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f61e547 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 | import OAI.Probability.DirectionalWalk.RenewalFactorization
namespace OAI
open MeasureTheory ProbabilityTheory Filter Preorder
open scoped ENNReal BigOperators Topology
namespace DirectionalZeroOne
open scoped Classical
def InCountWindow {α : Type*} (L : Bool → α → ℕ) (k n : ℕ) (a : TwoTapeList α) : Prop :=
BalancedList L a ∧ k ≤ listCommonCount L a ∧ listCommonCount L a < k+n
lemma windowCommonLaw_atom {α : Type*} [Countable α] [MeasurableSpace α]
[MeasurableSingletonClass α] (ν : Bool → Measure α) [∀ b, IsProbabilityMeasure (ν b)]
(L : Bool → α → ℕ) (hL : ∀ b, ∀ᵐ a ∂ν b, 0 < L b a)
(he : ∀ᵐ Z ∂twoTapeLaw ν, ∃ H, 0 < H ∧ Z ∈ commonCut L H)
(k n : ℕ) (a : TwoTapeList α) :
windowCommonLaw ν L k n {a} =
(by classical exact if InCountWindow L k n a then (n : ℝ≥0∞)⁻¹ * twoListWeight ν a else 0) := by
classical
rw [windowCommonLaw,Measure.smul_apply,smul_eq_mul,Measure.finsetSum_apply]
simp_rw [nthCommonLaw_atom ν L hL he]
by_cases ha : InCountWindow L k n a
· rw [ite_eq_left ha]
congr 1
let i : Fin n := ⟨listCommonCount L a-k,by obtain ⟨_,_,_⟩ := ha;omega⟩
rw [Finset.sum_eq_single i]
· exact ite_eq_left ⟨ha.1,by dsimp [i];obtain ⟨_,_,_⟩ := ha;omega⟩
· intro j _ hji
apply ite_eq_right
rintro ⟨_,hj⟩
apply hji
apply Fin.ext
dsimp [i]
omega
· simp
· rw [ite_eq_right ha]
have hh : ∑ i : Fin n, (if BalancedList L a ∧ listCommonCount L a = k+↑i then
(∏ b : Bool, ∏ j : Fin (a b).1, ν b {(a b).2 j}) else 0) = 0 := by
apply Finset.sum_eq_zero
intro i _
apply ite_eq_right
rintro ⟨hb,hc⟩
exact ha ⟨hb,by omega,by omega⟩
rw [hh,mul_zero]
lemma windowCommonLaw_ae_window {α : Type*} [Countable α] [MeasurableSpace α]
[MeasurableSingletonClass α] (ν : Bool → Measure α) [∀ b, IsProbabilityMeasure (ν b)]
(L : Bool → α → ℕ) (hL : ∀ b, ∀ᵐ a ∂ν b, 0 < L b a)
(he : ∀ᵐ Z ∂twoTapeLaw ν, ∃ H, 0 < H ∧ Z ∈ commonCut L H)
(k n : ℕ) : ∀ᵐ a ∂windowCommonLaw ν L k n, InCountWindow L k n a := by
classical
filter_upwards [ae_atom_pos (windowCommonLaw ν L k n)] with a ha
by_contra hn
rw [windowCommonLaw_atom ν L hL he,ite_eq_right hn] at ha
exact ha rfl
noncomputable def pairBridgeLaw {α : Type*} [MeasurableSpace α]
(ν : Bool → Measure α) [∀ b, IsProbabilityMeasure (ν b)]
(L : Bool → α → ℕ) (H : ℕ) : Measure (TwoTapeList α) :=
Measure.pi (fun b => bridgeListLaw (ν b) (L b) H)
lemma pairBridgeLaw_probability {α : Type*} [Countable α] [MeasurableSpace α]
[MeasurableSingletonClass α] (ν : Bool → Measure α) [∀ b, IsProbabilityMeasure (ν b)]
(L : Bool → α → ℕ) (H : ℕ)
(hH : ∀ b, Measure.infinitePi (fun _ : ℕ => ν b) (renewalCut (L b) H) ≠ 0) :
IsProbabilityMeasure (pairBridgeLaw ν L H) := by
let : ∀ b, IsProbabilityMeasure (bridgeListLaw (ν b) (L b) H) :=
fun b => bridgeListLaw_probability (ν b) (L b) H (hH b)
unfold pairBridgeLaw
infer_instance
instance bridgeListLaw_finite {α : Type*} [Countable α] [MeasurableSpace α]
[MeasurableSingletonClass α] (ν : Measure α) [IsProbabilityMeasure ν]
(L : α → ℕ) (H : ℕ) : IsFiniteMeasure (bridgeListLaw ν L H) := by
constructor
rw [bridgeListLaw,Measure.smul_apply,smul_eq_mul,bridgeListMeasure_mass]
exact (ENNReal.inv_mul_le_one _).trans_lt (by simp)
lemma pairBridgeLaw_atom {α : Type*} [Countable α] [MeasurableSpace α]
[MeasurableSingletonClass α] (ν : Bool → Measure α) [∀ b, IsProbabilityMeasure (ν b)]
(L : Bool → α → ℕ) (hL : ∀ b, ∀ᵐ a ∂ν b, 0 < L b a) (H : ℕ) (a : TwoTapeList α)
(ha : ∀ b, listHeight (L b) (a b) = H) :
pairBridgeLaw ν L H {a} =
(∏ b : Bool, Measure.infinitePi (fun _ : ℕ => ν b) (renewalCut (L b) H))⁻¹ * twoListWeight ν a := by
classical
have hs : ({a} : Set (TwoTapeList α)) = Set.univ.pi (fun b => {a b}) := by
ext x
