module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Analysis.Complex.Hadamard | {
"line": 331,
"column": 2
} | {
"line": 331,
"column": 46
} | {
"line": 333,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝ : NormedAddCommGroup E\nf : ℂ → E\nl u a : ℝ\nha : ∀ z ∈ re ⁻¹' {l}, ‖f z‖ ≤ a\nz : ℂ\nhz : z.re = 0\n⊢ ‖f (↑l + z * (↑u - ↑l))‖ ≤ a",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Real",
"Complex.mul_re",
"HMul.hMul",
"sub_self",
... | [] | exact ha (↑l + z * (↑u - ↑l)) (by simp [hz]) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Complex.PhragmenLindelof | {
"line": 139,
"column": 4
} | {
"line": 139,
"column": 33
} | {
"line": 139,
"column": 34
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nC✝ : ℝ\nf : ℂ → E\nz : ℂ\nC : ℝ\nhC₀ : 0 < C\na b : ℝ\nhza : a - b < z.im\nhle_a : ∀ (z : ℂ), z.im = a - b → ‖f z‖ ≤ C\nhzb : z.im < a + b\nhle_b : ∀ (z : ℂ), z.im = a + b → ‖f z‖ ≤ C\nhfd : DiffContOnCl ℂ f (im ⁻¹' Ioo (a - b) (a + ... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nC✝ : ℝ\nf : ℂ → E\nz : ℂ\nC : ℝ\nhC₀ : 0 < C\na b : ℝ\nhza : a - b < z.im\nhle_a : ∀ (z : ℂ), z.im = a - b → ‖f z‖ ≤ C\nhzb : z.im < a + b\nhle_b : ∀ (z : ℂ), z.im = a + b → ‖f z‖ ≤ C\nhfd : DiffContOnCl ℂ f (im ⁻¹' Ioo (a - b) (a + b))\nhab : a... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.PhragmenLindelof | {
"line": 142,
"column": 2
} | {
"line": 142,
"column": 72
} | {
"line": 145,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nC✝ : ℝ\nf : ℂ → E\nz : ℂ\nC : ℝ\nhC₀ : 0 < C\na b : ℝ\nhza : a - b < z.im\nhle_a : ∀ (z : ℂ), z.im = a - b → ‖f z‖ ≤ C\nhzb : z.im < a + b\nhle_b : ∀ (z : ℂ), z.im = a + b → ‖f z‖ ≤ C\nhfd : DiffContOnCl ℂ f (im ⁻¹' Ioo (a - b) (a + ... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nC✝ : ℝ\nf : ℂ → E\nz : ℂ\nC : ℝ\nhC₀ : 0 < C\na b : ℝ\nhza : a - b < z.im\nhle_a : ∀ (z : ℂ), z.im = a - b → ‖f z‖ ≤ C\nhzb : z.im < a + b\nhle_b : ∀ (z : ℂ), z.im = a + b → ‖f z‖ ≤ C\nhfd : DiffContOnCl ℂ f (im ⁻¹' Ioo (a - b) (a + b))\nhab : a... | set g := fun (ε : ℝ) (w : ℂ) => exp (ε * (exp (aff w) + exp (-aff w))) | Mathlib.Tactic._aux_Mathlib_Tactic_Set___elabRules_Mathlib_Tactic_setTactic_1 | Mathlib.Tactic.setTactic |
Mathlib.Analysis.Complex.Hadamard | {
"line": 369,
"column": 2
} | {
"line": 387,
"column": 11
} | {
"line": 389,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝ : NormedAddCommGroup E\nf : ℂ → E\nl u : ℝ\nhul : l < u\n⊢ sSupNormIm (scale f l u) 1 = sSupNormIm f u",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Set.ext",
"Real.instIsOrderedRing",
"Norm.norm",
"Not.intro",
"Mathlib.Tactic.R... | [] | simp_rw [sSupNormIm, image_comp]
have : scale f l u '' re ⁻¹' {1} = f '' re ⁻¹' {u} := by
ext e
simp only [scale, smul_eq_mul, mem_image, mem_preimage, mem_singleton_iff]
constructor
· intro h
obtain ⟨z, hz₁, hz₂⟩ := h
use ↑l + z * (↑u - ↑l)
simp only [add_re, ofReal_re, mul_re, hz₁,... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.Hadamard | {
"line": 369,
"column": 2
} | {
"line": 387,
"column": 11
} | {
"line": 389,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝ : NormedAddCommGroup E\nf : ℂ → E\nl u : ℝ\nhul : l < u\n⊢ sSupNormIm (scale f l u) 1 = sSupNormIm f u",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Set.ext",
"Real.instIsOrderedRing",
"Norm.norm",
"Not.intro",
"Mathlib.Tactic.R... | [] | simp_rw [sSupNormIm, image_comp]
have : scale f l u '' re ⁻¹' {1} = f '' re ⁻¹' {u} := by
ext e
simp only [scale, smul_eq_mul, mem_image, mem_preimage, mem_singleton_iff]
constructor
· intro h
obtain ⟨z, hz₁, hz₂⟩ := h
use ↑l + z * (↑u - ↑l)
simp only [add_re, ofReal_re, mul_re, hz₁,... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.Hadamard | {
"line": 408,
"column": 4
} | {
"line": 408,
"column": 68
} | {
"line": 409,
"column": 6
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\nf : ℂ → E\ninst✝ : NormedSpace ℂ E\nε : ℝ\nhε : ε > 0\nz : ℂ\nhB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)\nhd : DiffContOnCl ℂ f (verticalStrip 0 1)\nhz : z ∈ verticalClosedStrip 0 1\n⊢ ‖f z‖ * ((ε + sSupNormIm f 0) ^ (z.re - 1) * (ε + sSupNormIm f 1... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\nf : ℂ → E\ninst✝ : NormedSpace ℂ E\nε : ℝ\nhε : ε > 0\nz : ℂ\nhB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)\nhd : DiffContOnCl ℂ f (verticalStrip 0 1)\nhz : z ∈ verticalClosedStrip 0 1\n⊢ ‖f z‖ * ((ε + sSupNormIm f 0) ^ (z.re - 1) * (ε + sSupNormIm f 1) ^ (-z.re))... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Hadamard | {
"line": 438,
"column": 16
} | {
"line": 438,
"column": 27
} | {
"line": 438,
"column": 28
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\nf : ℂ → E\ninst✝ : NormedSpace ℂ E\nz : ℂ\nhd : DiffContOnCl ℂ f (verticalStrip 0 1)\nhB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)\nhz : z ∈ verticalStrip 0 1\nx : ℝ\nhx : x ∈ Ioi 0\n⊢ z.re ≠ 0",
"ppTerm": "?m.406",
"assigned": true,
"used... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\nf : ℂ → E\ninst✝ : NormedSpace ℂ E\nz : ℂ\nhd : DiffContOnCl ℂ f (verticalStrip 0 1)\nhB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)\nhz : z ∈ verticalStrip 0 1\nx : ℝ\nhx : x ∈ Ioi 0\n⊢ ¬z.re = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Homotopy.Lifting | {
"line": 399,
"column": 2
} | {
"line": 399,
"column": 48
} | {
"line": 400,
"column": 2
} | [
{
"pp": "case mk\nE : Type u_1\nX : Type u_2\ninst✝¹ : TopologicalSpace E\ninst✝ : TopologicalSpace X\np : E → X\ncov : IsCoveringMap p\nx y : X\nγ✝ : Path.Homotopic.Quotient x y\ne : ↑(p ⁻¹' {x})\nγ : Path x y\n⊢ (cov.liftPathQuotient (Quot.mk (⇑(Path.Homotopic.setoid x y)) γ) e).map { toFun := p, continuous_t... | [
"case mk\nE : Type u_1\nX : Type u_2\ninst✝¹ : TopologicalSpace E\ninst✝ : TopologicalSpace X\np : E → X\ncov : IsCoveringMap p\nx y : X\nγ✝ : Path.Homotopic.Quotient x y\ne : ↑(p ⁻¹' {x})\nγ : Path x y\n⊢ (fun q ↦ q.map ⋯) { toContinuousMap := cov.liftPath ↑γ ↑e ⋯, source' := ⋯, target' := ⋯ } = (fun p_1 ↦ p_1.cas... | refine congr_arg Path.Homotopic.Quotient.mk ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Complex.Hadamard | {
"line": 439,
"column": 16
} | {
"line": 439,
"column": 41
} | {
"line": 439,
"column": 42
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\nf : ℂ → E\ninst✝ : NormedSpace ℂ E\nz : ℂ\nhd : DiffContOnCl ℂ f (verticalStrip 0 1)\nhB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)\nhz : z ∈ verticalStrip 0 1\nx : ℝ\nhx : x ∈ Ioi 0\n⊢ (1 - z).re ≠ 0",
"ppTerm": "?m.394",
"assigned": true,
... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\nf : ℂ → E\ninst✝ : NormedSpace ℂ E\nz : ℂ\nhd : DiffContOnCl ℂ f (verticalStrip 0 1)\nhB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)\nhz : z ∈ verticalStrip 0 1\nx : ℝ\nhx : x ∈ Ioi 0\n⊢ ¬1 = z.re"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.PhragmenLindelof | {
"line": 163,
"column": 4
} | {
"line": 164,
"column": 41
} | {
"line": 165,
"column": 6
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nC✝ : ℝ\nf : ℂ → E\nz : ℂ\nC : ℝ\nhC₀ : 0 < C\na b : ℝ\nhza : a - b < z.im\nhle_a : ∀ (z : ℂ), z.im = a - b → ‖f z‖ ≤ C\nhzb : z.im < a + b\nhle_b : ∀ (z : ℂ), z.im = a + b → ‖f z‖ ≤ C\nhfd : DiffContOnCl ℂ f (im ⁻¹' Ioo (a - b) (a + ... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nC✝ : ℝ\nf : ℂ → E\nz : ℂ\nC : ℝ\nhC₀ : 0 < C\na b : ℝ\nhza : a - b < z.im\nhle_a : ∀ (z : ℂ), z.im = a - b → ‖f z‖ ≤ C\nhzb : z.im < a + b\nhle_b : ∀ (z : ℂ), z.im = a + b → ‖f z‖ ≤ C\nhfd : DiffContOnCl ℂ f (im ⁻¹' Ioo (a - b) (a + b))\nhab : a... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Homotopy.Lifting | {
"line": 474,
"column": 4
} | {
"line": 474,
"column": 26
} | {
"line": 474,
"column": 27
} | [
{
"pp": "case refine_1\nE : Type u_1\nX : Type u_2\nA : Type u_3\ninst✝⁴ : TopologicalSpace E\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace A\np : E → X\ncov : IsCoveringMap p\ninst✝¹ : SimplyConnectedSpace A\ninst✝ : LocallyPathConnectedSpace A\nf : C(A, X)\na₀ : A\ne₀ : E\nhe : p e₀ = f a₀\nγ : C(↑I... | [
