| import OAI.Probability.Renewal.Definitions |
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| namespace OAI |
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| noncomputable section |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped Topology ENNReal |
| namespace HoneycombLocal |
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| def indexWindow (x r : ℝ) : Finset ℕ := Finset.Icc ⌈x⌉₊ ⌊(1 + r) * x⌋₊ |
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| lemma mem_indexWindow {x r : ℝ} (hx : 0 ≤ x) (hr : 0 ≤ r) {k : ℕ} |
| (hk : k ∈ indexWindow x r) : x ≤ (k : ℝ) ∧ (k : ℝ) ≤ (1 + r) * x := by |
| obtain ⟨hl, hu⟩ := Finset.mem_Icc.mp hk |
| exact ⟨(Nat.le_ceil x).trans (by exact_mod_cast hl), |
| (by exact_mod_cast hu : (k : ℝ) ≤ (⌊(1 + r) * x⌋₊ : ℝ)).trans (Nat.floor_le (by positivity))⟩ |
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| lemma card_indexWindow {x r : ℝ} (hx : 0 ≤ x) (_hr : 0 ≤ r) (hrx : 2 ≤ r * x) : |
| r * x / 2 ≤ ((indexWindow x r).card : ℝ) := by |
| have hceil := Nat.ceil_lt_add_one hx |
| have hfloor := Nat.lt_floor_add_one ((1 + r) * x) |
| have hcf : ⌈x⌉₊ ≤ ⌊(1 + r) * x⌋₊ := Nat.le_floor (by nlinarith) |
| rw [indexWindow, Nat.card_Icc, Nat.cast_sub (hcf.trans (Nat.le_succ _)), Nat.cast_succ] |
| nlinarith |
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| lemma renewalMass_eq_tsum (p : PMF ℕ) {m : ℕ} (hm : 0 < m) : |
| renewalMass p m = ∑' k : ℕ, sumLaw p k m := by |
| rw [tsum_eq_zero_add' ENNReal.summable] |
| simp [sumLaw, PMF.pure_apply, hm.ne', renewalMass] |
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| lemma renewal_lower_finset (p : PMF ℕ) {m : ℕ} (hm : 0 < m) (s : Finset ℕ) {b : ℝ} |
| (hb : ∀ k ∈ s, b ≤ (sumLaw p k m).toReal) : |
| ENNReal.ofReal (b * (s.card : ℝ)) ≤ renewalMass p m := by |
| calc |
| ENNReal.ofReal (b * (s.card : ℝ)) = ENNReal.ofReal (∑ _k ∈ s, b) := by simp [mul_comm] |
| _ ≤ ENNReal.ofReal (∑ k ∈ s, (sumLaw p k m).toReal) := |
| ENNReal.ofReal_le_ofReal (Finset.sum_le_sum hb) |
| _ = ∑ k ∈ s, sumLaw p k m := by |
| rw [ENNReal.ofReal_sum_of_nonneg (fun _ _ => ENNReal.toReal_nonneg)] |
| simp only [ENNReal.ofReal_toReal (PMF.apply_ne_top _ _)] |
| _ ≤ renewalMass p m := by rw [renewalMass_eq_tsum p hm]; exact ENNReal.sum_le_tsum s |
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| lemma inverse_tilt_scale {a A m : ℝ} (hA : 0 < A) (hm : 0 < m) : |
| m / (A / m) ^ (a - 1) = A ^ (1 - a) * m ^ a := by |
| rw [Real.div_rpow hA.le hm.le, div_div_eq_mul_div] |
| have hM : m * m ^ (a - 1) = m ^ a := by |
| have hh := Real.rpow_add_one hm.ne' (a - 1) |
| simp only [sub_add_cancel] at hh |
| simpa only [mul_comm] using hh.symm |
| rw [hM] |
| have hA' : A ^ (1 - a) = (A ^ (a - 1))⁻¹ := by |
| rw [show (1 - a : ℝ) = -(a - 1) by ring, Real.rpow_neg hA.le] |
| rw [hA'] |
| ring |
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| end HoneycombLocal |
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| end |
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| end OAI |
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