module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
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.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.TsumDivisorsAntidiagonal | {
"line": 115,
"column": 2
} | {
"line": 118,
"column": 11
} | {
"line": 118,
"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": 123,
"column": 8
} | {
"line": 123,
"column": 19
} | {
"line": 123,
"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.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.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.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": 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.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": 211,
"column": 8
} | {
"line": 211,
"column": 45
} | {
"line": 211,
"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.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.Modular | {
"line": 660,
"column": 33
} | {
"line": 660,
"column": 44
} | {
"line": 660,
"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": 660,
"column": 53
} | {
"line": 660,
"column": 64
} | {
"line": 660,
"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.Modular | {
"line": 664,
"column": 4
} | {
"line": 664,
"column": 15
} | {
"line": 664,
"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": 664,
"column": 43
} | {
"line": 664,
"column": 54
} | {
"line": 664,
"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": 665,
"column": 10
} | {
"line": 665,
"column": 21
} | {
"line": 665,
"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": 665,
"column": 32
} | {
"line": 665,
"column": 43
} | {
"line": 665,
"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": 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.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.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.ModularForms.Discriminant | {
"line": 139,
"column": 41
} | {
"line": 139,
"column": 66
} | {
"line": 139,
"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": 137,
"column": 48
} | {
"line": 140,
"column": 51
} | {
"line": 142,
"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": 151,
"column": 4
} | {
"line": 151,
"column": 25
} | {
"line": 151,
"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.Discriminant | {
"line": 163,
"column": 10
} | {
"line": 163,
"column": 38
} | {
"line": 163,
"column": 39
} | [
{
"pp": "this :\n (Summable fun n ↦ (1 / 2) ^ (n + 1)) →\n (∀ (k : ℕ), Tendsto (fun x ↦ -x ^ (k + 1)) (𝓝 0) (𝓝 0)) →\n (∀ᶠ (n : ℂ) in 𝓝 0, ∀ (k : ℕ), ‖n‖ ^ (k + 1) ≤ (1 / 2) ^ (k + 1)) →\n Tendsto (fun n ↦ ∏' (k : ℕ), (1 + -n ^ (k + 1))) (𝓝 0) (𝓝 1)\n⊢ Summable fun n ↦ (1 / 2) ^ (n + 1)",
... | [
"this :\n (Summable fun n ↦ (1 / 2) ^ (n + 1)) →\n (∀ (k : ℕ), Tendsto (fun x ↦ -x ^ (k + 1)) (𝓝 0) (𝓝 0)) →\n (∀ᶠ (n : ℂ) in 𝓝 0, ∀ (k : ℕ), ‖n‖ ^ (k + 1) ≤ (1 / 2) ^ (k + 1)) →\n Tendsto (fun n ↦ ∏' (k : ℕ), (1 + -n ^ (k + 1))) (𝓝 0) (𝓝 1)\n⊢ Summable fun n ↦ 1 / 2 * (1 / 2) ^ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.Discriminant | {
"line": 164,
"column": 18
} | {
"line": 164,
"column": 29
} | {
"line": 164,
"column": 30
} | [
{
"pp": "this :\n (Summable fun n ↦ (1 / 2) ^ (n + 1)) →\n (∀ (k : ℕ), Tendsto (fun x ↦ -x ^ (k + 1)) (𝓝 0) (𝓝 0)) →\n (∀ᶠ (n : ℂ) in 𝓝 0, ∀ (k : ℕ), ‖n‖ ^ (k + 1) ≤ (1 / 2) ^ (k + 1)) →\n Tendsto (fun n ↦ ∏' (k : ℕ), (1 + -n ^ (k + 1))) (𝓝 0) (𝓝 1)\nk : ℕ\n⊢ Tendsto (fun x ↦ -x ^ (k + 1)) ... | [
