Context stringlengths 227 76.5k | target stringlengths 0 11.6k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 16 3.69k |
|---|---|---|---|---|
/-
Copyright (c) 2022 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed
import Mathlib.RingTheory.PowerBasis
/-!
# A predicate on adjoining root... | h.coeff (algebraMap R S x) = Pi.single 0 x := by
ext i
| Mathlib/RingTheory/IsAdjoinRoot.lean | 499 | 500 |
/-
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... |
@[simp]
lemma bitwise_zero_right (n : Nat) : bitwise f n 0 = if f true false then n else 0 := by
| Mathlib/Data/Nat/Bitwise.lean | 50 | 52 |
/-
Copyright (c) 2021 Arthur Paulino. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Arthur Paulino, Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Clique
import Mathlib.Data.ENat.Lattice
import Mathlib.Data.Nat.Lattice
import Mathlib.Data.Setoid.Partition
imp... | rw [← Ne, chromaticNumber_ne_top_iff_exists] at hc
obtain ⟨n, ⟨hn⟩⟩ := hc
exact not_injective_infinite_finite _ hn.injective_of_top_hom
/-- The bicoloring of a complete bipartite graph using whether a vertex
is on the left or on the right. -/
def CompleteBipartiteGraph.bicoloring (V W : Type*) : (completeBiparti... | Mathlib/Combinatorics/SimpleGraph/Coloring.lean | 386 | 401 |
/-
Copyright (c) 2024 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.Data.Matroid.Constructions
import Mathlib.Data.Set.Notation
/-!
# Maps between matroids
This file defines maps and comaps, which move a matroid on one ty... | variable {E X I : Set α} {M : Matroid α}
/-- Given `M : Matroid α` and `X : Set α`, the restriction of `M` to `X`,
viewed as a matroid on type `X` with ground set `univ`.
| Mathlib/Data/Matroid/Map.lean | 635 | 638 |
/-
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 | 1,057 | 1,079 | |
/-
Copyright (c) 2022 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Oleksandr Manzyuk
-/
import Mathlib.CategoryTheory.Bicategory.Basic
import Mathlib.CategoryTheory.Monoidal.Mon_
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Equali... | section
variable [∀ X : C, PreservesColimitsOfSize.{0, 0} (tensorLeft X)]
variable [∀ X : C, PreservesColimitsOfSize.{0, 0} (tensorRight X)]
| Mathlib/CategoryTheory/Monoidal/Bimod.lean | 309 | 313 |
/-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Algebra.Group.AddChar
import Mathlib.Analysis.Complex.Circle
import Mathlib.MeasureTheory.Group.Integral
import Mathlib.MeasureTheory.Integral.Prod
imp... | VectorFourier.fourierIntegral 𝐞 μ L.toLinearMap₂ (fun x ↦ f x m) w :=
VectorFourier.fourierIntegral_continuousMultilinearMap_apply continuous_fourierChar hf
end Apply
variable {V : Type*} [NormedAddCommGroup V]
[InnerProductSpace ℝ V] [MeasurableSpace V] [BorelSpace V]
| Mathlib/Analysis/Fourier/FourierTransform.lean | 348 | 354 |
/-
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... | simpa only [smul_zero] using smul_lt_smul_iff_of_neg_left ha (b₁ := (0 : β))
| Mathlib/Algebra/Order/Module/Defs.lean | 900 | 901 |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Edward Ayers
-/
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.HasPullback
import Mathlib.Data.Set.BooleanAlgebra
/-!
# Theory of sieves
- For an object `X` of a ca... | theorem le_pullback_bind (S : Presieve X) (R : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S f → Sieve Y) (f : Y ⟶ X)
(h : S f) : R h ≤ (bind S R).pullback f := by
rw [← galoisConnection f]
apply pushforward_le_bind_of_mem
| Mathlib/CategoryTheory/Sites/Sieves.lean | 601 | 604 |
/-
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.RingTheory.Flat.Basic
import Mathlib.RingTheory.IsTensorProduct
import Mathlib.LinearAlgebra.TensorProduct.Tower
import Mathlib.RingTheory.Localiza... | Mathlib/RingTheory/Flat/Stability.lean | 127 | 130 | |
/-
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 | 464 | 465 | |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Yaël Dillies
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Order.Closure
/-!
# Convex hull
This file defines the convex hull of a set `s` in a module. `con... | Mathlib/Analysis/Convex/Hull.lean | 220 | 222 | |
/-
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... |
theorem tendsto_logb_atTop : Tendsto (logb b) atTop atTop :=
| Mathlib/Analysis/SpecialFunctions/Log/Base.lean | 258 | 259 |
/-
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.LinearAlgebra.Matrix.ToLin
import Mathlib.LinearAlgebra.Quotient.Basic
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.RingTheory.Nilpotent.Defs
/-!
# N... | end CommSemiring
namespace LinearMap
| Mathlib/RingTheory/Nilpotent/Lemmas.lean | 66 | 68 |
/-
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... | [K.HasHomology j] [(K.sc' i j k).HasHomology]
/-- The cycles of a homological complex in degree `j` can be computed
by specifying a choice of `c.prev j` and `c.next j`. -/
noncomputable def cyclesIsoSc' : K.cycles j ≅ (K.sc' i j k).cycles :=
ShortComplex.cyclesMapIso (K.isoSc' i j k hi hk)
| Mathlib/Algebra/Homology/ShortComplex/HomologicalComplex.lean | 767 | 772 |
/-
Copyright (c) 2024 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.Data.Matroid.Minor.Restrict
/-!
