* Step 1: Sum WORST_CASE(Omega(n^1),O(n^1))
+ Considered Problem:
- Strict TRS:
div_active(x,y) -> div(x,y)
div_active(0(),s(y)) -> 0()
div_active(s(x),s(y)) -> if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0())
ge_active(x,y) -> ge(x,y)
ge_active(x,0()) -> true()
ge_active(0(),s(y)) -> false()
ge_active(s(x),s(y)) -> ge_active(x,y)
if_active(x,y,z) -> if(x,y,z)
if_active(false(),x,y) -> mark(y)
if_active(true(),x,y) -> mark(x)
mark(0()) -> 0()
mark(div(x,y)) -> div_active(mark(x),y)
mark(ge(x,y)) -> ge_active(x,y)
mark(if(x,y,z)) -> if_active(mark(x),y,z)
mark(minus(x,y)) -> minus_active(x,y)
mark(s(x)) -> s(mark(x))
minus_active(x,y) -> minus(x,y)
minus_active(0(),y) -> 0()
minus_active(s(x),s(y)) -> minus_active(x,y)
- Signature:
{div_active/2,ge_active/2,if_active/3,mark/1,minus_active/2} / {0/0,div/2,false/0,ge/2,if/3,minus/2,s/1
,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {div_active,ge_active,if_active,mark
,minus_active} and constructors {0,div,false,ge,if,minus,s,true}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
div_active(x,y) -> div(x,y)
div_active(0(),s(y)) -> 0()
div_active(s(x),s(y)) -> if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0())
ge_active(x,y) -> ge(x,y)
ge_active(x,0()) -> true()
ge_active(0(),s(y)) -> false()
ge_active(s(x),s(y)) -> ge_active(x,y)
if_active(x,y,z) -> if(x,y,z)
if_active(false(),x,y) -> mark(y)
if_active(true(),x,y) -> mark(x)
mark(0()) -> 0()
mark(div(x,y)) -> div_active(mark(x),y)
mark(ge(x,y)) -> ge_active(x,y)
mark(if(x,y,z)) -> if_active(mark(x),y,z)
mark(minus(x,y)) -> minus_active(x,y)
mark(s(x)) -> s(mark(x))
minus_active(x,y) -> minus(x,y)
minus_active(0(),y) -> 0()
minus_active(s(x),s(y)) -> minus_active(x,y)
- Signature:
{div_active/2,ge_active/2,if_active/3,mark/1,minus_active/2} / {0/0,div/2,false/0,ge/2,if/3,minus/2,s/1
,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {div_active,ge_active,if_active,mark
,minus_active} and constructors {0,div,false,ge,if,minus,s,true}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
ge_active(x,y){x -> s(x),y -> s(y)} =
ge_active(s(x),s(y)) ->^+ ge_active(x,y)
= C[ge_active(x,y) = ge_active(x,y){}]
** Step 1.b:1: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
div_active(x,y) -> div(x,y)
div_active(0(),s(y)) -> 0()
div_active(s(x),s(y)) -> if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0())
ge_active(x,y) -> ge(x,y)
ge_active(x,0()) -> true()
ge_active(0(),s(y)) -> false()
ge_active(s(x),s(y)) -> ge_active(x,y)
if_active(x,y,z) -> if(x,y,z)
if_active(false(),x,y) -> mark(y)
if_active(true(),x,y) -> mark(x)
mark(0()) -> 0()
mark(div(x,y)) -> div_active(mark(x),y)
mark(ge(x,y)) -> ge_active(x,y)
mark(if(x,y,z)) -> if_active(mark(x),y,z)
mark(minus(x,y)) -> minus_active(x,y)
mark(s(x)) -> s(mark(x))
minus_active(x,y) -> minus(x,y)
minus_active(0(),y) -> 0()
minus_active(s(x),s(y)) -> minus_active(x,y)
- Signature:
{div_active/2,ge_active/2,if_active/3,mark/1,minus_active/2} / {0/0,div/2,false/0,ge/2,if/3,minus/2,s/1
,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {div_active,ge_active,if_active,mark
,minus_active} and constructors {0,div,false,ge,if,minus,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_active) = {1},
uargs(if_active) = {1},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [4]
p(div) = [1] x1 + [0]
p(div_active) = [1] x1 + [0]
p(false) = [0]
p(ge) = [0]
p(ge_active) = [0]
p(if) = [1] x1 + [1] x2 + [1] x3 + [0]
p(if_active) = [1] x1 + [1] x2 + [1] x3 + [0]
p(mark) = [1] x1 + [0]
p(minus) = [1] x1 + [3]
p(minus_active) = [1] x1 + [0]
p(s) = [1] x1 + [1]
p(true) = [0]
Following rules are strictly oriented:
mark(minus(x,y)) = [1] x + [3]
> [1] x + [0]
= minus_active(x,y)
minus_active(s(x),s(y)) = [1] x + [1]
> [1] x + [0]
