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) 2019 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Lu-Ming Zhang
-/
import Mathlib.Data.Matrix.Invertible
import Mathlib.Data.Matrix.Kronecker
import Mathlib.LinearAlgebra.FiniteDimensional.Basic
import Mathlib.LinearAlgebra.... | Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean | 802 | 803 | |
/-
Copyright (c) 2023 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.Dual.Lemmas
import Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv
import Mathlib.LinearAlgebra.QuadraticForm.Prod
/-!
# Quadratic form stru... | let e := LinearEquiv.prodComm R _ _ ≪≫ₗ Module.dualProdDualEquivDual R (Module.Dual R M) M
let h_d := e.symm.toLinearMap.comp (dualProd R M)
refine (Function.Injective.of_comp_iff e.symm.injective
(dualProd R M)).symm.trans ?_
rw [← LinearEquiv.coe_toLinearMap, ← coe_comp]
change Function.Injective h_d ↔ ... | Mathlib/LinearAlgebra/QuadraticForm/Dual.lean | 52 | 72 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Finset.Fold
import Mathlib.Data.Multiset.Bind
import Mathlib.Order.SetNotation
/-!
# Unions of finite sets
This file defines the union of a fami... | ⟨a, ha, Finset.mem_of_subset (h a ha) hb⟩
simpa only [subset_iff, mem_biUnion, exists_imp, and_imp, exists_prop]
lemma biUnion_subset_biUnion_of_subset_left (t : α → Finset β) (h : s₁ ⊆ s₂) :
| Mathlib/Data/Finset/Union.lean | 194 | 197 |
/-
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... | @[simp]
theorem attachFin_Ioo_eq_Ioi : attachFin (Ioo a n) (fun _x hx ↦ (mem_Ioo.mp hx).2) = Ioi a := by
| Mathlib/Order/Interval/Finset/Fin.lean | 79 | 80 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Uni Marx
-/
import Mathlib.CategoryTheory.Iso
import Mathlib.CategoryTheory.EssentialImage
import Mathlib.CategoryTheory.Types
import Mathlib.CategoryTheory.Opposites
impor... | from the category of types and functions into the category of types and relations. -/
def graphFunctor : Type u ⥤ RelCat.{u} where
| Mathlib/CategoryTheory/Category/RelCat.lean | 65 | 66 |
/-
Copyright (c) 2022 Benjamin Davidson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Benjamin Davidson, Devon Tuma, Eric Rodriguez, Oliver Nash
-/
import Mathlib.Algebra.Order.Group.Pointwise.Interval
import Mathlib.Order.Filter.AtTopBot.Field
import Mathlib.Topolog... | theorem Filter.Tendsto.neg_mul_atBot {C : 𝕜} (hC : C < 0) (hf : Tendsto f l (𝓝 C))
(hg : Tendsto g l atBot) : Tendsto (fun x => f x * g x) l atTop := by
| Mathlib/Topology/Algebra/Order/Field.lean | 127 | 128 |
/-
Copyright (c) 2023 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Computability.AkraBazzi.GrowsPolynomially
import Mathlib.Analysis.Calculus.Deriv.Inv
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
/-!
# Divid... | positivity
| Mathlib/Computability/AkraBazzi/AkraBazzi.lean | 440 | 441 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Topology.ContinuousMap.Algebra
import Mathlib.Topology.Compactness.Paracompact
import Mathlib.Topology.Sh... |
/-- Partition of unity defined by a `BumpCovering`. We use this auxiliary definition to prove some
properties of the new family of functions before bundling it into a `PartitionOfUnity`. Do not use
this definition, use `BumpCovering.toPartitionOfUnity` instead.
The partition of unity is given by the formula `g i x = ... | Mathlib/Topology/PartitionOfUnity.lean | 482 | 488 |
/-
Copyright (c) 2022 Alex J. Best. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex J. Best, Yaël Dillies
-/
import Mathlib.Algebra.Order.Archimedean.Hom
import Mathlib.Algebra.Order.Group.Pointwise.CompleteLattice
/-!
# Conditionally complete linear ordered field... | variable (β) [IsStrictOrderedRing α] [Field β] [LinearOrder β] [IsStrictOrderedRing β]
{a a₁ a₂ : α} {b : β} {q : ℚ}
theorem cutMap_coe (q : ℚ) : cutMap β (q : α) = Rat.cast '' {r : ℚ | (r : β) < q} := by
simp_rw [cutMap, Rat.cast_lt]
variable [Archimedean α]
| Mathlib/Algebra/Order/CompleteField.lean | 121 | 127 |
/-
Copyright (c) 2022 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang, Andrew Yang
-/
import Mathlib.AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf
import Mathlib.AlgebraicGeometry.GammaSpecAdjunction
import Mathlib.RingTheory.GradedAlgeb... | Localization.mk_pow, pow_mul]
rfl
theorem num_mem_carrier_iff (hm : 0 < m) (q : Spec.T A⁰_ f)
(z : HomogeneousLocalization.NumDenSameDeg 𝒜 (.powers f)) :
z.num.1 ∈ carrier f_deg q ↔ HomogeneousLocalization.mk z ∈ q.asIdeal := by
obtain ⟨n, hn : f ^ n = _⟩ := z.den_mem
have : f ^ n ≠ 0 := fun e ↦ by
... | Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.lean | 313 | 323 |
/-
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.Card
import Mathlib.Data.Fintype.Basic
/-!
# Cardinalities of finite types
This file defines the cardinality `Fintype.card α` as the numb... | Mathlib/Data/Fintype/Card.lean | 1,184 | 1,186 | |
/-
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 | 446 | 458 | |
/-
Copyright (c) 2021 Noam Atar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Noam Atar
-/
import Mathlib.Order.Ideal
import Mathlib.Order.PFilter
/-!
