```* 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) = 
p(fac) =  x1 + 
p(p) =  x1 + 
p(s) =  x1 + 
p(times) =  x2 + 

Following rules are strictly oriented:
p(s(x)) =  x + 
>  x + 
= x

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

fac(s(x)) =   x + 
>=  x + 
=  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) = 
p(fac) =  x1 + 
p(p) =  x1 + 
p(s) =  x1 + 
p(times) =  x2 + 

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

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

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^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) = 


p(fac) = [1 0 1]       
[2 0 0] x_1 + 
[0 0 0]       
p(p) = [1 0 0]       
[2 0 4] x_1 + 
[0 1 0]       
p(s) = [1 1 4]       
[0 0 1] x_1 + 
[0 0 1]       
p(times) = [1 0 0]       
[0 0 0] x_2 + 
[0 0 0]       

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

Following rules are (at-least) weakly oriented:
fac(0()) =  


>= 


=  s(0())

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