WORST_CASE(?,O(n^3))
* Step 1: NaturalPI WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
a__div(X1,X2) -> div(X1,X2)
a__div(0(),s(Y)) -> 0()
a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
a__geq(X,0()) -> true()
a__geq(X1,X2) -> geq(X1,X2)
a__geq(0(),s(Y)) -> false()
a__geq(s(X),s(Y)) -> a__geq(X,Y)
a__if(X1,X2,X3) -> if(X1,X2,X3)
a__if(false(),X,Y) -> mark(Y)
a__if(true(),X,Y) -> mark(X)
a__minus(X1,X2) -> minus(X1,X2)
a__minus(0(),Y) -> 0()
a__minus(s(X),s(Y)) -> a__minus(X,Y)
mark(0()) -> 0()
mark(div(X1,X2)) -> a__div(mark(X1),X2)
mark(false()) -> false()
mark(geq(X1,X2)) -> a__geq(X1,X2)
mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
mark(minus(X1,X2)) -> a__minus(X1,X2)
mark(s(X)) -> s(mark(X))
mark(true()) -> true()
- Signature:
{a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1} / {0/0,div/2,false/0,geq/2,if/3,minus/2,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__div,a__geq,a__if,a__minus,mark} and constructors {0
,div,false,geq,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(a__div) = {1},
uargs(a__if) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{a__div,a__geq,a__if,a__minus,mark}
TcT has computed the following interpretation:
p(0) = 0
p(a__div) = 4 + x1 + 4*x2
p(a__geq) = 0
p(a__if) = x1 + 4*x2 + 4*x3
p(a__minus) = 0
p(div) = 1 + x1 + x2
p(false) = 0
p(geq) = 0
p(if) = x1 + x2 + x3
p(mark) = 4*x1
p(minus) = 0
p(s) = x1
p(true) = 0
Following rules are strictly oriented:
a__div(X1,X2) = 4 + X1 + 4*X2
> 1 + X1 + X2
= div(X1,X2)
a__div(0(),s(Y)) = 4 + 4*Y
> 0
= 0()
Following rules are (at-least) weakly oriented:
a__div(s(X),s(Y)) = 4 + X + 4*Y
>= 4 + 4*Y
= a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
a__geq(X,0()) = 0
>= 0
= true()
a__geq(X1,X2) = 0
>= 0
= geq(X1,X2)
a__geq(0(),s(Y)) = 0
>= 0
= false()
a__geq(s(X),s(Y)) = 0
>= 0
= a__geq(X,Y)
a__if(X1,X2,X3) = X1 + 4*X2 + 4*X3
>= X1 + X2 + X3
= if(X1,X2,X3)
a__if(false(),X,Y) = 4*X + 4*Y
>= 4*Y
= mark(Y)
a__if(true(),X,Y) = 4*X + 4*Y
>= 4*X
= mark(X)
a__minus(X1,X2) = 0
>= 0
= minus(X1,X2)
a__minus(0(),Y) = 0
>= 0
= 0()
a__minus(s(X),s(Y)) = 0
>= 0
= a__minus(X,Y)
mark(0()) = 0
>= 0
= 0()
mark(div(X1,X2)) = 4 + 4*X1 + 4*X2
>= 4 + 4*X1 + 4*X2
= a__div(mark(X1),X2)
mark(false()) = 0
>= 0
= false()
mark(geq(X1,X2)) = 0
>= 0
= a__geq(X1,X2)
mark(if(X1,X2,X3)) = 4*X1 + 4*X2 + 4*X3
>= 4*X1 + 4*X2 + 4*X3
= a__if(mark(X1),X2,X3)
mark(minus(X1,X2)) = 0
>= 0
= a__minus(X1,X2)
mark(s(X)) = 4*X
>= 4*X
= s(mark(X))
mark(true()) = 0
>= 0
= true()
* Step 2: NaturalPI WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
a__geq(X,0()) -> true()
a__geq(X1,X2) -> geq(X1,X2)
a__geq(0(),s(Y)) -> false()
a__geq(s(X),s(Y)) -> a__geq(X,Y)
a__if(X1,X2,X3) -> if(X1,X2,X3)
a__if(false(),X,Y) -> mark(Y)
a__if(true(),X,Y) -> mark(X)
a__minus(X1,X2) -> minus(X1,X2)
a__minus(0(),Y) -> 0()
a__minus(s(X),s(Y)) -> a__minus(X,Y)
mark(0()) -> 0()
mark(div(X1,X2)) -> a__div(mark(X1),X2)
mark(false()) -> false()
mark(geq(X1,X2)) -> a__geq(X1,X2)
mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
mark(minus(X1,X2)) -> a__minus(X1,X2)
mark(s(X)) -> s(mark(X))
mark(true()) -> true()
- Weak TRS:
a__div(X1,X2) -> div(X1,X2)
a__div(0(),s(Y)) -> 0()
- Signature:
{a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1} / {0/0,div/2,false/0,geq/2,if/3,minus/2,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__div,a__geq,a__if,a__minus,mark} and constructors {0
,div,false,geq,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(a__div) = {1},
uargs(a__if) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{a__div,a__geq,a__if,a__minus,mark}
TcT has computed the following interpretation:
p(0) = 0
p(a__div) = x1
p(a__geq) = x1
p(a__if) = x1 + x2 + x3
p(a__minus) = 0
p(div) = x1
p(false) = 0
p(geq) = x1
p(if) = x1 + x2 + x3
p(mark) = x1
p(minus) = 0
p(s) = 1 + x1
p(true) = 0
Following rules are strictly oriented:
a__geq(s(X),s(Y)) = 1 + X
> X
= a__geq(X,Y)
Following rules are (at-least) weakly oriented:
a__div(X1,X2) = X1
>= X1
= div(X1,X2)
a__div(0(),s(Y)) = 0
>= 0
= 0()
a__div(s(X),s(Y)) = 1 + X
>= 1 + X
= a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
a__geq(X,0()) = X
>= 0
= true()
a__geq(X1,X2) = X1
>= X1
= geq(X1,X2)
a__geq(0(),s(Y)) = 0
>= 0
= false()
a__if(X1,X2,X3) = X1 + X2 + X3
>= X1 + X2 + X3
= if(X1,X2,X3)
a__if(false(),X,Y) = X + Y
>= Y
= mark(Y)
a__if(true(),X,Y) = X + Y
>= X
= mark(X)
a__minus(X1,X2) = 0
>= 0
= minus(X1,X2)
a__minus(0(),Y) = 0
>= 0
= 0()
a__minus(s(X),s(Y)) = 0
>= 0
= a__minus(X,Y)
mark(0()) = 0
>= 0
= 0()
mark(div(X1,X2)) = X1
>= X1
= a__div(mark(X1),X2)
mark(false()) = 0
>= 0
= false()
mark(geq(X1,X2)) = X1
>= X1
= a__geq(X1,X2)
mark(if(X1,X2,X3)) = X1 + X2 + X3
>= X1 + X2 + X3
= a__if(mark(X1),X2,X3)
mark(minus(X1,X2)) = 0
>= 0
= a__minus(X1,X2)
mark(s(X)) = 1 + X
>= 1 + X
= s(mark(X))
mark(true()) = 0
>= 0
= true()
* Step 3: NaturalPI WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
a__geq(X,0()) -> true()
a__geq(X1,X2) -> geq(X1,X2)
a__geq(0(),s(Y)) -> false()
a__if(X1,X2,X3) -> if(X1,X2,X3)
a__if(false(),X,Y) -> mark(Y)
a__if(true(),X,Y) -> mark(X)
a__minus(X1,X2) -> minus(X1,X2)
a__minus(0(),Y) -> 0()
a__minus(s(X),s(Y)) -> a__minus(X,Y)
mark(0()) -> 0()
mark(div(X1,X2)) -> a__div(mark(X1),X2)
mark(false()) -> false()
mark(geq(X1,X2)) -> a__geq(X1,X2)
mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
mark(minus(X1,X2)) -> a__minus(X1,X2)
mark(s(X)) -> s(mark(X))
mark(true()) -> true()
- Weak TRS:
a__div(X1,X2) -> div(X1,X2)
a__div(0(),s(Y)) -> 0()
a__geq(s(X),s(Y)) -> a__geq(X,Y)
- Signature:
{a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1} / {0/0,div/2,false/0,geq/2,if/3,minus/2,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__div,a__geq,a__if,a__minus,mark} and constructors {0
,div,false,geq,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(a__div) = {1},
uargs(a__if) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{a__div,a__geq,a__if,a__minus,mark}
TcT has computed the following interpretation:
p(0) = 0
p(a__div) = 4 + x1 + x2
p(a__geq) = 0
p(a__if) = x1 + x2 + x3
p(a__minus) = x1
p(div) = 4 + x1 + x2
p(false) = 0
p(geq) = 0
p(if) = x1 + x2 + x3
p(mark) = x1
p(minus) = x1
p(s) = 4 + x1
p(true) = 0
Following rules are strictly oriented:
a__minus(s(X),s(Y)) = 4 + X
> X
= a__minus(X,Y)
Following rules are (at-least) weakly oriented:
a__div(X1,X2) = 4 + X1 + X2
>= 4 + X1 + X2
= div(X1,X2)
a__div(0(),s(Y)) = 8 + Y
>= 0
= 0()
a__div(s(X),s(Y)) = 12 + X + Y
>= 12 + X + Y
= a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
a__geq(X,0()) = 0
>= 0
= true()
a__geq(X1,X2) = 0
>= 0
= geq(X1,X2)
a__geq(0(),s(Y)) = 0
>= 0
= false()
a__geq(s(X),s(Y)) = 0
>= 0
= a__geq(X,Y)
a__if(X1,X2,X3) = X1 + X2 + X3
>= X1 + X2 + X3
= if(X1,X2,X3)
a__if(false(),X,Y) = X + Y
>= Y
= mark(Y)
a__if(true(),X,Y) = X + Y
>= X
= mark(X)
a__minus(X1,X2) = X1
>= X1
= minus(X1,X2)
a__minus(0(),Y) = 0
>= 0
= 0()
mark(0()) = 0
>= 0
= 0()
mark(div(X1,X2)) = 4 + X1 + X2
>= 4 + X1 + X2
= a__div(mark(X1),X2)
mark(false()) = 0
>= 0
= false()
mark(geq(X1,X2)) = 0
>= 0
= a__geq(X1,X2)
mark(if(X1,X2,X3)) = X1 + X2 + X3
>= X1 + X2 + X3
= a__if(mark(X1),X2,X3)
mark(minus(X1,X2)) = X1
>= X1
= a__minus(X1,X2)
mark(s(X)) = 4 + X
>= 4 + X
= s(mark(X))
mark(true()) = 0
>= 0
= true()
* Step 4: NaturalMI WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
a__geq(X,0()) -> true()
a__geq(X1,X2) -> geq(X1,X2)
a__geq(0(),s(Y)) -> false()
a__if(X1,X2,X3) -> if(X1,X2,X3)
a__if(false(),X,Y) -> mark(Y)
a__if(true(),X,Y) -> mark(X)
a__minus(X1,X2) -> minus(X1,X2)
a__minus(0(),Y) -> 0()
mark(0()) -> 0()
mark(div(X1,X2)) -> a__div(mark(X1),X2)
mark(false()) -> false()
mark(geq(X1,X2)) -> a__geq(X1,X2)
mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
mark(minus(X1,X2)) -> a__minus(X1,X2)
mark(s(X)) -> s(mark(X))
mark(true()) -> true()
- Weak TRS:
a__div(X1,X2) -> div(X1,X2)
a__div(0(),s(Y)) -> 0()
a__geq(s(X),s(Y)) -> a__geq(X,Y)
a__minus(s(X),s(Y)) -> a__minus(X,Y)
- Signature:
