| import Mathlib |
|
|
| namespace OAI |
|
|
| |
|
|
| namespace UniqueGames.Foundations.PCP.PoweringDecoding |
|
|
| open scoped BigOperators |
|
|
| variable {Ω A : Type*} [Fintype Ω] [Nonempty Ω] [Fintype A] [Nonempty A] |
|
|
| noncomputable def mass (opinion : Ω → A) (a : A) : ℝ := by |
| classical |
| exact Finset.univ.expect (fun ω => if opinion ω = a then 1 else 0) |
|
|
| omit [Nonempty A] in |
| theorem mass_sum (opinion : Ω → A) : (∑ a, mass opinion a) = 1 := by |
| classical |
| calc |
| (∑ a, mass opinion a) = |
| (Finset.univ : Finset Ω).expect |
| (fun ω => ∑ a : A, if opinion ω = a then (1 : ℝ) else 0) := |
| (Finset.expect_sum_comm (Finset.univ : Finset Ω) (Finset.univ : Finset A) |
| (fun ω a => if opinion ω = a then (1 : ℝ) else 0)).symm |
| _ = (Finset.univ : Finset Ω).expect (fun _ => (1 : ℝ)) := by |
| apply Finset.expect_congr rfl |
| intro ω _ |
| simp |
| _ = 1 := Finset.expect_const Finset.univ_nonempty 1 |
|
|
| omit [Nonempty Ω] in |
| theorem exists_mode (opinion : Ω → A) : |
| ∃ a : A, ∀ b : A, mass opinion b ≤ mass opinion a := by |
| obtain ⟨a, _, ha⟩ := Finset.exists_max_image (Finset.univ : Finset A) |
| (mass opinion) Finset.univ_nonempty |
| exact ⟨a, fun b => ha b (Finset.mem_univ b)⟩ |
|
|
| noncomputable def mode (opinion : Ω → A) : A := Classical.choose (exists_mode opinion) |
|
|
| omit [Nonempty Ω] in |
| theorem mode_max (opinion : Ω → A) (b : A) : |
| mass opinion b ≤ mass opinion (mode opinion) := |
| Classical.choose_spec (exists_mode opinion) b |
|
|
| theorem mode_mass_lower (opinion : Ω → A) : |
| 1 / (Fintype.card A : ℝ) ≤ mass opinion (mode opinion) := by |
| have hk : 0 < (Fintype.card A : ℝ) := Nat.cast_pos.mpr Fintype.card_pos |
| have hsum : (1 : ℝ) ≤ mass opinion (mode opinion) * (Fintype.card A : ℝ) := by |
| calc |
| (1 : ℝ) = ∑ a : A, mass opinion a := (mass_sum opinion).symm |
| _ ≤ ∑ _a : A, mass opinion (mode opinion) := |
| Finset.sum_le_sum (fun a _ => mode_max opinion a) |
| _ = mass opinion (mode opinion) * (Fintype.card A : ℝ) := by simp [mul_comm] |
| exact (div_le_iff₀ hk).2 hsum |
|
|
| theorem exists_letter_mass_lower (opinion : Ω → A) : |
| ∃ a : A, 1 / (Fintype.card A : ℝ) ≤ mass opinion a := |
| ⟨mode opinion, mode_mass_lower opinion⟩ |
|
|
| end UniqueGames.Foundations.PCP.PoweringDecoding |
|
|
| end OAI |
|
|
