| import Mathlib |
| import OAI.Probability.Ballisticity.Estimates.ShiftedCurveConcatenation |
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| namespace OAI |
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| section |
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| section |
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| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| namespace DirectionalTransience |
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| lemma curvePrefixAt_center_comparison {d : ℕ} (ℓ : Vector d) (f : Direction d) (x : Lattice d) |
| (b c : ℕ → ℝ) (B D : ℝ) (H n : ℕ) (X : Path d) |
| (hc : ∀ j ≤ H, |b j-c j| ≤ D) (hX : X ∈ CurvePrefixAt ℓ f x b B H n) : |
| X ∈ CurvePrefixAt ℓ f x c (B+D) H n := by |
| refine ⟨hX.1,?_⟩ |
| intro j hj |
| obtain ⟨a,ha,hr,hg⟩ := hX.2 j hj |
| refine ⟨a,ha,hr,?_⟩ |
| calc |
| _ = |(signedCoordinate f (X a-x)-b j)+(b j-c j)| := by congr 1; ring |
| _ ≤ |signedCoordinate f (X a-x)-b j|+|b j-c j| := abs_add_le _ _ |
| _ ≤ B+D := add_le_add hg (hc j hj) |
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| lemma curvePrefixAt_radius_mono {d : ℕ} (ℓ : Vector d) (f : Direction d) (x : Lattice d) |
| (b : ℕ → ℝ) {B C : ℝ} (hBC : B ≤ C) (H n : ℕ) (X : Path d) |
| (hX : X ∈ CurvePrefixAt ℓ f x b B H n) : X ∈ CurvePrefixAt ℓ f x b C H n := by |
| refine ⟨hX.1,?_⟩ |
| intro j hj |
| obtain ⟨a,ha,hr,hg⟩ := hX.2 j hj |
| exact ⟨a,ha,hr,hg.trans hBC⟩ |
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| lemma curveWordEvent_shifted_tube {d : ℕ} (e f : Direction d) (x : Lattice d) |
| (b : ℕ → ℝ) {B D : ℝ} (hB : 0 ≤ B) (hD : 0 ≤ D) {H : ℕ} (hH : 0 < H) |
| (θ a r : ℝ) (hb : ∀ j ≤ H, |b j-(j:ℝ)*θ| ≤ D) |
| (w : List (Lattice d)) |
| (hp : ∀ k ≤ w.length, |a+signedCoordinate f (w.take k).sum-(k:ℝ)*(H:ℝ)*θ| ≤ r) |
| (X : Path d) (h0 : X 0=x) (hnn : ∀ j, ∃ g, X (j+1)=X j+step g) |
| (hX : X ∈ curveWordEvent (realPosition (step e)) f b B H x w) : |
| ∃ n, X ∈ CurvePrefixAt (realPosition (step e)) f x |
| (fun j => (j:ℝ)*θ-a) (r+B+D) (w.length*H) n := by |
| induction w generalizing x X a with |
| | nil => |
| have ha : |a| ≤ r := by simpa [signedCoordinate] using hp 0 le_rfl |
| refine ⟨0,⟨?_,?_⟩,?_⟩ |
| · simp [Upper,h0] |
| · intro j hj; omega |
| · intro j hj |
| have hj0 : j=0 := by simpa using hj |
| subst j |
| refine ⟨0,le_rfl,?_,?_⟩ |
| · simp [RecordOrZeroPrefix] |
| · simpa [h0,signedCoordinate] using ha.trans (by linarith : r ≤ r+B+D) |
| | cons u w ih => |
| obtain ⟨n,hn,hnu,hnt⟩ := (curveWordEvent_cons_mem _ f b B H x u w X).mp hX |
| let a' := a+signedCoordinate f u-(H:ℝ)*θ |
| have hp' : ∀ k ≤ w.length, |a'+signedCoordinate f (w.take k).sum-(k:ℝ)*(H:ℝ)*θ| ≤ r := by |
| intro k hk |
| have hh := hp (k+1) (by simp only [List.length_cons]; omega) |
| simp only [List.take_succ_cons,List.sum_cons,signedCoordinate_add,Nat.cast_add,Nat.cast_one] at hh |
| dsimp [a'] |
| convert hh using 1; congr 1; ring |
| have hnt' : (fun j => X (n+j)) ∈ curveWordEvent (realPosition (step e)) f b B H (X n) w := by simpa only [hnu] using hnt |
| obtain ⟨m,hm⟩ := ih (X n) a' hp' (fun j => X (n+j)) (by simp) |
| (fun j => by simpa only [Nat.add_assoc] using hnn (n+j)) hnt' |
| have ha : |a| ≤ r := by simpa [signedCoordinate] using hp 0 (Nat.zero_le _) |
| have hc : ∀ j ≤ H, |b j-((j:ℝ)*θ-a)| ≤ D+r := by |
| intro j hj |
| calc |
| _ = |(b j-(j:ℝ)*θ)+a| := by congr 1; ring |
| _ ≤ |b j-(j:ℝ)*θ|+|a| := abs_add_le _ _ |
| _ ≤ D+r := add_le_add (hb j hj) ha |
| have hn' := curvePrefixAt_center_comparison _ f x b (fun j => (j:ℝ)*θ-a) B (D+r) H n X hc hn |
| have hn'' : X ∈ CurvePrefixAt (realPosition (step e)) f x (fun j => (j:ℝ)*θ-a) (r+B+D) H n := by |
| simpa only [add_comm,add_left_comm,add_assoc] using hn' |
| have ha' : a'=a+signedCoordinate f (X n-x)-(H:ℝ)*θ := by |
| simp only [a',hnu,add_sub_cancel_left] |
| rw [ha'] at hm |
| have hh := shiftedCurvePrefixAt_concatenate e f x θ a (r+B+D) hH n (w.length*H) m X h0 hnn hn'' hm |
| refine ⟨n+m,?_⟩ |
| simpa only [List.length_cons,Nat.add_mul,one_mul,Nat.add_comm] using hh |
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| end DirectionalTransience |
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| end |
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| end |
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| end OAI |
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