TIMEOUT

The TRS could not be proven terminating. The proof attempt took 60002 ms.

The following DP Processors were used


Problem 1 was processed with processor DependencyGraph (334ms).
 | – Problem 2 was processed with processor ForwardNarrowing (74ms).
 |    | – Problem 10 was processed with processor ForwardNarrowing (5ms).
 |    |    | – Problem 11 was processed with processor ForwardNarrowing (6ms).
 |    |    |    | – Problem 12 was processed with processor ForwardNarrowing (38ms).
 |    |    |    |    | – Problem 13 was processed with processor ForwardNarrowing (3ms).
 |    |    |    |    |    | – Problem 14 was processed with processor ForwardNarrowing (5ms).
 |    |    |    |    |    |    | – Problem 15 was processed with processor ForwardNarrowing (5ms).
 |    |    |    |    |    |    |    | – Problem 16 was processed with processor ForwardNarrowing (3ms).
 |    |    |    |    |    |    |    |    | – Problem 17 was processed with processor ForwardNarrowing (6ms).
 |    |    |    |    |    |    |    |    |    | – Problem 18 was processed with processor ForwardNarrowing (10ms).
 |    |    |    |    |    |    |    |    |    |    | – Problem 19 was processed with processor ForwardNarrowing (7ms).
 |    |    |    |    |    |    |    |    |    |    |    | – Problem 20 was processed with processor ForwardNarrowing (14ms).
 |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 21 was processed with processor ForwardNarrowing (7ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 22 was processed with processor ForwardNarrowing (10ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 23 was processed with processor ForwardNarrowing (10ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 24 was processed with processor ForwardNarrowing (10ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 25 was processed with processor ForwardNarrowing (9ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 26 was processed with processor ForwardNarrowing (13ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 27 was processed with processor ForwardNarrowing (9ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 28 was processed with processor ForwardNarrowing (9ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 29 was processed with processor ForwardNarrowing (30ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 30 was processed with processor ForwardNarrowing (230ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 31 was processed with processor ForwardNarrowing (357ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 32 remains open; application of the following processors failed [ForwardNarrowing (170ms), ForwardNarrowing (464ms), ForwardNarrowing (180ms), ForwardNarrowing (447ms), ForwardNarrowing (448ms), ForwardNarrowing (628ms), ForwardNarrowing (779ms), ForwardNarrowing (630ms), ForwardNarrowing (448ms), ForwardNarrowing (477ms), ForwardNarrowing (184ms), ForwardNarrowing (373ms), ForwardNarrowing (800ms), ForwardNarrowing (795ms), ForwardNarrowing (200ms), ForwardNarrowing (200ms), ForwardNarrowing (823ms), ForwardNarrowing (1267ms), ForwardNarrowing (238ms), ForwardNarrowing (1222ms), ForwardNarrowing (311ms), ForwardNarrowing (1270ms), ForwardNarrowing (timeout)].
 | – Problem 3 was processed with processor SubtermCriterion (0ms).
 | – Problem 4 was processed with processor SubtermCriterion (3ms).
 |    | – Problem 6 was processed with processor ReductionPairSAT (105ms).
 |    |    | – Problem 8 was processed with processor ReductionPairSAT (22ms).
 | – Problem 5 was processed with processor SubtermCriterion (1ms).
 |    | – Problem 7 was processed with processor ReductionPairSAT (47ms).
 |    |    | – Problem 9 was processed with processor ReductionPairSAT (82ms).

The following open problems remain:



Open Dependency Pair Problem 2

Dependency Pairs

mark#(a)active#(a)mark#(g(X1, X2))mark#(X1)
active#(g(a, X))mark#(f(b, X))active#(f(X, X))mark#(h(a))
mark#(b)active#(b)mark#(h(X))active#(h(mark(X)))
mark#(f(X1, X2))active#(f(mark(X1), X2))mark#(g(X1, X2))active#(g(mark(X1), X2))
active#(h(X))mark#(g(X, X))active#(a)mark#(b)
mark#(h(X))mark#(X)mark#(f(X1, X2))mark#(X1)

Rewrite Rules

active(h(X))mark(g(X, X))active(g(a, X))mark(f(b, X))
active(f(X, X))mark(h(a))active(a)mark(b)
mark(h(X))active(h(mark(X)))mark(g(X1, X2))active(g(mark(X1), X2))
mark(a)active(a)mark(f(X1, X2))active(f(mark(X1), X2))
mark(b)active(b)h(mark(X))h(X)
h(active(X))h(X)g(mark(X1), X2)g(X1, X2)
g(X1, mark(X2))g(X1, X2)g(active(X1), X2)g(X1, X2)
g(X1, active(X2))g(X1, X2)f(mark(X1), X2)f(X1, X2)
f(X1, mark(X2))f(X1, X2)f(active(X1), X2)f(X1, X2)
f(X1, active(X2))f(X1, X2)

Original Signature

Termination of terms over the following signature is verified: f, g, b, a, active, mark, h


Problem 1: DependencyGraph



Dependency Pair Problem

Dependency Pairs

h#(mark(X))h#(X)g#(X1, active(X2))g#(X1, X2)
mark#(a)active#(a)g#(active(X1), X2)g#(X1, X2)
mark#(f(X1, X2))f#(mark(X1), X2)active#(f(X, X))mark#(h(a))
f#(X1, mark(X2))f#(X1, X2)mark#(b)active#(b)
mark#(g(X1, X2))g#(mark(X1), X2)mark#(h(X))active#(h(mark(X)))
active#(f(X, X))h#(a)mark#(g(X1, X2))active#(g(mark(X1), X2))
f#(active(X1), X2)f#(X1, X2)active#(a)mark#(b)
active#(h(X))mark#(g(X, X))h#(active(X))h#(X)
g#(mark(X1), X2)g#(X1, X2)g#(X1, mark(X2))g#(X1, X2)
active#(h(X))g#(X, X)f#(X1, active(X2))f#(X1, X2)
mark#(g(X1, X2))mark#(X1)active#(g(a, X))mark#(f(b, X))
active#(g(a, X))f#(b, X)mark#(f(X1, X2))active#(f(mark(X1), X2))
f#(mark(X1), X2)f#(X1, X2)mark#(h(X))mark#(X)
mark#(f(X1, X2))mark#(X1)mark#(h(X))h#(mark(X))

Rewrite Rules

active(h(X))mark(g(X, X))active(g(a, X))mark(f(b, X))
active(f(X, X))mark(h(a))active(a)mark(b)
mark(h(X))active(h(mark(X)))mark(g(X1, X2))active(g(mark(X1), X2))
mark(a)active(a)mark(f(X1, X2))active(f(mark(X1), X2))
mark(b)active(b)h(mark(X))h(X)
h(active(X))h(X)g(mark(X1), X2)g(X1, X2)
g(X1, mark(X2))g(X1, X2)g(active(X1), X2)g(X1, X2)
g(X1, active(X2))g(X1, X2)f(mark(X1), X2)f(X1, X2)
f(X1, mark(X2))f(X1, X2)f(active(X1), X2)f(X1, X2)
f(X1, active(X2))f(X1, X2)

Original Signature

Termination of terms over the following signature is verified: f, g, b, a, active, mark, h

Strategy


The following SCCs where found

active#(f(X, X)) → mark#(h(a))mark#(b) → active#(b)
mark#(f(X1, X2)) → active#(f(mark(X1), X2))mark#(h(X)) → active#(h(mark(X)))
mark#(g(X1, X2)) → active#(g(mark(X1), X2))mark#(a) → active#(a)
active#(a) → mark#(b)active#(h(X)) → mark#(g(X, X))
mark#(g(X1, X2)) → mark#(X1)mark#(h(X)) → mark#(X)
active#(g(a, X)) → mark#(f(b, X))mark#(f(X1, X2)) → mark#(X1)

g#(X1, mark(X2)) → g#(X1, X2)g#(X1, active(X2)) → g#(X1, X2)
g#(active(X1), X2) → g#(X1, X2)g#(mark(X1), X2) → g#(X1, X2)

f#(X1, mark(X2)) → f#(X1, X2)f#(X1, active(X2)) → f#(X1, X2)
f#(active(X1), X2) → f#(X1, X2)f#(mark(X1), X2) → f#(X1, X2)

h#(mark(X)) → h#(X)h#(active(X)) → h#(X)

Problem 2: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

active#(f(X, X))mark#(h(a))mark#(b)active#(b)
mark#(f(X1, X2))active#(f(mark(X1), X2))mark#(h(X))active#(h(mark(X)))
mark#(g(X1, X2))active#(g(mark(X1), X2))mark#(a)active#(a)
active#(a)mark#(b)active#(h(X))mark#(g(X, X))
mark#(h(X))mark#(X)mark#(g(X1, X2))mark#(X1)
active#(g(a, X))mark#(f(b, X))mark#(f(X1, X2))mark#(X1)

Rewrite Rules

active(h(X))mark(g(X, X))active(g(a, X))mark(f(b, X))
active(f(X, X))mark(h(a))active(a)mark(b)
mark(h(X))active(h(mark(X)))mark(g(X1, X2))active(g(mark(X1), X2))
mark(a)active(a)mark(f(X1, X2))active(f(mark(X1), X2))
mark(b)active(b)h(mark(X))h(X)
h(active(X))h(X)g(mark(X1), X2)g(X1, X2)
g(X1, mark(X2))g(X1, X2)g(active(X1), X2)g(X1, X2)
g(X1, active(X2))g(X1, X2)f(mark(X1), X2)f(X1, X2)
f(X1, mark(X2))f(X1, X2)f(active(X1), X2)f(X1, X2)
f(X1, active(X2))f(X1, X2)

Original Signature

Termination of terms over the following signature is verified: f, g, b, a, active, mark, h

Strategy


The right-hand side of the rule mark#(b) → active#(b) 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 mark#(b) → active#(b) is deleted.

Problem 10: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

active#(f(X, X))mark#(h(a))mark#(h(X))active#(h(mark(X)))
mark#(f(X1, X2))active#(f(mark(X1), X2))mark#(g(X1, X2))active#(g(mark(X1), X2))
mark#(a)active#(a)active#(h(X))mark#(g(X, X))
active#(a)mark#(b)mark#(g(X1, X2))mark#(X1)
mark#(h(X))mark#(X)active#(g(a, X))mark#(f(b, X))
mark#(f(X1, X2))mark#(X1)

Rewrite Rules

active(h(X))mark(g(X, X))active(g(a, X))mark(f(b, X))
active(f(X, X))mark(h(a))active(a)mark(b)
mark(h(X))active(h(mark(X)))mark(g(X1, X2))active(g(mark(X1), X2))
mark(a)active(a)mark(f(X1, X2))active(f(mark(X1), X2))
mark(b)active(b)h(mark(X))h(X)
h(active(X))h(X)g(mark(X1), X2)g(X1, X2)
g(X1, mark(X2))g(X1, X2)g(active(X1), X2)g(X1, X2)
g(X1, active(X2))g(X1, X2)f(mark(X1), X2)f(X1, X2)
f(X1, mark(X2))f(X1, X2)f(active(X1), X2)f(X1, X2)
f(X1, active(X2))f(X1, X2)

Original Signature

Termination of terms over the following signature is verified: f, g, b, a, active, mark, h

Strategy


The right-hand side of the rule mark#(g(X1, X2)) → active#(g(mark(X1), X2)) 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
active#(g(active(f(mark(_x31), _x32)), X2)) 
active#(g(active(g(mark(_x31), _x32)), X2)) 
active#(g(_x21, _x22)) 
active#(g(mark(X1), _x22)) 
active#(g(active(h(mark(_x31))), X2)) 
active#(g(active(b), X2)) 
active#(g(active(a), X2)) 
Thus, the rule mark#(g(X1, X2)) → active#(g(mark(X1), X2)) is replaced by the following rules:
mark#(g(b, X2)) → active#(g(active(b), X2))mark#(g(g(_x31, _x32), X2)) → active#(g(active(g(mark(_x31), _x32)), X2))
mark#(g(a, X2)) → active#(g(active(a), X2))mark#(g(X1, mark(_x22))) → active#(g(mark(X1), _x22))
mark#(g(h(_x31), X2)) → active#(g(active(h(mark(_x31))), X2))mark#(g(f(_x31, _x32), X2)) → active#(g(active(f(mark(_x31), _x32)), X2))
mark#(g(_x21, _x22)) → active#(g(_x21, _x22))mark#(g(X1, active(_x22))) → active#(g(mark(X1), _x22))

Problem 11: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

mark#(g(b, X2))active#(g(active(b), X2))mark#(g(a, X2))active#(g(active(a), X2))
mark#(a)active#(a)mark#(g(f(_x31, _x32), X2))active#(g(active(f(mark(_x31), _x32)), X2))
mark#(g(X1, X2))mark#(X1)mark#(g(X1, active(_x22)))active#(g(mark(X1), _x22))
mark#(g(_x21, _x22))active#(g(_x21, _x22))active#(g(a, X))mark#(f(b, X))
active#(f(X, X))mark#(h(a))mark#(f(X1, X2))active#(f(mark(X1), X2))
mark#(h(X))active#(h(mark(X)))mark#(g(g(_x31, _x32), X2))active#(g(active(g(mark(_x31), _x32)), X2))
active#(h(X))mark#(g(X, X))active#(a)mark#(b)
mark#(g(X1, mark(_x22)))active#(g(mark(X1), _x22))mark#(g(h(_x31), X2))active#(g(active(h(mark(_x31))), X2))
mark#(h(X))mark#(X)mark#(f(X1, X2))mark#(X1)

Rewrite Rules

active(h(X))mark(g(X, X))active(g(a, X))mark(f(b, X))
active(f(X, X))mark(h(a))active(a)mark(b)
mark(h(X))active(h(mark(X)))mark(g(X1, X2))active(g(mark(X1), X2))
mark(a)active(a)mark(f(X1, X2))active(f(mark(X1), X2))
mark(b)active(b)h(mark(X))h(X)
h(active(X))h(X)g(mark(X1), X2)g(X1, X2)
g(X1, mark(X2))g(X1, X2)g(active(X1), X2)g(X1, X2)
g(X1, active(X2))g(X1, X2)f(mark(X1), X2)f(X1, X2)
f(X1, mark(X2))f(X1, X2)f(active(X1), X2)f(X1, X2)
f(X1, active(X2))f(X1, X2)

Original Signature

Termination of terms over the following signature is verified: f, g, b, a, active, mark, h

Strategy


The right-hand side of the rule mark#(g(b, X2)) → active#(g(active(b), X2)) 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
active#(g(b, _x22)) 
active#(g(active(b), _x22)) 
Thus, the rule mark#(g(b, X2)) → active#(g(active(b), X2)) is replaced by the following rules:
mark#(g(b, mark(_x22))) → active#(g(active(b), _x22))mark#(g(b, active(_x22))) → active#(g(active(b), _x22))
mark#(g(b, _x22)) → active#(g(b, _x22))

Problem 12: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

mark#(g(a, X2))active#(g(active(a), X2))mark#(a)active#(a)
mark#(g(f(_x31, _x32), X2))active#(g(active(f(mark(_x31), _x32)), X2))mark#(g(X1, X2))mark#(X1)
mark#(g(X1, active(_x22)))active#(g(mark(X1), _x22))mark#(g(_x21, _x22))active#(g(_x21, _x22))
active#(g(a, X))mark#(f(b, X))mark#(g(b, mark(_x22)))active#(g(active(b), _x22))
active#(f(X, X))mark#(h(a))mark#(g(b, active(_x22)))active#(g(active(b), _x22))
mark#(g(b, _x22))active#(g(b, _x22))mark#(f(X1, X2))active#(f(mark(X1), X2))
mark#(h(X))active#(h(mark(X)))mark#(g(g(_x31, _x32), X2))active#(g(active(g(mark(_x31), _x32)), X2))
mark#(g(h(_x31), X2))active#(g(active(h(mark(_x31))), X2))mark#(g(X1, mark(_x22)))active#(g(mark(X1), _x22))
active#(a)mark#(b)active#(h(X))mark#(g(X, X))
mark#(h(X))mark#(X)mark#(f(X1, X2))mark#(X1)

Rewrite Rules

active(h(X))mark(g(X, X))active(g(a, X))mark(f(b, X))
active(f(X, X))mark(h(a))active(a)mark(b)
mark(h(X))active(h(mark(X)))mark(g(X1, X2))active(g(mark(X1), X2))
mark(a)active(a)mark(f(X1, X2))active(f(mark(X1), X2))
mark(b)active(b)h(mark(X))h(X)
h(active(X))h(X)g(mark(X1), X2)g(X1, X2)
g(X1, mark(X2))g(X1, X2)g(active(X1), X2)g(X1, X2)
g(X1, active(X2))g(X1, X2)f(mark(X1), X2)f(X1, X2)
f(X1, mark(X2))f(X1, X2)f(active(X1), X2)f(X1, X2)
f(X1, active(X2))f(X1, X2)

Original Signature

Termination of terms over the following signature is verified: f, g, b, a, active, mark, h

Strategy


The right-hand side of the rule mark#(g(a, X2)) → active#(g(active(a), X2)) 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
active#(g(active(a), _x22)) 
active#(g(a, _x22)) 
active#(g(mark(b), X2)) 
Thus, the rule mark#(g(a, X2)) → active#(g(active(a), X2)) is replaced by the following rules:
mark#(g(a, active(_x22))) → active#(g(active(a), _x22))mark#(g(a, _x22)) → active#(g(a, _x22))
mark#(g(a, mark(_x22))) → active#(g(active(a), _x22))mark#(g(a, X2)) → active#(g(mark(b), X2))

Problem 13: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

mark#(g(a, _x22))active#(g(a, _x22))mark#(g(a, active(_x22)))active#(g(active(a), _x22))
mark#(a)active#(a)mark#(g(f(_x31, _x32), X2))active#(g(active(f(mark(_x31), _x32)), X2))
mark#(g(X1, X2))mark#(X1)mark#(g(X1, active(_x22)))active#(g(mark(X1), _x22))
mark#(g(_x21, _x22))active#(g(_x21, _x22))active#(g(a, X))mark#(f(b, X))
mark#(g(b, mark(_x22)))active#(g(active(b), _x22))active#(f(X, X))mark#(h(a))
mark#(g(b, active(_x22)))active#(g(active(b), _x22))mark#(h(X))active#(h(mark(X)))
mark#(g(b, _x22))active#(g(b, _x22))mark#(f(X1, X2))active#(f(mark(X1), X2))
mark#(g(g(_x31, _x32), X2))active#(g(active(g(mark(_x31), _x32)), X2))active#(h(X))mark#(g(X, X))
active#(a)mark#(b)mark#(g(X1, mark(_x22)))active#(g(mark(X1), _x22))
mark#(g(h(_x31), X2))active#(g(active(h(mark(_x31))), X2))mark#(g(a, mark(_x22)))active#(g(active(a), _x22))
mark#(h(X))mark#(X)mark#(g(a, X2))active#(g(mark(b), X2))
mark#(f(X1, X2))mark#(X1)

Rewrite Rules

active(h(X))mark(g(X, X))active(g(a, X))mark(f(b, X))
active(f(X, X))mark(h(a))active(a)mark(b)
mark(h(X))active(h(mark(X)))mark(g(X1, X2))active(g(mark(X1), X2))
mark(a)active(a)mark(f(X1, X2))active(f(mark(X1), X2))
mark(b)active(b)h(mark(X))h(X)
h(active(X))h(X)g(mark(X1), X2)g(X1, X2)
g(X1, mark(X2))g(X1, X2)g(active(X1), X2)g(X1, X2)
g(X1, active(X2))g(X1, X2)f(mark(X1), X2)f(X1, X2)
f(X1, mark(X2))f(X1, X2)f(active(X1), X2)f(X1, X2)
f(X1, active(X2))f(X1, X2)

Original Signature

Termination of terms over the following signature is verified: f, g, b, a, active, mark, h

Strategy


The right-hand side of the rule mark#(g(a, active(_x22))) → active#(g(active(a), _x22)) 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
active#(g(mark(b), _x22)) 
active#(g(a, _x32)) 
active#(g(active(a), _x32)) 
Thus, the rule mark#(g(a, active(_x22))) → active#(g(active(a), _x22)) is replaced by the following rules:
mark#(g(a, active(active(_x32)))) → active#(g(active(a), _x32))mark#(g(a, active(_x32))) → active#(g(a, _x32))
mark#(g(a, active(mark(_x32)))) → active#(g(active(a), _x32))mark#(g(a, active(_x22))) → active#(g(mark(b), _x22))

Problem 14: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

mark#(g(a, _x22))active#(g(a, _x22))mark#(g(a, active(active(_x32))))active#(g(active(a), _x32))
mark#(g(a, active(mark(_x32))))active#(g(active(a), _x32))mark#(a)active#(a)
mark#(g(f(_x31, _x32), X2))active#(g(active(f(mark(_x31), _x32)), X2))mark#(g(b, mark(_x22)))active#(g(active(b), _x22))
active#(f(X, X))mark#(h(a))mark#(h(X))active#(h(mark(X)))
mark#(g(b, _x22))active#(g(b, _x22))active#(h(X))mark#(g(X, X))
active#(a)mark#(b)mark#(g(h(_x31), X2))active#(g(active(h(mark(_x31))), X2))
mark#(g(a, X2))active#(g(mark(b), X2))mark#(g(a, active(_x32)))active#(g(a, _x32))
mark#(g(a, active(_x22)))active#(g(mark(b), _x22))mark#(g(X1, X2))mark#(X1)
mark#(g(_x21, _x22))active#(g(_x21, _x22))mark#(g(X1, active(_x22)))active#(g(mark(X1), _x22))
active#(g(a, X))mark#(f(b, X))mark#(g(b, active(_x22)))active#(g(active(b), _x22))
mark#(f(X1, X2))active#(f(mark(X1), X2))mark#(g(g(_x31, _x32), X2))active#(g(active(g(mark(_x31), _x32)), X2))
mark#(g(X1, mark(_x22)))active#(g(mark(X1), _x22))mark#(g(a, mark(_x22)))active#(g(active(a), _x22))
mark#(h(X))mark#(X)mark#(f(X1, X2))mark#(X1)

Rewrite Rules

active(h(X))mark(g(X, X))active(g(a, X))mark(f(b, X))
active(f(X, X))mark(h(a))active(a)mark(b)
mark(h(X))active(h(mark(X)))mark(g(X1, X2))active(g(mark(X1), X2))
mark(a)active(a)mark(f(X1, X2))active(f(mark(X1), X2))
mark(b)active(b)h(mark(X))h(X)
h(active(X))h(X)g(mark(X1), X2)g(X1, X2)
g(X1, mark(X2))g(X1, X2)g(active(X1), X2)g(X1, X2)
g(X1, active(X2))g(X1, X2)f(mark(X1), X2)f(X1, X2)
f(X1, mark(X2))f(X1, X2)f(active(X1), X2)f(X1, X2)
f(X1, active(X2))f(X1, X2)

Original Signature

Termination of terms over the following signature is verified: f, g, b, a, active, mark, h

Strategy


The right-hand side of the rule mark#(g(a, active(active(_x32)))) → active#(g(active(a), _x32)) 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
active#(g(active(a), _x22)) 
active#(g(mark(b), _x32)) 
active#(g(a, _x22)) 
Thus, the rule mark#(g(a, active(active(_x32)))) → active#(g(active(a), _x32)) is replaced by the following rules:
mark#(g(a, active(active(_x22)))) → active#(g(a, _x22))mark#(g(a, active(active(_x32)))) → active#(g(mark(b), _x32))
mark#(g(a, active(active(mark(_x22))))) → active#(g(active(a), _x22))mark#(g(a, active(active(active(_x22))))) → active#(g(active(a), _x22))

Problem 15: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

mark#(g(a, _x22))active#(g(a, _x22))mark#(g(a, active(active(mark(_x22)))))active#(g(active(a), _x22))
mark#(a)active#(a)mark#(g(a, active(mark(_x32))))active#(g(active(a), _x32))
mark#(g(f(_x31, _x32), X2))active#(g(active(f(mark(_x31), _x32)), X2))mark#(g(a, active(active(active(_x22)))))active#(g(active(a), _x22))
active#(f(X, X))mark#(h(a))mark#(g(b, mark(_x22)))active#(g(active(b), _x22))
mark#(g(b, _x22))active#(g(b, _x22))mark#(h(X))active#(h(mark(X)))
mark#(g(a, active(active(_x32))))active#(g(mark(b), _x32))active#(a)mark#(b)
active#(h(X))mark#(g(X, X))mark#(g(h(_x31), X2))active#(g(active(h(mark(_x31))), X2))
mark#(g(a, X2))active#(g(mark(b), X2))mark#(g(a, active(_x32)))active#(g(a, _x32))
mark#(g(a, active(_x22)))active#(g(mark(b), _x22))mark#(g(X1, X2))mark#(X1)
mark#(g(_x21, _x22))active#(g(_x21, _x22))mark#(g(X1, active(_x22)))active#(g(mark(X1), _x22))
active#(g(a, X))mark#(f(b, X))mark#(g(a, active(active(_x22))))active#(g(a, _x22))
mark#(g(b, active(_x22)))active#(g(active(b), _x22))mark#(f(X1, X2))active#(f(mark(X1), X2))
mark#(g(g(_x31, _x32), X2))active#(g(active(g(mark(_x31), _x32)), X2))mark#(g(X1, mark(_x22)))active#(g(mark(X1), _x22))
mark#(g(a, mark(_x22)))active#(g(active(a), _x22))mark#(h(X))mark#(X)
mark#(f(X1, X2))mark#(X1)

Rewrite Rules

active(h(X))mark(g(X, X))active(g(a, X))mark(f(b, X))
active(f(X, X))mark(h(a))active(a)mark(b)
mark(h(X))active(h(mark(X)))mark(g(X1, X2))active(g(mark(X1), X2))
mark(a)active(a)mark(f(X1, X2))active(f(mark(X1), X2))
mark(b)active(b)h(mark(X))h(X)
h(active(X))h(X)g(mark(X1), X2)g(X1, X2)
g(X1, mark(X2))g(X1, X2)g(active(X1), X2)g(X1, X2)
g(X1, active(X2))g(X1, X2)f(mark(X1), X2)f(X1, X2)
f(X1, mark(X2))f(X1, X2)f(active(X1), X2)f(X1, X2)
f(X1, active(X2))f(X1, X2)

Original Signature

Termination of terms over the following signature is verified: f, g, b, a, active, mark, h

Strategy


The right-hand side of the rule mark#(g(a, active(active(mark(_x22))))) → active#(g(active(a), _x22)) 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
active#(g(mark(b), _x22)) 
active#(g(a, _x32)) 
active#(g(active(a), _x32)) 
Thus, the rule mark#(g(a, active(active(mark(_x22))))) → active#(g(active(a), _x22)) is replaced by the following rules:
mark#(g(a, active(active(mark(active(_x32)))))) → active#(g(active(a), _x32))mark#(g(a, active(active(mark(_x22))))) → active#(g(mark(b), _x22))
mark#(g(a, active(active(mark(mark(_x32)))))) → active#(g(active(a), _x32))mark#(g(a, active(active(mark(_x32))))) → active#(g(a, _x32))

Problem 16: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

mark#(g(a, _x22))active#(g(a, _x22))mark#(g(a, active(active(mark(active(_x32))))))active#(g(active(a), _x32))
mark#(g(a, active(mark(_x32))))active#(g(active(a), _x32))mark#(a)active#(a)
mark#(g(a, active(active(mark(_x32)))))active#(g(a, _x32))mark#(g(f(_x31, _x32), X2))active#(g(active(f(mark(_x31), _x32)), X2))
mark#(g(a, active(active(active(_x22)))))active#(g(active(a), _x22))mark#(g(b, mark(_x22)))active#(g(active(b), _x22))
active#(f(X, X))mark#(h(a))mark#(h(X))active#(h(mark(X)))
mark#(g(b, _x22))active#(g(b, _x22))mark#(g(a, active(active(_x32))))active#(g(mark(b), _x32))
active#(a)mark#(b)active#(h(X))mark#(g(X, X))
mark#(g(h(_x31), X2))active#(g(active(h(mark(_x31))), X2))mark#(g(a, X2))active#(g(mark(b), X2))
mark#(g(a, active(active(mark(_x22)))))active#(g(mark(b), _x22))mark#(g(a, active(_x32)))active#(g(a, _x32))
mark#(g(a, active(_x22)))active#(g(mark(b), _x22))mark#(g(X1, X2))mark#(X1)
mark#(g(_x21, _x22))active#(g(_x21, _x22))mark#(g(X1, active(_x22)))active#(g(mark(X1), _x22))
active#(g(a, X))mark#(f(b, X))mark#(g(a, active(active(_x22))))active#(g(a, _x22))
mark#(g(b, active(_x22)))active#(g(active(b), _x22))mark#(f(X1, X2))active#(f(mark(X1), X2))
mark#(g(g(_x31, _x32), X2))active#(g(active(g(mark(_x31), _x32)), X2))mark#(g(X1, mark(_x22)))active#(g(mark(X1), _x22))
mark#(g(a, active(active(mark(mark(_x32))))))active#(g(active(a), _x32))mark#(g(a, mark(_x22)))active#(g(active(a), _x22))
mark#(h(X))mark#(X)mark#(f(X1, X2))mark#(X1)

Rewrite Rules

active(h(X))mark(g(X, X))active(g(a, X))mark(f(b, X))
active(f(X, X))mark(h(a))active(a)mark(b)
mark(h(X))active(h(mark(X)))mark(g(X1, X2))active(g(mark(X1), X2))
mark(a)active(a)mark(f(X1, X2))active(f(mark(X1), X2))
mark(b)active(b)h(mark(X))h(X)
h(active(X))h(X)g(mark(X1), X2)g(X1, X2)
g(X1, mark(X2))g(X1, X2)g(active(X1), X2)g(X1, X2)
g(X1, active(X2))g(X1, X2)f(mark(X1), X2)f(X1, X2)
f(X1, mark(X2))f(X1, X2)f(active(X1), X2)f(X1, X2)
f(X1, active(X2))f(X1, X2)

Original Signature

Termination of terms over the following signature is verified: f, g, b, a, active, mark, h

Strategy


The right-hand side of the rule mark#(g(a, active(active(mark(active(_x32)))))) → active#(g(active(a), _x32)) 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
active#(g(active(a), _x22)) 
active#(g(mark(b), _x32)) 
active#(g(a, _x22)) 
Thus, the rule mark#(g(a, active(active(mark(active(_x32)))))) → active#(g(active(a), _x32)) is replaced by the following rules:
mark#(g(a, active(active(mark(active(_x22)))))) → active#(g(a, _x22))mark#(g(a, active(active(mark(active(_x32)))))) → active#(g(mark(b), _x32))
mark#(g(a, active(active(mark(active(mark(_x22))))))) → active#(g(active(a), _x22))mark#(g(a, active(active(mark(active(active(_x22))))))) → active#(g(active(a), _x22))

Problem 17: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

mark#(g(a, active(active(mark(active(_x22))))))active#(g(a, _x22))mark#(g(a, _x22))active#(g(a, _x22))
mark#(a)active#(a)mark#(g(a, active(mark(_x32))))active#(g(active(a), _x32))
mark#(g(a, active(active(mark(_x32)))))active#(g(a, _x32))mark#(g(f(_x31, _x32), X2))active#(g(active(f(mark(_x31), _x32)), X2))
mark#(g(a, active(active(active(_x22)))))active#(g(active(a), _x22))active#(f(X, X))mark#(h(a))
mark#(g(b, mark(_x22)))active#(g(active(b), _x22))mark#(g(b, _x22))active#(g(b, _x22))
mark#(h(X))active#(h(mark(X)))mark#(g(a, active(active(_x32))))active#(g(mark(b), _x32))
active#(a)mark#(b)active#(h(X))mark#(g(X, X))
mark#(g(h(_x31), X2))active#(g(active(h(mark(_x31))), X2))mark#(g(a, X2))active#(g(mark(b), X2))
mark#(g(a, active(active(mark(active(_x32))))))active#(g(mark(b), _x32))mark#(g(a, active(active(mark(_x22)))))active#(g(mark(b), _x22))
mark#(g(a, active(_x32)))active#(g(a, _x32))mark#(g(a, active(_x22)))active#(g(mark(b), _x22))
mark#(g(a, active(active(mark(active(active(_x22)))))))active#(g(active(a), _x22))mark#(g(X1, X2))mark#(X1)
mark#(g(_x21, _x22))active#(g(_x21, _x22))mark#(g(X1, active(_x22)))active#(g(mark(X1), _x22))
active#(g(a, X))mark#(f(b, X))mark#(g(a, active(active(_x22))))active#(g(a, _x22))
mark#(g(b, active(_x22)))active#(g(active(b), _x22))mark#(f(X1, X2))active#(f(mark(X1), X2))
mark#(g(g(_x31, _x32), X2))active#(g(active(g(mark(_x31), _x32)), X2))mark#(g(a, active(active(mark(mark(_x32))))))active#(g(active(a), _x32))
mark#(g(X1, mark(_x22)))active#(g(mark(X1), _x22))mark#(g(a, mark(_x22)))active#(g(active(a), _x22))
mark#(g(a, active(active(mark(active(mark(_x22)))))))active#(g(active(a), _x22))mark#(h(X))mark#(X)
mark#(f(X1, X2))mark#(X1)

