| import OAI.Probability.DirectionalWalk.ContactBins |
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| namespace OAI |
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| open MeasureTheory ProbabilityTheory Filter Preorder |
| open scoped ENNReal BigOperators Topology |
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| namespace DirectionalZeroOne |
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| open scoped Classical |
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| noncomputable def dyad (j : ℕ) : ℝ≥0∞ := ENNReal.ofReal ((1/2 : ℝ)^(j+1)) |
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| lemma dyad_pos (j : ℕ) : 0 < dyad j := ENNReal.ofReal_pos.mpr (pow_pos (by norm_num) _) |
| lemma dyad_ne_top (j : ℕ) : dyad j ≠ ∞ := ENNReal.ofReal_ne_top |
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| lemma exists_dyad_bin {q : ℝ≥0∞} (hq : 0 < q) (h1 : q < 1) : |
| ∃ j : ℕ, dyad j ≤ q ∧ q < 2*dyad j := by |
| have hqf : q ≠ ∞ := ne_of_lt (h1.trans (by norm_num)) |
| have hqr : 0 < q.toReal := ENNReal.toReal_pos (ne_of_gt hq) hqf |
| have hqr1 : q.toReal < 1 := by exact_mod_cast (ENNReal.toReal_lt_toReal hqf (by norm_num)).mpr h1 |
| have ht := tendsto_pow_atTop_nhds_zero_of_lt_one (by norm_num : 0 ≤ (1/2 : ℝ)) (by norm_num : (1/2 : ℝ) < 1) |
| have hex : ∃ n : ℕ, (1/2 : ℝ)^n ≤ q.toReal := by |
| obtain ⟨n,hn⟩ := (ht.eventually (eventually_lt_nhds hqr)).exists |
| exact ⟨n,hn.le⟩ |
| let n := Nat.find hex |
| have hn : (1/2 : ℝ)^n ≤ q.toReal := Nat.find_spec hex |
| have hn0 : 0 < n := by |
| by_contra h |
| have he : n = 0 := by omega |
| rw [he,pow_zero] at hn |
| linarith |
| refine ⟨n-1,?_,?_⟩ |
| · unfold dyad |
| rw [Nat.sub_add_cancel hn0] |
| exact (ENNReal.ofReal_le_iff_le_toReal hqf).mpr hn |
| · have hp : q.toReal < (1/2 : ℝ)^(n-1) := lt_of_not_ge (Nat.find_min hex (by omega)) |
| have he : (2 : ℝ) * (1/2 : ℝ)^n = (1/2 : ℝ)^(n-1) := by |
| conv_lhs => rw [← Nat.sub_add_cancel hn0,pow_succ] |
| ring |
| rw [dyad,Nat.sub_add_cancel hn0,← ENNReal.ofReal_ofNat,← ENNReal.ofReal_mul (by norm_num),he] |
| exact (ENNReal.lt_ofReal_iff_toReal_lt hqf).mpr hp |
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|
| lemma return_bins_cover {d : ℕ} (e : Step d) (k N : ℤ) (ω : Environment d) |
| (hω : ∀ x f, 0 < (ω x).val f) (y : Site d) |
| (hy : k ≤ axisHeight e y ∧ axisHeight e y < N) : |
| ∃ j, y ∈ returnBin e k N (dyad j) ω := by |
| obtain ⟨j,hj,hj'⟩ := exists_dyad_bin (quenchedReturn_pos_lt_one e k N ω hω y hy).1 |
| (quenchedReturn_pos_lt_one e k N ω hω y hy).2 |
| exact ⟨j,⟨hy.1,hy.2,hj⟩,hj'⟩ |
