YES

The TRS could be proven terminating. The proof took 43091 ms.

The following DP Processors were used


Problem 1 was processed with processor DependencyGraph (739ms).
 | – Problem 2 was processed with processor SubtermCriterion (0ms).
 |    | – Problem 9 was processed with processor SubtermCriterion (0ms).
 | – Problem 3 was processed with processor SubtermCriterion (1ms).
 |    | – Problem 10 was processed with processor SubtermCriterion (1ms).
 | – Problem 4 was processed with processor SubtermCriterion (0ms).
 | – Problem 5 was processed with processor PolynomialLinearRange4 (295ms).
 |    | – Problem 12 was processed with processor PolynomialLinearRange4 (349ms).
 |    |    | – Problem 13 was processed with processor PolynomialLinearRange4 (237ms).
 |    |    |    | – Problem 14 was processed with processor PolynomialLinearRange4 (227ms).
 |    |    |    |    | – Problem 15 was processed with processor PolynomialLinearRange4 (203ms).
 |    |    |    |    |    | – Problem 16 was processed with processor PolynomialLinearRange4 (235ms).
 |    |    |    |    |    |    | – Problem 17 was processed with processor PolynomialLinearRange4 (163ms).
 |    |    |    |    |    |    |    | – Problem 18 was processed with processor PolynomialLinearRange4 (146ms).
 |    |    |    |    |    |    |    |    | – Problem 19 was processed with processor DependencyGraph (2ms).
 |    |    |    |    |    |    |    |    |    | – Problem 20 was processed with processor PolynomialLinearRange4 (9ms).
 |    |    |    |    |    |    |    |    |    |    | – Problem 21 was processed with processor PolynomialLinearRange4 (8ms).
 | – Problem 6 was processed with processor SubtermCriterion (1ms).
 | – Problem 7 was processed with processor SubtermCriterion (0ms).
 | – Problem 8 was processed with processor SubtermCriterion (1ms).
 |    | – Problem 11 was processed with processor SubtermCriterion (0ms).

Problem 1: DependencyGraph



Dependency Pair Problem

Dependency Pairs

mark#(cons(X1, X2))active#(cons(mark(X1), X2))mark#(zWadr(X1, X2))mark#(X2)
active#(app(nil, YS))mark#(YS)mark#(s(X))s#(mark(X))
app#(mark(X1), X2)app#(X1, X2)active#(zWadr(nil, YS))mark#(nil)
active#(prefix(L))mark#(cons(nil, zWadr(L, prefix(L))))active#(zWadr(cons(X, XS), cons(Y, YS)))cons#(X, nil)
active#(prefix(L))prefix#(L)mark#(s(X))mark#(X)
mark#(app(X1, X2))active#(app(mark(X1), mark(X2)))active#(app(cons(X, XS), YS))cons#(X, app(XS, YS))
active#(zWadr(cons(X, XS), cons(Y, YS)))zWadr#(XS, YS)mark#(from(X))mark#(X)
mark#(cons(X1, X2))mark#(X1)mark#(app(X1, X2))app#(mark(X1), mark(X2))
active#(from(X))mark#(cons(X, from(s(X))))active#(app(cons(X, XS), YS))app#(XS, YS)
zWadr#(X1, mark(X2))zWadr#(X1, X2)active#(from(X))s#(X)
cons#(X1, active(X2))cons#(X1, X2)active#(zWadr(cons(X, XS), cons(Y, YS)))app#(Y, cons(X, nil))
zWadr#(active(X1), X2)zWadr#(X1, X2)mark#(from(X))from#(mark(X))
prefix#(mark(X))prefix#(X)mark#(prefix(X))active#(prefix(mark(X)))
zWadr#(mark(X1), X2)zWadr#(X1, X2)mark#(from(X))active#(from(mark(X)))
active#(zWadr(cons(X, XS), cons(Y, YS)))cons#(app(Y, cons(X, nil)), zWadr(XS, YS))from#(active(X))from#(X)
mark#(zWadr(X1, X2))zWadr#(mark(X1), mark(X2))mark#(zWadr(X1, X2))active#(zWadr(mark(X1), mark(X2)))
mark#(app(X1, X2))mark#(X2)app#(X1, active(X2))app#(X1, X2)
active#(prefix(L))zWadr#(L, prefix(L))prefix#(active(X))prefix#(X)
cons#(mark(X1), X2)cons#(X1, X2)zWadr#(X1, active(X2))zWadr#(X1, X2)
from#(mark(X))from#(X)mark#(prefix(X))mark#(X)
mark#(nil)active#(nil)mark#(app(X1, X2))mark#(X1)
active#(from(X))cons#(X, from(s(X)))cons#(X1, mark(X2))cons#(X1, X2)
app#(X1, mark(X2))app#(X1, X2)mark#(cons(X1, X2))cons#(mark(X1), X2)
mark#(s(X))active#(s(mark(X)))active#(zWadr(XS, nil))mark#(nil)
cons#(active(X1), X2)cons#(X1, X2)mark#(prefix(X))prefix#(mark(X))
s#(mark(X))s#(X)active#(zWadr(cons(X, XS), cons(Y, YS)))mark#(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active#(prefix(L))cons#(nil, zWadr(L, prefix(L)))mark#(zWadr(X1, X2))mark#(X1)
s#(active(X))s#(X)active#(app(cons(X, XS), YS))mark#(cons(X, app(XS, YS)))
app#(active(X1), X2)app#(X1, X2)active#(from(X))from#(s(X))

Rewrite Rules

active(app(nil, YS))mark(YS)active(app(cons(X, XS), YS))mark(cons(X, app(XS, YS)))
active(from(X))mark(cons(X, from(s(X))))active(zWadr(nil, YS))mark(nil)
active(zWadr(XS, nil))mark(nil)active(zWadr(cons(X, XS), cons(Y, YS)))mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L))mark(cons(nil, zWadr(L, prefix(L))))mark(app(X1, X2))active(app(mark(X1), mark(X2)))
mark(nil)active(nil)mark(cons(X1, X2))active(cons(mark(X1), X2))
mark(from(X))active(from(mark(X)))mark(s(X))active(s(mark(X)))
mark(zWadr(X1, X2))active(zWadr(mark(X1), mark(X2)))mark(prefix(X))active(prefix(mark(X)))
app(mark(X1), X2)app(X1, X2)app(X1, mark(X2))app(X1, X2)
app(active(X1), X2)app(X1, X2)app(X1, active(X2))app(X1, X2)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
from(mark(X))from(X)from(active(X))from(X)
s(mark(X))s(X)s(active(X))s(X)
zWadr(mark(X1), X2)zWadr(X1, X2)zWadr(X1, mark(X2))zWadr(X1, X2)
zWadr(active(X1), X2)zWadr(X1, X2)zWadr(X1, active(X2))zWadr(X1, X2)
prefix(mark(X))prefix(X)prefix(active(X))prefix(X)

Original Signature

Termination of terms over the following signature is verified: app, zWadr, s, prefix, active, mark, from, nil, cons

Strategy


The following SCCs where found

prefix#(mark(X)) → prefix#(X)prefix#(active(X)) → prefix#(X)

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

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

from#(active(X)) → from#(X)from#(mark(X)) → from#(X)

