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/-
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 | 723 | 726 | |
/-
Copyright (c) 2023 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calculus.Deriv.Add
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calcul... |
theorem lineDifferentiableWithinAt_smul_iff {c : 𝕜} (hc : c ≠ 0) :
LineDifferentiableWithinAt 𝕜 f s x (c • v) ↔ LineDifferentiableWithinAt 𝕜 f s x v :=
| Mathlib/Analysis/Calculus/LineDeriv/Basic.lean | 529 | 531 |
/-
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.Complex
/-!
# The `arctan` function.
Inequalit... | theorem tan_int_mul_pi_div_two (n : ℤ) : tan (n * π / 2) = 0 :=
tan_eq_zero_iff.mpr (by use n)
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Arctan.lean | 47 | 49 |
/-
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.Real
/-!
# Power function... | theorem rpow_ne_top_of_nonneg {x : ℝ≥0∞} {y : ℝ} (hy0 : 0 ≤ y) (h : x ≠ ⊤) : x ^ y ≠ ⊤ :=
mt (ENNReal.rpow_eq_top_of_nonneg x hy0) h
| Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | 564 | 565 |
/-
Copyright (c) 2020 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Polynomial.Degree.TrailingDegree
import Mathlib.Algebra.Polynomial.EraseLead
/-!
# Reverse of a univariate polynomial
The main definition is `r... | reverse (p + C t) = reverse p + C t * X ^ p.natDegree := by
simp [reverse]
@[simp] lemma reverse_C_add (p : R[X]) (t : R) :
| Mathlib/Algebra/Polynomial/Reverse.lean | 329 | 332 |
/-
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.Analysis.Normed.Group.Quotient
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Topology.Instances.AddCircle
/-!
# The additive circle as a norm... | rw [round_eq_zero_iff]
obtain ⟨hx₁, hx₂⟩ := abs_le.mp hx
replace hx₂ := Ne.lt_of_le hx' hx₂
constructor
· rwa [← mul_le_mul_left hp, ← mul_assoc, mul_inv_cancel₀ hp.ne.symm, one_mul, mul_neg, ←
| Mathlib/Analysis/Normed/Group/AddCircle.lean | 120 | 124 |
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.BorelSpace.Metric
import Mathlib.MeasureTheory.Constructions.BorelSpace.Real
import Mathlib.Topology.Metrizable.Real
im... | /-- If the indicator functions of a.e.-measurable sets `Aᵢ` converge a.e. to the indicator function
of a set `A` along a nontrivial countably generated filter, then `A` is also a.e.-measurable. -/
lemma nullMeasurableSet_of_tendsto_indicator [NeBot L] {μ : Measure α}
(As_mble : ∀ i, NullMeasurableSet (As i) μ)
... | Mathlib/MeasureTheory/Constructions/BorelSpace/Metrizable.lean | 146 | 153 |
/-
Copyright (c) 2020 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.Geometry.Manifold.MFDeriv.Defs
import Mathlib.Geometry.Manifold.ContMDiff.Defs
/-!
# Basic properties of the manifold Fréchet ... | ContinuousWithinAt f s x :=
and_iff_left_of_imp <| (continuousAt_extChartAt _).comp_continuousWithinAt
simp_rw [cont, DifferentiableWithinAtProp, extChartAt, PartialHomeomorph.extend,
PartialEquiv.coe_trans,
ModelWithCorners.toPartialEquiv_coe, PartialHomeomorph.coe_coe, modelWithCornersSelf_coe... | Mathlib/Geometry/Manifold/MFDeriv/Basic.lean | 218 | 223 |
/-
Copyright (c) 2024 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Cardinal.Arithmetic
import Mathlib.SetTheory.Ordinal.Principal
/-!
# Ordinal arithmetic with cardinals
This file co... | Mathlib/SetTheory/Cardinal/Ordinal.lean | 262 | 265 | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Joey van Langen, Casper Putz
-/
import Mathlib.Algebra.CharP.Algebra
import Mathlib.Algebra.CharP.Reduced
import Mathlib.Algebra.Field.ZMod
import Mathlib.Data.Nat.Prime.In... | zero_mul, zero_sub]
variable {K}
theorem frobenius_pow {p : ℕ} [Fact p.Prime] [CharP K p] {n : ℕ} (hcard : q = p ^ n) :
frobenius K p ^ n = 1 := by
| Mathlib/FieldTheory/Finite/Basic.lean | 442 | 447 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Joey van Langen, Casper Putz
-/
import Mathlib.Algebra.CharP.Algebra
import Mathlib.Algebra.CharP.Reduced
import Mathlib.Algebra.Field.ZMod
import Mathlib.Data.Nat.Prime.In... | (∑ x : Kˣ, (x ^ i : K)) = if q - 1 ∣ i then -1 else 0 := by
let φ : Kˣ →* K :=
{ toFun := fun x => x ^ i
map_one' := by simp
map_mul' := by intros; simp [mul_pow] }
have : Decidable (φ = 1) := by classical infer_instance
calc (∑ x : Kˣ, φ x) = if φ = 1 then Fintype.card Kˣ else 0 := sum_hom_un... | Mathlib/FieldTheory/Finite/Basic.lean | 280 | 297 |
/-
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.QuadraticDiscriminant
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
/-!
... | theorem cos_eq_iff_quadratic {z w : ℂ} :
cos z = w ↔ exp (z * I) ^ 2 - 2 * w * exp (z * I) + 1 = 0 := by
rw [← sub_eq_zero]
field_simp [cos, exp_neg, exp_ne_zero]
refine Eq.congr ?_ rfl
ring
theorem cos_surjective : Function.Surjective cos := by
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Complex.lean | 166 | 173 |
/-
Copyright (c) 2023 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang, Fangming Li
-/
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Data.Fintype.Pigeonhole
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.Fintype.Sigma
import Mathlib.... | /--
Given a series `a₀ -r→ a₁ -r→ ... -r→ aₙ` and an `a` such that `a₀ -r→ a` holds, there is
a series of length `n+1`: `a -r→ a₀ -r→ a₁ -r→ ... -r→ aₙ`.
| Mathlib/Order/RelSeries.lean | 410 | 412 |
/-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
import Mathlib.Topology.Category.Profinite.Nobeling.Basic
import Mathlib.Topology.Category.Profinite.Nobeling.Induction
import Mathlib.Topology.Category.Profinite... | Mathlib/Topology/Category/Profinite/Nobeling.lean | 557 | 570 | |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Moritz Doll
-/
import Mathlib.LinearAlgebra.Prod
/-!
