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.Complex.Conformal | {
"line": 80,
"column": 8
} | {
"line": 80,
"column": 37
} | [
{
"pp": "case refine_1\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace ℂ E\nmap : ℂ →L[ℂ] E\nnonzero : map ≠ 0\nminor₁ : ‖map 1‖ ≠ 0\nx : ℂ\nthis : x = x • 1\n⊢ ‖‖map 1‖⁻¹ • (↑ℝ ↑map) (x • 1)‖ = ‖x‖",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"No... | LinearMap.coe_restrictScalars | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.Conformal | {
"line": 73,
"column": 2
} | {
"line": 84,
"column": 17
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace ℂ E\nmap : ℂ →L[ℂ] E\nnonzero : map ≠ 0\n⊢ IsConformalMap (restrictScalars ℝ map)",
"usedConstants": [
"LinearIsometry",
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"Mathlib.Tactic.Fi... | have minor₁ : ‖map 1‖ ≠ 0 := by
simpa only [ContinuousLinearMap.ext_ring_iff, Ne, norm_eq_zero] using nonzero
refine ⟨‖map 1‖, minor₁, ⟨‖map 1‖⁻¹ • ((map : ℂ →ₗ[ℂ] E) : ℂ →ₗ[ℝ] E), ?_⟩, ?_⟩
· intro x
simp only [LinearMap.smul_apply]
have : x = x • (1 : ℂ) := by rw [smul_eq_mul, mul_one]
nth_rw 1 [th... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.Hadamard | {
"line": 369,
"column": 2
} | {
"line": 387,
"column": 11
} | [
{
"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",
"usedConstants": [
"Set.ext",
"Real.instIsOrderedRing",
"Norm.norm",
"Not.intro",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"... | 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.Conformal | {
"line": 73,
"column": 2
} | {
"line": 84,
"column": 17
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace ℂ E\nmap : ℂ →L[ℂ] E\nnonzero : map ≠ 0\n⊢ IsConformalMap (restrictScalars ℝ map)",
"usedConstants": [
"LinearIsometry",
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"Mathlib.Tactic.Fi... | have minor₁ : ‖map 1‖ ≠ 0 := by
simpa only [ContinuousLinearMap.ext_ring_iff, Ne, norm_eq_zero] using nonzero
refine ⟨‖map 1‖, minor₁, ⟨‖map 1‖⁻¹ • ((map : ℂ →ₗ[ℂ] E) : ℂ →ₗ[ℝ] E), ?_⟩, ?_⟩
· intro x
simp only [LinearMap.smul_apply]
have : x = x • (1 : ℂ) := by rw [smul_eq_mul, mul_one]
nth_rw 1 [th... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.Hadamard | {
"line": 369,
"column": 2
} | {
"line": 387,
"column": 11
} | [
{
"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",
"usedConstants": [
"Set.ext",
"Real.instIsOrderedRing",
"Norm.norm",
"Not.intro",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"... | 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
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Hadamard | {
"line": 438,
"column": 16
} | {
"line": 438,
"column": 27
} | [
{
"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",
"usedConstants": [
"Real",
"Real.instZe... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Hadamard | {
"line": 439,
"column": 16
} | {
"line": 439,
"column": 41
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"Rea... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Hadamard | {
"line": 455,
"column": 8
} | {
"line": 455,
"column": 19
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Hadamard | {
"line": 456,
"column": 8
} | {
"line": 456,
"column": 33
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.HasPrimitives | {
"line": 86,
"column": 4
} | {
"line": 86,
"column": 15
} | [
{
"pp": "case hb₂\nc z : ℂ\nr : ℝ\nw : ℂ\nhw : w ∈ ball z (r - dist z c)\n⊢ ↑w.re + ↑w.im * I ∈ ball z (r - dist z c)",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"HMul.hMul",
"congrArg",
"Complex.im",
"Real.instSub",
"Complex.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.PhragmenLindelof | {
"line": 133,
"column": 24
} | {
"line": 133,
"column": 95
} | [
{
"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 + ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.HasPrimitives | {
"line": 121,
"column": 42
} | {
"line": 121,
"column": 53
