* Step 1: Sum WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
goal(xs,ys) -> overlap(xs,ys)
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()
overlap(Cons(x,xs),ys) -> overlap[Ite][True][Ite](member(x,ys),Cons(x,xs),ys)
overlap(Nil(),ys) -> 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()
overlap[Ite][True][Ite](False(),Cons(x,xs),ys) -> overlap(xs,ys)
overlap[Ite][True][Ite](True(),xs,ys) -> True()
- Signature:
{!EQ/2,goal/2,member/2,member[Ite][True][Ite]/3,notEmpty/1,overlap/2,overlap[Ite][True][Ite]/3} / {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,overlap
,overlap[Ite][True][Ite]} and constructors {0,Cons,False,Nil,S,True}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
* Step 2: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
goal(xs,ys) -> overlap(xs,ys)
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()
overlap(Cons(x,xs),ys) -> overlap[Ite][True][Ite](member(x,ys),Cons(x,xs),ys)
overlap(Nil(),ys) -> 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()
overlap[Ite][True][Ite](False(),Cons(x,xs),ys) -> overlap(xs,ys)
overlap[Ite][True][Ite](True(),xs,ys) -> True()
- Signature:
{!EQ/2,goal/2,member/2,member[Ite][True][Ite]/3,notEmpty/1,overlap/2,overlap[Ite][True][Ite]/3} / {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,overlap
,overlap[Ite][True][Ite]} 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},
uargs(overlap[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 + [0]
p(False) = [0]
p(Nil) = [0]
p(S) = [1] x1 + [0]
p(True) = [0]
p(goal) = [4] x1 + [0]
p(member) = [0]
p(member[Ite][True][Ite]) = [1] x1 + [0]
p(notEmpty) = [0]
p(overlap) = [4] x1 + [7]
p(overlap[Ite][True][Ite]) = [1] x1 + [4] x2 + [7]
Following rules are strictly oriented:
overlap(Nil(),ys) = [7]
> [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(xs,ys) = [4] xs + [0]
>= [4] xs + [7]
= overlap(xs,ys)
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()
overlap(Cons(x,xs),ys) = [4] x + [4] xs + [7]
>= [4] x + [4] xs + [7]
= overlap[Ite][True][Ite](member(x,ys),Cons(x,xs),ys)
overlap[Ite][True][Ite](False(),Cons(x,xs),ys) = [4] x + [4] xs + [7]
>= [4] xs + [7]
= overlap(xs,ys)
overlap[Ite][True][Ite](True(),xs,ys) = [4] xs + [7]
>= [0]
= True()
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
goal(xs,ys) -> overlap(xs,ys)
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()
overlap(Cons(x,xs),ys) -> overlap[Ite][True][Ite](member(x,ys),Cons(x,xs),ys)
- 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()
overlap(Nil(),ys) -> False()
overlap[Ite][True][Ite](False(),Cons(x,xs),ys) -> overlap(xs,ys)
overlap[Ite][True][Ite](True(),xs,ys) -> True()
- Signature:
{!EQ/2,goal/2,member/2,member[Ite][True][Ite]/3,notEmpty/1,overlap/2,overlap[Ite][True][Ite]/3} / {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,overlap
,overlap[Ite][True][Ite]} 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},
uargs(overlap[Ite][True][Ite]) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(!EQ) = [0]
p(0) = [0]
p(Cons) = [4]
p(False) = [0]
p(Nil) = [0]
p(S) = [0]
p(True) = [0]
p(goal) = [4] x1 + [5]
p(member) = [0]
p(member[Ite][True][Ite]) = [1] x1 + [1] x3 + [0]
p(notEmpty) = [0]
p(overlap) = [3]
p(overlap[Ite][True][Ite]) = [1] x1 + [1] x2 + [0]
Following rules are strictly oriented:
goal(xs,ys) = [4] xs + [5]
> [3]
= overlap(xs,ys)
