| import OAI.Probability.DirectionalWalk.ReversedContacts |
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| namespace OAI |
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| open MeasureTheory ProbabilityTheory Filter Preorder |
| open scoped ENNReal BigOperators Topology |
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| namespace DirectionalZeroOne |
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| open scoped Classical |
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| 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) |
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| 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⟩ |
|
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| 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)))) |
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| end DirectionalZeroOne |
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| end OAI |
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