| import OAI.Probability.Localization.Cells |
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| namespace OAI |
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| open MeasureTheory ProbabilityTheory Set |
| open scoped BigOperators ENNReal |
|
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| namespace RepeatedLocalization |
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| namespace FiniteEntropy |
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| lemma Descendant.refl {n : ℕ} (Q : Cell n) : Descendant Q Q := ⟨le_rfl,Subset.rfl⟩ |
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| lemma Descendant.trans {n : ℕ} {Q R S : Cell n} (hQR : Descendant Q R) (hRS : Descendant R S) : |
| Descendant Q S := ⟨hQR.1.trans hRS.1,hRS.2.trans hQR.2⟩ |
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| lemma nested_of_point {n : ℕ} (hn : 0 < n) {Q R : Cell n} (h : Q.1 ≤ R.1) |
| {z : Coordinate n} (hzQ : z ∈ cellSet Q) (hzR : z ∈ cellSet R) : Descendant Q R := by |
| have ha := cellSet_subset_ancestor hn R Q.1 h |
| have h₁ := cellAt_eq_of_mem hn Q hzQ |
| have h₂ := cellAt_eq_of_mem hn (ancestorCell R Q.1 h) (ha hzR) |
| have he : ancestorCell R Q.1 h = Q := h₂.symm.trans h₁ |
| exact ⟨h,by simpa only [he] using ha⟩ |
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| lemma cell_eq_of_level_point {n : ℕ} (hn : 0 < n) {Q R : Cell n} (h : Q.1 = R.1) |
| {z : Coordinate n} (hzQ : z ∈ cellSet Q) (hzR : z ∈ cellSet R) : Q = R := by |
| have h₁ := cellAt_eq_of_mem hn Q hzQ |
| have h₂ := cellAt_eq_of_mem hn R hzR |
| rw [h] at h₁ |
| exact h₁.symm.trans h₂ |
|
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| lemma Descendant.eq_of_level {n : ℕ} (hn : 0 < n) {Q R : Cell n} |
| (h : Descendant Q R) (hlev : Q.1 = R.1) : Q = R := |
| cell_eq_of_level_point hn hlev (h.2 (cellCenter_mem hn R)) (cellCenter_mem hn R) |
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| lemma firstHeavy_disjoint {n : ℕ} (hn : 0 < n) (P : Measure (Coordinate n)) (Q : Cell n) |
| {R S : Cell n} (hR : FirstHeavy P Q R) (hS : FirstHeavy P Q S) (hne : R ≠ S) : |
| Disjoint (cellSet R) (cellSet S) := by |
| apply Set.disjoint_left.mpr |
| intro z hzR hzS |
| rcases le_total R.1 S.1 with h | h |
| · exact hne (hS.2 R hR.1 (nested_of_point hn h hzR hzS)) |
| · exact hne (hR.2 S hS.1 (nested_of_point hn h hzS hzR)).symm |
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| lemma exists_firstHeavy_ancestor {n : ℕ} (hn : 0 < n) (P : Measure (Coordinate n)) |
| {Q R : Cell n} (hR : Heavy P Q R) : |
| ∃ E : Cell n,FirstHeavy P Q E ∧ Descendant E R := by |
| classical |
| have hex : ∃ J : ℕ,∃ E : Cell n,Heavy P Q E ∧ Descendant E R ∧ E.1 = J := |
| ⟨R.1,R,hR,Descendant.refl R,rfl⟩ |
| obtain ⟨E,hE,hER,hlev⟩ := Nat.find_spec hex |
| refine ⟨E,⟨hE,?_⟩,hER⟩ |
| intro F hF hFE |
| have hmin := Nat.find_min' hex (show ∃ E : Cell n,Heavy P Q E ∧ Descendant E R ∧ E.1 = F.1 from |
| ⟨F,hF,hFE.trans hER,rfl⟩) |
| rw [← hlev] at hmin |
| exact hFE.eq_of_level hn (le_antisymm hFE.1 hmin) |
|
|
| lemma measurableSet_residual {n : ℕ} (P : Measure (Coordinate n)) (Q : Cell n) : |
| MeasurableSet (residual P Q) := |
| (measurableSet_cellSet Q).diff (MeasurableSet.iUnion (fun R => measurableSet_cellSet R.val)) |
|
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| lemma measurableSet_branch {n : ℕ} (P : Measure (Coordinate n)) (Q : Cell n) (a : Letter n) : |
| MeasurableSet (branch P Q a) := by |
| classical |
| cases a with |
| | none => exact measurableSet_residual P Q |
| | some R => by_cases h : FirstHeavy P Q R <;> simp [branch,h,measurableSet_cellSet] |
|
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| lemma branch_subset {n : ℕ} (P : Measure (Coordinate n)) (Q : Cell n) (a : Letter n) : |
| branch P Q a ⊆ cellSet Q := by |
| classical |
| cases a with |
| | none => exact Set.sdiff_subset |
| | some R => by_cases h : FirstHeavy P Q R <;> simp only [branch,h,ite_true,ite_false] |
| · exact h.1.1.2 |
| · exact Set.empty_subset _ |
|
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| lemma branch_disjoint {n : ℕ} (hn : 0 < n) (P : Measure (Coordinate n)) (Q : Cell n) |
| {a b : Letter n} (hab : a ≠ b) : Disjoint (branch P Q a) (branch P Q b) := by |
