```* 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) = [0]
p(add) = [1] x1 + [1] x2 + [0]
p(eq) = [5]
p(false) = [0]
p(ifrm) = [1] x1 + [1] x3 + [5]
p(nil) = [0]
p(purge) = [1] x1 + [5]
p(rm) = [1] x2 + [3]
p(s) = [1] x1 + [0]
p(true) = [0]

Following rules are strictly oriented:
eq(0(),0()) = [5]
> [0]
= true()

eq(0(),s(X)) = [5]
> [0]
= false()

eq(s(X),0()) = [5]
> [0]
= false()

ifrm(false(),N,add(M,X)) = [1] M + [1] X + [5]
> [1] M + [1] X + [3]

ifrm(true(),N,add(M,X)) = [1] M + [1] X + [5]
> [1] X + [3]
= rm(N,X)

purge(nil()) = [5]
> [0]
= nil()

rm(N,nil()) = [3]
> [0]
= nil()

Following rules are (at-least) weakly oriented:
eq(s(X),s(Y)) =  [5]
>= [5]
=  eq(X,Y)

purge(add(N,X)) =  [1] N + [1] X + [5]
>= [1] N + [1] X + [8]

rm(N,add(M,X)) =  [1] M + [1] X + [3]
>= [1] M + [1] X + [10]

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) = [1]
p(add) = [1] x_1 + [1] x_2 + [3]
p(eq) = [0]
p(false) = [0]
p(ifrm) = [8] x_1 + [1] x_3 + [1]
p(nil) = [0]
p(purge) = [2] x_1 + [0]
p(rm) = [1] x_2 + [1]
p(s) = [4]
p(true) = [0]

Following rules are strictly oriented:
purge(add(N,X)) = [2] N + [2] X + [6]
> [1] N + [2] X + [5]

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

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

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

eq(s(X),s(Y)) =  [0]
>= [0]
=  eq(X,Y)

ifrm(false(),N,add(M,X)) =  [1] M + [1] X + [4]
>= [1] M + [1] X + [4]

ifrm(true(),N,add(M,X)) =  [1] M + [1] X + [4]
>= [1] X + [1]
=  rm(N,X)

purge(nil()) =  [0]
>= [0]
=  nil()

rm(N,add(M,X)) =  [1] M + [1] X + [4]
>= [1] M + [1] X + [4]

rm(N,nil()) =  [1]
>= [0]
=  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) = [0]
[0]
p(add) = [1 4] x_2 + [0]
[0 1]       [2]
p(eq) = [0]
[2]
p(false) = [0]
[2]
p(ifrm) = [2 1] x_1 + [1 2] x_3 + [0]
[0 0]       [0 1]       [0]
p(nil) = [0]
[0]
p(purge) = [2 4] x_1 + [4]
[0 1]       [0]
p(rm) = [1 2] x_2 + [3]
[0 1]       [0]
p(s) = [1 0] x_1 + [0]
[0 1]       [0]
p(true) = [0]
[2]

Following rules are strictly oriented:
rm(N,add(M,X)) = [1 6] X + [7]
[0 1]     [2]
> [1 6] X + [6]
[0 1]     [2]

Following rules are (at-least) weakly oriented:
eq(0(),0()) =  [0]
[2]
>= [0]
[2]
=  true()

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

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

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

ifrm(false(),N,add(M,X)) =  [1 6] X + [6]
[0 1]     [2]
>= [1 6] X + [3]
[0 1]     [2]

ifrm(true(),N,add(M,X)) =  [1 6] X + [6]
[0 1]     [2]
>= [1 2] X + [3]
[0 1]     [0]
=  rm(N,X)

purge(add(N,X)) =  [2 12] X + [12]
[0  1]     [2]
>= [2 12] X + [10]
[0  1]     [2]

purge(nil()) =  [4]
[0]
>= [0]
[0]
=  nil()

rm(N,nil()) =  [3]
[0]
>= [0]
[0]
=  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) = [0]
[1]
[0]
[1]
p(add) = [0 0 0 0]       [1 0 2 0]       [0]
[0 0 0 0] x_1 + [0 0 1 0] x_2 + [0]
[0 0 0 1]       [0 0 1 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 1] x_1 + [0 0 0 1] x_2 + [0]
[0 0 0 0]       [0 0 0 0]       [0]
[1 1 0 2]       [0 0 0 0]       [2]
p(false) = [0]
[0]
[0]
[0]
p(ifrm) = [1 0 0 0]       [0 0 0 0]       [1 1 0 0]       [0]
[0 0 0 0] x_1 + [0 0 2 0] x_2 + [1 0 0 0] x_3 + [2]
[0 0 0 0]       [0 0 0 0]       [0 0 1 0]       [0]
[0 0 0 0]       [0 0 0 0]       [0 0 0 0]       [2]
p(nil) = [2]
[0]
[0]
[2]
p(purge) = [2 0 0 0]       [0]
[0 1 0 0] x_1 + [0]
[0 0 1 0]       [0]
[2 0 0 0]       [0]
p(rm) = [0 0 0 0]       [1 0 1 0]       [0]
[0 0 2 0] x_1 + [1 0 0 0] x_2 + [2]
[0 0 0 0]       [0 0 1 0]       [0]
[0 0 0 0]       [0 0 0 0]       [2]
p(s) = [1 1 2 0]       [0]
[0 0 0 0] x_1 + [2]
[0 0 0 0]       [3]
[0 0 0 1]       [1]
p(true) = [0]
[3]
[0]
[1]

Following rules are strictly oriented:
eq(s(X),s(Y)) = [0 0 0 0]     [0 0 0 1]     [2]
[1 1 2 1] X + [0 0 0 1] Y + [7]
[0 0 0 0]     [0 0 0 0]     [0]
[1 1 2 2]     [0 0 0 0]     [6]
> [0 0 0 0]     [0 0 0 1]     [1]
[1 1 1 1] X + [0 0 0 1] Y + [0]
[0 0 0 0]     [0 0 0 0]     [0]
[1 1 0 2]     [0 0 0 0]     [2]
= eq(X,Y)

Following rules are (at-least) weakly oriented:
eq(0(),0()) =  [2]
[3]
[0]
[5]
>= [0]
[3]
[0]
[1]
=  true()

eq(0(),s(X)) =  [0 0 0 1]     [2]
[0 0 0 1] X + [3]
[0 0 0 0]     [0]
[0 0 0 0]     [5]
>= [0]
[0]
[0]
[0]
=  false()

eq(s(X),0()) =  [0 0 0 0]     [2]
[1 1 2 1] X + [7]
[0 0 0 0]     [0]
[1 1 2 2]     [6]
>= [0]
[0]
[0]
[0]
=  false()

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

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

purge(nil()) =  [4]
[0]
[0]
[4]
>= [2]
[0]
[0]
[2]
=  nil()

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

rm(N,nil()) =  [0 0 0 0]     [2]
[0 0 2 0] N + [4]
[0 0 0 0]     [0]
[0 0 0 0]     [2]
>= [2]
[0]
[0]
[2]
=  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()