project
stringclasses 1
value | name
stringlengths 7
39
| informal
stringlengths 127
4.37k
| formal
stringlengths 82
14.6k
| informal_source
stringlengths 36
53
| formal_source
stringlengths 21
46
|
|---|---|---|---|---|---|
PFR | multiDist_of_perm | \begin{lemma}[Relabeling]\label{multidist-perm}\lean{multiDist_of_perm}\uses{multidist-def}\leanok If $\phi: \{1,\dots,m\} \to \{1,\dots,m\}$ is a bijection, then $D[X_{[m]}] = D[(X_{\phi(j)})_{1 \leq j \leq m}]$.
\end{lemma}
\begin{proof}\leanok Trivial.
\end{proof} | /-- If `φ : {1, ..., m} → {1, ...,m}` is a bijection, then `D[X_[m]] = D[(X_φ(1), ..., X_φ(m))]`-/
lemma multiDist_of_perm {m : ℕ} {Ω : Fin m → Type*}
(hΩ : ∀ i, MeasureSpace (Ω i)) (hΩprob: ∀ i, IsProbabilityMeasure (hΩ i).volume)
(X : ∀ i, (Ω i) → G) (φ : Equiv.Perm (Fin m)) :
D[fun i ↦ X (φ i); fun i ↦ hΩ (φ i)] = D[X ; hΩ] := by
simp [multiDist]
congr 1
· apply IdentDistrib.entropy_eq
exact {
aemeasurable_fst := by
apply Measurable.aemeasurable
apply Finset.measurable_sum
intro i _
exact measurable_pi_apply i
aemeasurable_snd := by
apply Measurable.aemeasurable
apply Finset.measurable_sum
intro i _
exact measurable_pi_apply i
map_eq := by
let sum := fun x : Fin m → G ↦ ∑ i, x i
let perm := MeasurableEquiv.piCongrLeft (fun _ ↦ G) φ
have perm_apply : ∀ (i : Fin m) (x : Fin m → G), perm x i = x (φ.symm i) := by
intro i x
simp only [perm]
rw [MeasurableEquiv.coe_piCongrLeft, Equiv.piCongrLeft_apply]
simp only [eq_rec_constant]
have invar : sum ∘ perm = sum := by
ext x
rw [comp_apply]
convert Finset.sum_bijective φ.symm (Equiv.bijective φ.symm) ?_ ?_
· simp only [Finset.mem_univ, implies_true]
intro i _
rw [perm_apply i x]
calc
_ = Measure.map (sum ∘ perm) (Measure.pi fun i ↦ Measure.map (X (φ i)) ℙ) := by rw [invar]
_ = Measure.map sum (Measure.map perm (Measure.pi fun i ↦ Measure.map (X (φ i)) ℙ)) := by
rw [Measure.map_map]
· apply Finset.measurable_sum
intro i _
exact measurable_pi_apply i
apply measurable_pi_lambda
intro i
have : (fun x : Fin m → G ↦ perm x i) = (fun x : Fin m → G ↦ x (φ.symm i)) := by
ext x
exact perm_apply i x
rw [this]
exact measurable_pi_apply ((Equiv.symm φ) i)
_ = _ := by
congr
exact (MeasureTheory.measurePreserving_piCongrLeft (fun i ↦ Measure.map (X i) ℙ) φ).map_eq
}
congr 1
convert Finset.sum_bijective φ (Equiv.bijective φ) ?_ ?_
· simp only [Finset.mem_univ, implies_true]
simp only [Finset.mem_univ, imp_self, implies_true]
-- The condition m ≥ 2 is likely not needed here.
/-- Let `m ≥ 2`, and let `X_[m]` be a tuple of `G`-valued random variables. Then
`∑ (1 ≤ j, k ≤ m, j ≠ k), d[X_j; -X_k] ≤ m(m-1) D[X_[m]].` -/ | pfr/blueprint/src/chapter/torsion.tex:191 | pfr/PFR/MoreRuzsaDist.lean:770 |
PFR | multiRefPackage | \begin{definition}[$\eta$]\label{eta-def-multi}\lean{multiRefPackage}\leanok
We set $\eta := \frac{1}{32m^3}$.
\end{definition} | /-- A structure that packages all the fixed information in the main argument. See https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/Problem.20when.20instances.20are.20inside.20a.20structure for more discussion of the design choices here. -/
structure multiRefPackage (G Ω₀ : Type*) [MeasureableFinGroup G] [MeasureSpace Ω₀] where
/-- The torsion index of the group we are considering. -/
(m : ℕ)
(hm : m ≥ 2)
(htorsion : ∀ x : G, m • x = 0)
(hprob : IsProbabilityMeasure (ℙ : Measure Ω₀))
/-- The random variable -/
(X₀ : Ω₀ → G)
(hmeas : Measurable X₀)
/-- A small constant. The argument will only work for suitably small `η`. -/
(η : ℝ)
(hη : 0 < η)
(hη': η ≤ 1)
/-- If $(X_i)_{1 \leq i \leq m}$ is a tuple, we define its $\tau$-functional
$$ \tau[ (X_i)_{1 \leq i \leq m}] := D[(X_i)_{1 \leq i \leq m}] + \eta \sum_{i=1}^m d[X_i; X^0].$$
-/
noncomputable def multiTau {G Ω₀ : Type*} [MeasureableFinGroup G] [MeasureSpace Ω₀] (p : multiRefPackage G Ω₀) (Ω : Fin p.m → Type*) (hΩ : ∀ i, MeasureSpace (Ω i)) (X : ∀ i, Ω i → G) : ℝ :=
D[X; hΩ] + p.η * ∑ i, d[ X i # p.X₀ ]
-- I can't figure out how to make a τ notation due to the dependent types in the arguments. But perhaps we don't need one. Also it may be better to define multiTau in terms of probability measures on G, rather than G-valued random variables, again to avoid dependent type issues.
-- had to force objects to lie in a fixed universe `u` here to avoid a compilation error | pfr/blueprint/src/chapter/torsion.tex:273 | pfr/PFR/MultiTauFunctional.lean:25 |
PFR | multiTau_continuous | \begin{proposition}[Existence of $\tau$-minimizer]\label{tau-min-exist-multi}\lean{multiTau_continuous, multiTau_min_exists}\leanok If $G$ is finite, then a $\tau$-minimizer exists.
\end{proposition}
\begin{proof}\uses{tau-def-multi} This is similar to the proof of \Cref{tau-min}.
\end{proof} | /-- If $G$ is finite, then a $\tau$ is continuous. -/
lemma multiTau_continuous {G Ω₀ : Type u} [MeasureableFinGroup G] [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G] [MeasureSpace Ω₀] (p : multiRefPackage G Ω₀) : Continuous
(fun (μ : Fin p.m → ProbabilityMeasure G) ↦ multiTau p (fun _ ↦ G) (fun i ↦ ⟨ μ i ⟩) (fun _ ↦ id)) := by sorry | pfr/blueprint/src/chapter/torsion.tex:284 | pfr/PFR/MultiTauFunctional.lean:52 |
PFR | multiTau_min_exists | \begin{proposition}[Existence of $\tau$-minimizer]\label{tau-min-exist-multi}\lean{multiTau_continuous, multiTau_min_exists}\leanok If $G$ is finite, then a $\tau$-minimizer exists.
\end{proposition}
\begin{proof}\uses{tau-def-multi} This is similar to the proof of \Cref{tau-min}.
\end{proof} | /-- If $G$ is finite, then a $\tau$-minimizer exists. -/
lemma multiTau_min_exists {G Ω₀ : Type u} [MeasureableFinGroup G] [MeasureSpace Ω₀] (p : multiRefPackage G Ω₀) : ∃ (Ω : Fin p.m → Type u) (hΩ : ∀ i, MeasureSpace (Ω i)) (X : ∀ i, Ω i → G), multiTauMinimizes p Ω hΩ X := by sorry | pfr/blueprint/src/chapter/torsion.tex:284 | pfr/PFR/MultiTauFunctional.lean:56 |
PFR | multiTau_min_sum_le | \begin{proposition}[Minimizer close to reference variables]\label{tau-ref}\lean{multiTau_min_sum_le}\leanok If $(X_i)_{1 \leq i \leq m}$ is a $\tau$-minimizer, then $\sum_{i=1}^m d[X_i; X^0] \leq \frac{2m}{\eta} d[X^0; X^0]$.
\end{proposition}
\begin{proof}\uses{tau-min-multi, multidist-nonneg, multidist-ruzsa-III}\leanok By \Cref{tau-min-multi} we have
$$ \tau[ (X_i)_{1 \leq i \leq m}] \leq \tau[ (X^0)_{1 \leq i \leq m}]$$
and hence by \Cref{tau-def-multi} and \Cref{multidist-nonneg}
$$ \eta \sum_{i=1}^m d[X_i; X^0] \leq D[(X^0)_{1 \leq i \leq m}] + m d[X^0; X^0].$$
The claim now follows from \Cref{multidist-ruzsa-III}.
\end{proof} | /-- If $(X_i)_{1 \leq i \leq m}$ is a $\tau$-minimizer, then $\sum_{i=1}^m d[X_i; X^0] \leq \frac{2m}{\eta} d[X^0; X^0]$. -/
lemma multiTau_min_sum_le {G Ω₀ : Type u} [hG: MeasureableFinGroup G] [hΩ₀: MeasureSpace Ω₀] (p : multiRefPackage G Ω₀) (Ω : Fin p.m → Type u) (hΩ : ∀ i, MeasureSpace (Ω i)) (hprobΩ : ∀ i, IsProbabilityMeasure (ℙ : Measure (Ω i))) (X : ∀ i, Ω i → G) (hX : ∀ i, Measurable (X i)) (h_min : multiTauMinimizes p Ω hΩ X):
∑ i, d[X i # p.X₀] ≤ 2 * p.m * p.η⁻¹ * d[p.X₀ # p.X₀] := by
have hη : p.η > 0 := p.hη
have hm : p.m > 0 := by linarith [p.hm]
have hprob := p.hprob
calc
_ = p.η⁻¹ * (0 + p.η * ∑ i, d[X i # p.X₀]) := by
field_simp
_ ≤ p.η⁻¹ * (D[X ; hΩ] + p.η * ∑ i, d[X i # p.X₀]) := by
gcongr
exact multiDist_nonneg hΩ hprobΩ X hX
_ ≤ p.η⁻¹ * (D[fun _ ↦ p.X₀ ; fun _ ↦ hΩ₀] + p.η * (p.m * d[p.X₀ # p.X₀])) := by
apply mul_le_mul_of_nonneg_left
· have ineq := h_min (fun _ ↦ Ω₀) (fun _ ↦ hΩ₀) (fun _ ↦ p.X₀)
simp [multiTau] at ineq
exact ineq
exact inv_nonneg_of_nonneg (le_of_lt hη)
_ ≤ p.η⁻¹ * (p.m * d[p.X₀ # p.X₀] + 1 * (p.m * d[p.X₀ # p.X₀])) := by
gcongr
· have : NeZero p.m := ⟨hm.ne'⟩
apply multidist_ruzsa_III p.hm (fun _ ↦ hΩ₀) (fun _ ↦ p.X₀) _ 0
intro _ _
simp
exact ProbabilityTheory.IdentDistrib.refl ( Measurable.aemeasurable p.hmeas)
· have : 0 ≤ d[p.X₀ # p.X₀] := rdist_nonneg p.hmeas p.hmeas
positivity
exact p.hη'
_ = _ := by
field_simp
ring
/-- If $(X_i)_{1 \leq i \leq m}$ is a $\tau$-minimizer, and $k := D[(X_i)_{1 \leq i \leq m}]$, then for any other tuple $(X'_i)_{1 \leq i \leq m}$, one has
$$ k - D[(X'_i)_{1 \leq i \leq m}] \leq \eta \sum_{i=1}^m d[X_i; X'_i].$$
-/ | pfr/blueprint/src/chapter/torsion.tex:290 | pfr/PFR/MultiTauFunctional.lean:59 |
PFR | multidist_eq_zero | \begin{proposition}[Vanishing]\label{multi-zero}\lean{multidist_eq_zero}\uses{multidist-def}\leanok
If $D[X_{[m]}]=0$, then for each $1 \leq i \leq m$ there is a finite subgroup $H_i \leq G$ such that $d[X_i; U_{H_i}] = 0$.
\end{proposition}
\begin{proof}\uses{multidist-ruzsa-I, ruzsa-nonneg, lem:100pc} From \Cref{multidist-ruzsa-I} and \Cref{ruzsa-nonneg} we have $d[X_j; X_{-k}]=0$ for all $1 \leq j,k \leq m$. The claim now follows from \Cref{lem:100pc}.
\end{proof} | lemma multidist_eq_zero {m:ℕ} (hm: m ≥ 2) {Ω: Fin m → Type*} (hΩ : ∀ i, MeasureSpace (Ω i)) (X : ∀ i, (Ω i) → G) (hvanish : D[X; hΩ] = 0) : ∀ i, ∃ H : AddSubgroup G, ∃ U : (Ω i) → G, Measurable U ∧ IsUniform H U ∧ d[X i # U] = 0 := by sorry
-- This is probably not the optimal spelling. For instance one could use the `μ "[|" t "]"` notation from Mathlib.ProbabilityTheory.ConditionalProbability to simplify the invocation of `ProbabilityTheory.cond`
/-- If `X_[m] = (X_1, ..., X_m)` and `Y_[m] = (Y_1, ..., Y_m)` are tuples of random variables,
with the `X_i` being `G`-valued (but the `Y_i` need not be), then we define
`D[X_[m] | Y_[m]] = ∑_{(y_i)_{1 \leq i \leq m}} (∏ i, p_{Y_i}(y_i)) D[(X_i | Y_i = y_i)_{i=1}^m]`
where each `y_i` ranges over the support of `p_{Y_i}` for `1 ≤ i ≤ m`.
-/
noncomputable | pfr/blueprint/src/chapter/torsion.tex:260 | pfr/PFR/MoreRuzsaDist.lean:853 |
PFR | multidist_ruzsa_I | \begin{lemma}[Multidistance and Ruzsa distance, I]\label{multidist-ruzsa-I}\lean{multidist_ruzsa_I}\uses{multidist-def}\leanok
Let $m \ge 2$, and let $X_{[m]}$ be a tuple of $G$-valued random variables. Then
$$\sum_{1 \leq j,k \leq m: j \neq k} d[X_j; -X_k] \leq m(m-1) D[X_{[m]}].$$
\end{lemma}
\begin{proof}\uses{ruz-copy, sumset-lower, ruz-indep, multidist-indep, multidist-copy}
By \Cref{ruz-copy}, \Cref{multidist-copy} we may take the $X_i$ to be jointly independent. From \Cref{sumset-lower}, we see that for any distinct $1 \leq j,k \leq m$, we have
\[
\bbH[X_j+X_k] \leq \bbH[\sum_{i=1}^m X_i],
\]
and hence by \Cref{ruz-indep}
\[
d[X_j;-X_k] \leq \bbH[\sum_{i=1}^m X_i] - \frac{1}{2} \bbH[X_j] - \frac{1}{2} \bbH[X_k].
\]
Summing this over all pairs $(j,k)$, $j \neq k$ and using Lemma \ref{multidist-indep}, we obtain the claim.
\end{proof} | lemma multidist_ruzsa_I {m:ℕ} (hm: m ≥ 2) {Ω: Fin m → Type*} (hΩ : ∀ i, MeasureSpace (Ω i))
(X : ∀ i, (Ω i) → G): ∑ j, ∑ k, (if j = k then (0:ℝ) else d[X j # X k]) ≤ m * (m-1) * D[X; hΩ] := by sorry
/-- Let `m ≥ 2`, and let `X_[m]` be a tuple of `G`-valued random variables. Then
`∑ j, d[X_j;X_j] ≤ 2 m D[X_[m]]`. -/ | pfr/blueprint/src/chapter/torsion.tex:197 | pfr/PFR/MoreRuzsaDist.lean:832 |
PFR | multidist_ruzsa_II | \begin{lemma}[Multidistance and Ruzsa distance, II]\label{multidist-ruzsa-II}\lean{multidist_ruzsa_II}\uses{multidist-def}\leanok
Let $m \ge 2$, and let $X_{[m]}$ be a tuple of $G$-valued random variables. Then
$$\sum_{j=1}^m d[X_j;X_j] \leq 2 m D[X_{[m]}].$$
\end{lemma}
\begin{proof}\uses{ruzsa-triangle,multidist-ruzsa-I}
From \Cref{ruzsa-triangle} we have $\dist{X_j}{X_j} \leq 2 \dist{X_j}{-X_k}$, and applying this to every summand in \Cref{multidist-ruzsa-I}, we obtain the claim.
\end{proof} | lemma multidist_ruzsa_II {m:ℕ} (hm: m ≥ 2) {Ω: Fin m → Type*} (hΩ : ∀ i, MeasureSpace (Ω i))
(X : ∀ i, (Ω i) → G): ∑ j, d[X j # X j] ≤ 2 * m * D[X; hΩ] := by sorry
/-- Let `I` be an indexing set of size `m ≥ 2`, and let `X_[m]` be a tuple of `G`-valued random
variables. If the `X_i` all have the same distribution, then `D[X_[m]] ≤ m d[X_i;X_i]` for any
`1 ≤ i ≤ m`. -/ | pfr/blueprint/src/chapter/torsion.tex:214 | pfr/PFR/MoreRuzsaDist.lean:837 |
PFR | multidist_ruzsa_III | \begin{lemma}[Multidistance and Ruzsa distance, III]\label{multidist-ruzsa-III}\lean{multidist_ruzsa_III}\uses{multidist-def}\leanok
Let $m \ge 2$, and let $X_{[m]}$ be a tuple of $G$-valued random variables. If the $X_i$ all have the same distribution, then $D[X_{[m]}] \leq m d[X_i;X_i]$ for any $1 \leq i \leq m$.
\end{lemma}
\begin{proof}\uses{klm-1, neg-ent, ruz-indep, sumset-lower, ruz-copy, multidist-copy}
By \Cref{ruz-copy}, \Cref{multidist-copy} we may take the $X_i$ to be jointly independent. Let $X_0$ be a further independent copy of the $X_i$.
From \Cref{klm-1}, we have
$$ \bbH[-X_0 + \sum_{i=1}^m X_i] - \bbH[-X_0] \leq \sum_{i=1}^m \bbH[X_0 - X_i] - \bbH[-X_0]$$
and hence by \Cref{neg-ent} and \Cref{ruz-indep}
$$ \bbH[-X_0 + \sum_{i=1}^m X_i] - \bbH[X_0] \leq m d[X_i,X_i].$$
On the other hand, by \Cref{sumset-lower} we have
$$ \bbH[\sum_{i=1}^m X_i] \leq \bbH[-X_0 + \sum_{i=1}^m X_i]$$
and the claim follows.
\end{proof} | lemma multidist_ruzsa_III {m:ℕ} (hm: m ≥ 2) {Ω: Fin m → Type*} (hΩ : ∀ i, MeasureSpace (Ω i))
(X : ∀ i, (Ω i) → G) (hidenT : ∀ j k, IdentDistrib (X j) (X k)): ∀ i, D[X; hΩ] ≤ m * d[X i # X i] := by sorry
/-- Let `m ≥ 2`, and let `X_[m]` be a tuple of `G`-valued random
variables. Let `W := ∑ X_i`. Then `d[W;-W] ≤ 2 D[X_i]`. -/ | pfr/blueprint/src/chapter/torsion.tex:223 | pfr/PFR/MoreRuzsaDist.lean:843 |
PFR | multidist_ruzsa_IV | \begin{lemma}[Multidistance and Ruzsa distance, IV]\label{multidist-ruzsa-IV}\lean{multidist_ruzsa_IV}\uses{multidist-def}\leanok
Let $m \ge 2$, and let $X_{[m]}$ be a tuple of independent $G$-valued random variables. Let $W := \sum_{i=1}^m X_i$. Then
$$ d[W;-W] \leq 2 D[X_i].$$
\end{lemma}
\begin{proof}\uses{independent-exist, kv, sumset-lower, ruz-indep}
Take $(X'_i)_{1 \leq i \leq m}$ to be further independent copies of $(X_i)_{1 \leq i \leq m}$ (which exist by \Cref{independent-exist}), and write $W' := \sum_{i=1}^m X'_i$.
Fix any distinct $a,b \in I$.
From \Cref{kv} one has
\begin{equation}\label{7922}
\bbH[W + W'] \leq \bbH[W] + \bbH[X_{a} + W'] - \bbH[X_{a}]
\end{equation}
and also
\[ \bbH[X_a + W'] \leq \bbH[X_a + X_b] + \bbH[W'] - \bbH[X'_b].\]
Combining this with~\eqref{7922} and then applying \Cref{sumset-lower} we have
\begin{align*} \bbH[W + W'] & \leq 2\bbH[W] + \bbH[X_a + X_b] - \bbH[X_a] - \bbH[X_b] \\ & \leq
3 \bbH[W] - \bbH[X_a] - \bbH[X_b].
\end{align*}
Averaging this over all choices of $(a,b)$ gives $\bbH[W] + 2 D[X_{[m]}]$, and the claim follows from \Cref{ruz-indep}.
\end{proof} | lemma multidist_ruzsa_IV {m:ℕ} (hm: m ≥ 2) {Ω : Type*} (hΩ : MeasureSpace Ω) (X : Fin m → Ω → G)
(h_indep : iIndepFun X) : d[∑ i, X i # ∑ i, X i] ≤ 2 * D[X; fun _ ↦ hΩ] := by sorry
/-- If `D[X_[m]]=0`, then for each `i ∈ I` there is a finite subgroup `H_i ≤ G` such that
`d[X_i; U_{H_i}] = 0`. -/ | pfr/blueprint/src/chapter/torsion.tex:238 | pfr/PFR/MoreRuzsaDist.lean:848 |
PFR | mutual_information_le | \begin{proposition}[Bounding mutual information]\label{key}\lean{mutual_information_le}\leanok
Suppose that $X_{i,j}$, $1 \leq i,j \leq m$, are jointly independent $G$-valued random variables, such that for each $j = 1,\dots,m$, the random variables $(X_{i,j})_{i = 1}^m$ coincide in distribution with some permutation of $X_{[m]}$.
Write
\[
{\mathcal I} := \bbI[ \bigl(\sum_{i=1}^m X_{i,j}\bigr)_{j =1}^{m} : \bigl(\sum_{j=1}^m X_{i,j}\bigr)_{i = 1}^m \; \big| \; \sum_{i=1}^m \sum_{j = 1}^m X_{i,j} ].
\]
Then
\begin{equation}\label{I-ineq}
{\mathcal I} \leq 4 m^2 \eta k.
\end{equation}
\end{proposition}
\begin{proof}\uses{cor-multid, multidist-perm, cond-multidist-lower-II, first-useful, klm-3, sumset-lower, ruzsa-triangle, compare-sums, multidist-ruzsa-II}
For each $j \in \{1,\dots,m\}$ we call the tuple $(X_{i,j})_{i = 1}^m$ a \emph{column} and for each $i \in \{1,\dots, m\}$ we call the tuple $(X_{i,j})_{j = 1}^m$ a \emph{row}. Hence, by hypothesis, each column is a permutation of $X_{[m]} = (X_i)_{i=1}^m$.
From \Cref{cor-multid} we have
\begin{equation}\label{441} {\mathcal I} \leq \sum_{j=1}^{m-1} A_j + B,\end{equation}
where
\[
A_j := D[ (X_{i, j})_{i = 1}^m] - D[ (X_{i, j})_{i = 1}^m \; \big| \; (X_{i,j} + \cdots + X_{i,m})_{i =1}^m ]
\]
and
\[
B := D[ (X_{i,m})_{i=1}^m ] - D[ \bigl(\sum_{j=1}^m X_{i,j}\bigr)_{i=1}^m ].
\]
We first consider the $A_j$, for fixed $j \in \{1,\dots, m-1\}$.
By \Cref{multidist-perm} and and our hypothesis on columns, we have
\[
D[ (X_{i, j})_{i = 1}^m ]= D[ (X_i)_{i=1}^m ] = k.
\]
Let $\sigma = \sigma_j \colon I \to I$ be a permutation such that $X_{i,j} = X_{\sigma(i)}$, and write $X'_i := X_{i,j}$ and $Y_i := X_{i,j} + \cdots + X_{i,m}$.
\Cref{cond-multidist-lower-II}, we have
\begin{align}
A_j & \leq \eta \sum_{i = 1}^m d[X_{i,j}; X_{i, j}|X_{i, j} + \cdots + X_{i,m}].\label{54a}
\end{align}
We similarly consider $B$. By \Cref{multidist-perm} applied to the $m$-th column,
\[
D[ (X_{i, m})_{i = 1}^m ] = D[X_{[m]}] = k.
\]
For $1 \leq i \leq m$, denote the sum of row $i$ by
\[
V_i := \sum_{j=1}^m X_{i,j};
\]
if we apply \Cref{cond-multidist-lower-II} again, now with $X_{\sigma(i)} = X_{i,m}$, $X'_i := V_i$, and with the variable $Y_i$ being trivial, we obtain
\begin{equation}\label{55a}
B \leq \eta \sum_{i = 1}^m d[X_{i,m}; V_i].
\end{equation}
It remains to bound the distances appearing in~\eqref{54a} and~\eqref{55a} further using Ruzsa calculus.
For $1 \leq j \leq m-1$ and $1 \leq i \leq m$, by \Cref{first-useful} we have
\begin{align*} &d[ X_{i,j}; X_{i,j}| X_{i,j}+\cdots+X_{i,m}]
\leq d[X_{i,j}; X_{i,j}] \\
&\quad + \tfrac{1}{2} \bigl(\bbH[X_{i,j}+\cdots+X_{i,m}] - \bbH[X_{i,{j+1}}+\cdots+X_{i,m}]\bigr).
\end{align*}
For each $i$, summing over $j = 1,\dots, m-1$ gives
\begin{align}
\nonumber
&\sum_{j=1}^{m-1} d[X_{i,j}; X_{i,j}| X_{i,j}+\cdots+X_{i,m}] \\
&\qquad \leq \sum_{j=1}^{m-1} d[X_{i,j}; X_{i,j}] + \frac12 \bigl( \bbH[V_i] - \bbH[X_{i,m}] \bigr).
\label{eq:distbnd1}
\end{align}
On the other hand, by \Cref{klm-3} (since $X_{i,m}$ appears in the sum $V_i$) we have
\begin{align}
d[X_{i,m}; V_i]
&\leq d[X_{i,m}; X_{i,m}] + \frac12 \bigl( \bbH[V_i] - \bbH[X_{i,m}] \bigr).
\label{eq:distbnd2}
\end{align}
Combining~\eqref{441},~\eqref{54a} and~\eqref{55a} with~\eqref{eq:distbnd1} and~\eqref{eq:distbnd2} (the latter two summed over $i$), we get
\begin{align}
\nonumber
\frac1{\eta} {\mathcal I} &\leq \sum_{i,j=1}^m d[X_{i,j};X_{i,j}] + \sum_{i=1}^m (\bbH[V_i] - \bbH[X_{i,m}]) \\
&= m \sum_{i=1}^m d[X_i; X_i] + \sum_{i=1}^m \bbH[V_i] - \sum_{i=1}^m \bbH[X_i].
\label{eq:distbnd3}
\end{align}
By \Cref{compare-sums} (with $f$ taking each $j$ to the index $j'$ such that $X_{i,j}$ is a copy of $X_{j'}$) we obtain the bound
\[
\bbH[V_i] \leq \bbH[\sum_{j=1}^m X_j] + \sum_{j=1}^m d[X_{i,j}; X_{i,j}].
\]
Finally, summing over $i$ and using $D[X_{[m]}] = k$ gives
\begin{align*}
\sum_{i=1}^m \bbH[V_i] - \sum_{i=1}^m \bbH[X_i] & \leq \sum_{i,j=1}^m d[X_{i,j}; X_{i,j}] + m k \\ & = m\sum_{i = 1}^m d[X_i; X_i] + mk,
\end{align*}
where in the second step we used the permutation hypothesis. Combining this with~\eqref{eq:distbnd3} gives the
$$
{\mathcal I} \leq 2\eta m \biggl( \sum_{i=1}^m d[X_i;X_i] \biggr).$$
The claim \eqref{I-ineq} is now immediate from \Cref{multidist-ruzsa-II}.
\end{proof} | lemma mutual_information_le {G Ωₒ : Type u} [MeasureableFinGroup G] [MeasureSpace Ωₒ]
(p : multiRefPackage G Ωₒ) (Ω : Type u) [hΩ: MeasureSpace Ω] (X : ∀ i, Ω → G)
(h_indep : iIndepFun X)
(h_min : multiTauMinimizes p (fun _ ↦ Ω) (fun _ ↦ hΩ) X) (Ω' : Type*) [MeasureSpace Ω']
(X' : Fin p.m × Fin p.m → Ω' → G) (h_indep': iIndepFun X')
(hperm : ∀ j, ∃ e : Fin p.m ≃ Fin p.m, IdentDistrib (fun ω ↦ (fun i ↦ X' (i, j) ω))
(fun ω ↦ (fun i ↦ X (e i) ω))) :
I[ fun ω ↦ ( fun j ↦ ∑ i, X' (i, j) ω) : fun ω ↦ ( fun i ↦ ∑ j, X' (i, j) ω) |
fun ω ↦ ∑ i, ∑ j, X' (i, j) ω ] ≤ 4 * p.m^2 * p.η * D[ X; (fun _ ↦ hΩ)] := sorry | pfr/blueprint/src/chapter/torsion.tex:486 | pfr/PFR/BoundingMutual.lean:28 |
PFR | mutual_information_le_t_12 | \begin{proposition}[Mutual information bound]\label{prop:52}\lean{mutual_information_le_t_12, mutual_information_le_t_13, mutual_information_le_t_23}\uses{more-random}\leanok
We have
\[
\bbI[Z_1 : Z_2\, |\, W],\
\bbI[Z_2 : Z_3\, |\, W],\
\bbI[Z_1 : Z_3\, |\, W] \leq t
\]
where
\begin{equation}\label{t-def}
t := 4m^2 \eta k.
\end{equation}
\end{proposition}
\begin{proof}\uses{key, data-process}
We analyze these variables by \Cref{key} in several different ways.
In the first application, take $X_{i,j}=Y_{i,j}$.
Note that each column $(X_{i,j})_{i=1}^m$ is indeed a permutation of $X_1,\dots,X_m$; in fact, the trivial permutation.
Note also that for each $i \in \Z/m\Z$, the row sum is
\[
\sum_{j=1}^m X_{i,j} = \sum_{j \in \Z/m\Z} Y_{i,j} = P_i
\]
and for each $j \in \Z/m\Z$, the column sum is
\[
\sum_{i=1}^m X_{i,j} = \sum_{i \in \Z/m\Z} Y_{i,j} = Q_j.
\]
Finally note that $\sum_{i,j=1}^m X_{i,j} = W$.
From \Cref{key} we then have
\[
\bbI[ (P_i)_{i \in \Z/m\Z} : (Q_j)_{j \in \Z/m\Z } \,\big|\, W ] \leq t,
\]
with $t$ as in~\eqref{t-def}.
Since $Z_1$ is a function of $(P_i)_{i \in \Z/m\Z}$ by~\eqref{pqr-defs}, and similarly $Z_2$ is a function of $(Q_j)_{j \in \Z/m\Z}$, it follows immediately from \Cref{data-process} that
\[
\bbI[ Z_1 : Z_2 \,|\, W ] \leq t.
\]
In the second application of \Cref{key}, we instead consider $X'_{i,j} = Y_{i-j,j}$.
Again, for each fixed $j$, the tuple $(X'_{i,j})_{i=1}^m$ is a permutation of $X_1,\dots,X_m$.
This time the row sums for $i \in \{1,\dots, m\}$ are
\[
\sum_{j=1}^m X'_{i,j} = \sum_{j \in \Z/m\Z} Y_{i-j,j} = R_{-i}.
\]
Similarly, the column sums for $j \in \{1,\dots, m\}$ are
\[
\sum_{i=1}^m X'_{i,j} = \sum_{i \in \Z/m\Z} Y_{i-j,j} = Q_j.
\]
As before, $\sum_{i,j=1}^m X'_{i,j} = W$.
Hence, using~\eqref{pqr-defs} and \Cref{data-process} again, \Cref{key} tells us
\[
\bbI[ Z_3 : Z_2 \,|\, W] \leq \bbI[ (R_i)_{i \in \Z/m\Z} : (Q_j)_{j \in \Z/m\Z } \,\big|\, W ] \leq t.
\]
In the third application\footnote{In fact, by permuting the variables $(Y_{i,j})_{i,j \in \Z/m\Z}$, one can see that the random variables $(W, Z_1, Z_2)$ and $(W, Z_1, Z_3)$ have the same distribution, so this is in some sense identical to -- and can be deduced from -- the first application.} of \Cref{key}, take $X''_{i,j} = Y_{i,j-i}$.
The column and row sums are respectively
\[
\sum_{j=1}^m X''_{i,j} = \sum_{j \in \Z/m\Z} Y_{i,j-i} = P_i \] and
\[ \sum_{i=1}^m X''_{i,j} = \sum_{i \in \Z/m\Z} Y_{i,j-i} = R_{-j}.
