module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.NumberTheory.Height.MvPolynomial | {
"line": 332,
"column": 9
} | {
"line": 332,
"column": 40
} | [
{
"pp": "case inr.inr.h₁\nK : Type u_4\ninst✝³ : Field K\nι : Type u_5\nι' : Type u_6\ninst✝² : AdmissibleAbsValues K\ninst✝¹ : Finite ι'\ninst✝ : Finite ι\nN : ℕ\np : ι' → MvPolynomial ι K\nhp : ∀ (i : ι'), (p i).IsHomogeneous N\nx : ι → K\nhx : x ≠ 0\nh₀ : (fun j ↦ (eval x) (p j)) ≠ 0\nH₀ : ∀ (v : AbsoluteVal... | v.eval_mvPolynomial_le (hp j) x | Mathlib.Tactic.evalGRewriteSeq | null |
Mathlib.NumberTheory.LSeries.ZMod | {
"line": 196,
"column": 6
} | {
"line": 196,
"column": 61
} | [
{
"pp": "case neg\nN : ℕ\ninst✝ : NeZero N\nj : ZMod N\ns : ℂ\nhjs : j ≠ 0 ∨ s ≠ 1\nU : Set ℂ := if j = 0 then {z | z ≠ 1} else Set.univ\nV : Set ℂ := {z | 1 < z.re}\nhUo : IsOpen U\nf : ℂ → ℂ := LFunction fun k ↦ 𝕖 (j * k)\ng : ℂ → ℂ := expZeta (toAddCircle j)\nhU : ∀ {u : ℂ}, u ∈ U ↔ u ≠ 1 ∨ j ≠ 0\nhf : Anal... | simpa only [h, ↓reduceIte, U] using isPreconnected_univ | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.NumberTheory.LSeries.ZMod | {
"line": 196,
"column": 6
} | {
"line": 196,
"column": 61
} | [
{
"pp": "case neg\nN : ℕ\ninst✝ : NeZero N\nj : ZMod N\ns : ℂ\nhjs : j ≠ 0 ∨ s ≠ 1\nU : Set ℂ := if j = 0 then {z | z ≠ 1} else Set.univ\nV : Set ℂ := {z | 1 < z.re}\nhUo : IsOpen U\nf : ℂ → ℂ := LFunction fun k ↦ 𝕖 (j * k)\ng : ℂ → ℂ := expZeta (toAddCircle j)\nhU : ∀ {u : ℂ}, u ∈ U ↔ u ≠ 1 ∨ j ≠ 0\nhf : Anal... | simpa only [h, ↓reduceIte, U] using isPreconnected_univ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.LSeries.ZMod | {
"line": 196,
"column": 6
} | {
"line": 196,
"column": 61
} | [
{
"pp": "case neg\nN : ℕ\ninst✝ : NeZero N\nj : ZMod N\ns : ℂ\nhjs : j ≠ 0 ∨ s ≠ 1\nU : Set ℂ := if j = 0 then {z | z ≠ 1} else Set.univ\nV : Set ℂ := {z | 1 < z.re}\nhUo : IsOpen U\nf : ℂ → ℂ := LFunction fun k ↦ 𝕖 (j * k)\ng : ℂ → ℂ := expZeta (toAddCircle j)\nhU : ∀ {u : ℂ}, u ∈ U ↔ u ≠ 1 ∨ j ≠ 0\nhf : Anal... | simpa only [h, ↓reduceIte, U] using isPreconnected_univ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Height.MvPolynomial | {
"line": 553,
"column": 94
} | {
"line": 575,
"column": 26
} | [
{
"pp": "K : Type u_4\ninst✝¹ : Field K\ninst✝ : AdmissibleAbsValues K\n⊢ ∃ C > 0, ∀ (a b c d : K), mulHeight ![a * c, a * d + b * c, b * d] ≤ C * mulHeight ![a, b] * mulHeight ![c, d]",
"usedConstants": [
"Height.mulHeight_mul_mulHeight",
"Finsupp.instAddZeroClass",
"Real.instIsOrderedRin... | by
let p : Fin 3 → MvPolynomial (Fin 4) K := ![X 0, X 1 + X 2, X 3]
have hom i : (p i).IsHomogeneous 1 := by
fin_cases i <;> simp [p, isHomogeneous_X, IsHomogeneous.add]
obtain ⟨C, hC₀, hC⟩ := mulHeight_eval_le' hom
simp only [pow_one] at hC
refine ⟨max C 1, by grind, fun a b c d ↦ ?_⟩
by_cases hab : ![... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.LSeries.DirichletContinuation | {
"line": 133,
"column": 2
} | {
"line": 134,
"column": 97
} | [
{
"pp": "M N : ℕ\ninst✝¹ : NeZero M\ninst✝ : NeZero N\nhMN : M ∣ N\nχ : DirichletCharacter ℂ M\ns : ℂ\nhs : s ≠ 1\nhpc : IsPreconnected {1}ᶜ\nhne : 2 ∈ {1}ᶜ\n⊢ LFunction ((changeLevel hMN) χ) s = LFunction χ s * ∏ p ∈ N.primeFactors, (1 - χ ↑p * ↑p ^ (-s))",
"usedConstants": [
"InnerProductSpace.toNor... | refine AnalyticOnNhd.eqOn_of_preconnected_of_eventuallyEq (𝕜 := ℂ)
(g := fun s ↦ LFunction χ s * ∏ p ∈ N.primeFactors, (1 - χ p * p ^ (-s))) ?_ ?_ hpc hne ?_ hs | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.LSeries.DirichletContinuation | {
"line": 140,
"column": 4
} | {
"line": 140,
"column": 60
} | [
{
"pp": "case refine_2\nM N : ℕ\ninst✝¹ : NeZero M\ninst✝ : NeZero N\nhMN : M ∣ N\nχ : DirichletCharacter ℂ M\ns✝ : ℂ\nhs✝ : s✝ ≠ 1\nhpc : IsPreconnected {1}ᶜ\nhne : 2 ∈ {1}ᶜ\ns : ℂ\nhs : s ∈ {1}ᶜ\ni : ℕ\nh : i ∈ N.primeFactors\n⊢ DifferentiableAt ℂ (fun s ↦ 1 - χ ↑i * ↑i ^ (-s)) s",
"usedConstants": [
... | have : NeZero i := ⟨(Nat.pos_of_mem_primeFactors h).ne'⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.Height.MvPolynomial | {
"line": 597,
"column": 2
} | {
"line": 597,
"column": 45
} | [
{
"pp": "K : Type u_4\ninst✝¹ : Field K\ninst✝ : AdmissibleAbsValues K\n⊢ ∃ C, ∀ (a b c d : K), logHeight ![a * c, a * d + b * c, b * d] ≤ C + logHeight ![a, b] + logHeight ![c, d]",
"usedConstants": [
"Height.mulHeight_sym2_le"
]
}
] | obtain ⟨C', hC₀, hC⟩ := mulHeight_sym2_le K | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.NumberTheory.LSeries.DirichletContinuation | {
"line": 352,
"column": 81
} | {
"line": 352,
"column": 89
} | [
{
"pp": "case funProp.discharger\nn : ℕ\ninst✝ : NeZero n\nx✝ : ℂ\nhs : x✝ ∈ Set.univ \\ {1}\n⊢ x✝ ≠ 1 ∨ 1 ≠ 1",
"usedConstants": [
"False",
"ZMod.commRing",
"eq_false",
"MulChar.hasOne",
"congrArg",
"Set.mem_univ._simp_1",
"Set.univ",
"Membership.mem",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.NumberTheory.LSeries.DirichletContinuation | {
"line": 352,
"column": 81
} | {
"line": 352,
"column": 89
} | [
{
"pp": "case funProp.discharger\nn : ℕ\ninst✝ : NeZero n\nx✝ : ℂ\nhs : x✝ ∈ Set.univ \\ {1}\n⊢ x✝ ≠ 1 ∨ 1 ≠ 1",
"usedConstants": [
"False",
"ZMod.commRing",
"eq_false",
"MulChar.hasOne",
"congrArg",
"Set.mem_univ._simp_1",
"Set.univ",
"Membership.mem",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.LSeries.DirichletContinuation | {
"line": 352,
"column": 81
} | {
"line": 352,
"column": 89
} | [
{
"pp": "case funProp.discharger\nn : ℕ\ninst✝ : NeZero n\nx✝ : ℂ\nhs : x✝ ∈ Set.univ \\ {1}\n⊢ x✝ ≠ 1 ∨ 1 ≠ 1",
"usedConstants": [
"False",
"ZMod.commRing",
"eq_false",
"MulChar.hasOne",
"congrArg",
"Set.mem_univ._simp_1",
"Set.univ",
"Membership.mem",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.LSeries.ZMod | {
"line": 512,
"column": 2
} | {
"line": 512,
"column": 60
} | [
{
"pp": "N : ℕ\ninst✝ : NeZero N\nΦ : ZMod N → ℂ\nhΦ : Function.Odd Φ\ns : ℂ\nF : ℂ → ℂ := fun t ↦ completedLFunction Φ (1 - t)\n⊢ completedLFunction Φ (1 - s) = ↑N ^ (s - 1) * I * completedLFunction (𝓕 Φ) s",
"usedConstants": [
"ZMod.completedLFunction",
"Pi.Function.module",
"Semiring.t... | let G (t) := ↑N ^ (t - 1) * I * completedLFunction (𝓕 Φ) t | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.NumberTheory.ZetaValues | {
