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 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Ines Wright, Joachim Breitner
-/
import Mathlib.GroupTheory.Solvable
import Mathlib.GroupTheory.Sylow
import Mathlib.Algebra.Group.Subgroup.Order
import Mathlib.GroupTheo... | subsingleton_iff_bot_eq_top, Subgroup.subsingleton_iff]
/-- Quotienting the `center G` reduces the nilpotency class by 1 -/
theorem nilpotencyClass_quotient_center [hH : IsNilpotent G] :
Group.nilpotencyClass (G ⧸ center G) = Group.nilpotencyClass G - 1 := by
generalize hn : Group.nilpotencyClass G = n
rca... | Mathlib/GroupTheory/Nilpotent.lean | 590 | 605 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Batteries.Tactic.Congr
import Mathlib.Data.Option.Basic
import Mathlib.Data.Prod.Basic
import Mathlib.Data.Set.Subsingleton
import Mathlib.Dat... | @[simp]
theorem image_surjective : Surjective (image f) ↔ Surjective f := by
| Mathlib/Data/Set/Image.lean | 1,289 | 1,290 |
/-
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... | Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 1,574 | 1,578 | |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Emily Riehl
-/
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryTheory.Functor.TwoSquare
import Mathlib.CategoryTheory.HomCongr
import Mathlib.Tactic.... | theorem mateEquiv_square (α : TwoSquare G₁ L₁ L₃ H₁) (β : TwoSquare H₁ L₂ L₄ K₁)
(γ : TwoSquare G₂ L₃ L₅ H₂) (δ : TwoSquare H₂ L₄ L₆ K₂) :
(mateEquiv (adj₁.comp adj₂) (adj₅.comp adj₆)) ((α ≫ᵥ β) ≫ₕ (γ ≫ᵥ δ)) =
| Mathlib/CategoryTheory/Adjunction/Mates.lean | 218 | 220 |
/-
Copyright (c) 2023 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.Portmanteau
import Mathlib.MeasureTheory.Integral.DominatedConvergence
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | ≤ (∫ t in Ioc 0 ‖f‖, ν.real (thickening ε {a | t ≤ f a})) + ε * ‖f‖ := by
rw [BoundedContinuousFunction.integral_eq_integral_meas_le f μ f_nn]
have key : (fun (t : ℝ) ↦ μ.real {a | t ≤ f a})
≤ (fun (t : ℝ) ↦ ν.real (thickening ε {a | t ≤ f a}) + ε) := by
intro t
simp only [measureReal_de... | Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.lean | 360 | 385 |
/-
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.VecNotation
import Mathlib.Logic.Small.Basic
import Mathlib.SetTheory.ZFC.PSet
/-!
# A model of ZFC
In this file, we model Zermelo-Fraenkel ... | Mathlib/SetTheory/ZFC/Basic.lean | 1,570 | 1,573 | |
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Analysis.Normed.Ring.Lemmas
/-!
# The integers as normed ring
This file contains basic facts about the integers as normed ring.
Recall that `‖n‖` de... | @[simp]
theorem nnnorm_natCast (n : ℕ) : ‖(n : ℤ)‖₊ = n :=
| Mathlib/Analysis/NormedSpace/Int.lean | 29 | 30 |
/-
Copyright (c) 2022 Jake Levinson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jake Levinson
-/
import Mathlib.Data.Finset.Preimage
import Mathlib.Data.Finset.Prod
import Mathlib.Data.SetLike.Basic
import Mathlib.Order.UpperLower.Basic
/-!
# Young diagrams
A You... | Mathlib/Combinatorics/Young/YoungDiagram.lean | 500 | 502 | |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Jakob von Raumer
-/
import Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
/-!
# Biproducts and binary biproducts
... | inv := biproduct.lift fun i => biproduct.lift fun x => biproduct.π _ (⟨i, x⟩ : Σ i, f i)
section πKernel
section
variable (f : J → C) [HasBiproduct f]
variable (p : J → Prop) [HasBiproduct (Subtype.restrict p f)]
/-- The canonical morphism from the biproduct over a restricted index type to the biproduct of
the fu... | Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean | 700 | 716 |
/-
Copyright (c) 2023 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Order.OmegaCompletePartialOrder
/-!
# Complete Partial Orders
This file considers complete partial orders (sometimes called directedly comple... | /-- Scott-continuity takes on a simpler form in complete partial orders. -/
lemma CompletePartialOrder.scottContinuous {f : α → β} :
ScottContinuous f ↔
∀ ⦃d : Set α⦄, d.Nonempty → DirectedOn (· ≤ ·) d → IsLUB (f '' d) (f (sSup d)) := by
refine ⟨fun h d hd₁ hd₂ ↦ h hd₁ hd₂ hd₂.isLUB_sSup, fun h d hne hd a hda... | Mathlib/Order/CompletePartialOrder.lean | 63 | 69 |
/-
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... | Mathlib/Data/Set/Function.lean | 1,876 | 1,877 | |
/-
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.Calculus.InverseFunctionTheorem.Deriv
import Mathlib.Analysis.Calculus.LogDeriv... |
theorem HasFDerivWithinAt.clog {f : E → ℂ} {f' : E →L[ℂ] ℂ} {s : Set E} {x : E}
(h₁ : HasFDerivWithinAt f f' s x) (h₂ : f x ∈ slitPlane) :
HasFDerivWithinAt (fun t => log (f t)) ((f x)⁻¹ • f') s x :=
| Mathlib/Analysis/SpecialFunctions/Complex/LogDeriv.lean | 110 | 113 |
/-
Copyright (c) 2018 Rohan Mitta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rohan Mitta, Kevin Buzzard, Alistair Tucker, Johannes Hölzl, Yury Kudryashov, Winston Yin
-/
import Mathlib.Algebra.Group.End
import Mathlib.Topology.EMetricSpace.Diam
/-!