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

s#(mark(X)) → s#(X)s#(active(X)) → s#(X)

mark#(cons(X1, X2)) → active#(cons(mark(X1), X2))mark#(s(X)) → active#(s(mark(X)))
mark#(zWadr(X1, X2)) → mark#(X2)mark#(from(X)) → mark#(X)
mark#(cons(X1, X2)) → mark#(X1)active#(from(X)) → mark#(cons(X, from(s(X))))
active#(app(nil, YS)) → mark#(YS)mark#(prefix(X)) → mark#(X)
active#(prefix(L)) → mark#(cons(nil, zWadr(L, prefix(L))))mark#(prefix(X)) → active#(prefix(mark(X)))
mark#(from(X)) → active#(from(mark(X)))active#(zWadr(cons(X, XS), cons(Y, YS))) → mark#(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
mark#(app(X1, X2)) → mark#(X1)mark#(zWadr(X1, X2)) → mark#(X1)
mark#(s(X)) → mark#(X)active#(app(cons(X, XS), YS)) → mark#(cons(X, app(XS, YS)))
mark#(app(X1, X2)) → active#(app(mark(X1), mark(X2)))mark#(zWadr(X1, X2)) → active#(zWadr(mark(X1), mark(X2)))
mark#(app(X1, X2)) → mark#(X2)

Problem 2: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

active(app(nil, YS))mark(YS)active(app(cons(X, XS), YS))mark(cons(X, app(XS, YS)))
active(from(X))mark(cons(X, from(s(X))))active(zWadr(nil, YS))mark(nil)
active(zWadr(XS, nil))mark(nil)active(zWadr(cons(X, XS), cons(Y, YS)))mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L))mark(cons(nil, zWadr(L, prefix(L))))mark(app(X1, X2))active(app(mark(X1), mark(X2)))
mark(nil)active(nil)mark(cons(X1, X2))active(cons(mark(X1), X2))
mark(from(X))active(from(mark(X)))mark(s(X))active(s(mark(X)))
mark(zWadr(X1, X2))active(zWadr(mark(X1), mark(X2)))mark(prefix(X))active(prefix(mark(X)))
app(mark(X1), X2)app(X1, X2)app(X1, mark(X2))app(X1, X2)
app(active(X1), X2)app(X1, X2)app(X1, active(X2))app(X1, X2)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
from(mark(X))from(X)from(active(X))from(X)
s(mark(X))s(X)s(active(X))s(X)
zWadr(mark(X1), X2)zWadr(X1, X2)zWadr(X1, mark(X2))zWadr(X1, X2)
zWadr(active(X1), X2)zWadr(X1, X2)zWadr(X1, active(X2))zWadr(X1, X2)
prefix(mark(X))prefix(X)prefix(active(X))prefix(X)

Original Signature

Termination of terms over the following signature is verified: app, zWadr, s, prefix, active, mark, from, nil, cons

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

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

Problem 9: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

active(app(nil, YS))mark(YS)active(app(cons(X, XS), YS))mark(cons(X, app(XS, YS)))
active(from(X))mark(cons(X, from(s(X))))active(zWadr(nil, YS))mark(nil)
active(zWadr(XS, nil))mark(nil)active(zWadr(cons(X, XS), cons(Y, YS)))mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L))mark(cons(nil, zWadr(L, prefix(L))))mark(app(X1, X2))active(app(mark(X1), mark(X2)))
mark(nil)active(nil)mark(cons(X1, X2))active(cons(mark(X1), X2))
mark(from(X))active(from(mark(X)))mark(s(X))active(s(mark(X)))
mark(zWadr(X1, X2))active(zWadr(mark(X1), mark(X2)))mark(prefix(X))active(prefix(mark(X)))
app(mark(X1), X2)app(X1, X2)app(X1, mark(X2))app(X1, X2)
app(active(X1), X2)app(X1, X2)app(X1, active(X2))app(X1, X2)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
from(mark(X))from(X)from(active(X))from(X)
s(mark(X))s(X)s(active(X))s(X)
zWadr(mark(X1), X2)zWadr(X1, X2)zWadr(X1, mark(X2))zWadr(X1, X2)
zWadr(active(X1), X2)zWadr(X1, X2)zWadr(X1, active(X2))zWadr(X1, X2)
prefix(mark(X))prefix(X)prefix(active(X))prefix(X)

Original Signature

Termination of terms over the following signature is verified: app, zWadr, s, prefix, active, mark, from, cons, nil

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

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

Problem 3: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

active(app(nil, YS))mark(YS)active(app(cons(X, XS), YS))mark(cons(X, app(XS, YS)))
active(from(X))mark(cons(X, from(s(X))))active(zWadr(nil, YS))mark(nil)
active(zWadr(XS, nil))mark(nil)active(zWadr(cons(X, XS), cons(Y, YS)))mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L))mark(cons(nil, zWadr(L, prefix(L))))mark(app(X1, X2))active(app(mark(X1), mark(X2)))
mark(nil)active(nil)mark(cons(X1, X2))active(cons(mark(X1), X2))
mark(from(X))active(from(mark(X)))mark(s(X))active(s(mark(X)))
mark(zWadr(X1, X2))active(zWadr(mark(X1), mark(X2)))mark(prefix(X))active(prefix(mark(X)))
app(mark(X1), X2)app(X1, X2)app(X1, mark(X2))app(X1, X2)
app(active(X1), X2)app(X1, X2)app(X1, active(X2))app(X1, X2)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
from(mark(X))from(X)from(active(X))from(X)
s(mark(X))s(X)s(active(X))s(X)
zWadr(mark(X1), X2)zWadr(X1, X2)zWadr(X1, mark(X2))zWadr(X1, X2)
zWadr(active(X1), X2)zWadr(X1, X2)zWadr(X1, active(X2))zWadr(X1, X2)
prefix(mark(X))prefix(X)prefix(active(X))prefix(X)

Original Signature

Termination of terms over the following signature is verified: app, zWadr, s, prefix, active, mark, from, nil, cons

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

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

Problem 10: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

active(app(nil, YS))mark(YS)active(app(cons(X, XS), YS))mark(cons(X, app(XS, YS)))
active(from(X))mark(cons(X, from(s(X))))active(zWadr(nil, YS))mark(nil)
active(zWadr(XS, nil))mark(nil)active(zWadr(cons(X, XS), cons(Y, YS)))mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L))mark(cons(nil, zWadr(L, prefix(L))))mark(app(X1, X2))active(app(mark(X1), mark(X2)))
mark(nil)active(nil)mark(cons(X1, X2))active(cons(mark(X1), X2))
mark(from(X))active(from(mark(X)))mark(s(X))active(s(mark(X)))
mark(zWadr(X1, X2))active(zWadr(mark(X1), mark(X2)))mark(prefix(X))active(prefix(mark(X)))
app(mark(X1), X2)app(X1, X2)app(X1, mark(X2))app(X1, X2)
app(active(X1), X2)app(X1, X2)app(X1, active(X2))app(X1, X2)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
from(mark(X))from(X)from(active(X))from(X)
s(mark(X))s(X)s(active(X))s(X)
zWadr(mark(X1), X2)zWadr(X1, X2)zWadr(X1, mark(X2))zWadr(X1, X2)
zWadr(active(X1), X2)zWadr(X1, X2)zWadr(X1, active(X2))zWadr(X1, X2)
prefix(mark(X))prefix(X)prefix(active(X))prefix(X)

Original Signature

Termination of terms over the following signature is verified: app, zWadr, s, prefix, active, mark, from, cons, nil

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

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

Problem 4: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

prefix#(mark(X))prefix#(X)prefix#(active(X))prefix#(X)

Rewrite Rules

active(app(nil, YS))mark(YS)active(app(cons(X, XS), YS))mark(cons(X, app(XS, YS)))
active(from(X))mark(cons(X, from(s(X))))active(zWadr(nil, YS))mark(nil)
active(zWadr(XS, nil))mark(nil)active(zWadr(cons(X, XS), cons(Y, YS)))mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L))mark(cons(nil, zWadr(L, prefix(L))))mark(app(X1, X2))active(app(mark(X1), mark(X2)))
mark(nil)active(nil)mark(cons(X1, X2))active(cons(mark(X1), X2))
mark(from(X))active(from(mark(X)))mark(s(X))active(s(mark(X)))
mark(zWadr(X1, X2))active(zWadr(mark(X1), mark(X2)))mark(prefix(X))active(prefix(mark(X)))
app(mark(X1), X2)app(X1, X2)app(X1, mark(X2))app(X1, X2)
app(active(X1), X2)app(X1, X2)app(X1, active(X2))app(X1, X2)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
from(mark(X))from(X)from(active(X))from(X)
s(mark(X))s(X)s(active(X))s(X)
zWadr(mark(X1), X2)zWadr(X1, X2)zWadr(X1, mark(X2))zWadr(X1, X2)
zWadr(active(X1), X2)zWadr(X1, X2)zWadr(X1, active(X2))zWadr(X1, X2)
prefix(mark(X))prefix(X)prefix(active(X))prefix(X)

