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) 2021 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Data.Fintype.Shrink
import Mathlib.Data.Fintype.Sum
import Mathlib.Data.Finite.Prod
import Mathlib.Algebra.BigOperators.Grou... | -- Note: This is not made a `FunLike` instance to avoid having two syntactically different coercions
lemma coe_injective [Nontrivial α] : Injective ((⇑) : Line α ι → α → ι → α) := by
rintro l m hlm
ext i a
| Mathlib/Combinatorics/HalesJewett.lean | 190 | 193 |
/-
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.BoxIntegral.Partition.Filter
import Mathlib.Analysis.BoxIntegral.Partition.Measure
import Mathlib.Analysis.Oscillation
import Mathlib.Data.B... | have H0 : 0 < (#π₀.boxes + 1 : ℝ) := Nat.cast_add_one_pos _
have δ'0 : 0 < δ' := div_pos δ0 H0
set C := max π₀.distortion π₀.compl.distortion
/- Next we choose a tagged partition of each `J ∈ π₀` such that the integral sum of `f` over this
partition is `δ'`-close to the integral of `f` over `J`. -/
have :... | Mathlib/Analysis/BoxIntegral/Basic.lean | 517 | 522 |
/-
Copyright (c) 2022 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Order.ToIntervalMod
import Mathlib.Algebra.Ring.AddAut
import Mathlib.Data.Nat.Totient
import Mathlib.GroupTheory.Divisible
import Mathlib.Topology.C... | @[continuity, nolint unusedArguments]
protected theorem continuous_mk' [TopologicalSpace 𝕜] :
| Mathlib/Topology/Instances/AddCircle.lean | 152 | 153 |
/-
Copyright (c) 2020 Jalex Stark. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jalex Stark, Kim Morrison, Eric Wieser, Oliver Nash, Wen Yang
-/
import Mathlib.Data.Matrix.Basic
/-!
# Matrices with a single non-zero element.
This file provides `Matrix.stdBasisMatri... | theorem smul_stdBasisMatrix [SMulZeroClass R α] (r : R) (i : m) (j : n) (a : α) :
r • stdBasisMatrix i j a = stdBasisMatrix i j (r • a) := by
unfold stdBasisMatrix
ext
| Mathlib/Data/Matrix/Basis.lean | 45 | 48 |
/-
Copyright (c) 2020 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Ira Fesefeldt
-/
import Mathlib.Control.Monad.Basic
import Mathlib.Dynamics.FixedPoints.Basic
import Mathlib.Order.CompleteLattice.Basic
import Mathlib.Order.Iterate
import M... |
open OmegaCompletePartialOrder
| Mathlib/Order/OmegaCompletePartialOrder.lean | 155 | 156 |
/-
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.BoxIntegral.Partition.Filter
import Mathlib.Analysis.BoxIntegral.Partition.Measure
import Mathlib.Analysis.Oscillation
import Mathlib.Data.B... | countable set `s` with the following property.
For every `c : ℝ≥0`, a point `x ∈ I.Icc`, and a positive `ε` there exists `δ > 0` such that for any
box `J ≤ I` such that
- `x ∈ J.Icc ⊆ Metric.closedBall x δ`;
- if `l.bDistortion` (i.e., `l = ⊥`), then the distortion of `J` is less than or equal to `c`,
| Mathlib/Analysis/BoxIntegral/Basic.lean | 839 | 845 |
/-
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... | ∀ a l, foldl f a l = foldr f a l
| _, nil => rfl
| Mathlib/Data/List/Basic.lean | 911 | 912 |
/-
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.Single
import Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex
/-!
# The homology of single complexes
The main definition in this file ... | assoc, iCyclesIso_inv_hom_id, comp_id, single_map_f_self]
@[reassoc (attr := simp)]
lemma singleObjCyclesSelfIso_inv_naturality :
(singleObjCyclesSelfIso c j A).inv ≫ cyclesMap ((single C c j).map f) j =
f ≫ (singleObjCyclesSelfIso c j B).inv := by
rw [← cancel_epi (singleObjCyclesSelfIso c j A).hom, I... | Mathlib/Algebra/Homology/SingleHomology.lean | 136 | 142 |
/-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Algebra.MvPolynomial.Equiv
import Mathlib.Algebra.P... | imp_true_iff]
· simp +contextual only [Nat.lt_add_one_iff, Nat.choose_eq_zero_of_lt,
| Mathlib/RingTheory/Polynomial/Basic.lean | 265 | 266 |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.RingTheory.Nilpotent.Basic
import Mathlib.RingTheory.UniqueFactorizationDomain.GCDMonoid
import Mathlib.RingTheory.UniqueFactorizationDomain.Multiplici... | /-- `x * y` is square-free iff `x` and `y` have no common factors and are themselves square-free. -/
theorem squarefree_mul_iff : Squarefree (x * y) ↔ IsRelPrime x y ∧ Squarefree x ∧ Squarefree y :=
⟨fun h ↦ ⟨IsRelPrime.of_squarefree_mul h, h.of_mul_left, h.of_mul_right⟩,
fun ⟨hp, sqx, sqy⟩ _ dvd ↦ hp (sqy.dvd_of... | Mathlib/Algebra/Squarefree/Basic.lean | 210 | 220 |
/-
Copyright (c) 2023 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.MvPolynomial.Eval
import Mathlib.Analysis.Analytic.Constructions
import Mathlib.Topology.Algebra.Module.Fini... | theorem AnalyticWithinAt.aeval_polynomial (hf : AnalyticWithinAt 𝕜 f s z) (p : A[X]) :
AnalyticWithinAt 𝕜 (fun x ↦ aeval (f x) p) s z := by
refine p.induction_on (fun k ↦ ?_) (fun p q hp hq ↦ ?_) fun p i hp ↦ ?_
· simp_rw [aeval_C]; apply analyticWithinAt_const
· simp_rw [aeval_add]; exact hp.add hq
· con... | Mathlib/Analysis/Analytic/Polynomial.lean | 26 | 32 |
/-
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... | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 588 | 589 | |
/-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Martin Dvorak
-/
import Mathlib.Algebra.Order.Kleene
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.Data.Set.Lattice
import Mathlib.Tactic.DeriveFintype
/-!