# Prime ideals
## Main definitions
Throughout this file, `P` is at least a preorder, but some sections require mo... |
section DistribLattice
variable [DistribLattice P] {I : Ideal P}
| Mathlib/Order/PrimeIdeal.lean | 124 | 128 |
/-
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.MonoidAlgebra.Degree
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.Polynomial.Basi... | rcases le_max_iff.1 (degree_add_le p q) with h | h <;> simp [natDegree_le_natDegree h]
theorem natDegree_add_le_of_degree_le {p q : R[X]} {n : ℕ} (hp : natDegree p ≤ n)
(hq : natDegree q ≤ n) : natDegree (p + q) ≤ n :=
(natDegree_add_le p q).trans <| max_le hp hq
| Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 338 | 342 |
/-
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.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.Symmetric
/-!
# Integer powers of square matrices
In this file, we defi... | Mathlib/LinearAlgebra/Matrix/ZPow.lean | 332 | 334 | |
/-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.LinearAlgebra.Matrix.ZPow
import Mathlib.Data.Matrix.ConjTranspose
/-! # Hermitian matrices
Th... |
/-- A block matrix `A.from_blocks B C D` is hermitian,
if `A` and `D` are hermitian and `Bᴴ = C`. -/
| Mathlib/LinearAlgebra/Matrix/Hermitian.lean | 94 | 96 |
/-
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, Sabbir Rahman
-/
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Algebra.Order.Ring.Pow
import Mathlib.Algeb... | let _a ← synthInstanceQ q(GroupWithZero $α)
assumeInstancesCommute
pure (.nonzero q(zpow_ne_zero $b $pa))
| Mathlib/Algebra/Order/Field/Power.lean | 150 | 152 |
/-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Algebra.Ring.Int.Defs
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.Cast.Order.Basic
import Math... | · apply IH
change Nat.shiftRight m n = Nat.shiftRight (m + m + 1) (n + 1)
rw [add_comm n 1, @Nat.shiftRight_eq _ (1 + n), Nat.shiftRight_add]
apply congr_arg fun x => Nat.shiftRight x n
simp [-add_assoc, Nat.shiftRight_succ, Nat.shiftRight_zero, ← Nat.div2_val, hdiv2]
· trans
| Mathlib/Data/Num/Lemmas.lean | 839 | 844 |
/-
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... | ext j
simpa only [colimit.ι_desc_assoc, Discrete.functor_obj, Cofan.mk_pt, Cofan.mk_ι_app,
Category.assoc, extendCofan_pt, Functor.const_obj_obj, NatTrans.comp_app, extendCofan_ι_app,
Fin.cases_succ, Functor.const_map_app] using congr_app e ⟨j.succ⟩
· let F' : Fin (n + 1) → C := F.obj ∘ Di... | Mathlib/CategoryTheory/Limits/VanKampen.lean | 660 | 727 |
/-
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
-/
import Mathlib.LinearAlgebra.Quotient.Basic
/-!
# Isomorphism theorems for modules.
* The Noether's first, second, a... |
end LinearMap
/-! The third isomorphism theorem for modules. -/
| Mathlib/LinearAlgebra/Isomorphisms.lean | 132 | 136 |
/-
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.Group.Basic
import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
import Mathlib.Order.Lattice
/-!
# Ordered Subtraction
This file proves l... |
protected theorem le_add_tsub_swap (hb : AddLECancellable b) : a ≤ b + a - b :=
hb le_add_tsub
protected theorem le_add_tsub (hb : AddLECancellable b) : a ≤ a + b - b := by
| Mathlib/Algebra/Order/Sub/Defs.lean | 174 | 178 |
/-
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.NoZeroSMulDivisors.Basic
import Mathlib.Algebra.Order.GroupWithZero.Action.Synonym
import Mathlib.Tactic.GCongr
import Mathlib.Tactic.Positivity.Co... |
lemma SMulPosReflectLT.lift [SMulPosReflectLT α γ]
(hf : ∀ {b₁ b₂}, f b₁ ≤ f b₂ ↔ b₁ ≤ b₂)
| Mathlib/Algebra/Order/Module/Defs.lean | 1,114 | 1,116 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.List.Sublists
import Mathlib.Data.List.Zip
import Mathlib.Data.Multiset.Bind
import Mathlib.Data.Multiset.Range
/-!
# The powerset of a multiset
... | theorem powersetCard_card_add (s : Multiset α) {i : ℕ} (hi : 0 < i) :
s.powersetCard (card s + i) = 0 :=
powersetCard_eq_empty _ (Nat.lt_add_of_pos_right hi)
| Mathlib/Data/Multiset/Powerset.lean | 255 | 257 |
/-
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
-/
import Mathlib.Data.Finset.Basic
import Mathlib.Data.Finset.Image
/-!
# Cardinality of a finite set
This defines the cardinality of a `Fins... | @[elab_as_elim]
def strongInductionOn {p : Finset α → Sort*} (s : Finset α) :
(∀ s, (∀ t ⊂ s, p t) → p s) → p s := fun H => strongInduction H s
| Mathlib/Data/Finset/Card.lean | 745 | 747 |
/-
Copyright (c) 2024 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Order.Hom.Ring
import Mathlib.Data.ENat.Basic
import Mathlib.SetTheory.Cardinal.Basic
/-!
# Conversion between `Cardinal` and `ℕ∞`
In this ... | | ⊤ => lift_aleph0
| Mathlib/SetTheory/Cardinal/ENat.lean | 124 | 125 |
/-
Copyright (c) 2022 Mario Carneiro, Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Heather Macbeth, Yaël Dillies
-/
import Mathlib.Tactic.NormNum.Core
import Mathlib.Tactic.HaveI
import Mathlib.Algebra.Order.Invertible
import Mathlib.... | rw [NormNum.IsNat.to_eq h rfl]
apply Nat.cast_pos.2
simpa using w
lemma nonneg_of_isNat {n : ℕ} [Semiring A] [PartialOrder A] [IsOrderedRing A]
| Mathlib/Tactic/Positivity/Core.lean | 125 | 129 |
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Data.List.Rotate
import Mathlib.GroupTheory.Perm.Support
/-!