Original Signature

Termination of terms over the following signature is verified: app, zWadr, s, prefix, active, mark, from, nil, cons

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

prefix#(mark(X))prefix#(X)prefix#(active(X))prefix#(X)

Problem 5: PolynomialLinearRange4



Dependency Pair Problem

Dependency Pairs

mark#(cons(X1, X2))active#(cons(mark(X1), X2))mark#(s(X))active#(s(mark(X)))
mark#(zWadr(X1, X2))mark#(X2)mark#(from(X))mark#(X)
mark#(cons(X1, X2))mark#(X1)active#(app(nil, YS))mark#(YS)
active#(from(X))mark#(cons(X, from(s(X))))mark#(prefix(X))mark#(X)
active#(prefix(L))mark#(cons(nil, zWadr(L, prefix(L))))mark#(prefix(X))active#(prefix(mark(X)))
active#(zWadr(cons(X, XS), cons(Y, YS)))mark#(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))mark#(from(X))active#(from(mark(X)))
mark#(app(X1, X2))mark#(X1)mark#(zWadr(X1, X2))mark#(X1)
mark#(s(X))mark#(X)active#(app(cons(X, XS), YS))mark#(cons(X, app(XS, YS)))
mark#(app(X1, X2))active#(app(mark(X1), mark(X2)))mark#(zWadr(X1, X2))active#(zWadr(mark(X1), mark(X2)))
mark#(app(X1, X2))mark#(X2)

Rewrite Rules

active(app(nil, YS))mark(YS)active(app(cons(X, XS), YS))mark(cons(X, app(XS, YS)))
active(from(X))mark(cons(X, from(s(X))))active(zWadr(nil, YS))mark(nil)
active(zWadr(XS, nil))mark(nil)active(zWadr(cons(X, XS), cons(Y, YS)))mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L))mark(cons(nil, zWadr(L, prefix(L))))mark(app(X1, X2))active(app(mark(X1), mark(X2)))
mark(nil)active(nil)mark(cons(X1, X2))active(cons(mark(X1), X2))
mark(from(X))active(from(mark(X)))mark(s(X))active(s(mark(X)))
mark(zWadr(X1, X2))active(zWadr(mark(X1), mark(X2)))mark(prefix(X))active(prefix(mark(X)))
app(mark(X1), X2)app(X1, X2)app(X1, mark(X2))app(X1, X2)
app(active(X1), X2)app(X1, X2)app(X1, active(X2))app(X1, X2)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
from(mark(X))from(X)from(active(X))from(X)
s(mark(X))s(X)s(active(X))s(X)
zWadr(mark(X1), X2)zWadr(X1, X2)zWadr(X1, mark(X2))zWadr(X1, X2)
zWadr(active(X1), X2)zWadr(X1, X2)zWadr(X1, active(X2))zWadr(X1, X2)
prefix(mark(X))prefix(X)prefix(active(X))prefix(X)

Original Signature

Termination of terms over the following signature is verified: app, zWadr, s, prefix, active, mark, from, nil, cons

Strategy


Polynomial Interpretation

Standard Usable rules

cons(active(X1), X2)cons(X1, X2)from(mark(X))from(X)
zWadr(X1, mark(X2))zWadr(X1, X2)mark(s(X))active(s(mark(X)))
prefix(active(X))prefix(X)zWadr(active(X1), X2)zWadr(X1, X2)
cons(X1, mark(X2))cons(X1, X2)app(X1, mark(X2))app(X1, X2)
active(zWadr(nil, YS))mark(nil)active(zWadr(XS, nil))mark(nil)
mark(from(X))active(from(mark(X)))app(X1, active(X2))app(X1, X2)
app(active(X1), X2)app(X1, X2)active(from(X))mark(cons(X, from(s(X))))
mark(zWadr(X1, X2))active(zWadr(mark(X1), mark(X2)))mark(nil)active(nil)
s(active(X))s(X)prefix(mark(X))prefix(X)
active(app(nil, YS))mark(YS)cons(X1, active(X2))cons(X1, X2)
from(active(X))from(X)mark(app(X1, X2))active(app(mark(X1), mark(X2)))
zWadr(X1, active(X2))zWadr(X1, X2)zWadr(mark(X1), X2)zWadr(X1, X2)
mark(prefix(X))active(prefix(mark(X)))active(app(cons(X, XS), YS))mark(cons(X, app(XS, YS)))
cons(mark(X1), X2)cons(X1, X2)active(zWadr(cons(X, XS), cons(Y, YS)))mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L))mark(cons(nil, zWadr(L, prefix(L))))mark(cons(X1, X2))active(cons(mark(X1), X2))
s(mark(X))s(X)app(mark(X1), X2)app(X1, X2)

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

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

Problem 12: PolynomialLinearRange4



Dependency Pair Problem

Dependency Pairs

mark#(s(X))active#(s(mark(X)))mark#(zWadr(X1, X2))mark#(X2)
mark#(from(X))mark#(X)mark#(cons(X1, X2))mark#(X1)
active#(app(nil, YS))mark#(YS)active#(from(X))mark#(cons(X, from(s(X))))
mark#(prefix(X))mark#(X)active#(prefix(L))mark#(cons(nil, zWadr(L, prefix(L))))
mark#(prefix(X))active#(prefix(mark(X)))active#(zWadr(cons(X, XS), cons(Y, YS)))mark#(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
mark#(from(X))active#(from(mark(X)))mark#(app(X1, X2))mark#(X1)
mark#(zWadr(X1, X2))mark#(X1)mark#(s(X))mark#(X)
active#(app(cons(X, XS), YS))mark#(cons(X, app(XS, YS)))mark#(app(X1, X2))active#(app(mark(X1), mark(X2)))
mark#(zWadr(X1, X2))active#(zWadr(mark(X1), mark(X2)))mark#(app(X1, X2))mark#(X2)

Rewrite Rules

active(app(nil, YS))mark(YS)active(app(cons(X, XS), YS))mark(cons(X, app(XS, YS)))
active(from(X))mark(cons(X, from(s(X))))active(zWadr(nil, YS))mark(nil)
active(zWadr(XS, nil))mark(nil)active(zWadr(cons(X, XS), cons(Y, YS)))mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L))mark(cons(nil, zWadr(L, prefix(L))))mark(app(X1, X2))active(app(mark(X1), mark(X2)))
mark(nil)active(nil)mark(cons(X1, X2))active(cons(mark(X1), X2))
mark(from(X))active(from(mark(X)))mark(s(X))active(s(mark(X)))
mark(zWadr(X1, X2))active(zWadr(mark(X1), mark(X2)))mark(prefix(X))active(prefix(mark(X)))
app(mark(X1), X2)app(X1, X2)app(X1, mark(X2))app(X1, X2)
app(active(X1), X2)app(X1, X2)app(X1, active(X2))app(X1, X2)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
from(mark(X))from(X)from(active(X))from(X)
s(mark(X))s(X)s(active(X))s(X)
zWadr(mark(X1), X2)zWadr(X1, X2)zWadr(X1, mark(X2))zWadr(X1, X2)
zWadr(active(X1), X2)zWadr(X1, X2)zWadr(X1, active(X2))zWadr(X1, X2)
prefix(mark(X))prefix(X)prefix(active(X))prefix(X)

