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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 | import OAI.Computability.DepthThree.EntropyBits
import OAI.Computability.DepthThree.EntropyThreshold
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.Ring
namespace OAI
universe uDepth1
noncomputable section
namespace DepthThreeLowerBound
open scoped BigOperators
theorem finite_card_le_of_uniform_weight_lower_bound {α : Type uDepth1} [Fintype α]
(A : Finset α) (w : α → ℝ) {δ : ℝ} (hδ : 0 < δ)
(hw : ∀ a, 0 ≤ w a) (hsum : ∑ a, w a ≤ 1)
(hlower : ∀ a ∈ A, δ ≤ w a) : (A.card : ℝ) ≤ 1 / δ := by
classical
have hcard : (A.card : ℝ) * δ ≤ 1 := by
calc
(A.card : ℝ) * δ = ∑ _a ∈ A, δ := by simp
_ ≤ ∑ a ∈ A, w a := Finset.sum_le_sum hlower
_ ≤ ∑ a, w a := Finset.sum_le_sum_of_subset_of_nonneg
(Finset.subset_univ _) (fun a _ _ => hw a)
_ ≤ 1 := hsum
exact (le_div_iff₀ hδ).2 hcard
def bernoulliStringWeight {N : ℕ} (x : ℝ) (z : Fin N → Bool) : ℝ :=
∏ i, if z i = true then x else 1 - x
theorem bernoulliStringWeight_pos {N : ℕ} {x : ℝ} (hx : 0 < x)
(hx1 : x < 1) (z : Fin N → Bool) : 0 < bernoulliStringWeight x z := by
apply Finset.prod_pos
intro i _
split_ifs
· exact hx
· exact sub_pos.mpr hx1
theorem sum_bernoulliStringWeight (N : ℕ) (x : ℝ) :
∑ z : Fin N → Bool, bernoulliStringWeight x z = 1 := by
have hbit : (∑ b : Bool, if b = true then x else 1 - x) = 1 := by
simp [Fintype.univ_bool]
calc
(∑ z : Fin N → Bool, bernoulliStringWeight x z) =
∏ _i : Fin N, ∑ b : Bool, if b = true then x else 1 - x :=
(Fintype.prod_sum (fun (_i : Fin N) (b : Bool) =>
if b = true then x else 1 - x)).symm
_ = 1 := by simp only [hbit, Finset.prod_const_one]
theorem log_bernoulliStringWeight {N : ℕ} {x : ℝ} (hx : 0 < x)
(hx1 : x < 1) (z : Fin N → Bool) :
Real.log (bernoulliStringWeight x z) =
(N : ℝ) * Real.log (1 - x) +
(trueCount z : ℝ) * (Real.log x - Real.log (1 - x)) := by
have hne : ∀ i ∈ (Finset.univ : Finset (Fin N)),
(if z i = true then x else 1 - x) ≠ 0 := by
intro i _
split_ifs
· exact ne_of_gt hx
· exact ne_of_gt (sub_pos.mpr hx1)
rw [bernoulliStringWeight, Real.log_prod hne]
calc
(∑ i : Fin N, Real.log (if z i = true then x else 1 - x)) =
∑ i : Fin N, (Real.log (1 - x) +
(if z i = true then (1 : ℝ) else 0) *
(Real.log x - Real.log (1 - x))) := by
apply Finset.sum_congr rfl
intro i _
split_ifs <;> ring
_ = (N : ℝ) * Real.log (1 - x) +
(trueCount z : ℝ) * (Real.log x - Real.log (1 - x)) := by
rw [Finset.sum_add_distrib, ← Finset.sum_mul, ← trueCount_cast]
simp
theorem bernoulliStringWeight_lower_bound {N : ℕ} {x : ℝ}
(hx : 0 < x) (hxhalf : x ≤ 1 / 2) (z : Fin N → Bool)
(hz : (trueCount z : ℝ) ≤ (N : ℝ) * x) :
Real.exp (-((N : ℝ) * binaryEntropy x * Real.log 2)) ≤
bernoulliStringWeight x z := by
have hx1 : x < 1 := by linarith
have hc : Real.log x - Real.log (1 - x) ≤ 0 := by
apply sub_nonpos.mpr
apply Real.log_le_log hx
linarith
have hmul := mul_le_mul_of_nonpos_right hz hc
have hlog : -((N : ℝ) * binaryEntropy x * Real.log 2) ≤
Real.log (bernoulliStringWeight x z) := by
rw [log_bernoulliStringWeight hx hx1]
calc
-((N : ℝ) * binaryEntropy x * Real.log 2) =
(N : ℝ) * Real.log (1 - x) +
((N : ℝ) * x) * (Real.log x - Real.log (1 - x)) := by
rw [mul_assoc, binaryEntropy_mul_log_two]
ring
_ ≤ (N : ℝ) * Real.log (1 - x) +
(trueCount z : ℝ) * (Real.log x - Real.log (1 - x)) :=
add_le_add_right hmul _
calc
Real.exp (-((N : ℝ) * binaryEntropy x * Real.log 2)) ≤
Real.exp (Real.log (bernoulliStringWeight x z)) :=
Real.exp_le_exp.mpr hlog
_ = bernoulliStringWeight x z :=
Real.exp_log (bernoulliStringWeight_pos hx hx1 z)
def sparseBinaryStrings (N : ℕ) (x : ℝ) : Finset (Fin N → Bool) := by
classical
exact Finset.univ.filter fun z => (trueCount z : ℝ) ≤ (N : ℝ) * x
theorem card_sparse_binary_strings_le (N : ℕ) {x : ℝ}
(hx : 0 < x) (hxhalf : x ≤ 1 / 2) :
((sparseBinaryStrings N x).card : ℝ) ≤
(2 : ℝ) ^ ((N : ℝ) * binaryEntropy x) := by
classical
have hx1 : x < 1 := by linarith
have h := finite_card_le_of_uniform_weight_lower_bound
(sparseBinaryStrings N x) (bernoulliStringWeight x)
(Real.exp_pos (-((N : ℝ) * binaryEntropy x * Real.log 2)))
(fun z => (bernoulliStringWeight_pos hx hx1 z).le)
(sum_bernoulliStringWeight N x).le
(fun z hz => bernoulliStringWeight_lower_bound hx hxhalf z
(Finset.mem_filter.mp hz).2)
calc
((sparseBinaryStrings N x).card : ℝ) ≤
1 / Real.exp (-((N : ℝ) * binaryEntropy x * Real.log 2)) := h
_ = Real.exp ((N : ℝ) * binaryEntropy x * Real.log 2) := by
rw [one_div, Real.exp_neg, inv_inv]
_ = (2 : ℝ) ^ ((N : ℝ) * binaryEntropy x) := by
rw [Real.rpow_def_of_pos (show (0 : ℝ) < 2 by norm_num)]
congr 1
ring
end DepthThreeLowerBound
end
end OAI
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