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → E\ns : Set ℂ\nx₀ : ℂ\ny : E\nη : ℂ → E\nhη : ∀ z ∈ s, HasDerivAt η (f z) z\n⊢ ∀ x ∈ s, HasDerivAt (fun z ↦ η z - η x₀ + y) (f x) x",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"N... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.PhragmenLindelof | {
"line": 139,
"column": 4
} | {
"line": 139,
"column": 33
} | [
{
"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 + ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.HasPrimitives | {
"line": 155,
"column": 56
} | {
"line": 155,
"column": 67
} | [
{
"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)\n⊢ r - dist z c > 0",
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.PhragmenLindelof | {
"line": 142,
"column": 2
} | {
"line": 142,
"column": 72
} | [
{
"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 + ... | 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.PhragmenLindelof | {
"line": 163,
"column": 4
} | {
"line": 164,
"column": 41
} | [
{
"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 + ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Hadamard | {
"line": 507,
"column": 6
} | {
"line": 508,
"column": 48
} | [
{
"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\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Hadamard | {
"line": 514,
"column": 8
} | {
"line": 515,
"column": 50
} | [
{
"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 : ‖... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.HasPrimitives | {
"line": 185,
"column": 6
} | {
"line": 185,
"column": 17
} | [
{
"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 :=... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.PhragmenLindelof | {
"line": 192,
"column": 6
} | {
"line": 194,
"column": 13
} | [
{
"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 + ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.HasPrimitives | {
"line": 192,
"column": 12
} | {
"line": 192,
"column": 23
} | [
{
"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 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.HasPrimitives | {
"line": 192,
"column": 32
} | {
"line": 192,
"column": 43
} | [
{
"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 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.HasPrimitives | {
"line": 193,
"column": 4
} | {
"line": 193,
"column": 78
} | [
{
"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 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Harmonic.Basic | {
"line": 149,
"column": 4
} | {
"line": 149,
"column": 15
} | [
{
"pp": "case left\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\nh : HarmonicAt f x\n⊢ ContDiffAt ℝ 2 (-f) x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Harmonic.Basic | {
"line": 201,
"column": 4
} | {
"line": 201,
"column": 35
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.HasPrimitives | {
"line": 205,
"column": 29
} | {
"line": 205,
"column": 69
} | [
{
"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",
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.HasPrimitives | {
"line": 207,
"column": 2
} | {
"line": 207,
"column": 50
} | [
{
"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 (... | 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
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.HasPrimitives | {
"line": 228,
"column": 58
} | {
"line": 228,
"column": 69
} | [
{
"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",
"use... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.PhragmenLindelof | {
"line": 280,
"column": 35
} | {
"line": 280,
"column": 58
} | [
{
"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_... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.PhragmenLindelof | {
"line": 281,
"column": 78
} | {
"line": 281,
"column": 89
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Harmonic.Liouville | {
"line": 54,
"column": 4
