Loogle!
Result
Found 171 definitions mentioning Complex.re and Complex.ofReal'. Of these, 26 match your pattern(s).
- Complex.re_add_im Mathlib.Data.Complex.Basic
∀ (z : ℂ), ↑z.re + ↑z.im * Complex.I = z - Complex.conj_eq_iff_re Mathlib.Data.Complex.Basic
∀ {z : ℂ}, (starRingEnd ℂ) z = z ↔ ↑z.re = z - Complex.re_eq_add_conj Mathlib.Data.Complex.Basic
∀ (z : ℂ), ↑z.re = (z + (starRingEnd ℂ) z) / 2 - Complex.coe_realPart Mathlib.Data.Complex.Module
∀ (z : ℂ), ↑(realPart z) = ↑z.re - Complex.eq_re_of_ofReal_le Mathlib.Data.Complex.Order
∀ {r : ℝ} {z : ℂ}, ↑r ≤ z → z = ↑z.re - Complex.ofReal_cos_ofReal_re Mathlib.Data.Complex.Exponential
∀ (x : ℝ), ↑(↑x).cos.re = (↑x).cos - Complex.ofReal_cosh_ofReal_re Mathlib.Data.Complex.Exponential
∀ (x : ℝ), ↑(↑x).cosh.re = (↑x).cosh - Complex.ofReal_exp_ofReal_re Mathlib.Data.Complex.Exponential
∀ (x : ℝ), ↑(↑x).exp.re = (↑x).exp - Complex.ofReal_sin_ofReal_re Mathlib.Data.Complex.Exponential
∀ (x : ℝ), ↑(↑x).sin.re = (↑x).sin - Complex.ofReal_sinh_ofReal_re Mathlib.Data.Complex.Exponential
∀ (x : ℝ), ↑(↑x).sinh.re = (↑x).sinh - Complex.ofReal_tan_ofReal_re Mathlib.Data.Complex.Exponential
∀ (x : ℝ), ↑(↑x).tan.re = (↑x).tan - Complex.ofReal_tanh_ofReal_re Mathlib.Data.Complex.Exponential
∀ (x : ℝ), ↑(↑x).tanh.re = (↑x).tanh - Complex.exp_eq_exp_re_mul_sin_add_cos Mathlib.Data.Complex.Exponential
∀ (x : ℂ), x.exp = (↑x.re).exp * ((↑x.im).cos + (↑x.im).sin * Complex.I) - Complex.cos_eq Mathlib.Data.Complex.Exponential
∀ (z : ℂ), z.cos = (↑z.re).cos * (↑z.im).cosh - (↑z.re).sin * (↑z.im).sinh * Complex.I - Complex.sin_eq Mathlib.Data.Complex.Exponential
∀ (z : ℂ), z.sin = (↑z.re).sin * (↑z.im).cosh + (↑z.re).cos * (↑z.im).sinh * Complex.I - Complex.integral_boundary_rect_eq_zero_of_differentiableOn Mathlib.Analysis.Complex.CauchyIntegral
∀ {E : Type u} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℂ E] [inst_2 : CompleteSpace E] (f : ℂ → E) (z w : ℂ), DifferentiableOn ℂ f (Set.uIcc z.re w.re ×ℂ Set.uIcc z.im w.im) → (((∫ (x : ℝ) in z.re..w.re, f (↑x + ↑z.im * Complex.I)) - ∫ (x : ℝ) in z.re..w.re, f (↑x + ↑w.im * Complex.I)) + Complex.I • ∫ (y : ℝ) in z.im..w.im, f (↑w.re + ↑y * Complex.I)) - Complex.I • ∫ (y : ℝ) in z.im..w.im, f (↑z.re + ↑y * Complex.I) = 0 - Complex.integral_boundary_rect_eq_zero_of_continuousOn_of_differentiableOn Mathlib.Analysis.Complex.CauchyIntegral
∀ {E : Type u} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℂ E] [inst_2 : CompleteSpace E] (f : ℂ → E) (z w : ℂ), ContinuousOn f (Set.uIcc z.re w.re ×ℂ Set.uIcc z.im w.im) → DifferentiableOn ℂ f (Set.Ioo (min z.re w.re) (max z.re w.re) ×ℂ Set.Ioo (min z.im w.im) (max z.im w.im)) → (((∫ (x : ℝ) in z.re..w.re, f (↑x + ↑z.im * Complex.I)) - ∫ (x : ℝ) in z.re..w.re, f (↑x + ↑w.im * Complex.I)) + Complex.I • ∫ (y : ℝ) in z.im..w.im, f (↑w.re + ↑y * Complex.I)) - Complex.I • ∫ (y : ℝ) in z.im..w.im, f (↑z.re + ↑y * Complex.I) = 0 - Complex.integral_boundary_rect_eq_zero_of_differentiable_on_off_countable Mathlib.Analysis.Complex.CauchyIntegral