# Lipschitz co... | (K⁻¹ : ℝ≥0∞) * edist (f x) (f y) ≤ edist x y := by
rw [mul_comm, ← div_eq_mul_inv]
exact ENNReal.div_le_of_le_mul' (h x y)
protected theorem of_edist_le (h : ∀ x y, edist (f x) (f y) ≤ edist x y) : LipschitzWith 1 f :=
| Mathlib/Topology/EMetricSpace/Lipschitz.lean | 155 | 159 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Polynomial.Degree.Domain
import Mathlib.Algebra.Polynomial.Degree.Support
import Mathlib.Algebra.Poly... |
@[simp]
| Mathlib/Algebra/Polynomial/Derivative.lean | 125 | 126 |
/-
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... | -/
theorem bijOn_image_image {p₁ : α → γ} {p₂ : β → δ} {g : γ → δ} (comm : ∀ a, p₂ (f a) = g (p₁ a))
(hbij : BijOn f s t) (hinj: InjOn g (p₁ '' s)) : BijOn g (p₁ '' s) (p₂ '' t) := by
obtain ⟨h1, h2, h3⟩ := hbij
| Mathlib/Data/Set/Function.lean | 661 | 664 |
/-
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... |
theorem subsingleton_iff_exists_singleton_mem [Nonempty α] : l.Subsingleton ↔ ∃ a, {a} ∈ l := by
| Mathlib/Order/Filter/Subsingleton.lean | 70 | 71 |
/-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.Order.Floor.Semiring
import Mathlib.Data.Nat.Log
/-!
# Integer logarithms in a field with respect to a natural base
This file defines two `ℤ`-value... | Int.ofNat_zero, neg_zero]
· exact if_neg hr.not_le
@[simp, norm_cast]
theorem log_natCast (b : ℕ) (n : ℕ) : log b (n : R) = Nat.log b n := by
| Mathlib/Data/Int/Log.lean | 66 | 70 |
/-
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.Algebra.ZMod
import Mathlib.Data.Nat.Multiplicity
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.WittVector.Basic
import Mathlib.... | refine
-- Porting note: removing the placeholders give an error
IsPoly.ext (@IsPoly.comp p _ _ (frobeniusFun_isPoly p) WittVector.oneIsPoly)
(@IsPoly.comp p _ _ WittVector.oneIsPoly (frobeniusFun_isPoly p)) ?_ _ 0
| Mathlib/RingTheory/WittVector/Frobenius.lean | 236 | 239 |
/-
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.Algebra.BigOperators.Fin
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Finset.Sort
/-!
# Compositions
A compositio... |
theorem length_pos_of_mem_splitWrtComposition {l l' : List α} {c : Composition l.length}
(h : l' ∈ l.splitWrtComposition c) : 0 < length l' := by
have : l'.length ∈ (l.splitWrtComposition c).map List.length :=
| Mathlib/Combinatorics/Enumerative/Composition.lean | 739 | 742 |
/-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.SetTheory.Cardinal.Finite
/-!
# Cardinality of finite types
The cardinality of a finite type `α` is given by `Nat.card α`. This function has
the "junk val... |
@[simp]
theorem card_option [Finite α] : Nat.card (Option α) = Nat.card α + 1 := by
| Mathlib/Data/Finite/Card.lean | 78 | 80 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Data.Set.Constructions
import Mathlib.Order.Filter.AtTopBot.CountablyGenerated
import Mathlib.Topology.Constructions
import Mathlib.Top... |
/-- A second countable space is mapped by an open quotient map to a second countable space. -/
theorem _root_.Topology.IsQuotientMap.secondCountableTopology [SecondCountableTopology X]
(h' : IsQuotientMap π) (h : IsOpenMap π) : SecondCountableTopology Y where
is_open_generated_countable := by
obtain ⟨V, V_co... | Mathlib/Topology/Bases.lean | 931 | 937 |
/-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Dagur Asgeirsson
-/
import Mathlib.Topology.Category.CompHaus.Basic
import Mathlib.Topology.Category.CompHausLike.Limits
/-!
# Explicit limits and colimits
This file applies ... | Mathlib/Topology/Category/CompHaus/Limits.lean | 205 | 207 | |
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Measure.Restrict
/-! # Mutually singular measures
Two measures `μ`, `ν` are said to be mutually singular (`MeasureTheory.Mea... | · rwa [compl_iInter, measure_iUnion_null_iff]
@[simp]
theorem sum_right {ι : Type*} [Countable ι] {ν : ι → Measure α} : μ ⟂ₘ sum ν ↔ ∀ i, μ ⟂ₘ ν i :=
comm.trans <| sum_left.trans <| forall_congr' fun _ => comm
@[simp]
| Mathlib/MeasureTheory/Measure/MutuallySingular.lean | 114 | 120 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | constructor
| Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean | 233 | 233 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker, Johan Commelin
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Div
import Mathlib.RingTheory... | Mathlib/Algebra/Polynomial/RingDivision.lean | 528 | 531 | |
/-
Copyright (c) 2019 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.List.FinRange
import Mathlib.Data.List.Perm.Basic
import Mathlib.Data.List.Lex
import Mathlib.Data.List.Induc... | exact mem_append_left _ IH
| cons₂ a s IH =>
rw [length, sublistsLen_succ_cons]
exact mem_append_right _ (mem_map.2 ⟨_, IH, rfl⟩)
| Mathlib/Data/List/Sublists.lean | 275 | 278 |
/-
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.LinearAlgebra.FreeModule.Basic
import Mathlib.MeasureTheory.Measure.Decomposition.Exhaustion
import Mathlib.Probability.ConditionalProbability
/-!
# s-fin... | theorem isFiniteMeasure_toFiniteAux [SFinite μ] : IsFiniteMeasure μ.toFiniteAux := by
rw [Measure.toFiniteAux]
split_ifs
| Mathlib/MeasureTheory/Measure/WithDensityFinite.lean | 68 | 70 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.RingTheory.WittVector.StructurePolynomial
/-!
