* Step 1: Sum WORST_CASE(Omega(n^1),O(n^2))
+ 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:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ 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:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
a__geq(x,y){x -> s(x),y -> s(y)} =
a__geq(s(x),s(y)) ->^+ a__geq(x,y)
= C[a__geq(x,y) = a__geq(x,y){}]
** Step 1.b:1: WeightGap WORST_CASE(?,O(n^2))
+ 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:
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(a__div) = {1},
uargs(a__if) = {1},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(a__div) = [1] x1 + [1]
p(a__geq) = [0]
p(a__if) = [1] x1 + [3]
p(a__minus) = [0]
p(div) = [0]
p(false) = [0]
p(geq) = [0]
p(if) = [1] x1 + [0]
p(mark) = [0]
p(minus) = [0]
p(s) = [1] x1 + [0]
p(true) = [0]
Following rules are strictly oriented:
a__div(X1,X2) = [1] X1 + [1]
> [0]
= div(X1,X2)
a__div(0(),s(Y)) = [1]
> [0]
= 0()
a__if(X1,X2,X3) = [1] X1 + [3]
> [1] X1 + [0]
= if(X1,X2,X3)
a__if(false(),X,Y) = [3]
> [0]
= mark(Y)
a__if(true(),X,Y) = [3]
> [0]
= mark(X)
Following rules are (at-least) weakly oriented:
a__div(s(X),s(Y)) = [1] X + [1]
>= [3]
= 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__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)) = [0]
>= [1]
= 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)) = [0]
>= [3]
= a__if(mark(X1),X2,X3)
mark(minus(X1,X2)) = [0]
>= [0]
= a__minus(X1,X2)
mark(s(X)) = [0]
>= [0]
= s(mark(X))
mark(true()) = [0]
>= [0]
= true()
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:2: WeightGap WORST_CASE(?,O(n^2))
+ 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__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__if(X1,X2,X3) -> if(X1,X2,X3)
a__if(false(),X,Y) -> mark(Y)
a__if(true(),X,Y) -> mark(X)
- 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:
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(a__div) = {1},
uargs(a__if) = {1},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [2]
p(a__div) = [1] x1 + [7]
p(a__geq) = [6]
p(a__if) = [1] x1 + [4]
p(a__minus) = [0]
p(div) = [1] x1 + [5]
p(false) = [7]
p(geq) = [0]
p(if) = [1] x1 + [1]
p(mark) = [2]
p(minus) = [1]
p(s) = [1] x1 + [0]
p(true) = [5]
Following rules are strictly oriented:
a__geq(X,0()) = [6]
> [5]
= true()
a__geq(X1,X2) = [6]
> [0]
= geq(X1,X2)
mark(minus(X1,X2)) = [2]
> [0]
= a__minus(X1,X2)
Following rules are (at-least) weakly oriented:
a__div(X1,X2) = [1] X1 + [7]
>= [1] X1 + [5]
= div(X1,X2)
a__div(0(),s(Y)) = [9]
>= [2]
= 0()
a__div(s(X),s(Y)) = [1] X + [7]
>= [10]
= a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
a__geq(0(),s(Y)) = [6]
>= [7]
= false()
a__geq(s(X),s(Y)) = [6]
>= [6]
= a__geq(X,Y)
a__if(X1,X2,X3) = [1] X1 + [4]
>= [1] X1 + [1]
= if(X1,X2,X3)
a__if(false(),X,Y) = [11]
>= [2]
= mark(Y)
a__if(true(),X,Y) = [9]
>= [2]
= mark(X)
a__minus(X1,X2) = [0]
>= [1]
= minus(X1,X2)
a__minus(0(),Y) = [0]
>= [2]
= 0()
a__minus(s(X),s(Y)) = [0]
>= [0]
= a__minus(X,Y)
mark(0()) = [2]
>= [2]
= 0()
mark(div(X1,X2)) = [2]
>= [9]
= a__div(mark(X1),X2)
mark(false()) = [2]
>= [7]
= false()
mark(geq(X1,X2)) = [2]
>= [6]
= a__geq(X1,X2)
mark(if(X1,X2,X3)) = [2]
>= [6]
= a__if(mark(X1),X2,X3)
mark(s(X)) = [2]
>= [2]
= s(mark(X))
mark(true()) = [2]
>= [5]
= true()
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:
a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
