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) 2020 SΓ©bastien GouΓ«zel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: SΓ©bastien GouΓ«zel, Floris van Doorn
-/
import Mathlib.Geometry.Manifold.MFDeriv.FDeriv
/-!
# Differentiability of specific functions
In this file, we establish differentiability r... | section prodMap
variable {f : M β M'} {g : N β N'} {r : Set N} {y : N}
/-- The product map of two `C^n` functions within a set at a point is `C^n`
within the product set at the product point. -/
theorem MDifferentiableWithinAt.prodMap' {p : M Γ N} (hf : MDifferentiableWithinAt I I' f s p.1)
(hg : MDifferentiableW... | Mathlib/Geometry/Manifold/MFDeriv/SpecificFunctions.lean | 403 | 419 |
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, SΓ©bastien GouΓ«zel, Yury Kudryashov
-/
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calc... | rw [β iso.symm_apply_apply x] at H
have A : f = (f β iso) β iso.symm := by
rw [Function.comp_assoc, iso.self_comp_symm]
rfl
have B : f' = (f'.comp (iso : E βL[π] F)).comp (iso.symm : F βL[π] E) := by
rw [ContinuousLinearMap.comp_assoc, iso.coe_comp_coe_symm, ContinuousLinearMap.comp_id]
rw [A, B]
... | Mathlib/Analysis/Calculus/FDeriv/Equiv.lean | 191 | 199 |
/-
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.Algebra.Group.Units.Equiv
import Mathlib.CategoryTheory.Endomorphism
import Mathlib.CategoryTheory.HomCongr
/-!
# Conjugate morphisms by isomorphism... | Ξ±.symm.symm_self_conj f
@[simp]
| Mathlib/CategoryTheory/Conj.lean | 64 | 66 |
/-
Copyright (c) 2019 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, Patrick Massot, Casper Putz, Anne Baanen
-/
import Mathlib.Algebra.Algebra.Subalgebra.Tower
import Mathlib.Data.Finite.Sum
import Mathlib.Data.Matrix.Block
import Mathl... | Mathlib/LinearAlgebra/Matrix/ToLin.lean | 1,049 | 1,055 | |
/-
Copyright (c) 2021 Stuart Presnell. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stuart Presnell
-/
import Mathlib.Data.Nat.PrimeFin
import Mathlib.Data.Nat.Factorization.Defs
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Order.Interval.Finset.Nat
import... | exact Nat.div_pos (ordProj_le p hn) (ordProj_pos n p)
| Mathlib/Data/Nat/Factorization/Basic.lean | 120 | 120 |
/-
Copyright (c) 2019 SΓ©bastien GouΓ«zel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: SΓ©bastien GouΓ«zel
-/
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Complex.Module
import Mathlib.Data.Complex.Order
import Mathli... | @[simp]
theorem imCLM_apply (z : β) : (imCLM : β β β) z = z.im :=
| Mathlib/Analysis/Complex/Basic.lean | 180 | 181 |
/-
Copyright (c) 2021 Apurva Nakade. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Apurva Nakade
-/
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Ring.Regular
import Mathlib.GroupTheory.MonoidLocalization.Away
impo... | theorem dyadic_aux {mβ mβ : β€} {yβ yβ : β} (hβ : mβ * 2 ^ yβ = mβ * 2 ^ yβ) :
mβ * powHalf yβ = mβ * powHalf yβ := by
revert mβ mβ
wlog h : yβ β€ yβ
Β· intro mβ mβ aux; exact (this (le_of_not_le h) aux.symm).symm
| Mathlib/SetTheory/Surreal/Dyadic.lean | 187 | 191 |
/-
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.Order.ConditionallyCompleteLattice.Indexed
import Mathlib.Order.SuccPred.Limit
/-!
# Relation between `IsSuccPrelimit` and `iSup` in (conditionally) comple... | (hne : s.Nonempty) (hbdd : BddBelow s) (hlim : Β¬ IsPredPrelimit (sInf s)) : sInf s β s := by
obtain β¨y, hyβ© := not_forall_not.mp hlim
obtain β¨i, his, hiβ© := exists_lt_of_csInf_lt hne hy.lt
exact eq_of_le_of_not_lt (csInf_le hbdd his) (hy.2 Β· hi) βΈ his
lemma exists_eq_ciSup_of_not_isSuccPrelimit
| Mathlib/Order/SuccPred/CompleteLinearOrder.lean | 29 | 34 |
/-
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.Convex.Between
import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathli... | -- We mostly unfold the definitions but we need to switch the order of `β'` and `β¨
`
simp only [β OuterMeasure.coe_mkMetric, OuterMeasure.mkMetric, OuterMeasure.mkMetric',
OuterMeasure.iSup_apply, OuterMeasure.mkMetric'.pre, OuterMeasure.boundedBy_apply, extend]
| Mathlib/MeasureTheory/Measure/Hausdorff.lean | 469 | 471 |
/-
Copyright (c) 2023 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, JoΓ«l Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.ModuleCat
import Mathlib.RepresentationTheory.GroupCohomology.Basic
import Mathlib.RepresentationTheory.... | @[simp]
theorem oneCocycles.coe_mk (f : G β A) (hf) : ((β¨f, hfβ© : oneCocycles A) : G β A) = f := rfl
@[simp]
theorem oneCocycles.val_eq_coe (f : oneCocycles A) : f.1 = f := rfl
| Mathlib/RepresentationTheory/GroupCohomology/LowDegree.lean | 250 | 255 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes HΓΆlzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Polynomial.Eval.Degree
import Mathlib.Algebra.Prime.Lemmas
/-!
# Theory of degrees of polynomials
S... | by_cases h : p = 0
Β· simp [h]
rw [Polynomial.leadingCoeff, natDegree_comp_eq_of_mul_ne_zero, coeff_comp_degree_mul_degree] <;>
| Mathlib/Algebra/Polynomial/Degree/Lemmas.lean | 376 | 378 |
/-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.GroupWithZero.Divisibili... | Mathlib/Algebra/MvPolynomial/Basic.lean | 1,360 | 1,377 | |
/-
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.FieldTheory.Finite.Polynomial
import Mathlib.NumberTheory.Basic
import Mathlib.RingTheory.WittVector.WittPolynomial
/-!