= minus_active(x,y)
Following rules are (at-least) weakly oriented:
div_active(x,y) = [1] x + [0]
>= [1] x + [0]
= div(x,y)
div_active(0(),s(y)) = [4]
>= [4]
= 0()
div_active(s(x),s(y)) = [1] x + [1]
>= [1] x + [8]
= if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0())
ge_active(x,y) = [0]
>= [0]
= ge(x,y)
ge_active(x,0()) = [0]
>= [0]
= true()
ge_active(0(),s(y)) = [0]
>= [0]
= false()
ge_active(s(x),s(y)) = [0]
>= [0]
= ge_active(x,y)
if_active(x,y,z) = [1] x + [1] y + [1] z + [0]
>= [1] x + [1] y + [1] z + [0]
= if(x,y,z)
if_active(false(),x,y) = [1] x + [1] y + [0]
>= [1] y + [0]
= mark(y)
if_active(true(),x,y) = [1] x + [1] y + [0]
>= [1] x + [0]
= mark(x)
mark(0()) = [4]
>= [4]
= 0()
mark(div(x,y)) = [1] x + [0]
>= [1] x + [0]
= div_active(mark(x),y)
mark(ge(x,y)) = [0]
>= [0]
= ge_active(x,y)
mark(if(x,y,z)) = [1] x + [1] y + [1] z + [0]
>= [1] x + [1] y + [1] z + [0]
= if_active(mark(x),y,z)
mark(s(x)) = [1] x + [1]
>= [1] x + [1]
= s(mark(x))
minus_active(x,y) = [1] x + [0]
>= [1] x + [3]
= minus(x,y)
minus_active(0(),y) = [4]
>= [4]
= 0()
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:
div_active(x,y) -> div(x,y)
div_active(0(),s(y)) -> 0()
div_active(s(x),s(y)) -> if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0())
ge_active(x,y) -> ge(x,y)
ge_active(x,0()) -> true()
ge_active(0(),s(y)) -> false()
ge_active(s(x),s(y)) -> ge_active(x,y)
if_active(x,y,z) -> if(x,y,z)
if_active(false(),x,y) -> mark(y)
if_active(true(),x,y) -> mark(x)
mark(0()) -> 0()
mark(div(x,y)) -> div_active(mark(x),y)
mark(ge(x,y)) -> ge_active(x,y)
mark(if(x,y,z)) -> if_active(mark(x),y,z)
mark(s(x)) -> s(mark(x))
minus_active(x,y) -> minus(x,y)
minus_active(0(),y) -> 0()
- Weak TRS:
mark(minus(x,y)) -> minus_active(x,y)
minus_active(s(x),s(y)) -> minus_active(x,y)
- Signature:
{div_active/2,ge_active/2,if_active/3,mark/1,minus_active/2} / {0/0,div/2,false/0,ge/2,if/3,minus/2,s/1
,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {div_active,ge_active,if_active,mark
,minus_active} and constructors {0,div,false,ge,if,minus,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_active) = {1},
uargs(if_active) = {1},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [4]
p(div) = [1] x1 + [0]
p(div_active) = [1] x1 + [0]
p(false) = [0]
p(ge) = [1] x1 + [0]
p(ge_active) = [1] x1 + [0]
p(if) = [1] x1 + [1] x2 + [1] x3 + [0]
p(if_active) = [1] x1 + [1] x2 + [1] x3 + [0]
p(mark) = [1] x1 + [7]
p(minus) = [0]
p(minus_active) = [7]
p(s) = [1] x1 + [5]
p(true) = [0]
Following rules are strictly oriented:
ge_active(0(),s(y)) = [4]
> [0]
= false()
ge_active(s(x),s(y)) = [1] x + [5]
> [1] x + [0]
= ge_active(x,y)
mark(0()) = [11]
> [4]
= 0()
mark(ge(x,y)) = [1] x + [7]
> [1] x + [0]
= ge_active(x,y)
minus_active(x,y) = [7]
> [0]
= minus(x,y)
minus_active(0(),y) = [7]
> [4]
= 0()
Following rules are (at-least) weakly oriented:
div_active(x,y) = [1] x + [0]
>= [1] x + [0]
= div(x,y)
div_active(0(),s(y)) = [4]
>= [4]
= 0()
div_active(s(x),s(y)) = [1] x + [5]
>= [1] x + [9]
= if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0())
ge_active(x,y) = [1] x + [0]
>= [1] x + [0]
= ge(x,y)
ge_active(x,0()) = [1] x + [0]
>= [0]
= true()
if_active(x,y,z) = [1] x + [1] y + [1] z + [0]
>= [1] x + [1] y + [1] z + [0]
= if(x,y,z)
if_active(false(),x,y) = [1] x + [1] y + [0]
>= [1] y + [7]
= mark(y)
if_active(true(),x,y) = [1] x + [1] y + [0]
>= [1] x + [7]
= mark(x)
mark(div(x,y)) = [1] x + [7]
>= [1] x + [7]
= div_active(mark(x),y)
mark(if(x,y,z)) = [1] x + [1] y + [1] z + [7]
>= [1] x + [1] y + [1] z + [7]
= if_active(mark(x),y,z)
mark(minus(x,y)) = [7]
>= [7]
= minus_active(x,y)
mark(s(x)) = [1] x + [12]
>= [1] x + [12]
= s(mark(x))
minus_active(s(x),s(y)) = [7]
>= [7]
= minus_active(x,y)
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:
div_active(x,y) -> div(x,y)
div_active(0(),s(y)) -> 0()
div_active(s(x),s(y)) -> if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0())
ge_active(x,y) -> ge(x,y)
ge_active(x,0()) -> true()
if_active(x,y,z) -> if(x,y,z)
if_active(false(),x,y) -> mark(y)
if_active(true(),x,y) -> mark(x)
mark(div(x,y)) -> div_active(mark(x),y)
mark(if(x,y,z)) -> if_active(mark(x),y,z)
mark(s(x)) -> s(mark(x))
- Weak TRS:
ge_active(0(),s(y)) -> false()
ge_active(s(x),s(y)) -> ge_active(x,y)
mark(0()) -> 0()
mark(ge(x,y)) -> ge_active(x,y)
mark(minus(x,y)) -> minus_active(x,y)
minus_active(x,y) -> minus(x,y)
minus_active(0(),y) -> 0()
minus_active(s(x),s(y)) -> minus_active(x,y)
- Signature:
{div_active/2,ge_active/2,if_active/3,mark/1,minus_active/2} / {0/0,div/2,false/0,ge/2,if/3,minus/2,s/1
,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {div_active,ge_active,if_active,mark
,minus_active} and constructors {0,div,false,ge,if,minus,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_active) = {1},
uargs(if_active) = {1},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(div) = [0]
p(div_active) = [1] x1 + [5]
p(false) = [0]
p(ge) = [0]
p(ge_active) = [0]
p(if) = [1] x1 + [0]
p(if_active) = [1] x1 + [0]
p(mark) = [0]
p(minus) = [0]
p(minus_active) = [0]
p(s) = [1] x1 + [0]
p(true) = [0]
Following rules are strictly oriented:
div_active(x,y) = [1] x + [5]
> [0]
= div(x,y)
div_active(0(),s(y)) = [5]
> [0]
= 0()
div_active(s(x),s(y)) = [1] x + [5]
> [0]
= if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0())
Following rules are (at-least) weakly oriented:
ge_active(x,y) = [0]
>= [0]
= ge(x,y)
ge_active(x,0()) = [0]
>= [0]
= true()
ge_active(0(),s(y)) = [0]
>= [0]
= false()
ge_active(s(x),s(y)) = [0]
>= [0]
= ge_active(x,y)
if_active(x,y,z) = [1] x + [0]
>= [1] x + [0]
= if(x,y,z)
if_active(false(),x,y) = [0]
>= [0]
= mark(y)
if_active(true(),x,y) = [0]
>= [0]
= mark(x)
mark(0()) = [0]
>= [0]
= 0()
mark(div(x,y)) = [0]
>= [5]
= div_active(mark(x),y)
mark(ge(x,y)) = [0]
>= [0]
= ge_active(x,y)
mark(if(x,y,z)) = [0]
>= [0]
= if_active(mark(x),y,z)
mark(minus(x,y)) = [0]
>= [0]
= minus_active(x,y)
mark(s(x)) = [0]
>= [0]
= s(mark(x))
minus_active(x,y) = [0]
>= [0]
= minus(x,y)
minus_active(0(),y) = [0]
>= [0]
= 0()
minus_active(s(x),s(y)) = [0]
>= [0]
= minus_active(x,y)
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:
ge_active(x,y) -> ge(x,y)
ge_active(x,0()) -> true()
if_active(x,y,z) -> if(x,y,z)
if_active(false(),x,y) -> mark(y)
if_active(true(),x,y) -> mark(x)
mark(div(x,y)) -> div_active(mark(x),y)
mark(if(x,y,z)) -> if_active(mark(x),y,z)
mark(s(x)) -> s(mark(x))
- Weak TRS:
div_active(x,y) -> div(x,y)
div_active(0(),s(y)) -> 0()
div_active(s(x),s(y)) -> if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0())
ge_active(0(),s(y)) -> false()
ge_active(s(x),s(y)) -> ge_active(x,y)
mark(0()) -> 0()
mark(ge(x,y)) -> ge_active(x,y)
mark(minus(x,y)) -> minus_active(x,y)
minus_active(x,y) -> minus(x,y)
minus_active(0(),y) -> 0()
minus_active(s(x),s(y)) -> minus_active(x,y)
- Signature:
{div_active/2,ge_active/2,if_active/3,mark/1,minus_active/2} / {0/0,div/2,false/0,ge/2,if/3,minus/2,s/1
,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {div_active,ge_active,if_active,mark
,minus_active} and constructors {0,div,false,ge,if,minus,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_active) = {1},
uargs(if_active) = {1},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(div) = [1] x1 + [1] x2 + [1]
p(div_active) = [1] x1 + [1] x2 + [2]
p(false) = [0]
p(ge) = [1]
p(ge_active) = [1]
p(if) = [1] x1 + [1] x2 + [1] x3 + [0]
p(if_active) = [1] x1 + [1] x2 + [1] x3 + [0]
p(mark) = [1] x1 + [0]
p(minus) = [0]
p(minus_active) = [0]
p(s) = [1] x1 + [2]
p(true) = [0]
