```* 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) = 
p(a__b) = 
p(a__f) =  x1 +  x2 + 
p(b) = 
p(f) =  x1 +  x2 + 
p(mark) =  x1 + 

Following rules are strictly oriented:
a__f(a(),X,X) =  X + 
>  X + 
= a__f(X,a__b(),b())

mark(a()) = 
> 
= a()

mark(b()) = 
> 
= a__b()

mark(f(X1,X2,X3)) =  X1 +  X2 + 
>  X1 +  X2 + 
= a__f(X1,mark(X2),X3)

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

a__b() =  
>= 
=  b()

a__f(X1,X2,X3) =   X1 +  X2 + 
>=  X1 +  X2 + 
=  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) = 
p(a__b) = 
p(a__f) =  x1 +  x2 +  x3 + 
p(b) = 
p(f) =  x1 +  x2 +  x3 + 
p(mark) =  x1 + 

Following rules are strictly oriented:
a__b() = 
> 
= b()

a__f(X1,X2,X3) =  X1 +  X2 +  X3 + 
>  X1 +  X2 +  X3 + 
= f(X1,X2,X3)

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

a__f(a(),X,X) =   X + 
>=  X + 
=  a__f(X,a__b(),b())

mark(a()) =  
>= 
=  a()

mark(b()) =  
>= 
=  a__b()

mark(f(X1,X2,X3)) =   X1 +  X2 +  X3 + 
>=  X1 +  X2 +  X3 + 
=  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) = 
p(a__b) = 
p(a__f) =  x_1 +  x_2 +  x_3 + 
p(b) = 
p(f) =  x_1 +  x_2 +  x_3 + 
p(mark) =  x_1 + 

Following rules are strictly oriented:
a__b() = 
> 
= a()

Following rules are (at-least) weakly oriented:
a__b() =  
>= 
=  b()

a__f(X1,X2,X3) =   X1 +  X2 +  X3 + 
>=  X1 +  X2 +  X3 + 
=  f(X1,X2,X3)

a__f(a(),X,X) =   X + 
>=  X + 
=  a__f(X,a__b(),b())

mark(a()) =  
>= 
=  a()

mark(b()) =  
>= 
=  a__b()

mark(f(X1,X2,X3)) =   X1 +  X2 +  X3 + 
>=  X1 +  X2 +  X3 + 
=  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))
```