| import Mathlib |
| import OAI.Geometry.HyperbolicGroups.Moment |
| import OAI.Geometry.HyperbolicGroups.Cost |
|
|
| namespace OAI |
|
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| section |
| section |
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| noncomputable section |
| open scoped BigOperators |
| namespace HyperbolicObstruction |
| namespace TableModel |
| variable {F R A : Type*} [Fintype F] [Fintype R] [Fintype A] |
| [DecidableEq F] [DecidableEq R] [DecidableEq A] |
|
|
| lemma powerProbability_mem {N : ℝ} (hN : 1 ≤ N) {c : ℝ} (hc : 0 ≤ c) : |
| N ^ (-c) ∈ Set.Icc (0 : ℝ) 1 := by |
| constructor |
| · exact (Real.rpow_pos_of_pos (by linarith) _).le |
| · simpa using Real.rpow_le_rpow_of_exponent_le hN (show -c ≤ 0 by linarith) |
|
|
| lemma powerProbability_pos {N : ℝ} (hN : 1 ≤ N) (c : ℝ) : 0 < N ^ (-c) := |
| Real.rpow_pos_of_pos (by linarith) _ |
|
|
| omit [Fintype R] [DecidableEq R] in |
| lemma inverse_cost_product {N : ℝ} (hN : 1 ≤ N) (c : R → ℝ) (hc : ∀ r, 0 ≤ c r) |
| (J : Finset R) : |
| (∏ r ∈ J, ((N ^ (-c r))⁻¹ - 1)) ≤ N ^ (∑ r ∈ J, c r) := by |
| have hN0 : 0 < N := by linarith |
| have he (r : R) : (N ^ (-c r))⁻¹ = N ^ (c r) := by |
| rw [Real.rpow_neg hN0.le, inv_inv] |
| simp_rw [he] |
| rw [Real.rpow_sum_of_pos hN0] |
| apply Finset.prod_le_prod₀ |
| · intro r _ |
| exact sub_nonneg.mpr (Real.one_le_rpow hN (hc r)) |
| · intro r _ |
| linarith |
|
|
| omit [Fintype R] [DecidableEq R] in |
| lemma cost_term_bound {N ε : ℝ} (hN : 1 ≤ N) (c : R → ℝ) (hc : ∀ r, 0 ≤ c r) |
| (support : R → Finset F) (I : Finset F) (J : Finset R) |
| (gap : ε ≤ ((J.biUnion support \ I).card : ℝ) - ∑ r ∈ J, c r) : |
| N ^ (Fintype.card F - (J.biUnion support ∪ I).card) * |
| (∏ r ∈ J, ((N ^ (-c r))⁻¹ - 1)) ≤ |
| N ^ (Fintype.card F - I.card) * N ^ (-ε) := by |
| have hN0 : 0 < N := by linarith |
| have hi := Finset.card_le_univ I |
| have hu := Finset.card_le_univ (J.biUnion support ∪ I) |
| have hcard := Finset.card_union_add_card_inter (J.biUnion support) I |
| have hs := Finset.card_sdiff_add_card_inter (J.biUnion support) I |
| have he : (Fintype.card F - (J.biUnion support ∪ I).card) + |
| (J.biUnion support \ I).card = Fintype.card F - I.card := by omega |
| have he' : ((Fintype.card F - (J.biUnion support ∪ I).card : ℕ) : ℝ) + |
| (J.biUnion support \ I).card = (Fintype.card F - I.card : ℕ) := by exact_mod_cast he |
| calc |
| _ ≤ N ^ (Fintype.card F - (J.biUnion support ∪ I).card) * N ^ (∑ r ∈ J, c r) := |
| mul_le_mul_of_nonneg_left (inverse_cost_product hN c hc J) (by positivity) |
| _ = N ^ (((Fintype.card F - (J.biUnion support ∪ I).card : ℕ) : ℝ) + ∑ r ∈ J, c r) := by |
| rw [Real.rpow_add hN0, Real.rpow_natCast] |
| _ ≤ N ^ (((Fintype.card F - I.card : ℕ) : ℝ) + -ε) := |
| Real.rpow_le_rpow_of_exponent_le hN (by linarith) |
| _ = _ := by rw [Real.rpow_add hN0, Real.rpow_natCast] |
|
|
| /-- A uniform, explicit finite relative-variance bound for pinned master matches. |
| No numerical values of the pinned letters occur in this bound. -/ |
| theorem variance_pinned_power [Nonempty A] (support : R → Finset F) (c : R → ℝ) |
| (hc : ∀ r, 0 ≤ c r) (I : Finset F) (ξ : F → A) {ε : ℝ} |
| (gap : ∀ J : Finset R, J.Nonempty → |
| ε ≤ ((J.biUnion support \ I).card : ℝ) - ∑ r ∈ J, c r) : |
| let N : ℝ := Fintype.card A |
