TIMEOUT

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

The following DP Processors were used


Problem 1 was processed with processor DependencyGraph (16268ms).
 | – Problem 2 was processed with processor SubtermCriterion (2ms).
 |    | – Problem 16 was processed with processor PolynomialLinearRange4iUR (42ms).
 |    |    | – Problem 24 was processed with processor PolynomialLinearRange4iUR (16ms).
 | – Problem 3 was processed with processor SubtermCriterion (4ms).
 | – Problem 4 was processed with processor SubtermCriterion (1ms).
 |    | – Problem 17 was processed with processor PolynomialLinearRange4iUR (126ms).
 |    |    | – Problem 23 was processed with processor PolynomialLinearRange4iUR (76ms).
 | – Problem 5 was processed with processor SubtermCriterion (1ms).
 | – Problem 6 was processed with processor SubtermCriterion (1ms).
 | – Problem 7 was processed with processor SubtermCriterion (2ms).
 | – Problem 8 was processed with processor SubtermCriterion (3ms).
 |    | – Problem 18 was processed with processor PolynomialLinearRange4iUR (69ms).
 |    |    | – Problem 26 remains open; application of the following processors failed [DependencyGraph (3ms), PolynomialLinearRange4iUR (5ms), DependencyGraph (2ms), PolynomialLinearRange4iUR (5ms), DependencyGraph (3ms), PolynomialLinearRange4iUR (0ms), DependencyGraph (2ms), PolynomialLinearRange8NegiUR (8ms), DependencyGraph (3ms)].
 | – Problem 9 remains open; application of the following processors failed [SubtermCriterion (1ms), DependencyGraph (2746ms), PolynomialLinearRange4iUR (1283ms), DependencyGraph (2749ms), PolynomialLinearRange4iUR (1428ms), DependencyGraph (2762ms), PolynomialLinearRange4iUR (2000ms), DependencyGraph (2857ms), PolynomialLinearRange4iUR (2512ms), DependencyGraph (2841ms), PolynomialLinearRange8NegiUR (7500ms), DependencyGraph (2915ms), ReductionPairSAT (timeout)].
 | – Problem 10 was processed with processor SubtermCriterion (1ms).
 |    | – Problem 19 was processed with processor PolynomialLinearRange4iUR (51ms).
 |    |    | – Problem 25 was processed with processor PolynomialLinearRange4iUR (29ms).
 | – Problem 11 was processed with processor SubtermCriterion (2ms).
 | – Problem 12 was processed with processor SubtermCriterion (3ms).
 |    | – Problem 20 was processed with processor PolynomialLinearRange4iUR (77ms).
 | – Problem 13 was processed with processor SubtermCriterion (2ms).
 |    | – Problem 21 was processed with processor PolynomialLinearRange4iUR (28ms).
 |    |    | – Problem 27 remains open; application of the following processors failed [DependencyGraph (2ms), PolynomialLinearRange4iUR (4ms), DependencyGraph (2ms), PolynomialLinearRange4iUR (21ms), DependencyGraph (2ms), PolynomialLinearRange4iUR (9ms), DependencyGraph (2ms), PolynomialLinearRange8NegiUR (16ms), DependencyGraph (3ms)].
 | – Problem 14 was processed with processor SubtermCriterion (1ms).
 | – Problem 15 was processed with processor SubtermCriterion (2ms).
 |    | – Problem 22 was processed with processor PolynomialLinearRange4iUR (57ms).
 |    |    | – Problem 28 remains open; application of the following processors failed [DependencyGraph (24ms), PolynomialLinearRange4iUR (3ms), DependencyGraph (26ms), PolynomialLinearRange4iUR (6ms), DependencyGraph (23ms), PolynomialLinearRange4iUR (10ms), DependencyGraph (26ms), PolynomialLinearRange8NegiUR (6ms), DependencyGraph (24ms)].

The following open problems remain:



Open Dependency Pair Problem 9

Dependency Pairs

active#(take(N, XS))mark#(fst(splitAt(N, XS)))mark#(fst(X))active#(fst(mark(X)))
mark#(head(X))active#(head(mark(X)))mark#(cons(X1, X2))active#(cons(mark(X1), X2))
mark#(take(X1, X2))mark#(X1)active#(u(pair(YS, ZS), N, X, XS))mark#(pair(cons(X, YS), ZS))
active#(splitAt(0, XS))mark#(pair(nil, XS))mark#(pair(X1, X2))mark#(X2)
mark#(u(X1, X2, X3, X4))active#(u(mark(X1), X2, X3, X4))mark#(take(X1, X2))active#(take(mark(X1), mark(X2)))
active#(head(cons(N, XS)))mark#(N)mark#(tail(X))active#(tail(mark(X)))
mark#(head(X))mark#(X)mark#(nil)active#(nil)
mark#(splitAt(X1, X2))mark#(X1)active#(natsFrom(N))mark#(cons(N, natsFrom(s(N))))
active#(tail(cons(N, XS)))mark#(XS)mark#(snd(X))active#(snd(mark(X)))
mark#(sel(X1, X2))mark#(X2)mark#(s(X))mark#(X)
mark#(natsFrom(X))active#(natsFrom(mark(X)))mark#(sel(X1, X2))mark#(X1)
mark#(fst(X))mark#(X)active#(afterNth(N, XS))mark#(snd(splitAt(N, XS)))
mark#(0)active#(0)mark#(s(X))active#(s(mark(X)))
mark#(cons(X1, X2))mark#(X1)mark#(splitAt(X1, X2))mark#(X2)
mark#(snd(X))mark#(X)mark#(afterNth(X1, X2))mark#(X1)
mark#(natsFrom(X))mark#(X)mark#(splitAt(X1, X2))active#(splitAt(mark(X1), mark(X2)))
mark#(sel(X1, X2))active#(sel(mark(X1), mark(X2)))mark#(afterNth(X1, X2))active#(afterNth(mark(X1), mark(X2)))
mark#(pair(X1, X2))mark#(X1)active#(splitAt(s(N), cons(X, XS)))mark#(u(splitAt(N, XS), N, X, XS))
mark#(tail(X))mark#(X)mark#(u(X1, X2, X3, X4))mark#(X1)
active#(sel(N, XS))mark#(head(afterNth(N, XS)))mark#(take(X1, X2))mark#(X2)
active#(snd(pair(XS, YS)))mark#(YS)active#(fst(pair(XS, YS)))mark#(XS)
mark#(afterNth(X1, X2))mark#(X2)mark#(pair(X1, X2))active#(pair(mark(X1), mark(X2)))

Rewrite Rules

active(natsFrom(N))mark(cons(N, natsFrom(s(N))))active(fst(pair(XS, YS)))mark(XS)
active(snd(pair(XS, YS)))mark(YS)active(splitAt(0, XS))mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS)))mark(u(splitAt(N, XS), N, X, XS))active(u(pair(YS, ZS), N, X, XS))mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS)))mark(N)active(tail(cons(N, XS)))mark(XS)
active(sel(N, XS))mark(head(afterNth(N, XS)))active(take(N, XS))mark(fst(splitAt(N, XS)))
active(afterNth(N, XS))mark(snd(splitAt(N, XS)))mark(natsFrom(X))active(natsFrom(mark(X)))
mark(cons(X1, X2))active(cons(mark(X1), X2))mark(s(X))active(s(mark(X)))
mark(fst(X))active(fst(mark(X)))mark(pair(X1, X2))active(pair(mark(X1), mark(X2)))
mark(snd(X))active(snd(mark(X)))mark(splitAt(X1, X2))active(splitAt(mark(X1), mark(X2)))
mark(0)active(0)mark(nil)active(nil)
mark(u(X1, X2, X3, X4))active(u(mark(X1), X2, X3, X4))mark(head(X))active(head(mark(X)))
mark(tail(X))active(tail(mark(X)))mark(sel(X1, X2))active(sel(mark(X1), mark(X2)))
mark(afterNth(X1, X2))active(afterNth(mark(X1), mark(X2)))mark(take(X1, X2))active(take(mark(X1), mark(X2)))
natsFrom(mark(X))natsFrom(X)natsFrom(active(X))natsFrom(X)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
s(mark(X))s(X)s(active(X))s(X)
fst(mark(X))fst(X)fst(active(X))fst(X)
pair(mark(X1), X2)pair(X1, X2)pair(X1, mark(X2))pair(X1, X2)
pair(active(X1), X2)pair(X1, X2)pair(X1, active(X2))pair(X1, X2)
snd(mark(X))snd(X)snd(active(X))snd(X)
splitAt(mark(X1), X2)splitAt(X1, X2)splitAt(X1, mark(X2))splitAt(X1, X2)
splitAt(active(X1), X2)splitAt(X1, X2)splitAt(X1, active(X2))splitAt(X1, X2)
u(mark(X1), X2, X3, X4)u(X1, X2, X3, X4)u(X1, mark(X2), X3, X4)u(X1, X2, X3, X4)
u(X1, X2, mark(X3), X4)u(X1, X2, X3, X4)u(X1, X2, X3, mark(X4))u(X1, X2, X3, X4)
u(active(X1), X2, X3, X4)u(X1, X2, X3, X4)u(X1, active(X2), X3, X4)u(X1, X2, X3, X4)
u(X1, X2, active(X3), X4)u(X1, X2, X3, X4)u(X1, X2, X3, active(X4))u(X1, X2, X3, X4)
head(mark(X))head(X)head(active(X))head(X)
tail(mark(X))tail(X)tail(active(X))tail(X)
sel(mark(X1), X2)sel(X1, X2)sel(X1, mark(X2))sel(X1, X2)
sel(active(X1), X2)sel(X1, X2)sel(X1, active(X2))sel(X1, X2)
afterNth(mark(X1), X2)afterNth(X1, X2)afterNth(X1, mark(X2))afterNth(X1, X2)
afterNth(active(X1), X2)afterNth(X1, X2)afterNth(X1, active(X2))afterNth(X1, X2)
take(mark(X1), X2)take(X1, X2)take(X1, mark(X2))take(X1, X2)
take(active(X1), X2)take(X1, X2)take(X1, active(X2))take(X1, X2)

Original Signature

Termination of terms over the following signature is verified: pair, natsFrom, mark, tail, splitAt, fst, u, 0, s, take, active, afterNth, head, sel, cons, snd, nil




Open Dependency Pair Problem 27

Dependency Pairs

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

Rewrite Rules

active(natsFrom(N))mark(cons(N, natsFrom(s(N))))active(fst(pair(XS, YS)))mark(XS)
active(snd(pair(XS, YS)))mark(YS)active(splitAt(0, XS))mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS)))mark(u(splitAt(N, XS), N, X, XS))active(u(pair(YS, ZS), N, X, XS))mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS)))mark(N)active(tail(cons(N, XS)))mark(XS)
active(sel(N, XS))mark(head(afterNth(N, XS)))active(take(N, XS))mark(fst(splitAt(N, XS)))
active(afterNth(N, XS))mark(snd(splitAt(N, XS)))mark(natsFrom(X))active(natsFrom(mark(X)))
mark(cons(X1, X2))active(cons(mark(X1), X2))mark(s(X))active(s(mark(X)))
mark(fst(X))active(fst(mark(X)))mark(pair(X1, X2))active(pair(mark(X1), mark(X2)))
mark(snd(X))active(snd(mark(X)))mark(splitAt(X1, X2))active(splitAt(mark(X1), mark(X2)))
mark(0)active(0)mark(nil)active(nil)
mark(u(X1, X2, X3, X4))active(u(mark(X1), X2, X3, X4))mark(head(X))active(head(mark(X)))
mark(tail(X))active(tail(mark(X)))mark(sel(X1, X2))active(sel(mark(X1), mark(X2)))
mark(afterNth(X1, X2))active(afterNth(mark(X1), mark(X2)))mark(take(X1, X2))active(take(mark(X1), mark(X2)))
natsFrom(mark(X))natsFrom(X)natsFrom(active(X))natsFrom(X)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
s(mark(X))s(X)s(active(X))s(X)
fst(mark(X))fst(X)fst(active(X))fst(X)
pair(mark(X1), X2)pair(X1, X2)pair(X1, mark(X2))pair(X1, X2)
pair(active(X1), X2)pair(X1, X2)pair(X1, active(X2))pair(X1, X2)
snd(mark(X))snd(X)snd(active(X))snd(X)
splitAt(mark(X1), X2)splitAt(X1, X2)splitAt(X1, mark(X2))splitAt(X1, X2)
splitAt(active(X1), X2)splitAt(X1, X2)splitAt(X1, active(X2))splitAt(X1, X2)
u(mark(X1), X2, X3, X4)u(X1, X2, X3, X4)u(X1, mark(X2), X3, X4)u(X1, X2, X3, X4)
u(X1, X2, mark(X3), X4)u(X1, X2, X3, X4)u(X1, X2, X3, mark(X4))u(X1, X2, X3, X4)
u(active(X1), X2, X3, X4)u(X1, X2, X3, X4)u(X1, active(X2), X3, X4)u(X1, X2, X3, X4)
u(X1, X2, active(X3), X4)u(X1, X2, X3, X4)u(X1, X2, X3, active(X4))u(X1, X2, X3, X4)
head(mark(X))head(X)head(active(X))head(X)
tail(mark(X))tail(X)tail(active(X))tail(X)
sel(mark(X1), X2)sel(X1, X2)sel(X1, mark(X2))sel(X1, X2)
sel(active(X1), X2)sel(X1, X2)sel(X1, active(X2))sel(X1, X2)
afterNth(mark(X1), X2)afterNth(X1, X2)afterNth(X1, mark(X2))afterNth(X1, X2)
afterNth(active(X1), X2)afterNth(X1, X2)afterNth(X1, active(X2))afterNth(X1, X2)
take(mark(X1), X2)take(X1, X2)take(X1, mark(X2))take(X1, X2)
take(active(X1), X2)take(X1, X2)take(X1, active(X2))take(X1, X2)

Original Signature

Termination of terms over the following signature is verified: pair, natsFrom, mark, tail, splitAt, fst, u, 0, s, take, active, afterNth, head, sel, cons, snd, nil




Open Dependency Pair Problem 26

Dependency Pairs

pair#(X1, mark(X2))pair#(X1, X2)

