module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.NumberTheory.Modular | {
"line": 318,
"column": 48
} | {
"line": 318,
"column": 59
} | {
"line": 318,
"column": 60
} | [
{
"pp": "z : ℍ\n⊢ (T • z).re = z.re + 1",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Real",
"instHSMul",
"Matrix.SpecialLinearGroup",
"UpperHalfPlane.SLAction",
"instDecidableEqFin",
"ModularGroup.T",
"id",
"Real.instRing",
"instOfN... | [
"z : ℍ\n⊢ (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) T) • z).re = z.re + 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Modular | {
"line": 320,
"column": 44
} | {
"line": 320,
"column": 55
} | {
"line": 320,
"column": 56
} | [
{
"pp": "z : ℍ\n⊢ (T • z).im = z.im",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Real",
"instHSMul",
"Matrix.SpecialLinearGroup",
"UpperHalfPlane.SLAction",
"instDecidableEqFin",
"ModularGroup.T",
"id",
"Real.instRing",
"instOfNatNa... | [
"z : ℍ\n⊢ (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) T) • z).im = z.im"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Modular | {
"line": 324,
"column": 50
} | {
"line": 324,
"column": 61
} | {
"line": 324,
"column": 62
} | [
{
"pp": "z : ℍ\n⊢ (T⁻¹ • z).im = z.im",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"UpperHalfPlane.glAction",
"Eq.mpr",
"MonoidHom.instMonoidHomClass",
"Real",
"DivInvMonoid.toInv",
"instHSMul",
"Matrix.SpecialLinearGroup",
"MonoidHom.inst... | [
"z : ℍ\n⊢ ((toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) T))⁻¹ • z).im = z.im"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.Defs | {
"line": 156,
"column": 2
} | {
"line": 156,
"column": 13
} | {
"line": 156,
"column": 14
} | [
{
"pp": "r : ℕ\ninst✝ : NeZero r\nv : Fin 2 → ℤ\nhab : finGcdMap v = r\nA : SL(2, ℤ)\nhvr : v = r • (v / ↑r)\n⊢ finGcdMap (r • (v / ↑r) ᵥ* ↑A) = r",
"ppTerm": "?m.307",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Int.instAddCommMonoid",
"NonAssocSemiring.toAddCommMonoidWithOne"... | [
"r : ℕ\ninst✝ : NeZero r\nv : Fin 2 → ℤ\nhab : finGcdMap v = r\nA : SL(2, ℤ)\nhvr : v = r • (v / ↑r)\n⊢ finGcdMap (↑r * (v / ↑r) ᵥ* ↑A) = r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.Bounds | {
"line": 81,
"column": 6
} | {
"line": 81,
"column": 16
} | {
"line": 81,
"column": 16
} | [
{
"pp": "E : Type u_1\ninst✝ : SeminormedAddCommGroup E\nf : ℍ → E\nhf_cont : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f\nt : ℝ\nht : 0 ≤ t\nhf_infinity : f =O[atImInfty] fun z ↦ z.im ^ t\nhf_inv : ∀ (g : SL(2, ℤ)) (τ : ℍ), f (g • τ) = f τ\nF : ℝ\nτ : ℍ\ng : SL(2, ℤ)\nhF𝒟 : ‖f (g • τ)... | [
"E : Type u_1\ninst✝ : SeminormedAddCommGroup E\nf : ℍ → E\nhf_cont : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f\nt : ℝ\nht : 0 ≤ t\nhf_infinity : f =O[atImInfty] fun z ↦ z.im ^ t\nhf_inv : ∀ (g : SL(2, ℤ)) (τ : ℍ), f (g • τ) = f τ\nF : ℝ\nτ : ℍ\ng : SL(2, ℤ)\nhF𝒟 : ‖f τ‖ ≤ F * (g • τ).im... | hf_inv g τ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.ModularForms.QExpansion | {
"line": 316,
"column": 70
} | {
"line": 316,
"column": 81
} | {
"line": 316,
"column": 82
} | [
{
"pp": "k : ℤ\nF : Type u_1\ninst✝ : FunLike F ℍ ℂ\nΓ : Subgroup (GL (Fin 2) ℝ)\nh : ℝ\nf✝ : F\nf : ℍ → ℂ\nhh : 0 < h\nhfper : Periodic (f ∘ ↑ofComplex) ↑h\nhfhol : MDiff f\nhfbdd : IsBoundedAtImInfty f\nn : ℕ\nt : ℝ\nht : 0 < t\nu : ℝ\n⊢ 0 < (↑u + ↑t * ↑I).im",
"ppTerm": "?m.83",
"assigned": true,
... | [
"k : ℤ\nF : Type u_1\ninst✝ : FunLike F ℍ ℂ\nΓ : Subgroup (GL (Fin 2) ℝ)\nh : ℝ\nf✝ : F\nf : ℍ → ℂ\nhh : 0 < h\nhfper : Periodic (f ∘ ↑ofComplex) ↑h\nhfhol : MDiff f\nhfbdd : IsBoundedAtImInfty f\nn : ℕ\nt : ℝ\nht : 0 < t\nu : ℝ\n⊢ 0 < t"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Modular | {
"line": 339,
"column": 45
} | {
"line": 339,
"column": 79
} | {
"line": 340,
"column": 4
} | [
{
"pp": "g : SL(2, ℤ)\nhc : ↑g 1 0 = 0\nhad : ↑g 0 0 * ↑g 1 1 = 1\nha : ↑g 0 0 = -1\nhd : ↑g 1 1 = -1\nthis : g = -T ^ (-↑g 0 1)\nz : ℍ\n⊢ g • z = T ^ (-↑g 0 1) • z",
"ppTerm": "?m.915",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Fintype.card_fin_two",
"instHSMul... | [] | conv_lhs => rw [this, SL_neg_smul] | Mathlib.Tactic.Conv._aux_Mathlib_Tactic_Conv___macroRules_Mathlib_Tactic_Conv_convLHS_1 | Mathlib.Tactic.Conv.convLHS |
Mathlib.NumberTheory.ModularForms.QExpansion | {
"line": 320,
"column": 50
} | {
"line": 320,
"column": 71
} | {
"line": 320,
"column": 72
} | [
{
"pp": "h : ℝ\nf : ℍ → ℂ\nhh : 0 < h\nhfper : Periodic (f ∘ ↑ofComplex) ↑h\nhfhol : MDiff f\nhfbdd : IsBoundedAtImInfty f\nn : ℕ\nt : ℝ\nht : 0 < t\nR : ℝ := rexp (-2 * π * t / h)\nhR0 : 0 < R\n⊢ -2 * π * t / h < 0",
"ppTerm": "?m.125",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtrac... | [
"h : ℝ\nf : ℍ → ℂ\nhh : 0 < h\nhfper : Periodic (f ∘ ↑ofComplex) ↑h\nhfhol : MDiff f\nhfbdd : IsBoundedAtImInfty f\nn : ℕ\nt : ℝ\nht : 0 < t\nR : ℝ := rexp (-2 * π * t / h)\nhR0 : 0 < R\n⊢ 0 < 2 * π * t / h"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.Defs | {
"line": 198,
"column": 2
} | {
"line": 198,
"column": 13
} | {
"line": 198,
"column": 14
} | [
{
"pp": "k : ℤ\ni : Fin 2 → ℤ\nA : SL(2, ℤ)\nz : ℍ\nh :\n ∀ (a b c d u v : ℂ),\n c * ↑z + d ≠ 0 →\n (u * ((a * ↑z + b) / (c * ↑z + d)) + v) ^ (-k) =\n (c * ↑z + d) ^ k * ((u * a + v * c) * ↑z + (u * b + v * d)) ^ (-k)\n⊢ (↑(i 0) *\n ((↑((algebraMap ℤ ℝ) (↑A 0 0)) * ↑z + ↑((algebraMap ℤ ... | [
"k : ℤ\ni : Fin 2 → ℤ\nA : SL(2, ℤ)\nz : ℍ\nh :\n ∀ (a b c d u v : ℂ),\n c * ↑z + d ≠ 0 →\n (u * ((a * ↑z + b) / (c * ↑z + d)) + v) ^ (-k) =\n (c * ↑z + d) ^ k * ((u * a + v * c) * ↑z + (u * b + v * d)) ^ (-k)\n⊢ ((↑(i 0) * ((↑(↑A 0 0) * ↑z + ↑(↑A 0 1)) / (↑(↑A 1 0) * ↑z + ↑(↑A 1 1))) + ↑(i 1)) ^ k)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Modular | {
"line": 355,
"column": 2
} | {
"line": 355,
"column": 33
} | {
"line": 355,
"column": 34
} | [
{
"pp": "z : ℍ\nh : 1 < normSq ↑z\n⊢ normSq\n ((↑((algebraMap ℤ ℝ) (↑S 0 0)) * ↑z + ↑((algebraMap ℤ ℝ) (↑S 0 1))) /\n (↑((algebraMap ℤ ℝ) (↑S 1 0)) * ↑z + ↑((algebraMap ℤ ℝ) (↑S 1 1)))) <\n 1",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid... | [
"z : ℍ\nh : 1 < normSq ↑z\n⊢ (normSq ↑z)⁻¹ < 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.Bounds | {
"line": 93,
"column": 6
} | {
"line": 93,
"column": 82
} | {
"line": 93,
"column": 83
} | [
{
"pp": "E : Type u_1\ninst✝ : SeminormedAddCommGroup E\nf : ℍ → E\nhf_cont : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f\nt : ℝ\nht : 0 ≤ t\nhf_infinity : f =O[atImInfty] fun z ↦ z.im ^ t\nhf_inv : ∀ (g : SL(2, ℤ)) (τ : ℍ), f (g • τ) = f τ\nF : ℝ\nτ : ℍ\ng : SL(2, ℤ)\nhF𝒟 : ‖f τ‖ ≤ F ... | [
"E : Type u_1\ninst✝ : SeminormedAddCommGroup E\nf : ℍ → E\nhf_cont : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f\nt : ℝ\nht : 0 ≤ t\nhf_infinity : f =O[atImInfty] fun z ↦ z.im ^ t\nhf_inv : ∀ (g : SL(2, ℤ)) (τ : ℍ), f (g • τ) = f τ\nF : ℝ\nτ : ℍ\ng : SL(2, ℤ)\nhF𝒟 : ‖f τ‖ ≤ F * (g • τ).im... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Modular | {
"line": 395,
"column": 45
} | {
"line": 395,
"column": 79
} | {
"line": 395,
"column": 80
} | [
{
"pp": "z : ℍ\nh : z ∈ 𝒟ᵒ\n⊢ 1 < z.re * z.re + z.im * z.im",
"ppTerm": "?m.47",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"z : ℍ\nh : z ∈ 𝒟ᵒ\n⊢ 1 < z.re * z.re + z.im * z.im"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.Bounds | {
"line": 97,
"column": 4
} | {
"line": 97,
"column": 23
} | {
"line": 97,
"column": 24
} | [
{
