The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).

(0) Obligation:

The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following 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)))
active(natsFrom(X)) → natsFrom(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(fst(X)) → fst(active(X))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(u(X1, X2, X3, X4)) → u(active(X1), X2, X3, X4)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
natsFrom(mark(X)) → mark(natsFrom(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
fst(mark(X)) → mark(fst(X))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
u(mark(X1), X2, X3, X4) → mark(u(X1, X2, X3, X4))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(fst(X)) → fst(proper(X))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(u(X1, X2, X3, X4)) → u(proper(X1), proper(X2), proper(X3), proper(X4))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
natsFrom(ok(X)) → ok(natsFrom(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
fst(ok(X)) → ok(fst(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
u(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(u(X1, X2, X3, X4))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Rewrite Strategy: FULL

(1) NestedDefinedSymbolProof (BOTH BOUNDS(ID, ID) transformation)

The following defined symbols can occur below the 0th argument of top: proper, active
The following defined symbols can occur below the 0th argument of proper: proper, active
The following defined symbols can occur below the 0th argument of active: proper, active

Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
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)))
active(natsFrom(X)) → natsFrom(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(fst(X)) → fst(active(X))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(u(X1, X2, X3, X4)) → u(active(X1), X2, X3, X4)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(fst(X)) → fst(proper(X))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(u(X1, X2, X3, X4)) → u(proper(X1), proper(X2), proper(X3), proper(X4))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))

(2) Obligation:

The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

top(ok(X)) → top(active(X))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
tail(ok(X)) → ok(tail(X))
snd(ok(X)) → ok(snd(X))
pair(X1, mark(X2)) → mark(pair(X1, X2))
snd(mark(X)) → mark(snd(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(nil) → ok(nil)
take(X1, mark(X2)) → mark(take(X1, X2))
head(ok(X)) → ok(head(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
head(mark(X)) → mark(head(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
natsFrom(ok(X)) → ok(natsFrom(X))
fst(mark(X)) → mark(fst(X))
u(mark(X1), X2, X3, X4) → mark(u(X1, X2, X3, X4))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
tail(mark(X)) → mark(tail(X))
s(ok(X)) → ok(s(X))
s(mark(X)) → mark(s(X))
u(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(u(X1, X2, X3, X4))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
proper(0) → ok(0)
cons(mark(X1), X2) → mark(cons(X1, X2))
top(mark(X)) → top(proper(X))

Rewrite Strategy: FULL

(3) RcToIrcProof (BOTH BOUNDS(ID, ID) transformation)

Converted rc-obligation to irc-obligation.

As the TRS is a non-duplicating overlay system, we have rc = irc.

(4) Obligation:

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

top(ok(X)) → top(active(X))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
tail(ok(X)) → ok(tail(X))
snd(ok(X)) → ok(snd(X))
pair(X1, mark(X2)) → mark(pair(X1, X2))
snd(mark(X)) → mark(snd(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(nil) → ok(nil)
take(X1, mark(X2)) → mark(take(X1, X2))
head(ok(X)) → ok(head(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
head(mark(X)) → mark(head(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
natsFrom(ok(X)) → ok(natsFrom(X))
fst(mark(X)) → mark(fst(X))
u(mark(X1), X2, X3, X4) → mark(u(X1, X2, X3, X4))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
tail(mark(X)) → mark(tail(X))
s(ok(X)) → ok(s(X))
s(mark(X)) → mark(s(X))
u(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(u(X1, X2, X3, X4))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
proper(0) → ok(0)
cons(mark(X1), X2) → mark(cons(X1, X2))
top(mark(X)) → top(proper(X))

Rewrite Strategy: INNERMOST

(5) CpxTrsMatchBoundsTAProof (EQUIVALENT transformation)

A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 2.

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by:
final states : [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]
transitions:
ok0(0) → 0
active0(0) → 0
mark0(0) → 0
nil0() → 0
00() → 0
top0(0) → 1
afterNth0(0, 0) → 2
take0(0, 0) → 3
cons0(0, 0) → 4
tail0(0) → 5
snd0(0) → 6
pair0(0, 0) → 7
sel0(0, 0) → 8
splitAt0(0, 0) → 9
proper0(0) → 10
head0(0) → 11
natsFrom0(0) → 12
fst0(0) → 13
u0(0, 0, 0, 0) → 14
s0(0) → 15
active1(0) → 16
top1(16) → 1
afterNth1(0, 0) → 17
mark1(17) → 2
take1(0, 0) → 18
mark1(18) → 3
cons1(0, 0) → 19
ok1(19) → 4
tail1(0) → 20
ok1(20) → 5
snd1(0) → 21
ok1(21) → 6
pair1(0, 0) → 22
mark1(22) → 7
snd1(0) → 23
mark1(23) → 6
take1(0, 0) → 24
ok1(24) → 3
sel1(0, 0) → 25
ok1(25) → 8
splitAt1(0, 0) → 26
mark1(26) → 9
sel1(0, 0) → 27
mark1(27) → 8
nil1() → 28
ok1(28) → 10
head1(0) → 29
ok1(29) → 11
pair1(0, 0) → 30
ok1(30) → 7
head1(0) → 31
mark1(31) → 11
natsFrom1(0) → 32
mark1(32) → 12
natsFrom1(0) → 33
ok1(33) → 12
fst1(0) → 34
mark1(34) → 13
u1(0, 0, 0, 0) → 35
mark1(35) → 14
afterNth1(0, 0) → 36
ok1(36) → 2
tail1(0) → 37
mark1(37) → 5
s1(0) → 38
ok1(38) → 15
s1(0) → 39
mark1(39) → 15
u1(0, 0, 0, 0) → 40
ok1(40) → 14
fst1(0) → 41
ok1(41) → 13
splitAt1(0, 0) → 42
ok1(42) → 9
01() → 43
ok1(43) → 10
cons1(0, 0) → 44
mark1(44) → 4
proper1(0) → 45
top1(45) → 1
mark1(17) → 17
mark1(17) → 36
mark1(18) → 18
mark1(18) → 24
ok1(19) → 19
ok1(19) → 44
ok1(20) → 20
ok1(20) → 37
ok1(21) → 21
ok1(21) → 23
mark1(22) → 22
mark1(22) → 30
mark1(23) → 21
mark1(23) → 23
ok1(24) → 18
ok1(24) → 24
ok1(25) → 25
ok1(25) → 27
mark1(26) → 26
mark1(26) → 42
mark1(27) → 25
mark1(27) → 27
ok1(28) → 45
ok1(29) → 29
ok1(29) → 31
ok1(30) → 22
ok1(30) → 30
mark1(31) → 29
mark1(31) → 31
mark1(32) → 32
mark1(32) → 33
ok1(33) → 32
ok1(33) → 33
mark1(34) → 34
mark1(34) → 41
mark1(35) → 35
mark1(35) → 40
ok1(36) → 17
ok1(36) → 36
mark1(37) → 20
mark1(37) → 37
ok1(38) → 38
ok1(38) → 39
mark1(39) → 38
mark1(39) → 39
ok1(40) → 35
ok1(40) → 40
ok1(41) → 34
ok1(41) → 41
ok1(42) → 26
ok1(42) → 42
ok1(43) → 45
mark1(44) → 19
mark1(44) → 44
active2(28) → 46
top2(46) → 1
active2(43) → 46

(6) BOUNDS(1, n^1)