import OAI.Geometry.HeilbronnTriangle.AnisotropicGeometry import OAI.Geometry.HeilbronnTriangle.AnisotropicDyadic import OAI.Geometry.HeilbronnTriangle.PlaneFiberAssembly import OAI.Geometry.HeilbronnTriangle.IntegerNormalGroups import OAI.Geometry.HeilbronnTriangle.AnisotropicPlaneData namespace OAI noncomputable section namespace Problem355.Anisotropic open Matrix open scoped Matrix open PrimitiveNormal abbrev Triple := PlaneFiberAssembly.Triple theorem thick_fiber_card_le (S : Finset Triple) (y : IntVector) (d : PlaneData y) (t : ℤ) (R₁ R₂ R₃ : ℝ) (h₂₁ : R₂ ≤ R₁) (h₃₂ : R₃ ≤ R₂) (hlarge : d.second ≤ R₃) (hdet : ∀ x ∈ S, x.1 ⬝ᵥ (x.2.1 ⨯₃ x.2.2) = t) (hcross : ∀ x ∈ S, ∃ a : ℤ, a ≠ 0 ∧ x.2.1 ⨯₃ x.2.2 = a • y) (hplane : ∀ x ∈ S, x.2.1 ⬝ᵥ y = 0 ∧ x.2.2 ⬝ᵥ y = 0) (hnorm : ∀ x ∈ S, length x.1 ≤ R₁ ∧ length x.2.1 ≤ R₂ ∧ length x.2.2 ≤ R₃) : (S.card : ℝ) ≤ 2916 * Real.pi ^ 3 * (R₁ * R₂ * R₃) ^ 2 / length y ^ 3 := by have h := PlaneFiberAssembly.triple_count_of_plane_counts S y t R₁ R₂ R₃ (36 * Real.pi * R₁ ^ 2 / length y) (9 * Real.pi * R₂ ^ 2 / length y) (9 * Real.pi * R₃ ^ 2 / length y) (by positivity) (by positivity) hdet hcross hplane hnorm (fun T center hT => affine_plane_card_le y d T center R₁ (hlarge.trans (h₃₂.trans h₂₁)) (fun v hv => (hT v hv).2) (fun v hv w hw => by rw [sub_dotProduct, (hT v hv).1, (hT w hw).1, sub_self])) (fun T hT => d.large_count T R₂ (hlarge.trans h₃₂) hT) (fun T hT => d.large_count T R₃ hlarge hT) convert h using 1; ring theorem thin_fiber_card_le (S : Finset Triple) (y : IntVector) (d : PlaneData y) (t : ℤ) (R₁ R₂ R₃ : ℝ) (h₂₁ : R₂ ≤ R₁) (hlarge : d.second ≤ R₂) (hfirst : d.first ≤ R₃) (hthin : R₃ < d.second) (hdet : ∀ x ∈ S, x.1 ⬝ᵥ (x.2.1 ⨯₃ x.2.2) = t) (hcross : ∀ x ∈ S, ∃ a : ℤ, a ≠ 0 ∧ x.2.1 ⨯₃ x.2.2 = a • y) (hplane : ∀ x ∈ S, x.2.1 ⬝ᵥ y = 0 ∧ x.2.2 ⬝ᵥ y = 0) (hnorm : ∀ x ∈ S, length x.1 ≤ R₁ ∧ length x.2.1 ≤ R₂ ∧ length x.2.2 ≤ R₃) : (S.card : ℝ) ≤ 972 * Real.pi ^ 2 * R₁ ^ 2 * R₂ ^ 2 * R₃ / (length y ^ 2 * d.first) := by have h := PlaneFiberAssembly.triple_count_of_plane_counts S y t R₁ R₂ R₃ (36 * Real.pi * R₁ ^ 2 / length y) (9 * Real.pi * R₂ ^ 2 / length y) (3 * R₃ / d.first) (by positivity) (by positivity) hdet hcross hplane hnorm (fun T center hT => affine_plane_card_le y d T center R₁ (hlarge.trans h₂₁) (fun v hv => (hT v hv).2) (fun v hv w hw => by rw [sub_dotProduct, (hT v hv).1, (hT w hw).1, sub_self])) (fun T hT => d.large_count T R₂ hlarge hT) (fun T hT => d.small_count