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) 2015, 2017 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
import Mathlib.Topology.MetricSpace.Pseudo.Constructions
import Mathlib.Topology.Order.Densely... | using uniformity_basis_dist.lebesgue_number_lemma hs hc₁ hc₂
theorem lebesgue_number_lemma_of_metric_sUnion {s : Set α} {c : Set (Set α)} (hs : IsCompact s)
(hc₁ : ∀ t ∈ c, IsOpen t) (hc₂ : s ⊆ ⋃₀ c) : ∃ δ > 0, ∀ x ∈ s, ∃ t ∈ c, ball x δ ⊆ t := by
rw [sUnion_eq_iUnion] at hc₂; simpa using lebesgue_number_lem... | Mathlib/Topology/MetricSpace/Pseudo/Lemmas.lean | 118 | 122 |
/-
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.NumberTheory.LSeries.HurwitzZeta
import Mathlib.Analysis.PSeriesComplex
/-!
# Definition of the Riemann zeta function
## Main definitions:
* `rieman... |
/-- The completed Riemann zeta function `Λ(s)` is differentiable away from `s = 0` and `s = 1`. -/
theorem differentiableAt_completedZeta {s : ℂ} (hs : s ≠ 0) (hs' : s ≠ 1) :
| Mathlib/NumberTheory/LSeries/RiemannZeta.lean | 87 | 89 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calculus.Deriv.Mul
import ... | exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by
simpa only [sub_self] using (derivf y hy).sub (derivg y hy)
/-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere
on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b... | Mathlib/Analysis/Calculus/MeanValue.lean | 378 | 382 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Cardinal.Arithmetic
import Mathlib.SetTheory.Ordinal.FixedPoint
/-!
# Cofinality
This file co... | Mathlib/SetTheory/Cardinal/Cofinality.lean | 1,114 | 1,116 | |
/-
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.Data.Bundle
import Mathlib.Data.Set.Image
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.PartialHomeomorph
import Mathlib.Topology.O... | /-- Natural identification as a `Pretrivialization`. -/
def toPretrivialization : Pretrivialization F proj :=
{ e with }
| Mathlib/Topology/FiberBundle/Trivialization.lean | 288 | 290 |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov, Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
import Mathlib.MeasureTheory.Measure.Typeclasses.Probability
import... | · have A : Disjoint {a} (Ioo a b) := by simp
simp [← Icc_union_Ioo_eq_Ico le_rfl hab, -singleton_union, hab.ne, f.mono.leftLim_le,
measure_union A measurableSet_Ioo, f.mono.le_leftLim hab, ← ENNReal.ofReal_add]
theorem measure_Iic {l : ℝ} (hf : Tendsto f atBot (𝓝 l)) (x : ℝ) :
f.measure (Iic x) = ofRe... | Mathlib/MeasureTheory/Measure/Stieltjes.lean | 427 | 434 |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Ring.Divisibility.Lemmas
import Mathlib.Algebra.Lie.Nilpotent
import Mathlib.Algebra.Lie.Engel
import Mathlib.LinearAlgebra.Eigenspace.Pi
import Math... | -- Eliminate the binomial coefficients from the picture.
suffices (f₁ ^ i * f₂ ^ j) (m₁ ⊗ₜ m₂) = 0 by rw [this]; apply smul_zero
-- Finish off with appropriate case analysis.
rcases Nat.le_or_le_of_add_eq_add_pred (Finset.mem_antidiagonal.mp hij) with hi | hj
· rw [(hf_comm.pow_pow i j).eq, Module.End.mul_app... | Mathlib/Algebra/Lie/Weights/Basic.lean | 136 | 143 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.SymmDiff
import Mathlib.Order.SuccPred.Relation
import Mathlib.Topology.Irreducible
/-!
# Connected subsets ... | Mathlib/Topology/Connected/Basic.lean | 1,303 | 1,306 | |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Kappelmann
-/
import Mathlib.Algebra.Order.Floor.Defs
import Mathlib.Algebra.Order.Floor.Ring
import Mathlib.Algebra.Order.Floor.Semiring
deprecated_module (sinc... | Mathlib/Algebra/Order/Floor.lean | 1,302 | 1,304 | |
/-
Copyright (c) 2020 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Algebra.Group.Conj
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.Group.Subgroup.Ker
/-!
# Basic results on subgroups
We prove basic results... | Mathlib/Algebra/Group/Subgroup/Basic.lean | 2,509 | 2,509 | |
/-
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 | 497 | 501 | |
/-
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.Analysis.Analytic.Composition
import Mathlib.Analysis.Analytic.Linear
import Mathlib.Tactic.Positivity
/-!
# Inverse of analytic functions
We ... | a * I +
∑ k ∈ Ico 2 (n + 1),
a ^ k *
‖(i.symm : F →L[𝕜] E).compContinuousMultilinearMap
(∑ c ∈ ({c | 1 < Composition.length c}.toFinset : Finset (Composition k)),
p.compAlongComposition (p.rightInv i x) c)‖ := by
congr! 2 with j hj... | Mathlib/Analysis/Analytic/Inverse.lean | 442 | 496 |
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.Fintype.List
import Mathlib.Data.Fintype.OfMap
/-!
