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989cdb8 | 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 | {"id":"E8Packing","label":"E8Packing","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["E8-Lattice","E8-vector-norms"],"nlp_text":"The \\(E_8\\) sphere packing is the (periodic) sphere packing with separation \\(\\sqrt{2}\\), whose set of centres is \\(\\Lambda _8\\).","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/E8Packing","latex_link":"sect0001.html#E8Packing","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/E8Packing","lean_decl_name":"E8Packing","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"SpherePacking/Basic/E8.lean","lean4_code":"noncomputable def E8Packing : PeriodicSpherePacking 8 where\n separation := √2\n lattice := E8Lattice\n centers := E8Lattice\n centers_dist := by\n simp only [Pairwise, E8Lattice, Submodule.map_coe, WithLp.linearEquiv_symm_apply, ne_eq,\n Subtype.forall, Subtype.mk.injEq, Set.mem_image, SetLike.mem_coe, forall_exists_index,\n forall_and_index]\n intro _ a ha rfl _ b hb rfl hab\n rw [(WithLp.toLp_injective _).eq_iff] at hab\n have : a - b ∈ Submodule.E8 ℝ := Submodule.sub_mem _ ha hb\n simp only [AddEquiv.toEquiv_eq_coe, Equiv.invFun_as_coe, AddEquiv.coe_toEquiv_symm,\n WithLp.addEquiv_symm_apply] at hab\n simpa [Subtype.dist_eq, WithLp.ofLp_eq_zero, sub_eq_zero, hab] using\n E8_norm_lower_bound _ this\n lattice_action x y := add_mem\n","line":602,"end_line":617,"proof_status":"formalized"}
{"id":"E8Packing-covol","label":"E8Packing-covol","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["E8Packing"],"nlp_text":"\\(\\operatorname {Vol}\\! \\left(\\Lambda _8\\right) = \\mathrm{Covol}(\\mathbb {R}^8 / \\Lambda _8) = 1\\).","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/E8Basis_volume","latex_link":"sect0001.html#E8Packing-covol","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/E8Basis_volume","lean_decl_name":"E8Basis_volume","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"SpherePacking/Basic/E8.lean","lean4_code":"lemma E8Basis_volume : volume (fundamentalDomain (E8Basis ℝ)) = 1 := by\n rw [volume_fundamentalDomain', of_basis_eq_matrix, E8Matrix_myDet_eq_one]\n simp\n","line":653,"end_line":655,"proof_status":"formalized"}
{"id":"E8Packing-density","label":"E8Packing-density","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["E8Packing-covol","theorem:psp-density"],"nlp_text":"We have \\(\\Delta _{\\mathcal{P}(E_8)} = \\frac{\\pi ^4}{384}\\).","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/E8Packing_density","latex_link":"sect0001.html#E8Packing-density","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/E8Packing_density","lean_decl_name":"E8Packing_density","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"SpherePacking/Basic/E8.lean","lean4_code":"theorem E8Packing_density : E8Packing.density = ENNReal.ofReal π ^ 4 / 384 := by\n rw [PeriodicSpherePacking.density_eq E8_ℤBasis ?_ (by omega) (L := 16)]\n · rw [E8Packing_numReps, Nat.cast_one, one_mul, volume_ball, finrank_euclideanSpace,\n Fintype.card_fin, Nat.cast_ofNat]\n simp only [E8Packing]\n have {x : ℝ} (hx : 0 ≤ x := by positivity) : √x ^ 8 = x ^ 4 := calc\n √x ^ 8 = (√x ^ 2) ^ 4 := by rw [← pow_mul]\n _ = x ^ 4 := by rw [Real.sq_sqrt hx]\n rw [← ENNReal.ofReal_pow, ← ENNReal.ofReal_mul, div_pow, this, this, ← mul_div_assoc,\n div_mul_eq_mul_div, mul_comm, mul_div_assoc, mul_div_assoc]\n · norm_num [Nat.factorial, mul_one_div]\n convert div_one _\n · rw [E8_Basis_volume]\n · rw [← ENNReal.ofReal_pow, ENNReal.ofReal_div_of_pos, ENNReal.ofReal_ofNat] <;> positivity\n · positivity\n · positivity\n · intro x hx\n trans ∑ i, ‖E8_ℤBasis i‖\n · rw [← fract_eq_self.mpr hx]\n convert norm_fract_le (K := ℝ) _ _\n simp; rfl\n · refine (Finset.sum_le_sum (fun i hi ↦ E8_ℤBasis_apply_norm i)).trans ?_\n norm_num\n","line":681,"end_line":703,"proof_status":"formalized"}
{"id":"theorem:CE_Main","label":"CE_Main","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["E8Packing-density","SpherePackingConstant","thm:g","thm:Cohn-Elkies-general"],"nlp_text":"All periodic packing \\(\\mathcal{P} \\subseteq \\mathbb {R}^8\\) has density satisfying \\(\\Delta _{\\mathcal{P}} \\leq \\Delta _{E_8} = \\frac{\\pi ^4}{384}\\), the density of the \\(E_8\\) sphere packing (see definition 3.9).","doc_url":null,"latex_link":"sec-sphere-packings.html#theorem:CE_Main","lean_link":null,"lean_decl_name":"CE_Main","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"lemma:inv-power-summable","label":"inv-power-summable","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(X \\subset \\mathbb {R}^d\\) be a set of sphere packing centres of separation \\(1\\) that is periodic with some lattice \\(\\Lambda \\subset \\mathbb {R}^d\\). Then, there exists \\(k \\in \\mathbb {N}\\) sufficiently large such that \\[ \\sum _{x \\in X} \\frac{1}{\\left\\lVert x \\right\\rVert ^{k}} {\\lt} \\infty . \\]","doc_url":null,"latex_link":"sect0002.html#lemma:inv-power-summable","lean_link":null,"lean_decl_name":"inv-power-summable","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"thm:Poisson-summation-formula","label":"Poisson-summation-formula","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["lemma:inv-power-summable","def:Fourier-Transform","lemma:Schwartz-summable","def:dual-lattice"],"nlp_text":"Let \\(\\Lambda \\) be a lattice in \\(\\mathbb {R}^d\\), and let \\(f:\\mathbb {R}^d\\to \\mathbb {R}\\) be a Schwartz function. Then, for all \\(v \\in \\mathbb {R}^d\\), \\[ \\sum _{\\ell \\in \\Lambda }f(\\ell + v) = \\frac{1}{\\operatorname {Vol}\\! \\left(\\mathbb {R}^d/\\Lambda \\right)} \\sum _{m\\in \\Lambda ^*}\\widehat{f}(m) e^{-2\\pi i \\left\\langle v, m \\right\\rangle }. \\]","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/SchwartzMap.PoissonSummation_Lattices","latex_link":"sect0002.html#thm:Poisson-summation-formula","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/SchwartzMap.PoissonSummation_Lattices","lean_decl_name":"SchwartzMap.PoissonSummation_Lattices","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"SpherePacking/CohnElkies/Prereqs.lean","lean4_code":"theorem PoissonSummation_Lattices (f : SchwartzMap (EuclideanSpace ℝ (Fin d)) ℂ)\n (v : EuclideanSpace ℝ (Fin d)) : ∑' ℓ : Λ, f (v + ℓ) = (1 / ZLattice.covolume Λ) *\n ∑' m : bilinFormOfRealInner.dualSubmodule Λ, (𝓕 ⇑f m) * exp (2 * π * I * ⟪v, m⟫_[ℝ]) := by\n sorry\n","line":117,"end_line":120,"proof_status":"formalized"}
{"id":"thm:Cohn-Elkies-periodic","label":"Cohn-Elkies-periodic","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["thm:Poisson-summation-formula","SpherePacking.density"],"nlp_text":"Let \\(X\\subset \\mathbb {R}^d\\) be a discrete subset such that \\(\\| x-y\\| \\geq 1\\) for any distinct \\(x,y\\in X\\). Suppose that \\(X\\) is \\(\\Lambda \\)-periodic with respect to some lattice \\(\\Lambda \\subset \\mathbb {R}^d\\). Let \\(f:\\mathbb {R}^d\\to \\mathbb {R}\\) be a Schwartz function that is not identically zero and satisfies the following conditions: \\begin{equation} \\label{eqn:Cohn-Elkies-condition-1}f(x)\\leq 0\\mbox{ for } \\| x\\| \\geq 1\\end{equation}","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/LinearProgrammingBound'","latex_link":"sect0003.html#thm:Cohn-Elkies-periodic","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/LinearProgrammingBound'","lean_decl_name":"LinearProgrammingBound'","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/thefundamentaltheor3m/Sphere-Packing-Lean/2928857eab3e21dc6c3b5bfea025528145fe2113/SpherePacking/CohnElkies/LPBound.lean","lean4_code":"theorem LinearProgrammingBound' (hd : 0 < d) (hf : Summable f) :\n P.density ≤ (f 0).re.toNNReal / (𝓕 f 0).re.toNNReal *\n volume (ball (0 : EuclideanSpace ℝ (Fin d)) (1 / 2)) := by\n -- HUGE TODO: Get the periodic density formula in terms of some `D`.\n have : Fact (0 < d) := ⟨hd⟩\n rw [P.density_eq' hd]\n suffices hCalc : (P.numReps' hd hD_isBounded) * (f 0).re ≥\n (P.numReps' hd hD_isBounded)^2 * (𝓕 f 0).re / ZLattice.covolume P.lattice by\n rw [hP]\n rw [ge_iff_le] at hCalc\n have vol_pos := EuclideanSpace.volume_ball_pos (0 : EuclideanSpace ℝ (Fin d)) one_half_pos\n have vol_ne_zero : volume (ball (0 : EuclideanSpace ℝ (Fin d)) (1 / 2)) ≠ 0 :=\n Ne.symm (ne_of_lt vol_pos)\n have vol_ne_top : volume (ball (0 : EuclideanSpace ℝ (Fin d)) (1 / 2)) ≠ ∞ := by\n rw [← lt_top_iff_ne_top]\n exact EuclideanSpace.volume_ball_lt_top 0\n cases eq_or_ne (𝓕 f 0) 0\n · case inl h𝓕f =>\n rw [h𝓕f, zero_re]\n -- For `ENNReal.div_zero`, we need `f 0 ≠ 0`. This can be deduced from the fact that\n -- `𝓕 f ≥ 0` and `f ≠ 0`.\n have ne_zero_at_zero : ((f 0).re.toNNReal : ENNReal) ≠ 0 :=\n ENNReal.coe_ne_zero.mpr (Ne.symm (ne_of_lt (toNNReal_pos.mpr\n (f_zero_pos hne_zero hReal hRealFourier hCohnElkies₂))))\n -- Now we can safely divide by zero!\n rw [ENat.toENNReal_coe, toNNReal_zero, ENNReal.coe_zero, ENNReal.div_zero ne_zero_at_zero]\n -- We now need to multiply by ⊤.\n rw [ENNReal.top_mul vol_ne_zero]\n exact le_top\n · case inr h𝓕f =>\n -- First, we shift things around and cancel volumes on the right\n rw [ENat.toENNReal_coe, mul_div_assoc, div_eq_mul_inv (volume _), mul_comm (volume _),\n ← mul_assoc, ENNReal.mul_le_mul_iff_left vol_ne_zero vol_ne_top]\n -- Next, we simplify `hCalc` by replacing `numReps'` with `numReps`\n rw [← P.numReps_eq_numReps' Fact.out hD_isBounded hD_unique_covers] at hCalc\n -- Next, we multiply both sides by `(𝓕 (⇑f) 0).re.toNNReal`, cancelling accordingly.\n have hfouaux₁ : ((𝓕 f 0).re.toNNReal : ENNReal) ≠ 0 := by\n intro hContra\n apply h𝓕f\n simp only [ENNReal.coe_eq_zero, toNNReal_eq_zero] at hContra\n specialize hCohnElkies₂ 0\n rw [ge_iff_le] at hCohnElkies₂\n -- We can't simply do antisymm because we have an equality in ℂ, not ℝ!\n rw [← re_add_im (𝓕 f 0), le_antisymm hContra hCohnElkies₂,\n hFourierImZero hRealFourier 0, ofReal_zero, zero_mul, add_zero]\n have hfouaux₂ : ((𝓕 (⇑f) 0).re.toNNReal : ENNReal) ≠ ⊤ := ENNReal.coe_ne_top\n rw [← ENNReal.mul_le_mul_iff_left hfouaux₁ hfouaux₂,\n div_eq_mul_inv ((f 0).re.toNNReal : ENNReal) _,\n mul_assoc ((f 0).re.toNNReal : ENNReal) _ _, ENNReal.inv_mul_cancel hfouaux₁ hfouaux₂]\n -- We put it in a more desirable form and consolidate.\n rw [mul_one, mul_assoc, ← ENNReal.div_eq_inv_mul]\n -- Next, we multiply both sides on the left by `↑P.numReps`.\n have hnRaux₁ : ENat.toENNReal (P.numReps : ENat) ≠ 0 := by\n rw [ENat.toENNReal_coe, ne_eq, Nat.cast_eq_zero, ← ne_eq]\n -- intro hContra\n -- rw [← P.card_centers_inter_isFundamentalDomain D hD_isBounded hD_unique_covers Fact.out]\n unfold PeriodicSpherePacking.numReps\n haveI : Nonempty (Quotient (AddAction.orbitRel ↥P.lattice ↑P.centers)) := by\n rw [nonempty_quotient_iff]\n assumption\n exact Fintype.card_ne_zero\n have hnRaux₂ : ENat.toENNReal (P.numReps : ENat) ≠ ⊤ := Ne.symm (ne_of_beq_false rfl)\n rw [← ENNReal.mul_le_mul_iff_right hnRaux₁ hnRaux₂]\n -- We put it in a more desirable form and consolidate.\n rw [ENat.toENNReal_coe, ← mul_assoc, ← pow_two, ← mul_div_assoc]\n -- Now, we use the nonnegativity of... everything... to get the `toNNReal`s to the outside.\n have hRHSCast : (P.numReps : ENNReal) * ↑(f 0).re.toNNReal = (P.numReps * (f 0).re).toNNReal\n := by\n -- rw [ENNReal.coe_mul, ENNReal.coe_natCast]\n norm_cast\n refine NNReal.eq ?_\n have haux₁ : 0 ≤ ↑P.numReps * (f 0).re := mul_nonneg (Nat.cast_nonneg' P.numReps)\n (f_nonneg_at_zero hCohnElkies₂)\n rw [Real.toNNReal_of_nonneg (f_nonneg_at_zero hCohnElkies₂),\n Real.toNNReal_of_nonneg haux₁]\n push_cast\n rfl\n have hLHSCast : (P.numReps : ENNReal) ^ 2 * ((𝓕 f 0).re.toNNReal : ENNReal) /\n ((ZLattice.covolume P.lattice volume).toNNReal : ENNReal) = ((P.numReps) ^ 2 *\n (𝓕 f 0).re / ZLattice.covolume P.lattice volume).toNNReal := by\n simp only [div_eq_mul_inv]\n have haux₁ : 0 ≤ ↑P.numReps ^ 2 * (𝓕 f 0).re * (ZLattice.covolume P.lattice volume)⁻¹\n := by\n refine mul_nonneg (mul_nonneg (sq_nonneg (P.numReps : ℝ)) (hCohnElkies₂ 0)) ?_\n rw [inv_nonneg]\n exact LT.lt.le (ZLattice.covolume_pos P.lattice volume)\n rw [Real.toNNReal_of_nonneg haux₁]\n have haux₂ : (ZLattice.covolume P.lattice volume).toNNReal ≠ 0 := by\n apply LT.lt.ne'\n rw [Real.toNNReal_pos]\n exact ZLattice.covolume_pos P.lattice volume\n rw [← ENNReal.coe_inv haux₂]\n norm_cast\n rw [Real.toNNReal_of_nonneg (hCohnElkies₂ 0),\n Real.toNNReal_of_nonneg (LT.lt.le (ZLattice.covolume_pos P.lattice volume))]\n refine NNReal.eq ?_\n push_cast\n rfl\n -- We can now get rid of the `toNNReal`s and use `hCalc` to finish the proof!\n rw [hRHSCast, hLHSCast, ENNReal.coe_le_coe]\n exact Real.toNNReal_le_toNNReal hCalc\n exact calc_steps hne_zero hReal hRealFourier hCohnElkies₁ hCohnElkies₂ hP hD_isBounded hd hf","line":526,"end_line":627,"proof_status":"formalized"}
{"id":"thm:Cohn-Elkies-general","label":"Cohn-Elkies-general","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["thm:Cohn-Elkies-periodic","thm:periodic-packing-optimal"],"nlp_text":"Let \\(f:\\mathbb {R}^d\\to \\mathbb {R}\\) be a Schwartz function that is not identically zero and satisfies 2 and 3. Then the density of any \\(\\Lambda \\)-periodic sphere packing is bounded above by \\[ \\frac{f(0)}{\\widehat{f}(0)}\\cdot \\mathrm{vol}(B_d(0,1/2)). \\]","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/LinearProgrammingBound","latex_link":"sect0003.html#thm:Cohn-Elkies-general","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/LinearProgrammingBound","lean_decl_name":"LinearProgrammingBound","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"SpherePacking/CohnElkies/LPBound.lean","lean4_code":"/-\nI think the only sustainable fix is to show that the volume of a sphere is finite and then turn\n`SpherePacking.density` into an element of `ℝ` instead of `ℝ≥0∞`.\n-/\n\nend Fundamental_Domain_Dependent\n\nsection Main_Theorem_For_One_Packing\n\nvariable {P : PeriodicSpherePacking d} (hP : P.separation = 1) [Nonempty P.centers]\nvariable {D : Set (EuclideanSpace ℝ (Fin d))} (hD_isBounded : IsBounded D)\nvariable (hD_unique_covers : ∀ x, ∃! g : P.lattice, g +ᵥ x ∈ D)\n\ninclude d f hne_zero hReal hRealFourier hCohnElkies₁ hCohnElkies₂ P hP D hD_isBounded\n hD_unique_covers\n\ntheorem LinearProgrammingBound' (hd : 0 < d) (hf : Summable f) :\n P.density ≤ (f 0).re.toNNReal / (𝓕 f 0).re.toNNReal *\n volume (ball (0 : EuclideanSpace ℝ (Fin d)) (1 / 2)) := by\n -- HUGE TODO: Get the periodic density formula in terms of some `D`.\n have : Fact (0 < d) := ⟨hd⟩\n rw [P.density_eq' hd]\n suffices hCalc : (P.numReps' hd hD_isBounded) * (f 0).re ≥\n (P.numReps' hd hD_isBounded)^2 * (𝓕 f 0).re / ZLattice.covolume P.lattice by\n rw [hP]\n rw [ge_iff_le] at hCalc\n have vol_pos := EuclideanSpace.volume_ball_pos (0 : EuclideanSpace ℝ (Fin d)) one_half_pos\n have vol_ne_zero : volume (ball (0 : EuclideanSpace ℝ (Fin d)) (1 / 2)) ≠ 0 :=\n Ne.symm (ne_of_lt vol_pos)\n have vol_ne_top : volume (ball (0 : EuclideanSpace ℝ (Fin d)) (1 / 2)) ≠ ∞ := by\n rw [← lt_top_iff_ne_top]\n exact EuclideanSpace.volume_ball_lt_top 0\n cases eq_or_ne (𝓕 f 0) 0\n · case inl h𝓕f =>\n rw [h𝓕f, zero_re]\n -- For `ENNReal.div_zero`, we need `f 0 ≠ 0`. This can be deduced from the fact that\n -- `𝓕 f ≥ 0` and `f ≠ 0`.\n have ne_zero_at_zero : ((f 0).re.toNNReal : ENNReal) ≠ 0 :=\n ENNReal.coe_ne_zero.mpr (Ne.symm (ne_of_lt (toNNReal_pos.mpr\n (f_zero_pos hne_zero hReal hRealFourier hCohnElkies₂))))\n -- Now we can safely divide by zero!\n rw [ENat.toENNReal_coe, toNNReal_zero, ENNReal.coe_zero, ENNReal.div_zero ne_zero_at_zero]\n -- We now need to multiply by ⊤.\n rw [ENNReal.top_mul vol_ne_zero]\n exact le_top\n · case inr h𝓕f =>\n -- First, we shift things around and cancel volumes on the right\n rw [ENat.toENNReal_coe, mul_div_assoc, div_eq_mul_inv (volume _), mul_comm (volume _),\n ← mul_assoc, ENNReal.mul_le_mul_iff_left vol_ne_zero vol_ne_top]\n -- Next, we simplify `hCalc` by replacing `numReps'` with `numReps`\n rw [← P.numReps_eq_numReps' Fact.out hD_isBounded hD_unique_covers] at hCalc\n -- Next, we multiply both sides by `(𝓕 (⇑f) 0).re.toNNReal`, cancelling accordingly.\n have hfouaux₁ : ((𝓕 f 0).re.toNNReal : ENNReal) ≠ 0 := by\n intro hContra\n apply h𝓕f\n simp only [ENNReal.coe_eq_zero, toNNReal_eq_zero] at hContra\n specialize hCohnElkies₂ 0\n rw [ge_iff_le] at hCohnElkies₂\n -- We can't simply do antisymm because we have an equality in ℂ, not ℝ!\n rw [← re_add_im (𝓕 f 0), le_antisymm hContra hCohnElkies₂,\n hFourierImZero hRealFourier 0, ofReal_zero, zero_mul, add_zero]\n have hfouaux₂ : ((𝓕 (⇑f) 0).re.toNNReal : ENNReal) ≠ ⊤ := ENNReal.coe_ne_top\n rw [← ENNReal.mul_le_mul_iff_left hfouaux₁ hfouaux₂,\n div_eq_mul_inv ((f 0).re.toNNReal : ENNReal) _,\n mul_assoc ((f 0).re.toNNReal : ENNReal) _ _, ENNReal.inv_mul_cancel hfouaux₁ hfouaux₂]\n -- We put it in a more desirable form and consolidate.\n rw [mul_one, mul_assoc, ← ENNReal.div_eq_inv_mul]\n -- Next, we multiply both sides on the left by `↑P.numReps`.\n have hnRaux₁ : ENat.toENNReal (P.numReps : ENat) ≠ 0 := by\n rw [ENat.toENNReal_coe, ne_eq, Nat.cast_eq_zero, ← ne_eq]\n -- intro hContra\n -- rw [← P.card_centers_inter_isFundamentalDomain D hD_isBounded hD_unique_covers Fact.out]\n unfold PeriodicSpherePacking.numReps\n haveI : Nonempty (Quotient (AddAction.orbitRel ↥P.lattice ↑P.centers)) := by\n rw [nonempty_quotient_iff]\n assumption\n exact Fintype.card_ne_zero\n have hnRaux₂ : ENat.toENNReal (P.numReps : ENat) ≠ ⊤ := Ne.symm (ne_of_beq_false rfl)\n rw [← ENNReal.mul_le_mul_iff_right hnRaux₁ hnRaux₂]\n -- We put it in a more desirable form and consolidate.\n rw [ENat.toENNReal_coe, ← mul_assoc, ← pow_two, ← mul_div_assoc]\n -- Now, we use the nonnegativity of... everything... to get the `toNNReal`s to the outside.\n have hRHSCast : (P.numReps : ENNReal) * ↑(f 0).re.toNNReal = (P.numReps * (f 0).re).toNNReal\n := by\n -- rw [ENNReal.coe_mul, ENNReal.coe_natCast]\n norm_cast\n refine NNReal.eq ?_\n have haux₁ : 0 ≤ ↑P.numReps * (f 0).re := mul_nonneg (Nat.cast_nonneg' P.numReps)\n (f_nonneg_at_zero hCohnElkies₂)\n rw [Real.toNNReal_of_nonneg (f_nonneg_at_zero hCohnElkies₂),\n Real.toNNReal_of_nonneg haux₁]\n push_cast\n rfl\n have hLHSCast : (P.numReps : ENNReal) ^ 2 * ((𝓕 f 0).re.toNNReal : ENNReal) /\n ((ZLattice.covolume P.lattice volume).toNNReal : ENNReal) = ((P.numReps) ^ 2 *\n (𝓕 f 0).re / ZLattice.covolume P.lattice volume).toNNReal := by\n simp only [div_eq_mul_inv]\n have haux₁ : 0 ≤ ↑P.numReps ^ 2 * (𝓕 f 0).re * (ZLattice.covolume P.lattice volume)⁻¹\n := by\n refine mul_nonneg (mul_nonneg (sq_nonneg (P.numReps : ℝ)) (hCohnElkies₂ 0)) ?_\n rw [inv_nonneg]\n exact LT.lt.le (ZLattice.covolume_pos P.lattice volume)\n","line":526,"end_line":627,"proof_status":"formalized"}
