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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 |
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