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namespace OAI
open MeasureTheory ProbabilityTheory Filter Preorder
open scoped ENNReal BigOperators Topology
namespace DirectionalZeroOne
open scoped Classical
lemma property_of_atom {α : Type*} [MeasurableSpace α] (ν : Measure α)
(P : α → Prop) (hP : ∀ᵐ a ∂ν, P a) (a : α) (ha : ν {a} ≠ 0) : P a := by
by_contra h
apply ha
exact measure_mono_null (Set.singleton_subset_iff.mpr h) (ae_iff.mp hP)
lemma windowCommonLaw_support {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)
(P : Bool → α → Prop) (hP : ∀ b, ∀ᵐ a ∂ν b, P b a) (k n : ℕ) :
∀ᵐ a ∂windowCommonLaw ν L k n, InCountWindow L k n a ∧ ∀ b i, P b ((a b).2 i) := by
classical
filter_upwards [ae_atom_pos (windowCommonLaw ν L k n),windowCommonLaw_ae_window ν L hL he k n] with a ha hw
rw [windowCommonLaw_atom ν L hL he,ite_eq_left hw] at ha
have hp : twoListWeight ν a ≠ 0 := right_ne_zero_of_mul ha
refine ⟨hw,fun b i => property_of_atom (ν b) (P b) (hP b) _ ?_⟩
exact (Finset.prod_ne_zero_iff.mp (Finset.prod_ne_zero_iff.mp hp b (Finset.mem_univ b))) i (Finset.mem_univ i)
lemma listCommonCount_le_height {α : Type*} (L : Bool → α → ℕ) (a : TwoTapeList α) :
listCommonCount L a ≤ chunkHeight L a := by
classical
unfold listCommonCount chunkHeight
calc
_ ≤ ∑ _j ∈ Finset.range (listHeight (L false) (a false)), 1 := by
apply Finset.sum_le_sum
intro j hj
split_ifs <;> omega
_ = _ := by simp
lemma experimentLaw_fourWindows {α G : Type*} [Countable α] [MeasurableSpace α]
[MeasurableSingletonClass α] [AddCommGroup G] [Countable G] [MeasurableSpace G]
[MeasurableSingletonClass G] (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)
(D : Bool → α → G) (n : ℕ) :
experimentLaw ν L D n = ((windowCommonLaw ν L n n).prod (threeChunkLaw ν L n)).map
(fun p => ((threeHeight L p.2,pairTotal D p.1+signedTotal (D false) (splitThree p.2).1),splitThree p.2)) := by
rw [experimentLaw,← windowCommonLaw_marked ν L hL he D n n,signedChunkLaw,
Measure.map_prod_map _ _ (measurable_of_countable _) (measurable_of_countable _),
Measure.map_map (measurable_of_countable _) (measurable_of_countable _)]
congr 1
lemma axisHeight_listTotal {d : ℕ} (e : Step d) (b : Bool) (a : TapeList (Word d))
(ha : ∀ i, RegenerationWord (axisDirection (placedAxis e b)) (a.2 i)) :
axisHeight e (listTotal wordEnd a) = if b then -(listHeight (placedWidth e b) a : ℤ)
else (listHeight (placedWidth e b) a : ℤ) := by
rw [listTotal,axisHeight_sum]
cases b with
| false =>
simp only [Bool.false_eq_true,↓reduceIte,listHeight,Nat.cast_sum]
apply Finset.sum_congr rfl
intro i hi
exact axisHeight_wordEnd e _ (ha i)
| true =>
simp only [↓reduceIte,listHeight,Nat.cast_sum,← Finset.sum_neg_distrib]
