* Step 1: Sum WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
goal(xs) -> ordered(xs)
notEmpty(Cons(x,xs)) -> True()
notEmpty(Nil()) -> False()
ordered(Cons(x,Nil())) -> True()
ordered(Cons(x',Cons(x,xs))) -> ordered[Ite](<(x',x),Cons(x',Cons(x,xs)))
ordered(Nil()) -> True()
- Weak TRS:
<(x,0()) -> False()
<(0(),S(y)) -> True()
<(S(x),S(y)) -> <(x,y)
ordered[Ite](False(),xs) -> False()
ordered[Ite](True(),Cons(x,xs)) -> ordered(xs)
- Signature:
{ ordered(xs)
notEmpty(Cons(x,xs)) -> True()
notEmpty(Nil()) -> False()
ordered(Cons(x,Nil())) -> True()
ordered(Cons(x',Cons(x,xs))) -> ordered[Ite](<(x',x),Cons(x',Cons(x,xs)))
ordered(Nil()) -> True()
- Weak TRS:
<(x,0()) -> False()
<(0(),S(y)) -> True()
<(S(x),S(y)) -> <(x,y)
ordered[Ite](False(),xs) -> False()
ordered[Ite](True(),Cons(x,xs)) -> ordered(xs)
- Signature:
{ [0]
= True()
Following rules are (at-least) weakly oriented:
<(x,0()) = [0]
>= [0]
= False()
<(0(),S(y)) = [0]
>= [0]
= True()
<(S(x),S(y)) = [0]
>= [0]
= <(x,y)
goal(xs) = [2] xs + [0]
>= [2] xs + [0]
= ordered(xs)
notEmpty(Cons(x,xs)) = [0]
>= [0]
= True()
notEmpty(Nil()) = [0]
>= [0]
= False()
ordered(Cons(x',Cons(x,xs))) = [2] xs + [16]
>= [2] xs + [16]
= ordered[Ite](<(x',x),Cons(x',Cons(x,xs)))
ordered(Nil()) = [0]
>= [0]
= True()
ordered[Ite](False(),xs) = [2] xs + [0]
>= [0]
= False()
ordered[Ite](True(),Cons(x,xs)) = [2] xs + [8]
>= [2] xs + [0]
= ordered(xs)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
goal(xs) -> ordered(xs)
notEmpty(Cons(x,xs)) -> True()
notEmpty(Nil()) -> False()
ordered(Cons(x',Cons(x,xs))) -> ordered[Ite](<(x',x),Cons(x',Cons(x,xs)))
ordered(Nil()) -> True()
- Weak TRS:
<(x,0()) -> False()
<(0(),S(y)) -> True()
<(S(x),S(y)) -> <(x,y)
ordered(Cons(x,Nil())) -> True()
ordered[Ite](False(),xs) -> False()
ordered[Ite](True(),Cons(x,xs)) -> ordered(xs)
- Signature:
{ [2] xs + [3]
= ordered(xs)
notEmpty(Nil()) = [28]
> [0]
= False()
ordered(Nil()) = [9]
> [5]
= True()
Following rules are (at-least) weakly oriented:
<(x,0()) = [11]
>= [0]
= False()
<(0(),S(y)) = [11]
>= [5]
= True()
<(S(x),S(y)) = [11]
>= [11]
= <(x,y)
notEmpty(Cons(x,xs)) = [8] x + [8] xs + [4]
>= [5]
= True()
ordered(Cons(x,Nil())) = [2] x + [9]
>= [5]
= True()
ordered(Cons(x',Cons(x,xs))) = [2] x + [2] x' + [2] xs + [3]
>= [2] x + [2] x' + [2] xs + [12]
= ordered[Ite](<(x',x),Cons(x',Cons(x,xs)))
ordered[Ite](False(),xs) = [2] xs + [1]
>= [0]
= False()
ordered[Ite](True(),Cons(x,xs)) = [2] x + [2] xs + [6]
>= [2] xs + [3]
