TIMEOUT

The TRS could not be proven terminating. The proof attempt took 60001 ms.

The following DP Processors were used


Problem 1 was processed with processor DependencyGraph (21419ms).
 | – Problem 2 was processed with processor SubtermCriterion (5ms).
 | – Problem 3 was processed with processor SubtermCriterion (2ms).
 | – Problem 4 was processed with processor SubtermCriterion (5ms).
 | – Problem 5 was processed with processor SubtermCriterion (2ms).
 | – Problem 6 was processed with processor SubtermCriterion (1ms).
 | – Problem 7 was processed with processor SubtermCriterion (41ms).
 |    | – Problem 20 was processed with processor ReductionPairSAT (70ms).
 | – Problem 8 was processed with processor SubtermCriterion (1ms).
 | – Problem 9 was processed with processor SubtermCriterion (3ms).
 | – Problem 10 was processed with processor SubtermCriterion (3ms).
 | – Problem 11 was processed with processor SubtermCriterion (2ms).
 |    | – Problem 21 was processed with processor ReductionPairSAT (59ms).
 | – Problem 12 was processed with processor SubtermCriterion (1ms).
 | – Problem 13 was processed with processor SubtermCriterion (4ms).
 | – Problem 14 was processed with processor SubtermCriterion (2ms).
 |    | – Problem 22 was processed with processor ReductionPairSAT (32ms).
 | – Problem 15 was processed with processor SubtermCriterion (1ms).
 |    | – Problem 23 was processed with processor ReductionPairSAT (69ms).
 | – Problem 16 was processed with processor SubtermCriterion (1ms).
 | – Problem 17 remains open; application of the following processors failed [SubtermCriterion (4ms), DependencyGraph (8ms), PolynomialLinearRange4iUR (1428ms), DependencyGraph (8ms), PolynomialLinearRange4iUR (1667ms), DependencyGraph (8ms), PolynomialLinearRange8NegiUR (5000ms), DependencyGraph (6ms), ReductionPairSAT (11192ms), DependencyGraph (7ms), ReductionPairSAT (11006ms), DependencyGraph (8ms), SizeChangePrinciple (timeout)].
 | – Problem 18 was processed with processor SubtermCriterion (4ms).
 |    | – Problem 25 was processed with processor ReductionPairSAT (114ms).
 | – Problem 19 was processed with processor SubtermCriterion (2ms).
 |    | – Problem 24 was processed with processor PolynomialLinearRange4iUR (56ms).

The following open problems remain:

Open Dependency Pair Problem 17

Dependency Pairs

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

Rewrite Rules

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))proper(0) → ok(0)
proper(first(X1, X2)) → first(proper(X1), proper(X2))proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01) → ok(01)proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))top(ok(X)) → top(active(X))

Original Signature

Termination of terms over the following signature is verified: s1, cons1, mark, fcons, from, 01, first1, quote1, unquote, 0, s, sel1, unquote1, quote, active, ok, proper, nil1, sel, first, top, cons, nil

