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

Strict DPs
g#(X,s(Y)) -> c_1(g#(X,Y))
g#(0(),Y) -> c_2()
h#(X,Z) -> c_3()
Weak DPs

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

2:W:g#(0(),Y) -> c_2()

3:W:h#(X,Z) -> c_3()

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

The strictly oriented rules are moved into the weak component.
** Step 5.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
g#(X,s(Y)) -> c_1(g#(X,Y))
- Signature:
{g/2,h/2,g#/2,h#/2} / {0/0,f/3,s/1,c_1/1,c_2/0,c_3/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {g#,h#} and constructors {0,f,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:
{g#,h#}
TcT has computed the following interpretation:
p(0) = [0]
p(f) = [1] x3 + [0]
p(g) = [2] x2 + [1]
p(h) = [2] x1 + [4] x2 + [1]
p(s) = [1] x1 + [6]
p(g#) = [2] x2 + [12]
p(h#) = [4] x2 + [0]
p(c_1) = [1] x1 + [6]
p(c_2) = [0]
p(c_3) = [1]

Following rules are strictly oriented:
g#(X,s(Y)) = [2] Y + [24]
> [2] Y + [18]
= c_1(g#(X,Y))

Following rules are (at-least) weakly oriented:

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

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

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

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