WORST_CASE(Omega(n^1),O(n^1))
* Step 1: Sum WORST_CASE(Omega(n^1),O(n^1))
+ Considered Problem:
- Strict TRS:
even(Cons(x,Nil())) -> False()
even(Cons(x',Cons(x,xs))) -> even(xs)
even(Nil()) -> True()
goal(x,y) -> and(lte(x,y),even(x))
lte(Cons(x,xs),Nil()) -> False()
lte(Cons(x',xs'),Cons(x,xs)) -> lte(xs',xs)
lte(Nil(),y) -> True()
notEmpty(Cons(x,xs)) -> True()
notEmpty(Nil()) -> False()
- Weak TRS:
and(False(),False()) -> False()
and(False(),True()) -> False()
and(True(),False()) -> False()
and(True(),True()) -> True()
- Signature:
{and/2,even/1,goal/2,lte/2,notEmpty/1} / {Cons/2,False/0,Nil/0,True/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {and,even,goal,lte,notEmpty} and constructors {Cons,False
,Nil,True}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
even(Cons(x,Nil())) -> False()
even(Cons(x',Cons(x,xs))) -> even(xs)
even(Nil()) -> True()
goal(x,y) -> and(lte(x,y),even(x))
lte(Cons(x,xs),Nil()) -> False()
lte(Cons(x',xs'),Cons(x,xs)) -> lte(xs',xs)
lte(Nil(),y) -> True()
notEmpty(Cons(x,xs)) -> True()
notEmpty(Nil()) -> False()
- Weak TRS:
and(False(),False()) -> False()
and(False(),True()) -> False()
and(True(),False()) -> False()
and(True(),True()) -> True()
- Signature:
{and/2,even/1,goal/2,lte/2,notEmpty/1} / {Cons/2,False/0,Nil/0,True/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {and,even,goal,lte,notEmpty} and constructors {Cons,False
,Nil,True}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
even(z){z -> Cons(x,Cons(y,z))} =
even(Cons(x,Cons(y,z))) ->^+ even(z)
= C[even(z) = even(z){}]
** Step 1.b:1: NaturalPI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
even(Cons(x,Nil())) -> False()
even(Cons(x',Cons(x,xs))) -> even(xs)
even(Nil()) -> True()
goal(x,y) -> and(lte(x,y),even(x))
lte(Cons(x,xs),Nil()) -> False()
lte(Cons(x',xs'),Cons(x,xs)) -> lte(xs',xs)
lte(Nil(),y) -> True()
notEmpty(Cons(x,xs)) -> True()
notEmpty(Nil()) -> False()
- Weak TRS:
and(False(),False()) -> False()
and(False(),True()) -> False()
and(True(),False()) -> False()
and(True(),True()) -> True()
- Signature:
{and/2,even/1,goal/2,lte/2,notEmpty/1} / {Cons/2,False/0,Nil/0,True/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {and,even,goal,lte,notEmpty} and constructors {Cons,False
,Nil,True}
+ Applied Processor:
NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a polynomial interpretation of kind constructor-based(linear):
The following argument positions are considered usable:
uargs(and) = {1,2}
Following symbols are considered usable:
{and,even,goal,lte,notEmpty}
TcT has computed the following interpretation:
p(Cons) = 2 + x2
p(False) = 0
p(Nil) = 8
p(True) = 2
p(and) = 4 + 8*x1 + 4*x2
p(even) = x1
p(goal) = 10 + 12*x1
p(lte) = x1
p(notEmpty) = x1
Following rules are strictly oriented:
even(Cons(x,Nil())) = 10
> 0
= False()
even(Cons(x',Cons(x,xs))) = 4 + xs
> xs
= even(xs)
even(Nil()) = 8
> 2
= True()
goal(x,y) = 10 + 12*x
> 4 + 12*x
= and(lte(x,y),even(x))
lte(Cons(x,xs),Nil()) = 2 + xs
> 0
= False()
lte(Cons(x',xs'),Cons(x,xs)) = 2 + xs'
> xs'
= lte(xs',xs)
lte(Nil(),y) = 8
> 2
= True()
notEmpty(Nil()) = 8
> 0
= False()
Following rules are (at-least) weakly oriented:
and(False(),False()) = 4
>= 0
= False()
and(False(),True()) = 12
>= 0
= False()
and(True(),False()) = 20
>= 0
= False()
and(True(),True()) = 28
>= 2
= True()
notEmpty(Cons(x,xs)) = 2 + xs
>= 2
= True()
** Step 1.b:2: NaturalPI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
notEmpty(Cons(x,xs)) -> True()
- Weak TRS:
and(False(),False()) -> False()
and(False(),True()) -> False()
and(True(),False()) -> False()
and(True(),True()) -> True()
even(Cons(x,Nil())) -> False()
even(Cons(x',Cons(x,xs))) -> even(xs)
even(Nil()) -> True()
goal(x,y) -> and(lte(x,y),even(x))
lte(Cons(x,xs),Nil()) -> False()
lte(Cons(x',xs'),Cons(x,xs)) -> lte(xs',xs)
lte(Nil(),y) -> True()
notEmpty(Nil()) -> False()
- Signature:
{and/2,even/1,goal/2,lte/2,notEmpty/1} / {Cons/2,False/0,Nil/0,True/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {and,even,goal,lte,notEmpty} and constructors {Cons,False
,Nil,True}
+ Applied Processor:
NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a polynomial interpretation of kind constructor-based(linear):
The following argument positions are considered usable:
uargs(and) = {1,2}
Following symbols are considered usable:
{and,even,goal,lte,notEmpty}
TcT has computed the following interpretation:
p(Cons) = 4
p(False) = 0
p(Nil) = 1
p(True) = 0
p(and) = 8*x1 + x2
p(even) = 0
p(goal) = 0
p(lte) = 0
p(notEmpty) = 4*x1
Following rules are strictly oriented:
notEmpty(Cons(x,xs)) = 16
> 0
= True()
Following rules are (at-least) weakly oriented:
and(False(),False()) = 0
>= 0
= False()
and(False(),True()) = 0
>= 0
= False()
and(True(),False()) = 0
>= 0
= False()
and(True(),True()) = 0
>= 0
= True()
even(Cons(x,Nil())) = 0
>= 0
= False()
even(Cons(x',Cons(x,xs))) = 0
>= 0
= even(xs)
even(Nil()) = 0
>= 0
= True()
goal(x,y) = 0
>= 0
= and(lte(x,y),even(x))
lte(Cons(x,xs),Nil()) = 0
>= 0
= False()
lte(Cons(x',xs'),Cons(x,xs)) = 0
>= 0
= lte(xs',xs)
lte(Nil(),y) = 0
>= 0
= True()
notEmpty(Nil()) = 4
>= 0
= False()
** Step 1.b:3: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
and(False(),False()) -> False()
and(False(),True()) -> False()
and(True(),False()) -> False()
and(True(),True()) -> True()
even(Cons(x,Nil())) -> False()
even(Cons(x',Cons(x,xs))) -> even(xs)
even(Nil()) -> True()
goal(x,y) -> and(lte(x,y),even(x))
lte(Cons(x,xs),Nil()) -> False()
lte(Cons(x',xs'),Cons(x,xs)) -> lte(xs',xs)
lte(Nil(),y) -> True()
notEmpty(Cons(x,xs)) -> True()
notEmpty(Nil()) -> False()
- Signature:
{and/2,even/1,goal/2,lte/2,notEmpty/1} / {Cons/2,False/0,Nil/0,True/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {and,even,goal,lte,notEmpty} and constructors {Cons,False
,Nil,True}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(Omega(n^1),O(n^1))