Lirem

Lirem is a number based on the repetition of cheats on cheats.

## Terms

Define BC as "all concepts that will increase the size of a given number".

## Hierarchy of Strength

Begin with the definition "greater than all other numbers", this is SH0 = A0 (GF).

Next in the hierarchy is SH1, defined as "SH0 augmented by the statement "all of BC is used for this entire definition"", which makes SH1 ~ Reduk (f[sd2->/d5/]->GF)

You may not have noticed this, but "augmented by all concepts that will increase the size of it", doesn't actually do what it claims, so the part giving cheat-levels, needs cheats itself... but if we give that a reasonably high amount of cheats... the cheats will also need cheats, so we need a way to solve that.

SH2 is defined as "SH1 augmented by the statement "all of BC is used for this entire definition"", which makes SH2 ~ (f[sd2->/d5/]<->f[sd2->/d5/])->GF

What's going on here? If you looked at the analysis you probably already know, but what's happening is that there are cheats added at SH2 which help SH1's cheats, but the newly added cheats are getting helped by the cheats that they themselves helped, which creates an infinite feedback loop of making the cheats more powerful. It doesn't end there though, this isn't the conventional "more powerful" that you may be thinking of, these get more powerful by including more, different, undiscovered cheats. I do think it is important to note that f[sd2->/d5/]<->(f[sd2->/d5/]<->f[sd2->/d5/]) is not a naive extension of f[sd2->/d5/]<->f[sd2->/d5/] for the reason above.

SHn, n>1, is defined as "SHn-1 augmented by the cheats in the definition of SHn-1", this in essence doubles the amount of cheats, but note that this cannot be represented as c+c where c is the cheats, in HGAL. Rather it is represented as c<->c.

In SHn, if n-1 does not exist, then just do the supremum of {SH0,SH1,.. to SHn exclusive}

## Lirem

Lirem (L[]) is simply defined as a number beyond all outputs to the function f(x) n -> SHn. This makes Lirem approximately f[<->,f[sd2->/d5/]], where f[a,b] is the strongest combination of any (possibly infinite) combination of a and b. In this case, the only possible combination is f[sd2->/d5/]<->f[sd2->/d5/]<->f[sd2->/d5/]<->..<->f[sd2->/d5/], so it will just be that with very many <->'s.

## Naivety

I talked a little about this previously, but here I want to dive into how "naive" the extensions may be. I will be using HGAL since it is much faster to write out than writing the definitions themselves. Let's look at the jump from f[sd2->/d5/] to f[sd2->/d5/]->f[sd2->/d5/]... this includes cheats that cheats in BC to make themselves more powerful, call those cheats BC2. f[sd2->/d5/]->(f[sd2->/d5/]->f[sd2->/d5/]) allows cheats that use cheats in BC2 to make themselves more powerful. From now on I will be writing f[sd2->/d5/] as CA. CA<->CA >>>>...>>> f[->,CA] >>>>>>>...>>> CA->CA->CA->..->CA with w CA's. This is because CA<->CA will start off with CA->CA, next it will become CA<-(CA->CA), then (CA<-(CA->CA)) -> (CA->CA), and it will continue to expand similarly, until we reach a fixed point of a->b, although we can still get further with -> using cheats, despite it being a fixed point. CA->(CA<->CA) can even be regarded as a non-naive extension, keyword "can". It's already taken long enough to get here, so I'll let you imagine things like : CA<->(CA<->CA), and that's not even at SH3. So, we so far have the series A0, Reduk, (CA<->CA)->GF, (CA<->CA)<->(CA<->CA)->GF,... Judging by the jump from A0 to Reduk, this is a fairly powerful series. Something to note is that L[] is greater than even SHL[].