chars int64 40 24k | content_type stringclasses 9
values | id stringlengths 13 64 | source stringclasses 5
values | subset stringclasses 31
values | text stringlengths 40 24k |
|---|---|---|---|---|---|
17,097 | lean_file_chunk | 26d9a984a427b4e9e578bf519b842f88c6bf55fd190ac2f3976a193b6e581e82 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
module
public import Mathlib.Analysis.Calculus.MeanValue
public import Mathlib.MeasureTheory.Integral.DominatedConvergence
public import Mathlib.MeasureTheory.Integr... |
7,054 | lean_file_chunk | f0f94bcee175f2089c6a05a3e29ac802ca0e77789089fc1bcff8a43851fb10db | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
module
public import Mathlib.Analysis.Calculus.Deriv.Add
public import Mathlib.Analysis.Calculus.Deriv.Mul
public import Mathlib.Analysis.Calculus.LocalExtr.Basic
... |
1,194 | lean_file_chunk | 1c919b13896af3fc588ffd0dd208d58befc065567bd98424c94743d20b32e50b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
module
public import Mathlib.Analysis.Complex.LocallyUniformLimit
public import Mathlib.Topology.Algebra.InfiniteSum.UniformOn
/-!
# The Logarithmic derivative of a... |
6,517 | lean_file_chunk | 2261df2f575e767374a469d2b6c3fb3c1fd30bf265f958c71dcb88135f25d354 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
module
public import Mathlib.Analysis.Calculus.Deriv.ZPow
public import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Analysis.Analytic.IsolatedZeros
import Ma... |
13,801 | lean_file_chunk | a560732c726f3cf5873db1440bc99f236eda27aad8992f5874757bc86008d13a | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
module
public import Mathlib.Analysis.Calculus.Deriv.Slope
public import Mathlib.MeasureTheory.Covering.OneDim
public import Mathlib.Order.Monotone.Extension
... |
23,950 | lean_file_chunk | 6c30aab7e9bb9a60d9703ee5f019b7760d89c82c5f64cc8ac8bead14b2104561 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
module
public import Mathlib.Analysis.Calculus.Deriv.AffineMap
public import Mathlib.Analysis.Calculus.Deriv.Comp
public import Mathlib.Analys... |
21,370 | lean_file_chunk | 6ed44381c4ca30885c250c04476d6125304e4005256fd69d6f48d410f2522b30 | lean_source_file | pkg_mathlib_source_file_chunks | `K`-Lipschitz on some neighborhood of `x` within `s`. See also
`Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt` for a version that claims
existence of `K` instead of an explicit estimate. -/
theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs : Convex ℝ s)
{f : E → G} (hde... |
15,558 | lean_file_chunk | 03c59d51962f525f8be1e4546373a0aa1ea72cddbc91af9d7695154eb2ce2d2f | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
module
public import Mathlib.Analysis.Calculus.ContDiff.Operations
public import Mathlib.Analysis.Calculus.UniformLimitsDeriv
public import Mathlib.Topology.Al... |
17,378 | lean_file_chunk | 59824577cc6d6833ceb38293d2a667936e961e98adf7eb0ea94574891e3d4fa0 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
module
public import Mathlib.Analysis.Normed.Module.Multilinear.Curry
/-!
# Formal multilinear series
In this file we define `FormalMultilinearSeries 𝕜 E F`... |
23,877 | lean_file_chunk | 95c44fed4acba75c40fd965f318feb1618c076a32f3047f3eedcd529717451a4 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
module
public import Mathlib.Analysis.Calculus.Deriv.Inv
public import Mathlib.Analysis.Calculus.Deriv.MeanValue
/-!
# L'Hôpital's rule for 0/0 indeterminate fo... |
5,493 | lean_file_chunk | 5ae416011b348e86c869527e0e10ae5a9f95f26a00bc4dabc5c0c19f3051c08c | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
module
public import Mathlib.Analysis.Normed.Module.RCLike.Real
public import Mathlib.Analysis.Calculus.FDeriv.Add
public import Mathlib.Analysis.Calculus.FDeriv.M... |
476 | lean_file_chunk | 74cfff8704d223d241192bbf752033e64d3b92dc796e268c822feb3b3dcb0d2f | lean_source_file | pkg_mathlib_source_file_chunks | module -- shake: keep-all
public import Mathlib.Analysis.Calculus.TangentCone.Basic
public import Mathlib.Analysis.Calculus.TangentCone.Defs
public import Mathlib.Analysis.Calculus.TangentCone.DimOne
public import Mathlib.Analysis.Calculus.TangentCone.Pi
public import Mathlib.Analysis.Calculus.TangentCone.Prod
public ... |
7,100 | lean_file_chunk | df3d51cc5179b2f1dc6506004cb75a436861af61f20479a88992a98e87fc3afb | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
module
public import Mathlib.Analysis.Normed.Field.Basic
public import Mathlib.Data.ENNReal.Action
public import Mathlib.Topology.Algebra.UniformMulAction
public import Ma... |
4,901 | lean_file_chunk | 58c8421454f4f00d45383484d0d450b30d8dcef2015e59d0174ea800b3956766 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2026 Kevin H. Wilson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin H. Wilson
-/
module
public import Mathlib.Analysis.Fourier.AddCircle
public import Mathlib.MeasureTheory.Integral.CircleAverage
/-!