# Cycles of a list
Lists have an equivalence relation of whether they are rotational permut... | Mathlib/Data/List/Cycle.lean | 1,004 | 1,017 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Order.Filter.Bases.Finit... | Mathlib/Topology/Algebra/Group/Basic.lean | 1,744 | 1,761 | |
/-
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.NumberTheory.ModularForms.JacobiTheta.TwoVariable
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
/-! # Jacobi's theta function
This file define... | rexp (-π * τ.im) / (1 - rexp (-π * τ.im)) by
calc
‖jacobiTheta τ - 1‖ = ↑2 * ‖∑' n : ℕ, cexp (π * I * ((n : ℂ) + 1) ^ 2 * τ)‖ := by
rw [sub_eq_iff_eq_add'.mpr (jacobiTheta_eq_tsum_nat hτ), norm_mul, Complex.norm_two]
| Mathlib/NumberTheory/ModularForms/JacobiTheta/OneVariable.lean | 89 | 92 |
/-
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... | exact Or.inr hσ
lemma mem_orbit_iff [IsGalois k K] {w w' : InfinitePlace K} :
w' ∈ MulAction.orbit (K ≃ₐ[k] K) w ↔ w.comap (algebraMap k K) = w'.comap (algebraMap k K) := by
| Mathlib/NumberTheory/NumberField/Embeddings.lean | 771 | 774 |
/-
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.BigOperators.Group.Finset.Piecewise
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Pi
import Mathlib.Algebra.O... | omit [LinearOrder β] [IsStrictOrderedRing β] in
lemma collapse_eq (ha : a ∉ s) (𝒜 : Finset (Finset α)) (f : Finset α → β) :
collapse 𝒜 a f s = (if s ∈ 𝒜 then f s else 0) +
if insert a s ∈ 𝒜 then f (insert a s) else 0 := by
rw [collapse, filter_collapse_eq ha]
| Mathlib/Combinatorics/SetFamily/FourFunctions.lean | 106 | 110 |
/-
Copyright (c) 2020 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.Group.Action.Pi
import Mathlib.Data.Finset.Prod
import Mathlib.Data.SetLike.Basic
import Mathlib.Data.Sym.Basic
import Mathlib.Data.Sym.Sym2.Init
/-... | clear h z
intro x y h
| Mathlib/Data/Sym/Sym2.lean | 701 | 702 |
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Independence.Kernel
import Mathlib.Probability.Kernel.Condexp
/-!
# Conditional Independence
We define conditional independence of sets/σ-alg... | simp only [CondIndepSets, Kernel.IndepSets]
have hs1_eq : ∀ s ∈ s1, (fun ω ↦ ENNReal.toReal (condExpKernel μ m' ω s)) =ᵐ[μ] μ⟦s | m'⟧ :=
fun s hs ↦ condExpKernel_ae_eq_condExp hm' (hs1 s hs)
have hs2_eq : ∀ s ∈ s2, (fun ω ↦ ENNReal.toReal (condExpKernel μ m' ω s)) =ᵐ[μ] μ⟦s | m'⟧ :=
fun s hs ↦ condExpKern... | Mathlib/Probability/Independence/Conditional.lean | 193 | 219 |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Ring.Divisibility.Lemmas
import Mathlib.Algebra.Lie.Nilpotent
import Mathlib.Algebra.Lie.Engel
import Mathlib.LinearAlgebra.Eigenspace.Pi
import Math... | posFittingComp R L M = ⊥ := by
simp [posFittingComp]
section map_comap
variable {R L M}
variable
{M₂ : Type*} [AddCommGroup M₂] [Module R M₂] [LieRingModule L M₂] [LieModule R L M₂]
{χ : L → R} (f : M →ₗ⁅R,L⁆ M₂)
| Mathlib/Algebra/Lie/Weights/Basic.lean | 492 | 501 |
/-
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*}
/-! ###... | variable [SemilatticeInf α] [OrderTop α]
/-- Infimum of a multiset: `inf {a, b, c} = a ⊓ b ⊓ c` -/
def inf (s : Multiset α) : α :=
s.fold (· ⊓ ·) ⊤
@[simp]
| Mathlib/Data/Multiset/Lattice.lean | 93 | 99 |
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jujian Zhang
-/
import Mathlib.Algebra.GroupWithZero.Subgroup
import Mathlib.Algebra.Order.Group.Action
import Mathlib.LinearAlgebra.Finsupp.Supported
import Mathlib.LinearAl... | Mathlib/Algebra/Module/Submodule/Pointwise.lean | 569 | 584 | |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.Filter.Prod
/-!
# N-ary maps of filter
This file defines the binary and ternary maps of filters. This is mostly useful to define pointwise
operatio... | -/
theorem map₂_assoc {m : δ → γ → ε} {n : α → β → δ} {m' : α → ε' → ε} {n' : β → γ → ε'}
| Mathlib/Order/Filter/NAry.lean | 167 | 169 |
/-
Copyright (c) 2022 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Algebra.Ring.Idempotent
import Mathlib.Analysis.Normed.Group.Basic
import Mathlib.Order.Basic
import Mathlib.Tactic.NoncommRing
/-!
# M-struct... | sub_self, add_zero]
le_sup_right P Q := by
| Mathlib/Analysis/NormedSpace/MStructure.lean | 259 | 260 |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.LinearAlgebra.Finsupp.Span
/-!
# Lie submodules of a Lie algebra
In this file we define Lie submodules, we construct ... | lie_mem := fun h ↦ mem_inter (N.lie_mem h.1) (N'.lie_mem h.2) }⟩
instance : InfSet (LieSubmodule R L M) :=
⟨fun S ↦
{ toSubmodule := sInf {(s : Submodule R M) | s ∈ S}
| Mathlib/Algebra/Lie/Submodule.lean | 322 | 326 |
/-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Algebra.Ring.Int.Defs
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.Cast.Order.Basic
import Math... | apply castNum_eq_bitwise PosNum.lxor <;> intros <;> (try cases_type* Bool) <;> rfl
| Mathlib/Data/Num/Lemmas.lean | 812 | 812 |
/-
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.Data.List.Iterate
import Mathlib.GroupTheory.Perm.Cycle.Basic
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Tactic.Group
/-!