# Witt vectors
In this file we define the type of `p`-typical Witt vectors and ring op... |
theorem mul_coeff (x y : 𝕎 R) (n : ℕ) :
(x * y).coeff n = peval (wittMul p n) ![x.coeff, y.coeff] := by
| Mathlib/RingTheory/WittVector/Defs.lean | 316 | 318 |
/-
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
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.LogDeriv
import Mathlib.Analysis.SpecialFuncti... | · simp [hasSum_zero]
· convert hasSum_log_one_add_inv (inv_pos.mpr (lt_of_le_of_ne h ha0.symm)) using 4
all_goals field_simp [add_comm]
lemma le_log_one_add_of_nonneg {x : ℝ} (hx : 0 ≤ x) : 2 * x / (x + 2) ≤ log (1 + x) := by
convert le_hasSum (hasSum_log_one_add hx) 0 (by intros; positivity) using 1
field... | Mathlib/Analysis/SpecialFunctions/Log/Deriv.lean | 326 | 343 |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.CategoryTheory.Limits.Shapes.Terminal
import Mathlib.CategoryTheory.Limits.Preserves.Basic
/-!
# Preserving terminal object
Constructions to relate the n... | PreservesColimit (Functor.empty.{0} C) G := by
rw [Subsingleton.elim f (initialComparison G)] at i
exact PreservesInitial.of_iso_comparison G
/-- If there is any isomorphism `⊥ ≅ G.obj ⊥`, then `G` preserves initial objects. -/
| Mathlib/CategoryTheory/Limits/Preserves/Shapes/Terminal.lean | 192 | 196 |
/-
Copyright (c) 2020 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Algebra.GroupWithZero.Basic
/-!
# Indicator functions and support of a function in groups with z... |
lemma mulSupport_one_add [AddLeftCancelMonoid R] (f : ι → R) :
| Mathlib/Algebra/GroupWithZero/Indicator.lean | 141 | 142 |
/-
Copyright (c) 2023 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.DirectSum.LinearMap
import Mathlib.Algebra.Lie.InvariantForm
import Mathlib.Algebra.Lie.Weights.Cartan
import Mathlib.Algebra.Lie.Weights.Linear
impo... | variable (I : LieIdeal R L)
/-- The orthogonal complement of an ideal with respect to the killing form is an ideal. -/
| Mathlib/Algebra/Lie/TraceForm.lean | 357 | 359 |
/-
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.Functor.Currying
import Mathlib.CategoryTheory.Subobject.FactorThru
import Mathlib.CategoryTheory.Subobject.WellPowered
import... | (X ⊔ Y).Factors f :=
factors_of_le f le_sup_right P
variable [HasInitial C] [InitialMonoClass C]
theorem finset_sup_factors {I : Type*} {A B : C} {s : Finset I} {P : I → Subobject B} {f : A ⟶ B}
| Mathlib/CategoryTheory/Subobject/Lattice.lean | 473 | 478 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.MvPolynomial.Rename
/-!
# Degrees of polynomials
This file establ... | @[simp]
theorem totalDegree_monomial (s : σ →₀ ℕ) {c : R} (hc : c ≠ 0) :
(monomial s c : MvPolynomial σ R).totalDegree = s.sum fun _ e => e := by
| Mathlib/Algebra/MvPolynomial/Degrees.lean | 432 | 434 |
/-
Copyright (c) 2019 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.List.FinRange
import Mathlib.Data.List.Perm.Basic
import Mathlib.Data.List.Lex
import Mathlib.Data.List.Induc... | | nil => simp
| cons a l₁ ih =>
rw [foldr_cons, ih]
simp [List.flatMap, flatten_flatten, Function.comp_def]
theorem sublists_cons (a : α) (l : List α) :
sublists (a :: l) = sublists l >>= (fun x => [x, a :: x]) :=
show sublists ([a] ++ l) = _ by
| Mathlib/Data/List/Sublists.lean | 131 | 138 |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.AlgebraicGeometry.Morphisms.Constructors
import Mathlib.AlgebraicGeometry.Morphisms.QuasiCompact
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.Equaliz... | ∃ (n : ℕ) (y : Γ(X, U)), y |_ X.basicOpen f = (f |_ X.basicOpen f) ^ n * x := by
dsimp only [TopCat.Presheaf.restrictOpenCommRingCat_apply]
revert hU' f x
refine compact_open_induction_on U hU ?_ ?_
· intro _ f x
use 0, f
| Mathlib/AlgebraicGeometry/Morphisms/QuasiSeparated.lean | 247 | 252 |
/-
Copyright (c) 2022 María Inés de Frutos-Fernández. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: María Inés de Frutos-Fernández
-/
import Mathlib.RingTheory.DedekindDomain.Ideal
import Mathlib.RingTheory.Valuation.ExtendToLocalization
import Mathlib.Topology.Algebr... | · rw [WithZero.coe_le_coe, ofAdd_le, neg_le_neg_iff, Int.ofNat_le, Ideal.dvd_span_singleton, ←
Associates.le_singleton_iff,
Associates.prime_pow_dvd_iff_le (Associates.mk_ne_zero'.mpr hr)
(by apply v.associates_irreducible)]
/-- The `v`-adic valuation of `0 : R` equals 0. -/
theorem intValuation.... | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | 139 | 147 |
/-
Copyright (c) 2019 Abhimanyu Pallavi Sudhir. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Abhimanyu Pallavi Sudhir
-/
import Mathlib.Order.Filter.FilterProduct
import Mathlib.Analysis.SpecificLimits.Basic
/-!
# Construction of the hyperreal numbers as an ultrapro... | Mathlib/Data/Real/Hyperreal.lean | 789 | 790 | |
/-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Eric Wieser, Jeremy Avigad, Johan Commelin
-/
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.Linear... | end Matrix
| Mathlib/LinearAlgebra/Matrix/SchurComplement.lean | 532 | 548 |
/-
Copyright (c) 2022 Joachim Breitner. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joachim Breitner
-/
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Order.SupIndep
/-!