Rewrite Rules

active(h(X))mark(g(X, X))active(g(a, X))mark(f(b, X))
active(f(X, X))mark(h(a))active(a)mark(b)
mark(h(X))active(h(mark(X)))mark(g(X1, X2))active(g(mark(X1), X2))
mark(a)active(a)mark(f(X1, X2))active(f(mark(X1), X2))
mark(b)active(b)h(mark(X))h(X)
h(active(X))h(X)g(mark(X1), X2)g(X1, X2)
g(X1, mark(X2))g(X1, X2)g(active(X1), X2)g(X1, X2)
g(X1, active(X2))g(X1, X2)f(mark(X1), X2)f(X1, X2)
f(X1, mark(X2))f(X1, X2)f(active(X1), X2)f(X1, X2)
f(X1, active(X2))f(X1, X2)

Original Signature

Termination of terms over the following signature is verified: f, g, b, a, active, mark, h

Strategy


The right-hand side of the rule mark#(g(a, active(mark(_x32)))) → active#(g(active(a), _x32)) 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
active#(g(active(a), _x22)) 
active#(g(mark(b), _x32)) 
active#(g(a, _x22)) 
Thus, the rule mark#(g(a, active(mark(_x32)))) → active#(g(active(a), _x32)) is replaced by the following rules:
mark#(g(a, active(mark(mark(_x22))))) → active#(g(active(a), _x22))mark#(g(a, active(mark(_x32)))) → active#(g(mark(b), _x32))
mark#(g(a, active(mark(active(_x22))))) → active#(g(active(a), _x22))mark#(g(a, active(mark(_x22)))) → active#(g(a, _x22))

Problem 18: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

mark#(g(a, active(active(mark(active(_x22))))))active#(g(a, _x22))mark#(g(a, _x22))active#(g(a, _x22))
mark#(a)active#(a)mark#(g(a, active(active(mark(_x32)))))active#(g(a, _x32))
mark#(g(a, active(mark(_x22))))active#(g(a, _x22))mark#(g(f(_x31, _x32), X2))active#(g(active(f(mark(_x31), _x32)), X2))
mark#(g(a, active(active(active(_x22)))))active#(g(active(a), _x22))mark#(g(b, mark(_x22)))active#(g(active(b), _x22))
active#(f(X, X))mark#(h(a))mark#(h(X))active#(h(mark(X)))
mark#(g(b, _x22))active#(g(b, _x22))mark#(g(a, active(active(_x32))))active#(g(mark(b), _x32))
active#(a)mark#(b)active#(h(X))mark#(g(X, X))
mark#(g(h(_x31), X2))active#(g(active(h(mark(_x31))), X2))mark#(g(a, X2))active#(g(mark(b), X2))
mark#(g(a, active(active(mark(active(_x32))))))active#(g(mark(b), _x32))mark#(g(a, active(active(mark(_x22)))))active#(g(mark(b), _x22))
mark#(g(a, active(_x32)))active#(g(a, _x32))mark#(g(a, active(mark(active(_x22)))))active#(g(active(a), _x22))
mark#(g(a, active(_x22)))active#(g(mark(b), _x22))mark#(g(a, active(active(mark(active(active(_x22)))))))active#(g(active(a), _x22))
mark#(g(X1, X2))mark#(X1)mark#(g(_x21, _x22))active#(g(_x21, _x22))
mark#(g(X1, active(_x22)))active#(g(mark(X1), _x22))active#(g(a, X))mark#(f(b, X))
mark#(g(a, active(active(_x22))))active#(g(a, _x22))mark#(g(b, active(_x22)))active#(g(active(b), _x22))
mark#(f(X1, X2))active#(f(mark(X1), X2))mark#(g(a, active(mark(mark(_x22)))))active#(g(active(a), _x22))
mark#(g(g(_x31, _x32), X2))active#(g(active(g(mark(_x31), _x32)), X2))mark#(g(a, active(mark(_x32))))active#(g(mark(b), _x32))
mark#(g(X1, mark(_x22)))active#(g(mark(X1), _x22))mark#(g(a, active(active(mark(mark(_x32))))))active#(g(active(a), _x32))
mark#(g(a, active(active(mark(active(mark(_x22)))))))active#(g(active(a), _x22))mark#(g(a, mark(_x22)))active#(g(active(a), _x22))
mark#(h(X))mark#(X)mark#(f(X1, X2))mark#(X1)

Rewrite Rules

active(h(X))mark(g(X, X))active(g(a, X))mark(f(b, X))
active(f(X, X))mark(h(a))active(a)mark(b)
mark(h(X))active(h(mark(X)))mark(g(X1, X2))active(g(mark(X1), X2))
mark(a)active(a)mark(f(X1, X2))active(f(mark(X1), X2))
mark(b)active(b)h(mark(X))h(X)
h(active(X))h(X)g(mark(X1), X2)g(X1, X2)
g(X1, mark(X2))g(X1, X2)g(active(X1), X2)g(X1, X2)
g(X1, active(X2))g(X1, X2)f(mark(X1), X2)f(X1, X2)
f(X1, mark(X2))f(X1, X2)f(active(X1), X2)f(X1, X2)
f(X1, active(X2))f(X1, X2)

Original Signature

Termination of terms over the following signature is verified: f, g, b, a, active, mark, h

Strategy


The right-hand side of the rule mark#(g(f(_x31, _x32), X2)) → active#(g(active(f(mark(_x31), _x32)), X2)) 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
active#(g(active(f(active(a), _x32)), X2)) 
active#(g(f(mark(_x31), _x32), _x22)) 
active#(g(mark(h(a)), X2)) 
active#(g(active(f(mark(_x31), _x32)), _x22)) 
active#(g(active(f(active(h(mark(_x61))), _x32)), X2)) 
active#(g(active(f(active(g(mark(_x61), _x62)), _x32)), X2)) 
active#(g(active(f(active(b), _x32)), X2)) 
active#(g(active(f(active(f(mark(_x61), _x62)), _x32)), X2)) 
active#(g(active(f(_x51, _x52)), X2)) 
active#(g(active(f(mark(_x31), _x52)), X2)) 
Thus, the rule mark#(g(f(_x31, _x32), X2)) → active#(g(active(f(mark(_x31), _x32)), X2)) is replaced by the following rules:
mark#(g(f(_x31, _x32), mark(_x22))) → active#(g(active(f(mark(_x31), _x32)), _x22))mark#(g(f(_x31, active(_x52)), X2)) → active#(g(active(f(mark(_x31), _x52)), X2))
mark#(g(f(h(_x61), _x32), X2)) → active#(g(active(f(active(h(mark(_x61))), _x32)), X2))mark#(g(f(g(_x61, _x62), _x32), X2)) → active#(g(active(f(active(g(mark(_x61), _x62)), _x32)), X2))
mark#(g(f(_x31, mark(_x31)), X2)) → active#(g(mark(h(a)), X2))mark#(g(f(_x31, mark(_x52)), X2)) → active#(g(active(f(mark(_x31), _x52)), X2))
mark#(g(f(_x31, _x32), active(_x22))) → active#(g(active(f(mark(_x31), _x32)), _x22))mark#(g(f(_x51, _x52), X2)) → active#(g(active(f(_x51, _x52)), X2))
mark#(g(f(b, _x32), X2)) → active#(g(active(f(active(b), _x32)), X2))mark#(g(f(a, _x32), X2)) → active#(g(active(f(active(a), _x32)), X2))
mark#(g(f(_x31, _x32), _x22)) → active#(g(f(mark(_x31), _x32), _x22))mark#(g(f(f(_x61, _x62), _x32), X2)) → active#(g(active(f(active(f(mark(_x61), _x62)), _x32)), X2))

Problem 19: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

mark#(g(a, active(active(mark(active(_x22))))))active#(g(a, _x22))mark#(g(a, _x22))active#(g(a, _x22))
mark#(g(f(_x31, _x32), mark(_x22)))active#(g(active(f(mark(_x31), _x32)), _x22))mark#(a)active#(a)
mark#(g(f(b, _x32), X2))active#(g(active(f(active(b), _x32)), X2))mark#(g(a, active(active(mark(_x32)))))active#(g(a, _x32))
mark#(g(f(f(_x61, _x62), _x32), X2))active#(g(active(f(active(f(mark(_x61), _x62)), _x32)), X2))active#(f(X, X))mark#(h(a))
mark#(g(b, mark(_x22)))active#(g(active(b), _x22))mark#(g(f(_x31, active(_x52)), X2))active#(g(active(f(mark(_x31), _x52)), X2))
mark#(g(b, _x22))active#(g(b, _x22))mark#(g(h(_x31), X2))active#(g(active(h(mark(_x31))), X2))
active#(h(X))mark#(g(X, X))mark#(g(a, X2))active#(g(mark(b), X2))
mark#(g(a, active(active(mark(_x22)))))active#(g(mark(b), _x22))mark#(g(a, active(_x32)))active#(g(a, _x32))
mark#(g(a, active(mark(active(_x22)))))active#(g(active(a), _x22))mark#(g(a, active(_x22)))active#(g(mark(b), _x22))
mark#(g(f(_x31, _x32), _x22))active#(g(f(mark(_x31), _x32), _x22))mark#(g(a, active(active(mark(active(active(_x22)))))))active#(g(active(a), _x22))
mark#(g(_x21, _x22))active#(g(_x21, _x22))active#(g(a, X))mark#(f(b, X))
mark#(g(a, active(mark(mark(_x22)))))active#(g(active(a), _x22))mark#(g(f(_x31, mark(_x31)), X2))active#(g(mark(h(a)), X2))
mark#(g(a, active(active(mark(mark(_x32))))))active#(g(active(a), _x32))mark#(g(X1, mark(_x22)))active#(g(mark(X1), _x22))
mark#(g(a, mark(_x22)))active#(g(active(a), _x22))mark#(g(f(a, _x32), X2))active#(g(active(f(active(a), _x32)), X2))
mark#(h(X))mark#(X)mark#(f(X1, X2))mark#(X1)
mark#(g(a, active(mark(_x22))))active#(g(a, _x22))mark#(g(a, active(active(active(_x22)))))active#(g(active(a), _x22))
mark#(h(X))active#(h(mark(X)))mark#(g(a, active(active(_x32))))active#(g(mark(b), _x32))
active#(a)mark#(b)mark#(g(a, active(active(mark(active(_x32))))))active#(g(mark(b), _x32))
mark#(g(f(_x51, _x52), X2))active#(g(active(f(_x51, _x52)), X2))mark#(g(X1, X2))mark#(X1)
mark#(g(X1, active(_x22)))active#(g(mark(X1), _x22))mark#(g(a, active(active(_x22))))active#(g(a, _x22))
mark#(g(b, active(_x22)))active#(g(active(b), _x22))mark#(g(f(h(_x61), _x32), X2))active#(g(active(f(active(h(mark(_x61))), _x32)), X2))
mark#(f(X1, X2))active#(f(mark(X1), X2))mark#(g(g(_x31, _x32), X2))active#(g(active(g(mark(_x31), _x32)), X2))
mark#(g(f(g(_x61, _x62), _x32), X2))active#(g(active(f(active(g(mark(_x61), _x62)), _x32)), X2))mark#(g(a, active(mark(_x32))))active#(g(mark(b), _x32))
mark#(g(f(_x31, mark(_x52)), X2))active#(g(active(f(mark(_x31), _x52)), X2))mark#(g(f(_x31, _x32), active(_x22)))active#(g(active(f(mark(_x31), _x32)), _x22))
mark#(g(a, active(active(mark(active(mark(_x22)))))))active#(g(active(a), _x22))

Rewrite Rules

active(h(X))mark(g(X, X))active(g(a, X))mark(f(b, X))
active(f(X, X))mark(h(a))active(a)mark(b)
mark(h(X))active(h(mark(X)))mark(g(X1, X2))active(g(mark(X1), X2))
mark(a)active(a)mark(f(X1, X2))active(f(mark(X1), X2))
mark(b)active(b)h(mark(X))h(X)
h(active(X))h(X)g(mark(X1), X2)g(X1, X2)
g(X1, mark(X2))g(X1, X2)g(active(X1), X2)g(X1, X2)
g(X1, active(X2))g(X1, X2)f(mark(X1), X2)f(X1, X2)
f(X1, mark(X2))f(X1, X2)f(active(X1), X2)f(X1, X2)
f(X1, active(X2))f(X1, X2)

Original Signature

Termination of terms over the following signature is verified: f, g, b, a, active, mark, h

Strategy


The right-hand side of the rule mark#(g(f(_x31, _x32), mark(_x22))) → active#(g(active(f(mark(_x31), _x32)), _x22)) 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
active#(g(active(f(active(b), _x32)), _x22)) 
active#(g(active(f(active(g(mark(_x71), _x72)), _x32)), _x22)) 
active#(g(active(f(_x61, _x62)), _x22)) 
active#(g(mark(h(a)), _x22)) 
active#(g(f(mark(_x31), _x32), _x42)) 
active#(g(active(f(active(a), _x32)), _x22)) 
active#(g(active(f(mark(_x31), _x62)), _x22)) 
active#(g(active(f(mark(_x31), _x32)), _x42)) 
active#(g(active(f(active(f(mark(_x71), _x72)), _x32)), _x22)) 
active#(g(active(f(active(h(mark(_x71))), _x32)), _x22)) 
Thus, the rule mark#(g(f(_x31, _x32), mark(_x22))) → active#(g(active(f(mark(_x31), _x32)), _x22)) is replaced by the following rules:
mark#(g(f(g(_x71, _x72), _x32), mark(_x22))) → active#(g(active(f(active(g(mark(_x71), _x72)), _x32)), _x22))mark#(g(f(_x31, _x32), mark(active(_x42)))) → active#(g(active(f(mark(_x31), _x32)), _x42))
mark#(g(f(_x31, mark(_x62)), mark(_x22))) → active#(g(active(f(mark(_x31), _x62)), _x22))mark#(g(f(_x31, active(_x62)), mark(_x22))) → active#(g(active(f(mark(_x31), _x62)), _x22))
mark#(g(f(_x31, _x32), mark(mark(_x42)))) → active#(g(active(f(mark(_x31), _x32)), _x42))mark#(g(f(h(_x71), _x32), mark(_x22))) → active#(g(active(f(active(h(mark(_x71))), _x32)), _x22))
mark#(g(f(_x31, mark(_x31)), mark(_x22))) → active#(g(mark(h(a)), _x22))mark#(g(f(a, _x32), mark(_x22))) → active#(g(active(f(active(a), _x32)), _x22))
mark#(g(f(b, _x32), mark(_x22))) → active#(g(active(f(active(b), _x32)), _x22))mark#(g(f(_x61, _x62), mark(_x22))) → active#(g(active(f(_x61, _x62)), _x22))
mark#(g(f(f(_x71, _x72), _x32), mark(_x22))) → active#(g(active(f(active(f(mark(_x71), _x72)), _x32)), _x22))mark#(g(f(_x31, _x32), mark(_x42))) → active#(g(f(mark(_x31), _x32), _x42))

Problem 20: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

mark#(g(a, active(active(mark(active(_x22))))))active#(g(a, _x22))mark#(g(f(_x31, mark(_x62)), mark(_x22)))active#(g(active(f(mark(_x31), _x62)), _x22))
mark#(g(a, _x22))active#(g(a, _x22))mark#(g(f(_x31, _x32), mark(mark(_x42))))active#(g(active(f(mark(_x31), _x32)), _x42))
mark#(a)active#(a)mark#(g(f(b, _x32), X2))active#(g(active(f(active(b), _x32)), X2))
mark#(g(a, active(active(mark(_x32)))))active#(g(a, _x32))mark#(g(f(f(_x61, _x62), _x32), X2))active#(g(active(f(active(f(mark(_x61), _x62)), _x32)), X2))
mark#(g(f(_x31, _x32), mark(active(_x42))))active#(g(active(f(mark(_x31), _x32)), _x42))active#(f(X, X))mark#(h(a))
mark#(g(b, mark(_x22)))active#(g(active(b), _x22))mark#(g(f(_x31, active(_x52)), X2))active#(g(active(f(mark(_x31), _x52)), X2))
mark#(g(b, _x22))active#(g(b, _x22))active#(h(X))mark#(g(X, X))
mark#(g(h(_x31), X2))active#(g(active(h(mark(_x31))), X2))mark#(g(a, X2))active#(g(mark(b), X2))
mark#(g(a, active(active(mark(_x22)))))active#(g(mark(b), _x22))mark#(g(a, active(_x32)))active#(g(a, _x32))
mark#(g(a, active(mark(active(_x22)))))active#(g(active(a), _x22))mark#(g(a, active(_x22)))active#(g(mark(b), _x22))
mark#(g(a, active(active(mark(active(active(_x22)))))))active#(g(active(a), _x22))mark#(g(f(_x31, _x32), _x22))active#(g(f(mark(_x31), _x32), _x22))
mark#(g(_x21, _x22))active#(g(_x21, _x22))active#(g(a, X))mark#(f(b, X))
mark#(g(f(_x31, mark(_x31)), mark(_x22)))active#(g(mark(h(a)), _x22))mark#(g(a, active(mark(mark(_x22)))))active#(g(active(a), _x22))
mark#(g(f(_x31, mark(_x31)), X2))active#(g(mark(h(a)), X2))mark#(g(a, active(active(mark(mark(_x32))))))active#(g(active(a), _x32))
mark#(g(X1, mark(_x22)))active#(g(mark(X1), _x22))mark#(g(f(_x31, _x32), mark(_x42)))active#(g(f(mark(_x31), _x32), _x42))
mark#(g(a, mark(_x22)))active#(g(active(a), _x22))mark#(g(f(a, _x32), X2))active#(g(active(f(active(a), _x32)), X2))
mark#(h(X))mark#(X)mark#(f(X1, X2))mark#(X1)
mark#(g(f(h(_x71), _x32), mark(_x22)))active#(g(active(f(active(h(mark(_x71))), _x32)), _x22))mark#(g(a, active(mark(_x22))))active#(g(a, _x22))
mark#(g(a, active(active(active(_x22)))))active#(g(active(a), _x22))mark#(h(X))active#(h(mark(X)))
mark#(g(a, active(active(_x32))))active#(g(mark(b), _x32))mark#(g(f(b, _x32), mark(_x22)))active#(g(active(f(active(b), _x32)), _x22))
active#(a)mark#(b)mark#(g(f(_x31, active(_x62)), mark(_x22)))active#(g(active(f(mark(_x31), _x62)), _x22))
mark#(g(a, active(active(mark(active(_x32))))))active#(g(mark(b), _x32))mark#(g(f(_x51, _x52), X2))active#(g(active(f(_x51, _x52)), X2))
mark#(g(f(_x61, _x62), mark(_x22)))active#(g(active(f(_x61, _x62)), _x22))mark#(g(X1, X2))mark#(X1)
mark#(g(X1, active(_x22)))active#(g(mark(X1), _x22))mark#(g(f(g(_x71, _x72), _x32), mark(_x22)))active#(g(active(f(active(g(mark(_x71), _x72)), _x32)), _x22))
mark#(g(a, active(active(_x22))))active#(g(a, _x22))mark#(g(f(h(_x61), _x32), X2))active#(g(active(f(active(h(mark(_x61))), _x32)), X2))
mark#(g(b, active(_x22)))active#(g(active(b), _x22))mark#(f(X1, X2))active#(f(mark(X1), X2))
mark#(g(f(g(_x61, _x62), _x32), X2))active#(g(active(f(active(g(mark(_x61), _x62)), _x32)), X2))mark#(g(g(_x31, _x32), X2))active#(g(active(g(mark(_x31), _x32)), X2))
mark#(g(a, active(mark(_x32))))active#(g(mark(b), _x32))mark#(g(f(a, _x32), mark(_x22)))active#(g(active(f(active(a), _x32)), _x22))
mark#(g(f(_x31, _x32), active(_x22)))active#(g(active(f(mark(_x31), _x32)), _x22))mark#(g(f(_x31, mark(_x52)), X2))active#(g(active(f(mark(_x31), _x52)), X2))
mark#(g(f(f(_x71, _x72), _x32), mark(_x22)))active#(g(active(f(active(f(mark(_x71), _x72)), _x32)), _x22))mark#(g(a, active(active(mark(active(mark(_x22)))))))active#(g(active(a), _x22))

Rewrite Rules

active(h(X))mark(g(X, X))active(g(a, X))mark(f(b, X))
active(f(X, X))mark(h(a))active(a)mark(b)
mark(h(X))active(h(mark(X)))mark(g(X1, X2))active(g(mark(X1), X2))
mark(a)active(a)mark(f(X1, X2))active(f(mark(X1), X2))
mark(b)active(b)h(mark(X))h(X)
h(active(X))h(X)g(mark(X1), X2)g(X1, X2)
g(X1, mark(X2))g(X1, X2)g(active(X1), X2)g(X1, X2)
g(X1, active(X2))g(X1, X2)f(mark(X1), X2)f(X1, X2)
f(X1, mark(X2))f(X1, X2)f(active(X1), X2)f(X1, X2)
f(X1, active(X2))f(X1, X2)

Original Signature

Termination of terms over the following signature is verified: f, g, b, a, active, mark, h

Strategy


The right-hand side of the rule mark#(g(f(_x31, mark(_x62)), mark(_x22))) → active#(g(active(f(mark(_x31), _x62)), _x22)) 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
active#(g(active(f(active(f(mark(_x81), _x82)), _x62)), _x22)) 
active#(g(active(f(active(g(mark(_x81), _x82)), _x62)), _x22)) 
active#(g(mark(h(a)), _x22)) 
active#(g(active(f(mark(_x31), _x72)), _x22)) 
active#(g(active(f(active(h(mark(_x81))), _x62)), _x22)) 
active#(g(f(mark(_x31), _x62), _x42)) 
active#(g(active(f(_x71, _x72)), _x22)) 
active#(g(active(f(mark(_x31), _x62)), _x42)) 
active#(g(active(f(active(a), _x62)), _x22)) 
active#(g(active(f(active(b), _x62)), _x22)) 
Thus, the rule mark#(g(f(_x31, mark(_x62)), mark(_x22))) → active#(g(active(f(mark(_x31), _x62)), _x22)) is replaced by the following rules:
mark#(g(f(_x31, mark(_x62)), mark(_x42))) → active#(g(f(mark(_x31), _x62), _x42))mark#(g(f(_x71, mark(_x72)), mark(_x22))) → active#(g(active(f(_x71, _x72)), _x22))
mark#(g(f(_x31, mark(mark(_x31))), mark(_x22))) → active#(g(mark(h(a)), _x22))mark#(g(f(_x31, mark(mark(_x72))), mark(_x22))) → active#(g(active(f(mark(_x31), _x72)), _x22))
mark#(g(f(_x31, mark(active(_x72))), mark(_x22))) → active#(g(active(f(mark(_x31), _x72)), _x22))mark#(g(f(_x31, mark(_x62)), mark(mark(_x42)))) → active#(g(active(f(mark(_x31), _x62)), _x42))
mark#(g(f(b, mark(_x62)), mark(_x22))) → active#(g(active(f(active(b), _x62)), _x22))mark#(g(f(h(_x81), mark(_x62)), mark(_x22))) → active#(g(active(f(active(h(mark(_x81))), _x62)), _x22))
mark#(g(f(f(_x81, _x82), mark(_x62)), mark(_x22))) → active#(g(active(f(active(f(mark(_x81), _x82)), _x62)), _x22))mark#(g(f(g(_x81, _x82), mark(_x62)), mark(_x22))) → active#(g(active(f(active(g(mark(_x81), _x82)), _x62)), _x22))
mark#(g(f(a, mark(_x62)), mark(_x22))) → active#(g(active(f(active(a), _x62)), _x22))mark#(g(f(_x31, mark(_x62)), mark(active(_x42)))) → active#(g(active(f(mark(_x31), _x62)), _x42))

Problem 21: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

mark#(g(a, active(active(mark(active(_x22))))))active#(g(a, _x22))mark#(g(f(_x31, mark(_x62)), mark(_x42)))active#(g(f(mark(_x31), _x62), _x42))
mark#(g(a, _x22))active#(g(a, _x22))mark#(g(f(_x31, _x32), mark(mark(_x42))))active#(g(active(f(mark(_x31), _x32)), _x42))
mark#(a)active#(a)mark#(g(f(b, _x32), X2))active#(g(active(f(active(b), _x32)), X2))
mark#(g(a, active(active(mark(_x32)))))active#(g(a, _x32))mark#(g(f(f(_x61, _x62), _x32), X2))active#(g(active(f(active(f(mark(_x61), _x62)), _x32)), X2))
mark#(g(f(_x31, _x32), mark(active(_x42))))active#(g(active(f(mark(_x31), _x32)), _x42))active#(f(X, X))mark#(h(a))
mark#(g(b, mark(_x22)))active#(g(active(b), _x22))mark#(g(f(_x31, active(_x52)), X2))active#(g(active(f(mark(_x31), _x52)), X2))
mark#(g(b, _x22))active#(g(b, _x22))mark#(g(h(_x31), X2))active#(g(active(h(mark(_x31))), X2))
active#(h(X))mark#(g(X, X))mark#(g(f(f(_x81, _x82), mark(_x62)), mark(_x22)))active#(g(active(f(active(f(mark(_x81), _x82)), _x62)), _x22))
mark#(g(a, X2))active#(g(mark(b), X2))mark#(g(a, active(active(mark(_x22)))))active#(g(mark(b), _x22))
mark#(g(a, active(_x32)))active#(g(a, _x32))mark#(g(a, active(mark(active(_x22)))))active#(g(active(a), _x22))
mark#(g(a, active(_x22)))active#(g(mark(b), _x22))mark#(g(f(_x31, mark(active(_x72))), mark(_x22)))active#(g(active(f(mark(_x31), _x72)), _x22))
mark#(g(f(_x31, _x32), _x22))active#(g(f(mark(_x31), _x32), _x22))mark#(g(a, active(active(mark(active(active(_x22)))))))active#(g(active(a), _x22))
mark#(g(_x21, _x22))active#(g(_x21, _x22))active#(g(a, X))mark#(f(b, X))
mark#(g(f(_x31, mark(_x31)), mark(_x22)))active#(g(mark(h(a)), _x22))mark#(g(a, active(mark(mark(_x22)))))active#(g(active(a), _x22))
mark#(g(f(_x31, mark(mark(_x72))), mark(_x22)))active#(g(active(f(mark(_x31), _x72)), _x22))mark#(g(f(_x31, mark(_x31)), X2))active#(g(mark(h(a)), X2))
mark#(g(a, active(active(mark(mark(_x32))))))active#(g(active(a), _x32))mark#(g(X1, mark(_x22)))active#(g(mark(X1), _x22))
mark#(g(f(_x31, _x32), mark(_x42)))active#(g(f(mark(_x31), _x32), _x42))mark#(g(f(h(_x81), mark(_x62)), mark(_x22)))active#(g(active(f(active(h(mark(_x81))), _x62)), _x22))
mark#(g(a, mark(_x22)))active#(g(active(a), _x22))mark#(g(f(a, _x32), X2))active#(g(active(f(active(a), _x32)), X2))
mark#(h(X))mark#(X)mark#(f(X1, X2))mark#(X1)
mark#(g(f(h(_x71), _x32), mark(_x22)))active#(g(active(f(active(h(mark(_x71))), _x32)), _x22))mark#(g(a, active(mark(_x22))))active#(g(a, _x22))
mark#(g(f(g(_x81, _x82), mark(_x62)), mark(_x22)))active#(g(active(f(active(g(mark(_x81), _x82)), _x62)), _x22))mark#(g(a, active(active(active(_x22)))))active#(g(active(a), _x22))
mark#(h(X))active#(h(mark(X)))mark#(g(a, active(active(_x32))))active#(g(mark(b), _x32))
mark#(g(f(_x31, mark(mark(_x31))), mark(_x22)))active#(g(mark(h(a)), _x22))mark#(g(f(b, _x32), mark(_x22)))active#(g(active(f(active(b), _x32)), _x22))
active#(a)mark#(b)mark#(g(f(b, mark(_x62)), mark(_x22)))active#(g(active(f(active(b), _x62)), _x22))
mark#(g(f(a, mark(_x62)), mark(_x22)))active#(g(active(f(active(a), _x62)), _x22))mark#(g(f(_x31, active(_x62)), mark(_x22)))active#(g(active(f(mark(_x31), _x62)), _x22))
mark#(g(a, active(active(mark(active(_x32))))))active#(g(mark(b), _x32))mark#(g(f(_x51, _x52), X2))active#(g(active(f(_x51, _x52)), X2))
mark#(g(f(_x61, _x62), mark(_x22)))active#(g(active(f(_x61, _x62)), _x22))mark#(g(X1, X2))mark#(X1)
mark#(g(X1, active(_x22)))active#(g(mark(X1), _x22))mark#(g(f(_x31, mark(_x62)), mark(active(_x42))))active#(g(active(f(mark(_x31), _x62)), _x42))
mark#(g(a, active(active(_x22))))active#(g(a, _x22))mark#(g(f(g(_x71, _x72), _x32), mark(_x22)))active#(g(active(f(active(g(mark(_x71), _x72)), _x32)), _x22))
mark#(g(b, active(_x22)))active#(g(active(b), _x22))mark#(g(f(h(_x61), _x32), X2))active#(g(active(f(active(h(mark(_x61))), _x32)), X2))
mark#(f(X1, X2))active#(f(mark(X1), X2))mark#(g(g(_x31, _x32), X2))active#(g(active(g(mark(_x31), _x32)), X2))
mark#(g(f(g(_x61, _x62), _x32), X2))active#(g(active(f(active(g(mark(_x61), _x62)), _x32)), X2))mark#(g(f(a, _x32), mark(_x22)))active#(g(active(f(active(a), _x32)), _x22))
mark#(g(a, active(mark(_x32))))active#(g(mark(b), _x32))mark#(g(f(_x71, mark(_x72)), mark(_x22)))active#(g(active(f(_x71, _x72)), _x22))
mark#(g(f(_x31, mark(_x52)), X2))active#(g(active(f(mark(_x31), _x52)), X2))mark#(g(f(_x31, _x32), active(_x22)))active#(g(active(f(mark(_x31), _x32)), _x22))
mark#(g(f(f(_x71, _x72), _x32), mark(_x22)))active#(g(active(f(active(f(mark(_x71), _x72)), _x32)), _x22))mark#(g(a, active(active(mark(active(mark(_x22)))))))active#(g(active(a), _x22))
mark#(g(f(_x31, mark(_x62)), mark(mark(_x42))))active#(g(active(f(mark(_x31), _x62)), _x42))

Rewrite Rules

active(h(X))mark(g(X, X))active(g(a, X))mark(f(b, X))
active(f(X, X))mark(h(a))active(a)mark(b)
mark(h(X))active(h(mark(X)))mark(g(X1, X2))active(g(mark(X1), X2))
mark(a)active(a)mark(f(X1, X2))active(f(mark(X1), X2))
mark(b)active(b)h(mark(X))h(X)
h(active(X))h(X)g(mark(X1), X2)g(X1, X2)
g(X1, mark(X2))g(X1, X2)g(active(X1), X2)g(X1, X2)
g(X1, active(X2))g(X1, X2)f(mark(X1), X2)f(X1, X2)
f(X1, mark(X2))f(X1, X2)f(active(X1), X2)f(X1, X2)
f(X1, active(X2))f(X1, X2)

Original Signature

Termination of terms over the following signature is verified: f, g, b, a, active, mark, h

Strategy


The right-hand side of the rule mark#(g(f(_x31, mark(_x62)), mark(_x42))) → active#(g(f(mark(_x31), _x62), _x42)) 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
active#(g(f(mark(_x31), _x42), _x42)) 
active#(g(f(active(a), _x62), _x42)) 
active#(g(f(active(f(mark(_x51), _x52)), _x62), _x42)) 
active#(g(f(active(g(mark(_x51), _x52)), _x62), _x42)) 
active#(g(f(active(h(mark(_x51))), _x62), _x42)) 
active#(g(f(mark(_x31), _x62), _x22)) 
active#(g(f(active(b), _x62), _x42)) 
active#(g(f(_x41, _x42), _x42)) 
Thus, the rule mark#(g(f(_x31, mark(_x62)), mark(_x42))) → active#(g(f(mark(_x31), _x62), _x42)) is replaced by the following rules:
mark#(g(f(h(_x51), mark(_x62)), mark(_x42))) → active#(g(f(active(h(mark(_x51))), _x62), _x42))mark#(g(f(a, mark(_x62)), mark(_x42))) → active#(g(f(active(a), _x62), _x42))
mark#(g(f(_x31, mark(mark(_x42))), mark(_x42))) → active#(g(f(mark(_x31), _x42), _x42))mark#(g(f(g(_x51, _x52), mark(_x62)), mark(_x42))) → active#(g(f(active(g(mark(_x51), _x52)), _x62), _x42))
mark#(g(f(_x31, mark(active(_x42))), mark(_x42))) → active#(g(f(mark(_x31), _x42), _x42))mark#(g(f(f(_x51, _x52), mark(_x62)), mark(_x42))) → active#(g(f(active(f(mark(_x51), _x52)), _x62), _x42))
mark#(g(f(_x41, mark(_x42)), mark(_x42))) → active#(g(f(_x41, _x42), _x42))mark#(g(f(_x31, mark(_x62)), mark(active(_x22)))) → active#(g(f(mark(_x31), _x62), _x22))
mark#(g(f(_x31, mark(_x62)), mark(mark(_x22)))) → active#(g(f(mark(_x31), _x62), _x22))mark#(g(f(b, mark(_x62)), mark(_x42))) → active#(g(f(active(b), _x62), _x42))