T R₃ hfirst hthin hT) convert h using 1; ring theorem scales_le_of_nonempty (S : Finset Triple) (y : IntVector) (d : PlaneData y) (t : ℤ) (ht : t ≠ 0) (R₂ R₃ : ℝ) (h₃₂ : R₃ ≤ R₂) (hS : S.Nonempty) (hdet : ∀ x ∈ S, x.1 ⬝ᵥ (x.2.1 ⨯₃ x.2.2) = t) (hplane : ∀ x ∈ S, x.2.1 ⬝ᵥ y = 0 ∧ x.2.2 ⬝ᵥ y = 0) (hnorm : ∀ x ∈ S, length x.2.1 ≤ R₂ ∧ length x.2.2 ≤ R₃) : d.first ≤ R₃ ∧ d.second ≤ R₂ := by obtain ⟨x, hx⟩ := hS have hc : x.2.1 ⨯₃ x.2.2 ≠ 0 := by intro hc apply ht have h := hdet x hx simpa [hc] using h.symm have hv : x.2.2 ≠ 0 := by intro hv apply hc simp [hv] refine ⟨(d.first_le _ (hplane x hx).2 hv).trans (hnorm x hx).2, ?_⟩ exact (d.second_le _ _ (hplane x hx).1 (hplane x hx).2 hc).trans (max_le (hnorm x hx).1 ((hnorm x hx).2.trans h₃₂)) theorem thin_shell_sum_le (S : Finset IntVector) (first weight : IntVector → ℝ) (R₁ R₂ R₃ U Y C₀ C₁ : ℝ) (hU : 0 < U) (hY : 0 < Y) (hR₃ : 0 ≤ R₃) (hUR : U ≤ R₃) (hC₀ : 0 ≤ C₀) (hC₁ : 0 ≤ C₁) (hcard : (S.card : ℝ) ≤ C₀ * U ^ 2 * Y ^ 2) (hrange : ∀ y ∈ S, U ≤ first y ∧ Y ≤ length y) (hweight : ∀ y ∈ S, weight y ≤ C₁ * R₁ ^ 2 * R₂ ^ 2 * R₃ / (length y ^ 2 * first y)) : ∑ y ∈ S, weight y ≤ C₀ * C₁ * (R₁ * R₂ * R₃) ^ 2 := by have hB : 0 ≤ C₁ * R₁ ^ 2 * R₂ ^ 2 * R₃ / (Y ^ 2 * U) := by positivity calc ∑ y ∈ S, weight y ≤ ∑ _y ∈ S, C₁ * R₁ ^ 2 * R₂ ^ 2 * R₃ / (Y ^ 2 * U) := by apply Finset.sum_le_sum intro y hy apply (hweight y hy).trans apply div_le_div_of_nonneg_left (by positivity) (by positivity) exact mul_le_mul (pow_le_pow_left₀ hY.le (hrange y hy).2 2) (hrange y hy).1 hU.le (sq_nonneg _) _ = (S.card : ℝ) * (C₁ * R₁ ^ 2 * R₂ ^ 2 * R₃ / (Y ^ 2 * U)) := by simp _ ≤ (C₀ * U ^ 2 * Y ^ 2) * (C₁ * R₁ ^ 2 * R₂ ^ 2 * R₃ / (Y ^ 2 * U)) := mul_le_mul_of_nonneg_right hcard hB _ = (C₀ * C₁) * U ^ 2 * Y ^ 2 * (R₁ ^ 2 / Y) * (R₂ ^ 2 / Y) * (R₃ / U) := by field_simp _ ≤ (C₀ * C₁) * (R₁ * R₂ * R₃) ^ 2 := thin_normal_group_numeric hU hY hR₃ hUR (mul_nonneg hC₀ hC₁) theorem thin_normal_sum_le (S : Finset IntVector) (first weight : IntVector → ℝ) (R₁ R₂ R₃ C₀ C₁ : ℝ) (h₂ : 1 ≤ R₂) (h₃ : 1 ≤ R₃) (h₂₁ : R₂ ≤ R₁) (h₃₁ : R₃ ≤ R₁) (hC₀ : 0 ≤ C₀) (hC₁ : 0 ≤ C₁) (hweight0 : ∀ y ∈ S, 0 ≤ weight y) (hrange : ∀ y ∈ S, 1 ≤ first y ∧ first y ≤ R₃ ∧ 1 ≤ length y ∧ length y ≤ R₂ * R₃) (hweight : ∀ y ∈ S, weight y ≤ C₁ * R₁ ^ 2 * R₂ ^ 2 * R₃ / (length y ^ 2 * first y)) (hgroup : ∀ (T : Finset IntVector), T ⊆ S → ∀ U Y : ℝ, 1 ≤ U → 1 ≤ Y → (∀ y ∈ T, U ≤ first y ∧ first y ≤ 2 * U ∧ Y ≤ length y ∧ length y ≤ 2 * Y) → (T.card : ℝ) ≤ C₀ * U ^ 2 * Y ^ 2) : ∑ y ∈ S, weight y ≤ 2 * C₀ * C₁ * (R₁ * R₂ * R₃) ^ 2 * (Real.log (2 * R₁) / Real.log 2) ^ 2 := by classical have hprod : 1 ≤ R₂ * R₃ := by nlinarith have hA : 0 ≤ C₀ * C₁ * (R₁ * R₂ * R₃) ^ 2 := by positivity have h := sum_le_of_dyadic_shell_estimates S first length weight h₃ hprod hA hweight0 hrange (by intro i j let T := S.filter fun y => (2 : ℝ) ^ i ≤ first y ∧ first y < 2 ^ (i + 1) ∧ (2 : ℝ) ^ j ≤ length y ∧ length y < 2 ^ (j + 1) have hU : 1 ≤ (2 : ℝ) ^ i := one_le_pow₀ (by norm_num) have hY : 1 ≤ (2 : ℝ) ^ j := one_le_pow₀ (by norm_num) have hTS : T ⊆ S := Finset.filter_subset _ _ have hT (y : IntVector) (hy : y ∈ T) : (2 : ℝ) ^ i ≤ first y ∧ first y ≤ 2 * 2 ^ i ∧ (2 : ℝ) ^ j ≤ length y ∧ length y ≤ 2 * 2 ^ j := by obtain ⟨_, h₁, h₂, h₃, h₄⟩ := Finset.mem_filter.mp hy exact ⟨h₁, by simpa [pow_succ, mul_comm] using h₂.le, h₃, by simpa [pow_succ, mul_comm] using h₄.le⟩ change ∑ y ∈ T, weight y ≤ _ by_cases hne : T.Nonempty · obtain ⟨y, hy⟩ := hne exact thin_shell_sum_le T first weight R₁ R₂ R₃ _ _ C₀ C₁ (by positivity) (by positivity) (by linarith) ((hT y hy).1.trans (hrange y (hTS hy)).2.1) hC₀ hC₁ (hgroup T hTS _ _ hU hY hT) (fun z hz => ⟨(hT z hz).1, (hT z hz).2.2.1⟩) (fun z hz => hweight z (hTS hz)) · rw [Finset.not_nonempty_iff_eq_empty.mp hne] simpa using hA) calc ∑ y ∈ S, weight y ≤ (C₀ * C₁ * (R₁ * R₂ * R₃) ^ 2) * ((Real.log (2 * R₃) / Real.log 2) * (Real.log (2 * (R₂ * R₃)) / Real.log 2)) := by simpa only [mul_assoc] using h _ ≤ (C₀ * C₁ * (R₁ * R₂ * R₃) ^ 2) * (2 * (Real.log (2 * R₁) / Real.log 2) ^ 2) := mul_le_mul_of_nonneg_left (dyadic_log_product_le h₂ h₃ h₂₁ h₃₁) hA _ = _ := by ring theorem triple_card_le_of_normal_assignment (hplanes : ∀ y : IntVector, IsPrimitive y → Nonempty (PlaneData y)) (S : Finset Triple) (normal : Triple → IntVector) (t : ℤ) (ht : t ≠ 0) (R₁ R₂ R₃ : ℝ) (h₃ : 1 ≤ R₃) (h₃₂ : R₃ ≤ R₂) (h₂₁ : R₂ ≤ R₁) (hdet : ∀ x ∈ S, x.1 ⬝ᵥ (x.2.1 ⨯₃ x.2.2) = t) (hprim : ∀ x ∈ S, IsPrimitive (normal x)) (hcross : ∀ x ∈ S, ∃ a : ℤ, a ≠ 0 ∧ x.2.1 ⨯₃ x.2.2 = a • normal x) (hplane : ∀ x ∈ S, x.2.1 ⬝ᵥ normal x = 0 ∧ x.2.2 ⬝ᵥ normal x = 0) (hnormal : ∀ x ∈ S, length (normal x) ≤ R₂ * R₃) (hnorm : ∀ x ∈ S, length x.1 ≤ R₁ ∧ length x.2.1 ≤ R₂ ∧ length x.2.2 ≤ R₃) : (S.card : ℝ) ≤ 35721000 * Real.pi ^ 3 * (R₁ * R₂ * R₃) ^ 2 * (Real.log (2 * R₁) / Real.log 2) ^ 2 := by classical let Y := S.image normal let F (y : IntVector) := S.filter (fun x => normal x = y) let