Context stringlengths 227 76.5k | target stringlengths 0 11.6k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 16 3.69k |
|---|---|---|---|---|
/-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.Sieves
import Mathlib.CategoryTheory.EffectiveEpi.Basic
/-!
# Effective epimorphic sieves
We define the notion of effective epimorphic (... | lemma Sieve.generateSingleton_eq {X Y : C} (f : Y ⟶ X) :
Sieve.generate (Presieve.singleton f) = Sieve.generateSingleton f := by
ext Z g
constructor
· rintro ⟨W,i,p,⟨⟩,rfl⟩
exact ⟨i,rfl⟩
· rintro ⟨g,h⟩
exact ⟨Y,g,f,⟨⟩,h⟩
| Mathlib/CategoryTheory/Sites/EffectiveEpimorphic.lean | 48 | 55 |
/-
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.Data.Nat.Sqrt
import Mathlib.Tactic.Common
import Mathlib.Data.Int.Init
/-!
# Square root of integers
This file defines the square root function on integers. `... | theorem exists_mul_self (x : ℤ) : (∃ n, n * n = x) ↔ sqrt x * sqrt x = x :=
⟨fun ⟨n, hn⟩ => by rw [← hn, sqrt_eq, ← Int.natCast_mul, natAbs_mul_self], fun h => ⟨sqrt x, h⟩⟩
| Mathlib/Data/Int/Sqrt.lean | 30 | 31 |
/-
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.Star.Basic
import Mathlib.Algebra.Ring.Pi
/-!
# `star` on pi types
We put a `Star` structure on pi types that operates elementwise, such that it de... | end Function
| Mathlib/Algebra/Star/Pi.lean | 79 | 81 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
-/
import Mathlib.MeasureTheory.Integral.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Lebesgue.Countable
import Mathlib.MeasureTheory.Integral.Le... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 1,000 | 1,002 | |
/-
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.Finset.Card
import Mathlib.Data.Finset.Lattice.Union
import Mathlib.Data.Multiset.Powerset
import Mathlib.Data.Set.Pairwise.Lattice
/-!
# The pow... | alias ⟨_, powersetCard_nonempty_of_le⟩ := powersetCard_nonempty
| Mathlib/Data/Finset/Powerset.lean | 250 | 251 |
/-
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... |
theorem stepBound_mono : Monotone stepBound := fun _ _ h =>
| Mathlib/Combinatorics/SimpleGraph/Regularity/Bound.lean | 43 | 44 |
/-
Copyright (c) 2022 Julian Berman. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Julian Berman
-/
import Mathlib.GroupTheory.PGroup
import Mathlib.LinearAlgebra.Quotient.Defs
/-!
# Torsion groups
This file defines torsion groups, i.e. groups where all elements hav... | section Module
-- A (semi/)ring of scalars and a commutative monoid of elements
| Mathlib/GroupTheory/Torsion.lean | 130 | 132 |
/-
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... | theorem ofFn_mul' {m n} (f : Fin (m * n) → α) :
List.ofFn f = List.flatten (List.ofFn fun i : Fin n => List.ofFn fun j : Fin m => f ⟨m * i + j,
calc
| Mathlib/Data/List/OfFn.lean | 86 | 88 |
/-
Copyright (c) 2024 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.Data.Matroid.Constructions
import Mathlib.Data.Set.Notation
/-!
# Maps between matroids
This file defines maps and comaps, which move a matroid on one ty... | inferInstanceAs (M.map f f.injective.injOn).RankPos
end mapEmbedding
| Mathlib/Data/Matroid/Map.lean | 580 | 582 |
/-
Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
import Mathlib.Topology.Order.Compact
import Mathlib.Topology.MetricSpace.ProperSpace
import M... |
theorem disjoint_nhdsSet_cobounded {s : Set α} (hs : IsCompact s) : Disjoint (𝓝ˢ s) (cobounded α) :=
| Mathlib/Topology/MetricSpace/Bounded.lean | 223 | 224 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.Group.Action.Units
import Mathlib.Algebra.Group.Nat.Units
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra... | theorem isCoprime_zero_right : IsCoprime x 0 ↔ IsUnit x :=
isCoprime_comm.trans isCoprime_zero_left
theorem not_isCoprime_zero_zero [Nontrivial R] : ¬IsCoprime (0 : R) 0 :=
| Mathlib/RingTheory/Coprime/Basic.lean | 62 | 65 |
/-
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 | 1,889 | 1,892 | |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.InnerProductSpace.Defs
impor... | Mathlib/Analysis/InnerProductSpace/Basic.lean | 1,642 | 1,652 | |
/-
Copyright (c) 2022 Wrenna Robson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Wrenna Robson
-/
import Mathlib.Topology.MetricSpace.Basic
/-!
# Infimum separation
This file defines the extended infimum separation of a set. This is approximately dual to the
diame... | Mathlib/Topology/MetricSpace/Infsep.lean | 509 | 517 | |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.GDelta.Basic
/-!
