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/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Functor.ReflectsIso.Basic
import Mathlib.CategoryTheory.MorphismProperty.Basic
/-!
# Morphism properties that are inverted by a functor
In this ... | instance (W : MorphismProperty C) (D : Type*) [Category D] : Category (FunctorsInverting W D) :=
ObjectProperty.FullSubcategory.category _
@[ext]
lemma FunctorsInverting.hom_ext {W : MorphismProperty C} {F₁ F₂ : FunctorsInverting W D}
{α β : F₁ ⟶ F₂} (h : α.app = β.app) : α = β :=
NatTrans.ext h
| Mathlib/CategoryTheory/MorphismProperty/IsInvertedBy.lean | 103 | 109 |
/-
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 | 1,721 | 1,731 | |
/-
Copyright (c) 2023 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth, Adam Topaz
-/
import Mathlib.Algebra.Category.Ring.Colimits
import Mathlib.Algebra.Category.Ring.FilteredColimits
import Mathlib.Algebra.Category.Ring.Limits
import M... | ((smoothSheafCommRing IM I M R).presheaf.germ U x hx f)
= f ⟨x, hx⟩ :=
congr_arg (fun a ↦ a f) <| smoothSheafCommRing.ι_evalHom IM I M R x ⟨U, hx⟩
@[simp, reassoc, elementwise] lemma smoothSheafCommRing.forgetStalk_inv_comp_eval
(x : TopCat.of M) :
(smoothSheafCommRing.forgetStalk IM I M R x).inv ≫... | Mathlib/Geometry/Manifold/Sheaf/Smooth.lean | 343 | 353 |
/-
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.Algebra.BigOperators.Group.Finset.Piecewise
import Mathlib.Algebra.Group.Ext
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.BinaryProducts
import Ma... | /-- A preadditive category with finite coproducts has finite biproducts. -/
theorem HasFiniteBiproducts.of_hasFiniteCoproducts [HasFiniteCoproducts C] :
HasFiniteBiproducts C :=
⟨fun _ => { has_biproduct := fun _ => HasBiproduct.of_hasCoproduct _ }⟩
| Mathlib/CategoryTheory/Preadditive/Biproducts.lean | 192 | 196 |
/-
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.FieldTheory.SplittingField.Construction
import Mathlib.RingTheory.Localization.Integral
import Mathlib.RingTheory.IntegralClosure.IntegrallyClosed
impo... | or_iff_not_imp_left.mpr fun hₐ => ?_⟩,
fun hp =>
h.isPrimitive.irreducible_of_irreducible_map_of_injective (IsFractionRing.injective R K) hp⟩
obtain ⟨a', ha⟩ := eq_map_mul_C_of_dvd K h (dvd_of_mul_right_eq b H.symm)
obtain ⟨b', hb⟩ := eq_map_mul_C_of_dvd K h (dvd_of_mul_left_eq a H.symm)
hav... | Mathlib/RingTheory/Polynomial/GaussLemma.lean | 152 | 184 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Bhavik Mehta, Stuart Presnell
-/
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Order.Monotone.Defs
/-!
# Binomial coefficients
This file defines binomial coeffic... | (i + j).choose j * i ! * j ! = (i + j)! := by
rw [← choose_mul_factorial_mul_factorial (Nat.le_add_left _ _), Nat.add_sub_cancel_right,
Nat.mul_right_comm]
| Mathlib/Data/Nat/Choose/Basic.lean | 175 | 178 |
/-
Copyright (c) 2021 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.Algebra.Group.Defs
import Mathlib.Topology.Separation.Regular
/-!
# Idempotents in topological semigroups
This file provides a sufficient condition for a sem... | let M' := { m // m ∈ s }
letI : Semigroup M' :=
{ mul := fun p q => ⟨p.1 * q.1, s_add _ p.2 _ q.2⟩
mul_assoc := fun p q r => Subtype.eq (mul_assoc _ _ _) }
haveI : CompactSpace M' := isCompact_iff_compactSpace.mp s_compact
haveI : Nonempty M' := nonempty_subtype.mpr snemp
have : ∀ p : M', Continuous... | Mathlib/Topology/Algebra/Semigroup.lean | 82 | 95 |
/-
Copyright (c) 2018 Rohan Mitta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rohan Mitta, Kevin Buzzard, Alistair Tucker, Johannes Hölzl, Yury Kudryashov, Winston Yin
-/
import Mathlib.Algebra.Group.End
import Mathlib.Topology.EMetricSpace.Diam
/-!
# Lipschitz co... | (hg : LocallyLipschitz g) : LocallyLipschitz (f * g : Function.End α) := hf.comp hg
protected theorem pow_end {f : Function.End α} (h : LocallyLipschitz f) :
∀ n : ℕ, LocallyLipschitz (f ^ n : Function.End α)
| 0 => by simpa only [pow_zero] using LocallyLipschitz.id
| n + 1 => by
| Mathlib/Topology/EMetricSpace/Lipschitz.lean | 405 | 410 |
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Analysis.Normed.Group.Int
import Mathlib.Analysis.Normed.Group.Subgroup
import Mathlib.Analysis.Normed.Group.Uniform
/-!
# Normed groups homomorphisms... | ‖f‖ ≤ K :=
f.opNorm_le_bound K.2 fun x => by simpa only [dist_zero_right, map_zero] using hf.dist_le_mul x 0
| Mathlib/Analysis/Normed/Group/Hom.lean | 249 | 250 |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Set.Basic
import Mathlib.Tactic.Order
/-!
