```* Step 1: Bounds WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
__(X1,mark(X2)) -> mark(__(X1,X2))
__(mark(X1),X2) -> mark(__(X1,X2))
__(ok(X1),ok(X2)) -> ok(__(X1,X2))
active(__(X,nil())) -> mark(X)
active(__(X1,X2)) -> __(X1,active(X2))
active(__(X1,X2)) -> __(active(X1),X2)
active(__(__(X,Y),Z)) -> mark(__(X,__(Y,Z)))
active(__(nil(),X)) -> mark(X)
active(and(X1,X2)) -> and(active(X1),X2)
active(and(tt(),X)) -> mark(X)
active(isList(V)) -> mark(isNeList(V))
active(isList(__(V1,V2))) -> mark(and(isList(V1),isList(V2)))
active(isList(nil())) -> mark(tt())
active(isNeList(V)) -> mark(isQid(V))
active(isNeList(__(V1,V2))) -> mark(and(isList(V1),isNeList(V2)))
active(isNeList(__(V1,V2))) -> mark(and(isNeList(V1),isList(V2)))
active(isNePal(V)) -> mark(isQid(V))
active(isNePal(__(I,__(P,I)))) -> mark(and(isQid(I),isPal(P)))
active(isPal(V)) -> mark(isNePal(V))
active(isPal(nil())) -> mark(tt())
active(isQid(a())) -> mark(tt())
active(isQid(e())) -> mark(tt())
active(isQid(i())) -> mark(tt())
active(isQid(o())) -> mark(tt())
active(isQid(u())) -> mark(tt())
and(mark(X1),X2) -> mark(and(X1,X2))
and(ok(X1),ok(X2)) -> ok(and(X1,X2))
isList(ok(X)) -> ok(isList(X))
isNeList(ok(X)) -> ok(isNeList(X))
isNePal(ok(X)) -> ok(isNePal(X))
isPal(ok(X)) -> ok(isPal(X))
isQid(ok(X)) -> ok(isQid(X))
proper(__(X1,X2)) -> __(proper(X1),proper(X2))
proper(a()) -> ok(a())
proper(and(X1,X2)) -> and(proper(X1),proper(X2))
proper(e()) -> ok(e())
proper(i()) -> ok(i())
proper(isList(X)) -> isList(proper(X))
proper(isNeList(X)) -> isNeList(proper(X))
proper(isNePal(X)) -> isNePal(proper(X))
proper(isPal(X)) -> isPal(proper(X))
proper(isQid(X)) -> isQid(proper(X))
proper(nil()) -> ok(nil())
proper(o()) -> ok(o())
proper(tt()) -> ok(tt())
proper(u()) -> ok(u())
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
- Signature:
{__/2,active/1,and/2,isList/1,isNeList/1,isNePal/1,isPal/1,isQid/1,proper/1,top/1} / {a/0,e/0,i/0,mark/1
,nil/0,o/0,ok/1,tt/0,u/0}
- Obligation:
runtime complexity wrt. defined symbols {__,active,and,isList,isNeList,isNePal,isPal,isQid,proper
,top} and constructors {a,e,i,mark,nil,o,ok,tt,u}
+ Applied Processor:
Bounds {initialAutomaton = perSymbol, enrichment = match}
+ Details:
The problem is match-bounded by 2.
The enriched problem is compatible with follwoing automaton.
