WORST_CASE(?,O(n^3))
* Step 1: NaturalPI WORST_CASE(?,O(n^3))
+ 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:
NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a polynomial interpretation of kind constructor-based(linear):
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
p(div) = 2 + x1 + x2
p(div_active) = 4 + x1 + 2*x2
p(false) = 0
p(ge) = x1
p(ge_active) = x1
p(if) = x1 + x2 + x3
p(if_active) = x1 + 2*x2 + 2*x3
p(mark) = 2*x1
p(minus) = 0
p(minus_active) = 0
p(s) = x1
p(true) = 0
Following rules are strictly oriented:
div_active(x,y) = 4 + x + 2*y
> 2 + x + y
= div(x,y)
div_active(0(),s(y)) = 4 + 2*y
> 0
= 0()
Following rules are (at-least) weakly oriented:
div_active(s(x),s(y)) = 4 + x + 2*y
>= 4 + x + 2*y
= if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0())
ge_active(x,y) = x
>= x
= ge(x,y)
ge_active(x,0()) = x
>= 0
= true()
ge_active(0(),s(y)) = 0
>= 0
= false()
ge_active(s(x),s(y)) = x
>= x
= ge_active(x,y)
if_active(x,y,z) = x + 2*y + 2*z
>= x + y + z
= if(x,y,z)
if_active(false(),x,y) = 2*x + 2*y
>= 2*y
= mark(y)
if_active(true(),x,y) = 2*x + 2*y
>= 2*x
= mark(x)
mark(0()) = 0
>= 0
= 0()
mark(div(x,y)) = 4 + 2*x + 2*y
>= 4 + 2*x + 2*y
= div_active(mark(x),y)
mark(ge(x,y)) = 2*x
>= x
= ge_active(x,y)
mark(if(x,y,z)) = 2*x + 2*y + 2*z
>= 2*x + 2*y + 2*z
= if_active(mark(x),y,z)
mark(minus(x,y)) = 0
>= 0
= minus_active(x,y)
mark(s(x)) = 2*x
>= 2*x
= 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)
* Step 2: NaturalPI WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
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)
- Weak TRS:
div_active(x,y) -> div(x,y)
div_active(0(),s(y)) -> 0()
- 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:
NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a polynomial interpretation of kind constructor-based(linear):
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
p(div) = x1
p(div_active) = x1
p(false) = 0
p(ge) = 0
p(ge_active) = 0
p(if) = x1 + x2 + x3
p(if_active) = x1 + x2 + x3
p(mark) = x1
p(minus) = x1
p(minus_active) = x1
p(s) = 4 + x1
p(true) = 0
Following rules are strictly oriented:
minus_active(s(x),s(y)) = 4 + x
> x
= minus_active(x,y)
Following rules are (at-least) weakly oriented:
div_active(x,y) = x
>= x
= div(x,y)
div_active(0(),s(y)) = 0
>= 0
= 0()
div_active(s(x),s(y)) = 4 + x
>= 4 + x
= 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) = x + y + z
>= x + y + z
= if(x,y,z)
if_active(false(),x,y) = x + y
>= y
= mark(y)
if_active(true(),x,y) = x + y
>= x
= mark(x)
mark(0()) = 0
>= 0
= 0()
mark(div(x,y)) = x
>= x
= div_active(mark(x),y)
mark(ge(x,y)) = 0
>= 0
= ge_active(x,y)
mark(if(x,y,z)) = x + y + z
>= x + y + z
= if_active(mark(x),y,z)
mark(minus(x,y)) = x
>= x
= minus_active(x,y)
mark(s(x)) = 4 + x
>= 4 + x
= s(mark(x))
minus_active(x,y) = x
>= x
= minus(x,y)
minus_active(0(),y) = 0
>= 0
= 0()
* Step 3: NaturalPI WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
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()
- Weak TRS:
div_active(x,y) -> div(x,y)
div_active(0(),s(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:
NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a polynomial interpretation of kind constructor-based(linear):
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
p(div) = x1 + x2
p(div_active) = x1 + x2
p(false) = 0
p(ge) = x1
p(ge_active) = x1
p(if) = x1 + x2 + x3
p(if_active) = x1 + x2 + x3
p(mark) = x1
p(minus) = 0
p(minus_active) = 0
p(s) = 4 + x1
p(true) = 0
Following rules are strictly oriented:
ge_active(s(x),s(y)) = 4 + x
> x
= ge_active(x,y)
Following rules are (at-least) weakly oriented:
div_active(x,y) = x + y
>= x + y
= div(x,y)
div_active(0(),s(y)) = 4 + y
>= 0
= 0()
div_active(s(x),s(y)) = 8 + x + y
>= 8 + x + y
= if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0())
ge_active(x,y) = x
>= x
= ge(x,y)
ge_active(x,0()) = x
>= 0
= true()
ge_active(0(),s(y)) = 0
>= 0
= false()
if_active(x,y,z) = x + y + z
>= x + y + z
= if(x,y,z)
if_active(false(),x,y) = x + y
>= y
= mark(y)
if_active(true(),x,y) = x + y
>= x
= mark(x)
mark(0()) = 0
>= 0
= 0()
mark(div(x,y)) = x + y
>= x + y
= div_active(mark(x),y)
mark(ge(x,y)) = x
>= x
= ge_active(x,y)
mark(if(x,y,z)) = x + y + z
>= x + y + z
= if_active(mark(x),y,z)
mark(minus(x,y)) = 0
>= 0
= minus_active(x,y)
mark(s(x)) = 4 + x
>= 4 + x
= 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)
* Step 4: NaturalMI WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
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()
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()
- Weak TRS:
div_active(x,y) -> div(x,y)
div_active(0(),s(y)) -> 0()
ge_active(s(x),s(y)) -> ge_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:
NaturalMI {miDimension = 3, miDegree = 3, 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_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]
[0]
p(div) = [0 0 0] [0]
[0 1 1] x1 + [0]
[0 0 0] [1]
p(div_active) = [1 0 0] [0]
[0 1 1] x1 + [0]
[0 0 0] [1]
p(false) = [0]
[0]
[0]
p(ge) = [0]
[0]
[0]
p(ge_active) = [0]
[0]
[0]
p(if) = [0 0 0] [0 0 0] [0 1 0] [0]
[0 1 0] x1 + [0 1 0] x2 + [0 1 0] x3 + [1]
[0 0 0] [0 0 1] [0 0 1] [0]
p(if_active) = [1 0 0] [0 1 0] [0 1 0] [0]
[0 1 0] x1 + [0 1 0] x2 + [0 1 0] x3 + [1]
[0 0 0] [0 0 1] [0 0 1] [0]
p(mark) = [0 1 0] [0]
[0 1 0] x1 + [0]
[0 0 1] [0]
p(minus) = [0]
[0]
[0]
p(minus_active) = [0]
[0]
[0]
p(s) = [1 0 0] [0]
[0 1 0] x1 + [0]
[0 0 0] [1]
p(true) = [0]
[0]
[0]
Following rules are strictly oriented:
mark(if(x,y,z)) = [0 1 0] [0 1 0] [0 1 0] [1]
[0 1 0] x + [0 1 0] y + [0 1 0] z + [1]
[0 0 0] [0 0 1] [0 0 1] [0]
> [0 1 0] [0 1 0] [0 1 0] [0]
[0 1 0] x + [0 1 0] y + [0 1 0] z + [1]
[0 0 0] [0 0 1] [0 0 1] [0]
= if_active(mark(x),y,z)
Following rules are (at-least) weakly oriented:
div_active(x,y) = [1 0 0] [0]
[0 1 1] x + [0]
[0 0 0] [1]
>= [0 0 0] [0]
[0 1 1] x + [0]
[0 0 0] [1]
= div(x,y)