# Fourier Coefficients of Polynomials
We d... |
3,803 | lean_file_chunk | 951ef10bcd311b7e353bdc0c215c7eef147dece250e79dc5aa1dc1c82ba84d7f | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Kevin H. Wilson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin H. Wilson
-/
module
public import Mathlib.RingTheory.Polynomial.GaussNorm
public import Mathlib.Analysis.Normed.Unbundled.RingSeminorm
public import Mathlib.Algebra.Order.Hom.B... |
6,102 | lean_file_chunk | fceb180516d6b72c7eebdcf8e4b1972c369892c393ac4dfca288393ee9553f39 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2026 Yuval Filmus. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuval Filmus
-/
module
public import Mathlib.Analysis.Polynomial.Basic
public import Mathlib.Topology.Algebra.Polynomial
/-!
# Eventual sign of polynomials
This file proves that a pol... |
17,522 | lean_file_chunk | 368d57d913792ba531160dc84f6f1a7e0d7c061792cff226f4652ff223b32150 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Devon Tuma
-/
module
public import Mathlib.Algebra.Polynomial.Roots
public import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
public import Mathlib.Analysis.... |
3,522 | lean_file_chunk | c506db41f0cb88d295edeed1be15b559b0e2340f4f60cb3dc96e3bca966b6738 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
module
public import Mathlib.Algebra.Polynomial.Degree.IsMonicOfDegree
public import Mathlib.Analysis.Complex.Polynomial.Basic
/-!
# Factorization of monic polynomial... |
23,967 | lean_file_chunk | ef4932fb256a8df442a4a5cf5c1b31cd42b846f80b538d6f9f22d96ed8eaf434 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Fabrizio Barroero. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fabrizio Barroero, Kevin H. Wilson
-/
module
public import Mathlib.Analysis.Analytic.Polynomial
public import Mathlib.Analysis.Complex.Polynomial.Basic
public import Mathlib.Algebr... |
156 | lean_file_chunk | 577b08d5cbb312377411daf9d100279a24723f188a143266427354ecc65d5972 | lean_source_file | pkg_mathlib_source_file_chunks | grw [hv_norm, hcoeff, norm_coeff_le_choose_mul_mahlerMeasure,
natDegree_map_eq_of_injective hv.injective, mapMahlerMeasure]
end Polynomial
end generic |
4,624 | lean_file_chunk | 28f5a91a3c25126e9c42fcb9fed678cf3c89f6d838904d9793fb879c51527af9 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Daniel Weber. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Daniel Weber
-/
module
public import Mathlib.Algebra.Order.Field.GeomSum
public import Mathlib.Algebra.Polynomial.Monic
public import Mathlib.Analysis.Normed.Field.Basic
/-!
# Cauchy's... |
23,969 | lean_file_chunk | ab35139c0efefc4fbfb30e7773b9b84ae47ec70fbcc0d00d8d4cab512701057d | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth, Eric Wieser
-/
module
public import Mathlib.Analysis.InnerProductSpace.PiL2
/-!
# Matrices as a normed space
In this file we provide the following non-instances fo... |
2,864 | lean_file_chunk | 099154392ae63911a52077588f0984c15e76a4528ed8732d840d0f72f051fbbf | lean_source_file | pkg_mathlib_source_file_chunks | simp_rw [diagonal_apply_eq, NNReal.rpow_two]
· suffices i.1 ≠ i.2 by rw [diagonal_apply_ne _ this, nnnorm_zero, NNReal.zero_rpow two_ne_zero]
intro h
exact Finset.mem_map.not.mp his ⟨i.1, Finset.mem_univ _, Prod.ext rfl h⟩
@[simp]
lemma frobenius_norm_diagonal [DecidableEq n] (v : n → α) : ‖diagonal v‖ = ‖to... |
4,828 | lean_file_chunk | d7a61abd8345ee5323931622e492a525e98a98be02867e455b8dcee472e7a03c | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp
-/
module
public import Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho
public import Mathlib.Analysis.Matrix.PosDef
/-! # LDL decomposition
This file pro... |
5,764 | lean_file_chunk | 6b6ab00fa7c39828b91d7f5318b748c29e08d01dce2ed84b044e461e1122fc0a | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
module
public import Mathlib.Analysis.Matrix.Spectrum
public import Mathlib.LinearAlgebra.Matrix.PosDef
/-!