Problem 22: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

mark#(g(a, active(active(mark(active(_x22))))))active#(g(a, _x22))mark#(g(a, _x22))active#(g(a, _x22))
mark#(g(f(_x31, _x32), mark(mark(_x42))))active#(g(active(f(mark(_x31), _x32)), _x42))mark#(a)active#(a)
mark#(g(f(b, _x32), X2))active#(g(active(f(active(b), _x32)), X2))mark#(g(a, active(active(mark(_x32)))))active#(g(a, _x32))
mark#(g(f(f(_x61, _x62), _x32), X2))active#(g(active(f(active(f(mark(_x61), _x62)), _x32)), X2))mark#(g(f(_x31, _x32), mark(active(_x42))))active#(g(active(f(mark(_x31), _x32)), _x42))
active#(f(X, X))mark#(h(a))mark#(g(b, mark(_x22)))active#(g(active(b), _x22))
mark#(g(f(_x31, active(_x52)), X2))active#(g(active(f(mark(_x31), _x52)), X2))mark#(g(b, _x22))active#(g(b, _x22))
active#(h(X))mark#(g(X, X))mark#(g(h(_x31), X2))active#(g(active(h(mark(_x31))), X2))
mark#(g(f(f(_x81, _x82), mark(_x62)), mark(_x22)))active#(g(active(f(active(f(mark(_x81), _x82)), _x62)), _x22))mark#(g(f(_x31, mark(_x62)), mark(active(_x22))))active#(g(f(mark(_x31), _x62), _x22))
mark#(g(a, X2))active#(g(mark(b), X2))mark#(g(f(b, mark(_x62)), mark(_x42)))active#(g(f(active(b), _x62), _x42))
mark#(g(a, active(active(mark(_x22)))))active#(g(mark(b), _x22))mark#(g(a, active(_x32)))active#(g(a, _x32))
mark#(g(f(g(_x51, _x52), mark(_x62)), mark(_x42)))active#(g(f(active(g(mark(_x51), _x52)), _x62), _x42))mark#(g(a, active(mark(active(_x22)))))active#(g(active(a), _x22))
mark#(g(f(_x31, mark(active(_x72))), mark(_x22)))active#(g(active(f(mark(_x31), _x72)), _x22))mark#(g(a, active(_x22)))active#(g(mark(b), _x22))
mark#(g(f(_x41, mark(_x42)), mark(_x42)))active#(g(f(_x41, _x42), _x42))mark#(g(f(_x31, mark(_x62)), mark(mark(_x22))))active#(g(f(mark(_x31), _x62), _x22))
mark#(g(f(_x31, _x32), _x22))active#(g(f(mark(_x31), _x32), _x22))mark#(g(a, active(active(mark(active(active(_x22)))))))active#(g(active(a), _x22))
mark#(g(_x21, _x22))active#(g(_x21, _x22))active#(g(a, X))mark#(f(b, X))
mark#(g(f(_x31, mark(mark(_x42))), mark(_x42)))active#(g(f(mark(_x31), _x42), _x42))mark#(g(f(_x31, mark(_x31)), mark(_x22)))active#(g(mark(h(a)), _x22))
mark#(g(a, active(mark(mark(_x22)))))active#(g(active(a), _x22))mark#(g(f(_x31, mark(mark(_x72))), mark(_x22)))active#(g(active(f(mark(_x31), _x72)), _x22))
mark#(g(f(_x31, mark(_x31)), X2))active#(g(mark(h(a)), X2))mark#(g(a, active(active(mark(mark(_x32))))))active#(g(active(a), _x32))
mark#(g(X1, mark(_x22)))active#(g(mark(X1), _x22))mark#(g(f(_x31, _x32), mark(_x42)))active#(g(f(mark(_x31), _x32), _x42))
mark#(g(f(h(_x81), mark(_x62)), mark(_x22)))active#(g(active(f(active(h(mark(_x81))), _x62)), _x22))mark#(g(a, mark(_x22)))active#(g(active(a), _x22))
mark#(g(f(a, _x32), X2))active#(g(active(f(active(a), _x32)), X2))mark#(h(X))mark#(X)
mark#(f(X1, X2))mark#(X1)mark#(g(f(h(_x71), _x32), mark(_x22)))active#(g(active(f(active(h(mark(_x71))), _x32)), _x22))
mark#(g(f(_x31, mark(active(_x42))), mark(_x42)))active#(g(f(mark(_x31), _x42), _x42))mark#(g(a, active(mark(_x22))))active#(g(a, _x22))
mark#(g(f(g(_x81, _x82), mark(_x62)), mark(_x22)))active#(g(active(f(active(g(mark(_x81), _x82)), _x62)), _x22))mark#(g(a, active(active(active(_x22)))))active#(g(active(a), _x22))
mark#(h(X))active#(h(mark(X)))mark#(g(f(a, mark(_x62)), mark(_x42)))active#(g(f(active(a), _x62), _x42))
mark#(g(a, active(active(_x32))))active#(g(mark(b), _x32))mark#(g(f(_x31, mark(mark(_x31))), mark(_x22)))active#(g(mark(h(a)), _x22))
active#(a)mark#(b)mark#(g(f(b, _x32), mark(_x22)))active#(g(active(f(active(b), _x32)), _x22))
mark#(g(f(b, mark(_x62)), mark(_x22)))active#(g(active(f(active(b), _x62)), _x22))mark#(g(f(a, mark(_x62)), mark(_x22)))active#(g(active(f(active(a), _x62)), _x22))
mark#(g(f(_x31, active(_x62)), mark(_x22)))active#(g(active(f(mark(_x31), _x62)), _x22))mark#(g(a, active(active(mark(active(_x32))))))active#(g(mark(b), _x32))
mark#(g(f(_x51, _x52), X2))active#(g(active(f(_x51, _x52)), X2))mark#(g(f(_x61, _x62), mark(_x22)))active#(g(active(f(_x61, _x62)), _x22))
mark#(g(X1, X2))mark#(X1)mark#(g(X1, active(_x22)))active#(g(mark(X1), _x22))
mark#(g(f(_x31, mark(_x62)), mark(active(_x42))))active#(g(active(f(mark(_x31), _x62)), _x42))mark#(g(f(g(_x71, _x72), _x32), mark(_x22)))active#(g(active(f(active(g(mark(_x71), _x72)), _x32)), _x22))
mark#(g(a, active(active(_x22))))active#(g(a, _x22))mark#(g(f(h(_x51), mark(_x62)), mark(_x42)))active#(g(f(active(h(mark(_x51))), _x62), _x42))
mark#(g(f(h(_x61), _x32), X2))active#(g(active(f(active(h(mark(_x61))), _x32)), X2))mark#(g(b, active(_x22)))active#(g(active(b), _x22))
mark#(f(X1, X2))active#(f(mark(X1), X2))mark#(g(f(g(_x61, _x62), _x32), X2))active#(g(active(f(active(g(mark(_x61), _x62)), _x32)), X2))
mark#(g(g(_x31, _x32), X2))active#(g(active(g(mark(_x31), _x32)), X2))mark#(g(f(_x71, mark(_x72)), mark(_x22)))active#(g(active(f(_x71, _x72)), _x22))
mark#(g(a, active(mark(_x32))))active#(g(mark(b), _x32))mark#(g(f(a, _x32), mark(_x22)))active#(g(active(f(active(a), _x32)), _x22))
mark#(g(f(_x31, _x32), active(_x22)))active#(g(active(f(mark(_x31), _x32)), _x22))mark#(g(f(_x31, mark(_x52)), X2))active#(g(active(f(mark(_x31), _x52)), X2))
mark#(g(f(f(_x51, _x52), mark(_x62)), mark(_x42)))active#(g(f(active(f(mark(_x51), _x52)), _x62), _x42))mark#(g(f(f(_x71, _x72), _x32), mark(_x22)))active#(g(active(f(active(f(mark(_x71), _x72)), _x32)), _x22))
mark#(g(f(_x31, mark(_x62)), mark(mark(_x42))))active#(g(active(f(mark(_x31), _x62)), _x42))mark#(g(a, active(active(mark(active(mark(_x22)))))))active#(g(active(a), _x22))

Rewrite Rules

active(h(X))mark(g(X, X))active(g(a, X))mark(f(b, X))
active(f(X, X))mark(h(a))active(a)mark(b)
mark(h(X))active(h(mark(X)))mark(g(X1, X2))active(g(mark(X1), X2))
mark(a)active(a)mark(f(X1, X2))active(f(mark(X1), X2))
mark(b)active(b)h(mark(X))h(X)
h(active(X))h(X)g(mark(X1), X2)g(X1, X2)
g(X1, mark(X2))g(X1, X2)g(active(X1), X2)g(X1, X2)
g(X1, active(X2))g(X1, X2)f(mark(X1), X2)f(X1, X2)
f(X1, mark(X2))f(X1, X2)f(active(X1), X2)f(X1, X2)
f(X1, active(X2))f(X1, X2)

Original Signature

Termination of terms over the following signature is verified: f, g, b, a, active, mark, h

Strategy


The right-hand side of the rule mark#(g(f(_x31, _x32), mark(mark(_x42)))) → active#(g(active(f(mark(_x31), _x32)), _x42)) 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
active#(g(active(f(active(a), _x32)), _x42)) 
active#(g(active(f(_x51, _x52)), _x42)) 
active#(g(active(f(active(b), _x32)), _x42)) 
active#(g(f(mark(_x31), _x32), _x22)) 
active#(g(active(f(active(g(mark(_x61), _x62)), _x32)), _x42)) 
active#(g(active(f(mark(_x31), _x32)), _x22)) 
active#(g(active(f(active(h(mark(_x61))), _x32)), _x42)) 
active#(g(active(f(mark(_x31), _x52)), _x42)) 
active#(g(active(f(active(f(mark(_x61), _x62)), _x32)), _x42)) 
active#(g(mark(h(a)), _x42)) 
Thus, the rule mark#(g(f(_x31, _x32), mark(mark(_x42)))) → active#(g(active(f(mark(_x31), _x32)), _x42)) is replaced by the following rules:
mark#(g(f(_x31, mark(_x31)), mark(mark(_x42)))) → active#(g(mark(h(a)), _x42))mark#(g(f(g(_x61, _x62), _x32), mark(mark(_x42)))) → active#(g(active(f(active(g(mark(_x61), _x62)), _x32)), _x42))
mark#(g(f(_x51, _x52), mark(mark(_x42)))) → active#(g(active(f(_x51, _x52)), _x42))mark#(g(f(_x31, _x32), mark(mark(active(_x22))))) → active#(g(active(f(mark(_x31), _x32)), _x22))
mark#(g(f(_x31, _x32), mark(mark(_x22)))) → active#(g(f(mark(_x31), _x32), _x22))mark#(g(f(a, _x32), mark(mark(_x42)))) → active#(g(active(f(active(a), _x32)), _x42))
mark#(g(f(b, _x32), mark(mark(_x42)))) → active#(g(active(f(active(b), _x32)), _x42))mark#(g(f(f(_x61, _x62), _x32), mark(mark(_x42)))) → active#(g(active(f(active(f(mark(_x61), _x62)), _x32)), _x42))
mark#(g(f(_x31, _x32), mark(mark(mark(_x22))))) → active#(g(active(f(mark(_x31), _x32)), _x22))mark#(g(f(_x31, mark(_x52)), mark(mark(_x42)))) → active#(g(active(f(mark(_x31), _x52)), _x42))
mark#(g(f(_x31, active(_x52)), mark(mark(_x42)))) → active#(g(active(f(mark(_x31), _x52)), _x42))mark#(g(f(h(_x61), _x32), mark(mark(_x42)))) → active#(g(active(f(active(h(mark(_x61))), _x32)), _x42))

Problem 23: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

mark#(g(a, active(active(mark(active(_x22))))))active#(g(a, _x22))mark#(g(a, _x22))active#(g(a, _x22))
mark#(g(f(_x31, _x32), mark(mark(_x22))))active#(g(f(mark(_x31), _x32), _x22))mark#(g(f(_x51, _x52), mark(mark(_x42))))active#(g(active(f(_x51, _x52)), _x42))
mark#(a)active#(a)mark#(g(f(b, _x32), X2))active#(g(active(f(active(b), _x32)), X2))
mark#(g(a, active(active(mark(_x32)))))active#(g(a, _x32))mark#(g(f(_x31, _x32), mark(mark(mark(_x22)))))active#(g(active(f(mark(_x31), _x32)), _x22))
mark#(g(f(f(_x61, _x62), _x32), X2))active#(g(active(f(active(f(mark(_x61), _x62)), _x32)), X2))mark#(g(f(_x31, mark(_x31)), mark(mark(_x42))))active#(g(mark(h(a)), _x42))
mark#(g(f(_x31, _x32), mark(active(_x42))))active#(g(active(f(mark(_x31), _x32)), _x42))active#(f(X, X))mark#(h(a))
mark#(g(b, mark(_x22)))active#(g(active(b), _x22))mark#(g(f(_x31, active(_x52)), X2))active#(g(active(f(mark(_x31), _x52)), X2))
mark#(g(b, _x22))active#(g(b, _x22))mark#(g(h(_x31), X2))active#(g(active(h(mark(_x31))), X2))
active#(h(X))mark#(g(X, X))mark#(g(f(_x31, mark(_x62)), mark(active(_x22))))active#(g(f(mark(_x31), _x62), _x22))
mark#(g(f(f(_x81, _x82), mark(_x62)), mark(_x22)))active#(g(active(f(active(f(mark(_x81), _x82)), _x62)), _x22))mark#(g(f(b, mark(_x62)), mark(_x42)))active#(g(f(active(b), _x62), _x42))
mark#(g(a, X2))active#(g(mark(b), X2))mark#(g(f(g(_x61, _x62), _x32), mark(mark(_x42))))active#(g(active(f(active(g(mark(_x61), _x62)), _x32)), _x42))
mark#(g(a, active(active(mark(_x22)))))active#(g(mark(b), _x22))mark#(g(f(g(_x51, _x52), mark(_x62)), mark(_x42)))active#(g(f(active(g(mark(_x51), _x52)), _x62), _x42))
mark#(g(a, active(_x32)))active#(g(a, _x32))mark#(g(a, active(mark(active(_x22)))))active#(g(active(a), _x22))
mark#(g(f(_x31, mark(active(_x72))), mark(_x22)))active#(g(active(f(mark(_x31), _x72)), _x22))mark#(g(a, active(_x22)))active#(g(mark(b), _x22))
mark#(g(f(b, _x32), mark(mark(_x42))))active#(g(active(f(active(b), _x32)), _x42))mark#(g(f(_x41, mark(_x42)), mark(_x42)))active#(g(f(_x41, _x42), _x42))
mark#(g(f(_x31, mark(_x52)), mark(mark(_x42))))active#(g(active(f(mark(_x31), _x52)), _x42))mark#(g(f(_x31, mark(_x62)), mark(mark(_x22))))active#(g(f(mark(_x31), _x62), _x22))
mark#(g(f(_x31, _x32), _x22))active#(g(f(mark(_x31), _x32), _x22))mark#(g(a, active(active(mark(active(active(_x22)))))))active#(g(active(a), _x22))
mark#(g(_x21, _x22))active#(g(_x21, _x22))active#(g(a, X))mark#(f(b, X))
mark#(g(f(_x31, mark(mark(_x42))), mark(_x42)))active#(g(f(mark(_x31), _x42), _x42))mark#(g(f(_x31, mark(_x31)), mark(_x22)))active#(g(mark(h(a)), _x22))
mark#(g(a, active(mark(mark(_x22)))))active#(g(active(a), _x22))mark#(g(f(_x31, mark(mark(_x72))), mark(_x22)))active#(g(active(f(mark(_x31), _x72)), _x22))
mark#(g(f(_x31, mark(_x31)), X2))active#(g(mark(h(a)), X2))mark#(g(a, active(active(mark(mark(_x32))))))active#(g(active(a), _x32))
mark#(g(X1, mark(_x22)))active#(g(mark(X1), _x22))mark#(g(f(_x31, _x32), mark(_x42)))active#(g(f(mark(_x31), _x32), _x42))
mark#(g(f(h(_x81), mark(_x62)), mark(_x22)))active#(g(active(f(active(h(mark(_x81))), _x62)), _x22))mark#(g(a, mark(_x22)))active#(g(active(a), _x22))
mark#(g(f(a, _x32), X2))active#(g(active(f(active(a), _x32)), X2))mark#(h(X))mark#(X)
mark#(f(X1, X2))mark#(X1)mark#(g(f(h(_x71), _x32), mark(_x22)))active#(g(active(f(active(h(mark(_x71))), _x32)), _x22))
mark#(g(f(_x31, mark(active(_x42))), mark(_x42)))active#(g(f(mark(_x31), _x42), _x42))mark#(g(a, active(mark(_x22))))active#(g(a, _x22))
mark#(g(a, active(active(active(_x22)))))active#(g(active(a), _x22))mark#(g(f(g(_x81, _x82), mark(_x62)), mark(_x22)))active#(g(active(f(active(g(mark(_x81), _x82)), _x62)), _x22))
mark#(g(f(a, mark(_x62)), mark(_x42)))active#(g(f(active(a), _x62), _x42))mark#(h(X))active#(h(mark(X)))
mark#(g(a, active(active(_x32))))active#(g(mark(b), _x32))mark#(g(f(_x31, mark(mark(_x31))), mark(_x22)))active#(g(mark(h(a)), _x22))
mark#(g(f(b, _x32), mark(_x22)))active#(g(active(f(active(b), _x32)), _x22))active#(a)mark#(b)
mark#(g(f(b, mark(_x62)), mark(_x22)))active#(g(active(f(active(b), _x62)), _x22))mark#(g(f(a, mark(_x62)), mark(_x22)))active#(g(active(f(active(a), _x62)), _x22))
mark#(g(f(f(_x61, _x62), _x32), mark(mark(_x42))))active#(g(active(f(active(f(mark(_x61), _x62)), _x32)), _x42))mark#(g(f(_x31, active(_x52)), mark(mark(_x42))))active#(g(active(f(mark(_x31), _x52)), _x42))
mark#(g(f(h(_x61), _x32), mark(mark(_x42))))active#(g(active(f(active(h(mark(_x61))), _x32)), _x42))mark#(g(f(_x31, active(_x62)), mark(_x22)))active#(g(active(f(mark(_x31), _x62)), _x22))
mark#(g(a, active(active(mark(active(_x32))))))active#(g(mark(b), _x32))mark#(g(f(_x51, _x52), X2))active#(g(active(f(_x51, _x52)), X2))
mark#(g(f(_x61, _x62), mark(_x22)))active#(g(active(f(_x61, _x62)), _x22))mark#(g(X1, X2))mark#(X1)
mark#(g(X1, active(_x22)))active#(g(mark(X1), _x22))mark#(g(f(_x31, mark(_x62)), mark(active(_x42))))active#(g(active(f(mark(_x31), _x62)), _x42))
mark#(g(f(h(_x51), mark(_x62)), mark(_x42)))active#(g(f(active(h(mark(_x51))), _x62), _x42))mark#(g(a, active(active(_x22))))active#(g(a, _x22))
mark#(g(f(g(_x71, _x72), _x32), mark(_x22)))active#(g(active(f(active(g(mark(_x71), _x72)), _x32)), _x22))mark#(g(b, active(_x22)))active#(g(active(b), _x22))
mark#(g(f(h(_x61), _x32), X2))active#(g(active(f(active(h(mark(_x61))), _x32)), X2))mark#(f(X1, X2))active#(f(mark(X1), X2))
mark#(g(f(_x31, _x32), mark(mark(active(_x22)))))active#(g(active(f(mark(_x31), _x32)), _x22))mark#(g(g(_x31, _x32), X2))active#(g(active(g(mark(_x31), _x32)), X2))
mark#(g(f(g(_x61, _x62), _x32), X2))active#(g(active(f(active(g(mark(_x61), _x62)), _x32)), X2))mark#(g(f(a, _x32), mark(_x22)))active#(g(active(f(active(a), _x32)), _x22))
mark#(g(a, active(mark(_x32))))active#(g(mark(b), _x32))mark#(g(f(_x71, mark(_x72)), mark(_x22)))active#(g(active(f(_x71, _x72)), _x22))
mark#(g(f(a, _x32), mark(mark(_x42))))active#(g(active(f(active(a), _x32)), _x42))mark#(g(f(_x31, mark(_x52)), X2))active#(g(active(f(mark(_x31), _x52)), X2))
mark#(g(f(_x31, _x32), active(_x22)))active#(g(active(f(mark(_x31), _x32)), _x22))mark#(g(f(f(_x51, _x52), mark(_x62)), mark(_x42)))active#(g(f(active(f(mark(_x51), _x52)), _x62), _x42))
mark#(g(f(f(_x71, _x72), _x32), mark(_x22)))active#(g(active(f(active(f(mark(_x71), _x72)), _x32)), _x22))mark#(g(a, active(active(mark(active(mark(_x22)))))))active#(g(active(a), _x22))
mark#(g(f(_x31, mark(_x62)), mark(mark(_x42))))active#(g(active(f(mark(_x31), _x62)), _x42))

Rewrite Rules

active(h(X))mark(g(X, X))active(g(a, X))mark(f(b, X))
active(f(X, X))mark(h(a))active(a)mark(b)
mark(h(X))active(h(mark(X)))mark(g(X1, X2))active(g(mark(X1), X2))
mark(a)active(a)mark(f(X1, X2))active(f(mark(X1), X2))
mark(b)active(b)h(mark(X))h(X)
h(active(X))h(X)g(mark(X1), X2)g(X1, X2)
g(X1, mark(X2))g(X1, X2)g(active(X1), X2)g(X1, X2)
g(X1, active(X2))g(X1, X2)f(mark(X1), X2)f(X1, X2)
f(X1, mark(X2))f(X1, X2)f(active(X1), X2)f(X1, X2)
f(X1, active(X2))f(X1, X2)

Original Signature

Termination of terms over the following signature is verified: f, g, b, a, active, mark, h

Strategy


The right-hand side of the rule mark#(g(f(_x31, _x32), mark(mark(_x22)))) → active#(g(f(mark(_x31), _x32), _x22)) 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
active#(g(f(active(f(mark(_x61), _x62)), _x32), _x22)) 
active#(g(f(_x51, _x52), _x22)) 
active#(g(f(active(b), _x32), _x22)) 
active#(g(f(mark(_x31), _x32), _x42)) 
active#(g(f(mark(_x31), _x52), _x22)) 
active#(g(f(active(g(mark(_x61), _x62)), _x32), _x22)) 
active#(g(f(active(a), _x32), _x22)) 
active#(g(f(active(h(mark(_x61))), _x32), _x22)) 
Thus, the rule mark#(g(f(_x31, _x32), mark(mark(_x22)))) → active#(g(f(mark(_x31), _x32), _x22)) is replaced by the following rules:
mark#(g(f(a, _x32), mark(mark(_x22)))) → active#(g(f(active(a), _x32), _x22))mark#(g(f(h(_x61), _x32), mark(mark(_x22)))) → active#(g(f(active(h(mark(_x61))), _x32), _x22))
mark#(g(f(b, _x32), mark(mark(_x22)))) → active#(g(f(active(b), _x32), _x22))mark#(g(f(f(_x61, _x62), _x32), mark(mark(_x22)))) → active#(g(f(active(f(mark(_x61), _x62)), _x32), _x22))
mark#(g(f(_x31, active(_x52)), mark(mark(_x22)))) → active#(g(f(mark(_x31), _x52), _x22))mark#(g(f(_x31, _x32), mark(mark(mark(_x42))))) → active#(g(f(mark(_x31), _x32), _x42))
mark#(g(f(g(_x61, _x62), _x32), mark(mark(_x22)))) → active#(g(f(active(g(mark(_x61), _x62)), _x32), _x22))mark#(g(f(_x31, mark(_x52)), mark(mark(_x22)))) → active#(g(f(mark(_x31), _x52), _x22))
mark#(g(f(_x31, _x32), mark(mark(active(_x42))))) → active#(g(f(mark(_x31), _x32), _x42))mark#(g(f(_x51, _x52), mark(mark(_x22)))) → active#(g(f(_x51, _x52), _x22))

Problem 24: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

mark#(g(a, active(active(mark(active(_x22))))))active#(g(a, _x22))mark#(g(a, _x22))active#(g(a, _x22))
mark#(a)active#(a)mark#(g(f(b, _x32), X2))active#(g(active(f(active(b), _x32)), X2))
mark#(g(f(_x31, _x32), mark(mark(mark(_x22)))))active#(g(active(f(mark(_x31), _x32)), _x22))mark#(g(f(_x31, mark(_x52)), mark(mark(_x22))))active#(g(f(mark(_x31), _x52), _x22))
mark#(g(f(_x31, mark(_x31)), mark(mark(_x42))))active#(g(mark(h(a)), _x42))mark#(g(f(_x31, _x32), mark(active(_x42))))active#(g(active(f(mark(_x31), _x32)), _x42))
mark#(g(b, mark(_x22)))active#(g(active(b), _x22))mark#(g(f(_x31, active(_x52)), X2))active#(g(active(f(mark(_x31), _x52)), X2))
mark#(g(h(_x31), X2))active#(g(active(h(mark(_x31))), X2))active#(h(X))mark#(g(X, X))
mark#(g(f(f(_x81, _x82), mark(_x62)), mark(_x22)))active#(g(active(f(active(f(mark(_x81), _x82)), _x62)), _x22))mark#(g(f(_x31, mark(_x62)), mark(active(_x22))))active#(g(f(mark(_x31), _x62), _x22))
mark#(g(f(b, mark(_x62)), mark(_x42)))active#(g(f(active(b), _x62), _x42))mark#(g(f(g(_x51, _x52), mark(_x62)), mark(_x42)))active#(g(f(active(g(mark(_x51), _x52)), _x62), _x42))
mark#(g(a, active(_x22)))active#(g(mark(b), _x22))mark#(g(a, active(active(mark(active(active(_x22)))))))active#(g(active(a), _x22))
mark#(g(f(_x31, _x32), _x22))active#(g(f(mark(_x31), _x32), _x22))mark#(g(f(_x31, mark(_x52)), mark(mark(_x42))))active#(g(active(f(mark(_x31), _x52)), _x42))
active#(g(a, X))mark#(f(b, X))mark#(g(f(_x51, _x52), mark(mark(_x22))))active#(g(f(_x51, _x52), _x22))
mark#(g(f(_x31, mark(_x31)), mark(_x22)))active#(g(mark(h(a)), _x22))mark#(g(f(_x31, mark(_x31)), X2))active#(g(mark(h(a)), X2))
mark#(g(a, active(active(mark(mark(_x32))))))active#(g(active(a), _x32))mark#(g(X1, mark(_x22)))active#(g(mark(X1), _x22))
mark#(g(f(_x31, _x32), mark(_x42)))active#(g(f(mark(_x31), _x32), _x42))mark#(g(f(h(_x81), mark(_x62)), mark(_x22)))active#(g(active(f(active(h(mark(_x81))), _x62)), _x22))
mark#(g(f(a, _x32), X2))active#(g(active(f(active(a), _x32)), X2))mark#(h(X))mark#(X)
mark#(f(X1, X2))mark#(X1)mark#(g(f(h(_x71), _x32), mark(_x22)))active#(g(active(f(active(h(mark(_x71))), _x32)), _x22))
mark#(g(a, active(mark(_x22))))active#(g(a, _x22))mark#(g(f(g(_x81, _x82), mark(_x62)), mark(_x22)))active#(g(active(f(active(g(mark(_x81), _x82)), _x62)), _x22))
mark#(g(f(a, mark(_x62)), mark(_x42)))active#(g(f(active(a), _x62), _x42))mark#(h(X))active#(h(mark(X)))
mark#(g(a, active(active(_x32))))active#(g(mark(b), _x32))mark#(g(f(b, _x32), mark(_x22)))active#(g(active(f(active(b), _x32)), _x22))
mark#(g(f(b, mark(_x62)), mark(_x22)))active#(g(active(f(active(b), _x62)), _x22))mark#(g(f(h(_x61), _x32), mark(mark(_x42))))active#(g(active(f(active(h(mark(_x61))), _x32)), _x42))
mark#(g(f(_x31, active(_x62)), mark(_x22)))active#(g(active(f(mark(_x31), _x62)), _x22))mark#(g(f(_x51, _x52), X2))active#(g(active(f(_x51, _x52)), X2))
mark#(g(f(g(_x61, _x62), _x32), mark(mark(_x22))))active#(g(f(active(g(mark(_x61), _x62)), _x32), _x22))mark#(g(X1, active(_x22)))active#(g(mark(X1), _x22))
mark#(g(f(h(_x51), mark(_x62)), mark(_x42)))active#(g(f(active(h(mark(_x51))), _x62), _x42))mark#(g(a, active(active(_x22))))active#(g(a, _x22))
mark#(g(g(_x31, _x32), X2))active#(g(active(g(mark(_x31), _x32)), X2))mark#(g(f(_x71, mark(_x72)), mark(_x22)))active#(g(active(f(_x71, _x72)), _x22))
mark#(g(f(a, _x32), mark(mark(_x42))))active#(g(active(f(active(a), _x32)), _x42))mark#(g(f(b, _x32), mark(mark(_x22))))active#(g(f(active(b), _x32), _x22))
mark#(g(f(f(_x71, _x72), _x32), mark(_x22)))active#(g(active(f(active(f(mark(_x71), _x72)), _x32)), _x22))mark#(g(a, active(active(mark(active(mark(_x22)))))))active#(g(active(a), _x22))
mark#(g(f(_x51, _x52), mark(mark(_x42))))active#(g(active(f(_x51, _x52)), _x42))mark#(g(a, active(active(mark(_x32)))))active#(g(a, _x32))
mark#(g(f(f(_x61, _x62), _x32), X2))active#(g(active(f(active(f(mark(_x61), _x62)), _x32)), X2))active#(f(X, X))mark#(h(a))
mark#(g(b, _x22))active#(g(b, _x22))mark#(g(f(a, _x32), mark(mark(_x22))))active#(g(f(active(a), _x32), _x22))
mark#(g(f(_x31, active(_x52)), mark(mark(_x22))))active#(g(f(mark(_x31), _x52), _x22))mark#(g(a, X2))active#(g(mark(b), X2))
mark#(g(f(g(_x61, _x62), _x32), mark(mark(_x42))))active#(g(active(f(active(g(mark(_x61), _x62)), _x32)), _x42))mark#(g(a, active(active(mark(_x22)))))active#(g(mark(b), _x22))
mark#(g(a, active(_x32)))active#(g(a, _x32))mark#(g(a, active(mark(active(_x22)))))active#(g(active(a), _x22))
mark#(g(f(_x31, mark(active(_x72))), mark(_x22)))active#(g(active(f(mark(_x31), _x72)), _x22))mark#(g(f(b, _x32), mark(mark(_x42))))active#(g(active(f(active(b), _x32)), _x42))
mark#(g(f(_x41, mark(_x42)), mark(_x42)))active#(g(f(_x41, _x42), _x42))mark#(g(f(f(_x61, _x62), _x32), mark(mark(_x22))))active#(g(f(active(f(mark(_x61), _x62)), _x32), _x22))
mark#(g(f(_x31, mark(_x62)), mark(mark(_x22))))active#(g(f(mark(_x31), _x62), _x22))mark#(g(_x21, _x22))active#(g(_x21, _x22))
mark#(g(a, active(mark(mark(_x22)))))active#(g(active(a), _x22))mark#(g(f(_x31, mark(mark(_x42))), mark(_x42)))active#(g(f(mark(_x31), _x42), _x42))
mark#(g(f(_x31, mark(mark(_x72))), mark(_x22)))active#(g(active(f(mark(_x31), _x72)), _x22))mark#(g(a, mark(_x22)))active#(g(active(a), _x22))
mark#(g(f(_x31, mark(active(_x42))), mark(_x42)))active#(g(f(mark(_x31), _x42), _x42))mark#(g(f(_x31, _x32), mark(mark(mark(_x42)))))active#(g(f(mark(_x31), _x32), _x42))
mark#(g(a, active(active(active(_x22)))))active#(g(active(a), _x22))mark#(g(f(_x31, _x32), mark(mark(active(_x42)))))active#(g(f(mark(_x31), _x32), _x42))
mark#(g(f(_x31, mark(mark(_x31))), mark(_x22)))active#(g(mark(h(a)), _x22))active#(a)mark#(b)
mark#(g(f(_x31, active(_x52)), mark(mark(_x42))))active#(g(active(f(mark(_x31), _x52)), _x42))mark#(g(f(f(_x61, _x62), _x32), mark(mark(_x42))))active#(g(active(f(active(f(mark(_x61), _x62)), _x32)), _x42))
mark#(g(f(a, mark(_x62)), mark(_x22)))active#(g(active(f(active(a), _x62)), _x22))mark#(g(a, active(active(mark(active(_x32))))))active#(g(mark(b), _x32))
mark#(g(f(_x61, _x62), mark(_x22)))active#(g(active(f(_x61, _x62)), _x22))mark#(g(X1, X2))mark#(X1)
mark#(g(f(_x31, mark(_x62)), mark(active(_x42))))active#(g(active(f(mark(_x31), _x62)), _x42))mark#(g(f(g(_x71, _x72), _x32), mark(_x22)))active#(g(active(f(active(g(mark(_x71), _x72)), _x32)), _x22))
mark#(g(f(h(_x61), _x32), X2))active#(g(active(f(active(h(mark(_x61))), _x32)), X2))mark#(g(b, active(_x22)))active#(g(active(b), _x22))
mark#(f(X1, X2))active#(f(mark(X1), X2))mark#(g(f(_x31, _x32), mark(mark(active(_x22)))))active#(g(active(f(mark(_x31), _x32)), _x22))
mark#(g(f(g(_x61, _x62), _x32), X2))active#(g(active(f(active(g(mark(_x61), _x62)), _x32)), X2))mark#(g(a, active(mark(_x32))))active#(g(mark(b), _x32))
mark#(g(f(a, _x32), mark(_x22)))active#(g(active(f(active(a), _x32)), _x22))mark#(g(f(h(_x61), _x32), mark(mark(_x22))))active#(g(f(active(h(mark(_x61))), _x32), _x22))
mark#(g(f(_x31, mark(_x52)), X2))active#(g(active(f(mark(_x31), _x52)), X2))mark#(g(f(_x31, _x32), active(_x22)))active#(g(active(f(mark(_x31), _x32)), _x22))
mark#(g(f(f(_x51, _x52), mark(_x62)), mark(_x42)))active#(g(f(active(f(mark(_x51), _x52)), _x62), _x42))mark#(g(f(_x31, mark(_x62)), mark(mark(_x42))))active#(g(active(f(mark(_x31), _x62)), _x42))