# Circular order hierarchy
This file defines circular preorders, circular partial orders and circular orde... | See note [reducible non-instances]. -/
abbrev PartialOrder.toCircularPartialOrder (α : Type*) [PartialOrder α] : CircularPartialOrder α :=
{ Preorder.toCircularPreorder α with
| Mathlib/Order/Circular.lean | 374 | 376 |
/-
Copyright (c) 2022 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.Polynomial.Mirror
import Mathlib.Algebra.Ring.Regular
import Mathlib.Data.Int.Order.Units
import Mathlib.RingTheory.Coprime.Basic
/-!
# Unit... | (h : IsRelPrime p p.mirror) : Irreducible p := by
refine irreducible_of_mirror hp.not_isUnit (fun q hpq => ?_) h
have hq : IsUnitTrinomial q := (isUnitTrinomial_iff'' hpq).mp hp
obtain ⟨k, m, n, hkm, hmn, u, v, w, hp⟩ := hp
obtain ⟨k', m', n', hkm', hmn', x, y, z, hq⟩ := hq
have hk : k = k' := by
rw [... | Mathlib/Algebra/Polynomial/UnitTrinomial.lean | 282 | 308 |
/-
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.AlgebraicGeometry.Cover.MorphismProperty
/-!
# Open covers of schemes
This file provides the basic API for open covers of schemes.
## Main definition
- `A... | instance [H : CompactSpace X] : Fintype 𝒰.finiteSubcover.J := by
delta OpenCover.finiteSubcover; infer_instance
| Mathlib/AlgebraicGeometry/Cover/Open.lean | 94 | 96 |
/-
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... | rcases h with ⟨p, hp⟩
rw [hp.fderiv_eq]
exact hp.hasStrictFDerivAt
| Mathlib/Analysis/Calculus/FDeriv/Analytic.lean | 139 | 142 |
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.RingTheory.DedekindDomain.Ideal
/-!
# The ideal class group
This file defines the ideal class group `ClassGroup R` of fractional ideals of `R`
inside its f... |
theorem mem_principal_ideals_iff {I : (FractionalIdeal R⁰ K)ˣ} :
I ∈ (toPrincipalIdeal R K).range ↔ ∃ x : K, spanSingleton R⁰ x = I := by
| Mathlib/RingTheory/ClassGroup.lean | 61 | 63 |
/-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.GCDMonoid.Multiset
import Mathlib.Algebra.GCDMonoid.Nat
import Mathlib.Algebra.Group.TypeTags.Finite
import Mathlib.Combinatorics.Enumerative... | theorem sign (h : IsThreeCycle σ) : sign σ = 1 := by
rw [Equiv.Perm.sign_of_cycleType, h.cycleType]
| Mathlib/GroupTheory/Perm/Cycle/Type.lean | 617 | 618 |
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel, Adam Topaz, Johan Commelin, Jakob von Raumer
-/
import Mathlib.Algebra.Homology.ImageToKernel
import Mathlib.Algebra.Homology.ShortComplex.Exact
import Mathlib.CategoryTh... | variable [L.PreservesZeroMorphisms]
variable (hL : ∀ (S : ShortComplex A), S.Exact → (S.map L).Exact)
include hL
open ZeroObject
| Mathlib/CategoryTheory/Abelian/Exact.lean | 223 | 228 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Independence.Kernel
import Mathlib.MeasureTheory.Constructions.Pi
/-!
# Independence of sets of sets and measure spaces (σ-algebras)
* A fami... |
@[deprecated (since := "2025-03-18")] alias IndepFun.ae_eq := IndepFun.congr
theorem IndepFun.comp {_mβ : MeasurableSpace β} {_mβ' : MeasurableSpace β'}
{_mγ : MeasurableSpace γ} {_mγ' : MeasurableSpace γ'} {φ : β → γ} {ψ : β' → γ'}
(hfg : IndepFun f g μ) (hφ : Measurable φ) (hψ : Measurable ψ) :
| Mathlib/Probability/Independence/Basic.lean | 654 | 659 |
/-
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 ... | interpStrip f z = sSupNormIm f 0 ^ (1 - z) * sSupNormIm f 1 ^ z := by
simp only [ne_of_gt h0, ne_of_gt h1, interpStrip, if_false, or_false]
/-- Rewrite for `InterpStrip` when `0 = sSupNormIm f 0` or `0 = sSupNormIm f 1`. -/
lemma interpStrip_eq_of_zero (z : ℂ) (h : sSupNormIm f 0 = 0 ∨ sSupNormIm f 1 = 0) :
... | Mathlib/Analysis/Complex/Hadamard.lean | 254 | 272 |
/-
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.LinearAlgebra.Quotient.Basic
import Mathlib.LinearAlgebra.Prod
/-!
# Projection to a subspace
In this file we define
* `Submodule.linearProjOfIsCom... | theorem ofIsCompl_eq' (h : IsCompl p q) {φ : p →ₗ[R] F} {ψ : q →ₗ[R] F} {χ : E →ₗ[R] F}
(hφ : φ = χ.comp p.subtype) (hψ : ψ = χ.comp q.subtype) : ofIsCompl h φ ψ = χ :=
ofIsCompl_eq h (fun _ => hφ.symm ▸ rfl) fun _ => hψ.symm ▸ rfl
@[simp]
| Mathlib/LinearAlgebra/Projection.lean | 242 | 246 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Sébastien Gouëzel, Patrick Massot
-/
import Mathlib.Topology.UniformSpace.Cauchy
import Mathlib.Topology.UniformSpace.Separation
import Mathlib.Topology.DenseEmbedding
... | have ho : IsOpen (UniformSpace.ball (e a) U) := UniformSpace.isOpen_ball (e a) hUo
refine mem_of_superset (ho.mem_nhds <| (UniformSpace.mem_ball_symmetry hsymm).2 ha) fun y hy => ?_
refine mem_closure_iff_nhds.2 fun V hV => ?_
rcases he₂.dense.mem_nhds (inter_mem hV (ho.mem_nhds hy)) with ⟨x, hxV, hxU⟩
exact ... | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | 258 | 264 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Anne Baanen
-/
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
import Mathlib.Algebra.Algebra.Tower
/-!