{a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1} / {0/0,div/2,false/0,geq/2,if/3,minus/2,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__div,a__geq,a__if,a__minus,mark} and constructors {0
,div,false,geq,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(a__div) = {1},
uargs(a__if) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{a__div,a__geq,a__if,a__minus,mark}
TcT has computed the following interpretation:
p(0) = [0]
[0]
[0]
p(a__div) = [1 0 0] [0]
[0 1 1] x1 + [0]
[0 0 1] [0]
p(a__geq) = [0]
[1]
[0]
p(a__if) = [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(a__minus) = [0]
[0]
[0]
p(div) = [0 0 0] [0]
[0 1 1] x1 + [0]
[0 0 1] [0]
p(false) = [0]
[1]
[0]
p(geq) = [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(mark) = [0 1 0] [0]
[0 1 0] x1 + [1]
[0 0 1] [0]
p(minus) = [0]
[0]
[0]
p(s) = [1 0 0] [0]
[0 1 0] x1 + [0]
[0 0 0] [1]
p(true) = [0]
[1]
[0]
Following rules are strictly oriented:
mark(false()) = [1]
[2]
[0]
> [0]
[1]
[0]
= false()
mark(true()) = [1]
[2]
[0]
> [0]
[1]
[0]
= true()
Following rules are (at-least) weakly oriented:
a__div(X1,X2) = [1 0 0] [0]
[0 1 1] X1 + [0]
[0 0 1] [0]
>= [0 0 0] [0]
[0 1 1] X1 + [0]
[0 0 1] [0]
= div(X1,X2)
a__div(0(),s(Y)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= 0()
a__div(s(X),s(Y)) = [1 0 0] [0]
[0 1 0] X + [1]
[0 0 0] [1]
>= [0]
[1]
[1]
= a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
a__geq(X,0()) = [0]
[1]
[0]
>= [0]
[1]
[0]
= true()
a__geq(X1,X2) = [0]
[1]
[0]
>= [0]
[0]
[0]
= geq(X1,X2)
a__geq(0(),s(Y)) = [0]
[1]
[0]
>= [0]
[1]
[0]
= false()
a__geq(s(X),s(Y)) = [0]
[1]
[0]
>= [0]
[1]
[0]
= a__geq(X,Y)
a__if(X1,X2,X3) = [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]
>= [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]
= if(X1,X2,X3)
a__if(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 + [1]
[0 0 1] [0]
= mark(Y)
a__if(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 + [1]
[0 0 1] [0]
= mark(X)
a__minus(X1,X2) = [0]
[0]
[0]
>= [0]
[0]
[0]
= minus(X1,X2)
a__minus(0(),Y) = [0]
[0]
[0]
>= [0]
[0]
[0]
= 0()
a__minus(s(X),s(Y)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= a__minus(X,Y)
mark(0()) = [0]
[1]
[0]
>= [0]
[0]
[0]
= 0()
mark(div(X1,X2)) = [0 1 1] [0]
[0 1 1] X1 + [1]
[0 0 1] [0]
>= [0 1 0] [0]
[0 1 1] X1 + [1]
[0 0 1] [0]
= a__div(mark(X1),X2)
mark(geq(X1,X2)) = [0]
[1]
[0]
>= [0]
[1]
[0]
= a__geq(X1,X2)
mark(if(X1,X2,X3)) = [0 1 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]
>= [0 1 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]
= a__if(mark(X1),X2,X3)
mark(minus(X1,X2)) = [0]
[1]
[0]
>= [0]
[0]
[0]
= a__minus(X1,X2)
mark(s(X)) = [0 1 0] [0]
[0 1 0] X + [1]
[0 0 0] [1]
>= [0 1 0] [0]
[0 1 0] X + [1]
[0 0 0] [1]
= s(mark(X))
* Step 5: NaturalMI WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
a__geq(X,0()) -> true()
a__geq(X1,X2) -> geq(X1,X2)
a__geq(0(),s(Y)) -> false()
a__if(X1,X2,X3) -> if(X1,X2,X3)
a__if(false(),X,Y) -> mark(Y)
a__if(true(),X,Y) -> mark(X)
a__minus(X1,X2) -> minus(X1,X2)
a__minus(0(),Y) -> 0()
mark(0()) -> 0()
mark(div(X1,X2)) -> a__div(mark(X1),X2)
mark(geq(X1,X2)) -> a__geq(X1,X2)
mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
mark(minus(X1,X2)) -> a__minus(X1,X2)
mark(s(X)) -> s(mark(X))
- Weak TRS:
a__div(X1,X2) -> div(X1,X2)
a__div(0(),s(Y)) -> 0()
a__geq(s(X),s(Y)) -> a__geq(X,Y)
a__minus(s(X),s(Y)) -> a__minus(X,Y)
mark(false()) -> false()
mark(true()) -> true()
- Signature:
{a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1} / {0/0,div/2,false/0,geq/2,if/3,minus/2,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__div,a__geq,a__if,a__minus,mark} and constructors {0
,div,false,geq,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(a__div) = {1},
uargs(a__if) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{a__div,a__geq,a__if,a__minus,mark}
TcT has computed the following interpretation:
p(0) = [0]
[0]
[0]
p(a__div) = [1 1 0] [0 0 1] [0]
[0 1 0] x1 + [0 0 0] x2 + [0]
[0 0 1] [0 0 1] [0]
p(a__geq) = [0 0 0] [1]
[0 0 0] x1 + [0]
[0 0 1] [0]
p(a__if) = [1 0 0] [1 0 1] [1 0 1] [0]
[0 0 0] x1 + [0 1 0] x2 + [0 1 0] x3 + [0]
[0 0 1] [0 0 1] [0 0 1] [0]
p(a__minus) = [1 1 0] [0]
[0 0 0] x1 + [0]
[0 0 0] [0]
p(div) = [1 1 0] [0 0 0] [0]
[0 1 0] x1 + [0 0 0] x2 + [0]
[0 0 1] [0 0 1] [0]
p(false) = [0]
[0]
[0]
p(geq) = [0 0 0] [1]
[0 0 0] x1 + [0]
[0 0 1] [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 1] [0 0 1] [0 0 1] [0]
p(mark) = [1 0 1] [0]
[0 1 0] x1 + [0]
[0 0 1] [0]
p(minus) = [1 1 0] [0]
[0 0 0] x1 + [0]
[0 0 0] [0]
p(s) = [1 1 0] [0]
[0 0 0] x1 + [1]
[0 0 1] [0]
p(true) = [0]
[0]
[0]
Following rules are strictly oriented:
a__geq(X,0()) = [0 0 0] [1]
[0 0 0] X + [0]
[0 0 1] [0]
> [0]
[0]
[0]
= true()
a__geq(0(),s(Y)) = [1]
[0]
[0]
> [0]
[0]
[0]
= false()
Following rules are (at-least) weakly oriented:
a__div(X1,X2) = [1 1 0] [0 0 1] [0]
[0 1 0] X1 + [0 0 0] X2 + [0]
[0 0 1] [0 0 1] [0]
>= [1 1 0] [0 0 0] [0]
[0 1 0] X1 + [0 0 0] X2 + [0]
[0 0 1] [0 0 1] [0]
= div(X1,X2)
a__div(0(),s(Y)) = [0 0 1] [0]
[0 0 0] Y + [0]
[0 0 1] [0]
>= [0]
[0]
[0]
= 0()
a__div(s(X),s(Y)) = [1 1 0] [0 0 1] [1]
[0 0 0] X + [0 0 0] Y + [1]
[0 0 1] [0 0 1] [0]
>= [1 1 0] [0 0 1] [1]
[0 0 0] X + [0 0 0] Y + [1]
[0 0 1] [0 0 1] [0]
= a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
a__geq(X1,X2) = [0 0 0] [1]
[0 0 0] X1 + [0]
[0 0 1] [0]
>= [0 0 0] [1]
[0 0 0] X1 + [0]
[0 0 1] [0]
= geq(X1,X2)
a__geq(s(X),s(Y)) = [0 0 0] [1]
[0 0 0] X + [0]
[0 0 1] [0]
>= [0 0 0] [1]
[0 0 0] X + [0]
[0 0 1] [0]
= a__geq(X,Y)
a__if(X1,X2,X3) = [1 0 0] [1 0 1] [1 0 1] [0]
[0 0 0] X1 + [0 1 0] X2 + [0 1 0] X3 + [0]
[0 0 1] [0 0 1] [0 0 1] [0]
>= [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 1] [0 0 1] [0 0 1] [0]
= if(X1,X2,X3)
a__if(false(),X,Y) = [1 0 1] [1 0 1] [0]
[0 1 0] X + [0 1 0] Y + [0]
[0 0 1] [0 0 1] [0]
>= [1 0 1] [0]
[0 1 0] Y + [0]
[0 0 1] [0]
= mark(Y)
a__if(true(),X,Y) = [1 0 1] [1 0 1] [0]
[0 1 0] X + [0 1 0] Y + [0]
[0 0 1] [0 0 1] [0]
>= [1 0 1] [0]
[0 1 0] X + [0]
[0 0 1] [0]
= mark(X)
a__minus(X1,X2) = [1 1 0] [0]
[0 0 0] X1 + [0]
[0 0 0] [0]
>= [1 1 0] [0]
[0 0 0] X1 + [0]
[0 0 0] [0]
= minus(X1,X2)
a__minus(0(),Y) = [0]
[0]
[0]
>= [0]
[0]
[0]
= 0()
a__minus(s(X),s(Y)) = [1 1 0] [1]
[0 0 0] X + [0]
[0 0 0] [0]
>= [1 1 0] [0]
[0 0 0] X + [0]
[0 0 0] [0]
= a__minus(X,Y)
mark(0()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= 0()
mark(div(X1,X2)) = [1 1 1] [0 0 1] [0]
[0 1 0] X1 + [0 0 0] X2 + [0]
[0 0 1] [0 0 1] [0]
>= [1 1 1] [0 0 1] [0]
[0 1 0] X1 + [0 0 0] X2 + [0]
[0 0 1] [0 0 1] [0]
= a__div(mark(X1),X2)
mark(false()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= false()
mark(geq(X1,X2)) = [0 0 1] [1]
[0 0 0] X1 + [0]
[0 0 1] [0]
>= [0 0 0] [1]
[0 0 0] X1 + [0]
[0 0 1] [0]
= a__geq(X1,X2)
mark(if(X1,X2,X3)) = [1 0 1] [1 0 1] [1 0 1] [0]
[0 0 0] X1 + [0 1 0] X2 + [0 1 0] X3 + [0]
[0 0 1] [0 0 1] [0 0 1] [0]
>= [1 0 1] [1 0 1] [1 0 1] [0]
[0 0 0] X1 + [0 1 0] X2 + [0 1 0] X3 + [0]
[0 0 1] [0 0 1] [0 0 1] [0]
= a__if(mark(X1),X2,X3)
mark(minus(X1,X2)) = [1 1 0] [0]
[0 0 0] X1 + [0]
[0 0 0] [0]
>= [1 1 0] [0]
[0 0 0] X1 + [0]
[0 0 0] [0]
= a__minus(X1,X2)
mark(s(X)) = [1 1 1] [0]
[0 0 0] X + [1]
[0 0 1] [0]
>= [1 1 1] [0]
[0 0 0] X + [1]
[0 0 1] [0]
= s(mark(X))
mark(true()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= true()
* Step 6: NaturalMI WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
a__geq(X1,X2) -> geq(X1,X2)
a__if(X1,X2,X3) -> if(X1,X2,X3)
a__if(false(),X,Y) -> mark(Y)
a__if(true(),X,Y) -> mark(X)
a__minus(X1,X2) -> minus(X1,X2)
a__minus(0(),Y) -> 0()
mark(0()) -> 0()
mark(div(X1,X2)) -> a__div(mark(X1),X2)
mark(geq(X1,X2)) -> a__geq(X1,X2)
mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
mark(minus(X1,X2)) -> a__minus(X1,X2)
mark(s(X)) -> s(mark(X))
- Weak TRS:
a__div(X1,X2) -> div(X1,X2)
a__div(0(),s(Y)) -> 0()
a__geq(X,0()) -> true()
a__geq(0(),s(Y)) -> false()
a__geq(s(X),s(Y)) -> a__geq(X,Y)
a__minus(s(X),s(Y)) -> a__minus(X,Y)
mark(false()) -> false()
mark(true()) -> true()
- Signature:
{a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1} / {0/0,div/2,false/0,geq/2,if/3,minus/2,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__div,a__geq,a__if,a__minus,mark} and constructors {0
,div,false,geq,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(a__div) = {1},
uargs(a__if) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{a__div,a__geq,a__if,a__minus,mark}
TcT has computed the following interpretation:
p(0) = [0]
[0]
[0]
p(a__div) = [1 0 1] [0 1 1] [1]
[0 1 1] x1 + [0 1 1] x2 + [1]
[0 0 1] [0 0 0] [0]
p(a__geq) = [0 0 0] [0]
[0 1 0] x1 + [0]
[0 0 0] [0]
p(a__if) = [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(a__minus) = [1]
[1]
[0]
p(div) = [1 0 0] [0 0 0] [1]
[0 1 1] x1 + [0 1 1] x2 + [1]
[0 0 1] [0 0 0] [0]
p(false) = [0]
[0]
[0]
p(geq) = [0 0 0] [0]
[0 1 0] x1 + [0]
[0 0 0] [0]
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 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) = [1]
[1]
[0]
p(s) = [1 0 0] [0]
[0 1 1] x1 + [0]
[0 0 1] [1]
p(true) = [0]
[0]
[0]
Following rules are strictly oriented:
a__minus(0(),Y) = [1]
[1]
[0]
> [0]
[0]
[0]
= 0()
Following rules are (at-least) weakly oriented:
a__div(X1,X2) = [1 0 1] [0 1 1] [1]
[0 1 1] X1 + [0 1 1] X2 + [1]
[0 0 1] [0 0 0] [0]
>= [1 0 0] [0 0 0] [1]
[0 1 1] X1 + [0 1 1] X2 + [1]
[0 0 1] [0 0 0] [0]
= div(X1,X2)
a__div(0(),s(Y)) = [0 1 2] [2]
[0 1 2] Y + [2]
[0 0 0] [0]
>= [0]
[0]
[0]
= 0()
a__div(s(X),s(Y)) = [1 0 1] [0 1 2] [3]
[0 1 2] X + [0 1 2] Y + [3]
[0 0 1] [0 0 0] [1]
>= [0 0 0] [0 1 2] [3]
[0 1 0] X + [0 1 2] Y + [3]
[0 0 0] [0 0 0] [1]
= a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
a__geq(X,0()) = [0 0 0] [0]
[0 1 0] X + [0]
[0 0 0] [0]
>= [0]
[0]
[0]
= true()
a__geq(X1,X2) = [0 0 0] [0]
[0 1 0] X1 + [0]
[0 0 0] [0]
>= [0 0 0] [0]
[0 1 0] X1 + [0]
[0 0 0] [0]
= geq(X1,X2)
a__geq(0(),s(Y)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= false()
a__geq(s(X),s(Y)) = [0 0 0] [0]
[0 1 1] X + [0]
[0 0 0] [0]
>= [0 0 0] [0]
[0 1 0] X + [0]
[0 0 0] [0]
= a__geq(X,Y)
a__if(X1,X2,X3) = [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]
>= [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 0] [0 0 1] [0 0 1] [0]
= if(X1,X2,X3)
a__if(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)
a__if(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)
a__minus(X1,X2) = [1]
[1]
[0]
>= [1]
[1]
[0]
= minus(X1,X2)
a__minus(s(X),s(Y)) = [1]
[1]
[0]
>= [1]
[1]
[0]
= a__minus(X,Y)
mark(0()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= 0()
mark(div(X1,X2)) = [0 1 1] [0 1 1] [1]
[0 1 1] X1 + [0 1 1] X2 + [1]
[0 0 1] [0 0 0] [0]
>= [0 1 1] [0 1 1] [1]
[0 1 1] X1 + [0 1 1] X2 + [1]
[0 0 1] [0 0 0] [0]
= a__div(mark(X1),X2)
mark(false()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= false()
mark(geq(X1,X2)) = [0 1 0] [0]
[0 1 0] X1 + [0]
[0 0 0] [0]
>= [0 0 0] [0]
[0 1 0] X1 + [0]
[0 0 0] [0]
= a__geq(X1,X2)
mark(if(X1,X2,X3)) = [0 1 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]
>= [0 1 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]
= a__if(mark(X1),X2,X3)
mark(minus(X1,X2)) = [1]
[1]
[0]
>= [1]
[1]
[0]
= a__minus(X1,X2)
mark(s(X)) = [0 1 1] [0]
[0 1 1] X + [0]
[0 0 1] [1]
>= [0 1 0] [0]
[0 1 1] X + [0]
[0 0 1] [1]
= s(mark(X))
mark(true()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= true()
* Step 7: NaturalMI WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
a__geq(X1,X2) -> geq(X1,X2)
a__if(X1,X2,X3) -> if(X1,X2,X3)
a__if(false(),X,Y) -> mark(Y)
a__if(true(),X,Y) -> mark(X)
a__minus(X1,X2) -> minus(X1,X2)
mark(0()) -> 0()
mark(div(X1,X2)) -> a__div(mark(X1),X2)
mark(geq(X1,X2)) -> a__geq(X1,X2)
mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
mark(minus(X1,X2)) -> a__minus(X1,X2)
mark(s(X)) -> s(mark(X))
- Weak TRS:
a__div(X1,X2) -> div(X1,X2)
a__div(0(),s(Y)) -> 0()
a__geq(X,0()) -> true()
a__geq(0(),s(Y)) -> false()
a__geq(s(X),s(Y)) -> a__geq(X,Y)
a__minus(0(),Y) -> 0()
a__minus(s(X),s(Y)) -> a__minus(X,Y)
mark(false()) -> false()
mark(true()) -> true()
- Signature:
{a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1} / {0/0,div/2,false/0,geq/2,if/3,minus/2,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__div,a__geq,a__if,a__minus,mark} and constructors {0
,div,false,geq,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(a__div) = {1},
uargs(a__if) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{a__div,a__geq,a__if,a__minus,mark}
TcT has computed the following interpretation:
p(0) = [0]
[0]
[0]
p(a__div) = [1 0 1] [0]
[0 1 1] x1 + [0]
[0 0 1] [0]
p(a__geq) = [0]
[0]
[0]
p(a__if) = [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(a__minus) = [0]
[0]
[0]
p(div) = [0 0 0] [0]
[0 1 1] x1 + [0]
[0 0 1] [0]
p(false) = [0]
[0]
[0]
p(geq) = [0]
[0]
[0]
p(if) = [1 0 0] [0 0 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(s) = [1 0 0] [0]
[0 1 0] x1 + [0]
[0 0 0] [1]
p(true) = [0]
[0]
[0]
Following rules are strictly oriented:
a__div(s(X),s(Y)) = [1 0 0] [1]
[0 1 0] X + [1]
[0 0 0] [1]
> [0]
[0]
[1]
= a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
Following rules are (at-least) weakly oriented:
a__div(X1,X2) = [1 0 1] [0]
[0 1 1] X1 + [0]
[0 0 1] [0]
>= [0 0 0] [0]
[0 1 1] X1 + [0]
[0 0 1] [0]
= div(X1,X2)
a__div(0(),s(Y)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= 0()
a__geq(X,0()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= true()
a__geq(X1,X2) = [0]
[0]
[0]
>= [0]
[0]
[0]
= geq(X1,X2)
a__geq(0(),s(Y)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= false()
a__geq(s(X),s(Y)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= a__geq(X,Y)
a__if(X1,X2,X3) = [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]
>= [1 0 0] [0 0 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]
= if(X1,X2,X3)
a__if(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)
a__if(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)
a__minus(X1,X2) = [0]
[0]
[0]
>= [0]
[0]
[0]
= minus(X1,X2)
a__minus(0(),Y) = [0]
[0]
[0]
>= [0]
[0]
[0]
= 0()
a__minus(s(X),s(Y)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= a__minus(X,Y)
mark(0()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= 0()
mark(div(X1,X2)) = [0 1 1] [0]
[0 1 1] X1 + [0]
[0 0 1] [0]
>= [0 1 1] [0]
[0 1 1] X1 + [0]
[0 0 1] [0]
= a__div(mark(X1),X2)
mark(false()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= false()
mark(geq(X1,X2)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= a__geq(X1,X2)
mark(if(X1,X2,X3)) = [0 1 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]
>= [0 1 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]
= a__if(mark(X1),X2,X3)
mark(minus(X1,X2)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= a__minus(X1,X2)
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))
mark(true()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= true()
* Step 8: NaturalMI WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
a__geq(X1,X2) -> geq(X1,X2)
a__if(X1,X2,X3) -> if(X1,X2,X3)
a__if(false(),X,Y) -> mark(Y)
a__if(true(),X,Y) -> mark(X)
a__minus(X1,X2) -> minus(X1,X2)
mark(0()) -> 0()
mark(div(X1,X2)) -> a__div(mark(X1),X2)
mark(geq(X1,X2)) -> a__geq(X1,X2)
mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
mark(minus(X1,X2)) -> a__minus(X1,X2)
mark(s(X)) -> s(mark(X))
- Weak TRS:
a__div(X1,X2) -> div(X1,X2)
a__div(0(),s(Y)) -> 0()
a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
a__geq(X,0()) -> true()
a__geq(0(),s(Y)) -> false()
a__geq(s(X),s(Y)) -> a__geq(X,Y)
a__minus(0(),Y) -> 0()
a__minus(s(X),s(Y)) -> a__minus(X,Y)
mark(false()) -> false()
mark(true()) -> true()
- Signature:
{a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1} / {0/0,div/2,false/0,geq/2,if/3,minus/2,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__div,a__geq,a__if,a__minus,mark} and constructors {0
,div,false,geq,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(a__div) = {1},
uargs(a__if) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{a__div,a__geq,a__if,a__minus,mark}
TcT has computed the following interpretation:
p(0) = [0]
[0]
[0]