# ... |
theorem isCycleOn_support_cycleOf [DecidableEq α] [Fintype α] (f : Perm α) (x : α) :
f.IsCycleOn (f.cycleOf x).support :=
isCycleOn_support_cycleOf_aux f x
theorem SameCycle.exists_pow_eq_of_mem_support {f} [DecidableEq α] [Fintype α] (h : SameCycle f x y)
(hx : x ∈ f.support) : ∃ i < #(f.cycleOf x).support... | Mathlib/GroupTheory/Perm/Cycle/Factors.lean | 285 | 296 |
/-
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.Sites.Sheaf
/-!
# 1-hypercovers
Given a Grothendieck topology `J` on a category `C`, we define the type of
`1`-hypercovers of an object `S : C`.... | toPreOneHypercover.sieve₁ p₁ p₂ ∈ J W
variable {J}
lemma OneHypercover.mem_sieve₁' {S : C} (E : J.OneHypercover S)
| Mathlib/CategoryTheory/Sites/OneHypercover.lean | 154 | 158 |
/-
Copyright (c) 2022 Antoine Labelle. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Labelle
-/
import Mathlib.RepresentationTheory.FDRep
import Mathlib.LinearAlgebra.Trace
import Mathlib.RepresentationTheory.Invariants
/-!
# Characters of representations
Th... | ext g
simp only [character, FDRep.Iso.conj_ρ i]
| Mathlib/RepresentationTheory/Character.lean | 64 | 65 |
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym
import Mathlib.MeasureTheory.Measure.Haar.OfBasis
import Mathlib.Probability.Independence.Basic
/-!
# Proba... | Mathlib/Probability/Density.lean | 354 | 370 | |
/-
Copyright (c) 2021 Justus Springer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Justus Springer
-/
import Mathlib.Topology.Sheaves.Forget
import Mathlib.Topology.Sheaves.SheafCondition.PairwiseIntersections
import Mathlib.CategoryTheory.Limits.Types.Shapes
/-!
#... | apply gl_uniq
intro i
rw [← h]
/-- In this version of the lemma, the inclusion homs `iUV` can be specified directly by the user,
which can be more convenient in practice.
-/
theorem eq_of_locally_eq' (V : Opens X) (iUV : ∀ i : ι, U i ⟶ V) (hcover : V ≤ iSup U)
(s t : ToType (F.1.obj (op V))) (h : ∀ i, ... | Mathlib/Topology/Sheaves/SheafCondition/UniqueGluing.lean | 216 | 231 |
/-
Copyright (c) 2014 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Leonardo de Moura, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Ord... | Mathlib/Algebra/Order/Field/Basic.lean | 934 | 940 | |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Function.L1Space.Integrable
import Mathlib.MeasureTheory.Function.LpSpace.Indicator
/-! # Functions integrable on a set an... | theorem IntegrableAtFilter.filter_mono (hl : l ≤ l') (hl' : IntegrableAtFilter f l' μ) :
IntegrableAtFilter f l μ :=
let ⟨s, hs, hsf⟩ := hl'
⟨s, hl hs, hsf⟩
theorem IntegrableAtFilter.inf_of_left (hl : IntegrableAtFilter f l μ) :
IntegrableAtFilter f (l ⊓ l') μ :=
| Mathlib/MeasureTheory/Integral/IntegrableOn.lean | 432 | 438 |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kenny Lau, Johan Commelin, Mario Carneiro, Kevin Buzzard,
Amelia Livingston, Yury Kudryashov, Yakov Pechersky
-/
import Mathlib.Algebra.Group.Subsemigroup.Defs
import M... | Mathlib/Algebra/Group/Subsemigroup/Basic.lean | 446 | 447 | |
/-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Algebra.Order.Pi
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Data.Real.Pointwise
/... | @[simp]
theorem closedBall_normSeminorm : (normSeminorm 𝕜 E).closedBall = Metric.closedBall := by
ext x r y
simp only [Seminorm.mem_closedBall, Metric.mem_closedBall, coe_normSeminorm, dist_eq_norm]
variable {𝕜 E} {x : E}
/-- Balls at the origin are absorbent. -/
theorem absorbent_ball_zero (hr : 0 < r) : Absor... | Mathlib/Analysis/Seminorm.lean | 1,299 | 1,325 |
/-
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.Data.Matrix.Basis
import Mathlib.Data.Matrix.DMatrix
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.Re... | rw [reverse_inv_prod_mul_prod, prod_mul_reverse_inv_prod, Matrix.one_mul, Matrix.mul_one]
end Pivot
open Pivot TransvectionStruct
| Mathlib/LinearAlgebra/Matrix/Transvection.lean | 683 | 688 |
/-
Copyright (c) 2019 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Eric Wieser
-/
import Mathlib.Data.Matrix.ConjTranspose
/-!
# Row and column matrices
This file provides results about row and column matrices.