"case refine_1\nE : Type u_1\nX : Type u_2\nA : Type u_3\ninst✝⁴ : TopologicalSpace E\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace A\np : E → X\ncov : IsCoveringMap p\ninst✝¹ : SimplyConnectedSpace A\ninst✝ : LocallyPathConnectedSpace A\nf : C(A, X)\na₀ : A\ne₀ : E\nhe : p e₀ = f a₀\nγ : C(↑I, A)\nγ_0 : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Homotopy.Lifting | {
"line": 497,
"column": 4
} | {
"line": 497,
"column": 26
} | {
"line": 497,
"column": 27
} | [
{
"pp": "case refine_1\nE : Type u_1\nX : Type u_2\nA : Type u_3\ninst✝⁴ : TopologicalSpace E\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace A\np : E → X\ncov : IsCoveringMap p\ninst✝¹ : PathConnectedSpace A\ninst✝ : LocallyPathConnectedSpace A\nf : C(A, X)\na₀ : A\ne₀ : E\nhe : p e₀ = f a₀\nle : (Fund... | [
"case refine_1\nE : Type u_1\nX : Type u_2\nA : Type u_3\ninst✝⁴ : TopologicalSpace E\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace A\np : E → X\ncov : IsCoveringMap p\ninst✝¹ : PathConnectedSpace A\ninst✝ : LocallyPathConnectedSpace A\nf : C(A, X)\na₀ : A\ne₀ : E\nhe : p e₀ = f a₀\nle : (FundamentalGroup... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Hadamard | {
"line": 455,
"column": 8
} | {
"line": 455,
"column": 19
} | {
"line": 455,
"column": 20
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\nf : ℂ → E\ninst✝ : NormedSpace ℂ E\nz : ℂ\nhd : DiffContOnCl ℂ f (verticalStrip 0 1)\nhB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)\nhz : z ∈ verticalStrip 0 1\nthis :\n ∀ x ∈ Ioi 0,\n (x + sSupNormIm f 0) ^ (1 - z.re) * (x + sSupNormIm f 1) ^ z.re... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\nf : ℂ → E\ninst✝ : NormedSpace ℂ E\nz : ℂ\nhd : DiffContOnCl ℂ f (verticalStrip 0 1)\nhB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)\nhz : z ∈ verticalStrip 0 1\nthis :\n ∀ x ∈ Ioi 0,\n (x + sSupNormIm f 0) ^ (1 - z.re) * (x + sSupNormIm f 1) ^ z.re =\n ‖↑... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Hadamard | {
"line": 456,
"column": 8
} | {
"line": 456,
"column": 33
} | {
"line": 456,
"column": 34
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\nf : ℂ → E\ninst✝ : NormedSpace ℂ E\nz : ℂ\nhd : DiffContOnCl ℂ f (verticalStrip 0 1)\nhB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)\nhz : z ∈ verticalStrip 0 1\nthis :\n ∀ x ∈ Ioi 0,\n (x + sSupNormIm f 0) ^ (1 - z.re) * (x + sSupNormIm f 1) ^ z.re... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\nf : ℂ → E\ninst✝ : NormedSpace ℂ E\nz : ℂ\nhd : DiffContOnCl ℂ f (verticalStrip 0 1)\nhB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)\nhz : z ∈ verticalStrip 0 1\nthis :\n ∀ x ∈ Ioi 0,\n (x + sSupNormIm f 0) ^ (1 - z.re) * (x + sSupNormIm f 1) ^ z.re =\n ‖↑... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.HasPrimitives | {
"line": 185,
"column": 6
} | {
"line": 185,
"column": 17
} | {
"line": 185,
"column": 18
} | [
{
"pp": "case hbc.hb₂\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nc : ℂ\nr : ℝ\nf : ℂ → E\nf_cont : ContinuousOn f (ball c r)\nz : ℂ\nhz : z ∈ ball c r\nhf : IsConservativeOn f (ball c r)\nw : ℂ\nw_in_z_ball : w ∈ ball z (r - dist z c)\nI₁ : E := ⋯\nI₂ : E := ⋯\nI₃ : E := ⋯\nI₄ : E :=... | [
"case hbc.hb₂\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nc : ℂ\nr : ℝ\nf : ℂ → E\nf_cont : ContinuousOn f (ball c r)\nz : ℂ\nhz : z ∈ ball c r\nhf : IsConservativeOn f (ball c r)\nw : ℂ\nw_in_z_ball : w ∈ ball z (r - dist z c)\nI₁ : E := ∫ (x : ℝ) in c.re..w.re, f (↑x + ↑c.im * I)\nI₂ : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Homotopy.Lifting | {
"line": 606,
"column": 2
} | {
"line": 606,
"column": 57
} | {
"line": 607,
"column": 2
} | [
{
"pp": "E : Type u_1\nX : Type u_2\ninst✝⁴ : TopologicalSpace E\ninst✝³ : TopologicalSpace X\np : E → X\nG : Type u_4\ninst✝² : Group G\ninst✝¹ : MulAction G E\nhp : IsQuotientCoveringMap p G\nx : X\ninst✝ : SimplyConnectedSpace E\ne : ↑(p ⁻¹' {x}) := ⟨⋯.choose, ⋯⟩\n⊢ Injective ⇑(⋯.monodromyPerm x)",
"ppTe... | [
"E : Type u_1\nX : Type u_2\ninst✝⁴ : TopologicalSpace E\ninst✝³ : TopologicalSpace X\np : E → X\nG : Type u_4\ninst✝² : Group G\ninst✝¹ : MulAction G E\nhp : IsQuotientCoveringMap p G\nx : X\ninst✝ : SimplyConnectedSpace E\ne : ↑(p ⁻¹' {x}) := ⟨⋯.choose, ⋯⟩\n⊢ (FundamentalGroup.mapOfEq { toFun := p, continuous_toF... | rw [← MonoidHom.ker_eq_bot_iff, hp.ker_monodromyPerm e] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Complex.HasPrimitives | {
"line": 192,
"column": 12
} | {
"line": 192,
"column": 23
} | {
"line": 192,
"column": 24
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nc : ℂ\nr : ℝ\nf : ℂ → E\nf_cont : ContinuousOn f (ball c r)\nz : ℂ\nhz : z ∈ ball c r\nhf : IsConservativeOn f (ball c r)\nw : ℂ\nw_in_z_ball : w ∈ ball z (r - dist z c)\nI₁ : E := ∫ (x : ℝ) in c.re..w.re, f (↑x + ↑c.im * I)\nI₂ : E ... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nc : ℂ\nr : ℝ\nf : ℂ → E\nf_cont : ContinuousOn f (ball c r)\nz : ℂ\nhz : z ∈ ball c r\nhf : IsConservativeOn f (ball c r)\nw : ℂ\nw_in_z_ball : w ∈ ball z (r - dist z c)\nI₁ : E := ∫ (x : ℝ) in c.re..w.re, f (↑x + ↑c.im * I)\nI₂ : E := I • ∫ (y ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.PhragmenLindelof | {
"line": 192,
"column": 6
} | {
"line": 194,
"column": 13
} | {
"line": 194,
"column": 14
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nC✝ : ℝ\nf : ℂ → E\nz : ℂ\nC : ℝ\nhC₀ : 0 < C\na b : ℝ\nhza : a - b < z.im\nhle_a : ∀ (z : ℂ), z.im = a - b → ‖f z‖ ≤ C\nhzb : z.im < a + b\nhle_b : ∀ (z : ℂ), z.im = a + b → ‖f z‖ ≤ C\nhfd : DiffContOnCl ℂ f (im ⁻¹' Ioo (a - b) (a + ... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nC✝ : ℝ\nf : ℂ → E\nz : ℂ\nC : ℝ\nhC₀ : 0 < C\na b : ℝ\nhza : a - b < z.im\nhle_a : ∀ (z : ℂ), z.im = a - b → ‖f z‖ ≤ C\nhzb : z.im < a + b\nhle_b : ∀ (z : ℂ), z.im = a + b → ‖f z‖ ≤ C\nhfd : DiffContOnCl ℂ f (im ⁻¹' Ioo (a - b) (a + b))\nhab : a... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Homotopy.Lifting | {
"line": 631,
"column": 96
} | {
"line": 635,
"column": 7
} | {
"line": 637,
"column": 0
} | [
{
"pp": "E : Type u_1\nX : Type u_2\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace X\np : E → X\nG : Type u_4\ninst✝¹ : Group G\ninst✝ : MulAction G E\nhp : IsQuotientCoveringMap p G\nx : X\ne : ↑(p ⁻¹' {x})\nγ : FundamentalGroup X x\ng : Gᵐᵒᵖ\n⊢ (hp.fundamentalGroupToMulOpposite e) γ = g ↔ MulOpposite... | [] | by
rw [fundamentalGroupToMulOpposite, ← MulOpposite.unop_injective.eq_iff, iff_comm, eq_comm,
← hp.fiberEquivGroup_smul_self e]
have := hp.isCancelSMul.right_cancel'
aesop | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Complex.HasPrimitives | {
"line": 192,
"column": 32
} | {
"line": 192,
"column": 43
} | {
"line": 192,
"column": 44
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nc : ℂ\nr : ℝ\nf : ℂ → E\nf_cont : ContinuousOn f (ball c r)\nz : ℂ\nhz : z ∈ ball c r\nhf : IsConservativeOn f (ball c r)\nw : ℂ\nw_in_z_ball : w ∈ ball z (r - dist z c)\nI₁ : E := ∫ (x : ℝ) in c.re..w.re, f (↑x + ↑c.im * I)\nI₂ : E ... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nc : ℂ\nr : ℝ\nf : ℂ → E\nf_cont : ContinuousOn f (ball c r)\nz : ℂ\nhz : z ∈ ball c r\nhf : IsConservativeOn f (ball c r)\nw : ℂ\nw_in_z_ball : w ∈ ball z (r - dist z c)\nI₁ : E := ∫ (x : ℝ) in c.re..w.re, f (↑x + ↑c.im * I)\nI₂ : E := I • ∫ (y ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Homotopy.Lifting | {
"line": 664,
"column": 8
} | {
"line": 664,
"column": 19
} | {
"line": 664,
"column": 20
} | [
{
"pp": "E : Type u_1\nX : Type u_2\ninst✝⁴ : TopologicalSpace E\ninst✝³ : TopologicalSpace X\np : E → X\nG : Type u_4\ninst✝² : Group G\ninst✝¹ : MulAction G E\nhp : IsQuotientCoveringMap p G\nx : X\ne : ↑(p ⁻¹' {x})\ninst✝ : PathConnectedSpace E\ng : Gᵐᵒᵖ\ne' : ↑(p ⁻¹' {x}) := ⟨MulOpposite.unop g • ↑e, ⋯⟩\nhe... | [
