* Step 1: Sum WORST_CASE(Omega(n^1),O(n^1))
+ Considered Problem:
- Strict TRS:
f(x,Cons(x',xs)) -> f[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))
f(x,Nil()) -> Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil()))))
g(x,Cons(x',xs)) -> g[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))
g(x,Nil()) -> Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil()))))
goal(x,y) -> Cons(f(x,y),Cons(g(x,y),Nil()))
lt0(x,Nil()) -> False()
lt0(Cons(x',xs'),Cons(x,xs)) -> lt0(xs',xs)
lt0(Nil(),Cons(x',xs)) -> True()
notEmpty(Cons(x,xs)) -> True()
notEmpty(Nil()) -> False()
number4(n) -> Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil()))))
- Weak TRS:
f[Ite][False][Ite](False(),Cons(x,xs),y) -> f(xs,Cons(Cons(Nil(),Nil()),y))
f[Ite][False][Ite](True(),x',Cons(x,xs)) -> f(x',xs)
g[Ite][False][Ite](False(),Cons(x,xs),y) -> g(xs,Cons(Cons(Nil(),Nil()),y))
g[Ite][False][Ite](True(),x',Cons(x,xs)) -> g(x',xs)
- Signature:
{f/2,f[Ite][False][Ite]/3,g/2,g[Ite][False][Ite]/3,goal/2,lt0/2,notEmpty/1,number4/1} / {Cons/2,False/0
,Nil/0,True/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {f,f[Ite][False][Ite],g,g[Ite][False][Ite],goal,lt0
,notEmpty,number4} 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:
f(x,Cons(x',xs)) -> f[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))
f(x,Nil()) -> Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil()))))
g(x,Cons(x',xs)) -> g[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))
g(x,Nil()) -> Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil()))))
goal(x,y) -> Cons(f(x,y),Cons(g(x,y),Nil()))
lt0(x,Nil()) -> False()
lt0(Cons(x',xs'),Cons(x,xs)) -> lt0(xs',xs)
lt0(Nil(),Cons(x',xs)) -> True()
notEmpty(Cons(x,xs)) -> True()
notEmpty(Nil()) -> False()
number4(n) -> Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil()))))
- Weak TRS:
f[Ite][False][Ite](False(),Cons(x,xs),y) -> f(xs,Cons(Cons(Nil(),Nil()),y))
f[Ite][False][Ite](True(),x',Cons(x,xs)) -> f(x',xs)
g[Ite][False][Ite](False(),Cons(x,xs),y) -> g(xs,Cons(Cons(Nil(),Nil()),y))
g[Ite][False][Ite](True(),x',Cons(x,xs)) -> g(x',xs)
- Signature:
{f/2,f[Ite][False][Ite]/3,g/2,g[Ite][False][Ite]/3,goal/2,lt0/2,notEmpty/1,number4/1} / {Cons/2,False/0
,Nil/0,True/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {f,f[Ite][False][Ite],g,g[Ite][False][Ite],goal,lt0
,notEmpty,number4} and constructors {Cons,False,Nil,True}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
lt0(y,u){y -> Cons(x,y),u -> Cons(z,u)} =
lt0(Cons(x,y),Cons(z,u)) ->^+ lt0(y,u)
= C[lt0(y,u) = lt0(y,u){}]
** Step 1.b:1: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
f(x,Cons(x',xs)) -> f[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))
f(x,Nil()) -> Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil()))))
g(x,Cons(x',xs)) -> g[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))
g(x,Nil()) -> Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil()))))
goal(x,y) -> Cons(f(x,y),Cons(g(x,y),Nil()))
lt0(x,Nil()) -> False()
lt0(Cons(x',xs'),Cons(x,xs)) -> lt0(xs',xs)
lt0(Nil(),Cons(x',xs)) -> True()
notEmpty(Cons(x,xs)) -> True()
notEmpty(Nil()) -> False()
number4(n) -> Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil()))))