"line": 331,
"column": 4
} | {
"line": 332,
"column": 77
} | [
{
"pp": "case h.e'_6.h.e'_6\nk : ℕ\nhk : 2 ≤ k\nx : ℝ\nhx : x ∈ Icc 0 1\ny : ℝ\nhy : y ∈ Ico 0 1\nB : C(𝕌, ℂ) := { toFun := ofReal ∘ periodizedBernoulli k, continuous_toFun := ⋯ }\nstep1 : ∀ (n : ℤ), fourierCoeff (⇑B) n = -↑k ! / (2 * ↑π * I * ↑n) ^ k\nstep2 : HasSum (fun i ↦ (-↑k ! / (2 * ↑π * I * ↑i) ^ k) • ... | rw [ContinuousMap.coe_mk, Function.comp_apply, ofReal_inj, periodizedBernoulli,
AddCircle.liftIco_coe_apply (show y ∈ Ico 0 (0 + 1) by rwa [zero_add])] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.ZetaValues | {
"line": 331,
"column": 4
} | {
"line": 332,
"column": 77
} | [
{
"pp": "case h.e'_6.h.e'_6\nk : ℕ\nhk : 2 ≤ k\nx : ℝ\nhx : x ∈ Icc 0 1\ny : ℝ\nhy : y ∈ Ico 0 1\nB : C(𝕌, ℂ) := { toFun := ofReal ∘ periodizedBernoulli k, continuous_toFun := ⋯ }\nstep1 : ∀ (n : ℤ), fourierCoeff (⇑B) n = -↑k ! / (2 * ↑π * I * ↑n) ^ k\nstep2 : HasSum (fun i ↦ (-↑k ! / (2 * ↑π * I * ↑i) ^ k) • ... | rw [ContinuousMap.coe_mk, Function.comp_apply, ofReal_inj, periodizedBernoulli,
AddCircle.liftIco_coe_apply (show y ∈ Ico 0 (0 + 1) by rwa [zero_add])] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.ZetaValues | {
"line": 331,
"column": 4
} | {
"line": 332,
"column": 77
} | [
{
"pp": "case h.e'_6.h.e'_6\nk : ℕ\nhk : 2 ≤ k\nx : ℝ\nhx : x ∈ Icc 0 1\ny : ℝ\nhy : y ∈ Ico 0 1\nB : C(𝕌, ℂ) := { toFun := ofReal ∘ periodizedBernoulli k, continuous_toFun := ⋯ }\nstep1 : ∀ (n : ℤ), fourierCoeff (⇑B) n = -↑k ! / (2 * ↑π * I * ↑n) ^ k\nstep2 : HasSum (fun i ↦ (-↑k ! / (2 * ↑π * I * ↑i) ^ k) • ... | rw [ContinuousMap.coe_mk, Function.comp_apply, ofReal_inj, periodizedBernoulli,
AddCircle.liftIco_coe_apply (show y ∈ Ico 0 (0 + 1) by rwa [zero_add])] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.LSeries.PrimesInAP | {
"line": 89,
"column": 68
} | {
"line": 96,
"column": 73
} | [
{
"pp": "α : Type u_1\ninst✝⁴ : CommGroup α\ninst✝³ : UniformSpace α\ninst✝² : IsUniformGroup α\ninst✝¹ : CompleteSpace α\ninst✝ : T0Space α\nf : ℕ → α\nhfm : Multipliable f\nhf : Function.mulSupport f ⊆ {n | IsPrimePow n}\n⊢ ∏' (n : ℕ), f n = ∏' (p : Nat.Primes) (k : ℕ), f (↑p ^ (k + 1))",
"usedConstants":... | by
have hfm' : Multipliable fun pk : Nat.Primes × ℕ ↦ f (pk.fst ^ (pk.snd + 1)) :=
prodNatEquiv.symm.multipliable_iff.mp <| by
simpa only [← coe_prodNatEquiv_apply, Prod.eta, Function.comp_def, Equiv.apply_symm_apply]
using hfm.subtype _
simp only [← tprod_subtype_eq_of_mulSupport_subset hf, Set.c... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.LSeries.Nonvanishing | {
"line": 176,
"column": 2
} | {
"line": 181,
"column": 70
} | [
{
"pp": "case inr.refine_2\nN : ℕ\ninst✝ : NeZero N\nB : BadChar N\nG : ℂ → ℂ := Function.update (fun s ↦ (s - 1) * riemannZeta s) 1 1\nH : ℂ → ℂ := Function.update (fun s ↦ (LFunction B.χ s - LFunction B.χ 1) / (s - 1)) 1 (deriv (LFunction B.χ) 1)\n⊢ ContinuousAt B.F 1",
"usedConstants": [
"Iff.mpr",... | have : B.F = G * H := by
ext1 t
rcases eq_or_ne t 1 with rfl | ht
· simp only [F, G, H, Pi.mul_apply, one_mul, Function.update_self]
· simp only [F, G, H, Function.update_of_ne ht, mul_comm _ (riemannZeta _), B.hχ, sub_zero,
Pi.mul_apply, mul_assoc, mul_div_cancel₀ _ (sub_ne_zero.mpr ht)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.LSeries.Injectivity | {
"line": 184,
"column": 38
} | {
"line": 184,
"column": 51
} | [
{
"pp": "f : ℕ → ℂ\nhf : f 0 = 0\nh : ¬abscissaOfAbsConv f = ⊤\nH : LSeries f = 0\nH' : f = 0\nx✝¹ : ℕ\nx✝ : x✝¹ ≠ 0\n⊢ 0 x✝¹ = 0",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Complex.instZero",
"Pi.zero_apply",
"id",
"Pi.instZero",
"Nat",
"Zero.toOfNat0",
... | Pi.zero_apply | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LSeries.PrimesInAP | {
"line": 317,
"column": 4
} | {
"line": 317,
"column": 45
} | [
{
"pp": "case refine_1\nq : ℕ\na : ZMod q\ninst✝ : NeZero q\ns : ℂ\nhs : s ∈ {s | s = 1 ∨ ∀ (χ : DirichletCharacter ℂ q), LFunction χ s ≠ 0}\n⊢ s ∈ {s | s = 1 ∨ LFunctionTrivChar q s ≠ 0}",
"usedConstants": []
}
] | simp only [ne_eq, Set.mem_setOf_eq] at hs | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.LSeries.Injectivity | {
"line": 212,
"column": 71
} | {
"line": 212,
"column": 84
} | [
{
"pp": "f g : ℕ → ℂ\nhf : abscissaOfAbsConv f < ⊤\nhg : abscissaOfAbsConv g < ⊤\nx₀ : ℝ\nhx₀ : ∀ b ≥ x₀, LSeries f ↑b = LSeries g ↑b\nyf : ℝ\nhyf₁ : abscissaOfAbsConv f < ↑yf\nhyf₂ : ↑yf < ⊤\nyg : ℝ\nhyg₁ : abscissaOfAbsConv g < ↑yg\nhyg₂ : ↑yg < ⊤\nx : ℝ\nhx : x ≥ max x₀ (max yf yg)\nHf : LSeriesSummable f ↑x... | Pi.zero_apply | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LSeries.PrimesInAP | {
"line": 323,
"column": 4
} | {
"line": 323,
"column": 45
} | [
{
"pp": "case refine_2\nq : ℕ\na : ZMod q\ninst✝ : NeZero q\nχ : DirichletCharacter ℂ q\nhχ : χ ≠ 1\ns : ℂ\nhs : s ∈ {s | s = 1 ∨ ∀ (χ : DirichletCharacter ℂ q), LFunction χ s ≠ 0}\n⊢ s ∈ {s | LFunction χ s ≠ 0}",
"usedConstants": []
}
] | simp only [ne_eq, Set.mem_setOf_eq] at hs | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.LSeries.ZetaZeros | {
"line": 54,
"column": 2
} | {
"line": 55,
"column": 64
} | [
{
"pp": "case inr\nthis : ∀ (x : ℂ), ¬x = 1 → ({1}ᶜ ∩ riemannZetaZeros)ᶜ ∈ nhdsWithin x {x}ᶜ\nx : ℂ\nhx : x ≠ 1\n⊢ riemannZetaZerosᶜ ∈ nhdsWithin x {x}ᶜ",
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"Compl.compl",
"nhdsWithin",
"Complex.instNormedField",
"PseudoMe... | · exact Filter.mem_of_superset (this x hx)
(by grind [riemannZeta_one_ne_zero, mem_riemannZetaZeros]) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.LSeries.SumCoeff | {
"line": 146,
"column": 2
} | {
"line": 146,
"column": 10
} | [
{
"pp": "f : ℕ → ℂ\nr : ℝ\nhr : 0 ≤ r\ns : ℂ\nhs : r < s.re\nhS : LSeriesSummable (fun n ↦ if n = 0 then 0 else f n) s\nhO : (fun n ↦ ∑ k ∈ Icc 1 n, f k) =O[atTop] fun n ↦ ↑n ^ r\nthis : ∀ (n : ℕ), (∑ k ∈ Icc 1 n, if k = 0 then 0 else f k) = ∑ k ∈ Icc 1 n, f k\n⊢ s * ∫ (t : ℝ) in Set.Ioi 1, (∑ k ∈ Icc 1 ⌊t⌋₊, i... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.NumberTheory.LSeries.SumCoeff | {
"line": 146,
"column": 2
} | {
"line": 146,
"column": 10
} | [
{
