Context stringlengths 227 76.5k | target stringlengths 0 11.6k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 16 3.69k |
|---|---|---|---|---|
/-
Copyright (c) 2023 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck, Ruben Van de Velde
-/
import Mathlib.Analysis.Calculus.ContDiff.Operations
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Shift
impor... | iteratedDerivWithin n (f + g) s x =
iteratedDerivWithin n f s x + iteratedDerivWithin n g s x := by
simp_rw [iteratedDerivWithin, iteratedFDerivWithin_add_apply hf hg h hx,
ContinuousMultilinearMap.add_apply]
theorem iteratedDerivWithin_const_add (hn : 0 < n) (c : F) :
iteratedDerivWithin n (fun z ... | Mathlib/Analysis/Calculus/IteratedDeriv/Lemmas.lean | 40 | 46 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Function.L1Space.Integrable
import Mathlib.MeasureTheory.Function.LpSpace.Indicator
/-! # Functions integrable on a set an... | filter_upwards [hu] with x hx using (and_iff_left hx).symm
alias ⟨IntegrableAtFilter.of_inf_ae, _⟩ := IntegrableAtFilter.inf_ae_iff
@[simp]
| Mathlib/MeasureTheory/Integral/IntegrableOn.lean | 451 | 455 |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Geometry.Euclidean.Altitude
import Mathlib.Geometry.Euclidean.Circumcenter
/-!
# Monge point and orthocenter
This file defines the orthocenter of a trian... | (∃ i₁ i₂ i₃ j₂ j₃ : Fin 3,
i₁ ≠ i₂ ∧ i₁ ≠ i₃ ∧ i₂ ≠ i₃ ∧ (∀ i : Fin 3, i = i₁ ∨ i = i₂ ∨ i = i₃) ∧
p i₁ = t.orthocenter ∧ j₂ ≠ j₃ ∧ t.points j₂ = p i₂ ∧ t.points j₃ = p i₃) ∨
Set.range p = Set.range t.points := by
by_cases h : t.orthocenter ∈ Set.range p
· left
rcases h with ⟨i₁, h₁⟩
... | Mathlib/Geometry/Euclidean/MongePoint.lean | 498 | 516 |
/-
Copyright (c) 2022 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Group.Nat.Even
import Mathlib.Data.Nat.Cast.Basic
import Mathlib.Data.Nat.Cast.Commute
import Mathlib.Data.Set.Operations
import Mathlib.Logic.Fu... | Mathlib/Algebra/Ring/Parity.lean | 441 | 442 | |
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Functor.ReflectsIso.Basic
import Mathlib.CategoryTheory.MorphismProperty.Basic
/-!
# Morphism properties that are inverted by a functor
In this ... | haveI := h f.unop hf
dsimp
infer_instance
theorem rightOp {W : MorphismProperty C} {L : Cᵒᵖ ⥤ D} (h : W.op.IsInvertedBy L) :
| Mathlib/CategoryTheory/MorphismProperty/IsInvertedBy.lean | 49 | 53 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Operations
import Mathlib.Order.Basic
import Mathlib.Order.BooleanAlgebra
import Mathlib.Tactic.Tauto
import Mathlib.Tactic.B... | Mathlib/Data/Set/Basic.lean | 2,442 | 2,444 | |
/-
Copyright (c) 2018 Michael Jendrusch. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Jendrusch, Kim Morrison, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Monoidal.Category
import Mathlib.CategoryTheory.Adjunction.FullyFaithful
import Mathlib.CategoryTheory... | end
end LaxMonoidal
| Mathlib/CategoryTheory/Monoidal/Functor.lean | 237 | 240 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Integral.Bochner.Set
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory... | theorem integral_comp_abs {f : ℝ → ℝ} :
∫ x, f |x| = 2 * ∫ x in Ioi (0 : ℝ), f x := by
have eq : ∫ (x : ℝ) in Ioi 0, f |x| = ∫ (x : ℝ) in Ioi 0, f x := by
refine setIntegral_congr_fun measurableSet_Ioi (fun _ hx => ?_)
| Mathlib/MeasureTheory/Measure/Lebesgue/Integral.lean | 96 | 99 |
/-
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... | hasseDeriv_monomial, taylor_apply, monomial_comp, C_1, (commute_X (C r)).add_pow i,
| Mathlib/Algebra/Polynomial/Taylor.lean | 66 | 66 |
/-
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.Order.Filter.Curry
import Mathlib.Data.Set.Countable
/-!
