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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 $\phi[X_1;X_2]\le \phi[X_1|X_1+X_2;\tilde X_2|\tilde X_1+\tilde X_2]$, and then apply \Cref{first-fibre} to get an upper bound for $I_1$.
\end{proof} | /-- $I_1\le 2\eta d[X_1;X_2]$ -/
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] - ρ[X₁ # A])
- η * (ρ[X₂ | X₂ + X₁' # A] - ρ[X₂ # A]) ≤ d[X₁ | X₁ + X₂' # X₂ | X₂ + X₁'] :=
condRho_le_condRuzsaDist_of_phiMinimizes h_min hX₁ hX₂ (by fun_prop) (by fun_prop)
have : k - η * (ρ[X₁ + X₂' # A] - ρ[X₁ # A])
- η * (ρ[X₂ + X₁' # A] - ρ[X₂ # A]) ≤ d[X₁ + X₂' # X₂ + X₁'] :=
le_rdist_of_phiMinimizes h_min (hX₁.add hX₂') (hX₂.add hX₁')
have : ρ[X₁ + X₂' # A] ≤ (ρ[X₁ # A] + ρ[X₂ # A] + d[ X₁ # X₂ ]) / 2 := by
rw [rho_eq_of_identDistrib h₂, (IdentDistrib.refl hX₁.aemeasurable).rdist_eq h₂]
apply rho_of_sum_le hX₁ hX₂' hA
simpa using h_indep.indepFun (show (0 : Fin 4) ≠ 3 by decide)
have : ρ[X₂ + X₁' # A] ≤ (ρ[X₁ # A] + ρ[X₂ # A] + d[ X₁ # X₂ ]) / 2 := by
rw [add_comm, rho_eq_of_identDistrib h₁, h₁.rdist_eq (IdentDistrib.refl hX₂.aemeasurable)]
apply rho_of_sum_le hX₁' hX₂ hA
simpa using h_indep.indepFun (show (2 : Fin 4) ≠ 1 by decide)
have : ρ[X₁ | X₁ + X₂' # A] ≤ (ρ[X₁ # A] + ρ[X₂ # A] + d[ X₁ # X₂ ]) / 2 := by
rw [rho_eq_of_identDistrib h₂, (IdentDistrib.refl hX₁.aemeasurable).rdist_eq h₂]
apply condRho_of_sum_le hX₁ hX₂' hA
simpa using h_indep.indepFun (show (0 : Fin 4) ≠ 3 by decide)
have : ρ[X₂ | X₂ + X₁' # A] ≤ (ρ[X₁ # A] + ρ[X₂ # A] + d[ X₁ # X₂ ]) / 2 := by
have : ρ[X₂ | X₂ + X₁' # A] ≤ (ρ[X₂ # A] + ρ[X₁' # A] + d[ X₂ # X₁' ]) / 2 := by
apply condRho_of_sum_le hX₂ hX₁' hA
simpa using h_indep.indepFun (show (1 : Fin 4) ≠ 2 by decide)
have I : ρ[X₁' # A] = ρ[X₁ # A] := rho_eq_of_identDistrib h₁.symm
have J : d[X₂ # X₁'] = d[X₁ # X₂] := by
rw [rdist_symm, h₁.rdist_eq (IdentDistrib.refl hX₂.aemeasurable)]
linarith
nlinarith
/- *****************************************
Second estimate
********************************************* -/
include hX₁ hX₂ hX₁' hX₂' h₁ h₂ h_indep in | 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 $\phi[X_1;X_2]\le \phi[X_1+\tilde X_1;X_2+\tilde X_2]$, $\phi[X_1;X_2]\le \phi[X_1|X_1+\tilde X_1;X_2|X_2+\tilde X_2]$, and the fibring identity obtained by applying \Cref{cor-fibre} on $(X_1,X_2,\tilde X_1,\tilde X_2)$,
we have $I_2\le \eta (d[X_1;X_1]+d[X_2;X_2])$. Then apply \Cref{I1-I2-diff} to get $I_2\le 2\eta d[X_1;X_2] +\eta(I_2-I_1)$, and rearrange.
\end{proof} | /-- $I_2\le 2\eta d[X_1;X_2] + \frac{\eta}{1-\eta}(2\eta d[X_1;X_2]-I_1)$. -/
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_of_phiMinimizes h_min (hX₁.add hX₁') (hX₂.add hX₂') (μ₁ := ℙ) (μ₂ := ℙ)
have W' : k - η * (ρ[X₁ | X₁ + X₁' # A] - ρ[X₁ # A])
- η * (ρ[X₂ | X₂ + X₂' # A] - ρ[X₂ # A]) ≤ d[X₁ | X₁ + X₁' # X₂ | X₂ + X₂'] :=
condRho_le_condRuzsaDist_of_phiMinimizes h_min hX₁ hX₂ (hX₁.add hX₁') (hX₂.add hX₂')
have Z : 2 * k = d[X₁ + X₁' # X₂ + X₂'] + d[X₁ | X₁ + X₁' # X₂ | X₂ + X₂'] + I₂ :=
I_two_aux' h₁ h₂ h_indep hX₁ hX₂ hX₁' hX₂'
have : ρ[X₁ + X₁' # A] ≤ (ρ[X₁ # A] + ρ[X₁ # A] + d[ X₁ # X₁ ]) / 2 := by
refine (rho_of_sum_le hX₁ hX₁' hA
(by simpa using h_indep.indepFun (show (0 : Fin 4) ≠ 2 by decide))).trans_eq ?_
rw [rho_eq_of_identDistrib h₁.symm,
IdentDistrib.rdist_eq (IdentDistrib.refl hX₁.aemeasurable) h₁]
have : ρ[X₂ + X₂' # A] ≤ (ρ[X₂ # A] + ρ[X₂ # A] + d[ X₂ # X₂ ]) / 2 := by
refine (rho_of_sum_le hX₂ hX₂' hA
(by simpa using h_indep.indepFun (show (1 : Fin 4) ≠ 3 by decide))).trans_eq ?_
rw [rho_eq_of_identDistrib h₂.symm,
IdentDistrib.rdist_eq (IdentDistrib.refl hX₂.aemeasurable) h₂]
have : ρ[X₁ | X₁ + X₁' # A] ≤ (ρ[X₁ # A] + ρ[X₁ # A] + d[ X₁ # X₁ ]) / 2 := by
refine (condRho_of_sum_le hX₁ hX₁' hA
(by simpa using h_indep.indepFun (show (0 : Fin 4) ≠ 2 by decide))).trans_eq ?_
rw [rho_eq_of_identDistrib h₁.symm,
IdentDistrib.rdist_eq (IdentDistrib.refl hX₁.aemeasurable) h₁]
have : ρ[X₂ | X₂ + X₂' # A] ≤ (ρ[X₂ # A] + ρ[X₂ # A] + d[ X₂ # X₂ ]) / 2 := by
refine (condRho_of_sum_le hX₂ hX₂' hA
(by simpa using h_indep.indepFun (show (1 : Fin 4) ≠ 3 by decide))).trans_eq ?_
rw [rho_eq_of_identDistrib h₂.symm,
IdentDistrib.rdist_eq (IdentDistrib.refl hX₂.aemeasurable) h₂]
have : I₂ ≤ η * (d[X₁ # X₁] + d[X₂ # X₂]) := by nlinarith
rw [rdist_add_rdist_eq h₁ h₂ h_indep hX₁ hX₂ hX₁' hX₂'] at this
have one_eta : 0 < 1 - η := by linarith
apply (mul_le_mul_iff_of_pos_left one_eta).1
have : (1 - η) * I₂ ≤ 2 * η * k - I₁ * η := by linarith
apply this.trans_eq
field_simp
ring
/- ****************************************
End Game
******************************************* -/
include h_min in
omit [IsProbabilityMeasure (ℙ : Measure Ω)] in
/-- If $G$-valued random variables $T_1,T_2,T_3$ satisfy $T_1+T_2+T_3=0$, then
$$d[X_1;X_2]\le 3\bbI[T_1:T_2\mid T_3] + (2\bbH[T_3]-\bbH[T_1]-\bbH[T_2])+ \eta(\rho(T_1|T_3)+\rho(T_2|T_3)-\rho(X_1)-\rho(X_2)).$$ -/ | 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 $z$, $D_{KL}(X+z\Vert Y+z)=D_{KL}(X\Vert Y)$ by \Cref{kl-div-inj}. Then apply \Cref{kl-div-convex} with $w_z=\mathbf{P}(Z=z)$.
\end{proof} | 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 + Z ; μ] ≤ KL[X ; μ # Y ; μ] := by
rcases eq_zero_or_isProbabilityMeasure μ with rfl | hμ
· simp [KLDiv]
set X' : G → Ω → G := fun s ↦ (· + s) ∘ X with hX'
set Y' : G → Ω → G := fun s ↦ (· + s) ∘ Y with hY'
have AX' x i : μ.map (X' i) {x} = μ.map X {x - i} := by
rw [hX', ← Measure.map_map (by fun_prop) (by fun_prop),
Measure.map_apply (by fun_prop) (measurableSet_singleton x)]
congr
ext y
simp [sub_eq_add_neg]
have AY' x i : μ.map (Y' i) {x} = μ.map Y {x - i} := by
rw [hY', ← Measure.map_map (by fun_prop) (by fun_prop),
Measure.map_apply (by fun_prop) (measurableSet_singleton x)]
congr
ext y
simp [sub_eq_add_neg]
let w : G → ℝ := fun s ↦ (μ.map Z {s}).toReal
have sum_w : ∑ s, w s = 1 := by
have : IsProbabilityMeasure (μ.map Z) := isProbabilityMeasure_map hZ.aemeasurable
simp [w]
have A x : (μ.map (X + Z) {x}).toReal = ∑ s, w s * (μ.map (X' s) {x}).toReal := by
have : IndepFun X Z μ := h_indep.comp (φ := Prod.fst) (ψ := id) measurable_fst measurable_id
rw [this.map_add_singleton_eq_sum hX hZ, ENNReal.toReal_sum (by simp [ENNReal.mul_eq_top])]
simp only [ENNReal.toReal_mul]
congr with i
congr 1
rw [AX']
have B x : (μ.map (Y + Z) {x}).toReal = ∑ s, w s * (μ.map (Y' s) {x}).toReal := by
have : IndepFun Y Z μ := h_indep.comp (φ := Prod.snd) (ψ := id) measurable_snd measurable_id
rw [this.map_add_singleton_eq_sum hY hZ, ENNReal.toReal_sum (by simp [ENNReal.mul_eq_top])]
simp only [ENNReal.toReal_mul]
congr with i
congr 1
rw [AY']
have : KL[X + Z ; μ # Y + Z; μ] ≤ ∑ s, w s * KL[X' s ; μ # Y' s ; μ] := by
apply KLDiv_of_convex (fun s _ ↦ by simp [w])
· exact A
· exact B
· intro s _ x
rw [AX', AY']
exact habs _
apply this.trans_eq
have C s : KL[X' s ; μ # Y' s ; μ] = KL[X ; μ # Y ; μ] :=
KLDiv_of_comp_inj (add_left_injective s) hX hY
simp_rw [C, ← Finset.sum_mul, sum_w, one_mul]
/-- If $X,Y,Z$ are random variables, with $X,Z$ defined on the same sample space, we define
$$ D_{KL}(X|Z \Vert Y) := \sum_z \mathbf{P}(Z=z) D_{KL}( (X|Z=z) \Vert Y).$$ -/
noncomputable def condKLDiv {S : Type*} (X : Ω → G) (Y : Ω' → G) (Z : Ω → S)
(μ : Measure Ω := by volume_tac) (μ' : Measure Ω' := by volume_tac) : ℝ :=
∑' z, (μ (Z⁻¹' {z})).toReal * KL[X ; (ProbabilityTheory.cond μ (Z⁻¹' {z})) # Y ; μ']
@[inherit_doc condKLDiv]
notation3:max "KL[" X " | " Z " ; " μ " # " Y " ; " μ' "]" => condKLDiv X Y Z μ μ'
@[inherit_doc condKLDiv]
notation3:max "KL[" X " | " Z " # " Y "]" => condKLDiv X Y Z volume volume
/-- If $X, Y$ are $G$-valued random variables, and $Z$ is another random variable
defined on the same sample space as $X$, then
$$D_{KL}((X|Z)\Vert Y) = D_{KL}(X\Vert Y) + \bbH[X] - \bbH[X|Z].$$ -/ | 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 ↔ IdentDistrib X Y μ μ' := by
refine ⟨fun h ↦ ?_, fun h ↦ by simp [KLDiv, h.map_eq]⟩
let νY := μ'.map Y
have : IsProbabilityMeasure νY := isProbabilityMeasure_map hY.aemeasurable
let νX := μ.map X
have : IsProbabilityMeasure νX := isProbabilityMeasure_map hX.aemeasurable
obtain ⟨r, hr⟩ : ∃ (r : ℝ), ∀ x ∈ Finset.univ, (νX {x}).toReal = r * (νY {x}).toReal := by
apply sum_mul_log_div_eq_iff (by simp) (by simp) (fun i _ hi ↦ ?_)
· rw [KLDiv_eq_sum] at h
simpa using h
· simp only [ENNReal.toReal_eq_zero_iff, measure_ne_top, or_false] at hi
simp [habs i hi, νX]
have r_eq : r = 1 := by
have : r * ∑ x, (νY {x}).toReal = ∑ x, (νX {x}).toReal := by
simp only [Finset.mul_sum, Finset.mem_univ, hr]
simpa using this
have : νX = νY := by
apply Measure.ext_iff_singleton.mpr (fun x ↦ ?_)
simpa [r_eq, ENNReal.toReal_eq_toReal] using hr x (Finset.mem_univ _)
exact ⟨hX.aemeasurable, hY.aemeasurable, this⟩
/-- If $S$ is a finite set, $w_s$ is non-negative,
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
$$D_{KL}(X\Vert Y) \le \sum_{s\in S} w_s D_{KL}(X_s\Vert Y_s).$$ -/ | 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_isProbabilityMeasure μ with rfl | hμ
· simp
rcases eq_zero_or_isProbabilityMeasure μ' with rfl | hμ'
· simp
apply le_trans ?_ (sum_mul_log_div_leq (by simp) (by simp) ?_)
· have : IsProbabilityMeasure (μ'.map Y) := isProbabilityMeasure_map hY.aemeasurable
have : IsProbabilityMeasure (μ.map X) := isProbabilityMeasure_map hX.aemeasurable
simp
· intro i _ hi
simp only [ENNReal.toReal_eq_zero_iff, measure_ne_top, or_false] at hi
simp [habs i hi] | 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'] = KL[X ; μ # Y ; μ'] := by
simp only [KLDiv]
rw [← hf.tsum_eq]
· symm
congr with x
have : (Measure.map X μ) {x} = (Measure.map (f ∘ X) μ) {f x} := by
rw [Measure.map_apply, Measure.map_apply]
· rw [Set.preimage_comp, ← Set.image_singleton, Set.preimage_image_eq _ hf]
· exact .comp .of_discrete hX
· exact measurableSet_singleton (f x)
· exact hX
· exact measurableSet_singleton x
have : (Measure.map Y μ') {x} = (Measure.map (f ∘ Y) μ') {f x} := by
rw [Measure.map_apply, Measure.map_apply]
· congr
exact Set.Subset.antisymm (fun ⦃a⦄ ↦ congrArg f) fun ⦃a⦄ a_1 ↦ hf a_1
· exact .comp .of_discrete hY
· exact measurableSet_singleton (f x)
· exact hY
· exact measurableSet_singleton x
congr
· intro y hy
have : Measure.map (f ∘ X) μ {y} ≠ 0 := by
intro h
simp [h] at hy
rw [Measure.map_apply (.comp .of_discrete hX) (measurableSet_singleton y)] at this
have : f ∘ X ⁻¹' {y} ≠ ∅ := by
intro h
simp [h] at this
obtain ⟨z, hz⟩ := Set.nonempty_iff_ne_empty.2 this
simp only [Set.mem_preimage, Function.comp_apply, Set.mem_singleton_iff] at hz
exact Set.mem_range.2 ⟨X z, hz⟩ | 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
$$D_{KL}(X\Vert Y) \le \sum_{s\in S} w_s D_{KL}(X_s\Vert Y_s).$$
\end{lemma}
\begin{proof}\leanok
\uses{kl-div,log-sum}
For each $x$, replace $\log \frac{\mathbf{P}(X_s=x)}{\mathbf{P}(Y_s=x)}$ in the definition with $\log \frac{w_s\mathbf{P}(X_s=x)}{w_s\mathbf{P}(Y_s=x)}$ for each $s$, and apply \Cref{log-sum}.
\end{proof} | 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'''.map (Y' s) {x}).toReal)
(habs : ∀ s ∈ S, ∀ x, μ'''.map (Y' s) {x} = 0 → μ''.map (X' s) {x} = 0) :
KL[X ; μ # Y ; μ'] ≤ ∑ s ∈ S, w s * KL[X' s ; μ'' # Y' s ; μ'''] := by
conv_lhs => rw [KLDiv_eq_sum]
have A x : (μ.map X {x}).toReal * log ((μ.map X {x}).toReal / (μ'.map Y {x}).toReal)
≤ ∑ s ∈ S, (w s * (μ''.map (X' s) {x}).toReal) *
log ((w s * (μ''.map (X' s) {x}).toReal) / (w s * (μ'''.map (Y' s) {x}).toReal)) := by
rw [hconvex, hconvex']
apply sum_mul_log_div_leq (fun s hs ↦ ?_) (fun s hs ↦ ?_) (fun s hs h's ↦ ?_)
· exact mul_nonneg (by simp [hw s hs]) (by simp)
· exact mul_nonneg (by simp [hw s hs]) (by simp)
· rcases mul_eq_zero.1 h's with h | h
· simp [h]
· simp only [ENNReal.toReal_eq_zero_iff, measure_ne_top, or_false] at h
simp [habs s hs x h]
have B x : (μ.map X {x}).toReal * log ((μ.map X {x}).toReal / (μ'.map Y {x}).toReal)
≤ ∑ s ∈ S, (w s * (μ''.map (X' s) {x}).toReal) *
log ((μ''.map (X' s) {x}).toReal / (μ'''.map (Y' s) {x}).toReal) := by
apply (A x).trans_eq
apply Finset.sum_congr rfl (fun s _ ↦ ?_)
rcases eq_or_ne (w s) 0 with h's | h's
· simp [h's]
· congr 2
rw [mul_div_mul_left _ _ h's]
apply (Finset.sum_le_sum (fun x _ ↦ B x)).trans_eq
rw [Finset.sum_comm]
simp_rw [mul_assoc, ← Finset.mul_sum, KLDiv_eq_sum] | 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}.
If $|H| \leq |A|$ then we are already done thanks to~\eqref{ah}. If $|H| > |A|$ then we can cover $H$ by at most $2 |H|/|A|$ translates of a subspace $H'$ of $H$ with $|H'| \leq |A|$. We can thus cover $A$ by at most
\[2K^{13/2} \frac{|H|^{1/2}}{|A|^{1/2}}\]
translates of $H'$, and the claim again follows from~\eqref{ah}.
\end{proof} | 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_aux' h₀A hA
-- consider the subgroup `H` given by Lemma `PFR_conjecture_aux`.
obtain ⟨H, c, hc, IHA, IAH, A_subs_cH⟩ : ∃ (H : Submodule (ZMod 2) G) (c : Set G),
Nat.card c ≤ K ^ (13/2) * Nat.card A ^ (1/2) * Nat.card H ^ (-1/2)
∧ Nat.card H ≤ K ^ 11 * Nat.card A ∧ Nat.card A ≤ K ^ 11 * Nat.card H
∧ A ⊆ c + H :=
PFR_conjecture_aux h₀A hA
have H_pos : (0 : ℝ) < Nat.card H := by
have : 0 < Nat.card H := Nat.card_pos; positivity
rcases le_or_lt (Nat.card H) (Nat.card A) with h|h
-- If `#H ≤ #A`, then `H` satisfies the conclusion of the theorem
· refine ⟨H, c, ?_, h, A_subs_cH⟩
calc
Nat.card c ≤ K ^ (13/2 : ℝ) * Nat.card A ^ (1/2 : ℝ) * Nat.card H ^ (-1/2 : ℝ) := hc
_ ≤ K ^ (13/2 : ℝ) * (K ^ 11 * Nat.card H) ^ (1/2 : ℝ) * Nat.card H ^ (-1/2 : ℝ) := by gcongr
_ = K ^ 12 := by rpow_ring; norm_num
_ < 2 * K ^ 12 := by linarith [show 0 < K ^ 12 by positivity]
-- otherwise, we decompose `H` into cosets of one of its subgroups `H'`, chosen so that
-- `#A / 2 < #H' ≤ #A`. This `H'` satisfies the desired conclusion.
· obtain ⟨H', IH'A, IAH', H'H⟩ : ∃ H' : Submodule (ZMod 2) G, Nat.card H' ≤ Nat.card A
∧ Nat.card A < 2 * Nat.card H' ∧ H' ≤ H := by
have A_pos' : 0 < Nat.card A := mod_cast A_pos
exact ZModModule.exists_submodule_subset_card_le Nat.prime_two H h.le A_pos'.ne'
have : (Nat.card A / 2 : ℝ) < Nat.card H' := by
rw [div_lt_iff₀ zero_lt_two, mul_comm]; norm_cast
have H'_pos : (0 : ℝ) < Nat.card H' := by
have : 0 < Nat.card H' := Nat.card_pos; positivity
obtain ⟨u, HH'u, hu⟩ :=
H'.toAddSubgroup.exists_left_transversal_of_le (H := H.toAddSubgroup) H'H
dsimp at HH'u
refine ⟨H', c + u, ?_, IH'A, by rwa [add_assoc, HH'u]⟩
calc
(Nat.card (c + u) : ℝ)
≤ Nat.card c * Nat.card u := mod_cast natCard_add_le
_ ≤ (K ^ (13/2 : ℝ) * Nat.card A ^ (1 / 2 : ℝ) * (Nat.card H ^ (-1 / 2 : ℝ)))
* (Nat.card H / Nat.card H') := by
gcongr
apply le_of_eq
rw [eq_div_iff H'_pos.ne']
norm_cast
_ < (K ^ (13/2) * Nat.card A ^ (1 / 2) * (Nat.card H ^ (-1 / 2)))
* (Nat.card H / (Nat.card A / 2)) := by
gcongr
_ = 2 * K ^ (13/2) * Nat.card A ^ (-1/2) * Nat.card H ^ (1/2) := by
field_simp
rpow_ring
norm_num
_ ≤ 2 * K ^ (13/2) * Nat.card A ^ (-1/2) * (K ^ 11 * Nat.card A) ^ (1/2) := by
gcongr
_ = 2 * K ^ 12 := by
rpow_ring
norm_num
/-- Corollary of `PFR_conjecture` in which the ambient group is not required to be finite (but) then
`H` and `c` are finite. -/ | 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{proof}\uses{pfr}\leanok Apply \Cref{pfr} to the group generated by $A$, which is isomorphic to $\F_2^n$ for some $n$.
\end{proof} | 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 ⊆ c + H := by
let G' := Submodule.span (ZMod 2) A
let G'fin : Fintype G' := (Afin.submoduleSpan _).fintype
let ι : G'→ₗ[ZMod 2] G := G'.subtype
have ι_inj : Injective ι := G'.toAddSubgroup.subtype_injective
let A' : Set G' := ι ⁻¹' A
have A_rg : A ⊆ range ι := by
simp only [AddMonoidHom.coe_coe, Submodule.coe_subtype, Subtype.range_coe_subtype, G', ι]
exact Submodule.subset_span
have cardA' : Nat.card A' = Nat.card A := Nat.card_preimage_of_injective ι_inj A_rg
have hA' : Nat.card (A' + A') ≤ K * Nat.card A' := by
rwa [cardA', ← preimage_add _ ι_inj A_rg A_rg,
Nat.card_preimage_of_injective ι_inj (add_subset_range _ A_rg A_rg)]
rcases PFR_conjecture (h₀A.preimage' A_rg) hA' with ⟨H', c', hc', hH', hH'₂⟩
refine ⟨H'.map ι , ι '' c', toFinite _, toFinite (ι '' H'), ?_, ?_, fun x hx ↦ ?_⟩
· rwa [Nat.card_image_of_injective ι_inj]
· erw [Nat.card_image_of_injective ι_inj, ← cardA']
exact hH'
· erw [← image_add]
exact ⟨⟨x, Submodule.subset_span hx⟩, hH'₂ hx, rfl⟩ | 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}].
\end{equation}
\end{lemma}
\begin{proof}
\uses{entropy-pfr, unif-exist, uniform-entropy-II, jensen-bound,ruz-dist-def,ruzsa-diff,bound-conc,ruz-cov}\leanok
Let $U_A$ be the uniform distribution on $A$ (which exists by \Cref{unif-exist}), thus $\bbH[U_A] = \log |A|$ by \Cref{uniform-entropy-II}. By \Cref{jensen-bound} and the fact that $U_A + U_A$ is supported on $A + A$, $\bbH[U_A + U_A] \leq \log|A+A|$. By \Cref{ruz-dist-def}, the doubling condition $|A+A| \leq K|A|$ therefore gives
\[d[U_A;U_A] \leq \log K.\]
By \Cref{entropy-pfr}, we may thus find a subspace $H$ of $\F_2^n$ such that
\begin{equation}\label{uauh} d[U_A;U_H] \leq \tfrac{1}{2} C' \log K\end{equation}
with $C' = 11$.
By \Cref{ruzsa-diff} we conclude that
\begin{equation*}
|\log |H| - \log |A|| \leq C' \log K,
\end{equation*}
proving~\eqref{ah}.
From \Cref{ruz-dist-def},~\eqref{uauh} is equivalent to
\[\bbH[U_A - U_H] \leq \log( |A|^{1/2} |H|^{1/2}) + \tfrac{1}{2} C' \log K.\]
By \Cref{bound-conc} we conclude the existence of a point $x_0 \in \F_p^n$ such that
\[p_{U_A-U_H}(x_0) \geq |A|^{-1/2} |H|^{-1/2} K^{-C'/2},\]
or equivalently
\[|A \cap (H + x_0)| \geq K^{-C'/2} |A|^{1/2} |H|^{1/2}.\]
Applying \Cref{ruz-cov}, we may thus cover $A$ by at most
\[\frac{|A + (A \cap (H+x_0))|}{|A \cap (H + x_0)|} \leq \frac{K|A|}{K^{-C'/2} |A|^{1/2} |H|^{1/2}} = K^{C'/2+1} \frac{|A|^{1/2}}{|H|^{1/2}}\]
translates of
\[\bigl(A \cap (H + x_0)\bigr) - \bigl(A \cap (H + x_0)\bigr) \subseteq H.\]
This proves the claim.
\end{proof} | 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
classical
have A_fin : Finite A := by infer_instance
let _mG : MeasurableSpace G := ⊤
rw [sumset_eq_sub] at hA
have : MeasurableSingletonClass G := ⟨λ _ ↦ trivial⟩
obtain ⟨A_pos, -, K_pos⟩ : (0 : ℝ) < Nat.card A ∧ (0 : ℝ) < Nat.card (A - A) ∧ 0 < K :=
PFR_conjecture_pos_aux h₀A hA
let A' := A.toFinite.toFinset
have h₀A' : Finset.Nonempty A' := by simpa [Finset.Nonempty, A'] using h₀A
have hAA' : A' = A := Finite.coe_toFinset (toFinite A)
rcases exists_isUniform_measureSpace A' h₀A' with ⟨Ω₀, mΩ₀, UA, hP₀, UAmeas, UAunif, -, -⟩
rw [hAA'] at UAunif
have : d[UA # UA] ≤ log K := rdist_le_of_isUniform_of_card_add_le h₀A hA UAunif UAmeas
rw [← sumset_eq_sub] at hA
let p : refPackage Ω₀ Ω₀ G := ⟨UA, UA, UAmeas, UAmeas, 1/9, (by norm_num), (by norm_num)⟩
-- entropic PFR gives a subgroup `H` which is close to `A` for the Rusza distance
rcases entropic_PFR_conjecture p (by norm_num) with ⟨H, Ω₁, mΩ₁, UH, hP₁, UHmeas, UHunif, hUH⟩
have H_fin : (H : Set G).Finite := (H : Set G).toFinite
rcases independent_copies_two UAmeas UHmeas
with ⟨Ω, mΩ, VA, VH, hP, VAmeas, VHmeas, Vindep, idVA, idVH⟩
have VAunif : IsUniform A VA := UAunif.of_identDistrib idVA.symm .of_discrete
have VA'unif := VAunif
rw [← hAA'] at VA'unif
have VHunif : IsUniform H VH := UHunif.of_identDistrib idVH.symm .of_discrete
let H' := (H : Set G).toFinite.toFinset
have hHH' : H' = (H : Set G) := (toFinite (H : Set G)).coe_toFinset
have VH'unif := VHunif
rw [← hHH'] at VH'unif
have : d[VA # VH] ≤ 11/2 * log K := by rw [idVA.rdist_eq idVH]; linarith
have H_pos : (0 : ℝ) < Nat.card H := by
have : 0 < Nat.card H := Nat.card_pos
positivity
have VA_ent : H[VA] = log (Nat.card A) := VAunif.entropy_eq' A_fin VAmeas
have VH_ent : H[VH] = log (Nat.card H) := VHunif.entropy_eq' H_fin VHmeas
have Icard : |log (Nat.card A) - log (Nat.card H)| ≤ 11 * log K := by
rw [← VA_ent, ← VH_ent]
apply (diff_ent_le_rdist VAmeas VHmeas).trans
linarith
have IAH : Nat.card A ≤ K ^ 11 * Nat.card H := by
have : log (Nat.card A) ≤ log K * 11 + log (Nat.card H) := by
linarith [(le_abs_self _).trans Icard]
convert exp_monotone this using 1
· exact (exp_log A_pos).symm
· rw [exp_add, exp_log H_pos, ← rpow_def_of_pos K_pos]
have IHA : Nat.card H ≤ K ^ 11 * Nat.card A := by
have : log (Nat.card H) ≤ log K * 11 + log (Nat.card A) := by
linarith [(neg_le_abs _).trans Icard]
convert exp_monotone this using 1
· exact (exp_log H_pos).symm
· rw [exp_add, exp_log A_pos, ← rpow_def_of_pos K_pos]
-- entropic PFR shows that the entropy of `VA - VH` is small
have I : log K * (-11/2) + log (Nat.card A) * (-1/2) + log (Nat.card H) * (-1/2)
≤ - H[VA - VH] := by
rw [Vindep.rdist_eq VAmeas VHmeas] at this
linarith
-- therefore, there exists a point `x₀` which is attained by `VA - VH` with a large probability
obtain ⟨x₀, h₀⟩ : ∃ x₀ : G, rexp (- H[VA - VH]) ≤ (ℙ : Measure Ω).real ((VA - VH) ⁻¹' {x₀}) :=
prob_ge_exp_neg_entropy' _ ((VAmeas.sub VHmeas).comp measurable_id')
-- massage the previous inequality to get that `A ∩ (H + {x₀})` is large
have J : K ^ (-11/2 : ℝ) * Nat.card A ^ (1/2) * Nat.card H ^ (1/2 : ℝ) ≤
Nat.card (A ∩ (H + {x₀}) : Set G) := by
rw [VA'unif.measureReal_preimage_sub VAmeas VH'unif VHmeas Vindep] at h₀
have := (Real.exp_monotone I).trans h₀
have hAA'_card : Nat.card A' = Nat.card A := congrArg Nat.card (congrArg Subtype hAA')
have hHH'_card : Nat.card H' = Nat.card H := congrArg Nat.card (congrArg Subtype hHH')
rw [hAA'_card, hHH'_card, le_div_iff₀] at this
convert this using 1
· rw [exp_add, exp_add, ← rpow_def_of_pos K_pos, ← rpow_def_of_pos A_pos,
← rpow_def_of_pos H_pos]
rpow_ring
norm_num
· rw [hAA', hHH']
positivity
have Hne : (A ∩ (H + {x₀} : Set G)).Nonempty := by
by_contra h'
have : (0 : ℝ) < Nat.card (A ∩ (H + {x₀}) : Set G) := lt_of_lt_of_le (by positivity) J
simp only [Nat.card_eq_fintype_card, card_of_isEmpty, CharP.cast_eq_zero, lt_self_iff_false,
not_nonempty_iff_eq_empty.1 h'] at this
/- use Rusza covering lemma to cover `A` by few translates of `A ∩ (H + {x₀}) - A ∩ (H + {x₀})`
(which is contained in `H`). The number of translates is at most
`#(A + (A ∩ (H + {x₀}))) / #(A ∩ (H + {x₀}))`, where the numerator is controlled as this is
a subset of `A + A`, and the denominator is bounded below by the previous inequality`. -/
have Z3 :
(Nat.card (A + A ∩ (↑H + {x₀})) : ℝ) ≤ (K ^ (13/2 : ℝ) * Nat.card A ^ (1/2 : ℝ) *
Nat.card H ^ (-1/2 : ℝ)) * Nat.card ↑(A ∩ (↑H + {x₀})) := by
calc
(Nat.card (A + A ∩ (↑H + {x₀})) : ℝ)
_ ≤ Nat.card (A + A) := by
gcongr; exact Nat.card_mono (toFinite _) <| add_subset_add_left inter_subset_left
_ ≤ K * Nat.card A := hA
_ = (K ^ (13/2 : ℝ) * Nat.card A ^ (1/2 : ℝ) * Nat.card H ^ (-1/2 : ℝ)) *
(K ^ (-11/2 : ℝ) * Nat.card A ^ (1/2 : ℝ) * Nat.card H ^ (1/2 : ℝ)) := by
rpow_ring; norm_num
_ ≤ (K ^ (13/2 : ℝ) * Nat.card A ^ (1/2 : ℝ) * Nat.card H ^ (-1/2 : ℝ)) *
Nat.card ↑(A ∩ (↑H + {x₀})) := by gcongr
obtain ⟨u, huA, hucard, hAu, -⟩ :=
Set.ruzsa_covering_add (toFinite A) (toFinite (A ∩ ((H + {x₀} : Set G)))) Hne (by convert Z3)
have A_subset_uH : A ⊆ u + H := by
refine hAu.trans $ add_subset_add_left $
(sub_subset_sub (inter_subset_right ..) (inter_subset_right ..)).trans ?_
rw [add_sub_add_comm, singleton_sub_singleton, sub_self]
simp
exact ⟨H, u, hucard, IHA, IAH, A_subset_uH⟩
/-- The polynomial Freiman-Ruzsa (PFR) conjecture: if `A` is a subset of an elementary abelian
2-group of doubling constant at most `K`, then `A` can be covered by at most `2 * K ^ 12` cosets of
a subgroup of cardinality at most `|A|`. -/ | 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 repeating the proof of \Cref{pfr} and using \Cref{pfr_aux-improv} one can obtain the claim with $11$ replaced by $10$.
\end{proof} | 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_conjecture_pos_aux' h₀A hA
-- consider the subgroup `H` given by Lemma `PFR_conjecture_aux`.
obtain ⟨H, c, hc, IHA, IAH, A_subs_cH⟩ : ∃ (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 :=
PFR_conjecture_improv_aux h₀A hA
have H_pos : (0 : ℝ) < Nat.card H := by
have : 0 < Nat.card H := Nat.card_pos; positivity
rcases le_or_lt (Nat.card H) (Nat.card A) with h|h
-- If `#H ≤ #A`, then `H` satisfies the conclusion of the theorem
· refine ⟨H, c, ?_, h, A_subs_cH⟩
calc
Nat.card c ≤ K ^ 6 * Nat.card A ^ (1/2) * Nat.card H ^ (-1/2) := hc
_ ≤ K ^ 6 * (K ^ 10 * Nat.card H) ^ (1/2) * Nat.card H ^ (-1/2) := by
gcongr
_ = K ^ 11 := by rpow_ring; norm_num
_ < 2 * K ^ 11 := by linarith [show 0 < K ^ 11 by positivity]
-- otherwise, we decompose `H` into cosets of one of its subgroups `H'`, chosen so that
-- `#A / 2 < #H' ≤ #A`. This `H'` satisfies the desired conclusion.
· obtain ⟨H', IH'A, IAH', H'H⟩ : ∃ H' : Submodule (ZMod 2) G, Nat.card H' ≤ Nat.card A
∧ Nat.card A < 2 * Nat.card H' ∧ H' ≤ H := by
have A_pos' : 0 < Nat.card A := mod_cast A_pos
exact ZModModule.exists_submodule_subset_card_le Nat.prime_two H h.le A_pos'.ne'
have : (Nat.card A / 2 : ℝ) < Nat.card H' := by
rw [div_lt_iff₀ zero_lt_two, mul_comm]; norm_cast
have H'_pos : (0 : ℝ) < Nat.card H' := by
have : 0 < Nat.card H' := Nat.card_pos; positivity
obtain ⟨u, HH'u, hu⟩ :=
H'.toAddSubgroup.exists_left_transversal_of_le (H := H.toAddSubgroup) H'H
dsimp at HH'u
refine ⟨H', c + u, ?_, IH'A, by rwa [add_assoc, HH'u]⟩
calc
(Nat.card (c + u) : ℝ)
≤ Nat.card c * Nat.card u := mod_cast natCard_add_le
_ ≤ (K ^ 6 * Nat.card A ^ (1 / 2) * (Nat.card H ^ (-1 / 2)))
* (Nat.card H / Nat.card H') := by
gcongr
apply le_of_eq
rw [eq_div_iff H'_pos.ne']
norm_cast
_ < (K ^ 6 * Nat.card A ^ (1 / 2) * (Nat.card H ^ (-1 / 2)))
* (Nat.card H / (Nat.card A / 2)) := by
gcongr
_ = 2 * K ^ 6 * Nat.card A ^ (-1/2) * Nat.card H ^ (1/2) := by
field_simp
rpow_ring
norm_num
_ ≤ 2 * K ^ 6 * Nat.card A ^ (-1/2) * (K ^ 10 * Nat.card A) ^ (1/2) := by
gcongr
_ = 2 * K ^ 11 := by
rpow_ring
norm_num
/-- Corollary of `PFR_conjecture_improv` in which the ambient group is not required to be finite
(but) then $H$ and $c$ are finite. -/ | 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{entropy-pfr-improv, unif-exist, uniform-entropy-II, jensen-bound,ruz-dist-def,ruzsa-diff,bound-conc,ruz-cov}\leanok
By repeating the proof of \Cref{pfr_aux} and using \Cref{entropy-pfr-improv} one can obtain the claim with $13/2$
replaced with $6$ and $11$ replaced by $10$.