"pp": "f : ℕ → ℂ\nr : ℝ\nhr : 0 ≤ r\ns : ℂ\nhs : r < s.re\nhS : LSeriesSummable (fun n ↦ if n = 0 then 0 else f n) s\nhO : (fun n ↦ ∑ k ∈ Icc 1 n, f k) =O[atTop] fun n ↦ ↑n ^ r\nthis : ∀ (n : ℕ), (∑ k ∈ Icc 1 n, if k = 0 then 0 else f k) = ∑ k ∈ Icc 1 n, f k\n⊢ (fun n ↦ ∑ k ∈ Icc 1 n, if k = 0 then 0 else f k... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.NumberTheory.LSeries.SumCoeff | {
"line": 146,
"column": 2
} | {
"line": 146,
"column": 10
} | [
{
"pp": "f : ℕ → ℂ\nr : ℝ\nhr : 0 ≤ r\ns : ℂ\nhs : r < s.re\nhS : LSeriesSummable (fun n ↦ if n = 0 then 0 else f n) s\nhO : (fun n ↦ ∑ k ∈ Icc 1 n, f k) =O[atTop] fun n ↦ ↑n ^ r\nthis : ∀ (n : ℕ), (∑ k ∈ Icc 1 n, if k = 0 then 0 else f k) = ∑ k ∈ Icc 1 n, f k\n⊢ ∀ {n : ℕ}, n ≠ 0 → ¬n = 0",
"usedConstants":... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.NumberTheory.LSeries.SumCoeff | {
"line": 178,
"column": 4
} | {
"line": 178,
"column": 22
} | [
{
"pp": "f : ℕ → ℂ\nl : ℂ\nhlim : Tendsto (fun n ↦ (∑ k ∈ Icc 1 n, f k) / ↑n) atTop (𝓝 l)\ns : ℝ\nhs : 1 < s\n⊢ LocallyIntegrableOn (fun t ↦ (∑ k ∈ Icc 1 ⌊t⌋₊, f k) * ↑t ^ (-↑s - 1)) (Set.Ici 1) volume",
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Real.instI... | simp_rw [mul_comm] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.NumberTheory.LSeries.SumCoeff | {
"line": 193,
"column": 44
} | {
"line": 193,
"column": 58
} | [
{
"pp": "case refine_2\nf : ℕ → ℂ\nl : ℂ\nhlim : Tendsto (fun n ↦ (∑ k ∈ Icc 1 n, f k) / ↑n) atTop (𝓝 l)\ns : ℝ\nhs : 1 < s\nh₁ : LocallyIntegrableOn (fun t ↦ (∑ k ∈ Icc 1 ⌊t⌋₊, f k) * ↑t ^ (-↑s - 1)) (Set.Ici 1) volume\nh₂ : (fun t ↦ ∑ k ∈ Icc 1 ⌊t⌋₊, f k) =O[atTop] fun t ↦ t ^ 1\n⊢ -s < -1",
"usedConstan... | neg_lt_neg_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LSeries.SumCoeff | {
"line": 249,
"column": 43
} | {
"line": 249,
"column": 57
} | [
{
"pp": "l : ℂ\ns T ε : ℝ\nS : ℝ → ℂ\nhS : LocallyIntegrableOn (fun t ↦ S t - l * ↑t) (Set.Ici 1) volume\nhε : 0 < ε\nhs : 1 < s\nhT₁ : 1 ≤ T\nhT : ∀ t ≥ T, ‖S t - l * ↑t‖ ≤ ε * t\nhT₀ : 0 < T\nh : ∀ {t : ℝ}, 0 < t → t ^ (-s) = t * t ^ (-s - 1)\n⊢ -s < -1",
"usedConstants": [
"IsRightCancelAdd.addRigh... | neg_lt_neg_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LSeries.SumCoeff | {
"line": 280,
"column": 49
} | {
"line": 280,
"column": 63
} | [
{
"pp": "f : ℕ → ℂ\nl : ℂ\nhlim : Tendsto (fun n ↦ (∑ k ∈ Icc 1 n, f k) / ↑n) atTop (𝓝 l)\nhfS : ∀ (s : ℝ), 1 < s → LSeriesSummable f ↑s\nε : ℝ\nhε : ε > 0\nT : ℝ\nhT₁ : 1 ≤ T\nhT : ∀ (y : ℝ), T ≤ y → ‖∑ k ∈ Icc 1 ⌊y⌋₊, f k - l * ↑y‖ < ε * y\nS : ℝ → ℂ := fun t ↦ ∑ k ∈ Icc 1 ⌊t⌋₊, f k\nC : ℝ := ∫ (t : ℝ) in Se... | neg_lt_neg_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LSeries.SumCoeff | {
"line": 302,
"column": 38
} | {
"line": 302,
"column": 52
} | [
{
"pp": "f : ℕ → ℂ\nl : ℂ\nhlim : Tendsto (fun n ↦ (∑ k ∈ Icc 1 n, f k) / ↑n) atTop (𝓝 l)\nhfS : ∀ (s : ℝ), 1 < s → LSeriesSummable f ↑s\nε : ℝ\nhε : ε > 0\nT : ℝ\nhT₁ : 1 ≤ T\nhT : ∀ (y : ℝ), T ≤ y → ‖∑ k ∈ Icc 1 ⌊y⌋₊, f k - l * ↑y‖ < ε * y\nS : ℝ → ℂ := fun t ↦ ∑ k ∈ Icc 1 ⌊t⌋₊, f k\nC : ℝ := ∫ (t : ℝ) in Se... | neg_lt_neg_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol | {
"line": 263,
"column": 6
} | {
"line": 263,
"column": 38
} | [
{
"pp": "a : ℤ\nn : ℕ\nh : J(a | n) = -1\nhn₀ : n ≠ 0\nhf₀ : ∀ p ∈ n.primeFactorsList, p ≠ 0\n⊢ ∃ p, Nat.Prime p ∧ p ∣ n ∧ J(a | p) = -1",
"usedConstants": [
"Nat.instOne",
"congrArg",
"Eq.mp",
"Int.instNegInt",
"instMulNat",
"Int",
"instOfNat",
"Nat",
"... | ← Nat.prod_primeFactorsList hn₀, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LSeries.SumCoeff | {
"line": 355,
"column": 4
} | {
"line": 355,
"column": 48
} | [
{
"pp": "case refine_2\nf : ℕ → ℂ\nl : ℂ\nhlim : Tendsto (fun n ↦ (∑ k ∈ Icc 1 n, f k) / ↑n) atTop (𝓝 l)\nhfS : ∀ (s : ℝ), 1 < s → LSeriesSummable f ↑s\nh₁ : ∀ {C ε : ℝ}, Tendsto (fun s ↦ (s - 1) * s * C + s * ε) (𝓝[>] 1) (𝓝 ε)\nh₂ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) (𝓝[>] 1) fun s ↦ ‖(↑s - 1) * LSeries ... | refine le_of_forall_pos_le_add fun ε hε ↦ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.Algebra.Valued.LocallyCompact | {
"line": 193,
"column": 10
} | {
"line": 193,
"column": 32
} | [
{
"pp": "case inl\nK : Type u_1\nΓ₀ : Type u_2\ninst✝² : Field K\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : Valued K Γ₀\nhc : IsCompact ↑𝒪[K]\nthis : ∀ (z : (↥(MonoidHom.mrange v))ˣ), (Set.Icc z 1).Finite\nx y : (↥(MonoidHom.mrange v))ˣ\nhxy : y < x\n⊢ (Set.Icc x y).Finite",
"usedConstants": [
... | Set.Icc_eq_empty_of_lt | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Algebra.Valued.LocallyCompact | {
"line": 218,
"column": 8
} | {
"line": 218,
"column": 30
} | [
{
"pp": "case val.inl\nK : Type u_1\nΓ₀ : Type u_2\ninst✝² : Field K\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : Valued K Γ₀\nhc : IsCompact ↑𝒪[K]\nz : (↥(MonoidHom.mrange v))ˣ\na : K\nha : v a = ↑↑z\nhz1 : 1 < z\n⊢ (Set.Icc z 1).Finite",
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toM... | Set.Icc_eq_empty_of_lt | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Algebra.Valued.LocallyCompact | {
"line": 234,
"column": 4
} | {
"line": 234,
"column": 21
} | [
{
"pp": "case val.inr.specialize_1\nK : Type u_1\nΓ₀ : Type u_2\ninst✝² : Field K\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : Valued K Γ₀\nhc : IsCompact ↑𝒪[K]\nz : (↥(MonoidHom.mrange v))ˣ\na : K\nha : v a = ↑↑z\nhz1 : z ≤ 1\nz0' : 0 < ↑z\nz0 : 0 < ↑↑z\na0 : 0 < v a\nU : K → Set K := fun y ↦ if v y ≤... | split_ifs with hw | Mathlib.Tactic._aux_Mathlib_Tactic_SplitIfs___elabRules_Mathlib_Tactic_splitIfs_1 | Mathlib.Tactic.splitIfs |
Mathlib.Topology.Algebra.Valued.LocallyCompact | {
"line": 252,
"column": 4
} | {
"line": 252,
"column": 12
} | [
{
"pp": "case pos\nK : Type u_1\nΓ₀ : Type u_2\ninst✝² : Field K\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : Valued K Γ₀\nhc : IsCompact ↑𝒪[K]\nz : (↥(MonoidHom.mrange v))ˣ\na : K\nha : v a = ↑↑z\nhz1 : z ≤ 1\nz0' : 0 < ↑z\nz0 : 0 < ↑↑z\na0 : 0 < v a\nU : K → Set K := fun y ↦ if v y ≤ ↑↑z then {w | v ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Topology.Algebra.Valued.LocallyCompact | {
"line": 252,
"column": 4
} | {
"line": 252,
"column": 12
} | [
{
