* Step 1: Sum 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:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
* Step 2: MI 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:
MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity Nothing)), miDimension = 1, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity Nothing)):
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) = [0]
p(False) = [0]
p(Nil) = [0]
p(S) = [0]
p(True) = [0]
p(goal) = [4]
p(member) = [0]
p(member[Ite][True][Ite]) = [1] x_1 + [0]
p(notEmpty) = [0]
Following rules are strictly oriented:
goal(x,xs) = [4]
> [0]
= member(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)
member(x,Nil()) = [0]
>= [0]
= False()
member(x',Cons(x,xs)) = [0]
>= [0]
= member[Ite][True][Ite](!EQ(x',x),x',Cons(x,xs))
member[Ite][True][Ite](False(),x',Cons(x,xs)) = [0]
>= [0]
= member(x',xs)
member[Ite][True][Ite](True(),x,xs) = [0]
>= [0]
= True()
notEmpty(Cons(x,xs)) = [0]
>= [0]
= True()
notEmpty(Nil()) = [0]
>= [0]
= False()
* Step 3: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
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)
goal(x,xs) -> member(x,xs)
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) = [0]
p(0) = [0]
p(Cons) = [1] x1 + [0]
p(False) = [0]
p(Nil) = [0]
p(S) = [1] x1 + [0]
p(True) = [0]
p(goal) = [0]
p(member) = [0]
p(member[Ite][True][Ite]) = [1] x1 + [0]
p(notEmpty) = [1]
Following rules are strictly oriented:
notEmpty(Cons(x,xs)) = [1]
> [0]
= True()
notEmpty(Nil()) = [1]
> [0]
= False()
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) = [0]
>= [0]
= member(x,xs)
member(x,Nil()) = [0]
>= [0]
= False()
member(x',Cons(x,xs)) = [0]
>= [0]
= member[Ite][True][Ite](!EQ(x',x),x',Cons(x,xs))
member[Ite][True][Ite](False(),x',Cons(x,xs)) = [0]
>= [0]
= member(x',xs)
member[Ite][True][Ite](True(),x,xs) = [0]
>= [0]
= True()
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 4: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
member(x,Nil()) -> False()
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[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) = [9]
p(0) = [0]
p(Cons) = [1] x1 + [1] x2 + [0]
p(False) = [5]
p(Nil) = [0]
p(S) = [1] x1 + [0]
p(True) = [8]
p(goal) = [9] x2 + [6]
p(member) = [6]
p(member[Ite][True][Ite]) = [1] x1 + [3]
p(notEmpty) = [4] x1 + [10]
Following rules are strictly oriented:
member(x,Nil()) = [6]
> [5]
= False()
Following rules are (at-least) weakly oriented:
!EQ(0(),0()) = [9]
>= [8]
= True()
!EQ(0(),S(y)) = [9]
>= [5]
= False()
!EQ(S(x),0()) = [9]
>= [5]
= False()
!EQ(S(x),S(y)) = [9]
>= [9]
= !EQ(x,y)
goal(x,xs) = [9] xs + [6]
>= [6]
= member(x,xs)
member(x',Cons(x,xs)) = [6]
>= [12]
= member[Ite][True][Ite](!EQ(x',x),x',Cons(x,xs))
member[Ite][True][Ite](False(),x',Cons(x,xs)) = [8]
>= [6]
= member(x',xs)
member[Ite][True][Ite](True(),x,xs) = [11]
>= [8]
= True()
notEmpty(Cons(x,xs)) = [4] x + [4] xs + [10]
>= [8]
= True()
notEmpty(Nil()) = [10]
>= [5]
= False()
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 5: WeightGap 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:
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) = [0]
p(0) = [0]
p(Cons) = [1] x1 + [1] x2 + [1]
p(False) = [0]
p(Nil) = [1]
p(S) = [1] x1 + [0]
p(True) = [0]
p(goal) = [1] x2 + [10]
p(member) = [1] x2 + [10]
p(member[Ite][True][Ite]) = [1] x1 + [1] x3 + [9]
p(notEmpty) = [12] x1 + [0]
Following rules are strictly oriented:
member(x',Cons(x,xs)) = [1] x + [1] xs + [11]
> [1] x + [1] xs + [10]
= 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] xs + [10]
>= [1] xs + [10]
= member(x,xs)
member(x,Nil()) = [11]
>= [0]
= False()
member[Ite][True][Ite](False(),x',Cons(x,xs)) = [1] x + [1] xs + [10]
>= [1] xs + [10]
= member(x',xs)
member[Ite][True][Ite](True(),x,xs) = [1] xs + [9]
>= [0]
= True()
notEmpty(Cons(x,xs)) = [12] x + [12] xs + [12]
>= [0]
= True()
notEmpty(Nil()) = [12]
>= [0]
= False()
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 6: 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))