div_active(0(),s(y)) = [0]
[0]
[1]
>= [0]
[0]
[0]
= 0()
div_active(s(x),s(y)) = [1 0 0] [0]
[0 1 0] x + [1]
[0 0 0] [1]
>= [0]
[1]
[1]
= if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0())
ge_active(x,y) = [0]
[0]
[0]
>= [0]
[0]
[0]
= ge(x,y)
ge_active(x,0()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= true()
ge_active(0(),s(y)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= false()
ge_active(s(x),s(y)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= ge_active(x,y)
if_active(x,y,z) = [1 0 0] [0 1 0] [0 1 0] [0]
[0 1 0] x + [0 1 0] y + [0 1 0] z + [1]
[0 0 0] [0 0 1] [0 0 1] [0]
>= [0 0 0] [0 0 0] [0 1 0] [0]
[0 1 0] x + [0 1 0] y + [0 1 0] z + [1]
[0 0 0] [0 0 1] [0 0 1] [0]
= if(x,y,z)
if_active(false(),x,y) = [0 1 0] [0 1 0] [0]
[0 1 0] x + [0 1 0] y + [1]
[0 0 1] [0 0 1] [0]
>= [0 1 0] [0]
[0 1 0] y + [0]
[0 0 1] [0]
= mark(y)
if_active(true(),x,y) = [0 1 0] [0 1 0] [0]
[0 1 0] x + [0 1 0] y + [1]
[0 0 1] [0 0 1] [0]
>= [0 1 0] [0]
[0 1 0] x + [0]
[0 0 1] [0]
= mark(x)
mark(0()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= 0()
mark(div(x,y)) = [0 1 1] [0]
[0 1 1] x + [0]
[0 0 0] [1]
>= [0 1 0] [0]
[0 1 1] x + [0]
[0 0 0] [1]
= div_active(mark(x),y)
mark(ge(x,y)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= ge_active(x,y)
mark(minus(x,y)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= minus_active(x,y)
mark(s(x)) = [0 1 0] [0]
[0 1 0] x + [0]
[0 0 0] [1]
>= [0 1 0] [0]
[0 1 0] x + [0]
[0 0 0] [1]
= s(mark(x))
minus_active(x,y) = [0]
[0]
[0]
>= [0]
[0]
[0]
= minus(x,y)
minus_active(0(),y) = [0]
[0]
[0]
>= [0]
[0]
[0]
= 0()
minus_active(s(x),s(y)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= minus_active(x,y)
* Step 5: NaturalMI WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
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()
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(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()
- Weak TRS:
div_active(x,y) -> div(x,y)
div_active(0(),s(y)) -> 0()
ge_active(s(x),s(y)) -> ge_active(x,y)
mark(if(x,y,z)) -> if_active(mark(x),y,z)
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:
NaturalMI {miDimension = 3, miDegree = 3, 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_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]
[0]
p(div) = [0 0 0] [0 0 0] [0]
[0 1 1] x1 + [0 0 0] x2 + [0]
[0 0 1] [0 0 1] [1]
p(div_active) = [1 0 0] [0 0 0] [0]
[0 1 1] x1 + [0 0 0] x2 + [0]
[0 0 1] [0 0 1] [1]
p(false) = [0]
[1]
[0]
p(ge) = [0]
[1]
[1]
p(ge_active) = [0]
[1]
[1]
p(if) = [0 0 0] [0 1 0] [0 0 0] [0]
[0 1 0] x1 + [0 1 0] x2 + [0 1 0] x3 + [0]
[0 0 1] [0 0 1] [0 0 1] [1]
p(if_active) = [1 0 0] [0 1 0] [0 1 0] [0]
[0 1 0] x1 + [0 1 0] x2 + [0 1 0] x3 + [0]
[0 0 1] [0 0 1] [0 0 1] [1]
p(mark) = [0 1 0] [0]
[0 1 0] x1 + [0]
[0 0 1] [0]
p(minus) = [0]
[0]
[0]
p(minus_active) = [0]
[0]
[0]
p(s) = [1 0 0] [0]
[0 1 0] x1 + [0]
[0 0 0] [1]
p(true) = [0]
[0]
[1]
Following rules are strictly oriented:
mark(ge(x,y)) = [1]
[1]
[1]
> [0]
[1]
[1]
= ge_active(x,y)
Following rules are (at-least) weakly oriented:
div_active(x,y) = [1 0 0] [0 0 0] [0]
[0 1 1] x + [0 0 0] y + [0]
[0 0 1] [0 0 1] [1]
>= [0 0 0] [0 0 0] [0]
[0 1 1] x + [0 0 0] y + [0]
[0 0 1] [0 0 1] [1]
= div(x,y)
div_active(0(),s(y)) = [0]
[0]
[2]
>= [0]
[0]
[0]
= 0()
div_active(s(x),s(y)) = [1 0 0] [0]
[0 1 0] x + [1]
[0 0 0] [3]
>= [0]
[1]
[3]
= if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0())
ge_active(x,y) = [0]
[1]
[1]
>= [0]
[1]
[1]
= ge(x,y)
ge_active(x,0()) = [0]
[1]
[1]
>= [0]
[0]
[1]
= true()
ge_active(0(),s(y)) = [0]
[1]
[1]
>= [0]
[1]
[0]
= false()
ge_active(s(x),s(y)) = [0]
[1]
[1]
>= [0]
[1]
[1]
= ge_active(x,y)
if_active(x,y,z) = [1 0 0] [0 1 0] [0 1 0] [0]
[0 1 0] x + [0 1 0] y + [0 1 0] z + [0]
[0 0 1] [0 0 1] [0 0 1] [1]
>= [0 0 0] [0 1 0] [0 0 0] [0]
[0 1 0] x + [0 1 0] y + [0 1 0] z + [0]
[0 0 1] [0 0 1] [0 0 1] [1]
= if(x,y,z)
if_active(false(),x,y) = [0 1 0] [0 1 0] [0]
[0 1 0] x + [0 1 0] y + [1]
[0 0 1] [0 0 1] [1]
>= [0 1 0] [0]
[0 1 0] y + [0]
[0 0 1] [0]
= mark(y)
if_active(true(),x,y) = [0 1 0] [0 1 0] [0]
[0 1 0] x + [0 1 0] y + [0]
[0 0 1] [0 0 1] [2]
>= [0 1 0] [0]
[0 1 0] x + [0]
[0 0 1] [0]
= mark(x)
mark(0()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= 0()
mark(div(x,y)) = [0 1 1] [0 0 0] [0]
[0 1 1] x + [0 0 0] y + [0]
[0 0 1] [0 0 1] [1]
>= [0 1 0] [0 0 0] [0]
[0 1 1] x + [0 0 0] y + [0]
[0 0 1] [0 0 1] [1]
= div_active(mark(x),y)
mark(if(x,y,z)) = [0 1 0] [0 1 0] [0 1 0] [0]
[0 1 0] x + [0 1 0] y + [0 1 0] z + [0]
[0 0 1] [0 0 1] [0 0 1] [1]
>= [0 1 0] [0 1 0] [0 1 0] [0]
[0 1 0] x + [0 1 0] y + [0 1 0] z + [0]
[0 0 1] [0 0 1] [0 0 1] [1]
= if_active(mark(x),y,z)
mark(minus(x,y)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= minus_active(x,y)
mark(s(x)) = [0 1 0] [0]
[0 1 0] x + [0]
[0 0 0] [1]
>= [0 1 0] [0]
[0 1 0] x + [0]
[0 0 0] [1]
= s(mark(x))
minus_active(x,y) = [0]
[0]
[0]
>= [0]
[0]
[0]
= minus(x,y)
minus_active(0(),y) = [0]
[0]
[0]
>= [0]
[0]
[0]
= 0()
minus_active(s(x),s(y)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= minus_active(x,y)
* Step 6: NaturalMI WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
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()
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(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()
- Weak TRS:
div_active(x,y) -> div(x,y)
div_active(0(),s(y)) -> 0()
ge_active(s(x),s(y)) -> ge_active(x,y)
mark(ge(x,y)) -> ge_active(x,y)
mark(if(x,y,z)) -> if_active(mark(x),y,z)
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:
NaturalMI {miDimension = 3, miDegree = 3, 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_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]
[0]
p(div) = [1 0 1] [0 1 1] [0]
[0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 1] [0 0 0] [0]
p(div_active) = [1 0 1] [0 1 1] [0]
[0 1 0] x1 + [0 0 0] x2 + [0]