# Spectrum of positive (semi)defi... |
14,589 | lean_file_chunk | 72f703e9fa43475cc52f4acc53d796d8c676426232dea9a155bae655653670ad | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Monica Omar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Monica Omar
-/
module
public import Mathlib.Algebra.Order.Module.PositiveLinearMap
public import Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Instances
public import Mathli... |
6,917 | lean_file_chunk | 0bd4b4a774f90fd5fb72a77b29f5c539fefbcb42618669bab679259ebacab552 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Jon Bannon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jon Bannon, Jireh Loreaux
-/
module
public import Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique
public import Mathlib.Analysis.Matrix.Spectrum
public import Mathlib.To... |
2,964 | lean_file_chunk | 1dccef3655f927e3d0bd3764a37a653bc1fe3d2ddbc0a75c161310ff430eeb19 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp
-/
module
public import Mathlib.Analysis.InnerProductSpace.PiL2
public import Mathlib.LinearAlgebra.Matrix.Hermitian
import Mathlib.Analysis.InnerProductSpace.A... |
11,039 | lean_file_chunk | 8d801862d6c400ebb8786f9e8fa694f4ae81df493a0457923b3d8d9ac3b9bbf4 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp
-/
module
public import Mathlib.Algebra.Star.UnitaryStarAlgAut
public import Mathlib.Analysis.InnerProductSpace.Spectrum
public import Mathlib.Analysis.Matrix.H... |
2,818 | lean_file_chunk | e7f83b3f3e52f147636bb4a94425ad450eb46693ad156223ff1075c189213f03 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
module
public import Mathlib.Analysis.Analytic.ConvergenceRadius
public import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
/-!
# Representation of `FormalMultili... |
23,995 | lean_file_chunk | f7c06a7e8a3586eb23bc286d9451b77107d1b2b8bf910e1e58b208e6009ac4db | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
module
public import Mathlib.Analysis.Analytic.Composition
public import Mathlib.Analysis.Analytic.Linear
public import Mathlib.Tactic.Positivity
/-!
# Inver... |
9,537 | lean_file_chunk | 367dfeec15f003c2f185918d2c721be86ece0ea5a0a7b68e9ca7af6736f04b72 | lean_source_file | pkg_mathlib_source_file_chunks | have IRec : ∀ n, 1 ≤ n → S n ≤ (I + 1) * a := by
apply Nat.le_induction
· simp only [S]
rw [Ico_eq_empty_of_le (le_refl 1), sum_empty]
positivity
· intro n one_le_n hn
have In : 2 ≤ n + 1 := by lia
have rSn : r * S n ≤ 1 / 2 :=
calc
r * S n ≤ r * ((I + 1) * a) := by... |
20,920 | lean_file_chunk | e8060da5421b24b23d97160e3ef4a8d9b279376a6707e81b51d80f05e4b742c7 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
module
public import Mathlib.Analysis.Calculus.FormalMultilinearSeries
public import Mathlib.Analysis.SpecificLimits.Normed
/-!
# Radius of c... |
23,965 | lean_file_chunk | db8a9e1496c3defe92f42d87a4f41c440db5ff153d76fd8f66585ba85ed05bf6 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Johan Commelin
-/
module
public import Mathlib.Analysis.Analytic.Basic
public import Mathlib.Analysis.Analytic.CPolynomialDef
public import Mathlib.Combinatorics... |
23,954 | lean_file_chunk | 8ebff3bf7a59046a96cb995ddeafd83c9702c60745f744e3ac7b05124fe08bf0 | lean_source_file | pkg_mathlib_source_file_chunks | ∑' c : Composition n, ‖compAlongComposition q p c‖₊ * r ^ n := by
rw [tsum_fintype, ← Finset.sum_mul]
exact mul_le_mul' (nnnorm_sum_le _ _) le_rfl
_ ≤ ∑' i : Σ n : ℕ, Composition n, ‖compAlongComposition q p i.snd‖₊ * r ^ i.fst :=
NNReal.tsum_comp_le_tsum_of_inj hr sigma_mk_injective
/-!
### Comp... |
16,142 | lean_file_chunk | 11f70a4917e8b829154d470cfb11ef8bda209b0740b780ac6d05adfa7ce99b5e | lean_source_file | pkg_mathlib_source_file_chunks | = ∑_{i₁ + ... + iₖ = n} (r.comp q)ₖ (p_{i₁} (v₀, ..., v_{i₁ -1}), p_{i₂} (...), ..., p_{iₖ}(...))
= ∑_{a : Composition n} (r.comp q) a.length (applyComposition p a v)
```
decomposing `r.comp q` in the same way, we get
```
(r.comp q).comp p n v
= ∑_{a : Composition n} ∑_{b : Composition a.length}
r b.length (applyComp... |
8,326 | lean_file_chunk | 50f84fe021d65a03c63a07a9ff881ec808a96a5b30a0a53599194451b6bf37b5 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
module
public import Mathlib.Analysis.Analytic.Basic
public import Mathlib.Analysis.Analytic.CPolynomialDef
/-!