# Witt struct... | rw [β constantCoeff_map, map_wittStructureInt, constantCoeff_wittStructureRat, RingHom.map_zero]
rw [constantCoeff_map, h, RingHom.map_zero]
variable (R)
| Mathlib/RingTheory/WittVector/StructurePolynomial.lean | 362 | 366 |
/-
Copyright (c) 2023 JoΓ«l Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: JoΓ«l Riou
-/
import Mathlib.Algebra.Homology.HomotopyCategory.HomComplex
import Mathlib.Algebra.Homology.HomotopyCofiber
/-! # The mapping cone of a morphism of cochain complexes
In this ... | exact ext_to Ο i j hij hβ hβ
lemma ext_from (i j : β€) (hij : j + 1 = i) {A : C} {f g : (mappingCone Ο).X j βΆ A}
(hβ : (inl Ο).v i j (by omega) β« f = (inl Ο).v i j (by omega) β« g)
(hβ : (inr Ο).f j β« f = (inr Ο).f j β« g) :
| Mathlib/Algebra/Homology/HomotopyCategory/MappingCone.lean | 162 | 166 |
/-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, Yury Kudryashov
-/
import Mathlib.Data.ENNReal.Operations
/-!
# Results about division in extended non-negative reals
This file establishes basic properties related t... | @[simp] protected theorem div_self_le_one : a / a β€ 1 := div_le_of_le_mul <| by rw [one_mul]
| Mathlib/Data/ENNReal/Inv.lean | 368 | 369 |
/-
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 SΓΆnne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
import Mathlib.Analysis.SpecialFunctions... | rintro h rfl
have : sin β arcsin =αΆ [π[β₯] (-1 : β)] id := by
filter_upwards [Icc_mem_nhdsGE (neg_lt_self zero_lt_one)] with x using sin_arcsin'
have := h.hasDerivWithinAt.sin.congr_of_eventuallyEq this.symm (by simp)
simpa using (uniqueDiffOn_Ici _ _ left_mem_Ici).eq_deriv _ this (hasDerivWithinAt_id _ _)
| Mathlib/Analysis/SpecialFunctions/Trigonometric/InverseDeriv.lean | 74 | 79 |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark, Kyle Miller, Lu-Ming Zhang
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Combinatorics.SimpleGraph.Connectivity.WalkCounting
import Math... | Β· norm_cast
simp only [Nat.cast_sum, card_map, neighborFinset_def]
| Mathlib/Combinatorics/SimpleGraph/AdjMatrix.lean | 244 | 245 |
/-
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.LSeries.AbstractFuncEq
import Mathlib.NumberTheory.ModularForms.JacobiTheta.Bounds
import Mathlib.Analysis.SpecialFunctions.Gamma.Deligne
... |
lemma completedCosZeta_eq (a : UnitAddCircle) (s : β) :
completedCosZeta a s =
completedCosZetaβ a s - 1 / s - (if a = 0 then 1 else 0) / (1 - s) := by
rw [completedCosZeta, WeakFEPair.Ξ, sub_div, sub_div]
congr 1
Β· rw [completedCosZetaβ, WeakFEPair.symm, hurwitzEvenFEPair, smul_eq_mul, mul_one, div_div,... | Mathlib/NumberTheory/LSeries/HurwitzZetaEven.lean | 325 | 335 |
/-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.Shapes.StrongEpi
import Mathlib.CategoryTheory.LiftingProperties.Adjunction
/-!
# Preservati... |
theorem strongEpi_map_of_strongEpi (adj : F β£ F') (f : A βΆ B) [F'.PreservesMonomorphisms]
[F.PreservesEpimorphisms] [StrongEpi f] : StrongEpi (F.map f) :=
β¨inferInstance, fun X Y Z => by
intro
rw [adj.hasLiftingProperty_iff]
| Mathlib/CategoryTheory/Functor/EpiMono.lean | 255 | 260 |
/-
Copyright (c) 2023 JoΓ«l Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: JoΓ«l Riou
-/
import Mathlib.Algebra.Homology.HomotopyCategory.HomComplex
import Mathlib.Algebra.Homology.HomotopyCofiber
/-! # The mapping cone of a morphism of cochain complexes
In this ... | exact β¨hβ, hββ©
lemma ext_cochain_from_iff (i j : β€) (hij : i + 1 = j)
{K : CochainComplex C β€} {Ξ³β Ξ³β : Cochain (mappingCone Ο) K j} :
Ξ³β = Ξ³β β
(inl Ο).comp Ξ³β (show _ = i by omega) = (inl Ο).comp Ξ³β (by omega) β§
(Cochain.ofHom (inr Ο)).comp Ξ³β (zero_add j) =
(Cochain.ofHom (inr Ο)... | Mathlib/Algebra/Homology/HomotopyCategory/MappingCone.lean | 204 | 220 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes HΓΆlzl
-/
import Mathlib.MeasureTheory.Integral.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Lebesgue.Countable
import Mathlib.MeasureTheory.Integral.Le... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 1,010 | 1,028 | |
/-
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.Group.Nat
import Mathlib.Data.List.Rotate
import Mathlib.GroupTheory.Perm.Support
/-!
# Permutations from a list
A list `l : List Ξ±` ... | | [_] => rfl
| a::b::l => by
simp [formPerm_append_pair, swap_comm, β formPerm_reverse (b::l)]
| Mathlib/GroupTheory/Perm/List.lean | 244 | 247 |
/-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Data.Countable.Small
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Powerset
import Mathlib.Dat... | Mathlib/SetTheory/Cardinal/Basic.lean | 2,077 | 2,083 | |
/-
Copyright (c) 2022 JoΓ«l Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: JoΓ«l Riou
-/
import Mathlib.CategoryTheory.Localization.Construction
/-!
# Predicate for localized categories
In this file, a predicate `L.IsLocalization W` is introduced for a functor `... | is commutative up to an isomorphism. -/
class Lifting (W : MorphismProperty C) (F : C β₯€ E) (F' : D β₯€ E) where
/-- the isomorphism relating the localization functor and the two other given functors -/
iso' : L β F' β
F
| Mathlib/CategoryTheory/Localization/Predicate.lean | 279 | 283 |
/-
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 HΓΆlzl
-/
import Mathlib.Algebra.Order.Group.Unbundled.Basic
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Or... | Mathlib/Algebra/Order/Group/Defs.lean | 418 | 419 | |
/-
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.LinearAlgebra.Dimension.Free
import Mathlib.Algebra.Homology.ShortComplex.ModuleCat
/-!
