* 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: WeightGap 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:
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(and) = {1,2}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(Cons) = [1] x2 + [0]
p(False) = [0]
p(Nil) = [5]
p(True) = [2]
p(and) = [1] x1 + [1] x2 + [14]
p(even) = [5] x1 + [1]
p(goal) = [9] x1 + [9] x2 + [2]
p(lte) = [4] x1 + [2] x2 + [0]
p(notEmpty) = [6] x1 + [1]
Following rules are strictly oriented:
even(Cons(x,Nil())) = [26]
> [0]
= False()
even(Nil()) = [26]
> [2]
= True()
lte(Cons(x,xs),Nil()) = [4] xs + [10]
> [0]
= False()
lte(Nil(),y) = [2] y + [20]
> [2]
= True()
notEmpty(Nil()) = [31]
> [0]
= False()
Following rules are (at-least) weakly oriented:
and(False(),False()) = [14]
>= [0]
= False()
and(False(),True()) = [16]
>= [0]
= False()
and(True(),False()) = [16]
>= [0]
= False()
and(True(),True()) = [18]
>= [2]
= True()
even(Cons(x',Cons(x,xs))) = [5] xs + [1]
>= [5] xs + [1]
= even(xs)
goal(x,y) = [9] x + [9] y + [2]
>= [9] x + [2] y + [15]
= and(lte(x,y),even(x))
lte(Cons(x',xs'),Cons(x,xs)) = [2] xs + [4] xs' + [0]
>= [2] xs + [4] xs' + [0]
= lte(xs',xs)
notEmpty(Cons(x,xs)) = [6] xs + [1]
>= [2]
= True()
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:2: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
even(Cons(x',Cons(x,xs))) -> even(xs)
goal(x,y) -> and(lte(x,y),even(x))
lte(Cons(x',xs'),Cons(x,xs)) -> lte(xs',xs)
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(Nil()) -> True()
lte(Cons(x,xs),Nil()) -> False()
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:
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(and) = {1,2}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(Cons) = [1] x1 + [9]
p(False) = [8]
p(Nil) = [12]
p(True) = [7]
p(and) = [1] x1 + [1] x2 + [2]
p(even) = [8]
p(goal) = [1] x1 + [8]
p(lte) = [11]
p(notEmpty) = [2] x1 + [0]
Following rules are strictly oriented:
notEmpty(Cons(x,xs)) = [2] x + [18]
> [7]
= True()
Following rules are (at-least) weakly oriented:
and(False(),False()) = [18]
>= [8]
= False()
and(False(),True()) = [17]
>= [8]
= False()
and(True(),False()) = [17]
>= [8]
= False()
and(True(),True()) = [16]
>= [7]
= True()
even(Cons(x,Nil())) = [8]
>= [8]
= False()
even(Cons(x',Cons(x,xs))) = [8]
>= [8]
= even(xs)
even(Nil()) = [8]
>= [7]
= True()
goal(x,y) = [1] x + [8]
>= [21]
= and(lte(x,y),even(x))
lte(Cons(x,xs),Nil()) = [11]
>= [8]
= False()
lte(Cons(x',xs'),Cons(x,xs)) = [11]
>= [11]
= lte(xs',xs)
lte(Nil(),y) = [11]
>= [7]
= True()
notEmpty(Nil()) = [24]
>= [8]
= False()
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:3: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
even(Cons(x',Cons(x,xs))) -> even(xs)
goal(x,y) -> and(lte(x,y),even(x))
lte(Cons(x',xs'),Cons(x,xs)) -> lte(xs',xs)
- Weak TRS:
and(False(),False()) -> False()
and(False(),True()) -> False()
and(True(),False()) -> False()
and(True(),True()) -> True()
even(Cons(x,Nil())) -> False()
even(Nil()) -> True()
lte(Cons(x,xs),Nil()) -> False()
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:
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(and) = {1,2}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(Cons) = [1] x1 + [1] x2 + [2]
p(False) = [2]
p(Nil) = [2]
p(True) = [12]
p(and) = [1] x1 + [1] x2 + [4]
p(even) = [4] x1 + [6]
p(goal) = [9] x1 + [3] x2 + [1]
p(lte) = [5] x1 + [12]
p(notEmpty) = [12]
Following rules are strictly oriented:
even(Cons(x',Cons(x,xs))) = [4] x + [4] x' + [4] xs + [22]
> [4] xs + [6]
= even(xs)
lte(Cons(x',xs'),Cons(x,xs)) = [5] x' + [5] xs' + [22]
> [5] xs' + [12]
= lte(xs',xs)
Following rules are (at-least) weakly oriented:
and(False(),False()) = [8]
>= [2]
= False()
and(False(),True()) = [18]
>= [2]
= False()
and(True(),False()) = [18]
>= [2]
= False()
and(True(),True()) = [28]
>= [12]
= True()
even(Cons(x,Nil())) = [4] x + [22]
>= [2]
= False()
even(Nil()) = [14]
>= [12]
= True()
goal(x,y) = [9] x + [3] y + [1]
>= [9] x + [22]
= and(lte(x,y),even(x))
lte(Cons(x,xs),Nil()) = [5] x + [5] xs + [22]
>= [2]
= False()
lte(Nil(),y) = [22]
>= [12]
= True()
notEmpty(Cons(x,xs)) = [12]
>= [12]
= True()
notEmpty(Nil()) = [12]
>= [2]
= False()
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:4: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
goal(x,y) -> and(lte(x,y),even(x))
- 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()
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:
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(and) = {1,2}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(Cons) = [1] x1 + [1] x2 + [1]
p(False) = [0]
p(Nil) = [3]
p(True) = [8]
p(and) = [1] x1 + [1] x2 + [1]
p(even) = [4] x1 + [4]
p(goal) = [10] x1 + [8] x2 + [8]
p(lte) = [2] x1 + [3] x2 + [2]
p(notEmpty) = [8] x1 + [6]
Following rules are strictly oriented:
goal(x,y) = [10] x + [8] y + [8]
> [6] x + [3] y + [7]
= and(lte(x,y),even(x))
Following rules are (at-least) weakly oriented:
and(False(),False()) = [1]
>= [0]
= False()
and(False(),True()) = [9]
>= [0]
= False()
and(True(),False()) = [9]
>= [0]
= False()
and(True(),True()) = [17]
>= [8]
= True()
even(Cons(x,Nil())) = [4] x + [20]
>= [0]
= False()
even(Cons(x',Cons(x,xs))) = [4] x + [4] x' + [4] xs + [12]
>= [4] xs + [4]
= even(xs)
even(Nil()) = [16]
>= [8]
= True()
lte(Cons(x,xs),Nil()) = [2] x + [2] xs + [13]
>= [0]
= False()
lte(Cons(x',xs'),Cons(x,xs)) = [3] x + [2] x' + [3] xs + [2] xs' + [7]
>= [3] xs + [2] xs' + [2]
= lte(xs',xs)
lte(Nil(),y) = [3] y + [8]
>= [8]
= True()
notEmpty(Cons(x,xs)) = [8] x + [8] xs + [14]
>= [8]
= True()
notEmpty(Nil()) = [30]
>= [0]
= False()
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:5: 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))