```* Step 1: DependencyPairs WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
f(x,c(y)) -> f(x,s(f(y,y)))
f(s(x),y) -> f(x,s(c(y)))
- Signature:
{f/2} / {c/1,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f} and constructors {c,s}
+ Applied Processor:
DependencyPairs {dpKind_ = DT}
+ Details:
We add the following dependency tuples:

Strict DPs
f#(x,c(y)) -> c_1(f#(x,s(f(y,y))),f#(y,y))
f#(s(x),y) -> c_2(f#(x,s(c(y))))
Weak DPs

and mark the set of starting terms.
* Step 2: PredecessorEstimationCP WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
f#(x,c(y)) -> c_1(f#(x,s(f(y,y))),f#(y,y))
f#(s(x),y) -> c_2(f#(x,s(c(y))))
- Weak TRS:
f(x,c(y)) -> f(x,s(f(y,y)))
f(s(x),y) -> f(x,s(c(y)))
- Signature:
{f/2,f#/2} / {c/1,s/1,c_1/2,c_2/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#} and constructors {c,s}
+ 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:
2: f#(s(x),y) -> c_2(f#(x,s(c(y))))

The strictly oriented rules are moved into the weak component.
** Step 2.a:1: NaturalMI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
f#(x,c(y)) -> c_1(f#(x,s(f(y,y))),f#(y,y))
f#(s(x),y) -> c_2(f#(x,s(c(y))))
- Weak TRS:
f(x,c(y)) -> f(x,s(f(y,y)))
f(s(x),y) -> f(x,s(c(y)))
- Signature:
{f/2,f#/2} / {c/1,s/1,c_1/2,c_2/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#} and constructors {c,s}
+ 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_1) = {1,2},
uargs(c_2) = {1}

Following symbols are considered usable:
{f,f#}
TcT has computed the following interpretation:
p(c) = [1 0 1]      
[0 0 0] x1 + 
[0 0 1]      
p(f) = [0 0 0]      [1 0 0]      
[0 0 0] x1 + [0 0 0] x2 + 
[0 0 1]      [0 0 0]      
p(s) = [0 0 0]      
[0 0 1] x1 + 
[0 0 1]      
p(f#) = [0 0 1]      [1 0 0]      
[1 0 0] x1 + [0 0 1] x2 + 
[0 1 0]      [1 0 1]      
p(c_1) = [1 0 0]      [1 0 0]      
[0 0 0] x1 + [0 0 0] x2 + 
[0 0 1]      [0 1 0]      
p(c_2) = [1 0 0]      
[0 0 0] x1 + 
[0 0 0]      

Following rules are strictly oriented:
f#(s(x),y) = [0 0 1]     [1 0 0]     
[0 0 0] x + [0 0 1] y + 
[0 0 1]     [1 0 1]     
> [0 0 1]     
[0 0 0] x + 
[0 0 0]     
= c_2(f#(x,s(c(y))))

Following rules are (at-least) weakly oriented:
f#(x,c(y)) =  [0 0 1]     [1 0 1]     
[1 0 0] x + [0 0 1] y + 
[0 1 0]     [1 0 2]     
>= [0 0 1]     [1 0 1]     
[0 0 0] x + [0 0 0] y + 
[0 1 0]     [1 0 2]     
=  c_1(f#(x,s(f(y,y))),f#(y,y))

f(x,c(y)) =  [0 0 0]     [1 0 1]     
[0 0 0] x + [0 0 0] y + 
[0 0 1]     [0 0 0]     
>= [0 0 0]     
[0 0 0] x + 
[0 0 1]     
=  f(x,s(f(y,y)))

f(s(x),y) =  [0 0 0]     [1 0 0]     
[0 0 0] x + [0 0 0] y + 
[0 0 1]     [0 0 0]     
>= [0 0 0]     
[0 0 0] x + 
[0 0 1]     
=  f(x,s(c(y)))

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

** Step 2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f#(x,c(y)) -> c_1(f#(x,s(f(y,y))),f#(y,y))
- Weak DPs:
f#(s(x),y) -> c_2(f#(x,s(c(y))))
- Weak TRS:
f(x,c(y)) -> f(x,s(f(y,y)))
f(s(x),y) -> f(x,s(c(y)))
- Signature:
{f/2,f#/2} / {c/1,s/1,c_1/2,c_2/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#} and constructors {c,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:f#(x,c(y)) -> c_1(f#(x,s(f(y,y))),f#(y,y))
-->_2 f#(s(x),y) -> c_2(f#(x,s(c(y)))):2
-->_1 f#(s(x),y) -> c_2(f#(x,s(c(y)))):2
