project stringclasses 1
value | name stringlengths 7 39 | informal stringlengths 127 4.37k | formal stringlengths 82 14.6k | informal_source stringlengths 36 53 | formal_source stringlengths 21 46 |
|---|---|---|---|---|---|
PFR | I_one_le | \begin{lemma}\label{phi-first-estimate}\lean{I_one_le}\leanok
$I_1\le 2\eta d[X_1;X_2]$
\end{lemma}
\begin{proof}\leanok
\uses{phi-min-def,first-fibre}
Similar to \Cref{first-estimate}: get upper bounds for $d[X_1;X_2]$ by $\phi[X_1;X_2]\le \phi[X_1+X_2;\tilde X_1+\tilde X_2]$ and $\phi[X_1;X_2]\le \phi[X_1|X_1+X_2;\... | /-- $I_1\le 2\eta d[X_1;X_2]$ -/
lemma I_one_le (hA : A.Nonempty) : I₁ ≤ 2 * η * d[ X₁ # X₂ ] := by
have : d[X₁ + X₂' # X₂ + X₁'] + d[X₁ | X₁ + X₂' # X₂ | X₂ + X₁'] + I₁ = 2 * k :=
rdist_add_rdist_add_condMutual_eq _ _ _ _ hX₁ hX₂ hX₁' hX₂' h₁ h₂ h_indep.reindex_four_abdc
have : k - η * (ρ[X₁ | X₁ + X₂' # A] - ... | pfr/blueprint/src/chapter/further_improvement.tex:227 | pfr/PFR/RhoFunctional.lean:1294 |
PFR | I_two_le | \begin{lemma}\label{phi-second-estimate}\lean{I_two_le}\leanok
$I_2\le 2\eta d[X_1;X_2] + \frac{\eta}{1-\eta}(2\eta d[X_1;X_2]-I_1)$.
\end{lemma}
\begin{proof}\leanok
\uses{phi-min-def,cor-fibre,I1-I2-diff}
First of all, by $\phi[X_1;X_2]\le \phi[X_1+\tilde X_1;X_2+\tilde X_2]$, $\phi[X_1;X_2]\le \phi[X_1|X_1+\tilde ... | /-- $I_2\le 2\eta d[X_1;X_2] + \frac{\eta}{1-\eta}(2\eta d[X_1;X_2]-I_1)$. -/
lemma I_two_le (hA : A.Nonempty) (h'η : η < 1) :
I₂ ≤ 2 * η * k + (η / (1 - η)) * (2 * η * k - I₁) := by
have W : k - η * (ρ[X₁ + X₁' # A] - ρ[X₁ # A]) - η * (ρ[X₂ + X₂' # A] - ρ[X₂ # A]) ≤
d[X₁ + X₁' # X₂ + X₂'] :=
le_rdist_o... | pfr/blueprint/src/chapter/further_improvement.tex:244 | pfr/PFR/RhoFunctional.lean:1407 |
PFR | KLDiv_add_le_KLDiv_of_indep | \begin{lemma}[Kullback--Leibler and sums]\label{kl-sums}\lean{KLDiv_add_le_KLDiv_of_indep}\leanok If $X, Y, Z$ are independent $G$-valued random variables, then
$$D_{KL}(X+Z\Vert Y+Z) \leq D_{KL}(X\Vert Y).$$
\end{lemma}
\begin{proof}\leanok
\uses{kl-div-inj,kl-div-convex}
For each $z$, $D_{KL}(X+z\Vert Y+z)=D_{KL}... | lemma KLDiv_add_le_KLDiv_of_indep [Fintype G] [AddCommGroup G] [DiscreteMeasurableSpace G]
{X Y Z : Ω → G} [IsZeroOrProbabilityMeasure μ]
(h_indep : IndepFun (⟨X, Y⟩) Z μ)
(hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z)
(habs : ∀ x, μ.map Y {x} = 0 → μ.map X {x} = 0) :
KL[X + Z ; μ # Y +... | pfr/blueprint/src/chapter/further_improvement.tex:51 | pfr/PFR/Kullback.lean:265 |
PFR | KLDiv_eq_zero_iff_identDistrib | \begin{lemma}[Converse Gibbs inequality]\label{Gibbs-converse}\lean{KLDiv_eq_zero_iff_identDistrib}\leanok If $D_{KL}(X\Vert Y) = 0$, then $Y$ is a copy of $X$.
\end{lemma}
\begin{proof}\leanok
\uses{converse-log-sum}
Apply \Cref{converse-log-sum}.
\end{proof} | /-- `KL(X ‖ Y) = 0` if and only if `Y` is a copy of `X`. -/
lemma KLDiv_eq_zero_iff_identDistrib [Fintype G] [MeasurableSingletonClass G]
[IsProbabilityMeasure μ] [IsProbabilityMeasure μ'] (hX : Measurable X) (hY : Measurable Y)
(habs : ∀ x, μ'.map Y {x} = 0 → μ.map X {x} = 0) :
KL[X ; μ # Y ; μ'] = 0 ↔ Ide... | pfr/blueprint/src/chapter/further_improvement.tex:25 | pfr/PFR/Kullback.lean:89 |
PFR | KLDiv_nonneg | \begin{lemma}[Gibbs inequality]\label{Gibbs}\uses{kl-div}\lean{KLDiv_nonneg}\leanok $D_{KL}(X\Vert Y) \geq 0$.
\end{lemma}
\begin{proof}\leanok
\uses{log-sum}
Apply \Cref{log-sum} on the definition.
\end{proof} | /-- `KL(X ‖ Y) ≥ 0`.-/
lemma KLDiv_nonneg [Fintype G] [MeasurableSingletonClass G] [IsZeroOrProbabilityMeasure μ]
[IsZeroOrProbabilityMeasure μ'] (hX : Measurable X) (hY : Measurable Y)
(habs : ∀ x, μ'.map Y {x} = 0 → μ.map X {x} = 0) : 0 ≤ KL[X ; μ # Y ; μ'] := by
rw [KLDiv_eq_sum]
rcases eq_zero_or_isProb... | pfr/blueprint/src/chapter/further_improvement.tex:17 | pfr/PFR/Kullback.lean:71 |
PFR | KLDiv_of_comp_inj | \begin{lemma}[Kullback--Leibler and injections]\label{kl-div-inj}\lean{KLDiv_of_comp_inj}\leanok If $f:G \to H$ is an injection, then $D_{KL}(f(X)\Vert f(Y)) = D_{KL}(X\Vert Y)$.
\end{lemma}
\begin{proof}\leanok\uses{kl-div} Clear from definition.
\end{proof} | /-- If $f:G \to H$ is an injection, then $D_{KL}(f(X)\Vert f(Y)) = D_{KL}(X\Vert Y)$. -/
lemma KLDiv_of_comp_inj {H : Type*} [MeasurableSpace H] [DiscreteMeasurableSpace G]
[MeasurableSingletonClass H] {f : G → H}
(hf : Function.Injective f) (hX : Measurable X) (hY : Measurable Y) :
KL[f ∘ X ; μ # f ∘ Y ; μ... | pfr/blueprint/src/chapter/further_improvement.tex:43 | pfr/PFR/Kullback.lean:150 |
PFR | KLDiv_of_convex | \begin{lemma}[Convexity of Kullback--Leibler]\label{kl-div-convex}\lean{KLDiv_of_convex}\leanok If $S$ is a finite set, $\sum_{s \in S} w_s = 1$ for some non-negative $w_s$, and ${\bf P}(X=x) = \sum_{s\in S} w_s {\bf P}(X_s=x)$, ${\bf P}(Y=x) = \sum_{s\in S} w_s {\bf P}(Y_s=x)$ for all $x$, then
$$D_{KL}(X\Vert Y) \... | lemma KLDiv_of_convex [Fintype G] [IsFiniteMeasure μ''']
{ι : Type*} {S : Finset ι} {w : ι → ℝ} (hw : ∀ s ∈ S, 0 ≤ w s)
(X' : ι → Ω'' → G) (Y' : ι → Ω''' → G)
(hconvex : ∀ x, (μ.map X {x}).toReal = ∑ s ∈ S, (w s) * (μ''.map (X' s) {x}).toReal)
(hconvex' : ∀ x, (μ'.map Y {x}).toReal = ∑ s ∈ S, (w s) * (μ... | pfr/blueprint/src/chapter/further_improvement.tex:33 | pfr/PFR/Kullback.lean:118 |
PFR | PFR_conjecture | \begin{theorem}[PFR]\label{pfr}
\lean{PFR_conjecture}\leanok
If $A \subset {\bf F}_2^n$ is non-empty and $|A+A| \leq K|A|$, then $A$ can be covered by most $2K^{12}$ translates of a subspace $H$ of ${\bf F}_2^n$ with $|H| \leq |A|$.
\end{theorem}
\begin{proof}
\uses{pfr_aux}\leanok
Let $H$ be given by \Cref{pfr_aux... | theorem PFR_conjecture (h₀A : A.Nonempty) (hA : Nat.card (A + A) ≤ K * Nat.card A) :
∃ (H : Submodule (ZMod 2) G) (c : Set G),
Nat.card c < 2 * K ^ 12 ∧ Nat.card H ≤ Nat.card A ∧ A ⊆ c + H := by
obtain ⟨A_pos, -, K_pos⟩ : (0 : ℝ) < Nat.card A ∧ (0 : ℝ) < Nat.card (A + A) ∧ 0 < K :=
PFR_conjecture_pos_a... | pfr/blueprint/src/chapter/pfr.tex:50 | pfr/PFR/Main.lean:276 |
PFR | PFR_conjecture' | \begin{corollary}[PFR in infinite groups]\label{pfr-cor}
\lean{PFR_conjecture'}\leanok
If $G$ is an abelian $2$-torsion group, $A \subset G$ is non-empty finite, and $|A+A| \leq K|A|
$, then $A$ can be covered by most $2K^{12}$ translates of a finite group $H$ of $G$ with $|H| \leq |A|$.
\end{corollary}
\begin... | theorem PFR_conjecture' {G : Type*} [AddCommGroup G] [Module (ZMod 2) G]
{A : Set G} {K : ℝ} (h₀A : A.Nonempty) (Afin : A.Finite)
(hA : Nat.card (A + A) ≤ K * Nat.card A) :
∃ (H : Submodule (ZMod 2) G) (c : Set G), c.Finite ∧ (H : Set G).Finite ∧
Nat.card c < 2 * K ^ 12 ∧ Nat.card H ≤ Nat.card A ∧ A ⊆... | pfr/blueprint/src/chapter/pfr.tex:63 | pfr/PFR/Main.lean:335 |
PFR | PFR_conjecture_aux | \begin{lemma}\label{pfr_aux}
\lean{PFR_conjecture_aux}\leanok If $A \subset {\bf F}_2^n$ is non-empty and
$|A+A| \leq K|A|$, then $A$ can be covered by at most $K ^
{13/2}|A|^{1/2}/|H|^{1/2}$ translates of a subspace $H$ of ${\bf F}_2^n$ with
\begin{equation}
\label{ah}
|H|/|A| \in [K^{-11}, K^{11}].
... | lemma PFR_conjecture_aux (h₀A : A.Nonempty) (hA : Nat.card (A + A) ≤ K * Nat.card A) :
∃ (H : Submodule (ZMod 2) G) (c : Set G),
Nat.card c ≤ K ^ (13/2 : ℝ) * Nat.card A ^ (1/2 : ℝ) * Nat.card H ^ (-1/2 : ℝ)
∧ Nat.card H ≤ K ^ 11 * Nat.card A ∧ Nat.card A ≤ K ^ 11 * Nat.card H ∧ A ⊆ c + H := by
classica... | pfr/blueprint/src/chapter/pfr.tex:14 | pfr/PFR/Main.lean:163 |
PFR | PFR_conjecture_improv | \begin{theorem}[Improved PFR]\label{pfr-improv}\lean{PFR_conjecture_improv}\leanok
If $A \subset {\bf F}_2^n$ is non-empty and $|A+A| \leq K|A|$, then $A$ can be covered by most $2K^{11}$ translates of a subspace $H$ of ${\bf F}_2^n$ with $|H| \leq |A|$.
\end{theorem}
\begin{proof}\uses{pfr_aux-improv}\leanok
By repe... | theorem PFR_conjecture_improv (h₀A : A.Nonempty) (hA : Nat.card (A + A) ≤ K * Nat.card A) :
∃ (H : Submodule (ZMod 2) G) (c : Set G),
Nat.card c < 2 * K ^ 11 ∧ Nat.card H ≤ Nat.card A ∧ A ⊆ c + H := by
obtain ⟨A_pos, -, K_pos⟩ : (0 : ℝ) < Nat.card A ∧ (0 : ℝ) < Nat.card (A + A) ∧ 0 < K :=
PFR_conjectur... | pfr/blueprint/src/chapter/improved_exponent.tex:229 | pfr/PFR/ImprovedPFR.lean:982 |
PFR | PFR_conjecture_improv_aux | \begin{lemma}\label{pfr_aux-improv}\lean{PFR_conjecture_improv_aux}\leanok
If $A \subset {\bf F}_2^n$ is non-empty and
$|A+A| \leq K|A|$, then $A$ can be covered by at most $K^6 |A|^{1/2}/|H|^{1/2}$ translates of a subspace $H$ of ${\bf F}_2^n$ with
$$
|H|/|A| \in [K^{-10}, K^{10}].
