WORST_CASE(?,O(n^2))
* Step 1: InnermostRuleRemoval WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
0() -> n__0()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__div(X1,X2)) -> div(activate(X1),X2)
activate(n__minus(X1,X2)) -> minus(X1,X2)
activate(n__s(X)) -> s(activate(X))
div(X1,X2) -> n__div(X1,X2)
div(0(),n__s(Y)) -> 0()
div(s(X),n__s(Y)) -> if(geq(X,activate(Y)),n__s(n__div(n__minus(X,activate(Y)),n__s(activate(Y)))),n__0())
geq(X,n__0()) -> true()
geq(n__0(),n__s(Y)) -> false()
geq(n__s(X),n__s(Y)) -> geq(activate(X),activate(Y))
if(false(),X,Y) -> activate(Y)
if(true(),X,Y) -> activate(X)
minus(X1,X2) -> n__minus(X1,X2)
minus(n__0(),Y) -> 0()
minus(n__s(X),n__s(Y)) -> minus(activate(X),activate(Y))
s(X) -> n__s(X)
- Signature:
{0/0,activate/1,div/2,geq/2,if/3,minus/2,s/1} / {false/0,n__0/0,n__div/2,n__minus/2,n__s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0,activate,div,geq,if,minus,s} and constructors {false
,n__0,n__div,n__minus,n__s,true}
+ Applied Processor:
InnermostRuleRemoval
+ Details:
Arguments of following rules are not normal-forms.
div(0(),n__s(Y)) -> 0()
div(s(X),n__s(Y)) -> if(geq(X,activate(Y)),n__s(n__div(n__minus(X,activate(Y)),n__s(activate(Y)))),n__0())
All above mentioned rules can be savely removed.
* Step 2: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
0() -> n__0()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__div(X1,X2)) -> div(activate(X1),X2)
activate(n__minus(X1,X2)) -> minus(X1,X2)
activate(n__s(X)) -> s(activate(X))
div(X1,X2) -> n__div(X1,X2)
geq(X,n__0()) -> true()
geq(n__0(),n__s(Y)) -> false()
geq(n__s(X),n__s(Y)) -> geq(activate(X),activate(Y))
if(false(),X,Y) -> activate(Y)
if(true(),X,Y) -> activate(X)
minus(X1,X2) -> n__minus(X1,X2)
minus(n__0(),Y) -> 0()
minus(n__s(X),n__s(Y)) -> minus(activate(X),activate(Y))
s(X) -> n__s(X)
- Signature:
{0/0,activate/1,div/2,geq/2,if/3,minus/2,s/1} / {false/0,n__0/0,n__div/2,n__minus/2,n__s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0,activate,div,geq,if,minus,s} and constructors {false
,n__0,n__div,n__minus,n__s,true}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(div) = {1},
uargs(geq) = {1,2},
uargs(minus) = {1,2},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(activate) = [1] x1 + [5]
p(div) = [1] x1 + [1] x2 + [7]
p(false) = [0]
p(geq) = [1] x1 + [1] x2 + [8]
p(if) = [7] x1 + [1] x2 + [3] x3 + [1]
p(minus) = [1] x1 + [1] x2 + [8]
p(n__0) = [11]
p(n__div) = [1] x1 + [1] x2 + [3]
p(n__minus) = [1] x1 + [1] x2 + [2]
p(n__s) = [1] x1 + [0]
p(s) = [1] x1 + [0]
p(true) = [4]
Following rules are strictly oriented:
activate(X) = [1] X + [5]
> [1] X + [0]
= X
activate(n__0()) = [16]
> [0]
= 0()
div(X1,X2) = [1] X1 + [1] X2 + [7]
> [1] X1 + [1] X2 + [3]
= n__div(X1,X2)
geq(X,n__0()) = [1] X + [19]
> [4]
= true()
geq(n__0(),n__s(Y)) = [1] Y + [19]
> [0]
= false()
if(true(),X,Y) = [1] X + [3] Y + [29]
> [1] X + [5]
= activate(X)
minus(X1,X2) = [1] X1 + [1] X2 + [8]
> [1] X1 + [1] X2 + [2]
= n__minus(X1,X2)
minus(n__0(),Y) = [1] Y + [19]