# Baire spaces
A topological space is called a *Baire space*
if a countable intersection of dense open subsets is den... |
/-- Baire theorem: a countable intersection of dense open sets is dense. Formulated here with
an index set which is a countable type. -/
theorem dense_iInter_of_isOpen [Countable ι] {f : ι → Set X} (ho : ∀ i, IsOpen (f i))
| Mathlib/Topology/Baire/Lemmas.lean | 60 | 63 |
/-
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.compl_mem_sets_of_nhdsWithin (hs : IsLindelof s) {f : Filter X}
[CountableInterFilter f] (hf : ∀ x ∈ s, ∃ t ∈ 𝓝[s] x, tᶜ ∈ f) : sᶜ ∈ f := by
refine hs.compl_mem_sets fun x hx ↦ ?_
rw [← disjoint_principal_right, disjoint_right_comm, (basis_sets _).disjoint_iff_left]
exact hf x hx
| Mathlib/Topology/Compactness/Lindelof.lean | 60 | 64 |
/-
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.BigOperators.Expect
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Field.Canonical
import Mathlib.Algebra.O... | Mathlib/Data/NNReal/Basic.lean | 721 | 722 | |
/-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin, Lu-Ming Zhang
-/
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Algebra.Pi
import Mathlib.Algebra.BigOp... | Mathlib/Data/Matrix/Basic.lean | 1,991 | 1,992 | |
/-
Copyright (c) 2017 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Keeley Hoek
-/
import Mathlib.Algebra.NeZero
import Mathlib.Data.Int.DivMod
import Mathlib.Logic.Embedding.Basic
import Mathlib.Logic.Equiv.Set
import Mathlib.Tactic.... | cases hi <| Nat.lt_trans i.castSucc_lt_succ hj
· exact succ_injective _ hij
| Mathlib/Data/Fin/Basic.lean | 1,010 | 1,011 |
/-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa, Junyan Xu
-/
import Mathlib.Data.DFinsupp.Defs
/-!
# Locus of unequal values of finitely supported dependent functions
Let `N : α → Type*` be a type family, assume that `N ... | theorem nonempty_neLocus_iff {f g : Π₀ a, N a} : (f.neLocus g).Nonempty ↔ f ≠ g :=
Finset.nonempty_iff_ne_empty.trans neLocus_eq_empty.not
theorem neLocus_comm : f.neLocus g = g.neLocus f := by
| Mathlib/Data/DFinsupp/NeLocus.lean | 56 | 59 |
/-
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.Multiset.Dedup
import Mathlib.Data.List.Infix
/-!
# Preparations for defining operations on `Finset`.
The operations here ignore multiplicities,... | @[simp]
theorem coe_ndunion (l₁ l₂ : List α) : @ndunion α _ l₁ l₂ = (l₁ ∪ l₂ : List α) :=
rfl
| Mathlib/Data/Multiset/FinsetOps.lean | 127 | 129 |
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Order.Filter.CountablyGenerated
import Mathlib.Order.Filter.Prod
import Mathlib.Order.Filter.Ultrafilter.Defs
/-!
# Subsingleton filters
We say that... | /-- A filter is a subsingleton iff it is equal to `⊥` or to `pure a` for some `a`. -/
theorem subsingleton_iff_bot_or_pure : l.Subsingleton ↔ l = ⊥ ∨ ∃ a, l = pure a := by
refine ⟨fun hl ↦ ?_, ?_⟩
· exact (eq_or_neBot l).imp_right (@Subsingleton.exists_eq_pure _ _ · hl)
| Mathlib/Order/Filter/Subsingleton.lean | 58 | 61 |
/-
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.Countable
import Mathlib.MeasureTheory.Measure.Decomposition.Exhaustion
import Mathlib.MeasureTheory.Me... |
theorem withDensity_smul_measure (r : ℝ≥0∞) (f : α → ℝ≥0∞) :
(r • μ).withDensity f = r • μ.withDensity f := by
ext s hs
simp [withDensity_apply, hs]
| Mathlib/MeasureTheory/Measure/WithDensity.lean | 130 | 135 |
/-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Data.Set.Finite.Lemmas
import Mathlib.ModelTheory.Substructures
/-!
# Finitely Generated First-Order Structures
This file defines what it means for a... | theorem FG.cg {N : L.Substructure M} (h : N.FG) : N.CG := by
obtain ⟨s, hf, rfl⟩ := fg_def.1 h
exact ⟨s, hf.countable, rfl⟩
theorem cg_iff_empty_or_exists_nat_generating_family {N : L.Substructure M} :
N.CG ↔ N = (∅ : Set M) ∨ ∃ s : ℕ → M, closure L (range s) = N := by
rw [cg_def]
constructor
· rintro ⟨S... | Mathlib/ModelTheory/FinitelyGenerated.lean | 116 | 135 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Eric Wieser
-/
import Mathlib.Algebra.DirectSum.Internal
import Mathlib.Algebra.GradedMonoid
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Algebra.MvPolyn... | (X (R := R) i ^ n).IsHomogeneous n := by
simpa only [one_mul] using (isHomogeneous_X _ _).pow n
| Mathlib/RingTheory/MvPolynomial/Homogeneous.lean | 201 | 203 |
/-
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.Combinatorics.SimpleGraph.Path
import Mathlib.Combinatorics.SimpleGraph.Operations
import Mathlib.Data.Finset.Pairwise
import M... | fun t ht hsub => by
by_contra hc
have fint : Fintype t := ofFinite ↑t
have ne : s ≠ t.toFinset := fun a ↦ by subst a; simp_all[Set.coe_toFinset, not_true_eq_false]
| Mathlib/Combinatorics/SimpleGraph/Clique.lean | 657 | 660 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Robert Y. Lewis
-/
import Mathlib.Algebra.Order.CauSeq.Basic
import Mathlib.Algebra.Ring.Action.Rat
import Mathlib.Tactic.FastInstance
/-!