"pp": "case pos\nE : Type u_1\ninst✝ : SeminormedAddCommGroup E\nf : ℍ → E\nhf_cont : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f\nt : ℝ\nht : 0 ≤ t\nhf_infinity : f =O[atImInfty] fun z ↦ z.im ^ t\nhf_inv : ∀ (g : SL(2, ℤ)) (τ : ℍ), f (g • τ) = f τ\nF : ℝ\nτ : ℍ\ng : SL(2, ℤ)\nhF𝒟 : ... | [
"case pos\nE : Type u_1\ninst✝ : SeminormedAddCommGroup E\nf : ℍ → E\nhf_cont : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f\nt : ℝ\nht : 0 ≤ t\nhf_infinity : f =O[atImInfty] fun z ↦ z.im ^ t\nhf_inv : ∀ (g : SL(2, ℤ)) (τ : ℍ), f (g • τ) = f τ\nF : ℝ\nτ : ℍ\ng : SL(2, ℤ)\nhF𝒟 : ‖f τ‖ ≤ F * ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Modular | {
"line": 401,
"column": 45
} | {
"line": 401,
"column": 79
} | {
"line": 401,
"column": 80
} | [
{
"pp": "τ : ℍ\nh : τ ∈ 𝒟\n⊢ 1 ≤ τ.re * τ.re + τ.im * τ.im",
"ppTerm": "?m.47",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"τ : ℍ\nh : τ ∈ 𝒟\n⊢ 1 ≤ τ.re * τ.re + τ.im * τ.im"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.QExpansion | {
"line": 329,
"column": 30
} | {
"line": 329,
"column": 41
} | {
"line": 329,
"column": 42
} | [
{
"pp": "h : ℝ\nf : ℍ → ℂ\nhh : 0 < h\nhfper : Periodic (f ∘ ↑ofComplex) ↑h\nhfhol : MDiff f\nhfbdd : IsBoundedAtImInfty f\nn : ℕ\nt : ℝ\nht : 0 < t\nR : ℝ := rexp (-2 * π * t / h)\nhR0 : 0 < R\nhR1 : R < 1\nu : ℝ\n⊢ 0 < (↑u + ↑t * ↑I).im",
"ppTerm": "?m.248",
"assigned": true,
"usedConstants": [
... | [
"h : ℝ\nf : ℍ → ℂ\nhh : 0 < h\nhfper : Periodic (f ∘ ↑ofComplex) ↑h\nhfhol : MDiff f\nhfbdd : IsBoundedAtImInfty f\nn : ℕ\nt : ℝ\nht : 0 < t\nR : ℝ := rexp (-2 * π * t / h)\nhR0 : 0 < R\nhR1 : R < 1\nu : ℝ\n⊢ 0 < t"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.UniformConvergence | {
"line": 54,
"column": 2
} | {
"line": 55,
"column": 23
} | {
"line": 55,
"column": 24
} | [
{
"pp": "k : ℤ\nhk : 3 ≤ k\nN : ℕ\na : Fin 2 → ZMod N\nhk' : 2 < ↑k\np_sum : Summable fun x ↦ ‖↑x‖ ^ (-k)\nK : Set ℍ\nhK : IsCompact K\nA B : ℝ\nhB : 0 < B\nHABK : K ⊆ verticalStrip A B\np : ↑(gammaSet N 1 a)\nz : ℍ\nhz : z ∈ verticalStrip A B\n⊢ ‖eisSummand k (↑p) z‖ ≤ r { coe := { re := A, im := B }, coe_im_p... | [
"k : ℤ\nhk : 3 ≤ k\nN : ℕ\na : Fin 2 → ZMod N\nhk' : 2 < ↑k\np_sum : Summable fun x ↦ ‖↑x‖ ^ (-k)\nK : Set ℍ\nhK : IsCompact K\nA B : ℝ\nhB : 0 < B\nHABK : K ⊆ verticalStrip A B\np : ↑(gammaSet N 1 a)\nz : ℍ\nhz : z ∈ verticalStrip A B\n⊢ ‖↑(↑p 0) * ↑z + ↑(↑p 1)‖ ^ (-↑k) ≤ r { coe := { re := A, im := B }, coe_im_po... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Modular | {
"line": 437,
"column": 4
} | {
"line": 437,
"column": 27
} | {
"line": 437,
"column": 28
} | [
{
"pp": "z : ℍ\ng₀ : SL(2, ℤ)\nhg₀ : ∀ (g' : SL(2, ℤ)), (g' • z).im ≤ (g₀ • z).im\ng : SL(2, ℤ)\nhg : ↑g 1 = ↑g₀ 1\nhg' : ∀ (g' : SL(2, ℤ)), ↑g 1 = ↑g' 1 → |(g • z).re| ≤ |(g' • z).re|\nhg'' : (g • z).im = (g₀ • z).im\n⊢ ∀ (g' : SL(2, ℤ)), (g' • z).im ≤ (g • z).im",
"ppTerm": "?m.81",
"assigned": true,
... | [
"z : ℍ\ng₀ : SL(2, ℤ)\nhg₀ : ∀ (g' : SL(2, ℤ)), (g' • z).im ≤ (g₀ • z).im\ng : SL(2, ℤ)\nhg : ↑g 1 = ↑g₀ 1\nhg' : ∀ (g' : SL(2, ℤ)), ↑g 1 = ↑g' 1 → |(g • z).re| ≤ |(g' • z).re|\nhg'' : (g • z).im = (g₀ • z).im\n⊢ ∀ (g' : SL(2, ℤ)), (g' • z).im ≤ (g₀ • z).im"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.QExpansion | {
"line": 350,
"column": 12
} | {
"line": 350,
"column": 23
} | {
"line": 350,
"column": 24
} | [
{
"pp": "h : ℝ\nf : ℍ → ℂ\nhh : 0 < h\nhfper : Periodic (f ∘ ↑ofComplex) ↑h\nhfhol : MDiff f\nhfbdd : IsBoundedAtImInfty f\nthis : (f ∘ ↑ofComplex) =O[I∞] (1 ∘ ↑ofComplex)\nz : ℂ\nhz : z ∈ Complex.im ⁻¹' Ioi 0\n⊢ DifferentiableAt ℂ (f ∘ ↑ofComplex) z",
"ppTerm": "?m.86",
"assigned": false,
"usedCons... | [
"h : ℝ\nf : ℍ → ℂ\nhh : 0 < h\nhfper : Periodic (f ∘ ↑ofComplex) ↑h\nhfhol : MDiff f\nhfbdd : IsBoundedAtImInfty f\nthis : (f ∘ ↑ofComplex) =O[I∞] (1 ∘ ↑ofComplex)\nz : ℂ\nhz : z ∈ Complex.im ⁻¹' Ioi 0\n⊢ DifferentiableAt ℂ (f ∘ ↑ofComplex) z"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.QExpansion | {
"line": 353,
"column": 2
} | {
"line": 353,
"column": 28
} | {
"line": 354,
"column": 4
} | [
{
"pp": "case e'_7\nh : ℝ\nf : ℍ → ℂ\nhh : 0 < h\nhfper : Periodic (f ∘ ↑ofComplex) ↑h\nhfhol : MDiff f\nhfbdd : IsBoundedAtImInfty f\nthis : (f ∘ ↑ofComplex) =O[I∞] (1 ∘ ↑ofComplex)\nx✝ : ℍ\n⊢ f x✝ - valueAtInfty f =\n ((fun z ↦ (f ∘ ↑ofComplex) z - Periodic.cuspFunction h (f ∘ ↑ofComplex) 0) ∘ UpperHalfPla... | [
"case e'_7\nh : ℝ\nf : ℍ → ℂ\nhh : 0 < h\nhfper : Periodic (f ∘ ↑ofComplex) ↑h\nhfhol : MDiff f\nhfbdd : IsBoundedAtImInfty f\nthis : (f ∘ ↑ofComplex) =O[I∞] (1 ∘ ↑ofComplex)\nx✝ : ℍ\n⊢ valueAtInfty f = Periodic.cuspFunction h (f ∘ ↑ofComplex) 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.QExpansion | {
"line": 371,
"column": 2
} | {
"line": 371,
"column": 39
} | {
"line": 371,
"column": 40
} | [
{
"pp": "h : ℝ\nf : ℍ → ℂ\nhf : IsZeroAtImInfty f\nhh : 0 < h\nhfper : Periodic (f ∘ ↑ofComplex) ↑h\nhfhol : MDiff f\nhfbdd : IsBoundedAtImInfty f\n⊢ f =O[atImInfty] fun τ ↦ rexp (-2 * π * τ.im / h)",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"instHD... | [
"h : ℝ\nf : ℍ → ℂ\nhf : IsZeroAtImInfty f\nhh : 0 < h\nhfper : Periodic (f ∘ ↑ofComplex) ↑h\nhfhol : MDiff f\nhfbdd : IsBoundedAtImInfty f\n⊢ f =O[atImInfty] fun τ ↦ rexp (-(2 * π * τ.im) / h)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.QExpansion | {
"line": 380,
"column": 70
} | {
"line": 380,
"column": 81
} | {
"line": 380,
"column": 82
} | [
{
"pp": "k : ℤ\nF : Type u_1\ninst✝¹ : FunLike F ℍ ℂ\nΓ : Subgroup (GL (Fin 2) ℝ)\nh : ℝ\nf : F\ninst✝ : ModularFormClass F Γ k\nhh : 0 < h\nhΓ : h ∈ Γ.strictPeriods\nn : ℕ\nt : ℝ\nht : 0 < t\nu : ℝ\n⊢ 0 < (↑u + ↑t * I).im",
"ppTerm": "?m.78",
"assigned": true,
"usedConstants": [
"Complex.mul_... | [
"k : ℤ\nF : Type u_1\ninst✝¹ : FunLike F ℍ ℂ\nΓ : Subgroup (GL (Fin 2) ℝ)\nh : ℝ\nf : F\ninst✝ : ModularFormClass F Γ k\nhh : 0 < h\nhΓ : h ∈ Γ.strictPeriods\nn : ℕ\nt : ℝ\nht : 0 < t\nu : ℝ\n⊢ 0 < t"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.QExpansion | {
"line": 406,
"column": 2
} | {
"line": 406,
"column": 39
} | {
"line": 406,
"column": 40
} | [
{
"pp": "k : ℤ\nF : Type u_1\ninst✝⁴ : FunLike F ℍ ℂ\nΓ : Subgroup (GL (Fin 2) ℝ)\nf : F\ninst✝³ : ModularFormClass F Γ k\ninst✝² : Γ.HasDetPlusMinusOne\ninst✝¹ : DiscreteTopology ↥Γ\ninst✝ : Fact (IsCusp OnePoint.infty Γ)\nhf : IsZeroAtImInfty ⇑f\n⊢ ∃ c > 0, ⇑f =O[atImInfty] fun τ ↦ rexp (-c * τ.im)",
"ppT... | [
"k : ℤ\nF : Type u_1\ninst✝⁴ : FunLike F ℍ ℂ\nΓ : Subgroup (GL (Fin 2) ℝ)\nf : F\ninst✝³ : ModularFormClass F Γ k\ninst✝² : Γ.HasDetPlusMinusOne\ninst✝¹ : DiscreteTopology ↥Γ\ninst✝ : Fact (IsCusp OnePoint.infty Γ)\nhf : IsZeroAtImInfty ⇑f\n⊢ ∃ c, 0 < c ∧ ⇑f =O[atImInfty] fun τ ↦ rexp (-(c * τ.im))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.QExpansion | {
"line": 450,
"column": 65
} | {
"line": 452,
"column": 96
} | {
"line": 454,
"column": 0
} | [
{
"pp": "h : ℝ\nf g : ℂ → ℂ\nhfcts : ContinuousAt (Periodic.cuspFunction h f) 0\nhgcts : ContinuousAt (Periodic.cuspFunction h g) 0\n⊢ Periodic.cuspFunction h (f * g) 0 = Periodic.cuspFunction h f 0 * Periodic.cuspFunction h g 0",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Eq.mpr... | [] | by
rw [Periodic.cuspFunction, update_self]
exact (Periodic.tendsto_nhds_zero hfcts).mul (Periodic.tendsto_nhds_zero hgcts) |>.limUnder_eq | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.ModularForms.QExpansion | {
"line": 504,
"column": 2
} | {
"line": 504,
"column": 13
} | {
"line": 504,
"column": 14
} | [
{
"pp": "h : ℝ\nf : ℍ → ℂ\nhf : AnalyticAt ℂ (cuspFunction h f) 0\n⊢ qExpansion h (-f) = -qExpansion h f",
