```* Step 1: ToInnermost WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
a__b() -> a()
a__b() -> b()
a__f(X1,X2,X3) -> f(X1,X2,X3)
a__f(a(),X,X) -> a__f(X,a__b(),b())
mark(a()) -> a()
mark(b()) -> a__b()
mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
- Signature:
{a__b/0,a__f/3,mark/1} / {a/0,b/0,f/3}
- 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: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
a__b() -> a()
a__b() -> b()
a__f(X1,X2,X3) -> f(X1,X2,X3)
a__f(a(),X,X) -> a__f(X,a__b(),b())
mark(a()) -> a()
mark(b()) -> a__b()
mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
- Signature:
{a__b/0,a__f/3,mark/1} / {a/0,b/0,f/3}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__b,a__f,mark} and constructors {a,b,f}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(a__f) = {2}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(a) = [4]
p(a__b) = [0]
p(a__f) = [1] x1 + [1] x2 + [0]
p(b) = [10]
p(f) = [1] x1 + [1] x2 + [12]
p(mark) = [2] x1 + [0]

Following rules are strictly oriented:
a__f(a(),X,X) = [1] X + [4]
> [1] X + [0]
= a__f(X,a__b(),b())

mark(a()) = [8]
> [4]
= a()

mark(b()) = [20]
> [0]
= a__b()

mark(f(X1,X2,X3)) = [2] X1 + [2] X2 + [24]
> [1] X1 + [2] X2 + [0]
= a__f(X1,mark(X2),X3)

Following rules are (at-least) weakly oriented:
a__b() =  [0]
>= [4]
=  a()

a__b() =  [0]
>= [10]
=  b()

a__f(X1,X2,X3) =  [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [12]
=  f(X1,X2,X3)

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
a__b() -> a()
a__b() -> b()
a__f(X1,X2,X3) -> f(X1,X2,X3)
- Weak TRS:
a__f(a(),X,X) -> a__f(X,a__b(),b())
mark(a()) -> a()
mark(b()) -> a__b()
mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
- Signature:
{a__b/0,a__f/3,mark/1} / {a/0,b/0,f/3}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__b,a__f,mark} and constructors {a,b,f}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(a__f) = {2}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(a) = [1]
p(a__b) = [1]
p(a__f) = [1] x1 + [1] x2 + [1] x3 + [3]
p(b) = [0]
p(f) = [1] x1 + [1] x2 + [1] x3 + [2]
p(mark) = [9] x1 + [1]

Following rules are strictly oriented:
a__b() = [1]
> [0]
= b()

a__f(X1,X2,X3) = [1] X1 + [1] X2 + [1] X3 + [3]
> [1] X1 + [1] X2 + [1] X3 + [2]
= f(X1,X2,X3)

Following rules are (at-least) weakly oriented:
a__b() =  [1]
>= [1]
=  a()

a__f(a(),X,X) =  [2] X + [4]
>= [1] X + [4]
=  a__f(X,a__b(),b())

mark(a()) =  [10]
>= [1]
=  a()

mark(b()) =  [1]
>= [1]
=  a__b()

mark(f(X1,X2,X3)) =  [9] X1 + [9] X2 + [9] X3 + [19]
>= [1] X1 + [9] X2 + [1] X3 + [4]
=  a__f(X1,mark(X2),X3)

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 4: MI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
a__b() -> a()
- Weak TRS:
a__b() -> b()
a__f(X1,X2,X3) -> f(X1,X2,X3)
a__f(a(),X,X) -> a__f(X,a__b(),b())
mark(a()) -> a()
mark(b()) -> a__b()
mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
- Signature:
{a__b/0,a__f/3,mark/1} / {a/0,b/0,f/3}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__b,a__f,mark} and constructors {a,b,f}
+ Applied Processor:
MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity Nothing)), miDimension = 1, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity Nothing)):

The following argument positions are considered usable:
uargs(a__f) = {2}

Following symbols are considered usable:
{a__b,a__f,mark}
TcT has computed the following interpretation:
p(a) = [4]
p(a__b) = [6]
p(a__f) = [4] x_1 + [1] x_2 + [4] x_3 + [8]
p(b) = [2]
p(f) = [1] x_1 + [1] x_2 + [1] x_3 + [4]
p(mark) = [4] x_1 + [4]

Following rules are strictly oriented:
a__b() = [6]
> [4]
= a()

Following rules are (at-least) weakly oriented:
a__b() =  [6]
>= [2]
=  b()

a__f(X1,X2,X3) =  [4] X1 + [1] X2 + [4] X3 + [8]
>= [1] X1 + [1] X2 + [1] X3 + [4]
=  f(X1,X2,X3)

a__f(a(),X,X) =  [5] X + [24]
>= [4] X + [22]
=  a__f(X,a__b(),b())

mark(a()) =  [20]
>= [4]
=  a()

mark(b()) =  [12]
>= [6]
=  a__b()

mark(f(X1,X2,X3)) =  [4] X1 + [4] X2 + [4] X3 + [20]
>= [4] X1 + [4] X2 + [4] X3 + [12]
=  a__f(X1,mark(X2),X3)

* Step 5: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
a__b() -> a()
a__b() -> b()
a__f(X1,X2,X3) -> f(X1,X2,X3)
a__f(a(),X,X) -> a__f(X,a__b(),b())
mark(a()) -> a()
mark(b()) -> a__b()
mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
- Signature:
{a__b/0,a__f/3,mark/1} / {a/0,b/0,f/3}
- 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))
```