apply Finset.sum_congr rfl
intro i hi
have h := axisHeight_wordEnd (oppositeStep e) _ (ha i)
rw [axisHeight_opposite] at h
exact neg_eq_iff_eq_neg.mp h
lemma axisHeight_pairTotal {d : ℕ} (e : Step d) (a : TwoTapeList (Word d))
(hb : BalancedList (placedWidth e) a)
(ha : ∀ b i, RegenerationWord (axisDirection (placedAxis e b)) ((a b).2 i)) :
axisHeight e (pairTotal (fun _ => wordEnd) a) = 0 := by
rw [pairTotal,axisHeight_add,axisHeight_listTotal e false _ (ha false),
axisHeight_listTotal e true _ (ha true)]
simp only [Bool.false_eq_true,↓reduceIte,hb true,add_neg_cancel]
lemma axisHeight_signedTotal {d : ℕ} (e : Step d) (a : ThreeChunk (Word d))
(ha : ∀ i, RegenerationWord (axisDirection e) ((a.1 false).2 i))
(hb : ∀ i, RegenerationWord (axisDirection e) ((a.2.1 false).2 i))
(hc : ∀ i, RegenerationWord (axisDirection e) ((a.2.2 false).2 i)) :
axisHeight e (signedTotal wordEnd (splitThree a).1) =
((threeHeight (placedWidth e) a).1+(threeHeight (placedWidth e) a).2.1+
(threeHeight (placedWidth e) a).2.2 : ℕ) := by
simp only [signedTotal,splitThree,axisHeight_add,axisHeight_listTotal e false _ ha,
axisHeight_listTotal e false _ hb,axisHeight_listTotal e false _ hc,Bool.false_eq_true,↓reduceIte]
simp only [threeHeight,chunkHeight,Nat.cast_add,add_assoc]
lemma experimentLaw_axis_support {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))))
(he : letI : ∀ b, IsProbabilityMeasure (slabLaw μ (axisDirection (placedAxis e b))) :=
fun b => slabLaw_probability μ _ (hp b)
∀ᵐ Z ∂twoTapeLaw (fun b => slabLaw μ (axisDirection (placedAxis e b))),
∃ H, 0 < H ∧ Z ∈ commonCut (placedWidth e) H)
(n : ℕ) [NeZero n] :
let : ∀ b, IsProbabilityMeasure (slabLaw μ (axisDirection (placedAxis e b))) :=
fun b => slabLaw_probability μ _ (hp b)
∀ᵐ x ∂experimentLaw (fun b => slabLaw μ (axisDirection (placedAxis e b))) (placedWidth e)
(fun _ => wordEnd) n,
n ≤ x.1.1.1 ∧ n ≤ x.1.1.2.2 ∧
axisHeight e x.1.2 = ((x.1.1.1+x.1.1.2.1+x.1.1.2.2 : ℕ) : ℤ) := by
dsimp only
let : ∀ b, IsProbabilityMeasure (slabLaw μ (axisDirection (placedAxis e b))) :=
fun b => slabLaw_probability μ _ (hp b)
have : Countable (Word d) := inferInstance
have : Countable (TapeList (Word d)) := inferInstance
have : Countable (TwoTapeList (Word d)) := inferInstance
have : Countable (ThreeChunk (Word d)) := inferInstance
have : Countable (ThreeLists (Word d)) := inferInstance
let ν := fun b => slabLaw μ (axisDirection (placedAxis e b))
have hs (b : Bool) : ∀ᵐ a ∂ν b, RegenerationWord (axisDirection (placedAxis e b)) a :=
ae_slabLaw_regeneration μ hell _ (axisDirection_ne_zero (placedAxis e b)) (hp b)
have hL (b : Bool) : ∀ᵐ a ∂ν b, 0 < placedWidth e b a := by