# Languages... | exact le_trans (le_mul_congr le_rfl h) ih,
mul_kstar_le_self := fun l m h ↦ by
| Mathlib/Computability/Language.lean | 280 | 281 |
/-
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, Mario Carneiro
-/
import Mathlib.Data.Countable.Defs
import Mathlib.Data.Nat.Factors
import Mathlib.Data.Nat.Prime.Infinite
import Mathlib.Data... | · contrapose!
rintro hn
obtain ⟨p, hp, hpn⟩ := exists_prime_and_dvd hn.2
exact Nonempty.ne_empty <| ⟨_, mem_primeFactors.2 ⟨hp, hpn, hn.1⟩⟩
· rintro (rfl | rfl) <;> simp
@[simp]
| Mathlib/Data/Nat/PrimeFin.lean | 67 | 73 |
/-
Copyright (c) 2023 Andrew Yang, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots
import Mathlib.FieldTheory.Galois.Basic
import Mathlib.FieldTheory.KummerPolynomial
import Mathlib.Linea... | include hζ H in
lemma isCyclic_of_isSplittingField_X_pow_sub_C [NeZero n] : IsCyclic (L ≃ₐ[K] L) :=
have hn := Nat.pos_iff_ne_zero.mpr (ne_zero_of_irreducible_X_pow_sub_C H)
isCyclic_of_surjective _
| Mathlib/FieldTheory/KummerExtension.lean | 429 | 432 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Chris Hughes
-/
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.FieldTheory.Minpoly.Basic
import Mathlib.RingTheory.A... | Mathlib/RingTheory/AdjoinRoot.lean | 930 | 937 | |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Malo Jaffré
-/
import Mathlib.Analysis.Convex.Function
import Mathlib.Tactic.AdaptationNote
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Linarith
/-!
# Slop... | · convert hf.secant_mono_aux3 hx ha hxy hya using 1 <;> rw [← neg_div_neg_eq] <;> field_simp
· convert hf.slope_mono_adjacent hx hy hxa hya using 1
rw [← neg_div_neg_eq]; field_simp
· exact hf.secant_mono_aux2 ha hy hxa hxy
theorem StrictConvexOn.secant_strict_mono_aux1 (hf : StrictConvexOn 𝕜 s f) {x ... | Mathlib/Analysis/Convex/Slope.lean | 260 | 265 |
/-
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
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Data.Set.Function
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Algebra.Order.Monoid.Defs... |
/-!
### Images under `x ↦ x + a`
-/
| Mathlib/Algebra/Order/Interval/Set/Monoid.lean | 65 | 69 |
/-
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.Convex.Function
import Mathlib.Analysis.Convex.StrictConvexSpace
import Mathlib.MeasureTheory.Function.AEEqOfIntegral
import Mathlib.Measure... | set `s`, `μ` is a finite non-zero measure on `α`, and `f : α → E` is a function sending
`μ`-a.e. points to `s`, then the value of `g` at the average value of `f` is less than or equal to
the average value of `g ∘ f` provided that both `f` and `g ∘ f` are integrable. See also
`ConvexOn.map_centerMass_le` for a finite su... | Mathlib/Analysis/Convex/Integral.lean | 122 | 127 |
/-
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.Basic
import Mathlib.Topology.Order.ProjIcc
/-!... |
theorem arcsin_projIcc (x : ℝ) :
arcsin (projIcc (-1) 1 (neg_le_self zero_le_one) x) = arcsin x := by
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean | 45 | 47 |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Manuel Candales
-/
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
import Mathlib.Tactic.IntervalCases
/-... | simpa only [neg_vsub_eq_vsub_rev] using
positiveOrientation.oangle_add_cyc3_neg_left (vsub_ne_zero.mpr h21) (vsub_ne_zero.mpr h32)
(vsub_ne_zero.mpr h13)
/-- **Stewart's Theorem**. -/
theorem dist_sq_mul_dist_add_dist_sq_mul_dist (a b c p : P) (h : ∠ b p c = π) :
dist a b ^ 2 * dist c p + dist a c ^ 2 ... | Mathlib/Geometry/Euclidean/Triangle.lean | 300 | 308 |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Logic.Relator
import Mathlib.Tactic.Use
import Mathlib.Tactic.MkIffOfInductiveProp
import Mathlib.Tactic.SimpRw
import Mathlib.Logic.Basic
import Mathl... | | @tail f b _ hfb ih =>
rcases ih with ⟨a, hea, hfa⟩
cases hfa with
| Mathlib/Logic/Relation.lean | 655 | 657 |
/-
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
-/
import Mathlib.Analysis.Complex.Asymptotics
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Data.Complex.Trig... |
/-- `Real.exp (f x)` is bounded away from zero along a filter if and only if this filter is bounded
from below under `f`. -/
@[simp]
| Mathlib/Analysis/SpecialFunctions/Exp.lean | 402 | 405 |
/-
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 | 360 | 361 | |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou, Adam Topaz
-/
import Mathlib.AlgebraicGeometry.Sites.MorphismProperty
import Mathlib.CategoryTheory.Sites.Canonical
/-!