Rewrite Rules

active(natsFrom(N))mark(cons(N, natsFrom(s(N))))active(fst(pair(XS, YS)))mark(XS)
active(snd(pair(XS, YS)))mark(YS)active(splitAt(0, XS))mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS)))mark(u(splitAt(N, XS), N, X, XS))active(u(pair(YS, ZS), N, X, XS))mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS)))mark(N)active(tail(cons(N, XS)))mark(XS)
active(sel(N, XS))mark(head(afterNth(N, XS)))active(take(N, XS))mark(fst(splitAt(N, XS)))
active(afterNth(N, XS))mark(snd(splitAt(N, XS)))mark(natsFrom(X))active(natsFrom(mark(X)))
mark(cons(X1, X2))active(cons(mark(X1), X2))mark(s(X))active(s(mark(X)))
mark(fst(X))active(fst(mark(X)))mark(pair(X1, X2))active(pair(mark(X1), mark(X2)))
mark(snd(X))active(snd(mark(X)))mark(splitAt(X1, X2))active(splitAt(mark(X1), mark(X2)))
mark(0)active(0)mark(nil)active(nil)
mark(u(X1, X2, X3, X4))active(u(mark(X1), X2, X3, X4))mark(head(X))active(head(mark(X)))
mark(tail(X))active(tail(mark(X)))mark(sel(X1, X2))active(sel(mark(X1), mark(X2)))
mark(afterNth(X1, X2))active(afterNth(mark(X1), mark(X2)))mark(take(X1, X2))active(take(mark(X1), mark(X2)))
natsFrom(mark(X))natsFrom(X)natsFrom(active(X))natsFrom(X)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
s(mark(X))s(X)s(active(X))s(X)
fst(mark(X))fst(X)fst(active(X))fst(X)
pair(mark(X1), X2)pair(X1, X2)pair(X1, mark(X2))pair(X1, X2)
pair(active(X1), X2)pair(X1, X2)pair(X1, active(X2))pair(X1, X2)
snd(mark(X))snd(X)snd(active(X))snd(X)
splitAt(mark(X1), X2)splitAt(X1, X2)splitAt(X1, mark(X2))splitAt(X1, X2)
splitAt(active(X1), X2)splitAt(X1, X2)splitAt(X1, active(X2))splitAt(X1, X2)
u(mark(X1), X2, X3, X4)u(X1, X2, X3, X4)u(X1, mark(X2), X3, X4)u(X1, X2, X3, X4)
u(X1, X2, mark(X3), X4)u(X1, X2, X3, X4)u(X1, X2, X3, mark(X4))u(X1, X2, X3, X4)
u(active(X1), X2, X3, X4)u(X1, X2, X3, X4)u(X1, active(X2), X3, X4)u(X1, X2, X3, X4)
u(X1, X2, active(X3), X4)u(X1, X2, X3, X4)u(X1, X2, X3, active(X4))u(X1, X2, X3, X4)
head(mark(X))head(X)head(active(X))head(X)
tail(mark(X))tail(X)tail(active(X))tail(X)
sel(mark(X1), X2)sel(X1, X2)sel(X1, mark(X2))sel(X1, X2)
sel(active(X1), X2)sel(X1, X2)sel(X1, active(X2))sel(X1, X2)
afterNth(mark(X1), X2)afterNth(X1, X2)afterNth(X1, mark(X2))afterNth(X1, X2)
afterNth(active(X1), X2)afterNth(X1, X2)afterNth(X1, active(X2))afterNth(X1, X2)
take(mark(X1), X2)take(X1, X2)take(X1, mark(X2))take(X1, X2)
take(active(X1), X2)take(X1, X2)take(X1, active(X2))take(X1, X2)

Original Signature

Termination of terms over the following signature is verified: pair, natsFrom, mark, tail, splitAt, fst, u, 0, s, take, active, afterNth, head, sel, cons, snd, nil




Open Dependency Pair Problem 28

Dependency Pairs

u#(X1, mark(X2), X3, X4)u#(X1, X2, X3, X4)u#(X1, X2, active(X3), X4)u#(X1, X2, X3, X4)
u#(X1, X2, X3, active(X4))u#(X1, X2, X3, X4)u#(X1, active(X2), X3, X4)u#(X1, X2, X3, X4)
u#(X1, X2, mark(X3), X4)u#(X1, X2, X3, X4)

Rewrite Rules

active(natsFrom(N))mark(cons(N, natsFrom(s(N))))active(fst(pair(XS, YS)))mark(XS)
active(snd(pair(XS, YS)))mark(YS)active(splitAt(0, XS))mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS)))mark(u(splitAt(N, XS), N, X, XS))active(u(pair(YS, ZS), N, X, XS))mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS)))mark(N)active(tail(cons(N, XS)))mark(XS)
active(sel(N, XS))mark(head(afterNth(N, XS)))active(take(N, XS))mark(fst(splitAt(N, XS)))
active(afterNth(N, XS))mark(snd(splitAt(N, XS)))mark(natsFrom(X))active(natsFrom(mark(X)))
mark(cons(X1, X2))active(cons(mark(X1), X2))mark(s(X))active(s(mark(X)))
mark(fst(X))active(fst(mark(X)))mark(pair(X1, X2))active(pair(mark(X1), mark(X2)))
mark(snd(X))active(snd(mark(X)))mark(splitAt(X1, X2))active(splitAt(mark(X1), mark(X2)))
mark(0)active(0)mark(nil)active(nil)
mark(u(X1, X2, X3, X4))active(u(mark(X1), X2, X3, X4))mark(head(X))active(head(mark(X)))
mark(tail(X))active(tail(mark(X)))mark(sel(X1, X2))active(sel(mark(X1), mark(X2)))
mark(afterNth(X1, X2))active(afterNth(mark(X1), mark(X2)))mark(take(X1, X2))active(take(mark(X1), mark(X2)))
natsFrom(mark(X))natsFrom(X)natsFrom(active(X))natsFrom(X)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
s(mark(X))s(X)s(active(X))s(X)
fst(mark(X))fst(X)fst(active(X))fst(X)
pair(mark(X1), X2)pair(X1, X2)pair(X1, mark(X2))pair(X1, X2)
pair(active(X1), X2)pair(X1, X2)pair(X1, active(X2))pair(X1, X2)
snd(mark(X))snd(X)snd(active(X))snd(X)
splitAt(mark(X1), X2)splitAt(X1, X2)splitAt(X1, mark(X2))splitAt(X1, X2)
splitAt(active(X1), X2)splitAt(X1, X2)splitAt(X1, active(X2))splitAt(X1, X2)
u(mark(X1), X2, X3, X4)u(X1, X2, X3, X4)u(X1, mark(X2), X3, X4)u(X1, X2, X3, X4)
u(X1, X2, mark(X3), X4)u(X1, X2, X3, X4)u(X1, X2, X3, mark(X4))u(X1, X2, X3, X4)
u(active(X1), X2, X3, X4)u(X1, X2, X3, X4)u(X1, active(X2), X3, X4)u(X1, X2, X3, X4)
u(X1, X2, active(X3), X4)u(X1, X2, X3, X4)u(X1, X2, X3, active(X4))u(X1, X2, X3, X4)
head(mark(X))head(X)head(active(X))head(X)
tail(mark(X))tail(X)tail(active(X))tail(X)
sel(mark(X1), X2)sel(X1, X2)sel(X1, mark(X2))sel(X1, X2)
sel(active(X1), X2)sel(X1, X2)sel(X1, active(X2))sel(X1, X2)
afterNth(mark(X1), X2)afterNth(X1, X2)afterNth(X1, mark(X2))afterNth(X1, X2)
afterNth(active(X1), X2)afterNth(X1, X2)afterNth(X1, active(X2))afterNth(X1, X2)
take(mark(X1), X2)take(X1, X2)take(X1, mark(X2))take(X1, X2)
take(active(X1), X2)take(X1, X2)take(X1, active(X2))take(X1, X2)

Original Signature

Termination of terms over the following signature is verified: pair, natsFrom, mark, tail, splitAt, fst, u, 0, s, take, active, afterNth, head, sel, cons, snd, nil


Problem 1: DependencyGraph



Dependency Pair Problem

Dependency Pairs

mark#(fst(X))active#(fst(mark(X)))active#(u(pair(YS, ZS), N, X, XS))mark#(pair(cons(X, YS), ZS))
active#(take(N, XS))splitAt#(N, XS)mark#(pair(X1, X2))mark#(X2)
mark#(take(X1, X2))active#(take(mark(X1), mark(X2)))u#(X1, X2, X3, active(X4))u#(X1, X2, X3, X4)
splitAt#(mark(X1), X2)splitAt#(X1, X2)active#(afterNth(N, XS))snd#(splitAt(N, XS))
mark#(snd(X))active#(snd(mark(X)))active#(splitAt(0, XS))pair#(nil, XS)
mark#(s(X))mark#(X)mark#(sel(X1, X2))mark#(X1)
afterNth#(active(X1), X2)afterNth#(X1, X2)mark#(cons(X1, X2))mark#(X1)
pair#(X1, mark(X2))pair#(X1, X2)splitAt#(X1, mark(X2))splitAt#(X1, X2)
mark#(afterNth(X1, X2))mark#(X1)fst#(active(X))fst#(X)
sel#(X1, mark(X2))sel#(X1, X2)active#(sel(N, XS))head#(afterNth(N, XS))
u#(active(X1), X2, X3, X4)u#(X1, X2, X3, X4)mark#(pair(X1, X2))mark#(X1)
sel#(X1, active(X2))sel#(X1, X2)mark#(tail(X))mark#(X)
active#(u(pair(YS, ZS), N, X, XS))cons#(X, YS)head#(active(X))head#(X)
mark#(afterNth(X1, X2))mark#(X2)active#(take(N, XS))mark#(fst(splitAt(N, XS)))
pair#(mark(X1), X2)pair#(X1, X2)active#(afterNth(N, XS))splitAt#(N, XS)
natsFrom#(mark(X))natsFrom#(X)cons#(mark(X1), X2)cons#(X1, X2)
active#(splitAt(0, XS))mark#(pair(nil, XS))mark#(tail(X))tail#(mark(X))
mark#(splitAt(X1, X2))splitAt#(mark(X1), mark(X2))mark#(u(X1, X2, X3, X4))u#(mark(X1), X2, X3, X4)
mark#(head(X))mark#(X)mark#(splitAt(X1, X2))mark#(X1)
u#(X1, X2, active(X3), X4)u#(X1, X2, X3, X4)splitAt#(X1, active(X2))splitAt#(X1, X2)
cons#(X1, mark(X2))cons#(X1, X2)mark#(fst(X))mark#(X)
active#(afterNth(N, XS))mark#(snd(splitAt(N, XS)))mark#(0)active#(0)
mark#(s(X))active#(s(mark(X)))snd#(mark(X))snd#(X)
head#(mark(X))head#(X)pair#(X1, active(X2))pair#(X1, X2)
mark#(snd(X))mark#(X)mark#(natsFrom(X))mark#(X)
cons#(active(X1), X2)cons#(X1, X2)mark#(afterNth(X1, X2))active#(afterNth(mark(X1), mark(X2)))
snd#(active(X))snd#(X)u#(mark(X1), X2, X3, X4)u#(X1, X2, X3, X4)
active#(natsFrom(N))natsFrom#(s(N))mark#(cons(X1, X2))active#(cons(mark(X1), X2))
mark#(fst(X))fst#(mark(X))mark#(take(X1, X2))mark#(X1)
active#(splitAt(s(N), cons(X, XS)))u#(splitAt(N, XS), N, X, XS)mark#(snd(X))snd#(mark(X))
afterNth#(X1, mark(X2))afterNth#(X1, X2)mark#(u(X1, X2, X3, X4))active#(u(mark(X1), X2, X3, X4))
mark#(s(X))s#(mark(X))mark#(tail(X))active#(tail(mark(X)))
active#(sel(N, XS))afterNth#(N, XS)active#(natsFrom(N))mark#(cons(N, natsFrom(s(N))))
active#(tail(cons(N, XS)))mark#(XS)afterNth#(mark(X1), X2)afterNth#(X1, X2)
mark#(natsFrom(X))active#(natsFrom(mark(X)))pair#(active(X1), X2)pair#(X1, X2)
mark#(sel(X1, X2))sel#(mark(X1), mark(X2))tail#(mark(X))tail#(X)
take#(X1, mark(X2))take#(X1, X2)mark#(splitAt(X1, X2))active#(splitAt(mark(X1), mark(X2)))
mark#(sel(X1, X2))active#(sel(mark(X1), mark(X2)))sel#(active(X1), X2)sel#(X1, X2)
cons#(X1, active(X2))cons#(X1, X2)mark#(u(X1, X2, X3, X4))mark#(X1)
active#(snd(pair(XS, YS)))mark#(YS)mark#(afterNth(X1, X2))afterNth#(mark(X1), mark(X2))
active#(splitAt(s(N), cons(X, XS)))splitAt#(N, XS)sel#(mark(X1), X2)sel#(X1, X2)
active#(u(pair(YS, ZS), N, X, XS))pair#(cons(X, YS), ZS)mark#(head(X))active#(head(mark(X)))
take#(mark(X1), X2)take#(X1, X2)active#(natsFrom(N))s#(N)
mark#(pair(X1, X2))pair#(mark(X1), mark(X2))u#(X1, X2, X3, mark(X4))u#(X1, X2, X3, X4)
u#(X1, active(X2), X3, X4)u#(X1, X2, X3, X4)splitAt#(active(X1), X2)splitAt#(X1, X2)
u#(X1, X2, mark(X3), X4)u#(X1, X2, X3, X4)active#(head(cons(N, XS)))mark#(N)
mark#(nil)active#(nil)take#(X1, active(X2))take#(X1, X2)
tail#(active(X))tail#(X)mark#(sel(X1, X2))mark#(X2)
fst#(mark(X))fst#(X)active#(take(N, XS))fst#(splitAt(N, XS))
afterNth#(X1, active(X2))afterNth#(X1, X2)mark#(cons(X1, X2))cons#(mark(X1), X2)
natsFrom#(active(X))natsFrom#(X)u#(X1, mark(X2), X3, X4)u#(X1, X2, X3, X4)
mark#(splitAt(X1, X2))mark#(X2)mark#(head(X))head#(mark(X))
mark#(take(X1, X2))take#(mark(X1), mark(X2))s#(mark(X))s#(X)
active#(splitAt(s(N), cons(X, XS)))mark#(u(splitAt(N, XS), N, X, XS))mark#(take(X1, X2))mark#(X2)
active#(sel(N, XS))mark#(head(afterNth(N, XS)))s#(active(X))s#(X)
active#(natsFrom(N))cons#(N, natsFrom(s(N)))active#(fst(pair(XS, YS)))mark#(XS)
mark#(natsFrom(X))natsFrom#(mark(X))mark#(pair(X1, X2))active#(pair(mark(X1), mark(X2)))
take#(active(X1), X2)take#(X1, X2)

Rewrite Rules

active(natsFrom(N))mark(cons(N, natsFrom(s(N))))active(fst(pair(XS, YS)))mark(XS)
active(snd(pair(XS, YS)))mark(YS)active(splitAt(0, XS))mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS)))mark(u(splitAt(N, XS), N, X, XS))active(u(pair(YS, ZS), N, X, XS))mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS)))mark(N)active(tail(cons(N, XS)))mark(XS)
active(sel(N, XS))mark(head(afterNth(N, XS)))active(take(N, XS))mark(fst(splitAt(N, XS)))
active(afterNth(N, XS))mark(snd(splitAt(N, XS)))mark(natsFrom(X))active(natsFrom(mark(X)))
mark(cons(X1, X2))active(cons(mark(X1), X2))mark(s(X))active(s(mark(X)))
mark(fst(X))active(fst(mark(X)))mark(pair(X1, X2))active(pair(mark(X1), mark(X2)))
mark(snd(X))active(snd(mark(X)))mark(splitAt(X1, X2))active(splitAt(mark(X1), mark(X2)))
mark(0)active(0)mark(nil)active(nil)
mark(u(X1, X2, X3, X4))active(u(mark(X1), X2, X3, X4))mark(head(X))active(head(mark(X)))
mark(tail(X))active(tail(mark(X)))mark(sel(X1, X2))active(sel(mark(X1), mark(X2)))
mark(afterNth(X1, X2))active(afterNth(mark(X1), mark(X2)))mark(take(X1, X2))active(take(mark(X1), mark(X2)))
natsFrom(mark(X))natsFrom(X)natsFrom(active(X))natsFrom(X)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
s(mark(X))s(X)s(active(X))s(X)
fst(mark(X))fst(X)fst(active(X))fst(X)
pair(mark(X1), X2)pair(X1, X2)pair(X1, mark(X2))pair(X1, X2)
pair(active(X1), X2)pair(X1, X2)pair(X1, active(X2))pair(X1, X2)
snd(mark(X))snd(X)snd(active(X))snd(X)
splitAt(mark(X1), X2)splitAt(X1, X2)splitAt(X1, mark(X2))splitAt(X1, X2)
splitAt(active(X1), X2)splitAt(X1, X2)splitAt(X1, active(X2))splitAt(X1, X2)
u(mark(X1), X2, X3, X4)u(X1, X2, X3, X4)u(X1, mark(X2), X3, X4)u(X1, X2, X3, X4)
u(X1, X2, mark(X3), X4)u(X1, X2, X3, X4)u(X1, X2, X3, mark(X4))u(X1, X2, X3, X4)
u(active(X1), X2, X3, X4)u(X1, X2, X3, X4)u(X1, active(X2), X3, X4)u(X1, X2, X3, X4)
u(X1, X2, active(X3), X4)u(X1, X2, X3, X4)u(X1, X2, X3, active(X4))u(X1, X2, X3, X4)
head(mark(X))head(X)head(active(X))head(X)
tail(mark(X))tail(X)tail(active(X))tail(X)
sel(mark(X1), X2)sel(X1, X2)sel(X1, mark(X2))sel(X1, X2)
sel(active(X1), X2)sel(X1, X2)sel(X1, active(X2))sel(X1, X2)
afterNth(mark(X1), X2)afterNth(X1, X2)afterNth(X1, mark(X2))afterNth(X1, X2)
afterNth(active(X1), X2)afterNth(X1, X2)afterNth(X1, active(X2))afterNth(X1, X2)
take(mark(X1), X2)take(X1, X2)take(X1, mark(X2))take(X1, X2)
take(active(X1), X2)take(X1, X2)take(X1, active(X2))take(X1, X2)