Following rules are strictly oriented:
ge_active(x,0()) = [1]
> [0]
= true()
Following rules are (at-least) weakly oriented:
div_active(x,y) = [1] x + [1] y + [2]
>= [1] x + [1] y + [1]
= div(x,y)
div_active(0(),s(y)) = [1] y + [4]
>= [0]
= 0()
div_active(s(x),s(y)) = [1] x + [1] y + [6]
>= [1] y + [6]
= if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0())
ge_active(x,y) = [1]
>= [1]
= ge(x,y)
ge_active(0(),s(y)) = [1]
>= [0]
= false()
ge_active(s(x),s(y)) = [1]
>= [1]
= ge_active(x,y)
if_active(x,y,z) = [1] x + [1] y + [1] z + [0]
>= [1] x + [1] y + [1] z + [0]
= if(x,y,z)
if_active(false(),x,y) = [1] x + [1] y + [0]
>= [1] y + [0]
= mark(y)
if_active(true(),x,y) = [1] x + [1] y + [0]
>= [1] x + [0]
= mark(x)
mark(0()) = [0]
>= [0]
= 0()
mark(div(x,y)) = [1] x + [1] y + [1]
>= [1] x + [1] y + [2]
= div_active(mark(x),y)
mark(ge(x,y)) = [1]
>= [1]
= ge_active(x,y)
mark(if(x,y,z)) = [1] x + [1] y + [1] z + [0]
>= [1] x + [1] y + [1] z + [0]
= if_active(mark(x),y,z)
mark(minus(x,y)) = [0]
>= [0]
= minus_active(x,y)
mark(s(x)) = [1] x + [2]
>= [1] x + [2]
= s(mark(x))
minus_active(x,y) = [0]
>= [0]
= minus(x,y)
minus_active(0(),y) = [0]
>= [0]
= 0()
minus_active(s(x),s(y)) = [0]
>= [0]
= minus_active(x,y)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:5: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
ge_active(x,y) -> ge(x,y)
if_active(x,y,z) -> if(x,y,z)
if_active(false(),x,y) -> mark(y)
if_active(true(),x,y) -> mark(x)
mark(div(x,y)) -> div_active(mark(x),y)
mark(if(x,y,z)) -> if_active(mark(x),y,z)
mark(s(x)) -> s(mark(x))
- Weak TRS:
div_active(x,y) -> div(x,y)
div_active(0(),s(y)) -> 0()
div_active(s(x),s(y)) -> if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0())
ge_active(x,0()) -> true()
ge_active(0(),s(y)) -> false()
ge_active(s(x),s(y)) -> ge_active(x,y)
mark(0()) -> 0()
mark(ge(x,y)) -> ge_active(x,y)
mark(minus(x,y)) -> minus_active(x,y)
minus_active(x,y) -> minus(x,y)
minus_active(0(),y) -> 0()
minus_active(s(x),s(y)) -> minus_active(x,y)
- Signature:
{div_active/2,ge_active/2,if_active/3,mark/1,minus_active/2} / {0/0,div/2,false/0,ge/2,if/3,minus/2,s/1
,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {div_active,ge_active,if_active,mark
,minus_active} and constructors {0,div,false,ge,if,minus,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_active) = {1},
uargs(if_active) = {1},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [1]
p(div) = [1] x1 + [1]
p(div_active) = [1] x1 + [4]
p(false) = [0]
p(ge) = [1]
p(ge_active) = [1]
p(if) = [1] x1 + [1] x2 + [1] x3 + [1]
p(if_active) = [1] x1 + [1] x2 + [1] x3 + [0]
p(mark) = [1] x1 + [0]
p(minus) = [1] x1 + [0]
p(minus_active) = [1] x1 + [0]
p(s) = [1] x1 + [0]
p(true) = [0]
Following rules are strictly oriented:
mark(if(x,y,z)) = [1] x + [1] y + [1] z + [1]
> [1] x + [1] y + [1] z + [0]
= if_active(mark(x),y,z)
Following rules are (at-least) weakly oriented:
div_active(x,y) = [1] x + [4]
>= [1] x + [1]
= div(x,y)
div_active(0(),s(y)) = [5]
>= [1]
= 0()
div_active(s(x),s(y)) = [1] x + [4]
>= [1] x + [3]
= if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0())
ge_active(x,y) = [1]
>= [1]
= ge(x,y)
ge_active(x,0()) = [1]
>= [0]
= true()
ge_active(0(),s(y)) = [1]
>= [0]
= false()
ge_active(s(x),s(y)) = [1]
>= [1]
= ge_active(x,y)
if_active(x,y,z) = [1] x + [1] y + [1] z + [0]
>= [1] x + [1] y + [1] z + [1]
= if(x,y,z)
if_active(false(),x,y) = [1] x + [1] y + [0]
>= [1] y + [0]
= mark(y)
if_active(true(),x,y) = [1] x + [1] y + [0]
>= [1] x + [0]
= mark(x)
mark(0()) = [1]
>= [1]
= 0()
mark(div(x,y)) = [1] x + [1]
>= [1] x + [4]
= div_active(mark(x),y)
mark(ge(x,y)) = [1]
>= [1]
= ge_active(x,y)
mark(minus(x,y)) = [1] x + [0]
>= [1] x + [0]
= minus_active(x,y)
mark(s(x)) = [1] x + [0]
>= [1] x + [0]
= s(mark(x))