# Partially defined linear maps
A `LinearPMap R E F` or `E →ₗ.[R] F` is a linear map from a submodule of `E` to... | simp [h]
rcases h with ⟨⟨x', hx'⟩, ⟨h1, h2⟩⟩
| Mathlib/LinearAlgebra/LinearPMap.lean | 793 | 794 |
/-
Copyright (c) 2022 Praneeth Kolichala. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Praneeth Kolichala
-/
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Nat.BinaryRec
import Mathlib.Data.List.Defs
import Mathlib.Tactic.Convert
import Mathlib.Tactic.GeneralizePr... | Mathlib/Data/Nat/Bits.lean | 444 | 448 | |
/-
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.TangentCone
import Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics
import Mathlib.Analysis.As... | HasFDerivWithinAt f (0 : E →L[𝕜] F) s x₀ := by
simp_rw [HasFDerivWithinAt, hasFDerivAtFilter_iff_isLittleO,
h.eq_zero_of_norm_pow_within hx₀ hn.ne_bot, zero_apply, sub_zero,
h.trans_isLittleO ((isLittleO_pow_sub_sub x₀ hn).mono nhdsWithin_le_nhds)]
theorem Asymptotics.IsBigO.hasFDerivAt {x₀ : E} {n : ℕ}... | Mathlib/Analysis/Calculus/FDeriv/Basic.lean | 701 | 707 |
/-
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.ChangeOfRings
/-!
# Pushforward of presheaves of modules
If `F : C ⥤ D`, the precomposition `F.op ⋙ _` induces a functor fr... | /-- The pushforward of presheaves of modules commutes with the forgetful functor
to presheaves of abelian groups. -/
noncomputable def pushforwardCompToPresheaf :
pushforward.{v} φ ⋙ toPresheaf _ ≅ toPresheaf _ ⋙ (whiskeringLeft _ _ _).obj F.op :=
Iso.refl _
| Mathlib/Algebra/Category/ModuleCat/Presheaf/Pushforward.lean | 72 | 76 |
/-
Copyright (c) 2018 Michael Jendrusch. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Jendrusch, Kim Morrison, Bhavik Mehta, Jakob von Raumer
-/
import Mathlib.CategoryTheory.EqToHom
import Mathlib.CategoryTheory.Functor.Trifunctor
import Mathlib.CategoryTheo... | 𝟙_ C ◁ f = (λ_ X).hom ≫ f ≫ (λ_ Y).inv := by
rw [← assoc, ← leftUnitor_naturality]; simp [id_tensorHom]
| Mathlib/CategoryTheory/Monoidal/Category.lean | 235 | 237 |
/-
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... |
/-- **First moment method**. The maximum of a measurable function is greater than its integral,
while avoiding a null set. -/
theorem exists_not_mem_null_lintegral_le (hint : ∫⁻ a, f a ∂μ ≠ ∞) (hN : μ N = 0) :
| Mathlib/MeasureTheory/Integral/Average.lean | 746 | 749 |
/-
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.Data.Set.Constructions
import Mathlib.Order.Filter.AtTopBot.CountablyGenerated
import Mathlib.Topology.Constructions
import Mathlib.Top... | sUnion_eq := sUnion_eq_univ_iff.2 fun x ↦ (h_nhds x).ex_mem
eq_generateFrom := ext_nhds fun x ↦ by
simpa only [nhds_generateFrom, and_comm] using (h_nhds x).eq_biInf
/-- If a family of open sets `s` is such that every open neighbourhood contains some
member of `s`, then `s` is a topological basis. -/
theorem i... | Mathlib/Topology/Bases.lean | 108 | 119 |
/-
Copyright (c) 2018 Guy Leroy. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sangwoo Jo (aka Jason), Guy Leroy, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.GroupWithZero.Semiconj
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Data.Nat.GCD.Bas... | Mathlib/Data/Int/GCD.lean | 454 | 459 | |
/-
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.Partition.Filter
import Mathlib.Analysis.BoxIntegral.Partition.Measure
import Mathlib.Analysis.Oscillation
import Mathlib.Data.B... |
/-!
# Henstock-Sacks inequality and integrability on subboxes
| Mathlib/Analysis/BoxIntegral/Basic.lean | 358 | 360 |
/-
Copyright (c) 2023 Antoine Chambert-Loir. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir
-/
import Mathlib.Algebra.Exact
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.RingTheory.Ideal.Quotient.Defs
import Mathlib.RingTheory.TensorProduct... | rw [← comap_coe_toLinearMap, ← toLinearMap_eq_coe, comap_equiv_eq_map_symm, toLinearMap_eq_coe,
map_coe_toLinearMap, map_symm_eq_iff, map_range_rTensor_subtype_lid]
variable {M' N' P' : Type*}
[AddCommGroup M'] [AddCommGroup N'] [AddCommGroup P']
[Module R M'] [Module R N'] [Module R P']
{f' : M' →ₗ[... | Mathlib/LinearAlgebra/TensorProduct/RightExactness.lean | 406 | 418 |
/-
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.Algebra.Group.Embedding
import Mathlib.Order.Interval.Multiset
/-!
