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) 2021 Hunter Monroe. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hunter Monroe, Kyle Miller, Alena Gusakov
-/
import Mathlib.Combinatorics.SimpleGraph.DeleteEdges
import Mathlib.Data.Fintype.Powerset
/-!
# Subgraphs of a simple graph
A subgraph of ... |
@[simp]
theorem edgeSet_iInf (f : ι → G.Subgraph) :
(⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by
| Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | 548 | 551 |
/-
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.Order.Ring.Rat
import Mathlib.Data.Multiset.Sort
import Mathlib.Data.PNat.Basic
import Mathlib.Data.PNat.Interval
import Mathlib.Tactic.NormNum... | rw [sumInv_pqr]
have h4r := H.trans hqr
have hq : (q : ℚ)⁻¹ ≤ 4⁻¹ := by
rw [inv_le_inv₀ _ h4]
· assumption_mod_cast
· norm_num
have hr : (r : ℚ)⁻¹ ≤ 4⁻¹ := by
rw [inv_le_inv₀ _ h4]
· assumption_mod_cast
· norm_num
calc
(2⁻¹ + (q : ℚ)⁻¹ + (r : ℚ)⁻¹) ≤ 2⁻¹ + 4⁻¹ + 4⁻¹ := add_le_add (... | Mathlib/NumberTheory/ADEInequality.lean | 175 | 195 |
/-
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.Control.Combinators
import Mathlib.Data.Option.Defs
import Mathlib.Logic.IsEmpty
import Mathlib.Logic.Relator
import Mathlib.Util.CompileInductive
impo... | Mathlib/Data/Option/Basic.lean | 342 | 343 | |
/-
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.Abelian
import Mathlib.Algebra.Lie.BaseChange
import Mathlib.Algebra.Lie.IdealOperations
import Mathlib.Order.Hom.Basic
import Mathlib.RingTheory... | theorem derivedSeries_eq_derivedSeriesOfIdeal_comap (k : ℕ) :
derivedSeries R I k = (derivedSeriesOfIdeal R L k I).comap I.incl := by
induction k with
| zero => simp only [derivedSeries_def, comap_incl_self, derivedSeriesOfIdeal_zero]
| succ k ih =>
simp only [derivedSeries_def, derivedSeriesOfIdeal_succ]... | Mathlib/Algebra/Lie/Solvable.lean | 149 | 155 |
/-
Copyright (c) 2021 Arthur Paulino. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Arthur Paulino, Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Clique
import Mathlib.Data.ENat.Lattice
import Mathlib.Data.Nat.Lattice
import Mathlib.Data.Setoid.Partition
imp... | exact le_top
· have hc' := hc
rw [chromaticNumber_ne_top_iff_exists] at hc'
obtain ⟨n, c⟩ := hc'
rw [← ENat.coe_toNat_eq_self] at hc
rw [← hc, Nat.cast_le]
| Mathlib/Combinatorics/SimpleGraph/Coloring.lean | 432 | 437 |
/-
Copyright (c) 2021 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Johan Commelin
-/
import Mathlib.Algebra.Category.ModuleCat.Monoidal.Basic
import Mathlib.CategoryTheory.Monoidal.Types.Basic
import Mathlib.LinearAlgebra.DirectSum.Finsupp... | rw [add_comp]
rw [Finsupp.sum_add_index', Finsupp.sum_add_index']
· simp only [w₁, w₂, add_comp]
| Mathlib/Algebra/Category/ModuleCat/Adjunctions.lean | 305 | 307 |
/-
Copyright (c) 2020 Paul van Wamelen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Paul van Wamelen
-/
import Mathlib.Data.Int.NatPrime
import Mathlib.Data.ZMod.Basic
import Mathlib.RingTheory.Int.Basic
import Mathlib.Tactic.FieldSimp
/-!
# Pythagorean Triples
Th... | theorem symm (h : PythagoreanTriple x y z) : PythagoreanTriple y x z := by
rwa [pythagoreanTriple_comm]
| Mathlib/NumberTheory/PythagoreanTriples.lean | 60 | 61 |
/-
Copyright (c) 2021 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn, Kim Morrison
-/
import Mathlib.Data.Opposite
/-!
# Quivers
This module defines quivers. A quiver on a type `V` of vertices assigns to every
pair `a b : V` of vertices a type ... | @[simp]
theorem empty_arrow {V : Type u} (a b : Empty V) : (a ⟶ b) = PEmpty := rfl
/-- A quiver is thin if it has no parallel arrows. -/
abbrev IsThin (V : Type u) [Quiver V] : Prop := ∀ a b : V, Subsingleton (a ⟶ b)
section
variable {V : Type*} [Quiver V]
/-- An arrow in a quiver can be transported across equalit... | Mathlib/Combinatorics/Quiver/Basic.lean | 76 | 87 |
/-
Copyright (c) 2024 Fabrizio Barroero. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fabrizio Barroero, Laura Capuano, Amos Turchet
-/
import Mathlib.Analysis.Matrix
import Mathlib.Data.Pi.Interval
import Mathlib.Tactic.Rify
/-!
# Siegel's Lemma
In this file we in... |
private lemma card_S_lt_card_T [DecidableEq α] [DecidableEq β]
(hn : Fintype.card α < Fintype.card β) (hm : 0 < Fintype.card α) :
#S < #T := by
zify -- This is necessary to use card_S_eq
rw [card_T_eq A, card_S_eq]
rify -- This is necessary because ‖A‖ is a real number
calc
∏ x : α, (∑ x_1 : β, ↑B * ... | Mathlib/NumberTheory/SiegelsLemma.lean | 133 | 167 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | @[simp]
lemma add_inv_app_obj (a b : A) (X : C) :
(add F s i a b).inv.app (F.obj X) =
(s b).map ((i a).hom.app X) ≫ (i b).hom.app ((shiftFunctor C a).obj X) ≫
F.map ((shiftFunctorAdd C a b).inv.app X) ≫ (i (a + b)).inv.app X := by
have h : whiskerLeft F (add F s i a b).inv = _ :=
((whiskeringLef... | Mathlib/CategoryTheory/Shift/Induced.lean | 75 | 82 |
/-
Copyright (c) 2019 Neil Strickland. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Neil Strickland
-/
import Mathlib.Tactic.Ring
import Mathlib.Data.PNat.Prime
/-!