# Permutations from a list
A list `l : List α` ... | theorem formPerm_apply_head (x y : α) (xs : List α) (h : Nodup (x :: y :: xs)) :
formPerm (x :: y :: xs) x = y := by simp [formPerm_apply_of_not_mem h.not_mem]
theorem formPerm_apply_getElem_zero (l : List α) (h : Nodup l) (hl : 1 < l.length) :
formPerm l l[0] = l[1] := by
| Mathlib/GroupTheory/Perm/List.lean | 142 | 146 |
/-
Copyright (c) 2023 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.Algebra.Polynomial.Bivariate
import Mathlib.AlgebraicGeometry.EllipticCurve.Weierstrass
import Mathlib.AlgebraicGeometry.EllipticCu... |
lemma polynomial_ne_zero [Nontrivial R] : W'.polynomial ≠ 0 := by
rw [polynomial_eq]
exact Cubic.ne_zero_of_b_ne_zero one_ne_zero
| Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean | 144 | 147 |
/-
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... | Mathlib/LinearAlgebra/Lagrange.lean | 691 | 698 | |
/-
Copyright (c) 2024 Raghuram Sundararajan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Raghuram Sundararajan
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Ext
/-!
# Extensionality lemmas for rings and similar structures
In this file we prove e... | /-! ### Ring -/
namespace Ring
@[ext] theorem ext ⦃inst₁ inst₂ : Ring R⦄
(h_add : local_hAdd[R, inst₁] = local_hAdd[R, inst₂])
| Mathlib/Algebra/Ring/Ext.lean | 306 | 310 |
/-
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.Data.Set.Finite.Basic
import Mathlib.Order.Atoms
import Mathlib.Order.Grade
import Mathlib.Order.Nat
/-!
# Finsets and multisets form a graded order
This... | lemma covBy_iff_card_sdiff_eq_one : t ⋖ s ↔ t ⊆ s ∧ (s \ t).card = 1 := by
rw [covBy_iff_exists_insert]
constructor
· rintro ⟨a, ha, rfl⟩
simp [*]
· simp_rw [card_eq_one]
rintro ⟨hts, a, ha⟩
refine ⟨a, (mem_sdiff.1 <| superset_of_eq ha <| mem_singleton_self _).2, ?_⟩
rw [insert_eq, ← ha, sdiff_u... | Mathlib/Data/Finset/Grade.lean | 116 | 124 |
/-
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.LinearAlgebra.Matrix.Adjugate
import Mathlib.LinearAlgebra.Matrix.Block
import Mathlib.RingTheory.MatrixPolynomialAlgebra
/-!
# Characteristic polynomials... | @[simp]
lemma charpoly_fromBlocks_zero₂₁ :
(fromBlocks M₁₁ M₁₂ 0 M₂₂).charpoly = (M₁₁.charpoly * M₂₂.charpoly) := by
simp only [charpoly, charmatrix_fromBlocks, Matrix.map_zero _ (Polynomial.C_0), neg_zero,
det_fromBlocks_zero₂₁]
| Mathlib/LinearAlgebra/Matrix/Charpoly/Basic.lean | 120 | 124 |
/-
Copyright (c) 2022 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Order.Filter.Cofinite
/-!
# Basic theory of bornology
We develop the basic theory of bornologies. Instead of axiomatizing bounded sets and defining
bor... | cases t'
congr
/-- A constructor for bornologies by specifying the bounded sets,
and showing that they satisfy the appropriate conditions. -/
@[simps]
def Bornology.ofBounded {α : Type*} (B : Set (Set α))
| Mathlib/Topology/Bornology/Basic.lean | 75 | 81 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Data.FunLike.Equiv
import Mathlib.Data.Quot
import Mathlib.Data.Subtype
import Mathlib.Logic.U... | /-- A version of `Equiv.arrowCongr` in `Type`, rather than `Sort`.
| Mathlib/Logic/Equiv/Defs.lean | 450 | 451 |
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, María Inés de Frutos-Fernández, Filippo A. E. Nuccio
-/
import Mathlib.FieldTheory.RatFunc.Basic
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.DedekindD... |
section AdicValuation
variable (K : Type*) [Field K]
namespace Polynomial
open IsDedekindDomain.HeightOneSpectrum
/-- This is the principal ideal generated by `X` in the ring of polynomials over a field K,
regarded as an element of the height-one-spectrum. -/
def idealX : IsDedekindDomain.HeightOneSpectrum K[X] ... | Mathlib/FieldTheory/RatFunc/AsPolynomial.lean | 179 | 190 |
/-
Copyright (c) 2021 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.Group.End
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Fintype.Sum
import Mathlib.Data.Prod.Lex
import Mathlib.Order.Interval.Finset.Fin
/-!