# Subalgebras in towers of algebras
In this file we prove facts about ... |
namespace IsScalarTower
open Subalgebra
variable [CommSemiring R] [CommSemiring S] [CommSemiring A]
variable [Algebra R S] [Algebra S A] [Algebra R A] [IsScalarTower R S A]
| Mathlib/Algebra/Algebra/Subalgebra/Tower.lean | 129 | 136 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.AEEqOfIntegral
import Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable
/-!
# Uniqueness of the conditional expect... | rw [IntegrableOn, integrable_congr (ae_restrict_of_ae (Lp.coeFn_sub f g))]
exact (hf_int_finite s hs hμs).sub (hg_int_finite s hs hμs)
exact Lp.ae_eq_zero_of_forall_setIntegral_eq_zero' 𝕜 hm (f - g) hp_ne_zero hp_ne_top hfg_int hfg'
<| (hf_meas.sub hg_meas).congr (Lp.coeFn_sub f g).symm
variable {𝕜}
t... | Mathlib/MeasureTheory/Function/ConditionalExpectation/Unique.lean | 101 | 122 |
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.Group.Action.Pointwise.Set.Basic
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Order.BigOperators.Group.List
import Mathli... | Mathlib/Algebra/Group/Submonoid/Pointwise.lean | 682 | 684 | |
/-
Copyright (c) 2016 Leonardo de Moura. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Mathlib.Control.Basic
import Mathlib.Data.Set.Defs
import Mathlib.Data.Set.Lattice.Image
import Mathlib.Data.Set.Notation
/-!
# Functoriality of `Set`
... | ext; simp
| Mathlib/Data/Set/Functor.lean | 128 | 129 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.Lattice
import Mathlib.RingTheory.Coprime.Lemmas
import Mathlib... | Mathlib/RingTheory/Ideal/Operations.lean | 739 | 739 | |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.CategoryTheory.Comma.Over.Pullback
import Mathlib.CategoryTheory.Limits.Shapes.KernelPair
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq
import ... | @[reassoc (attr := simp)]
theorem diagonalObjPullbackFstIso_hom_snd {X Y Z : C} (f : X ⟶ Z) (g : Y ⟶ Z) :
(diagonalObjPullbackFstIso f g).hom ≫ pullback.snd _ _ =
pullback.fst _ _ ≫ pullback.fst _ _ := by
| Mathlib/CategoryTheory/Limits/Shapes/Diagonal.lean | 323 | 326 |
/-
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.DominatedConvergence
import Mathlib.Probability.Kernel.MeasurableLIntegral
/-!
# Measurability of the integral against a kernel
Th... | simpa using this
end MeasureTheory
| Mathlib/Probability/Kernel/MeasurableIntegral.lean | 151 | 181 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Floris van Doorn
-/
import Mathlib.CategoryTheory.Limits.Filtered
import Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts
import Mathlib.CategoryTheory.Limits.Shapes.Ker... | _ ≅ (walkingCospanOpEquiv.functor ⋙ walkingCospanOpEquiv.inverse) ⋙ (cospan f g).op :=
(isoWhiskerRight walkingCospanOpEquiv.unitIso _)
_ ≅ walkingCospanOpEquiv.functor ⋙ walkingCospanOpEquiv.inverse ⋙ (cospan f g).op :=
(Functor.associator _ _ _)
_ ≅ walkingCospanOpEquiv.functor ⋙ span f.op g.o... | Mathlib/CategoryTheory/Limits/Opposites.lean | 845 | 850 |
/-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.Orientation
import Mathlib.Data.Complex.FiniteDimensional
import Mathlib.Da... | Mathlib/Analysis/InnerProductSpace/TwoDim.lean | 579 | 583 | |
/-
Copyright (c) 2024 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Kernel.Disintegration.Density
import Mathlib.Probability.Kernel.WithDensity
/-!
# Radon-Nikodym derivative and Lebesgue decomposition for kern... | refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩
· rw [h, add_zero]
exact withDensity_absolutelyContinuous _ _
rw [Measure.AbsolutelyContinuous.add_left_iff] at h
exact Measure.eq_zero_of_absolutelyContinuous_of_mutuallySingular h.2
(mutuallySingular_singularPart _ _ _)
| Mathlib/Probability/Kernel/RadonNikodym.lean | 416 | 422 |
/-
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.Nat.ModEq
import Mathlib.Data.Nat.Prime.Basic
import Mathlib.NumberTheory.Zsqrtd.Basic
/-!
# Pell's equation and Matiyasevic's theorem
This file... | theorem n_lt_xn (n) : n < xn a1 n :=
lt_of_lt_of_le (Nat.lt_pow_self a1) (xn_ge_a_pow a1 n)
theorem x_pos (n) : 0 < xn a1 n :=
lt_of_le_of_lt (Nat.zero_le n) (n_lt_xn a1 n)
| Mathlib/NumberTheory/PellMatiyasevic.lean | 227 | 231 |
/-
Copyright (c) 2024 Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Merten
-/
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
import Mathlib.CategoryTheory.Limits.FintypeCat
import Mathlib.CategoryTheory.Lim... | rfl
lemma mulAction_naturality {X Y : C} (σ : Aut F) (f : X ⟶ Y) (x : F.obj X) :
σ • F.map f x = F.map f (σ • x) :=
FunctorToFintypeCat.naturality F F σ.hom f x
| Mathlib/CategoryTheory/Galois/Basic.lean | 192 | 197 |
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Simon Hudon
-/
import Mathlib.Control.Functor.Multivariate
import Mathlib.Data.PFunctor.Univariate.Basic
/-!