\]
Hence, \Cref{key} and \Cref{data-process} give
\[
\bbI[ Z_1 : Z_3 \,|\, W] \leq \bbI[ (P_i)_{i \in \Z/m\Z} : (R_j)_{j \in \Z/m\Z } \,\big|\, W ] \leq t,
\]
which completes the proof.
\end{proof} | lemma mutual_information_le_t_12 : I[Z1 : Z2 | W] ≤ 4 * p.m ^ 2 * p.η * k := sorry | pfr/blueprint/src/chapter/torsion.tex:608 | pfr/PFR/TorsionEndgame.lean:50 |
PFR | mutual_information_le_t_13 | \begin{proposition}[Mutual information bound]\label{prop:52}\lean{mutual_information_le_t_12, mutual_information_le_t_13, mutual_information_le_t_23}\uses{more-random}\leanok
We have
\[
\bbI[Z_1 : Z_2\, |\, W],\
\bbI[Z_2 : Z_3\, |\, W],\
\bbI[Z_1 : Z_3\, |\, W] \leq t
\]
where
\begin{equation}\label{t-def}
t := 4m^2 \eta k.
\end{equation}
\end{proposition}
\begin{proof}\uses{key, data-process}
We analyze these variables by \Cref{key} in several different ways.
In the first application, take $X_{i,j}=Y_{i,j}$.
Note that each column $(X_{i,j})_{i=1}^m$ is indeed a permutation of $X_1,\dots,X_m$; in fact, the trivial permutation.
Note also that for each $i \in \Z/m\Z$, the row sum is
\[
\sum_{j=1}^m X_{i,j} = \sum_{j \in \Z/m\Z} Y_{i,j} = P_i
\]
and for each $j \in \Z/m\Z$, the column sum is
\[
\sum_{i=1}^m X_{i,j} = \sum_{i \in \Z/m\Z} Y_{i,j} = Q_j.
\]
Finally note that $\sum_{i,j=1}^m X_{i,j} = W$.
From \Cref{key} we then have
\[
\bbI[ (P_i)_{i \in \Z/m\Z} : (Q_j)_{j \in \Z/m\Z } \,\big|\, W ] \leq t,
\]
with $t$ as in~\eqref{t-def}.
Since $Z_1$ is a function of $(P_i)_{i \in \Z/m\Z}$ by~\eqref{pqr-defs}, and similarly $Z_2$ is a function of $(Q_j)_{j \in \Z/m\Z}$, it follows immediately from \Cref{data-process} that
\[
\bbI[ Z_1 : Z_2 \,|\, W ] \leq t.
\]
In the second application of \Cref{key}, we instead consider $X'_{i,j} = Y_{i-j,j}$.
Again, for each fixed $j$, the tuple $(X'_{i,j})_{i=1}^m$ is a permutation of $X_1,\dots,X_m$.
This time the row sums for $i \in \{1,\dots, m\}$ are
\[
\sum_{j=1}^m X'_{i,j} = \sum_{j \in \Z/m\Z} Y_{i-j,j} = R_{-i}.
\]
Similarly, the column sums for $j \in \{1,\dots, m\}$ are
\[
\sum_{i=1}^m X'_{i,j} = \sum_{i \in \Z/m\Z} Y_{i-j,j} = Q_j.
\]
As before, $\sum_{i,j=1}^m X'_{i,j} = W$.
Hence, using~\eqref{pqr-defs} and \Cref{data-process} again, \Cref{key} tells us
\[
\bbI[ Z_3 : Z_2 \,|\, W] \leq \bbI[ (R_i)_{i \in \Z/m\Z} : (Q_j)_{j \in \Z/m\Z } \,\big|\, W ] \leq t.
\]
In the third application\footnote{In fact, by permuting the variables $(Y_{i,j})_{i,j \in \Z/m\Z}$, one can see that the random variables $(W, Z_1, Z_2)$ and $(W, Z_1, Z_3)$ have the same distribution, so this is in some sense identical to -- and can be deduced from -- the first application.} of \Cref{key}, take $X''_{i,j} = Y_{i,j-i}$.
The column and row sums are respectively
\[
\sum_{j=1}^m X''_{i,j} = \sum_{j \in \Z/m\Z} Y_{i,j-i} = P_i \] and
\[ \sum_{i=1}^m X''_{i,j} = \sum_{i \in \Z/m\Z} Y_{i,j-i} = R_{-j}.
\]
Hence, \Cref{key} and \Cref{data-process} give
\[
\bbI[ Z_1 : Z_3 \,|\, W] \leq \bbI[ (P_i)_{i \in \Z/m\Z} : (R_j)_{j \in \Z/m\Z } \,\big|\, W ] \leq t,
\]
which completes the proof.
\end{proof} | lemma mutual_information_le_t_13 : I[Z1 : Z3 | W] ≤ 4 * p.m ^ 2 * p.η * k := sorry | pfr/blueprint/src/chapter/torsion.tex:608 | pfr/PFR/TorsionEndgame.lean:52 |
PFR | mutual_information_le_t_23 | \begin{proposition}[Mutual information bound]\label{prop:52}\lean{mutual_information_le_t_12, mutual_information_le_t_13, mutual_information_le_t_23}\uses{more-random}\leanok
We have
\[
\bbI[Z_1 : Z_2\, |\, W],\
\bbI[Z_2 : Z_3\, |\, W],\
\bbI[Z_1 : Z_3\, |\, W] \leq t
\]
where
\begin{equation}\label{t-def}
t := 4m^2 \eta k.
\end{equation}
\end{proposition}
\begin{proof}\uses{key, data-process}
We analyze these variables by \Cref{key} in several different ways.
In the first application, take $X_{i,j}=Y_{i,j}$.
Note that each column $(X_{i,j})_{i=1}^m$ is indeed a permutation of $X_1,\dots,X_m$; in fact, the trivial permutation.
Note also that for each $i \in \Z/m\Z$, the row sum is
\[
\sum_{j=1}^m X_{i,j} = \sum_{j \in \Z/m\Z} Y_{i,j} = P_i
\]
and for each $j \in \Z/m\Z$, the column sum is
\[
\sum_{i=1}^m X_{i,j} = \sum_{i \in \Z/m\Z} Y_{i,j} = Q_j.
\]
Finally note that $\sum_{i,j=1}^m X_{i,j} = W$.
From \Cref{key} we then have
\[
\bbI[ (P_i)_{i \in \Z/m\Z} : (Q_j)_{j \in \Z/m\Z } \,\big|\, W ] \leq t,
\]
with $t$ as in~\eqref{t-def}.
Since $Z_1$ is a function of $(P_i)_{i \in \Z/m\Z}$ by~\eqref{pqr-defs}, and similarly $Z_2$ is a function of $(Q_j)_{j \in \Z/m\Z}$, it follows immediately from \Cref{data-process} that
\[
\bbI[ Z_1 : Z_2 \,|\, W ] \leq t.
\]
In the second application of \Cref{key}, we instead consider $X'_{i,j} = Y_{i-j,j}$.
Again, for each fixed $j$, the tuple $(X'_{i,j})_{i=1}^m$ is a permutation of $X_1,\dots,X_m$.
This time the row sums for $i \in \{1,\dots, m\}$ are
\[
\sum_{j=1}^m X'_{i,j} = \sum_{j \in \Z/m\Z} Y_{i-j,j} = R_{-i}.
\]
Similarly, the column sums for $j \in \{1,\dots, m\}$ are
\[
\sum_{i=1}^m X'_{i,j} = \sum_{i \in \Z/m\Z} Y_{i-j,j} = Q_j.
\]
As before, $\sum_{i,j=1}^m X'_{i,j} = W$.
Hence, using~\eqref{pqr-defs} and \Cref{data-process} again, \Cref{key} tells us
\[
\bbI[ Z_3 : Z_2 \,|\, W] \leq \bbI[ (R_i)_{i \in \Z/m\Z} : (Q_j)_{j \in \Z/m\Z } \,\big|\, W ] \leq t.
\]
In the third application\footnote{In fact, by permuting the variables $(Y_{i,j})_{i,j \in \Z/m\Z}$, one can see that the random variables $(W, Z_1, Z_2)$ and $(W, Z_1, Z_3)$ have the same distribution, so this is in some sense identical to -- and can be deduced from -- the first application.} of \Cref{key}, take $X''_{i,j} = Y_{i,j-i}$.
The column and row sums are respectively
\[
\sum_{j=1}^m X''_{i,j} = \sum_{j \in \Z/m\Z} Y_{i,j-i} = P_i \] and
\[ \sum_{i=1}^m X''_{i,j} = \sum_{i \in \Z/m\Z} Y_{i,j-i} = R_{-j}.
\]
Hence, \Cref{key} and \Cref{data-process} give
\[
\bbI[ Z_1 : Z_3 \,|\, W] \leq \bbI[ (P_i)_{i \in \Z/m\Z} : (R_j)_{j \in \Z/m\Z } \,\big|\, W ] \leq t,
\]
which completes the proof.
\end{proof} | lemma mutual_information_le_t_23 : I[Z2 : Z3 | W] ≤ 4 * p.m ^ 2 * p.η * k := sorry | pfr/blueprint/src/chapter/torsion.tex:608 | pfr/PFR/TorsionEndgame.lean:54 |
PFR | mutual_of_W_Z_two_le | \begin{lemma}[Mutual information bound]\label{mutual-w-z2}\lean{mutual_of_W_Z_two_le}\leanok We have $\bbI[W : Z_2] \leq 2 (m-1) k$.
\end{lemma}
\begin{proof}\uses{alternative-mutual, ent-w}
From \Cref{alternative-mutual} we have $\bbI[W : Z_2] = \bbH[W] - \bbH[W | Z_2]$, and since $Z_2 = \sum_{j=1}^{m-1} j Q_j$ and $W = \sum_{j=1}^m Q_j$,
\[
\bbH[W | Z_2] \geq \bbH[W \,|\, Q_1,\dots,Q_{m-1}] = \bbH[Q_m] = \bbH[S].
\]
Hence, by \Cref{ent-w},
\[
\bbI[W : Z_2] \leq \bbH[W] - \bbH[S] \leq 2 (m-1) k,
\]
as claimed.
\end{proof} | /-- We have $\bbI[W : Z_2] \leq 2 (m-1) k$. -/
lemma mutual_of_W_Z_two_le : I[W : Z2] ≤ 2 * (p.m-1) * k := sorry | pfr/blueprint/src/chapter/torsion.tex:720 | pfr/PFR/TorsionEndgame.lean:63 |
PFR | phiMinimizes | \begin{definition}\label{phi-min-def}\lean{phiMinimizes}\leanok Given $G$-valued random variables $X,Y$, define
$$ \phi[X;Y] := d[X;Y] + \eta(\rho(X) + \rho(Y))$$
and define a \emph{$\phi$-minimizer} to be a pair of random variables $X,Y$ which minimizes $\phi[X;Y]$.
\end{definition} | def phiMinimizes {Ω : Type*} [MeasurableSpace Ω] (X Y : Ω → G) (η : ℝ) (A : Finset G)
(μ : Measure Ω) : Prop :=
∀ (Ω' : Type uG) (_ : MeasureSpace Ω') (X' Y' : Ω' → G),
IsProbabilityMeasure (ℙ : Measure Ω') → Measurable X' → Measurable Y' →
phi X Y η A μ ≤ phi X' Y' η A ℙ | pfr/blueprint/src/chapter/further_improvement.tex:211 | pfr/PFR/RhoFunctional.lean:1131 |
PFR | phi_min_exists | \begin{lemma}[$\phi$-minimizers exist]\label{phi-min-exist}\lean{phi_min_exists}\leanok There exists a $\phi$-minimizer.
\end{lemma}
\begin{proof}\leanok\uses{rho-cts} Clear from compactness.
\end{proof} | /-- There exists a $\phi$-minimizer. -/
lemma phi_min_exists (hA : A.Nonempty) : ∃ (μ : Measure (G × G)), IsProbabilityMeasure μ ∧
phiMinimizes Prod.fst Prod.snd η A μ := by
let _i : TopologicalSpace G := (⊥ : TopologicalSpace G)
have : DiscreteTopology G := ⟨rfl⟩
let iG : Inhabited G := ⟨0⟩
have T : Continuous (fun (μ : ProbabilityMeasure (G × G)) ↦ phi Prod.fst Prod.snd η A μ) := by
apply continuous_iff_continuousAt.2 (fun μ ↦ ?_)
apply Tendsto.add
· apply tendsto_rdist_probabilityMeasure continuous_fst continuous_snd tendsto_id
apply Tendsto.const_mul
apply Tendsto.add
· apply tendsto_rho_probabilityMeasure continuous_fst hA tendsto_id
· apply tendsto_rho_probabilityMeasure continuous_snd hA tendsto_id
obtain ⟨μ, _, hμ⟩ := @IsCompact.exists_isMinOn ℝ (ProbabilityMeasure (G × G))
_ _ _ _ Set.univ isCompact_univ ⟨default, trivial⟩ _ T.continuousOn
refine ⟨μ, by infer_instance, ?_⟩
intro Ω' mΩ' X' Y' hP hX' hY'
let ν : Measure (G × G) := Measure.map (⟨X', Y'⟩) ℙ
have : IsProbabilityMeasure ν := isProbabilityMeasure_map (by fun_prop)
let ν' : ProbabilityMeasure (G × G) := ⟨ν, this⟩
have : phi Prod.fst Prod.snd η A ↑μ ≤ phi Prod.fst Prod.snd η A ↑ν' := hμ (mem_univ _)
apply this.trans_eq
have h₁ : IdentDistrib Prod.fst X' (ν' : Measure (G × G)) ℙ := by
refine ⟨measurable_fst.aemeasurable, hX'.aemeasurable, ?_⟩
simp only [ProbabilityMeasure.coe_mk, ν', ν]
rw [Measure.map_map measurable_fst (by fun_prop)]
rfl
have h₂ : IdentDistrib Prod.snd Y' (ν' : Measure (G × G)) ℙ := by
refine ⟨measurable_snd.aemeasurable, hY'.aemeasurable, ?_⟩
simp only [ProbabilityMeasure.coe_mk, ν', ν]
rw [Measure.map_map measurable_snd (by fun_prop)]
rfl
simp [phi, h₁.rdist_eq h₂, rho_eq_of_identDistrib h₁, rho_eq_of_identDistrib h₂]
-- Let $(X_1, X_2)$ be a $\phi$-minimizer, and $\tilde X_1, \tilde X_2$ be independent copies
-- of $X_1,X_2$ respectively.
variable {X₁ X₂ X₁' X₂' : Ω → G} (h_min : phiMinimizes X₁ X₂ η A ℙ)
(h₁ : IdentDistrib X₁ X₁')
(h₂ : IdentDistrib X₂ X₂')
(h_indep : iIndepFun ![X₁, X₂, X₁', X₂'])
(hX₁ : Measurable X₁) (hX₂ : Measurable X₂) (hX₁' : Measurable X₁') (hX₂' : Measurable X₂')
local notation3 "I₁" => I[X₁ + X₂ : X₁' + X₂ | X₁ + X₂ + X₁' + X₂']
local notation3 "I₂" => I[X₁ + X₂ : X₁ + X₁' | X₁ + X₂ + X₁' + X₂']
/-- `k := d[X₁ # X₂]`, the Ruzsa distance `rdist` between X₁ and X₂. -/
local notation3 "k" => d[X₁ # X₂] | pfr/blueprint/src/chapter/further_improvement.tex:216 | pfr/PFR/RhoFunctional.lean:1156 |
PFR | rdist_add_rdist_add_condMutual_eq | \begin{lemma}[Fibring identity for first estimate]\label{first-fibre}
\lean{rdist_add_rdist_add_condMutual_eq}\leanok
We have
\begin{align*}
& d[X_1+\tilde X_2;X_2+\tilde X_1] + d[X_1|X_1+\tilde X_2; X_2|X_2+\tilde X_1] \\
&\quad + \bbI[X_1+ X_2 : \tilde X_1 + X_2 \,|\, X_1 + X_2 + \tilde X_1 + \tilde X_2] = 2k.
\end{align*}
\end{lemma}
\begin{proof}\uses{cor-fibre} \leanok Immediate from \Cref{cor-fibre}.
\end{proof} | lemma rdist_add_rdist_add_condMutual_eq [Module (ZMod 2) G] :
d[X₁ + X₂' # X₂ + X₁'] + d[X₁ | X₁ + X₂' # X₂ | X₂ + X₁']
+ I[X₁ + X₂ : X₁' + X₂ | X₁ + X₂ + X₁' + X₂'] = 2 * k := by
have h0 : ![X₁, X₂, X₂', X₁'] 0 = X₁ := rfl
have h1 : ![X₁, X₂, X₂', X₁'] 1 = X₂ := rfl
have h2 : ![X₁, X₂, X₂', X₁'] 2 = X₂' := rfl
have h3 : ![X₁, X₂, X₂', X₁'] 3 = X₁' := rfl
have h := sum_of_rdist_eq_char_2 ![X₁, X₂, X₂', X₁'] h_indep
(fun i => by fin_cases i <;> assumption)
rw [h0, h1, h2, h3] at h
have heq : d[X₂' # X₁'] = k := by
rw [rdist_symm]
apply h₁.symm.rdist_eq h₂.symm
rw [heq] at h
convert h.symm using 1
· congr 2 <;> abel
· ring
include h_min hX₁ hX₂ hX₁' hX₂' in
/-- The distance $d[X_1+\tilde X_2; X_2+\tilde X_1]$ is at least
$$ k - \eta (d[X^0_1; X_1+\tilde X_2] - d[X^0_1; X_1]) - \eta (d[X^0_2; X_2+\tilde X_1] - d[X^0_2; X_2]).$$ -/ | pfr/blueprint/src/chapter/entropy_pfr.tex:79 | pfr/PFR/FirstEstimate.lean:53 |
PFR | rdist_add_rdist_eq | \begin{lemma}\label{I1-I2-diff}\lean{rdist_add_rdist_eq}\leanok
$d[X_1;X_1]+d[X_2;X_2]= 2d[X_1;X_2]+(I_2-I_1)$.
\end{lemma}
\begin{proof}\leanok
\uses{first-fibre,cor-fibre}
Compare \Cref{first-fibre} with the identity obtained from applying \Cref{cor-fibre} on $(X_1,\tilde X_1, X_2, \tilde X_2)$.
\end{proof} | /-- $d[X_1;X_1]+d[X_2;X_2]= 2d[X_1;X_2]+(I_2-I_1)$. -/
lemma rdist_add_rdist_eq :
d[ X₁ # X₁ ] + d[ X₂ # X₂ ] = 2 * k + (I₂ - I₁) := 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 : d[X₁ # X₁] + d[X₂ # X₂]
= d[X₁ + X₂' # X₂ + X₁'] + d[X₁ | X₁ + X₂' # X₂ | X₂ + X₁'] + I₂ :=
I_two_aux h₁ h₂ h_indep hX₁ hX₂ hX₁' hX₂'
linarith
include hX₁ hX₂ hX₁' hX₂' h₁ h₂ h_indep in | pfr/blueprint/src/chapter/further_improvement.tex:236 | pfr/PFR/RhoFunctional.lean:1380 |
PFR | rdist_def | \begin{definition}[Ruzsa distance]\label{ruz-dist-def}
\uses{entropy-def, independent-exist}
\lean{rdist_def}\leanok
Let $X,Y$ be $G$-valued random variables (not necessarily on the same sample space). The \emph{Ruzsa distance} $d[X ;Y]$ between $X$ and $Y$ is defined to be
$$ d[X ;Y] := \bbH[X' - Y'] - \bbH[X']/2 - \bbH[Y']/2$$
where $X',Y'$ are (the canonical) independent copies of $X,Y$ from \Cref{independent-exist}.
\end{definition} | /-- Explicit formula for the Ruzsa distance. -/
lemma rdist_def (X : Ω → G) (Y : Ω' → G) (μ : Measure Ω) (μ' : Measure Ω') :
d[X ; μ # Y ; μ']
= H[fun x ↦ x.1 - x.2 ; (μ.map X).prod (μ'.map Y)] - H[X ; μ]/2 - H[Y ; μ']/2 := rfl | pfr/blueprint/src/chapter/distance.tex:80 | pfr/PFR/ForMathlib/Entropy/RuzsaDist.lean:72 |
PFR | rdist_le_sum_fibre | \begin{proposition}[General fibring identity]\label{fibring-ident}
\lean{rdist_of_indep_eq_sum_fibre, rdist_le_sum_fibre}\leanok
Let $\pi : H \to H'$ be a homomorphism additive groups, and let $Z_1,Z_2$ be $H$-valued random variables. Then we have
\[
d[Z_1; Z_2] \geq d[\pi(Z_1);\pi(Z_2)] + d[Z_1|\pi(Z_1); Z_2 |\pi(Z_2)].
\]
Moreover, if $Z_1,Z_2$ are taken to be independent, then the difference between the two sides is
$$I( Z_1 - Z_2 : (\pi(Z_1), \pi(Z_2)) | \pi(Z_1 - Z_2) ).$$
\end{proposition}
\begin{proof}\uses{ruz-copy, independent-exist, submodularity,conditional-mutual-alt,chain-rule,relabeled-entropy, cond-dist-alt}\leanok
Let $Z_1,Z_2$ be independent throughout (this is possible by \Cref{ruz-copy} and \Cref{independent-exist}). By \Cref{cond-dist-alt}, We have
\begin{align*}
& d[Z_1 |\pi(Z_1); Z_2 |\pi(Z_2)] \\
& = \bbH[Z_1 - Z_2 | \pi(Z_1),\pi(Z_2)] - \tfrac{1}{2} \bbH[Z_1 | \pi(Z_1)] - \tfrac{1}{2} \bbH[Z_2 | \pi(Z_2)] \\
& \leq \bbH[Z_1 - Z_2 | \pi(Z_1+Z_2)] - \tfrac{1}{2} \bbH[Z_1 | \pi(Z_1)] - \tfrac{1}{2}H[Z_2 | \pi(Z_2)] \\
& = d[Z_1;Z_2] - d[\pi(Z_1);\pi(Z_2)].
\end{align*}
In the middle step, we used \Cref{submodularity}, and in the last step we used the fact that
\[\bbH[Z_1 - Z_2 | \pi(Z_1-Z_2)] = \bbH[Z_1 - Z_2] - \bbH[\pi(Z_1-Z_2)]\]
(thanks to \Cref{chain-rule} and \Cref{relabeled-entropy}) and that
\[\bbH[Z_i| \pi(Z_i)] = \bbH[Z_i] - \bbH[\pi(Z_i)]\] (since $Z_i$ determines $\pi(Z_i)$).
This gives the claimed inequality. The difference between the two sides is precisely
\[\bbH[Z_1 - Z_2 | \pi(Z_1 - Z_2)] - \bbH[Z_1 - Z_2 | \pi(Z_1),\pi(Z_2)].\]
To rewrite this in terms of (conditional) mutual information, we use the identity
\[\bbH[A|B] - \bbH[A | B,C] = \bbI[A : C | B],\]
(which follows \Cref{conditional-mutual-alt})
taking
$A := Z_1 - Z_2$, $B := \pi(Z_1 - Z_2)$ and $C := (\pi(Z_1),\pi(Z_{2}))$, and noting that in this case $\bbH[A | B,C] = \bbH[A | C]$ since $C$ uniquely determines $B$ (this may require another helper lemma about entropy).
This completes the proof.
\end{proof} | lemma rdist_le_sum_fibre {Z_1 : Ω → H} {Z_2 : Ω' → H}
(h1 : Measurable Z_1) (h2 : Measurable Z_2) [FiniteRange Z_1] [FiniteRange Z_2] :
d[π ∘ Z_1; μ # π ∘ Z_2; μ'] + d[Z_1|π∘Z_1; μ # Z_2|π∘Z_2; μ'] ≤ d[Z_1; μ # Z_2; μ']:= by
obtain ⟨ν, W_1, W_2, hν, m1, m2, hi, hi1, hi2, _, _⟩ := ProbabilityTheory.independent_copies_finiteRange h1 h2 μ μ'
have hπ : Measurable π := .of_discrete
have hφ : Measurable (fun x ↦ (x, π x)) := .of_discrete
have hπ1 : IdentDistrib (⟨Z_1, π ∘ Z_1⟩) (⟨W_1, π ∘ W_1⟩) μ ν := hi1.symm.comp hφ
have hπ2 : IdentDistrib (⟨Z_2, π ∘ Z_2⟩) (⟨W_2, π ∘ W_2⟩) μ' ν := hi2.symm.comp hφ
rw [← hi1.rdist_eq hi2, ← (hi1.comp hπ).rdist_eq (hi2.comp hπ),
rdist_of_indep_eq_sum_fibre π hi m1 m2,
condRuzsaDist_of_copy h1 (hπ.comp h1) h2 (hπ.comp h2) m1 (hπ.comp m1) m2 (hπ.comp m2) hπ1 hπ2]
exact le_add_of_nonneg_right (condMutualInfo_nonneg (by fun_prop) (by fun_prop)) | pfr/blueprint/src/chapter/fibring.tex:3 | pfr/PFR/Fibring.lean:62 |
PFR | rdist_of_hom_le | \begin{corollary}\label{fibring-ineq}
\lean{rdist_of_hom_le} \leanok
If $\pi:G\to H$ is a homomorphism of additive groups and $X,Y$ are $G$-valued random variables then
\[d[X;Y]\geq d[\pi(X);\pi(Y)].\]
\end{corollary}
\begin{proof}
\uses{fibring-ident, ruzsa-nonneg}\leanok
By \Cref{fibring-ident} and the nonnegativity of conditional Ruzsa distance (from \Cref{ruzsa-nonneg}) we have
\[d[X;Y]\geq d[\pi(X);\pi(Y)]+d[X\mid \pi(X);Y\mid \pi(Y)].\]
The inequality follows from $d[X\mid \pi(X);Y\mid \pi(Y)]\geq 0$ (\Cref{ruzsa-nonneg}).
\end{proof} | /-- \[d[X;Y]\geq d[\pi(X);\pi(Y)].\] -/
lemma rdist_of_hom_le {Z_1 : Ω → H} {Z_2 : Ω' → H}
(h1 : Measurable Z_1) (h2 : Measurable Z_2) [FiniteRange Z_1] [FiniteRange Z_2] :
d[π ∘ Z_1; μ # π ∘ Z_2; μ'] ≤ d[Z_1; μ # Z_2; μ'] := by
apply le_trans _ (rdist_le_sum_fibre π h1 h2 (μ := μ) (μ' := μ'))
rw [le_add_iff_nonneg_right]
exact condRuzsaDist_nonneg h1 (by fun_prop) h2 (by fun_prop)
end GeneralFibring
variable {G : Type*} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G]
[MeasurableSingletonClass G]
variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {μ : Measure Ω} [IsProbabilityMeasure μ] | pfr/blueprint/src/chapter/fibring.tex:37 | pfr/PFR/Fibring.lean:75 |
PFR | rdist_of_indep_eq_sum_fibre | \begin{proposition}[General fibring identity]\label{fibring-ident}
\lean{rdist_of_indep_eq_sum_fibre, rdist_le_sum_fibre}\leanok
Let $\pi : H \to H'$ be a homomorphism additive groups, and let $Z_1,Z_2$ be $H$-valued random variables. Then we have
\[
d[Z_1; Z_2] \geq d[\pi(Z_1);\pi(Z_2)] + d[Z_1|\pi(Z_1); Z_2 |\pi(Z_2)].
\]
Moreover, if $Z_1,Z_2$ are taken to be independent, then the difference between the two sides is
$$I( Z_1 - Z_2 : (\pi(Z_1), \pi(Z_2)) | \pi(Z_1 - Z_2) ).$$
\end{proposition}
\begin{proof}\uses{ruz-copy, independent-exist, submodularity,conditional-mutual-alt,chain-rule,relabeled-entropy, cond-dist-alt}\leanok
Let $Z_1,Z_2$ be independent throughout (this is possible by \Cref{ruz-copy} and \Cref{independent-exist}). By \Cref{cond-dist-alt}, We have
\begin{align*}
& d[Z_1 |\pi(Z_1); Z_2 |\pi(Z_2)] \\
& = \bbH[Z_1 - Z_2 | \pi(Z_1),\pi(Z_2)] - \tfrac{1}{2} \bbH[Z_1 | \pi(Z_1)] - \tfrac{1}{2} \bbH[Z_2 | \pi(Z_2)] \\
& \leq \bbH[Z_1 - Z_2 | \pi(Z_1+Z_2)] - \tfrac{1}{2} \bbH[Z_1 | \pi(Z_1)] - \tfrac{1}{2}H[Z_2 | \pi(Z_2)] \\
& = d[Z_1;Z_2] - d[\pi(Z_1);\pi(Z_2)].
\end{align*}
In the middle step, we used \Cref{submodularity}, and in the last step we used the fact that
\[\bbH[Z_1 - Z_2 | \pi(Z_1-Z_2)] = \bbH[Z_1 - Z_2] - \bbH[\pi(Z_1-Z_2)]\]
(thanks to \Cref{chain-rule} and \Cref{relabeled-entropy}) and that
\[\bbH[Z_i| \pi(Z_i)] = \bbH[Z_i] - \bbH[\pi(Z_i)]\] (since $Z_i$ determines $\pi(Z_i)$).
This gives the claimed inequality. The difference between the two sides is precisely
\[\bbH[Z_1 - Z_2 | \pi(Z_1 - Z_2)] - \bbH[Z_1 - Z_2 | \pi(Z_1),\pi(Z_2)].\]
To rewrite this in terms of (conditional) mutual information, we use the identity
\[\bbH[A|B] - \bbH[A | B,C] = \bbI[A : C | B],\]
(which follows \Cref{conditional-mutual-alt})
taking
$A := Z_1 - Z_2$, $B := \pi(Z_1 - Z_2)$ and $C := (\pi(Z_1),\pi(Z_{2}))$, and noting that in this case $\bbH[A | B,C] = \bbH[A | C]$ since $C$ uniquely determines $B$ (this may require another helper lemma about entropy).
This completes the proof.
\end{proof} | lemma rdist_of_indep_eq_sum_fibre {Z_1 Z_2 : Ω → H} (h : IndepFun Z_1 Z_2 μ)
(h1 : Measurable Z_1) (h2 : Measurable Z_2) [FiniteRange Z_1] [FiniteRange Z_2]:
d[Z_1; μ # Z_2; μ] = d[π ∘ Z_1; μ # π ∘ Z_2; μ] + d[Z_1|π∘Z_1; μ # Z_2|π∘Z_2; μ] + I[Z_1-Z_2 : ⟨π∘Z_1, π∘Z_2⟩ | π∘(Z_1 - Z_2); μ] := by
have hπ : Measurable π := .of_discrete
have step1 : d[Z_1; μ # Z_2; μ] = d[π ∘ Z_1; μ # π ∘ Z_2; μ] +
H[(Z_1 - Z_2)| π ∘ (Z_1 - Z_2); μ] - H[Z_1 | π ∘ Z_1; μ] / 2 - H[Z_2 | π ∘ Z_2; μ] / 2 := by
have hsub : H[(Z_1 - Z_2)| π ∘ (Z_1 - Z_2); μ] = H[(Z_1 - Z_2); μ] - H[π ∘ (Z_1 - Z_2); μ] :=
condEntropy_comp_self (by fun_prop) hπ
rw [h.rdist_eq h1 h2, (h.comp hπ hπ).rdist_eq (hπ.comp h1) (hπ.comp h2),
condEntropy_comp_self h1 hπ, condEntropy_comp_self h2 hπ, hsub, map_comp_sub π]
ring_nf
have m0 : Measurable (fun x ↦ (x, π x)) := .of_discrete
have h' : IndepFun (⟨Z_1, π ∘ Z_1⟩) (⟨Z_2, π ∘ Z_2⟩) μ := h.comp m0 m0
have m1 : Measurable (Z_1 - Z_2) := h1.sub h2
have m2 : Measurable (⟨↑π ∘ Z_1, ↑π ∘ Z_2⟩) := (hπ.comp h1).prodMk (hπ.comp h2)
have m3 : Measurable (↑π ∘ (Z_1 - Z_2)) := hπ.comp m1
have entroplem : H[Z_1 - Z_2|⟨⟨↑π ∘ Z_1, ↑π ∘ Z_2⟩, ↑π ∘ (Z_1 - Z_2)⟩; μ]
= H[Z_1 - Z_2|⟨↑π ∘ Z_1, ↑π ∘ Z_2⟩; μ] := by
rw [map_comp_sub π]
let f : H' × H' → (H' × H') × H' := fun (x,y) ↦ ((x,y), x - y)
have hf : Injective f := fun _ _ h ↦ (Prod.ext_iff.1 h).1
have mf : Measurable f := measurable_id.prodMk measurable_sub
refine condEntropy_of_injective' μ m1 m2 f hf (mf.comp m2)
rw [step1, condMutualInfo_eq' m1 m2 m3, entroplem,
condRuzsaDist_of_indep h1 (hπ.comp h1) h2 (hπ.comp h2) μ h']
ring_nf | pfr/blueprint/src/chapter/fibring.tex:3 | pfr/PFR/Fibring.lean:35 |
PFR | rdist_of_neg_le | \begin{lemma}[Flipping a sign]\label{sign-flip}\lean{rdist_of_neg_le}\leanok
If $X,Y$ are $G$-valued, then
$$ d[X ; -Y] \leq 3 d[X;Y].$$
\end{lemma}
\begin{proof}\uses{ruz-copy, independent-exist, cond-indep-exist, alt-submodularity, ruz-indep, neg-ent, relabeled-entropy, add-entropy, subadditive, data-process-single}\leanok
Without loss of generality (using \Cref{ruz-copy} and \Cref{independent-exist}) we may take $X,Y$ to be independent.