Problem 1: DependencyGraph

Dependency Pair Problem

Dependency Pairs

proper#(sel1(X1, X2)) → proper#(X1)proper#(cons(X1, X2)) → proper#(X1)
active#(unquote(s1(X))) → s#(unquote(X))active#(sel1(X1, X2)) → active#(X2)
proper#(s1(X)) → proper#(X)active#(unquote1(X)) → active#(X)
active#(sel1(X1, X2)) → active#(X1)unquote#(ok(X)) → unquote#(X)
fcons#(mark(X1), X2) → fcons#(X1, X2)active#(first(s(X), cons(Y, Z))) → cons#(Y, first(X, Z))
active#(s1(X)) → active#(X)proper#(quote(X)) → quote#(proper(X))
active#(s1(X)) → s1#(active(X))cons1#(X1, mark(X2)) → cons1#(X1, X2)
top#(mark(X)) → proper#(X)active#(first(X1, X2)) → active#(X2)
first1#(mark(X1), X2) → first1#(X1, X2)active#(sel(s(X), cons(Y, Z))) → sel#(X, Z)
s1#(ok(X)) → s1#(X)proper#(first(X1, X2)) → first#(proper(X1), proper(X2))
active#(quote(s(X))) → quote#(X)active#(first1(X1, X2)) → active#(X2)
active#(fcons(X1, X2)) → fcons#(active(X1), X2)active#(cons1(X1, X2)) → active#(X2)
sel#(X1, mark(X2)) → sel#(X1, X2)active#(first1(X1, X2)) → active#(X1)
proper#(first(X1, X2)) → proper#(X2)active#(quote1(cons(X, Z))) → quote#(X)
proper#(cons1(X1, X2)) → proper#(X2)proper#(cons1(X1, X2)) → cons1#(proper(X1), proper(X2))
proper#(sel1(X1, X2)) → proper#(X2)sel#(ok(X1), ok(X2)) → sel#(X1, X2)
active#(fcons(X1, X2)) → active#(X1)active#(cons1(X1, X2)) → active#(X1)
active#(quote(s(X))) → s1#(quote(X))unquote1#(mark(X)) → unquote1#(X)
cons#(mark(X1), X2) → cons#(X1, X2)active#(first(X1, X2)) → first#(X1, active(X2))
from#(mark(X)) → from#(X)active#(unquote(X)) → active#(X)
top#(ok(X)) → active#(X)sel1#(X1, mark(X2)) → sel1#(X1, X2)
active#(sel(X1, X2)) → active#(X1)proper#(s1(X)) → s1#(proper(X))
proper#(from(X)) → from#(proper(X))proper#(sel(X1, X2)) → proper#(X2)
active#(first1(X1, X2)) → first1#(active(X1), X2)active#(quote1(first(X, Z))) → first1#(X, Z)
first#(X1, mark(X2)) → first#(X1, X2)active#(sel(X1, X2)) → sel#(X1, active(X2))
active#(sel(X1, X2)) → sel#(active(X1), X2)active#(cons1(X1, X2)) → cons1#(X1, active(X2))
fcons#(ok(X1), ok(X2)) → fcons#(X1, X2)proper#(quote1(X)) → proper#(X)
active#(s(X)) → s#(active(X))s#(ok(X)) → s#(X)
proper#(sel(X1, X2)) → proper#(X1)first#(ok(X1), ok(X2)) → first#(X1, X2)
active#(first(X1, X2)) → first#(active(X1), X2)cons1#(mark(X1), X2) → cons1#(X1, X2)
proper#(s(X)) → s#(proper(X))proper#(first(X1, X2)) → proper#(X1)
first1#(X1, mark(X2)) → first1#(X1, X2)active#(from(X)) → from#(s(X))
top#(ok(X)) → top#(active(X))unquote1#(ok(X)) → unquote1#(X)
cons#(ok(X1), ok(X2)) → cons#(X1, X2)from#(ok(X)) → from#(X)
active#(cons1(X1, X2)) → cons1#(active(X1), X2)active#(cons(X1, X2)) → cons#(active(X1), X2)
active#(fcons(X, Z)) → cons#(X, Z)active#(quote1(cons(X, Z))) → cons1#(quote(X), quote1(Z))
active#(first1(s(X), cons(Y, Z))) → first1#(X, Z)active#(sel(X1, X2)) → active#(X2)
proper#(cons1(X1, X2)) → proper#(X1)sel1#(ok(X1), ok(X2)) → sel1#(X1, X2)
proper#(unquote(X)) → proper#(X)first#(mark(X1), X2) → first#(X1, X2)
proper#(from(X)) → proper#(X)proper#(quote1(X)) → quote1#(proper(X))
top#(mark(X)) → top#(proper(X))proper#(cons(X1, X2)) → proper#(X2)
active#(first1(s(X), cons(Y, Z))) → cons1#(quote(Y), first1(X, Z))active#(first1(s(X), cons(Y, Z))) → quote#(Y)
active#(first(X1, X2)) → active#(X1)active#(unquote1(cons1(X, Z))) → unquote1#(Z)
active#(fcons(X1, X2)) → fcons#(X1, active(X2))active#(from(X)) → s#(X)
active#(unquote(X)) → unquote#(active(X))proper#(s(X)) → proper#(X)
sel1#(mark(X1), X2) → sel1#(X1, X2)quote1#(ok(X)) → quote1#(X)
proper#(quote(X)) → proper#(X)proper#(unquote(X)) → unquote#(proper(X))
active#(cons(X1, X2)) → active#(X1)sel#(mark(X1), X2) → sel#(X1, X2)
proper#(fcons(X1, X2)) → fcons#(proper(X1), proper(X2))first1#(ok(X1), ok(X2)) → first1#(X1, X2)
active#(from(X)) → from#(active(X))active#(sel1(X1, X2)) → sel1#(X1, active(X2))
active#(fcons(X1, X2)) → active#(X2)active#(quote1(cons(X, Z))) → quote1#(Z)
proper#(sel(X1, X2)) → sel#(proper(X1), proper(X2))proper#(unquote1(X)) → proper#(X)
active#(first(s(X), cons(Y, Z))) → first#(X, Z)quote#(ok(X)) → quote#(X)
active#(sel1(s(X), cons(Y, Z))) → sel1#(X, Z)active#(from(X)) → cons#(X, from(s(X)))
active#(sel1(X1, X2)) → sel1#(active(X1), X2)proper#(first1(X1, X2)) → proper#(X1)
proper#(fcons(X1, X2)) → proper#(X1)unquote#(mark(X)) → unquote#(X)
s1#(mark(X)) → s1#(X)proper#(first1(X1, X2)) → proper#(X2)
active#(from(X)) → active#(X)cons1#(ok(X1), ok(X2)) → cons1#(X1, X2)
active#(quote(sel(X, Z))) → sel1#(X, Z)proper#(fcons(X1, X2)) → proper#(X2)
proper#(unquote1(X)) → unquote1#(proper(X))proper#(sel1(X1, X2)) → sel1#(proper(X1), proper(X2))
s#(mark(X)) → s#(X)active#(unquote1(cons1(X, Z))) → unquote#(X)
proper#(first1(X1, X2)) → first1#(proper(X1), proper(X2))proper#(cons(X1, X2)) → cons#(proper(X1), proper(X2))
active#(unquote1(X)) → unquote1#(active(X))active#(s(X)) → active#(X)
active#(first1(X1, X2)) → first1#(X1, active(X2))active#(unquote1(cons1(X, Z))) → fcons#(unquote(X), unquote1(Z))
active#(sel1(0, cons(X, Z))) → quote#(X)fcons#(X1, mark(X2)) → fcons#(X1, X2)
active#(unquote(s1(X))) → unquote#(X)

Rewrite Rules

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))proper(0) → ok(0)
proper(first(X1, X2)) → first(proper(X1), proper(X2))proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01) → ok(01)proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))top(ok(X)) → top(active(X))