# Cauchy completion
This fi... |
theorem le_lim {f : CauSeq α abs} {x : α} (h : CauSeq.const abs x ≤ f) : x ≤ lim f :=
CauSeq.const_le.1 <| CauSeq.le_of_le_of_eq h (equiv_lim f)
theorem lt_lim {f : CauSeq α abs} {x : α} (h : CauSeq.const abs x < f) : x < lim f :=
CauSeq.const_lt.1 <| CauSeq.lt_of_lt_of_eq h (equiv_lim f)
theorem lim_lt {f : Cau... | Mathlib/Algebra/Order/CauSeq/Completion.lean | 397 | 405 |
/-
Copyright (c) 2023 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash, Deepro Choudhury, Scott Carnahan
-/
import Mathlib.LinearAlgebra.PerfectPairing.Basic
import Mathlib.LinearAlgebra.Reflection
/-!
# Root data and root systems
This file con... | Module.bijOn_reflection_of_mapsTo _ <| P.mapsTo_reflection_root i
| Mathlib/LinearAlgebra/RootSystem/Defs.lean | 288 | 289 |
/-
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... | Mathlib/Data/Finset/PImage.lean | 106 | 108 | |
/-
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.Topology.UniformSpace.CompleteSeparated
import Mathlib.Topology.EMetricSpace.Lipschitz
import Mathlib.Topology.MetricSpace.Basic
import Mathlib.Topol... | theorem isClosed_range {α β : Type*} [PseudoEMetricSpace α] [EMetricSpace β] [CompleteSpace α]
{f : α → β} {K : ℝ≥0} (hf : AntilipschitzWith K f) (hfc : UniformContinuous f) :
IsClosed (range f) :=
(hf.isComplete_range hfc).isClosed
| Mathlib/Topology/MetricSpace/Antilipschitz.lean | 157 | 161 |
/-
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.Logic.Basic
import Mathlib.Tactic.Positivity.Basic
/-!
# Algebraic order homomorphism classes
This file defines hom classes for common properties at the ... | extends AddGroupSeminormClass F α β, SubmultiplicativeHomClass F α β
/-- `RingNormClass F α` states that `F` is a type of `β`-valued norms on the ring `α`.
| Mathlib/Algebra/Order/Hom/Basic.lean | 288 | 291 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Group.Action.Pi
import Mathlib.Algebra.Module.Hom
import Mathlib.GroupTheory.GroupAction.Ri... | end NonUnitalNonAssocSemiring
section NonUnitalNonAssocCommSemiring
variable [NonUnitalNonAssocCommSemiring α]
/-
Left and right multiplication coincide as α is commutative
-/
local notation "L" => AddMonoid.End.mulLeft
| Mathlib/Algebra/Ring/CentroidHom.lean | 499 | 509 |
/-
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 | 391 | 393 | |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Frédéric Dupuis, Heather Macbeth
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Analysis.InnerProductSpace.Orthogonal
import Mathlib.Analysis.InnerProductSpace.Sy... |
/-- The orthogonal projection onto a complete subspace. -/
def orthogonalProjection : E →L[𝕜] K :=
LinearMap.mkContinuous
{ toFun := fun v => ⟨K.orthogonalProjectionFn v, orthogonalProjectionFn_mem v⟩
map_add' := fun x y => by
have hm : K.orthogonalProjectionFn x + K.orthogonalProjectionFn y ∈ K :... | Mathlib/Analysis/InnerProductSpace/Projection.lean | 440 | 447 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Johan Commelin, Patrick Massot
-/
import Mathlib.Algebra.GroupWithZero.InjSurj
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.GroupWithZero.WithZero
impo... |
lemma zero_eq_bot : (0 : WithZero α) = ⊥ := rfl
@[simp, norm_cast] lemma coe_lt_coe : (a : WithZero α) < b ↔ a < b := WithBot.coe_lt_coe
| Mathlib/Algebra/Order/GroupWithZero/Canonical.lean | 261 | 264 |
/-
Copyright (c) 2024 Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Merten
-/
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
import Mathlib.CategoryTheory.Limits.FintypeCat
import Mathlib.CategoryTheory.Lim... | simp [fiberPullbackEquiv]
change ((Types.pullbackIsoPullback _ _).inv ≫ _ ≫
(F ⋙ FintypeCat.incl).map (pullback.snd f g)) _ = _
erw [PreservesPullback.iso_inv_snd, Types.pullbackIsoPullback_inv_snd]
/-- The fiber of the binary product is the binary product of the fibers. -/
noncomputable def fiberBinaryProdu... | Mathlib/CategoryTheory/Galois/Basic.lean | 273 | 279 |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Solvable
import Mathlib.Algebra.Lie.Quotient
import Mathlib.Algebra.Lie.Normalizer
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.... | have h' : (⊤ : LieIdeal R L).map f = ⊤ := by
rw [← f.idealRange_eq_map]
exact f.idealRange_eq_top_of_surjective h
induction k with
| zero => simp only [LieModule.lowerCentralSeries_zero]; exact h'
| succ k ih => simp only [LieModule.lowerCentralSeries_succ, LieIdeal.map_bracket_eq f h, ih, h']
| Mathlib/Algebra/Lie/Nilpotent.lean | 835 | 840 |
/-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.AlgebraicTopology.SimplicialObject.Basic
import Mathlib.CategoryTheory.Comma.Arrow
import Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks
import Mathlib.Cat... | Mathlib/AlgebraicTopology/CechNerve.lean | 427 | 430 | |
/-
Copyright (c) 2021 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Bhavik Mehta
-/
import Mathlib.Analysis.Calculus.Deriv.Support
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
import Mathlib.MeasureTheory.Function.Jacobian
imp... | Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean | 1,332 | 1,349 | |
/-
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.Data.Finset.BooleanAlgebra
import Mathlib.Data.Set.Piecewise
import Mathlib.Order.Interval.Set.Basic
/-!