weight (y : IntVector) : ℝ := (F y).card let data (y : IntVector) (hy : IsPrimitive y) : PlaneData y := Classical.choice (hplanes y hy) let first (y : IntVector) := if hy : IsPrimitive y then (data y hy).first else 1 let second (y : IntVector) := if hy : IsPrimitive y then (data y hy).second else 1 have hY (y : IntVector) (hy : y ∈ Y) : IsPrimitive y ∧ 1 ≤ length y ∧ length y ≤ R₂ * R₃ ∧ (F y).Nonempty := by obtain ⟨x, hx, rfl⟩ := Finset.mem_image.mp hy exact ⟨hprim x hx, LatticeBox.one_le_norm_realVector (hprim x hx).ne_zero, hnormal x hx, ⟨x, Finset.mem_filter.mpr ⟨hx, rfl⟩⟩⟩ have hFdet (y : IntVector) : ∀ x ∈ F y, x.1 ⬝ᵥ (x.2.1 ⨯₃ x.2.2) = t := fun x hx => hdet x (Finset.mem_filter.mp hx).1 have hFcross (y : IntVector) : ∀ x ∈ F y, ∃ a : ℤ, a ≠ 0 ∧ x.2.1 ⨯₃ x.2.2 = a • y := by intro x hx obtain ⟨hxS, hxy⟩ := Finset.mem_filter.mp hx simpa only [hxy] using hcross x hxS have hFplane (y : IntVector) : ∀ x ∈ F y, x.2.1 ⬝ᵥ y = 0 ∧ x.2.2 ⬝ᵥ y = 0 := by intro x hx obtain ⟨hxS, hxy⟩ := Finset.mem_filter.mp hx simpa only [hxy] using hplane x hxS have hFnorm (y : IntVector) : ∀ x ∈ F y, length x.1 ≤ R₁ ∧ length x.2.1 ≤ R₂ ∧ length x.2.2 ≤ R₃ := fun x hx => hnorm x (Finset.mem_filter.mp hx).1 have hscale (y : IntVector) (hy : y ∈ Y) : 1 ≤ first y ∧ first y ≤ R₃ ∧ second y ≤ R₂ := by have hp := (hY y hy).1 have hs := scales_le_of_nonempty (F y) y (data y hp) t ht R₂ R₃ h₃₂ (hY y hy).2.2.2 (hFdet y) (hFplane y) (fun x hx => (hFnorm y x hx).2) simpa only [first, second, dite_eq_left hp] using And.intro (data y hp).one_le_first hs let E := Y.filter (fun y => second y ≤ R₃) let T := Y.filter (fun y => ¬ second y ≤ R₃) have h₂ : 1 ≤ R₂ := h₃.trans h₃₂ have h₁ : 1 ≤ R₁ := h₂.trans h₂₁ have hRprod : 1 ≤ R₂ * R₃ := by nlinarith have hthin : ∑ y ∈ T, weight y ≤ 2 * (18000 * Real.pi) * (972 * Real.pi ^ 2) * (R₁ * R₂ * R₃) ^ 2 * (Real.log (2 * R₁) / Real.log 2) ^ 2 := by apply thin_normal_sum_le T first weight R₁ R₂ R₃ _ _ h₂ h₃ h₂₁ (h₃₂.trans h₂₁) (by positivity) (by positivity) · intro y hy exact Nat.cast_nonneg _ · intro y hy have hyY := (Finset.mem_filter.mp hy).1 exact ⟨(hscale y hyY).1, (hscale y hyY).2.1, (hY y hyY).2.1, (hY y hyY).2.2.1⟩ · intro y hy obtain ⟨hyY, hyt⟩ := Finset.mem_filter.mp hy have hp := (hY y hyY).1 have hlarge : (data y hp).second ≤ R₂ := by simpa only [second, dite_eq_left hp] using (hscale y hyY).2.2 have hfirst : (data y hp).first ≤ R₃ := by simpa only [first, dite_eq_left hp] using (hscale y hyY).2.1 