| import OAI.Combinatorics.ContingencyTables.Dense.CheegerCoarea |
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| namespace OAI |
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| /-! |
| namespace ContingencyTables.CheegerProof |
| open CommonBasesFPRAS |
| open scoped BigOperators Classical |
| noncomputable section |
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| private lemma weighted_cauchy {ι : Type*} [Fintype ι] |
| (w a b : ι → ℝ) (hw : ∀ i, 0 ≤ w i) : |
| (∑ i, w i*a i*b i)^2 ≤ (∑ i, w i*(a i)^2)*(∑ i, w i*(b i)^2) := by |
| have h := Finset.sum_mul_sq_le_sq_mul_sq Finset.univ |
| (fun i => Real.sqrt (w i)*a i) (fun i => Real.sqrt (w i)*b i) |
| have hmul (i : ι) : (Real.sqrt (w i)*a i)*(Real.sqrt (w i)*b i) = w i*a i*b i := by |
| calc |
| _ = (Real.sqrt (w i))^2*a i*b i := by ring |
| _ = _ := by rw [Real.sq_sqrt (hw i)] |
| have hsq (f : ι → ℝ) (i : ι) : (Real.sqrt (w i)*f i)^2 = w i*(f i)^2 := by |
| rw [mul_pow,Real.sq_sqrt (hw i)] |
| simpa only [hmul,hsq] using h |
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| variable {X : Type*} [Fintype X] (C : FiniteChain X) |
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| lemma sum_pair_squares (g : X → ℝ) : |
| (∑ x, ∑ y, C.π x*C.P x y*((g x)^2+(g y)^2)) = 2*(∑ x, C.π x*(g x)^2) := by |
| simp_rw [mul_add,Finset.sum_add_distrib] |
| have hleft : (∑ x, ∑ y, C.π x*C.P x y*(g x)^2) = ∑ x, C.π x*(g x)^2 := by |
| simp_rw [← Finset.sum_mul,← Finset.mul_sum,C.stochastic,mul_one] |
| have hright : (∑ x, ∑ y, C.π x*C.P x y*(g y)^2) = ∑ y, C.π y*(g y)^2 := by |
| rw [Finset.sum_comm] |
| simp_rw [← Finset.sum_mul,C.stationary] |
| rw [hleft,hright] |
| ring |
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| lemma sum_pair_add_square_le (g : X → ℝ) : |
| (∑ x, ∑ y, C.π x*C.P x y*(g x+g y)^2) ≤ 4*(∑ x, C.π x*(g x)^2) := by |
| calc |
| _ ≤ ∑ x, ∑ y, 2*(C.π x*C.P x y*((g x)^2+(g y)^2)) := by |
| apply Finset.sum_le_sum |
| intro x _ |
| apply Finset.sum_le_sum |
| intro y _ |
| have hQ := mul_nonneg (C.positive x).le (C.nonnegative x y) |
| have hsq : (g x+g y)^2 ≤ 2*((g x)^2+(g y)^2) := by nlinarith [sq_nonneg (g x-g y)] |
| nlinarith [mul_le_mul_of_nonneg_left hsq hQ] |
| _ = _ := by simp_rw [← Finset.mul_sum]; rw [sum_pair_squares]; ring |
| |
| theorem positive_energy (κ : ℝ) (hk : 0 < κ) |
| (hcut : ∀ S : Finset X, κ*min (mass C S) (mass C Sᶜ) ≤ cut C S) |
| (g : X → ℝ) (hg : ∀ x, 0 ≤ g x) |
| (hsmall : mass C (Finset.univ.filter fun x => 0 < g x) ≤ 1/2) : |
| κ^2*(∑ x, C.π x*(g x)^2) ≤ 2*C.energy g := by |
| let N := ∑ x, C.π x*(g x)^2 |
| let W := ∑ x, ∑ y, C.π x*C.P x y*|(g x)^2-(g y)^2| |
| have hE : 0 ≤ C.energy g := C.energy_nonneg g |
| have hN : 0 ≤ N := Finset.sum_nonneg fun x _ => mul_nonneg (C.positive x).le (sq_nonneg _) |
| have hW : 0 ≤ W := Finset.sum_nonneg fun x _ => Finset.sum_nonneg fun y _ => |
| mul_nonneg (mul_nonneg (C.positive x).le (C.nonnegative x y)) (abs_nonneg _) |
| have hsquare : mass C (Finset.univ.filter fun x => 0 < (g x)^2) ≤ 1/2 := by |
| apply le_trans (mass_mono C ?_) hsmall |
| intro x hx |
| simp only [Finset.mem_filter,Finset.mem_univ,true_and] at hx ⊢ |
| nlinarith [hg x] |
| have hc : 2*κ*N ≤ W := coarea C κ hcut (fun x => (g x)^2) (fun _ => sq_nonneg _) hsquare |
| have hsqW : (2*κ*N)^2 ≤ W^2 := (sq_le_sq₀ (by positivity) hW).mpr hc |
| have ha (x y : X) : |(g x)^2-(g y)^2| = |g x-g y| * (g x+g y) := by |
| rw [show (g x)^2-(g y)^2 = (g x-g y)*(g x+g y) by ring, |
| abs_mul,abs_of_nonneg (add_nonneg (hg x) (hg y))] |
| have hcs := weighted_cauchy |
| (fun z : X×X => C.π z.1*C.P z.1 z.2) |
| (fun z => |g z.1-g z.2|) (fun z => g z.1+g z.2) |
| (fun z => mul_nonneg (C.positive _).le (C.nonnegative _ _)) |
| simp only [Fintype.sum_prod_type,sq_abs] at hcs |
| have he : (∑ x, ∑ y, C.π x*C.P x y*(g x-g y)^2) = 2*C.energy g := by |
| rw [C.energy_eq]; ring |
| have hW' : W = ∑ x, ∑ y, C.π x*C.P x y*|g x-g y| * (g x+g y) := by |
| dsimp [W] |
| simp_rw [ha,mul_assoc] |
| rw [← hW',he] at hcs |
| have hb := sum_pair_add_square_le C g |
| have hall : (2*κ*N)^2 ≤ (2*C.energy g)*(4*N) := |
| (hsqW.trans hcs).trans (mul_le_mul_of_nonneg_left hb (by positivity : 0 ≤ 2*C.energy g)) |
| rcases hN.eq_or_lt with hz | hp |
| · change κ^2*N ≤ _ |
| rw [← hz,mul_zero] |
| positivity |
| · change κ^2*N ≤ _ |
| have h : 4*N*(κ^2*N) ≤ 4*N*(2*C.energy g) := by nlinarith [hall] |
| exact (mul_le_mul_iff_right₀ (by positivity : 0 < 4*N)).mp h |
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| end |
| end ContingencyTables.CheegerProof |
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| end OAI |
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