# Filters with countable intersection property
In this file we define `CountableInterFilte... |
theorem eventually_countable_forall [Countable ι] {p : α → ι → Prop} :
(∀ᶠ x in l, ∀ i, p x i) ↔ ∀ i, ∀ᶠ x in l, p x i := by
simpa only [Filter.Eventually, setOf_forall] using
@countable_iInter_mem _ _ l _ _ fun i => { x | p x i }
| Mathlib/Order/Filter/CountableInter.lean | 58 | 62 |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Jakob von Raumer
-/
import Mathlib.Data.List.Chain
import Mathlib.CategoryTheory.PUnit
import Mathlib.CategoryTheory.Groupoid
import Mathlib.CategoryTheory.Category.ULift
... |
/-- If `J` is preconnected, then `Jᵒᵖ` is preconnected as well. -/
instance isPreconnected_op [IsPreconnected J] : IsPreconnected Jᵒᵖ where
iso_constant := fun {α} F X =>
| Mathlib/CategoryTheory/IsConnected.lean | 248 | 251 |
/-
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.Computability.Tape
import Mathlib.Data.Fintype.Option
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.PFun
import M... | Mathlib/Computability/TuringMachine.lean | 1,132 | 1,133 | |
/-
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.CharP.Invertible
import Mathlib.Algebra.MvPolynomial.Variables
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Alg... | rw [add_comm, add_tsub_assoc_of_le (Nat.lt_succ_iff.mp hk), ← add_comm]
section PPrime
variable [hp : NeZero p]
theorem wittPolynomial_vars [CharZero R] (n : ℕ) : (wittPolynomial p R n).vars = range (n + 1) := by
have : ∀ i, (monomial (Finsupp.single i (p ^ (n - i))) ((p : R) ^ i)).vars = {i} := by
intro i
... | Mathlib/RingTheory/WittVector/WittPolynomial.lean | 154 | 163 |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Indicator
import Mathlib.Algebra.Module.BigOperators
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic
import... |
/-- `centroidWeights` at any point. -/
@[simp]
theorem centroidWeights_apply (i : ι) : s.centroidWeights k i = (#s : k)⁻¹ :=
| Mathlib/LinearAlgebra/AffineSpace/Combination.lean | 728 | 731 |
/-
Copyright (c) 2020 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Algebra.Algebra.Field
import Mathlib.Algebra.BigOperators.Balance
import Mathlib.Algebra.Order.BigOperators.Expect
import Mathlib.Algebra.Order.Star.... | norm_sq_eq_def_ax z
theorem normSq_eq_def' (z : K) : normSq z = ‖z‖ ^ 2 :=
| Mathlib/Analysis/RCLike/Basic.lean | 385 | 387 |
/-
Copyright (c) 2021 Arthur Paulino. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Arthur Paulino, Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Clique
import Mathlib.Data.ENat.Lattice
import Mathlib.Data.Nat.Lattice
import Mathlib.Data.Setoid.Partition
imp... | G.chromaticNumber ≤ 1 := by
rw [← Nat.cast_one, chromaticNumber_le_iff_colorable]
refine ⟨Coloring.mk (fun _ => 0) ?_⟩
intros v w
cases Subsingleton.elim v w
simp
| Mathlib/Combinatorics/SimpleGraph/Coloring.lean | 284 | 289 |
/-
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... | theorem derivative_X_add_C (c : R) : derivative (X + C c) = 1 := by
rw [derivative_add, derivative_X, derivative_C, add_zero]
| Mathlib/Algebra/Polynomial/Derivative.lean | 130 | 131 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Polynomial.Expand
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.FieldTheory.IntermediateField.Basic
imp... |
variable (F L K : Type*) [CommRing F] [Ring K] [Algebra F K]
-- TODO: refactor to allow transcendental extensions?
-- See: https://en.wikipedia.org/wiki/Separable_extension#Separability_of_transcendental_extensions
-- Note that right now a Galois extension (class `IsGalois`) is defined to be an extension which
-- is ... | Mathlib/FieldTheory/Separable.lean | 531 | 540 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Polynomial.Expand
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.FieldTheory.IntermediateField.Basic
imp... | ((hf.mul_right h).add_mul_left_right _).mul_left ((h.symm.mul_right hg).mul_add_right_right _)
theorem separable_prod' {ι : Sort _} {f : ι → R[X]} {s : Finset ι} :
| Mathlib/FieldTheory/Separable.lean | 204 | 206 |
/-
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... | rw [hausdorffEdist_comm] at fin
| Mathlib/Topology/MetricSpace/HausdorffDistance.lean | 755 | 755 |
/-
Copyright (c) 2020 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Kim Morrison
-/
import Mathlib.Algebra.Order.Interval.Set.Instances
import Mathlib.Order.Interval.Set.ProjIcc
import Mathlib.Topology.Algebra.Ring.Real
/-!
# The unit ... | example : CompactSpace I := by infer_instance
| Mathlib/Topology/UnitInterval.lean | 174 | 174 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kenny Lau, Yury Kudryashov
-/
import Mathlib.Data.List.Forall2
import Mathlib.Data.List.Lex
import Mathlib.Logic.Function.Iterate
import Mathlib.Logic.Relation
/-!
# R... | simp only [chain_cons, Chain.nil, and_true]
theorem chain_split {a b : α} {l₁ l₂ : List α} :
Chain R a (l₁ ++ b :: l₂) ↔ Chain R a (l₁ ++ [b]) ∧ Chain R b l₂ := by
induction' l₁ with x l₁ IH generalizing a <;>
simp only [*, nil_append, cons_append, Chain.nil, chain_cons, and_true, and_assoc]
| Mathlib/Data/List/Chain.lean | 49 | 55 |
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Shapes.NormalMono.Eq... | rw [sub_sub_sub, sub_zero, sub_self, sub_zero]
theorem add_comm {X Y : C} (a b : X ⟶ Y) : a + b = b + a := by
rw [add_def]
conv_lhs => rw [← neg_neg a]
rw [neg_def, neg_def, neg_def, sub_sub_sub]
| Mathlib/CategoryTheory/Abelian/NonPreadditive.lean | 350 | 355 |
/-
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.Analytic.Within
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries
/-!
# Higher d... | Mathlib/Analysis/Calculus/ContDiff/Defs.lean | 1,698 | 1,701 | |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Multiset.Dedup
/-!
# The fold operation for a commutative associative operation over a multiset.
-/
namespace Multiset
variable {α β : Type*}
... | Mathlib/Data/Multiset/Fold.lean | 108 | 110 | |
/-
Copyright (c) 2023 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.NoZeroSMulDivisors.Basic
import Mathlib.Data.Int.ModEq
import Mathlib.GroupTheory.QuotientGroup.Defs
import Math... |
alias ⟨ModEq.symm, _⟩ := modEq_comm
| Mathlib/Algebra/ModEq.lean | 62 | 63 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.Measure.Comap
import Mathlib.MeasureTheory.Measure.QuasiMeasurePreserving
/-!
# Restricting a measure to a subset or a s... | refine ext_of_generateFrom_of_cover h_gen hc h_inter hU htop ?_ fun t ht => h_eq t (h_sub ht)
intro t ht s hs; rcases (s ∩ t).eq_empty_or_nonempty with H | H
· simp only [H, measure_empty]
| Mathlib/MeasureTheory/Measure/Restrict.lean | 447 | 449 |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Thomas Read, Andrew Yang, Dagur Asgeirsson, Joël Riou
-/
import Mathlib.CategoryTheory.Adjunction.Mates
/-!
