| import Mathlib |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory Filter Set |
| open scoped ENNReal Topology BigOperators |
| open scoped BigOperators ComplexConjugate |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped ENNReal NNReal Topology BigOperators |
| open MeasureTheory ProbabilityTheory Real Complex |
| open scoped ENNReal ComplexConjugate |
| open ProbabilityTheory |
| namespace HardSphere |
| open MeasureTheory Set Filter |
|
|
| lemma finset_erase_recursion {S : Type*} [DecidableEq S] (a : Finset S → ℝ) (q : ℝ) |
| (hq : 0 ≤ q) (h0 : a ∅ ≤ 1) |
| (hr : ∀ A : Finset S, A.Nonempty → (A.card : ℝ)*a A ≤ q*∑ s∈A, a (A.erase s)) : |
| ∀ A : Finset S, a A ≤ q^A.card := by |
| intro A |
| induction A using Finset.strongInductionOn with |
| | _ A ih => |
| by_cases hA : A.Nonempty |
| · have hc : 0 < (A.card : ℝ) := by exact_mod_cast hA.card_pos |
| have hs : (∑ s∈A, a (A.erase s)) ≤ (A.card : ℝ)*q^(A.card-1) := by |
| calc |
| _ ≤ ∑ _s∈A, q^(A.card-1) := Finset.sum_le_sum fun s hs ↦ by |
| have hh := ih (A.erase s) (Finset.erase_ssubset hs) |
| rwa [Finset.card_erase_of_mem hs] at hh |
| _ = _ := by simp |
| have he : q*((A.card : ℝ)*q^(A.card-1))=(A.card : ℝ)*q^A.card := by |
| rw [show q^A.card=q^(A.card-1)*q by |
| conv_lhs => rw [← Nat.sub_add_cancel hA.card_pos] |
| rw [pow_succ]] |
| ring |
| have ht := (hr A hA).trans ((mul_le_mul_of_nonneg_left hs hq).trans_eq he) |
| exact (mul_le_mul_iff_right₀ hc).mp (by simpa only [mul_comm (A.card : ℝ)] using ht) |
| · have he : A=∅ := Finset.not_nonempty_iff_eq_empty.mp hA |
| simpa [he] using h0 |
|
|
| end HardSphere |
|
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| end |
|
|
| end OAI |
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|
