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/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Prime.Basic
import Mathlib.Data.Real.Archimedean
import Mathlib.NumberTheory.Zsqrtd.Basic
/-!
# Gaussian intege... | Mathlib/NumberTheory/Zsqrtd/GaussianInt.lean | 270 | 275 | |
/-
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
-/
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Logic.Equiv.Fin.Basic
imp... | (hf : HasFPowerSeriesAt f (0 : FormalMultilinearSeries 𝕜 E F) x) : ∀ᶠ z in 𝓝 x, f z = 0 :=
let ⟨_, hr⟩ := hf
hr.eventually_eq_zero
@[simp] lemma hasFPowerSeriesWithinOnBall_univ :
HasFPowerSeriesWithinOnBall f p univ x r ↔ HasFPowerSeriesOnBall f p x r := by
constructor
· intro h
refine ⟨h.r_le, ... | Mathlib/Analysis/Analytic/Basic.lean | 625 | 634 |
/-
Copyright (c) 2020 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky, Anthony DeRossi
-/
import Mathlib.Data.List.Basic
/-!
# Properties of `List.reduceOption`
In this file we prove basic lemmas about `List.reduceOption`.
-/
namespac... |
theorem reduceOption_eq_concat_iff (l : List (Option α)) (l' : List α) (a : α) :
l.reduceOption = l'.concat a ↔
∃ l₁ l₂, l = l₁ ++ some a :: l₂ ∧ l₁.reduceOption = l' ∧ l₂.reduceOption = [] := by
rw [concat_eq_append]
constructor
| Mathlib/Data/List/ReduceOption.lean | 80 | 85 |
/-
Copyright (c) 2024 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Basic
import Mathlib.Tactic.ComputeDegree
/-!
# Division polynomials of Weierst... | lemma coeff_preΨ₄_ne_zero (h : (2 : R) ≠ 0) : W.preΨ₄.coeff 6 ≠ 0 := by
rwa [coeff_preΨ₄]
| Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Degree.lean | 129 | 130 |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Semisimple.Defs
import Mathlib.Order.BooleanGenerators
/-!
# Semisimple Lie algebras
The famous Cartan-Dynkin-Killing classification of semisim... | rw [isLieAbelian_iff_center_eq_top R L, center_eq_bot] at h
exact (LieSubmodule.subsingleton_iff R L L).mp (subsingleton_of_bot_eq_top h)
theorem abelian_radical_of_hasTrivialRadical [HasTrivialRadical R L] :
| Mathlib/Algebra/Lie/Semisimple/Basic.lean | 302 | 305 |
/-
Copyright (c) 2024 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Basic
import Mathlib.Tactic.ComputeDegree
/-!
# Division polynomials of Weierst... | private lemma expDegree_cast {n : ℕ} (hn : n ≠ 0) :
2 * (expDegree n : ℤ) = n ^ 2 - if Even n then 4 else 1 := by
rcases n.even_or_odd' with ⟨n, rfl | rfl⟩
· rcases n with _ | n
· contradiction
push_cast [expDegree, show (2 * (n + 1)) ^ 2 = 2 * (2 * n * (n + 2)) + 4 by ring1, even_two_mul,
Nat.add... | Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Degree.lean | 153 | 162 |
/-
Copyright (c) 2014 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Jeremy Avigad
-/
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Algebra.Order.Ring.Canonical
/-!
# Distance function on ℕ
This file defines a simple dista... | Mathlib/Data/Nat/Dist.lean | 99 | 100 | |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.CategoryTheory.Category.GaloisConnection
import Mathlib.CategoryTheory.EqToHom
import Mathlib.Topology.Category.TopCat.EpiMono
import Mathlib.Topology.Sets... | @[simps obj_coe]
def IsOpenMap.functor {X Y : TopCat} {f : X ⟶ Y} (hf : IsOpenMap f) : Opens X ⥤ Opens Y where
obj U := ⟨f '' (U : Set X), hf (U : Set X) U.2⟩
| Mathlib/Topology/Category/TopCat/Opens.lean | 284 | 286 |
/-
Copyright (c) 2020 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap
import Mathlib.MeasureTheory.Integral.Bochner.FundThmCalculus
import Mathlib.MeasureT... | Mathlib/MeasureTheory/Integral/SetIntegral.lean | 121 | 124 | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Order.CauSeq.BigOperators
import Mathlib.Algebra.Order.Star.Basic
import Mathlib.Data.C... | Mathlib/Data/Complex/Exponential.lean | 1,414 | 1,423 | |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno
import Mathlib.Analysis.Calculus.FDeriv.Ad... | (hf.fderiv (contDiff_zero.mpr hg) hn).continuous
/-- `x ↦ fderiv 𝕜 (f x) (g x) (k x)` is smooth. -/
theorem ContDiff.fderiv_apply {f : E → F → G} {g k : E → F}
(hf : ContDiff 𝕜 m <| Function.uncurry f) (hg : ContDiff 𝕜 n g) (hk : ContDiff 𝕜 n k)
(hnm : n + 1 ≤ m) : ContDiff 𝕜 n fun x => fderiv 𝕜 (f x) ... | Mathlib/Analysis/Calculus/ContDiff/Basic.lean | 1,201 | 1,206 |
/-
Copyright (c) 2023 Joachim Breitner. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joachim Breitner
-/
import Mathlib.Probability.ProbabilityMassFunction.Basic
import Mathlib.Probability.ProbabilityMassFunction.Constructions
import Mathlib.MeasureTheory.Integral.Bo... | theorem integral_eq_sum [Fintype α] (p : PMF α) (f : α → E) :
∫ a, f a ∂(p.toMeasure) = ∑ a, (p a).toReal • f a := by
rw [integral_fintype _ .of_finite]
congr with x
rw [measureReal_def]
| Mathlib/Probability/ProbabilityMassFunction/Integrals.lean | 43 | 47 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Group.Action.End
import Mathlib.Algebra.Group.Action.Pointwise.Set.Basic
import Mathlib.Algebra.Group.Action.Prod
import Mathlib.Algebra.Group.Subg... | Mathlib/GroupTheory/GroupAction/Basic.lean | 441 | 459 | |
