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/-
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 | 268 | 271 | |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Yury Kudryashov
-/
import Mathlib.Data.Option.Basic
import Mathlib.Topology.Separation.Regular
/-!
# Paracompact topological spaces
A topological space `X` is said to b... | apply Subset.antisymm (subset_univ _)
· simp_rw [← compl_union_self s, Option.elim', iUnion_option]
apply union_subset_union_right sᶜ us
rcases precise_refinement (Option.elim' sᶜ u) (Option.forall.2 ⟨isOpen_compl_iff.2 hs, uo⟩)
uc with
⟨v, vo, vc, vf, vu⟩
refine ⟨v ∘ some, fun i ↦ vo _, ?_,... | Mathlib/Topology/Compactness/Paracompact.lean | 99 | 112 |
/-
Copyright (c) 2020 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Utensil Song
-/
import Mathlib.Algebra.RingQuot
import Mathlib.LinearAlgebra.TensorAlgebra.Basic
import Mathlib.LinearAlgebra.QuadraticForm.Isometry
import Mathlib.LinearAlge... | apply (lift Q).symm.injective
rw [lift_symm_apply, lift_symm_apply]
simp only [h]
-- This proof closely follows `TensorAlgebra.induction`
| Mathlib/LinearAlgebra/CliffordAlgebra/Basic.lean | 175 | 179 |
/-
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... | inst₁ = inst₂ := by
have : inst₁.toNonUnitalNonAssocRing = inst₂.toNonUnitalNonAssocRing := by
ext : 1 <;> assumption
-- Split into fields and prove they are equal using the above.
cases inst₁; cases inst₂
congr
| Mathlib/Algebra/Ring/Ext.lean | 187 | 193 |
/-
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.Fin.VecNotation
import Mathlib.SetTheory.Cardinal.Basic
/-!
# Basics on... |
@[simp]
theorem comp_refl (g : M ≃[L] N) : g.comp (refl L M) = g :=
rfl
@[simp]
theorem refl_comp (g : M ≃[L] N) : (refl L N).comp g = g :=
| Mathlib/ModelTheory/Basic.lean | 647 | 653 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.RingTheory.WittVector.StructurePolynomial
/-!
# Witt vectors
In this file we define the type of `p`-typical Witt vectors and ring op... | simp only [wittAdd, wittStructureRat, map_add, rename_X, xInTermsOfW_zero, map_X,
wittPolynomial_zero, bind₁_X_right, map_wittStructureInt]
@[simp]
| Mathlib/RingTheory/WittVector/Defs.lean | 232 | 235 |
/-
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
-/
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.Matrix.Composition
import Mathlib.Data.Matrix.ConjTranspose
/-!... | blockDiagonal (M - N) = blockDiagonal M - blockDiagonal N :=
map_sub (blockDiagonalAddMonoidHom m n o α) M N
@[simp]
| Mathlib/Data/Matrix/Block.lean | 409 | 412 |
/-
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... | (m m' : ℤ) (hm' : m + a = m') :
δ n m (γ.leftUnshift n hn) = a.negOnePow • (δ n' m' γ).leftUnshift m hm' := by
obtain ⟨γ', rfl⟩ := (leftShiftAddEquiv K L n a n' hn).surjective γ
dsimp
simp only [leftUnshift_leftShift, γ'.δ_leftShift a n' m' hn m hm', leftUnshift_units_smul,
smul_smul, Int.units_mul_se... | Mathlib/Algebra/Homology/HomotopyCategory/HomComplexShift.lean | 447 | 459 |
/-
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.Data.ENNReal.Operations
/-!
# Results about division in extended non-negative reals
This file establishes basic properties related t... | /-- See `ENNReal.div_mul_cancel'` for a stronger version. -/
protected lemma div_mul_cancel (ha₀ : a ≠ 0) (ha : a ≠ ∞) : b / a * a = b :=
ENNReal.div_mul_cancel' (by simp [ha₀]) (by simp [ha])
| Mathlib/Data/ENNReal/Inv.lean | 161 | 163 |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.FieldTheory.Finiteness
import Mathlib.LinearAlgebra.AffineSpace.Basis
import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
/-!
# Finite-dimensional subsp... | Module.rank k (vectorSpan k s) ≤ 2
/-- The `vectorSpan` of coplanar points is finite-dimensional. -/
| Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean | 614 | 616 |
/-
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... | Mathlib/Data/Option/NAry.lean | 212 | 216 | |
/-
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.Algebra.Module.Torsion
import Mathlib.Algebra.Ring.Idempotent
import Mathlib.LinearAlgebra.Dimension.FreeAndStrongRankCondition
import Mathlib.LinearAlgebra.... | theorem toCotangent_surjective : Function.Surjective I.toCotangent := Submodule.mkQ_surjective _
theorem toCotangent_range : LinearMap.range I.toCotangent = ⊤ := Submodule.range_mkQ _
| Mathlib/RingTheory/Ideal/Cotangent.lean | 74 | 76 |
/-
Copyright (c) 2024 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.DirectSum.LinearMap
import Mathlib.Algebra.Lie.Weights.Cartan
import Mathlib.Data.Int.Interval
import Mathlib.LinearAlgebra.Trace
import Mathlib.Ring... | { toFun := fun ⟨m, hm⟩ ↦ ⟨⁅(y : L), m⁆,
lie_mem_genWeightSpaceChain_of_genWeightSpace_eq_bot_right M α χ p q hq hy hm⟩
map_add' := fun _ _ ↦ by simp
map_smul' := fun t m ↦ by simp }
let g : Module.End R (genWeightSpaceChain M α χ p q) :=
{ toFun := fun ⟨m, hm⟩ ↦ ⟨⁅(z : L), m⁆,
... | Mathlib/Algebra/Lie/Weights/Chain.lean | 181 | 214 |
/-
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.Data.List.Duplicate
import Mathlib.Data.List.Sort
/-!