} | {
"line": 54,
"column": 28
} | [
{
"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))",
"usedConstants": [
"Set.range_comp",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.PhragmenLindelof | {
"line": 290,
"column": 6
} | {
"line": 290,
"column": 37
} | [
{
"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 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.PhragmenLindelof | {
"line": 291,
"column": 4
} | {
"line": 291,
"column": 35
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.PhragmenLindelof | {
"line": 455,
"column": 4
} | {
"line": 455,
"column": 77
} | [
{
"pp": "E : 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)\nhB : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 (Iio 0 ×ℂ Ioi 0)] fun z ↦ expR (B * ‖z‖ ^ c)\nhre : ∀ x ≤ 0, ‖f ↑x‖ ≤ C\nhim : ∀ (x : ℝ), 0 ≤ x → ‖f (↑x * I)‖ ≤ C\nz : ℂ\nhz_re : 0 ≤... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.PhragmenLindelof | {
"line": 459,
"column": 4
} | {
"line": 460,
"column": 11
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.PhragmenLindelof | {
"line": 518,
"column": 4
} | {
"line": 518,
"column": 73
} | [
{
"pp": "E : 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)\nhB : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 (Iio 0 ×ℂ Iio 0)] fun z ↦ expR (B * ‖z‖ ^ c)\nhre : ∀ x ≤ 0, ‖f ↑x‖ ≤ C\nhim : ∀ x ≤ 0, ‖f (↑x * I)‖ ≤ C\nz : ℂ\nhz_re : 0 ≤ z.re\nhz_... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.PhragmenLindelof | {
"line": 524,
"column": 4
} | {
"line": 525,
"column": 11
} | [
{
"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 :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.PhragmenLindelof | {
"line": 583,
"column": 4
} | {
"line": 583,
"column": 91
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.PhragmenLindelof | {
"line": 588,
"column": 4
} | {
"line": 589,
"column": 11
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalAverage | {
"line": 90,
"column": 35
} | {
"line": 90,
"column": 46
} | [
{
"pp": "f : ℝ → ℝ\na b : ℝ\nμ : Measure ℝ\ninst✝ : NoAtoms μ\nhf : ContinuousOn f [[a, b]]\nhμfin : μ (Ι a b) ≠ ⊤\nhμ0 : μ (Ι a b) ≠ 0\nhint : IntegrableOn f (Ι a b) μ\nh : a ≠ b\ns : Set ℝ := uIoo a b\nhs' : s ⊆ Ι a b\n⊢ s =ᶠ[ae μ] Ι a b",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalAverage | {
"line": 104,
"column": 57
} | {
"line": 104,
"column": 68
} | [
{
"pp": "f : ℝ → ℝ\na b : ℝ\nhab : a ≠ b\nhf : ContinuousOn f [[a, b]]\n⊢ volume (Ι a b) ≠ 0",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"Real.instLE",
"Real",
"MeasureTheory.Measure",
"Real.lattice",
"Real.instZero",
"ENNReal.ofReal",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.PhragmenLindelof | {
"line": 678,
"column": 6
} | {
"line": 678,
"column": 36
} | [
{
"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' ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.PhragmenLindelof | {
"line": 684,
"column": 4
} | {
"line": 684,
"column": 36
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.PhragmenLindelof | {
"line": 688,
"column": 6
} | {
"line": 688,
"column": 37
} | [
{
"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' ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.CircleAverage | {
"line": 184,
"column": 4
} | {
"line": 184,
"column": 15
} | [
{
"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)",
"usedConstants": [
"Eq.mpr",
"Real",
"Meas... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.PhragmenLindelof | {
"line": 705,
"column": 4
} | {
"line": 705,
"column": 15
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.MeanValue | {
"line": 75,
"column": 4
} | {
"line": 75,
"column": 57
} | [
{
"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|)",
"usedConstants": [
"instInnerProductSpaceRealComplex",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.PhragmenLindelof | {
"line": 734,
"column": 2
} | {
"line": 734,
"column": 75