## Main definitions
* `Mat... | theorem updateCol_apply [DecidableEq n] {j' : n} :
updateCol M j c i j' = if j' = j then c i else M i j' := by
by_cases h : j' = j
· rw [h, updateCol_self, if_pos rfl]
· rw [updateCol_ne h, if_neg h]
| Mathlib/Data/Matrix/RowCol.lean | 267 | 271 |
/-
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... | ∃ i, testBit n i = true ∧ ∀ j, i < j → testBit n j = false := by
induction n using Nat.binaryRec with | z => exact False.elim (h rfl) | f b n hn => ?_
by_cases h' : n = 0
· subst h'
rw [show b = true by
revert h
cases b <;> simp]
refine ⟨0, ⟨by rw [testBit_bit_zero], fun j hj => ?_⟩⟩
... | Mathlib/Data/Nat/Bitwise.lean | 188 | 203 |
/-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl, Simon Hudon, Kenny Lau
-/
import Mathlib.Data.Multiset.Bind
import Mathlib.Control.Traversable.Lemmas
import Mathlib.Control.Traversable.Instances
/-!
# F... | [CommApplicative H] {α β γ : Type _} (g : α → G β) (h : β → H γ) (x : Multiset α) :
traverse (Comp.mk ∘ Functor.map h ∘ g) x =
Comp.mk (Functor.map (traverse h) (traverse g x)) := by
refine Quotient.inductionOn x ?_
| Mathlib/Data/Multiset/Functor.lean | 96 | 99 |
/-
Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Data.Set.Equitable
import Mathlib.Logic.Equiv.Fin.Basic
import Mathlib.Order.Partition.Fi... | Mathlib/Order/Partition/Equipartition.lean | 172 | 173 | |
/-
Copyright (c) 2020 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Halting
import Mathlib.Computability.TuringMachine
import Mathlib.Data.Num.Lemmas
import Mathlib.Tactic.DeriveFintype
import Mathlib.Comp... | | apply Decidable.isFalse; rintro ⟨⟨⟩⟩; done
| exact decidable_of_iff' _ (by simp [funext_iff]; rfl)
| Mathlib/Computability/TMToPartrec.lean | 211 | 212 |
/-
Copyright (c) 2023 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Integral.Bochner.Basic
import Mathlib.Topology.ContinuousMap.Bounded.Normed
/-!
# Integration of bounded continuous functions
In this file,... |
variable [OpensMeasurableSpace X] [SecondCountableTopology E] [MeasurableSpace E] [BorelSpace E]
lemma integrable [IsFiniteMeasure μ] (f : X →ᵇ E) :
Integrable f μ := by
| Mathlib/MeasureTheory/Integral/BoundedContinuousFunction.lean | 83 | 87 |
/-
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,
Kim Morrison
-/
import Mathlib.Data.List.Basic
/-!
# Lattice structure of lists
This files pro... | by_cases ba : b = a
· simp only [beq_iff_eq]
rw [if_pos ba, Nat.sub_add_cancel]
rwa [succ_le_iff, count_pos_iff, ← ba]
· simp only [beq_iff_eq]
| Mathlib/Data/List/Lattice.lean | 203 | 207 |
/-
Copyright (c) 2021 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.Convex.Topology
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.Seminorm
import Mathlib.Analysis... | obtain ⟨b, hb, hb', y, hy, rfl⟩ :=
exists_lt_of_gauge_lt absorbs (lt_add_of_pos_right (gauge s y) (half_pos hε))
calc
gauge s (a • x + b • y) ≤ a + b := gauge_le_of_mem (by positivity) <| by
rw [hs.add_smul ha.le hb.le]
| Mathlib/Analysis/Convex/Gauge.lean | 184 | 188 |
/-
Copyright (c) 2017 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Keeley Hoek
-/
import Mathlib.Algebra.NeZero
import Mathlib.Data.Int.DivMod
import Mathlib.Logic.Embedding.Basic
import Mathlib.Logic.Equiv.Set
import Mathlib.Tactic.... | Mathlib/Data/Fin/Basic.lean | 1,733 | 1,738 | |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.LinearAlgebra.Finsupp.Span
/-!
# Lie submodules of a Lie algebra
In this file we define Lie submodules, we construct ... | @[deprecated (since := "2024-11-24")]
alias independent_iff_toSubmodule := iSupIndep_iff_toSubmodule
| Mathlib/Algebra/Lie/Submodule.lean | 514 | 516 |
/-
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 | 2,306 | 2,307 | |
/-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Topology.UniformSpace.UniformConvergenceTopology
/-!
# Equicontinuity of a family of functions
Let `X` be a topological space and `α` a `UniformS... | filter, and if the family `𝓕` is equicontinuous at `x₀ : X` within `S`, then the limit is
continuous at `x₀` within `S`. -/
theorem Filter.Tendsto.continuousWithinAt_of_equicontinuousWithinAt {l : Filter ι} [l.NeBot]
{F : ι → X → α} {f : X → α} {S : Set X} {x₀ : X} (h₁ : ∀ x ∈ S, Tendsto (F · x) l (𝓝 (f x)))
... | Mathlib/Topology/UniformSpace/Equicontinuity.lean | 897 | 901 |
/-
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.Algebra.BigOperators.Group.Multiset.Basic
/-!