# Euclidean algorithm for ℕ
This file sets up a version of the Euclidean algorithm that only works w... | rw [← ha, hr]
ring
/-- We can now define the full reduction function, which applies
step as long as possible, and then applies finish. Note that the
"have" statement puts a fact in the local context, and the
equation compiler uses this fact to help construct the full
definition in terms of well-founded rec... | Mathlib/Data/PNat/Xgcd.lean | 286 | 296 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Shapes.Equalizers
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.Mono
import Mathlib.CategoryTheory.Limits.Shape... | /-- The functor from the arrow category of `C` to `C` itself that maps a morphism to its image
and a commutative square to the induced morphism on images. -/
@[simps]
def im : Arrow C ⥤ C where
obj f := image f.hom
map st := image.map st
| Mathlib/CategoryTheory/Limits/Shapes/Images.lean | 772 | 778 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Algebra.Category.Ring.Basic
import Mathlib.CategoryTheory.Limits.HasLimits
/-!
# The category of commutative rings has all colimits.
This file uses a "pr... | | mul_one x => dsimp; rw [mul_one]
| neg_add_cancel x => dsimp; rw [neg_add_cancel]
| add_comm x y => dsimp; rw [add_comm]
| add_assoc x y z => dsimp; rw [add_assoc]
| mul_assoc x y z => dsimp; rw [mul_assoc]
| Mathlib/Algebra/Category/Ring/Colimits.lean | 245 | 249 |
/-
Copyright (c) 2020 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.NormedSpace.IndicatorFunction
import Mathlib.Data.Fintype.Order
import Mathlib.MeasureTheory.Function.AEEqFun
import Mathlib.Me... | lemma eLpNormEssSup_indicator_const_le (s : Set α) (c : ε) :
eLpNormEssSup (s.indicator fun _ : α => c) μ ≤ ‖c‖ₑ := by
by_cases hμ0 : μ = 0
· rw [hμ0, eLpNormEssSup_measure_zero]
exact zero_le _
· exact (eLpNormEssSup_indicator_le s fun _ => c).trans (eLpNormEssSup_const c hμ0).le
| Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean | 676 | 682 |
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes Hölzl, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Continuity
import Mathlib.Topology.Algebra.IsUniformGroup.Basic
import Mathlib.Topology.MetricSpace... | rw [show (fun n => ∏ k ∈ range (n + 1), u k) = d * fun n => ∏ k ∈ range (n + 1), v k
by ext n; simp [d]]
| Mathlib/Analysis/Normed/Group/Uniform.lean | 395 | 396 |
/-
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.MeasurableSpace.MeasurablyGenerated
import Mathlib.MeasureTheory.Measure.NullMeasurable
import Mathlib.Order.Interval.Set... | @[simp]
theorem union_ae_eq_left_iff_ae_subset : (s ∪ t : Set α) =ᵐ[μ] s ↔ t ≤ᵐ[μ] s := by
rw [ae_le_set]
| Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 304 | 306 |
/-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.Algebra.Order.Group.OrderIso
import Mathlib.SetTheory.Game.Ordinal
import Mathlib.SetTheory.Ordinal.NaturalOps
/-!
# Birthdays... | @[simp]
theorem birthday_zero : birthday 0 = 0 := by simp [inferInstanceAs (IsEmpty PEmpty)]
| Mathlib/SetTheory/Game/Birthday.lean | 97 | 99 |
/-
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... | refine I.finite_or_infinite.elim (hI _ Subset.rfl) (fun h ↦ False.elim ?_)
obtain ⟨B, hB⟩ := M.exists_isBase
obtain ⟨I₀, hI₀I, hI₀fin, hI₀card⟩ := h.exists_subset_ncard_eq (B.ncard + 1)
obtain ⟨B', hB', hI₀B'⟩ := (hI _ hI₀I hI₀fin).exists_isBase_superset
have hle := ncard_le_ncard hI₀B' hB'.finite
... | Mathlib/Data/Matroid/Basic.lean | 755 | 760 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou, Johan Commelin
-/
import Mathlib.Algebra.Ring.Int.Parity
import Mathlib.Algebra.Ring.Int.Units
import Mathlib.Data.ZMod.IntUnitsPower
/-!
# Integer powers of (-1)
This file def... | rw [negOnePow_odd _ h₂, Int.even_sub, negOnePow_eq_neg_one_iff,
← Int.not_odd_iff_even, ← Int.not_odd_iff_even]
tauto
| Mathlib/Algebra/Ring/NegOnePow.lean | 100 | 102 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Wen Yang
-/
import Mathlib.LinearAlgebra.Matrix.Adjugate
import Mathlib.LinearAlgebra.Matrix.ToLin
import Mathlib.LinearAlgebra.Matrix.Transvection
import Mathlib.RingTheory.... | Mathlib/LinearAlgebra/Matrix/SpecialLinearGroup.lean | 536 | 537 | |
/-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.Add
/-!
# Local extrema of differentiable functions
## Main definitions
In a real normed space `E` we define `posTangentCo... | rw [eventually_nhdsWithin_iff]
filter_upwards [ge_mem_nhds one_pos] with t ht₁ ht₀
apply h
| Mathlib/Analysis/Calculus/LocalExtr/Basic.lean | 99 | 101 |
/-
Copyright (c) 2024 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir, Oliver Nash
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Identities
import Mathlib.RingTheory.Nilpotent.Lemmas
import Mathlib.R... | theorem newtonMap_apply_of_isUnit (h : IsUnit <| aeval x (derivative P)) :
P.newtonMap x = x - h.unit⁻¹ * aeval x P := by
simp [newtonMap_apply, Ring.inverse, h]
| Mathlib/Dynamics/Newton.lean | 53 | 55 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Andrew Zipperer, Haitao Zhang, Minchao Wu, Yury Kudryashov
-/
import Mathlib.Data.Set.Prod
import Mathlib.Data.Set.Restrict
/-!
# Functions over sets
This file contains... | @[simp] theorem bijOn_empty_iff_left : BijOn f s ∅ ↔ s = ∅ :=
⟨fun h ↦ by simpa using h.mapsTo, by rintro rfl; exact bijOn_empty f⟩
| Mathlib/Data/Set/Function.lean | 582 | 583 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.Images
import Mathlib.CategoryTheor... |
/-- A category with a zero object has zero morphisms.