minus_active(x,y) = [1] x + [0]
>= [1] x + [0]
= minus(x,y)
minus_active(0(),y) = [1]
>= [1]
= 0()
minus_active(s(x),s(y)) = [1] x + [0]
>= [1] x + [0]
= minus_active(x,y)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:6: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
ge_active(x,y) -> ge(x,y)
if_active(x,y,z) -> if(x,y,z)
if_active(false(),x,y) -> mark(y)
if_active(true(),x,y) -> mark(x)
mark(div(x,y)) -> div_active(mark(x),y)
mark(s(x)) -> s(mark(x))
- Weak TRS:
div_active(x,y) -> div(x,y)
div_active(0(),s(y)) -> 0()
div_active(s(x),s(y)) -> if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0())
ge_active(x,0()) -> true()
ge_active(0(),s(y)) -> false()
ge_active(s(x),s(y)) -> ge_active(x,y)
mark(0()) -> 0()
mark(ge(x,y)) -> ge_active(x,y)
mark(if(x,y,z)) -> if_active(mark(x),y,z)
mark(minus(x,y)) -> minus_active(x,y)
minus_active(x,y) -> minus(x,y)
minus_active(0(),y) -> 0()
minus_active(s(x),s(y)) -> minus_active(x,y)
- Signature:
{div_active/2,ge_active/2,if_active/3,mark/1,minus_active/2} / {0/0,div/2,false/0,ge/2,if/3,minus/2,s/1
,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {div_active,ge_active,if_active,mark
,minus_active} and constructors {0,div,false,ge,if,minus,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_active) = {1},
uargs(if_active) = {1},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(div) = [0]
p(div_active) = [1] x1 + [2]
p(false) = [0]
p(ge) = [0]
p(ge_active) = [2]
p(if) = [0]
p(if_active) = [1] x1 + [0]
p(mark) = [3]
p(minus) = [0]
p(minus_active) = [0]
p(s) = [1] x1 + [0]
p(true) = [0]
Following rules are strictly oriented:
ge_active(x,y) = [2]
> [0]
= ge(x,y)
Following rules are (at-least) weakly oriented:
div_active(x,y) = [1] x + [2]
>= [0]
= div(x,y)
div_active(0(),s(y)) = [2]
>= [0]
= 0()
div_active(s(x),s(y)) = [1] x + [2]
>= [2]
= if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0())
ge_active(x,0()) = [2]
>= [0]
= true()
ge_active(0(),s(y)) = [2]
>= [0]
= false()
ge_active(s(x),s(y)) = [2]
>= [2]
= ge_active(x,y)
if_active(x,y,z) = [1] x + [0]
>= [0]
= if(x,y,z)
if_active(false(),x,y) = [0]
>= [3]
= mark(y)
if_active(true(),x,y) = [0]
>= [3]
= mark(x)
mark(0()) = [3]
>= [0]
= 0()
mark(div(x,y)) = [3]
>= [5]
= div_active(mark(x),y)
mark(ge(x,y)) = [3]
>= [2]
= ge_active(x,y)
mark(if(x,y,z)) = [3]
>= [3]
= if_active(mark(x),y,z)
mark(minus(x,y)) = [3]
>= [0]
= minus_active(x,y)
mark(s(x)) = [3]
>= [3]
= s(mark(x))
minus_active(x,y) = [0]
>= [0]
= minus(x,y)
minus_active(0(),y) = [0]
>= [0]
= 0()
minus_active(s(x),s(y)) = [0]
>= [0]
= minus_active(x,y)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:7: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
if_active(x,y,z) -> if(x,y,z)
if_active(false(),x,y) -> mark(y)
if_active(true(),x,y) -> mark(x)
mark(div(x,y)) -> div_active(mark(x),y)
mark(s(x)) -> s(mark(x))
- Weak TRS:
div_active(x,y) -> div(x,y)
div_active(0(),s(y)) -> 0()
div_active(s(x),s(y)) -> if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0())
ge_active(x,y) -> ge(x,y)
ge_active(x,0()) -> true()
ge_active(0(),s(y)) -> false()
ge_active(s(x),s(y)) -> ge_active(x,y)
mark(0()) -> 0()
mark(ge(x,y)) -> ge_active(x,y)
mark(if(x,y,z)) -> if_active(mark(x),y,z)
mark(minus(x,y)) -> minus_active(x,y)
minus_active(x,y) -> minus(x,y)
minus_active(0(),y) -> 0()
minus_active(s(x),s(y)) -> minus_active(x,y)
- Signature:
{div_active/2,ge_active/2,if_active/3,mark/1,minus_active/2} / {0/0,div/2,false/0,ge/2,if/3,minus/2,s/1
,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {div_active,ge_active,if_active,mark
,minus_active} and constructors {0,div,false,ge,if,minus,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_active) = {1},
uargs(if_active) = {1},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(div) = [1] x1 + [0]
