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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 | import OAI.Probability.DirectionalWalk.ReversedContacts
namespace OAI
open MeasureTheory ProbabilityTheory Filter Preorder
open scoped ENNReal BigOperators Topology
namespace DirectionalZeroOne
open scoped Classical
def finiteLateContact {d : ℕ} (e : Step d) (k r N : ℕ) (a : Word d) (Y : Path d) : Prop :=
∃ t τ, t ≤ τ ∧ τ < a.1 ∧ AxisBridgeWord e k (prefixWord t (wordPath a)) ∧
(∀ j, t ≤ j → j ≤ τ → (k : ℤ) ≤ axisHeight e (wordPath a j) ∧ axisHeight e (wordPath a j) < (N : ℤ)) ∧
(r : ℤ) ≤ axisHeight e (wordPath a τ) ∧
(∃ j, Y j+(wordEnd a-stepVector e) = wordPath a τ) ∧
(∀ i, t ≤ i → i < τ → ∀ j, Y j+(wordEnd a-stepVector e) ≠ wordPath a i)
lemma uncovered_bridge_late {d : ℕ} (e : Step d) (Z : TwoTape (Word d))
(hZ : ∀ b i, RegenerationWord (axisDirection (placedAxis e b)) (Z (b,i)))
(Y : Path d) (hY : GoodSlabPath (axisDirection (oppositeStep e)) Y)
(hZY : ∀ i, Z (true,i) = slabs (axisDirection (oppositeStep e)) Y i)
(m n low high : ℕ) (hm : 0 < m) (hmn : m ≤ n)
(hslo : low ≤ tapeHeight (placedWidth e false) (fun j => Z (false,j)) m)
(hshi : tapeHeight (placedWidth e false) (fun j => Z (false,j)) m < high)
(hno : ∀ h, low ≤ h → h < high → ¬contactDepth e Z h) :
let H := tapeHeight (placedWidth e false) (fun j => Z (false,j)) n
let s := tapeHeight (placedWidth e false) (fun j => Z (false,j)) m
finiteLateContact e (H-s) (H-low) H
(concatenateList (reverseTapeList (tapePrefix n (fun j => Z (false,j))))) Y := by
classical
let H := tapeHeight (placedWidth e false) (fun j => Z (false,j)) n
let s := tapeHeight (placedWidth e false) (fun j => Z (false,j)) m
let a := concatenateList (reverseTapeList (tapePrefix n (fun j => Z (false,j))))
let D := ∑ j ∈ Finset.range n, wordEnd (Z (false,j))
have hl : ∀ j, 0 < placedWidth e false (Z (false,j)) := fun j => slabRecords_pos _ _ (hZ false j)
have hsH : s ≤ H := (tapeHeight_strictMono _ _ hl).monotone hmn
have hsp : 0 < s := hm.trans_le (le_tapeHeight _ _ hl m)
have hH : 1 ≤ H := (Nat.succ_le_of_lt hsp).trans hsH
have hD : wordEnd a = D := by
rw [concatenateList_end]
change (∑ j : Fin n, wordEnd (Z (false,j.rev))) = _
have hh := Equiv.sum_comp Fin.revPerm (fun j : Fin n => wordEnd (Z (false,j)))
apply (show (∑ j : Fin n, wordEnd (Z (false,j.rev))) = ∑ j : Fin n, wordEnd (Z (false,j)) by
simpa only [Fin.revPerm_apply] using hh).trans
rw [Fin.sum_univ_eq_sum_range (fun j => wordEnd (Z (false,j)))]
have hDh : axisHeight e D = (H : ℤ) := axisHeight_tapeTotal e (fun j => Z (false,j)) (hZ false) n
have ha : AxisBridgeWord e H a := by
have hh := concatenateWords_axis_bridge e (fun j : Fin n => Z (false,j.rev)) (fun j => hZ false j.rev)
have he : (∑ j : Fin n, slabRecords (axisDirection e) (Z (false,j.rev))) = H := by
have hh := Equiv.sum_comp Fin.revPerm (fun j : Fin n => slabRecords (axisDirection e) (Z (false,j)))
apply (show (∑ j : Fin n, slabRecords (axisDirection e) (Z (false,j.rev))) =
∑ j : Fin n, slabRecords (axisDirection e) (Z (false,j)) by
simpa only [Fin.revPerm_apply] using hh).trans
rw [Fin.sum_univ_eq_sum_range (fun j => slabRecords (axisDirection e) (Z (false,j)))]
rfl
exact he ▸ hh
obtain ⟨t,ht,htbridge,htband⟩ := reversePrefix_cut_geometry e Z hZ m n hm hmn
change t < a.1 at ht
have hforce := (opposed_forced_contact e Z hZ).1.1
obtain ⟨v,hv,hvsite⟩ := opposed_positive_in_finite_bridge e Z hZ n 1 hH (-stepVector e) hforce
(by rw [axisHeight_neg,axisHeight_step_self];norm_num)
change v < a.1 at hv
change wordPath a v-D = -stepVector e at hvsite
have hvheight : axisHeight e (wordPath a v) = (H : ℤ)-1 := by
have hh := congrArg (axisHeight e) hvsite