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Mathlib.Control.Basic
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Option.Basic
im... |
theorem head?_append_of_ne_nil :
| Mathlib/Data/List/Basic.lean | 439 | 440 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Powerset
import Mathlib.Algebra.NoZeroSMulDivisors.Pi
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Finty... |
open Finset in
/-- This expresses the difference between the values of a multilinear map
at two points "close to `x`" in terms of the "derivative" of the multilinear map at `x`
| Mathlib/LinearAlgebra/Multilinear/Basic.lean | 1,334 | 1,337 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.NaturalTransformation
import Mathlib.CategoryTheory.Monoidal.Opposite
import Mathli... | ((β_ (𝟙_ C) X).inv ▷ 𝟙_ C) ≫ (α_ _ X _).hom ≫ (_ ◁ (ρ_ X).hom) =
(X ◁ (ρ_ _).hom) ≫ (β_ (𝟙_ C) X).inv := by
| Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean | 295 | 296 |
/-
Copyright (c) 2024 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.HasPullbac... | Sigma.ι (functorObjSrcFamily f πY)
(FunctorObjIndex.mk i t' b' (by
have := τ.w
dsimp at this
| Mathlib/CategoryTheory/SmallObject/Construction.lean | 207 | 210 |
/-
Copyright (c) 2023 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.ModuleCat
import Mathlib.RepresentationTheory.GroupCohomology.Basic
import Mathlib.RepresentationTheory.... | simpa only [mul_one, add_left_inj, this]
lemma twoCocycles_ρ_map_inv_sub_map_inv (f : twoCocycles A) (g : G) :
| Mathlib/RepresentationTheory/GroupCohomology/LowDegree.lean | 345 | 347 |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Manuel Candales
-/
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
import Mat... | ∠ p₃ (midpoint ℝ p₁ p₂) p₂ = π / 2 := by
rw [midpoint_comm p₁ p₂, angle_left_midpoint_eq_pi_div_two_of_dist_eq h.symm]
/-- If the second of three points is strictly between the other two, the angle at that point
| Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean | 242 | 245 |
/-
Copyright (c) 2023 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Algebra.Group.Action.Pointwise.Finset
import Mathlib.GroupTheory.QuotientGroup.Defs
import Mathlib.Order.ConditionallyCompleteLattice.Basic
/-!
# Stabiliz... |
section Subgroup
-- TODO: Is there a lemma that could unify the following three very similar lemmas?
@[to_additive (attr := simp)]
lemma stabilizer_subgroup (s : Subgroup G) : stabilizer G (s : Set G) = s := by
| Mathlib/Algebra/Pointwise/Stabilizer.lean | 135 | 141 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl, Patrick Massot
-/
import Mathlib.Data.Set.Image
import Mathlib.Data.SProd
/-!
# Sets in product and pi types
This file proves basic properties of prod... | theorem piMap_mapsTo_pi {I : Set ι} {f : ∀ i, α i → β i} {s : ∀ i, Set (α i)} {t : ∀ i, Set (β i)}
(h : ∀ i ∈ I, MapsTo (f i) (s i) (t i)) :
MapsTo (Pi.map f) (I.pi s) (I.pi t) :=
| Mathlib/Data/Set/Prod.lean | 814 | 816 |
/-
Copyright (c) 2024 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Lie.Weights.Killing
import Mathlib.LinearAlgebra.RootSystem.Basic
import Mathlib.LinearAlgebra.RootSystem.Reduced
import Mathlib.LinearAlgebra.RootSy... | add_assoc, ← coe_chainTop, Nat.cast_smul_eq_nsmul]
exact rootSpace_neg_nsmul_add_chainTop_of_lt α β hα (Nat.lt_succ_self _)
lemma chainBotCoeff_add_chainTopCoeff :
chainBotCoeff α β + chainTopCoeff α β = chainLength α β := by
by_cases hα : α.IsZero
· rw [hα.eq, chainTopCoeff_zero, chainBotCoeff_zero, zer... | Mathlib/Algebra/Lie/Weights/RootSystem.lean | 142 | 162 |
/-
Copyright (c) 2017 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Kim Morrison, Mario Carneiro, Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.EpiMono
import Mathlib.Topology.Category.TopCat.Limits.Basic
import Mathlib.CategoryT... | · rw [continuousOn_iff_continuous_restrict]
convert_to Continuous (f ∘ h₁.isEmbedding.toHomeomorph.symm)
· ext ⟨x, hx⟩
exact dif_pos hx
continuity
· exact h₁.isOpen_range
| Mathlib/Topology/Category/TopCat/Limits/Products.lean | 287 | 292 |
/-
Copyright (c) 2021 Stuart Presnell. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stuart Presnell
-/
import Mathlib.Data.Nat.PrimeFin
import Mathlib.Data.Nat.Factorization.Defs
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Order.Interval.Finset.Nat
import... |
@[deprecated (since := "2024-10-24")]
alias ord_proj_mul_ord_compl_eq_self := ordProj_mul_ordCompl_eq_self
| Mathlib/Data/Nat/Factorization/Basic.lean | 133 | 135 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.Normed.Module.Basic
import Mathlib.Analysis.Seminorm
import Mathlib.Analysis.SpecificLimits.Basi... |
@[deprecated (since := "2025-04-27")] alias mapsTo_tangentCone_pi := mapsTo_tangentConeAt_pi
| Mathlib/Analysis/Calculus/TangentCone.lean | 239 | 241 |
/-
Copyright (c) 2020 Google LLC. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Wong
-/
import Mathlib.Data.List.Induction
/-!
# Palindromes
This module defines *palindromes*, lists which are equal to their reverse.