> [0]
= 0()
Following rules are (at-least) weakly oriented:
0() = [0]
>= [11]
= n__0()
activate(n__div(X1,X2)) = [1] X1 + [1] X2 + [8]
>= [1] X1 + [1] X2 + [12]
= div(activate(X1),X2)
activate(n__minus(X1,X2)) = [1] X1 + [1] X2 + [7]
>= [1] X1 + [1] X2 + [8]
= minus(X1,X2)
activate(n__s(X)) = [1] X + [5]
>= [1] X + [5]
= s(activate(X))
geq(n__s(X),n__s(Y)) = [1] X + [1] Y + [8]
>= [1] X + [1] Y + [18]
= geq(activate(X),activate(Y))
if(false(),X,Y) = [1] X + [3] Y + [1]
>= [1] Y + [5]
= activate(Y)
minus(n__s(X),n__s(Y)) = [1] X + [1] Y + [8]
>= [1] X + [1] Y + [18]
= minus(activate(X),activate(Y))
s(X) = [1] X + [0]
>= [1] X + [0]
= n__s(X)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
0() -> n__0()
activate(n__div(X1,X2)) -> div(activate(X1),X2)
activate(n__minus(X1,X2)) -> minus(X1,X2)
activate(n__s(X)) -> s(activate(X))
geq(n__s(X),n__s(Y)) -> geq(activate(X),activate(Y))
if(false(),X,Y) -> activate(Y)
minus(n__s(X),n__s(Y)) -> minus(activate(X),activate(Y))
s(X) -> n__s(X)
- Weak TRS:
activate(X) -> X
activate(n__0()) -> 0()
div(X1,X2) -> n__div(X1,X2)
geq(X,n__0()) -> true()
geq(n__0(),n__s(Y)) -> false()
if(true(),X,Y) -> activate(X)
minus(X1,X2) -> n__minus(X1,X2)
minus(n__0(),Y) -> 0()
- Signature:
{0/0,activate/1,div/2,geq/2,if/3,minus/2,s/1} / {false/0,n__0/0,n__div/2,n__minus/2,n__s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0,activate,div,geq,if,minus,s} and constructors {false
,n__0,n__div,n__minus,n__s,true}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(div) = {1},
uargs(geq) = {1,2},
uargs(minus) = {1,2},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [10]
p(activate) = [1] x1 + [14]
p(div) = [1] x1 + [0]
p(false) = [15]
p(geq) = [1] x1 + [1] x2 + [3]
p(if) = [1] x2 + [1] x3 + [14]
p(minus) = [1] x1 + [1] x2 + [2]
p(n__0) = [8]
p(n__div) = [1] x1 + [0]
p(n__minus) = [1] x1 + [1] x2 + [2]
p(n__s) = [1] x1 + [4]
p(s) = [1] x1 + [0]
p(true) = [11]
Following rules are strictly oriented:
0() = [10]
> [8]
= n__0()
activate(n__minus(X1,X2)) = [1] X1 + [1] X2 + [16]
> [1] X1 + [1] X2 + [2]
= minus(X1,X2)
activate(n__s(X)) = [1] X + [18]
> [1] X + [14]
= s(activate(X))
Following rules are (at-least) weakly oriented:
activate(X) = [1] X + [14]
>= [1] X + [0]
= X
activate(n__0()) = [22]
>= [10]
= 0()
activate(n__div(X1,X2)) = [1] X1 + [14]
>= [1] X1 + [14]
= div(activate(X1),X2)
div(X1,X2) = [1] X1 + [0]
>= [1] X1 + [0]
= n__div(X1,X2)
geq(X,n__0()) = [1] X + [11]
>= [11]
= true()
geq(n__0(),n__s(Y)) = [1] Y + [15]
>= [15]
= false()
geq(n__s(X),n__s(Y)) = [1] X + [1] Y + [11]
>= [1] X + [1] Y + [31]
= geq(activate(X),activate(Y))
if(false(),X,Y) = [1] X + [1] Y + [14]
>= [1] Y + [14]
= activate(Y)
if(true(),X,Y) = [1] X + [1] Y + [14]
>= [1] X + [14]
= activate(X)
minus(X1,X2) = [1] X1 + [1] X2 + [2]
>= [1] X1 + [1] X2 + [2]
= n__minus(X1,X2)
minus(n__0(),Y) = [1] Y + [10]
>= [10]
= 0()
minus(n__s(X),n__s(Y)) = [1] X + [1] Y + [10]
>= [1] X + [1] Y + [30]
= minus(activate(X),activate(Y))
s(X) = [1] X + [0]