# Equivalence between `Fin (length l)` and elements of a list
Given a list `l`,
* if `l` has... | constructor
· rintro ⟨f, hf⟩
refine ⟨f ⟨0, by simp⟩, f ⟨1, by simp⟩, f.lt_iff_lt.2 (Nat.zero_lt_one), ?_⟩
rw [← hf, ← hf]; simp
· rintro ⟨n, m, hnm, h, h'⟩
refine ⟨OrderEmbedding.ofStrictMono (fun i => if (i : ℕ) = 0 then n else m) ?_, ?_⟩
· rintro ⟨⟨_ | i⟩, hi⟩ ⟨⟨_ | j⟩, hj⟩
... | Mathlib/Data/List/NodupEquivFin.lean | 214 | 235 |
/-
Copyright (c) 2023 Antoine Chambert-Loir and María Inés de Frutos-Fernández. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir, María Inés de Frutos-Fernández, Bhavik Mehta, Eric Wieser
-/
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import ... | ext ⟨x, y⟩
simp
| Mathlib/Algebra/Order/Antidiag/Prod.lean | 130 | 132 |
/-
Copyright (c) 2021 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kim Morrison
-/
import Mathlib.Algebra.Homology.ComplexShape
import Mathlib.CategoryTheory.Subobject.Limits
import Mathlib.CategoryTheory.GradedObject
import Mathlib.Alge... | Mathlib/Algebra/Homology/HomologicalComplex.lean | 1,085 | 1,087 | |
/-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.Topology.MetricSpace.HausdorffDistance
/-!
# Thickenings in pseudo-metric spaces
## Main definitions
* `Metric.thickening δ s`, the open thickening by ra... | `Metric.cthickening δ E` with the same radius. -/
theorem thickening_subset_cthickening (δ : ℝ) (E : Set α) : thickening δ E ⊆ cthickening δ E := by
intro x hx
rw [thickening, mem_setOf_eq] at hx
| Mathlib/Topology/MetricSpace/Thickening.lean | 264 | 267 |
/-
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.Ring.Semiconj
import Mathlib.Algebra.Ring.Units
import Mathlib.Algebra.Gro... | theorem mul_self_sub_mul_self [NonUnitalNonAssocCommRing R] (a b : R) :
a * a - b * b = (a + b) * (a - b) :=
(Commute.all a b).mul_self_sub_mul_self_eq
theorem mul_self_sub_one [NonAssocRing R] (a : R) : a * a - 1 = (a + 1) * (a - 1) := by
| Mathlib/Algebra/Ring/Commute.lean | 166 | 170 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot, Yury Kudryashov, Rémy Degenne
-/
import Mathlib.Data.Set.Subsingleton
import Mathlib.Order.Interval.Set.Defs
/-!
# Intervals
In any pr... | Mathlib/Order/Interval/Set/Basic.lean | 1,735 | 1,744 | |
/-
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 | 793 | 795 | |
/-
Copyright (c) 2014 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Leonardo de Moura, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Ord... | Mathlib/Algebra/Order/Field/Basic.lean | 980 | 985 | |
/-
Copyright (c) 2021 Luke Kershaw. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Luke Kershaw, Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.Basic
import Mathlib.CategoryTheory.Limits.Constructions.FiniteProductsOfBinaryProducts
import Mathlib.CategoryThe... | biproduct of `T.obj₁` and `T.obj₃`.
See also `exists_iso_binaryBiproduct_of_distTriang`. -/
@[simps]
def binaryBiproductData (T : Triangle C) (hT : T ∈ distTriang C) (hT₀ : T.mor₃ = 0)
| Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean | 465 | 468 |
/-
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.Order.Disjoint
import Mathlib.Order.RelIso.Basic
import Mathlib.Tactic.Monotonicity.Attr
/-!
# Order homomorphisms
This file defines order homomorphi... | variable [Preorder α] [Preorder β] [EquivLike F α β] [OrderIsoClass F α β]
theorem map_lt_map_iff (f : F) {a b : α} : f a < f b ↔ a < b :=
| Mathlib/Order/Hom/Basic.lean | 187 | 189 |
/-
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.Analysis.Convex.Exposed
import Mathlib.Analysis.NormedSpace.HahnBanach.Separation
import Mathlib.Topology.Algebra.ContinuousAffineMap
/-!
# The Krein-Milm... | rintro w hw
-- The fiber of `w` is nonempty and compact
have ht : IsCompact {x ∈ s | f x = w} :=
hs.inter_right <| isClosed_singleton.preimage f.continuous
have ht₀ : {x ∈ s | f x = w}.Nonempty := by simpa using extremePoints_subset hw
-- Hence by the Krein-Milman lemma it has an extreme point `x`
obtai... | Mathlib/Analysis/Convex/KreinMilman.lean | 109 | 125 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Algebra.MvPolynomial.Eval
/-!