# The big Zariski site of schemes
In this file, we define... | Mathlib/AlgebraicGeometry/Sites/BigZariski.lean | 73 | 90 | |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Heather Macbeth
-/
import Mathlib.Analysis.SpecialFunctions.Complex.Circle
import Mathlib.Geometry.Euclidean.Angle.Oriented.Basic
/-!
# Rotations by oriented angles.
This... | o.oangle (o.rotation θ x) y = o.oangle x y - θ := by
simp only [oangle, o.kahler_rotation_left']
rw [Complex.arg_mul_coe_angle, Real.Angle.arg_toCircle]
| Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean | 198 | 200 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Minchao Wu, Mario Carneiro
-/
import Mathlib.Data.Finset.Attach
import Mathlib.Data.Finset.Disjoint
import Mathlib.Data.Finset.Erase
import Mat... | Mathlib/Data/Finset/Basic.lean | 1,487 | 1,488 | |
/-
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... | theorem comp_right_top (r : Rel α β) : r • (⊤ : Rel β γ) = fun x _ ↦ x ∈ r.dom := by
ext x z
simp [comp, Top.top, dom]
| Mathlib/Data/Rel.lean | 119 | 122 |
/-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
-- Some proofs and docs came from mathlib3 `src/algebra/commute.lean` (c) Neil Strickland
import Mathlib.Algebra.Group.Defs
import Mathlib.Order.Defs.Unbundled
/-... | Mathlib/Algebra/Group/Semiconj/Defs.lean | 141 | 142 | |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Vector.Defs
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.OfFn
import Mathlib.Data.List.Scan
import Mathlib.Control.Applicative
import M... | @[simp]
theorem toList_set (v : Vector α n) (i : Fin n) (a : α) :
(v.set i a).toList = v.toList.set i a :=
rfl
@[simp]
theorem get_set_same (v : Vector α n) (i : Fin n) (a : α) : (v.set i a).get i = a := by
cases v; cases i; simp [Vector.set, get_eq_get_toList]
theorem get_set_of_ne {v : Vector α n} {i j : Fi... | Mathlib/Data/Vector/Basic.lean | 581 | 599 |
/-
Copyright (c) 2024 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Measure.GiryMonad
import Mathlib.MeasureTheory.Measure.Stieltjes
import Mathlib.Analysis.Normed.Order.Lattice
import Mathlib.MeasureTheory.Fu... | suffices {a | IsRatStieltjesPoint f a}
= ({a | Monotone (f a)} ∩ {a | Tendsto (f a) atTop (𝓝 1)} ∩ {a | Tendsto (f a) atBot (𝓝 0)}
∩ {a | ∀ t : ℚ, ⨅ r : Ioi t, f a r = f a t}) by
rw [this]
exact (((h1.inter h2).inter h3).inter h4)
ext a
simp only [mem_setOf_eq, mem_inter_iff]
refine ⟨fun... | Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean | 100 | 108 |
/-
Copyright (c) 2024 Frédéric Marbach. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Marbach
-/
import Mathlib.Algebra.Lie.Abelian
import Mathlib.Algebra.Lie.Derivation.Basic
import Mathlib.Algebra.Lie.OfAssociative
/-!
# Adjoint action of a Lie algebra on ... | /-- The commutator of a derivation `D` and a derivation of the form `ad x` is `ad (D x)`. -/
lemma lie_der_ad_eq_ad_der (D : LieDerivation R L L) (x : L) : ⁅D, ad R L x⁆ = ad R L (D x) := by
rw [ad_apply_lieDerivation, ← lie_ad, lie_skew]
| Mathlib/Algebra/Lie/Derivation/AdjointAction.lean | 72 | 74 |
/-
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.MeasureTheory.Measure.AEMeasurable
/-!
# Typeclasses for measurability of operations
In this file we define classes `MeasurableMul` etc and prove d... | Mathlib/MeasureTheory/Group/Arithmetic.lean | 946 | 949 | |
/-
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... | @[simp]
theorem length_comp (p : Path a b) : ∀ {c} (q : Path b c), (p.comp q).length = p.length + q.length
| _, nil => rfl
| Mathlib/Combinatorics/Quiver/Path.lean | 105 | 107 |
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel, Johan Commelin, Kim Morrison
-/
import Mathlib.CategoryTheory.Limits.Constructions.Pullbacks
import Mathlib.CategoryTheory.Preadditive.Biproducts
import Mathlib.CategoryT... | def coimageStrongEpiMonoFactorisation : StrongEpiMonoFactorisation f where
I := Abelian.coimage f
| Mathlib/CategoryTheory/Abelian/Basic.lean | 344 | 345 |
/-
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... | refine ⟨x.re, x₁, by simp⟩
| Mathlib/Data/Complex/Basic.lean | 823 | 823 |
/-
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.MeasureTheory.Constructions.BorelSpace.Order
import Mathlib.MeasureTheory.MeasurableSpace.Prod
import Mathlib.MeasureTheory.Measure.Ty... |
/-- The set of finite `EReal` numbers is `MeasurableEquiv` to `ℝ`. -/
def MeasurableEquiv.erealEquivReal : ({⊥, ⊤}ᶜ : Set EReal) ≃ᵐ ℝ :=
EReal.neBotTopHomeomorphReal.toMeasurableEquiv
| Mathlib/MeasureTheory/Constructions/BorelSpace/Real.lean | 395 | 399 |
/-
Copyright (c) 2024 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.GradedObject.Associator
import Mathlib.CategoryTheory.GradedObject.Single
/-!