# Multivariate polynomial functors.
Multivariate polynomial... | /-- Split polynomial functor, get an n-ary functor
from an `n+1`-ary functor -/
def drop : MvPFunctor n where
A := P.A
B a := (P.B a).drop
/-- Split polynomial functor, get a univariate functor
from an `n+1`-ary functor -/
def last : PFunctor where
A := P.A
B a := (P.B a).last
| Mathlib/Data/PFunctor/Multivariate/Basic.lean | 206 | 217 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
import Mathlib.Analysis.Normed.Module.Basic
import Mathlib.MeasureTheory.Function.SimpleFuncDense
/-!
# Strongly measurable and finitely strongly meas... | refine ⟨fun n => -hf.approx n, fun n => ?_, fun x => (hf.tendsto_approx x).neg⟩
suffices μ (Function.support fun x => -(hf.approx n) x) < ∞ by convert this
rw [Function.support_neg (hf.approx n)]
exact hf.fin_support_approx n
@[measurability]
protected theorem sub [SubtractionMonoid β] [ContinuousSub β] (hf : ... | Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean | 1,086 | 1,092 |
/-
Copyright (c) 2021 Eric Rodriguez. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Rodriguez
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.Tactic.ByContra
import Mathlib.Topology.Algebra.Polynomial
import Mathlib.NumberTheory.Padics.Pad... | simpa only [map_cyclotomic, Int.cast_one, h, eq_intCast] using this
· exfalso
linarith [cyclotomic_nonneg n (le_refl (1 : ℤ))]
rw [← Int.natAbs_eq_natAbs_iff, Int.natAbs_one, Nat.eq_one_iff_not_exists_prime_dvd]
intro p hp hpe
haveI := Fact.mk hp
have := prod_cyclotomic_eq_geom_sum hn' ℤ
apply_fun e... | Mathlib/RingTheory/Polynomial/Cyclotomic/Eval.lean | 139 | 170 |
/-
Copyright (c) 2022 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Group.Embedding
import Mathlib.Algebra.MonoidAlgebra.Defs
import Mathlib.LinearAlgebra.Finsupp.Supported
import Mathlib.Algebra.Group.Pointwise.F... | end Span
end AddMonoidAlgebra
| Mathlib/Algebra/MonoidAlgebra/Support.lean | 136 | 138 |
/-
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 | 703 | 704 | |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.MkIffOfInductiveProp
/-!
# Additional lemmas about sum types
Most of the former contents ... | @[simp]
theorem update_inl_apply_inl [DecidableEq α] [DecidableEq (α ⊕ β)] {f : α ⊕ β → γ} {i j : α}
| Mathlib/Data/Sum/Basic.lean | 81 | 82 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Morenikeji Neri
-/
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.Algebra.EuclideanDomain.Field
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.RingTheor... | Mathlib/RingTheory/PrincipalIdealDomain.lean | 532 | 542 | |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Casper Putz, Anne Baanen
-/
import Mathlib.Algebra.Order.Star.Basic
import Mathlib.Data.Matrix.RowCol
/-!
# Dot product of two vectors
This file conta... | protected alias Matrix.dotProduct_star_self_eq_zero := dotProduct_star_self_eq_zero
/-- Note that this applies to `ℂ` via `RCLike.toStarOrderedRing`. -/
@[simp]
theorem dotProduct_self_star_eq_zero {v : n → R} : dotProduct v (star v) = 0 ↔ v = 0 :=
(Fintype.sum_eq_zero_iff_of_nonneg fun _ => mul_star_self_nonneg _).... | Mathlib/LinearAlgebra/Matrix/DotProduct.lean | 130 | 135 |
/-
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... | @[simp]
lemma image_fst_graphOn (f : α → β) (s : Set α) : Prod.fst '' graphOn f s = s := by
simp [graphOn, image_image]
@[simp] lemma image_snd_graphOn (f : α → β) : Prod.snd '' s.graphOn f = f '' s := by ext x; simp
lemma fst_injOn_graph : (s.graphOn f).InjOn Prod.fst := by aesop (add simp InjOn)
lemma graphOn_co... | Mathlib/Data/Set/Prod.lean | 962 | 971 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.InnerProductSpace.Defs
impor... | Mathlib/Analysis/InnerProductSpace/Basic.lean | 1,211 | 1,220 | |
/-
Copyright (c) 2021 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Floris van Doorn
-/
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Analysis.Normed.Module.Convex
import Mathlib.Analysis.NormedSpace.Connected
import Mathli... | /-- Pre-images of ample sets under continuous affine equivalences are ample. -/
theorem preimage {s : Set F} (h : AmpleSet s) (L : E ≃ᴬ[ℝ] F) : AmpleSet (L ⁻¹' s) := by
rw [← L.image_symm_eq_preimage]
| Mathlib/Analysis/Convex/AmpleSet.lean | 94 | 96 |
/-
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, Benjamin Davidson
-/
import Mathlib.Algebra.Field.NegOnePow
import Mathlib.Algebra.Field.Periodic
import Mathlib.Algebra.Qua... | π / 2 ≤ 2 := pi_div_two_le_two
_ = 4 / 2 := by norm_num)
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | 147 | 149 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.GroupWithZero.NeZero
import Mathlib.Logic.Unique
import Mathlib.Tactic.Conv
/-!