Rewrite Rules

active(h(X))mark(g(X, X))active(g(a, X))mark(f(b, X))
active(f(X, X))mark(h(a))active(a)mark(b)
mark(h(X))active(h(mark(X)))mark(g(X1, X2))active(g(mark(X1), X2))
mark(a)active(a)mark(f(X1, X2))active(f(mark(X1), X2))
mark(b)active(b)h(mark(X))h(X)
h(active(X))h(X)g(mark(X1), X2)g(X1, X2)
g(X1, mark(X2))g(X1, X2)g(active(X1), X2)g(X1, X2)
g(X1, active(X2))g(X1, X2)f(mark(X1), X2)f(X1, X2)
f(X1, mark(X2))f(X1, X2)f(active(X1), X2)f(X1, X2)
f(X1, active(X2))f(X1, X2)

Original Signature

Termination of terms over the following signature is verified: f, g, b, a, active, mark, h

Strategy


The right-hand side of the rule mark#(g(f(b, _x32), X2)) → active#(g(active(f(active(b), _x32)), X2)) 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
active#(g(active(f(active(b), _x32)), _x22)) 
active#(g(f(active(b), _x32), _x22)) 
active#(g(active(f(b, _x52)), X2)) 
active#(g(active(f(active(b), _x52)), X2)) 
active#(g(mark(h(a)), X2)) 
Thus, the rule mark#(g(f(b, _x32), X2)) → active#(g(active(f(active(b), _x32)), X2)) is replaced by the following rules:
mark#(g(f(b, _x32), active(_x22))) → active#(g(active(f(active(b), _x32)), _x22))mark#(g(f(b, _x32), _x22)) → active#(g(f(active(b), _x32), _x22))
mark#(g(f(b, _x32), mark(_x22))) → active#(g(active(f(active(b), _x32)), _x22))mark#(g(f(b, mark(_x52)), X2)) → active#(g(active(f(active(b), _x52)), X2))
mark#(g(f(b, active(b)), X2)) → active#(g(mark(h(a)), X2))mark#(g(f(b, active(_x52)), X2)) → active#(g(active(f(active(b), _x52)), X2))
mark#(g(f(b, _x52), X2)) → active#(g(active(f(b, _x52)), X2))

Problem 25: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

mark#(g(a, active(active(mark(active(_x22))))))active#(g(a, _x22))mark#(g(a, _x22))active#(g(a, _x22))
mark#(a)active#(a)mark#(g(f(_x31, _x32), mark(mark(mark(_x22)))))active#(g(active(f(mark(_x31), _x32)), _x22))
mark#(g(f(b, mark(_x52)), X2))active#(g(active(f(active(b), _x52)), X2))mark#(g(f(_x31, mark(_x52)), mark(mark(_x22))))active#(g(f(mark(_x31), _x52), _x22))
mark#(g(b, mark(_x22)))active#(g(active(b), _x22))mark#(g(f(_x31, _x32), mark(active(_x42))))active#(g(active(f(mark(_x31), _x32)), _x42))
mark#(g(f(_x31, mark(_x31)), mark(mark(_x42))))active#(g(mark(h(a)), _x42))mark#(g(f(_x31, active(_x52)), X2))active#(g(active(f(mark(_x31), _x52)), X2))
active#(h(X))mark#(g(X, X))mark#(g(h(_x31), X2))active#(g(active(h(mark(_x31))), X2))
mark#(g(f(f(_x81, _x82), mark(_x62)), mark(_x22)))active#(g(active(f(active(f(mark(_x81), _x82)), _x62)), _x22))mark#(g(f(_x31, mark(_x62)), mark(active(_x22))))active#(g(f(mark(_x31), _x62), _x22))
mark#(g(f(b, mark(_x62)), mark(_x42)))active#(g(f(active(b), _x62), _x42))mark#(g(f(g(_x51, _x52), mark(_x62)), mark(_x42)))active#(g(f(active(g(mark(_x51), _x52)), _x62), _x42))
mark#(g(a, active(_x22)))active#(g(mark(b), _x22))mark#(g(a, active(active(mark(active(active(_x22)))))))active#(g(active(a), _x22))
mark#(g(f(_x31, _x32), _x22))active#(g(f(mark(_x31), _x32), _x22))mark#(g(f(_x31, mark(_x52)), mark(mark(_x42))))active#(g(active(f(mark(_x31), _x52)), _x42))
mark#(g(f(b, active(_x52)), X2))active#(g(active(f(active(b), _x52)), X2))active#(g(a, X))mark#(f(b, X))
mark#(g(f(_x51, _x52), mark(mark(_x22))))active#(g(f(_x51, _x52), _x22))mark#(g(f(_x31, mark(_x31)), mark(_x22)))active#(g(mark(h(a)), _x22))
mark#(g(f(_x31, mark(_x31)), X2))active#(g(mark(h(a)), X2))mark#(g(X1, mark(_x22)))active#(g(mark(X1), _x22))
mark#(g(a, active(active(mark(mark(_x32))))))active#(g(active(a), _x32))mark#(g(f(_x31, _x32), mark(_x42)))active#(g(f(mark(_x31), _x32), _x42))
mark#(g(f(a, _x32), X2))active#(g(active(f(active(a), _x32)), X2))mark#(g(f(h(_x81), mark(_x62)), mark(_x22)))active#(g(active(f(active(h(mark(_x81))), _x62)), _x22))
mark#(h(X))mark#(X)mark#(f(X1, X2))mark#(X1)
mark#(g(f(b, _x32), active(_x22)))active#(g(active(f(active(b), _x32)), _x22))mark#(g(f(h(_x71), _x32), mark(_x22)))active#(g(active(f(active(h(mark(_x71))), _x32)), _x22))
mark#(g(a, active(mark(_x22))))active#(g(a, _x22))mark#(g(f(g(_x81, _x82), mark(_x62)), mark(_x22)))active#(g(active(f(active(g(mark(_x81), _x82)), _x62)), _x22))
mark#(g(f(b, _x52), X2))active#(g(active(f(b, _x52)), X2))mark#(h(X))active#(h(mark(X)))
mark#(g(f(a, mark(_x62)), mark(_x42)))active#(g(f(active(a), _x62), _x42))mark#(g(a, active(active(_x32))))active#(g(mark(b), _x32))
mark#(g(f(b, _x32), mark(_x22)))active#(g(active(f(active(b), _x32)), _x22))mark#(g(f(b, mark(_x62)), mark(_x22)))active#(g(active(f(active(b), _x62)), _x22))
mark#(g(f(h(_x61), _x32), mark(mark(_x42))))active#(g(active(f(active(h(mark(_x61))), _x32)), _x42))mark#(g(f(_x31, active(_x62)), mark(_x22)))active#(g(active(f(mark(_x31), _x62)), _x22))
mark#(g(f(_x51, _x52), X2))active#(g(active(f(_x51, _x52)), X2))mark#(g(f(g(_x61, _x62), _x32), mark(mark(_x22))))active#(g(f(active(g(mark(_x61), _x62)), _x32), _x22))
mark#(g(X1, active(_x22)))active#(g(mark(X1), _x22))mark#(g(a, active(active(_x22))))active#(g(a, _x22))
mark#(g(f(h(_x51), mark(_x62)), mark(_x42)))active#(g(f(active(h(mark(_x51))), _x62), _x42))mark#(g(g(_x31, _x32), X2))active#(g(active(g(mark(_x31), _x32)), X2))
mark#(g(f(_x71, mark(_x72)), mark(_x22)))active#(g(active(f(_x71, _x72)), _x22))mark#(g(f(a, _x32), mark(mark(_x42))))active#(g(active(f(active(a), _x32)), _x42))
mark#(g(f(b, _x32), mark(mark(_x22))))active#(g(f(active(b), _x32), _x22))mark#(g(f(f(_x71, _x72), _x32), mark(_x22)))active#(g(active(f(active(f(mark(_x71), _x72)), _x32)), _x22))
mark#(g(a, active(active(mark(active(mark(_x22)))))))active#(g(active(a), _x22))mark#(g(f(_x51, _x52), mark(mark(_x42))))active#(g(active(f(_x51, _x52)), _x42))
mark#(g(f(b, _x32), _x22))active#(g(f(active(b), _x32), _x22))mark#(g(a, active(active(mark(_x32)))))active#(g(a, _x32))
mark#(g(f(f(_x61, _x62), _x32), X2))active#(g(active(f(active(f(mark(_x61), _x62)), _x32)), X2))active#(f(X, X))mark#(h(a))
mark#(g(b, _x22))active#(g(b, _x22))mark#(g(f(a, _x32), mark(mark(_x22))))active#(g(f(active(a), _x32), _x22))
mark#(g(f(_x31, active(_x52)), mark(mark(_x22))))active#(g(f(mark(_x31), _x52), _x22))mark#(g(f(b, active(b)), X2))active#(g(mark(h(a)), X2))
mark#(g(a, X2))active#(g(mark(b), X2))mark#(g(f(g(_x61, _x62), _x32), mark(mark(_x42))))active#(g(active(f(active(g(mark(_x61), _x62)), _x32)), _x42))
mark#(g(a, active(active(mark(_x22)))))active#(g(mark(b), _x22))mark#(g(a, active(_x32)))active#(g(a, _x32))
mark#(g(a, active(mark(active(_x22)))))active#(g(active(a), _x22))mark#(g(f(_x31, mark(active(_x72))), mark(_x22)))active#(g(active(f(mark(_x31), _x72)), _x22))
mark#(g(f(b, _x32), mark(mark(_x42))))active#(g(active(f(active(b), _x32)), _x42))mark#(g(f(_x41, mark(_x42)), mark(_x42)))active#(g(f(_x41, _x42), _x42))
mark#(g(f(f(_x61, _x62), _x32), mark(mark(_x22))))active#(g(f(active(f(mark(_x61), _x62)), _x32), _x22))mark#(g(f(_x31, mark(_x62)), mark(mark(_x22))))active#(g(f(mark(_x31), _x62), _x22))
mark#(g(_x21, _x22))active#(g(_x21, _x22))mark#(g(a, active(mark(mark(_x22)))))active#(g(active(a), _x22))
mark#(g(f(_x31, mark(mark(_x42))), mark(_x42)))active#(g(f(mark(_x31), _x42), _x42))mark#(g(f(_x31, mark(mark(_x72))), mark(_x22)))active#(g(active(f(mark(_x31), _x72)), _x22))
mark#(g(a, mark(_x22)))active#(g(active(a), _x22))mark#(g(f(_x31, mark(active(_x42))), mark(_x42)))active#(g(f(mark(_x31), _x42), _x42))
mark#(g(f(_x31, _x32), mark(mark(mark(_x42)))))active#(g(f(mark(_x31), _x32), _x42))mark#(g(a, active(active(active(_x22)))))active#(g(active(a), _x22))
mark#(g(f(_x31, _x32), mark(mark(active(_x42)))))active#(g(f(mark(_x31), _x32), _x42))mark#(g(f(_x31, mark(mark(_x31))), mark(_x22)))active#(g(mark(h(a)), _x22))
active#(a)mark#(b)mark#(g(f(_x31, active(_x52)), mark(mark(_x42))))active#(g(active(f(mark(_x31), _x52)), _x42))
mark#(g(f(f(_x61, _x62), _x32), mark(mark(_x42))))active#(g(active(f(active(f(mark(_x61), _x62)), _x32)), _x42))mark#(g(f(a, mark(_x62)), mark(_x22)))active#(g(active(f(active(a), _x62)), _x22))
mark#(g(a, active(active(mark(active(_x32))))))active#(g(mark(b), _x32))mark#(g(f(_x61, _x62), mark(_x22)))active#(g(active(f(_x61, _x62)), _x22))
mark#(g(X1, X2))mark#(X1)mark#(g(f(_x31, mark(_x62)), mark(active(_x42))))active#(g(active(f(mark(_x31), _x62)), _x42))
mark#(g(f(g(_x71, _x72), _x32), mark(_x22)))active#(g(active(f(active(g(mark(_x71), _x72)), _x32)), _x22))mark#(g(f(h(_x61), _x32), X2))active#(g(active(f(active(h(mark(_x61))), _x32)), X2))
mark#(g(b, active(_x22)))active#(g(active(b), _x22))mark#(f(X1, X2))active#(f(mark(X1), X2))
mark#(g(f(_x31, _x32), mark(mark(active(_x22)))))active#(g(active(f(mark(_x31), _x32)), _x22))mark#(g(f(g(_x61, _x62), _x32), X2))active#(g(active(f(active(g(mark(_x61), _x62)), _x32)), X2))
mark#(g(f(a, _x32), mark(_x22)))active#(g(active(f(active(a), _x32)), _x22))mark#(g(a, active(mark(_x32))))active#(g(mark(b), _x32))
mark#(g(f(h(_x61), _x32), mark(mark(_x22))))active#(g(f(active(h(mark(_x61))), _x32), _x22))mark#(g(f(_x31, _x32), active(_x22)))active#(g(active(f(mark(_x31), _x32)), _x22))
mark#(g(f(_x31, mark(_x52)), X2))active#(g(active(f(mark(_x31), _x52)), X2))mark#(g(f(f(_x51, _x52), mark(_x62)), mark(_x42)))active#(g(f(active(f(mark(_x51), _x52)), _x62), _x42))
mark#(g(f(_x31, mark(_x62)), mark(mark(_x42))))active#(g(active(f(mark(_x31), _x62)), _x42))

Rewrite Rules

active(h(X))mark(g(X, X))active(g(a, X))mark(f(b, X))
active(f(X, X))mark(h(a))active(a)mark(b)
mark(h(X))active(h(mark(X)))mark(g(X1, X2))active(g(mark(X1), X2))
mark(a)active(a)mark(f(X1, X2))active(f(mark(X1), X2))
mark(b)active(b)h(mark(X))h(X)
h(active(X))h(X)g(mark(X1), X2)g(X1, X2)
g(X1, mark(X2))g(X1, X2)g(active(X1), X2)g(X1, X2)
g(X1, active(X2))g(X1, X2)f(mark(X1), X2)f(X1, X2)
f(X1, mark(X2))f(X1, X2)f(active(X1), X2)f(X1, X2)
f(X1, active(X2))f(X1, X2)

Original Signature

Termination of terms over the following signature is verified: f, g, b, a, active, mark, h

Strategy


The right-hand side of the rule mark#(g(f(b, mark(_x52)), X2)) → active#(g(active(f(active(b), _x52)), X2)) 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
active#(g(active(f(b, _x42)), X2)) 
active#(g(active(f(active(b), _x42)), X2)) 
active#(g(mark(h(a)), X2)) 
active#(g(active(f(active(b), _x52)), _x22)) 
active#(g(f(active(b), _x52), _x22)) 
Thus, the rule mark#(g(f(b, mark(_x52)), X2)) → active#(g(active(f(active(b), _x52)), X2)) is replaced by the following rules:
mark#(g(f(b, mark(_x42)), X2)) → active#(g(active(f(b, _x42)), X2))mark#(g(f(b, mark(active(_x42))), X2)) → active#(g(active(f(active(b), _x42)), X2))
mark#(g(f(b, mark(active(b))), X2)) → active#(g(mark(h(a)), X2))mark#(g(f(b, mark(mark(_x42))), X2)) → active#(g(active(f(active(b), _x42)), X2))
mark#(g(f(b, mark(_x52)), active(_x22))) → active#(g(active(f(active(b), _x52)), _x22))mark#(g(f(b, mark(_x52)), _x22)) → active#(g(f(active(b), _x52), _x22))
mark#(g(f(b, mark(_x52)), mark(_x22))) → active#(g(active(f(active(b), _x52)), _x22))

Problem 26: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

mark#(g(a, active(active(mark(active(_x22))))))active#(g(a, _x22))mark#(g(a, _x22))active#(g(a, _x22))
mark#(a)active#(a)mark#(g(f(_x31, _x32), mark(mark(mark(_x22)))))active#(g(active(f(mark(_x31), _x32)), _x22))
mark#(g(f(_x31, mark(_x52)), mark(mark(_x22))))active#(g(f(mark(_x31), _x52), _x22))mark#(g(f(_x31, mark(_x31)), mark(mark(_x42))))active#(g(mark(h(a)), _x42))
mark#(g(f(_x31, _x32), mark(active(_x42))))active#(g(active(f(mark(_x31), _x32)), _x42))mark#(g(b, mark(_x22)))active#(g(active(b), _x22))
mark#(g(f(_x31, active(_x52)), X2))active#(g(active(f(mark(_x31), _x52)), X2))mark#(g(h(_x31), X2))active#(g(active(h(mark(_x31))), X2))
active#(h(X))mark#(g(X, X))mark#(g(f(_x31, mark(_x62)), mark(active(_x22))))active#(g(f(mark(_x31), _x62), _x22))
mark#(g(f(f(_x81, _x82), mark(_x62)), mark(_x22)))active#(g(active(f(active(f(mark(_x81), _x82)), _x62)), _x22))mark#(g(f(b, mark(_x62)), mark(_x42)))active#(g(f(active(b), _x62), _x42))
mark#(g(f(g(_x51, _x52), mark(_x62)), mark(_x42)))active#(g(f(active(g(mark(_x51), _x52)), _x62), _x42))mark#(g(a, active(_x22)))active#(g(mark(b), _x22))
mark#(g(a, active(active(mark(active(active(_x22)))))))active#(g(active(a), _x22))mark#(g(f(_x31, _x32), _x22))active#(g(f(mark(_x31), _x32), _x22))
mark#(g(f(_x31, mark(_x52)), mark(mark(_x42))))active#(g(active(f(mark(_x31), _x52)), _x42))mark#(g(f(b, active(_x52)), X2))active#(g(active(f(active(b), _x52)), X2))
active#(g(a, X))mark#(f(b, X))mark#(g(f(_x51, _x52), mark(mark(_x22))))active#(g(f(_x51, _x52), _x22))
mark#(g(f(b, mark(active(b))), X2))active#(g(mark(h(a)), X2))mark#(g(f(_x31, mark(_x31)), mark(_x22)))active#(g(mark(h(a)), _x22))
mark#(g(f(_x31, mark(_x31)), X2))active#(g(mark(h(a)), X2))mark#(g(a, active(active(mark(mark(_x32))))))active#(g(active(a), _x32))
mark#(g(X1, mark(_x22)))active#(g(mark(X1), _x22))mark#(g(f(_x31, _x32), mark(_x42)))active#(g(f(mark(_x31), _x32), _x42))
mark#(g(f(h(_x81), mark(_x62)), mark(_x22)))active#(g(active(f(active(h(mark(_x81))), _x62)), _x22))mark#(g(f(a, _x32), X2))active#(g(active(f(active(a), _x32)), X2))
mark#(h(X))mark#(X)mark#(f(X1, X2))mark#(X1)
mark#(g(f(b, _x32), active(_x22)))active#(g(active(f(active(b), _x32)), _x22))mark#(g(f(h(_x71), _x32), mark(_x22)))active#(g(active(f(active(h(mark(_x71))), _x32)), _x22))
mark#(g(a, active(mark(_x22))))active#(g(a, _x22))mark#(g(f(b, mark(_x52)), _x22))active#(g(f(active(b), _x52), _x22))
mark#(g(f(g(_x81, _x82), mark(_x62)), mark(_x22)))active#(g(active(f(active(g(mark(_x81), _x82)), _x62)), _x22))mark#(g(f(b, _x52), X2))active#(g(active(f(b, _x52)), X2))
mark#(g(f(a, mark(_x62)), mark(_x42)))active#(g(f(active(a), _x62), _x42))mark#(h(X))active#(h(mark(X)))
mark#(g(a, active(active(_x32))))active#(g(mark(b), _x32))mark#(g(f(b, _x32), mark(_x22)))active#(g(active(f(active(b), _x32)), _x22))
mark#(g(f(b, mark(_x62)), mark(_x22)))active#(g(active(f(active(b), _x62)), _x22))mark#(g(f(h(_x61), _x32), mark(mark(_x42))))active#(g(active(f(active(h(mark(_x61))), _x32)), _x42))
mark#(g(f(_x31, active(_x62)), mark(_x22)))active#(g(active(f(mark(_x31), _x62)), _x22))mark#(g(f(_x51, _x52), X2))active#(g(active(f(_x51, _x52)), X2))
mark#(g(f(g(_x61, _x62), _x32), mark(mark(_x22))))active#(g(f(active(g(mark(_x61), _x62)), _x32), _x22))mark#(g(X1, active(_x22)))active#(g(mark(X1), _x22))
mark#(g(a, active(active(_x22))))active#(g(a, _x22))mark#(g(f(h(_x51), mark(_x62)), mark(_x42)))active#(g(f(active(h(mark(_x51))), _x62), _x42))
mark#(g(g(_x31, _x32), X2))active#(g(active(g(mark(_x31), _x32)), X2))mark#(g(f(_x71, mark(_x72)), mark(_x22)))active#(g(active(f(_x71, _x72)), _x22))
mark#(g(f(a, _x32), mark(mark(_x42))))active#(g(active(f(active(a), _x32)), _x42))mark#(g(f(b, _x32), mark(mark(_x22))))active#(g(f(active(b), _x32), _x22))
mark#(g(f(f(_x71, _x72), _x32), mark(_x22)))active#(g(active(f(active(f(mark(_x71), _x72)), _x32)), _x22))mark#(g(a, active(active(mark(active(mark(_x22)))))))active#(g(active(a), _x22))
mark#(g(f(_x51, _x52), mark(mark(_x42))))active#(g(active(f(_x51, _x52)), _x42))mark#(g(f(b, _x32), _x22))active#(g(f(active(b), _x32), _x22))
mark#(g(a, active(active(mark(_x32)))))active#(g(a, _x32))mark#(g(f(f(_x61, _x62), _x32), X2))active#(g(active(f(active(f(mark(_x61), _x62)), _x32)), X2))
active#(f(X, X))mark#(h(a))mark#(g(b, _x22))active#(g(b, _x22))
mark#(g(f(a, _x32), mark(mark(_x22))))active#(g(f(active(a), _x32), _x22))mark#(g(f(_x31, active(_x52)), mark(mark(_x22))))active#(g(f(mark(_x31), _x52), _x22))
mark#(g(f(b, active(b)), X2))active#(g(mark(h(a)), X2))mark#(g(a, X2))active#(g(mark(b), X2))
mark#(g(f(b, mark(active(_x42))), X2))active#(g(active(f(active(b), _x42)), X2))mark#(g(f(g(_x61, _x62), _x32), mark(mark(_x42))))active#(g(active(f(active(g(mark(_x61), _x62)), _x32)), _x42))
mark#(g(a, active(active(mark(_x22)))))active#(g(mark(b), _x22))mark#(g(a, active(_x32)))active#(g(a, _x32))
mark#(g(a, active(mark(active(_x22)))))active#(g(active(a), _x22))mark#(g(f(_x31, mark(active(_x72))), mark(_x22)))active#(g(active(f(mark(_x31), _x72)), _x22))
mark#(g(f(b, _x32), mark(mark(_x42))))active#(g(active(f(active(b), _x32)), _x42))mark#(g(f(_x41, mark(_x42)), mark(_x42)))active#(g(f(_x41, _x42), _x42))
mark#(g(f(f(_x61, _x62), _x32), mark(mark(_x22))))active#(g(f(active(f(mark(_x61), _x62)), _x32), _x22))mark#(g(f(_x31, mark(_x62)), mark(mark(_x22))))active#(g(f(mark(_x31), _x62), _x22))
mark#(g(_x21, _x22))active#(g(_x21, _x22))mark#(g(a, active(mark(mark(_x22)))))active#(g(active(a), _x22))
mark#(g(f(_x31, mark(mark(_x42))), mark(_x42)))active#(g(f(mark(_x31), _x42), _x42))mark#(g(f(_x31, mark(mark(_x72))), mark(_x22)))active#(g(active(f(mark(_x31), _x72)), _x22))
mark#(g(a, mark(_x22)))active#(g(active(a), _x22))mark#(g(f(_x31, mark(active(_x42))), mark(_x42)))active#(g(f(mark(_x31), _x42), _x42))
mark#(g(f(_x31, _x32), mark(mark(mark(_x42)))))active#(g(f(mark(_x31), _x32), _x42))mark#(g(a, active(active(active(_x22)))))active#(g(active(a), _x22))
mark#(g(f(_x31, _x32), mark(mark(active(_x42)))))active#(g(f(mark(_x31), _x32), _x42))mark#(g(f(b, mark(_x42)), X2))active#(g(active(f(b, _x42)), X2))
mark#(g(f(_x31, mark(mark(_x31))), mark(_x22)))active#(g(mark(h(a)), _x22))active#(a)mark#(b)
mark#(g(f(b, mark(_x52)), mark(_x22)))active#(g(active(f(active(b), _x52)), _x22))mark#(g(f(_x31, active(_x52)), mark(mark(_x42))))active#(g(active(f(mark(_x31), _x52)), _x42))
mark#(g(f(f(_x61, _x62), _x32), mark(mark(_x42))))active#(g(active(f(active(f(mark(_x61), _x62)), _x32)), _x42))mark#(g(f(a, mark(_x62)), mark(_x22)))active#(g(active(f(active(a), _x62)), _x22))
mark#(g(a, active(active(mark(active(_x32))))))active#(g(mark(b), _x32))mark#(g(f(_x61, _x62), mark(_x22)))active#(g(active(f(_x61, _x62)), _x22))
mark#(g(X1, X2))mark#(X1)mark#(g(f(_x31, mark(_x62)), mark(active(_x42))))active#(g(active(f(mark(_x31), _x62)), _x42))
mark#(g(f(g(_x71, _x72), _x32), mark(_x22)))active#(g(active(f(active(g(mark(_x71), _x72)), _x32)), _x22))mark#(g(b, active(_x22)))active#(g(active(b), _x22))
mark#(g(f(h(_x61), _x32), X2))active#(g(active(f(active(h(mark(_x61))), _x32)), X2))mark#(f(X1, X2))active#(f(mark(X1), X2))
mark#(g(f(b, mark(mark(_x42))), X2))active#(g(active(f(active(b), _x42)), X2))mark#(g(f(_x31, _x32), mark(mark(active(_x22)))))active#(g(active(f(mark(_x31), _x32)), _x22))
mark#(g(f(g(_x61, _x62), _x32), X2))active#(g(active(f(active(g(mark(_x61), _x62)), _x32)), X2))mark#(g(a, active(mark(_x32))))active#(g(mark(b), _x32))
mark#(g(f(a, _x32), mark(_x22)))active#(g(active(f(active(a), _x32)), _x22))mark#(g(f(h(_x61), _x32), mark(mark(_x22))))active#(g(f(active(h(mark(_x61))), _x32), _x22))
mark#(g(f(_x31, mark(_x52)), X2))active#(g(active(f(mark(_x31), _x52)), X2))mark#(g(f(_x31, _x32), active(_x22)))active#(g(active(f(mark(_x31), _x32)), _x22))
mark#(g(f(f(_x51, _x52), mark(_x62)), mark(_x42)))active#(g(f(active(f(mark(_x51), _x52)), _x62), _x42))mark#(g(f(_x31, mark(_x62)), mark(mark(_x42))))active#(g(active(f(mark(_x31), _x62)), _x42))
mark#(g(f(b, mark(_x52)), active(_x22)))active#(g(active(f(active(b), _x52)), _x22))

Rewrite Rules

active(h(X))mark(g(X, X))active(g(a, X))mark(f(b, X))
active(f(X, X))mark(h(a))active(a)mark(b)
mark(h(X))active(h(mark(X)))mark(g(X1, X2))active(g(mark(X1), X2))
mark(a)active(a)mark(f(X1, X2))active(f(mark(X1), X2))
mark(b)active(b)h(mark(X))h(X)
h(active(X))h(X)g(mark(X1), X2)g(X1, X2)
g(X1, mark(X2))g(X1, X2)g(active(X1), X2)g(X1, X2)
g(X1, active(X2))g(X1, X2)f(mark(X1), X2)f(X1, X2)
f(X1, mark(X2))f(X1, X2)f(active(X1), X2)f(X1, X2)
f(X1, active(X2))f(X1, X2)

Original Signature

Termination of terms over the following signature is verified: f, g, b, a, active, mark, h

Strategy


The right-hand side of the rule mark#(g(f(_x31, _x32), mark(mark(mark(_x22))))) → active#(g(active(f(mark(_x31), _x32)), _x22)) 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
active#(g(active(f(active(b), _x32)), _x22)) 
active#(g(active(f(active(g(mark(_x71), _x72)), _x32)), _x22)) 
active#(g(active(f(_x61, _x62)), _x22)) 
active#(g(mark(h(a)), _x22)) 
active#(g(f(mark(_x31), _x32), _x42)) 
active#(g(active(f(active(a), _x32)), _x22)) 
active#(g(active(f(mark(_x31), _x62)), _x22)) 
active#(g(active(f(mark(_x31), _x32)), _x42)) 
active#(g(active(f(active(f(mark(_x71), _x72)), _x32)), _x22)) 
active#(g(active(f(active(h(mark(_x71))), _x32)), _x22)) 
Thus, the rule mark#(g(f(_x31, _x32), mark(mark(mark(_x22))))) → active#(g(active(f(mark(_x31), _x32)), _x22)) is replaced by the following rules:
mark#(g(f(_x31, mark(_x62)), mark(mark(mark(_x22))))) → active#(g(active(f(mark(_x31), _x62)), _x22))mark#(g(f(b, _x32), mark(mark(mark(_x22))))) → active#(g(active(f(active(b), _x32)), _x22))
mark#(g(f(g(_x71, _x72), _x32), mark(mark(mark(_x22))))) → active#(g(active(f(active(g(mark(_x71), _x72)), _x32)), _x22))mark#(g(f(a, _x32), mark(mark(mark(_x22))))) → active#(g(active(f(active(a), _x32)), _x22))
mark#(g(f(_x61, _x62), mark(mark(mark(_x22))))) → active#(g(active(f(_x61, _x62)), _x22))mark#(g(f(_x31, active(_x62)), mark(mark(mark(_x22))))) → active#(g(active(f(mark(_x31), _x62)), _x22))
mark#(g(f(h(_x71), _x32), mark(mark(mark(_x22))))) → active#(g(active(f(active(h(mark(_x71))), _x32)), _x22))mark#(g(f(_x31, _x32), mark(mark(mark(active(_x42)))))) → active#(g(active(f(mark(_x31), _x32)), _x42))
mark#(g(f(_x31, mark(_x31)), mark(mark(mark(_x22))))) → active#(g(mark(h(a)), _x22))mark#(g(f(_x31, _x32), mark(mark(mark(_x42))))) → active#(g(f(mark(_x31), _x32), _x42))
mark#(g(f(_x31, _x32), mark(mark(mark(mark(_x42)))))) → active#(g(active(f(mark(_x31), _x32)), _x42))mark#(g(f(f(_x71, _x72), _x32), mark(mark(mark(_x22))))) → active#(g(active(f(active(f(mark(_x71), _x72)), _x32)), _x22))