___0(2,2) -> 1
___0(2,5) -> 1
___0(2,6) -> 1
___0(2,12) -> 1
___0(2,13) -> 1
___0(2,14) -> 1
___0(2,15) -> 1
___0(2,18) -> 1
___0(2,19) -> 1
___0(5,2) -> 1
___0(5,5) -> 1
___0(5,6) -> 1
___0(5,12) -> 1
___0(5,13) -> 1
___0(5,14) -> 1
___0(5,15) -> 1
___0(5,18) -> 1
___0(5,19) -> 1
___0(6,2) -> 1
___0(6,5) -> 1
___0(6,6) -> 1
___0(6,12) -> 1
___0(6,13) -> 1
___0(6,14) -> 1
___0(6,15) -> 1
___0(6,18) -> 1
___0(6,19) -> 1
___0(12,2) -> 1
___0(12,5) -> 1
___0(12,6) -> 1
___0(12,12) -> 1
___0(12,13) -> 1
___0(12,14) -> 1
___0(12,15) -> 1
___0(12,18) -> 1
___0(12,19) -> 1
___0(13,2) -> 1
___0(13,5) -> 1
___0(13,6) -> 1
___0(13,12) -> 1
___0(13,13) -> 1
___0(13,14) -> 1
___0(13,15) -> 1
___0(13,18) -> 1
___0(13,19) -> 1
___0(14,2) -> 1
___0(14,5) -> 1
___0(14,6) -> 1
___0(14,12) -> 1
___0(14,13) -> 1
___0(14,14) -> 1
___0(14,15) -> 1
___0(14,18) -> 1
___0(14,19) -> 1
___0(15,2) -> 1
___0(15,5) -> 1
___0(15,6) -> 1
___0(15,12) -> 1
___0(15,13) -> 1
___0(15,14) -> 1
___0(15,15) -> 1
___0(15,18) -> 1
___0(15,19) -> 1
___0(18,2) -> 1
___0(18,5) -> 1
___0(18,6) -> 1
___0(18,12) -> 1
___0(18,13) -> 1
___0(18,14) -> 1
___0(18,15) -> 1
___0(18,18) -> 1
___0(18,19) -> 1
___0(19,2) -> 1
___0(19,5) -> 1
___0(19,6) -> 1
___0(19,12) -> 1
___0(19,13) -> 1
___0(19,14) -> 1
___0(19,15) -> 1
___0(19,18) -> 1
___0(19,19) -> 1
___1(2,2) -> 20
___1(2,5) -> 20
___1(2,6) -> 20
___1(2,12) -> 20
___1(2,13) -> 20
___1(2,14) -> 20
___1(2,15) -> 20
___1(2,18) -> 20
___1(2,19) -> 20
___1(5,2) -> 20
___1(5,5) -> 20
___1(5,6) -> 20
___1(5,12) -> 20
___1(5,13) -> 20
___1(5,14) -> 20
___1(5,15) -> 20
___1(5,18) -> 20
___1(5,19) -> 20
___1(6,2) -> 20
___1(6,5) -> 20
___1(6,6) -> 20
___1(6,12) -> 20
___1(6,13) -> 20
___1(6,14) -> 20
___1(6,15) -> 20
___1(6,18) -> 20
___1(6,19) -> 20
___1(12,2) -> 20
___1(12,5) -> 20
___1(12,6) -> 20
___1(12,12) -> 20
___1(12,13) -> 20
___1(12,14) -> 20
___1(12,15) -> 20
___1(12,18) -> 20
___1(12,19) -> 20
___1(13,2) -> 20
___1(13,5) -> 20
___1(13,6) -> 20
___1(13,12) -> 20
___1(13,13) -> 20
___1(13,14) -> 20
___1(13,15) -> 20
___1(13,18) -> 20
___1(13,19) -> 20
___1(14,2) -> 20
___1(14,5) -> 20
___1(14,6) -> 20
___1(14,12) -> 20
___1(14,13) -> 20
___1(14,14) -> 20
___1(14,15) -> 20
___1(14,18) -> 20
___1(14,19) -> 20
___1(15,2) -> 20
___1(15,5) -> 20
___1(15,6) -> 20
___1(15,12) -> 20
___1(15,13) -> 20
___1(15,14) -> 20
___1(15,15) -> 20
___1(15,18) -> 20
___1(15,19) -> 20
___1(18,2) -> 20
___1(18,5) -> 20
___1(18,6) -> 20
___1(18,12) -> 20
___1(18,13) -> 20
___1(18,14) -> 20
___1(18,15) -> 20
___1(18,18) -> 20
___1(18,19) -> 20
___1(19,2) -> 20
___1(19,5) -> 20
___1(19,6) -> 20
___1(19,12) -> 20
___1(19,13) -> 20
___1(19,14) -> 20
___1(19,15) -> 20
___1(19,18) -> 20
___1(19,19) -> 20
a_0() -> 2
a_1() -> 27
active_0(2) -> 3
