| import OAI.Geometry.CylinderCovering.Convexity.NormalFrameDefinitions |
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| namespace OAI |
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| noncomputable section |
| open Finset |
| open scoped BigOperators InnerProductSpace |
| namespace CylinderNormalFrame |
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|
| theorem positive_mass_eq_negative_mass {ι : Type*} [Fintype ι] |
| (x : ι → ℝ) (hx : ∑ i, x i = 0) : |
| ∑ i, max (x i) 0 = ∑ i, max (-x i) 0 := by |
| have hpoint (i : ι) : max (x i) 0 - max (-x i) 0 = x i := by |
| by_cases hi : 0 ≤ x i |
| · rw [max_eq_left hi, max_eq_right (by linarith)] |
| ring |
| · rw [max_eq_right (le_of_not_ge hi), max_eq_left (by linarith)] |
| ring |
| have hsum := congrArg (fun f : ι → ℝ => ∑ i, f i) (funext hpoint) |
| simp only [sum_sub_distrib, hx] at hsum |
| linarith |
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|
| theorem sum_abs_eq_twice_positive_mass {ι : Type*} [Fintype ι] |
| (x : ι → ℝ) (hx : ∑ i, x i = 0) : |
| ∑ i, |x i| = 2 * ∑ i, max (x i) 0 := by |
| have hpoint (i : ι) : |x i| = max (x i) 0 + max (-x i) 0 := by |
| by_cases hi : 0 ≤ x i |
| · rw [abs_of_nonneg hi, max_eq_left hi, max_eq_right (by linarith)] |
| ring |
| · rw [abs_of_nonpos (le_of_not_ge hi), max_eq_right (le_of_not_ge hi), |
| max_eq_left (by linarith)] |
| ring |
| simp_rw [hpoint, sum_add_distrib] |
| rw [positive_mass_eq_negative_mass x hx] |
| ring |
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|
| theorem sum_sq_le_half_sq_sum_abs {ι : Type*} [Fintype ι] |
| (x : ι → ℝ) (hx : ∑ i, x i = 0) : |
| ∑ i, x i ^ 2 ≤ (∑ i, |x i|) ^ 2 / 2 := by |
| have hpoint (i : ι) : x i ^ 2 = (max (x i) 0) ^ 2 + (max (-x i) 0) ^ 2 := by |
| by_cases hi : 0 ≤ x i |
| · rw [max_eq_left hi, max_eq_right (by linarith)] |
| ring |
| · rw [max_eq_right (le_of_not_ge hi), max_eq_left (by linarith)] |
| ring |
| have hp : ∑ i, (max (x i) 0) ^ 2 ≤ (∑ i, max (x i) 0) ^ 2 := |
| sum_sq_le_sq_sum_of_nonneg (fun i _ => le_max_right _ _) |
| have hn : ∑ i, (max (-x i) 0) ^ 2 ≤ (∑ i, max (-x i) 0) ^ 2 := |
| sum_sq_le_sq_sum_of_nonneg (fun i _ => le_max_right _ _) |
| rw [← positive_mass_eq_negative_mass x hx] at hn |
| rw [sum_abs_eq_twice_positive_mass x hx] |
| simp_rw [hpoint, sum_add_distrib] |
| nlinarith |
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|
| theorem sum_normal : ∑ i : Fin 4, normal i = 0 := by |
| ext j |
| fin_cases j <;> simp [normal, Fin.sum_univ_succ] |
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| theorem inner_normal_zero (u : Space) : |
| inner ℝ u (normal 0) = (Real.sqrt 2 * u 1 - u 2) / Real.sqrt 3 := by |
| simp [normal, PiLp.inner_apply, Fin.sum_univ_succ, div_eq_mul_inv] |
| ring |
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|
| theorem inner_normal_one (u : Space) : |