$$
\end{lemma}
\begin{proof}\uses{... | lemma PFR_conjecture_improv_aux (h₀A : A.Nonempty) (hA : Nat.card (A + A) ≤ K * Nat.card A) :
∃ (H : Submodule (ZMod 2) G) (c : Set G),
Nat.card c ≤ K ^ 6 * Nat.card A ^ (1/2) * Nat.card H ^ (-1/2)
∧ Nat.card H ≤ K ^ 10 * Nat.card A ∧ Nat.card A ≤ K ^ 10 * Nat.card H ∧ A ⊆ c + H := by
have A_fin : Finit... | pfr/blueprint/src/chapter/improved_exponent.tex:214 | pfr/PFR/ImprovedPFR.lean:864 |
PFR | PFR_projection | \begin{lemma}\label{pfr-projection}\lean{PFR_projection}\leanok
If $G=\mathbb{F}_2^d$ and $\alpha\in (0,1)$ and $X,Y$ are $G$-valued random
variables then there is a subgroup $H\leq \mathbb{F}_2^d$ such that
\[\log \lvert H\rvert \leq 2 (\mathbb{H}(X)+\mathbb{H}(Y))\]
and if $\psi:G \to G/H$ is the natural projection t... | lemma PFR_projection (hX : Measurable X) (hY : Measurable Y) :
∃ H : Submodule (ZMod 2) G, log (Nat.card H) ≤ 2 * (H[X ; μ] + H[Y;μ']) ∧
H[H.mkQ ∘ X ; μ] + H[H.mkQ ∘ Y; μ'] ≤
34 * d[H.mkQ ∘ X ;μ # H.mkQ ∘ Y;μ'] := by
rcases PFR_projection' X Y μ μ' ((3 : ℝ) / 5) hX hY (by norm_num) (by norm_num) with ⟨H... | pfr/blueprint/src/chapter/weak_pfr.tex:127 | pfr/PFR/WeakPFR.lean:397 |
PFR | PFR_projection' | \begin{lemma}\label{pfr-projection'}\lean{PFR_projection'}\leanok
If $G=\mathbb{F}_2^d$ and $\alpha\in (0,1)$ and $X,Y$ are $G$-valued random
variables then there is a subgroup $H\leq \mathbb{F}_2^d$ such that
\[\log \lvert H\rvert \leq \frac{1+\alpha}{2(1-\alpha)} (\mathbb{H}(X)+\mathbb{H}(Y))\]
and if $\psi:G \to G/H... | lemma PFR_projection'
(α : ℝ) (hX : Measurable X) (hY : Measurable Y) (αpos : 0 < α) (αone : α < 1) :
∃ H : Submodule (ZMod 2) G, log (Nat.card H) ≤ (1 + α) / (2 * (1 - α)) * (H[X ; μ] + H[Y ; μ']) ∧
α * (H[H.mkQ ∘ X ; μ] + H[H.mkQ ∘ Y ; μ']) ≤
20 * d[H.mkQ ∘ X ; μ # H.mkQ ∘ Y ; μ'] := by
let S := {... | pfr/blueprint/src/chapter/weak_pfr.tex:86 | pfr/PFR/WeakPFR.lean:300 |
PFR | ProbabilityTheory.IdentDistrib.rdist_eq | \begin{lemma}[Copy preserves Ruzsa distance]\label{ruz-copy}
\uses{ruz-dist-def}
\lean{ProbabilityTheory.IdentDistrib.rdist_eq}\leanok
If $X',Y'$ are copies of $X,Y$ respectively then $d[X';Y']=d[X ;Y]$.
\end{lemma}
\begin{proof} \uses{copy-ent}\leanok Immediate from Definitions \ref{ruz-dist-def} and \Cref{copy-... | /-- If `X', Y'` are copies of `X, Y` respectively then `d[X' ; Y'] = d[X ; Y]`. -/
lemma ProbabilityTheory.IdentDistrib.rdist_eq {X' : Ω'' → G} {Y' : Ω''' → G}
(hX : IdentDistrib X X' μ μ'') (hY : IdentDistrib Y Y' μ' μ''') :
d[X ; μ # Y ; μ'] = d[X' ; μ'' # Y' ; μ'''] := by
simp [rdist, hX.map_eq, hY.map_eq,... | pfr/blueprint/src/chapter/distance.tex:99 | pfr/PFR/ForMathlib/Entropy/RuzsaDist.lean:129 |
PFR | ProbabilityTheory.IdentDistrib.tau_eq | \begin{lemma}[$\tau$ depends only on distribution]\label{tau-copy}\leanok
\uses{tau-def}
\lean{ProbabilityTheory.IdentDistrib.tau_eq} If $X'_1, X'_2$ are copies of $X_1,X_2$, then $\tau[X'_1;X'_2] = \tau[X_1;X_2]$.
\end{lemma}
\begin{proof}\uses{copy-ent}\leanok Immediate from \Cref{copy-ent}.
\end{proof} | /-- If $X'_1, X'_2$ are copies of $X_1,X_2$, then $\tau[X'_1;X'_2] = \tau[X_1;X_2]$. -/
lemma ProbabilityTheory.IdentDistrib.tau_eq [MeasurableSpace Ω₁] [MeasurableSpace Ω₂]
[MeasurableSpace Ω'₁] [MeasurableSpace Ω'₂]
{μ₁ : Measure Ω₁} {μ₂ : Measure Ω₂} {μ'₁ : Measure Ω'₁} {μ'₂ : Measure Ω'₂}
{X₁ : Ω₁ → G} ... | pfr/blueprint/src/chapter/entropy_pfr.tex:17 | pfr/PFR/TauFunctional.lean:90 |
PFR | ProbabilityTheory.IndepFun.rdist_eq | \begin{lemma}[Ruzsa distance in independent case]\label{ruz-indep}
\uses{ruz-dist-def}
\lean{ProbabilityTheory.IndepFun.rdist_eq}\leanok
If $X,Y$ are independent $G$-random variables then
$$ d[X ;Y] := \bbH[X - Y] - \bbH[X]/2 - \bbH[Y]/2.$$
\end{lemma}
\begin{proof} \uses{relabeled-entropy, copy-ent}\leanok Imm... | /-- If `X, Y` are independent `G`-random variables then `d[X ; Y] = H[X - Y] - H[X]/2 - H[Y]/2`. -/
lemma ProbabilityTheory.IndepFun.rdist_eq [IsFiniteMeasure μ]
{Y : Ω → G} (h : IndepFun X Y μ) (hX : Measurable X) (hY : Measurable Y) :
d[X ; μ # Y ; μ] = H[X - Y ; μ] - H[X ; μ]/2 - H[Y ; μ]/2 := by
rw [rdist... | pfr/blueprint/src/chapter/distance.tex:108 | pfr/PFR/ForMathlib/Entropy/RuzsaDist.lean:161 |
PFR | app_ent_PFR | \begin{lemma}\label{app-ent-pfr}\lean{app_ent_PFR}\leanok
Let $G=\mathbb{F}_2^n$ and $\alpha\in (0,1)$ and let $X,Y$ be $G$-valued
random variables such that
\[\mathbb{H}(X)+\mathbb{H}(Y)> \frac{20}{\alpha} d[X;Y].\]
There is a non-trivial subgroup $H\leq G$ such that
\[\log \lvert H\rvert <\frac{1+\alpha}{2}(\mathbb{H... | lemma app_ent_PFR (α : ℝ) (hent : 20 * d[X ;μ # Y;μ'] < α * (H[X ; μ] + H[Y; μ'])) (hX : Measurable X)
(hY : Measurable Y) :
∃ H : Submodule (ZMod 2) G, log (Nat.card H) < (1 + α) / 2 * (H[X ; μ] + H[Y;μ']) ∧
H[H.mkQ ∘ X ; μ] + H[H.mkQ ∘ Y; μ'] < α * (H[ X ; μ] + H[Y; μ']) :=
app_ent_PFR' (mΩ := .mk μ) (m... | pfr/blueprint/src/chapter/weak_pfr.tex:52 | pfr/PFR/WeakPFR.lean:288 |
PFR | approx_hom_pfr | \begin{theorem}[Approximate homomorphism form of PFR]\label{approx-hom-pfr}\lean{approx_hom_pfr}\leanok Let $G,G'$ be finite abelian $2$-groups.
Let $f: G \to G'$ be a function, and suppose that there are at least $|G|^2 / K$ pairs $(x,y) \in G^2$ such that
$$ f(x+y) = f(x) + f(y).$$
Then there exists a homomorphism ... | theorem approx_hom_pfr (f : G → G') (K : ℝ) (hK : K > 0)
(hf : Nat.card G ^ 2 / K ≤ Nat.card {x : G × G | f (x.1 + x.2) = f x.1 + f x.2}) :
∃ (φ : G →+ G') (c : G'), Nat.card {x | f x = φ x + c} ≥ Nat.card G / (2 ^ 144 * K ^ 122) := by
let A := (Set.univ.graphOn f).toFinite.toFinset
have hA : #A = Nat.card ... | pfr/blueprint/src/chapter/approx_hom_pfr.tex:27 | pfr/PFR/ApproxHomPFR.lean:33 |
PFR | averaged_construct_good | \begin{lemma}[Constructing good variables, III']\label{averaged-construct-good}\lean{averaged_construct_good}\leanok
One has
\begin{align*} k & \leq I(U : V \, | \, S) + I(V : W \, | \,S) + I(W : U \, | \, S) + \frac{\eta}{6} \sum_{i=1}^2 \sum_{A,B \in \{U,V,W\}: A \neq B} (d[X^0_i;A|B,S] - d[X^0_i; X_i]).
\e... | lemma averaged_construct_good : k ≤ (I[U : V | S] + I[V : W | S] + I[W : U | S])
+ (p.η / 6) * (((d[p.X₀₁ # U | ⟨V, S⟩] - d[p.X₀₁ # X₁]) + (d[p.X₀₁ # U | ⟨W, S⟩] - d[p.X₀₁ # X₁])
+ (d[p.X₀₁ # V | ⟨U, S⟩] - d[p.X₀₁ # X₁]) + (d[p.X₀₁ # V | ⟨W, S⟩] - d[p.X₀₁ # X₁])
+ (d[p.X₀₁ # W | ... | pfr/blueprint/src/chapter/improved_exponent.tex:77 | pfr/PFR/ImprovedPFR.lean:436 |
PFR | better_PFR_conjecture | \begin{theorem}[PFR with \texorpdfstring{$C=9$}{C=9}]\label{pfr-9}\lean{better_PFR_conjecture}\leanok 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 $H$.
\end{theor... | lemma better_PFR_conjecture {A : Set G} (h₀A : A.Nonempty) {K : ℝ}
(hA : Nat.card (A + A) ≤ K * Nat.card A) :
∃ (H : Submodule (ZMod 2) G) (c : Set G),
Nat.card c < 2 * K ^ 9 ∧ Nat.card H ≤ Nat.card A ∧ A ⊆ c + H := by
obtain ⟨A_pos, -, K_pos⟩ : (0 : ℝ) < Nat.card A ∧ (0 : ℝ) < Nat.card (A + A) ∧ 0 < K ... | pfr/blueprint/src/chapter/further_improvement.tex:371 | pfr/PFR/RhoFunctional.lean:2074 |
PFR | better_PFR_conjecture_aux | \begin{corollary}\label{pfr-9-aux'}\lean{better_PFR_conjecture_aux}\leanok
If $|A+A| \leq K|A|$, then there exist a subgroup $H$ and a subset $c$ of $G$
with $A \subseteq c + H$, such that $|c| \leq K^{5} |A|^{1/2}/|H|^{1/2}$ and
$|H|/|A|\in[K^{-8},K^8]$.
\end{corollary}
\begin{proof}\leanok
\uses{pfr-9-aux, ruz-cov}... | lemma better_PFR_conjecture_aux {A : Set G} (h₀A : A.Nonempty) {K : ℝ}
(hA : Nat.card (A + A) ≤ K * Nat.card A) :
∃ (H : Submodule (ZMod 2) G) (c : Set G),
Nat.card c ≤ K ^ 5 * Nat.card A ^ (1 / 2 : ℝ) * (Nat.card H : ℝ) ^ (-1 / 2 : ℝ)
∧ Nat.card H ≤ K ^ 8 * Nat.card A ∧ Nat.card A ≤ K ^ 8 * Nat.card ... | pfr/blueprint/src/chapter/further_improvement.tex:358 | pfr/PFR/RhoFunctional.lean:2028 |
PFR | better_PFR_conjecture_aux0 | \begin{corollary}\label{pfr-9-aux}\lean{better_PFR_conjecture_aux0}\leanok
If $|A+A| \leq K|A|$, then there exists a subgroup $H$ and $t\in G$ such that
$|A \cap (H+t)| \geq K^{-4} \sqrt{|A||H|}$, and $|H|/|A|\in[K^{-8},K^8]$.
\end{corollary}
\begin{proof}\leanok
\uses{pfr-rho,rho-init,rho-subgroup}
Apply \Cref{pfr... | lemma better_PFR_conjecture_aux0 {A : Set G} (h₀A : A.Nonempty) {K : ℝ}
(hA : Nat.card (A + A) ≤ K * Nat.card A) :
∃ (H : Submodule (ZMod 2) G) (t : G),
K ^ (-4 : ℤ) * Nat.card A ^ (1 / 2 : ℝ) * Nat.card H ^ (1 / 2 : ℝ) ≤ Nat.card ↑(A ∩ (H + {t})) ∧
Nat.card A ≤ K ^ 8 * Nat.card H ∧ Nat.card H ≤ K ^ 8... | pfr/blueprint/src/chapter/further_improvement.tex:347 | pfr/PFR/RhoFunctional.lean:1977 |
PFR | condKLDiv_eq | \begin{lemma}[Kullback--Leibler and conditioning]\label{kl-cond}\lean{condKLDiv_eq}\leanok If $X, Y$ are independent $G$-valued random variables, and $Z$ is another random variable defined on the same sample space as $X$, then
$$D_{KL}((X|Z)\Vert Y) = D_{KL}(X\Vert Y) + \bbH[X] - \bbH[X|Z].$$
\end{lemma}
\begin{proof... | lemma condKLDiv_eq {S : Type*} [MeasurableSpace S] [Fintype S] [MeasurableSingletonClass S]
[Fintype G] [IsZeroOrProbabilityMeasure μ] [IsFiniteMeasure μ']
{X : Ω → G} {Y : Ω' → G} {Z : Ω → S}
(hX : Measurable X) (hZ : Measurable Z)
(habs : ∀ x, μ'.map Y {x} = 0 → μ.map X {x} = 0) :
KL[ X | Z ; μ # ... | pfr/blueprint/src/chapter/further_improvement.tex:65 | pfr/PFR/Kullback.lean:332 |
PFR | condKLDiv_nonneg | \begin{lemma}[Conditional Gibbs inequality]\label{Conditional-Gibbs}\lean{condKLDiv_nonneg}\leanok $D_{KL}((X|W)\Vert Y) \geq 0$.
\end{lemma}
\begin{proof}\leanok \uses{Gibbs, ckl-div} Clear from Definition \ref{ckl-div} and Lemma \ref{Gibbs}.