>= [1] X + [4]
= n__s(X)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 4: NaturalMI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
activate(n__div(X1,X2)) -> div(activate(X1),X2)
geq(n__s(X),n__s(Y)) -> geq(activate(X),activate(Y))
if(false(),X,Y) -> activate(Y)
minus(n__s(X),n__s(Y)) -> minus(activate(X),activate(Y))
s(X) -> n__s(X)
- Weak TRS:
0() -> n__0()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__minus(X1,X2)) -> minus(X1,X2)
activate(n__s(X)) -> s(activate(X))
div(X1,X2) -> n__div(X1,X2)
geq(X,n__0()) -> true()
geq(n__0(),n__s(Y)) -> false()
if(true(),X,Y) -> activate(X)
minus(X1,X2) -> n__minus(X1,X2)
minus(n__0(),Y) -> 0()
- Signature:
{0/0,activate/1,div/2,geq/2,if/3,minus/2,s/1} / {false/0,n__0/0,n__div/2,n__minus/2,n__s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0,activate,div,geq,if,minus,s} and constructors {false
,n__0,n__div,n__minus,n__s,true}
+ Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(div) = {1},
uargs(geq) = {1,2},
uargs(minus) = {1,2},
uargs(s) = {1}
Following symbols are considered usable:
{0,activate,div,geq,if,minus,s}
TcT has computed the following interpretation:
p(0) = [3]
p(activate) = [1] x1 + [0]
p(div) = [1] x1 + [8]
p(false) = [2]
p(geq) = [1] x1 + [4] x2 + [0]
p(if) = [8] x1 + [1] x2 + [1] x3 + [0]
p(minus) = [1] x1 + [1] x2 + [1]
p(n__0) = [3]
p(n__div) = [1] x1 + [8]
p(n__minus) = [1] x1 + [1] x2 + [1]
p(n__s) = [1] x1 + [0]
p(s) = [1] x1 + [0]
p(true) = [0]
Following rules are strictly oriented:
if(false(),X,Y) = [1] X + [1] Y + [16]
> [1] Y + [0]
= activate(Y)
Following rules are (at-least) weakly oriented:
0() = [3]
>= [3]
= n__0()
activate(X) = [1] X + [0]
>= [1] X + [0]
= X
activate(n__0()) = [3]
>= [3]
= 0()
activate(n__div(X1,X2)) = [1] X1 + [8]
>= [1] X1 + [8]
= div(activate(X1),X2)
activate(n__minus(X1,X2)) = [1] X1 + [1] X2 + [1]
>= [1] X1 + [1] X2 + [1]
= minus(X1,X2)
activate(n__s(X)) = [1] X + [0]
>= [1] X + [0]
= s(activate(X))
div(X1,X2) = [1] X1 + [8]
>= [1] X1 + [8]
= n__div(X1,X2)
geq(X,n__0()) = [1] X + [12]
>= [0]
= true()
geq(n__0(),n__s(Y)) = [4] Y + [3]
>= [2]
= false()
geq(n__s(X),n__s(Y)) = [1] X + [4] Y + [0]
>= [1] X + [4] Y + [0]
= geq(activate(X),activate(Y))
if(true(),X,Y) = [1] X + [1] Y + [0]
>= [1] X + [0]
= activate(X)
minus(X1,X2) = [1] X1 + [1] X2 + [1]
>= [1] X1 + [1] X2 + [1]
= n__minus(X1,X2)
minus(n__0(),Y) = [1] Y + [4]
>= [3]
= 0()
minus(n__s(X),n__s(Y)) = [1] X + [1] Y + [1]
>= [1] X + [1] Y + [1]
= minus(activate(X),activate(Y))
s(X) = [1] X + [0]
>= [1] X + [0]
= n__s(X)
* Step 5: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
activate(n__div(X1,X2)) -> div(activate(X1),X2)
geq(n__s(X),n__s(Y)) -> geq(activate(X),activate(Y))
minus(n__s(X),n__s(Y)) -> minus(activate(X),activate(Y))
s(X) -> n__s(X)
- Weak TRS:
0() -> n__0()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__minus(X1,X2)) -> minus(X1,X2)
activate(n__s(X)) -> s(activate(X))
div(X1,X2) -> n__div(X1,X2)
geq(X,n__0()) -> true()
geq(n__0(),n__s(Y)) -> false()