active_0(5) -> 3
active_0(6) -> 3
active_0(12) -> 3
active_0(13) -> 3
active_0(14) -> 3
active_0(15) -> 3
active_0(18) -> 3
active_0(19) -> 3
active_1(2) -> 28
active_1(5) -> 28
active_1(6) -> 28
active_1(12) -> 28
active_1(13) -> 28
active_1(14) -> 28
active_1(15) -> 28
active_1(18) -> 28
active_1(19) -> 28
active_2(27) -> 29
and_0(2,2) -> 4
and_0(2,5) -> 4
and_0(2,6) -> 4
and_0(2,12) -> 4
and_0(2,13) -> 4
and_0(2,14) -> 4
and_0(2,15) -> 4
and_0(2,18) -> 4
and_0(2,19) -> 4
and_0(5,2) -> 4
and_0(5,5) -> 4
and_0(5,6) -> 4
and_0(5,12) -> 4
and_0(5,13) -> 4
and_0(5,14) -> 4
and_0(5,15) -> 4
and_0(5,18) -> 4
and_0(5,19) -> 4
and_0(6,2) -> 4
and_0(6,5) -> 4
and_0(6,6) -> 4
and_0(6,12) -> 4
and_0(6,13) -> 4
and_0(6,14) -> 4
and_0(6,15) -> 4
and_0(6,18) -> 4
and_0(6,19) -> 4
and_0(12,2) -> 4
and_0(12,5) -> 4
and_0(12,6) -> 4
and_0(12,12) -> 4
and_0(12,13) -> 4
and_0(12,14) -> 4
and_0(12,15) -> 4
and_0(12,18) -> 4
and_0(12,19) -> 4
and_0(13,2) -> 4
and_0(13,5) -> 4
and_0(13,6) -> 4
and_0(13,12) -> 4
and_0(13,13) -> 4
and_0(13,14) -> 4
and_0(13,15) -> 4
and_0(13,18) -> 4
and_0(13,19) -> 4
and_0(14,2) -> 4
and_0(14,5) -> 4
and_0(14,6) -> 4
and_0(14,12) -> 4
and_0(14,13) -> 4
and_0(14,14) -> 4
and_0(14,15) -> 4
and_0(14,18) -> 4
and_0(14,19) -> 4
and_0(15,2) -> 4
and_0(15,5) -> 4
and_0(15,6) -> 4
and_0(15,12) -> 4
and_0(15,13) -> 4
and_0(15,14) -> 4
and_0(15,15) -> 4
and_0(15,18) -> 4
and_0(15,19) -> 4
and_0(18,2) -> 4
and_0(18,5) -> 4
and_0(18,6) -> 4
and_0(18,12) -> 4
and_0(18,13) -> 4
and_0(18,14) -> 4
and_0(18,15) -> 4
and_0(18,18) -> 4
and_0(18,19) -> 4
and_0(19,2) -> 4
and_0(19,5) -> 4
and_0(19,6) -> 4
and_0(19,12) -> 4
and_0(19,13) -> 4
and_0(19,14) -> 4
and_0(19,15) -> 4
and_0(19,18) -> 4
and_0(19,19) -> 4
and_1(2,2) -> 21
and_1(2,5) -> 21
and_1(2,6) -> 21
and_1(2,12) -> 21
and_1(2,13) -> 21
and_1(2,14) -> 21
and_1(2,15) -> 21
and_1(2,18) -> 21
and_1(2,19) -> 21
and_1(5,2) -> 21
and_1(5,5) -> 21
and_1(5,6) -> 21
and_1(5,12) -> 21
and_1(5,13) -> 21
and_1(5,14) -> 21
and_1(5,15) -> 21
and_1(5,18) -> 21
and_1(5,19) -> 21
and_1(6,2) -> 21
and_1(6,5) -> 21
and_1(6,6) -> 21
and_1(6,12) -> 21
and_1(6,13) -> 21
and_1(6,14) -> 21
and_1(6,15) -> 21
and_1(6,18) -> 21
and_1(6,19) -> 21
and_1(12,2) -> 21
and_1(12,5) -> 21
and_1(12,6) -> 21
and_1(12,12) -> 21
and_1(12,13) -> 21
and_1(12,14) -> 21
and_1(12,15) -> 21
and_1(12,18) -> 21
and_1(12,19) -> 21
and_1(13,2) -> 21
and_1(13,5) -> 21
and_1(13,6) -> 21
and_1(13,12) -> 21
and_1(13,13) -> 21
and_1(13,14) -> 21
and_1(13,15) -> 21
and_1(13,18) -> 21
and_1(13,19) -> 21
and_1(14,2) -> 21
and_1(14,5) -> 21
and_1(14,6) -> 21
and_1(14,12) -> 21
and_1(14,13) -> 21
and_1(14,14) -> 21
and_1(14,15) -> 21
and_1(14,18) -> 21
and_1(14,19) -> 21
and_1(15,2) -> 21
and_1(15,5) -> 21
and_1(15,6) -> 21