filter_upwards [hs b] with a ha
exact slabRecords_pos _ _ ha
have hw (k : ℕ) := windowCommonLaw_support ν (placedWidth e) hL he _ hs k n
rw [experimentLaw_fourWindows ν (placedWidth e) hL he,
ae_map_iff (measurable_of_countable _).aemeasurable (Set.to_countable _).measurableSet]
apply (Measure.ae_prod_iff_ae_ae (Set.to_countable _).measurableSet).mpr
filter_upwards [hw n] with a ha
change ∀ᵐ c ∂(windowCommonLaw ν (placedWidth e) n n).prod
((windowCommonLaw ν (placedWidth e) (7*n) n).prod (windowCommonLaw ν (placedWidth e) n n)), _
apply (Measure.ae_prod_iff_ae_ae (Set.to_countable _).measurableSet).mpr
filter_upwards [hw n] with b hb
apply (Measure.ae_prod_iff_ae_ae (Set.to_countable _).measurableSet).mpr
filter_upwards [hw (7*n)] with c hc
filter_upwards [hw n] with f hf
refine ⟨hb.1.2.1.trans (listCommonCount_le_height _ b),
hf.1.2.1.trans (listCommonCount_le_height _ f),?_⟩
rw [axisHeight_add,axisHeight_pairTotal e a ha.1.1 ha.2,zero_add]
exact axisHeight_signedTotal e (b,c,f) (hb.2 false) (hc.2 false) (hf.2 false)
lemma experiment_height_moment {α G : Type*} [Countable α] [MeasurableSpace α]
[MeasurableSingletonClass α] [AddCommGroup G] [Countable G] [MeasurableSpace G]
[MeasurableSingletonClass G] (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))
(D : Bool → α → G) (n : ℕ) [NeZero n] :
Integrable (fun x => ((x.1.1.1+x.1.1.2.1+x.1.1.2.2 : ℕ) : ℝ)) (experimentLaw ν L D n) ∧
(∫ x, ((x.1.1.1+x.1.1.2.1+x.1.1.2.2 : ℕ) : ℝ) ∂experimentLaw ν L D n) ≤
12*n*(∫ Z, (firstCommonWidth L Z : ℝ) ∂twoTapeLaw ν) := by
have h0 := windowCommonLaw_height_integrable ν L hL he hi n n
have h1 := windowCommonLaw_height_integrable ν L hL he hi (7*n) n
have hv : Integrable (fun a : ThreeChunk α =>
(chunkHeight L a.1 : ℝ)+(chunkHeight L a.2.1 : ℝ)+(chunkHeight L a.2.2 : ℝ))
(threeChunkLaw ν L n) :=
((h0.comp_fst _).add ((h1.comp_fst _).comp_snd _)).add ((h0.comp_snd _).comp_snd _)
rw [experimentLaw_fourWindows ν L hL he]
constructor
· rw [integrable_map_measure (measurable_of_countable _).aestronglyMeasurable
(measurable_of_countable _).aemeasurable]
simpa only [Function.comp_def,threeHeight,Nat.cast_add] using hv.comp_snd (windowCommonLaw ν L n n)
· rw [integral_map (measurable_of_countable _).aemeasurable
(measurable_of_countable _).aestronglyMeasurable]
simp only [threeHeight,Nat.cast_add]
rw [integral_prod _ (hv.comp_snd _)]
simp only [integral_const,probReal_univ,one_smul]
have hs : (∫ a : ThreeChunk α, (chunkHeight L a.1 : ℝ)+(chunkHeight L a.2.1 : ℝ)+(chunkHeight L a.2.2 : ℝ) ∂threeChunkLaw ν L n) =
(∫ a : ThreeChunk α, (chunkHeight L a.1 : ℝ) ∂threeChunkLaw ν L n) +
(∫ a : ThreeChunk α, (chunkHeight L a.2.1 : ℝ) ∂threeChunkLaw ν L n) +