rw [sub_eq_add_neg,axisHeight_add,axisHeight_neg,hDh,axisHeight_neg,axisHeight_step_self] at hh
omega
have htv : t ≤ v := by
by_contra h
have hvt : v < t := Nat.lt_of_not_ge h
have hh := (htbridge.2 v hvt).2
rw [wordPath_prefixWord t _ hvt.le,hvheight,Nat.cast_sub hsH] at hh
omega
have hex : ∃ τ, t ≤ τ ∧ τ < a.1 ∧ ∃ j, Y j+(wordEnd a-stepVector e) = wordPath a τ := by
refine ⟨v,htv,hv,0,?_⟩
rw [hY.1,zero_add,hD]
calc
D-stepVector e = -stepVector e+D := by abel
_ = wordPath a v := (sub_eq_iff_eq_add.mp hvsite).symm
let τ := Nat.find hex
have hτ := Nat.find_spec hex
have hcontact : contactAt e Z (wordPath a τ-D) := by
constructor
· exact reversePrefix_departure_in_opposed e Z (fun i => (hZ false i).2.1) τ hτ.2.1
· rw [opposed_negative_slabs e Z Y hY hZY]
obtain ⟨j,hj⟩ := hτ.2.2
refine ⟨j,?_⟩
change Y j+(wordEnd a-stepVector e) = wordPath a τ at hj
rw [hD] at hj
rw [← hj]
abel
have hτband : ((H-s : ℕ) : ℤ) ≤ axisHeight e (wordPath a τ) ∧ axisHeight e (wordPath a τ) < (H : ℤ) :=
⟨htband τ hτ.1 hτ.2.1,(ha.2 τ hτ.2.1).2⟩
let δ : ℤ := (H : ℤ)-axisHeight e (wordPath a τ)
have hδ0 : 0 < δ := sub_pos.mpr hτband.2
have hδs : δ ≤ (s : ℤ) := by dsimp [δ];rw [Nat.cast_sub hsH] at hτband;omega
have hδ : (Int.toNat δ : ℤ) = δ := Int.toNat_of_nonneg hδ0.le
have hdep : contactDepth e Z (Int.toNat δ) := by
refine ⟨wordPath a τ-D,hcontact,?_⟩
rw [sub_eq_add_neg,axisHeight_add,axisHeight_neg,hDh,hδ]
dsimp [δ]
omega
have hsmall : Int.toNat δ < low := by
by_contra h
apply hno _ (Nat.le_of_not_gt h) ?_ hdep
have hh : Int.toNat δ ≤ s := by
rw [← hδ] at hδs
exact_mod_cast hδs
exact hh.trans_lt hshi
have hr : ((H-low : ℕ) : ℤ) ≤ axisHeight e (wordPath a τ) := by
rw [Nat.cast_sub (hslo.trans hsH)]
have hh : δ < (low : ℤ) := by
rw [← hδ]
exact_mod_cast hsmall
dsimp [δ] at hh
omega
refine ⟨t,τ,hτ.1,hτ.2.1,htbridge,?_,hr,hτ.2.2,?_⟩
· intro j htj hjτ
exact ⟨htband j htj (hjτ.trans_lt hτ.2.1),(ha.2 j (hjτ.trans_lt hτ.2.1)).2⟩
· intro i hti hiτ j hj
exact Nat.find_min hex hiτ ⟨hti,hiτ.trans hτ.2.1,j,hj⟩
lemma measurableSet_finiteLateContact {d : ℕ} (e : Step d) (k r N : ℕ) :
MeasurableSet {p : Word d × Path d | finiteLateContact e k r N p.1 p.2} := by
classical
have hword (P : Word d → Prop) : MeasurableSet {p : Word d × Path d | P p.1} :=
measurable_fst (Set.to_countable _).measurableSet
have heq (i j : ℕ) : MeasurableSet {p : Word d × Path d |
p.2 j + (wordEnd p.1-stepVector e) = wordPath p.1 i} := by
apply measurableSet_eq_fun
· exact ((measurable_pi_apply j).comp measurable_snd).add
((measurable_of_countable (fun a : Word d => wordEnd a-stepVector e)).comp measurable_fst)
· exact (measurable_of_countable (fun a : Word d => wordPath a i)).comp measurable_fst
simp only [finiteLateContact,Set.ofPred_exists]
apply MeasurableSet.iUnion;intro t
apply MeasurableSet.iUnion;intro τ
refine (hword (fun _ => t ≤ τ)).inter ((hword (fun a => τ < a.1)).inter
((hword (fun a => AxisBridgeWord e k (prefixWord t (wordPath a)))).inter
((hword (fun a => ∀ j, t ≤ j → j ≤ τ → (k : ℤ) ≤ axisHeight e (wordPath a j) ∧
axisHeight e (wordPath a j) < (N : ℤ))).inter
((hword (fun a => (r : ℤ) ≤ axisHeight e (wordPath a τ))).inter ?_))))
apply MeasurableSet.inter
· change MeasurableSet {p : Word d × Path d | ∃ j, p.2 j+(wordEnd p.1-stepVector e) = wordPath p.1 τ}
simpa only [Set.ofPred_exists] using MeasurableSet.iUnion (fun j => heq τ j)
· change MeasurableSet {p : Word d × Path d | ∀ i, t ≤ i → i < τ → ∀ j,
p.2 j+(wordEnd p.1-stepVector e) ≠ wordPath p.1 i}
simp only [Set.ofPred_forall]
exact MeasurableSet.iInter (fun i => MeasurableSet.iInter (fun _ =>
MeasurableSet.iInter (fun _ => MeasurableSet.iInter (fun j => (heq i j).compl))))
end DirectionalZeroOne
end OAI
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