It is rarely a good idea to use this. Many categories that have a zero object have zero
| Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean | 241 | 244 |
/-
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... | @[simp] lemma biUnion_val (s : Finset α) (t : α → Finset β) :
(s.biUnion t).1 = (s.1.bind fun a ↦ (t a).1).dedup := rfl
@[simp] lemma biUnion_empty : Finset.biUnion ∅ t = ∅ := rfl
| Mathlib/Data/Finset/Union.lean | 138 | 142 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Independence.Basic
import Mathlib.Probability.Independence.Conditional
/-!
# Kolmogorov's 0-1 law
Let `s : ι → MeasurableSpace Ω` be an indep... | theorem Kernel.indep_biSup_limsup (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s κ μα)
(hf : ∀ t, p t → tᶜ ∈ f) {t : Set ι} (ht : p t) :
Indep (⨆ n ∈ t, s n) (limsup s f) κ μα := by
refine indep_of_indep_of_le_right (indep_biSup_compl h_le h_indep t) ?_
refine limsSup_le_of_le (by isBoundedDefault) ?_
simp on... | Mathlib/Probability/Independence/ZeroOne.lean | 109 | 115 |
/-
Copyright (c) 2020 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.Algebra.FreeAlgebra
import Mathlib.Algebra.RingQuot
import Mathlib.Algebra.TrivSqZeroExt
import Mathlib.Algebra.Algebra.Operations
import Mathlib.LinearAlgebra... | rw [← (lift R).symm_apply_eq]
simp only [lift, Equiv.coe_fn_symm_mk]
-- Marking `TensorAlgebra` irreducible makes `Ring` instances inaccessible on quotients.
| Mathlib/LinearAlgebra/TensorAlgebra/Basic.lean | 159 | 162 |
/-
Copyright (c) 2020 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.Group.End
import Mathlib.Data.ZMod.Defs
import Mathlib.Tactic.Ring
/-!
# Racks and Quandles
This file defines racks and quandles, algebraic structu... | instance inhabited (S : Type*) [Shelf S] : Inhabited (S →◃ S) :=
⟨id S⟩
| Mathlib/Algebra/Quandle.lean | 336 | 338 |
/-
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... | theorem sin_nat_mul_pi (n : ℕ) : sin (n * π) = 0 :=
sin_antiperiodic.nat_mul_eq_of_eq_zero sin_zero n
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | 1,038 | 1,039 |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.CategoryTheory.Comma.Arrow
import Mathlib.Order.CompleteBooleanAlgebra
/-!
# Properties of morphisms
We provide the basic framework for talking about prope... | (hX : X = X') (hY : Y = Y') (h : f' = eqToHom hX.symm ≫ f ≫ eqToHom hY) :
P f' := by
rw [← P.arrow_mk_mem_toSet_iff] at hf ⊢
| Mathlib/CategoryTheory/MorphismProperty/Basic.lean | 143 | 145 |
/-
Copyright (c) 2021 Bryan Gin-ge Chen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Bryan Gin-ge Chen, Yaël Dillies
-/
import Mathlib.Order.BooleanAlgebra
import Mathlib.Logic.Equiv.Basic
/-!
# Symmetric difference and bi-implication
This file defines... |
@[simp]
| Mathlib/Order/SymmDiff.lean | 264 | 265 |
/-
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.MeasureTheory.Function.AEEqFun.DomAct
import Mathlib.MeasureTheory.Function.LpSpace.Indicator
/-!
# Action of `Mᵈᵐᵃ` on `Lᵖ` spaces
In this file we... | SMulCommClass 𝕜 Mᵈᵐᵃ (Lp E p μ) :=
.symm _ _ _
| Mathlib/MeasureTheory/Function/LpSpace/DomAct/Basic.lean | 70 | 71 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel, Kim Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Abelian.Basic
i... | by_contra h'
exact cokernel_not_iso_of_nonzero hf ((h _).mpr h')⟩
/-- A nonzero epimorphism from a simple object is an isomorphism. -/
theorem isIso_of_epi_of_nonzero {X Y : C} [Simple X] {f : X ⟶ Y} [Epi f] (w : f ≠ 0) : IsIso f :=
-- `f ≠ 0` means that `kernel.ι f` is not an iso, and hence zero, an... | Mathlib/CategoryTheory/Simple.lean | 145 | 159 |
/-
Copyright (c) 2020 Alexander Bentkamp, Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Sébastien Gouëzel, Eric Wieser
-/
import Mathlib.Algebra.Algebra.Rat
import Mathlib.Data.Complex.Cardinality
import Mathlib.Data.Complex.Modu... | lemma Complex.rank_rat_complex : Module.rank ℚ ℂ = continuum := by
refine (Free.rank_eq_mk_of_infinite_lt ℚ ℂ ?_).trans Cardinal.mk_complex
simpa using aleph0_lt_continuum
| Mathlib/Data/Complex/FiniteDimensional.lean | 68 | 71 |
/-
Copyright (c) 2020 Zhangir Azerbayev. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Zhangir Azerbayev
-/
import Mathlib.GroupTheory.Perm.Sign
import Mathlib.LinearAlgebra.LinearIndependent.Defs
import Mathlib.LinearAlgebra.Multilinear.Basis
/-!
# Alte... |
end DomLcongr
/-- Composing an alternating map with the same linear equiv on each argument gives the zero map
if and only if the alternating map is the zero map. -/
@[simp]
| Mathlib/LinearAlgebra/Alternating/Basic.lean | 571 | 576 |
/-
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.Ordering.Basic
import Mathlib.Order.Synonym
/-!
# Comparison
This file provides basic results about orderings and comparison in linear orders.
... | rcases lt_trichotomy a b with hab | hab | hab
· have hab : Compares Ordering.lt a b := hab
rwa [ho.inj (h hab)]
· have hab : Compares Ordering.eq a b := hab
| Mathlib/Order/Compare.lean | 94 | 97 |
/-
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.WSeq.Basic
import Mathlib.Data.WSeq.Defs
import Mathlib.Data.WSeq.Productive
import Mathlib.Data.WSeq.Relation
deprecated_module (since :=... | Mathlib/Data/Seq/WSeq.lean | 1,694 | 1,709 | |
/-
Copyright (c) 2021 Hunter Monroe. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hunter Monroe, Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Dart
import Mathlib.Data.FunLike.Fintype
import Mathlib.Logic.Embedding.Set
/-!