# The left and right unitors
Given a bifunctor `F : C ⥤ D ⥤ D`, an ... | variable {Y}
@[reassoc]
lemma mapBifunctorRightUnitor_inv_naturality :
φ ≫ (mapBifunctorRightUnitor F Y e p hp X').inv =
(mapBifunctorRightUnitor F Y e p hp X).inv ≫ mapBifunctorMapMap F p φ (𝟙 _) := by
ext j
dsimp
rw [mapBifunctorRightUnitor_inv_apply, mapBifunctorRightUnitor_inv_apply, assoc, assoc,... | Mathlib/CategoryTheory/GradedObject/Unitor.lean | 233 | 244 |
/-
Copyright (c) 2021 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn, Joachim Breitner
-/
import Mathlib.Algebra.Group.Action.End
import Mathlib.Algebra.Group.Action.Pointwise.Set.Basic
import Mathlib.Algebra.Group.Submonoid.Membership
import Mat... | chain_ne := by
induction w
· exact List.chain'_singleton _
· refine List.Chain'.append (by assumption) (by assumption) ?_
intro x hx y hy
| Mathlib/GroupTheory/CoprodI.lean | 679 | 683 |
/-
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 | 884 | 885 | |
/-
Copyright (c) 2023 Ashvni Narayanan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ashvni Narayanan, Moritz Firsching, Michael Stoll
-/
import Mathlib.Algebra.Group.EvenFunction
import Mathlib.Data.ZMod.Units
import Mathlib.NumberTheory.MulChar.Basic
/-!
# Dirichl... | lemma changeLevel_self_toUnitHom : (changeLevel (dvd_refl n) χ).toUnitHom = χ.toUnitHom := by
rw [changeLevel_self]
| Mathlib/NumberTheory/DirichletCharacter/Basic.lean | 87 | 89 |
/-
Copyright (c) 2024 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Lie.Weights.Killing
import Mathlib.LinearAlgebra.RootSystem.Basic
import Mathlib.LinearAlgebra.RootSystem.Reduced
import Mathlib.LinearAlgebra.RootSy... | · simp only [tsub_le_iff_right, le_add_iff_nonneg_right, Nat.cast_nonneg, neg_sub, true_and]
rw [← Nat.cast_add, chainBotCoeff_add_chainTopCoeff, hn]
omega
rw [h, hk, ← Int.cast_smul_eq_zsmul K, ← add_smul] at this
simp only [Int.cast_sub, Int.cast_natCast,
sub_add_sub_cancel', add_sub_can... | Mathlib/Algebra/Lie/Weights/RootSystem.lean | 337 | 345 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Johan Commelin
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Combinatorics.Enumerative.Composition
/-!
# Composition of analytic functions
In this fi... | equivalence `Composition.sigmaEquivSigmaPi n`, and we deduce finally the associativity of
composition of formal multilinear series in `FormalMultilinearSeries.comp_assoc`.
-/
namespace Composition
variable {n : ℕ}
/-- Rewriting equality in the dependent type `Σ (a : Composition n), Composition a.length)` in
non-dep... | Mathlib/Analysis/Analytic/Composition.lean | 950 | 974 |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Algebra.Order.Group.Int
import Mathlib.Algebra.Order.Group.Unbundled.Int
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Ring.Int.Parity
im... | Mathlib/Algebra/Order/Ring/Int.lean | 55 | 56 | |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory
deprecated_module (since := "2025-04-06")
| Mathlib/MeasureTheory/Integral/VitaliCaratheodory.lean | 421 | 447 | |
/-
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.Combinatorics.SimpleGraph.Path
import Mathlib.Combinatorics.SimpleGraph.Operations
import Mathlib.Data.Finset.Pairwise
import M... | @[simp]
lemma isNClique_singleton : G.IsNClique n {a} ↔ n = 1 := by simp [isNClique_iff, eq_comm]
| Mathlib/Combinatorics/SimpleGraph/Clique.lean | 206 | 208 |
/-
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.BigOperators.Group.Finset.Basic
import Mathlib.Algebra.Group.Commute.Hom
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Data.Fintype.B... |
@[to_additive]
theorem noncommProd_eq_prod {β : Type*} [CommMonoid β] (s : Finset α) (f : α → β) :
(noncommProd s f fun _ _ _ _ _ => Commute.all _ _) = s.prod f := by
induction' s using Finset.cons_induction_on with a s ha IH
· simp
| Mathlib/Data/Finset/NoncommProd.lean | 347 | 352 |
/-
Copyright (c) 2024 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Group.Measure
import Mathlib.Tactic.Group
import Mathlib.Topology.UrysohnsLemma
/-!
# Everywhere positive sets in measure spaces
... |
/-- If two measures coincide locally, then a set which is everywhere positive for the former is
also everywhere positive for the latter. -/
lemma IsEverywherePos.of_forall_exists_nhds_eq (hs : IsEverywherePos μ s)
(h : ∀ x ∈ s, ∃ t ∈ 𝓝 x, ∀ u ⊆ t, ν u = μ u) : IsEverywherePos ν s := by
intro x hx n hn
rcases ... | Mathlib/MeasureTheory/Measure/EverywherePos.lean | 173 | 181 |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
/-!
# Right-angled triangles
This file proves ba... | rw [sub_eq_add_neg, tan_angle_add_of_inner_eq_zero h, norm_neg]
/-- The cosine of an angle in a right-angled triangle multiplied by the hypotenuse equals the
adjacent side, version subtracting vectors. -/
theorem cos_angle_sub_mul_norm_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) :
| Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean | 266 | 270 |
/-
Copyright (c) 2014 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Field.Defs
import Mathlib.Tactic.Common
/-!
