```* Step 1: MI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
-(x,0()) -> x
-(0(),s(y)) -> 0()
-(s(x),s(y)) -> -(x,y)
div(x,0()) -> 0()
div(0(),y) -> 0()
div(s(x),s(y)) -> if(lt(x,y),0(),s(div(-(x,y),s(y))))
if(false(),x,y) -> y
if(true(),x,y) -> x
lt(x,0()) -> false()
lt(0(),s(y)) -> true()
lt(s(x),s(y)) -> lt(x,y)
- Signature:
{-/2,div/2,if/3,lt/2} / {0/0,false/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {-,div,if,lt} and constructors {0,false,s,true}
+ Applied Processor:
MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity Nothing)), miDimension = 1, 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(if) = {1,3},
uargs(s) = {1}

Following symbols are considered usable:
{-,div,if,lt}
TcT has computed the following interpretation:
p(-) =  x_1 + 
p(0) = 
p(div) =  x_1 + 
p(false) = 
p(if) =  x_1 +  x_2 +  x_3 + 
p(lt) = 
p(s) =  x_1 + 
p(true) = 

Following rules are strictly oriented:
-(x,0()) =  x + 
>  x + 
= x

-(0(),s(y)) = 
> 
= 0()

-(s(x),s(y)) =  x + 
>  x + 
= -(x,y)

lt(x,0()) = 
> 
= false()

lt(0(),s(y)) = 
> 
= true()

Following rules are (at-least) weakly oriented:
div(x,0()) =   x + 
>= 
=  0()

div(0(),y) =  
>= 
=  0()

div(s(x),s(y)) =   x + 
>=  x + 
=  if(lt(x,y),0(),s(div(-(x,y),s(y))))

if(false(),x,y) =   x +  y + 
>=  y + 
=  y

if(true(),x,y) =   x +  y + 
>=  x + 
=  x

lt(s(x),s(y)) =  
>= 
=  lt(x,y)

* Step 2: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
div(x,0()) -> 0()
div(0(),y) -> 0()
div(s(x),s(y)) -> if(lt(x,y),0(),s(div(-(x,y),s(y))))
if(false(),x,y) -> y
if(true(),x,y) -> x
lt(s(x),s(y)) -> lt(x,y)
- Weak TRS:
-(x,0()) -> x
-(0(),s(y)) -> 0()
-(s(x),s(y)) -> -(x,y)
lt(x,0()) -> false()
lt(0(),s(y)) -> true()
- Signature:
{-/2,div/2,if/3,lt/2} / {0/0,false/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {-,div,if,lt} and constructors {0,false,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(if) = {1,3},
uargs(s) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(-) =  x1 + 
p(0) = 
p(div) =  x1 +  x2 + 
p(false) = 
p(if) =  x1 +  x2 +  x3 + 
p(lt) = 
p(s) =  x1 + 
p(true) = 

Following rules are strictly oriented:
div(x,0()) =  x + 
> 
= 0()

div(0(),y) =  y + 
> 
= 0()

if(false(),x,y) =  x +  y + 
>  y + 
= y

if(true(),x,y) =  x +  y + 
>  x + 
= x

Following rules are (at-least) weakly oriented:
-(x,0()) =   x + 
>=  x + 
=  x

-(0(),s(y)) =  
>= 
=  0()

-(s(x),s(y)) =   x + 
>=  x + 
=  -(x,y)

div(s(x),s(y)) =   x +  y + 
>=  x +  y + 
=  if(lt(x,y),0(),s(div(-(x,y),s(y))))

lt(x,0()) =  
>= 
=  false()

lt(0(),s(y)) =  
>= 
=  true()

lt(s(x),s(y)) =  
>= 
=  lt(x,y)

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: MI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
div(s(x),s(y)) -> if(lt(x,y),0(),s(div(-(x,y),s(y))))
lt(s(x),s(y)) -> lt(x,y)
- Weak TRS:
-(x,0()) -> x
-(0(),s(y)) -> 0()
-(s(x),s(y)) -> -(x,y)
div(x,0()) -> 0()
div(0(),y) -> 0()
if(false(),x,y) -> y
if(true(),x,y) -> x
lt(x,0()) -> false()