# Finite intervals of naturals
This file proves that `ℕ` is a `LocallyFiniteOrder` and... | Mathlib/Order/Interval/Finset/Nat.lean | 333 | 339 | |
/-
Copyright (c) 2018 Louis Carlin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Louis Carlin, Mario Carneiro
-/
import Mathlib.Algebra.EuclideanDomain.Defs
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Regular
import Mathlib.Algebra.Grou... | theorem lcm_zero_left (x : R) : lcm 0 x = 0 := by rw [lcm, zero_mul, zero_div]
@[simp]
theorem lcm_zero_right (x : R) : lcm x 0 = 0 := by rw [lcm, mul_zero, zero_div]
@[simp]
theorem lcm_eq_zero_iff {x y : R} : lcm x y = 0 ↔ x = 0 ∨ y = 0 := by
constructor
· intro hxy
rw [lcm, mul_div_assoc _ (gcd_dvd_right _... | Mathlib/Algebra/EuclideanDomain/Basic.lean | 268 | 288 |
/-
Copyright (c) 2023 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Data.Int.Order.Units
import Mathlib.Data.ZMod.IntUnitsPower
import Mathlib.RingTheory.TensorProduct.Basic
import Mathlib.LinearAlgebra.DirectSum.TensorProduc... |
theorem gradedMul_one (x : (⨁ i, 𝒜 i) ⊗[R] (⨁ i, ℬ i)) :
gradedMul R 𝒜 ℬ x 1 = x := by
-- Note: https://github.com/leanprover-community/mathlib4/pull/8386 had to specialize `map_one` to avoid timeouts.
simpa only [RingHom.map_one, one_smul] using gradedMul_algebraMap 𝒜 ℬ x 1
theorem gradedMul_assoc (x y z ... | Mathlib/LinearAlgebra/TensorProduct/Graded/External.lean | 231 | 246 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro
-/
import Mathlib.Algebra.Module.Submodule.Bilinear
import Mathlib.Algebra.Module.Equiv.Basic
import Mathlib.GroupTheory.Congruence.Hom
import Mathlib.Tactic.Abel
... | Mathlib/LinearAlgebra/TensorProduct/Basic.lean | 1,629 | 1,631 | |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
import Mathlib.Algebra.Notation.Prod
import Mathlib.Data.Nat.Sqrt
import Mathlib.Data.Set.Lattice.Image
/-!
# Naturals pairing function
Th... | Mathlib/Data/Nat/Pairing.lean | 197 | 200 | |
/-
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.Calculus.ContDiff.RCLike
import Mathlib.MeasureTheory.Measure.Hausdorff
/-!
# Hausdorff dimension
The Hausdorff dimension of a set `X` in ... | have := hf.hausdorffMeasure_preimage_le d.coe_nonneg s
rw [hd, top_le_iff] at this
contrapose! this
exact ENNReal.mul_ne_top (by simp) this
theorem le_dimH_image (hf : AntilipschitzWith K f) (s : Set X) : dimH s ≤ dimH (f '' s) :=
calc
dimH s ≤ dimH (f ⁻¹' (f '' s)) := dimH_mono (subset_preimage_image _ ... | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | 357 | 364 |
/-
Copyright (c) 2021 Aaron Anderson, Jesse Michael Han, Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jesse Michael Han, Floris van Doorn
-/
import Mathlib.Data.Set.Prod
import Mathlib.Logic.Equiv.Fin.Basic
import Mathlib.ModelTheory... | φ.not
/-- The implication between formulas, as a formula. -/
| Mathlib/ModelTheory/Syntax.lean | 738 | 740 |
/-
Copyright (c) 2022 Yaël Dillies, Sara Rousta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Sara Rousta
-/
import Mathlib.Logic.Equiv.Set
import Mathlib.Order.Interval.Set.OrderEmbedding
import Mathlib.Order.SetNotation
/-!
# Properties of unbundled ... | Mathlib/Order/UpperLower/Basic.lean | 1,207 | 1,209 | |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jens Wagemaker, Aaron Anderson
-/
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.Algebra.EuclideanDomain.Int
import Mathlib.Algebra.GCDMonoid.Nat
import Ma... |
namespace Int
theorem span_natAbs (a : ℤ) : Ideal.span ({(a.natAbs : ℤ)} : Set ℤ) = Ideal.span {a} := by
rw [Ideal.span_singleton_eq_span_singleton]
exact (associated_natAbs _).symm
theorem eq_pow_of_mul_eq_pow_odd_left {a b c : ℤ} (hab : IsCoprime a b) {k : ℕ} (hk : Odd k)
| Mathlib/RingTheory/Int/Basic.lean | 111 | 118 |
/-
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.MeasureTheory.Measure.AEMeasurable
import Mathlib.Order.Filter.EventuallyConst
/-!
# Measure preserving maps
We say that `f : α → β` is a measure p... |
theorem restrict_image_emb {f : α → β} (hf : MeasurePreserving f μa μb) (h₂ : MeasurableEmbedding f)
(s : Set α) : MeasurePreserving f (μa.restrict s) (μb.restrict (f '' s)) := by
| Mathlib/Dynamics/Ergodic/MeasurePreserving.lean | 76 | 78 |
/-
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.LpSeminorm.Basic
import Mathlib.MeasureTheory.Integral.MeanInequalities
/-!
# Triangle inequality for `Lp`-seminorm
In this file w... |
variable {μ E}
theorem eLpNorm_sub_le' (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ)
(p : ℝ≥0∞) : eLpNorm (f - g) p μ ≤ LpAddConst p * (eLpNorm f p μ + eLpNorm g p μ) := by
simpa only [sub_eq_add_neg, eLpNorm_neg] using eLpNorm_add_le' hf hg.neg p
theorem eLpNorm_sub_le {f g : α → E} (hf : AE... | Mathlib/MeasureTheory/Function/LpSeminorm/TriangleInequality.lean | 118 | 134 |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Data.Matrix.Basis
import Mathlib.Data.Matrix.DMatrix
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.Re... | have A : (listTransvecRow M).length = r := by simp [listTransvecRow]
rw [← List.take_length (l := listTransvecRow M), A]
have : ¬r ≤ i := by simp
simpa only [this, ite_eq_right_iff] using H r le_rfl
intro k hk
| Mathlib/LinearAlgebra/Matrix/Transvection.lean | 455 | 459 |
/-
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... | If `E` has codimension at least two, its complement is ample. -/
theorem of_one_lt_codim [IsTopologicalAddGroup F] [ContinuousSMul ℝ F] {E : Submodule ℝ F}
(hcodim : 1 < Module.rank ℝ (F ⧸ E)) :
AmpleSet (Eᶜ : Set F) := fun x hx ↦ by
rw [E.connectedComponentIn_eq_self_of_one_lt_codim hcodim hx, eq_univ_iff_fo... | Mathlib/Analysis/Convex/AmpleSet.lean | 120 | 132 |
/-
Copyright (c) 2020 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.Group.Action.Pi
import Mathlib.Data.Finset.Prod
import Mathlib.Data.SetLike.Basic
import Mathlib.Data.Sym.Basic
import Mathlib.Data.Sym.Sym2.Init
/-... |
/-- `Sym2.map` as an embedding. -/
@[simps]
| Mathlib/Data/Sym/Sym2.lean | 265 | 267 |
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Data.Prod.Lex
import Mathlib.Data.Sigma.Lex
import Mathlib.Order.RelIso.Set
import Mathlib.Order.WellQuasiOrder
import Mathlib.Tactic.TFAE
/-!