# Some constructions of matroids
This file defines some very elementary examples of matroids, namely those with at most o... |
section uniqueBaseOn
/-- The matroid on `E` whose unique base is the subset `I` of `E`.
Intended for use when `I ⊆ E`; if this not not the case, then the base is `I ∩ E`. -/
| Mathlib/Data/Matroid/Constructions.lean | 198 | 202 |
/-
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.Nat.ModEq
import Mathlib.Data.Nat.Prime.Basic
import Mathlib.NumberTheory.Zsqrtd.Basic
/-!
# Pell's equation and Matiyasevic's theorem
This file... | exact _root_.ne_of_lt (strictMono_y a1 hp) (eq_zero_of_zero_dvd h).symm
rw [← Nat.mod_add_div n m, yn_add] at h
exact
not_le_of_gt (strictMono_y _ <| Nat.mod_lt n m0)
| Mathlib/NumberTheory/PellMatiyasevic.lean | 378 | 381 |
/-
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.Data.List.Nodup
import Mathlib.Data.Set.Prod
/-!
# Finite products of types
This file defines the product of types over a list. For `l : List ι` ... | v.elim (mem_cons_of_mem _ hj) = TProd.elim v.2 hj := by
| Mathlib/Data/Prod/TProd.lean | 90 | 90 |
/-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Ordinal.Enum
import Mathlib.Tactic.TFAE
import Mathlib.Topology.Order.Monotone
/-!
### Topology of ordinals
We prov... | · obtain ⟨b, c, hbc, hbc'⟩ :=
(mem_nhds_iff_exists_Ioo_subset' ⟨0, ha.pos⟩ ⟨_, lt_succ a⟩).1
(h.mem_nhds rfl)
have hba := ha.succ_lt hbc.1
exact hba.ne (hbc' ⟨lt_succ b, hba.trans hbc.2⟩)
· rcases zero_or_succ_or_limit a with (rfl | ⟨b, rfl⟩ | ha')
· rw [← bot_eq_zero, ← Set.Iic_bot, ← Iio... | Mathlib/SetTheory/Ordinal/Topology.lean | 41 | 53 |
/-
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.Algebra.Order.Pi
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
/-!
# Simple functions
A function `f` from a measurable ... |
-- A special form of `pair_preimage`
theorem pair_preimage_singleton (f : α →ₛ β) (g : α →ₛ γ) (b : β) (c : γ) :
| Mathlib/MeasureTheory/Function/SimpleFunc.lean | 351 | 353 |
/-
Copyright (c) 2021 François Sunatori. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: François Sunatori
-/
import Mathlib.Analysis.Complex.Circle
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Basic
/-!
# Isometries o... | rw [← LinearMap.det_toMatrix basisOneI, toMatrix_rotation, Matrix.det_fin_two]
simp [← normSq_apply]
/-- The determinant of `rotation` (as a linear equiv) is equal to `1`. -/
@[simp]
theorem linearEquiv_det_rotation (a : Circle) : LinearEquiv.det (rotation a).toLinearEquiv = 1 := by
rw [← Units.eq_iff, LinearEqu... | Mathlib/Analysis/Complex/Isometry.lean | 154 | 162 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Moritz Doll
-/
import Mathlib.LinearAlgebra.Prod
/-!
# Partially defined linear maps
A `LinearPMap R E F` or `E →ₗ.[R] F` is a linear map from a submodule of `E` to... | simp only at h
| Mathlib/LinearAlgebra/LinearPMap.lean | 779 | 779 |
/-
Copyright (c) 2020 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Yury Kudryashov
-/
import Mathlib.Topology.Instances.NNReal.Lemmas
import Mathlib.Topology.Order.MonotoneContinuity
/-!
# Square root of a real numbe... | Mathlib/Data/Real/Sqrt.lean | 83 | 83 | |
/-
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... |
theorem MeasureTheory.measure_unitBall_eq_integral_div_gamma {E : Type*} {p : ℝ}
[NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E]
[BorelSpace E] (μ : Measure E) [IsAddHaarMeasure μ] (hp : 0 < p) :
μ (Metric.ball 0 1) =
.ofReal ((∫ (x : E), Real.exp (- ‖x‖ ^ p) ∂μ) ... | Mathlib/MeasureTheory/Measure/Lebesgue/VolumeOfBalls.lean | 45 | 65 |
/-
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, Heather Macbeth
-/
import Mathlib.MeasureTheory.Group.Action
import Mathlib.MeasureTheory.Group.Pointwise
import Mathlib.MeasureTheory.Integral.Lebe... | Mathlib/MeasureTheory/Group/FundamentalDomain.lean | 955 | 978 | |
/-
Copyright (c) 2024 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.Analysis.Normed.Group.Basic
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Instances.ENNReal.Lemmas
/-!
