WORST_CASE(?,O(n^1))
* Step 1: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
goal(x,xs) -> member(x,xs)
member(x,Nil()) -> False()
member(x',Cons(x,xs)) -> member[Ite][True][Ite](!EQ(x',x),x',Cons(x,xs))
notEmpty(Cons(x,xs)) -> True()
notEmpty(Nil()) -> False()
- Weak TRS:
!EQ(0(),0()) -> True()
!EQ(0(),S(y)) -> False()
!EQ(S(x),0()) -> False()
!EQ(S(x),S(y)) -> !EQ(x,y)
member[Ite][True][Ite](False(),x',Cons(x,xs)) -> member(x',xs)
member[Ite][True][Ite](True(),x,xs) -> True()
- Signature:
{!EQ/2,goal/2,member/2,member[Ite][True][Ite]/3,notEmpty/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {!EQ,goal,member,member[Ite][True][Ite]
,notEmpty} and constructors {0,Cons,False,Nil,S,True}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(member[Ite][True][Ite]) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(!EQ) = [3]
p(0) = [0]
p(Cons) = [1] x1 + [1] x2 + [0]
p(False) = [3]
p(Nil) = [13]
p(S) = [1] x1 + [0]
p(True) = [0]
p(goal) = [1] x2 + [0]
p(member) = [1] x2 + [3]
p(member[Ite][True][Ite]) = [1] x1 + [1] x3 + [0]
p(notEmpty) = [5]
Following rules are strictly oriented:
member(x,Nil()) = [16]
> [3]
= False()
notEmpty(Cons(x,xs)) = [5]
> [0]
= True()
notEmpty(Nil()) = [5]
> [3]
= False()
Following rules are (at-least) weakly oriented:
!EQ(0(),0()) = [3]
>= [0]
= True()
!EQ(0(),S(y)) = [3]
>= [3]
= False()
!EQ(S(x),0()) = [3]
>= [3]
= False()
!EQ(S(x),S(y)) = [3]
>= [3]
= !EQ(x,y)
goal(x,xs) = [1] xs + [0]
>= [1] xs + [3]
= member(x,xs)
member(x',Cons(x,xs)) = [1] x + [1] xs + [3]
>= [1] x + [1] xs + [3]
= member[Ite][True][Ite](!EQ(x',x),x',Cons(x,xs))
member[Ite][True][Ite](False(),x',Cons(x,xs)) = [1] x + [1] xs + [3]
>= [1] xs + [3]
= member(x',xs)
member[Ite][True][Ite](True(),x,xs) = [1] xs + [0]
>= [0]
= True()
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
goal(x,xs) -> member(x,xs)
member(x',Cons(x,xs)) -> member[Ite][True][Ite](!EQ(x',x),x',Cons(x,xs))
- Weak TRS:
!EQ(0(),0()) -> True()
!EQ(0(),S(y)) -> False()
!EQ(S(x),0()) -> False()
!EQ(S(x),S(y)) -> !EQ(x,y)
member(x,Nil()) -> False()
member[Ite][True][Ite](False(),x',Cons(x,xs)) -> member(x',xs)
member[Ite][True][Ite](True(),x,xs) -> True()
notEmpty(Cons(x,xs)) -> True()
notEmpty(Nil()) -> False()
- Signature:
{!EQ/2,goal/2,member/2,member[Ite][True][Ite]/3,notEmpty/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {!EQ,goal,member,member[Ite][True][Ite]
,notEmpty} and constructors {0,Cons,False,Nil,S,True}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(member[Ite][True][Ite]) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(!EQ) = [6]
p(0) = [2]
p(Cons) = [1] x2 + [8]
p(False) = [0]
p(Nil) = [0]
p(S) = [0]
p(True) = [6]
p(goal) = [4] x1 + [2] x2 + [4]
p(member) = [4] x1 + [2] x2 + [2]
p(member[Ite][True][Ite]) = [1] x1 + [4] x2 + [2] x3 + [5]
p(notEmpty) = [2] x1 + [1]
Following rules are strictly oriented:
goal(x,xs) = [4] x + [2] xs + [4]
> [4] x + [2] xs + [2]
= member(x,xs)
Following rules are (at-least) weakly oriented:
!EQ(0(),0()) = [6]
>= [6]
= True()
!EQ(0(),S(y)) = [6]
>= [0]
= False()
!EQ(S(x),0()) = [6]