Original Signature

Termination of terms over the following signature is verified: s1, cons1, mark, fcons, from, 01, first1, quote1, unquote, 0, s, sel1, unquote1, quote, active, ok, nil1, proper, first, sel, nil, cons, top

Strategy

The following SCCs where found

sel#(mark(X1), X2) → sel#(X1, X2)sel#(ok(X1), ok(X2)) → sel#(X1, X2)
sel#(X1, mark(X2)) → sel#(X1, X2)
sel1#(X1, mark(X2)) → sel1#(X1, X2)sel1#(mark(X1), X2) → sel1#(X1, X2)
sel1#(ok(X1), ok(X2)) → sel1#(X1, X2)
first1#(mark(X1), X2) → first1#(X1, X2)first1#(ok(X1), ok(X2)) → first1#(X1, X2)
first1#(X1, mark(X2)) → first1#(X1, X2)
proper#(fcons(X1, X2)) → proper#(X1)proper#(first1(X1, X2)) → proper#(X1)
proper#(sel1(X1, X2)) → proper#(X1)proper#(cons(X1, X2)) → proper#(X1)
proper#(cons(X1, X2)) → proper#(X2)proper#(first1(X1, X2)) → proper#(X2)
proper#(s1(X)) → proper#(X)proper#(fcons(X1, X2)) → proper#(X2)
proper#(quote1(X)) → proper#(X)proper#(unquote1(X)) → proper#(X)
proper#(sel(X1, X2)) → proper#(X1)proper#(first(X1, X2)) → proper#(X2)
proper#(cons1(X1, X2)) → proper#(X2)proper#(s(X)) → proper#(X)
proper#(sel1(X1, X2)) → proper#(X2)proper#(first(X1, X2)) → proper#(X1)
proper#(cons1(X1, X2)) → proper#(X1)proper#(quote(X)) → proper#(X)
proper#(sel(X1, X2)) → proper#(X2)proper#(unquote(X)) → proper#(X)
proper#(from(X)) → proper#(X)
s#(mark(X)) → s#(X)s#(ok(X)) → s#(X)
quote#(ok(X)) → quote#(X)
cons#(mark(X1), X2) → cons#(X1, X2)cons#(ok(X1), ok(X2)) → cons#(X1, X2)
s1#(mark(X)) → s1#(X)s1#(ok(X)) → s1#(X)
quote1#(ok(X)) → quote1#(X)
unquote#(mark(X)) → unquote#(X)unquote#(ok(X)) → unquote#(X)
fcons#(ok(X1), ok(X2)) → fcons#(X1, X2)fcons#(X1, mark(X2)) → fcons#(X1, X2)
fcons#(mark(X1), X2) → fcons#(X1, X2)
active#(first(X1, X2)) → active#(X2)active#(sel1(X1, X2)) → active#(X2)
active#(from(X)) → active#(X)active#(first1(X1, X2)) → active#(X2)
active#(unquote1(X)) → active#(X)active#(sel1(X1, X2)) → active#(X1)
active#(cons1(X1, X2)) → active#(X2)active#(first1(X1, X2)) → active#(X1)
active#(fcons(X1, X2)) → active#(X2)active#(unquote(X)) → active#(X)
active#(first(X1, X2)) → active#(X1)active#(unquote1(cons1(X, Z))) → unquote#(X)
active#(sel(X1, X2)) → active#(X2)active#(s1(X)) → active#(X)
active#(s(X)) → active#(X)active#(sel(X1, X2)) → active#(X1)
active#(fcons(X1, X2)) → active#(X1)active#(cons1(X1, X2)) → active#(X1)
active#(cons(X1, X2)) → active#(X1)
active#(first(X1, X2)) → active#(X2)active#(sel1(X1, X2)) → active#(X2)
active#(from(X)) → active#(X)active#(unquote1(X)) → active#(X)
active#(first1(X1, X2)) → active#(X2)active#(sel1(X1, X2)) → active#(X1)
active#(cons1(X1, X2)) → active#(X2)active#(fcons(X1, X2)) → active#(X2)
active#(first1(X1, X2)) → active#(X1)active#(first(X1, X2)) → active#(X1)
active#(unquote(X)) → active#(X)active#(sel(X1, X2)) → active#(X2)
active#(s1(X)) → active#(X)active#(sel(X1, X2)) → active#(X1)
active#(s(X)) → active#(X)active#(fcons(X1, X2)) → active#(X1)
active#(cons1(X1, X2)) → active#(X1)active#(cons(X1, X2)) → active#(X1)
unquote1#(mark(X)) → unquote1#(X)unquote1#(ok(X)) → unquote1#(X)
from#(mark(X)) → from#(X)from#(ok(X)) → from#(X)
top#(mark(X)) → top#(proper(X))top#(ok(X)) → top#(active(X))
cons1#(mark(X1), X2) → cons1#(X1, X2)cons1#(ok(X1), ok(X2)) → cons1#(X1, X2)
cons1#(X1, mark(X2)) → cons1#(X1, X2)
first#(ok(X1), ok(X2)) → first#(X1, X2)first#(mark(X1), X2) → first#(X1, X2)
first#(X1, mark(X2)) → first#(X1, X2)

