WORST_CASE(?,O(n^1))
* Step 1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
append(l1,l2) -> ifappend(l1,l2,l1)
hd(cons(x,l)) -> x
ifappend(l1,l2,cons(x,l)) -> cons(x,append(l,l2))
ifappend(l1,l2,nil()) -> l2
is_empty(cons(x,l)) -> false()
is_empty(nil()) -> true()
tl(cons(x,l)) -> l
- Signature:
{append/2,hd/1,ifappend/3,is_empty/1,tl/1} / {cons/2,false/0,nil/0,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {append,hd,ifappend,is_empty,tl} and constructors {cons
,false,nil,true}
+ Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(cons) = {2}
Following symbols are considered usable:
{append,hd,ifappend,is_empty,tl}
TcT has computed the following interpretation:
p(append) = [2] x1 + [1] x2 + [0]
p(cons) = [1] x1 + [1] x2 + [0]
p(false) = [5]
p(hd) = [1] x1 + [4]
p(ifappend) = [1] x2 + [2] x3 + [0]
p(is_empty) = [2] x1 + [9]
p(nil) = [5]
p(tl) = [4] x1 + [5]
p(true) = [1]
Following rules are strictly oriented:
hd(cons(x,l)) = [1] l + [1] x + [4]
> [1] x + [0]
= x
ifappend(l1,l2,nil()) = [1] l2 + [10]
> [1] l2 + [0]
= l2
is_empty(cons(x,l)) = [2] l + [2] x + [9]
> [5]
= false()
is_empty(nil()) = [19]
> [1]
= true()
tl(cons(x,l)) = [4] l + [4] x + [5]
> [1] l + [0]
= l
Following rules are (at-least) weakly oriented:
append(l1,l2) = [2] l1 + [1] l2 + [0]
>= [2] l1 + [1] l2 + [0]
= ifappend(l1,l2,l1)
ifappend(l1,l2,cons(x,l)) = [2] l + [1] l2 + [2] x + [0]
>= [2] l + [1] l2 + [1] x + [0]
= cons(x,append(l,l2))
* Step 2: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
append(l1,l2) -> ifappend(l1,l2,l1)
ifappend(l1,l2,cons(x,l)) -> cons(x,append(l,l2))
- Weak TRS:
hd(cons(x,l)) -> x
ifappend(l1,l2,nil()) -> l2
is_empty(cons(x,l)) -> false()
is_empty(nil()) -> true()
tl(cons(x,l)) -> l
- Signature:
{append/2,hd/1,ifappend/3,is_empty/1,tl/1} / {cons/2,false/0,nil/0,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {append,hd,ifappend,is_empty,tl} and constructors {cons
,false,nil,true}
+ Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(cons) = {2}
Following symbols are considered usable:
{append,hd,ifappend,is_empty,tl}
TcT has computed the following interpretation:
p(append) = [2] x1 + [2] x2 + [1]
p(cons) = [1] x1 + [1] x2 + [8]
p(false) = [0]
p(hd) = [1] x1 + [0]
p(ifappend) = [2] x2 + [2] x3 + [1]
p(is_empty) = [0]
p(nil) = [5]
p(tl) = [1] x1 + [8]
p(true) = [0]
Following rules are strictly oriented:
ifappend(l1,l2,cons(x,l)) = [2] l + [2] l2 + [2] x + [17]
> [2] l + [2] l2 + [1] x + [9]
= cons(x,append(l,l2))
Following rules are (at-least) weakly oriented:
append(l1,l2) = [2] l1 + [2] l2 + [1]
>= [2] l1 + [2] l2 + [1]
= ifappend(l1,l2,l1)
hd(cons(x,l)) = [1] l + [1] x + [8]
>= [1] x + [0]
= x
ifappend(l1,l2,nil()) = [2] l2 + [11]
>= [1] l2 + [0]
= l2
is_empty(cons(x,l)) = [0]
>= [0]
= false()
is_empty(nil()) = [0]
>= [0]
= true()
tl(cons(x,l)) = [1] l + [1] x + [16]
>= [1] l + [0]
= l
* Step 3: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
append(l1,l2) -> ifappend(l1,l2,l1)
- Weak TRS:
hd(cons(x,l)) -> x
ifappend(l1,l2,cons(x,l)) -> cons(x,append(l,l2))
ifappend(l1,l2,nil()) -> l2
is_empty(cons(x,l)) -> false()
is_empty(nil()) -> true()
tl(cons(x,l)) -> l
- Signature:
{append/2,hd/1,ifappend/3,is_empty/1,tl/1} / {cons/2,false/0,nil/0,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {append,hd,ifappend,is_empty,tl} and constructors {cons
,false,nil,true}
+ Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(cons) = {2}
Following symbols are considered usable:
{append,hd,ifappend,is_empty,tl}
TcT has computed the following interpretation:
p(append) = [8] x1 + [1] x2 + [7]
p(cons) = [1] x1 + [1] x2 + [1]
p(false) = [0]
p(hd) = [14] x1 + [0]
p(ifappend) = [1] x2 + [8] x3 + [5]
p(is_empty) = [1] x1 + [15]
p(nil) = [1]
p(tl) = [5] x1 + [7]
p(true) = [0]
Following rules are strictly oriented:
append(l1,l2) = [8] l1 + [1] l2 + [7]
> [8] l1 + [1] l2 + [5]
= ifappend(l1,l2,l1)
Following rules are (at-least) weakly oriented:
hd(cons(x,l)) = [14] l + [14] x + [14]
>= [1] x + [0]
= x
ifappend(l1,l2,cons(x,l)) = [8] l + [1] l2 + [8] x + [13]
>= [8] l + [1] l2 + [1] x + [8]
= cons(x,append(l,l2))
ifappend(l1,l2,nil()) = [1] l2 + [13]
>= [1] l2 + [0]
= l2
is_empty(cons(x,l)) = [1] l + [1] x + [16]
>= [0]
= false()
is_empty(nil()) = [16]
>= [0]
= true()
tl(cons(x,l)) = [5] l + [5] x + [12]
>= [1] l + [0]
= l
* Step 4: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
append(l1,l2) -> ifappend(l1,l2,l1)
hd(cons(x,l)) -> x
ifappend(l1,l2,cons(x,l)) -> cons(x,append(l,l2))
ifappend(l1,l2,nil()) -> l2
is_empty(cons(x,l)) -> false()
is_empty(nil()) -> true()
tl(cons(x,l)) -> l
- Signature:
{append/2,hd/1,ifappend/3,is_empty/1,tl/1} / {cons/2,false/0,nil/0,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {append,hd,ifappend,is_empty,tl} and constructors {cons
,false,nil,true}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(?,O(n^1))