# `IsField` predicate
Predicate on a (semi)r... | calc
y = y * (x * z) := by rw [hxz, mul_one]
_ = x * y * z := by rw [← mul_assoc, hf.mul_comm y x]
_ = z := by rw [hxy, one_mul]
end IsField
| Mathlib/Algebra/Field/IsField.lean | 84 | 93 |
/-
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... | apply (trans hj).ext
intro Y f hf
rw [← FunctorToTypes.map_comp_apply, ← op_comp, hS.valid_glue (hx.restrict h) _ hf,
FamilyOfElements.restrict, ← hx (𝟙 _) f (h _ hf) _ (id_comp _)]
simp
/--
If `P` is a sheaf for every pullback of the sieve `S`, then `P` is a sheaf for any presieve which
contain... | Mathlib/CategoryTheory/Sites/IsSheafFor.lean | 663 | 674 |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Casper Putz, Anne Baanen
-/
import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.L... | LinearMap.det (0 : M →ₗ[A] M) = (0 : A) ^ Module.finrank A M := by
simp only [← zero_smul A (1 : M →ₗ[A] M), det_smul, mul_one, MonoidHom.map_one]
theorem det_eq_one_of_not_module_finite (h : ¬Module.Finite R M) (f : M →ₗ[R] M) : f.det = 1 := by
rw [LinearMap.det, dif_neg, MonoidHom.one_apply]
exact fun ⟨_, ... | Mathlib/LinearAlgebra/Determinant.lean | 256 | 267 |
/-
Copyright (c) 2020 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Polynomial.Degree.Support
import Mathlib.Data.ENat.Basic
/-!
# Trailing degree of univariate polynomials
## Main definitions
* `trailingDegree... |
@[simp]
| Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean | 237 | 238 |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Casper Putz, Anne Baanen
-/
import Mathlib.Algebra.Algebra.Subalgebra.Tower
import Mathlib.Data.Finite.Sum
import Mathlib.Data.Matrix.Block
import Mathl... |
theorem leftMulMatrix_apply (x : S) : leftMulMatrix b x = LinearMap.toMatrix b b (lmul R S x) :=
rfl
theorem leftMulMatrix_eq_repr_mul (x : S) (i j) : leftMulMatrix b x i j = b.repr (x * b j) i := by
| Mathlib/LinearAlgebra/Matrix/ToLin.lean | 820 | 824 |
/-
Copyright (c) 2020 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp
-/
import Mathlib.LinearAlgebra.Eigenspace.Basic
import Mathlib.FieldTheory.IsAlgClosed.Spectrum
import Mathlib.LinearAlgebra.FreeModule.Finite.Matrix
/-!
# Tri... | apply (f.genEigenspace μ').mono
rintro k rfl
simp only [ENat.some_eq_coe, Nat.cast_inj, exists_eq_left']
apply Finset.sup_le
| Mathlib/LinearAlgebra/Eigenspace/Triangularizable.lean | 194 | 197 |
/-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller, Vincent Beffara, Rida Hamadani
-/
import Mathlib.Combinatorics.SimpleGraph.Path
import Mathlib.Data.ENat.Lattice
/-!
# Graph metric
This module defines the `SimpleGraph.edi... | end edist
section dist
| Mathlib/Combinatorics/SimpleGraph/Metric.lean | 155 | 158 |
/-
Copyright (c) 2018 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp
-/
import Mathlib.Data.Nat.Find
import Mathlib.Algebra.Module.Pi
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
/-!
# Basic properties of holors
Holors... | rw [Nat.add_comm]
apply cprankMax_add
· exact cprankMax_1 (CPRankMax1.cons _ _ hy₁)
| Mathlib/Data/Holor.lean | 277 | 279 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Tape
import Mathlib.Data.Fintype.Option
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.PFun
import M... | Mathlib/Computability/TuringMachine.lean | 1,918 | 1,959 | |
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kenny Lau
-/
import Mathlib.Algebra.Order.Antidiag.Finsupp
import Mathlib.Data.Finsupp.Weight
import Mathlib.Tactic.Linarith
import Mathlib.LinearAlgebra.Pi
import Mat... |
theorem coeff_zero_mul_X (φ : MvPowerSeries σ R) (s : σ) : coeff R (0 : σ →₀ ℕ) (φ * X s) = 0 := by
have : ¬single s 1 ≤ 0 := fun h => by simpa using h s
| Mathlib/RingTheory/MvPowerSeries/Basic.lean | 418 | 420 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl, Patrick Massot
-/
import Mathlib.Data.Set.Image
import Mathlib.Data.SProd
/-!
