openAI-math-dataset / lean /OAI /Probability /FreeIsing /Martingales /PredictableMeshApproximation.lean
| import Mathlib | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory Filter Set | |
| open scoped NNReal Topology | |
| namespace Problem367.PredictableMesh | |
| def point (n k : ℕ) : ℝ≥0 := (k : ℝ≥0) / (n + 1 : ℕ) | |
| def left (n : ℕ) (t : ℝ≥0) : ℝ≥0 := | |
| point n (⌈(n + 1 : ℕ) * t⌉₊ - 1) | |
| @[simp] theorem point_zero (n : ℕ) : point n 0 = 0 := by simp [point] | |
| @[simp] theorem left_zero (n : ℕ) : left n 0 = 0 := by simp [left] | |
| theorem point_strictMono (n : ℕ) : StrictMono (point n) := by | |
| intro i j hij | |
| exact div_lt_div_of_pos_right (by exact_mod_cast hij) (by positivity) | |
| theorem left_eq_of_mem_Ioc {n k : ℕ} {t : ℝ≥0} | |
| (ht : t ∈ Ioc (point n k) (point n (k + 1))) : | |
| left n t = point n k := by | |
| have hN : (0 : ℝ≥0) < (n + 1 : ℕ) := by positivity | |
| have hc : ⌈(n + 1 : ℕ) * t⌉₊ = k + 1 := by | |
| apply (Nat.ceil_eq_iff (Nat.succ_ne_zero k)).2 | |
| constructor | |
| · have := (div_lt_iff₀ hN).1 ht.1 | |
| simpa only [mul_comm, Nat.succ_sub_one] using this | |
| · have := (le_div_iff₀ hN).1 ht.2 | |
| simpa only [mul_comm, Nat.succ_sub_one] using this | |
| change point n (⌈(n + 1 : ℕ) * t⌉₊ - 1) = point n k | |
| rw [hc, Nat.add_sub_cancel] | |
| theorem mem_mesh_Ioc {n : ℕ} {t : ℝ≥0} (ht : 0 < t) : | |
| t ∈ Ioc (left n t) | |
| (point n (⌈(n + 1 : ℕ) * t⌉₊ - 1 + 1)) := by | |
| have hN : (0 : ℝ≥0) < (n + 1 : ℕ) := by positivity | |
| have hc : 1 ≤ ⌈(n + 1 : ℕ) * t⌉₊ := | |
| Nat.one_le_ceil_iff.mpr (mul_pos hN ht) | |
| have hh := (Nat.ceil_eq_iff (show ⌈(n + 1 : ℕ) * t⌉₊ ≠ 0 by omega)).mp rfl | |
| constructor | |
| · apply (div_lt_iff₀ hN).2 | |
| simpa only [mul_comm] using hh.1 | |
| · dsimp [point] | |
| rw [Nat.sub_add_cancel hc] | |
| apply (le_div_iff₀ hN).2 | |
| simpa only [mul_comm] using hh.2 | |
| theorem left_le (n : ℕ) (t : ℝ≥0) : left n t ≤ t := by | |
| by_cases ht : t = 0 | |
| · simp [ht] | |
| · exact (mem_mesh_Ioc (n := n) (pos_iff_ne_zero.mpr ht)).1.le | |
| theorem le_left_add_width (n : ℕ) (t : ℝ≥0) : | |
| t ≤ left n t + 1 / (n + 1 : ℕ) := by | |
| by_cases ht : t = 0 | |
| · simp [ht] | |
| · have h := (mem_mesh_Ioc (n := n) (pos_iff_ne_zero.mpr ht)).2 | |
| simpa only [point, left, Nat.cast_add, Nat.cast_one, add_div] using h | |
| theorem tendsto_left (t : ℝ≥0) : Tendsto (fun n => left n t) atTop (𝓝 t) := by | |
| apply NNReal.tendsto_coe.mp | |
| apply tendsto_iff_norm_sub_tendsto_zero.mpr | |
| apply squeeze_zero (fun n => norm_nonneg _) | |
| (fun n => show ‖(left n t : ℝ) - (t : ℝ)‖ ≤ 1 / ((n : ℝ) + 1) from ?_) | |
| tendsto_one_div_add_atTop_nhds_zero_nat | |
| have hl : (left n t : ℝ) ≤ (t : ℝ) := left_le n t | |
| have hu := le_left_add_width n t | |
| have hu' : (t : ℝ) ≤ (left n t : ℝ) + 1 / ((n : ℝ) + 1) := by | |
| exact_mod_cast hu | |
| rw [Real.norm_eq_abs, abs_of_nonpos (sub_nonpos.mpr hl)] | |
| linarith | |
| theorem measurable_left (n : ℕ) : Measurable (left n) := by | |
| exact (measurable_of_countable (fun k : ℕ => point n (k - 1))).comp | |
| ((measurable_const.mul measurable_id).nat_ceil) | |
| def sample {Ω : Type*} (u : ℝ≥0 → Ω → ℝ) (n : ℕ) (t : ℝ≥0) (ω : Ω) : ℝ := | |