# Exact sequences with free modules
This file proves resu... | Module.Free.of_basis (Basis.ofShortExact hS' (Module.Free.chooseBasis R S.Xβ)
(Module.Free.chooseBasis R S.Xβ))
theorem free_shortExact_rank_add [Module.Free R S.Xβ] [Module.Free R S.Xβ]
[StrongRankCondition R] :
Module.rank R S.Xβ = Module.rank R S.Xβ + Module.rank R S.Xβ := by
haveI := free_shortExac... | Mathlib/Algebra/Category/ModuleCat/Free.lean | 166 | 174 |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Heather Macbeth
-/
import Mathlib.Analysis.SpecialFunctions.Complex.Circle
import Mathlib.Geometry.Euclidean.Angle.Oriented.Basic
/-!
# Rotations by oriented angles.
This... | rw [Complex.arg_mul_coe_angle, Real.Angle.arg_toCircle]
Β· abel
Β· exact Circle.coe_ne_zero _
| Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean | 210 | 212 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Mario Carneiro, Sean Leather
-/
import Mathlib.Data.Finset.Card
import Mathlib.Data.Finset.Union
/-!
# Finite sets in `Option Ξ±`
In this file we define
* `Option.t... | lemma insertNone_nonempty {s : Finset Ξ±} : insertNone s |>.Nonempty := β¨none, none_mem_insertNoneβ©
| Mathlib/Data/Finset/Option.lean | 75 | 76 |
/-
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.Data.Finite.Sum
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.GroupTheory.Perm.Support
import Mathlib.Logic.Equiv.Fintype
/-!
# Permutations on... | theorem support_pow_coprime {Ο : Perm Ξ±} {n : β} (h : Nat.Coprime n (orderOf Ο)) :
(Ο ^ n).support = Ο.support := by
obtain β¨m, hmβ© := exists_pow_eq_self_of_coprime h
exact
le_antisymm (support_pow_le Ο n)
(le_trans (ge_of_eq (congr_arg support hm)) (support_pow_le (Ο ^ n) m))
| Mathlib/GroupTheory/Perm/Finite.lean | 241 | 246 |
/-
Copyright (c) 2019 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Rat
import Mathlib.Data.Rat.Cast.CharZero
import Mathlib.Tactic.Positivity.Core
/-!
# Casts of rational numbers in... | @[simp, norm_cast]
| Mathlib/Data/Rat/Cast/Order.lean | 79 | 79 |
/-
Copyright (c) 2023 YaΓ«l Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: YaΓ«l Dillies
-/
import Mathlib.Data.Finset.Lattice.Fold
/-!
# Irreducible and prime elements in an order
This file defines irreducible and prime elements in an order and shows that in ... | induction' s using Finset.induction with i s _ ih
| Mathlib/Order/Irreducible.lean | 104 | 104 |
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Kim Morrison
-/
import Mathlib.CategoryTheory.Opposites
import Mathlib.CategoryTheory.Groupoid
/-!
# Facts about epimorphisms and monomorphisms.
The definitions of `Epi` an... | instance (priority := 100) IsSplitMono.mono {X Y : C} (f : X βΆ Y) [hf : IsSplitMono f] : Mono f :=
hf.exists_splitMono.some.mono
| Mathlib/CategoryTheory/EpiMono.lean | 149 | 150 |
/-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.GCDMonoid.Multiset
import Mathlib.Algebra.GCDMonoid.Nat
import Mathlib.Algebra.Group.TypeTags.Finite
import Mathlib.Combinatorics.Enumerative... | add_comm]
@[simp]
theorem cycleType_conj {Ο Ο : Perm Ξ±} : (Ο * Ο * Οβ»ΒΉ).cycleType = Ο.cycleType := by
induction Ο using cycle_induction_on with
| base_one => simp
| Mathlib/GroupTheory/Perm/Cycle/Type.lean | 139 | 144 |
/-
Copyright (c) 2021 RΓ©my Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: RΓ©my Degenne
-/
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
import Mathlib.MeasureTheory.Measure.Count
import Mathlib.Order.Filter.ENNReal
import Mathlib.Probability.Unif... |
variable {f : Ξ± β ββ₯0β}
| Mathlib/MeasureTheory/Function/EssSup.lean | 262 | 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 HΓΆlzl, Yury Kudryashov, Winston Yin
-/
import Mathlib.Algebra.Group.End
import Mathlib.Topology.EMetricSpace.Diam
/-!
# Lipschitz co... |
protected theorem eval {Ξ± : ΞΉ β Type u} [β i, PseudoEMetricSpace (Ξ± i)] [Fintype ΞΉ] (i : ΞΉ) :
LipschitzWith 1 (Function.eval i : (β i, Ξ± i) β Ξ± i) :=
| Mathlib/Topology/EMetricSpace/Lipschitz.lean | 204 | 206 |
/-
Copyright (c) 2022 Kyle Miller, Adam Topaz, RΓ©mi Bottinelli, Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller, Adam Topaz, RΓ©mi Bottinelli, Junyan Xu
-/
import Mathlib.Topology.Category.TopCat.Limits.Konig
/-!