"pp": "case neg\nK : Type u_1\nΓ₀ : Type u_2\ninst✝² : Field K\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : Valued K Γ₀\nhc : IsCompact ↑𝒪[K]\nz : (↥(MonoidHom.mrange v))ˣ\na : K\nha : v a = ↑↑z\nhz1 : z ≤ 1\nz0' : 0 < ↑z\nz0 : 0 < ↑↑z\na0 : 0 < v a\nU : K → Set K := fun y ↦ if v y ≤ ↑↑z then {w | v ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Topology.Algebra.Valued.LocallyCompact | {
"line": 252,
"column": 4
} | {
"line": 252,
"column": 12
} | [
{
"pp": "case neg\nK : Type u_1\nΓ₀ : Type u_2\ninst✝² : Field K\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : Valued K Γ₀\nhc : IsCompact ↑𝒪[K]\nz : (↥(MonoidHom.mrange v))ˣ\na : K\nha : v a = ↑↑z\nhz1 : z ≤ 1\nz0' : 0 < ↑z\nz0 : 0 < ↑↑z\na0 : 0 < v a\nU : K → Set K := fun y ↦ if v y ≤ ↑↑z then {w | v ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Topology.Algebra.Valued.LocallyCompact | {
"line": 308,
"column": 43
} | {
"line": 308,
"column": 71
} | [
{
"pp": "K : Type u_3\nΓ : Type u_4\ninst✝¹ : Field K\ninst✝ : LinearOrderedCommGroupWithZero Γ\nv : Valuation K Γ\nx✝ : ∃ x, x ≠ 0 ∧ v x < 1\nx : K\nhx0 : x ≠ 0\nhx1 : v x < 1\n⊢ v ↑⟨x, ⋯⟩ < 1 ∧ ¬⟨x, ⋯⟩ = 0",
"usedConstants": [
"LinearOrderedCommGroupWithZero.toLinearOrderedCommMonoidWithZero",
... | by simp [Subtype.ext_iff, *] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.LucasLehmer | {
"line": 205,
"column": 4
} | {
"line": 209,
"column": 39
} | [
{
"pp": "case mp\np : ℕ\nw : 1 < p\n⊢ ↑(sMod p (p - 2)) = 0 → sMod p (p - 2) = 0",
"usedConstants": [
"Int.cast",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Nat.instMulZeroClass",
"Preorder.toLT",
"Dvd.dvd",
"LinearOrderedCommMonoidWithZero.toIsBotZeroClass",
"ZMod... | intro h
apply Int.eq_zero_of_dvd_of_nonneg_of_lt _ _
(by simpa [ZMod.intCast_zmod_eq_zero_iff_dvd] using h) <;> clear h
· exact sMod_nonneg _ (by positivity) _
· exact sMod_lt _ (by positivity) _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.LucasLehmer | {
"line": 205,
"column": 4
} | {
"line": 209,
"column": 39
} | [
{
"pp": "case mp\np : ℕ\nw : 1 < p\n⊢ ↑(sMod p (p - 2)) = 0 → sMod p (p - 2) = 0",
"usedConstants": [
"Int.cast",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Nat.instMulZeroClass",
"Preorder.toLT",
"Dvd.dvd",
"LinearOrderedCommMonoidWithZero.toIsBotZeroClass",
"ZMod... | intro h
apply Int.eq_zero_of_dvd_of_nonneg_of_lt _ _
(by simpa [ZMod.intCast_zmod_eq_zero_iff_dvd] using h) <;> clear h
· exact sMod_nonneg _ (by positivity) _
· exact sMod_lt _ (by positivity) _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.MaricaSchoenheim | {
"line": 56,
"column": 4
} | {
"line": 57,
"column": 53
} | [
{
"pp": "case calc_2\nn : ℕ\nf : ℕ → ℕ\nhf' : ∀ k < n, Squarefree (f k)\nhn : n ≠ 0\nhf : StrictMonoOn f (Set.Iio n)\nthis : ∀ i < n, ∀ j < n, f i < (f i).gcd (f j) * n\n𝒜 : Finset (Finset ℕ) := image (fun n ↦ (f n).primeFactors) (Iio n)\nhf'' : ∀ i < n, ∀ (j : ℕ), Squarefree (f i / (f i).gcd (f j))\ni : ℕ\nhi... | rw [← primeFactors_div_gcd (hf' _ hi) (hf' _ hj).ne_zero,
prod_primeFactors_of_squarefree <| hf'' _ hi _] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.MaricaSchoenheim | {
"line": 64,
"column": 6
} | {
"line": 64,
"column": 51
} | [
{
"pp": "case calc_3\nn : ℕ\nf : ℕ → ℕ\nhf' : ∀ k < n, Squarefree (f k)\nhn : n ≠ 0\nhf : StrictMonoOn f (Set.Iio n)\nthis : ∀ i < n, ∀ j < n, f i < (f i).gcd (f j) * n\n𝒜 : Finset (Finset ℕ) := image (fun n ↦ (f n).primeFactors) (Iio n)\nhf'' : ∀ i < n, ∀ (j : ℕ), Squarefree (f i / (f i).gcd (f j))\na : ℕ\nha... | prod_primeFactors_of_squarefree (hf'' _ hc _) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.MahlerMeasure | {
"line": 143,
"column": 4
} | {
"line": 143,
"column": 12
} | [
{
"pp": "case inl\np : ℤ[X]\nh : (map (castRingHom ℂ) p).mahlerMeasure = 1\nh✝ : p = 0\n⊢ ‖(map (castRingHom ℂ) p).leadingCoeff‖ = 1",
"usedConstants": [
"Norm.norm",
"False",
"Real",
"Complex.commRing",
"Real.instZero",
"congrArg",
"False.elim",
"Complex.inst... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.NumberTheory.MahlerMeasure | {
"line": 143,
"column": 4
} | {
"line": 143,
"column": 12
} | [
{
"pp": "case inl\np : ℤ[X]\nh : (map (castRingHom ℂ) p).mahlerMeasure = 1\nh✝ : p = 0\n⊢ ‖(map (castRingHom ℂ) p).leadingCoeff‖ = 1",
"usedConstants": [
"Norm.norm",
"False",
"Real",
"Complex.commRing",
"Real.instZero",
"congrArg",
"False.elim",
"Complex.inst... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.MahlerMeasure | {
"line": 143,
"column": 4
} | {
"line": 143,
"column": 12
} | [
{
"pp": "case inl\np : ℤ[X]\nh : (map (castRingHom ℂ) p).mahlerMeasure = 1\nh✝ : p = 0\n⊢ ‖(map (castRingHom ℂ) p).leadingCoeff‖ = 1",
"usedConstants": [
"Norm.norm",
"False",
"Real",
"Complex.commRing",
"Real.instZero",
"congrArg",
"False.elim",
"Complex.inst... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Order.ArchimedeanDiscrete | {
"line": 48,
"column": 4
} | {
"line": 48,
"column": 12
} | [
{
"pp": "case inr.h.mpr\nG✝ : Type u_1\ninst✝⁶ : CommGroup G✝\ninst✝⁵ : TopologicalSpace G✝\nG : Type u_1\ninst✝⁴ : CommGroup G\ninst✝³ : LinearOrder G\ninst✝² : IsOrderedMonoid G\ninst✝¹ : TopologicalSpace G\ninst✝ : OrderTopology G\ng : G\nha✝ : 1 ≤ g\nha : 1 < g\nn : ℤ\n⊢ ⟨(fun x ↦ g ^ x) n, ⋯⟩ ∈ {1} → ⟨(fun... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Topology.Algebra.Order.ArchimedeanDiscrete | {
"line": 48,
"column": 4
} | {
"line": 48,
"column": 12
} | [
{
"pp": "case inr.h.mpr\nG✝ : Type u_1\ninst✝⁶ : CommGroup G✝\ninst✝⁵ : TopologicalSpace G✝\nG : Type u_1\ninst✝⁴ : CommGroup G\ninst✝³ : LinearOrder G\ninst✝² : IsOrderedMonoid G\ninst✝¹ : TopologicalSpace G\ninst✝ : OrderTopology G\ng : G\nha✝ : 1 ≤ g\nha : 1 < g\nn : ℤ\n⊢ ⟨(fun x ↦ g ^ x) n, ⋯⟩ ∈ {1} → ⟨(fun... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.Order.ArchimedeanDiscrete | {
"line": 48,
"column": 4
} | {
"line": 48,
"column": 12
} | [
{
"pp": "case inr.h.mpr\nG✝ : Type u_1\ninst✝⁶ : CommGroup G✝\ninst✝⁵ : TopologicalSpace G✝\nG : Type u_1\ninst✝⁴ : CommGroup G\ninst✝³ : LinearOrder G\ninst✝² : IsOrderedMonoid G\ninst✝¹ : TopologicalSpace G\ninst✝ : OrderTopology G\ng : G\nha✝ : 1 ≤ g\nha : 1 < g\nn : ℤ\n⊢ ⟨(fun x ↦ g ^ x) n, ⋯⟩ ∈ {1} → ⟨(fun... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Compactification.OnePoint.ProjectiveLine | {