Problem 2: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

active#(first(X1, X2)) → active#(X2)active#(sel1(X1, X2)) → active#(X2)
active#(from(X)) → active#(X)active#(first1(X1, X2)) → active#(X2)
active#(unquote1(X)) → active#(X)active#(sel1(X1, X2)) → active#(X1)
active#(cons1(X1, X2)) → active#(X2)active#(first1(X1, X2)) → active#(X1)
active#(fcons(X1, X2)) → active#(X2)active#(unquote(X)) → active#(X)
active#(first(X1, X2)) → active#(X1)active#(s1(X)) → active#(X)
active#(sel(X1, X2)) → active#(X2)active#(sel(X1, X2)) → active#(X1)
active#(s(X)) → active#(X)active#(fcons(X1, X2)) → active#(X1)
active#(cons1(X1, X2)) → active#(X1)active#(cons(X1, X2)) → active#(X1)

Rewrite Rules

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))proper(0) → ok(0)
proper(first(X1, X2)) → first(proper(X1), proper(X2))proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01) → ok(01)proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))top(ok(X)) → top(active(X))

Original Signature

Termination of terms over the following signature is verified: s1, cons1, mark, fcons, from, 01, first1, quote1, unquote, 0, s, sel1, unquote1, quote, active, ok, nil1, proper, first, sel, nil, cons, top

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

active#(first(X1, X2)) → active#(X2)active#(sel1(X1, X2)) → active#(X2)
active#(from(X)) → active#(X)active#(first1(X1, X2)) → active#(X2)
active#(unquote1(X)) → active#(X)active#(sel1(X1, X2)) → active#(X1)
active#(cons1(X1, X2)) → active#(X2)active#(fcons(X1, X2)) → active#(X2)
active#(first1(X1, X2)) → active#(X1)active#(first(X1, X2)) → active#(X1)
active#(unquote(X)) → active#(X)active#(sel(X1, X2)) → active#(X2)
active#(s1(X)) → active#(X)active#(s(X)) → active#(X)
active#(sel(X1, X2)) → active#(X1)active#(fcons(X1, X2)) → active#(X1)
active#(cons1(X1, X2)) → active#(X1)active#(cons(X1, X2)) → active#(X1)

Problem 3: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

proper#(sel1(X1, X2)) → proper#(X1)proper#(first1(X1, X2)) → proper#(X1)
proper#(fcons(X1, X2)) → proper#(X1)proper#(cons(X1, X2)) → proper#(X1)
proper#(first1(X1, X2)) → proper#(X2)proper#(cons(X1, X2)) → proper#(X2)
proper#(s1(X)) → proper#(X)proper#(fcons(X1, X2)) → proper#(X2)
proper#(quote1(X)) → proper#(X)proper#(unquote1(X)) → proper#(X)
proper#(first(X1, X2)) → proper#(X2)proper#(sel(X1, X2)) → proper#(X1)
proper#(cons1(X1, X2)) → proper#(X2)proper#(s(X)) → proper#(X)
proper#(sel1(X1, X2)) → proper#(X2)proper#(first(X1, X2)) → proper#(X1)
proper#(cons1(X1, X2)) → proper#(X1)proper#(quote(X)) → proper#(X)
proper#(sel(X1, X2)) → proper#(X2)proper#(unquote(X)) → proper#(X)
proper#(from(X)) → proper#(X)

Rewrite Rules

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))proper(0) → ok(0)
proper(first(X1, X2)) → first(proper(X1), proper(X2))proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01) → ok(01)proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))top(ok(X)) → top(active(X))

Original Signature

Termination of terms over the following signature is verified: s1, cons1, mark, fcons, from, 01, first1, quote1, unquote, 0, s, sel1, unquote1, quote, active, ok, nil1, proper, first, sel, nil, cons, top

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

proper#(sel1(X1, X2)) → proper#(X1)proper#(first1(X1, X2)) → proper#(X1)
proper#(fcons(X1, X2)) → proper#(X1)proper#(cons(X1, X2)) → proper#(X1)
proper#(first1(X1, X2)) → proper#(X2)proper#(cons(X1, X2)) → proper#(X2)
proper#(s1(X)) → proper#(X)proper#(fcons(X1, X2)) → proper#(X2)
proper#(quote1(X)) → proper#(X)proper#(unquote1(X)) → proper#(X)
proper#(sel(X1, X2)) → proper#(X1)proper#(first(X1, X2)) → proper#(X2)
proper#(cons1(X1, X2)) → proper#(X2)proper#(s(X)) → proper#(X)
proper#(sel1(X1, X2)) → proper#(X2)proper#(first(X1, X2)) → proper#(X1)
proper#(cons1(X1, X2)) → proper#(X1)proper#(quote(X)) → proper#(X)
proper#(sel(X1, X2)) → proper#(X2)proper#(unquote(X)) → proper#(X)
proper#(from(X)) → proper#(X)

Problem 4: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

unquote1#(mark(X)) → unquote1#(X)unquote1#(ok(X)) → unquote1#(X)

Rewrite Rules

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))proper(0) → ok(0)
proper(first(X1, X2)) → first(proper(X1), proper(X2))proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01) → ok(01)proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))top(ok(X)) → top(active(X))

Original Signature

Termination of terms over the following signature is verified: s1, cons1, mark, fcons, from, 01, first1, quote1, unquote, 0, s, sel1, unquote1, quote, active, ok, nil1, proper, first, sel, nil, cons, top

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

unquote1#(mark(X)) → unquote1#(X)unquote1#(ok(X)) → unquote1#(X)