have hlow : R₃ < (data y hp).second := by simpa only [second, dite_eq_left hp] using lt_of_not_ge hyt simpa only [first, dite_eq_left hp, weight] using thin_fiber_card_le (F y) y (data y hp) t R₁ R₂ R₃ h₂₁ hlarge hfirst hlow (hFdet y) (hFcross y) (hFplane y) (hFnorm y) · intro A hAT U Z hU hZ hA apply normal_shell_card_le hplanes hU (by linarith) A intro y hy have hyY := (Finset.mem_filter.mp (hAT hy)).1 have hp := (hY y hyY).1 refine ⟨hp, (hA y hy).2.2.2, data y hp, ?_, ?_⟩ · simpa only [first, dite_eq_left hp] using (hA y hy).1 · simpa only [first, dite_eq_left hp] using (hA y hy).2.1 have heasy : ∑ y ∈ E, weight y ≤ (2916 * Real.pi ^ 3 * (R₁ * R₂ * R₃) ^ 2) * ((125 / Real.log 2) * Real.log (2 * (R₂ * R₃))) := by calc ∑ y ∈ E, weight y ≤ ∑ y ∈ E, (2916 * Real.pi ^ 3 * (R₁ * R₂ * R₃) ^ 2) / length y ^ 3 := by apply Finset.sum_le_sum intro y hy obtain ⟨hyY, hye⟩ := Finset.mem_filter.mp hy have hp := (hY y hyY).1 have hlarge : (data y hp).second ≤ R₃ := by simpa only [second, dite_eq_left hp] using hye exact thick_fiber_card_le (F y) y (data y hp) t R₁ R₂ R₃ h₂₁ h₃₂ hlarge (hFdet y) (hFcross y) (hFplane y) (hFnorm y) _ = (2916 * Real.pi ^ 3 * (R₁ * R₂ * R₃) ^ 2) * ∑ y ∈ E, 1 / length y ^ 3 := by rw [Finset.mul_sum] apply Finset.sum_congr rfl intro y hy ring _ ≤ _ := mul_le_mul_of_nonneg_left (LatticeBox.reciprocal_cube_sum_le_log hRprod E (by intro y hy have h := hY y (Finset.mem_filter.mp hy).1 exact ⟨h.1.ne_zero, h.2.2.1⟩)) (by positivity) have hlog2 : 0 < Real.log 2 := Real.log_pos (by norm_num) have hlogprod : 0 ≤ Real.log (2 * (R₂ * R₃)) / Real.log 2 := by apply div_nonneg _ hlog2.le exact Real.log_nonneg (by nlinarith) have hlog3 : 1 ≤ Real.log (2 * R₃) / Real.log 2 := by apply (le_div_iff₀ hlog2).mpr simpa using Real.log_le_log (by norm_num : (0 : ℝ) < 2) (by linarith : 2 ≤ 2 * R₃) have hlogbound : Real.log (2 * (R₂ * R₃)) / Real.log 2 ≤ 2 * (Real.log (2 * R₁) / Real.log 2) ^ 2 := by exact (le_mul_of_one_le_left hlogprod hlog3).trans (dyadic_log_product_le h₂ h₃ h₂₁ (h₃₂.trans h₂₁)) have heasy' : ∑ y ∈ E, weight y ≤ 250 * 2916 * Real.pi ^ 3 * (R₁ * R₂ * R₃) ^ 2 * (Real.log (2 * R₁) / Real.log 2) ^ 2 := by calc ∑ y ∈ E, weight y ≤ _ := heasy _ = (125 * 2916 * Real.pi ^ 3 * (R₁ * R₂ * R₃) ^ 2) * (Real.log (2 * (R₂ * R₃)) / Real.log 2) := by ring _ ≤ (125 * 2916 * Real.pi ^ 3 * (R₁ * R₂ * R₃) ^ 2) * (2 * (Real.log (2 * R₁) / Real.log 2) ^ 2) := mul_le_mul_of_nonneg_left hlogbound (by positivity) _ = _ := by ring