```* Step 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:
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 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:
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 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:
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:
all
TcT has computed the following interpretation:
p(+) = x1 + 2*x2
p(0) = 0
p(double) = 1 + 2*x1
p(s) = 1 + x1
p(sqr) = 2*x1 + 2*x1^2

Following rules are strictly oriented:
+(x,s(y)) = 2 + x + 2*y
> 1 + x + 2*y
= s(+(x,y))

Following rules are (at-least) weakly oriented:
+(x,0()) =  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)) =  4 + 6*x + 2*x^2
>= 4 + 6*x + 2*x^2
=  +(sqr(x),s(double(x)))

sqr(s(x)) =  4 + 6*x + 2*x^2
>= 3 + 6*x + 2*x^2
=  s(+(sqr(x),double(x)))

* Step 4: NaturalPI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
double(s(x)) -> s(s(double(x)))
- Weak TRS:
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
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:
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:
all
TcT has computed the following interpretation:
p(+) = x1 + x2
p(0) = 0
p(double) = 3*x1
p(s) = 1 + x1
p(sqr) = 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()) =  0
>= 0
=  0()

sqr(s(x)) =  2 + 4*x + 2*x^2
>= 1 + 3*x + 2*x^2
=  +(sqr(x),s(double(x)))

sqr(s(x)) =  2 + 4*x + 2*x^2
>= 1 + 3*x + 2*x^2
=  s(+(sqr(x),double(x)))

* Step 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:
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(?,O(n^2))
```