"ppTerm": "?m.21",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"h : ℝ\nf : ℍ → ℂ\nhf : AnalyticAt ℂ (cuspFunction h f) 0\n⊢ qExpansion h (-f) = -qExpansion h f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.QExpansion | {
"line": 516,
"column": 4
} | {
"line": 516,
"column": 50
} | {
"line": 516,
"column": 51
} | [
{
"pp": "h : ℝ\nf g : ℍ → ℂ\nhf : AnalyticAt ℂ (cuspFunction h f) 0\nhg : AnalyticAt ℂ (cuspFunction h g) 0\n⊢ AnalyticAt ℂ (cuspFunction h (-g)) 0",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Pi.instNeg",
"Complex.in... | [
"h : ℝ\nf g : ℍ → ℂ\nhf : AnalyticAt ℂ (cuspFunction h f) 0\nhg : AnalyticAt ℂ (cuspFunction h g) 0\n⊢ AnalyticAt ℂ (cuspFunction h g) 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.QExpansion | {
"line": 517,
"column": 2
} | {
"line": 517,
"column": 49
} | {
"line": 517,
"column": 50
} | [
{
"pp": "h : ℝ\nf g : ℍ → ℂ\nhf : AnalyticAt ℂ (cuspFunction h f) 0\nhg : AnalyticAt ℂ (cuspFunction h g) 0\nhg' : AnalyticAt ℂ (cuspFunction h (-g)) 0\n⊢ qExpansion h (f - g) = qExpansion h f - qExpansion h g",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"UpperHalf... | [
"h : ℝ\nf g : ℍ → ℂ\nhf : AnalyticAt ℂ (cuspFunction h f) 0\nhg : AnalyticAt ℂ (cuspFunction h g) 0\nhg' : AnalyticAt ℂ (cuspFunction h (-g)) 0\n⊢ qExpansion h (f + -g) = qExpansion h f + -qExpansion h g"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.QExpansion | {
"line": 532,
"column": 4
} | {
"line": 532,
"column": 19
} | {
"line": 532,
"column": 20
} | [
{
"pp": "case mpr\nh : ℝ\nf : ℍ → ℂ\nhh : 0 < h\nhfper : Periodic (f ∘ ↑ofComplex) ↑h\nhfhol : MDiff f\nhfbdd : IsBoundedAtImInfty f\nH : f = 0\n⊢ qExpansion h f = 0",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MvPowerSeries.instZero",
"UpperHalfPlane.qExpans... | [
"case mpr\nh : ℝ\nf : ℍ → ℂ\nhh : 0 < h\nhfper : Periodic (f ∘ ↑ofComplex) ↑h\nhfhol : MDiff f\nhfbdd : IsBoundedAtImInfty f\nH : f = 0\n⊢ qExpansion h 0 = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.Bounds | {
"line": 136,
"column": 2
} | {
"line": 136,
"column": 89
} | {
"line": 137,
"column": 4
} | [
{
"pp": "E : Type u_1\ninst✝¹ : SeminormedAddCommGroup E\nf : ℍ → E\nhf_cont : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f\nt : ℝ\nht : 0 ≤ t\nhf_infinity : ∀ (g : SL(2, ℤ)), (fun τ ↦ f (g • τ)) =O[atImInfty] fun z ↦ z.im ^ t\nΓ : Subgroup SL(2, ℤ)\ninst✝ : Γ.FiniteIndex\nhf_inv : ∀ g ∈... | [
"E : Type u_1\ninst✝¹ : SeminormedAddCommGroup E\nf : ℍ → E\nhf_cont : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f\nt : ℝ\nht : 0 ≤ t\nhf_infinity : ∀ (g : SL(2, ℤ)), (fun τ ↦ f (g • τ)) =O[atImInfty] fun z ↦ z.im ^ t\nΓ : Subgroup SL(2, ℤ)\ninst✝ : Γ.FiniteIndex\nhf_inv : ∀ g ∈ Γ, ∀ (τ : ℍ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.QExpansion | {
"line": 615,
"column": 2
} | {
"line": 615,
"column": 23
} | {
"line": 617,
"column": 0
} | [
{
"pp": "Γ : Subgroup (GL (Fin 2) ℝ)\nh : ℝ\ninst✝ : Γ.HasDetPlusMinusOne\n⊢ qExpansion h ⇑1 = 1",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"ModularForm",
"UpperHalfPlane.qExpansion_one",
"UpperHalfPlane.qExpansion",
"congrArg",
"Int",
"Unit",
... | [] | simp [qExpansion_one] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.ModularForms.QExpansion | {
"line": 615,
"column": 2
} | {
"line": 615,
"column": 23
} | {
"line": 617,
"column": 0
} | [
{
"pp": "Γ : Subgroup (GL (Fin 2) ℝ)\nh : ℝ\ninst✝ : Γ.HasDetPlusMinusOne\n⊢ qExpansion h ⇑1 = 1",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"ModularForm",
"UpperHalfPlane.qExpansion_one",
"UpperHalfPlane.qExpansion",
"congrArg",
"Int",
"Unit",
... | [] | simp [qExpansion_one] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.ModularForms.QExpansion | {
"line": 615,
"column": 2
} | {
"line": 615,
"column": 23
} | {
"line": 617,
"column": 0
} | [
{
"pp": "Γ : Subgroup (GL (Fin 2) ℝ)\nh : ℝ\ninst✝ : Γ.HasDetPlusMinusOne\n⊢ qExpansion h ⇑1 = 1",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"ModularForm",
"UpperHalfPlane.qExpansion_one",
"UpperHalfPlane.qExpansion",
"congrArg",
"Int",
"Unit",
... | [] | simp [qExpansion_one] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.ModularForms.QExpansion | {
"line": 706,
"column": 4
} | {
"line": 707,
"column": 11
} | {
"line": 707,
"column": 12
} | [
{
"pp": "F : Type u_1\ninst✝ : FunLike F ℍ ℂ\nh : ℝ\nf : F\nc : ℕ → ℂ\nhh : 0 < h\nhfanalytic : AnalyticAt ℂ (cuspFunction h ⇑f) 0\nhf : ∀ (τ : ℍ), HasSum (fun m ↦ c m • 𝕢 h ↑τ ^ m) (f τ)\nh1 : HasFPowerSeriesAt (cuspFunction h ⇑f) (FormalMultilinearSeries.ofScalars ℂ c) 0\n⊢ HasFPowerSeriesAt (cuspFunction h ... | [
"F : Type u_1\ninst✝ : FunLike F ℍ ℂ\nh : ℝ\nf : F\nc : ℕ → ℂ\nhh : 0 < h\nhfanalytic : AnalyticAt ℂ (cuspFunction h ⇑f) 0\nhf : ∀ (τ : ℍ), HasSum (fun m ↦ c m • 𝕢 h ↑τ ^ m) (f τ)\nh1 : HasFPowerSeriesAt (cuspFunction h ⇑f) (FormalMultilinearSeries.ofScalars ℂ c) 0\n⊢ HasFPowerSeriesAt (cuspFunction h ⇑f)\n (Fo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.QExpansion | {
"line": 708,
"column": 2
} | {
"line": 708,
"column": 48
} | {
"line": 708,
"column": 49
} | [
{
"pp": "F : Type u_1\ninst✝ : FunLike F ℍ ℂ\nh : ℝ\nf : F\nc : ℕ → ℂ\nhh : 0 < h\nhfanalytic : AnalyticAt ℂ (cuspFunction h ⇑f) 0\nhf : ∀ (τ : ℍ), HasSum (fun m ↦ c m • 𝕢 h ↑τ ^ m) (f τ)\nh1 : HasFPowerSeriesAt (cuspFunction h ⇑f) (FormalMultilinearSeries.ofScalars ℂ c) 0\nh2 : HasFPowerSeriesAt (cuspFunction... | [
"F : Type u_1\ninst✝ : FunLike F ℍ ℂ\nh : ℝ\nf : F\nc : ℕ → ℂ\nhh : 0 < h\nhfanalytic : AnalyticAt ℂ (cuspFunction h ⇑f) 0\nhf : ∀ (τ : ℍ), HasSum (fun m ↦ c m • 𝕢 h ↑τ ^ m) (f τ)\nh1 : HasFPowerSeriesAt (cuspFunction h ⇑f) (FormalMultilinearSeries.ofScalars ℂ c) 0\nh2 : HasFPowerSeriesAt (cuspFunction h ⇑f) (qExp... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.QExpansion | {
"line": 716,
"column": 2
} | {
"line": 716,
"column": 13
} | {
"line": 716,
"column": 14
} | [
{
"pp": "F : Type u_1\ninst✝ : FunLike F ℍ ℂ\nh : ℝ\nf : F\nc : ℕ → ℂ\nhh : 0 < h\nhfanalytic : AnalyticAt ℂ (cuspFunction h ⇑f) 0\nhf : ∀ (τ : ℍ), HasSum (fun m ↦ c m • 𝕢 h ↑τ ^ m) (f τ)\nm : ℕ\nh1 : HasFPowerSeriesAt (cuspFunction h ⇑f) (FormalMultilinearSeries.ofScalars ℂ c) 0\nh2 : HasFPowerSeriesAt (cuspF... | [
"F : Type u_1\ninst✝ : FunLike F ℍ ℂ\nh : ℝ\nf : F\nc : ℕ → ℂ\nhh : 0 < h\nhfanalytic : AnalyticAt ℂ (cuspFunction h ⇑f) 0\nhf : ∀ (τ : ℍ), HasSum (fun m ↦ c m • 𝕢 h ↑τ ^ m) (f τ)\nm : ℕ\nh1 : HasFPowerSeriesAt (cuspFunction h ⇑f) (FormalMultilinearSeries.ofScalars ℂ c) 0\nh2 : HasFPowerSeriesAt (cuspFunction h ⇑f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.Bounds | {
"line": 181,
"column": 2
} | {
"line": 182,
"column": 9
} | {
"line": 182,
"column": 10
} | [
{
"pp": "k : ℤ\nhk : 0 ≤ k\nΓ : Subgroup (GL (Fin 2) ℝ)\ninst✝⁴ : Γ.IsArithmetic\nF : Type u_2\nF' : Type u_3\nf : F\nf' : F'\ninst✝³ : FunLike F ℍ ℂ\ninst✝² : FunLike F' ℍ ℂ\ninst✝¹ : ModularFormClass F Γ k\ninst✝ : ModularFormClass F' Γ k\ng : SL(2, ℤ)\n⊢ (fun τ ↦ ‖petersson k (⇑f ∣[k] g) (⇑f' ∣[k] g) τ‖) =O[... | [
"k : ℤ\nhk : 0 ≤ k\nΓ : Subgroup (GL (Fin 2) ℝ)\ninst✝⁴ : Γ.IsArithmetic\nF : Type u_2\nF' : Type u_3\nf : F\nf' : F'\ninst✝³ : FunLike F ℍ ℂ\ninst✝² : FunLike F' ℍ ℂ\ninst✝¹ : ModularFormClass F Γ k\ninst✝ : ModularFormClass F' Γ k\ng : SL(2, ℤ)\n⊢ (fun τ ↦ ‖(⇑f ∣[k] g) τ‖ * ‖(⇑f' ∣[k] g) τ‖ * τ.im ^ k) =O[atImInf... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Modular | {
"line": 488,
"column": 60
} | {
"line": 488,
"column": 71
} | {
"line": 488,
"column": 72
} | [
{
