```* Step 1: WeightGap WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
eq(0(),0()) -> true()
eq(0(),s(X)) -> false()
eq(s(X),0()) -> false()
eq(s(X),s(Y)) -> eq(X,Y)
purge(nil()) -> nil()
rm(N,nil()) -> nil()
- Signature:
- Obligation:
runtime complexity wrt. defined symbols {eq,ifrm,purge,rm} and constructors {0,add,false,nil,s,true}
+ 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(ifrm) = {1,3},
uargs(purge) = {1},
uargs(rm) = {2}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = 
p(add) =  x1 +  x2 + 
p(eq) = 
p(false) = 
p(ifrm) =  x1 +  x3 + 
p(nil) = 
p(purge) =  x1 + 
p(rm) =  x2 + 
p(s) =  x1 + 
p(true) = 

Following rules are strictly oriented:
eq(0(),0()) = 
> 
= true()

eq(0(),s(X)) = 
> 
= false()

eq(s(X),0()) = 
> 
= false()

ifrm(false(),N,add(M,X)) =  M +  X + 
>  M +  X + 

ifrm(true(),N,add(M,X)) =  M +  X + 
>  X + 
= rm(N,X)

purge(nil()) = 
> 
= nil()

rm(N,nil()) = 
> 
= nil()

Following rules are (at-least) weakly oriented:
eq(s(X),s(Y)) =  
>= 
=  eq(X,Y)

purge(add(N,X)) =   N +  X + 
>=  N +  X + 

rm(N,add(M,X)) =   M +  X + 
>=  M +  X + 

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: MI WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
eq(s(X),s(Y)) -> eq(X,Y)
- Weak TRS:
eq(0(),0()) -> true()
eq(0(),s(X)) -> false()
eq(s(X),0()) -> false()
purge(nil()) -> nil()
rm(N,nil()) -> nil()
- Signature:
- Obligation:
runtime complexity wrt. defined symbols {eq,ifrm,purge,rm} and constructors {0,add,false,nil,s,true}
+ 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(ifrm) = {1,3},
uargs(purge) = {1},
uargs(rm) = {2}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = 
p(add) =  x_1 +  x_2 + 
p(eq) = 
p(false) = 
p(ifrm) =  x_1 +  x_3 + 
p(nil) = 
p(purge) =  x_1 + 
p(rm) =  x_2 + 
p(s) = 
p(true) = 

Following rules are strictly oriented:
purge(add(N,X)) =  N +  X + 
>  N +  X + 

Following rules are (at-least) weakly oriented:
eq(0(),0()) =  
>= 
=  true()

eq(0(),s(X)) =  
>= 
=  false()

eq(s(X),0()) =  
>= 
=  false()

eq(s(X),s(Y)) =  
>= 
=  eq(X,Y)

ifrm(false(),N,add(M,X)) =   M +  X + 
>=  M +  X + 

ifrm(true(),N,add(M,X)) =   M +  X + 
>=  X + 
=  rm(N,X)

purge(nil()) =  
>= 
=  nil()

rm(N,add(M,X)) =   M +  X + 
>=  M +  X + 

rm(N,nil()) =  
>= 
=  nil()

* Step 3: MI WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
eq(s(X),s(Y)) -> eq(X,Y)
- Weak TRS:
eq(0(),0()) -> true()
eq(0(),s(X)) -> false()
eq(s(X),0()) -> false()
purge(nil()) -> nil()
rm(N,nil()) -> nil()
- Signature:
- Obligation:
runtime complexity wrt. defined symbols {eq,ifrm,purge,rm} and constructors {0,add,false,nil,s,true}
+ Applied Processor:
MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity Nothing)), miDimension = 2, 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(ifrm) = {1,3},
uargs(purge) = {1},
uargs(rm) = {2}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = 

p(add) = [1 4] x_2 + 
[0 1]       
p(eq) = 

p(false) = 

p(ifrm) = [2 1] x_1 + [1 2] x_3 + 
[0 0]       [0 1]       
p(nil) = 

p(purge) = [2 4] x_1 + 
[0 1]       
p(rm) = [1 2] x_2 + 
[0 1]       
p(s) = [1 0] x_1 + 
[0 1]       
p(true) = 


Following rules are strictly oriented:
rm(N,add(M,X)) = [1 6] X + 
[0 1]     
> [1 6] X + 
[0 1]     

Following rules are (at-least) weakly oriented:
eq(0(),0()) =  

>= 

=  true()

eq(0(),s(X)) =  

>= 

=  false()

eq(s(X),0()) =  

>= 

=  false()

eq(s(X),s(Y)) =  

>= 

=  eq(X,Y)

ifrm(false(),N,add(M,X)) =  [1 6] X + 
[0 1]     
>= [1 6] X + 
[0 1]     

ifrm(true(),N,add(M,X)) =  [1 6] X + 
[0 1]     
>= [1 2] X + 
[0 1]     
=  rm(N,X)

purge(add(N,X)) =  [2 12] X + 
[0  1]     
>= [2 12] X + 
[0  1]     

purge(nil()) =  

>= 

=  nil()

rm(N,nil()) =  

