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

Strict DPs
g#(e(x),e(y)) -> c_1(g#(x,y))
h#(z,e(x)) -> c_2(h#(c(z),d(z,x)))
Weak DPs

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

2:W:h#(z,e(x)) -> c_2(h#(c(z),d(z,x)))

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
2: h#(z,e(x)) -> c_2(h#(c(z),d(z,x)))
* Step 5: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
g#(e(x),e(y)) -> c_1(g#(x,y))
- Signature:
{g/2,h/2,g#/2,h#/2} / {c/1,d/2,e/1,c_1/1,c_2/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {g#,h#} and constructors {c,d,e}
+ 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#(e(x),e(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#(e(x),e(y)) -> c_1(g#(x,y))
- Signature:
{g/2,h/2,g#/2,h#/2} / {c/1,d/2,e/1,c_1/1,c_2/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {g#,h#} and constructors {c,d,e}
+ 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(c) = [0]
p(d) = [1] x2 + [0]
p(e) = [1] x1 + [1]
p(g) = [1] x1 + [1] x2 + [8]
p(h) = [1] x1 + [1] x2 + [2]
p(g#) = [1] x1 + [0]
p(h#) = [2] x1 + [1] x2 + [1]
p(c_1) = [1] x1 + [0]
p(c_2) = [1]

Following rules are strictly oriented:
g#(e(x),e(y)) = [1] x + [1]
> [1] x + [0]
= 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#(e(x),e(y)) -> c_1(g#(x,y))
- Signature:
{g/2,h/2,g#/2,h#/2} / {c/1,d/2,e/1,c_1/1,c_2/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {g#,h#} and constructors {c,d,e}
+ 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#(e(x),e(y)) -> c_1(g#(x,y))
- Signature:
{g/2,h/2,g#/2,h#/2} / {c/1,d/2,e/1,c_1/1,c_2/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {g#,h#} and constructors {c,d,e}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:g#(e(x),e(y)) -> c_1(g#(x,y))
-->_1 g#(e(x),e(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#(e(x),e(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} / {c/1,d/2,e/1,c_1/1,c_2/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {g#,h#} and constructors {c,d,e}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

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