A unique deletion when the extra label is opposite to a target label

Proved

ProofsInTheBook.Chapter39.sigmaDoorSet_card_opposite_of_door

by xiangyazi24 · Sep 12, 2026 · Mathlib c5ea003 (Lean v4.30.0)

auxiliary-lemmabook-chapter-43combinatoricsgraph-theorylean4proofs-from-the-book

Write $[a]=\{0,\ldots,a-1\}$ for $a\in\mathbb N$ (empty when $a=0$ ). Let $d\in\mathbb N$ , let $L:[d+1]\to\{+,-\}\times[d]$ , and put $a_j=((-1)^j,j)$ and $A_d=\{a_j:j\in[d]\}$ . For $i\in[d+1]$ write $H(i)$ for $\forall j\in[d],\ \exists t\in[d+1]\setminus\{i\},\ L(t)=a_j$ , and put $D_A(L)=\{i:H(i)\}$ . Let $e\in[d+1]$ and $j\in[d]$ . If $H(e)$ and $L(e)=-a_j$ , then

|D_A(L)|=1.

Preamble

import Init
import Mathlib
import Mathlib.Data.Fin.Tuple.Sort
import Definitions.Def_P2MAssembly_Chapter39
set_option autoImplicit true
open ProofsInTheBook.Chapter39
open SignedPermutation

Formal statement

theorem ProofsInTheBook.Chapter39.sigmaDoorSet_card_opposite_of_door {d : ℕ}
    {sigmaLabel : Fin (d + 1) → SignedLabel d} {extra : Fin (d + 1)} {k : Fin d}
    (hdoorExtra : SigmaDeletionHasAlternatingLabelSet sigmaLabel extra)
    (hextra : sigmaLabel extra = (alternatingLabel k).neg) :
    (sigmaDoorSet sigmaLabel).card = 1 := by sorry

Source

Original formalization: https://github.com/xiangyazi24/proof_in_the_book/blob/88d88d141768cded75e782c525ef1bf04b8fe220/ProofsInTheBook/Chapter39Tucker.lean#L2593. Topic: Aigner and Ziegler, Proofs from THE BOOK, 6th edition, Chapter 43, “The chromatic number of Kneser graphs”, pp. 301–305 (https://doi.org/10.1007/978-3-662-57265-8_43).

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