# Uniqueness of adjoints
This file shows that adjoints are uni... |
@[reassoc (attr := simp)]
theorem rightAdjointUniq_hom_counit {F : C ⥤ D} {G G' : D ⥤ C} (adj1 : F ⊣ G) (adj2 : F ⊣ G') :
whiskerRight (rightAdjointUniq adj1 adj2).hom F ≫ adj2.counit = adj1.counit := by
ext
| Mathlib/CategoryTheory/Adjunction/Unique.lean | 123 | 127 |
/-
Copyright (c) 2018 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
/-!
# Equalizers and coequalizers
This file defines (co)equalizers a... | id := id
comp := comp
comp_id := comp_id
| Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean | 101 | 103 |
/-
Copyright (c) 2020 Kevin Buzzard, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Equalizers
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Products
import Mathl... | def IsSheaf.amalgamate {A : Type u₂} [Category.{v₂} A] {E : A} {X : C} {P : Cᵒᵖ ⥤ A}
(hP : Presheaf.IsSheaf J P) (S : J.Cover X) (x : ∀ I : S.Arrow, E ⟶ P.obj (op I.Y))
(hx : ∀ ⦃I₁ I₂ : S.Arrow⦄ (r : I₁.Relation I₂),
x I₁ ≫ P.map r.g₁.op = x I₂ ≫ P.map r.g₂.op) : E ⟶ P.obj (op X) :=
(hP _ _ S.condition... | Mathlib/CategoryTheory/Sites/Sheaf.lean | 223 | 231 |
/-
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.Data.Fintype.List
import Mathlib.Data.Fintype.OfMap
/-!
# Cycles of a list
Lists have an equivalence relation of whether they are rotational permut... | theorem chain_coe_cons (r : α → α → Prop) (a : α) (l : List α) :
Chain r (a :: l) ↔ List.Chain r a (l ++ [a]) :=
| Mathlib/Data/List/Cycle.lean | 814 | 815 |
/-
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.HomologicalComplexLimits
import Mathlib.Algebra.Homology.Additive
/-! Binary biproducts of homological complexes
In this file, it is shown tha... | @[reassoc (attr := simp)]
lemma biprod_inl_snd_f (i : ι) :
(biprod.inl : K ⟶ K ⊞ L).f i ≫ (biprod.snd : K ⊞ L ⟶ L).f i = 0 := by
rw [← comp_f, biprod.inl_snd, zero_f]
| Mathlib/Algebra/Homology/HomologicalComplexBiprod.lean | 70 | 73 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.CoreAttrs
/-!
# (Semi-)lattices
Semilatti... | @sup_sup_distrib_right αᵒᵈ _ _ _ _
| Mathlib/Order/Lattice.lean | 404 | 405 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kenny Lau, Yury Kudryashov
-/
import Mathlib.Data.List.Forall2
import Mathlib.Data.List.Lex
import Mathlib.Logic.Function.Iterate
import Mathlib.Logic.Relation
/-!
# R... | trivial
| Mathlib/Data/List/Chain.lean | 170 | 171 |
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Eric Wieser
-/
import Mathlib.Data.Real.Basic
import Mathlib.Tactic.NormNum.Inv
/-!
# Real sign function
This file introduces and contains some results about `Real.sign` wh... | Mathlib/Data/Real/Sign.lean | 109 | 116 | |
/-
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 | 1,170 | 1,174 | |
/-
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 | 1,291 | 1,292 | |
/-
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.Asymptotics
/-!
# Continu... | refine (continuousAt_log ?_).comp continuous_fst.continuousAt
exact hp.lt.ne.symm
theorem continuousAt_rpow_of_pos (p : ℝ × ℝ) (hp : 0 < p.2) :
ContinuousAt (fun p : ℝ × ℝ => p.1 ^ p.2) p := by
obtain ⟨x, y⟩ := p
dsimp only at hp
| Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean | 193 | 199 |
/-
Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
import Mathlib.Topology.Bornology.Constructions
import Mathlib.Topology.MetricSpace.Pseudo.Def... | Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean | 404 | 408 | |
/-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Analysis.Convex.Topology
import Mathlib.Topology.Connected.LocPathConnected
/-!
# Locally convex topological modules
A `LocallyConvexSpace` is a ... | variable {𝕜 E}
/-- In a locally convex space, if `s`, `t` are disjoint convex sets, `s` is compact and `t` is
closed, then we can find open disjoint convex sets containing them. -/
theorem Disjoint.exists_open_convexes [LocallyConvexSpace 𝕜 E] {s t : Set E} (disj : Disjoint s t)
(hs₁ : Convex 𝕜 s) (hs₂ : IsComp... | Mathlib/Topology/Algebra/Module/LocallyConvex.lean | 130 | 142 |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Solvable
import Mathlib.Algebra.Lie.Quotient
import Mathlib.Algebra.Lie.Normalizer
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.... |
instance (priority := 100) trivialIsNilpotent [IsTrivial L M] : IsNilpotent L M :=
⟨by use 1; simp⟩
instance instIsNilpotentSup (M₁ M₂ : LieSubmodule R L M) [IsNilpotent L M₁] [IsNilpotent L M₂] :
IsNilpotent L (M₁ ⊔ M₂ : LieSubmodule R L M) := by
| Mathlib/Algebra/Lie/Nilpotent.lean | 305 | 310 |
/-
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.Computability.Tape
import Mathlib.Data.Fintype.Option
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.PFun
import M... | Mathlib/Computability/TuringMachine.lean | 2,195 | 2,196 | |
/-
Copyright (c) 2023 Alex Keizer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Keizer
-/
import Mathlib.Data.Vector.Basic
import Mathlib.Data.Vector.Snoc
/-!