Original Signature

Termination of terms over the following signature is verified: pair, natsFrom, mark, tail, splitAt, fst, u, 0, s, take, active, afterNth, head, sel, nil, snd, cons

Strategy


The following SCCs where found

active#(take(N, XS)) → mark#(fst(splitAt(N, XS)))mark#(cons(X1, X2)) → active#(cons(mark(X1), X2))
mark#(head(X)) → active#(head(mark(X)))mark#(fst(X)) → active#(fst(mark(X)))
mark#(take(X1, X2)) → mark#(X1)active#(u(pair(YS, ZS), N, X, XS)) → mark#(pair(cons(X, YS), ZS))
active#(splitAt(0, XS)) → mark#(pair(nil, XS))mark#(pair(X1, X2)) → mark#(X2)
mark#(u(X1, X2, X3, X4)) → active#(u(mark(X1), X2, X3, X4))mark#(take(X1, X2)) → active#(take(mark(X1), mark(X2)))
active#(head(cons(N, XS))) → mark#(N)mark#(tail(X)) → active#(tail(mark(X)))
mark#(nil) → active#(nil)mark#(head(X)) → mark#(X)
active#(natsFrom(N)) → mark#(cons(N, natsFrom(s(N))))mark#(splitAt(X1, X2)) → mark#(X1)
active#(tail(cons(N, XS))) → mark#(XS)mark#(snd(X)) → active#(snd(mark(X)))
mark#(sel(X1, X2)) → mark#(X2)mark#(s(X)) → mark#(X)
mark#(sel(X1, X2)) → mark#(X1)mark#(natsFrom(X)) → active#(natsFrom(mark(X)))
mark#(fst(X)) → mark#(X)active#(afterNth(N, XS)) → mark#(snd(splitAt(N, XS)))
mark#(0) → active#(0)mark#(s(X)) → active#(s(mark(X)))
mark#(cons(X1, X2)) → mark#(X1)mark#(snd(X)) → mark#(X)
mark#(splitAt(X1, X2)) → mark#(X2)mark#(natsFrom(X)) → mark#(X)
mark#(afterNth(X1, X2)) → mark#(X1)mark#(sel(X1, X2)) → active#(sel(mark(X1), mark(X2)))
mark#(splitAt(X1, X2)) → active#(splitAt(mark(X1), mark(X2)))mark#(afterNth(X1, X2)) → active#(afterNth(mark(X1), mark(X2)))
mark#(pair(X1, X2)) → mark#(X1)active#(splitAt(s(N), cons(X, XS))) → mark#(u(splitAt(N, XS), N, X, XS))
mark#(tail(X)) → mark#(X)mark#(u(X1, X2, X3, X4)) → mark#(X1)
mark#(take(X1, X2)) → mark#(X2)active#(sel(N, XS)) → mark#(head(afterNth(N, XS)))
active#(snd(pair(XS, YS))) → mark#(YS)mark#(afterNth(X1, X2)) → mark#(X2)
active#(fst(pair(XS, YS))) → mark#(XS)mark#(pair(X1, X2)) → active#(pair(mark(X1), mark(X2)))

u#(mark(X1), X2, X3, X4) → u#(X1, X2, X3, X4)u#(active(X1), X2, X3, X4) → u#(X1, X2, X3, X4)
u#(X1, mark(X2), X3, X4) → u#(X1, X2, X3, X4)u#(X1, X2, active(X3), X4) → u#(X1, X2, X3, X4)
u#(X1, X2, X3, mark(X4)) → u#(X1, X2, X3, X4)u#(X1, active(X2), X3, X4) → u#(X1, X2, X3, X4)
u#(X1, X2, X3, active(X4)) → u#(X1, X2, X3, X4)u#(X1, X2, mark(X3), X4) → u#(X1, X2, X3, X4)

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

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

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

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

fst#(mark(X)) → fst#(X)fst#(active(X)) → fst#(X)

natsFrom#(mark(X)) → natsFrom#(X)natsFrom#(active(X)) → natsFrom#(X)

tail#(active(X)) → tail#(X)tail#(mark(X)) → tail#(X)

snd#(active(X)) → snd#(X)snd#(mark(X)) → snd#(X)

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

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

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

head#(mark(X)) → head#(X)head#(active(X)) → head#(X)

Problem 2: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

active(natsFrom(N))mark(cons(N, natsFrom(s(N))))active(fst(pair(XS, YS)))mark(XS)
active(snd(pair(XS, YS)))mark(YS)active(splitAt(0, XS))mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS)))mark(u(splitAt(N, XS), N, X, XS))active(u(pair(YS, ZS), N, X, XS))mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS)))mark(N)active(tail(cons(N, XS)))mark(XS)
active(sel(N, XS))mark(head(afterNth(N, XS)))active(take(N, XS))mark(fst(splitAt(N, XS)))
active(afterNth(N, XS))mark(snd(splitAt(N, XS)))mark(natsFrom(X))active(natsFrom(mark(X)))
mark(cons(X1, X2))active(cons(mark(X1), X2))mark(s(X))active(s(mark(X)))
mark(fst(X))active(fst(mark(X)))mark(pair(X1, X2))active(pair(mark(X1), mark(X2)))
mark(snd(X))active(snd(mark(X)))mark(splitAt(X1, X2))active(splitAt(mark(X1), mark(X2)))
mark(0)active(0)mark(nil)active(nil)
mark(u(X1, X2, X3, X4))active(u(mark(X1), X2, X3, X4))mark(head(X))active(head(mark(X)))
mark(tail(X))active(tail(mark(X)))mark(sel(X1, X2))active(sel(mark(X1), mark(X2)))
mark(afterNth(X1, X2))active(afterNth(mark(X1), mark(X2)))mark(take(X1, X2))active(take(mark(X1), mark(X2)))
natsFrom(mark(X))natsFrom(X)natsFrom(active(X))natsFrom(X)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
s(mark(X))s(X)s(active(X))s(X)
fst(mark(X))fst(X)fst(active(X))fst(X)
pair(mark(X1), X2)pair(X1, X2)pair(X1, mark(X2))pair(X1, X2)
pair(active(X1), X2)pair(X1, X2)pair(X1, active(X2))pair(X1, X2)
snd(mark(X))snd(X)snd(active(X))snd(X)
splitAt(mark(X1), X2)splitAt(X1, X2)splitAt(X1, mark(X2))splitAt(X1, X2)
splitAt(active(X1), X2)splitAt(X1, X2)splitAt(X1, active(X2))splitAt(X1, X2)
u(mark(X1), X2, X3, X4)u(X1, X2, X3, X4)u(X1, mark(X2), X3, X4)u(X1, X2, X3, X4)
u(X1, X2, mark(X3), X4)u(X1, X2, X3, X4)u(X1, X2, X3, mark(X4))u(X1, X2, X3, X4)
u(active(X1), X2, X3, X4)u(X1, X2, X3, X4)u(X1, active(X2), X3, X4)u(X1, X2, X3, X4)
u(X1, X2, active(X3), X4)u(X1, X2, X3, X4)u(X1, X2, X3, active(X4))u(X1, X2, X3, X4)
head(mark(X))head(X)head(active(X))head(X)
tail(mark(X))tail(X)tail(active(X))tail(X)
sel(mark(X1), X2)sel(X1, X2)sel(X1, mark(X2))sel(X1, X2)
sel(active(X1), X2)sel(X1, X2)sel(X1, active(X2))sel(X1, X2)
afterNth(mark(X1), X2)afterNth(X1, X2)afterNth(X1, mark(X2))afterNth(X1, X2)
afterNth(active(X1), X2)afterNth(X1, X2)afterNth(X1, active(X2))afterNth(X1, X2)
take(mark(X1), X2)take(X1, X2)take(X1, mark(X2))take(X1, X2)
take(active(X1), X2)take(X1, X2)take(X1, active(X2))take(X1, X2)

Original Signature

Termination of terms over the following signature is verified: pair, natsFrom, mark, tail, splitAt, fst, u, 0, s, take, active, afterNth, head, sel, nil, snd, cons

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

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

Problem 16: PolynomialLinearRange4iUR



Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

active(natsFrom(N))mark(cons(N, natsFrom(s(N))))active(fst(pair(XS, YS)))mark(XS)
active(snd(pair(XS, YS)))mark(YS)active(splitAt(0, XS))mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS)))mark(u(splitAt(N, XS), N, X, XS))active(u(pair(YS, ZS), N, X, XS))mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS)))mark(N)active(tail(cons(N, XS)))mark(XS)
active(sel(N, XS))mark(head(afterNth(N, XS)))active(take(N, XS))mark(fst(splitAt(N, XS)))
active(afterNth(N, XS))mark(snd(splitAt(N, XS)))mark(natsFrom(X))active(natsFrom(mark(X)))
mark(cons(X1, X2))active(cons(mark(X1), X2))mark(s(X))active(s(mark(X)))
mark(fst(X))active(fst(mark(X)))mark(pair(X1, X2))active(pair(mark(X1), mark(X2)))
mark(snd(X))active(snd(mark(X)))mark(splitAt(X1, X2))active(splitAt(mark(X1), mark(X2)))
mark(0)active(0)mark(nil)active(nil)
mark(u(X1, X2, X3, X4))active(u(mark(X1), X2, X3, X4))mark(head(X))active(head(mark(X)))
mark(tail(X))active(tail(mark(X)))mark(sel(X1, X2))active(sel(mark(X1), mark(X2)))
mark(afterNth(X1, X2))active(afterNth(mark(X1), mark(X2)))mark(take(X1, X2))active(take(mark(X1), mark(X2)))
natsFrom(mark(X))natsFrom(X)natsFrom(active(X))natsFrom(X)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
s(mark(X))s(X)s(active(X))s(X)
fst(mark(X))fst(X)fst(active(X))fst(X)
pair(mark(X1), X2)pair(X1, X2)pair(X1, mark(X2))pair(X1, X2)
pair(active(X1), X2)pair(X1, X2)pair(X1, active(X2))pair(X1, X2)
snd(mark(X))snd(X)snd(active(X))snd(X)
splitAt(mark(X1), X2)splitAt(X1, X2)splitAt(X1, mark(X2))splitAt(X1, X2)
splitAt(active(X1), X2)splitAt(X1, X2)splitAt(X1, active(X2))splitAt(X1, X2)
u(mark(X1), X2, X3, X4)u(X1, X2, X3, X4)u(X1, mark(X2), X3, X4)u(X1, X2, X3, X4)
u(X1, X2, mark(X3), X4)u(X1, X2, X3, X4)u(X1, X2, X3, mark(X4))u(X1, X2, X3, X4)
u(active(X1), X2, X3, X4)u(X1, X2, X3, X4)u(X1, active(X2), X3, X4)u(X1, X2, X3, X4)
u(X1, X2, active(X3), X4)u(X1, X2, X3, X4)u(X1, X2, X3, active(X4))u(X1, X2, X3, X4)
head(mark(X))head(X)head(active(X))head(X)
tail(mark(X))tail(X)tail(active(X))tail(X)
sel(mark(X1), X2)sel(X1, X2)sel(X1, mark(X2))sel(X1, X2)
sel(active(X1), X2)sel(X1, X2)sel(X1, active(X2))sel(X1, X2)
afterNth(mark(X1), X2)afterNth(X1, X2)afterNth(X1, mark(X2))afterNth(X1, X2)
afterNth(active(X1), X2)afterNth(X1, X2)afterNth(X1, active(X2))afterNth(X1, X2)
take(mark(X1), X2)take(X1, X2)take(X1, mark(X2))take(X1, X2)
take(active(X1), X2)take(X1, X2)take(X1, active(X2))take(X1, X2)

Original Signature

Termination of terms over the following signature is verified: pair, natsFrom, mark, tail, splitAt, fst, u, 0, s, take, active, afterNth, head, sel, cons, snd, 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:

afterNth#(X1, active(X2))afterNth#(X1, X2)

Problem 24: PolynomialLinearRange4iUR



Dependency Pair Problem

Dependency Pairs

afterNth#(X1, mark(X2))afterNth#(X1, X2)

Rewrite Rules

active(natsFrom(N))mark(cons(N, natsFrom(s(N))))active(fst(pair(XS, YS)))mark(XS)
active(snd(pair(XS, YS)))mark(YS)active(splitAt(0, XS))mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS)))mark(u(splitAt(N, XS), N, X, XS))active(u(pair(YS, ZS), N, X, XS))mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS)))mark(N)active(tail(cons(N, XS)))mark(XS)
active(sel(N, XS))mark(head(afterNth(N, XS)))active(take(N, XS))mark(fst(splitAt(N, XS)))
active(afterNth(N, XS))mark(snd(splitAt(N, XS)))mark(natsFrom(X))active(natsFrom(mark(X)))
mark(cons(X1, X2))active(cons(mark(X1), X2))mark(s(X))active(s(mark(X)))
mark(fst(X))active(fst(mark(X)))mark(pair(X1, X2))active(pair(mark(X1), mark(X2)))
mark(snd(X))active(snd(mark(X)))mark(splitAt(X1, X2))active(splitAt(mark(X1), mark(X2)))
mark(0)active(0)mark(nil)active(nil)
mark(u(X1, X2, X3, X4))active(u(mark(X1), X2, X3, X4))mark(head(X))active(head(mark(X)))
mark(tail(X))active(tail(mark(X)))mark(sel(X1, X2))active(sel(mark(X1), mark(X2)))
mark(afterNth(X1, X2))active(afterNth(mark(X1), mark(X2)))mark(take(X1, X2))active(take(mark(X1), mark(X2)))
natsFrom(mark(X))natsFrom(X)natsFrom(active(X))natsFrom(X)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
s(mark(X))s(X)s(active(X))s(X)
fst(mark(X))fst(X)fst(active(X))fst(X)
pair(mark(X1), X2)pair(X1, X2)pair(X1, mark(X2))pair(X1, X2)
pair(active(X1), X2)pair(X1, X2)pair(X1, active(X2))pair(X1, X2)
snd(mark(X))snd(X)snd(active(X))snd(X)
splitAt(mark(X1), X2)splitAt(X1, X2)splitAt(X1, mark(X2))splitAt(X1, X2)
splitAt(active(X1), X2)splitAt(X1, X2)splitAt(X1, active(X2))splitAt(X1, X2)
u(mark(X1), X2, X3, X4)u(X1, X2, X3, X4)u(X1, mark(X2), X3, X4)u(X1, X2, X3, X4)
u(X1, X2, mark(X3), X4)u(X1, X2, X3, X4)u(X1, X2, X3, mark(X4))u(X1, X2, X3, X4)
u(active(X1), X2, X3, X4)u(X1, X2, X3, X4)u(X1, active(X2), X3, X4)u(X1, X2, X3, X4)
u(X1, X2, active(X3), X4)u(X1, X2, X3, X4)u(X1, X2, X3, active(X4))u(X1, X2, X3, X4)
head(mark(X))head(X)head(active(X))head(X)
tail(mark(X))tail(X)tail(active(X))tail(X)
sel(mark(X1), X2)sel(X1, X2)sel(X1, mark(X2))sel(X1, X2)
sel(active(X1), X2)sel(X1, X2)sel(X1, active(X2))sel(X1, X2)
afterNth(mark(X1), X2)afterNth(X1, X2)afterNth(X1, mark(X2))afterNth(X1, X2)
afterNth(active(X1), X2)afterNth(X1, X2)afterNth(X1, active(X2))afterNth(X1, X2)
take(mark(X1), X2)take(X1, X2)take(X1, mark(X2))take(X1, X2)
take(active(X1), X2)take(X1, X2)take(X1, active(X2))take(X1, X2)

