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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 | import OAI.Probability.DirectionalWalk.MixedTapes
namespace OAI
open MeasureTheory ProbabilityTheory Filter Preorder
open scoped ENNReal BigOperators Topology
namespace DirectionalZeroOne
open scoped Classical
def coveredDepthBand {d : ℕ} (e : Step d) (low high : ℕ) : Set (TwoTape (Word d)) :=
{Z | ∃ h, low ≤ h ∧ h < high ∧ contactDepth e Z h}
lemma measurableSet_coveredDepthBand {d : ℕ} (e : Step d) (low high : ℕ) :
MeasurableSet (coveredDepthBand e low high) := by
simp only [coveredDepthBand,Set.ofPred_exists]
exact MeasurableSet.iUnion (fun h => (MeasurableSet.const _).inter
((MeasurableSet.const _).inter (measurableSet_contactDepth e h)))
lemma two_cuts_uncovered_mixed {d : ℕ} (e : Step d) (p : (ℕ → Word d) × Path d)
(hpos : ∀ i, RegenerationWord (axisDirection e) (p.1 i))
(hneg : GoodSlabPath (axisDirection (oppositeStep e)) p.2)
(s N low high : ℕ) (hs : 0 < s) (hsN : s ≤ N) (hls : low ≤ s) (hsh : s < high)
(hcs : p.1 ∈ tapeCut (axisDirection e) s) (hcN : p.1 ∈ tapeCut (axisDirection e) N)
(hno : mixedTapes e p ∉ coveredDepthBand e low high) :
finiteLateContact e (N-s) (N-low) N
(concatenateList (reverseTapeList (cutList (slabRecords (axisDirection e)) N p.1))) p.2 := by
obtain ⟨m,hm⟩ := hcs
obtain ⟨n,hn⟩ := hcN
change tapeHeight (slabRecords (axisDirection e)) p.1 m = s at hm
change tapeHeight (slabRecords (axisDirection e)) p.1 n = N at hn
have hL : ∀ i, 0 < slabRecords (axisDirection e) (p.1 i) := fun i => slabRecords_pos _ _ (hpos i)
have hm0 : 0 < m := by
by_contra hh
have hmz : m = 0 := by omega
rw [hmz] at hm
change 0 = s at hm
omega
have hmn : m ≤ n := (tapeHeight_strictMono _ _ hL).le_iff_le.mp (by rw [hm,hn];exact hsN)
have hgood : ∀ b i, RegenerationWord (axisDirection (placedAxis e b)) (mixedTapes e p (b,i)) := by
intro b i
cases b
· exact hpos i
· exact hneg.2 i
have hh := uncovered_bridge_late e (mixedTapes e p) hgood p.2 hneg (fun _ => rfl)
m n low high hm0 hmn (by change low ≤ tapeHeight (slabRecords (axisDirection e)) p.1 m;rw [hm];exact hls)
(by change tapeHeight (slabRecords (axisDirection e)) p.1 m < high;rw [hm];exact hsh)
(fun h hl hh hc => hno ⟨h,hl,hh,hc⟩)
change finiteLateContact e (tapeHeight (slabRecords (axisDirection e)) p.1 n-tapeHeight (slabRecords (axisDirection e)) p.1 m)
(tapeHeight (slabRecords (axisDirection e)) p.1 n-low) (tapeHeight (slabRecords (axisDirection e)) p.1 n)
(concatenateList (reverseTapeList (tapePrefix n p.1))) p.2 at hh
rw [hm,hn] at hh
rw [cutList,cutListIndex_eq _ N p.1 hL hn]
exact hh
lemma nthCommon_suffix_map {α β : Type*} [Countable α] [Countable β]
[MeasurableSpace α] [MeasurableSingletonClass α] [MeasurableSpace β] [MeasurableSingletonClass β]
(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)
(r : ℕ) (f : TwoTape α → β) (hf : Measurable f) :
(twoTapeLaw ν).map (fun Z => (nthCommonList L r Z,f ((commonRestart L)^[r] Z))) =
(nthCommonLaw ν L r).prod ((twoTapeLaw ν).map f) := by
apply Measure.ext_of_singleton
intro a
have hpair : Measurable (fun Z => (nthCommonList L r Z,f ((commonRestart L)^[r] Z))) :=
(measurable_nthCommonList L r).prodMk (hf.comp ((measurable_commonRestart L).iterate r))