# Sets in product and pi types
This file proves basic properties of prod... | ⟨fun h => or_iff_not_imp_right.2 ?_, fun h => h.elim pi_mono fun h' => h'.symm ▸ empty_subset _⟩
rw [← Ne, ← nonempty_iff_ne_empty]
| Mathlib/Data/Set/Prod.lean | 844 | 845 |
/-
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.Algebra.Field.NegOnePow
import Mathlib.Algebra.Field.Periodic
import Mathlib.Algebra.Qua... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | 1,423 | 1,435 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kim Morrison
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Algebra.Group.InjSurj
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Tactic.FastInstance
impo... | Mathlib/Data/Finsupp/Defs.lean | 984 | 986 | |
/-
Copyright (c) 2024 Brendan Murphy. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Brendan Murphy
-/
import Mathlib.Algebra.Module.Torsion
import Mathlib.RingTheory.Flat.Basic
import Mathlib.RingTheory.Ideal.AssociatedPrime.Basic
import Mathlib.RingTheory.QuotSMulTop... | variable {N} in
lemma smul_top_inf_eq_smul_of_isSMulRegular_on_quot :
IsSMulRegular (M ⧸ N) r → r • ⊤ ⊓ N ≤ r • N := by
convert map_mono ∘ (isSMulRegular_on_quot_iff_lsmul_comap_le N r).mp using 2
| Mathlib/RingTheory/Regular/IsSMulRegular.lean | 182 | 185 |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Module.BigOperators
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.Nat.Squarefree
imp... | cardFactors_mul hn hm, pow_add]
theorem IsMultiplicative.prodPrimeFactors_one_add_of_squarefree [CommSemiring R]
| Mathlib/NumberTheory/ArithmeticFunction.lean | 1,037 | 1,039 |
/-
Copyright (c) 2020 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca, Johan Commelin
-/
import Mathlib.Algebra.GCDMonoid.IntegrallyClosed
import Mathlib.FieldTheory.Finite.Basic
import Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed
impo... | rcases n.eq_zero_or_pos with (rfl | h0)
· simp
apply minpoly.isIntegrallyClosed_dvd (isIntegral h h0)
simp only [((IsPrimitiveRoot.iff_def μ n).mp h).left, aeval_X_pow, eq_intCast, Int.cast_one,
aeval_one, map_sub, sub_self]
| Mathlib/RingTheory/RootsOfUnity/Minpoly.lean | 53 | 58 |
/-
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... | Mathlib/Algebra/Ring/Ext.lean | 484 | 488 | |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.CategoryTheory.Adjunction.FullyFaithful
import Mathlib.CategoryTheory.Elementwise
import Mathlib.Topology.Sheaves.Presheaf
/-!
# Presheafed spaces
Introd... | (F.mapPresheaf.obj X : TopCat) = (X : TopCat) :=
rfl
@[simp]
theorem mapPresheaf_obj_presheaf (F : C ⥤ D) (X : PresheafedSpace C) :
(F.mapPresheaf.obj X).presheaf = X.presheaf ⋙ F :=
| Mathlib/Geometry/RingedSpace/PresheafedSpace.lean | 412 | 417 |
/-
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.Topology.Order.IsLUB
/-!
# Order topology on a densely ordered set
-/
open Set Filter TopologicalSpace Topology Func... | frontier_Iio' nonempty_Iio
| Mathlib/Topology/Order/DenselyOrdered.lean | 160 | 161 |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Joël Riou
-/
import Mathlib.CategoryTheory.Sites.Subsheaf
import Mathlib.CategoryTheory.Sites.CompatibleSheafification
import Mathlib.CategoryTheory.Sites.LocallyInjective
/-... | imageSieve_mem s := by apply hJK; exact h.1 _
lemma isLocallyInjective_of_isLocallyInjective_of_isLocallySurjective
| Mathlib/CategoryTheory/Sites/LocallySurjective.lean | 186 | 188 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Alex J. Best
-/
import Mathlib.MeasureTheory.Group.Arithmetic
/-!
# Pointwise set operations on `MeasurableSet`s
In this file we prove several versions of the follo... | exacts [(subsingleton_zero_smul_set s).measurableSet, hs.const_smul_of_ne_zero ha]
| Mathlib/MeasureTheory/Group/Pointwise.lean | 39 | 44 |
/-
Copyright (c) 2020 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
import Mathlib.Algebra.ContinuedFractions.Computation.Approximations
import Mathlib.Algebra.ContinuedFractions.Computation.CorrectnessTerminating
import Mathlib.D... | theorem exists_rat_eq_nth_conv : ∃ q : ℚ, (of v).convs n = (q : K) := by
rcases exists_rat_eq_nth_num v n with ⟨Aₙ, nth_num_eq⟩
rcases exists_rat_eq_nth_den v n with ⟨Bₙ, nth_den_eq⟩
use Aₙ / Bₙ
| Mathlib/Algebra/ContinuedFractions/Computation/TerminatesIffRat.lean | 112 | 115 |
/-
Copyright (c) 2021 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno
-/
import Mathlib.CategoryTheory.NatIso
/-!
# Bicategories
In this file we define typeclass for bicategories.
A bicategory `B` consists of
* objects `a : B`,
* 1-morphisms ... | (ρ_ (f ≫ g)).inv = f ◁ (ρ_ g).inv ≫ (α_ f g (𝟙 c)).inv := by simp
| Mathlib/CategoryTheory/Bicategory/Basic.lean | 408 | 409 |
/-
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 | 1,363 | 1,365 | |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.CharP.Reduced
import Mathlib.RingTheory.IntegralDomain
-- TODO: remove Mathlib.Algebra.CharP.Reduced and move the last two lemmas to Lemmas
/-... | Mathlib/RingTheory/RootsOfUnity/Basic.lean | 615 | 626 | |
/-
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.Multiset.FinsetOps
import Mathlib.Data.Multiset.Fold
/-!
# Lattice operations on multisets
-/
namespace Multiset
variable {α : Type*}
/-! ###... | theorem inf_ndunion (s₁ s₂ : Multiset α) : (ndunion s₁ s₂).inf = s₁.inf ⊓ s₂.inf := by
rw [← inf_dedup, dedup_ext.2, inf_dedup, inf_add]; simp
| Mathlib/Data/Multiset/Lattice.lean | 137 | 138 |
/-
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... | attribute [mono] edgeFinset_mono edgeFinset_strict_mono
| Mathlib/Combinatorics/SimpleGraph/Finite.lean | 78 | 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
-/
import Mathlib.MeasureTheory.Measure.Comap
import Mathlib.MeasureTheory.Measure.QuasiMeasurePreserving
/-!