"E : Type u_1\nX : Type u_2\ninst✝⁴ : TopologicalSpace E\ninst✝³ : TopologicalSpace X\np : E → X\nG : Type u_4\ninst✝² : Group G\ninst✝¹ : MulAction G E\nhp : IsQuotientCoveringMap p G\nx : X\ne : ↑(p ⁻¹' {x})\ninst✝ : PathConnectedSpace E\ng : Gᵐᵒᵖ\ne' : ↑(p ⁻¹' {x}) := ⟨MulOpposite.unop g • ↑e, ⋯⟩\nhe' : e' = ⟨Mu... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Homotopy.Lifting | {
"line": 664,
"column": 41
} | {
"line": 664,
"column": 52
} | {
"line": 664,
"column": 53
} | [
{
"pp": "E : Type u_1\nX : Type u_2\ninst✝⁴ : TopologicalSpace E\ninst✝³ : TopologicalSpace X\np : E → X\nG : Type u_4\ninst✝² : Group G\ninst✝¹ : MulAction G E\nhp : IsQuotientCoveringMap p G\nx : X\ne : ↑(p ⁻¹' {x})\ninst✝ : PathConnectedSpace E\ng : Gᵐᵒᵖ\ne' : ↑(p ⁻¹' {x}) := ⟨MulOpposite.unop g • ↑e, ⋯⟩\nhe... | [
"E : Type u_1\nX : Type u_2\ninst✝⁴ : TopologicalSpace E\ninst✝³ : TopologicalSpace X\np : E → X\nG : Type u_4\ninst✝² : Group G\ninst✝¹ : MulAction G E\nhp : IsQuotientCoveringMap p G\nx : X\ne : ↑(p ⁻¹' {x})\ninst✝ : PathConnectedSpace E\ng : Gᵐᵒᵖ\ne' : ↑(p ⁻¹' {x}) := ⟨MulOpposite.unop g • ↑e, ⋯⟩\nhe' : e' = ⟨Mu... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.HasPrimitives | {
"line": 193,
"column": 4
} | {
"line": 193,
"column": 78
} | {
"line": 194,
"column": 6
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nc : ℂ\nr : ℝ\nf : ℂ → E\nf_cont : ContinuousOn f (ball c r)\nz : ℂ\nhz : z ∈ ball c r\nhf : IsConservativeOn f (ball c r)\nw : ℂ\nw_in_z_ball : w ∈ ball z (r - dist z c)\nI₁ : E := ∫ (x : ℝ) in c.re..w.re, f (↑x + ↑c.im * I)\nI₂ : E ... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nc : ℂ\nr : ℝ\nf : ℂ → E\nf_cont : ContinuousOn f (ball c r)\nz : ℂ\nhz : z ∈ ball c r\nhf : IsConservativeOn f (ball c r)\nw : ℂ\nw_in_z_ball : w ∈ ball z (r - dist z c)\nI₁ : E := ∫ (x : ℝ) in c.re..w.re, f (↑x + ↑c.im * I)\nI₂ : E := I • ∫ (y ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.HasPrimitives | {
"line": 205,
"column": 29
} | {
"line": 205,
"column": 69
} | {
"line": 205,
"column": 70
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\nc : ℂ\nr : ℝ\nf : ℂ → E\ninst✝ : CompleteSpace E\nf_cont : ContinuousOn f (ball c r)\nz : ℂ\nhz : z ∈ ball c r\nr₁ : ℝ := r - dist z c\n⊢ 0 < r₁",
"ppTerm": "?m.136",
"assigned": true,
"usedConstants": [
"IsRightCa... | [
"E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\nc : ℂ\nr : ℝ\nf : ℂ → E\ninst✝ : CompleteSpace E\nf_cont : ContinuousOn f (ball c r)\nz : ℂ\nhz : z ∈ ball c r\nr₁ : ℝ := r - dist z c\n⊢ dist z c < r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Hadamard | {
"line": 507,
"column": 6
} | {
"line": 508,
"column": 48
} | {
"line": 508,
"column": 49
} | [
{
"pp": "case h₁\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → E\nz : ℂ\na b : ℝ\nhz : z ∈ verticalClosedStrip 0 1\nhd : DiffContOnCl ℂ f (verticalStrip 0 1)\nhB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)\nha : ∀ z ∈ re ⁻¹' {0}, ‖f z‖ ≤ a\nhb : ∀ z ∈ re ⁻¹' {1}, ‖f z‖ ≤ b\... | [
"case h₁\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → E\nz : ℂ\na b : ℝ\nhz : z ∈ verticalClosedStrip 0 1\nhd : DiffContOnCl ℂ f (verticalStrip 0 1)\nhB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)\nha : ∀ z ∈ re ⁻¹' {0}, ‖f z‖ ≤ a\nhb : ∀ z ∈ re ⁻¹' {1}, ‖f z‖ ≤ b\nthis : ‖int... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.HasPrimitives | {
"line": 207,
"column": 2
} | {
"line": 207,
"column": 50
} | {
"line": 208,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\nc : ℂ\nr : ℝ\nf : ℂ → E\ninst✝ : CompleteSpace E\nf_cont : ContinuousOn f (ball c r)\nz : ℂ\nhz : z ∈ ball c r\nr₁ : ℝ := r - dist z c\nr₁_pos : 0 < r₁\ns : Set ℝ := Ioo (z.re - r₁) (z.re + r₁)\n⊢ (fun x ↦ (∫ (t : ℝ) in z.re..x, f (... | [
"E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\nc : ℂ\nr : ℝ\nf : ℂ → E\ninst✝ : CompleteSpace E\nf_cont : ContinuousOn f (ball c r)\nz : ℂ\nhz : z ∈ ball c r\nr₁ : ℝ := r - dist z c\nr₁_pos : 0 < r₁\ns : Set ℝ := Ioo (z.re - r₁) (z.re + r₁)\nzRe_mem_s : z.re ∈ s\n⊢ (fun x ↦ (∫ (t : ℝ) in z.... | have zRe_mem_s : z.re ∈ s := by simp [s, r₁_pos] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Complex.HasPrimitives | {
"line": 217,
"column": 2
} | {
"line": 217,
"column": 13
} | {
"line": 217,
"column": 14
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\nc : ℂ\nr : ℝ\nf : ℂ → E\ninst✝ : CompleteSpace E\nf_cont : ContinuousOn f (ball c r)\nz : ℂ\nhz : z ∈ ball c r\nr₁ : ℝ := r - dist z c\nr₁_pos : 0 < r₁\ns : Set ℝ := Ioo (z.re - r₁) (z.re + r₁)\nzRe_mem_s : z.re ∈ s\nf_contOn : Cont... | [
"E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\nc : ℂ\nr : ℝ\nf : ℂ → E\ninst✝ : CompleteSpace E\nf_cont : ContinuousOn f (ball c r)\nz : ℂ\nhz : z ∈ ball c r\nr₁ : ℝ := r - dist z c\nr₁_pos : 0 < r₁\ns : Set ℝ := Ioo (z.re - r₁) (z.re + r₁)\nzRe_mem_s : z.re ∈ s\nf_contOn : ContinuousOn (fu... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.HasPrimitives | {
"line": 228,
"column": 58
} | {
"line": 228,
"column": 69
} | {
"line": 228,
"column": 70
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\nc : ℂ\nr : ℝ\nf : ℂ → E\ninst✝ : CompleteSpace E\nf_cont : ContinuousOn f (ball c r)\nz : ℂ\nhz : z ∈ ball c r\nthis : (fun w ↦ ∫ (y : ℝ) in z.im..w.im, f (↑w.re + ↑y * I) - f z) =o[𝓝 z] fun w ↦ w - z\n⊢ r - dist z c > 0",
"ppT... | [
"E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\nc : ℂ\nr : ℝ\nf : ℂ → E\ninst✝ : CompleteSpace E\nf_cont : ContinuousOn f (ball c r)\nz : ℂ\nhz : z ∈ ball c r\nthis : (fun w ↦ ∫ (y : ℝ) in z.im..w.im, f (↑w.re + ↑y * I) - f z) =o[𝓝 z] fun w ↦ w - z\n⊢ dist z c < r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Hadamard | {
"line": 514,
"column": 8
} | {
"line": 515,
"column": 50
} | {
"line": 515,
"column": 51
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → E\nz : ℂ\na b : ℝ\nhz : z ∈ verticalClosedStrip 0 1\nhd : DiffContOnCl ℂ f (verticalStrip 0 1)\nhB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)\nha : ∀ z ∈ re ⁻¹' {0}, ‖f z‖ ≤ a\nhb : ∀ z ∈ re ⁻¹' {1}, ‖f z‖ ≤ b\nthis : ‖... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → E\nz : ℂ\na b : ℝ\nhz : z ∈ verticalClosedStrip 0 1\nhd : DiffContOnCl ℂ f (verticalStrip 0 1)\nhB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)\nha : ∀ z ∈ re ⁻¹' {0}, ‖f z‖ ≤ a\nhb : ∀ z ∈ re ⁻¹' {1}, ‖f z‖ ≤ b\nthis : ‖interpStrip ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Harmonic.Basic | {
"line": 201,
"column": 4
} | {
"line": 201,
"column": 35
} | {
"line": 201,
"column": 36
} | [
{
"pp": "case mp\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\nl : F ≃L[ℝ] G\nh : Harmoni... | [
"case mp\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\nl : F ≃L[ℝ] G\nh : HarmonicAt (⇑l ∘ f)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Harmonic.Liouville | {
"line": 55,
"column": 4
} | {
"line": 55,
"column": 28
} | {
"line": 55,
"column": 29
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℂ → E\nh_harm : HarmonicOnNhd f univ\nh_bound : IsBounded (range f)\nz w : ℂ\nℓ : StrongDual ℝ E\nh₁ℓ : ‖ℓ‖ ≤ 1\nh₂ℓ : ℓ (f z) - ℓ (f w) = ‖f z - f w‖\n⊢ IsBounded (range (⇑ℓ ∘ f))",
"ppTerm": "?m.80",
"assigned": true,
... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℂ → E\nh_harm : HarmonicOnNhd f univ\nh_bound : IsBounded (range f)\nz w : ℂ\nℓ : StrongDual ℝ E\nh₁ℓ : ‖ℓ‖ ≤ 1\nh₂ℓ : ℓ (f z) - ℓ (f w) = ‖f z - f w‖\n⊢ IsBounded (⇑ℓ '' range f)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalAverage | {
"line": 108,
"column": 8
} | {
"line": 108,
"column": 19
} | {
"line": 108,
"column": 20
} | [
{
"pp": "f : ℝ → ℝ\na b : ℝ\nhab : a ≠ b\nhf : ContinuousOn f [[a, b]]\n⊢ volume (Ι a b) ≠ 0",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"Real.instLE",
"Real",
"MeasureTheory.Measure",
"Real.lattice",
"... | [
"f : ℝ → ℝ\na b : ℝ\nhab : a ≠ b\nhf : ContinuousOn f [[a, b]]\n⊢ ¬b - a = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.PhragmenLindelof | {
