File size: 1,820 Bytes
dd14c26 | 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 | import Mathlib
namespace OAI
noncomputable section
open scoped BigOperators
namespace SparseWitness
attribute [local instance] Classical.propDecidable
local instance sparseEntropyDecidableEq (carrier : Sort*) : DecidableEq carrier :=
Classical.typeDecidableEq carrier
def pathSubsets {E : Type*} [DecidableEq E] (k : ℕ) (F : Finset E) : Finset (Finset E) :=
F.powerset.filter (fun S => F.card ≤ 2*S.card ∧ k ≤ F.card)
lemma path_entropy {E : Type*} [DecidableEq E] (k : ℕ) (F : Finset E)
(q : ℝ) (hq : 0 ≤ q) (hqsmall : q ≤ 1/16) :
(∑ S ∈ pathSubsets k F, q^S.card) ≤ (1/2:ℝ)^k := by
by_cases hk : k ≤ F.card
· have heach (S : Finset E) (hS : S ∈ pathSubsets k F) :
q^S.card ≤ (1/4:ℝ)^F.card := by
have hsize := (Finset.mem_filter.mp hS).2.1
calc
q^S.card ≤ (1/16:ℝ)^S.card := pow_le_pow_left₀ hq hqsmall _
_ = (1/4:ℝ)^(2*S.card) := by rw [pow_mul]; norm_num
_ ≤ _ := pow_le_pow_of_le_one (by norm_num) (by norm_num) hsize
calc
_ ≤ ∑ _S ∈ pathSubsets k F, (1/4:ℝ)^F.card := Finset.sum_le_sum heach
_ = ((pathSubsets k F).card : ℝ)*(1/4:ℝ)^F.card := by simp
_ ≤ (2:ℝ)^F.card*(1/4:ℝ)^F.card := by
apply mul_le_mul_of_nonneg_right _ (by positivity)
have hc : (pathSubsets k F).card ≤ 2^F.card :=
(Finset.card_filter_le _ _).trans_eq (Finset.card_powerset F)
exact_mod_cast hc
_ = (1/2:ℝ)^F.card := by rw [← mul_pow];norm_num
_ ≤ _ := pow_le_pow_of_le_one (by norm_num) (by norm_num) hk
· have he : pathSubsets k F = ∅ := by
apply Finset.eq_empty_iff_forall_notMem.mpr
intro S hS
exact hk (Finset.mem_filter.mp hS).2.2
rw [he,Finset.sum_empty]
positivity
end SparseWitness
end
end OAI
|