This file establishes a set of normalization lemmas for `map`/`mapAccumr` operations on vectors
-/
... |
/-!
## Commutativity
| Mathlib/Data/Vector/MapLemmas.lean | 369 | 371 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Topology.UniformSpace.CompleteSeparated
import Mathlib.Topology.EMetricSpace.Lipschitz
import Mathlib.Topology.MetricSpace.Basic
import Mathlib.Topol... | exact (hf.image_preimage _).symm
| Mathlib/Topology/MetricSpace/Antilipschitz.lean | 206 | 207 |
/-
Copyright (c) 2020 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
import Mathlib.Analysis.NormedSpace.IndicatorFunction
import Mathlib.Data.Fintype.Order
import Mathlib.MeasureTheory.Function.AEEqFun
import Mathlib.Me... | simp_rw [eLpNorm_eq_lintegral_rpow_enorm hp_zero hp_top]
rw [lintegral_map' (hg.enorm.pow_const p.toReal) hf]
rfl
theorem memLp_map_measure_iff (hg : AEStronglyMeasurable g (Measure.map f μ))
(hf : AEMeasurable f μ) : MemLp g p (Measure.map f μ) ↔ MemLp (g ∘ f) p μ := by
simp [MemLp, eLpNorm_map_measure hg... | Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean | 1,069 | 1,075 |
/-
Copyright (c) 2024 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Group.Measure
import Mathlib.Tactic.Group
import Mathlib.Topology.UrysohnsLemma
/-!
# Everywhere positive sets in measure spaces
... |
/-- In a space with an inner regular measure, any measurable set coincides almost everywhere with
its everywhere positive subset. -/
lemma everywherePosSubset_ae_eq [OpensMeasurableSpace α] [InnerRegular μ] (hs : MeasurableSet s) :
μ.everywherePosSubset s =ᵐ[μ] s := by
simp only [ae_eq_set, diff_eq_empty.mpr (ev... | Mathlib/MeasureTheory/Measure/EverywherePos.lean | 108 | 115 |
/-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.MeasureTheory.Measure.Regular
import Mathlib.MeasureTheory.Function.SimpleFuncDenseLp
import Mathlib.Topology.UrysohnsLemma
import Mathlib.MeasureThe... | [R1Space α] [WeaklyLocallyCompactSpace α] [μ.Regular]
{p : ℝ} (hp : 0 < p) {f : α → E} (hf : MemLp f (ENNReal.ofReal p) μ) {ε : ℝ} (hε : 0 < ε) :
∃ g : α → E,
HasCompactSupport g ∧
(∫ x, ‖f x - g x‖ ^ p ∂μ) ≤ ε ∧ Continuous g ∧ MemLp g (ENNReal.ofReal p) μ := by
have I : 0 < ε ^ (1 / p) := R... | Mathlib/MeasureTheory/Function/ContinuousMapDense.lean | 193 | 210 |
/-
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, 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 | 1,039 | 1,047 | |
/-
Copyright (c) 2024 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.Data.Matroid.Constructions
import Mathlib.Data.Set.Notation
/-!
# Maps between matroids
This file defines maps and comaps, which move a matroid on one ty... | instance [M.RankFinite] {f : α ≃ β} : (M.mapEquiv f).RankFinite :=
inferInstanceAs (M.map f f.injective.injOn).RankFinite
| Mathlib/Data/Matroid/Map.lean | 625 | 627 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Johan Commelin
-/
import Mathlib.Algebra.Algebra.RestrictScalars
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
import Mathlib.Algebra.Module.Rat
import Mathlib.GroupThe... | theorem productMap_left : (productMap f g).comp includeLeft = f :=
lift_comp_includeLeft _ _ (fun _ _ => Commute.all _ _)
theorem productMap_right_apply (b : B) :
productMap f g (1 ⊗ₜ b) = g b := by simp
| Mathlib/RingTheory/TensorProduct/Basic.lean | 1,240 | 1,244 |
/-
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.Group.Submonoid.Operations
import Mathlib.Algebra.MonoidAlgebra.Defs
import Mathlib.Algebra.Order.Mon... | cases a
rw [← ofFinsupp_pow]
theorem _root_.IsSMulRegular.polynomial {S : Type*} [SMulZeroClass S R] {a : S}
| Mathlib/Algebra/Polynomial/Basic.lean | 233 | 236 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Order.Filter.Bases.Finit... | Mathlib/Topology/Algebra/Group/Basic.lean | 1,628 | 1,631 | |
/-
Copyright (c) 2024 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.GradedObject
/-!
# The graded object in a single degree
In this file, we define the functor `GradedObject.single j : C ⥤ GradedObject J C`
which... | @[reassoc (attr := simp)]
lemma single_map_singleObjApplyIso_hom (j : J) {X Y : C} (f : X ⟶ Y) :
(single j).map f j ≫ (singleObjApplyIso j Y).hom = (singleObjApplyIso j X).hom ≫ f := by
apply single_map_singleObjApplyIsoOfEq_hom
| Mathlib/CategoryTheory/GradedObject/Single.lean | 70 | 73 |
/-
Copyright (c) 2020 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.FieldTheory.RatFunc.AsPolynomial
import Mathlib.NumberTheory.ArithmeticFunction
import Mathlib.RingTheory.Ro... | simpa only [Polynomial.map_prod, map_cyclotomic_int, Polynomial.map_sum, Polynomial.map_pow,
Polynomial.map_X] using congr_arg (map (Int.castRingHom R)) this
rw [← mul_left_inj' (cyclotomic_ne_zero 1 ℤ), prod_erase_mul _ _ (Nat.one_mem_divisors.2 h.ne'),
| Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean | 349 | 351 |
/-
Copyright (c) 2021 Hunter Monroe. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hunter Monroe, Kyle Miller, Alena Gusakov
-/
import Mathlib.Combinatorics.SimpleGraph.DeleteEdges
import Mathlib.Data.Fintype.Powerset
/-!