- Weak TRS:
f[Ite][False][Ite](False(),Cons(x,xs),y) -> f(xs,Cons(Cons(Nil(),Nil()),y))
f[Ite][False][Ite](True(),x',Cons(x,xs)) -> f(x',xs)
g[Ite][False][Ite](False(),Cons(x,xs),y) -> g(xs,Cons(Cons(Nil(),Nil()),y))
g[Ite][False][Ite](True(),x',Cons(x,xs)) -> g(x',xs)
- Signature:
{f/2,f[Ite][False][Ite]/3,g/2,g[Ite][False][Ite]/3,goal/2,lt0/2,notEmpty/1,number4/1} / {Cons/2,False/0
,Nil/0,True/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {f,f[Ite][False][Ite],g,g[Ite][False][Ite],goal,lt0
,notEmpty,number4} 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(Cons) = {1,2},
uargs(f[Ite][False][Ite]) = {1},
uargs(g[Ite][False][Ite]) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(Cons) = [1] x1 + [1] x2 + [2]
p(False) = [4]
p(Nil) = [1]
p(True) = [6]
p(f) = [1] x1 + [6]
p(f[Ite][False][Ite]) = [1] x1 + [1] x2 + [0]
p(g) = [2]
p(g[Ite][False][Ite]) = [1] x1 + [0]
p(goal) = [4] x1 + [1] x2 + [0]
p(lt0) = [0]
p(notEmpty) = [4] x1 + [0]
p(number4) = [0]
Following rules are strictly oriented:
f(x,Cons(x',xs)) = [1] x + [6]
> [1] x + [0]
= f[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))
g(x,Cons(x',xs)) = [2]
> [0]
= g[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))
notEmpty(Cons(x,xs)) = [4] x + [4] xs + [8]
> [6]
= True()
Following rules are (at-least) weakly oriented:
f(x,Nil()) = [1] x + [6]
>= [13]
= Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil()))))
f[Ite][False][Ite](False(),Cons(x,xs),y) = [1] x + [1] xs + [6]
>= [1] xs + [6]
= f(xs,Cons(Cons(Nil(),Nil()),y))
f[Ite][False][Ite](True(),x',Cons(x,xs)) = [1] x' + [6]
>= [1] x' + [6]
= f(x',xs)
g(x,Nil()) = [2]
>= [13]
= Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil()))))
g[Ite][False][Ite](False(),Cons(x,xs),y) = [4]
>= [2]
= g(xs,Cons(Cons(Nil(),Nil()),y))
g[Ite][False][Ite](True(),x',Cons(x,xs)) = [6]
>= [2]
= g(x',xs)
goal(x,y) = [4] x + [1] y + [0]
>= [1] x + [13]
= Cons(f(x,y),Cons(g(x,y),Nil()))
lt0(x,Nil()) = [0]
>= [4]
= False()
lt0(Cons(x',xs'),Cons(x,xs)) = [0]
>= [0]
= lt0(xs',xs)
lt0(Nil(),Cons(x',xs)) = [0]
>= [6]
= True()
notEmpty(Nil()) = [4]
>= [4]
= False()
number4(n) = [0]
>= [13]
= Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil()))))
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:
f(x,Nil()) -> Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil()))))
g(x,Nil()) -> Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil()))))
goal(x,y) -> Cons(f(x,y),Cons(g(x,y),Nil()))
lt0(x,Nil()) -> False()
lt0(Cons(x',xs'),Cons(x,xs)) -> lt0(xs',xs)
lt0(Nil(),Cons(x',xs)) -> True()
notEmpty(Nil()) -> False()
number4(n) -> Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil()))))
- Weak TRS:
f(x,Cons(x',xs)) -> f[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))
f[Ite][False][Ite](False(),Cons(x,xs),y) -> f(xs,Cons(Cons(Nil(),Nil()),y))
f[Ite][False][Ite](True(),x',Cons(x,xs)) -> f(x',xs)
g(x,Cons(x',xs)) -> g[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))
g[Ite][False][Ite](False(),Cons(x,xs),y) -> g(xs,Cons(Cons(Nil(),Nil()),y))
g[Ite][False][Ite](True(),x',Cons(x,xs)) -> g(x',xs)
notEmpty(Cons(x,xs)) -> True()
- Signature:
{f/2,f[Ite][False][Ite]/3,g/2,g[Ite][False][Ite]/3,goal/2,lt0/2,notEmpty/1,number4/1} / {Cons/2,False/0
,Nil/0,True/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {f,f[Ite][False][Ite],g,g[Ite][False][Ite],goal,lt0
,notEmpty,number4} 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(Cons) = {1,2},
uargs(f[Ite][False][Ite]) = {1},
uargs(g[Ite][False][Ite]) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(Cons) = [1] x1 + [1] x2 + [0]
p(False) = [7]
p(Nil) = [2]
p(True) = [2]
p(f) = [3] x1 + [1] x2 + [1]
p(f[Ite][False][Ite]) = [1] x1 + [3] x2 + [1] x3 + [0]
p(g) = [3]
p(g[Ite][False][Ite]) = [1] x1 + [2]
p(goal) = [4] x1 + [1] x2 + [7]
p(lt0) = [0]
p(notEmpty) = [2] x1 + [5]
p(number4) = [2]
Following rules are strictly oriented:
goal(x,y) = [4] x + [1] y + [7]
> [3] x + [1] y + [6]
= Cons(f(x,y),Cons(g(x,y),Nil()))
notEmpty(Nil()) = [9]
> [7]
= False()
Following rules are (at-least) weakly oriented:
f(x,Cons(x',xs)) = [3] x + [1] x' + [1] xs + [1]
>= [3] x + [1] x' + [1] xs + [0]
= f[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))
f(x,Nil()) = [3] x + [3]
>= [10]
= Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil()))))
f[Ite][False][Ite](False(),Cons(x,xs),y) = [3] x + [3] xs + [1] y + [7]
>= [3] xs + [1] y + [5]
= f(xs,Cons(Cons(Nil(),Nil()),y))
f[Ite][False][Ite](True(),x',Cons(x,xs)) = [1] x + [3] x' + [1] xs + [2]
>= [3] x' + [1] xs + [1]
= f(x',xs)
g(x,Cons(x',xs)) = [3]
>= [2]
= g[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))
g(x,Nil()) = [3]
>= [10]
= Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil()))))
g[Ite][False][Ite](False(),Cons(x,xs),y) = [9]
>= [3]
= g(xs,Cons(Cons(Nil(),Nil()),y))
g[Ite][False][Ite](True(),x',Cons(x,xs)) = [4]
>= [3]
= g(x',xs)
lt0(x,Nil()) = [0]
>= [7]
= False()
lt0(Cons(x',xs'),Cons(x,xs)) = [0]
>= [0]
= lt0(xs',xs)
lt0(Nil(),Cons(x',xs)) = [0]
>= [2]
= True()
notEmpty(Cons(x,xs)) = [2] x + [2] xs + [5]
>= [2]
= True()
number4(n) = [2]
>= [10]
= Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil()))))
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:
f(x,Nil()) -> Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil()))))
g(x,Nil()) -> Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil()))))
lt0(x,Nil()) -> False()
lt0(Cons(x',xs'),Cons(x,xs)) -> lt0(xs',xs)
lt0(Nil(),Cons(x',xs)) -> True()
number4(n) -> Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil()))))
- Weak TRS:
f(x,Cons(x',xs)) -> f[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))
f[Ite][False][Ite](False(),Cons(x,xs),y) -> f(xs,Cons(Cons(Nil(),Nil()),y))
f[Ite][False][Ite](True(),x',Cons(x,xs)) -> f(x',xs)
g(x,Cons(x',xs)) -> g[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))
g[Ite][False][Ite](False(),Cons(x,xs),y) -> g(xs,Cons(Cons(Nil(),Nil()),y))
g[Ite][False][Ite](True(),x',Cons(x,xs)) -> g(x',xs)
goal(x,y) -> Cons(f(x,y),Cons(g(x,y),Nil()))
notEmpty(Cons(x,xs)) -> True()
notEmpty(Nil()) -> False()
- Signature:
{f/2,f[Ite][False][Ite]/3,g/2,g[Ite][False][Ite]/3,goal/2,lt0/2,notEmpty/1,number4/1} / {Cons/2,False/0
,Nil/0,True/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {f,f[Ite][False][Ite],g,g[Ite][False][Ite],goal,lt0
