openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /Recovery /InheritedStatisticWindows.lean
| import OAI.LinearAlgebra.MatrixMultiplication.Recovery.InheritedMasks | |
| import Mathlib.Algebra.BigOperators.Field | |
| namespace OAI | |
| /-! Finite orbit symmetries, masks and exact recovery operations. -/ | |
| namespace MatrixMultiplication.InheritedMasks | |
| open MatrixMultiplication.Foundation | |
| open scoped BigOperators | |
| section Pushforward | |
| variable {A B : Type*} [Fintype A] [DecidableEq B] | |
| def pushforwardLaw (statistic : A → B) (ν : A → ℝ) (b : B) : ℝ := | |
| ∑ a with statistic a = b, ν a | |
| theorem pushforwardLaw_nonneg (statistic : A → B) (ν : A → ℝ) | |
| (hν : ∀ a, 0 ≤ ν a) (b : B) : 0 ≤ pushforwardLaw statistic ν b := by | |
| exact Finset.sum_nonneg fun a _ => hν a | |
| theorem pushforwardLaw_le_one (statistic : A → B) (ν : A → ℝ) | |
| (hν : ∀ a, 0 ≤ ν a) (hsum : ∑ a, ν a = 1) (b : B) : | |
| pushforwardLaw statistic ν b ≤ 1 := by | |
| calc | |
| pushforwardLaw statistic ν b ≤ ∑ a, ν a := | |
| Finset.sum_le_sum_of_subset_of_nonneg (Finset.filter_subset _ _) | |
| (fun a _ _ => hν a) | |
| _ = 1 := hsum | |
| end Pushforward | |
| section Windows | |
| variable {P A B : Type*} [Fintype P] | |
| [Fintype A] [DecidableEq A] [DecidableEq B] | |
| theorem wordPopulation_statistic (w : P → A) (statistic : A → B) (b : B) : | |
| wordPopulation (statistic ∘ w) b = | |
| ∑ a with statistic a = b, wordPopulation w a := by | |
| simpa only [wordPopulation, Fintype.card_subtype, Finset.mem_filter, | |
| Finset.mem_univ, true_and, Function.comp_apply] using | |
| (Finset.sum_card_fiberwise_eq_card_filter (Finset.univ : Finset P) | |
| (Finset.univ.filter fun a => statistic a = b) w).symm | |
| theorem empiricalLaw_statistic (w : P → A) (statistic : A → B) (b : B) : | |
| empiricalLaw (statistic ∘ w) b = | |
| pushforwardLaw statistic (empiricalLaw w) b := by | |
| simp only [empiricalLaw, wordPopulation_statistic, pushforwardLaw, | |
| Nat.cast_sum, Finset.sum_div] | |
| omit [Fintype A] in | |
| theorem typeWindow_mono (ν : A → ℝ) (w : P → A) {τ η : ℝ} | |
| (hτη : τ ≤ η) (hw : typeWindow ν τ w) : typeWindow ν η w := by | |
| intro a | |
| exact (hw a).trans hτη | |
| theorem typeWindow_statistic_coordinate (ν : A → ℝ) (τ : ℝ) (w : P → A) | |
| (statistic : A → B) (hw : typeWindow ν τ w) (b : B) : | |
| |empiricalLaw (statistic ∘ w) b - pushforwardLaw statistic ν b| ≤ | |
| ((Finset.univ.filter fun a => statistic a = b).card : ℝ) * τ := by | |
| rw [empiricalLaw_statistic] | |
| unfold pushforwardLaw | |
| rw [← Finset.sum_sub_distrib] | |
| calc | |
| _ ≤ ∑ a with statistic a = b, |empiricalLaw w a - ν a| := | |
| Finset.abs_sum_le_sum_abs _ _ | |
| _ ≤ ∑ _a ∈ Finset.univ.filter (fun a => statistic a = b), τ := | |
| Finset.sum_le_sum fun a _ => hw a | |
| _ = _ := by simp only [Finset.sum_const, nsmul_eq_mul] | |
| theorem typeWindow_statistic (ν : A → ℝ) (τ : ℝ) (w : P → A) | |
| (statistic : A → B) (hτ : 0 ≤ τ) (hw : typeWindow ν τ w) : | |
| typeWindow (pushforwardLaw statistic ν) (Fintype.card A * τ) | |
| (statistic ∘ w) := by | |
| intro b | |
| calc | |
| _ ≤ ((Finset.univ.filter fun a => statistic a = b).card : ℝ) * τ := | |
| typeWindow_statistic_coordinate ν τ w statistic hw b | |
| _ ≤ Fintype.card A * τ := | |
| mul_le_mul_of_nonneg_right (Nat.cast_le.mpr (Finset.card_le_univ _)) hτ | |
| end Windows | |
| end MatrixMultiplication.InheritedMasks | |
| end OAI | |