"this :\n (Summable fun n ↦ (1 / 2) ^ (n + 1)) →\n (∀ (k : ℕ), Tendsto (fun x ↦ -x ^ (k + 1)) (𝓝 0) (𝓝 0)) →\n (∀ᶠ (n : ℂ) in 𝓝 0, ∀ (k : ℕ), ‖n‖ ^ (k + 1) ≤ (1 / 2) ^ (k + 1)) →\n Tendsto (fun n ↦ ∏' (k : ℕ), (1 + -n ^ (k + 1))) (𝓝 0) (𝓝 1)\nk : ℕ\n⊢ Tendsto (fun x ↦ -x ^ (k + 1)) (𝓝 0) (𝓝 0... | 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.Discriminant | {
"line": 206,
"column": 10
} | {
"line": 206,
"column": 39
} | {
"line": 206,
"column": 40
} | [
{
"pp": "q : ℂ\nhq : q ∈ Metric.ball 0 1\nhq0 : ¬q = 0\n⊢ ‖q‖ < 1",
"ppTerm": "?m.138",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"q : ℂ\nhq : q ∈ Metric.ball 0 1\nhq0 : ¬q = 0\n⊢ ‖q‖ < 1"
] | 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.Discriminant | {
"line": 263,
"column": 8
} | {
"line": 263,
"column": 19
} | {
"line": 263,
"column": 20
} | [
{
"pp": "k : ℤ\nf : CuspForm (Matrix.SpecialLinearGroup.mapGL ℝ).range k\n⊢ (fun τ ↦ rexp (-2 * π * τ.im / 1)) =O[atImInfty] Δ",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"instHDiv",
"NonUnitalCommRi... | [
"k : ℤ\nf : CuspForm (Matrix.SpecialLinearGroup.mapGL ℝ).range k\n⊢ (fun τ ↦ rexp (-(2 * π * τ.im))) =O[atImInfty] Δ"
] | 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.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.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.LevelOne.DimensionFormula | {
"line": 121,
"column": 2
} | {
"line": 121,
"column": 13
} | {
"line": 121,
"column": 14
} | [
{
"pp": "hi : 0 < 1\n⊢ (PowerSeries.coeff 0) (qExpansion 1 Δ) = 0",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Semiring.toModule",
"UpperHalfPlane.qExpansion",
"congrArg",
"LinearMap.instFunLike",
"RingHom",
"id",
... | [
"hi : 0 < 1\n⊢ PowerSeries.constantCoeff (qExpansion 1 Δ) = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Modular | {
"line": 708,
"column": 4
} | {
"line": 708,
"column": 44
} | {
"line": 708,
"column": 45
} | [
{
"pp": "case inr\ng : SL(2, ℤ)\nz : ℍ\nhz : z ∈ 𝒟\nhg : g • z ∈ 𝒟\nhim : (g • z).im = z.im\nthis :\n ∀ {g : SL(2, ℤ)} {z : ℍ},\n z ∈ 𝒟 →\n g • z ∈ 𝒟 →\n (g • z).im = z.im →\n 0 ≤ ↑g 1 0 →\n (g = 1 ∨ g = -1) ∨\n (g = T ∨ g = -T) ∧ z.re = -1 / 2 ∨\n ... | [
"case inr\ng : SL(2, ℤ)\nz : ℍ\nhz : z ∈ 𝒟\nhg : g • z ∈ 𝒟\nhim : (g • z).im = z.im\nthis :\n ∀ {g : SL(2, ℤ)} {z : ℍ},\n z ∈ 𝒟 →\n g • z ∈ 𝒟 →\n (g • z).im = z.im →\n 0 ≤ ↑g 1 0 →\n (g = 1 ∨ g = -1) ∨\n (g = T ∨ g = -T) ∧ z.re = -1 / 2 ∨\n (g = T⁻... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Modular | {
"line": 708,
"column": 65
} | {
"line": 708,
"column": 76
} | {
"line": 708,
"column": 77
} | [
{
"pp": "g : SL(2, ℤ)\nz : ℍ\nhz : z ∈ 𝒟\nhg : g • z ∈ 𝒟\nhim : (g • z).im = z.im\nthis :\n ∀ {g : SL(2, ℤ)} {z : ℍ},\n z ∈ 𝒟 →\n g • z ∈ 𝒟 →\n (g • z).im = z.im →\n 0 ≤ ↑g 1 0 →\n (g = 1 ∨ g = -1) ∨\n (g = T ∨ g = -T) ∧ z.re = -1 / 2 ∨\n (g = ... | [
"g : SL(2, ℤ)\nz : ℍ\nhz : z ∈ 𝒟\nhg : g • z ∈ 𝒟\nhim : (g • z).im = z.im\nthis :\n ∀ {g : SL(2, ℤ)} {z : ℍ},\n z ∈ 𝒟 →\n g • z ∈ 𝒟 →\n (g • z).im = z.im →\n 0 ≤ ↑g 1 0 →\n (g = 1 ∨ g = -1) ∨\n (g = T ∨ g = -T) ∧ z.re = -1 / 2 ∨\n (g = T⁻¹ ∨ g = -T... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Modular | {
"line": 709,
"column": 10
} | {