p(div_active) = [1] x1 + [2]
p(false) = [0]
p(ge) = [1] x1 + [2]
p(ge_active) = [1] x1 + [2]
p(if) = [1] x1 + [1] x2 + [1] x3 + [0]
p(if_active) = [1] x1 + [1] x2 + [1] x3 + [0]
p(mark) = [1] x1 + [0]
p(minus) = [0]
p(minus_active) = [0]
p(s) = [1] x1 + [0]
p(true) = [2]
Following rules are strictly oriented:
if_active(true(),x,y) = [1] x + [1] y + [2]
> [1] x + [0]
= mark(x)
Following rules are (at-least) weakly oriented:
div_active(x,y) = [1] x + [2]
>= [1] x + [0]
= div(x,y)
div_active(0(),s(y)) = [2]
>= [0]
= 0()
div_active(s(x),s(y)) = [1] x + [2]
>= [1] x + [2]
= if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0())
ge_active(x,y) = [1] x + [2]
>= [1] x + [2]
= ge(x,y)
ge_active(x,0()) = [1] x + [2]
>= [2]
= true()
ge_active(0(),s(y)) = [2]
>= [0]
= false()
ge_active(s(x),s(y)) = [1] x + [2]
>= [1] x + [2]
= ge_active(x,y)
if_active(x,y,z) = [1] x + [1] y + [1] z + [0]
>= [1] x + [1] y + [1] z + [0]
= if(x,y,z)
if_active(false(),x,y) = [1] x + [1] y + [0]
>= [1] y + [0]
= mark(y)
mark(0()) = [0]
>= [0]
= 0()
mark(div(x,y)) = [1] x + [0]
>= [1] x + [2]
= div_active(mark(x),y)
mark(ge(x,y)) = [1] x + [2]
>= [1] x + [2]
= ge_active(x,y)
mark(if(x,y,z)) = [1] x + [1] y + [1] z + [0]
>= [1] x + [1] y + [1] z + [0]
= if_active(mark(x),y,z)
mark(minus(x,y)) = [0]
>= [0]
= minus_active(x,y)
mark(s(x)) = [1] x + [0]
>= [1] x + [0]
= s(mark(x))
minus_active(x,y) = [0]
>= [0]
= minus(x,y)
minus_active(0(),y) = [0]
>= [0]
= 0()
minus_active(s(x),s(y)) = [0]
>= [0]
= minus_active(x,y)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:8: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
if_active(x,y,z) -> if(x,y,z)
if_active(false(),x,y) -> mark(y)
mark(div(x,y)) -> div_active(mark(x),y)
mark(s(x)) -> s(mark(x))
- Weak TRS:
div_active(x,y) -> div(x,y)
div_active(0(),s(y)) -> 0()
div_active(s(x),s(y)) -> if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0())
ge_active(x,y) -> ge(x,y)
ge_active(x,0()) -> true()
ge_active(0(),s(y)) -> false()
ge_active(s(x),s(y)) -> ge_active(x,y)
if_active(true(),x,y) -> mark(x)
mark(0()) -> 0()
mark(ge(x,y)) -> ge_active(x,y)
mark(if(x,y,z)) -> if_active(mark(x),y,z)
mark(minus(x,y)) -> minus_active(x,y)
minus_active(x,y) -> minus(x,y)
minus_active(0(),y) -> 0()
minus_active(s(x),s(y)) -> minus_active(x,y)
- Signature:
{div_active/2,ge_active/2,if_active/3,mark/1,minus_active/2} / {0/0,div/2,false/0,ge/2,if/3,minus/2,s/1
,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {div_active,ge_active,if_active,mark
,minus_active} and constructors {0,div,false,ge,if,minus,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_active) = {1},
uargs(if_active) = {1},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(div) = [1] x1 + [1] x2 + [0]
p(div_active) = [1] x1 + [4] x2 + [6]
p(false) = [3]
p(ge) = [2]
p(ge_active) = [3]
p(if) = [1] x1 + [1] x2 + [1] x3 + [0]
p(if_active) = [1] x1 + [4] x2 + [4] x3 + [0]
p(mark) = [4] x1 + [0]
p(minus) = [0]
p(minus_active) = [0]
p(s) = [1] x1 + [1]
p(true) = [3]
Following rules are strictly oriented:
if_active(false(),x,y) = [4] x + [4] y + [3]
> [4] y + [0]
= mark(y)
mark(s(x)) = [4] x + [4]
> [4] x + [1]
= s(mark(x))
Following rules are (at-least) weakly oriented:
div_active(x,y) = [1] x + [4] y + [6]
>= [1] x + [1] y + [0]
= div(x,y)
div_active(0(),s(y)) = [4] y + [10]
>= [0]
= 0()
div_active(s(x),s(y)) = [1] x + [4] y + [11]
>= [4] y + [11]
= if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0())
ge_active(x,y) = [3]
>= [2]
= ge(x,y)
ge_active(x,0()) = [3]
>= [3]
= true()
ge_active(0(),s(y)) = [3]
>= [3]
= false()
ge_active(s(x),s(y)) = [3]
>= [3]
= ge_active(x,y)
if_active(x,y,z) = [1] x + [4] y + [4] z + [0]
>= [1] x + [1] y + [1] z + [0]
= if(x,y,z)
if_active(true(),x,y) = [4] x + [4] y + [3]
>= [4] x + [0]