[0 0 1] [0 0 0] [0]
p(false) = [1]
[0]
[0]
p(ge) = [1]
[0]
[0]
p(ge_active) = [1]
[0]
[0]
p(if) = [1 0 0] [1 0 0] [1 0 0] [0]
[0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 1] [0 0 1] [0 0 1] [0]
p(if_active) = [1 0 0] [1 0 0] [1 0 0] [0]
[0 1 0] x1 + [0 0 1] x2 + [0 0 1] x3 + [0]
[0 0 1] [0 0 1] [0 0 1] [0]
p(mark) = [1 0 0] [0]
[0 0 1] x1 + [0]
[0 0 1] [0]
p(minus) = [0]
[0]
[0]
p(minus_active) = [0]
[0]
[0]
p(s) = [1 0 0] [0]
[0 0 0] x1 + [1]
[0 0 0] [1]
p(true) = [0]
[0]
[0]
Following rules are strictly oriented:
ge_active(x,0()) = [1]
[0]
[0]
> [0]
[0]
[0]
= true()
if_active(false(),x,y) = [1 0 0] [1 0 0] [1]
[0 0 1] x + [0 0 1] y + [0]
[0 0 1] [0 0 1] [0]
> [1 0 0] [0]
[0 0 1] y + [0]
[0 0 1] [0]
= mark(y)
Following rules are (at-least) weakly oriented:
div_active(x,y) = [1 0 1] [0 1 1] [0]
[0 1 0] x + [0 0 0] y + [0]
[0 0 1] [0 0 0] [0]
>= [1 0 1] [0 1 1] [0]
[0 0 0] x + [0 0 0] y + [0]
[0 0 1] [0 0 0] [0]
= div(x,y)
div_active(0(),s(y)) = [2]
[0]
[0]
>= [0]
[0]
[0]
= 0()
div_active(s(x),s(y)) = [1 0 0] [3]
[0 0 0] x + [1]
[0 0 0] [1]
>= [3]
[1]
[1]
= if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0())
ge_active(x,y) = [1]
[0]
[0]
>= [1]
[0]
[0]
= ge(x,y)
ge_active(0(),s(y)) = [1]
[0]
[0]
>= [1]
[0]
[0]
= false()
ge_active(s(x),s(y)) = [1]
[0]
[0]
>= [1]
[0]
[0]
= ge_active(x,y)
if_active(x,y,z) = [1 0 0] [1 0 0] [1 0 0] [0]
[0 1 0] x + [0 0 1] y + [0 0 1] z + [0]
[0 0 1] [0 0 1] [0 0 1] [0]
>= [1 0 0] [1 0 0] [1 0 0] [0]
[0 0 0] x + [0 0 0] y + [0 0 0] z + [0]
[0 0 1] [0 0 1] [0 0 1] [0]
= if(x,y,z)
if_active(true(),x,y) = [1 0 0] [1 0 0] [0]
[0 0 1] x + [0 0 1] y + [0]
[0 0 1] [0 0 1] [0]
>= [1 0 0] [0]
[0 0 1] x + [0]
[0 0 1] [0]
= mark(x)
mark(0()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= 0()
mark(div(x,y)) = [1 0 1] [0 1 1] [0]
[0 0 1] x + [0 0 0] y + [0]
[0 0 1] [0 0 0] [0]
>= [1 0 1] [0 1 1] [0]
[0 0 1] x + [0 0 0] y + [0]
[0 0 1] [0 0 0] [0]
= div_active(mark(x),y)
mark(ge(x,y)) = [1]
[0]
[0]
>= [1]
[0]
[0]
= ge_active(x,y)
mark(if(x,y,z)) = [1 0 0] [1 0 0] [1 0 0] [0]
[0 0 1] x + [0 0 1] y + [0 0 1] z + [0]
[0 0 1] [0 0 1] [0 0 1] [0]
>= [1 0 0] [1 0 0] [1 0 0] [0]
[0 0 1] x + [0 0 1] y + [0 0 1] z + [0]
[0 0 1] [0 0 1] [0 0 1] [0]
= if_active(mark(x),y,z)
mark(minus(x,y)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= minus_active(x,y)
mark(s(x)) = [1 0 0] [0]
[0 0 0] x + [1]
[0 0 0] [1]
>= [1 0 0] [0]
[0 0 0] x + [1]
[0 0 0] [1]
= s(mark(x))
minus_active(x,y) = [0]
[0]
[0]
>= [0]
[0]
[0]
= minus(x,y)
minus_active(0(),y) = [0]
[0]
[0]
>= [0]
[0]
[0]
= 0()
minus_active(s(x),s(y)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= minus_active(x,y)
* Step 7: NaturalMI WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
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(0(),s(y)) -> false()
if_active(x,y,z) -> if(x,y,z)
if_active(true(),x,y) -> mark(x)
mark(0()) -> 0()
mark(div(x,y)) -> div_active(mark(x),y)
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()
- Weak TRS:
div_active(x,y) -> div(x,y)
div_active(0(),s(y)) -> 0()
ge_active(x,0()) -> true()
ge_active(s(x),s(y)) -> ge_active(x,y)
if_active(false(),x,y) -> mark(y)
mark(ge(x,y)) -> ge_active(x,y)
mark(if(x,y,z)) -> if_active(mark(x),y,z)
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:
NaturalMI {miDimension = 3, miDegree = 3, 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_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]
[0]
p(div) = [0 0 0] [0]
[0 1 1] x1 + [0]
[0 0 1] [0]
p(div_active) = [1 0 1] [0]
[0 1 1] x1 + [0]
[0 0 1] [0]
p(false) = [0]
[0]
[0]
p(ge) = [0]
[0]
[0]
p(ge_active) = [0]
[0]
[0]
p(if) = [0 0 0] [0 0 0] [0 0 0] [0]
[0 1 0] x1 + [0 1 0] x2 + [0 1 0] x3 + [0]
[0 0 0] [0 0 1] [0 0 1] [0]
p(if_active) = [1 0 0] [0 1 0] [0 1 0] [0]
[0 1 0] x1 + [0 1 0] x2 + [0 1 0] x3 + [0]
[0 0 0] [0 0 1] [0 0 1] [0]
p(mark) = [0 1 0] [0]
[0 1 0] x1 + [0]
[0 0 1] [0]
p(minus) = [0]
[0]
[0]
p(minus_active) = [0]
[0]
[0]
p(s) = [1 0 0] [0]
[0 1 0] x1 + [0]
[0 0 0] [1]
p(true) = [0]
[0]
[0]
Following rules are strictly oriented:
div_active(s(x),s(y)) = [1 0 0] [1]
[0 1 0] x + [1]
[0 0 0] [1]
> [0]
[0]
[1]
= if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0())
Following rules are (at-least) weakly oriented:
div_active(x,y) = [1 0 1] [0]
[0 1 1] x + [0]
[0 0 1] [0]
>= [0 0 0] [0]
[0 1 1] x + [0]
[0 0 1] [0]
= div(x,y)
div_active(0(),s(y)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= 0()
ge_active(x,y) = [0]
[0]
[0]
>= [0]
[0]
[0]
= ge(x,y)
ge_active(x,0()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= true()
ge_active(0(),s(y)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= false()
ge_active(s(x),s(y)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= ge_active(x,y)
if_active(x,y,z) = [1 0 0] [0 1 0] [0 1 0] [0]
[0 1 0] x + [0 1 0] y + [0 1 0] z + [0]
[0 0 0] [0 0 1] [0 0 1] [0]
>= [0 0 0] [0 0 0] [0 0 0] [0]
[0 1 0] x + [0 1 0] y + [0 1 0] z + [0]
[0 0 0] [0 0 1] [0 0 1] [0]
= if(x,y,z)
if_active(false(),x,y) = [0 1 0] [0 1 0] [0]
[0 1 0] x + [0 1 0] y + [0]
[0 0 1] [0 0 1] [0]
>= [0 1 0] [0]
[0 1 0] y + [0]
[0 0 1] [0]
= mark(y)
if_active(true(),x,y) = [0 1 0] [0 1 0] [0]
[0 1 0] x + [0 1 0] y + [0]
[0 0 1] [0 0 1] [0]
>= [0 1 0] [0]
[0 1 0] x + [0]
[0 0 1] [0]
= mark(x)
mark(0()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= 0()
mark(div(x,y)) = [0 1 1] [0]
[0 1 1] x + [0]
[0 0 1] [0]
>= [0 1 1] [0]
[0 1 1] x + [0]
[0 0 1] [0]
= div_active(mark(x),y)
mark(ge(x,y)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= ge_active(x,y)
mark(if(x,y,z)) = [0 1 0] [0 1 0] [0 1 0] [0]
[0 1 0] x + [0 1 0] y + [0 1 0] z + [0]
[0 0 0] [0 0 1] [0 0 1] [0]
>= [0 1 0] [0 1 0] [0 1 0] [0]
[0 1 0] x + [0 1 0] y + [0 1 0] z + [0]
[0 0 0] [0 0 1] [0 0 1] [0]
= if_active(mark(x),y,z)
mark(minus(x,y)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= minus_active(x,y)
mark(s(x)) = [0 1 0] [0]
[0 1 0] x + [0]