and_1(15,12) -> 21
and_1(15,13) -> 21
and_1(15,14) -> 21
and_1(15,15) -> 21
and_1(15,18) -> 21
and_1(15,19) -> 21
and_1(18,2) -> 21
and_1(18,5) -> 21
and_1(18,6) -> 21
and_1(18,12) -> 21
and_1(18,13) -> 21
and_1(18,14) -> 21
and_1(18,15) -> 21
and_1(18,18) -> 21
and_1(18,19) -> 21
and_1(19,2) -> 21
and_1(19,5) -> 21
and_1(19,6) -> 21
and_1(19,12) -> 21
and_1(19,13) -> 21
and_1(19,14) -> 21
and_1(19,15) -> 21
and_1(19,18) -> 21
and_1(19,19) -> 21
e_0() -> 5
e_1() -> 27
i_0() -> 6
i_1() -> 27
isList_0(2) -> 7
isList_0(5) -> 7
isList_0(6) -> 7
isList_0(12) -> 7
isList_0(13) -> 7
isList_0(14) -> 7
isList_0(15) -> 7
isList_0(18) -> 7
isList_0(19) -> 7
isList_1(2) -> 22
isList_1(5) -> 22
isList_1(6) -> 22
isList_1(12) -> 22
isList_1(13) -> 22
isList_1(14) -> 22
isList_1(15) -> 22
isList_1(18) -> 22
isList_1(19) -> 22
isNeList_0(2) -> 8
isNeList_0(5) -> 8
isNeList_0(6) -> 8
isNeList_0(12) -> 8
isNeList_0(13) -> 8
isNeList_0(14) -> 8
isNeList_0(15) -> 8
isNeList_0(18) -> 8
isNeList_0(19) -> 8
isNeList_1(2) -> 23
isNeList_1(5) -> 23
isNeList_1(6) -> 23
isNeList_1(12) -> 23
isNeList_1(13) -> 23
isNeList_1(14) -> 23
isNeList_1(15) -> 23
isNeList_1(18) -> 23
isNeList_1(19) -> 23
isNePal_0(2) -> 9
isNePal_0(5) -> 9
isNePal_0(6) -> 9
isNePal_0(12) -> 9
isNePal_0(13) -> 9
isNePal_0(14) -> 9
isNePal_0(15) -> 9
isNePal_0(18) -> 9
isNePal_0(19) -> 9
isNePal_1(2) -> 24
isNePal_1(5) -> 24
isNePal_1(6) -> 24
isNePal_1(12) -> 24
isNePal_1(13) -> 24
isNePal_1(14) -> 24
isNePal_1(15) -> 24
isNePal_1(18) -> 24
isNePal_1(19) -> 24
isPal_0(2) -> 10
isPal_0(5) -> 10
isPal_0(6) -> 10
isPal_0(12) -> 10
isPal_0(13) -> 10
isPal_0(14) -> 10
isPal_0(15) -> 10
isPal_0(18) -> 10
isPal_0(19) -> 10
isPal_1(2) -> 25
isPal_1(5) -> 25
isPal_1(6) -> 25
isPal_1(12) -> 25
isPal_1(13) -> 25
isPal_1(14) -> 25
isPal_1(15) -> 25
isPal_1(18) -> 25
isPal_1(19) -> 25
isQid_0(2) -> 11
isQid_0(5) -> 11
isQid_0(6) -> 11
isQid_0(12) -> 11
isQid_0(13) -> 11
isQid_0(14) -> 11
isQid_0(15) -> 11
isQid_0(18) -> 11
isQid_0(19) -> 11
isQid_1(2) -> 26
isQid_1(5) -> 26
isQid_1(6) -> 26
isQid_1(12) -> 26
isQid_1(13) -> 26
isQid_1(14) -> 26
isQid_1(15) -> 26
isQid_1(18) -> 26
isQid_1(19) -> 26
mark_0(2) -> 12
mark_0(5) -> 12
mark_0(6) -> 12
mark_0(12) -> 12
mark_0(13) -> 12
mark_0(14) -> 12
mark_0(15) -> 12
mark_0(18) -> 12
mark_0(19) -> 12
mark_1(20) -> 1
mark_1(20) -> 20
mark_1(21) -> 4
mark_1(21) -> 21
nil_0() -> 13
nil_1() -> 27
o_0() -> 14
o_1() -> 27
ok_0(2) -> 15
ok_0(5) -> 15
ok_0(6) -> 15
ok_0(12) -> 15
ok_0(13) -> 15
ok_0(14) -> 15
ok_0(15) -> 15
ok_0(18) -> 15
ok_0(19) -> 15
ok_1(20) -> 1
ok_1(20) -> 20
ok_1(21) -> 4
ok_1(21) -> 21
ok_1(22) -> 7
ok_1(22) -> 22
ok_1(23) -> 8
ok_1(23) -> 23
ok_1(24) -> 9
ok_1(24) -> 24
ok_1(25) -> 10
ok_1(25) -> 25
ok_1(26) -> 11
ok_1(26) -> 26