"line": 280,
"column": 35
} | {
"line": 280,
"column": 58
} | {
"line": 280,
"column": 59
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\na b C : ℝ\nf : ℂ → E\nz : ℂ\nhfd : DiffContOnCl ℂ f (re ⁻¹' Ioo a b)\nhB : ∃ c < π / (b - a), ∃ B, f =O[comap (abs ∘ im) atTop ⊓ 𝓟 (re ⁻¹' Ioo a b)] fun z ↦ expR (B * expR (c * |z.im|))\nhle_a : ∀ (z : ℂ), z.re = a → ‖f z‖ ≤ C\nhle_... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\na b C : ℝ\nf : ℂ → E\nz : ℂ\nhfd : DiffContOnCl ℂ f (re ⁻¹' Ioo a b)\nhB : ∃ c < π / (b - a), ∃ B, f =O[comap (abs ∘ im) atTop ⊓ 𝓟 (re ⁻¹' Ioo a b)] fun z ↦ expR (B * expR (c * |z.im|))\nhle_a : ∀ (z : ℂ), z.re = a → ‖f z‖ ≤ C\nhle_b : ∀ (z : ℂ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.PhragmenLindelof | {
"line": 281,
"column": 78
} | {
"line": 281,
"column": 89
} | {
"line": 281,
"column": 90
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\na b C : ℝ\nf : ℂ → E\nz✝ : ℂ\nhfd : DiffContOnCl ℂ f (re ⁻¹' Ioo a b)\nhB : ∃ c < π / (b - a), ∃ B, f =O[comap (abs ∘ im) atTop ⊓ 𝓟 (re ⁻¹' Ioo a b)] fun z ↦ expR (B * expR (c * |z.im|))\nhle_a : ∀ (z : ℂ), z.re = a → ‖f z‖ ≤ C\nhle... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\na b C : ℝ\nf : ℂ → E\nz✝ : ℂ\nhfd : DiffContOnCl ℂ f (re ⁻¹' Ioo a b)\nhB : ∃ c < π / (b - a), ∃ B, f =O[comap (abs ∘ im) atTop ⊓ 𝓟 (re ⁻¹' Ioo a b)] fun z ↦ expR (B * expR (c * |z.im|))\nhle_a : ∀ (z : ℂ), z.re = a → ‖f z‖ ≤ C\nhle_b : ∀ (z : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.PhragmenLindelof | {
"line": 290,
"column": 6
} | {
"line": 290,
"column": 37
} | {
"line": 290,
"column": 38
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\na b C : ℝ\nf : ℂ → E\nz : ℂ\nhfd : DiffContOnCl ℂ f (re ⁻¹' Ioo a b)\nhle_a : ∀ (z : ℂ), z.re = a → ‖f z‖ ≤ C\nhle_b : ∀ (z : ℂ), z.re = b → ‖f z‖ ≤ C\nhza : a ≤ z.re\nhzb : z.re ≤ b\nH : MapsTo (fun x ↦ x * -I) (im ⁻¹' Ioo a b) (re ... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\na b C : ℝ\nf : ℂ → E\nz : ℂ\nhfd : DiffContOnCl ℂ f (re ⁻¹' Ioo a b)\nhle_a : ∀ (z : ℂ), z.re = a → ‖f z‖ ≤ C\nhle_b : ∀ (z : ℂ), z.re = b → ‖f z‖ ≤ C\nhza : a ≤ z.re\nhzb : z.re ≤ b\nH : MapsTo (fun x ↦ x * -I) (im ⁻¹' Ioo a b) (re ⁻¹' Ioo a b)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.PhragmenLindelof | {
"line": 291,
"column": 4
} | {
"line": 291,
"column": 35
} | {
"line": 291,
"column": 36
} | [
{
"pp": "case refine_1\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\na b C : ℝ\nf : ℂ → E\nz : ℂ\nhfd : DiffContOnCl ℂ f (re ⁻¹' Ioo a b)\nhle_a : ∀ (z : ℂ), z.re = a → ‖f z‖ ≤ C\nhle_b : ∀ (z : ℂ), z.re = b → ‖f z‖ ≤ C\nhza : a ≤ z.re\nhzb : z.re ≤ b\nH : MapsTo (fun x ↦ x * -I) (im ⁻¹... | [
"case refine_1\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\na b C : ℝ\nf : ℂ → E\nz : ℂ\nhfd : DiffContOnCl ℂ f (re ⁻¹' Ioo a b)\nhle_a : ∀ (z : ℂ), z.re = a → ‖f z‖ ≤ C\nhle_b : ∀ (z : ℂ), z.re = b → ‖f z‖ ≤ C\nhza : a ≤ z.re\nhzb : z.re ≤ b\nH : MapsTo (fun x ↦ x * -I) (im ⁻¹' Ioo a b) (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.MeanValue | {
"line": 75,
"column": 4
} | {
"line": 75,
"column": 57
} | {
"line": 75,
"column": 58
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf : ℂ → E\nR : ℝ\nc w : ℂ\nhf : DiffContOnCl ℂ f (ball c |R|)\nhw : w ∈ ball c |R|\nhR : ¬|R| ≤ 0\n⊢ ContinuousOn f (closedBall c |R|)",
"ppTerm": "?m.72",
"assigned": true,
"usedConstants": [
... | [
"E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf : ℂ → E\nR : ℝ\nc w : ℂ\nhf : DiffContOnCl ℂ f (ball c |R|)\nhw : w ∈ ball c |R|\nhR : ¬|R| ≤ 0\n⊢ ContinuousOn f (closure (ball c |R|))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.CircleAverage | {
"line": 183,
"column": 4
} | {
"line": 183,
"column": 15
} | {
"line": 183,
"column": 16
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℂ → E\nt₀ : Function.Periodic (fun w ↦ f (circleMap 0 1 w)) (2 * π)\n⊢ ∫ (x : ℝ) in -(2 * π)..-0, f (circleMap 0 1 x) = ∫ (θ : ℝ) in 0..2 * π, f (circleMap 0 1 θ)",
"ppTerm": "?m.149",
"assigned": true,
"usedConstants... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℂ → E\nt₀ : Function.Periodic (fun w ↦ f (circleMap 0 1 w)) (2 * π)\n⊢ ∫ (x : ℝ) in -(2 * π)..0, f (circleMap 0 1 x) = ∫ (x : ℝ) in 0..2 * π, f (circleMap 0 1 x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Poisson | {
"line": 62,
"column": 2
} | {
"line": 62,
"column": 83
} | {
"line": 63,
"column": 2
} | [
{
"pp": "a b : ℂ\n⊢ ((a + b) / (a - b)).re = (‖a‖ ^ 2 - ‖b‖ ^ 2) / ‖a - b‖ ^ 2",
"ppTerm": "?m.67",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Complex.add_im",
"Real",
"instHDiv",
"HMul.hMul",
"GroupWithZero.toDivInvMonoid",
"congrArg... | [
"a b : ℂ\n⊢ ((a.re + b.re) * (a.re - b.re) + (a.im + b.im) * (a.im - b.im)) / ‖a - b‖ ^ 2 = (‖a‖ ^ 2 - ‖b‖ ^ 2) / ‖a - b‖ ^ 2"
] | rw [div_re, normSq_eq_norm_sq (a - b), ← add_div, add_re, sub_re, add_im, sub_im] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Complex.Poisson | {
"line": 107,
"column": 2
} | {
"line": 107,
"column": 13
} | {
"line": 107,
"column": 14
} | [
{
"pp": "case neg\nw c z : ℂ\nη₂ : z - c ≠ 0\nhz : z ∈ sphere c ‖z - c‖\nhw : w ∈ ball c ‖z - c‖\nη₀ : 0 < ‖z - c‖\nh₁w : ¬‖w - c‖ = 0\n⊢ ((z - c + (w - c)) / (z - c - (w - c))).re ≤ (‖z - c‖ + ‖w - c‖) / (‖z - c‖ - ‖w - c‖)",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"sub_sub_su... | [
"case neg\nw c z : ℂ\nη₂ : z - c ≠ 0\nhz : z ∈ sphere c ‖z - c‖\nhw : w ∈ ball c ‖z - c‖\nη₀ : 0 < ‖z - c‖\nh₁w : ¬‖w - c‖ = 0\n⊢ ((z - c + (w - c)) / (z - w)).re ≤ (‖z - c‖ + ‖w - c‖) / (‖z - c‖ - ‖w - c‖)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Poisson | {
"line": 108,
"column": 8
} | {
"line": 108,
"column": 19
} | {
"line": 108,
"column": 20
} | [
{
"pp": "w c z : ℂ\nη₂ : z - c ≠ 0\nhz : z ∈ sphere c ‖z - c‖\nhw : w ∈ ball c ‖z - c‖\nη₀ : 0 < ‖z - c‖\nh₁w : ¬‖w - c‖ = 0\n⊢ 0 < ‖w - c‖",
"ppTerm": "?m.154",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"Eq.mpr",
"Real",
"Complex... | [
"w c z : ℂ\nη₂ : z - c ≠ 0\nhz : z ∈ sphere c ‖z - c‖\nhw : w ∈ ball c ‖z - c‖\nη₀ : 0 < ‖z - c‖\nh₁w : ¬‖w - c‖ = 0\n⊢ ¬w - c = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.PhragmenLindelof | {
"line": 459,
"column": 4
} | {
"line": 460,
"column": 11
} | {
"line": 460,
"column": 12
} | [
{
"pp": "case refine_1\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nC : ℝ\nf : ℂ → E\nhd : DiffContOnCl ℂ f (Iio 0 ×ℂ Ioi 0)\nhre : ∀ x ≤ 0, ‖f ↑x‖ ≤ C\nhim : ∀ (x : ℝ), 0 ≤ x → ‖f (↑x * I)‖ ≤ C\nz : ℂ\nhz_re : 0 ≤ z.im\nhz_im : 0 ≤ z.re\nH : MapsTo (fun x ↦ x * I) (Ioi 0 ×ℂ Ioi 0) (Ii... | [
"case refine_1\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nC : ℝ\nf : ℂ → E\nhd : DiffContOnCl ℂ f (Iio 0 ×ℂ Ioi 0)\nhre : ∀ x ≤ 0, ‖f ↑x‖ ≤ C\nhim : ∀ (x : ℝ), 0 ≤ x → ‖f (↑x * I)‖ ≤ C\nz : ℂ\nhz_re : 0 ≤ z.im\nhz_im : 0 ≤ z.re\nH : MapsTo (fun x ↦ x * I) (Ioi 0 ×ℂ Ioi 0) (Iio 0 ×ℂ Ioi 0... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Harmonic.Poisson | {
"line": 94,
"column": 2
} | {
"line": 94,
"column": 56
} | {
"line": 95,
"column": 2
} | [
{
"pp": "f : ℂ → ℝ\nc w : ℂ\nR : ℝ\nhf : HarmonicOnNhd f (closedBall c R)\nhw : w ∈ ball c R\n⊢ circleAverage (poissonKernel c w • f) c R = f w",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Real",
"instHSMul",
"i... | [
"f : ℂ → ℝ\nc w : ℂ\nR : ℝ\nhf : HarmonicOnNhd f (closedBall c R)\nhw : w ∈ ball c R\n⊢ circleAverage (poissonKernel c w • f) c R = circleAverage (re ∘ herglotzRieszKernel c w • f) c R"
] | rw [← hf.circleAverage_re_herglotzRieszKernel_smul hw] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Complex.Harmonic.Poisson | {
"line": 105,
"column": 2
} | {
"line": 105,
"column": 56
} | {
"line": 106,
"column": 2
} | [
{
"pp": "f : ℂ → ℝ\nc w : ℂ\nR : ℝ\nhf : HarmonicContOnCl f (ball c R)\nhw : w ∈ ball c R\n⊢ circleAverage (poissonKernel c w • f) c R = f w",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Real",
"instHSMul",
"inst... | [