Original Signature

Termination of terms over the following signature is verified: app, zWadr, s, prefix, active, mark, from, cons, nil

Strategy


Polynomial Interpretation

Standard Usable rules

cons(active(X1), X2)cons(X1, X2)from(mark(X))from(X)
zWadr(X1, mark(X2))zWadr(X1, X2)mark(s(X))active(s(mark(X)))
prefix(active(X))prefix(X)zWadr(active(X1), X2)zWadr(X1, X2)
cons(X1, mark(X2))cons(X1, X2)app(X1, mark(X2))app(X1, X2)
active(zWadr(nil, YS))mark(nil)active(zWadr(XS, nil))mark(nil)
mark(from(X))active(from(mark(X)))app(X1, active(X2))app(X1, X2)
app(active(X1), X2)app(X1, X2)active(from(X))mark(cons(X, from(s(X))))
mark(zWadr(X1, X2))active(zWadr(mark(X1), mark(X2)))mark(nil)active(nil)
s(active(X))s(X)prefix(mark(X))prefix(X)
active(app(nil, YS))mark(YS)cons(X1, active(X2))cons(X1, X2)
from(active(X))from(X)mark(app(X1, X2))active(app(mark(X1), mark(X2)))
zWadr(X1, active(X2))zWadr(X1, X2)zWadr(mark(X1), X2)zWadr(X1, X2)
mark(prefix(X))active(prefix(mark(X)))active(app(cons(X, XS), YS))mark(cons(X, app(XS, YS)))
cons(mark(X1), X2)cons(X1, X2)active(zWadr(cons(X, XS), cons(Y, YS)))mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L))mark(cons(nil, zWadr(L, prefix(L))))mark(cons(X1, X2))active(cons(mark(X1), X2))
s(mark(X))s(X)app(mark(X1), X2)app(X1, X2)

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

mark#(s(X))active#(s(mark(X)))

Problem 13: PolynomialLinearRange4



Dependency Pair Problem

Dependency Pairs

mark#(zWadr(X1, X2))mark#(X2)mark#(from(X))mark#(X)
mark#(cons(X1, X2))mark#(X1)active#(app(nil, YS))mark#(YS)
active#(from(X))mark#(cons(X, from(s(X))))mark#(prefix(X))mark#(X)
active#(prefix(L))mark#(cons(nil, zWadr(L, prefix(L))))mark#(prefix(X))active#(prefix(mark(X)))
active#(zWadr(cons(X, XS), cons(Y, YS)))mark#(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))mark#(from(X))active#(from(mark(X)))
mark#(app(X1, X2))mark#(X1)mark#(zWadr(X1, X2))mark#(X1)
mark#(s(X))mark#(X)active#(app(cons(X, XS), YS))mark#(cons(X, app(XS, YS)))
mark#(app(X1, X2))active#(app(mark(X1), mark(X2)))mark#(zWadr(X1, X2))active#(zWadr(mark(X1), mark(X2)))
mark#(app(X1, X2))mark#(X2)

Rewrite Rules

active(app(nil, YS))mark(YS)active(app(cons(X, XS), YS))mark(cons(X, app(XS, YS)))
active(from(X))mark(cons(X, from(s(X))))active(zWadr(nil, YS))mark(nil)
active(zWadr(XS, nil))mark(nil)active(zWadr(cons(X, XS), cons(Y, YS)))mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L))mark(cons(nil, zWadr(L, prefix(L))))mark(app(X1, X2))active(app(mark(X1), mark(X2)))
mark(nil)active(nil)mark(cons(X1, X2))active(cons(mark(X1), X2))
mark(from(X))active(from(mark(X)))mark(s(X))active(s(mark(X)))
mark(zWadr(X1, X2))active(zWadr(mark(X1), mark(X2)))mark(prefix(X))active(prefix(mark(X)))
app(mark(X1), X2)app(X1, X2)app(X1, mark(X2))app(X1, X2)
app(active(X1), X2)app(X1, X2)app(X1, active(X2))app(X1, X2)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
from(mark(X))from(X)from(active(X))from(X)
s(mark(X))s(X)s(active(X))s(X)
zWadr(mark(X1), X2)zWadr(X1, X2)zWadr(X1, mark(X2))zWadr(X1, X2)
zWadr(active(X1), X2)zWadr(X1, X2)zWadr(X1, active(X2))zWadr(X1, X2)
prefix(mark(X))prefix(X)prefix(active(X))prefix(X)

Original Signature

Termination of terms over the following signature is verified: app, zWadr, s, prefix, active, mark, from, nil, cons

Strategy


Polynomial Interpretation

Standard Usable rules

cons(active(X1), X2)cons(X1, X2)from(mark(X))from(X)
zWadr(X1, mark(X2))zWadr(X1, X2)mark(s(X))active(s(mark(X)))
prefix(active(X))prefix(X)zWadr(active(X1), X2)zWadr(X1, X2)
cons(X1, mark(X2))cons(X1, X2)app(X1, mark(X2))app(X1, X2)
active(zWadr(nil, YS))mark(nil)active(zWadr(XS, nil))mark(nil)
mark(from(X))active(from(mark(X)))app(X1, active(X2))app(X1, X2)
app(active(X1), X2)app(X1, X2)active(from(X))mark(cons(X, from(s(X))))
mark(zWadr(X1, X2))active(zWadr(mark(X1), mark(X2)))mark(nil)active(nil)
s(active(X))s(X)prefix(mark(X))prefix(X)
active(app(nil, YS))mark(YS)cons(X1, active(X2))cons(X1, X2)
from(active(X))from(X)mark(app(X1, X2))active(app(mark(X1), mark(X2)))
zWadr(X1, active(X2))zWadr(X1, X2)zWadr(mark(X1), X2)zWadr(X1, X2)
mark(prefix(X))active(prefix(mark(X)))active(app(cons(X, XS), YS))mark(cons(X, app(XS, YS)))
cons(mark(X1), X2)cons(X1, X2)active(zWadr(cons(X, XS), cons(Y, YS)))mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L))mark(cons(nil, zWadr(L, prefix(L))))mark(cons(X1, X2))active(cons(mark(X1), X2))
s(mark(X))s(X)app(mark(X1), X2)app(X1, X2)

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

active#(app(nil, YS))mark#(YS)mark#(prefix(X))mark#(X)

Problem 14: PolynomialLinearRange4



Dependency Pair Problem

Dependency Pairs

mark#(zWadr(X1, X2))mark#(X2)mark#(from(X))mark#(X)
mark#(cons(X1, X2))mark#(X1)active#(from(X))mark#(cons(X, from(s(X))))
active#(prefix(L))mark#(cons(nil, zWadr(L, prefix(L))))mark#(prefix(X))active#(prefix(mark(X)))
active#(zWadr(cons(X, XS), cons(Y, YS)))mark#(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))mark#(from(X))active#(from(mark(X)))
mark#(app(X1, X2))mark#(X1)mark#(zWadr(X1, X2))mark#(X1)
mark#(s(X))mark#(X)active#(app(cons(X, XS), YS))mark#(cons(X, app(XS, YS)))
mark#(app(X1, X2))active#(app(mark(X1), mark(X2)))mark#(zWadr(X1, X2))active#(zWadr(mark(X1), mark(X2)))
mark#(app(X1, X2))mark#(X2)