= mark(x)
mark(0()) = [0]
>= [0]
= 0()
mark(div(x,y)) = [4] x + [4] y + [0]
>= [4] x + [4] y + [6]
= div_active(mark(x),y)
mark(ge(x,y)) = [8]
>= [3]
= ge_active(x,y)
mark(if(x,y,z)) = [4] x + [4] y + [4] z + [0]
>= [4] x + [4] y + [4] z + [0]
= if_active(mark(x),y,z)
mark(minus(x,y)) = [0]
>= [0]
= minus_active(x,y)
minus_active(x,y) = [0]
>= [0]
= minus(x,y)
minus_active(0(),y) = [0]
>= [0]
= 0()
minus_active(s(x),s(y)) = [0]
>= [0]
= minus_active(x,y)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:9: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
if_active(x,y,z) -> if(x,y,z)
mark(div(x,y)) -> div_active(mark(x),y)
- Weak TRS:
div_active(x,y) -> div(x,y)
div_active(0(),s(y)) -> 0()
div_active(s(x),s(y)) -> if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0())
ge_active(x,y) -> ge(x,y)
ge_active(x,0()) -> true()
ge_active(0(),s(y)) -> false()
ge_active(s(x),s(y)) -> ge_active(x,y)
if_active(false(),x,y) -> mark(y)
if_active(true(),x,y) -> mark(x)
mark(0()) -> 0()
mark(ge(x,y)) -> ge_active(x,y)
mark(if(x,y,z)) -> if_active(mark(x),y,z)
mark(minus(x,y)) -> minus_active(x,y)
mark(s(x)) -> s(mark(x))
minus_active(x,y) -> minus(x,y)
minus_active(0(),y) -> 0()
minus_active(s(x),s(y)) -> minus_active(x,y)
- Signature:
{div_active/2,ge_active/2,if_active/3,mark/1,minus_active/2} / {0/0,div/2,false/0,ge/2,if/3,minus/2,s/1
,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {div_active,ge_active,if_active,mark
,minus_active} and constructors {0,div,false,ge,if,minus,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_active) = {1},
uargs(if_active) = {1},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(div) = [1] x1 + [0]
p(div_active) = [1] x1 + [2]
p(false) = [0]
p(ge) = [0]
p(ge_active) = [0]
p(if) = [1] x1 + [1] x2 + [1] x3 + [1]
p(if_active) = [1] x1 + [4] x2 + [4] x3 + [2]
p(mark) = [4] x1 + [0]
p(minus) = [0]
p(minus_active) = [0]
p(s) = [1] x1 + [0]
p(true) = [0]
Following rules are strictly oriented:
if_active(x,y,z) = [1] x + [4] y + [4] z + [2]
> [1] x + [1] y + [1] z + [1]
= if(x,y,z)
Following rules are (at-least) weakly oriented:
div_active(x,y) = [1] x + [2]
>= [1] x + [0]
= div(x,y)
div_active(0(),s(y)) = [2]
>= [0]
= 0()
div_active(s(x),s(y)) = [1] x + [2]
>= [2]
= if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0())
ge_active(x,y) = [0]
>= [0]
= ge(x,y)
ge_active(x,0()) = [0]
>= [0]
= true()
ge_active(0(),s(y)) = [0]
>= [0]
= false()
ge_active(s(x),s(y)) = [0]
>= [0]
= ge_active(x,y)
if_active(false(),x,y) = [4] x + [4] y + [2]
>= [4] y + [0]
= mark(y)
if_active(true(),x,y) = [4] x + [4] y + [2]
>= [4] x + [0]
= mark(x)
mark(0()) = [0]
>= [0]
= 0()
mark(div(x,y)) = [4] x + [0]
>= [4] x + [2]
= div_active(mark(x),y)
mark(ge(x,y)) = [0]
>= [0]
= ge_active(x,y)
mark(if(x,y,z)) = [4] x + [4] y + [4] z + [4]
>= [4] x + [4] y + [4] z + [2]
= if_active(mark(x),y,z)
mark(minus(x,y)) = [0]
>= [0]
= minus_active(x,y)
mark(s(x)) = [4] x + [0]
>= [4] x + [0]
= s(mark(x))
minus_active(x,y) = [0]
>= [0]
= minus(x,y)
minus_active(0(),y) = [0]
>= [0]
= 0()
minus_active(s(x),s(y)) = [0]
>= [0]
= minus_active(x,y)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:10: MI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
mark(div(x,y)) -> div_active(mark(x),y)
- Weak TRS:
div_active(x,y) -> div(x,y)
div_active(0(),s(y)) -> 0()
div_active(s(x),s(y)) -> if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0())
ge_active(x,y) -> ge(x,y)
ge_active(x,0()) -> true()
ge_active(0(),s(y)) -> false()
ge_active(s(x),s(y)) -> ge_active(x,y)
if_active(x,y,z) -> if(x,y,z)
if_active(false(),x,y) -> mark(y)
if_active(true(),x,y) -> mark(x)
mark(0()) -> 0()
mark(ge(x,y)) -> ge_active(x,y)
mark(if(x,y,z)) -> if_active(mark(x),y,z)
mark(minus(x,y)) -> minus_active(x,y)