\end{proof} | /-- `KL(X|Z ‖ Y) ≥ 0`.-/
lemma condKLDiv_nonneg {S : Type*} [MeasurableSingletonClass G] [Fintype G]
{X : Ω → G} {Y : Ω' → G} {Z : Ω → S}
[IsZeroOrProbabilityMeasure μ']
(hX : Measurable X) (hY : Measurable Y)
(habs : ∀ x, μ'.map Y {x} = 0 → μ.map X {x} = 0) :
0 ≤ KL[X | Z; μ # Y ; μ'] := by
rw [c... | pfr/blueprint/src/chapter/further_improvement.tex:73 | pfr/PFR/Kullback.lean:376 |
PFR | condMultiDist | \begin{definition}[Conditional multidistance]\label{cond-multidist-def}\uses{multidist-def}\lean{condMultiDist}
\leanok If $X_{[m]} = (X_i)_{1 \leq i \leq m}$ and $Y_{[m]} = (Y_i)_{1 \leq i \leq m}$ are tuples of random variables, with the $X_i$ being $G$-valued (but the $Y_i$ need not be), then we define
\begin{eq... | def condMultiDist {m : ℕ} {Ω : Fin m → Type*} (hΩ : ∀ i, MeasureSpace (Ω i)) {S : Type*} [Fintype S]
(X : ∀ i, (Ω i) → G) (Y : ∀ i, (Ω i) → S) : ℝ := ∑ ω : Fin m → S, (∏ i, ((hΩ i).volume ((Y i) ⁻¹' {ω i})).toReal) * D[X; fun i ↦ ⟨cond (hΩ i).volume (Y i ⁻¹' {ω i})⟩]
@[inherit_doc multiDist] notation3:max "D[" X "... | pfr/blueprint/src/chapter/torsion.tex:314 | pfr/PFR/MoreRuzsaDist.lean:862 |
PFR | condMultiDist_eq | \begin{lemma}[Alternate form of conditional multidistance]\label{cond-multidist-alt}\lean{condMultiDist_eq}\leanok
If the $(X_i,Y_i)$ are independent,
\begin{equation}\label{multi-def-cond}
D[ X_{[m]} | Y_{[m]}] := \bbH[\sum_{i=1}^m X_i \big| (Y_j)_{1 \leq j \leq m} ] - \frac{1}{m} \sum_{i=1}^m \bbH[ X_i | Y_i].
... | lemma condMultiDist_eq {m : ℕ}
{Ω : Type*} [hΩ : MeasureSpace Ω]
{S : Type*} [Fintype S] [hS : MeasurableSpace S] [MeasurableSingletonClass S]
{X : Fin m → Ω → G} (hX : ∀ i, Measurable (X i))
{Y : Fin m → Ω → S} (hY : ∀ i, Measurable (Y i))
(h_indep: iIndepFun (fun i ↦ ⟨X i, Y i⟩)) :
D[X | Y ; f... | pfr/blueprint/src/chapter/torsion.tex:322 | pfr/PFR/MoreRuzsaDist.lean:999 |
PFR | condMultiDist_nonneg | \begin{lemma}[Conditional multidistance nonnegative]\label{cond-multidist-nonneg}\uses{cond-multidist-def}\lean{condMultiDist_nonneg}\leanok If $X_{[m]} = (X_i)_{1 \leq i \leq m}$ and $Y_{[m]} = (Y_i)_{1 \leq i \leq m}$ are tuples of random variables, then $D[ X_{[m]} | Y_{[m]} ] \geq 0$.
\end{lemma}
\begin{proof}\uses... | /--Conditional multidistance is nonnegative. -/
theorem condMultiDist_nonneg [Fintype G] {m : ℕ} {Ω : Fin m → Type*} (hΩ : ∀ i, MeasureSpace (Ω i)) (hprob : ∀ i, IsProbabilityMeasure (ℙ : Measure (Ω i))) {S : Type*} [Fintype S] (X : ∀ i, (Ω i) → G) (Y : ∀ i, (Ω i) → S) (hX : ∀ i, Measurable (X i)) : 0 ≤ D[X | Y; hΩ] :=... | pfr/blueprint/src/chapter/torsion.tex:333 | pfr/PFR/MoreRuzsaDist.lean:921 |
PFR | condRhoMinus_le | \begin{lemma}[Rho and conditioning]\label{rho-cond}\lean{condRhoMinus_le, condRhoPlus_le, condRho_le}\leanok If $X,Z$ are defined on the same space, one has
$$ \rho^-(X|Z) \leq \rho^-(X) + \bbH[X] - \bbH[X|Z]$$
$$ \rho^+(X|Z) \leq \rho^+(X)$$
and
$$ \rho(X|Z) \leq \rho(X) + \frac{1}{2}( \bbH[X] - \bbH[X|Z] ).$$
... | /-- $$ \rho^-(X|Z) \leq \rho^-(X) + \bbH[X] - \bbH[X|Z]$$ -/
lemma condRhoMinus_le [IsZeroOrProbabilityMeasure μ] {S : Type*} [MeasurableSpace S]
[Fintype S] [MeasurableSingletonClass S]
{Z : Ω → S} (hX : Measurable X) (hZ : Measurable Z) (hA : A.Nonempty) :
ρ⁻[X | Z ; μ # A] ≤ ρ⁻[X ; μ # A] + H[X ; μ] - H[... | pfr/blueprint/src/chapter/further_improvement.tex:176 | pfr/PFR/RhoFunctional.lean:937 |
PFR | condRhoPlus_le | \begin{lemma}[Rho and conditioning]\label{rho-cond}\lean{condRhoMinus_le, condRhoPlus_le, condRho_le}\leanok If $X,Z$ are defined on the same space, one has
$$ \rho^-(X|Z) \leq \rho^-(X) + \bbH[X] - \bbH[X|Z]$$
$$ \rho^+(X|Z) \leq \rho^+(X)$$
and
$$ \rho(X|Z) \leq \rho(X) + \frac{1}{2}( \bbH[X] - \bbH[X|Z] ).$$
... | /-- $$ \rho^+(X|Z) \leq \rho^+(X)$$ -/
lemma condRhoPlus_le [IsProbabilityMeasure μ] {S : Type*} [MeasurableSpace S]
[Fintype S] [MeasurableSingletonClass S]
{Z : Ω → S} (hX : Measurable X) (hZ : Measurable Z) (hA : A.Nonempty) :
ρ⁺[X | Z ; μ # A] ≤ ρ⁺[X ; μ # A] := by
have : IsProbabilityMeasure (Measure... | pfr/blueprint/src/chapter/further_improvement.tex:176 | pfr/PFR/RhoFunctional.lean:964 |
PFR | condRho_le | \begin{lemma}[Rho and conditioning]\label{rho-cond}\lean{condRhoMinus_le, condRhoPlus_le, condRho_le}\leanok If $X,Z$ are defined on the same space, one has
$$ \rho^-(X|Z) \leq \rho^-(X) + \bbH[X] - \bbH[X|Z]$$
$$ \rho^+(X|Z) \leq \rho^+(X)$$
and
$$ \rho(X|Z) \leq \rho(X) + \frac{1}{2}( \bbH[X] - \bbH[X|Z] ).$$
... | /-- $$ \rho(X|Z) \leq \rho(X) + \frac{1}{2}( \bbH[X] - \bbH[X|Z] )$$ -/
lemma condRho_le [IsProbabilityMeasure μ] {S : Type*} [MeasurableSpace S]
[Fintype S] [MeasurableSingletonClass S]
{Z : Ω → S} (hX : Measurable X) (hZ : Measurable Z) (hA : A.Nonempty) :
ρ[X | Z ; μ # A] ≤ ρ[X ; μ # A] + (H[X ; μ] - H[X... | pfr/blueprint/src/chapter/further_improvement.tex:176 | pfr/PFR/RhoFunctional.lean:987 |
PFR | condRho_of_injective | \begin{lemma}[Conditional rho and relabeling]\label{rho-cond-relabeled}\lean{condRho_of_injective}\leanok
If $f$ is injective, then $\rho(X|f(Y))=\rho(X|Y)$.
\end{lemma}
\begin{proof}\leanok
\uses{rho-cond-def}
Clear from the definition.
\end{proof} | /-- If $f$ is injective, then $\rho(X|f(Y))=\rho(X|Y)$. -/
lemma condRho_of_injective {S T : Type*}
(Y : Ω → S) {A : Finset G} {f : S → T} (hf : Function.Injective f) :
ρ[X | f ∘ Y ; μ # A] = ρ[X | Y ; μ # A] := by
simp only [condRho]
rw [← hf.tsum_eq]
· have I c : f ∘ Y ⁻¹' {f c} = Y ⁻¹' {c} := by
... | pfr/blueprint/src/chapter/further_improvement.tex:168 | pfr/PFR/RhoFunctional.lean:895 |
PFR | condRho_of_sum_le | \begin{lemma}[Rho and conditioning, symmetrized]\label{rho-cond-sym}\lean{condRho_of_sum_le}\leanok
If $X,Y$ are independent, then
$$ \rho(X | X+Y) \leq \frac{1}{2}(\rho(X)+\rho(Y) + d[X;Y]).$$
\end{lemma}
\begin{proof}\leanok
\uses{rho-invariant,rho-cond}
First apply \Cref{rho-cond} to get $\rho(X|X+Y)\le \rho... | lemma condRho_of_sum_le [IsProbabilityMeasure μ]
(hX : Measurable X) (hY : Measurable Y) (hA : A.Nonempty) (h_indep : IndepFun X Y μ) :
ρ[X | X + Y ; μ # A] ≤ (ρ[X ; μ # A] + ρ[Y ; μ # A] + d[ X ; μ # Y ; μ ]) / 2 := by
have I : ρ[X | X + Y ; μ # A] ≤ ρ[X ; μ # A] + (H[X ; μ] - H[X | X + Y ; μ]) / 2 :=
co... | pfr/blueprint/src/chapter/further_improvement.tex:198 | pfr/PFR/RhoFunctional.lean:1075 |
PFR | condRho_of_translate | \begin{lemma}[Conditional rho and translation]\label{rho-cond-invariant}\lean{condRho_of_translate}\leanok
For any $s\in G$, $\rho(X+s|Y)=\rho(X|Y)$.
\end{lemma}
\begin{proof}
\uses{rho-cond-def,rho-invariant}\leanok
Direct corollary of \Cref{rho-invariant}.
\end{proof} | /-- For any $s\in G$, $\rho(X+s|Y)=\rho(X|Y)$. -/
lemma condRho_of_translate {S : Type*}
{Y : Ω → S} (hX : Measurable X) (hA : A.Nonempty) (s : G) :
ρ[fun ω ↦ X ω + s | Y ; μ # A] = ρ[X | Y ; μ # A] := by
simp [condRho, rho_of_translate hX hA]
omit [Fintype G] [DiscreteMeasurableSpace G] in
variable (X) in | pfr/blueprint/src/chapter/further_improvement.tex:160 | pfr/PFR/RhoFunctional.lean:887 |
PFR | condRho_sum_le | \begin{lemma}\label{rho-increase}\lean{condRho_sum_le}\leanok
For independent random variables $Y_1,Y_2,Y_3,Y_4$ over $G$, define $S:=Y_1+Y_2+Y_3+Y_4$, $T_1:=Y_1+Y_2$, $T_2:=Y_1+Y_3$. Then
$$\rho(T_1|T_2,S)+\rho(T_2|T_1,S) - \frac{1}{2}\sum_{i} \rho(Y_i)\le \frac{1}{2}(d[Y_1;Y_2]+d[Y_3;Y_4]+d[Y_1;Y_3]+d[Y_2;Y_4]).$... | lemma condRho_sum_le {Y₁ Y₂ Y₃ Y₄ : Ω → G}
(hY₁ : Measurable Y₁) (hY₂ : Measurable Y₂) (hY₃ : Measurable Y₃) (hY₄ : Measurable Y₄)
(h_indep : iIndepFun ![Y₁, Y₂, Y₃, Y₄]) (hA : A.Nonempty) :
ρ[Y₁ + Y₂ | ⟨Y₁ + Y₃, Y₁ + Y₂ + Y₃ + Y₄⟩ # A] + ρ[Y₁ + Y₃ | ⟨Y₁ + Y₂, Y₁ + Y₂ + Y₃ + Y₄⟩ # A] -
(ρ[Y₁ # A] + ρ[... | pfr/blueprint/src/chapter/further_improvement.tex:276 | pfr/PFR/RhoFunctional.lean:1710 |
PFR | condRho_sum_le' | \begin{lemma}\label{rho-increase-symmetrized}\lean{condRho_sum_le'}\leanok
For independent random variables $Y_1,Y_2,Y_3,Y_4$ over $G$, define $T_1:=Y_1+Y_2,T_2:=Y_1+Y_3,T_3:=Y_2+Y_3$ and $S:=Y_1+Y_2+Y_3+Y_4$. Then
$$\sum_{1 \leq i<j \leq 3} (\rho(T_i|T_j,S) + \rho(T_j|T_i,S) - \frac{1}{2}\sum_{i} \rho(Y_i))\le \s... | lemma condRho_sum_le' {Y₁ Y₂ Y₃ Y₄ : Ω → G}
(hY₁ : Measurable Y₁) (hY₂ : Measurable Y₂) (hY₃ : Measurable Y₃) (hY₄ : Measurable Y₄)
(h_indep : iIndepFun ![Y₁, Y₂, Y₃, Y₄]) (hA : A.Nonempty) :
let S := Y₁ + Y₂ + Y₃ + Y₄
let T₁ := Y₁ + Y₂
let T₂ := Y₁ + Y₃
let T₃ := Y₂ + Y₃
ρ[T₁ | ⟨T₂, S⟩ ... | pfr/blueprint/src/chapter/further_improvement.tex:306 | pfr/PFR/RhoFunctional.lean:1764 |
PFR | condRuzsaDist | \begin{definition}[Conditioned Ruzsa distance]\label{cond-dist-def}
\uses{ruz-dist-def}
\lean{condRuzsaDist}\leanok
If $(X, Z)$ and $(Y, W)$ are random variables (where $X$ and $Y$ are $G$-valued) we define
$$ d[X | Z; Y | W] := \sum_{z,w} \bbP[Z=z] \bbP[W=w] d[(X|Z=z); (Y|(W=w))].$$
similarly
$$ d[X ; Y | W] :=... | def condRuzsaDist (X : Ω → G) (Z : Ω → S) (Y : Ω' → G) (W : Ω' → T)
(μ : Measure Ω := by volume_tac) [IsFiniteMeasure μ]
(μ' : Measure Ω' := by volume_tac) [IsFiniteMeasure μ'] : ℝ :=
dk[condDistrib X Z μ ; μ.map Z # condDistrib Y W μ' ; μ'.map W]
@[inherit_doc condRuzsaDist]
notation3:max "d[" X " | " Z " ;... | pfr/blueprint/src/chapter/distance.tex:217 | pfr/PFR/ForMathlib/Entropy/RuzsaDist.lean:455 |
PFR | condRuzsaDist'_of_copy | \begin{lemma}[Alternate form of distance]\label{cond-dist-alt}
\lean{condRuzsaDist_of_copy, condRuzsaDist'_of_copy, condRuzsaDist_of_indep, condRuzsaDist'_of_indep}\leanok
The expression $d[X | Z;Y | W]$ is unchanged if $(X,Z)$ or $(Y,W)$ is replaced by a copy. Furthermore, if $(X,Z)$ and $(Y,W)$ are independent, t... | lemma condRuzsaDist'_of_copy (X : Ω → G) {Y : Ω' → G} (hY : Measurable Y)