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]
>= [4]
= member[Ite][True][Ite](!EQ(x,x'),x',Cons(x,xs))
member[Ite][True][Ite](False(),x',Cons(x,xs)) = [4]
>= [0]
= member(x',xs)
member[Ite][True][Ite](True(),x,xs) = [1] xs + [0]
>= [0]
= True()
notEmpty(Cons(x,xs)) = [0]
>= [0]
= True()
notEmpty(Nil()) = [0]
>= [0]
= False()
overlap(Cons(x,xs),ys) = [3]
>= [4]
= overlap[Ite][True][Ite](member(x,ys),Cons(x,xs),ys)
overlap(Nil(),ys) = [3]
>= [0]
= False()
overlap[Ite][True][Ite](False(),Cons(x,xs),ys) = [4]
>= [3]
= overlap(xs,ys)
overlap[Ite][True][Ite](True(),xs,ys) = [1] 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^2))
+ 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()
overlap(Cons(x,xs),ys) -> overlap[Ite][True][Ite](member(x,ys),Cons(x,xs),ys)
- 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(xs,ys) -> overlap(xs,ys)
member[Ite][True][Ite](False(),x',Cons(x,xs)) -> member(x',xs)
member[Ite][True][Ite](True(),x,xs) -> True()
overlap(Nil(),ys) -> False()
overlap[Ite][True][Ite](False(),Cons(x,xs),ys) -> overlap(xs,ys)
overlap[Ite][True][Ite](True(),xs,ys) -> True()
- Signature:
{!EQ/2,goal/2,member/2,member[Ite][True][Ite]/3,notEmpty/1,overlap/2,overlap[Ite][True][Ite]/3} / {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,overlap
,overlap[Ite][True][Ite]} 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},
uargs(overlap[Ite][True][Ite]) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(!EQ) = [2]
p(0) = [0]
p(Cons) = [1] x1 + [1] x2 + [0]
p(False) = [1]
p(Nil) = [0]
p(S) = [1] x1 + [0]
p(True) = [2]
p(goal) = [6] x2 + [5]
p(member) = [0]
p(member[Ite][True][Ite]) = [1] x1 + [0]
p(notEmpty) = [3]
p(overlap) = [6] x2 + [2]
p(overlap[Ite][True][Ite]) = [1] x1 + [6] x3 + [4]
Following rules are strictly oriented:
notEmpty(Cons(x,xs)) = [3]
> [2]
= True()
notEmpty(Nil()) = [3]
> [1]
= False()
Following rules are (at-least) weakly oriented:
!EQ(0(),0()) = [2]
>= [2]
= True()
!EQ(0(),S(y)) = [2]
>= [1]
= False()
!EQ(S(x),0()) = [2]
>= [1]
= False()
!EQ(S(x),S(y)) = [2]
>= [2]
= !EQ(x,y)
goal(xs,ys) = [6] ys + [5]
>= [6] ys + [2]
= overlap(xs,ys)
member(x,Nil()) = [0]
>= [1]
= False()
member(x',Cons(x,xs)) = [0]
>= [2]
= member[Ite][True][Ite](!EQ(x,x'),x',Cons(x,xs))
member[Ite][True][Ite](False(),x',Cons(x,xs)) = [1]
>= [0]
= member(x',xs)
member[Ite][True][Ite](True(),x,xs) = [2]
>= [2]
= True()
overlap(Cons(x,xs),ys) = [6] ys + [2]
>= [6] ys + [4]
= overlap[Ite][True][Ite](member(x,ys),Cons(x,xs),ys)
overlap(Nil(),ys) = [6] ys + [2]
>= [1]
= False()
overlap[Ite][True][Ite](False(),Cons(x,xs),ys) = [6] ys + [5]
>= [6] ys + [2]
= overlap(xs,ys)
overlap[Ite][True][Ite](True(),xs,ys) = [6] ys + [6]
>= [2]
= True()
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 5: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
member(x,Nil()) -> False()
member(x',Cons(x,xs)) -> member[Ite][True][Ite](!EQ(x,x'),x',Cons(x,xs))
overlap(Cons(x,xs),ys) -> overlap[Ite][True][Ite](member(x,ys),Cons(x,xs),ys)
- 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(xs,ys) -> overlap(xs,ys)
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()
overlap(Nil(),ys) -> False()
overlap[Ite][True][Ite](False(),Cons(x,xs),ys) -> overlap(xs,ys)
overlap[Ite][True][Ite](True(),xs,ys) -> True()
- Signature:
{!EQ/2,goal/2,member/2,member[Ite][True][Ite]/3,notEmpty/1,overlap/2,overlap[Ite][True][Ite]/3} / {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,overlap