By $(X_1,Y_1)$, $(X_2,Y_2)$ be copies of $(X,Y)$ that are conditionally independent over $X_1-Y_1=X_2-Y_2$ (this exists thanks to \Cref{cond-indep-exist}). By \Cref{independent-exist}, we can also find another copy $(X_3,Y_3)$ of $(X,Y)$ that is independent of $X_1,Y_1,X_2,Y_2$. From \Cref{alt-submodularity}, one has
$$ \bbH[X_3-Y_2, X_1-Y_3, X_2, Y_1, X_3, Y_3, X_3+Y_3] + \bbH[X_3+Y_3] \leq \bbH[X_3-Y_2, X_1-Y_3, X_2, Y_1, X_3+Y_3] + \bbH[X_3, Y_3, X_3+Y_3].$$
From \Cref{ruz-indep}, \Cref{neg-ent}, \Cref{ruz-copy} we have
$$ \bbH[X_3+Y_3] = \frac{1}{2} \bbH[X_3] + \frac{1}{2} \bbH[-Y_3] + d[X_3;-Y_3] = \frac{1}{2} \bbH[X] + \frac{1}{2} \bbH[Y] + d[X;-Y].$$
Since $X_3+Y_3$ is a function of $X_3,Y_3$, we see from \Cref{relabeled-entropy} and \Cref{add-entropy} that
$$ \bbH[X_3,Y_3,X_3+Y_3] = \bbH[X_3,Y_3] = \bbH[X,Y] = \bbH[X]+\bbH[Y].$$
Because $X_1-Y_1=X_2-Y_2$, we have
$$ X_3+Y_3 = (X_3-Y_2) - (X_1-Y_3) + (X_2+Y_1)$$
and thus by \Cref{relabeled-entropy}
$$ \bbH[X_3-Y_2, X_1-Y_3, X_2, Y_1, X_3+Y_3] = \bbH[X_3-Y_2, X_1-Y_3, X_2, Y_1]$$
and hence by \Cref{subadditive}
$$ \bbH[X_3-Y_2, X_1-Y_3, X_2, Y_1, X_3+Y_3] \leq \bbH[X_3-Y_2] + \bbH[X_1-Y_3] + \bbH[X_2] + \bbH[Y_1].$$
Since $X_3,Y_2$ are independent, we see from \Cref{ruz-indep}, \Cref{ruz-copy} that
$$\bbH[X_3-Y_2] = \frac{1}{2} \bbH[X] + \frac{1}{2} \bbH[Y] + d[X; Y].$$
Similarly
$$ \bbH[X_1-Y_3] = \frac{1}{2} \bbH[X] + \frac{1}{2} \bbH[Y] + d[X; Y].$$
We conclude that
$$ \bbH[X_3-Y_2, X_1-Y_3, X_2, Y_1, X_3+Y_3] \leq 2\bbH[X] + 2\bbH[Y] + 2d[X; Y].$$
Finally, from \Cref{data-process-single} we have
$$ \bbH[X_1,Y_1,X_2,Y_2,X_3,Y_3] \leq \bbH[X_3-Y_2, X_1-Y_3, X_2, Y_1, X_3, Y_3, X_3+Y_3].$$
From \Cref{add-entropy} followed by \Cref{cond-trial-ent}, we have
$$\bbH[X_1,Y_1,X_2,Y_2,X_3,Y_3] = \bbH[X_1,Y_1,X_1-Y_1] + \bbH[X_2,Y_2,X_2-Y_2] - \bbH[X_1-Y_1] + \bbH[X_3,Y_3]$$
and thus by \Cref{ruz-indep}, \Cref{ruz-copy}, \Cref{relabeled-entropy}, \Cref{add-entropy}
$$\bbH[X_1,Y_1,X_2,Y_2,X_3,Y_3] = \bbH[X] + \bbH[Y] + \bbH[X] + \bbH[Y] -\left(\frac{1}{2}\bbH[X] + \frac{1}{2}\bbH[Y] + d[X; Y]\right) + \bbH[X] + \bbH[Y].$$
Applying all of these estimates, the claim now follows from linear arithmetic.
\end{proof} | /-- If `X, Y` are `G`-valued, then `d[X;-Y] ≤ 3 d[X;Y]`. -/
lemma rdist_of_neg_le [IsProbabilityMeasure μ] [IsProbabilityMeasure μ'] (hX : Measurable X)
(hY : Measurable Y) [Fintype G] :
d[X ; μ # -Y ; μ'] ≤ 3 * d[X ; μ # Y ; μ'] := by
obtain ⟨ν, X', Y', hν, mX', mY', h_indep', hXX', hYY'⟩ := independent_copies hX hY μ μ'
rw [← IdentDistrib.rdist_eq hXX' hYY', ← IdentDistrib.rdist_eq hXX' (IdentDistrib.neg hYY')]
obtain ⟨Ω₀, mΩ₀, XY'₁, XY'₂, Z', ν'₀, hν'₀, hXY'₁, hXY'₂, hZ', h_condIndep, h_id1sub, h_id2sub⟩
:= condIndep_copies (⟨X', Y'⟩) (X' - Y') (mX'.prodMk mY') (by fun_prop) ν
let X₁' := fun ω ↦ (XY'₁ ω).fst
let Y'₁ := fun ω ↦ (XY'₁ ω).snd
let X₂' := fun ω ↦ (XY'₂ ω).fst
let Y'₂ := fun ω ↦ (XY'₂ ω).snd
have mX₁' : Measurable X₁' := by fun_prop
have mY'₁ : Measurable Y'₁ := by fun_prop
have Z'eq1 : Z' =ᵐ[ν'₀] X₁' - Y'₁ :=
(IdentDistrib.ae_snd h_id1sub.symm (MeasurableSet.of_discrete (s := {x | x.2 = x.1.1 - x.1.2}))
(Eventually.of_forall fun ω ↦ rfl) :)
obtain ⟨ν₀, XY₁XY₂Z, XY₃, hν₀, hXY₁XY₂Z, hXY₃, h_indep, h_idXY₁XY₂Z, h_idXY₃⟩ :=
independent_copies (hXY'₁.prodMk hXY'₂ |>.prodMk hZ') (mX'.prodMk mY') ν'₀ ν
let X₁ := fun ω ↦ (XY₁XY₂Z ω).fst.fst.fst
let Y₁ := fun ω ↦ (XY₁XY₂Z ω).fst.fst.snd
let X₂ := fun ω ↦ (XY₁XY₂Z ω).fst.snd.fst
let Y₂ := fun ω ↦ (XY₁XY₂Z ω).fst.snd.snd
let Z := fun ω ↦ (XY₁XY₂Z ω).snd
let X₃ := fun ω ↦ (XY₃ ω).fst
let Y₃ := fun ω ↦ (XY₃ ω).snd
have mX₁ : Measurable X₁ := by fun_prop
have mY₁ : Measurable Y₁ := by fun_prop
have mX₂ : Measurable X₂ := by fun_prop
have mY₂ : Measurable Y₂ := by fun_prop
have mX₃ : Measurable X₃ := by fun_prop
have mY₃ : Measurable Y₃ := by fun_prop
have mZ : Measurable Z := by fun_prop
have idXY₁Z : IdentDistrib (⟨⟨X₁, Y₁⟩, Z⟩) (⟨⟨X', Y'⟩, X' - Y'⟩) ν₀ ν :=
h_idXY₁XY₂Z.comp (.of_discrete (f := fun x ↦ (x.1.1, x.2))) |>.trans h_id1sub
have idXY₂Z : IdentDistrib (⟨⟨X₂, Y₂⟩, Z⟩) (⟨⟨X', Y'⟩, X' - Y'⟩) ν₀ ν :=
h_idXY₁XY₂Z.comp (.of_discrete (f := fun x ↦ (x.1.2, x.2))) |>.trans h_id2sub
have idXY₁ : IdentDistrib (⟨X₁, Y₁⟩) (⟨X', Y'⟩) ν₀ ν := by
convert h_idXY₁XY₂Z.comp (.of_discrete (f := fun x ↦ x.1.1)) |>.trans ?_
exact h_id1sub.comp (.of_discrete (f := fun ((x, y), _) ↦ (x, y)))
have idXY₂ : IdentDistrib (⟨X₂, Y₂⟩) (⟨X', Y'⟩) ν₀ ν := by
convert h_idXY₁XY₂Z.comp (.of_discrete (f := fun x ↦ x.1.2)) |>.trans ?_
exact h_id2sub.comp (.of_discrete (f := fun ((x, y), _) ↦ (x, y)))
have idXY₃ : IdentDistrib (⟨X₃, Y₃⟩) (⟨X', Y'⟩) ν₀ ν := h_idXY₃
have idX₁ : IdentDistrib X₁ X' ν₀ ν := idXY₁.comp (by fun_prop)
have idY₁ : IdentDistrib Y₁ Y' ν₀ ν := idXY₁.comp (by fun_prop)
have idX₂ : IdentDistrib X₂ X' ν₀ ν := idXY₂.comp (by fun_prop)
have idY₂ : IdentDistrib Y₂ Y' ν₀ ν := idXY₂.comp (by fun_prop)
have idX₃ : IdentDistrib X₃ X' ν₀ ν := idXY₃.comp (by fun_prop)
have idY₃ : IdentDistrib Y₃ Y' ν₀ ν := idXY₃.comp (by fun_prop)
have idXY₁₂XY'₁₂ : IdentDistrib (⟨⟨X₁, Y₁⟩, ⟨X₂, Y₂⟩⟩) (⟨⟨X₁', Y'₁⟩, ⟨X₂', Y'₂⟩⟩) ν₀ ν'₀ :=
h_idXY₁XY₂Z.comp (.of_discrete (f := fun x ↦ x.1))
have idXY₁ZXY'₁Z' : IdentDistrib (⟨⟨X₁, Y₁⟩, Z⟩) (⟨⟨X₁', Y'₁⟩, Z'⟩) ν₀ ν'₀ :=
h_idXY₁XY₂Z.comp (.of_discrete (f := fun x ↦ (x.1.1, x.2)))
have idXY₂ZXY'₂Z' : IdentDistrib (⟨⟨X₂, Y₂⟩, Z⟩) (⟨⟨X₂', Y'₂⟩, Z'⟩) ν₀ ν'₀ :=
h_idXY₁XY₂Z.comp (.of_discrete (f := fun x ↦ (x.1.2, x.2)))
have idZZ' : IdentDistrib Z Z' ν₀ ν'₀ :=
h_idXY₁XY₂Z.comp .of_discrete
have Zeq1 : Z =ᵐ[ν₀] X₁ - Y₁ := (IdentDistrib.ae_snd idXY₁Z.symm
(MeasurableSet.of_discrete (s := {x | x.2 = x.1.1 - x.1.2}))
(Eventually.of_forall fun ω ↦ rfl) :)
have Zeq2 : Z =ᵐ[ν₀] X₂ - Y₂ :=
(IdentDistrib.ae_snd idXY₂Z.symm (MeasurableSet.of_discrete (s := {x | x.2 = x.1.1 - x.1.2}))
(Eventually.of_forall fun ω ↦ rfl) :)
have iX₁Y₃ : IndepFun X₁ Y₃ ν₀ := by
convert h_indep.comp (.of_discrete (f := fun x ↦ x.1.1.1)) (.of_discrete (f := fun x ↦ x.2))
have iX₃Y₂ : IndepFun X₃ Y₂ ν₀ := by
convert h_indep.symm.comp (.of_discrete (f := fun x ↦ x.1)) (.of_discrete (f := fun x ↦ x.1.2.2))
have iX₁Y₁ : IndepFun X₁ Y₁ ν₀ := indepFun_of_identDistrib_pair h_indep' idXY₁.symm
have iX₂Y₂ : IndepFun X₂ Y₂ ν₀ := indepFun_of_identDistrib_pair h_indep' idXY₂.symm
have iX₃Y₃ : IndepFun X₃ Y₃ ν₀ := indepFun_of_identDistrib_pair h_indep' idXY₃.symm
have iX₃negY₃ : IndepFun X₃ (-Y₃) ν₀ := iX₃Y₃.comp measurable_id measurable_neg
have i112233 : IndepFun (⟨⟨X₁, Y₁⟩, ⟨X₂, Y₂⟩⟩) (⟨X₃, Y₃⟩) ν₀ :=
h_indep.comp (.of_discrete (f := fun (xy, _) ↦ xy)) measurable_id
have hX1 : H[X' ; ν] = H[X₁ ; ν₀] := idX₁.entropy_eq.symm
have hX2 : H[X' ; ν] = H[X₂ ; ν₀] := idX₂.entropy_eq.symm
have hX3 : H[X' ; ν] = H[X₃ ; ν₀] := idX₃.entropy_eq.symm
have hY1 : H[Y' ; ν] = H[Y₁ ; ν₀] := idY₁.entropy_eq.symm
have hY2 : H[Y' ; ν] = H[Y₂ ; ν₀] := idY₂.entropy_eq.symm
have hY3 : H[Y' ; ν] = H[Y₃ ; ν₀] := idY₃.entropy_eq.symm
have hnegY3 : H[Y₃ ; ν₀] = H[-Y₃ ; ν₀] := (entropy_neg mY₃).symm
have hX1Y1 : H[⟨X₁, Y₁⟩; ν₀] = H[X'; ν] + H[Y'; ν] :=
hX1.symm ▸ hY1.symm ▸ (entropy_pair_eq_add mX₁ mY₁).mpr iX₁Y₁
have hX2Y2 : H[⟨X₂, Y₂⟩; ν₀] = H[X'; ν] + H[Y'; ν] :=
hX2.symm ▸ hY2.symm ▸ (entropy_pair_eq_add mX₂ mY₂).mpr iX₂Y₂
have hX3Y3 : H[⟨X₃, Y₃⟩; ν₀] = H[X'; ν] + H[Y'; ν] :=
hX3.symm ▸ hY3.symm ▸ (entropy_pair_eq_add mX₃ mY₃).mpr iX₃Y₃
have dX3negY3 : d[X' ; ν # -Y' ; ν] = d[X₃ ; ν₀ # -Y₃ ; ν₀] := (idX₃.rdist_eq idY₃.neg).symm
have dX1Y1 : d[X' ; ν # Y' ; ν] = d[X₁ ; ν₀ # Y₁ ; ν₀] := (idX₁.rdist_eq idY₁).symm
have dX1Y3 : d[X' ; ν # Y' ; ν] = d[X₁ ; ν₀ # Y₃ ; ν₀] := (idX₁.rdist_eq idY₃).symm
have dX3Y2 : d[X' ; ν # Y' ; ν] = d[X₃ ; ν₀ # Y₂ ; ν₀] := (idX₃.rdist_eq idY₂).symm
have meas1321 : Measurable (⟨X₁ - Y₃, ⟨X₂, Y₁⟩⟩) := (mX₁.sub mY₃).prodMk <| mX₂.prodMk mY₁
have meas321321 : Measurable (⟨X₃ - Y₂, ⟨X₁ - Y₃, ⟨X₂, Y₁⟩⟩⟩) := (mX₃.sub mY₂).prodMk meas1321
have meas11 : Measurable (⟨X₁, Y₁⟩) := mX₁.prodMk mY₁
have meas22 : Measurable (⟨X₂, Y₂⟩) := mX₂.prodMk mY₂
have meas1122 : Measurable (⟨⟨X₁, Y₁⟩, ⟨X₂, Y₂⟩⟩) := meas11.prodMk meas22
have meas33 : Measurable (⟨X₃, Y₃⟩) := mX₃.prodMk mY₃
have meas1neg1 : Measurable (X₁ - Y₁) := mX₁.sub mY₁
have in1 : H[⟨⟨X₃ - Y₂, ⟨X₁ - Y₃, ⟨X₂, Y₁⟩⟩⟩, ⟨⟨X₃, Y₃⟩, X₃ + Y₃⟩⟩ ; ν₀] + H[X₃ + Y₃; ν₀]
≤ H[⟨⟨X₃ - Y₂, ⟨X₁ - Y₃, ⟨X₂, Y₁⟩⟩⟩, X₃ + Y₃⟩ ; ν₀] + H[⟨⟨X₃, Y₃⟩, X₃ + Y₃⟩ ; ν₀] :=
entropy_triple_add_entropy_le _ (by fun_prop) meas33 (mX₃.add mY₃)
have eq2 : H[X₃ + Y₃; ν₀] = 1/2 * H[X'; ν] + 1/2 * H[Y'; ν] + d[X'; ν # -Y'; ν] := by
rw [hX3, hY3, dX3negY3, hnegY3, IndepFun.rdist_eq iX₃negY₃ mX₃ mY₃.neg, sub_neg_eq_add]
ring
have eq3 : H[⟨⟨X₃, Y₃⟩, X₃ + Y₃⟩ ; ν₀] = H[X'; ν] + H[Y'; ν] :=
hX3Y3 ▸ entropy_of_comp_eq_of_comp ν₀ (meas33 |>.prodMk <| mX₃.add mY₃) meas33
(fun ((x3, y3), _) ↦ (x3, y3)) (fun (x3, y3) ↦ ((x3, y3), x3 + y3)) rfl rfl
have eq4' : X₁ =ᵐ[ν₀] X₂ - Y₂ + Y₁ := by
filter_upwards [Zeq1, Zeq2] with ω hZ hZ'
simp only [Pi.add_apply, ← hZ', hZ, Pi.sub_apply, sub_add_cancel]
have eq4 : X₃ + Y₃ =ᵐ[ν₀] (X₃ - Y₂) - (X₁ - Y₃) + X₂ + Y₁ := by
filter_upwards [eq4'] with ω h
simp only [Pi.add_apply, sub_eq_add_neg, neg_add_rev, neg_neg, add_assoc, Pi.neg_apply, h,
neg_add_cancel, add_zero, neg_add_cancel_comm_assoc]
have eq5 : H[⟨⟨X₃ - Y₂, ⟨X₁ - Y₃, ⟨X₂, Y₁⟩⟩⟩, X₃ + Y₃⟩ ; ν₀]
= H[⟨X₃ - Y₂, ⟨X₁ - Y₃, ⟨X₂, Y₁⟩⟩⟩ ; ν₀] :=
calc
_ = H[⟨⟨X₃ - Y₂, ⟨X₁ - Y₃, ⟨X₂, Y₁⟩⟩⟩, (X₃ - Y₂) - (X₁ - Y₃) + X₂ + Y₁⟩ ; ν₀] := by
refine IdentDistrib.entropy_eq <|
IdentDistrib.of_ae_eq (meas321321.prodMk <| mX₃.add mY₃).aemeasurable ?_
filter_upwards [eq4] with ω h
simp only [Prod.mk.injEq, h, Pi.add_apply, Pi.sub_apply, and_self]
_ = _ := by
refine entropy_of_comp_eq_of_comp ν₀
(meas321321.prodMk <| (((mX₃.sub mY₂).sub (mX₁.sub mY₃)).add mX₂).add mY₁) meas321321
(fun ((x3y2, (x1y3, (x2, y1))), _) ↦ (x3y2, (x1y3, (x2, y1))))
(fun (x3y2, (x1y3, (x2, y1))) ↦ ((x3y2, (x1y3, (x2, y1))), x3y2 - x1y3 + x2 + y1))
rfl rfl
have in6 : H[⟨⟨X₃ - Y₂, ⟨X₁ - Y₃, ⟨X₂, Y₁⟩⟩⟩, X₃ + Y₃⟩ ; ν₀]
≤ H[X₃ - Y₂; ν₀] + H[X₁ - Y₃; ν₀] + H[X₂; ν₀] + H[Y₁; ν₀] := by
rw [eq5]
refine (entropy_pair_le_add ?_ meas1321 ν₀).trans ?_
· exact (mX₃.sub mY₂)
simp only [add_assoc, add_le_add_iff_left]
refine (entropy_pair_le_add ?_ ?_ ν₀).trans ?_
· exact (mX₁.sub mY₃)
· exact (mX₂.prodMk mY₁)
simp only [add_assoc, add_le_add_iff_left]
exact entropy_pair_le_add mX₂ mY₁ ν₀
have eq7 : H[X₃ - Y₂; ν₀] = 1/2 * (H[X'; ν] + H[Y'; ν]) + d[X'; ν # Y'; ν] := by
rw [dX3Y2, IndepFun.rdist_eq iX₃Y₂ mX₃ mY₂, hX3, hY2]
ring_nf
have eq8 : H[X₁ - Y₃; ν₀] = 1/2 * (H[X'; ν] + H[Y'; ν]) + d[X'; ν # Y'; ν] := by
rw [dX1Y3, IndepFun.rdist_eq iX₁Y₃ mX₁ mY₃, hX1, hY3]
ring_nf
have eq8' : H[X₁ - Y₁; ν₀] = 1/2 * (H[X'; ν] + H[Y'; ν]) + d[X'; ν # Y'; ν] := by
rw [dX1Y1, IndepFun.rdist_eq iX₁Y₁ mX₁ mY₁, hX1, hY1]
ring_nf
have in9 : H[⟨⟨X₃ - Y₂, ⟨X₁ - Y₃, ⟨X₂, Y₁⟩⟩⟩, X₃ + Y₃⟩ ; ν₀]
≤ 2 * H[X'; ν] + 2 * H[Y'; ν] + 2 * d[X'; ν # Y'; ν] := by
rw [eq7, eq8, ← hX2, ← hY1] at in6
ring_nf at in6 ⊢
exact in6
have in10 : H[⟨X₁, ⟨Y₁, ⟨X₂, ⟨Y₂, ⟨X₃, Y₃⟩⟩⟩⟩⟩ ; ν₀]
≤ H[⟨⟨X₃ - Y₂, ⟨X₁ - Y₃, ⟨X₂, Y₁⟩⟩⟩, ⟨⟨X₃, Y₃⟩, X₃ + Y₃⟩⟩ ; ν₀] := by
convert entropy_comp_le ν₀
(meas321321.prodMk <| meas33.prodMk <| mX₃.add mY₃)
(fun ((x3y2, (x1y3, (x2, y1))), ((x3, y3), _))
↦ (x1y3 + y3, (y1, (x2, (x3 - x3y2, (x3, y3))))))
<;> simp only [comp_apply, Pi.sub_apply, sub_add_cancel, sub_sub_cancel]
have eq11 : H[⟨X₁, ⟨Y₁, ⟨X₂, ⟨Y₂, ⟨X₃, Y₃⟩⟩⟩⟩⟩ ; ν₀]
= H[⟨X₁, ⟨Y₁, X₁ - Y₁⟩⟩ ; ν₀] + H[⟨X₂, ⟨Y₂, X₂ - Y₂⟩⟩ ; ν₀]
- H[X₁ - Y₁; ν₀] + H[⟨X₃, Y₃⟩ ; ν₀] := by
calc
_ = H[⟨⟨X₁', Y'₁⟩, ⟨X₂', Y'₂⟩⟩ ; ν'₀] + H[⟨X₃, Y₃⟩ ; ν₀] := by
rw [← idXY₁₂XY'₁₂.entropy_eq, ← (entropy_pair_eq_add meas1122 meas33).mpr i112233]
exact entropy_of_comp_eq_of_comp ν₀
(mX₁.prodMk <| mY₁.prodMk <| mX₂.prodMk <| mY₂.prodMk <| meas33)
(meas1122.prodMk meas33)
(fun (x1, (y1, (x2, (y2, (x3, y3))))) ↦ (((x1, y1), (x2, y2)), (x3, y3)))
(fun (((x1, y1), (x2, y2)), (x3, y3)) ↦ (x1, (y1, (x2, (y2, (x3, y3)))))) rfl rfl
_ = H[⟨⟨X₁', Y'₁⟩, ⟨⟨X₂', Y'₂⟩, X₁' - Y'₁⟩⟩ ; ν'₀] + H[⟨X₃, Y₃⟩ ; ν₀] := by
congr 1
exact entropy_of_comp_eq_of_comp ν'₀ (hXY'₁.prodMk hXY'₂)
(hXY'₁.prodMk <| hXY'₂.prodMk <| mX₁'.sub mY'₁)
(fun ((x1, y1), (x2, y2)) ↦ ((x1, y1), ((x2, y2), x1 - y1)))
(fun ((x1, y1), ((x2, y2), _)) ↦ ((x1, y1), (x2, y2))) rfl rfl
_ = H[⟨⟨X₁', Y'₁⟩, ⟨⟨X₂', Y'₂⟩, Z'⟩⟩ ; ν'₀] + H[⟨X₃, Y₃⟩ ; ν₀] := by
congr 1
refine IdentDistrib.entropy_eq <| IdentDistrib.of_ae_eq
(hXY'₁.prodMk <| hXY'₂.prodMk <| mX₁'.sub mY'₁).aemeasurable ?_
filter_upwards [Z'eq1] with ω h
simp only [Prod.mk.injEq, Pi.sub_apply, h, and_self]
_ = H[⟨⟨X₁, Y₁⟩, Z⟩ ; ν₀] + H[⟨⟨X₂, Y₂⟩, Z⟩ ; ν₀] - H[Z ; ν₀]
+ H[⟨X₃, Y₃⟩ ; ν₀] := by
rw [ent_of_cond_indep (μ := ν'₀) hXY'₁ hXY'₂ hZ' h_condIndep, idXY₁ZXY'₁Z'.entropy_eq,
idXY₂ZXY'₂Z'.entropy_eq, idZZ'.entropy_eq]
_ = H[⟨⟨X₁, Y₁⟩, X₁ - Y₁⟩ ; ν₀] + H[⟨⟨X₂, Y₂⟩, X₂ - Y₂⟩ ; ν₀] - H[X₁ - Y₁ ; ν₀]
+ H[⟨X₃, Y₃⟩ ; ν₀] := by
rw [IdentDistrib.entropy_eq <| IdentDistrib.of_ae_eq mZ.aemeasurable Zeq1]
congr 3
· refine IdentDistrib.entropy_eq <| IdentDistrib.of_ae_eq
(((mX₁.prodMk mY₁).prodMk mZ).aemeasurable) ?_
filter_upwards [Zeq1] with ω h
simp only [Prod.mk.injEq, h, Pi.sub_apply, and_self]
· refine IdentDistrib.entropy_eq <| IdentDistrib.of_ae_eq
((mX₂.prodMk mY₂).prodMk mZ).aemeasurable ?_
filter_upwards [Zeq2] with ω h
simp only [Prod.mk.injEq, h, Pi.sub_apply, and_self]
_ = H[⟨X₁, ⟨Y₁, X₁ - Y₁⟩⟩ ; ν₀] + H[⟨X₂, ⟨Y₂, X₂ - Y₂⟩⟩ ; ν₀]
- H[X₁ - Y₁; ν₀] + H[⟨X₃, Y₃⟩ ; ν₀] := by
congr 3
· exact entropy_of_comp_eq_of_comp ν₀ (meas11.prodMk meas1neg1)
(mX₁.prodMk <| mY₁.prodMk <| mX₁.sub mY₁)
(fun ((x1, y1),x1y1) ↦ (x1, (y1, x1y1))) (fun (x1, (y1, x1y1)) ↦ ((x1, y1),x1y1))
rfl rfl
· exact entropy_of_comp_eq_of_comp ν₀ (meas22.prodMk <| (mX₂).sub (mY₂))
(mX₂.prodMk <| mY₂.prodMk <| mX₂.sub mY₂)
(fun ((x1, y1),x1y1) ↦ (x1, (y1, x1y1))) (fun (x1, (y1, x1y1)) ↦ ((x1, y1),x1y1))
rfl rfl
have eq12_aux1 : H[⟨X₁, ⟨Y₁, X₁ - Y₁⟩⟩ ; ν₀] = H[⟨X₁, Y₁⟩ ; ν₀] :=
entropy_of_comp_eq_of_comp ν₀
(mX₁.prodMk <| mY₁.prodMk <| mX₁.sub mY₁) meas11
(fun (x1, (y1, _)) ↦ (x1, y1)) (fun (x1, y1) ↦ (x1, (y1, x1 - y1))) rfl rfl
have eq12_aux2 : H[⟨X₂, ⟨Y₂, X₂ - Y₂⟩⟩ ; ν₀] = H[⟨X₂, Y₂⟩ ; ν₀] :=
entropy_of_comp_eq_of_comp ν₀
(mX₂.prodMk <| mY₂.prodMk <| mX₂.sub mY₂) meas22
(fun (x1, (y1, _)) ↦ (x1, y1)) (fun (x1, y1) ↦ (x1, (y1, x1 - y1))) rfl rfl
have eq12 : H[⟨X₁, ⟨Y₁, ⟨X₂, ⟨Y₂, ⟨X₃, Y₃⟩⟩⟩⟩⟩ ; ν₀]
= 5/2 * (H[X'; ν] + H[Y'; ν]) - d[X'; ν # Y'; ν] := by
rw [eq11, eq8', eq12_aux1, eq12_aux2, hX1Y1, hX2Y2, hX3Y3]
ring_nf
suffices h : 3 * (H[X'; ν] + H[Y'; ν]) - d[X'; ν # Y'; ν] + d[X'; ν # -Y'; ν]
≤ 3 * (H[X'; ν] + H[Y'; ν]) + 2 * d[X'; ν # Y'; ν] by
simp only [sub_eq_add_neg, add_assoc, add_le_add_iff_left, neg_add_le_iff_le_add] at h
ring_nf at h ⊢
exact h
calc
_ = 5/2 * (H[X' ; ν] + H[Y' ; ν]) - d[X' ; ν # Y' ; ν]
+ 1/2 * (H[X' ; ν] + H[Y' ; ν]) + d[X' ; ν # -Y' ; ν] := by
ring
_ ≤ H[⟨⟨X₃ - Y₂, ⟨X₁ - Y₃, ⟨X₂, Y₁⟩⟩⟩, ⟨⟨X₃, Y₃⟩, X₃ + Y₃⟩⟩ ; ν₀]
+ 1/2 * (H[X' ; ν] + H[Y' ; ν]) + d[X' ; ν # -Y' ; ν] := by
simp only [one_div, add_le_add_iff_right, eq12 ▸ in10]
_ = H[⟨⟨X₃ - Y₂, ⟨X₁ - Y₃, ⟨X₂, Y₁⟩⟩⟩, ⟨⟨X₃, Y₃⟩, X₃ + Y₃⟩⟩ ; ν₀] + H[X₃ + Y₃ ; ν₀] := by
simp only [one_div, eq2]
ring
_ ≤ H[⟨⟨X₃ - Y₂, ⟨X₁ - Y₃, ⟨X₂, Y₁⟩⟩⟩, X₃ + Y₃⟩ ; ν₀] + H[⟨⟨X₃, Y₃⟩, X₃ + Y₃⟩ ; ν₀] := in1
_ ≤ 2 * (H[X' ; ν] + H[Y' ; ν]) + 2 * d[X' ; ν # Y' ; ν] + H[⟨⟨X₃, Y₃⟩, X₃ + Y₃⟩ ; ν₀] := by
gcongr
ring_nf at in9 ⊢
simp only [in9]
_ = 3 * (H[X' ; ν] + H[Y' ; ν]) + 2 * d[X' ; ν # Y' ; ν] := by
simp only [eq3]
ring
/-- If `n ≥ 0` and `X, Y₁, ..., Yₙ` are jointly independent `G`-valued random variables,
then `H[Y i₀ + ∑ i ∈ s, Y i; μ] - H[Y i₀; μ] ≤ ∑ i ∈ s, (H[Y i₀ + Y i; μ] - H[Y i₀; μ])`.
The spelling here is tentative.
Feel free to modify it to make the proof easier, or the application easier. -/ | pfr/blueprint/src/chapter/torsion.tex:39 | pfr/PFR/MoreRuzsaDist.lean:107 |
PFR | rdist_of_sums_ge | \begin{lemma}[Lower bound on distances]\label{first-dist-sum}
\lean{rdist_of_sums_ge}\leanok
We have
\begin{align*}
d[X_1+\tilde X_2; X_2+\tilde X_1] \geq k &- \eta (d[X^0_1; X_1+\tilde X_2] - d[X^0_1; X_1]) \\& \qquad- \eta (d[X^0_2; X_2+\tilde X_1] - d[X^0_2; X_2])
\end{align*}
\end{lemma}
\begin{proof}\uses{distance-lower}\leanok Immediate from \Cref{distance-lower}.
