```* Step 1: DependencyPairs WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
f(0(),y) -> 0()
f(s(x),y) -> f(f(x,y),y)
- Signature:
{f/2} / {0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f} and constructors {0,s}
+ Applied Processor:
DependencyPairs {dpKind_ = WIDP}
+ Details:
We add the following weak innermost dependency pairs:

Strict DPs
f#(0(),y) -> c_1()
f#(s(x),y) -> c_2(f#(f(x,y),y))
Weak DPs

and mark the set of starting terms.
* Step 2: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f#(0(),y) -> c_1()
f#(s(x),y) -> c_2(f#(f(x,y),y))
- Strict TRS:
f(0(),y) -> 0()
f(s(x),y) -> f(f(x,y),y)
- Signature:
{f/2,f#/2} / {0/0,s/1,c_1/0,c_2/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#} and constructors {0,s}
+ 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(f) = {1},
uargs(f#) = {1},
uargs(c_2) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [2]
p(f) = [1] x1 + [2]
p(s) = [1] x1 + [3]
p(f#) = [1] x1 + [0]
p(c_1) = [0]
p(c_2) = [1] x1 + [0]

Following rules are strictly oriented:
f#(0(),y) = [2]
> [0]
= c_1()

f#(s(x),y) = [1] x + [3]
> [1] x + [2]
= c_2(f#(f(x,y),y))

f(0(),y) = [4]
> [2]
= 0()

f(s(x),y) = [1] x + [5]
> [1] x + [4]
= f(f(x,y),y)

Following rules are (at-least) weakly oriented:

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
f#(0(),y) -> c_1()
f#(s(x),y) -> c_2(f#(f(x,y),y))
- Weak TRS:
f(0(),y) -> 0()
f(s(x),y) -> f(f(x,y),y)
- Signature:
{f/2,f#/2} / {0/0,s/1,c_1/0,c_2/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#} and constructors {0,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:f#(0(),y) -> c_1()

2:W:f#(s(x),y) -> c_2(f#(f(x,y),y))
-->_1 f#(s(x),y) -> c_2(f#(f(x,y),y)):2
-->_1 f#(0(),y) -> c_1():1

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#(f(x,y),y))
1: f#(0(),y) -> c_1()
* Step 4: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
f(0(),y) -> 0()
f(s(x),y) -> f(f(x,y),y)
- Signature:
{f/2,f#/2} / {0/0,s/1,c_1/0,c_2/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#} and constructors {0,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

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