```* Step 1: Bounds WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
U11(mark(X1),X2) -> mark(U11(X1,X2))
U11(ok(X1),ok(X2)) -> ok(U11(X1,X2))
U12(mark(X1),X2) -> mark(U12(X1,X2))
U12(ok(X1),ok(X2)) -> ok(U12(X1,X2))
active(U11(X1,X2)) -> U11(active(X1),X2)
active(U11(tt(),L)) -> mark(U12(tt(),L))
active(U12(X1,X2)) -> U12(active(X1),X2)
active(U12(tt(),L)) -> mark(s(length(L)))
active(cons(X1,X2)) -> cons(active(X1),X2)
active(length(X)) -> length(active(X))
active(length(cons(N,L))) -> mark(U11(tt(),L))
active(length(nil())) -> mark(0())
active(s(X)) -> s(active(X))
active(zeros()) -> mark(cons(0(),zeros()))
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(U11(X1,X2)) -> U11(proper(X1),proper(X2))
proper(U12(X1,X2)) -> U12(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(tt()) -> ok(tt())
proper(zeros()) -> ok(zeros())
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
- Signature:
{U11/2,U12/2,active/1,cons/2,length/1,proper/1,s/1,top/1} / {0/0,mark/1,nil/0,ok/1,tt/0,zeros/0}
- Obligation:
runtime complexity wrt. defined symbols {U11,U12,active,cons,length,proper,s,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() -> 18
0_2() -> 26
0_3() -> 37
U11_0(1,1) -> 2
U11_0(1,7) -> 2
U11_0(1,8) -> 2
U11_0(1,9) -> 2
U11_0(1,13) -> 2
U11_0(1,14) -> 2
U11_0(7,1) -> 2
U11_0(7,7) -> 2
U11_0(7,8) -> 2
U11_0(7,9) -> 2
U11_0(7,13) -> 2
U11_0(7,14) -> 2
U11_0(8,1) -> 2
U11_0(8,7) -> 2
U11_0(8,8) -> 2
U11_0(8,9) -> 2
U11_0(8,13) -> 2
U11_0(8,14) -> 2
U11_0(9,1) -> 2
U11_0(9,7) -> 2
U11_0(9,8) -> 2
U11_0(9,9) -> 2
U11_0(9,13) -> 2
U11_0(9,14) -> 2
U11_0(13,1) -> 2
U11_0(13,7) -> 2
U11_0(13,8) -> 2
U11_0(13,9) -> 2
U11_0(13,13) -> 2
U11_0(13,14) -> 2
U11_0(14,1) -> 2
U11_0(14,7) -> 2
U11_0(14,8) -> 2
U11_0(14,9) -> 2
U11_0(14,13) -> 2
U11_0(14,14) -> 2
U11_1(1,1) -> 15
U11_1(1,7) -> 15
U11_1(1,8) -> 15
U11_1(1,9) -> 15
U11_1(1,13) -> 15
U11_1(1,14) -> 15
U11_1(7,1) -> 15
U11_1(7,7) -> 15
U11_1(7,8) -> 15
U11_1(7,9) -> 15
U11_1(7,13) -> 15
U11_1(7,14) -> 15
U11_1(8,1) -> 15
U11_1(8,7) -> 15
U11_1(8,8) -> 15
U11_1(8,9) -> 15
U11_1(8,13) -> 15
U11_1(8,14) -> 15
U11_1(9,1) -> 15
U11_1(9,7) -> 15
U11_1(9,8) -> 15
U11_1(9,9) -> 15
U11_1(9,13) -> 15
U11_1(9,14) -> 15
U11_1(13,1) -> 15
U11_1(13,7) -> 15
U11_1(13,8) -> 15
U11_1(13,9) -> 15
U11_1(13,13) -> 15
U11_1(13,14) -> 15
U11_1(14,1) -> 15
U11_1(14,7) -> 15
U11_1(14,8) -> 15
U11_1(14,9) -> 15
U11_1(14,13) -> 15
U11_1(14,14) -> 15
U12_0(1,1) -> 3
U12_0(1,7) -> 3
U12_0(1,8) -> 3
U12_0(1,9) -> 3
U12_0(1,13) -> 3
U12_0(1,14) -> 3
U12_0(7,1) -> 3
U12_0(7,7) -> 3
U12_0(7,8) -> 3
U12_0(7,9) -> 3
U12_0(7,13) -> 3
U12_0(7,14) -> 3
U12_0(8,1) -> 3
U12_0(8,7) -> 3
U12_0(8,8) -> 3
U12_0(8,9) -> 3
