| 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 | |