# Renaming variables of polynomials
This file establishes the `rename` operation on mul... | lemma map_comp_rename (f : R →+* S) (g : σ → τ) :
(map f).comp (rename g).toRingHom = (rename g).toRingHom.comp (map f) :=
RingHom.ext fun p ↦ map_rename f g p
@[simp]
theorem rename_rename (f : σ → τ) (g : τ → α) (p : MvPolynomial σ R) :
| Mathlib/Algebra/MvPolynomial/Rename.lean | 67 | 72 |
/-
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.Data.Finset.Option
import Mathlib.Data.PFun
import Mathlib.Data.Part
/-!
# Image of a `Finset α` under a partially defined function
In this file we... | Set.ext fun _ => mem_toFinset
| Mathlib/Data/Finset/PImage.lean | 44 | 45 |
/-
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.Constructions.BorelSpace.Basic
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.Combinatorics.Pigeonhole
/-!
# Conserv... | .of_absolutelyContinuous hf h.ge.absolutelyContinuous_of_ae <|
hf.toQuasiMeasurePreserving.mono h.ge.absolutelyContinuous_of_ae h.le.absolutelyContinuous_of_ae
theorem _root_.MeasureTheory.conservative_congr {ν : Measure α} (h : ae μ = ae ν) :
Conservative f μ ↔ Conservative f ν :=
⟨(congr_ae · h), (congr_... | Mathlib/Dynamics/Ergodic/Conservative.lean | 83 | 106 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Minchao Wu, Mario Carneiro
-/
import Mathlib.Data.Finset.Attach
import Mathlib.Data.Finset.Disjoint
import Mathlib.Data.Finset.Erase
import Mat... | Mathlib/Data/Finset/Basic.lean | 3,024 | 3,025 | |
/-
Copyright (c) 2017 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Tim Baumann, Stephen Morgan, Kim Morrison, Floris van Doorn
-/
import Mathlib.Tactic.CategoryTheory.Reassoc
/-!
# Isomorphisms
This file defines isomorphisms between objects of a categ... | rfl
| Mathlib/CategoryTheory/Iso.lean | 207 | 208 |
/-
Copyright (c) 2019 Calle Sönne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology... |
/-- Convert a `Real.Angle` to a real number in the interval `Ioc (-π) π`. -/
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 427 | 428 |
/-
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.MeasureTheory.Integral.IntegralEqImproper
/-!
# Integrals against peak functions
A sequence of peak functions is a sequence of functions with a... | `∫ (c ^ d * φ (c • x)) • g x` converges to `g 0` as `c → ∞` if `g` is continuous at `0`
and integrable. -/
theorem tendsto_integral_comp_smul_smul_of_integrable
{φ : F → ℝ} (hφ : ∀ x, 0 ≤ φ x) (h'φ : ∫ x, φ x ∂μ = 1)
(h : Tendsto (fun x ↦ ‖x‖ ^ finrank ℝ F * φ x) (cobounded F) (𝓝 0))
{g : F → E} (hg : Inte... | Mathlib/MeasureTheory/Integral/PeakFunction.lean | 388 | 396 |
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Data.Set.Finite.Lattice
import Mathlib.Data.Matroid.Basic
/-!
# Matroid Independence and Basis axioms
Matroids i... | obtain ⟨J, ⟨hB₀J, hJ, hJss⟩, hJmax⟩ := Finite.exists_maximal_wrt (f := id)
(s := {J | B₀ ⊆ J ∧ Indep J ∧ J ⊆ E₀})
(hE₀fin.finite_subsets.subset (by simp))
⟨B₀, Subset.rfl, hB₀, subset_union_right.trans (subset_insert _ _)⟩
have heI₀ : e ∉ I₀ := not_mem_subset hI₀I heI
have heI... | Mathlib/Data/Matroid/IndepAxioms.lean | 265 | 281 |
/-
Copyright (c) 2021 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.GroupWithZero.Subgroup
import Mathlib.Data.Finite.Card
import Mathlib.Data.Finite.Pr... | @[to_additive]
theorem relindex_ne_zero_trans (hHK : H.relindex K ≠ 0) (hKL : K.relindex L ≠ 0) :
H.relindex L ≠ 0 := fun h =>
mul_ne_zero (mt (relindex_eq_zero_of_le_right (show K ⊓ L ≤ K from inf_le_left)) hHK) hKL
| Mathlib/GroupTheory/Index.lean | 311 | 314 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yaël Dillies
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Algebra.Order.CauSeq.Basic
/-!
# Cauchy sequences and big oper... | IsCauSeq abs (fun n ↦ ∑ i ∈ range n, a i) → IsCauSeq abv fun n ↦ ∑ i ∈ range n, f i := by
intro hg ε ε0
obtain ⟨i, hi⟩ := hg (ε / 2) (div_pos ε0 (by norm_num))
exists max n i
intro j ji
have hi₁ := hi j (le_trans (le_max_right n i) ji)
have hi₂ := hi (max n i) (le_max_right n i)
have sub_le :=
abs... | Mathlib/Algebra/Order/CauSeq/BigOperators.lean | 24 | 49 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Cardinal.Arithmetic
import Mathlib.SetTheory.Ordinal.FixedPoint
/-!
# Cofinality
This file co... | namespace IsFundamentalSequence
variable {a o : Ordinal.{u}} {f : ∀ b < o, Ordinal.{u}}
| Mathlib/SetTheory/Cardinal/Cofinality.lean | 435 | 437 |
/-
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.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Analysis.SumOverResidueClass
/-!