"f : ℂ → ℝ\nc w : ℂ\nR : ℝ\nhf : HarmonicContOnCl f (ball c R)\nhw : w ∈ ball c R\n⊢ circleAverage (poissonKernel c w • f) c R = circleAverage (re ∘ herglotzRieszKernel c w • f) c R"
] | rw [← hf.circleAverage_re_herglotzRieszKernel_smul hw] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Complex.UpperHalfPlane.Basic | {
"line": 140,
"column": 2
} | {
"line": 141,
"column": 72
} | {
"line": 141,
"column": 73
} | [
{
"pp": "τ τ' : ℍ\nhre : τ.re = τ'.re\nhnorm : ‖↑τ‖ ^ 2 = ‖↑τ'‖ ^ 2\n⊢ τ = τ'",
"ppTerm": "?m.91",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"UpperHalfPlane.coe",
"_private.Mathlib.Analysis.Complex.UpperHalfPlane.Basic.0.UpperHalfPlane.eq_of_re_of_norm._simp_1_1"... | [
"τ τ' : ℍ\nhre : τ.re = τ'.re\nhnorm : ‖↑τ‖ ^ 2 = ‖↑τ'‖ ^ 2\n⊢ τ.im = τ'.im"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.UpperHalfPlane.Basic | {
"line": 180,
"column": 2
} | {
"line": 180,
"column": 23
} | {
"line": 180,
"column": 24
} | [
{
"pp": "z : ℍ\n⊢ 0 < (-↑z)⁻¹.im",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"neg_div",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"CommRing.toNonUnitalCommRing",
"DivisionCommMonoid.toDivisionMonoid",
"... | [
"z : ℍ\n⊢ 0 < z.im / normSq ↑z"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Poisson | {
"line": 120,
"column": 15
} | {
"line": 120,
"column": 52
} | {
"line": 121,
"column": 4
} | [
{
"pp": "θ φ r R : ℝ\nh₁ : 0 < r\nh₂ : r < R\n⊢ 0 < ‖↑R * cexp (↑θ * I) - ↑r * cexp (↑φ * I)‖ ^ 2",
"ppTerm": "?m.83",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Analysis.Complex.Poisson.0.le_re_herglotzRieszKernel_aux._simp_1_9",
"AddGroup.toSubtractionMonoid",
"NonUni... | [
"θ φ r R : ℝ\nh₁ : 0 < r\nh₂ : r < R\n⊢ ¬↑R * cexp (↑θ * I) = ↑r * cexp (↑φ * I)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Poisson | {
"line": 121,
"column": 34
} | {
"line": 121,
"column": 75
} | {
"line": 121,
"column": 76
} | [
{
"pp": "θ φ r R : ℝ\nh₁ : 0 < r\nh₂ : r < R\n⊢ ¬‖↑R * cexp (↑θ * I)‖ = ‖↑r * cexp (↑φ * I)‖",
"ppTerm": "?m.113",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Norm.norm",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
... | [
"θ φ r R : ℝ\nh₁ : 0 < r\nh₂ : r < R\n⊢ ¬R = r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.UpperHalfPlane.Basic | {
"line": 200,
"column": 43
} | {
"line": 200,
"column": 54
} | {
"line": 200,
"column": 55
} | [
{
"pp": "x : { x // 0 < x }\nz : ℍ\n⊢ 0 < (↑x • ↑z).im",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Complex.mul_im",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"instHSMul",
"RCLike.toNormedAlgebra",
"HMul.hMul",
"UpperHalfPl... | [
"x : { x // 0 < x }\nz : ℍ\n⊢ 0 < ↑x * z.im"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.UpperHalfPlane.Basic | {
"line": 228,
"column": 31
} | {
"line": 228,
"column": 42
} | {
"line": 228,
"column": 43
} | [
{
"pp": "x : ℝ\nz : ℍ\n⊢ 0 < (↑x + ↑z).im",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"UpperHalfPlane.coe",
"Real.instZero",
"Real.instAddMonoid",
"congrArg",
"Complex.im",
"AddMonoid.toAddZeroClass",
"Real.instLT"... | [
"x : ℝ\nz : ℍ\n⊢ 0 < z.im"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Poisson | {
"line": 125,
"column": 2
} | {
"line": 125,
"column": 13
} | {
"line": 125,
"column": 14
} | [
{
"pp": "θ φ r R : ℝ\nh₁ : 0 < r\nh₂ : r < R\nkey : (-(↑R * cexp (↑θ * I) * (starRingEnd ℂ) (↑r * cexp (↑φ * I)))).re ≤ R * r\n⊢ 1 *\n (‖↑R * cexp (↑θ * I)‖ ^ 2 + ‖↑r * cexp (↑φ * I)‖ ^ 2 -\n 2 * (↑R * cexp (↑θ * I) * (starRingEnd ℂ) (↑r * cexp (↑φ * I))).re) ≤\n (R + r) * (R + r)",
"ppTerm":... | [
"θ φ r R : ℝ\nh₁ : 0 < r\nh₂ : r < R\nkey : (-(↑R * cexp (↑θ * I) * (starRingEnd ℂ) (↑r * cexp (↑φ * I)))).re ≤ R * r\n⊢ R ^ 2 + r ^ 2 ≤ (R + r) * (R + r) + 2 * (R * Real.cos θ * (r * Real.cos φ) + R * Real.sin θ * (r * Real.sin φ))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.UpperHalfPlane.Basic | {
"line": 229,
"column": 17
} | {
"line": 229,
"column": 38
} | {
"line": 230,
"column": 2
} | [
{
"pp": "x✝ : ℍ\n⊢ 0 +ᵥ x✝ = x✝",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Real",
"UpperHalfPlane.instAddActionReal._proof_1",
"UpperHalfPlane.coe",
"Real.instAddMonoid",
"congrArg",
"UpperHalfPlane.mk.congr_simp",
"AddMonoid.toAddZeroClass",... | [] | by simp [HVAdd.hVAdd] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Complex.Poisson | {
"line": 140,
"column": 2
} | {
"line": 140,
"column": 13
} | {
"line": 140,
"column": 14
} | [
{
"pp": "case neg\nw c z : ℂ\nη₂ : z - c ≠ 0\nhz : z ∈ sphere c ‖z - c‖\nhw : w ∈ ball c ‖z - c‖\nη₀ : 0 < ‖z - c‖\nh₁w : ¬‖w - c‖ = 0\n⊢ (‖z - c‖ - ‖w - c‖) / (‖z - c‖ + ‖w - c‖) ≤ ((z - c + (w - c)) / (z - c - (w - c))).re",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"sub_sub_su... | [
"case neg\nw c z : ℂ\nη₂ : z - c ≠ 0\nhz : z ∈ sphere c ‖z - c‖\nhw : w ∈ ball c ‖z - c‖\nη₀ : 0 < ‖z - c‖\nh₁w : ¬‖w - c‖ = 0\n⊢ (‖z - c‖ - ‖w - c‖) / (‖z - c‖ + ‖w - c‖) ≤ ((z - c + (w - c)) / (z - w)).re"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Poisson | {
"line": 141,
"column": 8
} | {
"line": 141,
"column": 19
} | {
"line": 141,
"column": 20
} | [
{
"pp": "w c z : ℂ\nη₂ : z - c ≠ 0\nhz : z ∈ sphere c ‖z - c‖\nhw : w ∈ ball c ‖z - c‖\nη₀ : 0 < ‖z - c‖\nh₁w : ¬‖w - c‖ = 0\n⊢ 0 < ‖w - c‖",
"ppTerm": "?m.154",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"Eq.mpr",
"Real",
"Complex... | [
"w c z : ℂ\nη₂ : z - c ≠ 0\nhz : z ∈ sphere c ‖z - c‖\nhw : w ∈ ball c ‖z - c‖\nη₀ : 0 < ‖z - c‖\nh₁w : ¬‖w - c‖ = 0\n⊢ ¬w - c = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.PhragmenLindelof | {
"line": 524,
"column": 4
} | {
"line": 525,
"column": 11
} | {
"line": 525,
"column": 12
} | [
{
"pp": "case refine_1\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nC : ℝ\nf : ℂ → E\nhd : DiffContOnCl ℂ f (Iio 0 ×ℂ Iio 0)\nhre : ∀ x ≤ 0, ‖f ↑x‖ ≤ C\nhim : ∀ x ≤ 0, ‖f (↑x * I)‖ ≤ C\nz : ℂ\nhz_re : 0 ≤ z.re\nhz_im : 0 ≤ z.im\nH : MapsTo Neg.neg (Ioi 0 ×ℂ Ioi 0) (Iio 0 ×ℂ Iio 0)\nc :... | [
"case refine_1\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nC : ℝ\nf : ℂ → E\nhd : DiffContOnCl ℂ f (Iio 0 ×ℂ Iio 0)\nhre : ∀ x ≤ 0, ‖f ↑x‖ ≤ C\nhim : ∀ x ≤ 0, ‖f (↑x * I)‖ ≤ C\nz : ℂ\nhz_re : 0 ≤ z.re\nhz_im : 0 ≤ z.im\nH : MapsTo Neg.neg (Ioi 0 ×ℂ Ioi 0) (Iio 0 ×ℂ Iio 0)\nc : ℝ\nhc : c <... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Poisson | {
"line": 173,
"column": 25
} | {
"line": 173,
"column": 54
} | {
"line": 173,
"column": 55
} | [
{
"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\n⊢ q < 1",
"ppTerm": "?m.158",
"a... | [
"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\n⊢ q < 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.PhragmenLindelof | {
"line": 583,
"column": 4
} | {
"line": 583,
"column": 91
} | {
"line": 583,
"column": 92
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nC : ℝ\nf : ℂ → E\nhd : DiffContOnCl ℂ f (Ioi 0 ×ℂ Iio 0)\nhB : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 (Ioi 0 ×ℂ Iio 0)] fun z ↦ expR (B * ‖z‖ ^ c)\nhre : ∀ (x : ℝ), 0 ≤ x → ‖f ↑x‖ ≤ C\nhim : ∀ x ≤ 0, ‖f (↑x * I)‖ ≤ C\nz : ℂ\nhz_re : z.r... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nC : ℝ\nf : ℂ → E\nhd : DiffContOnCl ℂ f (Ioi 0 ×ℂ Iio 0)\nhB : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 (Ioi 0 ×ℂ Iio 0)] fun z ↦ expR (B * ‖z‖ ^ c)\nhre : ∀ (x : ℝ), 0 ≤ x → ‖f ↑x‖ ≤ C\nhim : ∀ x ≤ 0, ‖f (↑x * I)‖ ≤ C\nz : ℂ\nhz_re : z.re ≤ 0\nhz_im... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Poisson | {
"line": 197,
"column": 31
} | {
"line": 197,
"column": 58
} | {
"line": 197,
"column": 59
} | [
{
"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 → ↑... | [
"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 → ↑q * x - W ≠ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.PhragmenLindelof | {
"line": 588,
"column": 4
} | {
"line": 589,
"column": 11
} | {
"line": 589,
"column": 12
} | [
{