# Maps between graphs
This file ... | invFun := Hom.mapEdgeSet f.symm
left_inv := by
rintro ⟨e, h⟩
| Mathlib/Combinatorics/SimpleGraph/Maps.lean | 512 | 514 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Star.SelfAdjoint
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
/-!
# Quate... | Mathlib/Algebra/Quaternion.lean | 1,546 | 1,547 | |
/-
Copyright (c) 2023 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.GroupTheory.Torsion
import Mathlib.Data.ENat.Lattice
/-!
# Minimum order of an element
This file defines the minimum order of an element of a monoid.
##... |
@[simp]
lemma minOrder_of_prime {p : ℕ} (hp : p.Prime) : minOrder (ZMod p) = p := by
| Mathlib/GroupTheory/Order/Min.lean | 92 | 94 |
/-
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... | · apply cof_type_le
refine fun a => ⟨Sum.inr PUnit.unit, Set.mem_singleton _, ?_⟩
rcases a with (a | ⟨⟨⟨⟩⟩⟩) <;> simp [EmptyRelation]
| Mathlib/SetTheory/Cardinal/Cofinality.lean | 389 | 391 |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Rat
import Mathlib.Data.Rat.Cast.CharZero
import Mathlib.Tactic.Positivity.Core
/-!
# Casts of rational numbers in... | (castOrderEmbedding (K := K)).preimage_uIoc p q
end LinearOrderedField
end Rat
| Mathlib/Data/Rat/Cast/Order.lean | 139 | 143 |
/-
Copyright (c) 2024 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.Topology.Separation.CompletelyRegular
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
/-!
# Dirac deltas as probability measures and embedding of ... | /-- The assignment `x ↦ diracProba x` is injective if all singletons are measurable. -/
lemma injective_diracProba {X : Type*} [MeasurableSpace X] [MeasurableSpace.SeparatesPoints X] :
Function.Injective (fun (x : X) ↦ diracProba x) := by
intro x y x_eq_y
rw [← dirac_eq_dirac_iff]
rwa [Subtype.ext_iff] at x_e... | Mathlib/MeasureTheory/Measure/DiracProba.lean | 51 | 56 |
/-
Copyright (c) 2020 Bhavik Mehta, Edward Ayers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Edward Ayers
-/
import Mathlib.CategoryTheory.Sites.Sieves
import Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer
import Mathlib.CategoryTheory.Category.P... | apply J.transitive h
intro Z g hg
rw [← Sieve.pullback_comp]
| Mathlib/CategoryTheory/Sites/Grothendieck.lean | 191 | 193 |
/-
Copyright (c) 2020 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Ira Fesefeldt
-/
import Mathlib.Control.Monad.Basic
import Mathlib.Dynamics.FixedPoints.Basic
import Mathlib.Order.CompleteLattice.Basic
import Mathlib.Order.Iterate
import M... |
theorem mem_ωSup (x : α) (c : Chain (Part α)) : x ∈ ωSup c ↔ some x ∈ c := by
simp only [ωSup, Part.ωSup]
constructor
· split_ifs with h
swap
· rintro ⟨⟨⟩⟩
| Mathlib/Order/OmegaCompletePartialOrder.lean | 354 | 360 |
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kenny Lau
-/
import Mathlib.Algebra.Group.Units.Basic
import Mathlib.RingTheory.MvPowerSeries.Basic
import Mathlib.RingTheory.MvPowerSeries.NoZeroDivisors
import Mathl... | φ * φ⁻¹ = 1 := by rw [← invOfUnit_eq φ h, mul_invOfUnit φ (Units.mk0 _ h) rfl]
| Mathlib/RingTheory/MvPowerSeries/Inverse.lean | 243 | 244 |
/-
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 mem_coclosed_Lindelof' : s ∈ coclosedLindelof X ↔
∃ t, IsClosed t ∧ IsLindelof t ∧ sᶜ ⊆ t := by
simp only [mem_coclosedLindelof, compl_subset_comm]
| Mathlib/Topology/Compactness/Lindelof.lean | 466 | 468 |
/-
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.Conj
import Mathlib.Algebra.Group.Subgroup.Lattice
import Mathlib.Algebra.Group.Submonoid.BigOperators
import Mathlib.Data.Finset.Fin
import ... | exact ne_of_lt hab
rw [if_pos h₁, if_neg (h₁.lt_of_ne this).not_le]
rfl
· rw [if_neg h₁, if_pos (lt_of_not_ge h₁).le]
rfl
private theorem signAux_swap_zero_one' (n : ℕ) : signAux (swap (0 : Fin (n + 2)) 1) = -1 :=
show _ = ∏ x ∈ {(⟨1, 0⟩ : Σ _ : Fin (n + 2), Fin (n + 2))},
if (Equ... | Mathlib/GroupTheory/Perm/Sign.lean | 244 | 264 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Data.Nat.Log
import Mathlib.Data.Nat.Pri... | simp only [Nat.add_div (pow_pos hp.pos _)]
_ = _ := by simp [sum_add_distrib, sum_boole]
/-- The multiplicity of `p` in `choose (n + k) k` is the number of carries when `k` and `n`
are added in base `p`. The set is expressed by filtering `Ico 1 b` where `b`
is any bound greater than `log p (n + k)`. -/
t... | Mathlib/Data/Nat/Multiplicity.lean | 185 | 195 |
/-
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.Operations
import Mathlib.Order.Basic
import Mathlib.Order.BooleanAlgebra
import Mathlib.Tactic.Tauto
import Mathlib.Tactic.B... | theorem ne_univ_iff_exists_not_mem {α : Type*} (s : Set α) : s ≠ univ ↔ ∃ a, a ∉ s := by
rw [← not_forall, ← eq_univ_iff_forall]
| Mathlib/Data/Set/Basic.lean | 585 | 586 |
/-
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... | IsLindelof t := hs t ht
lemma IsHereditarilyLindelof.isLindelof (hs : IsHereditarilyLindelof s) :
IsLindelof s := hs.isLindelof_subset Subset.rfl
| Mathlib/Topology/Compactness/Lindelof.lean | 710 | 713 |
/-
Copyright (c) 2024 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.NumberTheory.ZetaValues
import Mathlib.NumberTheory.LSeries.RiemannZeta
/-!