# Restricting a measure to a subset or a s... | Mathlib/MeasureTheory/Measure/Restrict.lean | 1,076 | 1,082 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Order.SuccPred
import Mathlib.Data.Sum.Order
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Tactic.PPWithUniv
/-!
# ... | theorem ord_inj {a b : Cardinal} : a.ord = b.ord ↔ a = b :=
ord_injective.eq_iff
@[simp]
| Mathlib/SetTheory/Ordinal/Basic.lean | 1,152 | 1,155 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Minchao Wu, Mario Carneiro
-/
import Mathlib.Algebra.NeZero
import Mathlib.Data.Finset.Attach
import Mathlib.Data.Finset.Disjoint
import Mathli... | (fun ⟨i, hi⟩ =>
have : 0 ≤ i % ↑n := Int.emod_nonneg _ (ne_of_gt hn')
⟨Int.toNat (i % n), by
rw [← Int.ofNat_lt, Int.toNat_of_nonneg this]; exact ⟨Int.emod_lt_of_pos i hn', hi⟩⟩)
| Mathlib/Data/Finset/Image.lean | 480 | 483 |
/-
Copyright (c) 2022 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex J. Best, Xavier Roblot
-/
import Mathlib.Algebra.Algebra.Hom.Rat
import Mathlib.Analysis.Complex.Polynomial.Basic
import Mathlib.NumberTheory.NumberField.Norm
import Mathlib.RingTh... | ← not_isReal_iff_isComplex, comap_mk, isReal_mk_iff, isReal_mk_iff, eq_true h.isReal_comp,
and_true]
lemma isUnramified_mk_iff_forall_isConj [IsGalois k K] {φ : K →+* ℂ} :
IsUnramified k (mk φ) ↔ ∀ σ : K ≃ₐ[k] K, ComplexEmbedding.IsConj φ σ → σ = 1 := by
refine ⟨fun H σ hσ ↦ hσ.isUnramified_mk_iff.mp H,
... | Mathlib/NumberTheory/NumberField/Embeddings.lean | 861 | 876 |
/-
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, David Loeffler
-/
import Mathlib.Analysis.Convex.Slope
import Mathlib.Analysis.Calculus.Deriv.MeanValue
/-!
# Convexity of functions and derivat... |
end ConvexOn
namespace StrictConvexOn
| Mathlib/Analysis/Convex/Deriv.lean | 723 | 727 |
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Data.SetLike.Basic
import Mathlib.ModelTheory.Semantics
/-!
# Definable Sets
This file defines what it means for a set over a first-order structure t... | (((h.image_comp_equiv (Equiv.Set.sumCompl (range f))).image_comp_equiv
(Equiv.sumCongr (Equiv.ofInjective f f.injective)
(Fintype.equivFin (↥(range f)ᶜ)).symm)).image_comp_sumInl_fin
_)
simp only [mem_preimage, mem_image, exists_exists_and_eq_and]
refine exists_co... | Mathlib/ModelTheory/Definability.lean | 214 | 226 |
/-
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 ... | Mathlib/SetTheory/Cardinal/ENat.lean | 112 | 112 | |
/-
Copyright (c) 2022 Julian Berman. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Julian Berman
-/
import Mathlib.GroupTheory.PGroup
import Mathlib.LinearAlgebra.Quotient.Defs
/-!
# Torsion groups
This file defines torsion groups, i.e. groups where all elements hav... | section CommGroup
| Mathlib/GroupTheory/Torsion.lean | 367 | 368 |
/-
Copyright (c) 2019 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Yury Kudryashov, Yaël Dillies
-/
import Mathlib.Algebra.Order.Invertible
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.LinearAlgebra.AffineSpac... |
section OrderedRing
variable (𝕜) [Ring 𝕜] [PartialOrder 𝕜] [AddRightMono 𝕜]
| Mathlib/Analysis/Convex/Segment.lean | 152 | 155 |
/-
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 | 1,667 | 1,674 | |
/-
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.Shift.CommShift
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
/-! Shifted morphisms
Given a category `C` endowed with a shift by an ... | lemma comp_assoc {a₁ a₂ a₃ a₁₂ a₂₃ a : M}
(α : ShiftedHom X Y a₁) (β : ShiftedHom Y Z a₂) (γ : ShiftedHom Z T a₃)
(h₁₂ : a₂ + a₁ = a₁₂) (h₂₃ : a₃ + a₂ = a₂₃) (h : a₃ + a₂ + a₁ = a) :
(α.comp β h₁₂).comp γ (show a₃ + a₁₂ = a by rw [← h₁₂, ← add_assoc, h]) =
α.comp (β.comp γ h₂₃) (by rw [← h₂₃, h]) := b... | Mathlib/CategoryTheory/Shift/ShiftedHom.lean | 48 | 55 |
/-
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.Pow.NNReal
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Analysis.SumOverResidueClass
/-!
# Conve... | /-- Summability of the `p`-series over `ℤ`. -/
theorem summable_one_div_int_pow {p : ℕ} :
(Summable fun n : ℤ ↦ 1 / (n : ℝ) ^ p) ↔ 1 < p := by
| Mathlib/Analysis/PSeries.lean | 314 | 316 |
/-
Copyright (c) 2019 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Yury Kudryashov, Yaël Dillies
-/
import Mathlib.Algebra.Order.Invertible
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.LinearAlgebra.AffineSpac... |
@[simp]
theorem vadd_segment [AddTorsor G E] [VAddCommClass G E E] (a : G) (b c : E) :
a +ᵥ [b -[𝕜] c] = [a +ᵥ b -[𝕜] a +ᵥ c] :=
| Mathlib/Analysis/Convex/Segment.lean | 221 | 224 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Multiset.ZeroCons
/-!
