(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(and(tt, X)) → mark(X)
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(s(length(L)))
active(cons(X1, X2)) → cons(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(length(X)) → length(active(X))
active(s(X)) → s(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
length(mark(X)) → mark(length(X))
s(mark(X)) → mark(s(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
proper(s(X)) → s(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
length(ok(X)) → ok(length(X))
s(ok(X)) → ok(s(X))
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]
transitions:
zeros0() → 0
mark0(0) → 0
00() → 0
tt0() → 0
nil0() → 0
ok0(0) → 0
active0(0) → 1
cons0(0, 0) → 2
and0(0, 0) → 3
length0(0) → 4
s0(0) → 5
proper0(0) → 6
top0(0) → 7
01() → 9
zeros1() → 10
cons1(9, 10) → 8
mark1(8) → 1
cons1(0, 0) → 11
mark1(11) → 2
and1(0, 0) → 12
mark1(12) → 3
length1(0) → 13
mark1(13) → 4
s1(0) → 14
mark1(14) → 5
zeros1() → 15
ok1(15) → 6
01() → 16
ok1(16) → 6
tt1() → 17
ok1(17) → 6
nil1() → 18
ok1(18) → 6
cons1(0, 0) → 19
ok1(19) → 2
and1(0, 0) → 20
ok1(20) → 3
length1(0) → 21
ok1(21) → 4
s1(0) → 22
ok1(22) → 5
proper1(0) → 23
top1(23) → 7
active1(0) → 24
top1(24) → 7
mark1(8) → 24
mark1(11) → 11
mark1(11) → 19
mark1(12) → 12
mark1(12) → 20
mark1(13) → 13
mark1(13) → 21
mark1(14) → 14
mark1(14) → 22
ok1(15) → 23
ok1(16) → 23
ok1(17) → 23
ok1(18) → 23
ok1(19) → 11
ok1(19) → 19
ok1(20) → 12
ok1(20) → 20
ok1(21) → 13
ok1(21) → 21
ok1(22) → 14
ok1(22) → 22
proper2(8) → 25
top2(25) → 7
active2(15) → 26
top2(26) → 7
active2(16) → 26
active2(17) → 26
active2(18) → 26
02() → 28
zeros2() → 29
cons2(28, 29) → 27
mark2(27) → 26
proper2(9) → 30
proper2(10) → 31
cons2(30, 31) → 25
zeros2() → 32
ok2(32) → 31
02() → 33
ok2(33) → 30
proper3(27) → 34
top3(34) → 7
proper3(28) → 35
proper3(29) → 36
cons3(35, 36) → 34
cons3(33, 32) → 37
ok3(37) → 25
zeros3() → 38
ok3(38) → 36
03() → 39
ok3(39) → 35
active3(37) → 40
top3(40) → 7
cons4(39, 38) → 41
ok4(41) → 34
active4(33) → 42
cons4(42, 32) → 40
active4(41) → 43
top4(43) → 7
active5(39) → 44
cons5(44, 38) → 43
(2) BOUNDS(O(1), O(n^1))