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|
| lemma tsum_subtype_cover {α : Type*} [Countable α] (A : Set α) (B : ℕ → Set α) |
| (h : A ⊆ ⋃ n, B n) (f : α → ℝ≥0∞) : |
| (∑' a : A, f a) ≤ ∑' n, ∑' a : B n, f a := by |
| classical |
| have hex (a : A) : ∃ n, a.val ∈ B n := Set.mem_iUnion.mp (h a.property) |
| let g : A → (Σ n, B n) := fun a => ⟨(hex a).choose,⟨a.val,(hex a).choose_spec⟩⟩ |
| have hg : Function.Injective g := by |
| intro a b hab |
| have hv : ((g a).2 : α) = ((g b).2 : α) := congrArg (fun i : Σ n, B n => i.2.val) hab |
| exact Subtype.ext hv |
| calc |
| _ ≤ ∑' i : Σ n, B n, f i.2 := Summable.tsum_le_tsum_of_inj g hg (fun _ _ => zero_le) (fun _ => le_rfl) ENNReal.summable ENNReal.summable |
| _ = _ := ENNReal.tsum_sigma (fun n (a : B n) => f a) |
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|
| lemma half_geometric_sum_le (s : Finset ℕ) : (∑ j ∈ s, (1/2 : ℝ)^(j+1)) ≤ 1 := by |
| have hs : HasSum (fun j : ℕ => (1/2 : ℝ)^(j+1)) 1 := by |
| have hh := (hasSum_geometric_of_norm_lt_one (by norm_num : ‖(1/2 : ℝ)‖ < 1)).mul_right (1/2) |
| norm_num [pow_succ] at hh ⊢ |
| exact hh |
| exact (hs.summable.sum_le_tsum s (fun j _ => by positivity)).trans_eq hs.tsum_eq |
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|
| lemma half_geometric_tail_sum_le (s : Finset ℕ) (m : ℕ) : |
| (∑ j ∈ s.filter (m ≤ ·), (1/2 : ℝ)^(j+1)) ≤ (1/2 : ℝ)^m := by |
| classical |
| let t := (s.filter (m ≤ ·)).image (fun j => j-m) |
| have he : (∑ j ∈ s.filter (m ≤ ·), (1/2 : ℝ)^(j+1)) = |
| (1/2 : ℝ)^m * ∑ j ∈ t, (1/2 : ℝ)^(j+1) := by |
| rw [Finset.mul_sum,Finset.sum_image] |
| · apply Finset.sum_congr rfl |
| intro j hj |
| rw [← pow_add] |
| congr 1 |
| have := (Finset.mem_filter.mp hj).2 |
| omega |
| · intro a ha b hb hab |
| have := (Finset.mem_filter.mp ha).2 |
| have := (Finset.mem_filter.mp hb).2 |
| change a-m=b-m at hab |
| omega |
| rw [he] |
| exact (mul_le_mul_of_nonneg_left (half_geometric_sum_le t) (by positivity)).trans_eq (mul_one _) |
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| lemma dyadic_weighted_mass_sum {v : ℕ → ℝ} (_hv0 : ∀ j, 0 ≤ v j) (hv1 : ∀ j, v j ≤ 1) |
| (m : ℕ) (hv : ∀ j, (1/2 : ℝ)^(j+1)*v j ≤ (1/2 : ℝ)^m) (s : Finset ℕ) : |
| (∑ j ∈ s, (1/2 : ℝ)^(j+1)*v j) ≤ (m+1)*(1/2 : ℝ)^m := by |
| classical |
| have hsmall : (∑ j ∈ s.filter (· < m), (1/2 : ℝ)^(j+1)*v j) ≤ m*(1/2 : ℝ)^m := by |
| calc |
| _ ≤ ∑ _ ∈ s.filter (· < m), (1/2 : ℝ)^m := Finset.sum_le_sum (fun j _ => hv j) |
| _ = ((s.filter (· < m)).card : ℝ)*(1/2 : ℝ)^m := by simp |
| _ ≤ m*(1/2 : ℝ)^m := by |
| apply mul_le_mul_of_nonneg_right _ (by positivity) |