,notEmpty,number4} 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(Cons) = {1,2},
uargs(f[Ite][False][Ite]) = {1},
uargs(g[Ite][False][Ite]) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(Cons) = [1] x1 + [1] x2 + [0]
p(False) = [0]
p(Nil) = [0]
p(True) = [0]
p(f) = [1] x2 + [5]
p(f[Ite][False][Ite]) = [1] x1 + [1] x3 + [5]
p(g) = [0]
p(g[Ite][False][Ite]) = [1] x1 + [0]
p(goal) = [1] x2 + [5]
p(lt0) = [1] x2 + [0]
p(notEmpty) = [0]
p(number4) = [1] x1 + [1]
Following rules are strictly oriented:
f(x,Nil()) = [5]
> [0]
= Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil()))))
number4(n) = [1] n + [1]
> [0]
= Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil()))))
Following rules are (at-least) weakly oriented:
f(x,Cons(x',xs)) = [1] x' + [1] xs + [5]
>= [1] x' + [1] xs + [5]
= f[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))
f[Ite][False][Ite](False(),Cons(x,xs),y) = [1] y + [5]
>= [1] y + [5]
= f(xs,Cons(Cons(Nil(),Nil()),y))
f[Ite][False][Ite](True(),x',Cons(x,xs)) = [1] x + [1] xs + [5]
>= [1] xs + [5]
= f(x',xs)
g(x,Cons(x',xs)) = [0]
>= [0]
= g[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))
g(x,Nil()) = [0]
>= [0]
= Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil()))))
g[Ite][False][Ite](False(),Cons(x,xs),y) = [0]
>= [0]
= g(xs,Cons(Cons(Nil(),Nil()),y))
g[Ite][False][Ite](True(),x',Cons(x,xs)) = [0]
>= [0]
= g(x',xs)
goal(x,y) = [1] y + [5]
>= [1] y + [5]
= Cons(f(x,y),Cons(g(x,y),Nil()))
lt0(x,Nil()) = [0]
>= [0]
= False()
lt0(Cons(x',xs'),Cons(x,xs)) = [1] x + [1] xs + [0]
>= [1] xs + [0]
= lt0(xs',xs)
lt0(Nil(),Cons(x',xs)) = [1] x' + [1] xs + [0]
>= [0]
= True()
notEmpty(Cons(x,xs)) = [0]
>= [0]
= True()
notEmpty(Nil()) = [0]
>= [0]
= 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:
g(x,Nil()) -> Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil()))))
lt0(x,Nil()) -> False()
lt0(Cons(x',xs'),Cons(x,xs)) -> lt0(xs',xs)
lt0(Nil(),Cons(x',xs)) -> True()
- Weak TRS:
f(x,Cons(x',xs)) -> f[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))
f(x,Nil()) -> Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil()))))
f[Ite][False][Ite](False(),Cons(x,xs),y) -> f(xs,Cons(Cons(Nil(),Nil()),y))
f[Ite][False][Ite](True(),x',Cons(x,xs)) -> f(x',xs)
g(x,Cons(x',xs)) -> g[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))
g[Ite][False][Ite](False(),Cons(x,xs),y) -> g(xs,Cons(Cons(Nil(),Nil()),y))
g[Ite][False][Ite](True(),x',Cons(x,xs)) -> g(x',xs)
goal(x,y) -> Cons(f(x,y),Cons(g(x,y),Nil()))
notEmpty(Cons(x,xs)) -> True()
notEmpty(Nil()) -> False()
number4(n) -> Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil()))))
- Signature:
{f/2,f[Ite][False][Ite]/3,g/2,g[Ite][False][Ite]/3,goal/2,lt0/2,notEmpty/1,number4/1} / {Cons/2,False/0
,Nil/0,True/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {f,f[Ite][False][Ite],g,g[Ite][False][Ite],goal,lt0
,notEmpty,number4} 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(Cons) = {1,2},
uargs(f[Ite][False][Ite]) = {1},
uargs(g[Ite][False][Ite]) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(Cons) = [1] x1 + [1] x2 + [0]
p(False) = [0]
p(Nil) = [0]
p(True) = [0]
p(f) = [1]
p(f[Ite][False][Ite]) = [1] x1 + [1]
p(g) = [1]