# Functions defined piecewise on a fins... |
lemma piecewise_le_of_le_of_le (hf : f ≤ h) (hg : g ≤ h) : s.piecewise f g ≤ h := fun x =>
piecewise_cases s f g (· ≤ h x) (hf x) (hg x)
lemma le_piecewise_of_le_of_le (hf : h ≤ f) (hg : h ≤ g) : h ≤ s.piecewise f g := fun x =>
| Mathlib/Data/Finset/Piecewise.lean | 151 | 155 |
/-
Copyright (c) 2024 Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Merten
-/
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
import Mathlib.CategoryTheory.Limits.FintypeCat
import Mathlib.CategoryTheory.Lim... | (PreservesLimitPair.iso (F ⋙ FintypeCat.incl) X Y ≪≫
Types.binaryProductIso (F.obj X) (F.obj Y)).toEquiv
@[simp]
lemma fiberBinaryProductEquiv_symm_fst_apply {X Y : C} (x : F.obj X) (y : F.obj Y) :
F.map prod.fst ((fiberBinaryProductEquiv F X Y).symm (x, y)) = x := by
| Mathlib/CategoryTheory/Galois/Basic.lean | 281 | 286 |
/-
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.Order.Atoms
import Mathlib.Order.OrderIsoNat
import Mathlib.Order.RelIso.Set
import Mathlib.Order.SupClosed
import Mathlib.Order.SupIndep
import Mathlib.Orde... | rw [iSup, h.directedOn_range.inf_sSup_eq, iSup_range]
protected theorem Directed.iSup_inf_eq (h : Directed (· ≤ ·) f) :
| Mathlib/Order/CompactlyGenerated/Basic.lean | 399 | 401 |
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Floris van Doorn
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Order.Filter.AtTopBot.Basic
import Mathlib.Order.Filter.Subsingleton
/-!
# Functions that ar... |
end EventuallyConst
lemma eventuallyConst_atTop [SemilatticeSup α] [Nonempty α] :
EventuallyConst f atTop ↔ (∃ i, ∀ j, i ≤ j → f j = f i) :=
(atTop_basis.eventuallyConst_iff' fun _ _ ↦ left_mem_Ici).trans <| by
simp only [true_and, mem_Ici]
| Mathlib/Order/Filter/EventuallyConst.lean | 156 | 163 |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Violeta Hernández Palacios, Grayson Burton, Floris van Doorn
-/
import Mathlib.Order.Antisymmetrization
import Mathlib.Order.Hom.WithTopBot
import Mathlib.Order.Interval.Se... |
/-- If `a < b` then there exist `a' > a` and `b' < b` such that `Set.Iio a'` is strictly to the left
| Mathlib/Order/Cover.lean | 417 | 418 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Function.L1Space.Integrable
import Mathlib.MeasureTheory.Function.LpSpace.Indicator
/-! # Functions integrable on a set an... | Mathlib/MeasureTheory/Integral/IntegrableOn.lean | 730 | 735 | |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Devon Tuma
-/
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.RingTheory.Coprime.Basic
import Mathlib.Tactic.... | lemma scaleRoots_zero (p : R[X]) :
p.scaleRoots 0 = p.leadingCoeff • X ^ p.natDegree := by
ext n
simp only [coeff_scaleRoots, ne_eq, tsub_eq_zero_iff_le, not_le, zero_pow_eq, mul_ite,
| Mathlib/RingTheory/Polynomial/ScaleRoots.lean | 98 | 101 |
/-
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 | 2,240 | 2,242 | |
/-
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.Ring.Regular
import Mathlib.Algebra.Equiv.TransferInstance
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.Algebra.BigOperators.Ring.Finset... | lemma map_neg_eq_inv (ψ : AddChar A M) (a : A) : ψ (-a) = (ψ a)⁻¹ := by
apply eq_inv_of_mul_eq_one_left
simp only [← map_add_eq_mul, neg_add_cancel, map_zero_eq_one]
| Mathlib/Algebra/Group/AddChar.lean | 381 | 384 |
/-
Copyright (c) 2021 Julian Kuelshammer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Julian Kuelshammer
-/
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.Algebra.GCDMonoid.Finset
import Mathlib.Algebra.GCDMonoid.Nat
import Mathlib.Data.Nat.Factorization.B... | exact dvd_trans (Monoid.order_dvd_exponent (g.1)) <| dvd_lcm_left _ _
· rw [Prod.pow_snd, Prod.snd_one, ← orderOf_dvd_iff_pow_eq_one]
exact dvd_trans (Monoid.order_dvd_exponent (g.2)) <| dvd_lcm_right _ _
· exact MonoidHom.exponent_dvd (f := MonoidHom.fst M₁ M₂) Prod.fst_surjective
· exact MonoidHom... | Mathlib/GroupTheory/Exponent.lean | 583 | 590 |
/-
Copyright (c) 2022 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Group.Hom.End
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Group.Subsemigroup.Membership
import Mathlib.Algebra.Group... | SetLike.coe_injective <| (coe_srange _).trans Subtype.range_coe
variable [NonUnitalNonAssocSemiring S]
@[simp]
theorem range_fst : NonUnitalRingHom.srange (fst R S) = ⊤ :=
NonUnitalRingHom.srange_eq_top_of_surjective (fst R S) Prod.fst_surjective
@[simp]
| Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | 695 | 703 |
/-
Copyright (c) 2020 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Eric Wieser
-/
import Mathlib.Data.ENNReal.Holder
import Mathlib.MeasureTheory.Function.LpSeminorm.Basic
import Mathlib.MeasureTheory.Integral.MeanInequalities
import Mathl... | rw [ENNReal.mul_rpow_of_nonneg (hz := hp.le), ENNReal.rpow_inv_rpow hp.ne',
← lintegral_const_mul'' _ (by fun_prop)]
simp only [← ENNReal.mul_rpow_of_nonneg (hz := hp.le)]
apply lintegral_mono_ae
filter_upwards [h, enorm_ae_le_eLpNormEssSup f μ] with x hb hf
refine ENNReal.rpow_le_rpow ?_ hp.le
gcongr
| Mathlib/MeasureTheory/Function/LpSeminorm/CompareExp.lean | 199 | 205 |
/-
Copyright (c) 2019 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.Module.BigOperators