# Well-... | · exact PSigma.subtype_ext (Subtype.ext h') rfl
· dsimp only [Subtype.coe_mk, Subrel, Order.Preimage] at *
revert h'
generalize f c = d
rintro rfl h''
exact h''
theorem Set.WellFoundedOn.sigma_lex_of_wellFoundedOn_fiber (hι : s.WellFoundedOn (rι on f))
(hπ : ∀ i, (s ∩ f ⁻¹' {i}).Wel... | Mathlib/Order/WellFoundedSet.lean | 904 | 920 |
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.MeasureTheory.I... | _ = |(R ^ finrank ℝ E)⁻¹| • ∫ y in e.symm ⁻¹' s, f y ∂μ := by
simp [map_addHaar_smul μ hR, integral_smul_measure, ENNReal.toReal_ofReal, abs_nonneg]
_ = |(R ^ finrank ℝ E)⁻¹| • ∫ x in R • s, f x ∂μ := by
| Mathlib/MeasureTheory/Measure/Haar/NormedSpace.lean | 135 | 137 |
/-
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.Order.Bounds.Defs
import Mathlib.Order.Directed
import Mathlib.Order.BoundedOrder.Monotone
import Mathlib.Order.Interval.Set.Basic
/-... | Mathlib/Order/Bounds/Basic.lean | 980 | 982 | |
/-
Copyright (c) 2024 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.Algebra.BigOperators.Field
import Mathlib.NumberTheory.LSeries.Basic
/-!
# Linearity of the L-series of `f` as a function of `f`
We show that the `LSer... | ext ⟨- | n⟩ <;>
simp [neg_div]
lemma LSeries.term_neg_apply (f : ℕ → ℂ) (s : ℂ) (n : ℕ) : term (-f) s n = -term f s n := by
| Mathlib/NumberTheory/LSeries/Linearity.lean | 50 | 53 |
/-
Copyright (c) 2023 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Ring.CharZero
import Mathlib.Algebra.Ring.Int.Units
import Mathlib.GroupTheory.Coprod.Basic
import Mathlib.GroupTheory.Complement
/-!
## HNN Exte... | unfold unitsSMul
have : Cancels 1 ((g : G) • w) ↔ Cancels 1 w := by
simp [Cancels, Subgroup.mul_mem_cancel_left]
by_cases hcan : Cancels 1 w
· simp [unitsSMulWithCancel, dif_pos (this.2 hcan), dif_pos hcan]
cases w using consRecOn
· simp [Cancels] at hcan
· simp only [smul_cons, consRecOn_cons, ... | Mathlib/GroupTheory/HNNExtension.lean | 458 | 480 |
/-
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
-/
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
import Mathlib.LinearAlgebra.Basis.Prod
impo... | lift_eq_nat_iff] at this)
theorem finrank_span_set_eq_card {s : Set M} [Fintype s] (hs : LinearIndepOn R id s) :
finrank R (span R s) = s.toFinset.card :=
finrank_eq_of_rank_eq
(by
| Mathlib/LinearAlgebra/Dimension/Constructions.lean | 475 | 480 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Yury Kudryashov
-/
import Mathlib.Algebra.Algebra.Hom
import Mathlib.Algebra.Ring.Action.Group
/-!
# Isomorphisms of `R`-algebras
This file defines bundled isomorphisms of `R`-... |
end simps
| Mathlib/Algebra/Algebra/Equiv.lean | 356 | 358 |
/-
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
-/
import Mathlib.Algebra.Order.GroupWithZero.Synonym
import Mathlib.Algebra.Order.Ring.Canonical
import Mathlib.Algebra.Ring.Hom.Defs
i... |
lemma coe_mul_eq_bind {a : α} (ha : a ≠ 0) : ∀ b, (a * b : WithTop α) = b.bind fun b ↦ ↑(a * b)
| ⊤ => by simp [top_mul, ha]; rfl
| Mathlib/Algebra/Order/Ring/WithTop.lean | 64 | 66 |
/-
Copyright (c) 2018 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Mario Carneiro, Kim Morrison, Floris van Doorn
-/
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryTheory.Limits.Cones
import Batteries.Tactic.Congr
/-... | This is the most general form of uniqueness of cone points,
allowing relabelling of both the indexing category (up to equivalence)
and the functor (up to natural isomorphism).
