File size: 2,385 Bytes
9b6e014 | 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 | 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
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