f(cons(s(cons(cons(0, 0), f(p(p(_x101))))), 0)))) → top#(f(cons(s(cons(cons(ok(0), ok(0)), f(p(p(proper(_x101)))))), ok(0))))top#(mark(f(cons(f(0), f(_x51))))) → top#(f(cons(f(ok(0)), f(proper(_x51)))))
top#(ok(s(f(p(p(cons(p(_x81), _x72))))))) → top#(s(f(p(p(cons(p(active(_x81)), _x72))))))top#(mark(f(cons(s(cons(cons(_x71, f(f(_x101))), f(p(_x91)))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), f(f(proper(_x101)))), f(p(proper(_x91))))), ok(0))))
top#(mark(f(cons(s(p(_x61)), _x42)))) → top#(f(cons(s(p(proper(_x61))), proper(_x42))))top#(mark(f(cons(s(cons(cons(_x71, p(_x81)), f(p(cons(cons(_x111, _x112), _x102))))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), p(proper(_x81))), f(p(cons(cons(proper(_x111), proper(_x112)), proper(_x102)))))), ok(0))))
top#(mark(f(s(_x41)))) → top#(f(s(proper(_x41))))top#(mark(f(cons(s(cons(cons(_x71, cons(_x81, _x82)), f(p(p(p(_x111)))))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), cons(proper(_x81), proper(_x82))), f(p(p(p(proper(_x111))))))), ok(0))))
top#(mark(f(cons(s(cons(cons(s(_x81), f(_x81)), f(p(_x91)))), 0)))) → top#(f(cons(s(cons(cons(s(proper(_x81)), f(proper(_x81))), f(p(proper(_x91))))), ok(0))))top#(ok(cons(f(s(0)), _x22))) → top#(cons(mark(f(p(s(0)))), _x22))
top#(mark(f(cons(s(cons(cons(0, _x72), f(p(p(p(_x111)))))), 0)))) → top#(f(cons(s(cons(cons(ok(0), proper(_x72)), f(p(p(p(proper(_x111))))))), ok(0))))top#(mark(f(cons(_x41, p(_x51))))) → top#(f(cons(proper(_x41), p(proper(_x51)))))
top#(ok(s(f(p(p(p(s(_x71)))))))) → top#(s(f(p(p(mark(_x71))))))top#(mark(f(cons(s(cons(cons(_x71, cons(_x81, f(_x111))), f(p(p(_x101))))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), cons(proper(_x81), f(proper(_x111)))), f(p(p(proper(_x101)))))), ok(0))))
top#(mark(f(cons(s(f(_x61)), _x42)))) → top#(f(cons(s(f(proper(_x61))), proper(_x42))))top#(mark(f(cons(_x41, s(0))))) → top#(f(cons(proper(_x41), s(ok(0)))))
top#(mark(f(cons(s(cons(cons(cons(_x81, _x82), p(_x81)), f(p(p(_x101))))), 0)))) → top#(f(cons(s(cons(cons(cons(proper(_x81), proper(_x82)), p(proper(_x81))), f(p(p(proper(_x101)))))), ok(0))))top#(mark(f(cons(s(cons(cons(_x71, p(_x81)), f(p(p(s(_x111)))))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), p(proper(_x81))), f(p(p(s(proper(_x111))))))), ok(0))))
top#(mark(f(cons(s(cons(cons(_x71, p(cons(_x101, _x102))), f(p(p(_x101))))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), p(cons(proper(_x101), proper(_x102)))), f(p(p(proper(_x101)))))), ok(0))))top#(mark(f(cons(f(s(_x61)), 0)))) → top#(f(cons(f(s(proper(_x61))), ok(0))))
top#(mark(f(cons(s(cons(cons(f(_x81), cons(_x81, _x82)), f(p(p(_x101))))), 0)))) → top#(f(cons(s(cons(cons(f(proper(_x81)), cons(proper(_x81), proper(_x82))), f(p(p(proper(_x101)))))), ok(0))))top#(mark(f(cons(s(cons(cons(_x71, cons(f(_x101), _x82)), f(p(p(_x101))))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), cons(f(proper(_x101)), proper(_x82))), f(p(p(proper(_x101)))))), ok(0))))
top#(ok(cons(p(s(_x41)), _x22))) → top#(cons(mark(_x41), _x22))top#(mark(f(cons(s(cons(cons(_x71, cons(_x81, _x82)), f(p(p(0))))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), cons(proper(_x81), proper(_x82))), f(p(p(ok(0)))))), ok(0))))
top#(mark(p(_x21))) → top#(p(proper(_x21)))top#(mark(f(cons(s(cons(cons(_x71, _x72), p(0))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), proper(_x72)), p(ok(0)))), ok(0))))
top#(mark(f(cons(s(0), _x42)))) → top#(f(cons(s(ok(0)), proper(_x42))))top#(mark(f(cons(s(cons(cons(_x71, _x72), f(p(cons(s(_x111), _x102))))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), proper(_x72)), f(p(cons(s(proper(_x111)), proper(_x102)))))), ok(0))))