{"id":"SpherePacking.balls","label":"SpherePacking.balls","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Given a set \\(X \\subset \\mathbb {R}^d\\) and a real number \\(r {\\gt} 0\\) (known as the separation radius) such that \\(\\| x - y\\| \\geq r\\) for all distinct \\(x, y \\in X\\), we define the sphere packing \\(\\mathcal{P}(X)\\) with centres at \\(X\\) to be the union of all open balls of radius \\(r\\) centred at points in \\(X\\): \\[ \\mathcal{P}(X) := \\bigcup _{x \\in X} B_d(x, r) \\]","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/SpherePacking.balls","latex_link":"sec-sphere-packings.html#SpherePacking.balls","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/SpherePacking.balls","lean_decl_name":"SpherePacking.balls","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Algebra/Order/Interval/Set/Group.lean","lean4_code":"-- for instance when considering metric balls in `ℝ`.\n","line":106,"end_line":106,"proof_status":"formalized"}
{"id":"SpherePacking.finiteDensity","label":"SpherePacking.finiteDensity","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["SpherePacking.balls"],"nlp_text":"The finite density of a packing \\(\\mathcal{P}\\) is defined as \\[ \\Delta _{\\mathcal{P}}(R):=\\frac{\\mathrm{Vol}(\\mathcal{P}\\cap B_d(0,R))}{\\mathrm{Vol}(B_d(0,R))},\\quad R{\\gt}0. \\]","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/SpherePacking.finiteDensity","latex_link":"sec-sphere-packings.html#SpherePacking.finiteDensity","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/SpherePacking.finiteDensity","lean_decl_name":"SpherePacking.finiteDensity","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"SpherePacking/Basic/SpherePacking.lean","lean4_code":"noncomputable def SpherePacking.finiteDensity (S : SpherePacking d) (R : ℝ) : ℝ≥0∞ :=\n volume (S.balls ∩ ball 0 R) / (volume (ball (0 : EuclideanSpace ℝ (Fin d)) R))\n","line":99,"end_line":100,"proof_status":"formalized"}
{"id":"lemma:sp-finite-density-bound","label":"sp-finite-density-bound","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["SpherePacking.finiteDensity"],"nlp_text":"For any \\(R {\\gt} 0\\), \\[ \\left|X \\cap \\mathcal{B}_d\\left(R - \\frac{r}{2}\\right)\\right| \\cdot \\frac{\\mathrm{Vol}\\left(\\mathcal{B}_d\\left(\\frac{r}{2}\\right)\\right)}{\\mathrm{Vol}(\\mathcal{B}_d(R))} \\leq \\Delta _{\\mathcal{P}}(R) \\leq \\left|X \\cap \\mathcal{B}_d\\left(R + \\frac{r}{2}\\right)\\right| \\cdot \\frac{\\mathrm{Vol}\\left(\\mathcal{B}_d\\left(\\frac{r}{2}\\right)\\right)}{\\mathrm{Vol}(\\mathcal{B}_d(R))} \\]","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/SpherePacking.finiteDensity_le","latex_link":"sec-packings-density.html#lemma:sp-finite-density-bound","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/SpherePacking.finiteDensity_le","lean_decl_name":"SpherePacking.finiteDensity_le","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/thefundamentaltheor3m/Sphere-Packing-Lean/2928857eab3e21dc6c3b5bfea025528145fe2113/SpherePacking/Basic/SpherePacking.lean","lean4_code":"theorem SpherePacking.finiteDensity_le (hd : 0 < d) (R : ℝ) :\n S.finiteDensity R\n ≤ (S.centers ∩ ball 0 (R + S.separation / 2)).encard\n * volume (ball (0 : EuclideanSpace ℝ (Fin d)) (S.separation / 2))\n / volume (ball (0 : EuclideanSpace ℝ (Fin d)) R) := by\n haveI : Nonempty (Fin d) := Fin.pos_iff_nonempty.mp hd\n rw [finiteDensity, balls]\n apply ENNReal.div_le_div_right\n rw [← ENNReal.div_le_iff_le_mul] <;> try left\n · have := S.inter_ball_encard_ge hd (R + S.separation / 2)\n rwa [add_sub_cancel_right] at this\n · exact (volume_ball_pos _ (by linarith [S.separation_pos])).ne.symm\n · exact (volume_ball_lt_top _).ne","line":455,"end_line":467,"proof_status":"formalized"}
{"id":"SpherePacking.scale_finiteDensity","label":"SpherePacking.scale_finiteDensity","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["SpherePacking.finiteDensity","SpherePacking.scale"],"nlp_text":"Let \\(\\mathcal{P}(X)\\) be a sphere packing and \\(c\\) a positive real number. Then, for all \\(R {\\gt} 0\\), \\[ \\Delta _{\\mathcal{P}(cX)}(cR) = \\Delta _{\\mathcal{P}(X)}(R). \\]","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/SpherePacking.scale_finiteDensity","latex_link":"sec-sphere-packings.html#SpherePacking.scale_finiteDensity","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/SpherePacking.scale_finiteDensity","lean_decl_name":"SpherePacking.scale_finiteDensity","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"SpherePacking/Basic/SpherePacking.lean","lean4_code":"/-- Finite density of a scaled packing. -/\n@[simp]\nlemma scale_finiteDensity {d : ℕ} (_ : 0 < d) (S : SpherePacking d) {c : ℝ} (hc : 0 < c) (R : ℝ) :\n (S.scale hc).finiteDensity (c * R) = S.finiteDensity R := by\n -- haveI : Nonempty (Fin d) := Fin.pos_iff_nonempty.mp hd -- (_ : 0 < d) unnecessary\n have : ball (0 : EuclideanSpace ℝ (Fin d)) (c * R) = c • ball 0 R := by\n convert (_root_.smul_ball hc.ne.symm (0 : EuclideanSpace ℝ (Fin d)) R).symm\n · exact Eq.symm (DistribMulAction.smul_zero c)\n · rw [Real.norm_eq_abs, abs_eq_self.mpr hc.le]\n rw [finiteDensity, scale_balls, this, ← Set.smul_set_inter₀ hc.ne.symm]\n repeat rw [Measure.addHaar_smul_of_nonneg _ hc.le]\n rw [ENNReal.mul_div_mul_left, finiteDensity]\n · rw [ne_eq, ENNReal.ofReal_eq_zero, not_le, finrank_euclideanSpace_fin]\n positivity\n · apply ENNReal.ofReal_ne_top\n","line":272,"end_line":286,"proof_status":"formalized"}
{"id":"SpherePacking.density","label":"SpherePacking.density","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["SpherePacking.finiteDensity"],"nlp_text":"We define the density of a packing \\(\\mathcal{P}\\) as the limit superior \\[ \\Delta _{\\mathcal{P}}:=\\limsup \\limits _{R\\to \\infty }\\Delta _{\\mathcal{P}}(R). \\]","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/SpherePacking.density","latex_link":"sec-sphere-packings.html#SpherePacking.density","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/SpherePacking.density","lean_decl_name":"SpherePacking.density","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"SpherePacking/Basic/SpherePacking.lean","lean4_code":"noncomputable def SpherePacking.density (S : SpherePacking d) : ℝ≥0∞ :=\n limsup S.finiteDensity atTop\n","line":102,"end_line":103,"proof_status":"formalized"}
{"id":"theorem:psp-density","label":"psp-density","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["lemma:sp-finite-density-bound","SpherePacking.density","lemma:lattice-points-bound","lemma:periodic-points-bounds","lemma:volume-ball-ratio-limit"],"nlp_text":"For a periodic sphere packing \\(\\mathcal{P} = \\mathcal{P}(X)\\) with centers \\(X\\) periodic to the lattice \\(\\Lambda \\) and separation \\(r\\), \\[ \\Delta _{\\mathcal{P}} = |X / \\Lambda | \\cdot \\frac{\\operatorname {Vol}\\! \\left(\\mathcal{B}_d(r / 2)\\right)}{\\operatorname {Vol}\\! \\left(\\mathbb {R}^d / \\Lambda \\right)} \\]","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/PeriodicSpherePacking.density_eq","latex_link":"sec-packings-density.html#theorem:psp-density","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/PeriodicSpherePacking.density_eq","lean_decl_name":"PeriodicSpherePacking.density_eq","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/thefundamentaltheor3m/Sphere-Packing-Lean/2928857eab3e21dc6c3b5bfea025528145fe2113/SpherePacking/Basic/PeriodicPacking.lean","lean4_code":"theorem PeriodicSpherePacking.density_eq\n (hL : ∀ x ∈ fundamentalDomain (b.ofZLatticeBasis ℝ _), ‖x‖ ≤ L) (hd : 0 < d) :\n S.density\n = S.numReps * volume (ball (0 : EuclideanSpace ℝ (Fin d)) (S.separation / 2))\n / volume (fundamentalDomain (b.ofZLatticeBasis ℝ _)) :=\n limsSup_eq_of_le_nhds (S.tendsto_finiteDensity b hL hd)","line":932,"end_line":937,"proof_status":"formalized"}
{"id":"SpherePacking.scale_density","label":"SpherePacking.scale_density","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["SpherePacking.scale_finiteDensity"],"nlp_text":"Let \\(\\mathcal{P}(X)\\) be a sphere packing and \\(c\\) a positive real number. Then, the density of the scaled packing \\(\\mathcal{P}(cX)\\) is equal to the density of the original packing \\(\\mathcal{P}(X)\\).","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/SpherePacking.scale_density","latex_link":"sec-sphere-packings.html#SpherePacking.scale_density","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/SpherePacking.scale_density","lean_decl_name":"SpherePacking.scale_density","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"SpherePacking/Basic/SpherePacking.lean","lean4_code":"/-- Density of a scaled packing. -/\nlemma scale_density {d : ℕ} (hd : 0 < d) (S : SpherePacking d) {c : ℝ} (hc : 0 < c) :\n (S.scale hc).density = S.density := by\n simp only [density, limsup, limsSup, eventually_map, eventually_atTop]\n apply le_antisymm\n -- The following are almost identical. Can we condense the proof?\n · simp only [sInf_le_iff, le_sInf_iff, Set.mem_setOf_eq, lowerBounds]\n intro x hx y hy\n rcases hx with ⟨a, ha⟩\n apply hy\n use c * a\n intro b' hb'\n rw [scale_finiteDensity' hd S hc]\n apply ha\n exact (le_div_iff₀' hc).mpr hb'\n · simp only [sInf_le_iff, le_sInf_iff, Set.mem_setOf_eq, lowerBounds]\n intro x hx y hy\n rcases hx with ⟨a, ha⟩\n apply hy\n use a / c\n intro b' hb'\n rw [← scale_finiteDensity hd S hc]\n apply ha\n exact (div_le_iff₀' hc).mp hb'\n","line":294,"end_line":317,"proof_status":"formalized"}
{"id":"SpherePackingConstant","label":"SpherePackingConstant","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["SpherePacking.density"],"nlp_text":"The sphere packing constant is defined as supremum of packing densities over all possible packings: \\[ \\Delta _d:=\\sup \\limits _{\\substack {\\mathcal{P}\\subset \\mathbb {R}^d\\\\ \\scriptscriptstyle \\mathrm{sphere}\\; \\mathrm{packing}}}\\Delta _{\\mathcal{P}}. \\]","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/SpherePackingConstant","latex_link":"sec-sphere-packings.html#SpherePackingConstant","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/SpherePackingConstant","lean_decl_name":"SpherePackingConstant","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"SpherePacking/Basic/SpherePacking.lean","lean4_code":"/-- The `PeriodicSpherePackingConstant` in dimension d is the supremum of the density of all\nperiodic packings. See also `<TODO>` for specifying the separation radius of the packings. -/\ndef PeriodicSpherePackingConstant (d : ℕ) : ℝ≥0∞ :=\n ⨆ S : PeriodicSpherePacking d, S.density\n","line":244,"end_line":247,"proof_status":"formalized"}
{"id":"def:Periodic-sphere-packing-constant","label":"Periodic-sphere-packing-constant","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["SpherePacking.density","PeriodicSpherePacking"],"nlp_text":"The periodic sphere packing constant is defined to be \\[ \\Delta _{d}^{\\text{periodic}} := \\sup _{\\substack {P \\subset \\mathbb {R}^d \\\\ \\text{periodic packing}}} \\Delta _P \\]","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/PeriodicSpherePackingConstant","latex_link":"sec-sphere-packings.html#def:Periodic-sphere-packing-constant","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/PeriodicSpherePackingConstant","lean_decl_name":"PeriodicSpherePackingConstant","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"SpherePacking/Basic/SpherePacking.lean","lean4_code":"/-- The `PeriodicSpherePackingConstant` in dimension d is the supremum of the density of all\nperiodic packings. See also `<TODO>` for specifying the separation radius of the packings. -/\ndef PeriodicSpherePackingConstant (d : ℕ) : ℝ≥0∞ :=\n ⨆ S : PeriodicSpherePacking d, S.density\n","line":244,"end_line":247,"proof_status":"formalized"}
{"id":"SpherePacking.constant_eq_constant_normalized","label":"SpherePacking.constant_eq_constant_normalized","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["SpherePackingConstant","SpherePacking.scale_density"],"nlp_text":"\\[ \\Delta _d = \\sup \\limits _{\\substack {\\mathcal{P}\\subset \\mathbb {R}^d \\\\ \\text{sphere packing} \\\\ \\text{sep.~ rad.} = 1}} \\Delta _{\\mathcal{P}} \\]","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/SpherePacking.constant_eq_constant_normalized","latex_link":"sec-sphere-packings.html#SpherePacking.constant_eq_constant_normalized","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/SpherePacking.constant_eq_constant_normalized","lean_decl_name":"SpherePacking.constant_eq_constant_normalized","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"SpherePacking/Basic/SpherePacking.lean","lean4_code":"theorem constant_eq_constant_normalized {d : ℕ} (hd : 0 < d) :\n SpherePackingConstant d = ⨆ (S : SpherePacking d) (_ : S.separation = 1), S.density := by\n rw [iSup_subtype', SpherePackingConstant]\n apply le_antisymm\n · apply iSup_le\n intro S\n have h := inv_mul_cancel₀ S.separation_pos.ne.symm\n have := le_iSup (fun S : { S : SpherePacking d // S.separation = 1 } ↦ S.val.density)\n ⟨S.scale (inv_pos.mpr S.separation_pos), h⟩\n simpa only [scale_density hd]\n · apply iSup_le\n intro ⟨S, _⟩\n exact le_iSup density S\n","line":319,"end_line":331,"proof_status":"formalized"}
{"id":"corollary:upper-bound-E8","label":"upper-bound-E8","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["theorem:CE_Main"],"nlp_text":"All packing \\(\\mathcal{P} \\subseteq \\mathbb {R}^8\\) has density satisfying \\(\\Delta _{\\mathcal{P}} \\leq \\Delta _{E_8}\\).","doc_url":null,"latex_link":"sec-sphere-packings.html#corollary:upper-bound-E8","lean_link":null,"lean_decl_name":"upper-bound-E8","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"IsZLattice","label":"IsZLattice","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"We say that an additive subgroup \\(\\Lambda \\leq \\mathbb {R}^d\\) is a lattice if it is discrete and its \\(\\mathbb {R}\\)-span contains all the elements of \\(\\mathbb {R}^d\\).","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/IsZLattice","latex_link":"sec-sphere-packings.html#IsZLattice","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/IsZLattice","lean_decl_name":"IsZLattice","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"./SpherePacking/Basic/E8.lean","lean4_code":"instance instIsZLatticeE8Lattice : IsZLattice ℝ E8Lattice where\n span_top := by rw [span_E8_eq_top']\n\n","line":561,"end_line":563,"proof_status":"formalized"}
{"id":"PeriodicSpherePacking","label":"PeriodicSpherePacking","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["IsZLattice"],"nlp_text":"We say that a sphere packing \\(\\mathcal{P}(X)\\) is (\\(\\Lambda \\)-)periodic if there exists a lattice \\(\\Lambda \\subset \\mathbb {R}^d\\) such that for all \\(x \\in X\\) and \\(y \\in \\Lambda \\), \\(x + y \\in X\\) (ie, \\(X\\) is \\(\\Lambda \\)-periodic).","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/PeriodicSpherePacking","latex_link":"sec-sphere-packings.html#PeriodicSpherePacking","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/PeriodicSpherePacking","lean_decl_name":"PeriodicSpherePacking","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"./SpherePacking/Basic/PeriodicPacking.lean","lean4_code":"theorem PeriodicSpherePacking.fract_centers\n {ι : Type*} [Fintype ι] (b : Basis ι ℤ S.lattice) (s : S.centers) :\n fract (b.ofZLatticeBasis ℝ _) s.val ∈ S.centers := by\n have := (floor (b.ofZLatticeBasis ℝ _) s).prop\n simp_rw [S.basis_Z_span] at this\n rw [fract_apply, sub_eq_add_neg, add_comm]\n apply S.lattice_action (neg_mem this) s.prop\n\n-- TODO: rename + move\n","line":128,"end_line":136,"proof_status":"formalized"}
{"id":"thm:periodic-packing-optimal","label":"periodic-packing-optimal","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["def:Periodic-sphere-packing-constant"],"nlp_text":"For all \\(d\\), the periodic sphere packing constant in \\(\\mathbb {R}^d\\) is equal to the sphere packing constant in \\(\\mathbb {R}^d\\).","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/periodic_constant_eq_constant","latex_link":"sec-sphere-packings.html#thm:periodic-packing-optimal","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/periodic_constant_eq_constant","lean_decl_name":"periodic_constant_eq_constant","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"SpherePacking/Basic/PeriodicPacking.lean","lean4_code":"theorem periodic_constant_eq_constant (hd : 0 < d) :\n PeriodicSpherePackingConstant d = SpherePackingConstant d := by\n sorry\n","line":1186,"end_line":1188,"proof_status":"statement_formalized"}