Problem 27: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

mark#(g(a, active(active(mark(active(_x22))))))active#(g(a, _x22))mark#(g(a, _x22))active#(g(a, _x22))
mark#(a)active#(a)mark#(g(f(_x31, mark(_x52)), mark(mark(_x22))))active#(g(f(mark(_x31), _x52), _x22))
mark#(g(b, mark(_x22)))active#(g(active(b), _x22))mark#(g(f(_x31, _x32), mark(active(_x42))))active#(g(active(f(mark(_x31), _x32)), _x42))
mark#(g(f(_x31, mark(_x31)), mark(mark(_x42))))active#(g(mark(h(a)), _x42))mark#(g(f(_x31, active(_x52)), X2))active#(g(active(f(mark(_x31), _x52)), X2))
active#(h(X))mark#(g(X, X))mark#(g(h(_x31), X2))active#(g(active(h(mark(_x31))), X2))
mark#(g(f(f(_x81, _x82), mark(_x62)), mark(_x22)))active#(g(active(f(active(f(mark(_x81), _x82)), _x62)), _x22))mark#(g(f(_x31, mark(_x62)), mark(active(_x22))))active#(g(f(mark(_x31), _x62), _x22))
mark#(g(f(b, mark(_x62)), mark(_x42)))active#(g(f(active(b), _x62), _x42))mark#(g(f(g(_x51, _x52), mark(_x62)), mark(_x42)))active#(g(f(active(g(mark(_x51), _x52)), _x62), _x42))
mark#(g(a, active(_x22)))active#(g(mark(b), _x22))mark#(g(a, active(active(mark(active(active(_x22)))))))active#(g(active(a), _x22))
mark#(g(f(_x31, _x32), _x22))active#(g(f(mark(_x31), _x32), _x22))mark#(g(f(_x31, mark(_x52)), mark(mark(_x42))))active#(g(active(f(mark(_x31), _x52)), _x42))
mark#(g(f(b, active(_x52)), X2))active#(g(active(f(active(b), _x52)), X2))active#(g(a, X))mark#(f(b, X))
mark#(g(f(_x51, _x52), mark(mark(_x22))))active#(g(f(_x51, _x52), _x22))mark#(g(f(b, mark(active(b))), X2))active#(g(mark(h(a)), X2))
mark#(g(f(_x31, mark(_x31)), mark(_x22)))active#(g(mark(h(a)), _x22))mark#(g(f(_x31, mark(_x31)), X2))active#(g(mark(h(a)), X2))
mark#(g(X1, mark(_x22)))active#(g(mark(X1), _x22))mark#(g(a, active(active(mark(mark(_x32))))))active#(g(active(a), _x32))
mark#(g(f(_x31, _x32), mark(_x42)))active#(g(f(mark(_x31), _x32), _x42))mark#(g(f(a, _x32), X2))active#(g(active(f(active(a), _x32)), X2))
mark#(g(f(h(_x81), mark(_x62)), mark(_x22)))active#(g(active(f(active(h(mark(_x81))), _x62)), _x22))mark#(h(X))mark#(X)
mark#(f(X1, X2))mark#(X1)mark#(g(f(b, _x32), active(_x22)))active#(g(active(f(active(b), _x32)), _x22))
mark#(g(f(h(_x71), _x32), mark(_x22)))active#(g(active(f(active(h(mark(_x71))), _x32)), _x22))mark#(g(a, active(mark(_x22))))active#(g(a, _x22))
mark#(g(f(b, mark(_x52)), _x22))active#(g(f(active(b), _x52), _x22))mark#(g(f(g(_x81, _x82), mark(_x62)), mark(_x22)))active#(g(active(f(active(g(mark(_x81), _x82)), _x62)), _x22))
mark#(g(f(b, _x52), X2))active#(g(active(f(b, _x52)), X2))mark#(h(X))active#(h(mark(X)))
mark#(g(f(a, mark(_x62)), mark(_x42)))active#(g(f(active(a), _x62), _x42))mark#(g(a, active(active(_x32))))active#(g(mark(b), _x32))
mark#(g(f(b, _x32), mark(_x22)))active#(g(active(f(active(b), _x32)), _x22))mark#(g(f(b, mark(_x62)), mark(_x22)))active#(g(active(f(active(b), _x62)), _x22))
mark#(g(f(h(_x61), _x32), mark(mark(_x42))))active#(g(active(f(active(h(mark(_x61))), _x32)), _x42))mark#(g(f(_x31, mark(_x62)), mark(mark(mark(_x22)))))active#(g(active(f(mark(_x31), _x62)), _x22))
mark#(g(f(b, _x32), mark(mark(mark(_x22)))))active#(g(active(f(active(b), _x32)), _x22))mark#(g(f(_x31, active(_x62)), mark(_x22)))active#(g(active(f(mark(_x31), _x62)), _x22))
mark#(g(f(_x51, _x52), X2))active#(g(active(f(_x51, _x52)), X2))mark#(g(f(g(_x61, _x62), _x32), mark(mark(_x22))))active#(g(f(active(g(mark(_x61), _x62)), _x32), _x22))
mark#(g(X1, active(_x22)))active#(g(mark(X1), _x22))mark#(g(a, active(active(_x22))))active#(g(a, _x22))
mark#(g(f(h(_x51), mark(_x62)), mark(_x42)))active#(g(f(active(h(mark(_x51))), _x62), _x42))mark#(g(g(_x31, _x32), X2))active#(g(active(g(mark(_x31), _x32)), X2))
mark#(g(f(_x71, mark(_x72)), mark(_x22)))active#(g(active(f(_x71, _x72)), _x22))mark#(g(f(a, _x32), mark(mark(_x42))))active#(g(active(f(active(a), _x32)), _x42))
mark#(g(f(b, _x32), mark(mark(_x22))))active#(g(f(active(b), _x32), _x22))mark#(g(f(f(_x71, _x72), _x32), mark(_x22)))active#(g(active(f(active(f(mark(_x71), _x72)), _x32)), _x22))
mark#(g(f(_x31, _x32), mark(mark(mark(mark(_x42))))))active#(g(active(f(mark(_x31), _x32)), _x42))mark#(g(a, active(active(mark(active(mark(_x22)))))))active#(g(active(a), _x22))
mark#(g(f(_x51, _x52), mark(mark(_x42))))active#(g(active(f(_x51, _x52)), _x42))mark#(g(f(g(_x71, _x72), _x32), mark(mark(mark(_x22)))))active#(g(active(f(active(g(mark(_x71), _x72)), _x32)), _x22))
mark#(g(f(b, _x32), _x22))active#(g(f(active(b), _x32), _x22))mark#(g(a, active(active(mark(_x32)))))active#(g(a, _x32))
mark#(g(f(f(_x61, _x62), _x32), X2))active#(g(active(f(active(f(mark(_x61), _x62)), _x32)), X2))active#(f(X, X))mark#(h(a))
mark#(g(b, _x22))active#(g(b, _x22))mark#(g(f(a, _x32), mark(mark(_x22))))active#(g(f(active(a), _x32), _x22))
mark#(g(f(_x31, _x32), mark(mark(mark(active(_x42))))))active#(g(active(f(mark(_x31), _x32)), _x42))mark#(g(f(_x31, active(_x52)), mark(mark(_x22))))active#(g(f(mark(_x31), _x52), _x22))
mark#(g(f(b, active(b)), X2))active#(g(mark(h(a)), X2))mark#(g(a, X2))active#(g(mark(b), X2))
mark#(g(f(b, mark(active(_x42))), X2))active#(g(active(f(active(b), _x42)), X2))mark#(g(f(g(_x61, _x62), _x32), mark(mark(_x42))))active#(g(active(f(active(g(mark(_x61), _x62)), _x32)), _x42))
mark#(g(a, active(active(mark(_x22)))))active#(g(mark(b), _x22))mark#(g(a, active(_x32)))active#(g(a, _x32))
mark#(g(a, active(mark(active(_x22)))))active#(g(active(a), _x22))mark#(g(f(_x31, mark(active(_x72))), mark(_x22)))active#(g(active(f(mark(_x31), _x72)), _x22))
mark#(g(f(b, _x32), mark(mark(_x42))))active#(g(active(f(active(b), _x32)), _x42))mark#(g(f(_x41, mark(_x42)), mark(_x42)))active#(g(f(_x41, _x42), _x42))
mark#(g(f(f(_x61, _x62), _x32), mark(mark(_x22))))active#(g(f(active(f(mark(_x61), _x62)), _x32), _x22))mark#(g(f(_x31, mark(_x62)), mark(mark(_x22))))active#(g(f(mark(_x31), _x62), _x22))
mark#(g(_x21, _x22))active#(g(_x21, _x22))mark#(g(a, active(mark(mark(_x22)))))active#(g(active(a), _x22))
mark#(g(f(_x31, mark(mark(_x42))), mark(_x42)))active#(g(f(mark(_x31), _x42), _x42))mark#(g(f(a, _x32), mark(mark(mark(_x22)))))active#(g(active(f(active(a), _x32)), _x22))
mark#(g(f(_x31, mark(mark(_x72))), mark(_x22)))active#(g(active(f(mark(_x31), _x72)), _x22))mark#(g(f(h(_x71), _x32), mark(mark(mark(_x22)))))active#(g(active(f(active(h(mark(_x71))), _x32)), _x22))
mark#(g(a, mark(_x22)))active#(g(active(a), _x22))mark#(g(f(_x31, active(_x62)), mark(mark(mark(_x22)))))active#(g(active(f(mark(_x31), _x62)), _x22))
mark#(g(f(_x31, mark(active(_x42))), mark(_x42)))active#(g(f(mark(_x31), _x42), _x42))mark#(g(f(_x31, _x32), mark(mark(mark(_x42)))))active#(g(f(mark(_x31), _x32), _x42))
mark#(g(a, active(active(active(_x22)))))active#(g(active(a), _x22))mark#(g(f(_x31, _x32), mark(mark(active(_x42)))))active#(g(f(mark(_x31), _x32), _x42))
mark#(g(f(b, mark(_x42)), X2))active#(g(active(f(b, _x42)), X2))mark#(g(f(_x31, mark(mark(_x31))), mark(_x22)))active#(g(mark(h(a)), _x22))
active#(a)mark#(b)mark#(g(f(_x31, mark(_x31)), mark(mark(mark(_x22)))))active#(g(mark(h(a)), _x22))
mark#(g(f(a, mark(_x62)), mark(_x22)))active#(g(active(f(active(a), _x62)), _x22))mark#(g(f(f(_x61, _x62), _x32), mark(mark(_x42))))active#(g(active(f(active(f(mark(_x61), _x62)), _x32)), _x42))
mark#(g(f(_x31, active(_x52)), mark(mark(_x42))))active#(g(active(f(mark(_x31), _x52)), _x42))mark#(g(f(b, mark(_x52)), mark(_x22)))active#(g(active(f(active(b), _x52)), _x22))
mark#(g(a, active(active(mark(active(_x32))))))active#(g(mark(b), _x32))mark#(g(f(_x61, _x62), mark(_x22)))active#(g(active(f(_x61, _x62)), _x22))
mark#(g(X1, X2))mark#(X1)mark#(g(f(f(_x71, _x72), _x32), mark(mark(mark(_x22)))))active#(g(active(f(active(f(mark(_x71), _x72)), _x32)), _x22))
mark#(g(f(_x31, mark(_x62)), mark(active(_x42))))active#(g(active(f(mark(_x31), _x62)), _x42))mark#(g(f(g(_x71, _x72), _x32), mark(_x22)))active#(g(active(f(active(g(mark(_x71), _x72)), _x32)), _x22))
mark#(g(f(h(_x61), _x32), X2))active#(g(active(f(active(h(mark(_x61))), _x32)), X2))mark#(g(b, active(_x22)))active#(g(active(b), _x22))
mark#(g(f(b, mark(mark(_x42))), X2))active#(g(active(f(active(b), _x42)), X2))mark#(f(X1, X2))active#(f(mark(X1), X2))
mark#(g(f(_x31, _x32), mark(mark(active(_x22)))))active#(g(active(f(mark(_x31), _x32)), _x22))mark#(g(f(g(_x61, _x62), _x32), X2))active#(g(active(f(active(g(mark(_x61), _x62)), _x32)), X2))
mark#(g(f(a, _x32), mark(_x22)))active#(g(active(f(active(a), _x32)), _x22))mark#(g(a, active(mark(_x32))))active#(g(mark(b), _x32))
mark#(g(f(h(_x61), _x32), mark(mark(_x22))))active#(g(f(active(h(mark(_x61))), _x32), _x22))mark#(g(f(_x61, _x62), mark(mark(mark(_x22)))))active#(g(active(f(_x61, _x62)), _x22))
mark#(g(f(_x31, _x32), active(_x22)))active#(g(active(f(mark(_x31), _x32)), _x22))mark#(g(f(_x31, mark(_x52)), X2))active#(g(active(f(mark(_x31), _x52)), X2))
mark#(g(f(f(_x51, _x52), mark(_x62)), mark(_x42)))active#(g(f(active(f(mark(_x51), _x52)), _x62), _x42))mark#(g(f(b, mark(_x52)), active(_x22)))active#(g(active(f(active(b), _x52)), _x22))
mark#(g(f(_x31, mark(_x62)), mark(mark(_x42))))active#(g(active(f(mark(_x31), _x62)), _x42))

Rewrite Rules

active(h(X))mark(g(X, X))active(g(a, X))mark(f(b, X))
active(f(X, X))mark(h(a))active(a)mark(b)
mark(h(X))active(h(mark(X)))mark(g(X1, X2))active(g(mark(X1), X2))
mark(a)active(a)mark(f(X1, X2))active(f(mark(X1), X2))
mark(b)active(b)h(mark(X))h(X)
h(active(X))h(X)g(mark(X1), X2)g(X1, X2)
g(X1, mark(X2))g(X1, X2)g(active(X1), X2)g(X1, X2)
g(X1, active(X2))g(X1, X2)f(mark(X1), X2)f(X1, X2)
f(X1, mark(X2))f(X1, X2)f(active(X1), X2)f(X1, X2)
f(X1, active(X2))f(X1, X2)

Original Signature

Termination of terms over the following signature is verified: f, g, b, a, active, mark, h

Strategy


The right-hand side of the rule mark#(g(f(_x31, mark(_x52)), mark(mark(_x22)))) → active#(g(f(mark(_x31), _x52), _x22)) 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
active#(g(f(active(h(mark(_x71))), _x52), _x22)) 
active#(g(f(active(g(mark(_x71), _x72)), _x52), _x22)) 
active#(g(f(mark(_x31), _x62), _x22)) 
active#(g(f(active(a), _x52), _x22)) 
active#(g(f(active(f(mark(_x71), _x72)), _x52), _x22)) 
active#(g(f(active(b), _x52), _x22)) 
active#(g(f(_x61, _x62), _x22)) 
active#(g(f(mark(_x31), _x52), _x42)) 
Thus, the rule mark#(g(f(_x31, mark(_x52)), mark(mark(_x22)))) → active#(g(f(mark(_x31), _x52), _x22)) is replaced by the following rules:
mark#(g(f(_x31, mark(_x52)), mark(mark(mark(_x42))))) → active#(g(f(mark(_x31), _x52), _x42))mark#(g(f(_x31, mark(active(_x62))), mark(mark(_x22)))) → active#(g(f(mark(_x31), _x62), _x22))
mark#(g(f(g(_x71, _x72), mark(_x52)), mark(mark(_x22)))) → active#(g(f(active(g(mark(_x71), _x72)), _x52), _x22))mark#(g(f(_x61, mark(_x62)), mark(mark(_x22)))) → active#(g(f(_x61, _x62), _x22))
mark#(g(f(f(_x71, _x72), mark(_x52)), mark(mark(_x22)))) → active#(g(f(active(f(mark(_x71), _x72)), _x52), _x22))mark#(g(f(h(_x71), mark(_x52)), mark(mark(_x22)))) → active#(g(f(active(h(mark(_x71))), _x52), _x22))
mark#(g(f(a, mark(_x52)), mark(mark(_x22)))) → active#(g(f(active(a), _x52), _x22))mark#(g(f(_x31, mark(_x52)), mark(mark(active(_x42))))) → active#(g(f(mark(_x31), _x52), _x42))
mark#(g(f(_x31, mark(mark(_x62))), mark(mark(_x22)))) → active#(g(f(mark(_x31), _x62), _x22))mark#(g(f(b, mark(_x52)), mark(mark(_x22)))) → active#(g(f(active(b), _x52), _x22))

Problem 28: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

mark#(g(a, active(active(mark(active(_x22))))))active#(g(a, _x22))mark#(g(a, _x22))active#(g(a, _x22))
mark#(g(f(_x31, mark(_x52)), mark(mark(mark(_x42)))))active#(g(f(mark(_x31), _x52), _x42))mark#(g(f(g(_x71, _x72), mark(_x52)), mark(mark(_x22))))active#(g(f(active(g(mark(_x71), _x72)), _x52), _x22))
mark#(a)active#(a)mark#(g(f(_x31, mark(_x31)), mark(mark(_x42))))active#(g(mark(h(a)), _x42))
mark#(g(f(_x31, _x32), mark(active(_x42))))active#(g(active(f(mark(_x31), _x32)), _x42))mark#(g(b, mark(_x22)))active#(g(active(b), _x22))
mark#(g(f(_x31, active(_x52)), X2))active#(g(active(f(mark(_x31), _x52)), X2))mark#(g(f(f(_x71, _x72), mark(_x52)), mark(mark(_x22))))active#(g(f(active(f(mark(_x71), _x72)), _x52), _x22))
mark#(g(h(_x31), X2))active#(g(active(h(mark(_x31))), X2))active#(h(X))mark#(g(X, X))
mark#(g(f(f(_x81, _x82), mark(_x62)), mark(_x22)))active#(g(active(f(active(f(mark(_x81), _x82)), _x62)), _x22))mark#(g(f(_x31, mark(_x62)), mark(active(_x22))))active#(g(f(mark(_x31), _x62), _x22))
mark#(g(f(b, mark(_x62)), mark(_x42)))active#(g(f(active(b), _x62), _x42))mark#(g(f(g(_x51, _x52), mark(_x62)), mark(_x42)))active#(g(f(active(g(mark(_x51), _x52)), _x62), _x42))
mark#(g(a, active(_x22)))active#(g(mark(b), _x22))mark#(g(a, active(active(mark(active(active(_x22)))))))active#(g(active(a), _x22))
mark#(g(f(_x31, _x32), _x22))active#(g(f(mark(_x31), _x32), _x22))mark#(g(f(_x31, mark(_x52)), mark(mark(_x42))))active#(g(active(f(mark(_x31), _x52)), _x42))
mark#(g(f(b, active(_x52)), X2))active#(g(active(f(active(b), _x52)), X2))mark#(g(f(b, mark(_x52)), mark(mark(_x22))))active#(g(f(active(b), _x52), _x22))
active#(g(a, X))mark#(f(b, X))mark#(g(f(_x51, _x52), mark(mark(_x22))))active#(g(f(_x51, _x52), _x22))
mark#(g(f(b, mark(active(b))), X2))active#(g(mark(h(a)), X2))mark#(g(f(_x31, mark(_x31)), mark(_x22)))active#(g(mark(h(a)), _x22))
mark#(g(f(_x31, mark(_x31)), X2))active#(g(mark(h(a)), X2))mark#(g(a, active(active(mark(mark(_x32))))))active#(g(active(a), _x32))
mark#(g(X1, mark(_x22)))active#(g(mark(X1), _x22))mark#(g(f(_x31, _x32), mark(_x42)))active#(g(f(mark(_x31), _x32), _x42))
mark#(g(f(h(_x81), mark(_x62)), mark(_x22)))active#(g(active(f(active(h(mark(_x81))), _x62)), _x22))mark#(g(f(a, _x32), X2))active#(g(active(f(active(a), _x32)), X2))
mark#(h(X))mark#(X)mark#(f(X1, X2))mark#(X1)
mark#(g(f(b, _x32), active(_x22)))active#(g(active(f(active(b), _x32)), _x22))mark#(g(f(h(_x71), _x32), mark(_x22)))active#(g(active(f(active(h(mark(_x71))), _x32)), _x22))
mark#(g(f(_x61, mark(_x62)), mark(mark(_x22))))active#(g(f(_x61, _x62), _x22))mark#(g(a, active(mark(_x22))))active#(g(a, _x22))
mark#(g(f(b, mark(_x52)), _x22))active#(g(f(active(b), _x52), _x22))mark#(g(f(g(_x81, _x82), mark(_x62)), mark(_x22)))active#(g(active(f(active(g(mark(_x81), _x82)), _x62)), _x22))
mark#(g(f(b, _x52), X2))active#(g(active(f(b, _x52)), X2))mark#(g(f(a, mark(_x62)), mark(_x42)))active#(g(f(active(a), _x62), _x42))
mark#(h(X))active#(h(mark(X)))mark#(g(a, active(active(_x32))))active#(g(mark(b), _x32))
mark#(g(f(b, _x32), mark(_x22)))active#(g(active(f(active(b), _x32)), _x22))mark#(g(f(b, mark(_x62)), mark(_x22)))active#(g(active(f(active(b), _x62)), _x22))
mark#(g(f(h(_x61), _x32), mark(mark(_x42))))active#(g(active(f(active(h(mark(_x61))), _x32)), _x42))mark#(g(f(_x31, mark(_x62)), mark(mark(mark(_x22)))))active#(g(active(f(mark(_x31), _x62)), _x22))
mark#(g(f(b, _x32), mark(mark(mark(_x22)))))active#(g(active(f(active(b), _x32)), _x22))mark#(g(f(_x31, active(_x62)), mark(_x22)))active#(g(active(f(mark(_x31), _x62)), _x22))
mark#(g(f(_x51, _x52), X2))active#(g(active(f(_x51, _x52)), X2))mark#(g(f(g(_x61, _x62), _x32), mark(mark(_x22))))active#(g(f(active(g(mark(_x61), _x62)), _x32), _x22))
mark#(g(f(_x31, mark(mark(_x62))), mark(mark(_x22))))active#(g(f(mark(_x31), _x62), _x22))mark#(g(X1, active(_x22)))active#(g(mark(X1), _x22))
mark#(g(a, active(active(_x22))))active#(g(a, _x22))mark#(g(f(h(_x51), mark(_x62)), mark(_x42)))active#(g(f(active(h(mark(_x51))), _x62), _x42))
mark#(g(g(_x31, _x32), X2))active#(g(active(g(mark(_x31), _x32)), X2))mark#(g(f(_x71, mark(_x72)), mark(_x22)))active#(g(active(f(_x71, _x72)), _x22))
mark#(g(f(a, _x32), mark(mark(_x42))))active#(g(active(f(active(a), _x32)), _x42))mark#(g(f(h(_x71), mark(_x52)), mark(mark(_x22))))active#(g(f(active(h(mark(_x71))), _x52), _x22))
mark#(g(f(b, _x32), mark(mark(_x22))))active#(g(f(active(b), _x32), _x22))mark#(g(f(f(_x71, _x72), _x32), mark(_x22)))active#(g(active(f(active(f(mark(_x71), _x72)), _x32)), _x22))
mark#(g(f(_x31, _x32), mark(mark(mark(mark(_x42))))))active#(g(active(f(mark(_x31), _x32)), _x42))mark#(g(a, active(active(mark(active(mark(_x22)))))))active#(g(active(a), _x22))
mark#(g(f(_x51, _x52), mark(mark(_x42))))active#(g(active(f(_x51, _x52)), _x42))mark#(g(f(g(_x71, _x72), _x32), mark(mark(mark(_x22)))))active#(g(active(f(active(g(mark(_x71), _x72)), _x32)), _x22))
mark#(g(f(b, _x32), _x22))active#(g(f(active(b), _x32), _x22))mark#(g(a, active(active(mark(_x32)))))active#(g(a, _x32))
mark#(g(f(_x31, mark(_x52)), mark(mark(active(_x42)))))active#(g(f(mark(_x31), _x52), _x42))mark#(g(f(f(_x61, _x62), _x32), X2))active#(g(active(f(active(f(mark(_x61), _x62)), _x32)), X2))
active#(f(X, X))mark#(h(a))mark#(g(b, _x22))active#(g(b, _x22))
mark#(g(f(a, _x32), mark(mark(_x22))))active#(g(f(active(a), _x32), _x22))mark#(g(f(_x31, _x32), mark(mark(mark(active(_x42))))))active#(g(active(f(mark(_x31), _x32)), _x42))
mark#(g(f(_x31, active(_x52)), mark(mark(_x22))))active#(g(f(mark(_x31), _x52), _x22))mark#(g(f(b, active(b)), X2))active#(g(mark(h(a)), X2))
mark#(g(a, X2))active#(g(mark(b), X2))mark#(g(f(b, mark(active(_x42))), X2))active#(g(active(f(active(b), _x42)), X2))
mark#(g(f(g(_x61, _x62), _x32), mark(mark(_x42))))active#(g(active(f(active(g(mark(_x61), _x62)), _x32)), _x42))mark#(g(a, active(active(mark(_x22)))))active#(g(mark(b), _x22))
mark#(g(a, active(_x32)))active#(g(a, _x32))mark#(g(a, active(mark(active(_x22)))))active#(g(active(a), _x22))
mark#(g(f(_x31, mark(active(_x72))), mark(_x22)))active#(g(active(f(mark(_x31), _x72)), _x22))mark#(g(f(b, _x32), mark(mark(_x42))))active#(g(active(f(active(b), _x32)), _x42))
mark#(g(f(_x41, mark(_x42)), mark(_x42)))active#(g(f(_x41, _x42), _x42))mark#(g(f(f(_x61, _x62), _x32), mark(mark(_x22))))active#(g(f(active(f(mark(_x61), _x62)), _x32), _x22))
mark#(g(f(_x31, mark(_x62)), mark(mark(_x22))))active#(g(f(mark(_x31), _x62), _x22))mark#(g(_x21, _x22))active#(g(_x21, _x22))
mark#(g(a, active(mark(mark(_x22)))))active#(g(active(a), _x22))mark#(g(f(_x31, mark(mark(_x42))), mark(_x42)))active#(g(f(mark(_x31), _x42), _x42))
mark#(g(f(a, _x32), mark(mark(mark(_x22)))))active#(g(active(f(active(a), _x32)), _x22))mark#(g(f(_x31, mark(mark(_x72))), mark(_x22)))active#(g(active(f(mark(_x31), _x72)), _x22))
mark#(g(f(h(_x71), _x32), mark(mark(mark(_x22)))))active#(g(active(f(active(h(mark(_x71))), _x32)), _x22))mark#(g(a, mark(_x22)))active#(g(active(a), _x22))
mark#(g(f(_x31, active(_x62)), mark(mark(mark(_x22)))))active#(g(active(f(mark(_x31), _x62)), _x22))mark#(g(f(_x31, mark(active(_x42))), mark(_x42)))active#(g(f(mark(_x31), _x42), _x42))
mark#(g(f(a, mark(_x52)), mark(mark(_x22))))active#(g(f(active(a), _x52), _x22))mark#(g(f(_x31, _x32), mark(mark(mark(_x42)))))active#(g(f(mark(_x31), _x32), _x42))
mark#(g(a, active(active(active(_x22)))))active#(g(active(a), _x22))mark#(g(f(_x31, _x32), mark(mark(active(_x42)))))active#(g(f(mark(_x31), _x32), _x42))
mark#(g(f(b, mark(_x42)), X2))active#(g(active(f(b, _x42)), X2))mark#(g(f(_x31, mark(mark(_x31))), mark(_x22)))active#(g(mark(h(a)), _x22))
active#(a)mark#(b)mark#(g(f(_x31, mark(_x31)), mark(mark(mark(_x22)))))active#(g(mark(h(a)), _x22))
mark#(g(f(b, mark(_x52)), mark(_x22)))active#(g(active(f(active(b), _x52)), _x22))mark#(g(f(_x31, active(_x52)), mark(mark(_x42))))active#(g(active(f(mark(_x31), _x52)), _x42))
mark#(g(f(f(_x61, _x62), _x32), mark(mark(_x42))))active#(g(active(f(active(f(mark(_x61), _x62)), _x32)), _x42))mark#(g(f(a, mark(_x62)), mark(_x22)))active#(g(active(f(active(a), _x62)), _x22))
mark#(g(a, active(active(mark(active(_x32))))))active#(g(mark(b), _x32))mark#(g(f(_x31, mark(active(_x62))), mark(mark(_x22))))active#(g(f(mark(_x31), _x62), _x22))
mark#(g(f(_x61, _x62), mark(_x22)))active#(g(active(f(_x61, _x62)), _x22))mark#(g(f(f(_x71, _x72), _x32), mark(mark(mark(_x22)))))active#(g(active(f(active(f(mark(_x71), _x72)), _x32)), _x22))
mark#(g(X1, X2))mark#(X1)mark#(g(f(_x31, mark(_x62)), mark(active(_x42))))active#(g(active(f(mark(_x31), _x62)), _x42))
mark#(g(f(g(_x71, _x72), _x32), mark(_x22)))active#(g(active(f(active(g(mark(_x71), _x72)), _x32)), _x22))mark#(g(b, active(_x22)))active#(g(active(b), _x22))
mark#(g(f(h(_x61), _x32), X2))active#(g(active(f(active(h(mark(_x61))), _x32)), X2))mark#(f(X1, X2))active#(f(mark(X1), X2))
mark#(g(f(b, mark(mark(_x42))), X2))active#(g(active(f(active(b), _x42)), X2))mark#(g(f(_x31, _x32), mark(mark(active(_x22)))))active#(g(active(f(mark(_x31), _x32)), _x22))
mark#(g(f(g(_x61, _x62), _x32), X2))active#(g(active(f(active(g(mark(_x61), _x62)), _x32)), X2))mark#(g(a, active(mark(_x32))))active#(g(mark(b), _x32))
mark#(g(f(a, _x32), mark(_x22)))active#(g(active(f(active(a), _x32)), _x22))mark#(g(f(_x61, _x62), mark(mark(mark(_x22)))))active#(g(active(f(_x61, _x62)), _x22))
mark#(g(f(h(_x61), _x32), mark(mark(_x22))))active#(g(f(active(h(mark(_x61))), _x32), _x22))mark#(g(f(_x31, mark(_x52)), X2))active#(g(active(f(mark(_x31), _x52)), X2))
mark#(g(f(_x31, _x32), active(_x22)))active#(g(active(f(mark(_x31), _x32)), _x22))mark#(g(f(f(_x51, _x52), mark(_x62)), mark(_x42)))active#(g(f(active(f(mark(_x51), _x52)), _x62), _x42))
mark#(g(f(_x31, mark(_x62)), mark(mark(_x42))))active#(g(active(f(mark(_x31), _x62)), _x42))mark#(g(f(b, mark(_x52)), active(_x22)))active#(g(active(f(active(b), _x52)), _x22))

Rewrite Rules

active(h(X))mark(g(X, X))active(g(a, X))mark(f(b, X))
active(f(X, X))mark(h(a))active(a)mark(b)
mark(h(X))active(h(mark(X)))mark(g(X1, X2))active(g(mark(X1), X2))
mark(a)active(a)mark(f(X1, X2))active(f(mark(X1), X2))
mark(b)active(b)h(mark(X))h(X)
h(active(X))h(X)g(mark(X1), X2)g(X1, X2)
g(X1, mark(X2))g(X1, X2)g(active(X1), X2)g(X1, X2)
g(X1, active(X2))g(X1, X2)f(mark(X1), X2)f(X1, X2)
f(X1, mark(X2))f(X1, X2)f(active(X1), X2)f(X1, X2)
f(X1, active(X2))f(X1, X2)

Original Signature

Termination of terms over the following signature is verified: f, g, b, a, active, mark, h

Strategy


The right-hand side of the rule mark#(g(f(_x31, mark(_x52)), mark(mark(mark(_x42))))) → active#(g(f(mark(_x31), _x52), _x42)) 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
active#(g(f(mark(_x31), _x42), _x42)) 
active#(g(f(active(h(mark(_x51))), _x52), _x42)) 
active#(g(f(active(f(mark(_x51), _x52)), _x52), _x42)) 
active#(g(f(active(b), _x52), _x42)) 
active#(g(f(mark(_x31), _x52), _x22)) 
active#(g(f(active(a), _x52), _x42)) 
active#(g(f(active(g(mark(_x51), _x52)), _x52), _x42)) 
active#(g(f(_x41, _x42), _x42)) 
Thus, the rule mark#(g(f(_x31, mark(_x52)), mark(mark(mark(_x42))))) → active#(g(f(mark(_x31), _x52), _x42)) is replaced by the following rules:
mark#(g(f(f(_x51, _x52), mark(_x52)), mark(mark(mark(_x42))))) → active#(g(f(active(f(mark(_x51), _x52)), _x52), _x42))mark#(g(f(_x31, mark(_x52)), mark(mark(mark(mark(_x22)))))) → active#(g(f(mark(_x31), _x52), _x22))
mark#(g(f(a, mark(_x52)), mark(mark(mark(_x42))))) → active#(g(f(active(a), _x52), _x42))mark#(g(f(_x31, mark(_x52)), mark(mark(mark(active(_x22)))))) → active#(g(f(mark(_x31), _x52), _x22))
mark#(g(f(b, mark(_x52)), mark(mark(mark(_x42))))) → active#(g(f(active(b), _x52), _x42))mark#(g(f(g(_x51, _x52), mark(_x52)), mark(mark(mark(_x42))))) → active#(g(f(active(g(mark(_x51), _x52)), _x52), _x42))
mark#(g(f(_x31, mark(active(_x42))), mark(mark(mark(_x42))))) → active#(g(f(mark(_x31), _x42), _x42))mark#(g(f(_x41, mark(_x42)), mark(mark(mark(_x42))))) → active#(g(f(_x41, _x42), _x42))
mark#(g(f(_x31, mark(mark(_x42))), mark(mark(mark(_x42))))) → active#(g(f(mark(_x31), _x42), _x42))mark#(g(f(h(_x51), mark(_x52)), mark(mark(mark(_x42))))) → active#(g(f(active(h(mark(_x51))), _x52), _x42))