"pp": "g : SL(2, ℤ)\nz : ℍ\nhz : z ∈ 𝒟\nhg : g • z ∈ 𝒟\nhc : ↑g 1 0 = 0\nthis :\n ∀ {g : SL(2, ℤ)} {z : ℍ},\n z ∈ 𝒟 →\n g • z ∈ 𝒟 →\n ↑g 1 0 = 0 →\n 0 ≤ ↑g 1 1 → (g = T ∨ g = -T) ∧ z.re = -1 / 2 ∨ (g = T⁻¹ ∨ g = -T⁻¹) ∧ z.re = 1 / 2 ∨ g = 1 ∨ g = -1\nhd : ¬0 ≤ ↑g 1 1\n⊢ ↑(-g) 1 0 ... | [
"g : SL(2, ℤ)\nz : ℍ\nhz : z ∈ 𝒟\nhg : g • z ∈ 𝒟\nhc : ↑g 1 0 = 0\nthis :\n ∀ {g : SL(2, ℤ)} {z : ℍ},\n z ∈ 𝒟 →\n g • z ∈ 𝒟 →\n ↑g 1 0 = 0 →\n 0 ≤ ↑g 1 1 → (g = T ∨ g = -T) ∧ z.re = -1 / 2 ∨ (g = T⁻¹ ∨ g = -T⁻¹) ∧ z.re = 1 / 2 ∨ g = 1 ∨ g = -1\nhd : ¬0 ≤ ↑g 1 1\n⊢ ↑g 1 0 = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Modular | {
"line": 489,
"column": 6
} | {
"line": 489,
"column": 17
} | {
"line": 489,
"column": 18
} | [
{
"pp": "case inr\ng : SL(2, ℤ)\nz : ℍ\nhz : z ∈ 𝒟\nhg : g • z ∈ 𝒟\nhc : ↑g 1 0 = 0\nthis :\n ∀ {g : SL(2, ℤ)} {z : ℍ},\n z ∈ 𝒟 →\n g • z ∈ 𝒟 →\n ↑g 1 0 = 0 →\n 0 ≤ ↑g 1 1 → (g = T ∨ g = -T) ∧ z.re = -1 / 2 ∨ (g = T⁻¹ ∨ g = -T⁻¹) ∧ z.re = 1 / 2 ∨ g = 1 ∨ g = -1\nhd : ¬0 ≤ ↑g 1 1\n⊢ ... | [
"case inr\ng : SL(2, ℤ)\nz : ℍ\nhz : z ∈ 𝒟\nhg : g • z ∈ 𝒟\nhc : ↑g 1 0 = 0\nthis :\n ∀ {g : SL(2, ℤ)} {z : ℍ},\n z ∈ 𝒟 →\n g • z ∈ 𝒟 →\n ↑g 1 0 = 0 →\n 0 ≤ ↑g 1 1 → (g = T ∨ g = -T) ∧ z.re = -1 / 2 ∨ (g = T⁻¹ ∨ g = -T⁻¹) ∧ z.re = 1 / 2 ∨ g = 1 ∨ g = -1\nhd : ¬0 ≤ ↑g 1 1\n⊢ ↑g 1 1 ≤ 0"
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Modular | {
"line": 492,
"column": 4
} | {
"line": 492,
"column": 57
} | {
"line": 492,
"column": 58
} | [
{
"pp": "g✝ : SL(2, ℤ)\nz✝ : ℍ\ng : SL(2, ℤ)\nz : ℍ\nhz : z ∈ 𝒟\nhg : g • z ∈ 𝒟\nhc : ↑g 1 0 = 0\nhd : 0 ≤ ↑g 1 1\n⊢ ↑g 1 1 = 1 ∨ ↑g 1 1 = -1",
"ppTerm": "?m.392",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"g✝ : SL(2, ℤ)\nz✝ : ℍ\ng : SL(2, ℤ)\nz : ℍ\nhz : z ∈ 𝒟\nhg : g • z ∈ 𝒟\nhc : ↑g 1 0 = 0\nhd : 0 ≤ ↑g 1 1\n⊢ ↑g 1 1 = 1 ∨ ↑g 1 1 = -1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.Bounds | {
"line": 208,
"column": 2
} | {
"line": 208,
"column": 35
} | {
"line": 208,
"column": 36
} | [
{
"pp": "k : ℤ\nΓ : Subgroup (GL (Fin 2) ℝ)\ninst✝⁴ : Γ.IsArithmetic\nF : Type u_2\nF' : Type u_3\nf : F\nf' : F'\ninst✝³ : FunLike F ℍ ℂ\ninst✝² : FunLike F' ℍ ℂ\ninst✝¹ : ModularFormClass F Γ k\ninst✝ : CuspFormClass F' Γ k\n⊢ ∃ C, ∀ (τ : ℍ), ‖petersson k (⇑f) (⇑f') τ‖ ≤ C",
"ppTerm": "?m.18",
"assign... | [
"k : ℤ\nΓ : Subgroup (GL (Fin 2) ℝ)\ninst✝⁴ : Γ.IsArithmetic\nF : Type u_2\nF' : Type u_3\nf : F\nf' : F'\ninst✝³ : FunLike F ℍ ℂ\ninst✝² : FunLike F' ℍ ℂ\ninst✝¹ : ModularFormClass F Γ k\ninst✝ : CuspFormClass F' Γ k\n⊢ ∃ C, ∀ (τ : ℍ), ‖petersson k (⇑f) (⇑f') τ‖ ≤ C"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.TsumDivisorsAntidiagonal | {
"line": 114,
"column": 2
} | {
"line": 117,
"column": 11
} | {
"line": 117,
"column": 12
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : CompleteSpace 𝕜\ninst✝ : NormSMulClass ℤ 𝕜\nk : ℕ\nr : 𝕜\nhr : ‖r‖ < 1\nn : ℕ+\n⊢ ∑' (c : ↥(↑n).divisorsAntidiagonal), ↑(↑c).2 ^ k * r ^ ((↑c).1 * (↑c).2) =\n (∑ x ∈ (↑n).divisors, ↑(↑n / x) ^ k) * r ^ ↑n",
"ppTerm": "?m.123",
"... | [
"𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : CompleteSpace 𝕜\ninst✝ : NormSMulClass ℤ 𝕜\nk : ℕ\nr : 𝕜\nhr : ‖r‖ < 1\nn : ℕ+\n⊢ ∑ i ∈ (↑n).divisors, ↑(↑n / i) ^ k * r ^ (i * (↑n / i)) = ∑ i ∈ (↑n).divisors, ↑(↑n / i) ^ k * r ^ ↑n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.TsumDivisorsAntidiagonal | {
"line": 122,
"column": 8
} | {
"line": 122,
"column": 19
} | {
"line": 122,
"column": 20
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : CompleteSpace 𝕜\ninst✝¹ : NormSMulClass ℤ 𝕜\nr : 𝕜\nhr : ‖r‖ < 1\nk m : ℕ\ninst✝ : NeZero m\n⊢ ‖r ^ m‖ < 1",
"ppTerm": "?m.68",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toSem... | [
"𝕜 : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : CompleteSpace 𝕜\ninst✝¹ : NormSMulClass ℤ 𝕜\nr : 𝕜\nhr : ‖r‖ < 1\nk m : ℕ\ninst✝ : NeZero m\n⊢ ‖r‖ ^ m < 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.AtTopBot.Interval | {
"line": 31,
"column": 2
} | {
"line": 32,
"column": 9
} | {
"line": 32,
"column": 10
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Preorder α\ninst✝ : LocallyFiniteOrder α\n⊢ Tendsto (fun p ↦ Icc p.1 p.2) (atBot ×ˢ atTop) atTop",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Order.Filter.AtTopBot.Interval.0.Finset.tendsto_Icc_atBot_prod_atTop._simp_1_1",
"E... | [
"α : Type u_1\ninst✝¹ : Preorder α\ninst✝ : LocallyFiniteOrder α\n⊢ ∀ (b : Finset α), ∀ i ∈ b, ∀ᶠ (x : α × α) in atBot ×ˢ atTop, x.1 ≤ i ∧ i ≤ x.2"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.AtTopBot.Interval | {
"line": 36,
"column": 2
} | {
"line": 37,
"column": 9
} | {
"line": 37,
"column": 10
} | [
{
"pp": "α : Type u_1\ninst✝² : Preorder α\ninst✝¹ : LocallyFiniteOrder α\ninst✝ : NoBotOrder α\n⊢ Tendsto (fun p ↦ Ioc p.1 p.2) (atBot ×ˢ atTop) atTop",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.Ioc",
"SetLike.mem_coe._simp_1",
"Preorder.toLT",
... | [
"α : Type u_1\ninst✝² : Preorder α\ninst✝¹ : LocallyFiniteOrder α\ninst✝ : NoBotOrder α\n⊢ ∀ (b : Finset α), ∀ i ∈ b, ∀ᶠ (x : α × α) in atBot ×ˢ atTop, x.1 < i ∧ i ≤ x.2"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.AtTopBot.Interval | {
"line": 41,
"column": 2
} | {
"line": 42,
"column": 9
} | {
"line": 42,
"column": 10
} | [
{
"pp": "α : Type u_1\ninst✝² : Preorder α\ninst✝¹ : LocallyFiniteOrder α\ninst✝ : NoTopOrder α\n⊢ Tendsto (fun p ↦ Ico p.1 p.2) (atBot ×ˢ atTop) atTop",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SetLike.mem_coe._simp_1",
"Finset.coe_Ico",
"Preorder.t... | [
"α : Type u_1\ninst✝² : Preorder α\ninst✝¹ : LocallyFiniteOrder α\ninst✝ : NoTopOrder α\n⊢ ∀ (b : Finset α), ∀ i ∈ b, ∀ᶠ (x : α × α) in atBot ×ˢ atTop, x.1 ≤ i ∧ i < x.2"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.AtTopBot.Interval | {
"line": 46,
"column": 2
} | {
"line": 47,
"column": 9
} | {
"line": 47,
"column": 10
} | [
{
"pp": "α : Type u_1\ninst✝³ : Preorder α\ninst✝² : LocallyFiniteOrder α\ninst✝¹ : NoBotOrder α\ninst✝ : NoTopOrder α\n⊢ Tendsto (fun p ↦ Ioo p.1 p.2) (atBot ×ˢ atTop) atTop",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SetLike.mem_coe._simp_1",
"Preorder.to... | [
"α : Type u_1\ninst✝³ : Preorder α\ninst✝² : LocallyFiniteOrder α\ninst✝¹ : NoBotOrder α\ninst✝ : NoTopOrder α\n⊢ ∀ (b : Finset α), ∀ i ∈ b, ∀ᶠ (x : α × α) in atBot ×ˢ atTop, x.1 < i ∧ i < x.2"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Modular | {
"line": 521,
"column": 37
} | {
"line": 521,
"column": 60
} | {
"line": 521,
"column": 61
} | [
{
"pp": "g : SL(2, ℤ)\nz : ℍ\nhz : z ∈ 𝒟\nhg' : ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) g)) ↑z‖ ≤ 1\nhc : ↑g 1 0 = 1\n⊢ ‖↑z + ↑(↑g 1 1)‖ ≤ 1",
"ppTerm": "?m.51",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"g : SL(2, ℤ)\nz : ℍ\nhz : z ∈ 𝒟\nhg' : ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) g)) ↑z‖ ≤ 1\nhc : ↑g 1 0 = 1\n⊢ ‖↑z + ↑(↑g 1 1)‖ ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.ConditionalInt | {
"line": 131,
"column": 2
} | {
"line": 132,
"column": 9
} | {
"line": 132,
"column": 10
} | [
{
"pp": "α : Type u_1\nf : ℤ → α\ninst✝² : CommGroup α\ninst✝¹ : TopologicalSpace α\ninst✝ : ContinuousMul α\na : α\nhf2 : Tendsto (fun N ↦ (f ↑N)⁻¹) atTop (𝓝 1)\nhf : Tendsto ((fun s ↦ ∏ b ∈ s, f b) ∘ fun N ↦ Icc (-↑N) ↑N) atTop (𝓝 a)\n⊢ Tendsto (((fun s ↦ ∏ b ∈ s, f b) ∘ fun N ↦ Ico (-N) N) ∘ Nat.cast / (fu... | [
"α : Type u_1\nf : ℤ → α\ninst✝² : CommGroup α\ninst✝¹ : TopologicalSpace α\ninst✝ : ContinuousMul α\na : α\nhf2 : Tendsto (fun N ↦ (f ↑N)⁻¹) atTop (𝓝 1)\nhf : Tendsto ((fun s ↦ ∏ b ∈ s, f b) ∘ fun N ↦ Icc (-↑N) ↑N) atTop (𝓝 a)\n⊢ Tendsto (fun i ↦ (f ↑i)⁻¹) atTop (𝓝 1)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Modular | {