# Cofiltered systems
This file de... | β j, β (i) (f : i βΆ j), (F.map f).Injective := by
haveI : β j, Fintype (F.obj j) := fun j => Fintype.ofFinite (F.obj j)
haveI : Fintype F.sections := Fintype.ofFinite F.sections
have card_le : β j, Fintype.card (F.obj j) β€ Fintype.card F.sections :=
fun j => Fintype.card_le_of_surjective _ (F.eval_section... | Mathlib/CategoryTheory/CofilteredSystem.lean | 333 | 341 |
/-
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.Convex.Between
import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathli... | /-- Given a function `m : Set X β ββ₯0β`, `mkMetric' m` is the supremum of `ΞΌ r`
over `r > 0`, where `ΞΌ r` is the maximal outer measure `ΞΌ` such that `ΞΌ s β€ m s`
for all `s`. While each `ΞΌ r` is an *outer* measure, the supremum is a measure. -/
def mkMetric' (m : Set X β ββ₯0β) : Measure X :=
(OuterMeasure.mkMetric' m)... | Mathlib/MeasureTheory/Measure/Hausdorff.lean | 412 | 417 |
/-
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.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Eval.SMul
import Mathlib.Algebra.Polyno... | split_ifs with h; Β· rfl
push_neg at h; rw [Nat.choose_eq_zero_of_lt h, Nat.cast_zero, mul_zero]
| Mathlib/Algebra/Polynomial/Taylor.lean | 70 | 71 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Vector.Defs
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.OfFn
import Mathlib.Data.List.Scan
import Mathlib.Control.Applicative
import M... | @[simp]
theorem get_append_cons_succ {i : Fin (n + m)} {h} :
get (append (x ::α΅₯ xs) ys) β¨i+1, hβ© = get (append xs ys) i :=
rfl
@[simp]
| Mathlib/Data/Vector/Basic.lean | 733 | 738 |
/-
Copyright (c) 2020 RΓ©my Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: RΓ©my Degenne, SΓ©bastien GouΓ«zel
-/
import Mathlib.Analysis.NormedSpace.IndicatorFunction
import Mathlib.Data.Fintype.Order
import Mathlib.MeasureTheory.Function.AEEqFun
import Mathlib.Me... | MemLp (s.indicator fun _ => c) p ΞΌ := by
rw [memLp_indicator_iff_restrict hs]
| Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean | 750 | 751 |
/-
Copyright (c) 2019 Calle SΓΆnne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Calle SΓΆnne
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 1,042 | 1,047 | |
/-
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 HernΓ‘ndez Palacios
-/
import Mathlib.SetTheory.Cardinal.Arithmetic
import Mathlib.SetTheory.Ordinal.FixedPoint
/-!
# Cofinality
This file co... | Mathlib/SetTheory/Cardinal/Cofinality.lean | 879 | 888 | |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.FiniteStability
import Mathlib.RingTheory.Ideal.Quotient.Nilpotent
import Mathlib.RingTheory.Kaehler.Basic
import Mathlib.RingTheory.Localization.... | variable {A B : Type*} [CommRing A] [Algebra R A] [CommRing B] [Algebra R B]
/-- This holds in general for epimorphisms. -/
theorem of_surjective [FormallyUnramified R A] (f : A ββ[R] B) (H : Function.Surjective f) :
FormallyUnramified R B := by
rw [iff_comp_injective]
| Mathlib/RingTheory/Unramified/Basic.lean | 221 | 226 |
/-
Copyright (c) 2017 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Tim Baumann, Stephen Morgan, Kim Morrison, Floris van Doorn
-/
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.ObjectProperty.FullSubcategory
import Mat... | ext X
dsimp
simp only [funInvIdAssoc_inv_app, id_obj, comp_obj, invFunIdAssoc_hom_app,
Functor.comp_map, β F.map_comp, unit_inverse_comp, map_id]
| Mathlib/CategoryTheory/Equivalence.lean | 351 | 354 |
/-
Copyright (c) 2024 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.Algebra.Ring.NegOnePow
/-!
# Miscellaneous results about determinant
In this file, we collect var... | /-- Let `M` be a `(n+1) Γ n` matrix whose row sums to zero. Then all the matrices obtained by
deleting one row have the same determinant up to a sign. -/
theorem submatrix_succAbove_det_eq_negOnePow_submatrix_succAbove_det {n : β}
(M : Matrix (Fin (n + 1)) (Fin n) R) (hv : β j, M j = 0) (jβ jβ : Fin (n + 1)) :
... | Mathlib/LinearAlgebra/Matrix/Determinant/Misc.lean | 21 | 47 |
/-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, Yury Kudryashov
-/
import Mathlib.Data.Finset.NatAntidiagonal
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Data.Multiset.Antidiagonal
/-!
# The `Finsupp` counte... | intro i
replace h := DFunLike.congr_fun h i
simp_rw [single_apply, Finsupp.add_apply] at h β’
obtain rfl | hai := Decidable.eq_or_ne a i
Β· exact β¨if_pos rfl, if_pos rflβ©
Β· simp_rw [if_neg hai, add_eq_zero] at h β’
exact h.imp Eq.symm Eq.symm
Β· rintro β¨a, b, rfl, rfl, rflβ©
exact (single... | Mathlib/Data/Finsupp/Antidiagonal.lean | 61 | 79 |
/-
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 HΓΆlzl, Mario Carneiro
-/
import Mathlib.Logic.Pairwise
import Mathlib.Data.Set.BooleanAlgebra
/-!
# The set lattice
This file is a collectio... | Mathlib/Data/Set/Lattice.lean | 2,184 | 2,192 | |
/-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, SΓ©bastien GouΓ«zel, Yury Kudryashov
-/
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Dia... |
theorem volume_pi_Ioc {a b : ΞΉ β β} :
volume (pi univ fun i => Ioc (a i) (b i)) = β i, ENNReal.ofReal (b i - a i) :=
| Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | 240 | 242 |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.ModEq
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Algebra.Ring.Periodic
import Mathlib.Data.Int.SuccPred
import Mathlib.Order.Cir... | theorem toIocMod_add_left (a b : Ξ±) : toIocMod hp a (p + b) = toIocMod hp a b := by
rw [add_comm, toIocMod_add_right]
| Mathlib/Algebra/Order/ToIntervalMod.lean | 419 | 421 |
/-
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.Algebra.Group.AddChar
import Mathlib.Analysis.Complex.Circle
import Mathlib.MeasureTheory.Group.Integral
import Mathlib.MeasureTheory.Integral.Prod
imp... |
lemma fourierIntegralInv_comp_linearIsometry (A : W ββα΅’[β] V) (f : V β E) (w : W) :
πβ» (f β A) w = (πβ» f) (A w) := by
| Mathlib/Analysis/Fourier/FourierTransform.lean | 416 | 418 |
/-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Order.SuccPred
import Mathlib.Data.Sum.Order
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Tactic.PPWithUniv
/-!