p(g[Ite][False][Ite]) = [1] x1 + [1]
p(goal) = [3] x1 + [4]
p(lt0) = [1] x2 + [0]
p(notEmpty) = [1]
p(number4) = [1] x1 + [5]
Following rules are strictly oriented:
g(x,Nil()) = [1]
> [0]
= Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil()))))
Following rules are (at-least) weakly oriented:
f(x,Cons(x',xs)) = [1]
>= [1]
= f[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))
f(x,Nil()) = [1]
>= [0]
= Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil()))))
f[Ite][False][Ite](False(),Cons(x,xs),y) = [1]
>= [1]
= f(xs,Cons(Cons(Nil(),Nil()),y))
f[Ite][False][Ite](True(),x',Cons(x,xs)) = [1]
>= [1]
= f(x',xs)
g(x,Cons(x',xs)) = [1]
>= [1]
= g[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))
g[Ite][False][Ite](False(),Cons(x,xs),y) = [1]
>= [1]
= g(xs,Cons(Cons(Nil(),Nil()),y))
g[Ite][False][Ite](True(),x',Cons(x,xs)) = [1]
>= [1]
= g(x',xs)
goal(x,y) = [3] x + [4]
>= [2]
= Cons(f(x,y),Cons(g(x,y),Nil()))
lt0(x,Nil()) = [0]
>= [0]
= False()
lt0(Cons(x',xs'),Cons(x,xs)) = [1] x + [1] xs + [0]
>= [1] xs + [0]
= lt0(xs',xs)
lt0(Nil(),Cons(x',xs)) = [1] x' + [1] xs + [0]
>= [0]
= True()
notEmpty(Cons(x,xs)) = [1]
>= [0]
= True()
notEmpty(Nil()) = [1]
>= [0]
= False()
number4(n) = [1] n + [5]
>= [0]
= Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil()))))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:5: MI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
lt0(x,Nil()) -> False()
lt0(Cons(x',xs'),Cons(x,xs)) -> lt0(xs',xs)
lt0(Nil(),Cons(x',xs)) -> True()
- Weak TRS:
f(x,Cons(x',xs)) -> f[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))
f(x,Nil()) -> Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil()))))
f[Ite][False][Ite](False(),Cons(x,xs),y) -> f(xs,Cons(Cons(Nil(),Nil()),y))
f[Ite][False][Ite](True(),x',Cons(x,xs)) -> f(x',xs)
g(x,Cons(x',xs)) -> g[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))
g(x,Nil()) -> Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil()))))
g[Ite][False][Ite](False(),Cons(x,xs),y) -> g(xs,Cons(Cons(Nil(),Nil()),y))
g[Ite][False][Ite](True(),x',Cons(x,xs)) -> g(x',xs)
goal(x,y) -> Cons(f(x,y),Cons(g(x,y),Nil()))
notEmpty(Cons(x,xs)) -> True()
notEmpty(Nil()) -> False()
number4(n) -> Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil()))))
- Signature:
{f/2,f[Ite][False][Ite]/3,g/2,g[Ite][False][Ite]/3,goal/2,lt0/2,notEmpty/1,number4/1} / {Cons/2,False/0
,Nil/0,True/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {f,f[Ite][False][Ite],g,g[Ite][False][Ite],goal,lt0
,notEmpty,number4} and constructors {Cons,False,Nil,True}
+ Applied Processor:
MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity Nothing)), miDimension = 1, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity Nothing)):
The following argument positions are considered usable:
uargs(Cons) = {1,2},
uargs(f[Ite][False][Ite]) = {1},
uargs(g[Ite][False][Ite]) = {1}
Following symbols are considered usable:
{f,f[Ite][False][Ite],g,g[Ite][False][Ite],goal,lt0,notEmpty,number4}
TcT has computed the following interpretation:
p(Cons) = [1] x_1 + [1] x_2 + [1]
p(False) = [0]
p(Nil) = [0]
p(True) = [2]
p(f) = [2] x_1 + [4]
p(f[Ite][False][Ite]) = [1] x_1 + [2] x_2 + [2]