} | [
{
"pp": "case h\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... | refine right_half_plane_of_tendsto_zero_on_real hd ?_ ?_ (fun y => ?_) hz | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Complex.PhragmenLindelof | {
"line": 767,
"column": 4
} | {
"line": 767,
"column": 42
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\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 : SuperpolynomialDecay atTop expR fun x ↦ ‖f ↑x‖\nC : ℝ\nhC : ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C\nth... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.PhragmenLindelof | {
"line": 822,
"column": 4
} | {
"line": 822,
"column": 69
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.UpperHalfPlane.Basic | {
"line": 140,
"column": 2
} | {
"line": 141,
"column": 72
} | [
{
"pp": "τ τ' : ℍ\nhre : τ.re = τ'.re\nhnorm : ‖↑τ‖ ^ 2 = ‖↑τ'‖ ^ 2\n⊢ τ = τ'",
"usedConstants": [
"Eq.mpr",
"Real",
"UpperHalfPlane.coe",
"_private.Mathlib.Analysis.Complex.UpperHalfPlane.Basic.0.UpperHalfPlane.eq_of_re_of_norm._simp_1_1",
"congrArg",
"Complex.im",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.UpperHalfPlane.Basic | {
"line": 178,
"column": 2
} | {
"line": 178,
"column": 23
} | [
{
"pp": "z : ℍ\n⊢ 0 < (-↑z)⁻¹.im",
"usedConstants": [
"Eq.mpr",
"Real",
"neg_div",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"DivisionCommMonoid.toDivisionMonoid",
"DivInvOneMonoid.toInvOneClass",
"UpperHalfPlane.coe",
"Real.instZero",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.UpperHalfPlane.Basic | {
"line": 198,
"column": 43
} | {
"line": 198,
"column": 54
} | [
{
"pp": "x : { x // 0 < x }\nz : ℍ\n⊢ 0 < (↑x • ↑z).im",
"usedConstants": [
"Complex.mul_im",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"instHSMul",
"RCLike.toNormedAlgebra",
"HMul.hMul",
"UpperHalfPlane.coe",
"Real.instZero",
"Real.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.UpperHalfPlane.Basic | {
"line": 226,
"column": 31
} | {
"line": 226,
"column": 42
} | [
{
"pp": "x : ℝ\nz : ℍ\n⊢ 0 < (↑x + ↑z).im",
"usedConstants": [
"Eq.mpr",
"Real",
"UpperHalfPlane.coe",
"Real.instZero",
"Real.instAddMonoid",
"congrArg",
"Complex.im",
"AddMonoid.toAddZeroClass",
"Real.instLT",
"id",
"zero_add",
"Comple... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.UpperHalfPlane.Basic | {
"line": 227,
"column": 17
} | {
"line": 227,
"column": 38
} | [
{
"pp": "x✝ : ℍ\n⊢ 0 +ᵥ x✝ = x✝",
"usedConstants": [
"Real",
"UpperHalfPlane.instAddActionReal._proof_1",
"UpperHalfPlane.coe",
"Real.instAddMonoid",
"congrArg",
"UpperHalfPlane.mk.congr_simp",
"AddMonoid.toAddZeroClass",
"AddGroupWithOne.toAddMonoidWithOne",
... | by simp [HVAdd.hVAdd] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.InnerProductSpace.Harmonic.Constructions | {
"line": 94,
"column": 8
} | {
"line": 94,
"column": 40
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"RingHom.instR... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Harmonic.Constructions | {
"line": 76,
"column": 2
} | {
"line": 102,
"column": 45
} | [
{
"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",
"usedConstants": [
"instInnerProductSpaceRealComplex",
"Filter.instMembership",
"Iff.mpr",
"IsModuleTopology.toContinuousSMul",
"AddGroup.toSubtrac... | 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
} | [
{
"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",
"usedConstants": [
"instInnerProductSpaceRealComplex",
"Filter.instMembership",
"Iff.mpr",
"IsModuleTopology.toContinuousSMul",
"AddGroup.toSubtrac... | 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.Log.NegMulLog | {
"line": 57,
"column": 4
} | {
"line": 57,
"column": 31
} | [
{
"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)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.NegMulLog | {
"line": 175,
"column": 2
} | {
"line": 175,
"column": 49
} | [
{
