```* Step 1: ToInnermost WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
a__b() -> a()
a__b() -> b()
a__f(X,X) -> a__f(a(),b())
a__f(X1,X2) -> f(X1,X2)
mark(a()) -> a()
mark(b()) -> a__b()
mark(f(X1,X2)) -> a__f(mark(X1),X2)
- Signature:
{a__b/0,a__f/2,mark/1} / {a/0,b/0,f/2}
- Obligation:
runtime complexity wrt. defined symbols {a__b,a__f,mark} and constructors {a,b,f}
+ Applied Processor:
ToInnermost
+ Details:
switch to innermost, as the system is overlay and right linear and does not contain weak rules
* Step 2: DependencyPairs WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
a__b() -> a()
a__b() -> b()
a__f(X,X) -> a__f(a(),b())
a__f(X1,X2) -> f(X1,X2)
mark(a()) -> a()
mark(b()) -> a__b()
mark(f(X1,X2)) -> a__f(mark(X1),X2)
- Signature:
{a__b/0,a__f/2,mark/1} / {a/0,b/0,f/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__b,a__f,mark} and constructors {a,b,f}
+ Applied Processor:
DependencyPairs {dpKind_ = DT}
+ Details:
We add the following dependency tuples:

Strict DPs
a__b#() -> c_1()
a__b#() -> c_2()
a__f#(X,X) -> c_3(a__f#(a(),b()))
a__f#(X1,X2) -> c_4()
mark#(a()) -> c_5()
mark#(b()) -> c_6(a__b#())
mark#(f(X1,X2)) -> c_7(a__f#(mark(X1),X2),mark#(X1))
Weak DPs

and mark the set of starting terms.
* Step 3: PredecessorEstimation WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
a__b#() -> c_1()
a__b#() -> c_2()
a__f#(X,X) -> c_3(a__f#(a(),b()))
a__f#(X1,X2) -> c_4()
mark#(a()) -> c_5()
mark#(b()) -> c_6(a__b#())
mark#(f(X1,X2)) -> c_7(a__f#(mark(X1),X2),mark#(X1))
- Weak TRS:
a__b() -> a()
a__b() -> b()
a__f(X,X) -> a__f(a(),b())
a__f(X1,X2) -> f(X1,X2)
mark(a()) -> a()
mark(b()) -> a__b()
mark(f(X1,X2)) -> a__f(mark(X1),X2)
- Signature:
{a__b/0,a__f/2,mark/1,a__b#/0,a__f#/2,mark#/1} / {a/0,b/0,f/2,c_1/0,c_2/0,c_3/1,c_4/0,c_5/0,c_6/1,c_7/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__b#,a__f#,mark#} and constructors {a,b,f}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{1,2,4,5}
by application of
Pre({1,2,4,5}) = {3,6,7}.
Here rules are labelled as follows:
1: a__b#() -> c_1()
2: a__b#() -> c_2()
3: a__f#(X,X) -> c_3(a__f#(a(),b()))
4: a__f#(X1,X2) -> c_4()
5: mark#(a()) -> c_5()
6: mark#(b()) -> c_6(a__b#())
7: mark#(f(X1,X2)) -> c_7(a__f#(mark(X1),X2),mark#(X1))
* Step 4: PredecessorEstimation WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
a__f#(X,X) -> c_3(a__f#(a(),b()))
mark#(b()) -> c_6(a__b#())
mark#(f(X1,X2)) -> c_7(a__f#(mark(X1),X2),mark#(X1))
- Weak DPs:
a__b#() -> c_1()
a__b#() -> c_2()
a__f#(X1,X2) -> c_4()
mark#(a()) -> c_5()
- Weak TRS:
a__b() -> a()
a__b() -> b()
a__f(X,X) -> a__f(a(),b())
a__f(X1,X2) -> f(X1,X2)
mark(a()) -> a()
mark(b()) -> a__b()
mark(f(X1,X2)) -> a__f(mark(X1),X2)
- Signature:
{a__b/0,a__f/2,mark/1,a__b#/0,a__f#/2,mark#/1} / {a/0,b/0,f/2,c_1/0,c_2/0,c_3/1,c_4/0,c_5/0,c_6/1,c_7/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__b#,a__f#,mark#} and constructors {a,b,f}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{1,2}
by application of
Pre({1,2}) = {3}.