"line": 709,
"column": 21
} | {
"line": 709,
"column": 22
} | [
{
"pp": "g : SL(2, ℤ)\nz : ℍ\nhz : z ∈ 𝒟\nhg : g • z ∈ 𝒟\nhim : (g • z).im = z.im\nthis :\n ∀ {g : SL(2, ℤ)} {z : ℍ},\n z ∈ 𝒟 →\n g • z ∈ 𝒟 →\n (g • z).im = z.im →\n 0 ≤ ↑g 1 0 →\n (g = 1 ∨ g = -1) ∨\n (g = T ∨ g = -T) ∧ z.re = -1 / 2 ∨\n (g = ... | [
"g : SL(2, ℤ)\nz : ℍ\nhz : z ∈ 𝒟\nhg : g • z ∈ 𝒟\nhim : (g • z).im = z.im\nthis :\n ∀ {g : SL(2, ℤ)} {z : ℍ},\n z ∈ 𝒟 →\n g • z ∈ 𝒟 →\n (g • z).im = z.im →\n 0 ≤ ↑g 1 0 →\n (g = 1 ∨ g = -1) ∨\n (g = T ∨ g = -T) ∧ z.re = -1 / 2 ∨\n (g = T⁻¹ ∨ g = -T... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Modular | {
"line": 709,
"column": 31
} | {
"line": 709,
"column": 42
} | {
"line": 709,
"column": 43
} | [
{
"pp": "g : SL(2, ℤ)\nz : ℍ\nhz : z ∈ 𝒟\nhg : g • z ∈ 𝒟\nhim : (g • z).im = z.im\nthis :\n ∀ {g : SL(2, ℤ)} {z : ℍ},\n z ∈ 𝒟 →\n g • z ∈ 𝒟 →\n (g • z).im = z.im →\n 0 ≤ ↑g 1 0 →\n (g = 1 ∨ g = -1) ∨\n (g = T ∨ g = -T) ∧ z.re = -1 / 2 ∨\n (g = ... | [
"g : SL(2, ℤ)\nz : ℍ\nhz : z ∈ 𝒟\nhg : g • z ∈ 𝒟\nhim : (g • z).im = z.im\nthis :\n ∀ {g : SL(2, ℤ)} {z : ℍ},\n z ∈ 𝒟 →\n g • z ∈ 𝒟 →\n (g • z).im = z.im →\n 0 ≤ ↑g 1 0 →\n (g = 1 ∨ g = -1) ∨\n (g = T ∨ g = -T) ∧ z.re = -1 / 2 ∨\n (g = T⁻¹ ∨ g = -T... | 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 |
Mathlib.NumberTheory.ModularForms.LevelOne.DimensionFormula | {
"line": 244,
"column": 2
} | {
"line": 245,
"column": 9
} | {
"line": 245,
"column": 10
} | [
{
"pp": "⊢ Module.rank ℂ (ModularForm (Matrix.SpecialLinearGroup.mapGL ℝ).range 2) = 0",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"ModularForm",
"Eq.mpr",
"Subgroup.instHasDetOneRangeSpecialLinearGroupGeneralLinearGroupMapGL",
"MonoidHom.range",
"Real",
... | [
"⊢ ∀ (x : ModularForm (Matrix.SpecialLinearGroup.mapGL ℝ).range 2), x = 0"
] | 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.LevelOne.DimensionFormula | {
"line": 258,
"column": 6
} | {
"line": 258,
"column": 29
} | {
"line": 258,
"column": 30
} | [
{
"pp": "case h.inl.«2»\nk : ℕ\nihn :\n ∀ m < 2,\n Even m →\n Module.rank ℂ (ModularForm (Matrix.SpecialLinearGroup.mapGL ℝ).range ↑m) =\n ↑(if m ≡ 2 [MOD 12] then m / 12 else m / 12 + 1)\nhk2 : Even 2\nhk : 2 < 3\n⊢ Module.rank ℂ (ModularForm (Matrix.SpecialLinearGroup.mapGL ℝ).range ↑2) =\n ... | [
"case h.inl.«2»\nk : ℕ\nihn :\n ∀ m < 2,\n Even m →\n Module.rank ℂ (ModularForm (Matrix.SpecialLinearGroup.mapGL ℝ).range ↑m) =\n ↑(if m ≡ 2 [MOD 12] then m / 12 else m / 12 + 1)\nhk2 : Even 2\nhk : 2 < 3\n⊢ Module.rank ℂ (ModularForm (Matrix.SpecialLinearGroup.mapGL ℝ).range 2) = 0"
] | 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": 176,
"column": 50
} | {
"line": 189,
"column": 20
} | {
"line": 191,
"column": 0
} | [
{
"pp": "k : ℕ\nhk : 1 ≤ k\nz : ℍ\n⊢ ∑' (n : ℤ), 1 / (↑z + ↑n) ^ (k + 1) = (-2 * ↑π * I) ^ (k + 1) / ↑k ! * ∑' (n : ℕ), ↑n ^ k * cexp (2 * ↑π * I * ↑z) ^ n",
"ppTerm": "?m.93",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Iff.mpr",
"one_pow",
... | [] | by
have : (-1) ^ k * k ! * ∑' n : ℤ, 1 / ((z : ℂ) + n) ^ (k + 1) =
-(2 * π * I) ^ (k + 1) * ∑' n : ℕ, n ^ k * cexp (2 * π * I * z) ^ n := by
rw [← iteratedDerivWithin_tsum_exp_aux_eq hk z,
← iteratedDerivWithin_cot_pi_mul_eq_mul_tsum_div_pow hk (by simpa using z.2)]
exact iteratedDerivWithin_congr (... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.QExpansion | {
