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 |
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/-
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, Kevin Buzzard, Yury Kudryashov, Frédéric Dupuis,
Heather Macbeth
-/
import Mathlib.Algebra.Module.Submodule.Ker
import Mathlib.Algebra.Module.Submodul... | Mathlib/Algebra/Module/Submodule/Range.lean | 437 | 438 | |
/-
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, Yaël Dillies, Yuyang Zhao
-/
import Mathlib.Algebra.Order.Ring.Unbundled.Basic
import Mathlib.Algebra.CharZero.Defs
import Mathlib.Alge... | Mathlib/Algebra/Order/Ring/Defs.lean | 753 | 756 | |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.Exact
/-!
# Refinements
In order to prove injectivity/surjectivity/exactness properties for diagrams
in the category of abelian g... | lemma ShortComplex.exact_iff_exact_up_to_refinements :
S.Exact ↔ ∀ ⦃A : C⦄ (x₂ : A ⟶ S.X₂) (_ : x₂ ≫ S.g = 0),
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x₁ : A' ⟶ S.X₁), π ≫ x₂ = x₁ ≫ S.f := by
rw [S.exact_iff_epi_toCycles, epi_iff_surjective_up_to_refinements]
constructor
· intro hS A a ha
obtain ⟨A', π... | Mathlib/CategoryTheory/Abelian/Refinements.lean | 93 | 103 |
/-
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.Countable.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Order.Disjointed
import Mathlib.MeasureTheory.OuterMeasure.Defs... | exact ge_of_tendsto' B A
/-- If a set has zero measure in a neighborhood of each of its points, then it has zero measure
in a second-countable space. -/
theorem measure_null_of_locally_null [TopologicalSpace α] [SecondCountableTopology α]
(s : Set α) (hs : ∀ x ∈ s, ∃ u ∈ 𝓝[s] x, μ u = 0) : μ s = 0 := by
choos... | Mathlib/MeasureTheory/OuterMeasure/Basic.lean | 138 | 148 |
/-
Copyright (c) 2023 Adrian Wüthrich. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adrian Wüthrich
-/
import Mathlib.Combinatorics.SimpleGraph.AdjMatrix
import Mathlib.LinearAlgebra.Matrix.PosDef
/-!
# Laplacian Matrix
This module defines the Laplacian matrix of a... | The family of these maps indexed by the connected components of `G` proves to be a basis
of the kernel of `lapMatrix G R` -/
def lapMatrix_ker_basis_aux (c : G.ConnectedComponent) :
LinearMap.ker (Matrix.toLin' (G.lapMatrix ℝ)) :=
⟨fun i ↦ if G.connectedComponentMk i = c then (1 : ℝ) else 0,
mem_ker_toLin'_l... | Mathlib/Combinatorics/SimpleGraph/LapMatrix.lean | 148 | 161 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Data.ENNReal.Basic
/-!
# Maps between real and extended non-negative real numbers
This file focuses on the functions `ENNReal.toReal... |
theorem toNNReal_sSup (s : Set ℝ≥0∞) (hs : ∀ r ∈ s, r ≠ ∞) :
(sSup s).toNNReal = sSup (ENNReal.toNNReal '' s) := by
| Mathlib/Data/ENNReal/Real.lean | 394 | 396 |
/-
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, Heather Macbeth
-/
import Mathlib.MeasureTheory.Function.SimpleFunc
import Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable
/-!
# Density of si... | theorem nearestPtInd_le (e : ℕ → α) (N : ℕ) (x : α) : nearestPtInd e N x ≤ N := by
induction' N with N ihN; · simp
simp only [nearestPtInd_succ]
split_ifs
exacts [le_rfl, ihN.trans N.le_succ]
| Mathlib/MeasureTheory/Function/SimpleFuncDense.lean | 87 | 92 |
/-
Copyright (c) 2020 Bhavik Mehta, Edward Ayers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Edward Ayers
-/
import Mathlib.CategoryTheory.Sites.Sieves
import Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer
import Mathlib.CategoryTheory.Category.P... | @[simp]
theorem mem_sieves_iff_coe : S ∈ J.sieves X ↔ S ∈ J X :=
Iff.rfl
/-- Also known as the maximality axiom. -/
| Mathlib/CategoryTheory/Sites/Grothendieck.lean | 105 | 109 |
/-
Copyright (c) 2019 Calle Sönne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology... | ⟨(div_lt_iff₀' (zero_lt_two' ℝ)).1 h, by linarith [pi_pos, toReal_le_pi θ]⟩⟩
theorem two_zsmul_toReal_eq_two_mul_sub_two_pi {θ : Angle} :
((2 : ℤ) • θ).toReal = 2 * θ.toReal - 2 * π ↔ π / 2 < θ.toReal := by
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 579 | 582 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Algebra.MvPolynomial.Eval
/-!
# Renaming variables of polynomials
This file establishes the `rename` operation on mul... | theorem rename_eq (f : σ → τ) (p : MvPolynomial σ R) :
rename f p = Finsupp.mapDomain (Finsupp.mapDomain f) p := by
simp only [rename, aeval_def, eval₂, Finsupp.mapDomain, algebraMap_eq, comp_apply,
X_pow_eq_monomial, ← monomial_finsupp_sum_index]
rfl
| Mathlib/Algebra/MvPolynomial/Rename.lean | 102 | 106 |
/-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib... | exact log_lt_log_iff hx hy
| Mathlib/Analysis/SpecialFunctions/Log/Base.lean | 195 | 196 |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.Cover
import Mathlib.Order.Iterate
/-!
# Successor and predecessor
This file defines succes... | Mathlib/Order/SuccPred/Basic.lean | 740 | 740 | |
/-
Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Data.Fintype.Lattice
import Mathlib.Data.Fintype.Sum
import Mathlib.Topology.Homeomorph.Lemmas
import Mathlib.Topology.MetricSpace.Antilipschitz
... | Mathlib/Topology/MetricSpace/Isometry.lean | 658 | 662 | |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Independence.Kernel
import Mathlib.MeasureTheory.Constructions.Pi
/-!