| let hp := fun r => powerProbability_mem (show 1 ≤ N by change (1 : ℝ) ≤ Fintype.card A; exact_mod_cast (show 1 ≤ Fintype.card A from Fintype.card_pos)) (hc r) |
| let μ := law support (fun r => N ^ (-c r)) hp |
| let X := fun w => (candidateCount support (pinCandidates I ξ) w : ℝ) |
| μ.variance X ≤ Fintype.card (Finset R) * N ^ (-ε) * μ.expect X ^ 2 := by |
| classical |
| dsimp only |
| have hn : (1 : ℝ) ≤ Fintype.card A := by |
| exact_mod_cast (show 1 ≤ Fintype.card A from Fintype.card_pos) |
| let p (r : R) := (Fintype.card A : ℝ) ^ (-c r) |
| let hp := fun r => powerProbability_mem hn (hc r) |
| change (law support p hp).variance _ ≤ _ |
| rw [variance_pinCount support p hp (fun r => powerProbability_pos hn (c r)), |
| expect_candidateCount, card_pinCandidates] |
| have hs : (∑ J ∈ (Finset.univ : Finset (Finset R)).erase ∅, |
| (Fintype.card A : ℝ) ^ (Fintype.card F - (J.biUnion support ∪ I).card) * |
| ∏ r ∈ J, ((p r)⁻¹ - 1)) ≤ |
| Fintype.card (Finset R) * (Fintype.card A : ℝ) ^ (Fintype.card F - I.card) * |
| (Fintype.card A : ℝ) ^ (-ε) := by |
| calc |
| _ ≤ ∑ _J ∈ (Finset.univ : Finset (Finset R)).erase ∅, |
| (Fintype.card A : ℝ) ^ (Fintype.card F - I.card) * (Fintype.card A : ℝ) ^ (-ε) := by |
| apply Finset.sum_le_sum |
| intro J hJ |
| exact cost_term_bound hn c hc support I J |
| (gap J (Finset.nonempty_iff_ne_empty.mpr (Finset.mem_erase.mp hJ).1)) |
| _ ≤ ∑ _J : Finset R, |
| (Fintype.card A : ℝ) ^ (Fintype.card F - I.card) * (Fintype.card A : ℝ) ^ (-ε) := |
| Finset.sum_le_sum_of_subset_of_nonneg (Finset.erase_subset _ _) |
| (fun _ _ _ => by positivity) |
| _ = _ := by simp; ring |
| calc |
| _ ≤ (∏ r, p r) ^ 2 * (Fintype.card A : ℝ) ^ (Fintype.card F - I.card) * |
| (Fintype.card (Finset R) * (Fintype.card A : ℝ) ^ (Fintype.card F - I.card) * |
| (Fintype.card A : ℝ) ^ (-ε)) := mul_le_mul_of_nonneg_left hs (by positivity) |
| _ = _ := by dsimp [p]; ring |
|
|
| end TableModel |
| namespace IncidenceTemplate |
|
|
| /-- Source single-copy concentration, also for every singleton-pinned count, |
| with a common actual-parameter half-unit margin. -/ |
| theorem master_variance_bound {A : Type*} [Fintype A] [Nonempty A] [DecidableEq A] |
| {α β δ : ℝ} (ha : 0 ≤ α) (hb : 7 * α ≤ β) (hd : 0 ≤ δ) |
| (hsmall : β + 90 * δ ≤ 1 / 2) (I : Finset Field) (hI : I.card ≤ 1) (ξ : Field → A) : |
| let N : ℝ := Fintype.card A |
| let c := actualRowCost α β δ |
| let hp := fun r => TableModel.powerProbability_mem |
| (show 1 ≤ N by change (1 : ℝ) ≤ Fintype.card A; exact_mod_cast (show 1 ≤ Fintype.card A from Fintype.card_pos)) (actualRowCost_nonneg ha hb hd r) |
| let μ := TableModel.law rowSupport (fun r => N ^ (-c r)) hp |
| let X := fun w => (TableModel.candidateCount rowSupport (TableModel.pinCandidates I ξ) w : ℝ) |
| μ.variance X ≤ Fintype.card (Finset Row) * N ^ (-(1 / 2 : ℝ)) * μ.expect X ^ 2 := by |
| apply TableModel.variance_pinned_power rowSupport (actualRowCost α β δ) |
| (actualRowCost_nonneg ha hb hd) I ξ |
| intro J hJ |
| have hh := actual_pinned_overlap ha hb hd hJ I hI |
| linarith |
|
|
| end IncidenceTemplate |
| end HyperbolicObstruction |
| end |
| end |
|
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| section |
|
|
| end |
| end |
|
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| end OAI |
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|