∀ {E : Type u} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℂ E] [inst_2 : CompleteSpace E] (f : ℂ → E) (z w : ℂ) (s : Set ℂ), s.Countable → ContinuousOn f (Set.uIcc z.re w.re ×ℂ Set.uIcc z.im w.im) → (∀ x ∈ Set.Ioo (min z.re w.re) (max z.re w.re) ×ℂ Set.Ioo (min z.im w.im) (max z.im w.im) \ s, DifferentiableAt ℂ f x) → (((∫ (x : ℝ) in z.re..w.re, f (↑x + ↑z.im * Complex.I)) - ∫ (x : ℝ) in z.re..w.re, f (↑x + ↑w.im * Complex.I)) + Complex.I • ∫ (y : ℝ) in z.im..w.im, f (↑w.re + ↑y * Complex.I)) - Complex.I • ∫ (y : ℝ) in z.im..w.im, f (↑z.re + ↑y * Complex.I) = 0 - Complex.integral_boundary_rect_of_continuousOn_of_hasFDerivAt_real Mathlib.Analysis.Complex.CauchyIntegral
∀ {E : Type u} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℂ E] [inst_2 : CompleteSpace E] (f : ℂ → E) (f' : ℂ → ℂ →L[ℝ] E) (z w : ℂ), ContinuousOn f (Set.uIcc z.re w.re ×ℂ Set.uIcc z.im w.im) → (∀ x ∈ Set.Ioo (min z.re w.re) (max z.re w.re) ×ℂ Set.Ioo (min z.im w.im) (max z.im w.im), HasFDerivAt f (f' x) x) → MeasureTheory.IntegrableOn (fun z => Complex.I • (f' z) 1 - (f' z) Complex.I) (Set.uIcc z.re w.re ×ℂ Set.uIcc z.im w.im) MeasureTheory.volume → (((∫ (x : ℝ) in z.re..w.re, f (↑x + ↑z.im * Complex.I)) - ∫ (x : ℝ) in z.re..w.re, f (↑x + ↑w.im * Complex.I)) + Complex.I • ∫ (y : ℝ) in z.im..w.im, f (↑w.re + ↑y * Complex.I)) - Complex.I • ∫ (y : ℝ) in z.im..w.im, f (↑z.re + ↑y * Complex.I) = ∫ (x : ℝ) in z.re..w.re, ∫ (y : ℝ) in z.im..w.im, Complex.I • (f' (↑x + ↑y * Complex.I)) 1 - (f' (↑x + ↑y * Complex.I)) Complex.I - Complex.integral_boundary_rect_of_hasFDerivAt_real_off_countable Mathlib.Analysis.Complex.CauchyIntegral
∀ {E : Type u} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℂ E] [inst_2 : CompleteSpace E] (f : ℂ → E) (f' : ℂ → ℂ →L[ℝ] E) (z w : ℂ) (s : Set ℂ), s.Countable → ContinuousOn f (Set.uIcc z.re w.re ×ℂ Set.uIcc z.im w.im) → (∀ x ∈ Set.Ioo (min z.re w.re) (max z.re w.re) ×ℂ Set.Ioo (min z.im w.im) (max z.im w.im) \ s, HasFDerivAt f (f' x) x) → MeasureTheory.IntegrableOn (fun z => Complex.I • (f' z) 1 - (f' z) Complex.I) (Set.uIcc z.re w.re ×ℂ Set.uIcc z.im w.im) MeasureTheory.volume → (((∫ (x : ℝ) in z.re..w.re, f (↑x + ↑z.im * Complex.I)) - ∫ (x : ℝ) in z.re..w.re, f (↑x + ↑w.im * Complex.I)) + Complex.I • ∫ (y : ℝ) in z.im..w.im, f (↑w.re + ↑y * Complex.I)) - Complex.I • ∫ (y : ℝ) in z.im..w.im, f (↑z.re + ↑y * Complex.I) = ∫ (x : ℝ) in z.re..w.re, ∫ (y : ℝ) in z.im..w.im, Complex.I • (f' (↑x + ↑y * Complex.I)) 1 - (f' (↑x + ↑y * Complex.I)) Complex.I - Complex.integral_boundary_rect_of_differentiableOn_real Mathlib.Analysis.Complex.CauchyIntegral
∀ {E : Type u} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℂ E] [inst_2 : CompleteSpace E] (f : ℂ → E) (z w : ℂ), DifferentiableOn ℝ f (Set.uIcc z.re w.re ×ℂ Set.uIcc z.im w.im) → MeasureTheory.IntegrableOn (fun z => Complex.I • (fderiv ℝ f z) 1 - (fderiv ℝ f z) Complex.I) (Set.uIcc z.re w.re ×ℂ Set.uIcc z.im w.im) MeasureTheory.volume → (((∫ (x : ℝ) in z.re..w.re, f (↑x + ↑z.im * Complex.I)) - ∫ (x : ℝ) in z.re..w.re, f (↑x + ↑w.im * Complex.I)) + Complex.I • ∫ (y : ℝ) in z.im..w.im, f (↑w.re + ↑y * Complex.I)) - Complex.I • ∫ (y : ℝ) in z.im..w.im, f (↑z.re + ↑y * Complex.I) = ∫ (x : ℝ) in z.re..w.re, ∫ (y : ℝ) in z.im..w.im, Complex.I • (fderiv ℝ f (↑x + ↑y * Complex.I)) 1 - (fderiv ℝ f (↑x + ↑y * Complex.I)) Complex.I - Complex.tan_eq Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex
∀ {z : ℂ}, (((∀ (k : ℤ), ↑z.re ≠ (2 * ↑k + 1) * ↑Real.pi / 2) ∧ ∀ (l : ℤ), ↑z.im * Complex.I ≠ (2 * ↑l + 1) * ↑Real.pi / 2) ∨ (∃ k, ↑z.re = (2 * ↑k + 1) * ↑Real.pi / 2) ∧ ∃ l, ↑z.im * Complex.I = (2 * ↑l + 1) * ↑Real.pi / 2) → z.tan = ((↑z.re).tan + (↑z.im).tanh * Complex.I) / (1 - (↑z.re).tan * (↑z.im).tanh * Complex.I) - SpectrumRestricts.real_iff Mathlib.Analysis.NormedSpace.Spectrum
∀ {A : Type u_3} [inst : Ring A] [inst_1 : Algebra ℂ A] {a : A}, SpectrumRestricts a ⇑Complex.reCLM ↔ ∀ x ∈ spectrum ℂ a, x = ↑x.re - QuasispectrumRestricts.real_iff Mathlib.Analysis.NormedSpace.Spectrum
∀ {A : Type u_3} [inst : NonUnitalRing A] [inst_1 : Module ℂ A] [inst_2 : IsScalarTower ℂ A A] [inst_3 : SMulCommClass ℂ A A] {a : A}, QuasispectrumRestricts a ⇑Complex.reCLM ↔ ∀ x ∈ quasispectrum ℂ a, x = ↑x.re - IsSelfAdjoint.mem_spectrum_eq_re Mathlib.Analysis.NormedSpace.Star.Spectrum
∀ {A : Type u_1} [inst : NormedRing A] [inst_1 : NormedAlgebra ℂ A] [inst_2 : CompleteSpace A] [inst_3 : StarRing A] [inst_4 : CstarRing A] [inst_5 : StarModule ℂ A] {a : A}, IsSelfAdjoint a → ∀ {z : ℂ}, z ∈ spectrum ℂ a → z = ↑z.re - selfAdjoint.mem_spectrum_eq_re Mathlib.Analysis.NormedSpace.Star.Spectrum
∀ {A : Type u_1} [inst : NormedRing A] [inst_1 : NormedAlgebra ℂ A] [inst_2 : CompleteSpace A] [inst_3 : StarRing A] [inst_4 : CstarRing A] [inst_5 : StarModule ℂ A] (a : ↥(selfAdjoint A)) {z : ℂ}, z ∈ spectrum ℂ ↑a → z = ↑z.re
Did you maybe mean
About
Loogle searches of Lean and Mathlib definitions and theorems.
You may also want to try the CLI version, the VS
Code extension, the lean.nvim
integration or the Zulip bot.
Usage
Loogle finds definitions and lemmas in various ways:
By constant:
🔍Real.sin
finds all lemmas whose statement somehow mentions the sine function.By lemma name substring:
🔍"differ"
finds all lemmas that have"differ"
somewhere in their lemma name.By subexpression:
🔍_ * (_ ^ _)
finds all lemmas whose statements somewhere include a product where the second argument is raised to some power.The pattern can also be non-linear, as in
🔍Real.sqrt ?a * Real.sqrt ?a
If the pattern has parameters, they are matched in any order. Both of these will find
List.map
:
🔍(?a -> ?b) -> List ?a -> List ?b
🔍List ?a -> (?a -> ?b) -> List ?b
By main conclusion:
🔍|- tsum _ = _ * tsum _
finds all lemmas where the conclusion (the subexpression to the right of all→
and∀
) has the given shape.As before, if the pattern has parameters, they are matched against the hypotheses of the lemma in any order; for example,
🔍|- _ < _ → tsum _ < tsum _
will findtsum_lt_tsum
even though the hypothesisf i < g i
is not the last.
If you pass more than one such search filter, separated by commas
Loogle will return lemmas which match all of them. The
search
🔍 Real.sin, "two", tsum, _ * _, _ ^ _, |- _ < _ → _
woould find all lemmas which mention the constants Real.sin
and tsum
, have "two"
as a substring of the
lemma name, include a product and a power somewhere in the type,
and have a hypothesis of the form _ < _
(if
there were any such lemmas). Metavariables (?a
) are
assigned independently in each filter.
The #lucky
button will directly send you to the
documentation of the first hit.
Source code
You can find the source code for this service at https://github.com/nomeata/loogle. The https://loogle.lean-lang.org/ service is currently provided by Joachim Breitner <mail@joachim-breitner.de>.
This is Loogle revision fa2ddf5
serving mathlib revision c44d42e