WORST_CASE(?,O(n^1))
* Step 1: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
div(0(),s(Y)) -> 0()
div(s(X),s(Y)) -> if(geq(X,Y),s(div(minus(X,Y),s(Y))),0())
geq(X,0()) -> true()
geq(0(),s(Y)) -> false()
geq(s(X),s(Y)) -> geq(X,Y)
if(false(),X,Y) -> Y
if(true(),X,Y) -> X
minus(0(),Y) -> 0()
minus(s(X),s(Y)) -> minus(X,Y)
- Signature:
{div/2,geq/2,if/3,minus/2} / {0/0,false/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {div,geq,if,minus} and constructors {0,false,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(div) = {1},
uargs(if) = {1,2},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [5]
p(div) = [1] x1 + [9] x2 + [0]
p(false) = [0]
p(geq) = [15]
p(if) = [1] x1 + [1] x2 + [2] x3 + [0]
p(minus) = [1] x1 + [1]
p(s) = [1] x1 + [0]
p(true) = [8]
Following rules are strictly oriented:
geq(X,0()) = [15]
> [8]
= true()
geq(0(),s(Y)) = [15]
> [0]
= false()
if(true(),X,Y) = [1] X + [2] Y + [8]
> [1] X + [0]
= X
minus(0(),Y) = [6]
> [5]
= 0()
Following rules are (at-least) weakly oriented:
div(0(),s(Y)) = [9] Y + [5]
>= [5]
= 0()
div(s(X),s(Y)) = [1] X + [9] Y + [0]
>= [1] X + [9] Y + [26]
= if(geq(X,Y),s(div(minus(X,Y),s(Y))),0())
geq(s(X),s(Y)) = [15]
>= [15]
= geq(X,Y)
if(false(),X,Y) = [1] X + [2] Y + [0]
>= [1] Y + [0]
= Y
minus(s(X),s(Y)) = [1] X + [1]
>= [1] X + [1]
= minus(X,Y)
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:
div(0(),s(Y)) -> 0()
div(s(X),s(Y)) -> if(geq(X,Y),s(div(minus(X,Y),s(Y))),0())
geq(s(X),s(Y)) -> geq(X,Y)
if(false(),X,Y) -> Y
minus(s(X),s(Y)) -> minus(X,Y)
- Weak TRS:
geq(X,0()) -> true()
geq(0(),s(Y)) -> false()
if(true(),X,Y) -> X
minus(0(),Y) -> 0()
- Signature:
{div/2,geq/2,if/3,minus/2} / {0/0,false/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {div,geq,if,minus} and constructors {0,false,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(div) = {1},
uargs(if) = {1,2},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [2]
p(div) = [1] x1 + [2] x2 + [10]
p(false) = [1]
p(geq) = [3]
p(if) = [1] x1 + [1] x2 + [6] x3 + [0]
p(minus) = [5]
p(s) = [1] x1 + [0]
p(true) = [0]
Following rules are strictly oriented:
div(0(),s(Y)) = [2] Y + [12]
> [2]
= 0()
if(false(),X,Y) = [1] X + [6] Y + [1]
> [1] Y + [0]
= Y
Following rules are (at-least) weakly oriented:
div(s(X),s(Y)) = [1] X + [2] Y + [10]
>= [2] Y + [30]
= if(geq(X,Y),s(div(minus(X,Y),s(Y))),0())
geq(X,0()) = [3]
>= [0]
= true()
geq(0(),s(Y)) = [3]
>= [1]
= false()
geq(s(X),s(Y)) = [3]
>= [3]
= geq(X,Y)
if(true(),X,Y) = [1] X + [6] Y + [0]
>= [1] X + [0]
= X
minus(0(),Y) = [5]
>= [2]
= 0()
minus(s(X),s(Y)) = [5]
>= [5]
= minus(X,Y)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
div(s(X),s(Y)) -> if(geq(X,Y),s(div(minus(X,Y),s(Y))),0())
geq(s(X),s(Y)) -> geq(X,Y)
minus(s(X),s(Y)) -> minus(X,Y)
- Weak TRS:
div(0(),s(Y)) -> 0()
geq(X,0()) -> true()
geq(0(),s(Y)) -> false()
if(false(),X,Y) -> Y
if(true(),X,Y) -> X
minus(0(),Y) -> 0()
- Signature:
{div/2,geq/2,if/3,minus/2} / {0/0,false/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {div,geq,if,minus} and constructors {0,false,s,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(div) = {1},
uargs(if) = {1,2},
uargs(s) = {1}
Following symbols are considered usable:
{div,geq,if,minus}
TcT has computed the following interpretation:
p(0) = [0]
p(div) = [8] x1 + [4]
p(false) = [0]
p(geq) = [1]
p(if) = [1] x1 + [1] x2 + [4] x3 + [1]
p(minus) = [2]
p(s) = [1] x1 + [3]
p(true) = [1]
Following rules are strictly oriented:
div(s(X),s(Y)) = [8] X + [28]
> [25]
= if(geq(X,Y),s(div(minus(X,Y),s(Y))),0())
Following rules are (at-least) weakly oriented:
div(0(),s(Y)) = [4]
>= [0]
= 0()
geq(X,0()) = [1]
>= [1]
= true()
geq(0(),s(Y)) = [1]
>= [0]
= false()
geq(s(X),s(Y)) = [1]
>= [1]
= geq(X,Y)
if(false(),X,Y) = [1] X + [4] Y + [1]
>= [1] Y + [0]
= Y
if(true(),X,Y) = [1] X + [4] Y + [2]
>= [1] X + [0]
= X
minus(0(),Y) = [2]
>= [0]
= 0()
minus(s(X),s(Y)) = [2]
>= [2]
= minus(X,Y)
* Step 4: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
geq(s(X),s(Y)) -> geq(X,Y)
minus(s(X),s(Y)) -> minus(X,Y)
- Weak TRS:
div(0(),s(Y)) -> 0()
div(s(X),s(Y)) -> if(geq(X,Y),s(div(minus(X,Y),s(Y))),0())
geq(X,0()) -> true()
geq(0(),s(Y)) -> false()
if(false(),X,Y) -> Y
if(true(),X,Y) -> X
minus(0(),Y) -> 0()
- Signature:
{div/2,geq/2,if/3,minus/2} / {0/0,false/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {div,geq,if,minus} and constructors {0,false,s,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(div) = {1},
uargs(if) = {1,2},
uargs(s) = {1}
Following symbols are considered usable:
{div,geq,if,minus}
TcT has computed the following interpretation:
p(0) = [0]
p(div) = [12] x1 + [1]
p(false) = [0]
p(geq) = [4] x1 + [0]
p(if) = [2] x1 + [8] x2 + [2] x3 + [1]
p(minus) = [0]
p(s) = [1] x1 + [2]
p(true) = [0]
Following rules are strictly oriented:
geq(s(X),s(Y)) = [4] X + [8]
> [4] X + [0]
= geq(X,Y)
Following rules are (at-least) weakly oriented:
div(0(),s(Y)) = [1]
>= [0]
= 0()
div(s(X),s(Y)) = [12] X + [25]
>= [8] X + [25]
= if(geq(X,Y),s(div(minus(X,Y),s(Y))),0())
geq(X,0()) = [4] X + [0]
>= [0]
= true()
geq(0(),s(Y)) = [0]
>= [0]
= false()
if(false(),X,Y) = [8] X + [2] Y + [1]
>= [1] Y + [0]
= Y
if(true(),X,Y) = [8] X + [2] Y + [1]
>= [1] X + [0]
= X
minus(0(),Y) = [0]
>= [0]
= 0()
minus(s(X),s(Y)) = [0]
>= [0]
= minus(X,Y)
* Step 5: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
minus(s(X),s(Y)) -> minus(X,Y)
- Weak TRS:
div(0(),s(Y)) -> 0()
div(s(X),s(Y)) -> if(geq(X,Y),s(div(minus(X,Y),s(Y))),0())
geq(X,0()) -> true()
geq(0(),s(Y)) -> false()
geq(s(X),s(Y)) -> geq(X,Y)
if(false(),X,Y) -> Y
if(true(),X,Y) -> X
minus(0(),Y) -> 0()
- Signature:
{div/2,geq/2,if/3,minus/2} / {0/0,false/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {div,geq,if,minus} and constructors {0,false,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(div) = {1},
uargs(if) = {1,2},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(div) = [1] x1 + [0]
p(false) = [0]
p(geq) = [0]
p(if) = [1] x1 + [1] x2 + [8] x3 + [0]
p(minus) = [1] x1 + [0]
p(s) = [1] x1 + [8]
p(true) = [0]
Following rules are strictly oriented:
minus(s(X),s(Y)) = [1] X + [8]
> [1] X + [0]
= minus(X,Y)
Following rules are (at-least) weakly oriented:
div(0(),s(Y)) = [0]
>= [0]
= 0()
div(s(X),s(Y)) = [1] X + [8]
>= [1] X + [8]
= if(geq(X,Y),s(div(minus(X,Y),s(Y))),0())
geq(X,0()) = [0]
>= [0]
= true()
geq(0(),s(Y)) = [0]
>= [0]
= false()
geq(s(X),s(Y)) = [0]
>= [0]
= geq(X,Y)
if(false(),X,Y) = [1] X + [8] Y + [0]
>= [1] Y + [0]
= Y
if(true(),X,Y) = [1] X + [8] Y + [0]
>= [1] X + [0]
= X
minus(0(),Y) = [0]
>= [0]
= 0()
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 6: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
div(0(),s(Y)) -> 0()
div(s(X),s(Y)) -> if(geq(X,Y),s(div(minus(X,Y),s(Y))),0())
geq(X,0()) -> true()
geq(0(),s(Y)) -> false()
geq(s(X),s(Y)) -> geq(X,Y)
if(false(),X,Y) -> Y
if(true(),X,Y) -> X
minus(0(),Y) -> 0()
minus(s(X),s(Y)) -> minus(X,Y)
- Signature:
{div/2,geq/2,if/3,minus/2} / {0/0,false/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {div,geq,if,minus} and constructors {0,false,s,true}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(?,O(n^1))