{"id":"def:psiI-psiT-psiS","label":"psiI-psiT-psiS","background_color":"white","border_color":"black","kind":"box","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["def: h"],"nlp_text":"We define the following three functions \\begin{align} \\psi _I\\, :=\\, & h-h|_{-2}ST \\label{eqn:psiI-define}\\\\ \\psi _T\\, :=\\, & \\psi _I|_{-2}T \\label{eqn:psiT-define}\\\\ \\psi _S\\, :=\\, & \\psi _I|_{-2}S. \\label{eqn:psiS-define} \\end{align}","doc_url":null,"latex_link":"sec--fourier-double-zeroes.html#def:psiI-psiT-psiS","lean_link":null,"lean_decl_name":"psiI-psiT-psiS","kind_description":"definitions","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"cor:psiI-near-0-infty","label":"psiI-near-0-infty","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["def:psiI-psiT-psiS","lemma:psi-bound"],"nlp_text":"We have \\begin{align} \\psi _I(it) & = O(t^2 e^{\\pi /t}) \\quad \\text{as } t \\to 0 \\label{eqn:psiI-near-0} \\\\ \\psi _I(it) & = O(e^{2 \\pi t}) \\quad \\text{as } t \\to \\infty . \\label{eqn:psiI-near-infty} \\end{align}","doc_url":null,"latex_link":"sec--fourier-double-zeroes.html#cor:psiI-near-0-infty","lean_link":null,"lean_decl_name":"psiI-near-0-infty","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"lemma:ineqABnew-equiv","label":"ineqABnew-equiv","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["def:psiI-psiT-psiS","lemma:F-G-phi-psi-identities","cor:disc-pos"],"nlp_text":"Inequality 204 and 206 are equivalent to \\begin{align} F(it) + \\frac{18}{\\pi ^2} G(it) {\\gt} 0 \\label{eqn:ineqAnew} \\\\ F(it) - \\frac{18}{\\pi ^2} G(it) {\\gt} 0 \\label{eqn:ineqBnew} \\end{align}","doc_url":null,"latex_link":"sec--g.html#lemma:ineqABnew-equiv","lean_link":null,"lean_decl_name":"ineqABnew-equiv","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"def:b-definition","label":"b-definition","background_color":"white","border_color":"black","kind":"box","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["def:psiI-psiT-psiS"],"nlp_text":"Define \\(b_\\mathrm{rad}: \\mathbb {R}\\to \\mathbb {C}\\) by \\begin{equation} b_\\mathrm{rad}(r) := J_1(r) + J_2(r) + J_3(r) + J_4(r) + J_5(r) + J_6(r) \\end{equation}","doc_url":null,"latex_link":"sec--fourier-double-zeroes.html#def:b-definition","lean_link":null,"lean_decl_name":"b-definition","kind_description":"definitions","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"prop:b-double-zeros","label":"b-double-zeros","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["cor:psiI-near-0-infty","cor:disc-nonvanishing"],"nlp_text":"For \\(r{\\gt}\\sqrt{2}\\) function \\(b(r)\\) can be expressed as \\begin{equation} \\label{eqn: b double zeroes} b(r)=-4\\sin (\\pi r^2/2)^2\\, \\int \\limits _{0}^{i\\infty }\\psi _I(z)\\, e^{\\pi i r^2 \\, z}\\, dz. \\end{equation}","doc_url":null,"latex_link":"sec--fourier-double-zeroes.html#prop:b-double-zeros","lean_link":null,"lean_decl_name":"b-double-zeros","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"prop:ineqB","label":"ineqB","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["lemma:ineqABnew-equiv","cor:ineqBnew"],"nlp_text":"Consider the function \\(B:(0,\\infty )\\to \\mathbb {C}\\) defined as \\begin{equation} \\label{eqn:defB} B(t) := -t^2\\phi _0(i/t)+\\frac{36}{\\pi ^2}\\, \\psi _I(it) \\end{equation}","doc_url":null,"latex_link":"sec--g.html#prop:ineqB","lean_link":null,"lean_decl_name":"ineqB","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"prop:ineqA","label":"ineqA","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["lemma:ineqABnew-equiv","cor:ineqAnew"],"nlp_text":"Consider the function \\(A:(0,\\infty )\\to \\mathbb {C}\\) defined as \\begin{equation} \\label{eqn:defA} A(t):=-t^2\\phi _0(i/t)-\\frac{36}{\\pi ^2}\\, \\psi _I(it). \\end{equation}","doc_url":null,"latex_link":"sec--g.html#prop:ineqA","lean_link":null,"lean_decl_name":"ineqA","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"prop:b-fourier","label":"b-fourier","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["def:b-definition","lemma:Gaussian-Fourier","prop:b-schwartz"],"nlp_text":"\\(b(x)\\) satisfies 114.","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/MagicFunction.b.Fourier.eig_b","latex_link":"sec--fourier-double-zeroes.html#prop:b-fourier","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/MagicFunction.b.Fourier.eig_b","lean_decl_name":"MagicFunction.b.Fourier.eig_b","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"SpherePacking/MagicFunction/b/Eigenfunction.lean","lean4_code":"theorem eig_b : fourierTransformCLE ℂ b = -b := by\n rw [b_eq_sum_integrals_SchwartzIntegrals]\n have hrw : J₁ + J₂ + J₃ + J₄ + J₅ + J₆ = (J₁ + J₂) + (J₃ + J₄) + J₅ + J₆ := by ac_rfl\n rw [hrw, map_add, map_add, map_add, perm_J₁_J₂, perm_J₅, perm_₃_J₄, perm_J₆]\n abel\n","line":55,"end_line":59,"proof_status":"formalized"}
{"id":"prop:b-another-integral","label":"b-another-integral","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["def:b-definition","prop:b-double-zeros"],"nlp_text":"For \\(r\\geq 0\\) we have \\begin{equation} \\label{eqn:b-another-integral}b(r)=4i\\, \\sin (\\pi r^2/2)^2\\, \\left(\\frac{144}{\\pi \\, r^2}+\\frac{1}{\\pi \\, (r^2-2)}+\\int \\limits _0^\\infty \\, \\left(\\psi _I(it)-144-e^{2\\pi t}\\right)\\, e^{-\\pi r^2 t}\\, dt\\right).\\end{equation}","doc_url":null,"latex_link":"sec--fourier-double-zeroes.html#prop:b-another-integral","lean_link":null,"lean_decl_name":"b-another-integral","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"thm:g1","label":"g1","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["prop:b-fourier","prop:ineqB","prop:ineqA","prop:a-fourier","prop:b0","prop:a0"],"nlp_text":"The function \\[ g(x):=\\frac{\\pi \\, i}{8640}a(x)+\\frac{i}{240\\pi }\\, b(x) \\]","doc_url":null,"latex_link":"sec--g.html#thm:g1","lean_link":null,"lean_decl_name":"g1","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"thm:g","label":"g","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["thm:g1"],"nlp_text":"There exists a radial Schwartz function \\(g:\\mathbb {R}^8\\to \\mathbb {R}\\) which satisfies: \\begin{align} g(x)& \\leq 0\\mbox{ for } \\| x\\| \\geq \\sqrt{2} \\label{eqn:g1}\\\\ \\widehat{g}(x)& \\geq 0\\mbox{ for all } x\\in \\mathbb {R}^8\\label{eqn:g2}\\\\ g(0)& =\\widehat{g}(0)=1.\\label{eqn:g3} \\end{align}","doc_url":null,"latex_link":"sect0003.html#thm:g","lean_link":null,"lean_decl_name":"g","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"./SpherePacking/MagicFunction/g/Basic.lean","lean4_code":"noncomputable def g : 𝓢(ℝ⁸, ℂ) := ((π * I) / 8640) • a + (I / (240 * π)) • b\n\n-- Note that in the proof, we need `g` to be Real-valued. We need to decide how we want to state\n-- this: either `g ∘ Complex.im = 0` or we actually construct an element of `𝓢(ℝ⁸, ℝ)`...\n\nsection Zero\n\n","line":26,"end_line":32,"proof_status":"statement_formalized"}
{"id":"prop:b0","label":"b0","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["prop:b-another-integral"],"nlp_text":"We have \\(b(0) = 0\\).","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/MagicFunction.b.SpecialValues.b_zero","latex_link":"sec--fourier-double-zeroes.html#prop:b0","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/MagicFunction.b.SpecialValues.b_zero","lean_decl_name":"MagicFunction.b.SpecialValues.b_zero","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"SpherePacking/MagicFunction/b/SpecialValues.lean","lean4_code":"theorem b_zero : b 0 = 0 := by sorry\n","line":15,"end_line":15,"proof_status":"formalized"}
{"id":"E8-Lattice","label":"E8-Lattice","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["E8-defs-equivalent"],"nlp_text":"\\(\\Lambda _8\\) is an additive subgroup of \\(\\mathbb {R}^8\\).","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/E8Lattice","latex_link":"sect0001.html#E8-Lattice","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/E8Lattice","lean_decl_name":"E8Lattice","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"SpherePacking/Basic/E8.lean","lean4_code":"noncomputable abbrev E8Lattice : Submodule ℤ (EuclideanSpace ℝ (Fin 8)) :=\n (Submodule.E8 ℝ).map (WithLp.linearEquiv 2 _ (Fin 8 → ℝ)).symm\n","line":521,"end_line":522,"proof_status":"formalized"}
{"id":"MainTheorem","label":"MainTheorem","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["corollary:upper-bound-E8"],"nlp_text":"\\(\\Delta _8 = \\Delta _{E_8}\\).","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/SpherePacking.MainTheorem","latex_link":"sec-sphere-packings.html#MainTheorem","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/SpherePacking.MainTheorem","lean_decl_name":"SpherePacking.MainTheorem","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/thefundamentaltheor3m/Sphere-Packing-Lean/2928857eab3e21dc6c3b5bfea025528145fe2113/SpherePacking/MainTheorem.lean","lean4_code":"theorem SpherePacking.MainTheorem : SpherePackingConstant 8 = E8Packing.density :=\n sorry","line":5,"end_line":6,"proof_status":"statement_formalized"}
{"id":"lemma:lattice-points-bound","label":"lattice-points-bound","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"For all \\(R\\), we have the following inequality relating the number of lattice points from \\(\\Lambda \\) in a ball with the volume of the ball and the fundamental region \\(\\mathcal{D}\\): \\[ \\frac{\\mathrm{Vol}(\\mathcal{B}_d(R - L))}{\\mathrm{Vol}(\\mathcal{D})} \\leq \\left|\\Lambda \\cap \\mathcal{B}_d(R)\\right| \\leq \\frac{\\mathrm{Vol}(\\mathcal{B}_d(R + L))}{\\mathrm{Vol}(\\mathcal{D})} \\]","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/PeriodicSpherePacking.aux2_ge'","latex_link":"sec-packings-density.html#lemma:lattice-points-bound","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/PeriodicSpherePacking.aux2_ge'","lean_decl_name":"PeriodicSpherePacking.aux2_ge'","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/thefundamentaltheor3m/Sphere-Packing-Lean/2928857eab3e21dc6c3b5bfea025528145fe2113/SpherePacking/Basic/PeriodicPacking.lean","lean4_code":"theorem PeriodicSpherePacking.aux2_ge'\n (hL : ∀ x ∈ fundamentalDomain (b.ofZLatticeBasis ℝ _), ‖x‖ ≤ L) (hd : 0 < d) :\n (↑S.lattice ∩ ball (0 : EuclideanSpace ℝ (Fin d)) R).encard\n ≥ volume (ball (0 : EuclideanSpace ℝ (Fin d)) (R - L))\n / volume (fundamentalDomain (b.ofZLatticeBasis ℝ _)) := by\n refine S.aux2_ge _ R ?_ (fundamentalDomain_measurableSet _) hL hd\n intro x\n obtain ⟨⟨v, hv⟩, hv'⟩ := exist_unique_vadd_mem_fundamentalDomain (b.ofZLatticeBasis ℝ _) x\n simp only [S.basis_Z_span] at hv hv' ⊢\n use ⟨v, hv⟩, hv'.left, ?_\n intro ⟨y, hy⟩ hy'\n have := hv'.right ⟨y, ?_⟩ hy'\n · rwa [Subtype.ext_iff] at this ⊢\n · rw [S.basis_Z_span]\n exact hy","line":674,"end_line":688,"proof_status":"formalized"}
{"id":"E8-vector-norms","label":"E8-vector-norms","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["E8-defs-equivalent"],"nlp_text":"All vectors in \\(\\Lambda _8\\) have norm of the form \\(\\sqrt{2n}\\), where \\(n\\) is a nonnegative integer.","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/E8_norm_eq_sqrt_even","latex_link":"sect0001.html#E8-vector-norms","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/E8_norm_eq_sqrt_even","lean_decl_name":"E8_norm_eq_sqrt_even","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"SpherePacking/Basic/E8.lean","lean4_code":"lemma E8_norm_eq_sqrt_even\n (v : Fin 8 → ℝ) (hv : v ∈ Submodule.E8 ℝ) :\n ∃ n : ℤ, Even n ∧ n = ‖WithLp.toLp 2 v‖ ^ 2 := by\n rw [← real_inner_self_eq_norm_sq, EuclideanSpace.inner_toLp_toLp, star_trivial]\n exact E8_integral_self _ hv\n","line":502,"end_line":506,"proof_status":"formalized"}
{"id":"instDiscreteE8Lattice","label":"instDiscreteE8Lattice","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["E8-vector-norms"],"nlp_text":"\\(c\\Lambda _8\\) is discrete, i.e. that the subspace topology induced by its inclusion into \\(\\mathbb {R}^8\\) is the discrete topology.","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/instDiscreteE8Lattice","latex_link":"sect0001.html#instDiscreteE8Lattice","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/instDiscreteE8Lattice","lean_decl_name":"instDiscreteE8Lattice","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/thefundamentaltheor3m/Sphere-Packing-Lean/2928857eab3e21dc6c3b5bfea025528145fe2113/SpherePacking/Basic/E8.lean","lean4_code":"instance instDiscreteE8Lattice : DiscreteTopology E8Lattice := by\n rw [discreteTopology_iff_isOpen_singleton_zero, Metric.isOpen_singleton_iff]\n use 1, by norm_num\n rintro ⟨v, hv⟩ h\n simp only [dist_zero_right, AddSubgroupClass.coe_norm] at h\n simp only [Submodule.mk_eq_zero]\n simp only [Submodule.mem_map, WithLp.linearEquiv_symm_apply] at hv\n obtain ⟨v, hv, rfl⟩ := hv\n simp only [AddEquiv.toEquiv_eq_coe, Equiv.invFun_as_coe, AddEquiv.coe_toEquiv_symm,\n WithLp.addEquiv_symm_apply] at h\n suffices v = 0 from congrArg (WithLp.toLp 2) this\n refine (E8_norm_lower_bound v hv).resolve_right ?_\n have : 1 < √2 := by rw [Real.lt_sqrt zero_le_one, sq, mul_one]; exact one_lt_two\n linarith","line":524,"end_line":537,"proof_status":"formalized"}
{"id":"instLatticeE8","label":"instLatticeE8","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["instDiscreteE8Lattice","E8-is-basis"],"nlp_text":"\\(c\\Lambda _8\\) is a \\(\\mathbb {Z}\\)-lattice, i.e. it is discrete and spans \\(\\mathbb {R}^8\\) over \\(\\mathbb {R}\\).","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/instIsZLatticeE8Lattice","latex_link":"sect0001.html#instLatticeE8","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/instIsZLatticeE8Lattice","lean_decl_name":"instIsZLatticeE8Lattice","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/thefundamentaltheor3m/Sphere-Packing-Lean/2928857eab3e21dc6c3b5bfea025528145fe2113/SpherePacking/Basic/E8.lean","lean4_code":"instance instIsZLatticeE8Lattice : IsZLattice ℝ E8Lattice where\n span_top := by rw [span_E8_eq_top']","line":561,"end_line":562,"proof_status":"formalized"}
{"id":"thm:ramanujan-formula","label":"ramanujan-formula","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["thm:serre-der-modularity","cor:dim-mf"],"nlp_text":"We have \\begin{align} E_2’ & = \\frac{E_2^2 - E_4}{12} \\label{eqn:DE2} \\\\ E_4’ & = \\frac{E_2 E_4 - E_6}{3} \\label{eqn:DE4} \\\\ E_6’ & = \\frac{E_2 E_6 - E_4^2}{2} \\label{eqn:DE6} \\end{align}","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/ramanujan_E₂","latex_link":"sect0004.html#thm:ramanujan-formula","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/ramanujan_E₂","lean_decl_name":"ramanujan_E₂","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/thefundamentaltheor3m/Sphere-Packing-Lean/2928857eab3e21dc6c3b5bfea025528145fe2113/SpherePacking/ModularForms/Derivative.lean","lean4_code":"@[simp]\ntheorem ramanujan_E₂ : D E₂ = 12⁻¹ * (E₂ * E₂ - E₄.toFun) := by\n ext z\n have h := ramanujan_E₂'\n unfold serre_D at h\n have h1 := congrFun h z\n simp [field]\n field_simp at h1\n simpa [add_comm, sub_eq_iff_eq_add] using h1","line":609,"end_line":617,"proof_status":"statement_formalized"}
{"id":"lemma:FG-de","label":"FG-de","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["thm:ramanujan-formula","thm:serre-der-prod-rule","prop:theta-der","lemma:lv1-lv2-identities"],"nlp_text":"\\(F\\) and \\(G\\) satisfy the following differential equations: \\begin{align} \\partial _{12}\\partial _{10} F - \\frac{5}{6} E_{4} F & = 7200 \\Delta (-E_{2}’) \\label{eqn:ddf} \\\\ \\partial _{12}\\partial _{10} G - \\frac{5}{6} E_{4} G & = -640 \\Delta H_{2} \\label{eqn:ddg} \\end{align}","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/MLDE_F","latex_link":"sec--g.html#lemma:FG-de","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/MLDE_F","lean_decl_name":"MLDE_F","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"SpherePacking/ModularForms/FG.lean","lean4_code":"Modular linear differential equation satisfied by $F$.\n-/\ntheorem MLDE_F : serre_D 12 (serre_D 10 F) = 5 * 6⁻¹ * F + 7200 * Δ_fun * negDE₂ := by\n ext x\n rw [negDE₂, Δ_fun, serre_D, serre_D, F_aux]\n unfold serre_D\n rw [F_aux]\n sorry\n\n","line":110,"end_line":118,"proof_status":"formalized"}
{"id":"cor:logder-disc-E2","label":"logder-disc-E2","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["thm:ramanujan-formula"],"nlp_text":"\\begin{equation} \\label{eqn:logder-disc-E2} \\Delta ' = E_2 \\Delta . \\end{equation}","doc_url":null,"latex_link":"sect0004.html#cor:logder-disc-E2","lean_link":null,"lean_decl_name":"logder-disc-E2","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"lemma:F-G-pos","label":"F-G-pos","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["thm:ramanujan-formula","cor:theta-pos"],"nlp_text":"For all \\(t {\\gt} 0\\), we have \\(F(it) {\\gt} 0\\) and \\(G(it) {\\gt} 0\\).","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/F_imag_axis_pos","latex_link":"sec--g.html#lemma:F-G-pos","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/F_imag_axis_pos","lean_decl_name":"F_imag_axis_pos","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"SpherePacking/ModularForms/FG.lean","lean4_code":" rw [negDE₂_qexp z]\n have hsum : Summable fun n : ℕ+ => (n : ℂ) * (ArithmeticFunction.sigma 1 n : ℂ) *\n Complex.exp (2 * ↑Real.pi * Complex.I * n * z) := negDE₂_summable t ht\n have hsum_re : Summable fun n : ℕ+ =>\n ((n : ℂ) * (ArithmeticFunction.sigma 1 n : ℂ) *\n Complex.exp (2 * ↑Real.pi * Complex.I * n * z)).re := ⟨_, Complex.hasSum_re hsum.hasSum⟩\n have hpos : ∀ n : ℕ+, 0 < ((n : ℂ) * (ArithmeticFunction.sigma 1 n : ℂ) *\n Complex.exp (2 * ↑Real.pi * Complex.I * n * z)).re := negDE₂_term_re_pos t ht\n","line":515,"end_line":522,"proof_status":"formalized"}
{"id":"cor:phi0-bound","label":"phi0-bound","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["thm:ramanujan-formula","lemma:Ek-Fourier","lemma:mod-div-disc-bound","lemma:mod_form_poly_growth"],"nlp_text":"There exists a constant \\(C_0 {\\gt} 0\\) such that \\begin{equation} \\label{eqn:phi0-bound} |\\phi _0(z)| \\le C_0 e^{-2 \\pi \\Im z} \\end{equation}","doc_url":null,"latex_link":"sec--fourier-double-zeroes.html#cor:phi0-bound","lean_link":null,"lean_decl_name":"phi0-bound","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"cor:MLDE-pos","label":"MLDE-pos","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["lemma:FG-de","cor:disc-pos","cor:theta-pos"],"nlp_text":"216 (resp. 217) is positive (resp. negative) on the (positive) imaginary axis.","doc_url":null,"latex_link":"sec--g.html#cor:MLDE-pos","lean_link":null,"lean_decl_name":"MLDE-pos","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"thm:anti-serre-der-pos","label":"anti-serre-der-pos","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["cor:logder-disc-E2"],"nlp_text":"Let \\(F\\) be a holomorphic quasimodular cusp form with real Fourier coefficients. Assume that there exists \\(k\\) such that \\((\\partial _{k}F)(it) {\\gt} 0\\) for all \\(t {\\gt} 0\\). If the first Fourier coefficient of \\(F\\) is positive, then \\(F(it) {\\gt} 0\\) for all \\(t {\\gt} 0\\).","doc_url":null,"latex_link":"sect0004.html#thm:anti-serre-der-pos","lean_link":null,"lean_decl_name":"anti-serre-der-pos","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"cor:ineqAnew","label":"ineqAnew","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["lemma:F-G-pos"],"nlp_text":"211 holds.","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/FG_inequality_1","latex_link":"sec--g.html#cor:ineqAnew","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/FG_inequality_1","lean_decl_name":"FG_inequality_1","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"SpherePacking/ModularForms/FG.lean","lean4_code":" have hProd : ResToImagAxis.Real (E₂ * E₄.toFun) :=\n ResToImagAxis.Real.mul E₂_imag_axis_real E₄_imag_axis_real\n have hNeg : ResToImagAxis.Real ((-1 : ℝ) • E₆.toFun) :=\n ResToImagAxis.Real.smul E₆_imag_axis_real\n have hSub : ResToImagAxis.Real (E₂ * E₄.toFun - E₆.toFun) := by\n have hEq : E₂ * E₄.toFun - E₆.toFun = E₂ * E₄.toFun + (-1 : ℝ) • E₆.toFun := by\n","line":595,"end_line":600,"proof_status":"statement_formalized"}
{"id":"cor:ineqBnew","label":"ineqBnew","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["lemma:F-G-pos","prop:Qdec","lemma:Qlim"],"nlp_text":"212 holds.","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/FG_inequality_2","latex_link":"sec--g.html#cor:ineqBnew","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/FG_inequality_2","lean_decl_name":"FG_inequality_2","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"SpherePacking/ModularForms/FG.lean","lean4_code":" simp [sub_eq_add_neg]\n simpa [hEq] using ResToImagAxis.Real.add hProd hNeg\n simpa [pow_two] using ResToImagAxis.Real.mul hSub hSub\n","line":602,"end_line":604,"proof_status":"statement_formalized"}