# Bind operation for multisets
This file defines a few basic operations on `Multiset`, notably the mona... | theorem bind_singleton (f : α → β) : (s.bind fun x => ({f x} : Multiset β)) = map f s :=
| Mathlib/Data/Multiset/Bind.lean | 130 | 130 |
/-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll, Anatole Dedecker
-/
import Mathlib.Analysis.LocallyConvex.Bounded
import Mathlib.Analysis.Seminorm
import Mathlib.Data.Real.Sqrt
import Mathlib.Topology.Algebra.Equicontinuit... | are exactly the sets which contain seminorm balls around `x`. -/
theorem WithSeminorms.mem_nhds_iff (hp : WithSeminorms p) (x : E) (U : Set E) :
U ∈ 𝓝 x ↔ ∃ s : Finset ι, ∃ r > 0, (s.sup p).ball x r ⊆ U := by
rw [hp.hasBasis_ball.mem_iff, Prod.exists]
/-- The open sets of a space whose topology is induced by a ... | Mathlib/Analysis/LocallyConvex/WithSeminorms.lean | 309 | 319 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Johannes Hölzl, Yury Kudryashov, Patrick Massot
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Order.Filter.AtTopBot.Archimedean
import Mathlib.Order.Iterate
impor... | · simp [_root_.pow_succ', mul_assoc, le_refl]
theorem le_geom {u : ℕ → ℝ} {c : ℝ} (hc : 0 ≤ c) (n : ℕ) (h : ∀ k < n, u (k + 1) ≤ c * u k) :
u n ≤ c ^ n * u 0 := by
apply (monotone_mul_left_of_nonneg hc).seq_le_seq n _ h _ <;>
| Mathlib/Analysis/SpecificLimits/Basic.lean | 222 | 226 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Sébastien Gouëzel, Yury Kudryashov, Dylan MacKenzie, Patrick Massot
-/
import Mathlib.Algebra.BigOperators.Module
import Mathlib.Algebra.Order.Field.Power
import M... | theorem Real.summable_pow_div_factorial (x : ℝ) : Summable (fun n ↦ x ^ n / n ! : ℕ → ℝ) := by
-- We start with trivial estimates
have A : (0 : ℝ) < ⌊‖x‖⌋₊ + 1 := zero_lt_one.trans_le (by simp)
have B : ‖x‖ / (⌊‖x‖⌋₊ + 1) < 1 := (div_lt_one A).2 (Nat.lt_floor_add_one _)
-- Then we apply the ratio test. The esti... | Mathlib/Analysis/SpecificLimits/Normed.lean | 814 | 827 |
/-
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, norm_cast]
theorem coeFn_le [Preorder β] {f g : α →ₘ[μ] β} : (f : α → β) ≤ᵐ[μ] g ↔ f ≤ g :=
liftRel_iff_coeFn.symm
instance instPartialOrder [PartialOrder β] : PartialOrder (α →ₘ[μ] β) :=
PartialOrder.lift toGerm toGerm_injective
| Mathlib/MeasureTheory/Function/AEEqFun.lean | 484 | 490 |
/-
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.Calculus.BumpFunction.FiniteDimension
import Mathlib.Geometry.Manifold.ContMDiff.Atlas
import Mathlib.Geometry.Manifold.ContMDiff.NormedSpac... |
theorem eventuallyEq_one : f =ᶠ[𝓝 c] 1 :=
f.eventuallyEq_one_of_dist_lt (mem_chart_source _ _) <| by rw [dist_self]; exact f.rIn_pos
@[simp]
| Mathlib/Geometry/Manifold/BumpFunction.lean | 163 | 167 |
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca, Johan Commelin
-/
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.Algebra.Polynomial.Lifts
import Mathlib.FieldTheory.Minpoly.Basic
import Mathlib.RingT... | suffices (f : E →ₗ[F] K) = (g : E →ₗ[F] K) by rwa [DFunLike.ext'_iff] at this ⊢
rw [funext_iff] at h
| Mathlib/FieldTheory/Minpoly/Field.lean | 220 | 221 |
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.AlgebraicTopology.DoldKan.Homotopies
import Mathlib.Tactic.Ring
/-!
# Study of face maps for the Dold-Kan correspondence
In this file, we obtain the technical... | · simp only [hj₂, Fin.eta, δ_comp_σ_succ, comp_id]
rfl
-- now, we assume j ≠ a (i.e. a < j)
have haj : a < j := (Ne.le_iff_lt hj₂).mp (by omega)
have ham : a ≤ m := by omega
rw [X.δ_comp_σ_of_gt', j.pred_succ]
swap
· rw [Fin.lt_iff_val_lt_val]
simpa only [Fin.val_mk, Fin.val_succ, add_lt_add_iff_r... | Mathlib/AlgebraicTopology/DoldKan/Faces.lean | 172 | 219 |
/-
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... | rw [← vecMulLinear_transpose, range_vecMulLinear, row_transpose]
theorem Matrix.mulVec_injective_iff {R : Type*} [CommRing R] {M : Matrix m n R} :
| Mathlib/LinearAlgebra/Matrix/ToLin.lean | 261 | 263 |
/-
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... | · rintro (⟨_, h, rfl⟩ | hm')
· simp only [Sigma.ext_iff, heq_eq_eq, true_and] at h
subst h
rfl
· simp only [fstIdx, Option.map_eq_some_iff, Sigma.exists,
| Mathlib/GroupTheory/CoprodI.lean | 535 | 539 |
/-
Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
import Mathlib.Topology.Bornology.Constructions
import Mathlib.Topology.MetricSpace.Pseudo.Def... | Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean | 429 | 433 | |
/-
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... |
variable (f)
/-- In an abelian category, any morphism that turns to zero when precomposed with the kernel of an
epimorphism factors through that epimorphism. -/
def epiDesc [Epi f] {T : C} (g : X ⟶ T) (hg : kernel.ι f ≫ g = 0) : Y ⟶ T :=
(epiIsCokernelOfKernel _ (limit.isLimit _)).desc (CokernelCofork.ofπ _ hg)... | Mathlib/CategoryTheory/Abelian/Basic.lean | 442 | 448 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.HomotopyCategory.HomComplex
import Mathlib.Algebra.Homology.HomotopyCategory.Shift
/-! Shifting cochains
Let `C` be a preadditive category. Gi... |
@[simp]
lemma leftShift_comp_zero_cochain (a n' : ℤ) (hn' : n + a = n') (γ' : Cochain L M 0) :
(γ.comp γ' (add_zero n)).leftShift a n' hn' =
(γ.leftShift a n' hn').comp γ' (add_zero n') := by
rw [leftShift_comp γ a n' hn' γ' (add_zero _) hn', mul_zero, Int.negOnePow_zero, one_smul]
lemma δ_rightShift (a n... | Mathlib/Algebra/Homology/HomotopyCategory/HomComplexShift.lean | 387 | 406 |
/-
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
-/
import Mathlib.Data.Set.Lattice.Image
import Mathlib.Order.Interval.Set.LinearOrder
/-!
