When I was working on proving the Bruck Ryser Chowla theorem, I stumbled upon this false theorem:
theory Bruck_Ryser_Chowla_Improved imports
Fishers_Inequality.Fishers_Inequality
begin
sublocale ordered_bibd โ ordered_regular_pairwise_balance ๐ฑs โฌs ฮ ๐
by unfold_locales
context ordered_sym_bibd
begin
lemma brc_v_1_mod_4:
assumes "๐ = 1 mod 4"
shows "(1 - ฮ) * ๐^2 = (๐ - ฮ)"
proof -
have "(1 - ฮ) * ๐^2 = (๐ - ฮ)"
using assms t_design_min_v by linarith
then show ?thesis by blast
qed
end
According my understanding of the Fisher's Inequality library, this means that if there is a symmetric block design (v,k,ฮ) where v = 1 mod 4, the equation (1 - ฮ) * ๐^2 = (๐ - ฮ) must hold true. However, the projective plane of order 3 exists, where (v,k,ฮ)=(13,4,1), yet the equation (1 - 1) * 13^2 = (4 - 1) is false. What is going on?
Interestingly if one assumes that v=3 mod 4, Isabelle will not accept the proof.
I think you want to assume v mod 4 = 1
, here you're assuming that v=1
Which gives you the obvious contradiction
Thank you! Glad to know Isabelle is right. That really threw me off.
Last updated: Apr 03 2025 at 20:22 UTC