Rewrite Rules

active(app(nil, YS))mark(YS)active(app(cons(X, XS), YS))mark(cons(X, app(XS, YS)))
active(from(X))mark(cons(X, from(s(X))))active(zWadr(nil, YS))mark(nil)
active(zWadr(XS, nil))mark(nil)active(zWadr(cons(X, XS), cons(Y, YS)))mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L))mark(cons(nil, zWadr(L, prefix(L))))mark(app(X1, X2))active(app(mark(X1), mark(X2)))
mark(nil)active(nil)mark(cons(X1, X2))active(cons(mark(X1), X2))
mark(from(X))active(from(mark(X)))mark(s(X))active(s(mark(X)))
mark(zWadr(X1, X2))active(zWadr(mark(X1), mark(X2)))mark(prefix(X))active(prefix(mark(X)))
app(mark(X1), X2)app(X1, X2)app(X1, mark(X2))app(X1, X2)
app(active(X1), X2)app(X1, X2)app(X1, active(X2))app(X1, X2)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
from(mark(X))from(X)from(active(X))from(X)
s(mark(X))s(X)s(active(X))s(X)
zWadr(mark(X1), X2)zWadr(X1, X2)zWadr(X1, mark(X2))zWadr(X1, X2)
zWadr(active(X1), X2)zWadr(X1, X2)zWadr(X1, active(X2))zWadr(X1, X2)
prefix(mark(X))prefix(X)prefix(active(X))prefix(X)

Original Signature

Termination of terms over the following signature is verified: app, zWadr, s, prefix, active, mark, from, cons, nil

Strategy


Polynomial Interpretation

Standard Usable rules

cons(active(X1), X2)cons(X1, X2)from(mark(X))from(X)
zWadr(X1, mark(X2))zWadr(X1, X2)mark(s(X))active(s(mark(X)))
prefix(active(X))prefix(X)zWadr(active(X1), X2)zWadr(X1, X2)
cons(X1, mark(X2))cons(X1, X2)app(X1, mark(X2))app(X1, X2)
active(zWadr(nil, YS))mark(nil)active(zWadr(XS, nil))mark(nil)
mark(from(X))active(from(mark(X)))app(X1, active(X2))app(X1, X2)
app(active(X1), X2)app(X1, X2)active(from(X))mark(cons(X, from(s(X))))
mark(zWadr(X1, X2))active(zWadr(mark(X1), mark(X2)))mark(nil)active(nil)
s(active(X))s(X)prefix(mark(X))prefix(X)
active(app(nil, YS))mark(YS)cons(X1, active(X2))cons(X1, X2)
from(active(X))from(X)mark(app(X1, X2))active(app(mark(X1), mark(X2)))
zWadr(X1, active(X2))zWadr(X1, X2)zWadr(mark(X1), X2)zWadr(X1, X2)
mark(prefix(X))active(prefix(mark(X)))active(app(cons(X, XS), YS))mark(cons(X, app(XS, YS)))
cons(mark(X1), X2)cons(X1, X2)active(zWadr(cons(X, XS), cons(Y, YS)))mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L))mark(cons(nil, zWadr(L, prefix(L))))mark(cons(X1, X2))active(cons(mark(X1), X2))
s(mark(X))s(X)app(mark(X1), X2)app(X1, X2)

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

active#(prefix(L))mark#(cons(nil, zWadr(L, prefix(L))))

Problem 15: PolynomialLinearRange4



Dependency Pair Problem

Dependency Pairs

mark#(zWadr(X1, X2))mark#(X2)mark#(from(X))mark#(X)
mark#(cons(X1, X2))mark#(X1)active#(from(X))mark#(cons(X, from(s(X))))
mark#(prefix(X))active#(prefix(mark(X)))active#(zWadr(cons(X, XS), cons(Y, YS)))mark#(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
mark#(from(X))active#(from(mark(X)))mark#(app(X1, X2))mark#(X1)
mark#(zWadr(X1, X2))mark#(X1)mark#(s(X))mark#(X)
active#(app(cons(X, XS), YS))mark#(cons(X, app(XS, YS)))mark#(app(X1, X2))active#(app(mark(X1), mark(X2)))
mark#(zWadr(X1, X2))active#(zWadr(mark(X1), mark(X2)))mark#(app(X1, X2))mark#(X2)

Rewrite Rules

active(app(nil, YS))mark(YS)active(app(cons(X, XS), YS))mark(cons(X, app(XS, YS)))
active(from(X))mark(cons(X, from(s(X))))active(zWadr(nil, YS))mark(nil)
active(zWadr(XS, nil))mark(nil)active(zWadr(cons(X, XS), cons(Y, YS)))mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L))mark(cons(nil, zWadr(L, prefix(L))))mark(app(X1, X2))active(app(mark(X1), mark(X2)))
mark(nil)active(nil)mark(cons(X1, X2))active(cons(mark(X1), X2))
mark(from(X))active(from(mark(X)))mark(s(X))active(s(mark(X)))
mark(zWadr(X1, X2))active(zWadr(mark(X1), mark(X2)))mark(prefix(X))active(prefix(mark(X)))
app(mark(X1), X2)app(X1, X2)app(X1, mark(X2))app(X1, X2)
app(active(X1), X2)app(X1, X2)app(X1, active(X2))app(X1, X2)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
from(mark(X))from(X)from(active(X))from(X)
s(mark(X))s(X)s(active(X))s(X)
zWadr(mark(X1), X2)zWadr(X1, X2)zWadr(X1, mark(X2))zWadr(X1, X2)
zWadr(active(X1), X2)zWadr(X1, X2)zWadr(X1, active(X2))zWadr(X1, X2)
prefix(mark(X))prefix(X)prefix(active(X))prefix(X)

Original Signature

Termination of terms over the following signature is verified: app, zWadr, s, prefix, active, mark, from, nil, cons

Strategy


Polynomial Interpretation

Standard Usable rules

cons(active(X1), X2)cons(X1, X2)from(mark(X))from(X)
zWadr(X1, mark(X2))zWadr(X1, X2)mark(s(X))active(s(mark(X)))
prefix(active(X))prefix(X)zWadr(active(X1), X2)zWadr(X1, X2)
cons(X1, mark(X2))cons(X1, X2)app(X1, mark(X2))app(X1, X2)
active(zWadr(nil, YS))mark(nil)active(zWadr(XS, nil))mark(nil)
mark(from(X))active(from(mark(X)))app(X1, active(X2))app(X1, X2)
app(active(X1), X2)app(X1, X2)active(from(X))mark(cons(X, from(s(X))))
mark(zWadr(X1, X2))active(zWadr(mark(X1), mark(X2)))mark(nil)active(nil)
s(active(X))s(X)prefix(mark(X))prefix(X)
active(app(nil, YS))mark(YS)cons(X1, active(X2))cons(X1, X2)
from(active(X))from(X)mark(app(X1, X2))active(app(mark(X1), mark(X2)))
zWadr(X1, active(X2))zWadr(X1, X2)zWadr(mark(X1), X2)zWadr(X1, X2)
mark(prefix(X))active(prefix(mark(X)))active(app(cons(X, XS), YS))mark(cons(X, app(XS, YS)))
cons(mark(X1), X2)cons(X1, X2)active(zWadr(cons(X, XS), cons(Y, YS)))mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L))mark(cons(nil, zWadr(L, prefix(L))))mark(cons(X1, X2))active(cons(mark(X1), X2))
s(mark(X))s(X)app(mark(X1), X2)app(X1, X2)

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

mark#(prefix(X))active#(prefix(mark(X)))

Problem 16: PolynomialLinearRange4



Dependency Pair Problem

Dependency Pairs

mark#(zWadr(X1, X2))mark#(X2)mark#(from(X))mark#(X)
mark#(cons(X1, X2))mark#(X1)active#(from(X))mark#(cons(X, from(s(X))))
active#(zWadr(cons(X, XS), cons(Y, YS)))mark#(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))mark#(from(X))active#(from(mark(X)))
mark#(app(X1, X2))mark#(X1)mark#(zWadr(X1, X2))mark#(X1)
mark#(s(X))mark#(X)active#(app(cons(X, XS), YS))mark#(cons(X, app(XS, YS)))
mark#(app(X1, X2))active#(app(mark(X1), mark(X2)))mark#(zWadr(X1, X2))active#(zWadr(mark(X1), mark(X2)))
mark#(app(X1, X2))mark#(X2)