>= 

=  nil()

* Step 4: MI WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
eq(s(X),s(Y)) -> eq(X,Y)
- Weak TRS:
eq(0(),0()) -> true()
eq(0(),s(X)) -> false()
eq(s(X),0()) -> false()
purge(nil()) -> nil()
rm(N,nil()) -> nil()
- Signature:
- Obligation:
runtime complexity wrt. defined symbols {eq,ifrm,purge,rm} and constructors {0,add,false,nil,s,true}
+ Applied Processor:
MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity (Just 3))), miDimension = 4, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity (Just 3))):

The following argument positions are considered usable:
uargs(ifrm) = {1,3},
uargs(purge) = {1},
uargs(rm) = {2}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = 



p(add) = [0 0 0 0]       [1 0 2 0]       
[0 0 0 0] x_1 + [0 0 1 0] x_2 + 
[0 0 0 1]       [0 0 1 0]       
[0 0 0 0]       [0 0 0 0]       
p(eq) = [0 0 0 0]       [0 0 0 1]       
[1 1 1 1] x_1 + [0 0 0 1] x_2 + 
[0 0 0 0]       [0 0 0 0]       
[1 1 0 2]       [0 0 0 0]       
p(false) = 



p(ifrm) = [1 0 0 0]       [0 0 0 0]       [1 1 0 0]       
[0 0 0 0] x_1 + [0 0 2 0] x_2 + [1 0 0 0] x_3 + 
[0 0 0 0]       [0 0 0 0]       [0 0 1 0]       
[0 0 0 0]       [0 0 0 0]       [0 0 0 0]       
p(nil) = 



p(purge) = [2 0 0 0]       
[0 1 0 0] x_1 + 
[0 0 1 0]       
[2 0 0 0]       
p(rm) = [0 0 0 0]       [1 0 1 0]       
[0 0 2 0] x_1 + [1 0 0 0] x_2 + 
[0 0 0 0]       [0 0 1 0]       
[0 0 0 0]       [0 0 0 0]       
p(s) = [1 1 2 0]       
[0 0 0 0] x_1 + 
[0 0 0 0]       
[0 0 0 1]       
p(true) = 




Following rules are strictly oriented:
eq(s(X),s(Y)) = [0 0 0 0]     [0 0 0 1]     
[1 1 2 1] X + [0 0 0 1] Y + 
[0 0 0 0]     [0 0 0 0]     
[1 1 2 2]     [0 0 0 0]     
> [0 0 0 0]     [0 0 0 1]     
[1 1 1 1] X + [0 0 0 1] Y + 
[0 0 0 0]     [0 0 0 0]     
[1 1 0 2]     [0 0 0 0]     
= eq(X,Y)

Following rules are (at-least) weakly oriented:
eq(0(),0()) =  



>= 



=  true()

eq(0(),s(X)) =  [0 0 0 1]     
[0 0 0 1] X + 
[0 0 0 0]     
[0 0 0 0]     
>= 



=  false()

eq(s(X),0()) =  [0 0 0 0]     
[1 1 2 1] X + 
[0 0 0 0]     
[1 1 2 2]     
>= 



=  false()

ifrm(false(),N,add(M,X)) =  [0 0 0 0]     [0 0 0 0]     [1 0 3 0]     
[0 0 0 0] M + [0 0 2 0] N + [1 0 2 0] X + 
[0 0 0 1]     [0 0 0 0]     [0 0 1 0]     
[0 0 0 0]     [0 0 0 0]     [0 0 0 0]     
>= [0 0 0 0]     [1 0 3 0]     
[0 0 0 0] M + [0 0 1 0] X + 
[0 0 0 1]     [0 0 1 0]     
[0 0 0 0]     [0 0 0 0]     

ifrm(true(),N,add(M,X)) =  [0 0 0 0]     [0 0 0 0]     [1 0 3 0]     
[0 0 0 0] M + [0 0 2 0] N + [1 0 2 0] X + 
[0 0 0 1]     [0 0 0 0]     [0 0 1 0]     
[0 0 0 0]     [0 0 0 0]     [0 0 0 0]     
>= [0 0 0 0]     [1 0 1 0]     
[0 0 2 0] N + [1 0 0 0] X + 
[0 0 0 0]     [0 0 1 0]     
[0 0 0 0]     [0 0 0 0]     
=  rm(N,X)

purge(add(N,X)) =  [0 0 0 0]     [2 0 4 0]     
[0 0 0 0] N + [0 0 1 0] X + 
[0 0 0 1]     [0 0 1 0]     
[0 0 0 0]     [2 0 4 0]     
>= [0 0 0 0]     [2 0 4 0]     
[0 0 0 0] N + [0 0 1 0] X + 
[0 0 0 1]     [0 0 1 0]     
[0 0 0 0]     [0 0 0 0]     

purge(nil()) =  



>= 



=  nil()

rm(N,add(M,X)) =  [0 0 0 1]     [0 0 0 0]     [1 0 3 0]     
[0 0 0 0] M + [0 0 2 0] N + [1 0 2 0] X + 
[0 0 0 1]     [0 0 0 0]     [0 0 1 0]     
[0 0 0 0]     [0 0 0 0]     [0 0 0 0]     
>= [0 0 0 1]     [0 0 0 0]     [1 0 3 0]     
[0 0 0 0] M + [0 0 2 0] N + [1 0 2 0] X + 
[0 0 0 1]     [0 0 0 0]     [0 0 1 0]     
[0 0 0 0]     [0 0 0 0]     [0 0 0 0]     

rm(N,nil()) =  [0 0 0 0]     
[0 0 2 0] N + 
[0 0 0 0]     
[0 0 0 0]     
>= 



=  nil()

* Step 5: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
eq(0(),0()) -> true()
eq(0(),s(X)) -> false()
eq(s(X),0()) -> false()
eq(s(X),s(Y)) -> eq(X,Y)
purge(nil()) -> nil()