"line": 98,
"column": 51
} | {
"line": 98,
"column": 59
} | [
{
"pp": "K : Type u_1\ninst✝¹ : DivisionRing K\ninst✝ : DecidableEq K\nw : Fin 2 → K\nhw : w ≠ 0\nh₀ : w 1 = 0\nh : w 0 = 0\n⊢ ∀ (x : Fin 2), w x = 0 x",
"usedConstants": [
"congrArg",
"and_self",
"AddMonoid.toAddZeroClass",
"AddCommGroup.toAddGroup",
"DivisionRing.toDivisionSe... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Topology.Compactification.OnePoint.ProjectiveLine | {
"line": 98,
"column": 51
} | {
"line": 98,
"column": 59
} | [
{
"pp": "K : Type u_1\ninst✝¹ : DivisionRing K\ninst✝ : DecidableEq K\nw : Fin 2 → K\nhw : w ≠ 0\nh₀ : w 1 = 0\nh : w 0 = 0\n⊢ ∀ (x : Fin 2), w x = 0 x",
"usedConstants": [
"congrArg",
"and_self",
"AddMonoid.toAddZeroClass",
"AddCommGroup.toAddGroup",
"DivisionRing.toDivisionSe... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Compactification.OnePoint.ProjectiveLine | {
"line": 98,
"column": 51
} | {
"line": 98,
"column": 59
} | [
{
"pp": "K : Type u_1\ninst✝¹ : DivisionRing K\ninst✝ : DecidableEq K\nw : Fin 2 → K\nhw : w ≠ 0\nh₀ : w 1 = 0\nh : w 0 = 0\n⊢ ∀ (x : Fin 2), w x = 0 x",
"usedConstants": [
"congrArg",
"and_self",
"AddMonoid.toAddZeroClass",
"AddCommGroup.toAddGroup",
"DivisionRing.toDivisionSe... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Compactification.OnePoint.ProjectiveLine | {
"line": 101,
"column": 22
} | {
"line": 101,
"column": 30
} | [
{
"pp": "case h.h.«0»\nK : Type u_1\ninst✝¹ : DivisionRing K\ninst✝ : DecidableEq K\nw : Fin 2 → K\nhw : w ≠ 0\nh₀ : w 1 = 0\nthis : w 0 ≠ 0\n⊢ ((w 0)⁻¹ • w) ((fun i ↦ i) ⟨0, ⋯⟩) = ![1, 0] ((fun i ↦ i) ⟨0, ⋯⟩)",
"usedConstants": [
"Pi.Function.module",
"GroupWithZero.toMonoidWithZero",
"Mu... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Topology.Compactification.OnePoint.ProjectiveLine | {
"line": 101,
"column": 22
} | {
"line": 101,
"column": 30
} | [
{
"pp": "case h.h.«1»\nK : Type u_1\ninst✝¹ : DivisionRing K\ninst✝ : DecidableEq K\nw : Fin 2 → K\nhw : w ≠ 0\nh₀ : w 1 = 0\nthis : w 0 ≠ 0\n⊢ ((w 0)⁻¹ • w) ((fun i ↦ i) ⟨1, ⋯⟩) = ![1, 0] ((fun i ↦ i) ⟨1, ⋯⟩)",
"usedConstants": [
"Pi.Function.module",
"instNeZeroNatHAdd_1",
"instHSMul",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Topology.Compactification.OnePoint.ProjectiveLine | {
"line": 179,
"column": 4
} | {
"line": 181,
"column": 9
} | [
{
"pp": "case pos\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : DecidableEq K\nc : K\ng : GL (Fin 2) K\nh : ↑g 1 0 * c + ↑g 1 1 = 0\n⊢ ↑g 1 0 * c ^ 2 + (↑g 1 1 - ↑g 0 0) * c - ↑g 0 1 = 0 ↔ ∞ = ↑c",
"usedConstants": [
"Units.val",
"Eq.mpr",
"HMul.hMul",
"OnePoint.infty",
"AddGroupWit... | refine ⟨fun hg ↦ (g.det_ne_zero ?_).elim, fun hg ↦ (infty_ne_coe _ hg).elim⟩
rw [det_fin_two]
grind | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Compactification.OnePoint.ProjectiveLine | {
"line": 179,
"column": 4
} | {
"line": 181,
"column": 9
} | [
{
"pp": "case pos\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : DecidableEq K\nc : K\ng : GL (Fin 2) K\nh : ↑g 1 0 * c + ↑g 1 1 = 0\n⊢ ↑g 1 0 * c ^ 2 + (↑g 1 1 - ↑g 0 0) * c - ↑g 0 1 = 0 ↔ ∞ = ↑c",
"usedConstants": [
"Units.val",
"Eq.mpr",
"HMul.hMul",
"OnePoint.infty",
"AddGroupWit... | refine ⟨fun hg ↦ (g.det_ne_zero ?_).elim, fun hg ↦ (infty_ne_coe _ hg).elim⟩
rw [det_fin_two]
grind | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.ModularForms.SlashActions | {
"line": 203,
"column": 2
} | {
"line": 203,
"column": 29
} | [
{
"pp": "case h\nk1 k2 : ℤ\nA : GL (Fin 2) ℝ\nf g : ℍ → ℂ\nx : ℍ\n⊢ (σ A) (f (A • x)) * (σ A) (g (A • x)) * ↑|↑(Matrix.GeneralLinearGroup.det A)| ^ (k1 + k2 - 1) *\n (denom A ↑x ^ (-k1) * denom A ↑x ^ (-k2)) =\n ↑|↑(Matrix.GeneralLinearGroup.det A)| *\n ((σ A) (f (A • x)) * ↑|↑(Matrix.GeneralLinear... | set d := (↑|A.det.val| : ℂ) | Mathlib.Tactic._aux_Mathlib_Tactic_Set___elabRules_Mathlib_Tactic_setTactic_1 | Mathlib.Tactic.setTactic |
Mathlib.NumberTheory.ModularForms.SlashInvariantForms | {
"line": 167,
"column": 29
} | {
"line": 167,
"column": 47
} | [
{
"pp": "F : Type u_1\nΓ : Subgroup (GL (Fin 2) ℝ)\nk : ℤ\ninst✝³ : FunLike F ℍ ℂ\nα : Type u_2\ninst✝² : SMul α ℂ\ninst✝¹ : SMul α ℝ\ninst✝ : IsScalarTower α ℝ ℂ\nc : α\nf : SlashInvariantForm Γ k\nγ : GL (Fin 2) ℝ\nhγ : γ ∈ Γ\n⊢ ((c • 1) • ⇑f) ∣[k] γ = (c • 1) • ⇑f",
"usedConstants": [
"SlashInvaria... | ← smul_one_smul ℂ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.ModularForms.BoundedAtCusp | {
"line": 37,
"column": 2
} | {
"line": 40,
"column": 68
} | [
{
"pp": "g : GL (Fin 2) ℝ\nf : ℍ → ℂ\nk : ℤ\nhg : ↑g 1 0 = 0\nhf : IsBoundedAtImInfty f\n⊢ IsBoundedAtImInfty (f ∣[k] g)",
"usedConstants": [
"UpperHalfPlane.glAction",
"UpperHalfPlane.IsBoundedAtImInfty.eq_1",
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Norm.norm",... | rw [IsBoundedAtImInfty, BoundedAtFilter, ← Asymptotics.isBigO_norm_left] at hf ⊢
suffices (fun x ↦ (‖g.det.val ^ (k - 1)‖ * ‖g 1 1 ^ (-k)‖) * ‖f (g • x)‖) =O[atImInfty] 1 by
simpa [ModularForm.slash_def, denom, hg, mul_assoc, mul_comm ‖f _‖]
apply (hf.comp_tendsto (tendsto_smul_atImInfty hg)).const_mul_left | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.ModularForms.BoundedAtCusp | {
"line": 37,
"column": 2
} | {
"line": 40,
"column": 68
} | [
{
"pp": "g : GL (Fin 2) ℝ\nf : ℍ → ℂ\nk : ℤ\nhg : ↑g 1 0 = 0\nhf : IsBoundedAtImInfty f\n⊢ IsBoundedAtImInfty (f ∣[k] g)",
"usedConstants": [
"UpperHalfPlane.glAction",
"UpperHalfPlane.IsBoundedAtImInfty.eq_1",
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Norm.norm",... | rw [IsBoundedAtImInfty, BoundedAtFilter, ← Asymptotics.isBigO_norm_left] at hf ⊢
suffices (fun x ↦ (‖g.det.val ^ (k - 1)‖ * ‖g 1 1 ^ (-k)‖) * ‖f (g • x)‖) =O[atImInfty] 1 by
simpa [ModularForm.slash_def, denom, hg, mul_assoc, mul_comm ‖f _‖]
apply (hf.comp_tendsto (tendsto_smul_atImInfty hg)).const_mul_left | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.ModularForms.Cusps | {
"line": 102,
"column": 2
} | {
"line": 102,
"column": 27
} | [
{
"pp": "𝒢 ℋ : Subgroup (GL (Fin 2) ℝ)\nc : OnePoint ℝ\ninst✝ : 𝒢.IsFiniteRelIndex ℋ\nhc : IsCusp c (𝒢 ⊓ ℋ)\nhGH : (𝒢 ⊓ ℋ).relIndex ℋ ≠ 0\n⊢ IsCusp c 𝒢",