a__geq(0(),s(Y)) -> false()
a__geq(s(X),s(Y)) -> a__geq(X,Y)
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(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(X,0()) -> true()
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)
mark(minus(X1,X2)) -> a__minus(X1,X2)
- 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:
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(a__div) = {1},
uargs(a__if) = {1},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(a__div) = [1] x1 + [0]
p(a__geq) = [0]
p(a__if) = [1] x1 + [1]
p(a__minus) = [1]
p(div) = [0]
p(false) = [0]
p(geq) = [0]
p(if) = [1]
p(mark) = [1]
p(minus) = [0]
p(s) = [1] x1 + [0]
p(true) = [0]
Following rules are strictly oriented:
a__minus(X1,X2) = [1]
> [0]
= minus(X1,X2)
a__minus(0(),Y) = [1]
> [0]
= 0()
mark(0()) = [1]
> [0]
= 0()
mark(false()) = [1]
> [0]
= false()
mark(geq(X1,X2)) = [1]
> [0]
= a__geq(X1,X2)
mark(true()) = [1]
> [0]
= true()
Following rules are (at-least) weakly oriented:
a__div(X1,X2) = [1] X1 + [0]
>= [0]
= div(X1,X2)
a__div(0(),s(Y)) = [0]
>= [0]
= 0()
a__div(s(X),s(Y)) = [1] X + [0]
>= [1]
= 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) = [1] X1 + [1]
>= [1]
= if(X1,X2,X3)
a__if(false(),X,Y) = [1]
>= [1]
= mark(Y)
a__if(true(),X,Y) = [1]
>= [1]
= mark(X)
a__minus(s(X),s(Y)) = [1]
>= [1]
= a__minus(X,Y)
mark(div(X1,X2)) = [1]
>= [1]
= a__div(mark(X1),X2)
mark(if(X1,X2,X3)) = [1]
>= [2]
= a__if(mark(X1),X2,X3)
mark(minus(X1,X2)) = [1]
>= [1]
= a__minus(X1,X2)
mark(s(X)) = [1]
>= [1]
= s(mark(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:
a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
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(div(X1,X2)) -> a__div(mark(X1),X2)
mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
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(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(false()) -> false()
mark(geq(X1,X2)) -> a__geq(X1,X2)
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:
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(a__div) = {1},
uargs(a__if) = {1},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(a__div) = [1] x1 + [1]
p(a__geq) = [0]
p(a__if) = [1] x1 + [0]
p(a__minus) = [0]
p(div) = [0]
p(false) = [0]
p(geq) = [0]
p(if) = [0]
p(mark) = [0]
p(minus) = [0]
p(s) = [1] x1 + [1]
p(true) = [0]
Following rules are strictly oriented:
a__div(s(X),s(Y)) = [1] X + [2]
> [0]
= 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] X1 + [1]
>= [0]
= div(X1,X2)
a__div(0(),s(Y)) = [1]
>= [0]
= 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) = [1] X1 + [0]
>= [0]
= if(X1,X2,X3)
a__if(false(),X,Y) = [0]
>= [0]
= mark(Y)
a__if(true(),X,Y) = [0]
>= [0]
= 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)) = [0]
>= [1]
= 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)) = [0]
>= [0]
= a__if(mark(X1),X2,X3)
mark(minus(X1,X2)) = [0]
>= [0]
= a__minus(X1,X2)
mark(s(X)) = [0]
>= [1]
= s(mark(X))
mark(true()) = [0]
>= [0]
= true()
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:
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(div(X1,X2)) -> a__div(mark(X1),X2)
mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
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(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(false()) -> false()
mark(geq(X1,X2)) -> a__geq(X1,X2)