# Conve... | convert add_le_add_left (nsmul_le_nsmul_right (sum_condensed_le' hf n) 2) (f 1)
simp [sum_range_succ', add_comm, pow_succ', mul_nsmul', sum_nsmul]
end Finset
| Mathlib/Analysis/PSeries.lean | 127 | 130 |
/-
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.Equivalence
/-!
# Morphisms of localizers
A morphism of localizers consists of a functor `F : C₁ ⥤ C₂` between
two categories equip... | lemma IsLocalizedEquivalence.mk' [CatCommSq Φ.functor L₁ L₂ G] [G.IsEquivalence] :
Φ.IsLocalizedEquivalence where
isEquivalence := by
rw [Φ.isEquivalence_iff W₁.Q W₂.Q (Φ.localizedFunctor W₁.Q W₂.Q) L₁ L₂ G]
exact inferInstance
| Mathlib/CategoryTheory/Localization/LocalizerMorphism.lean | 133 | 137 |
/-
Copyright (c) 2021 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Order.Lattice
import Mathlib.Data.List.Sort
import Mathlib.Logic.Equiv.Fin.Basic
import Mathlib.Logic.Equiv.Functor
import Mathlib.Data.Fintype.Pigeonhole
... | Mathlib/Order/JordanHolder.lean | 448 | 459 | |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
import Mathlib.LinearAlgebra.FreeModule.Basic
import Mathlib.LinearAlgebra.Matrix.ToLin
/-! # Free modules over ... | Module.basisOfFiniteTypeTorsionFree Module.Finite.exists_fin.choose_spec.choose_spec
instance Module.free_of_finite_type_torsion_free' [Module.Finite R M] [NoZeroSMulDivisors R M] :
Module.Free R M := by
obtain ⟨n, b⟩ : Σn, Basis (Fin n) R M := Module.basisOfFiniteTypeTorsionFree'
| Mathlib/LinearAlgebra/FreeModule/PID.lean | 390 | 394 |
/-
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.Typeclasses.Finite
import Mathlib.MeasureTheory.Measure.Typeclasses.NoAtoms
import Mathlib.MeasureTheory.Measure.... | Mathlib/MeasureTheory/Measure/Typeclasses.lean | 491 | 498 | |
/-
Copyright (c) 2021 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.GroupWithZero.Subgroup
import Mathlib.Data.Finite.Card
import Mathlib.Data.Finite.Pr... | @[to_additive]
theorem dvd_index_map {f : G →* G'} (hf : f.ker ≤ H) :
| Mathlib/GroupTheory/Index.lean | 247 | 248 |
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Algebra.Group.Commute.Hom
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Data.Fintype.B... |
end assoc
variable [Monoid α] [Monoid β]
| Mathlib/Data/Finset/NoncommProd.lean | 97 | 100 |
/-
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, Frédéric Dupuis,
Heather Macbeth
-/
import Mathlib.Algebra.Module.Submodule.Ker
import Mathlib.Algebra.Module.Submodul... | [AddCommMonoid M] [AddCommGroup M₂] [Module R M] [Module R₂ M₂] {τ₁₂ : R →+* R₂}
[RingHomSurjective τ₁₂] (f : M →ₛₗ[τ₁₂] M₂) : LinearMap.range (-f) = LinearMap.range f := by
| Mathlib/Algebra/Module/Submodule/Range.lean | 104 | 105 |
/-
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.Finset.Card
import Mathlib.Data.Fintype.Basic
/-!
# Cardinalities of finite types
This file defines the cardinality `Fintype.card α` as the numb... | Mathlib/Data/Fintype/Card.lean | 612 | 617 | |
/-
Copyright (c) 2024 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.AlgebraicGeometry.EllipticCurve.Group
import Mathlib.NumberTheory.EllipticDivisibilitySequence
/-!
# Division polynomials of Weier... | @[simp]
lemma Ψ_one : W.Ψ 1 = 1 := by
rw [← Nat.cast_one, Ψ_ofNat, preΨ'_one, C_1, if_neg Nat.not_even_one, mul_one]
@[simp]
lemma Ψ_two : W.Ψ 2 = W.ψ₂ := by
rw [← Nat.cast_two, Ψ_ofNat, preΨ'_two, C_1, one_mul, if_pos even_two]
| Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Basic.lean | 325 | 332 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Fin.Tuple.Basic
/-!
# Lists from functions
Theorems and lemmas for dealing with `List.ofFn`, which converts a function on `Fin n` to a list
of l... | @[deprecated getLast_ofFn_succ (since := "2024-11-06")]
theorem last_ofFn_succ {n : ℕ} (f : Fin n.succ → α)
(h : ofFn f ≠ [] := mt ofFn_eq_nil_iff.mp (Nat.succ_ne_zero _)) :
| Mathlib/Data/List/OfFn.lean | 150 | 152 |
/-
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.... | Mathlib/MeasureTheory/Constructions/Pi.lean | 931 | 935 | |
/-
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,
Kim Morrison
-/
import Mathlib.Data.List.Basic
/-!
# Lattice structure of lists
This files pro... | | b :: l₁, l₂ => by
| Mathlib/Data/List/Lattice.lean | 199 | 199 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kim Morrison
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Algebra.Group.InjSurj
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Tactic.FastInstance
impo... | @[simp]
lemma mapRange_mapRange (e₁ : N → P) (e₂ : M → N) (he₁ he₂) (f : α →₀ M) :
mapRange e₁ he₁ (mapRange e₂ he₂ f) = mapRange (e₁ ∘ e₂) (by simp [*]) f := ext fun _ ↦ rfl
| Mathlib/Data/Finsupp/Defs.lean | 313 | 315 |
/-
Copyright (c) 2021 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.SpecificLimits.Basic
/-!