"usedConstants": [
"Real",
"Matrix",
"instDecidableEqFin",
"CompleteLattice.toConditionallyCompleteLattice",
"Real.semi... | exact hc.mono inf_le_left | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.ModularForms.LevelOne.Basic | {
"line": 83,
"column": 4
} | {
"line": 83,
"column": 70
} | [
{
"pp": "case refine_3.inl\nF : Type u_1\ninst✝¹ : FunLike F ℍ ℂ\nk : ℤ\ninst✝ : ModularFormClass F (Matrix.SpecialLinearGroup.mapGL ℝ).range k\nhk : k ≤ 0\nf : F\nhq : 0 ∈ Metric.ball 0 1\n⊢ ∃ w, ‖w‖ ≤ rexp (-π) ∧ ‖UpperHalfPlane.cuspFunction 1 (⇑f) 0‖ ≤ ‖UpperHalfPlane.cuspFunction 1 (⇑f) w‖",
"usedConsta... | · refine ⟨0, by simpa only [norm_zero] using exp_nonneg _, le_rfl⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.ModularForms.Bounds | {
"line": 58,
"column": 2
} | {
"line": 58,
"column": 33
} | [
{
"pp": "E✝ : Type u_1\ninst✝ : SeminormedAddCommGroup E✝\nf : ℍ → E✝\nhf_cont : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f\nt : ℝ\nhf_infinity : f =O[atImInfty] fun z ↦ z.im ^ t\nD : ℝ\nhD : D > 0\ny : ℝ\nhy : ∀ b ≥ y, ∀ (a : ℍ), a.im = b → ‖f a‖ ≤ D * a.im ^ t\nhfm : ContinuousOn (fu... | refine ⟨max D E, fun τ hτ ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.ModularForms.Bounds | {
"line": 83,
"column": 4
} | {
"line": 83,
"column": 45
} | [
{
"pp": "case ha\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)\nhg : g •... | rw [← div_le_iff₀ (by positivity)] at hF𝒟 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.Modular | {
"line": 506,
"column": 4
} | {
"line": 507,
"column": 8
} | [
{
"pp": "case inl\ng✝ : SL(2, ℤ)\nz✝ : ℍ\ng : SL(2, ℤ)\nz : ℍ\nhz : z ∈ 𝒟\nhg : g • z ∈ 𝒟\nhc : ↑g 1 0 = 0\nhd' : ↑g 1 1 = 1 ∨ ↑g 1 1 = -1\nhd : ↑g 1 1 = 1\nha : ↑g 0 0 = 1\nb : ℤ := ↑g 0 1\nhgz : g = T ^ 0\nhre : (g • z).re = ↑b + z.re\nhb : b = 0\n⊢ (g = T ∨ g = -T) ∧ z.re = -1 / 2 ∨ (g = T⁻¹ ∨ g = -T⁻¹) ∧ ... | rw [hgz]
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Modular | {
"line": 506,
"column": 4
} | {
"line": 507,
"column": 8
} | [
{
"pp": "case inl\ng✝ : SL(2, ℤ)\nz✝ : ℍ\ng : SL(2, ℤ)\nz : ℍ\nhz : z ∈ 𝒟\nhg : g • z ∈ 𝒟\nhc : ↑g 1 0 = 0\nhd' : ↑g 1 1 = 1 ∨ ↑g 1 1 = -1\nhd : ↑g 1 1 = 1\nha : ↑g 0 0 = 1\nb : ℤ := ↑g 0 1\nhgz : g = T ^ 0\nhre : (g • z).re = ↑b + z.re\nhb : b = 0\n⊢ (g = T ∨ g = -T) ∧ z.re = -1 / 2 ∨ (g = T⁻¹ ∨ g = -T⁻¹) ∧ ... | rw [hgz]
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Modular | {
"line": 514,
"column": 13
} | {
"line": 514,
"column": 26
} | [
{
"pp": "case inr.inr.h.h\ng✝ : SL(2, ℤ)\nz✝ : ℍ\ng : SL(2, ℤ)\nz : ℍ\nhz : z ∈ 𝒟\nhg : g • z ∈ 𝒟\nhc : ↑g 1 0 = 0\nhd' : ↑g 1 1 = 1 ∨ ↑g 1 1 = -1\nhd : ↑g 1 1 = 1\nha : ↑g 0 0 = 1\nb : ℤ := ↑g 0 1\nhgz : g = T ^ (-1)\nhre : (g • z).re = ↑b + z.re\nhb : b = -1\n⊢ (T ^ (-1) = T⁻¹ ∨ T ^ (-1) = -T⁻¹) ∧ z.re = 1 ... | zpow_neg_one, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.TsumDivisorsAntidiagonal | {
"line": 87,
"column": 10
} | {
"line": 87,
"column": 83
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : CompleteSpace 𝕜\ninst✝ : NormSMulClass ℤ 𝕜\nk : ℕ\nr : 𝕜\nhr : ‖r‖ < 1\n⊢ Summable fun c ↦ ‖↑↑c ^ (k + 1) * r ^ ↑c‖",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Norm.norm",
"Real",
"NormedRing.toRing",
... | apply (summable_norm_pow_mul_geometric_of_norm_lt_one (k + 1) hr).subtype | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.NumberTheory.TsumDivisorsAntidiagonal | {
"line": 87,
"column": 10
} | {
"line": 87,
"column": 83
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : CompleteSpace 𝕜\ninst✝ : NormSMulClass ℤ 𝕜\nk : ℕ\nr : 𝕜\nhr : ‖r‖ < 1\n⊢ Summable fun c ↦ ‖↑↑c ^ (k + 1) * r ^ ↑c‖",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Norm.norm",
"Real",
"NormedRing.toRing",
... | apply (summable_norm_pow_mul_geometric_of_norm_lt_one (k + 1) hr).subtype | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.TsumDivisorsAntidiagonal | {
"line": 87,
"column": 10
} | {
"line": 87,
"column": 83
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : CompleteSpace 𝕜\ninst✝ : NormSMulClass ℤ 𝕜\nk : ℕ\nr : 𝕜\nhr : ‖r‖ < 1\n⊢ Summable fun c ↦ ‖↑↑c ^ (k + 1) * r ^ ↑c‖",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Norm.norm",
"Real",
"NormedRing.toRing",
... | apply (summable_norm_pow_mul_geometric_of_norm_lt_one (k + 1) hr).subtype | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.TsumDivisorsAntidiagonal | {
"line": 93,
"column": 8
} | {
"line": 93,
"column": 76
} | [
{
"pp": "case h.h₂\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : CompleteSpace 𝕜\ninst✝ : NormSMulClass ℤ 𝕜\nk : ℕ\nr : 𝕜\nhr : ‖r‖ < 1\nb : ℕ+\ni : ℕ\nhi : i ∈ (↑b).divisors\n⊢ ‖r‖ ^ (i * (↑b / i)) ≤ ‖r ^ ↑b‖",
"usedConstants": [
"PNat.val",
"Norm.norm",
"Eq.mpr",
... | rw [norm_pow, mul_comm, Nat.div_mul_cancel (dvd_of_mem_divisors hi)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.TsumDivisorsAntidiagonal | {
"line": 93,
"column": 8
} | {
"line": 93,
"column": 76
} | [
{
"pp": "case h.h₂\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : CompleteSpace 𝕜\ninst✝ : NormSMulClass ℤ 𝕜\nk : ℕ\nr : 𝕜\nhr : ‖r‖ < 1\nb : ℕ+\ni : ℕ\nhi : i ∈ (↑b).divisors\n⊢ ‖r‖ ^ (i * (↑b / i)) ≤ ‖r ^ ↑b‖",
"usedConstants": [
"PNat.val",
"Norm.norm",
"Eq.mpr",
... | rw [norm_pow, mul_comm, Nat.div_mul_cancel (dvd_of_mem_divisors hi)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.TsumDivisorsAntidiagonal | {
"line": 93,
"column": 8
} | {
"line": 93,
"column": 76
} | [
{
"pp": "case h.h₂\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : CompleteSpace 𝕜\ninst✝ : NormSMulClass ℤ 𝕜\nk : ℕ\nr : 𝕜\nhr : ‖r‖ < 1\nb : ℕ+\ni : ℕ\nhi : i ∈ (↑b).divisors\n⊢ ‖r‖ ^ (i * (↑b / i)) ≤ ‖r ^ ↑b‖",
"usedConstants": [
"PNat.val",
"Norm.norm",
"Eq.mpr",
... | rw [norm_pow, mul_comm, Nat.div_mul_cancel (dvd_of_mem_divisors hi)] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Modular | {
"line": 627,
"column": 4
} | {
"line": 627,
"column": 31
} | [
{
"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\nthis✝ : ↑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 : normSq (↑z - 1) = normSq ↑z ... | have : 1 ≤ normSq z := hz.1 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.ModularForms.DedekindEta | {