-/
| Mathlib/CategoryTheory/Limits/IsLimit.lean | 328 | 331 |
/-
Copyright (c) 2021 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn, Joachim Breitner
-/
import Mathlib.Algebra.Group.Action.End
import Mathlib.Algebra.Group.Action.Pointwise.Set.Basic
import Mathlib.Algebra.Group.Submonoid.Membership
import Mat... | congr 1
| Mathlib/GroupTheory/CoprodI.lean | 777 | 778 |
/-
Copyright (c) 2022 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.Algebra.BigOperators.Field
import Mathlib.Algebra.Order.Chebyshev
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Math... | (4 ^ #P.parts - a) * m + a * (m + 1) = #u := by
rw [hucard, mul_add, mul_one, ← add_assoc, ← add_mul,
Nat.sub_add_cancel ((Nat.le_succ _).trans a_add_one_le_four_pow_parts_card), mul_comm]
theorem card_aux₂ (hP : P.IsEquipartition) (hu : u ∈ P.parts) (hucard : #u ≠ m * 4 ^ #P.parts + a) :
| Mathlib/Combinatorics/SimpleGraph/Regularity/Bound.lean | 138 | 142 |
/-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Junyan Xu
-/
import Mathlib.Data.Fintype.Order
import Mathlib.FieldTheory.IntermediateField.Adjoin.Basic
/-!
# Extension of field embeddings
`IntermediateField.exis... | def IsExtendible (σ : Lifts F E K) : Prop :=
∀ S : Finset E, ∃ τ ≥ σ, (S : Set E) ⊆ τ.carrier
section Chain
variable (c : Set (Lifts F E K)) (hc : IsChain (· ≤ ·) c)
/-- The union of a chain of lifts. -/
noncomputable def union : Lifts F E K :=
let t (i : ↑(insert ⊥ c)) := i.val.carrier
have hc := hc.insert fun... | Mathlib/FieldTheory/Extension.lean | 100 | 112 |
/-
Copyright (c) 2020 Heather Macbeth, Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth, Patrick Massot
-/
import Mathlib.Algebra.Group.Subgroup.Order
import Mathlib.Algebra.Order.Archimedean.Basic
/-!
# Archimedean groups
This file prov... | _ ≤ a ^ (m + 1) := pow_le_pow_right' h₀.le (add_le_add_right hnm _)
_ = a ^ m * a := pow_succ _ _
_ < y * a := mul_lt_mul_right' hm.1 _
/-- If a subgroup of a linear ordered commutative group is disjoint with the
| Mathlib/GroupTheory/Archimedean.lean | 91 | 95 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Alex Kontorovich, Heather Macbeth
-/
import Mathlib.MeasureTheory.Group.Action
import Mathlib.MeasureTheory.Group.Pointwise
import Mathlib.MeasureTheory.Integral.Lebe... | Mathlib/MeasureTheory/Group/FundamentalDomain.lean | 905 | 912 | |
/-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Limits.Preserves.Finite
import Mathlib.CategoryTheory.Sites.Canonical
import Mathlib.Category... | theorem extensiveTopology.isSheaf_yoneda_obj (W : C) : Presieve.IsSheaf (extensiveTopology C)
(yoneda.obj W) := by
rw [extensiveTopology, isSheaf_coverage]
intro X R ⟨Y, α, Z, π, hR, hi⟩
have : IsIso (Sigma.desc (Cofan.inj (Cofan.mk X π))) := hi
have : R.Extensive := ⟨Y, α, Z, π, hR, ⟨Cofan.isColimitOfIsIso... | Mathlib/CategoryTheory/Sites/Coherent/ExtensiveSheaves.lean | 63 | 69 |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Kevin Kappelmann
-/
import Mathlib.Algebra.Order.Round
import Mathlib.Data.Rat.Cast.Order
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Ring
/-... | rw [← Int.ofNat_inj, Int.natCast_floor_eq_floor (by positivity)]
push_cast
| Mathlib/Data/Rat/Floor.lean | 86 | 87 |
/-
Copyright (c) 2022 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.InnerProductSpace.Convex
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Combinatorics.Additive.AP.Three... | Mathlib/Combinatorics/Additive/AP/Three/Behrend.lean | 514 | 518 | |
/-
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.Basic
import Mathlib.Probability.Independence.Conditional
/-!
# Kolmogorov's 0-1 law
Let `s : ι → MeasurableSpace Ω` be an indep... | Kernel.measure_eq_zero_or_one_of_indepSet_self' (ae_of_all μα h) h_indep
theorem measure_eq_zero_or_one_of_indepSet_self [IsFiniteMeasure μ] {t : Set Ω}
(h_indep : IndepSet t t μ) : μ t = 0 ∨ μ t = 1 := by
| Mathlib/Probability/Independence/ZeroOne.lean | 58 | 61 |
/-
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, Yury Kudryashov
-/
import Mathlib.Topology.Order.IsLUB
/-!
# Order topology on a densely ordered set
-/
open Set Filter TopologicalSpace Topology Func... |
@[simp]
| Mathlib/Topology/Order/DenselyOrdered.lean | 101 | 102 |
/-
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.Data.Fintype.Card
import Mathlib.Algebra.Order.BigOperators.Group.Multiset
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Data.Multiset.OrderedM... | Mathlib/Algebra/Order/BigOperators/Group/Finset.lean | 670 | 675 | |
/-
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.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calc... | theorem HasFDerivWithinAt.mapsTo_tangent_cone {x : E} (h : HasFDerivWithinAt f f' s x) :
MapsTo f' (tangentConeAt 𝕜 s x) (tangentConeAt 𝕜 (f '' s) (f x)) := by
rintro v ⟨c, d, dtop, clim, cdlim⟩
refine
⟨c, fun n => f (x + d n) - f x, mem_of_superset dtop ?_, clim, h.lim atTop dtop clim cdlim⟩
simp +cont... | Mathlib/Analysis/Calculus/FDeriv/Equiv.lean | 459 | 465 |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Group.TypeTags.Basic
import Mathlib.Topology.Bornology.Basic
/-!
# Bornology structure on products and subtypes
In this file we define `Bor... | ### `Additive`, `Multiplicative`
The bornology on those type synonyms is inherited without change.
| Mathlib/Topology/Bornology/Constructions.lean | 164 | 166 |
/-
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.Order.Bounds.Basic
import Mathlib.Order.Preorder.Chain
/-!