Problem 29: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

mark#(g(a, active(active(mark(active(_x22))))))active#(g(a, _x22))mark#(g(f(_x31, mark(mark(_x42))), mark(mark(mark(_x42)))))active#(g(f(mark(_x31), _x42), _x42))
mark#(g(f(b, active(_x52)), _x22))active#(g(f(active(b), _x52), _x22))mark#(g(f(g(a, _x72), _x32), mark(active(active(_x22)))))active#(g(active(f(active(g(active(a), _x72)), _x32)), _x22))
mark#(g(f(h(_x71), _x32), mark(active(mark(active(_x42))))))active#(g(active(f(active(h(mark(_x71))), _x32)), _x42))mark#(g(f(_x51, _x52), mark(mark(_x22))))active#(g(f(_x51, _x52), _x22))
mark#(g(f(h(f(_x111, _x112)), _x32), mark(active(mark(_x22)))))active#(g(active(f(active(h(active(f(_x111, _x112)))), _x32)), _x22))mark#(g(f(g(h(g(_x131, _x132)), _x72), _x32), mark(active(active(mark(_x42))))))active#(g(active(f(active(g(active(h(active(g(mark(_x131), _x132)))), _x72)), _x32)), _x42))
mark#(g(f(f(_x71, _x72), active(_x52)), X2))active#(g(active(f(active(f(mark(_x71), _x72)), _x52)), X2))mark#(g(f(b, _x32), mark(active(mark(_x22)))))active#(g(active(f(active(b), _x32)), _x22))
mark#(g(f(_x31, mark(_x31)), X2))active#(g(mark(h(a)), X2))mark#(g(f(h(_x71), active(h(mark(_x71)))), mark(active(mark(_x22)))))active#(g(mark(h(a)), _x22))
mark#(g(f(g(g(_x101, _x102), _x82), mark(_x52)), mark(mark(mark(_x22)))))active#(g(active(f(active(g(active(g(mark(_x101), _x102)), _x82)), _x52)), _x22))mark#(g(f(f(g(a, _x92), _x82), mark(_x62)), mark(_x22)))active#(g(active(f(active(f(active(g(active(a), _x92)), _x82)), _x62)), _x22))
mark#(g(f(_x51, _x52), mark(active(mark(_x22)))))active#(g(active(f(_x51, _x52)), _x22))mark#(g(h(h(_x61)), X2))active#(g(active(h(active(h(mark(_x61))))), X2))
mark#(g(h(_x31), X2))active#(g(mark(g(mark(_x31), mark(_x31))), X2))mark#(g(f(h(f(g(_x121, _x122), _x102)), _x32), mark(active(mark(_x22)))))active#(g(active(f(active(h(active(f(active(g(mark(_x121), _x122)), _x102)))), _x32)), _x22))
mark#(g(h(f(_x61, mark(_x61))), X2))active#(g(mark(g(mark(h(a)), mark(h(a)))), X2))mark#(g(f(_x31, mark(mark(_x62))), mark(mark(mark(active(_x42))))))active#(g(active(f(mark(_x31), _x62)), _x42))
mark#(g(f(h(_x71), _x32), mark(_x22)))active#(g(active(f(active(h(mark(_x71))), _x32)), _x22))mark#(g(f(f(g(_x91, _x92), mark(_x72)), mark(_x62)), mark(_x22)))active#(g(active(f(active(f(active(g(mark(_x91), _x92)), _x72)), _x62)), _x22))
mark#(g(f(mark(_x41), _x42), mark(active(_x42))))active#(g(active(f(_x41, _x42)), _x42))mark#(g(h(f(_x61, mark(_x72))), X2))active#(g(active(h(active(f(mark(_x61), _x72)))), X2))
mark#(g(f(g(_x71, _x72), mark(mark(_x42))), mark(mark(_x22))))active#(g(f(active(g(mark(_x71), _x72)), _x42), _x22))mark#(g(f(b, _x32), mark(_x22)))active#(g(active(f(active(b), _x32)), _x22))
mark#(g(f(b, mark(_x62)), mark(_x22)))active#(g(active(f(active(b), _x62)), _x22))mark#(g(f(g(h(g(_x131, _x132)), _x72), _x32), mark(active(active(active(_x42))))))active#(g(active(f(active(g(active(h(active(g(mark(_x131), _x132)))), _x72)), _x32)), _x42))
mark#(g(f(h(_x61), _x32), mark(mark(_x42))))active#(g(active(f(active(h(mark(_x61))), _x32)), _x42))mark#(g(f(g(_x71, mark(_x62)), _x62), mark(active(active(_x22)))))active#(g(active(f(g(mark(_x71), _x62), _x62)), _x22))
mark#(g(f(h(_x91), active(_x62)), mark(active(mark(_x22)))))active#(g(active(f(active(h(_x91)), _x62)), _x22))mark#(g(f(_x31, mark(_x62)), mark(mark(mark(_x22)))))active#(g(active(f(mark(_x31), _x62)), _x22))
mark#(g(f(_x31, mark(mark(_x31))), mark(mark(mark(_x22)))))active#(g(mark(h(a)), _x22))mark#(g(f(f(_x71, _x72), mark(_x62)), mark(_x22)))active#(g(active(f(active(f(_x71, _x72)), _x62)), _x22))
mark#(g(f(a, mark(_x52)), mark(mark(mark(_x42)))))active#(g(f(active(a), _x52), _x42))mark#(g(f(g(h(_x101), mark(_x92)), _x32), mark(active(active(_x22)))))active#(g(active(f(active(g(active(h(mark(_x101))), _x92)), _x32)), _x22))
mark#(g(f(g(_x71, active(_x92)), _x32), mark(active(active(_x22)))))active#(g(active(f(active(g(mark(_x71), _x92)), _x32)), _x22))mark#(g(f(g(h(f(_x131, _x132)), _x72), _x32), mark(active(active(_x22)))))active#(g(active(f(active(g(active(h(active(f(mark(_x131), _x132)))), _x72)), _x32)), _x22))
mark#(g(f(g(h(g(_x131, _x132)), _x72), _x32), mark(active(active(_x42)))))active#(g(f(active(g(active(h(active(g(mark(_x131), _x132)))), _x72)), _x32), _x42))mark#(g(f(_x31, _x32), mark(active(mark(mark(_x42))))))active#(g(active(f(mark(_x31), _x32)), _x42))
mark#(g(X1, active(_x22)))active#(g(mark(X1), _x22))mark#(g(f(g(_x71, _x72), mark(active(_x42))), mark(mark(_x22))))active#(g(f(active(g(mark(_x71), _x72)), _x42), _x22))
mark#(g(f(f(f(_x91, _x92), _x82), mark(_x62)), mark(_x22)))active#(g(active(f(active(f(active(f(mark(_x91), _x92)), _x82)), _x62)), _x22))mark#(g(f(_x31, active(_x52)), mark(_x22)))active#(g(active(f(mark(_x31), _x52)), _x22))
mark#(g(f(g(_x91, _x92), mark(_x52)), mark(mark(mark(_x22)))))active#(g(active(f(active(g(_x91, _x92)), _x52)), _x22))mark#(g(f(h(_x71), mark(_x52)), mark(mark(_x22))))active#(g(f(active(h(mark(_x71))), _x52), _x22))
mark#(g(f(_x31, _x32), mark(mark(mark(mark(_x42))))))active#(g(active(f(mark(_x31), _x32)), _x42))mark#(g(h(_x31), _x22))active#(g(h(mark(_x31)), _x22))
mark#(g(f(b, _x32), mark(active(_x42))))active#(g(active(f(active(b), _x32)), _x42))mark#(g(f(g(h(_x81), _x72), _x62), mark(active(active(_x22)))))active#(g(active(f(g(active(h(mark(_x81))), _x72), _x62)), _x22))
mark#(g(f(g(b, _x82), mark(_x52)), mark(mark(mark(_x22)))))active#(g(active(f(active(g(active(b), _x82)), _x52)), _x22))mark#(g(f(b, mark(_x52)), mark(mark(mark(_x22)))))active#(g(active(f(active(b), _x52)), _x22))
mark#(g(f(_x31, mark(_x62)), mark(active(mark(_x22)))))active#(g(active(f(mark(_x31), _x62)), _x22))mark#(g(f(f(g(_x101, _x102), _x82), mark(_x62)), mark(_x22)))active#(g(active(f(active(f(active(g(_x101, _x102)), _x82)), _x62)), _x22))
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active#(g(a, X))mark#(f(b, X))mark#(g(f(b, active(active(_x42))), X2))active#(g(active(f(active(b), _x42)), X2))
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mark#(g(h(_x31), mark(_x22)))active#(g(active(h(mark(_x31))), _x22))mark#(g(f(g(h(g(a, _x132)), _x72), _x62), mark(active(active(_x22)))))active#(g(active(f(g(active(h(active(g(active(a), _x132)))), _x72), _x62)), _x22))
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mark#(g(a, active(mark(_x22))))active#(g(a, _x22))mark#(g(f(b, mark(_x52)), _x22))active#(g(f(active(b), _x52), _x22))
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mark#(g(f(g(_x51, _x52), mark(_x52)), mark(mark(mark(_x42)))))active#(g(f(active(g(mark(_x51), _x52)), _x52), _x42))mark#(g(f(_x41, mark(_x42)), mark(mark(mark(_x42)))))active#(g(f(_x41, _x42), _x42))
mark#(g(f(f(g(h(_x111), _x92), _x82), mark(_x62)), mark(_x22)))active#(g(active(f(active(f(active(g(active(h(mark(_x111))), _x92)), _x82)), _x62)), _x22))mark#(g(f(b, active(active(b))), X2))active#(g(mark(h(a)), X2))
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mark#(g(f(g(f(_x81, _x82), _x72), mark(_x52)), mark(mark(_x22))))active#(g(f(active(g(active(f(mark(_x81), _x82)), _x72)), _x52), _x22))mark#(g(f(g(_x91, _x92), _x32), mark(active(active(active(_x42))))))active#(g(active(f(active(g(_x91, _x92)), _x32)), _x42))
mark#(g(f(h(_x71), _x32), mark(active(mark(_x22)))))active#(g(active(f(mark(g(mark(_x71), mark(_x71))), _x32)), _x22))mark#(g(f(g(_x71, _x72), mark(_x62)), mark(active(active(_x22)))))active#(g(active(f(active(g(mark(_x71), _x72)), _x62)), _x22))
mark#(g(f(_x31, mark(_x52)), mark(mark(mark(active(active(_x22)))))))active#(g(active(f(mark(_x31), _x52)), _x22))mark#(g(f(_x31, _x32), mark(mark(mark(active(_x42))))))active#(g(active(f(mark(_x31), _x32)), _x42))
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mark#(g(f(_x31, active(mark(_x62))), X2))active#(g(active(f(mark(_x31), _x62)), X2))mark#(g(_x21, _x22))active#(g(_x21, _x22))
mark#(g(f(_x31, mark(_x31)), mark(active(mark(_x22)))))active#(g(mark(h(a)), _x22))mark#(g(f(g(b, _x72), _x32), mark(active(active(_x22)))))active#(g(active(f(active(g(active(b), _x72)), _x32)), _x22))
mark#(g(f(f(_x81, _x82), mark(_x52)), mark(_x22)))active#(g(active(f(f(mark(_x81), _x82), _x52)), _x22))mark#(g(f(b, mark(_x32)), mark(_x42)))active#(g(f(b, _x32), _x42))
mark#(g(a, active(mark(mark(_x22)))))active#(g(active(a), _x22))mark#(g(f(_x31, mark(mark(_x31))), mark(mark(mark(active(_x42))))))active#(g(mark(h(a)), _x42))
mark#(g(f(h(_x71), active(_x52)), X2))active#(g(active(f(active(h(mark(_x71))), _x52)), X2))mark#(g(f(g(_x91, _x92), _x32), mark(active(active(mark(_x42))))))active#(g(active(f(active(g(_x91, _x92)), _x32)), _x42))
mark#(g(f(h(_x81), mark(_x52)), mark(mark(mark(_x22)))))active#(g(active(f(active(h(mark(_x81))), _x52)), _x22))mark#(g(f(g(h(_x101), _x72), _x32), mark(active(active(_x22)))))active#(g(active(f(active(g(mark(g(mark(_x101), mark(_x101))), _x72)), _x32)), _x22))
mark#(g(f(_x31, _x32), mark(active(_x22))))active#(g(f(mark(_x31), _x32), _x22))mark#(g(b, mark(_x32)))active#(g(b, _x32))
mark#(g(f(g(a, _x82), mark(_x52)), mark(mark(mark(_x22)))))active#(g(active(f(active(g(active(a), _x82)), _x52)), _x22))mark#(g(f(g(h(g(_x131, _x132)), _x72), _x62), mark(active(active(_x22)))))active#(g(active(f(g(active(h(active(g(mark(_x131), _x132)))), _x72), _x62)), _x22))
mark#(g(f(b, mark(_x42)), X2))active#(g(active(f(b, _x42)), X2))mark#(g(f(b, mark(_x52)), mark(mark(_x42))))active#(g(active(f(active(b), _x52)), _x42))
mark#(g(f(_x31, mark(mark(_x31))), mark(_x22)))active#(g(mark(h(a)), _x22))mark#(g(f(a, active(_x52)), X2))active#(g(active(f(active(a), _x52)), X2))
mark#(g(f(a, _x32), mark(active(_x42))))active#(g(active(f(active(a), _x32)), _x42))mark#(g(f(f(_x81, _x82), mark(_x62)), mark(mark(_x32))))active#(g(active(f(active(f(mark(_x81), _x82)), _x62)), _x32))
mark#(g(a, active(active(mark(active(_x32))))))active#(g(mark(b), _x32))mark#(g(f(_x31, mark(active(_x62))), mark(mark(_x22))))active#(g(f(mark(_x31), _x62), _x22))
mark#(g(f(g(h(g(a, active(_x142))), _x72), _x32), mark(active(active(_x22)))))active#(g(active(f(active(g(active(h(active(g(active(a), _x142)))), _x72)), _x32)), _x22))mark#(g(f(g(_x71, _x72), _x32), mark(active(active(active(_x42))))))active#(g(active(f(active(g(mark(_x71), _x72)), _x32)), _x42))
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mark#(g(f(g(h(g(a, _x132)), _x72), _x32), mark(active(active(_x22)))))active#(g(active(f(active(g(active(h(g(active(a), _x132))), _x72)), _x32)), _x22))mark#(g(h(f(_x61, mark(_x61))), X2))active#(g(active(h(active(h(mark(a))))), X2))
mark#(g(f(f(_x81, mark(_x81)), mark(_x62)), mark(_x22)))active#(g(active(f(mark(h(a)), _x62)), _x22))mark#(g(f(b, mark(mark(_x42))), X2))active#(g(active(f(active(b), _x42)), X2))
mark#(g(f(_x61, _x62), mark(active(active(_x22)))))active#(g(active(f(_x61, _x62)), _x22))mark#(g(f(h(_x61), _x32), mark(mark(_x22))))active#(g(f(active(h(mark(_x61))), _x32), _x22))
mark#(g(f(_x61, _x62), mark(mark(mark(_x22)))))active#(g(active(f(_x61, _x62)), _x22))mark#(g(f(_x31, _x32), active(_x22)))active#(g(active(f(mark(_x31), _x32)), _x22))
mark#(g(f(b, mark(_x52)), active(_x22)))active#(g(active(f(active(b), _x52)), _x22))mark#(g(f(h(f(_x101, _x102)), active(_x62)), mark(active(mark(_x22)))))active#(g(active(f(active(h(active(f(mark(_x101), _x102)))), _x62)), _x22))
mark#(g(f(g(_x71, active(_x62)), _x62), mark(active(active(_x22)))))active#(g(active(f(g(mark(_x71), _x62), _x62)), _x22))mark#(g(f(h(_x91), active(h(_x91))), mark(active(mark(_x22)))))active#(g(mark(h(a)), _x22))
mark#(g(f(g(b, _x72), _x62), mark(active(active(_x22)))))active#(g(active(f(g(active(b), _x72), _x62)), _x22))mark#(g(f(h(f(_x101, _x102)), mark(_x62)), mark(active(mark(_x22)))))active#(g(active(f(active(h(active(f(mark(_x101), _x102)))), _x62)), _x22))
mark#(g(f(_x31, mark(_x52)), mark(mark(_x22))))active#(g(f(mark(_x31), _x52), _x22))mark#(g(f(_x61, mark(_x62)), mark(mark(_x42))))active#(g(active(f(_x61, _x62)), _x42))
mark#(g(f(g(_x71, _x72), _x32), mark(active(active(mark(_x42))))))active#(g(active(f(active(g(mark(_x71), _x72)), _x32)), _x42))mark#(g(f(_x61, _x62), mark(active(mark(_x22)))))active#(g(active(f(_x61, _x62)), _x22))
mark#(g(f(f(_x81, _x82), mark(active(f(mark(_x81), _x82)))), mark(_x22)))active#(g(mark(h(a)), _x22))mark#(g(f(g(h(g(_x131, _x132)), _x72), active(_x62)), mark(active(active(_x22)))))active#(g(active(f(active(g(active(h(active(g(mark(_x131), _x132)))), _x72)), _x62)), _x22))
mark#(g(f(g(h(b), _x72), _x32), mark(active(active(_x22)))))active#(g(active(f(active(g(active(h(active(b))), _x72)), _x32)), _x22))mark#(g(f(_x31, mark(_x31)), mark(active(active(_x22)))))active#(g(mark(h(a)), _x22))
mark#(g(f(g(h(g(a, _x132)), _x92), _x32), mark(active(active(_x22)))))active#(g(active(f(active(g(h(active(g(active(a), _x132))), _x92)), _x32)), _x22))mark#(g(a, active(active(mark(active(active(_x22)))))))active#(g(active(a), _x22))
mark#(g(f(h(f(_x101, _x102)), _x62), mark(active(mark(_x22)))))active#(g(active(f(h(active(f(mark(_x101), _x102))), _x62)), _x22))mark#(g(f(b, mark(_x52)), mark(mark(_x22))))active#(g(f(active(b), _x52), _x22))
mark#(g(f(b, active(mark(_x42))), X2))active#(g(active(f(active(b), _x42)), X2))mark#(g(h(f(_x61, mark(_x61))), active(_x22)))active#(g(active(h(mark(h(a)))), _x22))
mark#(g(a, active(active(mark(mark(_x32))))))active#(g(active(a), _x32))mark#(h(X))mark#(X)
mark#(g(f(g(h(_x101), active(_x92)), _x32), mark(active(active(_x22)))))active#(g(active(f(active(g(active(h(mark(_x101))), _x92)), _x32)), _x22))mark#(g(f(b, _x32), active(_x22)))active#(g(active(f(active(b), _x32)), _x22))
mark#(g(h(a), X2))active#(g(active(h(active(a))), X2))mark#(g(h(_x51), X2))active#(g(active(h(_x51)), X2))
mark#(g(f(b, _x52), X2))active#(g(active(f(b, _x52)), X2))mark#(g(f(b, mark(_x52)), mark(mark(mark(active(_x22))))))active#(g(f(active(b), _x52), _x22))
mark#(h(X))active#(h(mark(X)))mark#(g(f(a, mark(_x62)), mark(_x42)))active#(g(f(active(a), _x62), _x42))
mark#(g(f(g(_x61, _x62), _x32), mark(active(_x42))))active#(g(active(f(active(g(mark(_x61), _x62)), _x32)), _x42))mark#(g(a, active(active(_x32))))active#(g(mark(b), _x32))
mark#(g(f(h(f(f(_x121, _x122), _x102)), _x32), mark(active(mark(_x22)))))active#(g(active(f(active(h(active(f(active(f(mark(_x121), _x122)), _x102)))), _x32)), _x22))mark#(g(f(g(_x81, _x82), mark(_x52)), mark(mark(mark(_x32)))))active#(g(f(active(g(mark(_x81), _x82)), _x52), _x32))
mark#(g(f(f(_x71, _x72), _x32), mark(active(active(_x22)))))active#(g(active(f(active(f(mark(_x71), _x72)), _x32)), _x22))mark#(g(f(h(_x91), mark(_x62)), mark(active(mark(_x22)))))active#(g(active(f(active(h(_x91)), _x62)), _x22))
mark#(g(f(h(_x91), _x32), mark(active(mark(_x42)))))active#(g(f(active(h(_x91)), _x32), _x42))mark#(g(f(g(_x71, _x72), active(_x52)), X2))active#(g(active(f(active(g(mark(_x71), _x72)), _x52)), X2))
mark#(g(a, active(active(_x22))))active#(g(a, _x22))mark#(g(f(g(h(g(_x131, mark(_x142))), _x72), _x32), mark(active(active(_x22)))))active#(g(active(f(active(g(active(h(active(g(mark(_x131), _x142)))), _x72)), _x32)), _x22))
mark#(g(f(_x61, active(_x62)), X2))active#(g(active(f(_x61, _x62)), X2))mark#(g(f(_x31, mark(mark(_x31))), mark(mark(_x42))))active#(g(mark(h(a)), _x42))
mark#(g(f(f(g(_x91, _x92), active(g(mark(_x91), _x92))), mark(_x62)), mark(_x22)))active#(g(active(f(mark(h(a)), _x62)), _x22))mark#(g(f(f(_x71, _x72), _x32), mark(active(mark(_x22)))))active#(g(active(f(active(f(mark(_x71), _x72)), _x32)), _x22))
mark#(g(f(_x31, mark(_x31)), mark(active(_x42))))active#(g(mark(h(a)), _x42))mark#(g(f(_x31, mark(active(_x62))), mark(mark(mark(active(_x42))))))active#(g(active(f(mark(_x31), _x62)), _x42))
mark#(g(f(a, _x32), mark(active(active(_x22)))))active#(g(active(f(active(a), _x32)), _x22))mark#(g(f(b, _x32), _x22))active#(g(f(active(b), _x32), _x22))
mark#(g(h(f(_x71, _x72)), X2))active#(g(active(h(active(f(_x71, _x72)))), X2))mark#(g(a, active(active(mark(_x32)))))active#(g(a, _x32))
mark#(g(f(_x31, mark(_x52)), mark(mark(active(_x42)))))active#(g(f(mark(_x31), _x52), _x42))mark#(g(f(f(_x61, _x62), _x32), X2))active#(g(active(f(active(f(mark(_x61), _x62)), _x32)), X2))
active#(f(X, X))mark#(h(a))mark#(g(f(h(active(_x81)), _x32), mark(active(mark(_x22)))))active#(g(active(f(active(h(_x81)), _x32)), _x22))
mark#(g(h(f(_x61, _x62)), mark(_x22)))active#(g(active(h(active(f(mark(_x61), _x62)))), _x22))mark#(g(f(g(_x81, mark(_x82)), _x32), mark(active(active(_x22)))))active#(g(active(f(active(g(_x81, _x82)), _x32)), _x22))
mark#(g(f(b, mark(active(_x42))), X2))active#(g(active(f(active(b), _x42)), X2))mark#(g(h(f(_x61, active(_x72))), X2))active#(g(active(h(active(f(mark(_x61), _x72)))), X2))
mark#(g(a, active(mark(active(_x22)))))active#(g(active(a), _x22))mark#(g(f(h(_x71), _x62), mark(active(mark(_x22)))))active#(g(active(f(h(mark(_x71)), _x62)), _x22))
mark#(g(f(g(h(g(_x141, _x142)), _x72), _x32), mark(active(active(_x22)))))active#(g(active(f(active(g(active(h(active(g(_x141, _x142)))), _x72)), _x32)), _x22))mark#(g(f(b, mark(_x32)), mark(mark(mark(_x42)))))active#(g(f(b, _x32), _x42))
mark#(g(f(g(h(_x101), _x92), _x32), mark(active(active(_x22)))))active#(g(active(f(active(g(h(mark(_x101)), _x92)), _x32)), _x22))mark#(g(f(g(h(_x81), _x72), mark(_x52)), mark(mark(_x22))))active#(g(f(active(g(active(h(mark(_x81))), _x72)), _x52), _x22))
mark#(g(f(b, mark(_x62)), mark(active(_x22))))active#(g(f(active(b), _x62), _x22))mark#(g(f(h(_x51), mark(_x52)), mark(mark(mark(_x42)))))active#(g(f(active(h(mark(_x51))), _x52), _x42))
mark#(g(f(_x31, mark(mark(_x42))), mark(_x42)))active#(g(f(mark(_x31), _x42), _x42))mark#(g(f(_x41, active(_x42)), mark(active(_x42))))active#(g(active(f(_x41, _x42)), _x42))
mark#(g(f(_x31, mark(_x52)), mark(mark(mark(active(_x22))))))active#(g(f(mark(_x31), _x52), _x22))mark#(g(a, mark(_x22)))active#(g(active(a), _x22))
mark#(g(h(f(_x61, mark(_x61))), X2))active#(g(active(h(h(a))), X2))mark#(g(f(g(h(g(a, _x132)), mark(_x92)), _x32), mark(active(active(_x22)))))active#(g(active(f(active(g(active(h(active(g(active(a), _x132)))), _x92)), _x32)), _x22))
mark#(g(f(b, active(_x52)), active(_x22)))active#(g(active(f(active(b), _x52)), _x22))mark#(g(f(_x31, mark(active(_x42))), mark(_x42)))active#(g(f(mark(_x31), _x42), _x42))
mark#(g(f(g(_x81, active(_x92)), mark(_x52)), mark(mark(mark(_x22)))))active#(g(active(f(active(g(mark(_x81), _x92)), _x52)), _x22))mark#(g(f(_x31, active(mark(_x31))), X2))active#(g(mark(h(a)), X2))
mark#(g(f(f(g(_x91, _x92), _x82), mark(_x62)), mark(mark(_x32))))active#(g(active(f(active(f(active(g(mark(_x91), _x92)), _x82)), _x62)), _x32))mark#(g(b, mark(mark(_x32))))active#(g(active(b), _x32))
mark#(g(f(g(b, _x72), mark(_x52)), mark(mark(_x22))))active#(g(f(active(g(active(b), _x72)), _x52), _x22))mark#(g(f(f(g(g(_x111, _x112), _x92), _x82), mark(_x62)), mark(_x22)))active#(g(active(f(active(f(active(g(active(g(mark(_x111), _x112)), _x92)), _x82)), _x62)), _x22))
mark#(g(f(a, mark(_x62)), mark(_x22)))active#(g(active(f(active(a), _x62)), _x22))mark#(g(f(_x31, active(_x52)), mark(mark(_x42))))active#(g(active(f(mark(_x31), _x52)), _x42))
mark#(g(f(b, mark(_x52)), mark(_x22)))active#(g(active(f(active(b), _x52)), _x22))mark#(g(f(g(_x71, _x72), mark(_x52)), mark(mark(active(_x32)))))active#(g(f(active(g(mark(_x71), _x72)), _x52), _x32))
mark#(g(f(h(_x61), _x32), mark(active(_x42))))active#(g(active(f(active(h(mark(_x61))), _x32)), _x42))mark#(g(f(g(_x71, _x72), active(_x62)), mark(active(active(_x22)))))active#(g(active(f(active(g(mark(_x71), _x72)), _x62)), _x22))
mark#(g(f(g(_x71, _x72), g(mark(_x71), _x72)), mark(active(active(_x22)))))active#(g(mark(h(a)), _x22))mark#(g(f(h(h(_x101)), _x32), mark(active(mark(_x22)))))active#(g(active(f(active(h(active(h(mark(_x101))))), _x32)), _x22))
mark#(g(f(_x31, mark(_x62)), mark(active(active(_x22)))))active#(g(active(f(mark(_x31), _x62)), _x22))mark#(g(f(_x31, active(_x62)), mark(active(mark(_x22)))))active#(g(active(f(mark(_x31), _x62)), _x22))
mark#(g(f(g(_x71, _x72), _x62), mark(active(active(mark(_x32))))))active#(g(active(f(g(mark(_x71), _x72), _x62)), _x32))mark#(g(f(g(_x71, _x72), mark(_x52)), mark(mark(_x42))))active#(g(active(f(active(g(mark(_x71), _x72)), _x52)), _x42))
mark#(g(f(f(_x71, _x72), _x32), mark(mark(mark(_x22)))))active#(g(active(f(active(f(mark(_x71), _x72)), _x32)), _x22))mark#(g(f(_x31, _x32), mark(active(active(active(_x42))))))active#(g(active(f(mark(_x31), _x32)), _x42))
mark#(g(f(_x31, mark(_x62)), mark(active(_x42))))active#(g(active(f(mark(_x31), _x62)), _x42))mark#(g(f(g(h(g(a, _x132)), _x72), _x32), mark(active(active(active(_x42))))))active#(g(active(f(active(g(active(h(active(g(active(a), _x132)))), _x72)), _x32)), _x42))
mark#(g(h(g(_x61, _x62)), X2))active#(g(active(h(active(g(mark(_x61), _x62)))), X2))mark#(g(f(f(g(_x91, _x92), _x82), mark(mark(_x52))), mark(_x22)))active#(g(active(f(active(f(active(g(mark(_x91), _x92)), _x82)), _x52)), _x22))
mark#(g(f(a, _x32), mark(_x22)))active#(g(active(f(active(a), _x32)), _x22))mark#(g(f(a, mark(a)), mark(mark(_x42))))active#(g(active(h(mark(a))), _x42))
mark#(g(a, active(_x32)))active#(g(b, _x32))

Rewrite Rules

active(h(X))mark(g(X, X))active(g(a, X))mark(f(b, X))
active(f(X, X))mark(h(a))active(a)mark(b)
mark(h(X))active(h(mark(X)))mark(g(X1, X2))active(g(mark(X1), X2))
mark(a)active(a)mark(f(X1, X2))active(f(mark(X1), X2))
mark(b)active(b)h(mark(X))h(X)
h(active(X))h(X)g(mark(X1), X2)g(X1, X2)
g(X1, mark(X2))g(X1, X2)g(active(X1), X2)g(X1, X2)
g(X1, active(X2))g(X1, X2)f(mark(X1), X2)f(X1, X2)
f(X1, mark(X2))f(X1, X2)f(active(X1), X2)f(X1, X2)
f(X1, active(X2))f(X1, X2)

Original Signature

Termination of terms over the following signature is verified: f, g, b, a, active, mark, h

Strategy


The right-hand side of the rule mark#(g(f(b, active(_x52)), _x22)) → active#(g(f(active(b), _x52), _x22)) 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
active#(g(f(b, _x42), _x22)) 
active#(g(f(active(b), _x52), _x32)) 
active#(g(f(active(b), _x42), _x22)) 
Thus, the rule mark#(g(f(b, active(_x52)), _x22)) → active#(g(f(active(b), _x52), _x22)) is replaced by the following rules:
mark#(g(f(b, active(_x52)), active(_x32))) → active#(g(f(active(b), _x52), _x32))mark#(g(f(b, active(mark(_x42))), _x22)) → active#(g(f(active(b), _x42), _x22))
mark#(g(f(b, active(active(_x42))), _x22)) → active#(g(f(active(b), _x42), _x22))mark#(g(f(b, active(_x52)), mark(_x32))) → active#(g(f(active(b), _x52), _x32))
mark#(g(f(b, active(_x42)), _x22)) → active#(g(f(b, _x42), _x22))