>= [0]
= False()
!EQ(S(x),S(y)) = [6]
>= [6]
= !EQ(x,y)
member(x,Nil()) = [4] x + [2]
>= [0]
= False()
member(x',Cons(x,xs)) = [4] x' + [2] xs + [18]
>= [4] x' + [2] xs + [27]
= member[Ite][True][Ite](!EQ(x',x),x',Cons(x,xs))
member[Ite][True][Ite](False(),x',Cons(x,xs)) = [4] x' + [2] xs + [21]
>= [4] x' + [2] xs + [2]
= member(x',xs)
member[Ite][True][Ite](True(),x,xs) = [4] x + [2] xs + [11]
>= [6]
= True()
notEmpty(Cons(x,xs)) = [2] xs + [17]
>= [6]
= True()
notEmpty(Nil()) = [1]
>= [0]
= False()
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
member(x',Cons(x,xs)) -> member[Ite][True][Ite](!EQ(x',x),x',Cons(x,xs))
- Weak TRS:
!EQ(0(),0()) -> True()
!EQ(0(),S(y)) -> False()
!EQ(S(x),0()) -> False()
!EQ(S(x),S(y)) -> !EQ(x,y)
goal(x,xs) -> member(x,xs)
member(x,Nil()) -> False()
member[Ite][True][Ite](False(),x',Cons(x,xs)) -> member(x',xs)
member[Ite][True][Ite](True(),x,xs) -> True()
notEmpty(Cons(x,xs)) -> True()
notEmpty(Nil()) -> False()
- Signature:
{!EQ/2,goal/2,member/2,member[Ite][True][Ite]/3,notEmpty/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {!EQ,goal,member,member[Ite][True][Ite]
,notEmpty} and constructors {0,Cons,False,Nil,S,True}
+ Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(member[Ite][True][Ite]) = {1}
Following symbols are considered usable:
{!EQ,goal,member,member[Ite][True][Ite],notEmpty}
TcT has computed the following interpretation:
p(!EQ) = [0]
p(0) = [0]
p(Cons) = [1] x2 + [12]
p(False) = [0]
p(Nil) = [5]
p(S) = [0]
p(True) = [0]
p(goal) = [1] x1 + [1] x2 + [12]
p(member) = [1] x2 + [12]
p(member[Ite][True][Ite]) = [2] x1 + [1] x3 + [0]
p(notEmpty) = [4]
Following rules are strictly oriented:
member(x',Cons(x,xs)) = [1] xs + [24]
> [1] xs + [12]
= member[Ite][True][Ite](!EQ(x',x),x',Cons(x,xs))
Following rules are (at-least) weakly oriented:
!EQ(0(),0()) = [0]
>= [0]
= True()
!EQ(0(),S(y)) = [0]
>= [0]
= False()
!EQ(S(x),0()) = [0]
>= [0]
= False()
!EQ(S(x),S(y)) = [0]
>= [0]
= !EQ(x,y)
goal(x,xs) = [1] x + [1] xs + [12]
>= [1] xs + [12]
= member(x,xs)
member(x,Nil()) = [17]
>= [0]
= False()
member[Ite][True][Ite](False(),x',Cons(x,xs)) = [1] xs + [12]
>= [1] xs + [12]
= member(x',xs)
member[Ite][True][Ite](True(),x,xs) = [1] xs + [0]
>= [0]
= True()
notEmpty(Cons(x,xs)) = [4]
>= [0]
= True()
notEmpty(Nil()) = [4]
>= [0]
= False()
* Step 4: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
!EQ(0(),0()) -> True()
!EQ(0(),S(y)) -> False()
!EQ(S(x),0()) -> False()
!EQ(S(x),S(y)) -> !EQ(x,y)
goal(x,xs) -> member(x,xs)
member(x,Nil()) -> False()
member(x',Cons(x,xs)) -> member[Ite][True][Ite](!EQ(x',x),x',Cons(x,xs))
member[Ite][True][Ite](False(),x',Cons(x,xs)) -> member(x',xs)
member[Ite][True][Ite](True(),x,xs) -> True()
notEmpty(Cons(x,xs)) -> True()
notEmpty(Nil()) -> False()
- Signature:
{!EQ/2,goal/2,member/2,member[Ite][True][Ite]/3,notEmpty/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {!EQ,goal,member,member[Ite][True][Ite]
,notEmpty} and constructors {0,Cons,False,Nil,S,True}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(?,O(n^1))