Original Signature

Termination of terms over the following signature is verified: pair, natsFrom, mark, tail, splitAt, fst, u, 0, s, take, active, afterNth, head, sel, nil, snd, cons

Strategy


Polynomial Interpretation

There are no usable rules

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

afterNth#(X1, mark(X2))afterNth#(X1, X2)

Problem 3: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

tail#(active(X))tail#(X)tail#(mark(X))tail#(X)

Rewrite Rules

active(natsFrom(N))mark(cons(N, natsFrom(s(N))))active(fst(pair(XS, YS)))mark(XS)
active(snd(pair(XS, YS)))mark(YS)active(splitAt(0, XS))mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS)))mark(u(splitAt(N, XS), N, X, XS))active(u(pair(YS, ZS), N, X, XS))mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS)))mark(N)active(tail(cons(N, XS)))mark(XS)
active(sel(N, XS))mark(head(afterNth(N, XS)))active(take(N, XS))mark(fst(splitAt(N, XS)))
active(afterNth(N, XS))mark(snd(splitAt(N, XS)))mark(natsFrom(X))active(natsFrom(mark(X)))
mark(cons(X1, X2))active(cons(mark(X1), X2))mark(s(X))active(s(mark(X)))
mark(fst(X))active(fst(mark(X)))mark(pair(X1, X2))active(pair(mark(X1), mark(X2)))
mark(snd(X))active(snd(mark(X)))mark(splitAt(X1, X2))active(splitAt(mark(X1), mark(X2)))
mark(0)active(0)mark(nil)active(nil)
mark(u(X1, X2, X3, X4))active(u(mark(X1), X2, X3, X4))mark(head(X))active(head(mark(X)))
mark(tail(X))active(tail(mark(X)))mark(sel(X1, X2))active(sel(mark(X1), mark(X2)))
mark(afterNth(X1, X2))active(afterNth(mark(X1), mark(X2)))mark(take(X1, X2))active(take(mark(X1), mark(X2)))
natsFrom(mark(X))natsFrom(X)natsFrom(active(X))natsFrom(X)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
s(mark(X))s(X)s(active(X))s(X)
fst(mark(X))fst(X)fst(active(X))fst(X)
pair(mark(X1), X2)pair(X1, X2)pair(X1, mark(X2))pair(X1, X2)
pair(active(X1), X2)pair(X1, X2)pair(X1, active(X2))pair(X1, X2)
snd(mark(X))snd(X)snd(active(X))snd(X)
splitAt(mark(X1), X2)splitAt(X1, X2)splitAt(X1, mark(X2))splitAt(X1, X2)
splitAt(active(X1), X2)splitAt(X1, X2)splitAt(X1, active(X2))splitAt(X1, X2)
u(mark(X1), X2, X3, X4)u(X1, X2, X3, X4)u(X1, mark(X2), X3, X4)u(X1, X2, X3, X4)
u(X1, X2, mark(X3), X4)u(X1, X2, X3, X4)u(X1, X2, X3, mark(X4))u(X1, X2, X3, X4)
u(active(X1), X2, X3, X4)u(X1, X2, X3, X4)u(X1, active(X2), X3, X4)u(X1, X2, X3, X4)
u(X1, X2, active(X3), X4)u(X1, X2, X3, X4)u(X1, X2, X3, active(X4))u(X1, X2, X3, X4)
head(mark(X))head(X)head(active(X))head(X)
tail(mark(X))tail(X)tail(active(X))tail(X)
sel(mark(X1), X2)sel(X1, X2)sel(X1, mark(X2))sel(X1, X2)
sel(active(X1), X2)sel(X1, X2)sel(X1, active(X2))sel(X1, X2)
afterNth(mark(X1), X2)afterNth(X1, X2)afterNth(X1, mark(X2))afterNth(X1, X2)
afterNth(active(X1), X2)afterNth(X1, X2)afterNth(X1, active(X2))afterNth(X1, X2)
take(mark(X1), X2)take(X1, X2)take(X1, mark(X2))take(X1, X2)
take(active(X1), X2)take(X1, X2)take(X1, active(X2))take(X1, X2)

Original Signature

Termination of terms over the following signature is verified: pair, natsFrom, mark, tail, splitAt, fst, u, 0, s, take, active, afterNth, head, sel, nil, snd, cons

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

tail#(active(X))tail#(X)tail#(mark(X))tail#(X)

Problem 4: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

active(natsFrom(N))mark(cons(N, natsFrom(s(N))))active(fst(pair(XS, YS)))mark(XS)
active(snd(pair(XS, YS)))mark(YS)active(splitAt(0, XS))mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS)))mark(u(splitAt(N, XS), N, X, XS))active(u(pair(YS, ZS), N, X, XS))mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS)))mark(N)active(tail(cons(N, XS)))mark(XS)
active(sel(N, XS))mark(head(afterNth(N, XS)))active(take(N, XS))mark(fst(splitAt(N, XS)))
active(afterNth(N, XS))mark(snd(splitAt(N, XS)))mark(natsFrom(X))active(natsFrom(mark(X)))
mark(cons(X1, X2))active(cons(mark(X1), X2))mark(s(X))active(s(mark(X)))
mark(fst(X))active(fst(mark(X)))mark(pair(X1, X2))active(pair(mark(X1), mark(X2)))
mark(snd(X))active(snd(mark(X)))mark(splitAt(X1, X2))active(splitAt(mark(X1), mark(X2)))
mark(0)active(0)mark(nil)active(nil)
mark(u(X1, X2, X3, X4))active(u(mark(X1), X2, X3, X4))mark(head(X))active(head(mark(X)))
mark(tail(X))active(tail(mark(X)))mark(sel(X1, X2))active(sel(mark(X1), mark(X2)))
mark(afterNth(X1, X2))active(afterNth(mark(X1), mark(X2)))mark(take(X1, X2))active(take(mark(X1), mark(X2)))
natsFrom(mark(X))natsFrom(X)natsFrom(active(X))natsFrom(X)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
s(mark(X))s(X)s(active(X))s(X)
fst(mark(X))fst(X)fst(active(X))fst(X)
pair(mark(X1), X2)pair(X1, X2)pair(X1, mark(X2))pair(X1, X2)
pair(active(X1), X2)pair(X1, X2)pair(X1, active(X2))pair(X1, X2)
snd(mark(X))snd(X)snd(active(X))snd(X)
splitAt(mark(X1), X2)splitAt(X1, X2)splitAt(X1, mark(X2))splitAt(X1, X2)
splitAt(active(X1), X2)splitAt(X1, X2)splitAt(X1, active(X2))splitAt(X1, X2)
u(mark(X1), X2, X3, X4)u(X1, X2, X3, X4)u(X1, mark(X2), X3, X4)u(X1, X2, X3, X4)
u(X1, X2, mark(X3), X4)u(X1, X2, X3, X4)u(X1, X2, X3, mark(X4))u(X1, X2, X3, X4)
u(active(X1), X2, X3, X4)u(X1, X2, X3, X4)u(X1, active(X2), X3, X4)u(X1, X2, X3, X4)
u(X1, X2, active(X3), X4)u(X1, X2, X3, X4)u(X1, X2, X3, active(X4))u(X1, X2, X3, X4)
head(mark(X))head(X)head(active(X))head(X)
tail(mark(X))tail(X)tail(active(X))tail(X)
sel(mark(X1), X2)sel(X1, X2)sel(X1, mark(X2))sel(X1, X2)
sel(active(X1), X2)sel(X1, X2)sel(X1, active(X2))sel(X1, X2)
afterNth(mark(X1), X2)afterNth(X1, X2)afterNth(X1, mark(X2))afterNth(X1, X2)
afterNth(active(X1), X2)afterNth(X1, X2)afterNth(X1, active(X2))afterNth(X1, X2)
take(mark(X1), X2)take(X1, X2)take(X1, mark(X2))take(X1, X2)
take(active(X1), X2)take(X1, X2)take(X1, active(X2))take(X1, X2)

Original Signature

Termination of terms over the following signature is verified: pair, natsFrom, mark, tail, splitAt, fst, u, 0, s, take, active, afterNth, head, sel, nil, snd, cons

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

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

Problem 17: PolynomialLinearRange4iUR



Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

active(natsFrom(N))mark(cons(N, natsFrom(s(N))))active(fst(pair(XS, YS)))mark(XS)
active(snd(pair(XS, YS)))mark(YS)active(splitAt(0, XS))mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS)))mark(u(splitAt(N, XS), N, X, XS))active(u(pair(YS, ZS), N, X, XS))mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS)))mark(N)active(tail(cons(N, XS)))mark(XS)
active(sel(N, XS))mark(head(afterNth(N, XS)))active(take(N, XS))mark(fst(splitAt(N, XS)))
active(afterNth(N, XS))mark(snd(splitAt(N, XS)))mark(natsFrom(X))active(natsFrom(mark(X)))
mark(cons(X1, X2))active(cons(mark(X1), X2))mark(s(X))active(s(mark(X)))
mark(fst(X))active(fst(mark(X)))mark(pair(X1, X2))active(pair(mark(X1), mark(X2)))
mark(snd(X))active(snd(mark(X)))mark(splitAt(X1, X2))active(splitAt(mark(X1), mark(X2)))
mark(0)active(0)mark(nil)active(nil)
mark(u(X1, X2, X3, X4))active(u(mark(X1), X2, X3, X4))mark(head(X))active(head(mark(X)))
mark(tail(X))active(tail(mark(X)))mark(sel(X1, X2))active(sel(mark(X1), mark(X2)))
mark(afterNth(X1, X2))active(afterNth(mark(X1), mark(X2)))mark(take(X1, X2))active(take(mark(X1), mark(X2)))
natsFrom(mark(X))natsFrom(X)natsFrom(active(X))natsFrom(X)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
s(mark(X))s(X)s(active(X))s(X)
fst(mark(X))fst(X)fst(active(X))fst(X)
pair(mark(X1), X2)pair(X1, X2)pair(X1, mark(X2))pair(X1, X2)
pair(active(X1), X2)pair(X1, X2)pair(X1, active(X2))pair(X1, X2)
snd(mark(X))snd(X)snd(active(X))snd(X)
splitAt(mark(X1), X2)splitAt(X1, X2)splitAt(X1, mark(X2))splitAt(X1, X2)
splitAt(active(X1), X2)splitAt(X1, X2)splitAt(X1, active(X2))splitAt(X1, X2)
u(mark(X1), X2, X3, X4)u(X1, X2, X3, X4)u(X1, mark(X2), X3, X4)u(X1, X2, X3, X4)
u(X1, X2, mark(X3), X4)u(X1, X2, X3, X4)u(X1, X2, X3, mark(X4))u(X1, X2, X3, X4)
u(active(X1), X2, X3, X4)u(X1, X2, X3, X4)u(X1, active(X2), X3, X4)u(X1, X2, X3, X4)
u(X1, X2, active(X3), X4)u(X1, X2, X3, X4)u(X1, X2, X3, active(X4))u(X1, X2, X3, X4)
head(mark(X))head(X)head(active(X))head(X)
tail(mark(X))tail(X)tail(active(X))tail(X)
sel(mark(X1), X2)sel(X1, X2)sel(X1, mark(X2))sel(X1, X2)
sel(active(X1), X2)sel(X1, X2)sel(X1, active(X2))sel(X1, X2)
afterNth(mark(X1), X2)afterNth(X1, X2)afterNth(X1, mark(X2))afterNth(X1, X2)
afterNth(active(X1), X2)afterNth(X1, X2)afterNth(X1, active(X2))afterNth(X1, X2)
take(mark(X1), X2)take(X1, X2)take(X1, mark(X2))take(X1, X2)
take(active(X1), X2)take(X1, X2)take(X1, active(X2))take(X1, X2)

Original Signature

Termination of terms over the following signature is verified: pair, natsFrom, mark, tail, splitAt, fst, u, 0, s, take, active, afterNth, head, sel, cons, snd, 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:

take#(X1, active(X2))take#(X1, X2)

Problem 23: PolynomialLinearRange4iUR



Dependency Pair Problem

Dependency Pairs

take#(X1, mark(X2))take#(X1, X2)

Rewrite Rules

active(natsFrom(N))mark(cons(N, natsFrom(s(N))))active(fst(pair(XS, YS)))mark(XS)
active(snd(pair(XS, YS)))mark(YS)active(splitAt(0, XS))mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS)))mark(u(splitAt(N, XS), N, X, XS))active(u(pair(YS, ZS), N, X, XS))mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS)))mark(N)active(tail(cons(N, XS)))mark(XS)
active(sel(N, XS))mark(head(afterNth(N, XS)))active(take(N, XS))mark(fst(splitAt(N, XS)))
active(afterNth(N, XS))mark(snd(splitAt(N, XS)))mark(natsFrom(X))active(natsFrom(mark(X)))
mark(cons(X1, X2))active(cons(mark(X1), X2))mark(s(X))active(s(mark(X)))
mark(fst(X))active(fst(mark(X)))mark(pair(X1, X2))active(pair(mark(X1), mark(X2)))
mark(snd(X))active(snd(mark(X)))mark(splitAt(X1, X2))active(splitAt(mark(X1), mark(X2)))
mark(0)active(0)mark(nil)active(nil)
mark(u(X1, X2, X3, X4))active(u(mark(X1), X2, X3, X4))mark(head(X))active(head(mark(X)))
mark(tail(X))active(tail(mark(X)))mark(sel(X1, X2))active(sel(mark(X1), mark(X2)))
mark(afterNth(X1, X2))active(afterNth(mark(X1), mark(X2)))mark(take(X1, X2))active(take(mark(X1), mark(X2)))
natsFrom(mark(X))natsFrom(X)natsFrom(active(X))natsFrom(X)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
s(mark(X))s(X)s(active(X))s(X)
fst(mark(X))fst(X)fst(active(X))fst(X)
pair(mark(X1), X2)pair(X1, X2)pair(X1, mark(X2))pair(X1, X2)
pair(active(X1), X2)pair(X1, X2)pair(X1, active(X2))pair(X1, X2)
snd(mark(X))snd(X)snd(active(X))snd(X)
splitAt(mark(X1), X2)splitAt(X1, X2)splitAt(X1, mark(X2))splitAt(X1, X2)
splitAt(active(X1), X2)splitAt(X1, X2)splitAt(X1, active(X2))splitAt(X1, X2)
u(mark(X1), X2, X3, X4)u(X1, X2, X3, X4)u(X1, mark(X2), X3, X4)u(X1, X2, X3, X4)
u(X1, X2, mark(X3), X4)u(X1, X2, X3, X4)u(X1, X2, X3, mark(X4))u(X1, X2, X3, X4)
u(active(X1), X2, X3, X4)u(X1, X2, X3, X4)u(X1, active(X2), X3, X4)u(X1, X2, X3, X4)
u(X1, X2, active(X3), X4)u(X1, X2, X3, X4)u(X1, X2, X3, active(X4))u(X1, X2, X3, X4)
head(mark(X))head(X)head(active(X))head(X)
tail(mark(X))tail(X)tail(active(X))tail(X)
sel(mark(X1), X2)sel(X1, X2)sel(X1, mark(X2))sel(X1, X2)
sel(active(X1), X2)sel(X1, X2)sel(X1, active(X2))sel(X1, X2)
afterNth(mark(X1), X2)afterNth(X1, X2)afterNth(X1, mark(X2))afterNth(X1, X2)
afterNth(active(X1), X2)afterNth(X1, X2)afterNth(X1, active(X2))afterNth(X1, X2)
take(mark(X1), X2)take(X1, X2)take(X1, mark(X2))take(X1, X2)
take(active(X1), X2)take(X1, X2)take(X1, active(X2))take(X1, X2)