{W : Ω' → T} (hW : Measurable W)
(X' : Ω'' → G) {Y' : Ω''' → G} (hY' : Measurable Y') {W' : Ω''' → T} (hW' : Measurable W')
[IsFiniteMeasure μ'] [IsFiniteMeasure μ''']
(h1 : IdentDistrib X X' μ μ'') (h2 : IdentDistrib (⟨Y, W⟩) (⟨Y... | pfr/blueprint/src/chapter/distance.tex:226 | pfr/PFR/ForMathlib/Entropy/RuzsaDist.lean:901 |
PFR | condRuzsaDist'_of_indep | \begin{lemma}[Alternate form of distance]\label{cond-dist-alt}
\lean{condRuzsaDist_of_copy, condRuzsaDist'_of_copy, condRuzsaDist_of_indep, condRuzsaDist'_of_indep}\leanok
The expression $d[X | Z;Y | W]$ is unchanged if $(X,Z)$ or $(Y,W)$ is replaced by a copy. Furthermore, if $(X,Z)$ and $(Y,W)$ are independent, t... | /-- Formula for conditional Ruzsa distance for independent sets of variables. -/
lemma condRuzsaDist'_of_indep {X : Ω → G} {Y : Ω → G} {W : Ω → T}
(hX : Measurable X) (hY : Measurable Y) (hW : Measurable W)
(μ : Measure Ω) [IsProbabilityMeasure μ]
(h : IndepFun X (⟨Y, W⟩) μ) [FiniteRange W] :
d[X ; μ # ... | pfr/blueprint/src/chapter/distance.tex:226 | pfr/PFR/ForMathlib/Entropy/RuzsaDist.lean:757 |
PFR | condRuzsaDist_diff_le | \begin{lemma}[Comparison of Ruzsa distances, I]\label{first-useful}
\lean{condRuzsaDist_diff_le, condRuzsaDist_diff_le', condRuzsaDist_diff_le'', condRuzsaDist_diff_le'''}\leanok
Let $X, Y, Z$ be random variables taking values in some abelian group of characteristic $2$, and with $Y, Z$ independent. Then we have
... | lemma condRuzsaDist_diff_le [IsProbabilityMeasure μ] [IsProbabilityMeasure μ']
{X : Ω → G} {Y : Ω' → G} {Z : Ω' → G}
(hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) (h : IndepFun Y Z μ')
[FiniteRange X] [FiniteRange Z] [FiniteRange Y] :
d[X ; μ # Y+ Z ; μ'] - d[X ; μ # Y ; μ'] ≤ (H[Y + Z; μ'... | pfr/blueprint/src/chapter/distance.tex:322 | pfr/PFR/ForMathlib/Entropy/RuzsaDist.lean:1386 |
PFR | condRuzsaDist_diff_le' | \begin{lemma}[Comparison of Ruzsa distances, I]\label{first-useful}
\lean{condRuzsaDist_diff_le, condRuzsaDist_diff_le', condRuzsaDist_diff_le'', condRuzsaDist_diff_le'''}\leanok
Let $X, Y, Z$ be random variables taking values in some abelian group of characteristic $2$, and with $Y, Z$ independent. Then we have
... | lemma condRuzsaDist_diff_le' [IsProbabilityMeasure μ] [IsProbabilityMeasure μ']
{X : Ω → G} {Y : Ω' → G} {Z : Ω' → G}
(hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) (h : IndepFun Y Z μ')
[FiniteRange X] [FiniteRange Z] [FiniteRange Y] :
d[X ; μ # Y + Z; μ'] - d[X ; μ # Y; μ'] ≤
d[Y; μ' ... | pfr/blueprint/src/chapter/distance.tex:322 | pfr/PFR/ForMathlib/Entropy/RuzsaDist.lean:1402 |
PFR | condRuzsaDist_diff_le'' | \begin{lemma}[Comparison of Ruzsa distances, I]\label{first-useful}
\lean{condRuzsaDist_diff_le, condRuzsaDist_diff_le', condRuzsaDist_diff_le'', condRuzsaDist_diff_le'''}\leanok
Let $X, Y, Z$ be random variables taking values in some abelian group of characteristic $2$, and with $Y, Z$ independent. Then we have
... | lemma condRuzsaDist_diff_le'' [IsProbabilityMeasure μ] [IsProbabilityMeasure μ']
{X : Ω → G} {Y : Ω' → G} {Z : Ω' → G}
(hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) (h : IndepFun Y Z μ')
[FiniteRange X] [FiniteRange Z] [FiniteRange Y] :
d[X ; μ # Y|Y+ Z ; μ'] - d[X ; μ # Y ; μ'] ≤ (H[Y+ Z ... | pfr/blueprint/src/chapter/distance.tex:322 | pfr/PFR/ForMathlib/Entropy/RuzsaDist.lean:1411 |
PFR | condRuzsaDist_diff_le''' | \begin{lemma}[Comparison of Ruzsa distances, I]\label{first-useful}
\lean{condRuzsaDist_diff_le, condRuzsaDist_diff_le', condRuzsaDist_diff_le'', condRuzsaDist_diff_le'''}\leanok
Let $X, Y, Z$ be random variables taking values in some abelian group of characteristic $2$, and with $Y, Z$ independent. Then we have
... | lemma condRuzsaDist_diff_le''' [IsProbabilityMeasure μ] [IsProbabilityMeasure μ']
{X : Ω → G} {Y : Ω' → G} {Z : Ω' → G}
(hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) (h : IndepFun Y Z μ')
[FiniteRange X] [FiniteRange Z] [FiniteRange Y] :
d[X ; μ # Y|Y+ Z ; μ'] - d[X ; μ # Y ; μ'] ≤
d[Y... | pfr/blueprint/src/chapter/distance.tex:322 | pfr/PFR/ForMathlib/Entropy/RuzsaDist.lean:1420 |
PFR | condRuzsaDist_diff_ofsum_le | \begin{lemma}[Comparison of Ruzsa distances, II]\label{second-useful}
\lean{condRuzsaDist_diff_ofsum_le}\leanok
Let $X, Y, Z, Z'$ be random variables taking values in some abelian group, and with $Y, Z, Z'$ independent. Then we have
\begin{align}\nonumber
& d[X ;Y + Z | Y + Z + Z'] - d[X ;Y] \\ & \qquad \leq \t... | lemma condRuzsaDist_diff_ofsum_le [IsProbabilityMeasure μ] [IsProbabilityMeasure μ']
{X : Ω → G} {Y Z Z' : Ω' → G}
(hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) (hZ' : Measurable Z')
(h : iIndepFun ![Y, Z, Z'] μ')
[FiniteRange X] [FiniteRange Z] [FiniteRange Y] [FiniteRange Z'] :
d[X ;... | pfr/blueprint/src/chapter/distance.tex:344 | pfr/PFR/ForMathlib/Entropy/RuzsaDist.lean:1429 |
PFR | condRuzsaDist_le | \begin{lemma}[Upper bound on conditioned Ruzsa distance]\label{cond-dist-fact}
\uses{cond-dist-def, information-def}
\lean{condRuzsaDist_le, condRuzsaDist_le'}\leanok
Suppose that $(X, Z)$ and $(Y, W)$ are random variables, where $X, Y$ take values in an abelian group. Then
\[ d[X | Z;Y | W] \leq d[X ; Y] + ... | lemma condRuzsaDist_le [Countable T] {X : Ω → G} {Z : Ω → S} {Y : Ω' → G} {W : Ω' → T}
[IsProbabilityMeasure μ] [IsProbabilityMeasure μ']
(hX : Measurable X) (hZ : Measurable Z) (hY : Measurable Y) (hW : Measurable W)
[FiniteRange X] [FiniteRange Z] [FiniteRange Y] [FiniteRange W] :
d[X | Z ; μ # Y|W ... | pfr/blueprint/src/chapter/distance.tex:302 | pfr/PFR/ForMathlib/Entropy/RuzsaDist.lean:1289 |
PFR | condRuzsaDist_le' | \begin{lemma}[Upper bound on conditioned Ruzsa distance]\label{cond-dist-fact}
\uses{cond-dist-def, information-def}
\lean{condRuzsaDist_le, condRuzsaDist_le'}\leanok
Suppose that $(X, Z)$ and $(Y, W)$ are random variables, where $X, Y$ take values in an abelian group. Then
\[ d[X | Z;Y | W] \leq d[X ; Y] + ... | lemma condRuzsaDist_le' [Countable T] {X : Ω → G} {Y : Ω' → G} {W : Ω' → T}
[IsProbabilityMeasure μ] [IsProbabilityMeasure μ']
(hX : Measurable X) (hY : Measurable Y) (hW : Measurable W)
[FiniteRange X] [FiniteRange Y] [FiniteRange W] :
d[X ; μ # Y|W ; μ'] ≤ d[X ; μ # Y ; μ'] + I[Y : W ; μ']/2 := by
r... | pfr/blueprint/src/chapter/distance.tex:302 | pfr/PFR/ForMathlib/Entropy/RuzsaDist.lean:1324 |
PFR | condRuzsaDist_of_copy | \begin{lemma}[Alternate form of distance]\label{cond-dist-alt}
\lean{condRuzsaDist_of_copy, condRuzsaDist'_of_copy, condRuzsaDist_of_indep, condRuzsaDist'_of_indep}\leanok
The expression $d[X | Z;Y | W]$ is unchanged if $(X,Z)$ or $(Y,W)$ is replaced by a copy. Furthermore, if $(X,Z)$ and $(Y,W)$ are independent, t... | lemma condRuzsaDist_of_copy {X : Ω → G} (hX : Measurable X) {Z : Ω → S} (hZ : Measurable Z)
{Y : Ω' → G} (hY : Measurable Y) {W : Ω' → T} (hW : Measurable W)
{X' : Ω'' → G} (hX' : Measurable X') {Z' : Ω'' → S} (hZ' : Measurable Z')
{Y' : Ω''' → G} (hY' : Measurable Y') {W' : Ω''' → T} (hW' : Measurable W')
... | pfr/blueprint/src/chapter/distance.tex:226 | pfr/PFR/ForMathlib/Entropy/RuzsaDist.lean:826 |
PFR | condRuzsaDist_of_indep | \begin{lemma}[Alternate form of distance]\label{cond-dist-alt}
\lean{condRuzsaDist_of_copy, condRuzsaDist'_of_copy, condRuzsaDist_of_indep, condRuzsaDist'_of_indep}\leanok
The expression $d[X | Z;Y | W]$ is unchanged if $(X,Z)$ or $(Y,W)$ is replaced by a copy. Furthermore, if $(X,Z)$ and $(Y,W)$ are independent, t... | lemma condRuzsaDist_of_indep
{X : Ω → G} {Z : Ω → S} {Y : Ω → G} {W : Ω → T}
(hX : Measurable X) (hZ : Measurable Z) (hY : Measurable Y) (hW : Measurable W)
(μ : Measure Ω) [IsProbabilityMeasure μ]
(h : IndepFun (⟨X, Z⟩) (⟨Y, W⟩) μ) [FiniteRange Z] [FiniteRange W] :
d[X | Z ; μ # Y | W ; μ] = H[X - ... | pfr/blueprint/src/chapter/distance.tex:226 | pfr/PFR/ForMathlib/Entropy/RuzsaDist.lean:729 |
PFR | condRuzsaDist_of_sums_ge | \begin{lemma}[Lower bound on conditional distances]\label{first-cond}
\lean{condRuzsaDist_of_sums_ge}\leanok
We have
\begin{align*}
& d[X_1|X_1+\tilde X_2; X_2|X_2+\tilde X_1] \\ & \qquad\quad \geq k - \eta (d[X^0_1; X_1 | X_1 + \tilde X_2] - d[X^0_1; X_1]) \\
& \qquad\qquad\qquad\qquad - \eta(d[X^0_2;... | lemma condRuzsaDist_of_sums_ge :
d[X₁ | X₁ + X₂' # X₂ | X₂ + X₁'] ≥
k - p.η * (d[p.X₀₁ # X₁ | X₁ + X₂'] - d[p.X₀₁ # X₁])
- p.η * (d[p.X₀₂ # X₂ | X₂ + X₁'] - d[p.X₀₂ # X₂]) :=
condRuzsaDistance_ge_of_min _ h_min hX₁ hX₂ _ _ (by fun_prop) (by fun_prop) | pfr/blueprint/src/chapter/entropy_pfr.tex:103 | pfr/PFR/FirstEstimate.lean:84 |
PFR | condRuzsaDistance_ge_of_min | \begin{lemma}[Conditional distance lower bound]\label{cond-distance-lower}
\uses{tau-min-def, cond-dist-def}
\lean{condRuzsaDistance_ge_of_min}\leanok
For any $G$-valued random variables $X'_1,X'_2$ and random variables $Z,W$, one has
$$ d[X'_1|Z;X'_2|W] \geq k - \eta (d[X^0_1;X'_1|Z] - d[X^0_1;X_1] ) - \eta (d[X... | lemma condRuzsaDistance_ge_of_min [MeasurableSingletonClass G]
[Fintype S] [MeasurableSpace S] [MeasurableSingletonClass S]
[Fintype T] [MeasurableSpace T] [MeasurableSingletonClass T]
(h : tau_minimizes p X₁ X₂) (h1 : Measurable X₁') (h2 : Measurable X₂')
(Z : Ω'₁ → S) (W : Ω'₂ → T) (hZ : Measurable Z)... | pfr/blueprint/src/chapter/entropy_pfr.tex:60 | pfr/PFR/TauFunctional.lean:207 |
PFR | cond_multiDist_chainRule | \begin{lemma}[Conditional multidistance chain rule]\label{multidist-chain-rule-cond}\lean{cond_multiDist_chainRule}\leanok
Let $\pi \colon G \to H$ be a homomorphism of abelian groups.