... | /-- A permutation `σ` equals `sort f` if and only if `f ∘ σ` is monotone and whenever `i < j`
and `f (σ i) = f (σ j)`, then `σ i < σ j`. This means that `sort f` is the lexicographically
smallest permutation `σ` such that `f ∘ σ` is monotone. -/
theorem eq_sort_iff :
σ = sort f ↔ Monotone (f ∘ σ) ∧ ∀ i j, i < j → f... | Mathlib/Data/Fin/Tuple/Sort.lean | 164 | 170 |
/-
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... |
/-- A functor between monoidal categories is monoidal if it is lax and oplax monoidals,
| Mathlib/CategoryTheory/Monoidal/Functor.lean | 376 | 377 |
/-
Copyright (c) 2019 Jan-David Salchow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jan-David Salchow, Sébastien Gouëzel, Jean Lo
-/
import Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear
import Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm
import Mathlib.Ana... | (hc : 1 < ‖c‖) (hf : ∀ x, ε / ‖c‖ ≤ ‖x‖ → ‖x‖ < ε → ‖f x‖ ≤ C * ‖x‖) (x : E) :
‖f x‖ ≤ C * ‖x‖ := by
by_cases hx : x = 0; · simp [hx]
exact SemilinearMapClass.bound_of_shell_semi_normed f ε_pos hc hf (norm_ne_zero_iff.2 hx)
| Mathlib/Analysis/NormedSpace/OperatorNorm/NormedSpace.lean | 42 | 46 |
/-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.Sheaf
/-!
# The plus construction for presheaves.
This file contains the construction of `P⁺`, for a presheaf `P : Cᵒᵖ ⥤ D`
where `C` i... | obj P := J.diagram P X
map η := J.diagramNatTrans η X
variable [∀ X : C, HasColimitsOfShape (J.Cover X)ᵒᵖ D]
/-- The plus construction, associating a presheaf to any presheaf.
| Mathlib/CategoryTheory/Sites/Plus.lean | 90 | 95 |
/-
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 | 1,334 | 1,340 | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex
/-!
# The `arctan` function.
Inequalit... | tendsto_Ioo_atTop.mp tanOrderIso.symm.tendsto_atTop
theorem tendsto_arctan_atBot : Tendsto arctan atBot (𝓝[>] (-(π / 2))) :=
tendsto_Ioo_atBot.mp tanOrderIso.symm.tendsto_atBot
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Arctan.lean | 162 | 165 |
/-
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.GroupTheory.MonoidLocalization.Away
import Mathlib.Algebra.Algebra.Pi
import Mathlib.RingTheory.I... | @atUnit R _ S _ _ (1 : R) isUnit_one _
theorem away_of_isUnit_of_bijective {R : Type*} (S : Type*) [CommSemiring R] [CommSemiring S]
[Algebra R S] {r : R} (hr : IsUnit r) (H : Function.Bijective (algebraMap R S)) :
IsLocalization.Away r S :=
{ map_units' := by
rintro ⟨_, n, rfl⟩
exact (algebraM... | Mathlib/RingTheory/Localization/Away/Basic.lean | 299 | 321 |
/-
Copyright (c) 2023 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.Analysis.Normed.Ring.InfiniteSum
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.NumberTheory.ArithmeticFunction
import Mathlib.NumberTheory... |
namespace EulerProduct
-- a helper lemma that is useful below
lemma one_sub_inv_eq_geometric_of_summable_norm {f : ℕ →*₀ F} {p : ℕ} (hp : p.Prime)
(hsum : Summable fun x ↦ ‖f x‖) :
(1 - f p)⁻¹ = ∑' (e : ℕ), f (p ^ e) := by
simp only [map_pow]
refine (tsum_geometric_of_norm_lt_one <| summable_geometric_iff... | Mathlib/NumberTheory/EulerProduct/Basic.lean | 277 | 296 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Independence.Basic
import Mathlib.Probability.Independence.Conditional
/-!
# Kolmogorov's 0-1 law
Let `s : ι → MeasurableSpace Ω` be an indep... | simpa only [ae_dirac_eq, Filter.eventually_pure]
using Kernel.measure_eq_zero_or_one_or_top_of_indepSet_self h_indep
theorem Kernel.measure_eq_zero_or_one_of_indepSet_self' (h : ∀ᵐ a ∂μα, IsFiniteMeasure (κ a))
| Mathlib/Probability/Independence/ZeroOne.lean | 46 | 49 |
/-
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, Kim Morrison
-/
import Mathlib.Algebra.Algebra.Equiv
import Mathlib.Algebra.Algebra.NonUnitalHom
import Mathlib.Algebra.Module.BigOperators
import Math... | Mathlib/Algebra/MonoidAlgebra/Basic.lean | 932 | 933 | |
/-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Applicative
import Mathlib.Control.Traversable.Basic
import Mathlib.Data.List.Forall2
import Mathlib.Data.Set.Functor
/-!
# LawfulTraversable instan... | Mathlib/Control/Traversable/Instances.lean | 182 | 186 | |
/-
Copyright (c) 2021 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Alena Gusakov, Yaël Dillies
-/
import Mathlib.Data.Finset.Grade
import Mathlib.Data.Finset.Sups
import Mathlib.Logic.Function.Iterate
/-!
# Shadows
This file defines shad... | simp_rw [mem_shadow_iff, ← covBy_iff_card_sdiff_eq_one, covBy_iff_exists_erase]
/-- `t` is in the shadow of `𝒜` iff we can add an element to it so that the resulting finset is in
`𝒜`. -/
lemma mem_shadow_iff_insert_mem : t ∈ ∂ 𝒜 ↔ ∃ a ∉ t, insert a t ∈ 𝒜 := by
| Mathlib/Combinatorics/SetFamily/Shadow.lean | 100 | 104 |
/-
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, Simon Hudon, Mario Carneiro
-/
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Init
import Mathlib.Data.Int.Init
import Mathlib.... | Mathlib/Algebra/Group/Basic.lean | 1,345 | 1,346 | |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.HomotopyCategory.HomComplex
import Mathlib.Algebra.Homology.HomotopyCofiber
/-! # The mapping cone of a morphism of cochain complexes
In this ... | (inr φ).f p₁ ≫ (descCochain φ α β h).v p₁ p₂ h₁₂ = β.v p₁ p₂ h₁₂ := by
simpa only [Cochain.comp_v _ _ (zero_add n) p₁ p₁ p₂ (add_zero p₁) h₁₂, Cochain.ofHom_v]
using Cochain.congr_v (inr_descCochain φ α β h) p₁ p₂ (by omega)
lemma δ_descCochain (n' : ℤ) (hn' : n + 1 = n') :
δ n n' (descCochain φ α β h) =... | Mathlib/Algebra/Homology/HomotopyCategory/MappingCone.lean | 321 | 332 |
/-
Copyright (c) 2021 Vladimir Goryachev. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Vladimir Goryachev, Kyle Miller, Kim Morrison, Eric Rodriguez
-/
import Mathlib.Algebra.Group.Nat.Range
import Mathlib.Data.Set.Finite.Basic
/-!
