```* Step 1: DependencyPairs WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
f(x,y,s(z)) -> s(f(0(),1(),z))
f(0(),1(),x) -> f(s(x),x,x)
g(x,y) -> x
g(x,y) -> y
- Signature:
{f/3,g/2} / {0/0,1/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f,g} and constructors {0,1,s}
+ Applied Processor:
DependencyPairs {dpKind_ = WIDP}
+ Details:
We add the following weak innermost dependency pairs:

Strict DPs
f#(x,y,s(z)) -> c_1(f#(0(),1(),z))
f#(0(),1(),x) -> c_2(f#(s(x),x,x))
g#(x,y) -> c_3()
g#(x,y) -> c_4()
Weak DPs

and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f#(x,y,s(z)) -> c_1(f#(0(),1(),z))
f#(0(),1(),x) -> c_2(f#(s(x),x,x))
g#(x,y) -> c_3()
g#(x,y) -> c_4()
- Strict TRS:
f(x,y,s(z)) -> s(f(0(),1(),z))
f(0(),1(),x) -> f(s(x),x,x)
g(x,y) -> x
g(x,y) -> y
- Signature:
{f/3,g/2,f#/3,g#/2} / {0/0,1/0,s/1,c_1/1,c_2/1,c_3/0,c_4/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,1,s}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
f#(x,y,s(z)) -> c_1(f#(0(),1(),z))
f#(0(),1(),x) -> c_2(f#(s(x),x,x))
g#(x,y) -> c_3()
g#(x,y) -> c_4()
* Step 3: PredecessorEstimation WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f#(x,y,s(z)) -> c_1(f#(0(),1(),z))
f#(0(),1(),x) -> c_2(f#(s(x),x,x))
g#(x,y) -> c_3()
g#(x,y) -> c_4()
- Signature:
{f/3,g/2,f#/3,g#/2} / {0/0,1/0,s/1,c_1/1,c_2/1,c_3/0,c_4/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,1,s}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{3,4}
by application of
Pre({3,4}) = {}.
Here rules are labelled as follows:
1: f#(x,y,s(z)) -> c_1(f#(0(),1(),z))
2: f#(0(),1(),x) -> c_2(f#(s(x),x,x))
3: g#(x,y) -> c_3()
4: g#(x,y) -> c_4()
* Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f#(x,y,s(z)) -> c_1(f#(0(),1(),z))
f#(0(),1(),x) -> c_2(f#(s(x),x,x))
- Weak DPs:
g#(x,y) -> c_3()
g#(x,y) -> c_4()
- Signature:
{f/3,g/2,f#/3,g#/2} / {0/0,1/0,s/1,c_1/1,c_2/1,c_3/0,c_4/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,1,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:f#(x,y,s(z)) -> c_1(f#(0(),1(),z))
-->_1 f#(0(),1(),x) -> c_2(f#(s(x),x,x)):2
-->_1 f#(x,y,s(z)) -> c_1(f#(0(),1(),z)):1

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

3:W:g#(x,y) -> c_3()

4:W:g#(x,y) -> c_4()

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
4: g#(x,y) -> c_4()
3: g#(x,y) -> c_3()
* Step 5: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f#(x,y,s(z)) -> c_1(f#(0(),1(),z))
f#(0(),1(),x) -> c_2(f#(s(x),x,x))
- Signature:
{f/3,g/2,f#/3,g#/2} / {0/0,1/0,s/1,c_1/1,c_2/1,c_3/0,c_4/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,1,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,y,s(z)) -> c_1(f#(0(),1(),z))

Consider the set of all dependency pairs
1: f#(x,y,s(z)) -> c_1(f#(0(),1(),z))
2: f#(0(),1(),x) -> c_2(f#(s(x),x,x))
Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1))
SPACE(?,?)on application of the dependency pairs
{1}
These cover all (indirect) predecessors of dependency pairs
{1,2}
their number of applications is equally bounded.
The dependency pairs are shifted into the weak component.
** Step 5.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f#(x,y,s(z)) -> c_1(f#(0(),1(),z))
f#(0(),1(),x) -> c_2(f#(s(x),x,x))
- Signature:
{f/3,g/2,f#/3,g#/2} / {0/0,1/0,s/1,c_1/1,c_2/1,c_3/0,c_4/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,1,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},
uargs(c_2) = {1}

Following symbols are considered usable:
{f#,g#}
TcT has computed the following interpretation:
p(0) = [1]
p(1) = [1]
p(f) = [1] x1 + [0]
p(g) = [1] x1 + [0]
p(s) = [1] x1 + [6]
p(f#) = [4] x3 + [1]
p(g#) = [2] x2 + [8]
p(c_1) = [1] x1 + [7]
p(c_2) = [1] x1 + [0]
p(c_3) = [1]
p(c_4) = [0]

Following rules are strictly oriented:
f#(x,y,s(z)) = [4] z + [25]
> [4] z + [8]
= c_1(f#(0(),1(),z))

Following rules are (at-least) weakly oriented:
f#(0(),1(),x) =  [4] x + [1]
>= [4] x + [1]
=  c_2(f#(s(x),x,x))

** Step 5.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
f#(0(),1(),x) -> c_2(f#(s(x),x,x))
- Weak DPs:
f#(x,y,s(z)) -> c_1(f#(0(),1(),z))
- Signature:
{f/3,g/2,f#/3,g#/2} / {0/0,1/0,s/1,c_1/1,c_2/1,c_3/0,c_4/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,1,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()

** Step 5.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
f#(x,y,s(z)) -> c_1(f#(0(),1(),z))
f#(0(),1(),x) -> c_2(f#(s(x),x,x))
- Signature:
{f/3,g/2,f#/3,g#/2} / {0/0,1/0,s/1,c_1/1,c_2/1,c_3/0,c_4/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,1,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:f#(x,y,s(z)) -> c_1(f#(0(),1(),z))
-->_1 f#(0(),1(),x) -> c_2(f#(s(x),x,x)):2
-->_1 f#(x,y,s(z)) -> c_1(f#(0(),1(),z)):1

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

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: f#(x,y,s(z)) -> c_1(f#(0(),1(),z))
2: f#(0(),1(),x) -> c_2(f#(s(x),x,x))
** Step 5.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:

- Signature:
{f/3,g/2,f#/3,g#/2} / {0/0,1/0,s/1,c_1/1,c_2/1,c_3/0,c_4/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,1,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

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