# Canonical homomorphism from a finite famil... | end MonoidHom
end FamilyOfGroups
| Mathlib/GroupTheory/NoncommPiCoprod.lean | 282 | 285 |
/-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.JapaneseBracket
import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.MeasureTheory.Group.Integral
import Mathlib... | theorem integral_exp_neg_Ioi_zero : (∫ x : ℝ in Ioi 0, exp (-x)) = 1 := by
simpa only [neg_zero, exp_zero] using integral_exp_neg_Ioi 0
| Mathlib/Analysis/SpecialFunctions/ImproperIntegrals.lean | 57 | 58 |
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Topology.Bases
import Mathlib.Topology.Separation.Regular
/-!
# Dense embeddings
This file defines three properties of f... | variable {f : s → β}
theorem extend_eq_of_tendsto [T2Space β] (hs : Dense s) {a : α} {b : β}
(hf : Tendsto f (comap (↑) (𝓝 a)) (𝓝 b)) : hs.extend f a = b :=
hs.isDenseInducing_val.extend_eq_of_tendsto hf
| Mathlib/Topology/DenseEmbedding.lean | 220 | 224 |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Module.BigOperators
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.Nat.Squarefree
imp... | ∀ n > 0, n ∈ s →
(∑ x ∈ n.divisorsAntidiagonal, (μ x.fst : R) * g x.snd) = f n := by
rw [sum_eq_iff_sum_smul_moebius_eq_on s hs]
apply forall_congr'
intro a; refine imp_congr_right ?_
refine fun _ => imp_congr_right fun _ => (sum_congr rfl fun x _hx => ?_).congr_left
rw [zsmul_eq_mul]
/-- Möbiu... | Mathlib/NumberTheory/ArithmeticFunction.lean | 1,232 | 1,264 |
/-
Copyright (c) 2018 Rohan Mitta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rohan Mitta, Kevin Buzzard, Alistair Tucker, Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Order.Interval.Set.ProjIcc
import Mathlib.Topology.Algebra.Order.Field
import Mathlib.Topolo... | LipschitzWith (Real.toNNReal K) f :=
have I : ∀ x y, f x - f y ≤ K * dist x y := fun x y => sub_le_iff_le_add'.2 (h x y)
| Mathlib/Topology/MetricSpace/Lipschitz.lean | 76 | 77 |
/-
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 | 775 | 777 | |
/-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.Algebra.EuclideanDomain.Field
import Mathlib.Algebra.Polynomial.Module.Basic
import Mathlib.Analysis.Calculus.Co... | (gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt g (g' x_1) x_1)
(g'_ne : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → g' x_1 ≠ 0) :
∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x =
((x - x') ^ n / n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x₀ x) x' := by
-- We apply the mean value th... | Mathlib/Analysis/Calculus/Taylor.lean | 294 | 307 |
/-
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.InnerProductSpace.Calculus
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Topology.MetricSpace.ProperSpace.Lemmas
/-!
# Eucl... |
nonrec theorem closure_ball (x : E) {r : ℝ} (h : r ≠ 0) : closure (ball x r) = closedBall x r := by
rw [ball_eq_preimage, ← toEuclidean.preimage_closure, closure_ball (toEuclidean x) h,
| Mathlib/Analysis/InnerProductSpace/EuclideanDist.lean | 82 | 84 |
/-
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.List.TakeDrop
import Mathlib.Data.List.Induction
/-!
# Prefixes, suffixes, infixes
This file proves properties about
* `List.isPrefix`: `l₁` is ... | rw [tails_eq_inits l]
simp [reverse_involutive.comp_self, ← map_reverse]
| Mathlib/Data/List/Infix.lean | 211 | 212 |
/-
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.Module.Basic
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
/-!
# Midpoint of a segment
## Main definitions
* `midpoint R x y`: midp... | theorem midpoint_comm (x y : P) : midpoint R x y = midpoint R y x := by
rw [midpoint, ← lineMap_apply_one_sub, one_sub_invOf_two, midpoint]
theorem AffineEquiv.pointReflection_midpoint_right (x y : P) :
| Mathlib/LinearAlgebra/AffineSpace/Midpoint.lean | 68 | 71 |
/-
Copyright (c) 2014 Floris van Doorn (c) 2016 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Leonardo de Moura, Jeremy Avigad, Mario Carneiro
-/
import Mathlib.Algebra.Group.Nat.Defs
import Mathlib.Algebra.Group.Basic
import M... | end
@[simp]
theorem size_shiftLeft' {b m n} (h : shiftLeft' b m n ≠ 0) :
size (shiftLeft' b m n) = size m + n := by
induction n with
| zero => simp [shiftLeft']
| Mathlib/Data/Nat/Size.lean | 55 | 61 |
/-
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.Topology.Category.TopCat.Limits.Pullbacks
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
/-!
# Open immersions of structured spaces
We say that a m... | Mathlib/Geometry/RingedSpace/OpenImmersion.lean | 1,298 | 1,302 | |
/-
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.NumberTheory.LegendreSymbol.AddCharacter
import Mathlib.NumberTheory.LegendreSymbol.ZModChar
import Mathlib.Algebra.CharP.CharAndCard
/-!
# Gauss sums
... | ### Values of quadratic characters
-/
section GaussSumValues
variable {R : Type u} [CommRing R] [Fintype R] {R' : Type v} [CommRing R'] [IsDomain R']
/-- If the square of the Gauss sum of a quadratic character is `χ(-1) * #R`,
then we get, for all `n : ℕ`, the relation `(χ(-1) * #R) ^ (p^n/2) = χ(p^n)`,
where `p` is... | Mathlib/NumberTheory/GaussSum.lean | 213 | 229 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Batteries.Data.Nat.Gcd
import Mathlib.Algebra.Group.Nat.Units
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.GroupWi... | /-!
| Mathlib/Data/Nat/GCD/Basic.lean | 85 | 85 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.Algebra.MvPolynomial.Counit
import Mathlib.Algebra.MvPolynomial.Invertible
import Mathlib.RingTheory.WittVector.Defs
/-!