# Special values of Hurwitz and Riemann zeta functions
This file gives t... | have : (2 * k + 1)! = (2 * k + 1) * Complex.Gamma (2 * k + 1) := by
rw [(by simp : Complex.Gamma (2 * k + 1) = Complex.Gamma (↑(2 * k) + 1)),
Complex.Gamma_nat_eq_factorial, ← Nat.cast_ofNat (R := ℂ), ← Nat.cast_mul,
← Nat.cast_add_one, ← Nat.cast_mul, ← Nat.factorial_succ]
simp_rw [this, Gammaℂ, c... | Mathlib/NumberTheory/LSeries/HurwitzZetaValues.lean | 113 | 124 |
/-
Copyright (c) 2015 Nathaniel Thomas. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nathaniel Thomas, Jeremy Avigad, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.GroupWithZero.Action.Defs
import Mathlib.Algebra.Ring.Defs
/-!
# Modules over a ring
In th... | Mathlib/Algebra/Module/Defs.lean | 643 | 644 | |
/-
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.Logic.OpClass
import Mathlib.Order.Lattice
/-!
# `max` and `min`
This file proves basic properties about maxima and minima on a `LinearOrder`.
## Ta... | -- short-circuit type class inference
theorem min_lt_max : min a b < max a b ↔ a ≠ b :=
inf_lt_sup
theorem max_lt_max (h₁ : a < c) (h₂ : b < d) : max a b < max c d :=
max_lt (lt_max_of_lt_left h₁) (lt_max_of_lt_right h₂)
| Mathlib/Order/MinMax.lean | 165 | 170 |
/-
Copyright (c) 2018 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Limits.HasLimits
import Mathlib.CategoryTheory.Discrete.Basic
/-!
# Categorical (co)products
This file defines (co)products ... |
/-- A cofan `c` on `f` such that the induced map `∐ f ⟶ c.pt` is an iso, is a coproduct. -/
def Cofan.isColimitOfIsIsoSigmaDesc {f : β → C} [HasCoproduct f] (c : Cofan f)
| Mathlib/CategoryTheory/Limits/Shapes/Products.lean | 273 | 275 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Ordmap.Invariants
/-!
# Verification of `Ordnode`
This file uses the invariants defined in `Mathlib.Data.Ordmap.Invariants` to construct `Ordset... | Mathlib/Data/Ordmap/Ordset.lean | 1,383 | 1,387 | |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yourong Zang
-/
import Mathlib.Analysis.Calculus.ContDiff.Operations
import Mathlib.Analysis.Calculus.Deriv.Linear
import Mathlib.Analysis.Complex.Basic
/-! # Re... | Mathlib/Analysis/Complex/RealDeriv.lean | 162 | 166 | |
/-
Copyright (c) 2023 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.Algebra.Order.Group.Indicator
import Mathlib.Analysis.PSeries
import Mathlib.NumberTheory.SmoothNumbers
/-!
# The sum of the reciprocals of the primes d... | /-- The series over `p^r` for primes `p` converges if and only if `r < -1`. -/
theorem Nat.Primes.summable_rpow {r : ℝ} :
Summable (fun p : Nat.Primes ↦ (p : ℝ) ^ r) ↔ r < -1 := by
by_cases h : r < -1
· -- case `r < -1`
simp only [h, iff_true]
exact (Real.summable_nat_rpow.mpr h).subtype _
· -- case `... | Mathlib/NumberTheory/SumPrimeReciprocals.lean | 86 | 97 |
/-
Copyright (c) 2024 Calle Sönne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Paul Lezeau, Calle Sönne
-/
import Mathlib.CategoryTheory.FiberedCategory.HomLift
/-!
# Cartesian morphisms
This file defines cartesian resp. strongly cartesian morphisms with respect ... | ```
such that `φ'` lifts `g ≫ f`, there exists a lift `χ` of `g` such that `φ' = χ ≫ φ`. -/
@[stacks 02XK]
class IsStronglyCartesian : Prop extends IsHomLift p f φ where
universal_property' {a' : 𝒳} (g : p.obj a' ⟶ R) (φ' : a' ⟶ b) [IsHomLift p (g ≫ f) φ'] :
∃! χ : a' ⟶ a, IsHomLift p g χ ∧ χ ≫ φ = φ'
| Mathlib/CategoryTheory/FiberedCategory/Cartesian.lean | 67 | 72 |
/-
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.Composition.MeasureComp
import Mathlib.Probability.Kernel.CondDistrib
import Mathlib.Probability.ConditionalProbability
/-!
# Kernel as... |
lemma condExpKernel_apply_eq_condDistrib [Nonempty Ω] {ω : Ω} :
condExpKernel μ m ω = @condDistrib Ω Ω Ω mΩ _ _ mΩ (m ⊓ mΩ) id id μ _ (id ω) := by
simp [condExpKernel_eq, Kernel.comap_apply]
@[deprecated (since := "2025-01-21")]
| Mathlib/Probability/Kernel/Condexp.lean | 87 | 92 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Order.Filter.Bases.Finit... | Mathlib/Topology/Algebra/Group/Basic.lean | 1,794 | 1,803 | |
/-
Copyright (c) 2024 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Sites.OneHypercover
/-!
# Characterization of sheaves using 1-hypercovers
In this file, given a Grothendieck topology `J` on a category `C`,
we ... | {x y : T ⟶ P.obj (Opposite.op X)}
(h : ∀ ⦃Y : C⦄ (f : Y ⟶ X) (_ : S f), x ≫ P.map f.op = y ≫ P.map f.op) :
x = y := by
obtain ⟨E, hE, le⟩ := H.exists_oneHypercover S hS
exact Multifork.IsLimit.hom_ext (hP E hE).some (fun j => h _ (le _ (Sieve.ofArrows_mk _ _ _)))
| Mathlib/CategoryTheory/Sites/IsSheafOneHypercover.lean | 74 | 79 |
/-
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.Finite.Defs
import Mathlib.Data.Finset.BooleanAlgebra
import Mathlib.Data.Finset.Image
import Mathlib.Data.Fintype.Defs
import Mathlib.Data.Fintyp... | Mathlib/Data/Fintype/Basic.lean | 329 | 330 | |
/-
Copyright (c) 2022 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.Tactic.IntervalCases
/-!