"line": 535,
"column": 4
} | {
"line": 535,
"column": 85
} | {
"line": 535,
"column": 86
} | [
{
"pp": "g : SL(2, ℤ)\nz : ℍ\nhz : z ∈ 𝒟\nhg : g • z ∈ 𝒟\nhg' : ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) g)) ↑z‖ ≤ 1\nhc : ↑g 1 0 = 1\nhd : ↑g 1 1 = 0\n⊢ ↑g 0 1 = -1",
"ppTerm": "?m.75",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"g : SL(2, ℤ)\nz : ℍ\nhz : z ∈ 𝒟\nhg : g • z ∈ 𝒟\nhg' : ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) g)) ↑z‖ ≤ 1\nhc : ↑g 1 0 = 1\nhd : ↑g 1 1 = 0\n⊢ ↑g 0 1 = -1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Modular | {
"line": 537,
"column": 20
} | {
"line": 537,
"column": 47
} | {
"line": 537,
"column": 48
} | [
{
"pp": "g : SL(2, ℤ)\nz : ℍ\nhz : z ∈ 𝒟\nhg : g • z ∈ 𝒟\nhg' : ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) g)) ↑z‖ ≤ 1\nhc : ↑g 1 0 = 1\nhd : ↑g 1 1 = 0\nhb : ↑g 0 1 = -1\n⊢ ‖↑z‖ ≤ 1",
"ppTerm": "?m.98",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": ... | [
"g : SL(2, ℤ)\nz : ℍ\nhz : z ∈ 𝒟\nhg : g • z ∈ 𝒟\nhg' : ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) g)) ↑z‖ ≤ 1\nhc : ↑g 1 0 = 1\nhd : ↑g 1 1 = 0\nhb : ↑g 0 1 = -1\n⊢ ‖↑z‖ ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.E2.Defs | {
"line": 84,
"column": 4
} | {
"line": 84,
"column": 35
} | {
"line": 84,
"column": 36
} | [
{
"pp": "A B : SL(2, ℤ)\nz : ℍ\n⊢ ↑((↑A * ↑B) 1 0) * denom (toGL (φ B)) ↑z - ↑(↑B 1 0) * denom (toGL (φ A) * toGL (φ B)) ↑z = ↑(↑A 1 0)",
"ppTerm": "?m.103",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Int.instAddCommMonoid",
"Real",
"Matrix.SpecialLinea... | [
"A B : SL(2, ℤ)\nz : ℍ\n⊢ ↑((↑A * ↑B) 1 0) * denom (toGL (φ B)) ↑z = ↑(↑A 1 0) + ↑(↑B 1 0) * denom (toGL (φ A) * toGL (φ B)) ↑z"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.E2.Defs | {
"line": 95,
"column": 2
} | {
"line": 95,
"column": 38
} | {
"line": 95,
"column": 39
} | [
{
"pp": "A : SL(2, ℤ)\n⊢ D2 A ∣[2] A⁻¹ = -D2 A⁻¹",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"Matrix.SpecialLinearGroup",
"Pi.instNeg",
"Pi.addGroup",
"AddGroupWithOne.toAddGroup",
"AddMonoid.toAddZeroC... | [
"A : SL(2, ℤ)\n⊢ D2 A ∣[2] A⁻¹ + D2 A⁻¹ = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.DedekindEta | {
"line": 56,
"column": 2
} | {
"line": 56,
"column": 32
} | {
"line": 56,
"column": 33
} | [
{
"pp": "n : ℕ\nz : ℂ\nhz : z ∈ ℍₒ\nh : 1 = cexp (2 * ↑π * I * (↑n + 1) * z)\n⊢ False",
"ppTerm": "?m.22",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"n : ℕ\nz : ℂ\nhz : z ∈ ℍₒ\nh : 1 = cexp (2 * ↑π * I * (↑n + 1) * z)\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.DedekindEta | {
"line": 57,
"column": 4
} | {
"line": 57,
"column": 54
} | {
"line": 57,
"column": 55
} | [
{
"pp": "n : ℕ\nz : ℂ\nhz : z ∈ ℍₒ\nh : 1 = cexp (2 * ↑π * I * (↑n + 1) * z)\n⊢ 0 < ((n + 1) • z).im",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Complex.mul_im",
"Real.instIsOrderedRing",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.partialO... | [
"n : ℕ\nz : ℂ\nhz : z ∈ ℍₒ\nh : 1 = cexp (2 * ↑π * I * (↑n + 1) * z)\n⊢ 0 < z.im"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.DedekindEta | {
"line": 69,
"column": 2
} | {
"line": 69,
"column": 13
} | {
"line": 69,
"column": 14
} | [
{
"pp": "q : ℂ\nhq : ‖q‖ < 1\n⊢ Summable fun i ↦ ‖-q ^ (i + 1)‖",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"NegZeroClass.toNeg",
... | [
"q : ℂ\nhq : ‖q‖ < 1\n⊢ Summable fun i ↦ ‖q‖ ^ (i + 1)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.DedekindEta | {
"line": 80,
"column": 4
} | {
"line": 80,
"column": 63
} | {
"line": 80,
"column": 64
} | [
{
"pp": "case h.inl\nK : Set ℂ\nhK : K ⊆ Metric.ball 0 1\nhcK : IsCompact K\nhN : K = ∅\n⊢ HasProdUniformlyOn (fun n q ↦ 1 + -q ^ (n + 1)) (fun q ↦ ∏' (n : ℕ), (1 + -q ^ (n + 1))) K",
"ppTerm": "?h.inl",
"assigned": true,
"usedConstants": [
"UniformSpace",
"Eq.mpr",
"NormedCommRing... | [
"case h.inl\nK : Set ℂ\nhK : K ⊆ Metric.ball 0 1\nhcK : IsCompact K\nhN : K = ∅\n⊢ TendstoUniformlyOn (fun x1 x2 ↦ ∏ i ∈ x1, (1 + -x2 ^ (i + 1))) (fun q ↦ ∏' (n : ℕ), (1 + -q ^ (n + 1))) Filter.atTop ∅"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.DedekindEta | {
"line": 84,
"column": 10
} | {
"line": 84,
"column": 56
} | {
"line": 84,
"column": 57
} | [
{
"pp": "K : Set ℂ\nhK : K ⊆ Metric.ball 0 1\nhcK : IsCompact K\nhN : K.Nonempty\nq₀ : ℂ\nhq₀ : q₀ ∈ K\nleft✝ : sSup ((fun q ↦ ‖q‖) '' K) = ‖q₀‖\nHB : ∀ y ∈ K, ‖y‖ ≤ ‖q₀‖\n⊢ ‖?m.238‖ < 1",
"ppTerm": "?m.239",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"K : Set ℂ\nhK : K ⊆ Metric.ball 0 1\nhcK : IsCompact K\nhN : K.Nonempty\nq₀ : ℂ\nhq₀ : q₀ ∈ K\nleft✝ : sSup ((fun q ↦ ‖q‖) '' K) = ‖q₀‖\nHB : ∀ y ∈ K, ‖y‖ ≤ ‖q₀‖\n⊢ ‖?m.238‖ < 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.DedekindEta | {
"line": 86,
"column": 4
} | {
"line": 86,
"column": 15
} | {
"line": 86,
"column": 16
} | [
{
"pp": "case h.inr\nK : Set ℂ\nhK : K ⊆ Metric.ball 0 1\nhcK : IsCompact K\nhN : K.Nonempty\nq₀ : ℂ\nhq₀ : q₀ ∈ K\nleft✝ : sSup ((fun q ↦ ‖q‖) '' K) = ‖q₀‖\nHB : ∀ y ∈ K, ‖y‖ ≤ ‖q₀‖\nn : ℕ\nx : ℂ\nhx : x ∈ K\n⊢ ‖-x ^ (n + 1)‖ ≤ ‖q₀‖ ^ (n + 1)",
"ppTerm": "?h.inr",
"assigned": true,
"usedConstants":... | [
"case h.inr\nK : Set ℂ\nhK : K ⊆ Metric.ball 0 1\nhcK : IsCompact K\nhN : K.Nonempty\nq₀ : ℂ\nhq₀ : q₀ ∈ K\nleft✝ : sSup ((fun q ↦ ‖q‖) '' K) = ‖q₀‖\nHB : ∀ y ∈ K, ‖y‖ ≤ ‖q₀‖\nn : ℕ\nx : ℂ\nhx : x ∈ K\n⊢ ‖x‖ ^ (n + 1) ≤ ‖q₀‖ ^ (n + 1)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.DedekindEta | {
"line": 92,
"column": 27
} | {
"line": 92,
"column": 55
} | {
"line": 93,
"column": 6
} | [
{
"pp": "x✝ : Finset ℕ\n⊢ DifferentiableOn ℂ (fun x2 ↦ ∏ i ∈ x✝, (fun n q ↦ 1 - q ^ (n + 1)) i x2) (Metric.ball 0 1)",
"ppTerm": "?m.105",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"NonUnitalCommCStarAlgebra.toNonUnitalCStarAlgebra",
... | [
"x✝ : Finset ℕ\n⊢ DifferentiableOn ℂ (fun x2 ↦ ∏ x ∈ x✝, (1 - x2 ^ (x + 1))) (Metric.ball 0 1)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.DedekindEta | {
"line": 100,
"column": 34
} | {
"line": 100,
"column": 45
} | {
"line": 100,
"column": 46
} | [
{
"pp": "k : ℕ\nx✝ : ℂ\nhq : x✝ ∈ Metric.ball 0 1\n⊢ ‖x✝‖ < 1",
"ppTerm": "?m.117",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"k : ℕ\nx✝ : ℂ\nhq : x✝ ∈ Metric.ball 0 1\n⊢ ‖x✝‖ < 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.DedekindEta | {
"line": 103,
"column": 2
} | {
"line": 103,
"column": 36
} | {
"line": 104,
"column": 4
} | [
{
"pp": "z : ℍ\n⊢ Summable fun n ↦ ‖-eta_q n ↑z‖",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"summable_geometric_iff_norm_lt_one._simp_1",
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"NegZeroClass.toNeg",
... | [
"z : ℍ\n⊢ ‖𝕢 1 ↑z‖ < 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.DedekindEta | {
"line": 109,
"column": 19
} | {
"line": 109,
"column": 30
} | {
"line": 109,
"column": 31
} | [
{
"pp": "z : ℂ\nhz : z ∈ ℍₒ\n⊢ 𝕢 1 z ∈ Metric.ball 0 1",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"Function.Periodic.qParam",
"congrArg",
... | [
"z : ℂ\nhz : z ∈ ℍₒ\n⊢ ‖𝕢 1 z‖ < 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Modular | {
"line": 580,
"column": 48
} | {
"line": 580,
"column": 85
} | {
"line": 580,
"column": 86
} | [
{
"pp": "g : SL(2, ℤ)\nz : ℍ\nhz : z ∈ 𝒟\nhg : g • z ∈ 𝒟\nhg' : ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) g)) ↑z‖ ≤ 1\nhc : ↑g 1 0 = 1\nhd : ↑g 1 1 = 1\nhgeq : g = T ^ ↑g 0 0 * S * T\n⊢ normSq (↑z + 1) ≤ 1",
"ppTerm": "?m.913",