"pp": "x : ℝ\nh1 : 0 ≤ x\nh2 : x ≤ 1\n⊢ 0 ≤ x.negMulLog",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"Real.instLE",
"Real",
"HMul.hMul",
"Real.instZero",
"congrArg",
"id",
"_private.Mathlib.Analysis.SpecialFunctions.Log.NegMulLog.0.Re... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.NegMulLog | {
"line": 187,
"column": 2
} | {
"line": 187,
"column": 37
} | [
{
"pp": "⊢ Continuous negMulLog",
"usedConstants": [
"Eq.mpr",
"Real",
"Continuous",
"HMul.hMul",
"congrArg",
"PseudoMetricSpace.toUniformSpace",
"id",
"Real.negMulLog",
"Real.log",
"Real.instMul",
"Real.negMulLog_eq_neg",
"Real.instNeg... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.NegMulLog | {
"line": 190,
"column": 2
} | {
"line": 190,
"column": 37
} | [
{
"pp": "⊢ DifferentiableOn ℝ negMulLog {0}ᶜ",
"usedConstants": [
"Eq.mpr",
"Real",
"Semiring.toModule",
"HMul.hMul",
"Real.denselyNormedField",
"Real.instZero",
"congrArg",
"Compl.compl",
"PseudoMetricSpace.toUniformSpace",
"NormedField.toField",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.NegMulLog | {
"line": 225,
"column": 2
} | {
"line": 225,
"column": 37
} | [
{
"pp": "⊢ StrictConcaveOn ℝ (Set.Ici 0) negMulLog",
"usedConstants": [
"Eq.mpr",
"StrictConcaveOn",
"Real.partialOrder",
"Real",
"instSMulOfMul",
"HMul.hMul",
"Set.Ici",
"Real.instZero",
"congrArg",
"Real.semiring",
"id",
"Real.negMulL... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Poisson | {
"line": 62,
"column": 2
} | {
"line": 62,
"column": 83
} | [
{
"pp": "a b : ℂ\n⊢ ((a + b) / (a - b)).re = (‖a‖ ^ 2 - ‖b‖ ^ 2) / ‖a - b‖ ^ 2",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Complex.add_im",
"Real",
"instHDiv",
"HMul.hMul",
"GroupWithZero.toDivInvMonoid",
"congrArg",
"Complex.sub_im",
"Complex.im"... | 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
} | [
{
"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‖)",
"usedConstants": [
"sub_sub_sub_cancel_right",
"Norm.norm",
"Eq... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Poisson | {
"line": 108,
"column": 8
} | {
"line": 108,
"column": 19
} | [
{
"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‖",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"Eq.mpr",
"Real",
"Complex.instNormedAddCommGroup",
"Real.instZero... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Poisson | {
"line": 120,
"column": 15
} | {
"line": 120,
"column": 52
} | [
{
"pp": "θ φ r R : ℝ\nh₁ : 0 < r\nh₂ : r < R\n⊢ 0 < ‖↑R * cexp (↑θ * I) - ↑r * cexp (↑φ * I)‖ ^ 2",
"usedConstants": [
"_private.Mathlib.Analysis.Complex.Poisson.0.le_re_herglotzRieszKernel_aux._simp_1_9",
"AddGroup.toSubtractionMonoid",
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSe... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Poisson | {
"line": 121,
"column": 34
} | {
"line": 121,
"column": 75
} | [
{
"pp": "θ φ r R : ℝ\nh₁ : 0 < r\nh₂ : r < R\n⊢ ¬‖↑R * cexp (↑θ * I)‖ = ‖↑r * cexp (↑φ * I)‖",
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Norm.norm",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"Real",
"NonUnitalCommRing.toNonUnitalNo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Poisson | {
"line": 125,
"column": 2
} | {
"line": 125,
"column": 13
} | [
{
"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)",
"usedCons... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Sinc | {
"line": 71,
"column": 29
} | {
"line": 71,
"column": 39
} | [
{
"pp": "case inl\nx : ℝ\nhx : x < 0\n⊢ sin x / x ≤ (-x)⁻¹",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NegZeroClass.toNeg",
"MulOne.toOne",
"Real.instLE",
"Real",
"DivInvMonoid.toInv",
"instHDiv",
"Monoid.toMulOneClass",
"congrA... | ← one_div, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Sinc | {
"line": 75,
"column": 6
} | {
"line": 75,
"column": 17
} | [
{
"pp": "case inl\nx : ℝ\nhx : x < 0\n⊢ 0 < -x",
"usedConstants": [
"Left.neg_pos_iff._simp_1",
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"NegZeroClass.toNeg",