The main result is the `Palindrome` inducti... | rw [head_eq_of_cons_eq hr]
have : Palindrome l := hp (append_inj_left' (tail_eq_of_cons_eq hr) rfl)
exact Palindrome.cons_concat x this
theorem iff_reverse_eq {l : List α} : Palindrome l ↔ reverse l = l :=
Iff.intro reverse_eq of_reverse_eq
| Mathlib/Data/List/Palindrome.lean | 55 | 61 |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Yaël Dillies
-/
import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap
/-!
# Integral average of a function
In this file we define `MeasureTheory.average... | have mu_ai : μ (a i) < ∞ := by
rw [lt_top_iff_ne_top]
intro h
| Mathlib/MeasureTheory/Integral/Average.lean | 794 | 796 |
/-
Copyright (c) 2022 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.MeasureTheory.Integral.Prod
import Mathlib.Meas... | rwa [neg_sub, sub_add_cancel]
· have : t ∉ support fun t => L (f t) (g (x₀ - t)) := mt (fun h => h2s h) ht
rw [nmem_support.mp this, norm_zero]
refine Integrable.mono' ?_ ?_ this
· rw [integrable_indicator_iff hs]; exact ((hf.norm.const_mul _).mul_const _).integrableOn
· exact hf.aestronglyMeasura... | Mathlib/Analysis/Convolution.lean | 216 | 235 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Group.Equiv.Basic
import Mathlib.Data.ENat.Lattice
import Mathlib.Data.Part
import Mathlib.Tactic.NormNum
/-!
# Natural numbers with infinity
The... | theorem find_eq_top_iff : find P = ⊤ ↔ ∀ n, ¬P n :=
(eq_top_iff_forall_lt _).trans
⟨fun h n => (lt_find_iff P n).mp (h n) _ le_rfl, fun h n => lt_find P n fun _ _ => h _⟩
end Find
noncomputable instance : LinearOrderedAddCommMonoidWithTop PartENat :=
| Mathlib/Data/Nat/PartENat.lean | 730 | 736 |
/-
Copyright (c) 2022 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex J. Best, Xavier Roblot
-/
import Mathlib.Algebra.Algebra.Hom.Rat
import Mathlib.Analysis.Complex.Polynomial.Basic
import Mathlib.NumberTheory.NumberField.Norm
import Mathlib.RingTh... | obtain ⟨σ, rfl⟩ := exists_smul_eq_of_comap_eq hv
rwa [isUnramified_smul_iff] at H
lemma even_card_aut_of_not_isUnramifiedIn [IsGalois k K] [FiniteDimensional k K]
{w : InfinitePlace k} (hw : ¬ w.IsUnramifiedIn K) :
Even (Fintype.card <| K ≃ₐ[k] K) := by
obtain ⟨v, rfl⟩ := comap_surjective (K := K) w
rw... | Mathlib/NumberTheory/NumberField/Embeddings.lean | 980 | 999 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.GDelta.Basic
/-!
# Baire spaces
A topological space is called a *Baire space*
if a countable intersection of dense open subsets is den... | exact id
refine (hd.inter_of_Gδ hs (.iInter_of_isOpen fun i => (hgo i)) hgd).mono ?_
rintro x ⟨hxs, hxg⟩
rw [mem_iInter] at hxg
| Mathlib/Topology/Baire/Lemmas.lean | 123 | 126 |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.ModEq
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Algebra.Ring.Periodic
import Mathlib.Data.Int.SuccPred
import Mathlib.Order.Cir... | theorem modEq_iff_toIcoDiv_eq_toIocDiv_add_one :
a ≡ b [PMOD p] ↔ toIcoDiv hp a b = toIocDiv hp a b + 1 := by
rw [modEq_iff_toIcoMod_add_period_eq_toIocMod hp, toIcoMod, toIocMod, ← eq_sub_iff_add_eq,
| Mathlib/Algebra/Order/ToIntervalMod.lean | 561 | 563 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Batteries.Tactic.Congr
import Mathlib.Data.Option.Basic
import Mathlib.Data.Prod.Basic
import Mathlib.Data.Set.Subsingleton
import Mathlib.Dat... | @[simp]
theorem InjOn.image_nontrivial_iff (hf : s.InjOn f) :
(f '' s).Nontrivial ↔ s.Nontrivial :=
⟨nontrivial_of_image f s, fun h ↦ h.image_of_injOn hf⟩
/-- If the preimage of a set under an injective map is nontrivial, the set is nontrivial. -/
theorem nontrivial_of_preimage (hf : Function.Injective f) (s : S... | Mathlib/Data/Set/Image.lean | 1,055 | 1,061 |
/-
Copyright (c) 2020 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Algebra.Group.Conj
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.Group.Subgroup.Ker
/-!
# Basic results on subgroups
We prove basic results... | Mathlib/Algebra/Group/Subgroup/Basic.lean | 3,510 | 3,513 | |
/-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Bilinear
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Group.Pointwise.Finset.Basic
import Mathlib.Algebra.Group.Pointwise.Set.B... | variable (R A)
/-- R-submodules of the R-algebra A are a module over `Set A`. -/
| Mathlib/Algebra/Algebra/Operations.lean | 744 | 746 |
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Constructions.Polish.Basic
import Mathlib.MeasureTheory.Function.UniformIntegrable
import Mathlib.Probability.Martingale.Upcrossing
/-!
