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9dbc64e | 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 | 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
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