Let $I$ be a finite index set and let $X_{[m]}$ be a tuple of $G$-valued random variables.
Let $Y_{[m]}$ be another tuple o... | lemma cond_multiDist_chainRule {G H : Type*} [hG : MeasurableSpace G] [MeasurableSingletonClass G]
[AddCommGroup G] [Fintype G]
[hH : MeasurableSpace H] [MeasurableSingletonClass H] [AddCommGroup H]
[Fintype H] (π : G →+ H)
{S : Type*} [Fintype S] [hS : MeasurableSpace S] [MeasurableSingletonClass S]
... | pfr/blueprint/src/chapter/torsion.tex:390 | pfr/PFR/MoreRuzsaDist.lean:1190 |
PFR | construct_good_improved' | \begin{lemma}[Constructing good variables, II']\label{construct-good-improv}\lean{construct_good_improved'}\leanok
One has
\begin{align*} k & \leq \delta + \frac{\eta}{6} \sum_{i=1}^2 \sum_{1 \leq j,l \leq 3; j \neq l} (d[X^0_i;T_j|T_l] - d[X^0_i; X_i])
\end{align*}
\end{lemma}
\begin{proof}
\uses{construct-good-p... | lemma construct_good_improved' :
k ≤ δ + (p.η / 6) *
((d[p.X₀₁ # T₁ | T₂] - d[p.X₀₁ # X₁]) + (d[p.X₀₁ # T₁ | T₃] - d[p.X₀₁ # X₁])
+ (d[p.X₀₁ # T₂ | T₁] - d[p.X₀₁ # X₁]) + (d[p.X₀₁ # T₂ | T₃] - d[p.X₀₁ # X₁])
+ (d[p.X₀₁ # T₃ | T₁] - d[p.X₀₁ # X₁]) + (d[p.X₀₁ # T₃ | T₂] - d[p.X₀₁ # X₁])
+ (d[p.X₀₂ # ... | pfr/blueprint/src/chapter/improved_exponent.tex:57 | pfr/PFR/ImprovedPFR.lean:384 |
PFR | construct_good_prelim | \begin{lemma}[Constructing good variables, I]\label{construct-good-prelim}
\lean{construct_good_prelim}\leanok
One has
\begin{align*} k \leq
\delta + \eta (& d[X^0_1;T_1]-d[X^0_1;X_1])
+ \eta (d[X^0_2;T_2]-d[X^0_2;X_2]) \\ & + \tfrac12 \eta \bbI[T_1:T_3] + \tfrac12 \eta \bbI[T_2:T_3].
\end{align*}
\e... | lemma construct_good_prelim :
k ≤ δ + p.η * c[T₁ # T₂] + p.η * (I[T₁: T₃] + I[T₂ : T₃])/2 := by
let sum1 : ℝ := (Measure.map T₃ ℙ)[fun t ↦ d[T₁; ℙ[|T₃ ⁻¹' {t}] # T₂; ℙ[|T₃ ⁻¹' {t}]]]
let sum2 : ℝ := (Measure.map T₃ ℙ)[fun t ↦ d[p.X₀₁; ℙ # T₁; ℙ[|T₃ ⁻¹' {t}]] - d[p.X₀₁ # X₁]]
let sum3 : ℝ := (Measure.map T₃ ℙ)... | pfr/blueprint/src/chapter/entropy_pfr.tex:327 | pfr/PFR/Endgame.lean:367 |
PFR | construct_good_prelim' | \begin{lemma}[Constructing good variables, I']\label{construct-good-prelim-improv}\lean{construct_good_prelim'}\leanok
One has
\begin{align*} k \leq
\delta + \eta (& d[X^0_1;T_1|T_3]-d[X^0_1;X_1])
+ \eta (d[X^0_2;T_2|T_3]-d[X^0_2;X_2]).
\end{align*}
\end{lemma}
\begin{proof} \uses{entropic-bsg,distance... | lemma construct_good_prelim' : k ≤ δ + p.η * c[T₁ | T₃ # T₂ | T₃] := by
let sum1 : ℝ := (Measure.map T₃ ℙ)[fun t ↦ d[T₁; ℙ[|T₃ ⁻¹' {t}] # T₂; ℙ[|T₃ ⁻¹' {t}]]]
let sum2 : ℝ := (Measure.map T₃ ℙ)[fun t ↦ d[p.X₀₁; ℙ # T₁; ℙ[|T₃ ⁻¹' {t}]] - d[p.X₀₁ # X₁]]
let sum3 : ℝ := (Measure.map T₃ ℙ)[fun t ↦ d[p.X₀₂; ℙ # T₂; ℙ[... | pfr/blueprint/src/chapter/improved_exponent.tex:19 | pfr/PFR/ImprovedPFR.lean:326 |
PFR | cor_multiDist_chainRule | \begin{corollary}\label{cor-multid}\lean{cor_multiDist_chainRule}\leanok Let $G$ be an abelian group and let $m \geq 2$. Suppose that $X_{i,j}$, $1 \leq i, j \leq m$, are independent $G$-valued random variables.
Then
\begin{align*}
&\bbI[ \bigl(\sum_{i=1}^m X_{i,j}\bigr)_{j =1}^{m} : \bigl(\sum_{j=1}^m X... | lemma cor_multiDist_chainRule [Fintype G] {m:ℕ} (hm: m ≥ 1) {Ω : Type*} (hΩ : MeasureSpace Ω)
(X : Fin (m + 1) × Fin (m + 1) → Ω → G) (h_indep : iIndepFun X) :
I[fun ω ↦ (fun j ↦ ∑ i, X (i, j) ω) : fun ω ↦ (fun i ↦ ∑ j, X (i, j) ω) | ∑ p, X p]
≤ ∑ j, (D[fun i ↦ X (i, j); fun _ ↦ hΩ] - D[fun i ↦ X (i, j) |... | pfr/blueprint/src/chapter/torsion.tex:440 | pfr/PFR/MoreRuzsaDist.lean:1489 |
PFR | diff_ent_le_rdist | \begin{lemma}[Distance controls entropy difference]\label{ruzsa-diff}
\uses{ruz-dist-def}
\lean{diff_ent_le_rdist}\leanok
If $X,Y$ are $G$-valued random variables, then
$$|\bbH[X]-H[Y]| \leq 2 d[X ;Y].$$
\end{lemma}
\begin{proof} \uses{sumset-lower, neg-ent} \leanok Immediate from \Cref{sumset-lower} and \Cref{ru... | /-- `|H[X] - H[Y]| ≤ 2 d[X ; Y]`. -/
lemma diff_ent_le_rdist [IsProbabilityMeasure μ] [IsProbabilityMeasure μ']
(hX : Measurable X) (hY : Measurable Y) :
|H[X ; μ] - H[Y ; μ']| ≤ 2 * d[X ; μ # Y ; μ'] := by
obtain ⟨ν, X', Y', _, hX', hY', hind, hIdX, hIdY, _, _⟩ := independent_copies_finiteRange hX hY μ μ'
... | pfr/blueprint/src/chapter/distance.tex:128 | pfr/PFR/ForMathlib/Entropy/RuzsaDist.lean:243 |
PFR | diff_ent_le_rdist' | \begin{lemma}[Distance controls entropy growth]\label{ruzsa-growth}
\uses{ruz-dist-def}
\lean{diff_ent_le_rdist', diff_ent_le_rdist''}\leanok
If $X,Y$ are independent $G$-valued random variables, then
$$ \bbH[X-Y] - \bbH[X], \bbH[X-Y] - \bbH[Y] \leq 2d[X ;Y].$$
\end{lemma}
\begin{proof} \uses{sumset-lower, neg-e... | /-- `H[X - Y] - H[X] ≤ 2d[X ; Y]`. -/
lemma diff_ent_le_rdist' [IsProbabilityMeasure μ] {Y : Ω → G}
(hX : Measurable X) (hY : Measurable Y) (h : IndepFun X Y μ) [FiniteRange Y]:
H[X - Y ; μ] - H[X ; μ] ≤ 2 * d[X ; μ # Y ; μ] := by
rw [h.rdist_eq hX hY]
linarith[max_entropy_le_entropy_sub hX hY h, le_max_rig... | pfr/blueprint/src/chapter/distance.tex:138 | pfr/PFR/ForMathlib/Entropy/RuzsaDist.lean:254 |
PFR | diff_ent_le_rdist'' | \begin{lemma}[Distance controls entropy growth]\label{ruzsa-growth}
\uses{ruz-dist-def}
\lean{diff_ent_le_rdist', diff_ent_le_rdist''}\leanok
If $X,Y$ are independent $G$-valued random variables, then
$$ \bbH[X-Y] - \bbH[X], \bbH[X-Y] - \bbH[Y] \leq 2d[X ;Y].$$
\end{lemma}
\begin{proof} \uses{sumset-lower, neg-e... | /-- `H[X - Y] - H[Y] ≤ 2d[X ; Y]`. -/
lemma diff_ent_le_rdist'' [IsProbabilityMeasure μ] {Y : Ω → G}
(hX : Measurable X) (hY : Measurable Y) (h : IndepFun X Y μ) [FiniteRange Y]:
H[X - Y ; μ] - H[Y ; μ] ≤ 2 * d[X ; μ # Y ; μ] := by
rw [h.rdist_eq hX hY]
linarith[max_entropy_le_entropy_sub hX hY h, le_max_le... | pfr/blueprint/src/chapter/distance.tex:138 | pfr/PFR/ForMathlib/Entropy/RuzsaDist.lean:261 |
PFR | diff_rdist_le_1 | \begin{lemma}[Upper bound on distance differences]\label{first-upper}\leanok
\lean{diff_rdist_le_1, diff_rdist_le_2, diff_rdist_le_3, diff_rdist_le_4}
We have
\begin{align*}
d[X^0_1; X_1+\tilde X_2] - d[X^0_1; X_1] &\leq \tfrac{1}{2} k + \tfrac{1}{4} \bbH[X_2] - \tfrac{1}{4} \bbH[X_1]\\
d[X^0_2;X_2+\tilde... | /--`d[X₀₁ # X₁ + X₂'] - d[X₀₁ # X₁] ≤ k/2 + H[X₂]/4 - H[X₁]/4`. -/
lemma diff_rdist_le_1 [IsProbabilityMeasure (ℙ : Measure Ω₀₁)] :
d[p.X₀₁ # X₁ + X₂'] - d[p.X₀₁ # X₁] ≤ k/2 + H[X₂]/4 - H[X₁]/4 := by
have h : IndepFun X₁ X₂' := by simpa using h_indep.indepFun (show (0 : Fin 4) ≠ 2 by decide)
convert condRuzsaDi... | pfr/blueprint/src/chapter/entropy_pfr.tex:115 | pfr/PFR/FirstEstimate.lean:93 |
PFR | diff_rdist_le_2 | \begin{lemma}[Upper bound on distance differences]\label{first-upper}\leanok
\lean{diff_rdist_le_1, diff_rdist_le_2, diff_rdist_le_3, diff_rdist_le_4}
We have
\begin{align*}
d[X^0_1; X_1+\tilde X_2] - d[X^0_1; X_1] &\leq \tfrac{1}{2} k + \tfrac{1}{4} \bbH[X_2] - \tfrac{1}{4} \bbH[X_1]\\
d[X^0_2;X_2+\tilde... | /-- $$ d[X^0_2;X_2+\tilde X_1] - d[X^0_2; X_2] \leq \tfrac{1}{2} k + \tfrac{1}{4} \mathbb{H}[X_1] - \tfrac{1}{4} \mathbb{H}[X_2].$$ -/
lemma diff_rdist_le_2 [IsProbabilityMeasure (ℙ : Measure Ω₀₂)] :
d[p.X₀₂ # X₂ + X₁'] - d[p.X₀₂ # X₂] ≤ k/2 + H[X₁]/4 - H[X₂]/4 := by
have h : IndepFun X₂ X₁' := by simpa using h_i... | pfr/blueprint/src/chapter/entropy_pfr.tex:115 | pfr/PFR/FirstEstimate.lean:102 |
PFR | diff_rdist_le_3 | \begin{lemma}[Upper bound on distance differences]\label{first-upper}\leanok
\lean{diff_rdist_le_1, diff_rdist_le_2, diff_rdist_le_3, diff_rdist_le_4}
We have
\begin{align*}
d[X^0_1; X_1+\tilde X_2] - d[X^0_1; X_1] &\leq \tfrac{1}{2} k + \tfrac{1}{4} \bbH[X_2] - \tfrac{1}{4} \bbH[X_1]\\
d[X^0_2;X_2+\tilde... | lemma diff_rdist_le_3 [IsProbabilityMeasure (ℙ : Measure Ω₀₁)] :
d[p.X₀₁ # X₁ | X₁ + X₂'] - d[p.X₀₁ # X₁] ≤ k/2 + H[X₁]/4 - H[X₂]/4 := by
have h : IndepFun X₁ X₂' := by simpa using h_indep.indepFun (show (0 : Fin 4) ≠ 2 by decide)
convert condRuzsaDist_diff_le''' ℙ p.hmeas1 hX₁ hX₂' h using 3
· rw [(IdentDist... | pfr/blueprint/src/chapter/entropy_pfr.tex:115 | pfr/PFR/FirstEstimate.lean:114 |
PFR | diff_rdist_le_4 | \begin{lemma}[Upper bound on distance differences]\label{first-upper}\leanok
\lean{diff_rdist_le_1, diff_rdist_le_2, diff_rdist_le_3, diff_rdist_le_4}
We have
\begin{align*}
d[X^0_1; X_1+\tilde X_2] - d[X^0_1; X_1] &\leq \tfrac{1}{2} k + \tfrac{1}{4} \bbH[X_2] - \tfrac{1}{4} \bbH[X_1]\\
d[X^0_2;X_2+\tilde... | lemma diff_rdist_le_4 [IsProbabilityMeasure (ℙ : Measure Ω₀₂)] :
d[p.X₀₂ # X₂ | X₂ + X₁'] - d[p.X₀₂ # X₂] ≤ k/2 + H[X₂]/4 - H[X₁]/4 := by
have h : IndepFun X₂ X₁' := by simpa using h_indep.indepFun (show (1 : Fin 4) ≠ 3 by decide)
convert condRuzsaDist_diff_le''' ℙ p.hmeas2 hX₂ hX₁' h using 3
· rw [rdist_symm... | pfr/blueprint/src/chapter/entropy_pfr.tex:115 | pfr/PFR/FirstEstimate.lean:124 |
PFR | dist_diff_bound_1 | \begin{lemma}[Bound on distance differences]\label{dist-diff-bound}\lean{dist_diff_bound_1, dist_diff_bound_2}\leanok We have
\begin{align*} &\sum_{i=1}^2 \sum_{A,B \in \{U,V,W\}: A \neq B} d[X_i^0;A|B, S] - d[X_i^0;X_i]\\
&\qquad \leq 12 k + \frac{4(2 \eta k - I_1)}{1-\eta}.