# Witt vecto... | section Tactic
open Lean Elab Tactic
| Mathlib/RingTheory/WittVector/Basic.lean | 137 | 139 |
/-
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.Fin.Fin2
import Mathlib.Data.PFun
import Mathlib.Data.Vector3
import Mathlib.NumberTheory.PellMatiyasevic
/-!
# Diophantine functions and Matiyas... | ((D&1 D= D.0 D∧ D&0 D= D.0) D∨ (D.0 D< D&1 D∧
((D∃) 3 <| (D∃) 4 <| (D∃) 5 <| (D∃) 6 <| (D∃) 7 <| (D∃) 8 <|
pell_dioph.reindex_dioph (Fin2 9) [&4, &8, &1, &0] D∧
(D≡ (D&1) (D&0 D* (D&4 D- D&7) D+ D&6) (D&3)) D∧
D.2 D* D&4 D* D&7 D= D&3 D+ (D&7 D* D&7 D+ D.1) D∧
D&6 D< D&3 D∧ D&7 D≤ D&5 D∧ D&8 D≤ ... | Mathlib/NumberTheory/Dioph.lean | 707 | 732 |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.IsPrimePow
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Ring.Int
import Mathlib.Algebra.Ring.CharZero
im... | exact ⟨fun ⟨_, hkm⟩ ↦ ⟨hkm, ne_zero_of_dvd_ne_zero hn hm⟩, fun ⟨hk, _⟩ ↦ ⟨⟨hk.trans hm, hn⟩, hk⟩⟩
@[simp]
theorem divisors_zero : divisors 0 = ∅ := by
ext
simp
| Mathlib/NumberTheory/Divisors.lean | 224 | 229 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.LinearAlgebra.Quotient.Basic
import Mathlib.LinearAlgebra.Prod
/-!
# Projection to a subspace
In this file we define
* `Submodule.linearProjOfIsCom... |
section
| Mathlib/LinearAlgebra/Projection.lean | 260 | 262 |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic
import Mathlib.Algebra.Ring.Pointwise.Set
import Mathlib.Topology.MetricSpace.Isometry
import Mathlib.Topol... | (IsometryEquiv.inv G).dist_eq a b
@[to_additive (attr := simp)]
| Mathlib/Topology/MetricSpace/IsometricSMul.lean | 294 | 296 |
/-
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
-/
import Mathlib.Tactic.Attr.Register
import Mathlib.Tactic.Basic
import Batteries.Logic
import Batteries.Tactic.Trans
import Batteries.Util.LibraryNot... | Mathlib/Logic/Basic.lean | 1,164 | 1,167 | |
/-
Copyright (c) 2024 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
/-!
# Angle between complex numbers
This file rel... | obtain rfl | ha := eq_or_ne a 0
· simp
· rw [← angle_mul_right ha, div_mul_cancel₀ _ ha]
| Mathlib/Analysis/Complex/Angle.lean | 56 | 59 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Heather Macbeth
-/
import Mathlib.Analysis.Convex.Cone.Extension
import Mathlib.Analysis.NormedSpace.RCLike
import Mathlib.Analysis.NormedSpace.Extend
import Mathlib.... | letI : Module ℝ E := RestrictScalars.module ℝ 𝕜 E
letI : IsScalarTower ℝ 𝕜 E := RestrictScalars.isScalarTower _ _ _
letI : NormedSpace ℝ E := NormedSpace.restrictScalars _ 𝕜 _
-- Let `fr: p →L[ℝ] ℝ` be the real part of `f`.
let fr := reCLM.comp (f.restrictScalars ℝ)
-- Use the real version to get a norm-... | Mathlib/Analysis/NormedSpace/HahnBanach/Extension.lean | 73 | 109 |
/-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Gamma.Deriv
import Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral
/-! # Convexity properties of the Gamma funct... | · rw [one_div_one_div, one_div_one_div]
congr 2 <;> exact setIntegral_congr_fun measurableSet_Ioi fun x hx => fpow (by assumption) _ hx
theorem convexOn_log_Gamma : ConvexOn ℝ (Ioi 0) (log ∘ Gamma) := by
refine convexOn_iff_forall_pos.mpr ⟨convex_Ioi _, fun x hx y hy a b ha hb hab => ?_⟩
have : b = 1 - a := ... | Mathlib/Analysis/SpecialFunctions/Gamma/BohrMollerup.lean | 106 | 161 |
/-
Copyright (c) 2018 Sean Leather. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sean Leather, Mario Carneiro
-/
import Mathlib.Data.List.AList
import Mathlib.Data.Finset.Sigma
import Mathlib.Data.Part
/-!
# Finite maps over `Multiset`
-/
universe u v w
open List
... | @[simp]
theorem lookup_union_left {a} {s₁ s₂ : Finmap β} : a ∈ s₁ → lookup a (s₁ ∪ s₂) = lookup a s₁ :=
induction_on₂ s₁ s₂ fun _ _ => AList.lookup_union_left
| Mathlib/Data/Finmap.lean | 508 | 510 |
/-
Copyright (c) 2023 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.DedekindDomain.Ideal
import Mathlib.RingTheory.Discriminant
import Mathlib.RingTheory.DedekindDomain.IntegralClosure
import Mathlib.NumberTheory.K... | have hKx : IsIntegral K x := Algebra.IsIntegral.isIntegral x
let pb := (Algebra.adjoin.powerBasis' hKx).map
((Subalgebra.equivOfEq _ _ hx).trans (Subalgebra.topEquiv))
have pbgen : pb.gen = x := by simp [pb]
have hpb : ⇑(LinearMap.BilinForm.dualBasis (traceForm K L) _ pb.basis) = _ :=
_root_.funext (tra... | Mathlib/RingTheory/DedekindDomain/Different.lean | 487 | 525 |
/-
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.ModularForms.JacobiTheta.TwoVariable
/-!