# Cubics and discriminants
This file defines cubic polynomials ... | theorem degree_of_a_ne_zero' (ha : a ≠ 0) : (toPoly ⟨a, b, c, d⟩).degree = 3 :=
degree_of_a_ne_zero ha
| Mathlib/Algebra/CubicDiscriminant.lean | 259 | 261 |
/-
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.CharP.Basic
import Mathlib.Algebra.Module.End
import Mathlib.Algebra.Ring.Prod
import Mathlib.Data.Fintype.Units
import Mathlib.GroupTheory.GroupAc... | ZMod.val_one_eq_one_mod, pow_mod_natCard, pow_one]
@[to_additive existing (attr := simp) zmod_val_inv_nsmul_nsmul]
lemma pow_pow_zmod_val_inv (hn : (Nat.card α).Coprime n) (a : α) :
(a ^ n) ^ (n⁻¹ : ZMod (Nat.card α)).val = a := by rw [pow_right_comm, pow_zmod_val_inv_pow hn]
end Group
open ZMod
| Mathlib/Data/ZMod/Basic.lean | 1,250 | 1,259 |
/-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Gabin Kolly
-/
import Mathlib.Data.Finite.Sum
import Mathlib.Data.Fintype.Order
import Mathlib.ModelTheory.FinitelyGenerated
import Mathlib.ModelTheory.Quotients
import... | (liftInclusion S) (of L ι _ (fun _ _ h ↦ Substructure.inclusion (S.monotone h)) i x)
= Substructure.subtype (S i) x := rfl
lemma rangeLiftInclusion : (liftInclusion S).toHom.range = ⨆ i, S i := by
simp_rw [liftInclusion, range_lift, Substructure.range_subtype]
/-- The isomorphism between a direct limit of a... | Mathlib/ModelTheory/DirectLimit.lean | 447 | 461 |
/-
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.Sum
import Mathlib.Data.Fintype.EquivFin
import Mathlib.Logic.Embedding.Set
/-!
## Instances
We provide the `Fintype` instance for the su... | classical
let s' : Finset α := s.toFinset
have hfst' : s'.image f ⊆ t := by simpa [s', ← Finset.coe_subset] using hfst
have hfs' : Set.InjOn f s' := by simpa [s'] using hfs
obtain ⟨g, hg⟩ := Finset.exists_equiv_extend_of_card_eq hαt hfst' hfs'
refine ⟨g, fun i hi => ?_⟩
| Mathlib/Data/Fintype/Sum.lean | 126 | 131 |
/-
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 ... | fun x ↦ mem_closure_iff_ultrafilter.mpr
⟨x.map pure, range_mem_map, ultrafilter_converges_iff.mpr (bind_pure x).symm⟩
/-- The map `pure : α → Ultrafilter α` induces on `α` the discrete topology. -/
theorem induced_topology_pure :
| Mathlib/Topology/StoneCech.lean | 140 | 144 |
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.AlgebraicGeometry.Cover.Open
import Mathlib.AlgebraicGeometry.Over
/-!
# Restriction of Schemes and Morphisms
## Main definition
- `AlgebraicGeometry.Schem... | @[reassoc (attr := simp)]
lemma Scheme.isoOfEq_hom_ι (X : Scheme.{u}) {U V : X.Opens} (e : U = V) :
(X.isoOfEq e).hom ≫ V.ι = U.ι :=
IsOpenImmersion.isoOfRangeEq_hom_fac _ _ _
| Mathlib/AlgebraicGeometry/Restrict.lean | 362 | 365 |
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.LocalProperties.Basic
import Mathlib.RingTheory.Localization.Integral
/-!
# The meta properties of integral ring homomorphisms.
-/
namespace ... | · apply isIntegral_zero
· intro x y; exact IsIntegral.tmul x (h y)
· intro x y hx hy; exact IsIntegral.add hx hy
open Polynomial in
/-- `S` is an integral `R`-algebra if there exists a set `{ r }` that
spans `R` such that each `Sᵣ` is an integral `Rᵣ`-algebra. -/
| Mathlib/RingTheory/RingHom/Integral.lean | 35 | 41 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.HomotopyCategory.HomComplex
import Mathlib.Algebra.Homology.HomotopyCofiber
/-! # The mapping cone of a morphism of cochain complexes
In this ... | simp [inr, fst]
@[reassoc (attr := simp)]
lemma inr_f_snd_v (p : ℤ) :
| Mathlib/Algebra/Homology/HomotopyCategory/MappingCone.lean | 87 | 90 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Algebra.Algebra.Hom
import Mathlib.RingTheory.Congruence.Basic
import Mathlib.RingTheory.Ideal.Quotient.Defs
import Mathlib.RingTheory.Ideal.Span
/-!
# Qu... | fun _ w ↦ w ⟨x, y, h, sub_add_cancel x y⟩⟩
@[simp]
theorem ringQuotToIdealQuotient_apply (r : B → B → Prop) (x : B) :
| Mathlib/Algebra/RingQuot.lean | 480 | 483 |
/-
Copyright (c) 2022 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.List.Induction
import Mathlib.Data.List.TakeWhile
/-!
# Dropping or taking from lists on the right
Taking or removing element from the tail e... | Mathlib/Data/List/DropRight.lean | 249 | 250 | |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Localization.Opposite
/-!
# Calculus of fractions
Following the definitions by [Gabriel and Zisman][gabriel-zisman-1967],
given a morphism prope... | φ.map L (Localization.inverts _ _) = ψ.map L (Localization.inverts _ _) ↔
RightFractionRel φ ψ := by
rw [← leftFractionRel_op_iff, ← LeftFraction.map_eq_iff L.op W.op φ.op ψ.op,
← φ.op_map L (Localization.inverts _ _), ← ψ.op_map L (Localization.inverts _ _)]
constructor
· apply Quiver.Hom.unop_inj
... | Mathlib/CategoryTheory/Localization/CalculusOfFractions.lean | 950 | 958 |
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel, Kim Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
/-!