# Basic results on multisets
-/
-- No algebra should be required
assert_not_exists Monoid
universe v
open List S... | Mathlib/Data/Multiset/Basic.lean | 308 | 327 | |
/-
Copyright (c) 2021 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Ines Wright, Joachim Breitner
-/
import Mathlib.GroupTheory.Solvable
import Mathlib.GroupTheory.Sylow
import Mathlib.Algebra.Group.Subgroup.Order
import Mathlib.GroupTheo... | exact of_quotient_center_nilpotent hnq
variable [Finite G]
/-- If a finite group is the direct product of its Sylow groups, it is nilpotent -/
theorem isNilpotent_of_product_of_sylow_group
(e : (∀ p : (Nat.card G).primeFactors, ∀ P : Sylow p G, (↑P : Subgroup G)) ≃* G) :
IsNilpotent G := by
classical
... | Mathlib/GroupTheory/Nilpotent.lean | 823 | 839 |
/-
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, Johan Commelin
-/
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib... | rw [mem_nthRootsFinset hn, one_pow]
end NthRoots
theorem zero_of_eval_zero [Infinite R] (p : R[X]) (h : ∀ x, p.eval x = 0) : p = 0 := by
classical
by_contra hp
refine @Fintype.false R _ ?_
exact ⟨p.roots.toFinset, fun x => Multiset.mem_toFinset.mpr ((mem_roots hp).mpr (h _))⟩
| Mathlib/Algebra/Polynomial/Roots.lean | 369 | 377 |
/-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Calle Sönne, Adam Topaz
-/
import Mathlib.Data.Setoid.Partition
import Mathlib.Topology.LocallyConstant.Basic
import Mathlib.Topology.Separation.Regular
import Mathlib.Topology.Connected.T... | rcases a with ⟨⟩
| Mathlib/Topology/DiscreteQuotient.lean | 297 | 297 |
/-
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.Convex.Topology
import Mathlib.Analysis.Normed.Module.Basic
import Mathlib.Analysis.Seminorm
import Mathlib.Analysis.SpecificLimits.Basi... | variable {x y : E} {s t : Set E}
theorem mem_tangentConeAt_of_pow_smul {r : 𝕜} (hr₀ : r ≠ 0) (hr : ‖r‖ < 1)
(hs : ∀ᶠ n : ℕ in atTop, x + r ^ n • y ∈ s) : y ∈ tangentConeAt 𝕜 s x := by
refine ⟨fun n ↦ (r ^ n)⁻¹, fun n ↦ r ^ n • y, hs, ?_, ?_⟩
· simp only [norm_inv, norm_pow, ← inv_pow]
| Mathlib/Analysis/Calculus/TangentCone.lean | 85 | 90 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Multiset.ZeroCons
/-!
# Basic results on multisets
-/
-- No algebra should be required
assert_not_exists Monoid
universe v
open List S... | Mathlib/Data/Multiset/Basic.lean | 831 | 832 | |
/-
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.Order.Filter.SmallSets
import Mathlib.Topology.UniformSpace.Defs
import Mathlib.Topology.ContinuousOn
/-!
# Basic resu... | Mathlib/Topology/UniformSpace/Basic.lean | 1,842 | 1,846 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
/-!
# (Generalized) Boolean algebras
A Boolean algebra is a bounded distributive lattice with a complement ope... | __ := instHeytingAlgebra
himp_eq x y := by ext <;> simp [himp_eq]
| Mathlib/Order/BooleanAlgebra.lean | 722 | 723 |
/-
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.Analysis.InnerProductSpace.Convex
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Combinatorics.Additive.AP.Three... | theorem dValue_pos (hN₃ : 8 ≤ N) : 0 < dValue N := by
have hN₀ : 0 < (N : ℝ) := cast_pos.2 (succ_pos'.trans_le hN₃)
rw [dValue, floor_pos, ← log_le_log_iff zero_lt_one, log_one, log_div _ two_ne_zero, log_rpow hN₀,
inv_mul_eq_div, sub_nonneg, le_div_iff₀]
· have : (nValue N : ℝ) ≤ 2 * √(log N) := by
app... | Mathlib/Combinatorics/Additive/AP/Three/Behrend.lean | 363 | 371 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.Preadditive
import Mathlib.CategoryTheory.Linear.LinearFunctor
/-!
# Homology of linear categories
In this file, it is shown that... | @[simp]
lemma rightHomologyMap'_smul :
rightHomologyMap' (a • φ) h₁ h₂ = a • rightHomologyMap' φ h₁ h₂ := by
have γ : RightHomologyMapData φ h₁ h₂ := default
simp only [(γ.smul a).rightHomologyMap'_eq, RightHomologyMapData.smul_φH, γ.rightHomologyMap'_eq]
| Mathlib/Algebra/Homology/ShortComplex/Linear.lean | 124 | 128 |
/-
Copyright (c) 2019 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.List.FinRange
import Mathlib.Data.List.Perm.Basic
import Mathlib.Data.List.Lex
import Mathlib.Data.List.Induc... | apply (perm_ext_iff_of_nodup _ _).mpr
· simp
· exact nodup_sublists.mpr (nodup_finRange _)
· exact (nodup_sublists'.mpr (nodup_finRange _))
| Mathlib/Data/List/Sublists.lean | 350 | 354 |
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