"line": 194,
"column": 4
} | {
"line": 194,
"column": 15
} | {
"line": 194,
"column": 16
} | [
{
"pp": "k : ℕ\ne : ℕ+\nz : ℍ\n⊢ 0 < (↑↑e * ↑z).im",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"PNat.val",
"Complex.mul_im",
"Real.instIsOrderedRing",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.partialOrder",
"Real",
"Preo... | [
"k : ℕ\ne : ℕ+\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.Modular | {
"line": 716,
"column": 4
} | {
"line": 719,
"column": 53
} | {
"line": 721,
"column": 0
} | [
{
"pp": "case inr\ng✝ : SL(2, ℤ)\nz✝ : ℍ\ng : SL(2, ℤ)\nz : ℍ\nhz : z ∈ 𝒟\nhg : g • z ∈ 𝒟\nhim : ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) g)) ↑z‖ = 1\nhc✝ : 0 ≤ ↑g 1 0\nhc : ↑g 1 0 = 1\n⊢ (g = 1 ∨ g = -1) ∨\n (g = T ∨ g = -T) ∧ z.re = -1 / 2 ∨\n (g = T⁻¹ ∨ g = -T⁻¹) ∧ z.re = 1 / 2 ∨\... | [] | rcases Int.abs_le_one_iff.mp (cases_d_of_c_eq_one hz him.le hc) with hd | hd | hd
· grind [cases_c_one_d_zero hz hg him.le hc hd] -- ± S, T⁻¹S, TS
· grind [case_c_one_d_one hz hg him.le hc hd] -- ± ST, TST
· grind [case_c_one_d_neg_one hz hg him.le hc hd] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Modular | {
"line": 716,
"column": 4
} | {
"line": 719,
"column": 53
} | {
"line": 721,
"column": 0
} | [
{
"pp": "case inr\ng✝ : SL(2, ℤ)\nz✝ : ℍ\ng : SL(2, ℤ)\nz : ℍ\nhz : z ∈ 𝒟\nhg : g • z ∈ 𝒟\nhim : ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) g)) ↑z‖ = 1\nhc✝ : 0 ≤ ↑g 1 0\nhc : ↑g 1 0 = 1\n⊢ (g = 1 ∨ g = -1) ∨\n (g = T ∨ g = -T) ∧ z.re = -1 / 2 ∨\n (g = T⁻¹ ∨ g = -T⁻¹) ∧ z.re = 1 / 2 ∨\... | [] | rcases Int.abs_le_one_iff.mp (cases_d_of_c_eq_one hz him.le hc) with hd | hd | hd
· grind [cases_c_one_d_zero hz hg him.le hc hd] -- ± S, T⁻¹S, TS
· grind [case_c_one_d_one hz hg him.le hc hd] -- ± ST, TST
· grind [case_c_one_d_neg_one hz hg him.le hc hd] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.QExpansion | {
"line": 226,
"column": 10
} | {
"line": 226,
"column": 21
} | {
"line": 226,
"column": 22
} | [
{
"pp": "k : ℕ\nhk : 3 ≤ k\nhk2 : Even k\nz : ℍ\n⊢ Summable fun b ↦ ∑' (c : ℤ), eisSummand ↑k ![(b, c).1, (b, c).2] z",
"ppTerm": "?m.127",
"assigned": true,
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"AddCommGroup.toAddCommMonoid",
"PseudoMetricSpace.toUniformSpace"... | [
"k : ℕ\nhk : 3 ≤ k\nhk2 : Even k\nz : ℍ\n⊢ Summable fun b ↦ ∑' (c : ℤ), eisSummand ↑k ![b, c] z"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.QExpansion | {
"line": 230,
"column": 81
} | {
"line": 230,
"column": 92
} | {
"line": 230,
"column": 93
} | [
{
"pp": "k : ℕ\nhk : 3 ≤ k\nhk2 : Even k\nz : ℍ\nb : ℕ+\n⊢ 0 < (↑↑b * ↑z).im",
"ppTerm": "?m.174",
"assigned": true,
"usedConstants": [
"PNat.val",
"Complex.mul_im",
"Real.instIsOrderedRing",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.partialOrder"... | [
"k : ℕ\nhk : 3 ≤ k\nhk2 : Even k\nz : ℍ\nb : ℕ+\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.Modular | {
"line": 745,
"column": 4
} | {
"line": 745,
"column": 47
} | {
"line": 746,
"column": 4
} | [
{
"pp": "case mpr\ng : SL(2, ℤ)\n⊢ S • I = I",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"instHSMul",
"Matrix.SpecialLinearGroup",
"UpperHalfPlane.SLAction",
"UpperHalfPlane.coe",
"congrArg",
"UpperHalfPlane.ext_iff",
... | [