# Subgraphs of a simple graph
A subgraph of ... | theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ :=
rfl
@[simp]
theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ :=
Sym2.fromRel_bot
| Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | 855 | 860 |
/-
Copyright (c) 2014 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Field.Defs
import Mathlib.Algebra.Ring.Commute
import Mathlib.Algebra.Ring.Invertible
impo... | a⁻¹ + b⁻¹ = (a + b) / (a * b) := by
rw [inv_eq_one_div, inv_eq_one_div, hab.one_div_add_one_div ha hb]
| Mathlib/Algebra/Field/Basic.lean | 71 | 72 |
/-
Copyright (c) 2024 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Kernel.Disintegration.Density
import Mathlib.Probability.Kernel.WithDensity
/-!
# Radon-Nikodym derivative and Lebesgue decomposition for kern... | lemma measurableSet_mutuallySingularSetSlice (κ η : Kernel α γ) (a : α) :
MeasurableSet (mutuallySingularSetSlice κ η a) :=
measurable_prodMk_left (measurableSet_mutuallySingularSet κ η)
| Mathlib/Probability/Kernel/RadonNikodym.lean | 205 | 207 |
/-
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.RCLike.Basic
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Complex.Module
import Mathlib.Data.Complex.Order
import Mathli... | lemma _root_.RCLike.isUniformEmbedding_ofReal {𝕜 : Type*} [RCLike 𝕜] :
IsUniformEmbedding ((↑) : ℝ → 𝕜) :=
RCLike.ofRealLI.isometry.isUniformEmbedding
| Mathlib/Analysis/Complex/Basic.lean | 265 | 267 |
/-
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.NatTrans
import Mathlib.CategoryTheory.Iso
/-!
# The category of functors and natural transf... | theorem hcomp_id_app {H : D ⥤ E} (α : F ⟶ G) (X : C) : (α ◫ 𝟙 H).app X = H.map (α.app X) := by
simp
| Mathlib/CategoryTheory/Functor/Category.lean | 121 | 122 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Notation.Pi
import Mathlib.Data.Set.Lattice
import Mathlib.Order.Filter.Defs
/-!
# Theory of... | @[simp]
theorem eventually_false_iff_eq_bot {f : Filter α} : (∀ᶠ _ in f, False) ↔ f = ⊥ :=
| Mathlib/Order/Filter/Basic.lean | 595 | 596 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Finset.Max
import Mathlib.Data.Fintype.Basic
/-!
# Lemmas relating fintypes and order/lattice structure.
-/
open Function
open Nat
universe u... | simpa using exists_max_image univ f univ_nonempty
| Mathlib/Data/Fintype/Lattice.lean | 55 | 56 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.GaloisConnection.Basic
/-!
# Heyting regular elements
This file defines Heyting regular elements, elements of a Heyting algebra that are their own ... | compl_compl_compl _
| Mathlib/Order/Heyting/Regular.lean | 66 | 67 |
/-
Copyright (c) 2020 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Logic.IsEmpty
import Mathlib.Order.Basic
import Mathlib.Tactic.MkIffOfInductiveProp
import Batteries.WF
/-!
# Unbundled... | alias HasSSubset.SSubset.trans_eq := ssubset_of_ssubset_of_eq
| Mathlib/Order/RelClasses.lean | 557 | 558 |
/-
Copyright (c) 2021 Justus Springer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Justus Springer
-/
import Mathlib.Topology.Sheaves.Forget
import Mathlib.Topology.Sheaves.SheafCondition.PairwiseIntersections
import Mathlib.CategoryTheory.Limits.Types.Shapes
/-!
#... | Types.isLimit_iff, IsSheafUniqueGluing, isGluing_iff_pairwise]
refine forall₂_congr fun ι U ↦ ⟨fun h sf cpt ↦ ?_, fun h s hs ↦ ?_⟩
· exact h _ cpt.sectionPairwise.prop
· specialize h (fun i ↦ s <| op <| Pairwise.single i) fun i j ↦
(hs <| op <| Pairwise.Hom.left i j).trans (hs <| op <| Pairwise.Hom.righ... | Mathlib/Topology/Sheaves/SheafCondition/UniqueGluing.lean | 125 | 134 |
/-
Copyright (c) 2024 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.HasPullbac... | isColimit₂ := coproductIsCoproduct _
m := functorObjLeft f πX
g₁ := functorObjTop f πX
g₂ := ρFunctorObj f πX
isPushout := IsPushout.of_hasPushout (functorObjTop f πX) (functorObjLeft f πX)
cofan₁ := _
cofan₂ := _
| Mathlib/CategoryTheory/SmallObject/Construction.lean | 150 | 157 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Alex Kontorovich
-/
import Mathlib.Data.Set.Piecewise
import Mathlib.Order.Filter.Tendsto
import Mathlib.Order.Filter.Bases.Finite
/-!