Original Signature

Termination of terms over the following signature is verified: pair, natsFrom, mark, tail, splitAt, fst, u, 0, s, take, active, afterNth, head, sel, nil, snd, cons

Strategy


Polynomial Interpretation

There are no usable rules

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

take#(X1, mark(X2))take#(X1, X2)

Problem 5: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

natsFrom#(mark(X))natsFrom#(X)natsFrom#(active(X))natsFrom#(X)

Rewrite Rules

active(natsFrom(N))mark(cons(N, natsFrom(s(N))))active(fst(pair(XS, YS)))mark(XS)
active(snd(pair(XS, YS)))mark(YS)active(splitAt(0, XS))mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS)))mark(u(splitAt(N, XS), N, X, XS))active(u(pair(YS, ZS), N, X, XS))mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS)))mark(N)active(tail(cons(N, XS)))mark(XS)
active(sel(N, XS))mark(head(afterNth(N, XS)))active(take(N, XS))mark(fst(splitAt(N, XS)))
active(afterNth(N, XS))mark(snd(splitAt(N, XS)))mark(natsFrom(X))active(natsFrom(mark(X)))
mark(cons(X1, X2))active(cons(mark(X1), X2))mark(s(X))active(s(mark(X)))
mark(fst(X))active(fst(mark(X)))mark(pair(X1, X2))active(pair(mark(X1), mark(X2)))
mark(snd(X))active(snd(mark(X)))mark(splitAt(X1, X2))active(splitAt(mark(X1), mark(X2)))
mark(0)active(0)mark(nil)active(nil)
mark(u(X1, X2, X3, X4))active(u(mark(X1), X2, X3, X4))mark(head(X))active(head(mark(X)))
mark(tail(X))active(tail(mark(X)))mark(sel(X1, X2))active(sel(mark(X1), mark(X2)))
mark(afterNth(X1, X2))active(afterNth(mark(X1), mark(X2)))mark(take(X1, X2))active(take(mark(X1), mark(X2)))
natsFrom(mark(X))natsFrom(X)natsFrom(active(X))natsFrom(X)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
s(mark(X))s(X)s(active(X))s(X)
fst(mark(X))fst(X)fst(active(X))fst(X)
pair(mark(X1), X2)pair(X1, X2)pair(X1, mark(X2))pair(X1, X2)
pair(active(X1), X2)pair(X1, X2)pair(X1, active(X2))pair(X1, X2)
snd(mark(X))snd(X)snd(active(X))snd(X)
splitAt(mark(X1), X2)splitAt(X1, X2)splitAt(X1, mark(X2))splitAt(X1, X2)
splitAt(active(X1), X2)splitAt(X1, X2)splitAt(X1, active(X2))splitAt(X1, X2)
u(mark(X1), X2, X3, X4)u(X1, X2, X3, X4)u(X1, mark(X2), X3, X4)u(X1, X2, X3, X4)
u(X1, X2, mark(X3), X4)u(X1, X2, X3, X4)u(X1, X2, X3, mark(X4))u(X1, X2, X3, X4)
u(active(X1), X2, X3, X4)u(X1, X2, X3, X4)u(X1, active(X2), X3, X4)u(X1, X2, X3, X4)
u(X1, X2, active(X3), X4)u(X1, X2, X3, X4)u(X1, X2, X3, active(X4))u(X1, X2, X3, X4)
head(mark(X))head(X)head(active(X))head(X)
tail(mark(X))tail(X)tail(active(X))tail(X)
sel(mark(X1), X2)sel(X1, X2)sel(X1, mark(X2))sel(X1, X2)
sel(active(X1), X2)sel(X1, X2)sel(X1, active(X2))sel(X1, X2)
afterNth(mark(X1), X2)afterNth(X1, X2)afterNth(X1, mark(X2))afterNth(X1, X2)
afterNth(active(X1), X2)afterNth(X1, X2)afterNth(X1, active(X2))afterNth(X1, X2)
take(mark(X1), X2)take(X1, X2)take(X1, mark(X2))take(X1, X2)
take(active(X1), X2)take(X1, X2)take(X1, active(X2))take(X1, X2)

Original Signature

Termination of terms over the following signature is verified: pair, natsFrom, mark, tail, splitAt, fst, u, 0, s, take, active, afterNth, head, sel, nil, snd, cons

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

natsFrom#(mark(X))natsFrom#(X)natsFrom#(active(X))natsFrom#(X)

Problem 6: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

active(natsFrom(N))mark(cons(N, natsFrom(s(N))))active(fst(pair(XS, YS)))mark(XS)
active(snd(pair(XS, YS)))mark(YS)active(splitAt(0, XS))mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS)))mark(u(splitAt(N, XS), N, X, XS))active(u(pair(YS, ZS), N, X, XS))mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS)))mark(N)active(tail(cons(N, XS)))mark(XS)
active(sel(N, XS))mark(head(afterNth(N, XS)))active(take(N, XS))mark(fst(splitAt(N, XS)))
active(afterNth(N, XS))mark(snd(splitAt(N, XS)))mark(natsFrom(X))active(natsFrom(mark(X)))
mark(cons(X1, X2))active(cons(mark(X1), X2))mark(s(X))active(s(mark(X)))
mark(fst(X))active(fst(mark(X)))mark(pair(X1, X2))active(pair(mark(X1), mark(X2)))
mark(snd(X))active(snd(mark(X)))mark(splitAt(X1, X2))active(splitAt(mark(X1), mark(X2)))
mark(0)active(0)mark(nil)active(nil)
mark(u(X1, X2, X3, X4))active(u(mark(X1), X2, X3, X4))mark(head(X))active(head(mark(X)))
mark(tail(X))active(tail(mark(X)))mark(sel(X1, X2))active(sel(mark(X1), mark(X2)))
mark(afterNth(X1, X2))active(afterNth(mark(X1), mark(X2)))mark(take(X1, X2))active(take(mark(X1), mark(X2)))
natsFrom(mark(X))natsFrom(X)natsFrom(active(X))natsFrom(X)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
s(mark(X))s(X)s(active(X))s(X)
fst(mark(X))fst(X)fst(active(X))fst(X)
pair(mark(X1), X2)pair(X1, X2)pair(X1, mark(X2))pair(X1, X2)
pair(active(X1), X2)pair(X1, X2)pair(X1, active(X2))pair(X1, X2)
snd(mark(X))snd(X)snd(active(X))snd(X)
splitAt(mark(X1), X2)splitAt(X1, X2)splitAt(X1, mark(X2))splitAt(X1, X2)
splitAt(active(X1), X2)splitAt(X1, X2)splitAt(X1, active(X2))splitAt(X1, X2)
u(mark(X1), X2, X3, X4)u(X1, X2, X3, X4)u(X1, mark(X2), X3, X4)u(X1, X2, X3, X4)
u(X1, X2, mark(X3), X4)u(X1, X2, X3, X4)u(X1, X2, X3, mark(X4))u(X1, X2, X3, X4)
u(active(X1), X2, X3, X4)u(X1, X2, X3, X4)u(X1, active(X2), X3, X4)u(X1, X2, X3, X4)
u(X1, X2, active(X3), X4)u(X1, X2, X3, X4)u(X1, X2, X3, active(X4))u(X1, X2, X3, X4)
head(mark(X))head(X)head(active(X))head(X)
tail(mark(X))tail(X)tail(active(X))tail(X)
sel(mark(X1), X2)sel(X1, X2)sel(X1, mark(X2))sel(X1, X2)
sel(active(X1), X2)sel(X1, X2)sel(X1, active(X2))sel(X1, X2)
afterNth(mark(X1), X2)afterNth(X1, X2)afterNth(X1, mark(X2))afterNth(X1, X2)
afterNth(active(X1), X2)afterNth(X1, X2)afterNth(X1, active(X2))afterNth(X1, X2)
take(mark(X1), X2)take(X1, X2)take(X1, mark(X2))take(X1, X2)
take(active(X1), X2)take(X1, X2)take(X1, active(X2))take(X1, X2)

Original Signature

Termination of terms over the following signature is verified: pair, natsFrom, mark, tail, splitAt, fst, u, 0, s, take, active, afterNth, head, sel, nil, snd, cons

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

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

Problem 7: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

head#(mark(X))head#(X)head#(active(X))head#(X)

Rewrite Rules

active(natsFrom(N))mark(cons(N, natsFrom(s(N))))active(fst(pair(XS, YS)))mark(XS)
active(snd(pair(XS, YS)))mark(YS)active(splitAt(0, XS))mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS)))mark(u(splitAt(N, XS), N, X, XS))active(u(pair(YS, ZS), N, X, XS))mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS)))mark(N)active(tail(cons(N, XS)))mark(XS)
active(sel(N, XS))mark(head(afterNth(N, XS)))active(take(N, XS))mark(fst(splitAt(N, XS)))
active(afterNth(N, XS))mark(snd(splitAt(N, XS)))mark(natsFrom(X))active(natsFrom(mark(X)))
mark(cons(X1, X2))active(cons(mark(X1), X2))mark(s(X))active(s(mark(X)))
mark(fst(X))active(fst(mark(X)))mark(pair(X1, X2))active(pair(mark(X1), mark(X2)))
mark(snd(X))active(snd(mark(X)))mark(splitAt(X1, X2))active(splitAt(mark(X1), mark(X2)))
mark(0)active(0)mark(nil)active(nil)
mark(u(X1, X2, X3, X4))active(u(mark(X1), X2, X3, X4))mark(head(X))active(head(mark(X)))
mark(tail(X))active(tail(mark(X)))mark(sel(X1, X2))active(sel(mark(X1), mark(X2)))
mark(afterNth(X1, X2))active(afterNth(mark(X1), mark(X2)))mark(take(X1, X2))active(take(mark(X1), mark(X2)))
natsFrom(mark(X))natsFrom(X)natsFrom(active(X))natsFrom(X)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
s(mark(X))s(X)s(active(X))s(X)
fst(mark(X))fst(X)fst(active(X))fst(X)
pair(mark(X1), X2)pair(X1, X2)pair(X1, mark(X2))pair(X1, X2)
pair(active(X1), X2)pair(X1, X2)pair(X1, active(X2))pair(X1, X2)
snd(mark(X))snd(X)snd(active(X))snd(X)
splitAt(mark(X1), X2)splitAt(X1, X2)splitAt(X1, mark(X2))splitAt(X1, X2)
splitAt(active(X1), X2)splitAt(X1, X2)splitAt(X1, active(X2))splitAt(X1, X2)
u(mark(X1), X2, X3, X4)u(X1, X2, X3, X4)u(X1, mark(X2), X3, X4)u(X1, X2, X3, X4)
u(X1, X2, mark(X3), X4)u(X1, X2, X3, X4)u(X1, X2, X3, mark(X4))u(X1, X2, X3, X4)
u(active(X1), X2, X3, X4)u(X1, X2, X3, X4)u(X1, active(X2), X3, X4)u(X1, X2, X3, X4)
u(X1, X2, active(X3), X4)u(X1, X2, X3, X4)u(X1, X2, X3, active(X4))u(X1, X2, X3, X4)
head(mark(X))head(X)head(active(X))head(X)
tail(mark(X))tail(X)tail(active(X))tail(X)
sel(mark(X1), X2)sel(X1, X2)sel(X1, mark(X2))sel(X1, X2)
sel(active(X1), X2)sel(X1, X2)sel(X1, active(X2))sel(X1, X2)
afterNth(mark(X1), X2)afterNth(X1, X2)afterNth(X1, mark(X2))afterNth(X1, X2)
afterNth(active(X1), X2)afterNth(X1, X2)afterNth(X1, active(X2))afterNth(X1, X2)
take(mark(X1), X2)take(X1, X2)take(X1, mark(X2))take(X1, X2)
take(active(X1), X2)take(X1, X2)take(X1, active(X2))take(X1, X2)

Original Signature

Termination of terms over the following signature is verified: pair, natsFrom, mark, tail, splitAt, fst, u, 0, s, take, active, afterNth, head, sel, nil, snd, cons

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

head#(mark(X))head#(X)head#(active(X))head#(X)

Problem 8: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

active(natsFrom(N))mark(cons(N, natsFrom(s(N))))active(fst(pair(XS, YS)))mark(XS)
active(snd(pair(XS, YS)))mark(YS)active(splitAt(0, XS))mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS)))mark(u(splitAt(N, XS), N, X, XS))active(u(pair(YS, ZS), N, X, XS))mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS)))mark(N)active(tail(cons(N, XS)))mark(XS)
active(sel(N, XS))mark(head(afterNth(N, XS)))active(take(N, XS))mark(fst(splitAt(N, XS)))
active(afterNth(N, XS))mark(snd(splitAt(N, XS)))mark(natsFrom(X))active(natsFrom(mark(X)))
mark(cons(X1, X2))active(cons(mark(X1), X2))mark(s(X))active(s(mark(X)))
mark(fst(X))active(fst(mark(X)))mark(pair(X1, X2))active(pair(mark(X1), mark(X2)))
mark(snd(X))active(snd(mark(X)))mark(splitAt(X1, X2))active(splitAt(mark(X1), mark(X2)))
mark(0)active(0)mark(nil)active(nil)
mark(u(X1, X2, X3, X4))active(u(mark(X1), X2, X3, X4))mark(head(X))active(head(mark(X)))
mark(tail(X))active(tail(mark(X)))mark(sel(X1, X2))active(sel(mark(X1), mark(X2)))
mark(afterNth(X1, X2))active(afterNth(mark(X1), mark(X2)))mark(take(X1, X2))active(take(mark(X1), mark(X2)))
natsFrom(mark(X))natsFrom(X)natsFrom(active(X))natsFrom(X)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
s(mark(X))s(X)s(active(X))s(X)
fst(mark(X))fst(X)fst(active(X))fst(X)
pair(mark(X1), X2)pair(X1, X2)pair(X1, mark(X2))pair(X1, X2)
pair(active(X1), X2)pair(X1, X2)pair(X1, active(X2))pair(X1, X2)
snd(mark(X))snd(X)snd(active(X))snd(X)
splitAt(mark(X1), X2)splitAt(X1, X2)splitAt(X1, mark(X2))splitAt(X1, X2)
splitAt(active(X1), X2)splitAt(X1, X2)splitAt(X1, active(X2))splitAt(X1, X2)
u(mark(X1), X2, X3, X4)u(X1, X2, X3, X4)u(X1, mark(X2), X3, X4)u(X1, X2, X3, X4)
u(X1, X2, mark(X3), X4)u(X1, X2, X3, X4)u(X1, X2, X3, mark(X4))u(X1, X2, X3, X4)
u(active(X1), X2, X3, X4)u(X1, X2, X3, X4)u(X1, active(X2), X3, X4)u(X1, X2, X3, X4)
u(X1, X2, active(X3), X4)u(X1, X2, X3, X4)u(X1, X2, X3, active(X4))u(X1, X2, X3, X4)
head(mark(X))head(X)head(active(X))head(X)
tail(mark(X))tail(X)tail(active(X))tail(X)
sel(mark(X1), X2)sel(X1, X2)sel(X1, mark(X2))sel(X1, X2)
sel(active(X1), X2)sel(X1, X2)sel(X1, active(X2))sel(X1, X2)
afterNth(mark(X1), X2)afterNth(X1, X2)afterNth(X1, mark(X2))afterNth(X1, X2)
afterNth(active(X1), X2)afterNth(X1, X2)afterNth(X1, active(X2))afterNth(X1, X2)
take(mark(X1), X2)take(X1, X2)take(X1, mark(X2))take(X1, X2)
take(active(X1), X2)take(X1, X2)take(X1, active(X2))take(X1, X2)

Original Signature

Termination of terms over the following signature is verified: pair, natsFrom, mark, tail, splitAt, fst, u, 0, s, take, active, afterNth, head, sel, nil, snd, cons