ok_1(27) -> 16
ok_1(27) -> 28
proper_0(2) -> 16
proper_0(5) -> 16
proper_0(6) -> 16
proper_0(12) -> 16
proper_0(13) -> 16
proper_0(14) -> 16
proper_0(15) -> 16
proper_0(18) -> 16
proper_0(19) -> 16
proper_1(2) -> 28
proper_1(5) -> 28
proper_1(6) -> 28
proper_1(12) -> 28
proper_1(13) -> 28
proper_1(14) -> 28
proper_1(15) -> 28
proper_1(18) -> 28
proper_1(19) -> 28
top_0(2) -> 17
top_0(5) -> 17
top_0(6) -> 17
top_0(12) -> 17
top_0(13) -> 17
top_0(14) -> 17
top_0(15) -> 17
top_0(18) -> 17
top_0(19) -> 17
top_1(28) -> 17
top_2(29) -> 17
tt_0() -> 18
tt_1() -> 27
u_0() -> 19
u_1() -> 27
* Step 2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
__(X1,mark(X2)) -> mark(__(X1,X2))
__(mark(X1),X2) -> mark(__(X1,X2))
__(ok(X1),ok(X2)) -> ok(__(X1,X2))
active(__(X,nil())) -> mark(X)
active(__(X1,X2)) -> __(X1,active(X2))
active(__(X1,X2)) -> __(active(X1),X2)
active(__(__(X,Y),Z)) -> mark(__(X,__(Y,Z)))
active(__(nil(),X)) -> mark(X)
active(and(X1,X2)) -> and(active(X1),X2)
active(and(tt(),X)) -> mark(X)
active(isList(V)) -> mark(isNeList(V))
active(isList(__(V1,V2))) -> mark(and(isList(V1),isList(V2)))
active(isList(nil())) -> mark(tt())
active(isNeList(V)) -> mark(isQid(V))
active(isNeList(__(V1,V2))) -> mark(and(isList(V1),isNeList(V2)))
active(isNeList(__(V1,V2))) -> mark(and(isNeList(V1),isList(V2)))
active(isNePal(V)) -> mark(isQid(V))
active(isNePal(__(I,__(P,I)))) -> mark(and(isQid(I),isPal(P)))
active(isPal(V)) -> mark(isNePal(V))
active(isPal(nil())) -> mark(tt())
active(isQid(a())) -> mark(tt())
active(isQid(e())) -> mark(tt())
active(isQid(i())) -> mark(tt())
active(isQid(o())) -> mark(tt())
active(isQid(u())) -> mark(tt())
and(mark(X1),X2) -> mark(and(X1,X2))
and(ok(X1),ok(X2)) -> ok(and(X1,X2))
isList(ok(X)) -> ok(isList(X))
isNeList(ok(X)) -> ok(isNeList(X))
isNePal(ok(X)) -> ok(isNePal(X))
isPal(ok(X)) -> ok(isPal(X))
isQid(ok(X)) -> ok(isQid(X))
proper(__(X1,X2)) -> __(proper(X1),proper(X2))
proper(a()) -> ok(a())
proper(and(X1,X2)) -> and(proper(X1),proper(X2))
proper(e()) -> ok(e())
proper(i()) -> ok(i())
proper(isList(X)) -> isList(proper(X))
proper(isNeList(X)) -> isNeList(proper(X))
proper(isNePal(X)) -> isNePal(proper(X))
proper(isPal(X)) -> isPal(proper(X))
proper(isQid(X)) -> isQid(proper(X))
proper(nil()) -> ok(nil())
proper(o()) -> ok(o())
proper(tt()) -> ok(tt())
proper(u()) -> ok(u())
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
- Signature:
{__/2,active/1,and/2,isList/1,isNeList/1,isNePal/1,isPal/1,isQid/1,proper/1,top/1} / {a/0,e/0,i/0,mark/1
,nil/0,o/0,ok/1,tt/0,u/0}
- Obligation:
runtime complexity wrt. defined symbols {__,active,and,isList,isNeList,isNePal,isPal,isQid,proper
,top} and constructors {a,e,i,mark,nil,o,ok,tt,u}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))
```