top#(mark(f(cons(s(cons(_x61, _x62)), _x42)))) → top#(f(cons(s(cons(proper(_x61), proper(_x62))), proper(_x42))))top#(mark(f(cons(s(cons(cons(_x71, p(_x81)), f(p(cons(_x101, p(_x111)))))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), p(proper(_x81))), f(p(cons(proper(_x101), p(proper(_x111))))))), ok(0))))
top#(mark(f(cons(s(cons(cons(_x71, f(p(_x101))), f(p(_x91)))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), f(p(proper(_x101)))), f(p(proper(_x91))))), ok(0))))top#(mark(f(cons(s(cons(0, _x62)), 0)))) → top#(f(cons(s(cons(ok(0), proper(_x62))), ok(0))))
top#(mark(f(cons(s(cons(cons(_x71, 0), f(p(_x91)))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), ok(0)), f(p(proper(_x91))))), ok(0))))top#(mark(f(cons(f(0), s(0))))) → top#(f(cons(f(ok(0)), s(ok(0)))))
top#(mark(f(cons(s(cons(cons(p(_x81), _x72), f(p(cons(_x101, _x102))))), 0)))) → top#(f(cons(s(cons(cons(p(proper(_x81)), proper(_x72)), f(p(cons(proper(_x101), proper(_x102)))))), ok(0))))top#(mark(cons(_x21, _x22))) → top#(cons(proper(_x21), proper(_x22)))
top#(mark(f(cons(s(cons(cons(_x71, f(_x81)), f(p(p(_x101))))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), f(proper(_x81))), f(p(p(proper(_x101)))))), ok(0))))top#(mark(f(cons(s(cons(cons(_x71, f(_x81)), f(p(s(_x101))))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), f(proper(_x81))), f(p(s(proper(_x101)))))), ok(0))))
top#(mark(f(cons(s(cons(cons(_x71, 0), f(p(cons(_x101, _x102))))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), ok(0)), f(p(cons(proper(_x101), proper(_x102)))))), ok(0))))top#(ok(f(s(0)))) → top#(mark(f(p(s(0)))))
top#(mark(f(cons(s(cons(s(_x71), f(p(_x91)))), 0)))) → top#(f(cons(s(cons(s(proper(_x71)), f(p(proper(_x91))))), ok(0))))top#(mark(f(cons(f(p(_x61)), 0)))) → top#(f(cons(f(p(proper(_x61))), ok(0))))
top#(mark(f(cons(s(cons(cons(_x71, _x72), f(p(p(s(_x111)))))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), proper(_x72)), f(p(p(s(proper(_x111))))))), ok(0))))top#(mark(f(cons(s(cons(cons(_x71, cons(_x81, _x82)), f(p(_x91)))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), cons(proper(_x81), proper(_x82))), f(p(proper(_x91))))), ok(0))))
top#(mark(f(cons(p(_x51), s(_x51))))) → top#(f(cons(p(proper(_x51)), s(proper(_x51)))))top#(mark(f(cons(s(cons(cons(0, _x72), f(p(p(cons(_x111, _x112)))))), 0)))) → top#(f(cons(s(cons(cons(ok(0), proper(_x72)), f(p(p(cons(proper(_x111), proper(_x112))))))), ok(0))))
top#(mark(f(cons(s(cons(cons(_x71, _x72), f(p(s(_x101))))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), proper(_x72)), f(p(s(proper(_x101)))))), ok(0))))top#(ok(cons(f(0), _x32))) → top#(mark(cons(cons(0, f(s(0))), _x32)))
top#(mark(f(cons(s(cons(cons(_x71, p(_x81)), f(p(p(f(_x111)))))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), p(proper(_x81))), f(p(p(f(proper(_x111))))))), ok(0))))top#(mark(f(cons(s(cons(cons(_x71, cons(_x81, _x82)), f(p(cons(_x101, _x102))))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), cons(proper(_x81), proper(_x82))), f(p(cons(proper(_x101), proper(_x102)))))), ok(0))))
top#(mark(f(cons(s(cons(cons(_x71, _x72), f(p(p(0))))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), proper(_x72)), f(p(p(ok(0)))))), ok(0))))top#(mark(f(cons(s(cons(cons(_x71, cons(_x81, 0)), f(p(p(_x101))))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), cons(proper(_x81), ok(0))), f(p(p(proper(_x101)))))), ok(0))))