p(a__div) = [1 0 0] [1]
[0 1 1] x1 + [1]
[0 0 1] [0]
p(a__geq) = [0]
[0]
[0]
p(a__if) = [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(a__minus) = [0]
[0]
[0]
p(div) = [0 0 0] [1]
[0 1 1] x1 + [1]
[0 0 1] [0]
p(false) = [0]
[0]
[0]
p(geq) = [0]
[0]
[0]
p(if) = [1 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(mark) = [0 1 0] [0]
[0 1 0] x1 + [1]
[0 0 1] [0]
p(minus) = [0]
[0]
[0]
p(s) = [1 0 0] [1]
[0 1 0] x1 + [1]
[0 0 0] [1]
p(true) = [0]
[0]
[0]
Following rules are strictly oriented:
mark(if(X1,X2,X3)) = [0 1 0] [0 1 0] [0 1 0] [1]
[0 1 0] X1 + [0 1 0] X2 + [0 1 0] X3 + [2]
[0 0 0] [0 0 1] [0 0 1] [0]
> [0 1 0] [0 1 0] [0 1 0] [0]
[0 1 0] X1 + [0 1 0] X2 + [0 1 0] X3 + [2]
[0 0 0] [0 0 1] [0 0 1] [0]
= a__if(mark(X1),X2,X3)
Following rules are (at-least) weakly oriented:
a__div(X1,X2) = [1 0 0] [1]
[0 1 1] X1 + [1]
[0 0 1] [0]
>= [0 0 0] [1]
[0 1 1] X1 + [1]
[0 0 1] [0]
= div(X1,X2)
a__div(0(),s(Y)) = [1]
[1]
[0]
>= [0]
[0]
[0]
= 0()
a__div(s(X),s(Y)) = [1 0 0] [2]
[0 1 0] X + [3]
[0 0 0] [1]
>= [2]
[3]
[1]
= a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
a__geq(X,0()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= true()
a__geq(X1,X2) = [0]
[0]
[0]
>= [0]
[0]
[0]
= geq(X1,X2)
a__geq(0(),s(Y)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= false()
a__geq(s(X),s(Y)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= a__geq(X,Y)
a__if(X1,X2,X3) = [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]
>= [1 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]
= if(X1,X2,X3)
a__if(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 + [1]
[0 0 1] [0]
= mark(Y)
a__if(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 + [1]
[0 0 1] [0]
= mark(X)
a__minus(X1,X2) = [0]
[0]
[0]
>= [0]
[0]
[0]
= minus(X1,X2)
a__minus(0(),Y) = [0]
[0]
[0]
>= [0]
[0]
[0]
= 0()
a__minus(s(X),s(Y)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= a__minus(X,Y)
mark(0()) = [0]
[1]
[0]
>= [0]
[0]
[0]
= 0()
mark(div(X1,X2)) = [0 1 1] [1]
[0 1 1] X1 + [2]
[0 0 1] [0]
>= [0 1 0] [1]
[0 1 1] X1 + [2]
[0 0 1] [0]
= a__div(mark(X1),X2)
mark(false()) = [0]
[1]
[0]
>= [0]
[0]
[0]
= false()
mark(geq(X1,X2)) = [0]
[1]
[0]
>= [0]
[0]
[0]
= a__geq(X1,X2)
mark(minus(X1,X2)) = [0]
[1]
[0]
>= [0]
[0]
[0]
= a__minus(X1,X2)
mark(s(X)) = [0 1 0] [1]
[0 1 0] X + [2]
[0 0 0] [1]
>= [0 1 0] [1]
[0 1 0] X + [2]
[0 0 0] [1]
= s(mark(X))
mark(true()) = [0]
[1]
[0]
>= [0]
[0]
[0]
= true()
* Step 9: NaturalMI WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
a__geq(X1,X2) -> geq(X1,X2)
a__if(X1,X2,X3) -> if(X1,X2,X3)
a__if(false(),X,Y) -> mark(Y)
a__if(true(),X,Y) -> mark(X)
a__minus(X1,X2) -> minus(X1,X2)
mark(0()) -> 0()
mark(div(X1,X2)) -> a__div(mark(X1),X2)
mark(geq(X1,X2)) -> a__geq(X1,X2)
mark(minus(X1,X2)) -> a__minus(X1,X2)
mark(s(X)) -> s(mark(X))
- Weak TRS:
a__div(X1,X2) -> div(X1,X2)
a__div(0(),s(Y)) -> 0()
a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
a__geq(X,0()) -> true()
a__geq(0(),s(Y)) -> false()
a__geq(s(X),s(Y)) -> a__geq(X,Y)
a__minus(0(),Y) -> 0()
a__minus(s(X),s(Y)) -> a__minus(X,Y)
mark(false()) -> false()
mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
mark(true()) -> true()
- Signature:
{a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1} / {0/0,div/2,false/0,geq/2,if/3,minus/2,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__div,a__geq,a__if,a__minus,mark} and constructors {0
,div,false,geq,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(a__div) = {1},
uargs(a__if) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{a__div,a__geq,a__if,a__minus,mark}
TcT has computed the following interpretation:
p(0) = [0]
[0]
[0]
p(a__div) = [1 0 0] [0]
[0 1 1] x1 + [0]
[0 0 1] [0]
p(a__geq) = [0]
[1]
[0]
p(a__if) = [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(a__minus) = [0]
[0]
[0]
p(div) = [0 0 0] [0]
[0 1 1] x1 + [0]
[0 0 1] [0]
p(false) = [0]
[0]
[0]
p(geq) = [0]
[1]
[0]
p(if) = [1 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(mark) = [0 1 0] [0]
[0 1 0] x1 + [0]
[0 0 1] [0]
p(minus) = [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(geq(X1,X2)) = [1]
[1]
[0]
> [0]
[1]
[0]
= a__geq(X1,X2)
Following rules are (at-least) weakly oriented:
a__div(X1,X2) = [1 0 0] [0]
[0 1 1] X1 + [0]
[0 0 1] [0]
>= [0 0 0] [0]
[0 1 1] X1 + [0]
[0 0 1] [0]
= div(X1,X2)
a__div(0(),s(Y)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= 0()
a__div(s(X),s(Y)) = [1 0 0] [0]
[0 1 0] X + [1]
[0 0 0] [1]
>= [0]
[1]
[1]
= a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
a__geq(X,0()) = [0]
[1]
[0]
>= [0]
[0]
[0]
= true()
a__geq(X1,X2) = [0]
[1]
[0]
>= [0]
[1]
[0]
= geq(X1,X2)
a__geq(0(),s(Y)) = [0]
[1]
[0]
>= [0]
[0]
[0]
= false()
a__geq(s(X),s(Y)) = [0]
[1]
[0]
>= [0]
[1]
[0]
= a__geq(X,Y)
a__if(X1,X2,X3) = [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]
>= [1 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]
= if(X1,X2,X3)
a__if(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)
a__if(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)
a__minus(X1,X2) = [0]
[0]
[0]
>= [0]
[0]
[0]
= minus(X1,X2)
a__minus(0(),Y) = [0]
[0]
[0]
>= [0]
[0]
[0]
= 0()
a__minus(s(X),s(Y)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= a__minus(X,Y)
mark(0()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= 0()
mark(div(X1,X2)) = [0 1 1] [0]
[0 1 1] X1 + [0]
[0 0 1] [0]
>= [0 1 0] [0]
[0 1 1] X1 + [0]
[0 0 1] [0]
= a__div(mark(X1),X2)
mark(false()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= false()
mark(if(X1,X2,X3)) = [0 1 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]
>= [0 1 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]
= a__if(mark(X1),X2,X3)
mark(minus(X1,X2)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= a__minus(X1,X2)
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))
mark(true()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= true()
* Step 10: NaturalMI WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
a__geq(X1,X2) -> geq(X1,X2)
a__if(X1,X2,X3) -> if(X1,X2,X3)
a__if(false(),X,Y) -> mark(Y)
a__if(true(),X,Y) -> mark(X)
a__minus(X1,X2) -> minus(X1,X2)
mark(0()) -> 0()
mark(div(X1,X2)) -> a__div(mark(X1),X2)
mark(minus(X1,X2)) -> a__minus(X1,X2)
mark(s(X)) -> s(mark(X))
- Weak TRS:
a__div(X1,X2) -> div(X1,X2)
a__div(0(),s(Y)) -> 0()
a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
a__geq(X,0()) -> true()
a__geq(0(),s(Y)) -> false()
a__geq(s(X),s(Y)) -> a__geq(X,Y)
a__minus(0(),Y) -> 0()
a__minus(s(X),s(Y)) -> a__minus(X,Y)
mark(false()) -> false()
mark(geq(X1,X2)) -> a__geq(X1,X2)
mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
mark(true()) -> true()
- Signature:
{a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1} / {0/0,div/2,false/0,geq/2,if/3,minus/2,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__div,a__geq,a__if,a__minus,mark} and constructors {0
,div,false,geq,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(a__div) = {1},
uargs(a__if) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{a__div,a__geq,a__if,a__minus,mark}
TcT has computed the following interpretation:
p(0) = [0]
[0]
[0]
p(a__div) = [1 1 0] [0 0 0] [1]
[0 1 0] x1 + [0 1 0] x2 + [1]
[0 0 1] [0 0 0] [1]
p(a__geq) = [1]
[0]
[0]
p(a__if) = [1 0 0] [1 0 0] [1 0 0] [0]
[0 0 0] x1 + [0 1 1] x2 + [0 1 0] x3 + [0]
[0 0 0] [0 0 1] [0 0 1] [1]
p(a__minus) = [0]
[0]
[1]
p(div) = [1 1 0] [0 0 0] [1]
[0 1 0] x1 + [0 1 0] x2 + [1]
[0 0 1] [0 0 0] [1]
p(false) = [0]
[0]
[0]
p(geq) = [1]
[0]
[0]
p(if) = [1 0 0] [1 0 0] [1 0 0] [0]
[0 0 0] x1 + [0 1 1] x2 + [0 1 0] x3 + [0]
[0 0 0] [0 0 1] [0 0 1] [1]
p(mark) = [1 0 0] [0]
[0 1 0] x1 + [0]
[0 0 1] [0]
p(minus) = [0]
[0]
[1]
p(s) = [1 0 0] [0]
[0 0 1] x1 + [1]
[0 0 0] [0]
p(true) = [1]
[0]
[0]
Following rules are strictly oriented:
a__if(true(),X,Y) = [1 0 0] [1 0 0] [1]
[0 1 1] X + [0 1 0] Y + [0]
[0 0 1] [0 0 1] [1]
> [1 0 0] [0]
[0 1 0] X + [0]
[0 0 1] [0]
= mark(X)
Following rules are (at-least) weakly oriented:
a__div(X1,X2) = [1 1 0] [0 0 0] [1]
[0 1 0] X1 + [0 1 0] X2 + [1]