"case mpr\ng : SL(2, ℤ)\n⊢ ↑{ coe := (-↑I)⁻¹, coe_im_pos := ⋯ } = ↑I"
] | rw [modular_S_smul, UpperHalfPlane.ext_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
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.EisensteinSeries.E2.Summable | {
"line": 255,
"column": 10
} | {
"line": 255,
"column": 21
} | {
"line": 255,
"column": 22
} | [
{
"pp": "z : ℍ\nN : ℕ+\nh2 :\n ↑π * I - 2 * ↑π * I * ∑' (n : ℕ), cexp (2 * ↑π * I * ↑{ coe := -↑↑N / ↑z, coe_im_pos := ⋯ }) ^ n - ↑z / -↑↑N =\n ↑π * I - 2 * ↑π * I * ∑' (n : ℕ), cexp (2 * ↑π * I * ↑{ coe := -↑↑N / ↑z, coe_im_pos := ⋯ }) ^ n - ↑z / -↑↑N\n⊢ Summable (HPow.hPow (cexp (2 * ↑π * I * ↑{ coe := -↑... | [
"z : ℍ\nN : ℕ+\nh2 :\n ↑π * I - 2 * ↑π * I * ∑' (n : ℕ), cexp (2 * ↑π * I * ↑{ coe := -↑↑N / ↑z, coe_im_pos := ⋯ }) ^ n - ↑z / -↑↑N =\n ↑π * I - 2 * ↑π * I * ∑' (n : ℕ), cexp (2 * ↑π * I * ↑{ coe := -↑↑N / ↑z, coe_im_pos := ⋯ }) ^ n - ↑z / -↑↑N\n⊢ ‖cexp (2 * ↑π * I * (-↑↑N / ↑z))‖ < 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.E2.Summable | {
"line": 255,
"column": 10
} | {
"line": 255,
"column": 21
} | {
"line": 255,
"column": 22
} | [
{
"pp": "z : ℍ\nN : ℕ+\nh2 :\n ↑π * I - 2 * ↑π * I * ∑' (n : ℕ), cexp (2 * ↑π * I * ↑{ coe := -↑↑N / ↑z, coe_im_pos := ⋯ }) ^ n - ↑z / -↑↑N =\n ↑π * I - 2 * ↑π * I * ∑' (n : ℕ), cexp (2 * ↑π * I * ↑{ coe := -↑↑N / ↑z, coe_im_pos := ⋯ }) ^ n - ↑z / -↑↑N\n⊢ Summable (HPow.hPow (cexp (2 * ↑π * I * ↑{ coe := -↑... | [
"z : ℍ\nN : ℕ+\nh2 :\n ↑π * I - 2 * ↑π * I * ∑' (n : ℕ), cexp (2 * ↑π * I * ↑{ coe := -↑↑N / ↑z, coe_im_pos := ⋯ }) ^ n - ↑z / -↑↑N =\n ↑π * I - 2 * ↑π * I * ∑' (n : ℕ), cexp (2 * ↑π * I * ↑{ coe := -↑↑N / ↑z, coe_im_pos := ⋯ }) ^ n - ↑z / -↑↑N\n⊢ ‖cexp (2 * ↑π * I * (-↑↑N / ↑z))‖ < 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Modular | {
"line": 785,
"column": 4
} | {
"line": 785,
"column": 15
} | {
"line": 786,
"column": 2
} | [
{
"pp": "g : SL(2, ℤ)\nz : ℍ\nhz : z ∈ 𝒟ᵒ\nhg : g • z ∈ 𝒟\nthis✝ : ρ ∉ 𝒟ᵒ\nh : 1 +ᵥ ρ ∈ 𝒟ᵒ\nthis : (1 +ᵥ ρ).re = 1 / 2\n⊢ False",
"ppTerm": "?m.64",
"assigned": true,
"usedConstants": [
"_private.Mathlib.NumberTheory.Modular.0.ModularGroup.eq_one_or_neg_one_of_mem_fdo_mem_fd._proof_1_3"
... | [] | grind [h.2] | Lean.Elab.Tactic.evalGrind | Lean.Parser.Tactic.grind |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.QExpansion | {
"line": 266,
"column": 4
} | {
"line": 266,
"column": 15
} | {
"line": 266,
"column": 16
} | [
{
"pp": "k : ℕ\nhk : 3 ≤ k\nz : ℍ\nhk1 : 1 < k\nhk2 : 3 ≤ ↑k\nb : ℕ\n⊢ Summable fun c ↦ (↑(b, c).1 ^ k)⁻¹ * eisSummand (↑k) (↑(b, c).2) z",
"ppTerm": "?m.110",
"assigned": true,
"usedConstants": [
"DivInvMonoid.toInv",
"NormedRing.toRing",
"HMul.hMul",
"ZMod.commRing",
... | [
"k : ℕ\nhk : 3 ≤ k\nz : ℍ\nhk1 : 1 < k\nhk2 : 3 ≤ ↑k\nb : ℕ\n⊢ Summable fun c ↦ (↑b ^ k)⁻¹ * eisSummand (↑k) (↑c) z"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.QExpansion | {
"line": 271,
"column": 4
} | {
"line": 272,
"column": 11
} | {
"line": 272,
"column": 12
} | [
{
"pp": "case inr\nk : ℕ\nhk : 3 ≤ k\nz : ℍ\nhk1 : 1 < k\nhk2 : 3 ≤ ↑k\nb : ℕ\nhb : b ≠ 0\nthis : NeZero b\n⊢ ∑' (c : ↑(gammaSet 1 b 0)), eisSummand (↑k) (↑c) z =\n ∑' (c : { x // x ∈ gammaSet 1 1 0 }), (↑(b, c).1 ^ k)⁻¹ * eisSummand (↑k) (↑(b, c).2) z",
"ppTerm": "?inr",
"assigned": true,
"usedC... | [
