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

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(fac) = [1] x1 + [0]
p(p) = [1] x1 + [3]
p(s) = [1] x1 + [0]
p(times) = [1] x2 + [0]

Following rules are strictly oriented:
p(s(x)) = [1] x + [3]
> [1] x + [0]
= x

Following rules are (at-least) weakly oriented:
fac(0()) =  [0]
>= [0]
=  s(0())

fac(s(x)) =  [1] x + [0]
>= [1] x + [3]
=  times(s(x),fac(p(s(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:
fac(0()) -> s(0())
fac(s(x)) -> times(s(x),fac(p(s(x))))
- Weak TRS:
p(s(x)) -> x
- Signature:
{fac/1,p/1} / {0/0,s/1,times/2}
- Obligation:
runtime complexity wrt. defined symbols {fac,p} and constructors {0,s,times}
+ 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(fac) = {1},
uargs(times) = {2}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(fac) = [1] x1 + [2]
p(p) = [1] x1 + [0]
p(s) = [1] x1 + [0]
p(times) = [1] x2 + [13]

Following rules are strictly oriented:
fac(0()) = [2]
> [0]
= s(0())

Following rules are (at-least) weakly oriented:
fac(s(x)) =  [1] x + [2]
>= [1] x + [15]
=  times(s(x),fac(p(s(x))))

p(s(x)) =  [1] x + [0]
>= [1] x + [0]
=  x

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:
fac(s(x)) -> times(s(x),fac(p(s(x))))
- Weak TRS:
fac(0()) -> s(0())
p(s(x)) -> x
- Signature:
{fac/1,p/1} / {0/0,s/1,times/2}
- Obligation:
runtime complexity wrt. defined symbols {fac,p} and constructors {0,s,times}
+ Applied Processor:
MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity (Just 2))), miDimension = 3, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity (Just 2))):

The following argument positions are considered usable:
uargs(fac) = {1},
uargs(times) = {2}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
[4]
[0]
p(fac) = [1 0 1]       [4]
[2 0 0] x_1 + [0]
[0 0 0]       [4]
p(p) = [1 0 0]       [2]
[2 0 4] x_1 + [3]
[0 1 0]       [0]
p(s) = [1 1 4]       [0]
[0 0 1] x_1 + [0]
[0 0 1]       [3]
p(times) = [1 0 0]       [0]
[0 0 0] x_2 + [0]
[0 0 0]       [4]

Following rules are strictly oriented:
fac(s(x)) = [1 1 5]     [7]
[2 2 8] x + [0]
[0 0 0]     [4]
> [1 1 5]     [6]
[0 0 0] x + [0]
[0 0 0]     [4]
= times(s(x),fac(p(s(x))))

Following rules are (at-least) weakly oriented:
fac(0()) =  [4]
[0]
[4]
>= [4]
[0]
[3]
=  s(0())

p(s(x)) =  [1 1  4]     [2]
[2 2 12] x + [15]
[0 0  1]     [0]
>= [1 0 0]     [0]
[0 1 0] x + [0]
[0 0 1]     [0]
=  x

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

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