| classical |
| cases a with |
| | none => |
| cases b with |
| | none => exact False.elim (hab rfl) |
| | some R => |
| by_cases h : FirstHeavy P Q R |
| · simp only [branch,ite_eq_left h] |
| apply Set.disjoint_left.mpr |
| intro z hz hzR |
| exact hz.2 (Set.mem_iUnion.mpr ⟨⟨R,h⟩,hzR⟩) |
| · simp [branch,h] |
| | some R => |
| cases b with |
| | none => |
| by_cases h : FirstHeavy P Q R |
| · simp only [branch,ite_eq_left h] |
| apply Set.disjoint_right.mpr |
| intro z hz hzR |
| exact hz.2 (Set.mem_iUnion.mpr ⟨⟨R,h⟩,hzR⟩) |
| · simp [branch,h] |
| | some S => |
| by_cases hR : FirstHeavy P Q R |
| · by_cases hS : FirstHeavy P Q S |
| · simpa only [branch,ite_eq_left hR,ite_eq_left hS] using |
| firstHeavy_disjoint hn P Q hR hS (fun h => hab (congrArg some h)) |
| · simp [branch,hS] |
| · simp [branch,hR] |
|
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| lemma attemptLetter_mem_branch {n : ℕ} (P : Measure (Coordinate n)) (Q : Cell n) |
| {z : Coordinate n} (hz : z ∈ cellSet Q) : z ∈ branch P Q (attemptLetter P Q z) := by |
| classical |
| by_cases h : ∃ R : Cell n,FirstHeavy P Q R ∧ z ∈ cellSet R |
| · simp only [attemptLetter,dite_eq_left h,branch,ite_eq_left h.choose_spec.1] |
| exact h.choose_spec.2 |
| · simp only [attemptLetter,dite_eq_right h,branch,residual] |
| refine ⟨hz,?_⟩ |
| intro hm |
| obtain ⟨R,hR⟩ := Set.mem_iUnion.mp hm |
| exact h ⟨R.val,R.property,hR⟩ |
|
|
| lemma attemptLetter_eq_iff {n : ℕ} (hn : 0 < n) (P : Measure (Coordinate n)) (Q : Cell n) |
| {z : Coordinate n} (hz : z ∈ cellSet Q) (a : Letter n) : |
| attemptLetter P Q z = a ↔ z ∈ branch P Q a := by |
| constructor |
| · intro he; rw [← he]; exact attemptLetter_mem_branch P Q hz |
| · intro hm |
| by_contra he |
| exact Set.disjoint_left.mp (branch_disjoint hn P Q he) (attemptLetter_mem_branch P Q hz) hm |
|
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| lemma heavy_disjoint_residual {n : ℕ} (hn : 0 < n) (P : Measure (Coordinate n)) |
| {Q R : Cell n} (hR : Heavy P Q R) : Disjoint (residual P Q) (cellSet R) := by |
| obtain ⟨E,hE,hER⟩ := exists_firstHeavy_ancestor hn P hR |
| apply Set.disjoint_left.mpr |
| intro z hz hzR |
| exact hz.2 (Set.mem_iUnion.mpr ⟨⟨E,hE⟩,hER.2 hzR⟩) |
|
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| lemma stopped_regular {n : ℕ} (hn : 0 < n) (P : Measure (Coordinate n)) [IsProbabilityMeasure P] |
| (Q : Cell n) (hq : (1/2:ℝ) ≤ (P (residual P Q)).toReal) : |
| Regular (ProbabilityTheory.cond P (residual P Q)) Q := by |
| have hq0 : P (residual P Q) ≠ 0 := by |
| intro hzero |
| rw [hzero, ENNReal.toReal_zero] at hq |
| norm_num at hq |
| have hp := ProbabilityTheory.cond_isProbabilityMeasure (μ := P) hq0 |
| let := hp |
| refine ⟨?_,?_⟩ |
| · apply le_antisymm prob_le_one |
| have hself := ProbabilityTheory.cond_apply_self (μ := P) hq0 (measure_ne_top _ _) |
| rw [← hself] |
| exact measure_mono Set.sdiff_subset |
| · intro R hQR hlev |
| by_cases hR : Heavy P Q R |
| · have hd := heavy_disjoint_residual hn P hR |
| rw [ProbabilityTheory.cond_apply' (measurableSet_cellSet R),Set.disjoint_iff_inter_eq_empty.mp hd] |
| simp only [measure_empty,mul_zero,ENNReal.toReal_zero] |
| exact mul_nonneg (by norm_num) (Real.rpow_pos_of_pos (by norm_num) _).le |
| · have hm : (P (cellSet R)).toReal ≤ threshold n (R.1-Q.1) := by |
| exact le_of_not_gt (fun h => hR ⟨hQR,hlev,h⟩) |
| rw [ProbabilityTheory.cond_apply' (measurableSet_cellSet R),ENNReal.toReal_mul,ENNReal.toReal_inv] |
| have hinter : (P (residual P Q ∩ cellSet R)).toReal ≤ (P (cellSet R)).toReal := |
| ENNReal.toReal_mono (measure_ne_top _ _) (measure_mono Set.inter_subset_right) |
| have hpos : 0 < (P (residual P Q)).toReal := by linarith |
| have hinv : ((P (residual P Q)).toReal)⁻¹ ≤ 2 := by |
| apply (inv_le_comm₀ hpos (by norm_num)).mpr |
| simpa only [one_div] using hq |
| exact (mul_le_mul_of_nonneg_left (hinter.trans hm) (inv_nonneg.mpr hpos.le)).trans |
| (mul_le_mul_of_nonneg_right hinv (Real.rpow_pos_of_pos (by norm_num) _).le) |
|
|
| end FiniteEntropy |
|
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| end RepeatedLocalization |
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| end OAI |
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