# Independence of sets of sets and measure spaces (σ-algebras)
* A fami... |
section FromIndepToIndep
| Mathlib/Probability/Independence/Basic.lean | 351 | 354 |
/-
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.Logic.Function.Conjugate
/-!
# Iterations of a function
In this file we prove simple properties of `Nat.iterate f n` a.k.a. `f^[n]`:
* `iterate_ze... | theorem iterate_add_apply (m n : ℕ) (x : α) : f^[m + n] x = f^[m] (f^[n] x) := by
rw [iterate_add f m n]
| Mathlib/Logic/Function/Iterate.lean | 71 | 72 |
/-
Copyright (c) 2024 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Heather Macbeth
-/
import Mathlib.Analysis.Calculus.Deriv.Pi
import Mathlib.Analysis.InnerProductSpace.EuclideanDist
import Mathlib.Analysis.InnerProductSpace.NormP... | /-- The **Gagliardo-Nirenberg-Sobolev inequality**. Let `u` be a continuously differentiable
compactly-supported function `u` on `ℝⁿ`, for `n ≥ 2`. (More literally we encode `ℝⁿ` as
`ι → ℝ` where `n := #ι` is finite and at least 2.) Then the Lebesgue integral of the pointwise
expression `|u x| ^ (n / (n - 1))` is bo... | Mathlib/Analysis/FunctionalSpaces/SobolevInequality.lean | 292 | 338 |
/-
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... | there is a neighborhood of `x` where `f` admits derivatives up to order `n`, which are continuous.
-/
def ContDiffAt (n : WithTop ℕ∞) (f : E → F) (x : E) : Prop :=
ContDiffWithinAt 𝕜 n f univ x
theorem contDiffWithinAt_univ : ContDiffWithinAt 𝕜 n f univ x ↔ ContDiffAt 𝕜 n f x :=
Iff.rfl
| Mathlib/Analysis/Calculus/ContDiff/Defs.lean | 916 | 923 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
-/
import Mathlib.MeasureTheory.Integral.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Lebesgue.Countable
import Mathlib.MeasureTheory.Integral.Le... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 716 | 733 | |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Algebra.Algebra.Basic
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Eval.Algebra
impo... |
@[simp, norm_cast]
theorem ascPochhammer_eval_cast (n k : ℕ) :
(((ascPochhammer ℕ n).eval k : ℕ) : S) = ((ascPochhammer S n).eval k : S) := by
rw [← ascPochhammer_map (algebraMap ℕ S), eval_map, ← eq_natCast (algebraMap ℕ S),
| Mathlib/RingTheory/Polynomial/Pochhammer.lean | 95 | 99 |
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.SetTheory.Cardinal.Finite
import Mathlib.Data.Set.Finite.Powerset
/-!
# Noncomputable Set Cardinality
We define the cardinality of set `s` as a term `Set... | (ht : t.Finite := by toFinite_tac) : s.ncard ≤ t.ncard ↔ (s \ t).ncard ≤ (t \ s).ncard := by
rw [← ncard_inter_add_ncard_diff_eq_ncard s t hs, ← ncard_inter_add_ncard_diff_eq_ncard t s ht,
inter_comm, add_le_add_iff_left]
| Mathlib/Data/Set/Card.lean | 922 | 925 |
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.TangentCone
import Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics
import Mathlib.Analysis.As... |
theorem DifferentiableAt.differentiableWithinAt (h : DifferentiableAt 𝕜 f x) :
DifferentiableWithinAt 𝕜 f s x :=
| Mathlib/Analysis/Calculus/FDeriv/Basic.lean | 631 | 633 |
/-
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 | 1,327 | 1,329 | |
/-
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.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Order.Interval.Finset.Na... | Mathlib/Topology/EMetricSpace/Basic.lean | 750 | 751 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.Order.Sub.WithTop
import Mathlib.Data.NNReal.Defs
import Mathlib.Order.Interval.Set.... | theorem toReal_ne_zero : a.toReal ≠ 0 ↔ a ≠ 0 ∧ a ≠ ∞ :=
a.toReal_eq_zero_iff.not.trans not_or
| Mathlib/Data/ENNReal/Basic.lean | 313 | 314 |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Topology
import Mathlib.Analysis.SpecialFunctions.Arsinh
import Mathlib.Geometry.Euclidean.Inversion.Basic
/-!
# Met... | @[simp]
theorem dist_self_center (z : ℍ) (r : ℝ) :
dist (z : ℂ) (z.center r) = z.im * (Real.cosh r - 1) := by
| Mathlib/Analysis/Complex/UpperHalfPlane/Metric.lean | 171 | 173 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.QuasiIso
import Mathlib.CategoryTheory.Limits.Preserves.Finite
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels
/-!
... | Functor.map_id, Functor.mapShortComplex_obj]
lemma RightHomologyData.mapHomologyIso'_eq [S.HasHomology]
[(S.map F).HasHomology] [F.PreservesRightHomologyOf S] :
S.mapHomologyIso' F = (hr.map F).homologyIso ≪≫ F.mapIso hr.homologyIso.symm := by
ext
dsimp only [Iso.trans, Iso.symm, Iso.refl, Functor.mapI... | Mathlib/Algebra/Homology/ShortComplex/PreservesHomology.lean | 485 | 491 |
/-
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.Data.Set.SymmDiff
import Mathlib.Order.SuccPred.Relation
import Mathlib.Topology.Irreducible
/-!
# Connected subsets ... | lemma connectedSpace_iff_univ : ConnectedSpace α ↔ IsConnected (univ : Set α) :=
⟨fun h ↦ ⟨univ_nonempty, h.1.1⟩,
fun h ↦ ConnectedSpace.mk (toPreconnectedSpace := ⟨h.2⟩) ⟨h.1.some⟩⟩
| Mathlib/Topology/Connected/Basic.lean | 631 | 634 |
/-
Copyright (c) 2016 Leonardo de Moura. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Mathlib.Control.Basic
import Mathlib.Data.Set.Defs
import Mathlib.Data.Set.Lattice.Image
import Mathlib.Data.Set.Notation
/-!
