File size: 2,408 Bytes
92416a2 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 | 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
|