Here rules are labelled as follows:
1: a__f#(X,X) -> c_3(a__f#(a(),b()))
2: mark#(b()) -> c_6(a__b#())
3: mark#(f(X1,X2)) -> c_7(a__f#(mark(X1),X2),mark#(X1))
4: a__b#() -> c_1()
5: a__b#() -> c_2()
6: a__f#(X1,X2) -> c_4()
7: mark#(a()) -> c_5()
* Step 5: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
mark#(f(X1,X2)) -> c_7(a__f#(mark(X1),X2),mark#(X1))
- Weak DPs:
a__b#() -> c_1()
a__b#() -> c_2()
a__f#(X,X) -> c_3(a__f#(a(),b()))
a__f#(X1,X2) -> c_4()
mark#(a()) -> c_5()
mark#(b()) -> c_6(a__b#())
- Weak TRS:
a__b() -> a()
a__b() -> b()
a__f(X,X) -> a__f(a(),b())
a__f(X1,X2) -> f(X1,X2)
mark(a()) -> a()
mark(b()) -> a__b()
mark(f(X1,X2)) -> a__f(mark(X1),X2)
- Signature:
{a__b/0,a__f/2,mark/1,a__b#/0,a__f#/2,mark#/1} / {a/0,b/0,f/2,c_1/0,c_2/0,c_3/1,c_4/0,c_5/0,c_6/1,c_7/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__b#,a__f#,mark#} and constructors {a,b,f}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:mark#(f(X1,X2)) -> c_7(a__f#(mark(X1),X2),mark#(X1))
-->_2 mark#(b()) -> c_6(a__b#()):7
-->_1 a__f#(X,X) -> c_3(a__f#(a(),b())):4
-->_2 mark#(a()) -> c_5():6
-->_1 a__f#(X1,X2) -> c_4():5
-->_2 mark#(f(X1,X2)) -> c_7(a__f#(mark(X1),X2),mark#(X1)):1

2:W:a__b#() -> c_1()

3:W:a__b#() -> c_2()

4:W:a__f#(X,X) -> c_3(a__f#(a(),b()))
-->_1 a__f#(X1,X2) -> c_4():5

5:W:a__f#(X1,X2) -> c_4()

6:W:mark#(a()) -> c_5()

7:W:mark#(b()) -> c_6(a__b#())
-->_1 a__b#() -> c_2():3
-->_1 a__b#() -> c_1():2

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
6: mark#(a()) -> c_5()
4: a__f#(X,X) -> c_3(a__f#(a(),b()))
5: a__f#(X1,X2) -> c_4()
7: mark#(b()) -> c_6(a__b#())
2: a__b#() -> c_1()
3: a__b#() -> c_2()
* Step 6: SimplifyRHS WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
mark#(f(X1,X2)) -> c_7(a__f#(mark(X1),X2),mark#(X1))
- Weak TRS:
a__b() -> a()
a__b() -> b()
a__f(X,X) -> a__f(a(),b())
a__f(X1,X2) -> f(X1,X2)
mark(a()) -> a()
mark(b()) -> a__b()
mark(f(X1,X2)) -> a__f(mark(X1),X2)
- Signature:
{a__b/0,a__f/2,mark/1,a__b#/0,a__f#/2,mark#/1} / {a/0,b/0,f/2,c_1/0,c_2/0,c_3/1,c_4/0,c_5/0,c_6/1,c_7/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__b#,a__f#,mark#} and constructors {a,b,f}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:mark#(f(X1,X2)) -> c_7(a__f#(mark(X1),X2),mark#(X1))
-->_2 mark#(f(X1,X2)) -> c_7(a__f#(mark(X1),X2),mark#(X1)):1

Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
mark#(f(X1,X2)) -> c_7(mark#(X1))
* Step 7: UsableRules WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
mark#(f(X1,X2)) -> c_7(mark#(X1))
- Weak TRS:
a__b() -> a()
a__b() -> b()
a__f(X,X) -> a__f(a(),b())
a__f(X1,X2) -> f(X1,X2)
mark(a()) -> a()
mark(b()) -> a__b()
mark(f(X1,X2)) -> a__f(mark(X1),X2)
- Signature:
{a__b/0,a__f/2,mark/1,a__b#/0,a__f#/2,mark#/1} / {a/0,b/0,f/2,c_1/0,c_2/0,c_3/1,c_4/0,c_5/0,c_6/1,c_7/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__b#,a__f#,mark#} and constructors {a,b,f}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
mark#(f(X1,X2)) -> c_7(mark#(X1))
* Step 8: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
mark#(f(X1,X2)) -> c_7(mark#(X1))
- Signature:
{a__b/0,a__f/2,mark/1,a__b#/0,a__f#/2,mark#/1} / {a/0,b/0,f/2,c_1/0,c_2/0,c_3/1,c_4/0,c_5/0,c_6/1,c_7/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__b#,a__f#,mark#} and constructors {a,b,f}
+ 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: mark#(f(X1,X2)) -> c_7(mark#(X1))

The strictly oriented rules are moved into the weak component.
** Step 8.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
mark#(f(X1,X2)) -> c_7(mark#(X1))
- Signature:
{a__b/0,a__f/2,mark/1,a__b#/0,a__f#/2,mark#/1} / {a/0,b/0,f/2,c_1/0,c_2/0,c_3/1,c_4/0,c_5/0,c_6/1,c_7/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__b#,a__f#,mark#} and constructors {a,b,f}
+ 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_7) = {1}

Following symbols are considered usable:
{a__b#,a__f#,mark#}
TcT has computed the following interpretation:
p(a) = [2]
p(a__b) = [1]
p(a__f) = [1]
p(b) = [0]
p(f) = [1] x1 + [1]
p(mark) = [2]
p(a__b#) = [1]
p(a__f#) = [8] x2 + [1]
p(mark#) = [1] x1 + [10]
p(c_1) = [1]
p(c_2) = [0]
p(c_3) = [1] x1 + [1]
p(c_4) = [0]
p(c_5) = [1]
p(c_6) = [2] x1 + [4]
p(c_7) = [1] x1 + [0]

Following rules are strictly oriented:
mark#(f(X1,X2)) = [1] X1 + [11]
> [1] X1 + [10]
= c_7(mark#(X1))

Following rules are (at-least) weakly oriented:

** Step 8.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
mark#(f(X1,X2)) -> c_7(mark#(X1))
- Signature:
{a__b/0,a__f/2,mark/1,a__b#/0,a__f#/2,mark#/1} / {a/0,b/0,f/2,c_1/0,c_2/0,c_3/1,c_4/0,c_5/0,c_6/1,c_7/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__b#,a__f#,mark#} and constructors {a,b,f}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()

** Step 8.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
mark#(f(X1,X2)) -> c_7(mark#(X1))
- Signature:
{a__b/0,a__f/2,mark/1,a__b#/0,a__f#/2,mark#/1} / {a/0,b/0,f/2,c_1/0,c_2/0,c_3/1,c_4/0,c_5/0,c_6/1,c_7/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__b#,a__f#,mark#} and constructors {a,b,f}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:mark#(f(X1,X2)) -> c_7(mark#(X1))
-->_1 mark#(f(X1,X2)) -> c_7(mark#(X1)):1

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: mark#(f(X1,X2)) -> c_7(mark#(X1))
** Step 8.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:

- Signature:
{a__b/0,a__f/2,mark/1,a__b#/0,a__f#/2,mark#/1} / {a/0,b/0,f/2,c_1/0,c_2/0,c_3/1,c_4/0,c_5/0,c_6/1,c_7/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__b#,a__f#,mark#} and constructors {a,b,f}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

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