| import OAI.Geometry.CylinderCovering.Caps.CapsDefinitions |
| import OAI.Geometry.CylinderCovering.Radial.CapCoverage |
|
|
| namespace OAI |
|
|
| open Set |
| noncomputable section |
| namespace CylinderCovering.HalfHeightCaps |
|
|
| theorem covered_of_padding {δ : ℝ} {R b : ℝ → ℝ} |
| (hδ : 0 < δ) |
| (hcap : ∀ r ∈ Icc (-1 : ℝ) 1, |
| 2 * R r = 1 + (δ / 2) ^ 2 * (RadialCaps.H r + b r)) |
| (hb : ∀ r ∈ Icc (-1 : ℝ) 1, 0 ≤ b r) |
| (hpadding : ∀ p ∈ Icc (-1 : ℝ) 1, ∀ q ∈ Icc (-1 : ℝ) 1, |
| 9 * (δ / 2) ≤ b p + b q) : Covered δ R := by |
| intro X Y Z hX hY |
| have hx : |2 * Y| ≤ 1 + (2 * Z - 1) := by |
| rw [abs_mul] |
| norm_num |
| linarith only [hY] |
| have hy : |2 * X| ≤ 1 - (2 * Z - 1) := by |
| rw [abs_mul] |
| norm_num |
| linarith only [hX] |
| have h := RadialCapCoverage.padding_coverage (eps := δ / 2) |
| (R := fun r => 2 * R r) (b := b) (by positivity) hcap hb hpadding hx hy |
| rcases h with ⟨q, ρ, hq, hρ, hR, heq, ht⟩ | ⟨p, σ, hp, hσ, hR, heq, ht⟩ |
| · right |
| refine ⟨q, ρ / 2, hq, by positivity, by linarith only [hR], ?_, ?_⟩ |
| · unfold alpha |
| unfold RadialCaps.G at heq |
| nlinarith only [heq] |
| · unfold beta |
| nlinarith only [ht] |
| · left |
| refine ⟨p, σ / 2, hp, by positivity, by linarith only [hR], ?_, ?_⟩ |
| · unfold alpha |
| unfold RadialCaps.G at heq |
| nlinarith only [heq] |
| · unfold beta |
| nlinarith only [ht] |
|
|
| theorem cubic_dictionary (δ r : ℝ) : |
| 2 * cubicCap δ r = RadialCaps.cubicCap 64 (δ / 2) r := by |
| unfold cubicCap F RadialCaps.cubicCap RadialCaps.H |
| ring |
|
|
| theorem cubic_covers {δ : ℝ} (hδ : 0 < δ) : Covered δ (cubicCap δ) := by |
| apply covered_of_padding (b := fun _ => 32 * δ) hδ |
| · intro r hr |
| unfold cubicCap F RadialCaps.H |
| ring |
| · intro r hr |
| positivity |
| · intro p hp q hq |
| linarith only [hδ] |
|
|
| theorem fractional_covers {δ : ℝ} (hδ : 0 < δ) (hsmall : δ ≤ 1) : |
| Covered δ (fractionalCap δ) := by |
| have hs : 0 ≤ Real.sqrt δ := Real.sqrt_nonneg δ |
| have hs1 : Real.sqrt δ ≤ 1 := by nlinarith [Real.sq_sqrt hδ.le] |
| have hδs : δ ≤ Real.sqrt δ := by nlinarith [Real.sq_sqrt hδ.le] |
| apply covered_of_padding (b := fun _ => 8 * Real.sqrt δ) hδ |
| · intro r hr |
| unfold fractionalCap F RadialCaps.H |
| rw [RadialCapCoverage.fractional_power δ hδ] |
| ring |
| · intro r hr |
| positivity |
| · intro p hp q hq |
| nlinarith only [hδs, hs] |
|
|
| theorem fractional_dictionary {δ : ℝ} (hδ : 0 < δ) (r : ℝ) : |
| 2 * fractionalCap δ r = RadialCaps.fractionalCap (8 * Real.sqrt 2) (δ / 2) r := by |
| unfold fractionalCap RadialCaps.fractionalCap F RadialCaps.H |
| rw [RadialCapCoverage.fractional_power δ hδ, |
| RadialCapCoverage.fractional_power (δ / 2) (by positivity), Real.sqrt_div hδ.le] |
| have hs2 : (Real.sqrt 2) ^ 2 = 2 := Real.sq_sqrt (by norm_num) |
| have hs0 : Real.sqrt 2 ≠ 0 := ne_of_gt (Real.sqrt_pos.2 (by norm_num)) |
| field_simp |
| nlinarith only [hs2] |
|
|
| theorem halfheight_coverage : CoverageStatement := by |
| refine ⟨fun δ hδ => cubic_covers hδ, ⟨1, by norm_num, ?_⟩, |
| cubic_dictionary, ?_⟩ |
| · exact fun δ hδ hs => fractional_covers hδ hs |
| · exact fun δ r hδ => fractional_dictionary hδ r |
|
|
| theorem compensation : CompensationStatement := by |
| intro δ X Y Z r s u v hY hZ hX hZv |
| unfold alpha beta at * |
| linear_combination δ * X * hY - δ * Y * hX - hZ - hZv |
|
|
| theorem physical_covers_of_covered {δ : ℝ} {R : ℝ → ℝ} (h : Covered δ R) : |
| physicalTetrahedron ⊆ firstPhysicalSweep δ R ∪ secondPhysicalSweep δ R := by |
| rintro z ⟨X, Y, Z, hX, hY, rfl⟩ |
| have hX1 : |X| ≤ 1 := by linarith only [hX, hY, abs_nonneg Y] |
| have hY1 : |Y| ≤ 1 := by linarith only [hX, hY, abs_nonneg X] |
| rcases h X Y Z hX hY with ⟨r, u, hr, hu, hR, hYr, hZr⟩ | |
| ⟨s, v, hs, hv, hR, hXs, hZs⟩ |
| · left |
| refine ⟨r, abs_le.mp hr, u, ⟨hu, hR⟩, X, hX1, ?_⟩ |
| rw [← hYr, ← hZr] |
| · right |
| refine ⟨s, abs_le.mp hs, v, ⟨hv, hR⟩, Y, hY1, ?_⟩ |
| have hZs' : Z = 1 - v + δ * Y * beta s := by linarith only [hZs] |
| rw [← hXs, ← hZs'] |
|
|
| theorem halfheight_physical_coverage : PhysicalCoverageStatement := by |
| constructor |
| · exact fun δ hδ => physical_covers_of_covered (cubic_covers hδ) |
| · exact fun δ hδ hs => physical_covers_of_covered (fractional_covers hδ hs) |
|
|
| theorem fractional_defect (a X Y : ℝ) : |
| X ^ 2 * (a / 2 + X ^ 2 / (2 * a)) + Y ^ 2 * (a / 2 + Y ^ 2 / (2 * a)) - |
| (a / 2 + X ^ 2 / (2 * a)) * Y ^ 2 - |
| (a / 2 + Y ^ 2 / (2 * a)) * X ^ 2 = (X ^ 2 - Y ^ 2) ^ 2 / (2 * a) := by |
| ring |
|
|
| end CylinderCovering.HalfHeightCaps |
|
|
| end |
|
|
| end OAI |
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|