# The abelian image and coimage.
In an abelian category we usually want the image of a morphism ... | end CategoryTheory.Abelian
| Mathlib/CategoryTheory/Abelian/Images.lean | 115 | 118 |
/-
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 | 160 | 164 | |
/-
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.... | replace hM : lowerCentralSeries R L M (2 * k) = ⊥ := by
rw [eq_bot_iff, ← hM]; exact antitone_lowerCentralSeries R L M (by omega)
use k
ext m
rw [Module.End.pow_apply, LinearMap.zero_apply, ← LieSubmodule.mem_bot (R := R) (L := L), ← hM]
exact iterate_toEnd_mem_lowerCentralSeries₂ R L M x y m k
@[simp] l... | Mathlib/Algebra/Lie/Nilpotent.lean | 340 | 348 |
/-
Copyright (c) 2024 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Kernel.Disintegration.Density
import Mathlib.Probability.Kernel.WithDensity
/-!
# Radon-Nikodym derivative and Lebesgue decomposition for kern... | lemma singularPart_self (κ : Kernel α γ) [IsFiniteKernel κ] : κ.singularPart κ = 0 := by
ext : 1; rw [zero_apply, singularPart_eq_zero_iff_absolutelyContinuous]
section Unique
variable {ξ : Kernel α γ} {f : α → γ → ℝ≥0∞} [IsFiniteKernel η]
omit hαγ in
| Mathlib/Probability/Kernel/RadonNikodym.lean | 480 | 487 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.MeasureTheory.Measure.AEMeasurable
import Mathlib.Order.Filter.EventuallyConst
/-!
# Measure preserving maps
We say that `f : α → β` is a measure p... | simp only [add_smul, one_smul, ← n.add_one]
refine le_trans (measure_symmDiff_le s (f^[n] ⁻¹' s) (f^[n+1] ⁻¹' s)) (add_le_add ih ?_)
replace hs : NullMeasurableSet (s ∆ (f ⁻¹' s)) μ :=
hs.symmDiff <| hs.preimage hf.quasiMeasurePreserving
rw [iterate_succ', preimage_comp, ← preimage_symmDiff, (hf.iterate n).... | Mathlib/Dynamics/Ergodic/MeasurePreserving.lean | 182 | 186 |
/-
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... | simp only [ofHom, ofHoms_v]
@[simp]
lemma ofHom_v_comp_d (φ : F ⟶ G) (p q q' : ℤ) (hpq : p + 0 = q) :
| Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean | 170 | 173 |
/-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Data.Complex.Basic
import Mathlib.MeasureTheory.Integral.CircleIntegral
/-!
# Circle integral transform
In this file we define the circle integral tr... | theorem continuous_circleTransform {R : ℝ} (hR : 0 < R) {f : ℂ → E} {z w : ℂ}
(hf : ContinuousOn f <| sphere z R) (hw : w ∈ ball z R) :
Continuous (circleTransform R z w f) := by
apply_rules [Continuous.smul, continuous_const]
· rw [funext <| deriv_circleMap _ _]
| Mathlib/MeasureTheory/Integral/CircleTransform.lean | 68 | 72 |
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.MetricSpace.HausdorffDistance
/-!
# Topological study of spaces `Π (n : ℕ), E n`
When `E n` are topological spaces, the space `Π (n : ... | ∃ v ∈ { S : Set (∀ i : ℕ, E i) | ∃ (U : ∀ i : ℕ, Set (E i)) (F : Finset ℕ),
(∀ i : ℕ, i ∈ F → U i ∈ { s : Set (E i) | IsOpen s }) ∧ S = (F : Set ℕ).pi U },
x ∈ v ∧ v ⊆ u :=
(isTopologicalBasis_pi fun n : ℕ => isTopologicalBasis_opens).exists_subset_of_mem_open hx
u_open
rcase... | Mathlib/Topology/MetricSpace/PiNat.lean | 339 | 354 |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Data.Set.Lattice
import Mathlib.Data.SetLike.Basic
import Mathlib.Order.ModularLattice
import Mathlib.Order.SuccPred.Basic
import Mathlib.Order.WellFou... | @[simp]
theorem isCoatom_iff [OrderTop α] [OrderTop β] (f : α ≃o β) (a : α) :
| Mathlib/Order/Atoms.lean | 990 | 991 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.CategoryTheory.Adjunction.FullyFaithful
import Mathlib.CategoryTheory.Elementwise
import Mathlib.Topology.Sheaves.Presheaf
/-!
# Presheafed spaces
Introd... | theorem mapPresheaf_map_f (F : C ⥤ D) {X Y : PresheafedSpace C} (f : X ⟶ Y) :
(F.mapPresheaf.map f).base = f.base :=
rfl
@[simp]
theorem mapPresheaf_map_c (F : C ⥤ D) {X Y : PresheafedSpace C} (f : X ⟶ Y) :
(F.mapPresheaf.map f).c = whiskerRight f.c F :=
rfl
end Functor
namespace NatTrans
/-- A natural ... | Mathlib/Geometry/RingedSpace/PresheafedSpace.lean | 421 | 434 |
/-
Copyright (c) 2023 Rémi Bottinelli. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémi Bottinelli
-/
import Mathlib.Data.Set.Function
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Topology.EMetricSpace.BoundedVariation
/-!