import Mathlib.GroupTheory.Perm.Basic
import Mathlib.GroupTheory.Perm.Finite
import Mathlib.GroupTheory.Perm.Lis... | ⟨-i, by rw [zpow_neg, ← hi, inv_apply_self]⟩
| Mathlib/GroupTheory/Perm/Cycle/Basic.lean | 64 | 64 |
/-
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.Order.Filter.Tendsto
import Mathlib.Data.Set.Accumulate
import Mathlib.Topology.Bornology.Basic
import Mathlib.Topolog... |
/-- If `X` is a compact topological space, then `Prod.snd : X × Y → Y` is a closed map. -/
theorem isClosedMap_snd_of_compactSpace [CompactSpace X] :
| Mathlib/Topology/Compactness/Compact.lean | 809 | 811 |
/-
Copyright (c) 2021 Jordan Brown, Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jordan Brown, Thomas Browning, Patrick Lutz
-/
import Mathlib.GroupTheory.Abelianization
import Mathlib.GroupTheory.Perm.ViaEmbedding
import Mathlib.GroupT... | IsSolvable (G ⧸ H) :=
solvable_of_surjective (QuotientGroup.mk'_surjective H)
instance solvable_prod {G' : Type*} [Group G'] [IsSolvable G] [IsSolvable G'] :
IsSolvable (G × G') :=
solvable_of_ker_le_range (MonoidHom.inl G G') (MonoidHom.snd G G') fun x hx =>
⟨x.1, Prod.ext rfl hx.symm⟩
variable (G) i... | Mathlib/GroupTheory/Solvable.lean | 135 | 145 |
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Constructions.Polish.Basic
import Mathlib.MeasureTheory.Function.UniformIntegrable
import Mathlib.Probability.Martingale.Upcrossing
/-!
# Mar... | integrable_condExp.tendsto_eLpNorm_condExp stronglyMeasurable_condExp
have heq : ∀ n, ∀ᵐ x ∂μ, (μ[μ[g|⨆ n, ℱ n]|ℱ n]) x = (μ[g|ℱ n]) x := fun n =>
condExp_condExp_of_le (le_iSup _ n) (iSup_le fun n => ℱ.le n)
refine ht.congr fun n => eLpNorm_congr_ae ?_
filter_upwards [heq n] with x hxeq
simp only [hxeq... | Mathlib/Probability/Martingale/Convergence.lean | 453 | 461 |
/-
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... | mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]
refine
| Mathlib/RingTheory/Ideal/Operations.lean | 852 | 853 |
/-
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.Complex.Arg
import Mathlib.Analysis.SpecialFunctions.Log.Basic... | theorem log_ofReal_re (x : ℝ) : (log (x : ℂ)).re = Real.log x := by simp [log_re]
theorem log_ofReal_mul {r : ℝ} (hr : 0 < r) {x : ℂ} (hx : x ≠ 0) :
| Mathlib/Analysis/SpecialFunctions/Complex/Log.lean | 70 | 72 |
/-
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, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Field.IsField
import Mathlib.Algebra.GroupWithZero.N... | Mathlib/RingTheory/Localization/Basic.lean | 1,371 | 1,379 | |
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Kernel.Disintegration.Unique
import Mathlib.Probability.Notation
/-!
# Regular conditional probability distribution
We define the regular con... | @[deprecated (since := "2025-01-24")]
alias aestronglyMeasurable'_integral_condDistrib := aestronglyMeasurable_integral_condDistrib
end Measurability
/-- `condDistrib` is a.e. uniquely defined as the kernel satisfying the defining property of
`condKernel`. -/
theorem condDistrib_ae_eq_of_measure_eq_compProd (hX : Mea... | Mathlib/Probability/Kernel/CondDistrib.lean | 120 | 130 |
/-
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 | 989 | 995 | |
/-
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... | rw [logb, logb, div_lt_div_iff_of_pos_right (log_pos hb)]
| Mathlib/Analysis/SpecialFunctions/Log/Base.lean | 189 | 189 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Kim Morrison
-/
import Mathlib.CategoryTheory.Subobject.Lattice
/-!
# Specific subobjects
We define `equalizerSubobject`, `kernelSubobject` and `imageSubobject`, which ar... |
@[simp]
theorem factorThruKernelSubobject_comp_arrow {W : C} (h : W ⟶ X) (w : h ≫ f = 0) :
| Mathlib/CategoryTheory/Subobject/Limits.lean | 115 | 117 |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Joël Riou
-/
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq
import Mathlib.CategoryTheory.Limits.Shapes.Diagonal
import Mathlib.CategoryTheory.Limits.Final
impor... | rw [this]
apply P.comp_mem <;> rw [P.cancel_left_of_respectsIso]
exacts [baseChange_map _ (Over.homMk _ e₂.symm : Over.mk g ⟶ Over.mk g') h₂,
baseChange_map _ (Over.homMk _ e₁.symm : Over.mk f ⟶ Over.mk f') h₁]
| Mathlib/CategoryTheory/MorphismProperty/Limits.lean | 231 | 235 |
/-
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.Trim
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner
import Mathlib.MeasureTheory.Function.StronglyMeasura... | @[elab_as_elim]
theorem Lp.induction_stronglyMeasurable (hm : m ≤ m0) (hp_ne_top : p ≠ ∞) (P : Lp F p μ → Prop)
(h_ind : ∀ (c : F) {s : Set α} (hs : MeasurableSet[m] s) (hμs : μ s < ∞),
P (Lp.simpleFunc.indicatorConst p (hm s hs) hμs.ne c))
(h_add : ∀ ⦃f g⦄, ∀ hf : MemLp f p μ, ∀ hg : MemLp g p μ, Strongl... | Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean | 539 | 549 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Morenikeji Neri
-/
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.Algebra.EuclideanDomain.Field
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.RingTheor... | simpa only [← mem_iff_generator_dvd S] using is_prime.2⟩
-- Note that the converse may not hold if `ϕ` is not injective.