| inner ℝ u (normal 1) = (-Real.sqrt 2 * u 1 - u 2) / Real.sqrt 3 := by |
| simp [normal, PiLp.inner_apply, Fin.sum_univ_succ, div_eq_mul_inv] |
| ring |
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|
| theorem inner_normal_two (u : Space) : |
| inner ℝ u (normal 2) = (Real.sqrt 2 * u 0 + u 2) / Real.sqrt 3 := by |
| simp [normal, PiLp.inner_apply, Fin.sum_univ_succ, div_eq_mul_inv] |
| ring |
|
|
| theorem inner_normal_three (u : Space) : |
| inner ℝ u (normal 3) = (-Real.sqrt 2 * u 0 + u 2) / Real.sqrt 3 := by |
| simp [normal, PiLp.inner_apply, Fin.sum_univ_succ, div_eq_mul_inv] |
| ring |
|
|
| theorem sum_sq_inner_normal (u : Space) : |
| ∑ i : Fin 4, (inner ℝ u (normal i)) ^ 2 = (4 / 3 : ℝ) * ‖u‖ ^ 2 := by |
| rw [EuclideanSpace.real_norm_sq_eq] |
| simp only [Fin.sum_univ_four, inner_normal_zero, inner_normal_one, |
| inner_normal_two, inner_normal_three] |
| simp only [Fin.sum_univ_three] |
| have h3 : (Real.sqrt 3) ^ 2 = 3 := Real.sq_sqrt (by norm_num) |
| have h2 : (Real.sqrt 2) ^ 2 = 2 := Real.sq_sqrt (by norm_num) |
| simp only [div_pow, h3] |
| nlinarith [sq_nonneg (u 0), sq_nonneg (u 1), sq_nonneg (u 2)] |
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| theorem normal_unit (i : Fin 4) : ‖normal i‖ = 1 := by |
| have h3 : (Real.sqrt 3) ^ 2 = 3 := Real.sq_sqrt (by norm_num) |
| have h2 : (Real.sqrt 2) ^ 2 = 2 := Real.sq_sqrt (by norm_num) |
| have hs : ‖normal i‖ ^ 2 = 1 := by |
| rw [EuclideanSpace.real_norm_sq_eq] |
| fin_cases i <;> norm_num [normal, Fin.sum_univ_succ, mul_pow, h3, h2] |
| nlinarith [norm_nonneg (normal i)] |
|
|
| theorem frameCost_lower_bound (u : Space) (hu : ‖u‖ = 1) : |
| Real.sqrt 2 ≤ frameCost u := by |
| have hsum : ∑ i : Fin 4, inner ℝ u (normal i) = 0 := by |
| rw [← inner_sum, sum_normal, inner_zero_right] |
| have hsq := sum_sq_le_half_sq_sum_abs (fun i : Fin 4 => inner ℝ u (normal i)) hsum |
| rw [sum_sq_inner_normal, hu] at hsq |
| have hnonneg : 0 ≤ ∑ i : Fin 4, |inner ℝ u (normal i)| := |
| sum_nonneg (fun i _ => abs_nonneg _) |
| have h3 : (Real.sqrt 3) ^ 2 = 3 := Real.sq_sqrt (by norm_num) |
| apply Real.sqrt_le_iff.mpr |
| constructor |
| · unfold frameCost |
| positivity |
| · unfold frameCost |
| nlinarith [sq_nonneg (∑ i : Fin 4, |inner ℝ u (normal i)|)] |
| |
| theorem edgeDirection_unit : ‖edgeDirection‖ = 1 := by |
| simp [edgeDirection, EuclideanSpace.norm_eq, Fin.sum_univ_succ] |
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| theorem frameCost_edgeDirection : frameCost edgeDirection = Real.sqrt 2 := by |
| have h3 : Real.sqrt 3 ≠ 0 := ne_of_gt (Real.sqrt_pos.2 (by norm_num)) |
| have h2 : 0 ≤ Real.sqrt 2 / Real.sqrt 3 := by positivity |
| simp only [frameCost, Fin.sum_univ_four, inner_normal_zero, inner_normal_one, |
| inner_normal_two, inner_normal_three] |
| simp [edgeDirection, abs_of_nonneg h2, neg_div] |
| field_simp |
| norm_num |