# (Co)product of a family of f... | theorem coprodᵢ_eq_bot_iff [∀ i, Nonempty (α i)] : Filter.coprodᵢ f = ⊥ ↔ f = ⊥ := by
simpa [funext_iff] using coprodᵢ_neBot_iff.not
@[simp] theorem coprodᵢ_bot' : Filter.coprodᵢ (⊥ : ∀ i, Filter (α i)) = ⊥ :=
coprodᵢ_eq_bot_iff'.2 (Or.inr rfl)
@[simp]
| Mathlib/Order/Filter/Pi.lean | 284 | 290 |
/-
Copyright (c) 2023 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.Data.Complex.FiniteDimensional
import Mathlib.MeasureTheory.Constructions.HaarToSphere
import Mathlib.MeasureTheory.Integral.Gamma
import Mathlib.Measure... | include h1 h2 h3 h4 h5
theorem MeasureTheory.measure_lt_one_eq_integral_div_gamma {p : ℝ} (hp : 0 < p) :
μ {x : E | g x < 1} =
.ofReal ((∫ (x : E), Real.exp (- (g x) ^ p) ∂μ) / Real.Gamma (finrank ℝ E / p + 1)) := by
-- We copy `E` to a new type `F` on which we will put the norm defined by `g`
letI F : T... | Mathlib/MeasureTheory/Measure/Lebesgue/VolumeOfBalls.lean | 73 | 115 |
/-
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.Group.Finset.Powerset
import Mathlib.Algebra.NoZeroSMulDivisors.Pi
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Finty... | Mathlib/LinearAlgebra/Multilinear/Basic.lean | 1,811 | 1,823 | |
/-
Copyright (c) 2021 Alex Kontorovich and Heather Macbeth and Marc Masdeu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Kontorovich, Heather Macbeth, Marc Masdeu
-/
import Mathlib.Analysis.Complex.Basic
import Mathlib.Data.Fintype.Parity
import Mathlib.LinearAl... | field_simp
linear_combination (-(z * (c : ℂ) ^ 2) - c * d) * h
end UpperHalfPlane
namespace ModularGroup -- results specific to `SL(2, ℤ)`
| Mathlib/Analysis/Complex/UpperHalfPlane/Basic.lean | 401 | 407 |
/-
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.Topology.Order.Lattice
import Mathlib.Order.PartialSups
/-!
# Continuity of `partialSups`
In this file we prove that `partialSups` of a sequence of... | protected lemma partialSups_apply (hf : ∀ k ≤ n, Tendsto (f k) l (𝓝 (g k))) :
Tendsto (fun a ↦ partialSups (f · a) n) l (𝓝 (partialSups g n)) := by
simpa only [← Pi.partialSups_apply] using Tendsto.partialSups hf
| Mathlib/Topology/Order/PartialSups.lean | 31 | 33 |
/-
Copyright (c) 2020 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa, Alex Meiburg
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Degree.Monomial
/-!
# Erase the... | Mathlib/Algebra/Polynomial/EraseLead.lean | 60 | 60 | |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Data.Finset.NatAntidiagonal
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
/-!
# Big operators for `NatAntidiagonal`
This file contains theor... |
/-- This lemma matches more generally than `Finset.Nat.prod_antidiagonal_eq_prod_range_succ_mk` when
using `rw ← `. -/
| Mathlib/Algebra/BigOperators/NatAntidiagonal.lean | 54 | 56 |
/-
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... | @[simp]
theorem rpow_zero (x : ℝ) : x ^ (0 : ℝ) = 1 := by simp [rpow_def]
theorem rpow_zero_pos (x : ℝ) : 0 < x ^ (0 : ℝ) := by simp
@[simp]
theorem zero_rpow {x : ℝ} (h : x ≠ 0) : (0 : ℝ) ^ x = 0 := by simp [rpow_def, *]
theorem zero_rpow_eq_iff {x : ℝ} {a : ℝ} : 0 ^ x = a ↔ x ≠ 0 ∧ a = 0 ∨ x = 0 ∧ a = 1 := by
co... | Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 100 | 112 |
/-
Copyright (c) 2021 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.Data.Stream.Init
import Mathlib.Topology.Algebra.Semigroup
import Mathlib.Topology.StoneCech
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
/-!
# Hind... | rintro p rfl
rw [Stream'.corec_eq, Stream'.head_cons]
exact Set.inter_subset_left (Set.Nonempty.some_mem _)
| tail b n h ih =>
rintro p rfl
refine Set.inter_subset_left (ih (succ p) ?_)
rw [Stream'.corec_eq, Stream'.tail_cons]
| cons b n h ih =>
rintro p rfl
have := Set.inter_subset_... | Mathlib/Combinatorics/Hindman.lean | 172 | 203 |
/-
Copyright (c) 2017 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.CategoryTheory.EqToHom
import Mathlib.CategoryTheory.Products.Basic
/-!
# Curry and uncurry, as functors.
We define `curry : ((C × D) ⥤ E) ⥤ (C ⥤ (D ⥤ E)... | dsimp
simp only [← F.map_comp, Category.id_comp, Category.comp_id, prod_comp])
| Mathlib/CategoryTheory/Functor/Currying.lean | 138 | 140 |
/-
Copyright (c) 2022 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Eisenstein.Criterion
import Mathlib.RingTheory.Polynomial.ScaleRoots
/-!
# Eisenstein polynomials
Given an ideal `𝓟` of a com... | (hp : p.Monic) (x y : R) (z : A) (h : p.eval₂ f z = 0) (hz : f x * z = f y) :
x ∣ y ^ p.natDegree := by
rw [← natDegree_scaleRoots p x, ← Ideal.mem_span_singleton]
refine
(scaleRoots.isWeaklyEisensteinAt _
(Ideal.mem_span_singleton.mpr <| dvd_refl x)).pow_natDegree_le_of_root_of_monic_mem
| Mathlib/RingTheory/Polynomial/Eisenstein/Basic.lean | 161 | 166 |
/-
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 [max_eq_right hxy, max_eq_right (rpow_le_rpow hx hxy hp)]
· rw [max_eq_left hxy, max_eq_left (rpow_le_rpow hy hxy hp)]
theorem self_le_rpow_of_le_one (h₁ : 0 ≤ x) (h₂ : x ≤ 1) (h₃ : y ≤ 1) : x ≤ x ^ y := by
| Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 727 | 730 |
/-
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... | dsimp
constructor
· intro h
exact le_of_comm (((underlyingIso _).inv ≫ ofLE _ _ (Subtype.mk_le_mk.mp h) ≫
(underlyingIso _).hom)) (by aesop_cat)
· intro h
exact Subtype.mk_le_mk.mpr (le_of_comm
((underlyingIso _).hom ≫ ofLE _ _ h ≫ (underlyingIso _).inv) (by simp))
end Subob... | Mathlib/CategoryTheory/Subobject/Lattice.lean | 701 | 713 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Johan Commelin
-/
import Mathlib.Algebra.Algebra.RestrictScalars
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
import Mathlib.Algebra.Module.Rat
import Mathlib.GroupThe... | instance instNonUnitalRing : NonUnitalRing (A ⊗[R] B) where
toAddCommGroup := TensorProduct.addCommGroup
| Mathlib/RingTheory/TensorProduct/Basic.lean | 540 | 541 |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash, Yury Kudryashov
-/
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.LocallyFinite
import Mathlib.Topology.Maps.Proper.Basic
import Mathlib.Topology.UniformSpace.Un... | constructor
· -- Let us prove that convergence in the compact-open topology
-- implies uniform convergence on compacts.