| u (left n t) ω | |
| theorem sample_tendsto {Ω : Type*} {u : ℝ≥0 → Ω → ℝ} | |
| (hu : ∀ ω, Continuous (fun t => u t ω)) (t : ℝ≥0) (ω : Ω) : | |
| Tendsto (fun n => sample u n t ω) atTop (𝓝 (u t ω)) := | |
| (hu ω).continuousAt.tendsto.comp (tendsto_left t) | |
| theorem sample_stronglyAdapted {Ω : Type*} [mΩ : MeasurableSpace Ω] | |
| {F : Filtration ℝ≥0 mΩ} {u : ℝ≥0 → Ω → ℝ} | |
| (hu : StronglyAdapted F u) (n : ℕ) : StronglyAdapted F (sample u n) := | |
| fun t => (hu (left n t)).mono (F.mono (left_le n t)) | |
| theorem sample_isStronglyPredictable {Ω : Type*} [mΩ : MeasurableSpace Ω] | |
| {F : Filtration ℝ≥0 mΩ} {u : ℝ≥0 → Ω → ℝ} | |
| (hu : StronglyAdapted F u) (n : ℕ) : IsStronglyPredictable F (sample u n) := by | |
| apply Measurable.stronglyMeasurable | |
| intro s hs | |
| have heq : (Function.uncurry (sample u n)) ⁻¹' s = | |
| ({0} ×ˢ (u 0 ⁻¹' s)) ∪ | |
| ⋃ k : ℕ, (Ioc (point n k) (point n (k + 1))) ×ˢ (u (point n k) ⁻¹' s) := by | |
| ext ⟨t, ω⟩ | |
| simp only [mem_preimage, Function.uncurry_apply_pair, mem_union, mem_prod, | |
| mem_singleton_iff, mem_iUnion] | |
| constructor | |
| · intro h | |
| by_cases ht : t = 0 | |
| · exact Or.inl ⟨ht, by simpa [ht, sample] using h⟩ | |
| · exact Or.inr ⟨⌈(n + 1 : ℕ) * t⌉₊ - 1, | |
| mem_mesh_Ioc (pos_iff_ne_zero.mpr ht), h⟩ | |
| · rintro (⟨ht, h⟩ | ⟨k, ht, h⟩) | |
| · simpa [ht, sample] using h | |
| · simpa only [sample, left_eq_of_mem_Ioc ht] using h | |
| rw [heq] | |
| apply MeasurableSet.union | |
| · exact measurableSet_predictable_singleton_bot_prod ((hu 0).measurable hs) | |
| · exact MeasurableSet.iUnion fun k => | |
| measurableSet_predictable_Ioc_prod _ _ ((hu (point n k)).measurable hs) | |
| theorem isStronglyPredictable_of_continuous {Ω : Type*} [mΩ : MeasurableSpace Ω] | |
| {F : Filtration ℝ≥0 mΩ} {u : ℝ≥0 → Ω → ℝ} | |
| (hu : StronglyAdapted F u) (hc : ∀ ω, Continuous (fun t => u t ω)) : | |
| IsStronglyPredictable F u := by | |
| apply stronglyMeasurable_of_tendsto atTop (f := fun n => Function.uncurry (sample u n)) | |
| (fun n => sample_isStronglyPredictable hu n) | |
| exact tendsto_pi_nhds.mpr fun p => sample_tendsto hc p.1 p.2 | |
| theorem stronglyMeasurable_uncurry {Ω : Type*} [mΩ : MeasurableSpace Ω] | |
| {F : Filtration ℝ≥0 mΩ} {u : ℝ≥0 → Ω → ℝ} | |
| (hu : StronglyAdapted F u) (hc : ∀ ω, Continuous (fun t => u t ω)) : | |
| StronglyMeasurable (Function.uncurry u) := | |
| stronglyMeasurable_uncurry_of_continuous_of_stronglyMeasurable hc | |
| (fun t => (hu t).mono (F.le t)) | |
| theorem stronglyMeasurable_sample {Ω : Type*} [mΩ : MeasurableSpace Ω] | |
| {F : Filtration ℝ≥0 mΩ} {u : ℝ≥0 → Ω → ℝ} | |
| (hu : StronglyAdapted F u) (hc : ∀ ω, Continuous (fun t => u t ω)) (n : ℕ) : | |
| StronglyMeasurable (Function.uncurry (sample u n)) := | |
| (stronglyMeasurable_uncurry hu hc).comp_measurable | |
| (((measurable_left n).comp measurable_fst).prodMk measurable_snd) | |
| theorem memLp_uncurry {Ω : Type*} [mΩ : MeasurableSpace Ω] | |
| {F : Filtration ℝ≥0 mΩ} {u : ℝ≥0 → Ω → ℝ} | |
| (hu : StronglyAdapted F u) (hc : ∀ ω, Continuous (fun t => u t ω)) | |
| (ν : Measure ℝ≥0) (P : Measure Ω) [IsFiniteMeasure ν] [IsFiniteMeasure P] | |
| (C : ℝ) (hb : ∀ t ω, |u t ω| ≤ C) : | |
| MemLp (Function.uncurry u) 2 (ν.prod P) := | |