(∫ a : ThreeChunk α, (chunkHeight L a.2.2 : ℝ) ∂threeChunkLaw ν L n) := by
calc
_ = (∫ a : ThreeChunk α, (chunkHeight L a.1 : ℝ)+(chunkHeight L a.2.1 : ℝ) ∂threeChunkLaw ν L n) + _ :=
integral_add ((h0.comp_fst _).add ((h1.comp_fst _).comp_snd _)) ((h0.comp_snd _).comp_snd _)
_ = _ := congrArg (fun t : ℝ => t+(∫ a : ThreeChunk α, (chunkHeight L a.2.2 : ℝ) ∂threeChunkLaw ν L n))
(integral_add (h0.comp_fst _) ((h1.comp_fst _).comp_snd _))
rw [hs]
simp only [threeChunkLaw]
rw [integral_prod _ (h0.comp_fst _),integral_prod _ ((h1.comp_fst _).comp_snd _),
integral_prod _ ((h0.comp_snd _).comp_snd _)]
simp only [integral_const,probReal_univ,one_smul]
rw [integral_prod _ (h1.comp_fst _),integral_prod _ (h0.comp_snd _)]
simp only [integral_const,probReal_univ,one_smul]
have hm0 := windowCommonLaw_height_mean ν L hL he hi n n
have hm1 := windowCommonLaw_height_mean ν L hL he hi (7*n) n
push_cast at hm0 hm1
linarith
lemma axis_renewal_tail_bound_ennreal {d : ℕ} (μ : Measure (Row d)) [IsProbabilityMeasure μ]
(hell : StrictEllipticity μ) (e : Step d)
(hp : 0 < annealed μ 0 (nonBacktracking (axisDirection e))) {n N : ℕ} (hnN : n ≤ N) :
axisReachProb μ e n-axisReachProb μ e N ≤ (N : ℝ≥0∞)*axisWidthTail μ e n := by
have := slabLaw_probability μ (axisDirection e) hp
apply (ENNReal.toReal_le_toReal (lt_of_le_of_lt tsub_le_self (measure_lt_top _ _)).ne
(ENNReal.mul_ne_top (by simp) (measure_ne_top _ _))).mp
change (axisReachProb μ e n-axisReachProb μ e N).toReal ≤ ((N : ℝ≥0∞)*axisWidthTail μ e n).toReal
rw [ENNReal.toReal_sub_of_le (axisReachProb_antitone μ e hnN) (measure_ne_top _ _),
ENNReal.toReal_mul,ENNReal.toReal_natCast]
refine (axis_renewal_tail_bound μ hell e hp hnN).2.trans ?_
exact mul_le_mul_of_nonneg_right (sub_le_self _ (Nat.cast_nonneg _)) ENNReal.toReal_nonneg
def experimentContact {d : ℕ} (e : Step d) :
Set (((ℕ × (ℕ × ℕ)) × Site d) × (ThreeLists (Word d) × ThreeLists (Word d))) :=
{x | x.2 ∈ middleContact (x.1.2-stepVector e)}
lemma experimentReference_contact_bound {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))))
(he : letI : ∀ b, IsProbabilityMeasure (slabLaw μ (axisDirection (placedAxis e b))) :=
fun b => slabLaw_probability μ _ (hp b)
∀ᵐ Z ∂twoTapeLaw (fun b => slabLaw μ (axisDirection (placedAxis e b))),
∃ H, 0 < H ∧ Z ∈ commonCut (placedWidth e) H)
(hi : letI : ∀ b, IsProbabilityMeasure (slabLaw μ (axisDirection (placedAxis e b))) :=
fun b => slabLaw_probability μ _ (hp b)
Integrable (fun Z => (firstCommonWidth (placedWidth e) Z : ℝ))
(twoTapeLaw (fun b => slabLaw μ (axisDirection (placedAxis e b)))))
(n : ℕ) [NeZero n] :
letI : ∀ b, IsProbabilityMeasure (slabLaw μ (axisDirection (placedAxis e b))) :=