[0 0 0] [1]
>= [0 1 0] [0]
[0 1 0] x + [0]
[0 0 0] [1]
= s(mark(x))
minus_active(x,y) = [0]
[0]
[0]
>= [0]
[0]
[0]
= minus(x,y)
minus_active(0(),y) = [0]
[0]
[0]
>= [0]
[0]
[0]
= 0()
minus_active(s(x),s(y)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= minus_active(x,y)
* Step 8: NaturalMI WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
ge_active(x,y) -> ge(x,y)
ge_active(0(),s(y)) -> false()
if_active(x,y,z) -> if(x,y,z)
if_active(true(),x,y) -> mark(x)
mark(0()) -> 0()
mark(div(x,y)) -> div_active(mark(x),y)
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()
- 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(s(x),s(y)) -> ge_active(x,y)
if_active(false(),x,y) -> mark(y)
mark(ge(x,y)) -> ge_active(x,y)
mark(if(x,y,z)) -> if_active(mark(x),y,z)
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:
NaturalMI {miDimension = 3, miDegree = 3, 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_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]
[0]
p(div) = [0 0 0] [0]
[0 1 1] x1 + [0]
[0 0 0] [1]
p(div_active) = [1 0 1] [0]
[0 1 1] x1 + [0]
[0 0 0] [1]
p(false) = [0]
[0]
[0]
p(ge) = [0]
[0]
[0]
p(ge_active) = [0]
[0]
[0]
p(if) = [0 0 0] [0 0 0] [0 0 0] [0]
[0 1 0] x1 + [0 1 0] x2 + [0 1 0] x3 + [0]
[0 0 0] [0 0 1] [0 0 1] [0]
p(if_active) = [1 0 0] [0 1 0] [0 1 0] [0]
[0 1 0] x1 + [0 1 0] x2 + [0 1 0] x3 + [0]
[0 0 0] [0 0 1] [0 0 1] [0]
p(mark) = [0 1 0] [0]
[0 1 0] x1 + [0]
[0 0 1] [0]
p(minus) = [0]
[1]
[0]
p(minus_active) = [0]
[1]
[0]
p(s) = [1 0 0] [0]
[0 1 0] x1 + [0]
[0 0 0] [1]
p(true) = [0]
[0]
[0]
Following rules are strictly oriented:
mark(minus(x,y)) = [1]
[1]
[0]
> [0]
[1]
[0]
= minus_active(x,y)
Following rules are (at-least) weakly oriented:
div_active(x,y) = [1 0 1] [0]
[0 1 1] x + [0]
[0 0 0] [1]
>= [0 0 0] [0]
[0 1 1] x + [0]
[0 0 0] [1]
= div(x,y)
div_active(0(),s(y)) = [0]
[0]
[1]
>= [0]
[0]
[0]
= 0()
div_active(s(x),s(y)) = [1 0 0] [1]
[0 1 0] x + [1]
[0 0 0] [1]
>= [1]
[1]
[1]
= if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0())
ge_active(x,y) = [0]
[0]
[0]
>= [0]
[0]
[0]
= ge(x,y)
ge_active(x,0()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= true()
ge_active(0(),s(y)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= false()
ge_active(s(x),s(y)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= ge_active(x,y)
if_active(x,y,z) = [1 0 0] [0 1 0] [0 1 0] [0]
[0 1 0] x + [0 1 0] y + [0 1 0] z + [0]
[0 0 0] [0 0 1] [0 0 1] [0]
>= [0 0 0] [0 0 0] [0 0 0] [0]
[0 1 0] x + [0 1 0] y + [0 1 0] z + [0]
[0 0 0] [0 0 1] [0 0 1] [0]
= if(x,y,z)
if_active(false(),x,y) = [0 1 0] [0 1 0] [0]
[0 1 0] x + [0 1 0] y + [0]
[0 0 1] [0 0 1] [0]
>= [0 1 0] [0]
[0 1 0] y + [0]
[0 0 1] [0]
= mark(y)
if_active(true(),x,y) = [0 1 0] [0 1 0] [0]
[0 1 0] x + [0 1 0] y + [0]
[0 0 1] [0 0 1] [0]
>= [0 1 0] [0]
[0 1 0] x + [0]
[0 0 1] [0]
= mark(x)
mark(0()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= 0()
mark(div(x,y)) = [0 1 1] [0]
[0 1 1] x + [0]
[0 0 0] [1]
>= [0 1 1] [0]
[0 1 1] x + [0]
[0 0 0] [1]
= div_active(mark(x),y)
mark(ge(x,y)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= ge_active(x,y)
mark(if(x,y,z)) = [0 1 0] [0 1 0] [0 1 0] [0]
[0 1 0] x + [0 1 0] y + [0 1 0] z + [0]
[0 0 0] [0 0 1] [0 0 1] [0]
>= [0 1 0] [0 1 0] [0 1 0] [0]
[0 1 0] x + [0 1 0] y + [0 1 0] z + [0]
[0 0 0] [0 0 1] [0 0 1] [0]
= if_active(mark(x),y,z)
mark(s(x)) = [0 1 0] [0]
[0 1 0] x + [0]
[0 0 0] [1]
>= [0 1 0] [0]
[0 1 0] x + [0]
[0 0 0] [1]
= s(mark(x))
minus_active(x,y) = [0]
[1]
[0]
>= [0]
[1]
[0]
= minus(x,y)
minus_active(0(),y) = [0]
[1]
[0]
>= [0]
[0]
[0]
= 0()
minus_active(s(x),s(y)) = [0]
[1]
[0]
>= [0]
[1]
[0]
= minus_active(x,y)
* Step 9: NaturalMI WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
ge_active(x,y) -> ge(x,y)
ge_active(0(),s(y)) -> false()
if_active(x,y,z) -> if(x,y,z)
if_active(true(),x,y) -> mark(x)
mark(0()) -> 0()
mark(div(x,y)) -> div_active(mark(x),y)
mark(s(x)) -> s(mark(x))
minus_active(x,y) -> minus(x,y)
minus_active(0(),y) -> 0()
- 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(s(x),s(y)) -> ge_active(x,y)
if_active(false(),x,y) -> mark(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)
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:
NaturalMI {miDimension = 3, miDegree = 3, 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_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]
[0]
p(div) = [0 0 0] [0]
[0 1 1] x1 + [0]
[0 0 1] [0]
p(div_active) = [1 0 1] [0]
[0 1 1] x1 + [0]
[0 0 1] [0]
p(false) = [1]
[0]
[0]
p(ge) = [1]
[0]
[0]
p(ge_active) = [1]
[0]
[0]
p(if) = [0 0 0] [0 0 0] [0 0 0] [0]
[0 1 0] x1 + [0 1 0] x2 + [0 1 0] x3 + [0]
[0 0 0] [0 0 1] [0 0 1] [0]
p(if_active) = [1 0 0] [0 1 0] [0 1 0] [0]
[0 1 0] x1 + [0 1 0] x2 + [0 1 0] x3 + [0]
[0 0 0] [0 0 1] [0 0 1] [0]
p(mark) = [0 1 0] [1]
[0 1 0] x1 + [0]
[0 0 1] [0]
p(minus) = [1]
[0]
[0]
p(minus_active) = [1]
[0]
[0]
p(s) = [1 0 0] [0]
[0 1 0] x1 + [0]
[0 0 0] [1]
p(true) = [1]
[0]
[0]
Following rules are strictly oriented:
mark(0()) = [1]
[0]
[0]
> [0]
[0]
[0]
= 0()
minus_active(0(),y) = [1]
[0]
[0]
> [0]
[0]
[0]
= 0()
Following rules are (at-least) weakly oriented:
div_active(x,y) = [1 0 1] [0]
[0 1 1] x + [0]
[0 0 1] [0]
>= [0 0 0] [0]
[0 1 1] x + [0]
[0 0 1] [0]
= div(x,y)
div_active(0(),s(y)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= 0()
div_active(s(x),s(y)) = [1 0 0] [1]
[0 1 0] x + [1]
[0 0 0] [1]
>= [1]
[0]
[1]
= if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0())
ge_active(x,y) = [1]
[0]
[0]
>= [1]
[0]
[0]
= ge(x,y)
ge_active(x,0()) = [1]
[0]
[0]
>= [1]
[0]
[0]
= true()
ge_active(0(),s(y)) = [1]
[0]
[0]
>= [1]
[0]
[0]
= false()
ge_active(s(x),s(y)) = [1]
[0]
[0]
>= [1]
[0]
[0]
= ge_active(x,y)
if_active(x,y,z) = [1 0 0] [0 1 0] [0 1 0] [0]
[0 1 0] x + [0 1 0] y + [0 1 0] z + [0]
[0 0 0] [0 0 1] [0 0 1] [0]
>= [0 0 0] [0 0 0] [0 0 0] [0]
[0 1 0] x + [0 1 0] y + [0 1 0] z + [0]