# Mar... | have heq : ∀ n, ∀ᵐ x ∂μ, (μ[μ[g|⨆ n, ℱ n]|ℱ n]) x = (μ[g|ℱ n]) x := fun n =>
condExp_condExp_of_le (le_iSup _ n) (iSup_le fun n => ℱ.le n)
rw [← ae_all_iff] at heq
filter_upwards [heq, ht] with x hxeq hxt
exact hxt.congr hxeq
@[deprecated (since := "2025-01-21")] alias tendsto_ae_condexp := tendsto_ae_cond... | Mathlib/Probability/Martingale/Convergence.lean | 440 | 448 |
/-
Copyright (c) 2019 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Mario Carneiro, Isabel Longbottom, Kim Morrison, Apurva Nakade, Yuyang Zhao
-/
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.SetTheory.PGame.Algebra
import Mathl... | (-(mk xl xr xL xR * mk yl yr yL yR)).moveRight (Sum.inr (i, j)) =
-(xR i * mk yl yr yL yR + mk xl xr xL xR * yR j - xR i * yR j) :=
rfl
theorem leftMoves_mul_cases {x y : PGame} (k) {P : (x * y).LeftMoves → Prop}
(hl : ∀ ix iy, P <| toLeftMovesMul (Sum.inl ⟨ix, iy⟩))
| Mathlib/SetTheory/Game/Basic.lean | 360 | 365 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Sébastien Gouëzel,
Rémy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics
/-!
# Continu... | (h : x ≠ 0 ∨ 0 ≤ p) : Tendsto (fun a => f a ^ p) l (𝓝 (x ^ p)) :=
if h0 : 0 = p then h0 ▸ by simp [tendsto_const_nhds]
else hf.rpow tendsto_const_nhds (h.imp id fun h' => h'.lt_of_ne h0)
variable [TopologicalSpace α] {f g : α → ℝ} {s : Set α} {x : α} {p : ℝ}
| Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean | 244 | 249 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Order.Module.Synonym
import Mathlib.LinearAlgebra.Aff... |
theorem right_lt_lineMap_iff_lt (h : r < 1) : b < lineMap a b r ↔ b < a :=
| Mathlib/LinearAlgebra/AffineSpace/Ordered.lean | 89 | 90 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.List.Sublists
import Mathlib.Data.List.Zip
import Mathlib.Data.Multiset.Bind
import Mathlib.Data.Multiset.Range
/-!
# The powerset of a multiset
... | theorem revzip_powersetAux_perm {l₁ l₂ : List α} (p : l₁ ~ l₂) :
revzip (powersetAux l₁) ~ revzip (powersetAux l₂) := by
haveI := Classical.decEq α
simp only [fun l : List α => revzip_powersetAux_lemma l revzip_powersetAux, coe_eq_coe.2 p]
exact (powersetAux_perm p).map _
/-! ### powersetCard -/
/-- Helper... | Mathlib/Data/Multiset/Powerset.lean | 140 | 151 |
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap
import Mathlib.Probability.Kernel.MeasurableLIntegral
/-!
# With Density
For an s-finite kernel `κ : K... | infer_instance
/-- Auxiliary lemma for `IsSFiniteKernel.withDensity`.
If a kernel `κ` is finite, then `withDensity κ f` is s-finite. -/
theorem isSFiniteKernel_withDensity_of_isFiniteKernel (κ : Kernel α β) [IsFiniteKernel κ]
(hf_ne_top : ∀ a b, f a b ≠ ∞) : IsSFiniteKernel (withDensity κ f) := by
-- We alre... | Mathlib/Probability/Kernel/WithDensity.lean | 191 | 203 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.Nat.SuccPred
import Mathlib.Order.SuccPred.Initial... | Mathlib/SetTheory/Ordinal/Arithmetic.lean | 2,288 | 2,294 | |
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Group.Center
import Mathlib.Algebra.GroupWithZero.Units.Basic
/-!
# Center of a group with zero
-/
assert_not_exists RelIso Finset R... | lemma center_units_eq : center G₀ˣ = ((↑) : G₀ˣ → G₀) ⁻¹' center G₀ :=
center_units_subset.antisymm subset_center_units
@[simp] lemma inv_mem_centralizer₀ (ha : a ∈ centralizer s) : a⁻¹ ∈ centralizer s := by
obtain rfl | ha₀ := eq_or_ne a 0
· rw [inv_zero]
exact zero_mem_centralizer
| Mathlib/Algebra/GroupWithZero/Center.lean | 42 | 48 |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Algebra.Hom
import Mathlib.Algebra.GroupWithZero.Action.Prod
/-!
# Morphisms of non-unital algebras
This file defines morphisms between two types, ... | { ‹AlgHomClass F R A B› with map_smulₛₗ := map_smul }
| Mathlib/Algebra/Algebra/NonUnitalHom.lean | 439 | 440 |
/-
Copyright (c) 2022 Mario Carneiro, Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Heather Macbeth, Yaël Dillies
-/
import Mathlib.Algebra.Order.Group.PosPart
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Data.Int.CharZero
im... | Mathlib/Tactic/Positivity/Basic.lean | 42 | 42 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Order.Preorder.Chain
import Mathlib.Tactic.Linter.DeprecatedModule
deprecated_module (since := "2025-04-13")
| Mathlib/Order/Chain.lean | 269 | 275 | |
/-
Copyright (c) 2019 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, François Dupuis
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Order.Filter.Extr
import Mathlib.Tactic.NormNum
/-!
# Convex and concave functions
This... | add_neg_lt_iff_lt_add] at h
exact h hx hy hxy ha hb hab
· rintro ⟨hconv, h⟩
refine ⟨hconv, fun x hx y hy hxy a b ha hb hab => ?_⟩
| Mathlib/Analysis/Convex/Function.lean | 805 | 808 |
/-
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.BigOperators.Expect
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Field.Canonical
import Mathlib.Algebra.O... | Mathlib/Data/NNReal/Basic.lean | 381 | 381 | |
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Kim Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.FiniteLimits
/-!
# Filtered categories
A category is filtered if every finite diagram admits a cocone.