\end{proof} | 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 : Finite A := by infer_instance
classical
let mG : MeasurableSpace G := ⊤
have : MeasurableSingletonClass G := ⟨λ _ ↦ trivial⟩
obtain ⟨A_pos, -, K_pos⟩ : (0 : ℝ) < Nat.card A ∧ (0 : ℝ) < Nat.card (A + A) ∧ 0 < K :=
PFR_conjecture_pos_aux' h₀A hA
let A' := A.toFinite.toFinset
have h₀A' : Finset.Nonempty A' := by
simp [A', Finset.Nonempty]
exact h₀A
have hAA' : A' = A := Finite.coe_toFinset (toFinite A)
rcases exists_isUniform_measureSpace A' h₀A' with ⟨Ω₀, mΩ₀, UA, hP₀, UAmeas, UAunif, -⟩
rw [hAA'] at UAunif
have hadd_sub : A + A = A - A := by ext; simp [mem_add, mem_sub, ZModModule.sub_eq_add]
rw [hadd_sub] at hA
have : d[UA # UA] ≤ log K := rdist_le_of_isUniform_of_card_add_le h₀A hA UAunif UAmeas
rw [← hadd_sub] at hA
let p : refPackage Ω₀ Ω₀ G := ⟨UA, UA, UAmeas, UAmeas, 1/8, (by norm_num), (by norm_num)⟩
-- entropic PFR gives a subgroup `H` which is close to `A` for the Rusza distance
rcases entropic_PFR_conjecture_improv p (by norm_num)
with ⟨H, Ω₁, mΩ₁, UH, hP₁, UHmeas, UHunif, hUH⟩
rcases independent_copies_two UAmeas UHmeas
with ⟨Ω, mΩ, VA, VH, hP, VAmeas, VHmeas, Vindep, idVA, idVH⟩
have VAunif : IsUniform A VA := UAunif.of_identDistrib idVA.symm .of_discrete
have VA'unif := VAunif
rw [← hAA'] at VA'unif
have VHunif : IsUniform H VH := UHunif.of_identDistrib idVH.symm .of_discrete
let H' := (H : Set G).toFinite.toFinset
have hHH' : H' = (H : Set G) := Finite.coe_toFinset (toFinite (H : Set G))
have VH'unif := VHunif
rw [← hHH'] at VH'unif
have H_fin : Finite (H : Set G) := by infer_instance
have : d[VA # VH] ≤ 5 * log K := by rw [idVA.rdist_eq idVH]; linarith
have H_pos : (0 : ℝ) < Nat.card H := by
have : 0 < Nat.card H := Nat.card_pos
positivity
have VA_ent : H[VA] = log (Nat.card A) := IsUniform.entropy_eq' A_fin VAunif VAmeas
have VH_ent : H[VH] = log (Nat.card H) := IsUniform.entropy_eq' H_fin VHunif VHmeas
have Icard : |log (Nat.card A) - log (Nat.card H)| ≤ 10 * log K := by
rw [← VA_ent, ← VH_ent]
apply (diff_ent_le_rdist VAmeas VHmeas).trans
linarith
have IAH : Nat.card A ≤ K ^ 10 * Nat.card H := by
have : log (Nat.card A) ≤ log K * 10 + log (Nat.card H) := by
linarith [(le_abs_self _).trans Icard]
convert exp_monotone this using 1
· exact (exp_log A_pos).symm
· rw [exp_add, exp_log H_pos, ← rpow_def_of_pos K_pos]
have IHA : Nat.card H ≤ K ^ 10 * Nat.card A := by
have : log (Nat.card H) ≤ log K * 10 + log (Nat.card A) := by
linarith [(neg_le_abs _).trans Icard]
convert exp_monotone this using 1
· exact (exp_log H_pos).symm
· rw [exp_add, exp_log A_pos, ← rpow_def_of_pos K_pos]
-- entropic PFR shows that the entropy of `VA - VH` is small
have I : log K * (-5) + log (Nat.card A) * (-1/2) + log (Nat.card H) * (-1/2)
≤ - H[VA - VH] := by
rw [Vindep.rdist_eq VAmeas VHmeas] at this
linarith
-- therefore, there exists a point `x₀` which is attained by `VA - VH` with a large probability
obtain ⟨x₀, h₀⟩ : ∃ x₀ : G, rexp (- H[VA - VH]) ≤ (ℙ : Measure Ω).real ((VA - VH) ⁻¹' {x₀}) :=
prob_ge_exp_neg_entropy' _ ((VAmeas.sub VHmeas).comp measurable_id')
-- massage the previous inequality to get that `A ∩ (H + {x₀})` is large
have J : K ^ (-5) * Nat.card A ^ (1/2) * Nat.card H ^ (1/2) ≤
Nat.card (A ∩ (H + {x₀}) : Set G) := by
rw [VA'unif.measureReal_preimage_sub VAmeas VH'unif VHmeas Vindep] at h₀
have := (Real.exp_monotone I).trans h₀
have hAA'_card : Nat.card A' = Nat.card A := congrArg Nat.card (congrArg Subtype hAA')
have hHH'_card : Nat.card H' = Nat.card H := congrArg Nat.card (congrArg Subtype hHH')
rw [hAA'_card, hHH'_card, le_div_iff₀] at this
convert this using 1
· rw [exp_add, exp_add, ← rpow_def_of_pos K_pos, ← rpow_def_of_pos A_pos,
← rpow_def_of_pos H_pos]
rpow_ring
norm_num
· rw [hAA', hHH']
positivity
have Hne : (A ∩ (H + {x₀} : Set G)).Nonempty := by
by_contra h'
have : (0 : ℝ) < Nat.card (A ∩ (H + {x₀}) : Set G) := lt_of_lt_of_le (by positivity) J
simp only [Nat.card_eq_fintype_card, card_of_isEmpty, CharP.cast_eq_zero, lt_self_iff_false,
not_nonempty_iff_eq_empty.1 h'] at this
/- use Rusza covering lemma to cover `A` by few translates of `A ∩ (H + {x₀}) - A ∩ (H + {x₀})`
(which is contained in `H`). The number of translates is at most
`#(A + (A ∩ (H + {x₀}))) / #(A ∩ (H + {x₀}))`, where the numerator is controlled as this is
a subset of `A + A`, and the denominator is bounded below by the previous inequality`. -/
have Z3 :
(Nat.card (A + A ∩ (↑H + {x₀})) : ℝ) ≤ (K ^ 6 * Nat.card A ^ (1/2 : ℝ) *
Nat.card H ^ (-1/2 : ℝ)) * Nat.card ↑(A ∩ (↑H + {x₀})) := by
calc
(Nat.card (A + A ∩ (↑H + {x₀})) : ℝ)
_ ≤ Nat.card (A + A) := by
gcongr; exact Nat.card_mono (toFinite _) <| add_subset_add_left inter_subset_left
_ ≤ K * Nat.card A := hA
_ = (K ^ 6 * Nat.card A ^ (1/2 : ℝ) * Nat.card H ^ (-1/2 : ℝ)) *
(K ^ (-5 : ℝ) * Nat.card A ^ (1/2 : ℝ) * Nat.card H ^ (1/2 : ℝ)) := by
rpow_ring; norm_num
_ ≤ (K ^ 6 * Nat.card A ^ (1/2 : ℝ) * Nat.card H ^ (-1/2 : ℝ)) *
Nat.card ↑(A ∩ (↑H + {x₀})) := by gcongr
obtain ⟨u, huA, hucard, hAu, -⟩ :=
Set.ruzsa_covering_add (toFinite A) (toFinite (A ∩ ((H + {x₀} : Set G)))) Hne (by convert Z3)
have A_subset_uH : A ⊆ u + H := by
refine hAu.trans $ add_subset_add_left $
(sub_subset_sub (inter_subset_right ..) (inter_subset_right ..)).trans ?_
rw [add_sub_add_comm, singleton_sub_singleton, sub_self]
simp
exact ⟨H, u, hucard, IHA, IAH, A_subset_uH⟩
/-- The polynomial Freiman-Ruzsa (PFR) conjecture: if $A$ is a subset of an elementary abelian
2-group of doubling constant at most $K$, then $A$ can be covered by at most $2K^{11$} cosets of
a subgroup of cardinality at most $|A|$. -/ | 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 then
\[\mathbb{H}(\psi(X))+\mathbb{H}(\psi(Y))\leq 34 d[\psi(X);\psi(Y)].\]
\end{lemma}
\begin{proof}
\uses{pfr-projection'}\leanok
Specialize \Cref{pfr-projection'} to $\alpha=3/5$. In the second
inequality, it gives a bound $100/3 < 34$.
\end{proof} | 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, h, h'⟩
refine ⟨H, ?_, ?_⟩
· convert h
norm_num
· have : 0 ≤ d[⇑H.mkQ ∘ X ; μ # ⇑H.mkQ ∘ Y ; μ'] :=
rdist_nonneg (.comp .of_discrete hX) (.comp .of_discrete hY)
linarith
end F2_projection
open MeasureTheory ProbabilityTheory Real Set | 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 $\psi:G \to G/H$ is the natural projection then
\[\mathbb{H}(\psi(X))+\mathbb{H}(\psi(Y))\leq \frac{20}{\alpha} d[\psi(X);\psi(Y)].\]
\end{lemma}
\begin{proof}
\uses{app-ent-pfr}\leanok
Let $H\leq \mathbb{F}_2^d$ be a maximal subgroup such that
\[\mathbb{H}(\psi(X))+\mathbb{H}(\psi(Y))> \frac{20}{\alpha} d[\psi(X);\psi(Y)]\]
and such that there exists $c \ge 0$ with
\[\log \lvert H\rvert \leq \frac{1+\alpha}{2(1-\alpha)}(1-c)(\mathbb{H}(X)+\mathbb{H}(Y))\]
and
\[\mathbb{H}(\psi(X))+\mathbb{H}(\psi(Y))\leq c (\mathbb{H}(X)+\mathbb{H}(Y)).\]
Note that this exists since $H=\{0\}$ is an example of such a subgroup or we are done with this choice of $H$.
We know that $G/H$ is a $2$-elementary group and so by Lemma
\ref{app-ent-pfr} there exists some non-trivial subgroup $H'\leq G/H$ such
that
\[\log \lvert H'\rvert < \frac{1+\alpha}{2}(\mathbb{H}(\psi(X))+\mathbb{H}(\psi(Y))\]
and
\[\mathbb{H}(\psi' \circ\psi(X))+\mathbb{H}(\psi' \circ \psi(Y))< \alpha(\mathbb{H}(\psi(X))+\mathbb{H}(\psi(Y)))\]
where $\psi':G/H\to (G/H)/H'$. By group isomorphism theorems we know that
there exists some $H''$ with $H\leq H''\leq G$ such that $H'\cong H''/H$ and
$\psi' \circ \psi(X)=\psi''(X)$ where $\psi'':G\to G/H''$ is the projection
homomorphism.
Since $H'$ is non-trivial we know that $H$ is a proper subgroup of $H''$. On the other hand we know that
\[\log \lvert H''\rvert=\log \lvert H'\rvert+\log \lvert H\rvert< \frac{1+\alpha}{2(1-\alpha)}(1-\alpha c)(\mathbb{H}(X)+\mathbb{H}(Y))\]
and
\[\mathbb{H}(\psi''(X))+\mathbb{H}(\psi''(Y))< \alpha (\mathbb{H}(\psi(X))+\mathbb{H}(\psi(Y)))\leq \alpha c (\mathbb{H}(X)+\mathbb{H}(Y)).\]
Therefore (using the maximality of $H$) it must be the first condition that fails, whence
\[\mathbb{H}(\psi''(X))+\mathbb{H}(\psi''(Y))\leq \frac{20}{\alpha}d[\psi''(X);\psi''(Y)].\]
\end{proof} | 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 := {H : Submodule (ZMod 2) G | (∃ (c : ℝ), 0 ≤ c ∧
log (Nat.card H) ≤ (1 + α) / (2 * (1 - α)) * (1 - c) * (H[X ; μ] + H[Y;μ']) ∧
H[H.mkQ ∘ X ; μ] + H[H.mkQ ∘ Y; μ'] ≤ c * (H[X ; μ] + H[Y;μ'])) ∧
20 * d[H.mkQ ∘ X ; μ # H.mkQ ∘ Y ; μ'] < α * (H[H.mkQ ∘ X ; μ ] + H[H.mkQ ∘ Y; μ'])}
have : 0 ≤ H[X ; μ] + H[Y ; μ'] := by linarith [entropy_nonneg X μ, entropy_nonneg Y μ']
have : 0 < 1 - α := sub_pos.mpr αone
by_cases hE : ⊥ ∈ S
· classical
obtain ⟨H, ⟨⟨c, hc, hlog, hup⟩, hent⟩, hMaxl⟩ :=
S.toFinite.exists_maximal_wrt id S (Set.nonempty_of_mem hE)
set G' := G ⧸ H
set ψ : G →ₗ[ZMod 2] G' := H.mkQ
have surj : Function.Surjective ψ := Submodule.Quotient.mk_surjective H
obtain ⟨H', hlog', hup'⟩ := app_ent_PFR _ _ _ _ α hent (.comp .of_discrete hX)
(.comp .of_discrete hY)
have H_ne_bot : H' ≠ ⊥ := by
by_contra!
rcases this with rfl
have inj : Function.Injective (Submodule.mkQ (⊥ : Submodule (ZMod 2) G')) :=
QuotientAddGroup.quotientBot.symm.injective
rw [entropy_comp_of_injective _ (.comp .of_discrete hX) _ inj,
entropy_comp_of_injective _ (.comp .of_discrete hY) _ inj] at hup'
nlinarith [entropy_nonneg (ψ ∘ X) μ, entropy_nonneg (ψ ∘ Y) μ']
let H'' := H'.comap ψ
use H''
rw [← (Submodule.map_comap_eq_of_surjective surj _ : H''.map ψ = H')] at hup' hlog'
set H' := H''.map ψ
have Hlt :=
calc
H = (⊥ : Submodule (ZMod 2) G').comap ψ := by simp [ψ]; rw [Submodule.ker_mkQ]
_ < H'' := by rw [Submodule.comap_lt_comap_iff_of_surjective surj]; exact H_ne_bot.bot_lt
let φ : (G' ⧸ H') ≃ₗ[ZMod 2] (G ⧸ H'') := Submodule.quotientQuotientEquivQuotient H H'' Hlt.le
set ψ' : G' →ₗ[ZMod 2] G' ⧸ H' := H'.mkQ
set ψ'' : G →ₗ[ZMod 2] G ⧸ H'' := H''.mkQ
have diag : ψ' ∘ ψ = φ.symm ∘ ψ'' := rfl
rw [← Function.comp_assoc, ← Function.comp_assoc, diag, Function.comp_assoc,
Function.comp_assoc] at hup'
have cond : log (Nat.card H'') ≤
(1 + α) / (2 * (1 - α)) * (1 - α * c) * (H[X ; μ] + H[Y;μ']) := by
have cardprod : Nat.card H'' = Nat.card H' * Nat.card H := by
have hcard₀ := Nat.card_congr <| (Submodule.comapSubtypeEquivOfLe Hlt.le).toEquiv
have hcard₁ := Nat.card_congr <| (ψ.domRestrict H'').quotKerEquivRange.toEquiv
have hcard₂ := (H.comap H''.subtype).card_eq_card_quotient_mul_card
rw [ψ.ker_domRestrict H'', Submodule.ker_mkQ, ψ.range_domRestrict H''] at hcard₁
simpa only [← Nat.card_eq_fintype_card, hcard₀, hcard₁, mul_comm] using hcard₂
calc
log (Nat.card H'')
_ = log (Nat.card H' * Nat.card H) := by rw [cardprod]; norm_cast
_ = log (Nat.card H') + log (Nat.card H) := by
rw [Real.log_mul (Nat.cast_ne_zero.2 (@Nat.card_pos H').ne')
(Nat.cast_ne_zero.2 (@Nat.card_pos H).ne')]
_ ≤ (1 + α) / 2 * (H[ψ ∘ X ; μ] + H[ψ ∘ Y ; μ']) + log (Nat.card H) := by gcongr
_ ≤ (1 + α) / 2 * (c * (H[X ; μ] + H[Y;μ'])) +
(1 + α) / (2 * (1 - α)) * (1 - c) * (H[X ; μ] + H[Y ; μ']) := by gcongr
_ = (1 + α) / (2 * (1 - α)) * (1 - α * c) * (H[X ; μ] + H[Y ; μ']) := by
field_simp; ring
have HS : H'' ∉ S := λ Hs => Hlt.ne (hMaxl H'' Hs Hlt.le)
simp only [S, Set.mem_setOf_eq, not_and, not_lt] at HS
refine ⟨?_, HS ⟨α * c, by positivity, cond, ?_⟩⟩
· calc
log (Nat.card H'')
_ ≤ (1 + α) / (2 * (1 - α)) * (1 - α * c) * (H[X ; μ] + H[Y;μ']) := cond
_ ≤ (1 + α) / (2 * (1 - α)) * 1 * (H[X ; μ] + H[Y;μ']) := by gcongr; simp; positivity
_ = (1 + α) / (2 * (1 - α)) * (H[X ; μ] + H[Y;μ']) := by simp only [mul_one]
· calc
H[ ψ'' ∘ X ; μ ] + H[ ψ'' ∘ Y; μ' ]
_ = H[ φ.symm ∘ ψ'' ∘ X ; μ ] + H[ φ.symm ∘ ψ'' ∘ Y; μ' ] := by
simp_rw [← entropy_comp_of_injective _ (.comp .of_discrete hX) _ φ.symm.injective,
← entropy_comp_of_injective _ (.comp .of_discrete hY) _ φ.symm.injective]
_ ≤ α * (H[ ψ ∘ X ; μ ] + H[ ψ ∘ Y; μ' ]) := hup'.le
_ ≤ α * (c * (H[X ; μ] + H[Y ; μ'])) := by gcongr
_ = (α * c) * (H[X ; μ] + H[Y ; μ']) := by ring
· use ⊥
constructor
· simp only [AddSubgroup.mem_bot, Nat.card_eq_fintype_card, Fintype.card_ofSubsingleton,
Nat.cast_one, log_one]
positivity
· simp only [S, Set.mem_setOf_eq, not_and, not_lt] at hE
exact hE ⟨1, by norm_num, by
norm_num; exact add_le_add (entropy_comp_le μ hX _) (entropy_comp_le μ' hY _)⟩
/-- If $G=\mathbb{F}_2^d$ 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 then
\[\mathbb{H}(\psi(X))+\mathbb{H}(\psi(Y))\leq 34 * d[\psi(X);\psi(Y)].\] -/ | 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-ent}.
\end{proof} | /-- 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, hX.entropy_eq, hY.entropy_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} {X₂ : Ω₂ → G} {X₁' : Ω'₁ → G} {X₂' : Ω'₂ → G}
(h₁ : IdentDistrib X₁ X₁' μ₁ μ'₁) (h₂ : IdentDistrib X₂ X₂' μ₂ μ'₂) :
τ[X₁ ; μ₁ # X₂ ; μ₂ | p] = τ[X₁' ; μ'₁ # X₂' ; μ'₂ | p] := by
simp only [tau]
rw [(IdentDistrib.refl p.hmeas1.aemeasurable).rdist_eq h₁,
(IdentDistrib.refl p.hmeas2.aemeasurable).rdist_eq h₂,
h₁.rdist_eq h₂]
/-- Property recording the fact that two random variables minimize the tau functional. Expressed
in terms of measures on the group to avoid quantifying over all spaces, but this implies comparison
with any pair of random variables, see Lemma `is_tau_min`. -/ | 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 Immediate from \Cref{ruz-dist-def} and Lemmas \ref{relabeled-entropy}, \ref{copy-ent}.
\end{proof} | /-- 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_def]
congr 2
have h_prod : (μ.map X).prod (μ.map Y) = μ.map (⟨X, Y⟩) :=
((indepFun_iff_map_prod_eq_prod_map_map hX.aemeasurable hY.aemeasurable).mp h).symm
rw [h_prod, entropy_def, Measure.map_map (measurable_fst.sub measurable_snd) (hX.prodMk hY)]
rfl | 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
\[\log \lvert H\rvert <\frac{1+\alpha}{2}(\mathbb{H}(X)+\mathbb{H}(Y))\] and
\[\mathbb{H}(\psi(X))+\mathbb{H}(\psi(Y))< \alpha (\mathbb{H}(X)+\mathbb{H}(Y))\]
where $\psi:G\to G/H$ is the natural projection homomorphism.
\end{lemma}
\begin{proof}
\uses{entropy-pfr-improv, ruzsa-diff, dist-projection, ruzsa-nonneg}\leanok
By \Cref{entropy-pfr-improv} there exists a subgroup $H$ such that
$d[X;U_H] + d[Y;U_H] \leq 10 d[X;Y]$. Using \Cref{dist-projection} we
deduce that $\mathbb{H}(\psi(X)) + \mathbb{H}(\psi(X)) \leq 20 d[X;Y]$. The
second claim follows adding these inequalities and using the assumption on
$\mathbb{H}(X)+\mathbb{H}(Y)$.
Furthermore we have by \Cref{ruzsa-diff}
\[\log \lvert H \rvert-\mathbb{H}(X)\leq 2d[X;U_H]\]
and similarly for $Y$ and thus
\begin{align*}
\log \lvert H\rvert
&\leq
\frac{\mathbb{H}(X)+\mathbb{H}(Y)}{2}+d[X;U_H] + d[Y;U_H] \leq
\frac{\mathbb{H}(X)+\mathbb{H}(Y)}{2}+ 10d[X;Y]
\\& <
\frac{1+\alpha}{2}(\mathbb{H}(X)+\mathbb{H}(Y)).
\end{align*}
Finally note that if $H$
were trivial then $\psi(X)=X$ and $\psi(Y)=Y$ and hence
$\mathbb{H}(X)+\mathbb{H}(Y)=0$, which contradicts \Cref{ruzsa-nonneg}.
\end{proof} | 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) (mΩ' := .mk μ') X Y hent hX hY
set_option maxHeartbeats 300000 in
/-- If $G=\mathbb{F}_2^d$ and `X, Y` are `G`-valued random variables and $\alpha < 1$ then there is
a subgroup $H\leq \mathbb{F}_2^d$ such that
\[\log \lvert H\rvert \leq (1 + α) / (2 * (1 - α)) * (\mathbb{H}(X)+\mathbb{H}(Y))\]
and if $\psi:G \to G/H$ is the natural projection then
\[\mathbb{H}(\psi(X))+\mathbb{H}(\psi(Y))\leq 20/\alpha * d[\psi(X);\psi(Y)].\] -/ | 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 $\phi: G \to G'$ and a constant $c \in G'$
such that $f(x) = \phi(x)+c$ for at least $|G| / (2 ^ {144} * K ^ {122})$
values of $x \in G$.
\end{theorem}
\begin{proof}\uses{goursat, cs-bound, bsg, pfr_aux-improv}\leanok Consider the graph $A \subset G \times G'$ defined by
$$ A := \{ (x,f(x)): x \in G \}.$$
Clearly, $|A| = |G|$. By hypothesis, we have $a+a' \in A$ for at least
$|A|^2/K$ pairs $(a,a') \in A^2$. By \Cref{cs-bound}, this implies that $E(A)
\geq |A|^3/K^2$. Applying \Cref{bsg}, we conclude that there exists a subset
$A' \subset A$ with $|A'| \geq |A|/C_1 K^{2C_2}$ and $|A'+A'| \leq C_1C_3
K^{2(C_2+C_4)} |A'|$. Applying \Cref{pfr-9-aux'}, we may find a subspace $H
\subset G \times G'$ such that $|H| / |A'| \in [L^{-8}, L^{8}]$ and a subset
$c$ of cardinality at most $L^5 |A'|^{1/2} / |H|^{1/2}$ such that $A'
\subseteq c + H$, where $L = C_1C_3 K^{2(C_2+C_4)}$. If we let $H_0,H_1$ be
as in \Cref{goursat}, this implies on taking projections the projection of
$A'$ to $G$ is covered by at most $|c|$ translates of $H_0$. This implies
that
$$ |c| |H_0| \geq |A'|;$$
since $|H_0| |H_1| = |H|$, we conclude that
$$ |H_1| \leq |c| |H|/|A'|.$$
By hypothesis, $A'$ is covered by at most $|c|$ translates of $H$, and hence
by at most $|c| |H_1|$ translates of $\{ (x,\phi(x)): x \in G \}$. As $\phi$
is a homomorphism, each such translate can be written in the form $\{
(x,\phi(x)+c): x \in G \}$ for some $c \in G'$. The number of translates is
bounded by
$$
|c|^2 \frac{|H|}{|A'|} \leq \left(L^5 \frac{|A'|^{1/2}}{|H|^{1/2}}\right)^2 \frac{|H|}{|A'|} = L^{10}.
$$
By the pigeonhole principle, one of these translates must then contain at
least $|A'|/L^{10} \geq |G| / (C_1C_3 K^{2(C_2+C_4)})^{10} (C_1 K^{2C_2})$
elements of $A'$ (and hence of $A$), and the claim follows.
\end{proof} | 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 G := by rw [Set.Finite.card_toFinset]; simp [← Nat.card_eq_fintype_card]
have hA_nonempty : A.Nonempty := by simp [-Set.Finite.toFinset_setOf, A]
have := calc
(#A ^ 3 / K ^ 2 : ℝ)
= (Nat.card G ^ 2 / K) ^ 2 / #A := by field_simp [hA]; ring
_ ≤ Nat.card {x : G × G | f (x.1 + x.2) = f x.1 + f x.2} ^ 2 / #A := by gcongr
_ = #{ab ∈ A ×ˢ A | ab.1 + ab.2 ∈ A} ^ 2 / #A := by
congr
rw [← Nat.card_eq_finsetCard, ← Finset.coe_sort_coe, Finset.coe_filter,
Set.Finite.toFinset_prod]
simp only [Set.Finite.mem_toFinset, A, Set.graphOn_prod_graphOn]
rw [← Set.natCard_graphOn _ (Prod.map f f),
← Nat.card_image_equiv (Equiv.prodProdProdComm G G' G G'), Set.image_equiv_eq_preimage_symm]
congr
aesop
_ ≤ #A * E[A] / #A := by gcongr; exact mod_cast card_sq_le_card_mul_addEnergy ..
_ = E[A] := by field_simp
obtain ⟨A', hA', hA'1, hA'2⟩ :=
BSG_self' (sq_nonneg K) hA_nonempty (by simpa only [inv_mul_eq_div] using this)
clear hf this
have hA'₀ : A'.Nonempty := Finset.card_pos.1 $ Nat.cast_pos.1 $ hA'1.trans_lt' $ by positivity
let A'' := A'.toSet
have hA''_coe : Nat.card A'' = #A' := Nat.card_eq_finsetCard A'
have hA''_pos : 0 < Nat.card A'' := by rw [hA''_coe]; exact hA'₀.card_pos
have hA''_nonempty : Set.Nonempty A'' := nonempty_subtype.mp (Finite.card_pos_iff.mp hA''_pos)
have : Finset.card (A' - A') = Nat.card (A'' + A'') := calc
_ = Nat.card (A' - A').toSet := (Nat.card_eq_finsetCard _).symm
_ = Nat.card (A'' + A'') := by rw [Finset.coe_sub, sumset_eq_sub]
replace : Nat.card (A'' + A'') ≤ 2 ^ 14 * K ^ 12 * Nat.card A'' := by
rewrite [← this, hA''_coe]
simpa [← pow_mul] using hA'2
obtain ⟨H, c, hc_card, hH_le, hH_ge, hH_cover⟩ := better_PFR_conjecture_aux hA''_nonempty this
clear hA'2 hA''_coe hH_le hH_ge
obtain ⟨H₀, H₁, φ, hH₀H₁, hH₀H₁_card⟩ := goursat H
have h_le_H₀ : Nat.card A'' ≤ Nat.card c * Nat.card H₀ := by
have h_le := Nat.card_mono (Set.toFinite _) (Set.image_subset Prod.fst hH_cover)
have h_proj_A'' : Nat.card A'' = Nat.card (Prod.fst '' A'') := Nat.card_congr
(Equiv.Set.imageOfInjOn Prod.fst A'' <|
Set.fst_injOn_graph.mono (Set.Finite.subset_toFinset.mp hA'))
have h_proj_c : Prod.fst '' (c + H : Set (G × G')) = (Prod.fst '' c) + H₀ := by
ext x ; constructor <;> intro hx
· obtain ⟨x, ⟨⟨c, hc, h, hh, hch⟩, hx⟩⟩ := hx
rewrite [← hx]
exact ⟨c.1, Set.mem_image_of_mem Prod.fst hc, h.1, ((hH₀H₁ h).mp hh).1, (Prod.ext_iff.mp hch).1⟩
· obtain ⟨_, ⟨c, hc⟩, h, hh, hch⟩ := hx
refine ⟨c + (h, φ h), ⟨⟨c, hc.1, (h, φ h), ?_⟩, by rwa [← hc.2] at hch⟩⟩
exact ⟨(hH₀H₁ ⟨h, φ h⟩).mpr ⟨hh, by rw [sub_self]; apply zero_mem⟩, rfl⟩
rewrite [← h_proj_A'', h_proj_c] at h_le
apply (h_le.trans Set.natCard_add_le).trans
gcongr
exact Nat.card_image_le c.toFinite
have hH₀_pos : (0 : ℝ) < Nat.card H₀ := Nat.cast_pos.mpr Nat.card_pos
have h_le_H₁ : (Nat.card H₁ : ℝ) ≤ (Nat.card c) * (Nat.card H) / Nat.card A'' := calc
_ = (Nat.card H : ℝ) / (Nat.card H₀) :=
(eq_div_iff <| ne_of_gt <| hH₀_pos).mpr <| by rw [mul_comm, ← Nat.cast_mul, hH₀H₁_card]
_ ≤ (Nat.card c : ℝ) * (Nat.card H) / Nat.card A'' := by
nth_rewrite 1 [← mul_one (Nat.card H : ℝ), mul_comm (Nat.card c : ℝ)]
repeat rewrite [mul_div_assoc]
refine mul_le_mul_of_nonneg_left ?_ (Nat.cast_nonneg _)
refine le_of_mul_le_mul_right ?_ hH₀_pos
refine le_of_mul_le_mul_right ?_ (Nat.cast_pos.mpr hA''_pos)
rewrite [div_mul_cancel₀ 1, mul_right_comm, one_mul, div_mul_cancel₀, ← Nat.cast_mul]
· exact Nat.cast_le.mpr h_le_H₀
· exact ne_of_gt (Nat.cast_pos.mpr hA''_pos)
· exact ne_of_gt hH₀_pos
clear h_le_H₀ hA''_pos hH₀_pos hH₀H₁_card
let translate (c : G × G') (h : G') := A'' ∩ ({c} + {(0, h)} + Set.univ.graphOn φ)
have h_translate (c : G × G') (h : G') :
Prod.fst '' translate c h ⊆ { x : G | f x = φ x + (-φ c.1 + c.2 + h) } := by
intro x hx
obtain ⟨x, ⟨hxA'', _, ⟨c', hc, h', hh, hch⟩, x', hx, hchx⟩, hxx⟩ := hx
show f _ = φ _ + (-φ c.1 + c.2 + h)
replace := by simpa [-Set.Finite.toFinset_setOf, A] using hA' hxA''
rewrite [← hxx, this, ← hchx, ← hch, hc, hh]
show c.2 + h + x'.2 = φ (c.1 + 0 + x'.1) + (-φ c.1 + c.2 + h)
replace : φ x'.1 = x'.2 := (Set.mem_graphOn.mp hx).2
rw [map_add, map_add, map_zero, add_zero, this, add_comm (φ c.1), add_assoc x'.2,
← add_assoc (φ c.1), ← add_assoc (φ c.1), ← sub_eq_add_neg, sub_self, zero_add, add_comm]
have h_translate_card c h : Nat.card (translate c h) = Nat.card (Prod.fst '' translate c h) :=
Nat.card_congr (Equiv.Set.imageOfInjOn Prod.fst (translate c h) <|
Set.fst_injOn_graph.mono fun _ hx ↦ Set.Finite.subset_toFinset.mp hA' hx.1)
let cH₁ := (c ×ˢ H₁).toFinite.toFinset
have A_nonempty : Nonempty A'' := Set.nonempty_coe_sort.mpr hA''_nonempty
replace hc : c.Nonempty := by
obtain ⟨x, hx, _, _, _⟩ := hH_cover (Classical.choice A_nonempty).property
exact ⟨x, hx⟩
replace : A' = Finset.biUnion cH₁ fun ch ↦ (translate ch.1 ch.2).toFinite.toFinset := by
ext x ; constructor <;> intro hx
· obtain ⟨c', hc, h, hh, hch⟩ := hH_cover hx
refine Finset.mem_biUnion.mpr ⟨(c', h.2 - φ h.1), ?_⟩
refine ⟨(Set.Finite.mem_toFinset _).mpr ⟨hc, ((hH₀H₁ h).mp hh).2⟩, ?_⟩
refine (Set.Finite.mem_toFinset _).mpr ⟨hx, c' + (0, h.2 - φ h.1), ?_⟩
refine ⟨⟨c', rfl, (0, h.2 - φ h.1), rfl, rfl⟩, (h.1, φ h.1), ⟨h.1, by simp⟩, ?_⟩
beta_reduce
rewrite [add_assoc]
show c' + (0 + h.1, h.2 - φ h.1 + φ h.1) = x
rewrite [zero_add, sub_add_cancel]
exact hch
· obtain ⟨ch, hch⟩ := Finset.mem_biUnion.mp hx
exact ((Set.Finite.mem_toFinset _).mp hch.2).1
replace : ∑ _ ∈ cH₁, ((2 ^ 4)⁻¹ * (K ^ 2)⁻¹ * #A / cH₁.card : ℝ) ≤
∑ ch ∈ cH₁, ((translate ch.1 ch.2).toFinite.toFinset.card : ℝ) := by
rewrite [Finset.sum_const, nsmul_eq_mul, ← mul_div_assoc, mul_div_right_comm, div_self, one_mul]
· apply hA'1.trans
norm_cast
exact (congrArg Finset.card this).trans_le Finset.card_biUnion_le
· symm
refine ne_of_lt <| Nat.cast_zero.symm.trans_lt <| Nat.cast_lt.mpr <| Finset.card_pos.mpr ?_
exact (Set.Finite.toFinset_nonempty _).mpr <| hc.prod H₁.nonempty
obtain ⟨c', h, hch⟩ : ∃ c' : G × G', ∃ h : G', (2 ^ 4 : ℝ)⁻¹ * (K ^ 2)⁻¹ * #A / cH₁.card ≤
Nat.card { x : G | f x = φ x + (-φ c'.1 + c'.2 + h) } := by
obtain ⟨ch, hch⟩ :=
Finset.exists_le_of_sum_le ((Set.Finite.toFinset_nonempty _).mpr (hc.prod H₁.nonempty)) this
refine ⟨ch.1, ch.2, hch.2.trans ?_⟩
rewrite [Set.Finite.card_toFinset, ← Nat.card_eq_fintype_card, h_translate_card]
exact Nat.cast_le.mpr <| Nat.card_mono (Set.toFinite _) (h_translate ch.1 ch.2)
clear! hA' hA'1 hH_cover hH₀H₁ translate h_translate h_translate_card
use φ, -φ c'.1 + c'.2 + h
calc
Nat.card G / (2 ^ 144 * K ^ 122)
_ = Nat.card G / (2 ^ 4 * K ^ 2 * (2 ^ 140 * K ^ 120)) := by ring
_ ≤ Nat.card G / (2 ^ 4 * K ^ 2 * #(c ×ˢ H₁).toFinite.toFinset) := ?_
_ = (2 ^ 4)⁻¹ * (K ^ 2)⁻¹ * ↑(#A) / ↑(#cH₁) := by rw [hA, ← mul_inv, inv_mul_eq_div, div_div]
_ ≤ _ := hch
have := (c ×ˢ H₁).toFinite.toFinset_nonempty.2 (hc.prod H₁.nonempty)
gcongr
calc
(#(c ×ˢ H₁).toFinite.toFinset : ℝ)
_ = #c.toFinite.toFinset * #(H₁ : Set G').toFinite.toFinset := by
rw [← Nat.cast_mul, ← Finset.card_product, Set.Finite.toFinset_prod]
_ = Nat.card c * Nat.card H₁ := by
simp_rw [Set.Finite.card_toFinset, ← Nat.card_eq_fintype_card]; norm_cast
_ ≤ Nat.card c * (Nat.card c * Nat.card H / Nat.card ↑A'') := by gcongr
_ = Nat.card c ^ 2 * Nat.card H / Nat.card ↑A'' := by ring
_ ≤ ((2 ^ 14 * K ^ 12) ^ 5 * Nat.card A'' ^ (1 / 2 : ℝ) * Nat.card H ^ (-1 / 2 : ℝ)) ^ 2 *
Nat.card H / Nat.card ↑A'' := by gcongr
_ = 2 ^ 140 * K ^ 120 := by field_simp; rpow_simp; norm_num | 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]).
\end{align*}
\end{lemma}
\begin{proof}\uses{construct-good-improv, key-ident}\leanok For each $s$ in the range of $S$, apply \Cref{construct-good-improv} with $T_1,T_2,T_3$ equal to $(U|S=s)$, $(V|S=s)$, $(W|S=s)$ respectively (which works thanks to \Cref{key-ident}), multiply by $\bbP[S=s]$, and sum in $s$ to conclude.
\end{proof} | 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 | ⟨U, S⟩] - d[p.X₀₁ # X₁]) + (d[p.X₀₁ # W | ⟨V, S⟩] - d[p.X₀₁ # X₁]))
+ ((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])))
:= by
have hS : Measurable S := by fun_prop
have hU : Measurable U := by fun_prop
have hV : Measurable V := by fun_prop
have hW : Measurable W := by fun_prop
have hUVW : U + V + W = 0 := sum_uvw_eq_zero X₁ X₂ X₁'
have hz (a : ℝ) : a = ∑ z, (ℙ (S ⁻¹' {z})).toReal * a := by
rw [← Finset.sum_mul, sum_measure_preimage_singleton' ℙ hS, one_mul]
rw [hz k, hz (d[p.X₀₁ # X₁]), hz (d[p.X₀₂ # X₂])]
simp only [condMutualInfo_eq_sum' hS, ← Finset.sum_add_distrib, ← mul_add,
condRuzsaDist'_prod_eq_sum', hU, hS, hV, hW, ← Finset.sum_sub_distrib, ← mul_sub, Finset.mul_sum,
← mul_assoc (p.η/6), mul_comm (p.η/6), mul_assoc _ _ (p.η/6)]
rw [Finset.sum_mul, ← Finset.sum_add_distrib]
apply Finset.sum_le_sum (fun i _hi ↦ ?_)
rcases eq_or_ne (ℙ (S ⁻¹' {i})) 0 with h'i|h'i
· simp [h'i]
rw [mul_assoc, ← mul_add]
gcongr
have : IsProbabilityMeasure (ℙ[|S ⁻¹' {i}]) := cond_isProbabilityMeasure h'i
linarith [construct_good_improved'' h_min (ℙ[|S ⁻¹' {i}]) hUVW hU hV hW]
variable (p)
include hX₁ hX₂ hX₁' hX₂' h_indep h₁ h₂ in
omit [IsProbabilityMeasure (ℙ : Measure Ω₀₂)] in | 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{theorem}
\begin{proof}\leanok
\uses{pfr-9-aux,ruz-cov}
Given \Cref{pfr-9-aux'}, the proof is the same as that of \Cref{pfr}.
\end{proof} | 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_conjecture_pos_aux' h₀A hA
-- consider the subgroup `H` given by Lemma `PFR_conjecture_aux`.
obtain ⟨H, c, hc, IHA, IAH, A_subs_cH⟩ : ∃ (H : Submodule (ZMod 2) G) (c : Set G),
Nat.card c ≤ K ^ 5 * Nat.card A ^ (1 / 2 : ℝ) * Nat.card H ^ (-1 / 2 : ℝ)
∧ Nat.card H ≤ K ^ 8 * Nat.card A ∧ Nat.card A ≤ K ^ 8 * Nat.card H
∧ A ⊆ c + H :=
better_PFR_conjecture_aux h₀A hA
have H_pos : (0 : ℝ) < Nat.card H := by
have : 0 < Nat.card H := Nat.card_pos; positivity
rcases le_or_lt (Nat.card H) (Nat.card A) with h|h
-- If `#H ≤ #A`, then `H` satisfies the conclusion of the theorem
· refine ⟨H, c, ?_, h, A_subs_cH⟩
calc
Nat.card c ≤ K ^ 5 * Nat.card A ^ (1 / 2 : ℝ) * Nat.card H ^ (-1 / 2 : ℝ) := hc
_ ≤ K ^ 5 * (K ^ 8 * Nat.card H) ^ (1 / 2 : ℝ) * Nat.card H ^ (-1 / 2 : ℝ) := by
gcongr
_ = K ^ 9 := by simp_rw [← rpow_natCast]; rpow_ring; norm_num
_ < 2 * K ^ 9 := by linarith [show 0 < K ^ 9 by positivity]
-- otherwise, we decompose `H` into cosets of one of its subgroups `H'`, chosen so that
-- `#A / 2 < #H' ≤ #A`. This `H'` satisfies the desired conclusion.
· obtain ⟨H', IH'A, IAH', H'H⟩ : ∃ H' : Submodule (ZMod 2) G, Nat.card H' ≤ Nat.card A
∧ Nat.card A < 2 * Nat.card H' ∧ H' ≤ H := by
have A_pos' : 0 < Nat.card A := mod_cast A_pos
exact ZModModule.exists_submodule_subset_card_le Nat.prime_two H h.le A_pos'.ne'
have : (Nat.card A / 2 : ℝ) < Nat.card H' := by
rw [div_lt_iff₀ zero_lt_two, mul_comm]; norm_cast
have H'_pos : (0 : ℝ) < Nat.card H' := by
have : 0 < Nat.card H' := Nat.card_pos; positivity
obtain ⟨u, HH'u, hu⟩ :=
H'.toAddSubgroup.exists_left_transversal_of_le (H := H.toAddSubgroup) H'H
dsimp at HH'u
refine ⟨H', c + u, ?_, IH'A, by rwa [add_assoc, HH'u]⟩
calc
(Nat.card (c + u) : ℝ)
≤ Nat.card c * Nat.card u := mod_cast natCard_add_le
_ ≤ (K ^ 5 * Nat.card A ^ (1 / 2 : ℝ) * (Nat.card H ^ (-1 / 2 : ℝ)))
* (Nat.card H / Nat.card H') := by
gcongr
apply le_of_eq
rw [eq_div_iff H'_pos.ne']
norm_cast
_ < (K ^ 5 * Nat.card A ^ (1 / 2 : ℝ) * (Nat.card H ^ (-1 / 2 : ℝ)))
* (Nat.card H / (Nat.card A / 2)) := by
gcongr
_ = 2 * K ^ 5 * Nat.card A ^ (-1 / 2 : ℝ) * Nat.card H ^ (1 / 2 : ℝ) := by
field_simp
simp_rw [← rpow_natCast]
rpow_ring
norm_num
_ ≤ 2 * K ^ 5 * Nat.card A ^ (-1 / 2 : ℝ) * (K ^ 8 * Nat.card A) ^ (1 / 2 : ℝ) := by
gcongr
_ = 2 * K ^ 9 := by
simp_rw [← rpow_natCast]
rpow_ring
norm_num
/-- Corollary of `better_PFR_conjecture` in which the ambient group is not required to be finite
(but) then $H$ and $c$ are finite. -/ | 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}
Apply \Cref{pfr-9-aux} and \Cref{ruz-cov} to get the result, as in the proof
of \Cref{pfr_aux}.