Rewrite Rules

active(app(nil, YS))mark(YS)active(app(cons(X, XS), YS))mark(cons(X, app(XS, YS)))
active(from(X))mark(cons(X, from(s(X))))active(zWadr(nil, YS))mark(nil)
active(zWadr(XS, nil))mark(nil)active(zWadr(cons(X, XS), cons(Y, YS)))mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L))mark(cons(nil, zWadr(L, prefix(L))))mark(app(X1, X2))active(app(mark(X1), mark(X2)))
mark(nil)active(nil)mark(cons(X1, X2))active(cons(mark(X1), X2))
mark(from(X))active(from(mark(X)))mark(s(X))active(s(mark(X)))
mark(zWadr(X1, X2))active(zWadr(mark(X1), mark(X2)))mark(prefix(X))active(prefix(mark(X)))
app(mark(X1), X2)app(X1, X2)app(X1, mark(X2))app(X1, X2)
app(active(X1), X2)app(X1, X2)app(X1, active(X2))app(X1, X2)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
from(mark(X))from(X)from(active(X))from(X)
s(mark(X))s(X)s(active(X))s(X)
zWadr(mark(X1), X2)zWadr(X1, X2)zWadr(X1, mark(X2))zWadr(X1, X2)
zWadr(active(X1), X2)zWadr(X1, X2)zWadr(X1, active(X2))zWadr(X1, X2)
prefix(mark(X))prefix(X)prefix(active(X))prefix(X)

Original Signature

Termination of terms over the following signature is verified: app, zWadr, s, prefix, active, mark, from, cons, nil

Strategy


Polynomial Interpretation

Standard Usable rules

cons(active(X1), X2)cons(X1, X2)from(mark(X))from(X)
zWadr(X1, mark(X2))zWadr(X1, X2)mark(s(X))active(s(mark(X)))
prefix(active(X))prefix(X)zWadr(active(X1), X2)zWadr(X1, X2)
cons(X1, mark(X2))cons(X1, X2)app(X1, mark(X2))app(X1, X2)
active(zWadr(XS, nil))mark(nil)active(zWadr(nil, YS))mark(nil)
mark(from(X))active(from(mark(X)))app(X1, active(X2))app(X1, X2)
app(active(X1), X2)app(X1, X2)active(from(X))mark(cons(X, from(s(X))))
mark(zWadr(X1, X2))active(zWadr(mark(X1), mark(X2)))mark(nil)active(nil)
prefix(mark(X))prefix(X)s(active(X))s(X)
active(app(nil, YS))mark(YS)cons(X1, active(X2))cons(X1, X2)
from(active(X))from(X)mark(app(X1, X2))active(app(mark(X1), mark(X2)))
zWadr(X1, active(X2))zWadr(X1, X2)zWadr(mark(X1), X2)zWadr(X1, X2)
mark(prefix(X))active(prefix(mark(X)))active(app(cons(X, XS), YS))mark(cons(X, app(XS, YS)))
cons(mark(X1), X2)cons(X1, X2)active(zWadr(cons(X, XS), cons(Y, YS)))mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L))mark(cons(nil, zWadr(L, prefix(L))))mark(cons(X1, X2))active(cons(mark(X1), X2))
s(mark(X))s(X)app(mark(X1), X2)app(X1, X2)

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

mark#(zWadr(X1, X2))mark#(X2)mark#(from(X))mark#(X)
active#(from(X))mark#(cons(X, from(s(X))))mark#(app(X1, X2))mark#(X1)
mark#(zWadr(X1, X2))mark#(X1)active#(app(cons(X, XS), YS))mark#(cons(X, app(XS, YS)))
mark#(app(X1, X2))mark#(X2)

Problem 17: PolynomialLinearRange4



Dependency Pair Problem

Dependency Pairs

mark#(from(X))active#(from(mark(X)))active#(zWadr(cons(X, XS), cons(Y, YS)))mark#(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
mark#(cons(X1, X2))mark#(X1)mark#(s(X))mark#(X)
mark#(app(X1, X2))active#(app(mark(X1), mark(X2)))mark#(zWadr(X1, X2))active#(zWadr(mark(X1), mark(X2)))

Rewrite Rules

active(app(nil, YS))mark(YS)active(app(cons(X, XS), YS))mark(cons(X, app(XS, YS)))
active(from(X))mark(cons(X, from(s(X))))active(zWadr(nil, YS))mark(nil)
active(zWadr(XS, nil))mark(nil)active(zWadr(cons(X, XS), cons(Y, YS)))mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L))mark(cons(nil, zWadr(L, prefix(L))))mark(app(X1, X2))active(app(mark(X1), mark(X2)))
mark(nil)active(nil)mark(cons(X1, X2))active(cons(mark(X1), X2))
mark(from(X))active(from(mark(X)))mark(s(X))active(s(mark(X)))
mark(zWadr(X1, X2))active(zWadr(mark(X1), mark(X2)))mark(prefix(X))active(prefix(mark(X)))
app(mark(X1), X2)app(X1, X2)app(X1, mark(X2))app(X1, X2)
app(active(X1), X2)app(X1, X2)app(X1, active(X2))app(X1, X2)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
from(mark(X))from(X)from(active(X))from(X)
s(mark(X))s(X)s(active(X))s(X)
zWadr(mark(X1), X2)zWadr(X1, X2)zWadr(X1, mark(X2))zWadr(X1, X2)
zWadr(active(X1), X2)zWadr(X1, X2)zWadr(X1, active(X2))zWadr(X1, X2)
prefix(mark(X))prefix(X)prefix(active(X))prefix(X)

Original Signature

Termination of terms over the following signature is verified: app, zWadr, s, prefix, active, mark, from, nil, cons

Strategy


Polynomial Interpretation

Standard Usable rules

cons(active(X1), X2)cons(X1, X2)from(mark(X))from(X)
zWadr(X1, mark(X2))zWadr(X1, X2)mark(s(X))active(s(mark(X)))
prefix(active(X))prefix(X)zWadr(active(X1), X2)zWadr(X1, X2)
cons(X1, mark(X2))cons(X1, X2)app(X1, mark(X2))app(X1, X2)
active(zWadr(XS, nil))mark(nil)active(zWadr(nil, YS))mark(nil)
mark(from(X))active(from(mark(X)))app(X1, active(X2))app(X1, X2)
app(active(X1), X2)app(X1, X2)active(from(X))mark(cons(X, from(s(X))))
mark(zWadr(X1, X2))active(zWadr(mark(X1), mark(X2)))mark(nil)active(nil)
s(active(X))s(X)prefix(mark(X))prefix(X)
active(app(nil, YS))mark(YS)cons(X1, active(X2))cons(X1, X2)
from(active(X))from(X)mark(app(X1, X2))active(app(mark(X1), mark(X2)))
zWadr(X1, active(X2))zWadr(X1, X2)zWadr(mark(X1), X2)zWadr(X1, X2)
mark(prefix(X))active(prefix(mark(X)))cons(mark(X1), X2)cons(X1, X2)
active(app(cons(X, XS), YS))mark(cons(X, app(XS, YS)))active(zWadr(cons(X, XS), cons(Y, YS)))mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
mark(cons(X1, X2))active(cons(mark(X1), X2))active(prefix(L))mark(cons(nil, zWadr(L, prefix(L))))
s(mark(X))s(X)app(mark(X1), X2)app(X1, X2)

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

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

Problem 18: PolynomialLinearRange4



Dependency Pair Problem

Dependency Pairs

active#(zWadr(cons(X, XS), cons(Y, YS)))mark#(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))mark#(from(X))active#(from(mark(X)))
mark#(cons(X1, X2))mark#(X1)mark#(s(X))mark#(X)
mark#(zWadr(X1, X2))active#(zWadr(mark(X1), mark(X2)))