"pp": "case refine_1\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nC : ℝ\nf : ℂ → E\nhd : DiffContOnCl ℂ f (Ioi 0 ×ℂ Iio 0)\nhre : ∀ (x : ℝ), 0 ≤ x → ‖f ↑x‖ ≤ C\nhim : ∀ x ≤ 0, ‖f (↑x * I)‖ ≤ C\nz : ℂ\nhz_re : z.re ≤ 0\nhz_im : 0 ≤ z.im\nH : MapsTo Neg.neg (Iio 0 ×ℂ Ioi 0) (Ioi 0 ×ℂ I... | [
"case refine_1\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nC : ℝ\nf : ℂ → E\nhd : DiffContOnCl ℂ f (Ioi 0 ×ℂ Iio 0)\nhre : ∀ (x : ℝ), 0 ≤ x → ‖f ↑x‖ ≤ C\nhim : ∀ x ≤ 0, ‖f (↑x * I)‖ ≤ C\nz : ℂ\nhz_re : z.re ≤ 0\nhz_im : 0 ≤ z.im\nH : MapsTo Neg.neg (Iio 0 ×ℂ Ioi 0) (Ioi 0 ×ℂ Iio 0)\nc : ℝ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Harmonic.Constructions | {
"line": 94,
"column": 8
} | {
"line": 94,
"column": 40
} | {
"line": 94,
"column": 41
} | [
{
"pp": "z : ℂ\ng : ℂ → ℂ\nh₁g : AnalyticAt ℂ g z\nh₂g : g z ≠ 0\nh₃g : g z ∈ slitPlane\nt₀ : g ⁻¹' (slitPlane ∩ {y | y ≠ 0}) ∈ 𝓝 z\nx : ℂ\nhx : x ∈ g ⁻¹' (slitPlane ∩ {y | y ≠ 0})\n⊢ (starRingEnd ℂ) (g x) ≠ 0",
"ppTerm": "?m.377",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWi... | [
"z : ℂ\ng : ℂ → ℂ\nh₁g : AnalyticAt ℂ g z\nh₂g : g z ≠ 0\nh₃g : g z ∈ slitPlane\nt₀ : g ⁻¹' (slitPlane ∩ {y | y ≠ 0}) ∈ 𝓝 z\nx : ℂ\nhx : x ∈ g ⁻¹' (slitPlane ∩ {y | y ≠ 0})\n⊢ ¬g x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Poisson | {
"line": 228,
"column": 36
} | {
"line": 228,
"column": 47
} | {
"line": 228,
"column": 48
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\nf : ℂ → E\nR : ℝ\nw : ℂ\ninst✝ : CompleteSpace E\nc : ℂ\nhf : DiffContOnCl ℂ f (ball c R)\nhw : w ∈ ball c R\nhR : 0 < R\nh₁g : DiffContOnCl ℂ (fun z ↦ f (z + c)) (ball 0 R)\n⊢ w - c ∈ ball 0 R",
"ppTerm": "?m.125",
"assigne... | [
"E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\nf : ℂ → E\nR : ℝ\nw : ℂ\ninst✝ : CompleteSpace E\nc : ℂ\nhf : DiffContOnCl ℂ f (ball c R)\nhw : w ∈ ball c R\nhR : 0 < R\nh₁g : DiffContOnCl ℂ (fun z ↦ f (z + c)) (ball 0 R)\n⊢ ‖w - c‖ < R"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Poisson | {
"line": 229,
"column": 2
} | {
"line": 230,
"column": 9
} | {
"line": 230,
"column": 10
} | [
{
"pp": "case inr\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\nf : ℂ → E\nR : ℝ\nw : ℂ\ninst✝ : CompleteSpace E\nc : ℂ\nhf : DiffContOnCl ℂ f (ball c R)\nhw : w ∈ ball c R\nhR : 0 < R\nh₁g : DiffContOnCl ℂ (fun z ↦ f (z + c)) (ball 0 R)\nh₂g : w - c ∈ ball 0 R\n⊢ Real.circleAverage (r... | [
"case inr\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\nf : ℂ → E\nR : ℝ\nw : ℂ\ninst✝ : CompleteSpace E\nc : ℂ\nhf : DiffContOnCl ℂ f (ball c R)\nhw : w ∈ ball c R\nhR : 0 < R\nh₁g : DiffContOnCl ℂ (fun z ↦ f (z + c)) (ball 0 R)\nh₂g : w - c ∈ ball 0 R\n⊢ Real.circleAverage (fun z ↦ ((z +... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Harmonic.Constructions | {
"line": 76,
"column": 2
} | {
"line": 102,
"column": 45
} | {
"line": 104,
"column": 0
} | [
{
"pp": "z : ℂ\ng : ℂ → ℂ\nh₁g : AnalyticAt ℂ g z\nh₂g : g z ≠ 0\nh₃g : g z ∈ slitPlane\n⊢ HarmonicAt (Real.log ∘ ⇑normSq ∘ g) z",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"instInnerProductSpaceRealComplex",
"Filter.instMembership",
"Iff.mpr",
"IsModuleTopology... | [] | rw [harmonicAt_congr_nhds (f₂ := reCLM ∘ (conjCLE ∘ log ∘ g + log ∘ g))]
· exact (((harmonicAt_comp_CLE_iff conjCLE).2 ((analyticAt_clog h₃g).comp h₁g).harmonicAt).add
((analyticAt_clog h₃g).comp h₁g).harmonicAt).comp_CLM reCLM
· have t₀ := h₁g.differentiableAt.continuousAt.preimage_mem_nhds
((isOpen_sl... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.Harmonic.Constructions | {
"line": 76,
"column": 2
} | {
"line": 102,
"column": 45
} | {
"line": 104,
"column": 0
} | [
{
"pp": "z : ℂ\ng : ℂ → ℂ\nh₁g : AnalyticAt ℂ g z\nh₂g : g z ≠ 0\nh₃g : g z ∈ slitPlane\n⊢ HarmonicAt (Real.log ∘ ⇑normSq ∘ g) z",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"instInnerProductSpaceRealComplex",
"Filter.instMembership",
"Iff.mpr",
"IsModuleTopology... | [] | rw [harmonicAt_congr_nhds (f₂ := reCLM ∘ (conjCLE ∘ log ∘ g + log ∘ g))]
· exact (((harmonicAt_comp_CLE_iff conjCLE).2 ((analyticAt_clog h₃g).comp h₁g).harmonicAt).add
((analyticAt_clog h₃g).comp h₁g).harmonicAt).comp_CLM reCLM
· have t₀ := h₁g.differentiableAt.continuousAt.preimage_mem_nhds
((isOpen_sl... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Sinc | {
"line": 71,
"column": 29
} | {
"line": 71,
"column": 39
} | {
"line": 71,
"column": 40
} | [
{
"pp": "case inl\nx : ℝ\nhx : x < 0\n⊢ sin x / x ≤ (-x)⁻¹",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NegZeroClass.toNeg",
"MulOne.toOne",
"Real.instLE",
"Real",
"DivInvMonoid.toInv",
"instHDiv",... | [
"case inl\nx : ℝ\nhx : x < 0\n⊢ sin x / x ≤ 1 / -x"
] | ← one_div, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Log.NegMulLog | {
"line": 57,
"column": 4
} | {
"line": 57,
"column": 31
} | {
"line": 57,
"column": 32
} | [
{
"pp": "case inr.refine_1\nthis : Set.univ = Set.Iio 0 ∪ Set.Ioi 0 ∪ {0}\n⊢ Filter.Tendsto (fun x ↦ log x * x) (𝓝[>] 0) (𝓝 0)",
"ppTerm": "?inr.refine_1",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case inr.refine_1\nthis : Set.univ = Set.Iio 0 ∪ Set.Ioi 0 ∪ {0}\n⊢ Filter.Tendsto (fun x ↦ log x * x) (𝓝[>] 0) (𝓝 0)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Sinc | {
"line": 75,
"column": 6
} | {
"line": 75,
"column": 17
} | {
"line": 75,
"column": 18
} | [
{
"pp": "case inl\nx : ℝ\nhx : x < 0\n⊢ 0 < -x",
"ppTerm": "?inl✝",
"assigned": true,
"usedConstants": [
"Left.neg_pos_iff._simp_1",
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"NegZeroClass.toNeg",
"Real.partialOrder",
"Real"... | [
"case inl\nx : ℝ\nhx : x < 0\n⊢ x < 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Sinc | {
"line": 81,
"column": 2
} | {
"line": 81,
"column": 25
} | {
"line": 82,
"column": 2
} | [
{
"pp": "x : ℝ\nhx : x ≠ 0\n⊢ sinc x ≤ |x|⁻¹",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"instHDiv",
"Real.lattice",
"abs",
"congrArg",
"Real.instInv",
"Real.instDivInvMonoid",
"id",
"HDi... | [
"x : ℝ\nhx : x ≠ 0\n⊢ sin x / x ≤ |x|⁻¹"
] | rw [sinc_of_ne_zero hx] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.Log.NegMulLog | {
"line": 175,
"column": 2
} | {
"line": 175,
"column": 49
} | {
"line": 175,
"column": 50
} | [
{
"pp": "x : ℝ\nh1 : 0 ≤ x\nh2 : x ≤ 1\n⊢ 0 ≤ x.negMulLog",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"Real.instLE",
"Real",
"HMul.hMul",
"Real.instZero",
"congrArg",
"id",
"_private.Mathlib... | [
"x : ℝ\nh1 : 0 ≤ x\nh2 : x ≤ 1\n⊢ x * log x ≤ 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.NegMulLog | {
"line": 187,
"column": 2
} | {
"line": 187,
"column": 37
} | {
"line": 187,
"column": 38
} | [
{
"pp": "⊢ Continuous negMulLog",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Continuous",
"HMul.hMul",
"congrArg",
"PseudoMetricSpace.toUniformSpace",
"id",
"Real.negMulLog",
"Real.log",
"Real.instMul",
... | [
"⊢ Continuous fun x ↦ -(x * log x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.InvLog | {
"line": 47,
"column": 34
} | {
"line": 47,
"column": 45
} | {
"line": 47,
"column": 46
} | [
{
"pp": "⊢ HasDerivAt log 1 1",
"ppTerm": "?m.65",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"⊢ HasDerivAt log 1 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.InvLog | {
"line": 51,
"column": 8
} | {
"line": 51,
"column": 41
} | {
"line": 51,
"column": 42
} | [
{
"pp": "H : ContinuousAt (fun x ↦ (log x)⁻¹) (-1)\n⊢ ContinuousAt (fun x ↦ (log x)⁻¹) 1",
"ppTerm": "?m.14",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"H : ContinuousAt (fun x ↦ (log x)⁻¹) (-1)\n⊢ ContinuousAt (fun x ↦ (log x)⁻¹) 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.InvLog | {
"line": 72,
"column": 2
} | {
"line": 72,
"column": 13
} | {
"line": 72,
"column": 14
} | [
{
"pp": "x : ℝ\nhx₀ : x ≠ 0\nhx₁ : x ≠ 1\nhx₂ : x ≠ -1\n⊢ HasDerivAt (fun x ↦ (log x)⁻¹) (-x⁻¹ / log x ^ 2) x",
"ppTerm": "?m.42",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"x : ℝ\nhx₀ : x ≠ 0\nhx₁ : x ≠ 1\nhx₂ : x ≠ -1\n⊢ HasDerivAt (fun x ↦ (log x)⁻¹) (-x⁻¹ / log x ^ 2) x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.InvLog | {
