File size: 2,965 Bytes
2fca478 | 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 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 | import OAI.Geometry.PeriodicTiling.TilingBasic
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.Algebra.Module.ZLattice.Basic
namespace OAI
noncomputable section
namespace PeriodicTilingThree
open MeasureTheory
@[simp] theorem castLattice_apply {d : ℕ} (z : Lattice d) (i : Fin d) :
castLattice z i = (z i : ℝ) := rfl
theorem castLattice_injective {d : ℕ} :
Function.Injective (castLattice : Lattice d → Space d) := by
intro z w h
funext i
exact Int.cast_injective (congrArg (fun x : Space d => x i) h)
def unitVoxel {d : ℕ} (c : Space d) : Set (Space d) :=
{x | x - c ∈ unitCube d}
theorem mem_unitCube {d : ℕ} {x : Space d} :
x ∈ unitCube d ↔ ∀ i, 0 ≤ x i ∧ x i ≤ 1 := by
constructor
· intro h i
exact ⟨h.1 i, h.2 i⟩
· intro h
exact ⟨fun i => (h i).1, fun i => (h i).2⟩
@[simp] theorem mem_unitVoxel {d : ℕ} {c x : Space d} :
x ∈ unitVoxel c ↔ x - c ∈ unitCube d := Iff.rfl
theorem unitVoxel_eq_Icc {d : ℕ} (c : Space d) :
unitVoxel c = Set.Icc c (c + 1) := by
ext x
constructor
· intro hx
have h := mem_unitCube.mp hx
constructor
· intro i
exact sub_nonneg.mp (h i).1
· intro i
have hi := (sub_le_iff_le_add.mp (h i).2)
simpa [add_comm] using hi
· intro hx
apply mem_unitCube.mpr
intro i
constructor
· exact sub_nonneg.mpr (hx.1 i)
· apply sub_le_iff_le_add.mpr
simpa [add_comm] using hx.2 i
@[simp] theorem mem_Thickening {d : ℕ} {F : Finset (Lattice d)} {x : Space d} :
x ∈ Thickening F ↔ ∃ z ∈ F, x - castLattice z ∈ unitCube d := Iff.rfl
theorem unitVoxel_subset_Thickening {d : ℕ} {F : Finset (Lattice d)}
{z : Lattice d} (hz : z ∈ F) : unitVoxel (castLattice z) ⊆ Thickening F := by
intro x hx
exact ⟨z, hz, hx⟩
def TypedAETiles {d : ℕ} (Ω0 Ω1 C C1 : Set (Space d)) : Prop := by
classical
exact ∀ᵐ x ∂volume, ∃! c : C,
x - (c : Space d) ∈ if (c : Space d) ∈ C1 then Ω1 else Ω0
def scaledCast {d : ℕ} (m : ℕ) : Lattice d →+ Space d where
toFun z := (m : ℝ) • castLattice z
map_zero' := by simp
map_add' z w := by simp [smul_add]
@[simp] theorem scaledCast_apply {d : ℕ} (m : ℕ) (z : Lattice d) :
scaledCast m z = (m : ℝ) • castLattice z := rfl
theorem scaledCast_injective {d : ℕ} {m : ℕ} (hm : m ≠ 0) :
Function.Injective (scaledCast m : Lattice d → Space d) := by
intro z w h
funext i
have hi := congrArg (fun x : Space d => x i) h
change (m : ℝ) * (z i : ℝ) = (m : ℝ) * (w i : ℝ) at hi
exact Int.cast_injective (mul_left_cancel₀ (Nat.cast_ne_zero.mpr hm) hi)
def scaledGrid (m : ℕ) : AddSubgroup (Space 3) :=
(scaledCast m : Lattice 3 →+ Space 3).range
theorem mem_scaledGrid {m : ℕ} {x : Space 3} :
x ∈ scaledGrid m ↔ ∃ z : Lattice 3, (m : ℝ) • castLattice z = x := Iff.rfl
end PeriodicTilingThree
end
end OAI
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