p(g) = [8] x_1 + [8]
p(g[Ite][False][Ite]) = [4] x_1 + [8] x_2 + [0]
p(goal) = [10] x_1 + [15]
p(lt0) = [2] x_2 + [0]
p(notEmpty) = [8] x_1 + [15]
p(number4) = [1] x_1 + [4]
Following rules are strictly oriented:
lt0(Cons(x',xs'),Cons(x,xs)) = [2] x + [2] xs + [2]
> [2] xs + [0]
= lt0(xs',xs)
Following rules are (at-least) weakly oriented:
f(x,Cons(x',xs)) = [2] x + [4]
>= [2] x + [4]
= f[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))
f(x,Nil()) = [2] x + [4]
>= [4]
= Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil()))))
f[Ite][False][Ite](False(),Cons(x,xs),y) = [2] x + [2] xs + [4]
>= [2] xs + [4]
= f(xs,Cons(Cons(Nil(),Nil()),y))
f[Ite][False][Ite](True(),x',Cons(x,xs)) = [2] x' + [4]
>= [2] x' + [4]
= f(x',xs)
g(x,Cons(x',xs)) = [8] x + [8]
>= [8] x + [8]
= g[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))
g(x,Nil()) = [8] x + [8]
>= [4]
= Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil()))))
g[Ite][False][Ite](False(),Cons(x,xs),y) = [8] x + [8] xs + [8]
>= [8] xs + [8]
= g(xs,Cons(Cons(Nil(),Nil()),y))
g[Ite][False][Ite](True(),x',Cons(x,xs)) = [8] x' + [8]
>= [8] x' + [8]
= g(x',xs)
goal(x,y) = [10] x + [15]
>= [10] x + [14]
= Cons(f(x,y),Cons(g(x,y),Nil()))
lt0(x,Nil()) = [0]
>= [0]
= False()
lt0(Nil(),Cons(x',xs)) = [2] x' + [2] xs + [2]
>= [2]
= True()
notEmpty(Cons(x,xs)) = [8] x + [8] xs + [23]
>= [2]
= True()
notEmpty(Nil()) = [15]
>= [0]
= False()
number4(n) = [1] n + [4]
>= [4]
= Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil()))))
** Step 1.b:6: MI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
lt0(x,Nil()) -> False()
lt0(Nil(),Cons(x',xs)) -> True()
- Weak TRS:
f(x,Cons(x',xs)) -> f[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))
f(x,Nil()) -> Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil()))))
f[Ite][False][Ite](False(),Cons(x,xs),y) -> f(xs,Cons(Cons(Nil(),Nil()),y))
f[Ite][False][Ite](True(),x',Cons(x,xs)) -> f(x',xs)
g(x,Cons(x',xs)) -> g[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))
g(x,Nil()) -> Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil()))))
g[Ite][False][Ite](False(),Cons(x,xs),y) -> g(xs,Cons(Cons(Nil(),Nil()),y))
g[Ite][False][Ite](True(),x',Cons(x,xs)) -> g(x',xs)
goal(x,y) -> Cons(f(x,y),Cons(g(x,y),Nil()))
lt0(Cons(x',xs'),Cons(x,xs)) -> lt0(xs',xs)
notEmpty(Cons(x,xs)) -> True()
notEmpty(Nil()) -> False()
number4(n) -> Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil()))))
- Signature:
{f/2,f[Ite][False][Ite]/3,g/2,g[Ite][False][Ite]/3,goal/2,lt0/2,notEmpty/1,number4/1} / {Cons/2,False/0
,Nil/0,True/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {f,f[Ite][False][Ite],g,g[Ite][False][Ite],goal,lt0
,notEmpty,number4} and constructors {Cons,False,Nil,True}
+ Applied Processor:
MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity Nothing)), miDimension = 1, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity Nothing)):
The following argument positions are considered usable:
uargs(Cons) = {1,2},
uargs(f[Ite][False][Ite]) = {1},
uargs(g[Ite][False][Ite]) = {1}
Following symbols are considered usable:
{f,f[Ite][False][Ite],g,g[Ite][False][Ite],goal,lt0,notEmpty,number4}
TcT has computed the following interpretation:
p(Cons) = [1] x_1 + [1] x_2 + [1]
p(False) = [0]
p(Nil) = [0]
p(True) = [1]
p(f) = [8] x_1 + [8]
p(f[Ite][False][Ite]) = [8] x_1 + [8] x_2 + [0]
p(g) = [1] x_1 + [5]
p(g[Ite][False][Ite]) = [1] x_1 + [1] x_2 + [4]
p(goal) = [10] x_1 + [15]
p(lt0) = [1]
p(notEmpty) = [4]
p(number4) = [9]
Following rules are strictly oriented:
lt0(x,Nil()) = [1]
> [0]
= False()
Following rules are (at-least) weakly oriented:
f(x,Cons(x',xs)) = [8] x + [8]
>= [8] x + [8]
= f[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))
f(x,Nil()) = [8] x + [8]
>= [4]
= Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil()))))
f[Ite][False][Ite](False(),Cons(x,xs),y) = [8] x + [8] xs + [8]
>= [8] xs + [8]
= f(xs,Cons(Cons(Nil(),Nil()),y))
f[Ite][False][Ite](True(),x',Cons(x,xs)) = [8] x' + [8]
>= [8] x' + [8]
= f(x',xs)
g(x,Cons(x',xs)) = [1] x + [5]
>= [1] x + [5]
= g[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))
g(x,Nil()) = [1] x + [5]
>= [4]
= Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil()))))
g[Ite][False][Ite](False(),Cons(x,xs),y) = [1] x + [1] xs + [5]
>= [1] xs + [5]
= g(xs,Cons(Cons(Nil(),Nil()),y))
g[Ite][False][Ite](True(),x',Cons(x,xs)) = [1] x' + [5]
>= [1] x' + [5]
= g(x',xs)
goal(x,y) = [10] x + [15]
>= [9] x + [15]
= Cons(f(x,y),Cons(g(x,y),Nil()))
lt0(Cons(x',xs'),Cons(x,xs)) = [1]
>= [1]
= lt0(xs',xs)
lt0(Nil(),Cons(x',xs)) = [1]
>= [1]
= True()
notEmpty(Cons(x,xs)) = [4]
>= [1]
= True()
notEmpty(Nil()) = [4]
>= [0]
= False()
number4(n) = [9]
>= [4]
= Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil()))))
** Step 1.b:7: MI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
lt0(Nil(),Cons(x',xs)) -> True()
- Weak TRS:
f(x,Cons(x',xs)) -> f[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))
f(x,Nil()) -> Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil()))))
f[Ite][False][Ite](False(),Cons(x,xs),y) -> f(xs,Cons(Cons(Nil(),Nil()),y))
f[Ite][False][Ite](True(),x',Cons(x,xs)) -> f(x',xs)
g(x,Cons(x',xs)) -> g[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))
g(x,Nil()) -> Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil()))))
g[Ite][False][Ite](False(),Cons(x,xs),y) -> g(xs,Cons(Cons(Nil(),Nil()),y))
g[Ite][False][Ite](True(),x',Cons(x,xs)) -> g(x',xs)
goal(x,y) -> Cons(f(x,y),Cons(g(x,y),Nil()))
lt0(x,Nil()) -> False()
lt0(Cons(x',xs'),Cons(x,xs)) -> lt0(xs',xs)
notEmpty(Cons(x,xs)) -> True()
notEmpty(Nil()) -> False()
number4(n) -> Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil()))))
- Signature:
{f/2,f[Ite][False][Ite]/3,g/2,g[Ite][False][Ite]/3,goal/2,lt0/2,notEmpty/1,number4/1} / {Cons/2,False/0
,Nil/0,True/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {f,f[Ite][False][Ite],g,g[Ite][False][Ite],goal,lt0
,notEmpty,number4} and constructors {Cons,False,Nil,True}
+ Applied Processor:
MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity Nothing)), miDimension = 1, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity Nothing)):
The following argument positions are considered usable:
uargs(Cons) = {1,2},
uargs(f[Ite][False][Ite]) = {1},
uargs(g[Ite][False][Ite]) = {1}
Following symbols are considered usable:
{f,f[Ite][False][Ite],g,g[Ite][False][Ite],goal,lt0,notEmpty,number4}
TcT has computed the following interpretation:
p(Cons) = [1] x_1 + [1] x_2 + [1]
p(False) = [0]
p(Nil) = [0]
p(True) = [1]
p(f) = [7] x_1 + [2] x_2 + [9]
p(f[Ite][False][Ite]) = [1] x_1 + [7] x_2 + [2] x_3 + [6]
p(g) = [8] x_1 + [2] x_2 + [4]
p(g[Ite][False][Ite]) = [2] x_1 + [8] x_2 + [2] x_3 + [0]
p(goal) = [15] x_1 + [4] x_2 + [15]
p(lt0) = [2] x_2 + [0]
p(notEmpty) = [3] x_1 + [0]
p(number4) = [9]
Following rules are strictly oriented:
lt0(Nil(),Cons(x',xs)) = [2] x' + [2] xs + [2]
> [1]
= True()
Following rules are (at-least) weakly oriented:
f(x,Cons(x',xs)) = [7] x + [2] x' + [2] xs + [11]
>= [7] x + [2] x' + [2] xs + [10]
= f[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))
f(x,Nil()) = [7] x + [9]
>= [4]
= Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil()))))
f[Ite][False][Ite](False(),Cons(x,xs),y) = [7] x + [7] xs + [2] y + [13]
>= [7] xs + [2] y + [13]
= f(xs,Cons(Cons(Nil(),Nil()),y))
f[Ite][False][Ite](True(),x',Cons(x,xs)) = [2] x + [7] x' + [2] xs + [9]
>= [7] x' + [2] xs + [9]
= f(x',xs)
g(x,Cons(x',xs)) = [8] x + [2] x' + [2] xs + [6]
>= [8] x + [2] x' + [2] xs + [6]
= g[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))
g(x,Nil()) = [8] x + [4]
>= [4]
= Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil()))))
g[Ite][False][Ite](False(),Cons(x,xs),y) = [8] x + [8] xs + [2] y + [8]
>= [8] xs + [2] y + [8]
= g(xs,Cons(Cons(Nil(),Nil()),y))
g[Ite][False][Ite](True(),x',Cons(x,xs)) = [2] x + [8] x' + [2] xs + [4]
>= [8] x' + [2] xs + [4]
= g(x',xs)
goal(x,y) = [15] x + [4] y + [15]
>= [15] x + [4] y + [15]
= Cons(f(x,y),Cons(g(x,y),Nil()))
lt0(x,Nil()) = [0]
>= [0]
= False()
lt0(Cons(x',xs'),Cons(x,xs)) = [2] x + [2] xs + [2]
>= [2] xs + [0]
= lt0(xs',xs)
notEmpty(Cons(x,xs)) = [3] x + [3] xs + [3]
>= [1]
= True()
notEmpty(Nil()) = [0]
>= [0]
= False()
number4(n) = [9]
>= [4]
= Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil()))))
** Step 1.b:8: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
f(x,Cons(x',xs)) -> f[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))
f(x,Nil()) -> Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil()))))
f[Ite][False][Ite](False(),Cons(x,xs),y) -> f(xs,Cons(Cons(Nil(),Nil()),y))
f[Ite][False][Ite](True(),x',Cons(x,xs)) -> f(x',xs)
g(x,Cons(x',xs)) -> g[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))
g(x,Nil()) -> Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil()))))
g[Ite][False][Ite](False(),Cons(x,xs),y) -> g(xs,Cons(Cons(Nil(),Nil()),y))
g[Ite][False][Ite](True(),x',Cons(x,xs)) -> g(x',xs)
goal(x,y) -> Cons(f(x,y),Cons(g(x,y),Nil()))
lt0(x,Nil()) -> False()
lt0(Cons(x',xs'),Cons(x,xs)) -> lt0(xs',xs)
lt0(Nil(),Cons(x',xs)) -> True()
notEmpty(Cons(x,xs)) -> True()
notEmpty(Nil()) -> False()
number4(n) -> Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil()))))
- Signature:
{f/2,f[Ite][False][Ite]/3,g/2,g[Ite][False][Ite]/3,goal/2,lt0/2,notEmpty/1,number4/1} / {Cons/2,False/0
,Nil/0,True/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {f,f[Ite][False][Ite],g,g[Ite][False][Ite],goal,lt0
,notEmpty,number4} 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))