# ... | theorem univ_umax : univ.{u, max (u + 1) v} = univ.{u, v} :=
congr_fun lift_umax _
theorem lift_lt_univ (c : Cardinal) : lift.{u + 1, u} c < univ.{u, u + 1} := by
simpa only [liftPrincipalSeg_coe, lift_ord, lift_succ, ord_le, succ_le_iff] using
| Mathlib/SetTheory/Ordinal/Basic.lean | 1,209 | 1,213 |
/-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, Jeremy Avigad
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Notation.Pi
import Mathlib.Data.Set.Lattice
import Mathlib.Order.Filter.Defs
/-!
# Theory of... | eventually_iff_exists_mem
| Mathlib/Order/Filter/Basic.lean | 889 | 890 |
/-
Copyright (c) 2022 Violeta HernΓ‘ndez Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta HernΓ‘ndez Palacios
-/
import Mathlib.Order.RelClasses
import Mathlib.Order.Interval.Set.Basic
/-!
# Bounded and unbounded sets
We prove miscellaneous lemmas about ... | @unbounded_lt_inter_le Ξ±α΅α΅ s _ a
theorem bounded_gt_inter_gt [LinearOrder Ξ±] [NoMinOrder Ξ±] (a : Ξ±) :
Bounded (Β· > Β·) (s β© { b | b < a }) β Bounded (Β· > Β·) s :=
| Mathlib/Order/Bounded.lean | 366 | 369 |
/-
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 HΓΆlzl
-/
import Mathlib.Algebra.Order.Group.Unbundled.Basic
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Or... | Mathlib/Algebra/Order/Group/Defs.lean | 382 | 384 | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes HΓΆlzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Polynomial.Eval.Degree
import Mathlib.Algebra.Prime.Lemmas
/-!
# Theory of degrees of polynomials
S... | Β· simpa only [h, ite_false] using coeff_eq_zero_of_natDegree_lt <|
lt_of_le_of_lt (natDegree_pow_le_of_le m pn) (lt_of_le_of_ne mno h.symm)
theorem coeff_add_eq_left_of_lt (qn : q.natDegree < n) : (p + q).coeff n = p.coeff n :=
(coeff_add _ _ _).trans <|
(congr_arg _ <| coeff_eq_zero_of_natDegree_lt <| q... | Mathlib/Algebra/Polynomial/Degree/Lemmas.lean | 172 | 178 |
/-
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 SΓΆnne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Topology.Order.ProjIcc
/-!... | @[simp]
theorem arccos_eq_pi {x} : arccos x = Ο β x β€ -1 := by
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean | 349 | 350 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Logic.Encodable.Pi
import Mathlib.Logic.Function.Iterate
/-!
# The primitive recursive functions
The primitive ... | can be smuggled in through a cunning choice of the subset of `β` into
which the type is encoded. -/
| Mathlib/Computability/Primrec.lean | 128 | 129 |
/-
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
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Order.CauSeq.BigOperators
import Mathlib.Algebra.Order.Star.Basic
import Mathlib.Data.C... | Mathlib/Data/Complex/Exponential.lean | 1,015 | 1,017 | |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Polynomial.Module.AEval
/-!
# Polynomial module
In this file, we define the polynomial module for an `R`-module `M`, i.e. the `R[X]`-module `M[X]`.... | rw [eq_iff_iff]
constructor
Β· rintro rfl
simp
Β· rintro β¨e, rflβ©
rw [add_comm, tsub_add_cancel_of_le e]
@[simp]
theorem smul_single_apply (i : β) (f : R[X]) (m : M) (n : β) :
(f β’ single R i m) n = ite (i β€ n) (f.coeff (n - i) β’ m) 0 := by
induction' f using Polynomial.induction_on' wi... | Mathlib/Algebra/Polynomial/Module/Basic.lean | 139 | 153 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes HΓΆlzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Group.Submonoid.Operations
import Mathlib.Algebra.MonoidAlgebra.Defs
import Mathlib.Algebra.Order.Mon... |
@[simp]
theorem toFinsupp_zsmul (a : β€) (b : R[X]) :
(a β’ b).toFinsupp = a β’ b.toFinsupp :=
rfl
| Mathlib/Algebra/Polynomial/Basic.lean | 1,080 | 1,085 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Minchao Wu, Mario Carneiro
-/
import Mathlib.Data.Finset.Attach
import Mathlib.Data.Finset.Disjoint
import Mathlib.Data.Finset.Erase
import Mat... | Mathlib/Data/Finset/Basic.lean | 2,818 | 2,819 | |
/-
Copyright (c) 2024 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Order.Hom.Ring
import Mathlib.Data.ENat.Basic
import Mathlib.SetTheory.Cardinal.Basic
/-!
# Conversion between `Cardinal` and `ββ`
In this ... | rw [β lift_ofENat.{u, v}, lift_lt]
| Mathlib/SetTheory/Cardinal/ENat.lean | 136 | 137 |
/-
Copyright (c) 2022 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Jujian Zhang
-/
import Mathlib.RingTheory.IsTensorProduct
import Mathlib.RingTheory.Localization.Module
import Mathlib.LinearAlgebra.DirectSum.Finsupp
import Mathlib.Algebra... | /-- The forward direction of `isLocalizedModule_iff_isBaseChange`. It is also used to prove the
other direction. -/
theorem IsLocalizedModule.isBaseChange [IsLocalizedModule S f] : IsBaseChange A f :=
.of_lift_unique _ fun Q _ _ _ _ g β¦ by
obtain β¨β, rfl, hββ© := IsLocalizedModule.is_universal S f g fun s β¦ by
... | Mathlib/RingTheory/Localization/BaseChange.lean | 28 | 33 |
/-
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 | 713 | 715 | |
/-
Copyright (c) 2020 NicolΓ² Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: NicolΓ² Cavalleri, Andrew Yang
-/
import Mathlib.RingTheory.Derivation.ToSquareZero
import Mathlib.RingTheory.Ideal.Cotangent
import Mathlib.RingTheory.IsTensorProduct
import Mathlib.... | refine (Ideal.quotEquivOfEq ?_).trans (RingHom.quotientKerEquivOfRightInverse this)
ext; rfl
/-- The quotient ring of `S β S β§Έ J ^ 2` by `Ξ©[SβR]` is isomorphic to `S` as an `S`-algebra. -/
def KaehlerDifferential.quotientCotangentIdeal :
| Mathlib/RingTheory/Kaehler/Basic.lean | 341 | 345 |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, Mario Carneiro, Oliver Nash
-/
import Mathlib.Data.Finset.Card
import Mathlib.Data.Finset.Union
/-!