# A collection of specific asymptotic results
This fil... |
theorem tendsto_pow_div_pow_atTop_atTop {p q : ℕ} (hpq : q < p) :
Tendsto (fun x : 𝕜 => x ^ p / x ^ q) atTop atTop := by
rw [tendsto_congr' pow_div_pow_eventuallyEq_atTop]
apply tendsto_zpow_atTop_atTop
| Mathlib/Analysis/Asymptotics/SpecificAsymptotics.lean | 49 | 53 |
/-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib... | exact hx
| Mathlib/Analysis/SpecialFunctions/Log/Base.lean | 215 | 216 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Multiset.ZeroCons
/-!
# Basic results on multisets
-/
-- No algebra should be required
assert_not_exists Monoid
universe v
open List S... | Mathlib/Data/Multiset/Basic.lean | 2,666 | 2,668 | |
/-
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
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Data.Set.Function
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Algebra.Order.Monoid.Defs... | theorem Ici_add_bij : BijOn (· + d) (Ici a) (Ici (a + d)) := by
refine
⟨fun x h => add_le_add_right (mem_Ici.mp h) _, (add_left_injective d).injOn, fun _ h => ?_⟩
obtain ⟨c, rfl⟩ := exists_add_of_le (mem_Ici.mp h)
rw [mem_Ici, add_right_comm, add_le_add_iff_right] at h
exact ⟨a + c, h, by rw [add_right_comm... | Mathlib/Algebra/Order/Interval/Set/Monoid.lean | 27 | 32 |
/-
Copyright (c) 2020 Kenji Nakagawa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenji Nakagawa, Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.LinearAlgebra.FreeModule.PID
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.Bilin... |
variable (A K) [IsDomain C]
/-- If `L` is a finite separable extension of `K = Frac(A)`, where `A` is a Dedekind domain,
the integral closure `C` of `A` in `L` is a Dedekind domain.
This cannot be an instance since `A`, `K` or `L` can't be inferred. See also the instance
`integralClosure.isDedekindDomain_fractionRin... | Mathlib/RingTheory/DedekindDomain/IntegralClosure.lean | 210 | 217 |
/-
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... | rw [stepBound, ← Nat.div_div_eq_div_mul]
exact Nat.div_mul_le_self _ _
rw [Nat.add_sub_of_le this] at hucard
rw [(hP.card_parts_eq_average hu).resolve_left hucard, mul_add, mul_one, ← add_assoc, ← add_mul,
Nat.sub_add_cancel a_add_one_le_four_pow_parts_card, ← add_assoc, mul_comm,
Nat.add_sub_of_le ... | Mathlib/Combinatorics/SimpleGraph/Regularity/Bound.lean | 145 | 150 |
/-
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.Composition.MapComap
import Mathlib.Probability.Martingale.Convergence
import Mathlib.Probability.Process.PartitionFiltration
/-!
# Ker... | (n : ℕ) (a : α) {s : Set β} (hs : MeasurableSet s) {A : Set γ}
(hA : MeasurableSet[countableFiltration γ n] A) :
∫ x in A, densityProcess κ ν n a x s ∂(ν a) = (κ a).real (A ×ˢ s) := by
have : IsFiniteKernel κ := isFiniteKernel_of_isFiniteKernel_fst (h := isFiniteKernel_of_le hκν)
obtain ⟨S, hS_subset, r... | Mathlib/Probability/Kernel/Disintegration/Density.lean | 241 | 268 |
/-
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.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Order.WithBot
/-!
# Intervals in `WithTop α` and `WithBot α`
In this file we ... | Mathlib/Order/Interval/Set/WithBotTop.lean | 171 | 171 | |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.GroupWithZero.Action.Defs
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.NoZeroSMulDiviso... | · exact Commute.sum_right _ _ _ (fun i hi ↦ h_comm _ _ (by simp) (by simp [hi]))
· apply hnp; simp
· exact ih (fun i hi ↦ hnp i (by simp [hi]))
(fun i j hi hj ↦ h_comm i j (by simp [hi]) (by simp [hj]))
theorem isNilpotent_finsum {ι : Type*} {f : ι → R}
| Mathlib/RingTheory/Nilpotent/Basic.lean | 161 | 166 |
/-
Copyright (c) 2018 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton
-/
import Mathlib.Topology.Bases
import Mathlib.Topology.DenseEmbedding
import Mathlib.Topology.Connected.TotallyDisconnected
/-! # Stone-Čech compactification
Construction ... | section Embedding
theorem ultrafilter_pure_injective : Function.Injective (pure : α → Ultrafilter α) := by
intro x y h
have : {x} ∈ (pure x : Ultrafilter α) := singleton_mem_pure
rw [h] at this
exact (mem_singleton_iff.mp (mem_pure.mp this)).symm
| Mathlib/Topology/StoneCech.lean | 128 | 135 |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Nat.Find
import Mathlib.Data.Stream.Init
import Mathlib.Tactic.Common
/-!
# Coinductive formalization of unbounded computations.
This fil... | LiftRel R (think ca) cb ↔ LiftRel R ca cb :=
and_congr (forall_congr' fun _ => imp_congr ⟨of_think_mem, think_mem⟩ Iff.rfl)
| Mathlib/Data/Seq/Computation.lean | 987 | 988 |
/-
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.Order.Filter.Prod
/-!