Problem 5: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))proper(0) → ok(0)
proper(first(X1, X2)) → first(proper(X1), proper(X2))proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01) → ok(01)proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))top(ok(X)) → top(active(X))

Original Signature

Termination of terms over the following signature is verified: s1, cons1, mark, fcons, from, 01, first1, quote1, unquote, 0, s, sel1, unquote1, quote, active, ok, nil1, proper, first, sel, nil, cons, top

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

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

Problem 6: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

unquote#(mark(X)) → unquote#(X)unquote#(ok(X)) → unquote#(X)

Rewrite Rules

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))proper(0) → ok(0)
proper(first(X1, X2)) → first(proper(X1), proper(X2))proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01) → ok(01)proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))top(ok(X)) → top(active(X))

Original Signature

Termination of terms over the following signature is verified: s1, cons1, mark, fcons, from, 01, first1, quote1, unquote, 0, s, sel1, unquote1, quote, active, ok, nil1, proper, first, sel, nil, cons, top

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

unquote#(mark(X)) → unquote#(X)unquote#(ok(X)) → unquote#(X)

Problem 7: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

cons1#(mark(X1), X2) → cons1#(X1, X2)cons1#(ok(X1), ok(X2)) → cons1#(X1, X2)
cons1#(X1, mark(X2)) → cons1#(X1, X2)

Rewrite Rules

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))proper(0) → ok(0)
proper(first(X1, X2)) → first(proper(X1), proper(X2))proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01) → ok(01)proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))top(ok(X)) → top(active(X))

Original Signature

Termination of terms over the following signature is verified: s1, cons1, mark, fcons, from, 01, first1, quote1, unquote, 0, s, sel1, unquote1, quote, active, ok, nil1, proper, first, sel, nil, cons, top

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

cons1#(mark(X1), X2) → cons1#(X1, X2)cons1#(ok(X1), ok(X2)) → cons1#(X1, X2)

Problem 20: ReductionPairSAT

Dependency Pair Problem

Dependency Pairs

cons1#(X1, mark(X2)) → cons1#(X1, X2)

Rewrite Rules

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))proper(0) → ok(0)
proper(first(X1, X2)) → first(proper(X1), proper(X2))proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01) → ok(01)proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))top(ok(X)) → top(active(X))

Original Signature

Termination of terms over the following signature is verified: s1, cons1, mark, fcons, from, 01, first1, quote1, unquote, 0, s, sel1, unquote1, quote, active, ok, proper, nil1, sel, first, top, cons, nil

Strategy

Function Precedence

mark < cons1# = cons1 = s1 = fcons = from = 01 = first1 = quote1 = unquote = 0 = s = sel1 = unquote1 = quote = active = ok = proper = nil1 = sel = first = top = nil = cons

Argument Filtering

cons1#: collapses to 2
cons1: all arguments are removed from cons1
s1: all arguments are removed from s1
mark: 1
fcons: all arguments are removed from fcons
from: all arguments are removed from from
01: all arguments are removed from 01
first1: 2
quote1: all arguments are removed from quote1
unquote: all arguments are removed from unquote
0: all arguments are removed from 0
s: all arguments are removed from s
sel1: all arguments are removed from sel1
unquote1: collapses to 1
quote: all arguments are removed from quote
active: all arguments are removed from active
ok: all arguments are removed from ok
proper: all arguments are removed from proper
nil1: all arguments are removed from nil1
sel: all arguments are removed from sel
first: 1 2
top: all arguments are removed from top
nil: all arguments are removed from nil
cons: 1 2

Status

cons1: multiset
s1: multiset
mark: multiset
fcons: multiset
from: multiset
01: multiset
first1: lexicographic with permutation 2 → 1
quote1: multiset
unquote: multiset
0: multiset
s: multiset
sel1: multiset
quote: multiset
active: multiset
ok: multiset
proper: multiset
nil1: multiset
sel: multiset
first: lexicographic with permutation 1 → 1 2 → 2
top: multiset
nil: multiset
cons: lexicographic with permutation 1 → 1 2 → 2

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.

cons1#(X1, mark(X2)) → cons1#(X1, X2)

Problem 8: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

s1#(mark(X)) → s1#(X)s1#(ok(X)) → s1#(X)

Rewrite Rules

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))proper(0) → ok(0)
proper(first(X1, X2)) → first(proper(X1), proper(X2))proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01) → ok(01)proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))top(ok(X)) → top(active(X))

Original Signature

Termination of terms over the following signature is verified: s1, cons1, mark, fcons, from, 01, first1, quote1, unquote, 0, s, sel1, unquote1, quote, active, ok, nil1, proper, first, sel, nil, cons, top

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

s1#(mark(X)) → s1#(X)s1#(ok(X)) → s1#(X)

Problem 9: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

active#(first(X1, X2)) → active#(X2)active#(sel1(X1, X2)) → active#(X2)
active#(from(X)) → active#(X)active#(unquote1(X)) → active#(X)
active#(first1(X1, X2)) → active#(X2)active#(sel1(X1, X2)) → active#(X1)
active#(cons1(X1, X2)) → active#(X2)active#(fcons(X1, X2)) → active#(X2)
active#(first1(X1, X2)) → active#(X1)active#(first(X1, X2)) → active#(X1)
active#(unquote(X)) → active#(X)active#(unquote1(cons1(X, Z))) → unquote#(X)
active#(sel(X1, X2)) → active#(X2)active#(s1(X)) → active#(X)
active#(s(X)) → active#(X)active#(sel(X1, X2)) → active#(X1)
active#(fcons(X1, X2)) → active#(X1)active#(cons1(X1, X2)) → active#(X1)
active#(cons(X1, X2)) → active#(X1)

