| 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 |
|
|