# Finsets in product types
This file defines finset constru... |
theorem subset_product_image_snd [DecidableEq Ξ²] : (s ΓΛ’ t).image Prod.snd β t := fun i => by
| Mathlib/Data/Finset/Prod.lean | 68 | 69 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yakov Pechersky
-/
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Infix
import Mathlib.Data.Quot
/-!
# List rotation
This file proves basic results about `List.r... | have : n % length l + m - length l < length l := by
rw [Nat.sub_lt_iff_lt_add hm']
| Mathlib/Data/List/Rotate.lean | 201 | 202 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes HΓΆlzl
-/
import Mathlib.Algebra.Order.Pi
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
/-!
# Simple functions
A function `f` from a measurable ... | f.FinMeasSupp ΞΌ :=
flip lt_of_le_of_lt h <| measure_mono <| support_subset_comp @(hg) _
| Mathlib/MeasureTheory/Function/SimpleFunc.lean | 1,081 | 1,083 |
/-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, Mario Carneiro
-/
import Mathlib.Data.Set.Constructions
import Mathlib.Order.Filter.AtTopBot.CountablyGenerated
import Mathlib.Topology.Constructions
import Mathlib.Top... | h.isInducing β¨rflβ©
section
variable {ΞΉ : Type*} {Ο : ΞΉ β Type*} [β i, TopologicalSpace (Ο i)]
lemma isOpenMap_eval (i : ΞΉ) : IsOpenMap (Function.eval i : (β i, Ο i) β Ο i) := by
classical
refine (isTopologicalBasis_pi fun _ β¦ isTopologicalBasis_opens).isOpenMap_iff.2 ?_
rintro _ β¨U, s, hU, rflβ©
obtain h | h... | Mathlib/Topology/Bases.lean | 570 | 586 |
/-
Copyright (c) 2024 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Kontorovich, David Loeffler, Heather Macbeth, SΓ©bastien GouΓ«zel
-/
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.Analysis.Calculus.ContDiff.CPolynomial
import... |
/-- If `βvβ^n * βf vβ` is integrable for all `n β€ N`, then the `n`-th derivative of the Fourier
transform of `f` is the Fourier transform of `fourierPowSMulRight L f v n`,
i.e., `(L v β¬) ^ n β’ f v`. -/
lemma iteratedFDeriv_fourierIntegral {N : ββ}
(hf : β (n : β), n β€ N β Integrable (fun v β¦ βvβ^n * βf vβ) ΞΌ)
... | Mathlib/Analysis/Fourier/FourierTransformDeriv.lean | 511 | 545 |
/-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.Shapes.StrongEpi
import Mathlib.CategoryTheory.LiftingProperties.Adjunction
/-!
# Preservati... | Mathlib/CategoryTheory/Functor/EpiMono.lean | 299 | 304 | |
/-
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 | 1,516 | 1,517 | |
/-
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.Tactic.Lemma
import Mathlib.Tactic.TypeStar
import Batteries.Tactic.Alias
/-!
# Extra definitions on `Option`
This file defines more operations invol... |
instance liftOrGet_isId (f : Ξ± β Ξ± β Ξ±) : Std.LawfulIdentity (liftOrGet f) none where
| Mathlib/Data/Option/Defs.lean | 96 | 97 |
/-
Copyright (c) 2021 Martin Zinkevich. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Martin Zinkevich, Vincent Beffara
-/
import Mathlib.MeasureTheory.Integral.Bochner.Set
import Mathlib.Probability.Independence.Basic
/-!
# Integration in Probability Theory
Integra... | simp only [β Finset.prod_apply, Finset.prod_cons, β ihs]
apply lintegral_mul_eq_lintegral_mul_lintegral_of_indepFun'
Β· exact (x_mea j).aemeasurable
Β· exact s.aemeasurable_prod' (fun i _ β¦ (x_mea i).aemeasurable)
Β· exact (iIndepFun.indepFun_finset_prod_of_not_mem hX x_mea hj).symm
/-- The product of... | Mathlib/Probability/Integration.lean | 138 | 153 |
/-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.Plus
import Mathlib.CategoryTheory.Limits.Shapes.ConcreteCategory
/-!
# Sheafification
We construct the sheafification of a presheaf ov... | ext : 3
refine colimit.hom_ext (fun j => ?_)
erw [colimit.ΞΉ_map, comp_zero]
simp
/-- The sheafification functor is left adjoint to the forgetful functor. -/
| Mathlib/CategoryTheory/Sites/ConcreteSheafification.lean | 571 | 576 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Patrick Massot, SΓ©bastien GouΓ«zel
-/
import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic
import Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalcul... | Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | 747 | 749 | |
/-
Copyright (c) 2024 RΓ©my Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: RΓ©my Degenne, Peter Pfaffelhuber
-/
import Mathlib.MeasureTheory.SetSemiring
import Mathlib.MeasureTheory.OuterMeasure.Induced
/-!
# Additive Contents
An additive content `m` on a set ... | exact Set.union_subset hI (hC.disjointOfDiffUnion_subset hs hI)
Β· rw [coe_union]
exact hC.pairwiseDisjoint_union_disjointOfDiffUnion hs hI h_dis
Β· rwa [hC.sUnion_union_disjointOfDiffUnion_of_subset hs hI hI_ss]
/-- For an `m : addContent C` on a `SetSemiring C`, if `I` is a `Finset` of pairwise disjoint
... | Mathlib/MeasureTheory/Measure/AddContent.lean | 134 | 141 |
/-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, SΓ©bastien GouΓ«zel, Patrick Massot
-/
import Mathlib.Topology.UniformSpace.Cauchy
import Mathlib.Topology.UniformSpace.Separation
import Mathlib.Topology.DenseEmbedding
... | theorem IsUniformInducing.uniformContinuous {f : Ξ± β Ξ²} (hf : IsUniformInducing f) :
UniformContinuous f := (isUniformInducing_iff'.1 hf).1
| Mathlib/Topology/UniformSpace/UniformEmbedding.lean | 88 | 90 |
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Mathlib.Control.Basic
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Option.Basic
im... | Mathlib/Data/List/Basic.lean | 1,536 | 1,539 | |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.RingTheory.Localization.Basic
import Mathlib.RingT... | constructor
Β· rintro β¨y, hy, _, β¨z, hz, rflβ©, eβ©
exact β¨β¨y, hyβ©, β¨z, hzβ©, e.symmβ©
Β· rintro β¨y, z, eβ©
exact β¨y, y.prop, _, β¨z, z.prop, rflβ©, e.symmβ©
variable (M N)
variable [IsLocalization M S]
| Mathlib/RingTheory/Localization/LocalizationLocalization.lean | 53 | 61 |
/-
Copyright (c) 2020 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Analysis.Calculus.Deriv.Inv
import Mathlib.Analysis.Calculus.Deriv.MeanValue
/-!
