| import OAI.Computability.DepthThree.FiniteProbability |
| import Mathlib.Algebra.BigOperators.GroupWithZero.Finset |
| import Mathlib.Tactic.NormNum |
|
|
| namespace OAI |
|
|
| universe uDepth1 |
|
|
| noncomputable section |
|
|
| open scoped BigOperators |
| attribute [local instance] Classical.propDecidable |
|
|
| namespace DepthThreeLowerBound |
|
|
| variable {V : Type uDepth1} [Fintype V] |
|
|
| theorem finiteAvg_eq_pattern (U : Finset V) (σ : Cube V) : |
| finiteAvg (fun x : Cube V => |
| if ∀ v ∈ U, x v = σ v then (1 : ℝ) else 0) = |
| ((2 : ℝ) ^ U.card)⁻¹ := by |
| classical |
| let q : V → Bool → ℝ := fun v b => if v ∈ U → b = σ v then 1 else 0 |
| have hpoint (x : Cube V) : |
| (if ∀ v ∈ U, x v = σ v then (1 : ℝ) else 0) = ∏ v, q v (x v) := by |
| dsimp [q] |
| rw [Fintype.prod_boole] |
| have hcoordinate (v : V) : |
| finiteAvg (q v) = if v ∈ U then (2 : ℝ)⁻¹ else 1 := by |
| by_cases hv : v ∈ U |
| · cases hσ : σ v <;> |
| norm_num [q, hv, hσ, finiteAvg, Fintype.sum_bool] |
| · simp [q, hv] |
| calc |
| _ = finiteAvg (fun x : Cube V => ∏ v, q v (x v)) := finiteAvg_congr hpoint |
| _ = ∏ v, finiteAvg (q v) := finiteAvg_pi_prod q |
| _ = ∏ v, if v ∈ U then (2 : ℝ)⁻¹ else 1 := by |
| exact Finset.prod_congr rfl (fun v _ => hcoordinate v) |
| _ = ((2 : ℝ) ^ U.card)⁻¹ := by |
| rw [← Finset.prod_filter] |
| have hfilter : Finset.univ.filter (fun v => v ∈ U) = U := by |
| ext v |
| simp |
| rw [hfilter, Finset.prod_const, inv_pow] |
|
|
| theorem finiteAvg_indicator_eq_pattern (U : Finset V) (σ : Cube V) : |
| finiteAvg (fun x : Cube V => indicator (decide (∀ v ∈ U, x v = σ v))) = |
| ((2 : ℝ) ^ U.card)⁻¹ := by |
| refine (finiteAvg_congr fun x => ?_).trans (finiteAvg_eq_pattern U σ) |
| by_cases hx : ∀ v ∈ U, x v = σ v <;> simp [indicator, hx] |
|
|
| end DepthThreeLowerBound |
|
|
| end |
|
|
| end OAI |
|
|