top#(mark(f(cons(s(cons(cons(_x71, p(_x81)), f(p(cons(_x101, f(_x111)))))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), p(proper(_x81))), f(p(cons(proper(_x101), f(proper(_x111))))))), ok(0))))top#(mark(f(cons(s(cons(cons(0, f(_x81)), f(p(p(_x101))))), 0)))) → top#(f(cons(s(cons(cons(ok(0), f(proper(_x81))), f(p(p(proper(_x101)))))), ok(0))))
top#(mark(f(cons(s(cons(cons(_x71, _x72), f(p(p(p(_x111)))))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), proper(_x72)), f(p(p(p(proper(_x111))))))), ok(0))))top#(mark(f(cons(s(cons(cons(f(_x81), p(_x81)), f(p(cons(_x101, _x102))))), 0)))) → top#(f(cons(s(cons(cons(f(proper(_x81)), p(proper(_x81))), f(p(cons(proper(_x101), proper(_x102)))))), ok(0))))
top#(ok(s(f(p(s(ok(_x51))))))) → top#(mark(s(ok(f(_x51)))))top#(ok(s(f(f(s(0)))))) → top#(mark(s(f(f(p(s(0)))))))
top#(mark(f(cons(f(p(_x61)), _x42)))) → top#(f(cons(f(p(proper(_x61))), proper(_x42))))top#(ok(s(p(p(s(_x51)))))) → top#(s(p(mark(_x51))))
top#(ok(s(p(p(_x51))))) → top#(s(p(p(active(_x51)))))top#(mark(f(cons(f(0), s(cons(_x71, _x72)))))) → top#(f(cons(f(ok(0)), s(cons(proper(_x71), proper(_x72))))))
top#(mark(f(cons(s(cons(_x61, s(_x71))), 0)))) → top#(f(cons(s(cons(proper(_x61), s(proper(_x71)))), ok(0))))top#(ok(s(s(cons(_x51, _x52))))) → top#(s(s(cons(active(_x51), _x52))))
top#(ok(s(f(p(p(s(_x71))))))) → top#(s(f(p(p(s(active(_x71)))))))top#(ok(cons(s(_x41), _x22))) → top#(cons(s(active(_x41)), _x22))
top#(mark(f(cons(s(cons(cons(p(_x81), cons(_x81, _x82)), f(p(p(_x101))))), 0)))) → top#(f(cons(s(cons(cons(p(proper(_x81)), cons(proper(_x81), proper(_x82))), f(p(p(proper(_x101)))))), ok(0))))top#(ok(s(f(p(p(f(_x71))))))) → top#(s(f(p(p(f(active(_x71)))))))
top#(ok(s(f(p(s(ok(ok(_x61)))))))) → top#(s(mark(ok(ok(f(_x61))))))top#(mark(f(cons(_x41, 0)))) → top#(f(cons(proper(_x41), ok(0))))
top#(mark(f(cons(cons(_x51, _x52), s(_x51))))) → top#(f(cons(cons(proper(_x51), proper(_x52)), s(proper(_x51)))))top#(ok(s(f(p(p(cons(f(s(_x91)), _x72))))))) → top#(s(f(p(p(cons(f(s(active(_x91))), _x72))))))
top#(mark(f(cons(s(cons(cons(_x71, f(_x81)), f(p(0)))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), f(proper(_x81))), f(p(ok(0))))), ok(0))))top#(mark(f(cons(s(cons(cons(_x71, cons(_x81, s(_x111))), f(p(p(_x101))))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), cons(proper(_x81), s(proper(_x111)))), f(p(p(proper(_x101)))))), ok(0))))
top#(mark(f(cons(s(cons(cons(_x71, p(_x81)), f(p(p(p(_x111)))))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), p(proper(_x81))), f(p(p(p(proper(_x111))))))), ok(0))))top#(mark(f(cons(s(cons(cons(s(_x81), p(_x81)), f(p(p(_x101))))), 0)))) → top#(f(cons(s(cons(cons(s(proper(_x81)), p(proper(_x81))), f(p(p(proper(_x101)))))), ok(0))))
top#(mark(f(cons(s(cons(cons(_x71, p(_x81)), f(p(p(0))))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), p(proper(_x81))), f(p(p(ok(0)))))), ok(0))))top#(mark(f(cons(s(cons(cons(s(_x81), p(_x81)), f(p(cons(_x101, _x102))))), 0)))) → top#(f(cons(s(cons(cons(s(proper(_x81)), p(proper(_x81))), f(p(cons(proper(_x101), proper(_x102)))))), ok(0))))
top#(mark(f(cons(s(cons(cons(0, p(_x81)), f(p(cons(_x101, _x102))))), 0)))) → top#(f(cons(s(cons(cons(ok(0), p(proper(_x81))), f(p(cons(proper(_x101), proper(_x102)))))), ok(0))))top#(ok(cons(cons(_x41, _x42), _x22))) → top#(cons(cons(active(_x41), _x42), _x22))
top#(mark(f(cons(s(cons(cons(_x71, f(s(_x101))), f(p(_x91)))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), f(s(proper(_x101)))), f(p(proper(_x91))))), ok(0))))top#(mark(f(cons(s(cons(cons(_x71, cons(cons(_x101, _x102), _x82)), f(p(p(_x101))))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), cons(cons(proper(_x101), proper(_x102)), proper(_x82))), f(p(p(proper(_x101)))))), ok(0))))
