* Step 1: WeightGap 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:
runtime complexity wrt. defined symbols {append,hd,ifappend,is_empty,tl} 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) = {2}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(append) = [1] x1 + [2] x2 + [0]
p(cons) = [1] x1 + [1] x2 + [9]
p(false) = [0]
p(hd) = [2] x1 + [0]
p(ifappend) = [2] x2 + [1] x3 + [0]
p(is_empty) = [0]
p(nil) = [0]
p(tl) = [2] x1 + [0]
p(true) = [0]
Following rules are strictly oriented:
hd(cons(x,l)) = [2] l + [2] x + [18]
> [1] x + [0]
= x
tl(cons(x,l)) = [2] l + [2] x + [18]
> [1] l + [0]
= l
Following rules are (at-least) weakly oriented:
append(l1,l2) = [1] l1 + [2] l2 + [0]
>= [1] l1 + [2] l2 + [0]
= ifappend(l1,l2,l1)
ifappend(l1,l2,cons(x,l)) = [1] l + [2] l2 + [1] x + [9]
>= [1] l + [2] l2 + [1] x + [9]
= cons(x,append(l,l2))
ifappend(l1,l2,nil()) = [2] l2 + [0]
>= [1] l2 + [0]
= l2
is_empty(cons(x,l)) = [0]
>= [0]
= false()
is_empty(nil()) = [0]
>= [0]
= true()
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: WeightGap 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))
ifappend(l1,l2,nil()) -> l2
is_empty(cons(x,l)) -> false()
is_empty(nil()) -> true()
- Weak TRS:
hd(cons(x,l)) -> x
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:
runtime complexity wrt. defined symbols {append,hd,ifappend,is_empty,tl} 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) = {2}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(append) = [1] x1 + [2] x2 + [0]
p(cons) = [1] x1 + [1] x2 + [0]
p(false) = [0]
p(hd) = [1] x1 + [0]
p(ifappend) = [2] x2 + [1] x3 + [0]
p(is_empty) = [1]
p(nil) = [0]
p(tl) = [2] x1 + [0]
p(true) = [0]
Following rules are strictly oriented:
is_empty(cons(x,l)) = [1]
> [0]
= false()
is_empty(nil()) = [1]
> [0]
= true()
Following rules are (at-least) weakly oriented:
append(l1,l2) = [1] l1 + [2] l2 + [0]
>= [1] l1 + [2] l2 + [0]
= ifappend(l1,l2,l1)
hd(cons(x,l)) = [1] l + [1] x + [0]
>= [1] x + [0]
= x
ifappend(l1,l2,cons(x,l)) = [1] l + [2] l2 + [1] x + [0]
>= [1] l + [2] l2 + [1] x + [0]
= cons(x,append(l,l2))
ifappend(l1,l2,nil()) = [2] l2 + [0]
>= [1] l2 + [0]
= l2
tl(cons(x,l)) = [2] l + [2] x + [0]
>= [1] l + [0]
= l
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:
append(l1,l2) -> ifappend(l1,l2,l1)
ifappend(l1,l2,cons(x,l)) -> cons(x,append(l,l2))
ifappend(l1,l2,nil()) -> l2
- Weak TRS:
hd(cons(x,l)) -> x
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:
runtime complexity wrt. defined symbols {append,hd,ifappend,is_empty,tl} 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) = {2}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(append) = [3] x_1 + [1] x_2 + [2]
p(cons) = [1] x_1 + [1] x_2 + [1]
p(false) = [1]
p(hd) = [1] x_1 + [2]
p(ifappend) = [1] x_2 + [3] x_3 + [1]
p(is_empty) = [1] x_1 + [8]
p(nil) = [8]
p(tl) = [10] x_1 + [1]
p(true) = [1]
Following rules are strictly oriented:
append(l1,l2) = [3] l1 + [1] l2 + [2]
> [3] l1 + [1] l2 + [1]
= ifappend(l1,l2,l1)
ifappend(l1,l2,cons(x,l)) = [3] l + [1] l2 + [3] x + [4]
> [3] l + [1] l2 + [1] x + [3]
= cons(x,append(l,l2))
ifappend(l1,l2,nil()) = [1] l2 + [25]
> [1] l2 + [0]
= l2
Following rules are (at-least) weakly oriented:
hd(cons(x,l)) = [1] l + [1] x + [3]
>= [1] x + [0]
= x
is_empty(cons(x,l)) = [1] l + [1] x + [9]
>= [1]
= false()
is_empty(nil()) = [16]
>= [1]
= true()
tl(cons(x,l)) = [10] l + [10] x + [11]
>= [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:
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))