simp only [Set.mem_singleton_iff,Set.mem_pi,Set.mem_univ,true_implies]
exact funext_iff
rw [pairBridgeLaw,hs,Measure.pi_pi]
simp_rw [bridgeListLaw_atom (ν _) (L _) (hL _) H _ (ha _)]
rw [Finset.prod_mul_distrib]
change _ * twoListWeight ν a = _ * twoListWeight ν a
congr 1
simp only [Fintype.prod_bool]
exact (ENNReal.mul_inv (Or.inr (measure_ne_top _ _)) (Or.inl (measure_ne_top _ _))).symm
noncomputable def chunkHeight {α : Type*} (L : Bool → α → ℕ) (a : TwoTapeList α) : ℕ :=
listHeight (L false) (a false)
noncomputable def pairBridgeKernel {α : Type*} [Countable α] [MeasurableSpace α]
[MeasurableSingletonClass α] (ν : Bool → Measure α) [∀ b, IsProbabilityMeasure (ν b)]
(L : Bool → α → ℕ) : Kernel ℕ (TwoTapeList α) where
toFun := pairBridgeLaw ν L
measurable' := measurable_of_countable _
lemma pairBridgeKernel_markov {α : Type*} [Countable α] [MeasurableSpace α]
[MeasurableSingletonClass α] (ν : Bool → Measure α) [∀ b, IsProbabilityMeasure (ν b)]
(L : Bool → α → ℕ)
(hu : ∀ H b, Measure.infinitePi (fun _ : ℕ => ν b) (renewalCut (L b) H) ≠ 0) :
IsMarkovKernel (pairBridgeKernel ν L) :=
⟨fun H => pairBridgeLaw_probability ν L H (hu H)⟩
lemma map_graph_atom {α β : Type*} [Countable α] [MeasurableSpace α] [MeasurableSpace β]
[MeasurableSingletonClass α] [MeasurableSingletonClass β]
(μ : Measure α) (f : α → β) (x : α) : μ.map (fun a => (f a,a)) {(f x,x)} = μ {x} := by
rw [Measure.map_apply (measurable_of_countable _) (measurableSet_singleton _)]
congr 1
ext a
simp only [Set.mem_preimage,Set.mem_singleton_iff,Prod.mk.injEq,and_iff_right_iff_imp]
rintro rfl
rfl
lemma map_graph_fst {α β : Type*} [Countable α] [MeasurableSpace α] [MeasurableSpace β]
[MeasurableSingletonClass α] [MeasurableSingletonClass β]
(μ : Measure α) (f : α → β) : (μ.map (fun a => (f a,a))).fst = μ.map f := by
rw [Measure.fst,Measure.map_map measurable_fst (measurable_of_countable _)]
rfl
lemma window_reference_ac {α : Type*} [Countable α] [MeasurableSpace α]
[MeasurableSingletonClass α] (ν : Bool → Measure α) [∀ b, IsProbabilityMeasure (ν b)]
(L : Bool → α → ℕ) (hL : ∀ b, ∀ᵐ a ∂ν b, 0 < L b a)
(he : ∀ᵐ Z ∂twoTapeLaw ν, ∃ H, 0 < H ∧ Z ∈ commonCut L H)
(k n : ℕ) [NeZero n]
(hu : ∀ H b, Measure.infinitePi (fun _ : ℕ => ν b) (renewalCut (L b) H) ≠ 0) :
(windowCommonLaw ν L k n).map (fun a => (chunkHeight L a,a)) ≪
(windowCommonLaw ν L k n).map (chunkHeight L) ⊗ₘ pairBridgeKernel ν L := by
classical
let P := windowCommonLaw ν L k n
let H := chunkHeight L
let κ := pairBridgeKernel ν L
let : IsMarkovKernel κ := pairBridgeKernel_markov ν L hu
have hP : IsProbabilityMeasure P := inferInstance
apply absolutelyContinuous_of_atoms
rintro ⟨h,a⟩ hz
by_contra hap
have hg : h = H a := by
by_contra hne
apply hap
rw [Measure.map_apply (measurable_of_countable _) (measurableSet_singleton _)]
have hh : (fun a => (H a,a)) ⁻¹' {(h,a)} = ∅ := by
ext b
simp only [Set.mem_preimage,Set.mem_singleton_iff,Prod.mk.injEq,Set.mem_empty_iff_false,iff_false,not_and]
intro hh hb
exact hne (hb ▸ hh.symm)
rw [hh,measure_empty]
subst h
rw [map_graph_atom] at hap
have hw : InCountWindow L k n a := by
by_contra hn
exact hap (by rw [windowCommonLaw_atom ν L hL he,ite_eq_right hn])
have hwa : twoListWeight ν a ≠ 0 := by
intro hz
apply hap
rw [windowCommonLaw_atom ν L hL he,ite_eq_left hw,hz,mul_zero]
have hx : (P.map H) {H a} ≠ 0 := fun hh => hap (le_antisymm ((atom_le_map P H (measurable_of_countable _) a).trans hh.le) bot_le)
have hκ : κ (H a) {a} ≠ 0 := by
change pairBridgeLaw ν L (H a) {a} ≠ 0
rw [pairBridgeLaw_atom ν L hL (H a) a hw.1]
exact mul_ne_zero (ENNReal.inv_ne_zero.mpr (ENNReal.prod_ne_top (fun _ _ => measure_ne_top _ _))) hwa
exact (mul_ne_zero hx hκ) (by simpa only [compProd_atom] using hz)
end DirectionalZeroOne
end OAI
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