# Asymptotic bounds for Jacobi theta functions
The goal of this file is to establish some tech... | lemma isBigO_atTop_F_int_zero_sub (a : UnitAddCircle) : ∃ p, 0 < p ∧
(fun t ↦ F_int 0 a t - (if a = 0 then 1 else 0)) =O[atTop] fun t ↦ exp (-p * t) := by
obtain ⟨a, ha, rfl⟩ := a.eq_coe_Ico
obtain ⟨p, hp, hp'⟩ := isBigO_atTop_F_nat_zero_sub ha.1
obtain ⟨q, hq, hq'⟩ := isBigO_atTop_F_nat_zero_sub (sub_nonneg.... | Mathlib/NumberTheory/ModularForms/JacobiTheta/Bounds.lean | 247 | 266 |
/-
Copyright (c) 2024 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.Algebra.BigOperators.Field
import Mathlib.NumberTheory.LSeries.Basic
/-!
# Linearity of the L-series of `f` as a function of `f`
We show that the `LSer... | simpa [LSeriesHasSum, term_neg] using HasSum.neg hf
lemma LSeriesSummable.neg {f : ℕ → ℂ} {s : ℂ} (hf : LSeriesSummable f s) :
| Mathlib/NumberTheory/LSeries/Linearity.lean | 58 | 60 |
/-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp
-/
import Mathlib.Analysis.InnerProductSpace.Spectrum
import Mathlib.Data.Matrix.Rank
import Mathlib.LinearAlgebra.Matrix.Diagonal
import Mathlib.LinearAlgebra.M... | theorem det_eq_prod_eigenvalues : det A = ∏ i, (hA.eigenvalues i : 𝕜) := by
convert congr_arg det hA.spectral_theorem
rw [det_mul_right_comm]
simp
/-- rank of a hermitian matrix is the rank of after diagonalization by the eigenvector unitary -/
| Mathlib/LinearAlgebra/Matrix/Spectrum.lean | 131 | 136 |
/-
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.Order.Group.Multiset
import Mathlib.Data.Setoid.Basic
import Mathlib.Data.Vector.Basic
import Mathlib.Logic.Nontrivial.Basic
import Mathlib.Tactic.Ap... |
theorem eq_replicate_of_subsingleton [Subsingleton α] (a : α) {n : ℕ} (s : Sym α n) :
s = replicate n a :=
| Mathlib/Data/Sym/Basic.lean | 302 | 304 |
/-
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.... | rw [le_pred_iff, succ_castPred_le_iff]
| Mathlib/Data/Fin/Basic.lean | 913 | 914 |
/-
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.Hom.ContinuousEval
import Mathlib.Topology.ContinuousMap.Basic
import Mathlib.Topology.Separation.Regular
/-!
# The compact-open topology
In this ... |
/-- If `g : C(Y, Z)` is a topological embedding,
then the composition `ContinuousMap.comp g : C(X, Y) → C(X, Z)` is an embedding too. -/
| Mathlib/Topology/CompactOpen.lean | 108 | 110 |
/-
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.Order.Monoid.Canonical.Defs
import Mathlib.Algebra.Order.Monoid.Unbundled.OrderDual
import Mathlib.Algebra.BigOperators.Group.List.Basic
/-!... | simp only [mem_cons, exists_eq_or_imp] at h₂
cases h₂
· exact mul_lt_mul_of_lt_of_le ‹_› (prod_le_prod' h₁.2)
· exact mul_lt_mul_of_le_of_lt h₁.1 <| ihl h₁.2 ‹_›
@[to_additive]
lemma prod_lt_prod_of_ne_nil [Preorder M] [MulLeftStrictMono M]
[MulLeftMono M] [MulRightStrictMono M]
[MulRightMono M] {l : L... | Mathlib/Algebra/Order/BigOperators/Group/List.lean | 72 | 83 |
/-
Copyright (c) 2020 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang, Johan Commelin
-/
import Mathlib.RingTheory.GradedAlgebra.Homogeneous.Ideal
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.Topology.Sets.Opens
import Mathlib.... |
/-!
## The specialization order
| Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Topology.lean | 402 | 404 |
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Johan Commelin, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Equivalence
import Mathlib.CategoryTheory.Yoneda
/-!
# Adjunctions between functors
`F ⊣ G` represents the dat... | (h : adj.unit = adj'.unit) : adj = adj' := by
suffices h' : adj.counit = adj'.counit by cases adj; cases adj'; aesop
ext X
apply (adj.homEquiv _ _).injective
| Mathlib/CategoryTheory/Adjunction/Basic.lean | 179 | 182 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn
/-!
# Inverse functio... | Mathlib/Analysis/Calculus/InverseFunctionTheorem/FDeriv.lean | 197 | 200 | |
/-
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.Analysis.NormedSpace.Multilinear.Basic
import Mathlib.LinearAlgebra.Multilinear.Curry
/-!
# Currying and uncurrying continuous multilinear maps
... | def currySumEquiv : ContinuousMultilinearMap 𝕜 (fun _ : ι ⊕ ι' => G) G' ≃ₗᵢ[𝕜]
ContinuousMultilinearMap 𝕜 (fun _ : ι => G) (ContinuousMultilinearMap 𝕜 (fun _ : ι' => G) G') :=
LinearIsometryEquiv.ofBounds
{ toFun := currySum
invFun := uncurrySum
| Mathlib/Analysis/NormedSpace/Multilinear/Curry.lean | 531 | 535 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.ConcreteCategory
import Mathlib.Algebra.Category.ModuleCat.Colimits
/-!