Rewrite Rules

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))proper(0) → ok(0)
proper(first(X1, X2)) → first(proper(X1), proper(X2))proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01) → ok(01)proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))top(ok(X)) → top(active(X))

Original Signature

Termination of terms over the following signature is verified: s1, cons1, mark, fcons, from, 01, first1, quote1, unquote, 0, s, sel1, unquote1, quote, active, ok, nil1, proper, first, sel, nil, cons, top

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

active#(first(X1, X2)) → active#(X2)active#(sel1(X1, X2)) → active#(X2)
active#(from(X)) → active#(X)active#(unquote1(X)) → active#(X)
active#(first1(X1, X2)) → active#(X2)active#(sel1(X1, X2)) → active#(X1)
active#(cons1(X1, X2)) → active#(X2)active#(fcons(X1, X2)) → active#(X2)
active#(first1(X1, X2)) → active#(X1)active#(first(X1, X2)) → active#(X1)
active#(unquote(X)) → active#(X)active#(unquote1(cons1(X, Z))) → unquote#(X)
active#(sel(X1, X2)) → active#(X2)active#(s1(X)) → active#(X)
active#(s(X)) → active#(X)active#(sel(X1, X2)) → active#(X1)
active#(fcons(X1, X2)) → active#(X1)active#(cons1(X1, X2)) → active#(X1)
active#(cons(X1, X2)) → active#(X1)

Problem 10: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

quote#(ok(X)) → quote#(X)

Rewrite Rules

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))proper(0) → ok(0)
proper(first(X1, X2)) → first(proper(X1), proper(X2))proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01) → ok(01)proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))top(ok(X)) → top(active(X))

Original Signature

Termination of terms over the following signature is verified: s1, cons1, mark, fcons, from, 01, first1, quote1, unquote, 0, s, sel1, unquote1, quote, active, ok, nil1, proper, first, sel, nil, cons, top

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

quote#(ok(X)) → quote#(X)

Problem 11: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))proper(0) → ok(0)
proper(first(X1, X2)) → first(proper(X1), proper(X2))proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01) → ok(01)proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))top(ok(X)) → top(active(X))

Original Signature

Termination of terms over the following signature is verified: s1, cons1, mark, fcons, from, 01, first1, quote1, unquote, 0, s, sel1, unquote1, quote, active, ok, nil1, proper, first, sel, nil, cons, top

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

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

Problem 21: ReductionPairSAT

Dependency Pair Problem

Dependency Pairs

first#(X1, mark(X2)) → first#(X1, X2)

Rewrite Rules

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))proper(0) → ok(0)
proper(first(X1, X2)) → first(proper(X1), proper(X2))proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01) → ok(01)proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))top(ok(X)) → top(active(X))

Original Signature

Termination of terms over the following signature is verified: s1, cons1, mark, fcons, from, 01, first1, quote1, unquote, 0, s, sel1, unquote1, quote, active, ok, proper, nil1, sel, first, top, cons, nil

Strategy

Function Precedence

cons1 = s1 = mark = fcons = from = 01 = first1 = quote1 = first# = unquote = 0 = s = sel1 = unquote1 = quote = active = ok = nil1 = proper = sel = first = top = nil = cons

Argument Filtering

cons1: 1 2
s1: collapses to 1
mark: 1
fcons: all arguments are removed from fcons
from: collapses to 1
01: all arguments are removed from 01
first1: 1 2
quote1: all arguments are removed from quote1
first#: collapses to 2
unquote: all arguments are removed from unquote
0: all arguments are removed from 0
s: all arguments are removed from s
sel1: all arguments are removed from sel1
unquote1: all arguments are removed from unquote1
quote: all arguments are removed from quote
active: all arguments are removed from active
ok: all arguments are removed from ok
nil1: all arguments are removed from nil1
proper: all arguments are removed from proper
sel: 1 2
first: 2
top: 1
nil: all arguments are removed from nil
cons: 1 2

Status

cons1: lexicographic with permutation 1 → 1 2 → 2
mark: lexicographic with permutation 1 → 1
fcons: multiset
01: multiset
first1: lexicographic with permutation 1 → 2 2 → 1
quote1: multiset
unquote: multiset
0: multiset
s: multiset
sel1: multiset
unquote1: multiset
quote: multiset
active: multiset
ok: multiset
nil1: multiset
proper: multiset
sel: lexicographic with permutation 1 → 1 2 → 2
first: lexicographic with permutation 2 → 1
top: lexicographic with permutation 1 → 1
nil: multiset
cons: lexicographic with permutation 1 → 1 2 → 2

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.

first#(X1, mark(X2)) → first#(X1, X2)

Problem 12: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

s#(mark(X)) → s#(X)s#(ok(X)) → s#(X)

Rewrite Rules

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))proper(0) → ok(0)
proper(first(X1, X2)) → first(proper(X1), proper(X2))proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01) → ok(01)proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))top(ok(X)) → top(active(X))

Original Signature

Termination of terms over the following signature is verified: s1, cons1, mark, fcons, from, 01, first1, quote1, unquote, 0, s, sel1, unquote1, quote, active, ok, nil1, proper, first, sel, nil, cons, top

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

s#(mark(X)) → s#(X)s#(ok(X)) → s#(X)