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:
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(a__div) = {1},
uargs(a__if) = {1},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(a__div) = [1] x1 + [7]
p(a__geq) = [1]
p(a__if) = [1] x1 + [2]
p(a__minus) = [1]
p(div) = [1] x1 + [7]
p(false) = [0]
p(geq) = [0]
p(if) = [0]
p(mark) = [1]
p(minus) = [1]
p(s) = [1] x1 + [0]
p(true) = [1]
Following rules are strictly oriented:
a__geq(0(),s(Y)) = [1]
> [0]
= false()
Following rules are (at-least) weakly oriented:
a__div(X1,X2) = [1] X1 + [7]
>= [1] X1 + [7]
= div(X1,X2)
a__div(0(),s(Y)) = [7]
>= [0]
= 0()
a__div(s(X),s(Y)) = [1] X + [7]
>= [3]
= a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
a__geq(X,0()) = [1]
>= [1]
= true()
a__geq(X1,X2) = [1]
>= [0]
= geq(X1,X2)
a__geq(s(X),s(Y)) = [1]
>= [1]
= a__geq(X,Y)
a__if(X1,X2,X3) = [1] X1 + [2]
>= [0]
= if(X1,X2,X3)
a__if(false(),X,Y) = [2]
>= [1]
= mark(Y)
a__if(true(),X,Y) = [3]
>= [1]
= mark(X)
a__minus(X1,X2) = [1]
>= [1]
= minus(X1,X2)
a__minus(0(),Y) = [1]
>= [0]
= 0()
a__minus(s(X),s(Y)) = [1]
>= [1]
= a__minus(X,Y)
mark(0()) = [1]
>= [0]
= 0()
mark(div(X1,X2)) = [1]
>= [8]
= a__div(mark(X1),X2)
mark(false()) = [1]
>= [0]
= false()
mark(geq(X1,X2)) = [1]
>= [1]
= a__geq(X1,X2)
mark(if(X1,X2,X3)) = [1]
>= [3]
= a__if(mark(X1),X2,X3)
mark(minus(X1,X2)) = [1]
>= [1]
= a__minus(X1,X2)
mark(s(X)) = [1]
>= [1]
= s(mark(X))
mark(true()) = [1]
>= [1]
= true()
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:
a__geq(s(X),s(Y)) -> a__geq(X,Y)
a__minus(s(X),s(Y)) -> a__minus(X,Y)
mark(div(X1,X2)) -> a__div(mark(X1),X2)
mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
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(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(false()) -> false()
mark(geq(X1,X2)) -> a__geq(X1,X2)
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:
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(a__div) = {1},
uargs(a__if) = {1},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(a__div) = [1] x1 + [3]
p(a__geq) = [1] x1 + [0]
p(a__if) = [1] x1 + [4] x2 + [4] x3 + [0]
p(a__minus) = [0]
p(div) = [1] x1 + [0]
p(false) = [0]
p(geq) = [1] x1 + [0]
p(if) = [1] x1 + [1] x2 + [1] x3 + [0]
p(mark) = [4] x1 + [0]
p(minus) = [0]
p(s) = [1] x1 + [1]
p(true) = [0]
Following rules are strictly oriented:
a__geq(s(X),s(Y)) = [1] X + [1]
> [1] X + [0]
= a__geq(X,Y)
mark(s(X)) = [4] X + [4]
> [4] X + [1]
= s(mark(X))
Following rules are (at-least) weakly oriented:
a__div(X1,X2) = [1] X1 + [3]
>= [1] X1 + [0]
= div(X1,X2)
a__div(0(),s(Y)) = [3]
>= [0]
= 0()
a__div(s(X),s(Y)) = [1] X + [4]
>= [1] X + [4]
= a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
a__geq(X,0()) = [1] X + [0]
>= [0]
= true()
a__geq(X1,X2) = [1] X1 + [0]
>= [1] X1 + [0]
= geq(X1,X2)
a__geq(0(),s(Y)) = [0]
>= [0]
= false()
a__if(X1,X2,X3) = [1] X1 + [4] X2 + [4] X3 + [0]
>= [1] X1 + [1] X2 + [1] X3 + [0]
= if(X1,X2,X3)
a__if(false(),X,Y) = [4] X + [4] Y + [0]
>= [4] Y + [0]
= mark(Y)
a__if(true(),X,Y) = [4] X + [4] Y + [0]
>= [4] X + [0]
= 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] X1 + [0]
>= [4] X1 + [3]
= a__div(mark(X1),X2)
mark(false()) = [0]
>= [0]
= false()
mark(geq(X1,X2)) = [4] X1 + [0]
>= [1] X1 + [0]
= a__geq(X1,X2)
mark(if(X1,X2,X3)) = [4] X1 + [4] X2 + [4] X3 + [0]
>= [4] X1 + [4] X2 + [4] X3 + [0]
