* Step 1: Sum WORST_CASE(Omega(n^1),O(n^2))
+ Considered Problem:
- Strict TRS:
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
double(0()) -> 0()
double(s(x)) -> s(s(double(x)))
sqr(0()) -> 0()
sqr(s(x)) -> +(sqr(x),s(double(x)))
sqr(s(x)) -> s(+(sqr(x),double(x)))
- Signature:
{+/2,double/1,sqr/1} / {0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {+,double,sqr} and constructors {0,s}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
double(0()) -> 0()
double(s(x)) -> s(s(double(x)))
sqr(0()) -> 0()
sqr(s(x)) -> +(sqr(x),s(double(x)))
sqr(s(x)) -> s(+(sqr(x),double(x)))
- Signature:
{+/2,double/1,sqr/1} / {0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {+,double,sqr} and constructors {0,s}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
+(x,y){y -> s(y)} =
+(x,s(y)) ->^+ s(+(x,y))
= C[+(x,y) = +(x,y){}]
** Step 1.b:1: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
double(0()) -> 0()
double(s(x)) -> s(s(double(x)))
sqr(0()) -> 0()
sqr(s(x)) -> +(sqr(x),s(double(x)))
sqr(s(x)) -> s(+(sqr(x),double(x)))
- Signature:
{+/2,double/1,sqr/1} / {0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {+,double,sqr} and constructors {0,s}
+ 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(+) = {1,2},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(+) = [1] x1 + [1] x2 + [6]
p(0) = [4]
p(double) = [10]
p(s) = [1] x1 + [4]
p(sqr) = [5] x1 + [2]
Following rules are strictly oriented:
+(x,0()) = [1] x + [10]
> [1] x + [0]
= x
double(0()) = [10]
> [4]
= 0()
sqr(0()) = [22]
> [4]
= 0()
Following rules are (at-least) weakly oriented:
+(x,s(y)) = [1] x + [1] y + [10]
>= [1] x + [1] y + [10]
= s(+(x,y))
double(s(x)) = [10]
>= [18]
= s(s(double(x)))
sqr(s(x)) = [5] x + [22]
>= [5] x + [22]
= +(sqr(x),s(double(x)))
sqr(s(x)) = [5] x + [22]
>= [5] x + [22]
= s(+(sqr(x),double(x)))
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:
+(x,s(y)) -> s(+(x,y))
double(s(x)) -> s(s(double(x)))
sqr(s(x)) -> +(sqr(x),s(double(x)))
sqr(s(x)) -> s(+(sqr(x),double(x)))
- Weak TRS:
+(x,0()) -> x
double(0()) -> 0()
sqr(0()) -> 0()
- Signature:
{+/2,double/1,sqr/1} / {0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {+,double,sqr} and constructors {0,s}
+ 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(+) = {1,2},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(+) = [1] x1 + [1] x2 + [0]
p(0) = [0]
p(double) = [1]
p(s) = [1] x1 + [2]
p(sqr) = [6] x1 + [1]
Following rules are strictly oriented:
sqr(s(x)) = [6] x + [13]
> [6] x + [4]
= +(sqr(x),s(double(x)))
sqr(s(x)) = [6] x + [13]
> [6] x + [4]
= s(+(sqr(x),double(x)))
Following rules are (at-least) weakly oriented:
+(x,0()) = [1] x + [0]
>= [1] x + [0]
= x
+(x,s(y)) = [1] x + [1] y + [2]
>= [1] x + [1] y + [2]
= s(+(x,y))
double(0()) = [1]
>= [0]
= 0()
double(s(x)) = [1]
>= [5]
= s(s(double(x)))
sqr(0()) = [1]
>= [0]
= 0()
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:3: NaturalPI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
+(x,s(y)) -> s(+(x,y))
double(s(x)) -> s(s(double(x)))
- Weak TRS:
+(x,0()) -> x
double(0()) -> 0()
sqr(0()) -> 0()
sqr(s(x)) -> +(sqr(x),s(double(x)))
sqr(s(x)) -> s(+(sqr(x),double(x)))
- Signature:
{+/2,double/1,sqr/1} / {0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {+,double,sqr} and constructors {0,s}
+ Applied Processor:
NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a polynomial interpretation of kind constructor-based(mixed(2)):
The following argument positions are considered usable:
uargs(+) = {1,2},
uargs(s) = {1}
Following symbols are considered usable:
{+,double,sqr}
TcT has computed the following interpretation:
p(+) = x1 + x2
p(0) = 0
p(double) = 3*x1
p(s) = 1 + x1
p(sqr) = 2 + 2*x1^2
Following rules are strictly oriented:
double(s(x)) = 3 + 3*x
> 2 + 3*x
= s(s(double(x)))
Following rules are (at-least) weakly oriented:
+(x,0()) = x
>= x
= x
+(x,s(y)) = 1 + x + y
>= 1 + x + y
= s(+(x,y))
double(0()) = 0
>= 0
= 0()
sqr(0()) = 2
>= 0
= 0()
sqr(s(x)) = 4 + 4*x + 2*x^2
>= 3 + 3*x + 2*x^2
= +(sqr(x),s(double(x)))
sqr(s(x)) = 4 + 4*x + 2*x^2
>= 3 + 3*x + 2*x^2
= s(+(sqr(x),double(x)))
** Step 1.b:4: NaturalPI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
+(x,s(y)) -> s(+(x,y))
- Weak TRS:
+(x,0()) -> x
double(0()) -> 0()
double(s(x)) -> s(s(double(x)))
sqr(0()) -> 0()
sqr(s(x)) -> +(sqr(x),s(double(x)))
sqr(s(x)) -> s(+(sqr(x),double(x)))
- Signature:
{+/2,double/1,sqr/1} / {0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {+,double,sqr} and constructors {0,s}
+ Applied Processor:
NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a polynomial interpretation of kind constructor-based(mixed(2)):
The following argument positions are considered usable:
uargs(+) = {1,2},
uargs(s) = {1}
Following symbols are considered usable:
{+,double,sqr}
TcT has computed the following interpretation:
p(+) = 1 + x1 + 2*x2
p(0) = 0
p(double) = 1 + 2*x1
p(s) = 1 + x1
p(sqr) = 3*x1 + 2*x1^2
Following rules are strictly oriented:
+(x,s(y)) = 3 + x + 2*y
> 2 + x + 2*y
= s(+(x,y))
Following rules are (at-least) weakly oriented:
+(x,0()) = 1 + x
>= x
= x
double(0()) = 1
>= 0
= 0()
double(s(x)) = 3 + 2*x
>= 3 + 2*x
= s(s(double(x)))
sqr(0()) = 0
>= 0
= 0()
sqr(s(x)) = 5 + 7*x + 2*x^2
>= 5 + 7*x + 2*x^2
= +(sqr(x),s(double(x)))
sqr(s(x)) = 5 + 7*x + 2*x^2
>= 4 + 7*x + 2*x^2
= s(+(sqr(x),double(x)))
** Step 1.b:5: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
double(0()) -> 0()
double(s(x)) -> s(s(double(x)))
sqr(0()) -> 0()
sqr(s(x)) -> +(sqr(x),s(double(x)))
sqr(s(x)) -> s(+(sqr(x),double(x)))
- Signature:
{+/2,double/1,sqr/1} / {0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {+,double,sqr} and constructors {0,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(Omega(n^1),O(n^2))