top#(mark(f(cons(s(cons(s(_x71), _x62)), 0)))) → top#(f(cons(s(cons(s(proper(_x71)), proper(_x62))), ok(0))))top#(mark(f(cons(s(cons(cons(_x71, _x72), f(p(cons(0, _x102))))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), proper(_x72)), f(p(cons(ok(0), proper(_x102)))))), ok(0))))
top#(mark(f(cons(s(cons(cons(p(_x81), _x72), f(p(p(_x101))))), 0)))) → top#(f(cons(s(cons(cons(p(proper(_x81)), proper(_x72)), f(p(p(proper(_x101)))))), ok(0))))top#(mark(f(cons(f(f(_x61)), _x42)))) → top#(f(cons(f(f(proper(_x61))), proper(_x42))))
top#(mark(f(cons(s(cons(cons(_x71, p(_x81)), f(p(cons(p(_x111), _x102))))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), p(proper(_x81))), f(p(cons(p(proper(_x111)), proper(_x102)))))), ok(0))))top#(mark(f(cons(s(cons(cons(_x71, _x72), _x62)), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), proper(_x72)), proper(_x62))), ok(0))))
top#(mark(f(cons(s(cons(cons(_x71, f(_x81)), f(p(f(_x101))))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), f(proper(_x81))), f(p(f(proper(_x101)))))), ok(0))))top#(mark(f(cons(s(cons(cons(0, s(_x81)), f(p(p(_x101))))), 0)))) → top#(f(cons(s(cons(cons(ok(0), s(proper(_x81))), f(p(p(proper(_x101)))))), ok(0))))
top#(mark(f(cons(s(cons(cons(cons(_x81, _x82), _x72), f(p(cons(_x101, _x102))))), 0)))) → top#(f(cons(s(cons(cons(cons(proper(_x81), proper(_x82)), proper(_x72)), f(p(cons(proper(_x101), proper(_x102)))))), ok(0))))top#(ok(s(f(p(p(cons(p(s(_x81)), _x72))))))) → top#(s(f(p(p(cons(mark(_x81), _x72))))))
top#(ok(s(f(p(p(cons(f(f(0)), _x72))))))) → top#(s(f(p(p(cons(f(mark(cons(0, f(s(0))))), _x72))))))top#(mark(f(cons(s(cons(cons(f(_x81), _x72), f(p(cons(_x101, _x102))))), 0)))) → top#(f(cons(s(cons(cons(f(proper(_x81)), proper(_x72)), f(p(cons(proper(_x101), proper(_x102)))))), ok(0))))
top#(mark(f(cons(s(cons(cons(_x71, p(_x81)), f(p(p(cons(_x111, _x112)))))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), p(proper(_x81))), f(p(p(cons(proper(_x111), proper(_x112))))))), ok(0))))top#(ok(s(f(p(p(cons(f(p(_x91)), _x72))))))) → top#(s(f(p(p(cons(f(p(active(_x91))), _x72))))))
top#(ok(cons(f(_x41), _x22))) → top#(cons(f(active(_x41)), _x22))top#(mark(f(cons(f(0), 0)))) → top#(f(cons(f(ok(0)), ok(0))))
top#(ok(s(f(p(s(ok(mark(_x61)))))))) → top#(s(mark(ok(mark(f(_x61))))))top#(mark(f(cons(s(cons(cons(_x71, _x72), f(p(cons(f(_x111), _x102))))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), proper(_x72)), f(p(cons(f(proper(_x111)), proper(_x102)))))), ok(0))))
top#(mark(f(cons(s(cons(cons(_x71, cons(0, _x82)), f(p(p(_x101))))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), cons(ok(0), proper(_x82))), f(p(p(proper(_x101)))))), ok(0))))top#(mark(f(cons(s(cons(cons(_x71, p(_x81)), f(p(cons(0, _x102))))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), p(proper(_x81))), f(p(cons(ok(0), proper(_x102)))))), ok(0))))
top#(mark(f(cons(s(cons(cons(_x71, 0), f(p(p(_x101))))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), ok(0)), f(p(p(proper(_x101)))))), ok(0))))top#(mark(f(cons(s(cons(cons(f(_x81), _x72), f(p(p(_x101))))), 0)))) → top#(f(cons(s(cons(cons(f(proper(_x81)), proper(_x72)), f(p(p(proper(_x101)))))), ok(0))))
top#(ok(s(f(s(0))))) → top#(s(mark(f(p(s(0))))))top#(mark(f(cons(s(cons(_x61, f(cons(_x91, _x92)))), 0)))) → top#(f(cons(s(cons(proper(_x61), f(cons(proper(_x91), proper(_x92))))), ok(0))))