# Antichains
This file defines antichains. An antichain is a set where any two distinct elemen... | hs.flip
theorem image (hs : IsAntichain r s) (f : α → β) (h : ∀ ⦃a b⦄, r' (f a) (f b) → r a b) :
IsAntichain r' (f '' s) := by
rintro _ ⟨b, hb, rfl⟩ _ ⟨c, hc, rfl⟩ hbc hr
exact hs hb hc (ne_of_apply_ne _ hbc) (h hr)
| Mathlib/Order/Antichain.lean | 73 | 78 |
/-
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... | obtain ⟨δ, δ_pos, hδr, hδ⟩ := exists_lt_of_gauge_lt hs₂ (h.trans_lt hr)
suffices (r⁻¹ * δ) • δ⁻¹ • x ∈ s by rwa [smul_smul, mul_inv_cancel_right₀ δ_pos.ne'] at this
| Mathlib/Analysis/Convex/Gauge.lean | 138 | 139 |
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kenny Lau, María Inés de Frutos-Fernández, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.RingTheory.DiscreteValuationRing.Basi... | end PowerSeries
| Mathlib/RingTheory/PowerSeries/Inverse.lean | 373 | 374 |
/-
Copyright (c) 2018 Louis Carlin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Louis Carlin, Mario Carneiro
-/
import Mathlib.Algebra.EuclideanDomain.Defs
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Regular
import Mathlib.Algebra.Grou... | (by dsimp [P]; rw [mul_one, mul_zero, zero_add])
rwa [xgcdAux_val, xgcd_val] at this
| Mathlib/Algebra/EuclideanDomain/Basic.lean | 203 | 204 |
/-
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.CatCommSq
import Mathlib.CategoryTheory.GuitartExact.Basic
/-!
# Vertical composition of Guitart exact squares
In this file, we show that the ve... | /-- A 2-square is Guitart exact iff it is so after replacing the left and right functors by
isomorphic functors. -/
@[simp]
lemma whiskerVertical_iff (α : L ≅ L') (β : R ≅ R') :
(w.whiskerVertical α.hom β.inv).GuitartExact ↔ w.GuitartExact := by
constructor
· intro h
have : w = (w.whiskerVertical α.hom β.in... | Mathlib/CategoryTheory/GuitartExact/VerticalComposition.lean | 55 | 70 |
/-
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, Floris van Doorn, Yury Kudryashov, Neil Strickland
-/
import Mathlib.Algebra.Group.Defs
import Mathlib.Algebra.GroupWithZero.Defs
import Mathlib.Data.I... | Mathlib/Algebra/Ring/Defs.lean | 357 | 357 | |
/-
Copyright (c) 2021 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, David Kurniadi Angdinata
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.CubicDiscriminant
import Mathlib.RingTheory.Nilpotent.Defs
import Mathlib.Tactic.Fiel... | Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean | 622 | 623 | |
/-
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 | 3,192 | 3,194 | |
/-
Copyright (c) 2022 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.InnerProductSpace.Convex
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Combinatorics.Additive.AP.Three... | sum_eq.trans_lt <| (Nat.div_le_self _ 2).trans_lt <| pred_lt (pow_pos (succ_pos _) _).ne'
theorem card_sphere_le_rothNumberNat (n d k : ℕ) :
#(sphere n d k) ≤ rothNumberNat ((2 * d - 1) ^ n) := by
cases n
· dsimp; refine (card_le_univ _).trans_eq ?_; rfl
cases d
· simp
apply threeAPFree_image_sphere.le... | Mathlib/Combinatorics/Additive/AP/Three/Behrend.lean | 205 | 216 |
/-
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.Data.Nat.Gcd
import Mathlib.Algebra.Group.Nat.Units
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.GroupWi... | ((Nat.dvd_add_iff_left (Nat.dvd_mul_left n b)).mpr
((congr_arg _ h).mpr (Nat.dvd_mul_left n m)))
| Mathlib/Data/Nat/GCD/Basic.lean | 235 | 236 |
/-
Copyright (c) 2020 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.CategoryTheory.NatIso
import Mathlib.CategoryTheory.EqToHom
/-!
# Quotient category
Constructs the quotient of a category by an arbitrary family of relations... | lemma compClosure_iff_self [h : Congruence r] {X Y : C} (f g : X ⟶ Y) :
CompClosure r f g ↔ r f g := by
constructor
· intro hfg
| Mathlib/CategoryTheory/Quotient.lean | 148 | 151 |
/-
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.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Reverse
import Mathlib.Algebra.Polynomial.Inductions
import Mathlib.RingTheory.Localizati... | @[simp]
theorem mk'_one_X :
IsLocalization.mk' R[T;T⁻¹] 1 (⟨X, 1, pow_one X⟩ : Submonoid.powers (X : R[X])) = T (-1) := by
| Mathlib/Algebra/Polynomial/Laurent.lean | 531 | 533 |
/-
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... | Mathlib/Data/List/Basic.lean | 2,069 | 2,085 | |
/-
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... | @[simp]
| Mathlib/Data/Real/Sqrt.lean | 305 | 305 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.Group.Pointwise.Set.Basic
import Mathlib.Algebra.G... | @[to_additive] lemma MonoidHomClass.isMulFreimanHom [FunLike F α β] [MonoidHomClass F α β] (f : F)
(hfAB : MapsTo f A B) : IsMulFreimanHom n A B f where
mapsTo := hfAB
map_prod_eq_map_prod s t _ _ _ _ h := by rw [← map_multiset_prod, h, map_multiset_prod]
@[to_additive] lemma MulEquivClass.isMulFreimanIso [Equ... | Mathlib/Combinatorics/Additive/FreimanHom.lean | 244 | 269 |
/-
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.Deriv.Basic
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Normed.Operator.BoundedLinear... | apply le_of_lt
exact hr' _ ((mem_closedBall.1 hy).trans_lt r'mem.1) _ ((mem_closedBall.1 hz).trans_lt r'mem.1)
theorem mem_A_of_differentiable {ε : ℝ} (hε : 0 < ε) {x : E} (hx : DifferentiableAt 𝕜 f x) :
| Mathlib/Analysis/Calculus/FDeriv/Measurable.lean | 149 | 152 |
/-
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.Analysis.Convex.Side
import Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
/-!
