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59d1c80 | 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 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 | import OAI.Geometry.IsometricImmersion.Curvature.DiskCurvatureProfiles
import OAI.Geometry.IsometricImmersion.Curvature.PatchCurvatureProfiles
import Mathlib.Basic.Countable.Basic
import Mathlib.Logic.Encodable.Basic
namespace OAI
noncomputable section
open scoped ContDiff Topology BigOperators Matrix
open Set Filter
namespace SmoothLocal.Geometry
abbrev PatchAddress := Σ n : ℕ, Σ k : ℕ, Σ R : OrientationLabel,
{c : Coord // c ∈ orientedExteriorCenters n k R}
instance : Countable PatchAddress := by
classical
let : ∀ (n k : ℕ) (R : OrientationLabel),
Finite {c : Coord // c ∈ orientedExteriorCenters n k R} :=
fun n k R => (orientedExteriorCenters n k R).finite_toSet.to_subtype
infer_instance
abbrev AssemblyProfileIndex := ℕ ⊕ PatchAddress
instance assemblyProfileEncodable : Encodable AssemblyProfileIndex :=
Encodable.ofCountable AssemblyProfileIndex
def patchAddressCarrier (a : PatchAddress) : Set Coord :=
orientedClosedPatch a.1 a.2.1 a.2.2.1 a.2.2.2.val
def patchAddressProfile (a : PatchAddress) : Coord → ℝ :=
orientedPatchProfile a.1 a.2.1 a.2.2.1 a.2.2.2.val
theorem patchAddressCarriers_pairwise_disjoint :
Pairwise (fun a b => Disjoint (patchAddressCarrier a) (patchAddressCarrier b)) := by
intro a b hab
rcases a with ⟨n, k, R, c, hc⟩
rcases b with ⟨m, l, S, d, hd⟩
by_cases hnm : n = m
· subst m
by_cases hkl : k = l
· subst l
by_cases hRS : R = S
· subst S
apply orientedClosedPatches_disjoint_same_layer n k R hc hd
intro hcd
subst d
exact hab rfl
· apply orientedClosedPatches_disjoint_different_layers n
(a := (k, R)) (b := (k, S)) _ hc hd
exact fun h => hRS (congrArg Prod.snd h)
· apply orientedClosedPatches_disjoint_different_layers n
(a := (k, R)) (b := (l, S)) _ hc hd
exact fun h => hkl (congrArg Prod.fst h)
· exact orientedClosedPatches_disjoint_different_disks hnm k l R S hc hd
def assemblyCarrier : AssemblyProfileIndex → Set Coord
| .inl n => accumulatingDisk n
| .inr a => patchAddressCarrier a
def assemblyProfile : AssemblyProfileIndex → Coord → ℝ
| .inl n => accumulatingDiskProfile n
| .inr a => patchAddressProfile a
theorem assemblyProfile_contDiff (i : AssemblyProfileIndex) : ContDiff ℝ ∞ (assemblyProfile i) := by
cases i with
| inl n => exact accumulatingDiskProfile_contDiff n
| inr a => exact orientedPatchProfile_contDiff _ _ _ _
theorem assemblyProfile_hasCompactSupport (i : AssemblyProfileIndex) :
HasCompactSupport (assemblyProfile i) := by
cases i with
| inl n => exact accumulatingDiskProfile_hasCompactSupport n
| inr a => exact orientedPatchProfile_hasCompactSupport _ _ _ _
theorem assemblyProfile_tsupport_subset (i : AssemblyProfileIndex) :
tsupport (assemblyProfile i) ⊆ assemblyCarrier i := by
cases i with
| inl n => exact accumulatingDiskProfile_tsupport_subset n
| inr a => exact orientedPatchProfile_tsupport_subset_patch _ _ _ _
theorem assemblyCarriers_pairwise_disjoint :
Pairwise (fun i j => Disjoint (assemblyCarrier i) (assemblyCarrier j)) := by