# N-ary maps of filter
This file defines the binary and ternary maps of filters. This is mostly useful to define pointwise
operatio... | Mathlib/Order/Filter/NAry.lean | 323 | 324 | |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.CategoryTheory.Sites.Sieves
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.Mono
/-!
# The sheaf condition for a presieve
We define what it means fo... |
/-- Given a family of elements of a sieve `S` on `X`, and a map `Y ⟶ X`, we can obtain a
family of elements of `S.pullback f` by taking the same elements.
-/
| Mathlib/CategoryTheory/Sites/IsSheafFor.lean | 300 | 303 |
/-
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.CharP.Reduced
import Mathlib.RingTheory.IntegralDomain
-- TODO: remove Mathlib.Algebra.CharP.Reduced and move the last two lemmas to Lemmas
/-... | Mathlib/RingTheory/RootsOfUnity/Basic.lean | 611 | 612 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.Nat.SuccPred
import Mathlib.Order.SuccPred.Initial... | Mathlib/SetTheory/Ordinal/Arithmetic.lean | 1,613 | 1,617 | |
/-
Copyright (c) 2024 Jeremy Tan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Tan
-/
import Mathlib.Data.Int.Interval
import Mathlib.Data.Int.ModEq
import Mathlib.Data.Nat.Count
import Mathlib.Data.Rat.Floor
import Mathlib.Order.Interval.Finset.Nat
/-!
# Cou... | theorem Ico_filter_modEq_card (v : ℤ) :
#{x ∈ Ico a b | x ≡ v [ZMOD r]} = max (⌈(b - v) / (r : ℚ)⌉ - ⌈(a - v) / (r : ℚ)⌉) 0 := by
simp [Ico_filter_modEq_eq, Ico_filter_dvd_eq, toNat_eq_max, hr]
| Mathlib/Data/Int/CardIntervalMod.lean | 71 | 73 |
/-
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.Homotopy
import Mathlib.Algebra.Ring.NegOnePow
import Mathlib.Algebra.Category.Grp.Preadditive
import Mathlib.Tactic.Linarith
import Mathlib.Cat... | (Cochain.mk v).v p q hpq = v p q hpq := rfl
| Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean | 88 | 89 |
/-
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, Kexing Ying
-/
import Mathlib.Topology.Semicontinuous
import Mathlib.MeasureTheory.Function.AEMeasurableSequence
import Mathlib.MeasureTheory.Order.Lat... | · refine sUnion_eq_univ_iff.2 fun x => ?_
rcases hsd.exists_le' hsb x with ⟨l, hls, hlx⟩
rcases hsd.exists_gt x with ⟨u, hus, hxu⟩
exact ⟨_, ⟨l, hls, u, hus, hlx.trans_lt hxu, rfl⟩, hlx, hxu⟩
· rintro _ ⟨l, -, u, -, hlt, rfl⟩
exact hμ hlt
· rintro _ ⟨l, u, hlt, rfl⟩
exact h hlt
/-- Two measur... | Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean | 382 | 403 |
/-
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... | def IsRadical (I : Ideal R) : Prop :=
I.radical ≤ I
| Mathlib/RingTheory/Ideal/Operations.lean | 710 | 711 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.UniformSpace.Cauchy
/-!
# Uniform convergence
A sequence of functions `Fₙ` (with values in a metric space) converges uniformly on a se... | Mathlib/Topology/UniformSpace/UniformConvergence.lean | 945 | 950 | |
/-
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.Divisibility.Hom
import Mathlib.Algebra.Group.Even
import Mathlib.Algebra.Group.Nat.Hom
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.Alg... |
variable {A F : Type*} [MulZeroOneClass A] [FunLike F ℕ A]
/-- If two `MonoidWithZeroHom`s agree on the positive naturals they are equal. -/
theorem ext_nat'' [ZeroHomClass F ℕ A] (f g : F) (h_pos : ∀ {n : ℕ}, 0 < n → f n = g n) :
f = g := by
apply DFunLike.ext
| Mathlib/Data/Nat/Cast/Basic.lean | 110 | 116 |
/-
Copyright (c) 2017 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Mario Carneiro
-/
import Mathlib.Algebra.Ring.CharZero
import Mathlib.Algebra.Star.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Order.Interval.Set.UnorderedInterva... | inv_zero := Complex.inv_zero
nnqsmul := (· • ·)
| Mathlib/Data/Complex/Basic.lean | 682 | 683 |
/-
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
-/
import Mathlib.Data.Set.Function
import Mathlib.Order.Interval.Set.OrdConnected
/-!
# Projection of a line onto a closed interval
Given a linearly... | theorem range_projIci : range (projIci a) = univ := projIci_surjective.range_eq
| Mathlib/Order/Interval/Set/ProjIcc.lean | 124 | 124 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Batteries.Data.String.Lemmas
import Mathlib.Data.List.Lex
import Mathlib.Data.Char
import Mathlib.Algebra.Order.Group.Nat
/-!
# Strings
Supplementary theorem... | lt_iff_le_not_le a b := by
simp only [lt_iff_toList_lt, le_iff_toList_le, lt_iff_le_not_le]
le_antisymm a b := by
simp only [le_iff_toList_le, ← toList_inj]
apply le_antisymm
le_total a b := by
simp only [le_iff_toList_le]
| Mathlib/Data/String/Basic.lean | 143 | 149 |
/-
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.Weights.Basic
import Mathlib.LinearAlgebra.Trace
import Mathlib.LinearAlgebra.FreeModule.PID
/-!