\end{proof} | 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 H ∧ 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_aux' h₀A hA
rcases better_PFR_conjecture_aux0 h₀A hA with ⟨H, x₀, J, IAH, IHA⟩
have H_pos : (0 : ℝ) < Nat.card H := by
have : 0 < Nat.card H := Nat.card_pos
positivity
have Hne : Set.Nonempty (A ∩ (H + {x₀})) := by
by_contra h'
have : 0 < Nat.card H := Nat.card_pos
have : (0 : ℝ) < Nat.card (A ∩ (H + {x₀}) : Set G) := lt_of_lt_of_le (by positivity) J
simp only [Nat.card_eq_fintype_card, Nat.card_of_isEmpty, CharP.cast_eq_zero, lt_self_iff_false,
not_nonempty_iff_eq_empty.1 h', Fintype.card_ofIsEmpty] at this
/- use Rusza covering lemma to cover `A` by few translates of `A ∩ (H + {x₀}) - A ∩ (H + {x₀})`
(which is contained in `H`). The number of translates is at most
`#(A + (A ∩ (H + {x₀}))) / #(A ∩ (H + {x₀}))`, where the numerator is controlled as this is
a subset of `A + A`, and the denominator is bounded below by the previous inequality`. -/
have Z3 :
(Nat.card (A + A ∩ (↑H + {x₀})) : ℝ) ≤ (K ^ 5 * Nat.card A ^ (1/2 : ℝ) *
Nat.card H ^ (-1/2 : ℝ)) * Nat.card ↑(A ∩ (↑H + {x₀})) := by
calc
(Nat.card (A + A ∩ (↑H + {x₀})) : ℝ)
_ ≤ Nat.card (A + A) := by
gcongr; exact Nat.card_mono (toFinite _) <| add_subset_add_left inter_subset_left
_ ≤ K * Nat.card A := hA
_ = (K ^ 5 * Nat.card A ^ (1/2 : ℝ) * Nat.card H ^ (-1/2 : ℝ)) *
(K ^ (-4 : ℤ) * Nat.card A ^ (1/2 : ℝ) * Nat.card H ^ (1/2 : ℝ)) := by
simp_rw [← rpow_natCast, ← rpow_intCast]; rpow_ring; norm_num
_ ≤ (K ^ 5 * Nat.card A ^ (1/2 : ℝ) * Nat.card H ^ (-1/2 : ℝ)) *
Nat.card ↑(A ∩ (↑H + {x₀})) := by gcongr
obtain ⟨u, huA, hucard, hAu, -⟩ :=
Set.ruzsa_covering_add (toFinite A) (toFinite (A ∩ ((H + {x₀} : Set G)))) Hne (by convert Z3)
have A_subset_uH : A ⊆ u + H := by
refine hAu.trans $ add_subset_add_left $
(sub_subset_sub (inter_subset_right ..) (inter_subset_right ..)).trans ?_
rw [add_sub_add_comm, singleton_sub_singleton, _root_.sub_self]
simp
exact ⟨H, u, hucard, IHA, IAH, A_subset_uH⟩
/-- 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 $H$. -/ | 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-rho} on $U_A,U_A$ to get a subspace such that $2\rho(U_H)\le 2\rho(U_A)+8d[U_A;U_A]$. Recall that $d[U_A;U_A]\le \log K$ as proved in \Cref{pfr_aux}, and $\rho(U_A)=0$ by \Cref{rho-init}. Therefore $\rho(U_H)\le 4\log(K)$. The claim then follows from \Cref{rho-subgroup}.
\end{proof} | 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 * Nat.card A := by
have A_fin : Finite A := by infer_instance
classical
let mG : MeasurableSpace G := ⊤
have : MeasurableSingletonClass G := ⟨λ _ ↦ trivial⟩
obtain ⟨A_pos, -, K_pos⟩ : (0 : ℝ) < Nat.card A ∧ (0 : ℝ) < Nat.card (A + A) ∧ 0 < K :=
PFR_conjecture_pos_aux' h₀A hA
let A' := A.toFinite.toFinset
have h₀A' : Finset.Nonempty A' := by
simp [A', Finset.Nonempty]
exact h₀A
have hAA' : A' = A := Finite.coe_toFinset (toFinite A)
rcases exists_isUniform_measureSpace A' h₀A' with ⟨Ω₀, mΩ₀, UA, hP₀, UAmeas, UAunif, -⟩
rw [hAA'] at UAunif
have hadd_sub : A + A = A - A := by ext; simp [Set.mem_add, Set.mem_sub, ZModModule.sub_eq_add]
rw [hadd_sub] at hA
have : d[UA # UA] ≤ log K := rdist_le_of_isUniform_of_card_add_le h₀A hA UAunif UAmeas
rw [← hadd_sub] at hA
-- entropic PFR gives a subgroup `H` which is close to `A` for the rho functional
rcases rho_PFR_conjecture UA UA UAmeas UAmeas A' h₀A'
with ⟨H, Ω₁, mΩ₁, UH, hP₁, UHmeas, UHunif, hUH⟩
have ineq : ρ[UH # A'] ≤ 4 * log K := by
rw [← hAA'] at UAunif
have : ρ[UA # A'] = 0 := rho_of_uniform UAunif UAmeas h₀A'
linarith
set r := 4 * log K with hr
have J : K ^ (-4 : ℤ) = exp (-r) := by
rw [hr, ← neg_mul, mul_comm, exp_mul, exp_log K_pos]
norm_cast
have J' : K ^ 8 = exp (2 * r) := by
have : 2 * r = 8 * log K := by ring
rw [this, mul_comm, exp_mul, exp_log K_pos]
norm_cast
rw [J, J']
refine ⟨H, ?_⟩
have Z := rho_of_submodule UHunif h₀A' UHmeas r ineq
have : Nat.card A = Nat.card A' := by simp [← hAA']
have I t : t +ᵥ (H : Set G) = (H : Set G) + {t} := by
ext z; simp [mem_vadd_set_iff_neg_vadd_mem, add_comm]
simp_rw [← I]
convert Z
exact hAA'.symm
/-- Auxiliary statement towards the polynomial Freiman-Ruzsa (PFR) conjecture: if $A$ is a subset of
an elementary abelian 2-group of doubling constant at most $K$, then there exists a subgroup $H$
such that $A$ can be covered by at most $K^5 |A|^{1/2} / |H|^{1/2}$ cosets of $H$, and $H$ has
the same cardinality as $A$ up to a multiplicative factor $K^8$. -/ | 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}\leanok
\uses{ckl-div} Compare the terms correspond to each $x\in G$ on both sides.
\end{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 # Y ; μ'] = KL[X ; μ # Y ; μ'] + H[X ; μ] - H[ X | Z ; μ] := by
rcases eq_zero_or_isProbabilityMeasure μ with rfl | hμ
· simp [condKLDiv, tsum_fintype, KLDiv_eq_sum, Finset.mul_sum, entropy_eq_sum]
simp only [condKLDiv, tsum_fintype, KLDiv_eq_sum, Finset.mul_sum, entropy_eq_sum]
rw [Finset.sum_comm, condEntropy_eq_sum_sum_fintype hZ, Finset.sum_comm (α := G),
← Finset.sum_add_distrib, ← Finset.sum_sub_distrib]
congr with g
simp only [negMulLog, neg_mul, Finset.sum_neg_distrib, mul_neg, sub_neg_eq_add, ← sub_eq_add_neg,
← mul_sub]
simp_rw [← Measure.map_apply hZ (measurableSet_singleton _)]
have : Measure.map X μ {g} = ∑ x, (Measure.map Z μ {x}) * (Measure.map X μ[|Z ⁻¹' {x}] {g}) := by
simp_rw [Measure.map_apply hZ (measurableSet_singleton _)]
have : Measure.map X μ {g} = Measure.map X (∑ x, μ (Z ⁻¹' {x}) • μ[|Z ⁻¹' {x}]) {g} := by
rw [sum_meas_smul_cond_fiber hZ μ]
rw [← MeasureTheory.Measure.sum_fintype, Measure.map_sum hX.aemeasurable] at this
simpa using this
nth_rewrite 1 [this]
rw [ENNReal.toReal_sum (by simp [ENNReal.mul_eq_top]), Finset.sum_mul, ← Finset.sum_add_distrib]
congr with s
rw [ENNReal.toReal_mul, mul_assoc, ← mul_add, ← mul_add]
rcases eq_or_ne (Measure.map Z μ {s}) 0 with hs | hs
· simp [hs]
rcases eq_or_ne (Measure.map X μ[|Z ⁻¹' {s}] {g}) 0 with hg | hg
· simp [hg]
have h'g : (Measure.map X μ[|Z ⁻¹' {s}] {g}).toReal ≠ 0 := by
simp [ENNReal.toReal_eq_zero_iff, hg]
congr
have hXg : μ.map X {g} ≠ 0 := by
intro h
rw [this, Finset.sum_eq_zero_iff] at h
specialize h s (Finset.mem_univ _)
rw [mul_eq_zero] at h
tauto
have hXg' : (μ.map X {g}).toReal ≠ 0 := by simp [ENNReal.toReal_eq_zero_iff, hXg]
have hYg : μ'.map Y {g} ≠ 0 := fun h ↦ hXg (habs _ h)
have hYg' : (μ'.map Y {g}).toReal ≠ 0 := by simp [ENNReal.toReal_eq_zero_iff, hYg]
rw [Real.log_div h'g hYg', Real.log_div hXg' hYg']
abel | 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 [condKLDiv]
refine tsum_nonneg (fun i ↦ mul_nonneg (by simp) ?_)
apply KLDiv_nonneg hX hY
intro s hs
specialize habs s hs
rw [Measure.map_apply hX (measurableSet_singleton s)] at habs ⊢
exact cond_absolutelyContinuous habs | 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{equation}\label{multi-def-cond-alt}
D[ X_{[m]} | Y_{[m]} ] = \sum_{(y_i)_{1 \leq i \leq m}} \biggl(\prod_{1 \leq i \leq m} p_{Y_i}(y_i)\biggr) D[ (X_i \,|\, Y_i \mathop{=}y_i)_{1 \leq i \leq m}]
\end{equation}
where each $y_i$ ranges over the support of $p_{Y_i}$ for $1 \leq i \leq m$.
\end{definition} | 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 " | " Y " ; " hΩ "]" => condMultiDist hΩ X Y | 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].
\end{equation}
\end{lemma}
\begin{proof}\uses{conditional-entropy-def, multidist-def, cond-multidist-def}\leanok
This is routine from \Cref{conditional-entropy-def} and Definitions \ref{multidist-def} and \ref{cond-multidist-def}.
\end{proof} | 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 ; fun _ ↦ hΩ] =
H[fun ω ↦ ∑ i, X i ω | fun ω ↦ (fun i ↦ Y i ω)] - (∑ i, H[X i | Y i])/m := by
have : IsProbabilityMeasure (ℙ : Measure Ω) := h_indep.isProbabilityMeasure
let E := fun i (yi:S) ↦ Y i ⁻¹' {yi}
let E' := fun (y : Fin m → S) ↦ ⋂ i, E i (y i)
let f := fun (y : Fin m → S) ↦ ∏ i, (ℙ (E i (y i))).toReal
calc
_ = ∑ y, (f y) * D[X; fun i ↦ ⟨cond ℙ (E i (y i))⟩] := by rfl
_ = ∑ y, (f y) * (H[∑ i, X i; cond ℙ (E' y)] - (∑ i, H[X i; cond ℙ (E' y)]) / m) := by
congr with y
by_cases hf : f y = 0
. simp only [hf, zero_mul]
congr 1
rw [multiDist_copy (fun i ↦ ⟨cond ℙ (E i (y i))⟩)
(fun _ ↦ ⟨cond ℙ (E' y)⟩) X X
(fun i ↦ ident_of_cond_of_indep hX hY h_indep y i (prob_nonzero_of_prod_prob_nonzero hf))]
exact multiDist_indep _ _ <|
h_indep.cond hY (prob_nonzero_of_prod_prob_nonzero hf) fun _ ↦ .singleton _
_ = ∑ y, (f y) * H[∑ i, X i; cond ℙ (E' y)] - (∑ i, ∑ y, (f y) * H[X i; cond ℙ (E' y)])/m := by
rw [Finset.sum_comm, Finset.sum_div, ← Finset.sum_sub_distrib]
congr with y
rw [← Finset.mul_sum, mul_div_assoc, ← mul_sub]
_ = _ := by
congr
· rw [condEntropy_eq_sum_fintype]
· congr with y
congr
· calc
_ = (∏ i, (ℙ (E i (y i)))).toReal := Eq.symm ENNReal.toReal_prod
_ = (ℙ (⋂ i, (E i (y i)))).toReal := by
congr
exact (iIndepFun.meas_iInter h_indep fun _ ↦ mes_of_comap (.singleton _)).symm
_ = _ := by
congr
ext x
simp only [Set.mem_iInter, Set.mem_preimage, Set.mem_singleton_iff, E,
Iff.symm funext_iff]
· exact Finset.sum_fn Finset.univ fun c ↦ X c
ext x
simp only [Set.mem_iInter, Set.mem_preimage, Set.mem_singleton_iff, E']
exact Iff.symm funext_iff
exact measurable_pi_lambda (fun ω i ↦ Y i ω) hY
ext i
calc
_ = ∑ y, f y * H[X i; cond ℙ (E i (y i))] := by
congr with y
by_cases hf : f y = 0
. simp only [hf, zero_mul]
congr 1
apply IdentDistrib.entropy_eq
exact (ident_of_cond_of_indep hX hY h_indep y i
(prob_nonzero_of_prod_prob_nonzero hf)).symm
_ = ∑ y ∈ Fintype.piFinset (fun _ ↦ Finset.univ), ∏ i', (ℙ (E i' (y i'))).toReal
* (if i'=i then H[X i; cond ℙ (E i (y i'))] else 1) := by
simp only [Fintype.piFinset_univ]
congr with y
rw [Finset.prod_mul_distrib]
congr
rw [Fintype.prod_ite_eq']
_ = _ := by
convert (Finset.prod_univ_sum (fun _ ↦ Finset.univ)
(fun (i' : Fin m) (s : S) ↦ (ℙ (E i' s)).toReal *
if i' = i then H[X i ; ℙ[|E i s]] else 1)).symm
calc
_ = ∏ i', if i' = i then H[X i' | Y i'] else 1 := by
simp only [Finset.prod_ite_eq', Finset.mem_univ, ↓reduceIte]
_ = _ := by
congr with i'
by_cases h : i' = i
· simp only [h, ↓reduceIte, E]
rw [condEntropy_eq_sum_fintype]
exact hY i
· simp only [h, ↓reduceIte, mul_one, E]
exact (sum_measure_preimage_singleton' _ (hY i')).symm
/-- If `(X_i, Y_i)`, `1 ≤ i ≤ m` are independent, then `D[X_[m] | Y_[m]] = ∑_{(y_i)_{1 ≤ i ≤ m}} P(Y_i=y_i ∀ i) D[(X_i | Y_i=y_i ∀ i)_{i=1}^m]`
-/ | 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{multidist-nonneg}\leanok Clear from \Cref{multidist-nonneg} and \Cref{cond-multidist-def}, except that some care may need to be taken to deal with the $y_i$ where $p_{Y_i}$ vanish.
\end{proof} | /--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] := by
dsimp [condMultiDist]
apply Finset.sum_nonneg
intro y _
by_cases h: ∀ i : Fin m, ℙ (Y i ⁻¹' {y i}) ≠ 0
. apply mul_nonneg
. apply Finset.prod_nonneg
intro i _
exact ENNReal.toReal_nonneg
exact multiDist_nonneg (fun i => ⟨ℙ[|Y i ⁻¹' {y i}]⟩)
(fun i => ProbabilityTheory.cond_isProbabilityMeasure (h i)) X hX
simp only [ne_eq, not_forall, Decidable.not_not] at h
obtain ⟨i, hi⟩ := h
apply le_of_eq
symm
convert zero_mul ?_
apply Finset.prod_eq_zero (Finset.mem_univ i)
simp only [hi, ENNReal.zero_toReal]
/-- A technical lemma: can push a constant into a product at a specific term -/
private lemma Finset.prod_mul {α β:Type*} [Fintype α] [DecidableEq α] [CommMonoid β] (f:α → β) (c: β) (i₀:α) : (∏ i, f i) * c = ∏ i, (if i=i₀ then f i * c else f i) := calc
_ = (∏ i, f i) * (∏ i, if i = i₀ then c else 1) := by
congr
simp only [prod_ite_eq', mem_univ, ↓reduceIte]
_ = _ := by
rw [← Finset.prod_mul_distrib]
apply Finset.prod_congr rfl
intro i _
by_cases h : i = i₀
. simp [h]
simp [h]
/-- A technical lemma: a preimage of a singleton of Y i is measurable with respect to the comap of <X i, Y i> -/
private lemma mes_of_comap {Ω S G : Type*} [hG : MeasurableSpace G] [hS : MeasurableSpace S]
{X : Ω → G} {Y : Ω → S} {s : Set S} (hs : MeasurableSet s) :
MeasurableSet[(hG.prod hS).comap fun ω ↦ (X ω, Y ω)] (Y ⁻¹' s) :=
⟨.univ ×ˢ s, MeasurableSet.univ.prod hs, by ext; simp [eq_comm]⟩
/-- A technical lemma: two different ways of conditioning independent variables gives identical distributions -/
private lemma ident_of_cond_of_indep
{G : Type*} [hG : MeasurableSpace G] [MeasurableSingletonClass G] [Countable G]
{m : ℕ} {Ω : Type*} [hΩ : MeasureSpace Ω]
{S : Type*} [Fintype S] [hS : MeasurableSpace S] [MeasurableSingletonClass S]
{X : Fin m → Ω → G} (hX : (i:Fin m) → Measurable (X i))
{Y : Fin m → Ω → S} (hY : (i:Fin m) → Measurable (Y i))
(h_indep : ProbabilityTheory.iIndepFun (fun i ↦ ⟨X i, Y i⟩))
(y : Fin m → S) (i : Fin m) (hy: ∀ i, ℙ (Y i ⁻¹' {y i}) ≠ 0) :
IdentDistrib (X i) (X i) (cond ℙ (Y i ⁻¹' {y i})) (cond ℙ (⋂ i, Y i ⁻¹' {y i})) where
aemeasurable_fst := Measurable.aemeasurable (hX i)
aemeasurable_snd := Measurable.aemeasurable (hX i)
map_eq := by
ext s hs
rw [Measure.map_apply (hX i) hs, Measure.map_apply (hX i) hs]
let s' : Finset (Fin m) := {i}
let f' := fun _ : Fin m ↦ X i ⁻¹' s
have hf' : ∀ i' ∈ s', MeasurableSet[hG.comap (X i')] (f' i') := by
intro i' hi'
simp only [Finset.mem_singleton.mp hi']
exact MeasurableSet.preimage hs (comap_measurable (X i))
have h := cond_iInter hY h_indep hf' (fun _ _ ↦ hy _) fun _ ↦ .singleton _
simp only [Finset.mem_singleton, Set.iInter_iInter_eq_left, Finset.prod_singleton,
s'] at h
exact h.symm
/-- A technical lemma: if a product of probabilities is nonzero, then each probability is
individually non-zero -/
private lemma prob_nonzero_of_prod_prob_nonzero {m : ℕ}
{Ω : Type*} [hΩ : MeasureSpace Ω]
{S : Type*} [Fintype S] [MeasurableSpace S] [MeasurableSingletonClass S]
{Y : Fin m → Ω → S} {y : Fin m → S} (hf : ∏ i, (ℙ (Y i ⁻¹' {y i})).toReal ≠ 0) :
∀ i, ℙ (Y i ⁻¹' {y i}) ≠ 0 := by
simp [Finset.prod_ne_zero_iff, ENNReal.toReal_eq_zero_iff, forall_and] at hf
exact hf.1
/-- If `(X_i, Y_i)`, `1 ≤ i ≤ m` are independent, then
`D[X_[m] | Y_[m]] = H[∑ i, X_i | (Y_1, ..., Y_m)] - 1/m * ∑ i, H[X_i | Y_i]`
-/ | 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
$$ \rho(X|Z) \leq \rho(X) + \frac{1}{2}( \bbH[X] - \bbH[X|Z] ).$$
\end{lemma}
\begin{proof}\leanok
\uses{kl-cond}
The first inequality follows from \Cref{kl-cond}. The second and third inequalities are direct corollaries of the first.
\end{proof} | /-- $$ \rho^-(X|Z) \leq \rho^-(X) + \bbH[X] - \bbH[X|Z]$$ -/
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[X | Z ; μ] := by
have : IsProbabilityMeasure (uniformOn (A : Set G)) := by
apply uniformOn_isProbabilityMeasure A.finite_toSet hA
suffices ρ⁻[X | Z ; μ # A] - H[X ; μ] + H[X | Z ; μ] ≤ ρ⁻[X ; μ # A] by linarith
apply le_csInf (nonempty_rhoMinusSet hA)
rintro - ⟨μ', hμ', habs, rfl⟩
rw [condRhoMinus, tsum_fintype]
let _ : MeasureSpace (G × G) := ⟨μ'.prod (uniformOn (A : Set G))⟩
have hP : (ℙ : Measure (G × G)) = μ'.prod (uniformOn (A : Set G)) := rfl
have : IsProbabilityMeasure (ℙ : Measure (G × G)) := by rw [hP]; infer_instance
have : ∑ b : S, (μ (Z ⁻¹' {b})).toReal * ρ⁻[X ; μ[|Z ← b] # A]
≤ KL[ X | Z ; μ # (Prod.fst + Prod.snd : G × G → G) ; ℙ] := by
rw [condKLDiv, tsum_fintype]
apply Finset.sum_le_sum (fun i hi ↦ ?_)
gcongr
apply rhoMinus_le_def hX (fun y hy ↦ ?_)
have T := habs y hy
rw [Measure.map_apply hX (measurableSet_singleton _)] at T ⊢
exact cond_absolutelyContinuous T
rw [condKLDiv_eq hX hZ (by exact habs)] at this
rw [← hP]
linarith | 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
$$ \rho(X|Z) \leq \rho(X) + \frac{1}{2}( \bbH[X] - \bbH[X|Z] ).$$
\end{lemma}
\begin{proof}\leanok
\uses{kl-cond}
The first inequality follows from \Cref{kl-cond}. The second and third inequalities are direct corollaries of the first.
\end{proof} | /-- $$ \rho^+(X|Z) \leq \rho^+(X)$$ -/
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.map Z μ) := isProbabilityMeasure_map hZ.aemeasurable
have I₁ := condRhoMinus_le hX hZ hA (μ := μ)
simp_rw [condRhoPlus, rhoPlus, tsum_fintype]
simp only [Nat.card_eq_fintype_card, Fintype.card_coe, mul_sub, mul_add, Finset.sum_sub_distrib,
Finset.sum_add_distrib, tsub_le_iff_right]
rw [← Finset.sum_mul, ← tsum_fintype, ← condRhoMinus, ← condEntropy_eq_sum_fintype _ _ _ hZ]
simp_rw [← Measure.map_apply hZ (measurableSet_singleton _)]
simp only [Finset.sum_toReal_measure_singleton, Finset.coe_univ, measure_univ, ENNReal.one_toReal,
one_mul, sub_add_cancel, ge_iff_le]
linarith
omit [Fintype G] [DiscreteMeasurableSpace G] in | 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
$$ \rho(X|Z) \leq \rho(X) + \frac{1}{2}( \bbH[X] - \bbH[X|Z] ).$$
\end{lemma}
\begin{proof}\leanok
\uses{kl-cond}
The first inequality follows from \Cref{kl-cond}. The second and third inequalities are direct corollaries of the first.
\end{proof} | /-- $$ \rho(X|Z) \leq \rho(X) + \frac{1}{2}( \bbH[X] - \bbH[X|Z] )$$ -/
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 | Z ; μ]) / 2 := by
rw [condRho_eq, rho]
linarith [condRhoMinus_le hX hZ hA (μ := μ), condRhoPlus_le hX hZ hA (μ := μ)]
omit [Fintype G] [DiscreteMeasurableSpace G] in | 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
ext z; simp [hf.eq_iff]
simp [I]
· intro y hy
have : f ∘ Y ⁻¹' {y} ≠ ∅ := by
intro h
simp [h] at hy
rcases Set.nonempty_iff_ne_empty.2 this with ⟨a, ha⟩
simp only [mem_preimage, Function.comp_apply, mem_singleton_iff] at ha
rw [← ha]
exact mem_range_self (Y a) | 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 $\rho(X|X+Y)\le \rho(X) + \frac{1}{2}(\bbH[X+Y]-\bbH[Y])$, and $\rho(Y|X+Y)\le \rho(Y)+\frac{1}{2}(\bbH[X+Y]-\bbH[X])$. Then apply \Cref{rho-invariant} to get $\rho(Y|X+Y)=\rho(X|X+Y)$ and take the average of the two inequalities.
\end{proof} | 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 :=
condRho_le hX (by fun_prop) hA
have I' : H[X ; μ] - H[X | X + Y ; μ] = H[X + Y ; μ] - H[Y ; μ] := by
rw [ProbabilityTheory.chain_rule'' _ hX (by fun_prop), entropy_add_right hX hY,
IndepFun.entropy_pair_eq_add hX hY h_indep]
abel
have J : ρ[Y | Y + X ; μ # A] ≤ ρ[Y ; μ # A] + (H[Y ; μ] - H[Y | Y + X ; μ]) / 2 :=
condRho_le hY (by fun_prop) hA
have J' : H[Y ; μ] - H[Y | Y + X ; μ] = H[Y + X ; μ] - H[X ; μ] := by
rw [ProbabilityTheory.chain_rule'' _ hY (by fun_prop), entropy_add_right hY hX,
IndepFun.entropy_pair_eq_add hY hX h_indep.symm]
abel
have : Y + X = X + Y := by abel
simp only [this] at J J'
have : ρ[X | X + Y ; μ # A] = ρ[Y | X + Y ; μ # A] := by
simp only [condRho]
congr with s
congr 1
have : ρ[X ; μ[|(X + Y) ⁻¹' {s}] # A] = ρ[fun ω ↦ X ω + s ; μ[|(X + Y) ⁻¹' {s}] # A] := by
rw [rho_of_translate hX hA]
rw [this]
apply rho_eq_of_identDistrib
apply IdentDistrib.of_ae_eq (by fun_prop)
have : MeasurableSet ((X + Y) ⁻¹' {s}) := by
have : Measurable (X + Y) := by fun_prop
exact this (measurableSet_singleton _)
filter_upwards [ae_cond_mem this] with a ha
simp only [mem_preimage, Pi.add_apply, mem_singleton_iff] at ha
rw [← ha]
nth_rewrite 1 [← ZModModule.neg_eq_self (X a)]
abel
have : X - Y = X + Y := ZModModule.sub_eq_add _ _
rw [h_indep.rdist_eq hX hY, sub_eq_add_neg, this]
linarith
end | 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
$$\rho(T_1|T_2,S)+\rho(T_2|T_1,S) - \frac{1}{2}\sum_{i} \rho(Y_i)\le \frac{1}{2}(d[Y_1;Y_2]+d[Y_3;Y_4]+d[Y_1;Y_3]+d[Y_2;Y_4]).$$
\end{lemma}
\begin{proof}\leanok\uses{rho-sums-sym, rho-cond, rho-cond-sym, rho-cond-relabeled, cor-fibre}
Let $T_1':=Y_3+Y_4$, $T_2':=Y_2+Y_4$.
First note that
\begin{align*}
\rho(T_1|T_2,S)
&\le \rho(T_1|S) + \frac{1}{2}\bbI(T_1:T_2\mid S) \\
&\le \frac{1}{2}(\rho(T_1)+\rho(T_1'))+\frac{1}{2}(d[T_1;T_1']+\bbI(T_1:T_2\mid S)) \\
&\le \frac{1}{4} \sum_{i} \rho(Y_i) +\frac{1}{4}(d[Y_1;Y_2]+d[Y_3;Y_4]) + \frac{1}{2}(d[T_1;T_1']+\bbI(T_1:T_2\mid S)).
\end{align*}
by \Cref{rho-cond}, \Cref{rho-cond-sym}, \Cref{rho-sums-sym} respectively.
On the other hand, observe that
\begin{align*}
\rho(T_1|T_2,S)
&=\rho(Y_1+Y_2|T_2,T_2') \\
&\le \frac{1}{2}(\rho(Y_1|T_2)+\rho(Y_2|T_2'))+\frac{1}{2}(d[Y_1|T_2;Y_2|T_2']) \\
&\le \frac{1}{4} \sum_{i} \rho(Y_i) +\frac{1}{4}(d[Y_1;Y_3]+d[Y_2;Y_4]) + \frac{1}{2}(d[Y_1|T_2;Y_2|T_2']).
\end{align*}
by \Cref{rho-cond-relabeled}, \Cref{rho-sums-sym}, \Cref{rho-cond-sym} respectively.
By replacing $(Y_1,Y_2,Y_3,Y_4)$ with $(Y_1,Y_3,Y_2,Y_4)$ in the above inequalities, one has
$$\rho(T_2|T_1,S) \le \frac{1}{4} \sum_{i} \rho(Y_i) +\frac{1}{4}(d[Y_1;Y_3]+d[Y_2;Y_4]) + \frac{1}{2}(d[T_2;T_2']+\bbI(T_1:T_2\mid S))$$
and
$$\rho(T_2|T_1,S) \le \frac{1}{4} \sum_{i} \rho(Y_i) +\frac{1}{4}(d[Y_1;Y_2]+d[Y_3;Y_4]) + \frac{1}{2}(d[Y_1|T_1;Y_3|T_1']).$$
Finally, take the sum of all four inequalities, apply \Cref{cor-fibre} on $(Y_1,Y_2,Y_3,Y_4)$ and $(Y_1,Y_3,Y_2,Y_4)$ to rewrite the sum of last terms in the four inequalities, and divide the result by $2$.
\end{proof} | 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[Y₂ # A] + ρ[Y₃ # A] + ρ[Y₄ # A]) / 2 ≤
(d[Y₁ # Y₂] + d[Y₃ # Y₄] + d[Y₁ # Y₃] + d[Y₂ # Y₄]) / 2 := by
set S := Y₁ + Y₂ + Y₃ + Y₄
set T₁ := Y₁ + Y₂
set T₂ := Y₁ + Y₃
set T₁' := Y₃ + Y₄
set T₂' := Y₂ + Y₄
have J : ρ[T₁ | ⟨T₂, S⟩ # A] ≤
(ρ[Y₁ # A] + ρ[Y₂ # A] + ρ[Y₃ # A] + ρ[Y₄ # A]) / 4
+ (d[Y₁ # Y₂] + d[Y₃ # Y₄] + d[Y₁ # Y₃] + d[Y₂ # Y₄]) / 8 + (d[Y₁ + Y₂ # Y₃ + Y₄]
+ I[Y₁ + Y₂ : Y₁ + Y₃ | Y₁ + Y₂ + Y₃ + Y₄] + d[Y₁ | Y₁ + Y₃ # Y₂ | Y₂ + Y₄]) / 4 :=
new_gen_ineq hY₁ hY₂ hY₃ hY₄ h_indep hA
have J' : ρ[T₂ | ⟨T₁, Y₁ + Y₃ + Y₂ + Y₄⟩ # A] ≤
(ρ[Y₁ # A] + ρ[Y₃ # A] + ρ[Y₂ # A] + ρ[Y₄ # A]) / 4
+ (d[Y₁ # Y₃] + d[Y₂ # Y₄] + d[Y₁ # Y₂] + d[Y₃ # Y₄]) / 8 + (d[Y₁ + Y₃ # Y₂ + Y₄]
+ I[Y₁ + Y₃ : Y₁ + Y₂|Y₁ + Y₃ + Y₂ + Y₄] + d[Y₁ | Y₁ + Y₂ # Y₃ | Y₃ + Y₄]) / 4 :=
new_gen_ineq hY₁ hY₃ hY₂ hY₄ h_indep.reindex_four_acbd hA
have : Y₁ + Y₃ + Y₂ + Y₄ = S := by simp only [S]; abel
rw [this] at J'
have : d[Y₁ + Y₂ # Y₃ + Y₄] + I[Y₁ + Y₂ : Y₁ + Y₃ | Y₁ + Y₂ + Y₃ + Y₄]
+ d[Y₁ | Y₁ + Y₃ # Y₂ | Y₂ + Y₄] + d[Y₁ + Y₃ # Y₂ + Y₄]
+ I[Y₁ + Y₃ : Y₁ + Y₂|S] + d[Y₁ | Y₁ + Y₂ # Y₃ | Y₃ + Y₄]
= (d[Y₁ # Y₂] + d[Y₃ # Y₄]) + (d[Y₁ # Y₃] + d[Y₂ # Y₄]) := by
have K : Y₁ + Y₃ + Y₂ + Y₄ = S := by simp only [S]; abel
have K' : I[Y₁ + Y₃ : Y₁ + Y₂|Y₁ + Y₂ + Y₃ + Y₄] = I[Y₁ + Y₃ : Y₃ + Y₄|Y₁ + Y₂ + Y₃ + Y₄] := by
have : Measurable (Y₁ + Y₃) := by fun_prop
rw [condMutualInfo_comm this (by fun_prop), condMutualInfo_comm this (by fun_prop)]
have B := condMutualInfo_of_inj_map (X := Y₃ + Y₄) (Y := Y₁ + Y₃) (Z := Y₁ + Y₂ + Y₃ + Y₄)
(by fun_prop) (by fun_prop) (by fun_prop) (fun a b ↦ a - b) (fun a ↦ sub_right_injective)
(μ := ℙ)
convert B with g
simp
have K'' : I[Y₁ + Y₂ : Y₁ + Y₃|Y₁ + Y₂ + Y₃ + Y₄] = I[Y₁ + Y₂ : Y₂ + Y₄|Y₁ + Y₂ + Y₃ + Y₄] := by
have : Measurable (Y₁ + Y₂) := by fun_prop
rw [condMutualInfo_comm this (by fun_prop), condMutualInfo_comm this (by fun_prop)]
have B := condMutualInfo_of_inj_map (X := Y₂ + Y₄) (Y := Y₁ + Y₂) (Z := Y₁ + Y₂ + Y₃ + Y₄)
(by fun_prop) (by fun_prop) (by fun_prop) (fun a b ↦ a - b) (fun a ↦ sub_right_injective)
(μ := ℙ)
convert B with g
simp
abel
rw [sum_of_rdist_eq_char_2' Y₁ Y₂ Y₃ Y₄ h_indep hY₁ hY₂ hY₃ hY₄,
sum_of_rdist_eq_char_2' Y₁ Y₃ Y₂ Y₄ h_indep.reindex_four_acbd hY₁ hY₃ hY₂ hY₄, K, K', K'']
abel
linarith
/-- 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
$$\sum_{1 \leq i < j \leq 3} (\rho(T_i|T_j,S) + \rho(T_j|T_i,S)
- \frac{1}{2}\sum_{i} \rho(Y_i))\le \sum_{1\leq i < j \leq 4}d[Y_i;Y_j]$$ -/ | 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
$$\sum_{1 \leq i<j \leq 3} (\rho(T_i|T_j,S) + \rho(T_j|T_i,S) - \frac{1}{2}\sum_{i} \rho(Y_i))\le \sum_{1\leq i < j \leq 4}d[Y_i;Y_j]$$
\end{lemma}
\begin{proof}\uses{rho-increase}\leanok
Apply Lemma \ref{rho-increase} on $(Y_i,Y_j,Y_k,Y_4)$ for $(i,j,k)=(1,2,3),(2,3,1),(1,3,2)$, and take the sum.
\end{proof} | 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⟩ # A] + ρ[T₂ | ⟨T₁, S⟩ # A] + ρ[T₁ | ⟨T₃, S⟩ # A] + ρ[T₃ | ⟨T₁, S⟩ # A]
+ ρ[T₂ | ⟨T₃, S⟩ # A] + ρ[T₃ | ⟨T₂, S⟩ # A]
- 3 * (ρ[Y₁ # A] + ρ[Y₂ # A] + ρ[Y₃ # A] + ρ[Y₄ # A]) / 2 ≤
d[Y₁ # Y₂] + d[Y₁ # Y₃] + d[Y₁ # Y₄] + d[Y₂ # Y₃] + d[Y₂ # Y₄] + d[Y₃ # Y₄] := by
have K₁ := condRho_sum_le hY₁ hY₂ hY₃ hY₄ h_indep hA
have K₂ := condRho_sum_le hY₂ hY₁ hY₃ hY₄ h_indep.reindex_four_bacd hA
have Y₂₁ : Y₂ + Y₁ = Y₁ + Y₂ := by abel
have dY₂₁ : d[Y₂ # Y₁] = d[Y₁ # Y₂] := rdist_symm
rw [Y₂₁, dY₂₁] at K₂
have K₃ := condRho_sum_le hY₃ hY₁ hY₂ hY₄ h_indep.reindex_four_cabd hA
have Y₃₁ : Y₃ + Y₁ = Y₁ + Y₃ := by abel
have Y₃₂ : Y₃ + Y₂ = Y₂ + Y₃ := by abel
have S₃ : Y₁ + Y₃ + Y₂ + Y₄ = Y₁ + Y₂ + Y₃ + Y₄ := by abel
have dY₃₁ : d[Y₃ # Y₁] = d[Y₁ # Y₃] := rdist_symm
have dY₃₂ : d[Y₃ # Y₂] = d[Y₂ # Y₃] := rdist_symm
rw [Y₃₁, Y₃₂, S₃, dY₃₁, dY₃₂] at K₃
linarith
include hX₁ hX₂ hX₁' hX₂' h₁ h₂ h_indep h_min hη in | 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
$$ d[X ; Y | W] := \sum_{w} \bbP[W=w] d[X ; (Y|(W=w))].$$
\end{definition} | 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 " ; " μ " # " Y " | " W " ; " μ'"]" => condRuzsaDist X Z Y W μ μ'
@[inherit_doc condRuzsaDist]
notation3:max "d[" X " | " Z " # " Y " | " W "]" => condRuzsaDist X Z Y W volume volume | 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, then
$$ d[X | Z;Y | W] = \bbH[X-Y|Z,W] - \bbH[X|Z]/2 - \bbH[Y|W]/2$$
and similarly
$$ d[X ;Y | W] = \bbH[X-Y|W] - \bbH[X]/2 - \bbH[Y|W]/2.$$
\end{lemma}
\begin{proof}\uses{copy-ent, ruz-copy, ruz-indep, cond-dist-def, conditional-entropy-def}\leanok Straightforward thanks to \Cref{copy-ent}, \Cref{ruz-copy}, \Cref{ruz-indep}, \Cref{cond-dist-def}, \Cref{conditional-entropy-def}.
\end{proof} | 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', W'⟩) μ' μ''')
[FiniteRange W] [FiniteRange W'] :
d[X ; μ # Y | W ; μ'] = d[X' ; μ'' # Y' | W' ; μ'''] := by
classical
set A := (FiniteRange.toFinset W) ∪ (FiniteRange.toFinset W')
have hfull : Measure.prod (dirac ()) (μ'.map W)
((Finset.univ (α := Unit) ×ˢ A : Finset (Unit × T)) : Set (Unit × T))ᶜ = 0 := by
apply Measure.prod_of_full_measure_finset
· simp
simp only [A]
rw [Measure.map_apply ‹_›]
convert measure_empty (μ := μ)
simp [← FiniteRange.range]
measurability
have hfull' : Measure.prod (dirac ()) (μ'''.map W')
((Finset.univ (α := Unit) ×ˢ A : Finset (Unit × T)) : Set (Unit × T))ᶜ = 0 := by
apply Measure.prod_of_full_measure_finset
· simp
simp only [A]
rw [Measure.map_apply ‹_›]
convert measure_empty (μ := μ)
simp [← FiniteRange.range]
measurability
rw [condRuzsaDist'_def, condRuzsaDist'_def, Kernel.rdist, Kernel.rdist,
integral_eq_setIntegral hfull, integral_eq_setIntegral hfull', integral_finset _ _ IntegrableOn.finset,
integral_finset _ _ IntegrableOn.finset]
have hWW' : μ'.map W = μ'''.map W' := (h2.comp measurable_snd).map_eq
simp_rw [Measure.prod_apply_singleton, ENNReal.toReal_mul, ← hWW',
Measure.map_apply hW (.singleton _)]
congr with x
by_cases hw : μ' (W ⁻¹' {x.2}) = 0
· simp only [smul_eq_mul, mul_eq_mul_left_iff, mul_eq_zero]
refine Or.inr (Or.inr ?_)
simp [ENNReal.toReal_eq_zero_iff, measure_ne_top, hw]
congr 2
· rw [Kernel.const_apply, Kernel.const_apply, h1.map_eq]
· have hWW'x : μ' (W ⁻¹' {x.2}) = μ''' (W' ⁻¹' {x.2}) := by
have : μ'.map W {x.2} = μ'''.map W' {x.2} := by rw [hWW']
rwa [Measure.map_apply hW (.singleton _),
Measure.map_apply hW' (.singleton _)] at this
ext s hs
rw [condDistrib_apply' hY hW _ _ hw hs, condDistrib_apply' hY' hW' _ _ _ hs]
swap; · rwa [hWW'x] at hw
congr
have : μ'.map (⟨Y, W⟩) (s ×ˢ {x.2}) = μ'''.map (⟨Y', W'⟩) (s ×ˢ {x.2}) := by rw [h2.map_eq]
rwa [Measure.map_apply (hY.prodMk hW) (hs.prod (.singleton _)),
Measure.map_apply (hY'.prodMk hW') (hs.prod (.singleton _)),
Set.mk_preimage_prod, Set.mk_preimage_prod, Set.inter_comm,
Set.inter_comm ((fun a ↦ Y' a) ⁻¹' s)] at this | 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, then
$$ d[X | Z;Y | W] = \bbH[X-Y|Z,W] - \bbH[X|Z]/2 - \bbH[Y|W]/2$$
and similarly
$$ d[X ;Y | W] = \bbH[X-Y|W] - \bbH[X]/2 - \bbH[Y|W]/2.$$
\end{lemma}
\begin{proof}\uses{copy-ent, ruz-copy, ruz-indep, cond-dist-def, conditional-entropy-def}\leanok Straightforward thanks to \Cref{copy-ent}, \Cref{ruz-copy}, \Cref{ruz-indep}, \Cref{cond-dist-def}, \Cref{conditional-entropy-def}.
