* Step 1: Sum WORST_CASE(Omega(n^1),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:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ 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:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
activate(x){x -> n__div(x,y)} =
activate(n__div(x,y)) ->^+ div(activate(x),y)
= C[activate(x) = activate(x){}]
** Step 1.b: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 1.b: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 + [2]
p(div) = [1] x1 + [1] x2 + [0]
p(false) = [5]
p(geq) = [1] x1 + [1] x2 + [7]
p(if) = [5] x1 + [1] x2 + [1] x3 + [0]
p(minus) = [1] x1 + [1] x2 + [7]
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 + [0]
p(s) = [1] x1 + [0]
p(true) = [0]
Following rules are strictly oriented:
activate(X) = [1] X + [2]
> [1] X + [0]
= X
activate(n__0()) = [2]
> [0]
= 0()
geq(X,n__0()) = [1] X + [7]
> [0]
= true()
geq(n__0(),n__s(Y)) = [1] Y + [7]
> [5]
= false()
if(false(),X,Y) = [1] X + [1] Y + [25]
> [1] Y + [2]
= activate(Y)
minus(X1,X2) = [1] X1 + [1] X2 + [7]
> [1] X1 + [1] X2 + [0]
= n__minus(X1,X2)
minus(n__0(),Y) = [1] Y + [7]
> [0]
= 0()
Following rules are (at-least) weakly oriented:
0() = [0]
>= [0]
= n__0()
activate(n__div(X1,X2)) = [1] X1 + [1] X2 + [2]
>= [1] X1 + [1] X2 + [2]
= div(activate(X1),X2)
activate(n__minus(X1,X2)) = [1] X1 + [1] X2 + [2]
>= [1] X1 + [1] X2 + [7]
= minus(X1,X2)
activate(n__s(X)) = [1] X + [2]
>= [1] X + [2]
= s(activate(X))
div(X1,X2) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]
= n__div(X1,X2)
geq(n__s(X),n__s(Y)) = [1] X + [1] Y + [7]
>= [1] X + [1] Y + [11]
= geq(activate(X),activate(Y))
if(true(),X,Y) = [1] X + [1] Y + [0]
>= [1] X + [2]
= activate(X)
minus(n__s(X),n__s(Y)) = [1] X + [1] Y + [7]
>= [1] X + [1] Y + [11]
= 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 1.b: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))
div(X1,X2) -> n__div(X1,X2)
geq(n__s(X),n__s(Y)) -> geq(activate(X),activate(Y))
if(true(),X,Y) -> activate(X)
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()
geq(X,n__0()) -> true()
geq(n__0(),n__s(Y)) -> false()
if(false(),X,Y) -> activate(Y)
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) = [2]
p(activate) = [1] x1 + [14]
p(div) = [1] x1 + [1] x2 + [0]
p(false) = [1]
p(geq) = [1] x1 + [1] x2 + [0]
p(if) = [13] x1 + [1] x2 + [1] x3 + [1]
p(minus) = [1] x1 + [1] x2 + [1]
p(n__0) = [1]
p(n__div) = [1] x1 + [1] x2 + [0]
p(n__minus) = [1] x1 + [1] x2 + [1]
p(n__s) = [1] x1 + [0]
p(s) = [1] x1 + [0]
p(true) = [1]
Following rules are strictly oriented:
0() = [2]
> [1]
= n__0()
activate(n__minus(X1,X2)) = [1] X1 + [1] X2 + [15]
> [1] X1 + [1] X2 + [1]
= minus(X1,X2)
Following rules are (at-least) weakly oriented:
activate(X) = [1] X + [14]
>= [1] X + [0]
= X
activate(n__0()) = [15]
>= [2]
= 0()
activate(n__div(X1,X2)) = [1] X1 + [1] X2 + [14]
>= [1] X1 + [1] X2 + [14]
= div(activate(X1),X2)
activate(n__s(X)) = [1] X + [14]
>= [1] X + [14]
= 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 + [1]
>= [1]
= true()
geq(n__0(),n__s(Y)) = [1] Y + [1]
>= [1]
= false()
geq(n__s(X),n__s(Y)) = [1] X + [1] Y + [0]
>= [1] X + [1] Y + [28]
= 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 + [1]
>= [1] X1 + [1] X2 + [1]
