* Step 1: Bounds WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
active(and(X1,X2)) -> and(active(X1),X2)
active(and(tt(),X)) -> mark(X)
active(cons(X1,X2)) -> cons(active(X1),X2)
active(length(X)) -> length(active(X))
active(length(cons(N,L))) -> mark(s(length(L)))
active(length(nil())) -> mark(0())
active(s(X)) -> s(active(X))
active(take(X1,X2)) -> take(X1,active(X2))
active(take(X1,X2)) -> take(active(X1),X2)
active(take(0(),IL)) -> mark(nil())
active(take(s(M),cons(N,IL))) -> mark(cons(N,take(M,IL)))
active(zeros()) -> mark(cons(0(),zeros()))
and(mark(X1),X2) -> mark(and(X1,X2))
and(ok(X1),ok(X2)) -> ok(and(X1,X2))
cons(mark(X1),X2) -> mark(cons(X1,X2))
cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
length(mark(X)) -> mark(length(X))
length(ok(X)) -> ok(length(X))
proper(0()) -> ok(0())
proper(and(X1,X2)) -> and(proper(X1),proper(X2))
proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
proper(length(X)) -> length(proper(X))
proper(nil()) -> ok(nil())
proper(s(X)) -> s(proper(X))
proper(take(X1,X2)) -> take(proper(X1),proper(X2))
proper(tt()) -> ok(tt())
proper(zeros()) -> ok(zeros())
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
take(X1,mark(X2)) -> mark(take(X1,X2))
take(mark(X1),X2) -> mark(take(X1,X2))
take(ok(X1),ok(X2)) -> ok(take(X1,X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
- Signature:
{active/1,and/2,cons/2,length/1,proper/1,s/1,take/2,top/1} / {0/0,mark/1,nil/0,ok/1,tt/0,zeros/0}
- Obligation:
runtime complexity wrt. defined symbols {active,and,cons,length,proper,s,take,top} and constructors {0,mark
,nil,ok,tt,zeros}
+ Applied Processor:
Bounds {initialAutomaton = perSymbol, enrichment = match}
+ Details:
The problem is match-bounded by 5.
The enriched problem is compatible with follwoing automaton.
0_0() -> 1
0_1() -> 16
0_2() -> 26
0_3() -> 37
active_0(1) -> 2
active_0(6) -> 2
active_0(7) -> 2
active_0(8) -> 2
active_0(13) -> 2
active_0(14) -> 2
active_1(1) -> 23
active_1(6) -> 23
active_1(7) -> 23
active_1(8) -> 23
active_1(13) -> 23
active_1(14) -> 23
active_2(16) -> 24
active_2(17) -> 24
active_3(32) -> 31
active_4(26) -> 36
active_4(30) -> 36
active_4(38) -> 39
active_5(37) -> 40
and_0(1,1) -> 3
and_0(1,6) -> 3
and_0(1,7) -> 3
and_0(1,8) -> 3
and_0(1,13) -> 3
and_0(1,14) -> 3
and_0(6,1) -> 3
and_0(6,6) -> 3
and_0(6,7) -> 3
and_0(6,8) -> 3
and_0(6,13) -> 3
and_0(6,14) -> 3
and_0(7,1) -> 3
and_0(7,6) -> 3
and_0(7,7) -> 3
and_0(7,8) -> 3
and_0(7,13) -> 3
and_0(7,14) -> 3
and_0(8,1) -> 3
and_0(8,6) -> 3
and_0(8,7) -> 3
and_0(8,8) -> 3
and_0(8,13) -> 3
and_0(8,14) -> 3
and_0(13,1) -> 3
and_0(13,6) -> 3
and_0(13,7) -> 3
and_0(13,8) -> 3
and_0(13,13) -> 3
and_0(13,14) -> 3
and_0(14,1) -> 3
and_0(14,6) -> 3
and_0(14,7) -> 3
and_0(14,8) -> 3
and_0(14,13) -> 3
and_0(14,14) -> 3
and_1(1,1) -> 18
and_1(1,6) -> 18
and_1(1,7) -> 18
and_1(1,8) -> 18
and_1(1,13) -> 18
and_1(1,14) -> 18
and_1(6,1) -> 18
and_1(6,6) -> 18
and_1(6,7) -> 18
and_1(6,8) -> 18
and_1(6,13) -> 18
and_1(6,14) -> 18
and_1(7,1) -> 18
and_1(7,6) -> 18
and_1(7,7) -> 18
and_1(7,8) -> 18
and_1(7,13) -> 18
and_1(7,14) -> 18
and_1(8,1) -> 18
and_1(8,6) -> 18
and_1(8,7) -> 18
and_1(8,8) -> 18
and_1(8,13) -> 18
and_1(8,14) -> 18