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

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

Problem 18: PolynomialLinearRange4iUR



Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

active(natsFrom(N))mark(cons(N, natsFrom(s(N))))active(fst(pair(XS, YS)))mark(XS)
active(snd(pair(XS, YS)))mark(YS)active(splitAt(0, XS))mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS)))mark(u(splitAt(N, XS), N, X, XS))active(u(pair(YS, ZS), N, X, XS))mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS)))mark(N)active(tail(cons(N, XS)))mark(XS)
active(sel(N, XS))mark(head(afterNth(N, XS)))active(take(N, XS))mark(fst(splitAt(N, XS)))
active(afterNth(N, XS))mark(snd(splitAt(N, XS)))mark(natsFrom(X))active(natsFrom(mark(X)))
mark(cons(X1, X2))active(cons(mark(X1), X2))mark(s(X))active(s(mark(X)))
mark(fst(X))active(fst(mark(X)))mark(pair(X1, X2))active(pair(mark(X1), mark(X2)))
mark(snd(X))active(snd(mark(X)))mark(splitAt(X1, X2))active(splitAt(mark(X1), mark(X2)))
mark(0)active(0)mark(nil)active(nil)
mark(u(X1, X2, X3, X4))active(u(mark(X1), X2, X3, X4))mark(head(X))active(head(mark(X)))
mark(tail(X))active(tail(mark(X)))mark(sel(X1, X2))active(sel(mark(X1), mark(X2)))
mark(afterNth(X1, X2))active(afterNth(mark(X1), mark(X2)))mark(take(X1, X2))active(take(mark(X1), mark(X2)))
natsFrom(mark(X))natsFrom(X)natsFrom(active(X))natsFrom(X)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
s(mark(X))s(X)s(active(X))s(X)
fst(mark(X))fst(X)fst(active(X))fst(X)
pair(mark(X1), X2)pair(X1, X2)pair(X1, mark(X2))pair(X1, X2)
pair(active(X1), X2)pair(X1, X2)pair(X1, active(X2))pair(X1, X2)
snd(mark(X))snd(X)snd(active(X))snd(X)
splitAt(mark(X1), X2)splitAt(X1, X2)splitAt(X1, mark(X2))splitAt(X1, X2)
splitAt(active(X1), X2)splitAt(X1, X2)splitAt(X1, active(X2))splitAt(X1, X2)
u(mark(X1), X2, X3, X4)u(X1, X2, X3, X4)u(X1, mark(X2), X3, X4)u(X1, X2, X3, X4)
u(X1, X2, mark(X3), X4)u(X1, X2, X3, X4)u(X1, X2, X3, mark(X4))u(X1, X2, X3, X4)
u(active(X1), X2, X3, X4)u(X1, X2, X3, X4)u(X1, active(X2), X3, X4)u(X1, X2, X3, X4)
u(X1, X2, active(X3), X4)u(X1, X2, X3, X4)u(X1, X2, X3, active(X4))u(X1, X2, X3, X4)
head(mark(X))head(X)head(active(X))head(X)
tail(mark(X))tail(X)tail(active(X))tail(X)
sel(mark(X1), X2)sel(X1, X2)sel(X1, mark(X2))sel(X1, X2)
sel(active(X1), X2)sel(X1, X2)sel(X1, active(X2))sel(X1, X2)
afterNth(mark(X1), X2)afterNth(X1, X2)afterNth(X1, mark(X2))afterNth(X1, X2)
afterNth(active(X1), X2)afterNth(X1, X2)afterNth(X1, active(X2))afterNth(X1, X2)
take(mark(X1), X2)take(X1, X2)take(X1, mark(X2))take(X1, X2)
take(active(X1), X2)take(X1, X2)take(X1, active(X2))take(X1, X2)

Original Signature

Termination of terms over the following signature is verified: pair, natsFrom, mark, tail, splitAt, fst, u, 0, s, take, active, afterNth, head, sel, cons, snd, 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:

pair#(X1, active(X2))pair#(X1, X2)

Problem 10: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

active(natsFrom(N))mark(cons(N, natsFrom(s(N))))active(fst(pair(XS, YS)))mark(XS)
active(snd(pair(XS, YS)))mark(YS)active(splitAt(0, XS))mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS)))mark(u(splitAt(N, XS), N, X, XS))active(u(pair(YS, ZS), N, X, XS))mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS)))mark(N)active(tail(cons(N, XS)))mark(XS)
active(sel(N, XS))mark(head(afterNth(N, XS)))active(take(N, XS))mark(fst(splitAt(N, XS)))
active(afterNth(N, XS))mark(snd(splitAt(N, XS)))mark(natsFrom(X))active(natsFrom(mark(X)))
mark(cons(X1, X2))active(cons(mark(X1), X2))mark(s(X))active(s(mark(X)))
mark(fst(X))active(fst(mark(X)))mark(pair(X1, X2))active(pair(mark(X1), mark(X2)))
mark(snd(X))active(snd(mark(X)))mark(splitAt(X1, X2))active(splitAt(mark(X1), mark(X2)))
mark(0)active(0)mark(nil)active(nil)
mark(u(X1, X2, X3, X4))active(u(mark(X1), X2, X3, X4))mark(head(X))active(head(mark(X)))
mark(tail(X))active(tail(mark(X)))mark(sel(X1, X2))active(sel(mark(X1), mark(X2)))
mark(afterNth(X1, X2))active(afterNth(mark(X1), mark(X2)))mark(take(X1, X2))active(take(mark(X1), mark(X2)))
natsFrom(mark(X))natsFrom(X)natsFrom(active(X))natsFrom(X)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
s(mark(X))s(X)s(active(X))s(X)
fst(mark(X))fst(X)fst(active(X))fst(X)
pair(mark(X1), X2)pair(X1, X2)pair(X1, mark(X2))pair(X1, X2)
pair(active(X1), X2)pair(X1, X2)pair(X1, active(X2))pair(X1, X2)
snd(mark(X))snd(X)snd(active(X))snd(X)
splitAt(mark(X1), X2)splitAt(X1, X2)splitAt(X1, mark(X2))splitAt(X1, X2)
splitAt(active(X1), X2)splitAt(X1, X2)splitAt(X1, active(X2))splitAt(X1, X2)
u(mark(X1), X2, X3, X4)u(X1, X2, X3, X4)u(X1, mark(X2), X3, X4)u(X1, X2, X3, X4)
u(X1, X2, mark(X3), X4)u(X1, X2, X3, X4)u(X1, X2, X3, mark(X4))u(X1, X2, X3, X4)
u(active(X1), X2, X3, X4)u(X1, X2, X3, X4)u(X1, active(X2), X3, X4)u(X1, X2, X3, X4)
u(X1, X2, active(X3), X4)u(X1, X2, X3, X4)u(X1, X2, X3, active(X4))u(X1, X2, X3, X4)
head(mark(X))head(X)head(active(X))head(X)
tail(mark(X))tail(X)tail(active(X))tail(X)
sel(mark(X1), X2)sel(X1, X2)sel(X1, mark(X2))sel(X1, X2)
sel(active(X1), X2)sel(X1, X2)sel(X1, active(X2))sel(X1, X2)
afterNth(mark(X1), X2)afterNth(X1, X2)afterNth(X1, mark(X2))afterNth(X1, X2)
afterNth(active(X1), X2)afterNth(X1, X2)afterNth(X1, active(X2))afterNth(X1, X2)
take(mark(X1), X2)take(X1, X2)take(X1, mark(X2))take(X1, X2)
take(active(X1), X2)take(X1, X2)take(X1, active(X2))take(X1, X2)

Original Signature

Termination of terms over the following signature is verified: pair, natsFrom, mark, tail, splitAt, fst, u, 0, s, take, active, afterNth, head, sel, nil, snd, cons

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

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

Problem 19: PolynomialLinearRange4iUR



Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

active(natsFrom(N))mark(cons(N, natsFrom(s(N))))active(fst(pair(XS, YS)))mark(XS)
active(snd(pair(XS, YS)))mark(YS)active(splitAt(0, XS))mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS)))mark(u(splitAt(N, XS), N, X, XS))active(u(pair(YS, ZS), N, X, XS))mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS)))mark(N)active(tail(cons(N, XS)))mark(XS)
active(sel(N, XS))mark(head(afterNth(N, XS)))active(take(N, XS))mark(fst(splitAt(N, XS)))
active(afterNth(N, XS))mark(snd(splitAt(N, XS)))mark(natsFrom(X))active(natsFrom(mark(X)))
mark(cons(X1, X2))active(cons(mark(X1), X2))mark(s(X))active(s(mark(X)))
mark(fst(X))active(fst(mark(X)))mark(pair(X1, X2))active(pair(mark(X1), mark(X2)))
mark(snd(X))active(snd(mark(X)))mark(splitAt(X1, X2))active(splitAt(mark(X1), mark(X2)))
mark(0)active(0)mark(nil)active(nil)
mark(u(X1, X2, X3, X4))active(u(mark(X1), X2, X3, X4))mark(head(X))active(head(mark(X)))
mark(tail(X))active(tail(mark(X)))mark(sel(X1, X2))active(sel(mark(X1), mark(X2)))
mark(afterNth(X1, X2))active(afterNth(mark(X1), mark(X2)))mark(take(X1, X2))active(take(mark(X1), mark(X2)))
natsFrom(mark(X))natsFrom(X)natsFrom(active(X))natsFrom(X)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
s(mark(X))s(X)s(active(X))s(X)
fst(mark(X))fst(X)fst(active(X))fst(X)
pair(mark(X1), X2)pair(X1, X2)pair(X1, mark(X2))pair(X1, X2)
pair(active(X1), X2)pair(X1, X2)pair(X1, active(X2))pair(X1, X2)
snd(mark(X))snd(X)snd(active(X))snd(X)
splitAt(mark(X1), X2)splitAt(X1, X2)splitAt(X1, mark(X2))splitAt(X1, X2)
splitAt(active(X1), X2)splitAt(X1, X2)splitAt(X1, active(X2))splitAt(X1, X2)
u(mark(X1), X2, X3, X4)u(X1, X2, X3, X4)u(X1, mark(X2), X3, X4)u(X1, X2, X3, X4)
u(X1, X2, mark(X3), X4)u(X1, X2, X3, X4)u(X1, X2, X3, mark(X4))u(X1, X2, X3, X4)
u(active(X1), X2, X3, X4)u(X1, X2, X3, X4)u(X1, active(X2), X3, X4)u(X1, X2, X3, X4)
u(X1, X2, active(X3), X4)u(X1, X2, X3, X4)u(X1, X2, X3, active(X4))u(X1, X2, X3, X4)
head(mark(X))head(X)head(active(X))head(X)
tail(mark(X))tail(X)tail(active(X))tail(X)
sel(mark(X1), X2)sel(X1, X2)sel(X1, mark(X2))sel(X1, X2)
sel(active(X1), X2)sel(X1, X2)sel(X1, active(X2))sel(X1, X2)
afterNth(mark(X1), X2)afterNth(X1, X2)afterNth(X1, mark(X2))afterNth(X1, X2)
afterNth(active(X1), X2)afterNth(X1, X2)afterNth(X1, active(X2))afterNth(X1, X2)
take(mark(X1), X2)take(X1, X2)take(X1, mark(X2))take(X1, X2)
take(active(X1), X2)take(X1, X2)take(X1, active(X2))take(X1, X2)

Original Signature

Termination of terms over the following signature is verified: pair, natsFrom, mark, tail, splitAt, fst, u, 0, s, take, active, afterNth, head, sel, cons, snd, 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:

splitAt#(X1, mark(X2))splitAt#(X1, X2)

Problem 25: PolynomialLinearRange4iUR



Dependency Pair Problem

Dependency Pairs

splitAt#(X1, active(X2))splitAt#(X1, X2)

Rewrite Rules

active(natsFrom(N))mark(cons(N, natsFrom(s(N))))active(fst(pair(XS, YS)))mark(XS)
active(snd(pair(XS, YS)))mark(YS)active(splitAt(0, XS))mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS)))mark(u(splitAt(N, XS), N, X, XS))active(u(pair(YS, ZS), N, X, XS))mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS)))mark(N)active(tail(cons(N, XS)))mark(XS)
active(sel(N, XS))mark(head(afterNth(N, XS)))active(take(N, XS))mark(fst(splitAt(N, XS)))
active(afterNth(N, XS))mark(snd(splitAt(N, XS)))mark(natsFrom(X))active(natsFrom(mark(X)))
mark(cons(X1, X2))active(cons(mark(X1), X2))mark(s(X))active(s(mark(X)))
mark(fst(X))active(fst(mark(X)))mark(pair(X1, X2))active(pair(mark(X1), mark(X2)))
mark(snd(X))active(snd(mark(X)))mark(splitAt(X1, X2))active(splitAt(mark(X1), mark(X2)))
mark(0)active(0)mark(nil)active(nil)
mark(u(X1, X2, X3, X4))active(u(mark(X1), X2, X3, X4))mark(head(X))active(head(mark(X)))
mark(tail(X))active(tail(mark(X)))mark(sel(X1, X2))active(sel(mark(X1), mark(X2)))
mark(afterNth(X1, X2))active(afterNth(mark(X1), mark(X2)))mark(take(X1, X2))active(take(mark(X1), mark(X2)))
natsFrom(mark(X))natsFrom(X)natsFrom(active(X))natsFrom(X)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
s(mark(X))s(X)s(active(X))s(X)
fst(mark(X))fst(X)fst(active(X))fst(X)
pair(mark(X1), X2)pair(X1, X2)pair(X1, mark(X2))pair(X1, X2)
pair(active(X1), X2)pair(X1, X2)pair(X1, active(X2))pair(X1, X2)
snd(mark(X))snd(X)snd(active(X))snd(X)
splitAt(mark(X1), X2)splitAt(X1, X2)splitAt(X1, mark(X2))splitAt(X1, X2)
splitAt(active(X1), X2)splitAt(X1, X2)splitAt(X1, active(X2))splitAt(X1, X2)
u(mark(X1), X2, X3, X4)u(X1, X2, X3, X4)u(X1, mark(X2), X3, X4)u(X1, X2, X3, X4)
u(X1, X2, mark(X3), X4)u(X1, X2, X3, X4)u(X1, X2, X3, mark(X4))u(X1, X2, X3, X4)
u(active(X1), X2, X3, X4)u(X1, X2, X3, X4)u(X1, active(X2), X3, X4)u(X1, X2, X3, X4)
u(X1, X2, active(X3), X4)u(X1, X2, X3, X4)u(X1, X2, X3, active(X4))u(X1, X2, X3, X4)
head(mark(X))head(X)head(active(X))head(X)
tail(mark(X))tail(X)tail(active(X))tail(X)
sel(mark(X1), X2)sel(X1, X2)sel(X1, mark(X2))sel(X1, X2)
sel(active(X1), X2)sel(X1, X2)sel(X1, active(X2))sel(X1, X2)
afterNth(mark(X1), X2)afterNth(X1, X2)afterNth(X1, mark(X2))afterNth(X1, X2)
afterNth(active(X1), X2)afterNth(X1, X2)afterNth(X1, active(X2))afterNth(X1, X2)
take(mark(X1), X2)take(X1, X2)take(X1, mark(X2))take(X1, X2)
take(active(X1), X2)take(X1, X2)take(X1, active(X2))take(X1, X2)

Original Signature

Termination of terms over the following signature is verified: pair, natsFrom, mark, tail, splitAt, fst, u, 0, s, take, active, afterNth, head, sel, nil, snd, cons

Strategy


Polynomial Interpretation

There are no usable rules

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

splitAt#(X1, active(X2))splitAt#(X1, X2)

Problem 11: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

fst#(mark(X))fst#(X)fst#(active(X))fst#(X)