= n__minus(X1,X2)
minus(n__0(),Y) = [1] Y + [2]
>= [2]
= 0()
minus(n__s(X),n__s(Y)) = [1] X + [1] Y + [1]
>= [1] X + [1] Y + [29]
= 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 1.b:4: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
activate(n__div(X1,X2)) -> div(activate(X1),X2)
activate(n__s(X)) -> s(activate(X))
div(X1,X2) -> n__div(X1,X2)
geq(n__s(X),n__s(Y)) -> geq(activate(X),activate(Y))
if(true(),X,Y) -> activate(X)
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)
geq(X,n__0()) -> true()
geq(n__0(),n__s(Y)) -> false()
if(false(),X,Y) -> activate(Y)
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]
p(false) = [4]
p(geq) = [1] x1 + [1] x2 + [4]
p(if) = [1] x1 + [1] x2 + [9] x3 + [5]
p(minus) = [1] x1 + [1] x2 + [1]
p(n__0) = [0]
p(n__div) = [1] x1 + [0]
p(n__minus) = [1] x1 + [1] x2 + [1]
p(n__s) = [1] x1 + [0]
p(s) = [1] x1 + [0]
p(true) = [4]
Following rules are strictly oriented:
div(X1,X2) = [1] X1 + [1]
> [1] X1 + [0]
= n__div(X1,X2)
if(true(),X,Y) = [1] X + [9] Y + [9]
> [1] X + [0]
= activate(X)
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 + [0]
>= [1] X1 + [1]
= 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))
geq(X,n__0()) = [1] X + [4]
>= [4]
= true()
geq(n__0(),n__s(Y)) = [1] Y + [4]
>= [4]
= false()
geq(n__s(X),n__s(Y)) = [1] X + [1] Y + [4]
>= [1] X + [1] Y + [4]
= geq(activate(X),activate(Y))
if(false(),X,Y) = [1] X + [9] Y + [9]
>= [1] Y + [0]
= activate(Y)
minus(X1,X2) = [1] X1 + [1] X2 + [1]
>= [1] X1 + [1] X2 + [1]
= n__minus(X1,X2)
minus(n__0(),Y) = [1] Y + [1]
>= [0]
= 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)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:5: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
activate(n__div(X1,X2)) -> div(activate(X1),X2)
activate(n__s(X)) -> s(activate(X))
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)
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 + [0]
p(false) = [2]
p(geq) = [1] x1 + [1] x2 + [2]
p(if) = [4] x1 + [1] x2 + [1] x3 + [5]
p(minus) = [1] x1 + [1] x2 + [0]
p(n__0) = [0]
p(n__div) = [1] x1 + [0]
p(n__minus) = [1] x1 + [1] x2 + [0]
p(n__s) = [1] x1 + [0]
p(s) = [1] x1 + [4]
p(true) = [1]
Following rules are strictly oriented:
s(X) = [1] X + [4]
> [1] X + [0]
= n__s(X)
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 + [0]
>= [1] X1 + [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 + [0]
>= [1] X + [4]
= s(activate(X))
div(X1,X2) = [1] X1 + [0]
>= [1] X1 + [0]
= n__div(X1,X2)
geq(X,n__0()) = [1] X + [2]
>= [1]
= true()
geq(n__0(),n__s(Y)) = [1] Y + [2]
>= [2]
= false()
geq(n__s(X),n__s(Y)) = [1] X + [1] Y + [2]
>= [1] X + [1] Y + [2]
= geq(activate(X),activate(Y))
if(false(),X,Y) = [1] X + [1] Y + [13]
>= [1] Y + [0]
= activate(Y)
if(true(),X,Y) = [1] X + [1] Y + [9]
>= [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()
minus(n__s(X),n__s(Y)) = [1] X + [1] Y + [0]
>= [1] X + [1] Y + [0]
= minus(activate(X),activate(Y))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:6: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
activate(n__div(X1,X2)) -> div(activate(X1),X2)
activate(n__s(X)) -> s(activate(X))
geq(n__s(X),n__s(Y)) -> geq(activate(X),activate(Y))
minus(n__s(X),n__s(Y)) -> minus(activate(X),activate(Y))
- Weak TRS:
0() -> n__0()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__minus(X1,X2)) -> minus(X1,X2)
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()
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 + [0]
p(div) = [1] x1 + [1] x2 + [5]
p(false) = [1]
p(geq) = [1] x1 + [1] x2 + [0]
p(if) = [1] x2 + [1] x3 + [5]
p(minus) = [1] x1 + [1] x2 + [0]
p(n__0) = [0]
p(n__div) = [1] x1 + [1] x2 + [5]
p(n__minus) = [1] x1 + [1] x2 + [0]
p(n__s) = [1] x1 + [1]
p(s) = [1] x1 + [4]
p(true) = [0]
Following rules are strictly oriented:
geq(n__s(X),n__s(Y)) = [1] X + [1] Y + [2]
> [1] X + [1] Y + [0]
= geq(activate(X),activate(Y))
minus(n__s(X),n__s(Y)) = [1] X + [1] Y + [2]
> [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 + [5]
>= [1] X1 + [1] X2 + [5]
= 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 + [1]
>= [1] X + [4]
= s(activate(X))
div(X1,X2) = [1] X1 + [1] X2 + [5]
>= [1] X1 + [1] X2 + [5]
= n__div(X1,X2)
geq(X,n__0()) = [1] X + [0]
>= [0]
= true()
geq(n__0(),n__s(Y)) = [1] Y + [1]
>= [1]
= false()
if(false(),X,Y) = [1] X + [1] Y + [5]
>= [1] Y + [0]
= activate(Y)
if(true(),X,Y) = [1] X + [1] Y + [5]
>= [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 + [4]
>= [1] X + [1]
= n__s(X)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:7: MI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
activate(n__div(X1,X2)) -> div(activate(X1),X2)
activate(n__s(X)) -> s(activate(X))
- Weak TRS:
0() -> n__0()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__minus(X1,X2)) -> minus(X1,X2)
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:
MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity Nothing)), miDimension = 2, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity Nothing)):
The following argument positions are considered usable:
uargs(div) = {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) = [4]
[1]
p(activate) = [1 1] x_1 + [0]
[0 1] [0]
p(div) = [1 0] x_1 + [0 3] x_2 + [0]
[0 1] [0 1] [0]
p(false) = [2]
[1]
p(geq) = [2 3] x_1 + [1 3] x_2 + [1]
[3 1] [0 4] [1]
p(if) = [0 1] x_1 + [1 4] x_2 + [1 1] x_3 + [5]
[4 4] [0 4] [4 2] [0]
p(minus) = [1 3] x_1 + [1 7] x_2 + [1]
[0 0] [0 0] [4]
p(n__0) = [3]
[1]
p(n__div) = [1 0] x_1 + [0 3] x_2 + [0]
[0 1] [0 1] [0]
p(n__minus) = [1 3] x_1 + [1 7] x_2 + [1]
[0 0] [0 0] [4]
p(n__s) = [1 4] x_1 + [0]
[0 1] [1]
p(s) = [1 4] x_1 + [0]
[0 1] [1]
p(true) = [2]
[1]
Following rules are strictly oriented:
activate(n__s(X)) = [1 5] X + [1]
[0 1] [1]
> [1 5] X + [0]
[0 1] [1]
= s(activate(X))
Following rules are (at-least) weakly oriented:
0() = [4]
[1]
>= [3]
[1]
= n__0()
activate(X) = [1 1] X + [0]
[0 1] [0]
>= [1 0] X + [0]
[0 1] [0]
= X
activate(n__0()) = [4]
[1]
>= [4]
[1]
= 0()
activate(n__div(X1,X2)) = [1 1] X1 + [0 4] X2 + [0]
[0 1] [0 1] [0]
>= [1 1] X1 + [0 3] X2 + [0]
[0 1] [0 1] [0]
= div(activate(X1),X2)
activate(n__minus(X1,X2)) = [1 3] X1 + [1 7] X2 + [5]
[0 0] [0 0] [4]
>= [1 3] X1 + [1 7] X2 + [1]
[0 0] [0 0] [4]
= minus(X1,X2)
div(X1,X2) = [1 0] X1 + [0 3] X2 + [0]
[0 1] [0 1] [0]
>= [1 0] X1 + [0 3] X2 + [0]
[0 1] [0 1] [0]
= n__div(X1,X2)
geq(X,n__0()) = [2 3] X + [7]
[3 1] [5]
>= [2]
[1]
= true()
geq(n__0(),n__s(Y)) = [1 7] Y + [13]
[0 4] [15]
>= [2]
[1]
= false()
geq(n__s(X),n__s(Y)) = [2 11] X + [1 7] Y + [7]