"assigned": false,
"usedConstants": [],
"usedFVar... | [
"g : SL(2, ℤ)\nz : ℍ\nhz : z ∈ 𝒟\nhg : g • z ∈ 𝒟\nhg' : ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) g)) ↑z‖ ≤ 1\nhc : ↑g 1 0 = 1\nhd : ↑g 1 1 = 1\nhgeq : g = T ^ ↑g 0 0 * S * T\n⊢ normSq (↑z + 1) ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.DedekindEta | {
"line": 147,
"column": 42
} | {
"line": 147,
"column": 53
} | {
"line": 147,
"column": 54
} | [
{
"pp": "z : ℂ\nhz : z ∈ ℍₒ\n⊢ 𝕢 1 z ∈ Metric.ball 0 1",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"Function.Periodic.qParam",
"congrArg",
... | [
"z : ℂ\nhz : z ∈ ℍₒ\n⊢ ‖𝕢 1 z‖ < 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.DedekindEta | {
"line": 155,
"column": 76
} | {
"line": 160,
"column": 21
} | {
"line": 162,
"column": 0
} | [
{
"pp": "h : ℝ\nz : ℂ\n⊢ logDeriv (𝕢 h) z = 2 * ↑π * I / ↑h",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"logDeriv",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"logDeriv_comp",
"NonUnitalCommCStarAlgebra.toNonUnitalCStarAlgebra",
"Function.Peri... | [] | by
have : 𝕢 h = cexp ∘ ((2 * π * I / h) * ·) := by
ext
grind [Periodic.qParam]
rw [this, logDeriv_comp (by fun_prop) (by fun_prop), deriv_const_mul_id]
simp [logDeriv_exp] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.E2.Transform | {
"line": 99,
"column": 4
} | {
"line": 99,
"column": 49
} | {
"line": 99,
"column": 50
} | [
{
"pp": "z : ℍ\n⊢ (fun m ↦ ((↑(m 0) * ↑z + ↑(m 1)) ^ 2 * (↑(m 0) * ↑z + ↑(m 1) + 1))⁻¹) =O[cofinite] fun n ↦ (‖n‖ ^ 3)⁻¹",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"Norm.norm",
"Int.cast",
"Eq.mpr",
"Real.instPow",
"Se... | [
"z : ℍ\n⊢ (fun m ↦ (↑(m 0) * ↑z + ↑(m 1) + 1)⁻¹ * (↑(m 0) * ↑z + ↑(m 1))⁻¹ * (↑(m 0) * ↑z + ↑(m 1))⁻¹) =O[cofinite] fun n ↦\n ‖n‖⁻¹ * ‖n‖⁻¹ * ‖n‖⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.E2.Transform | {
"line": 121,
"column": 8
} | {
"line": 121,
"column": 31
} | {
"line": 121,
"column": 32
} | [
{
"pp": "z : ℍ\nb n : ℤ\nh : b = 0 → ¬n = 0\nhb : ¬b = 0\n⊢ ↑b * ↑z + ↑n + 1 ≠ 0",
"ppTerm": "?m.193",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"HMul.hMul",
"AddMonoid.toAddSemigroup",
"UpperHalfPlane.coe",
"congrArg",
"add_assoc",
"A... | [
"z : ℍ\nb n : ℤ\nh : b = 0 → ¬n = 0\nhb : ¬b = 0\n⊢ ¬↑b * ↑z + (↑n + 1) = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.E2.Transform | {
"line": 137,
"column": 6
} | {
"line": 138,
"column": 13
} | {
"line": 138,
"column": 14
} | [
{
"pp": "z : ℍ\nt : ℂ := ∑' (m : ℤ) (n : ℤ), G2Term z ![m, n]\n⊢ Summable fun b ↦\n ∑' (n : ℤ), G2Term z ![b, n] + ∑'[symmetricIco ℤ] (n : ℤ), (1 / (↑b * ↑z + ↑n) - 1 / (↑b * ↑z + ↑n + 1))",
"ppTerm": "?m.85",
"assigned": true,
"usedConstants": [
"_private.Mathlib.NumberTheory.ModularForms.... | [
"z : ℍ\nt : ℂ := ∑' (m : ℤ) (n : ℤ), G2Term z ![m, n]\n⊢ Summable fun b ↦ ∑' (n : ℤ), G2Term z ![b, n]"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.E2.Transform | {
"line": 210,
"column": 8
} | {
"line": 210,
"column": 45
} | {
"line": 210,
"column": 46
} | [
{
"pp": "case mem.inr\nγ g : SL(2, ℤ)\nh2 : g = T\n⊢ G2 ∣[2] g = G2 - D2 g",
"ppTerm": "?mem.inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Matrix.SpecialLinearGroup",
"congrArg",
"sub_zero",
"instDecidableEqFin",
"EisensteinSeries.G2",
"AddGroupWithO... | [
"case mem.inr\nγ g : SL(2, ℤ)\nh2 : g = T\n⊢ G2 ∣[2] T = G2"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.Derivative | {
"line": 53,
"column": 4
} | {
"line": 53,
"column": 19
} | {
"line": 53,
"column": 20
} | [
{
"pp": "F : ℍ → ℂ\nhF : DifferentiableOn ℂ (F ∘ ↑ofComplex) {z | 0 < z.im}\nc : ℂ := (2 * ↑π * I)⁻¹\n⊢ DifferentiableOn ℂ (fun z ↦ c * deriv (F ∘ ↑ofComplex) z) upperHalfPlaneSet",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
"UpperHalfPlane.ofComplex",
"Eq.mpr",
"Norme... | [
"F : ℍ → ℂ\nhF : DifferentiableOn ℂ (F ∘ ↑ofComplex) {z | 0 < z.im}\nc : ℂ := (2 * ↑π * I)⁻¹\n⊢ DifferentiableOn ℂ (fun y ↦ I * ((↑π)⁻¹ * 2⁻¹) * deriv (F ∘ ↑ofComplex) y) upperHalfPlaneSet"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Modular | {
"line": 650,
"column": 4
} | {
"line": 652,
"column": 99
} | {
"line": 653,
"column": 6
} | [
{
"pp": "g : SL(2, ℤ)\nz : ℍ\nhz : z ∈ 𝒟\nhg : T ^ ↑g 0 0 • (1 +ᵥ ρ) ∈ 𝒟\nhg' : ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) g)) ↑z‖ ≤ 1\nhc : ↑g 1 0 = 1\nhd : ↑g 1 1 = -1\nthis✝² : ↑g 0 1 = -↑g 0 0 - 1\nhgeq : g = T ^ ↑g 0 0 * S * T⁻¹\nhnorm : normSq ↑z + (-2 * z.re + 1) ≤ 1\nthis✝¹ : normSq (↑... | [
"g : SL(2, ℤ)\nz : ℍ\nhz : z ∈ 𝒟\nhg : T ^ ↑g 0 0 • (1 +ᵥ ρ) ∈ 𝒟\nhg' : ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) g)) ↑z‖ ≤ 1\nhc : ↑g 1 0 = 1\nhd : ↑g 1 1 = -1\nthis✝² : ↑g 0 1 = -↑g 0 0 - 1\nhgeq : g = T ^ ↑g 0 0 * S * T⁻¹\nhnorm : normSq ↑z + (-2 * z.re + 1) ≤ 1\nthis✝¹ : normSq (↑z - 1) = nor... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Modular | {
"line": 659,
"column": 33
} | {
"line": 659,
"column": 44
} | {
"line": 659,
"column": 45
} | [
{
"pp": "g : SL(2, ℤ)\nz : ℍ\nhz : z ∈ 𝒟\nhg : g • z ∈ 𝒟\nthis : ∀ {g : SL(2, ℤ)} {z : ℍ}, z ∈ 𝒟 → g • z ∈ 𝒟 → z.im ≤ (g • z).im → (g • z).im = z.im\nhden : ¬z.im ≤ (g • z).im\n⊢ g⁻¹ • g • z ∈ 𝒟",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
"UpperHalfPlane.glAction",
"Eq... | [
"g : SL(2, ℤ)\nz : ℍ\nhz : z ∈ 𝒟\nhg : g • z ∈ 𝒟\nthis : ∀ {g : SL(2, ℤ)} {z : ℍ}, z ∈ 𝒟 → g • z ∈ 𝒟 → z.im ≤ (g • z).im → (g • z).im = z.im\nhden : ¬z.im ≤ (g • z).im\n⊢ z ∈ 𝒟"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Modular | {
"line": 659,
"column": 53
} | {
"line": 659,
"column": 64
} | {
"line": 659,
"column": 65
} | [
{
"pp": "g : SL(2, ℤ)\nz : ℍ\nhz : z ∈ 𝒟\nhg : g • z ∈ 𝒟\nthis : ∀ {g : SL(2, ℤ)} {z : ℍ}, z ∈ 𝒟 → g • z ∈ 𝒟 → z.im ≤ (g • z).im → (g • z).im = z.im\nhden : ¬z.im ≤ (g • z).im\n⊢ (g • z).im ≤ (g⁻¹ • g • z).im",
"ppTerm": "?m.56",
"assigned": true,
"usedConstants": [
"UpperHalfPlane.glActio... | [
"g : SL(2, ℤ)\nz : ℍ\nhz : z ∈ 𝒟\nhg : g • z ∈ 𝒟\nthis : ∀ {g : SL(2, ℤ)} {z : ℍ}, z ∈ 𝒟 → g • z ∈ 𝒟 → z.im ≤ (g • z).im → (g • z).im = z.im\nhden : ¬z.im ≤ (g • z).im\n⊢ (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) g) • z).im ≤ z.im"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.QExpansion | {
"line": 71,
"column": 4
} | {
"line": 71,
"column": 20
} | {
"line": 71,
"column": 21
} | [
{
"pp": "k l : ℕ\np : ℝ\nf : ℕ → ℂ\nhf : f =O[atTop] fun n ↦ ↑n ^ l\nh1 : (fun n ↦ (2 * ↑π * I * ↑n / ↑p) ^ k) = fun n ↦ (2 * ↑π * I / ↑p) ^ k * ↑n ^ k\n⊢ (fun n ↦ (2 * ↑π * I * ↑n / ↑p) ^ k) =O[atTop] fun n ↦ ↑n ^ k",
"ppTerm": "?m.148",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"R... | [
"k l : ℕ\np : ℝ\nf : ℕ → ℂ\nhf : f =O[atTop] fun n ↦ ↑n ^ l\nh1 : (fun n ↦ (2 * ↑π * I * ↑n / ↑p) ^ k) = fun n ↦ (2 * ↑π * I / ↑p) ^ k * ↑n ^ k\n⊢ (fun n ↦ (2 * ↑π * I / ↑p) ^ k * ↑n ^ k) =O[atTop] fun n ↦ ↑n ^ k"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Modular | {
"line": 663,
"column": 4
} | {
"line": 663,
"column": 15
} | {
"line": 663,
"column": 16
} | [
{
"pp": "case inr\ng✝ : SL(2, ℤ)\nz✝ : ℍ\ng : SL(2, ℤ)\nz : ℍ\nhz : z ∈ 𝒟\nhg : g • z ∈ 𝒟\nhden : z.im ≤ (g • z).im\nthis :\n ∀ {g : SL(2, ℤ)} {z : ℍ} {g : SL(2, ℤ)} {z : ℍ},\n z ∈ 𝒟 → g • z ∈ 𝒟 → z.im ≤ (g • z).im → 0 ≤ ↑g 1 0 → (g • z).im = z.im\nhc : ¬0 ≤ ↑g 1 0\n⊢ (g • z).im = z.im",
"ppTerm": "... | [
"case inr\ng✝ : SL(2, ℤ)\nz✝ : ℍ\ng : SL(2, ℤ)\nz : ℍ\nhz : z ∈ 𝒟\nhg : g • z ∈ 𝒟\nhden : z.im ≤ (g • z).im\nthis :\n ∀ {g : SL(2, ℤ)} {z : ℍ} {g : SL(2, ℤ)} {z : ℍ},\n z ∈ 𝒟 → g • z ∈ 𝒟 → z.im ≤ (g • z).im → 0 ≤ ↑g 1 0 → (g • z).im = z.im\nhc : ¬0 ≤ ↑g 1 0\n⊢ (toGL ((SpecialLinearGroup.map (Int.castRingHom... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Modular | {