"line": 91,
"column": 2
} | {
"line": 91,
"column": 73
} | {
"line": 93,
"column": 0
} | [
{
"pp": "this : Tendsto log atBot atTop\n⊢ Tendsto (fun x ↦ log (log x)) (nhdsWithin 0 {0}ᶜ) (cobounded ℝ)",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Real.partialOrder",
"Real",
"PseudoMetricSpace.toBornology",
"instNoMaxOrderOf... | [] | exact (this.mono_right atTop_le_cobounded).comp tendsto_log_nhdsNE_zero | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.SpecialFunctions.Log.InvLog | {
"line": 97,
"column": 34
} | {
"line": 97,
"column": 45
} | {
"line": 97,
"column": 46
} | [
{
"pp": "⊢ HasDerivAt log 1 1",
"ppTerm": "?m.69",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"⊢ HasDerivAt log 1 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.InvLog | {
"line": 101,
"column": 8
} | {
"line": 101,
"column": 41
} | {
"line": 101,
"column": 42
} | [
{
"pp": "H : ContinuousAt (fun x ↦ log (log x)) (-1)\n⊢ ContinuousAt (fun x ↦ log (log x)) 1",
"ppTerm": "?m.11",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"H : ContinuousAt (fun x ↦ log (log x)) (-1)\n⊢ ContinuousAt (fun x ↦ log (log x)) 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.InvLog | {
"line": 122,
"column": 2
} | {
"line": 122,
"column": 13
} | {
"line": 122,
"column": 14
} | [
{
"pp": "x : ℝ\nhx₀ : x ≠ 0\nhx₁ : x ≠ 1\nhx₂ : x ≠ -1\n⊢ HasDerivAt (fun x ↦ log (log x)) (x⁻¹ / log x) x",
"ppTerm": "?m.26",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"x : ℝ\nhx₀ : x ≠ 0\nhx₁ : x ≠ 1\nhx₂ : x ≠ -1\n⊢ HasDerivAt (fun x ↦ log (log x)) (x⁻¹ / log x) x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Integrals.Basic | {
"line": 124,
"column": 6
} | {
"line": 124,
"column": 55
} | {
"line": 125,
"column": 2
} | [
{
"pp": "case inr\na b : ℝ\nr : ℂ\nh : r ≠ -1 ∧ 0 ∉ [[a, b]]\n⊢ r + 1 ≠ 0",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"NegZeroClass.toNeg",
"Real",
"Real.lattice",
"Real.instZero",
"AddGroupWithOne.toAddGroup",
"c... | [] | rw [Ne, ← add_eq_zero_iff_eq_neg] at h; exact h.1 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Integrals.Basic | {
"line": 124,
"column": 6
} | {
"line": 124,
"column": 55
} | {
"line": 125,
"column": 2
} | [
{
"pp": "case inr\na b : ℝ\nr : ℂ\nh : r ≠ -1 ∧ 0 ∉ [[a, b]]\n⊢ r + 1 ≠ 0",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"NegZeroClass.toNeg",
"Real",
"Real.lattice",
"Real.instZero",
"AddGroupWithOne.toAddGroup",
"c... | [] | rw [Ne, ← add_eq_zero_iff_eq_neg] at h; exact h.1 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.PhragmenLindelof | {
"line": 684,
"column": 4
} | {
"line": 684,
"column": 36
} | {
"line": 684,
"column": 37
} | [
{
"pp": "case inl\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nC : ℝ\nf : ℂ → E\nhd : DiffContOnCl ℂ f {z | 0 < z.re}\nhexp : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z ↦ expR (B * ‖z‖ ^ c)\nhre : Tendsto (fun x ↦ f ↑x) atTop (𝓝 0)\nhim : ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C\nhl... | [
"case inl\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nC : ℝ\nf : ℂ → E\nhd : DiffContOnCl ℂ f {z | 0 < z.re}\nhexp : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z ↦ expR (B * ‖z‖ ^ c)\nhre : Tendsto (fun x ↦ f ↑x) atTop (𝓝 0)\nhim : ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C\nhle : ∀ (C' : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Integrability.Basic | {
"line": 197,
"column": 2
} | {
"line": 197,
"column": 28
} | {
"line": 197,
"column": 29
} | [
{
"pp": "a b : ℝ\nf : ℝ → ℝ\nμ : Measure ℝ\ninst✝ : IsLocallyFiniteMeasure μ\nh : ∀ x ∈ [[a, b]], f x ≠ 0\nhf : ContinuousOn f [[a, b]]\n⊢ IntervalIntegrable (fun x ↦ (f x)⁻¹) μ a b",
"ppTerm": "?m.26",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"a b : ℝ\nf : ℝ → ℝ\nμ : Measure ℝ\ninst✝ : IsLocallyFiniteMeasure μ\nh : ∀ x ∈ [[a, b]], f x ≠ 0\nhf : ContinuousOn f [[a, b]]\n⊢ IntervalIntegrable (fun x ↦ (f x)⁻¹) μ a b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.PhragmenLindelof | {
"line": 688,
"column": 6
} | {
"line": 688,
"column": 37
} | {
"line": 688,
"column": 38
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nC : ℝ\nf : ℂ → E\nhd : DiffContOnCl ℂ f {z | 0 < z.re}\nhexp : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z ↦ expR (B * ‖z‖ ^ c)\nhre : Tendsto (fun x ↦ f ↑x) atTop (𝓝 0)\nhim : ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C\nhle : ∀ (C' ... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nC : ℝ\nf : ℂ → E\nhd : DiffContOnCl ℂ f {z | 0 < z.re}\nhexp : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z ↦ expR (B * ‖z‖ ^ c)\nhre : Tendsto (fun x ↦ f ↑x) atTop (𝓝 0)\nhim : ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C\nhle : ∀ (C' : ℝ), (∀ (x ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.PosLog | {
"line": 189,
"column": 42
} | {
"line": 189,
"column": 66
} | {
"line": 189,
"column": 66
} | [
{
"pp": "α : Type u_1\ns : Finset α\nf : α → ℝ\nhs : s.Nonempty\nt_max : α\nht_max : t_max ∈ s ∧ ∀ x' ∈ s, |f x'| ≤ |f t_max|\n⊢ ∑ t ∈ s, |f t| ∈ Set.Ici 0",
"ppTerm": "?m.153",
"assigned": true,
"usedConstants": [
"abs_nonneg._simp_1",
"AddGroup.toSubtractionMonoid",
"Real",
... | [] | simp [Finset.sum_nonneg] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.SpecialFunctions.Log.PosLog | {
"line": 189,
"column": 42
} | {
"line": 189,
"column": 66
} | {
"line": 189,
"column": 66
} | [
{
"pp": "α : Type u_1\ns : Finset α\nf : α → ℝ\nhs : s.Nonempty\nt_max : α\nht_max : t_max ∈ s ∧ ∀ x' ∈ s, |f x'| ≤ |f t_max|\n⊢ ∑ t ∈ s, |f t| ∈ Set.Ici 0",
"ppTerm": "?m.153",
"assigned": true,
"usedConstants": [
"abs_nonneg._simp_1",
"AddGroup.toSubtractionMonoid",
"Real",
... | [] | simp [Finset.sum_nonneg] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Log.PosLog | {
"line": 189,
"column": 42
} | {
"line": 189,
"column": 66
} | {
"line": 189,
"column": 66
} | [
{
"pp": "α : Type u_1\ns : Finset α\nf : α → ℝ\nhs : s.Nonempty\nt_max : α\nht_max : t_max ∈ s ∧ ∀ x' ∈ s, |f x'| ≤ |f t_max|\n⊢ ∑ t ∈ s, |f t| ∈ Set.Ici 0",
"ppTerm": "?m.153",
"assigned": true,
"usedConstants": [
"abs_nonneg._simp_1",
"AddGroup.toSubtractionMonoid",
"Real",
... | [] | simp [Finset.sum_nonneg] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Log.PosLog | {
"line": 192,
"column": 32
} | {
"line": 192,
"column": 56
} | {
"line": 192,
"column": 56
} | [
{
"pp": "α : Type u_1\ns : Finset α\nf : α → ℝ\nhs : s.Nonempty\nt_max : α\nht_max : t_max ∈ s ∧ ∀ x' ∈ s, |f x'| ≤ |f t_max|\n⊢ ∑ t ∈ s, |f t| ∈ Set.Ici 0",
"ppTerm": "?m.156",
"assigned": true,
"usedConstants": [
"abs_nonneg._simp_1",
"AddGroup.toSubtractionMonoid",
"Real",
... | [] | simp [Finset.sum_nonneg] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.SpecialFunctions.Log.PosLog | {
"line": 192,
"column": 32
} | {
"line": 192,
"column": 56
} | {
"line": 192,
"column": 56
} | [
{
"pp": "α : Type u_1\ns : Finset α\nf : α → ℝ\nhs : s.Nonempty\nt_max : α\nht_max : t_max ∈ s ∧ ∀ x' ∈ s, |f x'| ≤ |f t_max|\n⊢ ∑ t ∈ s, |f t| ∈ Set.Ici 0",
"ppTerm": "?m.156",
"assigned": true,
"usedConstants": [
"abs_nonneg._simp_1",
"AddGroup.toSubtractionMonoid",
"Real",
... | [] | simp [Finset.sum_nonneg] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Log.PosLog | {
"line": 192,
"column": 32
} | {
"line": 192,
"column": 56
} | {
"line": 192,
"column": 56
} | [
{
"pp": "α : Type u_1\ns : Finset α\nf : α → ℝ\nhs : s.Nonempty\nt_max : α\nht_max : t_max ∈ s ∧ ∀ x' ∈ s, |f x'| ≤ |f t_max|\n⊢ ∑ t ∈ s, |f t| ∈ Set.Ici 0",
"ppTerm": "?m.156",
"assigned": true,
"usedConstants": [
"abs_nonneg._simp_1",
"AddGroup.toSubtractionMonoid",
"Real",
... | [] | simp [Finset.sum_nonneg] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.PhragmenLindelof | {
"line": 705,
"column": 4
} | {
"line": 705,
"column": 15
} | {
"line": 705,
"column": 16
} | [
{