theorem generator_map_dvd_of_mem {N : Submodule R M} (ϕ : M →ₗ[R] R) [(N.map ϕ).IsPrincipal] {x : M}
| Mathlib/RingTheory/PrincipalIdealDomain.lean | 141 | 144 |
/-
Copyright (c) 2022 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.MeasureTheory.Group.GeometryOfNumbers
import Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls
import Mathlib.NumberTheory.NumberField.CanonicalEmbedd... | _ = (convexBodyLTFactor K) * ∏ w, (f w) ^ (mult w) := by
simp_rw [prod_eq_prod_mul_prod, coe_mul, coe_finset_prod, mult_isReal, mult_isComplex,
pow_one, ENNReal.coe_pow, ofReal_coe_nnreal]
variable {f}
/-- This is a technical result: quite often, we want to impose conditions at all infinite places
b... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean | 108 | 130 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
import Mathlib.Probability.Notation
import Mathlib.Probability.Process.Stopping
/-!
# Martingales
A family of functions `f : ι → Ω → E` is a martingale wit... | Mathlib/Probability/Martingale/Basic.lean | 537 | 560 | |
/-
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 Mathlib.Data.Set.Insert
/-!
# Subsingleton
Defines the predicate `Subsingleton s : Prop`, saying that `s` has at most one element.
Also def... |
theorem subsingleton_of_univ_subsingleton (h : (univ : Set α).Subsingleton) : Subsingleton α :=
⟨fun a b => h (mem_univ a) (mem_univ b)⟩
| Mathlib/Data/Set/Subsingleton.lean | 68 | 71 |
/-
Copyright (c) 2019 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Yury Kudryashov, Yaël Dillies
-/
import Mathlib.Algebra.Order.Invertible
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.LinearAlgebra.AffineSpac... | simp [*]
· rintro ⟨a, b, ha, hb, hab, rfl⟩
refine ⟨a / (a + b), b / (a + b), by positivity, by positivity, ?_, rfl⟩
| Mathlib/Analysis/Convex/Segment.lean | 344 | 346 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Uniqueness
import Mathlib.Analysis.Calculus.DiffContOnCl
import Mathlib.Analysis.Calculus.DSlope
import Mathlib.Analysis.Calculus.F... | (hd : DifferentiableOn ℂ f (closedBall c R)) (hw : w ∈ ball c R) :
(∮ z in C(c, R), (z - w)⁻¹ • f z) = (2 * π * I : ℂ) • f w :=
(hd.mono closure_ball_subset_closedBall).diffContOnCl.circleIntegral_sub_inv_smul hw
/-- **Cauchy integral formula**: if `f : ℂ → ℂ` is continuous on a closed disc of radius `R` and... | Mathlib/Analysis/Complex/CauchyIntegral.lean | 518 | 525 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
import Mathlib.Analysis.Normed.Module.Basic
import Mathlib.MeasureTheory.Function.SimpleFuncDense
/-!
# Strongly measurable and finitely strongly meas... | [OpensMeasurableSpace α] [TopologicalSpace β] [PseudoMetrizableSpace β]
[h : SecondCountableTopologyEither α β] {f : α → β} (hf : Continuous f) :
StronglyMeasurable f := by
| Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean | 640 | 642 |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Algebra.Order.Group.Unbundled.Basic
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Or... | Mathlib/Algebra/Order/Group/Defs.lean | 1,177 | 1,177 | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Algebra.Field.NegOnePow
import Mathlib.Algebra.Field.Periodic
import Mathlib.Algebra.Qua... | (strictAntiOn_cos.le_iff_le ⟨hx₁.trans hxy, hy₂⟩ ⟨hx₁, hxy.trans hy₂⟩).2 hxy
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | 558 | 559 |
/-
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.Relation
import Mathlib.Logic.Unique
import Mathlib.Util.Notation3
/-!
# Quotient types
This module extends the core library's treatment of quo... | /-- Given a class of functions `q : @Quotient (∀ i, α i) _`, returns the class of `i`-th projection
`Quotient (S i)`. -/
def Quotient.eval {ι : Type*} {α : ι → Sort*} {S : ∀ i, Setoid (α i)}
(q : @Quotient (∀ i, α i) (by infer_instance)) (i : ι) : Quotient (S i) :=
| Mathlib/Data/Quot.lean | 398 | 401 |
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Tactic.FinCases
import Mathlib.Topology.Connected.LocallyConnected
import Mathlib.Topology.Sets.Closeds
/-!