# Extra lemmas about intervals
This file contains lemma... |
theorem IsLUB.iUnion_Iio_eq (h : IsLUB (range f) a) : ⋃ x, Iio (f x) = Iio a :=
h.dual.iUnion_Ioi_eq
| Mathlib/Order/Interval/Set/Disjoint.lean | 188 | 190 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Localization.Opposite
/-!
# Calculus of fractions
Following the definitions by [Gabriel and Zisman][gabriel-zisman-1967],
given a morphism prope... | homMk z₁ ≫ homMk z₂ = homMk (z₁.comp₀ z₂ z₃) := by
change Hom.comp _ _ = _
rw [Hom.comp_eq, comp_eq z₁ z₂ z₃ h₃]
lemma homMk_eq_of_leftFractionRel {X Y : C} (z₁ z₂ : W.LeftFraction X Y)
| Mathlib/CategoryTheory/Localization/CalculusOfFractions.lean | 495 | 499 |
/-
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.ULift
import Mathlib.Data.ZMod.Defs
import Mathlib.SetTheory.Cardinal.ToNat
import Mathlib.SetTheory.Cardinal.ENat
/-!
# Finite Cardinality Funct... |
@[simp]
| Mathlib/SetTheory/Cardinal/Finite.lean | 187 | 188 |
/-
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.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.GroupWithZero.Subgroup
import Mathlib.Data.Finite.Card
import Mathlib.Data.Finite.Pr... | · exact (le_of_eq (relindex_eq_zero_of_le_left inf_le_left h)).trans (zero_le _)
rw [← inf_relindex_right, inf_assoc, ← relindex_mul_relindex _ _ L inf_le_right inf_le_right,
inf_relindex_right, inf_relindex_right]
exact mul_le_mul_right' (relindex_le_of_le_right inf_le_right h) (K.relindex L)
| Mathlib/GroupTheory/Index.lean | 333 | 336 |
/-
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 | 1,553 | 1,555 | |
/-
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.Algebra.Operations
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.Lattice
import Mathlib.RingTheory.Coprime.Lemmas
import Mathlib... | Mathlib/RingTheory/Ideal/Operations.lean | 1,415 | 1,418 | |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Iterate
import Mathlib.Order.SemiconjSup
import Mathlib.Topology.Order.MonotoneContinuity
import M... | translationNumber_eq_of_tendsto₀ _ <| by simp [tendsto_const_nhds]
theorem translationNumber_eq_of_semiconjBy {f g₁ g₂ : CircleDeg1Lift} (H : SemiconjBy f g₁ g₂) :
τ g₁ = τ g₂ :=
| Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean | 615 | 618 |
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Johan Commelin, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Equivalence
import Mathlib.CategoryTheory.Yoneda
/-!
# Adjunctions between functors
`F ⊣ G` represents the dat... | /-- The property that describes how `homEquiv` transforms compositions `F X ⟶ Y ⟶ Y'` -/
homEquiv_naturality_right :
∀ {X Y Y'} (f : F.obj X ⟶ Y) (g : Y ⟶ Y'),
| Mathlib/CategoryTheory/Adjunction/Basic.lean | 334 | 336 |
/-
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, Yury Kudryashov, Rémy Degenne
-/
import Mathlib.Data.Set.Subsingleton
import Mathlib.Order.Interval.Set.Defs
/-!
# Intervals
In any pr... | Mathlib/Order/Interval/Set/Basic.lean | 1,116 | 1,116 | |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Independence.Kernel
import Mathlib.MeasureTheory.Constructions.Pi
/-!
# Independence of sets of sets and measure spaces (σ-algebras)
* A fami... | μ (⋂ i : ι, (f i ⁻¹' s i)) = μ.map (fun ω i ↦ f i ω) (univ.pi s) := by
constructor
· congr with x
rw [Measure.map_apply_of_aemeasurable (hf x) (hm x)]
· rw [Measure.map_apply_of_aemeasurable (aemeasurable_pi_lambda _ fun x ↦ hf x)
(.univ_pi hm)]
| Mathlib/Probability/Independence/Basic.lean | 626 | 631 |
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Mathlib.Control.Basic
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Option.Basic
im... | Mathlib/Data/List/Basic.lean | 2,611 | 2,621 | |
/-
Copyright (c) 2024 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.Cardinal.Arithmetic
import Mathlib.SetTheory.Ordinal.Principal
/-!