# Groups with an adjoined z... | Mathlib/Algebra/GroupWithZero/Basic.lean | 464 | 467 | |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
import Mathlib.MeasureTheory.Measure.Count
import Mathlib.Order.Filter.ENNReal
import Mathlib.Probability.Unif... | theorem essSup_measure_zero {m : MeasurableSpace α} {f : α → β} : essSup f (0 : Measure α) = ⊥ :=
le_bot_iff.mp (sInf_le (by simp [Set.mem_setOf_eq, EventuallyLE, ae_iff]))
| Mathlib/MeasureTheory/Function/EssSup.lean | 162 | 163 |
/-
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, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan
import Mathlib.Analysis.SpecialFunctions.... | (hasDerivAt_arctan x).differentiableAt
theorem differentiable_arctan : Differentiable ℝ arctan :=
differentiableAt_arctan
| Mathlib/Analysis/SpecialFunctions/Trigonometric/ArctanDeriv.lean | 82 | 85 |
/-
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, Sander Dahmen,
Kim Morrison, Chris Hughes, Anne Baanen, Junyan Xu
-/
import Mathlib.LinearAlgebra.Basis.VectorSpace
import Mathlib.LinearAlgebra.Dimensi... | Mathlib/LinearAlgebra/Dimension/DivisionRing.lean | 338 | 340 | |
/-
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, Simon Hudon, Mario Carneiro
-/
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Init
import Mathlib.Data.Int.Init
import Mathlib.... | Mathlib/Algebra/Group/Basic.lean | 814 | 814 | |
/-
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.Group.Unbundled.Basic
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Or... | @[to_additive]
instance (priority := 100) LinearOrderedCommGroup.to_noMinOrder [Nontrivial α] : NoMinOrder α :=
| Mathlib/Algebra/Order/Group/Defs.lean | 132 | 133 |
/-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGrou... | Mathlib/NumberTheory/ModularForms/SlashActions.lean | 200 | 208 | |
/-
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
-/
import Mathlib.Algebra.Order.SuccPred
import Mathlib.Data.Sum.Order
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Tactic.PPWithUniv
/-!
# ... |
In this paragraph, we introduce the addition on ordinals, and prove just enough properties to
| Mathlib/SetTheory/Ordinal/Basic.lean | 755 | 756 |
/-
Copyright (c) 2020 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Yury Kudryashov
-/
import Mathlib.Topology.Instances.NNReal.Lemmas
import Mathlib.Topology.Order.MonotoneContinuity
/-!
# Square root of a real numbe... | rw [division_def, sqrt_mul' x (inv_nonneg.2 hy), sqrt_inv, division_def]
| Mathlib/Data/Real/Sqrt.lean | 319 | 320 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kim Morrison, Adam Topaz
-/
import Mathlib.Topology.Sheaves.SheafOfFunctions
import Mathlib.Topology.Sheaves.Stalks
import Mathlib.Topology.Sheaves.SheafCondition.UniqueG... | simp only [stalkToFiber, Types.Colimit.ι_desc_apply'] at h
specialize w (unop U) (unop V) fU hU fV hV h
rcases w with ⟨W, iU, iV, w⟩
-- and put it back together again in the correct order.
refine ⟨op W, fun w ↦ fU (iU w : (unop U).1), P.res ?_ _ hU, ?_⟩
· rcases W with ⟨W, m⟩
exact iU
· exact ⟨colimit... | Mathlib/Topology/Sheaves/LocalPredicate.lean | 272 | 299 |
/-
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.Localization.Opposite
/-!
# Calculus of fractions
Following the definitions by [Gabriel and Zisman][gabriel-zisman-1967],
given a morphism prope... | /-- The equivalence class of `z₁.comp₀ z₂ z₃` does not depend on the choice of `z₃` provided
they satisfy the compatibility `z₂.f ≫ z₃.s = z₁.s ≫ z₃.f`. -/
lemma comp₀_rel [W.HasLeftCalculusOfFractions]
{X Y Z : C} (z₁ : W.LeftFraction X Y) (z₂ : W.LeftFraction Y Z)
(z₃ z₃' : W.LeftFraction z₁.Y' z₂.Y') (h₃ : z... | Mathlib/CategoryTheory/Localization/CalculusOfFractions.lean | 286 | 302 |
/-
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.Polynomial.Reverse
import Mathlib.Algebra.Regular.SMul
/-!
# Theory of monic polynomials
We give se... | variable [Semiring R] {p : R[X]}
| Mathlib/Algebra/Polynomial/Monic.lean | 482 | 483 |
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.FieldTheory.RatFunc.AsPolynomial
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Poly... | rw [intDegree, num_zero, natDegree_zero, denom_zero, natDegree_one, sub_self]
| Mathlib/FieldTheory/RatFunc/Degree.lean | 44 | 45 |
/-
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... | -/
theorem tendsto_integral_smul_of_tendsto_average_norm_sub
[CompleteSpace E]
{ι : Type*} {a : ι → Set α} {l : Filter ι} {f : α → E} {c : E} {g : ι → α → ℝ} (K : ℝ)
(hf : Tendsto (fun i ↦ ⨍ y in a i, ‖f y - c‖ ∂μ) l (𝓝 0))
(f_int : ∀ᶠ i in l, IntegrableOn f (a i) μ)
| Mathlib/MeasureTheory/Integral/Average.lean | 760 | 765 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Heather Macbeth
-/
import Mathlib.MeasureTheory.Function.SimpleFunc
import Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable
/-!
# Density of si... | exists_seq_strictAnti_tendsto (0 : ℝ)
have : ∀ n, ∃ (g : SimpleFunc (X × Y) α), ∀ x, dist (f x) (g x) < u n :=
fun n ↦ h'f.exists_simpleFunc_approx_of_prod hf (u_pos n)
choose g hg using this
have A : ∀ x, Tendsto (fun n ↦ g n x) atTop (𝓝 (f x)) := by
intro x
rw [tendsto_iff_dist_tendsto_zero]
... | Mathlib/MeasureTheory/Function/SimpleFuncDense.lean | 245 | 265 |
/-
Copyright (c) 2023 Ali Ramsey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ali Ramsey, Eric Wieser
-/
import Mathlib.Algebra.Algebra.Bilinear
import Mathlib.LinearAlgebra.DFinsupp
import Mathlib.LinearAlgebra.Prod
import Mathlib.LinearAlgebra.TensorProduct.Finite... | {a₁ : (i : repr.ι) → Repr R (repr.left i)} {a₂ : (i : repr.ι) → Repr R (repr.right i)} :
| Mathlib/RingTheory/Coalgebra/Basic.lean | 153 | 153 |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.Real
/-!