[0 0 1] [0 0 0] [1]
>= [1 1 0] [0 0 0] [1]
[0 1 0] X1 + [0 1 0] X2 + [1]
[0 0 1] [0 0 0] [1]
= div(X1,X2)
a__div(0(),s(Y)) = [0 0 0] [1]
[0 0 1] Y + [2]
[0 0 0] [1]
>= [0]
[0]
[0]
= 0()
a__div(s(X),s(Y)) = [1 0 1] [0 0 0] [2]
[0 0 1] X + [0 0 1] Y + [3]
[0 0 0] [0 0 0] [1]
>= [2]
[3]
[1]
= a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
a__geq(X,0()) = [1]
[0]
[0]
>= [1]
[0]
[0]
= true()
a__geq(X1,X2) = [1]
[0]
[0]
>= [1]
[0]
[0]
= geq(X1,X2)
a__geq(0(),s(Y)) = [1]
[0]
[0]
>= [0]
[0]
[0]
= false()
a__geq(s(X),s(Y)) = [1]
[0]
[0]
>= [1]
[0]
[0]
= a__geq(X,Y)
a__if(X1,X2,X3) = [1 0 0] [1 0 0] [1 0 0] [0]
[0 0 0] X1 + [0 1 1] X2 + [0 1 0] X3 + [0]
[0 0 0] [0 0 1] [0 0 1] [1]
>= [1 0 0] [1 0 0] [1 0 0] [0]
[0 0 0] X1 + [0 1 1] X2 + [0 1 0] X3 + [0]
[0 0 0] [0 0 1] [0 0 1] [1]
= if(X1,X2,X3)
a__if(false(),X,Y) = [1 0 0] [1 0 0] [0]
[0 1 1] X + [0 1 0] Y + [0]
[0 0 1] [0 0 1] [1]
>= [1 0 0] [0]
[0 1 0] Y + [0]
[0 0 1] [0]
= mark(Y)
a__minus(X1,X2) = [0]
[0]
[1]
>= [0]
[0]
[1]
= minus(X1,X2)
a__minus(0(),Y) = [0]
[0]
[1]
>= [0]
[0]
[0]
= 0()
a__minus(s(X),s(Y)) = [0]
[0]
[1]
>= [0]
[0]
[1]
= a__minus(X,Y)
mark(0()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= 0()
mark(div(X1,X2)) = [1 1 0] [0 0 0] [1]
[0 1 0] X1 + [0 1 0] X2 + [1]
[0 0 1] [0 0 0] [1]
>= [1 1 0] [0 0 0] [1]
[0 1 0] X1 + [0 1 0] X2 + [1]
[0 0 1] [0 0 0] [1]
= a__div(mark(X1),X2)
mark(false()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= false()
mark(geq(X1,X2)) = [1]
[0]
[0]
>= [1]
[0]
[0]
= a__geq(X1,X2)
mark(if(X1,X2,X3)) = [1 0 0] [1 0 0] [1 0 0] [0]
[0 0 0] X1 + [0 1 1] X2 + [0 1 0] X3 + [0]
[0 0 0] [0 0 1] [0 0 1] [1]
>= [1 0 0] [1 0 0] [1 0 0] [0]
[0 0 0] X1 + [0 1 1] X2 + [0 1 0] X3 + [0]
[0 0 0] [0 0 1] [0 0 1] [1]
= a__if(mark(X1),X2,X3)
mark(minus(X1,X2)) = [0]
[0]
[1]
>= [0]
[0]
[1]
= a__minus(X1,X2)
mark(s(X)) = [1 0 0] [0]
[0 0 1] X + [1]
[0 0 0] [0]
>= [1 0 0] [0]
[0 0 1] X + [1]
[0 0 0] [0]
= s(mark(X))
mark(true()) = [1]
[0]
[0]
>= [1]
[0]
[0]
= true()
* Step 11: NaturalMI WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
a__geq(X1,X2) -> geq(X1,X2)
a__if(X1,X2,X3) -> if(X1,X2,X3)
a__if(false(),X,Y) -> mark(Y)
a__minus(X1,X2) -> minus(X1,X2)
mark(0()) -> 0()
mark(div(X1,X2)) -> a__div(mark(X1),X2)
mark(minus(X1,X2)) -> a__minus(X1,X2)
mark(s(X)) -> s(mark(X))
- Weak TRS:
a__div(X1,X2) -> div(X1,X2)
a__div(0(),s(Y)) -> 0()
a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
a__geq(X,0()) -> true()
a__geq(0(),s(Y)) -> false()
a__geq(s(X),s(Y)) -> a__geq(X,Y)
a__if(true(),X,Y) -> mark(X)
a__minus(0(),Y) -> 0()
a__minus(s(X),s(Y)) -> a__minus(X,Y)
mark(false()) -> false()
mark(geq(X1,X2)) -> a__geq(X1,X2)
mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
mark(true()) -> true()
- Signature:
{a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1} / {0/0,div/2,false/0,geq/2,if/3,minus/2,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__div,a__geq,a__if,a__minus,mark} and constructors {0
,div,false,geq,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(a__div) = {1},
uargs(a__if) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{a__div,a__geq,a__if,a__minus,mark}
TcT has computed the following interpretation:
p(0) = [0]
[1]
[0]
p(a__div) = [1 0 1] [1]
[0 1 1] x1 + [0]
[0 0 0] [1]
p(a__geq) = [0]
[0]
[0]
p(a__if) = [1 0 0] [1 0 0] [1 0 0] [1]
[0 0 0] x1 + [0 1 0] x2 + [0 1 1] x3 + [0]
[0 0 0] [0 0 1] [0 0 1] [0]
p(a__minus) = [1 0 0] [0]
[0 0 0] x1 + [1]
[0 0 0] [0]
p(div) = [1 0 1] [1]
[0 1 1] x1 + [0]
[0 0 0] [1]
p(false) = [0]
[0]
[0]
p(geq) = [0]
[0]
[0]
p(if) = [1 0 0] [1 0 0] [1 0 0] [1]
[0 0 0] x1 + [0 1 0] x2 + [0 1 1] x3 + [0]
[0 0 0] [0 0 1] [0 0 1] [0]
p(mark) = [1 0 0] [0]
[0 1 0] x1 + [0]
[0 0 1] [0]
p(minus) = [1 0 0] [0]
[0 0 0] x1 + [1]
[0 0 0] [0]
p(s) = [1 0 0] [0]
[0 0 0] x1 + [0]
[0 0 0] [1]
p(true) = [0]
[0]
[0]
Following rules are strictly oriented:
a__if(false(),X,Y) = [1 0 0] [1 0 0] [1]
[0 1 0] X + [0 1 1] Y + [0]
[0 0 1] [0 0 1] [0]
> [1 0 0] [0]
[0 1 0] Y + [0]
[0 0 1] [0]
= mark(Y)
Following rules are (at-least) weakly oriented:
a__div(X1,X2) = [1 0 1] [1]
[0 1 1] X1 + [0]
[0 0 0] [1]
>= [1 0 1] [1]
[0 1 1] X1 + [0]
[0 0 0] [1]
= div(X1,X2)
a__div(0(),s(Y)) = [1]
[1]
[1]
>= [0]
[1]
[0]
= 0()
a__div(s(X),s(Y)) = [1 0 0] [2]
[0 0 0] X + [1]
[0 0 0] [1]
>= [1 0 0] [2]
[0 0 0] X + [1]
[0 0 0] [1]
= a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
a__geq(X,0()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= true()
a__geq(X1,X2) = [0]
[0]
[0]
>= [0]
[0]
[0]
= geq(X1,X2)
a__geq(0(),s(Y)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= false()
a__geq(s(X),s(Y)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= a__geq(X,Y)
a__if(X1,X2,X3) = [1 0 0] [1 0 0] [1 0 0] [1]
[0 0 0] X1 + [0 1 0] X2 + [0 1 1] X3 + [0]
[0 0 0] [0 0 1] [0 0 1] [0]
>= [1 0 0] [1 0 0] [1 0 0] [1]
[0 0 0] X1 + [0 1 0] X2 + [0 1 1] X3 + [0]
[0 0 0] [0 0 1] [0 0 1] [0]
= if(X1,X2,X3)
a__if(true(),X,Y) = [1 0 0] [1 0 0] [1]
[0 1 0] X + [0 1 1] Y + [0]
[0 0 1] [0 0 1] [0]
>= [1 0 0] [0]
[0 1 0] X + [0]
[0 0 1] [0]
= mark(X)
a__minus(X1,X2) = [1 0 0] [0]
[0 0 0] X1 + [1]
[0 0 0] [0]
>= [1 0 0] [0]
[0 0 0] X1 + [1]
[0 0 0] [0]
= minus(X1,X2)
a__minus(0(),Y) = [0]
[1]
[0]
>= [0]
[1]
[0]
= 0()
a__minus(s(X),s(Y)) = [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]
= a__minus(X,Y)
mark(0()) = [0]
[1]
[0]
>= [0]
[1]
[0]
= 0()
mark(div(X1,X2)) = [1 0 1] [1]
[0 1 1] X1 + [0]
[0 0 0] [1]
>= [1 0 1] [1]
[0 1 1] X1 + [0]
[0 0 0] [1]
= a__div(mark(X1),X2)
mark(false()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= false()
mark(geq(X1,X2)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= a__geq(X1,X2)
mark(if(X1,X2,X3)) = [1 0 0] [1 0 0] [1 0 0] [1]
[0 0 0] X1 + [0 1 0] X2 + [0 1 1] X3 + [0]
[0 0 0] [0 0 1] [0 0 1] [0]
>= [1 0 0] [1 0 0] [1 0 0] [1]
[0 0 0] X1 + [0 1 0] X2 + [0 1 1] X3 + [0]
[0 0 0] [0 0 1] [0 0 1] [0]
= a__if(mark(X1),X2,X3)
mark(minus(X1,X2)) = [1 0 0] [0]
[0 0 0] X1 + [1]
[0 0 0] [0]
>= [1 0 0] [0]
[0 0 0] X1 + [1]
[0 0 0] [0]
= a__minus(X1,X2)
mark(s(X)) = [1 0 0] [0]
[0 0 0] X + [0]
[0 0 0] [1]
>= [1 0 0] [0]
[0 0 0] X + [0]
[0 0 0] [1]
= s(mark(X))
mark(true()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= true()
* Step 12: NaturalMI WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
a__geq(X1,X2) -> geq(X1,X2)
a__if(X1,X2,X3) -> if(X1,X2,X3)
a__minus(X1,X2) -> minus(X1,X2)
mark(0()) -> 0()
mark(div(X1,X2)) -> a__div(mark(X1),X2)
mark(minus(X1,X2)) -> a__minus(X1,X2)
mark(s(X)) -> s(mark(X))
- Weak TRS:
a__div(X1,X2) -> div(X1,X2)
a__div(0(),s(Y)) -> 0()
a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
a__geq(X,0()) -> true()
a__geq(0(),s(Y)) -> false()
a__geq(s(X),s(Y)) -> a__geq(X,Y)
a__if(false(),X,Y) -> mark(Y)
a__if(true(),X,Y) -> mark(X)
a__minus(0(),Y) -> 0()
a__minus(s(X),s(Y)) -> a__minus(X,Y)
mark(false()) -> false()
mark(geq(X1,X2)) -> a__geq(X1,X2)
mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
mark(true()) -> true()
- Signature:
{a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1} / {0/0,div/2,false/0,geq/2,if/3,minus/2,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__div,a__geq,a__if,a__minus,mark} and constructors {0
,div,false,geq,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(a__div) = {1},
uargs(a__if) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{a__div,a__geq,a__if,a__minus,mark}
TcT has computed the following interpretation:
p(0) = [0]
[0]
[0]
p(a__div) = [1 0 1] [0 1 0] [0]
[0 1 1] x1 + [0 1 0] x2 + [1]
[0 0 0] [0 0 1] [0]
p(a__geq) = [0 0 0] [0]
[0 1 0] x1 + [0]
[0 0 0] [0]
p(a__if) = [1 0 0] [0 1 0] [0 1 0] [0]
[0 1 0] x1 + [0 1 0] x2 + [0 1 1] x3 + [0]
[0 0 0] [0 0 1] [0 0 1] [0]
p(a__minus) = [0]
[0]
[0]
p(div) = [0 0 0] [0 1 0] [0]
[0 1 1] x1 + [0 1 0] x2 + [1]
[0 0 0] [0 0 1] [0]
p(false) = [0]
[0]
[0]
p(geq) = [0 0 0] [0]
[0 1 0] x1 + [0]
[0 0 0] [0]
p(if) = [0 0 0] [0 1 0] [0 1 0] [0]
[0 1 0] x1 + [0 1 0] x2 + [0 1 1] 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(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(X1,X2)) = [0 1 1] [0 1 0] [1]