top#(mark(f(cons(s(_x51), s(_x51))))) → top#(f(cons(s(proper(_x51)), s(proper(_x51)))))top#(mark(f(cons(s(cons(cons(cons(_x81, _x82), f(_x81)), f(p(_x91)))), 0)))) → top#(f(cons(s(cons(cons(cons(proper(_x81), proper(_x82)), f(proper(_x81))), f(p(proper(_x91))))), ok(0))))
top#(mark(f(cons(_x41, cons(_x51, _x52))))) → top#(f(cons(proper(_x41), cons(proper(_x51), proper(_x52)))))top#(mark(f(cons(s(cons(cons(_x71, p(0)), f(p(cons(_x101, _x102))))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), p(ok(0))), f(p(cons(proper(_x101), proper(_x102)))))), ok(0))))
top#(ok(s(p(f(s(0)))))) → top#(s(p(mark(f(p(s(0)))))))top#(ok(s(f(p(p(cons(f(p(s(_x71))), _x72))))))) → top#(s(f(p(p(cons(mark(f(_x71)), _x72))))))
top#(mark(f(cons(s(cons(cons(p(_x81), _x72), f(p(_x91)))), 0)))) → top#(f(cons(s(cons(cons(p(proper(_x81)), proper(_x72)), f(p(proper(_x91))))), ok(0))))top#(mark(f(cons(_x41, s(cons(_x71, _x72)))))) → top#(f(cons(proper(_x41), s(cons(proper(_x71), proper(_x72))))))
top#(mark(f(cons(s(cons(f(_x71), p(_x71))), 0)))) → top#(f(cons(s(cons(f(proper(_x71)), p(proper(_x71)))), ok(0))))top#(mark(f(cons(s(cons(f(_x71), p(0))), 0)))) → top#(f(cons(s(cons(f(proper(_x71)), p(ok(0)))), ok(0))))
top#(mark(f(cons(s(cons(_x61, f(p(p(_x101))))), 0)))) → top#(f(cons(s(cons(proper(_x61), f(p(p(proper(_x101)))))), ok(0))))top#(mark(f(cons(f(0), s(_x51))))) → top#(f(cons(ok(f(0)), s(proper(_x51)))))
top#(mark(f(cons(s(cons(s(_x71), f(_x71))), 0)))) → top#(f(cons(s(cons(s(proper(_x71)), f(proper(_x71)))), ok(0))))top#(mark(f(cons(s(cons(cons(f(_x81), p(_x81)), f(p(p(_x101))))), 0)))) → top#(f(cons(s(cons(cons(f(proper(_x81)), p(proper(_x81))), f(p(p(proper(_x101)))))), ok(0))))
top#(mark(f(cons(s(cons(0, f(_x71))), 0)))) → top#(f(cons(s(cons(ok(0), f(proper(_x71)))), ok(0))))top#(mark(f(cons(s(cons(cons(_x71, _x72), f(p(cons(_x101, f(_x111)))))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), proper(_x72)), f(p(cons(proper(_x101), f(proper(_x111))))))), ok(0))))
top#(mark(f(cons(0, s(_x51))))) → top#(f(cons(ok(0), s(proper(_x51)))))top#(mark(f(cons(s(cons(p(_x71), p(_x71))), 0)))) → top#(f(cons(s(cons(p(proper(_x71)), p(proper(_x71)))), ok(0))))
top#(ok(s(f(p(p(f(0))))))) → top#(s(f(p(mark(p(cons(0, f(s(0)))))))))top#(mark(f(cons(s(0), 0)))) → top#(f(cons(s(ok(0)), ok(0))))
top#(mark(f(cons(s(cons(cons(f(_x81), _x72), f(p(_x91)))), 0)))) → top#(f(cons(s(cons(cons(f(proper(_x81)), proper(_x72)), f(p(proper(_x91))))), ok(0))))top#(mark(f(cons(s(cons(cons(p(_x81), p(_x81)), f(p(p(_x101))))), 0)))) → top#(f(cons(s(cons(cons(p(proper(_x81)), p(proper(_x81))), f(p(p(proper(_x101)))))), ok(0))))
top#(ok(s(s(f(_x51))))) → top#(s(s(f(active(_x51)))))top#(ok(s(f(s(_x51))))) → top#(s(f(s(active(_x51)))))
top#(mark(f(cons(s(cons(_x61, f(f(_x91)))), 0)))) → top#(f(cons(s(cons(proper(_x61), f(f(proper(_x91))))), ok(0))))top#(ok(s(p(s(_x41))))) → top#(s(mark(_x41)))
top#(ok(s(p(cons(_x51, _x52))))) → top#(s(p(cons(active(_x51), _x52))))top#(ok(s(f(p(p(s(_x61))))))) → top#(s(f(p(mark(_x61)))))
top#(mark(f(cons(s(cons(cons(_x71, _x72), f(p(p(f(_x111)))))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), proper(_x72)), f(p(p(f(proper(_x111))))))), ok(0))))top#(ok(s(p(f(_x51))))) → top#(s(p(f(active(_x51)))))
top#(mark(f(cons(s(cons(cons(_x71, f(0)), f(p(_x91)))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), f(ok(0))), f(p(proper(_x91))))), ok(0))))top#(mark(f(cons(f(_x51), cons(_x51, _x52))))) → top#(f(cons(f(proper(_x51)), cons(proper(_x51), proper(_x52)))))
