| import Mathlib |
| import OAI.Probability.Ballisticity.Geometry.SignedHeight |
|
|
| namespace OAI |
|
|
| section |
|
|
| section |
|
|
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal Topology Classical |
| namespace DirectionalTransience |
|
|
| lemma signedHeight_add {d : ℕ} (e : Direction d) (x y : Lattice d) : |
| signedHeight e (x+y) = signedHeight e x+signedHeight e y := by |
| simp only [signedHeight,Pi.add_apply] |
| split <;> omega |
|
|
| lemma signedHeight_zsmul_step {d : ℕ} (e : Direction d) (a : ℤ) : |
| signedHeight e (a • step e) = a := by |
| simp only [signedHeight,Pi.smul_apply,smul_eq_mul,step,ite_true] |
| split <;> simp_all |
|
|
| noncomputable def alignHeight {d : ℕ} (e : Direction d) (a : ℤ) (x : Lattice d) : Lattice d := |
| x+(a-signedHeight e x) • step e |
|
|
| lemma signedHeight_align {d : ℕ} (e : Direction d) (a : ℤ) (x : Lattice d) : |
| signedHeight e (alignHeight e a x) = a := by |
| rw [alignHeight,signedHeight_add,signedHeight_zsmul_step] |
| omega |
|
|
| lemma alignHeight_eq_self {d : ℕ} (e : Direction d) (a : ℤ) (x : Lattice d) |
| (hx : signedHeight e x = a) : alignHeight e a x = x := by |
| simp [alignHeight,hx] |
|
|
| def blockRows {d : ℕ} (e : Direction d) (H i : ℕ) : Set (Lattice d) := |
| {x | (i:ℤ)*H ≤ signedHeight e x ∧ signedHeight e x < (i:ℤ)*H+H} |
|
|
| lemma blockRows_pairwise {d : ℕ} (e : Direction d) (H : ℕ) : |
| Pairwise (fun i j => Disjoint (blockRows e H i) (blockRows e H j)) := by |
| intro i j hij |
| apply Set.disjoint_left.mpr |
| intro x hx hy |
| change (i:ℤ)*H ≤ signedHeight e x ∧ signedHeight e x < (i:ℤ)*H+H at hx |
| change (j:ℤ)*H ≤ signedHeight e x ∧ signedHeight e x < (j:ℤ)*H+H at hy |
| rcases lt_or_gt_of_ne hij with h | h |
| · have hh : (i:ℤ)+1 ≤ j := by exact_mod_cast h |
| nlinarith [Int.natCast_nonneg H] |
| · have hh : (j:ℤ)+1 ≤ i := by exact_mod_cast h |
| nlinarith [Int.natCast_nonneg H] |
|
|
| lemma strip_align_blockRows {d : ℕ} (e : Direction d) (H i : ℕ) (x : Lattice d) : |
| Strip (realPosition (step e)) (alignHeight e ((i:ℤ)*H) x) H = blockRows e H i := by |
| ext y |
| simp only [Strip,Set.mem_ofPred_eq,signedHeight_projection,signedHeight_align,blockRows] |
| constructor |
| · rintro ⟨h₁,h₂⟩ |
| exact ⟨by exact_mod_cast h₁,by exact_mod_cast h₂⟩ |
| · rintro ⟨h₁,h₂⟩ |
| exact ⟨by exact_mod_cast h₁,by exact_mod_cast h₂⟩ |
|
|
| end DirectionalTransience |
|
|
| end |
|
|
| end |
|
|
| end OAI |
|
|