[0 1 1] X1 + [0 1 0] X2 + [1]
[0 0 0] [0 0 1] [0]
> [0 1 1] [0 1 0] [0]
[0 1 1] X1 + [0 1 0] X2 + [1]
[0 0 0] [0 0 1] [0]
= a__div(mark(X1),X2)
Following rules are (at-least) weakly oriented:
a__div(X1,X2) = [1 0 1] [0 1 0] [0]
[0 1 1] X1 + [0 1 0] X2 + [1]
[0 0 0] [0 0 1] [0]
>= [0 0 0] [0 1 0] [0]
[0 1 1] X1 + [0 1 0] X2 + [1]
[0 0 0] [0 0 1] [0]
= div(X1,X2)
a__div(0(),s(Y)) = [0 1 0] [0]
[0 1 0] Y + [1]
[0 0 0] [1]
>= [0]
[0]
[0]
= 0()
a__div(s(X),s(Y)) = [1 0 0] [0 1 0] [1]
[0 1 0] X + [0 1 0] Y + [2]
[0 0 0] [0 0 0] [1]
>= [0 0 0] [0 1 0] [1]
[0 1 0] X + [0 1 0] Y + [1]
[0 0 0] [0 0 0] [1]
= a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
a__geq(X,0()) = [0 0 0] [0]
[0 1 0] X + [0]
[0 0 0] [0]
>= [0]
[0]
[0]
= true()
a__geq(X1,X2) = [0 0 0] [0]
[0 1 0] X1 + [0]
[0 0 0] [0]
>= [0 0 0] [0]
[0 1 0] X1 + [0]
[0 0 0] [0]
= geq(X1,X2)
a__geq(0(),s(Y)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= false()
a__geq(s(X),s(Y)) = [0 0 0] [0]
[0 1 0] X + [0]
[0 0 0] [0]
>= [0 0 0] [0]
[0 1 0] X + [0]
[0 0 0] [0]
= a__geq(X,Y)
a__if(X1,X2,X3) = [1 0 0] [0 1 0] [0 1 0] [0]
[0 1 0] X1 + [0 1 0] X2 + [0 1 1] X3 + [0]
[0 0 0] [0 0 1] [0 0 1] [0]
>= [0 0 0] [0 1 0] [0 1 0] [0]
[0 1 0] X1 + [0 1 0] X2 + [0 1 1] X3 + [0]
[0 0 0] [0 0 1] [0 0 1] [0]
= if(X1,X2,X3)
a__if(false(),X,Y) = [0 1 0] [0 1 0] [0]
[0 1 0] X + [0 1 1] Y + [0]
[0 0 1] [0 0 1] [0]
>= [0 1 0] [0]
[0 1 0] Y + [0]
[0 0 1] [0]
= mark(Y)
a__if(true(),X,Y) = [0 1 0] [0 1 0] [0]
[0 1 0] X + [0 1 1] Y + [0]
[0 0 1] [0 0 1] [0]
>= [0 1 0] [0]
[0 1 0] X + [0]
[0 0 1] [0]
= mark(X)
a__minus(X1,X2) = [0]
[0]
[0]
>= [0]
[0]
[0]
= minus(X1,X2)
a__minus(0(),Y) = [0]
[0]
[0]
>= [0]
[0]
[0]
= 0()
a__minus(s(X),s(Y)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= a__minus(X,Y)
mark(0()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= 0()
mark(false()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= false()
mark(geq(X1,X2)) = [0 1 0] [0]
[0 1 0] X1 + [0]
[0 0 0] [0]
>= [0 0 0] [0]
[0 1 0] X1 + [0]
[0 0 0] [0]
= a__geq(X1,X2)
mark(if(X1,X2,X3)) = [0 1 0] [0 1 0] [0 1 1] [0]
[0 1 0] X1 + [0 1 0] X2 + [0 1 1] X3 + [0]
[0 0 0] [0 0 1] [0 0 1] [0]
>= [0 1 0] [0 1 0] [0 1 0] [0]
[0 1 0] X1 + [0 1 0] X2 + [0 1 1] X3 + [0]
[0 0 0] [0 0 1] [0 0 1] [0]
= a__if(mark(X1),X2,X3)
mark(minus(X1,X2)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= a__minus(X1,X2)
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))
mark(true()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= true()
* Step 13: NaturalMI WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
a__geq(X1,X2) -> geq(X1,X2)
a__if(X1,X2,X3) -> if(X1,X2,X3)
a__minus(X1,X2) -> minus(X1,X2)
mark(0()) -> 0()
mark(minus(X1,X2)) -> a__minus(X1,X2)
mark(s(X)) -> s(mark(X))
- Weak TRS:
a__div(X1,X2) -> div(X1,X2)
a__div(0(),s(Y)) -> 0()
a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
a__geq(X,0()) -> true()
a__geq(0(),s(Y)) -> false()
a__geq(s(X),s(Y)) -> a__geq(X,Y)
a__if(false(),X,Y) -> mark(Y)
a__if(true(),X,Y) -> mark(X)
a__minus(0(),Y) -> 0()
a__minus(s(X),s(Y)) -> a__minus(X,Y)
mark(div(X1,X2)) -> a__div(mark(X1),X2)
mark(false()) -> false()
mark(geq(X1,X2)) -> a__geq(X1,X2)
mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
mark(true()) -> true()
- Signature:
{a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1} / {0/0,div/2,false/0,geq/2,if/3,minus/2,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__div,a__geq,a__if,a__minus,mark} and constructors {0
,div,false,geq,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(a__div) = {1},
uargs(a__if) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{a__div,a__geq,a__if,a__minus,mark}
TcT has computed the following interpretation:
p(0) = [0]
[0]
[0]
p(a__div) = [1 0 1] [1 1 0] [1]
[1 0 1] x1 + [1 1 0] x2 + [0]
[0 0 1] [0 0 0] [0]
p(a__geq) = [0]
[0]
[0]
p(a__if) = [1 0 0] [1 0 0] [1 0 0] [1]
[0 0 0] x1 + [1 0 0] x2 + [1 0 0] x3 + [0]
[0 0 1] [0 0 1] [0 0 1] [0]
p(a__minus) = [1 0 1] [0]
[0 0 0] x1 + [0]
[0 0 0] [0]
p(div) = [1 0 1] [1 1 0] [1]
[0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 1] [0 0 0] [0]
p(false) = [0]
[0]
[0]
p(geq) = [0]
[0]
[0]
p(if) = [1 0 0] [1 0 0] [1 0 0] [1]
[0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 1] [0 0 1] [0 0 1] [0]
p(mark) = [1 0 0] [1]
[1 0 0] x1 + [0]
[0 0 1] [0]
p(minus) = [1 0 1] [0]
[0 0 0] x1 + [0]
[0 0 0] [0]
p(s) = [1 0 1] [0]
[0 0 1] x1 + [0]
[0 0 0] [1]
p(true) = [0]
[0]
[0]
Following rules are strictly oriented:
mark(0()) = [1]
[0]
[0]
> [0]
[0]
[0]
= 0()
mark(minus(X1,X2)) = [1 0 1] [1]
[1 0 1] X1 + [0]
[0 0 0] [0]
> [1 0 1] [0]
[0 0 0] X1 + [0]
[0 0 0] [0]
= a__minus(X1,X2)
Following rules are (at-least) weakly oriented:
a__div(X1,X2) = [1 0 1] [1 1 0] [1]
[1 0 1] X1 + [1 1 0] X2 + [0]
[0 0 1] [0 0 0] [0]
>= [1 0 1] [1 1 0] [1]
[0 0 0] X1 + [0 0 0] X2 + [0]
[0 0 1] [0 0 0] [0]
= div(X1,X2)
a__div(0(),s(Y)) = [1 0 2] [1]
[1 0 2] Y + [0]
[0 0 0] [0]
>= [0]
[0]
[0]
= 0()
a__div(s(X),s(Y)) = [1 0 1] [1 0 2] [2]
[1 0 1] X + [1 0 2] Y + [1]
[0 0 0] [0 0 0] [1]
>= [1 0 1] [1 0 2] [2]
[1 0 1] X + [1 0 2] Y + [1]
[0 0 0] [0 0 0] [1]
= a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
a__geq(X,0()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= true()
a__geq(X1,X2) = [0]
[0]
[0]
>= [0]
[0]
[0]
= geq(X1,X2)
a__geq(0(),s(Y)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= false()
a__geq(s(X),s(Y)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= a__geq(X,Y)
a__if(X1,X2,X3) = [1 0 0] [1 0 0] [1 0 0] [1]
[0 0 0] X1 + [1 0 0] X2 + [1 0 0] X3 + [0]
[0 0 1] [0 0 1] [0 0 1] [0]
>= [1 0 0] [1 0 0] [1 0 0] [1]
[0 0 0] X1 + [0 0 0] X2 + [0 0 0] X3 + [0]
[0 0 1] [0 0 1] [0 0 1] [0]
= if(X1,X2,X3)
a__if(false(),X,Y) = [1 0 0] [1 0 0] [1]
[1 0 0] X + [1 0 0] Y + [0]
[0 0 1] [0 0 1] [0]
>= [1 0 0] [1]
[1 0 0] Y + [0]
[0 0 1] [0]
= mark(Y)
a__if(true(),X,Y) = [1 0 0] [1 0 0] [1]
[1 0 0] X + [1 0 0] Y + [0]
[0 0 1] [0 0 1] [0]
>= [1 0 0] [1]
[1 0 0] X + [0]
[0 0 1] [0]
= mark(X)
a__minus(X1,X2) = [1 0 1] [0]
[0 0 0] X1 + [0]
[0 0 0] [0]
>= [1 0 1] [0]
[0 0 0] X1 + [0]
[0 0 0] [0]
= minus(X1,X2)
a__minus(0(),Y) = [0]
[0]
[0]
>= [0]
[0]
[0]
= 0()
a__minus(s(X),s(Y)) = [1 0 1] [1]
[0 0 0] X + [0]
[0 0 0] [0]
>= [1 0 1] [0]
[0 0 0] X + [0]
[0 0 0] [0]
= a__minus(X,Y)
mark(div(X1,X2)) = [1 0 1] [1 1 0] [2]
[1 0 1] X1 + [1 1 0] X2 + [1]
[0 0 1] [0 0 0] [0]
>= [1 0 1] [1 1 0] [2]
[1 0 1] X1 + [1 1 0] X2 + [1]
[0 0 1] [0 0 0] [0]
= a__div(mark(X1),X2)
mark(false()) = [1]
[0]
[0]
>= [0]
[0]
[0]
= false()
mark(geq(X1,X2)) = [1]
[0]
[0]
>= [0]
[0]
[0]
= a__geq(X1,X2)
mark(if(X1,X2,X3)) = [1 0 0] [1 0 0] [1 0 0] [2]
[1 0 0] X1 + [1 0 0] X2 + [1 0 0] X3 + [1]
[0 0 1] [0 0 1] [0 0 1] [0]
>= [1 0 0] [1 0 0] [1 0 0] [2]
[0 0 0] X1 + [1 0 0] X2 + [1 0 0] X3 + [0]
[0 0 1] [0 0 1] [0 0 1] [0]
= a__if(mark(X1),X2,X3)
mark(s(X)) = [1 0 1] [1]
[1 0 1] X + [0]
[0 0 0] [1]
>= [1 0 1] [1]
[0 0 1] X + [0]
[0 0 0] [1]
= s(mark(X))
mark(true()) = [1]
[0]
[0]
>= [0]
[0]
[0]
= true()
* Step 14: NaturalMI WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
a__geq(X1,X2) -> geq(X1,X2)
a__if(X1,X2,X3) -> if(X1,X2,X3)
a__minus(X1,X2) -> minus(X1,X2)
mark(s(X)) -> s(mark(X))
- Weak TRS:
a__div(X1,X2) -> div(X1,X2)
a__div(0(),s(Y)) -> 0()
a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
a__geq(X,0()) -> true()
a__geq(0(),s(Y)) -> false()
a__geq(s(X),s(Y)) -> a__geq(X,Y)
a__if(false(),X,Y) -> mark(Y)
a__if(true(),X,Y) -> mark(X)
a__minus(0(),Y) -> 0()
a__minus(s(X),s(Y)) -> a__minus(X,Y)
mark(0()) -> 0()
mark(div(X1,X2)) -> a__div(mark(X1),X2)
mark(false()) -> false()
mark(geq(X1,X2)) -> a__geq(X1,X2)
mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
mark(minus(X1,X2)) -> a__minus(X1,X2)
mark(true()) -> true()
- Signature:
{a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1} / {0/0,div/2,false/0,geq/2,if/3,minus/2,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__div,a__geq,a__if,a__minus,mark} and constructors {0