# Counting on ℕ
Thi... | Mathlib/Data/Nat/Count.lean | 165 | 167 | |
/-
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, Jeremy Avigad
-/
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Data.Set.Finite.Range
import Mathlib.Data.Set.Lattice
import Mathlib.Topology.Defs.... | Mathlib/Topology/Basic.lean | 756 | 757 | |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Convex.Between
import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathli... |
namespace Measure
| Mathlib/MeasureTheory/Measure/Hausdorff.lean | 407 | 409 |
/-
Copyright (c) 2020 Aaron Anderson, Jalex Stark, Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark, Kyle Miller, Alena Gusakov
-/
import Mathlib.Combinatorics.SimpleGraph.Maps
import Mathlib.Data.Finset.Max
import Mathlib.Data.Sy... | constructor
· simpa
· rw [neighborFinset, Set.toFinset_subset_toFinset]
exact G.commonNeighbors_subset_neighborSet_left _ _
| Mathlib/Combinatorics/SimpleGraph/Finite.lean | 411 | 414 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Localization.Opposite
/-!
# Calculus of fractions
Following the definitions by [Gabriel and Zisman][gabriel-zisman-1967],
given a morphism prope... | lemma trans {X Y : C} {z₁ z₂ z₃ : W.LeftFraction X Y}
[HasLeftCalculusOfFractions W]
(h₁₂ : LeftFractionRel z₁ z₂) (h₂₃ : LeftFractionRel z₂ z₃) :
LeftFractionRel z₁ z₃ := by
obtain ⟨Z₄, t₁, t₂, hst, hft, ht⟩ := h₁₂
obtain ⟨Z₅, u₂, u₃, hsu, hfu, hu⟩ := h₂₃
obtain ⟨⟨v₄, v₅, hv₅⟩, fac⟩ := HasLeftCalculu... | Mathlib/CategoryTheory/Localization/CalculusOfFractions.lean | 242 | 259 |
/-
Copyright (c) 2021 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.Algebra.Category.ModuleCat.EpiMono
import Mathlib.Algebra.Category.ModuleCat.Kernels
import Mathlib.CategoryTheory.Subobject.WellPowered
import Mathlib.C... | The application is for homology:
two elements in homology are equal if they differ by a boundary.
-/
theorem cokernel_π_imageSubobject_ext {L M N : ModuleCat.{v} R} (f : L ⟶ M) [HasImage f]
(g : (imageSubobject f : ModuleCat.{v} R) ⟶ N) [HasCokernel g] {x y : N} (l : L)
(w : x = y + g (factorThruImageSubobject ... | Mathlib/Algebra/Category/ModuleCat/Subobject.lean | 89 | 96 |
/-
Copyright (c) 2022 Praneeth Kolichala. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Praneeth Kolichala
-/
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Nat.BinaryRec
import Mathlib.Data.List.Defs
import Mathlib.Tactic.Convert
import Mathlib.Tactic.GeneralizePr... | Mathlib/Data/Nat/Bits.lean | 588 | 588 | |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
-/
import Mathlib.MeasureTheory.Integral.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Lebesgue.Countable
import Mathlib.MeasureTheory.Integral.Le... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 1,578 | 1,581 | |
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Data.Set.BooleanAlgebra
import Mathlib.Tactic.AdaptationNote
/-!
# Relations
This file defines bundled relations. A relation between `α` and `β` is a f... | Set.ext fun _ => mem_preimage _ _ _
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : r.preimage s ⊆ r.preimage t :=
| Mathlib/Data/Rel.lean | 204 | 206 |
/-
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... | (hC : IsPiSystem C) (h1B : ⋃ i, B i = univ) (h2B : ∀ i, B i ∈ C) (hμB : ∀ i, μ (B i) ≠ ∞)
(h_eq : ∀ s ∈ C, μ s = ν s) : μ = ν := by
refine ext_of_generateFrom_of_cover_subset hA hC ?_ (countable_range B) h1B ?_ h_eq
| Mathlib/MeasureTheory/Measure/Restrict.lean | 457 | 459 |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.TwoDim
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
/-!
# Oriented angles.
This file defines orie... | Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | 1,064 | 1,083 | |
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Function.ConvergenceInMeasure
import Mathlib.MeasureTheory.Function.L1Space.Integrable
/-!
# Uniform integrability
This file contains the def... | by_cases hx : M ≤ ‖f x‖
· rw [Pi.add_apply, Set.indicator_of_mem, Set.indicator_of_not_mem, add_zero] <;> simpa
· rw [Pi.add_apply, Set.indicator_of_not_mem, Set.indicator_of_mem, zero_add] <;>
simpa using hx
/-- This lemma is superseded by `MeasureTheory.MemLp.eLpNorm_indicator_le` which does not ... | Mathlib/MeasureTheory/Function/UniformIntegrable.lean | 378 | 386 |
/-
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.Tendsto
import Mathlib.Data.Set.Accumulate
import Mathlib.Topology.Bornology.Basic
import Mathlib.Topolog... | IsCompact s ↔ IsCompact ((↑) '' s : Set X) :=
IsEmbedding.subtypeVal.isCompact_iff
theorem isCompact_iff_isCompact_univ : IsCompact s ↔ IsCompact (univ : Set s) := by
rw [Subtype.isCompact_iff, image_univ, Subtype.range_coe]
| Mathlib/Topology/Compactness/Compact.lean | 889 | 894 |
/-
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... | Mathlib/Data/Nat/Bitwise.lean | 419 | 420 | |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.CategoryTheory.Comma.Arrow
import Mathlib.Order.CompleteBooleanAlgebra
/-!