# Lie modules with linear weights
Given a Lie ... | rw [LieSubmodule.coe_bracket]
rfl
/-- By Engel's theorem, if `M` is Noetherian, the shifted action `⁅x, m⁆ - χ x • m` makes the
`χ`-weight space into a nilpotent Lie module. -/
instance [IsNoetherian R M] : IsNilpotent L (shiftedGenWeightSpace R L M χ) :=
LieModule.isNilpotent_iff_forall'.mpr fun x ↦ isNilpotent... | Mathlib/Algebra/Lie/Weights/Linear.lean | 217 | 228 |
/-
Copyright (c) 2020 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Algebra.Group.Conj
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.Group.Subgroup.Ker
/-!
# Basic results on subgroups
We prove basic results... | Mathlib/Algebra/Group/Subgroup/Basic.lean | 2,111 | 2,112 | |
/-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.LinearAlgebra.Matrix.ZPow
import Mathlib.Data.Matrix.ConjTranspose
/-! # Hermitian matrices
Th... |
end InvolutiveStar
section AddMonoid
variable [AddMonoid α] [StarAddMonoid α]
| Mathlib/LinearAlgebra/Matrix/Hermitian.lean | 112 | 117 |
/-
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.Logic.Encodable.Lattice
import Mathlib.Order.Filter.AtTopBot.Finset
import Mathlib.Topology.Algebra.InfiniteSum.Group
/-!
# Infinite sums and product... | namespace HasProd
section ContinuousMul
| Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | 62 | 65 |
/-
Copyright (c) 2024 Josha Dekker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Josha Dekker
-/
import Mathlib.Order.Filter.Tendsto
import Mathlib.Order.Filter.Finite
import Mathlib.Order.Filter.CountableInter
import Mathlib.SetTheory.Cardinal.Regular
import Mathlib... |
theorem EventuallyLE.cardinal_bUnion {S : Set ι} (hS : #S < c)
{s t : ∀ i ∈ S, Set α} (h : ∀ i hi, s i hi ≤ᶠ[l] t i hi) :
⋃ i ∈ S, s i ‹_› ≤ᶠ[l] ⋃ i ∈ S, t i ‹_› := by
simp only [biUnion_eq_iUnion]
| Mathlib/Order/Filter/CardinalInter.lean | 123 | 127 |
/-
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... |
@[norm_cast, simp]
| Mathlib/Algebra/Lie/Basic.lean | 829 | 830 |
/-
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.Homology.Embedding.IsSupported
import Mathlib.Algebra.Homology.Additive
import Mathlib.Algebra.Homology.Opposite
/-!
# The extension of a homological co... |
lemma mapX_none {i : Option ι} (hi : i = none) :
mapX φ i = 0 := by subst hi; rfl
end extend
/-- Given `K : HomologicalComplex C c` and `e : c.Embedding c'`,
| Mathlib/Algebra/Homology/Embedding/Extend.lean | 101 | 107 |
/-
Copyright (c) 2014 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Leonardo de Moura, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Field.Defs
/-!
# Linear ordered (semi)fields
A ... | Mathlib/Algebra/Order/Field/Defs.lean | 79 | 80 | |
/-
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.Inverse
import Mathlib.Analysis.SpecialFunctions... | Mathlib/Analysis/SpecialFunctions/Trigonometric/InverseDeriv.lean | 176 | 177 | |
/-
Copyright (c) 2019 Calle Sönne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology... | @[simp]
theorem two_nsmul_coe_pi : (2 : ℕ) • (π : Angle) = 0 := by simp [← natCast_mul_eq_nsmul]
@[simp]
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 134 | 137 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Ordinal.Family
/-! # Ordinal exponential
In this file we define the power function and the lo... |
@[simp, norm_cast]
theorem opow_natCast (a : Ordinal) (n : ℕ) : a ^ (n : Ordinal) = a ^ n := by
induction n with
| Mathlib/SetTheory/Ordinal/Exponential.lean | 109 | 112 |
/-
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.LinearAlgebra.BilinearForm.TensorProduct
import Mathlib.LinearAlgebra.QuadraticForm.Basic
/-!
# The quadratic form on a tensor product
## Main definitions
... |
end QuadraticMap
namespace QuadraticForm
variable (R A) in
| Mathlib/LinearAlgebra/QuadraticForm/TensorProduct.lean | 77 | 82 |
/-
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... | exact FaithfulSMul.algebraMap_injective T S
| Mathlib/Algebra/Polynomial/Roots.lean | 447 | 448 |
/-
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 closedBall_comp (p : Seminorm 𝕜₂ E₂) (f : E →ₛₗ[σ₁₂] E₂) (x : E) (r : ℝ) :
(p.comp f).closedBall x r = f ⁻¹' p.closedBall (f x) r := by
| Mathlib/Analysis/Seminorm.lean | 741 | 743 |
/-
Copyright (c) 2024 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.LinearAlgebra.DirectSum.Finsupp
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
/-!