{"id":"cor:phi0-near-0-infty","label":"phi0-near-0-infty","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["cor:phi0-bound","lemma:phi0-transform","cor:phi2-bound","cor:phi4-bound"],"nlp_text":"We have \\begin{align} \\phi _0\\left(\\frac{i}{t}\\right) & = O(e^{-2 \\pi / t}) \\quad \\text{as } t \\to 0 \\label{eqn:phi0-near-0} \\\\ \\phi _0\\left(\\frac{i}{t}\\right) & = O(t^{-2}e^{2 \\pi t}) \\quad \\text{as } t \\to \\infty . \\label{eqn:phi0-near-infty} \\\\ \\end{align}","doc_url":null,"latex_link":"sec--fourier-double-zeroes.html#cor:phi0-near-0-infty","lean_link":null,"lean_decl_name":"phi0-near-0-infty","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"prop:a-schwartz","label":"a-schwartz","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["cor:phi0-bound","def:a-definition","thm:smooth-fast-decay-schwartz","lem:bound-I2-I4-I6","lem:bound-I1-I3-I5","lem:integral-bound"],"nlp_text":"\\(a(x)\\) is a Schwartz function.","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/MagicFunction.FourierEigenfunctions.a","latex_link":"sec--fourier-double-zeroes.html#prop:a-schwartz","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/MagicFunction.FourierEigenfunctions.a","lean_decl_name":"MagicFunction.FourierEigenfunctions.a","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"SpherePacking/MagicFunction/a/Schwartz.lean","lean4_code":"/-- The radial component of the +1-Fourier Eigenfunction of Viazovska's Magic Function. -/\n@[simps!]\ndef a' : 𝓢(ℝ, ℂ) :=\n MagicFunction.a.SchwartzIntegrals.I₁'\n + MagicFunction.a.SchwartzIntegrals.I₂'\n + MagicFunction.a.SchwartzIntegrals.I₃'\n + MagicFunction.a.SchwartzIntegrals.I₄'\n + MagicFunction.a.SchwartzIntegrals.I₅'\n + MagicFunction.a.SchwartzIntegrals.I₆'\n","line":172,"end_line":180,"proof_status":"formalized"}
{"id":"prop:Qdec","label":"Qdec","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["cor:MLDE-pos","thm:anti-serre-der-pos","lemma:log-der-inf"],"nlp_text":"The function \\(t \\mapsto Q(t)\\) is monotone decreasing.","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/FmodG_antitone","latex_link":"sec--g.html#prop:Qdec","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/FmodG_antitone","lean_decl_name":"FmodG_antitone","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"SpherePacking/ModularForms/FG.lean","lean4_code":"/--\n`G(it)` is real for all `t > 0`.\nBlueprint: G = H₂³ (2H₂² + 5H₂H₄ + 5H₄²), product of real functions.\n-/\ntheorem G_imag_axis_real : ResToImagAxis.Real G :=\n","line":582,"end_line":586,"proof_status":"formalized"}
{"id":"prop:a-double-zeros","label":"a-double-zeros","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["cor:phi0-near-0-infty","def:a-definition","cor:disc-nonvanishing"],"nlp_text":"For \\(r{\\gt}\\sqrt{2}\\) we can express \\(a(r)\\) in the following form \\begin{equation} \\label{eqn: a double zeroes} a(r)=-4\\sin (\\pi r^2/2)^2\\, \\int \\limits _{0}^{i\\infty }\\phi _0\\Big(\\frac{-1}{z}\\Big)\\, z^2\\, e^{\\pi i r^2 \\, z}\\, dz. \\end{equation}","doc_url":null,"latex_link":"sec--fourier-double-zeroes.html#prop:a-double-zeros","lean_link":null,"lean_decl_name":"a-double-zeros","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"prop:a-fourier","label":"a-fourier","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["prop:a-schwartz","lemma:Gaussian-Fourier"],"nlp_text":"\\(a(x)\\) satisfies 113.","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/MagicFunction.a.Fourier.eig_a","latex_link":"sec--fourier-double-zeroes.html#prop:a-fourier","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/MagicFunction.a.Fourier.eig_a","lean_decl_name":"MagicFunction.a.Fourier.eig_a","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"SpherePacking/MagicFunction/a/Eigenfunction.lean","lean4_code":"theorem eig_a : fourierTransformCLE ℂ a = a := by\n rw [a_eq_sum_integrals_SchwartzIntegrals]\n have hrw : I₁ + I₂ + I₃ + I₄ + I₅ + I₆ = (I₁ + I₂) + (I₃ + I₄) + I₅ + I₆ := by ac_rfl\n rw [hrw, map_add, map_add, map_add, perm_I₁_I₂, perm_I₅, perm_₃_I₄, perm_I₆]\n ac_rfl\n","line":56,"end_line":60,"proof_status":"statement_formalized"}
{"id":"def:Fourier-Transform","label":"Fourier-Transform","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"The Fourier transform of an \\(L^1\\)-function \\(f:\\mathbb {R}^d\\to \\mathbb {C}\\) is defined as \\[ \\mathcal{F}(f)(y) = \\widehat{f}(y) := \\int _{\\mathbb {R}^d} f(x)e^{-2\\pi i \\langle x, y \\rangle } \\, \\mathrm{d}x, \\quad y \\in \\mathbb {R}^d \\]","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/Real.fourierIntegral","latex_link":"sect0002.html#def:Fourier-Transform","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/Real.fourierIntegral","lean_decl_name":"Real.fourierIntegral","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Analysis/Fourier/FourierTransform.lean","lean4_code":"def fourierIntegral (e : AddChar 𝕜 𝕊) (μ : Measure V) (L : V →ₗ[𝕜] W →ₗ[𝕜] 𝕜) (f : V → E)\n (w : W) : E :=\n ∫ v, e (-L v w) • f v ∂μ\n\n","line":77,"end_line":80,"proof_status":"formalized"}
{"id":"lemma:Fourier-transform-is-automorphism","label":"Fourier-transform-is-automorphism","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["def:Fourier-Transform","def:Schwartz-Space"],"nlp_text":"The Fourier transform is a continuous, linear automorphism of the space of Schwartz functions.","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/SchwartzMap.fourierTransformCLM","latex_link":"sect0002.html#lemma:Fourier-transform-is-automorphism","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/SchwartzMap.fourierTransformCLM","lean_decl_name":"SchwartzMap.fourierTransformCLM","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Analysis/Distribution/FourierSchwartz.lean","lean4_code":"noncomputable def fourierTransformCLM : 𝓢(V, E) →L[𝕜] 𝓢(V, E) := by\n refine mkCLM (fun (f : V → E) ↦ 𝓕 f) ?_ ?_ ?_ ?_\n · intro f g x\n simp only [fourierIntegral_eq, Pi.add_apply, smul_add]\n rw [integral_add]\n · exact (fourierIntegral_convergent_iff _).2 f.integrable\n · exact (fourierIntegral_convergent_iff _).2 g.integrable\n · intro c f x\n simp only [fourierIntegral_eq, Pi.smul_apply, RingHom.id_apply, smul_comm _ c, integral_smul]\n · intro f\n exact Real.contDiff_fourierIntegral (fun n _ ↦ integrable_pow_mul volume f n)\n · rintro ⟨k, n⟩\n refine ⟨Finset.range (n + integrablePower (volume : Measure V) + 1) ×ˢ Finset.range (k + 1),\n (2 * π) ^ n * (2 * ↑n + 2) ^ k * (Finset.range (n + 1) ×ˢ Finset.range (k + 1)).card\n * 2 ^ integrablePower (volume : Measure V) *\n (∫ (x : V), (1 + ‖x‖) ^ (- (integrablePower (volume : Measure V) : ℝ))) * 2,\n ⟨by positivity, fun f x ↦ ?_⟩⟩\n apply (pow_mul_norm_iteratedFDeriv_fourierIntegral_le (f.smooth ⊤)\n (fun k n _hk _hn ↦ integrable_pow_mul_iteratedFDeriv _ f k n) le_top le_top x).trans\n simp only [mul_assoc]\n gcongr\n calc\n ∑ p ∈ Finset.range (n + 1) ×ˢ Finset.range (k + 1),\n ∫ (v : V), ‖v‖ ^ p.1 * ‖iteratedFDeriv ℝ p.2 (⇑f) v‖\n ≤ ∑ p ∈ Finset.range (n + 1) ×ˢ Finset.range (k + 1),\n 2 ^ integrablePower (volume : Measure V) *\n (∫ (x : V), (1 + ‖x‖) ^ (- (integrablePower (volume : Measure V) : ℝ))) * 2 *\n ((Finset.range (n + integrablePower (volume : Measure V) + 1) ×ˢ Finset.range (k + 1)).sup\n (schwartzSeminormFamily 𝕜 V E)) f := by\n apply Finset.sum_le_sum (fun p hp ↦ ?_)\n simp only [Finset.mem_product, Finset.mem_range] at hp\n apply (f.integral_pow_mul_iteratedFDeriv_le 𝕜 _ _ _).trans\n simp only [mul_assoc]\n rw [two_mul]\n gcongr\n · apply Seminorm.le_def.1\n have : (0, p.2) ∈ (Finset.range (n + integrablePower (volume : Measure V) + 1)\n ×ˢ Finset.range (k + 1)) := by simp [hp.2]\n apply Finset.le_sup this (f := fun p ↦ SchwartzMap.seminorm 𝕜 p.1 p.2 (E := V) (F := E))\n · apply Seminorm.le_def.1\n have : (p.1 + integrablePower (volume : Measure V), p.2) ∈ (Finset.range\n (n + integrablePower (volume : Measure V) + 1) ×ˢ Finset.range (k + 1)) := by\n simp [hp.2]\n omega\n apply Finset.le_sup this (f := fun p ↦ SchwartzMap.seminorm 𝕜 p.1 p.2 (E := V) (F := E))\n _ = _ := by simp [mul_assoc]\n\n","line":31,"end_line":77,"proof_status":"formalized"}
{"id":"lemma:Gaussian-Fourier","label":"Gaussian-Fourier","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["def:Fourier-Transform"],"nlp_text":"\\begin{equation} \\mathcal{F}(e^{\\pi i \\| x\\| ^2 z})(y) = z^{-4}\\, e^{\\pi i \\| y\\| ^2 \\, (\\frac{-1}{z}) }. \\end{equation}","doc_url":null,"latex_link":"sect0002.html#lemma:Gaussian-Fourier","lean_link":null,"lean_decl_name":"Gaussian-Fourier","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"def:E2","label":"E2","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"We set \\begin{equation} \\label{eqn:E2} E_2(z):= 1-24\\sum _{n=1}^\\infty \\sigma _1(n)\\, e^{2\\pi i n z}. \\end{equation}","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/E₂_eq","latex_link":"sect0004.html#def:E2","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/E₂_eq","lean_decl_name":"E₂_eq","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"SpherePacking/ModularForms/E2.lean","lean4_code":" rw [E₂]\n simp\n rw [G2_q_exp]\n rw [mul_sub]\n congr 1\n · rw [riemannZeta_two]\n have hpi : (π : ℂ) ≠ 0 := by simp\n field_simp\n · rw [← mul_assoc]\n congr 1\n · rw [riemannZeta_two]\n have hpi : (π : ℂ) ≠ 0 := by simp\n grind\n · have hl := tsum_pnat_eq_tsum_succ3 (fun n => sigma 1 n * cexp (2 * π * Complex.I * n * z))\n have hr := tsum_pnat_eq_tsum_succ3 (fun n => n * cexp (2 * π * Complex.I * n * z) / (1 - cexp\n (2 * π * Complex.I * n * z)))\n rw [hl, hr]\n have ht := tsum_eq_tsum_sigma z\n simp at *\n rw [ht]\n","line":864,"end_line":886,"proof_status":"formalized"}
{"id":"def:serre-der","label":"serre-der","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["def:E2","def:derivative"],"nlp_text":"For \\(k \\in \\mathbb {R}\\), define the weight \\(k\\) Serre derivative \\(\\partial _{k}\\) of a modular form \\(F\\) as \\begin{equation} \\label{eqn:serre-der} \\partial _{k}F := F' - \\frac{k}{12} E_2 F. \\end{equation}","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/serre_D","latex_link":"sect0004.html#def:serre-der","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/serre_D","lean_decl_name":"serre_D","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"SpherePacking/ModularForms/Derivative.lean","lean4_code":"/--\nSerre derivative of weight $k$.\nNote that the definition makes sense for any analytic function $F : \\mathbb{H} \\to \\mathbb{C}$.\n-/\nnoncomputable def serre_D (k : ℂ) : (ℍ → ℂ) → (ℍ → ℂ) :=\n fun (F : ℍ → ℂ) => (fun z => D F z - k * 12⁻¹ * E₂ z * F z)\n","line":318,"end_line":323,"proof_status":"formalized"}
{"id":"lemma:E2-transform-S","label":"E2-transform-S","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["def:E2"],"nlp_text":"This function is not modular, however it satisfies \\begin{equation} \\label{eqn:E2-S-transform} z^{-2}\\, E_2\\left(-\\frac{1}{z}\\right) = E_2(z) -\\frac{6i}{\\pi }\\, \\frac{1}{z}. \\end{equation}","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/E₂_transform","latex_link":"sect0004.html#lemma:E2-transform-S","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/E₂_transform","lean_decl_name":"E₂_transform","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"SpherePacking/ModularForms/E2.lean","lean4_code":"lemma E₂_transform (z : ℍ) : (E₂ ∣[(2 : ℤ)] ModularGroup.S) z =\n E₂ z + 6 / ( π * Complex.I * z) := by\n rw [E₂]\n have := G₂_transform (ModularGroup.S)\n have hsm := ModularForm.SL_smul_slash (2 : ℤ) ModularGroup.S G₂ (1 / (2 * riemannZeta 2))\n rw [hsm]\n simp only [SL_slash, one_div, mul_inv_rev, Pi.smul_apply, smul_eq_mul] at *\n rw [this]\n simp only [Pi.sub_apply]\n rw [D2_S]\n ring_nf\n rw [sub_eq_add_neg]\n congr\n rw [riemannZeta_two]\n have hpi : (π : ℂ) ≠ 0 := by simp\n ring_nf\n simp only [inv_pow, inv_I, mul_neg, neg_mul, neg_inj, mul_eq_mul_right_iff, OfNat.ofNat_ne_zero,\n or_false]\n rw [← inv_pow, pow_two, ← mul_assoc, mul_inv_cancel₀ hpi, one_mul]\n ring\n","line":762,"end_line":781,"proof_status":"formalized"}
{"id":"def:FG-definition","label":"FG-definition","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["def:E2","def:H2-H3-H4","def:Ek"],"nlp_text":"Define two (quasi) modular forms as \\begin{align} F(z) & = (E_2(z) E_4(z) - E_6(z))^2 \\label{eqn:defF} \\\\ G(z) & = H_2(z)^{3} (2 H_{2}(z)^{2} + 5 H_{2}(z) H_{4}(z) + 5 H_{4}(z)^{2}). \\label{eqn:defG} \\end{align}","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/F","latex_link":"sec--g.html#def:FG-definition","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/F","lean_decl_name":"F","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"SpherePacking/ModularForms/FG.lean","lean4_code":"Definition of $F$ and $G$ and auxiliary functions for the inequality between them\non the imaginary axis.\n-/\nnoncomputable def F := (E₂ * E₄.toFun - E₆.toFun) ^ 2\n\n","line":12,"end_line":16,"proof_status":"formalized"}
{"id":"def:phi4-phi2-phi0","label":"phi4-phi2-phi0","background_color":"white","border_color":"black","kind":"box","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["def:E2","def:Ek"],"nlp_text":"\\begin{align} \\phi _{-4} & := \\frac{E_4^2}{\\Delta } \\label{eqn: def phi4} \\\\ \\phi _{-2} & := \\frac{E_4(E_2 E_4 - E_6)}{\\Delta } \\label{eqn: def phi2} \\\\ \\phi _{0} & := \\frac{(E_2 E_4 - E_6)^2}{\\Delta } \\label{eqn: def phi0} \\end{align}","doc_url":null,"latex_link":"sec--fourier-double-zeroes.html#def:phi4-phi2-phi0","lean_link":null,"lean_decl_name":"phi4-phi2-phi0","kind_description":"definitions","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"thm:serre-der-equiv-action","label":"serre-der-equiv-action","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["def:serre-der","lemma:E2-transform-general"],"nlp_text":"Serre derivative \\(\\partial _{k}\\) is equivariant with the slash action of \\(\\mathrm{SL}_{2}(\\mathbb {Z})\\) in the following sense: \\begin{equation} \\partial _{k} (F|_{k}\\gamma ) = (\\partial _{k} F)|_{k+2}\\gamma , \\quad \\forall \\gamma \\in \\mathrm{SL}_{2}(\\mathbb {Z}). \\end{equation}","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/serre_D_slash_equivariant","latex_link":"sect0004.html#thm:serre-der-equiv-action","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/serre_D_slash_equivariant","lean_decl_name":"serre_D_slash_equivariant","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"SpherePacking/ModularForms/Derivative.lean","lean4_code":"/--\nSerre derivative is equivariant under the slash action. More precisely, if `F` is invariant\nunder the slash action of weight `k`, then `serre_D k F` is invariant under the slash action\nof weight `k + 2`.\n-/\ntheorem serre_D_slash_equivariant (k : ℤ) (F : ℍ → ℂ) (hF : MDifferentiable 𝓘(ℂ) 𝓘(ℂ) F) :\n ∀ γ : SL(2, ℤ), serre_D k F ∣[k + 2] γ = serre_D k (F ∣[k] γ) := by sorry\n","line":585,"end_line":591,"proof_status":"statement_formalized"}
{"id":"thm:serre-der-prod-rule","label":"serre-der-prod-rule","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["def:serre-der"],"nlp_text":"The Serre derivative satisfies the following product rule: for any quasimodular forms \\(F\\) and \\(G\\), \\begin{equation} \\partial _{w_1 + w_2} (FG) = (\\partial _{w_1}F)G + F (\\partial _{w_2}G). \\end{equation}","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/serre_D_mul","latex_link":"sect0004.html#thm:serre-der-prod-rule","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/serre_D_mul","lean_decl_name":"serre_D_mul","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"SpherePacking/ModularForms/Derivative.lean","lean4_code":"theorem serre_D_mul (k₁ k₂ : ℤ) (F G : ℍ → ℂ) (hF : MDifferentiable 𝓘(ℂ) 𝓘(ℂ) F)\n (hG : MDifferentiable 𝓘(ℂ) 𝓘(ℂ) G) :\n serre_D (k₁ + k₂) (F * G) = (serre_D k₁ F) * G + F * (serre_D k₂ G) := by\n calc\n serre_D (k₁ + k₂) (F * G)\n _ = D (F * G) - (k₁ + k₂) * 12⁻¹ * E₂ * (F * G) := by rfl\n _ = (D F * G + F * D G) - (k₁ + k₂) * 12⁻¹ * E₂ * (F * G) := by\n rw [D_mul F G hF hG]\n _ = (D F - k₁ * 12⁻¹ * E₂ * F) * G\n + F * (D G - k₂ * 12⁻¹ * E₂ * G) := by ring_nf\n _ = (serre_D k₁ F) * G + F * (serre_D k₂ G) := by rfl\n","line":349,"end_line":359,"proof_status":"formalized"}
{"id":"prop:theta-der","label":"theta-der","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["def:serre-der","lemma:jacobi-identity"],"nlp_text":"We have \\begin{align} H_2’ & = \\frac{1}{6} (H_{2}^{2} + 2 H_{2} H_{4} + E_2 H_2) \\label{eqn:H2-der}\\\\ H_3’ & = \\frac{1}{6} (H_{2}^{2} - H_{4}^{2} + E_2 H_3) \\label{eqn:H3-der}\\\\ H_4’ & = -\\frac{1}{6} (2H_{2} H_{4} + H_{4}^{2} - E_2 H_4) \\label{eqn:H4-der} \\end{align}","doc_url":null,"latex_link":"sect0004.html#prop:theta-der","lean_link":null,"lean_decl_name":"theta-der","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"lemma:E2-transform-general","label":"E2-transform-general","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["lemma:E2-transform-S"],"nlp_text":"\\begin{equation} \\label{eqn:E2-transform-general} (cz + d)^{-2} E_2\\left(\\frac{az + b}{cx + d}\\right) = E_2(z) - \\frac{6ic}{\\pi (cz + d)}, \\quad \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} \\in \\mathrm{SL}_{2}(\\mathbb {Z}). \\end{equation}","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/G₂_transform","latex_link":"sect0004.html#lemma:E2-transform-general","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/G₂_transform","lean_decl_name":"G₂_transform","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"SpherePacking/ModularForms/E2.lean","lean4_code":"lemma G₂_transform (γ : SL(2, ℤ)) : (G₂ ∣[(2 : ℤ)] γ) = G₂ - (D₂ γ) := by\n have := Subgroup.closure_induction (G := SL(2, ℤ)) (p := fun γ _ ↦ G₂ ∣[(2 : ℤ)] γ = G₂ - (D₂ γ))\n (k := ({ModularGroup.S, ModularGroup.T})) ?_ ?_\n · apply this\n · intro a b ha hb HA HB\n rw [D2_mul, SlashAction.slash_mul, HA, sub_eq_add_neg, SlashAction.add_slash, HB]\n ext z\n simp only [SlashAction.neg_slash, SL_slash, Pi.add_apply, Pi.sub_apply, Pi.neg_apply]\n ring\n · intro g hg hg2\n have H1 : (G₂ ∣[(2 : ℤ)] g) ∣[(2 : ℤ)] g⁻¹ = (G₂ - D₂ g)∣[(2 : ℤ)] g⁻¹ := by\n rw [hg2]\n rw [← SlashAction.slash_mul, sub_eq_add_neg, SlashAction.add_slash] at H1\n simp only [mul_inv_cancel, SlashAction.slash_one, SL_slash, SlashAction.neg_slash] at H1\n nth_rw 2 [H1]\n rw [← sub_eq_add_neg]\n have := D2_inv g\n simp only [SL_slash] at this\n rw [this]\n simp only [SL_slash, sub_neg_eq_add, add_sub_cancel_right]\n · rw [SL2_gens]\n simp only [Subgroup.mem_top]\n · intro a ha\n simp only [mem_insert_iff, mem_singleton_iff, SL_slash] at *\n rcases ha with h1|h2\n · ext z\n simp only [Pi.sub_apply]\n rw [h1, D2_S z]\n have:= modular_slash_S_apply G₂ 2 z\n simp only [SL_slash, Int.reduceNeg, zpow_neg] at this\n rw [this, mul_comm]\n have := G2_transf_aux z\n rw [← this]\n ring_nf\n rw [modular_S_smul]\n congr\n · simp only [UpperHalfPlane.coe, inv_pow, inv_inj]\n norm_cast\n simp only [UpperHalfPlane.coe]\n ring\n · simpa only [h2, D2_T, sub_zero] using G2_periodic\n · simp only [SlashAction.slash_one, D2_one, sub_zero]\n","line":717,"end_line":758,"proof_status":"formalized"}