"case inr\nk : ℕ\nhk : 3 ≤ k\nz : ℍ\nhk1 : 1 < k\nhk2 : 3 ≤ ↑k\nb : ℕ\nhb : b ≠ 0\nthis : NeZero b\n⊢ ∑' (x : ↑(gammaSet 1 b 0)), eisSummand (↑k) (divIntMap ↑b ↑x) z =\n ∑' (x : { x // x ∈ gammaSet 1 1 0 }), eisSummand (↑k) (divIntMap 1 ↑x) z"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Modular | {
"line": 868,
"column": 8
} | {
"line": 868,
"column": 31
} | {
"line": 868,
"column": 32
} | [
{
"pp": "x : ℍ\nhxnorm : 1 < ‖↑x‖\nhxre : |x.re| ≤ 1 / 2\n⊢ ↑1 * ↑x ∈ ofComplex.source",
"ppTerm": "?m.110",
"assigned": true,
"usedConstants": [
"UpperHalfPlane.ofComplex",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"UpperHalfPlane.isOpenEmbedding_coe",
"Real",
... | [
"x : ℍ\nhxnorm : 1 < ‖↑x‖\nhxre : |x.re| ≤ 1 / 2\n⊢ 0 < x.im"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Modular | {
"line": 869,
"column": 6
} | {
"line": 869,
"column": 58
} | {
"line": 869,
"column": 59
} | [
{
"pp": "case h.refine_1\nx : ℍ\nhxnorm : 1 < ‖↑x‖\nhxre : |x.re| ≤ 1 / 2\nthis : ContinuousAt (fun a ↦ ↑(↑ofComplex (↑a * ↑x))) 1\n⊢ Filter.Tendsto (fun x_1 ↦ ‖↑(↑ofComplex (↑x_1 * ↑x))‖) (𝓝 1) (𝓝 ‖↑x‖)",
"ppTerm": "?h.refine_1",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"u... | [
"case h.refine_1\nx : ℍ\nhxnorm : 1 < ‖↑x‖\nhxre : |x.re| ≤ 1 / 2\nthis : ContinuousAt (fun a ↦ ↑(↑ofComplex (↑a * ↑x))) 1\n⊢ Filter.Tendsto (fun x_1 ↦ ‖↑(↑ofComplex (↑x_1 * ↑x))‖) (𝓝 1) (𝓝 ‖↑x‖)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.E2.Summable | {
"line": 287,
"column": 2
} | {
"line": 288,
"column": 9
} | {
"line": 288,
"column": 10
} | [
{
"pp": "z : ℍ\n⊢ HasSum (fun b ↦ ∑' (m : ℤ), (1 / (↑m * ↑z + ↑b) - 1 / (↑m * ↑z + ↑b + 1))) (-2 * ↑π * I / ↑z) (symmetricIco ℤ)",
"ppTerm": "?m.85",
"assigned": true,
"usedConstants": [
"PNat.val",
"Int.cast",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"_private.... | [
"z : ℍ\n⊢ Tendsto (fun x ↦ ∑ x ∈ Ico (-↑↑x) ↑↑x, ∑' (m : ℤ), ((↑m * ↑z + ↑x)⁻¹ - (↑m * ↑z + ↑x + 1)⁻¹)) atTop\n (𝓝 (-(2 * ↑π * I) / ↑z))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Modular | {
"line": 872,
"column": 50
} | {
"line": 872,
"column": 61
} | {
"line": 872,
"column": 62
} | [
{
"pp": "x : ℍ\nhxnorm : 1 < ‖↑x‖\nhxre : |x.re| ≤ 1 / 2\na : ℝ\nha : 0 < a\nha' : a ∈ Set.Iio 1\n⊢ 0 < (↑a * ↑x).im",
"ppTerm": "?m.207",
"assigned": true,
"usedConstants": [
"Complex.mul_im",
"Eq.mpr",
"Real",
"HMul.hMul",
"UpperHalfPlane.coe",
"Real.instZero",
... | [
"x : ℍ\nhxnorm : 1 < ‖↑x‖\nhxre : |x.re| ≤ 1 / 2\na : ℝ\nha : 0 < a\nha' : a ∈ Set.Iio 1\n⊢ 0 < a * x.im"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Modular | {
"line": 882,
"column": 38
} | {
"line": 882,
"column": 49
} | {
"line": 882,
"column": 50
} | [
{
"pp": "x : ℍ\nhxnorm : 1 < ‖↑x‖\nhxre : |x.re| ≤ 1 / 2\nthis : Filter.Tendsto (fun t ↦ ↑t * ↑x) (𝓝 1) (𝓝 ↑x)\na : ℝ\nha : 0 < a\n⊢ 0 < (↑a * ↑x).im",
"ppTerm": "?m.380",
"assigned": true,
"usedConstants": [
"Complex.mul_im",
"Eq.mpr",
"Real",
"HMul.hMul",
"UpperHalf... | [
"x : ℍ\nhxnorm : 1 < ‖↑x‖\nhxre : |x.re| ≤ 1 / 2\nthis : Filter.Tendsto (fun t ↦ ↑t * ↑x) (𝓝 1) (𝓝 ↑x)\na : ℝ\nha : 0 < a\n⊢ 0 < a * x.im"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Modular | {
"line": 928,
"column": 4
} | {
"line": 928,
"column": 66
} | {
"line": 928,
"column": 67
} | [
{
"pp": "ho1 : interior 𝒟 ⊆ UpperHalfPlane.re ⁻¹' interior (UpperHalfPlane.re '' 𝒟)\nho2 : interior 𝒟 ⊆ (fun τ ↦ ‖↑τ‖) ⁻¹' interior ((fun τ ↦ ‖↑τ‖) '' 𝒟)\nx : ℍ\nhx : x ∈ interior 𝒟\nξ : ℍ\nhξ : ξ ∈ 𝒟\n⊢ ξ ∈ (fun τ ↦ ‖↑τ‖) ⁻¹' Set.Ici 1",