# L'HΓ΄pital's rule for 0/0 indeterminate forms
In this file, we ... | @[deprecated (since := "2025-03-02")]
alias lhopital_zero_nhds_right := lhopital_zero_nhdsGT
/-- L'HΓ΄pital's rule for approaching a real from the left, `HasDerivAt` version -/
theorem lhopital_zero_nhdsLT (hff' : βαΆ x in π[<] a, HasDerivAt f (f' x) x)
(hgg' : βαΆ x in π[<] a, HasDerivAt g (g' x) x) (hg' : βαΆ x in... | Mathlib/Analysis/Calculus/LHopital.lean | 281 | 296 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Algebra.Group.Subgroup.Actions
import Mathlib.Algebra.Group.Subgroup.ZPowers.Lemmas
import Mathlib.Data.Fintype.BigOperators
import Mathli... | @[to_additive]
instance mulLeftCosetsCompSubtypeVal (H I : Subgroup Ξ±) : MulAction I (Ξ± β§Έ H) :=
MulAction.compHom (Ξ± β§Έ H) (Subgroup.subtype I)
| Mathlib/GroupTheory/GroupAction/Quotient.lean | 120 | 123 |
/-
Copyright (c) 2022 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.Polynomial.Mirror
import Mathlib.Algebra.Ring.Regular
import Mathlib.Data.Int.Order.Units
import Mathlib.RingTheory.Coprime.Basic
/-!
# Unit... | trailingDegree_le_of_ne_zero <| by rwa [trinomial_trailing_coeff' hkm hmn]).symm
replace h := support_trinomial' k m n u v w h
rw [mem_insert, mem_insert, mem_singleton] at h
rcases h with (rfl | rfl | rfl)
Β· exact le_rfl
Β· exact WithTop.coe_le_coe.mpr hkm.le
Β· exact WithTop.coe_le_coe.mpr (hkm.tr... | Mathlib/Algebra/Polynomial/UnitTrinomial.lean | 81 | 92 |
/-
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, Patrick Massot
-/
import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic
import Mathlib.MeasureTheory.Measure.Real
import Mathlib.Order.Filter.Indi... | (h_lim : βα΅ x βΞΌ, x β Ξ a b β Tendsto (fun n => F n x) l (π (f x))) :
Tendsto (fun n => β« x in a..b, F n x βΞΌ) l (π <| β« x in a..b, f x βΞΌ) := by
simp only [intervalIntegrable_iff, intervalIntegral_eq_integral_uIoc,
β ae_restrict_iff' (Ξ± := β) (ΞΌ := ΞΌ) measurableSet_uIoc] at *
exact tendsto_const_nhds... | Mathlib/MeasureTheory/Integral/DominatedConvergence.lean | 218 | 230 |
/-
Copyright (c) 2024 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.Data.Matroid.Minor.Restrict
/-!
# Some constructions of matroids
This file defines some very elementary examples of matroids, namely those with at most o... | theorem eq_loopyOn_or_rankPos (M : Matroid Ξ±) : M = loopyOn M.E β¨ RankPos M := by
rw [β empty_isBase_iff, rankPos_iff]; apply em
| Mathlib/Data/Matroid/Constructions.lean | 126 | 127 |
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Function.LpSeminorm.Basic
/-!
# Chebyshev-Markov inequality in terms of Lp seminorms
In this file we formulate several versions of the Chebys... | have : 1 / p.toReal * p.toReal = 1 := by
refine one_div_mul_cancel ?_
rw [Ne, ENNReal.toReal_eq_zero_iff]
exact not_or_intro hp_ne_zero hp_ne_top
rw [β ENNReal.rpow_one (Ξ΅ * ΞΌ { x | Ξ΅ β€ (βf xββ : ββ₯0β) ^ p.toReal }), β this, ENNReal.rpow_mul]
gcongr
exact pow_mul_meas_ge_le_eLpNorm ΞΌ hp_ne_zero hp_n... | Mathlib/MeasureTheory/Function/LpSeminorm/ChebyshevMarkov.lean | 31 | 40 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Patrick Massot, Eric Wieser, YaΓ«l Dillies
-/
import Mathlib.Analysis.Normed.Module.Basic
import Mathlib.LinearAlgebra.Basis.VectorSpace
/-!
# Basic facts about real ... | refine Subset.antisymm ?_ ball_subset_interior_closedBall
intro y hy
rcases (mem_closedBall.1 <| interior_subset hy).lt_or_eq with (hr | rfl)
Β· exact hr
set f : β β E := fun c : β => c β’ (y - x) + x
suffices f β»ΒΉ' closedBall x (dist y x) β Icc (-1) 1 by
have hfc : Continuous f := (continuous_id.smul con... | Mathlib/Analysis/NormedSpace/Real.lean | 81 | 98 |
/-
Copyright (c) 2020 SΓ©bastien GouΓ«zel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: SΓ©bastien GouΓ«zel
-/
import Mathlib.Topology.UniformSpace.Cauchy
/-!
# Uniform convergence
A sequence of functions `Fβ` (with values in a metric space) converges uniformly on a se... | theorem UniformCauchySeqOn.mono (hf : UniformCauchySeqOn F p s) (hss' : s' β s) :
UniformCauchySeqOn F p s' := by
rw [uniformCauchySeqOn_iff_uniformCauchySeqOnFilter] at hf β’
exact hf.mono_right (le_principal_iff.mpr <| mem_principal.mpr hss')
/-- Composing on the right by a function preserves uniform Cauchy s... | Mathlib/Topology/UniformSpace/UniformConvergence.lean | 438 | 464 |
/-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Data.Set.Function
import Mathlib.Logic.Pairwise
import Mathlib.Logic.Relation
/-!