{"id":"lemma:phi0-transform","label":"phi0-transform","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["lemma:E2-transform-S","def:phi4-phi2-phi0","lemma:disc-cuspform","lemma:Ek-is-modular-form"],"nlp_text":"We have \\begin{align} \\phi _0(z + 1) & = \\phi _0(z) \\label{eqn:phi0-trans-T} \\\\ \\phi _0\\left(-\\frac{1}{z}\\right) & = \\phi _0(z)-\\frac{12i}{\\pi }\\, \\frac{1}{z}\\, \\phi _{-2}(z)-\\frac{36}{\\pi ^2}\\, \\frac{1}{z^2}\\, \\phi _{-4}(z).\\label{eqn:phi0-trans-S} \\end{align}","doc_url":null,"latex_link":"sec--fourier-double-zeroes.html#lemma:phi0-transform","lean_link":null,"lean_decl_name":"phi0-transform","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"lemma:Qlim","label":"Qlim","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["lemma:E2-transform-S","lemma:theta-transform-S-T","lemma:Ek-is-modular-form"],"nlp_text":"We have \\begin{equation} \\lim _{t \\to 0^+} Q(t) = \\frac{18}{\\pi ^2}. \\end{equation}","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/FmodG_rightLimitAt_zero","latex_link":"sec--g.html#lemma:Qlim","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/FmodG_rightLimitAt_zero","lean_decl_name":"FmodG_rightLimitAt_zero","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"SpherePacking/ModularForms/FG.lean","lean4_code":"\n/--\n`F(it)` is real for all `t > 0`.\nBlueprint: Follows from E₂, E₄, E₆ having real values on the imaginary axis.\n-/\ntheorem F_imag_axis_real : ResToImagAxis.Real F := by\n","line":588,"end_line":593,"proof_status":"formalized"}
{"id":"lemma:F-G-phi-psi-identities","label":"F-G-phi-psi-identities","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["def:FG-definition","lemma:psi-new"],"nlp_text":"We have \\begin{align} \\phi _0 & = \\frac{F}{\\Delta } \\label{eqn:phi0-F} \\\\ \\psi _S & = -\\frac{1}{2} \\frac{G}{\\Delta }\\label{eqn:psiS-G} \\end{align}","doc_url":null,"latex_link":"sec--g.html#lemma:F-G-phi-psi-identities","lean_link":null,"lean_decl_name":"F-G-phi-psi-identities","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"cor:phi2-bound","label":"phi2-bound","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["def:phi4-phi2-phi0","lemma:Ek-Fourier","lemma:mod-div-disc-bound"],"nlp_text":"There exists a constant \\(C_{-2} {\\gt} 0\\) such that \\begin{equation} \\label{eqn:phi2-bound} |\\phi _{-2}(z)| \\le C_{-2} \\end{equation}","doc_url":null,"latex_link":"sec--fourier-double-zeroes.html#cor:phi2-bound","lean_link":null,"lean_decl_name":"phi2-bound","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"cor:phi4-bound","label":"phi4-bound","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["def:phi4-phi2-phi0","lemma:Ek-Fourier","lemma:mod-div-disc-bound"],"nlp_text":"There exists a constant \\(C_{-4} {\\gt} 0\\) such that \\begin{equation} \\label{eqn:phi4-bound} |\\phi _{-4}(z)| \\le C_{-4} e^{2 \\pi \\Im z} \\end{equation}","doc_url":null,"latex_link":"sec--fourier-double-zeroes.html#cor:phi4-bound","lean_link":null,"lean_decl_name":"phi4-bound","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"def:a-definition","label":"a-definition","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["def:phi4-phi2-phi0"],"nlp_text":"Define \\(a_{\\mathrm{rad}} : \\mathbb {R}\\to \\mathbb {C}\\) by \\begin{align} \\label{eqn:a-rad-definition} a_\\mathrm{rad}(r) := I_1(r) + I_2(r) + I_3(r) + I_4(r) + I_5(r) + I_6(r) \\end{align}","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/MagicFunction.a.RealIntegrals.a'","latex_link":"sec--fourier-double-zeroes.html#def:a-definition","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/MagicFunction.a.RealIntegrals.a'","lean_decl_name":"MagicFunction.a.RealIntegrals.a'","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/thefundamentaltheor3m/Sphere-Packing-Lean/2928857eab3e21dc6c3b5bfea025528145fe2113/SpherePacking/MagicFunction/a/Basic.lean","lean4_code":"def a' : ℝ → ℂ := fun x ↦ I₁' x + I₂' x + I₃' x + I₄' x + I₅' x + I₆' x","line":123,"end_line":123,"proof_status":"formalized"}
{"id":"thm:serre-der-modularity","label":"serre-der-modularity","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["thm:serre-der-equiv-action","def:holomorphic-modular-form"],"nlp_text":"Let \\(F\\) be a modular form of weight \\(k\\) and level \\(\\Gamma \\). Then, \\(\\partial _{k}F\\) is a modular form of weight \\(k + 2\\) of the same level.","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/serre_D_slash_invariant","latex_link":"sect0004.html#thm:serre-der-modularity","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/serre_D_slash_invariant","lean_decl_name":"serre_D_slash_invariant","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"SpherePacking/ModularForms/Derivative.lean","lean4_code":"theorem serre_D_slash_invariant (k : ℤ) (F : ℍ → ℂ) (hF : MDifferentiable 𝓘(ℂ) 𝓘(ℂ) F)\n (γ : SL(2, ℤ)) (h : F ∣[k] γ = F) :\n serre_D k F ∣[k + 2] γ = serre_D k F := by\n rw [serre_D_slash_equivariant, h]\n exact hF\n","line":593,"end_line":597,"proof_status":"formalized"}
{"id":"lemma:disc-E4E6","label":"disc-E4E6","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["def:disc-definition"],"nlp_text":"We have \\begin{equation} \\Delta (z) = (E_4^3-E_6^2)/1728. \\end{equation}","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/Delta_E4_eqn","latex_link":"sect0004.html#lemma:disc-E4E6","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/Delta_E4_eqn","lean_decl_name":"Delta_E4_eqn","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"SpherePacking/ModularForms/DimensionFormulas.lean","lean4_code":"theorem Delta_E4_eqn : Delta = Delta_E4_E6_aux := by\n ext z\n obtain ⟨c, H⟩ := delta_eq_E4E6_const\n suffices h2 : c = 1 by\n rw [h2] at H\n simp at H\n rw [H]\n · have h1 := Delta_q_one_term\n have h2 := Delta_E4_E6_aux_q_one_term\n have := qExpansion_smul 1 c Delta\n rw [← H] at h2\n rw [← Nat.cast_one (R := ℝ), ← this] at h2\n simp at h2\n rw [h1] at h2\n simpa using h2\n","line":168,"end_line":182,"proof_status":"formalized"}
{"id":"thm:lvl1_dims","label":"lvl1_dims","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["lemma:disc-E4E6","def:holomorphic-modular-form"],"nlp_text":"Let \\(k \\in \\mathbb {Z}\\) with \\(k \\ge 0\\) and even. Then \\(\\dim M_k(\\Gamma _1) = \\lfloor k / 12 \\rfloor \\) if \\(k \\equiv 2 \\mod 12\\) and \\(\\dim M_k(\\Gamma _1) = \\lfloor k / 12 \\rfloor + 1\\) if \\(k \\not\\equiv 2 \\mod 12\\).","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/ModularForm.dimension_level_one","latex_link":"sect0004.html#thm:lvl1_dims","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/ModularForm.dimension_level_one","lean_decl_name":"ModularForm.dimension_level_one","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/thefundamentaltheor3m/Sphere-Packing-Lean/2928857eab3e21dc6c3b5bfea025528145fe2113/SpherePacking/ModularForms/DimensionFormulas.lean","lean4_code":"lemma ModularForm.dimension_level_one (k : ℕ) (hk : 3 ≤ (k : ℤ)) (hk2 : Even k) :\n Module.rank ℂ (ModularForm (CongruenceSubgroup.Gamma 1) (k)) = if 12 ∣ ((k) : ℤ) - 2 then\n Nat.floor ((k : ℚ)/ 12) else Nat.floor ((k : ℚ) / 12) + 1 := by\n apply dim_modforms_lvl_one k hk hk2","line":511,"end_line":514,"proof_status":"formalized"}
{"id":"thm:dim-mf-general-level","label":"dim-mf-general-level","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["thm:lvl1_dims","def:Mk"],"nlp_text":"Let \\(\\Gamma \\) be a congruence subgroup. Then \\(M_k(\\Gamma )\\) is finite-dimensional.","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/dim_gen_cong_levels","latex_link":"sect0004.html#thm:dim-mf-general-level","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/dim_gen_cong_levels","lean_decl_name":"dim_gen_cong_levels","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"SpherePacking/ModularForms/DimensionFormulas.lean","lean4_code":"lemma dim_gen_cong_levels (k : ℤ) (Γ : Subgroup SL(2, ℤ)) (hΓ : Subgroup.index Γ ≠ 0) :\n FiniteDimensional ℂ (ModularForm Γ k) := by sorry\n","line":516,"end_line":517,"proof_status":"statement_formalized"}
{"id":"cor:dim-mf","label":"dim-mf","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["thm:lvl1_dims","def:Mk"],"nlp_text":"We have \\begin{align} \\dim M_2(\\mathrm{SL}_{2}(\\mathbb {Z})) & = 0, \\label{eqn:dimM2} \\\\ \\dim M_4(\\mathrm{SL}_{2}(\\mathbb {Z})) & = 1, \\label{eqn:dimM4} \\\\ \\dim M_6(\\mathrm{SL}_{2}(\\mathbb {Z})) & = 1, \\label{eqn:dimM6} \\\\ \\dim M_8(\\mathrm{SL}_{2}(\\mathbb {Z})) & = 1, \\label{eqn:dimM8} \\\\ \\dim S_4(\\mathrm{SL}_{2}(\\mathbb {Z})) & = 0, \\label{eqn:dimS4} \\\\ \\dim S_6(\\mathrm{SL}_{2}(\\mathbb {Z})) & = 0, \\label{eqn:dimS6} \\\\ \\dim S_8(\\mathrm{SL}_{2}(\\mathbb {Z})) & = 0. \\label{eqn:dimS8} \\end{align}","doc_url":null,"latex_link":"sect0004.html#cor:dim-mf","lean_link":null,"lean_decl_name":"dim-mf","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"lemma:jacobi-identity","label":"jacobi-identity","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["cor:dim-mf","lemma:theta-modular"],"nlp_text":"These three theta functions satisfy the Jacobi identity \\begin{equation} \\label{eqn:jacobi-identity} H_{2} + H_{4} = H_{3} \\Leftrightarrow \\Theta _{2}^4 + \\Theta _{4}^4 = \\Theta _{3}^4. \\end{equation}","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/jacobi_identity","latex_link":"sect0004.html#lemma:jacobi-identity","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/jacobi_identity","lean_decl_name":"jacobi_identity","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"SpherePacking/ModularForms/JacobiTheta.lean","lean4_code":" change (cexp (Real.pi * I * ((n : ℂ) + 1 / 2) ^ 2 * (I * t))).im = 0\n have hexpr : Real.pi * I * ((n : ℂ) + 1 / 2) ^ 2 * (I * ↑t) =\n (-(Real.pi * ((n : ℝ) + 1/2) ^ 2 * t) : ℝ) := by\n","line":900,"end_line":902,"proof_status":"formalized"}
{"id":"E8-defs-equivalent","label":"E8-defs-equivalent","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["E8-Matrix","E8-Set"],"nlp_text":"The two definitions above coincide, i.e. \\(\\Lambda _8 = \\mathrm{span}_{\\mathbb {Z}}(\\mathcal{B}_8)\\).","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/span_E8Matrix","latex_link":"sect0001.html#E8-defs-equivalent","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/span_E8Matrix","lean_decl_name":"span_E8Matrix","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"SpherePacking/Basic/E8.lean","lean4_code":"theorem span_E8Matrix (R : Type*) [Field R] [CharZero R] :\n Submodule.span ℤ (Set.range (E8Matrix R).row) = Submodule.E8 R := by\n apply Submodule.span_eq_of_le\n · exact range_E8Matrix_row_subset R\n intro v hv\n rw [Submodule.mem_span_range_iff_exists_fun]\n convert_to ∃ c : Fin 8 → ℤ, Matrix.vecMul (LinearMap.intCast R c) (E8Matrix R) = v\n using 3 with c\n · simp only [Matrix.vecMul_eq_sum, Matrix.row, LinearMap.intCast_apply, zsmul_eq_mul]\n rfl\n obtain ⟨c, hc⟩ := exists_cast_eq_vecMul_E8Inverse v hv\n use c\n rw [hc, Matrix.vecMul_vecMul, E8Inverse_mul_E8Matrix]\n simp\n","line":423,"end_line":436,"proof_status":"formalized"}
{"id":"prop:a-another-integral","label":"a-another-integral","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["prop:a-double-zeros"],"nlp_text":"For \\(r\\geq 0\\) we have \\begin{align} \\label{eqn:a-another-integral}a(r)=& 4i\\, \\sin (\\pi r^2/2)^2\\, \\Bigg(\\frac{36}{\\pi ^3\\, (r^2-2)}-\\frac{8640}{\\pi ^3\\, r^4}+\\frac{18144}{\\pi ^3\\, r^2}\\\\ +& \\int \\limits _0^\\infty \\, \\left(t^2\\, \\phi _0\\Big(\\frac{i}{t}\\Big)-\\frac{36}{\\pi ^2}\\, e^{2\\pi t}+\\frac{8640}{\\pi }\\, t-\\frac{18144}{\\pi ^2}\\right)\\, e^{-\\pi r^2 t}\\, dt \\Bigg) .\\notag \\end{align}","doc_url":null,"latex_link":"sec--fourier-double-zeroes.html#prop:a-another-integral","lean_link":null,"lean_decl_name":"a-another-integral","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"def:congruence-subgroup","label":"congruence-subgroup","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["def:level-N-princ-cong-subgp"],"nlp_text":"A subgroup \\(\\Gamma \\subset \\Gamma _1\\) is called a congruence subgroup if \\(\\Gamma (N)\\subset \\Gamma \\) for some \\(N\\in \\mathbb {N}\\).","doc_url":null,"latex_link":"sect0004.html#def:congruence-subgroup","lean_link":null,"lean_decl_name":"congruence-subgroup","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"def:holomorphic-modular-form","label":"holomorphic-modular-form","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["def:congruence-subgroup"],"nlp_text":"Let \\(\\Gamma \\) denote a subgroup of \\(\\mathrm{SL}_2(\\mathbb {Z})\\), then a modular form of level \\(\\Gamma \\) and weight \\(k \\in \\mathbb {Z}\\) is a function \\(f : \\mathbb {H} \\to \\mathbb {C}\\) such that: For all \\(\\gamma \\in \\Gamma \\) we have \\(f\\mid _k \\gamma = f\\) (such functions are called slash invariant). \\(f\\) is holomorphic on \\(\\mathbb {H}\\). For all \\(\\gamma \\in \\mathrm{SL}_2(\\mathbb {Z})\\), there exist \\(A, B \\in \\mathbb {R}\\) such that for all \\(z \\in \\mathbb {H}\\), with \\( A \\le \\mathrm{Im}(z)\\), we have \\(|(f \\mid _k \\gamma ) (z) |\\le B\\). Here \\(| - |\\) denotes the standard complex absolute value.","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/ModularForm","latex_link":"sect0004.html#def:holomorphic-modular-form","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/ModularForm","lean_decl_name":"ModularForm","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"./SpherePacking/ModularForms/DimensionFormulas.lean","lean4_code":"lemma ModularForm.dimension_level_one (k : ℕ) (hk : 3 ≤ (k : ℤ)) (hk2 : Even k) :\n Module.rank ℂ (ModularForm (CongruenceSubgroup.Gamma 1) (k)) = if 12 ∣ ((k) : ℤ) - 2 then\n Nat.floor ((k : ℚ)/ 12) else Nat.floor ((k : ℚ) / 12) + 1 := by\n apply dim_modforms_lvl_one k hk hk2\n\n","line":511,"end_line":515,"proof_status":"formalized"}
{"id":"def:Mk","label":"Mk","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["def:holomorphic-modular-form"],"nlp_text":"Let \\(M_k(\\Gamma )\\) be the space of modular forms of weight \\(k\\) and congruence subgroup \\(\\Gamma \\).","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/ModularForm","latex_link":"sect0004.html#def:Mk","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/ModularForm","lean_decl_name":"ModularForm","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"./SpherePacking/ModularForms/DimensionFormulas.lean","lean4_code":"lemma ModularForm.dimension_level_one (k : ℕ) (hk : 3 ≤ (k : ℤ)) (hk2 : Even k) :\n Module.rank ℂ (ModularForm (CongruenceSubgroup.Gamma 1) (k)) = if 12 ∣ ((k) : ℤ) - 2 then\n Nat.floor ((k : ℚ)/ 12) else Nat.floor ((k : ℚ) / 12) + 1 := by\n apply dim_modforms_lvl_one k hk hk2\n\n","line":511,"end_line":515,"proof_status":"formalized"}
{"id":"thm:nonpos_wt","label":"nonpos_wt","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["def:holomorphic-modular-form"],"nlp_text":"Let \\(k \\in \\mathbb {Z}\\) with \\(k {\\lt} 0\\). Then \\(M_k(\\Gamma _1) = \\{ 0\\} \\) and moreover \\(\\dim M_0(\\Gamma (1)) = 1\\).","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/ModularForm","latex_link":"sect0004.html#thm:nonpos_wt","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/ModularForm","lean_decl_name":"ModularForm","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"./SpherePacking/ModularForms/DimensionFormulas.lean","lean4_code":"lemma ModularForm.dimension_level_one (k : ℕ) (hk : 3 ≤ (k : ℤ)) (hk2 : Even k) :\n Module.rank ℂ (ModularForm (CongruenceSubgroup.Gamma 1) (k)) = if 12 ∣ ((k) : ℤ) - 2 then\n Nat.floor ((k : ℚ)/ 12) else Nat.floor ((k : ℚ) / 12) + 1 := by\n apply dim_modforms_lvl_one k hk hk2\n\n","line":511,"end_line":515,"proof_status":"formalized"}
{"id":"lemma:Ek-is-modular-form","label":"Ek-is-modular-form","background_color":"#1CAC78","border_color":"black","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["def:Mk","def:Ek"],"nlp_text":"For all \\(k\\), \\(E_k\\in M_k(\\Gamma _1)\\). Especially, we have \\begin{equation} \\label{eqn:Ek-trans-S} E_k \\left(-\\frac{1}{z}\\right) = z^k E_k(z). \\end{equation}","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/EisensteinSeries.eisensteinSeries_SIF","latex_link":"sect0004.html#lemma:Ek-is-modular-form","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/EisensteinSeries.eisensteinSeries_SIF","lean_decl_name":"EisensteinSeries.eisensteinSeries_SIF","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/NumberTheory/ModularForms/EisensteinSeries/Defs.lean","lean4_code":"def eisensteinSeries_SIF (k : ℤ) : SlashInvariantForm (Gamma N) k where\n toFun := eisensteinSeries a k\n slash_action_eq' A hA := by simp only [eisensteinSeries_slash_apply, Gamma_mem'.mp hA,\n SpecialLinearGroup.coe_one, vecMul_one]\n\n","line":120,"end_line":124,"proof_status":"formalized"}
{"id":"lemma:disc-cuspform","label":"disc-cuspform","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["lemma:dedekind_eta_transformation","def:disc-definition"],"nlp_text":"\\(\\Delta (z) \\in M_{12}(\\Gamma _1)\\). Especially, we have \\begin{equation} \\label{eqn:disc-trans-S} \\Delta \\left(-\\frac{1}{z}\\right) = z^{12} \\Delta (z). \\end{equation}","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/Delta","latex_link":"sect0004.html#lemma:disc-cuspform","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/Delta","lean_decl_name":"Delta","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"./SpherePacking/ModularForms/Delta.lean","lean4_code":"def Delta : CuspForm (CongruenceSubgroup.Gamma 1) 12 where\n toFun := Discriminant_SIF\n slash_action_eq' := Discriminant_SIF.slash_action_eq'\n holo' := by\n rw [mdifferentiable_iff]\n simp only [SlashInvariantForm.coe_mk]\n have := eta_DifferentiableAt_UpperHalfPlane\n have he2 : DifferentiableOn ℂ (fun z => (η z) ^ 24) {z | 0 < z.im} := by\n apply DifferentiableOn.pow\n intro x hx\n apply DifferentiableAt.differentiableWithinAt\n exact this ⟨x, hx⟩\n rw [Discriminant_SIF]\n simp only [SlashInvariantForm.coe_mk]\n apply he2.congr\n intro z hz\n have := Delta_eq_eta_pow (⟨z, hz⟩ : ℍ)\n simp only [coe_mk_subtype, comp_apply] at *\n rw [ofComplex_apply_of_im_pos hz]\n exact this\n zero_at_cusps' hc := zero_at_cusps_of_zero_at_infty hc Discriminant_zeroAtImInfty\n\n","line":317,"end_line":338,"proof_status":"formalized"}