We give a... | theorem of_isInitial {X : C} (h : IsInitial X) : IsCofiltered C :=
of_cone_nonempty.{v} _ fun {_} _ _ _ => ⟨⟨X, ⟨fun _ => h.to _, fun _ _ _ => h.hom_ext _ _⟩⟩⟩
instance (priority := 100) of_hasInitial [HasInitial C] : IsCofiltered C :=
of_isInitial _ initialIsInitial
/-- For every universe `w`, `C` is filtered if... | Mathlib/CategoryTheory/Filtered/Basic.lean | 806 | 822 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Convex.Between
import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathli... | · have Hle : volume ≤ (μH[Fintype.card ι] : Measure (ι → ℝ)) := by
refine le_hausdorffMeasure _ _ ∞ ENNReal.coe_lt_top fun s _ => ?_
rw [ENNReal.rpow_natCast]
exact Real.volume_pi_le_diam_pow s
rw [← volume_pi_pi fun i => Ioo (a i : ℝ) (b i)]
| Mathlib/MeasureTheory/Measure/Hausdorff.lean | 882 | 886 |
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Mario Carneiro, Simon Hudon
-/
import Mathlib.Data.PFunctor.Multivariate.Basic
import Mathlib.Data.PFunctor.Univariate.M
/-!
# The M construction as a multivariate polyn... | Mathlib/Data/PFunctor/Multivariate/M.lean | 317 | 324 | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker, Johan Commelin
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Div
import Mathlib.RingTheory... | ((-1) ^ p.natDegree * p.comp (-X)).leadingCoeff =
(p.comp (-X) * C ((-1) ^ p.natDegree)).leadingCoeff := by
simp [mul_comm]
| Mathlib/Algebra/Polynomial/RingDivision.lean | 178 | 180 |
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.Fintype.List
import Mathlib.Data.Fintype.OfMap
/-!
# Cycles of a list
Lists have an equivalence relation of whether they are rotational permut... |
theorem prev_reverse_eq_next (l : List α) (h : Nodup l) (x : α) (hx : x ∈ l) :
prev l.reverse x (mem_reverse.mpr hx) = next l x hx := by
obtain ⟨k, hk, rfl⟩ := getElem_of_mem hx
have lpos : 0 < l.length := k.zero_le.trans_lt hk
| Mathlib/Data/List/Cycle.lean | 357 | 361 |
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel, Alex Keizer
-/
import Mathlib.Algebra.Group.Nat.Even
import Mathlib.Algebra.NeZero
import Mathlib.Algebra.Ring.Nat
import Mathlib.Data.List.GetD
import Mathlib.Data.Nat.B... | induction n using Nat.binaryRec with | z => rfl | f b n hn => ?_
have : b = false := by simpa using h 0
| Mathlib/Data/Nat/Bitwise.lean | 168 | 169 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Group.Arithmetic
import Mathlib.Topology.GDelta.UniformSpace
import Mathlib.Topology.Instances.EReal.Lemmas
import Mathl... | instance NNReal.measurableSpace : MeasurableSpace ℝ≥0 :=
Subtype.instMeasurableSpace
instance NNReal.borelSpace : BorelSpace ℝ≥0 :=
Subtype.borelSpace _
instance ENNReal.measurableSpace : MeasurableSpace ℝ≥0∞ :=
borel ℝ≥0∞
| Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean | 677 | 685 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.Basic
/-!
# Functors which commute with shifts
Let `C` and `D` be two categories equipped with shifts by an additive monoid `A`. In this ... | simp only [← NatTrans.naturality, comp_obj, comp_map, map_comp, assoc]
lemma ofIso_compatibility :
letI := ofIso e A
NatTrans.CommShift e.hom A := by
letI := ofIso e A
refine ⟨fun a => ?_⟩
dsimp [commShiftIso, ofIso]
rw [← whiskerLeft_comp_assoc, e.hom_inv_id, whiskerLeft_id', id_comp]
end CommShi... | Mathlib/CategoryTheory/Shift/CommShift.lean | 380 | 419 |
/-
Copyright (c) 2020 Kevin Buzzard, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Equalizers
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Products
import Mathl... | def firstObj : A :=
∏ᶜ fun f : ΣV, { f : V ⟶ U // R f } => P.obj (op f.1)
/-- The left morphism of the fork diagram given in Equation (3) of [MM92], as well as the fork
diagram of the Stacks entry. -/
@[stacks 00VM "The left morphism the fork diagram there."]
def forkMap : P.obj (op U) ⟶ firstObj R P :=
Pi.lift fu... | Mathlib/CategoryTheory/Sites/Sheaf.lean | 596 | 611 |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Algebra.Order.Monoid.Unbundled.WithTop
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
/-! # Adjoi... | Mathlib/Algebra/Order/Monoid/WithTop.lean | 147 | 151 | |
/-
Copyright (c) 2020 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Data.Tree.Basic
import Mathlib.Logic.Basic
import Mathlib.Tactic.NormNum.Co... | theorem sub_subst {α} [Ring α] {n e1 e2 t1 t2 : α} (h1 : n * e1 = t1) (h2 : n * e2 = t2) :
n * (e1 - e2) = t1 - t2 := by simp [left_distrib, *, sub_eq_add_neg]
| Mathlib/Tactic/CancelDenoms/Core.lean | 55 | 56 |
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.Limits.Pullbacks
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
/-!
# Open immersions of structured spaces
We say that a m... | f.c.app (op (opensFunctor f |>.obj U)) ≫
X.presheaf.map
(eqToHom (by simp [Opens.map, Set.preimage_image_eq _ H.base_open.injective])) := by
rw [← cancel_epi (H.invApp _ U), IsIso.hom_inv_id]
delta invApp
simp [← Functor.map_comp]
@[simp, reassoc, elementwise]
theorem invApp_app (U : Open... | Mathlib/Geometry/RingedSpace/OpenImmersion.lean | 183 | 191 |
/-
Copyright (c) 2021 Hunter Monroe. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hunter Monroe, Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Dart
import Mathlib.Data.FunLike.Fintype
import Mathlib.Logic.Embedding.Set
/-!