"ppTerm": "?m.110",
"assigned": true,
"usedConstants": ... | [
"ho1 : interior 𝒟 ⊆ UpperHalfPlane.re ⁻¹' interior (UpperHalfPlane.re '' 𝒟)\nho2 : interior 𝒟 ⊆ (fun τ ↦ ‖↑τ‖) ⁻¹' interior ((fun τ ↦ ‖↑τ‖) '' 𝒟)\nx : ℍ\nhx : x ∈ interior 𝒟\nξ : ℍ\nhξ : ξ ∈ 𝒟\n⊢ 1 ≤ ‖↑ξ‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Modular | {
"line": 933,
"column": 4
} | {
"line": 933,
"column": 55
} | {
"line": 933,
"column": 56
} | [
{
"pp": "ho1 : interior 𝒟 ⊆ UpperHalfPlane.re ⁻¹' interior (UpperHalfPlane.re '' 𝒟)\nho2 : interior 𝒟 ⊆ (fun τ ↦ ‖↑τ‖) ⁻¹' interior ((fun τ ↦ ‖↑τ‖) '' 𝒟)\nx : ℍ\nhx : x ∈ interior 𝒟\nξ : ℍ\nhξ : ξ ∈ 𝒟\n⊢ ξ ∈ UpperHalfPlane.re ⁻¹' Set.Icc (-(1 / 2)) (1 / 2)",
"ppTerm": "?m.173",
"assigned": true,
... | [
"ho1 : interior 𝒟 ⊆ UpperHalfPlane.re ⁻¹' interior (UpperHalfPlane.re '' 𝒟)\nho2 : interior 𝒟 ⊆ (fun τ ↦ ‖↑τ‖) ⁻¹' interior ((fun τ ↦ ‖↑τ‖) '' 𝒟)\nx : ℍ\nhx : x ∈ interior 𝒟\nξ : ℍ\nhξ : ξ ∈ 𝒟\n⊢ -2⁻¹ ≤ ξ.re ∧ ξ.re ≤ 2⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Modular | {
"line": 950,
"column": 33
} | {
"line": 950,
"column": 72
} | {
"line": 950,
"column": 73
} | [
{
"pp": "y : ℝ\nz : ℂ\nhz : 0 < z.im\nh : { coe := z, coe_im_pos := hz } ∈ truncatedFundamentalDomain y\n⊢ 1 ≤ ‖↑{ coe := z, coe_im_pos := hz }‖",
"ppTerm": "?m.122",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Real.instLE",
"Real",
"UpperHalfPlane.coe",
"Complex... | [
"y : ℝ\nz : ℂ\nhz : 0 < z.im\nh : { coe := z, coe_im_pos := hz } ∈ truncatedFundamentalDomain y\n⊢ 1 ≤ ‖z‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Modular | {
"line": 955,
"column": 6
} | {
"line": 956,
"column": 25
} | {
"line": 956,
"column": 26
} | [
{
"pp": "y : ℝ\nz : ℂ\nhz : 0 ≤ z.im\nh1 : z.im ≤ y\nh2 : |z.re| ≤ 1 / 2\nh3 : 0 = z.im\n⊢ ‖z‖ < 1",
"ppTerm": "?m.175",
"assigned": true,
"usedConstants": [
"sq_lt_one_iff₀",
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
... | [
"y : ℝ\nz : ℂ\nhz : 0 ≤ z.im\nh1 : z.im ≤ y\nh2 : |z.re| ≤ 1 / 2\nh3 : 0 = z.im\n⊢ |z.re| < 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.QExpansion | {
"line": 349,
"column": 2
} | {
"line": 349,
"column": 13
} | {
"line": 349,
"column": 14
} | [
{
"pp": "k : ℕ\nhk : 3 ≤ k\nhk2 : Even k\n⊢ (PowerSeries.coeff 0) (qExpansion 1 ⇑(E hk)) = 1",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"ModularForm",
"Eq.mpr",
"MonoidHom.range",
"Real",
"Matrix.SpecialLinearGroup",
"Semiring.toModule",
"Uppe... | [
"k : ℕ\nhk : 3 ≤ k\nhk2 : Even k\n⊢ PowerSeries.constantCoeff (qExpansion 1 ⇑(E hk)) = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.LevelOne.GradedRing | {
"line": 69,
"column": 4
} | {
"line": 70,
"column": 51
} | {
"line": 70,
"column": 52
} | [
{
"pp": "z : ℍ\ng : CuspForm (Matrix.SpecialLinearGroup.mapGL ℝ).range 12\nhg : CuspForm.toModularFormₗ g = E₄CubeSubE₆SqForm\nc : ℂ\nhc : c • CuspForm.discriminant = g\nhgE : ⇑g = ⇑E₄CubeSubE₆SqForm\nhcΔ : c • ⇑CuspForm.discriminant = ⇑g\nhgΔ : qExpansion 1 ⇑E₄CubeSubE₆SqForm = c • qExpansion 1 ⇑CuspForm.discr... | [
"z : ℍ\ng : CuspForm (Matrix.SpecialLinearGroup.mapGL ℝ).range 12\nhg : CuspForm.toModularFormₗ g = E₄CubeSubE₆SqForm\nc : ℂ\nhc : c • CuspForm.discriminant = g\nhgE : ⇑g = ⇑E₄CubeSubE₆SqForm\nhcΔ : c • ⇑CuspForm.discriminant = ⇑g\nhgΔ : qExpansion 1 ⇑E₄CubeSubE₆SqForm = c • qExpansion 1 ⇑CuspForm.discriminant\n⊢ c... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Modular | {