| have hh : (s.filter (· < m)).card ≤ m := by |
| simpa using (Finset.card_le_card (show s.filter (· < m) ⊆ Finset.range m by |
| intro j hj;exact Finset.mem_range.mpr (Finset.mem_filter.mp hj).2)) |
| exact_mod_cast hh |
| have hlarge : (∑ j ∈ s.filter (m ≤ ·), (1/2 : ℝ)^(j+1)*v j) ≤ (1/2 : ℝ)^m := by |
| apply (Finset.sum_le_sum (s := s.filter (m ≤ ·)) (fun j _ => |
| mul_le_of_le_one_right (by positivity) (hv1 j))).trans |
| exact half_geometric_tail_sum_le s m |
| have he : (∑ j ∈ s.filter (· < m), (1/2 : ℝ)^(j+1)*v j) + |
| (∑ j ∈ s.filter (m ≤ ·), (1/2 : ℝ)^(j+1)*v j) = ∑ j ∈ s, (1/2 : ℝ)^(j+1)*v j := by |
| simpa only [not_lt] using Finset.sum_filter_add_sum_filter_not s (fun j => j < m) |
| (fun j => (1/2 : ℝ)^(j+1)*v j) |
| linarith |
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|
| lemma dyadic_event_weight_bound {Ω : Type*} [MeasurableSpace Ω] |
| (ν : Measure Ω) [IsProbabilityMeasure ν] (S : Ω → ℝ) (hS : Integrable S ν) |
| (hS0 : ∀ ξ, 0 ≤ S ξ) (c : ℝ) (hc : 0 < c) |
| (hExp : Integrable (fun ξ => Real.exp (c*S ξ)) ν) |
| (hExp12 : (∫ ξ, Real.exp (c*S ξ) ∂ν) ≤ 12) |
| (V : ℕ → Set Ω) (m : ℕ) |
| (hV : ∀ j, dyad j * ν (V j) ≤ ENNReal.ofReal ((1/2 : ℝ)^m)) : |
| (∑' j, 2*dyad j * ∫⁻ ξ in V j, ENNReal.ofReal (S ξ) ∂ν) ≤ |
| ENNReal.ofReal ((24/c)*(m+1)^2*(1/2 : ℝ)^m) := by |
| classical |
| let t := (1/2 : ℝ)^m |
| have ht : 0 < t := pow_pos (by norm_num) _ |
| have ht1 : t ≤ 1 := pow_le_one₀ (by norm_num) (by norm_num) |
| have hl : Real.log t ≤ 0 := Real.log_nonpos ht.le ht1 |
| have hv (j : ℕ) : (1/2 : ℝ)^(j+1)*ν.real (V j) ≤ t := by |
| have hh := ENNReal.toReal_mono (ENNReal.ofReal_ne_top) (hV j) |
| simpa only [ENNReal.toReal_mul,dyad,ENNReal.toReal_ofReal (pow_nonneg (by norm_num : (0 : ℝ) ≤ 1/2) _),Measure.real,t] using hh |
| have hlog : -Real.log t ≤ m := by |
| dsimp [t] |
| rw [Real.log_pow,Real.log_div (by norm_num) (by norm_num),Real.log_one] |
| have h := Real.log_le_sub_one_of_pos (by norm_num : (0 : ℝ) < 2) |
| nlinarith [mul_le_mul_of_nonneg_left h (Nat.cast_nonneg m)] |
| rw [ENNReal.tsum_eq_iSup_sum] |
| apply iSup_le |
| intro s |
| have hI (j : ℕ) : c*(∫ ξ in V j, S ξ ∂ν) ≤ 12*t - ν.real (V j)*Real.log t := by |
| have hp (ξ : Ω) : c*S ξ ≤ t*Real.exp (c*S ξ)-Real.log t := by |
| have h := Real.add_one_le_exp (c*S ξ + Real.log t) |
| rw [Real.exp_add,Real.exp_log ht] at h |
| nlinarith |
| have hi := integral_mono (hS.restrict (s := V j) |>.const_mul c) |
| ((hExp.restrict.const_mul t).sub (integrable_const (Real.log t))) hp |
| simp only [Pi.sub_apply] at hi |