\end{align*}
\end{lemma}
\begin{proof}\us... | lemma dist_diff_bound_1 :
(d[p.X₀₁ # U | ⟨V, S⟩] - d[p.X₀₁ # X₁]) + (d[p.X₀₁ # U | ⟨W, S⟩] - d[p.X₀₁ # X₁])
+ (d[p.X₀₁ # V | ⟨U, S⟩] - d[p.X₀₁ # X₁]) + (d[p.X₀₁ # V | ⟨W, S⟩] - d[p.X₀₁ # X₁])
+ (d[p.X₀₁ # W | ⟨U, S⟩] - d[p.X₀₁ # X₁]) + (d[p.X₀₁ # W | ⟨V, S⟩] - d[p.X₀₁ # X₁])
≤ (16 * k + 6 * d[X₁ # X₁]... | pfr/blueprint/src/chapter/improved_exponent.tex:139 | pfr/PFR/ImprovedPFR.lean:468 |
PFR | dist_diff_bound_2 | \begin{lemma}[Bound on distance differences]\label{dist-diff-bound}\lean{dist_diff_bound_1, dist_diff_bound_2}\leanok We have
\begin{align*} &\sum_{i=1}^2 \sum_{A,B \in \{U,V,W\}: A \neq B} d[X_i^0;A|B, S] - d[X_i^0;X_i]\\
&\qquad \leq 12 k + \frac{4(2 \eta k - I_1)}{1-\eta}.
\end{align*}
\end{lemma}
\begin{proof}\us... | lemma dist_diff_bound_2 :
((d[p.X₀₂ # U | ⟨V, S⟩] - d[p.X₀₂ # X₂]) + (d[p.X₀₂ # U | ⟨W, S⟩] - d[p.X₀₂ # X₂])
+ (d[p.X₀₂ # V | ⟨U, S⟩] - d[p.X₀₂ # X₂]) + (d[p.X₀₂ # V | ⟨W, S⟩] - d[p.X₀₂ # X₂])
+ (d[p.X₀₂ # W | ⟨U, S⟩] - d[p.X₀₂ # X₂]) + (d[p.X₀₂ # W | ⟨V, S⟩] - d[p.X₀₂ # X₂]))
≤ (16 * k + 6 * d[X₂ # X... | pfr/blueprint/src/chapter/improved_exponent.tex:139 | pfr/PFR/ImprovedPFR.lean:561 |
PFR | dist_le_of_sum_zero | \begin{lemma}\label{rho-BSG-triplet}\lean{dist_le_of_sum_zero}\leanok
If $G$-valued random variables $T_1,T_2,T_3$ satisfy $T_1+T_2+T_3=0$, then
$$d[X_1;X_2]\le 3\bbI[T_1:T_2] + (2\bbH[T_3]-\bbH[T_1]-\bbH[T_2])+ \eta(\rho(T_1|T_3)+\rho(T_2|T_3)-\rho(X_1)-\rho(X_2)).$$
\end{lemma}
\begin{proof}\leanok\uses{entropic-... | lemma dist_le_of_sum_zero {Ω' : Type*} [MeasurableSpace Ω'] {μ : Measure Ω'}
[IsProbabilityMeasure μ] {T₁ T₂ T₃ : Ω' → G}
(hsum : T₁ + T₂ + T₃ = 0) (hT₁ : Measurable T₁) (hT₂ : Measurable T₂) (hT₃ : Measurable T₃) :
k ≤ 3 * I[T₁ : T₂ ; μ] + (2 * H[T₃ ; μ] - H[T₁ ; μ] - H[T₂ ; μ])
+ η * (ρ[T₁ | T₃ ; μ ... | pfr/blueprint/src/chapter/further_improvement.tex:257 | pfr/PFR/RhoFunctional.lean:1455 |
PFR | dist_le_of_sum_zero' | \begin{lemma}\label{rho-BSG-triplet-symmetrized}\lean{dist_le_of_sum_zero'}\leanok
If $G$-valued random variables $T_1,T_2,T_3$ satisfy $T_1+T_2+T_3=0$, then
$$d[X_1;X_2] \leq \sum_{1 \leq i<j \leq 3} \bbI[T_i:T_j] + \frac{\eta}{3} \sum_{1 \leq i<j \leq 3} (\rho(T_i|T_j) + \rho(T_j|T_i) -\rho(X_1)-\rho(X_2))$$
\... | lemma dist_le_of_sum_zero' {Ω' : Type*} [MeasureSpace Ω']
[IsProbabilityMeasure (ℙ : Measure Ω')] {T₁ T₂ T₃ : Ω' → G} (hsum : T₁ + T₂ + T₃ = 0)
(hT₁ : Measurable T₁) (hT₂ : Measurable T₂) (hT₃ : Measurable T₃) :
k ≤ I[T₁ : T₂] + I[T₁ : T₃] + I[T₂ : T₃]
+ (η / 3) * ((ρ[T₁ | T₂ # A] + ρ[T₂ | T₁ # A] - ρ... | pfr/blueprint/src/chapter/further_improvement.tex:266 | pfr/PFR/RhoFunctional.lean:1527 |
PFR | dist_of_U_add_le | \begin{lemma}[Application of BSG]
\label{lem:get-better}\lean{dist_of_U_add_le}\leanok
Let $G$ be an abelian group, let $(T_1,T_2,T_3)$ be a $G^3$-valued random variable such that $T_1+T_2+T_3=0$ holds identically, and write
\[
\delta := \bbI[T_1 : T_2] + \bbI[T_1 : T_3] + \bbI[T_2 : T_3].
\]
Let $Y_1,\do... | lemma dist_of_U_add_le {G: Type*} [MeasureableFinGroup G] {Ω: Type*} [MeasureSpace Ω] (T₁ T₂ T₃ : Ω → G) (hsum: T₁ + T₂ + T₃ = 0) (n:ℕ) {Ω': Fin n → Type*} (hΩ': ∀ i, MeasureSpace (Ω' i)) (Y: ∀ i, (Ω' i) → G) {α:ℝ} (hα: α > 0): ∃ (Ω'':Type*) (hΩ'': MeasureSpace Ω'') (U: Ω'' → G), d[U # U] + α * ∑ i, d[Y i # U] ≤ (2 + α... | pfr/blueprint/src/chapter/torsion.tex:754 | pfr/PFR/TorsionEndgame.lean:79 |
PFR | dist_of_X_U_H_le | \begin{theorem}[Entropy form of PFR]\label{main-entropy}\lean{dist_of_X_U_H_le}\leanok Suppose that $G$ is a finite abelian group of torsion $m$. Suppose that $X$ is a $G$-valued random variable. Then there exists a subgroup $H \leq G$ such that \[ d[X;U_H] \leq 64 m^3 d[X;X].\]
\end{theorem}
\begin{proof}\uses{k-vani... | /-- Suppose that $G$ is a finite abelian group of torsion $m$. Suppose that $X$ is a $G$-valued random variable. Then there exists a subgroup $H \leq G$ such that \[ d[X;U_H] \leq 64 m^3 d[X;X].\] -/
lemma dist_of_X_U_H_le {G : Type*} [AddCommGroup G] [Fintype G] [MeasurableSpace G]
[MeasurableSingletonClass G] (m:ℕ)... | pfr/blueprint/src/chapter/torsion.tex:857 | pfr/PFR/TorsionEndgame.lean:86 |
PFR | dist_of_min_eq_zero | \begin{proposition}\label{phi-minimizer-zero-distance}\lean{dist_of_min_eq_zero}\leanok If $X_1,X_2$ is a $\phi$-minimizer, then $d[X_1;X_2] = 0$.
\end{proposition}
\begin{proof}\leanok
\uses{rho-BSG-triplet-symmetrized,rho-increase-symmetrized,I1-I2-diff,phi-first-estimate,phi-second-estimate}
Consider $T_1:=X_1+X_... | theorem dist_of_min_eq_zero (hA : A.Nonempty) (hη' : η < 1/8) : d[X₁ # X₂] = 0 := by
let ⟨Ω', m', μ, Y₁, Y₂, Y₁', Y₂', hμ, h_indep, hY₁, hY₂, hY₁', hY₂', h_id1, h_id2, h_id1', h_id2'⟩
:= independent_copies4_nondep hX₁ hX₂ hX₁ hX₂ ℙ ℙ ℙ ℙ
rw [← h_id1.rdist_eq h_id2]
let _ : MeasureSpace Ω' := ⟨μ⟩
have : IsPr... | pfr/blueprint/src/chapter/further_improvement.tex:315 | pfr/PFR/RhoFunctional.lean:1860 |
PFR | distance_ge_of_min | \begin{lemma}[Distance lower bound]\label{distance-lower}
\uses{tau-min-def}\leanok
\lean{distance_ge_of_min}
For any $G$-valued random variables $X'_1,X'_2$, one has
$$ d[X'_1;X'_2] \geq k - \eta (d[X^0_1;X'_1] - d[X^0_1;X_1] ) - \eta (d[X^0_2;X'_2] - d[X^0_2;X_2] ).$$
\end{lemma}
\begin{proof}
\uses{tau-def, ... | lemma distance_ge_of_min (h : tau_minimizes p X₁ X₂) (h1 : Measurable X₁') (h2 : Measurable X₂') :
d[X₁ # X₂] - p.η * (d[p.X₀₁ # X₁'] - d[p.X₀₁ # X₁]) - p.η * (d[p.X₀₂ # X₂'] - d[p.X₀₂ # X₂])
≤ d[X₁' # X₂'] := by
have Z := is_tau_min p h h1 h2
simp [tau] at Z
linarith
omit [IsProbabilityMeasure (ℙ : Me... | pfr/blueprint/src/chapter/entropy_pfr.tex:48 | pfr/PFR/TauFunctional.lean:181 |
PFR | ent_bsg | \begin{lemma}[Balog-Szemer\'edi-Gowers]\label{entropic-bsg}
\lean{ent_bsg}\leanok
Let $A,B$ be $G$-valued random variables on $\Omega$, and set $Z := A+B$.
Then
\begin{equation}\label{2-bsg-takeaway} \sum_{z} \bbP[Z=z] d[(A | Z = z); (B | Z = z)] \leq 3 \bbI[A:B] + 2 \bbH[Z] - \bbH[A] - \bbH[B]. \end{equation}
\e... | lemma ent_bsg [IsProbabilityMeasure μ] {A B : Ω → G} (hA : Measurable A) (hB : Measurable B)
[Fintype G] :
(μ.map (A + B))[fun z ↦ d[A ; μ[|(A + B) ⁻¹' {z}] # B ; μ[|(A + B) ⁻¹' {z}]]]
≤ 3 * I[A : B; μ] + 2 * H[A + B ; μ] - H[A ; μ] - H[B ; μ] := by
let Z := A + B
have hZ : Measurable Z := hA.add hB
... | pfr/blueprint/src/chapter/distance.tex:261 | pfr/PFR/ForMathlib/Entropy/RuzsaDist.lean:1165 |
PFR | ent_of_proj_le | \begin{lemma}[Projection entropy and distance]\label{dist-projection}\lean{ent_of_proj_le}\leanok
If $G$ is an additive group and $X$ is a $G$-valued random variable and $H\leq G$ is a finite subgroup then, with $\pi:G\to G/H$ the natural homomorphism we have (where $U_H$ is uniform on $H$)
\[\mathbb{H}(\pi(X))\leq 2d[... | lemma ent_of_proj_le {UH: Ω' → G} [FiniteRange UH]
[IsProbabilityMeasure μ] [IsProbabilityMeasure μ']
(hX : Measurable X) (hU : Measurable UH) {H : AddSubgroup G} (hH : Set.Finite (H : Set G)) -- TODO: infer from [FiniteRange UH]?
