```* Step 1: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
div(0(),s(Y)) -> 0()
div(s(X),s(Y)) -> s(div(minus(X,Y),s(Y)))
minus(X,0()) -> X
minus(s(X),s(Y)) -> p(minus(X,Y))
p(s(X)) -> X
- Signature:
{div/2,minus/2,p/1} / {0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {div,minus,p} 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(div) = {1},
uargs(p) = {1},
uargs(s) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = 
p(div) =  x1 +  x2 + 
p(minus) =  x1 + 
p(p) =  x1 + 
p(s) =  x1 + 

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

p(s(X)) =  X + 
>  X + 
= X

Following rules are (at-least) weakly oriented:
div(s(X),s(Y)) =   X +  Y + 
>=  X +  Y + 
=  s(div(minus(X,Y),s(Y)))

minus(X,0()) =   X + 
>=  X + 
=  X

minus(s(X),s(Y)) =   X + 
>=  X + 
=  p(minus(X,Y))

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
div(s(X),s(Y)) -> s(div(minus(X,Y),s(Y)))
minus(X,0()) -> X
minus(s(X),s(Y)) -> p(minus(X,Y))
- Weak TRS:
div(0(),s(Y)) -> 0()
p(s(X)) -> X
- Signature:
{div/2,minus/2,p/1} / {0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {div,minus,p} 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(div) = {1},
uargs(p) = {1},
uargs(s) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = 
p(div) =  x1 +  x2 + 
p(minus) =  x1 + 
p(p) =  x1 + 
p(s) =  x1 + 

Following rules are strictly oriented:
minus(X,0()) =  X + 
>  X + 
= X

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

div(s(X),s(Y)) =   X +  Y + 
>=  X +  Y + 
=  s(div(minus(X,Y),s(Y)))

minus(s(X),s(Y)) =   X + 
>=  X + 
=  p(minus(X,Y))

p(s(X)) =   X + 
>=  X + 
=  X

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: MI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
div(s(X),s(Y)) -> s(div(minus(X,Y),s(Y)))
minus(s(X),s(Y)) -> p(minus(X,Y))
- Weak TRS:
div(0(),s(Y)) -> 0()
minus(X,0()) -> X
p(s(X)) -> X
- Signature:
{div/2,minus/2,p/1} / {0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {div,minus,p} and constructors {0,s}
+ 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(p) = {1},
uargs(s) = {1}

Following symbols are considered usable:
{div,minus,p}
TcT has computed the following interpretation:
p(0) = 
p(div) =  x_1 +  x_2 + 
p(minus) =  x_1 + 
p(p) =  x_1 + 
p(s) =  x_1 + 

Following rules are strictly oriented:
div(s(X),s(Y)) =  X +  Y + 
>  X +  Y + 
= s(div(minus(X,Y),s(Y)))

minus(s(X),s(Y)) =  X + 
>  X + 
= p(minus(X,Y))

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

minus(X,0()) =   X + 
>=  X + 
=  X

p(s(X)) =   X + 
>=  X + 
=  X

* Step 4: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
div(0(),s(Y)) -> 0()
div(s(X),s(Y)) -> s(div(minus(X,Y),s(Y)))
minus(X,0()) -> X
minus(s(X),s(Y)) -> p(minus(X,Y))
p(s(X)) -> X
- Signature:
{div/2,minus/2,p/1} / {0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {div,minus,p} and constructors {0,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))
```