# Constant speed
This file defines... | rw [← φst] at hf
convert unique_unit_speed φm hfφ hf ⟨le_rfl, hs⟩ using 1
have : φ 0 = 0 := by
have hm : 0 ∈ φ '' Icc 0 s := by simp only [φst, ht, mem_Icc, le_refl, and_self]
obtain ⟨x, xs, hx⟩ := hm
| Mathlib/Analysis/ConstantSpeed.lean | 212 | 216 |
/-
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 | 776 | 778 | |
/-
Copyright (c) 2023 Hanneke Wiersema. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Hanneke Wiersema, Andrew Yang
-/
import Mathlib.Algebra.Ring.Aut
import Mathlib.NumberTheory.Padics.RingHoms
import Mathlib.RingTheory.RootsOfUnity.EnoughRootsOfUnity
... | (L ≃+* L) →* (ZMod n)ˣ :=
(Units.mapEquiv <| (ZMod.ringEquivCongr hn).toMulEquiv).toMonoidHom.comp
(ModularCyclotomicCharacter' L n)
namespace ModularCyclotomicCharacter
variable {n : ℕ} [NeZero n] (hn : Fintype.card { x // x ∈ rootsOfUnity n L } = n)
lemma spec (g : L ≃+* L) {t : Lˣ} (ht : t ∈ rootsOfUnity ... | Mathlib/NumberTheory/Cyclotomic/CyclotomicCharacter.lean | 209 | 219 |
/-
Copyright (c) 2020 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.Group.Action.Pi
import Mathlib.Data.Finset.Prod
import Mathlib.Data.SetLike.Basic
import Mathlib.Data.Sym.Basic
import Mathlib.Data.Sym.Sym2.Init
/-... | rw [rel_iff', Prod.mk.injEq, Prod.swap_prod_mk]
apply hpq.imp <;> rintro rfl <;> simp
theorem forall_mem_pair {P : α → Prop} {a b : α} : (∀ x ∈ s(a, b), P x) ↔ P a ∧ P b := by
simp only [mem_iff, forall_eq_or_imp, forall_eq]
| Mathlib/Data/Sym/Sym2.lean | 414 | 419 |
/-
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... | theorem mapMatrix_comp (f : β →ₐ[R] γ) (g : α →ₐ[R] β) :
f.mapMatrix.comp g.mapMatrix = ((f.comp g).mapMatrix : Matrix m m α →ₐ[R] _) :=
| Mathlib/Data/Matrix/Basic.lean | 581 | 582 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Yury Kudryashov
-/
import Mathlib.Algebra.Algebra.Equiv
import Mathlib.Algebra.Algebra.NonUnitalSubalgebra
import Mathlib.RingTheory.SimpleRing.Basic
/-!
# Subalgebras over Comm... | Mathlib/Algebra/Algebra/Subalgebra/Basic.lean | 1,290 | 1,294 | |
/-
Copyright (c) 2021 Hunter Monroe. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hunter Monroe, Kyle Miller, Alena Gusakov
-/
import Mathlib.Combinatorics.SimpleGraph.DeleteEdges
import Mathlib.Data.Fintype.Powerset
/-!
# Subgraphs of a simple graph
A subgraph of ... | symm _ _ h := ⟨h.2.1, h.1, G'.symm h.2.2⟩
theorem _root_.SimpleGraph.induce_eq_coe_induce_top (s : Set V) :
| Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | 1,130 | 1,132 |
/-
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, Eric Wieser, Bhavik Mehta,
Yaël Dillies
-/
import Mathlib.Algebra.Order.Antidiag.Pi
i... | apply Function.Embedding.equiv_symm_toEmbedding_trans_toEmbedding
end AddCommMonoid
section CanonicallyOrderedAddCommMonoid
variable [DecidableEq ι] [DecidableEq μ] [AddCommMonoid μ] [PartialOrder μ]
| Mathlib/Algebra/Order/Antidiag/Finsupp.lean | 133 | 138 |
/-
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 | 1,076 | 1,085 | |
/-
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.QuadraticForm.TensorProduct
import Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv
/-!
# Linear equivalences of tensor products as isometrie... |
@[simp]
theorem tmul_comp_tensorMap
{Q₁ : QuadraticForm R M₁} {Q₂ : QuadraticForm R M₂}
{Q₃ : QuadraticForm R M₃} {Q₄ : QuadraticForm R M₄}
(f : Q₁ →qᵢ Q₂) (g : Q₃ →qᵢ Q₄) :
(Q₂.tmul Q₄).comp (TensorProduct.map f.toLinearMap g.toLinearMap) = Q₁.tmul Q₃ := by
have h₁ : Q₁ = Q₂.comp f.toLinearMap := Qu... | Mathlib/LinearAlgebra/QuadraticForm/TensorProduct/Isometries.lean | 37 | 46 |
/-
Copyright (c) 2020 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.Topology.Path
/-!
# Path connectedness
Continuing from `Mathlib.Topology.Path`, this file defines path components and path-connected
spaces.
## Main... |
theorem exists_path_through_family' {n : ℕ} (p : Fin (n + 1) → X) :
∃ (γ : Path (p 0) (p n)) (t : Fin (n + 1) → I), ∀ i, γ (t i) = p i := by
have : IsPathConnected (univ : Set X) := pathConnectedSpace_iff_univ.mp (by infer_instance)
rcases this.exists_path_through_family' p fun _i => True.intro with ⟨γ, t, -, ... | Mathlib/Topology/Connected/PathConnected.lean | 518 | 522 |
/-
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.Order.Interval.Set.IsoIoo
import Mathlib.Topology.ContinuousMap.Bounded.Normed
import Mathlib.Topology.UrysohnsBounded
/-!
# Tietze extension theore... | /-- Any homeomorphism from a `TietzeExtension` space is one itself. -/
theorem TietzeExtension.of_homeo {Y : Type v} {Z : Type w} [TopologicalSpace Y]
[TopologicalSpace Z] [TietzeExtension.{u, w} Z] (e : Y ≃ₜ Z) :
TietzeExtension.{u, v} Y :=
| Mathlib/Topology/TietzeExtension.lean | 146 | 149 |
/-
Copyright (c) 2024 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.ObjectProperty.ClosedUnderIsomorphisms
import Mathlib.CategoryTheory.Localization.CalculusOfFractions
import Mathlib.CategoryTheory.Localization.T... | variable {S}
lemma W.shift {X₁ X₂ : C} {f : X₁ ⟶ X₂} (hf : S.W f) (n : ℤ) : S.W (f⟦n⟧') := by
rw [← smul_mem_W_iff _ _ (n.negOnePow)]
| Mathlib/CategoryTheory/Triangulated/Subcategory.lean | 180 | 183 |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.Real
/-!
# Properties of pointwise scalar multiplication of se... | · simp [hr, zero_smul_set, Set.singleton_zero, nonempty_closedBall]
· exact smul_closedBall' hc x r
| Mathlib/Analysis/NormedSpace/Pointwise.lean | 363 | 365 |
/-
Copyright (c) 2020 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Polynomial.Degree.TrailingDegree
import Mathlib.Algebra.Polynomial.EraseLead
/-!
# Reverse of a univariate polynomial
The main definition is `r... | rw [leadingCoeff, reverse_natDegree, ← revAt_le f.natTrailingDegree_le_natDegree,
| Mathlib/Algebra/Polynomial/Reverse.lean | 269 | 269 |
/-
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 | 1,074 | 1,078 |
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