\end{proof} | /-- 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 # Y | W ; μ] = H[X - Y | W ; μ] - H[X ; μ]/2 - H[Y | W ; μ]/2 := by
have : IsProbabilityMeasure (μ.map W) := isProbabilityMeasure_map hW.aemeasurable
rw [condRuzsaDist'_def, Kernel.rdist_eq', condEntropy_eq_kernel_entropy _ hW,
condEntropy_eq_kernel_entropy hY hW, entropy_eq_kernel_entropy]
rotate_left
· exact hX.sub hY
congr 2
let Z : Ω → Unit := fun _ ↦ ()
rw [← condDistrib_unit_right hX μ]
have h' : IndepFun (⟨X,Z⟩) (⟨Y, W⟩) μ := by
rw [indepFun_iff_measure_inter_preimage_eq_mul]
intro s t hs ht
have : ⟨X, Z⟩ ⁻¹' s = X ⁻¹' ((fun c ↦ (c, ())) ⁻¹' s) := by ext1 y; simp
rw [this]
rw [indepFun_iff_measure_inter_preimage_eq_mul] at h
exact h _ _ (measurable_prodMk_right hs) ht
have h_indep := condDistrib_eq_prod_of_indepFun hX measurable_const hY hW _ h'
have h_meas_eq : μ.map (⟨Z, W⟩) = (Measure.dirac ()).prod (μ.map W) := by
ext s hs
rw [Measure.map_apply (measurable_const.prodMk hW) hs, Measure.prod_apply hs, lintegral_dirac,
Measure.map_apply hW (measurable_prodMk_left hs)]
congr
rw [← h_meas_eq]
have : Kernel.map (Kernel.prodMkRight T (condDistrib X Z μ)
×ₖ Kernel.prodMkLeft Unit (condDistrib Y W μ)) (fun x ↦ x.1 - x.2)
=ᵐ[μ.map (⟨Z, W⟩)] Kernel.map (condDistrib (⟨X, Y⟩) (⟨Z, W⟩) μ) (fun x ↦ x.1 - x.2) := by
filter_upwards [h_indep] with y hy
conv_rhs => rw [Kernel.map_apply _ (by fun_prop), hy]
rw [← Kernel.mapOfMeasurable_eq_map _ (by fun_prop)]
rfl
rw [Kernel.entropy_congr this]
have : Kernel.map (condDistrib (⟨X, Y⟩) (⟨Z, W⟩) μ) (fun x ↦ x.1 - x.2)
=ᵐ[μ.map (⟨Z, W⟩)] condDistrib (X - Y) (⟨Z, W⟩) μ :=
(condDistrib_comp (hX.prodMk hY) (measurable_const.prodMk hW) _ _).symm
rw [Kernel.entropy_congr this]
have h_meas : μ.map (⟨Z, W⟩) = (μ.map W).map (Prod.mk ()) := by
ext s hs
rw [Measure.map_apply measurable_prodMk_left hs, h_meas_eq, Measure.prod_apply hs,
lintegral_dirac]
have h_ker : condDistrib (X - Y) (⟨Z, W⟩) μ
=ᵐ[μ.map (⟨Z, W⟩)] Kernel.prodMkLeft Unit (condDistrib (X - Y) W μ) := by
rw [Filter.EventuallyEq, ae_iff_of_countable]
intro x hx
rw [Measure.map_apply (measurable_const.prodMk hW) (.singleton _)] at hx
ext s hs
have h_preimage_eq : (fun a ↦ (PUnit.unit, W a)) ⁻¹' {x} = W ⁻¹' {x.2} := by
conv_lhs => rw [← Prod.eta x, ← Set.singleton_prod_singleton, Set.mk_preimage_prod]
ext1 y
simp
rw [Kernel.prodMkLeft_apply, condDistrib_apply' _ (measurable_const.prodMk hW) _ _ hx hs,
condDistrib_apply' _ hW _ _ _ hs]
rotate_left
· exact hX.sub hY
· convert hx
exact h_preimage_eq.symm
· exact hX.sub hY
congr
rw [Kernel.entropy_congr h_ker, h_meas, Kernel.entropy_prodMkLeft_unit]
end
omit [Countable S] in
/-- The conditional Ruzsa distance is unchanged if the sets of random variables are replaced with
copies. -/ | 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
\begin{align}\nonumber d[X ; Y + Z] -d[X ; Y] & \leq \tfrac{1}{2} (\bbH[Y+ Z] - \bbH[Y]) \\ & = \tfrac{1}{2} d[Y; Z] + \tfrac{1}{4} \bbH[Z] - \tfrac{1}{4} \bbH[Y]. \label{lem51-a} \end{align}
and
\begin{align}\nonumber
d[X ;Y|Y+ Z] - d[X ;Y] & \leq \tfrac{1}{2} \bigl(\bbH[Y+ Z] - \bbH[Z]\bigr) \\ & = \tfrac{1}{2} d[Y;Z] + \tfrac{1}{4} \bbH[Y] - \tfrac{1}{4} \bbH[Z].
\label{ruzsa-3}
\end{align}
\end{lemma}
\begin{proof}
\uses{ruz-copy, independent-exist, kv, ruz-indep, relabeled-entropy, cond-dist-fact}\leanok
We first prove~\eqref{lem51-a}. We may assume (taking an independent copy, using \Cref{independent-exist} and \Cref{ruz-copy}, \ref{ruz-indep}) that $X$ is independent of $Y, Z$. Then we have
\begin{align*} d[X ;Y+ Z] & - d[X ;Y] \\ & = \bbH[X + Y + Z] - \bbH[X + Y] - \tfrac{1}{2}\bbH[Y + Z] + \tfrac{1}{2} \bbH[Y].\end{align*}
Combining this with \Cref{kv} gives the required bound. The second form of the result is immediate \Cref{ruz-indep}.
Turning to~\eqref{ruzsa-3}, we have from \Cref{information-def} and \Cref{relabeled-entropy}
\begin{align*} \bbI[Y : Y+ Z] & = \bbH[Y] + \bbH[Y + Z] - \bbH[Y, Y + Z] \\ & = \bbH[Y] + \bbH[Y + Z] - \bbH[Y, Z] = \bbH[Y + Z] - \bbH[Z],\end{align*}
and so~\eqref{ruzsa-3} is a consequence of \Cref{cond-dist-fact}. Once again the second form of the result is immediate from \Cref{ruz-indep}.
\end{proof} | 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'] - H[Y; μ']) / 2 :=
(comparison_of_ruzsa_distances μ hX hY hZ h).1
variable (μ) [Module (ZMod 2) G] in | 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
\begin{align}\nonumber d[X ; Y + Z] -d[X ; Y] & \leq \tfrac{1}{2} (\bbH[Y+ Z] - \bbH[Y]) \\ & = \tfrac{1}{2} d[Y; Z] + \tfrac{1}{4} \bbH[Z] - \tfrac{1}{4} \bbH[Y]. \label{lem51-a} \end{align}
and
\begin{align}\nonumber
d[X ;Y|Y+ Z] - d[X ;Y] & \leq \tfrac{1}{2} \bigl(\bbH[Y+ Z] - \bbH[Z]\bigr) \\ & = \tfrac{1}{2} d[Y;Z] + \tfrac{1}{4} \bbH[Y] - \tfrac{1}{4} \bbH[Z].
\label{ruzsa-3}
\end{align}
\end{lemma}
\begin{proof}
\uses{ruz-copy, independent-exist, kv, ruz-indep, relabeled-entropy, cond-dist-fact}\leanok
We first prove~\eqref{lem51-a}. We may assume (taking an independent copy, using \Cref{independent-exist} and \Cref{ruz-copy}, \ref{ruz-indep}) that $X$ is independent of $Y, Z$. Then we have
\begin{align*} d[X ;Y+ Z] & - d[X ;Y] \\ & = \bbH[X + Y + Z] - \bbH[X + Y] - \tfrac{1}{2}\bbH[Y + Z] + \tfrac{1}{2} \bbH[Y].\end{align*}
Combining this with \Cref{kv} gives the required bound. The second form of the result is immediate \Cref{ruz-indep}.
Turning to~\eqref{ruzsa-3}, we have from \Cref{information-def} and \Cref{relabeled-entropy}
\begin{align*} \bbI[Y : Y+ Z] & = \bbH[Y] + \bbH[Y + Z] - \bbH[Y, Y + Z] \\ & = \bbH[Y] + \bbH[Y + Z] - \bbH[Y, Z] = \bbH[Y + Z] - \bbH[Z],\end{align*}
and so~\eqref{ruzsa-3} is a consequence of \Cref{cond-dist-fact}. Once again the second form of the result is immediate from \Cref{ruz-indep}.
\end{proof} | 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' # Z; μ'] / 2 + H[Z; μ'] / 4 - H[Y; μ'] / 4 := by
linarith [condRuzsaDist_diff_le μ hX hY hZ h, entropy_sub_entropy_eq_condRuzsaDist_add μ hX hY hZ h]
variable (μ) in | 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
\begin{align}\nonumber d[X ; Y + Z] -d[X ; Y] & \leq \tfrac{1}{2} (\bbH[Y+ Z] - \bbH[Y]) \\ & = \tfrac{1}{2} d[Y; Z] + \tfrac{1}{4} \bbH[Z] - \tfrac{1}{4} \bbH[Y]. \label{lem51-a} \end{align}
and
\begin{align}\nonumber
d[X ;Y|Y+ Z] - d[X ;Y] & \leq \tfrac{1}{2} \bigl(\bbH[Y+ Z] - \bbH[Z]\bigr) \\ & = \tfrac{1}{2} d[Y;Z] + \tfrac{1}{4} \bbH[Y] - \tfrac{1}{4} \bbH[Z].
\label{ruzsa-3}
\end{align}
\end{lemma}
\begin{proof}
\uses{ruz-copy, independent-exist, kv, ruz-indep, relabeled-entropy, cond-dist-fact}\leanok
We first prove~\eqref{lem51-a}. We may assume (taking an independent copy, using \Cref{independent-exist} and \Cref{ruz-copy}, \ref{ruz-indep}) that $X$ is independent of $Y, Z$. Then we have
\begin{align*} d[X ;Y+ Z] & - d[X ;Y] \\ & = \bbH[X + Y + Z] - \bbH[X + Y] - \tfrac{1}{2}\bbH[Y + Z] + \tfrac{1}{2} \bbH[Y].\end{align*}
Combining this with \Cref{kv} gives the required bound. The second form of the result is immediate \Cref{ruz-indep}.
Turning to~\eqref{ruzsa-3}, we have from \Cref{information-def} and \Cref{relabeled-entropy}
\begin{align*} \bbI[Y : Y+ Z] & = \bbH[Y] + \bbH[Y + Z] - \bbH[Y, Y + Z] \\ & = \bbH[Y] + \bbH[Y + Z] - \bbH[Y, Z] = \bbH[Y + Z] - \bbH[Z],\end{align*}
and so~\eqref{ruzsa-3} is a consequence of \Cref{cond-dist-fact}. Once again the second form of the result is immediate from \Cref{ruz-indep}.
\end{proof} | 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 ; μ'] - H[Z ; μ'])/2 := by
rw [← mutualInfo_add_right hY hZ h]
linarith [condRuzsaDist_le' (W := Y + Z) μ μ' hX hY (by fun_prop)]
variable (μ) [Module (ZMod 2) G] in | 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
\begin{align}\nonumber d[X ; Y + Z] -d[X ; Y] & \leq \tfrac{1}{2} (\bbH[Y+ Z] - \bbH[Y]) \\ & = \tfrac{1}{2} d[Y; Z] + \tfrac{1}{4} \bbH[Z] - \tfrac{1}{4} \bbH[Y]. \label{lem51-a} \end{align}
and
\begin{align}\nonumber
d[X ;Y|Y+ Z] - d[X ;Y] & \leq \tfrac{1}{2} \bigl(\bbH[Y+ Z] - \bbH[Z]\bigr) \\ & = \tfrac{1}{2} d[Y;Z] + \tfrac{1}{4} \bbH[Y] - \tfrac{1}{4} \bbH[Z].
\label{ruzsa-3}
\end{align}
\end{lemma}
\begin{proof}
\uses{ruz-copy, independent-exist, kv, ruz-indep, relabeled-entropy, cond-dist-fact}\leanok
We first prove~\eqref{lem51-a}. We may assume (taking an independent copy, using \Cref{independent-exist} and \Cref{ruz-copy}, \ref{ruz-indep}) that $X$ is independent of $Y, Z$. Then we have
\begin{align*} d[X ;Y+ Z] & - d[X ;Y] \\ & = \bbH[X + Y + Z] - \bbH[X + Y] - \tfrac{1}{2}\bbH[Y + Z] + \tfrac{1}{2} \bbH[Y].\end{align*}
Combining this with \Cref{kv} gives the required bound. The second form of the result is immediate \Cref{ruz-indep}.
Turning to~\eqref{ruzsa-3}, we have from \Cref{information-def} and \Cref{relabeled-entropy}
\begin{align*} \bbI[Y : Y+ Z] & = \bbH[Y] + \bbH[Y + Z] - \bbH[Y, Y + Z] \\ & = \bbH[Y] + \bbH[Y + Z] - \bbH[Y, Z] = \bbH[Y + Z] - \bbH[Z],\end{align*}
and so~\eqref{ruzsa-3} is a consequence of \Cref{cond-dist-fact}. Once again the second form of the result is immediate from \Cref{ruz-indep}.
\end{proof} | 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 ; μ' # Z ; μ']/2 + H[Y ; μ']/4 - H[Z ; μ']/4 := by
linarith [condRuzsaDist_diff_le'' μ hX hY hZ h, entropy_sub_entropy_eq_condRuzsaDist_add μ hX hY hZ h]
variable (μ) in | 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 \tfrac{1}{2} ( \bbH[Y + Z + Z'] + \bbH[Y + Z] - \bbH[Y] - \bbH[Z']).\label{7111}
\end{align}
\end{lemma}
\begin{proof}
\uses{first-useful}\leanok
By \Cref{first-useful} (with a change of variables) we have
\[d[X ; Y + Z | Y + Z + Z'] - d[X ; Y + Z] \leq \tfrac{1}{2}( \bbH[Y + Z + Z'] - \bbH[Z']).\]
Adding this to~\eqref{lem51-a} gives the result.
\end{proof} | 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 ; μ # Y + Z | Y + Z + Z'; μ'] - d[X ; μ # Y; μ'] ≤
(H[Y + Z + Z'; μ'] + H[Y + Z; μ'] - H[Y ; μ'] - H[Z' ; μ'])/2 := by
have hadd : IndepFun (Y + Z) Z' μ' :=
(h.indepFun_add_left (Fin.cases hY <| Fin.cases hZ <| Fin.cases hZ' Fin.rec0) 0 1 2
(show 0 ≠ 2 by decide) (show 1 ≠ 2 by decide))
have h1 := condRuzsaDist_diff_le'' μ hX (show Measurable (Y + Z) by fun_prop) hZ' hadd
have h2 := condRuzsaDist_diff_le μ hX hY hZ (h.indepFun (show 0 ≠ 1 by decide))
linarith [h1, h2] | 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
\[ d[X | Z;Y | W] \leq d[X ; Y] + \tfrac{1}{2} \bbI[X : Z] + \tfrac{1}{2} \bbI[Y : W].\]
In particular,
\[ d[X ;Y | W] \leq d[X ; Y] + \tfrac{1}{2} \bbI[Y : W].\]
\end{lemma}
\begin{proof}
\uses{cond-dist-alt, independent-exist, cond-reduce}\leanok
Using \Cref{cond-dist-alt} and \Cref{independent-exist}, if $(X',Z'), (Y',W')$ are independent copies of the variables $(X,Z)$, $(Y,W)$, we have
\begin{align*}
d[X | Z; Y | W]&= \bbH[X'-Y'|Z',W'] - \tfrac{1}{2} \bbH[X'|Z'] - \tfrac{1}{2}H[Y'|W'] \\
&\le \bbH[X'-Y']- \tfrac{1}{2} \bbH[X'|Z'] - \tfrac{1}{2}H[Y'|W'] \\
&= d[X';Y'] + \tfrac{1}{2} \bbI[X' : Z'] + \tfrac{1}{2} \bbI[Y' : W'].
\end{align*}
Here, in the middle step we used \Cref{cond-reduce}, and in the last step we used \Cref{ruz-dist-def} and \Cref{information-def}.
\end{proof} | 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 ; μ'] ≤ d[X ; μ # Y ; μ'] + I[X : Z ; μ]/2 + I[Y : W ; μ']/2 := by
have hXZ : Measurable (⟨X, Z⟩ : Ω → G × S):= hX.prodMk hZ
have hYW : Measurable (⟨Y, W⟩ : Ω' → G × T):= hY.prodMk hW
obtain ⟨ν, XZ', YW', _, hXZ', hYW', hind, hIdXZ, hIdYW, _, _⟩ :=
independent_copies_finiteRange hXZ hYW μ μ'
let X' := Prod.fst ∘ XZ'
let Z' := Prod.snd ∘ XZ'
let Y' := Prod.fst ∘ YW'
let W' := Prod.snd ∘ YW'
have hX' : Measurable X' := hXZ'.fst
have hZ' : Measurable Z' := hXZ'.snd
have hY' : Measurable Y' := hYW'.fst
have hW' : Measurable W' := hYW'.snd
have : FiniteRange W' := instFiniteRangeComp ..
have : FiniteRange X' := instFiniteRangeComp ..
have : FiniteRange Y' := instFiniteRangeComp ..
have : FiniteRange Z' := instFiniteRangeComp ..
have hind' : IndepFun X' Y' ν := hind.comp measurable_fst measurable_fst
rw [show XZ' = ⟨X', Z'⟩ by rfl] at hIdXZ hind
rw [show YW' = ⟨Y', W'⟩ by rfl] at hIdYW hind
rw [← condRuzsaDist_of_copy hX' hZ' hY' hW' hX hZ hY hW hIdXZ hIdYW,
condRuzsaDist_of_indep hX' hZ' hY' hW' _ hind]
have hIdX : IdentDistrib X X' μ ν := hIdXZ.symm.comp measurable_fst
have hIdY : IdentDistrib Y Y' μ' ν := hIdYW.symm.comp measurable_fst
rw [hIdX.rdist_eq hIdY, hIdXZ.symm.mutualInfo_eq, hIdYW.symm.mutualInfo_eq,
hind'.rdist_eq hX' hY', mutualInfo_eq_entropy_sub_condEntropy hX' hZ',
mutualInfo_eq_entropy_sub_condEntropy hY' hW']
have h := condEntropy_le_entropy ν (X := X' - Y') (hX'.sub hY') (hZ'.prodMk hW')
linarith [h, entropy_nonneg Z' ν, entropy_nonneg W' ν]
variable (μ μ') in | 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
\[ d[X | Z;Y | W] \leq d[X ; Y] + \tfrac{1}{2} \bbI[X : Z] + \tfrac{1}{2} \bbI[Y : W].\]
In particular,
\[ d[X ;Y | W] \leq d[X ; Y] + \tfrac{1}{2} \bbI[Y : W].\]
\end{lemma}
\begin{proof}
\uses{cond-dist-alt, independent-exist, cond-reduce}\leanok
Using \Cref{cond-dist-alt} and \Cref{independent-exist}, if $(X',Z'), (Y',W')$ are independent copies of the variables $(X,Z)$, $(Y,W)$, we have
\begin{align*}
d[X | Z; Y | W]&= \bbH[X'-Y'|Z',W'] - \tfrac{1}{2} \bbH[X'|Z'] - \tfrac{1}{2}H[Y'|W'] \\
&\le \bbH[X'-Y']- \tfrac{1}{2} \bbH[X'|Z'] - \tfrac{1}{2}H[Y'|W'] \\
&= d[X';Y'] + \tfrac{1}{2} \bbI[X' : Z'] + \tfrac{1}{2} \bbI[Y' : W'].
\end{align*}
Here, in the middle step we used \Cref{cond-reduce}, and in the last step we used \Cref{ruz-dist-def} and \Cref{information-def}.
\end{proof} | 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
rw [← condRuzsaDist_of_const hX _ _ (0 : Fin 1)]
refine (condRuzsaDist_le μ μ' hX measurable_const hY hW).trans ?_
simp [mutualInfo_const hX (0 : Fin 1)]
variable (μ μ') in | 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, then
$$ d[X | Z;Y | W] = \bbH[X-Y|Z,W] - \bbH[X|Z]/2 - \bbH[Y|W]/2$$
and similarly
$$ d[X ;Y | W] = \bbH[X-Y|W] - \bbH[X]/2 - \bbH[Y|W]/2.$$
\end{lemma}
\begin{proof}\uses{copy-ent, ruz-copy, ruz-indep, cond-dist-def, conditional-entropy-def}\leanok Straightforward thanks to \Cref{copy-ent}, \Cref{ruz-copy}, \Cref{ruz-indep}, \Cref{cond-dist-def}, \Cref{conditional-entropy-def}.
\end{proof} | 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')
[IsFiniteMeasure μ] [IsFiniteMeasure μ'] [IsFiniteMeasure μ''] [IsFiniteMeasure μ''']
(h1 : IdentDistrib (⟨X, Z⟩) (⟨X', Z'⟩) μ μ'') (h2 : IdentDistrib (⟨Y, W⟩) (⟨Y', W'⟩) μ' μ''')
[FiniteRange Z] [FiniteRange W] [FiniteRange Z'] [FiniteRange W'] :
d[X | Z ; μ # Y | W ; μ'] = d[X' | Z' ; μ'' # Y' | W' ; μ'''] := by
classical
set A := (FiniteRange.toFinset Z) ∪ (FiniteRange.toFinset Z')
set B := (FiniteRange.toFinset W) ∪ (FiniteRange.toFinset W')
have hfull : Measure.prod (μ.map Z) (μ'.map W) ((A ×ˢ B : Finset (S × T)): Set (S × T))ᶜ = 0 := by
simp only [A, B]
apply Measure.prod_of_full_measure_finset
all_goals {
rw [Measure.map_apply ‹_›]
convert measure_empty (μ := μ)
simp [← FiniteRange.range]
measurability
}
have hfull' : Measure.prod (μ''.map Z') (μ'''.map W')
((A ×ˢ B : Finset (S × T)): Set (S × T))ᶜ = 0 := by
simp only [A, B]
apply Measure.prod_of_full_measure_finset
all_goals {
rw [Measure.map_apply ‹_›]
convert measure_empty (μ := μ)
simp [← FiniteRange.range]
measurability
}
rw [condRuzsaDist_def, condRuzsaDist_def, Kernel.rdist, Kernel.rdist,
integral_eq_setIntegral hfull, integral_eq_setIntegral hfull', integral_finset _ _ IntegrableOn.finset,
integral_finset _ _ IntegrableOn.finset]
have hZZ' : μ.map Z = μ''.map Z' := (h1.comp measurable_snd).map_eq
have hWW' : μ'.map W = μ'''.map W' := (h2.comp measurable_snd).map_eq
simp_rw [Measure.prod_apply_singleton, ENNReal.toReal_mul, ← hZZ', ← hWW',
Measure.map_apply hZ (.singleton _),
Measure.map_apply hW (.singleton _)]
congr with x
by_cases hz : μ (Z ⁻¹' {x.1}) = 0
· simp only [smul_eq_mul, mul_eq_mul_left_iff, mul_eq_zero]
refine Or.inr (Or.inl ?_)
simp [ENNReal.toReal_eq_zero_iff, measure_ne_top, hz]
by_cases hw : μ' (W ⁻¹' {x.2}) = 0
· simp only [smul_eq_mul, mul_eq_mul_left_iff, mul_eq_zero]
refine Or.inr (Or.inr ?_)
simp [ENNReal.toReal_eq_zero_iff, measure_ne_top, hw]
congr 2
· have hZZ'x : μ (Z ⁻¹' {x.1}) = μ'' (Z' ⁻¹' {x.1}) := by
have : μ.map Z {x.1} = μ''.map Z' {x.1} := by rw [hZZ']
rwa [Measure.map_apply hZ (.singleton _),
Measure.map_apply hZ' (.singleton _)] at this
ext s hs
rw [condDistrib_apply' hX hZ _ _ hz hs, condDistrib_apply' hX' hZ' _ _ _ hs]
swap; · rwa [hZZ'x] at hz
congr
have : μ.map (⟨X, Z⟩) (s ×ˢ {x.1}) = μ''.map (⟨X', Z'⟩) (s ×ˢ {x.1}) := by rw [h1.map_eq]
rwa [Measure.map_apply (hX.prodMk hZ) (hs.prod (.singleton _)),
Measure.map_apply (hX'.prodMk hZ') (hs.prod (.singleton _)),
Set.mk_preimage_prod, Set.mk_preimage_prod, Set.inter_comm,
Set.inter_comm ((fun a ↦ X' a) ⁻¹' s)] at this
· have hWW'x : μ' (W ⁻¹' {x.2}) = μ''' (W' ⁻¹' {x.2}) := by
have : μ'.map W {x.2} = μ'''.map W' {x.2} := by rw [hWW']
rwa [Measure.map_apply hW (.singleton _),
Measure.map_apply hW' (.singleton _)] at this
ext s hs
rw [condDistrib_apply' hY hW _ _ hw hs, condDistrib_apply' hY' hW' _ _ _ hs]
swap; · rwa [hWW'x] at hw
congr
have : μ'.map (⟨Y, W⟩) (s ×ˢ {x.2}) = μ'''.map (⟨Y', W'⟩) (s ×ˢ {x.2}) := by rw [h2.map_eq]
rwa [Measure.map_apply (hY.prodMk hW) (hs.prod (.singleton _)),
Measure.map_apply (hY'.prodMk hW') (hs.prod (.singleton _)),
Set.mk_preimage_prod, Set.mk_preimage_prod, Set.inter_comm,
Set.inter_comm ((fun a ↦ Y' a) ⁻¹' s)] at this | 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, then
$$ d[X | Z;Y | W] = \bbH[X-Y|Z,W] - \bbH[X|Z]/2 - \bbH[Y|W]/2$$
and similarly
$$ d[X ;Y | W] = \bbH[X-Y|W] - \bbH[X]/2 - \bbH[Y|W]/2.$$
\end{lemma}
\begin{proof}\uses{copy-ent, ruz-copy, ruz-indep, cond-dist-def, conditional-entropy-def}\leanok Straightforward thanks to \Cref{copy-ent}, \Cref{ruz-copy}, \Cref{ruz-indep}, \Cref{cond-dist-def}, \Cref{conditional-entropy-def}.
\end{proof} | 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 - Y | ⟨Z, W⟩ ; μ] - H[X | Z ; μ]/2 - H[Y | W ; μ]/2 := by
have : IsProbabilityMeasure (μ.map Z) := isProbabilityMeasure_map hZ.aemeasurable
have : IsProbabilityMeasure (μ.map W) := isProbabilityMeasure_map hW.aemeasurable
rw [condRuzsaDist_def, Kernel.rdist_eq', condEntropy_eq_kernel_entropy _ (hZ.prodMk hW),
condEntropy_eq_kernel_entropy hX hZ, condEntropy_eq_kernel_entropy hY hW]
swap; · exact hX.sub hY
congr 2
have hZW : IndepFun Z W μ := h.comp measurable_snd measurable_snd
have hZW_map : μ.map (⟨Z, W⟩) = (μ.map Z).prod (μ.map W) :=
(indepFun_iff_map_prod_eq_prod_map_map hZ.aemeasurable hW.aemeasurable).mp hZW
rw [← hZW_map]
refine Kernel.entropy_congr ?_
have : Kernel.map (condDistrib (⟨X, Y⟩) (⟨Z, W⟩) μ) (fun x ↦ x.1 - x.2)
=ᵐ[μ.map (⟨Z, W⟩)] condDistrib (X - Y) (⟨Z, W⟩) μ :=
(condDistrib_comp (hX.prodMk hY) (hZ.prodMk hW) _ _).symm
refine (this.symm.trans ?_).symm
suffices Kernel.prodMkRight T (condDistrib X Z μ)
×ₖ Kernel.prodMkLeft S (condDistrib Y W μ)
=ᵐ[μ.map (⟨Z, W⟩)] condDistrib (⟨X, Y⟩) (⟨Z, W⟩) μ by
filter_upwards [this] with x hx
rw [Kernel.map_apply _ (by fun_prop), Kernel.map_apply _ (by fun_prop), hx]
exact (condDistrib_eq_prod_of_indepFun hX hZ hY hW μ h).symm | 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; X_2 | X_2 + \tilde X_1] - d[X^0_2; X_2]).
\end{align*}
\end{lemma}
\begin{proof}\uses{cond-distance-lower}\leanok Immediate from \Cref{cond-distance-lower}.
\end{proof} | 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
$$ d[X'_1|Z;X'_2|W] \geq k - \eta (d[X^0_1;X'_1|Z] - d[X^0_1;X_1] ) - \eta (d[X^0_2;X'_2|W] - d[X^0_2;X_2] ).$$
\end{lemma}
\begin{proof}\uses{distance-lower}\leanok Apply \Cref{distance-lower} to conditioned random variables and then average.
\end{proof} | 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) (hW : Measurable W) :
d[X₁ # X₂] - p.η * (d[p.X₀₁ # X₁' | Z] - d[p.X₀₁ # X₁])
- p.η * (d[p.X₀₂ # X₂' | W] - d[p.X₀₂ # X₂]) ≤ d[X₁' | Z # X₂' | W] := by
have hz (a : ℝ) : a = ∑ z ∈ FiniteRange.toFinset Z, (ℙ (Z ⁻¹' {z})).toReal * a := by
simp_rw [← Finset.sum_mul,← Measure.map_apply hZ (MeasurableSet.singleton _), Finset.sum_toReal_measure_singleton]
rw [FiniteRange.full hZ]
simp
have hw (a : ℝ) : a = ∑ w ∈ FiniteRange.toFinset W, (ℙ (W ⁻¹' {w})).toReal * a := by
simp_rw [← Finset.sum_mul,← Measure.map_apply hW (MeasurableSet.singleton _), Finset.sum_toReal_measure_singleton]
rw [FiniteRange.full hW]
simp
rw [condRuzsaDist_eq_sum h1 hZ h2 hW, condRuzsaDist'_eq_sum h1 hZ, hz d[X₁ # X₂],
hz d[p.X₀₁ # X₁], hz (p.η * (d[p.X₀₂ # X₂' | W] - d[p.X₀₂ # X₂])),
← Finset.sum_sub_distrib, Finset.mul_sum, ← Finset.sum_sub_distrib, ← Finset.sum_sub_distrib]
apply Finset.sum_le_sum
intro z _
rw [condRuzsaDist'_eq_sum h2 hW, hw d[p.X₀₂ # X₂],
hw ((ℙ (Z ⁻¹' {z})).toReal * d[X₁ # X₂] - p.η * ((ℙ (Z ⁻¹' {z})).toReal *
d[p.X₀₁ ; ℙ # X₁' ; ℙ[|Z ← z]] - (ℙ (Z ⁻¹' {z})).toReal * d[p.X₀₁ # X₁])),
← Finset.sum_sub_distrib, Finset.mul_sum, Finset.mul_sum, ← Finset.sum_sub_distrib]
apply Finset.sum_le_sum
intro w _
rcases eq_or_ne (ℙ (Z ⁻¹' {z})) 0 with hpz | hpz
· simp [hpz]
rcases eq_or_ne (ℙ (W ⁻¹' {w})) 0 with hpw | hpw
· simp [hpw]
set μ := (hΩ₁.volume)[|Z ← z]
have hμ : IsProbabilityMeasure μ := cond_isProbabilityMeasure hpz
set μ' := ℙ[|W ← w]
have hμ' : IsProbabilityMeasure μ' := cond_isProbabilityMeasure hpw
suffices d[X₁ # X₂] - p.η * (d[p.X₀₁; volume # X₁'; μ] - d[p.X₀₁ # X₁]) -
p.η * (d[p.X₀₂; volume # X₂'; μ'] - d[p.X₀₂ # X₂]) ≤ d[X₁' ; μ # X₂'; μ'] by
replace this := mul_le_mul_of_nonneg_left this (show 0 ≤ (ℙ (Z ⁻¹' {z})).toReal * (ℙ (W ⁻¹' {w})).toReal by positivity)
convert this using 1
ring
exact distance_ge_of_min' p h h1 h2 | 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 of random variables (not necessarily $G$-valued).
Suppose that the pairs $(X_i, Y_i)$ are jointly independent of one another (but $X_i$ need not be independent of $Y_i$).
Then
\begin{align}\nonumber
D[ X_{[m]} | Y_{[m]} ] &= D[ X_{[m]} \,|\, \pi(X_{[m]}), Y_{[m]}] + D[ \pi(X_{[m]}) \,|\, Y_{[m]}] \\
&\quad\qquad + \bbI[ \sum_{i=1}^m X_i : \pi(X_{[m]}) \; \big| \; \pi\bigl(\sum_{i=1}^m X_i \bigr), Y_{[m]} ].\label{chain-eq-cond}
\end{align}
\end{lemma}
\begin{proof}\uses{multidist-chain-rule}\leanok
For each $y_i$ in the support of $p_{Y_i}$, apply \Cref{multidist-chain-rule} with $X_i$ replaced by the conditioned random variable $(X_i|Y_i=y_i)$, and the claim~\eqref{chain-eq-cond} follows by averaging~\eqref{chain-eq} in the $y_i$ using the weights $p_{Y_i}$.
\end{proof} | 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]
{m : ℕ} {Ω : Type*} [hΩ : MeasureSpace Ω]
{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; fun _ ↦ hΩ] = D[X | fun i ↦ ⟨π ∘ X i, Y i⟩; fun _ ↦ hΩ]
+ D[fun i ↦ π ∘ X i | Y; fun _ ↦ hΩ]
+ I[∑ i, X i : fun ω ↦ (fun i ↦ π (X i ω)) |
⟨π ∘ (∑ i, X i), fun ω ↦ (fun i ↦ Y i ω)⟩] := by
have : IsProbabilityMeasure (ℙ : Measure Ω) := h_indep.isProbabilityMeasure
set E' := fun (y : Fin m → S) ↦ ⋂ i, Y i ⁻¹' {y i}
set f := fun (y : Fin m → S) ↦ (ℙ (E' y)).toReal
set hΩc : (Fin m → S) → MeasureSpace Ω := fun y ↦ ⟨cond ℙ (E' y)⟩
calc
_ = ∑ y, (f y) * D[X; fun _ ↦ hΩc y] := condMultiDist_eq' hX hY h_indep
_ = ∑ y, (f y) * D[X | fun i ↦ π ∘ X i; fun _ ↦ hΩc y]
+ ∑ y, (f y) * D[fun i ↦ π ∘ X i; fun _ ↦ hΩc y]
+ ∑ y, (f y) * I[∑ i, X i : fun ω ↦ (fun i ↦ π (X i ω)) |
π ∘ (∑ i, X i); (hΩc y).volume] := by
simp_rw [← Finset.sum_add_distrib, ← left_distrib]
congr with y
by_cases hf : f y = 0
. simp only [hf, zero_mul]
congr 1
convert multiDist_chainRule π (hΩc y) hX _
refine h_indep.cond hY ?_ fun _ ↦ .singleton _
apply prob_nonzero_of_prod_prob_nonzero
convert hf
rw [← ENNReal.toReal_prod]
congr
exact (iIndepFun.meas_iInter h_indep fun _ ↦ mes_of_comap <| .singleton _).symm
_ = _ := by
have hmes : Measurable (π ∘ ∑ i : Fin m, X i) := by
apply Measurable.comp .of_discrete
convert Finset.measurable_sum (f := X) Finset.univ _ with ω
. exact Fintype.sum_apply ω X
exact (fun i _ ↦ hX i)
have hpi_indep : iIndepFun (fun i ↦ ⟨π ∘ X i, Y i⟩) ℙ := by
set g : G × S → H × S := fun p ↦ ⟨π p.1, p.2⟩
convert iIndepFun.comp h_indep (fun _ ↦ g) _
intro i
exact .of_discrete
have hpi_indep' : iIndepFun (fun i ↦ ⟨X i, ⟨π ∘ X i, Y i⟩⟩) ℙ := by
set g : G × S → G × (H × S) := fun p ↦ ⟨p.1, ⟨π p.1, p.2⟩⟩
convert iIndepFun.comp h_indep (fun _ ↦ g) _
intro i
exact .of_discrete
have hey_mes : ∀ y, MeasurableSet (E' y) := by
intro y
apply MeasurableSet.iInter
intro i
exact MeasurableSet.preimage (MeasurableSet.singleton (y i)) (hY i)
congr 2
. rw [condMultiDist_eq' hX _ hpi_indep']
. rw [← Equiv.sum_comp (Equiv.arrowProdEquivProdArrow _ _ _).symm, Fintype.sum_prod_type, Finset.sum_comm]
congr with y
by_cases pey : ℙ (E' y) = 0
. simp only [pey, ENNReal.zero_toReal, zero_mul, f]
apply (Finset.sum_eq_zero _).symm
intro s _
convert zero_mul _
convert ENNReal.zero_toReal
apply measure_mono_null _ pey
intro ω hω
simp [E', Equiv.arrowProdEquivProdArrow] at hω ⊢
intro i
exact (hω i).2
rw [condMultiDist_eq' (hΩ := hΩc y) hX, Finset.mul_sum]
. congr with s
dsimp [f, E', Equiv.arrowProdEquivProdArrow]
rw [← mul_assoc, ← ENNReal.toReal_mul]
congr 2
. rw [mul_comm]
convert ProbabilityTheory.cond_mul_eq_inter (hey_mes y) ?_ _
. rw [← Set.iInter_inter_distrib]
apply Set.iInter_congr
intro i
ext ω
simp only [Set.mem_preimage, Set.mem_singleton_iff, Prod.mk.injEq, comp_apply, Set.mem_inter_iff]
exact And.comm
infer_instance
funext _
congr 1
dsimp [hΩc, E']
rw [ProbabilityTheory.cond_cond_eq_cond_inter (hey_mes y), ← Set.iInter_inter_distrib]
. congr 1
apply Set.iInter_congr
intro i
ext ω
simp only [Set.mem_inter_iff, Set.mem_preimage, Set.mem_singleton_iff, comp_apply, Prod.mk.injEq]
exact And.comm
apply MeasurableSet.iInter
intro i
apply MeasurableSet.preimage (MeasurableSet.singleton _)
exact Measurable.comp .of_discrete (hX i)
. intro i
exact Measurable.comp .of_discrete (hX i)
set g : G → G × H := fun x ↦ ⟨x, π x⟩
refine iIndepFun.comp ?_ (fun _ ↦ g) fun _ ↦ .of_discrete
. refine h_indep.cond hY ?_ fun _ ↦ .singleton _
rw [iIndepFun.meas_iInter h_indep fun _ ↦ mes_of_comap <| .singleton _] at pey
contrapose! pey
obtain ⟨i, hi⟩ := pey
exact Finset.prod_eq_zero (Finset.mem_univ i) hi
intro i
exact Measurable.prodMk (Measurable.comp .of_discrete (hX i)) (hY i)
. rw [condMultiDist_eq' _ hY hpi_indep]
intro i
apply Measurable.comp .of_discrete (hX i)
rw [condMutualInfo_eq_sum', Fintype.sum_prod_type, Finset.sum_comm]
. congr with y
by_cases pey : ℙ (E' y) = 0
. simp only [pey, ENNReal.zero_toReal, zero_mul, f]
apply (Finset.sum_eq_zero _).symm
intro s _
convert zero_mul _
convert ENNReal.zero_toReal
apply measure_mono_null _ pey
intro ω hω
simp [E'] at hω ⊢
rw [← hω.2]
simp only [implies_true]
have : IsProbabilityMeasure (hΩc y).volume := cond_isProbabilityMeasure pey
rw [condMutualInfo_eq_sum' hmes, Finset.mul_sum]
congr with x
dsimp [f, E']
rw [← mul_assoc, ← ENNReal.toReal_mul]
congr 2
. rw [mul_comm]
convert ProbabilityTheory.cond_mul_eq_inter (hey_mes y) ?_ _
. ext ω
simp only [Set.mem_preimage, Set.mem_singleton_iff, Prod.mk.injEq, comp_apply,
Finset.sum_apply, _root_.map_sum, Set.mem_inter_iff, Set.mem_iInter, E']
rw [and_comm]
apply and_congr_left
intro _
exact funext_iff
infer_instance
dsimp [hΩc, E']
rw [ProbabilityTheory.cond_cond_eq_cond_inter (hey_mes y)]
. congr
ext ω
simp only [Set.mem_inter_iff, Set.mem_iInter, Set.mem_preimage, Set.mem_singleton_iff,
comp_apply, Finset.sum_apply, _root_.map_sum, Prod.mk.injEq, E']
rw [and_comm]
apply and_congr_right
intro _
exact Iff.symm funext_iff
exact MeasurableSet.preimage (MeasurableSet.singleton x) hmes
exact Measurable.prodMk hmes (measurable_pi_lambda (fun ω i ↦ Y i ω) hY) | 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-prelim-improv}\leanok
Average \Cref{construct-good-prelim-improv} over all six permutations of $T_1,T_2,T_3$.