Problem 30: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

mark#(g(f(h(f(g(g(_x151, _x152), active(_x142)), _x102)), _x32), mark(active(mark(_x22)))))active#(g(active(f(mark(g(active(f(active(g(active(g(mark(_x151), _x152)), _x142)), _x102)), active(f(active(g(active(g(mark(_x151), _x152)), _x142)), _x102)))), _x32)), _x22))mark#(g(h(_x31), X2))active#(g(mark(g(mark(_x31), mark(_x31))), X2))
mark#(g(f(g(g(_x101, _x102), _x82), mark(_x52)), mark(mark(mark(active(_x32))))))active#(g(active(f(active(g(active(g(mark(_x101), _x102)), _x82)), _x52)), _x32))mark#(g(f(b, _x32), mark(_x22)))active#(g(active(f(active(b), _x32)), _x22))
mark#(g(f(h(f(g(g(_x151, _x152), active(_x142)), active(g(active(g(mark(_x151), _x152)), _x142)))), _x42), mark(active(mark(_x42)))))active#(g(f(h(mark(h(a))), _x42), _x42))mark#(g(f(h(f(_x91, _x91)), _x32), mark(active(mark(_x22)))))active#(g(active(f(active(h(mark(h(a)))), _x32)), _x22))
mark#(g(f(h(_x61), _x32), mark(mark(_x42))))active#(g(active(f(active(h(mark(_x61))), _x32)), _x42))mark#(g(f(g(_x71, mark(_x62)), _x62), mark(active(active(_x22)))))active#(g(active(f(g(mark(_x71), _x62), _x62)), _x22))
mark#(g(f(h(f(g(g(_x151, _x152), active(_x142)), active(g(active(g(mark(_x151), _x152)), _x142)))), _x32), mark(active(mark(_x22)))))active#(g(active(f(active(h(active(h(mark(a))))), _x32)), _x22))mark#(g(f(h(f(_x101, active(_x102))), _x62), mark(active(mark(_x22)))))active#(g(active(f(h(active(f(_x101, _x102))), _x62)), _x22))
mark#(g(f(g(h(g(_x141, mark(_x142))), active(_x82)), _x32), mark(active(active(mark(_x42))))))active#(g(active(f(active(g(active(h(active(g(_x141, _x142)))), _x82)), _x32)), _x42))mark#(g(f(h(h(_x101)), _x32), mark(active(mark(active(_x42))))))active#(g(active(f(active(h(active(h(_x101)))), _x32)), _x42))
mark#(g(f(f(_x71, _x72), mark(_x62)), mark(_x22)))active#(g(active(f(active(f(_x71, _x72)), _x62)), _x22))mark#(g(f(a, mark(_x52)), mark(mark(mark(_x42)))))active#(g(f(active(a), _x52), _x42))
mark#(g(f(h(f(_x101, _x102)), _x32), mark(active(mark(active(mark(_x22)))))))active#(g(active(f(active(h(active(f(mark(_x101), _x102)))), _x32)), _x22))mark#(g(f(g(h(f(_x131, _x132)), _x72), _x32), mark(active(active(_x22)))))active#(g(active(f(active(g(active(h(active(f(mark(_x131), _x132)))), _x72)), _x32)), _x22))
mark#(g(f(g(h(g(_x131, _x132)), _x72), _x32), mark(active(active(_x42)))))active#(g(f(active(g(active(h(active(g(mark(_x131), _x132)))), _x72)), _x32), _x42))mark#(g(X1, active(_x22)))active#(g(mark(X1), _x22))
mark#(g(f(g(g(_x101, _x102), _x72), mark(active(_x62))), mark(mark(mark(_x22)))))active#(g(active(f(active(g(g(mark(_x101), _x102), _x72)), _x62)), _x22))mark#(g(f(h(f(g(g(a, _x152), active(active(_x142))), _x102)), _x32), mark(active(mark(_x22)))))active#(g(active(f(active(h(active(f(active(g(g(active(a), _x152), _x142)), _x102)))), _x32)), _x22))
mark#(g(f(h(f(g(g(a, _x152), _x122), _x102)), _x32), mark(active(mark(_x22)))))active#(g(active(f(mark(g(active(f(active(g(active(g(active(a), _x152)), _x122)), _x102)), active(f(active(g(active(g(active(a), _x152)), _x122)), _x102)))), _x32)), _x22))mark#(g(f(f(f(_x91, _x92), _x82), mark(_x62)), mark(_x22)))active#(g(active(f(active(f(active(f(mark(_x91), _x92)), _x82)), _x62)), _x22))
mark#(g(f(b, active(_x52)), mark(_x32)))active#(g(f(active(b), _x52), _x32))mark#(g(f(h(f(g(g(_x151, _x152), _x122), _x102)), _x32), mark(active(mark(active(_x42))))))active#(g(active(f(active(h(active(f(active(g(active(g(mark(_x151), _x152)), _x122)), _x102)))), _x32)), _x42))
mark#(g(f(_x31, _x32), mark(mark(mark(mark(_x42))))))active#(g(active(f(mark(_x31), _x32)), _x42))mark#(g(f(h(f(g(g(a, _x152), _x122), _x102)), _x32), mark(active(mark(_x42)))))active#(g(f(active(h(active(f(active(g(active(g(active(a), _x152)), _x122)), _x102)))), _x32), _x42))
mark#(g(f(b, _x32), mark(active(_x42))))active#(g(active(f(active(b), _x32)), _x42))mark#(g(f(g(b, _x82), mark(_x52)), mark(mark(mark(_x22)))))active#(g(active(f(active(g(active(b), _x82)), _x52)), _x22))
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mark#(g(f(h(f(g(g(_x151, _x152), active(_x142)), _x102)), _x32), mark(active(mark(_x22)))))active#(g(active(f(active(h(f(active(g(active(g(mark(_x151), _x152)), _x142)), _x102))), _x32)), _x22))mark#(g(f(g(a, _x72), active(g(active(a), _x72))), mark(active(active(mark(_x32))))))active#(g(mark(h(a)), _x32))
mark#(g(a, active(mark(active(_x22)))))active#(g(active(a), _x22))mark#(g(f(h(f(g(g(a, _x152), _x142), active(_x112))), _x32), mark(active(mark(_x22)))))active#(g(active(f(active(h(active(f(active(g(g(active(a), _x152), _x142)), _x112)))), _x32)), _x22))
mark#(g(f(f(g(_x81, _x82), _x72), active(_x52)), X2))active#(g(active(f(active(f(active(g(mark(_x81), _x82)), _x72)), _x52)), X2))mark#(g(f(b, mark(_x32)), mark(mark(mark(_x42)))))active#(g(f(b, _x32), _x42))
mark#(g(f(b, mark(_x62)), mark(active(_x22))))active#(g(f(active(b), _x62), _x22))mark#(g(f(g(g(_x101, _x102), mark(_x92)), mark(_x52)), mark(mark(mark(_x22)))))active#(g(active(f(active(g(active(g(mark(_x101), _x102)), _x92)), _x52)), _x22))
mark#(g(f(h(_x51), mark(_x52)), mark(mark(mark(_x42)))))active#(g(f(active(h(mark(_x51))), _x52), _x42))mark#(g(f(h(f(g(g(a, _x152), _x122), mark(_x92))), _x32), mark(active(mark(mark(_x42))))))active#(g(active(f(active(h(active(f(active(g(active(g(active(a), _x152)), _x122)), _x92)))), _x32)), _x42))
mark#(g(f(h(f(g(f(_x141, _x142), _x122), active(_x112))), _x32), mark(active(mark(_x22)))))active#(g(active(f(active(h(active(f(active(g(active(f(mark(_x141), _x142)), _x122)), _x112)))), _x32)), _x22))mark#(g(f(_x31, mark(mark(_x42))), mark(_x42)))active#(g(f(mark(_x31), _x42), _x42))
mark#(g(f(g(g(_x101, _x102), _x82), mark(active(active(g(active(g(mark(_x101), _x102)), _x82))))), mark(mark(mark(_x22)))))active#(g(mark(h(a)), _x22))mark#(g(f(g(g(_x101, _x102), _x82), mark(_x52)), mark(mark(mark(mark(_x32))))))active#(g(active(f(active(g(active(g(mark(_x101), _x102)), _x82)), _x52)), _x32))
mark#(g(f(_x41, active(_x42)), mark(mark(_x22))))active#(g(f(_x41, _x42), _x22))mark#(g(f(g(g(_x101, _x102), active(_x92)), mark(_x52)), mark(mark(mark(_x22)))))active#(g(active(f(active(g(active(g(mark(_x101), _x102)), _x92)), _x52)), _x22))
mark#(g(f(g(h(g(_x141, mark(_x142))), mark(_x82)), _x32), mark(active(active(mark(_x22))))))active#(g(f(active(g(active(h(active(g(_x141, _x142)))), _x82)), _x32), _x22))mark#(g(f(_x41, active(_x42)), mark(active(_x42))))active#(g(active(f(_x41, _x42)), _x42))
mark#(g(f(_x31, mark(_x52)), mark(mark(mark(active(_x22))))))active#(g(f(mark(_x31), _x52), _x22))mark#(g(f(h(f(g(g(a, mark(_x162)), active(_x132)), _x102)), _x32), mark(active(mark(_x22)))))active#(g(active(f(active(h(active(f(active(g(g(active(a), _x162), _x132)), _x102)))), _x32)), _x22))
mark#(g(f(h(f(g(g(_x151, _x152), mark(mark(_x132))), _x102)), _x32), mark(active(mark(_x22)))))active#(g(active(f(active(h(active(f(active(g(active(g(mark(_x151), _x152)), _x132)), _x102)))), _x32)), _x22))mark#(g(f(f(g(a, _x92), _x82), mark(active(_x52))), mark(_x22)))active#(g(active(f(active(f(active(g(active(a), _x92)), _x82)), _x52)), _x22))
mark#(g(f(h(f(g(_x121, _x122), active(_x112))), _x32), mark(active(mark(_x42)))))active#(g(f(active(h(active(f(active(g(mark(_x121), _x122)), _x112)))), _x32), _x42))mark#(g(f(g(h(g(a, _x132)), mark(_x92)), _x32), mark(active(active(_x22)))))active#(g(active(f(active(g(active(h(active(g(active(a), _x132)))), _x92)), _x32)), _x22))
mark#(g(f(b, active(_x52)), active(_x22)))active#(g(active(f(active(b), _x52)), _x22))mark#(g(f(h(f(g(_x121, _x122), active(g(mark(_x121), _x122)))), _x32), mark(active(mark(_x22)))))active#(g(active(f(active(h(mark(h(a)))), _x32)), _x22))
mark#(g(f(h(f(g(g(a, _x152), _x122), _x102)), _x32), mark(active(mark(_x22)))))active#(g(active(f(active(h(active(f(active(g(active(g(mark(b), _x152)), _x122)), _x102)))), _x32)), _x22))mark#(g(f(h(f(g(g(a, _x152), active(mark(_x142))), _x102)), _x32), mark(active(mark(_x22)))))active#(g(active(f(active(h(active(f(active(g(g(active(a), _x152), _x142)), _x102)))), _x32)), _x22))
mark#(g(f(g(_x81, active(_x92)), mark(_x52)), mark(mark(mark(_x22)))))active#(g(active(f(active(g(mark(_x81), _x92)), _x52)), _x22))mark#(g(f(h(f(g(g(a, _x152), _x142), active(g(g(active(a), _x152), _x142)))), _x32), mark(active(mark(_x22)))))active#(g(active(f(active(h(mark(h(a)))), _x32)), _x22))
mark#(g(f(f(g(a, _x92), _x82), mark(_x52)), mark(_x22)))active#(g(active(f(f(active(g(active(a), _x92)), _x82), _x52)), _x22))mark#(g(f(a, mark(_x62)), mark(_x22)))active#(g(active(f(active(a), _x62)), _x22))
mark#(g(f(b, mark(_x52)), mark(_x22)))active#(g(active(f(active(b), _x52)), _x22))mark#(g(f(g(_x71, _x72), mark(_x52)), mark(mark(active(_x32)))))active#(g(f(active(g(mark(_x71), _x72)), _x52), _x32))
mark#(g(f(h(f(g(g(_x151, _x152), active(_x142)), _x102)), _x32), mark(active(mark(_x42)))))active#(g(f(active(h(active(f(active(g(active(g(mark(_x151), _x152)), _x142)), _x102)))), _x32), _x42))mark#(g(f(h(_x61), _x32), mark(active(_x42))))active#(g(active(f(active(h(mark(_x61))), _x32)), _x42))
mark#(g(f(g(_x71, _x72), _x62), mark(active(active(mark(_x32))))))active#(g(active(f(g(mark(_x71), _x72), _x62)), _x32))mark#(g(f(h(h(_x101)), _x32), mark(active(mark(_x22)))))active#(g(active(f(active(h(active(h(mark(_x101))))), _x32)), _x22))
mark#(g(f(_x31, mark(_x62)), mark(active(active(_x22)))))active#(g(active(f(mark(_x31), _x62)), _x22))mark#(g(f(_x31, active(_x62)), mark(active(mark(_x22)))))active#(g(active(f(mark(_x31), _x62)), _x22))
mark#(g(f(b, _x32), mark(active(mark(_x42)))))active#(g(f(active(b), _x32), _x42))mark#(g(f(g(_x71, _x72), mark(_x52)), mark(mark(_x42))))active#(g(active(f(active(g(mark(_x71), _x72)), _x52)), _x42))
mark#(g(f(h(f(_x111, _x112)), active(h(active(f(_x111, _x112))))), mark(active(mark(mark(_x32))))))active#(g(mark(h(a)), _x32))mark#(g(f(f(_x71, _x72), _x32), mark(mark(mark(_x22)))))active#(g(active(f(active(f(mark(_x71), _x72)), _x32)), _x22))
mark#(g(f(h(f(_x111, _x112)), active(_x62)), mark(active(mark(_x22)))))active#(g(active(f(active(h(active(f(_x111, _x112)))), _x62)), _x22))mark#(g(f(f(g(g(_x121, _x122), _x92), mark(_x72)), mark(_x62)), mark(_x22)))active#(g(active(f(active(f(active(g(active(g(mark(_x121), _x122)), _x92)), _x72)), _x62)), _x22))
mark#(g(f(_x61, active(_x62)), mark(active(mark(_x32)))))active#(g(f(_x61, _x62), _x32))mark#(g(f(f(g(f(_x121, _x122), _x92), mark(_x72)), mark(_x62)), mark(_x22)))active#(g(active(f(active(f(active(g(active(f(mark(_x121), _x122)), _x92)), _x72)), _x62)), _x22))
mark#(g(f(mark(_x51), active(_x52)), mark(active(mark(_x22)))))active#(g(active(f(_x51, _x52)), _x22))mark#(g(f(a, _x32), mark(_x22)))active#(g(active(f(active(a), _x32)), _x22))
mark#(g(f(h(h(b)), active(h(active(h(active(b)))))), mark(active(mark(active(_x42))))))active#(g(mark(h(a)), _x42))mark#(g(h(f(_x61, mark(_x61))), X2))active#(g(active(g(mark(h(a)), mark(h(a)))), X2))
mark#(g(f(g(h(g(_x141, mark(_x142))), _x72), _x32), mark(active(active(mark(active(_x22)))))))active#(g(active(f(active(g(active(h(active(g(_x141, _x142)))), _x72)), _x32)), _x22))mark#(g(a, active(_x32)))active#(g(b, _x32))

Rewrite Rules

active(h(X))mark(g(X, X))active(g(a, X))mark(f(b, X))
active(f(X, X))mark(h(a))active(a)mark(b)
mark(h(X))active(h(mark(X)))mark(g(X1, X2))active(g(mark(X1), X2))
mark(a)active(a)mark(f(X1, X2))active(f(mark(X1), X2))
mark(b)active(b)h(mark(X))h(X)
h(active(X))h(X)g(mark(X1), X2)g(X1, X2)
g(X1, mark(X2))g(X1, X2)g(active(X1), X2)g(X1, X2)
g(X1, active(X2))g(X1, X2)f(mark(X1), X2)f(X1, X2)
f(X1, mark(X2))f(X1, X2)f(active(X1), X2)f(X1, X2)
f(X1, active(X2))f(X1, X2)

Original Signature

Termination of terms over the following signature is verified: f, g, b, a, active, mark, h

Strategy


The right-hand side of the rule mark#(g(f(g(g(_x101, _x102), _x82), mark(_x52)), mark(mark(mark(active(_x32)))))) → active#(g(active(f(active(g(active(g(mark(_x101), _x102)), _x82)), _x52)), _x32)) 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
active#(g(active(f(active(g(active(g(_x81, _x82)), _x82)), _x52)), _x32)) 
active#(g(active(f(active(g(active(g(active(g(mark(_x91), _x92)), _x102)), _x82)), _x52)), _x32)) 
active#(g(active(f(active(g(active(g(mark(_x101), _x102)), _x62)), _x52)), _x32)) 
active#(g(active(f(active(g(active(g(active(a), _x102)), _x82)), _x52)), _x32)) 
active#(g(active(f(active(g(active(g(mark(_x101), _x102)), _x82)), _x52)), _x22)) 
active#(g(active(f(active(g(active(g(active(b), _x102)), _x82)), _x52)), _x32)) 
active#(g(active(f(active(g(active(g(mark(_x101), _x102)), _x82)), _x42)), _x32)) 
active#(g(f(active(g(active(g(mark(_x101), _x102)), _x82)), _x52), _x22)) 
active#(g(active(f(active(g(active(g(active(h(mark(_x91))), _x102)), _x82)), _x52)), _x32)) 
active#(g(active(f(active(g(active(g(mark(_x101), _x82)), _x82)), _x52)), _x32)) 
active#(g(active(f(active(g(active(g(active(f(mark(_x91), _x92)), _x102)), _x82)), _x52)), _x32)) 
active#(g(active(f(active(g(g(mark(_x101), _x102), _x62)), _x52)), _x32)) 
active#(g(active(f(g(active(g(mark(_x101), _x102)), _x82), _x42)), _x32)) 
active#(g(mark(h(a)), _x32)) 
Thus, the rule mark#(g(f(g(g(_x101, _x102), _x82), mark(_x52)), mark(mark(mark(active(_x32)))))) → active#(g(active(f(active(g(active(g(mark(_x101), _x102)), _x82)), _x52)), _x32)) is replaced by the following rules:
mark#(g(f(g(g(_x101, _x102), _x82), mark(mark(_x42))), mark(mark(mark(active(_x32)))))) → active#(g(active(f(active(g(active(g(mark(_x101), _x102)), _x82)), _x42)), _x32))mark#(g(f(g(g(_x101, _x102), _x82), mark(_x52)), mark(mark(mark(active(_x22)))))) → active#(g(f(active(g(active(g(mark(_x101), _x102)), _x82)), _x52), _x22))
mark#(g(f(g(g(b, _x102), _x82), mark(_x52)), mark(mark(mark(active(_x32)))))) → active#(g(active(f(active(g(active(g(active(b), _x102)), _x82)), _x52)), _x32))mark#(g(f(g(g(_x101, _x102), _x82), mark(active(_x42))), mark(mark(mark(active(_x32)))))) → active#(g(active(f(active(g(active(g(mark(_x101), _x102)), _x82)), _x42)), _x32))
mark#(g(f(g(g(_x101, _x102), _x82), mark(active(g(active(g(mark(_x101), _x102)), _x82)))), mark(mark(mark(active(_x32)))))) → active#(g(mark(h(a)), _x32))mark#(g(f(g(g(_x101, _x102), _x82), mark(_x52)), mark(mark(mark(active(active(_x22))))))) → active#(g(active(f(active(g(active(g(mark(_x101), _x102)), _x82)), _x52)), _x22))
mark#(g(f(g(g(_x81, _x82), _x82), mark(_x52)), mark(mark(mark(active(_x32)))))) → active#(g(active(f(active(g(active(g(_x81, _x82)), _x82)), _x52)), _x32))mark#(g(f(g(g(_x101, _x102), _x82), mark(_x52)), mark(mark(mark(active(mark(_x22))))))) → active#(g(active(f(active(g(active(g(mark(_x101), _x102)), _x82)), _x52)), _x22))
mark#(g(f(g(g(_x101, _x102), mark(_x62)), mark(_x52)), mark(mark(mark(active(_x32)))))) → active#(g(active(f(active(g(active(g(mark(_x101), _x102)), _x62)), _x52)), _x32))mark#(g(f(g(g(h(_x91), _x102), _x82), mark(_x52)), mark(mark(mark(active(_x32)))))) → active#(g(active(f(active(g(active(g(active(h(mark(_x91))), _x102)), _x82)), _x52)), _x32))
mark#(g(f(g(g(_x101, active(_x82)), _x82), mark(_x52)), mark(mark(mark(active(_x32)))))) → active#(g(active(f(active(g(active(g(mark(_x101), _x82)), _x82)), _x52)), _x32))mark#(g(f(g(g(f(_x91, _x92), _x102), _x82), mark(_x52)), mark(mark(mark(active(_x32)))))) → active#(g(active(f(active(g(active(g(active(f(mark(_x91), _x92)), _x102)), _x82)), _x52)), _x32))
mark#(g(f(g(g(_x101, mark(_x82)), _x82), mark(_x52)), mark(mark(mark(active(_x32)))))) → active#(g(active(f(active(g(active(g(mark(_x101), _x82)), _x82)), _x52)), _x32))mark#(g(f(g(g(_x101, _x102), active(_x62)), mark(_x52)), mark(mark(mark(active(_x32)))))) → active#(g(active(f(active(g(active(g(mark(_x101), _x102)), _x62)), _x52)), _x32))
mark#(g(f(g(g(a, _x102), _x82), mark(_x52)), mark(mark(mark(active(_x32)))))) → active#(g(active(f(active(g(active(g(active(a), _x102)), _x82)), _x52)), _x32))mark#(g(f(g(g(_x101, _x102), _x62), mark(_x52)), mark(mark(mark(active(_x32)))))) → active#(g(active(f(active(g(g(mark(_x101), _x102), _x62)), _x52)), _x32))
mark#(g(f(g(g(g(_x91, _x92), _x102), _x82), mark(_x52)), mark(mark(mark(active(_x32)))))) → active#(g(active(f(active(g(active(g(active(g(mark(_x91), _x92)), _x102)), _x82)), _x52)), _x32))mark#(g(f(g(g(_x101, _x102), _x82), mark(_x42)), mark(mark(mark(active(_x32)))))) → active#(g(active(f(g(active(g(mark(_x101), _x102)), _x82), _x42)), _x32))

Problem 31: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

mark#(g(f(h(f(g(g(g(h(g(_x261, _x262)), active(_x202)), _x152), active(_x112)), _x102)), _x32), mark(active(mark(active(_x42))))))active#(g(active(f(active(h(f(active(g(active(g(active(g(active(h(active(g(_x261, _x262)))), _x202)), _x152)), _x112)), _x102))), _x32)), _x42))mark#(g(f(h(f(g(g(_x151, _x152), active(_x142)), _x102)), _x32), mark(active(mark(_x22)))))active#(g(active(f(mark(g(active(f(active(g(active(g(mark(_x151), _x152)), _x142)), _x102)), active(f(active(g(active(g(mark(_x151), _x152)), _x142)), _x102)))), _x32)), _x22))
mark#(g(f(h(f(g(g(a, _x132), active(active(active(_x132)))), mark(_x112))), _x32), mark(active(mark(_x22)))))active#(g(active(f(active(h(active(f(active(g(g(a, _x132), _x132)), _x112)))), _x32)), _x22))mark#(g(f(h(f(g(g(a, active(_x122)), active(active(active(_x132)))), mark(_x112))), _x32), mark(active(mark(_x22)))))active#(g(active(f(mark(g(f(active(g(g(active(a), _x122), _x132)), _x112), f(active(g(g(active(a), _x122), _x132)), _x112))), _x32)), _x22))
mark#(g(h(_x31), X2))active#(g(mark(g(mark(_x31), mark(_x31))), X2))mark#(g(f(h(f(g(g(g(h(g(_x261, _x262)), mark(_x172)), _x152), active(_x142)), _x102)), _x32), mark(active(mark(active(_x42))))))active#(g(active(f(active(h(f(active(g(g(active(g(active(h(active(g(_x261, _x262)))), _x172)), _x152), _x142)), _x102))), _x32)), _x42))
mark#(g(f(g(h(f(_x131, _x132)), _x72), _x32), mark(active(active(_x22)))))active#(g(active(f(active(g(mark(g(active(f(mark(_x131), _x132)), active(f(mark(_x131), _x132)))), _x72)), _x32)), _x22))mark#(g(f(h(f(g(g(g(h(g(_x261, _x262)), _x162), _x152), active(_x142)), mark(_x112))), _x32), mark(active(mark(active(_x42))))))active#(g(active(f(active(h(f(active(g(g(active(g(active(h(active(g(_x261, _x262)))), _x162)), _x152), _x142)), _x112))), _x32)), _x42))
mark#(g(f(h(f(g(g(a, _x152), _x122), _x102)), _x32), mark(active(mark(_x22)))))active#(g(active(f(mark(g(active(f(active(g(active(g(active(a), _x152)), _x122)), _x102)), active(f(active(g(active(g(active(a), _x152)), _x122)), _x102)))), _x32)), _x22))mark#(g(f(h(f(g(g(a, _x152), active(active(active(_x132)))), _x102)), _x32), mark(active(mark(_x22)))))active#(g(active(f(mark(g(active(f(active(g(g(active(a), _x152), _x132)), _x102)), active(f(active(g(g(active(a), _x152), _x132)), _x102)))), _x32)), _x22))
mark#(g(f(h(f(g(g(g(h(g(a, _x241)), _x162), _x152), active(_x112)), _x102)), _x32), mark(active(mark(active(_x42))))))active#(g(active(f(active(h(active(f(active(g(active(g(active(g(active(h(mark(f(b, _x241)))), _x162)), _x152)), _x112)), _x102)))), _x32)), _x42))mark#(g(f(h(f(g(g(a, _x152), _x122), _x102)), _x32), mark(active(mark(_x42)))))active#(g(f(active(h(active(f(active(g(active(g(active(a), _x152)), _x122)), _x102)))), _x32), _x42))
mark#(g(f(g(b, _x82), mark(_x52)), mark(mark(mark(_x22)))))active#(g(active(f(active(g(active(b), _x82)), _x52)), _x22))mark#(g(f(g(h(g(_x131, _x132)), _x82), _x32), mark(active(active(mark(_x42))))))active#(g(active(f(active(g(h(active(g(mark(_x131), _x132))), _x82)), _x32)), _x42))
mark#(g(f(b, mark(_x52)), mark(mark(mark(_x22)))))active#(g(active(f(active(b), _x52)), _x22))mark#(g(f(h(f(g(g(g(h(g(_x211, _x212)), _x162), _x152), active(_x112)), _x102)), _x32), mark(active(mark(active(_x42))))))active#(g(active(f(mark(g(active(f(active(g(active(g(active(g(active(h(active(g(mark(_x211), _x212)))), _x162)), _x152)), _x112)), _x102)), active(f(active(g(active(g(active(g(active(h(active(g(mark(_x211), _x212)))), _x162)), _x152)), _x112)), _x102)))), _x32)), _x42))
mark#(g(f(h(f(g(g(a, _x152), active(_x132)), mark(_x112))), _x32), mark(active(mark(_x22)))))active#(g(active(f(active(h(active(f(active(g(g(active(a), _x152), _x132)), _x112)))), _x32)), _x22))mark#(g(f(f(a, _x82), mark(_x62)), mark(_x22)))active#(g(active(f(active(f(active(a), _x82)), _x62)), _x22))
mark#(g(h(f(_x61, mark(_x61))), _x22))active#(g(g(mark(h(a)), h(a)), _x22))mark#(g(f(h(f(g(g(_x151, _x152), mark(_x142)), _x102)), _x32), mark(active(mark(_x22)))))active#(g(active(f(active(h(f(active(g(active(g(mark(_x151), _x152)), _x142)), _x102))), _x32)), _x22))
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mark#(g(f(h(f(g(g(a, _x152), active(active(active(active(_x122))))), active(_x102))), _x32), mark(active(mark(_x22)))))active#(g(active(f(active(h(active(f(g(g(active(a), _x152), _x122), _x102)))), _x32)), _x22))mark#(g(a, active(active(mark(_x32)))))active#(g(a, _x32))
mark#(g(f(f(_x71, _x72), active(active(f(mark(_x71), _x72)))), X2))active#(g(mark(h(a)), X2))mark#(g(f(g(h(g(f(_x141, _x142), _x132)), _x72), _x32), mark(active(active(mark(_x42))))))active#(g(active(f(active(g(active(h(active(g(active(f(mark(_x141), _x142)), _x132)))), _x72)), _x32)), _x42))
mark#(g(f(h(f(_x101, mark(_x102))), _x32), mark(active(mark(_x22)))))active#(g(active(f(active(h(active(f(_x101, _x102)))), _x32)), _x22))mark#(g(f(h(f(g(g(a, _x152), active(active(_x142))), _x112)), _x32), mark(active(mark(_x22)))))active#(g(active(f(active(h(active(f(g(g(active(a), _x152), _x142), _x112)))), _x32)), _x22))
mark#(g(f(h(f(g(g(g(h(g(_x261, _x262)), _x162), _x152), active(_x142)), _x102)), _x72), mark(active(mark(active(_x42))))))active#(g(active(f(h(f(active(g(g(active(g(active(h(active(g(_x261, _x262)))), _x162)), _x152), _x142)), _x102)), _x72)), _x42))mark#(g(f(f(_x71, _x72), active(_x52)), _x22))active#(g(f(active(f(mark(_x71), _x72)), _x52), _x22))
mark#(g(f(h(f(g(g(a, active(_x122)), active(active(active(active(_x142))))), _x102)), mark(_x62)), mark(active(mark(_x22)))))active#(g(active(f(active(h(active(f(active(g(g(active(a), _x122), _x142)), _x102)))), _x62)), _x22))mark#(g(f(h(f(g(g(a, _x152), active(active(active(_x132)))), mark(active(_x72)))), _x42), mark(active(mark(_x42)))))active#(g(f(h(f(active(g(g(active(a), _x152), _x132)), _x72)), _x42), _x42))
mark#(g(f(g(_x81, mark(_x82)), _x32), mark(active(active(_x22)))))active#(g(active(f(active(g(_x81, _x82)), _x32)), _x22))mark#(g(f(h(f(g(g(a, _x152), active(_x132)), _x102)), _x32), mark(active(mark(active(_x42))))))active#(g(active(f(active(h(active(f(active(g(g(active(a), _x152), _x132)), _x102)))), _x32)), _x42))
mark#(g(f(b, mark(active(_x42))), X2))active#(g(active(f(active(b), _x42)), X2))mark#(g(f(h(f(g(g(_x151, _x152), active(_x142)), _x102)), _x32), mark(active(mark(_x22)))))active#(g(active(f(active(h(f(active(g(active(g(mark(_x151), _x152)), _x142)), _x102))), _x32)), _x22))
mark#(g(h(f(_x61, active(_x72))), X2))active#(g(active(h(active(f(mark(_x61), _x72)))), X2))mark#(g(f(h(f(g(g(a, _x152), active(active(active(active(_x122))))), active(_x112))), active(_x62)), mark(active(mark(_x22)))))active#(g(active(f(active(h(active(f(active(g(g(active(a), _x152), _x122)), _x112)))), _x62)), _x22))
mark#(g(f(h(f(g(g(_x151, _x152), active(_x142)), active(g(active(g(mark(_x151), _x152)), _x142)))), mark(_x62)), mark(active(mark(_x22)))))active#(g(active(f(active(h(active(h(mark(a))))), _x62)), _x22))mark#(g(f(g(a, _x72), active(g(active(a), _x72))), mark(active(active(mark(_x32))))))active#(g(mark(h(a)), _x32))
mark#(g(a, active(mark(active(_x22)))))active#(g(active(a), _x22))mark#(g(f(h(f(g(g(g(h(g(mark(_x211), _x212)), _x162), _x152), active(_x142)), _x102)), _x32), mark(active(mark(active(_x42))))))active#(g(active(f(active(h(f(active(g(g(active(g(active(h(active(g(_x211, _x212)))), _x162)), _x152), _x142)), _x102))), _x32)), _x42))
mark#(g(f(f(g(_x81, _x82), _x72), active(_x52)), X2))active#(g(active(f(active(f(active(g(mark(_x81), _x82)), _x72)), _x52)), X2))mark#(g(f(b, mark(_x32)), mark(mark(mark(_x42)))))active#(g(f(b, _x32), _x42))
mark#(g(f(h(f(g(g(g(mark(_x161), _x162), _x152), active(_x112)), _x102)), _x32), mark(active(mark(active(_x22))))))active#(g(f(active(h(active(f(active(g(active(g(active(g(_x161, _x162)), _x152)), _x112)), _x102)))), _x32), _x22))mark#(g(f(b, mark(_x62)), mark(active(_x22))))active#(g(f(active(b), _x62), _x22))
mark#(g(f(g(g(_x101, _x102), mark(_x92)), mark(_x52)), mark(mark(mark(_x22)))))active#(g(active(f(active(g(active(g(mark(_x101), _x102)), _x92)), _x52)), _x22))mark#(g(f(h(_x51), mark(_x52)), mark(mark(mark(_x42)))))active#(g(f(active(h(mark(_x51))), _x52), _x42))
mark#(g(f(h(f(g(g(a, _x152), _x122), mark(_x92))), _x32), mark(active(mark(mark(_x42))))))active#(g(active(f(active(h(active(f(active(g(active(g(active(a), _x152)), _x122)), _x92)))), _x32)), _x42))mark#(g(f(g(g(_x101, _x102), active(mark(_x72))), mark(_x52)), mark(mark(mark(active(_x32))))))active#(g(active(f(active(g(active(g(mark(_x101), _x102)), _x72)), _x52)), _x32))
mark#(g(f(_x31, mark(mark(_x42))), mark(_x42)))active#(g(f(mark(_x31), _x42), _x42))mark#(g(f(g(g(_x101, _x102), _x82), mark(_x52)), mark(mark(mark(mark(_x32))))))active#(g(active(f(active(g(active(g(mark(_x101), _x102)), _x82)), _x52)), _x32))
mark#(g(f(h(f(g(g(g(h(g(_x261, _x262)), _x162), _x152), active(_x142)), _x102)), mark(_x72)), mark(active(mark(active(_x42))))))active#(g(active(f(active(h(f(active(g(g(active(g(active(h(active(g(_x261, _x262)))), _x162)), _x152), _x142)), _x102))), _x72)), _x42))mark#(g(f(h(f(g(g(g(h(g(_x261, _x262)), active(_x212)), _x152), active(_x112)), _x102)), _x32), mark(active(mark(active(_x42))))))active#(g(active(f(active(h(active(f(active(g(active(g(active(g(active(h(active(g(_x261, _x262)))), _x212)), _x152)), _x112)), _x102)))), _x32)), _x42))
mark#(g(f(g(g(_x101, _x102), active(_x92)), mark(_x52)), mark(mark(mark(_x22)))))active#(g(active(f(active(g(active(g(mark(_x101), _x102)), _x92)), _x52)), _x22))mark#(g(f(_x41, active(_x42)), mark(active(_x42))))active#(g(active(f(_x41, _x42)), _x42))
mark#(g(f(h(f(g(g(_x151, _x152), _x122), active(g(active(g(mark(_x151), _x152)), _x122)))), _x32), mark(active(mark(active(_x42))))))active#(g(active(f(active(h(mark(h(a)))), _x32)), _x42))mark#(g(f(g(h(g(a, _x132)), mark(_x92)), _x32), mark(active(active(_x22)))))active#(g(active(f(active(g(active(h(active(g(active(a), _x132)))), _x92)), _x32)), _x22))
mark#(g(f(h(f(g(_x121, _x122), active(g(mark(_x121), _x122)))), _x32), mark(active(mark(_x22)))))active#(g(active(f(active(h(mark(h(a)))), _x32)), _x22))mark#(g(f(h(f(g(g(a, _x152), _x122), _x102)), _x32), mark(active(mark(_x22)))))active#(g(active(f(active(h(active(f(active(g(active(g(mark(b), _x152)), _x122)), _x102)))), _x32)), _x22))
mark#(g(f(h(f(g(g(a, _x152), _x142), active(g(g(active(a), _x152), _x142)))), _x32), mark(active(mark(_x22)))))active#(g(active(f(active(h(mark(h(a)))), _x32)), _x22))mark#(g(f(h(f(a, a)), _x32), mark(active(mark(_x22)))))active#(g(active(f(mark(g(h(mark(a)), mark(h(a)))), _x32)), _x22))
mark#(g(f(h(f(g(g(g(_x161, _x162), _x152), active(_x122)), _x102)), _x32), mark(active(mark(active(_x42))))))active#(g(active(f(active(h(active(f(active(g(g(active(g(mark(_x161), _x162)), _x152), _x122)), _x102)))), _x32)), _x42))mark#(g(f(h(f(g(g(a, _x152), active(active(active(active(_x142))))), active(g(g(active(a), _x152), _x142)))), mark(_x62)), mark(active(mark(mark(_x32))))))active#(g(active(f(mark(g(mark(h(a)), mark(h(a)))), _x62)), _x32))
mark#(g(f(h(f(g(g(g(h(h(_x211)), _x162), _x152), active(_x112)), _x102)), _x32), mark(active(mark(active(_x42))))))active#(g(active(f(active(h(active(f(active(g(active(g(active(g(active(h(active(h(mark(_x211))))), _x162)), _x152)), _x112)), _x102)))), _x32)), _x42))mark#(g(f(h(f(g(g(_x151, _x152), active(_x142)), _x102)), _x32), mark(active(mark(_x42)))))active#(g(f(active(h(active(f(active(g(active(g(mark(_x151), _x152)), _x142)), _x102)))), _x32), _x42))
mark#(g(f(h(_x61), _x32), mark(active(_x42))))active#(g(active(f(active(h(mark(_x61))), _x32)), _x42))mark#(g(f(g(_x71, _x72), _x62), mark(active(active(mark(_x32))))))active#(g(active(f(g(mark(_x71), _x72), _x62)), _x32))
mark#(g(f(_x31, active(_x62)), mark(active(mark(_x22)))))active#(g(active(f(mark(_x31), _x62)), _x22))mark#(g(f(_x31, mark(_x62)), mark(active(active(_x22)))))active#(g(active(f(mark(_x31), _x62)), _x22))
mark#(g(f(h(h(_x101)), _x32), mark(active(mark(_x22)))))active#(g(active(f(active(h(active(h(mark(_x101))))), _x32)), _x22))mark#(g(f(b, _x32), mark(active(mark(_x42)))))active#(g(f(active(b), _x32), _x42))
mark#(g(f(h(f(g(g(a, _x152), active(active(active(_x132)))), _x102)), _x32), mark(active(mark(_x42)))))active#(g(f(active(h(active(f(active(g(g(active(a), _x152), _x132)), _x102)))), _x32), _x42))mark#(g(f(h(f(_x111, _x112)), active(_x62)), mark(active(mark(_x22)))))active#(g(active(f(active(h(active(f(_x111, _x112)))), _x62)), _x22))
mark#(g(f(h(f(_x91, _x92)), _x32), mark(mark(_x42))))active#(g(active(f(active(h(active(f(mark(_x91), _x92)))), _x32)), _x42))mark#(g(f(f(g(g(_x121, _x122), _x92), mark(_x72)), mark(_x62)), mark(_x22)))active#(g(active(f(active(f(active(g(active(g(mark(_x121), _x122)), _x92)), _x72)), _x62)), _x22))
mark#(g(f(_x61, active(_x62)), mark(active(mark(_x32)))))active#(g(f(_x61, _x62), _x32))mark#(g(f(f(g(f(_x121, _x122), _x92), mark(_x72)), mark(_x62)), mark(_x22)))active#(g(active(f(active(f(active(g(active(f(mark(_x121), _x122)), _x92)), _x72)), _x62)), _x22))
mark#(g(f(mark(_x51), active(_x52)), mark(active(mark(_x22)))))active#(g(active(f(_x51, _x52)), _x22))mark#(g(f(h(f(g(g(g(f(_x181, _x182), _x162), _x152), active(_x112)), _x102)), _x32), mark(active(mark(active(_x42))))))active#(g(active(f(active(h(active(f(active(g(active(g(active(g(active(f(mark(_x181), _x182)), _x162)), _x152)), _x112)), _x102)))), _x32)), _x42))
mark#(g(f(a, _x32), mark(_x22)))active#(g(active(f(active(a), _x32)), _x22))mark#(g(f(h(h(b)), active(h(active(h(active(b)))))), mark(active(mark(active(_x42))))))active#(g(mark(h(a)), _x42))
mark#(g(h(f(_x61, mark(_x61))), X2))active#(g(active(g(mark(h(a)), mark(h(a)))), X2))mark#(g(f(h(h(_x101)), active(_x52)), mark(active(mark(active(_x42))))))active#(g(active(f(active(h(active(h(_x101)))), _x52)), _x42))
mark#(g(f(h(f(g(g(g(h(_x181), _x162), _x152), active(_x112)), _x102)), active(h(active(f(active(g(active(g(active(g(active(h(mark(_x181))), _x162)), _x152)), _x112)), _x102))))), mark(active(mark(active(_x42))))))active#(g(mark(h(a)), _x42))mark#(g(f(h(f(g(g(g(_x171, _x172), _x152), active(_x112)), _x92)), _x32), mark(active(mark(active(_x22))))))active#(g(f(active(h(active(f(g(active(g(active(g(_x171, _x172)), _x152)), _x112), _x92)))), _x32), _x22))
mark#(g(f(h(f(g(g(a, _x152), active(active(active(_x132)))), mark(_x112))), _x32), mark(active(mark(_x22)))))active#(g(active(f(mark(g(active(f(active(g(g(active(a), _x152), _x132)), _x112)), active(f(active(g(g(active(a), _x152), _x132)), _x112)))), _x32)), _x22))mark#(g(a, active(_x32)))active#(g(b, _x32))