(hunif : IsUniform H UH μ') :
H[(QuotientAddGroup.mk' H) ∘ X; μ] ≤ 2 * d[... | pfr/blueprint/src/chapter/distance.tex:161 | pfr/PFR/ForMathlib/Entropy/RuzsaDist.lean:276 |
PFR | ent_of_sub_smul | \begin{lemma}[Sums of dilates I]\label{sum-dilate-I}\lean{ent_of_sub_smul, ent_of_sub_smul'}\leanok Let $X,Y,X'$ be independent $G$-valued random variables, with $X'$ a copy of $X$, and let $a$ be an integer. Then
$$\bbH[X-(a+1)Y] \leq \bbH[X-aY] + \bbH[X-Y-X'] - \bbH[X]$$
and
$$\bbH[X-(a-1)Y] \leq \bbH[X-aY] + \bbH[... | lemma ent_of_sub_smul {Y : Ω → G} {X' : Ω → G} [FiniteRange X] [FiniteRange Y] [FiniteRange X']
[IsProbabilityMeasure μ] (hX : Measurable X) (hY : Measurable Y) (hX' : Measurable X')
(h_indep : iIndepFun ![X, Y, X'] μ) (hident : IdentDistrib X X' μ μ) {a : ℤ} :
H[X - (a+1) • Y; μ] ≤ H[X - a • Y; μ] + H[X - ... | pfr/blueprint/src/chapter/torsion.tex:125 | pfr/PFR/MoreRuzsaDist.lean:532 |
PFR | ent_of_sub_smul' | \begin{lemma}[Sums of dilates I]\label{sum-dilate-I}\lean{ent_of_sub_smul, ent_of_sub_smul'}\leanok Let $X,Y,X'$ be independent $G$-valued random variables, with $X'$ a copy of $X$, and let $a$ be an integer. Then
$$\bbH[X-(a+1)Y] \leq \bbH[X-aY] + \bbH[X-Y-X'] - \bbH[X]$$
and
$$\bbH[X-(a-1)Y] \leq \bbH[X-aY] + \bbH[... | lemma ent_of_sub_smul' {Y : Ω → G} {X' : Ω → G} [FiniteRange X] [FiniteRange Y] [FiniteRange X']
[IsProbabilityMeasure μ] (hX : Measurable X) (hY : Measurable Y) (hX': Measurable X')
(h_indep : iIndepFun ![X, Y, X'] μ) (hident : IdentDistrib X X' μ μ) {a : ℤ} :
H[X - (a-1) • Y; μ] ≤ H[X - a • Y; μ] + H[X - ... | pfr/blueprint/src/chapter/torsion.tex:125 | pfr/PFR/MoreRuzsaDist.lean:558 |
PFR | ent_of_sub_smul_le | \begin{lemma}[Sums of dilates II]\label{sum-dilate-II}\lean{ent_of_sub_smul_le}\leanok Let $X,Y$ be independent $G$-valued random variables, and let $a$ be an integer. Then
$$\bbH[X-aY] - \bbH[X] \leq 4 |a| d[X;Y].$$
\end{lemma}
\begin{proof}\uses{kv, ruz-indep, sign-flip, sum-dilate-I}\leanok From \Cref{kv} one ha... | lemma ent_of_sub_smul_le {Y : Ω → G} [IsProbabilityMeasure μ] [Fintype G]
(hX : Measurable X) (hY : Measurable Y) (h_indep : IndepFun X Y μ) {a : ℤ} :
H[X - a • Y; μ] - H[X; μ] ≤ 4 * |a| * d[X ; μ # Y ; μ] := by
obtain ⟨Ω', mΩ', μ', X₁', Y', X₂', hμ', h_indep', hX₁', hY', hX₂', idX₁, idY, idX₂⟩
:= indepen... | pfr/blueprint/src/chapter/torsion.tex:139 | pfr/PFR/MoreRuzsaDist.lean:600 |
PFR | ent_of_sum_le_ent_of_sum | \begin{lemma}[Comparing sums]\label{compare-sums}\lean{ent_of_sum_le_ent_of_sum}\leanok Let $(X_i)_{1 \leq i \leq m}$ and $(Y_j)_{1 \leq j \leq l}$ be tuples of jointly independent random variables (so the $X$'s and $Y$'s are also independent of each other), and let $f: \{1,\dots,l\} \to \{1,\dots,m\}$ be a function, ... | lemma ent_of_sum_le_ent_of_sum [IsProbabilityMeasure μ] {I : Type*} {s t : Finset I} (hdisj : Disjoint s t)
(hs : Finset.Nonempty s) (ht : Finset.Nonempty t) (X : I → Ω → G) (hX : (i : I) → Measurable (X i))
(hX' : (i : I) → FiniteRange (X i)) (h_indep : iIndepFun X μ) (f : I → I)
(hf : Finset.image f t ⊆ s... | pfr/blueprint/src/chapter/torsion.tex:112 | pfr/PFR/MoreRuzsaDist.lean:523 |
PFR | ent_ofsum_le | \begin{lemma}[Entropy bound on quadruple sum]\label{foursum-bound}
\lean{ent_ofsum_le}\leanok
With the same notation, we have
\begin{equation}
\label{HS-bound}
\bbH[X_1+X_2+\tilde X_1+\tilde X_2] \le \tfrac{1}{2} \bbH[X_1]+\tfrac{1}{2} \bbH[X_2] + (2 + \eta) k - I_1.
\end{equation}
\end{lemma}
\begin{pr... | lemma ent_ofsum_le
[IsProbabilityMeasure (ℙ : Measure Ω₀₁)] [IsProbabilityMeasure (ℙ : Measure Ω₀₂)] :
H[X₁ + X₂ + X₁' + X₂'] ≤ H[X₁]/2 + H[X₂]/2 + (2+p.η)*k - I₁ := by
let D := d[X₁ + X₂' # X₂ + X₁']
let Dcc := d[X₁ | X₁ + X₂' # X₂ | X₂ + X₁']
let D1 := d[p.X₀₁ # X₁]
let Dc1 := d[p.X₀₁ # X₁ | X₁ + X₂']... | pfr/blueprint/src/chapter/entropy_pfr.tex:138 | pfr/PFR/FirstEstimate.lean:150 |
PFR | entropic_PFR_conjecture | \begin{theorem}[Entropy version of PFR]\label{entropy-pfr}
\lean{entropic_PFR_conjecture}\leanok
Let $G = \F_2^n$, and suppose that $X^0_1, X^0_2$ are $G$-valued random variables.
Then there is some subgroup $H \leq G$ such that
\[
d[X^0_1;U_H] + d[X^0_2;U_H] \le 11 d[X^0_1;X^0_2],
\]
where $U_H$ is uni... | theorem entropic_PFR_conjecture (hpη : p.η = 1/9):
∃ H : Submodule (ZMod 2) G, ∃ Ω : Type uG, ∃ mΩ : MeasureSpace Ω, ∃ U : Ω → G,
IsProbabilityMeasure (ℙ : Measure Ω) ∧ Measurable U ∧
IsUniform H U ∧ d[p.X₀₁ # U] + d[p.X₀₂ # U] ≤ 11 * d[p.X₀₁ # p.X₀₂] := by
obtain ⟨Ω', mΩ', X₁, X₂, hX₁, hX₂, _, htau_min⟩ ... | pfr/blueprint/src/chapter/entropy_pfr.tex:417 | pfr/PFR/EntropyPFR.lean:46 |
PFR | entropic_PFR_conjecture_improv | \begin{theorem}[Improved entropy version of PFR]\label{entropy-pfr-improv}\lean{entropic_PFR_conjecture_improv}\leanok
Let $G = \F_2^n$, and suppose that $X^0_1, X^0_2$ are $G$-valued random variables.
Then there is some subgroup $H \leq G$ such that
\[
d[X^0_1;U_H] + d[X^0_2;U_H] \le 10 d[X^0_1;X^0_2],
\]
... | theorem entropic_PFR_conjecture_improv (hpη : p.η = 1/8) :
∃ (H : Submodule (ZMod 2) G) (Ω : Type uG) (mΩ : MeasureSpace Ω) (U : Ω → G),
IsProbabilityMeasure (ℙ : Measure Ω) ∧ Measurable U ∧
IsUniform H U ∧ d[p.X₀₁ # U] + d[p.X₀₂ # U] ≤ 10 * d[p.X₀₁ # p.X₀₂] := by
obtain ⟨Ω', mΩ', X₁, X₂, hX₁, hX₂, hP, ht... | pfr/blueprint/src/chapter/improved_exponent.tex:197 | pfr/PFR/ImprovedPFR.lean:814 |
PFR | entropy_of_W_le | \begin{lemma}[Entropy of $W$]\label{ent-w}\lean{entropy_of_W_le}\uses{more-random}\leanok We have $\bbH[W] \leq (2m-1)k + \frac1m \sum_{i=1}^m \bbH[X_i]$.
\end{lemma}
\begin{proof}\uses{multidist-def, multidist-ruzsa-IV, klm-1} Without loss of generality, we may take $X_1,\dots,X_m$ to be independent. Write $S = \sum_... | /-- We have $\bbH[W] \leq (2m-1)k + \frac1m \sum_{i=1}^m \bbH[X_i]$. -/
lemma entropy_of_W_le : H[W] ≤ (2*p.m - 1) * k + (m:ℝ)⁻¹ * ∑ i, H[X i] := sorry | pfr/blueprint/src/chapter/torsion.tex:673 | pfr/PFR/TorsionEndgame.lean:57 |
PFR | entropy_of_Z_two_le | \begin{lemma}[Entropy of $Z_2$]\label{ent-z2}\lean{entropy_of_Z_two_le}\uses{more-random}\leanok We have $\bbH[Z_2] \leq (8m^2-16m+1) k + \frac{1}{m} \sum_{i=1}^m \bbH[X_i]$.
\end{lemma}
\begin{proof}\uses{sum-dilate-II, klm-1}
We observe
\[
\bbH[Z_2] = \bbH[\sum_{j \in \Z/m\Z} j Q_j].
\]
Applying \Cref{kl... | /-- We have $\bbH[Z_2] \leq (8m^2-16m+1) k + \frac{1}{m} \sum_{i=1}^m \bbH[X_i]$. -/
lemma entropy_of_Z_two_le : H[Z2] ≤ (8 * p.m^2 - 16 * p.m + 1) * k + (m:ℝ)⁻¹ * ∑ i, H[X i] := sorry | pfr/blueprint/src/chapter/torsion.tex:699 | pfr/PFR/TorsionEndgame.lean:60 |
PFR | exists_isUniform_of_rdist_eq_zero | \begin{corollary}[General 100\% inverse theorem]\label{lem:100pc}
\lean{exists_isUniform_of_rdist_eq_zero}\leanok
Suppose that $X_1,X_2$ are $G$-valued random variables such that
$d[X_1;X_2]=0$. Then there exists a subgroup $H \leq G$ such that $d[X_1;U_H] = d[X_2;U_H] = 0$.
\end{corollary}
\begin{proof}\uses{lem... | theorem exists_isUniform_of_rdist_eq_zero
{Ω' : Type*} [MeasureSpace Ω'] [IsProbabilityMeasure (ℙ : Measure Ω')] {X' : Ω' → G}
(hX : Measurable X) (hX' : Measurable X') (hdist : d[X # X'] = 0) :
∃ H : AddSubgroup G, ∃ U : Ω → G,
Measurable U ∧ IsUniform H U ∧ d[X # U] = 0 ∧ d[X' # U] = 0 := by
have ... | pfr/blueprint/src/chapter/100_percent.tex:51 | pfr/PFR/HundredPercent.lean:160 |
PFR | exists_isUniform_of_rdist_self_eq_zero | \begin{lemma}[Symmetric 100\% inverse theorem]\label{lem:100pc-self}
\lean{exists_isUniform_of_rdist_self_eq_zero}\leanok
Suppose that $X$ is a $G$-valued random variable such that
$d[X ;X]=0$. Then there exists a subgroup $H \leq G$ such that $d[X ;U_H] = 0$.
\end{lemma}
\begin{proof}\uses{sym-group, sym-zero}\l... | /-- If $d[X ;X]=0$, then there exists a subgroup $H \leq G$ such that $d[X ;U_H] = 0$. -/
theorem exists_isUniform_of_rdist_self_eq_zero (hX : Measurable X) (hdist : d[X # X] = 0) :
∃ H : AddSubgroup G, ∃ U : Ω → G, Measurable U ∧ IsUniform H U ∧ d[X # U] = 0 := by
-- use for `U` a translate of `X` to make sure t... | pfr/blueprint/src/chapter/100_percent.tex:41 | pfr/PFR/HundredPercent.lean:136 |
PFR | first_estimate | \begin{lemma}[First estimate]\label{first-estimate}
\lean{first_estimate}\leanok We have $I_1 \leq 2 \eta k$.
\end{lemma}
\begin{proof}\uses{first-fibre, first-dist-sum, first-cond, first-upper}\leanok Take a suitable linear combination of \Cref{first-fibre}, \Cref{first-dist-sum}, \Cref{first-cond}, and \Cref{first... | /-- We have $I_1 \leq 2 \eta k$ -/
lemma first_estimate
[IsProbabilityMeasure (ℙ : Measure Ω₀₁)] [IsProbabilityMeasure (ℙ : Measure Ω₀₂)] :
I₁ ≤ 2 * p.η * k := by
have v1 := rdist_add_rdist_add_condMutual_eq X₁ X₂ X₁' X₂' ‹_› ‹_› ‹_› ‹_› ‹_› ‹_› ‹_›
have v2 := rdist_of_sums_ge p X₁ X₂ X₁' X₂' ‹_› ‹_› ‹_› ‹_... | pfr/blueprint/src/chapter/entropy_pfr.tex:129 | pfr/PFR/FirstEstimate.lean:132 |
PFR | gen_ineq_00 | \begin{lemma}[General inequality]\label{gen-ineq}\lean{gen_ineq_00} \leanok
Let $X_1, X_2, X_3, X_4$ be independent $G$-valued random variables, and let $Y$ be another $G$-valued random variable. Set $S := X_1+X_2+X_3+X_4$. Then
\begin{align*}
& d[Y; X_1+X_2|X_1 + X_3, S] - d[Y; X_1] \\
&\quad \leq \tfrac{1... | lemma gen_ineq_00 : d[Y # Z₁ + Z₂ | ⟨Z₁ + Z₃, Sum⟩] - d[Y # Z₁] ≤
(d[Z₁ # Z₂] + 2 * d[Z₁ # Z₃] + d[Z₂ # Z₄]) / 4
+ (d[Z₁ | Z₁ + Z₃ # Z₂ | Z₂ + Z₄] - d[Z₁ | Z₁ + Z₂ # Z₃ | Z₃ + Z₄]) / 4
+ (H[Z₁ + Z₂] - H[Z₃ + Z₄] + H[Z₂] - H[Z₃] + H[Z₂ | Z₂ + Z₄] - H[Z₁ | Z₁ + Z₃]) / 8 := by
have I1 := gen_ineq_aux1 Y hY Z... | pfr/blueprint/src/chapter/improved_exponent.tex:88 | pfr/PFR/ImprovedPFR.lean:202 |
PFR | goursat | \begin{lemma}[Goursat type theorem]\label{goursat}\lean{goursat}\leanok 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, $|H| = |H_0| |H_1|$.... | 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,... | pfr/blueprint/src/chapter/hom_pfr.tex:11 | pfr/PFR/HomPFR.lean:39 |
PFR | hahn_banach | \begin{lemma}[Hahn-Banach type theorem]\label{hb-thm}\lean{hahn_banach}\leanok Let $H_0$ be a subgroup of $G$. Then every homomorphism $\phi: H_0 \to G'$ can be extended to a homomorphism $\tilde \phi: G \to G'$.