# L... | ∀ x, ∃ U : Set X, IsOpen U ∧ x ∈ U ∧ ∀ x' ∈ U, f x' = f x] := by
tfae_have 1 → 4 := fun h y => h {y}
tfae_have 4 → 3 := fun h x => h (f x)
tfae_have 3 → 2 := fun h x => IsOpen.mem_nhds (h x) rfl
tfae_have 2 → 5
| h, x => by
rcases mem_nhds_iff.1 (h x) with ⟨U, eq, hU, hx⟩
exact ⟨U, hU, hx, eq⟩
... | Mathlib/Topology/LocallyConstant/Basic.lean | 42 | 63 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yaël Dillies
-/
import Mathlib.Logic.Equiv.Set
import Mathlib.Order.CompleteLattice.Lemmas
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection.Basic
/-... | iInf_sup_le_sup_sInf a s:= sup_sInf_eq.ge
| Mathlib/Order/CompleteBooleanAlgebra.lean | 222 | 223 |
/-
Copyright (c) 2024 Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Merten
-/
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.CategoryTheory.Limits.Preserves.Limits
import Mathlib.CategoryTheory.Limits.Yoneda
/-!
# Ind- and ... | (colimitHomIsoLimitYoneda F A).hom ≫ limit.π (F.op ⋙ yoneda.obj A) ⟨i⟩ =
(coyoneda.map (colimit.ι F i).op).app A := by
simp only [colimitHomIsoLimitYoneda, Iso.trans_hom, Iso.app_hom, Category.assoc]
erw [limitObjIsoLimitCompEvaluation_hom_π]
| Mathlib/CategoryTheory/Limits/IndYoneda.lean | 70 | 73 |
/-
Copyright (c) 2023 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang, Junyan Xu
-/
import Mathlib.Algebra.Category.ModuleCat.Basic
import Mathlib.Algebra.Category.Grp.Injective
import Mathlib.Topology.Instances.AddCircle
import Mathlib.Linea... |
lemma exists_character_apply_ne_zero_of_ne_zero {a : A} (ne_zero : a ≠ 0) :
∃ (c : CharacterModule A), c a ≠ 0 :=
| Mathlib/Algebra/Module/CharacterModule.lean | 211 | 213 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Morenikeji Neri
-/
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.Algebra.EuclideanDomain.Field
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.RingTheor... |
theorem gcd_dvd_iff_exists (a b : R) {z} : gcd a b ∣ z ↔ ∃ x y, z = a * x + b * y := by
simp_rw [mul_comm a, mul_comm b, @eq_comm _ z, ← Ideal.mem_span_pair, ← span_gcd,
| Mathlib/RingTheory/PrincipalIdealDomain.lean | 437 | 439 |
/-
Copyright (c) 2019 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Mario Carneiro, Isabel Longbottom, Kim Morrison, Apurva Nakade, Yuyang Zhao
-/
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.SetTheory.PGame.Algebra
import Mathl... | /-- A small family of games is bounded below. -/
lemma bddBelow_range_of_small {ι : Type*} [Small.{u} ι] (f : ι → Game.{u}) :
BddBelow (Set.range f) := by
obtain ⟨x, hx⟩ := PGame.bddBelow_range_of_small (Quotient.out ∘ f)
refine ⟨⟦x⟧, Set.forall_mem_range.2 fun i ↦ ?_⟩
simpa [PGame.le_iff_game_le] using hx <|... | Mathlib/SetTheory/Game/Basic.lean | 206 | 211 |
/-
Copyright (c) 2022 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Group.Nat.Even
import Mathlib.Data.Nat.Cast.Basic
import Mathlib.Data.Nat.Cast.Commute
import Mathlib.Data.Set.Operations
import Mathlib.Logic.Fu... |
lemma iterate_bit1 (hf : Involutive f) (n : ℕ) : f^[2 * n + 1] = f := by
| Mathlib/Algebra/Ring/Parity.lean | 315 | 316 |
/-
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.GroupWithZero.Action.Defs
import Mathlib.Algebra.Order.AddGroupWithTop
import Mathlib.Algebra.Order.Monoid.Unbundled.MinMax
import Mathlib.Al... | Mathlib/Algebra/Tropical/Basic.lean | 569 | 572 | |
/-
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 | 2,954 | 2,957 | |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Logic.Equiv.Fin.Basic
imp... | hasSum hy h'y := hf.hasSum (insert_subset_insert h hy) h'y
lemma HasFPowerSeriesOnBall.hasFPowerSeriesWithinOnBall (hf : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesWithinOnBall f p s x r := by
rw [← hasFPowerSeriesWithinOnBall_univ] at hf
exact hf.mono (subset_univ _)
lemma HasFPowerSeriesWithinAt.mono... | Mathlib/Analysis/Analytic/Basic.lean | 645 | 653 |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.TwoDim
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
/-!
# Oriented angles.
This file defines orie... | angle. -/
@[simp]
| Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | 247 | 248 |
/-
Copyright (c) 2022 Wrenna Robson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Wrenna Robson
-/
import Mathlib.Topology.MetricSpace.Basic
/-!
# Infimum separation
This file defines the extended infimum separation of a set. This is approximately dual to the
diame... | simp_rw [einfsep, iInf_lt_iff, exists_prop]
theorem einfsep_ne_top :
| Mathlib/Topology/MetricSpace/Infsep.lean | 64 | 66 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Michael Stoll
-/
import Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic
/-!
# Legendre symbol
This file contains results about Legendre symbols.
We define the Le... | theorem sq_one {a : ℤ} (ha : (a : ZMod p) ≠ 0) : legendreSym p a ^ 2 = 1 :=
quadraticChar_sq_one ha
| Mathlib/NumberTheory/LegendreSymbol/Basic.lean | 160 | 161 |
/-
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 | 277 | 287 | |
/-
Copyright (c) 2023 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Algebra.Group.Submonoid.BigOperators
import Mathlib.GroupTheory.Subsemigroup.Center
import Mathlib.RingTheory... | theorem closure_union (s t : Set R) : closure (s ∪ t) = closure s ⊔ closure t :=
(NonUnitalSubring.gi R).gc.l_sup
| Mathlib/RingTheory/NonUnitalSubring/Basic.lean | 523 | 524 |
/-
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.Data.List.Basic
/-!