top#(mark(f(cons(s(cons(s(_x71), p(_x71))), 0)))) → top#(f(cons(s(cons(s(proper(_x71)), p(proper(_x71)))), ok(0))))top#(mark(f(cons(s(cons(cons(_x71, p(cons(_x101, _x102))), f(p(cons(_x101, _x102))))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), p(cons(proper(_x101), proper(_x102)))), f(p(cons(proper(_x101), proper(_x102)))))), ok(0))))
top#(mark(f(cons(f(0), _x42)))) → top#(f(cons(ok(f(0)), proper(_x42))))top#(ok(s(f(p(p(cons(f(cons(_x91, _x92)), _x72))))))) → top#(s(f(p(p(cons(f(cons(active(_x91), _x92)), _x72))))))
top#(ok(s(s(f(s(0)))))) → top#(s(mark(s(f(p(s(0)))))))top#(mark(f(cons(s(s(_x61)), _x42)))) → top#(f(cons(s(s(proper(_x61))), proper(_x42))))
top#(mark(f(cons(_x41, f(_x51))))) → top#(f(cons(proper(_x41), f(proper(_x51)))))top#(mark(f(cons(s(cons(cons(0, _x72), f(p(p(f(_x111)))))), 0)))) → top#(f(cons(s(cons(cons(ok(0), proper(_x72)), f(p(p(f(proper(_x111))))))), ok(0))))
top#(mark(f(cons(cons(_x51, _x52), _x42)))) → top#(f(cons(cons(proper(_x51), proper(_x52)), proper(_x42))))top#(ok(s(f(p(p(p(_x71))))))) → top#(s(f(p(p(p(active(_x71)))))))
top#(ok(s(s(p(s(_x51)))))) → top#(s(s(mark(_x51))))top#(mark(f(cons(s(cons(cons(_x71, cons(_x81, _x82)), f(p(p(cons(_x111, _x112)))))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), cons(proper(_x81), proper(_x82))), f(p(p(cons(proper(_x111), proper(_x112))))))), ok(0))))
top#(mark(f(cons(s(f(_x61)), 0)))) → top#(f(cons(s(f(proper(_x61))), ok(0))))top#(ok(s(f(p(p(cons(f(f(s(0))), _x72))))))) → top#(s(f(p(p(cons(f(mark(f(p(s(0))))), _x72))))))
top#(mark(f(cons(p(_x51), _x42)))) → top#(f(cons(p(proper(_x51)), proper(_x42))))top#(mark(f(cons(s(cons(cons(_x71, _x72), f(p(cons(p(_x111), _x102))))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), proper(_x72)), f(p(cons(p(proper(_x111)), proper(_x102)))))), ok(0))))
top#(mark(f(cons(s(cons(cons(_x71, cons(_x81, _x82)), f(p(p(f(_x111)))))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), cons(proper(_x81), proper(_x82))), f(p(p(f(proper(_x111))))))), ok(0))))top#(mark(f(cons(s(cons(0, f(p(_x91)))), 0)))) → top#(f(cons(s(cons(ok(0), f(p(proper(_x91))))), ok(0))))
top#(mark(f(cons(s(cons(cons(_x71, _x72), f(p(f(_x101))))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), proper(_x72)), f(p(f(proper(_x101)))))), ok(0))))top#(ok(s(f(p(f(s(0))))))) → top#(s(f(p(mark(f(p(s(0))))))))
top#(mark(f(cons(_x41, s(p(_x71)))))) → top#(f(cons(proper(_x41), s(p(proper(_x71))))))top#(ok(s(f(p(p(cons(f(0), _x72))))))) → top#(s(f(p(p(cons(mark(cons(0, f(s(0)))), _x72))))))
top#(mark(f(cons(s(cons(cons(0, p(_x81)), f(p(p(_x101))))), 0)))) → top#(f(cons(s(cons(cons(ok(0), p(proper(_x81))), f(p(p(proper(_x101)))))), ok(0))))top#(mark(f(cons(s(cons(f(_x71), f(p(_x91)))), 0)))) → top#(f(cons(s(cons(f(proper(_x71)), f(p(proper(_x91))))), ok(0))))
top#(mark(f(cons(s(cons(cons(cons(_x81, _x82), cons(_x81, _x82)), f(p(p(_x101))))), 0)))) → top#(f(cons(s(cons(cons(cons(proper(_x81), proper(_x82)), cons(proper(_x81), proper(_x82))), f(p(p(proper(_x101)))))), ok(0))))top#(mark(f(cons(f(f(_x61)), 0)))) → top#(f(cons(f(f(proper(_x61))), ok(0))))
top#(mark(f(cons(s(cons(cons(0, cons(_x81, _x82)), f(p(p(_x101))))), 0)))) → top#(f(cons(s(cons(cons(ok(0), cons(proper(_x81), proper(_x82))), f(p(p(proper(_x101)))))), ok(0))))top#(mark(f(cons(s(_x51), f(_x51))))) → top#(f(cons(s(proper(_x51)), f(proper(_x51)))))
top#(mark(f(cons(s(cons(cons(_x71, p(_x81)), f(p(cons(_x101, cons(_x111, _x112)))))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), p(proper(_x81))), f(p(cons(proper(_x101), cons(proper(_x111), proper(_x112))))))), ok(0))))top#(mark(f(cons(s(cons(cons(s(_x81), _x72), f(p(_x91)))), 0)))) → top#(f(cons(s(cons(cons(s(proper(_x81)), proper(_x72)), f(p(proper(_x91))))), ok(0))))