U12_0(8,13) -> 3
U12_0(8,14) -> 3
U12_0(9,1) -> 3
U12_0(9,7) -> 3
U12_0(9,8) -> 3
U12_0(9,9) -> 3
U12_0(9,13) -> 3
U12_0(9,14) -> 3
U12_0(13,1) -> 3
U12_0(13,7) -> 3
U12_0(13,8) -> 3
U12_0(13,9) -> 3
U12_0(13,13) -> 3
U12_0(13,14) -> 3
U12_0(14,1) -> 3
U12_0(14,7) -> 3
U12_0(14,8) -> 3
U12_0(14,9) -> 3
U12_0(14,13) -> 3
U12_0(14,14) -> 3
U12_1(1,1) -> 16
U12_1(1,7) -> 16
U12_1(1,8) -> 16
U12_1(1,9) -> 16
U12_1(1,13) -> 16
U12_1(1,14) -> 16
U12_1(7,1) -> 16
U12_1(7,7) -> 16
U12_1(7,8) -> 16
U12_1(7,9) -> 16
U12_1(7,13) -> 16
U12_1(7,14) -> 16
U12_1(8,1) -> 16
U12_1(8,7) -> 16
U12_1(8,8) -> 16
U12_1(8,9) -> 16
U12_1(8,13) -> 16
U12_1(8,14) -> 16
U12_1(9,1) -> 16
U12_1(9,7) -> 16
U12_1(9,8) -> 16
U12_1(9,9) -> 16
U12_1(9,13) -> 16
U12_1(9,14) -> 16
U12_1(13,1) -> 16
U12_1(13,7) -> 16
U12_1(13,8) -> 16
U12_1(13,9) -> 16
U12_1(13,13) -> 16
U12_1(13,14) -> 16
U12_1(14,1) -> 16
U12_1(14,7) -> 16
U12_1(14,8) -> 16
U12_1(14,9) -> 16
U12_1(14,13) -> 16
U12_1(14,14) -> 16
active_0(1) -> 4
active_0(7) -> 4
active_0(8) -> 4
active_0(9) -> 4
active_0(13) -> 4
active_0(14) -> 4
active_1(1) -> 23
active_1(7) -> 23
active_1(8) -> 23
active_1(9) -> 23
active_1(13) -> 23
active_1(14) -> 23
active_2(18) -> 24
active_2(19) -> 24
active_3(32) -> 31
active_4(26) -> 36
active_4(30) -> 36
active_4(38) -> 39
active_5(37) -> 40
cons_0(1,1) -> 5
cons_0(1,7) -> 5
cons_0(1,8) -> 5
cons_0(1,9) -> 5
cons_0(1,13) -> 5
cons_0(1,14) -> 5
cons_0(7,1) -> 5
cons_0(7,7) -> 5
cons_0(7,8) -> 5
cons_0(7,9) -> 5
cons_0(7,13) -> 5
cons_0(7,14) -> 5
cons_0(8,1) -> 5
cons_0(8,7) -> 5
cons_0(8,8) -> 5
cons_0(8,9) -> 5
cons_0(8,13) -> 5
cons_0(8,14) -> 5
cons_0(9,1) -> 5
cons_0(9,7) -> 5
cons_0(9,8) -> 5
cons_0(9,9) -> 5
cons_0(9,13) -> 5
cons_0(9,14) -> 5
cons_0(13,1) -> 5
cons_0(13,7) -> 5
cons_0(13,8) -> 5
cons_0(13,9) -> 5
cons_0(13,13) -> 5
cons_0(13,14) -> 5
cons_0(14,1) -> 5
cons_0(14,7) -> 5
cons_0(14,8) -> 5
cons_0(14,9) -> 5
cons_0(14,13) -> 5
cons_0(14,14) -> 5
cons_1(1,1) -> 20
cons_1(1,7) -> 20
cons_1(1,8) -> 20
cons_1(1,9) -> 20
cons_1(1,13) -> 20
cons_1(1,14) -> 20
cons_1(7,1) -> 20
cons_1(7,7) -> 20
cons_1(7,8) -> 20
cons_1(7,9) -> 20
cons_1(7,13) -> 20
cons_1(7,14) -> 20
cons_1(8,1) -> 20
cons_1(8,7) -> 20
cons_1(8,8) -> 20
cons_1(8,9) -> 20
cons_1(8,13) -> 20
cons_1(8,14) -> 20
cons_1(9,1) -> 20
cons_1(9,7) -> 20
cons_1(9,8) -> 20
cons_1(9,9) -> 20
cons_1(9,13) -> 20
cons_1(9,14) -> 20
cons_1(13,1) -> 20
cons_1(13,7) -> 20
cons_1(13,8) -> 20
cons_1(13,9) -> 20
cons_1(13,13) -> 20
cons_1(13,14) -> 20
cons_1(14,1) -> 20
cons_1(14,7) -> 20
cons_1(14,8) -> 20
cons_1(14,9) -> 20
cons_1(14,13) -> 20
cons_1(14,14) -> 20
cons_1(18,19) -> 17
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) -> 6