# Functoriality of `Set`
... | /-- This is `Set` but with a `Monad` instance. -/
def SetM (α : Type u) := Set α
| Mathlib/Data/Set/Functor.lean | 134 | 135 |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Data.Set.Lattice
import Mathlib.Data.SetLike.Basic
import Mathlib.Order.ModularLattice
import Mathlib.Order.SuccPred.Basic
import Mathlib.Order.WellFou... | theorem isCoatomic_of_orderTop_gt_wellFounded [OrderTop α]
(h : WellFounded ((· > ·) : α → α → Prop)) : IsCoatomic α :=
isAtomic_dual_iff_isCoatomic.1 (@isAtomic_of_orderBot_wellFounded_lt αᵒᵈ _ _ h)
| Mathlib/Order/Atoms.lean | 460 | 462 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Order.Disjoint
import Mathlib.Order.RelIso.Basic
import Mathlib.Tactic.Monotonicity.Attr
/-!
# Order homomorphisms
This file defines order homomorphi... | Mathlib/Order/Hom/Basic.lean | 1,285 | 1,288 | |
/-
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.Algebra.Homology.Bifunctor
import Mathlib.Algebra.Homology.Homotopy
/-!
# The action of a bifunctor on homological complexes factors through homotopies
Given a... | (F.obj (L₁.X (c₁.prev i₁))).map (f₂.f i₂) ≫ ιMapBifunctorOrZero L₁ L₂ F c _ _ j')
@[reassoc]
lemma ιMapBifunctor_hom₁ (i₁ i₁' : I₁) (i₂ : I₂) (j j' : J)
(h : ComplexShape.π c₁ c₂ c (i₁', i₂) = j) (h' : c₁.prev i₁' = i₁) :
ιMapBifunctor K₁ K₂ F c i₁' i₂ j h ≫ hom₁ h₁ f₂ F c j j' = ComplexShape.ε₁ c₁ c₂ c ... | Mathlib/Algebra/Homology/BifunctorHomotopy.lean | 45 | 52 |
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel, Alex Keizer
-/
import Mathlib.Algebra.Group.Nat.Even
import Mathlib.Algebra.NeZero
import Mathlib.Algebra.Ring.Nat
import Mathlib.Data.List.GetD
import Mathlib.Data.Nat.B... | · exact Nat.lt_add_right 1 this'
· calc
2 * n + 1 < 2 * n + 2 := lt.base _
_ ≤ 2 * m := mul_le_mul_left 2 this
· exact Nat.succ_lt_succ this'
theorem bitwise_swap {f : Bool → Bool → Bool} :
bitwise (Function.swap f) = Function.swap (bitwise f) := by
funext m n
si... | Mathlib/Data/Nat/Bitwise.lean | 230 | 245 |
/-
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... |
variable {k}
| Mathlib/LinearAlgebra/AffineSpace/Combination.lean | 653 | 654 |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Yury Kudryashov
-/
import Mathlib.Data.Finset.Fin
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Order.Interval.Set.Fin
/-!
# Finite intervals in `Fin n`
This fi... | theorem map_valEmbedding_uIcc : (uIcc a b).map valEmbedding = uIcc (a : ℕ) b :=
map_valEmbedding_Icc _ _
@[deprecated (since := "2025-04-08")]
alias map_subtype_embedding_uIcc := map_valEmbedding_uIcc
@[simp]
theorem map_valEmbedding_Ici : (Ici a).map Fin.valEmbedding = Ico (a : ℕ) n := by
rw [← attachFin_Ico_eq_... | Mathlib/Order/Interval/Finset/Fin.lean | 190 | 201 |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Manuel Candales
-/
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
import Mathlib.Tactic.IntervalCases
/-... | rw [h, hab, hcb]; ring
_ = (r * dist a c) ^ 2 := by simp [pow_two, ← law_cos a b c, mul_pow]; ring
by_cases hab₁ : a = b
· have hab'₁ : a' = b' := by
rw [← dist_eq_zero, hab, dist_eq_zero.mpr hab₁, mul_zero r]
rw [hab₁, hab'₁, dist_comm b' c', dist_comm b c, hcb]
· have h1 : 0 ≤ r * dist a b :... | Mathlib/Geometry/Euclidean/Triangle.lean | 346 | 362 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.HomotopyCategory
import Mathlib.Algebra.Ring.NegOnePow
import Mathlib.CategoryTheory.Shift.Quotient
import Mathlib.CategoryTheory.Linear.LinearF... |
lemma shiftFunctorAdd_inv_app_f (K : CochainComplex C ℤ) (a b n : ℤ) :
((shiftFunctorAdd (CochainComplex C ℤ) a b).inv.app K).f n =
(K.XIsoOfEq (by dsimp; rw [add_comm a, add_assoc])).hom := rfl
lemma shiftFunctorAdd_hom_app_f (K : CochainComplex C ℤ) (a b n : ℤ) :
((shiftFunctorAdd (CochainComplex C ℤ)... | Mathlib/Algebra/Homology/HomotopyCategory/Shift.lean | 127 | 135 |
/-
Copyright (c) 2018 Mario Carneiro. 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.CauSeq.Completion
import Mathlib.Algebra.Order.Ring.Rat
import Mathlib.Data.Rat.Cast.Defs
/-!
# Real numbers from Cauc... | ⟨inf⟩
| Mathlib/Data/Real/Basic.lean | 388 | 389 |
/-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | have bound : ∀ i, ∫⁻ x, ENNReal.ofReal (f x) ∂(μs i) ≤ nndist 0 g := fun i ↦ by
simpa only [coe_nnreal_ennreal_nndist, measure_univ, mul_one, ge_iff_le] using
BoundedContinuousFunction.lintegral_le_edist_mul (μ := μs i) g
apply ENNReal.liminf_toReal_eq ENNReal.coe_ne_top (Eventually.of_forall ... | Mathlib/MeasureTheory/Measure/Portmanteau.lean | 496 | 509 |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Casper Putz, Anne Baanen, Wen Yang
-/
import Mathlib.LinearAlgebra.Matrix.Transvection
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mat... | (h : ∀ i, ¬p i → ∀ j, p j → M i j = 0) :
M.det = (toSquareBlockProp M p).det * (toSquareBlockProp M fun i => ¬p i).det := by
rw [det_toBlock M p]
convert det_fromBlocks_zero₂₁ (toBlock M p p) (toBlock M p fun j => ¬p j)
(toBlock M (fun j => ¬p j) fun j => ¬p j)
ext i j
exact h (↑i) i.2 (↑j) j.2
t... | Mathlib/LinearAlgebra/Matrix/Block.lean | 220 | 250 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Group.Units.Basic
import Mathlib.Algebra.GroupWithZero.Basic
import Mathlib.Data.Int.Basic
import Mathlib.Lean.Meta.CongrTheorems
import Mathli... | -- comm
variable [CommGroupWithZero G₀] {a b c d : G₀}
| Mathlib/Algebra/GroupWithZero/Units/Basic.lean | 392 | 393 |
/-
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.Data.Finset.NAry
import Mathlib.Data.Finset.Slice
import Mathlib.Data.Set.Sups
/-!