Problem 13: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

quote1#(ok(X)) → quote1#(X)

Rewrite Rules

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))proper(0) → ok(0)
proper(first(X1, X2)) → first(proper(X1), proper(X2))proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01) → ok(01)proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))top(ok(X)) → top(active(X))

Original Signature

Termination of terms over the following signature is verified: s1, cons1, mark, fcons, from, 01, first1, quote1, unquote, 0, s, sel1, unquote1, quote, active, ok, nil1, proper, first, sel, nil, cons, top

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

quote1#(ok(X)) → quote1#(X)

Problem 14: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))proper(0) → ok(0)
proper(first(X1, X2)) → first(proper(X1), proper(X2))proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01) → ok(01)proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))top(ok(X)) → top(active(X))

Original Signature

Termination of terms over the following signature is verified: s1, cons1, mark, fcons, from, 01, first1, quote1, unquote, 0, s, sel1, unquote1, quote, active, ok, nil1, proper, first, sel, nil, cons, top

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

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

Problem 22: ReductionPairSAT

Dependency Pair Problem

Dependency Pairs

fcons#(X1, mark(X2)) → fcons#(X1, X2)

Rewrite Rules

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))proper(0) → ok(0)
proper(first(X1, X2)) → first(proper(X1), proper(X2))proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01) → ok(01)proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))top(ok(X)) → top(active(X))

Original Signature

Termination of terms over the following signature is verified: s1, cons1, mark, fcons, from, 01, first1, quote1, unquote, 0, s, sel1, unquote1, quote, active, ok, proper, nil1, sel, first, top, cons, nil

Strategy

Function Precedence

cons1 = s1 = fcons# = mark = fcons = from = 01 = first1 = quote1 = unquote = 0 = s = sel1 = unquote1 = quote = active = ok = proper = nil1 = sel = first = top = nil = cons

Argument Filtering

cons1: all arguments are removed from cons1
s1: all arguments are removed from s1
fcons#: 2
mark: 1
fcons: all arguments are removed from fcons
from: all arguments are removed from from
01: all arguments are removed from 01
first1: all arguments are removed from first1
quote1: all arguments are removed from quote1
unquote: all arguments are removed from unquote
0: all arguments are removed from 0
s: all arguments are removed from s
sel1: all arguments are removed from sel1
unquote1: all arguments are removed from unquote1
quote: all arguments are removed from quote
active: all arguments are removed from active
ok: all arguments are removed from ok
proper: all arguments are removed from proper
nil1: all arguments are removed from nil1
sel: all arguments are removed from sel
first: all arguments are removed from first
top: 1
nil: all arguments are removed from nil
cons: 1 2

Status

cons1: multiset
s1: multiset
fcons#: lexicographic with permutation 2 → 1
mark: multiset
fcons: multiset
from: multiset
01: multiset
first1: multiset
quote1: multiset
unquote: multiset
0: multiset
s: multiset
sel1: multiset
unquote1: multiset
quote: multiset
active: multiset
ok: multiset
proper: multiset
nil1: multiset
sel: multiset
first: multiset
top: lexicographic with permutation 1 → 1
nil: multiset
cons: lexicographic with permutation 1 → 1 2 → 2

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.

fcons#(X1, mark(X2)) → fcons#(X1, X2)

Problem 15: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

first1#(mark(X1), X2) → first1#(X1, X2)first1#(ok(X1), ok(X2)) → first1#(X1, X2)
first1#(X1, mark(X2)) → first1#(X1, X2)

Rewrite Rules

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))proper(0) → ok(0)
proper(first(X1, X2)) → first(proper(X1), proper(X2))proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01) → ok(01)proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))top(ok(X)) → top(active(X))

Original Signature

Termination of terms over the following signature is verified: s1, cons1, mark, fcons, from, 01, first1, quote1, unquote, 0, s, sel1, unquote1, quote, active, ok, nil1, proper, first, sel, nil, cons, top

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

first1#(mark(X1), X2) → first1#(X1, X2)first1#(ok(X1), ok(X2)) → first1#(X1, X2)

Problem 23: ReductionPairSAT

Dependency Pair Problem

Dependency Pairs

first1#(X1, mark(X2)) → first1#(X1, X2)

Rewrite Rules

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))proper(0) → ok(0)
proper(first(X1, X2)) → first(proper(X1), proper(X2))proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01) → ok(01)proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))top(ok(X)) → top(active(X))

Original Signature

Termination of terms over the following signature is verified: s1, cons1, mark, fcons, from, 01, first1, quote1, unquote, 0, s, sel1, unquote1, quote, active, ok, proper, nil1, sel, first, top, cons, nil

Strategy

Function Precedence

first1# < mark < cons1 = s1 = fcons = from = 01 = first1 = quote1 = unquote = 0 = s = sel1 = unquote1 = quote = active = ok = proper = nil1 = sel = first = top = nil = cons

Argument Filtering

cons1: all arguments are removed from cons1
s1: all arguments are removed from s1
mark: 1
fcons: all arguments are removed from fcons
from: all arguments are removed from from
01: all arguments are removed from 01
first1: all arguments are removed from first1
quote1: all arguments are removed from quote1
unquote: all arguments are removed from unquote
0: all arguments are removed from 0
first1#: 1 2
s: all arguments are removed from s
sel1: collapses to 1
unquote1: collapses to 1
quote: collapses to 1
active: all arguments are removed from active
ok: all arguments are removed from ok
proper: all arguments are removed from proper
nil1: all arguments are removed from nil1
sel: 1 2
first: 1 2
top: all arguments are removed from top
nil: all arguments are removed from nil
cons: 1