\end{proof} | 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 # T₁ | T₂] - d[p.X₀₂ # X₂]) + (d[p.X₀₂ # T₁ | T₃] - d[p.X₀₂ # X₂])
+ (d[p.X₀₂ # T₂ | T₁] - d[p.X₀₂ # X₂]) + (d[p.X₀₂ # T₂ | T₃] - d[p.X₀₂ # X₂])
+ (d[p.X₀₂ # T₃ | T₁] - d[p.X₀₂ # X₂]) + (d[p.X₀₂ # T₃ | T₂] - d[p.X₀₂ # X₂])) := by
have I1 : I[T₂ : T₁] = I[T₁ : T₂] := mutualInfo_comm hT₂ hT₁ _
have I2 : I[T₃ : T₁] = I[T₁ : T₃] := mutualInfo_comm hT₃ hT₁ _
have I3 : I[T₃ : T₂] = I[T₂ : T₃] := mutualInfo_comm hT₃ hT₂ _
have Z123 := construct_good_prelim' h_min hT hT₁ hT₂ hT₃
have h132 : T₁ + T₃ + T₂ = 0 := by rw [← hT]; abel
have Z132 := construct_good_prelim' h_min h132 hT₁ hT₃ hT₂
have h213 : T₂ + T₁ + T₃ = 0 := by rw [← hT]; abel
have Z213 := construct_good_prelim' h_min h213 hT₂ hT₁ hT₃
have h231 : T₂ + T₃ + T₁ = 0 := by rw [← hT]; abel
have Z231 := construct_good_prelim' h_min h231 hT₂ hT₃ hT₁
have h312 : T₃ + T₁ + T₂ = 0 := by rw [← hT]; abel
have Z312 := construct_good_prelim' h_min h312 hT₃ hT₁ hT₂
have h321 : T₃ + T₂ + T₁ = 0 := by rw [← hT]; abel
have Z321 := construct_good_prelim' h_min h321 hT₃ hT₂ hT₁
simp only [I1, I2, I3] at Z123 Z132 Z213 Z231 Z312 Z321
linarith
include h_min in
omit [IsProbabilityMeasure (ℙ : Measure Ω₀₁)] [IsProbabilityMeasure (ℙ : Measure Ω₀₂)]
[IsProbabilityMeasure (ℙ : Measure Ω)] in
/-- Rephrase `construct_good_improved'` with an explicit probability measure, as we will
apply it to (varying) conditional measures. -/ | 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*}
\end{lemma} | 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)[fun t ↦ d[p.X₀₂; ℙ # T₂; ℙ[|T₃ ⁻¹' {t}]] - d[p.X₀₂ # X₂]]
let sum4 : ℝ := (Measure.map T₃ ℙ)[fun t ↦ ψ[T₁; ℙ[|T₃ ⁻¹' {t}] # T₂; ℙ[|T₃ ⁻¹' {t}]]]
have hp.η : 0 ≤ p.η := by linarith [p.hη]
have hP : IsProbabilityMeasure (Measure.map T₃ ℙ) := isProbabilityMeasure_map hT₃.aemeasurable
have h2T₃ : T₃ = T₁ + T₂ :=
calc T₃ = T₁ + T₂ + T₃ - T₃ := by rw [hT, zero_sub]; simp [ZModModule.neg_eq_self]
_ = T₁ + T₂ := by rw [add_sub_cancel_right]
have h2T₁ : T₁ = T₂ + T₃ := by simp [h2T₃, add_left_comm, ZModModule.add_self]
have h2T₂ : T₂ = T₃ + T₁ := by simp [h2T₁, add_left_comm, ZModModule.add_self]
have h1 : sum1 ≤ δ := by
have h1 : sum1 ≤ 3 * I[T₁ : T₂] + 2 * H[T₃] - H[T₁] - H[T₂] := by
subst h2T₃; exact ent_bsg hT₁ hT₂
have h2 : H[⟨T₂, T₃⟩] = H[⟨T₁, T₂⟩] := by
rw [h2T₃, entropy_add_right', entropy_comm] <;> assumption
have h3 : H[⟨T₁, T₂⟩] = H[⟨T₃, T₁⟩] := by
rw [h2T₃, entropy_add_left, entropy_comm] <;> assumption
simp_rw [mutualInfo_def] at h1 ⊢; linarith
have h2 : p.η * sum2 ≤ p.η * (d[p.X₀₁ # T₁] - d[p.X₀₁ # X₁] + I[T₁ : T₃] / 2) := by
have : sum2 = d[p.X₀₁ # T₁ | T₃] - d[p.X₀₁ # X₁] := by
simp only [integral_sub .of_finite .of_finite, integral_const, measure_univ,
ENNReal.one_toReal, smul_eq_mul, one_mul, sub_left_inj, sum2]
simp_rw [condRuzsaDist'_eq_sum hT₁ hT₃,
integral_eq_setIntegral (FiniteRange.null_of_compl _ T₃), integral_finset _ _ IntegrableOn.finset,
Measure.map_apply hT₃ (.singleton _), smul_eq_mul]
gcongr
linarith [condRuzsaDist_le' ℙ ℙ p.hmeas1 hT₁ hT₃]
have h3 : p.η * sum3 ≤ p.η * (d[p.X₀₂ # T₂] - d[p.X₀₂ # X₂] + I[T₂ : T₃] / 2) := by
have : sum3 = d[p.X₀₂ # T₂ | T₃] - d[p.X₀₂ # X₂] := by
simp only [integral_sub .of_finite .of_finite, integral_const, measure_univ,
ENNReal.one_toReal, smul_eq_mul, one_mul, sub_left_inj, sum3]
simp_rw [condRuzsaDist'_eq_sum hT₂ hT₃,
integral_eq_setIntegral (FiniteRange.null_of_compl _ T₃), integral_finset _ _ IntegrableOn.finset,
Measure.map_apply hT₃ (.singleton _), smul_eq_mul]
gcongr
linarith [condRuzsaDist_le' ℙ ℙ p.hmeas2 hT₂ hT₃]
have h4 : sum4 ≤ δ + p.η * c[T₁ # T₂] + p.η * (I[T₁ : T₃] + I[T₂ : T₃]) / 2 := by
suffices sum4 = sum1 + p.η * (sum2 + sum3) by linarith
simp only [sum1, sum2, sum3, sum4, integral_add .of_finite .of_finite, integral_mul_left]
have hk : k ≤ sum4 := by
suffices (Measure.map T₃ ℙ)[fun _ ↦ k] ≤ sum4 by simpa using this
refine integral_mono_ae .of_finite .of_finite $ ae_iff_of_countable.2 fun t ht ↦ ?_
have : IsProbabilityMeasure (ℙ[|T₃ ⁻¹' {t}]) :=
cond_isProbabilityMeasure (by simpa [hT₃] using ht)
dsimp only
linarith only [distance_ge_of_min' (μ := ℙ[|T₃ ⁻¹' {t}]) (μ' := ℙ[|T₃ ⁻¹' {t}]) p h_min hT₁ hT₂]
exact hk.trans h4
include hT₁ hT₂ hT₃ hT h_min in
/-- If $T_1, T_2, T_3$ are $G$-valued random variables with $T_1+T_2+T_3=0$ holds identically and
-
$$ \delta := \sum_{1 \leq i < j \leq 3} I[T_i;T_j]$$
Then there exist random variables $T'_1, T'_2$ such that
$$ d[T'_1;T'_2] + \eta (d[X_1^0;T'_1] - d[X_1^0;X _1]) + \eta(d[X_2^0;T'_2] - d[X_2^0;X_2])$$
is at most
$$\delta + \frac{\eta}{3} \biggl( \delta + \sum_{i=1}^2 \sum_{j = 1}^3
(d[X^0_i;T_j] - d[X^0_i; X_i]) \biggr).$$
-/ | 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-lower}\leanok
We apply \Cref{entropic-bsg} with $(A,B) = (T_1, T_2)$ there.
Since $T_1 + T_2 = T_3$, the conclusion is that
\begin{align} \nonumber \sum_{t_3} \bbP[T_3 = t_3] & d[(T_1 | T_3 = t_3); (T_2 | T_3 = t_3)] \\ & \leq 3 \bbI[T_1 : T_2] + 2 \bbH[T_3] - \bbH[T_1] - \bbH[T_2].\label{bsg-t1t2'}
\end{align}
The right-hand side in~\eqref{bsg-t1t2'} can be rearranged as
\begin{align*} & 2( \bbH[T_1] + \bbH[T_2] + \bbH[T_3]) - 3 \bbH[T_1,T_2] \\ & = 2(\bbH[T_1] + \bbH[T_2] + \bbH[T_3]) - \bbH[T_1,T_2] - \bbH[T_2,T_3] - \bbH[T_1, T_3] = \delta,\end{align*}
using the fact (from \Cref{relabeled-entropy}) that all three terms $\bbH[T_i,T_j]$ are equal to $\bbH[T_1,T_2,T_3]$ and hence to each other.
We also have
\begin{align*}
& \sum_{t_3} P[T_3 = t_3] \bigl(d[X^0_1; (T_1 | T_3=t_3)] - d[X^0_1;X_1]\bigr) \\
&\quad = d[X^0_1; T_1 | T_3] - d[X^0_1;X_1]
\end{align*}
and similarly
\begin{align*}
& \sum_{t_3} \bbP[T_3 = t_3] (d[X^0_2;(T_2 | T_3=t_3)] - d[X^0_2; X_2]) \\
&\quad\quad\quad\quad\quad\quad \leq d[X^0_2;T_2|T_3] - d[X^0_2;X_2].
\end{align*}
Putting the above observations together, we have
\begin{align*}
\sum_{t_3} \bbP[T_3=t_3] \psi[(T_1 | T_3=t_3); (T_2 | T_3=t_3)] \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*}
where we introduce the notation
\[\psi[Y_1; Y_2] := d[Y_1;Y_2] + \eta (d[X_1^0;Y_1] - d[X_1^0;X_1]) + \eta(d[X_2^0;Y_2] - d[X_2^0;X_2]).\]
On the other hand, from \Cref{distance-lower} we have $k \leq \psi[Y_1;Y_2]$, and the claim follows.
\end{proof} | 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[|T₃ ⁻¹' {t}]] - d[p.X₀₂ # X₂]]
let sum4 : ℝ := (Measure.map T₃ ℙ)[fun t ↦ ψ[T₁; ℙ[|T₃ ⁻¹' {t}] # T₂; ℙ[|T₃ ⁻¹' {t}]]]
have h2T₃ : T₃ = T₁ + T₂ := by
calc T₃ = T₁ + T₂ + T₃ - T₃ := by simp [hT, ZModModule.neg_eq_self]
_ = T₁ + T₂ := by rw [add_sub_cancel_right]
have hP : IsProbabilityMeasure (Measure.map T₃ ℙ) := isProbabilityMeasure_map hT₃.aemeasurable
-- control sum1 with entropic BSG
have h1 : sum1 ≤ δ := by
have h1 : sum1 ≤ 3 * I[T₁ : T₂] + 2 * H[T₃] - H[T₁] - H[T₂] := by
subst h2T₃; exact ent_bsg hT₁ hT₂
have h2 : H[⟨T₂, T₃⟩] = H[⟨T₁, T₂⟩] := by
rw [h2T₃, entropy_add_right', entropy_comm] <;> assumption
have h3 : H[⟨T₁, T₂⟩] = H[⟨T₃, T₁⟩] := by
rw [h2T₃, entropy_add_left, entropy_comm] <;> assumption
simp_rw [mutualInfo_def] at h1 ⊢; linarith
-- rewrite sum2 and sum3 as Rusza distances
have h2 : sum2 = d[p.X₀₁ # T₁ | T₃] - d[p.X₀₁ # X₁] := by
simp only [sum2, integral_sub .of_finite .of_finite, integral_const,
measure_univ, ENNReal.one_toReal, smul_eq_mul, one_mul, sub_left_inj]
simp_rw [condRuzsaDist'_eq_sum hT₁ hT₃,
integral_eq_setIntegral (FiniteRange.null_of_compl _ T₃), integral_finset _ _ .finset,
Measure.map_apply hT₃ (.singleton _), smul_eq_mul]
have h3 : sum3 = d[p.X₀₂ # T₂ | T₃] - d[p.X₀₂ # X₂] := by
simp only [sum3, integral_sub .of_finite .of_finite, integral_const,
measure_univ, ENNReal.one_toReal, smul_eq_mul, one_mul, sub_left_inj]
simp_rw [condRuzsaDist'_eq_sum hT₂ hT₃,
integral_eq_setIntegral (FiniteRange.null_of_compl _ T₃), integral_finset _ _ .finset,
Measure.map_apply hT₃ (.singleton _), smul_eq_mul]
-- put all these estimates together to bound sum4
have h4 : sum4 ≤ δ + p.η * ((d[p.X₀₁ # T₁ | T₃] - d[p.X₀₁ # X₁])
+ (d[p.X₀₂ # T₂ | T₃] - d[p.X₀₂ # X₂])) := by
have : sum4 = sum1 + p.η * (sum2 + sum3) := by
simp only [sum1, sum2, sum3, sum4, integral_add .of_finite .of_finite, integral_mul_left]
rw [this, h2, h3, add_assoc, mul_add]
linarith
have hk : k ≤ sum4 := by
suffices (Measure.map T₃ ℙ)[fun _ ↦ k] ≤ sum4 by simpa using this
refine integral_mono_ae .of_finite .of_finite $
ae_iff_of_countable.2 fun t ht ↦ ?_
have : IsProbabilityMeasure (ℙ[|T₃ ⁻¹' {t}]) :=
cond_isProbabilityMeasure (by simpa [hT₃] using ht)
dsimp only
linarith only [distance_ge_of_min' (μ := ℙ[|T₃ ⁻¹' {t}]) (μ' := ℙ[|T₃ ⁻¹' {t}]) p h_min hT₁ hT₂]
exact hk.trans h4
open Module
include hT hT₁ hT₂ hT₃ h_min in
omit [IsProbabilityMeasure (ℙ : Measure Ω₀₁)] [IsProbabilityMeasure (ℙ : Measure Ω₀₂)]
[IsProbabilityMeasure (ℙ : Measure Ω)] in
/-- In fact $k$ is at most
$$ \delta + \frac{\eta}{6} \sum_{i=1}^2 \sum_{1 \leq j,l \leq 3; j \neq l}
(d[X^0_i;T_j|T_l] - d[X^0_i; X_i]).$$
-/ | 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_{i,j}\bigr)_{i = 1}^m \; \big| \; \sum_{i=1}^m \sum_{j = 1}^m X_{i,j} ] \\
&\quad \leq \sum_{j=1}^{m-1} \Bigl(D[(X_{i, j})_{i = 1}^m] - D[ (X_{i, j})_{i = 1}^m \; \big| \; (X_{i,j} + \cdots + X_{i,m})_{i =1}^m ]\Bigr) \\ & \qquad\qquad\qquad\qquad + D[(X_{i,m})_{i=1}^m] - D[ \bigl(\sum_{j=1}^m X_{i,j}\bigr)_{i=1}^m ],
\end{align*}
where all the multidistances here involve the indexing set $\{1,\dots, m\}$.
\end{corollary}
\begin{proof}\uses{multidist-chain-rule-iter, add-entropy}
In \Cref{multidist-chain-rule-iter} we take $G_d := G^d$ with the maps $\pi_d \colon G^m \to G^d$ for $d=1,\dots,m$ defined by
\[
\pi_d(x_1,\dots,x_m) := (x_1,\dots,x_{d-1}, x_d + \cdots + x_m)
\]
with $\pi_0=0$. Since $\pi_{d-1}(x)$ can be obtained from $\pi_{d}(x)$ by applying a homomorphism, we obtain a sequence of the form~\eqref{g-seq}.
Now we apply \Cref{multidist-chain-rule-iter} with $I = \{1,\dots, m\}$ and $X_i := (X_{i,j})_{j = 1}^m$. Using joint independence and \Cref{add-entropy}, we find that
\[
D[ X_{[m]} ] = \sum_{j=1}^m D[ (X_{i,j})_{1 \leq i \leq m} ].
\]
On the other hand, for $1 \leq j \leq m-1$, we see that once $\pi_{j}(X_i)$ is fixed, $\pi_{j+1}(X_i)$ is determined by $X_{i, j}$ and vice versa, so
\[
D[ \pi_{j+1}(X_{[m]}) \; | \; \pi_{j}(X_{[m]}) ] = D[ (X_{i, j})_{1 \leq i \leq m} \; | \; \pi_{j}(X_{[m]} )].
\]
Since the $X_{i,j}$ are jointly independent, we may further simplify:
\[
D[ (X_{i, j})_{1 \leq i \leq m} \; | \; \pi_{j}(X_{[m]})] = D[ (X_{i,j})_{1 \leq i \leq m} \; | \; ( X_{i, j} + \cdots + X_{i, m})_{1 \leq i \leq m} ].
\]
Putting all this into the conclusion of \Cref{multidist-chain-rule-iter}, we obtain
\[
\sum_{j=1}^{m} D[ (X_{i,j})_{1 \leq i \leq m} ]
\geq
\begin{aligned}[t]
&\sum_{j=1}^{m-1} D[ (X_{i,j})_{1 \leq i \leq m} \; | \; (X_{i,j} + \cdots + X_{i,m})_{1 \leq i \leq m} ] \\
&\!\!\!+
D[ \bigl(\sum_{j=1}^m X_{i,j}\bigr)_{1 \leq i \leq m}] \\
&\!\!\!+\bbI[ \bigl(\sum_{i=1}^m X_{i,j}\bigr)_{j =1}^{m} : \bigl(\sum_{j=1}^m X_{i,j}\bigr)_{i = 1}^m \; \big| \; \sum_{i=1}^m \sum_{j = 1}^m X_{i,j} ]
\end{aligned}
\]
and the claim follows by rearranging.
\end{proof} | 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) |
fun i ↦ ∑ k ∈ Finset.Ici j, X (i, k); fun _ ↦ hΩ]) + D[fun i ↦ X (i, m); fun _ ↦ hΩ]
- D[fun i ↦ ∑ j, X (i, j); fun _ ↦ hΩ] := by sorry
end multiDistance_chainRule | 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{ruz-dist-def}, and also \Cref{neg-ent}.
\end{proof} | /-- `|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'
rw [← hIdX.rdist_eq hIdY, hind.rdist_eq hX' hY', ← hIdX.entropy_eq, ← hIdY.entropy_eq, abs_le]
have := max_entropy_le_entropy_sub hX' hY' hind
constructor
· linarith[le_max_right H[X'; ν] H[Y'; ν]]
· linarith[le_max_left H[X'; ν] H[Y'; ν]] | 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-ent} \leanok Immediate from \Cref{sumset-lower} and \Cref{ruz-dist-def}, and also \Cref{neg-ent}.
\end{proof} | /-- `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_right H[X ; μ] H[Y; μ]] | 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-ent} \leanok Immediate from \Cref{sumset-lower} and \Cref{ruz-dist-def}, and also \Cref{neg-ent}.
\end{proof} | /-- `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_left H[X ; μ] H[Y; μ]] | 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 X_1] - d[X^0_2; X_2] &\leq \tfrac{1}{2} k + \tfrac{1}{4} \bbH[X_1] - \tfrac{1}{4} \bbH[X_2], \\
d[X_1^0;X_1|X_1+\tilde X_2] - d[X_1^0;X_1] &\leq \tfrac{1}{2} k + \tfrac{1}{4} \bbH[X_1] - \tfrac{1}{4} \bbH[X_2] \\
d[X_2^0; X_2|X_2+\tilde X_1] - d[X_2^0; X_2] &\leq \tfrac{1}{2}k + \tfrac{1}{4} \bbH[X_2] - \tfrac{1}{4} \bbH[X_1].
\end{align*}
\end{lemma}
\begin{proof}\uses{first-useful} \leanok Immediate from \Cref{first-useful} (and recalling that $k$ is defined to be $d[X_1;X_2]$).
\end{proof} | /--`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 condRuzsaDist_diff_le' ℙ p.hmeas1 hX₁ hX₂' h using 4
· exact (IdentDistrib.refl hX₁.aemeasurable).rdist_eq h₂
· exact h₂.entropy_eq
include hX₁' hX₂ h_indep h₁ in | 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 X_1] - d[X^0_2; X_2] &\leq \tfrac{1}{2} k + \tfrac{1}{4} \bbH[X_1] - \tfrac{1}{4} \bbH[X_2], \\
d[X_1^0;X_1|X_1+\tilde X_2] - d[X_1^0;X_1] &\leq \tfrac{1}{2} k + \tfrac{1}{4} \bbH[X_1] - \tfrac{1}{4} \bbH[X_2] \\
d[X_2^0; X_2|X_2+\tilde X_1] - d[X_2^0; X_2] &\leq \tfrac{1}{2}k + \tfrac{1}{4} \bbH[X_2] - \tfrac{1}{4} \bbH[X_1].
\end{align*}
\end{lemma}
\begin{proof}\uses{first-useful} \leanok Immediate from \Cref{first-useful} (and recalling that $k$ is defined to be $d[X_1;X_2]$).
\end{proof} | /-- $$ d[X^0_2;X_2+\tilde X_1] - d[X^0_2; X_2] \leq \tfrac{1}{2} k + \tfrac{1}{4} \mathbb{H}[X_1] - \tfrac{1}{4} \mathbb{H}[X_2].$$ -/
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_indep.indepFun (show (1 : Fin 4) ≠ 3 by decide)
convert condRuzsaDist_diff_le' ℙ p.hmeas2 hX₂ hX₁' h using 4
· rw [rdist_symm]
exact (IdentDistrib.refl hX₂.aemeasurable).rdist_eq h₁
· exact h₁.entropy_eq
include h_indep hX₁ hX₂' h₂ in
/-- $$ d[X_1^0;X_1|X_1+\tilde X_2] - d[X_1^0;X_1] \leq
\tfrac{1}{2} k + \tfrac{1}{4} \mathbb{H}[X_1] - \tfrac{1}{4} \mathbb{H}[X_2].$$ -/ | 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 X_1] - d[X^0_2; X_2] &\leq \tfrac{1}{2} k + \tfrac{1}{4} \bbH[X_1] - \tfrac{1}{4} \bbH[X_2], \\
d[X_1^0;X_1|X_1+\tilde X_2] - d[X_1^0;X_1] &\leq \tfrac{1}{2} k + \tfrac{1}{4} \bbH[X_1] - \tfrac{1}{4} \bbH[X_2] \\
d[X_2^0; X_2|X_2+\tilde X_1] - d[X_2^0; X_2] &\leq \tfrac{1}{2}k + \tfrac{1}{4} \bbH[X_2] - \tfrac{1}{4} \bbH[X_1].
\end{align*}
\end{lemma}
\begin{proof}\uses{first-useful} \leanok Immediate from \Cref{first-useful} (and recalling that $k$ is defined to be $d[X_1;X_2]$).
\end{proof} | 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 [(IdentDistrib.refl hX₁.aemeasurable).rdist_eq h₂]
· apply h₂.entropy_eq
include h_indep hX₂ hX₁' h₁
/-- $$ d[X_2^0; X_2|X_2+\tilde X_1] - d[X_2^0; X_2] \leq
\tfrac{1}{2}k + \tfrac{1}{4} \mathbb{H}[X_2] - \tfrac{1}{4} \mathbb{H}[X_1].$$ -/ | 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 X_1] - d[X^0_2; X_2] &\leq \tfrac{1}{2} k + \tfrac{1}{4} \bbH[X_1] - \tfrac{1}{4} \bbH[X_2], \\
d[X_1^0;X_1|X_1+\tilde X_2] - d[X_1^0;X_1] &\leq \tfrac{1}{2} k + \tfrac{1}{4} \bbH[X_1] - \tfrac{1}{4} \bbH[X_2] \\
d[X_2^0; X_2|X_2+\tilde X_1] - d[X_2^0; X_2] &\leq \tfrac{1}{2}k + \tfrac{1}{4} \bbH[X_2] - \tfrac{1}{4} \bbH[X_1].
\end{align*}
\end{lemma}
\begin{proof}\uses{first-useful} \leanok Immediate from \Cref{first-useful} (and recalling that $k$ is defined to be $d[X_1;X_2]$).
\end{proof} | 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, (IdentDistrib.refl hX₂.aemeasurable).rdist_eq h₁]
· apply h₁.entropy_eq
include hX₁ hX₂ hX₁' hX₂' h₁ h₂ h_min in | 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}\uses{gen-ineq, relabeled-entropy-cond,second-estimate-aux}\leanok
If we apply \Cref{gen-ineq} with $X_1:=X_1$, $Y:=X_1^0$ and $(X_2,X_3,X_4)$ equal to the $3!$ permutations of $(X_2,\tilde X_1,\tilde X_2)$, and sums (using the symmetry $\bbH[X|X+Y] = \bbH[Y|X+Y]$, which follows from \Cref{relabeled-entropy-cond}), we can bound
$$ \sum_{A,B \in \{U,V,W\}: A \neq B} d[X_1^0;A|B, S] - d[X_1^0;X_1]$$
by
\begin{align*}
&\quad \tfrac{1}{4} (6d[X_1;X_2] + 6d[X_1;\tilde X_2]\\
&\qquad + 6d[X_1;\tilde X_1] + 2d[\tilde X_1;\tilde X_2] + 2 d[\tilde X_1;X_2] + 2d[X_2;\tilde X_2])\\
&\quad + \tfrac{1}{8} (2\bbH[X_1+X_2] + 2\bbH[X_1+\tilde X_1] + 2 \bbH[X_1+\tilde X_2] \\
&\qquad - 2\bbH[\tilde X_1+X_2] - 2\bbH[X_2+\tilde X_2] - 2\bbH[\tilde X_1+\tilde X_2])\\
&\qquad \qquad + \tfrac{1}{4} (\bbH[X_2|X_2+\tilde X_2] + \bbH[\tilde X_1|\tilde X_1+\tilde X_2] + \bbH[\tilde X_1|X_1+\tilde X_2] \\
&\qquad \qquad \qquad - \bbH[X_1|X_1+\tilde X_1] - \bbH[X_1|X_1+X_2] - \bbH[X_1|X_1+\tilde X_2]),
\end{align*}
which simplifies to
\begin{align*}
&\quad \tfrac{1}{4} (16k + 6d[X_1;X_1] + 2d[X_2;X_2])\\
&\qquad \qquad + \tfrac{1}{4} (H[X_1+\tilde X_1] - H[X_2+\tilde X_2] + d[X_2|X_2+\tilde X_2] - d[X_1|X_1+\tilde X_1]).
\end{align*}
A symmetric argument also bounds
$$ \sum_{A,B \in \{U,V,W\}: A \neq B} d[X_2^0;A|B, S] - d[X_2^0;X_2]$$
by
\begin{align*}
&\quad \tfrac{1}{4} (16k + 6d[X_2;X_2] + 2d[X_1;X_1])\\
&\qquad \qquad + \tfrac{1}{4} (H[X_2+\tilde X_2] - H[X_1+\tilde X_1] + d[X_1|X_1+\tilde X_1] - d[X_2|X_2+\tilde X_2]).
\end{align*}
On the other hand, from \Cref{second-estimate-aux} one has
$$ d[X_1;X_1] + d[X_2;X_2] \leq 2 k + \frac{2(2 \eta k - I_1)}{1-\eta}.$$
Summing the previous three estimates, we obtain the claim.
\end{proof} | 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] + 2 * d[X₂ # X₂]) / 4 + (H[X₁ + X₁'] - H[X₂ + X₂']) / 4
+ (H[X₂ | X₂ + X₂'] - H[X₁ | X₁ + X₁']) / 4 := by
have I1 := gen_ineq_01 p.X₀₁ p.hmeas1 X₁ X₂ X₂' X₁' hX₁ hX₂ hX₂' hX₁' h_indep.reindex_four_abcd
have I2 := gen_ineq_00 p.X₀₁ p.hmeas1 X₁ X₂ X₁' X₂' hX₁ hX₂ hX₁' hX₂' h_indep.reindex_four_abdc
have I3 := gen_ineq_10 p.X₀₁ p.hmeas1 X₁ X₂' X₂ X₁' hX₁ hX₂' hX₂ hX₁' h_indep.reindex_four_acbd
have I4 := gen_ineq_10 p.X₀₁ p.hmeas1 X₁ X₂' X₁' X₂ hX₁ hX₂' hX₁' hX₂ h_indep.reindex_four_acdb
have I5 := gen_ineq_00 p.X₀₁ p.hmeas1 X₁ X₁' X₂ X₂' hX₁ hX₁' hX₂ hX₂' h_indep.reindex_four_adbc
have I6 := gen_ineq_01 p.X₀₁ p.hmeas1 X₁ X₁' X₂' X₂ hX₁ hX₁' hX₂' hX₂ h_indep.reindex_four_adcb
have C1 : U + X₂' + X₁' = S := by abel
have C2 : W + X₂ + X₂' = S := by abel
have C3 : X₁ + X₂' + X₂ + X₁' = S := by abel
have C4 : X₁ + X₂' + X₁' + X₂ = S := by abel
have C5 : W + X₂' + X₂ = S := by abel
have C7 : X₂ + X₁' = V := by abel
have C8 : X₁ + X₁' = W := by abel
have C9 : d[X₁ # X₂'] = d[X₁ # X₂] := (IdentDistrib.refl hX₁.aemeasurable).rdist_eq h₂.symm
have C10 : d[X₂ # X₁'] = d[X₁' # X₂] := rdist_symm
have C11 : d[X₁ # X₁'] = d[X₁ # X₁] := (IdentDistrib.refl hX₁.aemeasurable).rdist_eq h₁.symm
have C12 : d[X₁' # X₂'] = d[X₁ # X₂] := h₁.symm.rdist_eq h₂.symm
have C13 : d[X₂ # X₂'] = d[X₂ # X₂] := (IdentDistrib.refl hX₂.aemeasurable).rdist_eq h₂.symm
have C14 : d[X₁' # X₂] = d[X₁ # X₂] := h₁.symm.rdist_eq (IdentDistrib.refl hX₂.aemeasurable)
have C15 : H[X₁' + X₂'] = H[U] := by
apply ProbabilityTheory.IdentDistrib.entropy_eq
have I : IdentDistrib (⟨X₁, X₂⟩) (⟨X₁', X₂'⟩) := h₁.prodMk h₂ (h_indep.indepFun zero_ne_one)
(h_indep.indepFun (show 3 ≠ 2 by decide))
exact I.symm.comp measurable_add
have C16 : H[X₂'] = H[X₂] := h₂.symm.entropy_eq
have C17 : H[X₁'] = H[X₁] := h₁.symm.entropy_eq
have C18 : d[X₂' # X₁'] = d[X₁' # X₂'] := rdist_symm
have C19 : H[X₂' + X₁'] = H[U] := by rw [add_comm]; exact C15
have C20 : d[X₂' # X₂] = d[X₂ # X₂] := h₂.symm.rdist_eq (IdentDistrib.refl hX₂.aemeasurable)
have C21 : H[V] = H[U] := by
apply ProbabilityTheory.IdentDistrib.entropy_eq
have I : IdentDistrib (⟨X₁', X₂⟩) (⟨X₁, X₂⟩) := by
apply h₁.symm.prodMk (IdentDistrib.refl hX₂.aemeasurable)
(h_indep.indepFun (show 3 ≠ 1 by decide)) (h_indep.indepFun zero_ne_one)
exact I.comp measurable_add
have C22 : H[X₁ + X₂'] = H[X₁ + X₂] := by
apply ProbabilityTheory.IdentDistrib.entropy_eq
have I : IdentDistrib (⟨X₁, X₂'⟩) (⟨X₁, X₂⟩) := by
apply (IdentDistrib.refl hX₁.aemeasurable).prodMk h₂.symm
(h_indep.indepFun (show 0 ≠ 2 by decide)) (h_indep.indepFun zero_ne_one)
exact I.comp measurable_add
have C23 : X₂' + X₂ = X₂ + X₂' := by abel
have C24 : H[X₁ | X₁ + X₂'] = H[X₁ | X₁ + X₂] := by
apply IdentDistrib.condEntropy_eq hX₁ (hX₁.add hX₂') hX₁ (hX₁.add hX₂)
have I : IdentDistrib (⟨X₁, X₂'⟩) (⟨X₁, X₂⟩) := by
exact (IdentDistrib.refl hX₁.aemeasurable).prodMk h₂.symm
(h_indep.indepFun (show 0 ≠ 2 by decide)) (h_indep.indepFun zero_ne_one)
exact I.comp (measurable_fst.prodMk measurable_add)
have C25 : H[X₂ | V] = H[X₂ | X₁ + X₂] := by
apply IdentDistrib.condEntropy_eq hX₂ (hX₁'.add hX₂) hX₂ (hX₁.add hX₂)
have I : IdentDistrib (⟨X₁', X₂⟩) (⟨X₁, X₂⟩) := by
exact h₁.symm.prodMk (IdentDistrib.refl hX₂.aemeasurable)
(h_indep.indepFun (show 3 ≠ 1 by decide)) (h_indep.indepFun zero_ne_one)
exact I.comp (measurable_snd.prodMk measurable_add)
have C26 : H[X₂' | X₂' + X₁'] = H[X₂ | X₁ + X₂] := by
rw [add_comm]
apply IdentDistrib.condEntropy_eq hX₂' (hX₁'.add hX₂') hX₂ (hX₁.add hX₂)
have I : IdentDistrib (⟨X₁', X₂'⟩) (⟨X₁, X₂⟩) := h₁.symm.prodMk h₂.symm
(h_indep.indepFun (show 3 ≠ 2 by decide)) (h_indep.indepFun zero_ne_one)
exact I.comp (measurable_snd.prodMk measurable_add)
have C27 : H[X₂' | X₂ + X₂'] = H[X₂ | X₂ + X₂'] := by
conv_lhs => rw [add_comm]
apply IdentDistrib.condEntropy_eq hX₂' (hX₂'.add hX₂) hX₂ (hX₂.add hX₂')
have I : IdentDistrib (⟨X₂', X₂⟩) (⟨X₂, X₂'⟩) := h₂.symm.prodMk h₂
(h_indep.indepFun (show 2 ≠ 1 by decide)) (h_indep.indepFun (show 1 ≠ 2 by decide))
exact I.comp (measurable_fst.prodMk measurable_add)
have C28 : H[X₁' | X₁' + X₂'] = H[X₁ | X₁ + X₂] := by
apply IdentDistrib.condEntropy_eq hX₁' (hX₁'.add hX₂') hX₁ (hX₁.add hX₂)
have I : IdentDistrib (⟨X₁', X₂'⟩) (⟨X₁, X₂⟩) := h₁.symm.prodMk h₂.symm
(h_indep.indepFun (show 3 ≠ 2 by decide)) (h_indep.indepFun zero_ne_one)
exact I.comp (measurable_fst.prodMk measurable_add)
have C29 : H[X₁' | V] = H[X₁ | X₁ + X₂] := by
apply IdentDistrib.condEntropy_eq hX₁' (hX₁'.add hX₂) hX₁ (hX₁.add hX₂)
have I : IdentDistrib (⟨X₁', X₂⟩) (⟨X₁, X₂⟩) :=
h₁.symm.prodMk (IdentDistrib.refl hX₂.aemeasurable)
(h_indep.indepFun (show 3 ≠ 1 by decide)) (h_indep.indepFun zero_ne_one)
exact I.comp (measurable_fst.prodMk measurable_add)
have C30 : H[X₂ | X₁ + X₂] = H[X₁ | X₁ + X₂] := by
have := condEntropy_of_injective ℙ hX₁ (hX₁.add hX₂) _ (fun p ↦ add_right_injective p)
convert this with ω
simp [add_comm (X₁ ω), add_assoc (X₂ ω), ZModModule.add_self]
simp only [C1, C2, C3, C4, C5, C7, C8, C9, C10, C11, C12, C13, C14, C15, C16, C17, C18, C19,
C20, C21, C22, C23, C24, C25, C26, C27, C28, C29, C30] at I1 I2 I3 I4 I5 I6 ⊢
linarith only [I1, I2, I3, I4, I5, I6]
include hX₁ hX₂ hX₁' hX₂' h₁ h₂ h_indep in
omit [IsProbabilityMeasure (ℙ : Measure Ω₀₁)] in | 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}\uses{gen-ineq, relabeled-entropy-cond,second-estimate-aux}\leanok
If we apply \Cref{gen-ineq} with $X_1:=X_1$, $Y:=X_1^0$ and $(X_2,X_3,X_4)$ equal to the $3!$ permutations of $(X_2,\tilde X_1,\tilde X_2)$, and sums (using the symmetry $\bbH[X|X+Y] = \bbH[Y|X+Y]$, which follows from \Cref{relabeled-entropy-cond}), we can bound
$$ \sum_{A,B \in \{U,V,W\}: A \neq B} d[X_1^0;A|B, S] - d[X_1^0;X_1]$$
by
\begin{align*}
&\quad \tfrac{1}{4} (6d[X_1;X_2] + 6d[X_1;\tilde X_2]\\
&\qquad + 6d[X_1;\tilde X_1] + 2d[\tilde X_1;\tilde X_2] + 2 d[\tilde X_1;X_2] + 2d[X_2;\tilde X_2])\\
&\quad + \tfrac{1}{8} (2\bbH[X_1+X_2] + 2\bbH[X_1+\tilde X_1] + 2 \bbH[X_1+\tilde X_2] \\
&\qquad - 2\bbH[\tilde X_1+X_2] - 2\bbH[X_2+\tilde X_2] - 2\bbH[\tilde X_1+\tilde X_2])\\
&\qquad \qquad + \tfrac{1}{4} (\bbH[X_2|X_2+\tilde X_2] + \bbH[\tilde X_1|\tilde X_1+\tilde X_2] + \bbH[\tilde X_1|X_1+\tilde X_2] \\
&\qquad \qquad \qquad - \bbH[X_1|X_1+\tilde X_1] - \bbH[X_1|X_1+X_2] - \bbH[X_1|X_1+\tilde X_2]),
\end{align*}
which simplifies to
\begin{align*}
&\quad \tfrac{1}{4} (16k + 6d[X_1;X_1] + 2d[X_2;X_2])\\
&\qquad \qquad + \tfrac{1}{4} (H[X_1+\tilde X_1] - H[X_2+\tilde X_2] + d[X_2|X_2+\tilde X_2] - d[X_1|X_1+\tilde X_1]).
\end{align*}
A symmetric argument also bounds
$$ \sum_{A,B \in \{U,V,W\}: A \neq B} d[X_2^0;A|B, S] - d[X_2^0;X_2]$$
by
\begin{align*}
&\quad \tfrac{1}{4} (16k + 6d[X_2;X_2] + 2d[X_1;X_1])\\
&\qquad \qquad + \tfrac{1}{4} (H[X_2+\tilde X_2] - H[X_1+\tilde X_1] + d[X_1|X_1+\tilde X_1] - d[X_2|X_2+\tilde X_2]).
\end{align*}
On the other hand, from \Cref{second-estimate-aux} one has
$$ d[X_1;X_1] + d[X_2;X_2] \leq 2 k + \frac{2(2 \eta k - I_1)}{1-\eta}.$$
Summing the previous three estimates, we obtain the claim.
