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.Analysis.InnerProductSpace.Harmonic.Basic | {
"line": 167,
"column": 4
} | {
"line": 167,
"column": 12
} | [
{
"pp": "case h\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace ℝ E\ninst✝² : FiniteDimensional ℝ E\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : E → F\nx : E\nc : ℝ\nh : HarmonicAt f x\n⊢ ∀ (a : E), Δ (c • f) a = (c • Δ f) a → Δ f a = 0 a → Δ (c • f) a = 0 ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.InnerProductSpace.Harmonic.Basic | {
"line": 187,
"column": 4
} | {
"line": 187,
"column": 12
} | [
{
"pp": "case h\nE : Type u_1\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : InnerProductSpace ℝ E\ninst✝⁴ : FiniteDimensional ℝ E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\nG : Type u_3\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\nf : E → F\nx : E\nh : HarmonicAt f x\nl : F →... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Complex.Harmonic.Analytic | {
"line": 33,
"column": 2
} | {
"line": 35,
"column": 13
} | [
{
"pp": "f : ℂ → ℝ\nx : ℂ\nhf : HarmonicAt f x\n⊢ DifferentiableAt ℂ (fun z ↦ ↑((fderiv ℝ f z) 1) - I * ↑((fderiv ℝ f z) I)) x",
"usedConstants": [
"instInnerProductSpaceRealComplex",
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSeminormedCommRing",
"Real",
"instHSMul",... | have : (fun z ↦ fderiv ℝ f z 1 - I * fderiv ℝ f z I) =
(ofRealCLM ∘ (fderiv ℝ f · 1) - I • ofRealCLM ∘ (fderiv ℝ f · I)) := by
ext; simp | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Complex.PhragmenLindelof | {
"line": 388,
"column": 14
} | {
"line": 388,
"column": 31
} | [
{
"pp": "case h\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nC : ℝ\nf : ℂ → E\nhd : DiffContOnCl ℂ f (Ioi 0 ×ℂ Ioi 0)\nhre : ∀ (x : ℝ), 0 ≤ x → ‖f ↑x‖ ≤ C\nhim : ∀ (x : ℝ), 0 ≤ x → ‖f (↑x * I)‖ ≤ C\nζ : ℂ\nhζ : ζ.im ∈ Icc 0 (π / 2)\nhz_re : 0 ≤ (cexp ζ).re\nhz_im : 0 ≤ (cexp ζ).im\nhzn... | abs_of_nonneg hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.Harmonic.Analytic | {
"line": 101,
"column": 12
} | {
"line": 101,
"column": 20
} | [
{
"pp": "case right.hx\nf : ℂ → ℝ\nz : ℂ\nR : ℝ\nhf : HarmonicOnNhd f (ball z R)\nhR : ¬R ≤ 0\ng : ℂ → ℂ := ⋯\nhg : DifferentiableOn ℂ g (ball z R)\nF : ℂ → ℂ\nhF : F z = ↑(f z) ∧ ∀ x ∈ ball z R, HasDerivAt F (g x) x\nh₁F : DifferentiableOn ℂ F (ball z R)\nh₂F : DifferentiableOn ℝ F (ball z R)\nx : ℂ\nhx : x ∈ ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Complex.Harmonic.Analytic | {
"line": 101,
"column": 12
} | {
"line": 101,
"column": 20
} | [
{
"pp": "case right.hx\nf : ℂ → ℝ\nz : ℂ\nR : ℝ\nhf : HarmonicOnNhd f (ball z R)\nhR : ¬R ≤ 0\ng : ℂ → ℂ := ⋯\nhg : DifferentiableOn ℂ g (ball z R)\nF : ℂ → ℂ\nhF : F z = ↑(f z) ∧ ∀ x ∈ ball z R, HasDerivAt F (g x) x\nh₁F : DifferentiableOn ℂ F (ball z R)\nh₂F : DifferentiableOn ℝ F (ball z R)\nx : ℂ\nhx : x ∈ ... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.Harmonic.Analytic | {
"line": 101,
"column": 12
} | {
"line": 101,
"column": 20
} | [
{
"pp": "case right.hx\nf : ℂ → ℝ\nz : ℂ\nR : ℝ\nhf : HarmonicOnNhd f (ball z R)\nhR : ¬R ≤ 0\ng : ℂ → ℂ := ⋯\nhg : DifferentiableOn ℂ g (ball z R)\nF : ℂ → ℂ\nhF : F z = ↑(f z) ∧ ∀ x ∈ ball z R, HasDerivAt F (g x) x\nh₁F : DifferentiableOn ℂ F (ball z R)\nh₂F : DifferentiableOn ℝ F (ball z R)\nx : ℂ\nhx : x ∈ ... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.Harmonic.Analytic | {
"line": 101,
"column": 12
} | {
"line": 101,
"column": 20
} | [
{
"pp": "case right.hfgx\nf : ℂ → ℝ\nz : ℂ\nR : ℝ\nhf : HarmonicOnNhd f (ball z R)\nhR : ¬R ≤ 0\ng : ℂ → ℂ := ⋯\nhg : DifferentiableOn ℂ g (ball z R)\nF : ℂ → ℂ\nhF : F z = ↑(f z) ∧ ∀ x ∈ ball z R, HasDerivAt F (g x) x\nh₁F : DifferentiableOn ℂ F (ball z R)\nh₂F : DifferentiableOn ℝ F (ball z R)\nx : ℂ\nhx : x ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Complex.Harmonic.Analytic | {
"line": 101,
"column": 12
} | {
"line": 101,
"column": 20
} | [
{
"pp": "case right.hfgx\nf : ℂ → ℝ\nz : ℂ\nR : ℝ\nhf : HarmonicOnNhd f (ball z R)\nhR : ¬R ≤ 0\ng : ℂ → ℂ := ⋯\nhg : DifferentiableOn ℂ g (ball z R)\nF : ℂ → ℂ\nhF : F z = ↑(f z) ∧ ∀ x ∈ ball z R, HasDerivAt F (g x) x\nh₁F : DifferentiableOn ℂ F (ball z R)\nh₂F : DifferentiableOn ℝ F (ball z R)\nx : ℂ\nhx : x ... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.Harmonic.Analytic | {
"line": 101,
"column": 12
} | {
"line": 101,
"column": 20
} | [
{
"pp": "case right.hfgx\nf : ℂ → ℝ\nz : ℂ\nR : ℝ\nhf : HarmonicOnNhd f (ball z R)\nhR : ¬R ≤ 0\ng : ℂ → ℂ := ⋯\nhg : DifferentiableOn ℂ g (ball z R)\nF : ℂ → ℂ\nhF : F z = ↑(f z) ∧ ∀ x ∈ ball z R, HasDerivAt F (g x) x\nh₁F : DifferentiableOn ℂ F (ball z R)\nh₂F : DifferentiableOn ℝ F (ball z R)\nx : ℂ\nhx : x ... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.Harmonic.Analytic | {
"line": 101,
"column": 12
} | {
"line": 101,
"column": 20
} | [
{
"pp": "case right.a\nf : ℂ → ℝ\nz : ℂ\nR : ℝ\nhf : HarmonicOnNhd f (ball z R)\nhR : ¬R ≤ 0\ng : ℂ → ℂ := ⋯\nhg : DifferentiableOn ℂ g (ball z R)\nF : ℂ → ℂ\nhF : F z = ↑(f z) ∧ ∀ x ∈ ball z R, HasDerivAt F (g x) x\nh₁F : DifferentiableOn ℂ F (ball z R)\nh₂F : DifferentiableOn ℝ F (ball z R)\nx : ℂ\nhx : x ∈ b... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Complex.Harmonic.Analytic | {
"line": 101,
"column": 12
} | {
"line": 101,
"column": 20
} | [
{
"pp": "case right.a\nf : ℂ → ℝ\nz : ℂ\nR : ℝ\nhf : HarmonicOnNhd f (ball z R)\nhR : ¬R ≤ 0\ng : ℂ → ℂ := ⋯\nhg : DifferentiableOn ℂ g (ball z R)\nF : ℂ → ℂ\nhF : F z = ↑(f z) ∧ ∀ x ∈ ball z R, HasDerivAt F (g x) x\nh₁F : DifferentiableOn ℂ F (ball z R)\nh₂F : DifferentiableOn ℝ F (ball z R)\nx : ℂ\nhx : x ∈ b... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.Harmonic.Analytic | {
"line": 101,
"column": 12
} | {
"line": 101,
"column": 20
} | [
{