# Sums over residue classes
We consider... | residue class of `k` mod `m` if and only if `f (m*n + k)` is summable. -/
lemma summable_indicator_mod_iff_summable {R : Type*} [AddCommGroup R] [TopologicalSpace R]
[IsTopologicalAddGroup R] (m : ℕ) [hm : NeZero m] (k : ℕ) (f : ℕ → R) :
Summable ({n : ℕ | (n : ZMod m) = k}.indicator f) ↔ Summable fun n ↦ f (m ... | Mathlib/Analysis/SumOverResidueClass.lean | 29 | 46 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Data.Countable.Small
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Powerset
import Mathlib.Dat... | Mathlib/SetTheory/Cardinal/Basic.lean | 1,623 | 1,624 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Notation.Pi
import Mathlib.Data.Set.Lattice
import Mathlib.Order.Filter.Defs
/-!
# Theory of... | Mathlib/Order/Filter/Basic.lean | 1,358 | 1,359 | |
/-
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.DirectSum.Basic
import Mathlib.LinearAlgebra.DFinsupp
import Mathlib.LinearAlgebra.Basis.Defs
/-!
# Direct sum of modules
The first part of the file pr... | (sigmaFinsuppLequivDFinsupp R).symm
@[simp]
theorem IsInternal.collectedBasis_coe (h : IsInternal A) {α : ι → Type*}
(v : ∀ i, Basis (α i) R (A i)) : ⇑(h.collectedBasis v) = fun a : Σ i, α i ↦ ↑(v a.1 a.2) := by
funext a
| Mathlib/Algebra/DirectSum/Module.lean | 440 | 445 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Robin Carlier
-/
import Mathlib.CategoryTheory.Equivalence
/-!
# Binary disjoint unions of categories
We define the category instance on `C ⊕ D` when `C` and `D` are cate... | | inr X => ULift.up (𝟙 X)
comp {X Y Z} f g :=
| Mathlib/CategoryTheory/Sums/Basic.lean | 62 | 63 |
/-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.SetTheory.Cardinal.Finite
/-!
# Cardinality of finite types
The cardinality of a finite type `α` is given by `Nat.card α`. This function has
the "junk val... | exact Set.card_lt_card hsub
theorem eq_of_subset_of_card_le (ht : t.Finite) (hsub : s ⊆ t) (hcard : Nat.card t ≤ Nat.card s) :
s = t :=
(eq_or_ssubset_of_subset hsub).elim id fun h ↦ absurd hcard <| not_le_of_lt <| ht.card_lt_card h
| Mathlib/Data/Finite/Card.lean | 195 | 199 |
/-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | @[simp]
theorem involute_v (m : M) : involute (v Q m) = -v Q m :=
involute_ι _
| Mathlib/LinearAlgebra/CliffordAlgebra/EvenEquiv.lean | 89 | 91 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Data.Countable.Small
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Powerset
import Mathlib.Dat... | Mathlib/SetTheory/Cardinal/Basic.lean | 1,494 | 1,501 | |
/-
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
... | distances. -/
theorem isometry_iff_nndist_eq [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β} :
Isometry f ↔ ∀ x y, nndist (f x) (f y) = nndist x y := by
| Mathlib/Topology/MetricSpace/Isometry.lean | 40 | 42 |
/-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.Algebra.Order.Group.OrderIso
import Mathlib.SetTheory.Game.Ordinal
import Mathlib.SetTheory.Ordinal.NaturalOps
/-!
# Birthdays... | noncomputable def birthday (x : Game.{u}) : Ordinal.{u} :=
| Mathlib/SetTheory/Game/Birthday.lean | 177 | 177 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Michael Howes, Antoine Chambert-Loir
-/
import Mathlib.Data.Finite.Card
import Mathlib.GroupTheory.Commutator.Basic
import Mathlib.GroupTheory.Coset.Basic
import Mathlib.GroupThe... |
theorem rank_commutator_le_card [Finite (commutatorSet G)] :
Group.rank (commutator G) ≤ Nat.card (commutatorSet G) := by
rw [Subgroup.rank_congr (commutator_eq_closure G)]
| Mathlib/GroupTheory/Abelianization.lean | 65 | 68 |
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.MetricSpace.HausdorffDistance
/-!
# Topological study of spaces `Π (n : ℕ), E n`
When `E n` are topological spaces, the space `Π (n : ... |
open List
/-- In the case where `E` has constant value `α`,
the cylinder `cylinder x n` can be identified with the element of `List α`
consisting of the first `n` entries of `x`. See `cylinder_eq_res`.
We call this list `res x n`, the restriction of `x` to `n`. -/
def res (x : ℕ → α) : ℕ → List α
| 0 => nil
| Nat... | Mathlib/Topology/MetricSpace/PiNat.lean | 175 | 186 |
/-
Copyright (c) 2021 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.GroupTheory.Index
/-!
# Complements
In this file we define the complement of a subgroup.
## Main definitions
- `Subgroup.IsComplement S T` where ... | obtain ⟨S, cmem, -⟩ := H''.exists_isComplement_left 1
refine ⟨H.subtype '' S, ?_, ?_⟩
· have : H.subtype '' (S * H'') = H.subtype '' S * H''.map H.subtype := image_mul H.subtype
rw [← this, cmem.mul_eq]
simp [Set.ext_iff]
· rw [← cmem.card_mul_card]
refine congr_arg₂ (· * ·) ?_ ?_ <;>
exact Na... | Mathlib/GroupTheory/Complement.lean | 415 | 429 |
/-
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.Algebra.Tower
import Mathlib.Algebra.Field.IsField
import Mathlib.Algebra.GroupWithZero.N... | Mathlib/RingTheory/Localization/Basic.lean | 1,137 | 1,137 | |
/-
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.Ordmap.Invariants
/-!