# Set family operations
This file defines a few binary operations on `... | @[simp] lemma diffs_eq_empty : s \\ t = ∅ ↔ s = ∅ ∨ t = ∅ := image₂_eq_empty_iff
| Mathlib/Data/Finset/Sups.lean | 557 | 558 |
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.Limits.Pullbacks
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
/-!
# Open immersions of structured spaces
We say that a m... | f.c.app (op U) ≫ invApp f ((Opens.map f.base).obj U) =
Y.presheaf.map
(eqToHom
(le_antisymm (Set.image_preimage_subset f.base U.1) <|
| Mathlib/Geometry/RingedSpace/OpenImmersion.lean | 207 | 210 |
/-
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.Group.Units.Equiv
import Mathlib.Algebra.Order.Group.End
import Mathlib.Logic.Function.Conjugate
import Mathlib.Order.Bounds.OrderIso
import ... | end IsOrderRightAdjoint
namespace Function
/-- If an order automorphism `fa` is semiconjugate to an order embedding `fb` by a function `g`
and `g'` is an order right adjoint of `g` (i.e. `g' y = sSup {x | f x ≤ y}`), then `fb` is
| Mathlib/Order/SemiconjSup.lean | 73 | 78 |
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Order.Interval.Finset.Fin
import Mathlib.Data.Vector.Basic
/-!
# The structure of `Fintype (Fin n)`
This file contains some basic results about the `Fintyp... | Mathlib/Data/Fintype/Fin.lean | 67 | 70 | |
/-
Copyright (c) 2021 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Algebra.Order.Monoid.Unbundled.WithTop
/-!
# Lemma about subtraction in ordered monoids with a top element a... |
protected
theorem map_sub [Sub β] [Bot β] {f : α → β} (h : ∀ x y, f (x - y) = f x - f y) (h₀ : f ⊥ = ⊥) :
∀ x y : WithTop α, (x - y).map f = x.map f - y.map f
| Mathlib/Algebra/Order/Sub/WithTop.lean | 58 | 61 |
/-
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.Order.Group.Finset
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Algebra.Polynomial.Eva... | (hf' : f.derivative.eval a ≠ 0) : IsCoprime (X - C a : K[X]) (f /ₘ (X - C a)) := by
classical
refine Or.resolve_left
(EuclideanDomain.dvd_or_coprime (X - C a) (f /ₘ (X - C a))
| Mathlib/Algebra/Polynomial/FieldDivision.lean | 614 | 617 |
/-
Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Data.Set.Lattice
import Mathlib.Order.ConditionallyCompleteLattice.Defs
/-!
# Theory of conditionally complete lattices
A conditionally complet... | csSup_image2_eq_csSup_csSup (α := αᵒᵈ) (β := βᵒᵈ) (γ := γᵒᵈ) (u₁ := l₁) (u₂ := l₂)
(fun _ => (h₁ _).dual) fun _ => (h₂ _).dual
| Mathlib/Order/ConditionallyCompleteLattice/Basic.lean | 848 | 849 |
/-
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... | simpa [dist_comm] using exists_dist_lt_of_hausdorffDist_lt h H fin
| Mathlib/Topology/MetricSpace/HausdorffDistance.lean | 735 | 735 |
/-
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.Topology.Coherent
import Mathlib.Topology.UniformSpace.Equiv
import Mathlib.Topology.UniformSpace.Pi
import Mathlib.Topology.UniformSpace.UniformAp... | `α →ᵤ[𝔖] β`. -/
theorem uniformContinuous_eval_of_mem {x : α} (hxs : x ∈ s) (hs : s ∈ 𝔖) :
UniformContinuous ((Function.eval x : (α → β) → β) ∘ toFun 𝔖) :=
(UniformFun.uniformContinuous_eval β (⟨x, hxs⟩ : s)).comp
(UniformOnFun.uniformContinuous_restrict α β 𝔖 hs)
| Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | 804 | 808 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Batteries.Data.String.Lemmas
import Mathlib.Data.List.Lex
import Mathlib.Data.Char
import Mathlib.Algebra.Order.Group.Nat
/-!
# Strings
Supplementary theorem... | Mathlib/Data/String/Basic.lean | 202 | 203 | |
/-
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, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Ord... | `lt_of_one_div_lt_one_div` -/
theorem one_div_lt_one_div_of_neg (ha : a < 0) (hb : b < 0) : 1 / a < 1 / b ↔ b < a :=
lt_iff_lt_of_le_iff_le (one_div_le_one_div_of_neg hb ha)
| Mathlib/Algebra/Order/Field/Basic.lean | 518 | 520 |
/-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll, Anatole Dedecker
-/
import Mathlib.Analysis.LocallyConvex.Bounded
import Mathlib.Analysis.Seminorm
import Mathlib.Data.Real.Sqrt
import Mathlib.Topology.Algebra.Equicontinuit... | use (s.sup p).ball 0 (r / ‖x‖)
exact ⟨p.basisSets_mem s (div_pos hr (norm_pos_iff.mpr h)), Subset.rfl⟩
refine ⟨(s.sup p).ball 0 r, p.basisSets_mem s hr, ?_⟩
simp only [not_ne_iff.mp h, Set.subset_def, mem_ball_zero, hr, mem_univ, map_zero, imp_true_iff,
preimage_const_of_mem, zero_smul]
/-- The `module... | Mathlib/Analysis/LocallyConvex/WithSeminorms.lean | 160 | 166 |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.CategoryTheory.Sites.Sieves
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.Mono
/-!