# Relations holding pairwise
This file develops pairwise relations ... | (h : f i β f j β β₯) : i = j :=
(hs.elim hi hj) fun hij => h hij.eq_bot
theorem PairwiseDisjoint.eq_of_le (hs : s.PairwiseDisjoint f) {i j : ΞΉ} (hi : i β s) (hj : j β s)
| Mathlib/Data/Set/Pairwise/Basic.lean | 313 | 316 |
/-
Copyright (c) 2019 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 HΓΆlzl, Yury Kudryashov
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.Setoid.Basic
import Mathlib.Dynamics.Fixed... | simp only [(Β· β Β·), MapsTo.iterate_restrict, MapsTo.val_restrict_apply]
Β· convert hle n
rw [MapsTo.iterate_restrict]
rfl
variable (f) in
-- avoid `efixedPoint _` in pretty printer
/-- Let `s` be a complete forward-invariant set of a self-map `f`. If `f` contracts on `s`
and `x β s` satisfies `edist x (f ... | Mathlib/Topology/MetricSpace/Contracting.lean | 155 | 165 |
/-
Copyright (c) 2020 Shing Tak Lam. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Shing Tak Lam
-/
import Mathlib.Data.Finite.Sum
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Exponent
import Mathlib.GroupTheory.GroupAction.CardCommute
import Mathlib.Grou... | exact absurd h2.symm (orderOf_pos _).ne
/-- If `0 < n`, then `i : ZMod n` has order `n / gcd n i`.
| Mathlib/GroupTheory/SpecificGroups/Dihedral.lean | 189 | 191 |
/-
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.Over.Pullback
import Mathlib.CategoryTheory.Limits.Shapes.KernelPair
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq
import ... | @[reassoc (attr := simp)]
theorem diagonalObjPullbackFstIso_hom_fst_snd {X Y Z : C} (f : X βΆ Z) (g : Y βΆ Z) :
(diagonalObjPullbackFstIso f g).hom β« pullback.fst _ _ β« pullback.snd _ _ =
pullback.snd _ _ β« pullback.snd _ _ := by
| Mathlib/CategoryTheory/Limits/Shapes/Diagonal.lean | 316 | 319 |
/-
Copyright (c) 2019 SΓ©bastien GouΓ«zel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: SΓ©bastien GouΓ«zel
-/
import Mathlib.Topology.Algebra.GroupCompletion
import Mathlib.Topology.Algebra.Ring.Real
import Mathlib.Topology.MetricSpace.Algebra
import Mathlib.Topology.Me... | Β· intro a b
rw [Completion.dist_eq, Completion.dist_eq, dist_comm]
protected theorem dist_triangle (x y z : Completion Ξ±) : dist x z β€ dist x y + dist y z := by
refine induction_onβ x y z ?_ ?_
Β· refine isClosed_le ?_ (Continuous.add ?_ ?_) <;>
| Mathlib/Topology/MetricSpace/Completion.lean | 68 | 73 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.Lattice
import Mathlib.RingTheory.Coprime.Lemmas
import Mathlib... | theorem isCoprime_iff_codisjoint : IsCoprime I J β Codisjoint I J := by
rw [IsCoprime, codisjoint_iff]
constructor
Β· rintro β¨x, y, hxyβ©
rw [eq_top_iff_one]
| Mathlib/RingTheory/Ideal/Operations.lean | 629 | 633 |
/-
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... | Β· rw [exch.antichain hBβ hBβ (diff_eq_empty.mp he)]
Β· exact le_top.trans_eq hinf.symm
| Mathlib/Data/Matroid/Basic.lean | 295 | 297 |
/-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Topology.Order.LeftRight
import Mathlib.Topology.Separation.Hausdorff
/-!
# Order-closed topologies
In this file we ... | t.isClosed_le'.inter (isClosed_le continuous_snd continuous_fst)
end PartialOrder
section LinearOrder
| Mathlib/Topology/Order/OrderClosed.lean | 792 | 797 |
/-
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... | refine
eq_Gamma_of_log_convex doublingGamma_log_convex_Ioi
(fun {y} hy => doublingGamma_add_one y hy.ne') (fun {y} hy => ?_) doublingGamma_one hs
apply_rules [mul_pos, Gamma_pos_of_pos, add_pos, inv_pos_of_pos, rpow_pos_of_pos, two_pos,
one_pos, sqrt_pos_of_pos pi_pos]
/-- Legendre's doubling formula... | Mathlib/Analysis/SpecialFunctions/Gamma/BohrMollerup.lean | 418 | 424 |
/-
Copyright (c) 2024 YaΓ«l Dillies, Kin Yau James Wong. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: YaΓ«l Dillies, Kin Yau James Wong, RΓ©my Degenne
-/
import Mathlib.MeasureTheory.Function.AEEqOfLIntegral
import Mathlib.Probability.Kernel.Composition.MeasureCompProd
... | Mathlib/Probability/Kernel/Disintegration/Basic.lean | 389 | 392 | |
/-
Copyright (c) 2021 RΓ©my Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, SΓ©bastien GouΓ«zel, RΓ©my Degenne
-/
import Mathlib.MeasureTheory.Integral.FinMeasAdditive
/-!
# Extension of a linear function from indicators to L1
Give... | theorem setToFun_mono_left {T T' : Set Ξ± β E βL[β] G''} {C C' : β}
(hT : DominatedFinMeasAdditive ΞΌ T C) (hT' : DominatedFinMeasAdditive ΞΌ T' C')
(hTT' : β s x, T s x β€ T' s x) (f : Ξ± ββ[ΞΌ] E) : setToFun ΞΌ T hT f β€ setToFun ΞΌ T' hT' f :=
setToFun_mono_left' hT hT' (fun s _ _ x => hTT' s x) f
theorem setToFun... | Mathlib/MeasureTheory/Integral/SetToL1.lean | 799 | 805 |
/-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, Mario Carneiro
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Data.Int.Cast.Pi
import Mathlib.Data.Nat.Cast.Basic
import Mathlib.MeasureTheory.MeasurableSpace... | Mathlib/MeasureTheory/MeasurableSpace/Basic.lean | 1,314 | 1,327 |
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