\end{proof} | 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 # X₂] + 2 * d[X₁ # X₁]) / 4 + (H[X₂ + X₂'] - H[X₁ + X₁']) / 4
+ (H[X₁ | X₁ + X₁'] - H[X₂ | X₂ + X₂']) / 4 := by
have I1 := gen_ineq_01 p.X₀₂ p.hmeas2 X₂ X₁ X₂' X₁' hX₂ hX₁ hX₂' hX₁' h_indep.reindex_four_bacd
have I2 := gen_ineq_00 p.X₀₂ p.hmeas2 X₂ X₁ X₁' X₂' hX₂ hX₁ hX₁' hX₂' h_indep.reindex_four_badc
have I3 := gen_ineq_10 p.X₀₂ p.hmeas2 X₂ X₂' X₁ X₁' hX₂ hX₂' hX₁ hX₁' h_indep.reindex_four_bcad
have I4 := gen_ineq_10 p.X₀₂ p.hmeas2 X₂ X₂' X₁' X₁ hX₂ hX₂' hX₁' hX₁ h_indep.reindex_four_bcda
have I5 := gen_ineq_00 p.X₀₂ p.hmeas2 X₂ X₁' X₁ X₂' hX₂ hX₁' hX₁ hX₂' h_indep.reindex_four_bdac
have I6 := gen_ineq_01 p.X₀₂ p.hmeas2 X₂ X₁' X₂' X₁ hX₂ hX₁' hX₂' hX₁ h_indep.reindex_four_bdca
have C1 : X₂ + X₁ = X₁ + X₂ := by abel
have C2 : X₁ + X₁' = W := by abel
have C3 : U + X₂' + X₁' = S := by abel
have C4 : X₂ + X₁' = V := by abel
have C5 : X₂ + X₂' + X₁ + X₁' = S := by abel
have C6 : X₂ + X₂' + X₁' + X₁ = S := by abel
have C7 : V + X₁ + X₂' = S := by abel
have C8 : V + X₂' + X₁ = S := by abel
have C9 : d[X₂ # X₁] = d[X₁ # X₂] := rdist_symm
have C10 : d[X₁ # X₂'] = d[X₁ # X₂] :=
ProbabilityTheory.IdentDistrib.rdist_eq (IdentDistrib.refl hX₁.aemeasurable) h₂.symm
have C11 : d[X₂ # X₁'] = d[X₁ # X₂] := by
rw [rdist_symm]
exact ProbabilityTheory.IdentDistrib.rdist_eq h₁.symm (IdentDistrib.refl hX₂.aemeasurable)
have C12 : d[X₂' # X₁'] = d[X₁' # X₂'] := rdist_symm
have C13 : d[X₂' # X₁] = d[X₁ # X₂'] := rdist_symm
have C14 : d[X₁' # X₁] = d[X₁ # X₁'] := rdist_symm
have C15 : d[X₁' # X₂'] = d[X₁ # X₂] :=
ProbabilityTheory.IdentDistrib.rdist_eq h₁.symm h₂.symm
have C16 : H[X₁' + X₂'] = H[X₁ + X₂] := by
apply ProbabilityTheory.IdentDistrib.entropy_eq
have I : IdentDistrib (⟨X₁, X₂⟩) (⟨X₁', X₂'⟩) := h₁.prodMk h₂ (h_indep.indepFun zero_ne_one)
(h_indep.indepFun (show 3 ≠ 2 by decide))
exact I.symm.comp measurable_add
have C17 : H[X₂' + X₁'] = H[X₁ + X₂] := by rw [add_comm]; exact C16
have C18 : H[X₁'] = H[X₁] := ProbabilityTheory.IdentDistrib.entropy_eq h₁.symm
have C19 : H[X₂'] = H[X₂] := ProbabilityTheory.IdentDistrib.entropy_eq h₂.symm
have C20 : H[X₁ + X₂'] = H[X₁ + X₂] := by
apply ProbabilityTheory.IdentDistrib.entropy_eq
have I : IdentDistrib (⟨X₁, X₂'⟩) (⟨X₁, X₂⟩) :=
(IdentDistrib.refl hX₁.aemeasurable).prodMk h₂.symm
(h_indep.indepFun (show 0 ≠ 2 by decide)) (h_indep.indepFun zero_ne_one)
exact I.comp measurable_add
have C21 : H[X₁' | W] = H[X₁ | W] := by
conv_rhs => rw [add_comm]
apply IdentDistrib.condEntropy_eq hX₁' (hX₁'.add hX₁) hX₁ (hX₁.add hX₁')
have I : IdentDistrib (⟨X₁', X₁⟩) (⟨X₁, X₁'⟩) := h₁.symm.prodMk h₁
(h_indep.indepFun (show 3 ≠ 0 by decide)) (h_indep.indepFun (show 0 ≠ 3 by decide))
exact I.comp (measurable_fst.prodMk measurable_add)
have C22 : H[X₂' | X₂' + X₁] = H[X₂ | X₁ + X₂] := by
rw [add_comm]
apply IdentDistrib.condEntropy_eq hX₂' (hX₁.add hX₂') hX₂ (hX₁.add hX₂)
have I : IdentDistrib (⟨X₁, X₂'⟩) (⟨X₁, X₂⟩) :=
(IdentDistrib.refl hX₁.aemeasurable).prodMk h₂.symm
(h_indep.indepFun (show 0 ≠ 2 by decide)) (h_indep.indepFun zero_ne_one)
exact I.comp (measurable_snd.prodMk measurable_add)
have C23 : H[X₁ | X₁ + X₂'] = H[X₁ | X₁ + X₂] := by
apply IdentDistrib.condEntropy_eq hX₁ (hX₁.add hX₂') hX₁ (hX₁.add hX₂)
have I : IdentDistrib (⟨X₁, X₂'⟩) (⟨X₁, X₂⟩) :=
(IdentDistrib.refl hX₁.aemeasurable).prodMk h₂.symm
(h_indep.indepFun (show 0 ≠ 2 by decide)) (h_indep.indepFun zero_ne_one)
exact I.comp (measurable_fst.prodMk measurable_add)
have C24 : H[X₂ | V] = H[X₂ | X₁ + X₂] := by
apply IdentDistrib.condEntropy_eq hX₂ (hX₁'.add hX₂) hX₂ (hX₁.add hX₂)
have I : IdentDistrib (⟨X₁', X₂⟩) (⟨X₁, X₂⟩) :=
h₁.symm.prodMk (IdentDistrib.refl hX₂.aemeasurable)
(h_indep.indepFun (show 3 ≠ 1 by decide)) (h_indep.indepFun zero_ne_one)
exact I.comp (measurable_snd.prodMk measurable_add)
have C25 : H[X₂' | X₂' + X₁'] = H[X₂ | X₁ + X₂] := by
rw [add_comm]
apply IdentDistrib.condEntropy_eq hX₂' (hX₁'.add hX₂') hX₂ (hX₁.add hX₂)
have I : IdentDistrib (⟨X₁', X₂'⟩) (⟨X₁, X₂⟩) := h₁.symm.prodMk h₂.symm
(h_indep.indepFun (show 3 ≠ 2 by decide)) (h_indep.indepFun zero_ne_one)
exact I.comp (measurable_snd.prodMk measurable_add)
have C26 : H[X₁' | X₁' + X₂'] = H[X₁ | X₁ + X₂] := by
apply IdentDistrib.condEntropy_eq hX₁' (hX₁'.add hX₂') hX₁ (hX₁.add hX₂)
have I : IdentDistrib (⟨X₁', X₂'⟩) (⟨X₁, X₂⟩) := h₁.symm.prodMk h₂.symm
(h_indep.indepFun (show 3 ≠ 2 by decide)) (h_indep.indepFun zero_ne_one)
exact I.comp (measurable_fst.prodMk measurable_add)
have C27 : H[X₂ | X₁ + X₂] = H[X₁ | X₁ + X₂] := by
have := condEntropy_of_injective ℙ hX₁ (hX₁.add hX₂) _ (fun p ↦ add_right_injective p)
convert this with ω
simp only [Pi.add_apply, add_comm (X₁ ω), add_assoc (X₂ ω), ZModModule.add_self, add_zero]
have C28 : H[V] = H[U] := by
apply ProbabilityTheory.IdentDistrib.entropy_eq
have I : IdentDistrib (⟨X₁', X₂⟩) (⟨X₁, X₂⟩) :=
h₁.symm.prodMk (IdentDistrib.refl hX₂.aemeasurable)
(h_indep.indepFun (show 3 ≠ 1 by decide)) (h_indep.indepFun zero_ne_one)
exact I.comp measurable_add
have C29 : H[X₂' + X₁] = H[X₁ + X₂] := by
rw [add_comm]
apply ProbabilityTheory.IdentDistrib.entropy_eq
have I : IdentDistrib (⟨X₁, X₂'⟩) (⟨X₁, X₂⟩) :=
(IdentDistrib.refl hX₁.aemeasurable).prodMk h₂.symm
(h_indep.indepFun (show 0 ≠ 2 by decide)) (h_indep.indepFun zero_ne_one)
exact I.comp measurable_add
have C30 : d[X₁ # X₁'] = d[X₁ # X₁] :=
ProbabilityTheory.IdentDistrib.rdist_eq (IdentDistrib.refl hX₁.aemeasurable) h₁.symm
have C31 : d[X₂ # X₂'] = d[X₂ # X₂] :=
ProbabilityTheory.IdentDistrib.rdist_eq (IdentDistrib.refl hX₂.aemeasurable) h₂.symm
simp only [C1, C2, C3, C4, C5, C6, C7, C8, C9, C10, C11, C12, C13, C14, C15, C16, C17, C18, C19,
C20, C21, C22, C23, C24, C25, C25, C26, C27, C28, C29, C30, C31]
at I1 I2 I3 I4 I5 I6 ⊢
linarith only [I1, I2, I3, I4, I5, I6]
include hX₁ hX₂ hX₁' hX₂' h₁ h₂ h_indep h_min in | 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-bsg,phi-min-def}
Conditioned on every $T_3=t$, $d[X_1;X_2]\le d[T_1|T_3=t;T_2|T_3=t]+\eta(\rho(T_1|T_3=t)+\rho(T_2|T_3=t)-\rho(X_1)-\rho(X_2))$ by \Cref{phi-min-def}. Then take the weighted average with weight $\mathbf{P}(T_3=t)$ and then apply \Cref{entropic-bsg} to bound the RHS.
\end{proof} | 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 # A] + ρ[T₂ | T₃ ; μ # A] - ρ[X₁ # A] - ρ[X₂ # A]) := by
let _ : MeasureSpace Ω' := ⟨μ⟩
have : μ = ℙ := rfl
simp only [this]
have : ∑ t, (ℙ (T₃ ⁻¹' {t})).toReal * d[ X₁ # X₂ ] ≤ ∑ t, (ℙ (T₃ ⁻¹' {t})).toReal *
(d[T₁ ; ℙ[|T₃ ← t] # T₂ ; ℙ[|T₃ ← t]]
+ η * (ρ[T₁ ; ℙ[|T₃ ← t] # A] - ρ[X₁ # A]) + η * (ρ[T₂ ; ℙ[|T₃ ← t] # A] - ρ[X₂ # A])) := by
apply Finset.sum_le_sum (fun t ht ↦ ?_)
rcases eq_or_ne (ℙ (T₃ ⁻¹' {t})) 0 with h't | h't
· simp [h't]
have : IsProbabilityMeasure (ℙ[|T₃ ← t]) := cond_isProbabilityMeasure h't
gcongr
exact le_rdist_of_phiMinimizes' h_min hT₁ hT₂
have : k ≤ ∑ x : G, (ℙ (T₃ ⁻¹' {x})).toReal * d[T₁ ; ℙ[|T₃ ← x] # T₂ ; ℙ[|T₃ ← x]] +
η * (ρ[T₁ | T₃ # A] - ρ[X₁ # A]) + η * (ρ[T₂ | T₃ # A] - ρ[X₂ # A]) := by
have S : ∑ i : G, (ℙ (T₃ ⁻¹' {i})).toReal = 1 := by
have : IsProbabilityMeasure (Measure.map T₃ ℙ) := isProbabilityMeasure_map hT₃.aemeasurable
simp [← Measure.map_apply hT₃ (measurableSet_singleton _)]
simp_rw [← Finset.sum_mul, S, mul_add, Finset.sum_add_distrib, ← mul_assoc, mul_comm _ η,
mul_assoc, ← Finset.mul_sum, mul_sub, Finset.sum_sub_distrib, mul_sub,
← Finset.sum_mul, S] at this
simpa [mul_sub, condRho, tsum_fintype] using this
have J : ∑ x : G, (ℙ (T₃ ⁻¹' {x})).toReal * d[T₁ ; ℙ[|T₃ ← x] # T₂ ; ℙ[|T₃ ← x]]
≤ 3 * I[T₁ : T₂] + 2 * H[T₃] - H[T₁] - H[T₂] := by
have h2T₃ : T₃ = T₁ + T₂ :=
calc T₃ = T₁ + T₂ + T₃ - T₃ := by rw [hsum, _root_.zero_sub]; simp [ZModModule.neg_eq_self]
_ = T₁ + T₂ := by rw [add_sub_cancel_right]
subst h2T₃
have := ent_bsg hT₁ hT₂ (μ := ℙ)
simp_rw [integral_fintype _ Integrable.of_finite,
Measure.map_apply hT₃ (measurableSet_singleton _)] at this
exact this
linarith
include h_min in
omit [IsProbabilityMeasure (ℙ : Measure Ω)] in
/-- If $G$-valued random variables $T_1,T_2,T_3$ satisfy $T_1+T_2+T_3=0$, then
$$d[X_1;X_2]\le 3\bbI[T_1:T_2\mid T_3] + (2\bbH[T_3]-\bbH[T_1]-\bbH[T_2])+ \eta(\rho(T_1|T_3)+\rho(T_2|T_3)-\rho(X_1)-\rho(X_2)).$$ -/ | 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
$$d[X_1;X_2] \leq \sum_{1 \leq i<j \leq 3} \bbI[T_i:T_j] + \frac{\eta}{3} \sum_{1 \leq i<j \leq 3} (\rho(T_i|T_j) + \rho(T_j|T_i) -\rho(X_1)-\rho(X_2))$$
\end{lemma}
\begin{proof}\leanok\uses{rho-BSG-triplet}
Take the average of \Cref{rho-BSG-triplet} over all $6$ permutations of $T_1,T_2,T_3$.
\end{proof} | 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[X₁ # A] - ρ[X₂ # A])
+ (ρ[T₁ | T₃ # A] + ρ[T₃ | T₁ # A] - ρ[X₁ # A] - ρ[X₂ # A])
+ (ρ[T₂ | T₃ # A] + ρ[T₃ | T₂ # A] - ρ[X₁ # A] - ρ[X₂ # A])) := by
have := dist_le_of_sum_zero h_min hsum hT₁ hT₂ hT₃ (μ := ℙ)
have : T₁ + T₃ + T₂ = 0 := by convert hsum using 1; abel
have := dist_le_of_sum_zero h_min this hT₁ hT₃ hT₂ (μ := ℙ)
have : T₂ + T₃ + T₁ = 0 := by convert hsum using 1; abel
have := dist_le_of_sum_zero h_min this hT₂ hT₃ hT₁ (μ := ℙ)
linarith
include h_min in
omit [IsProbabilityMeasure (ℙ : Measure Ω)] in
/-- If $G$-valued random variables $T_1,T_2,T_3$ satisfy $T_1+T_2+T_3=0$, then
$$d[X_1;X_2] \leq \sum_{1 \leq i < j \leq 3} \bbI[T_i:T_j]
+ \frac{\eta}{3} \sum_{1 \leq i < j \leq 3} (\rho(T_i|T_j) + \rho(T_j|T_i) -\rho(X_1)-\rho(X_2))$$
-/ | 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 $Y_1,\dots,Y_n$ be some further $G$-valued random variables and let $\alpha>0$ be a constant.
Then there exists a random variable $U$ such that
\begin{equation}
\label{eq:get-better}
d[U;U] + \alpha \sum_{i=1}^n d[Y_i;U] \leq \Bigl(2 + \frac{\alpha n}{2} \Bigr) \delta + \alpha \sum_{i=1}^n d[Y_i;T_2].
\end{equation}
\end{lemma}
\begin{proof}\uses{entropic-bsg, relabeled-entropy, first-useful}
We apply \Cref{entropic-bsg} with $X=T_1$ and $Y=T_2$.
Since $T_1+T_2=-T_3$, we find that
\begin{align}\nonumber
\sum_{z} p_{T_3}(z) & d[T_1 \,|\, T_3 \mathop{=} z;T_2 \,|\, T_3 \mathop{=} z] \\ \nonumber
&\leq 3 \bbI[T_1 : T_2] + 2 \bbH[T_3] - \bbH[T_1] - \bbH[T_2] \\
&\ = \bbI[T_1 : T_2] + \bbI[T_1 : T_3] + \bbI[T_2 : T_3]
= \delta,\label{514a}
\end{align}
where the last line follows from \Cref{relabeled-entropy} by observing
\[
\bbH[T_1,T_2] = \bbH[T_1,T_3] = \bbH[T_2,T_3] = \bbH[T_1,T_2,T_3]
\]
since any two of $T_1,T_2,T_3$ determine the third.
By~\eqref{514a} and the triangle inequality,
\[
\sum_z p_{T_3}(z) d[T_2 \,|\, T_3 \mathop{=} z; T_2 \,|\, T_3\mathop{=}z] \leq 2 \delta
\]
and by \Cref{first-useful}, for each $Y_i$,
\begin{align*}
&\sum_z p_{T_3}(z) d[Y_i; T_2 \,|\, T_3 \mathop{=} z] \\
&\qquad= d[Y_i; T_2 \,|\, T_3]
\leq d[Y_i;T_2] + \frac12 \bbI[T_2 : T_3]
\leq d[Y_i;T_2] + \frac{\delta}{2}.
\end{align*}
Hence,
\begin{align*}
&\sum_z p_{T_3}(z) \bigg( d[T_2 \,|\, T_3 \mathop{=} z; T_2 \,|\, T_3 \mathop{=} z] + \alpha \sum_{i=1}^n d[Y_i;T_2 \,|\, T_3 \mathop{=} z] \bigg) \\
&\qquad \leq
\Bigl(2 + \frac{\alpha n}{2} \Bigr) \delta + \alpha \sum_{i=1}^n d[Y_i; T_2],
\end{align*}
and the result follows by setting $U=(T_2 \,|\, T_3 \mathop{=} z)$ for some $z$ such that the quantity in parentheses on the left-hand side is at most the weighted average value.
\end{proof} | 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 * n / 2) * (I[T₁ : T₂] + I[T₁ : T₃] + I[T₂ : T₃]) + α * ∑ i, d[Y i # T₂] := sorry | 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-vanish, ruzsa-triangle, tau-def, multi-zero, eta-def-multi, tau-ref, tau-min-exist-multi} Set $X^0 := X$. By \Cref{tau-min-exist-multi}, there exists a $\tau$-minimizer $X_{[m]} = (X_i)_{1 \leq i \leq m}$. By \Cref{k-vanish}, we have $D[X_{[m]}]=0$. By \Cref{tau-ref} and the pigeonhole principle, there exists $1 \leq i \leq m$ such that $d[X_i; X] \leq \frac{2}{\eta} d[X;X]$. By \Cref{multi-zero}, we have $d[X_i;U_H]=0$ for some subgroup $H \leq G$, hence by \Cref{ruzsa-triangle} we have $d[U_H; X] \leq \frac{2}{\eta} d[X;X]$. The claim then follows from \Cref{eta-def-multi}.
\end{proof} | /-- 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) (hm: m ≥ 2) (htorsion: ∀ x:G, m • x = 0) (Ω: Type*) [MeasureSpace Ω] (X: Ω → G): ∃ H : AddSubgroup G, ∃ Ω' : Type*, ∃ mΩ : MeasureSpace Ω', ∃ U : Ω' → G,
IsUniform H U ∧ d[X # U] ≤ 64 * m^3 * d[X # X] := sorry
/-- Suppose that $G$ is a finite abelian group of torsion $m$. If $A \subset G$ is non-empty and
$|A+A| \leq K|A|$, then $A$ can be covered by at most $K ^
{(64m^3+2)/2}|A|^{1/2}/|H|^{1/2}$ translates of a subspace $H$ of $G$ with
$|H|/|A| \in [K^{-64m^3}, K^{64m^3}]$
-/ | 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 $T_1:=X_1+X_2,T_2:=X_1+\tilde X_1, T_3:=\tilde X_1 + X_2$, and $S=X_1+X_2+\tilde X_1 + \tilde X_2$. Note that $T_1+T_2+T_3=0$.
First apply \Cref{rho-BSG-triplet-symmetrized} on $(T_1,T_2,T_3)$ when conditioned on $S$ to get
\begin{align} \label{eq:further-bsg}
d[X_1;X_2] &\leq \sum_{1 \leq i<j \leq 3} \bbI[T_i:T_j\mid S] + \frac{\eta}{3} \sum_{1 \leq i<j \leq 3} (\rho(T_i|T_j,S) + \rho(T_j|T_i,S) -\rho(X_1)-\rho(X_2))\nonumber\\
&= (I_1+2I_2) + \frac{\eta}{3} \sum_{1 \leq i<j \leq 3} (\rho(T_i|T_j,S) + \rho(T_j|T_i,S) -\rho(X_1)-\rho(X_2)).
\end{align}
Then apply \Cref{rho-increase-symmetrized} on $(X_1,X_2,\tilde X_1,\tilde X_2)$ and get
$$\sum_{1 \leq i<j \leq 3} (\rho(T_i|T_j,S) + \rho(T_j|T_i,S) - \rho(X_1)-\rho(X_2))\le (4d[X_1;X_2]+d[X_1;X_2]+d[X_2;X_2])= 6 d[X_1;X_2]+(I_2-I_1)$$
by \Cref{I1-I2-diff}. Plug in the inequality above to (\ref{eq:further-bsg}), we get
$$d[X_1;X_2] \le (I_1+2I_2)+2\eta d[X_1;X_2]+\frac{\eta}{3}(I_2-I_1).$$
By \Cref{phi-second-estimate} we can conclude that
$$d[X_1;X_2] \le 8\eta d[X_1;X_2]-\frac{3-10\eta}{3-3\eta} (2\eta d[X_1;X_2]-I_1).$$
Finally by \Cref{phi-first-estimate} and $\eta<1$ we get that the second term is $\le 0$, and thus $d[X_1;X_2] \le 8\eta d[X_1;X_2]$. By the choice $\eta<1/8$ and the non-negativity of $d$ we have $d[X_1;X_2]=0$.
\end{proof} | 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 : IsProbabilityMeasure (ℙ : Measure Ω') := hμ
have h'_min : phiMinimizes Y₁ Y₂ η A ℙ := phiMinimizes_of_identDistrib h_min h_id1.symm h_id2.symm
exact dist_of_min_eq_zero' hη h'_min (h_id1.trans h_id1'.symm) (h_id2.trans h_id2'.symm)
h_indep hY₁ hY₂ hY₁' hY₂' hA hη'
open Filter
open scoped Topology
/-- For `η ≤ 1/8`, there exist phi-minimizers `X₁, X₂` at zero Rusza distance. For `η < 1/8`,
all minimizers are fine, by `dist_of_min_eq_zero`. For `η = 1/8`, we use a limit of
minimizers for `η < 1/8`, which exists by compactness. -/ | 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, tau-min}\leanok
Immediate from \Cref{tau-def} and \Cref{tau-min}.
\end{proof} | 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 : Measure Ω₀₁)] [IsProbabilityMeasure (ℙ : Measure Ω₀₂)] [Fintype G]
[IsProbabilityMeasure (ℙ : Measure Ω)] in | 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}
\end{lemma}
\begin{proof}
\uses{cond-indep-exist, cond-trial-ent, conditional-entropy-def,submodularity, copy-ent, relabeled-entropy, add-entropy, ruz-indep}
\leanok
Let $(A_1, B_1)$ and $(A_2, B_2)$ (and $Z'$, which by abuse of notation we call $Z$) be conditionally independent trials of $(A,B)$ relative to $Z$ as produced by \Cref{cond-indep-exist}, thus $(A_1,B_1)$ and $(A_2,B_2)$ are coupled through the random variable $A_1 + B_1 = A_2 + B_2$, which by abuse of notation we shall also call $Z$.
Observe from \Cref{ruz-indep} that the left-hand side of~\eqref{2-bsg-takeaway} is
\begin{equation}\label{lhs-to-bound}
\bbH[A_1 - B_2| Z] - \bbH[A_1 | Z]/2 - \bbH[B_2 | Z]/2.
\end{equation}
since, crucially, $(A_1 | Z=z)$ and $(B_2 | Z=z)$ are independent for all $z$.
Applying submodularity (\Cref{alt-submodularity}) gives
\begin{equation}\label{bsg-31} \begin{split}
&\bbH[A_1 - B_2] + \bbH[A_1 - B_2, A_1, B_1] \\
&\qquad \leq \bbH[A_1 - B_2, A_1] + \bbH[A_1 - B_2,B_1].
\end{split}\end{equation}
We estimate the second, third and fourth terms appearing here.
First note that, by \Cref{cond-trial-ent} and \Cref{relabeled-entropy} (noting that the tuple $(A_1 - B_2, A_1, B_1)$ determines the tuple $(A_1, A_2, B_1, B_2)$ since $A_1+B_1=A_2+B_2$)
\begin{equation}\label{bsg-24} \bbH[A_1 - B_2, A_1, B_1] = \bbH[A_1, B_1, A_2, B_2,Z] = 2\bbH[A,B] - \bbH[Z].\end{equation}
Next observe that
\begin{equation}\label{bsg-23} \bbH[A_1 - B_2, A_1] = \bbH[A_1, B_2] \leq \bbH[A] + \bbH[B].
\end{equation}
Finally, we have
\begin{equation}\label{bsg-25} \bbH[A_1 - B_2, B_1] = \bbH[A_2 - B_1, B_1] = \bbH[A_2, B_1] \leq \bbH[A] + \bbH[B].\end{equation}
Substituting~\eqref{bsg-24},~\eqref{bsg-23} and~\eqref{bsg-25} into~\eqref{bsg-31} yields
\[\bbH[A_1 - B_2] \leq 2 \bbI[A:B] + \bbH[Z]\] and so by \Cref{cond-reduce}
\[\bbH[A_1 - B_2 | Z] \leq 2 \bbI[A:B] + \bbH[Z].\]
Since
\begin{align*} \bbH[A_1 | Z] & = \bbH[A_1, A_1 + B_1] - \bbH[Z] \\ & = \bbH[A,B] - \bbH[Z] \\ & = \bbH[Z] - \bbI[A:B] - 2 \bbH[Z]- \bbH[A]-\bbH[B]\end{align*}
and similarly for $\bbH[B_2 | Z]$, we see that~\eqref{lhs-to-bound} is bounded by
$3 \bbI[A:B] + 2\bbH[Z]-\bbH[A]-\bbH[B]$ as claimed.
\end{proof} | 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
obtain ⟨Ω', _, AB₁, AB₂, Z', ν, _, hAB₁, hAB₂, hZ', hABZ, hABZ₁, hABZ₂, hZ₁, hZ₂⟩ :=
condIndep_copies' (⟨A, B⟩) Z (hA.prodMk hB) hZ μ (fun (a, b) c ↦ c = a + b)
.of_discrete (Eventually.of_forall fun _ ↦ rfl)
let A₁ := fun ω ↦ (AB₁ ω).1
let B₁ := fun ω ↦ (AB₁ ω).2
let A₂ := fun ω ↦ (AB₂ ω).1
let B₂ := fun ω ↦ (AB₂ ω).2
replace hZ₁ : Z' = A₁ + B₁ := funext hZ₁
replace hZ₂ : Z' = A₂ + B₂ := funext hZ₂
have hA₁ : Measurable A₁ := hAB₁.fst
have hB₁ : Measurable B₁ := hAB₁.snd
have hA₂ : Measurable A₂ := hAB₂.fst
have hB₂ : Measurable B₂ := hAB₂.snd
have hZZ' : IdentDistrib Z' Z ν μ := hABZ₁.comp measurable_snd
have :=
calc
H[⟨A₁, ⟨B₁, A₁ - B₂⟩⟩ ; ν]
= H[⟨⟨A₁, B₁⟩, ⟨⟨A₂, B₂⟩, Z'⟩⟩ ; ν] := entropy_of_comp_eq_of_comp _
(hA₁.prodMk $ hB₁.prodMk $ hA₁.sub hB₂) (hAB₁.prodMk $ hAB₂.prodMk hZ')
(fun (a, b, c) ↦ ((a, b), (b + c, a - c), a + b))
(fun ((a, b), (_c, d), _e) ↦ (a, b, a - d))
(by funext; simpa [sub_add_eq_add_sub, Prod.ext_iff, ← hZ₁, hZ₂, two_nsmul, ← add_sub_assoc,
add_comm, eq_sub_iff_add_eq] using congr_fun (hZ₂.symm.trans hZ₁) _) rfl
_ = H[⟨⟨A₁, B₁⟩, Z'⟩ ; ν] + H[⟨⟨A₂, B₂⟩, Z'⟩ ; ν] - H[Z' ; ν] :=
ent_of_cond_indep _ hAB₁ hAB₂ hZ' hABZ
_ = 2 * H[⟨⟨A, B⟩, Z⟩ ; μ] - H[Z ; μ] := by
rw [two_mul]
congr 1
congr 1 <;> exact IdentDistrib.entropy_eq ‹_›
exact hZZ'.entropy_eq
_ = 2 * H[⟨A, B⟩ ; μ] - H[Z ; μ] := by
congr 2
exact entropy_prod_comp (hA.prodMk hB) _ fun x ↦ x.1 + x.2
have :=
calc
H[⟨A₁, A₁ - B₂⟩ ; ν]
= H[⟨A₁, B₂⟩ ; ν] := entropy_sub_right hA₁ hB₂ _
_ ≤ H[A₁ ; ν] + H[B₂ ; ν] := entropy_pair_le_add hA₁ hB₂ _
_ = H[A ; μ] + H[B ; μ] := by
congr 1
exact (hABZ₁.comp measurable_fst.fst).entropy_eq
exact (hABZ₂.comp measurable_fst.snd).entropy_eq
have :=
calc
H[⟨B₁, A₁ - B₂⟩ ; ν]
= H[⟨A₂, B₁⟩ ; ν] := by
rw [entropy_comm hB₁ (show Measurable (A₁ - B₂) from hA₁.sub hB₂),
← entropy_sub_left' hA₂ hB₁, sub_eq_sub_iff_add_eq_add.2 $ hZ₁.symm.trans hZ₂]
_ ≤ H[A₂ ; ν] + H[B₁ ; ν] := entropy_pair_le_add hA₂ hB₁ _
_ = H[A ; μ] + H[B ; μ] := by
congr 1
exact (hABZ₂.comp measurable_fst.fst).entropy_eq
exact (hABZ₁.comp measurable_fst.snd).entropy_eq
have :=
calc
_ ≤ _ := entropy_triple_add_entropy_le ν hA₁ hB₁ (show Measurable (A₁ - B₂) from hA₁.sub hB₂)
_ ≤ _ := add_le_add ‹_› ‹_›
have :=
calc
H[A₁ - B₂ | Z' ; ν]
≤ H[A₁ - B₂ ; ν] := condEntropy_le_entropy _ (hA₁.sub hB₂) hZ'
_ ≤ _ := le_sub_iff_add_le'.2 ‹_›
_ = 2 * I[A : B ; μ] + H[Z ; μ] := by
rw [‹H[⟨A₁, ⟨B₁, A₁ - B₂⟩⟩ ; ν] = _›, mutualInfo_def]; ring
have hA₁Z :=
calc
H[A₁ | Z' ; ν]
_ = H[⟨A₁, B₁⟩ ; ν] - H[Z' ; ν] := by
rw [chain_rule'', hZ₁, entropy_add_right, entropy_comm] <;> assumption
_ = H[⟨A, B⟩ ; μ] - H[Z ; μ] := by
congr 1
exact (hABZ₁.comp measurable_fst).entropy_eq
exact hZZ'.entropy_eq
_ = H[A ; μ] + H[B ; μ] - I[A : B ; μ] - H[Z ; μ] := by
rw [← entropy_add_entropy_sub_mutualInfo]
have hB₂Z :=
calc
H[B₂ | Z' ; ν]
_ = H[⟨A₂, B₂⟩ ; ν] - H[Z' ; ν] := by
rw [chain_rule'', hZ₂, entropy_add_right', entropy_comm] <;> assumption
_ = H[⟨A, B⟩ ; μ] - H[Z ; μ] := by
congr 1
exact (hABZ₂.comp measurable_fst).entropy_eq
exact hZZ'.entropy_eq
_ = H[A ; μ] + H[B ; μ] - I[A : B ; μ] - H[Z ; μ] := by
rw [← entropy_add_entropy_sub_mutualInfo]
calc
(μ.map Z)[fun z ↦ d[A ; μ[|Z ← z] # B ; μ[|Z ← z]]]
= (ν.map Z')[fun z ↦ d[A₁ ; ν[|Z' ← z] # B₂ ; ν[|Z' ← z]]] := by
rw [hZZ'.map_eq]
refine integral_congr_ae $ Eventually.of_forall fun z ↦ ?_
have hAA₁ : IdentDistrib A₁ A (ν[|Z' ← z]) (μ[|Z ← z]) :=
(hABZ₁.comp $ measurable_fst.fst.prodMk measurable_snd).cond
(.singleton z) hZ' hZ
have hBB₂ : IdentDistrib B₂ B (ν[|Z' ← z]) (μ[|Z ← z]) :=
(hABZ₂.comp $ measurable_fst.snd.prodMk measurable_snd).cond
.of_discrete hZ' hZ
dsimp (config := {zeta := false}) [rdist]
rw [← hAA₁.entropy_eq, ← hBB₂.entropy_eq, hAA₁.map_eq, hBB₂.map_eq]
_ = (ν.map Z')[fun z ↦
H[A₁ - B₂ ; ν[|Z' ← z]] - H[A₁ ; ν[|Z' ← z]]/2 - H[B₂ ; ν[|Z' ← z]]/2] := by
apply integral_congr_ae
apply hABZ.mono
intro z hz
exact (hz.comp measurable_fst measurable_snd).rdist_eq hA₁ hB₂
_ = H[A₁ - B₂ | Z' ; ν] - H[A₁ | Z' ; ν] / 2 - H[B₂ | Z' ; ν] / 2 := by
rw [integral_sub, integral_sub, integral_div, integral_div]
rfl
all_goals exact .of_finite
_ ≤ 2 * I[A : B ; μ] + H[Z ; μ] - H[A₁ | Z' ; ν] / 2 - H[B₂ | Z' ; ν] / 2 :=
sub_le_sub_right (sub_le_sub_right ‹_› _) _
_ = _ := by rw [hA₁Z, hB₂Z]; ring
end BalogSzemerediGowers
variable (μ μ') in
/-- Suppose that $(X, Z)$ and $(Y, W)$ are random variables, where $X, Y$ take values in an abelian
group. Then $$d[X | Z ; Y | W] \leq d[X ; Y] + \tfrac{1}{2} I[X : Z] + \tfrac{1}{2} I[Y : W]$$ -/ | 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 $H$)
\[\mathbb{H}(\pi(X))\leq 2d[X;U_H].\]
\end{lemma}
\begin{proof}
\uses{independent-exist, ruzsa-diff, chain-rule, shear-ent, submodularity, jensen-bound}\leanok
WLOG, we make $X$, $U_H$ independent (\Cref{independent-exist}).
Now by Lemmas \ref{submodularity}, \ref{shear-ent}, \ref{jensen-bound}
\begin{align*}
&\mathbb{H}(X-U_H|\pi(X)) \geq \mathbb{H}(X-U_H|X) &= \mathbb{H}(U_H) \\
&\mathbb{H}(X-U_H|\pi(X)) \leq \log |H| &= \mathbb{H}(U_H)
\end{align*}
By \Cref{chain-rule}
\[\mathbb{H}(X-U_H)=\mathbb{H}(\pi(X))+\mathbb{H}(X-U_H|\pi(X))=\mathbb{H}(\pi(X))+\mathbb{H}(U_H)\]
and therefore
\[d[X;U_H]=\mathbb{H}(\pi(X))+\frac{\mathbb{H}(U_H)-\mathbb{H}(X)}{2}.\]
Furthermore by \Cref{ruzsa-diff}
\[d[X;U_H]\geq \frac{\lvert \mathbb{H}(X)-\mathbb{H}(U_H)\rvert}{2}.\]
Adding these inequalities gives the result.
\end{proof} | 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[X; μ # UH ; μ'] := by
obtain ⟨ν, X', UH', hν, hX', hUH', h_ind, h_id_X', h_id_UH', _, _⟩ :=
independent_copies_finiteRange hX hU μ μ'
replace hunif : IsUniform H UH' ν :=
IsUniform.of_identDistrib hunif h_id_UH'.symm .of_discrete
rewrite [← (h_id_X'.comp (by fun_prop)).entropy_eq, ← h_id_X'.rdist_eq h_id_UH']
let π := ⇑(QuotientAddGroup.mk' H)
let νq := Measure.map (π ∘ X') ν
have : Countable (HasQuotient.Quotient G H) := Quotient.countable
have : MeasurableSingletonClass (HasQuotient.Quotient G H) :=
{ measurableSet_singleton := fun _ ↦ measurableSet_quotient.mpr .of_discrete }
have : Finite H := hH
have : H[X' - UH' | π ∘ X' ; ν] = H[UH' ; ν] := by
have h_meas_le : ∀ y ∈ FiniteRange.toFinset (π ∘ X'),
(νq {y}).toReal * H[X' - UH' | (π ∘ X') ← y ; ν] ≤ (νq {y}).toReal * H[UH' ; ν] := by
intro x _
refine mul_le_mul_of_nonneg_left ?_ ENNReal.toReal_nonneg
let ν' := ν[|π ∘ X' ← x]
let π' := QuotientAddGroup.mk (s := H)
have h_card : Nat.card (π' ⁻¹' {x}) = Nat.card H := Nat.card_congr <|
(QuotientAddGroup.preimageMkEquivAddSubgroupProdSet H _).trans <| Equiv.prodUnique H _
have : Finite (π' ⁻¹' {x}) :=
Nat.finite_of_card_ne_zero <| h_card.trans_ne <| Nat.pos_iff_ne_zero.mp (Nat.card_pos)
let H_x := (π' ⁻¹' {x}).toFinite.toFinset
have h : ∀ᵐ ω ∂ν', (X' - UH') ω ∈ H_x := by
let T : Set (G × G) := ((π' ∘ X') ⁻¹' {x})ᶜ
let U : Set (G × G) := UH' ⁻¹' Hᶜ
have h_subset : (X' - UH') ⁻¹' H_xᶜ ⊆ T ∪ U :=
fun ω hω ↦ Classical.byContradiction fun h ↦ by simp_all [not_or, T, U, H_x, π']
refine MeasureTheory.mem_ae_iff.mpr (le_zero_iff.mp ?_)
calc
_ ≤ ν' T + ν' U := (measure_mono h_subset).trans (measure_union_le T U)
_ = ν' T + 0 := congrArg _ <| by
simp only [ν', ProbabilityTheory.cond, Measure.smul_apply, smul_eq_mul]
rw [le_zero_iff.mp <| (restrict_apply_le _ U).trans_eq hunif.measure_preimage_compl,
mul_zero]
_ = 0 := (add_zero _).trans <| by
have : restrict ν (π ∘ X' ⁻¹' {x}) T = 0 := by
simp [restrict_apply .of_discrete, T, π', π]
simp only [ν', ProbabilityTheory.cond, Measure.smul_apply, smul_eq_mul]
rw [this, mul_zero]
convert entropy_le_log_card_of_mem (Measurable.sub hX' hUH') h
simp_rw [hunif.entropy_eq' hH hUH', H_x, Set.Finite.mem_toFinset, h_card,
SetLike.coe_sort_coe]
have h_one : (∑ x ∈ FiniteRange.toFinset (π ∘ X'), (νq {x}).toReal) = 1 := by
rewrite [Finset.sum_toReal_measure_singleton]
apply (ENNReal.toReal_eq_one_iff _).mpr
have := isProbabilityMeasure_map (μ := ν) <| .of_discrete (f := π ∘ X')
rewrite [← measure_univ (μ := νq), ← FiniteRange.range]
let rng := Set.range (π ∘ X')
have h_compl : νq rngᶜ = 0 := ae_map_mem_range (π ∘ X') .of_discrete ν
rw [← MeasureTheory.measure_add_measure_compl (MeasurableSet.of_discrete (s := rng)),
h_compl, add_zero]
have := FiniteRange.sub X' UH'
have h_ge : H[X' - UH' | π ∘ X' ; ν] ≥ H[UH' ; ν] := calc
_ ≥ H[X' - UH' | X' ; ν] := condEntropy_comp_ge ν hX' (hX'.sub hUH') π
_ = H[UH' | X' ; ν] := condEntropy_sub_left hUH' hX'
_ = H[UH' ; ν] := h_ind.symm.condEntropy_eq_entropy hUH' hX'
have h_le : H[X' - UH' | π ∘ X' ; ν] ≤ H[UH' ; ν] := by
rewrite [condEntropy_eq_sum _ _ _ .of_discrete]
apply (Finset.sum_le_sum h_meas_le).trans
rewrite [← Finset.sum_mul, h_one, one_mul]
rfl
exact h_le.ge_iff_eq.mp h_ge
have : H[X' - UH' ; ν] = H[π ∘ X' ; ν] + H[UH' ; ν] := by calc
_ = H[⟨X' - UH', π ∘ (X' - UH')⟩ ; ν] := (entropy_prod_comp (hX'.sub hUH') ν π).symm
_ = H[⟨X' - UH', π ∘ X'⟩ ; ν] := by
apply IdentDistrib.entropy_eq <| IdentDistrib.of_ae_eq (Measurable.aemeasurable
.of_discrete) <| MeasureTheory.mem_ae_iff.mpr _
convert hunif.measure_preimage_compl
ext; simp [π]
_ = H[π ∘ X' ; ν] + H[UH' ; ν] := by
rewrite [chain_rule ν (by exact hX'.sub hUH') .of_discrete]
congr
have : d[X' ; ν # UH' ; ν] = H[π ∘ X' ; ν] + (H[UH' ; ν] - H[X' ; ν]) / 2 := by
rewrite [h_ind.rdist_eq hX' hUH']
linarith only [this]
linarith only [this, (abs_le.mp (diff_ent_le_rdist hX' hUH' (μ := ν) (μ' := ν))).2] | 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
$$\bbH[X-(a-1)Y] \leq \bbH[X-aY] + \bbH[X-Y-X'] - \bbH[X].$$
\end{lemma}
\begin{proof}\uses{ruzsa-triangle-improved, neg-ent}\leanok
From \Cref{ruzsa-triangle-improved} we have
$$ \bbH[(X-Y)-aY] \leq \bbH[(X-Y) - X'] + \bbH[X'-aY] - \bbH[X']$$
which gives the first inequality. Similarly from \Cref{ruzsa-triangle-improved} we have
$$ \bbH[(X+Y)-aY] \leq \bbH[(X+Y) - X'] + \bbH[X'-aY] - \bbH[X']$$
which (when combined with \Cref{neg-ent}) gives the second inequality.
