```* Step 1: DependencyPairs 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:
innermost runtime complexity wrt. defined symbols {fac,p} and constructors {0,s,times}
+ Applied Processor:
DependencyPairs {dpKind_ = WIDP}
+ Details:
We add the following weak innermost dependency pairs:

Strict DPs
fac#(0()) -> c_1()
fac#(s(x)) -> c_2(fac#(p(s(x))))
p#(s(x)) -> c_3()
Weak DPs

and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
fac#(0()) -> c_1()
fac#(s(x)) -> c_2(fac#(p(s(x))))
p#(s(x)) -> c_3()
- 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,fac#/1,p#/1} / {0/0,s/1,times/2,c_1/0,c_2/1,c_3/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {fac#,p#} and constructors {0,s,times}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
p(s(x)) -> x
fac#(0()) -> c_1()
fac#(s(x)) -> c_2(fac#(p(s(x))))
p#(s(x)) -> c_3()
* Step 3: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
fac#(0()) -> c_1()
fac#(s(x)) -> c_2(fac#(p(s(x))))
p#(s(x)) -> c_3()
- Strict TRS:
p(s(x)) -> x
- Signature:
{fac/1,p/1,fac#/1,p#/1} / {0/0,s/1,times/2,c_1/0,c_2/1,c_3/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {fac#,p#} and constructors {0,s,times}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnTrs}
+ Details:
The weightgap principle applies using the following constant 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(c_2) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = 
p(fac) = 
p(p) =  x1 + 
p(s) =  x1 + 
p(times) =  x1 +  x2 + 
p(fac#) =  x1 + 
p(p#) = 
p(c_1) = 
p(c_2) =  x1 + 
p(c_3) = 

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

Following rules are (at-least) weakly oriented:
fac#(0()) =  
>= 
=  c_1()

fac#(s(x)) =   x + 
>=  x + 
=  c_2(fac#(p(s(x))))

p#(s(x)) =  
>= 
=  c_3()

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 4: PredecessorEstimation WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
fac#(0()) -> c_1()
fac#(s(x)) -> c_2(fac#(p(s(x))))
p#(s(x)) -> c_3()
- Weak TRS:
p(s(x)) -> x
- Signature:
{fac/1,p/1,fac#/1,p#/1} / {0/0,s/1,times/2,c_1/0,c_2/1,c_3/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {fac#,p#} and constructors {0,s,times}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{1,3}
by application of
Pre({1,3}) = {2}.
Here rules are labelled as follows:
1: fac#(0()) -> c_1()
2: fac#(s(x)) -> c_2(fac#(p(s(x))))
3: p#(s(x)) -> c_3()
* Step 5: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
fac#(s(x)) -> c_2(fac#(p(s(x))))
- Weak DPs:
fac#(0()) -> c_1()
p#(s(x)) -> c_3()
- Weak TRS:
p(s(x)) -> x
- Signature:
{fac/1,p/1,fac#/1,p#/1} / {0/0,s/1,times/2,c_1/0,c_2/1,c_3/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {fac#,p#} and constructors {0,s,times}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:fac#(s(x)) -> c_2(fac#(p(s(x))))
-->_1 fac#(0()) -> c_1():2
-->_1 fac#(s(x)) -> c_2(fac#(p(s(x)))):1

2:W:fac#(0()) -> c_1()

3:W:p#(s(x)) -> c_3()

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
3: p#(s(x)) -> c_3()
2: fac#(0()) -> c_1()
* Step 6: PredecessorEstimationCP WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
fac#(s(x)) -> c_2(fac#(p(s(x))))
- Weak TRS:
p(s(x)) -> x
- Signature:
{fac/1,p/1,fac#/1,p#/1} / {0/0,s/1,times/2,c_1/0,c_2/1,c_3/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {fac#,p#} and constructors {0,s,times}
+ Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
+ Details:
We first use the processor NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
1: fac#(s(x)) -> c_2(fac#(p(s(x))))

The strictly oriented rules are moved into the weak component.
** Step 6.a:1: NaturalMI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
fac#(s(x)) -> c_2(fac#(p(s(x))))
- Weak TRS:
p(s(x)) -> x
- Signature:
{fac/1,p/1,fac#/1,p#/1} / {0/0,s/1,times/2,c_1/0,c_2/1,c_3/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {fac#,p#} and constructors {0,s,times}
+ Applied Processor:
NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 2 non-zero interpretation-entries in the diagonal of the component-wise maxima):
The following argument positions are considered usable:
uargs(c_2) = {1}

Following symbols are considered usable:
{p,fac#,p#}
TcT has computed the following interpretation:
p(0) = 


p(fac) = [4 1 4]      
[0 0 2] x1 + 
[1 0 0]      
p(p) = [4 4 3]      
[1 0 0] x1 + 
[0 1 0]      
p(s) = [1 1 0]      
[0 0 1] x1 + 
[0 0 1]      
p(times) = [1 0 1]      [0 0 0]      
[0 0 0] x1 + [0 0 0] x2 + 
[0 0 1]      [0 0 1]      
p(fac#) = [0 0 2]      
[0 0 0] x1 + 
[0 0 0]      
p(p#) = [0 0 0]      
[1 2 1] x1 + 
[1 1 1]      
p(c_1) = 


p(c_2) = [1 1 1]      
[0 0 1] x1 + 
[0 0 0]      
p(c_3) = 



Following rules are strictly oriented:
fac#(s(x)) = [0 0 2]     
[0 0 0] x + 
[0 0 0]     
> [0 0 2]     
[0 0 0] x + 
[0 0 0]     
= c_2(fac#(p(s(x))))

Following rules are (at-least) weakly oriented:
p(s(x)) =  [4 4 7]     
[1 1 0] x + 
[0 0 1]     
>= [1 0 0]     
[0 1 0] x + 
[0 0 1]     
=  x

** Step 6.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
fac#(s(x)) -> c_2(fac#(p(s(x))))
- Weak TRS:
p(s(x)) -> x
- Signature:
{fac/1,p/1,fac#/1,p#/1} / {0/0,s/1,times/2,c_1/0,c_2/1,c_3/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {fac#,p#} and constructors {0,s,times}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()

** Step 6.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
fac#(s(x)) -> c_2(fac#(p(s(x))))
- Weak TRS:
p(s(x)) -> x
- Signature:
{fac/1,p/1,fac#/1,p#/1} / {0/0,s/1,times/2,c_1/0,c_2/1,c_3/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {fac#,p#} and constructors {0,s,times}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:fac#(s(x)) -> c_2(fac#(p(s(x))))
-->_1 fac#(s(x)) -> c_2(fac#(p(s(x)))):1

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: fac#(s(x)) -> c_2(fac#(p(s(x))))
** Step 6.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
p(s(x)) -> x
- Signature:
{fac/1,p/1,fac#/1,p#/1} / {0/0,s/1,times/2,c_1/0,c_2/1,c_3/0}
- Obligation:
innermost 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))
```