# The sheaf condition for a presieve
We define what it means fo... |
/-- The restriction of an extension is the original. -/
@[simp]
theorem restrict_extend {x : FamilyOfElements P R} (t : x.Compatible) :
x.sieveExtend.restrict (le_generate R) = x := by
funext Y f hf
exact extend_agrees t hf
| Mathlib/CategoryTheory/Sites/IsSheafFor.lean | 195 | 202 |
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.Limits.Pullbacks
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
/-!
# Open immersions of structured spaces
We say that a m... |
instance ofRestrict {X : TopCat} (Y : PresheafedSpace C) {f : X ⟶ Y.carrier}
(hf : IsOpenEmbedding f) : IsOpenImmersion (Y.ofRestrict hf) where
base_open := hf
c_iso U := by
dsimp
have : (Opens.map f).obj (hf.isOpenMap.functor.obj U) = U := by
ext1
| Mathlib/Geometry/RingedSpace/OpenImmersion.lean | 224 | 231 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth, Mitchell Lee
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Algebra.Ri... | ring
| neg_add_one n ih1 ih2 =>
simp only [U_sub_one, eval_sub, eval_mul, eval_ofNat, eval_X, mul_one,
| Mathlib/RingTheory/Polynomial/Chebyshev.lean | 233 | 235 |
/-
Copyright (c) 2023 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Heather Macbeth
-/
import Mathlib.MeasureTheory.Constructions.Pi
/-!
# Marginals of multivariate functions
In this file, we define a convenient way to compute int... | (∫⋯∫⁻_s, f ∂μ) x = (∫⋯∫⁻_s, f ∂μ) y := by
dsimp [lmarginal, updateFinset_def]; rcongr; exact h _ ‹_›
theorem lmarginal_update_of_mem {i : δ} (hi : i ∈ s)
| Mathlib/MeasureTheory/Integral/Marginal.lean | 105 | 108 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Frédéric Dupuis, Heather Macbeth
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Analysis.InnerProductSpace.Orthogonal
import Mathlib.Analysis.InnerProductSpace.Sy... | simp [Submodule.finrank_add_inf_finrank_orthogonal h, h_dim]
/-- Given a finite-dimensional space `E` and subspace `K`, the dimensions of `K` and `Kᗮ` add to
| Mathlib/Analysis/InnerProductSpace/Projection.lean | 1,091 | 1,093 |
/-
Copyright (c) 2021 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Sébastien Gouëzel
-/
import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
import Mathlib.MeasureTheory.Constructions.BorelSpace.Metric
import Mathlib.MeasureTheory... | _ = ENNReal.ofReal (|r| ^ finrank ℝ E) * (ENNReal.ofReal (|r| ^ finrank ℝ E))⁻¹ *
(μ s * (μ t)⁻¹) := by
rw [ENNReal.mul_inv]
· ring
| Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean | 538 | 541 |
/-
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.Topology.UniformSpace.UniformConvergenceTopology
/-!
# Equicontinuity of a family of functions
Let `X` be a topological space and `α` a `UniformS... | {S : Set X'} {x₀ : X'} {k : κ} (hk : EquicontinuousWithinAt (tX := t k) F S x₀) :
EquicontinuousWithinAt (tX := ⨅ k, t k) F S x₀ := by
simp only [equicontinuousWithinAt_iff_continuousWithinAt (tX := _)] at hk ⊢
unfold ContinuousWithinAt nhdsWithin at hk ⊢
| Mathlib/Topology/UniformSpace/Equicontinuity.lean | 568 | 571 |
/-
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.Group.Subgroup.Ker
import Mathlib.Algebra.BigOperators.Group.List.Basic
/-!
# Free groups
This file defines free groups over a type. Furthermore, it is... | Mathlib/GroupTheory/FreeGroup/Basic.lean | 1,160 | 1,185 | |
/-
Copyright (c) 2023 Yaël Dillies, Vladimir Ivanov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Vladimir Ivanov
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.BigOperators.... | lemma truncatedSup_infs_of_not_mem (ha : a ∉ lowerClosure s ⊓ lowerClosure t) :
truncatedSup (s ⊼ t) a = ⊤ :=
truncatedSup_of_not_mem <| by rwa [coe_infs, lowerClosure_infs]
| Mathlib/Combinatorics/SetFamily/AhlswedeZhang.lean | 273 | 275 |
/-
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.Adjunction.FullyFaithful
import Mathlib.CategoryTheory.Adjunction.Limits
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq
import Ma... | exact (congr_app e j).symm
theorem IsUniversalColimit.precompose_isIso {F G : J ⥤ C} (α : F ⟶ G) [IsIso α]
{c : Cocone G} (hc : IsUniversalColimit c) :
IsUniversalColimit ((Cocones.precompose α).obj c) := by
intros F' c' α' f e hα H
apply (hc c' (α' ≫ α) f ((Category.assoc _ _ _).trans e)
(hα.comp (N... | Mathlib/CategoryTheory/Limits/VanKampen.lean | 167 | 180 |
/-
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.... | @[simp] lemma LieModule.isNilpotent_of_top_iff' :
IsNilpotent L {x // x ∈ (⊤ : LieSubmodule R L M)} ↔ IsNilpotent L M :=
Equiv.lieModule_isNilpotent_iff 1 (LinearEquiv.ofTop ⊤ rfl) fun _ _ ↦ rfl
end Morphisms
| Mathlib/Algebra/Lie/Nilpotent.lean | 703 | 708 |
/-
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.OfAssociative
import Mathlib.Algebra.Lie.IdealOperations
/-!
# Trivial Lie modules and Abelian Lie algebras
The action of a Lie algebra `L` on ... | map_lie' := fun {x m} => by
have hf : ⁅x, f.val⁆ m = 0 := by rw [f.property x, LinearMap.zero_apply]
rw [LieHom.lie_apply, sub_eq_zero, ← LinearMap.toFun_eq_coe] at hf; exact hf.symm}
| Mathlib/Algebra/Lie/Abelian.lean | 201 | 203 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Bhavik Mehta, Stuart Presnell
-/
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Order.Monotone.Defs
/-!