\end{proof} | lemma rdist_of_sums_ge :
d[X₁ + X₂' # X₂ + X₁'] ≥
k - p.η * (d[p.X₀₁ # X₁ + X₂'] - d[p.X₀₁ # X₁])
- p.η * (d[p.X₀₂ # X₂ + X₁'] - d[p.X₀₂ # X₂]) :=
distance_ge_of_min _ h_min (hX₁.add hX₂') (hX₂.add hX₁')
include h_min hX₁ hX₂ hX₁' hX₂' in
/-- The distance $d[X_1|X_1+\tilde X_2; X_2|X_2+\tilde X_1]$ is at least
$$ k - \eta (d[X^0_1; X_1 | X_1 + \tilde X_2] - d[X^0_1; X_1]) - \eta(d[X^0_2; X_2 | X_2 + \tilde X_1] - d[X^0_2; X_2]).$$
-/ | pfr/blueprint/src/chapter/entropy_pfr.tex:91 | pfr/PFR/FirstEstimate.lean:74 |
PFR | rdist_of_sums_ge' | \begin{lemma}[Distance between sums]\label{dist-sums}
\lean{rdist_of_sums_ge'}\leanok
We have
$$ d[X_1+\tilde X_1; X_2+\tilde X_2] \geq k - \frac{\eta}{2} ( d[X_1; X_1] + d[X_2;X_2] ).$$
\end{lemma}
\begin{proof}\uses{distance-lower, first-useful}\leanok From \Cref{distance-lower} one has
\begin{align*}
d[X_1+\tilde X_1; X_2+\tilde X_2] \geq k &- \eta(d[X^0_1;X_1] - d[X^0_1;X_1+\tilde X_1]) \\
&- \eta(d[X^0_2;X_2] - d[X^0_2;X_2+\tilde X_2]).
\end{align*}
Now \Cref{first-useful} gives
$$
d[X^0_1;X_1+\tilde X_1] - d[X^0_1;X_1] \leq \tfrac{1}{2} d[X_1;X_1]$$
and
$$
d[X^0_2;X_2+\tilde X_2] - d[X^0_2;X_2] \leq \tfrac{1}{2} d[X_2;X_2],
$$
and the claim follows.
\end{proof} | lemma rdist_of_sums_ge' : d[X₁ + X₁' # X₂ + X₂'] ≥ k - p.η * (d[X₁ # X₁] + d[X₂ # X₂]) / 2 := by
refine LE.le.ge (LE.le.trans ?_ (distance_ge_of_min p h_min (hX₁.add hX₁') (hX₂.add hX₂')))
rw [sub_sub, sub_le_sub_iff_left k, ← mul_add,mul_div_assoc]
refine mul_le_mul_of_nonneg_left ?_ (by linarith [p.hη])
have h₁' := condRuzsaDist_diff_le' ℙ p.hmeas1 hX₁ hX₁' (h_indep.indepFun (show 0 ≠ 2 by decide))
have h₂' := condRuzsaDist_diff_le' ℙ p.hmeas2 hX₂ hX₂' (h_indep.indepFun (show 1 ≠ 3 by decide))
rw [h₁.entropy_eq, add_sub_cancel_right, ← (IdentDistrib.refl hX₁.aemeasurable).rdist_eq h₁] at h₁'
rw [h₂.entropy_eq, add_sub_cancel_right, ← (IdentDistrib.refl hX₂.aemeasurable).rdist_eq h₂] at h₂'
linarith
include h_min hX₁ hX₁' hX₂ hX₂' h_indep h₁ h₂ in | pfr/blueprint/src/chapter/entropy_pfr.tex:162 | pfr/PFR/SecondEstimate.lean:57 |
PFR | rdist_zero_eq_half_ent | \begin{lemma}[Distance from zero]\label{dist-zero}
\uses{ruz-dist-def}\lean{rdist_zero_eq_half_ent}\leanok
If $X$ is a $G$-valued random variable and $0$ is the random variable taking the value $0$ everywhere then
\[d[X;0]=\mathbb{H}(X)/2.\]
\end{lemma}
\begin{proof}\leanok
This is an immediate consequence of the definitions and $X-0\equiv X$ and $\mathbb{H}(0)=0$.
\end{proof} | /-- `d[X ; 0] = H[X] / 2`. -/
lemma rdist_zero_eq_half_ent [IsFiniteMeasure μ] [IsProbabilityMeasure μ'] :
d[X ; μ # fun _ ↦ 0 ; μ'] = H[X ; μ]/2 := by
have aux : H[fun x => x.1 - x.2 ; Measure.prod (Measure.map X μ) (Measure.map (fun x => 0) μ')]
= H[X ; μ] := by
have h: Measure.map (fun x => x.1 - x.2)
(Measure.prod (Measure.map X μ) (Measure.map (fun x => 0) μ'))
= Measure.map X μ := by
simp [MeasureTheory.Measure.map_const, MeasureTheory.Measure.prod_dirac]
rw [Measure.map_map (by fun_prop) (by fun_prop)]
have helper : ((fun (x : G × G) => x.1 - x.2) ∘ fun x => (x, (0 : G))) = id := by
funext; simp
rw [helper, Measure.map_id]
simp [entropy_def, h]
simp [rdist_def, entropy_const (0 : G), aux]
ring | pfr/blueprint/src/chapter/distance.tex:89 | pfr/PFR/ForMathlib/Entropy/RuzsaDist.lean:213 |
PFR | rhoMinus_of_subgroup | \begin{lemma}[Rho minus of subgroup]\label{rhominus-subgroup}\lean{rhoMinus_of_subgroup}\uses{rhominus-def}\leanok If $H$ is a finite subgroup of $G$, then $\rho^-(U_H) = \log |A| - \log \max_t |A \cap (H+t)|$.
\end{lemma}
\begin{proof}\leanok
\uses{log-sum}
For every $G$-valued random variable $T$ that is independent of $Y$,
$$D_{KL}(U_H \Vert U_A+T) = \sum_{h\in H} \frac{1}{|H|}\log\frac{1/|H|}{\mathbf{P}[U_A+T=h]}\ge -\log(\mathbf{P}[U_A+T\in H]),$$
by \Cref{log-sum}. Then observe that $$-\log(\mathbf{P}[U_A+T\in H])=-\log(\mathbf{P}[U_A\in H-T])\ge -\log(\max_{t\in G} \mathbf{P}[U_A\in H+t]).$$ This proves $\ge$.
To get the equality, let $t^*:=\arg\max_t |A \cap (H+t)|$ and observe that $$\rho^-(U_H)\le D_{KL}(U_H \Vert U_A+(U_H-t^*))= \log |A| - \log \max_t|A \cap (H+t)|.$$
\end{proof} | lemma rhoMinus_of_subgroup [IsProbabilityMeasure μ] {H : AddSubgroup G}
{U : Ω → G} (hunif : IsUniform H U μ) {A : Finset G} (hA : A.Nonempty) (hU : Measurable U) :
ρ⁻[U ; μ # A] = log (Nat.card A) -
log (sSup {Nat.card (A ∩ (t +ᵥ (H : Set G)) : Set G) | t : G} : ℕ) := by
apply le_antisymm _ (le_rhoMinus_of_subgroup hunif hA hU)
rcases exists_card_inter_add_eq_sSup (A := A) H hA with ⟨t, ht, hpos⟩
rw [← ht]
have : Nonempty (A ∩ (t +ᵥ (H : Set G)) : Set G) := (Nat.card_pos_iff.1 hpos).1
exact rhoMinus_le_of_subgroup t hunif hA .of_subtype hU
/-- If $H$ is a finite subgroup of $G$, then
$\rho^+(U_H) = \log |H| - \log \max_t |A \cap (H+t)|$. -/ | pfr/blueprint/src/chapter/further_improvement.tex:95 | pfr/PFR/RhoFunctional.lean:635 |
PFR | rhoMinus_of_sum | \begin{lemma}[Rho and sums]\label{rho-sums}\lean{rhoMinus_of_sum, rhoPlus_of_sum, rho_of_sum}\leanok If $X,Y$ are independent, one has
$$ \rho^-(X+Y) \leq \rho^-(X)$$
$$ \rho^+(X+Y) \leq \rho^+(X) + \bbH[X+Y] - \bbH[X]$$
and
$$ \rho(X+Y) \leq \rho(X) + \frac{1}{2}( \bbH[X+Y] - \bbH[X] ).$$
\end{lemma}
\begin{proof}\leanok
\uses{kl-sums}
The first inequality follows from \Cref{kl-sums}. The second and third inequalities are direct corollaries of the first.
\end{proof} | lemma rhoMinus_of_sum [IsZeroOrProbabilityMeasure μ]
(hX : Measurable X) (hY : Measurable Y)
(hA : A.Nonempty) (h_indep : IndepFun X Y μ) :
ρ⁻[X + Y ; μ # A] ≤ ρ⁻[X ; μ # A] := by
rcases eq_zero_or_isProbabilityMeasure μ with hμ | hμ
· simp [rhoMinus_zero_measure hμ]
apply le_csInf (nonempty_rhoMinusSet hA)
have : IsProbabilityMeasure (uniformOn (A : Set G)) :=
uniformOn_isProbabilityMeasure A.finite_toSet hA
rintro - ⟨μ', μ'_prob, habs, rfl⟩
obtain ⟨Ω', hΩ', m, X', Y', T, U, hm, h_indep', hX', hY', hT, hU, hXX', hYY', hTμ, hU_unif⟩ :=
independent_copies4_nondep (X₁ := X) (X₂ := Y) (X₃ := id) (X₄ := id) hX hY measurable_id
measurable_id μ μ μ' (uniformOn (A : Set G))
let _ : MeasureSpace Ω' := ⟨m⟩
have hP : (ℙ : Measure Ω') = m := rfl
have hTU : IdentDistrib (T + U) (Prod.fst + Prod.snd) ℙ (μ'.prod (uniformOn (A : Set G))) := by
apply IdentDistrib.add
· exact hTμ.trans IdentDistrib.fst_id.symm
· exact hU_unif.trans IdentDistrib.snd_id.symm
· exact h_indep'.indepFun (i := 2) (j := 3) (by simp)
· exact indepFun_fst_snd
have hXY : IdentDistrib (X + Y) (X' + Y') μ ℙ := by
apply IdentDistrib.add hXX'.symm hYY'.symm h_indep
exact h_indep'.indepFun zero_ne_one
have hX'TUY' : IndepFun (⟨X', T + U⟩) Y' ℙ := by
have I : iIndepFun ![X', Y', T + U] m :=
ProbabilityTheory.iIndepFun.apply_two_last h_indep' hX' hY' hT hU
(phi := fun a b ↦ a + b) (by fun_prop)
exact (I.reindex_three_bac.pair_last_of_three hY' hX' (by fun_prop)).symm
have I₁ : ρ⁻[X + Y ; μ # A] ≤ KL[X + Y ; μ # (T + Y') + U ; ℙ] := by
apply rhoMinus_le (by fun_prop) hA _ (by fun_prop) (by fun_prop)
· have : iIndepFun ![U, X', T, Y'] := h_indep'.reindex_four_dacb
have : iIndepFun ![U, X', T + Y'] :=
this.apply_two_last (phi := fun a b ↦ a + b) hU hX' hT hY' (by fun_prop)
apply this.indepFun (i := 2) (j := 0)
simp
· rw [hXY.map_eq]
have : T + Y' + U = (T + U) + Y' := by abel
rw [this]
apply absolutelyContinuous_add_of_indep hX'TUY' hX' (by fun_prop) hY'
rw [hTU.map_eq, hP, hXX'.map_eq]
exact habs
· exact isUniform_uniformOn.of_identDistrib hU_unif.symm A.measurableSet
have I₂ : KL[X + Y ; μ # (T + Y') + U ; ℙ] = KL[X' + Y' # (T + U) + Y'] := by
apply IdentDistrib.KLDiv_eq _ _ hXY
have : T + Y' + U = T + U + Y' := by abel
rw [this]
apply IdentDistrib.refl
fun_prop
have I₃ : KL[X' + Y' # (T + U) + Y'] ≤ KL[X' # T + U] := by
apply KLDiv_add_le_KLDiv_of_indep _ (by fun_prop) (by fun_prop) (by fun_prop)
· rw [hTU.map_eq, hP, hXX'.map_eq]
exact habs
· exact hX'TUY'
have I₄ : KL[X' # T + U] = KL[X ; μ # Prod.fst + Prod.snd ; μ'.prod (uniformOn (A : Set G))] :=
IdentDistrib.KLDiv_eq _ _ hXX' hTU
exact ((I₁.trans_eq I₂).trans I₃).trans_eq I₄
/-- If $X,Y$ are independent, one has
$$ \rho^+(X+Y) \leq \rho^+(X) + \bbH[X+Y] - \bbH[X]$$ -/ | pfr/blueprint/src/chapter/further_improvement.tex:144 | pfr/PFR/RhoFunctional.lean:767 |
PFR | rhoPlus_of_subgroup | \begin{corollary}[Rho plus of subgroup]\label{rhoplus-subgroup}\lean{rhoPlus_of_subgroup}\uses{rhoplus-def}\leanok If $H$ is a finite subgroup of $G$, then $\rho^+(U_H) = \log |H| - \log \max_t |A \cap (H+t)|$.
\end{corollary}
\begin{proof}\leanok \uses{rhominus-subgroup} Straightforward by definition and \Cref{rhominus-subgroup}.
\end{proof} | lemma rhoPlus_of_subgroup [IsProbabilityMeasure μ] {H : AddSubgroup G}
{U : Ω → G} (hunif : IsUniform H U μ) {A : Finset G} (hA : A.Nonempty) (hU : Measurable U) :
ρ⁺[U ; μ # A] = log (Nat.card H) -
log (sSup {Nat.card (A ∩ (t +ᵥ (H : Set G)) : Set G) | t : G} : ℕ) := by
have : H[U ; μ] = log (Nat.card H) := hunif.entropy_eq' (toFinite _) hU
rw [rhoPlus, rhoMinus_of_subgroup hunif hA hU, this]
abel
/-- We define $\rho(X) := (\rho^+(X) + \rho^-(X))/2$. -/
noncomputable def rho (X : Ω → G) (A : Finset G) (μ : Measure Ω ) : ℝ :=
(ρ⁻[X ; μ # A] + ρ⁺[X ; μ # A]) / 2
@[inherit_doc rho] notation3:max "ρ[" X " ; " μ " # " A "]" => rho X A μ
@[inherit_doc rho] notation3:max "ρ[" X " # " A "]" => rho X A volume | pfr/blueprint/src/chapter/further_improvement.tex:107 | pfr/PFR/RhoFunctional.lean:647 |
PFR | rhoPlus_of_sum | \begin{lemma}[Rho and sums]\label{rho-sums}\lean{rhoMinus_of_sum, rhoPlus_of_sum, rho_of_sum}\leanok If $X,Y$ are independent, one has
$$ \rho^-(X+Y) \leq \rho^-(X)$$
$$ \rho^+(X+Y) \leq \rho^+(X) + \bbH[X+Y] - \bbH[X]$$
and
$$ \rho(X+Y) \leq \rho(X) + \frac{1}{2}( \bbH[X+Y] - \bbH[X] ).$$
\end{lemma}
\begin{proof}\leanok
\uses{kl-sums}
The first inequality follows from \Cref{kl-sums}. The second and third inequalities are direct corollaries of the first.
\end{proof} | lemma rhoPlus_of_sum [IsZeroOrProbabilityMeasure μ]
(hX : Measurable X) (hY : Measurable Y) (hA : A.Nonempty) (h_indep : IndepFun X Y μ) :
ρ⁺[X + Y ; μ # A] ≤ ρ⁺[X ; μ # A] + H[X + Y ; μ] - H[X ; μ] := by
simp [rhoPlus]
have := rhoMinus_of_sum hX hY hA h_indep
linarith
/-- If $X,Y$ are independent, one has
$$ \rho(X+Y) \leq \rho(X) + \frac{1}{2}( \bbH[X+Y] - \bbH[X] ).$$
-/ | pfr/blueprint/src/chapter/further_improvement.tex:144 | pfr/PFR/RhoFunctional.lean:827 |
PFR | rho_PFR_conjecture | \begin{proposition} \label{pfr-rho}\lean{rho_PFR_conjecture}\leanok For any random variables $Y_1,Y_2$, there exist a subgroup $H$ such that
$$ 2\rho(U_H) \leq \rho(Y_1) + \rho(Y_2) + 8 d[Y_1;Y_2].$$
\end{proposition}
\begin{proof}\leanok
\uses{phi-min-exist, phi-minimizer-zero-distance,phi-min-def,rho-invariant,sym-zero}
Let $X_1,X_2$ be a $\phi$-minimizer. By \Cref{phi-minimizer-zero-distance} $d[X_1;X_2]=0$, which by \Cref{phi-min-def} implies $\rho(X_1)+\rho(X_2)\le \rho(Y_1) + \rho(Y_2) + \frac{1}{\eta} d[Y_1;Y_2]$ for every $\eta<1/8$. Take the limit at $\eta=1/8$ to get $\rho(X_1)+\rho(X_2)\le \rho(Y_1) + \rho(Y_2) + 8 d[Y_1;Y_2]$.
By \Cref{ruzsa-triangle} and \Cref{ruzsa-nonneg} we have $d[X_1;X_1]=d[X_2;X_2]=0$, and by \Cref{sym-zero} there are $H_1:=\mathrm{Sym}[X_1],H_2:=\mathrm{Sym}[X_2]$ such that $X_1=U_{H_1}+x_1$ and $X_2=U_{H_2}+x_2$ for some $x_2$.
By \Cref{rho-invariant} we get $\rho(U_{H_1})+\rho(U_{H_2})\le \rho(Y_1) + \rho(Y_2) + 8 d[Y_1;Y_2]$, and thus the claim holds for $H=H_1$ or $H=H_2$.
\end{proof} | theorem rho_PFR_conjecture [MeasurableSpace G] [DiscreteMeasurableSpace G]
(Y₁ Y₂ : Ω → G) (hY₁ : Measurable Y₁) (hY₂ : Measurable Y₂) (A : Finset G) (hA : A.Nonempty) :
∃ (H : Submodule (ZMod 2) G) (Ω' : Type uG) (mΩ' : MeasureSpace Ω') (U : Ω' → G),
IsProbabilityMeasure (ℙ : Measure Ω') ∧ Measurable U ∧
IsUniform H U ∧ 2 * ρ[U # A] ≤ ρ[Y₁ # A] + ρ[Y₂ # A] + 8 * d[Y₁ # Y₂] := by
obtain ⟨Ω', mΩ', X₁, X₂, hX₁, hX₂, hP, htau_min, hdist⟩ := phiMinimizer_exists_rdist_eq_zero hA
wlog h : ρ[X₁ # A] ≤ ρ[X₂ # A] generalizing X₁ X₂
· rw [rdist_symm] at hdist
exact this X₂ X₁ hX₂ hX₁ (phiMinimizes_comm htau_min) hdist (by linarith)
-- use for `U` a translate of `X` to make sure that `0` is in its support.
obtain ⟨x₀, h₀⟩ : ∃ x₀, ℙ (X₁⁻¹' {x₀}) ≠ 0 := by
by_contra! h
have A a : (ℙ : Measure Ω').map X₁ {a} = 0 := by
rw [Measure.map_apply hX₁ .of_discrete]
exact h _
have B : (ℙ : Measure Ω').map X₁ = 0 := by
rw [← Measure.sum_smul_dirac (μ := (ℙ : Measure Ω').map X₁)]
simp [A]
have : IsProbabilityMeasure ((ℙ : Measure Ω').map X₁) :=
isProbabilityMeasure_map hX₁.aemeasurable
exact IsProbabilityMeasure.ne_zero _ B
have h_unif : IsUniform (symmGroup X₁ hX₁) (fun ω ↦ X₁ ω - x₀) := by
have h' : d[X₁ # X₁] = 0 := by
apply le_antisymm _ (rdist_nonneg hX₁ hX₁)
calc
d[X₁ # X₁] ≤ d[X₁ # X₂] + d[X₂ # X₁] := rdist_triangle hX₁ hX₂ hX₁
_ = 0 := by rw [hdist, rdist_symm, hdist, zero_add]
exact isUniform_sub_const_of_rdist_eq_zero hX₁ h' h₀
refine ⟨AddSubgroup.toZModSubmodule 2 (symmGroup X₁ hX₁), Ω', by infer_instance,
fun ω ↦ X₁ ω - x₀, by infer_instance, by fun_prop, by exact h_unif, ?_⟩
have J : d[X₁ # X₂] + (1/8) * (ρ[X₁ # A] + ρ[X₂ # A])
≤ d[Y₁ # Y₂] + (1/8) * (ρ[Y₁ # A] + ρ[Y₂ # A]) := by
have Z := le_rdist_of_phiMinimizes htau_min hY₁ hY₂ (μ₁ := ℙ) (μ₂ := ℙ)
linarith
rw [hdist, zero_add] at J
have : ρ[fun ω ↦ X₁ ω - x₀ # A] = ρ[X₁ # A] := by
simp_rw [sub_eq_add_neg, rho_of_translate hX₁ hA]
linarith
/-- If $|A+A| \leq K|A|$, then there exists a subgroup $H$ and $t\in G$ such that
$|A \cap (H+t)| \geq K^{-4} \sqrt{|A||V|}$, and $|H|/|A|\in[K^{-8},K^8]$. -/ | pfr/blueprint/src/chapter/further_improvement.tex:336 | pfr/PFR/RhoFunctional.lean:1936 |
PFR | rho_continuous | \begin{lemma}[Rho continuous]\label{rho-cts}\lean{rho_continuous}\leanok\uses{rho-def} $\rho(X)$ depends continuously on the distribution of $X$.
\end{lemma}
\begin{proof} \leanok Clear from definition.
\end{proof} | /-- \rho(X)$ depends continuously on the distribution of $X$. -/
lemma rho_continuous [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G]
{A : Finset G} (hA : A.Nonempty) :
Continuous fun μ : ProbabilityMeasure G ↦ ρ[(id : G → G) ; μ # A] :=
((rhoMinus_continuous hA).add (rhoPlus_continuous hA)).div_const _ | pfr/blueprint/src/chapter/further_improvement.tex:138 | pfr/PFR/RhoFunctional.lean:744 |
PFR | rho_of_subgroup | \begin{lemma}[Rho of subgroup]\label{rho-subgroup}\uses{rho-def}\lean{rho_of_subgroup}\leanok
If $H$ is a finite subgroup of $G$, and $\rho(U_H) \leq r$, then there exists
$t$ such that $|A \cap (H+t)| \geq e^{-r} \sqrt{|A||H|}$, and
$|H|/|A|\in[e^{-2r},e^{2r}]$.
\end{lemma}
\begin{proof}\leanok\uses{rhominus-subgroup, rhoplus-subgroup} The first claim is a direct corollary of \Cref{rhominus-subgroup} and \Cref{rhoplus-subgroup}. To see the second claim, observe that \Cref{rhominus-nonneg} and \Cref{rhoplus-subgroup} imply $\rho^-(U_H),\rho^+(U_H)\ge 0$. Therefore $$|H(U_A)-H(U_H)|=|\rho^+(U_H)-\rho^-(U_H)|\le \rho^-(U_H)+\rho^+(U_H)= 2\rho(U_H)\le 2r,$$
which implies the second claim.
\end{proof} | lemma rho_of_subgroup [IsProbabilityMeasure μ] {H : AddSubgroup G} {U : Ω → G}
(hunif : IsUniform H U μ) {A : Finset G} (hA : A.Nonempty) (hU : Measurable U)
(r : ℝ) (hr : ρ[U ; μ # A] ≤ r) :
∃ t : G,
exp (-r) * Nat.card A ^ (1/2 : ℝ) * Nat.card H ^ (1/2 : ℝ) ≤
Nat.card ↑(↑A ∩ (t +ᵥ (H : Set G)))
∧ Nat.card A ≤ exp (2 * r) * Nat.card H
∧ Nat.card H ≤ exp (2 * r) * Nat.card A := by
have hr' : ρ[U ; μ # A] ≤ r := hr
have Hpos : 0 < (Nat.card H : ℝ) := by exact_mod_cast Nat.card_pos
have : Nonempty A := hA.to_subtype
have Apos : 0 < (Nat.card A : ℝ) := by exact_mod_cast Nat.card_pos
simp only [rho] at hr
rw [rhoMinus_of_subgroup hunif hA hU, rhoPlus_of_subgroup hunif hA hU] at hr
rcases exists_card_inter_add_eq_sSup (A := A) H hA with ⟨t, ht, hpos⟩
rw [← ht] at hr
have Rm : 0 ≤ ρ⁻[U ; μ # A] := rhoMinus_nonneg hU
have RM : 0 ≤ ρ⁺[U ; μ # A] := by
rw [rhoPlus_of_subgroup hunif hA hU, ← ht, sub_nonneg]
apply log_le_log (mod_cast hpos)
norm_cast
have : Nat.card (t +ᵥ (H : Set G) : Set G) = Nat.card H := by
apply Nat.card_image_of_injective (add_right_injective t)
rw [← this]
exact Nat.card_mono (toFinite _) inter_subset_right
have I : |log (Nat.card H) - log (Nat.card A)| ≤ 2 * r := calc
|log (Nat.card H) - log (Nat.card A)|
_ = |H[U ; μ] - log (Nat.card A)| := by rw [hunif.entropy_eq' (toFinite _) hU]; rfl
_ = |ρ⁺[U ; μ # A] - ρ⁻[U ; μ # A]| := by congr 1; simp [rhoPlus]; abel
_ ≤ ρ⁺[U ; μ # A] + ρ⁻[U ; μ # A] :=
(abs_sub _ _).trans_eq (by simp [abs_of_nonneg, Rm, RM])
_ = 2 * ρ[U ; μ # A] := by simp [rho]; ring
_ ≤ 2 * r := by linarith
refine ⟨t, ?_, ?_, ?_⟩
· have : - r + (log (Nat.card A) + log (Nat.card H)) * (1 / 2 : ℝ) ≤
log (Nat.card (A ∩ (t +ᵥ (H : Set G)) : Set G)) := by linarith
have := exp_monotone this
rwa [exp_add, exp_log (mod_cast hpos), exp_mul, exp_add,
exp_log Hpos, exp_log Apos, mul_rpow, ← mul_assoc] at this <;> positivity
· have : log (Nat.card A) ≤ 2 * r + log (Nat.card H) := by
linarith [(abs_sub_le_iff.1 I).2]
have := exp_monotone this
rwa [exp_log Apos, exp_add, exp_log Hpos] at this
· have : log (Nat.card H) ≤ 2 * r + log (Nat.card A) := by
linarith [(abs_sub_le_iff.1 I).1]
have := exp_monotone this
rwa [exp_log Hpos, exp_add, exp_log Apos] at this
/-- If $H$ is a finite subgroup of $G$, and $\rho(U_H) \leq r$, then there exists $t$ such
that $|A \cap (H+t)| \geq e^{-r} \sqrt{|A||H|}$, and $|H|/|A| \in [e^{-2r}, e^{2r}]$. -/ | pfr/blueprint/src/chapter/further_improvement.tex:122 | pfr/PFR/RhoFunctional.lean:683 |
PFR | rho_of_sum | \begin{lemma}[Rho and sums]\label{rho-sums}\lean{rhoMinus_of_sum, rhoPlus_of_sum, rho_of_sum}\leanok If $X,Y$ are independent, one has
$$ \rho^-(X+Y) \leq \rho^-(X)$$
$$ \rho^+(X+Y) \leq \rho^+(X) + \bbH[X+Y] - \bbH[X]$$
and
$$ \rho(X+Y) \leq \rho(X) + \frac{1}{2}( \bbH[X+Y] - \bbH[X] ).$$
\end{lemma}
\begin{proof}\leanok
\uses{kl-sums}
The first inequality follows from \Cref{kl-sums}. The second and third inequalities are direct corollaries of the first.
\end{proof} | lemma rho_of_sum [IsZeroOrProbabilityMeasure μ]
(hX : Measurable X) (hY : Measurable Y) (hA : A.Nonempty) (h_indep : IndepFun X Y μ) :
ρ[X + Y ; μ # A] ≤ ρ[X ; μ # A] + (H[X+Y ; μ] - H[X ; μ])/2 := by
simp [rho, rhoPlus]
have := rhoMinus_of_sum hX hY hA h_indep
linarith
private lemma rho_le_translate [IsZeroOrProbabilityMeasure μ]
(hX : Measurable X) (hA : A.Nonempty) (s : G) :
ρ[(fun ω ↦ X ω + s) ; μ # A] ≤ ρ[X ; μ # A] := by
have : ρ[(fun ω ↦ X ω + s) ; μ # A] ≤ ρ[X ; μ # A] + (H[fun ω ↦ X ω + s ; μ] - H[X ; μ]) / 2 :=
rho_of_sum (Y := fun ω ↦ s) hX measurable_const hA (indepFun_const s)
have : H[fun ω ↦ X ω + s ; μ] = H[X ; μ] := entropy_add_const hX _
linarith | pfr/blueprint/src/chapter/further_improvement.tex:144 | pfr/PFR/RhoFunctional.lean:837 |
PFR | rho_of_sum_le | \begin{lemma}[Rho and sums, symmetrized]\label{rho-sums-sym}\lean{rho_of_sum_le}\leanok If $X,Y$ are independent, then
$$ \rho(X+Y) \leq \frac{1}{2}(\rho(X)+\rho(Y) + d[X;Y]).$$
\end{lemma}
\begin{proof}\leanok
\uses{rho-sums}
Apply \Cref{rho-sums} for $(X,Y)$ and $(Y,X)$ and take their average.
\end{proof} | lemma rho_of_sum_le [IsZeroOrProbabilityMeasure μ]
(hX : Measurable X) (hY : Measurable Y) (hA : A.Nonempty) (h_indep : IndepFun X Y μ) :
ρ[X + Y ; μ # A] ≤ (ρ[X ; μ # A] + ρ[Y ; μ # A] + d[ X ; μ # Y ; μ]) / 2 := by
have I : ρ[X + Y ; μ # A] ≤ ρ[X ; μ # A] + (H[X+Y ; μ] - H[X ; μ])/2 := rho_of_sum hX hY hA h_indep
have J : ρ[Y + X ; μ # A] ≤ ρ[Y ; μ # A] + (H[Y+X ; μ] - H[Y ; μ ])/2 :=
rho_of_sum hY hX hA h_indep.symm
have : Y + X = X + Y := by abel
rw [this] at J
have : X - Y = X + Y := ZModModule.sub_eq_add _ _
rw [h_indep.rdist_eq hX hY, sub_eq_add_neg, this]
linarith
/-- If $X,Y$ are independent, then
$$ \rho(X | X+Y) \leq \frac{1}{2}(\rho(X)+\rho(Y) + d[X;Y]).$$ -/ | pfr/blueprint/src/chapter/further_improvement.tex:190 | pfr/PFR/RhoFunctional.lean:1061 |
PFR | rho_of_translate | \begin{lemma}[Rho invariant]\label{rho-invariant}\lean{rho_of_translate}\leanok\uses{rho-def} For any $s \in G$, $\rho(X+s) = \rho(X)$.
\end{lemma}
\begin{proof}\leanok\uses{kl-div-inj} Observe that by \Cref{kl-div-inj}, $$\inf_T D_{KL}(X\Vert U_A+T)=\inf_T D_{KL}(X+s\Vert U_A+T+s)=\inf_{T'} D_{KL}(X+s\Vert U_A+T').$$
\end{proof} | lemma rho_of_translate [IsZeroOrProbabilityMeasure μ]
(hX : Measurable X) (hA : A.Nonempty) (s : G) :
ρ[(fun ω ↦ X ω + s) ; μ # A] = ρ[X ; μ # A] := by
apply le_antisymm (rho_le_translate hX hA s)
convert rho_le_translate (X := fun ω ↦ X ω + s) (by fun_prop) hA (-s) (μ := μ) with ω
abel
-- This may not be the optimal spelling for condRho, feel free to improve
/-- We define $\rho(X|Y) := \sum_y {\bf P}(Y=y) \rho(X|Y=y)$. -/
noncomputable def condRho {S : Type*}
(X : Ω → G) (Y : Ω → S) (A : Finset G) (μ : Measure Ω): ℝ :=
∑' s, (μ (Y ⁻¹' {s})).toReal * ρ[X ; μ[|Y ← s] # A]
/-- Average of rhoMinus along the fibers-/
noncomputable def condRhoMinus {S : Type*}
(X : Ω → G) (Y : Ω → S) (A : Finset G) (μ : Measure Ω) : ℝ :=
∑' s, (μ (Y ⁻¹' {s})).toReal * ρ⁻[X ; μ[|Y ← s] # A]
/-- Average of rhoPlus along the fibers-/
noncomputable def condRhoPlus {S : Type*}
(X : Ω → G) (Y : Ω → S) (A : Finset G) (μ : Measure Ω) : ℝ :=
∑' s, (μ (Y ⁻¹' {s})).toReal * ρ⁺[X ; μ[|Y ← s] # A]
@[inherit_doc condRho]
notation3:max "ρ[" X " | " Z " ; " μ " # " A "]" => condRho X Z A μ
@[inherit_doc condRho]
notation3:max "ρ[" X " | " Z " # " A "]" => condRho X Z A volume
@[inherit_doc condRhoMinus]
notation3:max "ρ⁻[" X " | " Z " ; " μ " # " A "]" => condRhoMinus X Z A μ
@[inherit_doc condRhoPlus]
notation3:max "ρ⁺[" X " | " Z " ; " μ " # " A "]" => condRhoPlus X Z A μ | pfr/blueprint/src/chapter/further_improvement.tex:132 | pfr/PFR/RhoFunctional.lean:852 |
PFR | second_estimate | \begin{lemma}[Second estimate]\label{second-estimate}
\lean{second_estimate}\leanok
We have
$$ I_2 \leq 2 \eta k + \frac{2 \eta (2 \eta k - I_1)}{1 - \eta}.$$
\end{lemma}
\begin{proof}
\uses{cor-fibre,distance-lower,first-useful,second-estimate-aux}\leanok
We apply \Cref{cor-fibre}, but now with the choice
\[
(Y_1,Y_2,Y_3,Y_4) := (X_2, X_1, \tilde X_2, \tilde X_1).