"Real.partialOrder",
"Real",
"Preorder.toLT",
"instIsLeftCan... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Poisson | {
"line": 140,
"column": 2
} | {
"line": 140,
"column": 13
} | [
{
"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",
"usedConstants": [
"sub_sub_sub_cancel_right",
"Norm.norm",
"Eq... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Poisson | {
"line": 141,
"column": 8
} | {
"line": 141,
"column": 19
} | [
{
"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‖",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"Eq.mpr",
"Real",
"Complex.instNormedAddCommGroup",
"Real.instZero... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Sinc | {
"line": 81,
"column": 2
} | {
"line": 81,
"column": 25
} | [
{
"pp": "x : ℝ\nhx : x ≠ 0\n⊢ sinc x ≤ |x|⁻¹",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"instHDiv",
"Real.lattice",
"abs",
"congrArg",
"Real.instInv",
"Real.instDivInvMonoid",
"id",
"HDiv.hDiv",
"Real.sinc",
"Real.instA... | rw [sinc_of_ne_zero hx] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Complex.Poisson | {
"line": 173,
"column": 25
} | {
"line": 173,
"column": 54
} | [
{
"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",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Harmonic.Poisson | {
"line": 94,
"column": 2
} | {
"line": 94,
"column": 56
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Real",
"instHSMul",
"instSMulOfMul",
"Real.instRCLike",
... | 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
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Real",
"instHSMul",
"instSMulOfMul",
"Real.instRCLike",
"c... | 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.SpecialFunctions.Log.PosLog | {
"line": 174,
"column": 42
} | {
"line": 174,
"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",
"usedConstants": [
"abs_nonneg._simp_1",
"AddGroup.toSubtractionMonoid",
"Real",
"Set.Ici",
"Real.lattice",
"Real.... | simp [Finset.sum_nonneg] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.SpecialFunctions.Log.PosLog | {
"line": 174,
"column": 42
} | {
"line": 174,
"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",
"usedConstants": [
"abs_nonneg._simp_1",
"AddGroup.toSubtractionMonoid",
"Real",
"Set.Ici",
"Real.lattice",
"Real.... | simp [Finset.sum_nonneg] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Log.PosLog | {
"line": 174,
"column": 42
} | {
"line": 174,
"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",
"usedConstants": [
"abs_nonneg._simp_1",
"AddGroup.toSubtractionMonoid",
"Real",
"Set.Ici",
"Real.lattice",
"Real.... | simp [Finset.sum_nonneg] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Log.PosLog | {
"line": 177,
"column": 32
} | {
"line": 177,
"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",
"usedConstants": [
"abs_nonneg._simp_1",
"AddGroup.toSubtractionMonoid",
"Real",
"Set.Ici",
"Real.lattice",
"Real.... | simp [Finset.sum_nonneg] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.SpecialFunctions.Log.PosLog | {
"line": 177,
"column": 32
} | {
"line": 177,
"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",
"usedConstants": [
"abs_nonneg._simp_1",
"AddGroup.toSubtractionMonoid",
"Real",
"Set.Ici",
"Real.lattice",
"Real.... | simp [Finset.sum_nonneg] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Log.PosLog | {
"line": 177,
"column": 32
} | {
"line": 177,
"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",
"usedConstants": [
"abs_nonneg._simp_1",
"AddGroup.toSubtractionMonoid",
"Real",
"Set.Ici",
"Real.lattice",
"Real.... | simp [Finset.sum_nonneg] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.Poisson | {
"line": 197,
"column": 31
} | {
"line": 197,
"column": 58
} | [
{
"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 → ↑... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Poisson | {
"line": 228,
"column": 36
} | {
"line": 228,