-->_2 f#(x,c(y)) -> c_1(f#(x,s(f(y,y))),f#(y,y)):1

2:W:f#(s(x),y) -> c_2(f#(x,s(c(y))))
-->_1 f#(s(x),y) -> c_2(f#(x,s(c(y)))):2

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
2: f#(s(x),y) -> c_2(f#(x,s(c(y))))
** Step 2.b:2: SimplifyRHS WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f#(x,c(y)) -> c_1(f#(x,s(f(y,y))),f#(y,y))
- Weak TRS:
f(x,c(y)) -> f(x,s(f(y,y)))
f(s(x),y) -> f(x,s(c(y)))
- Signature:
{f/2,f#/2} / {c/1,s/1,c_1/2,c_2/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#} and constructors {c,s}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:f#(x,c(y)) -> c_1(f#(x,s(f(y,y))),f#(y,y))
-->_2 f#(x,c(y)) -> c_1(f#(x,s(f(y,y))),f#(y,y)):1

Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
f#(x,c(y)) -> c_1(f#(y,y))
** Step 2.b:3: UsableRules WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f#(x,c(y)) -> c_1(f#(y,y))
- Weak TRS:
f(x,c(y)) -> f(x,s(f(y,y)))
f(s(x),y) -> f(x,s(c(y)))
- Signature:
{f/2,f#/2} / {c/1,s/1,c_1/1,c_2/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#} and constructors {c,s}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
f#(x,c(y)) -> c_1(f#(y,y))
** Step 2.b:4: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f#(x,c(y)) -> c_1(f#(y,y))
- Signature:
{f/2,f#/2} / {c/1,s/1,c_1/1,c_2/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#} and constructors {c,s}
+ Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
+ Details:
We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
1: f#(x,c(y)) -> c_1(f#(y,y))

The strictly oriented rules are moved into the weak component.
*** Step 2.b:4.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f#(x,c(y)) -> c_1(f#(y,y))
- Signature:
{f/2,f#/2} / {c/1,s/1,c_1/1,c_2/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#} and constructors {c,s}
+ Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, 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:
The following argument positions are considered usable:
uargs(c_1) = {1}

Following symbols are considered usable:
{f#}
TcT has computed the following interpretation:
p(c) =  x1 + 
p(f) = 
p(s) = 
p(f#) =  x2 + 
p(c_1) =  x1 + 
p(c_2) = 

Following rules are strictly oriented:
f#(x,c(y)) =  y + 
>  y + 
= c_1(f#(y,y))

Following rules are (at-least) weakly oriented:

*** Step 2.b:4.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
f#(x,c(y)) -> c_1(f#(y,y))
- Signature:
{f/2,f#/2} / {c/1,s/1,c_1/1,c_2/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#} and constructors {c,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()

*** Step 2.b:4.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
f#(x,c(y)) -> c_1(f#(y,y))
- Signature:
{f/2,f#/2} / {c/1,s/1,c_1/1,c_2/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#} and constructors {c,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:f#(x,c(y)) -> c_1(f#(y,y))
-->_1 f#(x,c(y)) -> c_1(f#(y,y)):1

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: f#(x,c(y)) -> c_1(f#(y,y))
*** Step 2.b:4.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:

- Signature:
{f/2,f#/2} / {c/1,s/1,c_1/1,c_2/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#} and constructors {c,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

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