= ordered(xs)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 4: MI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
notEmpty(Cons(x,xs)) -> True()
ordered(Cons(x',Cons(x,xs))) -> ordered[Ite](<(x',x),Cons(x',Cons(x,xs)))
- Weak TRS:
<(x,0()) -> False()
<(0(),S(y)) -> True()
<(S(x),S(y)) -> <(x,y)
goal(xs) -> ordered(xs)
notEmpty(Nil()) -> False()
ordered(Cons(x,Nil())) -> True()
ordered(Nil()) -> True()
ordered[Ite](False(),xs) -> False()
ordered[Ite](True(),Cons(x,xs)) -> ordered(xs)
- Signature:
{ [4]
= True()
Following rules are (at-least) weakly oriented:
<(x,0()) = [4]
>= [2]
= False()
<(0(),S(y)) = [4]
>= [4]
= True()
<(S(x),S(y)) = [4]
>= [4]
= <(x,y)
goal(xs) = [8] xs + [8]
>= [8]
= ordered(xs)
notEmpty(Nil()) = [10]
>= [2]
= False()
ordered(Cons(x,Nil())) = [8]
>= [4]
= True()
ordered(Cons(x',Cons(x,xs))) = [8]
>= [8]
= ordered[Ite](<(x',x),Cons(x',Cons(x,xs)))
ordered(Nil()) = [8]
>= [4]
= True()
ordered[Ite](False(),xs) = [4]
>= [2]
= False()
ordered[Ite](True(),Cons(x,xs)) = [8]
>= [8]
= ordered(xs)
* Step 5: MI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
ordered(Cons(x',Cons(x,xs))) -> ordered[Ite](<(x',x),Cons(x',Cons(x,xs)))
- Weak TRS:
<(x,0()) -> False()
<(0(),S(y)) -> True()
<(S(x),S(y)) -> <(x,y)
goal(xs) -> ordered(xs)
notEmpty(Cons(x,xs)) -> True()
notEmpty(Nil()) -> False()
ordered(Cons(x,Nil())) -> True()
ordered(Nil()) -> True()
ordered[Ite](False(),xs) -> False()
ordered[Ite](True(),Cons(x,xs)) -> ordered(xs)
- Signature:
{ [8] xs + [24]
= ordered[Ite](<(x',x),Cons(x',Cons(x,xs)))
Following rules are (at-least) weakly oriented:
<(x,0()) = [1]
>= [0]
= False()
<(0(),S(y)) = [1]
>= [0]
= True()
<(S(x),S(y)) = [1]
>= [1]
= <(x,y)
goal(xs) = [12] xs + [15]
>= [8] xs + [15]
= ordered(xs)
notEmpty(Cons(x,xs)) = [10] xs + [12]
>= [0]
= True()
notEmpty(Nil()) = [12]
>= [0]
= False()
ordered(Cons(x,Nil())) = [31]
>= [0]
= True()
ordered(Nil()) = [23]
>= [0]
= True()
ordered[Ite](False(),xs) = [8] xs + [7]
>= [0]
= False()
ordered[Ite](True(),Cons(x,xs)) = [8] xs + [15]
>= [8] xs + [15]
= ordered(xs)
* Step 6: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
<(x,0()) -> False()
<(0(),S(y)) -> True()
<(S(x),S(y)) -> <(x,y)
goal(xs) -> ordered(xs)
notEmpty(Cons(x,xs)) -> True()
notEmpty(Nil()) -> False()
ordered(Cons(x,Nil())) -> True()
ordered(Cons(x',Cons(x,xs))) -> ordered[Ite](<(x',x),Cons(x',Cons(x,xs)))
ordered(Nil()) -> True()
ordered[Ite](False(),xs) -> False()
ordered[Ite](True(),Cons(x,xs)) -> ordered(xs)
- Signature:
{