length_0(7) -> 6
length_0(8) -> 6
length_0(9) -> 6
length_0(13) -> 6
length_0(14) -> 6
length_1(1) -> 21
length_1(7) -> 21
length_1(8) -> 21
length_1(9) -> 21
length_1(13) -> 21
length_1(14) -> 21
mark_0(1) -> 7
mark_0(7) -> 7
mark_0(8) -> 7
mark_0(9) -> 7
mark_0(13) -> 7
mark_0(14) -> 7
mark_1(15) -> 2
mark_1(15) -> 15
mark_1(16) -> 3
mark_1(16) -> 16
mark_1(17) -> 4
mark_1(17) -> 23
mark_1(20) -> 5
mark_1(20) -> 20
mark_1(21) -> 6
mark_1(21) -> 21
mark_1(22) -> 11
mark_1(22) -> 22
mark_2(25) -> 24
nil_0() -> 8
nil_1() -> 18
nil_2() -> 30
ok_0(1) -> 9
ok_0(7) -> 9
ok_0(8) -> 9
ok_0(9) -> 9
ok_0(13) -> 9
ok_0(14) -> 9
ok_1(15) -> 2
ok_1(15) -> 15
ok_1(16) -> 3
ok_1(16) -> 16
ok_1(18) -> 10
ok_1(18) -> 23
ok_1(19) -> 10
ok_1(19) -> 23
ok_1(20) -> 5
ok_1(20) -> 20
ok_1(21) -> 6
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) -> 10
proper_0(7) -> 10
proper_0(8) -> 10
proper_0(9) -> 10
proper_0(13) -> 10
proper_0(14) -> 10
proper_1(1) -> 23
proper_1(7) -> 23
proper_1(8) -> 23
proper_1(9) -> 23
proper_1(13) -> 23
proper_1(14) -> 23
proper_2(17) -> 24
proper_2(18) -> 28
proper_2(19) -> 29
proper_3(25) -> 31
proper_3(26) -> 33
proper_3(27) -> 34
s_0(1) -> 11
s_0(7) -> 11
s_0(8) -> 11
s_0(9) -> 11
s_0(13) -> 11
s_0(14) -> 11
s_1(1) -> 22
s_1(7) -> 22
s_1(8) -> 22
s_1(9) -> 22
s_1(13) -> 22
s_1(14) -> 22
top_0(1) -> 12
top_0(7) -> 12
top_0(8) -> 12
top_0(9) -> 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() -> 18
tt_2() -> 30
zeros_0() -> 14
zeros_1() -> 19
zeros_2() -> 27
zeros_3() -> 35
* Step 2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
U11(mark(X1),X2) -> mark(U11(X1,X2))
U11(ok(X1),ok(X2)) -> ok(U11(X1,X2))
U12(mark(X1),X2) -> mark(U12(X1,X2))
U12(ok(X1),ok(X2)) -> ok(U12(X1,X2))
active(U11(X1,X2)) -> U11(active(X1),X2)
active(U11(tt(),L)) -> mark(U12(tt(),L))
active(U12(X1,X2)) -> U12(active(X1),X2)
active(U12(tt(),L)) -> mark(s(length(L)))
active(cons(X1,X2)) -> cons(active(X1),X2)
active(length(X)) -> length(active(X))
active(length(cons(N,L))) -> mark(U11(tt(),L))
active(length(nil())) -> mark(0())
active(s(X)) -> s(active(X))
active(zeros()) -> mark(cons(0(),zeros()))
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(U11(X1,X2)) -> U11(proper(X1),proper(X2))
proper(U12(X1,X2)) -> U12(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(tt()) -> ok(tt())
proper(zeros()) -> ok(zeros())
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
- Signature:
{U11/2,U12/2,active/1,cons/2,length/1,proper/1,s/1,top/1} / {0/0,mark/1,nil/0,ok/1,tt/0,zeros/0}
- Obligation:
runtime complexity wrt. defined symbols {U11,U12,active,cons,length,proper,s,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))
```