# Binomial coefficients
This file defines binomial coeffic... | and in compiled code. -/
def fast_choose n k := Nat.descFactorial n k / Nat.factorial k
@[csimp] lemma choose_eq_fast_choose : Nat.choose = fast_choose :=
funext (fun _ => funext (Nat.choose_eq_descFactorial_div_factorial _))
| Mathlib/Data/Nat/Choose/Basic.lean | 268 | 273 |
/-
Copyright (c) 2019 Calle Sönne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology... |
theorem abs_cos_eq_abs_sin_of_two_zsmul_add_two_zsmul_eq_pi {θ ψ : Angle}
(h : (2 : ℤ) • θ + (2 : ℤ) • ψ = π) : |cos θ| = |sin ψ| := by
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 632 | 634 |
/-
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... |
If the index type is infinite, then `mem_pi_pure` and `hasBasis_pi_pure` below may be useful. -/
@[simp]
theorem pi_pure [Finite ι] (f : (i : ι) → α i) : pi (pure <| f ·) = pure f := by
simp only [← principal_singleton, pi_principal, univ_pi_singleton]
| Mathlib/Order/Filter/Pi.lean | 157 | 161 |
/-
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_mutuallySingularSet (κ η : Kernel α γ) :
MeasurableSet (mutuallySingularSet κ η) :=
measurable_rnDerivAux κ (κ + η) measurableSet_Ici
| Mathlib/Probability/Kernel/RadonNikodym.lean | 201 | 203 |
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap
import Mathlib.Probability.Kernel.MeasurableLIntegral
/-!
# With Density
For an s-finite kernel `κ : K... | (hfg : ∀ a, g a ≤ᵐ[κ a] f a) :
withDensity κ (fun a x ↦ f a x - g a x) + withDensity κ g = withDensity κ f := by
rw [← withDensity_add_right _ hg]
swap; · exact hf.sub hg
refine withDensity_congr_ae κ ((hf.sub hg).add hg) hf (fun a ↦ ?_)
filter_upwards [hfg a] with x hx
rwa [Pi.add_apply, Pi.add_apply... | Mathlib/Probability/Kernel/WithDensity.lean | 147 | 154 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.RingTheory.FractionalIdeal.Basic
import Mathlib.RingTheory.IntegralClosure.IsIntegral.Basi... | simp only [mem_singleton_mul, eq_comm]
theorem spanSingleton_mul_le_iff {x : P} {I J : FractionalIdeal S P} :
spanSingleton _ x * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by
simp only [mul_le, mem_singleton_mul, mem_spanSingleton]
constructor
| Mathlib/RingTheory/FractionalIdeal/Operations.lean | 795 | 800 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Patrick Stevens
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.BigOperators.Ring.Finset
import ... | /-- The sum of entries in a row of Pascal's triangle -/
theorem sum_range_choose (n : ℕ) : (∑ m ∈ range (n + 1), n.choose m) = 2 ^ n := by
have := (add_pow 1 1 n).symm
| Mathlib/Data/Nat/Choose/Sum.lean | 89 | 91 |
/-
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.Algebra.Homology.HomotopyCategory.Shift
import Mathlib.Algebra.Homology.TotalComplex
/-!
# Behaviour of the total complex with respect to shifts
There are two ... |
@[reassoc]
lemma ι_totalShift₂Iso_hom_f (a b n : ℤ) (h : a + b = n) (b' : ℤ) (hb' : b' = b + y)
(n' : ℤ) (hn' : n' = n + y) :
((shiftFunctor₂ C y).obj K).ιTotal (up ℤ) a b n h ≫ (K.totalShift₂Iso y).hom.f n =
(a * y).negOnePow • (K.shiftFunctor₂XXIso a b y b' hb').hom ≫
K.ιTotal (up ℤ) a b' n' (b... | Mathlib/Algebra/Homology/TotalComplexShift.lean | 307 | 315 |
/-
Copyright (c) 2020 Alena Gusakov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alena Gusakov, Arthur Paulino, Kyle Miller, Pim Otte
-/
import Mathlib.Combinatorics.SimpleGraph.Clique
import Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph
import Mathlib.Com... | refine ⟨sup_adj.mpr (.inl hw.1), ?_⟩
intro y hy
cases hy with
| inl h => exact hw.2 y h
| inr h =>
rw [Set.disjoint_left] at hd
| Mathlib/Combinatorics/SimpleGraph/Matching.lean | 114 | 119 |
/-
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
-/
import Mathlib.Algebra.Order.Group.Abs
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Algebra.Order.Ring.Int
import Mathlib.A... | lemma pow_eq_pow_iff_cases : a ^ n = b ^ n ↔ n = 0 ∨ a = b ∨ a = -b ∧ Even n := by
rcases eq_or_ne n 0 with rfl | hn <;> simp [pow_eq_pow_iff_of_ne_zero, *]
lemma pow_eq_one_iff_of_ne_zero (hn : n ≠ 0) : a ^ n = 1 ↔ a = 1 ∨ a = -1 ∧ Even n := by
simp [← pow_eq_pow_iff_of_ne_zero hn]
| Mathlib/Algebra/Order/Ring/Abs.lean | 220 | 224 |
/-
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... | refine ext_get (by convert h) fun n h₁ h₂ => ?_
simp
congr
omega
| Mathlib/Data/List/Basic.lean | 686 | 689 |
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Logic.Encodable.Pi
import Mathlib.MeasureTheory.Group.Measure
import Mathlib.MeasureTheory.MeasurableSpace.... |
/-- `Measure.pi μ` is the finite product of the measures `{μ i | i : ι}`.
It is defined to be measure corresponding to `MeasureTheory.OuterMeasure.pi`. -/
protected irreducible_def pi : Measure (∀ i, α i) :=
toMeasure (OuterMeasure.pi fun i => (μ i).toOuterMeasure) (pi_caratheodory μ)
instance _root_.MeasureTheor... | Mathlib/MeasureTheory/Constructions/Pi.lean | 204 | 210 |
/-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth, Eric Wieser
-/
import Mathlib.Analysis.Normed.Lp.PiLp
import Mathlib.Analysis.InnerProductSpace.PiL2
/-!