# Linear functions are analytic
In this file we p... |
12,535 | lean_file_chunk | 625346e42802b77cf70b41e6fd6a54cb457c63df3850caf9ce73d636b1124ef4 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
module
public import Mathlib.Analysis.Analytic.Linear
public import Mathlib.Analysis.Analytic.Composition
public import Mathlib.Analysis.Analytic.Constructions... |
23,958 | lean_file_chunk | 337e5ec862fbd650da972cc8ff525207ab190847c06ea17d6501965cba77b2dd | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Vincent Beffara. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Vincent Beffara, Stefan Kebekus
-/
module
public import Mathlib.Analysis.Analytic.IsolatedZeros
public import Mathlib.Analysis.Calculus.Deriv.Mul
public import Mathlib.Analysis.Calcu... |
7,553 | lean_file_chunk | e927d53c1fac568cb4b8ff4064ea3f02f0b5abcc3e54e5019ae872746e773598 | lean_source_file | pkg_mathlib_source_file_chunks | · -- The interesting case: both orders are finite. First unpack the data:
rw [eventuallyConst_iff_analyticOrderAt_sub_eq_top] at hg_nc
obtain ⟨r, hr⟩ := ENat.ne_top_iff_exists.mp hf'
obtain ⟨s, hs⟩ := ENat.ne_top_iff_exists.mp hg_nc
rw [← hr, ← hs, ← ENat.coe_mul, (hf.comp hg).analyticOrderAt_eq_natCast... |
23,967 | lean_file_chunk | 3210739b6646effe9c8f55097fd3bfac37eac221f5f442747ed5d61ffef9e93b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
module
public import Mathlib.Analysis.Analytic.ConvergenceRadius
public import Mathlib.Topology.Algebra.InfiniteSum.Module
/-!
# Analytic fun... |
23,978 | lean_file_chunk | 59420e107adb48e8cd90c25a9a500082753a4051ad261b10f60906cff112a665 | lean_source_file | pkg_mathlib_source_file_chunks | g.hasSum (h.hasSum hy h'y)
/-- If a function `f` has a power series `p` on a ball and `g` is linear, then `g ∘ f` has the
power series `g ∘ p` on the same ball. -/
theorem ContinuousLinearMap.comp_hasFPowerSeriesOnBall (g : F →L[𝕜] G)
(h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (g ∘ f) (g.comp... |
6,710 | lean_file_chunk | 275fe3a6810bdbc476663ccbbc8be34d7015a6854e6da294fd202ce94b38f50b | lean_source_file | pkg_mathlib_source_file_chunks | /-- If a function admits a power series expansion at `x`, then it is the locally uniform limit of
the partial sums of this power series on the disk of convergence, i.e., `f y`
is the locally uniform limit of `p.partialSum n (y - x)` there. -/
theorem HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn' (hf : HasFPowerSerie... |
15,517 | lean_file_chunk | 1a0cb3db055b04fde042532f3ce4508299253343597e3aad4598f49392bdd898 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
module
public import Mathlib.Analysis.Calculus.ContDiff.Operations
public import Mathlib.Analysis.Calculus.ContDiff.CPolynomial
public import Mathlib.Data.Fint... |
23,993 | lean_file_chunk | fc3ba9c09b1fbd3577934c8d858376b4aaf0b51016a199a3fddaf4313587760a | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Geoffrey Irving. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler, Geoffrey Irving, Stefan Kebekus
-/
module
public import Mathlib.Analysis.Analytic.Composition
public import Mathlib.Analysis.Analytic.Linear
public import Mathlib.Ana... |
23,994 | lean_file_chunk | 14d84b33480939ed3e01be2b2b15b54eb3f268c2c548231b4d2b43ffea1d730a | lean_source_file | pkg_mathlib_source_file_chunks | AnalyticAt 𝕜 (fun x ↦ (f · x)) e ↔ ∀ i, AnalyticAt 𝕜 (f i) e := by
simp_rw [← analyticWithinAt_univ]
exact analyticWithinAt_pi_iff
lemma AnalyticOn.pi (hf : ∀ i, AnalyticOn 𝕜 (f i) s) :
AnalyticOn 𝕜 (fun x ↦ (f · x)) s :=
fun x hx ↦ AnalyticWithinAt.pi (fun i ↦ hf i x hx)
lemma analyticOn_pi_iff :
A... |
4,398 | lean_file_chunk | 22832f738a1625b8ef83335f625f5e8e56e67abe0622cf24daadeb89403d1012 | lean_source_file | pkg_mathlib_source_file_chunks | @[fun_prop]
theorem analyticAt_finprod {α : Type*} {A : Type*} [NormedCommRing A] [NormedAlgebra 𝕜 A]
{f : α → E → A} {c : E} (h : ∀ a, AnalyticAt 𝕜 (f a) c) :
AnalyticAt 𝕜 (∏ᶠ n, f n) c := by
by_cases hf : (Function.mulSupport f).Finite
· simp_all [finprod_eq_prod _ hf, Finset.analyticAt_prod]
· rw [f... |
14,469 | lean_file_chunk | 03635faa86f9ec4ccd3a7cf89b52c810815b86adde82dba991e2c56d893fb214 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Edward Watine. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Edward Watine
-/
module
public import Mathlib.Analysis.Analytic.ConvergenceRadius
/-!
# Scalar series
This file contains API for analytic functions `∑ cᵢ • xⁱ` defined in terms of sc... |
19,520 | lean_file_chunk | 71c29aaca3aaa641afdfc44efae8fb36601783f9ef194a7a5ad87a7e100f9149 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
module
public import Mathlib.Analysis.Analytic.Basic
/-!
