| {"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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