fun b => slabLaw_probability μ _ (hp b)
experimentReference (fun b => slabLaw μ (axisDirection (placedAxis e b))) (placedWidth e)
(fun _ => wordEnd) n (experimentContact e) ≤
((annealed μ 0 (nonBacktracking (axisDirection e)))⁻¹)^2 *
((annealed μ 0 (nonBacktracking (axisDirection (oppositeStep e))))⁻¹)^2 * 2 *
ENNReal.ofReal (12*n*(∫ Z, (firstCommonWidth (placedWidth e) Z : ℝ)
∂twoTapeLaw (fun b => slabLaw μ (axisDirection (placedAxis e b))))) *
(axisWidthTail μ e n+axisWidthTail μ (oppositeStep e) n) := by
let : ∀ b, IsProbabilityMeasure (slabLaw μ (axisDirection (placedAxis e b))) :=
fun b => slabLaw_probability μ _ (hp b)
have : Countable (Word d) := inferInstance
have : Countable (TapeList (Word d)) := inferInstance
have : Countable (ThreeLists (Word d)) := inferInstance
let ν := fun b => slabLaw μ (axisDirection (placedAxis e b))
let P := experimentLaw ν (placedWidth e) (fun _ => wordEnd) n
let C : ℝ≥0∞ := ((annealed μ 0 (nonBacktracking (axisDirection e)))⁻¹)^2 *
((annealed μ 0 (nonBacktracking (axisDirection (oppositeStep e))))⁻¹)^2 * 2
let F := axisWidthTail μ e n+axisWidthTail μ (oppositeStep e) n
have hL (b : Bool) : ∀ᵐ a ∂ν b, 0 < placedWidth e b a := by
filter_upwards [ae_slabLaw_regeneration μ hell _ (axisDirection_ne_zero _) (hp b)] with a ha
exact slabRecords_pos _ _ ha
have hu (b : Bool) (H : ℕ) : Measure.infinitePi (fun _ : ℕ => ν b)
(renewalCut (placedWidth e b) H) ≠ 0 := by
rw [show Measure.infinitePi (fun _ : ℕ => ν b) (renewalCut (placedWidth e b) H) =
axisReachProb μ (placedAxis e b) H from axis_tapeCut_mass μ hell _ (hp b) H]
exact ne_of_gt ((hp b).trans_le (axisReachProb_lower μ hell _ _))
let : IsMarkovKernel (signedThreeKernel (ν false) (placedWidth e false)) :=
signedThreeKernel_markov _ _ (hu false)
let : IsMarkovKernel (signedThreeKernel (ν true) (placedWidth e true)) :=
signedThreeKernel_markov _ _ (hu true)
have hs : ∀ᵐ x ∂P.fst, n ≤ x.1.1 ∧ n ≤ x.1.2.2 ∧
axisHeight e x.2 = ((x.1.1+x.1.2.1+x.1.2.2 : ℕ) : ℤ) := by
rw [Measure.fst,ae_map_iff measurable_fst.aemeasurable (Set.to_countable _).measurableSet]
exact experimentLaw_axis_support μ hell e hp he n
have hm := experiment_height_moment ν (placedWidth e) hL he hi (fun _ => wordEnd) n
have hm' : Integrable (fun x : (ℕ × (ℕ × ℕ)) × Site d =>
((x.1.1+x.1.2.1+x.1.2.2 : ℕ) : ℝ)) P.fst := by
rw [Measure.fst,integrable_map_measure (measurable_of_countable _).aestronglyMeasurable measurable_fst.aemeasurable]
exact hm.1
have hmean : (∫⁻ x : (ℕ × (ℕ × ℕ)) × Site d,
(x.1.1+x.1.2.1+x.1.2.2 : ℝ≥0∞) ∂P.fst) ≤
ENNReal.ofReal (12*n*(∫ Z, (firstCommonWidth (placedWidth e) Z : ℝ) ∂twoTapeLaw ν)) := by
have heq (x : (ℕ × (ℕ × ℕ)) × Site d) :