| rw [integral_const_mul,integral_sub (hExp.restrict (s := V j) |>.const_mul t) (integrable_const _), |
| integral_const_mul,integral_const,measureReal_restrict_apply_univ,smul_eq_mul] at hi |
| have he : (∫ ξ in V j, Real.exp (c*S ξ) ∂ν) ≤ 12 := |
| (setIntegral_le_integral hExp (Filter.Eventually.of_forall (fun ξ => Real.exp_nonneg _))).trans hExp12 |
| nlinarith [mul_le_mul_of_nonneg_left he ht.le] |
| let B := ∑ j ∈ s, (1/2 : ℝ)^(j+1)*(∫ ξ in V j, S ξ ∂ν) |
| have hB : c*B ≤ 12*t + m*((m+1)*t) := by |
| have hsum := Finset.sum_le_sum (s := s) (fun j _ => mul_le_mul_of_nonneg_left (hI j) |
| (pow_nonneg (by norm_num : (0 : ℝ) ≤ 1/2) (j+1))) |
| have he1 : (∑ j ∈ s, (1/2 : ℝ)^(j+1)*(c*(∫ ξ in V j, S ξ ∂ν))) = c*B := by |
| dsimp [B];rw [Finset.mul_sum];apply Finset.sum_congr rfl;intro j _;ring |
| have he2 : (∑ j ∈ s, (1/2 : ℝ)^(j+1)*(12*t-ν.real (V j)*Real.log t)) = |
| 12*t*(∑ j ∈ s, (1/2 : ℝ)^(j+1)) + (-Real.log t)*(∑ j ∈ s, (1/2 : ℝ)^(j+1)*ν.real (V j)) := by |
| rw [Finset.mul_sum,Finset.mul_sum,← Finset.sum_add_distrib] |
| apply Finset.sum_congr rfl;intro j _;ring |
| rw [he1,he2] at hsum |
| have hmass := dyadic_weighted_mass_sum (fun j => measureReal_nonneg) (fun j => measureReal_le_one) m hv s |
| have hh := mul_le_mul_of_nonneg_left (half_geometric_sum_le s) (show 0 ≤ 12*t by positivity) |
| have hh' := mul_le_mul hlog hmass (Finset.sum_nonneg (fun j _ => mul_nonneg (by positivity) measureReal_nonneg)) (Nat.cast_nonneg m) |
| dsimp [t] at * |
| nlinarith |
| have hbound : 2*B ≤ (24/c)*(m+1)^2*t := by |
| have hm : (0 : ℝ) ≤ m := Nat.cast_nonneg m |
| have hpoly : 12*t + m*((m+1)*t) ≤ 12*(m+1)^2*t := by |
| nlinarith [sq_nonneg (m : ℝ)] |
| have hcc : c*(2*B) ≤ c*((24/c)*(m+1)^2*t) := by |
| field_simp |
| nlinarith [hB.trans hpoly] |
| exact (mul_le_mul_iff_right₀ hc).mp hcc |
| have he (j : ℕ) : 2*dyad j * (∫⁻ ξ in V j, ENNReal.ofReal (S ξ) ∂ν) = |
| ENNReal.ofReal (2*(1/2 : ℝ)^(j+1)*(∫ ξ in V j, S ξ ∂ν)) := by |
| rw [← ofReal_integral_eq_lintegral_ofReal hS.restrict (Filter.Eventually.of_forall hS0),dyad] |
| rw [ENNReal.ofReal_mul (by positivity),ENNReal.ofReal_mul (by norm_num)] |
| norm_num |
| simp_rw [he] |
| rw [← ENNReal.ofReal_sum_of_nonneg (fun j _ => mul_nonneg (by positivity) |
| (integral_nonneg hS0))] |
| apply ENNReal.ofReal_le_ofReal |
| convert hbound using 1 |
| dsimp [B,t] |
| rw [Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro j _ |
| ring |
|
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| end DirectionalZeroOne |
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| end OAI |
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