| import OAI.Combinatorics.ContingencyTables.PublishedInputs |
| import Mathlib.Tactic.FieldSimp |
| import Mathlib.Tactic.Linarith |
| import Mathlib.Tactic.Ring |
| import Mathlib.MeasureTheory.Function.LocallyIntegrable |
| import Mathlib.Analysis.Convex.Basic |
|
|
| namespace OAI |
|
|
| /-! |
| # Finite-box geometry for the conditional conductance argument |
|
|
| Finite unions of grid boxes remain finite unions of boxes under weighted |
| Minkowski interpolation, giving the domain hypothesis of Prékopa–Leindler. |
| -/ |
|
|
| namespace ContingencyTables.PublishedInputs |
|
|
| open MeasureTheory |
|
|
| private theorem scalar_box_interpolation (t a b c d z : ℝ) |
| (ht : 0 < t) (ht1 : t < 1) (hab : a < b) (hcd : c < d) |
| (hz : (1 - t) * a + t * c < z ∧ z < (1 - t) * b + t * d) : |
| ∃ w : ℝ × ℝ, a < w.1 ∧ w.1 < b ∧ c < w.2 ∧ w.2 < d ∧ |
| z = (1 - t) * w.1 + t * w.2 := by |
| let D := (1 - t) * (b - a) + t * (d - c) |
| have hD : 0 < D := add_pos (mul_pos (sub_pos.mpr ht1) (sub_pos.mpr hab)) |
| (mul_pos ht (sub_pos.mpr hcd)) |
| let wgt := (z - ((1 - t) * a + t * c)) / D |
| have hwgt : 0 < wgt := div_pos (sub_pos.mpr hz.1) hD |
| have hwgt1 : wgt < 1 := by |
| apply (div_lt_one hD).mpr |
| dsimp [D] |
| linarith [hz.2] |
| refine ⟨(a + wgt * (b - a), c + wgt * (d - c)), ?_, ?_, ?_, ?_, ?_⟩ |
| · linarith [mul_pos hwgt (sub_pos.mpr hab)] |
| · nlinarith [mul_pos (sub_pos.mpr hwgt1) (sub_pos.mpr hab)] |
| · linarith [mul_pos hwgt (sub_pos.mpr hcd)] |
| · nlinarith [mul_pos (sub_pos.mpr hwgt1) (sub_pos.mpr hcd)] |
| · dsimp [wgt] |
| field_simp |
| dsimp [D] |
| ring |
|
|
| private theorem finite_box_union_of_fintype {q : ℕ} {K : Type*} [Fintype K] |
| (lower upper : K → Point q) : |
| IsFiniteBoxUnion {x | ∃ k : K, ∀ i, lower k i < x i ∧ x i < upper k i} := by |
| classical |
| let e := (Fintype.equivFin K).symm |
| refine ⟨Fintype.card K, (fun k => lower (e k)), (fun k => upper (e k)), ?_⟩ |
| ext x |
| constructor |
| · rintro ⟨k, hk⟩ |
| exact ⟨e.symm k, by simpa using hk⟩ |
| · rintro ⟨k, hk⟩ |
| exact ⟨e k, hk⟩ |
|
|
| /-- Interpolation preserves exactly the finite-box class appearing in |
| Prékopa's published integral inequality. -/ |
| theorem IsFiniteBoxUnion.interpolate {q : ℕ} {S T : Set (Point q)} |
| (hS : IsFiniteBoxUnion S) (hT : IsFiniteBoxUnion T) |
| (t : ℝ) (ht : 0 < t) (ht1 : t < 1) : |
| IsFiniteBoxUnion (interpolate t S T) := by |
| classical |
| obtain ⟨n, l, u, rfl⟩ := hS |
| obtain ⟨m, l', u', rfl⟩ := hT |
| let K := {p : Fin n × Fin m // |
| (∀ i, l p.1 i < u p.1 i) ∧ (∀ i, l' p.2 i < u' p.2 i)} |
| let lower : K → Point q := fun p i => (1 - t) * l p.val.1 i + t * l' p.val.2 i |
| let upper : K → Point q := fun p i => (1 - t) * u p.val.1 i + t * u' p.val.2 i |
| have heq : PublishedInputs.interpolate t {x | ∃ k : Fin n, ∀ i, l k i < x i ∧ x i < u k i} |
| {x | ∃ k : Fin m, ∀ i, l' k i < x i ∧ x i < u' k i} = |
| {z | ∃ k : K, ∀ i, lower k i < z i ∧ z i < upper k i} := by |
| ext z |
| constructor |
| · rintro ⟨x, ⟨k, hk⟩, y, ⟨k', hk'⟩, rfl⟩ |
| let p : K := ⟨(k, k'), |
| (fun i => (hk i).1.trans (hk i).2), |
| (fun i => (hk' i).1.trans (hk' i).2)⟩ |
| refine ⟨p, ?_⟩ |
| intro i |
| change (1 - t) * l k i + t * l' k' i < (1 - t) * x i + t * y i ∧ |
| (1 - t) * x i + t * y i < (1 - t) * u k i + t * u' k' i |
| constructor |
| · exact add_lt_add |
| (mul_lt_mul_of_pos_left (hk i).1 (sub_pos.mpr ht1)) |
| (mul_lt_mul_of_pos_left (hk' i).1 ht) |
| · exact add_lt_add |
| (mul_lt_mul_of_pos_left (hk i).2 (sub_pos.mpr ht1)) |
| (mul_lt_mul_of_pos_left (hk' i).2 ht) |
| · rintro ⟨p, hp⟩ |
| have hw (i : Fin q) := scalar_box_interpolation t |
| (l p.val.1 i) (u p.val.1 i) (l' p.val.2 i) (u' p.val.2 i) (z i) |