have hcard : (S.card : ℝ) = ∑ y ∈ Y, weight y := by dsimp only [weight, F] exact_mod_cast (Finset.card_eq_sum_card_fiberwise (s := S) (t := Y) (f := normal) (fun x hx => Finset.mem_image_of_mem normal hx)) have hsplit : (∑ y ∈ E, weight y) + (∑ y ∈ T, weight y) = ∑ y ∈ Y, weight y := Finset.sum_filter_add_sum_filter_not Y _ _ rw [hcard, ← hsplit] calc _ ≤ _ := add_le_add heasy' hthin _ = _ := by ring theorem anisotropic_count_of_planeData (hplanes : ∀ y : IntVector, IsPrimitive y → Nonempty (PlaneData y)) (S : Finset Triple) (t : ℤ) (ht : t ≠ 0) (R₁ R₂ R₃ : ℝ) (h₃ : 1 ≤ R₃) (h₃₂ : R₃ ≤ R₂) (h₂₁ : R₂ ≤ R₁) (hdet : ∀ x ∈ S, x.1 ⬝ᵥ (x.2.1 ⨯₃ x.2.2) = t) (hnorm : ∀ x ∈ S, length x.1 ≤ R₁ ∧ length x.2.1 ≤ R₂ ∧ length x.2.2 ≤ R₃) : (S.card : ℝ) ≤ (35721000 * Real.pi ^ 3 / (Real.log 2) ^ 2) * (R₁ * R₂ * R₃) ^ 2 * Real.log (2 * R₁) ^ 2 := by classical have hex : ∀ x : Triple, ∃ y : IntVector, x ∈ S → IsPrimitive y ∧ (∃ a : ℤ, a ≠ 0 ∧ x.2.1 ⨯₃ x.2.2 = a • y) ∧ (x.2.1 ⬝ᵥ y = 0 ∧ x.2.2 ⬝ᵥ y = 0) ∧ length y ≤ R₂ * R₃ := by intro x by_cases hx : x ∈ S · have hnz : x.2.1 ⨯₃ x.2.2 ≠ 0 := by intro hc apply ht have hd := hdet x hx simpa [hc] using hd.symm obtain ⟨a, ha, y, hyp, hfactor, horth₁, horth₂, hyl, _⟩ := CrossNormal.exists_bounded_primitive_cross_factor x.2.1 x.2.2 hnz refine ⟨y, fun _ => ⟨hyp, ⟨a, ne_of_gt ha, hfactor⟩, ⟨horth₁, horth₂⟩, ?_⟩⟩ exact hyl.trans (mul_le_mul (hnorm x hx).2.1 (hnorm x hx).2.2 (norm_nonneg _) ((norm_nonneg _).trans (hnorm x hx).2.1)) · exact ⟨0, fun h => (hx h).elim⟩ choose normal hnormal using hex have h := triple_card_le_of_normal_assignment hplanes S normal t ht R₁ R₂ R₃ h₃ h₃₂ h₂₁ hdet (fun x hx => (hnormal x hx).1) (fun x hx => (hnormal x hx).2.1) (fun x hx => (hnormal x hx).2.2.1) (fun x hx => (hnormal x hx).2.2.2) hnorm convert h using 1; simp only [div_pow]; ring theorem anisotropic_count (S : Finset Triple) (t : ℤ) (ht : t ≠ 0) (R₁ R₂ R₃ : ℝ) (h₃ : 1 ≤ R₃) (h₃₂ : R₃ ≤ R₂) (h₂₁ : R₂ ≤ R₁) (hdet : ∀ x ∈ S, x.1 ⬝ᵥ (x.2.1 ⨯₃ x.2.2) = t) (hnorm : ∀ x ∈ S, length x.1 ≤ R₁ ∧ length x.2.1 ≤ R₂ ∧ length x.2.2 ≤ R₃) : (S.card : ℝ) ≤ (35721000 * Real.pi ^ 3 / (Real.log 2) ^ 2) * (R₁ * R₂ * R₃) ^ 2 * Real.log (2 * R₁) ^ 2 := anisotropic_count_of_planeData exists_planeData S t ht R₁ R₂ R₃ h₃ h₃₂ h₂₁ hdet hnorm theorem anisotropic_constant_pos : 0 < 35721000 * Real.pi ^ 3 / (Real.log 2) ^ 2 := by have h : 0 < Real.log 2 := Real.log_pos (by norm_num) positivity end Problem355.Anisotropic end end OAI