| MemLp.of_bound (stronglyMeasurable_uncurry hu hc).aestronglyMeasurable C | |
| (Filter.Eventually.of_forall fun p => by simpa only [Real.norm_eq_abs, Function.uncurry] using hb p.1 p.2) | |
| theorem memLp_sample {Ω : Type*} [mΩ : MeasurableSpace Ω] | |
| {F : Filtration ℝ≥0 mΩ} {u : ℝ≥0 → Ω → ℝ} | |
| (hu : StronglyAdapted F u) (hc : ∀ ω, Continuous (fun t => u t ω)) | |
| (ν : Measure ℝ≥0) (P : Measure Ω) [IsFiniteMeasure ν] [IsFiniteMeasure P] | |
| (C : ℝ) (hb : ∀ t ω, |u t ω| ≤ C) (n : ℕ) : | |
| MemLp (Function.uncurry (sample u n)) 2 (ν.prod P) := | |
| MemLp.of_bound (stronglyMeasurable_sample hu hc n).aestronglyMeasurable C | |
| (Filter.Eventually.of_forall fun p => by | |
| simpa only [Real.norm_eq_abs, Function.uncurry, sample] using hb (left n p.1) p.2) | |
| theorem tendsto_integral_sq_sample_sub | |
| {A Ω : Type*} [MeasurableSpace A] [mΩ : MeasurableSpace Ω] | |
| {F : Filtration ℝ≥0 mΩ} {u : ℝ≥0 → Ω → ℝ} | |
| (hu : StronglyAdapted F u) (hc : ∀ ω, Continuous (fun t => u t ω)) | |
| (ν : Measure A) (P : Measure Ω) [IsFiniteMeasure ν] [IsFiniteMeasure P] | |
| (τ : A → ℝ≥0) (hτ : Measurable τ) | |
| (C : ℝ) (hC : 0 ≤ C) (hb : ∀ t ω, |u t ω| ≤ C) : | |
| Tendsto (fun n => ∫ p : A × Ω, | |
| (sample u n (τ p.1) p.2 - u (τ p.1) p.2) ^ 2 ∂ν.prod P) atTop (𝓝 0) := by | |
| have hcoord : Measurable (fun p : A × Ω => (τ p.1, p.2)) := | |
| (hτ.comp measurable_fst).prodMk measurable_snd | |
| have huM := (stronglyMeasurable_uncurry hu hc).comp_measurable hcoord | |
| have hsM (n : ℕ) := (stronglyMeasurable_sample hu hc n).comp_measurable hcoord | |
| have hlim := tendsto_integral_of_dominated_convergence | |
| (μ := ν.prod P) (F := fun n p => | |
| (sample u n (τ p.1) p.2 - u (τ p.1) p.2) ^ 2) | |
| (f := fun _ => (0 : ℝ)) (fun _ => (2 * C) ^ 2) | |
| (fun n => ((hsM n).sub huM).pow 2 |>.aestronglyMeasurable) | |
| (integrable_const _) ?_ ?_ | |
| · simpa only [integral_zero] using hlim | |
| · intro n | |
| exact Filter.Eventually.of_forall fun p => by | |
| have hn : |sample u n (τ p.1) p.2 - u (τ p.1) p.2| ≤ 2 * C := by | |
| calc | |
| _ ≤ |sample u n (τ p.1) p.2| + |u (τ p.1) p.2| := abs_sub _ _ | |
| _ ≤ 2 * C := by | |
| dsimp [sample] | |
| linarith [hb (left n (τ p.1)) p.2, hb (τ p.1) p.2] | |
| simpa only [Real.norm_eq_abs, abs_sq, sq_abs] using | |
| (sq_le_sq₀ (abs_nonneg _) (mul_nonneg (by norm_num) hC)).mpr hn | |
| · exact Filter.Eventually.of_forall fun p => by | |
| simpa only [sub_self, zero_pow (by decide : 2 ≠ 0)] using | |
| ((sample_tendsto hc (τ p.1) p.2).sub | |
| (tendsto_const_nhds (x := u (τ p.1) p.2))).pow 2 | |
| theorem tendsto_time_integral_sq_sample_sub | |
| {Ω : Type*} [mΩ : MeasurableSpace Ω] | |
| {F : Filtration ℝ≥0 mΩ} {u : ℝ≥0 → Ω → ℝ} | |
| (hu : StronglyAdapted F u) (hc : ∀ ω, Continuous (fun t => u t ω)) | |
| (P : Measure Ω) [IsFiniteMeasure P] (T C : ℝ) | |
| (hC : 0 ≤ C) (hb : ∀ t ω, |u t ω| ≤ C) : | |
| Tendsto (fun n => ∫ p : ℝ × Ω, | |
| (sample u n p.1.toNNReal p.2 - u p.1.toNNReal p.2) ^ 2 | |
| ∂(volume.restrict (Icc 0 T)).prod P) atTop (𝓝 0) := | |
| tendsto_integral_sq_sample_sub hu hc (volume.restrict (Icc 0 T)) P | |
| Real.toNNReal (by fun_prop) C hC hb | |
| end Problem367.PredictableMesh | |
| end | |
| end OAI | |