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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 | import OAI.Probability.DirectionalWalk.ShiftedBlocks
namespace OAI
open MeasureTheory ProbabilityTheory Filter Preorder
open scoped ENNReal BigOperators Topology
namespace DirectionalZeroOne
open scoped Classical
def commonBandContact {d : ℕ} (e : Step d) (n : ℕ) : Set (TwoTape (Word d)) :=
{Z | ∃ h : ℕ, commonDepth (placedWidth e) Z (4*n) < h ∧
h ≤ commonDepth (placedWidth e) Z (8*n) ∧ contactDepth e Z h}
lemma measurableSet_commonBandContact {d : ℕ} (e : Step d) (n : ℕ) :
MeasurableSet (commonBandContact e n) := by
simp only [commonBandContact,Set.ofPred_exists]
exact MeasurableSet.iUnion (fun h =>
(measurableSet_lt (measurable_commonDepth (placedWidth e) (4*n)) measurable_const).inter
((measurableSet_le measurable_const (measurable_commonDepth (placedWidth e) (8*n))).inter
(measurableSet_contactDepth e h)))
lemma commonBand_probability_le_experiment {d : ℕ} (e : Step d)
(ν : Bool → Measure (Word d)) [∀ b, IsProbabilityMeasure (ν b)]
(hreg : ∀ b, ∀ᵐ a ∂ν b, RegenerationWord (axisDirection (placedAxis e b)) a)
(he : ∀ᵐ Z ∂twoTapeLaw ν, ∃ H, 0 < H ∧ Z ∈ commonCut (placedWidth e) H)
(n : ℕ) [NeZero n] :
twoTapeLaw ν (commonBandContact e n) ≤
experimentLaw ν (placedWidth e) (fun _ => wordEnd) n (experimentContact e) := by
have : Countable (Word d) := inferInstance
have : Countable (TapeList (Word d)) := inferInstance
have : Countable (ThreeLists (Word d)) := inferInstance
have : Countable (TwoTapeList (Word d)) := inferInstance
let L := placedWidth e
have hL : ∀ b, ∀ᵐ a ∂ν b, 0 < L b a := by
intro b
filter_upwards [hreg b] with a ha
exact slabRecords_pos _ _ ha
rw [experimentLaw_fourWindows ν L hL he,Measure.map_apply (measurable_of_countable _)
(Set.to_countable _).measurableSet]
apply fourWindow_ge
intro u r s t
rw [← consecutiveFour_law ν L hL he,Measure.map_apply
(measurable_consecutiveFour L _ _ _ _) (Set.to_countable _).measurableSet]
apply measure_mono_ae
filter_upwards [ae_twoTape_prop ν _ (fun b => (Set.to_countable _).measurableSet) hreg,
ae_common_iterates ν L hL he] with Z hZ hi
intro hc
obtain ⟨h,ha,hb,hc⟩ := hc
have hm := (commonDepth_strictMono L Z hi).monotone
apply commonBand_experiment_contact e Z hZ hi (n+u) (n+r) (7*n+s) (n+t) h
· exact (hm (by have := u.isLt; have := r.isLt; omega)).trans_lt ha
· exact hb.trans (hm (by omega))
· exact hc
lemma reversed_cut_endpoint_height {d : ℕ} (e : Step d) (Z : ℕ → Word d)
(hZ : ∀ i, RegenerationWord (axisDirection e) (Z i)) (N : ℕ)
(hC : Z ∈ tapeCut (axisDirection e) N) :
axisHeight e (wordEnd (concatenateList (reverseTapeList
(cutList (slabRecords (axisDirection e)) N Z)))-stepVector e) = (N : ℤ)-1 := by
rw [sub_eq_add_neg,axisHeight_add,axisHeight_neg,axisHeight_step_self,
concatenateList_end,listTotal_reverse,axisHeight_listTotal e false (cutList (slabRecords (axisDirection e)) N Z) (fun i => hZ i)]
change (listHeight (slabRecords (axisDirection e)) (cutList (slabRecords (axisDirection e)) N Z) : ℤ)+ -1 = _
have hh : listHeight (slabRecords (axisDirection e)) (cutList (slabRecords (axisDirection e)) N Z) = N := by
rw [cutList,listHeight_prefix]
exact cutListIndex_spec _ _ Z (fun i => slabRecords_pos _ _ (hZ i)) hC
rw [hh]
rfl
lemma reversed_cut_endpoint_cube {d : ℕ} (e : Step d) (Z : ℕ → Word d)
(hZ : ∀ i, RegenerationWord (axisDirection e) (Z i)) (N : ℕ)
(hC : Z ∈ tapeCut (axisDirection e) N)
(hR : ∑ j ∈ Finset.range N, slabRadius (Z j) < (N : ℝ)^2) :