if(false(),X,Y) -> activate(Y)
if(true(),X,Y) -> activate(X)
minus(X1,X2) -> n__minus(X1,X2)
minus(n__0(),Y) -> 0()
- Signature:
{0/0,activate/1,div/2,geq/2,if/3,minus/2,s/1} / {false/0,n__0/0,n__div/2,n__minus/2,n__s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0,activate,div,geq,if,minus,s} and constructors {false
,n__0,n__div,n__minus,n__s,true}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(div) = {1},
uargs(geq) = {1,2},
uargs(minus) = {1,2},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(activate) = [1] x1 + [0]
p(div) = [1] x1 + [1] x2 + [0]
p(false) = [5]
p(geq) = [1] x1 + [1] x2 + [0]
p(if) = [3] x1 + [1] x2 + [1] x3 + [0]
p(minus) = [1] x1 + [1] x2 + [0]
p(n__0) = [0]
p(n__div) = [1] x1 + [1] x2 + [0]
p(n__minus) = [1] x1 + [1] x2 + [0]
p(n__s) = [1] x1 + [5]
p(s) = [1] x1 + [5]
p(true) = [0]
Following rules are strictly oriented:
geq(n__s(X),n__s(Y)) = [1] X + [1] Y + [10]
> [1] X + [1] Y + [0]
= geq(activate(X),activate(Y))
minus(n__s(X),n__s(Y)) = [1] X + [1] Y + [10]
> [1] X + [1] Y + [0]
= minus(activate(X),activate(Y))
Following rules are (at-least) weakly oriented:
0() = [0]
>= [0]
= n__0()
activate(X) = [1] X + [0]
>= [1] X + [0]
= X
activate(n__0()) = [0]
>= [0]
= 0()
activate(n__div(X1,X2)) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]
= div(activate(X1),X2)
activate(n__minus(X1,X2)) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]
= minus(X1,X2)
activate(n__s(X)) = [1] X + [5]
>= [1] X + [5]
= s(activate(X))
div(X1,X2) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]
= n__div(X1,X2)
geq(X,n__0()) = [1] X + [0]
>= [0]
= true()
geq(n__0(),n__s(Y)) = [1] Y + [5]
>= [5]
= false()
if(false(),X,Y) = [1] X + [1] Y + [15]
>= [1] Y + [0]
= activate(Y)
if(true(),X,Y) = [1] X + [1] Y + [0]
>= [1] X + [0]
= activate(X)
minus(X1,X2) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]
= n__minus(X1,X2)
minus(n__0(),Y) = [1] Y + [0]
>= [0]
= 0()
s(X) = [1] X + [5]
>= [1] X + [5]
= n__s(X)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 6: NaturalMI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
activate(n__div(X1,X2)) -> div(activate(X1),X2)
s(X) -> n__s(X)
- Weak TRS:
0() -> n__0()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__minus(X1,X2)) -> minus(X1,X2)
activate(n__s(X)) -> s(activate(X))
div(X1,X2) -> n__div(X1,X2)
geq(X,n__0()) -> true()
geq(n__0(),n__s(Y)) -> false()
geq(n__s(X),n__s(Y)) -> geq(activate(X),activate(Y))
if(false(),X,Y) -> activate(Y)
if(true(),X,Y) -> activate(X)
minus(X1,X2) -> n__minus(X1,X2)
minus(n__0(),Y) -> 0()
minus(n__s(X),n__s(Y)) -> minus(activate(X),activate(Y))
- Signature:
{0/0,activate/1,div/2,geq/2,if/3,minus/2,s/1} / {false/0,n__0/0,n__div/2,n__minus/2,n__s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0,activate,div,geq,if,minus,s} and constructors {false
,n__0,n__div,n__minus,n__s,true}
+ Applied Processor:
NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(div) = {1},
uargs(geq) = {1,2},
uargs(minus) = {1,2},
uargs(s) = {1}
Following symbols are considered usable:
{0,activate,div,geq,if,minus,s}
TcT has computed the following interpretation:
p(0) = [0]
[4]
p(activate) = [1 2] x1 + [0]
[0 1] [0]
p(div) = [1 0] x1 + [0]
[0 1] [1]
p(false) = [0]
[1]
p(geq) = [2 0] x1 + [1 1] x2 + [4]
[2 2] [0 2] [4]
p(if) = [4 0] x1 + [1 3] x2 + [4 4] x3 + [1]
[6 2] [4 1] [0 2] [0]
p(minus) = [1 2] x1 + [1 0] x2 + [1]
[0 1] [0 0] [2]
p(n__0) = [0]
[4]
p(n__div) = [1 0] x1 + [0]
[0 1] [1]
p(n__minus) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 0] [2]
p(n__s) = [1 4] x1 + [2]
[0 1] [0]
p(s) = [1 4] x1 + [2]
[0 1] [0]
p(true) = [2]
[1]
Following rules are strictly oriented:
activate(n__div(X1,X2)) = [1 2] X1 + [2]
[0 1] [1]
> [1 2] X1 + [0]
[0 1] [1]
= div(activate(X1),X2)
Following rules are (at-least) weakly oriented:
0() = [0]
[4]
>= [0]
[4]
= n__0()
activate(X) = [1 2] X + [0]
[0 1] [0]
>= [1 0] X + [0]
[0 1] [0]
= X
activate(n__0()) = [8]
[4]
>= [0]
[4]
= 0()
activate(n__minus(X1,X2)) = [1 2] X1 + [1 0] X2 + [4]
[0 1] [0 0] [2]
>= [1 2] X1 + [1 0] X2 + [1]
[0 1] [0 0] [2]
= minus(X1,X2)
activate(n__s(X)) = [1 6] X + [2]
[0 1] [0]
>= [1 6] X + [2]
[0 1] [0]
= s(activate(X))
div(X1,X2) = [1 0] X1 + [0]
[0 1] [1]
>= [1 0] X1 + [0]
[0 1] [1]
= n__div(X1,X2)
geq(X,n__0()) = [2 0] X + [8]
[2 2] [12]
>= [2]
[1]
= true()
geq(n__0(),n__s(Y)) = [1 5] Y + [6]
[0 2] [12]
>= [0]
[1]
= false()
geq(n__s(X),n__s(Y)) = [2 8] X + [1 5] Y + [10]
[2 10] [0 2] [8]
>= [2 4] X + [1 3] Y + [4]
[2 6] [0 2] [4]
= geq(activate(X),activate(Y))
if(false(),X,Y) = [1 3] X + [4 4] Y + [1]
[4 1] [0 2] [2]
>= [1 2] Y + [0]
[0 1] [0]
= activate(Y)
if(true(),X,Y) = [1 3] X + [4 4] Y + [9]
[4 1] [0 2] [14]
>= [1 2] X + [0]
[0 1] [0]
= activate(X)
minus(X1,X2) = [1 2] X1 + [1 0] X2 + [1]
[0 1] [0 0] [2]
>= [1 0] X1 + [1 0] X2 + [0]
[0 1] [0 0] [2]
= n__minus(X1,X2)
minus(n__0(),Y) = [1 0] Y + [9]
[0 0] [6]
>= [0]
[4]
= 0()
minus(n__s(X),n__s(Y)) = [1 6] X + [1 4] Y + [5]
[0 1] [0 0] [2]
>= [1 4] X + [1 2] Y + [1]
[0 1] [0 0] [2]
= minus(activate(X),activate(Y))
s(X) = [1 4] X + [2]
[0 1] [0]
>= [1 4] X + [2]
[0 1] [0]
= n__s(X)
* Step 7: NaturalMI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
s(X) -> n__s(X)
- Weak TRS:
0() -> n__0()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__div(X1,X2)) -> div(activate(X1),X2)
activate(n__minus(X1,X2)) -> minus(X1,X2)
activate(n__s(X)) -> s(activate(X))
div(X1,X2) -> n__div(X1,X2)
geq(X,n__0()) -> true()
geq(n__0(),n__s(Y)) -> false()
geq(n__s(X),n__s(Y)) -> geq(activate(X),activate(Y))
if(false(),X,Y) -> activate(Y)
if(true(),X,Y) -> activate(X)
minus(X1,X2) -> n__minus(X1,X2)
minus(n__0(),Y) -> 0()
minus(n__s(X),n__s(Y)) -> minus(activate(X),activate(Y))
- Signature:
{0/0,activate/1,div/2,geq/2,if/3,minus/2,s/1} / {false/0,n__0/0,n__div/2,n__minus/2,n__s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0,activate,div,geq,if,minus,s} and constructors {false
,n__0,n__div,n__minus,n__s,true}
+ Applied Processor:
NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(div) = {1},
uargs(geq) = {1,2},