\]
Now \Cref{cor-fibre} can be rewritten as
\begin{align*}
&d[X_1+\tilde X_1;X_2+\tilde X_2] + d[X_1|X_1+\tilde X_1; X_2|X_2+\tilde X_2] \\
&\quad + \bbI[X_1+X_2 : X_1 + \tilde X_1 \,|\, X_1+X_2+\tilde X_1+\tilde X_2] = 2k,
\end{align*}
recalling once again that $k := d[X_1;X_2]$. From \Cref{cond-distance-lower} one has
\begin{align*}
d[X_1|X_1+\tilde X_1; X_2|X_2+\tilde X_2] \geq k &- \eta (d[X^0_1;X_1] - d[X^0_1;X_1|X_1+\tilde X_1]) \\& - \eta (d[X^0_2;X_2] - d[X^0_2;X_2|X_2+\tilde X_2]) .
\end{align*}
while from \Cref{first-useful} we have
\[
d[X^0_1;X_1|X_1+\tilde X_1] - d[X^0_1;X_1] \leq \tfrac{1}{2} d[X_1;X_1],
\]
and
\[
d[X^0_2;X_2|X_2+\tilde X_2] - d[X^0_2;X_2] \leq \tfrac{1}{2} d[X_1;X_2].
\]
Combining all these inequalities with \Cref{dist-sums}, we have
\begin{equation}\label{combined}
\bbI[X_1+X_2 : X_1 + \tilde X_1 | X_1+X_2+\tilde X_1+\tilde X_2] \leq \eta ( d[X_1; X_1] + d[X_2; X_2] ).
\end{equation}
Together with \Cref{second-estimate-aux}, this gives the conclusion.
\end{proof} | /-- $$ I_2 \leq 2 \eta k + \frac{2 \eta (2 \eta k - I_1)}{1 - \eta}.$$ -/
lemma second_estimate : I₂ ≤ 2 * p.η * k + (2 * p.η * (2 * p.η * k - I₁)) / (1 - p.η) := by
have hX₁_indep : IndepFun X₁ X₁' (μ := ℙ) := h_indep.indepFun (show 0 ≠ 2 by decide)
have hX₂_indep : IndepFun X₂ X₂' (μ := ℙ) := h_indep.indepFun (show 1 ≠ 3 by decide)
let Y : Fin 4 → Ω → G := ![X₂, X₁, X₂', X₁']
have hY : ∀ i, Measurable (Y i) := fun i => by fin_cases i <;> assumption
have hY_indep : iIndepFun Y := by exact h_indep.reindex_four_badc
have h := sum_of_rdist_eq_char_2 Y hY_indep hY
rw [show Y 0 = X₂ by rfl, show Y 1 = X₁ by rfl, show Y 2 = X₂' by rfl, show Y 3 = X₁' by rfl] at h
rw [← h₂.rdist_eq h₁, rdist_symm, rdist_symm (X := X₂ + X₂'),
condRuzsaDist_symm (Z := X₂ + X₂') (W := X₁ + X₁') (hX₂.add hX₂') (hX₁.add hX₁'),
← two_mul] at h
replace h : 2 * k = d[X₁ + X₁' # X₂ + X₂'] + d[X₁ | X₁ + X₁' # X₂ | X₂ + X₂']
+ I[X₁ + X₂ : X₁ + X₁'|X₁ + X₂ + X₁' + X₂'] := by
convert h using 3 <;> abel
have h' := condRuzsaDistance_ge_of_min p h_min hX₁ hX₂ (X₁ + X₁') (X₂ + X₂') (hX₁.add hX₁') (hX₂.add hX₂')
have h₁' := condRuzsaDist_diff_le''' ℙ p.hmeas1 hX₁ hX₁' hX₁_indep
have h₂' := condRuzsaDist_diff_le''' ℙ p.hmeas2 hX₂ hX₂' hX₂_indep
rw [h₁.entropy_eq, add_sub_cancel_right, ← (IdentDistrib.refl hX₁.aemeasurable).rdist_eq h₁] at h₁'
rw [h₂.entropy_eq, add_sub_cancel_right, ← (IdentDistrib.refl hX₂.aemeasurable).rdist_eq h₂] at h₂'
have h'' : I₂ ≤ p.η * (d[X₁ # X₁] + d[X₂ # X₂]) := by
simp_rw [← add_comm X₁ X₁']
have h₁'' := mul_le_mul_of_nonneg_left h₁' (show 0 ≤ p.η by linarith [p.hη])
have h₂'' := mul_le_mul_of_nonneg_left h₂' (show 0 ≤ p.η by linarith [p.hη])
have := rdist_of_sums_ge' p _ _ _ _ hX₁ hX₂ hX₁' hX₂' h₁ h₂ h_indep h_min
linarith
nth_rewrite 1 [mul_div_assoc, ← mul_add, mul_assoc, mul_left_comm]
refine h''.trans (mul_le_mul_of_nonneg_left ?_ (show 0 ≤ p.η by linarith [p.hη]))
exact second_estimate_aux p X₁ X₂ X₁' X₂' hX₁ hX₂ hX₁' hX₂' h₁ h₂ h_indep h_min | pfr/blueprint/src/chapter/entropy_pfr.tex:202 | pfr/PFR/SecondEstimate.lean:95 |
PFR | second_estimate_aux | \begin{lemma}\label{second-estimate-aux}\lean{second_estimate_aux}\leanok
We have
\[d[X_1;X_1] + d[X_2;X_2] \leq 2 k + \frac{2(2 \eta k - I_1)}{1-\eta}. \]
\end{lemma}
\begin{proof}
\uses{ruz-indep, foursum-bound, dist-sums}\leanok
We may use \Cref{ruz-indep} to expand
\begin{align*}
& d[X_1+\tilde X_1;X_2+\tilde X_2] \\ &= \bbH[X_1+\tilde X_1 + X_2 + \tilde X_2] - \tfrac{1}{2} \bbH[X_1+\tilde X_1] - \tfrac{1}{2} \bbH[X_2+\tilde X_2] \\
&= \bbH[X_1+\tilde X_1 + X_2 + \tilde X_2] - \tfrac{1}{2} \bbH[X_1] - \tfrac{1}{2} \bbH[X_2] \\ & \qquad\qquad\qquad - \tfrac{1}{2} \left( d[X_1;X_1] + d[X_2; X_2] \right),
\end{align*}
and hence by \Cref{foursum-bound}
\[
d[X_1+\tilde X_1; X_2+\tilde X_2] \leq (2+\eta) k - \tfrac{1}{2} \left( d[X_1;X_1] + d[X_2;X_2] \right) - I_1.
\]
Combining this bound with \Cref{dist-sums} we obtain
the result.
\end{proof} | lemma second_estimate_aux :
d[X₁ # X₁] + d[X₂ # X₂] ≤ 2 * (k + (2 * p.η * k - I₁) / (1 - p.η)) := by
have hX₁_indep : IndepFun X₁ X₁' (μ := ℙ) := h_indep.indepFun (show 0 ≠ 2 by decide)
have hX₂_indep : IndepFun X₂ X₂' (μ := ℙ) := h_indep.indepFun (show 1 ≠ 3 by decide)
have hX_indep : IndepFun (X₁ + X₁') (X₂ + X₂') := by
exact h_indep.indepFun_add_add (ι := Fin 4) (by intro i; fin_cases i <;> assumption) 0 2 1 3
(by decide) (by decide) (by decide) (by decide)
have h : d[X₁ + X₁' # X₂+ X₂'] ≤ (2 + p.η) * k - (d[X₁# X₁] + d[X₂ # X₂]) / 2 - I₁ := by
have h := hX_indep.rdist_eq (hX₁.add hX₁') (hX₂.add hX₂')
rw [ZModModule.sub_eq_add (X₁ + X₁') (X₂ + X₂'), ← ZModModule.sub_eq_add X₁ X₁',
← ZModModule.sub_eq_add X₂ X₂',
sub_eq_iff_eq_add.mp (sub_eq_iff_eq_add.mp (hX₁_indep.rdist_eq hX₁ hX₁').symm),
sub_eq_iff_eq_add.mp (sub_eq_iff_eq_add.mp (hX₂_indep.rdist_eq hX₂ hX₂').symm),
← h₁.entropy_eq, ← h₂.entropy_eq, add_assoc, add_assoc, add_halves, add_halves,
← (IdentDistrib.refl hX₁.aemeasurable).rdist_eq h₁,
← (IdentDistrib.refl hX₂.aemeasurable).rdist_eq h₂,
ZModModule.sub_eq_add X₁ X₁', ZModModule.sub_eq_add X₂ X₂', ← add_assoc, add_right_comm _ X₁']
at h
have h_indep' : iIndepFun ![X₁, X₂, X₂', X₁'] :=
by exact h_indep.reindex_four_abdc
have h' := ent_ofsum_le p X₁ X₂ X₁' X₂' hX₁ hX₂ hX₁' hX₂' h₁ h₂ h_indep' h_min
convert (h.symm ▸ (sub_le_sub_right (sub_le_sub_right h' _) _)) using 1; ring
have h' := (rdist_of_sums_ge' p X₁ X₂ X₁' X₂' hX₁ hX₂ hX₁' hX₂' h₁ h₂ h_indep h_min).le.trans h
rw [← div_le_iff₀' two_pos, ← sub_le_iff_le_add', le_div_iff₀ (by linarith [p.hη'])]
linarith
include h_min hX₁ hX₁' hX₂ hX₂' h_indep h₁ h₂ in | pfr/blueprint/src/chapter/entropy_pfr.tex:183 | pfr/PFR/SecondEstimate.lean:68 |
PFR | single_fibres | \begin{lemma}\label{single-fibres}\lean{single_fibres}\leanok
Let $\phi:G\to H$ be a homomorphism and $A,B\subseteq G$ be finite subsets. If $x,y\in H$ then let $A_x=A\cap \phi^{-1}(x)$ and $B_y=B\cap \phi^{-1}(y)$. There exist $x,y\in H$ such that $A_x,B_y$ are both non-empty and
\[d[\phi(U_A);\phi(U_B)]\log \frac{\lvert A\rvert\lvert B\rvert}{\lvert A_x\rvert\lvert B_y\rvert}\leq (\mathbb{H}(\phi(U_A))+\mathbb{H}(\phi(U_B)))(d(U_A,U_B)-d(U_{A_x},U_{B_y})).\]
\end{lemma}
\begin{proof}
\uses{fibring-ident}\leanok
The random variables $(U_A\mid \phi(U_A)=x)$ and $(U_B\mid \phi(U_B)=y)$ are equal in distribution to $U_{A_x}$ and $U_{B_y}$ respectively (both are uniformly distributed over their respective fibres). It follows from \Cref{fibring-ident} that
\begin{align*}
\sum_{x,y\in H}\frac{\lvert A_x\rvert\lvert B_y\rvert}{\lvert A\rvert\lvert B\rvert}d[U_{A_x};U_{B_y}]
&=d[U_A\mid \phi(U_A); U_B\mid \phi(U_B)]\\
&\leq d[U_A;U_B]-d[\phi(U_A);\phi(U_B)].
\end{align*}
Therefore with $M:=\mathbb{H}(\phi(U_A))+\mathbb{H}(\phi(U_B))$ we have
\[\left(\sum_{x,y\in H}\frac{\lvert A_x\rvert\lvert B_y\rvert}{\lvert A\rvert\lvert B\rvert}Md[U_{A_x};U_{B_y}]\right)+Md[\phi(U_A);\phi(U_B)]\leq Md[U_A;U_B].\]
Since
\[M=\sum_{x,y\in H}\frac{\lvert A_x\rvert\lvert B_y\rvert}{\lvert A\rvert\lvert B\rvert}\log \frac{\lvert A\rvert\lvert B\rvert}{\lvert A_x\rvert\lvert B_y\rvert}\]
we have
\[\sum_{x,y\in H} \frac{\lvert A_x\rvert\lvert B_y\rvert}{\lvert A\rvert\lvert B\rvert}\left(Md[U_{A_x};U_{B_y}]+d[\phi(U_A);\phi(U_B)]\log \frac{\lvert A\rvert\lvert B\rvert}{\lvert A_x\rvert\lvert B_y\rvert}\right)\leq Md[U_A;U_B].\]
It follows that there exists some $x,y\in H$ such that $\lvert A_x\rvert,\lvert B_y\rvert\neq 0$ and
\[Md[U_{A_x};U_{B_y}]+d[\phi(U_A);\phi(U_B)]\log \frac{\lvert A\rvert\lvert B\rvert}{\lvert A_x\rvert\lvert B_y\rvert}\leq Md[U_A;U_B].\]
\end{proof} | lemma single_fibres {G H Ω Ω': Type*}
[AddCommGroup G] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G]
[AddCommGroup H] [Countable H] [MeasurableSpace H] [MeasurableSingletonClass H]
[MeasureSpace Ω] [MeasureSpace Ω']
[IsProbabilityMeasure (ℙ : Measure Ω)] [IsProbabilityMeasure (ℙ : Measure Ω')]
(φ : G →+ H)
{A B : Set G} [Finite A] [Finite B] {UA : Ω → G} {UB : Ω' → G} (hA : A.Nonempty) (hB : B.Nonempty)
(hUA': Measurable UA) (hUB': Measurable UB) (hUA : IsUniform A UA) (hUB : IsUniform B UB)
(hUA_mem : ∀ ω, UA ω ∈ A) (hUB_mem : ∀ ω, UB ω ∈ B) :
∃ (x y : H) (Ax By : Set G),
Ax = A ∩ φ.toFun ⁻¹' {x} ∧ By = B ∩ φ.toFun ⁻¹' {y} ∧ Ax.Nonempty ∧ By.Nonempty ∧
d[φ.toFun ∘ UA # φ.toFun ∘ UB]
* log (Nat.card A * Nat.card B / ((Nat.card Ax) * (Nat.card By))) ≤
(H[φ.toFun ∘ UA] + H[φ.toFun ∘ UB]) * (d[UA # UB] - dᵤ[Ax # By]) := by
have : Nonempty A := hA.to_subtype
have : Nonempty B := hB.to_subtype
have : FiniteRange UA := finiteRange_of_finset UA A.toFinite.toFinset (by simpa)
have : FiniteRange UB := finiteRange_of_finset UB B.toFinite.toFinset (by simpa)
have hUA_coe : IsUniform A.toFinite.toFinset.toSet UA := by rwa [Set.Finite.coe_toFinset]
have hUB_coe : IsUniform B.toFinite.toFinset.toSet UB := by rwa [Set.Finite.coe_toFinset]
let A_ (x : H) : Set G := A ∩ φ.toFun ⁻¹' {x}
let B_ (y : H) : Set G := B ∩ φ.toFun ⁻¹' {y}
let X : Finset H := FiniteRange.toFinset (φ.toFun ∘ UA)
let Y : Finset H := FiniteRange.toFinset (φ.toFun ∘ UB)
have h_Ax (x : X) : Nonempty (A_ x.val) := by
obtain ⟨ω, hω⟩ := (FiniteRange.mem_iff _ _).mp x.property
use UA ω; exact Set.mem_inter (hUA_mem ω) hω
have h_By (y : Y) : Nonempty (B_ y.val) := by
obtain ⟨ω, hω⟩ := (FiniteRange.mem_iff _ _).mp y.property
use UB ω; exact Set.mem_inter (hUB_mem ω) hω
have h_AX (a : A) : φ.toFun a.val ∈ X := by
obtain ⟨ω, hω⟩ := hUA_coe.nonempty_preimage_of_mem hUA' (A.toFinite.mem_toFinset.mpr a.property)
exact (FiniteRange.mem_iff _ (φ.toFun a.val)).mpr ⟨ω, congr_arg _ hω⟩
have h_BY (b : B) : φ.toFun b.val ∈ Y := by
obtain ⟨ω, hω⟩ := hUB_coe.nonempty_preimage_of_mem hUB' (B.toFinite.mem_toFinset.mpr b.property)
exact (FiniteRange.mem_iff _ (φ.toFun b.val)).mpr ⟨ω, congr_arg _ hω⟩
let φ_AX (a : A) : X := by use φ.toFun a.val; exact h_AX a
let φ_BY (b : B) : Y := by use φ.toFun b.val; exact h_BY b
have h_φ_AX (x : X) : A_ x.val = φ_AX ⁻¹' {x} := by ext; simp [A_, φ_AX]; simp [Subtype.ext_iff]
have h_φ_BY (y : Y) : B_ y.val = φ_BY ⁻¹' {y} := by ext; simp [B_, φ_BY]; simp [Subtype.ext_iff]
let p (x : H) (y : H) : ℝ :=
(Nat.card (A_ x).Elem) * (Nat.card (B_ y).Elem) / ((Nat.card A.Elem) * (Nat.card B.Elem))
have :
∑ x ∈ X, ∑ y ∈ Y, (p x y) * dᵤ[A_ x # B_ y] ≤ d[UA # UB] - d[φ.toFun ∘ UA # φ.toFun ∘ UB] :=
calc
_ = d[UA | φ.toFun ∘ UA # UB | φ.toFun ∘ UB] := by
rewrite [condRuzsaDist_eq_sum hUA' (.comp .of_discrete hUA')
hUB' (.comp .of_discrete hUB')]
refine Finset.sum_congr rfl <| fun x hx ↦ Finset.sum_congr rfl <| fun y hy ↦ ?_
have : Nonempty (A_ x) := h_Ax ⟨x, hx⟩
have : Nonempty (B_ y) := h_By ⟨y, hy⟩
let μx := (ℙ : Measure Ω)[|(φ.toFun ∘ UA) ⁻¹' {x}]
have hμx : IsProbabilityMeasure μx := by
apply ProbabilityTheory.cond_isProbabilityMeasure
rw [Set.preimage_comp]
apply hUA_coe.measure_preimage_ne_zero hUA'
rw [Set.inter_comm, Set.Finite.coe_toFinset]
exact .of_subtype
let μy := (ℙ : Measure Ω')[|(φ.toFun ∘ UB) ⁻¹' {y}]
have hμy : IsProbabilityMeasure μy := by
apply ProbabilityTheory.cond_isProbabilityMeasure
rw [Set.preimage_comp]
apply hUB_coe.measure_preimage_ne_zero hUB'
rw [Set.inter_comm, Set.Finite.coe_toFinset]
exact .of_subtype
have h_μ_unif : IsUniform (A_ x) UA μx ∧ IsUniform (B_ y) UB μy := by
have : _ ∧ _ := ⟨hUA.restrict hUA' (φ.toFun ⁻¹' {x}), hUB.restrict hUB' (φ.toFun ⁻¹' {y})⟩
rwa [Set.inter_comm _ A, Set.inter_comm _ B] at this
rw [setRuzsaDist_eq_rdist h_μ_unif.1 h_μ_unif.2 hUA' hUB']
show _ = (Measure.real _ (UA ⁻¹' (_ ⁻¹' _))) * (Measure.real _ (UB ⁻¹' (_ ⁻¹' _))) * _
rewrite [hUA_coe.measureReal_preimage hUA', hUB_coe.measureReal_preimage hUB']
simp_rw [p, A_, B_, IsProbabilityMeasure.measureReal_univ, one_mul]
rewrite [mul_div_mul_comm, Set.inter_comm A, Set.inter_comm B]
simp only [Set.Finite.coe_toFinset, Set.Finite.mem_toFinset, Finset.mem_val]; rfl
_ ≤ d[UA # UB] - d[φ.toFun ∘ UA # φ.toFun ∘ UB] := by
rewrite [ZeroHom.toFun_eq_coe, AddMonoidHom.toZeroHom_coe]
linarith only [rdist_le_sum_fibre φ hUA' hUB' (μ := ℙ) (μ' := ℙ)]
let M := H[φ.toFun ∘ UA] + H[φ.toFun ∘ UB]
have hM : M = ∑ x ∈ X, ∑ y ∈ Y, Real.negMulLog (p x y) := by
have h_compl {x y} (h_notin : (x, y) ∉ X ×ˢ Y) : Real.negMulLog (p x y) = 0 := by
unfold p
rewrite [Finset.mem_product, not_and_or] at h_notin
suffices A_ x = ∅ ∨ B_ y = ∅ by obtain h | h := this <;> rw [h] <;> simp
refine h_notin.imp ?_ ?_
· rw [← not_nonempty_iff_eq_empty]
rintro h ⟨a, ha, rfl⟩
exact h (h_AX ⟨a, ha⟩)
· rw [← not_nonempty_iff_eq_empty]
rintro h ⟨a, ha, rfl⟩
exact h (h_BY ⟨a, ha⟩)
unfold M
unfold entropy
have : IsProbabilityMeasure (.map (φ ∘ UA) ℙ) :=
isProbabilityMeasure_map (.comp_measurable .of_discrete hUA')
have : IsProbabilityMeasure (.map (φ ∘ UB) ℙ) :=
isProbabilityMeasure_map (.comp_measurable .of_discrete hUB')
rewrite [← Finset.sum_product', ← tsum_eq_sum fun _ ↦ h_compl, ← measureEntropy_prod]
apply tsum_congr; intro; congr
rewrite [← Set.singleton_prod_singleton, Measure.smul_apply, Measure.prod_prod,
Measure.map_apply (.comp .of_discrete hUA') (MeasurableSet.singleton _),
Measure.map_apply (.comp .of_discrete hUB') (MeasurableSet.singleton _),
Set.preimage_comp, hUA_coe.measure_preimage hUA',
Set.preimage_comp, hUB_coe.measure_preimage hUB']
simp [p, A_, B_, mul_div_mul_comm, Set.inter_comm, ENNReal.toReal_div]
have h_sum : ∑ x ∈ X, ∑ y ∈ Y,
(p x y) * (M * dᵤ[A_ x # B_ y] + d[φ.toFun ∘ UA # φ.toFun ∘ UB] * -Real.log (p x y)) ≤
M * d[UA # UB] :=
calc
_ = ∑ x ∈ X, ∑ y ∈ Y, (p x y) * M * dᵤ[A_ x # B_ y] + M * d[φ.toFun ∘ UA # φ.toFun ∘ UB] := by
simp_rw [hM, Finset.sum_mul, ← Finset.sum_add_distrib]
refine Finset.sum_congr rfl <| fun _ _ ↦ Finset.sum_congr rfl <| fun _ _ ↦ ?_
simp only [negMulLog, left_distrib, mul_assoc, Finset.sum_mul]
exact congrArg (HAdd.hAdd _) (by group)
_ = M * ∑ x ∈ X, ∑ y ∈ Y, (p x y) * dᵤ[A_ x # B_ y] + M * d[φ.toFun ∘ UA # φ.toFun ∘ UB] := by
simp_rw [Finset.mul_sum]
congr; ext; congr; ext; group
_ ≤ M * d[UA # UB] := by
rewrite [← left_distrib]
apply mul_le_mul_of_nonneg_left
· linarith
· unfold M
linarith only [entropy_nonneg (φ.toFun ∘ UA) ℙ, entropy_nonneg (φ.toFun ∘ UB) ℙ]
have : ∃ x : X, ∃ y : Y,
M * dᵤ[A_ x.val # B_ y.val] + d[φ.toFun ∘ UA # φ.toFun ∘ UB] * -Real.log (p x.val y.val) ≤
M * d[UA # UB] := by
let f (xy : H × H) := (p xy.1 xy.2) * (M * d[UA # UB])
let g (xy : H × H) := (p xy.1 xy.2) *
(M * dᵤ[A_ xy.1 # B_ xy.2] + d[φ.toFun ∘ UA # φ.toFun ∘ UB] * -Real.log (p xy.1 xy.2))
by_contra hc; push_neg at hc
replace hc : ∀ xy ∈ X ×ˢ Y, f xy < g xy := by
refine fun xy h ↦ mul_lt_mul_of_pos_left ?_ ?_
· exact hc ⟨xy.1, (Finset.mem_product.mp h).1⟩ ⟨xy.2, (Finset.mem_product.mp h).2⟩
· have : Nonempty _ := h_Ax ⟨xy.1, (Finset.mem_product.mp h).1⟩
have : Nonempty _ := h_By ⟨xy.2, (Finset.mem_product.mp h).2⟩
simp only [p, div_pos, mul_pos, Nat.cast_pos, Nat.card_pos]
have h_nonempty : Finset.Nonempty (X ×ˢ Y) := by
use ⟨φ.toFun <| UA <| Classical.choice <| ProbabilityMeasure.nonempty ⟨ℙ, inferInstance⟩,
φ.toFun <| UB <| Classical.choice <| ProbabilityMeasure.nonempty ⟨ℙ, inferInstance⟩⟩
exact Finset.mem_product.mpr ⟨FiniteRange.mem _ _, FiniteRange.mem _ _⟩
replace hc := Finset.sum_lt_sum_of_nonempty h_nonempty hc
have h_p_one : ∑ x ∈ X ×ˢ Y, p x.1 x.2 = 1 := by
simp_rw [Finset.sum_product, p, mul_div_mul_comm, ← Finset.mul_sum,
← sum_prob_preimage hA h_φ_AX, sum_prob_preimage hB h_φ_BY, mul_one]
rewrite [← Finset.sum_mul, h_p_one, one_mul, Finset.sum_product] at hc
exact not_le_of_gt hc h_sum
obtain ⟨x, y, hxy⟩ := this
refine ⟨x, y, A_ x.val, B_ y.val, rfl, rfl, .of_subtype, .of_subtype, ?_⟩
rewrite [← inv_div, Real.log_inv]
show _ * -log (p x.val y.val) ≤ M * _
linarith only [hxy]
variable {G : Type*} [AddCommGroup G] [Module.Free ℤ G]
open Real MeasureTheory ProbabilityTheory Pointwise Set Function
open QuotientAddGroup | pfr/blueprint/src/chapter/weak_pfr.tex:142 | pfr/PFR/WeakPFR.lean:437 |
PFR | sub_condMultiDistance_le | \begin{lemma}[Lower bound on conditional multidistance]\label{cond-multidist-lower}\lean{sub_condMultiDistance_le}\leanok If $(X_i)_{1 \leq i \leq m}$ is a $\tau$-minimizer, and $k := D[(X_i)_{1 \leq i \leq m}]$, then for any other tuples $(X'_i)_{1 \leq i \leq m}$ and $(Y_i)_{1 \leq i \leq m}$ with the $X'_i$ $G$-valued, one has
$$ k - D[(X'_i)_{1 \leq i \leq m} | (Y_i)_{1 \leq i \leq m}] \leq \eta \sum_{i=1}^m d[X_i; X'_i|Y_i].$$
\end{lemma}
\begin{proof}\uses{multidist-lower, cond-multidist-def, cond-multidist-alt}\leanok
Immediate from \Cref{multidist-lower}, \Cref{cond-multidist-alt}, and \Cref{cond-dist-def}.
\end{proof} | lemma sub_condMultiDistance_le {G Ω₀ : Type u} [MeasureableFinGroup G] [MeasureSpace Ω₀] (p : multiRefPackage G Ω₀) (Ω : Fin p.m → Type u) (hΩ : ∀ i, MeasureSpace (Ω i)) (hΩprob: ∀ i, IsProbabilityMeasure (hΩ i).volume) (X : ∀ i, Ω i → G) (hmeasX: ∀ i, Measurable (X i)) (h_min : multiTauMinimizes p Ω hΩ X) (Ω' : Fin p.m → Type u) (hΩ' : ∀ i, MeasureSpace (Ω' i)) (hΩ'prob: ∀ i, IsProbabilityMeasure (hΩ' i).volume) (X' : ∀ i, Ω' i → G) (hmeasX': ∀ i, Measurable (X' i)) {S : Type u} [Fintype S][MeasurableSpace S] [MeasurableSingletonClass S] (Y : ∀ i, Ω' i → S) (hY : ∀ i, Measurable (Y i)): D[X; hΩ] - D[X'|Y; hΩ'] ≤ p.η * ∑ i, d[X i ; (hΩ i).volume # X' i | Y i; (hΩ' i).volume ] := by
set μ := fun ω: Fin p.m → S ↦ ∏ i : Fin p.m, (ℙ (Y i ⁻¹' {ω i})).toReal
have probmes (i : Fin p.m) : ∑ ωi : S, (ℙ (Y i ⁻¹' {ωi})).toReal = 1 := by
convert Finset.sum_toReal_measure_singleton (s := Finset.univ) (Measure.map (Y i) ℙ) with ω _ i _
· exact (MeasureTheory.Measure.map_apply (hY i) ( .singleton ω)).symm
replace hΩ'prob := hΩ'prob i
rw [MeasureTheory.Measure.map_apply (hY i) (Finset.measurableSet _), Finset.coe_univ, Set.preimage_univ, measure_univ, ENNReal.one_toReal]
-- μ has total mass one
have total : ∑ (ω : Fin p.m → S), μ ω = 1 := calc
_ = ∏ i, ∑ ωi, (ℙ (Y i ⁻¹' {ωi})).toReal := by
convert Finset.sum_prod_piFinset Finset.univ _ with ω _ i _
rfl
_ = ∏ i, 1 := by
apply Finset.prod_congr rfl
intro i _
exact probmes i
_ = 1 := by
simp only [Finset.prod_const_one]
calc
_ = ∑ (ω: Fin p.m → S), μ ω * D[X; hΩ] - ∑ (ω: Fin p.m → S), μ ω * D[X' ; fun i ↦ MeasureSpace.mk ℙ[|Y i ⁻¹' {ω i}]] := by
congr
rw [← Finset.sum_mul, total, one_mul]
_ = ∑ (ω: Fin p.m → S), μ ω * (D[X; hΩ] - D[X' ; fun i ↦ MeasureSpace.mk ℙ[|Y i ⁻¹' {ω i}]]) := by
rw [← Finset.sum_sub_distrib]
apply Finset.sum_congr rfl
intro _ _
exact (mul_sub_left_distrib _ _ _).symm
_ ≤ ∑ (ω: Fin p.m → S), μ ω * (p.η * ∑ i, d[X i ; (hΩ i).volume # X' i; ℙ[|Y i ⁻¹' {ω i}] ]) := by
apply Finset.sum_le_sum
intro ω _
rcases eq_or_ne (μ ω) 0 with hω | hω
· simp [hω]
gcongr
let hΩ'_cond i := MeasureSpace.mk ℙ[|Y i ⁻¹' {ω i}]
have hΩ'prob_cond i : IsProbabilityMeasure (hΩ'_cond i).volume := by
refine cond_isProbabilityMeasure ?_
contrapose! hω
apply Finset.prod_eq_zero (Finset.mem_univ i)
simp only [hω, ENNReal.zero_toReal]
exact sub_multiDistance_le p Ω hΩ hΩprob X hmeasX h_min Ω' hΩ'_cond hΩ'prob_cond X' hmeasX'
_ = p.η * ∑ i, ∑ (ω: Fin p.m → S), μ ω * d[X i ; (hΩ i).volume # X' i; ℙ[|Y i ⁻¹' {ω i}] ] := by
rw [Finset.sum_comm, Finset.mul_sum]
apply Finset.sum_congr rfl
intro ω _
rw [Finset.mul_sum, Finset.mul_sum, Finset.mul_sum]
apply Finset.sum_congr rfl
intro i _
ring
_ = _ := by
congr with i
let f := fun j ↦ (fun ωj ↦ (ℙ (Y j ⁻¹' {ωj})).toReal * (if i=j then d[X i ; ℙ # X' i ; ℙ[|Y i ⁻¹' {ωj}]] else 1))
calc
_ = ∑ ω : Fin p.m → S, ∏ j, f j (ω j) := by
apply Finset.sum_congr rfl
intro ω _
rw [Finset.prod_mul_distrib]
congr
simp only [Finset.prod_ite_eq, Finset.mem_univ, ↓reduceIte]
_ = ∏ j, ∑ ωj, f j ωj := Finset.sum_prod_piFinset Finset.univ f
_ = ∏ j, if i = j then d[X i # X' i | Y i] else 1 := by
apply Finset.prod_congr rfl
intro j _
by_cases hij : i = j
· simp only [hij, mul_ite, mul_one, ↓reduceIte, f]
rw [condRuzsaDist'_eq_sum' (hmeasX' i) (hY i), ← hij]
simp only [mul_ite, mul_one, hij, ↓reduceIte, f]
exact probmes j
_ = _ := by
simp only [Finset.prod_ite_eq, Finset.mem_univ, ↓reduceIte]
/-- With the notation of the previous lemma, we have
\begin{equation}\label{5.3-conv}
k - D[ X'_{[m]} | Y_{[m]} ] \leq \eta \sum_{i=1}^m d[X_{\sigma(i)};X'_i|Y_i]
\end{equation}
for any permutation $\sigma : \{1,\dots,m\} \rightarrow \{1,\dots,m\}$. -/ | pfr/blueprint/src/chapter/torsion.tex:340 | pfr/PFR/MultiTauFunctional.lean:113 |
PFR | sub_condMultiDistance_le' | \begin{corollary}[Lower bound on conditional multidistance, II]\label{cond-multidist-lower-II}\lean{sub_condMultiDistance_le'}\leanok With the notation of the previous lemma, we have
\begin{equation}\label{5.3-conv}
k - D[ X'_{[m]} | Y_{[m]} ] \leq \eta \sum_{i=1}^m d[X_{\sigma(i)};X'_i|Y_i]
\end{equation}
for any permutation $\sigma : \{1,\dots,m\} \rightarrow \{1,\dots,m\}$.