"line": 77,
"column": 2
} | {
"line": 77,
"column": 70
} | [
{
"pp": "case h\n⊢ HasProdLocallyUniformlyOn (fun n q ↦ 1 + -q ^ (n + 1)) (fun q ↦ ∏' (n : ℕ), (1 + -q ^ (n + 1))) (Metric.ball 0 1)",
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"locallyCompact_of_proper",
"Real",
"hasProdLocallyUniformlyOn_of_forall_compact",
"C... | apply hasProdLocallyUniformlyOn_of_forall_compact Metric.isOpen_ball | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.E2.Transform | {
"line": 114,
"column": 8
} | {
"line": 115,
"column": 12
} | [
{
"pp": "case pos\nz : ℍ\nb n : ℤ\nh : b = 0 → ¬n = 0\nhb : b = 0\nhn : n = -1\n⊢ (↑b * ↑z + ↑n + 1)⁻¹ * ((↑b * ↑z + ↑n) ^ 2)⁻¹ + δ ![b, n] + ((↑b * ↑z + ↑n)⁻¹ - (↑b * ↑z + ↑n + 1)⁻¹) =\n ((↑b * ↑z + ↑n) ^ 2)⁻¹",
"usedConstants": [
"one_pow",
"AddGroup.toSubtractionMonoid",
"Int.cast_ne... | simp [hb, hn, δ_eq_two]
ring | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.E2.Transform | {
"line": 114,
"column": 8
} | {
"line": 115,
"column": 12
} | [
{
"pp": "case pos\nz : ℍ\nb n : ℤ\nh : b = 0 → ¬n = 0\nhb : b = 0\nhn : n = -1\n⊢ (↑b * ↑z + ↑n + 1)⁻¹ * ((↑b * ↑z + ↑n) ^ 2)⁻¹ + δ ![b, n] + ((↑b * ↑z + ↑n)⁻¹ - (↑b * ↑z + ↑n + 1)⁻¹) =\n ((↑b * ↑z + ↑n) ^ 2)⁻¹",
"usedConstants": [
"one_pow",
"AddGroup.toSubtractionMonoid",
"Int.cast_ne... | simp [hb, hn, δ_eq_two]
ring | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Modular | {
"line": 771,
"column": 4
} | {
"line": 771,
"column": 93
} | [
{
"pp": "case mpr\ng : SL(2, ℤ)\n⊢ g ∈ {1, -1, S * T, -(S * T), T⁻¹ * S, -(T⁻¹ * S)} → g • ρ = ρ",
"usedConstants": [
"Real",
"Fintype.card_fin_two",
"instHSMul",
"Matrix.SpecialLinearGroup",
"UpperHalfPlane.SLAction",
"HMul.hMul",
"UpperHalfPlane.ρ",
"congrAr... | suffices (S * T) • ρ = ρ ∧ (T⁻¹ * S) • ρ = ρ by simp +contextual [-sl_moeb, or_imp, this] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.NumberTheory.Modular | {
"line": 891,
"column": 2
} | {
"line": 893,
"column": 68
} | [
{
"pp": "x : ℍ\nhxnorm : ‖↑x‖ = 1\nhxre : |x.re| ≤ 1 / 2\n⊢ x ∈ closure (closure 𝒟ᵒ)",
"usedConstants": [
"NNReal.instTopologicalSpace",
"Complex.mul_im",
"Eq.mpr",
"Real.instLE",
"Real",
"Set.Ioi",
"mem_closure_of_frequently_of_tendsto",
"HMul.hMul",
"... | apply mem_closure_of_frequently_of_tendsto (b := 𝓝[>] 0)
(f := fun t : ℝ≥0 ↦ ⟨x + t * Complex.I, by
simpa using add_pos_of_pos_of_nonneg x.coe_im_pos t.property⟩) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.NumberTheory.Modular | {
"line": 986,
"column": 2
} | {
"line": 986,
"column": 40
} | [
{
"pp": "τ : ℍ\n⊢ ∃ γ, 1 / 2 ≤ (γ • τ).im",
"usedConstants": [
"ModularGroup.exists_smul_mem_fd"
]
}
] | obtain ⟨γ, hγ⟩ := exists_smul_mem_fd τ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.NumberTheory.ModularForms.Discriminant | {
"line": 69,
"column": 6
} | {
"line": 69,
"column": 20
} | [
{
"pp": "z : ℍ\n⊢ logDeriv (η ∘ fun x ↦ -x⁻¹) ↑z = (↑z ^ 2)⁻¹ * logDeriv η (-(↑z)⁻¹)",
"usedConstants": [
"logDeriv",
"NormedCommRing.toNormedRing",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"logDeriv_comp",
"DivInvMonoid.toInv",
"NonUnitalCommRing.toNonUnita... | logDeriv_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.ModularForms.Discriminant | {
"line": 97,
"column": 4
} | {
"line": 97,
"column": 68
} | [
{
"pp": "⊢ Set.EqOn (logDeriv (η ∘ fun z ↦ -1 / z)) (logDeriv (sqrt * η)) upperHalfPlaneSet",
"usedConstants": [
"Membership.mem",
"UpperHalfPlane.mk",
"UpperHalfPlane.upperHalfPlaneSet",
"_private.Mathlib.NumberTheory.ModularForms.Discriminant.0.ModularForm.logDeriv_eta_comp_eq_logD... | exact fun z hz ↦ logDeriv_eta_comp_eq_logDeriv_csqrt_eta ⟨z, hz⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.ModularForms.Discriminant | {
"line": 97,
"column": 4
} | {
"line": 97,
"column": 68
} | [
{
"pp": "⊢ Set.EqOn (logDeriv (η ∘ fun z ↦ -1 / z)) (logDeriv (sqrt * η)) upperHalfPlaneSet",
"usedConstants": [
"Membership.mem",
"UpperHalfPlane.mk",
"UpperHalfPlane.upperHalfPlaneSet",
"_private.Mathlib.NumberTheory.ModularForms.Discriminant.0.ModularForm.logDeriv_eta_comp_eq_logD... | exact fun z hz ↦ logDeriv_eta_comp_eq_logDeriv_csqrt_eta ⟨z, hz⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.ModularForms.Discriminant | {
"line": 97,
"column": 4
} | {
"line": 97,
"column": 68
} | [
{
"pp": "⊢ Set.EqOn (logDeriv (η ∘ fun z ↦ -1 / z)) (logDeriv (sqrt * η)) upperHalfPlaneSet",
"usedConstants": [
"Membership.mem",
"UpperHalfPlane.mk",
"UpperHalfPlane.upperHalfPlaneSet",
"_private.Mathlib.NumberTheory.ModularForms.Discriminant.0.ModularForm.logDeriv_eta_comp_eq_logD... | exact fun z hz ↦ logDeriv_eta_comp_eq_logDeriv_csqrt_eta ⟨z, hz⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.QExpansion | {
"line": 271,
"column": 4
} | {
"line": 273,
"column": 62
} | [
{
"pp": "case inr\nk : ℕ\nhk : 3 ≤ k\nz : ℍ\nhk1 : 1 < k\nhk2 : 3 ≤ ↑k\nb : ℕ\nhb : b ≠ 0\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",
"usedConstants": [
"zpow_natCast",
"one_pow",
"Eq... | have : NeZero b := ⟨hb⟩
simpa [eisSummand_of_gammaSet_eq_divIntMap k z, tsum_mul_left, hb]
using (gammaSetDivGcdEquiv b).tsum_eq (eisSummand k · z) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.QExpansion | {
"line": 271,
"column": 4
} | {
"line": 273,
"column": 62
} | [
{
"pp": "case inr\nk : ℕ\nhk : 3 ≤ k\nz : ℍ\nhk1 : 1 < k\nhk2 : 3 ≤ ↑k\nb : ℕ\nhb : b ≠ 0\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",
"usedConstants": [
"zpow_natCast",
"one_pow",
"Eq... | have : NeZero b := ⟨hb⟩
simpa [eisSummand_of_gammaSet_eq_divIntMap k z, tsum_mul_left, hb]
using (gammaSetDivGcdEquiv b).tsum_eq (eisSummand k · z) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.ModularForms.LevelOne.GradedRing | {
"line": 46,
"column": 77
} | {
"line": 49,
"column": 75
} | [
{
"pp": "⊢ E₄CubeSubE₆SqForm.IsCuspForm",
"usedConstants": [
"ModularForm.IsCuspForm",
"ModularForm",
"Nat.instCanonicallyOrderedAdd",
"Subgroup.instHasDetOneRangeSpecialLinearGroupGeneralLinearGroupMapGL",
"MonoidHom.range",
"Real",
"Matrix.SpecialLinearGroup",
... | by
simp [isCuspForm_iff_coeffZero_eq_zero, E₄CubeSubE₆SqForm_qExpansion_eq,
PowerSeries.coeff_mul, -PowerSeries.coeff_zero_eq_constantCoeff,