[0 0 0] [0 0 1] [0 0 1] [0]
= if(x,y,z)
if_active(false(),x,y) = [0 1 0] [0 1 0] [1]
[0 1 0] x + [0 1 0] y + [0]
[0 0 1] [0 0 1] [0]
>= [0 1 0] [1]
[0 1 0] y + [0]
[0 0 1] [0]
= mark(y)
if_active(true(),x,y) = [0 1 0] [0 1 0] [1]
[0 1 0] x + [0 1 0] y + [0]
[0 0 1] [0 0 1] [0]
>= [0 1 0] [1]
[0 1 0] x + [0]
[0 0 1] [0]
= mark(x)
mark(div(x,y)) = [0 1 1] [1]
[0 1 1] x + [0]
[0 0 1] [0]
>= [0 1 1] [1]
[0 1 1] x + [0]
[0 0 1] [0]
= div_active(mark(x),y)
mark(ge(x,y)) = [1]
[0]
[0]
>= [1]
[0]
[0]
= ge_active(x,y)
mark(if(x,y,z)) = [0 1 0] [0 1 0] [0 1 0] [1]
[0 1 0] x + [0 1 0] y + [0 1 0] z + [0]
[0 0 0] [0 0 1] [0 0 1] [0]
>= [0 1 0] [0 1 0] [0 1 0] [1]
[0 1 0] x + [0 1 0] y + [0 1 0] z + [0]
[0 0 0] [0 0 1] [0 0 1] [0]
= if_active(mark(x),y,z)
mark(minus(x,y)) = [1]
[0]
[0]
>= [1]
[0]
[0]
= minus_active(x,y)
mark(s(x)) = [0 1 0] [1]
[0 1 0] x + [0]
[0 0 0] [1]
>= [0 1 0] [1]
[0 1 0] x + [0]
[0 0 0] [1]
= s(mark(x))
minus_active(x,y) = [1]
[0]
[0]
>= [1]
[0]
[0]
= minus(x,y)
minus_active(s(x),s(y)) = [1]
[0]
[0]
>= [1]
[0]
[0]
= minus_active(x,y)
* Step 10: NaturalMI WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
ge_active(x,y) -> ge(x,y)
ge_active(0(),s(y)) -> false()
if_active(x,y,z) -> if(x,y,z)
if_active(true(),x,y) -> mark(x)
mark(div(x,y)) -> div_active(mark(x),y)
mark(s(x)) -> s(mark(x))
minus_active(x,y) -> minus(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,0()) -> true()
ge_active(s(x),s(y)) -> ge_active(x,y)
if_active(false(),x,y) -> mark(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(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:
NaturalMI {miDimension = 3, miDegree = 3, 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_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]
[0]
p(div) = [0 0 0] [0]
[0 1 1] x1 + [1]
[0 0 1] [0]
p(div_active) = [1 0 1] [1]
[0 1 1] x1 + [1]
[0 0 1] [0]
p(false) = [0]
[0]
[0]
p(ge) = [0]
[0]
[0]
p(ge_active) = [0]
[0]
[0]
p(if) = [0 0 0] [0 0 0] [0 0 0] [0]
[0 1 0] x1 + [0 1 0] x2 + [0 1 0] x3 + [0]
[0 0 0] [0 0 1] [0 0 1] [0]
p(if_active) = [1 0 0] [0 1 0] [0 1 0] [0]
[0 1 0] x1 + [0 1 0] x2 + [0 1 0] x3 + [0]
[0 0 0] [0 0 1] [0 0 1] [0]
p(mark) = [0 1 0] [0]
[0 1 0] x1 + [0]
[0 0 1] [0]
p(minus) = [0 0 0] [0]
[0 0 1] x1 + [1]
[0 0 0] [0]
p(minus_active) = [0 0 0] [1]
[0 0 1] x1 + [1]
[0 0 0] [0]
p(s) = [1 0 0] [0]
[0 1 0] x1 + [0]
[0 0 1] [1]
p(true) = [0]
[0]
[0]
Following rules are strictly oriented:
minus_active(x,y) = [0 0 0] [1]
[0 0 1] x + [1]
[0 0 0] [0]
> [0 0 0] [0]
[0 0 1] x + [1]
[0 0 0] [0]
= minus(x,y)
Following rules are (at-least) weakly oriented:
div_active(x,y) = [1 0 1] [1]
[0 1 1] x + [1]
[0 0 1] [0]
>= [0 0 0] [0]
[0 1 1] x + [1]
[0 0 1] [0]
= div(x,y)
div_active(0(),s(y)) = [1]
[1]
[0]
>= [0]
[0]
[0]
= 0()
div_active(s(x),s(y)) = [1 0 1] [2]
[0 1 1] x + [2]
[0 0 1] [1]
>= [0 0 1] [2]
[0 0 1] x + [2]
[0 0 0] [1]
= if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0())
ge_active(x,y) = [0]
[0]
[0]
>= [0]
[0]
[0]
= ge(x,y)
ge_active(x,0()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= true()
ge_active(0(),s(y)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= false()
ge_active(s(x),s(y)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= ge_active(x,y)
if_active(x,y,z) = [1 0 0] [0 1 0] [0 1 0] [0]
[0 1 0] x + [0 1 0] y + [0 1 0] z + [0]
[0 0 0] [0 0 1] [0 0 1] [0]
>= [0 0 0] [0 0 0] [0 0 0] [0]
[0 1 0] x + [0 1 0] y + [0 1 0] z + [0]
[0 0 0] [0 0 1] [0 0 1] [0]
= if(x,y,z)
if_active(false(),x,y) = [0 1 0] [0 1 0] [0]
[0 1 0] x + [0 1 0] y + [0]
[0 0 1] [0 0 1] [0]
>= [0 1 0] [0]
[0 1 0] y + [0]
[0 0 1] [0]
= mark(y)
if_active(true(),x,y) = [0 1 0] [0 1 0] [0]
[0 1 0] x + [0 1 0] y + [0]
[0 0 1] [0 0 1] [0]
>= [0 1 0] [0]
[0 1 0] x + [0]
[0 0 1] [0]
= mark(x)
mark(0()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= 0()
mark(div(x,y)) = [0 1 1] [1]
[0 1 1] x + [1]
[0 0 1] [0]
>= [0 1 1] [1]
[0 1 1] x + [1]
[0 0 1] [0]
= div_active(mark(x),y)
mark(ge(x,y)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= ge_active(x,y)
mark(if(x,y,z)) = [0 1 0] [0 1 0] [0 1 0] [0]
[0 1 0] x + [0 1 0] y + [0 1 0] z + [0]
[0 0 0] [0 0 1] [0 0 1] [0]
>= [0 1 0] [0 1 0] [0 1 0] [0]
[0 1 0] x + [0 1 0] y + [0 1 0] z + [0]
[0 0 0] [0 0 1] [0 0 1] [0]
= if_active(mark(x),y,z)
mark(minus(x,y)) = [0 0 1] [1]
[0 0 1] x + [1]
[0 0 0] [0]
>= [0 0 0] [1]
[0 0 1] x + [1]
[0 0 0] [0]
= minus_active(x,y)
mark(s(x)) = [0 1 0] [0]
[0 1 0] x + [0]
[0 0 1] [1]
>= [0 1 0] [0]
[0 1 0] x + [0]
[0 0 1] [1]
= s(mark(x))
minus_active(0(),y) = [1]
[1]
[0]
>= [0]
[0]
[0]
= 0()
minus_active(s(x),s(y)) = [0 0 0] [1]
[0 0 1] x + [2]
[0 0 0] [0]
>= [0 0 0] [1]
[0 0 1] x + [1]
[0 0 0] [0]
= minus_active(x,y)
* Step 11: NaturalMI WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
ge_active(x,y) -> ge(x,y)
ge_active(0(),s(y)) -> false()
if_active(x,y,z) -> if(x,y,z)
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(s(x),s(y)) -> ge_active(x,y)
if_active(false(),x,y) -> mark(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:
NaturalMI {miDimension = 3, miDegree = 3, 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_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]
[0]
p(div) = [1 1 1] [0]
[0 0 0] x1 + [0]
[0 0 0] [1]
p(div_active) = [1 1 1] [0]
[0 0 0] x1 + [0]
[0 0 0] [1]
p(false) = [1]
[0]
[0]
p(ge) = [1]
[0]
[0]
p(ge_active) = [1]
[0]
[0]
p(if) = [1 0 0] [1 0 0] [1 0 0] [0]
[0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 1] [0 0 1] [0]
p(if_active) = [1 0 0] [1 0 0] [1 0 0] [0]
[0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 1] [0 0 1] [0]
p(mark) = [1 0 0] [0]
[0 0 0] x1 + [0]
[0 0 1] [0]
p(minus) = [0]
[0]
[0]
p(minus_active) = [0]
[0]
[0]
p(s) = [1 0 0] [1]
[0 0 0] x1 + [0]
[0 0 0] [1]
p(true) = [1]
[0]
[0]
Following rules are strictly oriented:
if_active(true(),x,y) = [1 0 0] [1 0 0] [1]
[0 0 0] x + [0 0 0] y + [0]
[0 0 1] [0 0 1] [0]
> [1 0 0] [0]
[0 0 0] x + [0]