{"id":"lemma:lv1-lv2-identities","label":"lv1-lv2-identities","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["lemma:disc-cuspform","lemma:theta-modular"],"nlp_text":"We have \\begin{align} E_4 & = \\frac{1}{2}(H_{2}^{2} + H_{3}^{2} + H_{4}^{2}) = H_{2}^{2} + H_{2}H_{4} + H_{4}^{2} \\label{eqn:e4theta} \\\\ E_6 & = \\frac{1}{2} (H_{2} + H_{3})(H_{3} + H_{4}) (H_{4} - H_{2}) = \\frac{1}{2}(H_2 + 2H_4)(2H_2 + H_4)(H_4 - H_2) \\label{eqn:e6theta} \\\\ \\Delta & = \\frac{1}{256} (H_{2}H_{3}H_{4})^2. \\label{eqn:disctheta} \\end{align}","doc_url":null,"latex_link":"sect0004.html#lemma:lv1-lv2-identities","lean_link":null,"lean_decl_name":"lv1-lv2-identities","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"def:th00-th01-th10","label":"th00-th01-th10","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"We define three different theta functions (so called “Thetanullwerte”) as \\begin{align} \\Theta _{2}(z) = \\theta _{10}(z)\\, =\\, & \\sum _{n\\in \\mathbb {Z}}e^{\\pi i (n+\\frac12)^2 z}. \\notag \\\\ \\Theta _{3}(z) = \\theta _{00}(z)\\, =\\, & \\sum _{n\\in \\mathbb {Z}}e^{\\pi i n^2 z} \\notag \\\\ \\Theta _{4}(z) = \\theta _{01}(z)\\, =\\, & \\sum _{n\\in \\mathbb {Z}}(-1)^n\\, e^{\\pi i n^2 z} \\notag \\\\ \\end{align}","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/Θ₂","latex_link":"sect0004.html#def:th00-th01-th10","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/Θ₂","lean_decl_name":"Θ₂","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"SpherePacking/ModularForms/JacobiTheta.lean","lean4_code":"noncomputable def Θ₂ (τ : ℍ) : ℂ := ∑' n : ℤ, Θ₂_term n τ\n","line":39,"end_line":39,"proof_status":"formalized"}
{"id":"def:H2-H3-H4","label":"H2-H3-H4","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["def:th00-th01-th10"],"nlp_text":"Define \\begin{equation} H_2 = \\Theta _2^4, \\quad H_3 = \\Theta _3^4, \\quad H_4 = \\Theta _4^4. \\label{eqn:H2-H3-H4} \\end{equation}","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/H₂","latex_link":"sect0004.html#def:H2-H3-H4","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/H₂","lean_decl_name":"H₂","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"SpherePacking/ModularForms/JacobiTheta.lean","lean4_code":"noncomputable def H₂ (τ : ℍ) : ℂ := (Θ₂ τ) ^ 4\n","line":42,"end_line":42,"proof_status":"formalized"}
{"id":"lemma:theta-transform-S-T","label":"theta-transform-S-T","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["def:H2-H3-H4"],"nlp_text":"These elements act on the theta functions in the following way \\begin{align} H_2 | S & = -H_4 \\label{eqn:H2-transform-S} \\\\ H_3 | S & = -H_3 \\label{eqn:H3-transform-S} \\\\ H_4 | S & = -H_2 \\label{eqn:H4-transform-S} \\end{align}","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/H₂_T_action","latex_link":"sect0004.html#lemma:theta-transform-S-T","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/H₂_T_action","lean_decl_name":"H₂_T_action","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"SpherePacking/ModularForms/JacobiTheta.lean","lean4_code":"/-- These three transformation laws follow directly from tsum definition. -/\nlemma H₂_T_action : (H₂ ∣[(2 : ℤ)] T) = -H₂ := by\n ext x\n suffices hΘ₂ : Θ₂ ((1 : ℝ) +ᵥ x) = cexp (π * I / 4) * Θ₂ x by\n simp_rw [modular_slash_T_apply, Pi.neg_apply, H₂, hΘ₂, mul_pow, ← Complex.exp_nat_mul,\n mul_comm ((4 : ℕ) : ℂ), Nat.cast_ofNat, div_mul_cancel₀ (b := (4 : ℂ)) _ (by simp),\n Complex.exp_pi_mul_I, neg_one_mul]\n calc\n _ = ∑' (n : ℤ), cexp (π * I * (n + 1 / 2) ^ 2 * ((1 : ℝ) +ᵥ x)) := by\n simp_rw [Θ₂, Θ₂_term]\n _ = ∑' (n : ℤ), cexp (π * I / 4) * cexp (π * I * (n ^ 2 + n) + π * I * (n + 1 / 2) ^ 2 * x) := by\n apply tsum_congr fun b ↦ ?_\n rw [coe_vadd, ofReal_one]\n repeat rw [← Complex.exp_add]\n congr\n ring_nf\n _ = cexp (π * I / 4) * ∑' (n : ℤ), cexp (π * I * (n ^ 2 + n) + π * I * (n + 1 / 2) ^ 2 * x) := by\n rw [tsum_mul_left]\n _ = _ := by\n simp_rw [Θ₂, Θ₂_term]\n congr 1\n apply tsum_congr fun b ↦ ?_\n have : Even (b ^ 2 + b) := by\n convert Int.even_mul_succ_self b using 1\n ring_nf\n norm_cast\n rw [Complex.exp_add]\n rw [mul_comm (π * I), Complex.exp_int_mul, Complex.exp_pi_mul_I, this.neg_one_zpow, one_mul]\n","line":77,"end_line":104,"proof_status":"formalized"}
{"id":"prop:H2-fourier","label":"H2-fourier","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["def:H2-H3-H4"],"nlp_text":"\\(H_2\\) admits a Fourier series of the form \\begin{equation} H_2(z) = \\sum _{n \\ge 1} c_{H_2}(n) e^{\\pi i n z} \\end{equation}","doc_url":null,"latex_link":"sect0004.html#prop:H2-fourier","lean_link":null,"lean_decl_name":"H2-fourier","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"prop:H4-fourier","label":"H4-fourier","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["def:H2-H3-H4"],"nlp_text":"\\(H_4\\) admits a Fourier series of the form \\begin{equation} H_4(z) = \\sum _{n \\ge 0} c_{H_4}(n) e^{\\pi i n z} \\end{equation}","doc_url":null,"latex_link":"sect0004.html#prop:H4-fourier","lean_link":null,"lean_decl_name":"H4-fourier","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"def: h","label":"h","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["def:H2-H3-H4"],"nlp_text":"\\begin{equation} \\label{eqn: h define} h(z) := 128 \\frac{H_3(z) + H_4(z)}{H_2(z)^2}. \\end{equation}","doc_url":null,"latex_link":"sec--fourier-double-zeroes.html#def: h","lean_link":null,"lean_decl_name":"h","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"./SpherePacking/MagicFunction/b/psi.lean","lean4_code":"def h : ℍ → ℂ := 128 • (H₃_MF + H₄_MF) / (H₂_MF ^ 2)\n\n","line":41,"end_line":42,"proof_status":"statement_formalized"}
{"id":"prop:H3-fourier","label":"H3-fourier","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["def:H2-H3-H4"],"nlp_text":"\\(H_3\\) admits a Fourier series of the form \\begin{equation} H_3(z) = \\sum _{n \\ge 0} c_{H_3}(n) e^{\\pi i n z} \\end{equation}","doc_url":null,"latex_link":"sect0004.html#prop:H3-fourier","lean_link":null,"lean_decl_name":"H3-fourier","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"lemma:theta-slash-invariant","label":"theta-slash-invariant","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["lemma:theta-transform-S-T","lemma:slash-negI-even-weight","lemma:Gamma-2-generators","lemma:slash-operator-chain-rule"],"nlp_text":"\\(H_{2}\\), \\(H_{3}\\), and \\(H_{4}\\) are slash invariant under \\(\\Gamma (2)\\), i.e. for all \\(\\gamma \\in \\Gamma (2)\\) and \\(i \\in \\{ 2, 3, 4\\} \\), we have \\(H_i|\\gamma = H_i|\\gamma ^{-1} = H_i\\).","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/H₂_SIF","latex_link":"sect0004.html#lemma:theta-slash-invariant","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/H₂_SIF","lean_decl_name":"H₂_SIF","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"SpherePacking/ModularForms/JacobiTheta.lean","lean4_code":"/-- H₂, H₃, H₄ are modular forms of weight 2 and level Γ(2) -/\nnoncomputable def H₂_SIF : SlashInvariantForm (Γ 2) 2 where\n toFun := H₂\n slash_action_eq' := slashaction_generators_Γ2 H₂ (2 : ℤ) H₂_α_action H₂_β_action H₂_negI_action\n","line":251,"end_line":254,"proof_status":"formalized"}
{"id":"lemma:theta-bounded-im-infty","label":"theta-bounded-im-infty","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["lemma:theta-slash-invariant","lemma:Gamma-1-generators"],"nlp_text":"For all \\(\\gamma \\in \\Gamma _1\\), \\(H_{2}|_2 \\gamma \\), \\(H_{3}|_2 \\gamma \\), and \\(H_{4}|_2 \\gamma \\) are holomorphic at \\(i\\infty \\).","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/isBoundedAtImInfty_H_slash","latex_link":"sect0004.html#lemma:theta-bounded-im-infty","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/isBoundedAtImInfty_H_slash","lean_decl_name":"isBoundedAtImInfty_H_slash","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"SpherePacking/ModularForms/JacobiTheta.lean","lean4_code":" · intro x hx y _ h\n simp_rw [slash_mul]\n rcases hx with (rfl | rfl | rfl | _)\n · simp_rw [H₂_S_inv_action, H₃_S_inv_action, H₄_S_inv_action, neg_slash,\n isBoundedAtImInfty_neg_iff]\n use h.right.right, h.right.left, h.left\n · simp_rw [H₂_T_inv_action, H₃_T_inv_action, H₄_T_inv_action, neg_slash,\n isBoundedAtImInfty_neg_iff]\n use h.left, h.right.right, h.right.left\n · rw [← Subgroup.coe_inv, modular_negI_inv, SL_slash,\n modular_slash_negI_of_even _ 2 (by decide)]\n rw [H₃_negI_action, H₄_negI_action]\n exact h\n · intro s hs\n simp_rw [Set.mem_setOf_eq, Set.mem_range] at hs\n obtain ⟨s, rfl⟩ := hs\n rw [Set.mem_iInter, SetLike.mem_coe]\n intro hs\n have hs2 : {S, T} ⊆ (s : Set (SL(2, ℤ))) := by\n apply subset_trans _ hs\n simp only [Set.singleton_subset_iff, Set.mem_insert_iff, Set.mem_singleton_iff, true_or,\n Set.insert_subset_insert]\n simp only [top_le_iff.mp <| SL2Z_generate.symm ▸ (Subgroup.closure_le s).mpr hs2,\n Subgroup.mem_top]\n\ntheorem isBoundedAtImInfty_H₂_slash :\n ∀ A ∈ 𝒮ℒ, IsBoundedAtImInfty (H₂ ∣[(2 : ℤ)] (A : GL (Fin 2) ℝ)) := by\n intro A ⟨A', hA⟩\n exact hA.symm ▸ (isBoundedAtImInfty_H_slash A').left\n\ntheorem isBoundedAtImInfty_H₃_slash :\n ∀ A ∈ 𝒮ℒ, IsBoundedAtImInfty (H₃ ∣[(2 : ℤ)] (A : GL (Fin 2) ℝ)) := by\n intro A ⟨A', hA⟩\n exact hA.symm ▸ (isBoundedAtImInfty_H_slash A').right.left\n\ntheorem isBoundedAtImInfty_H₄_slash :\n ∀ A ∈ 𝒮ℒ, IsBoundedAtImInfty (H₄ ∣[(2 : ℤ)] (A : GL (Fin 2) ℝ)) := by\n intro A ⟨A', hA⟩\n exact hA.symm ▸ (isBoundedAtImInfty_H_slash A').right.right\n","line":503,"end_line":541,"proof_status":"formalized"}
{"id":"def:Gamma-1-Action","label":"Gamma-1-Action","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"The modular group \\(\\Gamma _1:=\\mathrm{SL}_2(\\mathbb {Z})\\) acts on \\(\\mathfrak {H}\\) by linear fractional transformations \\[ \\left(\\begin{smallmatrix} a & b \\\\ c & d \\end{smallmatrix}\\right)z:=\\frac{az+b}{cz+d}. \\]","doc_url":null,"latex_link":"sect0004.html#def:Gamma-1-Action","lean_link":null,"lean_decl_name":"Gamma-1-Action","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"def:slash-operator","label":"slash-operator","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["def:Gamma-1-Action","def:automorphy-factor"],"nlp_text":"Let \\(F\\) be a function on \\(\\mathfrak {H}\\) and \\(\\gamma \\in \\mathrm{SL}_2(\\mathbb {Z})\\). Then the slash operator acts on \\(F\\) by \\[ (F|_k\\gamma )(z):=j_k(z,\\gamma )\\, F(\\gamma z). \\]","doc_url":null,"latex_link":"sect0004.html#def:slash-operator","lean_link":null,"lean_decl_name":"slash-operator","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"lemma:slash-negI-even-weight","label":"slash-negI-even-weight","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["def:slash-operator"],"nlp_text":"For even \\(k\\), \\(F|_{k}(-I) = F\\).","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/modular_slash_negI_of_even","latex_link":"sect0004.html#lemma:slash-negI-even-weight","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/modular_slash_negI_of_even","lean_decl_name":"modular_slash_negI_of_even","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"SpherePacking/ModularForms/SlashActionAuxil.lean","lean4_code":"theorem modular_slash_negI_of_even (hk : Even k) : f ∣[k] negI.1 = f := by\n ext x; rw [slash_action_eq'_iff, modular_negI_smul]; simp [negI, hk.neg_one_zpow]\n","line":47,"end_line":48,"proof_status":"formalized"}
{"id":"lemma:dedekind_eta_transformation","label":"dedekind_eta_transformation","background_color":"#1CAC78","border_color":"blue","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["def:dedekind_eta"],"nlp_text":"The Dedekind eta function transforms as \\[ \\eta \\left(-\\frac{1}{z}\\right) = \\sqrt{-iz} \\eta (z). \\]","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/eta_equality","latex_link":"sect0004.html#lemma:dedekind_eta_transformation","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/eta_equality","lean_decl_name":"eta_equality","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"SpherePacking/ModularForms/eta.lean","lean4_code":"lemma eta_equality : {z : ℂ | 0 < z.im}.EqOn ((η ∘ (fun z : ℂ => -1/z)))\n ((csqrt (Complex.I))⁻¹ • ((csqrt) * η)) := by\n have h := eta_logderivs_const\n obtain ⟨z, hz, h⟩ := h\n intro x hx\n have h2 := h hx\n have hI : (Complex.I) ∈ {z : ℂ | 0 < z.im} := by\n simp only [mem_setOf_eq, Complex.I_im, zero_lt_one]\n have h3 := h hI\n simp at h3\n conv at h3 =>\n enter [2]\n rw [← mul_assoc]\n have he : η Complex.I ≠ 0 := by\n have h:= eta_nonzero_on_UpperHalfPlane UpperHalfPlane.I\n convert h\n have hcd := (mul_eq_right₀ he).mp (_root_.id (Eq.symm h3))\n rw [mul_eq_one_iff_inv_eq₀ hz] at hcd\n rw [@inv_eq_iff_eq_inv] at hcd\n rw [hcd] at h2\n exact h2\n","line":408,"end_line":428,"proof_status":"formalized"}
{"id":"thm:smooth-fast-decay-schwartz","label":"smooth-fast-decay-schwartz","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Assume \\(f : \\mathbb {R}\\to \\mathbb {C}\\) is smooth on \\([0, \\infty )\\) and for all \\(k, n \\in \\mathbb {N}\\), there exists \\(C \\in \\mathbb {R}\\) such that \\[ x^{\\frac{k}{2}} \\cdot |f^{(n)}(x)| \\leq C. \\]","doc_url":null,"latex_link":"sect0002.html#thm:smooth-fast-decay-schwartz","lean_link":null,"lean_decl_name":"smooth-fast-decay-schwartz","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"prop:b-schwartz","label":"b-schwartz","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["thm:smooth-fast-decay-schwartz","lemma:bound-J2-J4-J6","lemma:psi-bound","lemma:bound-J1-J3-J5"],"nlp_text":"\\(b(x)\\) is a Schwartz function.","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/MagicFunction.FourierEigenfunctions.b","latex_link":"sec--fourier-double-zeroes.html#prop:b-schwartz","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/MagicFunction.FourierEigenfunctions.b","lean_decl_name":"MagicFunction.FourierEigenfunctions.b","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"SpherePacking/MagicFunction/b/Schwartz.lean","lean4_code":"/-- The radial component of the -1-Fourier Eigenfunction of Viazovska's Magic Function. -/\n@[simps!]\ndef b' : 𝓢(ℝ, ℂ) :=\n MagicFunction.b.SchwartzIntegrals.J₁'\n + MagicFunction.b.SchwartzIntegrals.J₂'\n + MagicFunction.b.SchwartzIntegrals.J₃'\n + MagicFunction.b.SchwartzIntegrals.J₄'\n + MagicFunction.b.SchwartzIntegrals.J₅'\n + MagicFunction.b.SchwartzIntegrals.J₆'\n","line":154,"end_line":162,"proof_status":"formalized"}
{"id":"lemma:Schwartz-summable","label":"Schwartz-summable","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["def:Schwartz-Space"],"nlp_text":"Let \\(f : \\mathbb {R}^d \\to \\mathbb {C}\\) be a Schwartz function and let \\(X \\subset \\mathbb {R}^d\\) be periodic with respect to some lattice \\(\\Lambda \\subset \\mathbb {R}^d\\). Then, \\[ \\sum _{x \\in X} |f(x)| {\\lt} \\infty . \\]","doc_url":null,"latex_link":"sect0002.html#lemma:Schwartz-summable","lean_link":null,"lean_decl_name":"Schwartz-summable","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"lemma:Gamma-2-generators","label":"Gamma-2-generators","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["def:Gamma-generators"],"nlp_text":"We have \\(\\Gamma (2) = \\langle \\alpha , \\beta , -I \\rangle \\).","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/Γ2_generate","latex_link":"sect0004.html#lemma:Gamma-2-generators","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/Γ2_generate","lean_decl_name":"Γ2_generate","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"SpherePacking/ModularForms/SlashActionAuxil.lean","lean4_code":"theorem Γ2_generate : (⊤ : Subgroup (Γ 2)) = Subgroup.closure {α, β, negI} := by\n refine le_antisymm ?_ le_top\n intro A hA\n by_contra h_contra\n let d := sInf {n : ℕ | ∃ M ∈ Subgroup.closure {α, β, negI}, n = Int.natAbs ((M * A).val.val 1 0)}\n obtain ⟨M, hM₁, hM₂⟩ :\n ∃ M ∈ Subgroup.closure {α, β, negI}, Int.natAbs ((M * A).val.val 1 0) = d := by\n have h_nonempty :\n {n : ℕ | ∃ M ∈ Subgroup.closure {α, β, negI}, n\n = Int.natAbs ((M * A).val.val 1 0)}.Nonempty :=\n ⟨_, ⟨1, Subgroup.one_mem _, rfl⟩⟩\n exact (Nat.sInf_mem h_nonempty) |> fun ⟨M, hM₁, hM₂⟩ ↦ ⟨M, hM₁, hM₂.symm⟩\n by_cases hd_pos : d > 0\n · obtain ⟨M', hM'₁, hM'₂⟩ : ∃ M' ∈ Subgroup.closure {α, β, negI},\n Int.natAbs ((M' * (M * A)).val.val 1 0) < d := by\n convert Γ2_descent (M * A) fun h ↦ _ <;> simp_all [← hM₂, ← Int.ofNat_lt]\n have h_inf_le :\n ∀ n ∈ {n : ℕ | ∃ M ∈ Subgroup.closure {α, β, negI}, n\n = Int.natAbs ((M * A).val.val 1 0)}, d ≤ n :=\n fun n hn ↦ Nat.sInf_le hn\n exact not_lt_of_ge (h_inf_le _ ⟨M' * M, Subgroup.mul_mem _ hM'₁ hM₁, rfl⟩)\n (by simpa [mul_assoc] using hM'₂)\n · have hMA_zero : (M * A).val.val 1 0 = 0 :=\n Int.natAbs_eq_zero.mp (hM₂.trans (le_antisymm (le_of_not_gt hd_pos) (Nat.zero_le _)))\n exact h_contra\n (by simpa only [inv_mul_cancel_left] using\n Subgroup.mul_mem _ (Subgroup.inv_mem _ hM₁) (Γ2_c_eq_zero _ hMA_zero))\n","line":202,"end_line":228,"proof_status":"formalized"}
{"id":"lemma:theta-modular","label":"theta-modular","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["lemma:theta-bounded-im-infty"],"nlp_text":"\\(H_{2}\\), \\(H_{3}\\), and \\(H_{4}\\) belong to \\(M_2(\\Gamma (2))\\).","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/H₂_MF","latex_link":"sect0004.html#lemma:theta-modular","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/H₂_MF","lean_decl_name":"H₂_MF","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"SpherePacking/ModularForms/JacobiTheta.lean","lean4_code":" bdd_at_cusps' hc := bounded_at_cusps_of_bounded_at_infty hc isBoundedAtImInfty_H₄_slash\n}\n\n/-!\n## Jacobi identity\n","line":560,"end_line":564,"proof_status":"formalized"}
{"id":"prop:a0","label":"a0","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["prop:a-another-integral"],"nlp_text":"We have \\(a(0) = -\\frac{i}{8640}\\).","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/MagicFunction.a.SpecialValues.a_zero","latex_link":"sec--fourier-double-zeroes.html#prop:a0","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/MagicFunction.a.SpecialValues.a_zero","lean_decl_name":"MagicFunction.a.SpecialValues.a_zero","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"SpherePacking/MagicFunction/a/SpecialValues.lean","lean4_code":"theorem a_zero : a 0 = -8640 * I / π := by sorry\n","line":15,"end_line":15,"proof_status":"formalized"}