# Oriented an... | /-- An oriented angle is `π` if and only if the angle with the order of the points reversed is
`π`. -/
| Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean | 176 | 177 |
/-
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, Jeremy Avigad
-/
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Data.Set.Finite.Range
import Mathlib.Data.Set.Lattice
import Mathlib.Topology.Defs.... | Mathlib/Topology/Basic.lean | 1,231 | 1,233 | |
/-
Copyright (c) 2020 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.BigOperators.Fin
import Mathlib.Logic.Encodable.Pi
import Mathlib.MeasureTheory.Group.Measure
import Mathlib.MeasureTheory.MeasurableSpace.... | volume preserving. -/
theorem volume_preserving_arrowCongr' {α₁ β₁ α₂ β₂ : Type*} [Fintype α₁] [Fintype α₂]
[MeasureSpace β₁] [MeasureSpace β₂] [SigmaFinite (volume : Measure β₂)]
(hα : α₁ ≃ α₂) (hβ : β₁ ≃ᵐ β₂) (hm : MeasurePreserving hβ) :
MeasurePreserving (MeasurableEquiv.arrowCongr' hα hβ) :=
measure... | Mathlib/MeasureTheory/Constructions/Pi.lean | 881 | 892 |
/-
Copyright (c) 2024 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Cardinal.Arithmetic
import Mathlib.SetTheory.Ordinal.Principal
/-!
# Ordinal arithmetic with cardinals
This file co... | Mathlib/SetTheory/Cardinal/Ordinal.lean | 603 | 607 | |
/-
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.Polynomial.AlgebraMap
import Mathlib.RingTheory.IsTensorProduct
/-!
# Base change of polynomial algebras
Given `[CommSemiring R] [Semiring A] [Al... | mul_zero, ite_mul, zero_mul]
simp_rw [← ite_zero_mul (¬coeff p₁ _ = 0) (a₁ * (algebraMap R A) (coeff p₁ _))]
simp_rw [← mul_ite_zero (¬coeff p₂ _ = 0) _ (_ * _)]
simp_rw [toFunLinear_mul_tmul_mul_aux_1, toFunLinear_mul_tmul_mul_aux_2]
theorem toFunLinear_one_tmul_one :
toFunLinear R A (1 ⊗ₜ[R] 1)... | Mathlib/RingTheory/PolynomialAlgebra.lean | 85 | 91 |
/-
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.TrivSqZeroExt
/-!
# Dual numbers
The dual numbers over `R` are of the form `a + bε`, where `a` and `b` are typically elements of a
commutative ring... | lift fe (ε : A[ε]) = fe.val.2 := by
simp only [lift_apply_apply, fst_eps, map_zero, snd_eps, map_one, one_mul, zero_add]
/-- Lifting `DualNumber.eps` itself gives the identity. -/
| Mathlib/Algebra/DualNumber.lean | 179 | 182 |
/-
Copyright (c) 2019 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Module.Submodule.Equiv
import Mathlib.Algebra.Module.Equiv.Basic
import Mathlib.Algebra.Module.Rat
im... | instance : One (M →ₗ⁅R,L⁆ M) :=
⟨id⟩
| Mathlib/Algebra/Lie/Basic.lean | 760 | 761 |
/-
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... | rw [← coe_coe, ← intCoe_mul, moebius_mul_coe_zeta, intCoe_one]
@[simp]
| Mathlib/NumberTheory/ArithmeticFunction.lean | 1,088 | 1,090 |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Order.Filter.Prod
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.Filter.Finite
import Mathlib.Order.Filter.Bases.Basic
/... | lift_map_le.antisymm <| le_lift.2 fun _s hs => lift_le hs <| hg <| image_preimage_subset _ _
theorem lift_comm {g : Filter β} {h : Set α → Set β → Filter γ} :
(f.lift fun s => g.lift (h s)) = g.lift fun t => f.lift fun s => h s t :=
| Mathlib/Order/Filter/Lift.lean | 111 | 114 |
/-
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.Data.Finite.Sum
import Mathlib.RingTheory.FiniteType
import Mathlib.RingTheory.Finiteness.Ideal
import Mathlib.RingTheory.Ideal.Quotient.Operations
imp... |
/-- If `f : A →ₐ[R] B` is surjective with finitely generated kernel and `A` is finitely presented,
then so is `B`. -/
theorem of_surjective {f : A →ₐ[R] B} (hf : Function.Surjective f)
(hker : (RingHom.ker f.toRingHom).FG)
[FinitePresentation R A] : FinitePresentation R B :=
letI : FinitePresentation R (A ⧸ ... | Mathlib/RingTheory/FinitePresentation.lean | 130 | 137 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jan-David Salchow, Sébastien Gouëzel, Jean Lo, Yury Kudryashov, Frédéric Dupuis,
Heather Macbeth
-/
import Mathlib.Algebra.Module.Opposite
import Mathlib.Topology.Algebra.Group.Qu... | Mathlib/Topology/Algebra/Module/Basic.lean | 1,238 | 1,240 | |
/-
Copyright (c) 2024 Raghuram Sundararajan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Raghuram Sundararajan
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Ext
/-!
# Extensionality lemmas for rings and similar structures
In this file we prove e... | Function.Injective (@toNonUnitalSemiring R) := by
intro _ _ h
ext x y
· exact congrArg (·.toAdd.add x y) h
· exact congrArg (·.toMul.mul x y) h
theorem toNonUnitalNonAssocring_injective :
| Mathlib/Algebra/Ring/Ext.lean | 195 | 201 |
/-
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
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
/-! Sequences of functors from a category equipped with a shift
Let `F : C... |
@[reassoc]
lemma shiftMap_comp' {X Y Z : C} {n : M} (f : X ⟶ Y) (g : Y ⟶ Z⟦n⟧) (a a' : M) (ha' : n + a = a') :
F.shiftMap (f ≫ g) a a' ha' = (F.shift a).map f ≫ F.shiftMap g a a' ha' := by
| Mathlib/CategoryTheory/Shift/ShiftSequence.lean | 202 | 205 |
/-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Ordinal.Family
import Mathlib.Tactic.Abel
/-!
