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

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

and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f#(s(0()),g(x)) -> c_1(f#(x,g(x)))
g#(s(x)) -> c_2(g#(x))
- Strict TRS:
f(s(0()),g(x)) -> f(x,g(x))
g(s(x)) -> g(x)
- Signature:
{f/2,g/1,f#/2,g#/1} / {0/0,s/1,c_1/1,c_2/1}
- Obligation:
runtime complexity wrt. defined symbols {f#,g#} and constructors {0,s}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
g#(s(x)) -> c_2(g#(x))
* Step 3: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
g#(s(x)) -> c_2(g#(x))
- Signature:
{f/2,g/1,f#/2,g#/1} / {0/0,s/1,c_1/1,c_2/1}
- Obligation:
runtime complexity wrt. defined symbols {f#,g#} and constructors {0,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: g#(s(x)) -> c_2(g#(x))

The strictly oriented rules are moved into the weak component.
** Step 3.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
g#(s(x)) -> c_2(g#(x))
- Signature:
{f/2,g/1,f#/2,g#/1} / {0/0,s/1,c_1/1,c_2/1}
- Obligation:
runtime complexity wrt. defined symbols {f#,g#} and constructors {0,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_2) = {1}

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

Following rules are strictly oriented:
g#(s(x)) =  x + 
>  x + 
= c_2(g#(x))

Following rules are (at-least) weakly oriented:

** Step 3.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
g#(s(x)) -> c_2(g#(x))
- Signature:
{f/2,g/1,f#/2,g#/1} / {0/0,s/1,c_1/1,c_2/1}
- Obligation:
runtime complexity wrt. defined symbols {f#,g#} and constructors {0,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()

** Step 3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
g#(s(x)) -> c_2(g#(x))
- Signature:
{f/2,g/1,f#/2,g#/1} / {0/0,s/1,c_1/1,c_2/1}
- Obligation:
runtime complexity wrt. defined symbols {f#,g#} and constructors {0,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:g#(s(x)) -> c_2(g#(x))
-->_1 g#(s(x)) -> c_2(g#(x)):1

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: g#(s(x)) -> c_2(g#(x))
** Step 3.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:

- Signature:
{f/2,g/1,f#/2,g#/1} / {0/0,s/1,c_1/1,c_2/1}
- Obligation:
runtime complexity wrt. defined symbols {f#,g#} and constructors {0,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

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