# Some results on tensor product of s... | theorem rTensorOne_tmul (y : R) (m : M) : M.rTensorOne (m ⊗ₜ[R] algebraMap R _ y) = y • m :=
M.rTensorOne'_tmul y m
variable {M} in
@[simp]
theorem rTensorOne_tmul_one (m : M) : M.rTensorOne (m ⊗ₜ[R] 1) = m :=
| Mathlib/LinearAlgebra/TensorProduct/Submodule.lean | 218 | 223 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Logic.Encodable.Pi
import Mathlib.Logic.Function.Iterate
/-!
# The primitive recursive functions
The primitive ... | Mathlib/Computability/Primrec.lean | 1,558 | 1,563 | |
/-
Copyright (c) 2022 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.CharP.Lemmas
import Mathlib.Data.Fintype.Units
import Mathlib.GroupTheory.OrderOfElement
/-!
# Multiplicative... | exact (injective_ringHomComp hf).eq_iff
| Mathlib/NumberTheory/MulChar/Basic.lean | 448 | 449 |
/-
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.Quaternion
import Mathlib.Tactic.Ring
/-!
# Basis on a quaternion-like algebra
## Main definitions
* `QuaternionAlgebra.Basis A c₁ c₂ c₃`: a basis... | theorem lift_mul (x y : ℍ[R,c₁,c₂,c₃]) : q.lift (x * y) = q.lift x * q.lift y := by
simp only [lift, Algebra.algebraMap_eq_smul_one]
simp_rw [add_mul, mul_add, smul_mul_assoc, mul_smul_comm, one_mul, mul_one, smul_smul]
| Mathlib/Algebra/QuaternionBasis.lean | 120 | 122 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Vector.Defs
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.OfFn
import Mathlib.Data.List.Scan
import Mathlib.Control.Applicative
import M... |
@[simp] theorem length_val (v : Vector α n) : v.val.length = n := v.2
| Mathlib/Data/Vector/Basic.lean | 71 | 72 |
/-
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, Yury Kudryashov
-/
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Dia... | theorem volume_emetric_ball (a : ℝ) (r : ℝ≥0∞) : volume (EMetric.ball a r) = 2 * r := by
rcases eq_or_ne r ∞ with (rfl | hr)
| Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | 140 | 141 |
/-
Copyright (c) 2023 Josha Dekker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Josha Dekker
-/
import Mathlib.Topology.Bases
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.Compactness.SigmaCompact
/-!
# Lindelöf sets and Lindelöf spaces
## Mai... | theorem IsLindelof.elim_countable_subcover_image {b : Set ι} {c : ι → Set X} (hs : IsLindelof s)
(hc₁ : ∀ i ∈ b, IsOpen (c i)) (hc₂ : s ⊆ ⋃ i ∈ b, c i) :
∃ b', b' ⊆ b ∧ Set.Countable b' ∧ s ⊆ ⋃ i ∈ b', c i := by
simp only [Subtype.forall', biUnion_eq_iUnion] at hc₁ hc₂
rcases hs.elim_countable_subcover (fun... | Mathlib/Topology/Compactness/Lindelof.lean | 245 | 252 |
/-
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.LeftRight
import Mathlib.Topology.Separation.Hausdorff
/-!
# Order-closed topologies
In this file we ... | theorem interior_Ioo : interior (Ioo a b) = Ioo a b :=
isOpen_Ioo.interior_eq
theorem Ioo_subset_closure_interior : Ioo a b ⊆ closure (interior (Ioo a b)) := by
simp only [interior_Ioo, subset_closure]
theorem Ioo_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ioo a b ∈ 𝓝 x :=
IsOpen.mem_nhds isOpen_Ioo ⟨ha,... | Mathlib/Topology/Order/OrderClosed.lean | 813 | 820 |
/-
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.Data.Set.SymmDiff
import Mathlib.Order.SuccPred.Relation
import Mathlib.Topology.Irreducible
/-!
# Connected subsets ... | Mathlib/Topology/Connected/Basic.lean | 946 | 948 | |
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Eric Wieser
-/
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Dual.Lemmas
import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
import Mathlib.Li... | simpa [-lift_cRank_submatrix] using congr_arg Cardinal.toENat <| A.lift_cRank_submatrix em en
theorem eRank_reindex {m₀ : Type um} {n : Type un} (A : Matrix m n R) (em : m ≃ m₀) (en : n ≃ n₀) :
| Mathlib/Data/Matrix/Rank.lean | 244 | 246 |
/-
Copyright (c) 2019 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, François Dupuis
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Order.Filter.Extr
import Mathlib.Tactic.NormNum
/-!
# Convex and concave functions
This... | ConcaveOn 𝕜 s f :=
LinearOrder.convexOn_of_lt (β := βᵒᵈ) hs hf
/-- For a function on a convex set in a linearly ordered space (where the order and the algebraic
structures aren't necessarily compatible), in order to prove that it is strictly convex, it suffices
to verify the inequality `f (a • x + b • y) < a • ... | Mathlib/Analysis/Convex/Function.lean | 415 | 427 |
/-
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.Measure.Typeclasses.SFinite
/-!
# Restriction of a measure to a sub-σ-algebra
## Main definitions
* `MeasureTheory.Measure.trim`: restric... | simp_rw [Measure.trim]
exact @le_toMeasure_apply _ m _ _ _
| Mathlib/MeasureTheory/Measure/Trim.lean | 53 | 54 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Batteries.Tactic.Congr
import Mathlib.Data.Option.Basic
import Mathlib.Data.Prod.Basic
import Mathlib.Data.Set.Subsingleton
import Mathlib.Dat... | @[simp]
| Mathlib/Data/Set/Image.lean | 890 | 890 |
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