= a__if(mark(X1),X2,X3)
mark(minus(X1,X2)) = [0]
>= [0]
= a__minus(X1,X2)
mark(true()) = [0]
>= [0]
= true()
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:7: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
a__minus(s(X),s(Y)) -> a__minus(X,Y)
mark(div(X1,X2)) -> a__div(mark(X1),X2)
mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
- 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()
mark(0()) -> 0()
mark(false()) -> false()
mark(geq(X1,X2)) -> a__geq(X1,X2)
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:
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(a__div) = {1},
uargs(a__if) = {1},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(a__div) = [1] x1 + [2]
p(a__geq) = [0]
p(a__if) = [1] x1 + [4] x2 + [4] x3 + [2]
p(a__minus) = [0]
p(div) = [1] x1 + [0]
p(false) = [0]
p(geq) = [0]
p(if) = [1] x1 + [1] x2 + [1] x3 + [1]
p(mark) = [4] x1 + [0]
p(minus) = [0]
p(s) = [1] x1 + [0]
p(true) = [0]
Following rules are strictly oriented:
mark(if(X1,X2,X3)) = [4] X1 + [4] X2 + [4] X3 + [4]
> [4] X1 + [4] X2 + [4] X3 + [2]
= a__if(mark(X1),X2,X3)
Following rules are (at-least) weakly oriented:
a__div(X1,X2) = [1] X1 + [2]
>= [1] X1 + [0]
= div(X1,X2)
a__div(0(),s(Y)) = [2]
>= [0]
= 0()
a__div(s(X),s(Y)) = [1] X + [2]
>= [2]
= 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) = [1] X1 + [4] X2 + [4] X3 + [2]
>= [1] X1 + [1] X2 + [1] X3 + [1]
= if(X1,X2,X3)
a__if(false(),X,Y) = [4] X + [4] Y + [2]
>= [4] Y + [0]
= mark(Y)
a__if(true(),X,Y) = [4] X + [4] Y + [2]
>= [4] X + [0]
= 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] X1 + [0]
>= [4] X1 + [2]
= a__div(mark(X1),X2)
mark(false()) = [0]
>= [0]
= false()
mark(geq(X1,X2)) = [0]
>= [0]
= a__geq(X1,X2)
mark(minus(X1,X2)) = [0]
>= [0]
= a__minus(X1,X2)
mark(s(X)) = [4] X + [0]
>= [4] X + [0]
= s(mark(X))
mark(true()) = [0]
>= [0]
= true()
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:8: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
a__minus(s(X),s(Y)) -> a__minus(X,Y)
mark(div(X1,X2)) -> a__div(mark(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(X1,X2) -> minus(X1,X2)
a__minus(0(),Y) -> 0()
mark(0()) -> 0()
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:
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(a__div) = {1},
uargs(a__if) = {1},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [1]
p(a__div) = [1] x1 + [1]
p(a__geq) = [0]
p(a__if) = [1] x1 + [1] x2 + [1] x3 + [0]
p(a__minus) = [1] x1 + [0]
p(div) = [1] x1 + [0]
p(false) = [0]
p(geq) = [0]
p(if) = [1] x1 + [1] x2 + [1] x3 + [0]
p(mark) = [1] x1 + [0]
p(minus) = [1] x1 + [0]
p(s) = [1] x1 + [5]
p(true) = [0]
Following rules are strictly oriented:
a__minus(s(X),s(Y)) = [1] X + [5]
> [1] X + [0]
= a__minus(X,Y)
Following rules are (at-least) weakly oriented:
a__div(X1,X2) = [1] X1 + [1]
>= [1] X1 + [0]
= div(X1,X2)
a__div(0(),s(Y)) = [2]
>= [1]
= 0()
a__div(s(X),s(Y)) = [1] X + [6]
>= [1] X + [6]
= 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) = [1] X1 + [1] X2 + [1] X3 + [0]
>= [1] X1 + [1] X2 + [1] X3 + [0]
= if(X1,X2,X3)
a__if(false(),X,Y) = [1] X + [1] Y + [0]
>= [1] Y + [0]
= mark(Y)
a__if(true(),X,Y) = [1] X + [1] Y + [0]
>= [1] X + [0]
= mark(X)
a__minus(X1,X2) = [1] X1 + [0]
>= [1] X1 + [0]
= minus(X1,X2)
a__minus(0(),Y) = [1]
>= [1]
= 0()
mark(0()) = [1]
>= [1]
= 0()
mark(div(X1,X2)) = [1] X1 + [0]
