(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(0, IL)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X1, X2, X3, X4)) → U21(active(X1), X2, X3, X4)
active(U22(X1, X2, X3, X4)) → U22(active(X1), X2, X3, X4)
active(U23(X1, X2, X3, X4)) → U23(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X1), X2, X3, X4) → mark(U21(X1, X2, X3, X4))
U22(mark(X1), X2, X3, X4) → mark(U22(X1, X2, X3, X4))
U23(mark(X1), X2, X3, X4) → mark(U23(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X1, X2, X3, X4)) → U21(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U22(X1, X2, X3, X4)) → U22(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U23(X1, X2, X3, X4)) → U23(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U21(X1, X2, X3, X4))
U22(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U22(X1, X2, X3, X4))
U23(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U23(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Rewrite Strategy: INNERMOST
(1) 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 5.
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]
transitions:
zeros0() → 0
mark0(0) → 0
00() → 0
tt0() → 0
nil0() → 0
ok0(0) → 0
active0(0) → 1
cons0(0, 0) → 2
U110(0, 0) → 3
U120(0, 0) → 4
s0(0) → 5
length0(0) → 6
U210(0, 0, 0, 0) → 7
U220(0, 0, 0, 0) → 8
U230(0, 0, 0, 0) → 9
take0(0, 0) → 10
proper0(0) → 11
top0(0) → 12
01() → 14
zeros1() → 15
cons1(14, 15) → 13
mark1(13) → 1
cons1(0, 0) → 16
mark1(16) → 2
U111(0, 0) → 17
mark1(17) → 3
U121(0, 0) → 18
mark1(18) → 4
s1(0) → 19
mark1(19) → 5
length1(0) → 20
mark1(20) → 6
U211(0, 0, 0, 0) → 21
mark1(21) → 7
U221(0, 0, 0, 0) → 22
mark1(22) → 8
U231(0, 0, 0, 0) → 23
mark1(23) → 9
take1(0, 0) → 24
mark1(24) → 10
zeros1() → 25
ok1(25) → 11
01() → 26
ok1(26) → 11
tt1() → 27
ok1(27) → 11
nil1() → 28
ok1(28) → 11
cons1(0, 0) → 29
ok1(29) → 2
U111(0, 0) → 30
ok1(30) → 3
U121(0, 0) → 31
ok1(31) → 4
s1(0) → 32
ok1(32) → 5
length1(0) → 33
ok1(33) → 6
U211(0, 0, 0, 0) → 34
ok1(34) → 7
U221(0, 0, 0, 0) → 35
ok1(35) → 8
U231(0, 0, 0, 0) → 36
ok1(36) → 9
take1(0, 0) → 37
ok1(37) → 10
proper1(0) → 38
top1(38) → 12
active1(0) → 39
top1(39) → 12
mark1(13) → 39
mark1(16) → 16
mark1(16) → 29
mark1(17) → 17
mark1(17) → 30
mark1(18) → 18
mark1(18) → 31
mark1(19) → 19
mark1(19) → 32
mark1(20) → 20
mark1(20) → 33
mark1(21) → 21
mark1(21) → 34
mark1(22) → 22
mark1(22) → 35
mark1(23) → 23
mark1(23) → 36
mark1(24) → 24
mark1(24) → 37
ok1(25) → 38
ok1(26) → 38
ok1(27) → 38
ok1(28) → 38
ok1(29) → 16
ok1(29) → 29
ok1(30) → 17
ok1(30) → 30
ok1(31) → 18
ok1(31) → 31
ok1(32) → 19
ok1(32) → 32
ok1(33) → 20
ok1(33) → 33
ok1(34) → 21
ok1(34) → 34
ok1(35) → 22
ok1(35) → 35
ok1(36) → 23
ok1(36) → 36
ok1(37) → 24
ok1(37) → 37
proper2(13) → 40
top2(40) → 12
active2(25) → 41
top2(41) → 12
active2(26) → 41
active2(27) → 41
active2(28) → 41
02() → 43
zeros2() → 44
cons2(43, 44) → 42
mark2(42) → 41
proper2(14) → 45
proper2(15) → 46
cons2(45, 46) → 40
zeros2() → 47
ok2(47) → 46
02() → 48
ok2(48) → 45
proper3(42) → 49
top3(49) → 12
proper3(43) → 50
proper3(44) → 51
cons3(50, 51) → 49
cons3(48, 47) → 52
ok3(52) → 40
zeros3() → 53
ok3(53) → 51
03() → 54
ok3(54) → 50
active3(52) → 55
top3(55) → 12
cons4(54, 53) → 56
ok4(56) → 49
active4(48) → 57
cons4(57, 47) → 55
active4(56) → 58
top4(58) → 12
active5(54) → 59
cons5(59, 53) → 58
(2) BOUNDS(O(1), O(n^1))