# Matrices as a normed space
In this file we provide the fo... | ‖replicateCol ι v‖₊ = ‖(WithLp.equiv 2 _).symm v‖₊ :=
Subtype.ext <| frobenius_norm_replicateCol v
| Mathlib/Analysis/Matrix.lean | 567 | 569 |
/-
Copyright (c) 2024 Jeremy Tan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Tan
-/
import Mathlib.Analysis.SpecialFunctions.Complex.LogBounds
/-!
# Complex arctangent
This file defines the complex arctangent `Complex.arctan` as
$$\arctan z = -\frac i2 \lo... | theorem hasSum_arctan {z : ℂ} (hz : ‖z‖ < 1) :
HasSum (fun n : ℕ ↦ (-1) ^ n * z ^ (2 * n + 1) / ↑(2 * n + 1)) (arctan z) := by
have := ((hasSum_taylorSeries_log (z := z * I) (by simpa)).add
(hasSum_taylorSeries_neg_log (z := z * I) (by simpa))).mul_left (-I / 2)
simp_rw [← add_div, ← add_one_mul, hasSum_arc... | Mathlib/Analysis/SpecialFunctions/Complex/Arctan.lean | 115 | 132 |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic
import Mathlib.Algebra.Ring.Pointwise.Set
import Mathlib.Topology.MetricSpace.Isometry
import Mathlib.Topol... | theorem smul_ball (c : G) (x : X) (r : ℝ) : c • ball x r = ball (c • x) r :=
(IsometryEquiv.constSMul c).image_ball _ _
@[to_additive (attr := simp)]
| Mathlib/Topology/MetricSpace/IsometricSMul.lean | 325 | 328 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Eric Wieser
-/
import Mathlib.Algebra.Quaternion
import Mathlib.Analysis.InnerProductSpace.Continuous
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.To... | theorem norm_piLp_equiv_symm_equivTuple (x : ℍ) :
| Mathlib/Analysis/Quaternion.lean | 150 | 150 |
/-
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.Group.Action.Pi
import Mathlib.Algebra.Order.AbsoluteValue.Basic
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.Mi... |
@[simp]
protected theorem sup_idem (a : CauSeq α abs) : a ⊔ a = a := Subtype.ext (sup_idem _)
| Mathlib/Algebra/Order/CauSeq/Basic.lean | 780 | 782 |
/-
Copyright (c) 2020 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou
-/
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.Group.Support
import Mathlib.Data.Set.SymmDiff
/-!
# Indicator function
- `Set.indicator (s : Set α) (f ... | @[to_additive]
theorem mulIndicator_union_mul_inter_apply (f : α → M) (s t : Set α) (a : α) :
mulIndicator (s ∪ t) f a * mulIndicator (s ∩ t) f a
= mulIndicator s f a * mulIndicator t f a := by
by_cases hs : a ∈ s <;> by_cases ht : a ∈ t <;> simp [*]
| Mathlib/Algebra/Group/Indicator.lean | 286 | 290 |
/-
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.NumberTheory.LSeries.MellinEqDirichlet
i... | rw [this, Left.sign_neg, abs_neg, SignType.coe_neg, neg_div, neg_div]
/-- Formula for `hurwitzZetaOdd` as a Dirichlet series in the convergence range, with sum over `ℕ`
(version without absolute values, assuming `a ∈ Icc 0 1`) -/
lemma hasSum_nat_hurwitzZetaOdd_of_mem_Icc {a : ℝ} (ha : a ∈ Icc 0 1) {s : ℂ} (hs : 1 <... | Mathlib/NumberTheory/LSeries/HurwitzZetaOdd.lean | 493 | 497 |
/-
Copyright (c) 2024 Josha Dekker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Josha Dekker, Devon Tuma, Kexing Ying
-/
import Mathlib.Probability.Notation
import Mathlib.Probability.Density
import Mathlib.Probability.ConditionalProbability
import Mathlib.Probabili... | rw [ProbabilityTheory.cond_toMeasurable_eq]
protected theorem toMeasurable {X : Ω → E} {s : Set E} (hu : IsUniform X s ℙ μ) :
IsUniform X (toMeasurable μ s) ℙ μ := by
| Mathlib/Probability/Distributions/Uniform.lean | 95 | 98 |
/-
Copyright (c) 2022 Yaël Dillies, Ella Yu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Ella Yu
-/
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Fintype.Prod
import Mathlib.Algebra.Group.Poin... | ((s ×ˢ t).filter fun (a, b) ↦ a * b ∈ u).card ^ 2 ≤ u.card * Eₘ[s, t] := by
calc
_ = (∑ c ∈ u, ((s ×ˢ t).filter fun (a, b) ↦ a * b = c).card) ^ 2 := by
| Mathlib/Combinatorics/Additive/Energy.lean | 136 | 138 |
/-
Copyright (c) 2022 Paul A. Reichert. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Paul A. Reichert
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Analysis.Normed.Module.Basic
import Mathlib.Topology.MetricSpace.HausdorffDistance
/-!
# Convex bodies
This ... | rw [← NNReal.tendsto_coe, NormedAddCommGroup.tendsto_atTop] at hu
intro ε hε
obtain ⟨n, hn⟩ := hu (ε / C) (div_pos hε hC_pos)
| Mathlib/Analysis/Convex/Body.lean | 207 | 209 |
/-
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.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
import Mathlib.Analy... | exact (hf.sub hg).norm 𝕜 (sub_ne_zero.2 hne)
theorem DifferentiableOn.norm_sq (hf : DifferentiableOn ℝ f s) :
| Mathlib/Analysis/InnerProductSpace/Calculus.lean | 253 | 255 |
/-
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.Prod
import Mathlib.Data.Fintype.EquivFin
/-!
# fintype instance for the product of two fintypes.
-/
open Function
universe u v
varia... | Infinite (∀ i : ι, π i) := by
classical
choose m n hm using fun i => exists_pair_ne (π i)
refine Infinite.of_injective (fun i => update m i (n i)) fun x y h => of_not_not fun hne => ?_
simp_rw [update_eq_iff, update_of_ne hne] at h
exact (hm x h.1.symm).elim
/-- If at least one `π i` is infinite and the ... | Mathlib/Data/Fintype/Prod.lean | 69 | 76 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Yaël Dillies
-/
import Mathlib.Order.Cover
import Mathlib.Order.Interval.Finset.Defs
/-!