"line": 996,
"column": 4
} | {
"line": 997,
"column": 56
} | {
"line": 998,
"column": 2
} | [
{
"pp": "case inl\nτ : ℍ\nh : 1 / 2 ≤ τ.im\n⊢ ∃ γ, 1 / 2 ≤ (γ • τ).im ∧ ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) γ)) ↑τ‖ ≤ 1",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Norm.norm",
"MonoidHom.instMonoidHomClass",
"MulOne.toOne",
"Real.instLE",
... | [] | exact ⟨1, (one_smul SL(2, ℤ) τ).symm ▸ h, by
simp only [map_one, denom_one, norm_one, le_refl]⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.Modular | {
"line": 996,
"column": 4
} | {
"line": 997,
"column": 56
} | {
"line": 998,
"column": 2
} | [
{
"pp": "case inl\nτ : ℍ\nh : 1 / 2 ≤ τ.im\n⊢ ∃ γ, 1 / 2 ≤ (γ • τ).im ∧ ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) γ)) ↑τ‖ ≤ 1",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Norm.norm",
"MonoidHom.instMonoidHomClass",
"MulOne.toOne",
"Real.instLE",
... | [] | exact ⟨1, (one_smul SL(2, ℤ) τ).symm ▸ h, by
simp only [map_one, denom_one, norm_one, le_refl]⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Modular | {
"line": 996,
"column": 4
} | {
"line": 997,
"column": 56
} | {
"line": 998,
"column": 2
} | [
{
"pp": "case inl\nτ : ℍ\nh : 1 / 2 ≤ τ.im\n⊢ ∃ γ, 1 / 2 ≤ (γ • τ).im ∧ ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) γ)) ↑τ‖ ≤ 1",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Norm.norm",
"MonoidHom.instMonoidHomClass",
"MulOne.toOne",
"Real.instLE",
... | [] | exact ⟨1, (one_smul SL(2, ℤ) τ).symm ▸ h, by
simp only [map_one, denom_one, norm_one, le_refl]⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.ModularForms.LevelOne.GradedRing | {
"line": 84,
"column": 2
} | {
"line": 84,
"column": 72
} | {
"line": 85,
"column": 2
} | [
{
"pp": "⊢ (DirectSum.of (ModularForm (Matrix.SpecialLinearGroup.mapGL ℝ).range) 12) (modularForm CuspForm.discriminant) =\n (1 / 1728) •\n ((DirectSum.of (ModularForm (Matrix.SpecialLinearGroup.mapGL ℝ).range) 12) (E₄.pow 3) -\n (DirectSum.of (ModularForm (Matrix.SpecialLinearGroup.mapGL ℝ).rang... | [
"⊢ (DirectSum.of (ModularForm (Matrix.SpecialLinearGroup.mapGL ℝ).range) 12) (modularForm CuspForm.discriminant) =\n (DirectSum.of (ModularForm (Matrix.SpecialLinearGroup.mapGL ℝ).range) 12) ((1 / 1728) • (E₄.pow 3 - E₆.pow 2))"
] | rw [← map_sub (DirectSum.of (ModularForm 𝒮ℒ) 12), ← DirectSum.of_smul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.Modular | {
"line": 1002,
"column": 4
} | {
"line": 1002,
"column": 57
} | {
"line": 1003,
"column": 6
} | [
{
"pp": "case inr\nτ : ℍ\nh : τ.im ≤ 1 / 2\nγ : SL(2, ℤ)\nhγ : 1 / 2 ≤ (γ • τ).im\nh1 : τ.im * ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) γ)) ↑τ‖ ^ 2 ≤ τ.im\n⊢ ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) γ)) ↑τ‖ ≤ 1",
"ppTerm": "?inr",
"assigned": false,
"usedConstants... | [
"case inr\nτ : ℍ\nh : τ.im ≤ 1 / 2\nγ : SL(2, ℤ)\nhγ : 1 / 2 ≤ (γ • τ).im\nh1 : τ.im * ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) γ)) ↑τ‖ ^ 2 ≤ τ.im\n⊢ ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) γ)) ↑τ‖ ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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