and_1(13,1) -> 18
and_1(13,6) -> 18
and_1(13,7) -> 18
and_1(13,8) -> 18
and_1(13,13) -> 18
and_1(13,14) -> 18
and_1(14,1) -> 18
and_1(14,6) -> 18
and_1(14,7) -> 18
and_1(14,8) -> 18
and_1(14,13) -> 18
and_1(14,14) -> 18
cons_0(1,1) -> 4
cons_0(1,6) -> 4
cons_0(1,7) -> 4
cons_0(1,8) -> 4
cons_0(1,13) -> 4
cons_0(1,14) -> 4
cons_0(6,1) -> 4
cons_0(6,6) -> 4
cons_0(6,7) -> 4
cons_0(6,8) -> 4
cons_0(6,13) -> 4
cons_0(6,14) -> 4
cons_0(7,1) -> 4
cons_0(7,6) -> 4
cons_0(7,7) -> 4
cons_0(7,8) -> 4
cons_0(7,13) -> 4
cons_0(7,14) -> 4
cons_0(8,1) -> 4
cons_0(8,6) -> 4
cons_0(8,7) -> 4
cons_0(8,8) -> 4
cons_0(8,13) -> 4
cons_0(8,14) -> 4
cons_0(13,1) -> 4
cons_0(13,6) -> 4
cons_0(13,7) -> 4
cons_0(13,8) -> 4
cons_0(13,13) -> 4
cons_0(13,14) -> 4
cons_0(14,1) -> 4
cons_0(14,6) -> 4
cons_0(14,7) -> 4
cons_0(14,8) -> 4
cons_0(14,13) -> 4
cons_0(14,14) -> 4
cons_1(1,1) -> 19
cons_1(1,6) -> 19
cons_1(1,7) -> 19
cons_1(1,8) -> 19
cons_1(1,13) -> 19
cons_1(1,14) -> 19
cons_1(6,1) -> 19
cons_1(6,6) -> 19
cons_1(6,7) -> 19
cons_1(6,8) -> 19
cons_1(6,13) -> 19
cons_1(6,14) -> 19
cons_1(7,1) -> 19
cons_1(7,6) -> 19
cons_1(7,7) -> 19
cons_1(7,8) -> 19
cons_1(7,13) -> 19
cons_1(7,14) -> 19
cons_1(8,1) -> 19
cons_1(8,6) -> 19
cons_1(8,7) -> 19
cons_1(8,8) -> 19
cons_1(8,13) -> 19
cons_1(8,14) -> 19
cons_1(13,1) -> 19
cons_1(13,6) -> 19
cons_1(13,7) -> 19
cons_1(13,8) -> 19
cons_1(13,13) -> 19
cons_1(13,14) -> 19
cons_1(14,1) -> 19
cons_1(14,6) -> 19
cons_1(14,7) -> 19
cons_1(14,8) -> 19
cons_1(14,13) -> 19
cons_1(14,14) -> 19
cons_1(16,17) -> 15
cons_2(26,27) -> 25
cons_2(28,29) -> 24
cons_3(26,27) -> 32
cons_3(30,27) -> 32
cons_3(33,34) -> 31
cons_4(36,27) -> 31
cons_4(37,35) -> 38
cons_5(40,35) -> 39
length_0(1) -> 5
length_0(6) -> 5
length_0(7) -> 5
length_0(8) -> 5
length_0(13) -> 5
length_0(14) -> 5
length_1(1) -> 20
length_1(6) -> 20
length_1(7) -> 20
length_1(8) -> 20
length_1(13) -> 20
length_1(14) -> 20
mark_0(1) -> 6
mark_0(6) -> 6
mark_0(7) -> 6
mark_0(8) -> 6
mark_0(13) -> 6
mark_0(14) -> 6
mark_1(15) -> 2
mark_1(15) -> 23
mark_1(18) -> 3
mark_1(18) -> 18
mark_1(19) -> 4
mark_1(19) -> 19
mark_1(20) -> 5
mark_1(20) -> 20
mark_1(21) -> 10
mark_1(21) -> 21
mark_1(22) -> 11
mark_1(22) -> 22
mark_2(25) -> 24
nil_0() -> 7
nil_1() -> 16
nil_2() -> 30
ok_0(1) -> 8
ok_0(6) -> 8
ok_0(7) -> 8
ok_0(8) -> 8
ok_0(13) -> 8
ok_0(14) -> 8
ok_1(16) -> 9
ok_1(16) -> 23
ok_1(17) -> 9
ok_1(17) -> 23
ok_1(18) -> 3
ok_1(18) -> 18
ok_1(19) -> 4
ok_1(19) -> 19
ok_1(20) -> 5
ok_1(20) -> 20
ok_1(21) -> 10
ok_1(21) -> 21
ok_1(22) -> 11
ok_1(22) -> 22
ok_2(26) -> 28
ok_2(27) -> 29
ok_2(30) -> 28
ok_3(32) -> 24
ok_3(35) -> 34
ok_3(37) -> 33
ok_4(38) -> 31
proper_0(1) -> 9
proper_0(6) -> 9
proper_0(7) -> 9
proper_0(8) -> 9
proper_0(13) -> 9
proper_0(14) -> 9
proper_1(1) -> 23
proper_1(6) -> 23
proper_1(7) -> 23
proper_1(8) -> 23
proper_1(13) -> 23
proper_1(14) -> 23
proper_2(15) -> 24
proper_2(16) -> 28
proper_2(17) -> 29
proper_3(25) -> 31
proper_3(26) -> 33
proper_3(27) -> 34
s_0(1) -> 10
s_0(6) -> 10
s_0(7) -> 10
s_0(8) -> 10
s_0(13) -> 10
s_0(14) -> 10
s_1(1) -> 21
s_1(6) -> 21
s_1(7) -> 21
s_1(8) -> 21
s_1(13) -> 21
s_1(14) -> 21
take_0(1,1) -> 11
take_0(1,6) -> 11
take_0(1,7) -> 11