mark(s(x)) -> s(mark(x))
minus_active(x,y) -> minus(x,y)
minus_active(0(),y) -> 0()
minus_active(s(x),s(y)) -> minus_active(x,y)
- Signature:
{div_active/2,ge_active/2,if_active/3,mark/1,minus_active/2} / {0/0,div/2,false/0,ge/2,if/3,minus/2,s/1
,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {div_active,ge_active,if_active,mark
,minus_active} and constructors {0,div,false,ge,if,minus,s,true}
+ Applied Processor:
MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity Nothing)), miDimension = 2, 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(div_active) = {1},
uargs(if_active) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{div_active,ge_active,if_active,mark,minus_active}
TcT has computed the following interpretation:
p(0) = [0]
[0]
p(div) = [1 1] x_1 + [1]
[0 0] [2]
p(div_active) = [1 1] x_1 + [2]
[0 0] [4]
p(false) = [0]
[1]
p(ge) = [0]
[3]
p(ge_active) = [0]
[3]
p(if) = [1 0] x_1 + [1 0] x_2 + [1 0] x_3 + [0]
[0 0] [0 1] [0 1] [0]
p(if_active) = [1 0] x_1 + [4 0] x_2 + [4 0] x_3 + [0]
[0 0] [0 2] [0 2] [0]
p(mark) = [4 0] x_1 + [0]
[0 2] [0]
p(minus) = [0]
[0]
p(minus_active) = [0]
[0]
p(s) = [1 0] x_1 + [0]
[0 0] [2]
p(true) = [0]
[0]
Following rules are strictly oriented:
mark(div(x,y)) = [4 4] x + [4]
[0 0] [4]
> [4 2] x + [2]
[0 0] [4]
= div_active(mark(x),y)
Following rules are (at-least) weakly oriented:
div_active(x,y) = [1 1] x + [2]
[0 0] [4]
>= [1 1] x + [1]
[0 0] [2]
= div(x,y)
div_active(0(),s(y)) = [2]
[4]
>= [0]
[0]
= 0()
div_active(s(x),s(y)) = [1 0] x + [4]
[0 0] [4]
>= [4]
[4]
= if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0())
ge_active(x,y) = [0]
[3]
>= [0]
[3]
= ge(x,y)
ge_active(x,0()) = [0]
[3]
>= [0]
[0]
= true()
ge_active(0(),s(y)) = [0]
[3]
>= [0]
[1]
= false()
ge_active(s(x),s(y)) = [0]
[3]
>= [0]
[3]
= ge_active(x,y)
if_active(x,y,z) = [1 0] x + [4 0] y + [4 0] z + [0]
[0 0] [0 2] [0 2] [0]
>= [1 0] x + [1 0] y + [1 0] z + [0]
[0 0] [0 1] [0 1] [0]
= if(x,y,z)
if_active(false(),x,y) = [4 0] x + [4 0] y + [0]
[0 2] [0 2] [0]
>= [4 0] y + [0]
[0 2] [0]
= mark(y)
if_active(true(),x,y) = [4 0] x + [4 0] y + [0]
[0 2] [0 2] [0]
>= [4 0] x + [0]
[0 2] [0]
= mark(x)
mark(0()) = [0]
[0]
>= [0]
[0]
= 0()
mark(ge(x,y)) = [0]
[6]
>= [0]
[3]
= ge_active(x,y)
mark(if(x,y,z)) = [4 0] x + [4 0] y + [4 0] z + [0]
[0 0] [0 2] [0 2] [0]
>= [4 0] x + [4 0] y + [4 0] z + [0]
[0 0] [0 2] [0 2] [0]
= if_active(mark(x),y,z)
mark(minus(x,y)) = [0]
[0]
>= [0]
[0]
= minus_active(x,y)
mark(s(x)) = [4 0] x + [0]
[0 0] [4]
>= [4 0] x + [0]
[0 0] [2]
= s(mark(x))
minus_active(x,y) = [0]
[0]
>= [0]
[0]
= minus(x,y)
minus_active(0(),y) = [0]
[0]
>= [0]
[0]
= 0()
minus_active(s(x),s(y)) = [0]
[0]
>= [0]
[0]
= minus_active(x,y)
** Step 1.b:11: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
div_active(x,y) -> div(x,y)
div_active(0(),s(y)) -> 0()
div_active(s(x),s(y)) -> if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0())
ge_active(x,y) -> ge(x,y)
ge_active(x,0()) -> true()
ge_active(0(),s(y)) -> false()
ge_active(s(x),s(y)) -> ge_active(x,y)
if_active(x,y,z) -> if(x,y,z)
if_active(false(),x,y) -> mark(y)
if_active(true(),x,y) -> mark(x)
mark(0()) -> 0()
mark(div(x,y)) -> div_active(mark(x),y)
mark(ge(x,y)) -> ge_active(x,y)
mark(if(x,y,z)) -> if_active(mark(x),y,z)
mark(minus(x,y)) -> minus_active(x,y)
mark(s(x)) -> s(mark(x))
minus_active(x,y) -> minus(x,y)
minus_active(0(),y) -> 0()
minus_active(s(x),s(y)) -> minus_active(x,y)
- Signature:
{div_active/2,ge_active/2,if_active/3,mark/1,minus_active/2} / {0/0,div/2,false/0,ge/2,if/3,minus/2,s/1
,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {div_active,ge_active,if_active,mark
,minus_active} and constructors {0,div,false,ge,if,minus,s,true}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(Omega(n^1),O(n^1))