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|
| theorem barycentricCost_lower_bound (u : BarycentricSpace) |
| (hu : ‖u‖ = 1) (hsum : ∑ i : Fin 4, u i = 0) : |
| 1 / Real.sqrt 2 ≤ barycentricCost u := by |
| have hsq := sum_sq_le_half_sq_sum_abs (fun i : Fin 4 => u i) hsum |
| rw [← EuclideanSpace.real_norm_sq_eq, hu] at hsq |
| have hnonneg : 0 ≤ ∑ i : Fin 4, |u i| := sum_nonneg (fun i _ => abs_nonneg _) |
| have hbound : Real.sqrt 2 ≤ ∑ i : Fin 4, |u i| := by |
| apply Real.sqrt_le_iff.mpr |
| exact ⟨hnonneg, by nlinarith⟩ |
| have h2pos : 0 < Real.sqrt 2 := Real.sqrt_pos.2 (by norm_num) |
| have h2sq : (Real.sqrt 2) ^ 2 = 2 := Real.sq_sqrt (by norm_num) |
| have hid : 1 / Real.sqrt 2 = Real.sqrt 2 / 2 := by |
| apply (div_eq_iff h2pos.ne').2 |
| nlinarith |
| rw [hid] |
| unfold barycentricCost |
| linarith |
|
|
| theorem sum_normal_mul (j k : Fin 3) : |
| ∑ i : Fin 4, normal i j * normal i k = if j = k then (4 / 3 : ℝ) else 0 := by |
| have h3 : (Real.sqrt 3) ^ 2 = 3 := Real.sq_sqrt (by norm_num) |
| have h2 : (Real.sqrt 2) ^ 2 = 2 := Real.sq_sqrt (by norm_num) |
| have h3ne : Real.sqrt 3 ≠ 0 := ne_of_gt (Real.sqrt_pos.2 (by norm_num)) |
| fin_cases j <;> fin_cases k <;> |
| simp [normal, Fin.sum_univ_succ] <;> field_simp <;> nlinarith |
|
|
| theorem isLeast_frameCost : |
| IsLeast {a : ℝ | ∃ u : Space, ‖u‖ = 1 ∧ a = frameCost u} (Real.sqrt 2) := by |
| constructor |
| · exact ⟨edgeDirection, edgeDirection_unit, frameCost_edgeDirection.symm⟩ |
| · rintro a ⟨u, hu, rfl⟩ |
| exact frameCost_lower_bound u hu |
|
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| theorem barycentricEdge_sum : ∑ i : Fin 4, barycentricEdge i = 0 := by |
| simp [barycentricEdge, Fin.sum_univ_succ] |
|
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| theorem barycentricEdge_unit : ‖barycentricEdge‖ = 1 := by |
| have hs : ‖barycentricEdge‖ ^ 2 = 1 := by |
| rw [EuclideanSpace.real_norm_sq_eq] |
| norm_num [barycentricEdge, Fin.sum_univ_succ, div_pow, Real.sq_sqrt] |
| nlinarith [norm_nonneg barycentricEdge] |
|
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| theorem barycentricCost_edge : barycentricCost barycentricEdge = 1 / Real.sqrt 2 := by |
| simp [barycentricCost, barycentricEdge, Fin.sum_univ_succ, |
| abs_of_nonneg (Real.sqrt_nonneg (2 : ℝ))] |
| ring |
|
|
| theorem isLeast_barycentricCost : |
| IsLeast {a : ℝ | ∃ u : BarycentricSpace, |
| ‖u‖ = 1 ∧ ∑ i : Fin 4, u i = 0 ∧ a = barycentricCost u} (1 / Real.sqrt 2) := by |
| constructor |
| · exact ⟨barycentricEdge, barycentricEdge_unit, barycentricEdge_sum, |
| barycentricCost_edge.symm⟩ |
| · rintro a ⟨u, hu, hsum, rfl⟩ |
| exact barycentricCost_lower_bound u hu hsum |
|
|
| end CylinderNormalFrame |
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| end |
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| end OAI |
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