# Properties of pointwise scalar multiplication of se... | ball a ε - closedBall b δ = ball (a - b) (ε + δ) := by
simp_rw [sub_eq_add_neg, neg_closedBall, ball_add_closedBall hε hδ]
| Mathlib/Analysis/NormedSpace/Pointwise.lean | 334 | 336 |
/-
Copyright (c) 2024 Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Merten
-/
import Mathlib.CategoryTheory.Galois.GaloisObjects
import Mathlib.CategoryTheory.Limits.Shapes.CombinedProducts
import Mathlib.Data.Finite.Sum
/-!
# Decompositio... | which has at index `x : F.obj X` the element `g x`. -/
private noncomputable def mkSelfProdFib : F.obj (selfProd F X) :=
(PreservesProduct.iso F _).inv ((Concrete.productEquiv (fun _ : F.obj X ↦ F.obj X)).symm id)
@[simp]
private lemma mkSelfProdFib_map_π (t : F.obj X) : F.map (Pi.π _ t) (mkSelfProdFib F X) = t := b... | Mathlib/CategoryTheory/Galois/Decomposition.lean | 202 | 207 |
/-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Algebra.Ring.Int.Defs
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.Cast.Order.Basic
import Math... | Mathlib/Data/Num/Lemmas.lean | 1,758 | 1,759 | |
/-
Copyright (c) 2016 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Mathlib.Init
import Batteries.Util.ExtendedBinder
import Lean.Elab.Term
/-!
# Sets
This file sets up the theory of sets whose elements have a given... | else
throw ()
| _ => throw ()
| Mathlib/Data/Set/Defs.lean | 182 | 185 |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov, Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
import Mathlib.MeasureTheory.Measure.Typeclasses.Probability
import... | f.mono.leftLim_le, measure_union A measurableSet_Ioc, f.mono hab]
· simp only [hab, measure_empty, Icc_eq_empty, not_le]
symm
simp [ENNReal.ofReal_eq_zero, f.mono.le_leftLim hab]
@[simp]
theorem measure_Ioo {a b : ℝ} : f.measure (Ioo a b) = ofReal (leftLim f b - f a) := by
rcases le_or_lt b a with (h... | Mathlib/MeasureTheory/Measure/Stieltjes.lean | 400 | 423 |
/-
Copyright (c) 2023 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang, Junyan Xu
-/
import Mathlib.Algebra.Category.ModuleCat.Basic
import Mathlib.Algebra.Category.Grp.Injective
import Mathlib.Topology.Instances.AddCircle
import Mathlib.Linea... | (LinearEquiv.eq_symm_apply _).mp <| Subtype.ext (one_zsmul _).symm
/--
For an abelian group `A` and an element `a ∈ A`, there is a character `c : ℤ ∙ a → ℚ ⧸ ℤ` given by
`m • a ↦ m / n` where `n` is the smallest positive integer such that `n • a = 0` and when such `n`
does not exist, `c` is defined by `m • a ↦ m / 2... | Mathlib/Algebra/Module/CharacterModule.lean | 179 | 193 |
/-
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.Algebra.Homology.HomotopyCategory.HomComplex
import Mathlib.Algebra.Homology.HomotopyCofiber
/-! # The mapping cone of a morphism of cochain complexes
In this ... | (inr φ).f p ≫ (desc φ α β eq).f p = β.f p := by
simp [desc]
@[reassoc (attr := simp)]
| Mathlib/Algebra/Homology/HomotopyCategory/MappingCone.lean | 380 | 383 |
/-
Copyright (c) 2020 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.Topology.Path
/-!
# Path connectedness
Continuing from `Mathlib.Topology.Path`, this file defines path components and path-connected
spaces.
## Main... | Mathlib/Topology/Connected/PathConnected.lean | 1,322 | 1,326 | |
/-
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.Data.Bundle
import Mathlib.Data.Set.Image
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.PartialHomeomorph
import Mathlib.Topology.O... | e.toPretrivialization.symm_apply_mk_proj ex
| Mathlib/Topology/FiberBundle/Trivialization.lean | 359 | 359 |
/-
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.BigOperators
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib... | · simp [hp0]
exact WithBot.coe_le_coe.1 (le_trans (card_roots hp0) (le_of_eq <| degree_eq_natDegree hp0))
theorem card_roots_sub_C {p : R[X]} {a : R} (hp0 : 0 < degree p) :
| Mathlib/Algebra/Polynomial/Roots.lean | 76 | 79 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Johan Commelin, Patrick Massot
-/
import Mathlib.Algebra.Order.Hom.Monoid
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.Tac... | Finset.induction_on s (fun _ => v.map_zero ▸ zero_le')
(fun a s has ih hf => ?_) hf
rw [Finset.forall_mem_insert] at hf; rw [Finset.sum_insert has]
exact v.map_add_le hf.1 (ih hf.2)
| Mathlib/RingTheory/Valuation/Basic.lean | 174 | 177 |
/-
Copyright (c) 2014 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Group.Prod
/-!