# Maps between graphs
This file ... | protected def _root_.Equiv.simpleGraph (e : V ≃ W) : SimpleGraph V ≃ SimpleGraph W where
toFun := SimpleGraph.comap e.symm
| Mathlib/Combinatorics/SimpleGraph/Maps.lean | 154 | 155 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
-/
import Mathlib.MeasureTheory.Integral.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Lebesgue.Countable
import Mathlib.MeasureTheory.Integral.Le... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 586 | 595 | |
/-
Copyright (c) 2019 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Algebra.Order.Group.Pointwise.Interval
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.Rat.Cardinal
import Mathlib.SetTheory.Cardi... | refine add_lt_of_lt (cantor _).le ?_ h
refine add_lt_of_lt (cantor _).le (mk_image_le.trans_lt h) ?_
rw [mk_singleton]
| Mathlib/Data/Real/Cardinality.lean | 219 | 221 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Star.SelfAdjoint
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
/-!
# Quate... | inferInstanceAs (Nontrivial <| ℍ[R, -1, 0, -1])
| Mathlib/Algebra/Quaternion.lean | 756 | 757 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Data.Countable.Small
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Powerset
import Mathlib.Dat... | Mathlib/SetTheory/Cardinal/Basic.lean | 2,102 | 2,105 | |
/-
Copyright (c) 2020 Kexing Ying and Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Kevin Buzzard, Yury Kudryashov
-/
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.Algebra.Gro... | (h : mulSupport f ⊆ (s : Set α)) : ∏ᶠ i, f i = ∏ i ∈ s, f i :=
haveI h' : (s.finite_toSet.subset h).toFinset ⊆ s := by
simpa [← Finset.coe_subset, Set.coe_toFinset]
| Mathlib/Algebra/BigOperators/Finprod.lean | 361 | 363 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Ordmap.Invariants
/-!
# Verification of `Ordnode`
This file uses the invariants defined in `Mathlib.Data.Ordmap.Invariants` to construct `Ordset... | rw [insert'_eq_insertWith]; exact insertWith.valid _ _ (fun _ => id) h
theorem Valid'.map_aux {β} [Preorder β] {f : α → β} (f_strict_mono : StrictMono f) {t a₁ a₂}
(h : Valid' a₁ t a₂) :
Valid' (Option.map f a₁) (map f t) (Option.map f a₂) ∧ (map f t).size = t.size := by
| Mathlib/Data/Ordmap/Ordset.lean | 534 | 538 |
/-
Copyright (c) 2021 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.LinearAlgebra.Ray
import Mathlib.LinearAlgebra.Determinant
/-!
# Orientations of modules
This file defines orientations of modules.
## Main definitions
... | theorem orientation_eq_iff_det_pos (e₁ e₂ : Basis ι R M) :
e₁.orientation = e₂.orientation ↔ 0 < e₁.det e₂ :=
calc
e₁.orientation = e₂.orientation ↔ SameRay R e₁.det e₂.det := ray_eq_iff _ _
_ ↔ SameRay R (e₁.det e₂ • e₂.det) e₂.det := by rw [← e₁.det.eq_smul_basis_det e₂]
_ ↔ 0 < e₁.det e₂ := sameRay... | Mathlib/LinearAlgebra/Orientation.lean | 225 | 238 |
/-
Copyright (c) 2019 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard
-/
import Mathlib.Data.EReal.Basic
deprecated_module (since := "2025-04-13")
| Mathlib/Data/Real/EReal.lean | 1,658 | 1,665 | |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Data.Finset.NatAntidiagonal
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
/-!
# Big operators for `NatAntidiagonal`
This file contains theor... | = f (0, n + 1) * ∏ p ∈ antidiagonal n, f (p.1 + 1, p.2) := by
rw [antidiagonal_succ, prod_cons, prod_map]; rfl
theorem sum_antidiagonal_succ {n : ℕ} {f : ℕ × ℕ → N} :
| Mathlib/Algebra/BigOperators/NatAntidiagonal.lean | 23 | 26 |
/-
Copyright (c) 2023 Xavier Généreux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Généreux
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
import Mathlib.Analysis.Complex.PhragmenLindelof
/-!
# Hadamard three-lines Theorem
In this file we present a ... |
/-- An auxiliary function to prove the general statement of Hadamard's three lines theorem. -/
def scale (f : ℂ → E) (l u : ℝ) : ℂ → E := fun z ↦ f (l + z • (u - l))
/-- The transformation on ℂ that is used for `scale` maps the closed strip ``re ⁻¹' [l, u]``
to the closed strip ``re ⁻¹' [0, 1]``. -/
lemma scale_id_... | Mathlib/Analysis/Complex/Hadamard.lean | 310 | 316 |
/-
Copyright (c) 2021 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.Seminorm
import Mathlib.Analysis... | have hpx₂ : 0 < 2 * p x := mul_pos zero_lt_two hpx
refine hp.subset ⟨hpx₂, (2 * p x)⁻¹ • x, ?_, smul_inv_smul₀ hpx₂.ne' _⟩
rw [p.mem_ball_zero, map_smul_eq_mul, Real.norm_eq_abs, abs_of_pos (inv_pos.2 hpx₂),
inv_mul_lt_iff₀ hpx₂, mul_one]
| Mathlib/Analysis/Convex/Gauge.lean | 500 | 503 |
/-
Copyright (c) 2022 Alex J. Best. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex J. Best
-/
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.Algebra.EuclideanDomain.Int
import Mathlib.Data.ZMod.Basic
import Mathlib.RingTheory.Nilpotent.Lemmas
import Mathli... | (h : RingHom.ker f = RingHom.ker g) : f = g := by
have := f.liftOfRightInverse_comp _ (ZMod.ringHom_rightInverse f) ⟨g, le_of_eq h⟩
rw [Subtype.coe_mk] at this
rw [← this, RingHom.ext_zmod (f.liftOfRightInverse _ _ ⟨g, _⟩) _, RingHom.id_comp]
| Mathlib/RingTheory/ZMod.lean | 33 | 37 |
/-
Copyright (c) 2018 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton
-/
import Mathlib.Topology.Hom.ContinuousEval
import Mathlib.Topology.ContinuousMap.Basic
import Mathlib.Topology.Separation.Regular
/-!