\end{corollary}
\begin{proof}\uses{cond-multidist-lower, multidist-perm}\leanok This follows from \Cref{cond-multidist-lower} and \Cref{multidist-perm}.
\end{proof} | lemma sub_condMultiDistance_le' {G Ω₀ : Type u} [MeasureableFinGroup G] [MeasureSpace Ω₀] (p : multiRefPackage G Ω₀) (Ω : Fin p.m → Type u) (hΩ : ∀ i, MeasureSpace (Ω i)) (hΩprob: ∀ i, IsProbabilityMeasure (hΩ i).volume) (X : ∀ i, Ω i → G) (hmeasX: ∀ i, Measurable (X i)) (h_min : multiTauMinimizes p Ω hΩ X) (Ω' : Fin p.m → Type u) (hΩ' : ∀ i, MeasureSpace (Ω' i)) (hΩ'prob: ∀ i, IsProbabilityMeasure (hΩ' i).volume) (X' : ∀ i, Ω' i → G) (hmeasX': ∀ i, Measurable (X' i)) {S : Type u} [Fintype S] [MeasurableSpace S] [MeasurableSingletonClass S] (Y : ∀ i, Ω' i → S) (hY : ∀ i, Measurable (Y i)) (φ : Equiv.Perm (Fin p.m)) : D[X; hΩ] - D[X'|Y; hΩ'] ≤ p.η * ∑ i, d[X (φ i) ; (hΩ (φ i)).volume # X' i | Y i; (hΩ' i).volume ] := by
let Xφ := fun i => X (φ i)
let Ωφ := fun i => Ω (φ i)
let hΩφ := fun i => hΩ (φ i)
let hΩφprob := fun i => hΩprob (φ i)
let hmeasXφ := fun i => hmeasX (φ i)
calc
_ = D[Xφ; hΩφ] - D[X'|Y; hΩ'] := by
congr 1
rw [multiDist_of_perm hΩ hΩprob X φ]
_ ≤ _ := by
apply sub_condMultiDistance_le p Ωφ hΩφ hΩφprob Xφ hmeasXφ _ Ω' hΩ' hΩ'prob X' hmeasX' Y hY
intro Ω'' hΩ'' X''
calc
_ = multiTau p Ω hΩ X := by
dsimp [multiTau]
congr 1
· exact multiDist_of_perm hΩ hΩprob X φ
congr 1
exact Fintype.sum_equiv φ _ _ fun _ ↦ rfl
_ ≤ multiTau p Ω'' hΩ'' X'' := h_min Ω'' hΩ'' X'' | pfr/blueprint/src/chapter/torsion.tex:348 | pfr/PFR/MultiTauFunctional.lean:190 |
PFR | sub_mem_symmGroup | \begin{lemma}[Zero Ruzsa distance implies large symmetry group]\label{zero-large}
\lean{sub_mem_symmGroup}\leanok If $X$ is a $G$-valued random variable such that
$d[X ;X]=0$, and $x,y \in G$ are such that $P[X=x], P[X=y]>0$, then $x-y \in \mathrm{Sym}[X]$.
\end{lemma}
\begin{proof}
\uses{ruz-indep, ruz-copy, relabeled-entropy-cond,vanish-entropy, alternative-mutual, independent-exist,sym-group-def}\leanok
Let $X_1,X_2$ be independent copies of $X$ (from \Cref{independent-exist}). Let $A$ denote the range of $X$. From \Cref{ruz-indep} and \Cref{ruz-copy} we have
$$ \bbH[X_1-X_2] = \bbH[X_1].$$
Observe from \Cref{relabeled-entropy-cond} that
$$ \bbH[X_1-X_2|X_2] = \bbH[X_1|X_2] = \bbH[X_1]$$
and hence by \Cref{alternative-mutual}
$$ \bbI[X_1-X_2 : X_1] = 0.$$
By \Cref{vanish-entropy}, $X_1-X_2$ and $X_1$ are therefore independent, thus the law of $(X_1-X_2|X_1=x)$ does not depend on $x \in A$. The claim follows.
\end{proof} | lemma sub_mem_symmGroup (hX : Measurable X) (hdist : d[X # X] = 0)
{x y : G} (hx : ℙ (X⁻¹' {x}) ≠ 0) (hy : ℙ (X⁻¹' {y}) ≠ 0) : x - y ∈ symmGroup X hX := by
/- Consider two independent copies `X'` and `Y'` of `X`. The assumption on the Rusza distance
ensures that `H[X' - Y' | Y'] = H[X' - Y']`, i.e., `X' - Y'` and `Y'` are independent. Therefore,
the distribution of `X' - c` is independent of `c` for `c` in the support of `Y'`.
In particular, `X' - x` and `X' - y` have the same distribution, which is equivalent to the
claim of the lemma. -/
rcases ProbabilityTheory.independent_copies_two hX hX with
⟨Ω', mΩ', X', Y', hP, hX', hY', h_indep, hidX, hidY⟩
rw [hidX.symm.symmGroup_eq hX hX']
have A : H[X' - Y' | Y'] = H[X' - Y'] := calc
H[X' - Y' | Y'] = H[X' | Y'] := condEntropy_sub_right hX' hY'
_ = H[X'] := h_indep.condEntropy_eq_entropy hX' hY'
_ = H[X' - Y'] := by
have : d[X' # Y'] = 0 := by rwa [hidX.rdist_eq hidY]
rw [h_indep.rdist_eq hX' hY', ← (hidX.trans hidY.symm).entropy_eq] at this
linarith
have I : IndepFun (X' - Y') Y' := by
refine (mutualInfo_eq_zero (by fun_prop) hY').1 ?_
rw [mutualInfo_eq_entropy_sub_condEntropy (by fun_prop) hY', A, sub_self]
have M : ∀ c, ℙ (Y' ⁻¹' {c}) ≠ 0 → IdentDistrib (fun ω ↦ X' ω - c) (X' - Y') := by
intro c hc
let F := fun ω ↦ X' ω - c
refine ⟨by fun_prop, by fun_prop, ?_⟩
ext s hs
rw [Measure.map_apply (by fun_prop) hs, Measure.map_apply (by fun_prop) hs]
have : ℙ (F ⁻¹' s) * ℙ (Y' ⁻¹' {c}) = ℙ ((X' - Y') ⁻¹' s) * ℙ (Y' ⁻¹' {c}) := by calc
ℙ (F ⁻¹' s) * ℙ (Y' ⁻¹' {c}) = ℙ (F ⁻¹' s ∩ Y' ⁻¹' {c}) := by
have hFY' : IndepFun F Y' := by
have : Measurable (fun z ↦ z - c) := measurable_sub_const' c
apply h_indep.comp this measurable_id
rw [indepFun_iff_measure_inter_preimage_eq_mul.1 hFY' _ _ hs .of_discrete]
_ = ℙ ((X' - Y') ⁻¹' s ∩ Y' ⁻¹' {c}) := by
congr 1; ext z; simp (config := {contextual := true}) [F]
_ = ℙ ((X' - Y') ⁻¹' s) * ℙ (Y' ⁻¹' {c}) := by
rw [indepFun_iff_measure_inter_preimage_eq_mul.1 I _ _ hs .of_discrete]
rwa [ENNReal.mul_left_inj hc (measure_ne_top ℙ _)] at this
have J : IdentDistrib (fun ω ↦ X' ω - x) (fun ω ↦ X' ω - y) := by
have Px : ℙ (Y' ⁻¹' {x}) ≠ 0 := by
convert hx; exact hidY.measure_mem_eq .of_discrete
have Py : ℙ (Y' ⁻¹' {y}) ≠ 0 := by
convert hy; exact hidY.measure_mem_eq .of_discrete
exact (M x Px).trans (M y Py).symm
have : IdentDistrib X' (fun ω ↦ X' ω + (x - y)) := by
have : Measurable (fun c ↦ c + x) := measurable_add_const x
convert J.comp this using 1
· ext ω; simp
· ext ω; simp; abel
exact this
/-- If `d[X # X] = 0`, then `X - x₀` is the uniform distribution on the subgroup of `G`
stabilizing the distribution of `X`, for any `x₀` of positive probability. -/ | pfr/blueprint/src/chapter/100_percent.tex:16 | pfr/PFR/HundredPercent.lean:60 |
PFR | sub_multiDistance_le | \begin{lemma}[Lower bound on multidistance]\label{multidist-lower}\lean{sub_multiDistance_le}\leanok If $(X_i)_{1 \leq i \leq m}$ is a $\tau$-minimizer, and $k := D[(X_i)_{1 \leq i \leq m}]$, then for any other tuple $(X'_i)_{1 \leq i \leq m}$, one has
$$ k - D[(X'_i)_{1 \leq i \leq m}] \leq \eta \sum_{i=1}^m d[X_i; X'_i].$$
\end{lemma}
\begin{proof}\uses{tau-min-multi, ruzsa-triangle}\leanok
By \Cref{tau-min-multi} we have
$$ \tau[ (X_i)_{1 \leq i \leq m}] \leq \tau[ (X'_i)_{1 \leq i \leq m}]$$
and hence by \Cref{tau-def-multi}
$$ k + \eta \sum_{i=1}^m d[X_i; X^0] \leq D[(X'_i)_{1 \leq i \leq m}] + \eta \sum_{i=1}^m d[X'_i; X^0].$$
On the other hand, by \Cref{ruzsa-triangle} we have
$$ d[X'_i; X^0] \leq d[X_i; X^0] + d[X_i; X'_i].$$
The claim follows.
\end{proof} | lemma sub_multiDistance_le {G Ω₀ : Type u} [MeasureableFinGroup G] [hΩ₀: MeasureSpace Ω₀] (p : multiRefPackage G Ω₀) (Ω : Fin p.m → Type u) (hΩ : ∀ i, MeasureSpace (Ω i)) (hΩprob: ∀ i, IsProbabilityMeasure (hΩ i).volume) (X : ∀ i, Ω i → G) (hmeasX: ∀ i, Measurable (X i)) (h_min : multiTauMinimizes p Ω hΩ X) (Ω' : Fin p.m → Type u) (hΩ' : ∀ i, MeasureSpace (Ω' i)) (hΩprob': ∀ i, IsProbabilityMeasure (hΩ' i).volume) (X' : ∀ i, Ω' i → G) (hmeasX': ∀ i, Measurable (X' i)) : D[X; hΩ] - D[X'; hΩ'] ≤ p.η * ∑ i, d[X i ; (hΩ i).volume # X' i; (hΩ' i).volume ] := by
suffices D[X; hΩ] + p.η * ∑ i, d[X i ; (hΩ i).volume # p.X₀; hΩ₀.volume ] ≤ D[X'; hΩ'] + (p.η * ∑ i, d[X i ; (hΩ i).volume # p.X₀; hΩ₀.volume ] + p.η * ∑ i, d[X i ; (hΩ i).volume # X' i; (hΩ' i).volume ]) by
linarith
calc
_ ≤ D[X'; hΩ'] + p.η * ∑ i, d[X' i ; (hΩ' i).volume # p.X₀; hΩ₀.volume ] := h_min Ω' hΩ' X'
_ ≤ _ := by
have hη : p.η > 0 := p.hη
have hprob := p.hprob
rw [← mul_add, ← Finset.sum_add_distrib]
gcongr with i _
rw [add_comm, rdist_symm (Y := X' i)]
apply rdist_triangle (hmeasX' i) (hmeasX i) p.hmeas
/-- If $(X_i)_{1 \leq i \leq m}$ is a $\tau$-minimizer, and $k := D[(X_i)_{1 \leq i \leq m}]$, then for any other tuples $(X'_i)_{1 \leq i \leq m}$ and $(Y_i)_{1 \leq i \leq m}$ with the $X'_i$ $G$-valued, one has
$$ k - D[(X'_i)_{1 \leq i \leq m} | (Y_i)_{1 \leq i \leq m}] \leq \eta \sum_{i=1}^m d[X_i; X'_i|Y_i].$$ -/ | pfr/blueprint/src/chapter/torsion.tex:300 | pfr/PFR/MultiTauFunctional.lean:95 |
PFR | sum_condMutual_le | \begin{lemma}[Bound on conditional mutual informations]\label{uvw-s}
\uses{conditional-mutual-def}
\lean{I₃_eq,sum_condMutual_le}\leanok
We have
$$
I(U : V \, | \, S) + I(V : W \, | \,S) + I(W : U \, | \, S) \leq 6 \eta k - \frac{1 - 5 \eta}{1-\eta} (2 \eta k - I_1).
$$
\end{lemma}
\begin{proof}
\uses{second-estimate,symm-lemma}\leanok
From the definitions of $I_1,I_2$ and \Cref{symm-lemma}, we see that
\[
I_1 = I(U : V \, | \, S), \qquad I_2 = I(W : U \, | \, S), \qquad I_2 = I(V : W \, | \,S).
\]
Applying \Cref{first-estimate} and \Cref{second-estimate} we have the inequalities
\[ I_2 \leq 2 \eta k + \frac{2\eta(2 \eta k - I_1)}{1-\eta} .
\]
We conclude that
$$
I_1 + I_2 + I_2 \leq I_1+4\eta k+ \frac{4\eta(2 \eta k - I_1)}{1-\eta}
$$
and the claim follows from some calculation.
\end{proof} | lemma sum_condMutual_le [Module (ZMod 2) G] [IsProbabilityMeasure (ℙ : Measure Ω)] :
I[U : V | S] + I[V : W | S] + I[W : U | S]
≤ 6 * p.η * k - (1 - 5 * p.η) / (1 - p.η) * (2 * p.η * k - I₁) := by
have : I[W : U | S] = I₂ := condMutualInfo_comm (by fun_prop) (by fun_prop) ..
rw [I₃_eq, this]
have h₂ := second_estimate p X₁ X₂ X₁' X₂' hX₁ hX₂ hX₁' hX₂' h₁ h₂ h_indep h_min
have h := add_le_add (add_le_add_left h₂ I₁) h₂
convert h using 1
have : 1 - p.η > 0 := by linarith [p.hη']
field_simp [this]
ring
all_goals { simpa }
local notation3:max "c[" A "; " μ " # " B " ; " μ' "]" =>
d[p.X₀₁; ℙ # A; μ] - d[p.X₀₁ # X₁] + (d[p.X₀₂; ℙ # B; μ'] - d[p.X₀₂ # X₂])
local notation3:max "c[" A " # " B "]" =>
d[p.X₀₁ # A] - d[p.X₀₁ # X₁] + (d[p.X₀₂ # B] - d[p.X₀₂ # X₂])
local notation3:max "c[" A " | " B " # " C " | " D "]" =>
d[p.X₀₁ # A|B] - d[p.X₀₁ # X₁] + (d[p.X₀₂ # C|D] - d[p.X₀₂ # X₂])
include h_indep h₁ h₂ in | pfr/blueprint/src/chapter/entropy_pfr.tex:255 | pfr/PFR/Endgame.lean:132 |
PFR | sum_dist_diff_le | \begin{lemma}[Bound on distance increments]\label{total-dist}\leanok
\lean{sum_dist_diff_le}
We have
\begin{align*}
\sum_{i=1}^2 \sum_{A\in\{U,V,W\}} \big(d[X^0_i;A|S] & - d[X^0_i;X_i]\big) \\
&\leq (6 - 3\eta) k + 3(2 \eta k - I_1).
\end{align*}
\end{lemma}
\begin{proof}
\uses{second-useful, foursum-bound}\leanok
By \Cref{second-useful} (taking $X = X_1^0$, $Y = X_1$, $Z = X_2$ and $Z' = \tilde X_1 + \tilde X_2$, so that $Y + Z = U$ and $Y + Z + Z' = S$) we have, noting that $\bbH[Y+ Z] = \bbH[Z']$,
\[
d[X^0_1;U|S] - d[X^0_1;X_1] \leq \tfrac{1}{2} (\bbH[S] - \bbH[X_1]).
\]
Further applications of \Cref{second-useful} give
\begin{align*}
d[X^0_2;U|S] - d[X^0_2; X_2] &\leq \tfrac{1}{2} (\bbH[S] - \bbH[X_2]) \\
d[X^0_1;V|S] - d[X^0_1;X_1] &\leq \tfrac{1}{2} (\bbH[S] - \bbH[X_1])\\
d[X^0_2;V|S] - d[X^0_2;X_2] &\leq \tfrac{1}{2} (\bbH[S] - \bbH[X_2])
\end{align*}
and
\[d[X^0_1;W|S] - d[X^0_1;X_1] \leq \tfrac{1}{2} (\bbH[S] + \bbH[W] - \bbH[X_1] - \bbH[W']),\] where $W' := X_2 + \tilde X_2$.
To treat $d[X^0_2;W|S]$, first note that this equals $d[X^0_2;W'|S]$, since for a fixed choice $s$ of $S$ we have $W' = W + s$ (here we need some helper lemma about Ruzsa distance). Now we may apply \Cref{second-useful} to obtain
\[d[X^0_2;W'|S] - d[X^0_2;X_2] \leq \tfrac{1}{2} (\bbH[S] + \bbH[W'] - \bbH[X_2] - \bbH[W]).\]
Summing these six estimates and using \Cref{foursum-bound}, we conclude that
\begin{align*}
\sum_{i=1}^2 \sum_{A\in\{U,V,W\}} \big(d[X^0_i;A|S] & - d[X^0_i;X_i]\big) \\
&\leq 3 \bbH[S] - \tfrac{3}{2} \bbH[X_1] - \tfrac{3}{2} \bbH[X_2]\\
&\leq (6 - 3\eta) k + 3(2 \eta k - I_1)
\end{align*}
as required.
\end{proof} | lemma sum_dist_diff_le [IsProbabilityMeasure (ℙ : Measure Ω)] [Module (ZMod 2) G] :
c[U|S # U|S] + c[V|S # V|S] + c[W|S # W|S] ≤ (6 - 3 * p.η)*k + 3 * (2*p.η*k - I₁) := by
let X₀₁ := p.X₀₁
let X₀₂ := p.X₀₂
have ineq1 : d[X₀₁ # U | S] - d[X₀₁ # X₁] ≤ (H[S ; ℙ] - H[X₁ ; ℙ])/2 := by
have aux1 : H[S] + H[U] - H[X₁] - H[X₁' + X₂'] = H[S] - H[X₁] := by
rw [hU X₁ X₂ X₁' X₂' h₁ h₂ h_indep]; ring
have aux2 : d[X₀₁ # U | U + (X₁' + X₂')] - d[X₀₁ # X₁]
≤ (H[U + (X₁' + X₂')] + H[U] - H[X₁] - H[X₁' + X₂']) / 2 :=
condRuzsaDist_diff_ofsum_le ℙ (hX := p.hmeas1) (hY := hX₁) (hZ := hX₂)
(Measurable.add hX₁' hX₂') (independenceCondition1 hX₁ hX₂ hX₁' hX₂' h_indep)
rw [← add_assoc, aux1] at aux2
linarith [aux2]
have ineq2 : d[X₀₂ # U | S] - d[X₀₂ # X₂] ≤ (H[S ; ℙ] - H[X₂ ; ℙ])/2 := by
have aux1 : H[S] + H[U] - H[X₂] - H[X₁' + X₂'] = H[S] - H[X₂] := by
rw [hU X₁ X₂ X₁' X₂' h₁ h₂ h_indep] ; ring
have aux2 : d[X₀₂ # U | U + (X₁' + X₂')] - d[X₀₂ # X₂]
≤ (H[U + (X₁' + X₂')] + H[U] - H[X₂] - H[X₁' + X₂']) / 2 := by
rw [(show U = X₂ + X₁ from add_comm _ _)]
apply condRuzsaDist_diff_ofsum_le ℙ (p.hmeas2) (hX₂) (hX₁)
(Measurable.add hX₁' hX₂') (independenceCondition2 hX₁ hX₂ hX₁' hX₂' h_indep)
rw [← add_assoc, aux1] at aux2
linarith [aux2]
have V_add_eq : V + (X₁ + X₂') = S := by abel
have ineq3 : d[X₀₁ # V | S] - d[X₀₁ # X₁] ≤ (H[S ; ℙ] - H[X₁ ; ℙ])/2 := by
have aux2 : d[p.X₀₁ # V | V + (X₁ + X₂')] - d[p.X₀₁ # X₁']
≤ (H[V + (X₁ + X₂')] + H[V] - H[X₁'] - H[X₁ + X₂']) / 2 :=
condRuzsaDist_diff_ofsum_le ℙ (p.hmeas1) (hX₁') (hX₂) (Measurable.add hX₁ hX₂')
(independenceCondition3 hX₁ hX₂ hX₁' hX₂' h_indep)
have aux1 : H[S] + H[V] - H[X₁'] - H[X₁ + X₂'] = H[S ; ℙ] - H[X₁ ; ℙ] := by
rw [hV X₁ X₂ X₁' X₂' h₁ h₂ h_indep, h₁.entropy_eq]; ring
rw [← ProbabilityTheory.IdentDistrib.rdist_eq (IdentDistrib.refl p.hmeas1.aemeasurable) h₁,
V_add_eq, aux1] at aux2
linarith [aux2]
have ineq4 : d[X₀₂ # V | S] - d[X₀₂ # X₂] ≤ (H[S ; ℙ] - H[X₂ ; ℙ])/2 := by
have aux2 : d[p.X₀₂ # V | V + (X₁ + X₂')] - d[p.X₀₂ # X₂]
≤ (H[V + (X₁ + X₂')] + H[V] - H[X₂] - H[X₁ + X₂']) / 2 := by
rw [(show V = X₂ + X₁' from add_comm _ _)]
apply condRuzsaDist_diff_ofsum_le ℙ (p.hmeas2) (hX₂) (hX₁') (Measurable.add hX₁ hX₂')
(independenceCondition4 hX₁ hX₂ hX₁' hX₂' h_indep)
have aux1 : H[S] + H[V] - H[X₂] - H[X₁ + X₂'] = H[S ; ℙ] - H[X₂ ; ℙ] := by
rw [hV X₁ X₂ X₁' X₂' h₁ h₂ h_indep]; ring
rw [V_add_eq, aux1] at aux2
linarith [aux2]
let W' := X₂ + X₂'
have ineq5 : d[X₀₁ # W | S] - d[X₀₁ # X₁] ≤ (H[S ; ℙ] + H[W ; ℙ] - H[X₁ ; ℙ] - H[W' ; ℙ])/2 := by
have := condRuzsaDist_diff_ofsum_le ℙ p.hmeas1 hX₁ hX₁' (Measurable.add hX₂ hX₂')
(independenceCondition5 hX₁ hX₂ hX₁' hX₂' h_indep)
have S_eq : X₁ + X₁' + (fun a ↦ X₂ a + X₂' a) = S := by
rw [(show (fun a ↦ X₂ a + X₂' a) = X₂ + X₂' by rfl), ← add_assoc, add_assoc X₁, add_comm X₁',
← add_assoc]
rwa [S_eq, add_comm X₁ X₁'] at this
have ineq6 : d[X₀₂ # W' | S] - d[X₀₂ # X₂] ≤ (H[S ; ℙ] + H[W' ; ℙ] - H[X₂ ; ℙ] - H[W ; ℙ])/2 := by
have := condRuzsaDist_diff_ofsum_le ℙ p.hmeas2 hX₂ hX₂' (Measurable.add hX₁' hX₁)
(independenceCondition6 hX₁ hX₂ hX₁' hX₂' h_indep)
have S_eq : X₂ + X₂' + (fun a ↦ X₁' a + X₁ a) = S := by
rw [(show (fun a ↦ X₁' a + X₁ a) = X₁' + X₁ by rfl), add_comm, ← add_assoc, add_comm X₁',
add_assoc X₁, add_comm X₁', ← add_assoc]
rwa [S_eq] at this
have dist_eq : d[X₀₂ # W' | S] = d[X₀₂ # W | S] := by
have S_eq : S = (X₂ + X₂') + (X₁' + X₁) := by
rw [add_comm X₁' X₁, add_assoc _ X₂', add_comm X₂', ← add_assoc X₂, ← add_assoc X₂,
add_comm X₂]
rw [S_eq]
apply condRuzsaDist'_of_inj_map' p.hmeas2 (hX₂.add hX₂') (hX₁'.add hX₁)
-- Put everything together to bound the sum of the `c` terms
have ineq7 : c[U|S # U|S] + c[V|S # V|S] + c[W|S # W|S] ≤
3 * H[S ; ℙ] - 3/2 * H[X₁ ; ℙ] -3/2 * H[X₂ ; ℙ] := by
have step₁ : c[U|S # U|S] ≤ H[S ; ℙ] - (H[X₁ ; ℙ] + H[X₂ ; ℙ])/2 :=
calc
_ = (d[p.X₀₁ # U|S] - d[p.X₀₁ # X₁]) + (d[p.X₀₂ # U|S] - d[p.X₀₂ # X₂]) := by ring
_ ≤ (H[S ; ℙ] - H[X₁ ; ℙ])/2 + (H[S ; ℙ] - H[X₂ ; ℙ])/2 := add_le_add ineq1 ineq2
_ = H[S ; ℙ] - (H[X₁ ; ℙ] + H[X₂ ; ℙ])/2 := by ring
have step₂ : c[V|S # V|S] ≤ H[S ; ℙ] - (H[X₁ ; ℙ] + H[X₂ ; ℙ])/2 :=
calc c[V|S # V|S] =(d[p.X₀₁ # V|S] - d[p.X₀₁ # X₁]) + (d[p.X₀₂ # V|S] - d[p.X₀₂ # X₂]) := by ring
_ ≤ (H[S ; ℙ] - H[X₁ ; ℙ])/2 + (H[S ; ℙ] - H[X₂ ; ℙ])/2 := add_le_add ineq3 ineq4
_ = H[S ; ℙ] - (H[X₁ ; ℙ] + H[X₂ ; ℙ])/2 := by ring
have step₃ : c[W|S # W|S] ≤ H[S ; ℙ] - (H[X₁ ; ℙ] + H[X₂ ; ℙ])/2 :=
calc c[W|S # W|S] = (d[X₀₁ # W | S] - d[X₀₁ # X₁]) + (d[X₀₂ # W' | S] - d[X₀₂ # X₂]) :=
by rw [dist_eq]
_ ≤ (H[S ; ℙ] + H[W ; ℙ] - H[X₁ ; ℙ] - H[W' ; ℙ])/2 + (H[S ; ℙ] + H[W' ; ℙ] - H[X₂ ; ℙ] - H[W ; ℙ])/2
:= add_le_add ineq5 ineq6
_ = H[S ; ℙ] - (H[X₁ ; ℙ] + H[X₂ ; ℙ])/2 := by ring
calc c[U|S # U|S] + c[V|S # V|S] + c[W|S # W|S] ≤ (H[S ; ℙ] - (H[X₁ ; ℙ] + H[X₂ ; ℙ])/2) +
(H[S ; ℙ] - (H[X₁ ; ℙ] + H[X₂ ; ℙ])/2) + (H[S ; ℙ] - (H[X₁ ; ℙ] + H[X₂ ; ℙ])/2) :=
add_le_add (add_le_add step₁ step₂) step₃
_ = 3 * H[S ; ℙ] - 3/2 * H[X₁ ; ℙ] -3/2 * H[X₂ ; ℙ] := by ring
have h_indep' : iIndepFun ![X₁, X₂, X₂', X₁'] := by
refine .of_precomp (Equiv.swap (2 : Fin 4) 3).surjective ?_
convert h_indep using 1
ext x
fin_cases x ; all_goals { aesop }
have ineq8 : 3 * H[S ; ℙ] ≤ 3/2 * (H[X₁ ; ℙ] + H[X₂ ; ℙ]) + 3*(2+p.η)*k - 3*I₁ :=
calc 3 * H[S ; ℙ] ≤ 3 * (H[X₁ ; ℙ] / 2 + H[X₂ ; ℙ] / 2 + (2+p.η)*k - I₁) := by
apply (mul_le_mul_left (zero_lt_three' ℝ)).mpr
(ent_ofsum_le p X₁ X₂ X₁' X₂' hX₁ hX₂ hX₁' hX₂' h₁ h₂ h_indep' h_min)
_ = 3/2 * ( H[X₁ ; ℙ] + H[X₂ ; ℙ]) + 3*(2+p.η)*k - 3*I₁ := by ring
-- Final computation
calc c[U|S # U|S] + c[V|S # V|S] + c[W|S # W|S] ≤ 3 * H[S ; ℙ] - 3/2 * H[X₁ ; ℙ] -3/2 * H[X₂ ; ℙ] := ineq7
_ = 3 * H[S ; ℙ] - (3/2 *(H[X₁ ; ℙ] + H[X₂ ; ℙ])) := by ring
_ ≤ (3/2 * ( H[X₁ ; ℙ] + H[X₂ ; ℙ]) + 3*(2+p.η)*k - 3*I₁) - (3/2 *(H[X₁ ; ℙ] + H[X₂ ; ℙ])) :=
sub_le_sub_right ineq8 _
_ = (6 - 3 * p.η)*k + 3 * (2*p.η*k - I₁) := by ring
omit [Fintype G] hG [MeasurableSingletonClass G] mΩ in | pfr/blueprint/src/chapter/entropy_pfr.tex:280 | pfr/PFR/Endgame.lean:209 |
PFR | sum_of_conditional_distance_le | \begin{lemma}[Distance bound]\label{xi-z2-w-dist}\lean{sum_of_conditional_distance_le}\leanok We have $\sum_{i=1}^m d[X_i;Z_2|W] \leq 8(m^3-m^2) k$.
\end{lemma}
\begin{proof}\uses{klm-3, cond-dist-fact, mutual-w-z2, ent-z2, multidist-ruzsa-II}
For each $i \in \{1,\dots, m\}$, using \Cref{klm-3} (noting the sum $Z_2$ contains $X_i$ as a summand) we have
\begin{equation}\label{in-a-bit-6}
d[X_i;Z_2] \leq d[X_i;X_i] + \tfrac12 (\bbH[Z_2] - \bbH[X_i])
\end{equation}
and using \Cref{cond-dist-fact} we have
\[
d[X_i;Z_2 | W] \leq d[X_i;Z_2] + \tfrac12 \bbI[W : Z_2].
\]
Combining with~\eqref{in-a-bit-6} and \Cref{mutual-w-z2} gives
\[ d[X_i;Z_2 | W] \leq d[X_i;X_i] + \tfrac12 (\bbH[Z_2] - \bbH[X_i]) + (m-1)k.\]
Summing over $i$ and applying \Cref{ent-z2} gives
\[ \sum_{i = 1}^m d[X_i;Z_2 | W] \leq \sum_{i = 1}^m d[X_i;X_i] + m(8m^2-16m+1) k + m(m-1) k.\]
Finally, applying \Cref{multidist-ruzsa-II} (and dropping some lower order terms) gives the claim.
\end{proof} | /-- We have $\sum_{i=1}^m d[X_i;Z_2|W] \leq 8(m^3-m^2) k$. -/
lemma sum_of_conditional_distance_le : ∑ i, d[ X i # Z2 | W] ≤ 8 * (p.m^3 - p.m^2)*k := sorry
/-- Let $G$ be an abelian group, let $(T_1,T_2,T_3)$ be a $G^3$-valued random variable such that $T_1+T_2+T_3=0$ holds identically, and write
\[
\delta := \bbI[T_1 : T_2] + \bbI[T_1 : T_3] + \bbI[T_2 : T_3].
\]
Let $Y_1,\dots,Y_n$ be some further $G$-valued random variables and let $\alpha>0$ be a constant.
Then there exists a random variable $U$ such that
$$ d[U;U] + \alpha \sum_{i=1}^n d[Y_i;U] \leq \Bigl(2 + \frac{\alpha n}{2} \Bigr) \delta + \alpha \sum_{i=1}^n d[Y_i;T_2].
$$
-/ | pfr/blueprint/src/chapter/torsion.tex:735 | pfr/PFR/TorsionEndgame.lean:66 |
PFR | sum_of_rdist_eq | \begin{corollary}[Specific fibring identity]\label{cor-fibre}
\lean{sum_of_rdist_eq}\leanok
Let $Y_1,Y_2,Y_3$ and $Y_4$ be independent $G$-valued random variables.
Then
\begin{align*}
& d[Y_1+Y_3; Y_2+Y_4] + d[Y_1|Y_1+Y_3; Y_2|Y_2+Y_4] \\
&\qquad + \bbI[Y_1+Y_2 : Y_2 + Y_4 | Y_1+Y_2+Y_3+Y_4] = d[Y_1; Y_2] + d[Y_3; Y_4].
\end{align*}
\end{corollary}
\begin{proof}
\uses{fibring-ident, add-entropy, relabeled-entropy-cond}\leanok
We apply \Cref{fibring-ident} with $H := G \times G$, $H' := G$, $\pi$ the addition homomorphism $\pi(x,y) := x+y$, and with the random variables $Z_1 := (Y_1,Y_3)$ and $Z_2 := (Y_2,Y_4)$. Then by independence (\Cref{add-entropy})
\[
d[Z_1; Z_2] = d[Y_1; Y_2] + d[Y_3; Y_4]
\]
while by definition
\[
d[\pi(Z_1); \pi(Z_2)] = d[Y_1+Y_3; Y_2+Y_4].