# Changing origin in a power series
If a function is analytic in a disk `D(x, R)`, ... |
23,936 | lean_file_chunk | 5fd90502fa9d31e5cbec93b2757e02ec550fdc56a76bde41cb7fceb1193d746c | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Sophie Morel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sophie Morel
-/
module
public import Mathlib.Analysis.Analytic.ChangeOrigin
/-! We specialize the theory of analytic functions to the case of functions that admit a
development given b... |
340 | lean_file_chunk | 0428aa65f896546947cee8a7c44b3c59ddd3708b8f265450539b1eaf08444868 | lean_source_file | pkg_mathlib_source_file_chunks | /-- If `f` is continuously polynomial at a point, then it is continuously polynomial in a
nonempty ball around that point. -/
theorem CPolynomialAt.exists_ball_cpolynomialOn {f : E → F} {x : E} (h : CPolynomialAt 𝕜 f x) :
∃ r : ℝ, 0 < r ∧ CPolynomialOn 𝕜 f (Metric.ball x r) :=
Metric.isOpen_iff.mp (isOpen_cpoly... |
1,216 | lean_file_chunk | 1834fcbadee694b233f9d0c59fce35492fc9550fab0c3af157ddef434e6aadbb | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Etienne Marion. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Etienne Marion
-/
module
public import Mathlib.Analysis.Analytic.Linear
public import Mathlib.Analysis.Normed.Lp.PiLp
/-!
# Analyticity on `WithLp`
-/
public section
open WithLp
o... |
3,901 | lean_file_chunk | fd2d0de0a62c4eaccec3f6b4791e985f1a7fcf8c985f2c8213d62a2b55d43f2e | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
module
public import Mathlib.Algebra.Polynomial.AlgebraMap
public import Mathlib.Algebra.MvPolynomial.Eval
public import Mathlib.Analysis.Analytic.Constructions
public import ... |
9,201 | lean_file_chunk | e5512b706a966a603fa2a815f030be87d36eba07621480e427e1dd521e1f1e5a | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Geoffrey Irving. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Geoffrey Irving
-/
module
public import Mathlib.Analysis.Analytic.Constructions
public import Mathlib.Analysis.Analytic.ChangeOrigin
/-!
# Properties of analyticity restricted to a ... |
11,545 | lean_file_chunk | 1652695a14dbd9cdfac9af110f50755467849310be837ec90a4fe3e9280a1dd3 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Sophie Morel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sophie Morel
-/
module
public import Mathlib.Analysis.Analytic.Constructions
public import Mathlib.Analysis.Analytic.CPolynomialDef
/-! # Properties of continuously polynomial function... |
15,709 | lean_file_chunk | f97d5dd05f221089db0ce616da5b975a832b54b65dba0728ce2861bbeac54fd9 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Vasilii Nesterov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Vasilii Nesterov, Andrew Yang
-/
module
public import Mathlib.Analysis.Calculus.IteratedDeriv.ConvergenceOnBall
public import Mathlib.Analysis.Complex.OperatorNorm
public import Mat... |
18,773 | lean_file_chunk | ed9b5712c750960e2f2c66d66cbd9430eff27c2470922397f4828d5daaf4d0ac | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Vincent Beffara. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Vincent Beffara, Stefan Kebekus
-/
module
public import Mathlib.Analysis.Analytic.Constructions
public import Mathlib.Analysis.Calculus.DSlope
public import Mathlib.Analysis.Calculus... |
6,121 | lean_file_chunk | 43ea0705be8ef1845673229e7656922bcdd2e23f0a7d23c047c46c5841024efd | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Stefan Kebekus. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stefan Kebekus
-/
module
public import Mathlib.Analysis.InnerProductSpace.Laplacian
/-!
# Harmonic Functions
This file defines harmonic functions on real, finite-dimensional, inner ... |
5,154 | lean_file_chunk | ca3b3245293185cb9f074fe9bb7e2880647933bd3fc1af72049638a77009b2de | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Stefan Kebekus. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stefan Kebekus
-/
module
public import Mathlib.Analysis.InnerProductSpace.Harmonic.Basic
public import Mathlib.Analysis.Calculus.ContDiff.RestrictScalars
public import Mathlib.Analysi... |
3,734 | lean_file_chunk | 79a91d841d808ccdb3b58d496594502135f58a8d997e8376dd338ae2e02792d5 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2026 Stefan Kebekus. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stefan Kebekus, based on code by Yury Kudryashov
-/
module
public import Mathlib.Analysis.InnerProductSpace.Harmonic.Basic
/-!
# Functions Harmonic on a Domain and Continuous on Its ... |
6,249 | lean_file_chunk | b30fcef3fc4aedea0377516ecb5fbe1a5cb88109008d6901414292a02eec2013 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Frédéric Dupuis, Heather Macbeth
-/
module
public import Mathlib.Analysis.InnerProductSpace.Projection.Basic
/-!