top#(mark(f(cons(s(cons(cons(_x71, p(s(_x101))), f(p(p(_x101))))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), p(s(proper(_x101)))), f(p(p(proper(_x101)))))), ok(0))))top#(mark(f(cons(s(cons(cons(cons(_x81, _x82), _x72), f(p(_x91)))), 0)))) → top#(f(cons(s(cons(cons(cons(proper(_x81), proper(_x82)), proper(_x72)), f(p(proper(_x91))))), ok(0))))
top#(mark(f(cons(s(cons(cons(0, _x72), f(p(p(0))))), 0)))) → top#(f(cons(s(cons(cons(ok(0), proper(_x72)), f(p(p(ok(0)))))), ok(0))))top#(mark(f(cons(f(0), cons(_x51, _x52))))) → top#(f(cons(f(ok(0)), cons(proper(_x51), proper(_x52)))))
top#(mark(f(cons(s(cons(_x61, 0)), 0)))) → top#(f(cons(s(cons(proper(_x61), ok(0))), ok(0))))top#(mark(f(cons(s(cons(cons(_x71, cons(_x81, cons(_x111, _x112))), f(p(p(_x101))))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), cons(proper(_x81), cons(proper(_x111), proper(_x112)))), f(p(p(proper(_x101)))))), ok(0))))
top#(mark(f(cons(s(cons(cons(_x71, cons(p(_x101), _x82)), f(p(p(_x101))))), 0)))) → top#(f(cons(s(cons(cons(proper(_x71), cons(p(proper(_x101)), proper(_x82))), f(p(p(proper(_x101)))))), ok(0))))top#(mark(f(cons(s(cons(0, p(_x71))), 0)))) → top#(f(cons(s(cons(ok(0), p(proper(_x71)))), ok(0))))
top#(mark(f(cons(s(cons(_x61, cons(_x71, _x72))), 0)))) → top#(f(cons(s(cons(proper(_x61), cons(proper(_x71), proper(_x72)))), ok(0))))

Rewrite Rules

active(f(0)) → mark(cons(0, f(s(0))))active(f(s(0))) → mark(f(p(s(0))))
active(p(s(X))) → mark(X)active(f(X)) → f(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)active(s(X)) → s(active(X))
active(p(X)) → p(active(X))f(mark(X)) → mark(f(X))
cons(mark(X1), X2) → mark(cons(X1, X2))s(mark(X)) → mark(s(X))
p(mark(X)) → mark(p(X))proper(f(X)) → f(proper(X))
proper(0) → ok(0)proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))proper(p(X)) → p(proper(X))
f(ok(X)) → ok(f(X))cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))p(ok(X)) → ok(p(X))
top(mark(X)) → top(proper(X))top(ok(X)) → top(active(X))

Original Signature

Termination of terms over the following signature is verified: f, 0, s, p, active, mark, ok, proper, cons, top

Strategy

The right-hand side of the rule top#(mark(f(cons(s(cons(cons(0, _x72), f(p(cons(_x101, _x102))))), 0)))) → top#(f(cons(s(cons(cons(ok(0), proper(_x72)), f(p(cons(proper(_x101), proper(_x102)))))), ok(0)))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
top#(f(cons(s(cons(cons(ok(0), proper(_x72)), f(p(cons(cons(proper(_x111), proper(_x112)), proper(_x102)))))), ok(0)))) 
top#(f(cons(s(cons(cons(ok(0), proper(_x72)), f(p(cons(f(proper(_x111)), proper(_x102)))))), ok(0)))) 
top#(f(cons(s(cons(cons(ok(0), f(proper(_x81))), f(p(cons(proper(_x101), proper(_x102)))))), ok(0)))) 
top#(f(cons(s(cons(cons(ok(0), ok(0)), f(p(cons(proper(_x101), proper(_x102)))))), ok(0)))) 
top#(f(cons(s(cons(cons(ok(0), proper(_x72)), f(p(cons(proper(_x101), ok(0)))))), ok(0)))) 
top#(f(cons(s(cons(cons(ok(0), proper(_x72)), f(p(cons(proper(_x101), f(proper(_x111))))))), ok(0)))) 
top#(f(cons(s(cons(cons(ok(0), p(proper(_x81))), f(p(cons(proper(_x101), proper(_x102)))))), ok(0)))) 
top#(f(cons(s(cons(cons(ok(0), proper(_x72)), f(p(cons(p(proper(_x111)), proper(_x102)))))), ok(0)))) 
top#(f(cons(s(cons(cons(ok(0), s(proper(_x81))), f(p(cons(proper(_x101), proper(_x102)))))), ok(0)))) 
top#(f(cons(s(cons(cons(ok(0), proper(_x72)), f(p(cons(ok(0), proper(_x102)))))), ok(0)))) 