-- Consider a compact set `K`
intro h K hK
-- Since `K` is compact, it suffices to prove locally uniform convergence
rw [← tendstoLocallyUniformlyOn_iff_tendstoUniformlyOn_of_c... | Mathlib/Topology/UniformSpace/CompactConvergence.lean | 103 | 144 |
/-
Copyright (c) 2020 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.FieldTheory.RatFunc.AsPolynomial
import Mathlib.NumberTheory.ArithmeticFunction
import Mathlib.RingTheory.Ro... | theorem cyclotomic_prime_mul_X_sub_one (R : Type*) [Ring R] (p : ℕ) [hn : Fact (Nat.Prime p)] :
cyclotomic p R * (X - 1) = X ^ p - 1 := by rw [cyclotomic_prime, geom_sum_mul]
@[simp]
theorem cyclotomic_two (R : Type*) [Ring R] : cyclotomic 2 R = X + 1 := by simp [cyclotomic_prime]
@[simp]
theorem cyclotomic_three... | Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean | 363 | 371 |
/-
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.Algebra.NeZero
import Mathlib.Data.Finset.Attach
import Mathlib.Data.Finset.Disjoint
import Mathli... | theorem subtype_map_of_mem {p : α → Prop} [DecidablePred p] {s : Finset α} (h : ∀ x ∈ s, p x) :
(s.subtype p).map (Embedding.subtype _) = s := ext <| by simpa [subtype_map] using h
/-- If a `Finset` of a subtype is converted to the main type with
`Embedding.subtype`, all elements of the result have the property of... | Mathlib/Data/Finset/Image.lean | 600 | 611 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Wrenna Robson
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Pi
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.LinearAlgebra.Vandermonde
import Mathlib.RingT... | constructor <;> intro h
· exact eq_interpolate_of_eval_eq _ hvs h.1 h.2
· rw [h]
exact ⟨degree_interpolate_lt _ hvs, fun _ hi => eval_interpolate_at_node _ hvs hi⟩
/-- Lagrange interpolation induces isomorphism between functions from `s`
and polynomials of degree less than `Fintype.card ι`. -/
def funEquivDe... | Mathlib/LinearAlgebra/Lagrange.lean | 366 | 373 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Iterate
import Mathlib.Order.SemiconjSup
import Mathlib.Topology.Order.MonotoneContinuity
import M... | Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean | 905 | 913 | |
/-
Copyright (c) 2023 Sophie Morel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sophie Morel
-/
import Mathlib.Analysis.Analytic.Constructions
import Mathlib.Analysis.Analytic.CPolynomialDef
/-! # Properties of continuously polynomial functions
We expand the API a... | Mathlib/Analysis/Analytic/CPolynomial.lean | 194 | 197 | |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Nat.Find
import Mathlib.Data.Stream.Init
import Mathlib.Tactic.Common
/-!
# Coinductive formalization of unbounded computations.
This fil... | Mathlib/Data/Seq/Computation.lean | 1,105 | 1,108 | |
/-
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.Int.Interval
import Mathlib.Data.Int.ConditionallyCompleteOrder
import Mathlib.Topology.Instances.Discrete
import Mathlib.Topology... | rfl
| Mathlib/Topology/Instances/Int.lean | 35 | 35 |
/-
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.Computability.Tape
import Mathlib.Data.Fintype.Option
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.PFun
import M... | | TM2.Stmt.peek k f q => goto fun _ _ ↦ go k (StAct.peek f) q
| TM2.Stmt.pop k f q => goto fun _ _ ↦ go k (StAct.pop f) q
| TM2.Stmt.load a q => load (fun _ ↦ a) (trNormal q)
| TM2.Stmt.branch f q₁ q₂ => branch (fun _ ↦ f) (trNormal q₁) (trNormal q₂)
| TM2.Stmt.goto l => goto fun _ s ↦ normal (l s)
| TM2.St... | Mathlib/Computability/TuringMachine.lean | 496 | 501 |
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Functor.ReflectsIso.Basic
import Mathlib.CategoryTheory.MorphismProperty.Basic
/-!