"pp": "case right.a\nf : ℂ → ℝ\nz : ℂ\nR : ℝ\nhf : HarmonicOnNhd f (ball z R)\nhR : ¬R ≤ 0\ng : ℂ → ℂ := ⋯\nhg : DifferentiableOn ℂ g (ball z R)\nF : ℂ → ℂ\nhF : F z = ↑(f z) ∧ ∀ x ∈ ball z R, HasDerivAt F (g x) x\nh₁F : DifferentiableOn ℂ F (ball z R)\nh₂F : DifferentiableOn ℝ F (ball z R)\nx : ℂ\nhx : x ∈ b... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.Harmonic.Analytic | {
"line": 117,
"column": 48
} | {
"line": 117,
"column": 56
} | [
{
"pp": "f : ℂ → ℝ\nhf : HarmonicOnNhd f univ\ng : ℂ → ℂ := (⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) 1) - I • ⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) I\nhg : Differentiable ℂ g\nF : ℂ → ℂ\nhF : F 0 = ↑(f 0) ∧ ∀ x ∈ univ, HasDerivAt F (g x) x\n⊢ ∀ (z₁ : ℂ), HasDerivAt F (g z₁) z₁",
"usedConstants": [
"InnerPro... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Complex.Harmonic.Analytic | {
"line": 131,
"column": 6
} | {
"line": 132,
"column": 63
} | [
{
"pp": "f : ℂ → ℝ\nhf : HarmonicOnNhd f univ\ng : ℂ → ℂ := ⋯\nhg : Differentiable ℂ g\nF : ℂ → ℂ\nhF : F 0 = ↑(f 0) ∧ ∀ x ∈ univ, HasDerivAt F (g x) x\nh₁F : ∀ (z₁ : ℂ), HasDerivAt F (g z₁) z₁\nh₂F : Differentiable ℂ F\nh₃F : Differentiable ℝ F\nx✝ y : ℂ\nhy : y ∈ univ\nx : ℂ\n⊢ (fderiv ℝ (⇑reCLM) (F y) ∘SL fd... | simp only [map_smul, map_add]
simp [(h₁F y).hasFDerivAt.restrictScalars ℝ |>.fderiv, g] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.Harmonic.Analytic | {
"line": 131,
"column": 6
} | {
"line": 132,
"column": 63
} | [
{
"pp": "f : ℂ → ℝ\nhf : HarmonicOnNhd f univ\ng : ℂ → ℂ := ⋯\nhg : Differentiable ℂ g\nF : ℂ → ℂ\nhF : F 0 = ↑(f 0) ∧ ∀ x ∈ univ, HasDerivAt F (g x) x\nh₁F : ∀ (z₁ : ℂ), HasDerivAt F (g z₁) z₁\nh₂F : Differentiable ℂ F\nh₃F : Differentiable ℝ F\nx✝ y : ℂ\nhy : y ∈ univ\nx : ℂ\n⊢ (fderiv ℝ (⇑reCLM) (F y) ∘SL fd... | simp only [map_smul, map_add]
simp [(h₁F y).hasFDerivAt.restrictScalars ℝ |>.fderiv, g] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.Harmonic.Analytic | {
"line": 134,
"column": 12
} | {
"line": 134,
"column": 20
} | [
{
"pp": "case right.h.refine_1\nf : ℂ → ℝ\nhf : HarmonicOnNhd f univ\ng : ℂ → ℂ := (⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) 1) - I • ⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) I\nhg : Differentiable ℂ g\nF : ℂ → ℂ\nhF : F 0 = ↑(f 0) ∧ ∀ x ∈ univ, HasDerivAt F (g x) x\nh₁F : ∀ (z₁ : ℂ), HasDerivAt F (g z₁) z₁\nh₂F : Differ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Complex.Harmonic.Analytic | {
"line": 134,
"column": 12
} | {
"line": 134,
"column": 20
} | [
{
"pp": "case right.h.refine_1\nf : ℂ → ℝ\nhf : HarmonicOnNhd f univ\ng : ℂ → ℂ := (⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) 1) - I • ⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) I\nhg : Differentiable ℂ g\nF : ℂ → ℂ\nhF : F 0 = ↑(f 0) ∧ ∀ x ∈ univ, HasDerivAt F (g x) x\nh₁F : ∀ (z₁ : ℂ), HasDerivAt F (g z₁) z₁\nh₂F : Differ... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.Harmonic.Analytic | {
"line": 134,
"column": 12
} | {
"line": 134,
"column": 20
} | [
{
"pp": "case right.h.refine_1\nf : ℂ → ℝ\nhf : HarmonicOnNhd f univ\ng : ℂ → ℂ := (⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) 1) - I • ⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) I\nhg : Differentiable ℂ g\nF : ℂ → ℂ\nhF : F 0 = ↑(f 0) ∧ ∀ x ∈ univ, HasDerivAt F (g x) x\nh₁F : ∀ (z₁ : ℂ), HasDerivAt F (g z₁) z₁\nh₂F : Differ... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.Harmonic.Analytic | {
"line": 134,
"column": 12
} | {
"line": 134,
"column": 20
} | [
{
"pp": "case right.h.refine_2\nf : ℂ → ℝ\nhf : HarmonicOnNhd f univ\ng : ℂ → ℂ := (⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) 1) - I • ⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) I\nhg : Differentiable ℂ g\nF : ℂ → ℂ\nhF : F 0 = ↑(f 0) ∧ ∀ x ∈ univ, HasDerivAt F (g x) x\nh₁F : ∀ (z₁ : ℂ), HasDerivAt F (g z₁) z₁\nh₂F : Differ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Complex.Harmonic.Analytic | {
"line": 134,
"column": 12
} | {
"line": 134,
"column": 20
} | [
{
"pp": "case right.h.refine_2\nf : ℂ → ℝ\nhf : HarmonicOnNhd f univ\ng : ℂ → ℂ := (⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) 1) - I • ⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) I\nhg : Differentiable ℂ g\nF : ℂ → ℂ\nhF : F 0 = ↑(f 0) ∧ ∀ x ∈ univ, HasDerivAt F (g x) x\nh₁F : ∀ (z₁ : ℂ), HasDerivAt F (g z₁) z₁\nh₂F : Differ... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.Harmonic.Analytic | {
"line": 134,
"column": 12
} | {
"line": 134,
"column": 20
} | [
{
"pp": "case right.h.refine_2\nf : ℂ → ℝ\nhf : HarmonicOnNhd f univ\ng : ℂ → ℂ := (⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) 1) - I • ⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) I\nhg : Differentiable ℂ g\nF : ℂ → ℂ\nhF : F 0 = ↑(f 0) ∧ ∀ x ∈ univ, HasDerivAt F (g x) x\nh₁F : ∀ (z₁ : ℂ), HasDerivAt F (g z₁) z₁\nh₂F : Differ... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.Harmonic.Analytic | {
"line": 134,
"column": 12
} | {
"line": 134,
"column": 20
} | [
{
"pp": "case right.h.refine_3\nf : ℂ → ℝ\nhf : HarmonicOnNhd f univ\ng : ℂ → ℂ := (⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) 1) - I • ⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) I\nhg : Differentiable ℂ g\nF : ℂ → ℂ\nhF : F 0 = ↑(f 0) ∧ ∀ x ∈ univ, HasDerivAt F (g x) x\nh₁F : ∀ (z₁ : ℂ), HasDerivAt F (g z₁) z₁\nh₂F : Differ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Complex.Harmonic.Analytic | {
"line": 134,
"column": 12
} | {
"line": 134,
"column": 20
} | [
{
"pp": "case right.h.refine_3\nf : ℂ → ℝ\nhf : HarmonicOnNhd f univ\ng : ℂ → ℂ := (⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) 1) - I • ⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) I\nhg : Differentiable ℂ g\nF : ℂ → ℂ\nhF : F 0 = ↑(f 0) ∧ ∀ x ∈ univ, HasDerivAt F (g x) x\nh₁F : ∀ (z₁ : ℂ), HasDerivAt F (g z₁) z₁\nh₂F : Differ... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.Harmonic.Analytic | {