# Verification of `Ordnode`
This file uses the invariants defined in `Mathlib.Data.Ordmap.Invariants` to construct `Ordset... |
theorem eraseMin.valid : ∀ {t}, @Valid α _ t → Valid (eraseMin t)
| Mathlib/Data/Ordmap/Ordset.lean | 394 | 395 |
/-
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... | smul_left_cancel_iff, ← Functor.map_comp_assoc, ← Functor.map_comp_assoc, f₂.comm]
lemma comm₁ (j : J) :
(mapBifunctorMap f₁ f₂ F c).f j =
(mapBifunctor K₁ K₂ F c).d j (c.next j) ≫
mapBifunctorMapHomotopy.hom₁ h₁ f₂ F c (c.next j) j +
mapBifunctorMapHomotopy.hom₁ h₁ f₂ F c j (c.prev j) ≫
... | Mathlib/Algebra/Homology/BifunctorHomotopy.lean | 88 | 141 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Data.Set.Countable
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Tactic.FunProp.Attr
import Mathlib.Tactic.Mea... | MeasurableSet (s₁ ⇨ s₂) := by rw [himp_eq]; exact h₂.union h₁.compl
@[simp, measurability]
protected theorem MeasurableSet.symmDiff {s₁ s₂ : Set α} (h₁ : MeasurableSet s₁)
| Mathlib/MeasureTheory/MeasurableSpace/Defs.lean | 177 | 180 |
/-
Copyright (c) 2020 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.Basic
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Normed.Operator.BoundedLinear... | Mathlib/Analysis/Calculus/FDeriv/Measurable.lean | 972 | 976 | |
/-
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.BigOperators.Expect
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Field.Canonical
import Mathlib.Algebra.O... | Mathlib/Data/NNReal/Basic.lean | 215 | 215 | |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Analytic.Within
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries
/-!
# Higher d... | Mathlib/Analysis/Calculus/ContDiff/Defs.lean | 1,309 | 1,312 | |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Module.ULift
import Mathlib.RingTheory.TensorProduct.Basic
import Mathlib.Tactic.Ring
/-!
# The characteristic predicate of tensor product
## Main ... | · dsimp at *; rw [smul_add, map_add, map_add, smul_add, hx, hy]
simp_rw [DFunLike.ext_iff, LinearMap.comp_apply, LinearMap.restrictScalars_apply] at hg
let fe : S ⊗[R] M ≃ₗ[S] N :=
LinearEquiv.ofLinear f'' (ULift.moduleEquiv.toLinearMap.comp g) ?_ ?_
· exact fe.bijective
· rw [← LinearMap.cancel_left (U... | Mathlib/RingTheory/IsTensorProduct.lean | 285 | 294 |
/-
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.SpecialFunctions.ExpDeriv
/-!
# Grönwall's inequality
The main technical result of this file is the Grönwall-like inequality
`norm_le_gron... | `∀ x ∈ [a, b), liminf_{z→x+0} (f z - f x)/(z - x) ≤ K * (f x) + ε`, then `f x`
is bounded by `gronwallBound δ K ε (x - a)` on `[a, b]`.
| Mathlib/Analysis/ODE/Gronwall.lean | 92 | 93 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Sébastien Gouëzel,
Rémy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Complex.Log
/-! # Power funct... | theorem cpow_one (x : ℂ) : x ^ (1 : ℂ) = x :=
if hx : x = 0 then by simp [hx, cpow_def]
| Mathlib/Analysis/SpecialFunctions/Pow/Complex.lean | 75 | 76 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing
/-!
# Stalks
For a presheaf `F` on a topol... | ext; delta stalkFunctor; simpa [stalkSpecializes] using by rfl
-- See https://github.com/leanprover-community/batteries/issues/365 for the simpNF issue.
-- It seems the side condition `h` is not applied by `simpNF`.
@[reassoc, elementwise, simp, nolint simpNF]
theorem stalkSpecializes_stalkPushforward (f : X ⟶ Y) (F... | Mathlib/Topology/Sheaves/Stalks.lean | 363 | 370 |
/-
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.Order.AddGroupWithTop
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.Order.Sub.WithTop
import Mathlib.Data.ENat.Defs
import... |
@[simp]
| Mathlib/Data/ENat/Basic.lean | 209 | 210 |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.WSeq.Basic
import Mathlib.Data.WSeq.Defs
import Mathlib.Data.WSeq.Productive
import Mathlib.Data.WSeq.Relation
deprecated_module (since :=... | Mathlib/Data/Seq/WSeq.lean | 893 | 896 | |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.FinMeasAdditive
/-!
# Extension of a linear function from indicators to L1
Give... | Mathlib/MeasureTheory/Integral/SetToL1.lean | 1,128 | 1,132 | |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Patrick Massot, Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic
import Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalcul... | Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | 487 | 488 | |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle
/-!
# Oriented angles in right-angled triangles.
T... | rcases lt_trichotomy r 0 with (hr | rfl | hr)
· have ha : o.oangle x (r • o.rotation (π / 2 : ℝ) x) = -(π / 2 : ℝ) := by
rw [o.oangle_smul_right_of_neg _ _ hr, o.oangle_neg_right h, o.oangle_rotation_self_right h, ←
sub_eq_zero, add_comm, sub_neg_eq_add, ← Real.Angle.coe_add, ← Real.Angle.coe_add,
| Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 448 | 451 |
/-
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... |
/--
If `P` is a sheaf for the sieve `S` on `X`, then if two natural transformations from `yoneda.obj X`
to `P` agree when restricted to the subfunctor given by `S`, they are equal.