# Morphism properties that are inverted by a functor
In this ... | W.IsInvertedBy F ↔ W.map F ≤ isomorphisms D := by
rw [iff_le_inverseImage_isomorphisms, map_le_iff]
lemma IsInvertedBy.map_iff {C₁ C₂ C₃ : Type*} [Category C₁] [Category C₂] [Category C₃]
(W : MorphismProperty C₁) (F : C₁ ⥤ C₂) (G : C₂ ⥤ C₃) :
(W.map F).IsInvertedBy G ↔ W.IsInvertedBy (F ⋙ G) := by
sim... | Mathlib/CategoryTheory/MorphismProperty/IsInvertedBy.lean | 147 | 156 |
/-
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... | /-- If the Hausdorff distance is `< r`, then any point in one of the sets has
a corresponding point at distance `< r` in the other set. -/
theorem exists_edist_lt_of_hausdorffEdist_lt {r : ℝ≥0∞} (h : x ∈ s) (H : hausdorffEdist s t < r) :
| Mathlib/Topology/MetricSpace/HausdorffDistance.lean | 287 | 289 |
/-
Copyright (c) 2022 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Sébastien Gouëzel, Heather Macbeth, Floris van Doorn
-/
import Mathlib.Topology.FiberBundle.Constructions
import Mathlib.Topology.VectorBundle.Basic
import Mathlib.... | variable (R 𝕜 : Type*) {B : Type*} (F : Type*) (E : B → Type*) {B' : Type*} (f : B' → B)
instance [i : ∀ x : B, AddCommMonoid (E x)] (x : B') : AddCommMonoid ((f *ᵖ E) x) := i _
instance [Semiring R] [∀ x : B, AddCommMonoid (E x)] [i : ∀ x, Module R (E x)] (x : B') :
Module R ((f *ᵖ E) x) := i _
variable {E F} ... | Mathlib/Topology/VectorBundle/Constructions.lean | 153 | 161 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Order.Preorder.Chain
import Mathlib.Tactic.Linter.DeprecatedModule
deprecated_module (since := "2025-04-13")
| Mathlib/Order/Chain.lean | 95 | 98 | |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Aurélien Saue, Anne Baanen
-/
import Mathlib.Tactic.NormNum.Inv
import Mathlib.Tactic.NormNum.Pow
import Mathlib.Util.AtomM
/-!
# `ring` tactic
A tactic for solving e... | let ⟨b, (pb : Q((Int.negOfNat (nat_lit 1)).rawCast * $a = $b))⟩ := rb.toRawEq
| Mathlib/Tactic/Ring/Basic.lean | 598 | 598 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
-/
import Mathlib.MeasureTheory.Integral.Lebesgue.Countable
import Mathlib.MeasureTheory.Measure.Decomposition.Exhaustion
import Mathlib.MeasureTheory.Me... | simpa only [add_comm] using withDensity_add_left hg f
theorem withDensity_add_measure {m : MeasurableSpace α} (μ ν : Measure α) (f : α → ℝ≥0∞) :
| Mathlib/MeasureTheory/Measure/WithDensity.lean | 105 | 107 |
/-
Copyright (c) 2020 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Ira Fesefeldt
-/
import Mathlib.Control.Monad.Basic
import Mathlib.Dynamics.FixedPoints.Basic
import Mathlib.Order.CompleteLattice.Basic
import Mathlib.Order.Iterate
import M... | · apply h.2
exact fun a ↦ le_ωSup c a
· rw [ωSup_le_iff]
apply h.1
| Mathlib/Order/OmegaCompletePartialOrder.lean | 224 | 227 |
/-
Copyright (c) 2017 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Kim Morrison, Mario Carneiro
-/
import Mathlib.CategoryTheory.Elementwise
import Mathlib.Topology.ContinuousMap.Basic
/-!
# Category instance for topological spaces
We ... | /-- Any homeomorphisms induces an isomorphism in `Top`. -/
@[simps]
def isoOfHomeo {X Y : TopCat.{u}} (f : X ≃ₜ Y) : X ≅ Y where
hom := ofHom f
inv := ofHom f.symm
| Mathlib/Topology/Category/TopCat/Basic.lean | 175 | 179 |
/-
Copyright (c) 2024 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Elements
import Mathlib.CategoryTheory.Limits.Types.Limits
import Mathlib.CategoryTheory.Limits.Creates
import Mathlib.CategoryTheory.Limi... |
@[simp]
lemma map_lift_mapCone (c : Cone F) :
A.map (limit.lift (F ⋙ π A) ((π A).mapCone c)) c.pt.snd = liftedConeElement F := by
apply (preservesLimitIso A (F ⋙ π A)).toEquiv.injective
ext i
have h₁ := congrFun (preservesLimitIso_hom_π A (F ⋙ π A) i)
(A.map (limit.lift (F ⋙ π A) ((π A).mapCone c)) c.pt.... | Mathlib/CategoryTheory/Limits/Elements.lean | 50 | 58 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Kexing Ying, Eric Wieser
-/
import Mathlib.Data.Finset.Sym
import Mathlib.LinearAlgebra.BilinearMap
import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
import Mathlib.Linea... | refine ⟨-B x y / B x x, fun z hz => ?_⟩
obtain ⟨c, rfl⟩ := Submodule.mem_span_singleton.1 hz
rw [IsOrtho, map_smul, smul_apply, map_add, map_smul, smul_eq_mul, smul_eq_mul,
div_mul_cancel₀ _ hx, add_neg_cancel, mul_zero])
refine ⟨b, ?_⟩
rw [Basis.coe_mkFinCons]
intro j i
refine... | Mathlib/LinearAlgebra/QuadraticForm/Basic.lean | 1,306 | 1,320 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Order.Filter.Interval
import Mathlib.Order.Interval.Set.Pi
import Mathlib.Tactic.TFAE
import Mathlib.Tactic.NormNum
im... | · exact (hy rfl).elim
· obtain ⟨l, lx, hl⟩ : ∃ l, l < x ∧ Ioc l x ⊆ s :=
exists_Ioc_subset_of_mem_nhds (hs.mem_nhds hx) ⟨y, H⟩
exact ⟨l, x, lx, Ioo_subset_Ioc_self.trans hl⟩
| Mathlib/Topology/Order/Basic.lean | 433 | 436 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.Measure.Comap
import Mathlib.MeasureTheory.Measure.QuasiMeasurePreserving
/-!
# Restricting a measure to a subset or a s... | · rwa [Pi.one_apply, Pi.div_apply, one_le_div'] at ha
· rwa [Pi.one_apply, Pi.div_apply, one_le_div']
theorem le_ae_restrict : ae μ ⊓ 𝓟 s ≤ ae (μ.restrict s) := fun _s hs =>
eventually_inf_principal.2 (ae_imp_of_ae_restrict hs)
| Mathlib/MeasureTheory/Measure/Restrict.lean | 641 | 646 |
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