# The compact-open topology
In this ... | end Curry
end CompactOpen
end ContinuousMap
| Mathlib/Topology/CompactOpen.lean | 405 | 410 |
/-
Copyright (c) 2019 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.Regular.Basic
import Mathlib.GroupTheory.MonoidLocalization.Basic
import Mathlib.LinearAlgebra.Matrix.MvPolynomial
import Mathlib.LinearAlgebra.Matri... | haveI : Subsingleton n := Fintype.card_le_one_iff_subsingleton.mp h.le
adjugate_subsingleton _
@[simp]
| Mathlib/LinearAlgebra/Matrix/Adjugate.lean | 288 | 291 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.CPolynomial
import Mathlib.Analysis.Analytic.Inverse
import Mathlib.Analysis.Analytic.Within
import Mathlib.Analysis.Calculus.Deriv... | rw [← h.fderiv_eq, add_sub_cancel]
simpa only [edist_eq_enorm_sub, EMetric.mem_ball] using hz
/-- If a function has a finite power series on a ball, then so does its derivative.
This is a variant of `HasFiniteFPowerSeriesOnBall.fderiv` where the degree of `f` is `< n`
and not `< n + 1`. -/
theorem HasFiniteFPowerS... | Mathlib/Analysis/Calculus/FDeriv/Analytic.lean | 490 | 496 |
/-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.RingTheory.AlgebraTower
import Mathli... | whenever `#R < #M`. -/
lemma rank_eq_mk_of_infinite_lt [Infinite R] (h_lt : lift.{v} #R < lift.{u} #M) :
Module.rank R M = #M := by
| Mathlib/LinearAlgebra/Dimension/Free.lean | 88 | 90 |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Neil Strickland
-/
import Mathlib.Data.Nat.Prime.Defs
import Mathlib.Data.PNat.Basic
/-!
# Primality and GCD on pnat
This file extends the theory of `ℕ+` with ... |
theorem Coprime.gcd_mul_right_cancel_right (m : ℕ+) {n k : ℕ+} :
k.Coprime m → m.gcd (n * k) = m.gcd n := by
rw [mul_comm]
| Mathlib/Data/PNat/Prime.lean | 198 | 201 |
/-
Copyright (c) 2019 Calle Sönne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology... |
theorem cos_pos_iff_abs_toReal_lt_pi_div_two {θ : Angle} : 0 < cos θ ↔ |θ.toReal| < π / 2 := by
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 618 | 619 |
/-
Copyright (c) 2022 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Algebra.NonUnitalHom
import Mathlib.Algebra.Star.Module
import Mathlib.Algebra.Star.NonUnitalSubalgebra
impor... | lemma starMap_id : starMap (NonUnitalStarAlgHom.id R B) = StarAlgHom.id R (Unitization R B) := by
ext; all_goals simp
| Mathlib/Algebra/Algebra/Unitization.lean | 804 | 806 |
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.FDeriv.Basic
/-!
# The derivative of a composition (chain rule)
For detailed documentation of the... | @[fun_prop]
theorem Differentiable.comp_differentiableOn {g : F → G} (hg : Differentiable 𝕜 g)
(hf : DifferentiableOn 𝕜 f s) : DifferentiableOn 𝕜 (g ∘ f) s :=
hg.differentiableOn.comp hf (mapsTo_univ _ _)
/-- The chain rule for derivatives in the sense of strict differentiability. -/
| Mathlib/Analysis/Calculus/FDeriv/Comp.lean | 191 | 196 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.BigOperators.Option
import Mathlib.Analysis.BoxIntegral.Box.Basic
import Mathlib.Data.Set.Pairwise.Lattice
/-!
# Partitions of rectangular b... | Mathlib/Analysis/BoxIntegral/Partition/Basic.lean | 773 | 774 | |
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral
import Mathlib.Analysis.Complex.CauchyIntegral
import Mathlib.MeasureTheory.Integral.Pi
impor... | · simp [PiLp.inner_apply, EuclideanSpace.measurableEquiv, Finset.mul_sum, mul_assoc]
simp_rw [mul_comm]
· simp only [EuclideanSpace.norm_eq, Real.norm_eq_abs, sq_abs, mul_pow, ← Finset.mul_sum]
congr
norm_cast
rw [sq_sqrt]
exact Finset.sum_nonneg (fun i _hi ↦ by positivity)
| Mathlib/Analysis/SpecialFunctions/Gaussian/FourierTransform.lean | 316 | 322 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.Measure.Comap
import Mathlib.MeasureTheory.Measure.QuasiMeasurePreserving
/-!
# Restricting a measure to a subset or a s... | theorem mem_map_indicator_ae_iff_of_zero_nmem [Zero β] {t : Set β} (ht : (0 : β) ∉ t) :
t ∈ Filter.map (s.indicator f) (ae μ) ↔ μ ((f ⁻¹' t)ᶜ ∪ sᶜ) = 0 := by
classical
| Mathlib/MeasureTheory/Measure/Restrict.lean | 916 | 918 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Yury Kudryashov, Yaël Dillies
-/
import Mathlib.Order.Synonym
/-!
