```* Step 1: DependencyPairs WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
f(a()) -> b()
f(c()) -> d()
f(h(x,y)) -> g(h(y,f(x)),h(x,f(y)))
g(x,x) -> h(e(),x)
- Signature:
{f/1,g/2} / {a/0,b/0,c/0,d/0,e/0,h/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {f,g} and constructors {a,b,c,d,e,h}
+ Applied Processor:
DependencyPairs {dpKind_ = DT}
+ Details:
We add the following dependency tuples:

Strict DPs
f#(a()) -> c_1()
f#(c()) -> c_2()
f#(h(x,y)) -> c_3(g#(h(y,f(x)),h(x,f(y))),f#(x),f#(y))
g#(x,x) -> c_4()
Weak DPs

and mark the set of starting terms.
* Step 2: PredecessorEstimation WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f#(a()) -> c_1()
f#(c()) -> c_2()
f#(h(x,y)) -> c_3(g#(h(y,f(x)),h(x,f(y))),f#(x),f#(y))
g#(x,x) -> c_4()
- Weak TRS:
f(a()) -> b()
f(c()) -> d()
f(h(x,y)) -> g(h(y,f(x)),h(x,f(y)))
g(x,x) -> h(e(),x)
- Signature:
{f/1,g/2,f#/1,g#/2} / {a/0,b/0,c/0,d/0,e/0,h/2,c_1/0,c_2/0,c_3/3,c_4/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {a,b,c,d,e,h}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{1,2,4}
by application of
Pre({1,2,4}) = {3}.
Here rules are labelled as follows:
1: f#(a()) -> c_1()
2: f#(c()) -> c_2()
3: f#(h(x,y)) -> c_3(g#(h(y,f(x)),h(x,f(y))),f#(x),f#(y))
4: g#(x,x) -> c_4()
* Step 3: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f#(h(x,y)) -> c_3(g#(h(y,f(x)),h(x,f(y))),f#(x),f#(y))
- Weak DPs:
f#(a()) -> c_1()
f#(c()) -> c_2()
g#(x,x) -> c_4()
- Weak TRS:
f(a()) -> b()
f(c()) -> d()
f(h(x,y)) -> g(h(y,f(x)),h(x,f(y)))
g(x,x) -> h(e(),x)
- Signature:
{f/1,g/2,f#/1,g#/2} / {a/0,b/0,c/0,d/0,e/0,h/2,c_1/0,c_2/0,c_3/3,c_4/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {a,b,c,d,e,h}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:f#(h(x,y)) -> c_3(g#(h(y,f(x)),h(x,f(y))),f#(x),f#(y))
-->_1 g#(x,x) -> c_4():4
-->_3 f#(c()) -> c_2():3
-->_2 f#(c()) -> c_2():3
-->_3 f#(a()) -> c_1():2
-->_2 f#(a()) -> c_1():2
-->_3 f#(h(x,y)) -> c_3(g#(h(y,f(x)),h(x,f(y))),f#(x),f#(y)):1
-->_2 f#(h(x,y)) -> c_3(g#(h(y,f(x)),h(x,f(y))),f#(x),f#(y)):1

2:W:f#(a()) -> c_1()

3:W:f#(c()) -> c_2()

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

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
2: f#(a()) -> c_1()
3: f#(c()) -> c_2()
4: g#(x,x) -> c_4()
* Step 4: SimplifyRHS WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f#(h(x,y)) -> c_3(g#(h(y,f(x)),h(x,f(y))),f#(x),f#(y))
- Weak TRS:
f(a()) -> b()
f(c()) -> d()
f(h(x,y)) -> g(h(y,f(x)),h(x,f(y)))
g(x,x) -> h(e(),x)
- Signature:
{f/1,g/2,f#/1,g#/2} / {a/0,b/0,c/0,d/0,e/0,h/2,c_1/0,c_2/0,c_3/3,c_4/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {a,b,c,d,e,h}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:f#(h(x,y)) -> c_3(g#(h(y,f(x)),h(x,f(y))),f#(x),f#(y))
-->_3 f#(h(x,y)) -> c_3(g#(h(y,f(x)),h(x,f(y))),f#(x),f#(y)):1
-->_2 f#(h(x,y)) -> c_3(g#(h(y,f(x)),h(x,f(y))),f#(x),f#(y)):1

Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
f#(h(x,y)) -> c_3(f#(x),f#(y))
* Step 5: UsableRules WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f#(h(x,y)) -> c_3(f#(x),f#(y))
- Weak TRS:
f(a()) -> b()
f(c()) -> d()
f(h(x,y)) -> g(h(y,f(x)),h(x,f(y)))
g(x,x) -> h(e(),x)
- Signature:
{f/1,g/2,f#/1,g#/2} / {a/0,b/0,c/0,d/0,e/0,h/2,c_1/0,c_2/0,c_3/2,c_4/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {a,b,c,d,e,h}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
f#(h(x,y)) -> c_3(f#(x),f#(y))
* Step 6: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f#(h(x,y)) -> c_3(f#(x),f#(y))
- Signature:
{f/1,g/2,f#/1,g#/2} / {a/0,b/0,c/0,d/0,e/0,h/2,c_1/0,c_2/0,c_3/2,c_4/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {a,b,c,d,e,h}
+ 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#(h(x,y)) -> c_3(f#(x),f#(y))

The strictly oriented rules are moved into the weak component.
** Step 6.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f#(h(x,y)) -> c_3(f#(x),f#(y))
- Signature:
{f/1,g/2,f#/1,g#/2} / {a/0,b/0,c/0,d/0,e/0,h/2,c_1/0,c_2/0,c_3/2,c_4/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {a,b,c,d,e,h}
+ 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_3) = {1,2}

Following symbols are considered usable:
{f#,g#}
TcT has computed the following interpretation:
p(a) = 
p(b) = 
p(c) = 
p(d) = 
p(e) = 
p(f) = 
p(g) =  x1 + 
p(h) =  x1 +  x2 + 
p(f#) =  x1 + 
p(g#) =  x1 + 
p(c_1) = 
p(c_2) = 
p(c_3) =  x1 +  x2 + 
p(c_4) = 

Following rules are strictly oriented:
f#(h(x,y)) =  x +  y + 
>  x +  y + 
= c_3(f#(x),f#(y))

Following rules are (at-least) weakly oriented:

** Step 6.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
f#(h(x,y)) -> c_3(f#(x),f#(y))
- Signature:
{f/1,g/2,f#/1,g#/2} / {a/0,b/0,c/0,d/0,e/0,h/2,c_1/0,c_2/0,c_3/2,c_4/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {a,b,c,d,e,h}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()

** Step 6.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
f#(h(x,y)) -> c_3(f#(x),f#(y))
- Signature:
{f/1,g/2,f#/1,g#/2} / {a/0,b/0,c/0,d/0,e/0,h/2,c_1/0,c_2/0,c_3/2,c_4/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {a,b,c,d,e,h}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:f#(h(x,y)) -> c_3(f#(x),f#(y))
-->_2 f#(h(x,y)) -> c_3(f#(x),f#(y)):1
-->_1 f#(h(x,y)) -> c_3(f#(x),f#(y)):1

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: f#(h(x,y)) -> c_3(f#(x),f#(y))
** Step 6.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:

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

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