"line": 663,
"column": 43
} | {
"line": 663,
"column": 54
} | {
"line": 663,
"column": 55
} | [
{
"pp": "g✝ : SL(2, ℤ)\nz✝ : ℍ\ng : SL(2, ℤ)\nz : ℍ\nhz : z ∈ 𝒟\nhg : g • z ∈ 𝒟\nhden : z.im ≤ (g • z).im\nthis :\n ∀ {g : SL(2, ℤ)} {z : ℍ} {g : SL(2, ℤ)} {z : ℍ},\n z ∈ 𝒟 → g • z ∈ 𝒟 → z.im ≤ (g • z).im → 0 ≤ ↑g 1 0 → (g • z).im = z.im\nhc : ¬0 ≤ ↑g 1 0\n⊢ -g • z ∈ 𝒟",
"ppTerm": "?m.116",
"as... | [
"g✝ : SL(2, ℤ)\nz✝ : ℍ\ng : SL(2, ℤ)\nz : ℍ\nhz : z ∈ 𝒟\nhg : g • z ∈ 𝒟\nhden : z.im ≤ (g • z).im\nthis :\n ∀ {g : SL(2, ℤ)} {z : ℍ} {g : SL(2, ℤ)} {z : ℍ},\n z ∈ 𝒟 → g • z ∈ 𝒟 → z.im ≤ (g • z).im → 0 ≤ ↑g 1 0 → (g • z).im = z.im\nhc : ¬0 ≤ ↑g 1 0\n⊢ toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) g) • z... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Modular | {
"line": 664,
"column": 10
} | {
"line": 664,
"column": 21
} | {
"line": 664,
"column": 22
} | [
{
"pp": "g✝ : SL(2, ℤ)\nz✝ : ℍ\ng : SL(2, ℤ)\nz : ℍ\nhz : z ∈ 𝒟\nhg : g • z ∈ 𝒟\nhden : z.im ≤ (g • z).im\nthis :\n ∀ {g : SL(2, ℤ)} {z : ℍ} {g : SL(2, ℤ)} {z : ℍ},\n z ∈ 𝒟 → g • z ∈ 𝒟 → z.im ≤ (g • z).im → 0 ≤ ↑g 1 0 → (g • z).im = z.im\nhc : ¬0 ≤ ↑g 1 0\n⊢ z.im ≤ (-g • z).im",
"ppTerm": "?m.117",
... | [
"g✝ : SL(2, ℤ)\nz✝ : ℍ\ng : SL(2, ℤ)\nz : ℍ\nhz : z ∈ 𝒟\nhg : g • z ∈ 𝒟\nhden : z.im ≤ (g • z).im\nthis :\n ∀ {g : SL(2, ℤ)} {z : ℍ} {g : SL(2, ℤ)} {z : ℍ},\n z ∈ 𝒟 → g • z ∈ 𝒟 → z.im ≤ (g • z).im → 0 ≤ ↑g 1 0 → (g • z).im = z.im\nhc : ¬0 ≤ ↑g 1 0\n⊢ z.im ≤ (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Modular | {
"line": 664,
"column": 32
} | {
"line": 664,
"column": 43
} | {
"line": 664,
"column": 44
} | [
{
"pp": "g✝ : SL(2, ℤ)\nz✝ : ℍ\ng : SL(2, ℤ)\nz : ℍ\nhz : z ∈ 𝒟\nhg : g • z ∈ 𝒟\nhden : z.im ≤ (g • z).im\nthis :\n ∀ {g : SL(2, ℤ)} {z : ℍ} {g : SL(2, ℤ)} {z : ℍ},\n z ∈ 𝒟 → g • z ∈ 𝒟 → z.im ≤ (g • z).im → 0 ≤ ↑g 1 0 → (g • z).im = z.im\nhc : ¬0 ≤ ↑g 1 0\n⊢ 0 ≤ ↑(-g) 1 0",
"ppTerm": "?m.118",
"... | [
"g✝ : SL(2, ℤ)\nz✝ : ℍ\ng : SL(2, ℤ)\nz : ℍ\nhz : z ∈ 𝒟\nhg : g • z ∈ 𝒟\nhden : z.im ≤ (g • z).im\nthis :\n ∀ {g : SL(2, ℤ)} {z : ℍ} {g : SL(2, ℤ)} {z : ℍ},\n z ∈ 𝒟 → g • z ∈ 𝒟 → z.im ≤ (g • z).im → 0 ≤ ↑g 1 0 → (g • z).im = z.im\nhc : ¬0 ≤ ↑g 1 0\n⊢ ↑g 1 0 ≤ 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.Derivative | {
"line": 199,
"column": 4
} | {
"line": 199,
"column": 15
} | {
"line": 199,
"column": 16
} | [
{
"pp": "k : ℤ\nF : ℍ → ℂ\nhF : MDiff F\ng : GL (Fin 2) ℝ\nhg : 0 < (↑g).det\nhdet : ↑(Matrix.GeneralLinearGroup.det g) = (↑g).det\nhdetℂ : ↑(↑g).det ≠ 0\nhσ : ∀ (x : ℂ), (σ g) x = x\nz : ℍ\nhz : denom g ↑z ≠ 0\nh_smul : HasDerivAt (fun w ↦ ↑(g • ↑ofComplex w)) (↑(↑g).det / denom g ↑z ^ 2) ↑z\nh_F : HasDerivAt ... | [
"k : ℤ\nF : ℍ → ℂ\nhF : MDiff F\ng : GL (Fin 2) ℝ\nhg : 0 < (↑g).det\nhdet : ↑(Matrix.GeneralLinearGroup.det g) = (↑g).det\nhdetℂ : ↑(↑g).det ≠ 0\nhσ : ∀ (x : ℂ), (σ g) x = x\nz : ℍ\nhz : denom g ↑z ≠ 0\nh_smul : HasDerivAt (fun w ↦ ↑(g • ↑ofComplex w)) (↑(↑g).det / denom g ↑z ^ 2) ↑z\nh_F : HasDerivAt (F ∘ ↑ofComp... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.QExpansion | {
"line": 95,
"column": 4
} | {
"line": 95,
"column": 15
} | {
"line": 95,
"column": 16
} | [
{
"pp": "k l : ℕ\nf : ℕ → ℂ\np : ℝ\nhp : 0 < p\nhf : f =O[atTop] fun n ↦ ↑n ^ l\nK : Set ℂ\nhK : K ⊆ ℍₒ\nhKc : IsCompact K\nthis : CompactSpace ↑K\nc : C(↑K, ℂ) := { toFun := fun r ↦ cexp (2 * ↑π * I * ↑r / ↑p), continuous_toFun := ⋯ }\nr : ℝ := ‖mkOfCompact c‖\nx : ↑K\nh1 : cexp (2 * ↑π * I * (↑x / ↑p)) = cexp... | [
"k l : ℕ\nf : ℕ → ℂ\np : ℝ\nhp : 0 < p\nhf : f =O[atTop] fun n ↦ ↑n ^ l\nK : Set ℂ\nhK : K ⊆ ℍₒ\nhKc : IsCompact K\nthis : CompactSpace ↑K\nc : C(↑K, ℂ) := { toFun := fun r ↦ cexp (2 * ↑π * I * ↑r / ↑p), continuous_toFun := ⋯ }\nr : ℝ := ‖mkOfCompact c‖\nx : ↑K\nh1 : cexp (2 * ↑π * I * (↑x / ↑p)) = cexp (2 * ↑π * I... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.QExpansion | {
"line": 96,
"column": 16
} | {
"line": 96,
"column": 27
} | {
"line": 96,
"column": 28
} | [
{
"pp": "k l : ℕ\nf : ℕ → ℂ\np : ℝ\nhp : 0 < p\nhf : f =O[atTop] fun n ↦ ↑n ^ l\nK : Set ℂ\nhK : K ⊆ ℍₒ\nhKc : IsCompact K\nthis : CompactSpace ↑K\nc : C(↑K, ℂ) := { toFun := fun r ↦ cexp (2 * ↑π * I * ↑r / ↑p), continuous_toFun := ⋯ }\nr : ℝ := ‖mkOfCompact c‖\nhr : ‖r‖ < 1\n⊢ Summable ?m.229",
"ppTerm": "... | [
"k l : ℕ\nf : ℕ → ℂ\np : ℝ\nhp : 0 < p\nhf : f =O[atTop] fun n ↦ ↑n ^ l\nK : Set ℂ\nhK : K ⊆ ℍₒ\nhKc : IsCompact K\nthis : CompactSpace ↑K\nc : C(↑K, ℂ) := { toFun := fun r ↦ cexp (2 * ↑π * I * ↑r / ↑p), continuous_toFun := ⋯ }\nr : ℝ := ‖mkOfCompact c‖\nhr : ‖r‖ < 1\n⊢ Summable ?m.229"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.E2.Summable | {
"line": 67,
"column": 21
} | {
"line": 67,
"column": 71
} | {
"line": 67,
"column": 72
} | [
{
"pp": "z : ℍ\nN a : ℕ\n⊢ 0 < ((↑a + 1) * ↑z).im",
"ppTerm": "?m.183",
"assigned": true,
"usedConstants": [
"Complex.mul_im",
"Real.instIsOrderedRing",
"Eq.mpr",
"Real.partialOrder",
"Real",
"Preorder.toLT",
"HMul.hMul",
"UpperHalfPlane.coe",
"M... | [
"z : ℍ\nN a : ℕ\n⊢ 0 < z.im"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.QExpansion | {
"line": 116,
"column": 2
} | {
"line": 116,
"column": 13
} | {
"line": 116,
"column": 14
} | [
{
"pp": "k : ℕ\nh0 : (fun n ↦ 1) =O[atTop] fun n ↦ ↑n ^ 1\n⊢ SummableLocallyUniformlyOn (fun n ↦ iteratedDerivWithin k (fun z ↦ cexp (2 * ↑π * I * z) ^ n) ℍₒ) ℍₒ",
"ppTerm": "?m.79",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"k : ℕ\nh0 : (fun n ↦ 1) =O[atTop] fun n ↦ ↑n ^ 1\n⊢ SummableLocallyUniformlyOn (fun n ↦ iteratedDerivWithin k (fun z ↦ cexp (2 * ↑π * I * z) ^ n) ℍₒ) ℍₒ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.QExpansion | {
"line": 130,
"column": 62
} | {
"line": 130,
"column": 73
} | {
"line": 130,
"column": 74
} | [
{
"pp": "k : ℕ\nz : ℍ\n⊢ ↑z ∈ ℍₒ",
"ppTerm": "?m.72",
"assigned": true,
"usedConstants": [
"UpperHalfPlane.coe",
"Membership.mem",
"id",
"UpperHalfPlane.upperHalfPlaneSet",
"Complex",
"Set.instMembership",
"Set"
],
"usedFVars": [
"z"
],
... | [
"k : ℕ\nz : ℍ\n⊢ 0 < z.im"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.Derivative | {
"line": 221,
"column": 2
} | {
"line": 221,
"column": 36
} | {
"line": 221,
"column": 37
} | [
{
"pp": "k : ℤ\nF : ℍ → ℂ\nhF : MDiff F\nγ : SL(2, ℤ)\nhdet : (↑(Matrix.SpecialLinearGroup.toGL ((Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)) γ))).det = 1\nz : ℍ\nthis :\n D (F ∣[k] Matrix.SpecialLinearGroup.toGL ((Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)) γ)) z =\n (↑1)⁻¹ * (D F ∣[k + 2] Ma... | [
"k : ℤ\nF : ℍ → ℂ\nhF : MDiff F\nγ : SL(2, ℤ)\nhdet : (↑(Matrix.SpecialLinearGroup.toGL ((Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)) γ))).det = 1\nz : ℍ\nthis :\n D (F ∣[k] Matrix.SpecialLinearGroup.toGL ((Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)) γ)) z =\n (↑1)⁻¹ * (D F ∣[k + 2] Matrix.Special... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.E2.Summable | {
"line": 121,
"column": 4
} | {
"line": 121,
"column": 68
} | {
"line": 121,
"column": 69
} | [
{
"pp": "τ : ℍ\n⊢ HasSum (fun m ↦ ?m.5 m • Function.Periodic.qParam 1 ↑τ ^ m) (E2 τ)",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"instHSMul",
"Function.Periodic.qParam",
"instHDiv",
"inst... | [
"τ : ℍ\n⊢ HasSum (fun m ↦ ?m.5 m • cexp (2 * ↑π * I * ↑τ) ^ m) (E2 τ)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.E2.Summable | {
"line": 148,
"column": 4
} | {
"line": 148,