# Intervals as finsets
This file provides basic results about all the `Finset.Ixx... | theorem Ioo_subset_Iio_self : Ioo a b ⊆ Iio b := by
simpa [← coe_subset] using Set.Ioo_subset_Iio_self
| Mathlib/Order/Interval/Finset/Basic.lean | 449 | 450 |
/-
Copyright (c) 2023 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.ModuleCat
import Mathlib.RepresentationTheory.GroupCohomology.Basic
import Mathlib.RepresentationTheory.... | DFunLike.ext f₁ f₂ (Prod.forall.mpr h)
variable (A) in
| Mathlib/RepresentationTheory/GroupCohomology/LowDegree.lean | 419 | 421 |
/-
Copyright (c) 2024 Jeremy Tan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Tan
-/
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Tactic.Peel
import Mathlib.Tactic.Positivity
/-!
# Abel's limit theorem
I... | ∃ M, 𝓝[stolzCone s] 1 ≤ 𝓝[stolzSet M] 1 := by
obtain ⟨M, ε, _, hε, H⟩ := stolzCone_subset_stolzSet_aux hs
use M
rw [nhdsWithin_le_iff, mem_nhdsWithin]
refine ⟨{w | 1 - ε < w.re}, isOpen_lt continuous_const continuous_re, ?_, H⟩
simp only [Set.mem_setOf_eq, one_re, sub_lt_self_iff, hε]
| Mathlib/Analysis/Complex/AbelLimit.lean | 110 | 116 |
/-
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... | simp only [Subtype.mk_eq_mk, Subtype.coe_mk, dite_eq_ite]
rw [mem_degreeLT] at hf
conv => rhs; rw [eq_interpolate hvs hf]
| Mathlib/LinearAlgebra/Lagrange.lean | 382 | 384 |
/-
Copyright (c) 2022 Jiale Miao. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jiale Miao, Kevin Buzzard, Alexander Bentkamp
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.LinearAlgebra.Matrix.Block
/-!
# Gram-Schmidt Orthogonalization and Orthonor... | have hki : k < i := by simpa using hki'
have : ⟪b j, b k⟫ = 0 := gramSchmidt_orthogonal 𝕜 v (hki.trans hij).ne'
simp [this]
open Submodule Set Order
theorem mem_span_gramSchmidt (f : ι → E) {i j : ι} (hij : i ≤ j) :
f i ∈ span 𝕜 (gramSchmidt 𝕜 f '' Set.Iic j) := by
rw [gramSchmidt_def' 𝕜 f i]
simp_r... | Mathlib/Analysis/InnerProductSpace/GramSchmidtOrtho.lean | 117 | 128 |
/-
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.Group.Basic
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.Algebra.Notation.Prod
import Mathlib.Data.Set.Image
/-!
# Support of a func... | @[to_additive]
theorem mulSupport_comp_subset {g : M → N} (hg : g 1 = 1) (f : α → M) :
mulSupport (g ∘ f) ⊆ mulSupport f := fun x => mt fun h => by simp only [(· ∘ ·), *]
@[to_additive]
| Mathlib/Algebra/Group/Support.lean | 156 | 160 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Logic.Function.Defs
import Mathlib.Logic.Function.Iterate
import Aesop
import Mathlib.Tactic.Inhabit
/-!
# Extra facts about `Prod`
This file proves ... |
theorem Surjective.prodMap (hf : Surjective f) (hg : Surjective g) : Surjective (map f g) :=
| Mathlib/Data/Prod/Basic.lean | 220 | 221 |
/-
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.PropInstances
import Mathlib.Order.GaloisConnection.Defs
/-!
# Heyting algebras
This file defines Heyting, co-Heyting and bi-Heyting algebras.
A H... | theorem inf_himp (a b : α) : a ⊓ (a ⇨ b) = a ⊓ b :=
| Mathlib/Order/Heyting/Basic.lean | 261 | 261 |
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.C... | Mathlib/FieldTheory/RatFunc/Basic.lean | 1,085 | 1,092 | |
/-
Copyright (c) 2023 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Yaël Dillies, Jineon Baek
-/
import Mathlib.Algebra.EuclideanDomain.Int
import Mathlib.Algebra.GCDMonoid.Finset
import Mathlib.Algebra.GCDMonoid.Nat
import Mathlib.Algebr... | tfae_have 2 → 3
| h, a, b, c, ha, hb, hc, habc => by
rw [← Rat.num_ne_zero] at ha hb hc
refine h (a.num * b.den * c.den) (a.den * b.num * c.den) (a.den * b.den * c.num)
(by positivity) (by positivity) (by positivity) ?_
have : (a.den * b.den * c.den : ℚ) ^ n ≠ 0 := by positivity
refine Int.cas... | Mathlib/NumberTheory/FLT/Basic.lean | 126 | 145 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.Polynomial.Lifts
import Mathlib.Grou... | end IsFractionRing
@[deprecated (since := "2025-03-23")]
alias isAlgebraic_of_isLocalization := IsLocalization.isAlgebraic
open nonZeroDivisors in
lemma isAlgebraic_of_isFractionRing {R S} (K L) [CommRing R] [CommRing S] [Field K] [CommRing L]
[Algebra R S] [Algebra R K] [Algebra R L] [Algebra S L] [Algebra K L] ... | Mathlib/RingTheory/Localization/Integral.lean | 429 | 458 |
/-
Copyright (c) 2024 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn,
Mario Carneiro
-/
import Mathlib.Data.List.Defs
import Mathlib.Data.Option.Basic
import Mathlib.Util... | theorem getD_map {n : ℕ} (f : α → β) : (map f l).getD n (f d) = f (l.getD n d) := by simp
theorem getD_eq_default {n : ℕ} (hn : l.length ≤ n) : l.getD n d = d := by
induction l generalizing n with
| nil => exact getD_nil
| cons head tail ih =>
cases n
| Mathlib/Data/List/GetD.lean | 38 | 44 |
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