{"id":"cor:theta-pos","label":"theta-pos","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["lemma:jacobi-identity"],"nlp_text":"All three functions \\(t \\mapsto H_2(it), H_3(it), H_4(it)\\) are positive for \\(t {\\gt} 0\\).","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/H₂_imag_axis_pos","latex_link":"sect0004.html#cor:theta-pos","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/H₂_imag_axis_pos","lean_decl_name":"H₂_imag_axis_pos","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"SpherePacking/ModularForms/JacobiTheta.lean","lean4_code":" · simp\n · simp [hz_real]\n rw [hz_eq]\n norm_cast\n rw [hpow]\n exact pow_pos hΘ₂_re_pos 4\n\n/--\n`H₄(it)` is real for all `t > 0`.\nBlueprint: Corollary 6.43 - follows from Θ₄ being real on the imaginary axis.\n-/\ntheorem H₄_imag_axis_real : ResToImagAxis.Real H₄ := by\n intro t ht\n simp only [Function.resToImagAxis, ResToImagAxis, ht, ↓reduceDIte, H₄]\n have hΘ₄_im := Θ₄_imag_axis_real t ht\n exact Complex.im_pow_eq_zero_of_im_eq_zero hΘ₄_im 4\n\n/--\n`H₄(it) > 0` for all `t > 0`.\nBlueprint: Corollary 6.43 - H₄ is positive on the imaginary axis.\n\nProof strategy: Use the modular S-transformation relating H₄ and H₂.\nFrom H₄_S_action: (H₄ ∣[2] S) = -H₂\nFrom ResToImagAxis.SlashActionS: relates values at t and 1/t.\nThis gives H₂(i/t) = t² * H₄(it), so H₄(it) > 0 follows from H₂(i/t) > 0.\n-/\ntheorem H₄_imag_axis_pos : ResToImagAxis.Pos H₄ := by\n constructor\n · exact H₄_imag_axis_real\n","line":1053,"end_line":1081,"proof_status":"statement_formalized"}
{"id":"lemma:bound-J2-J4-J6","label":"bound-J2-J4-J6","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"There exist \\(C_1, C_2 {\\gt} 0\\) such that \\begin{align} |J_2(r)|, |J_4(r)| & \\le C_1 e^{-\\pi r} \\\\ |J_6(r)| & \\le C_2 \\frac{e^{\\pi (r + 1)}}{r + 1} \\end{align}","doc_url":null,"latex_link":"sec--fourier-double-zeroes.html#lemma:bound-J2-J4-J6","lean_link":null,"lean_decl_name":"bound-J2-J4-J6","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"lemma:periodic-points-bounds","label":"periodic-points-bounds","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"For all \\(R\\), we have the following inequality relating the number of points from \\(X\\) (periodic w.r.t. \\(\\Lambda \\)) in a ball with the number of points from \\(\\Lambda \\): \\[ \\left|\\Lambda \\cap \\mathcal{B}_d(R - L)\\right|\\left|X / \\Lambda \\right| \\leq \\left|X \\cap \\mathcal{B}_d(R)\\right| \\leq \\left|\\Lambda \\cap \\mathcal{B}_d(R + L)\\right|\\left|X / \\Lambda \\right| \\]","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/PeriodicSpherePacking.aux2_ge'","latex_link":"sec-packings-density.html#lemma:periodic-points-bounds","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/PeriodicSpherePacking.aux2_ge'","lean_decl_name":"PeriodicSpherePacking.aux2_ge'","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/thefundamentaltheor3m/Sphere-Packing-Lean/2928857eab3e21dc6c3b5bfea025528145fe2113/SpherePacking/Basic/PeriodicPacking.lean","lean4_code":"theorem PeriodicSpherePacking.aux2_ge'\n (hL : ∀ x ∈ fundamentalDomain (b.ofZLatticeBasis ℝ _), ‖x‖ ≤ L) (hd : 0 < d) :\n (↑S.lattice ∩ ball (0 : EuclideanSpace ℝ (Fin d)) R).encard\n ≥ volume (ball (0 : EuclideanSpace ℝ (Fin d)) (R - L))\n / volume (fundamentalDomain (b.ofZLatticeBasis ℝ _)) := by\n refine S.aux2_ge _ R ?_ (fundamentalDomain_measurableSet _) hL hd\n intro x\n obtain ⟨⟨v, hv⟩, hv'⟩ := exist_unique_vadd_mem_fundamentalDomain (b.ofZLatticeBasis ℝ _) x\n simp only [S.basis_Z_span] at hv hv' ⊢\n use ⟨v, hv⟩, hv'.left, ?_\n intro ⟨y, hy⟩ hy'\n have := hv'.right ⟨y, ?_⟩ hy'\n · rwa [Subtype.ext_iff] at this ⊢\n · rw [S.basis_Z_span]\n exact hy","line":674,"end_line":688,"proof_status":"formalized"}
{"id":"def:Gamma-generators","label":"Gamma-generators","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["def:level-N-princ-cong-subgp"],"nlp_text":"Define the matrices \\[ S = \\begin{pmatrix} 0 & -1 \\\\ 1 & 0 \\end{pmatrix} \\in \\Gamma _1, T = \\begin{pmatrix} 1 & 1 \\\\ 0 & 1 \\end{pmatrix} \\in \\Gamma _1, \\alpha = \\begin{pmatrix} 1 & 2 \\\\ 0 & 1 \\end{pmatrix} \\in \\Gamma _2 \\subset \\Gamma _1, \\beta = \\begin{pmatrix} 1 & 0 \\\\ 2 & 1 \\end{pmatrix} \\in \\Gamma _2 \\subset \\Gamma _1. \\]","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/ModularGroup.S","latex_link":"sect0004.html#def:Gamma-generators","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/ModularGroup.S","lean_decl_name":"ModularGroup.S","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"./SpherePacking/Basic/PeriodicPacking.lean","lean4_code":"instance (S : PeriodicSpherePacking 0) : Subsingleton S.centers := inferInstance\n","line":259,"end_line":259,"proof_status":"formalized"}
{"id":"lemma:Gamma-1-generators","label":"Gamma-1-generators","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["def:Gamma-generators"],"nlp_text":"We have \\(\\Gamma (1) = \\langle S, T, -I \\rangle \\).","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/SL2Z_generate","latex_link":"sect0004.html#lemma:Gamma-1-generators","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/SL2Z_generate","lean_decl_name":"SL2Z_generate","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"SpherePacking/ModularForms/SlashActionAuxil.lean","lean4_code":"theorem SL2Z_generate : (⊤ : Subgroup SL(2, ℤ)) = Subgroup.closure {S, T} :=\n SpecialLinearGroup.SL2Z_generators.symm\n","line":64,"end_line":65,"proof_status":"formalized"}
{"id":"lemma:psi-new","label":"psi-new","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"\\(\\psi _I(z), \\psi _S(z), \\psi _T(z)\\) can be written as \\begin{align} \\psi _I(z) & = \\frac{H_4^3 (5 H_2^2 + 5 H_2 H_4 + 2 H_4^2)}{2\\Delta }, \\label{eqn:psiI-new} \\\\ \\psi _S(z) & = -\\frac{H_2^3 (2 H_2^2 + 5 H_2 H_4 + 5 H_4^2)}{2 \\Delta }. \\label{eqn:psiS-new} \\\\ \\psi _T(z) & = \\frac{H_3^3 (5 H_2^2 - 5 H_2 H_3 + 2 H_3^2)}{2 \\Delta }\\label{eqn:psiT-new} \\end{align}","doc_url":null,"latex_link":"sec--fourier-double-zeroes.html#lemma:psi-new","lean_link":null,"lean_decl_name":"psi-new","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"E8-is-basis","label":"E8-is-basis","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["E8-Matrix"],"nlp_text":"\\(B_8\\) is a \\(\\mathbb {R}\\)-basis of \\(\\mathbb {R}^8\\).","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/span_E8Matrix_eq_top","latex_link":"sect0001.html#E8-is-basis","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/span_E8Matrix_eq_top","lean_decl_name":"span_E8Matrix_eq_top","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"SpherePacking/Basic/E8.lean","lean4_code":"lemma span_E8Matrix_eq_top (R : Type*) [Field R] [NeZero (2 : R)] :\n Submodule.span R (Set.range (E8Matrix R).row) = ⊤ := (E8Matrix_is_basis _).2\n","line":311,"end_line":312,"proof_status":"formalized"}
{"id":"lemma:Ek-Fourier","label":"Ek-Fourier","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["def:Ek"],"nlp_text":"The Eisenstein series possesses the Fourier expansion \\begin{equation} \\label{eqn:Ek-Fourier}E_k(z)=1+\\frac{2}{\\zeta (1-k)}\\sum _{n=1}^\\infty \\sigma _{k-1}(n)\\, e^{2\\pi i z}, \\end{equation}","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/E_k_q_expansion","latex_link":"sect0004.html#lemma:Ek-Fourier","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/E_k_q_expansion","lean_decl_name":"E_k_q_expansion","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"SpherePacking/ModularForms/Eisensteinqexpansions.lean","lean4_code":"lemma E_k_q_expansion (k : ℕ) (hk : 3 ≤ (k : ℤ)) (hk2 : Even k) (z : ℍ) :\n (E k hk) z = 1 +\n (1 / (riemannZeta (k))) * ((-2 * ↑π * Complex.I) ^ k / (k - 1)!) *\n ∑' n : ℕ+, σ (k - 1) n * Complex.exp (2 * ↑π * Complex.I * z * n) := by\n rw [_root_.E]\n rw [IsGLPos.smul_apply]\n have : (eisensteinSeries_MF hk standardcongruencecondition) z =\n (eisensteinSeries_SIF standardcongruencecondition k) z := rfl\n rw [this]\n have := eisensteinSeries_SIF_apply standardcongruencecondition k z\n rw [this, eisensteinSeries, standardcongruencecondition]\n simp\n simp_rw [eisSummand]\n have HE1 := EQ1 k hk hk2 z\n have HE2 := EQ2 k hk z\n have z2 : (riemannZeta (k)) ≠ 0 := by\n refine riemannZeta_ne_zero_of_one_lt_re ?_\n simp\n omega\n rw [← inv_mul_eq_iff_eq_mul₀ z2 ] at HE2\n simp [UpperHalfPlane.coe] at *\n conv =>\n enter [1,2]\n rw [← HE2]\n simp_rw [← mul_assoc]\n rw [HE1, mul_add]\n have : 2⁻¹ * (riemannZeta (k))⁻¹ * (2 * riemannZeta (k)) = 1 := by\n field_simp\n rw [this]\n ring\n","line":302,"end_line":331,"proof_status":"formalized"}
{"id":"def:disc-definition","label":"disc-definition","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["def:dedekind_eta"],"nlp_text":"The discriminant form \\(\\Delta (z)\\) is given by \\begin{equation} \\label{eqn:disc-definition} \\Delta (z) = e^{2 \\pi i z} \\prod _{n \\ge 1} (1 - e^{2 \\pi i n z})^{24}. \\end{equation}","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/Delta","latex_link":"sect0004.html#def:disc-definition","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/Delta","lean_decl_name":"Delta","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"./SpherePacking/ModularForms/Delta.lean","lean4_code":"def Delta : CuspForm (CongruenceSubgroup.Gamma 1) 12 where\n toFun := Discriminant_SIF\n slash_action_eq' := Discriminant_SIF.slash_action_eq'\n holo' := by\n rw [mdifferentiable_iff]\n simp only [SlashInvariantForm.coe_mk]\n have := eta_DifferentiableAt_UpperHalfPlane\n have he2 : DifferentiableOn ℂ (fun z => (η z) ^ 24) {z | 0 < z.im} := by\n apply DifferentiableOn.pow\n intro x hx\n apply DifferentiableAt.differentiableWithinAt\n exact this ⟨x, hx⟩\n rw [Discriminant_SIF]\n simp only [SlashInvariantForm.coe_mk]\n apply he2.congr\n intro z hz\n have := Delta_eq_eta_pow (⟨z, hz⟩ : ℍ)\n simp only [coe_mk_subtype, comp_apply] at *\n rw [ofComplex_apply_of_im_pos hz]\n exact this\n zero_at_cusps' hc := zero_at_cusps_of_zero_at_infty hc Discriminant_zeroAtImInfty\n\n","line":317,"end_line":338,"proof_status":"formalized"}
{"id":"lemma:mod-div-disc-bound","label":"mod-div-disc-bound","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["def:disc-definition"],"nlp_text":"Let \\(f(z)\\) be a holomorphic function with a Fourier expansion \\begin{equation} f(z) = \\sum _{n \\ge n_0} c_f(n) e^{\\pi i n z} \\end{equation}","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/MagicFunction.PolyFourierCoeffBound.DivDiscBoundOfPolyFourierCoeff","latex_link":"sec--fourier-double-zeroes.html#lemma:mod-div-disc-bound","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/MagicFunction.PolyFourierCoeffBound.DivDiscBoundOfPolyFourierCoeff","lean_decl_name":"MagicFunction.PolyFourierCoeffBound.DivDiscBoundOfPolyFourierCoeff","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"SpherePacking/MagicFunction/PolyFourierCoeffBound.lean","lean4_code":"theorem DivDiscBoundOfPolyFourierCoeff : norm ((f z) / (Δ z)) ≤\n (DivDiscBound c n₀) * rexp (-π * (n₀ - 2) * z.im) := calc\n _ = norm ((∑' (n : ℕ), c (n + n₀) * cexp (π * I * (n + n₀) * z)) /\n (cexp (2 * π * I * z) * ∏' (n : ℕ+),\n (1 - cexp (2 * π * I * n * z)) ^ 24)) := step_1 z c n₀ f hf\n _ = norm ((cexp (π * I * n₀ * z) * ∑' (n : ℕ), c (n + n₀) * cexp (π * I * n * z)) /\n (cexp (2 * π * I * z) * ∏' (n : ℕ+), (1 - cexp (2 * π * I * n * z)) ^ 24)) := step_2 z c n₀\n _ = norm ((cexp (π * I * n₀ * z) / cexp (2 * π * I * z)) *\n (∑' (n : ℕ), c (n + n₀) * cexp (π * I * n * z)) /\n (∏' (n : ℕ+), (1 - cexp (2 * π * I * n * z)) ^ 24)) := step_3 z c n₀\n _ = norm ((cexp (π * I * (n₀ - 2) * z)) *\n (∑' (n : ℕ), c (n + n₀) * cexp (π * I * n * z)) /\n (∏' (n : ℕ+), (1 - cexp (2 * π * I * n * z)) ^ 24)) := step_4 z c n₀\n _ = norm (cexp (π * I * (n₀ - 2) * z)) *\n norm (∑' (n : ℕ), c (n + n₀) * cexp (π * I * n * z)) /\n norm (∏' (n : ℕ+), (1 - cexp (2 * π * I * n * z)) ^ 24) := step_5 z c n₀\n _ = norm (cexp (π * I * (n₀ - 2) * z)) * norm (∑' (n : ℕ), c (n + n₀) * cexp (π * I * n * z)) /\n ∏' (n : ℕ+), norm (1 - cexp (2 * π * I * n * z)) ^ 24 := step_6 z c n₀\n _ ≤ rexp (-π * (n₀ - 2) * z.im) * norm (∑' (n : ℕ), c (n + n₀) * cexp (π * I * n * z)) /\n (∏' (n : ℕ+), norm (1 - cexp (2 * π * I * n * z)) ^ 24) := step_7 z c n₀\n _ ≤ rexp (-π * (n₀ - 2) * z.im) * (∑' (n : ℕ), norm (c (n + n₀)) * norm (cexp (π * I * n * z))) /\n (∏' (n : ℕ+), norm (1 - cexp (2 * π * I * n * z)) ^ 24) := step_8 z c n₀ hcsum\n _ ≤ rexp (-π * (n₀ - 2) * z.im) * (∑' (n : ℕ), norm (c (n + n₀)) * rexp (-π * n * z.im)) /\n (∏' (n : ℕ+), norm (1 - cexp (2 * π * I * n * z)) ^ 24) := step_9 z c n₀ hcsum\n _ ≤ rexp (-π * (n₀ - 2) * z.im) * (∑' (n : ℕ), norm (c (n + n₀)) * rexp (-π * n * z.im)) /\n (∏' (n : ℕ+), (1 - rexp (-2 * π * n * z.im)) ^ 24) := step_10 z c n₀\n _ ≤ rexp (-π * (n₀ - 2) * z.im) * (∑' (n : ℕ), norm (c (n + n₀)) * rexp (-π * n / 2)) /\n (∏' (n : ℕ+), (1 - rexp (-2 * π * n * z.im)) ^ 24) := step_11 z hz c n₀ hcsum k hpoly\n _ ≤ rexp (-π * (n₀ - 2) * z.im) * (∑' (n : ℕ), norm (c (n + n₀)) * rexp (-π * n / 2)) /\n (∏' (n : ℕ+), (1 - rexp (-π * n)) ^ 24) := step_12 z hz c n₀\n _ = (DivDiscBound c n₀) * rexp (-π * (n₀ - 2) * z.im) := step_13 z c n₀\n","line":449,"end_line":479,"proof_status":"formalized"}
{"id":"cor:disc-pos","label":"disc-pos","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["def:disc-definition"],"nlp_text":"\\(\\Delta (it) {\\gt} 0\\) for all \\(t {\\gt} 0\\).","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/Delta_imag_axis_pos","latex_link":"sect0004.html#cor:disc-pos","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/Delta_imag_axis_pos","lean_decl_name":"Delta_imag_axis_pos","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"SpherePacking/ModularForms/Delta.lean","lean4_code":"lemma Delta_imag_axis_pos : ResToImagAxis.Pos Δ := by\n rw [ResToImagAxis.Pos]\n refine And.intro Delta_imag_axis_real ?_\n intro t ht\n have hprod :\n 0 < ∏' (n : ℕ), (1 - Real.exp (-(2 * π * ((n + 1) : ℝ) * t))) ^ 24 := by\n let z : ℍ := ⟨Complex.I * t, by simp [ht]⟩\n have hz : z.im = t := by simp [UpperHalfPlane.im, z]\n simpa [hz] using tprod_pos_nat_im z\n rw [re_ResToImagAxis_Delta_eq_real_prod t ht]\n exact mul_pos (Real.exp_pos _) hprod\n","line":597,"end_line":607,"proof_status":"formalized"}
{"id":"cor:disc-nonvanishing","label":"disc-nonvanishing","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["def:disc-definition"],"nlp_text":"\\(\\Delta (z) \\neq 0\\) for all \\(z \\in \\mathfrak {H}\\).","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/Δ_ne_zero","latex_link":"sect0004.html#cor:disc-nonvanishing","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/Δ_ne_zero","lean_decl_name":"Δ_ne_zero","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"SpherePacking/ModularForms/Delta.lean","lean4_code":"lemma Δ_ne_zero (z : UpperHalfPlane) : Δ z ≠ 0 := by\n rw [Delta_eq_eta_pow]\n simpa using eta_nonzero_on_UpperHalfPlane z\n","line":45,"end_line":47,"proof_status":"formalized"}
{"id":"lemma:log-der-inf","label":"log-der-inf","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["lemma:der-q-series"],"nlp_text":"Let \\(F\\) be a quasimodular form where the vanishing order of \\(F\\) at infinity is \\(n_0 {\\gt} 0\\), i.e. \\(F(z) = \\sum _{n \\geq n_0} a_n q^{n}\\) with \\(a_{n_0} \\ne 0\\). Then \\begin{equation} \\lim _{t \\to \\infty } \\frac{F'(it)}{F(it)} = n_0. \\end{equation}","doc_url":null,"latex_link":"sec--g.html#lemma:log-der-inf","lean_link":null,"lean_decl_name":"log-der-inf","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"lem:bound-I2-I4-I6","label":"bound-I2-I4-I6","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"There exist \\(C_1, C_2 {\\gt} 0\\) such that for all \\(r \\geq 0\\), \\begin{equation} |I_2(r)|, |I_4(r)| \\leq C_1 e^{-\\pi r} \\end{equation}","doc_url":null,"latex_link":"sec--fourier-double-zeroes.html#lem:bound-I2-I4-I6","lean_link":null,"lean_decl_name":"bound-I2-I4-I6","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"SpherePacking.scale","label":"SpherePacking.scale","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Given a sphere packing \\(\\mathcal{P}(X)\\) with separation radius \\(r\\), we defined the scaled packing with respect to a real number \\(c {\\gt} 0\\) to be the packing \\(\\mathcal{P}(cX)\\), where \\(cX = \\left\\{ cx \\in V \\; \\middle | \\; x \\in X \\right\\} \\) has separation radius \\(cr\\).","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/SpherePacking.scale","latex_link":"sec-sphere-packings.html#SpherePacking.scale","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/SpherePacking.scale","lean_decl_name":"SpherePacking.scale","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"SpherePacking/Basic/SpherePacking.lean","lean4_code":"def SpherePacking.scale (S : SpherePacking d) {c : ℝ} (hc : 0 < c) : SpherePacking d where\n centers := c • S.centers\n separation := c * S.separation\n separation_pos := mul_pos hc S.separation_pos\n centers_dist := fun ⟨x, hx⟩ ⟨y, hy⟩ _ ↦ by\n change c * S.separation ≤ ‖x - y‖\n obtain ⟨x', ⟨hx', rfl⟩⟩ := Set.mem_smul_set.mp hx\n obtain ⟨y', ⟨hy', rfl⟩⟩ := Set.mem_smul_set.mp hy\n rw [← smul_sub, norm_smul, norm_eq_abs, abs_eq_self.mpr hc.le]\n rw [ne_eq, Subtype.mk.injEq] at *\n have : x' ≠ y' := by rintro rfl; tauto\n have : (⟨x', hx'⟩ : S.centers) ≠ ⟨y', hy'⟩ := by simp [this]\n have := S.centers_dist this\n exact (mul_le_mul_iff_right₀ hc).mpr this\n","line":140,"end_line":153,"proof_status":"formalized"}
{"id":"def:dual-lattice","label":"dual-lattice","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"The dual lattice of a lattice \\(\\Lambda \\) is the set \\[ \\Lambda ^* := \\left\\{ v \\in \\mathbb {R}^d \\; \\middle | \\; \\forall l \\in \\Lambda , \\left\\langle v,l \\right\\rangle \\in \\mathbb {Z} \\right\\} \\]","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/LinearMap.BilinForm.dualSubmodule","latex_link":"sec-sphere-packings.html#def:dual-lattice","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/LinearMap.BilinForm.dualSubmodule","lean_decl_name":"LinearMap.BilinForm.dualSubmodule","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"./SpherePacking/CohnElkies/Prereqs.lean","lean4_code":"-- def bilinFormOfRealInner.dualSubmodule : AddSubgroup (EuclideanSpace ℝ (Fin d)) where\n-- carrier := { x | ∀ l : Λ, ∃ n : ℤ, ⟪x, l⟫_[ℝ] = ↑n }\n-- zero_mem' := by\n-- simp only [Subtype.forall, Set.mem_setOf_eq, inner_zero_left]\n-- intro a _\n-- use 0\n-- rw [Int.cast_zero]\n-- add_mem' := by\n-- intros x y hx hy l\n-- obtain ⟨n, hn⟩ := hx l\n-- obtain ⟨m, hm⟩ := hy l\n-- use n + m\n-- simp only [inner_add_left, hn, hm, Int.cast_add]\n-- neg_mem' := by\n-- intros x hx l\n-- obtain ⟨n, hn⟩ := hx l\n-- use -n\n-- simp only [inner_neg_left, hn, Int.cast_neg]\n\n-- end Dual_Lattice\n\nsection Euclidean_Space\n\n","line":39,"end_line":61,"proof_status":"formalized"}