[0 0 1] [0]
= mark(x)
Following rules are (at-least) weakly oriented:
div_active(x,y) = [1 1 1] [0]
[0 0 0] x + [0]
[0 0 0] [1]
>= [1 1 1] [0]
[0 0 0] x + [0]
[0 0 0] [1]
= div(x,y)
div_active(0(),s(y)) = [0]
[0]
[1]
>= [0]
[0]
[0]
= 0()
div_active(s(x),s(y)) = [1 0 0] [2]
[0 0 0] x + [0]
[0 0 0] [1]
>= [2]
[0]
[1]
= if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0())
ge_active(x,y) = [1]
[0]
[0]
>= [1]
[0]
[0]
= ge(x,y)
ge_active(x,0()) = [1]
[0]
[0]
>= [1]
[0]
[0]
= true()
ge_active(0(),s(y)) = [1]
[0]
[0]
>= [1]
[0]
[0]
= false()
ge_active(s(x),s(y)) = [1]
[0]
[0]
>= [1]
[0]
[0]
= ge_active(x,y)
if_active(x,y,z) = [1 0 0] [1 0 0] [1 0 0] [0]
[0 0 0] x + [0 0 0] y + [0 0 0] z + [0]
[0 0 0] [0 0 1] [0 0 1] [0]
>= [1 0 0] [1 0 0] [1 0 0] [0]
[0 0 0] x + [0 0 0] y + [0 0 0] z + [0]
[0 0 0] [0 0 1] [0 0 1] [0]
= if(x,y,z)
if_active(false(),x,y) = [1 0 0] [1 0 0] [1]
[0 0 0] x + [0 0 0] y + [0]
[0 0 1] [0 0 1] [0]
>= [1 0 0] [0]
[0 0 0] y + [0]
[0 0 1] [0]
= mark(y)
mark(0()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= 0()
mark(div(x,y)) = [1 1 1] [0]
[0 0 0] x + [0]
[0 0 0] [1]
>= [1 0 1] [0]
[0 0 0] x + [0]
[0 0 0] [1]
= div_active(mark(x),y)
mark(ge(x,y)) = [1]
[0]
[0]
>= [1]
[0]
[0]
= ge_active(x,y)
mark(if(x,y,z)) = [1 0 0] [1 0 0] [1 0 0] [0]
[0 0 0] x + [0 0 0] y + [0 0 0] z + [0]
[0 0 0] [0 0 1] [0 0 1] [0]
>= [1 0 0] [1 0 0] [1 0 0] [0]
[0 0 0] x + [0 0 0] y + [0 0 0] z + [0]
[0 0 0] [0 0 1] [0 0 1] [0]
= if_active(mark(x),y,z)
mark(minus(x,y)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= minus_active(x,y)
mark(s(x)) = [1 0 0] [1]
[0 0 0] x + [0]
[0 0 0] [1]
>= [1 0 0] [1]
[0 0 0] x + [0]
[0 0 0] [1]
= s(mark(x))
minus_active(x,y) = [0]
[0]
[0]
>= [0]
[0]
[0]
= minus(x,y)
minus_active(0(),y) = [0]
[0]
[0]
>= [0]
[0]
[0]
= 0()
minus_active(s(x),s(y)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= minus_active(x,y)
* Step 12: NaturalMI WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
ge_active(x,y) -> ge(x,y)
ge_active(0(),s(y)) -> false()
if_active(x,y,z) -> if(x,y,z)
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(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)
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:
NaturalMI {miDimension = 3, miDegree = 3, 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_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]
[0]
p(div) = [1 1 0] [0]
[0 1 0] x1 + [0]
[0 0 0] [0]
p(div_active) = [1 1 0] [0]
[0 1 0] x1 + [0]
[0 0 0] [0]
p(false) = [0]
[0]
[0]
p(ge) = [1]
[0]
[0]
p(ge_active) = [1]
[0]
[0]
p(if) = [1 0 0] [1 0 0] [1 0 0] [0]
[0 0 0] x1 + [0 1 0] x2 + [0 1 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
p(if_active) = [1 0 0] [1 0 0] [1 0 0] [0]
[0 0 0] x1 + [0 1 0] x2 + [0 1 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
p(mark) = [1 0 0] [0]
[0 1 0] x1 + [0]
[0 0 0] [0]
p(minus) = [0]
[0]
[0]
p(minus_active) = [0]
[0]
[0]
p(s) = [1 0 0] [0]
[0 0 0] x1 + [1]
[0 0 0] [0]
p(true) = [1]
[0]
[0]
Following rules are strictly oriented:
ge_active(0(),s(y)) = [1]
[0]
[0]
> [0]
[0]
[0]
= false()
Following rules are (at-least) weakly oriented:
div_active(x,y) = [1 1 0] [0]
[0 1 0] x + [0]
[0 0 0] [0]
>= [1 1 0] [0]
[0 1 0] x + [0]
[0 0 0] [0]
= div(x,y)
div_active(0(),s(y)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= 0()
div_active(s(x),s(y)) = [1 0 0] [1]
[0 0 0] x + [1]
[0 0 0] [0]
>= [1]
[1]
[0]
= if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0())
ge_active(x,y) = [1]
[0]
[0]
>= [1]
[0]
[0]
= ge(x,y)
ge_active(x,0()) = [1]
[0]
[0]
>= [1]
[0]
[0]
= true()
ge_active(s(x),s(y)) = [1]
[0]
[0]
>= [1]
[0]
[0]
= ge_active(x,y)
if_active(x,y,z) = [1 0 0] [1 0 0] [1 0 0] [0]
[0 0 0] x + [0 1 0] y + [0 1 0] z + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
>= [1 0 0] [1 0 0] [1 0 0] [0]
[0 0 0] x + [0 1 0] y + [0 1 0] z + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
= if(x,y,z)
if_active(false(),x,y) = [1 0 0] [1 0 0] [0]
[0 1 0] x + [0 1 0] y + [0]
[0 0 0] [0 0 0] [0]
>= [1 0 0] [0]
[0 1 0] y + [0]
[0 0 0] [0]
= mark(y)
if_active(true(),x,y) = [1 0 0] [1 0 0] [1]
[0 1 0] x + [0 1 0] y + [0]
[0 0 0] [0 0 0] [0]
>= [1 0 0] [0]
[0 1 0] x + [0]
[0 0 0] [0]
= mark(x)
mark(0()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= 0()
mark(div(x,y)) = [1 1 0] [0]
[0 1 0] x + [0]
[0 0 0] [0]
>= [1 1 0] [0]
[0 1 0] x + [0]
[0 0 0] [0]
= div_active(mark(x),y)
mark(ge(x,y)) = [1]
[0]
[0]
>= [1]
[0]
[0]
= ge_active(x,y)
mark(if(x,y,z)) = [1 0 0] [1 0 0] [1 0 0] [0]
[0 0 0] x + [0 1 0] y + [0 1 0] z + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
>= [1 0 0] [1 0 0] [1 0 0] [0]
[0 0 0] x + [0 1 0] y + [0 1 0] z + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
= if_active(mark(x),y,z)
mark(minus(x,y)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= minus_active(x,y)
mark(s(x)) = [1 0 0] [0]
[0 0 0] x + [1]
[0 0 0] [0]
>= [1 0 0] [0]
[0 0 0] x + [1]
[0 0 0] [0]
= s(mark(x))
minus_active(x,y) = [0]
[0]
[0]
>= [0]
[0]
[0]
= minus(x,y)
minus_active(0(),y) = [0]
[0]
[0]
>= [0]
[0]
[0]
= 0()
minus_active(s(x),s(y)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= minus_active(x,y)
* Step 13: NaturalMI WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
ge_active(x,y) -> ge(x,y)
if_active(x,y,z) -> if(x,y,z)
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)
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)
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:
NaturalMI {miDimension = 3, miDegree = 3, 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_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]
[0]
p(div) = [0 0 0] [0]
[0 1 1] x1 + [0]
[0 0 1] [0]
p(div_active) = [1 0 1] [0]
[0 1 1] x1 + [0]
[0 0 1] [0]
p(false) = [0]
[1]
[0]
p(ge) = [0]
[1]
[0]
p(ge_active) = [1]
[1]
[0]
p(if) = [0 0 0] [0 0 0] [0 0 0] [0]
[0 1 1] x1 + [0 1 0] x2 + [0 1 0] x3 + [0]
[0 0 0] [0 0 1] [0 0 1] [0]
p(if_active) = [1 0 0] [0 1 0] [0 1 0] [0]