# Double universal quantification on a list
This file provides an API for `List.Forall₂` (definition in `Data.Lis... | theorem rel_map : ((R ⇒ P) ⇒ Forall₂ R ⇒ Forall₂ P) map map
| _, _, _, [], [], Forall₂.nil => Forall₂.nil
| _, _, h, _ :: _, _ :: _, Forall₂.cons h₁ h₂ => Forall₂.cons (h h₁) (rel_map (@h) h₂)
theorem rel_append : (Forall₂ R ⇒ Forall₂ R ⇒ Forall₂ R) (· ++ ·) (· ++ ·)
| [], [], _, _, _, hl => hl
| _, _, Forall₂... | Mathlib/Data/List/Forall2.lean | 203 | 216 |
/-
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.Algebra.Order.Group.Abs
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.... | Mathlib/Algebra/Order/Interval/Set/Group.lean | 282 | 283 | |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Sébastien Gouëzel, Yury Kudryashov, Dylan MacKenzie, Patrick Massot
-/
import Mathlib.Algebra.BigOperators.Module
import Mathlib.Algebra.Order.Field.Power
import M... |
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
variable {b : ℝ} {f : ℕ → ℝ} {z : ℕ → E}
/-- **Dirichlet's test** for monotone sequences. -/
theorem Monotone.cauchySeq_series_mul_of_tendsto_zero_of_bounded (hfa : Monotone f)
(hf0 : Tendsto f atTop (𝓝 0)) (hgb : ∀ n, ‖∑ i ∈ range n, z i‖ ≤ b) :
... | Mathlib/Analysis/SpecificLimits/Normed.lean | 673 | 689 |
/-
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.Data.Finite.Prod
import Mathlib.Data.Matroid.Init
import Mathlib.Data.Set.Card
import Mathlib.Data.Set.Finite.Powerset
import Mathlib.Order.UpperLower.Clos... | rw [ncard_def B₁, hB₁.encard_eq_encard_of_isBase hB₂, ← ncard_def]
| Mathlib/Data/Matroid/Basic.lean | 427 | 428 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Algebra.Pi
import Mathlib.Algebra.Algebra.Prod
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
impo... | theorem aeval_prod_apply (x : A × B) (p : Polynomial R) :
p.aeval x = (p.aeval x.1, p.aeval x.2) := by simp [aeval_prod]
| Mathlib/Algebra/Polynomial/AlgebraMap.lean | 370 | 372 |
/-
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 | 129 | 139 | |
/-
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.RingTheory.PrincipalIdealDomain
/-!
# Bézout rings
A Bézout ring (Bezout ring) is a ring whose finitely generated ideals are principal.
Notable examples i... | tfae_have 1 → 2
| _ => inferInstance
tfae_have 2 → 3
| _ => inferInstance
tfae_have 3 → 4
| _ => inferInstance
tfae_have 4 → 1
| ⟨h⟩ => by
rw [isNoetherianRing_iff, isNoetherian_iff_fg_wellFounded]
refine ⟨RelEmbedding.wellFounded ?_ h⟩
have : ∀ I : { J : Ideal R // J.F... | Mathlib/RingTheory/Bezout.lean | 53 | 78 |
/-
Copyright (c) 2019 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Yury Kudryashov, Sébastien Gouëzel, Chris Hughes
-/
import Mathlib.Data.Fin.Rev
import Mathlib.Data.Nat.Find
/-!
# Operation on tuples
We interpret maps `∀ i : Fi... | g ∘ snoc q y = snoc (g ∘ q) (g y) := by
ext j
by_cases h : j.val < n
· simp [h, snoc, castSucc_castLT]
· rw [eq_last_of_not_lt h]
simp
| Mathlib/Data/Fin/Tuple/Basic.lean | 624 | 630 |
/-
Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
import Mathlib.Topology.Order.Compact
import Mathlib.Topology.MetricSpace.ProperSpace
import M... | /-- A totally bounded set is bounded -/
theorem _root_.TotallyBounded.isBounded {s : Set α} (h : TotallyBounded s) : IsBounded s :=
-- We cover the totally bounded set by finitely many balls of radius 1,
| Mathlib/Topology/MetricSpace/Bounded.lean | 142 | 144 |
/-
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.Fold
import Mathlib.Data.Multiset.Bind
import Mathlib.Order.SetNotation
/-!
# Unions of finite sets
This file defines the union of a fami... | obtain ⟨xb, hfb, hgb⟩ := mem_disjiUnion.mp hxb
refine disjoint_left.mp
(h2 (mem_disjiUnion.mpr ⟨_, a.prop, hfa⟩) (mem_disjiUnion.mpr ⟨_, b.prop, hfb⟩) ?_) hga
hgb
rintro rfl
exact disjoint_left.mp (h1 a.prop b.prop <| Subtype.coe_injective.ne hab) hfa hfb ... | Mathlib/Data/Finset/Union.lean | 76 | 90 |
/-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Devon Tuma
-/
import Mathlib.Probability.ProbabilityMassFunction.Basic
/-!
# Monad Operations for Probability Mass Functions
This file constructs two operations on `PMF` ... | rw [toPMF_eq_iff_toMeasure_eq, toMeasure_pure]
| Mathlib/Probability/ProbabilityMassFunction/Monad.lean | 92 | 93 |
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