(x.1.1+x.1.2.1+x.1.2.2 : ℝ≥0∞) = ENNReal.ofReal ((x.1.1+x.1.2.1+x.1.2.2 : ℕ) : ℝ) := by
norm_cast
simp_rw [heq]
rw [← ofReal_integral_eq_lintegral_ofReal hm' (Filter.Eventually.of_forall (by intro x; positivity)),
Measure.fst,integral_map measurable_fst.aemeasurable (measurable_of_countable _).aestronglyMeasurable]
exact ENNReal.ofReal_le_ofReal hm.2
unfold experimentReference
rw [Measure.compProd_apply (Set.to_countable _).measurableSet]
change (∫⁻ x, _ ∂P.fst) ≤ _
calc
_ ≤ ∫⁻ x : (ℕ × (ℕ × ℕ)) × Site d, C*(x.1.1+x.1.2.1+x.1.2.2 : ℝ≥0∞)*F ∂P.fst := by
apply lintegral_mono_ae
filter_upwards [hs] with x hx
have hz : axisHeight e (x.2-stepVector e) = ((x.1.1+x.1.2.1+x.1.2.2 : ℕ) : ℤ)-1 := by
rw [sub_eq_add_neg,axisHeight_add,axisHeight_neg,axisHeight_step_self,hx.2.2]; rfl
have hh := four_bridge_contact_le μ hell e (hp false) (hp true) x.1 (x.2-stepVector e) n hx.1 hx.2.1 hz
rw [Kernel.comap_apply,Kernel.prod_apply]
change ((signedThreeKernel (ν false) (placedWidth e false) x.1).prod
(signedThreeKernel (ν true) (placedWidth e true) x.1)) (middleContact (x.2-stepVector e)) ≤ _
refine hh.trans ?_
have hp' := axis_renewal_tail_bound_ennreal μ hell e (hp false)
(show n ≤ x.1.1+x.1.2.1+x.1.2.2 by omega)
have hm' := axis_renewal_tail_bound_ennreal μ hell (oppositeStep e) (hp true)
(show n ≤ x.1.1+x.1.2.1+x.1.2.2 by omega)
calc
_ ≤ _ * _ * (2*((x.1.1+x.1.2.1+x.1.2.2 : ℝ≥0∞)*axisWidthTail μ e n+
(x.1.1+x.1.2.1+x.1.2.2 : ℝ≥0∞)*axisWidthTail μ (oppositeStep e) n)) := by
gcongr
· simpa only [Nat.cast_add] using hp'
· simpa only [Nat.cast_add] using hm'
_ = _ := by dsimp only [C,F];ring
_ = C*(∫⁻ x : (ℕ × (ℕ × ℕ)) × Site d, (x.1.1+x.1.2.1+x.1.2.2 : ℝ≥0∞) ∂P.fst)*F := by
rw [lintegral_mul_const _ (measurable_of_countable _),lintegral_const_mul _ (measurable_of_countable _)]
_ ≤ _ := by
change C*_ *F ≤ C*_ *F
exact mul_le_mul_left (mul_le_mul_right hmean C) F
lemma axisBridgeWords_prefixFree {d : ℕ} (e : Step d) (k : ℕ) :
PrefixFreeWords {a : Word d | AxisBridgeWord e k a} := by
intro a b hab
exact (firstHitWords_prefixFree (axisUpper e k))
(i := ⟨a,a.property.firstHit⟩) (j := ⟨b,b.property.firstHit⟩)
(fun h => hab (Subtype.ext (congrArg (fun x : {a | firstHitWord (axisUpper e k) a} => x.val) h)))
lemma prefix_contact_total_bound {d ι : ℕ} [NeZero ι]
(μ : Measure (Row d)) [IsProbabilityMeasure μ] (hell : StrictEllipticity μ)
(e : Step d) (k r N m : ℕ) (hrN : r ≤ N)
(E : Word d → Fin ι → Set (Path d)) (hE : ∀ a q, MeasurableSet (E a q))
(hdis : ∀ a, Pairwise (fun i j => Disjoint (E a i) (E a j)))
(start : Word d → Fin ι → Site d)
(hΔ : axisReachProb μ e r-axisReachProb μ e N ≤ ENNReal.ofReal ((1/2 : ℝ)^m)) :
(∑' a : {a : Word d | AxisBridgeWord e k a}, ∑ q,