# Properties of morphisms
We provide the basic framework for talking about prope... | lemma top_apply {X Y : C} (f : X ⟶ Y) : (⊤ : MorphismProperty C) f := by
simp only [top_eq]
| Mathlib/CategoryTheory/MorphismProperty/Basic.lean | 61 | 63 |
/-
Copyright (c) 2021 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Devon Tuma
-/
import Mathlib.Algebra.Polynomial.Eval.Defs
import Mathlib.Analysis.Asymptotics.Lemmas
/-!
# Super-Polynomial Function Decay
This file defines a predicate `Asymptotics.Supe... | tendsto_nhds.2 fun s hs hs0 =>
l.sets_of_superset ((tendsto_nhds.1 (hf <| z + 1)) s hs hs0) fun x hx => by
simpa only [Set.mem_preimage, Pi.mul_apply, ← mul_assoc, ← pow_succ] using hx
| Mathlib/Analysis/Asymptotics/SuperpolynomialDecay.lean | 94 | 96 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.Bochner.Basic
import Mathlib.MeasureTheory.Integral.Bochner.L1
import Mathlib.Me... | Mathlib/MeasureTheory/Integral/Bochner.lean | 209 | 210 | |
/-
Copyright (c) 2022 Vincent Beffara. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Vincent Beffara, Stefan Kebekus
-/
import Mathlib.Analysis.Analytic.Constructions
import Mathlib.Analysis.Calculus.DSlope
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathl... | end HasSum
namespace HasFPowerSeriesAt
theorem has_fpower_series_dslope_fslope (hp : HasFPowerSeriesAt f p z₀) :
HasFPowerSeriesAt (dslope f z₀) p.fslope z₀ := by
have hpd : deriv f z₀ = p.coeff 1 := hp.deriv
have hp0 : p.coeff 0 = f z₀ := hp.coeff_zero 1
simp only [hasFPowerSeriesAt_iff, apply_eq_pow_smul_... | Mathlib/Analysis/Analytic/IsolatedZeros.lean | 69 | 80 |
/-
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.Dynamics.Ergodic.MeasurePreserving
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Dia... | theorem _root_.Filter.Eventually.volume_pos_of_nhds_real {p : ℝ → Prop} {a : ℝ}
(h : ∀ᶠ x in 𝓝 a, p x) : (0 : ℝ≥0∞) < volume { x | p x } := by
rcases h.exists_Ioo_subset with ⟨l, u, hx, hs⟩
| Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | 212 | 214 |
/-
Copyright (c) 2022 Rémy Degenne, Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
import Mathlib.MeasureTheory.Function.Egorov
import Mathlib.MeasureTheory.Function.LpSpace.Complete
/-!
# Convergence in measure
We define con... |
theorem seqTendstoAeSeq_spec (hfg : TendstoInMeasure μ f atTop g) (n k : ℕ)
(hn : seqTendstoAeSeq hfg n ≤ k) :
μ { x | (2 : ℝ)⁻¹ ^ n ≤ dist (f k x) (g x) } ≤ (2 : ℝ≥0∞)⁻¹ ^ n := by
cases n
| Mathlib/MeasureTheory/Function/ConvergenceInMeasure.lean | 178 | 182 |
/-
Copyright (c) 2024 Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Merten
-/
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
import Mathlib.CategoryTheory.Limits.FintypeCat
import Mathlib.CategoryTheory.Lim... | /-- If `F` is a fiber functor and `E` is an equivalence between categories of finite types,
then `F ⋙ E` is again a fiber functor. -/
lemma comp_right (E : FintypeCat.{w} ⥤ FintypeCat.{t}) [E.IsEquivalence] :
FiberFunctor (F ⋙ E) where
preservesQuotientsByFiniteGroups _ := comp_preservesColimitsOfShape F E
end
... | Mathlib/CategoryTheory/Galois/Basic.lean | 171 | 180 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Logic.OpClass
import Mathlib.Order.Lattice
/-!
# `max` and `min`
This file proves basic properties about maxima and minima on a `LinearOrder`.