# Ordinal arithmetic with cardinals
This file co... | Mathlib/SetTheory/Cardinal/Ordinal.lean | 793 | 810 | |
/-
Copyright (c) 2020 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Algebra.Group.Conj
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.Group.Subgroup.Ker
/-!
# Basic results on subgroups
We prove basic results... | Mathlib/Algebra/Group/Subgroup/Basic.lean | 2,158 | 2,162 | |
/-
Copyright (c) 2022 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.Finsupp.Defs
import Mathlib.Data.Finset.Pairwise
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
/-!
# Sums of collections of Finsupp, ... | dsimp only [Function.comp_def]
simp only [quot_mk_to_coe'', map_coe, sup_coe, Finset.le_eq_subset,
Finset.sup_eq_union, Finset.bot_eq_empty, List.foldr_map]
simp only [Multiset.quot_mk_to_coe'', Multiset.map_coe, Multiset.coe_eq_coe] at hl
exact hl.symm.pairwise hd fun h ↦ _root_.Disjoint.symm h
theo... | Mathlib/Data/Finsupp/BigOperators.lean | 99 | 111 |
/-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Order.Interval.Set.OrdConnected
/-! # Subtypes of cond... | Mathlib/Order/CompleteLatticeIntervals.lean | 273 | 276 | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Michael Stoll
-/
import Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic
/-!
# Legendre symbol
This file contains results about Legendre symbols.
We define the Le... | map_one' := at_one p
| Mathlib/NumberTheory/LegendreSymbol/Basic.lean | 156 | 156 |
/-
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.GCDMonoid.Basic
import Mathlib.Algebra.Order.Group.Multiset
import Mathlib.Data.Multiset.FinsetOps
import Mathlib.Data.Multiset.Fold
/-!
# GCD... |
theorem gcd_map_mul (a : α) (s : Multiset α) : (s.map (a * ·)).gcd = normalize a * s.gcd := by
refine s.induction_on ?_ fun b s ih ↦ ?_
| Mathlib/Algebra/GCDMonoid/Multiset.lean | 153 | 155 |
/-
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.AffineSpace.Independent
import Mathlib.LinearAlgebra.AffineSpace.Pointwise
import Mathlib.LinearAlgebra.Basis.SMul
/-!
# Affine bases and bary... | theorem coord_apply_combination_of_mem (hi : i ∈ s) {w : ι → k} (hw : s.sum w = 1) :
b.coord i (s.affineCombination k b w) = w i := by
classical simp only [coord_apply, hi, Finset.affineCombination_eq_linear_combination, if_true,
| Mathlib/LinearAlgebra/AffineSpace/Basis.lean | 162 | 164 |
/-
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
... | theorem ediam_image (hf : Isometry f) (s : Set α) : EMetric.diam (f '' s) = EMetric.diam s :=
eq_of_forall_ge_iff fun d => by simp only [EMetric.diam_le_iff, forall_mem_image, hf.edist_eq]
| Mathlib/Topology/MetricSpace/Isometry.lean | 134 | 135 |
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Data.List.Rotate
import Mathlib.GroupTheory.Perm.Support
/-!
# Permutations from a list
A list `l : List α` ... | | succ n hn =>
rw [← rotate_rotate, formPerm_rotate_one, hn]
rwa [IsRotated.nodup_iff]
| Mathlib/GroupTheory/Perm/List.lean | 225 | 227 |
/-
Copyright (c) 2021 Jakob Scholbach. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob Scholbach
-/
import Mathlib.Algebra.Algebra.Defs
import Mathlib.FieldTheory.Separable
/-!
# Separable degree
This file contains basics about the separable degree of a polynom... | theorem IsSeparableContraction.degree_eq [hF : ExpChar F q] (g : F[X])
(hg : IsSeparableContraction q f g) : g.natDegree = hf.degree := by
cases hF
· rcases hg with ⟨_, m, hm⟩
rw [one_pow, expand_one] at hm
rw [hf.eq_degree, hm]
· rcases hg with ⟨hg, m, hm⟩
let g' := Classical.choose hf
obtain... | Mathlib/RingTheory/Polynomial/SeparableDegree.lean | 121 | 131 |
/-
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... | section Left
variable [Preorder α] [Preorder β] [SMul α β] [Zero α]
| Mathlib/Algebra/Order/Module/Defs.lean | 757 | 758 |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Asymptotics.SpecificAsympt... | theorem abs_im_pow_eventuallyLE_exp_re (hl : IsExpCmpFilter l) (n : ℕ) :
(fun z : ℂ => |z.im| ^ n) ≤ᶠ[l] fun z => Real.exp z.re := by
simpa using (hl.isLittleO_im_pow_exp_re n).bound zero_lt_one
/-- If `l : Filter ℂ` is an "exponential comparison filter", then $\log |z| =o(ℜ z)$ along `l`.
This is the main lemma... | Mathlib/Analysis/SpecialFunctions/CompareExp.lean | 107 | 116 |
/-
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.Basic
import Mathlib.Algebra.ContinuedFractions.Translations
import Mathlib.Algebra.Order.Floor.Ring
/-!
# ... | #### Translation of the Values of the Sequence
Now let's show how the values of the sequences correspond to one another.
-/
| Mathlib/Algebra/ContinuedFractions/Computation/Translations.lean | 209 | 212 |
/-
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.Typeclasses.Finite
import Mathlib.MeasureTheory.Measure.Typeclasses.NoAtoms
import Mathlib.MeasureTheory.Measure.... | Mathlib/MeasureTheory/Measure/Typeclasses.lean | 711 | 715 | |
/-
Copyright (c) 2022 Niels Voss. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Niels Voss
-/
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.FieldTheory.Finite.Basic
import Mathlib.Order.Filter.Cofinite
/-!
