User talk:Username5243/BMS vs UNOCF

Proof of some stuff :
I'll prove that PrSS (Primitive Sequence System), a.k.a 1-row BMS behaves like Username describes it. That is, given two matrices X and Y, each one being possibly empty, and assuming standardness of the following expression, X(n)Y(n+1)[m] expands to X(n)Y...(n)Y, where "(n)Y" is repeated m times.

First, we need to assume that BM2.3 and Nish's ruleset behave the same way in PrSS

Theorem 1 : Delta is always equal to (0)

Proof: Any PrSS element is of the form (n). The case for n=0 is trivial. Delta is of the form (d1,...,d(x-1),0,..,0), where x is the index of the last nonzero entry of the rightmost element. Thus, Delta = (0) for any rightmost element (n). 

Theorem 2 : If the rightmost element is (n+1), then the bad root is the rightmost occurence of (n).

Proof : Given some rightmost element (m+1), the sequence reduction algorithm discards any element (a) iff a >= m+1. Thus, the rightmost occurence of (m) is the rightmost element that is not discarded, which makes it the second-to-last non-discarded element (the last being the rightmost element). Hence, the bad root is guaranteed to be the rightmost occurence of (m).

Theorem 3 : If a matrix M is equal to (a_0)(a_1)...(a_i)...(a_(n-1))(a_n), where i is the smallest i such that a_i < a_n, then M[x] expands to (a_0)(a_1)...(a_i)...(a_(n-1))...(a_i)...(a_(n-1)), where "(a_i)...(a_(n-1))" is repeated x times

Proof : By Theorem 2, (a_i) is the bad root, and by definition, "(a_i)...(a_(n-1))" is the bad part. Since delta is always (0) by Theorem 1, nothing is added to the elements of the bad part. If we let B be the bad part and G the good part (everything to the left of the bad part), then M[x] = G+B+...+B, where + is sequence concatenation, and where B is concatenated x times by definition.

(This is probably overly formal. Or for all I know this is actually too informal, fuck anything I say. This is my first time actually writing a proof, so please give me tips to improve)

Syst3ms (debate about whether funky kong is god incarnate) 17:51, August 22, 2018 (UTC)