Rewrite Rules

active(app(nil, YS))mark(YS)active(app(cons(X, XS), YS))mark(cons(X, app(XS, YS)))
active(from(X))mark(cons(X, from(s(X))))active(zWadr(nil, YS))mark(nil)
active(zWadr(XS, nil))mark(nil)active(zWadr(cons(X, XS), cons(Y, YS)))mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L))mark(cons(nil, zWadr(L, prefix(L))))mark(app(X1, X2))active(app(mark(X1), mark(X2)))
mark(nil)active(nil)mark(cons(X1, X2))active(cons(mark(X1), X2))
mark(from(X))active(from(mark(X)))mark(s(X))active(s(mark(X)))
mark(zWadr(X1, X2))active(zWadr(mark(X1), mark(X2)))mark(prefix(X))active(prefix(mark(X)))
app(mark(X1), X2)app(X1, X2)app(X1, mark(X2))app(X1, X2)
app(active(X1), X2)app(X1, X2)app(X1, active(X2))app(X1, X2)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
from(mark(X))from(X)from(active(X))from(X)
s(mark(X))s(X)s(active(X))s(X)
zWadr(mark(X1), X2)zWadr(X1, X2)zWadr(X1, mark(X2))zWadr(X1, X2)
zWadr(active(X1), X2)zWadr(X1, X2)zWadr(X1, active(X2))zWadr(X1, X2)
prefix(mark(X))prefix(X)prefix(active(X))prefix(X)

Original Signature

Termination of terms over the following signature is verified: app, zWadr, s, prefix, active, mark, from, cons, nil

Strategy


Polynomial Interpretation

Standard Usable rules

cons(active(X1), X2)cons(X1, X2)from(mark(X))from(X)
zWadr(X1, mark(X2))zWadr(X1, X2)mark(s(X))active(s(mark(X)))
prefix(active(X))prefix(X)zWadr(active(X1), X2)zWadr(X1, X2)
cons(X1, mark(X2))cons(X1, X2)app(X1, mark(X2))app(X1, X2)
active(zWadr(XS, nil))mark(nil)active(zWadr(nil, YS))mark(nil)
mark(from(X))active(from(mark(X)))app(X1, active(X2))app(X1, X2)
app(active(X1), X2)app(X1, X2)active(from(X))mark(cons(X, from(s(X))))
mark(zWadr(X1, X2))active(zWadr(mark(X1), mark(X2)))mark(nil)active(nil)
s(active(X))s(X)prefix(mark(X))prefix(X)
active(app(nil, YS))mark(YS)cons(X1, active(X2))cons(X1, X2)
from(active(X))from(X)mark(app(X1, X2))active(app(mark(X1), mark(X2)))
zWadr(X1, active(X2))zWadr(X1, X2)zWadr(mark(X1), X2)zWadr(X1, X2)
mark(prefix(X))active(prefix(mark(X)))cons(mark(X1), X2)cons(X1, X2)
active(app(cons(X, XS), YS))mark(cons(X, app(XS, YS)))active(zWadr(cons(X, XS), cons(Y, YS)))mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
mark(cons(X1, X2))active(cons(mark(X1), X2))active(prefix(L))mark(cons(nil, zWadr(L, prefix(L))))
s(mark(X))s(X)app(mark(X1), X2)app(X1, X2)

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

active#(zWadr(cons(X, XS), cons(Y, YS)))mark#(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))

Problem 19: DependencyGraph



Dependency Pair Problem

Dependency Pairs

mark#(from(X))active#(from(mark(X)))mark#(cons(X1, X2))mark#(X1)
mark#(s(X))mark#(X)mark#(zWadr(X1, X2))active#(zWadr(mark(X1), mark(X2)))

Rewrite Rules

active(app(nil, YS))mark(YS)active(app(cons(X, XS), YS))mark(cons(X, app(XS, YS)))
active(from(X))mark(cons(X, from(s(X))))active(zWadr(nil, YS))mark(nil)
active(zWadr(XS, nil))mark(nil)active(zWadr(cons(X, XS), cons(Y, YS)))mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L))mark(cons(nil, zWadr(L, prefix(L))))mark(app(X1, X2))active(app(mark(X1), mark(X2)))
mark(nil)active(nil)mark(cons(X1, X2))active(cons(mark(X1), X2))
mark(from(X))active(from(mark(X)))mark(s(X))active(s(mark(X)))
mark(zWadr(X1, X2))active(zWadr(mark(X1), mark(X2)))mark(prefix(X))active(prefix(mark(X)))
app(mark(X1), X2)app(X1, X2)app(X1, mark(X2))app(X1, X2)
app(active(X1), X2)app(X1, X2)app(X1, active(X2))app(X1, X2)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
from(mark(X))from(X)from(active(X))from(X)
s(mark(X))s(X)s(active(X))s(X)
zWadr(mark(X1), X2)zWadr(X1, X2)zWadr(X1, mark(X2))zWadr(X1, X2)
zWadr(active(X1), X2)zWadr(X1, X2)zWadr(X1, active(X2))zWadr(X1, X2)
prefix(mark(X))prefix(X)prefix(active(X))prefix(X)

Original Signature

Termination of terms over the following signature is verified: app, zWadr, s, prefix, active, mark, from, nil, cons

Strategy


The following SCCs where found

mark#(cons(X1, X2)) → mark#(X1)mark#(s(X)) → mark#(X)

Problem 20: PolynomialLinearRange4



Dependency Pair Problem

Dependency Pairs

mark#(cons(X1, X2))mark#(X1)mark#(s(X))mark#(X)

Rewrite Rules

active(app(nil, YS))mark(YS)active(app(cons(X, XS), YS))mark(cons(X, app(XS, YS)))
active(from(X))mark(cons(X, from(s(X))))active(zWadr(nil, YS))mark(nil)
active(zWadr(XS, nil))mark(nil)active(zWadr(cons(X, XS), cons(Y, YS)))mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L))mark(cons(nil, zWadr(L, prefix(L))))mark(app(X1, X2))active(app(mark(X1), mark(X2)))
mark(nil)active(nil)mark(cons(X1, X2))active(cons(mark(X1), X2))
mark(from(X))active(from(mark(X)))mark(s(X))active(s(mark(X)))
mark(zWadr(X1, X2))active(zWadr(mark(X1), mark(X2)))mark(prefix(X))active(prefix(mark(X)))
app(mark(X1), X2)app(X1, X2)app(X1, mark(X2))app(X1, X2)
app(active(X1), X2)app(X1, X2)app(X1, active(X2))app(X1, X2)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
from(mark(X))from(X)from(active(X))from(X)
s(mark(X))s(X)s(active(X))s(X)
zWadr(mark(X1), X2)zWadr(X1, X2)zWadr(X1, mark(X2))zWadr(X1, X2)
zWadr(active(X1), X2)zWadr(X1, X2)zWadr(X1, active(X2))zWadr(X1, X2)
prefix(mark(X))prefix(X)prefix(active(X))prefix(X)

Original Signature

Termination of terms over the following signature is verified: app, zWadr, s, prefix, active, mark, from, nil, cons

Strategy


Polynomial Interpretation

There are no usable rules

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

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

Problem 21: PolynomialLinearRange4



Dependency Pair Problem

Dependency Pairs

mark#(s(X))mark#(X)

