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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 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 | import OAI.Computability.DepthThree.FiniteProbability
import Mathlib.Data.Finset.Max
import Mathlib.Tactic.Linarith
namespace OAI
universe uDepth1 uDepth2 uDepth3 uDepth4 uDepth5
noncomputable section
open scoped BigOperators Classical
namespace DepthThreeLowerBound
def finiteProb {Ω : Type uDepth1} [Fintype Ω] (P : Ω → Prop) : ℝ :=
finiteAvg (fun ω => if P ω then 1 else 0)
variable {Ω : Type uDepth2} {Λ : Type uDepth3} [Fintype Ω] [Fintype Λ]
theorem finiteProb_nonneg (P : Ω → Prop) : 0 ≤ finiteProb P := by
apply finiteAvg_nonneg
intro ω
split_ifs <;> norm_num
theorem finiteProb_le_one [Nonempty Ω] (P : Ω → Prop) : finiteProb P ≤ 1 := by
have h : finiteProb P ≤ finiteAvg (fun _ : Ω => (1 : ℝ)) := by
apply finiteAvg_mono
intro ω
split_ifs <;> norm_num
simpa using h
theorem finiteProb_mono {P Q : Ω → Prop} (h : ∀ ω, P ω → Q ω) :
finiteProb P ≤ finiteProb Q := by
apply finiteAvg_mono
intro ω
by_cases hp : P ω
· simp [hp, h ω hp]
· simp only [hp, ite_false]
split_ifs <;> norm_num
theorem finiteAvg_exists_le [Nonempty Ω] (f : Ω → ℝ) :
∃ ω, f ω ≤ finiteAvg f := by
obtain ⟨ω, hω, hmin⟩ := Finset.exists_min_image Finset.univ f Finset.univ_nonempty
refine ⟨ω, ?_⟩
have h := finiteAvg_mono (fun x => hmin x (Finset.mem_univ x))
simpa using h
theorem finiteAvg_exists_le_of_le [Nonempty Ω] (f : Ω → ℝ) (c : ℝ)
(h : finiteAvg f ≤ c) : ∃ ω, f ω ≤ c := by
obtain ⟨ω, hω⟩ := finiteAvg_exists_le f
exact ⟨ω, hω.trans h⟩
theorem finiteProb_gt_le {f : Ω → ℝ} (hf : ∀ ω, 0 ≤ f ω)
{a : ℝ} (ha : 0 < a) :
finiteProb (fun ω => a < f ω) ≤ finiteAvg f / a := by
have h : finiteAvg (fun ω => a * (if a < f ω then 1 else 0)) ≤ finiteAvg f := by
apply finiteAvg_mono
intro ω
by_cases hω : a < f ω
· simpa [hω] using hω.le
· simpa [hω] using hf ω
rw [finiteAvg_const_mul] at h
apply (le_div_iff₀ ha).mpr
simpa only [finiteProb, mul_comm] using h
theorem finiteProb_exists_le_sum {ι : Type uDepth4} (s : Finset ι) (P : ι → Ω → Prop) :
finiteProb (fun ω => ∃ i ∈ s, P i ω) ≤ ∑ i ∈ s, finiteProb (P i) := by
have h : finiteProb (fun ω => ∃ i ∈ s, P i ω) ≤
finiteAvg (fun ω => ∑ i ∈ s, if P i ω then (1 : ℝ) else 0) := by
apply finiteAvg_mono
intro ω
by_cases he : ∃ i ∈ s, P i ω
· obtain ⟨i, hi, hPi⟩ := he
have hs : (if P i ω then (1 : ℝ) else 0) ≤
∑ j ∈ s, if P j ω then (1 : ℝ) else 0 := by
apply Finset.single_le_sum _ hi
intro j hj
split_ifs <;> norm_num
simpa [show ∃ i ∈ s, P i ω from ⟨i, hi, hPi⟩, hPi] using hs
· simp only [he, ite_false]
apply Finset.sum_nonneg
intro i hi
split_ifs <;> norm_num
simpa only [finiteAvg_sum, finiteProb] using h
theorem finiteProb_abs_gt_le_moment (f : Ω → ℝ) (t : ℕ) (ht : 0 < t)
(a C : ℝ) (ha : 0 < a)
(hM : finiteAvg (fun ω => |f ω| ^ t) ≤ C) :
finiteProb (fun ω => a < |f ω|) ≤ C / a ^ t := by
have he : finiteProb (fun ω => a < |f ω|) ≤
finiteProb (fun ω => a ^ t < |f ω| ^ t) := by
apply finiteProb_mono
intro ω hω
exact pow_lt_pow_left₀ hω ha.le (Nat.ne_of_gt ht)
have hmark := finiteProb_gt_le (fun ω => pow_nonneg (abs_nonneg (f ω)) t)
(pow_pos ha t)
exact he.trans (hmark.trans (div_le_div_of_nonneg_right hM (pow_nonneg ha.le t)))
theorem finiteAvg_restrictionAvg {V : Type uDepth5} [Fintype V] (p : ℝ)
(F : Ω → Restriction V → ℝ) :
finiteAvg (fun ω => restrictionAvg p (F ω)) =
restrictionAvg p (fun σ => finiteAvg (fun ω => F ω σ)) := by
unfold restrictionAvg
rw [finiteAvg_sum]
apply Finset.sum_congr rfl
intro σ hσ
exact finiteAvg_const_mul _ _
end DepthThreeLowerBound
end
end OAI
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