Project-declaredLean 4.33.0-rc1 · mathlib@79d0395a1825

Goursat

goursat

additive combinatorics entropy probability hom pfr

Project documentation

Let $H$ be a subgroup of $G \times G'$ . Then there exists a subgroup $H_0$ of $G$ , a subgroup $H_1$ of $G'$ , and a homomorphism $\phi: G \to G'$ such that $H := \{ (x, \phi(x) + y): x \in H_0, y \in H_1 \}.$ In particular, $|H| = |H_0| |H_1|$ .

Exact Lean statement

lemma goursat (H : Submodule (ZMod 2) (G × G')) :
    ∃ (H₀ : Submodule (ZMod 2) G) (H₁ : Submodule (ZMod 2) G') (φ : G →+ G'),
      (∀ x : G × G', x ∈ H ↔ (x.1 ∈ H₀ ∧ x.2 - φ x.1 ∈ H₁)) ∧
        Nat.card H = Nat.card H₀ * Nat.card H₁

Formal artifact

Lean source

Canonical source

Full Lean sourceLean 4

lemma goursat (H : Submodule (ZMod 2) (G × G')) :    ∃ (H₀ : Submodule (ZMod 2) G) (H₁ : Submodule (ZMod 2) G') (φ : G →+ G'),      (∀ x : G × G', x ∈ H ↔ (x.1 ∈ H₀ ∧ x.2 - φ x.1 ∈ H₁)) ∧        Nat.card H = Nat.card H₀ * Nat.card H₁ := by  obtain ⟨S₁, S₂, f, φ, hf, hf_inv⟩ := H.exists_equiv_fst_sndModFst  use S₁, S₂, φ  constructor ; swap  · show Nat.card H = _    exact Eq.trans (Nat.card_eq_of_bijective f f.bijective) (Nat.card_prod S₁ S₂)  · intro x    · constructor      · intro hx        let x : H := { val := x, property := hx }        · constructor          · exact Set.mem_of_eq_of_mem (hf x).1.symm (f x).1.property          · exact Set.mem_of_eq_of_mem (hf x).2.symm (f x).2.property      · intro hx        · let x₁ : S₁ := { val := x.1, property := hx.1 }          let x₂ : S₂ := { val := x.2 - φ x.1, property := hx.2 }          exact Set.mem_of_eq_of_mem (by rw [hf_inv, sub_add_cancel]) (f.symm (x₁, x₂)).property

Project: Polynomial Freiman-Ruzsa project
License: Apache-2.0
Commit: a177b2e4abe4
Source: PFR/HomPFR.lean:46-65

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Related declarations

Project-declaredLean 4.33.0-rc1

Approx hom pfr

approx_hom_pfr

Project documentation

An approximate-homomorphism theorem for finite elementary abelian $2$ -groups. Let $f:G\to G'$ and $K>0$ . If at least a proportion $K^{-1}$ of pairs $(x,y)\in G^2$ satisfy $f(x+y)=f(x)+f(y)$ , then there are an additive homomorphism $\varphi:G\to G'$ and a constant $c\in G'$ such that $f(x)=\varphi(x)+c$ for at least $|G|/(2^{144}K^{122})$ values of $x$ .

additive combinatoricsentropyprobability

Source project: Polynomial Freiman-Ruzsa project

Person-level attribution pending.

View proof record

Project-declaredLean 4.33.0-rc1

Better PFR conjecture

better_PFR_conjecture

Plain-language statement

If $A \subset {\bf F}_2^n$ is finite non-empty with $|A+A| \leq K|A|$ , then there exists a subgroup $H$ of ${\bf F}_2^n$ with $|H| \leq |A|$ such that $A$ can be covered by at most $2K^9$ translates of $H$ .

additive combinatoricsentropyprobability

Source project: Polynomial Freiman-Ruzsa project

Person-level attribution pending.

View proof record

Project-declaredLean 4.33.0-rc1

Better PFR conjecture

better_PFR_conjecture'

Project documentation

Polynomial Freiman-Ruzsa theorem with exponent $9$ , without a finite ambient-group assumption. Let $A$ be a nonempty finite subset of an elementary abelian $2$ -group. If $|A+A|\le K|A|$ , then there are a finite subspace $H$ and a finite set $c$ such that $A\subseteq c+H$ , $|c|<2K^9$ , and $|H|\le|A|$ .

additive combinatoricsentropyprobability

Source project: Polynomial Freiman-Ruzsa project

Person-level attribution pending.

View proof record