# The product of two `AddMonoidWithOne`s.
-/
assert_not_exists MonoidWithZero
variable {α β : Type*}
namespace Prod
variabl... | @[simp]
| Mathlib/Data/Nat/Cast/Prod.lean | 29 | 29 |
/-
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.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Module.BigOperators
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.Nat.Squarefree
imp... | ⟨fun f g => ⟨fun n => f n + g n, by simp⟩⟩
@[simp]
theorem add_apply {f g : ArithmeticFunction R} {n : ℕ} : (f + g) n = f n + g n :=
| Mathlib/NumberTheory/ArithmeticFunction.lean | 202 | 205 |
/-
Copyright (c) 2023 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.CartanSubalgebra
import Mathlib.Algebra.Lie.Weights.Basic
/-!
# Weights and roots of Lie modules and Lie algebras with respect to Cartan subalge... |
end LieAlgebra
| Mathlib/Algebra/Lie/Weights/Cartan.lean | 298 | 314 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.InnerProductSpace.Defs
impor... | Mathlib/Analysis/InnerProductSpace/Basic.lean | 1,128 | 1,131 | |
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Tactic.FinCases
import Mathlib.Topology.Connected.LocallyConnected
import Mathlib.Topology.Sets.Closeds
/-!
# L... |
@[to_additive]
theorem div [Div Y] ⦃f g : X → Y⦄ (hf : IsLocallyConstant f) (hg : IsLocallyConstant g) :
IsLocallyConstant (f / g) :=
hf.comp₂ hg (· / ·)
| Mathlib/Topology/LocallyConstant/Basic.lean | 170 | 174 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Johan Commelin
-/
import Mathlib.Algebra.Algebra.RestrictScalars
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
import Mathlib.Algebra.Module.Rat
import Mathlib.GroupThe... |
variable (R M) in
@[simp]
| Mathlib/RingTheory/TensorProduct/Basic.lean | 101 | 103 |
/-
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.Geometry.Manifold.Algebra.Structures
import Mathlib.Geometry.Manifold.BumpFunction
import Mathlib.Topology.MetricSpace.PartitionOfUnity
import Mathli... | Mathlib/Geometry/Manifold/PartitionOfUnity.lean | 824 | 831 | |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Callum Sutton, Yury Kudryashov
-/
import Mathlib.Algebra.Group.Equiv.Defs
import Mathlib.Algebra.Group.Hom.Basic
import Mathlib.Logic.Equiv.Basic
/-!
# Multiplicative ... | Mathlib/Algebra/Group/Equiv/Basic.lean | 571 | 573 | |
/-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin, Lu-Ming Zhang
-/
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Algebra.Pi
import Mathlib.Algebra.BigOp... |
variable {n α R}
| Mathlib/Data/Matrix/Basic.lean | 132 | 134 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Yakov Pechersky, Eric Wieser
-/
import Mathlib.Data.List.Basic
/-!
# Properties of `List.enum`
## Deprecation note
Many lemmas in this file have been replaced by the... | Mathlib/Data/List/Enum.lean | 89 | 90 | |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Yakov Pechersky, Eric Wieser
-/
import Mathlib.Data.List.Basic
/-!
# Properties of `List.enum`
## Deprecation note
Many lemmas in this file have been replaced by the... | Mathlib/Data/List/Enum.lean | 155 | 157 | |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Batteries.Tactic.Init
import Mathlib.Logic.Function.Defs
/-!
# Binary map of options
This file defines the binary map of `Option`. This is mostly useful to defin... | /-!
The following symmetric restatement are needed because unification has a hard time figuring all the
functions if you symmetrize on the spot. This is also how the other n-ary APIs do it.
-/
| Mathlib/Data/Option/NAry.lean | 134 | 137 |
/-
Copyright (c) 2024 Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Merten
-/
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
import Mathlib.CategoryTheory.Limits.FintypeCat
import Mathlib.CategoryTheory.Lim... |
open ConcreteCategory
/-- A mono between objects with equally sized fibers is an iso. -/
lemma isIso_of_mono_of_eq_card_fiber {X Y : C} (f : X ⟶ Y) [Mono f]
(h : Nat.card (F.obj X) = Nat.card (F.obj Y)) : IsIso f := by
| Mathlib/CategoryTheory/Galois/Basic.lean | 345 | 350 |
/-
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... | have A : Tendsto (fun p : ℝ × ℝ => exp (log p.1 * p.2)) (𝓝[≠] 0 ×ˢ 𝓝 y) (𝓝 0) :=
tendsto_exp_atBot.comp
((tendsto_log_nhdsNE_zero.comp tendsto_fst).atBot_mul_pos hp tendsto_snd)
have B : Tendsto (fun p : ℝ × ℝ => p.1 ^ p.2) (𝓝[≠] 0 ×ˢ 𝓝 y) (𝓝 0) :=
squeeze_zero_norm (fun p => abs_rpow_le_exp_log... | Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean | 202 | 217 |
/-
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... | /-- In a measure space, one can restrict the measure to a subtype to get a new measure space.
Not registered as an instance, as there are other natural choices such as the normalized restriction
for a probability measure, or the subspace measure when restricting to a vector subspace. Enable
| Mathlib/MeasureTheory/Measure/Restrict.lean | 743 | 745 |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.AlgebraicGeometry.Cover.Open
import Mathlib.AlgebraicGeometry.GammaSpecAdjunction
import Mathlib.AlgebraicGeometry.Restrict
import Mathlib.CategoryTheory.Lim... | · rintro ⟨_, ⟨_, ⟨x, rfl⟩, rfl⟩, rfl⟩
refine ⟨_, ⟨x, rfl⟩, ?_⟩
exact congr_arg Opens.carrier (Scheme.toSpecΓ_preimage_basicOpen _ _).symm
/-- The canonical map `U ⟶ Spec Γ(X, U)` for an open `U ⊆ X`. -/
noncomputable
| Mathlib/AlgebraicGeometry/AffineScheme.lean | 284 | 289 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yury Kudryashov
-/
import Mathlib.Algebra.Group.Commute.Defs
import Mathlib.Algebra.Opposites
import Mathlib.Tactic.Spread
/-!