uargs(minus) = {1,2},
uargs(s) = {1}
Following symbols are considered usable:
{0,activate,div,geq,if,minus,s}
TcT has computed the following interpretation:
p(0) = [1]
[0]
p(activate) = [1 4] x1 + [0]
[0 1] [0]
p(div) = [1 0] x1 + [4]
[0 1] [1]
p(false) = [0]
[2]
p(geq) = [1 1] x1 + [1 0] x2 + [0]
[0 1] [2 3] [7]
p(if) = [0 4] x1 + [4 6] x2 + [1 4] x3 + [5]
[1 0] [4 4] [0 4] [4]
p(minus) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
p(n__0) = [1]
[0]
p(n__div) = [1 0] x1 + [0]
[0 1] [1]
p(n__minus) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
p(n__s) = [1 4] x1 + [0]
[0 1] [1]
p(s) = [1 4] x1 + [4]
[0 1] [1]
p(true) = [0]
[0]
Following rules are strictly oriented:
s(X) = [1 4] X + [4]
[0 1] [1]
> [1 4] X + [0]
[0 1] [1]
= n__s(X)
Following rules are (at-least) weakly oriented:
0() = [1]
[0]
>= [1]
[0]
= n__0()
activate(X) = [1 4] X + [0]
[0 1] [0]
>= [1 0] X + [0]
[0 1] [0]
= X
activate(n__0()) = [1]
[0]
>= [1]
[0]
= 0()
activate(n__div(X1,X2)) = [1 4] X1 + [4]
[0 1] [1]
>= [1 4] X1 + [4]
[0 1] [1]
= div(activate(X1),X2)
activate(n__minus(X1,X2)) = [1 0] X1 + [1 0] X2 + [0]
[0 0] [0 0] [0]
>= [1 0] X1 + [1 0] X2 + [0]
[0 0] [0 0] [0]
= minus(X1,X2)
activate(n__s(X)) = [1 8] X + [4]
[0 1] [1]
>= [1 8] X + [4]
[0 1] [1]
= s(activate(X))
div(X1,X2) = [1 0] X1 + [4]
[0 1] [1]
>= [1 0] X1 + [0]
[0 1] [1]
= n__div(X1,X2)
geq(X,n__0()) = [1 1] X + [1]
[0 1] [9]
>= [0]
[0]
= true()
geq(n__0(),n__s(Y)) = [1 4] Y + [1]
[2 11] [10]
>= [0]
[2]
= false()
geq(n__s(X),n__s(Y)) = [1 5] X + [1 4] Y + [1]
[0 1] [2 11] [11]
>= [1 5] X + [1 4] Y + [0]
[0 1] [2 11] [7]
= geq(activate(X),activate(Y))
if(false(),X,Y) = [4 6] X + [1 4] Y + [13]
[4 4] [0 4] [4]
>= [1 4] Y + [0]
[0 1] [0]
= activate(Y)
if(true(),X,Y) = [4 6] X + [1 4] Y + [5]
[4 4] [0 4] [4]
>= [1 4] X + [0]
[0 1] [0]
= activate(X)
minus(X1,X2) = [1 0] X1 + [1 0] X2 + [0]
[0 0] [0 0] [0]
>= [1 0] X1 + [1 0] X2 + [0]
[0 0] [0 0] [0]
= n__minus(X1,X2)
minus(n__0(),Y) = [1 0] Y + [1]
[0 0] [0]
>= [1]
[0]
= 0()
minus(n__s(X),n__s(Y)) = [1 4] X + [1 4] Y + [0]
[0 0] [0 0] [0]
>= [1 4] X + [1 4] Y + [0]
[0 0] [0 0] [0]
= minus(activate(X),activate(Y))
* Step 8: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
0() -> n__0()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__div(X1,X2)) -> div(activate(X1),X2)
activate(n__minus(X1,X2)) -> minus(X1,X2)
activate(n__s(X)) -> s(activate(X))
div(X1,X2) -> n__div(X1,X2)
geq(X,n__0()) -> true()
geq(n__0(),n__s(Y)) -> false()
geq(n__s(X),n__s(Y)) -> geq(activate(X),activate(Y))
if(false(),X,Y) -> activate(Y)
if(true(),X,Y) -> activate(X)
minus(X1,X2) -> n__minus(X1,X2)
minus(n__0(),Y) -> 0()
minus(n__s(X),n__s(Y)) -> minus(activate(X),activate(Y))
s(X) -> n__s(X)
- Signature:
{0/0,activate/1,div/2,geq/2,if/3,minus/2,s/1} / {false/0,n__0/0,n__div/2,n__minus/2,n__s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0,activate,div,geq,if,minus,s} and constructors {false
,n__0,n__div,n__minus,n__s,true}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(?,O(n^2))