have hprod : ((nthCommonLaw ν L r).prod ((twoTapeLaw ν).map f)) {a} =
nthCommonLaw ν L r {a.1} * ((twoTapeLaw ν).map f) {a.2} := by
rw [show {a} = {a.1} ×ˢ {a.2} from (Set.singleton_prod_singleton).symm,Measure.prod_prod]
rw [Measure.map_apply hpair (measurableSet_singleton a),hprod,
Measure.map_apply hf (measurableSet_singleton _)]
have heq : (fun Z => (nthCommonList L r Z,f ((commonRestart L)^[r] Z))) ⁻¹' {a} =
{Z | nthCommonList L r Z = a.1 ∧ (commonRestart L)^[r] Z ∈ f ⁻¹' {a.2}} := by
rcases a with ⟨a,b⟩
ext Z;simp only [Set.mem_preimage,Set.mem_singleton_iff,Prod.mk.injEq,Set.mem_ofPred_eq]
rw [heq]
exact nthCommon_suffix_factorization ν L hL he r a.1 _ (hf (measurableSet_singleton _))
noncomputable def consecutiveThree {α : Type*} (L : Bool → α → ℕ)
(r s t : ℕ) (Z : TwoTape α) : ThreeChunk α :=
(nthCommonList L r Z,nthCommonList L s ((commonRestart L)^[r] Z),
nthCommonList L t ((commonRestart L)^[r+s] Z))
lemma measurable_consecutiveThree {α : Type*} [Countable α] [MeasurableSpace α]
[MeasurableSingletonClass α] (L : Bool → α → ℕ) (r s t : ℕ) :
Measurable (consecutiveThree L r s t) :=
(measurable_nthCommonList L r).prodMk
(((measurable_nthCommonList L s).comp ((measurable_commonRestart L).iterate r)).prodMk
((measurable_nthCommonList L t).comp ((measurable_commonRestart L).iterate (r+s))))
lemma consecutiveThree_law {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)
(r s t : ℕ) :
(twoTapeLaw ν).map (consecutiveThree L r s t) =
(nthCommonLaw ν L r).prod ((nthCommonLaw ν L s).prod (nthCommonLaw ν L t)) := by
have hm := nthCommon_suffix_map ν L hL he s (nthCommonList L t) (measurable_nthCommonList L t)
have hr := nthCommon_suffix_map ν L hL he r
(fun Z => (nthCommonList L s Z,nthCommonList L t ((commonRestart L)^[s] Z)))
((measurable_nthCommonList L s).prodMk
((measurable_nthCommonList L t).comp ((measurable_commonRestart L).iterate s)))
rw [hm] at hr
have heq : consecutiveThree L r s t = (fun Z =>
(nthCommonList L r Z,nthCommonList L s ((commonRestart L)^[r] Z),
nthCommonList L t ((commonRestart L)^[s] ((commonRestart L)^[r] Z)))) := by
funext Z
simp only [consecutiveThree]
rw [Nat.add_comm r s,Function.iterate_add_apply]
rw [heq]
exact hr
noncomputable def consecutiveFour {α : Type*} (L : Bool → α → ℕ)
(u r s t : ℕ) (Z : TwoTape α) : TwoTapeList α × ThreeChunk α :=
(nthCommonList L u Z,consecutiveThree L r s t ((commonRestart L)^[u] Z))
lemma measurable_consecutiveFour {α : Type*} [Countable α] [MeasurableSpace α]
[MeasurableSingletonClass α] (L : Bool → α → ℕ) (u r s t : ℕ) :
Measurable (consecutiveFour L u r s t) :=
(measurable_nthCommonList L u).prodMk
((measurable_consecutiveThree L r s t).comp ((measurable_commonRestart L).iterate u))
lemma consecutiveFour_law {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)
(u r s t : ℕ) :
(twoTapeLaw ν).map (consecutiveFour L u r s t) =
(nthCommonLaw ν L u).prod ((nthCommonLaw ν L r).prod
((nthCommonLaw ν L s).prod (nthCommonLaw ν L t))) := by
change (twoTapeLaw ν).map (fun Z => (nthCommonList L u Z,
consecutiveThree L r s t ((commonRestart L)^[u] Z))) = _
rw [nthCommon_suffix_map ν L hL he u _ (measurable_consecutiveThree L r s t),
consecutiveThree_law ν L hL he]
end DirectionalZeroOne
end OAI
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