>= [1] X1 + [1]
= 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)) = [1] X1 + [1] X2 + [1] X3 + [0]
>= [1] X1 + [1] X2 + [1] X3 + [0]
= a__if(mark(X1),X2,X3)
mark(minus(X1,X2)) = [1] X1 + [0]
>= [1] X1 + [0]
= a__minus(X1,X2)
mark(s(X)) = [1] X + [5]
>= [1] X + [5]
= s(mark(X))
mark(true()) = [0]
>= [0]
= true()
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:9: MI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
mark(div(X1,X2)) -> a__div(mark(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(X1,X2) -> minus(X1,X2)
a__minus(0(),Y) -> 0()
a__minus(s(X),s(Y)) -> a__minus(X,Y)
mark(0()) -> 0()
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:
MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity (Just 2))), miDimension = 3, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity (Just 2))):
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 2 0] [1]
[0 0 0] x_1 + [0 1 0] x_2 + [2]
[0 0 0] [0 0 0] [0]
p(a__geq) = [0]
[0]
[0]
p(a__if) = [1 0 0] [2 0 0] [2 0 0] [0]
[0 0 2] x_1 + [0 1 0] x_2 + [0 1 0] x_3 + [2]
[0 0 0] [0 0 0] [0 0 0] [0]
p(a__minus) = [0]
[0]
[0]
p(div) = [1 1 0] [0 1 0] [1]
[0 0 0] x_1 + [0 1 0] x_2 + [2]
[0 0 0] [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] [0]
[0 0 2] x_1 + [0 1 0] x_2 + [0 1 0] x_3 + [2]
[0 0 0] [0 0 0] [0 0 0] [0]
p(mark) = [2 0 0] [0]
[0 1 0] x_1 + [0]
[0 0 0] [0]
p(minus) = [0]
[0]
[0]
p(s) = [1 0 0] [0]
[0 0 1] x_1 + [2]
[0 0 0] [0]
p(true) = [0]
[0]
[0]
Following rules are strictly oriented:
mark(div(X1,X2)) = [2 2 0] [0 2 0] [2]
[0 0 0] X1 + [0 1 0] X2 + [2]
[0 0 0] [0 0 0] [0]
> [2 1 0] [0 2 0] [1]
[0 0 0] X1 + [0 1 0] X2 + [2]
[0 0 0] [0 0 0] [0]
= a__div(mark(X1),X2)
Following rules are (at-least) weakly oriented:
a__div(X1,X2) = [1 1 0] [0 2 0] [1]
[0 0 0] X1 + [0 1 0] X2 + [2]
[0 0 0] [0 0 0] [0]
>= [1 1 0] [0 1 0] [1]
[0 0 0] X1 + [0 1 0] X2 + [2]
[0 0 0] [0 0 0] [0]
= div(X1,X2)
a__div(0(),s(Y)) = [0 0 2] [5]
[0 0 1] Y + [4]
[0 0 0] [0]
>= [0]
[0]
[0]
= 0()
a__div(s(X),s(Y)) = [1 0 1] [0 0 2] [7]
[0 0 0] X + [0 0 1] Y + [4]
[0 0 0] [0 0 0] [0]
>= [0 0 2] [6]
[0 0 0] Y + [4]
[0 0 0] [0]
= 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] [2 0 0] [2 0 0] [0]
[0 0 2] X1 + [0 1 0] X2 + [0 1 0] X3 + [2]
[0 0 0] [0 0 0] [0 0 0] [0]
>= [1 0 0] [1 0 0] [1 0 0] [0]
[0 0 2] X1 + [0 1 0] X2 + [0 1 0] X3 + [2]
[0 0 0] [0 0 0] [0 0 0] [0]
= if(X1,X2,X3)
a__if(false(),X,Y) = [2 0 0] [2 0 0] [0]
[0 1 0] X + [0 1 0] Y + [2]
[0 0 0] [0 0 0] [0]
>= [2 0 0] [0]
[0 1 0] Y + [0]
[0 0 0] [0]
= mark(Y)
a__if(true(),X,Y) = [2 0 0] [2 0 0] [0]
[0 1 0] X + [0 1 0] Y + [2]
[0 0 0] [0 0 0] [0]
>= [2 0 0] [0]
[0 1 0] X + [0]
[0 0 0] [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]
[0]
[0]
>= [0]
[0]
[0]
= a__geq(X1,X2)
mark(if(X1,X2,X3)) = [2 0 0] [2 0 0] [2 0 0] [0]
[0 0 2] X1 + [0 1 0] X2 + [0 1 0] X3 + [2]
[0 0 0] [0 0 0] [0 0 0] [0]
>= [2 0 0] [2 0 0] [2 0 0] [0]
[0 0 0] X1 + [0 1 0] X2 + [0 1 0] X3 + [2]
[0 0 0] [0 0 0] [0 0 0] [0]
= a__if(mark(X1),X2,X3)
mark(minus(X1,X2)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= a__minus(X1,X2)
mark(s(X)) = [2 0 0] [0]
[0 0 1] X + [2]
[0 0 0] [0]
>= [2 0 0] [0]
[0 0 0] X + [2]
[0 0 0] [0]
= s(mark(X))
mark(true()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= true()
** Step 1.b:10: 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(Omega(n^1),O(n^2))