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

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(minus) = [1] x1 + [0]
p(quot) = [1] x1 + [4] x2 + [1]
p(s) = [1] x1 + [2]

Following rules are strictly oriented:
minus(s(x),s(y)) = [1] x + [2]
> [1] x + [0]
= minus(x,y)

quot(0(),s(y)) = [4] y + [9]
> [0]
= 0()

Following rules are (at-least) weakly oriented:
minus(x,0()) =  [1] x + [0]
>= [1] x + [0]
=  x

quot(s(x),s(y)) =  [1] x + [4] y + [11]
>= [1] x + [4] y + [11]
=  s(quot(minus(x,y),s(y)))

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

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [5]
p(minus) = [1] x_1 + [1]
p(quot) = [4] x_1 + [2] x_2 + [0]
p(s) = [1] x_1 + [4]

Following rules are strictly oriented:
minus(x,0()) = [1] x + [1]
> [1] x + [0]
= x

quot(s(x),s(y)) = [4] x + [2] y + [24]
> [4] x + [2] y + [16]
= s(quot(minus(x,y),s(y)))

Following rules are (at-least) weakly oriented:
minus(s(x),s(y)) =  [1] x + [5]
>= [1] x + [1]
=  minus(x,y)

quot(0(),s(y)) =  [2] y + [28]
>= [5]
=  0()

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

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