| import Mathlib | |
| import OAI.Combinatorics.IndependentSets.Repetition.Basic | |
| import OAI.Combinatorics.IndependentSets.Repetition.ProfileCorrection | |
| import OAI.Combinatorics.IndependentSets.Repetition.Value | |
| namespace OAI | |
| namespace IndependentSetsGames.Foundations.Repetition | |
| open scoped BigOperators | |
| open Games | |
| noncomputable section | |
| variable {X Y : Type*} [Fintype X] [Fintype Y] | |
| def normalizeOr (w : X → ℝ) (hw : ∀ x, 0 ≤ w x) | |
| (fallback : FiniteDistribution X) : FiniteDistribution X := | |
| if h : 0 < ∑ x, w x then | |
| { weight := normalizedWeight w | |
| nonnegative := (normalizedWeight_isProbability w hw h).1 | |
| normalized := (normalizedWeight_isProbability w hw h).2 } | |
| else fallback | |
| theorem sum_mul_normalizeOr (w : X → ℝ) (hw : ∀ x, 0 ≤ w x) | |
| (fallback : FiniteDistribution X) (x : X) : | |
| (∑ z, w z) * (normalizeOr w hw fallback).weight x = w x := by | |
| classical | |
| have hs : 0 ≤ ∑ z, w z := Finset.sum_nonneg (fun z _ => hw z) | |
| by_cases h : 0 < ∑ z, w z | |
| · simp only [normalizeOr, dite_eq_left h, normalizedWeight] | |
| field_simp | |
| · have hz : ∑ z, w z = 0 := le_antisymm (le_of_not_gt h) hs | |
| have hx : w x = 0 := by | |
| apply le_antisymm _ (hw x) | |
| have hl := Finset.single_le_sum (fun z _ => hw z) (Finset.mem_univ x) | |
| simpa only [hz] using hl | |
| simp [normalizeOr, hz, hx] | |
| theorem local_completion_recombination | |
| (left : X → ℝ) (right : Y → ℝ) | |
| (hl : ∀ x, 0 ≤ left x) (hr : ∀ y, 0 ≤ right y) | |
| (defaultLeft : FiniteDistribution X) (defaultRight : FiniteDistribution Y) | |
| (c : ℝ) (x : X) (y : Y) : | |
| (c * (∑ z, left z) * (∑ z, right z)) * | |
| ((normalizeOr left hl defaultLeft).weight x * | |
| (normalizeOr right hr defaultRight).weight y) = c * left x * right y := by | |
| calc | |
| _ = c * ((∑ z, left z) * (normalizeOr left hl defaultLeft).weight x) * | |
| ((∑ z, right z) * (normalizeOr right hr defaultRight).weight y) := by ring | |
| _ = _ := by rw [sum_mul_normalizeOr, sum_mul_normalizeOr] | |
| theorem factorized_completion_row | |
| (left : X → ℝ) (right : Y → ℝ) | |
| (hl : ∀ x, 0 ≤ left x) (hr : ∀ y, 0 ≤ right y) | |
| (defaultLeft : FiniteDistribution X) (defaultRight : FiniteDistribution Y) | |
| (c : ℝ) (xy : X × Y) : | |
| (∑ z : X × Y, c * left z.1 * right z.2) * | |
| ((normalizeOr left hl defaultLeft).product | |
| (normalizeOr right hr defaultRight)).weight xy = c * left xy.1 * right xy.2 := by | |
| have hm : (∑ z : X × Y, c * left z.1 * right z.2) = | |
| c * (∑ z, left z) * (∑ z, right z) := by | |
| simp_rw [mul_assoc, ← Finset.mul_sum] | |
| rw [productWeight_mass] | |
| rw [hm] | |
| exact local_completion_recombination left right hl hr defaultLeft defaultRight c xy.1 xy.2 | |
| end | |
| end IndependentSetsGames.Foundations.Repetition | |
| end OAI | |