"line": 134,
"column": 12
} | {
"line": 134,
"column": 20
} | [
{
"pp": "case right.h.refine_3\nf : ℂ → ℝ\nhf : HarmonicOnNhd f univ\ng : ℂ → ℂ := (⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) 1) - I • ⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) I\nhg : Differentiable ℂ g\nF : ℂ → ℂ\nhF : F 0 = ↑(f 0) ∧ ∀ x ∈ univ, HasDerivAt F (g x) x\nh₁F : ∀ (z₁ : ℂ), HasDerivAt F (g z₁) z₁\nh₂F : Differ... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.Harmonic.Analytic | {
"line": 134,
"column": 12
} | {
"line": 134,
"column": 20
} | [
{
"pp": "case right.h.refine_4\nf : ℂ → ℝ\nhf : HarmonicOnNhd f univ\ng : ℂ → ℂ := (⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) 1) - I • ⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) I\nhg : Differentiable ℂ g\nF : ℂ → ℂ\nhF : F 0 = ↑(f 0) ∧ ∀ x ∈ univ, HasDerivAt F (g x) x\nh₁F : ∀ (z₁ : ℂ), HasDerivAt F (g z₁) z₁\nh₂F : Differ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Complex.Harmonic.Analytic | {
"line": 134,
"column": 12
} | {
"line": 134,
"column": 20
} | [
{
"pp": "case right.h.refine_4\nf : ℂ → ℝ\nhf : HarmonicOnNhd f univ\ng : ℂ → ℂ := (⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) 1) - I • ⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) I\nhg : Differentiable ℂ g\nF : ℂ → ℂ\nhF : F 0 = ↑(f 0) ∧ ∀ x ∈ univ, HasDerivAt F (g x) x\nh₁F : ∀ (z₁ : ℂ), HasDerivAt F (g z₁) z₁\nh₂F : Differ... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.Harmonic.Analytic | {
"line": 134,
"column": 12
} | {
"line": 134,
"column": 20
} | [
{
"pp": "case right.h.refine_4\nf : ℂ → ℝ\nhf : HarmonicOnNhd f univ\ng : ℂ → ℂ := (⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) 1) - I • ⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) I\nhg : Differentiable ℂ g\nF : ℂ → ℂ\nhF : F 0 = ↑(f 0) ∧ ∀ x ∈ univ, HasDerivAt F (g x) x\nh₁F : ∀ (z₁ : ℂ), HasDerivAt F (g z₁) z₁\nh₂F : Differ... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.Harmonic.MeanValue | {
"line": 54,
"column": 4
} | {
"line": 54,
"column": 12
} | [
{
"pp": "case pos\nf : ℂ → ℝ\nc : ℂ\nR : ℝ\nh₁f : HarmonicContOnCl f (ball c |R|)\nhR : R = 0\n⊢ circleAverage f c R = f c",
"usedConstants": [
"InnerProductSpace.toNormedSpace",
"Real",
"Real.instZero",
"Real.instRCLike",
"congrArg",
"Real.circleAverage_zero",
"Rea... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Complex.Harmonic.MeanValue | {
"line": 54,
"column": 4
} | {
"line": 54,
"column": 12
} | [
{
"pp": "case pos\nf : ℂ → ℝ\nc : ℂ\nR : ℝ\nh₁f : HarmonicContOnCl f (ball c |R|)\nhR : R = 0\n⊢ circleAverage f c R = f c",
"usedConstants": [
"InnerProductSpace.toNormedSpace",
"Real",
"Real.instZero",
"Real.instRCLike",
"congrArg",
"Real.circleAverage_zero",
"Rea... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.Harmonic.MeanValue | {
"line": 54,
"column": 4
} | {
"line": 54,
"column": 12
} | [
{
"pp": "case pos\nf : ℂ → ℝ\nc : ℂ\nR : ℝ\nh₁f : HarmonicContOnCl f (ball c |R|)\nhR : R = 0\n⊢ circleAverage f c R = f c",
"usedConstants": [
"InnerProductSpace.toNormedSpace",
"Real",
"Real.instZero",
"Real.instRCLike",
"congrArg",
"Real.circleAverage_zero",
"Rea... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.MeanValue | {
"line": 104,
"column": 4
} | {
"line": 104,
"column": 12
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf : ℂ → E\nR : ℝ\nc : ℂ\ns : Set ℂ\nhs : s.Countable\nh₁f : ContinuousOn f (closedBall c |R|)\nh₂f : ∀ z ∈ ball c |R| \\ s, DifferentiableAt ℂ f z\nhR : ¬R = 0\nz : ℂ\nhz : z ∈ sphere c |R|\nthis : z - c ≠ 0... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Complex.MeanValue | {
"line": 118,
"column": 4
} | {
"line": 118,
"column": 12
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf : ℂ → E\nR : ℝ\nc : ℂ\nhf : DiffContOnCl ℂ f (ball c |R|)\nhR : ¬R = 0\nz : ℂ\nhz : z ∈ sphere c |R|\nthis : z - c ≠ 0\n⊢ f z = ((z - c) / (z - c)) • f z",
"usedConstants": [
"GroupWithZero.toMon... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.SpecialFunctions.Log.NegMulLog | {
"line": 56,
"column": 2
} | {
"line": 56,
"column": 53
} | [
{
"pp": "case inr\nthis : Set.univ = Set.Iio 0 ∪ Set.Ioi 0 ∪ {0}\n⊢ (Filter.Tendsto (fun x ↦ log x * x) (𝓝[<] 0) (𝓝 0) ∧ Filter.Tendsto (fun x ↦ log x * x) (𝓝[>] 0) (𝓝 0)) ∧\n Filter.Tendsto (fun x ↦ log x * x) (pure 0) (𝓝 0)",
"usedConstants": [
"Pure.pure",
"NonUnitalNonAssocCommRing.t... | refine ⟨⟨tendsto_log_mul_self_nhdsLT_zero, ?_⟩, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Complex.Poisson | {
"line": 118,
"column": 59
} | {
"line": 118,
"column": 79
} | [
{
"pp": "θ φ r R : ℝ\nh₁ : 0 < r\nh₂ : r < R\n⊢ 1 * ‖↑R * cexp (↑θ * I) - ↑r * cexp (↑φ * I)‖ ^ 2 ≤ (R + r) * (R + r)",
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Norm.norm",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"Real.partialOrder",
... | ← normSq_eq_norm_sq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Log.PosLog | {
"line": 114,
"column": 4
} | {
"line": 114,
"column": 12
} | [
{
"pp": "case pos\nn : ℕ\nx : ℝ\nhn : n = 0\n⊢ log⁺ (x ^ n) = ↑n * log⁺ x",
"usedConstants": [
"CharP.cast_eq_zero",
"MulOne.toOne",
"Real",
"HMul.hMul",
"Real.posLog",
"Real.instZero",
"Monoid.toMulOneClass",
"Real.instRCLike",
"congrArg",
"MulZer... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.SpecialFunctions.Log.PosLog | {
"line": 114,
"column": 4
} | {
"line": 114,
"column": 12
} | [
{
"pp": "case pos\nn : ℕ\nx : ℝ\nhn : n = 0\n⊢ log⁺ (x ^ n) = ↑n * log⁺ x",
"usedConstants": [
"CharP.cast_eq_zero",
"MulOne.toOne",
"Real",
"HMul.hMul",
"Real.posLog",
"Real.instZero",
"Monoid.toMulOneClass",
"Real.instRCLike",
"congrArg",