"pp": "case inr\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nC : ℝ\nf : ℂ → E\nhd : DiffContOnCl ℂ f {z | 0 < z.re}\nhexp : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z ↦ expR (B * ‖z‖ ^ c)\nhre : Tendsto (fun x ↦ f ↑x) atTop (𝓝 0)\nhim : ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C\nhl... | [
"case inr\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nC : ℝ\nf : ℂ → E\nhd : DiffContOnCl ℂ f {z | 0 < z.re}\nhexp : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z ↦ expR (B * ‖z‖ ^ c)\nhre : Tendsto (fun x ↦ f ↑x) atTop (𝓝 0)\nhim : ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C\nhle : ∀ (C' : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Integrals.LogTrigonometric | {
"line": 64,
"column": 8
} | {
"line": 64,
"column": 19
} | {
"line": 64,
"column": 20
} | [
{
"pp": "⊢ IntervalIntegrable (fun x ↦ log (sin (2 * x))) MeasureTheory.volume 0 (π / 2)",
"ppTerm": "?m.374",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"⊢ IntervalIntegrable (fun x ↦ log (sin (2 * x))) MeasureTheory.volume 0 (π / 2)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Integrals.LogTrigonometric | {
"line": 66,
"column": 8
} | {
"line": 66,
"column": 19
} | {
"line": 66,
"column": 20
} | [
{
"pp": "⊢ IntervalIntegrable (fun x ↦ log (sin (2 * x))) MeasureTheory.volume 0 (π / 2)",
"ppTerm": "?m.421",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"⊢ IntervalIntegrable (fun x ↦ log (sin (2 * x))) MeasureTheory.volume 0 (π / 2)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.PhragmenLindelof | {
"line": 734,
"column": 2
} | {
"line": 734,
"column": 75
} | {
"line": 735,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nC : ℝ\nf : ℂ → E\nz : ℂ\nhexp : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z ↦ expR (B * ‖z‖ ^ c)\nhre : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) atTop fun x ↦ ‖f ↑x‖\nhim : ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C\nhz : 0 ≤ z.re\nε : ℝ\... | [
"case refine_1\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nC : ℝ\nf : ℂ → E\nz : ℂ\nhexp : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z ↦ expR (B * ‖z‖ ^ c)\nhre : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) atTop fun x ↦ ‖f ↑x‖\nhim : ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C\nhz : 0 ≤ z.re\nε :... | refine right_half_plane_of_tendsto_zero_on_real hd ?_ ?_ (fun y => ?_) hz | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.SpecialFunctions.Integrals.Basic | {
"line": 338,
"column": 2
} | {
"line": 338,
"column": 59
} | {
"line": 339,
"column": 4
} | [
{
"pp": "a b : ℝ\n⊢ ∫ (x : ℝ) in a..b, cos x ^ 2 - sin x ^ 2 = sin b * cos b - sin a * cos a",
"ppTerm": "?m.65",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Real",
"MeasureTheory.Measure",
"NonUnitalCommRing.toNonUnitalNonAssocCom... | [
"a b : ℝ\n⊢ ∫ (x : ℝ) in a..b, cos x * cos x + sin x * -sin x = sin b * cos b + sin a * -cos a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Integrals.Basic | {
"line": 440,
"column": 4
} | {
"line": 440,
"column": 51
} | {
"line": 441,
"column": 4
} | [
{
"pp": "case convert_3\na b t : ℝ\nht : t ≠ -1\nthis : ∀ (x s : ℝ), ↑((1 + x ^ 2) ^ s) = (1 + ↑x ^ 2) ^ ↑s\n⊢ ↑t ≠ -1",
"ppTerm": "?convert_3",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"congrArg",
"Complex.ofReal_neg",
"id",
"Complex.ofReal_one",
... | [
"case convert_3\na b t : ℝ\nht : t ≠ -1\nthis : ∀ (x s : ℝ), ↑((1 + x ^ 2) ^ s) = (1 + ↑x ^ 2) ^ ↑s\n⊢ ¬t = -1"
] | rw [← ofReal_one, ← ofReal_neg, Ne, ofReal_inj] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Complex.OpenMapping | {
"line": 67,
"column": 37
} | {
"line": 67,
"column": 76
} | {
"line": 67,
"column": 77
} | [
{
"pp": "f : ℂ → ℂ\nz₀ : ℂ\nε r : ℝ\nh : DiffContOnCl ℂ f (ball z₀ r)\nhr : 0 < r\nhf : ∀ z ∈ sphere z₀ r, ε ≤ ‖f z - f z₀‖\nhz₀ : ∃ᶠ (z : ℂ) in 𝓝 z₀, f z ≠ f z₀\nv : ℂ\nhv : v ∈ ball (f z₀) (ε / 2)\nh1 : DiffContOnCl ℂ (fun z ↦ f z - v) (ball z₀ r)\nh2 : ContinuousOn (fun z ↦ ‖f z - v‖) (closedBall z₀ r)\nh3 ... | [
"f : ℂ → ℂ\nz₀ : ℂ\nε r : ℝ\nh : DiffContOnCl ℂ f (ball z₀ r)\nhr : 0 < r\nhf : ∀ z ∈ sphere z₀ r, ε ≤ ‖f z - f z₀‖\nhz₀ : ∃ᶠ (z : ℂ) in 𝓝 z₀, f z ≠ f z₀\nv : ℂ\nhv : v ∈ ball (f z₀) (ε / 2)\nh1 : DiffContOnCl ℂ (fun z ↦ f z - v) (ball z₀ r)\nh2 : ContinuousOn (fun z ↦ ‖f z - v‖) (closedBall z₀ r)\nh3 : AnalyticOn... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Integrals.Basic | {
"line": 461,
"column": 4
} | {
"line": 469,
"column": 58
} | {
"line": 470,
"column": 2
} | [
{
"pp": "case refine_3\na b : ℝ\nn : ℕ\nC : ℝ := sin a ^ (n + 1) * cos a - sin b ^ (n + 1) * cos b\nh : ∀ (α β γ : ℝ), β * α * γ * α = β * (α * α * γ)\nhu : ∀ x ∈ [[a, b]], HasDerivAt (fun y ↦ sin y ^ (n + 1)) (↑(n + 1) * cos x * sin x ^ n) x\nhv : ∀ x ∈ [[a, b]], HasDerivAt (-cos) (sin x) x\nH :\n ∫ (x : ℝ) i... | [] | calc
(∫ x in a..b, sin x ^ (n + 2)) = ∫ x in a..b, sin x ^ (n + 1) * sin x := by
simp only [_root_.pow_succ]
_ = C + (↑n + 1) * ∫ x in a..b, cos x ^ 2 * sin x ^ n := by simp [H, h, sq]; ring
_ = C + (↑n + 1) * ∫ x in a..b, sin x ^ n - sin x ^ (n + 2) := by
simp [cos_sq', sub_mul, ← pow... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcTactic |
Mathlib.Analysis.SpecialFunctions.Integrals.Basic | {
"line": 461,
"column": 4
} | {
"line": 469,
"column": 58
} | {
"line": 470,
"column": 2
} | [
{
"pp": "case refine_3\na b : ℝ\nn : ℕ\nC : ℝ := sin a ^ (n + 1) * cos a - sin b ^ (n + 1) * cos b\nh : ∀ (α β γ : ℝ), β * α * γ * α = β * (α * α * γ)\nhu : ∀ x ∈ [[a, b]], HasDerivAt (fun y ↦ sin y ^ (n + 1)) (↑(n + 1) * cos x * sin x ^ n) x\nhv : ∀ x ∈ [[a, b]], HasDerivAt (-cos) (sin x) x\nH :\n ∫ (x : ℝ) i... | [] | calc
(∫ x in a..b, sin x ^ (n + 2)) = ∫ x in a..b, sin x ^ (n + 1) * sin x := by
simp only [_root_.pow_succ]
_ = C + (↑n + 1) * ∫ x in a..b, cos x ^ 2 * sin x ^ n := by simp [H, h, sq]; ring
_ = C + (↑n + 1) * ∫ x in a..b, sin x ^ n - sin x ^ (n + 2) := by
simp [cos_sq', sub_mul, ← pow... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Integrals.Basic | {
"line": 461,
"column": 4
} | {
"line": 469,
"column": 58
} | {
"line": 470,
"column": 2
} | [
{
"pp": "case refine_3\na b : ℝ\nn : ℕ\nC : ℝ := sin a ^ (n + 1) * cos a - sin b ^ (n + 1) * cos b\nh : ∀ (α β γ : ℝ), β * α * γ * α = β * (α * α * γ)\nhu : ∀ x ∈ [[a, b]], HasDerivAt (fun y ↦ sin y ^ (n + 1)) (↑(n + 1) * cos x * sin x ^ n) x\nhv : ∀ x ∈ [[a, b]], HasDerivAt (-cos) (sin x) x\nH :\n ∫ (x : ℝ) i... | [] | calc
(∫ x in a..b, sin x ^ (n + 2)) = ∫ x in a..b, sin x ^ (n + 1) * sin x := by
simp only [_root_.pow_succ]
_ = C + (↑n + 1) * ∫ x in a..b, cos x ^ 2 * sin x ^ n := by simp [H, h, sq]; ring
_ = C + (↑n + 1) * ∫ x in a..b, sin x ^ n - sin x ^ (n + 2) := by
simp [cos_sq', sub_mul, ← pow... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.PhragmenLindelof | {
"line": 822,
"column": 4
} | {
"line": 822,
"column": 69
} | {
"line": 822,
"column": 70
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf g : ℂ → E\nhfd : DiffContOnCl ℂ f {z | 0 < z.re}\nhgd : DiffContOnCl ℂ g {z | 0 < z.re}\nhfexp : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z ↦ expR (B * ‖z‖ ^ c)\nhgexp : ∃ c < 2, ∃ B, g =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf g : ℂ → E\nhfd : DiffContOnCl ℂ f {z | 0 < z.re}\nhgd : DiffContOnCl ℂ g {z | 0 < z.re}\nhfexp : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z ↦ expR (B * ‖z‖ ^ c)\nhgexp : ∃ c < 2, ∃ B, g =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.ZeroAndBoundedAtFilter | {
"line": 47,
"column": 2
} | {
"line": 47,
"column": 21
} | {
"line": 49,
"column": 0
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝² : TopologicalSpace β\ninst✝¹ : SubtractionMonoid β\ninst✝ : ContinuousNeg β\nl : Filter α\nf : α → β\nhf : l.ZeroAtFilter f\n⊢ l.ZeroAtFilter (-f)",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Filter.Tendsto.neg",
"NegZeroClass.toN... | [] | simpa using! hf.neg | Lean.Elab.Tactic.Simpa.evalSimpaUsingBang | Lean.Parser.Tactic.simpaUsingBang |
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