{"id":"lem:bound-I1-I3-I5","label":"bound-I1-I3-I5","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"There exists \\(C {\\gt} 0\\) such that for all \\(r \\geq 0\\), \\begin{equation} |I_1(r)|, |I_3(r)|, |I_5(r)| \\leq C \\int _1^{\\infty } e^{-2\\pi s} \\, e^{-\\pi r / s} \\, \\mathrm{d}s. \\end{equation}","doc_url":null,"latex_link":"sec--fourier-double-zeroes.html#lem:bound-I1-I3-I5","lean_link":null,"lean_decl_name":"bound-I1-I3-I5","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"def:Schwartz-Space","label":"Schwartz-Space","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"A \\(C^\\infty \\) function \\(f:\\mathbb {R}^d\\to \\mathbb {C}\\) is called a Schwartz function if it decays to zero as \\(\\| x\\| \\to \\infty \\) faster then any inverse power of \\(\\| x\\| \\), and the same holds for all partial derivatives of \\(f\\), ie, if for all \\(k, n \\in \\mathbb {N}\\), there exists a constant \\(C \\in \\mathbb {R}\\) such that for all \\(x \\in \\mathbb {R}^d\\), \\(\\left\\lVert x \\right\\rVert ^k \\cdot \\left\\lVert f^{(n)}(x) \\right\\rVert \\leq C\\), where \\(f^{(n)}\\) denotes the \\(n\\)-th derivative of \\(f\\) considered along with the appropriate operator norm. The set of all Schwartz functions from \\(\\mathbb {R}^d\\) to \\(\\mathbb {C}\\) is called the Schwartz space. It is an \\(\\mathbb {R}\\)-vector space.","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/SchwartzMap","latex_link":"sect0002.html#def:Schwartz-Space","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/SchwartzMap","lean_decl_name":"SchwartzMap","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"./SpherePacking/MagicFunction/a/Schwartz.lean","lean4_code":"noncomputable section SchwartzMap\n\nnamespace MagicFunction.a.SchwartzIntegrals\n\n","line":116,"end_line":119,"proof_status":"formalized"}
{"id":"lemma:der-q-series","label":"der-q-series","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["def:derivative"],"nlp_text":"We have an equality of operators \\(D = q \\frac{\\mathrm{d}}{\\mathrm{d}q}\\). In particular, the \\(q\\)-series of the derivative of a quasimodular form \\(F(z) = \\sum _{n \\ge n_0} a_n q^n\\) is \\(F'(z) = \\sum _{n \\ge n_0} n a_n q^n\\).","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/D_qexp_tsum_pnat","latex_link":"sect0004.html#lemma:der-q-series","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/D_qexp_tsum_pnat","lean_decl_name":"D_qexp_tsum_pnat","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"SpherePacking/ModularForms/Derivative.lean","lean4_code":"/--\nSimplified version of `D_qexp_tsum` for ℕ+-indexed series (starting from n=1).\nThis is the form most commonly used for Eisenstein series q-expansions.\n\n**Thin layer implementation:** Extends `a : ℕ+ → ℂ` to `ℕ → ℂ` with `a' 0 = 0`,\nuses `tsum_pNat` and `nat_pos_tsum2` to convert between sums,\nthen applies `D_qexp_tsum`.\n-/\ntheorem D_qexp_tsum_pnat (a : ℕ+ → ℂ) (z : ℍ)\n (hsum : Summable (fun n : ℕ+ => a n * cexp (2 * π * I * n * z)))\n (hsum_deriv : ∀ K : Set ℂ, K ⊆ {w : ℂ | 0 < w.im} → IsCompact K →\n ∃ u : ℕ+ → ℝ, Summable u ∧ ∀ n (k : K), ‖a n * (2 * π * I * n) *\n cexp (2 * π * I * n * k.1)‖ ≤ u n) :\n D (fun w => ∑' n : ℕ+, a n * cexp (2 * π * I * n * w)) z =\n ∑' n : ℕ+, (n : ℂ) * a n * cexp (2 * π * I * n * z) := by\n -- Extend a to ℕ with a' 0 = 0\n let a' : ℕ → ℂ := fun n => if h : 0 < n then a ⟨n, h⟩ else 0\n have ha' : ∀ n : ℕ+, a' n = a n := fun n => dif_pos n.pos\n -- Derivative bounds: extend u using nat_pos_tsum2\n have hsum_deriv' : ∀ K : Set ℂ, K ⊆ {w : ℂ | 0 < w.im} → IsCompact K →\n ∃ u : ℕ → ℝ, Summable u ∧ ∀ n (k : K), ‖a' n * (2 * π * I * n) *\n cexp (2 * π * I * n * k.1)‖ ≤ u n := fun K hK hKc => by\n obtain ⟨u, hu_sum, hu_bound⟩ := hsum_deriv K hK hKc\n let u' : ℕ → ℝ := fun n => if h : 0 < n then u ⟨n, h⟩ else 0\n have hu' : ∀ n : ℕ+, u' n = u n := fun n => dif_pos n.pos\n refine ⟨u', (nat_pos_tsum2 u' (by simp [u'])).mp (hu_sum.congr fun n => by rw [hu']),\n fun n k => ?_⟩\n by_cases hn : 0 < n\n · simp only [a', u', dif_pos hn]; exact hu_bound _ k\n · simp only [Nat.not_lt, Nat.le_zero] at hn; simp [a', u', hn]\n -- Apply D_qexp_tsum and convert sums via tsum_pNat\n have hD := D_qexp_tsum a' z ((nat_pos_tsum2 _ (by simp [a'])).mp\n (hsum.congr fun n => by rw [ha'])) hsum_deriv'\n calc D (fun w => ∑' n : ℕ+, a n * cexp (2 * π * I * n * w)) z\n = D (fun w : ℍ => ∑' n : ℕ, a' n * cexp (2 * π * I * n * (w : ℂ))) z := by\n congr 1; ext w; rw [← tsum_pNat _ (by simp [a'])]; exact tsum_congr fun n => by rw [ha']\n _ = ∑' n : ℕ, (n : ℂ) * a' n * cexp (2 * π * I * n * (z : ℂ)) := hD\n _ = ∑' n : ℕ+, (n : ℂ) * a n * cexp (2 * π * I * n * z) := by\n rw [← tsum_pNat _ (by simp [a'])]; exact tsum_congr fun n => by rw [ha']\n","line":278,"end_line":316,"proof_status":"formalized"}
{"id":"lemma:slash-operator-chain-rule","label":"slash-operator-chain-rule","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["lemma:automorphy-factor-chain-rule"],"nlp_text":"The chain rule implies \\[ F|_k\\gamma _1\\gamma _2=(F|_k\\gamma _1)|_k\\gamma _2. \\]","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/SlashAction.slash_mul","latex_link":"sect0004.html#lemma:slash-operator-chain-rule","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/SlashAction.slash_mul","lean_decl_name":"SlashAction.slash_mul","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/NumberTheory/ModularForms/SlashActions.lean","lean4_code":"private theorem slash_mul (k : ℤ) (A B : GL (Fin 2) ℝ) (f : ℍ → ℂ) :\n f ∣[k] (A * B) = (f ∣[k] A) ∣[k] B := by\n ext1 τ\n calc σ (A * B) (f ((A * B) • τ)) * ((A * B).det) ^ (k - 1) * denom (A * B) τ ^ (-k)\n _ = σ B (σ A (f (A • B • τ))) * (A.det ^ (k - 1) * B.det ^ (k - 1)) *\n (((σ B) (denom A ↑(B • τ) ^ (-k))) * denom B τ ^ (-k)) := by\n rw [σ_mul_comm, σ_mul, denom_cocycle_σ, mul_zpow, mul_smul, map_mul, Units.val_mul,\n ofReal_mul, mul_zpow, map_zpow₀]\n _ = σ B (σ A (f (A • B • τ)) * A.det ^ (k - 1) * (denom A ↑(B • τ) ^ (-k)))\n * B.det ^ (k - 1) * denom B τ ^ (-k) := by\n rw [map_mul, map_zpow₀, map_mul, map_zpow₀, σ_ofReal]\n ring\n _ = ((f ∣[k] A) ∣[k] B) τ := rfl\n\nprivate theorem add_slash (k : ℤ) (A : GL (Fin 2) ℝ) (f g : ℍ → ℂ) :\n (f + g) ∣[k] A = f ∣[k] A + g ∣[k] A := by\n ext1 τ\n simp [slash, add_mul]\n\nprivate theorem slash_one (k : ℤ) (f : ℍ → ℂ) : f ∣[k] 1 = f :=\n funext <| by simp [slash, σ, denom]\n\nprivate theorem zero_slash (k : ℤ) (A : GL (Fin 2) ℝ) : (0 : ℍ → ℂ) ∣[k] A = 0 :=\n funext fun _ => by simp [slash]\n\n","line":77,"end_line":101,"proof_status":"formalized"}
{"id":"E8-Matrix","label":"E8-Matrix","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"(\\(E_8\\)-lattice, Definition 2)We define the \\(E_8\\) basis vectors to be the set of vectors \\[ \\mathcal{B}_8 = \\left\\{ \\begin{bmatrix} 1 \\\\ -1 \\\\ 0 \\\\ 0 \\\\ 0 \\\\ 0 \\\\ 0 \\\\ 0 \\end{bmatrix}, \\begin{bmatrix} 0 \\\\ 1 \\\\ -1 \\\\ 0 \\\\ 0 \\\\ 0 \\\\ 0 \\\\ 0 \\end{bmatrix}, \\begin{bmatrix} 0 \\\\ 0 \\\\ 1 \\\\ -1 \\\\ 0 \\\\ 0 \\\\ 0 \\\\ 0 \\end{bmatrix}, \\begin{bmatrix} 0 \\\\ 0 \\\\ 0 \\\\ 1 \\\\ -1 \\\\ 0 \\\\ 0 \\\\ 0 \\end{bmatrix}, \\begin{bmatrix} 0 \\\\ 0 \\\\ 0 \\\\ 0 \\\\ 1 \\\\ -1 \\\\ 0 \\\\ 0 \\end{bmatrix}, \\begin{bmatrix} 0 \\\\ 0 \\\\ 0 \\\\ 0 \\\\ 0 \\\\ 1 \\\\ 1 \\\\ 0 \\end{bmatrix}, \\begin{bmatrix} -1/2 \\\\ -1/2 \\\\ -1/2 \\\\ -1/2 \\\\ -1/2 \\\\ -1/2 \\\\ -1/2 \\\\ -1/2 \\end{bmatrix}, \\begin{bmatrix} 0 \\\\ 0 \\\\ 0 \\\\ 0 \\\\ 0 \\\\ 1 \\\\ -1 \\\\ 0 \\end{bmatrix} \\right\\} \\]","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/E8Matrix","latex_link":"sect0001.html#E8-Matrix","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/E8Matrix","lean_decl_name":"E8Matrix","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"SpherePacking/Basic/E8.lean","lean4_code":"def E8Matrix (R : Type*) [Field R] : Matrix (Fin 8) (Fin 8) R := !![\n 2, 0, 0, 0, 0, 0, 0, 0;\n -1, 1, 0, 0, 0, 0, 0, 0;\n 0, -1, 1, 0, 0, 0, 0, 0;\n 0, 0, -1, 1, 0, 0, 0, 0;\n 0, 0, 0, -1, 1, 0, 0, 0;\n 0, 0, 0, 0, -1, 1, 0, 0;\n 0, 0, 0, 0, 0, -1, 1, 0;\n 2⁻¹, 2⁻¹, 2⁻¹, 2⁻¹, 2⁻¹, 2⁻¹, 2⁻¹, 2⁻¹]\n","line":261,"end_line":269,"proof_status":"formalized"}
{"id":"def:Ek","label":"Ek","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"For an even integer \\(k\\geq 4\\) we define the weight \\(k\\) Eisenstein series as \\begin{equation} \\label{eqn:Ek-definition} E_k(z):=\\frac{1}{2}\\sum _{(c,d)\\in \\mathbb {Z}^2, (c,d)=1}(cz+d)^{-k}.\\end{equation}","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/ModularForm.eisensteinSeries_MF","latex_link":"sect0004.html#def:Ek","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/ModularForm.eisensteinSeries_MF","lean_decl_name":"ModularForm.eisensteinSeries_MF","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/NumberTheory/ModularForms/EisensteinSeries/Basic.lean","lean4_code":"def eisensteinSeries_MF {k : ℤ} {N : ℕ+} (hk : 3 ≤ k) (a : Fin 2 → ZMod N) :\n ModularForm (Gamma N) k where\n toFun := eisensteinSeries_SIF a k\n slash_action_eq' := (eisensteinSeries_SIF a k).slash_action_eq'\n holo' := eisensteinSeries_SIF_MDifferentiable hk a\n bdd_at_infty' := isBoundedAtImInfty_eisensteinSeries_SIF a hk\n\n","line":30,"end_line":36,"proof_status":"statement_formalized"}
{"id":"lemma:volume-ball-ratio-limit","label":"volume-ball-ratio-limit","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"For any constant \\(C {\\gt} 0\\), we have \\[ \\lim _{R \\to \\infty } \\frac{\\mathrm{Vol}(\\mathcal{B}_d(R))}{\\mathrm{Vol}(\\mathcal{B}_d(R + C))} = 1 \\]","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/volume_ball_ratio_tendsto_nhds_one''","latex_link":"sec-packings-density.html#lemma:volume-ball-ratio-limit","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/volume_ball_ratio_tendsto_nhds_one''","lean_decl_name":"volume_ball_ratio_tendsto_nhds_one''","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/thefundamentaltheor3m/Sphere-Packing-Lean/2928857eab3e21dc6c3b5bfea025528145fe2113/SpherePacking/Basic/PeriodicPacking.lean","lean4_code":"theorem volume_ball_ratio_tendsto_nhds_one'' {d : ℕ} {C C' : ℝ} (hd : 0 < d) :\n Tendsto (fun R ↦ volume (ball (0 : EuclideanSpace ℝ (Fin d)) (R + C))\n / volume (ball (0 : EuclideanSpace ℝ (Fin d)) (R + C'))) atTop (𝓝 1) := by\n apply (Filter.map_add_atTop_eq' (max (-C) (-C')) _).mpr\n simp_rw [add_assoc]\n convert volume_ball_ratio_tendsto_nhds_one' hd ?_ ?_\n · trans (-C) + C\n · linarith\n · gcongr; simp\n · trans (-C') + C'\n · linarith\n · gcongr; simp","line":889,"end_line":900,"proof_status":"formalized"}
{"id":"lemma:mod_form_poly_growth","label":"mod_form_poly_growth","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":": Let \\(\\Gamma \\) be a finite index subgroup of \\(\\mathrm{SL}_2(\\mathbb {Z})\\) and \\(f \\in \\mathcal{M}_k(\\Gamma )\\) be a modular form of weight \\(k\\). Then the Fourier coefficients \\(a_n(f)\\) has a polynomial growth, i.e. \\(|a_n(f)| = O(n^k)\\).","doc_url":null,"latex_link":"sect0004.html#lemma:mod_form_poly_growth","lean_link":null,"lean_decl_name":"mod_form_poly_growth","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"def:automorphy-factor","label":"automorphy-factor","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"The automorphy factor of weight \\(k\\) is defined as \\[ j_k(z,\\left(\\begin{smallmatrix} a & b \\\\ c & d \\end{smallmatrix}\\right)):=(cz+d)^{-k}. \\]","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/UpperHalfPlane.denom","latex_link":"sect0004.html#def:automorphy-factor","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/UpperHalfPlane.denom","lean_decl_name":"UpperHalfPlane.denom","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Analysis/Complex/UpperHalfPlane/MoebiusAction.lean","lean4_code":"def denom (g : GL (Fin 2) ℝ) (z : ℂ) : ℂ := g 1 0 * z + g 1 1\n\n","line":46,"end_line":47,"proof_status":"formalized"}
{"id":"lemma:automorphy-factor-chain-rule","label":"automorphy-factor-chain-rule","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":["def:automorphy-factor"],"nlp_text":"The automorphy factor satisfies the chain rule \\[ j_k(z,\\gamma _1\\gamma _2)=j_k(z,\\gamma _1)\\, j_k(\\gamma _2z,\\gamma _1). \\]","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/UpperHalfPlane.denom_cocycle","latex_link":"sect0004.html#lemma:automorphy-factor-chain-rule","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/UpperHalfPlane.denom_cocycle","lean_decl_name":"UpperHalfPlane.denom_cocycle","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Analysis/Complex/UpperHalfPlane/MoebiusAction.lean","lean4_code":"lemma denom_cocycle (g h : GL (Fin 2) ℝ) {z : ℂ} (hz : z.im ≠ 0) :\n denom (g * h) z = denom g (num h z / denom h z) * denom h z := by\n change _ = (_ * (_ / _) + _) * _\n field_simp [denom_ne_zero_of_im h hz]\n simp only [denom, Units.val_mul, mul_apply, Fin.sum_univ_succ, Finset.univ_unique,\n Fin.default_eq_zero, Finset.sum_singleton, Fin.succ_zero_eq_one, Complex.ofReal_add,\n Complex.ofReal_mul, num]\n ring\n\n","line":87,"end_line":95,"proof_status":"formalized"}
{"id":"def:level-N-princ-cong-subgp","label":"level-N-princ-cong-subgp","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"The level \\(N\\) principal congruence subgroup of \\(\\Gamma _1\\) is \\[ \\Gamma (N):=\\left\\{ \\left.\\left(\\begin{smallmatrix} a & b \\\\ c & d \\end{smallmatrix}\\right)\\in \\Gamma _1\\right|\\left(\\begin{smallmatrix} a & b \\\\ c & d \\end{smallmatrix}\\right)\\equiv \\left(\\begin{smallmatrix} 1 & 0 \\\\ 0 & 1 \\end{smallmatrix}\\right)\\; \\mathrm{mod}\\; N\\right\\} . \\]","doc_url":null,"latex_link":"sect0004.html#def:level-N-princ-cong-subgp","lean_link":null,"lean_decl_name":"level-N-princ-cong-subgp","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"lem:integral-bound","label":"integral-bound","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"For all \\(n \\in \\mathbb {N}\\), there exists a constant \\(C'\\) such that for all \\(r \\geq 0\\), \\begin{align} r^n \\cdot \\int _{1}^{\\infty } e^{-2\\pi s} \\, e^{-\\pi r /s} \\, \\mathrm{d}s \\leq C’ \\end{align}","doc_url":null,"latex_link":"sec--fourier-double-zeroes.html#lem:integral-bound","lean_link":null,"lean_decl_name":"integral-bound","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"E8-Set","label":"E8-Set","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"(\\(E_8\\)-lattice, Definition 1)We define the \\(E_8\\)-lattice (as a subset of \\(\\mathbb {R}^8\\)) to be \\[ \\Lambda _8=\\{ (x_i)\\in \\mathbb {Z}^8\\cup (\\mathbb {Z}+\\textstyle \\frac12\\displaystyle )^8|\\; \\sum _{i=1}^8x_i\\equiv 0\\; (\\mathrm{mod\\; 2})\\} . \\]","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/Submodule.E8","latex_link":"sect0001.html#E8-Set","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/Submodule.E8","lean_decl_name":"Submodule.E8","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"./SpherePacking/Basic/E8.lean","lean4_code":"noncomputable def Submodule.E8 (R : Type*) [Field R] [NeZero (2 : R)] :\n Submodule ℤ (Fin 8 → R) where\n carrier :=\n {v | ((∀ i, ∃ n : ℤ, n = v i) ∨ (∀ i, ∃ n : ℤ, Odd n ∧ n = 2 • v i)) ∧ ∑ i, v i ≡ 0 [PMOD 2]}\n add_mem' := by\n simp only [Set.mem_setOf_eq, and_imp, nsmul_eq_mul, Nat.cast_ofNat, Pi.add_apply]\n rintro a b ha has hb hbs\n constructor\n · obtain ha | ha := ha\n · refine hb.imp ?_ ?_\n · intro hb i\n obtain ⟨a', ha⟩ := ha i\n obtain ⟨b', hb⟩ := hb i\n use a' + b'\n simp [ha, hb]\n · intro hb i\n obtain ⟨a', ha⟩ := ha i\n obtain ⟨b', hb', hb⟩ := hb i\n exact ⟨2 * a' + b', Even.add_odd (by simp) hb', by simp [← ha, ← hb, mul_add]⟩\n · refine hb.symm.imp ?_ ?_\n · intro hb i\n obtain ⟨a', ha', ha⟩ := ha i\n obtain ⟨b', hb', hb⟩ := hb i\n use (a' + b') / 2\n rw [Int.cast_div _ (by simpa using NeZero.ne 2), Int.cast_add, add_div (K := R), ha, hb,\n Int.cast_ofNat,\n mul_div_cancel_left₀ _ (NeZero.ne 2), mul_div_cancel_left₀ _ (NeZero.ne _)]\n rw [← even_iff_two_dvd]\n apply ha'.add_odd hb'\n · intro hb i\n obtain ⟨a', ha', ha⟩ := ha i\n obtain ⟨b', hb⟩ := hb i\n exact ⟨a' + 2 * b', ha'.add_even (by simp), by simp [ha, hb, mul_add]⟩\n · rw [Finset.sum_add_distrib]\n exact ((has.add_right _).trans (hbs.add_left _)).trans (by simp)\n zero_mem' := by\n simp only [nsmul_eq_mul, Nat.cast_ofNat, Set.mem_setOf_eq, Pi.zero_apply, forall_const,\n mul_zero, Finset.sum_const_zero, AddCommGroup.modEq_refl, and_true]\n refine Or.inl ⟨0, by simp⟩\n smul_mem' := by\n simp only [nsmul_eq_mul, Nat.cast_ofNat, Set.mem_setOf_eq, zsmul_eq_mul, Pi.mul_apply,\n Pi.intCast_apply, and_imp]\n intro c a ha has\n constructor\n · obtain ha | ha := ha\n · left\n intro i\n obtain ⟨a, ha⟩ := ha i\n simp only [← ha, ← Int.cast_mul]\n exact ⟨_, rfl⟩\n","line":112,"end_line":161,"proof_status":"formalized"}
{"id":"def:dedekind_eta","label":"dedekind_eta","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"The Dedekind eta function is defined as \\[ \\eta (z) = q^{1/24} \\prod _{n \\ge 1} (1 - q^n) \\]","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/η","latex_link":"sect0004.html#def:dedekind_eta","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/η","lean_decl_name":"η","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"SpherePacking/ModularForms/eta.lean","lean4_code":"noncomputable def η (z : ℂ) := cexp (2 * π * Complex.I * z / 24) * ∏' (n : ℕ),\n (1 - cexp (2 * π * Complex.I * (n + 1) * z))\n","line":16,"end_line":17,"proof_status":"formalized"}
{"id":"lemma:psi-bound","label":"psi-bound","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"There exist constants \\(C_I, C_S, C_T {\\gt} 0\\) such that \\begin{align} |\\psi _I(z)| & \\le C_I e^{2\\pi \\Im z}, \\label{eqn:psiI-bound} \\\\ |\\psi _T(z)| & \\le C_T e^{2\\pi \\Im z}, \\label{eqn:psiT-bound} \\\\ |\\psi _S(z)| & \\le C_S e^{- \\pi \\Im z} \\label{eqn:psiS-bound} \\end{align}","doc_url":null,"latex_link":"sec--fourier-double-zeroes.html#lemma:psi-bound","lean_link":null,"lean_decl_name":"psi-bound","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"def:derivative","label":"derivative","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(F\\) be a quasimodular form. We define the (normalized) derivative of \\(F\\) as \\begin{equation} \\label{eqn:derivative} F' = DF := \\frac{1}{2\\pi i} \\frac{\\mathrm{d}}{\\mathrm{d}z} F. \\end{equation}","doc_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/D","latex_link":"sect0004.html#def:derivative","lean_link":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/docs/find/#doc/D","lean_decl_name":"D","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Sphere-Packing-Lean","source_path":"SpherePacking/ModularForms/Derivative.lean","lean4_code":"noncomputable def D (F : ℍ → ℂ) : ℍ → ℂ :=\n fun (z : ℍ) => (2 * π * I)⁻¹ * ((deriv (F ∘ ofComplex)) z)\n","line":15,"end_line":16,"proof_status":"formalized"}
{"id":"lemma:bound-J1-J3-J5","label":"bound-J1-J3-J5","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"There exist a constant \\(C {\\gt} 0\\) such that \\begin{align} |J_1(r)|, |J_3(r)|, |J_5(r)| & \\le C \\int _1^{\\infty } e^{-\\pi s} e^{\\pi r / s}\\, \\mathrm{d}s. \\end{align}","doc_url":null,"latex_link":"sec--fourier-double-zeroes.html#lemma:bound-J1-J3-J5","lean_link":null,"lean_decl_name":"bound-J1-J3-J5","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
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