"MulZer... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Log.PosLog | {
"line": 114,
"column": 4
} | {
"line": 114,
"column": 12
} | [
{
"pp": "case pos\nn : ℕ\nx : ℝ\nhn : n = 0\n⊢ log⁺ (x ^ n) = ↑n * log⁺ x",
"usedConstants": [
"CharP.cast_eq_zero",
"MulOne.toOne",
"Real",
"HMul.hMul",
"Real.posLog",
"Real.instZero",
"Monoid.toMulOneClass",
"Real.instRCLike",
"congrArg",
"MulZer... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.Poisson | {
"line": 214,
"column": 6
} | {
"line": 215,
"column": 99
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\nf : ℂ → E\nR : ℝ\nw : ℂ\ninst✝ : CompleteSpace E\nhf : DiffContOnCl ℂ f (ball 0 R)\nhw : w ∈ ball 0 R\nhR : 0 < R\nh₁w : w ≠ 0\nW : ℂ := ↑R * cexp (↑w.arg * I)\nq : ℝ := ‖w‖ / R\nh₁q : 0 < q\nh₂q : q < 1\nη₀ : ∀ {x : ℂ}, ‖x‖ ≤ R → ↑... | rw [← abs_of_pos hR] at hw hf
simp [← hf.circleAverage_smul_div hw, circleAverage_eq_circleIntegral (ne_of_lt hR).symm, h0] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.Poisson | {
"line": 214,
"column": 6
} | {
"line": 215,
"column": 99
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\nf : ℂ → E\nR : ℝ\nw : ℂ\ninst✝ : CompleteSpace E\nhf : DiffContOnCl ℂ f (ball 0 R)\nhw : w ∈ ball 0 R\nhR : 0 < R\nh₁w : w ≠ 0\nW : ℂ := ↑R * cexp (↑w.arg * I)\nq : ℝ := ‖w‖ / R\nh₁q : 0 < q\nh₂q : q < 1\nη₀ : ∀ {x : ℂ}, ‖x‖ ≤ R → ↑... | rw [← abs_of_pos hR] at hw hf
simp [← hf.circleAverage_smul_div hw, circleAverage_eq_circleIntegral (ne_of_lt hR).symm, h0] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Integrability.LogMeromorphic | {
"line": 47,
"column": 4
} | {
"line": 48,
"column": 71
} | [
{
"pp": "case pos\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\nhf : MeromorphicOn f [[a, b]]\nt₀ : ∀ (u : ↑[[a, b]]), meromorphicOrderAt f ↑u ≠ ⊤\n⊢ IntervalIntegrable (fun x ↦ log ‖f x‖) volume a b",
"usedConstants": [
"NormedCommRing.toSeminormedCommRing... | obtain ⟨g, h₁g, h₂g, h₃g⟩ := hf.extract_zeros_poles t₀
((MeromorphicOn.divisor f [[a, b]]).finiteSupport isCompact_uIcc) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.SpecialFunctions.Integrability.LogMeromorphic | {
"line": 61,
"column": 6
} | {
"line": 61,
"column": 14
} | [
{
"pp": "case pos.hg.hu\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\nhf : MeromorphicOn f [[a, b]]\nt₀ : ∀ (u : ↑[[a, b]]), meromorphicOrderAt f ↑u ≠ ⊤\ng : ℝ → E\nh₁g : AnalyticOnNhd ℝ g [[a, b]]\nh₂g : ∀ (u : ↑[[a, b]]), g ↑u ≠ 0\nh₃g : f =ᶠ[codiscreteWithin [[a, ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.SpecialFunctions.Integrability.LogMeromorphic | {
"line": 98,
"column": 47
} | {
"line": 100,
"column": 38
} | [
{
"pp": "a b : ℝ\nf : ℝ → ℝ\nhf : MeromorphicOn f [[a, b]]\n⊢ IntervalIntegrable (log ∘ f) volume a b",
"usedConstants": [
"MeromorphicOn.intervalIntegrable_log_norm",
"Norm.norm",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Real",
"Real.log_abs",
"Real.lattice",
... | by
rw [(by aesop : log ∘ f = (log ‖f ·‖))]
exact hf.intervalIntegrable_log_norm | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Integrability.LogMeromorphic | {
"line": 148,
"column": 6
} | {
"line": 148,
"column": 28
} | [
{
"pp": "case pos.hg.hu\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nc : ℂ\nR : ℝ\nf : ℂ → E\nhf : MeromorphicOn f (sphere c |R|)\nt₀ : ∀ (u : ↑(sphere c |R|)), meromorphicOrderAt f ↑u ≠ ⊤\ng : ℂ → E\nh₁g : AnalyticOnNhd ℂ g (sphere c |R|)\nh₂g : ∀ (u : ↑(sphere c |R|)), g ↑u ≠ 0\nh₃g ... | apply ContinuousOn.log | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.SpecialFunctions.Integrals.PosLogEqCircleAverage | {
"line": 94,
"column": 30
} | {
"line": 94,
"column": 38
} | [
{
"pp": "this✝ : AnalyticOnNhd ℝ (fun x ↦ 4 * sin x ^ 2) Set.univ\nthis : ((fun x ↦ 4 * sin x ^ 2) ⁻¹' {0})ᶜ ∈ codiscrete ℝ\na : ℝ\nha : ¬sin a = 0\n⊢ sin a ^ 2 ≠ 0",
"usedConstants": [
"NormedCommRing.toNormedRing",
"False",
"Real",
"eq_false",
"Real.instZero",
"congrArg... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.SpecialFunctions.Integrals.PosLogEqCircleAverage | {
"line": 94,
"column": 30
} | {
"line": 94,
"column": 38
} | [
{
"pp": "this✝ : AnalyticOnNhd ℝ (fun x ↦ 4 * sin x ^ 2) Set.univ\nthis : ((fun x ↦ 4 * sin x ^ 2) ⁻¹' {0})ᶜ ∈ codiscrete ℝ\na : ℝ\nha : ¬sin a = 0\n⊢ sin a ^ 2 ≠ 0",
"usedConstants": [
"NormedCommRing.toNormedRing",
"False",
"Real",
"eq_false",
"Real.instZero",
"congrArg... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Integrals.PosLogEqCircleAverage | {
"line": 94,
"column": 30
} | {
"line": 94,
"column": 38
} | [
{
"pp": "this✝ : AnalyticOnNhd ℝ (fun x ↦ 4 * sin x ^ 2) Set.univ\nthis : ((fun x ↦ 4 * sin x ^ 2) ⁻¹' {0})ᶜ ∈ codiscrete ℝ\na : ℝ\nha : ¬sin a = 0\n⊢ sin a ^ 2 ≠ 0",
"usedConstants": [
"NormedCommRing.toNormedRing",
"False",
"Real",
"eq_false",
"Real.instZero",
"congrArg... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Integrals.PosLogEqCircleAverage | {
"line": 185,
"column": 4
} | {
"line": 185,
"column": 12
} | [
{
"pp": "case inr.inl\na : ℂ\nh : 1 = ‖a‖\n⊢ |‖a‖| ≤ 1",
"usedConstants": [
"Norm.norm",
"SeminormedAddGroup.toNorm",
"NormedCommRing.toSeminormedCommRing",
"Real.instLE",
"Real",
"Real.lattice",
"instReflLe",
"abs",
"congrArg",
"Complex.instNormed... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.SpecialFunctions.Integrals.PosLogEqCircleAverage | {
"line": 247,
"column": 4
} | {
"line": 247,
"column": 12
} | [
{
"pp": "case pos\na c : ℂ\nR : ℝ\nhu : a ∈ closedBall c |R|\nhR : R = 0\n⊢ circleAverage (fun x ↦ log ‖x - a‖) c R = log R",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSeminormedCommRing",
"Real",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.SpecialFunctions.Integrals.PosLogEqCircleAverage | {
"line": 247,
"column": 4
} | {
"line": 247,
"column": 12
} | [
{