"column": 15
} | {
"line": 148,
"column": 16
} | [
{
"pp": "case ha\nz : ℍ\na b m : ℤ\nhm : m ≠ 0 ∨ a ≠ 0 ∧ b ≠ 0\n⊢ ↑m * ↑z + ↑a ≠ 0",
"ppTerm": "?ha",
"assigned": true,
"usedConstants": [
"Int.cast",
"HMul.hMul",
"UpperHalfPlane.coe",
"Field.toDivisionRing",
"Complex.instMul",
"DivisionRing.toDivisionSemiring",
... | [
"case ha\nz : ℍ\na b m : ℤ\nhm : m ≠ 0 ∨ a ≠ 0 ∧ b ≠ 0\n⊢ ¬↑m * ↑z + ↑a = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.E2.Summable | {
"line": 149,
"column": 4
} | {
"line": 149,
"column": 15
} | {
"line": 149,
"column": 16
} | [
{
"pp": "case hb\nz : ℍ\na b m : ℤ\nhm : m ≠ 0 ∨ a ≠ 0 ∧ b ≠ 0\n⊢ ↑m * ↑z + ↑b ≠ 0",
"ppTerm": "?hb",
"assigned": true,
"usedConstants": [
"Int.cast",
"HMul.hMul",
"UpperHalfPlane.coe",
"Field.toDivisionRing",
"Complex.instMul",
"DivisionRing.toDivisionSemiring",
... | [
"case hb\nz : ℍ\na b m : ℤ\nhm : m ≠ 0 ∨ a ≠ 0 ∧ b ≠ 0\n⊢ ¬↑m * ↑z + ↑b = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.QExpansion | {
"line": 153,
"column": 40
} | {
"line": 153,
"column": 51
} | {
"line": 153,
"column": 52
} | [
{
"pp": "k : ℕ\nhk : 1 ≤ k\nz : ℍ\n⊢ ↑z ∈ ℍₒ",
"ppTerm": "?m.243",
"assigned": true,
"usedConstants": [
"UpperHalfPlane.coe",
"Membership.mem",
"id",
"UpperHalfPlane.upperHalfPlaneSet",
"Complex",
"Set.instMembership",
"Set"
],
"usedFVars": [
"... | [
"k : ℕ\nhk : 1 ≤ k\nz : ℍ\n⊢ 0 < z.im"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.E2.Summable | {
"line": 163,
"column": 2
} | {
"line": 163,
"column": 35
} | {
"line": 164,
"column": 4
} | [
{
"pp": "z : ℍ\nm : ℤ\nthis : Summable fun x ↦ ((↑m * ↑z + ↑↑x + 1) * (↑m * ↑z + ↑↑x))⁻¹\nb : { x // x ∉ {0, -1} }\n⊢ ((↑m * ↑z + ↑↑b + 1) * (↑m * ↑z + ↑↑b))⁻¹ = 1 / (↑m * ↑z + ↑↑b) - 1 / (↑m * ↑z + ↑↑b + 1)",
"ppTerm": "?m.99",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.t... | [
"z : ℍ\nm : ℤ\nthis : Summable fun x ↦ ((↑m * ↑z + ↑↑x + 1) * (↑m * ↑z + ↑↑x))⁻¹\nb : { x // x ∉ {0, -1} }\n⊢ (↑z * ↑m + ↑↑b)⁻¹ * (↑z * ↑m + (↑↑b + 1))⁻¹ = (↑z * ↑m + ↑↑b)⁻¹ - (↑z * ↑m + (↑↑b + 1))⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.QExpansion | {
"line": 155,
"column": 58
} | {
"line": 155,
"column": 69
} | {
"line": 155,
"column": 70
} | [
{
"pp": "k : ℕ\nhk : 1 ≤ k\nz : ℍ\n⊢ ↑z ∈ ℍₒ",
"ppTerm": "?m.281",
"assigned": true,
"usedConstants": [
"UpperHalfPlane.coe",
"Membership.mem",
"id",
"UpperHalfPlane.upperHalfPlaneSet",
"Complex",
"Set.instMembership",
"Set"
],
"usedFVars": [
"... | [
"k : ℕ\nhk : 1 ≤ k\nz : ℍ\n⊢ 0 < z.im"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.E2.Summable | {
"line": 205,
"column": 2
} | {
"line": 205,
"column": 13
} | {
"line": 205,
"column": 14
} | [
{
"pp": "z : ℍ\nm : ℤ\n⊢ Tendsto (fun N ↦ 1 / (↑m * ↑z - ↑N) - 1 / (↑m * ↑z + ↑N)) atTop (𝓝 0)",
"ppTerm": "?m.64",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"DivInvMonoid.toInv",
"instHDiv",
"HMul.hMul",
... | [
"z : ℍ\nm : ℤ\n⊢ Tendsto (fun N ↦ (↑m * ↑z - ↑N)⁻¹ - (↑m * ↑z + ↑N)⁻¹) atTop (𝓝 0)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.Discriminant | {
"line": 124,
"column": 2
} | {
"line": 124,
"column": 28
} | {
"line": 124,
"column": 29
} | [
{
"pp": "z : ℍ\n⊢ Δ z ≠ 0",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"UpperHalfPlane.coe",
"congrArg",
"NormedDivisionRing.toNormMulClass",
"Nat.instAtLeastTwoHAddOfNat",
"Complex.instNormedField",
"Complex.instZero",
... | [
"z : ℍ\n⊢ ¬η ↑z = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.E2.Summable | {
"line": 231,
"column": 33
} | {
"line": 231,
"column": 44
} | {
"line": 231,
"column": 45
} | [
{
"pp": "z : ℍ\nd : ℕ+\n⊢ ?m.161 ∈ integerComplement",
"ppTerm": "?m.162",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"z : ℍ\nd : ℕ+\n⊢ ?m.161 ∈ integerComplement"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.Discriminant | {
"line": 138,
"column": 41
} | {
"line": 138,
"column": 66
} | {
"line": 138,
"column": 67
} | [
{
"pp": "z : ℂ\nhz : z ≠ 0\nh : Set.EqOn (η ∘ fun z ↦ -1 / z) (z • (sqrt * η)) upperHalfPlaneSet\n⊢ η I = z * I.sqrt * η I",
"ppTerm": "?m.53",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"z : ℂ\nhz : z ≠ 0\nh : Set.EqOn (η ∘ fun z ↦ -1 / z) (z • (sqrt * η)) upperHalfPlaneSet\n⊢ η I = z * I.sqrt * η I"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.Discriminant | {
"line": 136,
"column": 48
} | {
"line": 139,
"column": 51
} | {
"line": 141,
"column": 0
} | [
{
"pp": "⊢ Set.EqOn (η ∘ fun x ↦ -1 / x) (I.sqrt⁻¹ • (sqrt * η)) upperHalfPlaneSet",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Real.partialOrder",
"Semigroup.toMul",
"Real",
"instHSMul",
"Preorder.toLT",
"instHDiv",
"NonUnitalCommRing.toNonUni... | [] | by
obtain ⟨z, hz, h⟩ := eta_comp_eqOn_const_mul_csqrt_eta
have h3 : η I = z * sqrt I * η I := by simpa [← mul_assoc] using h (show I ∈ _ by simp)
grind [sqrt, eta_ne_zero (show 0 < I.im by simp)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.ModularForms.Discriminant | {
"line": 149,
"column": 4
} | {
"line": 149,
"column": 25
} | {
"line": 149,
"column": 26
} | [
{
"pp": "z : ℍ\n⊢ η (-(↑z)⁻¹) = I.sqrt⁻¹ * ((↑z).sqrt * η ↑z)",
"ppTerm": "?m.90",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"z : ℍ\n⊢ η (-(↑z)⁻¹) = I.sqrt⁻¹ * ((↑z).sqrt * η ↑z)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.QExpansion | {
"line": 168,
"column": 2
} | {
"line": 168,
"column": 40
} | {
"line": 169,
"column": 4
} | [
{
"pp": "case e_f\nk : ℕ\nhk : 1 ≤ k\nz : ℍ\nthis✝ :\n iteratedDerivWithin k (fun z ↦ ↑π * I - 2 * ↑π * I * ∑' (n : ℕ), cexp (2 * ↑π * I * z) ^ n) ℍₒ ↑z =\n -(2 * ↑π * I) * ∑' (n : ℕ), iteratedDerivWithin k (fun s ↦ cexp (2 * ↑π * I * s) ^ n) ℍₒ ↑z\nh :\n -(2 * ↑π * I * (2 * ↑π * I) ^ k) * ∑' (n : ℕ), ↑n ^... | [
"case e_f\nk : ℕ\nhk : 1 ≤ k\nz : ℍ\nthis✝ :\n iteratedDerivWithin k (fun z ↦ ↑π * I - 2 * ↑π * I * ∑' (n : ℕ), cexp (2 * ↑π * I * z) ^ n) ℍₒ ↑z =\n -(2 * ↑π * I) * ∑' (n : ℕ), iteratedDerivWithin k (fun s ↦ cexp (2 * ↑π * I * s) ^ n) ℍₒ ↑z\nh :\n -(2 * ↑π * I * (2 * ↑π * I) ^ k) * ∑' (n : ℕ), ↑n ^ k * cexp (2... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.QExpansion | {
"line": 180,
"column": 66
} | {
"line": 180,
"column": 77
} | {
"line": 180,
"column": 78
} | [
{
"pp": "k : ℕ\nhk : 1 ≤ k\nz : ℍ\n⊢ ↑z ∈ ℍₒ",
"ppTerm": "?m.201",
"assigned": true,
"usedConstants": [
"UpperHalfPlane.coe",
"Membership.mem",
"id",
"UpperHalfPlane.upperHalfPlaneSet",
"Complex",
"Set.instMembership",
"Set"
],
"usedFVars": [
"... | [
"k : ℕ\nhk : 1 ≤ k\nz : ℍ\n⊢ 0 < z.im"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.QExpansion | {
"line": 181,
"column": 52
} | {
"line": 181,
"column": 63
} | {
"line": 181,
"column": 64
} | [
{
"pp": "k : ℕ\nhk : 1 ≤ k\nz : ℍ\nx : ℂ\nhx : x ∈ ℍₒ\n⊢ ↑π * (↑π * x).cot = ↑π * I - 2 * ↑π * I * ∑' (n : ℕ), cexp (2 * ↑π * I * x) ^ n",
"ppTerm": "?m.221",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"k : ℕ\nhk : 1 ≤ k\nz : ℍ\nx : ℂ\nhx : x ∈ ℍₒ\n⊢ ↑π * (↑π * x).cot = ↑π * I - 2 * ↑π * I * ∑' (n : ℕ), cexp (2 * ↑π * I * x) ^ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.QExpansion | {
"line": 182,
"column": 10
} | {
"line": 182,
"column": 21
} | {
"line": 182,
"column": 22
} | [
{
"pp": "k : ℕ\nhk : 1 ≤ k\nz : ℍ\n⊢ ↑z ∈ ℍₒ",
"ppTerm": "?m.224",
"assigned": true,
"usedConstants": [
"UpperHalfPlane.coe",
"Membership.mem",
"id",
"UpperHalfPlane.upperHalfPlaneSet",
"Complex",
"Set.instMembership",
"Set"
],
"usedFVars": [
"... | [
"k : ℕ\nhk : 1 ≤ k\nz : ℍ\n⊢ 0 < z.im"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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