∃ q, wordEnd (concatenateList (reverseTapeList (cutList (slabRecords (axisDirection e)) N Z))) -
stepVector e = axisCubeEnumeration e N (N^2+1) q := by
have hbd (i : Fin d) : |(wordEnd (concatenateList (reverseTapeList (cutList (slabRecords (axisDirection e)) N Z)))-stepVector e) i| ≤ ((N^2+1 : ℕ) : ℤ) := by
have hh := (reversed_cut_endpoint_radius e Z hZ N hC i).trans hR.le
have hv : |((stepVector e i : ℤ) : ℝ)| ≤ 1 := by
dsimp [stepVector]
split_ifs <;> norm_num
have habs : |((wordEnd (concatenateList (reverseTapeList (cutList (slabRecords (axisDirection e)) N Z))) i : ℝ)) - (stepVector e i : ℝ)| ≤ |((wordEnd (concatenateList (reverseTapeList (cutList (slabRecords (axisDirection e)) N Z))) i : ℝ))| + |(stepVector e i : ℝ)| := by
simpa only [sub_eq_add_neg,abs_neg] using abs_add_le ((wordEnd (concatenateList (reverseTapeList (cutList (slabRecords (axisDirection e)) N Z))) i : ℝ)) (-(stepVector e i : ℝ))
have ht := habs.trans (add_le_add hh hv)
exact_mod_cast ht
obtain ⟨q,hq⟩ := axisCubeEnumeration_surjective e N (N^2+1)
(wordEnd (concatenateList (reverseTapeList (cutList (slabRecords (axisDirection e)) N Z)))-stepVector e)
(reversed_cut_endpoint_height e Z hZ N hC) hbd
exact ⟨q,hq.symm⟩
lemma coveredDepthBand_lower {d : ℕ} (μ : Measure (Row d)) [IsProbabilityMeasure μ]
(hell : StrictEllipticity μ) (e : Step d)
(hp : ∀ b, 0 < annealed μ 0 (nonBacktracking (axisDirection (placedAxis e b))))
(s N low high m : ℕ) (hs : 0 < s) (hsN : s ≤ N) (hls : low ≤ s) (hsh : s < high)
(hΔ : axisReachProb μ e (s-low)-axisReachProb μ e s ≤ ENNReal.ofReal ((1/2 : ℝ)^m)) :
letI : ∀ b, IsProbabilityMeasure (slabLaw μ (axisDirection (placedAxis e b))) :=
fun b => slabLaw_probability μ _ (hp b)
(annealed μ 0 (nonBacktracking (axisDirection e)))^2 ≤
twoTapeLaw (fun b => slabLaw μ (axisDirection (placedAxis e b))) (coveredDepthBand e low high) +
(∫⁻ a, ENNReal.ofReal (slabRadius a) ∂slabLaw μ (axisDirection e)) / (N : ℝ≥0∞) +
(annealed μ 0 (nonBacktracking (axisDirection e)))⁻¹ *
(annealed μ 0 (nonBacktracking (axisDirection (oppositeStep e))))⁻¹ *
ENNReal.ofReal ((24 / posteriorExponent)*Real.log (Fintype.card (AxisCubeSite e N (N^2+1))+1)*(m+1)^2*(1/2 : ℝ)^m) := by
let : ∀ b, IsProbabilityMeasure (slabLaw μ (axisDirection (placedAxis e b))) :=
fun b => slabLaw_probability μ _ (hp b)
let := slabLaw_probability μ (axisDirection e) (hp false)
let := conditioned_probability μ (axisDirection (oppositeStep e)) (hp true)
have hN : 0 < N := lt_of_lt_of_le hs hsN
have : Nonempty (AxisCubeSite e N (N^2+1)) := axisCubeSite_nonempty e N (N^2+1) hN (by nlinarith)
have : NeZero (Fintype.card (AxisCubeSite e N (N^2+1))) := ⟨Fintype.card_ne_zero⟩
let A := Measure.infinitePi (fun _ : ℕ => slabLaw μ (axisDirection e))
let B := conditioned μ (axisDirection (oppositeStep e))
let M := A.prod B
let R : Set ((ℕ → Word d) × Path d) := {p | (N : ℝ)^2 ≤ ∑ j ∈ Finset.range N, slabRadius (p.1 j)}
let C := mixedTapes e ⁻¹' coveredDepthBand e low high
let F : Set ((ℕ → Word d) × Path d) := {p | p.1 ∈ tapeCut (axisDirection e) N ∧
(concatenateList (reverseTapeList (cutList (slabRecords (axisDirection e)) N p.1)),p.2) ∈
finiteLateEndpointEvent e (N-s) (N-low) N (axisCubeEnumeration e N (N^2+1))}
have hM : M (Prod.fst ⁻¹' (tapeCut (axisDirection e) s ∩ tapeCut (axisDirection e) N)) ≤ M C+M R+M F := by
refine (measure_mono_ae (t := C ∪ R ∪ F) ?_).trans ((measure_union_le _ _).trans (add_le_add (measure_union_le _ _) le_rfl))
filter_upwards [Measure.quasiMeasurePreserving_fst.ae