-/
theorem IsSheafFor.hom_ext {P : Cᵒᵖ ⥤ Type v₁} (h : IsSheafFor P (S : Presieve X))
(t₁ t₂ : yoneda.obj X ⟶ P) (ht : S.functorInclus... | Mathlib/CategoryTheory/Sites/IsSheafFor.lean | 516 | 522 |
/-
Copyright (c) 2021 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Order.Lattice
import Mathlib.Data.List.Sort
import Mathlib.Logic.Equiv.Fin.Basic
import Mathlib.Logic.Equiv.Functor
import Mathlib.Data.Fintype.Pigeonhole
... | ⟨Equiv.refl _, fun _ => (s.step _).iso_refl⟩
@[symm]
theorem symm {s₁ s₂ : CompositionSeries X} (h : Equivalent s₁ s₂) : Equivalent s₂ s₁ :=
⟨h.choose.symm, fun i => iso_symm (by simpa using h.choose_spec (h.choose.symm i))⟩
@[trans]
theorem trans {s₁ s₂ s₃ : CompositionSeries X} (h₁ : Equivalent s₁ s₂) (h₂ : Equ... | Mathlib/Order/JordanHolder.lean | 257 | 265 |
/-
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... | /-- **Pigeonhole principle** for two finsets inside an ambient finset. -/
theorem inter_nonempty_of_card_lt_card_add_card (hts : t ⊆ s) (hus : u ⊆ s)
| Mathlib/Data/Finset/Card.lean | 544 | 545 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yaël Dillies
-/
import Mathlib.Algebra.Module.BigOperators
import Mathlib.GroupTheory.Perm.Basic
import Mathlib.GroupTheory.Perm.Finite
import Mathlib.GroupTheory.Perm.Lis... | not_exists, not_and, forall_exists_index, and_imp, Prod.forall, Prod.mk_inj]
rintro _ _ _ - rfl rfl a ha rfl h
rw [hf.pow_apply_eq_pow_apply ha] at h
rw [mem_coe, mem_range] at hm hn
exact hmn.symm (h.eq_of_lt_of_lt hn hm)
/-- We can partition the square `s ×ˢ s` into shifted diagonals as such:
`... | Mathlib/GroupTheory/Perm/Cycle/Basic.lean | 925 | 935 |
/-
Copyright (c) 2020 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Algebra.Algebra.Field
import Mathlib.Algebra.BigOperators.Balance
import Mathlib.Algebra.Order.BigOperators.Expect
import Mathlib.Algebra.Order.Star.... | theorem lt_iff_re_im : z < w ↔ re z < re w ∧ im z = im w := by
simp_rw [lt_iff_le_and_ne, @RCLike.le_iff_re_im K]
| Mathlib/Analysis/RCLike/Basic.lean | 753 | 754 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.Algebra.Polynomial.Lifts
import Mathlib.Data.List.Prime
import Mathlib.RingTheory.Polynomial.Tower
/-!
# S... | (∀ j ∈ t, (s j).Splits i) → (∏ x ∈ t, s x).Splits i := by
classical
| Mathlib/Algebra/Polynomial/Splits.lean | 124 | 125 |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.AlgebraicGeometry.Cover.Open
import Mathlib.AlgebraicGeometry.GammaSpecAdjunction
import Mathlib.AlgebraicGeometry.Restrict
import Mathlib.CategoryTheory.Lim... | conv_rhs =>
rw [fromSpec, ← X.homOfLE_ι (V := U) f.unop.le, isoSpec_inv, Category.assoc,
← Scheme.isoSpec_inv_naturality_assoc,
| Mathlib/AlgebraicGeometry/AffineScheme.lean | 397 | 399 |
/-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
/-!
# Recursive computation rules for the Clifford algebra
This file provides API for a special case `CliffordAlg... | exact
{ toFun := fun m => (l m).prod (f m)
map_add' := fun v₂ v₂ =>
LinearMap.ext fun x =>
Prod.ext (LinearMap.congr_fun (l.map_add _ _) x) (LinearMap.congr_fun (f.map_add _ _) x)
map_smul' := fun c v =>
LinearMap.ext fun x =>
Prod.ext (LinearMap.congr_fun (l.map_sm... | Mathlib/LinearAlgebra/CliffordAlgebra/Fold.lean | 161 | 168 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Group.Submonoid.Operations
import Mathlib.Algebra.MonoidAlgebra.Defs
import Mathlib.Algebra.Order.Mon... | theorem C_mul_X_pow_eq_monomial : ∀ {n : ℕ}, C a * X ^ n = monomial n a
| Mathlib/Algebra/Polynomial/Basic.lean | 668 | 668 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Sébastien Gouëzel,
Rémy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
/-!
