| import OAI.NumberTheory.Jacobsthal.Paths.SampleWitnessAverage |
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| namespace OAI |
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| namespace Erdos970 |
| open scoped _root_.Erdos970 |
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| section |
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| namespace ErdosInverseSampling |
| attribute [local instance] Classical.propDecidable |
| attribute [local instance] Classical.decEq |
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| theorem exists_reward_cost_sample {Ω : Type*} (S : Finset Ω) (hS : S.Nonempty) |
| (reward cost : Ω → ℝ) (gamma tau : ℝ) (hg : 0 < gamma) (ht : 0 < tau) |
| (hr : ∀ s ∈ S,reward s ≤ 1) (hc : ∀ s ∈ S,0 ≤ cost s) |
| (hreward : gamma*(S.card : ℝ) ≤ ∑ s ∈ S,reward s) |
| (hcost : (∑ s ∈ S,cost s) ≤ (gamma*tau/4)*(S.card : ℝ)) : |
| ∃ s ∈ S,gamma/2 ≤ reward s ∧ cost s ≤ tau := by |
| by_contra h |
| push Not at h |
| have hpoint (s : Ω) (hs : s ∈ S) : reward s ≤ gamma/2+cost s/tau := by |
| by_cases hh : gamma/2 ≤ reward s |
| · have hct : tau < cost s := h s hs hh |
| have hone : (1 : ℝ) < cost s/tau := (lt_div_iff₀ ht).mpr (by simpa using hct) |
| linarith [hr s hs] |
| · have hcn : 0 ≤ cost s/tau := div_nonneg (hc s hs) ht.le |
| linarith |
| have hupper : (∑ s ∈ S,reward s) ≤ (S.card : ℝ)*(gamma/2)+(∑ s ∈ S,cost s)/tau := by |
| calc |
| _ ≤ ∑ s ∈ S,(gamma/2+cost s/tau) := Finset.sum_le_sum hpoint |
| _ = _ := by |
| rw [Finset.sum_add_distrib] |
| simp only [Finset.sum_const,nsmul_eq_mul,← Finset.sum_div] |
| have hsmall : (∑ s ∈ S,cost s)/tau ≤ (gamma/4)*(S.card : ℝ) := by |
| apply (div_le_iff₀ ht).mpr |
| nlinarith only [hcost] |
| have hN : (0 : ℝ) < S.card := by exact_mod_cast Finset.card_pos.mpr hS |
| have hpos : 0 < gamma*(S.card : ℝ) := mul_pos hg hN |
| nlinarith only [hreward,hupper,hsmall,hpos] |
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| theorem selectedFraction_le_one {α β : Type*} (A : Finset α) (U : Finset β) |
| (witness : α → β → Prop) (h : ℕ) (e : Sample U h) (hA : 0 < A.card) : |
| selectedFraction A U witness h e ≤ 1 := by |
| have hAR : (0 : ℝ) < A.card := by exact_mod_cast hA |
| apply (div_le_iff₀ hAR).mpr |
| have hh : (selectedPairs A U witness h e).card ≤ A.card := Finset.card_filter_le A _ |
| simpa only [one_mul] using (Nat.cast_le.mpr hh : ((selectedPairs A U witness h e).card : ℝ) ≤ A.card) |
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| theorem exists_good_distinct_sample {α β : Type*} (A : Finset α) (U : Finset β) |
| (witness : α → β → Prop) (h : ℕ) (beta tau : ℝ) (hb : 0 < beta) (hb1 : beta ≤ 1) |
| (ht : 0 < tau) (hA : 0 < A.card) (hU : 0 < U.card) |
| (hdense : beta*(A.card : ℝ)*(U.card : ℝ) ≤ ((witnessPairs A U witness).card : ℝ)) |
| (hsize : 4*(h : ℝ)/beta ≤ (U.card : ℝ)) (cost : Sample U h → ℝ) |
| (hc : ∀ e,0 ≤ cost e) |
| (hcost : sampleAverage U h cost ≤ (((beta/2)*(beta/4)^h)*tau/4)) : |
| ∃ e : Sample U h,((beta/2)*(beta/4)^h)/2 ≤ selectedFraction A U witness h e ∧ cost e ≤ tau := by |
| let gamma := (beta/2)*(beta/4)^h |
| have hg : 0 < gamma := by dsimp [gamma];positivity |
| have hlen := sample_length_le U.card h beta hb hb1 hsize |
| have hn : 0 < Fintype.card (Sample U h) := by rw [sample_card];exact Nat.descFactorial_pos.mpr hlen |
| have hnR : (0 : ℝ) < Fintype.card (Sample U h) := by exact_mod_cast hn |
| have hreward := sample_incidence_average_lower A U witness h beta hb hb1 hA hU hdense hsize |
| have hreward' : gamma*(Fintype.card (Sample U h) : ℝ) ≤ |
| ∑ e : Sample U h,selectedFraction A U witness h e := (le_div_iff₀ hnR).mp hreward |
| have hcost' : (∑ e : Sample U h,cost e) ≤ (gamma*tau/4)*(Fintype.card (Sample U h) : ℝ) := |
| (div_le_iff₀ hnR).mp hcost |
| obtain ⟨e,_he,her,hec⟩ := exists_reward_cost_sample (Finset.univ : Finset (Sample U h)) |
| (Finset.card_pos.mp (by simpa only [Finset.card_univ] using hn)) |
| (selectedFraction A U witness h) cost gamma tau hg ht |
| (fun e _ => selectedFraction_le_one A U witness h e hA) (fun e _ => hc e) |
| (by simpa only [Finset.card_univ] using hreward') (by simpa only [Finset.card_univ] using hcost') |
| exact ⟨e,her,hec⟩ |
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| end ErdosInverseSampling |
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| end |
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| end Erdos970 |
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| end OAI |
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