lt(0(),s(y)) -> true()
- Signature:
{-/2,div/2,if/3,lt/2} / {0/0,false/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {-,div,if,lt} and constructors {0,false,s,true}
+ Applied Processor:
MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity Nothing)), miDimension = 1, 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(if) = {1,3},
uargs(s) = {1}

Following symbols are considered usable:
{-,div,if,lt}
TcT has computed the following interpretation:
p(-) =  x_1 + 
p(0) = 
p(div) =  x_1 +  x_2 + 
p(false) = 
p(if) =  x_1 +  x_2 +  x_3 + 
p(lt) = 
p(s) =  x_1 + 
p(true) = 

Following rules are strictly oriented:
div(s(x),s(y)) =  x +  y + 
>  x +  y + 
= if(lt(x,y),0(),s(div(-(x,y),s(y))))

Following rules are (at-least) weakly oriented:
-(x,0()) =   x + 
>=  x + 
=  x

-(0(),s(y)) =  
>= 
=  0()

-(s(x),s(y)) =   x + 
>=  x + 
=  -(x,y)

div(x,0()) =   x + 
>= 
=  0()

div(0(),y) =   y + 
>= 
=  0()

if(false(),x,y) =   x +  y + 
>=  y + 
=  y

if(true(),x,y) =   x +  y + 
>=  x + 
=  x

lt(x,0()) =  
>= 
=  false()

lt(0(),s(y)) =  
>= 
=  true()

lt(s(x),s(y)) =  
>= 
=  lt(x,y)

* Step 4: NaturalPI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
lt(s(x),s(y)) -> lt(x,y)
- Weak TRS:
-(x,0()) -> x
-(0(),s(y)) -> 0()
-(s(x),s(y)) -> -(x,y)
div(x,0()) -> 0()
div(0(),y) -> 0()
div(s(x),s(y)) -> if(lt(x,y),0(),s(div(-(x,y),s(y))))
if(false(),x,y) -> y
if(true(),x,y) -> x
lt(x,0()) -> false()
lt(0(),s(y)) -> true()
- Signature:
{-/2,div/2,if/3,lt/2} / {0/0,false/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {-,div,if,lt} and constructors {0,false,s,true}
+ 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(div) = {1},
uargs(if) = {1,3},
uargs(s) = {1}

Following symbols are considered usable:
{-,div,if,lt}
TcT has computed the following interpretation:
p(-) = x1
p(0) = 0
p(div) = x1 + x1^2
p(false) = 1
p(if) = x1 + x1*x2 + x2 + x3
p(lt) = 1 + x1
p(s) = 1 + x1
p(true) = 0

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

Following rules are (at-least) weakly oriented:
-(x,0()) =  x
>= x
=  x

-(0(),s(y)) =  0
>= 0
=  0()

-(s(x),s(y)) =  1 + x
>= x
=  -(x,y)

div(x,0()) =  x + x^2
>= 0
=  0()

div(0(),y) =  0
>= 0
=  0()

div(s(x),s(y)) =  2 + 3*x + x^2
>= 2 + 2*x + x^2
=  if(lt(x,y),0(),s(div(-(x,y),s(y))))

if(false(),x,y) =  1 + 2*x + y
>= y
=  y

if(true(),x,y) =  x + y
>= x
=  x

lt(x,0()) =  1 + x
>= 1
=  false()

lt(0(),s(y)) =  1
>= 0
=  true()

* Step 5: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
-(x,0()) -> x
-(0(),s(y)) -> 0()
-(s(x),s(y)) -> -(x,y)
div(x,0()) -> 0()
div(0(),y) -> 0()
div(s(x),s(y)) -> if(lt(x,y),0(),s(div(-(x,y),s(y))))
if(false(),x,y) -> y
if(true(),x,y) -> x
lt(x,0()) -> false()
lt(0(),s(y)) -> true()
lt(s(x),s(y)) -> lt(x,y)
- Signature:
{-/2,div/2,if/3,lt/2} / {0/0,false/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {-,div,if,lt} and constructors {0,false,s,true}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^2))
```