# Power function... | `⊤ ^ x = 1 / 0 ^ x`). -/
noncomputable def rpow : ℝ≥0∞ → ℝ → ℝ≥0∞
| some x, y => if x = 0 ∧ y < 0 then ⊤ else (x ^ y : ℝ≥0)
| none, y => if 0 < y then ⊤ else if y = 0 then 1 else 0
| Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | 437 | 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.Analysis.Convex.Between
import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathli... | Mathlib/MeasureTheory/Measure/Hausdorff.lean | 1,136 | 1,142 | |
/-
Copyright (c) 2022 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.NumberTheory.Cyclotomic.Discriminant
import Mathlib.RingTheory.Polynomial.Eisenstein.IsIntegral
import Mathlib.RingTheory.Ideal.Norm.AbsNorm
import M... | ext
exact hζ'
rw [hζ', hp2]
exact ⟨-1, by ring⟩
suffices (hζ.toInteger - 1) ∣ (p : ℤ) by simpa
have := IsCyclotomicExtension.numberField {p ^ (k + 1)} ℚ K
have H := hζ.norm_toInteger_pow_sub_one_of_prime_pow_ne_two (zero_le _) htwo
rw [pow_zero, pow_one] at H
rw [← Ideal.norm_dvd_iff, H]
... | Mathlib/NumberTheory/Cyclotomic/Rat.lean | 522 | 544 |
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Kevin Buzzard, Jujian Zhang, Fangming Li
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.DirectSum.Algebra... | [AddSubgroupClass σ R] (A : ι → σ) [SetLike.GradedOne A] (z : ℤ) : (z : R) ∈ A 0 := by
cases z
· rw [Int.ofNat_eq_coe, Int.cast_natCast]
exact SetLike.natCast_mem_graded _ _
· rw [Int.cast_negSucc]
exact neg_mem (SetLike.natCast_mem_graded _ _)
| Mathlib/Algebra/DirectSum/Internal.lean | 72 | 78 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Data.Countable.Small
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Powerset
import Mathlib.Dat... | Mathlib/SetTheory/Cardinal/Basic.lean | 1,479 | 1,481 | |
/-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Kernel.Defs
/-!
# Basic kernels
This file contains basic results about kernels in general and definitions of some particular
kernels.
## Mai... |
lemma lintegral_id [MeasurableSingletonClass α] {f : α → ℝ≥0∞} (a : α) :
| Mathlib/Probability/Kernel/Basic.lean | 117 | 118 |
/-
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.... | ⟨fun h => by
rw [← h]
exact ofReal_ne_top, ofReal_toReal⟩
| Mathlib/Data/ENNReal/Basic.lean | 344 | 347 |
/-
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.PFunctor.Univariate.M
/-!
# Quotients of Polynomial Functors
We assume the following:
* `P`: a polynomial functor
* `W`: its W-type
* `M`: its M... | ⟨Wequiv, @Wequiv.refl _ _, @Wequiv.symm _ _, @Wequiv.trans _ _⟩
attribute [local instance] Wsetoid
/-- inductive type defined as initial algebra of a Quotient of Polynomial Functor -/
def Fix (F : Type u → Type u) [q : QPF F] :=
| Mathlib/Data/QPF/Univariate/Basic.lean | 216 | 221 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Fin.Tuple.Basic
/-!
# Lists from functions
Theorems and lemmas for dealing with `List.ofFn`, which converts a function on `Fin n` to a list
of l... | theorem map_ofFn {β : Type*} {n : ℕ} (f : Fin n → α) (g : α → β) :
map g (ofFn f) = ofFn (g ∘ f) :=
| Mathlib/Data/List/OfFn.lean | 39 | 40 |
/-
Copyright (c) 2023 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll, Ralf Stephan
-/
import Mathlib.Data.Nat.Factorization.Defs
import Mathlib.Data.Nat.Squarefree
/-!
# Smooth numbers
For `s : Finset ℕ` we define the set `Nat.factoredNum... |
@[simp]
lemma primesBelow_zero : primesBelow 0 = ∅ := by
| Mathlib/NumberTheory/SmoothNumbers.lean | 38 | 40 |
/-
Copyright (c) 2023 Felix Weilacher. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Felix Weilacher, Yury Kudryashov, Rémy Degenne
-/
import Mathlib.MeasureTheory.MeasurableSpace.Embedding
import Mathlib.Data.Set.MemPartition
import Mathlib.Order.Filter.CountableSepa... | {s : Set α} (hs : s ∈ countableGeneratingSet α) :
MeasurableSet s := by
rw [← generateFrom_countableGeneratingSet (α := α)]
exact measurableSet_generateFrom hs
| Mathlib/MeasureTheory/MeasurableSpace/CountablyGenerated.lean | 76 | 80 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.MeasurableSpace.MeasurablyGenerated
import Mathlib.MeasureTheory.Measure.NullMeasurable
import Mathlib.Order.Interval.Set... | Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 2,054 | 2,069 | |
/-
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, Mitchell Lee
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.Algebra.Group.Submonoid.Basic
import M... | variable [TopologicalSpace α] [Monoid α] [ContinuousMul α]
/-- If multiplication on a monoid is continuous, then multiplication on the units of the monoid,
with respect to the induced topology, is continuous.
| Mathlib/Topology/Algebra/Monoid.lean | 788 | 791 |
/-
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 | 953 | 956 | |
/-
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... | theorem rpow_logb (hx : 0 < x) : b ^ logb b x = x := by
rw [rpow_logb_eq_abs b_pos b_ne_one hx.ne']
exact abs_of_pos hx
theorem rpow_logb_of_neg (hx : x < 0) : b ^ logb b x = -x := by
rw [rpow_logb_eq_abs b_pos b_ne_one (ne_of_lt hx)]
exact abs_of_neg hx
| Mathlib/Analysis/SpecialFunctions/Log/Base.lean | 137 | 143 |
/-
Copyright (c) 2021 Stuart Presnell. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stuart Presnell
-/
import Mathlib.Data.Nat.PrimeFin
import Mathlib.Data.Nat.Factorization.Defs
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Order.Interval.Finset.Nat
import... | rcases eq_or_ne n 0 with (rfl | hn)
· simp
| Mathlib/Data/Nat/Factorization/Basic.lean | 353 | 354 |
/-
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.Data.ENat.Lattice
import Mathlib.Order.OrderIsoNat
import Mathlib.Tactic.TFAE
/-!