Status

cons1: multiset
s1: multiset
mark: multiset
fcons: multiset
from: multiset
01: multiset
first1: multiset
quote1: multiset
unquote: multiset
0: multiset
first1#: multiset
s: multiset
active: multiset
ok: multiset
proper: multiset
nil1: multiset
sel: lexicographic with permutation 1 → 1 2 → 2
first: lexicographic with permutation 1 → 1 2 → 2
top: multiset
nil: multiset
cons: lexicographic with permutation 1 → 1

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.

first1#(X1, mark(X2)) → first1#(X1, X2)

Problem 16: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

from#(mark(X)) → from#(X)from#(ok(X)) → from#(X)

Rewrite Rules

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))proper(0) → ok(0)
proper(first(X1, X2)) → first(proper(X1), proper(X2))proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01) → ok(01)proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))top(ok(X)) → top(active(X))

Original Signature

Termination of terms over the following signature is verified: s1, cons1, mark, fcons, from, 01, first1, quote1, unquote, 0, s, sel1, unquote1, quote, active, ok, nil1, proper, first, sel, nil, cons, top

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

from#(mark(X)) → from#(X)from#(ok(X)) → from#(X)

Problem 18: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))proper(0) → ok(0)
proper(first(X1, X2)) → first(proper(X1), proper(X2))proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01) → ok(01)proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))top(ok(X)) → top(active(X))

Original Signature

Termination of terms over the following signature is verified: s1, cons1, mark, fcons, from, 01, first1, quote1, unquote, 0, s, sel1, unquote1, quote, active, ok, nil1, proper, first, sel, nil, cons, top

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

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

Problem 25: ReductionPairSAT

Dependency Pair Problem

Dependency Pairs

sel#(X1, mark(X2)) → sel#(X1, X2)

Rewrite Rules

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))proper(0) → ok(0)
proper(first(X1, X2)) → first(proper(X1), proper(X2))proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01) → ok(01)proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))top(ok(X)) → top(active(X))

Original Signature

Termination of terms over the following signature is verified: s1, cons1, mark, fcons, from, 01, first1, quote1, unquote, 0, s, sel1, unquote1, quote, active, ok, proper, nil1, sel, first, top, cons, nil

Strategy

Function Precedence

mark < sel# < cons1 = s1 = fcons = from = 01 = first1 = quote1 = unquote = 0 = s = sel1 = unquote1 = quote = active = ok = proper = nil1 = sel = first = top = nil = cons

Argument Filtering

cons1: all arguments are removed from cons1
s1: all arguments are removed from s1
mark: 1
fcons: all arguments are removed from fcons
sel#: collapses to 2
from: all arguments are removed from from
01: all arguments are removed from 01
first1: all arguments are removed from first1
quote1: all arguments are removed from quote1
unquote: all arguments are removed from unquote
0: all arguments are removed from 0
s: all arguments are removed from s
sel1: all arguments are removed from sel1
unquote1: all arguments are removed from unquote1
quote: all arguments are removed from quote
active: all arguments are removed from active
ok: all arguments are removed from ok
proper: all arguments are removed from proper
nil1: all arguments are removed from nil1
sel: all arguments are removed from sel
first: all arguments are removed from first
top: all arguments are removed from top
nil: all arguments are removed from nil
cons: all arguments are removed from cons

Status

cons1: multiset
s1: multiset
mark: multiset
fcons: multiset
from: multiset
01: multiset
first1: multiset
quote1: multiset
unquote: multiset
0: multiset
s: multiset
sel1: multiset
unquote1: multiset
quote: multiset
active: multiset
ok: multiset
proper: multiset
nil1: multiset
sel: multiset
first: multiset
top: multiset
nil: multiset
cons: 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.

sel#(X1, mark(X2)) → sel#(X1, X2)

Problem 19: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

sel1#(X1, mark(X2)) → sel1#(X1, X2)sel1#(mark(X1), X2) → sel1#(X1, X2)
sel1#(ok(X1), ok(X2)) → sel1#(X1, X2)

Rewrite Rules

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))proper(0) → ok(0)
proper(first(X1, X2)) → first(proper(X1), proper(X2))proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01) → ok(01)proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))top(ok(X)) → top(active(X))

Original Signature

Termination of terms over the following signature is verified: s1, cons1, mark, fcons, from, 01, first1, quote1, unquote, 0, s, sel1, unquote1, quote, active, ok, nil1, proper, first, sel, nil, cons, top

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

sel1#(mark(X1), X2) → sel1#(X1, X2)sel1#(ok(X1), ok(X2)) → sel1#(X1, X2)

Problem 24: PolynomialLinearRange4iUR

Dependency Pair Problem

Dependency Pairs

sel1#(X1, mark(X2)) → sel1#(X1, X2)

Rewrite Rules

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))proper(0) → ok(0)
proper(first(X1, X2)) → first(proper(X1), proper(X2))proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01) → ok(01)proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))top(ok(X)) → top(active(X))

Original Signature

Termination of terms over the following signature is verified: s1, cons1, mark, fcons, from, 01, first1, quote1, unquote, 0, s, sel1, unquote1, quote, active, ok, proper, nil1, sel, first, top, 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:

sel1#(X1, mark(X2)) → sel1#(X1, X2)