\]
Furthermore,
\[
d[Z_1|\pi(Z_1); Z_2|\pi(Z_2)] = d[Y_1|Y_1+Y_3;Y_2|Y_2+Y_4],
\]
since $Z_1=(Y_1,Y_3)$ and $Y_1$ are linked by an invertible affine transformation once $\pi(Z_1)=Y_1+Y_3$ is fixed, and similarly for $Z_2$ and $Y_2$. (This has to do with \Cref{relabeled-entropy-cond})
Finally, we have
\begin{align*}
&\bbI[Z_1 + Z_2 : (\pi(Z_1),\pi(Z_2)) \,|\, \pi(Z_1) + \pi(Z_2)] \\
&\ = \bbI[(Y_1+Y_2, Y_3+Y_4) : (Y_1+Y_3, Y_2+Y_4) \,|\, Y_1+Y_2+Y_3+Y_4] \\
&\ = \bbI[Y_1+Y_2 : Y_2+Y_4 \,|\, Y_1+Y_2+Y_3+Y_4]
\end{align*}
where in the last line we used the fact that $(Y_1+Y_2, Y_1+Y_2+Y_3+Y_4)$ uniquely determine $Y_3+Y_4$ and similarly
$(Y_2+Y_4, Y_1+Y_2+Y_3+Y_4)$ uniquely determine $Y_1+Y_3$. (This requires another helper lemma about entropy.)
\end{proof} | lemma sum_of_rdist_eq (Y : Fin 4 → Ω → G) (h_indep : iIndepFun Y μ)
(h_meas : ∀ i, Measurable (Y i)) :
d[Y 0; μ # Y 1; μ] + d[Y 2; μ # Y 3; μ]
= d[(Y 0) - (Y 2); μ # (Y 1) - (Y 3); μ]
+ d[Y 0 | (Y 0) - (Y 2); μ # Y 1 | (Y 1) - (Y 3); μ]
+ I[(Y 0) - (Y 1) : (Y 1) - (Y 3) | (Y 0) - (Y 1) - (Y 2) + (Y 3); μ] := by
let π : G × G →+ G := (AddMonoidHom.fst G G) - (AddMonoidHom.snd G G)
have hπ {W_1 W_2 : Ω → G} : π ∘ ⟨W_1, W_2⟩ = W_1 - W_2 := rfl
let Z_1 : Ω → G × G := ⟨Y 0, Y 2⟩
let Z_2 : Ω → G × G := ⟨Y 1, Y 3⟩
have hZ : Z_1 - Z_2 = ⟨Y 0 - Y 1, Y 2 - Y 3⟩ := rfl
have m1 : Measurable Z_1 := (h_meas 0).prodMk (h_meas 2)
have m2 : Measurable Z_2 := (h_meas 1).prodMk (h_meas 3)
have h_indep_0 : IndepFun (Y 0) (Y 1) μ := h_indep.indepFun (by decide)
have h_indep_2 : IndepFun (Y 2) (Y 3) μ := h_indep.indepFun (by decide)
have h_indep_Z : IndepFun Z_1 Z_2 μ := h_indep.indepFun_prodMk_prodMk h_meas
0 2 1 3 (by decide) (by decide) (by decide) (by decide)
have h_indep_sub : IndepFun (Y 0 - Y 1) (Y 2 - Y 3) μ :=
h_indep.indepFun_sub_sub h_meas 0 1 2 3 (by decide) (by decide) (by decide) (by decide)
have msub (i j : Fin 4) : Measurable (Y i - Y j) := (h_meas i).sub (h_meas j)
have h_add : d[Z_1; μ # Z_2; μ] = d[Y 0; μ # Y 1; μ] + d[Y 2; μ # Y 3; μ] := by
rw [h_indep_0.rdist_eq (h_meas 0) (h_meas 1), h_indep_2.rdist_eq (h_meas 2) (h_meas 3),
h_indep_Z.rdist_eq m1 m2, hZ,
(entropy_pair_eq_add (h_meas 0) (h_meas 2)).2 (h_indep.indepFun (by decide)),
(entropy_pair_eq_add (h_meas 1) (h_meas 3)).2 (h_indep.indepFun (by decide)),
(entropy_pair_eq_add (msub 0 1) (msub 2 3)).2 h_indep_sub]
ring_nf
rw [← h_add, rdist_of_indep_eq_sum_fibre π h_indep_Z m1 m2]
simp only [hπ, hZ]
rw [sum_of_rdist_eq_step_condRuzsaDist h_indep h_meas,
sum_of_rdist_eq_step_condMutualInfo h_meas]
/-- Let $Y_1,Y_2,Y_3$ and $Y_4$ be independent $G$-valued random variables.
Then
$$d[Y_1+Y_3; Y_2+Y_4] + d[Y_1|Y_1+Y_3; Y_2|Y_2+Y_4] $$
$$ + I[Y_1+Y_2 : Y_2 + Y_4 | Y_1+Y_2+Y_3+Y_4] = d[Y_1; Y_2] + d[Y_3; Y_4].$$
-/ | pfr/blueprint/src/chapter/fibring.tex:51 | pfr/PFR/Fibring.lean:152 |
PFR | sum_of_z_eq_zero | \begin{lemma}[Zero-sum]\label{Zero-sum}\lean{sum_of_z_eq_zero}\leanok We have
\begin{equation}%
\label{eq:sum-zero}
Z_1+Z_2+Z_3= 0
\end{equation}
\end{lemma}
\begin{proof}\uses{more-random}\leanok Clear from definition.
\end{proof} | /-- Z_1+Z_2+Z_3= 0 -/
lemma sum_of_z_eq_zero :Z1 + Z2 + Z3 = 0 := by
rw [← Finset.sum_add_distrib, ← Finset.sum_add_distrib]
apply Finset.sum_eq_zero
intro i _
rw [← Finset.sum_add_distrib, ← Finset.sum_add_distrib]
apply Finset.sum_eq_zero
intro j _
rw [← add_zsmul, ← add_zsmul]
convert zero_zsmul ?_
simp | pfr/blueprint/src/chapter/torsion.tex:598 | pfr/PFR/TorsionEndgame.lean:33 |
PFR | sum_uvw_eq_zero | \begin{lemma}[Key identity]\label{key-ident}
\lean{sum_uvw_eq_zero}\leanok
We have $U+V+W=0$.
\end{lemma}
\begin{proof} \leanok Obvious because we are in characteristic two.
\end{proof} | /-- `U + V + W = 0`. -/
lemma sum_uvw_eq_zero [Module (ZMod 2) G] : U + V + W = 0 := by
simp [add_assoc, add_left_comm (a := X₁), add_left_comm (a := X₂), ← ZModModule.sub_eq_add X₁',
ZModModule.add_self] | pfr/blueprint/src/chapter/entropy_pfr.tex:314 | pfr/PFR/Endgame.lean:326 |
PFR | tau_minimizer_exists | \begin{proposition}[$\tau$ has minimum]\label{tau-min}\uses{tau-min-def}
\lean{tau_minimizer_exists}\leanok
A pair $X_1, X_2$ of $\tau$-minimizers exist.
\end{proposition}
\begin{proof}\uses{tau-copy}\leanok By \Cref{tau-copy}, $\tau$ only depends on the probability distributions of $X_1, X_2$. This ranges over a compact space, and $\tau$ is continuous. So $\tau$ has a minimum.
\end{proof} | /-- A pair of random variables minimizing $τ$ exists. -/
lemma tau_minimizer_exists [MeasurableSingletonClass G] :
∃ (Ω : Type uG) (_ : MeasureSpace Ω) (X₁ : Ω → G) (X₂ : Ω → G),
Measurable X₁ ∧ Measurable X₂ ∧ IsProbabilityMeasure (ℙ : Measure Ω) ∧
tau_minimizes p X₁ X₂ := by
let μ := (tau_min_exists_measure p).choose
have : IsProbabilityMeasure μ.1 := (tau_min_exists_measure p).choose_spec.1
have : IsProbabilityMeasure μ.2 := (tau_min_exists_measure p).choose_spec.2.1
have P : IsProbabilityMeasure (μ.1.prod μ.2) := by infer_instance
let M : MeasureSpace (G × G) := ⟨μ.1.prod μ.2⟩
refine ⟨G × G, M, Prod.fst, Prod.snd, measurable_fst, measurable_snd, P, ?_⟩
intro ν₁ ν₂ h₁ h₂
have A : τ[@Prod.fst G G # @Prod.snd G G | p] = τ[id ; μ.1 # id ; μ.2 | p] :=
ProbabilityTheory.IdentDistrib.tau_eq p IdentDistrib.fst_id IdentDistrib.snd_id
convert (tau_min_exists_measure p).choose_spec.2.2 ν₁ ν₂ h₁ h₂ | pfr/blueprint/src/chapter/entropy_pfr.tex:36 | pfr/PFR/TauFunctional.lean:141 |
PFR | tau_minimizer_exists_rdist_eq_zero | \begin{theorem}[Limiting improved $\tau$-decrement]\label{de-prop-lim-improv}\lean{tau_minimizer_exists_rdist_eq_zero}\leanok
For $\eta = 1/8$, there exist tau-minimizers $X_1, X_2$ satisfying $d[X_1;X_2] = 0$.
\end{theorem}
\begin{proof}\uses{de-prop-improv, tau-min}\leanok
For each $\eta<1/8$, consider minimizers $X_1^\eta$ and $X_2^\eta$ from \Cref{tau-min}.
By \Cref{de-prop-improv},
they satisfy $d[X_1^\eta; X_2^\eta]=0$. By compactness of the space of probability measures
on $G$, one may extract a converging subsequence of the distributions of $X_1^\eta$ and $X_2^\eta$
as $\eta \to 1/8$. By continuity of all the involved quantities, the limit is a pair of
tau-minimizers for $1/8$ satisfying additionally $d[X_1;X_2] = 0$.
\end{proof} | lemma tau_minimizer_exists_rdist_eq_zero :
∃ (Ω : Type uG) (_ : MeasureSpace Ω) (X₁ : Ω → G) (X₂ : Ω → G),
Measurable X₁ ∧ Measurable X₂ ∧ IsProbabilityMeasure (ℙ : Measure Ω) ∧ tau_minimizes p X₁ X₂
∧ d[X₁ # X₂] = 0 := by
-- let `uₙ` be a sequence converging from below to `η`. In particular, `uₙ < 1/8`.
obtain ⟨u, -, u_mem, u_lim⟩ :
∃ u, StrictMono u ∧ (∀ (n : ℕ), u n ∈ Set.Ioo 0 p.η) ∧ Tendsto u atTop (𝓝 p.η) :=
exists_seq_strictMono_tendsto' p.hη
-- For each `n`, consider a minimizer associated to `η = uₙ`.
let q : ℕ → refPackage Ω₀₁ Ω₀₂ G := fun n ↦
⟨p.X₀₁, p.X₀₂, p.hmeas1, p.hmeas2, u n, (u_mem n).1, by linarith [(u_mem n).2, p.hη']⟩
have : ∀ n, ∃ (μ : Measure G × Measure G),
IsProbabilityMeasure μ.1 ∧ IsProbabilityMeasure μ.2 ∧
∀ (ν₁ : Measure G) (ν₂ : Measure G), IsProbabilityMeasure ν₁ → IsProbabilityMeasure ν₂ →
τ[id ; μ.1 # id ; μ.2 | q n] ≤ τ[id ; ν₁ # id ; ν₂ | q n] :=
fun n ↦ tau_min_exists_measure (q n)
choose μ μ1_prob μ2_prob hμ using this
-- The minimizer associated to `uₙ` is at zero Rusza distance of itself, by
-- lemma `tau_strictly_decreases'`.
have I n : d[id ; (μ n).1 # id ; (μ n).2] = 0 := by
let M : MeasureSpace (G × G) := ⟨(μ n).1.prod (μ n).2⟩
have : IsProbabilityMeasure ((μ n).1.prod (μ n).2) := by infer_instance
have : d[@Prod.fst G G # @Prod.snd G G] = d[id ; (μ n).1 # id ; (μ n).2] :=
IdentDistrib.rdist_eq IdentDistrib.fst_id IdentDistrib.snd_id
rw [← this]
apply tau_strictly_decreases' (q n) measurable_fst measurable_snd ?_
(by linarith [(u_mem n).2, p.hη'])
intro ν₁ ν₂ h₁ h₂
have A : τ[@Prod.fst G G # @Prod.snd G G | q n] = τ[id ; (μ n).1 # id ; (μ n).2 | q n] :=
ProbabilityTheory.IdentDistrib.tau_eq (q n) IdentDistrib.fst_id IdentDistrib.snd_id
rw [A]
exact hμ n _ _ h₁ h₂
-- extract a converging subsequence of the sequence of minimizers, seen as pairs of probability
-- measures on `G` (which is a compact space).
let μ' : ℕ → ProbabilityMeasure G × ProbabilityMeasure G :=
fun n ↦ (⟨(μ n).1, μ1_prob n⟩, ⟨(μ n).2, μ2_prob n⟩)
let _i : TopologicalSpace G := (⊥ : TopologicalSpace G)
have : DiscreteTopology G := ⟨rfl⟩
-- The limiting pair of measures will be the desired minimizer.
rcases IsCompact.tendsto_subseq (x := μ') isCompact_univ (fun n ↦ mem_univ _)
with ⟨ν, -, φ, φmono, hν⟩
have φlim : Tendsto φ atTop atTop := φmono.tendsto_atTop
let M : MeasureSpace (G × G) := ⟨(ν.1 : Measure G).prod ν.2⟩
have P : IsProbabilityMeasure ((ν.1 : Measure G).prod (ν.2 : Measure G)) := by infer_instance
refine ⟨G × G, M, Prod.fst, Prod.snd, measurable_fst, measurable_snd, P, ?_, ?_⟩
-- check that it is indeed a minimizer, as a limit of minimizers.
· intro ν₁ ν₂ h₁ h₂
have A : τ[@Prod.fst G G # @Prod.snd G G | p] = τ[id ; ν.1 # id ; ν.2 | p] :=
ProbabilityTheory.IdentDistrib.tau_eq p IdentDistrib.fst_id IdentDistrib.snd_id
rw [A]
have L1 : Tendsto (fun n ↦ τ[id ; (μ (φ n)).1 # id ; (μ (φ n)).2 | q (φ n)]) atTop
(𝓝 (τ[id ; ν.1 # id ; ν.2 | p])) := by
apply Tendsto.add (Tendsto.add ?_ (Tendsto.mul (u_lim.comp φlim) ?_))
(Tendsto.mul (u_lim.comp φlim) ?_)
· apply Tendsto.comp (continuous_rdist_restrict_probabilityMeasure.tendsto _) hν
· have : Continuous
(fun (μ : ProbabilityMeasure G × ProbabilityMeasure G) ↦ d[p.X₀₁ ; ℙ # id ; μ.1]) :=
Continuous.comp (continuous_rdist_restrict_probabilityMeasure₁' _ _ p.hmeas1) continuous_fst
apply Tendsto.comp (this.tendsto _) hν
· have : Continuous
(fun (μ : ProbabilityMeasure G × ProbabilityMeasure G) ↦ d[p.X₀₂ ; ℙ # id ; μ.2]) :=
Continuous.comp (continuous_rdist_restrict_probabilityMeasure₁' _ _ p.hmeas2) continuous_snd
apply Tendsto.comp (this.tendsto _) hν
have L2 : Tendsto (fun n ↦ τ[id ; ν₁ # id ; ν₂ | q (φ n)]) atTop
(𝓝 (τ[id ; ν₁ # id ; ν₂ | p])) :=
Tendsto.add (Tendsto.add tendsto_const_nhds (Tendsto.mul (u_lim.comp φlim)
tendsto_const_nhds)) (Tendsto.mul (u_lim.comp φlim) tendsto_const_nhds)
exact le_of_tendsto_of_tendsto' L1 L2 (fun n ↦ hμ (φ n) _ _ h₁ h₂)
-- check that it has zero Rusza distance, as a limit of a sequence at zero Rusza distance.
· have : d[@Prod.fst G G # @Prod.snd G G] = d[id ; ν.1 # id ; ν.2] :=
IdentDistrib.rdist_eq IdentDistrib.fst_id IdentDistrib.snd_id
rw [this]
have L1 : Tendsto (fun n ↦ d[id ; (μ (φ n)).1 # id ; (μ (φ n)).2]) atTop
(𝓝 (d[id ; ν.1 # id ; (ν.2 : Measure G)])) := by
apply Tendsto.comp (continuous_rdist_restrict_probabilityMeasure.tendsto _) hν
have L2 : Tendsto (fun n ↦ d[id ; (μ (φ n)).1 # id ; (μ (φ n)).2]) atTop (𝓝 0) := by simp [I]
exact tendsto_nhds_unique L1 L2
/-- `entropic_PFR_conjecture_improv`: For two $G$-valued random variables $X^0_1, X^0_2$, there is some
subgroup $H \leq G$ such that $d[X^0_1;U_H] + d[X^0_2;U_H] \le 10 d[X^0_1;X^0_2]$. -/ | pfr/blueprint/src/chapter/improved_exponent.tex:185 | pfr/PFR/ImprovedPFR.lean:734 |
PFR | tau_minimizes | \begin{definition}[$\tau$-minimizer]\label{tau-min-def}
\uses{tau-def}
\lean{tau_minimizes}\leanok
A pair of $G$-valued random variables $X_1, X_2$ are said to be a $\tau$-minimizer if one has
$$\tau[X_1;X_2] \leq \tau[X'_1;X'_2]
$$
for all $G$-valued random variables $X'_1, X'_2$.
\end{definition} | def tau_minimizes {Ω : Type*} [MeasureSpace Ω] (X₁ : Ω → G) (X₂ : Ω → G) : Prop :=
∀ (ν₁ : Measure G) (ν₂ : Measure G), IsProbabilityMeasure ν₁ → IsProbabilityMeasure ν₂ →
τ[X₁ # X₂ | p] ≤ τ[id ; ν₁ # id ; ν₂ | p]
omit [IsProbabilityMeasure (ℙ : Measure Ω₀₁)] [IsProbabilityMeasure (ℙ : Measure Ω₀₂)]
[Fintype G] in | pfr/blueprint/src/chapter/entropy_pfr.tex:26 | pfr/PFR/TauFunctional.lean:105 |
PFR | tau_strictly_decreases | \begin{theorem}[$\tau$-decrement]\label{de-prop}
\lean{tau_strictly_decreases}\leanok
Let $X_1, X_2$ be tau-minimizers. Then $d[X_1;X_2] = 0$.
\end{theorem}
\begin{proof}
\uses{construct-good, key-ident, uvw-s, total-dist, first-estimate, eta-def}\leanok
Set $k := d[X_1;X_2]$. Applying \Cref{construct-good} with any random variables $(T_1,T_2,T_3)$ such that $T_1+T_2+T_3=0$ holds identically, we deduce that
\[
k \leq \delta + \frac{\eta}{3} \biggl( \delta + \sum_{i=1}^2 \sum_{j = 1}^3 (d[X^0_1;T_j] - d[X^0_i;X_i]) \biggr).
\]
Note that $\delta$ is still defined by~\eqref{delta-t1t2t3-def} and thus depends on $T_1,T_2,T_3$.
In particular we may apply this for
\[
T_1 = (U | S = s),\qquad T_2 = (V | S = s), \qquad T_3 = (W | S = s)
\]
for $s$ in the range of $S$ (which is a valid choice by \Cref{key-ident}) and then average over $s$ with weights $p_S(s)$, to obtain
\[k \leq \tilde \delta + \frac{\eta}{3} \biggl( \tilde \delta + \sum_{i=1}^2 \sum_{A\in\{U,V,W\}} \bigl( d[X^0_i;A|S] - d[X^0_i;X_i]\bigr) \biggr),\]
where
\[
\tilde \delta := \bbI[U : V | S] + \bbI[V : W | S] + \bbI[W : U | S].
\]
Putting this together with \Cref{uvw-s} and \Cref{total-dist}, we conclude that
\begin{align*}
k &\leq \Bigl(1+\frac{\eta}{3}\Bigr)\Bigl(6\eta k-\frac{1-5\eta}{1-\eta}(2\eta k-I_1)\Bigr)+\frac{\eta}{3}\Bigl((6-3\eta)k+3(2\eta k-I_1)\Bigr)\\
&= (8\eta + \eta^2) k - \biggl( \frac{1 - 5 \eta}{1-\eta}\Bigl(1 + \frac{\eta}{3}\Bigr) - \eta \biggr)(2 \eta k - I_1)\\
&\leq (8 \eta + \eta^2) k
\end{align*}
since the quantity $2 \eta k - I_1$ is non-negative (by \Cref{first-estimate}), and its coefficient in the above expression is non-positive provided that $\eta(2\eta + 17) \le 3$, which is certainly the case with \Cref{eta-def}.
Moreover, from \Cref{eta-def} we have $8 \eta + \eta^2 < 1$. It follows that $k=0$, as desired.
\end{proof} | theorem tau_strictly_decreases (h_min : tau_minimizes p X₁ X₂) (hpη : p.η = 1/9) :
d[X₁ # X₂] = 0 := by
let ⟨A, mA, μ, Y₁, Y₂, Y₁', Y₂', hμ, h_indep, hY₁, hY₂, hY₁', hY₂', h_id1, h_id2, h_id1', h_id2'⟩
:= independent_copies4_nondep hX₁ hX₂ hX₁ hX₂ ℙ ℙ ℙ ℙ
rw [← h_id1.rdist_eq h_id2]
let _ : MeasureSpace A := ⟨μ⟩
have : IsProbabilityMeasure (ℙ : Measure A) := hμ
rw [← h_id1.tau_minimizes p h_id2] at h_min
apply tau_strictly_decreases_aux p Y₁ Y₂ Y₁' Y₂' hY₁ hY₂ hY₁' hY₂' (h_id1.trans h_id1'.symm)
(h_id2.trans h_id2'.symm) h_indep h_min hpη
/-- `entropic_PFR_conjecture`: For two $G$-valued random variables $X^0_1, X^0_2$, there is some
subgroup $H \leq G$ such that $d[X^0_1;U_H] + d[X^0_2;U_H] \le 11 d[X^0_1;X^0_2]$. -/ | pfr/blueprint/src/chapter/entropy_pfr.tex:382 | pfr/PFR/EntropyPFR.lean:33 |
PFR | tau_strictly_decreases' | \begin{theorem}[Improved $\tau$-decrement]\label{de-prop-improv}\lean{tau_strictly_decreases'}\leanok
Suppose $0 < \eta < 1/8$. Let $X_1, X_2$ be tau-minimizers. Then $d[X_1;X_2] = 0$.
\end{theorem}
\begin{proof}\uses{averaged-construct-good, dist-diff-bound, uvw-s}\leanok
From \Cref{averaged-construct-good}, \Cref{dist-diff-bound}, and \Cref{uvw-s} one has
\[ k \leq 8\eta k - \frac{(1 -5 \eta - \frac{4}{6} \eta)(2 \eta k - I_1)}{(1-\eta)}.\]
For any $\eta < 1/8$, we see from \Cref{first-estimate} that the expression $\frac{(1 -5 \eta - \frac{4}{6} \eta)(2 \eta k - I_1)}{(1-\eta)}$ is nonnegative, and hence $k = 0$ as required.
\end{proof} | theorem tau_strictly_decreases' (hp : 8 * p.η < 1) : d[X₁ # X₂] = 0 := by
let ⟨A, mA, μ, Y₁, Y₂, Y₁', Y₂', hμ, h_indep, hY₁, hY₂, hY₁', hY₂', h_id1, h_id2, h_id1', h_id2'⟩
:= independent_copies4_nondep hX₁ hX₂ hX₁ hX₂ ℙ ℙ ℙ ℙ
rw [← h_id1.rdist_eq h_id2]
let _ : MeasureSpace A := ⟨μ⟩
have : IsProbabilityMeasure (ℙ : Measure A) := hμ
rw [← h_id1.tau_minimizes p h_id2] at h_min
exact tau_strictly_decreases_aux' p hY₁ hY₂ hY₁' hY₂' (h_id1.trans h_id1'.symm)
(h_id2.trans h_id2'.symm) h_indep.reindex_four_abdc h_min hp
end MainEstimates | pfr/blueprint/src/chapter/improved_exponent.tex:174 | pfr/PFR/ImprovedPFR.lean:704 |
PFR | torsion_PFR | \begin{theorem}[PFR]\label{pfr-torsion}\lean{torsion_PFR}\leanok Suppose that $G$ is a finite abelian group of torsion $m$.
If $A \subset G$ is non-empty and $|A+A| \leq K|A|$, then $A$ can be covered by most $mK^{96m^3+2}$ translates of a subspace $H$ of $G$ with $|H| \leq |A|$.
\end{theorem}
\begin{proof}\uses{pfr_aux_torsion}\leanok Repeat the proof of \Cref{pfr}, but with \Cref{pfr_aux_torsion} in place of \Cref{pfr_aux}.
\end{proof} | theorem torsion_PFR {G : Type*} [AddCommGroup G] [Fintype G] {m:ℕ} (hm: m ≥ 2)
(htorsion : ∀ x:G, m • x = 0) {A : Set G} [Finite A] {K : ℝ} (h₀A : A.Nonempty)
(hA : Nat.card (A + A) ≤ K * Nat.card A) :
∃ (H : AddSubgroup G) (c : Set G),
Nat.card c < m * K ^ (96*m^3+2) ∧ 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 `torsion_PFR_conjecture_aux`.
obtain ⟨H, c, hc, IHA, IAH, A_subs_cH⟩ : ∃ (H : AddSubgroup G) (c : Set G),
Nat.card c ≤ K ^ (64 * m^3+2) * Nat.card A ^ (1/2) * Nat.card H ^ (-1/2)
∧ Nat.card H ≤ K ^ (64*m^3) * Nat.card A ∧ Nat.card A ≤ K ^ (64*m^3) * Nat.card H
∧ A ⊆ c + H :=
torsion_PFR_conjecture_aux hm htorsion 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 ^ ((64*m^3+2)) * Nat.card A ^ (1/2) * Nat.card H ^ (-1/2) := hc
_ ≤ K ^ ((64*m^3+2)) * (K ^ (64*m^3) * Nat.card H) ^ (1/2) * Nat.card H ^ (-1/2) := by
gcongr
_ = K ^ (96*m^3+2) := by rpow_ring; norm_num; congr 1; ring
_ < m * K ^ (96*m^3+2) := by
apply (lt_mul_iff_one_lt_left _).mpr
· norm_num; linarith [hm]
positivity
-- otherwise, we decompose `H` into cosets of one of its subgroups `H'`, chosen so that
-- `#A / m < #H' ≤ #A`. This `H'` satisfies the desired conclusion.
· obtain ⟨H', IH'A, IAH', H'H⟩ : ∃ H' : AddSubgroup G, Nat.card H' ≤ Nat.card A
∧ Nat.card A < m * Nat.card H' ∧ H' ≤ H := by
have A_pos' : 0 < Nat.card A := mod_cast A_pos
exact torsion_exists_subgroup_subset_card_le hm htorsion H h.le A_pos'.ne'
have : (Nat.card A / m : ℝ) < Nat.card H' := by
rw [div_lt_iff₀, mul_comm]
· norm_cast
norm_cast; exact Nat.zero_lt_of_lt hm
have H'_pos : (0 : ℝ) < Nat.card H' := by
have : 0 < Nat.card H' := Nat.card_pos; positivity
obtain ⟨u, HH'u, hu⟩ := AddSubgroup.exists_left_transversal_of_le H'H
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 Set.natCard_add_le
_ ≤ (K ^ ((64*m^3+2)) * 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 ^ ((64*m^3+2)) * Nat.card A ^ (1 / 2) * (Nat.card H ^ (-1 / 2)))
* (Nat.card H / (Nat.card A / m)) := by
gcongr
_ = m * K ^ ((64*m^3+2)) * Nat.card A ^ (-1/2) * Nat.card H ^ (1/2) := by
field_simp
rpow_ring
norm_num
_ ≤ m * K ^ ((64*m^3+2)) * Nat.card A ^ (-1/2) * (K ^ (64*m^3) * Nat.card A) ^ (1/2) := by
gcongr
_ = m * K ^ (96*m^3+2) := by
rpow_ring
norm_num
left; congr 1
ring | pfr/blueprint/src/chapter/torsion.tex:875 | pfr/PFR/TorsionEndgame.lean:163 |
PFR | torsion_PFR_conjecture_aux | \begin{lemma}\label{pfr_aux_torsion}\lean{torsion_PFR_conjecture_aux}\leanok Suppose that $G$ is a finite abelian group of torsion $m$. If $A \subset G$ is non-empty and
$|A+A| \leq K|A|$, then $A$ can be covered by at most $K ^
{(64m^3+2)/2}|A|^{1/2}/|H|^{1/2}$ translates of a subspace $H$ of $G$ with
\begin{equation}
\label{ah2}
|H|/|A| \in [K^{-64m^3}, K^{64m^3}].
\end{equation}
\end{lemma}
\begin{proof}\uses{main-entropy, unif-exist, uniform-entropy-II, jensen-bound,ruz-dist-def,ruzsa-diff,bound-conc,ruz-cov} Repeat the proof of \Cref{pfr_aux}, but with \Cref{main-entropy} in place of \Cref{entropy-pfr}.
\end{proof} | lemma torsion_PFR_conjecture_aux {G : Type*} [AddCommGroup G] [Fintype G] {m:ℕ} (hm: m ≥ 2)
(htorsion: ∀ x:G, m • x = 0) {A : Set G} [Finite A] {K : ℝ} (h₀A : A.Nonempty)
(hA : Nat.card (A + A) ≤ K * Nat.card A) :
∃ (H : AddSubgroup G) (c : Set G),
Nat.card c ≤ K ^ (64 * m^3 + 2) * Nat.card A ^ (1/2) * Nat.card H ^ (-1/2 : ℝ )
∧ Nat.card H ≤ K ^ (64 * m^3) * Nat.card A
∧ Nat.card A ≤ K ^ (64 * m^3) * Nat.card H ∧ A ⊆ c + H := sorry | pfr/blueprint/src/chapter/torsion.tex:863 | pfr/PFR/TorsionEndgame.lean:96 |
PFR | torsion_dist_shrinking | \begin{lemma}\label{torsion-dist-shrinking}\lean{torsion_dist_shrinking}\leanok
If $G$ is a torsion-free group and $X,Y$ are $G$-valued random variables and $\phi:G\to \mathbb{F}_2^d$ is a homomorphism then
\[\mathbb{H}(\phi(X))\leq 10d[X;Y].\]
\end{lemma}
\begin{proof}
\uses{fibring-ineq, torsion-free-doubling, dist-zero}\leanok
By \Cref{fibring-ineq} and \Cref{torsion-free-doubling} we have
\[d[\phi(X);\phi(2Y)]\leq d[X;2Y]\leq 5d[X;Y]\]
and $\phi(2Y)=2\phi(Y)\equiv 0$ so the left-hand side is equal to $d[\phi(X);0]=\mathbb{H}(\phi(X))/2$ (using \Cref{dist-zero}).
\end{proof} | lemma torsion_dist_shrinking {H : Type*} [FiniteRange X] [FiniteRange Y] (hX : Measurable X)
(hY : Measurable Y) [AddCommGroup H] [Module (ZMod 2) H]
[MeasurableSpace H] [MeasurableSingletonClass H] [Countable H]
(hG : AddMonoid.IsTorsionFree G) (φ : G →+ H) :
H[φ ∘ X ; μ] ≤ 10 * d[X ; μ # Y ; μ'] :=
calc
H[φ ∘ X ; μ] = 2 * d[φ ∘ X ; μ # φ ∘ (Y + Y) ; μ'] := by
rw [map_comp_add, ZModModule.add_self, Pi.zero_def, rdist_zero_eq_half_ent, mul_div_cancel₀]
exact two_ne_zero
_ ≤ 2 * d[X ; μ # Y + Y ; μ'] := by gcongr; exact rdist_of_hom_le φ hX (hY.add hY)
_ ≤ 2 * (5 * d[X ; μ # Y ; μ']) := by gcongr; exact torsion_free_doubling X Y μ μ' hX hY hG
_ = 10 * d[X ; μ # Y ; μ'] := by ring
end Torsion | pfr/blueprint/src/chapter/weak_pfr.tex:41 | pfr/PFR/WeakPFR.lean:212 |
PFR | torsion_free_doubling | \begin{lemma}\label{torsion-free-doubling}\lean{torsion_free_doubling}\leanok
If $G$ is torsion-free and $X,Y$ are $G$-valued random variables then $d[X;2Y]\leq 5d[X;Y]$.