E_qExpansion_coeff_zero _ ⟨2, rfl⟩, E_qExpansion_coeff_zero _ ⟨3, rfl⟩] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.ModularForms.LevelOne.GradedRing | {
"line": 84,
"column": 6
} | {
"line": 84,
"column": 51
} | [
{
"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... | ← map_sub (DirectSum.of (ModularForm 𝒮ℒ) 12), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.ModularForms.NormTrace | {
"line": 48,
"column": 11
} | {
"line": 48,
"column": 39
} | [
{
"pp": "case h\n𝒢 ℋ : Subgroup (GL (Fin 2) ℝ)\nF : Type u_1\nf : F\ninst✝¹ : FunLike F ℍ ℂ\nk : ℤ\ninst✝ : SlashInvariantFormClass F 𝒢 k\nh : GL (Fin 2) ℝ\nhh : h ∈ ℋ\nr : ↥ℋ\n⊢ quotientFunc f ⟦r⟧ ∣[k] h = quotientFunc f (⟨h, hh⟩⁻¹ • ⟦r⟧)",
"usedConstants": [
"instMulActionSubtypeGeneralLinearGroup... | simp [SlashAction.slash_mul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.ModularForms.NormTrace | {
"line": 48,
"column": 11
} | {
"line": 48,
"column": 39
} | [
{
"pp": "case h\n𝒢 ℋ : Subgroup (GL (Fin 2) ℝ)\nF : Type u_1\nf : F\ninst✝¹ : FunLike F ℍ ℂ\nk : ℤ\ninst✝ : SlashInvariantFormClass F 𝒢 k\nh : GL (Fin 2) ℝ\nhh : h ∈ ℋ\nr : ↥ℋ\n⊢ quotientFunc f ⟦r⟧ ∣[k] h = quotientFunc f (⟨h, hh⟩⁻¹ • ⟦r⟧)",
"usedConstants": [
"instMulActionSubtypeGeneralLinearGroup... | simp [SlashAction.slash_mul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.ModularForms.NormTrace | {
"line": 48,
"column": 11
} | {
"line": 48,
"column": 39
} | [
{
"pp": "case h\n𝒢 ℋ : Subgroup (GL (Fin 2) ℝ)\nF : Type u_1\nf : F\ninst✝¹ : FunLike F ℍ ℂ\nk : ℤ\ninst✝ : SlashInvariantFormClass F 𝒢 k\nh : GL (Fin 2) ℝ\nhh : h ∈ ℋ\nr : ↥ℋ\n⊢ quotientFunc f ⟦r⟧ ∣[k] h = quotientFunc f (⟨h, hh⟩⁻¹ • ⟦r⟧)",
"usedConstants": [
"instMulActionSubtypeGeneralLinearGroup... | simp [SlashAction.slash_mul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Multiplicity | {
"line": 92,
"column": 15
} | {
"line": 92,
"column": 17
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\na b : R\np : ℕ\nhp : Odd p\nh1 :\n ∀ (i : ℕ),\n (Ideal.Quotient.mk (span {↑p ^ 2})) ((a + ↑p * b) ^ i) =\n (Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)\ns : R := ↑p ^ 2\n⊢ (Ideal.Quotient.mk (span {s})) (∑ i ∈ range p, (a + ↑p * b)... | s, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.Multiplicity | {
"line": 164,
"column": 15
} | {
"line": 164,
"column": 46
} | [
{
"pp": "case refine_2\nR : Type u_1\ninst✝¹ : CommRing R\nx y : R\np : ℕ\ninst✝ : IsDomain R\nhp : Prime ↑p\nhp1 : Odd p\nhx : ¬↑p ∣ y\nk : R\nhk : x - y = ↑p * k\n⊢ ¬↑p * ↑p ∣ ↑p * y ^ (p - 1)",
"usedConstants": [
"Eq.mpr",
"Dvd.dvd",
"HMul.hMul",
"CommRing.toNonUnitalCommRing",
... | mul_dvd_mul_iff_left hp.ne_zero | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Multiplicity | {
"line": 283,
"column": 2
} | {
"line": 283,
"column": 73
} | [
{
"pp": "case refine_2.succ\nx✝ y : ℤ\nhx✝ : ¬2 ∣ x✝\nhxy : 4 ∣ x✝ - y\nhx_odd : Odd x✝\nhxy_even : Even (x✝ - y)\nhy_odd : Odd y\ni : ℕ\nx : ℤ\nhx : Odd x\n⊢ x ^ 2 ^ (i + 1) % 4 = 1",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"Monoid.toMulOneClass",
"CommSemiring.toNonUnitalCommSemi... | rw [pow_succ', mul_comm, pow_mul, Int.sq_mod_four_eq_one_of_odd hx.pow] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.Multiplicity | {
"line": 298,
"column": 2
} | {
"line": 298,
"column": 94
} | [
{
"pp": "case succ\nx y : ℤ\nhxy : 4 ∣ x - y\nhx : ¬2 ∣ x\nhx_odd : Odd x\nhxy_even : Even (x - y)\nhy_odd : Odd y\nn : ℕ\n⊢ emultiplicity 2 (x ^ (n + 1) - y ^ (n + 1)) = emultiplicity 2 (x - y) + emultiplicity 2 ↑(n + 1)",
"usedConstants": [
"Iff.mpr",
"False",
"congrArg",
"Nat.inst... | have h : FiniteMultiplicity 2 n.succ := Nat.finiteMultiplicity_iff.mpr ⟨by simp, n.succ_pos⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.Algebra.RestrictedProduct.TopologicalSpace | {
"line": 257,
"column": 8
} | {
"line": 257,
"column": 25
} | [
{
"pp": "ι : Type u_1\nR : ι → Type u_2\nA : (i : ι) → Set (R i)\ninst✝¹ : (i : ι) → TopologicalSpace (R i)\nS : Set ι\ninst✝ : ∀ (i : ι), WeaklyLocallyCompactSpace (R i)\nhS : cofinite ≤ 𝓟 S\nhAcompact : ∀ i ∈ S, IsCompact (A i)\nx : Πʳ (i : ι), [R i, A i]_[𝓟 S]\n⊢ ∃ s, IsCompact s ∧ s ∈ 𝓝 x",
"usedCons... | le_principal_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Algebra.RestrictedProduct.Basic | {
"line": 487,
"column": 4
} | {
"line": 487,
"column": 12
} | [
{
"pp": "case h\nι : Type u_1\nR : ι → Type u_2\nA✝ : (i : ι) → Set (R i)\n𝓕 𝓖 : Filter ι\nS : ι → Type u_3\nG : ι → Type u_4\ninst✝³ : (i : ι) → SetLike (S i) (G i)\nA : (i : ι) → S i\ninst✝² : DecidableEq ι\ninst✝¹ : (i : ι) → One (G i)\ninst✝ : ∀ (i : ι), OneMemClass (S i) (G i)\ni : ι\nx : G i\n⊢ ∀ a ∈ {i... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.NumberTheory.NumberField.Completion.InfinitePlace | {
"line": 159,
"column": 2
} | {
"line": 159,
"column": 69
} | [
{
"pp": "case a\nK : Type u_1\ninst✝ : Field K\nv : InfinitePlace K\nhv : (extensionEmbedding v).fieldRange = Complex.ofRealHom.fieldRange\nx : K\nr : ℝ\nhr : Complex.ofRealHom r = v.embedding x\n⊢ (star v.embedding) x = v.embedding x",
"usedConstants": [
"Real",
"congrArg",
"CommSemiring.... | simp [ComplexEmbedding.conjugate_coe_eq, ← hr, Complex.conj_ofReal] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.PolarCoord | {
"line": 322,
"column": 46
} | {
"line": 322,
"column": 72
} | [
{
"pp": "case h\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx✝ : polarSpace K\n⊢ (∀ (a : InfinitePlace K) (b : a.IsComplex), (x✝.1 a, x✝.2 ⟨a, b⟩) ∈ Complex.polarCoord.target) ↔\n (∀ (i : InfinitePlace K), i.IsComplex → x✝.1 i ∈ Set.Ioi 0) ∧\n ∀ (a : InfinitePlace K) (b : a.IsComplex), x✝.2 ... | Complex.polarCoord_target, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.FundamentalCone | {
"line": 154,
"column": 33
} | {
"line": 154,
"column": 46
} | [
{
"pp": "case h\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nc : ℝ\nx✝ : { w // w ≠ w₀ }\n⊢ 0 = 0 x✝",
"usedConstants": [
"Eq.mpr",
"Real",
"Real.instZero",
"congrArg",
"Pi.zero_apply",
"Real.semiring",
"id",
"Subtype",
"Pi.instZero",
"N... | Pi.zero_apply | Lean.Elab.Tactic.evalRewriteSeq | null |
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