# Homology and exactness of short complexes of modules
In... |
variable {S}
lemma Exact.moduleCat_range_eq_ker (hS : S.Exact) :
LinearMap.range S.f.hom = LinearMap.ker S.g.hom := by
simpa only [moduleCat_exact_iff_range_eq_ker] using hS
lemma ShortExact.moduleCat_injective_f (hS : S.ShortExact) :
Function.Injective S.f :=
hS.injective_f
lemma ShortExact.moduleCat_s... | Mathlib/Algebra/Homology/ShortComplex/ModuleCat.lean | 57 | 70 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Sébastien Gouëzel,
Rémy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
import Qq
/-! # P... | rw [← ofReal_natCast, norm_cpow_eq_rpow_re_of_pos (Nat.cast_pos.mpr hn) _]
| Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 338 | 339 |
/-
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.Ideal
import Mathlib.Algebra.Lie.OfAssociative
import Mathlib.LinearAlgebra.Isomorphisms
import Mathlib.RingTheory.Noetherian.Basic
/-!
# Quotie... | end Quotient
| Mathlib/Algebra/Lie/Quotient.lean | 205 | 205 |
/-
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.Support
import Mathlib.Data.ENat.Basic
/-!
# Trailing degree of univariate polynomials
## Main definitions
* `trailingDegree... | natTrailingDegree_monomial one_ne_zero
@[simp]
lemma trailingDegree_X_pow (n : ℕ) :
(X ^ n : R[X]).trailingDegree = n := by
rw [X_pow_eq_monomial, trailingDegree_monomial one_ne_zero]
@[simp]
lemma natTrailingDegree_X_pow (n : ℕ) :
(X ^ n : R[X]).natTrailingDegree = n := by
rw [X_pow_eq_monomial, natTra... | Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean | 357 | 367 |
/-
Copyright (c) 2022. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Yuma Mizuno, Oleksandr Manzyuk
-/
import Mathlib.CategoryTheory.Monoidal.Free.Coherence
import Mathlib.Lean.Meta
import Mathlib.Tactic.CategoryTheory.BicategoryCoherence
import Mathlib.... |
lemma insert_id_lhs {C : Type*} [Category C] {X Y : C} (f g : X ⟶ Y) (w : f ≫ 𝟙 _ = g) :
f = g := by
| Mathlib/Tactic/CategoryTheory/Coherence.lean | 196 | 198 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov, Kexing Ying
-/
import Mathlib.Topology.Semicontinuous
import Mathlib.MeasureTheory.Function.AEMeasurableSequence
import Mathlib.MeasureTheory.Order.Lat... |
instance (priority := 100) ContinuousInf.measurableInf [Min γ] [ContinuousInf γ] :
MeasurableInf γ where
| Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean | 486 | 488 |
/-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Ring.Idempotent
import Mathlib.Order.Basic
import Mathlib.RingTheory.Finiteness.Nakayama
/-!
## Lemmas on idempo... | · intro H
obtain ⟨e, he, rfl⟩ := (I.isIdempotentElem_iff_of_fg h).mp H
simp only [Ideal.submodule_span_eq, Ideal.span_singleton_eq_bot]
apply Or.imp id _ (IsIdempotentElem.iff_eq_zero_or_one.mp he)
rintro rfl
simp
· rintro (rfl | rfl) <;> simp [IsIdempotentElem]
end Ideal
| Mathlib/RingTheory/Ideal/IdempotentFG.lean | 38 | 47 |
/-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Calculus.SmoothSeries
import Mathlib.Analysis.NormedSpace.OperatorNorm.Prod
import Mathlib.Analysis.SpecialFunctions.Gaussian.PoissonSummation... |
/-- A uniform upper bound for `jacobiTheta₂'_term` on compact subsets. -/
lemma norm_jacobiTheta₂'_term_le {S T : ℝ} (hT : 0 < T) {z τ : ℂ}
(hz : |im z| ≤ S) (hτ : T ≤ im τ) (n : ℤ) :
‖jacobiTheta₂'_term n z τ‖ ≤ 2 * π * |n| * rexp (-π * (T * n ^ 2 - 2 * S * |n|)) := by
rw [jacobiTheta₂'_term, norm_mul]
re... | Mathlib/NumberTheory/ModularForms/JacobiTheta/TwoVariable.lean | 90 | 98 |
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Algebra.Group.Commute.Hom
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Data.Fintype.B... | · rfl
exact comm hx hy hxy
theorem noncommProd_erase_mul [DecidableEq α] (s : Multiset α) {a : α} (h : a ∈ s) (comm)
(comm' := fun _ hx _ hy hxy ↦ comm (s.mem_of_mem_erase hx) (s.mem_of_mem_erase hy) hxy) :
| Mathlib/Data/Finset/NoncommProd.lean | 206 | 210 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Vector.Defs
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.OfFn
import Mathlib.Data.List.Scan
import Mathlib.Control.Applicative
import M... | (hp : ∀ x ∈ v.toList, p x) :
(v.pmap f hp).tail = v.tail.pmap f (fun x hx ↦ hp _ <| by
| Mathlib/Data/Vector/Basic.lean | 124 | 125 |
/-
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.Algebra.Module.ZLattice.Basic
import Mathlib.Analysis.InnerProductSpace.ProdL2
import Mathlib.MeasureTheory.Measure.Haar.Unique
import Mathlib.NumberTheo... | Basis (ChooseBasisIndex ℤ (𝓞 K)) ℝ (mixedSpace K) := by
classical
-- We construct an `ℝ`-linear independent family from the image of
-- `canonicalEmbedding.lattice_basis` by `commMap`
| Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean | 650 | 653 |
/-
Copyright (c) 2022 Bhavik Mehta, Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Alena Gusakov, Yaël Dillies
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.Field.Rat
import Mathlib.Combinatorics.Enumerative.DoubleCounting
imp... | /-- A bound on any top part of the sum in LYM in terms of the size of `falling k 𝒜`. -/
theorem le_card_falling_div_choose [Fintype α] (hk : k ≤ Fintype.card α)
(h𝒜 : IsAntichain (· ⊆ ·) (𝒜 : Set (Finset α))) :
(∑ r ∈ range (k + 1),
(#(𝒜 # (Fintype.card α - r)) : 𝕜) / (Fintype.card α).choose (Finty... | Mathlib/Combinatorics/SetFamily/LYM.lean | 170 | 177 |
/-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.MeasureTheory.Integral.PeakFunction
/-! # Euler's infinite product for the sine function
This file... | · nth_rw 1 [(by ring : 0 = π / 2 - π / 2)]
nth_rw 3 [(by ring : π / 2 = π / 2 - 0)]
rw [← integral_comp_sub_left]
refine integral_congr fun x _ => ?_
rw [cos_pi_div_two_sub]
· nth_rw 3 [(by ring : π = π / 2 + π / 2)]
nth_rw 2 [(by ring : π / 2 = 0 + π / 2)]
rw [← integral_comp_add_right]
... | Mathlib/Analysis/SpecialFunctions/Trigonometric/EulerSineProd.lean | 181 | 199 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.RingTheory.Localization.Integer
import Mathlib.RingTheory.Localization.Submodule
/-!