# Maximal length of chains
This file contains lemmas to work with the ma... | and_assoc]
@[simp]
theorem singleton_mem_subchain_iff : [a] ∈ s.subchain ↔ a ∈ s := by simp [cons_mem_subchain_iff]
| Mathlib/Order/Height.lean | 70 | 73 |
/-
Copyright (c) 2021 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.Topology.Algebra.FilterBasis
import Mathlib.Topology.Algebra.IsUniformGroup.Defs
/-!
# Uniform properties of neighborhood bases in topological algebra... | F.NeBot ∧ ∀ U ∈ B, ∃ M ∈ F, ∀ᵉ (x ∈ M) (y ∈ M), y - x ∈ U := by
letI := B.uniformSpace
haveI := B.isUniformAddGroup
suffices F ×ˢ F ≤ uniformity G ↔ ∀ U ∈ B, ∃ M ∈ F, ∀ᵉ (x ∈ M) (y ∈ M), y - x ∈ U by
constructor <;> rintro ⟨h', h⟩ <;> refine ⟨h', ?_⟩ <;> [rwa [← this]; rwa [this]]
rw [uniformity_eq_co... | Mathlib/Topology/Algebra/UniformFilterBasis.lean | 42 | 51 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno
import Mathlib.Analysis.Calculus.FDeriv.Ad... | Mathlib/Analysis/Calculus/ContDiff/Basic.lean | 1,722 | 1,725 | |
/-
Copyright (c) 2022 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed
import Mathlib.RingTheory.PowerBasis
/-!
# A predicate on adjoining root... |
theorem map_repr (h : IsAdjoinRoot S f) (x : S) : h.map (h.repr x) = x :=
(h.map_surjective x).choose_spec
/-- `repr` preserves zero, up to multiples of `f` -/
| Mathlib/RingTheory/IsAdjoinRoot.lean | 150 | 154 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Yuyang Zhao
-/
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Polynomial.AlgebraMap
/-!
# Algebra towers for polynomial
This file proves some basic results about t... |
@[simp]
theorem aeval_algebraMap_eq_zero_iff [NoZeroSMulDivisors A B] [Nontrivial B] (x : A) (p : R[X]) :
| Mathlib/RingTheory/Polynomial/Tower.lean | 54 | 56 |
/-
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... |
end Measure
end MeasureTheory
/-!
### Hausdorff measure, Hausdorff dimension, and Hölder or Lipschitz continuous maps
| Mathlib/MeasureTheory/Measure/Hausdorff.lean | 652 | 658 |
/-
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... | Mathlib/Topology/Compactness/Compact.lean | 1,115 | 1,117 | |
/-
Copyright (c) 2019 Johannes Hölzl, Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Zhouhang Zhou
-/
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable
import M... | @[simp]
theorem pair_mk_mk (f : α → β) (hf) (g : α → γ) (hg) :
(mk f hf : α →ₘ[μ] β).pair (mk g hg) = mk (fun x => (f x, g x)) (hf.prodMk hg) :=
rfl
| Mathlib/MeasureTheory/Function/AEEqFun.lean | 326 | 329 |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Ring.Rat
import Mathlib.Algebra.Ring.Int.Parity
import Mathlib.Data.PNat.Defs... | @[simp]
theorem inv_natCast_num (a : ℕ) : (a : ℚ)⁻¹.num = Int.sign a :=
inv_intCast_num a
@[simp]
| Mathlib/Data/Rat/Lemmas.lean | 260 | 264 |
/-
Copyright (c) 2019 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Yury Kudryashov, Sébastien Gouëzel, Chris Hughes
-/
import Mathlib.Data.Fin.Rev
import Mathlib.Data.Nat.Find
/-!
# Operation on tuples
We interpret maps `∀ i : Fi... | else
if hlt : j < i then @Eq.recOn _ _ (fun x _ ↦ α x) _ (succAbove_castPred_of_lt _ _ hlt) (p _)
else @Eq.recOn _ _ (fun x _ ↦ α x) _ (succAbove_pred_of_lt _ _ <|
(Fin.lt_or_lt_of_ne hj).resolve_left hlt) (p _)
-- This is a duplicate of `Fin.exists_fin_succ` in Core. We should upstream the name change.
... | Mathlib/Data/Fin/Tuple/Basic.lean | 740 | 752 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kenny Lau
-/
import Mathlib.Data.List.Forall2
/-!