Rewrite Rules

active(natsFrom(N))mark(cons(N, natsFrom(s(N))))active(fst(pair(XS, YS)))mark(XS)
active(snd(pair(XS, YS)))mark(YS)active(splitAt(0, XS))mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS)))mark(u(splitAt(N, XS), N, X, XS))active(u(pair(YS, ZS), N, X, XS))mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS)))mark(N)active(tail(cons(N, XS)))mark(XS)
active(sel(N, XS))mark(head(afterNth(N, XS)))active(take(N, XS))mark(fst(splitAt(N, XS)))
active(afterNth(N, XS))mark(snd(splitAt(N, XS)))mark(natsFrom(X))active(natsFrom(mark(X)))
mark(cons(X1, X2))active(cons(mark(X1), X2))mark(s(X))active(s(mark(X)))
mark(fst(X))active(fst(mark(X)))mark(pair(X1, X2))active(pair(mark(X1), mark(X2)))
mark(snd(X))active(snd(mark(X)))mark(splitAt(X1, X2))active(splitAt(mark(X1), mark(X2)))
mark(0)active(0)mark(nil)active(nil)
mark(u(X1, X2, X3, X4))active(u(mark(X1), X2, X3, X4))mark(head(X))active(head(mark(X)))
mark(tail(X))active(tail(mark(X)))mark(sel(X1, X2))active(sel(mark(X1), mark(X2)))
mark(afterNth(X1, X2))active(afterNth(mark(X1), mark(X2)))mark(take(X1, X2))active(take(mark(X1), mark(X2)))
natsFrom(mark(X))natsFrom(X)natsFrom(active(X))natsFrom(X)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
s(mark(X))s(X)s(active(X))s(X)
fst(mark(X))fst(X)fst(active(X))fst(X)
pair(mark(X1), X2)pair(X1, X2)pair(X1, mark(X2))pair(X1, X2)
pair(active(X1), X2)pair(X1, X2)pair(X1, active(X2))pair(X1, X2)
snd(mark(X))snd(X)snd(active(X))snd(X)
splitAt(mark(X1), X2)splitAt(X1, X2)splitAt(X1, mark(X2))splitAt(X1, X2)
splitAt(active(X1), X2)splitAt(X1, X2)splitAt(X1, active(X2))splitAt(X1, X2)
u(mark(X1), X2, X3, X4)u(X1, X2, X3, X4)u(X1, mark(X2), X3, X4)u(X1, X2, X3, X4)
u(X1, X2, mark(X3), X4)u(X1, X2, X3, X4)u(X1, X2, X3, mark(X4))u(X1, X2, X3, X4)
u(active(X1), X2, X3, X4)u(X1, X2, X3, X4)u(X1, active(X2), X3, X4)u(X1, X2, X3, X4)
u(X1, X2, active(X3), X4)u(X1, X2, X3, X4)u(X1, X2, X3, active(X4))u(X1, X2, X3, X4)
head(mark(X))head(X)head(active(X))head(X)
tail(mark(X))tail(X)tail(active(X))tail(X)
sel(mark(X1), X2)sel(X1, X2)sel(X1, mark(X2))sel(X1, X2)
sel(active(X1), X2)sel(X1, X2)sel(X1, active(X2))sel(X1, X2)
afterNth(mark(X1), X2)afterNth(X1, X2)afterNth(X1, mark(X2))afterNth(X1, X2)
afterNth(active(X1), X2)afterNth(X1, X2)afterNth(X1, active(X2))afterNth(X1, X2)
take(mark(X1), X2)take(X1, X2)take(X1, mark(X2))take(X1, X2)
take(active(X1), X2)take(X1, X2)take(X1, active(X2))take(X1, X2)

Original Signature

Termination of terms over the following signature is verified: pair, natsFrom, mark, tail, splitAt, fst, u, 0, s, take, active, afterNth, head, sel, nil, snd, cons

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

fst#(mark(X))fst#(X)fst#(active(X))fst#(X)

Problem 12: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

active(natsFrom(N))mark(cons(N, natsFrom(s(N))))active(fst(pair(XS, YS)))mark(XS)
active(snd(pair(XS, YS)))mark(YS)active(splitAt(0, XS))mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS)))mark(u(splitAt(N, XS), N, X, XS))active(u(pair(YS, ZS), N, X, XS))mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS)))mark(N)active(tail(cons(N, XS)))mark(XS)
active(sel(N, XS))mark(head(afterNth(N, XS)))active(take(N, XS))mark(fst(splitAt(N, XS)))
active(afterNth(N, XS))mark(snd(splitAt(N, XS)))mark(natsFrom(X))active(natsFrom(mark(X)))
mark(cons(X1, X2))active(cons(mark(X1), X2))mark(s(X))active(s(mark(X)))
mark(fst(X))active(fst(mark(X)))mark(pair(X1, X2))active(pair(mark(X1), mark(X2)))
mark(snd(X))active(snd(mark(X)))mark(splitAt(X1, X2))active(splitAt(mark(X1), mark(X2)))
mark(0)active(0)mark(nil)active(nil)
mark(u(X1, X2, X3, X4))active(u(mark(X1), X2, X3, X4))mark(head(X))active(head(mark(X)))
mark(tail(X))active(tail(mark(X)))mark(sel(X1, X2))active(sel(mark(X1), mark(X2)))
mark(afterNth(X1, X2))active(afterNth(mark(X1), mark(X2)))mark(take(X1, X2))active(take(mark(X1), mark(X2)))
natsFrom(mark(X))natsFrom(X)natsFrom(active(X))natsFrom(X)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
s(mark(X))s(X)s(active(X))s(X)
fst(mark(X))fst(X)fst(active(X))fst(X)
pair(mark(X1), X2)pair(X1, X2)pair(X1, mark(X2))pair(X1, X2)
pair(active(X1), X2)pair(X1, X2)pair(X1, active(X2))pair(X1, X2)
snd(mark(X))snd(X)snd(active(X))snd(X)
splitAt(mark(X1), X2)splitAt(X1, X2)splitAt(X1, mark(X2))splitAt(X1, X2)
splitAt(active(X1), X2)splitAt(X1, X2)splitAt(X1, active(X2))splitAt(X1, X2)
u(mark(X1), X2, X3, X4)u(X1, X2, X3, X4)u(X1, mark(X2), X3, X4)u(X1, X2, X3, X4)
u(X1, X2, mark(X3), X4)u(X1, X2, X3, X4)u(X1, X2, X3, mark(X4))u(X1, X2, X3, X4)
u(active(X1), X2, X3, X4)u(X1, X2, X3, X4)u(X1, active(X2), X3, X4)u(X1, X2, X3, X4)
u(X1, X2, active(X3), X4)u(X1, X2, X3, X4)u(X1, X2, X3, active(X4))u(X1, X2, X3, X4)
head(mark(X))head(X)head(active(X))head(X)
tail(mark(X))tail(X)tail(active(X))tail(X)
sel(mark(X1), X2)sel(X1, X2)sel(X1, mark(X2))sel(X1, X2)
sel(active(X1), X2)sel(X1, X2)sel(X1, active(X2))sel(X1, X2)
afterNth(mark(X1), X2)afterNth(X1, X2)afterNth(X1, mark(X2))afterNth(X1, X2)
afterNth(active(X1), X2)afterNth(X1, X2)afterNth(X1, active(X2))afterNth(X1, X2)
take(mark(X1), X2)take(X1, X2)take(X1, mark(X2))take(X1, X2)
take(active(X1), X2)take(X1, X2)take(X1, active(X2))take(X1, X2)

Original Signature

Termination of terms over the following signature is verified: pair, natsFrom, mark, tail, splitAt, fst, u, 0, s, take, active, afterNth, head, sel, nil, snd, cons

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

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

Problem 20: PolynomialLinearRange4iUR



Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

active(natsFrom(N))mark(cons(N, natsFrom(s(N))))active(fst(pair(XS, YS)))mark(XS)
active(snd(pair(XS, YS)))mark(YS)active(splitAt(0, XS))mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS)))mark(u(splitAt(N, XS), N, X, XS))active(u(pair(YS, ZS), N, X, XS))mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS)))mark(N)active(tail(cons(N, XS)))mark(XS)
active(sel(N, XS))mark(head(afterNth(N, XS)))active(take(N, XS))mark(fst(splitAt(N, XS)))
active(afterNth(N, XS))mark(snd(splitAt(N, XS)))mark(natsFrom(X))active(natsFrom(mark(X)))
mark(cons(X1, X2))active(cons(mark(X1), X2))mark(s(X))active(s(mark(X)))
mark(fst(X))active(fst(mark(X)))mark(pair(X1, X2))active(pair(mark(X1), mark(X2)))
mark(snd(X))active(snd(mark(X)))mark(splitAt(X1, X2))active(splitAt(mark(X1), mark(X2)))
mark(0)active(0)mark(nil)active(nil)
mark(u(X1, X2, X3, X4))active(u(mark(X1), X2, X3, X4))mark(head(X))active(head(mark(X)))
mark(tail(X))active(tail(mark(X)))mark(sel(X1, X2))active(sel(mark(X1), mark(X2)))
mark(afterNth(X1, X2))active(afterNth(mark(X1), mark(X2)))mark(take(X1, X2))active(take(mark(X1), mark(X2)))
natsFrom(mark(X))natsFrom(X)natsFrom(active(X))natsFrom(X)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
s(mark(X))s(X)s(active(X))s(X)
fst(mark(X))fst(X)fst(active(X))fst(X)
pair(mark(X1), X2)pair(X1, X2)pair(X1, mark(X2))pair(X1, X2)
pair(active(X1), X2)pair(X1, X2)pair(X1, active(X2))pair(X1, X2)
snd(mark(X))snd(X)snd(active(X))snd(X)
splitAt(mark(X1), X2)splitAt(X1, X2)splitAt(X1, mark(X2))splitAt(X1, X2)
splitAt(active(X1), X2)splitAt(X1, X2)splitAt(X1, active(X2))splitAt(X1, X2)
u(mark(X1), X2, X3, X4)u(X1, X2, X3, X4)u(X1, mark(X2), X3, X4)u(X1, X2, X3, X4)
u(X1, X2, mark(X3), X4)u(X1, X2, X3, X4)u(X1, X2, X3, mark(X4))u(X1, X2, X3, X4)
u(active(X1), X2, X3, X4)u(X1, X2, X3, X4)u(X1, active(X2), X3, X4)u(X1, X2, X3, X4)
u(X1, X2, active(X3), X4)u(X1, X2, X3, X4)u(X1, X2, X3, active(X4))u(X1, X2, X3, X4)
head(mark(X))head(X)head(active(X))head(X)
tail(mark(X))tail(X)tail(active(X))tail(X)
sel(mark(X1), X2)sel(X1, X2)sel(X1, mark(X2))sel(X1, X2)
sel(active(X1), X2)sel(X1, X2)sel(X1, active(X2))sel(X1, X2)
afterNth(mark(X1), X2)afterNth(X1, X2)afterNth(X1, mark(X2))afterNth(X1, X2)
afterNth(active(X1), X2)afterNth(X1, X2)afterNth(X1, active(X2))afterNth(X1, X2)
take(mark(X1), X2)take(X1, X2)take(X1, mark(X2))take(X1, X2)
take(active(X1), X2)take(X1, X2)take(X1, active(X2))take(X1, X2)

Original Signature

Termination of terms over the following signature is verified: pair, natsFrom, mark, tail, splitAt, fst, u, 0, s, take, active, afterNth, head, sel, cons, snd, 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:

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

Problem 13: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

active(natsFrom(N))mark(cons(N, natsFrom(s(N))))active(fst(pair(XS, YS)))mark(XS)
active(snd(pair(XS, YS)))mark(YS)active(splitAt(0, XS))mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS)))mark(u(splitAt(N, XS), N, X, XS))active(u(pair(YS, ZS), N, X, XS))mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS)))mark(N)active(tail(cons(N, XS)))mark(XS)
active(sel(N, XS))mark(head(afterNth(N, XS)))active(take(N, XS))mark(fst(splitAt(N, XS)))
active(afterNth(N, XS))mark(snd(splitAt(N, XS)))mark(natsFrom(X))active(natsFrom(mark(X)))
mark(cons(X1, X2))active(cons(mark(X1), X2))mark(s(X))active(s(mark(X)))
mark(fst(X))active(fst(mark(X)))mark(pair(X1, X2))active(pair(mark(X1), mark(X2)))
mark(snd(X))active(snd(mark(X)))mark(splitAt(X1, X2))active(splitAt(mark(X1), mark(X2)))
mark(0)active(0)mark(nil)active(nil)
mark(u(X1, X2, X3, X4))active(u(mark(X1), X2, X3, X4))mark(head(X))active(head(mark(X)))
mark(tail(X))active(tail(mark(X)))mark(sel(X1, X2))active(sel(mark(X1), mark(X2)))
mark(afterNth(X1, X2))active(afterNth(mark(X1), mark(X2)))mark(take(X1, X2))active(take(mark(X1), mark(X2)))
natsFrom(mark(X))natsFrom(X)natsFrom(active(X))natsFrom(X)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
s(mark(X))s(X)s(active(X))s(X)
fst(mark(X))fst(X)fst(active(X))fst(X)
pair(mark(X1), X2)pair(X1, X2)pair(X1, mark(X2))pair(X1, X2)
pair(active(X1), X2)pair(X1, X2)pair(X1, active(X2))pair(X1, X2)
snd(mark(X))snd(X)snd(active(X))snd(X)
splitAt(mark(X1), X2)splitAt(X1, X2)splitAt(X1, mark(X2))splitAt(X1, X2)
splitAt(active(X1), X2)splitAt(X1, X2)splitAt(X1, active(X2))splitAt(X1, X2)
u(mark(X1), X2, X3, X4)u(X1, X2, X3, X4)u(X1, mark(X2), X3, X4)u(X1, X2, X3, X4)
u(X1, X2, mark(X3), X4)u(X1, X2, X3, X4)u(X1, X2, X3, mark(X4))u(X1, X2, X3, X4)
u(active(X1), X2, X3, X4)u(X1, X2, X3, X4)u(X1, active(X2), X3, X4)u(X1, X2, X3, X4)
u(X1, X2, active(X3), X4)u(X1, X2, X3, X4)u(X1, X2, X3, active(X4))u(X1, X2, X3, X4)
head(mark(X))head(X)head(active(X))head(X)
tail(mark(X))tail(X)tail(active(X))tail(X)
sel(mark(X1), X2)sel(X1, X2)sel(X1, mark(X2))sel(X1, X2)
sel(active(X1), X2)sel(X1, X2)sel(X1, active(X2))sel(X1, X2)
afterNth(mark(X1), X2)afterNth(X1, X2)afterNth(X1, mark(X2))afterNth(X1, X2)
afterNth(active(X1), X2)afterNth(X1, X2)afterNth(X1, active(X2))afterNth(X1, X2)
take(mark(X1), X2)take(X1, X2)take(X1, mark(X2))take(X1, X2)
take(active(X1), X2)take(X1, X2)take(X1, active(X2))take(X1, X2)

Original Signature

Termination of terms over the following signature is verified: pair, natsFrom, mark, tail, splitAt, fst, u, 0, s, take, active, afterNth, head, sel, nil, snd, cons