\end{lemma}
\begin{proof}
\uses{alt-submodularity,ruzsa-triangle,ruzsa-diff}\leanok
Let $Y_1,Y_2$ be independent copies of $Y$ (also independent of $X$). Since $G$ is torsion-free we know $X,Y_1-Y_2,X-2Y_1$ uniquely determine $X,Y_1,Y_2$ and so
\[\mathbb{H}(X,Y_1,Y_2,X-2Y_1)=\mathbb{H}(X,Y_1,Y_2)=\mathbb{H}(X)+2\mathbb{H}(Y).\]
Similarly
\[\mathbb{H}(X,X-2Y_1)=\mathbb{H}(X)+\mathbb{H}(2Y_1)=\mathbb{H}(X)+\mathbb{H}(Y).\]
Furthermore
\[\mathbb{H}(Y_1-Y_2,X-2Y_1)=\mathbb{H}(Y_1-Y_2,X-Y_1-Y_2)\leq \mathbb{H}(Y_1-Y_2)+\mathbb{H}(X-Y_1-Y_2).\]
By submodularity (\Cref{alt-submodularity})
\[\mathbb{H}(X,Y_1,Y_2,X-2Y_1)+\mathbb{H}(X-2Y_1)\leq \mathbb{H}(X,X-2Y_1)+\mathbb{H}(Y_1-Y_2,X-2Y_1).\]
Combining these inequalities
\[\mathbb{H}(X-2Y_1)\leq \mathbb{H}(Y_1-Y_2)+\mathbb{H}(X-Y_1-Y_2)-\mathbb{H}(Y).\]
Similarly we have
\[\mathbb{H}(Y_1,Y_2,X-Y_1-Y_2)=\mathbb{H}(X)+2\mathbb{H}(Y),\]
\[\mathbb{H}(Y_1,X-Y_1-Y_2)=\mathbb{H}(Y)+\mathbb{H}(X-Y_2),\]
and
\[\mathbb{H}(Y_2,X-Y_1-Y_2)=\mathbb{H}(Y)+\mathbb{H}(X-Y_1)\]
and by submodularity (\Cref{alt-submodularity}) again
\[\mathbb{H}(Y_1,Y_2,X-Y_1-Y_2)+ \mathbb{H}(X-Y_1-Y_2)\leq \mathbb{H}(Y_1,X-Y_1-Y_2)+\mathbb{H}(Y_2,X-Y_1-Y_2).\]
Combining these inequalities (and recalling the definition of Ruzsa distance) gives
\[\mathbb{H}(X-Y_1-Y_2)\leq \mathbb{H}(X-Y_1)+\mathbb{H}(X-Y_2)-\mathbb{H}(X)=2d[X;Y]+\mathbb{H}(Y).\]
It follows that
\[\mathbb{H}(X-2Y_1)\leq \mathbb{H}(Y_1-Y_2)+2d[X;Y]\]
and so (using $\mathbb{H}(2Y)=\mathbb{H}(Y)$)
\begin{align*}
d[X;2Y]
&=\mathbb{H}(X-2Y_1)-\mathbb{H}(X)/2-\mathbb{H}(2Y)/2\\
&\leq \mathbb{H}(Y_1-Y_2)+2d[X;Y]-\mathbb{H}(X)/2-\mathbb{H}(Y)/2\\
&= d[Y_1;Y_2]+\frac{\mathbb{H}(Y)-\mathbb{H}(X)}{2}+2d[X;Y].
\end{align*}
Finally note that by the triangle inequality (\Cref{ruzsa-triangle}) we have
\[d[Y_1;Y_2]\leq d[Y_1;X]+d[X;Y_2]=2d[X;Y].\]
The result follows from $(\mathbb{H}(Y)-\mathbb{H}(X))/2\leq d[X;Y]$ (\Cref{ruzsa-diff}).
\end{proof} | /-- If `G` is torsion-free and `X, Y` are `G`-valued random variables then `d[X ; 2Y] ≤ 5d[X ; Y]`. -/
lemma torsion_free_doubling [FiniteRange X] [FiniteRange Y]
(hX : Measurable X) (hY : Measurable Y) (hG : AddMonoid.IsTorsionFree G) :
d[X ; μ # (Y + Y) ; μ'] ≤ 5 * d[X ; μ # Y ; μ'] := by
obtain ⟨A, mA, μA, X', Y'₁, Y'₂, hμA, h_indep, hX'_meas, hY'₁_meas, hY'₂_meas, hX'_ident,
hY'₁_ident, hY'₂_ident, _, _, _⟩ := independent_copies3_nondep_finiteRange hX hY hY μ μ' μ'
have h_meas (i : Fin 3) : Measurable (![X', Y'₁, Y'₂] i) := by fin_cases i <;> assumption
have : NoZeroSMulDivisors ℕ G := hG.noZeroSMulDivisors_nat
have : H[⟨X', ⟨Y'₁ - Y'₂, X' - 2 • Y'₁⟩⟩ ; μA] = H[X ; μ] + 2 * H[Y ; μ'] := calc
H[⟨X', ⟨Y'₁ - Y'₂, X' - 2 • Y'₁⟩⟩ ; μA] = H[⟨X', ⟨Y'₁, Y'₂⟩⟩ ; μA] := by
let f : G × G × G → G × G × G := fun ⟨x, y₁, y₂⟩ ↦ (x, y₁ - y₂, x - 2 • y₁)
show H[f ∘ ⟨X', ⟨Y'₁, Y'₂⟩⟩ ; μA] = _
refine entropy_comp_of_injective μA ?_ f ?_
· exact Measurable.prod hX'_meas <| Measurable.prod hY'₁_meas hY'₂_meas
· exact fun ⟨_, _, _⟩ _ h ↦ by simp [f] at h; obtain ⟨_, _, _⟩ := h; simp_all [smul_right_inj]
_ = H[X ; μ] + 2 * H[Y ; μ'] := by
have : IndepFun X' (prod Y'₁ Y'₂) μA := Indep.symm <|
h_indep.indepFun_prodMk h_meas 1 2 0 (by decide) (by decide)
rw [this.entropy_pair_eq_add hX'_meas (by exact Measurable.prod hY'₁_meas hY'₂_meas),
IndepFun.entropy_pair_eq_add hY'₁_meas hY'₂_meas (h_indep.indepFun (show 1 ≠ 2 by decide)),
hX'_ident.entropy_eq, hY'₁_ident.entropy_eq, hY'₂_ident.entropy_eq, two_mul]
have : H[⟨X', X' - 2 • Y'₁⟩ ; μA] = H[X ; μ] + H[Y ; μ'] := calc
H[⟨X', X' - 2 • Y'₁⟩ ; μA] = H[⟨X', Y'₁⟩ ; μA] := by
let f : G × G → G × G := fun ⟨x, y₁⟩ ↦ (x, x - 2 • y₁)
show H[f ∘ ⟨X', Y'₁⟩ ; μA] = _
apply entropy_comp_of_injective μA (by exact Measurable.prod hX'_meas hY'₁_meas) f
exact fun ⟨_, _⟩ _ h ↦ by simp [f] at h; obtain ⟨_, _⟩ := h; simp_all [smul_right_inj]
_ = H[X ; μ] + H[Y ; μ'] := by
rw [IndepFun.entropy_pair_eq_add hX'_meas hY'₁_meas (h_indep.indepFun (show 0 ≠ 1 by decide)),
hX'_ident.entropy_eq, hY'₁_ident.entropy_eq]
let f : G × G → G × G := fun ⟨x, y⟩ ↦ (x, y - x)
have hf : f.Injective := fun ⟨_, _⟩ _ h ↦ by simp [f] at h; obtain ⟨_, _⟩ := h; simp_all
have : H[⟨Y'₁ - Y'₂, X' - 2 • Y'₁⟩ ; μA] ≤ H[Y'₁ - Y'₂ ; μA] + H[X' - Y'₁ - Y'₂ ; μA] := calc
H[⟨Y'₁ - Y'₂, X' - 2 • Y'₁⟩ ; μA] = H[f ∘ ⟨Y'₁ - Y'₂, X' - Y'₁ - Y'₂⟩ ; μA] := by
show _ = H[⟨Y'₁ - Y'₂, X' - Y'₁ - Y'₂ - (Y'₁ - Y'₂)⟩ ; μA]
rw [sub_sub_sub_cancel_right, ← sub_add_eq_sub_sub, two_nsmul]
_ = H[⟨Y'₁ - Y'₂, X' - Y'₁ - Y'₂⟩ ; μA] := by
refine entropy_comp_of_injective μA (Measurable.prod ?_ ?_) f hf
· exact Measurable.sub hY'₁_meas hY'₂_meas
· exact Measurable.sub (Measurable.sub hX'_meas hY'₁_meas) hY'₂_meas
_ ≤ H[Y'₁ - Y'₂ ; μA] + H[X' - Y'₁ - Y'₂ ; μA] :=
entropy_pair_le_add (hY'₁_meas.sub hY'₂_meas) (hX'_meas.sub hY'₁_meas |>.sub hY'₂_meas) μA
have : H[⟨X', ⟨Y'₁ - Y'₂, X' - 2 • Y'₁⟩⟩ ; μA] + H[X' - 2 • Y'₁ ; μA] ≤
H[⟨X', X' - 2 • Y'₁⟩ ; μA] + H[⟨Y'₁ - Y'₂, X' - 2 • Y'₁⟩ ; μA] := by
have : FiniteRange (Y'₁ - Y'₂) := FiniteRange.sub Y'₁ Y'₂
have : FiniteRange (2 • Y'₁) := by show FiniteRange ((fun x ↦ 2 • x) ∘ Y'₁); infer_instance
apply entropy_triple_add_entropy_le μA hX'_meas (Measurable.sub hY'₁_meas hY'₂_meas)
exact Measurable.sub hX'_meas <| Measurable.const_smul hY'₁_meas 2
have : H[⟨Y'₁, ⟨Y'₂, X' - Y'₁ - Y'₂⟩⟩ ; μA] = H[X ; μ] + 2 * H[Y ; μ'] := calc
H[⟨Y'₁, ⟨Y'₂, X' - Y'₁ - Y'₂⟩⟩ ; μA] = H[⟨Y'₁, ⟨Y'₂, X'⟩⟩ ; μA] := by
let f : G × G × G → G × G × G := fun ⟨y₁, y₂, x⟩ ↦ (y₁, y₂, x - y₁ - y₂)
show H[f ∘ ⟨Y'₁, ⟨Y'₂, X'⟩⟩ ; μA] = H[⟨Y'₁, ⟨Y'₂, X'⟩⟩ ; μA]
refine entropy_comp_of_injective μA ?_ f ?_
· exact Measurable.prod hY'₁_meas <| Measurable.prod hY'₂_meas hX'_meas
· exact fun ⟨_, _, _⟩ _ h ↦ by simp [f] at h; obtain ⟨_, _, _⟩ := h; simp_all
_ = H[X ; μ] + 2 * H[Y ; μ'] := by
have : IndepFun Y'₁ (prod Y'₂ X') μA := Indep.symm <|
h_indep.indepFun_prodMk h_meas 2 0 1 (by decide) (by decide)
rw [this.entropy_pair_eq_add hY'₁_meas (by exact Measurable.prod hY'₂_meas hX'_meas),
IndepFun.entropy_pair_eq_add hY'₂_meas hX'_meas (h_indep.indepFun (show 2 ≠ 0 by decide)),
hX'_ident.entropy_eq, hY'₁_ident.entropy_eq, hY'₂_ident.entropy_eq]
group
have : H[⟨Y'₁, X' - Y'₁ - Y'₂⟩ ; μA] = H[Y ; μ'] + H[X' - Y'₂ ; μA] := calc
H[⟨Y'₁, X' - Y'₁ - Y'₂⟩ ; μA] = H[f ∘ ⟨Y'₁, X' - Y'₂⟩ ; μA] := by rw [sub_right_comm] ; rfl
_ = H[⟨Y'₁, X' - Y'₂⟩ ; μA] := entropy_comp_of_injective μA
(by exact Measurable.prod hY'₁_meas <| Measurable.sub hX'_meas hY'₂_meas) f hf
_ = H[Y ; μ'] + H[X' - Y'₂ ; μA] := by
have : FiniteRange (X' - Y'₂) := FiniteRange.sub X' Y'₂
convert IndepFun.entropy_pair_eq_add hY'₁_meas (hX'_meas.sub hY'₂_meas)
<| h_indep.indepFun_sub_right h_meas 1 0 2 (by decide) (by decide)
exact hY'₁_ident.entropy_eq.symm
have : H[⟨Y'₂, X' - Y'₁ - Y'₂⟩ ; μA] = H[Y ; μ'] + H[X' - Y'₁ ; μA] := calc
H[⟨Y'₂, X' - Y'₁ - Y'₂⟩ ; μA] = H[f ∘ ⟨Y'₂, X' - Y'₁⟩ ; μA] := rfl
_ = H[⟨Y'₂, X' - Y'₁⟩ ; μA] := entropy_comp_of_injective μA
(by exact Measurable.prod hY'₂_meas <| Measurable.sub hX'_meas hY'₁_meas) f hf
_ = H[Y ; μ'] + H[X' - Y'₁ ; μA] := by
have : FiniteRange (X' - Y'₁) := FiniteRange.sub X' Y'₁
convert IndepFun.entropy_pair_eq_add hY'₂_meas (hX'_meas.sub hY'₁_meas)
<| h_indep.indepFun_sub_right h_meas 2 0 1 (by decide) (by decide)
exact hY'₂_ident.entropy_eq.symm
have : H[⟨Y'₁, ⟨Y'₂, X' - Y'₁ - Y'₂⟩⟩ ; μA] + H[X' - Y'₁ - Y'₂ ; μA] ≤
H[⟨Y'₁, X' - Y'₁ - Y'₂⟩ ; μA] + H[⟨Y'₂, X' - Y'₁ - Y'₂⟩ ; μA] := by
apply entropy_triple_add_entropy_le μA hY'₁_meas hY'₂_meas
exact Measurable.sub (Measurable.sub hX'_meas hY'₁_meas) hY'₂_meas
have : H[X' - Y'₁ - Y'₂ ; μA] ≤ 2 * d[X ; μ # Y ; μ'] + H[Y ; μ'] := calc
H[X' - Y'₁ - Y'₂ ; μA] ≤ H[X' - Y'₁ ; μA] + H[X' - Y'₂ ; μA] - H[X ; μ] := by linarith
_ = 2 * d[X ; μ # Y ; μ'] + H[Y ; μ'] := by
nth_rw 1 [two_mul, ← hX'_ident.rdist_eq hY'₁_ident, ← hX'_ident.rdist_eq hY'₂_ident]
have h1 : d[X' ; μA # Y'₁ ; μA] = H[X' - Y'₁ ; μA] - H[X' ; μA] / 2 - H[Y'₁ ; μA] / 2 :=
(h_indep.indepFun (show 0 ≠ 1 by decide)).rdist_eq hX'_meas hY'₁_meas
have h2 : d[X' ; μA # Y'₂ ; μA] = H[X' - Y'₂ ; μA] - H[X' ; μA] / 2 - H[Y'₂ ; μA] / 2 :=
(h_indep.indepFun (show 0 ≠ 2 by decide)).rdist_eq hX'_meas hY'₂_meas
rw [h1, h2, hY'₁_ident.entropy_eq, hY'₂_ident.entropy_eq, hX'_ident.entropy_eq]
group
have : d[X ; μ # 2 • Y ; μ'] ≤
d[Y'₁ ; μA # Y'₂ ; μA] + (H[Y ; μ'] - H[X ; μ]) / 2 + 2 * d[X ; μ # Y ; μ'] := calc
d[X ; μ # 2 • Y ; μ'] = H[X' - 2 • Y'₁ ; μA] - H[X ; μ] / 2 - H[2 • Y ; μ'] / 2 := by
have h2Y_ident : IdentDistrib (2 • Y'₁) (2 • Y) (μ := μA) (ν := μ') := by
convert hY'₁_ident.comp <| .of_discrete (f := fun g ↦ 2 • g)
have h2Y_indep : IndepFun X' (2 • Y'₁) (μ := μA) := by
convert (h_indep.indepFun (show 0 ≠ 1 by decide)).comp measurable_id
(measurable_const_smul 2)
rw [← hX'_ident.rdist_eq h2Y_ident,
h2Y_indep.rdist_eq hX'_meas <| Measurable.const_smul hY'₁_meas 2,
hX'_ident.entropy_eq, h2Y_ident.entropy_eq]
_ ≤ H[Y'₁ - Y'₂ ; μA] + 2 * d[X ; μ # Y ; μ'] - H[X ; μ] / 2 - H[2 • Y ; μ'] / 2 := by linarith
_ = d[Y'₁ ; μA # Y'₂ ; μA] + (H[Y ; μ'] - H[X ; μ]) / 2 + 2 * d[X ; μ # Y ; μ'] := by
have H2Y : H[2 • Y ; μ'] = H[Y ; μ'] := by
let f (g : G) := 2 • g
exact entropy_comp_of_injective μ' hY f (fun _ _ ↦ by simp [f, smul_right_inj])
have : d[Y'₁ ; μA # Y'₂ ; μA] = H[Y'₁ - Y'₂ ; μA] - H[Y'₁ ; μA] / 2 - H[Y'₂ ; μA] / 2 :=
(h_indep.indepFun (show 1 ≠ 2 by decide)).rdist_eq hY'₁_meas hY'₂_meas
rw [this, hY'₁_ident.entropy_eq, hY'₂_ident.entropy_eq, H2Y]
group
have : d[Y'₁ ; μA # Y'₂ ; μA] ≤ 2 * d[X ; μ # Y ; μ'] := by
rw [two_mul]
convert rdist_triangle hY'₁_meas hX'_meas hY'₂_meas (μ := μA) (μ' := μA) (μ'' := μA)
· exact rdist_symm.trans (hY'₁_ident.rdist_eq hX'_ident).symm
· exact (hX'_ident.rdist_eq hY'₂_ident).symm
rw [← two_nsmul]
linarith [abs_le.mp <| diff_ent_le_rdist hX hY (μ := μ) (μ' := μ')]
/-- If `G` is a torsion-free group and `X, Y` are `G`-valued random variables and
`φ : G → 𝔽₂^d` is a homomorphism then `H[φ ∘ X ; μ] ≤ 10 * d[X ; μ # Y ; μ']`. -/ | pfr/blueprint/src/chapter/weak_pfr.tex:3 | pfr/PFR/WeakPFR.lean:88 |
PFR | weak_PFR_asymm | \begin{theorem}\label{weak-pfr-asymm}\lean{weak_PFR_asymm}\leanok
If $A,B\subseteq \mathbb{Z}^d$ are finite non-empty sets then there exist non-empty $A'\subseteq A$ and $B'\subseteq B$ such that
\[\log\frac{\lvert A\rvert\lvert B\rvert}{\lvert A'\rvert\lvert B'\rvert}\leq 34d[U_A;U_B]\]
such that $\max(\dim A',\dim B')\leq \frac{40}{\log 2} d[U_A;U_B]$.
\end{theorem}
\begin{proof}
\uses{torsion-dist-shrinking, pfr-projection, single-fibres, dimension-def}\leanok
Without loss of generality we can assume that $A$ and $B$ are not both inside (possibly distinct) cosets of the same subgroup of $\mathbb{Z}^d$, or we just replace $\mathbb{Z}^d$ with that subgroup. We prove the result by induction on $\lvert A\rvert+\lvert B\rvert$.
Let $\phi:\mathbb{Z}^d\to \mathbb{F}_2^d$ be the natural mod-2 homomorphism. By \Cref{torsion-dist-shrinking}
\[\max(\mathbb{H}(\phi(U_A)),\mathbb{H}(\phi(U_B)))\leq 10d[U_A;U_B].\]
We now apply \Cref{pfr-projection}, obtaining some subgroup $H\leq \mathbb{F}_2^d$ such that
\[\log \lvert H\rvert \leq 40d[U_A;U_B]\]
and
\[\mathbb{H}(\tilde{\phi}(U_A))+\mathbb{H}(\tilde{\phi}(U_B))\leq 34 d[\tilde{\phi}(U_A);\tilde{\phi}(U_B)]\]
where $\tilde{\phi}:\mathbb{Z}^d\to \mathbb{F}_2^d/H$ is $\phi$ composed with the projection onto $\mathbb{F}_2^d/H$.
By \Cref{single-fibres} there exist $x,y\in \mathbb{F}_2^d/H$ such that, with $A_x=A\cap \tilde{\phi}^{-1}(x)$ and similarly for $B_y$,
\[\log \frac{\lvert A\rvert\lvert B\rvert}{\lvert A_x\rvert\lvert B_y\rvert}\leq 34(d[U_A;U_B]-d[U_{A_x};U_{B_y}]).\]
Suppose first that $\lvert A_x\rvert+\lvert B_y\rvert=\lvert A\rvert+\lvert B\rvert$. This means that $\tilde{\phi}(A)=\{x\}$ and $\tilde{\phi}(B)=\{y\}$, and hence both $A$ and $B$ are in cosets of $\ker \tilde{\phi}$. Since by assumption $A,B$ are not in cosets of a proper subgroup of $\mathbb{Z}^d$ this means $\ker\tilde{\phi}=\mathbb{Z}^d$, and so (examining the definition of $\tilde{\phi}$) we must have $H=\mathbb{F}_2^d$. Then our bound on $\log\lvert H\rvert$ forces $d\leq \frac{40}{\log 2}d[U_A;U_B]$ and we are done with $A'=A$ and $B'=B$.
Otherwise,
\[\lvert A_x\rvert+\lvert B_y\rvert <\lvert A\rvert+\lvert B\rvert.\]
By induction we can find some $A'\subseteq A_x$ and $B'\subseteq B_y$ such that $\dim A',\dim B'\leq \frac{40}{\log 2} d[U_{A_x};U_{B_y}]\leq \frac{40}{\log 2}d[U_A;U_B]$ and
\[\log \frac{\lvert A_x\rvert\lvert B_y\rvert}{\lvert A'\rvert\lvert B'\rvert}\leq 34d[U_{A_x};U_{B_y}].\]
Adding these inequalities implies
\[\log\frac{\lvert A\rvert\lvert B\rvert}{\lvert A'\rvert\lvert B'\rvert}\leq 34d[U_A;U_B]\]
as required.
\end{proof} | lemma weak_PFR_asymm (A B : Set G) [Finite A] [Finite B] (hA : A.Nonempty) (hB : B.Nonempty) :
WeakPFRAsymmConclusion A B := by
let P : ℕ → Prop := fun M ↦ (∀ (G : Type _) (hG_comm : AddCommGroup G) (_hG_free : Module.Free ℤ G)
(_hG_fin : Module.Finite ℤ G) (_hG_count : Countable G) (hG_mes : MeasurableSpace G)
(_hG_sing : MeasurableSingletonClass G) (A B : Set G) (_hA_fin : Finite A) (_hB_fin : Finite B)
(_hA_non : A.Nonempty) (_hB_non : B.Nonempty)
(_hM : Nat.card A + Nat.card B ≤ M), WeakPFRAsymmConclusion A B)
suffices ∀ M, (∀ M', M' < M → P M') → P M by
set M := Nat.card A + Nat.card B
have hM : Nat.card A + Nat.card B ≤ M := Nat.le_refl _
convert (Nat.strong_induction_on (p := P) M this) G ‹_› ‹_› ‹_› ‹_› _ ‹_› A B ‹_› ‹_› ‹_› ‹_› hM
intro M h_induct
-- wlog we can assume A, B are not in cosets of a smaller subgroup
suffices ∀ (G : Type _) (hG_comm : AddCommGroup G) (_hG_free : Module.Free ℤ G)
(_hG_fin : Module.Finite ℤ G) (_hG_count : Countable G) (hG_mes : MeasurableSpace G)
(_hG_sing : MeasurableSingletonClass G) (A B : Set G) (_hA_fin : Finite A) (_hB_fin : Finite B)
(_hA_non : A.Nonempty) (_hB_non : B.Nonempty) (_hM : Nat.card A + Nat.card B ≤ M)
(_hnot : NotInCoset A B), WeakPFRAsymmConclusion A B by
intro G hG_comm hG_free hG_fin hG_count hG_mes hG_sing A B hA_fin hB_fin hA_non hB_non hM
obtain ⟨G', A', B', hAA', hBB', hnot'⟩ := wlog_notInCoset hA_non hB_non
have hG'_fin : Module.Finite ℤ G' :=
Module.Finite.iff_fg (N := AddSubgroup.toIntSubmodule G').2 (IsNoetherian.noetherian _)
have hG'_free : Module.Free ℤ G' := by
rcases Submodule.nonempty_basis_of_pid (Module.Free.chooseBasis ℤ G)
(AddSubgroup.toIntSubmodule G') with ⟨n, ⟨b⟩⟩
exact Module.Free.of_basis b
have hAA'_card : Nat.card A = Nat.card A' := (Nat.card_image_of_injective Subtype.val_injective _) ▸ hAA'.card_congr
have hBB'_card : Nat.card B = Nat.card B' := (Nat.card_image_of_injective Subtype.val_injective _) ▸ hBB'.card_congr
have hA_non' : Nonempty A := Set.nonempty_coe_sort.mpr hA_non
have hB_non' : Nonempty B := Set.nonempty_coe_sort.mpr hB_non
rw [hAA'_card, hBB'_card] at hM
have hA'_nonfin : A'.Nonempty ∧ Finite A' := by
simpa [-Subtype.exists, hAA'_card, Nat.card_pos_iff] using Nat.card_pos (α := A)
have hB'_nonfin : B'.Nonempty ∧ Finite B' := by
simpa [-Subtype.exists, hBB'_card, Nat.card_pos_iff] using Nat.card_pos (α := B)
obtain ⟨hA'_non, hA'_fin⟩ := hA'_nonfin
obtain ⟨hB'_non, hB'_fin⟩ := hB'_nonfin
replace this := this G' _ hG'_free hG'_fin (by infer_instance) (by infer_instance) (by infer_instance) A' B' hA'_fin hB'_fin hA'_non hB'_non hM hnot'
exact conclusion_transfers G' A' B' hAA' hBB' hA'_non hB'_non this
intro G hG_comm hG_free hG_fin hG_count hG_mes hG_sing A B hA_fin hB_fin hA_non hB_non hM hnot
rcases weak_PFR_asymm_prelim A B hA_non hB_non with ⟨N, x, y, Ax, By, hAx_non, hBy_non, hAx_fin, hBy_fin, hAx, hBy, hdim, hcard⟩
have hAxA : Ax ⊆ A := by rw [hAx]; simp
have hByB : By ⊆ B := by rw [hBy]; simp
have hA_pos : (0 : ℝ) < Nat.card A := Nat.cast_pos.mpr (@Nat.card_pos _ hA_non.to_subtype _)
have hB_pos : (0 : ℝ) < Nat.card B := Nat.cast_pos.mpr (@Nat.card_pos _ hB_non.to_subtype _)
rcases lt_or_ge (Nat.card Ax + Nat.card By) (Nat.card A + Nat.card B) with h | h
· replace h := h_induct (Nat.card Ax + Nat.card By) (h.trans_le hM) G hG_comm hG_free hG_fin hG_count hG_mes hG_sing Ax By (Set.finite_coe_iff.mpr hAx_fin) (Set.finite_coe_iff.mpr hBy_fin) hAx_non hBy_non (Eq.le rfl)
rcases h with ⟨A', B', hA', hB', hA'_non, hB'_non, hcard_ineq, hdim_ineq⟩
use A', B'
have hAx_fin' := Set.finite_coe_iff.mpr hAx_fin
have hBy_fin' := Set.finite_coe_iff.mpr hBy_fin
have hA'_fin' := Set.finite_coe_iff.mpr (Set.Finite.subset hAx_fin hA')
have hB'_fin' := Set.finite_coe_iff.mpr (Set.Finite.subset hBy_fin hB')
have hAx_non' := Set.nonempty_coe_sort.mpr hAx_non
have hBy_non' := Set.nonempty_coe_sort.mpr hBy_non
have hA'_non' := Set.nonempty_coe_sort.mpr hA'_non
have hB'_non' := Set.nonempty_coe_sort.mpr hB'_non
have hAx_pos : (0 : ℝ) < Nat.card Ax := Nat.cast_pos.mpr Nat.card_pos
have hBy_pos : (0 : ℝ) < Nat.card By := Nat.cast_pos.mpr Nat.card_pos
have hA'_pos : (0 : ℝ) < Nat.card A' := Nat.cast_pos.mpr Nat.card_pos
have hB'_pos : (0 : ℝ) < Nat.card B' := Nat.cast_pos.mpr Nat.card_pos
have hAxA_le : (Nat.card Ax : ℝ) ≤ (Nat.card A : ℝ) := Nat.cast_le.mpr (Nat.card_mono A.toFinite hAxA)
have hByB_le : (Nat.card By : ℝ) ≤ (Nat.card B : ℝ) := Nat.cast_le.mpr (Nat.card_mono B.toFinite hByB)
refine ⟨hA'.trans hAxA, hB'.trans hByB, hA'_non, hB'_non, ?_, ?_⟩
· rw [four_logs hA_pos hB_pos hA'_pos hB'_pos]
rw [four_logs hAx_pos hBy_pos hA'_pos hB'_pos] at hcard_ineq
linarith only [hcard, hcard_ineq]
apply hdim_ineq.trans
gcongr
linarith only [Real.log_le_log hAx_pos hAxA_le, Real.log_le_log hBy_pos hByB_le, hcard]
use A, B
refine ⟨Eq.subset rfl, Eq.subset rfl, hA_non, hB_non, ?_, ?_⟩
· have := hA_non.to_subtype
have := hB_non.to_subtype
apply LE.le.trans _ <| mul_nonneg (by norm_num) <| setRuzsaDist_nonneg A B
rw [div_self (by positivity)]
simp
have hAx_eq : Ax = A := by
apply Set.Finite.eq_of_subset_of_card_le A.toFinite hAxA
linarith only [h, Nat.card_mono B.toFinite hByB]
have hBy_eq : By = B := by
apply Set.Finite.eq_of_subset_of_card_le B.toFinite hByB
linarith only [h, Nat.card_mono A.toFinite hAxA]
have hN : N = ⊤ := by
have : (A-A) ∪ (B-B) ⊆ N := by
rw [← hAx_eq, ← hBy_eq, hAx, hBy]
intro z hz
simp only [mk'_apply, mem_union, mem_sub, mem_setOf_eq] at hz
convert (QuotientAddGroup.eq_zero_iff z).mp ?_
· infer_instance
rcases hz with ⟨a, ⟨-, ha⟩, a', ⟨-, ha'⟩, haa'⟩ | ⟨b, ⟨-, hb⟩, b', ⟨-,hb'⟩, hbb'⟩
· rw [← haa']; simp [ha, ha']
rw [← hbb']; simp [hb, hb']
rw [← AddSubgroup.closure_le, hnot] at this
exact top_le_iff.mp this
have : Nat.card (G ⧸ N) = 1 := by
rw [Nat.card_eq_one_iff_unique]
constructor
· rw [hN]
exact QuotientAddGroup.subsingleton_quotient_top
infer_instance
simp only [this, Nat.cast_one, log_one, zero_add] at hdim
rw [← le_div_iff₀' (by positivity)] at hdim
convert le_trans ?_ hdim using 1
· field_simp
simp only [Nat.cast_max, max_le_iff, Nat.cast_le]
exact ⟨AffineSpace.finrank_le_moduleFinrank, AffineSpace.finrank_le_moduleFinrank⟩
/-- If $A\subseteq \mathbb{Z}^d$ is a finite non-empty set with $d[U_A;U_A]\leq \log K$ then
there exists a non-empty $A'\subseteq A$ such that $\lvert A'\rvert\geq K^{-17}\lvert A\rvert$
and $\dim A'\leq \frac{40}{\log 2} \log K$. -/ | pfr/blueprint/src/chapter/weak_pfr.tex:170 | pfr/PFR/WeakPFR.lean:917 |
PFR | weak_PFR_int | \begin{theorem}\label{weak-pfr-int}\lean{weak_PFR_int}\leanok
Let $A\subseteq \mathbb{Z}^d$ and $\lvert A-A\rvert\leq K\lvert A\rvert$.
There exists $A'\subseteq A$ such that $\lvert A'\rvert \geq K^{-17}\lvert
A\rvert$ and $\dim A' \leq \frac{40}{\log 2}\log K$.
\end{theorem}
\begin{proof}\leanok
\uses{weak-pfr-symm,dimension-def}
As in the beginning of \Cref{pfr} the doubling condition forces $d[U_A;U_A]\leq \log K$, and then we apply \Cref{weak-pfr-symm}.
\end{proof} | theorem weak_PFR_int
{G : Type*} [AddCommGroup G] [Module.Free ℤ G] [Module.Finite ℤ G]
{A : Set G} [A_fin : Finite A] (hnA : A.Nonempty) {K : ℝ}
(hA : Nat.card (A-A) ≤ K * Nat.card A) :
∃ A' : Set G, A' ⊆ A ∧ Nat.card A' ≥ K ^ (-17 : ℝ) * Nat.card A ∧
AffineSpace.finrank ℤ A' ≤ (40 / log 2) * log K := by
have : Nonempty (A - A) := (hnA.sub hnA).coe_sort
have : Finite (A - A) := Set.Finite.sub A_fin A_fin
have hK : 0 < K := by
have : 0 < K * Nat.card A := lt_of_lt_of_le (mod_cast Nat.card_pos) hA
nlinarith
have : Countable G := countable_of_finiteDimensional ℤ G
let m : MeasurableSpace G := ⊤
apply weak_PFR hnA hK ((setRuzsaDist_le A A hnA hnA).trans _)
suffices log (Nat.card (A-A)) ≤ log K + log (Nat.card A) by linarith
rw [← log_mul (by positivity) _]
· apply log_le_log _ hA
norm_cast
exact Nat.card_pos
exact_mod_cast ne_of_gt (@Nat.card_pos _ hnA.to_subtype _) | pfr/blueprint/src/chapter/weak_pfr.tex:213 | pfr/PFR/WeakPFR.lean:1088 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.