take_0(1,8) -> 11
take_0(1,13) -> 11
take_0(1,14) -> 11
take_0(6,1) -> 11
take_0(6,6) -> 11
take_0(6,7) -> 11
take_0(6,8) -> 11
take_0(6,13) -> 11
take_0(6,14) -> 11
take_0(7,1) -> 11
take_0(7,6) -> 11
take_0(7,7) -> 11
take_0(7,8) -> 11
take_0(7,13) -> 11
take_0(7,14) -> 11
take_0(8,1) -> 11
take_0(8,6) -> 11
take_0(8,7) -> 11
take_0(8,8) -> 11
take_0(8,13) -> 11
take_0(8,14) -> 11
take_0(13,1) -> 11
take_0(13,6) -> 11
take_0(13,7) -> 11
take_0(13,8) -> 11
take_0(13,13) -> 11
take_0(13,14) -> 11
take_0(14,1) -> 11
take_0(14,6) -> 11
take_0(14,7) -> 11
take_0(14,8) -> 11
take_0(14,13) -> 11
take_0(14,14) -> 11
take_1(1,1) -> 22
take_1(1,6) -> 22
take_1(1,7) -> 22
take_1(1,8) -> 22
take_1(1,13) -> 22
take_1(1,14) -> 22
take_1(6,1) -> 22
take_1(6,6) -> 22
take_1(6,7) -> 22
take_1(6,8) -> 22
take_1(6,13) -> 22
take_1(6,14) -> 22
take_1(7,1) -> 22
take_1(7,6) -> 22
take_1(7,7) -> 22
take_1(7,8) -> 22
take_1(7,13) -> 22
take_1(7,14) -> 22
take_1(8,1) -> 22
take_1(8,6) -> 22
take_1(8,7) -> 22
take_1(8,8) -> 22
take_1(8,13) -> 22
take_1(8,14) -> 22
take_1(13,1) -> 22
take_1(13,6) -> 22
take_1(13,7) -> 22
take_1(13,8) -> 22
take_1(13,13) -> 22
take_1(13,14) -> 22
take_1(14,1) -> 22
take_1(14,6) -> 22
take_1(14,7) -> 22
take_1(14,8) -> 22
take_1(14,13) -> 22
take_1(14,14) -> 22
top_0(1) -> 12
top_0(6) -> 12
top_0(7) -> 12
top_0(8) -> 12
top_0(13) -> 12
top_0(14) -> 12
top_1(23) -> 12
top_2(24) -> 12
top_3(31) -> 12
top_4(39) -> 12
tt_0() -> 13
tt_1() -> 16
tt_2() -> 30
zeros_0() -> 14
zeros_1() -> 17
zeros_2() -> 27
zeros_3() -> 35
* Step 2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
active(and(X1,X2)) -> and(active(X1),X2)
active(and(tt(),X)) -> mark(X)
active(cons(X1,X2)) -> cons(active(X1),X2)
active(length(X)) -> length(active(X))
active(length(cons(N,L))) -> mark(s(length(L)))
active(length(nil())) -> mark(0())
active(s(X)) -> s(active(X))
active(take(X1,X2)) -> take(X1,active(X2))
active(take(X1,X2)) -> take(active(X1),X2)
active(take(0(),IL)) -> mark(nil())
active(take(s(M),cons(N,IL))) -> mark(cons(N,take(M,IL)))
active(zeros()) -> mark(cons(0(),zeros()))
and(mark(X1),X2) -> mark(and(X1,X2))
and(ok(X1),ok(X2)) -> ok(and(X1,X2))
cons(mark(X1),X2) -> mark(cons(X1,X2))
cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
length(mark(X)) -> mark(length(X))
length(ok(X)) -> ok(length(X))
proper(0()) -> ok(0())
proper(and(X1,X2)) -> and(proper(X1),proper(X2))
proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
proper(length(X)) -> length(proper(X))
proper(nil()) -> ok(nil())
proper(s(X)) -> s(proper(X))
proper(take(X1,X2)) -> take(proper(X1),proper(X2))
proper(tt()) -> ok(tt())
proper(zeros()) -> ok(zeros())
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
take(X1,mark(X2)) -> mark(take(X1,X2))
take(mark(X1),X2) -> mark(take(X1,X2))
take(ok(X1),ok(X2)) -> ok(take(X1,X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
- Signature:
{active/1,and/2,cons/2,length/1,proper/1,s/1,take/2,top/1} / {0/0,mark/1,nil/0,ok/1,tt/0,zeros/0}
- Obligation:
runtime complexity wrt. defined symbols {active,and,cons,length,proper,s,take,top} and constructors {0,mark
,nil,ok,tt,zeros}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))