,div,false,geq,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(a__div) = {1},
uargs(a__if) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{a__div,a__geq,a__if,a__minus,mark}
TcT has computed the following interpretation:
p(0) = [0]
[0]
[0]
p(a__div) = [1 0 1] [0 0 1] [1]
[0 1 1] x1 + [0 0 0] x2 + [1]
[0 0 1] [0 0 1] [0]
p(a__geq) = [0]
[0]
[0]
p(a__if) = [1 0 0] [0 1 1] [0 1 1] [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(a__minus) = [0]
[0]
[0]
p(div) = [0 0 0] [0 0 1] [0]
[0 1 1] x1 + [0 0 0] x2 + [1]
[0 0 1] [0 0 1] [0]
p(false) = [0]
[0]
[0]
p(geq) = [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 1] [0 0 1] [0 0 1] [1]
p(mark) = [0 1 1] [0]
[0 1 0] x1 + [0]
[0 0 1] [0]
p(minus) = [0]
[0]
[0]
p(s) = [1 0 0] [0]
[0 1 1] x1 + [0]
[0 0 0] [1]
p(true) = [0]
[0]
[0]
Following rules are strictly oriented:
mark(s(X)) = [0 1 1] [1]
[0 1 1] X + [0]
[0 0 0] [1]
> [0 1 1] [0]
[0 1 1] X + [0]
[0 0 0] [1]
= s(mark(X))
Following rules are (at-least) weakly oriented:
a__div(X1,X2) = [1 0 1] [0 0 1] [1]
[0 1 1] X1 + [0 0 0] X2 + [1]
[0 0 1] [0 0 1] [0]
>= [0 0 0] [0 0 1] [0]
[0 1 1] X1 + [0 0 0] X2 + [1]
[0 0 1] [0 0 1] [0]
= div(X1,X2)
a__div(0(),s(Y)) = [2]
[1]
[1]
>= [0]
[0]
[0]
= 0()
a__div(s(X),s(Y)) = [1 0 0] [3]
[0 1 1] X + [2]
[0 0 0] [2]
>= [3]
[2]
[2]
= a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
a__geq(X,0()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= true()
a__geq(X1,X2) = [0]
[0]
[0]
>= [0]
[0]
[0]
= geq(X1,X2)
a__geq(0(),s(Y)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= false()
a__geq(s(X),s(Y)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= a__geq(X,Y)
a__if(X1,X2,X3) = [1 0 0] [0 1 1] [0 1 1] [0]
[0 1 0] X1 + [0 1 0] X2 + [0 1 0] X3 + [0]
[0 0 1] [0 0 1] [0 0 1] [1]
>= [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 1] [0 0 1] [0 0 1] [1]
= if(X1,X2,X3)
a__if(false(),X,Y) = [0 1 1] [0 1 1] [0]
[0 1 0] X + [0 1 0] Y + [0]
[0 0 1] [0 0 1] [1]
>= [0 1 1] [0]
[0 1 0] Y + [0]
[0 0 1] [0]
= mark(Y)
a__if(true(),X,Y) = [0 1 1] [0 1 1] [0]
[0 1 0] X + [0 1 0] Y + [0]
[0 0 1] [0 0 1] [1]
>= [0 1 1] [0]
[0 1 0] X + [0]
[0 0 1] [0]
= mark(X)
a__minus(X1,X2) = [0]
[0]
[0]
>= [0]
[0]
[0]
= minus(X1,X2)
a__minus(0(),Y) = [0]
[0]
[0]
>= [0]
[0]
[0]
= 0()
a__minus(s(X),s(Y)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= a__minus(X,Y)
mark(0()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= 0()
mark(div(X1,X2)) = [0 1 2] [0 0 1] [1]
[0 1 1] X1 + [0 0 0] X2 + [1]
[0 0 1] [0 0 1] [0]
>= [0 1 2] [0 0 1] [1]
[0 1 1] X1 + [0 0 0] X2 + [1]
[0 0 1] [0 0 1] [0]
= a__div(mark(X1),X2)
mark(false()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= false()
mark(geq(X1,X2)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= a__geq(X1,X2)
mark(if(X1,X2,X3)) = [0 1 1] [0 1 1] [0 1 1] [1]
[0 1 0] X1 + [0 1 0] X2 + [0 1 0] X3 + [0]
[0 0 1] [0 0 1] [0 0 1] [1]
>= [0 1 1] [0 1 1] [0 1 1] [0]
[0 1 0] X1 + [0 1 0] X2 + [0 1 0] X3 + [0]
[0 0 1] [0 0 1] [0 0 1] [1]
= a__if(mark(X1),X2,X3)
mark(minus(X1,X2)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= a__minus(X1,X2)
mark(true()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= true()
* Step 15: NaturalMI WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
a__geq(X1,X2) -> geq(X1,X2)
a__if(X1,X2,X3) -> if(X1,X2,X3)
a__minus(X1,X2) -> minus(X1,X2)
- Weak TRS:
a__div(X1,X2) -> div(X1,X2)
a__div(0(),s(Y)) -> 0()
a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
a__geq(X,0()) -> true()
a__geq(0(),s(Y)) -> false()
a__geq(s(X),s(Y)) -> a__geq(X,Y)
a__if(false(),X,Y) -> mark(Y)
a__if(true(),X,Y) -> mark(X)
a__minus(0(),Y) -> 0()
a__minus(s(X),s(Y)) -> a__minus(X,Y)
mark(0()) -> 0()
mark(div(X1,X2)) -> a__div(mark(X1),X2)
mark(false()) -> false()
mark(geq(X1,X2)) -> a__geq(X1,X2)
mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
mark(minus(X1,X2)) -> a__minus(X1,X2)
mark(s(X)) -> s(mark(X))
mark(true()) -> true()
- Signature:
{a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1} / {0/0,div/2,false/0,geq/2,if/3,minus/2,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__div,a__geq,a__if,a__minus,mark} and constructors {0
,div,false,geq,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(a__div) = {1},
uargs(a__if) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{a__div,a__geq,a__if,a__minus,mark}
TcT has computed the following interpretation:
p(0) = [0]
[0]
[0]
p(a__div) = [1 0 1] [0]
[0 1 1] x1 + [0]
[0 0 1] [0]
p(a__geq) = [0 0 0] [0]
[0 1 0] x1 + [0]
[0 0 0] [0]
p(a__if) = [1 0 0] [0 1 0] [0 1 0] [1]
[0 1 0] x1 + [0 1 0] x2 + [0 1 0] x3 + [1]
[0 0 0] [0 0 1] [0 0 1] [0]
p(a__minus) = [0]
[0]
[0]
p(div) = [0 0 0] [0]
[0 1 1] x1 + [0]
[0 0 1] [0]
p(false) = [0]
[0]
[0]
p(geq) = [0 0 0] [0]
[0 1 0] x1 + [0]
[0 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 + [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(s) = [1 0 0] [0]
[0 1 0] x1 + [0]
[0 0 0] [1]
p(true) = [0]
[0]
[0]
Following rules are strictly oriented:
a__if(X1,X2,X3) = [1 0 0] [0 1 0] [0 1 0] [1]
[0 1 0] X1 + [0 1 0] X2 + [0 1 0] X3 + [1]
[0 0 0] [0 0 1] [0 0 1] [0]
> [0 0 0] [0 0 0] [0 0 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]
= if(X1,X2,X3)
Following rules are (at-least) weakly oriented:
a__div(X1,X2) = [1 0 1] [0]
[0 1 1] X1 + [0]
[0 0 1] [0]
>= [0 0 0] [0]
[0 1 1] X1 + [0]
[0 0 1] [0]
= div(X1,X2)
a__div(0(),s(Y)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= 0()
a__div(s(X),s(Y)) = [1 0 0] [1]
[0 1 0] X + [1]
[0 0 0] [1]
>= [0 0 0] [1]
[0 1 0] X + [1]
[0 0 0] [1]
= a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
a__geq(X,0()) = [0 0 0] [0]
[0 1 0] X + [0]
[0 0 0] [0]
>= [0]
[0]
[0]
= true()
a__geq(X1,X2) = [0 0 0] [0]
[0 1 0] X1 + [0]
[0 0 0] [0]
>= [0 0 0] [0]
[0 1 0] X1 + [0]
[0 0 0] [0]
= geq(X1,X2)
a__geq(0(),s(Y)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= false()
a__geq(s(X),s(Y)) = [0 0 0] [0]
[0 1 0] X + [0]
[0 0 0] [0]
>= [0 0 0] [0]
[0 1 0] X + [0]
[0 0 0] [0]
= a__geq(X,Y)
a__if(false(),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] Y + [0]
[0 0 1] [0]
= mark(Y)
a__if(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)
a__minus(X1,X2) = [0]
[0]
[0]
>= [0]
[0]
[0]
= minus(X1,X2)
a__minus(0(),Y) = [0]
[0]
[0]
>= [0]
[0]
[0]
= 0()
a__minus(s(X),s(Y)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= a__minus(X,Y)
mark(0()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= 0()
mark(div(X1,X2)) = [0 1 1] [0]
[0 1 1] X1 + [0]
[0 0 1] [0]
>= [0 1 1] [0]
[0 1 1] X1 + [0]
[0 0 1] [0]
= a__div(mark(X1),X2)
mark(false()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= false()
mark(geq(X1,X2)) = [0 1 0] [0]
[0 1 0] X1 + [0]
[0 0 0] [0]
>= [0 0 0] [0]
[0 1 0] X1 + [0]
[0 0 0] [0]
= a__geq(X1,X2)
mark(if(X1,X2,X3)) = [0 1 0] [0 1 0] [0 1 0] [1]
[0 1 0] X1 + [0 1 0] X2 + [0 1 0] X3 + [1]
[0 0 0] [0 0 1] [0 0 1] [0]
>= [0 1 0] [0 1 0] [0 1 0] [1]
[0 1 0] X1 + [0 1 0] X2 + [0 1 0] X3 + [1]
[0 0 0] [0 0 1] [0 0 1] [0]
= a__if(mark(X1),X2,X3)
mark(minus(X1,X2)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= a__minus(X1,X2)
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))
mark(true()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= true()
* Step 16: NaturalMI WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
a__geq(X1,X2) -> geq(X1,X2)
a__minus(X1,X2) -> minus(X1,X2)
- Weak TRS:
a__div(X1,X2) -> div(X1,X2)
a__div(0(),s(Y)) -> 0()
a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
a__geq(X,0()) -> true()
a__geq(0(),s(Y)) -> false()
a__geq(s(X),s(Y)) -> a__geq(X,Y)
a__if(X1,X2,X3) -> if(X1,X2,X3)
a__if(false(),X,Y) -> mark(Y)
a__if(true(),X,Y) -> mark(X)
a__minus(0(),Y) -> 0()
a__minus(s(X),s(Y)) -> a__minus(X,Y)
mark(0()) -> 0()
mark(div(X1,X2)) -> a__div(mark(X1),X2)
mark(false()) -> false()
mark(geq(X1,X2)) -> a__geq(X1,X2)
mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
mark(minus(X1,X2)) -> a__minus(X1,X2)
mark(s(X)) -> s(mark(X))
mark(true()) -> true()
- Signature:
{a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1} / {0/0,div/2,false/0,geq/2,if/3,minus/2,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__div,a__geq,a__if,a__minus,mark} and constructors {0
,div,false,geq,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(a__div) = {1},
uargs(a__if) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{a__div,a__geq,a__if,a__minus,mark}
TcT has computed the following interpretation:
p(0) = [0]
[0]
[0]
p(a__div) = [1 0 1] [0]
[0 1 1] x1 + [0]
[0 0 1] [0]
p(a__geq) = [0 0 0] [1]
[0 1 0] x1 + [1]
[0 0 0] [0]
p(a__if) = [1 0 0] [0 1 0] [0 1 0] [0]
[0 1 1] x1 + [0 1 0] x2 + [0 1 1] x3 + [0]
[0 0 0] [0 0 1] [0 0 1] [0]
p(a__minus) = [0]
[0]
[0]
p(div) = [0 0 0] [0]
[0 1 1] x1 + [0]
[0 0 1] [0]
p(false) = [0]
[1]
[0]
p(geq) = [0 0 0] [0]
[0 1 0] x1 + [1]
[0 0 0] [0]
p(if) = [0 0 0] [0 0 0] [0 1 0] [0]
[0 1 1] x1 + [0 1 0] x2 + [0 1 1] 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(s) = [1 0 0] [0]
[0 1 0] x1 + [0]
[0 0 0] [1]
p(true) = [0]
[0]
[0]
Following rules are strictly oriented:
a__geq(X1,X2) = [0 0 0] [1]
[0 1 0] X1 + [1]
[0 0 0] [0]
> [0 0 0] [0]
[0 1 0] X1 + [1]
[0 0 0] [0]
= geq(X1,X2)
Following rules are (at-least) weakly oriented:
a__div(X1,X2) = [1 0 1] [0]
[0 1 1] X1 + [0]
[0 0 1] [0]
>= [0 0 0] [0]
[0 1 1] X1 + [0]
[0 0 1] [0]
= div(X1,X2)
a__div(0(),s(Y)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= 0()
a__div(s(X),s(Y)) = [1 0 0] [1]
[0 1 0] X + [1]
[0 0 0] [1]
>= [0 0 0] [1]
[0 1 0] X + [1]
[0 0 0] [1]
= a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
a__geq(X,0()) = [0 0 0] [1]
[0 1 0] X + [1]
[0 0 0] [0]
>= [0]
[0]
[0]
= true()
a__geq(0(),s(Y)) = [1]
[1]
[0]
>= [0]
[1]
[0]
= false()
a__geq(s(X),s(Y)) = [0 0 0] [1]
[0 1 0] X + [1]
[0 0 0] [0]
>= [0 0 0] [1]
[0 1 0] X + [1]
[0 0 0] [0]
= a__geq(X,Y)
a__if(X1,X2,X3) = [1 0 0] [0 1 0] [0 1 0] [0]
[0 1 1] X1 + [0 1 0] X2 + [0 1 1] X3 + [0]
[0 0 0] [0 0 1] [0 0 1] [0]
>= [0 0 0] [0 0 0] [0 1 0] [0]
[0 1 1] X1 + [0 1 0] X2 + [0 1 1] X3 + [0]
[0 0 0] [0 0 1] [0 0 1] [0]
= if(X1,X2,X3)
a__if(false(),X,Y) = [0 1 0] [0 1 0] [0]
[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)
a__if(true(),X,Y) = [0 1 0] [0 1 0] [0]
[0 1 0] X + [0 1 1] Y + [0]
[0 0 1] [0 0 1] [0]
>= [0 1 0] [0]
[0 1 0] X + [0]
[0 0 1] [0]
= mark(X)
a__minus(X1,X2) = [0]
[0]
[0]
>= [0]
[0]
[0]
= minus(X1,X2)
a__minus(0(),Y) = [0]
[0]
[0]
>= [0]
[0]
[0]
= 0()
a__minus(s(X),s(Y)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= a__minus(X,Y)
mark(0()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= 0()
mark(div(X1,X2)) = [0 1 1] [0]
[0 1 1] X1 + [0]
[0 0 1] [0]
>= [0 1 1] [0]
[0 1 1] X1 + [0]
[0 0 1] [0]
= a__div(mark(X1),X2)
mark(false()) = [1]
[1]
[0]
>= [0]
[1]
[0]
= false()
mark(geq(X1,X2)) = [0 1 0] [1]
[0 1 0] X1 + [1]
[0 0 0] [0]
>= [0 0 0] [1]
[0 1 0] X1 + [1]
[0 0 0] [0]
= a__geq(X1,X2)
mark(if(X1,X2,X3)) = [0 1 1] [0 1 0] [0 1 1] [0]
[0 1 1] X1 + [0 1 0] X2 + [0 1 1] X3 + [0]
[0 0 0] [0 0 1] [0 0 1] [0]
>= [0 1 0] [0 1 0] [0 1 0] [0]
[0 1 1] X1 + [0 1 0] X2 + [0 1 1] X3 + [0]
[0 0 0] [0 0 1] [0 0 1] [0]
= a__if(mark(X1),X2,X3)
mark(minus(X1,X2)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= a__minus(X1,X2)
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))
mark(true()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= true()
* Step 17: NaturalMI WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
a__minus(X1,X2) -> minus(X1,X2)
- Weak TRS:
a__div(X1,X2) -> div(X1,X2)
a__div(0(),s(Y)) -> 0()
a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
a__geq(X,0()) -> true()
a__geq(X1,X2) -> geq(X1,X2)
a__geq(0(),s(Y)) -> false()
a__geq(s(X),s(Y)) -> a__geq(X,Y)
a__if(X1,X2,X3) -> if(X1,X2,X3)
a__if(false(),X,Y) -> mark(Y)
a__if(true(),X,Y) -> mark(X)
a__minus(0(),Y) -> 0()
a__minus(s(X),s(Y)) -> a__minus(X,Y)
mark(0()) -> 0()
mark(div(X1,X2)) -> a__div(mark(X1),X2)
mark(false()) -> false()
mark(geq(X1,X2)) -> a__geq(X1,X2)
mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
mark(minus(X1,X2)) -> a__minus(X1,X2)
mark(s(X)) -> s(mark(X))
mark(true()) -> true()
- Signature:
{a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1} / {0/0,div/2,false/0,geq/2,if/3,minus/2,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__div,a__geq,a__if,a__minus,mark} and constructors {0
,div,false,geq,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(a__div) = {1},
uargs(a__if) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{a__div,a__geq,a__if,a__minus,mark}
TcT has computed the following interpretation:
p(0) = [0]
[0]
[0]
p(a__div) = [1 0 1] [1]
[0 1 1] x1 + [1]
[0 0 1] [0]
p(a__geq) = [0 0 0] [0]
[0 1 0] x1 + [0]
[0 0 0] [0]
p(a__if) = [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(a__minus) = [1]
[1]
[0]
p(div) = [0 0 0] [1]
[0 1 1] x1 + [1]
[0 0 1] [0]
p(false) = [0]
[0]
[0]
p(geq) = [0 0 0] [0]
[0 1 0] x1 + [0]
[0 0 0] [0]
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 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(s) = [1 0 0] [0]
[0 1 0] x1 + [0]
[0 0 0] [1]
p(true) = [0]
[0]
[0]
Following rules are strictly oriented:
a__minus(X1,X2) = [1]
[1]
[0]
> [0]
[1]
[0]
= minus(X1,X2)
Following rules are (at-least) weakly oriented:
a__div(X1,X2) = [1 0 1] [1]
[0 1 1] X1 + [1]
[0 0 1] [0]
>= [0 0 0] [1]
[0 1 1] X1 + [1]
[0 0 1] [0]
= div(X1,X2)
a__div(0(),s(Y)) = [1]
[1]
[0]
>= [0]
[0]
[0]
= 0()
a__div(s(X),s(Y)) = [1 0 0] [2]
[0 1 0] X + [2]
[0 0 0] [1]
>= [0 0 0] [2]
[0 1 0] X + [2]
[0 0 0] [1]
= a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
a__geq(X,0()) = [0 0 0] [0]
[0 1 0] X + [0]
[0 0 0] [0]
>= [0]
[0]
[0]
= true()
a__geq(X1,X2) = [0 0 0] [0]
[0 1 0] X1 + [0]
[0 0 0] [0]
>= [0 0 0] [0]
[0 1 0] X1 + [0]
[0 0 0] [0]
= geq(X1,X2)
a__geq(0(),s(Y)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= false()
a__geq(s(X),s(Y)) = [0 0 0] [0]
[0 1 0] X + [0]
[0 0 0] [0]
>= [0 0 0] [0]
[0 1 0] X + [0]
[0 0 0] [0]
= a__geq(X,Y)
a__if(X1,X2,X3) = [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]
>= [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 0] [0 0 1] [0 0 1] [0]
= if(X1,X2,X3)
a__if(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)
a__if(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)
a__minus(0(),Y) = [1]
[1]
[0]
>= [0]
[0]
[0]
= 0()
a__minus(s(X),s(Y)) = [1]
[1]
[0]
>= [1]
[1]
[0]
= a__minus(X,Y)
mark(0()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= 0()
mark(div(X1,X2)) = [0 1 1] [1]
[0 1 1] X1 + [1]
[0 0 1] [0]
>= [0 1 1] [1]
[0 1 1] X1 + [1]
[0 0 1] [0]
= a__div(mark(X1),X2)
mark(false()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= false()
mark(geq(X1,X2)) = [0 1 0] [0]
[0 1 0] X1 + [0]
[0 0 0] [0]
>= [0 0 0] [0]
[0 1 0] X1 + [0]
[0 0 0] [0]
= a__geq(X1,X2)
mark(if(X1,X2,X3)) = [0 1 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]
>= [0 1 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]
= a__if(mark(X1),X2,X3)
mark(minus(X1,X2)) = [1]
[1]
[0]
>= [1]
[1]
[0]
= a__minus(X1,X2)
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))
mark(true()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= true()
* Step 18: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
a__div(X1,X2) -> div(X1,X2)
a__div(0(),s(Y)) -> 0()
a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
a__geq(X,0()) -> true()
a__geq(X1,X2) -> geq(X1,X2)
a__geq(0(),s(Y)) -> false()
a__geq(s(X),s(Y)) -> a__geq(X,Y)
a__if(X1,X2,X3) -> if(X1,X2,X3)
a__if(false(),X,Y) -> mark(Y)
a__if(true(),X,Y) -> mark(X)
a__minus(X1,X2) -> minus(X1,X2)
a__minus(0(),Y) -> 0()
a__minus(s(X),s(Y)) -> a__minus(X,Y)
mark(0()) -> 0()
mark(div(X1,X2)) -> a__div(mark(X1),X2)
mark(false()) -> false()
mark(geq(X1,X2)) -> a__geq(X1,X2)
mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
mark(minus(X1,X2)) -> a__minus(X1,X2)
mark(s(X)) -> s(mark(X))
mark(true()) -> true()
- Signature:
{a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1} / {0/0,div/2,false/0,geq/2,if/3,minus/2,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__div,a__geq,a__if,a__minus,mark} and constructors {0
,div,false,geq,if,minus,s,true}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(?,O(n^3))