intro i j hij
cases i with
| inl n =>
cases j with
| inl m => exact accumulatingDisks_pairwise_disjoint (fun h => hij (congrArg Sum.inl h))
| inr a =>
exact (orientedClosedPatch_disjoint_every_disk a.1 a.2.1 a.2.2.1 a.2.2.2.property n).symm
| inr a =>
cases j with
| inl n => exact orientedClosedPatch_disjoint_every_disk a.1 a.2.1 a.2.2.1 a.2.2.2.property n
| inr b => exact patchAddressCarriers_pairwise_disjoint (fun h => hij (congrArg Sum.inr h))
theorem zero_not_mem_assemblyCarrier (i : AssemblyProfileIndex) :
(0 : Coord) ∉ assemblyCarrier i := by
cases i with
| inl n => exact zero_not_mem_accumulatingDisk n
| inr a =>
intro h
exact zero_not_mem_enlargedAccumulatingDisk a.1
(orientedClosedPatch_subset_enlarged a.1 a.2.1 a.2.2.1 a.2.2.2.property h)
def enumeratedAssemblyProfile (j : ℕ) : Coord → ℝ :=
match Encodable.decode₂ AssemblyProfileIndex j with
| none => 0
| some i => assemblyProfile i
theorem enumeratedAssemblyProfile_encode (i : AssemblyProfileIndex) :
enumeratedAssemblyProfile (Encodable.encode i) = assemblyProfile i := by
simp only [enumeratedAssemblyProfile, Encodable.decode₂_encode]
theorem enumeratedAssemblyProfile_contDiff (j : ℕ) :
ContDiff ℝ ∞ (enumeratedAssemblyProfile j) := by
cases h : Encodable.decode₂ AssemblyProfileIndex j with
| none =>
simp only [enumeratedAssemblyProfile, h]
change ContDiff ℝ ∞ (fun _ : Coord => (0 : ℝ))
exact contDiff_const
| some i => simpa only [enumeratedAssemblyProfile, h] using assemblyProfile_contDiff i
theorem enumeratedAssemblyProfile_hasCompactSupport (j : ℕ) :
HasCompactSupport (enumeratedAssemblyProfile j) := by
cases h : Encodable.decode₂ AssemblyProfileIndex j with
| none => simpa only [enumeratedAssemblyProfile, h] using
(HasCompactSupport.zero : HasCompactSupport (0 : Coord → ℝ))
| some i => simpa only [enumeratedAssemblyProfile, h] using assemblyProfile_hasCompactSupport i
theorem zero_not_mem_enumeratedAssemblyProfile_tsupport (j : ℕ) :
(0 : Coord) ∉ tsupport (enumeratedAssemblyProfile j) := by
cases h : Encodable.decode₂ AssemblyProfileIndex j with
| none => simp [enumeratedAssemblyProfile, h]
| some i =>
simpa only [enumeratedAssemblyProfile, h] using
(fun hp => zero_not_mem_assemblyCarrier i (assemblyProfile_tsupport_subset i hp))
theorem enumeratedAssemblyProfile_zero_other (i : AssemblyProfileIndex) {p : Coord}
(hp : p ∈ assemblyCarrier i) {j : ℕ} (hj : j ≠ Encodable.encode i) :
enumeratedAssemblyProfile j p = 0 := by
cases hd : Encodable.decode₂ AssemblyProfileIndex j with
| none => simp [enumeratedAssemblyProfile, hd]
| some a =>
have ha : a ≠ i := by
intro hai
subst a
exact hj (Encodable.decode₂_eq_some.mp hd).symm
have hnot : p ∉ tsupport (assemblyProfile a) := fun hpa =>
Set.disjoint_left.mp (assemblyCarriers_pairwise_disjoint ha)
(assemblyProfile_tsupport_subset a hpa) hp
have hz : assemblyProfile a p = 0 := by
by_contra hne
exact hnot (subset_tsupport (assemblyProfile a) hne)
simpa only [enumeratedAssemblyProfile, hd] using hz
end SmoothLocal.Geometry
end
end OAI
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