# Reflection
A linear isometry equivalence `K.reflecti... |
23,913 | lean_file_chunk | d83acdd87efb0a9349abdd17223803823db883527b85e37c35ad618a39f679ff | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Frédéric Dupuis, Heather Macbeth
-/
module
public import Mathlib.Analysis.InnerProductSpace.Projection.Minimal
public import Mathlib.Analysis.InnerProductSpace.Symmetric... |
5,814 | lean_file_chunk | 19d2aa0d38af630f2b5e63f50d05b4fec49a7b04ee748745538e71cb9d77ee9b | lean_source_file | pkg_mathlib_source_file_chunks | /-- If the orthogonal projection to `K` is well-defined, then a vector splits as the sum of its
orthogonal projections onto a complete submodule `K` and onto the orthogonal complement of `K`. -/
theorem starProjection_add_starProjection_orthogonal [K.HasOrthogonalProjection]
(w : E) : K.starProjection w + Kᗮ.starPr... |
13,400 | lean_file_chunk | 2ff8d2e0dbf8b719c530f0f4ba6b5cd2e38f458f8619ed5ace128fa10e69cc1a | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Frédéric Dupuis, Heather Macbeth
-/
module
public import Mathlib.Analysis.InnerProductSpace.Basic
public import Mathlib.Analysis.SpecificLimits.Basic
/-!
# Existence of... |
10,687 | lean_file_chunk | ebae7c707c60d1ac10322bc037599629ccdd4c5c5c6f05b87b154eb6be12ffb3 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Frédéric Dupuis, Heather Macbeth
-/
module
public import Mathlib.Analysis.InnerProductSpace.Projection.Basic
/-!
# Subspaces associated with orthogonal projections
Her... |
18,048 | lean_file_chunk | 867d55455163efd944235a0992ed1fdb36ba1a41ea58466947aa3d3275428784 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Frédéric Dupuis, Heather Macbeth
-/
module
public import Mathlib.Analysis.InnerProductSpace.Projection.Basic
public import Mathlib.Analysis.InnerProductSpace.Projection.... |
2,858 | lean_file_chunk | 239b4005deb3414b4a265a6caae0da06e61554a4a62bb03eb9efe083db4ff88a | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2026 Jon Bannon, Monica Omar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jon Bannon, Monica Omar
-/
module
public import Mathlib.Analysis.CStarAlgebra.Basic
/-! # Unitary maps in C⋆-algebras
This file defines some basic maps by unitaries in C⋆-a... |
5,193 | lean_file_chunk | b01e8888ac2fbf30602ccd52eee1ed4919e348d4aa8481acfb96900ba262e2b4 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
module
public import Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order
public import Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unitary
... |
19,729 | lean_file_chunk | 7fc22e9c39f256d670d817e58b2e54d2d4301343f244e6ab8d1bdca121cfe96c | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
module
public import Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity
public import Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Ord... |
10,861 | lean_file_chunk | ea36b1e93c89a96f688e6633d9360ee0f278802202aa34ea69fb92dc9eb1d073 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
module
public import Mathlib.RingTheory.Finiteness.Defs
public import Mathlib.Topology.Bornology.Constructions
public import Mathlib.Topology.UniformSpace.Equiv
public... |
15,855 | lean_file_chunk | 5263ea4ecd132305d8aea147f9bf36a1e243500b08ee53569654ab1270849562 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
module
public import Mathlib.Analysis.CStarAlgebra.Module.Defs
public import Mathlib.Analysis.CStarAlgebra.Module.Synonym
public import Mathlib.Analysis.InnerProductSp... |
13,978 | lean_file_chunk | f04088b9acaf52aa54a2344ff70eed9e51f62b37e328309ef25dc4144b4f5af3 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
module
public import Mathlib.Analysis.InnerProductSpace.Defs
import Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order
public import Mathlib.Analysis... |
2,892 | lean_file_chunk | 63944adc799d6fe093f07f46615af629cdf83ed88e6cf2539a3d3aec6db4f223 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
module
public import Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Isometric
public import Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalC... |
23,938 | lean_file_chunk | a5f53eb832bf8f477b9277b3e688ddbc493ce80d702fc5d901ec2422d13ccd9b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
module
public import Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Instances
/-! # Isometric continuous functional calculus
This file adds a class for a... |
3,817 | lean_file_chunk | 7d6ae5efb0e46af0a07e7f3ae807508f1602fd6778ce5f5381abb16db0c4970c | lean_source_file | pkg_mathlib_source_file_chunks | refine cfcₙ_cases (‖·‖₊ ≤ c) a f (by simp) fun hf hf0 ha ↦ ?_
simp only [← cfcₙ_apply f a, isLUB_le_iff (IsGreatest.nnnorm_cfcₙ_nnreal f a hf hf0 ha |>.isLUB)]