(ae_tape_prop (slabLaw μ (axisDirection e)) (RegenerationWord (axisDirection e))
(Set.to_countable _).measurableSet (ae_slabLaw_regeneration μ hell _ (axisDirection_ne_zero e) (hp false))),
Measure.quasiMeasurePreserving_snd.ae
(ae_goodSlabPath μ hell _ (axisDirection_ne_zero _) (hp true))] with p hpos hneg
intro hcuts
by_cases hc : p ∈ C
· exact Or.inl (Or.inl hc)
by_cases hr : p ∈ R
· exact Or.inl (Or.inr hr)
apply Or.inr
refine ⟨hcuts.2,?_,two_cuts_uncovered_mixed e p hpos hneg s N low high hs hsN hls hsh hcuts.1 hcuts.2 hc⟩
exact reversed_cut_endpoint_cube e p.1 hpos N hcuts.2 (lt_of_not_ge hr)
have hC : M C = twoTapeLaw (fun b => slabLaw μ (axisDirection (placedAxis e b))) (coveredDepthBand e low high) := by
rw [← mixedTapes_law μ hell e hp,Measure.map_apply (measurable_mixedTapes e) (measurableSet_coveredDepthBand e low high)]
have hR : M R ≤ (∫⁻ a, ENNReal.ofReal (slabRadius a) ∂slabLaw μ (axisDirection e)) / (N : ℝ≥0∞) := by
change (A.prod B) (Prod.fst ⁻¹' {Z : ℕ → Word d | (N : ℝ)^2 ≤ ∑ j ∈ Finset.range N, slabRadius (Z j)}) ≤ _
have hm : MeasurableSet {Z : ℕ → Word d | (N : ℝ)^2 ≤ ∑ j ∈ Finset.range N, slabRadius (Z j)} :=
measurableSet_le measurable_const (Finset.measurable_sum _ (fun j _ =>
(measurable_of_countable slabRadius).comp (measurable_pi_apply j)))
rw [(measurePreserving_fst (μ := A) (ν := B)).measure_preimage hm.nullMeasurableSet]
exact iid_radius_large _ N (lt_of_lt_of_le hs hsN)
have hF := mixed_bridge_late_bound μ hell e (hp false) (N-s) (N-low) N m
(by omega) (by omega) (axisCubeEnumeration e N (N^2+1))
(axisCubeEnumeration_height e N (N^2+1)) (axisCubeEnumeration_injective e N (N^2+1))
(by simpa only [show N-low-(N-s) = s-low by omega,show N-(N-s) = s by omega] using hΔ)
have hcuts := axis_two_cuts_lower μ hell e (hp false) s N hsN
have hfrom : M (Prod.fst ⁻¹' (tapeCut (axisDirection e) s ∩ tapeCut (axisDirection e) N)) =
A (tapeCut (axisDirection e) s ∩ tapeCut (axisDirection e) N) := by
exact measurePreserving_fst.measure_preimage ((measurableSet_tapeCut _ _).inter (measurableSet_tapeCut _ _)).nullMeasurableSet
rw [hfrom,hC] at hM
exact hcuts.trans (hM.trans (add_le_add (add_le_add le_rfl hR) hF))
lemma commonDepth_markov {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)
(hi : Integrable (fun Z => (firstCommonWidth L Z : ℝ)) (twoTapeLaw ν))
(r low : ℕ) (hlow : 0 < low) :
twoTapeLaw ν {Z | low ≤ commonDepth L Z r} ≤
(r : ℝ≥0∞) * ENNReal.ofReal (∫ Z, (firstCommonWidth L Z : ℝ) ∂twoTapeLaw ν) / (low : ℝ≥0∞) := by
have hh := meas_ge_le_lintegral_div (μ := twoTapeLaw ν)
(((measurable_of_countable (fun n : ℕ => (n : ℝ≥0∞))).comp (measurable_commonDepth L r)).aemeasurable)
(ε := (low : ℝ≥0∞)) (by exact_mod_cast Nat.ne_zero_of_lt hlow) (by finiteness)
have hs : {Z : TwoTape α | (low : ℝ≥0∞) ≤ (commonDepth L Z r : ℝ≥0∞)} =
{Z | low ≤ commonDepth L Z r} := by ext Z;simp
simp only [Function.comp_def] at hh
rw [hs] at hh
have heq : (∫⁻ Z, (commonDepth L Z r : ℝ≥0∞) ∂twoTapeLaw ν) =
(r : ℝ≥0∞)*ENNReal.ofReal (∫ Z, (firstCommonWidth L Z : ℝ) ∂twoTapeLaw ν) := by
have hv := ofReal_integral_eq_lintegral_ofReal (commonDepth_integrable ν L hL he hi r)
(Filter.Eventually.of_forall (fun Z => Nat.cast_nonneg (commonDepth L Z r)))
simp only [ENNReal.ofReal_natCast] at hv
rw [← hv,commonDepth_integral ν L hL he hi,ENNReal.ofReal_mul (Nat.cast_nonneg r),ENNReal.ofReal_natCast]
exact hh.trans_eq (congrArg (fun x => x/(low : ℝ≥0∞)) heq)
end DirectionalZeroOne
end OAI
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