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

Strict DPs
f#(X,X) -> c_1(X)
f#(X,c(X)) -> c_2(f#(s(X),X))
f#(s(X),X) -> c_3(f#(X,a(X)))
Weak DPs

and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
f#(X,X) -> c_1(X)
f#(X,c(X)) -> c_2(f#(s(X),X))
f#(s(X),X) -> c_3(f#(X,a(X)))
- Strict TRS:
f(X,X) -> c(X)
f(X,c(X)) -> f(s(X),X)
f(s(X),X) -> f(X,a(X))
- Signature:
{f/2,f#/2} / {a/1,c/1,s/1,c_1/1,c_2/1,c_3/1}
- Obligation:
runtime complexity wrt. defined symbols {f#} and constructors {a,c,s}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
f#(X,X) -> c_1(X)
f#(X,c(X)) -> c_2(f#(s(X),X))
f#(s(X),X) -> c_3(f#(X,a(X)))
* Step 3: WeightGap WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
f#(X,X) -> c_1(X)
f#(X,c(X)) -> c_2(f#(s(X),X))
f#(s(X),X) -> c_3(f#(X,a(X)))
- Signature:
{f/2,f#/2} / {a/1,c/1,s/1,c_1/1,c_2/1,c_3/1}
- Obligation:
runtime complexity wrt. defined symbols {f#} and constructors {a,c,s}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 0, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
The weightgap principle applies using the following constant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 0 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:
all
TcT has computed the following interpretation:
p(a) = 
p(c) = 
p(f) =  x2 + 
p(s) = 
p(f#) = 
p(c_1) = 
p(c_2) =  x1 + 
p(c_3) = 

Following rules are strictly oriented:
f#(X,X) = 
> 
= c_1(X)

f#(s(X),X) = 
> 
= c_3(f#(X,a(X)))

Following rules are (at-least) weakly oriented:
f#(X,c(X)) =  
>= 
=  c_2(f#(s(X),X))

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 4: WeightGap WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
f#(X,c(X)) -> c_2(f#(s(X),X))
- Weak DPs:
f#(X,X) -> c_1(X)
f#(s(X),X) -> c_3(f#(X,a(X)))
- Signature:
{f/2,f#/2} / {a/1,c/1,s/1,c_1/1,c_2/1,c_3/1}
- Obligation:
runtime complexity wrt. defined symbols {f#} and constructors {a,c,s}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 0, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
The weightgap principle applies using the following constant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 0 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:
all
TcT has computed the following interpretation:
p(a) = 
p(c) = 
p(f) = 
p(s) = 
p(f#) =  x1 +  x2 + 
p(c_1) = 
p(c_2) =  x1 + 
p(c_3) =  x1 + 

Following rules are strictly oriented:
f#(X,c(X)) =  X + 
>  X + 
= c_2(f#(s(X),X))

Following rules are (at-least) weakly oriented:
f#(X,X) =   X + 
>= 
=  c_1(X)

f#(s(X),X) =   X + 
>=  X + 
=  c_3(f#(X,a(X)))

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 5: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
f#(X,X) -> c_1(X)
f#(X,c(X)) -> c_2(f#(s(X),X))
f#(s(X),X) -> c_3(f#(X,a(X)))
- Signature:
{f/2,f#/2} / {a/1,c/1,s/1,c_1/1,c_2/1,c_3/1}
- Obligation:
runtime complexity wrt. defined symbols {f#} and constructors {a,c,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

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