# Lists with no duplicates
`List.Nodup` is defined in `Data/List/Basic`. In this file we prove various properties of... | have : replicate 2 a <+ l ↔ 1 < count a l := (le_count_iff_replicate_sublist ..).symm
(not_congr this).trans not_lt
theorem nodup_iff_count_eq_one [DecidableEq α] : Nodup l ↔ ∀ a ∈ l, count a l = 1 :=
nodup_iff_count_le_one.trans <| forall_congr' fun _ =>
⟨fun H h => H.antisymm (count_pos_iff.mpr h),... | Mathlib/Data/List/Nodup.lean | 151 | 162 |
/-
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.Nontrivial.Basic
import Mathlib.Order.TypeTags
import Mathlib.Data.Option.NAry
import Mathlib.Tactic.Contrapose
import Mathlib.Tactic.Lift
import... |
@[deprecated (since := "2025-03-19")] alias forall_lt_iff_eq_bot := eq_bot_iff_forall_lt
@[deprecated (since := "2025-03-19")] alias forall_le_iff_eq_bot := eq_bot_iff_forall_le
| Mathlib/Order/WithBot.lean | 337 | 339 |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Edward Ayers
-/
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.HasPullback
import Mathlib.Data.Set.BooleanAlgebra
/-!
# Theory of sieves
- For an object `X` of a ca... |
theorem ofArrows_pUnit : (ofArrows _ fun _ : PUnit => f) = singleton f := by
funext Y
ext g
constructor
· rintro ⟨_⟩
apply singleton.mk
· rintro ⟨_⟩
exact ofArrows.mk PUnit.unit
theorem ofArrows_pullback [HasPullbacks C] {ι : Type*} (Z : ι → C) (g : ∀ i : ι, Z i ⟶ X) :
| Mathlib/CategoryTheory/Sites/Sieves.lean | 151 | 161 |
/-
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.Group.Action.Opposite
import Mathlib.Algebra.Group.Action.Units
import Mathlib.Algebra.Group.Invertible.Defs
import Mathlib.Algebra.GroupWithZero.U... | Mathlib/Algebra/Star/Basic.lean | 581 | 583 | |
/-
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.Data.Fintype.Parity
import Mathlib.NumberTheory.LegendreSymbol.ZModChar
import Mathlib.FieldTheory.Finite.Basic
/-!
# Quadratic characters of finite fie... | theorem quadraticCharFun_one : quadraticCharFun F 1 = 1 := by
simp only [quadraticCharFun, one_ne_zero, IsSquare.one, if_true, if_false]
| Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/Basic.lean | 75 | 76 |
/-
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... | ∫⋯∫⁻_s, f ^ (1 - (s.card - 1 : ℝ) * p) * ∏ i ∈ s, (∫⋯∫⁻_{i}, f ∂μ) ^ p ∂μ
variable {p : ℝ}
@[simp] lemma T_univ [Fintype ι] [∀ i, SigmaFinite (μ i)] (f : (∀ i, A i) → ℝ≥0∞) (x : ∀ i, A i) :
| Mathlib/Analysis/FunctionalSpaces/SobolevInequality.lean | 90 | 94 |
/-
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... | apply not_le_of_lt hN'
change _ ≤ abv (_ * _)
| Mathlib/Algebra/Order/CauSeq/Basic.lean | 486 | 487 |
/-
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, Moritz Doll
-/
import Mathlib.Algebra.GroupWithZero.Action.Opposite
import Mathlib.LinearAlgebra.Finsupp.VectorSpace
import Mathlib.LinearAlgebra.Matrix.Basis
im... | · intros
simp only [zero_smul, smul_zero]
· intros
simp only [zero_smul, Finsupp.sum_zero]
theorem LinearMap.toMatrix₂_comp (B : M₁ →ₗ[R] M₂ →ₗ[R] R) (f : M₁' →ₗ[R] M₁) :
LinearMap.toMatrix₂ b₁' b₂ (B.comp f) =
(toMatrix b₁' b₁ f)ᵀ * LinearMap.toMatrix₂ b₁ b₂ B := by
rw [← LinearMap.compl₂_... | Mathlib/LinearAlgebra/Matrix/SesquilinearForm.lean | 420 | 440 |
/-
Copyright (c) 2022 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.Analysis.SpecialFunctions.Pow.Real
/-!
# Results on discretized exponentials
We state several auxi... | (hlim : ∀ a : ℝ, 1 < a → ∃ c : ℕ → ℕ, (∀ᶠ n in atTop, (c (n + 1) : ℝ) ≤ a * c n) ∧
Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / c n) atTop (𝓝 l)) :
Tendsto (fun n => u n / n) atTop (𝓝 l) := by
/- To check the result up to some `ε > 0`, we use a sequence `c` for which the ratio
`c (N+1) / c ... | Mathlib/Analysis/SpecificLimits/FloorPow.lean | 28 | 182 |
/-
Copyright (c) 2019 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard
-/
import Mathlib.Data.EReal.Basic
deprecated_module (since := "2025-04-13")
| Mathlib/Data/Real/EReal.lean | 1,071 | 1,075 | |
/-
Copyright (c) 2021 David Wärn,. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn, Kim Morrison
-/
import Mathlib.Combinatorics.Quiver.Prefunctor
import Mathlib.Logic.Lemmas
import Batteries.Data.List.Basic
/-!
# Paths in quivers
Given a quiver `V`, we def... | /-- The length of a path is the number of arrows it uses. -/
def length {a : V} : ∀ {b : V}, Path a b → ℕ
| Mathlib/Combinatorics/Quiver/Path.lean | 54 | 55 |
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