# Definitions of group actions
This file de... | Mathlib/Algebra/Group/Action/Defs.lean | 609 | 620 | |
/-
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.Constructions.SumProd
/-!
# Maps equivariantly-homeomorphic to projection in a product
This file contains the definition `IsHomeomorph... | /-- The projection from a trivial fiber bundle to its base is open. -/
protected theorem isOpenMap_proj (h : IsHomeomorphicTrivialFiberBundle F proj) :
IsOpenMap proj := by
| Mathlib/Topology/FiberBundle/IsHomeomorphicTrivialBundle.lean | 50 | 52 |
/-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth, Yury Kudryashov, Frédéric Dupuis
-/
import Mathlib.Topology.Algebra.InfiniteSum.Constructions
import Mathlib.Topology.Algebra.Module.Equiv
/-! # Infinite sums in top... | use a
change ∑' a : α, g (π (a +ᵥ b)) • f (a +ᵥ b) = g (π b) • ∑' a : α, f (a +ᵥ b)
simp_rw [H₁]
exact tsum_const_smul'' _
attribute [to_additive existing MulAction.automorphize_smul_left] AddAction.automorphize_smul_left
section
variable {G : Type*} [Group G] {Γ : Subgroup G}
/-- Given a subgroup `Γ` of ... | Mathlib/Topology/Algebra/InfiniteSum/Module.lean | 233 | 250 |
/-
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.OuterMeasure.Operations
import Mathlib.Analysis.SpecificLimits.Basic
/-!
# Outer measures from functions
Given an arbit... | Mathlib/MeasureTheory/OuterMeasure/OfFunction.lean | 472 | 476 | |
/-
Copyright (c) 2024 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.MvPolynomial.Monad
import Mathlib.LinearAlgebra.Charpoly.ToMatrix
import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition
import Mathlib.Li... | rw [polyCharpoly, polyCharpolyAux, (charpoly.univ_monic _ _).natDegree_map,
charpoly.univ_natDegree, finrank_eq_card_chooseBasisIndex]
lemma polyCharpoly_coeff_isHomogeneous (i j : ℕ) (hij : i + j = finrank R M) [Nontrivial R] :
((polyCharpoly φ b).coeff i).IsHomogeneous j := by
| Mathlib/Algebra/Module/LinearMap/Polynomial.lean | 378 | 382 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.InnerProductSpace.Defs
impor... | (hr : 0 < r) : ⟪x, r • x⟫_ℝ / (‖x‖ * ‖r • x‖) = 1 := by
rw [real_inner_smul_self_right, norm_smul, Real.norm_eq_abs, ← mul_assoc ‖x‖, mul_comm _ |r|,
| Mathlib/Analysis/InnerProductSpace/Basic.lean | 651 | 652 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Group.Units.Basic
import Mathlib.Algebra.GroupWithZero.Basic
import Mathlib.Data.Int.Basic
import Mathlib.Lean.Meta.CongrTheorems
import Mathli... | conv_lhs => rw [← mul_left_inj' hb, div_mul_cancel₀ _ hb]
conv_rhs => rw [← mul_left_inj' hc, div_mul_cancel₀ _ hc]
| Mathlib/Algebra/GroupWithZero/Units/Basic.lean | 431 | 432 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Notation.Pi
import Mathlib.Data.Set.Lattice
import Mathlib.Order.Filter.Defs
/-!
# Theory of... | Mathlib/Order/Filter/Basic.lean | 1,946 | 1,947 | |
/-
Copyright (c) 2022 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.Set.Basic
import Mathlib.Algebra.Group.Pointwise.Set.Lattice
import Mathlib.Algebra.Order.Group.Defs
import Mathlib.Algebra.... | @[to_additive]
instance : Mul (UpperSet α) :=
⟨fun s t ↦ ⟨image2 (· * ·) s t, s.2.mul_right⟩⟩
| Mathlib/Algebra/Order/UpperLower.lean | 97 | 99 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kenny Lau
-/
import Mathlib.Data.List.Forall2
/-!
# zip & unzip
This file provides results about `List.zipWith`, `List.zip` and `List.unzip` (definitions are in
core ... | | [], _ => rfl
| _ :: _, [] => rfl
| _ :: as, _ :: bs => congr_arg (cons _) <| zipWith3_same_right f as bs
instance (f : α → α → β) [IsSymmOp f] : IsSymmOp (zipWith f) :=
⟨fun _ _ => zipWith_comm_of_comm IsSymmOp.symm_op⟩
| Mathlib/Data/List/Zip.lean | 91 | 96 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.Bochner.Basic
import Mathlib.MeasureTheory.Integral.Bochner.L1
import Mathlib.Me... | Mathlib/MeasureTheory/Integral/Bochner.lean | 181 | 182 | |
/-
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, Kevin Buzzard, Yury Kudryashov, Eric Wieser
-/
import Mathlib.Algebra.Algebra.Prod
import Mathlib.Algebra.Group.Graph
import Mathlib.LinearAlgebra.Span.... | · intro h
| Mathlib/LinearAlgebra/Prod.lean | 585 | 585 |
/-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Bhavik Mehta, Yaël Dillies
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Analysis.Convex.Hull
import Mathlib.Analysis.Normed.Module.Basic
import Mathlib.Topology.Bornology.A... |
variable [NontriviallyNormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {s : Set E}
variable [Module ℝ E] [SMulCommClass ℝ 𝕜 E]
| Mathlib/Analysis/LocallyConvex/Basic.lean | 266 | 269 |
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