[3 13] [0 4] [6]
>= [2 5] X + [1 4] Y + [1]
[3 4] [0 4] [1]
= geq(activate(X),activate(Y))
if(false(),X,Y) = [1 4] X + [1 1] Y + [6]
[0 4] [4 2] [12]
>= [1 1] Y + [0]
[0 1] [0]
= activate(Y)
if(true(),X,Y) = [1 4] X + [1 1] Y + [6]
[0 4] [4 2] [12]
>= [1 1] X + [0]
[0 1] [0]
= activate(X)
minus(X1,X2) = [1 3] X1 + [1 7] X2 + [1]
[0 0] [0 0] [4]
>= [1 3] X1 + [1 7] X2 + [1]
[0 0] [0 0] [4]
= n__minus(X1,X2)
minus(n__0(),Y) = [1 7] Y + [7]
[0 0] [4]
>= [4]
[1]
= 0()
minus(n__s(X),n__s(Y)) = [1 7] X + [1 11] Y + [11]
[0 0] [0 0] [4]
>= [1 4] X + [1 8] Y + [1]
[0 0] [0 0] [4]
= minus(activate(X),activate(Y))
s(X) = [1 4] X + [0]
[0 1] [1]
>= [1 4] X + [0]
[0 1] [1]
= n__s(X)
** Step 1.b:8: MI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
activate(n__div(X1,X2)) -> div(activate(X1),X2)
- 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))
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:
MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity Nothing)), miDimension = 2, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity Nothing)):
The following argument positions are considered usable:
uargs(div) = {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 2] x_1 + [0]
[0 1] [1]
p(div) = [1 0] x_1 + [4]
[0 1] [1]
p(false) = [4]
[1]
p(geq) = [2 1] x_1 + [2 0] x_2 + [6]
[0 2] [0 0] [1]
p(if) = [0 2] x_1 + [1 4] x_2 + [2 4] x_3 + [5]
[2 1] [1 4] [0 1] [5]
p(minus) = [1 0] x_1 + [1 1] x_2 + [3]
[0 0] [0 0] [0]
p(n__0) = [1]
[0]
p(n__div) = [1 0] x_1 + [4]
[0 1] [1]
p(n__minus) = [1 0] x_1 + [1 1] x_2 + [3]
[0 0] [0 0] [0]
p(n__s) = [1 2] x_1 + [0]
[0 1] [5]
p(s) = [1 2] x_1 + [4]
[0 1] [5]
p(true) = [0]
[1]
Following rules are strictly oriented:
activate(n__div(X1,X2)) = [1 2] X1 + [6]
[0 1] [2]
> [1 2] X1 + [4]
[0 1] [2]
= div(activate(X1),X2)
Following rules are (at-least) weakly oriented:
0() = [1]
[0]
>= [1]
[0]
= n__0()
activate(X) = [1 2] X + [0]
[0 1] [1]
>= [1 0] X + [0]
[0 1] [0]
= X
activate(n__0()) = [1]
[1]
>= [1]
[0]
= 0()
activate(n__minus(X1,X2)) = [1 0] X1 + [1 1] X2 + [3]
[0 0] [0 0] [1]
>= [1 0] X1 + [1 1] X2 + [3]
[0 0] [0 0] [0]
= minus(X1,X2)
activate(n__s(X)) = [1 4] X + [10]
[0 1] [6]
>= [1 4] X + [6]
[0 1] [6]
= s(activate(X))
div(X1,X2) = [1 0] X1 + [4]
[0 1] [1]
>= [1 0] X1 + [4]
[0 1] [1]
= n__div(X1,X2)
geq(X,n__0()) = [2 1] X + [8]
[0 2] [1]
>= [0]
[1]
= true()
geq(n__0(),n__s(Y)) = [2 4] Y + [8]
[0 0] [1]
>= [4]
[1]
= false()
geq(n__s(X),n__s(Y)) = [2 5] X + [2 4] Y + [11]
[0 2] [0 0] [11]
>= [2 5] X + [2 4] Y + [7]
[0 2] [0 0] [3]
= geq(activate(X),activate(Y))
if(false(),X,Y) = [1 4] X + [2 4] Y + [7]
[1 4] [0 1] [14]
>= [1 2] Y + [0]
[0 1] [1]
= activate(Y)
if(true(),X,Y) = [1 4] X + [2 4] Y + [7]
[1 4] [0 1] [6]
>= [1 2] X + [0]
[0 1] [1]
= activate(X)
minus(X1,X2) = [1 0] X1 + [1 1] X2 + [3]
[0 0] [0 0] [0]
>= [1 0] X1 + [1 1] X2 + [3]
[0 0] [0 0] [0]
= n__minus(X1,X2)
minus(n__0(),Y) = [1 1] Y + [4]
[0 0] [0]
>= [1]
[0]
= 0()
minus(n__s(X),n__s(Y)) = [1 2] X + [1 3] Y + [8]
[0 0] [0 0] [0]
>= [1 2] X + [1 3] Y + [4]
[0 0] [0 0] [0]
= minus(activate(X),activate(Y))
s(X) = [1 2] X + [4]
[0 1] [5]
>= [1 2] X + [0]
[0 1] [5]
= n__s(X)
** Step 1.b:9: 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(Omega(n^1),O(n^2))