## Ta... |
theorem AntitoneOn.map_max (hf : AntitoneOn f s) (ha : a ∈ s) (hb : b ∈ s) : f (max a b) =
min (f a) (f b) := hf.dual_right.map_max ha hb
| Mathlib/Order/MinMax.lean | 191 | 193 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Kappelmann
-/
import Mathlib.Algebra.Order.Floor.Defs
import Mathlib.Algebra.Order.Floor.Ring
import Mathlib.Algebra.Order.Floor.Semiring
deprecated_module (sinc... | Mathlib/Algebra/Order/Floor.lean | 1,417 | 1,419 | |
/-
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... |
end Nontrivial
lemma volume_ball_of_dim_odd {k : ℕ} (hk : finrank ℝ E = 2 * k + 1) (x : E) (r : ℝ) :
volume (ball x r) =
| Mathlib/MeasureTheory/Measure/Lebesgue/VolumeOfBalls.lean | 384 | 388 |
/-
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... | Q (x + y) = Q x + Q y
theorem isOrtho_def {Q : QuadraticMap R M N} {x y : M} : Q.IsOrtho x y ↔ Q (x + y) = Q x + Q y :=
Iff.rfl
theorem IsOrtho.all (x y : M) : IsOrtho (0 : QuadraticMap R M N) x y := (zero_add _).symm
| Mathlib/LinearAlgebra/QuadraticForm/Basic.lean | 1,031 | 1,036 |
/-
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, Simon Hudon, Mario Carneiro
-/
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Init
import Mathlib.Data.Int.Init
import Mathlib.... | @[to_additive sub_sub_self]
theorem div_div_self' (a b : G) : a / (a / b) = b := by simp
| Mathlib/Algebra/Group/Basic.lean | 944 | 945 |
/-
Copyright (c) 2019 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann, Kyle Miller, Mario Carneiro
-/
import Mathlib.Data.Finset.NatAntidiagonal
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Data.Nat.BinaryRec
import Mathlib.Logic.F... | theorem fib_dvd (m n : ℕ) (h : m ∣ n) : fib m ∣ fib n := by
rwa [← gcd_eq_left_iff_dvd, ← fib_gcd, gcd_eq_left_iff_dvd.mpr]
theorem fib_succ_eq_sum_choose :
∀ n : ℕ, fib (n + 1) = ∑ p ∈ Finset.antidiagonal n, choose p.1 p.2 :=
twoStepInduction rfl rfl fun n h1 h2 => by
rw [fib_add_two, h1, h2, Finset.Nat.a... | Mathlib/Data/Nat/Fib/Basic.lean | 240 | 246 |
/-
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.ConcreteCategory.Basic
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Limits.Shapes.RegularMon... | variable (hι : Function.Injective ι) (c : Cofan X) (c₁ : Cofan (X ∘ ι))
(hc : IsColimit c) (hc₁ : IsColimit c₁)
include hι
include hc in
lemma mono_of_injective [HasCoproduct (fun (k : ((Set.range ι)ᶜ : Set I)) => X k.1)] :
Mono (Cofan.IsColimit.desc hc₁ (fun i => c.inj (ι i))) :=
mono_of_injective_aux X ι hι ... | Mathlib/CategoryTheory/Limits/MonoCoprod.lean | 174 | 181 |
/-
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... | theorem U_neg_sub_one (n : ℤ) : U R (-n - 1) = -U R (n - 1) := by
induction n using Polynomial.Chebyshev.induct with
| zero => simp
| one => simp
| add_two n ih1 ih2 =>
have h₁ := U_add_one R n
have h₂ := U_sub_two R (-n - 1)
linear_combination (norm := ring_nf) 2 * (X : R[X]) * ih1 - ih2 + h₁ + h₂
... | Mathlib/RingTheory/Polynomial/Chebyshev.lean | 206 | 219 |
/-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Bryan Gin-ge Chen
-/
import Mathlib.Logic.Relation
import Mathlib.Order.CompleteLattice.Basic
import Mathlib.Order.GaloisConnection.Defs
/-!
# Equivalence relati... | Quotient.mk'' ∘ f
variable (r f)
| Mathlib/Data/Setoid/Basic.lean | 399 | 401 |
/-
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.ChosenFiniteProducts
import Mathlib.CategoryTheory.Limits.FunctorCategory.Basic
/-!
# Functor categories have chosen finite products
If `C` is a... |
@[simp]
lemma fst_app (F₁ F₂ : J ⥤ C) (j : J) : (fst F₁ F₂).app j = fst (F₁.obj j) (F₂.obj j) := rfl
@[simp]
| Mathlib/CategoryTheory/ChosenFiniteProducts/FunctorCategory.lean | 81 | 85 |
/-
Copyright (c) 2022 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.CharP.Lemmas
import Mathlib.Data.Fintype.Units
import Mathlib.GroupTheory.OrderOfElement
/-!
# Multiplicative... | lemma ringHomComp_pow (χ : MulChar R R') (f : R' →+* R'') (n : ℕ) :
χ.ringHomComp f ^ n = (χ ^ n).ringHomComp f := by
| Mathlib/NumberTheory/MulChar/Basic.lean | 433 | 434 |
/-
Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Data.Set.Equitable
import Mathlib.Logic.Equiv.Fin.Basic
import Mathlib.Order.Partition.Fi... | theorem IsEquipartition.card_large_parts_eq_mod (hP : P.IsEquipartition) :
#{p ∈ P.parts | #p = #s / #P.parts + 1} = #s % #P.parts := by
have z := P.sum_card_parts
rw [← sum_filter_add_sum_filter_not (s := P.parts) (p := fun x ↦ #x = #s / #P.parts + 1),
hP.filter_ne_average_add_one_eq_average, sum_const_nat... | Mathlib/Order/Partition/Equipartition.lean | 80 | 85 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calculus.Deriv.Mul
import ... | Mathlib/Analysis/Calculus/MeanValue.lean | 946 | 950 | |
/-
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.Analysis.SpecialFunctions.Pow.Real
import Mathlib.LinearAlgebra.FreeModule.PID
import Mathlib.LinearAlgebra.Matrix.AbsoluteValue
import Mathlib.NumberTheory.... | exact fun h => hij ((distinctElems bS adm).injective h)
section Real
open Real
attribute [-instance] Real.decidableEq
/-- We can approximate `a / b : L` with `q / r`, where `r` has finitely many options for `L`. -/
theorem exists_mem_finsetApprox (a : S) {b} (hb : b ≠ (0 : R)) :
∃ q : S,
∃ r ∈ finsetA... | Mathlib/NumberTheory/ClassNumber/Finite.lean | 176 | 188 |
/-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Applicative
import Mathlib.Control.Traversable.Basic
/-!