"pp": "case pos\na c : ℂ\nR : ℝ\nhu : a ∈ closedBall c |R|\nhR : R = 0\n⊢ circleAverage (fun x ↦ log ‖x - a‖) c R = log R",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSeminormedCommRing",
"Real",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Integrals.PosLogEqCircleAverage | {
"line": 247,
"column": 4
} | {
"line": 247,
"column": 12
} | [
{
"pp": "case pos\na c : ℂ\nR : ℝ\nhu : a ∈ closedBall c |R|\nhR : R = 0\n⊢ circleAverage (fun x ↦ log ‖x - a‖) c R = log R",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSeminormedCommRing",
"Real",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Filter.ZeroAndBoundedAtFilter | {
"line": 104,
"column": 2
} | {
"line": 104,
"column": 47
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝ : SeminormedRing β\nl : Filter α\nf g : α → β\nhf : l.BoundedAtFilter f\nhg : l.BoundedAtFilter g\n⊢ (fun x ↦ 1 x * 1 x) =O[l] 1",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
... | convert! Asymptotics.isBigO_refl (E := ℝ) _ l | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.Analysis.Complex.Polynomial.GaussLucas | {
"line": 46,
"column": 24
} | {
"line": 46,
"column": 32
} | [
{
"pp": "P : ℂ[X]\nhP : 0 < P.degree\nz : ℂ\nhP₀ : P ≠ 0\nhPz : ¬eval z P = 0\nw : ℂ\nhw : w ∈ P.roots.toFinset\n⊢ 0 < ↑(rootMultiplicity w P)",
"usedConstants": [
"Multiset.toFinset",
"Real.instIsOrderedRing",
"Polynomial.eval",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Fals... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Complex.Polynomial.GaussLucas | {
"line": 46,
"column": 24
} | {
"line": 46,
"column": 32
} | [
{
"pp": "P : ℂ[X]\nhP : 0 < P.degree\nz : ℂ\nhP₀ : P ≠ 0\nhPz : ¬eval z P = 0\nw : ℂ\nhw : w ∈ P.roots.toFinset\n⊢ 0 < ↑(rootMultiplicity w P)",
"usedConstants": [
"Multiset.toFinset",
"Real.instIsOrderedRing",
"Polynomial.eval",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Fals... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.Polynomial.GaussLucas | {
"line": 46,
"column": 24
} | {
"line": 46,
"column": 32
} | [
{
"pp": "P : ℂ[X]\nhP : 0 < P.degree\nz : ℂ\nhP₀ : P ≠ 0\nhPz : ¬eval z P = 0\nw : ℂ\nhw : w ∈ P.roots.toFinset\n⊢ 0 < ↑(rootMultiplicity w P)",
"usedConstants": [
"Multiset.toFinset",
"Real.instIsOrderedRing",
"Polynomial.eval",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Fals... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.Polynomial.GaussLucas | {
"line": 49,
"column": 6
} | {
"line": 49,
"column": 14
} | [
{
"pp": "case neg.h\nP : ℂ[X]\nhP : 0 < P.degree\nz : ℂ\nhP₀ : P ≠ 0\nhPz : ¬eval z P = 0\nhw : z ∈ P.roots.toFinset\n⊢ False",
"usedConstants": [
"Multiset.toFinset",
"Polynomial.eval",
"False",
"Polynomial.roots",
"eq_false",
"Complex.commRing",
"congrArg",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Complex.Polynomial.GaussLucas | {
"line": 75,
"column": 36
} | {
"line": 75,
"column": 44
} | [
{
"pp": "case pos\nP : ℂ[X]\nz : ℂ\nhP : 0 < P.degree\nhz : eval z (derivative P) = 0\nweight : ℂ → ℝ := P.derivRootWeight z\ns : Finset ℂ := P.roots.toFinset\nhzP : eval z P = 0\n⊢ Pi.single z 1 z • (z - z) = 0",
"usedConstants": [
"instInnerProductSpaceRealComplex",
"InnerProductSpace.toNormed... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Complex.Polynomial.GaussLucas | {
"line": 75,
"column": 36
} | {
"line": 75,
"column": 44
} | [
{
"pp": "case pos.h₀\nP : ℂ[X]\nz : ℂ\nhP : 0 < P.degree\nhz : eval z (derivative P) = 0\nweight : ℂ → ℝ := P.derivRootWeight z\ns : Finset ℂ := P.roots.toFinset\nhzP : eval z P = 0\n⊢ ∀ b ∈ s, b ≠ z → Pi.single z 1 b • (z - b) = 0",
"usedConstants": [
"instInnerProductSpaceRealComplex",
"InnerP... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Complex.Polynomial.GaussLucas | {
"line": 75,
"column": 36
} | {
"line": 75,
"column": 44
} | [
{
"pp": "case pos.h₁\nP : ℂ[X]\nz : ℂ\nhP : 0 < P.degree\nhz : eval z (derivative P) = 0\nweight : ℂ → ℝ := P.derivRootWeight z\ns : Finset ℂ := P.roots.toFinset\nhzP : eval z P = 0\n⊢ z ∉ s → Pi.single z 1 z • (z - z) = 0",
"usedConstants": [
"instInnerProductSpaceRealComplex",
"InnerProductSpa... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Complex.Polynomial.GaussLucas | {
"line": 77,
"column": 90
} | {
"line": 84,
"column": 39
} | [
{
"pp": "P : ℂ[X]\nz : ℂ\nhP : 0 < P.degree\nhz : eval z (derivative P) = 0\nweight : ℂ → ℝ := P.derivRootWeight z\ns : Finset ℂ := P.roots.toFinset\nhzP : ¬eval z P = 0\n⊢ ∑ x ∈ s, weight x • (z - x) = (starRingEnd ℂ) (∑ x ∈ s, rootMultiplicity x P • (1 / (z - x)))",
"usedConstants": [
"Multiset.toFi... | by
simp only [map_sum, weight, derivRootWeight, if_neg hzP]
refine Finset.sum_congr rfl fun x hx ↦ ?_
have : z - x ≠ 0 := by
rw [sub_ne_zero]
rintro rfl
simp_all [s]
simp [← Complex.conj_mul', field] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Complex.Positivity | {
"line": 55,
"column": 32
} | {
"line": 55,
"column": 53
} | [
{
"pp": "f : ℂ → ℂ\nhf : Differentiable ℂ f\nc : ℂ\nh : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f c\nz : ℂ\nhz : 0 ≤ z - c\n⊢ ‖z - c‖ < (z - c).re + 1",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Complex.eq_re_of_ofReal_le",
"Real",
"Real.instZero",
"AddGroupWithOne.toAddGroup",
... | eq_re_of_ofReal_le hz | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.JensenFormula | {
"line": 74,
"column": 22
} | {
"line": 74,
"column": 42
} | [
{
"pp": "r₀ r R : ℝ\nρ : ℂ\nhρ : ‖ρ‖ = R\nhr₀ : 0 < r₀\nhR : 0 < R\nhr₀r : r₀ ≤ r\nhrR : r ≤ R\nθ r₁ : ℝ\n⊢ ↑(‖circleMap 0 r₁ θ - ρ‖ ^ 2) = ↑(r₁ ^ 2 + R ^ 2 - 2 * r₁ * R * Real.cos (θ - ρ.arg))",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",
"HMul.hMul",
"Real.cos",
"... | ← normSq_eq_norm_sq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Integrals.Basic | {
"line": 618,
"column": 2
} | {
"line": 618,
"column": 84
} | [
{