# Natural operations on ordinals
The goal of this file is to define n... | rw [nmul_nadd, nmul_nadd] at this
simp only [nadd_assoc] at this
rwa [nadd_left_comm, nadd_comm (a ⨳ c), nadd_left_comm (a' ⨳ d), nadd_left_comm (a ⨳ c'),
| Mathlib/SetTheory/Ordinal/NaturalOps.lean | 563 | 565 |
/-
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.ShortComplex.QuasiIso
import Mathlib.CategoryTheory.Limits.Preserves.Finite
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels
/-!
... | (homologyData S₁).right (homologyData S₂).right]
@[reassoc]
lemma mapHomologyIso'_inv_naturality [S₁.HasHomology] [S₂.HasHomology]
[(S₁.map F).HasHomology] [(S₂.map F).HasHomology]
[F.PreservesRightHomologyOf S₁] [F.PreservesRightHomologyOf S₂] :
F.map (homologyMap φ) ≫ (S₂.mapHomologyIso' F).inv = (... | Mathlib/Algebra/Homology/ShortComplex/PreservesHomology.lean | 598 | 605 |
/-
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.Fold
import Mathlib.Algebra.GCDMonoid.Multiset
/-!
# GCD and LCM operations on finsets
## Main definitions
- `Finset.gcd` - the greatest... | ∃ g : β → α, (∀ b ∈ s, f b = s.gcd f * g b) ∧ s.gcd g = 1 := by
classical
by_cases h : ∀ x ∈ s, f x = (0 : α)
· refine ⟨fun _ ↦ 1, fun b hb ↦ by rw [h b hb, gcd_eq_zero_iff.2 h, mul_one], ?_⟩
rw [gcd_eq_gcd_image, image_const hs, gcd_singleton, id, normalize_one]
· choose g' hg using @gcd_dvd _ ... | Mathlib/Algebra/GCDMonoid/Finset.lean | 228 | 241 |
/-
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.ExactSequence
import Mathlib.Algebra.Homology.ShortComplex.Limits
import Mathlib.CategoryTheory.Abelian.Refinements
/-!
# The snake lemma
The ... | @[reassoc]
lemma L₀X₂ToP_comp_φ₁ : S.L₀X₂ToP ≫ S.φ₁ = 0 := by
simp only [← cancel_mono S.L₂.f, L₀X₂ToP, assoc, φ₂, φ₁_L₂_f,
pullback.lift_fst_assoc, w₀₂_τ₂, zero_comp]
| Mathlib/Algebra/Homology/ShortComplex/SnakeLemma.lean | 287 | 290 |
/-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Algebra.Order.Pi
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Data.Real.Pointwise
/... | theorem sub_mem_ball (p : Seminorm 𝕜 E) (x₁ x₂ y : E) (r : ℝ) :
x₁ - x₂ ∈ p.ball y r ↔ x₁ ∈ p.ball (x₂ + y) r := by simp_rw [mem_ball, sub_sub]
| Mathlib/Analysis/Seminorm.lean | 712 | 714 |
/-
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.HomotopyCategory.Shift
/-! Shifting cochains
Let `C` be a preadditive category. Gi... | simp only [rightShift_v _ a n' hn' p q hpq _ rfl, add_v, add_comp]
@[simp]
lemma leftShift_add (a n' : ℤ) (hn' : n + a = n') :
(γ₁ + γ₂).leftShift a n' hn' = γ₁.leftShift a n' hn' + γ₂.leftShift a n' hn' := by
ext p q hpq
| Mathlib/Algebra/Homology/HomotopyCategory/HomComplexShift.lean | 154 | 159 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Patrick Massot, Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic
import Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalcul... | Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | 527 | 529 | |
/-
Copyright (c) 2022 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
import Mathlib.CategoryTheory.Limits.Constructions.BinaryProducts
/-!
# Limits involving zero objects
Binary p... | theorem inl_pushoutZeroZeroIso_inv (X Y : C) [HasBinaryCoproduct X Y] :
coprod.inl ≫ (pushoutZeroZeroIso X Y).inv = pushout.inl _ _ := by simp [Iso.comp_inv_eq]
| Mathlib/CategoryTheory/Limits/Constructions/ZeroObjects.lean | 184 | 185 |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Jakob von Raumer
-/
import Mathlib.Data.List.Chain
import Mathlib.CategoryTheory.PUnit
import Mathlib.CategoryTheory.Groupoid
import Mathlib.CategoryTheory.Category.ULift
... | theorem IsPreconnected.of_any_functor_const_on_obj
(h : ∀ {α : Type u₁} (F : J ⥤ Discrete α), ∀ j j' : J, F.obj j = F.obj j') :
IsPreconnected J where
| Mathlib/CategoryTheory/IsConnected.lean | 115 | 117 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Justus Springer
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.AlgebraicGeometry.StructureSheaf
import Mathlib.RingTheory.Localization.Localizatio... | -- homomorphism.
#adaptation_note /-- nightly-2024-04-01
It's this `erw` that is blowing up. The implicit arguments differ significantly. -/
erw [← localRingHom_comp_stalkIso_apply' f p a] at ha
have : IsLocalHom (stalkIso (↑S) p).inv.hom := isLocalHom_of_isIso _
replace ha := (isU... | Mathlib/AlgebraicGeometry/Spec.lean | 246 | 260 |
/-
Copyright (c) 2022 Anand Rao, Rémi Bottinelli. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anand Rao, Rémi Bottinelli
-/
import Mathlib.CategoryTheory.CofilteredSystem
import Mathlib.Combinatorics.SimpleGraph.Path
import Mathlib.Data.Finite.Set
/-!
# Ends
This ... |
/-- For a locally finite preconnected graph, the number of components outside of any finite set
is finite. -/
instance componentCompl_finite [LocallyFinite G] [Gpc : Fact G.Preconnected] (K : Finset V) :
Finite (G.ComponentCompl K) := by
| Mathlib/Combinatorics/SimpleGraph/Ends/Defs.lean | 212 | 216 |
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