# Traversing collections
This file proves basic properties of traversable and applicative ... | Mathlib/Control/Traversable/Lemmas.lean | 129 | 132 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Algebra.Group.TypeTags.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Finset.Piecewise
import Mathlib.Or... |
theorem ULift.isOpen_iff [TopologicalSpace X] {s : Set (ULift.{v} X)} :
IsOpen s ↔ IsOpen (ULift.up ⁻¹' s) := by
| Mathlib/Topology/Constructions.lean | 1,141 | 1,143 |
/-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Eric Wieser, Jeremy Avigad, Johan Commelin
-/
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.Linear... | end Triangular
/-! ### 2×2 block matrices -/
section Block
/-! #### General 2×2 block matrices -/
/-- A block matrix is invertible if the bottom right corner and the corresponding schur complement
is. -/
def fromBlocks₂₂Invertible (A : Matrix m m α) (B : Matrix m n α) (C : Matrix n m α)
(D : Matrix n n α) [In... | Mathlib/LinearAlgebra/Matrix/SchurComplement.lean | 220 | 233 |
/-
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.Additive
import Mathlib.Algebra.Homology.ShortComplex.Exact
import Mathlib.Algebra.Homology.ShortComplex.Preadditive
import Mathlib.Tactic.Linar... | shortComplexFunctor' C c i j k ⋙ ShortComplex.homologyFunctor C :=
homologyFunctorIso C c j ≪≫ isoWhiskerRight (natIsoSc' C c i j k hi hk) _
| Mathlib/Algebra/Homology/ShortComplex/HomologicalComplex.lean | 471 | 473 |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Emily Riehl
-/
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryTheory.Functor.TwoSquare
import Mathlib.CategoryTheory.HomCongr
import Mathlib.Tactic.... | /-- The mates equivalence commutes with horizontal composition of squares. -/
theorem mateEquiv_hcomp (α : TwoSquare G L₁ L₂ H) (β : TwoSquare H L₃ L₄ K) :
(mateEquiv (adj₁.comp adj₃) (adj₂.comp adj₄)) (α ≫ᵥ β) =
| Mathlib/CategoryTheory/Adjunction/Mates.lean | 184 | 186 |
/-
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.Topology.MetricSpace.Pseudo.Basic
import Mathlib.Topology.MetricSpace.Pseudo.Lemmas
import Mathlib.Topology.MetricSpace.Pseudo.Pi
import Mathlib... | Mathlib/Topology/MetricSpace/ProperSpace.lean | 159 | 165 | |
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Joseph Myers
-/
import Mathlib.Analysis.InnerProductSpace.Orthogonal
import Mathlib.Analysis.Normed.Group.AddTorsor
/-!
# Perpendicular bisector of a segment
We def... | rw [perpBisector_comm, right_mem_perpBisector, eq_comm]
| Mathlib/Geometry/Euclidean/PerpBisector.lean | 103 | 104 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calculus.Deriv.Mul
import ... | exact this.mono fun y => le_of_lt
· rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩
specialize hf' x xab r hfr
have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z :=
(hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici
(Ioi_mem_nhds hrB)
obtain ⟨z, hfz, hzB, hz... | Mathlib/Analysis/Calculus/MeanValue.lean | 92 | 124 |
/-
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, Jeremy Avigad
-/
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Data.Set.Finite.Range
import Mathlib.Data.Set.Lattice
import Mathlib.Topology.Defs.... | Mathlib/Topology/Basic.lean | 718 | 719 | |
/-
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 | 306 | 306 | |
/-
Copyright (c) 2017 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Mario Carneiro
-/
import Mathlib.Algebra.Ring.CharZero
import Mathlib.Algebra.Star.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Order.Interval.Set.UnorderedInterva... | smul r x := ⟨r • x.re - 0 * x.im, r • x.im + 0 * x.re⟩
| Mathlib/Data/Complex/Basic.lean | 296 | 297 |
/-
Copyright (c) 2020 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Andrew Yang
-/
import Mathlib.RingTheory.Derivation.ToSquareZero
import Mathlib.RingTheory.Ideal.Cotangent
import Mathlib.RingTheory.IsTensorProduct
import Mathlib.... | (S ⊗[R] S ⧸ KaehlerDifferential.ideal R S ^ 2) := Ideal.Quotient.isScalarTower_right
/-- A shortcut instance to prevent timing out. Hopefully to be removed in the future. -/
@[nolint defLemma]
local instance isScalarTower_SS_right : IsScalarTower (S ⊗[R] S)
(S ⊗[R] S ⧸ KaehlerDifferential.ideal R S ^ 2) (S ⊗... | Mathlib/RingTheory/Kaehler/Basic.lean | 389 | 408 |
/-
Copyright (c) 2022 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Order.Filter.Cofinite
/-!
# Basic theory of bornology
We develop the basic theory of bornologies. Instead of axiomatizing bounded sets and defining
bor... | @[simp]
theorem isBounded_union : IsBounded (s ∪ t) ↔ IsBounded s ∧ IsBounded t := by
simp only [← isCobounded_compl_iff, compl_union, isCobounded_inter]
| Mathlib/Topology/Bornology/Basic.lean | 166 | 169 |
/-
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 | 160 | 160 | |
/-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.GroupTheory.GroupAction.Pointwise
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.LocallyConvex.BalancedCoreHull
import Mathlib.Analysis.... | {s : Set E} (hs : IsVonNBounded 𝕜₁ s) (f : E →SL[σ] F) : IsVonNBounded 𝕜₂ (f '' s) := by
have σ_iso : Isometry σ := AddMonoidHomClass.isometry_of_norm σ fun x => RingHomIsometric.is_iso
have : map σ (𝓝 0) = 𝓝 0 := by
rw [σ_iso.isEmbedding.map_nhds_eq, σ.surjective.range_eq, nhdsWithin_univ, map_zero]
... | Mathlib/Analysis/LocallyConvex/Bounded.lean | 195 | 202 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.MeasurableSpace.Constructions
import Mathlib.Tactic.FunProp
/-!
# Measurable embeddings and equivalences
A measurable e... | ⟨fun h => by simpa only [symm_preimage_preimage] using e.symm.measurable h, fun h =>
e.measurable h⟩
@[simp]
| Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean | 317 | 320 |
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