# Minimal/maximal and bottom/top elements
This file defines predicates for elements to be minimal/maxi... | ⟨fun h =>
haveI := h
| Mathlib/Order/Max.lean | 144 | 145 |
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel, Alex Keizer
-/
import Mathlib.Algebra.Group.Nat.Even
import Mathlib.Algebra.NeZero
import Mathlib.Algebra.Ring.Nat
import Mathlib.Data.List.GetD
import Mathlib.Data.Nat.B... | · simp only [h, mod_two_eq_zero_iff_testBit_zero, testBit_zero, xor_mod_two_eq_one, decide_not,
| Mathlib/Data/Nat/Bitwise.lean | 351 | 351 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel
-/
import Mathlib.Analysis.Normed.Operator.Banach
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
import Mathlib.Topology.PartialHomeom... | · haveI : Subsingleton F := (Equiv.subsingleton_congr f'.toEquiv).1 hE
exact surjective_to_subsingleton _
· apply forall_of_forall_mem_closedBall (fun y : F => ∃ a, f a = y) (f 0) _
have hc' : (0 : ℝ) < N⁻¹ - c := by rw [sub_pos]; exact hc
let p : ℝ → Prop := fun R => closedBall (f 0) R ⊆ Set.range f
... | Mathlib/Analysis/Calculus/InverseFunctionTheorem/ApproximatesLinearOn.lean | 323 | 328 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Tape
import Mathlib.Data.Fintype.Option
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.PFun
import M... | `q` once we arrive
Because of the shuttling, emulation overhead is `O(n)`, where `n` is the current maximum of the
length of all stacks. Therefore a program that takes `k` steps to run in TM2 takes `O((m+k)k)`
steps to run when emulated in TM1, where `m` is the length of the input.
-/
namespace TM2to1
-- A displa... | Mathlib/Computability/TuringMachine.lean | 321 | 336 |
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes Hölzl, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Continuity
import Mathlib.Topology.Algebra.IsUniformGroup.Basic
import Mathlib.Topology.MetricSpace... | simp only [lipschitzWith_iff_dist_le_mul, dist_eq_norm_div]
alias ⟨LipschitzWith.norm_div_le, _⟩ := lipschitzWith_iff_norm_div_le
| Mathlib/Analysis/Normed/Group/Uniform.lean | 81 | 83 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Algebra.Group.TypeTags.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Finset.Piecewise
import Mathlib.Or... | Mathlib/Topology/Constructions.lean | 1,319 | 1,321 | |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.BoxIntegral.Box.SubboxInduction
import Mathlib.Analysis.BoxIntegral.Partition.Tagged
/-!
# Induction on subboxes
In this file we prove (se... | Mathlib/Analysis/BoxIntegral/Partition/SubboxInduction.lean | 56 | 56 | |
/-
Copyright (c) 2024 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Category.ModuleCat.Presheaf.Abelian
import Mathlib.Algebra.Category.ModuleCat.Presheaf.Sheafify
import Mathlib.Algebra.Category.ModuleCat.Presheaf.Limits... | (sheafificationAdjunction J AddCommGrp).homEquiv P.presheaf
((SheafOfModules.toSheaf _).obj F) ((SheafOfModules.toSheaf _).map f) := by
rw [toPresheaf_map_sheafificationHomEquiv_def, Adjunction.homEquiv_unit]
dsimp
lemma toSheaf_map_sheafificationHomEquiv_symm
{P : PresheafOfModules.{v} R₀} {F : ... | Mathlib/Algebra/Category/ModuleCat/Presheaf/Sheafification.lean | 93 | 102 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.List.Dedup
import Mathlib.Data.Multiset.UnionInter
/-!
# Erasing duplicates in a multiset.
-/
assert_not_exists Monoid
namespace Multiset
open... | Mathlib/Data/Multiset/Dedup.lean | 131 | 132 | |
/-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Applicative
import Mathlib.Control.Traversable.Basic
/-!
# Traversing collections
This file proves basic properties of traversable and applicative ... |
@[functor_norm]
| Mathlib/Control/Traversable/Lemmas.lean | 89 | 90 |
/-
Copyright (c) 2020 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Eric Wieser
-/
import Mathlib.Data.ENNReal.Holder
import Mathlib.MeasureTheory.Function.LpSeminorm.Basic
import Mathlib.MeasureTheory.Integral.MeanInequalities
import Mathl... | rw [hpqr]; simp [r, g]
theorem eLpNorm'_le_eLpNormEssSup_mul_rpow_measure_univ {q : ℝ} (hq_pos : 0 < q) :
eLpNorm' f q μ ≤ eLpNormEssSup f μ * μ Set.univ ^ (1 / q) := by
have h_le : (∫⁻ a : α, ‖f a‖ₑ ^ q ∂μ) ≤ ∫⁻ _ : α, eLpNormEssSup f μ ^ q ∂μ := by
refine lintegral_mono_ae ?_
have h_nnnorm_le_eLp... | Mathlib/MeasureTheory/Function/LpSeminorm/CompareExp.lean | 48 | 58 |
/-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston
-/
import Mathlib.Algebra.Group.Submonoid.Operations
import Mathlib.Data.Setoid.Basic
import Mathlib.GroupTheory.Congruence.Hom
/-!
# Congruence relations
This f... | Mathlib/GroupTheory/Congruence/Basic.lean | 453 | 454 | |
/-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Group.Commute.Defs
import Mathlib.Algebra.Group.Units.Defs
import Mathlib.Algebra.GroupWithZero.Defs
import Mathlib.Algebra.Order.Monoid.Unbundle... |
end Monoid
| Mathlib/Algebra/Regular/Basic.lean | 332 | 333 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.Polynomial.Basi... | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 760 | 774 | |
/-
Copyright (c) 2022 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Dynamics.FixedPoints.Prufer
import Mathlib.Dynamics.Ergodic.Ergodic
import Mathlib.MeasureTheory.Covering.DensityTheore... | theorem ergodic_zsmul_add (x : AddCircle T) {n : ℤ} (h : 1 < |n|) : Ergodic fun y => n • y + x := by
set f : AddCircle T → AddCircle T := fun y => n • y + x
| Mathlib/Dynamics/Ergodic/AddCircle.lean | 123 | 124 |
/-
Copyright (c) 2023 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Algebra.Order.ToIntervalMod
import Mathli... |
protected lemma GrowsPolynomially.zpow (p : ℤ) (hf : GrowsPolynomially f)
(hf_nonneg : ∀ᶠ x in atTop, 0 ≤ f x) : GrowsPolynomially fun x => (f x) ^ p := by
simp_rw [← rpow_intCast]
| Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean | 619 | 622 |
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