"pp": "case h.e'_3\na b : ℝ\nh1 : ∀ (c : ℝ), (1 - c) / 2 * ((1 + c) / 2) = (1 - c ^ 2) / 4\nh2 : Continuous fun x ↦ cos (2 * x) ^ 2\n⊢ (b - a) / 8 - (sin (4 * b) - sin (4 * a)) / 32 =\n ∫ (x : ℝ) in a..b, ((1 - cos (2 * x)) / 2) ^ 1 * ((1 + cos (2 * x)) / 2) ^ 1",
"usedConstants": [
"Mathlib.Tact... | have h3 : ∀ x, cos x * sin x = sin (2 * x) / 2 := by intro; rw [sin_two_mul]; ring | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Complex.JensenFormula | {
"line": 221,
"column": 62
} | {
"line": 221,
"column": 70
} | [
{
"pp": "w ρ c : ℂ\nR : ℝ\nhρ : ρ ∈ sphere c R\nhw : w ∈ ball c R\nthis :\n (fun z ↦ (herglotzRieszKernel 0 (w - c) z).re * log ‖z - (ρ - c)‖) =\n Complex.re ∘ herglotzRieszKernel 0 (w - c) • fun x ↦ log ‖x - (ρ - c)‖\n⊢ ρ - c ∈ sphere 0 R",
"usedConstants": [
"Norm.norm",
"NormedCommRing.to... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Complex.JensenFormula | {
"line": 221,
"column": 62
} | {
"line": 221,
"column": 70
} | [
{
"pp": "w ρ c : ℂ\nR : ℝ\nhρ : ρ ∈ sphere c R\nhw : w ∈ ball c R\nthis :\n (fun z ↦ (herglotzRieszKernel 0 (w - c) z).re * log ‖z - (ρ - c)‖) =\n Complex.re ∘ herglotzRieszKernel 0 (w - c) • fun x ↦ log ‖x - (ρ - c)‖\n⊢ ρ - c ∈ sphere 0 R",
"usedConstants": [
"Norm.norm",
"NormedCommRing.to... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.JensenFormula | {
"line": 221,
"column": 62
} | {
"line": 221,
"column": 70
} | [
{
"pp": "w ρ c : ℂ\nR : ℝ\nhρ : ρ ∈ sphere c R\nhw : w ∈ ball c R\nthis :\n (fun z ↦ (herglotzRieszKernel 0 (w - c) z).re * log ‖z - (ρ - c)‖) =\n Complex.re ∘ herglotzRieszKernel 0 (w - c) • fun x ↦ log ‖x - (ρ - c)‖\n⊢ ρ - c ∈ sphere 0 R",
"usedConstants": [
"Norm.norm",
"NormedCommRing.to... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.JensenFormula | {
"line": 254,
"column": 4
} | {
"line": 254,
"column": 12
} | [
{
"pp": "case e_a.a\nR : ℝ\nc : ℂ\nD : Function.locallyFinsuppWithin (closedBall c |R|) ℤ\nh : D.support.Finite\nu : ℂ\nhu : u ∈ h.toFinset\n⊢ u ∈ D.support",
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"False",
"Real",
"Function.locallyFinsuppWithin.instFunLike",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Complex.JensenFormula | {
"line": 291,
"column": 6
} | {
"line": 291,
"column": 14
} | [
{
"pp": "case pos\nc : ℂ\nR : ℝ\nD : ℂ → ℤ\nhR : R ≠ 0\nhD : Function.HasFiniteSupport D\nh : D c = 0\nx : ℂ\nh₁ : c = x\n⊢ ↑(D x) * (log R - log ‖c - x‖) = ↑(D x) * log (R * ‖c - x‖⁻¹)",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"Int.cast",
"GroupWithZero.toMon... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Complex.JensenFormula | {
"line": 291,
"column": 6
} | {
"line": 291,
"column": 14
} | [
{
"pp": "case pos\nc : ℂ\nR : ℝ\nD : ℂ → ℤ\nhR : R ≠ 0\nhD : Function.HasFiniteSupport D\nh : D c = 0\nx : ℂ\nh₁ : c = x\n⊢ ↑(D x) * (log R - log ‖c - x‖) = ↑(D x) * log (R * ‖c - x‖⁻¹)",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"Int.cast",
"GroupWithZero.toMon... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.JensenFormula | {
"line": 291,
"column": 6
} | {
"line": 291,
"column": 14
} | [
{
"pp": "case pos\nc : ℂ\nR : ℝ\nD : ℂ → ℤ\nhR : R ≠ 0\nhD : Function.HasFiniteSupport D\nh : D c = 0\nx : ℂ\nh₁ : c = x\n⊢ ↑(D x) * (log R - log ‖c - x‖) = ↑(D x) * log (R * ‖c - x‖⁻¹)",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"Int.cast",
"GroupWithZero.toMon... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.JensenFormula | {
"line": 317,
"column": 61
} | {
"line": 317,
"column": 69
} | [
{
"pp": "c : ℂ\nR : ℝ\nf : ℂ → ℂ\nhR : R ≠ 0\nh₁f : MeromorphicOn f (closedBall c |R|)\nCB : Set ℂ := closedBall c |R|\nh₂f : ∀ u ∈ CB, meromorphicOrderAt f u ≠ ⊤\nh₃f : (divisor f CB).support.Finite\n⊢ ∀ (u : ↑(closedBall c |R|)), meromorphicOrderAt f ↑u ≠ ⊤",
"usedConstants": [
"InnerProductSpace.to... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Complex.JensenFormula | {
"line": 317,
"column": 61
} | {
"line": 317,
"column": 69
} | [
{
"pp": "c : ℂ\nR : ℝ\nf : ℂ → ℂ\nhR : R ≠ 0\nh₁f : MeromorphicOn f (closedBall c |R|)\nCB : Set ℂ := closedBall c |R|\nh₂f : ∀ u ∈ CB, meromorphicOrderAt f u ≠ ⊤\nh₃f : (divisor f CB).support.Finite\n⊢ ∀ (u : ↑(closedBall c |R|)), meromorphicOrderAt f ↑u ≠ ⊤",
"usedConstants": [
"InnerProductSpace.to... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.JensenFormula | {
"line": 317,
"column": 61
} | {
"line": 317,
"column": 69
} | [
{
"pp": "c : ℂ\nR : ℝ\nf : ℂ → ℂ\nhR : R ≠ 0\nh₁f : MeromorphicOn f (closedBall c |R|)\nCB : Set ℂ := closedBall c |R|\nh₂f : ∀ u ∈ CB, meromorphicOrderAt f u ≠ ⊤\nh₃f : (divisor f CB).support.Finite\n⊢ ∀ (u : ↑(closedBall c |R|)), meromorphicOrderAt f ↑u ≠ ⊤",
"usedConstants": [
"InnerProductSpace.to... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.JensenFormula | {
"line": 332,
"column": 29
} | {
"line": 332,
"column": 38
} | [
{
"pp": "c : ℂ\nR : ℝ\nf : ℂ → ℂ\nhR : R ≠ 0\nh₁f : MeromorphicOn f (closedBall c |R|)\nCB : Set ℂ := closedBall c |R|\nh₂f : ∀ u ∈ CB, meromorphicOrderAt f u ≠ ⊤\nh₃f : (divisor f CB).support.Finite\ng : ℂ → ℂ\nh₁g : AnalyticOnNhd ℂ g (closedBall c |R|)\nh₂g : ∀ (u : ↑(closedBall c |R|)), g ↑u ≠ 0\nh₃g : f =ᶠ[... | simp [CB] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Complex.JensenFormula | {
"line": 347,
"column": 43
} | {
"line": 347,
"column": 51
} | [
{
"pp": "c : ℂ\nR : ℝ\nf : ℂ → ℂ\nhR : R ≠ 0\nh₁f : MeromorphicOn f (closedBall c |R|)\nCB : Set ℂ := closedBall c |R|\nh₂f : ∀ u ∈ CB, meromorphicOrderAt f u ≠ ⊤\nh₃f : (divisor f CB).support.Finite\ng : ℂ → ℂ\nh₁g : AnalyticOnNhd ℂ g (closedBall c |R|)\nh₂g : ∀ (u : ↑(closedBall c |R|)), g ↑u ≠ 0\nh₃g : f =ᶠ[... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.RCLike.Sqrt | {
"line": 106,
"column": 19
} | {
"line": 106,
"column": 29
} | [
{