Rewrite Rules

active(app(nil, YS))mark(YS)active(app(cons(X, XS), YS))mark(cons(X, app(XS, YS)))
active(from(X))mark(cons(X, from(s(X))))active(zWadr(nil, YS))mark(nil)
active(zWadr(XS, nil))mark(nil)active(zWadr(cons(X, XS), cons(Y, YS)))mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L))mark(cons(nil, zWadr(L, prefix(L))))mark(app(X1, X2))active(app(mark(X1), mark(X2)))
mark(nil)active(nil)mark(cons(X1, X2))active(cons(mark(X1), X2))
mark(from(X))active(from(mark(X)))mark(s(X))active(s(mark(X)))
mark(zWadr(X1, X2))active(zWadr(mark(X1), mark(X2)))mark(prefix(X))active(prefix(mark(X)))
app(mark(X1), X2)app(X1, X2)app(X1, mark(X2))app(X1, X2)
app(active(X1), X2)app(X1, X2)app(X1, active(X2))app(X1, X2)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
from(mark(X))from(X)from(active(X))from(X)
s(mark(X))s(X)s(active(X))s(X)
zWadr(mark(X1), X2)zWadr(X1, X2)zWadr(X1, mark(X2))zWadr(X1, X2)
zWadr(active(X1), X2)zWadr(X1, X2)zWadr(X1, active(X2))zWadr(X1, X2)
prefix(mark(X))prefix(X)prefix(active(X))prefix(X)

Original Signature

Termination of terms over the following signature is verified: app, zWadr, s, prefix, active, mark, from, cons, nil

Strategy


Polynomial Interpretation

There are no usable rules

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

mark#(s(X))mark#(X)

Problem 6: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

from#(active(X))from#(X)from#(mark(X))from#(X)

Rewrite Rules

active(app(nil, YS))mark(YS)active(app(cons(X, XS), YS))mark(cons(X, app(XS, YS)))
active(from(X))mark(cons(X, from(s(X))))active(zWadr(nil, YS))mark(nil)
active(zWadr(XS, nil))mark(nil)active(zWadr(cons(X, XS), cons(Y, YS)))mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L))mark(cons(nil, zWadr(L, prefix(L))))mark(app(X1, X2))active(app(mark(X1), mark(X2)))
mark(nil)active(nil)mark(cons(X1, X2))active(cons(mark(X1), X2))
mark(from(X))active(from(mark(X)))mark(s(X))active(s(mark(X)))
mark(zWadr(X1, X2))active(zWadr(mark(X1), mark(X2)))mark(prefix(X))active(prefix(mark(X)))
app(mark(X1), X2)app(X1, X2)app(X1, mark(X2))app(X1, X2)
app(active(X1), X2)app(X1, X2)app(X1, active(X2))app(X1, X2)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
from(mark(X))from(X)from(active(X))from(X)
s(mark(X))s(X)s(active(X))s(X)
zWadr(mark(X1), X2)zWadr(X1, X2)zWadr(X1, mark(X2))zWadr(X1, X2)
zWadr(active(X1), X2)zWadr(X1, X2)zWadr(X1, active(X2))zWadr(X1, X2)
prefix(mark(X))prefix(X)prefix(active(X))prefix(X)

Original Signature

Termination of terms over the following signature is verified: app, zWadr, s, prefix, active, mark, from, nil, cons

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

from#(active(X))from#(X)from#(mark(X))from#(X)

Problem 7: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

s#(mark(X))s#(X)s#(active(X))s#(X)

Rewrite Rules

active(app(nil, YS))mark(YS)active(app(cons(X, XS), YS))mark(cons(X, app(XS, YS)))
active(from(X))mark(cons(X, from(s(X))))active(zWadr(nil, YS))mark(nil)
active(zWadr(XS, nil))mark(nil)active(zWadr(cons(X, XS), cons(Y, YS)))mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L))mark(cons(nil, zWadr(L, prefix(L))))mark(app(X1, X2))active(app(mark(X1), mark(X2)))
mark(nil)active(nil)mark(cons(X1, X2))active(cons(mark(X1), X2))
mark(from(X))active(from(mark(X)))mark(s(X))active(s(mark(X)))
mark(zWadr(X1, X2))active(zWadr(mark(X1), mark(X2)))mark(prefix(X))active(prefix(mark(X)))
app(mark(X1), X2)app(X1, X2)app(X1, mark(X2))app(X1, X2)
app(active(X1), X2)app(X1, X2)app(X1, active(X2))app(X1, X2)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
from(mark(X))from(X)from(active(X))from(X)
s(mark(X))s(X)s(active(X))s(X)
zWadr(mark(X1), X2)zWadr(X1, X2)zWadr(X1, mark(X2))zWadr(X1, X2)
zWadr(active(X1), X2)zWadr(X1, X2)zWadr(X1, active(X2))zWadr(X1, X2)
prefix(mark(X))prefix(X)prefix(active(X))prefix(X)

Original Signature

Termination of terms over the following signature is verified: app, zWadr, s, prefix, active, mark, from, nil, cons

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

s#(mark(X))s#(X)s#(active(X))s#(X)

Problem 8: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

active(app(nil, YS))mark(YS)active(app(cons(X, XS), YS))mark(cons(X, app(XS, YS)))
active(from(X))mark(cons(X, from(s(X))))active(zWadr(nil, YS))mark(nil)
active(zWadr(XS, nil))mark(nil)active(zWadr(cons(X, XS), cons(Y, YS)))mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L))mark(cons(nil, zWadr(L, prefix(L))))mark(app(X1, X2))active(app(mark(X1), mark(X2)))
mark(nil)active(nil)mark(cons(X1, X2))active(cons(mark(X1), X2))
mark(from(X))active(from(mark(X)))mark(s(X))active(s(mark(X)))
mark(zWadr(X1, X2))active(zWadr(mark(X1), mark(X2)))mark(prefix(X))active(prefix(mark(X)))
app(mark(X1), X2)app(X1, X2)app(X1, mark(X2))app(X1, X2)
app(active(X1), X2)app(X1, X2)app(X1, active(X2))app(X1, X2)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
from(mark(X))from(X)from(active(X))from(X)
s(mark(X))s(X)s(active(X))s(X)
zWadr(mark(X1), X2)zWadr(X1, X2)zWadr(X1, mark(X2))zWadr(X1, X2)
zWadr(active(X1), X2)zWadr(X1, X2)zWadr(X1, active(X2))zWadr(X1, X2)
prefix(mark(X))prefix(X)prefix(active(X))prefix(X)

Original Signature

Termination of terms over the following signature is verified: app, zWadr, s, prefix, active, mark, from, nil, cons

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

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

Problem 11: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

active(app(nil, YS))mark(YS)active(app(cons(X, XS), YS))mark(cons(X, app(XS, YS)))
active(from(X))mark(cons(X, from(s(X))))active(zWadr(nil, YS))mark(nil)
active(zWadr(XS, nil))mark(nil)active(zWadr(cons(X, XS), cons(Y, YS)))mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L))mark(cons(nil, zWadr(L, prefix(L))))mark(app(X1, X2))active(app(mark(X1), mark(X2)))
mark(nil)active(nil)mark(cons(X1, X2))active(cons(mark(X1), X2))
mark(from(X))active(from(mark(X)))mark(s(X))active(s(mark(X)))
mark(zWadr(X1, X2))active(zWadr(mark(X1), mark(X2)))mark(prefix(X))active(prefix(mark(X)))
app(mark(X1), X2)app(X1, X2)app(X1, mark(X2))app(X1, X2)
app(active(X1), X2)app(X1, X2)app(X1, active(X2))app(X1, X2)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
from(mark(X))from(X)from(active(X))from(X)
s(mark(X))s(X)s(active(X))s(X)
zWadr(mark(X1), X2)zWadr(X1, X2)zWadr(X1, mark(X2))zWadr(X1, X2)
zWadr(active(X1), X2)zWadr(X1, X2)zWadr(X1, active(X2))zWadr(X1, X2)
prefix(mark(X))prefix(X)prefix(active(X))prefix(X)

Original Signature

Termination of terms over the following signature is verified: app, zWadr, s, prefix, active, mark, from, cons, nil

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

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