\end{lemma}
\begin{proof}\leanok By induction it suffices to treat the case where $H_0$ has index $2$ in... | lemma hahn_banach (H₀ : AddSubgroup G) (φ : H₀ →+ G') : ∃ (φ' : G →+ G'), ∀ x : H₀, φ x = φ' x := by
let H₀ := AddSubgroup.toZModSubmodule 2 H₀
let φ := (show H₀ →+ G' from φ).toZModLinearMap 2
obtain ⟨φ', hφ'⟩ := φ.exists_extend
use φ'; intro x; show φ x = φ'.comp H₀.subtype x; rw [hφ']
/-- Let $H$ be a subgr... | pfr/blueprint/src/chapter/hom_pfr.tex:5 | pfr/PFR/HomPFR.lean:29 |
PFR | homomorphism_pfr | \begin{theorem}[Homomorphism form of PFR]\label{hom-pfr}\lean{homomorphism_pfr}\leanok Let $f: G \to G'$ be a function, and let $S$ denote the set
$$ S := \{ f(x+y)-f(x)-f(y): x,y \in G \}.$$
Then there exists a homomorphism $\phi: G \to G'$ such that
$$ |\{ f(x) - \phi(x): x \in G \}| \leq |S|^{10}.$$
\end{theorem}
\... | theorem homomorphism_pfr (f : G → G') (S : Set G') (hS : ∀ x y : G, f (x+y) - (f x) - (f y) ∈ S) :
∃ (φ : G →+ G') (T : Set G'), Nat.card T ≤ Nat.card S ^ 10 ∧ ∀ x : G, (f x) - (φ x) ∈ T := by
classical
have : 0 < Nat.card G := Nat.card_pos
let A := univ.graphOn f
have hA_le : (Nat.card ↥(A + A) : ℝ) ≤ Nat.ca... | pfr/blueprint/src/chapter/hom_pfr.tex:19 | pfr/PFR/HomPFR.lean:68 |
PFR | isUniform_sub_const_of_rdist_eq_zero | \begin{lemma}[Translate is uniform on symmetry group]\label{sym-zero}
\lean{isUniform_sub_const_of_rdist_eq_zero}\leanok
If $X$ is a $G$-valued random variable with $d[X ;X]=0$, and $x_0$ is a point with $P[X=x_0] > 0$, then $X-x_0$ is uniformly distributed on $\mathrm{Sym}[X]$.
\end{lemma}
\begin{proof}\uses{zero-la... | lemma isUniform_sub_const_of_rdist_eq_zero (hX : Measurable X) (hdist : d[X # X] = 0) {x₀ : G}
(hx₀ : ℙ (X⁻¹' {x₀}) ≠ 0) : IsUniform (symmGroup X hX) (fun ω ↦ X ω - x₀) where
eq_of_mem := by
have B c z : (fun ω ↦ X ω - c) ⁻¹' {z} = X ⁻¹' {c + z} := by
ext w; simp [sub_eq_iff_eq_add']
have A : ∀ (z :... | pfr/blueprint/src/chapter/100_percent.tex:32 | pfr/PFR/HundredPercent.lean:112 |
PFR | iter_multiDist_chainRule | \begin{lemma}\label{multidist-chain-rule-iter}\lean{iter_multiDist_chainRule,iter_multiDist_chainRule'}\leanok Let $m$ be a positive integer.
Suppose one has a sequence
\begin{equation}\label{g-seq}
G_m \to G_{m-1} \to \dots \to G_1 \to G_0 = \{0\}
\end{equation}
of homomorphisms between abelian ... | lemma iter_multiDist_chainRule {m : ℕ}
{G : Fin (m + 1) → Type*}
[hG : ∀ i, MeasurableSpace (G i)] [hGs : ∀ i, MeasurableSingletonClass (G i)]
[∀ i, AddCommGroup (G i)] [hGcounT : ∀ i, Fintype (G i)]
{φ : ∀ i : Fin m, G (i.succ) →+ G i.castSucc} {π : ∀ d, G m →+ G d}
(hcomp: ∀ i : Fin m, π i.castSuc... | pfr/blueprint/src/chapter/torsion.tex:408 | pfr/PFR/MoreRuzsaDist.lean:1358 |
PFR | iter_multiDist_chainRule' | \begin{lemma}\label{multidist-chain-rule-iter}\lean{iter_multiDist_chainRule,iter_multiDist_chainRule'}\leanok Let $m$ be a positive integer.
Suppose one has a sequence
\begin{equation}\label{g-seq}
G_m \to G_{m-1} \to \dots \to G_1 \to G_0 = \{0\}
\end{equation}
of homomorphisms between abelian ... | lemma iter_multiDist_chainRule' {m : ℕ} (hm : m > 0)
{G : Fin (m + 1) → Type*} [hG : ∀ i, MeasurableSpace (G i)]
[hGs : ∀ i, MeasurableSingletonClass (G i)] [hGa : ∀ i, AddCommGroup (G i)]
[hGcount : ∀ i, Fintype (G i)] {φ : ∀ i : Fin m, G (i.succ) →+ G i.castSucc}
{π : ∀ d, G m →+ G d} (hπ0 : π 0 = 0) ... | pfr/blueprint/src/chapter/torsion.tex:408 | pfr/PFR/MoreRuzsaDist.lean:1427 |
PFR | kaimanovich_vershik | \begin{lemma}[Kaimanovich-Vershik-Madiman inequality]\label{kv}
\lean{kaimanovich_vershik}\leanok
Suppose that $X, Y, Z$ are independent $G$-valued random variables. Then
\[
\bbH[X + Y + Z] - \bbH[X + Y] \leq \bbH[Y+ Z] - \bbH[Y].
\]
\end{lemma}
\begin{proof}\uses{submodularity, add-entropy, relabeled-entropy}\lean... | /-- The **Kaimanovich-Vershik inequality**. `H[X + Y + Z] - H[X + Y] ≤ H[Y + Z] - H[Y]`. -/
lemma kaimanovich_vershik {X Y Z : Ω → G} (h : iIndepFun ![X, Y, Z] μ)
(hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z)
[FiniteRange X] [FiniteRange Z] [FiniteRange Y] :
H[X + Y + Z ; μ] - H[X + Y ; μ] ≤ H... | pfr/blueprint/src/chapter/distance.tex:237 | pfr/PFR/ForMathlib/Entropy/RuzsaDist.lean:1111 |
PFR | kvm_ineq_I | \begin{lemma}[Kaimonovich--Vershik--Madiman inequality]\label{klm-1}\lean{kvm_ineq_I}\leanok If $n \geq 0$ and $X, Y_1, \dots, Y_n$ are jointly independent $G$-valued random variables, then
$$\bbH\left[X + \sum_{i=1}^n Y_i\right] - \bbH[X] \leq \sum_{i=1}^n \left(\bbH[X+Y_i] - \bbH[X]\right).$$
\end{lemma}
\begin{pr... | lemma kvm_ineq_I {I : Type*} {i₀ : I} {s : Finset I} (hs : ¬ i₀ ∈ s)
{Y : I → Ω → G} [∀ i, FiniteRange (Y i)] (hY : (i : I) → Measurable (Y i))
(h_indep : iIndepFun Y μ) :
H[Y i₀ + ∑ i ∈ s, Y i ; μ] - H[Y i₀ ; μ] ≤ ∑ i ∈ s, (H[Y i₀ + Y i ; μ] - H[Y i₀ ; μ]) := by
classical
induction s using Finset.induc... | pfr/blueprint/src/chapter/torsion.tex:73 | pfr/PFR/MoreRuzsaDist.lean:357 |
PFR | kvm_ineq_II | \begin{lemma}[Kaimonovich--Vershik--Madiman inequality, II]\label{klm-2}\lean{kvm_ineq_II}\leanok If $n \geq 1$ and $X, Y_1, \dots, Y_n$ are jointly independent $G$-valued random variables, then
$$ d[X; \sum_{i=1}^n Y_i] \leq 2 \sum_{i=1}^n d[X; Y_i].$$
\end{lemma}
\begin{proof}\uses{klm-1, neg-ent, ruz-indep, sumse... | lemma kvm_ineq_II {I : Type*} {i₀ : I} {s : Finset I} (hs : ¬ i₀ ∈ s)
(hs' : Finset.Nonempty s) {Y : I → Ω → G} [∀ i, FiniteRange (Y i)]
(hY : (i : I) → Measurable (Y i)) (h_indep : iIndepFun Y μ) :
d[Y i₀; μ # ∑ i ∈ s, Y i; μ] ≤ 2 * ∑ i ∈ s, d[Y i₀; μ # Y i; μ] := by
classical
have : IsProbabilityMeasu... | pfr/blueprint/src/chapter/torsion.tex:84 | pfr/PFR/MoreRuzsaDist.lean:398 |
PFR | kvm_ineq_III | \begin{lemma}[Kaimonovich--Vershik--Madiman inequality, III]\label{klm-3}\lean{kvm_ineq_III}\leanok If $n \geq 1$ and $X, Y_1, \dots, Y_n$ are jointly independent $G$-valued random variables, then
$$d\left[X; \sum_{i=1}^n Y_i\right] \leq d\left[X; Y_1\right] + \frac{1}{2}\left(\bbH\left[ \sum_{i=1}^n Y_i\right] - \b... | lemma kvm_ineq_III {I : Type*} {i₀ i₁ : I} {s : Finset I}
(hs₀ : ¬ i₀ ∈ s) (hs₁ : ¬ i₁ ∈ s) (h01 : i₀ ≠ i₁)
(Y : I → Ω → G) [∀ i, FiniteRange (Y i)]
(hY : ∀ i, Measurable (Y i)) (h_indep : iIndepFun Y μ) :
d[Y i₀; μ # Y i₁ + ∑ i ∈ s, Y i; μ]
≤ d[Y i₀; μ # Y i₁; μ] + (2 : ℝ)⁻¹ * (H[Y i₁ + ∑ i ∈ s, ... | pfr/blueprint/src/chapter/torsion.tex:102 | pfr/PFR/MoreRuzsaDist.lean:493 |
PFR | multiDist | \begin{definition}[Multidistance]\label{multidist-def}\lean{multiDist}\leanok Let $m$ be a positive integer, and let $X_{[m]} = (X_i)_{1 \leq i \leq m}$ be an $m$-tuple of $G$-valued random variables $X_i$. Then we define
\[
D[X_{[m]}] := \bbH[\sum_{i=1}^m \tilde X_i] - \frac{1}{m} \sum_{i=1}^m \bbH[\tilde X_i],
\]
... | def multiDist {m : ℕ} {Ω : Fin m → Type*} (hΩ : ∀ i, MeasureSpace (Ω i))
(X : ∀ i, (Ω i) → G) : ℝ :=
H[fun x ↦ ∑ i, x i; .pi (fun i ↦ (hΩ i).volume.map (X i))] - (m:ℝ)⁻¹ * ∑ i, H[X i]
@[inherit_doc multiDist] notation3:max "D[" X " ; " hΩ "]" => multiDist hΩ X | pfr/blueprint/src/chapter/torsion.tex:164 | pfr/PFR/MoreRuzsaDist.lean:717 |
PFR | multiDist_chainRule | \begin{lemma}[Multidistance chain rule]\label{multidist-chain-rule}\lean{multiDist_chainRule}\uses{multidist-def}\leanok Let $\pi \colon G \to H$ be a homomorphism of abelian groups and let $X_{[m]}$ be a tuple of jointly independent $G$-valued random variables. Then $D[X_{[m]}]$ is equal to
\begin{equation}
... | lemma multiDist_chainRule {G H : Type*} [hG : MeasurableSpace G] [MeasurableSingletonClass G]
[AddCommGroup G] [Fintype G] [hH : MeasurableSpace H]
[MeasurableSingletonClass H] [AddCommGroup H]
[Fintype H] (π : G →+ H) {m : ℕ} {Ω : Type*} (hΩ : MeasureSpace Ω)
{X : Fin m → Ω → G} (hmes : ∀ i, Measurable... | pfr/blueprint/src/chapter/torsion.tex:360 | pfr/PFR/MoreRuzsaDist.lean:1111 |
PFR | multiDist_copy | \begin{lemma}[Multidistance of copy]\label{multidist-copy}\lean{multiDist_copy}\leanok If $X_{[m]} = (X_i)_{1 \leq i \leq m}$ and $Y_{[m]} = (Y_i)_{1 \leq i \leq m}$ are such that $X_i$ and $Y_i$ have the same distribution for each $i$, then $D[X_{[m]}] = D[Y_{[m]}]$.
\end{lemma}
\begin{proof}\uses{multidist-def}\lean... | /-- If `X_i` has the same distribution as `Y_i` for each `i`, then `D[X_[m]] = D[Y_[m]]`. -/
lemma multiDist_copy {m : ℕ} {Ω : Fin m → Type*} {Ω' : Fin m → Type*}
(hΩ : ∀ i, MeasureSpace (Ω i)) (hΩ': ∀ i, MeasureSpace (Ω' i))
(X : ∀ i, (Ω i) → G) (X' : ∀ i, (Ω' i) → G)
(hident : ∀ i, IdentDistrib (X i) (X' ... | pfr/blueprint/src/chapter/torsion.tex:171 | pfr/PFR/MoreRuzsaDist.lean:723 |
PFR | multiDist_indep | \begin{lemma}[Multidistance of independent variables]\label{multidist-indep}\lean{multiDist_indep}\leanok If $X_{[m]} = (X_i)_{1 \leq i \leq m}$ are jointly independent, then $D[X_{[m]}] = \bbH[\sum_{i=1}^m X_i] - \frac{1}{m} \sum_{i=1}^m \bbH[X_i]$.
\end{lemma}
\begin{proof}\uses{multidist-def} Clear from definition... | /-- If `X_i` are independent, then `D[X_{[m]}] = H[∑_{i=1}^m X_i] - \frac{1}{m} \sum_{i=1}^m H[X_i]`. -/
lemma multiDist_indep {m : ℕ} {Ω : Type*} (hΩ : MeasureSpace Ω) (X : Fin m → Ω → G)
(h_indep : iIndepFun X) :
D[X ; fun _ ↦ hΩ] = H[∑ i, X i] - (∑ i, H[X i]) / m := by sorry | pfr/blueprint/src/chapter/torsion.tex:177 | pfr/PFR/MoreRuzsaDist.lean:733 |
PFR | multiDist_nonneg | \begin{lemma}[Nonnegativity]\label{multidist-nonneg}\lean{multiDist_nonneg}\uses{multidist-def}\leanok For any such tuple, we have $D[X_{[m]}] \geq 0$.
\end{lemma}
\begin{proof}\uses{sumset-lower} From \Cref{sumset-lower} one has
$$ \bbH[\sum_{i =1}^m \tilde X_i] \geq \bbH[\tilde X_i]$$
for each $1 \leq i \leq m$. A... | /-- We have `D[X_[m]] ≥ 0`. -/
lemma multiDist_nonneg [Fintype G] {m : ℕ} {Ω : Fin m → Type*} (hΩ : ∀ i, MeasureSpace (Ω i))
(hprob : ∀ i, IsProbabilityMeasure (ℙ : Measure (Ω i))) (X : ∀ i, (Ω i) → G)
(hX : ∀ i, Measurable (X i)) :
0 ≤ D[X ; hΩ] := by
obtain ⟨A, mA, μA, Y, isProb, h_indep, hY⟩ :=
Pro... | pfr/blueprint/src/chapter/torsion.tex:183 | pfr/PFR/MoreRuzsaDist.lean:759 |
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