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dd14c26 | 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 | import Mathlib
import OAI.Probability.LatticeSurface.Convexity.RadialExposure
import OAI.Probability.LatticeSurface.Geometry.BlockEdgeGeometry
import OAI.Probability.LatticeSurface.Geometry.BlockFacetGeometry
import OAI.Probability.LatticeSurface.Geometry.BlockVolume
import OAI.Probability.LatticeSurface.Geometry.EdgeCoordinates
import OAI.Probability.LatticeSurface.Geometry.FacetExposure
import OAI.Probability.LatticeSurface.Geometry.TriangularCover
import OAI.Probability.LatticeSurface.Geometry.TriangularGeometry
namespace OAI
open MeasureTheory
open scoped ENNReal
noncomputable section
namespace TriangularLipschitz
end TriangularLipschitz
namespace FacetExposure
end FacetExposure
namespace TriangularLipschitz
end TriangularLipschitz
namespace TriangularLipschitz
end TriangularLipschitz
namespace TriangularLipschitz
end TriangularLipschitz
open MeasureTheory Set
open scoped ENNReal
noncomputable section
namespace EdgeCoordinates
end EdgeCoordinates
namespace EdgeCoordinates
end EdgeCoordinates
namespace EdgeCoordinates
end EdgeCoordinates
namespace TriangularLipschitz
end TriangularLipschitz
namespace TriangularLipschitz
end TriangularLipschitz
namespace TriangularLipschitz
lemma radialPoint_zero (L : ℕ) (side : Fin 4) : radialPoint L side 0=faceCenter L side := by
fin_cases side <;> simp [radialPoint,faceCenter]
lemma radialPoint_center (m : ℕ) (side : Fin 4) : radialPoint (2*m) side m=((m:ℤ),(m:ℤ)) := by
fin_cases side <;> simp [radialPoint] <;> omega
lemma radialPoint_mem_box {m k : ℕ} (hk : k ≤ m) (side : Fin 4) :
radialPoint (2*m) side k ∈ box (2*m) := by
fin_cases side <;> simp [radialPoint,mem_box] <;> omega
lemma radialPoint_mem_interior {m k : ℕ} (hk0 : 0 < k) (hk : k ≤ m) (side : Fin 4) :
radialPoint (2*m) side k ∈ interior (2*m) := by
fin_cases side <;> simp [radialPoint,interior] <;> omega
lemma faceCenter_mem_shell {m : ℕ} (hm : 0 < m) (side : Fin 4) :
faceCenter (2*m) side ∈ shell (2*m) := by
fin_cases side <;> simp [faceCenter,mem_shell,mem_box] <;> omega
lemma radialPoint_fixed_iff {m k : ℕ} (hm : 0 < m) (hk : k ≤ m) (side : Fin 4) :
radialPoint (2*m) side k ∉ interior (2*m) ↔ k=0 := by
constructor
· intro hi
by_contra h
exact hi (radialPoint_mem_interior (Nat.pos_of_ne_zero h) hk side)
· rintro rfl
rw [radialPoint_zero]
exact (shell_eq_box_not_interior.mp (faceCenter_mem_shell hm side)).2
lemma radialPoint_adj (L k : ℕ) (side : Fin 4) :
Adj (radialPoint L side k) (radialPoint L side (k+1)) := by
fin_cases side <;> simp [radialPoint,Adj] <;> omega
lemma shifted_radial_facet {m k : ℕ} {σ : ℝ} {h χ g : Vertex → ℝ}
(hm : 0 < m) (hk : k < m) (side : Fin 4)
(hg : ∀ b ∈ shell (2*m), g b=h b)
(hgrad : |g (radialPoint (2*m) side k)-g (radialPoint (2*m) side (k+1))|=1) :
let x := radialPoint (2*m) side k
let y := radialPoint (2*m) side (k+1)
let τ := g y-g x
τ*(filledHeight (2*m) (fun b => h b+σ*χ b)
(fun z => g z+σ*χ (faceCenter (2*m) side)) y-
filledHeight (2*m) (fun b => h b+σ*χ b)
(fun z => g z+σ*χ (faceCenter (2*m) side)) x)=1 := by
dsimp only
have hy := radialPoint_mem_interior (by omega : 0 < k+1) (by omega : k+1 ≤ m) side
have hgabs : |g (radialPoint (2*m) side (k+1))-g (radialPoint (2*m) side k)|=1 := by
rwa [abs_sub_comm]
have hsquare : (g (radialPoint (2*m) side (k+1))-g (radialPoint (2*m) side k))^2=1 := by
nlinarith [sq_abs (g (radialPoint (2*m) side (k+1))-g (radialPoint (2*m) side k))]
rw [filledHeight, dite_eq_left hy]
by_cases hx : radialPoint (2*m) side k ∈ interior (2*m)
· rw [filledHeight,dite_eq_left hx]
nlinarith
· have hk0 := (radialPoint_fixed_iff hm (Nat.le_of_lt hk) side).mp hx
subst k
rw [filledHeight,dite_eq_right hx,radialPoint_zero]
have hg0 := hg (faceCenter (2*m) side) (faceCenter_mem_shell hm side)
rw [radialPoint_zero] at hsquare
rw [← hg0]
nlinarith
lemma shifted_shell_error {L : ℕ} {h χ g : Vertex → ℝ} {σ β : ℝ} {v : Vertex}
(_hβ : 0 < β) (hv : v ∈ shell L)
(hg : ∀ b ∈ shell L, g b=h b)
(hplus : 0 < fillingVolume L (fun x => h x+χ x))
(hminus : 0 < fillingVolume L (fun x => h x-χ x))
(hσ : |σ| * (2*(L:ℝ)) ≤ β/4) :
∀ b ∈ shell L, |(h b+σ*χ b)-(g b+σ*χ v)| ≤ β/4 := by
intro b hb
rw [hg b hb]
calc
_ = |σ| * |χ b-χ v| := by rw [← abs_mul]; congr 1; ring
_ ≤ |σ| * latticeDist b v := mul_le_mul_of_nonneg_left
(endpoint_shell_difference hplus hminus hb hv) (abs_nonneg σ)
_ ≤ |σ| * (2*(L:ℝ)) := mul_le_mul_of_nonneg_left
(latticeDist_box_le (shell_eq_box_not_interior.mp hb).1 (shell_eq_box_not_interior.mp hv).1)
(abs_nonneg σ)
_ ≤ β/4 := hσ
end TriangularLipschitz
end
end
end OAI
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