\end{proof} | 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 - Y - X'; μ] - H[X; μ] := by
rw [add_smul, one_smul, add_comm, sub_add_eq_sub_sub]
have iX'Y : IndepFun X' Y μ := h_indep.indepFun (show 2 ≠ 1 by simp)
have iXY : IndepFun X Y μ := h_indep.indepFun (show 0 ≠ 1 by simp)
have hident' : IdentDistrib (X' - a • Y) (X - a • Y) μ μ := by
simp_rw [sub_eq_add_neg]
apply hident.symm.add (IdentDistrib.refl (hY.const_smul a).neg.aemeasurable)
· convert iX'Y.comp measurable_id (.of_discrete (f := fun y ↦ -(a • y))) using 1
· convert iXY.comp measurable_id (.of_discrete (f := fun y ↦ -(a • y))) using 1
have iXY_X' : IndepFun (⟨X, Y⟩) X' μ :=
h_indep.indepFun_prodMk (fun i ↦ (by fin_cases i <;> assumption)) 0 1 2
(show 0 ≠ 2 by simp) (show 1 ≠ 2 by simp)
calc
_ ≤ H[X - Y - X' ; μ] + H[X' - a • Y ; μ] - H[X' ; μ] := by
refine ent_of_diff_le _ _ _ (hX.sub hY) (hY.const_smul a) hX' ?_
exact iXY_X'.comp (φ := fun (x, y) ↦ (x - y, a • y)) .of_discrete measurable_id
_ = _ := by
rw [hident.entropy_eq]
simp only [add_comm, sub_left_inj, _root_.add_left_inj]
exact hident'.entropy_eq
/-- Let `X,Y,X'` be independent `G`-valued random variables, with `X'` a copy of `X`,
and let `a` be an integer. Then `H[X - (a-1)Y] ≤ H[X - aY] + H[X - Y - X'] - 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
$$\bbH[X-(a-1)Y] \leq \bbH[X-aY] + \bbH[X-Y-X'] - \bbH[X].$$
\end{lemma}
\begin{proof}\uses{ruzsa-triangle-improved, neg-ent}\leanok
From \Cref{ruzsa-triangle-improved} we have
$$ \bbH[(X-Y)-aY] \leq \bbH[(X-Y) - X'] + \bbH[X'-aY] - \bbH[X']$$
which gives the first inequality. Similarly from \Cref{ruzsa-triangle-improved} we have
$$ \bbH[(X+Y)-aY] \leq \bbH[(X+Y) - X'] + \bbH[X'-aY] - \bbH[X']$$
which (when combined with \Cref{neg-ent}) gives the second inequality.
\end{proof} | 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 - Y - X'; μ] - H[X; μ] := by
rw [sub_smul, one_smul, sub_eq_add_neg, neg_sub, add_sub]
have iX'Y : IndepFun X' Y μ := h_indep.indepFun (show 2 ≠ 1 by simp)
have iXY : IndepFun X Y μ := h_indep.indepFun (show 0 ≠ 1 by simp)
have hident' : IdentDistrib (X' - a • Y) (X - a • Y) μ μ := by
simp_rw [sub_eq_add_neg]
apply hident.symm.add (IdentDistrib.refl (hY.const_smul a).neg.aemeasurable)
· convert iX'Y.comp measurable_id (.of_discrete (f := fun y ↦ -(a • y))) using 1
· convert iXY.comp measurable_id (.of_discrete (f := fun y ↦ -(a • y))) using 1
have hident'' : IdentDistrib (-(X + Y - X')) (X - Y - X') μ μ := by
simp_rw [neg_sub, ← sub_sub, sub_eq_add_neg, add_assoc]
refine hident.symm.add ?_ ?_ ?_
rotate_left
· rw [← neg_add]
apply IndepFun.comp _ measurable_id measurable_neg
refine h_indep.indepFun_add_right (fun i ↦ (by fin_cases i <;> assumption))
2 0 1 (by simp) (by simp)
· rw [← neg_add]
apply IndepFun.comp _ measurable_id measurable_neg
refine h_indep.indepFun_add_right (fun i ↦ (by fin_cases i <;> assumption))
0 1 2 (by simp) (by simp)
rw [add_comm, ← neg_add, ← neg_add]
exact (IdentDistrib.refl hY.aemeasurable).add hident iXY.symm iX'Y.symm |>.neg
have iXY_X' : IndepFun (⟨X, Y⟩) X' μ :=
h_indep.indepFun_prodMk (fun i ↦ (by fin_cases i <;> assumption)) 0 1 2
(show 0 ≠ 2 by simp) (show 1 ≠ 2 by simp)
calc
_ ≤ H[X + Y - X' ; μ] + H[X' - a • Y ; μ] - H[X' ; μ] := by
refine ent_of_diff_le _ _ _ (hX.add hY) (hY.const_smul a) hX' ?_
exact iXY_X'.comp (φ := fun (x, y) ↦ (x + y, a • y)) .of_discrete measurable_id
_ = H[- (X + Y - X') ; μ] + H[X - a • Y ; μ] - H[X ; μ] := by
rw [hident.entropy_eq]
simp only [hident'.entropy_eq, add_comm, sub_left_inj, _root_.add_right_inj]
exact entropy_neg (hX.add hY |>.sub hX') |>.symm
_ = _ := by
rw [add_comm, hident''.entropy_eq]
/-- Let `X,Y` be independent `G`-valued random variables, and let `a` be an integer. Then
`H[X - aY] - H[X] ≤ 4 |a| d[X ; Y]`. -/ | 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 has
$$\bbH[Y-X+X'] - \bbH[Y-X] \leq \bbH[Y+X'] - \bbH[Y] = \bbH[Y+X] - \bbH[Y]$$
which by \Cref{ruz-indep} gives
$$\bbH[X-Y-X'] -\bbH[X] \leq d[X;Y] + d[X;-Y]$$
and hence by \Cref{sign-flip}
$$\bbH[X-Y-X'] - \bbH[X] \leq 4d[X;Y].$$
From \Cref{sum-dilate-I} we then have
$$\bbH[X-(a\pm 1)Y] \leq \bbH[X-aY] + 4 d[X;Y]$$
and the claim now follows by an induction on $|a|$.
\end{proof} | 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⟩
:= independent_copies3_nondep hX hY hX μ μ μ
have iX₁Y : IndepFun X₁' Y' μ' := h_indep'.indepFun (show 0 ≠ 1 by simp)
have iYX₂ : IndepFun Y' X₂' μ' := h_indep'.indepFun (show 1 ≠ 2 by simp)
have iX₂nY : IndepFun X₂' (-Y') μ' := iYX₂.symm.comp measurable_id measurable_neg
have inX₁YX₂ : iIndepFun ![-X₁', Y', X₂'] μ' := by
convert h_indep'.comp ![-id, id, id] (by fun_prop) with i
match i with | 0 => rfl | 1 => rfl | 2 => rfl
have idX₁X₂' : IdentDistrib X₁' X₂' μ' μ' := idX₁.trans idX₂.symm
have idX₁Y : IdentDistrib (⟨X, Y⟩) (⟨X₁', Y'⟩) μ μ' :=
IdentDistrib.prodMk idX₁.symm idY.symm h_indep iX₁Y
have h1 : H[Y' - X₁' + X₂'; μ'] - H[Y' - X₁'; μ'] ≤ H[Y' + X₂'; μ'] - H[Y'; μ'] := by
simp_rw [sub_eq_add_neg Y', add_comm Y' (-X₁')]
exact kaimanovich_vershik inX₁YX₂ hX₁'.neg hY' hX₂'
have h2 : H[X₁' - Y' - X₂'; μ'] - H[X₁'; μ'] ≤ d[X₁' ; μ' # Y' ; μ'] + d[X₁' ; μ' # -Y' ; μ'] := by
rw [idX₁X₂'.rdist_eq (IdentDistrib.refl hY'.aemeasurable).neg, iX₁Y.rdist_eq hX₁' hY',
iX₂nY.rdist_eq hX₂' hY'.neg, entropy_neg hY', idX₁X₂'.entropy_eq.symm]
rw [show H[X₁' - Y' - X₂'; μ'] = H[-(X₁' - Y' - X₂'); μ']
from entropy_neg (hX₁'.sub hY' |>.sub hX₂') |>.symm]
rw [show H[X₁' - Y'; μ'] = H[-(X₁' - Y'); μ'] from entropy_neg (hX₁'.sub hY') |>.symm]
ring_nf
rw [sub_eq_add_neg, add_comm, add_assoc, sub_neg_eq_add]
gcongr
convert sub_le_iff_le_add'.mp h1 using 1
· simp [sub_eq_add_neg, add_comm]
· simp only [sub_eq_add_neg, neg_add_rev, neg_neg, add_comm, add_assoc]
linarith
have h3 : H[X₁' - Y' - X₂' ; μ'] - H[X₁'; μ'] ≤ 4 * d[X₁' ; μ' # Y' ; μ'] :=
calc
_ ≤ d[X₁' ; μ' # Y' ; μ'] + d[X₁' ; μ' # -Y' ; μ'] := h2
_ ≤ d[X₁' ; μ' # Y' ; μ'] + 3 * d[X₁' ; μ' # Y' ; μ'] := by
gcongr
exact rdist_of_neg_le hX₁' hY'
_ = _ := by ring_nf
have h4 (a : ℤ) : H[X - (a + 1) • Y; μ] ≤ H[X₁' - a • Y'; μ'] + 4 * d[X₁' ; μ' # Y' ; μ'] := by
calc
_ = H[X₁' - (a + 1) • Y'; μ'] :=
IdentDistrib.entropy_eq <|
idX₁Y.comp (show Measurable (fun xy ↦ (xy.1 - (a + 1) • xy.2)) by fun_prop)
_ ≤ H[X₁' - a • Y'; μ'] + H[X₁' - Y' - X₂'; μ'] - H[X₁'; μ'] :=
ent_of_sub_smul hX₁' hY' hX₂' h_indep' idX₁X₂'
_ ≤ H[X₁' - a • Y'; μ'] + 4 * d[X₁' ; μ' # Y' ; μ'] := by
rw [add_sub_assoc]
gcongr
have h4' (a : ℤ) : H[X - (a - 1) • Y; μ] ≤ H[X₁' - a • Y'; μ'] + 4 * d[X₁' ; μ' # Y' ; μ'] := by
calc
_ = H[X₁' - (a - 1) • Y'; μ'] :=
IdentDistrib.entropy_eq <|
idX₁Y.comp (show Measurable (fun xy ↦ (xy.1 - (a - 1) • xy.2)) by fun_prop)
_ ≤ H[X₁' - a • Y'; μ'] + H[X₁' - Y' - X₂'; μ'] - H[X₁'; μ'] :=
ent_of_sub_smul' hX₁' hY' hX₂' h_indep' idX₁X₂'
_ ≤ H[X₁' - a • Y'; μ'] + 4 * d[X₁' ; μ' # Y' ; μ'] := by
rw [add_sub_assoc]
gcongr
have (a : ℤ) : H[X₁' - a • Y'; μ'] = H[X - a • Y; μ] :=
idX₁Y.symm.comp (show Measurable (fun xy ↦ (xy.1 - a • xy.2)) by fun_prop) |>.entropy_eq
simp_rw [IdentDistrib.rdist_eq idX₁ idY, this] at h4 h4'
set! n := |a| with ha
have hn : 0 ≤ n := by simp [ha]
lift n to ℕ using hn with n
induction' n with n ih generalizing a
· rw [← ha, abs_eq_zero.mp ha.symm]
simp
· rename_i m _
have : a ≠ 0 := by
rw [ne_eq, ← abs_eq_zero, ← ha]
exact NeZero.natCast_ne (m + 1) ℤ
have : m = |a - 1| ∨ m = |a + 1| := by
rcases lt_or_gt_of_ne this with h | h
· right
rw [abs_of_neg h] at ha
rw [abs_of_nonpos (by exact h), neg_add, ← ha]
norm_num
· left
rw [abs_of_pos h] at ha
rw [abs_of_nonneg ?_, ← ha]
swap; exact Int.sub_nonneg_of_le h
norm_num
rcases this with h | h
· calc
_ ≤ H[X - (a - 1) • Y; μ] - H[X; μ] + 4 * d[X ; μ # Y ; μ] := by
nth_rw 1 [(a.sub_add_cancel 1).symm, sub_add_eq_add_sub _ H[X; μ]]
gcongr
exact h4 (a - 1)
_ ≤ 4 * |a - 1| * d[X ; μ # Y ; μ] + 4 * d[X ; μ # Y ; μ] := by
gcongr
exact ih h h
_ = 4 * |a| * d[X ; μ # Y ; μ] := by
nth_rw 2 [← mul_one 4]
rw [← add_mul, ← mul_add, ← ha, ← h]
norm_num
· calc
_ ≤ H[X - (a + 1) • Y; μ] - H[X; μ] + 4 * d[X ; μ # Y ; μ] := by
nth_rw 1 [(a.add_sub_cancel 1).symm, sub_add_eq_add_sub _ H[X; μ]]
gcongr
exact h4' (a + 1)
_ ≤ 4 * |a + 1| * d[X ; μ # Y ; μ] + 4 * d[X ; μ # Y ; μ] := by
gcongr
exact ih h h
_ = 4 * |a| * d[X ; μ # Y ; μ] := by
nth_rw 2 [← mul_one 4]
rw [← add_mul, ← mul_add, ← ha, ← h]
norm_num | 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, then
$$ \bbH[\sum_{j=1}^l Y_j] \leq \bbH[ \sum_{i=1}^m X_i ] + \sum_{j=1}^l (\bbH[ Y_j - X_{f(j)}] - \bbH[X_{f(j)}]).$$
\end{lemma}
\begin{proof}\uses{klm-1, kv, neg-ent, sumset-lower} Write $W := \sum_{i=1}^m X_i$. From \Cref{sumset-lower} we have
$$ \bbH[\sum_{j=1}^l Y_j] \leq \bbH[-W + \sum_{j=1}^l Y_j]$$
while from \Cref{klm-1} one has
$$ \bbH[-W + \sum_{j=1}^l Y_j] \leq \bbH[-W] + \sum_{j=1}^l \bbH[-W + Y_j] - \bbH[-W].$$
From \Cref{kv} one has
$$ \bbH[-W + Y_j] - \bbH[-W] \leq \bbH[-X_{f(j)} + Y_j] - \bbH[-X_{f(j)}].$$
The claim now follows from \Cref{neg-ent} and some elementary algebra.
\end{proof} | 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) :
H[∑ i ∈ t, X i; μ] ≤ H[∑ i ∈ s, X i; μ] + ∑ i ∈ t, (H[X i - X (f i); μ] - H[X (f i); μ]) := by
sorry
/-- Let `X,Y,X'` be independent `G`-valued random variables, with `X'` a copy of `X`,
and let `a` be an integer. Then `H[X - (a+1)Y] ≤ H[X - aY] + H[X - Y - X'] - H[X]` -/ | 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{proof}\uses{first-cond, first-fibre, first-upper, ruz-indep}\leanok
Subtracting \Cref{first-cond} from \Cref{first-fibre}, and combining the resulting inequality with \Cref{first-upper} gives the bound
\[
d[X_1+\tilde X_2;X_2+\tilde X_1] \le (1 + \eta) k - I_1,
\]
and the claim follows from \Cref{ruz-indep} and the definition of $k$.
\end{proof} | 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']
let D2 := d[p.X₀₂ # X₂]
let Dc2 := d[p.X₀₂ # X₂ | X₂ + X₁']
have lem68 : D + Dcc + I₁ = 2 * k :=
rdist_add_rdist_add_condMutual_eq _ _ _ _ hX₁ hX₂ hX₁' hX₂' h₁ h₂ h_indep
have lem610 : Dcc ≥ k - p.η * (Dc1 - D1) - p.η * (Dc2 - D2) :=
condRuzsaDist_of_sums_ge p X₁ X₂ X₁' X₂' hX₁ hX₂ (by fun_prop) (by aesop) h_min
have lem611c : Dc1 - D1 ≤ k / 2 + H[X₁] / 4 - H[X₂] / 4 :=
diff_rdist_le_3 p X₁ X₂ X₁' X₂' hX₁ hX₂' h₂ h_indep
have lem611d : Dc2 - D2 ≤ k / 2 + H[X₂] / 4 - H[X₁] / 4 :=
diff_rdist_le_4 p X₁ X₂ X₁' X₂' hX₂ hX₁' h₁ h_indep
have aux : D + I₁ ≤ (1 + p.η) * k := by
calc D + I₁
≤ k + p.η * (Dc1 - D1) + p.η * (Dc2 - D2) := ?_
_ ≤ k + p.η * (k / 2 + H[X₁] / 4 - H[X₂] / 4) + p.η * (k / 2 + H[X₂] / 4 - H[X₁] / 4) := ?_
_ = (1 + p.η) * k := by ring
· convert add_le_add lem68.le (neg_le_neg lem610) using 1 <;> ring
· refine add_le_add (add_le_add (le_refl _) ?_) ?_
· apply (mul_le_mul_left p.hη).mpr lem611c
· apply (mul_le_mul_left p.hη).mpr lem611d
have ent_sub_eq_ent_add : H[X₁ + X₂' - (X₂ + X₁')] = H[X₁ + X₂' + (X₂ + X₁')] := by
simp [ZModModule.sub_eq_add]
have rw₁ : X₁ + X₂' + (X₂ + X₁') = X₁ + X₂ + X₁' + X₂' := by abel
have ind_aux : IndepFun (X₁ + X₂') (X₂ + X₁') := by
exact iIndepFun.indepFun_add_add h_indep (fun i ↦ by fin_cases i <;> assumption) 0 2 1 3
(by decide) (by decide) (by decide) (by decide)
have ind : D = H[X₁ + X₂' - (X₂ + X₁')] - H[X₁ + X₂'] / 2 - H[X₂ + X₁'] / 2 :=
ind_aux.rdist_eq (by fun_prop) (by fun_prop)
rw [ind, ent_sub_eq_ent_add, rw₁] at aux
have obs : H[X₁ + X₂ + X₁' + X₂'] ≤ H[X₁ + X₂'] / 2 + H[X₂ + X₁'] / 2 + (1 + p.η) * k - I₁ := by
linarith
have rw₂ : H[X₁ + X₂'] = k + H[X₁]/2 + H[X₂]/2 := by
have HX₂_eq : H[X₂] = H[X₂'] :=
congr_arg (fun (μ : Measure G) ↦ measureEntropy (μ := μ)) h₂.map_eq
have k_eq : k = H[X₁ - X₂'] - H[X₁] / 2 - H[X₂'] / 2 := by
have k_eq_aux : k = d[X₁ # X₂'] := (IdentDistrib.refl hX₁.aemeasurable).rdist_eq h₂
rw [k_eq_aux]
exact (h_indep.indepFun (show (0 : Fin 4) ≠ 2 by decide)).rdist_eq hX₁ hX₂'
rw [k_eq, ← ZModModule.sub_eq_add, ← HX₂_eq]
ring
have rw₃ : H[X₂ + X₁'] = k + H[X₁]/2 + H[X₂]/2 := by
have HX₁_eq : H[X₁] = H[X₁'] :=
congr_arg (fun (μ : Measure G) ↦ measureEntropy (μ := μ)) h₁.map_eq
have k_eq' : k = H[X₁' - X₂] - H[X₁'] / 2 - H[X₂] / 2 := by
have k_eq_aux : k = d[X₁' # X₂] :=
IdentDistrib.rdist_eq h₁ (IdentDistrib.refl hX₂.aemeasurable)
rw [k_eq_aux]
exact IndepFun.rdist_eq (h_indep.indepFun (show (3 : Fin 4) ≠ 1 by decide)) hX₁' hX₂
rw [add_comm X₂ X₁', k_eq', ← ZModModule.sub_eq_add, ← HX₁_eq]
ring
calc H[X₁ + X₂ + X₁' + X₂']
≤ H[X₁ + X₂'] / 2 + H[X₂ + X₁'] / 2 + (1 + p.η) * k - I₁ := obs
_ = (k + H[X₁] / 2 + H[X₂] / 2) / 2
+ (k + H[X₁] / 2 + H[X₂] / 2) / 2 + (1 + p.η) * k - I₁ := by rw [rw₂, rw₃]
_ = H[X₁] / 2 + H[X₂] / 2 + (2 + p.η) * k - I₁ := by ring | 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 uniformly distributed on $H$.
Furthermore, both $d[X^0_1;U_H]$ and $d[X^0_2;U_H]$ are at most $6 d[X^0_1;X^0_2]$.
\end{theorem}
\begin{proof} \uses{de-prop, tau-min, lem:100pc, ruzsa-triangle} \leanok Let $X_1, X_2$ be the $\tau$-minimizer from \Cref{tau-min}. From \Cref{de-prop}, $d[X_1;X_2]=0$. From \Cref{lem:100pc}, $d[X_1;U_H] = d[X_2; U_H] = 0$. Also from $\tau$-minimization we have $\tau[X_1;X_2] \leq \tau[X^0_2;X^0_1]$. Using this and the Ruzsa triangle inequality we can conclude.
\end{proof} | 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⟩ := tau_minimizer_exists p
have hdist : d[X₁ # X₂] = 0 := tau_strictly_decreases p hX₁ hX₂ htau_min hpη
obtain ⟨H, U, hU, hH_unif, hdistX₁, hdistX₂⟩ := exists_isUniform_of_rdist_eq_zero hX₁ hX₂ hdist
refine ⟨AddSubgroup.toZModSubmodule _ H, Ω', inferInstance, U, inferInstance, hU, hH_unif , ?_⟩
have h : τ[X₁ # X₂ | p] ≤ τ[p.X₀₂ # p.X₀₁ | p] := is_tau_min p htau_min p.hmeas2 p.hmeas1
rw [tau, tau, hpη] at h
norm_num at h
have : d[p.X₀₁ # p.X₀₂] = d[p.X₀₂ # p.X₀₁] := rdist_symm
have : d[p.X₀₁ # U] ≤ d[p.X₀₁ # X₁] + d[X₁ # U] := rdist_triangle p.hmeas1 hX₁ hU
have : d[p.X₀₂ # U] ≤ d[p.X₀₂ # X₂] + d[X₂ # U] := rdist_triangle p.hmeas2 hX₂ hU
linarith | 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
\[
d[X^0_1;U_H] + d[X^0_2;U_H] \le 10 d[X^0_1;X^0_2],
\]
where $U_H$ is uniformly distributed on $H$.
Furthermore, both $d[X^0_1;U_H]$ and $d[X^0_2;U_H]$ are at most $6 d[X^0_1;X^0_2]$.
\end{theorem}
\begin{proof} \uses{de-prop-lim-improv, lem:100pc, ruzsa-triangle}\leanok
Let $X_1, X_2$ be the good $\tau$-minimizer from \Cref{de-prop-lim-improv}. By construction, $d[X_1;X_2]=0$.
From \Cref{lem:100pc}, $d[X_1;U_H] = d[X_2; U_H] = 0$. Also from $\tau$-minimization we have $\tau[X_1;X_2] \leq \tau[X^0_2;X^0_1]$. Using this and the Ruzsa triangle inequality we can conclude.
\end{proof} | 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, htau_min, hdist⟩ := tau_minimizer_exists_rdist_eq_zero p
obtain ⟨H, U, hU, hH_unif, hdistX₁, hdistX₂⟩ := exists_isUniform_of_rdist_eq_zero hX₁ hX₂ hdist
refine ⟨AddSubgroup.toZModSubmodule 2 H, Ω', inferInstance, U, inferInstance, hU, hH_unif , ?_⟩
have h : τ[X₁ # X₂ | p] ≤ τ[p.X₀₂ # p.X₀₁ | p] := is_tau_min p htau_min p.hmeas2 p.hmeas1
rw [tau, tau, hpη] at h
norm_num at h
have : d[p.X₀₁ # p.X₀₂] = d[p.X₀₂ # p.X₀₁] := rdist_symm
have : d[p.X₀₁ # U] ≤ d[p.X₀₁ # X₁] + d[X₁ # U] := rdist_triangle p.hmeas1 hX₁ hU
have : d[p.X₀₂ # U] ≤ d[p.X₀₂ # X₂] + d[X₂ # U] := rdist_triangle p.hmeas2 hX₂ hU
linarith
/-- `entropic_PFR_conjecture_improv'`: For two $G$-valued random variables $X^0_1, X^0_2$, there is
some subgroup $H \leq G$ such that $d[X^0_1;U_H] + d[X^0_2;U_H] \le 10 d[X^0_1;X^0_2]$., and
d[X^0_1; U_H] and d[X^0_2; U_H] are at most 5/2 * d[X^0_1;X^0_2] -/ | 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 $S = \sum_{i=1}^m X_i$.
Note that for each $j \in \Z/m\Z$, the sum $Q_j$ from~\eqref{pqr-defs} above has the same distribution as $S$.
By \Cref{klm-1} we have
\begin{align*}
\bbH[W] = \bbH[\sum_{j \in \Z/m\Z} Q_j] & \leq \bbH[S] + \sum_{j=2}^m (\bbH[Q_1+Q_j] - \bbH[S]) \\ & = \bbH[S] + (m-1) d[S;-S].
\end{align*}
By \Cref{multidist-ruzsa-IV}, we have
\begin{equation}
\label{eq:s-bound}
d[S; -S] \leq 2 k
\end{equation}
and hence
\[
\bbH[W] \leq 2 k (m-1) + \bbH[S].
\]
From \Cref{multidist-def} we have
\begin{equation}
\label{eq:ent-s}
\bbH[S] = k + \frac1m \sum_{i=1}^m \bbH[X_i],
\end{equation}
and the claim follows.
\end{proof} | /-- 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{klm-1} one has
\begin{align*}
\bbH[Z_2] &\leq \sum_{i=2}^{m-1} \bbH[Q_1 + i Q_i] - (m-2) \bbH[S].
\end{align*}
Using \Cref{sum-dilate-II} and~\eqref{eq:s-bound} we get
\begin{align*}
\bbH[Z_2]
&\leq \bbH[S] + 4m (m-2) d[S;-S] \\
&\leq \bbH[S] + 8m (m-2) k.
\end{align*}
Applying~\eqref{eq:ent-s} gives the claim.
\end{proof} | /-- 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:100pc-self,ruzsa-triangle, ruzsa-nonneg}\leanok
Using \Cref{ruzsa-triangle} and \Cref{ruzsa-nonneg} we have $d[X_1;X_1]=0$, hence by \Cref{lem:100pc-self} $d[X_1;U_H]=0$ for some subgroup $H$. By \Cref{ruzsa-triangle} and \Cref{ruzsa-nonneg} again we also have $d[X_2;U_H]$ as required.
\end{proof} | 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 h' : d[X # X] = 0 := by
apply le_antisymm _ (rdist_nonneg hX hX)
calc
d[X # X] ≤ d[X # X'] + d[X' # X] := rdist_triangle hX hX' hX
_ = 0 := by rw [hdist, rdist_symm, hdist, zero_add]
rcases exists_isUniform_of_rdist_self_eq_zero hX h' with ⟨H, U, hmeas, hunif, hd⟩
refine ⟨H, U, hmeas, hunif, hd, ?_⟩
apply le_antisymm _ (rdist_nonneg hX' hmeas)
calc
d[X' # U] ≤ d[X' # X] + d[X # U] := rdist_triangle hX' hX hmeas
_ = 0 := by rw [hd, rdist_symm, hdist, zero_add] | 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}\leanok
Take $H$ to be the symmetry group of $X$, which is a group by \Cref{sym-group}. From \Cref{sym-zero}, $X-x_0$ is uniform on $H$, and $d[X ;X-x_0] = d[X ;X] \leq 0$, and the claim follows.
\end{proof} | /-- 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 that `0` is in its support.
obtain ⟨x₀, h₀⟩ : ∃ x₀, ℙ (X⁻¹' {x₀}) ≠ 0 := by
by_contra! h
have A a : (ℙ : Measure Ω).map X {a} = 0 := by
rw [Measure.map_apply hX .of_discrete]
exact h _
have B : (ℙ : Measure Ω).map X = 0 := by
rw [← Measure.sum_smul_dirac (μ := (ℙ : Measure Ω).map X)]
simp [A]
have : IsProbabilityMeasure ((ℙ : Measure Ω).map X) :=
isProbabilityMeasure_map hX.aemeasurable
exact IsProbabilityMeasure.ne_zero _ B
refine ⟨symmGroup X hX, fun ω ↦ X ω - x₀, hX.sub_const _,
isUniform_sub_const_of_rdist_eq_zero hX hdist h₀, ?_⟩
simp_rw [sub_eq_add_neg]
suffices d[X # X + fun _ ↦ -x₀] = 0 by convert this
rw [rdist_add_const hX hX]
exact hdist
/-- If $d[X_1;X_2]=0$, then there exists a subgroup $H \leq G$ such that
$d[X_1;U_H] = d[X_2;U_H] = 0$. Follows from the preceding claim by the triangle inequality. -/ | 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-upper}.
\end{proof} | /-- 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' ‹_› ‹_› ‹_› ‹_› ‹_›
have v3 := condRuzsaDist_of_sums_ge p X₁ X₂ X₁' X₂' ‹_› ‹_› ‹_› (by fun_prop) (by aesop)
have v4 := (mul_le_mul_left p.hη).2 (diff_rdist_le_1 p X₁ X₂ X₁' X₂' ‹_› ‹_› ‹_› ‹_›)
have v5 := (mul_le_mul_left p.hη).2 (diff_rdist_le_2 p X₁ X₂ X₁' X₂' ‹_› ‹_› ‹_› ‹_›)
have v6 := (mul_le_mul_left p.hη).2 (diff_rdist_le_3 p X₁ X₂ X₁' X₂' ‹_› ‹_› ‹_› ‹_›)
have v7 := (mul_le_mul_left p.hη).2 (diff_rdist_le_4 p X₁ X₂ X₁' X₂' ‹_› ‹_› ‹_› ‹_›)
simp only [inv_eq_one_div] at *
linarith [v1, v2, v3, v4, v5, v6, v7]
include hX₁ hX₂ hX₁' hX₂' h₁ h₂ h_min in
/--
$$\mathbb{H}[X_1+X_2+\tilde X_1+\tilde X_2] \le \tfrac{1}{2} \mathbb{H}[X_1]+\tfrac{1}{2} \mathbb{H}[X_2] + (2 + \eta) k - I_1.$$
-/ | 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}{4} (d[X_1;X_2] + 2d[X_1;X_3] + d[X_2;X_4])\\
&\qquad \qquad + \tfrac{1}{4} (d[X_1|X_1+X_3;X_2|X_2+X_4] - d[X_3|X_3+X_4; X_1|X_1+X_2])\\
&\qquad \qquad + \tfrac{1}{8} (\bbH[X_1+X_2] - \bbH[X_3+X_4] + \bbH[X_2] - \bbH[X_3]\\
&\qquad \qquad \qquad + \bbH[X_2|X_2+X_4] - \bbH[X_1|X_1+X_3]).
\end{align*}
\end{lemma}
\begin{proof}\uses{cond-dist-fact, first-useful, cor-fibre}\leanok
On the one hand, by \Cref{cond-dist-fact} and two applications of \Cref{first-useful} we have
\begin{align*}
&d[Y;X_1+X_2|X_1 + X_3, S] \\
&\quad \leq d[Y;X_1+X_2|S] + \tfrac{1}{2} \bbI[X_1 + X_2 : X_1 + X_3|S] \\
&\quad \leq d[Y;X_1+X_2]\\
&\qquad + \tfrac{1}{2} (d[X_1+X_2;X_3+X_4] + \bbI[X_1 + X_2 : X_1 + X_3|S])\\
&\qquad + \tfrac{1}{4} (\bbH[X_1+X_2] - \bbH[X_3+X_4])\\
&\quad \leq d[Y;X_1] \\
&\qquad + \tfrac{1}{2} (d[X_1;X_2] + d[X_1+X_2;X_3+X_4] + \bbI[X_1 + X_2 : X_1 + X_3|S])\\
&\qquad + \tfrac{1}{4} (\bbH[X_1+X_2] - \bbH[X_3+X_4] + \bbH[X_2] - \bbH[X_1]).
\end{align*}
From \Cref{cor-fibre} (with $Y_1,Y_2,Y_3,Y_4$ set equal to $X_3, X_1, X_4, X_2$ respectively) one has
$$ d[X_3+X_4; X_1+X_2] + d[X_3|X_3+X_4; X_1|X_1+X_2] $$
$$ + \bbI[X_3 + X_1 : X_1 + X_2|S] = d[X_3;X_1] + d[X_4;X_2].$$
Rearranging the mutual information and Ruzsa distances slightly, we conclude that
\begin{align*}
&d[Y;X_1+X_2|X_1 + X_3, S] \\
&\quad \leq d[Y;X_1] \\
&\qquad + \tfrac{1}{2} (d[X_1;X_2] + d[X_1;X_3] + d[X_2;X_4] - d[X_3|X_3+X_4; X_1|X_1+X_2])\\
&\qquad + \tfrac{1}{4} (\bbH[X_1+X_2] - \bbH[X_3+X_4] + \bbH[X_2] - \bbH[X_1]).
\end{align*}
On the other hand, $(X_1+X_2|X_1 + X_3, S)$ has an identical distribution to the independent sum of $(X_1|X_1+X_3)$ and $(X_2|X_2+X_4)$. We may therefore apply \Cref{first-useful} to conditioned variables $(X_1|X_1+X_3=s)$ and $(X_2|X_2+X_4=t)$ and average in $s,t$ to
obtain the alternative bound
\begin{align*}
& d[Y;X_1+X_2|X_1 + X_3, S] \\
&\quad \leq d[Y;X_1|X_1+X_3] + \tfrac{1}{2} d[X_1|X_1+X_3; X_2|X_2+X_4] \\
&\qquad + \tfrac{1}{4} (\bbH[X_2|X_2+X_4] - \bbH[X_1|X_1+X_3]) \\
&\quad \leq d[Y;X_1] \\
&\qquad + \tfrac{1}{2} (d[X_1;X_3] + d[X_1|X_1+X_3;X_2|X_2+X_4])\\
&\qquad + \tfrac{1}{4} (\bbH[X_2|X_2+X_4] - \bbH[X_1|X_1+X_3] + \bbH[X_1] - \bbH[X_3]).
\end{align*}
If one takes the arithmetic mean of these two bounds and simplifies using \Cref{cor-fibre}, one obtains the claim.
\end{proof} | 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 Z₁ Z₂ Z₃ Z₄ hZ₁ hZ₂ hZ₃ hZ₄ h_indep
have I2 := gen_ineq_aux2 Y hY Z₁ Z₂ Z₃ Z₄ hZ₁ hZ₂ hZ₃ hZ₄ h_indep
linarith
include hY hZ₁ hZ₂ hZ₃ hZ₄ h_indep in | 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, $|H| = |H_0| |H_1|$.
\end{lemma}
\begin{proof}\uses{hb-thm}\leanok We can take $H_0$ to be the projection of $H$ to $G$, and $H_1$ to be the slice $H_1 := \{ y: (0,y) \in H \}$. One can construct $\phi$ on $H_0$ one generator at a time by the greedy algorithm, and then extend to $G$ by \Cref{hb-thm}. The cardinality bound is clear from direct counting.
\end{proof} | 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 | 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 $G$, but then the extension can be constructed by hand.
\end{proof} | 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 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, $|H| = |H_0| |H_1|$. -/ | 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}
\begin{proof}\uses{goursat, pfr_aux-improv}\leanok
Consider the graph $A \subset G \times G'$ defined by
$$ A := \{ (x,f(x)): x \in G \}.$$
Clearly, $|A| = |G|$. By hypothesis, we have
$$ A+A \subset \{ (x,f(x)+s): x \in G, s \in S\}$$
and hence $|A+A| \leq |S| |A|$. Applying \Cref{pfr-9-aux'}, we may find a
subspace $H \subset G \times G'$ such that $|H|/ |A| \in [|S|^{-8}, |S|^{8}]$
and $A$ is covered by $c + H$ with $|c| \le |S|^5|A|^{1/2} / |H|^{1/2}$. If
we let $H_0, H_1$ be as in \Cref{goursat}, this implies on taking projections
that $G$ is covered by at most $|c|$ translates of $H_0$. This implies that
$$ |c| |H_0| \geq |G|;$$
since $|H_0| |H_1| = |H|$, we conclude that
$$ |H_1| \leq |c| |H|/|G| = |c| |H|/|A|.$$
By hypothesis, $A$ is covered by at most $|c|$ translates of $H$, and hence
by at most $|c| |H_1|$ translates of $\{ (x,\phi(x)): x \in G \}$. As $\phi$
is a homomorphism, each such translate can be written in the form $\{
(x,\phi(x)+d): x \in G \}$ for some $d \in G'$. Since
$$
|c| |H_1| \le |c|^2 \frac{|H|}{|A|} \le \left(|S|^5 \frac{|A|^{1/2}}{|H|^{1/2}}\right)^2 \frac{|H|}{|A|}
= |S|^{10},
$$
the result follows.
\end{proof} | 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.card S * Nat.card A := by
let B := A - {0}×ˢS
have hAB : A + A ⊆ B := by
intro x hx
obtain ⟨a, ha, a', ha', haa'⟩ := Set.mem_add.mp hx
simp only [mem_graphOn, A] at ha ha'
rw [Set.mem_sub]
refine ⟨(x.1, f x.1), ?_, (0, f (a.1 + a'.1) - f a.1 - f a'.1), ?_⟩
· simp [A]
· simp only [singleton_prod, mem_image, Prod.mk.injEq, true_and,
exists_eq_right, Prod.mk_sub_mk, sub_zero]
exact ⟨hS a.1 a'.1,
by rw [← Prod.fst_add, ha.2, ha'.2, sub_sub, ← Prod.snd_add, haa', sub_sub_self]⟩
have hB_card : Nat.card B ≤ Nat.card S * Nat.card A :=
natCard_sub_le.trans_eq $ by simp only [mul_comm, Set.card_singleton_prod]
norm_cast
exact (Nat.card_mono (toFinite B) hAB).trans hB_card
have hA_nonempty : A.Nonempty := by simp [A]
obtain ⟨H, c, hcS, -, -, hAcH⟩ := better_PFR_conjecture_aux hA_nonempty hA_le
have : 0 < Nat.card c := by
have : c.Nonempty := by
by_contra! H
simp only [H, empty_add, subset_empty_iff] at hAcH
simp [hAcH] at hA_nonempty
exact this.natCard_pos c.toFinite
obtain ⟨H₀, H₁, φ, hH₀₁, hH_card⟩ := goursat H
have hG_card_le : Nat.card G ≤ Nat.card c * Nat.card H₀ := by
let c' := Prod.fst '' c
have hc'_card : Nat.card c' ≤ Nat.card c := Nat.card_image_le (toFinite c)
have h_fstH : Prod.fst '' (H : Set (G × G')) = H₀:= by
ext x; simpa [hH₀₁] using fun _ ↦ ⟨φ x, by simp⟩
have hG_cover : (univ : Set G) = c' + (H₀:Set G) := by
apply (eq_univ_of_forall (fun g ↦ ?_)).symm
have := image_subset Prod.fst hAcH
rw [← AddHom.coe_fst, Set.image_add, AddHom.coe_fst, image_fst_graphOn] at this
rw [← h_fstH]
exact this (mem_univ g)
apply_fun Nat.card at hG_cover
rw [Nat.card_coe_set_eq, Set.ncard_univ] at hG_cover
rw [hG_cover]
calc
Nat.card (c' + (H₀ : Set G)) ≤ Nat.card c' * Nat.card H₀ := natCard_add_le
_ ≤ Nat.card c * Nat.card H₀ := by gcongr
have : (Nat.card H₁ : ℝ) ≤ (Nat.card H / Nat.card A) * Nat.card c := by calc
(Nat.card H₁ : ℝ) = (Nat.card H : ℝ) / Nat.card H₀ := by field_simp [hH_card, mul_comm]
_ ≤ (Nat.card H : ℝ) / (Nat.card G / Nat.card c) := by
gcongr
rw [div_le_iff₀' (by positivity)]
exact_mod_cast hG_card_le
_ = (Nat.card H / Nat.card G : ℝ) * Nat.card c := by field_simp
_ = (Nat.card H / Nat.card A) * Nat.card c := by congr; simp [-Nat.card_eq_fintype_card, A]
let T := (fun p ↦ p.2 - φ p.1) '' (c + {0} ×ˢ (H₁: Set G'))
have :=
calc
A ⊆ c + H := hAcH
_ ⊆ c + (({0} ×ˢ (H₁ : Set G')) + {(x, φ x) | x : G}) := by
gcongr
rintro ⟨g, g'⟩ hg
simp only [SetLike.mem_coe, hH₀₁] at hg
refine ⟨(0, g' - φ g), ?_, (g, φ g), ?_⟩
· simp only [singleton_prod, mem_image, SetLike.mem_coe,
Prod.mk.injEq, true_and, exists_eq_right, hg.2]
· simp only [mem_setOf_eq, Prod.mk.injEq, exists_eq_left, Prod.mk_add_mk, zero_add, true_and,
sub_add_cancel]
_ = ⋃ (a ∈ T), {(x, a + φ x) | x : G} := by
rw [← add_assoc, ← vadd_eq_add, ← Set.iUnion_vadd_set, Set.biUnion_image]
congr! 3 with a
rw [← range, ← range, ← graphOn_univ_eq_range, ← graphOn_univ_eq_range, vadd_graphOn_univ]
refine ⟨φ, T, ?_, ?_⟩
· have : (Nat.card T : ℝ) ≤ (Nat.card S : ℝ) ^ (10 : ℝ) := by calc
(Nat.card T : ℝ) ≤ Nat.card (c + {(0 : G)} ×ˢ (H₁ : Set G')) := by
norm_cast; apply Nat.card_image_le (toFinite _)
_ ≤ Nat.card c * Nat.card H₁ := by
norm_cast
apply natCard_add_le.trans
rw [Set.card_singleton_prod] ; rfl
_ ≤ Nat.card c * ((Nat.card H / Nat.card A) * Nat.card c) := by gcongr
_ = Nat.card c ^ 2 * (Nat.card H / Nat.card A) := by ring
_ ≤ (Nat.card S ^ 5 * Nat.card A ^ (1 / 2 : ℝ) * Nat.card H ^ (-1 / 2 : ℝ)) ^ 2
* (Nat.card H / Nat.card A) := by gcongr
_ = (Nat.card S : ℝ) ^ (10 : ℝ) := by
rw [← Real.rpow_two, div_eq_mul_inv, div_eq_mul_inv, div_eq_mul_inv]
have : 0 < Nat.card S := by
have : S.Nonempty := ⟨f (0 + 0) - f 0 - f 0, hS 0 0⟩
exact this.natCard_pos S.toFinite
have : 0 < Nat.card A := hA_nonempty.natCard_pos A.toFinite
have : 0 < Nat.card H := H.nonempty.natCard_pos $ toFinite _
simp_rw [← Real.rpow_natCast]
rpow_ring
norm_num
exact_mod_cast this
· intro g
specialize this (⟨g, by simp⟩ : (g, f g) ∈ A)
simp only [mem_iUnion, mem_setOf_eq, Prod.mk.injEq, exists_eq_left] at this
obtain ⟨t, ht, h⟩ := this
rw [← h]
convert ht
abel | 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-large,sym-group-def,uniform-def}\leanok The law of $X-x_0$ is invariant under $\mathrm{Sym}[X]$, non-zero at the origin, and supported on $\mathrm{Sym}[X]$, giving the claim.
