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dd14c26 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 | import OAI.Probability.Localization.History
namespace OAI
open MeasureTheory ProbabilityTheory Set
open scoped BigOperators ENNReal
namespace RepeatedLocalization
namespace FiniteEntropy
lemma volume_cellSet_ne_zero {n : ℕ} (hn : 0 < n) (Q : Cell n) : volume (cellSet Q) ≠ 0 := by
rw [volume_cellSet]
exact ENNReal.ofReal_ne_zero_iff.mpr (pow_pos (cellWidth_pos hn) _)
lemma volume_cellSet_ne_top {n : ℕ} (Q : Cell n) : volume (cellSet Q) ≠ ∞ := by
rw [volume_cellSet]
exact ENNReal.ofReal_ne_top
lemma uniformCell_probability {n : ℕ} (hn : 0 < n) (Q : Cell n) : IsProbabilityMeasure (uniformCell Q) :=
ProbabilityTheory.cond_isProbabilityMeasure_of_finite (volume_cellSet_ne_zero hn Q) (volume_cellSet_ne_top Q)
lemma uniformCell_self {n : ℕ} (hn : 0 < n) (Q : Cell n) : uniformCell Q (cellSet Q) = 1 :=
ProbabilityTheory.cond_apply_self (volume_cellSet_ne_zero hn Q) (volume_cellSet_ne_top Q)
lemma uniformCell_ae_mem {n : ℕ} (Q : Cell n) : ∀ᵐ z ∂uniformCell Q,z ∈ cellSet Q :=
ProbabilityTheory.ae_cond_mem (measurableSet_cellSet Q)
lemma ae_mem_cell_of_ac {n : ℕ} {P : Measure (Coordinate n)} {Q : Cell n}
(hac : P ≪ uniformCell Q) : ∀ᵐ z ∂P,z ∈ cellSet Q := hac.ae_le (uniformCell_ae_mem Q)
lemma uniformCell_cond {n : ℕ} {Q R : Cell n} (hQR : Descendant Q R) :
ProbabilityTheory.cond (uniformCell Q) (cellSet R) = uniformCell R := by
unfold uniformCell
rw [ProbabilityTheory.cond_cond_eq_cond_inter' (measurableSet_cellSet Q)
(measurableSet_cellSet R) (volume_cellSet_ne_top Q),Set.inter_eq_right.mpr hQR.2]
lemma uniformCell_eq_cond_prior {n : ℕ} (hn : 0 < n) (Q : Cell n) :
uniformCell Q = ProbabilityTheory.cond (cubePrior n) (cellSet Q) := by
unfold cubePrior uniformCell
rw [ProbabilityTheory.cond_cond_eq_cond_inter' (measurable_initialCube n)
(measurableSet_cellSet Q) (volume_initialCube_ne_top n),
Set.inter_eq_right.mpr (cellSet_subset_initialCube hn Q)]
lemma uniformCell_descendant_mass {n : ℕ} (hn : 0 < n) {Q R : Cell n} (hQR : Descendant Q R) :
uniformCell Q (cellSet R) = ENNReal.ofReal ((1/2:ℝ)^(n*(R.1-Q.1))) := by
rw [uniformCell_eq_cond_prior hn Q,ProbabilityTheory.cond_apply (measurableSet_cellSet Q),
Set.inter_eq_right.mpr hQR.2,cubePrior_cellSet hn Q,cubePrior_cellSet hn R]
rw [← ENNReal.ofReal_inv_of_pos (by positivity : 0 < (1/2:ℝ)^(n*Q.1)),
← ENNReal.ofReal_mul (by positivity)]
congr 1
have he : n*R.1 = n*Q.1+n*(R.1-Q.1) := by
rw [← Nat.mul_add,Nat.add_sub_of_le hQR.1]
rw [he,pow_add]
exact inv_mul_cancel_left₀ (pow_ne_zero _ (by norm_num : (1 / 2 : ℝ) ≠ 0)) _
lemma uniformCell_descendant_pos {n : ℕ} (hn : 0 < n) {Q R : Cell n} (hQR : Descendant Q R) :
uniformCell Q (cellSet R) ≠ 0 := by
rw [uniformCell_descendant_mass hn hQR]
exact ENNReal.ofReal_ne_zero_iff.mpr (by positivity)
lemma uniformCell_log_mass {n : ℕ} (hn : 0 < n) {Q R : Cell n} (hQR : Descendant Q R) :
Real.log ((uniformCell Q (cellSet R)).toReal) = -(n:ℝ)*((R.1-Q.1:ℕ):ℝ)*Real.log 2 := by
rw [uniformCell_descendant_mass hn hQR,ENNReal.toReal_ofReal (by positivity),Real.log_pow]
simp only [one_div,Real.log_inv,Nat.cast_mul]
ring
lemma cond_congr_ae {α : Type*} [MeasurableSpace α] (P : Measure α) {E F : Set α}
(h : E =ᵐ[P] F) : ProbabilityTheory.cond P E = ProbabilityTheory.cond P F := by
unfold ProbabilityTheory.cond
rw [measure_congr h,Measure.restrict_congr_set h]
lemma run_fiber_ae_event {n : ℕ} (hn : 0 < n) (P : Measure (Coordinate n)) (Q : Cell n)
(hroot : ∀ᵐ z ∂P,z ∈ cellSet Q) (N : ℕ) {z : Coordinate n} (hz : z ∈ cellSet Q) :
{w | run P Q N w = run P Q N z} =ᵐ[P] (state P Q (run P Q N z)).event := by
filter_upwards [hroot] with w hw
exact propext (run_eq_iff_mem hn P Q N hz hw)
lemma terminal_fiber_ae_event {n : ℕ} (hn : 0 < n) (P : Measure (Coordinate n)) (Q : Cell n)
(hroot : ∀ᵐ z ∂P,z ∈ cellSet Q) {z : Coordinate n} (hz : z ∈ cellSet Q) (ht : Terminates P Q z) :
{w | terminalLog P Q w = terminalLog P Q z} =ᵐ[P] (state P Q (terminalLog P Q z)).event := by
have he := terminalLog_fiber_inter_initial hn P Q hz ht
filter_upwards [hroot] with w hw
apply propext
constructor
· intro hm
rw [← he]
exact ⟨hw,hm⟩
· intro hm
rw [← he] at hm
exact hm.2
end FiniteEntropy
end RepeatedLocalization
end OAI
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