[0 1 1] x1 + [0 1 0] x2 + [0 1 0] x3 + [0]
[0 0 0] [0 0 1] [0 0 1] [0]
p(mark) = [0 1 0] [0]
[0 1 0] x1 + [0]
[0 0 1] [0]
p(minus) = [0]
[0]
[0]
p(minus_active) = [0]
[0]
[0]
p(s) = [1 0 0] [0]
[0 1 0] x1 + [0]
[0 0 0] [1]
p(true) = [1]
[1]
[0]
Following rules are strictly oriented:
ge_active(x,y) = [1]
[1]
[0]
> [0]
[1]
[0]
= ge(x,y)
Following rules are (at-least) weakly oriented:
div_active(x,y) = [1 0 1] [0]
[0 1 1] x + [0]
[0 0 1] [0]
>= [0 0 0] [0]
[0 1 1] x + [0]
[0 0 1] [0]
= div(x,y)
div_active(0(),s(y)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= 0()
div_active(s(x),s(y)) = [1 0 0] [1]
[0 1 0] x + [1]
[0 0 0] [1]
>= [1]
[1]
[1]
= if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0())
ge_active(x,0()) = [1]
[1]
[0]
>= [1]
[1]
[0]
= true()
ge_active(0(),s(y)) = [1]
[1]
[0]
>= [0]
[1]
[0]
= false()
ge_active(s(x),s(y)) = [1]
[1]
[0]
>= [1]
[1]
[0]
= ge_active(x,y)
if_active(x,y,z) = [1 0 0] [0 1 0] [0 1 0] [0]
[0 1 1] x + [0 1 0] y + [0 1 0] z + [0]
[0 0 0] [0 0 1] [0 0 1] [0]
>= [0 0 0] [0 0 0] [0 0 0] [0]
[0 1 1] x + [0 1 0] y + [0 1 0] z + [0]
[0 0 0] [0 0 1] [0 0 1] [0]
= if(x,y,z)
if_active(false(),x,y) = [0 1 0] [0 1 0] [0]
[0 1 0] x + [0 1 0] y + [1]
[0 0 1] [0 0 1] [0]
>= [0 1 0] [0]
[0 1 0] y + [0]
[0 0 1] [0]
= mark(y)
if_active(true(),x,y) = [0 1 0] [0 1 0] [1]
[0 1 0] x + [0 1 0] y + [1]
[0 0 1] [0 0 1] [0]
>= [0 1 0] [0]
[0 1 0] x + [0]
[0 0 1] [0]
= mark(x)
mark(0()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= 0()
mark(div(x,y)) = [0 1 1] [0]
[0 1 1] x + [0]
[0 0 1] [0]
>= [0 1 1] [0]
[0 1 1] x + [0]
[0 0 1] [0]
= div_active(mark(x),y)
mark(ge(x,y)) = [1]
[1]
[0]
>= [1]
[1]
[0]
= ge_active(x,y)
mark(if(x,y,z)) = [0 1 1] [0 1 0] [0 1 0] [0]
[0 1 1] x + [0 1 0] y + [0 1 0] z + [0]
[0 0 0] [0 0 1] [0 0 1] [0]
>= [0 1 0] [0 1 0] [0 1 0] [0]
[0 1 1] x + [0 1 0] y + [0 1 0] z + [0]
[0 0 0] [0 0 1] [0 0 1] [0]
= if_active(mark(x),y,z)
mark(minus(x,y)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= minus_active(x,y)
mark(s(x)) = [0 1 0] [0]
[0 1 0] x + [0]
[0 0 0] [1]
>= [0 1 0] [0]
[0 1 0] x + [0]
[0 0 0] [1]
= s(mark(x))
minus_active(x,y) = [0]
[0]
[0]
>= [0]
[0]
[0]
= minus(x,y)
minus_active(0(),y) = [0]
[0]
[0]
>= [0]
[0]
[0]
= 0()
minus_active(s(x),s(y)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= minus_active(x,y)
* Step 14: NaturalMI WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
if_active(x,y,z) -> if(x,y,z)
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(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)
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:
NaturalMI {miDimension = 3, miDegree = 3, 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_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]
[0]
p(div) = [0 0 0] [0]
[0 1 1] x1 + [0]
[0 0 1] [0]
p(div_active) = [1 0 1] [0]
[0 1 1] x1 + [0]
[0 0 1] [0]
p(false) = [0]
[0]
[0]
p(ge) = [0]
[0]
[0]
p(ge_active) = [0]
[0]
[0]
p(if) = [0 0 0] [0 0 0] [0 0 0] [0]
[0 1 0] x1 + [0 1 0] x2 + [0 1 1] x3 + [1]
[0 0 0] [0 0 1] [0 0 1] [0]
p(if_active) = [1 0 0] [0 1 0] [0 1 0] [1]
[0 1 0] x1 + [0 1 0] x2 + [0 1 1] x3 + [1]
[0 0 0] [0 0 1] [0 0 1] [0]
p(mark) = [0 1 0] [0]
[0 1 0] x1 + [0]
[0 0 1] [0]
p(minus) = [0]
[0]
[0]
p(minus_active) = [0]
[0]
[0]
p(s) = [1 0 0] [0]
[0 1 0] x1 + [0]
[0 0 0] [1]
p(true) = [0]
[0]
[0]
Following rules are strictly oriented:
if_active(x,y,z) = [1 0 0] [0 1 0] [0 1 0] [1]
[0 1 0] x + [0 1 0] y + [0 1 1] z + [1]
[0 0 0] [0 0 1] [0 0 1] [0]
> [0 0 0] [0 0 0] [0 0 0] [0]
[0 1 0] x + [0 1 0] y + [0 1 1] z + [1]
[0 0 0] [0 0 1] [0 0 1] [0]
= if(x,y,z)
Following rules are (at-least) weakly oriented:
div_active(x,y) = [1 0 1] [0]
[0 1 1] x + [0]
[0 0 1] [0]
>= [0 0 0] [0]
[0 1 1] x + [0]
[0 0 1] [0]
= div(x,y)
div_active(0(),s(y)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= 0()
div_active(s(x),s(y)) = [1 0 0] [1]
[0 1 0] x + [1]
[0 0 0] [1]
>= [1]
[1]
[1]
= if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0())
ge_active(x,y) = [0]
[0]
[0]
>= [0]
[0]
[0]
= ge(x,y)
ge_active(x,0()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= true()
ge_active(0(),s(y)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= false()
ge_active(s(x),s(y)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= ge_active(x,y)
if_active(false(),x,y) = [0 1 0] [0 1 0] [1]
[0 1 0] x + [0 1 1] y + [1]
[0 0 1] [0 0 1] [0]
>= [0 1 0] [0]
[0 1 0] y + [0]
[0 0 1] [0]
= mark(y)
if_active(true(),x,y) = [0 1 0] [0 1 0] [1]
[0 1 0] x + [0 1 1] y + [1]
[0 0 1] [0 0 1] [0]
>= [0 1 0] [0]
[0 1 0] x + [0]
[0 0 1] [0]
= mark(x)
mark(0()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= 0()
mark(div(x,y)) = [0 1 1] [0]
[0 1 1] x + [0]
[0 0 1] [0]
>= [0 1 1] [0]
[0 1 1] x + [0]
[0 0 1] [0]
= div_active(mark(x),y)
mark(ge(x,y)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= ge_active(x,y)
mark(if(x,y,z)) = [0 1 0] [0 1 0] [0 1 1] [1]
[0 1 0] x + [0 1 0] y + [0 1 1] z + [1]
[0 0 0] [0 0 1] [0 0 1] [0]
>= [0 1 0] [0 1 0] [0 1 0] [1]
[0 1 0] x + [0 1 0] y + [0 1 1] z + [1]
[0 0 0] [0 0 1] [0 0 1] [0]
= if_active(mark(x),y,z)
mark(minus(x,y)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= minus_active(x,y)
mark(s(x)) = [0 1 0] [0]
[0 1 0] x + [0]
[0 0 0] [1]
>= [0 1 0] [0]
[0 1 0] x + [0]
[0 0 0] [1]
= s(mark(x))
minus_active(x,y) = [0]
[0]
[0]
>= [0]
[0]
[0]
= minus(x,y)
minus_active(0(),y) = [0]
[0]
[0]
>= [0]
[0]
[0]
= 0()
minus_active(s(x),s(y)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= minus_active(x,y)
* Step 15: NaturalMI WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
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(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)
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:
NaturalMI {miDimension = 3, miDegree = 3, 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_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]
[0]