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

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

Problem 21: PolynomialLinearRange4iUR



Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

active(natsFrom(N))mark(cons(N, natsFrom(s(N))))active(fst(pair(XS, YS)))mark(XS)
active(snd(pair(XS, YS)))mark(YS)active(splitAt(0, XS))mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS)))mark(u(splitAt(N, XS), N, X, XS))active(u(pair(YS, ZS), N, X, XS))mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS)))mark(N)active(tail(cons(N, XS)))mark(XS)
active(sel(N, XS))mark(head(afterNth(N, XS)))active(take(N, XS))mark(fst(splitAt(N, XS)))
active(afterNth(N, XS))mark(snd(splitAt(N, XS)))mark(natsFrom(X))active(natsFrom(mark(X)))
mark(cons(X1, X2))active(cons(mark(X1), X2))mark(s(X))active(s(mark(X)))
mark(fst(X))active(fst(mark(X)))mark(pair(X1, X2))active(pair(mark(X1), mark(X2)))
mark(snd(X))active(snd(mark(X)))mark(splitAt(X1, X2))active(splitAt(mark(X1), mark(X2)))
mark(0)active(0)mark(nil)active(nil)
mark(u(X1, X2, X3, X4))active(u(mark(X1), X2, X3, X4))mark(head(X))active(head(mark(X)))
mark(tail(X))active(tail(mark(X)))mark(sel(X1, X2))active(sel(mark(X1), mark(X2)))
mark(afterNth(X1, X2))active(afterNth(mark(X1), mark(X2)))mark(take(X1, X2))active(take(mark(X1), mark(X2)))
natsFrom(mark(X))natsFrom(X)natsFrom(active(X))natsFrom(X)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
s(mark(X))s(X)s(active(X))s(X)
fst(mark(X))fst(X)fst(active(X))fst(X)
pair(mark(X1), X2)pair(X1, X2)pair(X1, mark(X2))pair(X1, X2)
pair(active(X1), X2)pair(X1, X2)pair(X1, active(X2))pair(X1, X2)
snd(mark(X))snd(X)snd(active(X))snd(X)
splitAt(mark(X1), X2)splitAt(X1, X2)splitAt(X1, mark(X2))splitAt(X1, X2)
splitAt(active(X1), X2)splitAt(X1, X2)splitAt(X1, active(X2))splitAt(X1, X2)
u(mark(X1), X2, X3, X4)u(X1, X2, X3, X4)u(X1, mark(X2), X3, X4)u(X1, X2, X3, X4)
u(X1, X2, mark(X3), X4)u(X1, X2, X3, X4)u(X1, X2, X3, mark(X4))u(X1, X2, X3, X4)
u(active(X1), X2, X3, X4)u(X1, X2, X3, X4)u(X1, active(X2), X3, X4)u(X1, X2, X3, X4)
u(X1, X2, active(X3), X4)u(X1, X2, X3, X4)u(X1, X2, X3, active(X4))u(X1, X2, X3, X4)
head(mark(X))head(X)head(active(X))head(X)
tail(mark(X))tail(X)tail(active(X))tail(X)
sel(mark(X1), X2)sel(X1, X2)sel(X1, mark(X2))sel(X1, X2)
sel(active(X1), X2)sel(X1, X2)sel(X1, active(X2))sel(X1, X2)
afterNth(mark(X1), X2)afterNth(X1, X2)afterNth(X1, mark(X2))afterNth(X1, X2)
afterNth(active(X1), X2)afterNth(X1, X2)afterNth(X1, active(X2))afterNth(X1, X2)
take(mark(X1), X2)take(X1, X2)take(X1, mark(X2))take(X1, X2)
take(active(X1), X2)take(X1, X2)take(X1, active(X2))take(X1, X2)

Original Signature

Termination of terms over the following signature is verified: pair, natsFrom, mark, tail, splitAt, fst, u, 0, s, take, active, afterNth, head, sel, cons, snd, 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:

sel#(X1, active(X2))sel#(X1, X2)

Problem 14: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

snd#(active(X))snd#(X)snd#(mark(X))snd#(X)

Rewrite Rules

active(natsFrom(N))mark(cons(N, natsFrom(s(N))))active(fst(pair(XS, YS)))mark(XS)
active(snd(pair(XS, YS)))mark(YS)active(splitAt(0, XS))mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS)))mark(u(splitAt(N, XS), N, X, XS))active(u(pair(YS, ZS), N, X, XS))mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS)))mark(N)active(tail(cons(N, XS)))mark(XS)
active(sel(N, XS))mark(head(afterNth(N, XS)))active(take(N, XS))mark(fst(splitAt(N, XS)))
active(afterNth(N, XS))mark(snd(splitAt(N, XS)))mark(natsFrom(X))active(natsFrom(mark(X)))
mark(cons(X1, X2))active(cons(mark(X1), X2))mark(s(X))active(s(mark(X)))
mark(fst(X))active(fst(mark(X)))mark(pair(X1, X2))active(pair(mark(X1), mark(X2)))
mark(snd(X))active(snd(mark(X)))mark(splitAt(X1, X2))active(splitAt(mark(X1), mark(X2)))
mark(0)active(0)mark(nil)active(nil)
mark(u(X1, X2, X3, X4))active(u(mark(X1), X2, X3, X4))mark(head(X))active(head(mark(X)))
mark(tail(X))active(tail(mark(X)))mark(sel(X1, X2))active(sel(mark(X1), mark(X2)))
mark(afterNth(X1, X2))active(afterNth(mark(X1), mark(X2)))mark(take(X1, X2))active(take(mark(X1), mark(X2)))
natsFrom(mark(X))natsFrom(X)natsFrom(active(X))natsFrom(X)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
s(mark(X))s(X)s(active(X))s(X)
fst(mark(X))fst(X)fst(active(X))fst(X)
pair(mark(X1), X2)pair(X1, X2)pair(X1, mark(X2))pair(X1, X2)
pair(active(X1), X2)pair(X1, X2)pair(X1, active(X2))pair(X1, X2)
snd(mark(X))snd(X)snd(active(X))snd(X)
splitAt(mark(X1), X2)splitAt(X1, X2)splitAt(X1, mark(X2))splitAt(X1, X2)
splitAt(active(X1), X2)splitAt(X1, X2)splitAt(X1, active(X2))splitAt(X1, X2)
u(mark(X1), X2, X3, X4)u(X1, X2, X3, X4)u(X1, mark(X2), X3, X4)u(X1, X2, X3, X4)
u(X1, X2, mark(X3), X4)u(X1, X2, X3, X4)u(X1, X2, X3, mark(X4))u(X1, X2, X3, X4)
u(active(X1), X2, X3, X4)u(X1, X2, X3, X4)u(X1, active(X2), X3, X4)u(X1, X2, X3, X4)
u(X1, X2, active(X3), X4)u(X1, X2, X3, X4)u(X1, X2, X3, active(X4))u(X1, X2, X3, X4)
head(mark(X))head(X)head(active(X))head(X)
tail(mark(X))tail(X)tail(active(X))tail(X)
sel(mark(X1), X2)sel(X1, X2)sel(X1, mark(X2))sel(X1, X2)
sel(active(X1), X2)sel(X1, X2)sel(X1, active(X2))sel(X1, X2)
afterNth(mark(X1), X2)afterNth(X1, X2)afterNth(X1, mark(X2))afterNth(X1, X2)
afterNth(active(X1), X2)afterNth(X1, X2)afterNth(X1, active(X2))afterNth(X1, X2)
take(mark(X1), X2)take(X1, X2)take(X1, mark(X2))take(X1, X2)
take(active(X1), X2)take(X1, X2)take(X1, active(X2))take(X1, X2)

Original Signature

Termination of terms over the following signature is verified: pair, natsFrom, mark, tail, splitAt, fst, u, 0, s, take, active, afterNth, head, sel, nil, snd, cons

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

snd#(active(X))snd#(X)snd#(mark(X))snd#(X)

Problem 15: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

u#(mark(X1), X2, X3, X4)u#(X1, X2, X3, X4)u#(active(X1), X2, X3, X4)u#(X1, X2, X3, X4)
u#(X1, mark(X2), X3, X4)u#(X1, X2, X3, X4)u#(X1, X2, active(X3), X4)u#(X1, X2, X3, X4)
u#(X1, X2, X3, mark(X4))u#(X1, X2, X3, X4)u#(X1, active(X2), X3, X4)u#(X1, X2, X3, X4)
u#(X1, X2, X3, active(X4))u#(X1, X2, X3, X4)u#(X1, X2, mark(X3), X4)u#(X1, X2, X3, X4)

Rewrite Rules

active(natsFrom(N))mark(cons(N, natsFrom(s(N))))active(fst(pair(XS, YS)))mark(XS)
active(snd(pair(XS, YS)))mark(YS)active(splitAt(0, XS))mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS)))mark(u(splitAt(N, XS), N, X, XS))active(u(pair(YS, ZS), N, X, XS))mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS)))mark(N)active(tail(cons(N, XS)))mark(XS)
active(sel(N, XS))mark(head(afterNth(N, XS)))active(take(N, XS))mark(fst(splitAt(N, XS)))
active(afterNth(N, XS))mark(snd(splitAt(N, XS)))mark(natsFrom(X))active(natsFrom(mark(X)))
mark(cons(X1, X2))active(cons(mark(X1), X2))mark(s(X))active(s(mark(X)))
mark(fst(X))active(fst(mark(X)))mark(pair(X1, X2))active(pair(mark(X1), mark(X2)))
mark(snd(X))active(snd(mark(X)))mark(splitAt(X1, X2))active(splitAt(mark(X1), mark(X2)))
mark(0)active(0)mark(nil)active(nil)
mark(u(X1, X2, X3, X4))active(u(mark(X1), X2, X3, X4))mark(head(X))active(head(mark(X)))
mark(tail(X))active(tail(mark(X)))mark(sel(X1, X2))active(sel(mark(X1), mark(X2)))
mark(afterNth(X1, X2))active(afterNth(mark(X1), mark(X2)))mark(take(X1, X2))active(take(mark(X1), mark(X2)))
natsFrom(mark(X))natsFrom(X)natsFrom(active(X))natsFrom(X)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
s(mark(X))s(X)s(active(X))s(X)
fst(mark(X))fst(X)fst(active(X))fst(X)
pair(mark(X1), X2)pair(X1, X2)pair(X1, mark(X2))pair(X1, X2)
pair(active(X1), X2)pair(X1, X2)pair(X1, active(X2))pair(X1, X2)
snd(mark(X))snd(X)snd(active(X))snd(X)
splitAt(mark(X1), X2)splitAt(X1, X2)splitAt(X1, mark(X2))splitAt(X1, X2)
splitAt(active(X1), X2)splitAt(X1, X2)splitAt(X1, active(X2))splitAt(X1, X2)
u(mark(X1), X2, X3, X4)u(X1, X2, X3, X4)u(X1, mark(X2), X3, X4)u(X1, X2, X3, X4)
u(X1, X2, mark(X3), X4)u(X1, X2, X3, X4)u(X1, X2, X3, mark(X4))u(X1, X2, X3, X4)
u(active(X1), X2, X3, X4)u(X1, X2, X3, X4)u(X1, active(X2), X3, X4)u(X1, X2, X3, X4)
u(X1, X2, active(X3), X4)u(X1, X2, X3, X4)u(X1, X2, X3, active(X4))u(X1, X2, X3, X4)
head(mark(X))head(X)head(active(X))head(X)
tail(mark(X))tail(X)tail(active(X))tail(X)
sel(mark(X1), X2)sel(X1, X2)sel(X1, mark(X2))sel(X1, X2)
sel(active(X1), X2)sel(X1, X2)sel(X1, active(X2))sel(X1, X2)
afterNth(mark(X1), X2)afterNth(X1, X2)afterNth(X1, mark(X2))afterNth(X1, X2)
afterNth(active(X1), X2)afterNth(X1, X2)afterNth(X1, active(X2))afterNth(X1, X2)
take(mark(X1), X2)take(X1, X2)take(X1, mark(X2))take(X1, X2)
take(active(X1), X2)take(X1, X2)take(X1, active(X2))take(X1, X2)

Original Signature

Termination of terms over the following signature is verified: pair, natsFrom, mark, tail, splitAt, fst, u, 0, s, take, active, afterNth, head, sel, nil, snd, cons

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

u#(mark(X1), X2, X3, X4)u#(X1, X2, X3, X4)u#(active(X1), X2, X3, X4)u#(X1, X2, X3, X4)

Problem 22: PolynomialLinearRange4iUR



Dependency Pair Problem

Dependency Pairs

u#(X1, mark(X2), X3, X4)u#(X1, X2, X3, X4)u#(X1, X2, active(X3), X4)u#(X1, X2, X3, X4)
u#(X1, X2, X3, mark(X4))u#(X1, X2, X3, X4)u#(X1, X2, X3, active(X4))u#(X1, X2, X3, X4)
u#(X1, active(X2), X3, X4)u#(X1, X2, X3, X4)u#(X1, X2, mark(X3), X4)u#(X1, X2, X3, X4)

Rewrite Rules

active(natsFrom(N))mark(cons(N, natsFrom(s(N))))active(fst(pair(XS, YS)))mark(XS)
active(snd(pair(XS, YS)))mark(YS)active(splitAt(0, XS))mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS)))mark(u(splitAt(N, XS), N, X, XS))active(u(pair(YS, ZS), N, X, XS))mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS)))mark(N)active(tail(cons(N, XS)))mark(XS)
active(sel(N, XS))mark(head(afterNth(N, XS)))active(take(N, XS))mark(fst(splitAt(N, XS)))
active(afterNth(N, XS))mark(snd(splitAt(N, XS)))mark(natsFrom(X))active(natsFrom(mark(X)))
mark(cons(X1, X2))active(cons(mark(X1), X2))mark(s(X))active(s(mark(X)))
mark(fst(X))active(fst(mark(X)))mark(pair(X1, X2))active(pair(mark(X1), mark(X2)))
mark(snd(X))active(snd(mark(X)))mark(splitAt(X1, X2))active(splitAt(mark(X1), mark(X2)))
mark(0)active(0)mark(nil)active(nil)
mark(u(X1, X2, X3, X4))active(u(mark(X1), X2, X3, X4))mark(head(X))active(head(mark(X)))
mark(tail(X))active(tail(mark(X)))mark(sel(X1, X2))active(sel(mark(X1), mark(X2)))
mark(afterNth(X1, X2))active(afterNth(mark(X1), mark(X2)))mark(take(X1, X2))active(take(mark(X1), mark(X2)))
natsFrom(mark(X))natsFrom(X)natsFrom(active(X))natsFrom(X)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
s(mark(X))s(X)s(active(X))s(X)
fst(mark(X))fst(X)fst(active(X))fst(X)
pair(mark(X1), X2)pair(X1, X2)pair(X1, mark(X2))pair(X1, X2)
pair(active(X1), X2)pair(X1, X2)pair(X1, active(X2))pair(X1, X2)
snd(mark(X))snd(X)snd(active(X))snd(X)
splitAt(mark(X1), X2)splitAt(X1, X2)splitAt(X1, mark(X2))splitAt(X1, X2)
splitAt(active(X1), X2)splitAt(X1, X2)splitAt(X1, active(X2))splitAt(X1, X2)
u(mark(X1), X2, X3, X4)u(X1, X2, X3, X4)u(X1, mark(X2), X3, X4)u(X1, X2, X3, X4)
u(X1, X2, mark(X3), X4)u(X1, X2, X3, X4)u(X1, X2, X3, mark(X4))u(X1, X2, X3, X4)
u(active(X1), X2, X3, X4)u(X1, X2, X3, X4)u(X1, active(X2), X3, X4)u(X1, X2, X3, X4)
u(X1, X2, active(X3), X4)u(X1, X2, X3, X4)u(X1, X2, X3, active(X4))u(X1, X2, X3, X4)
head(mark(X))head(X)head(active(X))head(X)
tail(mark(X))tail(X)tail(active(X))tail(X)
sel(mark(X1), X2)sel(X1, X2)sel(X1, mark(X2))sel(X1, X2)
sel(active(X1), X2)sel(X1, X2)sel(X1, active(X2))sel(X1, X2)
afterNth(mark(X1), X2)afterNth(X1, X2)afterNth(X1, mark(X2))afterNth(X1, X2)
afterNth(active(X1), X2)afterNth(X1, X2)afterNth(X1, active(X2))afterNth(X1, X2)
take(mark(X1), X2)take(X1, X2)take(X1, mark(X2))take(X1, X2)
take(active(X1), X2)take(X1, X2)take(X1, active(X2))take(X1, X2)

Original Signature

Termination of terms over the following signature is verified: pair, natsFrom, mark, tail, splitAt, fst, u, 0, s, take, active, afterNth, head, sel, cons, snd, 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:

u#(X1, X2, X3, mark(X4))u#(X1, X2, X3, X4)