,overlap[Ite][True][Ite]} 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},
uargs(overlap[Ite][True][Ite]) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(!EQ) = [1]
p(0) = [0]
p(Cons) = [1]
p(False) = [1]
p(Nil) = [1]
p(S) = [0]
p(True) = [1]
p(goal) = [1]
p(member) = [2]
p(member[Ite][True][Ite]) = [1] x1 + [5]
p(notEmpty) = [2]
p(overlap) = [1]
p(overlap[Ite][True][Ite]) = [1] x1 + [2] x2 + [0]
Following rules are strictly oriented:
member(x,Nil()) = [2]
> [1]
= False()
Following rules are (at-least) weakly oriented:
!EQ(0(),0()) = [1]
>= [1]
= True()
!EQ(0(),S(y)) = [1]
>= [1]
= False()
!EQ(S(x),0()) = [1]
>= [1]
= False()
!EQ(S(x),S(y)) = [1]
>= [1]
= !EQ(x,y)
goal(xs,ys) = [1]
>= [1]
= overlap(xs,ys)
member(x',Cons(x,xs)) = [2]
>= [6]
= member[Ite][True][Ite](!EQ(x,x'),x',Cons(x,xs))
member[Ite][True][Ite](False(),x',Cons(x,xs)) = [6]
>= [2]
= member(x',xs)
member[Ite][True][Ite](True(),x,xs) = [6]
>= [1]
= True()
notEmpty(Cons(x,xs)) = [2]
>= [1]
= True()
notEmpty(Nil()) = [2]
>= [1]
= False()
overlap(Cons(x,xs),ys) = [1]
>= [4]
= overlap[Ite][True][Ite](member(x,ys),Cons(x,xs),ys)
overlap(Nil(),ys) = [1]
>= [1]
= False()
overlap[Ite][True][Ite](False(),Cons(x,xs),ys) = [3]
>= [1]
= overlap(xs,ys)
overlap[Ite][True][Ite](True(),xs,ys) = [2] xs + [1]
>= [1]
= True()
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 6: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
member(x',Cons(x,xs)) -> member[Ite][True][Ite](!EQ(x,x'),x',Cons(x,xs))
overlap(Cons(x,xs),ys) -> overlap[Ite][True][Ite](member(x,ys),Cons(x,xs),ys)
- 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(xs,ys) -> overlap(xs,ys)
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()
overlap(Nil(),ys) -> False()
overlap[Ite][True][Ite](False(),Cons(x,xs),ys) -> overlap(xs,ys)
overlap[Ite][True][Ite](True(),xs,ys) -> True()
- Signature:
{!EQ/2,goal/2,member/2,member[Ite][True][Ite]/3,notEmpty/1,overlap/2,overlap[Ite][True][Ite]/3} / {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,overlap
,overlap[Ite][True][Ite]} 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},
uargs(overlap[Ite][True][Ite]) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(!EQ) = [4]
p(0) = [0]
p(Cons) = [1] x1 + [1] x2 + [2]
p(False) = [0]
p(Nil) = [1]
p(S) = [1] x1 + [4]
p(True) = [2]
p(goal) = [5] x1 + [1] x2 + [7]
p(member) = [0]
p(member[Ite][True][Ite]) = [1] x1 + [0]
p(notEmpty) = [5]
p(overlap) = [5] x1 + [5]
p(overlap[Ite][True][Ite]) = [1] x1 + [5] x2 + [4]
Following rules are strictly oriented:
overlap(Cons(x,xs),ys) = [5] x + [5] xs + [15]
> [5] x + [5] xs + [14]
= overlap[Ite][True][Ite](member(x,ys),Cons(x,xs),ys)
Following rules are (at-least) weakly oriented:
!EQ(0(),0()) = [4]
>= [2]
= True()
!EQ(0(),S(y)) = [4]
>= [0]
= False()
!EQ(S(x),0()) = [4]
>= [0]
= False()
!EQ(S(x),S(y)) = [4]
>= [4]
= !EQ(x,y)
goal(xs,ys) = [5] xs + [1] ys + [7]
>= [5] xs + [5]
= overlap(xs,ys)
member(x,Nil()) = [0]
>= [0]
= False()
member(x',Cons(x,xs)) = [0]
>= [4]
= 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) = [2]
>= [2]
= True()
notEmpty(Cons(x,xs)) = [5]
>= [2]
= True()
notEmpty(Nil()) = [5]
>= [0]
= False()
overlap(Nil(),ys) = [10]
>= [0]
= False()
overlap[Ite][True][Ite](False(),Cons(x,xs),ys) = [5] x + [5] xs + [14]
>= [5] xs + [5]
= overlap(xs,ys)