p(div) = [0 0 0] [0]
[0 1 1] x1 + [0]
[0 0 1] [0]
p(div_active) = [1 0 1] [0]
[0 1 1] x1 + [0]
[0 0 1] [0]
p(false) = [0]
[0]
[0]
p(ge) = [0 0 0] [0]
[0 0 0] x1 + [0]
[0 0 1] [0]
p(ge_active) = [0 0 0] [0]
[0 0 0] x1 + [0]
[0 0 1] [0]
p(if) = [0 0 0] [0 0 0] [0 0 0] [0]
[0 1 0] x1 + [0 1 0] x2 + [0 1 1] x3 + [1]
[0 0 1] [0 0 1] [0 0 1] [0]
p(if_active) = [1 0 0] [0 1 1] [0 1 1] [0]
[0 1 0] x1 + [0 1 0] x2 + [0 1 1] x3 + [1]
[0 0 1] [0 0 1] [0 0 1] [0]
p(mark) = [0 1 1] [0]
[0 1 0] x1 + [0]
[0 0 1] [0]
p(minus) = [0]
[0]
[0]
p(minus_active) = [0]
[0]
[0]
p(s) = [1 0 0] [0]
[0 1 0] x1 + [0]
[0 0 1] [1]
p(true) = [0]
[0]
[0]
Following rules are strictly oriented:
mark(s(x)) = [0 1 1] [1]
[0 1 0] x + [0]
[0 0 1] [1]
> [0 1 1] [0]
[0 1 0] x + [0]
[0 0 1] [1]
= s(mark(x))
Following rules are (at-least) weakly oriented:
div_active(x,y) = [1 0 1] [0]
[0 1 1] x + [0]
[0 0 1] [0]
>= [0 0 0] [0]
[0 1 1] x + [0]
[0 0 1] [0]
= div(x,y)
div_active(0(),s(y)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= 0()
div_active(s(x),s(y)) = [1 0 1] [1]
[0 1 1] x + [1]
[0 0 1] [1]
>= [0 0 0] [1]
[0 0 0] x + [1]
[0 0 1] [1]
= if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0())
ge_active(x,y) = [0 0 0] [0]
[0 0 0] x + [0]
[0 0 1] [0]
>= [0 0 0] [0]
[0 0 0] x + [0]
[0 0 1] [0]
= ge(x,y)
ge_active(x,0()) = [0 0 0] [0]
[0 0 0] x + [0]
[0 0 1] [0]
>= [0]
[0]
[0]
= true()
ge_active(0(),s(y)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= false()
ge_active(s(x),s(y)) = [0 0 0] [0]
[0 0 0] x + [0]
[0 0 1] [1]
>= [0 0 0] [0]
[0 0 0] x + [0]
[0 0 1] [0]
= ge_active(x,y)
if_active(x,y,z) = [1 0 0] [0 1 1] [0 1 1] [0]
[0 1 0] x + [0 1 0] y + [0 1 1] z + [1]
[0 0 1] [0 0 1] [0 0 1] [0]
>= [0 0 0] [0 0 0] [0 0 0] [0]
[0 1 0] x + [0 1 0] y + [0 1 1] z + [1]
[0 0 1] [0 0 1] [0 0 1] [0]
= if(x,y,z)
if_active(false(),x,y) = [0 1 1] [0 1 1] [0]
[0 1 0] x + [0 1 1] y + [1]
[0 0 1] [0 0 1] [0]
>= [0 1 1] [0]
[0 1 0] y + [0]
[0 0 1] [0]
= mark(y)
if_active(true(),x,y) = [0 1 1] [0 1 1] [0]
[0 1 0] x + [0 1 1] y + [1]
[0 0 1] [0 0 1] [0]
>= [0 1 1] [0]
[0 1 0] x + [0]
[0 0 1] [0]
= mark(x)
mark(0()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= 0()
mark(div(x,y)) = [0 1 2] [0]
[0 1 1] x + [0]
[0 0 1] [0]
>= [0 1 2] [0]
[0 1 1] x + [0]
[0 0 1] [0]
= div_active(mark(x),y)
mark(ge(x,y)) = [0 0 1] [0]
[0 0 0] x + [0]
[0 0 1] [0]
>= [0 0 0] [0]
[0 0 0] x + [0]
[0 0 1] [0]
= ge_active(x,y)
mark(if(x,y,z)) = [0 1 1] [0 1 1] [0 1 2] [1]
[0 1 0] x + [0 1 0] y + [0 1 1] z + [1]
[0 0 1] [0 0 1] [0 0 1] [0]
>= [0 1 1] [0 1 1] [0 1 1] [0]
[0 1 0] x + [0 1 0] y + [0 1 1] z + [1]
[0 0 1] [0 0 1] [0 0 1] [0]
= if_active(mark(x),y,z)
mark(minus(x,y)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= minus_active(x,y)
minus_active(x,y) = [0]
[0]
[0]
>= [0]
[0]
[0]
= minus(x,y)
minus_active(0(),y) = [0]
[0]
[0]
>= [0]
[0]
[0]
= 0()
minus_active(s(x),s(y)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= minus_active(x,y)
* Step 16: NaturalMI WORST_CASE(?,O(n^3))
+ 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:
NaturalMI {miDimension = 3, miDegree = 3, 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_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]
[0]
p(div) = [0 0 0] [0]
[0 1 1] x1 + [1]
[0 0 1] [0]
p(div_active) = [1 0 1] [0]
[0 1 1] x1 + [1]
[0 0 1] [0]
p(false) = [0]
[0]
[0]
p(ge) = [0]
[1]
[0]
p(ge_active) = [0]
[1]
[0]
p(if) = [0 0 0] [0 0 0] [0 0 0] [0]
[0 1 1] x1 + [0 1 0] x2 + [0 1 0] x3 + [0]
[0 0 0] [0 0 1] [0 0 1] [0]
p(if_active) = [1 0 1] [0 1 0] [0 1 0] [0]
[0 1 1] x1 + [0 1 0] x2 + [0 1 0] x3 + [0]
[0 0 0] [0 0 1] [0 0 1] [0]
p(mark) = [0 1 0] [0]
[0 1 0] x1 + [0]
[0 0 1] [0]
p(minus) = [0]
[0]
[0]
p(minus_active) = [0]
[0]
[0]
p(s) = [1 0 0] [0]
[0 1 0] x1 + [0]
[0 0 0] [1]
p(true) = [0]
[0]
[0]
Following rules are strictly oriented:
mark(div(x,y)) = [0 1 1] [1]
[0 1 1] x + [1]
[0 0 1] [0]
> [0 1 1] [0]
[0 1 1] x + [1]
[0 0 1] [0]
= div_active(mark(x),y)
Following rules are (at-least) weakly oriented:
div_active(x,y) = [1 0 1] [0]
[0 1 1] x + [1]
[0 0 1] [0]
>= [0 0 0] [0]
[0 1 1] x + [1]
[0 0 1] [0]
= div(x,y)
div_active(0(),s(y)) = [0]
[1]
[0]
>= [0]
[0]
[0]
= 0()
div_active(s(x),s(y)) = [1 0 0] [1]
[0 1 0] x + [2]
[0 0 0] [1]
>= [1]
[2]
[1]
= if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0())
ge_active(x,y) = [0]
[1]
[0]
>= [0]
[1]
[0]
= ge(x,y)
ge_active(x,0()) = [0]
[1]
[0]
>= [0]
[0]
[0]
= true()
ge_active(0(),s(y)) = [0]
[1]
[0]
>= [0]
[0]
[0]
= false()
ge_active(s(x),s(y)) = [0]
[1]
[0]
>= [0]
[1]
[0]
= ge_active(x,y)
if_active(x,y,z) = [1 0 1] [0 1 0] [0 1 0] [0]
[0 1 1] x + [0 1 0] y + [0 1 0] z + [0]
[0 0 0] [0 0 1] [0 0 1] [0]
>= [0 0 0] [0 0 0] [0 0 0] [0]
[0 1 1] x + [0 1 0] y + [0 1 0] z + [0]
[0 0 0] [0 0 1] [0 0 1] [0]
= if(x,y,z)
if_active(false(),x,y) = [0 1 0] [0 1 0] [0]
[0 1 0] x + [0 1 0] y + [0]
[0 0 1] [0 0 1] [0]
>= [0 1 0] [0]
[0 1 0] y + [0]
[0 0 1] [0]
= mark(y)
if_active(true(),x,y) = [0 1 0] [0 1 0] [0]
[0 1 0] x + [0 1 0] y + [0]
[0 0 1] [0 0 1] [0]
>= [0 1 0] [0]
[0 1 0] x + [0]
[0 0 1] [0]
= mark(x)
mark(0()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= 0()
mark(ge(x,y)) = [1]
[1]
[0]
>= [0]
[1]
[0]
= ge_active(x,y)
mark(if(x,y,z)) = [0 1 1] [0 1 0] [0 1 0] [0]
[0 1 1] x + [0 1 0] y + [0 1 0] z + [0]
[0 0 0] [0 0 1] [0 0 1] [0]
>= [0 1 1] [0 1 0] [0 1 0] [0]
[0 1 1] x + [0 1 0] y + [0 1 0] z + [0]
[0 0 0] [0 0 1] [0 0 1] [0]
= if_active(mark(x),y,z)
mark(minus(x,y)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= minus_active(x,y)
mark(s(x)) = [0 1 0] [0]
[0 1 0] x + [0]
[0 0 0] [1]
>= [0 1 0] [0]
[0 1 0] x + [0]
[0 0 0] [1]
= s(mark(x))
minus_active(x,y) = [0]
[0]
[0]
>= [0]
[0]
[0]
= minus(x,y)
minus_active(0(),y) = [0]
[0]
[0]
>= [0]
[0]
[0]
= 0()
minus_active(s(x),s(y)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= minus_active(x,y)
* Step 17: 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(?,O(n^3))