* 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',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',Cons(x,xs))) -> ordered(xs)
- Signature:
{ [3]
= True()
notEmpty(Nil()) = [21]
> [0]
= False()
Following rules are (at-least) weakly oriented:
<(x,0()) = [8]
>= [0]
= False()
<(0(),S(y)) = [8]
>= [3]
= True()
<(S(x),S(y)) = [8]
>= [8]
= <(x,y)
goal(xs) = [1] xs + [0]
>= [1]
= ordered(xs)
ordered(Cons(x,Nil())) = [1]
>= [3]
= True()
ordered(Cons(x',Cons(x,xs))) = [1]
>= [12]
= ordered[Ite](<(x',x),Cons(x',Cons(x,xs)))
ordered(Nil()) = [1]
>= [3]
= True()
ordered[Ite](False(),xs) = [4]
>= [0]
= False()
ordered[Ite](True(),Cons(x',Cons(x,xs))) = [7]
>= [1]
= ordered(xs)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: MI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
goal(xs) -> ordered(xs)
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)
notEmpty(Cons(x,xs)) -> True()
notEmpty(Nil()) -> False()
ordered[Ite](False(),xs) -> False()
ordered[Ite](True(),Cons(x',Cons(x,xs))) -> ordered(xs)
- Signature:
{ [0]
= ordered(xs)
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)
notEmpty(Cons(x,xs)) = [0]
>= [0]
= True()
notEmpty(Nil()) = [0]
>= [0]
= False()
ordered(Cons(x,Nil())) = [0]
>= [0]
= True()
ordered(Cons(x',Cons(x,xs))) = [0]
>= [0]
= ordered[Ite](<(x',x),Cons(x',Cons(x,xs)))
ordered(Nil()) = [0]
>= [0]
= True()
ordered[Ite](False(),xs) = [0]
>= [0]
= False()
ordered[Ite](True(),Cons(x',Cons(x,xs))) = [0]
>= [0]
= ordered(xs)
* Step 4: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
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)
goal(xs) -> ordered(xs)
notEmpty(Cons(x,xs)) -> True()
notEmpty(Nil()) -> False()
ordered[Ite](False(),xs) -> False()
ordered[Ite](True(),Cons(x',Cons(x,xs))) -> ordered(xs)
- Signature:
{ [1]
= True()
ordered(Nil()) = [9]
> [1]
= True()
Following rules are (at-least) weakly oriented:
<(x,0()) = [1]
>= [1]
= False()
<(0(),S(y)) = [1]
>= [1]
= True()
<(S(x),S(y)) = [1]
>= [1]
= <(x,y)
goal(xs) = [5] xs + [9]
>= [9]
= ordered(xs)
notEmpty(Cons(x,xs)) = [23]
>= [1]
= True()
notEmpty(Nil()) = [2]
>= [1]
= False()
ordered(Cons(x',Cons(x,xs))) = [9]
>= [14]
= ordered[Ite](<(x',x),Cons(x',Cons(x,xs)))
ordered[Ite](False(),xs) = [14]
>= [1]
= False()
ordered[Ite](True(),Cons(x',Cons(x,xs))) = [14]
>= [9]
= ordered(xs)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* 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',Cons(x,xs))) -> ordered(xs)
- Signature:
{ [1] xs + [14]
= ordered[Ite](<(x',x),Cons(x',Cons(x,xs)))
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) = [4] xs + [13]
>= [1] xs + [11]
= ordered(xs)
notEmpty(Cons(x,xs)) = [4]
>= [0]
= True()
notEmpty(Nil()) = [4]
>= [0]
= False()
ordered(Cons(x,Nil())) = [15]
>= [0]
= True()
ordered(Nil()) = [13]
>= [0]
= True()
ordered[Ite](False(),xs) = [1] xs + [10]
>= [0]
= False()
ordered[Ite](True(),Cons(x',Cons(x,xs))) = [1] xs + [14]
>= [1] xs + [11]
= 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',Cons(x,xs))) -> ordered(xs)
- Signature:
{