"pp": "case inr\n𝕜 : Type u_1\ninst✝ : RCLike 𝕜\na : 𝕜\nha : 0 ≤ a\nh : im I = 1\n⊢ (if h : im I = 1 then (complexRingEquiv h).symm ((complexRingEquiv h) a).sqrt else ↑√(re a)) = ↑√(re a)",
"usedConstants": [
"Eq.mpr",
"Real",
"AddMonoid.toAddSemigroup",
"Real.instAddMonoid",
... | dif_pos h, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.JensenFormula | {
"line": 347,
"column": 43
} | {
"line": 347,
"column": 51
} | [
{
"pp": "c : ℂ\nR : ℝ\nf : ℂ → ℂ\nhR : R ≠ 0\nh₁f : MeromorphicOn f (closedBall c |R|)\nCB : Set ℂ := closedBall c |R|\nh₂f : ∀ u ∈ CB, meromorphicOrderAt f u ≠ ⊤\nh₃f : (divisor f CB).support.Finite\ng : ℂ → ℂ\nh₁g : AnalyticOnNhd ℂ g (closedBall c |R|)\nh₂g : ∀ (u : ↑(closedBall c |R|)), g ↑u ≠ 0\nh₃g : f =ᶠ[... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.JensenFormula | {
"line": 347,
"column": 43
} | {
"line": 347,
"column": 51
} | [
{
"pp": "c : ℂ\nR : ℝ\nf : ℂ → ℂ\nhR : R ≠ 0\nh₁f : MeromorphicOn f (closedBall c |R|)\nCB : Set ℂ := closedBall c |R|\nh₂f : ∀ u ∈ CB, meromorphicOrderAt f u ≠ ⊤\nh₃f : (divisor f CB).support.Finite\ng : ℂ → ℂ\nh₁g : AnalyticOnNhd ℂ g (closedBall c |R|)\nh₂g : ∀ (u : ↑(closedBall c |R|)), g ↑u ≠ 0\nh₃g : f =ᶠ[... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.RCLike.Sqrt | {
"line": 111,
"column": 2
} | {
"line": 111,
"column": 62
} | [
{
"pp": "a : ℂ\nha : 0 ≤ a\n⊢ (-a).sqrt = I * a.sqrt",
"usedConstants": [
"Real.instLE",
"Real",
"Real.instZero",
"PartialOrder.toPreorder",
"Preorder.toLE",
"NormedField.toField",
"Exists",
"RCLike.toPartialOrder",
"GE.ge",
"Complex.instRCLike",
... | obtain ⟨α, hα, rfl⟩ := RCLike.nonneg_iff_exists_ofReal.mp ha | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.RCLike.Sqrt | {
"line": 121,
"column": 19
} | {
"line": 121,
"column": 29
} | [
{
"pp": "case inr\n𝕜 : Type u_1\ninst✝ : RCLike 𝕜\na : 𝕜\nha : 0 ≤ a\nh : im I = 1\n⊢ (if h : im I = 1 then (complexRingEquiv h).symm ((complexRingEquiv h) (-a)).sqrt else ↑√(re (-a))) = I * sqrt a",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"NegZeroClass.toNeg",
... | dif_pos h, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.JensenFormula | {
"line": 347,
"column": 43
} | {
"line": 347,
"column": 51
} | [
{
"pp": "c : ℂ\nR : ℝ\nf : ℂ → ℂ\nhR : R ≠ 0\nh₁f : MeromorphicOn f (closedBall c |R|)\nCB : Set ℂ := closedBall c |R|\nh₂f : ∀ u ∈ CB, meromorphicOrderAt f u ≠ ⊤\nh₃f : (divisor f CB).support.Finite\ng : ℂ → ℂ\nh₁g : AnalyticOnNhd ℂ g (closedBall c |R|)\nh₂g : ∀ (u : ↑(closedBall c |R|)), g ↑u ≠ 0\nh₃g : f =ᶠ[... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Complex.JensenFormula | {
"line": 347,
"column": 43
} | {
"line": 347,
"column": 51
} | [
{
"pp": "c : ℂ\nR : ℝ\nf : ℂ → ℂ\nhR : R ≠ 0\nh₁f : MeromorphicOn f (closedBall c |R|)\nCB : Set ℂ := closedBall c |R|\nh₂f : ∀ u ∈ CB, meromorphicOrderAt f u ≠ ⊤\nh₃f : (divisor f CB).support.Finite\ng : ℂ → ℂ\nh₁g : AnalyticOnNhd ℂ g (closedBall c |R|)\nh₂g : ∀ (u : ↑(closedBall c |R|)), g ↑u ≠ 0\nh₃g : f =ᶠ[... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.JensenFormula | {
"line": 347,
"column": 43
} | {
"line": 347,
"column": 51
} | [
{
"pp": "c : ℂ\nR : ℝ\nf : ℂ → ℂ\nhR : R ≠ 0\nh₁f : MeromorphicOn f (closedBall c |R|)\nCB : Set ℂ := closedBall c |R|\nh₂f : ∀ u ∈ CB, meromorphicOrderAt f u ≠ ⊤\nh₃f : (divisor f CB).support.Finite\ng : ℂ → ℂ\nh₁g : AnalyticOnNhd ℂ g (closedBall c |R|)\nh₂g : ∀ (u : ↑(closedBall c |R|)), g ↑u ≠ 0\nh₃g : f =ᶠ[... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.JensenFormula | {
"line": 372,
"column": 4
} | {
"line": 372,
"column": 12
} | [
{
"pp": "case h\nc : ℂ\nR : ℝ\nf : ℂ → ℂ\nhR : R ≠ 0\nh₁f : MeromorphicOn f (closedBall c |R|)\nCB : Set ℂ := closedBall c |R|\nh₂f✝ : ¬∀ u ∈ CB, meromorphicOrderAt f u ≠ ⊤\nh₂f : ∀ (u : ↑(closedBall c |R|)), meromorphicOrderAt f ↑u = ⊤\nthis✝ : divisor f CB = 0\nthis : f =ᶠ[codiscreteWithin CB] 0\nz : ℂ\nhz : ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Complex.JensenFormula | {
"line": 417,
"column": 22
} | {
"line": 417,
"column": 30
} | [
{
"pp": "c : ℂ\nr R M : ℝ\nf : ℂ → ℂ\nr_pos : 0 < |r|\nr_lt_R : |r| < |R|\nhM : 1 ≤ M\nh₁f : AnalyticOnNhd ℂ f (closedBall c |R|)\nh₂f : f c ≠ 0\nf_bound : ∀ z ∈ sphere c |R|, ‖f z‖ ≤ M\nhrR : 1 < |R / r|\njensen :\n circleAverage (fun x ↦ Real.log ‖f x‖) c R =\n ∑ᶠ (u : ℂ), ↑((divisor f (closedBall c |R|))... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Complex.JensenFormula | {
"line": 417,
"column": 22
} | {
"line": 417,
"column": 30
} | [
{
"pp": "c : ℂ\nr R M : ℝ\nf : ℂ → ℂ\nr_pos : 0 < |r|\nr_lt_R : |r| < |R|\nhM : 1 ≤ M\nh₁f : AnalyticOnNhd ℂ f (closedBall c |R|)\nh₂f : f c ≠ 0\nf_bound : ∀ z ∈ sphere c |R|, ‖f z‖ ≤ M\nhrR : 1 < |R / r|\njensen :\n circleAverage (fun x ↦ Real.log ‖f x‖) c R =\n ∑ᶠ (u : ℂ), ↑((divisor f (closedBall c |R|))... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.JensenFormula | {
"line": 417,
"column": 22
} | {
"line": 417,
"column": 30
} | [
{
"pp": "c : ℂ\nr R M : ℝ\nf : ℂ → ℂ\nr_pos : 0 < |r|\nr_lt_R : |r| < |R|\nhM : 1 ≤ M\nh₁f : AnalyticOnNhd ℂ f (closedBall c |R|)\nh₂f : f c ≠ 0\nf_bound : ∀ z ∈ sphere c |R|, ‖f z‖ ≤ M\nhrR : 1 < |R / r|\njensen :\n circleAverage (fun x ↦ Real.log ‖f x‖) c R =\n ∑ᶠ (u : ℂ), ↑((divisor f (closedBall c |R|))... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.JensenFormula | {
"line": 419,
"column": 22
} | {
"line": 419,
"column": 30
} | [
{
"pp": "c : ℂ\nr R M : ℝ\nf : ℂ → ℂ\nr_pos : 0 < |r|\nr_lt_R : |r| < |R|\nhM : 1 ≤ M\nh₁f : AnalyticOnNhd ℂ f (closedBall c |R|)\nh₂f : f c ≠ 0\nf_bound : ∀ z ∈ sphere c |R|, ‖f z‖ ≤ M\nhrR : 1 < |R / r|\njensen :\n circleAverage (fun x ↦ Real.log ‖f x‖) c R =\n ∑ᶠ (u : ℂ), ↑((divisor f (closedBall c |R|))... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Complex.JensenFormula | {
"line": 419,
"column": 22
} | {
"line": 419,
"column": 30
} | [
{