# Fractional ideals
This file defines fractional... |
instance commSemiring : CommSemiring (FractionalIdeal S P) :=
| Mathlib/RingTheory/FractionalIdeal/Basic.lean | 579 | 580 |
/-
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.Analysis.SpecificLimits.Basic
import Mathlib.Topology.MetricSpace.IsometricSMul
/-!
# Hausdorff distance
The Hausdorff distance on subsets of a... | exists cs, hcs
have : dist x cs ≤ max rs rt := le_trans (hrt xt) (le_max_right _ _)
rwa [edist_dist, ENNReal.ofReal_le_ofReal_iff]
exact le_trans dist_nonneg this
| Mathlib/Topology/MetricSpace/HausdorffDistance.lean | 648 | 651 |
/-
Copyright (c) 2023 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.GroupTheory.GroupAction.R... |
lemma pow_dvd_add_pow_of_pow_eq_zero_right (hp : n + m ≤ p + 1) (h_comm : Commute x y)
(hy : y ^ n = 0) : x ^ m ∣ (x + y) ^ p := by
rw [h_comm.add_pow']
refine Finset.dvd_sum fun ⟨i, j⟩ hij ↦ ?_
replace hij : i + j = p := by simpa using hij
apply dvd_nsmul_of_dvd
rcases le_or_lt m i with (hi : m ≤ i) | (... | Mathlib/Algebra/Ring/Divisibility/Lemmas.lean | 40 | 48 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Data.ENNReal.Operations
/-!
# Results about division in extended non-negative reals
This file establishes basic properties related t... | Mathlib/Data/ENNReal/Inv.lean | 486 | 486 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot, Yury Kudryashov, Rémy Degenne
-/
import Mathlib.Data.Set.Subsingleton
import Mathlib.Order.Interval.Set.Defs
/-!
# Intervals
In any pr... | theorem Icc_diff_Ico_same (h : a ≤ b) : Icc a b \ Ico a b = {b} := by
rw [← Icc_diff_right, diff_diff_cancel_left (singleton_subset_iff.2 <| right_mem_Icc.2 h)]
| Mathlib/Order/Interval/Set/Basic.lean | 704 | 705 |
/-
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.Group.Nat.Defs
import Mathlib.CategoryTheory.Category.Preorder
import Mathlib.CategoryTheory.EqToHom
import Mathlib.CategoryTheory.Functor.Const
import M... | namespace Precomp
| Mathlib/CategoryTheory/ComposableArrows.lean | 284 | 285 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Algebra.Order.Ring.Canonical
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Combinatorics.SetFamily.Compression.Down
import Mathlib.Data.Fintype.Powe... | card_sdiff inter_subset_right, sdiff_inter_self_right, sdiff_compl,
_root_.inf_comm] at this
· exact mul_le_mul_left' (card_le_card inter_subset_right) _
· rw [← Fintype.card_finset]
exact mul_le_mul_right' (card_le_univ _) _
| Mathlib/Combinatorics/SetFamily/HarrisKleitman.lean | 122 | 132 |
/-
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... | TopologicalSpace.ext_nhds fun x ↦ by
rw [← @nhds_translation_mul_inv G t _ _ x, ← @nhds_translation_mul_inv G t' _ _ x, ← h]
@[to_additive]
theorem IsTopologicalGroup.ext_iff {G : Type*} [Group G] {t t' : TopologicalSpace G}
(tg : @IsTopologicalGroup G t _) (tg' : @IsTopologicalGroup G t' _) :
t = t' ↔ @... | Mathlib/Topology/Algebra/Group/Basic.lean | 817 | 823 |
/-
Copyright (c) 2017 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Mario Carneiro
-/
import Mathlib.Algebra.Ring.CharZero
import Mathlib.Algebra.Star.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Order.Interval.Set.UnorderedInterva... | (normSq_nonneg z).lt_iff_ne.trans <| not_congr (eq_comm.trans normSq_eq_zero)
| Mathlib/Data/Complex/Basic.lean | 548 | 548 |
/-
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, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Pairwise
import Mathlib.Data.Set.BooleanAlgebra
/-!
# The set lattice
This file is a collectio... | Mathlib/Data/Set/Lattice.lean | 1,484 | 1,486 | |
/-
Copyright (c) 2022 Eric Rodriguez. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Rodriguez, Eric Wieser
-/
import Mathlib.Data.List.Chain
/-!
# Destuttering of Lists
This file proves theorems about `List.destutter` (in `Data.List.Defs`), which greedily remov... |
theorem destutter'_of_chain (h : l.Chain R a) : l.destutter' R a = a :: l := by
induction' l with b l hb generalizing a
· simp
obtain ⟨h, hc⟩ := chain_cons.mp h
rw [l.destutter'_cons_pos h, hb hc]
| Mathlib/Data/List/Destutter.lean | 92 | 98 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.Bochner.Basic
import Mathlib.MeasureTheory.Integral.Bochner.L1
import Mathlib.Me... | Mathlib/MeasureTheory/Integral/Bochner.lean | 1,891 | 1,894 |
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