top#(f(cons(s(cons(cons(ok(0), cons(proper(_x81), proper(_x82))), f(p(cons(proper(_x101), proper(_x102)))))), ok(0)))) 
top#(f(cons(s(cons(cons(ok(0), proper(_x72)), f(p(cons(proper(_x101), s(proper(_x111))))))), ok(0)))) 
top#(f(cons(s(cons(cons(ok(0), proper(_x72)), f(p(cons(proper(_x101), cons(proper(_x111), proper(_x112))))))), ok(0)))) 
top#(f(cons(s(cons(cons(ok(0), proper(_x72)), f(p(cons(s(proper(_x111)), proper(_x102)))))), ok(0)))) 
top#(f(cons(s(cons(cons(ok(0), proper(_x72)), f(p(cons(proper(_x101), p(proper(_x111))))))), ok(0)))) 
Thus, the rule top#(mark(f(cons(s(cons(cons(0, _x72), f(p(cons(_x101, _x102))))), 0)))) → top#(f(cons(s(cons(cons(ok(0), proper(_x72)), f(p(cons(proper(_x101), proper(_x102)))))), ok(0)))) is replaced by the following rules:
top#(mark(f(cons(s(cons(cons(0, p(_x81)), f(p(cons(_x101, _x102))))), 0)))) → top#(f(cons(s(cons(cons(ok(0), p(proper(_x81))), f(p(cons(proper(_x101), proper(_x102)))))), ok(0))))top#(mark(f(cons(s(cons(cons(0, s(_x81)), f(p(cons(_x101, _x102))))), 0)))) → top#(f(cons(s(cons(cons(ok(0), s(proper(_x81))), f(p(cons(proper(_x101), proper(_x102)))))), ok(0))))
top#(mark(f(cons(s(cons(cons(0, _x72), f(p(cons(_x101, 0))))), 0)))) → top#(f(cons(s(cons(cons(ok(0), proper(_x72)), f(p(cons(proper(_x101), ok(0)))))), ok(0))))top#(mark(f(cons(s(cons(cons(0, _x72), f(p(cons(p(_x111), _x102))))), 0)))) → top#(f(cons(s(cons(cons(ok(0), proper(_x72)), f(p(cons(p(proper(_x111)), proper(_x102)))))), ok(0))))
top#(mark(f(cons(s(cons(cons(0, _x72), f(p(cons(_x101, f(_x111)))))), 0)))) → top#(f(cons(s(cons(cons(ok(0), proper(_x72)), f(p(cons(proper(_x101), f(proper(_x111))))))), ok(0))))top#(mark(f(cons(s(cons(cons(0, _x72), f(p(cons(_x101, cons(_x111, _x112)))))), 0)))) → top#(f(cons(s(cons(cons(ok(0), proper(_x72)), f(p(cons(proper(_x101), cons(proper(_x111), proper(_x112))))))), ok(0))))
top#(mark(f(cons(s(cons(cons(0, _x72), f(p(cons(0, _x102))))), 0)))) → top#(f(cons(s(cons(cons(ok(0), proper(_x72)), f(p(cons(ok(0), proper(_x102)))))), ok(0))))top#(mark(f(cons(s(cons(cons(0, _x72), f(p(cons(_x101, s(_x111)))))), 0)))) → top#(f(cons(s(cons(cons(ok(0), proper(_x72)), f(p(cons(proper(_x101), s(proper(_x111))))))), ok(0))))
top#(mark(f(cons(s(cons(cons(0, f(_x81)), f(p(cons(_x101, _x102))))), 0)))) → top#(f(cons(s(cons(cons(ok(0), f(proper(_x81))), f(p(cons(proper(_x101), proper(_x102)))))), ok(0))))top#(mark(f(cons(s(cons(cons(0, _x72), f(p(cons(f(_x111), _x102))))), 0)))) → top#(f(cons(s(cons(cons(ok(0), proper(_x72)), f(p(cons(f(proper(_x111)), proper(_x102)))))), ok(0))))
top#(mark(f(cons(s(cons(cons(0, 0), f(p(cons(_x101, _x102))))), 0)))) → top#(f(cons(s(cons(cons(ok(0), ok(0)), f(p(cons(proper(_x101), proper(_x102)))))), ok(0))))top#(mark(f(cons(s(cons(cons(0, _x72), f(p(cons(cons(_x111, _x112), _x102))))), 0)))) → top#(f(cons(s(cons(cons(ok(0), proper(_x72)), f(p(cons(cons(proper(_x111), proper(_x112)), proper(_x102)))))), ok(0))))
top#(mark(f(cons(s(cons(cons(0, _x72), f(p(cons(s(_x111), _x102))))), 0)))) → top#(f(cons(s(cons(cons(ok(0), proper(_x72)), f(p(cons(s(proper(_x111)), proper(_x102)))))), ok(0))))top#(mark(f(cons(s(cons(cons(0, cons(_x81, _x82)), f(p(cons(_x101, _x102))))), 0)))) → top#(f(cons(s(cons(cons(ok(0), cons(proper(_x81), proper(_x82))), f(p(cons(proper(_x101), proper(_x102)))))), ok(0))))
top#(mark(f(cons(s(cons(cons(0, _x72), f(p(cons(_x101, p(_x111)))))), 0)))) → top#(f(cons(s(cons(cons(ok(0), proper(_x72)), f(p(cons(proper(_x101), p(proper(_x111))))))), ok(0))))