| ht ht1 (p.property.1 i) (p.property.2 i) (hp i) |
| choose w hw using hw |
| refine ⟨(fun i => (w i).1), ⟨p.val.1, ?_⟩, |
| (fun i => (w i).2), ⟨p.val.2, ?_⟩, ?_⟩ |
| · exact fun i => ⟨(hw i).1, (hw i).2.1⟩ |
| · exact fun i => ⟨(hw i).2.2.1, (hw i).2.2.2.1⟩ |
| · funext i |
| exact (hw i).2.2.2.2 |
| rw [heq] |
| exact finite_box_union_of_fintype lower upper |
|
|
| theorem IsFiniteBoxUnion.isOpen {q : ℕ} {S : Set (Point q)} |
| (hS : IsFiniteBoxUnion S) : IsOpen S := by |
| obtain ⟨n, l, u, rfl⟩ := hS |
| have heq : {x : Point q | ∃ k : Fin n, ∀ i, l k i < x i ∧ x i < u k i} = |
| ⋃ k : Fin n, ⋂ i : Fin q, {x : Point q | l k i < x i ∧ x i < u k i} := by |
| ext x |
| simp |
| rw [heq] |
| apply isOpen_iUnion |
| intro k |
| apply isOpen_iInter_of_finite |
| intro i |
| exact (isOpen_lt continuous_const (continuous_apply i)).inter |
| (isOpen_lt (continuous_apply i) continuous_const) |
|
|
| theorem interpolation_closure_in_box {q : ℕ} {S T : Set (Point q)} |
| {lower upper : Point q} (hS : closure S ⊆ Set.Icc lower upper) |
| (hT : closure T ⊆ Set.Icc lower upper) (t : ℝ) (ht : 0 ≤ t) (ht1 : t ≤ 1) : |
| closure (interpolate t S T) ⊆ Set.Icc lower upper := by |
| apply isClosed_Icc.closure_subset_iff.mpr |
| rintro _ ⟨x, hx, y, hy, rfl⟩ |
| exact convex_Icc lower upper (hS (subset_closure hx)) (hT (subset_closure hy)) |
| (sub_nonneg.mpr ht1) ht (by ring) |
|
|
| private theorem weighted_geometric_doubling (a b t : ℝ) (ha : 0 < a) |
| (hb : 2 * a ≤ b) (ht : 0 ≤ t) : |
| (2 : ℝ) ^ t * a ≤ a ^ (1 - t) * b ^ t := by |
| have hab : 0 ≤ 2 * a := by positivity |
| have hr := Real.rpow_le_rpow hab hb ht |
| calc |
| (2 : ℝ) ^ t * a = a ^ (1 - t) * (2 * a) ^ t := by |
| rw [Real.mul_rpow (by norm_num) ha.le] |
| have heq := Real.rpow_add ha (1 - t) t |
| have hexp : (1 - t) + t = 1 := by ring |
| rw [hexp, Real.rpow_one] at heq |
| calc |
| (2 : ℝ) ^ t * a = (2 : ℝ) ^ t * (a ^ (1 - t) * a ^ t) := |
| congrArg (fun z => (2 : ℝ) ^ t * z) heq |
| _ = a ^ (1 - t) * ((2 : ℝ) ^ t * a ^ t) := by ring |
| _ ≤ a ^ (1 - t) * b ^ t := |
| mul_le_mul_of_nonneg_left hr (Real.rpow_nonneg ha.le _) |
|
|
| /-- The interpolation shell gains the mass used in the grid-cut proof. |
| This conditional result depends only on the precisely identified published |
| integral inequality; finite-box closure and integrability are proved here. -/ |
| theorem interpolation_shell_gain {q : ℕ} (hPL : FiniteBoxPrekopaLeindler q) |
| (t : ℝ) (ht : 0 < t) (ht1 : t < 1) |
| (lower upper : Point q) (S T : Set (Point q)) (ρ : Point q → ℝ) |
| (hS : IsFiniteBoxUnion S) (hT : IsFiniteBoxUnion T) |
| (hSc : closure S ⊆ Set.Icc lower upper) |
| (hTc : closure T ⊆ Set.Icc lower upper) |
| (hρ : ContinuousOn ρ (Set.Icc lower upper)) |
| (hpos : ∀ x ∈ Set.Icc lower upper, 0 < ρ x) |
| (hlc : LogConcaveOn ρ (Set.Icc lower upper)) |
| (hsub : S ⊆ interpolate t S T) |
| (hSmass : 0 < ∫ x in S, ρ x) |
| (hTmass : 2 * (∫ x in S, ρ x) ≤ ∫ x in T, ρ x) : |
| ((2 : ℝ) ^ t - 1) * (∫ x in S, ρ x) ≤ |
| ∫ x in (interpolate t S T) \ S, ρ x := by |
| have hMc := interpolation_closure_in_box hSc hTc t ht.le ht1.le |
| have hM := hS.interpolate hT t ht ht1 |
| have hint : IntegrableOn ρ (interpolate t S T) := |
| hρ.integrableOn_Icc.mono_set (fun _ hx => hMc (subset_closure hx)) |
| rw [MeasureTheory.setIntegral_sdiff hS.isOpen.measurableSet hint hsub] |
| have hPL' := density_interpolation hPL t ht ht1 lower upper S T ρ |
| hS hT hM hSc hTc hMc hρ hpos hlc |
| have hgain := weighted_geometric_doubling (∫ x in S, ρ x) (∫ x in T, ρ x) |
| t hSmass hTmass ht.le |
| nlinarith |
|
|
| end ContingencyTables.PublishedInputs |
|
|
| end OAI |
|
|
