lean/OAI/Probability/FirstPassage/CrossingTrialGeometry.lean

import OAI.Probability.FirstPassage.RareFastGeometry

namespace OAI

namespace PlanarFPP
open Set Finset

noncomputable def trialSpacing (R s : ℝ) : ℝ := (R+1)*(⌈s⌉₊ : ℝ)

noncomputable def trialWindow (M : ℕ) (R s : ℝ) : ℝ := 100*M*trialSpacing R s

noncomputable def trialOffset (M : ℕ) (R s : ℝ) (j : Fin M) : ℝ :=
  (2*((j.val : ℝ)+1)-(M : ℝ)-1)*3*trialSpacing R s

noncomputable def trialTranslate (b : ℝ) (X : ℤ) (Y : ℝ)
    (M : ℕ) (R s : ℝ) (j : Fin M) : Vertex :=
  (X,⌊b*X+Y+trialOffset M R s j+1/2⌋)

theorem nearest_integer_error (t : ℝ) : |(⌊t+1/2⌋ : ℝ)-t| ≤ 1/2 := by
  have hlo := Int.floor_le (t+1/2)
  have hhi := Int.lt_floor_add_one (t+1/2)
  apply abs_le.mpr
  constructor <;> linarith

theorem trialTranslate_error (b : ℝ) (X : ℤ) (Y : ℝ)
    (M : ℕ) (R s : ℝ) (j : Fin M) :
    |transverse b (trialTranslate b X Y M R s j)-(Y+trialOffset M R s j)| ≤ 1/2 := by
  have hh := nearest_integer_error (b*X+Y+trialOffset M R s j)
  convert hh using 1
  simp [trialTranslate,transverse]
  congr 1
  ring

theorem trialSpacing_bounds (R s : ℝ) (hR : 0 ≤ R) (hs : 1 ≤ s) :
    1 ≤ trialSpacing R s ∧ R*s ≤ trialSpacing R s := by
  have hc := Nat.le_ceil s
  have hn : (1 : ℝ) ≤ (⌈s⌉₊ : ℝ) := by linarith
  unfold trialSpacing
  constructor
  · nlinarith
  · nlinarith

theorem trialOffset_bound (M : ℕ) (R s : ℝ) (j : Fin M)
    (hR : 0 ≤ R) (hs : 1 ≤ s) :
    |trialOffset M R s j| ≤ 3*M*trialSpacing R s := by
  have hB := (trialSpacing_bounds R s hR hs).1
  have hj : (j.val : ℝ)+1 ≤ M := by exact_mod_cast j.isLt
  have hj0 : (0 : ℝ) ≤ j.val := Nat.cast_nonneg _
  have ha : |2*((j.val : ℝ)+1)-(M : ℝ)-1| ≤ M := by
    apply abs_le.mpr
    constructor <;> linarith
  unfold trialOffset
  rw [abs_mul,abs_mul,abs_of_pos (by norm_num : (0 : ℝ)<3),abs_of_nonneg (show 0 ≤ trialSpacing R s by linarith)]
  nlinarith

theorem trial_strip_in_window (b : ℝ) (X : ℤ) (Y : ℝ)
    (M : ℕ) (R s : ℝ) (j : Fin M) (hR : 0 ≤ R) (hs : 1 ≤ s)
    (p : Vertex) (hp : |transverse b p-transverse b (trialTranslate b X Y M R s j)| ≤ R*s) :
    |transverse b p-Y| ≤ trialWindow M R s/4 := by
  have he := trialTranslate_error b X Y M R s j
  have ho := trialOffset_bound M R s j hR hs
  have hB := trialSpacing_bounds R s hR hs
  have hM : (1 : ℝ) ≤ M := by exact_mod_cast j.pos
  have hh := abs_add_le
    (transverse b (trialTranslate b X Y M R s j)-(Y+trialOffset M R s j))
    (trialOffset M R s j)
  have heq : transverse b (trialTranslate b X Y M R s j)-(Y+trialOffset M R s j)+
      trialOffset M R s j = transverse b (trialTranslate b X Y M R s j)-Y := by ring
  rw [heq] at hh
  have ht := abs_add_le (transverse b p-transverse b (trialTranslate b X Y M R s j))
    (transverse b (trialTranslate b X Y M R s j)-Y)
  rw [sub_add_sub_cancel] at ht
  unfold trialWindow
  nlinarith [mul_le_mul_of_nonneg_right hM (by linarith : 0 ≤ trialSpacing R s)]

theorem trial_centers_separated (b : ℝ) (X : ℤ) (Y : ℝ)
    (M : ℕ) (R s : ℝ) (j l : Fin M) (hjl : j ≠ l) (hR : 0 ≤ R) (hs : 1 ≤ s) :
    2*R*s < |transverse b (trialTranslate b X Y M R s j)-
      transverse b (trialTranslate b X Y M R s l)| := by
  have hB := trialSpacing_bounds R s hR hs
  have he := trialTranslate_error b X Y M R s j
  have hf := trialTranslate_error b X Y M R s l
  have he' := abs_le.mp he
  have hf' := abs_le.mp hf
  have hdiff : trialOffset M R s j-trialOffset M R s l =
      6*((j.val : ℝ)-(l.val : ℝ))*trialSpacing R s := by unfold trialOffset; ring
  have hne : j.val ≠ l.val := fun hh => hjl (Fin.ext hh)
  rcases lt_or_gt_of_ne hne with hlt | hgt
  · have hn : (j.val : ℝ)+1 ≤ l.val := by exact_mod_cast hlt
    have ha := neg_le_abs (transverse b (trialTranslate b X Y M R s j)-
      transverse b (trialTranslate b X Y M R s l))
    nlinarith [mul_le_mul_of_nonneg_right hn (by linarith : 0 ≤ trialSpacing R s)]
  · have hn : (l.val : ℝ)+1 ≤ j.val := by exact_mod_cast hgt
    have ha := le_abs_self (transverse b (trialTranslate b X Y M R s j)-
      transverse b (trialTranslate b X Y M R s l))
    nlinarith [mul_le_mul_of_nonneg_right hn (by linarith : 0 ≤ trialSpacing R s)]

end PlanarFPP

end OAI