Rewrite Rules

active(h(X))mark(g(X, X))active(g(a, X))mark(f(b, X))
active(f(X, X))mark(h(a))active(a)mark(b)
mark(h(X))active(h(mark(X)))mark(g(X1, X2))active(g(mark(X1), X2))
mark(a)active(a)mark(f(X1, X2))active(f(mark(X1), X2))
mark(b)active(b)h(mark(X))h(X)
h(active(X))h(X)g(mark(X1), X2)g(X1, X2)
g(X1, mark(X2))g(X1, X2)g(active(X1), X2)g(X1, X2)
g(X1, active(X2))g(X1, X2)f(mark(X1), X2)f(X1, X2)
f(X1, mark(X2))f(X1, X2)f(active(X1), X2)f(X1, X2)
f(X1, active(X2))f(X1, X2)

Original Signature

Termination of terms over the following signature is verified: f, g, b, a, active, mark, h

Strategy


The right-hand side of the rule mark#(g(f(h(f(g(g(g(h(g(_x261, _x262)), active(_x202)), _x152), active(_x112)), _x102)), _x32), mark(active(mark(active(_x42)))))) → active#(g(active(f(active(h(f(active(g(active(g(active(g(active(h(active(g(_x261, _x262)))), _x202)), _x152)), _x112)), _x102))), _x32)), _x42)) 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
active#(g(active(f(active(h(f(active(g(active(g(active(g(mark(g(active(g(_x261, _x262)), active(g(_x261, _x262)))), _x202)), _x152)), _x112)), _x102))), _x32)), _x42)) 
active#(g(active(f(active(h(f(active(g(active(g(active(g(active(h(g(_x261, _x262))), _x202)), _x152)), _x112)), _x102))), _x32)), _x42)) 
active#(g(active(f(mark(g(f(active(g(active(g(active(g(active(h(active(g(_x261, _x262)))), _x202)), _x152)), _x112)), _x102), f(active(g(active(g(active(g(active(h(active(g(_x261, _x262)))), _x202)), _x152)), _x112)), _x102))), _x32)), _x42)) 
active#(g(f(active(h(f(active(g(active(g(active(g(active(h(active(g(_x261, _x262)))), _x202)), _x152)), _x112)), _x102))), _x32), _x52)) 
active#(g(active(f(active(h(f(active(g(active(g(active(g(active(h(active(g(_x231, _x232)))), _x202)), _x152)), _x112)), _x102))), _x32)), _x42)) 
active#(g(active(f(active(h(f(active(g(active(g(active(g(h(active(g(_x261, _x262))), _x192)), _x152)), _x112)), _x102))), _x32)), _x42)) 
active#(g(active(f(active(h(f(active(g(active(g(active(g(active(h(active(g(_x261, _x262)))), _x202)), _x172)), _x112)), _x102))), _x32)), _x42)) 
active#(g(active(f(active(h(f(active(g(active(g(active(g(active(h(mark(f(b, _x221)))), _x202)), _x152)), _x112)), _x102))), _x32)), _x42)) 
active#(g(active(f(h(f(active(g(active(g(active(g(active(h(active(g(_x261, _x262)))), _x202)), _x152)), _x112)), _x102)), _x72)), _x42)) 
active#(g(active(f(active(h(f(active(g(active(g(active(g(active(h(active(g(_x261, _x262)))), _x192)), _x152)), _x112)), _x102))), _x32)), _x42)) 
active#(g(mark(h(a)), _x42)) 
active#(g(active(f(active(h(f(active(g(g(active(g(active(h(active(g(_x261, _x262)))), _x202)), _x152), _x142)), _x102))), _x32)), _x42)) 
active#(g(active(f(active(h(f(active(g(active(g(active(g(active(h(active(g(_x261, _x262)))), _x202)), _x152)), _x112)), _x122))), _x32)), _x42)) 
active#(g(active(f(active(h(f(active(g(active(g(active(g(active(h(active(g(_x261, _x262)))), _x202)), _x152)), _x112)), _x102))), _x72)), _x42)) 
active#(g(active(f(active(h(f(active(g(active(g(g(active(h(active(g(_x261, _x262)))), _x202), _x172)), _x112)), _x102))), _x32)), _x42)) 
active#(g(active(f(active(h(f(active(g(active(g(active(g(active(h(active(g(_x261, _x262)))), _x202)), _x152)), _x142)), _x102))), _x32)), _x42)) 
active#(g(active(f(active(h(f(active(g(active(g(active(g(active(h(active(g(_x261, _x262)))), _x202)), _x152)), _x112)), _x102))), _x32)), _x52)) 
active#(g(active(f(active(h(f(g(active(g(active(g(active(h(active(g(_x261, _x262)))), _x202)), _x152)), _x112), _x122))), _x32)), _x42)) 
Thus, the rule mark#(g(f(h(f(g(g(g(h(g(_x261, _x262)), active(_x202)), _x152), active(_x112)), _x102)), _x32), mark(active(mark(active(_x42)))))) → active#(g(active(f(active(h(f(active(g(active(g(active(g(active(h(active(g(_x261, _x262)))), _x202)), _x152)), _x112)), _x102))), _x32)), _x42)) is replaced by the following rules:
mark#(g(f(h(f(g(g(g(h(g(_x261, _x262)), active(_x202)), _x152), active(_x112)), _x102)), _x32), mark(active(mark(active(_x42)))))) → active#(g(active(f(mark(g(f(active(g(active(g(active(g(active(h(active(g(_x261, _x262)))), _x202)), _x152)), _x112)), _x102), f(active(g(active(g(active(g(active(h(active(g(_x261, _x262)))), _x202)), _x152)), _x112)), _x102))), _x32)), _x42))mark#(g(f(h(f(g(g(g(h(g(_x261, _x262)), active(_x192)), _x152), active(_x112)), _x102)), _x32), mark(active(mark(active(_x42)))))) → active#(g(active(f(active(h(f(active(g(active(g(active(g(h(active(g(_x261, _x262))), _x192)), _x152)), _x112)), _x102))), _x32)), _x42))
mark#(g(f(h(f(g(g(g(h(g(_x261, _x262)), active(_x202)), _x152), active(_x112)), _x102)), active(h(f(active(g(active(g(active(g(active(h(active(g(_x261, _x262)))), _x202)), _x152)), _x112)), _x102)))), mark(active(mark(active(_x42)))))) → active#(g(mark(h(a)), _x42))mark#(g(f(h(f(g(g(g(h(g(_x261, _x262)), active(_x202)), _x152), active(_x112)), _x102)), _x32), mark(active(mark(active(mark(_x52))))))) → active#(g(active(f(active(h(f(active(g(active(g(active(g(active(h(active(g(_x261, _x262)))), _x202)), _x152)), _x112)), _x102))), _x32)), _x52))
mark#(g(f(h(f(g(g(g(h(g(_x261, _x262)), active(_x202)), _x152), active(_x112)), _x102)), _x32), mark(active(mark(active(_x52)))))) → active#(g(f(active(h(f(active(g(active(g(active(g(active(h(active(g(_x261, _x262)))), _x202)), _x152)), _x112)), _x102))), _x32), _x52))mark#(g(f(h(f(g(g(g(h(g(_x261, _x262)), active(active(_x192))), _x152), active(_x112)), _x102)), _x32), mark(active(mark(active(_x42)))))) → active#(g(active(f(active(h(f(active(g(active(g(active(g(active(h(active(g(_x261, _x262)))), _x192)), _x152)), _x112)), _x102))), _x32)), _x42))
mark#(g(f(h(f(g(g(g(h(g(_x261, _x262)), active(_x202)), _x152), active(mark(_x142))), _x102)), _x32), mark(active(mark(active(_x42)))))) → active#(g(active(f(active(h(f(active(g(active(g(active(g(active(h(active(g(_x261, _x262)))), _x202)), _x152)), _x142)), _x102))), _x32)), _x42))mark#(g(f(h(f(g(g(g(h(g(mark(_x231), _x232)), active(_x202)), _x152), active(_x112)), _x102)), _x32), mark(active(mark(active(_x42)))))) → active#(g(active(f(active(h(f(active(g(active(g(active(g(active(h(active(g(_x231, _x232)))), _x202)), _x152)), _x112)), _x102))), _x32)), _x42))
mark#(g(f(h(f(g(g(g(h(g(_x231, active(_x232))), active(_x202)), _x152), active(_x112)), _x102)), _x32), mark(active(mark(active(_x42)))))) → active#(g(active(f(active(h(f(active(g(active(g(active(g(active(h(active(g(_x231, _x232)))), _x202)), _x152)), _x112)), _x102))), _x32)), _x42))mark#(g(f(h(f(g(g(g(h(g(_x261, _x262)), active(_x202)), mark(_x172)), active(_x112)), _x102)), _x32), mark(active(mark(active(_x42)))))) → active#(g(active(f(active(h(f(active(g(active(g(active(g(active(h(active(g(_x261, _x262)))), _x202)), _x172)), _x112)), _x102))), _x32)), _x42))
mark#(g(f(h(f(g(g(g(h(g(a, _x221)), active(_x202)), _x152), active(_x112)), _x102)), _x32), mark(active(mark(active(_x42)))))) → active#(g(active(f(active(h(f(active(g(active(g(active(g(active(h(mark(f(b, _x221)))), _x202)), _x152)), _x112)), _x102))), _x32)), _x42))mark#(g(f(h(f(g(g(g(h(g(_x261, _x262)), active(mark(_x192))), _x152), active(_x112)), _x102)), _x32), mark(active(mark(active(_x42)))))) → active#(g(active(f(active(h(f(active(g(active(g(active(g(active(h(active(g(_x261, _x262)))), _x192)), _x152)), _x112)), _x102))), _x32)), _x42))
mark#(g(f(h(f(g(g(g(h(g(_x261, _x262)), active(_x202)), _x152), active(_x112)), _x102)), mark(_x72)), mark(active(mark(active(_x42)))))) → active#(g(active(f(active(h(f(active(g(active(g(active(g(active(h(active(g(_x261, _x262)))), _x202)), _x152)), _x112)), _x102))), _x72)), _x42))mark#(g(f(h(f(g(g(g(h(g(_x261, _x262)), active(_x202)), _x152), active(_x112)), _x102)), _x72), mark(active(mark(active(_x42)))))) → active#(g(active(f(h(f(active(g(active(g(active(g(active(h(active(g(_x261, _x262)))), _x202)), _x152)), _x112)), _x102)), _x72)), _x42))
mark#(g(f(h(f(g(g(g(h(g(_x261, _x262)), active(_x202)), _x152), active(_x142)), _x102)), _x32), mark(active(mark(active(_x42)))))) → active#(g(active(f(active(h(f(active(g(g(active(g(active(h(active(g(_x261, _x262)))), _x202)), _x152), _x142)), _x102))), _x32)), _x42))mark#(g(f(h(f(g(g(g(h(g(_x261, _x262)), active(_x202)), _x152), active(_x112)), mark(_x122))), _x32), mark(active(mark(active(_x42)))))) → active#(g(active(f(active(h(f(active(g(active(g(active(g(active(h(active(g(_x261, _x262)))), _x202)), _x152)), _x112)), _x122))), _x32)), _x42))
mark#(g(f(h(f(g(g(g(h(g(_x261, _x262)), active(_x202)), active(_x172)), active(_x112)), _x102)), _x32), mark(active(mark(active(_x42)))))) → active#(g(active(f(active(h(f(active(g(active(g(active(g(active(h(active(g(_x261, _x262)))), _x202)), _x172)), _x112)), _x102))), _x32)), _x42))mark#(g(f(h(f(g(g(g(h(g(_x261, _x262)), active(_x202)), _x152), active(_x112)), active(_x122))), _x32), mark(active(mark(active(_x42)))))) → active#(g(active(f(active(h(f(active(g(active(g(active(g(active(h(active(g(_x261, _x262)))), _x202)), _x152)), _x112)), _x122))), _x32)), _x42))
mark#(g(f(h(f(g(g(g(h(g(_x261, _x262)), active(_x202)), _x152), active(_x112)), _x102)), active(_x72)), mark(active(mark(active(_x42)))))) → active#(g(active(f(active(h(f(active(g(active(g(active(g(active(h(active(g(_x261, _x262)))), _x202)), _x152)), _x112)), _x102))), _x72)), _x42))mark#(g(f(h(f(g(g(g(h(g(_x261, _x262)), active(_x202)), _x152), active(active(_x142))), _x102)), _x32), mark(active(mark(active(_x42)))))) → active#(g(active(f(active(h(f(active(g(active(g(active(g(active(h(active(g(_x261, _x262)))), _x202)), _x152)), _x142)), _x102))), _x32)), _x42))
mark#(g(f(h(f(g(g(g(h(g(_x261, _x262)), active(_x202)), _x172), active(_x112)), _x102)), _x32), mark(active(mark(active(_x42)))))) → active#(g(active(f(active(h(f(active(g(active(g(g(active(h(active(g(_x261, _x262)))), _x202), _x172)), _x112)), _x102))), _x32)), _x42))mark#(g(f(h(f(g(g(g(h(g(_x261, _x262)), active(_x202)), _x152), active(_x112)), _x102)), _x32), mark(active(mark(active(_x42)))))) → active#(g(active(f(active(h(f(active(g(active(g(active(g(active(h(g(_x261, _x262))), _x202)), _x152)), _x112)), _x102))), _x32)), _x42))
mark#(g(f(h(f(g(g(g(h(g(active(_x231), _x232)), active(_x202)), _x152), active(_x112)), _x102)), _x32), mark(active(mark(active(_x42)))))) → active#(g(active(f(active(h(f(active(g(active(g(active(g(active(h(active(g(_x231, _x232)))), _x202)), _x152)), _x112)), _x102))), _x32)), _x42))mark#(g(f(h(f(g(g(g(h(g(_x231, mark(_x232))), active(_x202)), _x152), active(_x112)), _x102)), _x32), mark(active(mark(active(_x42)))))) → active#(g(active(f(active(h(f(active(g(active(g(active(g(active(h(active(g(_x231, _x232)))), _x202)), _x152)), _x112)), _x102))), _x32)), _x42))
mark#(g(f(h(f(g(g(g(h(g(_x261, _x262)), active(_x202)), _x152), active(_x112)), _x122)), _x32), mark(active(mark(active(_x42)))))) → active#(g(active(f(active(h(f(g(active(g(active(g(active(h(active(g(_x261, _x262)))), _x202)), _x152)), _x112), _x122))), _x32)), _x42))mark#(g(f(h(f(g(g(g(h(g(_x261, _x262)), active(_x202)), _x152), active(_x112)), _x102)), _x32), mark(active(mark(active(active(_x52))))))) → active#(g(active(f(active(h(f(active(g(active(g(active(g(active(h(active(g(_x261, _x262)))), _x202)), _x152)), _x112)), _x102))), _x32)), _x52))
mark#(g(f(h(f(g(g(g(h(g(_x261, _x262)), active(_x202)), _x152), active(_x112)), _x102)), _x32), mark(active(mark(active(_x42)))))) → active#(g(active(f(active(h(f(active(g(active(g(active(g(mark(g(active(g(_x261, _x262)), active(g(_x261, _x262)))), _x202)), _x152)), _x112)), _x102))), _x32)), _x42))

Problem 3: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

h#(mark(X))h#(X)h#(active(X))h#(X)

Rewrite Rules

active(h(X))mark(g(X, X))active(g(a, X))mark(f(b, X))
active(f(X, X))mark(h(a))active(a)mark(b)
mark(h(X))active(h(mark(X)))mark(g(X1, X2))active(g(mark(X1), X2))
mark(a)active(a)mark(f(X1, X2))active(f(mark(X1), X2))
mark(b)active(b)h(mark(X))h(X)
h(active(X))h(X)g(mark(X1), X2)g(X1, X2)
g(X1, mark(X2))g(X1, X2)g(active(X1), X2)g(X1, X2)
g(X1, active(X2))g(X1, X2)f(mark(X1), X2)f(X1, X2)
f(X1, mark(X2))f(X1, X2)f(active(X1), X2)f(X1, X2)
f(X1, active(X2))f(X1, X2)

Original Signature

Termination of terms over the following signature is verified: f, g, b, a, active, mark, h

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

h#(mark(X))h#(X)h#(active(X))h#(X)

Problem 4: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

f#(X1, mark(X2))f#(X1, X2)f#(X1, active(X2))f#(X1, X2)
f#(active(X1), X2)f#(X1, X2)f#(mark(X1), X2)f#(X1, X2)

Rewrite Rules

active(h(X))mark(g(X, X))active(g(a, X))mark(f(b, X))
active(f(X, X))mark(h(a))active(a)mark(b)
mark(h(X))active(h(mark(X)))mark(g(X1, X2))active(g(mark(X1), X2))
mark(a)active(a)mark(f(X1, X2))active(f(mark(X1), X2))
mark(b)active(b)h(mark(X))h(X)
h(active(X))h(X)g(mark(X1), X2)g(X1, X2)
g(X1, mark(X2))g(X1, X2)g(active(X1), X2)g(X1, X2)
g(X1, active(X2))g(X1, X2)f(mark(X1), X2)f(X1, X2)
f(X1, mark(X2))f(X1, X2)f(active(X1), X2)f(X1, X2)
f(X1, active(X2))f(X1, X2)

Original Signature

Termination of terms over the following signature is verified: f, g, b, a, active, mark, h

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

f#(active(X1), X2)f#(X1, X2)f#(mark(X1), X2)f#(X1, X2)

Problem 6: ReductionPairSAT



Dependency Pair Problem

Dependency Pairs

f#(X1, mark(X2))f#(X1, X2)f#(X1, active(X2))f#(X1, X2)

Rewrite Rules

active(h(X))mark(g(X, X))active(g(a, X))mark(f(b, X))
active(f(X, X))mark(h(a))active(a)mark(b)
mark(h(X))active(h(mark(X)))mark(g(X1, X2))active(g(mark(X1), X2))
mark(a)active(a)mark(f(X1, X2))active(f(mark(X1), X2))
mark(b)active(b)h(mark(X))h(X)
h(active(X))h(X)g(mark(X1), X2)g(X1, X2)
g(X1, mark(X2))g(X1, X2)g(active(X1), X2)g(X1, X2)
g(X1, active(X2))g(X1, X2)f(mark(X1), X2)f(X1, X2)
f(X1, mark(X2))f(X1, X2)f(active(X1), X2)f(X1, X2)
f(X1, active(X2))f(X1, X2)

Original Signature

Termination of terms over the following signature is verified: f, g, b, a, active, mark, h

Strategy


Function Precedence

active < f = g = f# = b = a = mark = h

Argument Filtering

f: all arguments are removed from f
g: all arguments are removed from g
f#: collapses to 2
b: all arguments are removed from b
a: all arguments are removed from a
active: 1
mark: collapses to 1
h: all arguments are removed from h

Status

f: multiset
g: multiset
b: multiset
a: multiset
active: multiset
h: multiset

Usable Rules

There are no usable rules.

The dependency pairs and usable rules are stronlgy conservative!

Eliminated dependency pairs

The following dependency pairs (at least) can be eliminated according to the given precedence.

f#(X1, active(X2)) → f#(X1, X2)

Problem 8: ReductionPairSAT



Dependency Pair Problem

Dependency Pairs

f#(X1, mark(X2))f#(X1, X2)

Rewrite Rules

active(h(X))mark(g(X, X))active(g(a, X))mark(f(b, X))
active(f(X, X))mark(h(a))active(a)mark(b)
mark(h(X))active(h(mark(X)))mark(g(X1, X2))active(g(mark(X1), X2))
mark(a)active(a)mark(f(X1, X2))active(f(mark(X1), X2))
mark(b)active(b)h(mark(X))h(X)
h(active(X))h(X)g(mark(X1), X2)g(X1, X2)
g(X1, mark(X2))g(X1, X2)g(active(X1), X2)g(X1, X2)
g(X1, active(X2))g(X1, X2)f(mark(X1), X2)f(X1, X2)
f(X1, mark(X2))f(X1, X2)f(active(X1), X2)f(X1, X2)
f(X1, active(X2))f(X1, X2)

Original Signature

Termination of terms over the following signature is verified: f, g, b, a, active, mark, h

Strategy


Function Precedence

f = g = f# = b = a = active = mark = h

Argument Filtering

f: 1 2
g: all arguments are removed from g
f#: 2
b: all arguments are removed from b
a: all arguments are removed from a
active: all arguments are removed from active
mark: 1
h: all arguments are removed from h

Status

f: lexicographic with permutation 1 → 2 2 → 1
g: multiset
f#: multiset
b: multiset
a: multiset
active: multiset
mark: multiset
h: multiset

Usable Rules

There are no usable rules.

The dependency pairs and usable rules are stronlgy conservative!

Eliminated dependency pairs

The following dependency pairs (at least) can be eliminated according to the given precedence.

f#(X1, mark(X2)) → f#(X1, X2)

Problem 5: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

g#(X1, mark(X2))g#(X1, X2)g#(X1, active(X2))g#(X1, X2)
g#(active(X1), X2)g#(X1, X2)g#(mark(X1), X2)g#(X1, X2)

Rewrite Rules

active(h(X))mark(g(X, X))active(g(a, X))mark(f(b, X))
active(f(X, X))mark(h(a))active(a)mark(b)
mark(h(X))active(h(mark(X)))mark(g(X1, X2))active(g(mark(X1), X2))
mark(a)active(a)mark(f(X1, X2))active(f(mark(X1), X2))
mark(b)active(b)h(mark(X))h(X)
h(active(X))h(X)g(mark(X1), X2)g(X1, X2)
g(X1, mark(X2))g(X1, X2)g(active(X1), X2)g(X1, X2)
g(X1, active(X2))g(X1, X2)f(mark(X1), X2)f(X1, X2)
f(X1, mark(X2))f(X1, X2)f(active(X1), X2)f(X1, X2)
f(X1, active(X2))f(X1, X2)

Original Signature

Termination of terms over the following signature is verified: f, g, b, a, active, mark, h

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

g#(mark(X1), X2)g#(X1, X2)g#(active(X1), X2)g#(X1, X2)

Problem 7: ReductionPairSAT



Dependency Pair Problem

Dependency Pairs

g#(X1, mark(X2))g#(X1, X2)g#(X1, active(X2))g#(X1, X2)

Rewrite Rules

active(h(X))mark(g(X, X))active(g(a, X))mark(f(b, X))
active(f(X, X))mark(h(a))active(a)mark(b)
mark(h(X))active(h(mark(X)))mark(g(X1, X2))active(g(mark(X1), X2))
mark(a)active(a)mark(f(X1, X2))active(f(mark(X1), X2))
mark(b)active(b)h(mark(X))h(X)
h(active(X))h(X)g(mark(X1), X2)g(X1, X2)
g(X1, mark(X2))g(X1, X2)g(active(X1), X2)g(X1, X2)
g(X1, active(X2))g(X1, X2)f(mark(X1), X2)f(X1, X2)
f(X1, mark(X2))f(X1, X2)f(active(X1), X2)f(X1, X2)
f(X1, active(X2))f(X1, X2)

Original Signature

Termination of terms over the following signature is verified: f, g, b, a, active, mark, h

Strategy


Function Precedence

active = mark < f = g = b = a = g# = h

Argument Filtering

f: 1 2
g: 1 2
b: all arguments are removed from b
a: all arguments are removed from a
active: 1
mark: collapses to 1
g#: collapses to 2
h: all arguments are removed from h

Status

f: lexicographic with permutation 1 → 1 2 → 2
g: lexicographic with permutation 1 → 1 2 → 2
b: multiset
a: multiset
active: multiset
h: multiset

Usable Rules

There are no usable rules.

The dependency pairs and usable rules are stronlgy conservative!

Eliminated dependency pairs

The following dependency pairs (at least) can be eliminated according to the given precedence.

g#(X1, active(X2)) → g#(X1, X2)

Problem 9: ReductionPairSAT



Dependency Pair Problem

Dependency Pairs

g#(X1, mark(X2))g#(X1, X2)

Rewrite Rules

active(h(X))mark(g(X, X))active(g(a, X))mark(f(b, X))
active(f(X, X))mark(h(a))active(a)mark(b)
mark(h(X))active(h(mark(X)))mark(g(X1, X2))active(g(mark(X1), X2))
mark(a)active(a)mark(f(X1, X2))active(f(mark(X1), X2))
mark(b)active(b)h(mark(X))h(X)
h(active(X))h(X)g(mark(X1), X2)g(X1, X2)
g(X1, mark(X2))g(X1, X2)g(active(X1), X2)g(X1, X2)
g(X1, active(X2))g(X1, X2)f(mark(X1), X2)f(X1, X2)
f(X1, mark(X2))f(X1, X2)f(active(X1), X2)f(X1, X2)
f(X1, active(X2))f(X1, X2)

Original Signature

Termination of terms over the following signature is verified: f, g, b, a, active, mark, h

Strategy


Function Precedence

mark < f = g = b = a = active = g# = h

Argument Filtering

f: all arguments are removed from f
g: 2
b: all arguments are removed from b
a: all arguments are removed from a
active: all arguments are removed from active
mark: 1
g#: collapses to 2
h: all arguments are removed from h

Status

f: multiset
g: lexicographic with permutation 2 → 1
b: multiset
a: multiset
active: multiset
mark: multiset
h: multiset

Usable Rules

There are no usable rules.

The dependency pairs and usable rules are stronlgy conservative!

Eliminated dependency pairs

The following dependency pairs (at least) can be eliminated according to the given precedence.

g#(X1, mark(X2)) → g#(X1, X2)