\end{proof} | 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 : G), z ∈ symmGroup X hX →
ℙ ((fun ω ↦ X ω - x₀) ⁻¹' {z}) = ℙ ((fun ω ↦ X ω - x₀) ⁻¹' {0}) := by
intro z hz
have : X ⁻¹' {x₀ + z} = (fun ω ↦ X ω - z) ⁻¹' {x₀} := by simp [B, add_comm]
simp_rw [B, add_zero, this]
have Z := (mem_symmGroup hX).1 (AddSubgroup.neg_mem (symmGroup X hX) hz)
simp [← sub_eq_add_neg] at Z
exact Z.symm.measure_mem_eq .of_discrete
intro x hx y hy
rw [A x hx, A y hy]
measure_preimage_compl := by
apply (measure_preimage_eq_zero_iff_of_countable (Set.to_countable _)).2
intro x hx
contrapose! hx
have B : (fun ω ↦ X ω - x₀) ⁻¹' {x} = X ⁻¹' {x₀ + x} := by
ext w; simp [sub_eq_iff_eq_add']
rw [B] at hx
simpa using sub_mem_symmGroup hX hdist hx hx₀ | 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 groups $G_0,\dots,G_m$, and for each $d=0,\dots,m$, let $\pi_d : G_m \to G_d$ be the homomorphism from $G_m$ to $G_d$ arising from this sequence by composition (so for instance $\pi_m$ is the identity homomorphism and $\pi_0$ is the zero homomorphism).
Let $X_{[m]} = (X_i)_{1 \leq i \leq m}$ be a jointly independent tuple of $G_m$-valued random variables.
Then
\begin{equation}
\begin{split}
D[ X_{[m]} ] &= \sum_{d=1}^m D[ \pi_d(X_{[m]}) \,|\, \pi_{d-1}(X_{[m]})] \\
&\quad + \sum_{d=1}^{m-1} \bbI[ \sum_i X_i : \pi_d(X_{[m]}) \; \big| \; \pi_d\big(\sum_i X_i\big), \pi_{d-1}(X_{[m]}) ].
\end{split}\label{chain-eq-cond'}
\end{equation}
In particular, by \Cref{conditional-nonneg},
\begin{align}\nonumber
D[ X_{[m]} ] \geq & \sum_{d=1}^m D[ \pi_d(X_{[m]})|\pi_{d-1}(X_{[m]}) ] \\
& + \bbI[ \sum_i X_i : \pi_1(X_{[m]}) \; \big| \; \pi_1\bigl(\sum_i X_i\bigr) ].\label{chain-eq-cond''}
\end{align}
\end{lemma}
\begin{proof}\uses{multidist-chain-rule-cond, conditional-nonneg}\leanok
From \Cref{multidist-chain-rule-cond} (taking $Y_{[m]} = \pi_{d-1}(X_{[m]})$ and $\pi = \pi_d$ there, and noting that $\pi_d(X_{[m]})$ determines $Y_{[m]}$) we have
\begin{align*}
D[ X_{[m]} \,|\, \pi_{d-1}(X_{[m]}) ] &= D[ X_{[m]} \,|\, \pi_d(X_{[m]}) ] + D[ \pi_d(X_{[m]})\,|\,\pi_{d-1}(X_{[m]}) ] \\
&\quad + \bbI[ \sum_{i=1}^m X_i : \pi_d(X_{[m]}) \; \big| \; \pi_d\bigl(\sum_{i=1}^m X_i\bigr), \pi_{d-1}(X_{[m]}) ]
\end{align*}
for $d=1,\dots,m$. The claim follows by telescoping series, noting that $D[X_{[m]} | \pi_0(X_{[m]})] = D[X_{[m]}]$ and that $\pi_m(X_{[m]})=X_{[m]}$ (and also $\pi_m( \sum_i X_i ) = \sum_i X_i$).
\end{proof} | 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.castSucc = (φ i) ∘ (π i.succ))
{Ω : Type*} [hΩ : MeasureSpace Ω] {X : Fin m → Ω → (G m)}
(hX : ∀ i, Measurable (X i)) (h_indep : iIndepFun X) (n : Fin (m + 1)) :
D[X | fun i ↦ (π 0) ∘ X i; fun _ ↦ hΩ] = D[X | fun i ↦ (π n) ∘ X i; fun _ ↦ hΩ]
+ ∑ d ∈ Finset.Iio n, (D[fun i ↦ (π (d+1)) ∘ X i | fun i ↦ (π d) ∘ X i; fun _ ↦ hΩ]
+ I[∑ i, X i : fun ω ↦ (fun i ↦ (π (d+1)) (X i ω)) |
⟨(π (d+1)) ∘ ∑ i, X i, fun ω ↦ (fun i ↦ (π d) (X i ω))⟩]) := by
set S := ∑ i, X i
set motive := fun n:Fin (m + 1) ↦ D[X | fun i ↦ (π 0) ∘ X i; fun _ ↦ hΩ]
= D[X | fun i ↦ (π n) ∘ X i; fun _ ↦ hΩ]
+ ∑ d ∈ Finset.Iio n, (D[fun i ↦ (π (d+1)) ∘ X i | fun i ↦ (π d) ∘ X i; fun _ ↦ hΩ]
+ I[S : fun ω ↦ (fun i ↦ (π (d+1)) (X i ω)) |
⟨(π (d+1)) ∘ S, fun ω ↦ (fun i ↦ (π d) (X i ω))⟩])
have zero : motive 0 := by
have : (Finset.Iio 0 : Finset (Fin (m + 1))) = ∅ := rfl
simp [motive, this]
have succ : (n : Fin m) → motive n.castSucc → motive n.succ := by
intro n hn
dsimp [motive] at hn ⊢
have h2 : n.castSucc ∈ Finset.Iio n.succ := by
simp only [Nat.succ_eq_add_one, Finset.mem_Iio, Fin.castSucc_lt_succ_iff, le_refl]
rw [hn, ← Finset.add_sum_erase _ _ h2, Iio_of_succ_eq_Iic_of_castSucc, Finset.Iic_erase, ← add_assoc, ← add_assoc, Fin.coeSucc_eq_succ]
congr 1
convert cond_multiDist_chainRule (X := X) (Y := fun i ↦ ⇑(π n.castSucc) ∘ X i) (π n.succ) hX ?_ ?_
. set g : G n.succ → G n.succ × G n.castSucc := fun x ↦ ⟨x, ⇑(φ n) x⟩
convert (condMultiDist_of_inj (f := g) (fun _ ↦ hΩ) X (fun i ↦ ⇑(π n.succ) ∘ X i) _).symm using 3 with i
. ext ω
. dsimp [g, prod]
rw [hcomp n]
simp [g, prod]
intro x x' h
simp [g] at h
exact h.1
. intro _
exact Measurable.comp .of_discrete (hX _)
set g : (G m) → (G m) × (G n.castSucc) := fun x ↦ ⟨x, ⇑(π n.castSucc) x⟩
convert iIndepFun.comp h_indep (fun _ ↦ g) _
intro _
exact .of_discrete
exact Fin.induction zero succ n | 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 groups $G_0,\dots,G_m$, and for each $d=0,\dots,m$, let $\pi_d : G_m \to G_d$ be the homomorphism from $G_m$ to $G_d$ arising from this sequence by composition (so for instance $\pi_m$ is the identity homomorphism and $\pi_0$ is the zero homomorphism).
Let $X_{[m]} = (X_i)_{1 \leq i \leq m}$ be a jointly independent tuple of $G_m$-valued random variables.
Then
\begin{equation}
\begin{split}
D[ X_{[m]} ] &= \sum_{d=1}^m D[ \pi_d(X_{[m]}) \,|\, \pi_{d-1}(X_{[m]})] \\
&\quad + \sum_{d=1}^{m-1} \bbI[ \sum_i X_i : \pi_d(X_{[m]}) \; \big| \; \pi_d\big(\sum_i X_i\big), \pi_{d-1}(X_{[m]}) ].
\end{split}\label{chain-eq-cond'}
\end{equation}
In particular, by \Cref{conditional-nonneg},
\begin{align}\nonumber
D[ X_{[m]} ] \geq & \sum_{d=1}^m D[ \pi_d(X_{[m]})|\pi_{d-1}(X_{[m]}) ] \\
& + \bbI[ \sum_i X_i : \pi_1(X_{[m]}) \; \big| \; \pi_1\bigl(\sum_i X_i\bigr) ].\label{chain-eq-cond''}
\end{align}
\end{lemma}
\begin{proof}\uses{multidist-chain-rule-cond, conditional-nonneg}\leanok
From \Cref{multidist-chain-rule-cond} (taking $Y_{[m]} = \pi_{d-1}(X_{[m]})$ and $\pi = \pi_d$ there, and noting that $\pi_d(X_{[m]})$ determines $Y_{[m]}$) we have
\begin{align*}
D[ X_{[m]} \,|\, \pi_{d-1}(X_{[m]}) ] &= D[ X_{[m]} \,|\, \pi_d(X_{[m]}) ] + D[ \pi_d(X_{[m]})\,|\,\pi_{d-1}(X_{[m]}) ] \\
&\quad + \bbI[ \sum_{i=1}^m X_i : \pi_d(X_{[m]}) \; \big| \; \pi_d\bigl(\sum_{i=1}^m X_i\bigr), \pi_{d-1}(X_{[m]}) ]
\end{align*}
for $d=1,\dots,m$. The claim follows by telescoping series, noting that $D[X_{[m]} | \pi_0(X_{[m]})] = D[X_{[m]}]$ and that $\pi_m(X_{[m]})=X_{[m]}$ (and also $\pi_m( \sum_i X_i ) = \sum_i X_i$).
\end{proof} | 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) (hcomp : ∀ i : Fin m, π i.castSucc = (φ i) ∘ (π i.succ))
{Ω : Type*} [hΩ : MeasureSpace Ω] {X : Fin m → Ω → (G m)}
(hX : ∀ i, Measurable (X i)) (h_indep : iIndepFun X) :
D[X; fun _ ↦ hΩ] ≥
∑ d : Fin m, D[fun i ↦ (π (d.succ)) ∘ X i | fun i ↦ (π d.castSucc) ∘ X i; fun _ ↦ hΩ]
+ I[∑ i : Fin m, X i : fun ω i ↦ (π 1) (X i ω)| ⇑(π 1) ∘ ∑ i : Fin m, X i] := by
have : IsProbabilityMeasure (ℙ : Measure Ω) := h_indep.isProbabilityMeasure
calc
_ = D[X | fun i ↦ ⇑(π 0) ∘ X i ; fun _x ↦ hΩ] := by
rw [hπ0]
convert (condMultiDist_of_const (fun _ ↦ (0: G 0)) X).symm
_ = D[X | fun i ↦ ⇑(π m) ∘ X i ; fun _ ↦ hΩ] +
∑ d ∈ Finset.Iio (m : Fin (m + 1)),
(D[fun i ↦ ⇑(π (d + 1)) ∘ X i | fun i ↦ π d ∘ X i ; fun _ ↦ hΩ] +
I[∑ i : Fin m, X i : fun ω i ↦ (π (d + 1)) (X i ω)|⟨⇑(π (d + 1)) ∘ ∑ i : Fin m, X i, fun ω i ↦ (π d) (X i ω)⟩]) :=
iter_multiDist_chainRule hcomp hX h_indep (m : Fin (m + 1))
_ ≥ ∑ d ∈ Finset.Iio (m : Fin (m + 1)),
(D[fun i ↦ ⇑(π (d + 1)) ∘ X i | fun i ↦ π d ∘ X i ; fun _ ↦ hΩ] +
I[∑ i : Fin m, X i : fun ω i ↦ (π (d + 1)) (X i ω)|⟨⇑(π (d + 1)) ∘ ∑ i : Fin m, X i, fun ω i ↦ (π d) (X i ω)⟩]) := by
apply le_add_of_nonneg_left (condMultiDist_nonneg _ (fun _ => this) X _ hX)
_ = ∑ d : Fin m,
(D[fun i ↦ ⇑(π (d.succ)) ∘ X i | fun i ↦ ⇑(π d.castSucc) ∘ X i ; fun _ ↦ hΩ] +
I[∑ i : Fin m, X i : fun ω i ↦ π d.succ (X i ω)|⟨π d.succ ∘ ∑ i : Fin m, X i, fun ω i ↦ π d (X i ω)⟩]) := by
rw [sum_of_iio_last]
congr with d
rw [Fin.coeSucc_eq_succ, Fin.coe_eq_castSucc]
_ ≥ _ := by
rw [Finset.sum_add_distrib]
gcongr
have : NeZero m := ⟨hm.ne'⟩
let f (i : Fin m) :=
I[∑ i', X i' : fun ω i' ↦ π i.succ (X i' ω)|
⟨π i.succ ∘ ∑ i', X i', fun ω i' ↦ π i (X i' ω)⟩]
have hf i : 0 ≤ f i := condMutualInfo_nonneg (by fun_prop) (by fun_prop)
let F (x : G 1) : G 1 × (Fin m → G 0) := (x, fun _ ↦ 0)
have hF : Injective F := fun x y ↦ congr_arg Prod.fst
calc
I[∑ i, X i : fun ω i ↦ π 1 (X i ω) | π 1 ∘ ∑ i, X i]
= I[∑ i, X i : fun ω i ↦ π 1 (X i ω) | F ∘ π 1 ∘ ∑ i, X i] := by
exact (condMutualInfo_of_inj (by fun_prop) (by fun_prop) (by fun_prop) _ hF).symm
_ = f 0 := ?_
_ ≤ ∑ j, f j := Finset.single_le_sum (f := f) (fun _ _ ↦ hf _) (Finset.mem_univ _)
. simp [f]
simp [hπ0, AddMonoidHom.zero_apply, comp_apply, Finset.sum_apply, _root_.map_sum, F,
Fin.succ_zero_eq_one']
congr
any_goals congr!
any_goals simp [Fin.succ_zero_eq_one']
simp [Function.comp_def]
sorry
/-- Let `G` be an abelian group and let `m ≥ 2`. Suppose that `X_{i,j}`, `1 ≤ i, j ≤ m`, are
independent `G`-valued random variables. Then
`I[(∑ i, X_{i,j})_{j=1}^m : (∑ j, X_{i,j})_{i=1}^m | ∑ i j, X_{i,j}]`
is less than
`∑_{j=1}^{m-1} (D[(X_{i, j})_{i=1}^m] - D[(X_{i, j})_{i = 1}^m | (X_{i,j} + ... + X_{i,m})_{i=1}^m])`
`+ D[(X_{i,m})_{i=1}^m] - D[(∑ j, X_{i,j})_{i=1}^m],`
where all the multidistances here involve the indexing set `{1, ..., m}`. -/ | 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}\leanok
From \Cref{submodularity} we have
$$ \bbH[X, X + Y+ Z] + \bbH[Z, X + Y+ Z] \geq \bbH[X, Z, X + Y+ Z] + \bbH[X + Y+ Z].$$
However, using Lemmas \ref{add-entropy}, \ref{relabeled-entropy} repeatedly we have $\bbH[X, X + Y+ Z] = \bbH[X, Y+ Z] = \bbH[X] + \bbH[Y+ Z]$, $\bbH[Z, X + Y + Z] = \bbH[Z, X + Y] = \bbH[Z] + \bbH[X + Y]$ and $\bbH[X, Z, X + Y+ Z] = \bbH[X, Y, Z] = \bbH[X] + \bbH[Y] + \bbH[Z]$. The claim then follows from a calculation.
\end{proof} | /-- 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[Y + Z ; μ] - H[Y ; μ] := by
have : IsProbabilityMeasure μ := h.isProbabilityMeasure
suffices (H[X ; μ] + H[Y ; μ] + H[Z ; μ]) + H[X + Y + Z ; μ]
≤ (H[X ; μ] + H[Y + Z ; μ]) + (H[Z ; μ] + H[X + Y ; μ]) by linarith
have : ∀ (i : Fin 3), Measurable (![X, Y, Z] i) := fun i ↦ by fin_cases i <;> assumption
convert entropy_triple_add_entropy_le μ hX hZ (show Measurable (X + (Y + Z)) by fun_prop)
using 2
· calc
H[X ; μ] + H[Y ; μ] + H[Z ; μ] = H[⟨X, Y⟩ ; μ] + H[Z ; μ] := by
rw [IndepFun.entropy_pair_eq_add hX hY]
convert h.indepFun (show 0 ≠ 1 by decide)
_ = H[⟨⟨X, Y⟩, Z⟩ ; μ] := by
rw [IndepFun.entropy_pair_eq_add (hX.prodMk hY) hZ]
exact h.indepFun_prodMk this 0 1 2 (by decide) (by decide)
_ = H[⟨X, ⟨Z , X + (Y + Z)⟩⟩ ; μ] := by
apply entropy_of_comp_eq_of_comp μ (by fun_prop) (by fun_prop)
(fun ((x, y), z) ↦ (x, z, x + y + z)) (fun (a, b, c) ↦ ((a, c - a - b), b))
all_goals { funext ω; dsimp [prod]; ext <;> dsimp; abel }
· rw [add_assoc]
· symm
refine (entropy_add_right hX (by fun_prop) _).trans $
IndepFun.entropy_pair_eq_add hX (by fun_prop) ?_
exact h.indepFun_add_right this 0 1 2 (by decide) (by decide)
· rw [eq_comm, ← add_assoc]
refine (entropy_add_right' hZ (by fun_prop) _).trans $
IndepFun.entropy_pair_eq_add hZ (by fun_prop) ?_
exact h.indepFun_add_right this 2 0 1 (by decide) (by decide) | 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{proof}\uses{kv}\leanok This is trivial for $n=0,1$, while the $n=2$ case is \Cref{kv}. Now suppose inductively that $n > 2$, and the claim was already proven for $n-1$. By a further application of \Cref{kv} one has
$$ \bbH\left[X + \sum_{i=1}^n Y_i\right] - \bbH\left[X + \sum_{i=1}^{n-1} Y_i\right] \leq \bbH[X+Y_n] - \bbH[X].$$
By induction hypothesis one has
$$ \bbH\left[X + \sum_{i=1}^{n-1} Y_i\right] - \bbH[X] \leq \sum_{i=1}^{n-1} \bbH[X+Y_i] - \bbH[X].$$
Summing the two inequalities, we obtain the claim.
\end{proof} | 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.induction_on with
| empty => simp
| @insert i s hi IH =>
simp_rw [Finset.sum_insert hi]
have his : i₀ ∉ s := fun h ↦ hs (Finset.mem_insert_of_mem h)
have hii₀ : i ≠ i₀ := fun h ↦ hs (h ▸ Finset.mem_insert_self i s)
let J := Fin 3
let S : J → Finset I := ![s, {i₀}, {i}]
have h_dis : Set.univ.PairwiseDisjoint S := by
intro j _ k _ hjk
change Disjoint (S j) (S k)
fin_cases j <;> fin_cases k <;> try exact (hjk rfl).elim
all_goals
simp_all [Fin.isValue, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.head_cons,
Finset.disjoint_singleton_right, S, his, hi, hjk, hs]
let φ : (j : J) → ((_ : S j) → G) → G
| 0 => fun Ys ↦ ∑ i : s, Ys ⟨i.1, i.2⟩
| 1 => fun Ys ↦ Ys ⟨i₀, by simp [S]⟩
| 2 => fun Ys ↦ Ys ⟨i, by simp [S]⟩
have hφ : (j : J) → Measurable (φ j) := fun j ↦ .of_discrete
have h_ind : iIndepFun ![∑ j ∈ s, Y j, Y i₀, Y i] μ := by
convert iIndepFun.finsets_comp S h_dis h_indep hY φ hφ with j x
fin_cases j <;> simp [φ, (s.sum_attach _).symm]
have measSum : Measurable (∑ j ∈ s, Y j) := by
convert Finset.measurable_sum s (fun j _ ↦ hY j)
simp
have hkv := kaimanovich_vershik h_ind measSum (hY i₀) (hY i)
convert add_le_add (IH his) hkv using 1
· nth_rw 2 [add_comm (Y i₀)]
norm_num
congr 1
rw [add_comm _ (Y i₀), add_comm (Y i), add_assoc]
· ring
/-- If `n ≥ 1` and `X, Y₁, ..., Yₙ`$ are jointly independent `G`-valued random variables,
then `d[Y i₀; μ # ∑ i ∈ s, Y i; μ] ≤ 2 * ∑ i ∈ s, d[Y i₀; μ # Y i; μ]`.-/ | 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, sumset-lower, ruzsa-diff}\leanok
Applying \Cref{klm-1} with all the $Y_i$ replaced by $-Y_i$, and using \Cref{neg-ent} and \Cref{ruz-indep}, we obtain after some rearranging
$$ d[X; \sum_{i=1}^n Y_i] + \frac{1}{2} (\bbH[\sum_{i=1}^n Y_i] - \bbH[X]) \leq \sum_{i=1}^n \left(d[X;Y_i] + \frac{1}{2} (\bbH[Y_i] - \bbH[X])\right).$$
From \Cref{sumset-lower} we have
$$ \bbH[\sum_{i=1}^n Y_i] \geq \bbH[Y_i]$$
for all $i$; subtracting $\bbH[X]$ and averaging, we conclude that
$$ \bbH[\sum_{i=1}^n Y_i] - \bbH[X] \geq \frac{1}{n} \sum_{i=1}^n \bbH[Y_i] - \bbH[X]$$
and thus
$$ d[X; \sum_{i=1}^n Y_i] \leq \sum_{i=1}^n d[X;Y_i] + \frac{n-1}{2n} (\bbH[Y_i] - \bbH[X]).$$
From \Cref{ruzsa-diff} we have
$$ \bbH[Y_i] - \bbH[X] \leq 2 d[X;Y_i].$$
Since $0 \leq \frac{n-1}{2n} \leq \frac{1}{2}$, the claim follows.
\end{proof} | 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 : IsProbabilityMeasure μ := h_indep.isProbabilityMeasure
let φ i : G → G := if i = i₀ then id else - id
have hφ i : Measurable (φ i) := .of_discrete
let Y' i : Ω → G := φ i ∘ Y i
have mnY : (i : I) → Measurable (Y' i) := fun i ↦ (hφ i).comp (hY i)
have h_indep2 : IndepFun (Y i₀) (∑ i ∈ s, Y i) μ :=
iIndepFun.indepFun_finset_sum_of_not_mem h_indep (fun i ↦ hY i) hs |>.symm
have ineq4 : d[Y i₀; μ # ∑ i ∈ s, Y i; μ] + 1/2 * (H[∑ i ∈ s, Y i; μ] - H[Y i₀; μ])
≤ ∑ i ∈ s, (d[Y i₀; μ # Y i; μ] + 1/2 * (H[Y i; μ] - H[Y i₀; μ])) := by
calc
_ = H[Y i₀ - ∑ i ∈ s, Y i ; μ] - H[Y i₀ ; μ] := by
rw [IndepFun.rdist_eq h_indep2 (hY i₀) (by fun_prop)]
ring
_ = H[Y' i₀ + ∑ x ∈ s, Y' x ; μ] - H[Y' i₀ ; μ] := by
simp [Y', φ, sub_eq_add_neg, ← Finset.sum_neg_distrib]
congr! 3 with i hi
simp [ne_of_mem_of_not_mem hi hs, Pi.neg_comp]
_ ≤ ∑ x ∈ s, (H[Y' i₀ + Y' x ; μ] - H[Y' i₀ ; μ]) := kvm_ineq_I hs mnY (h_indep.comp φ hφ)
_ = ∑ i ∈ s, (H[Y i₀ - Y i ; μ] - H[Y i₀ ; μ]) := by
congr! 1 with i hi; simp [Y', φ, ne_of_mem_of_not_mem hi hs, Pi.neg_comp, sub_eq_add_neg]
_ = _ := by
refine Finset.sum_congr rfl fun i hi ↦ ?_
rw [IndepFun.rdist_eq (h_indep.indepFun (ne_of_mem_of_not_mem hi hs).symm) (hY i₀) (hY i)]
ring
replace ineq4 : d[Y i₀; μ # ∑ i ∈ s, Y i; μ] ≤ ∑ i ∈ s, (d[Y i₀; μ # Y i; μ]
+ 1/2 * (H[Y i; μ] - H[Y i₀; μ])) - 1/2 * (H[∑ i ∈ s, Y i; μ] - H[Y i₀; μ]) :=
le_tsub_of_add_le_right ineq4
have ineq5 (j : I) (hj : j ∈ s) : H[Y j ; μ] ≤ H[∑ i ∈ s, Y i; μ] :=
max_entropy_le_entropy_sum hj hY h_indep
have ineq6 : (s.card : ℝ)⁻¹ * ∑ i ∈ s, (H[Y i; μ] - H[Y i₀; μ]) ≤ H[∑ i ∈ s, Y i; μ] - H[Y i₀; μ] := by
rw [inv_mul_le_iff₀ (by exact_mod_cast Finset.card_pos.mpr hs'), ← smul_eq_mul,
Nat.cast_smul_eq_nsmul, ← Finset.sum_const]
refine Finset.sum_le_sum fun i hi ↦ ?_
gcongr
exact ineq5 i hi
have ineq7 : d[Y i₀; μ # ∑ i ∈ s, Y i; μ]
≤ ∑ i ∈ s, (d[Y i₀; μ # Y i; μ] + (s.card - 1) / (2 * s.card) * (H[Y i; μ] - H[Y i₀; μ])) := by
calc
_ ≤ ∑ i ∈ s, (d[Y i₀; μ # Y i; μ] + 1/2 * (H[Y i; μ] - H[Y i₀; μ]))
- 1/2 * (H[∑ i ∈ s, Y i; μ] - H[Y i₀; μ]) := ineq4
_ ≤ ∑ i ∈ s, (d[Y i₀; μ # Y i; μ] + 1/2 * (H[Y i; μ] - H[Y i₀; μ]))
- 1/2 * ((s.card : ℝ)⁻¹ * ∑ i ∈ s, (H[Y i; μ] - H[Y i₀; μ])) := by gcongr
_ = ∑ i ∈ s, (d[Y i₀; μ # Y i; μ] + 1/2 * (H[Y i; μ] - H[Y i₀; μ])
- 1/2 * ((s.card : ℝ)⁻¹ * (H[Y i; μ] - H[Y i₀; μ]))) := by
rw [Finset.mul_sum, Finset.mul_sum, ← Finset.sum_sub_distrib]
_ = ∑ i ∈ s, (d[Y i₀; μ # Y i; μ] + (s.card - 1) / (2 * s.card) * (H[Y i; μ] - H[Y i₀; μ])) := by
refine Finset.sum_congr rfl fun i _ ↦ ?_
rw [add_sub_assoc, ← mul_assoc, ← sub_mul]
field_simp
have ineq8 (i : I) : H[Y i; μ] - H[Y i₀; μ] ≤ 2 * d[Y i₀; μ # Y i; μ] := by
calc
_ ≤ |H[Y i₀ ; μ] - H[Y i ; μ]| := by
rw [← neg_sub]
exact neg_le_abs _
_ ≤ 2 * d[Y i₀; μ # Y i; μ] := diff_ent_le_rdist (hY i₀) (hY i)
calc
_ ≤ ∑ i ∈ s, (d[Y i₀; μ # Y i; μ] + (s.card - 1) / (2 * s.card) * (H[Y i; μ] - H[Y i₀; μ])) :=
ineq7
_ ≤ ∑ i ∈ s, (d[Y i₀; μ # Y i; μ] + (s.card - 1) / s.card * d[Y i₀; μ # Y i; μ]) := by
simp_rw [div_eq_mul_inv, mul_inv, mul_comm (2 : ℝ)⁻¹, mul_assoc]
gcongr ∑ i ∈ s, (d[Y i₀ ; μ # Y i ; μ] + (s.card - 1) * ((s.card : ℝ)⁻¹ * ?_))
· simp only [sub_nonneg, Nat.one_le_cast]
exact Nat.one_le_iff_ne_zero.mpr <| Finset.card_ne_zero.mpr hs'
· exact (inv_mul_le_iff₀ zero_lt_two).mpr (ineq8 _)
_ ≤ ∑ i ∈ s, (d[Y i₀; μ # Y i; μ] + d[Y i₀; μ # Y i; μ]) := by
gcongr ∑ i ∈ s, (d[Y i₀ ; μ # Y i ; μ] + ?_) with i
refine mul_le_of_le_one_left (rdist_nonneg (hY i₀) (hY i)) ?_
exact (div_le_one (Nat.cast_pos.mpr <| Finset.card_pos.mpr hs')).mpr (by simp)
_ = 2 * ∑ i ∈ s, d[Y i₀ ; μ # Y i ; μ] := by
ring_nf
exact (Finset.sum_mul _ _ _).symm | 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
$$d\left[X; \sum_{i=1}^n Y_i\right] \leq d\left[X; Y_1\right] + \frac{1}{2}\left(\bbH\left[ \sum_{i=1}^n Y_i\right] - \bbH[Y_1]\right).$$
\end{lemma}
\begin{proof}\uses{kv, ruz-indep}\leanok
From \Cref{kv} one has
$$ \bbH\left[-X + \sum_{i=1}^n Y_i\right] \leq \bbH[ - X + Y_1 ] + \bbH\left[ \sum_{i=1}^n Y_i \right] - \bbH[Y_1].$$
The claim then follows from \Cref{ruz-indep} and some elementary algebra.
\end{proof} | 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, Y i; μ] - H[Y i₁; μ]) := by
let J := Fin 3
let S : J → Finset I := ![{i₀}, {i₁}, s]
have h_dis : Set.univ.PairwiseDisjoint S := by
intro j _ k _ hjk
change Disjoint (S j) (S k)
fin_cases j <;> fin_cases k <;> try exact (hjk rfl).elim
all_goals
simp_all [Fin.isValue, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.head_cons,
Finset.disjoint_singleton_right, S, hs₀, hs₁, h01, h01.symm]
let φ : (j : J) → ((_ : S j) → G) → G
| 0 => fun Ys ↦ Ys ⟨i₀, by simp [S]⟩
| 1 => fun Ys ↦ Ys ⟨i₁, by simp [S]⟩
| 2 => fun Ys ↦ ∑ i : s, Ys ⟨i.1, i.2⟩
have hφ : (j : J) → Measurable (φ j) := fun j ↦ .of_discrete
have h_indep' : iIndepFun ![Y i₀, Y i₁, ∑ i ∈ s, Y i] μ := by
convert iIndepFun.finsets_comp S h_dis h_indep hY φ hφ with j x
fin_cases j <;> simp [φ, (s.sum_attach _).symm]
exact kvm_ineq_III_aux (hY i₀) (hY i₁) (by fun_prop) h_indep'
open Classical in
/-- Let `X₁, ..., Xₘ` and `Y₁, ..., Yₗ` be tuples of jointly independent random variables (so the
`X`'s and `Y`'s are also independent of each other), and let `f: {1,..., l} → {1,... ,m}` be a
function, then `H[∑ j, Y j] ≤ H[∑ i, X i] + ∑ j, H[Y j - X f(j)] - H[X_{f(j)}]`.-/ | 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
\[
D[X_{[m]}] := \bbH[\sum_{i=1}^m \tilde X_i] - \frac{1}{m} \sum_{i=1}^m \bbH[\tilde X_i],
\]
where the $\tilde X_i$ are independent copies of the $X_i$.
\end{definition} | 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}
D[ X_{[m]} | \pi(X_{[m]}) ] +D[ \pi(X_{[m]}) ] + \bbI[ \sum_{i=1}^m X_i : \pi(X_{[m]}) \; \big| \; \pi\bigl(\sum_{i=1}^m X_i\bigr) ]
\label{chain-eq}
\end{equation}
where $\pi(X_{[m]}) := (\pi(X_i))_{1 \leq i \leq m}$.
\end{lemma}
\begin{proof}\uses{conditional-mutual-alt, relabeled-entropy, chain-rule}\leanok For notational brevity during this proof, write $S := \sum_{i=1}^m X_i$.
From \Cref{conditional-mutual-alt} and \Cref{relabeled-entropy}, noting that $\pi(S)$ is determined both by $S$ and by $\pi(X_{[m]})$, we have
\begin{equation*}
\bbI[S:\pi(X_{[m]})|\pi(S)] = \bbH[S]+\bbH[\pi(X_{[m]})]-\bbH[S,\pi(X_{[m]})]-\bbH[\pi(S)],
\end{equation*}
and by \Cref{chain-rule} the right-hand side is equal to
\begin{equation*}
\bbH[S]-\bbH[S|\pi(X_{[m]})]-\bbH[\pi(S)].
\end{equation*}
Therefore,
\begin{equation}\label{chain-1}
\bbH[S]=\bbH[S|\pi(X_{[m]})]+\bbH[\pi(S)]+\bbI[S:\pi(X_{[m]})|\pi(S)]. \end{equation}
From a further application of \Cref{chain-rule} and \Cref{relabeled-entropy} we have
\begin{equation}\label{chain-2}
\bbH[X_i] = \bbH[X_i \, | \, \pi(X_i)] + \bbH[\pi(X_i)]
\end{equation}
for all $1 \leq i \leq m$. Averaging~\eqref{chain-2} in $i$ and subtracting this from~\eqref{chain-1}, we obtain the claim from \Cref{multidist-def}.
\end{proof} | 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 (X i))
(h_indep : iIndepFun X) :
D[X; fun _ ↦ hΩ] = D[X | fun i ↦ π ∘ X i; fun _ ↦ hΩ]
+ D[fun i ↦ π ∘ X i; fun _ ↦ hΩ]
+ I[∑ i, X i : fun ω ↦ (fun i ↦ π (X i ω)) | π ∘ (∑ i, X i)] := by
have : IsProbabilityMeasure (ℙ : Measure Ω) := h_indep.isProbabilityMeasure
set S := ∑ i, X i
set piX := fun ω ↦ (fun i ↦ π (X i ω))
set avg_HX := (∑ i, H[X i]) / m
set avg_HpiX := (∑ i, H[π ∘ X i])/m
set avg_HXpiX := (∑ i, H[X i | π ∘ X i])/m
have hSmes : Measurable S := by fun_prop
have hpiXmes : Measurable piX := by
rw [measurable_pi_iff]
intro i
exact Measurable.comp .of_discrete (hmes i)
have eq1 : I[S : piX | π ∘ S] = H[S | π ∘ S] + H[piX | π ∘ S] - H[⟨S, piX⟩ | π ∘ S] := by
rw [condMutualInfo_eq hSmes hpiXmes (Measurable.comp .of_discrete hSmes)]
have eq1a : H[S | π ∘ S] = H[S] - H[π ∘ S] :=
condEntropy_comp_self hSmes .of_discrete
have eq1b : H[piX | π ∘ S] = H[piX] - H[π ∘ S] := by
set g := fun (y : Fin m → H) ↦ ∑ i, y i
have : π ∘ S = g ∘ piX := by
ext x
simp only [comp_apply, Finset.sum_apply, _root_.map_sum, S, g, piX]
rw [this]
exact condEntropy_comp_self hpiXmes .of_discrete
have eq1c : H[⟨S, piX⟩ | π ∘ S] = H[⟨S, piX⟩] - H[π ∘ S] := by
set g := fun (x : G × (Fin m → H)) ↦ π x.1
have : π ∘ S = g ∘ ⟨S, piX⟩ := by
ext x
simp only [comp_apply, Finset.sum_apply, _root_.map_sum, S, g, piX]
rw [this]
apply condEntropy_comp_self (Measurable.prodMk hSmes hpiXmes) .of_discrete
have eq2 : H[⟨S, piX⟩] = H[piX] + H[S | piX] := chain_rule _ hSmes hpiXmes
have eq3 : D[X; fun _ ↦ hΩ] = H[S] - avg_HX := multiDist_indep _ _ h_indep
have eq4 : D[X | fun i ↦ π ∘ X i; fun _ ↦ hΩ] = H[S | piX] - avg_HXpiX := by
dsimp [S, piX]
convert condMultiDist_eq (S := H) hmes _ _
. exact Fintype.sum_apply _ _
. intro i
exact Measurable.comp .of_discrete (hmes i)
set g : G → G × H := fun x ↦ ⟨x, π x⟩
change iIndepFun (fun i ↦ g ∘ X i) ℙ
exact h_indep.comp _ fun _ ↦ .of_discrete
have eq5: D[fun i ↦ π ∘ X i; fun _ ↦ hΩ] = H[π ∘ S] - avg_HpiX := by
convert multiDist_indep _ (fun i ↦ π ∘ X i) _
. ext _
simp only [comp_apply, Finset.sum_apply, _root_.map_sum, S]
apply iIndepFun.comp h_indep
exact fun _ ↦ .of_discrete
have eq6: avg_HX = avg_HpiX + avg_HXpiX := by
dsimp [avg_HX, avg_HpiX, avg_HXpiX]
rw [← add_div, ← Finset.sum_add_distrib]
congr with i
rw [condEntropy_comp_self (hmes i) .of_discrete]
abel
linarith only [eq1, eq1a, eq1b, eq1c, eq2, eq3, eq4, eq5, eq6]
/-- Let `π : G → 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 of random variables
(not necessarily `G`-valued). Suppose that the pairs `(X_i, Y_i)` are jointly independent of one
another (but `X_i` need not be independent of `Y_i`). Then
`D[X_[m] | Y_[m]] = D[X_[m] ,|, π(X_[m]), Y_[m]] + D[π(X_[m]) ,| , Y_[m]]`
`+ I[∑ i, X_i : π(X_[m]) ; | ; π(∑ i, X_i), Y_[m]]`. -/ | 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}\leanok Clear from Lemma \ref{copy-ent}.
\end{proof} | /-- 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' i)) :
D[X ; hΩ] = D[X' ; hΩ'] := by
simp_rw [multiDist, IdentDistrib.entropy_eq (hident _), (hident _).map_eq] | 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.
\end{proof} | /-- 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 $1 \leq i \leq m$. Averaging over $i$, we obtain the claim.
\end{proof} | /-- 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⟩ :=
ProbabilityTheory.independent_copies' X hX (fun i => ℙ)
convert multiDist_nonneg_of_indep ⟨μA⟩ Y (fun i => (hY i).1) h_indep using 1
apply multiDist_copy
exact fun i => (hY i).2.symm | pfr/blueprint/src/chapter/torsion.tex:183 | pfr/PFR/MoreRuzsaDist.lean:759 |
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