overlap[Ite][True][Ite](True(),xs,ys) = [5] xs + [6]
>= [2]
= True()
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 7: NaturalPI WORST_CASE(?,O(n^2))
+ 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(xs,ys) -> overlap(xs,ys)
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()
overlap(Cons(x,xs),ys) -> overlap[Ite][True][Ite](member(x,ys),Cons(x,xs),ys)
overlap(Nil(),ys) -> False()
overlap[Ite][True][Ite](False(),Cons(x,xs),ys) -> overlap(xs,ys)
overlap[Ite][True][Ite](True(),xs,ys) -> True()
- Signature:
{!EQ/2,goal/2,member/2,member[Ite][True][Ite]/3,notEmpty/1,overlap/2,overlap[Ite][True][Ite]/3} / {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,overlap
,overlap[Ite][True][Ite]} and constructors {0,Cons,False,Nil,S,True}
+ Applied Processor:
NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a polynomial interpretation of kind constructor-based(mixed(2)):
The following argument positions are considered usable:
uargs(member[Ite][True][Ite]) = {1},
uargs(overlap[Ite][True][Ite]) = {1}
Following symbols are considered usable:
{!EQ,goal,member,member[Ite][True][Ite],notEmpty,overlap,overlap[Ite][True][Ite]}
TcT has computed the following interpretation:
p(!EQ) = 1
p(0) = 0
p(Cons) = 1 + x2
p(False) = 0
p(Nil) = 0
p(S) = 0
p(True) = 0
p(goal) = 4 + 5*x1 + 6*x1*x2 + 5*x1^2 + 6*x2 + 2*x2^2
p(member) = 2 + 2*x2
p(member[Ite][True][Ite]) = x1 + 2*x3
p(notEmpty) = 7*x1^2
p(overlap) = 1 + 4*x1 + 6*x1*x2 + 4*x1^2 + 6*x2 + 2*x2^2
p(overlap[Ite][True][Ite]) = x1 + 3*x2 + 6*x2*x3 + 4*x2^2 + 4*x3 + 2*x3^2
Following rules are strictly oriented:
member(x',Cons(x,xs)) = 4 + 2*xs
> 3 + 2*xs
= member[Ite][True][Ite](!EQ(x,x'),x',Cons(x,xs))
Following rules are (at-least) weakly oriented:
!EQ(0(),0()) = 1
>= 0
= True()
!EQ(0(),S(y)) = 1
>= 0
= False()
!EQ(S(x),0()) = 1
>= 0
= False()
!EQ(S(x),S(y)) = 1
>= 1
= !EQ(x,y)
goal(xs,ys) = 4 + 5*xs + 6*xs*ys + 5*xs^2 + 6*ys + 2*ys^2
>= 1 + 4*xs + 6*xs*ys + 4*xs^2 + 6*ys + 2*ys^2
= overlap(xs,ys)
member(x,Nil()) = 2
>= 0
= False()
member[Ite][True][Ite](False(),x',Cons(x,xs)) = 2 + 2*xs
>= 2 + 2*xs
= member(x',xs)
member[Ite][True][Ite](True(),x,xs) = 2*xs
>= 0
= True()
notEmpty(Cons(x,xs)) = 7 + 14*xs + 7*xs^2
>= 0
= True()
notEmpty(Nil()) = 0
>= 0
= False()
overlap(Cons(x,xs),ys) = 9 + 12*xs + 6*xs*ys + 4*xs^2 + 12*ys + 2*ys^2
>= 9 + 11*xs + 6*xs*ys + 4*xs^2 + 12*ys + 2*ys^2
= overlap[Ite][True][Ite](member(x,ys),Cons(x,xs),ys)
overlap(Nil(),ys) = 1 + 6*ys + 2*ys^2
>= 0
= False()
overlap[Ite][True][Ite](False(),Cons(x,xs),ys) = 7 + 11*xs + 6*xs*ys + 4*xs^2 + 10*ys + 2*ys^2
>= 1 + 4*xs + 6*xs*ys + 4*xs^2 + 6*ys + 2*ys^2
= overlap(xs,ys)
overlap[Ite][True][Ite](True(),xs,ys) = 3*xs + 6*xs*ys + 4*xs^2 + 4*ys + 2*ys^2
>= 0
= True()
* Step 8: 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(xs,ys) -> overlap(xs,ys)
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()
overlap(Cons(x,xs),ys) -> overlap[Ite][True][Ite](member(x,ys),Cons(x,xs),ys)
overlap(Nil(),ys) -> False()
overlap[Ite][True][Ite](False(),Cons(x,xs),ys) -> overlap(xs,ys)
overlap[Ite][True][Ite](True(),xs,ys) -> True()
- Signature:
{!EQ/2,goal/2,member/2,member[Ite][True][Ite]/3,notEmpty/1,overlap/2,overlap[Ite][True][Ite]/3} / {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,overlap
,overlap[Ite][True][Ite]} 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^2))