"pp": "c : ℂ\nr R M : ℝ\nf : ℂ → ℂ\nr_pos : 0 < |r|\nr_lt_R : |r| < |R|\nhM : 1 ≤ M\nh₁f : AnalyticOnNhd ℂ f (closedBall c |R|)\nh₂f : f c ≠ 0\nf_bound : ∀ z ∈ sphere c |R|, ‖f z‖ ≤ M\nhrR : 1 < |R / r|\njensen :\n circleAverage (fun x ↦ Real.log ‖f x‖) c R =\n ∑ᶠ (u : ℂ), ↑((divisor f (closedBall c |R|))... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.JensenFormula | {
"line": 419,
"column": 22
} | {
"line": 419,
"column": 30
} | [
{
"pp": "c : ℂ\nr R M : ℝ\nf : ℂ → ℂ\nr_pos : 0 < |r|\nr_lt_R : |r| < |R|\nhM : 1 ≤ M\nh₁f : AnalyticOnNhd ℂ f (closedBall c |R|)\nh₂f : f c ≠ 0\nf_bound : ∀ z ∈ sphere c |R|, ‖f z‖ ≤ M\nhrR : 1 < |R / r|\njensen :\n circleAverage (fun x ↦ Real.log ‖f x‖) c R =\n ∑ᶠ (u : ℂ), ↑((divisor f (closedBall c |R|))... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Projective | {
"line": 56,
"column": 53
} | {
"line": 58,
"column": 6
} | [
{
"pp": "n : Type u_1\nR : Type u_2\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommRing R\nu : Rˣ\n⊢ mk ((GeneralLinearGroup.scalar n) u) = 1",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"MonoidHom.range",
"MonoidHom.instFunLike",
"InvOneClass.toOne",
"DivInvO... | by
rw [← MonoidHom.mem_ker, ker_mk, GeneralLinearGroup.center_eq_range_scalar]
simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Complex.Tietze | {
"line": 117,
"column": 68
} | {
"line": 117,
"column": 76
} | [
{
"pp": "X : Type u\ninst✝⁵ : TopologicalSpace X\ninst✝⁴ : NormalSpace X\ns : Set X\nhs : IsClosed s\n𝕜 : Type v\ninst✝³ : RCLike 𝕜\nE : Type w\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : FiniteDimensional 𝕜 E\nf : ↑s →ᵇ E\nhf : ¬‖f‖ = 0\n⊢ 0 < ‖f‖",
"usedConstants": [
"AddGr... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Complex.Tietze | {
"line": 117,
"column": 68
} | {
"line": 117,
"column": 76
} | [
{
"pp": "X : Type u\ninst✝⁵ : TopologicalSpace X\ninst✝⁴ : NormalSpace X\ns : Set X\nhs : IsClosed s\n𝕜 : Type v\ninst✝³ : RCLike 𝕜\nE : Type w\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : FiniteDimensional 𝕜 E\nf : ↑s →ᵇ E\nhf : ¬‖f‖ = 0\n⊢ 0 < ‖f‖",
"usedConstants": [
"AddGr... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.Tietze | {
"line": 117,
"column": 68
} | {
"line": 117,
"column": 76
} | [
{
"pp": "X : Type u\ninst✝⁵ : TopologicalSpace X\ninst✝⁴ : NormalSpace X\ns : Set X\nhs : IsClosed s\n𝕜 : Type v\ninst✝³ : RCLike 𝕜\nE : Type w\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : FiniteDimensional 𝕜 E\nf : ↑s →ᵇ E\nhf : ¬‖f‖ = 0\n⊢ 0 < ‖f‖",
"usedConstants": [
"AddGr... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.JensenFormula | {
"line": 444,
"column": 10
} | {
"line": 444,
"column": 18
} | [
{
"pp": "c : ℂ\nr R M : ℝ\nf : ℂ → ℂ\nr_pos : 0 < |r|\nr_lt_R : |r| < |R|\nhM : 1 ≤ M\nh₁f : AnalyticOnNhd ℂ f (closedBall c |R|)\nh₂f : f c ≠ 0\nf_bound : ∀ z ∈ sphere c |R|, ‖f z‖ ≤ M\nhrR : 1 < |R / r|\njensen :\n circleAverage (fun x ↦ Real.log ‖f x‖) c R =\n ∑ᶠ (u : ℂ), ↑((divisor f (closedBall c |R|))... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Topology.TietzeExtension | {
"line": 471,
"column": 2
} | {
"line": 478,
"column": 35
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : NormalSpace Y\nf : C(X, ℝ)\nt : Set ℝ\ne : X → Y\nhs : t.OrdConnected\nhf : ∀ (x : X), f x ∈ t\nhne : t.Nonempty\nhe : IsClosedEmbedding e\nh : ℝ ≃o ↑(Ioo (-1) 1)\nF : X →ᵇ ℝ := { toFun := Subtype.val ∘ ⇑h ∘ ⇑... | have : OrdConnected t' := by
constructor
rintro _ ⟨x, hx, rfl⟩ _ ⟨y, hy, rfl⟩ z hz
lift z to Ioo (-1 : ℝ) 1 using Icc_subset_Ioo (h x).2.1 (h y).2.2 hz
change z ∈ Icc (h x) (h y) at hz
rw [← h.image_Icc] at hz
rcases hz with ⟨z, hz, rfl⟩
exact ⟨z, hs.out hx hy hz, rfl⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Complex.UpperHalfPlane.FixedPoints | {
"line": 157,
"column": 2
} | {
"line": 158,
"column": 74
} | [
{
"pp": "case inr\ng : GL (Fin 2) ℝ\nz : ℍ\nhpos : 0 < (↑g).det\nhell : g.IsElliptic\nthis : ∀ {g : GL (Fin 2) ℝ}, 0 < (↑g).det → ∀ (hell : g.IsElliptic), 0 < ↑g 1 0 → (g • z = z ↔ z = fixedPt g hell)\nhc : ¬0 < ↑g 1 0\n⊢ g • z = z ↔ z = fixedPt g hell",
"usedConstants": [
"UpperHalfPlane.glAction",
... | · replace hc := hell.c_ne_zero.lt_or_gt.resolve_right hc
simpa using @this (-g) (by simpa [Matrix.det_neg]) hell.neg (by simpa) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Complex.UpperHalfPlane.Topology | {
"line": 155,
"column": 2
} | {
"line": 155,
"column": 19
} | [
{
"pp": "w : ℂ\n⊢ ↑ofComplex w = if hw : 0 < w.im then { coe := w, coe_im_pos := hw } else Classical.choice ⋯",
"usedConstants": [
"UpperHalfPlane.ofComplex",
"NormedCommRing.toSeminormedCommRing",
"Real",
"Real.instZero",
"Complex.im",
"Complex.instNormedField",
"R... | split_ifs with hw | Mathlib.Tactic._aux_Mathlib_Tactic_SplitIfs___elabRules_Mathlib_Tactic_splitIfs_1 | Mathlib.Tactic.splitIfs |
Mathlib.Analysis.Complex.UpperHalfPlane.Topology | {
"line": 218,
"column": 48
} | {
"line": 218,
"column": 59
} | [
{
"pp": "τ : ℍ\nU : Set ℝ\nhU : U ∈ map (fun τ ↦ ‖↑τ‖) (𝓝 τ)\ns : Set ℍ\nhs' : (fun τ ↦ ‖↑τ‖) '' s ⊆ U\nε : ℝ\nhεpos : ε > 0\nhεs : Metric.ball (↑τ) ε ⊆ UpperHalfPlane.coe '' s\nr : ℝ\nhr : r ∈ Metric.ball ‖↑τ‖ ε\nhr' : 0 ≤ r\n⊢ ‖r / ‖‖↑τ‖‖ - 1‖ * ‖‖↑τ‖‖ < ε",
"usedConstants": [
"Norm.norm",
"E... | ← norm_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.UpperHalfPlane.FixedPoints | {
"line": 196,
"column": 8
} | {
"line": 196,
"column": 16
} | [
{
"pp": "case mp.inl.inl\ng : GL (Fin 2) ℝ\nhgc : g ∉ Subgroup.center (GL (Fin 2) ℝ)\nhlt : (↑g).det < 0\nha : (↑g).trace = 0\nhb : ↑g 0 1 = ↑g 1 0\nhc : ↑g 1 0 = 0\nhg : { coe := 1 + Complex.I, coe_im_pos := ⋯ }.re = ↑g 0 1 / (2 * ↑g 1 1)\n⊢ False",
"usedConstants": [
"Units.val",
"GroupWithZer... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
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