∑' b : {b | upperContactWord 0 (axisUpper e r) (axisLower e 0 ∪ axisUpper e N) b},
annealed μ 0 (wordCylinder a ∩ (shiftPath (-wordEnd a) ∘ tailPath a.val.1) ⁻¹'
(E a q ∩ wordCylinder b)) *
annealed μ (start a q) (avoids (axisUpper e N) ∩ contactAvoid b)) ≤
ENNReal.ofReal ((24 / posteriorExponent)*Real.log (ι+1)*(m+1)^2*(1/2 : ℝ)^m) := by
let C := ENNReal.ofReal ((24 / posteriorExponent)*Real.log (ι+1)*(m+1)^2*(1/2 : ℝ)^m)
calc
_ ≤ ∑' a : {a : Word d | AxisBridgeWord e k a}, annealed μ 0 (wordCylinder a)*C := by
apply ENNReal.tsum_le_tsum
intro a
by_cases ha : annealed μ 0 (wordCylinder a) = 0
· have hz (q : Fin ι) (b : {b | upperContactWord 0 (axisUpper e r) (axisLower e 0 ∪ axisUpper e N) b}) :
annealed μ 0 (wordCylinder a ∩ (shiftPath (-wordEnd a) ∘ tailPath a.val.1) ⁻¹'
(E a q ∩ wordCylinder b)) = 0 := measure_mono_null Set.inter_subset_left ha
simp only [hz,zero_mul,tsum_zero,Finset.sum_const_zero,ha,le_refl]
· exact posterior_old_prefix_weighted_bound μ hell e a ha
(fun i hi => (a.property.2 i hi).2.trans_eq a.property.1.symm)
(E a) (hE a) (hdis a) (start a) r N m hrN hΔ
_ = annealed μ 0 (⋃ a : {a : Word d | AxisBridgeWord e k a}, wordCylinder a)*C := by
rw [ENNReal.tsum_mul_right,measure_iUnion (axisBridgeWords_prefixFree e k)
(fun _ => measurableSet_pathCylinder _ _)]
_ ≤ C := by simpa only [one_mul] using mul_le_mul_left (prob_le_one (μ := annealed μ 0)) C
lemma conditioned_negative_shift {d : ℕ} (μ : Measure (Row d)) [IsProbabilityMeasure μ]
(e : Step d) (z : Site d) (N : ℕ) (hz : axisHeight e z = (N : ℤ)-1)
(E : Set (Path d)) (hE : MeasurableSet E) :
conditioned μ (axisDirection (oppositeStep e)) (shiftPath z ⁻¹' E) =
(annealed μ 0 (nonBacktracking (axisDirection (oppositeStep e))))⁻¹ *
annealed μ z (avoids (axisUpper e N) ∩ E) := by
have hb : shiftPath z ⁻¹' avoids (axisUpper e N) = nonBacktracking (axisDirection (oppositeStep e)) := by
ext X
change (∀ j, ¬(N : ℤ) ≤ axisHeight e (X j+z)) ↔
∀ j, 0 ≤ height (axisDirection (oppositeStep e)) (X j)
simp only [axisHeight_add,hz,height_axisDirection,axisHeight_opposite]
constructor
· intro h j
have hj := h j
have hx : axisHeight e (X j) ≤ 0 := by omega
exact_mod_cast (neg_nonneg.mpr hx)
· intro h j
have hx : -(axisHeight e (X j)) ≥ 0 := by exact_mod_cast h j
omega
rw [conditioned,cond_apply (measurableSet_nonBacktracking _)]
have hh := congrArg (fun M : Measure (Path d) => M (avoids (axisUpper e N) ∩ E)) (annealed_shift μ 0 z)
rw [zero_add,Measure.map_apply (measurable_shiftPath z) ((measurableSet_avoids _).inter hE),
Set.preimage_inter,hb] at hh
rw [hh]
end DirectionalZeroOne
end OAI
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