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From: Pierre Abbat <phma@webjockey.net>
To: lojban-list@lojban.org
Subject: [lojban] Lemma and conjecture
Date: Thu, 19 Dec 2002 06:52:50 -0500
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Lemma: All brivla contain a consonant followed two letters later, ignoring
apostrophes, by a vowel. If a brivla contains 'y', it contains such a
consonant after the 'y'.

Proof: If a brivla contains no 'y', it contains two adjacent consonants
in the first five letters, ignoring apostrophes. Find the first vowel
after this consonant cluster. Two letters before it is a consonant.
If a brivla contains 'y', there is at least one rafsi after 'y'. Consider
the last rafsi. Either it is a CVV or CCV rafsi of a gismu, in which case
the first and last letters of the rafsi are the consonant and the vowel
two letters later, or it is the final long rafsi of a gismu or fu'ivla,
in which case, being identical to the selrafsi, it has such a consonant
by the first part.

Conjecture: If two lerpoi R and S which both lack 'y' are such that for all i
R[i:i+1] is a valid initial consonant pair, valid consonant pair, valid lujvo
diphthong, fa'u valid fu'ivla diphthong iff S[i:i+1], and R[i] is a vowel,
consonant, fa'u y'ybu iff S[i] is, then R is a valid brivla iff S is,
regardless of whether for some i R[i] is 'n', 'r', or 'l' and S[i] isn't.

I am trying to prove the conjecture, but having trouble with words like 
{paske'usazri}, where removing the first rafsi results in {ke'usazri}, which 
falls apart, while {ke'ursazri}, which is a lujvo, when {pas-} is added 
becomes {paske'ursazri}, which falls apart.

phma