"column": 47
} | [
{
"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",
"usedConstants": [
"Norm.n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Poisson | {
"line": 229,
"column": 2
} | {
"line": 230,
"column": 9
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Integrals.LogTrigonometric | {
"line": 64,
"column": 8
} | {
"line": 64,
"column": 19
} | [
{
"pp": "⊢ IntervalIntegrable (fun x ↦ log (sin (2 * x))) MeasureTheory.volume 0 (π / 2)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Integrals.LogTrigonometric | {
"line": 66,
"column": 8
} | {
"line": 66,
"column": 19
} | [
{
"pp": "⊢ IntervalIntegrable (fun x ↦ log (sin (2 * x))) MeasureTheory.volume 0 (π / 2)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Integrability.Basic | {
"line": 197,
"column": 2
} | {
"line": 197,
"column": 28
} | [
{
"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",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Integrability.Basic | {
"line": 246,
"column": 6
} | {
"line": 246,
"column": 17
} | [
{
"pp": "case hab.h.hderiv\na b x : ℝ\nhx : 0 < x\ns : ℝ\nhs : 0 < s\nright✝ : s < 1\n⊢ HasDerivAt (fun x ↦ x - x * log x) (-log s) s",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Integrals.Basic | {
"line": 123,
"column": 6
} | {
"line": 123,
"column": 55
} | [
{
"pp": "case inr\na b : ℝ\nr : ℂ\nh : r ≠ -1 ∧ 0 ∉ [[a, b]]\n⊢ r + 1 ≠ 0",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"NegZeroClass.toNeg",
"Real",
"Real.lattice",
"Real.instZero",
"AddGroupWithOne.toAddGroup",
"congrArg",
"AddMonoid.toAddZeroClass",
... | 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": 123,
"column": 6
} | {
"line": 123,
"column": 55
} | [
{
"pp": "case inr\na b : ℝ\nr : ℂ\nh : r ≠ -1 ∧ 0 ∉ [[a, b]]\n⊢ r + 1 ≠ 0",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"NegZeroClass.toNeg",
"Real",
"Real.lattice",
"Real.instZero",
"AddGroupWithOne.toAddGroup",
"congrArg",
"AddMonoid.toAddZeroClass",
... | rw [Ne, ← add_eq_zero_iff_eq_neg] at h; exact h.1 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Integrals.Basic | {
"line": 173,
"column": 2
} | {
"line": 173,
"column": 53
} | [
{
"pp": "a b : ℝ\nn : ℕ\n⊢ ∫ (x : ℝ) in a..b, x ^ n = (b ^ (n + 1) - a ^ (n + 1)) / (↑n + 1)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Integrals.Basic | {
"line": 248,
"column": 4
} | {
"line": 248,
"column": 30
} | [
{
"pp": "a b : ℝ\nc : ℂ\nhc : c ≠ 0\nx : ℝ\n⊢ HasDerivAt (fun x ↦ c * ↑x) c x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Integrals.Basic | {
"line": 285,
"column": 4
} | {
"line": 285,
"column": 15
} | [
{
"pp": "case hderiv\nb : ℝ\nht : 0 < b\ns : ℝ\nhs : 0 < s\nright✝ : s < b\n⊢ HasDerivAt (fun x ↦ x * log x - x) (log s) s",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Integrals.Basic | {
"line": 286,
"column": 4
} | {
"line": 286,
"column": 26
} | [
{
"pp": "case ha\nb : ℝ\nht : 0 < b\n⊢ Filter.Tendsto (fun x ↦ x * log x - x) (nhdsWithin 0 (Set.Ioi 0)) (nhds 0)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Integrals.Basic | {
"line": 337,
"column": 2
} | {
"line": 337,
"column": 28
} | [
{
"pp": "case hderiv\nz : ℂ\nhz : z ≠ 0\na✝¹ b✝ x : ℝ\na✝ : x ∈ [[a✝¹, b✝]]\na : HasDerivAt Complex.sin (Complex.cos (↑x * z)) (↑x * z)\nb : HasDerivAt (fun y ↦ y * z) z ↑x\nc : HasDerivAt (Complex.sin ∘ fun y ↦ y * z) (Complex.cos (↑x * z) * z) ↑x\nd : HasDerivAt (fun y ↦ (Complex.sin ∘ fun y ↦ y * z) ↑y / z) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Integrals.Basic | {
"line": 341,
"column": 2
} | {
"line": 341,
"column": 59
} | [
{
"pp": "a b : ℝ\n⊢ ∫ (x : ℝ) in a..b, cos x ^ 2 - sin x ^ 2 = sin b * cos b - sin a * cos a",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Real",
"MeasureTheory.Measure",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"Real.cos",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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