rintro - ⟨x, hx, rfl⟩
exact h x hx
lemma nnnorm_cfcₙ_nnreal_le_iff (f : ℝ≥0 → ℝ≥0) (a : A) (c : ℝ≥0)
(hf : ContinuousOn f (σₙ ℝ≥0 a) := by cfc_cont_t... |
19,944 | lean_file_chunk | 55c26aea9febaeddeb4dc105ec7002b7a50f6e0c27a0239c14b97d40f70696a3 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
module
public import Mathlib.Analysis.Complex.Spectrum
public import Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Restrict
public import Mathlib.Analysis... |
7,822 | lean_file_chunk | 3d595186a6ac5ed63158229c6c291e0f21d0bc6dad5843ef4af4f0ee2535d13f | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2026 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
module
public import Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Instances
/-! # Interactions of the continuous functional calculus with the real and i... |
18,079 | lean_file_chunk | 0fd5b10b5c0e8e883ebe6a397851e5c959dfaaeb84a2bead3f4c950d4dd77301 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis, Anatole Dedecker
-/
module
public import Mathlib.Analysis.Normed.Algebra.Spectrum
public import Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital
... |
23,993 | lean_file_chunk | 172e2c1d6be6f59d735427d3a0361e93bf14d6b7aa4b8b29b862ff22ffbc5377 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
module
public import Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic
public import Mathlib.Analysis.CStarAlgebra.Unitization
public import Mathlib... |
473 | lean_file_chunk | 2ffe79480b3cc126baa6939e5efaf2ed9e60d2ca92931398b124d737df6b9b1a | lean_source_file | pkg_mathlib_source_file_chunks | rw [CFC.one_le_iff (R := ℝ) a] at ha
peel ha with x hx _
exact pow_le_pow_right₀ (ha x hx) hnm
lemma pow_antitone {a : A} (ha₀ : 0 ≤ a := by cfc_tac) (ha₁ : a ≤ 1) :
Antitone (a ^ · : ℕ → A) := by
intro n m hnm
simp only
rw [← cfc_pow_id (R := ℝ) a, ← cfc_pow_id (R := ℝ) a, cfc_le_iff ..]
rw [CFC.le_on... |
5,594 | lean_file_chunk | baf4a40071b9acdd70fdfa2e6341a1fb8aee70a8f5ce15f0cb9ad9bfc37bb5d2 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
module
public import Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique
public import Mathlib.Algebra.Algebra.Spectrum.Pi
public import Mathlib.Alge... |
18,344 | lean_file_chunk | c01c4fdddf580b7d3f7fa4ce9a20050a4fe59787ceeefba0fa0d3a35135de977 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
module
public import Mathlib.Topology.Algebra.Algebra
public import Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital
/-! # Restriction of the cont... |
8,098 | lean_file_chunk | 319abe497f3690d7beb0229b8c0623e879d726c6c7e0fe1bcf5eaf39d94133fc | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
module
public import Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Instances
/-! # Commuting with applications of the continuous functional calculus
Thi... |
23,212 | lean_file_chunk | 707274af4e10ea2133f35d9ad17fcb7931750e9cb6df3908a35a8600c8c149b9 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
module
public import Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric
public import Mathlib.Analysis.CStarAlgebra.GelfandDuality
public import Math... |
23,991 | lean_file_chunk | 6d77829a107ad85714bc13cd2e4eec4cfa4fe3e990f89231d7eca3a583c99c55 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
module
public import Mathlib.Algebra.Algebra.Spectrum.Quasispectrum
public import Mathlib.Algebra.Algebra.StrictPositivity
public import Mathlib.Tactic.ContinuousFunct... |
23,995 | lean_file_chunk | a5e61bf86de3ace6beda7f37ec66de757e983e617a856a1f8ee0b1ea7b695b26 | lean_source_file | pkg_mathlib_source_file_chunks | · rw [cfc_apply f a, cfc_apply g a, ← map_add, cfc_apply _ a]
congr
· simp [cfc_apply_of_not_predicate a ha]
lemma cfc_const_add (r : R) (f : R → R) (a : A)
(hf : ContinuousOn f (spectrum R a) := by cfc_cont_tac) (ha : p a := by cfc_tac) :
cfc (fun x => r + f x) a = algebraMap R A r + cfc f a := by
hav... |
3,493 | lean_file_chunk | f81862fa17fd0a191a74c169e26d1982aa02a10863d1f1381743ada1c1f39001 | lean_source_file | pkg_mathlib_source_file_chunks | algebraMap R A r ≤ a ↔ ∀ x ∈ spectrum R a, r ≤ x := by
nth_rw 1 [← cfc_id R a]
exact algebraMap_le_cfc_iff id r a
lemma cfc_le_one_iff (f : R → R) (a : A)
(hf : ContinuousOn f (spectrum R a) := by cfc_cont_tac) (ha : p a := by cfc_tac) :
cfc f a ≤ 1 ↔ ∀ x ∈ spectrum R a, f x ≤ 1 := by
simpa using cfc_le_... |
19,353 | lean_file_chunk | 89be3b501d0571aa9aae7a34d2b65b456921551c1ffe2be9737511136dfd92e7 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
module
public import Mathlib.Analysis.Complex.Basic
public import Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital
public import Mathlib.Topology.C... |
23,927 | lean_file_chunk | d46b4e3415205607e945b0890f7c387a632ba12df673d525237f0166e77ddc28 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
module
public import Mathlib.Algebra.Algebra.Spectrum.Quasispectrum
public import Mathlib.Topology.ContinuousMap.Compact
public import Mathlib.Analysis.CStarAlgebra.Co... |
15,826 | lean_file_chunk | d3fc9435d28eb74c2d32c44119daf13c76bc01e6981f6709c51a88a8395c8d97 | lean_source_file | pkg_mathlib_source_file_chunks | [NonUnitalContinuousFunctionalCalculus R A IsSelfAdjoint] {f : R → R} {a : A} :
IsSelfAdjoint (cfcₙ f a) :=
cfcₙ_predicate _ _
@[simp]
lemma cfcₙ_nonneg_of_predicate [LE A]
[NonUnitalContinuousFunctionalCalculus R A (0 ≤ ·)] {f : R → R} {a : A} :
0 ≤ cfcₙ f a :=
cfcₙ_predicate _ _
end CFCn
end Main
... |
23,991 | lean_file_chunk | 0a192bd761ef7f773cd816edcf626fa775943ee753305383c1ea489157ccd213 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
module
public import Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric
public import Mathlib.Topology.MetricSpace.UniformConvergence
public import M... |
23,945 | lean_file_chunk | 3782a576d2232513ecc42fd9c0bd3bf25dec51726f1a6a80539eb1c932c096fc | lean_source_file | pkg_mathlib_source_file_chunks | (ha : ∀ᶠ x in 𝓝 x₀, spectrum ℝ≥0 (a x) ⊆ s) (ha' : ∀ᶠ x in 𝓝 x₀, 0 ≤ a x)
(hf : ContinuousOn f s := by cfc_cont_tac) :
ContinuousAt (fun x ↦ cfc f (a x)) x₀ :=
ha_cont.tendsto.cfc_nnreal hs f ha ha' ha.self_of_nhds ha'.self_of_nhds
/-- If `f : ℝ≥0 → ℝ≥0` is continuous on a compact set `s` and `a : X → A` i... |
13,068 | lean_file_chunk | 7a02e9d498885839b9be840e1523ad0a280c6423a7685d816809421e2f47f43e | lean_source_file | pkg_mathlib_source_file_chunks | (ha_cont : Continuous a) (hs : ∀ x, IsCompact (s x))
(ha : ∀ x₀, ∀ᶠ x in 𝓝 x₀, quasispectrum 𝕜 (a x) ⊆ s x₀)
(hf : ∀ x, ContinuousOn f (s x) := by cfc_cont_tac) (hf0 : f 0 = 0 := by cfc_zero_tac)
(ha' : ∀ x, p (a x) := by cfc_tac) :
Continuous (fun x ↦ cfcₙ f (a x)) := by
rw [← continuousOn_univ] at... |
2,224 | lean_file_chunk | 9e9e07dfa004a27061aee99d18e9f95cb0e39f83aa6096bace06867be4624336 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
module
public import Mathlib.Tactic.Peel
public import Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital
public import Mathlib.Analysis.Complex.Basic
... |
10,917 | lean_file_chunk | fa28a187abf868d568e4239630eaa2be2500707c86b502003a949b2795085694 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
module
public import Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Instances
public import Mathlib.Topology.ContinuousMap.ContinuousSqrt
/-! # Range of t... |
1,795 | lean_file_chunk | 807031750cdc851b891d24e851595bbf61cc1cbadfafdccf2ea8c8c7cbbe77e3 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2026 Monica Omar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Monica Omar
-/
module
public import Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital
import Mathlib.FieldTheory.IsAlgClosed.Spectrum
/-! # Continuous functional calc... |
9,906 | lean_file_chunk | dfd701f2149147f4919b2c622a7010a611b5c7000af1ce40fd6a821d9bb0a937 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
module
public import Mathlib.Init
public import Batteries.Util.LibraryNote
/-!
# Documentation concerning the continuous functional calculus
A library note giving ad... |
7,046 | lean_file_chunk | 653e62569bda36806522642bde49cc911a12bc1fcb40e6c709aa8d6503408868 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Stefan Kebekus. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stefan Kebekus
-/
module
public import Mathlib.Analysis.Calculus.FDeriv.Symmetric
public import Mathlib.Analysis.Complex.Conformal
public import Mathlib.Analysis.Complex.HasPrimitives... |
2,177 | lean_file_chunk | 2f0e4cfb1ba3b8f90cea5c9a3ecf3e8f0830aeab363b4516c7dd2cd3a46cafd3 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2026 Stefan Kebekus. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stefan Kebekus
-/
module
public import Mathlib.Analysis.Complex.Liouville
public import Mathlib.Analysis.Complex.Harmonic.Analytic
/-!
# Liouville's Theorem for Harmonic Functions on... |
2,573 | lean_file_chunk | de3f29209fa230fbf8a2ee9f88e1f035780f5e3531918bff3584211766675101 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Stefan Kebekus. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stefan Kebekus
-/
module
public import Mathlib.Analysis.Complex.Harmonic.Analytic
public import Mathlib.Analysis.Complex.MeanValue
public import Mathlib.Analysis.InnerProductSpace.Har... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.