# Fermat Pseudoprimes
In this file we define... | private theorem bp_helper {b p : ℕ} (hb : 0 < b) (hp : 1 ≤ p) :
b ^ (2 * p) - 1 - (b ^ 2 - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b) :=
have hi_bsquared : 1 ≤ b ^ 2 := Nat.one_le_pow _ _ hb
calc
b ^ (2 * p) - 1 - (b ^ 2 - 1) = b ^ (2 * p) - (1 + (b ^ 2 - 1)) := by rw [Nat.sub_sub]
_ = b ^ (2 * p) - (1 + ... | Mathlib/NumberTheory/FermatPsp.lean | 150 | 155 |
/-
Copyright (c) 2023 Adrian Wüthrich. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adrian Wüthrich
-/
import Mathlib.Combinatorics.SimpleGraph.AdjMatrix
import Mathlib.LinearAlgebra.Matrix.PosDef
/-!
# Laplacian Matrix
This module defines the Laplacian matrix of a... | simp_rw [lapMatrix, sub_mulVec, Pi.sub_apply, degMatrix_mulVec_apply, adjMatrix_mulVec_apply]
theorem lapMatrix_mulVec_const_eq_zero [NonAssocRing R] :
| Mathlib/Combinatorics/SimpleGraph/LapMatrix.lean | 62 | 64 |
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Mathlib.Control.Basic
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Option.Basic
im... | Mathlib/Data/List/Basic.lean | 1,861 | 1,863 | |
/-
Copyright (c) 2023 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Topology.AlexandrovDiscrete
import Mathlib.Topology.ContinuousMap.Basic
import Mathlib.Topology.Order.LowerUpperTopology
/-!
# Upper and lower... | def WithLowerSetHomeomorph : WithLowerSet α ≃ₜ α :=
| Mathlib/Topology/Order/UpperLowerSetTopology.lean | 307 | 307 |
/-
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... | theorem birthday_neg : ∀ x : PGame, (-x).birthday = x.birthday
| Mathlib/SetTheory/Game/Birthday.lean | 107 | 107 |
/-
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.Data.Finset.Sort
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Sign
import Mathlib.LinearAlgebra.AffineSpace.Combination
import Mathlib.LinearAlg... | theorem centroid_eq_of_range_eq {n : ℕ} {s₁ s₂ : Simplex k P n}
(h : Set.range s₁.points = Set.range s₂.points) :
Finset.univ.centroid k s₁.points = Finset.univ.centroid k s₂.points := by
rw [← Set.image_univ, ← Set.image_univ, ← Finset.coe_univ] at h
exact
Finset.univ.centroid_eq_of_inj_on_of_image_eq ... | Mathlib/LinearAlgebra/AffineSpace/Independent.lean | 949 | 969 |
/-
Copyright (c) 2021 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Sébastien Gouëzel
-/
import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
import Mathlib.MeasureTheory.Constructions.BorelSpace.Metric
import Mathlib.MeasureTheory... | exact measure_iUnion_le _
_ = 0 := by simp only [H, tsum_zero]
intro R
apply addHaar_eq_zero_of_disjoint_translates_aux μ u
(isBounded_closedBall.subset inter_subset_right) hu _ (h's.inter measurableSet_closedBall)
refine pairwise_disjoint_mono hs fun n => ?_
exact add_subset_add Subset.rfl in... | Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean | 160 | 175 |
/-
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.FreeAlgebra
import Mathlib.RingTheory.Adjoin.Polynomial
import Mathlib.RingTheory.Adjoin.Tower
import Mathlib.RingTheory.Ideal.Quotient.Operati... | instance prod [hA : FiniteType R A] [hB : FiniteType R B] : FiniteType R (A × B) :=
⟨by rw [← Subalgebra.prod_top]; exact hA.1.prod hB.1⟩
theorem isNoetherianRing (R S : Type*) [CommRing R] [CommRing S] [Algebra R S]
[h : Algebra.FiniteType R S] [IsNoetherianRing R] : IsNoetherianRing S := by
obtain ⟨s, hs⟩ :=... | Mathlib/RingTheory/FiniteType.lean | 188 | 196 |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.CategoryTheory.Adjunction.FullyFaithful
import Mathlib.CategoryTheory.Adjunction.Limits
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq
import Ma... | theorem IsUniversalColimit.of_iso {F : J ⥤ C} {c c' : Cocone F} (hc : IsUniversalColimit c)
(e : c ≅ c') : IsUniversalColimit c' := by
intro F' c'' α f h hα H
have : c'.ι ≫ (Functor.const J).map e.inv.hom = c.ι := by
ext j
exact e.inv.2 j
apply hc c'' α (f ≫ e.inv.1) (by rw [Functor.map_comp, ← reasso... | Mathlib/CategoryTheory/Limits/VanKampen.lean | 129 | 136 |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Yaël Dillies
-/
import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap
/-!
# Integral average of a function
In this file we define `MeasureTheory.average... | (ht : NullMeasurableSet t μ) (hsμ : μ s ≠ ∞) (htμ : μ t ≠ ∞) (hfs : IntegrableOn f s μ)
(hft : IntegrableOn f t μ) :
⨍ x in s ∪ t, f x ∂μ ∈ [⨍ x in s, f x ∂μ -[ℝ] ⨍ x in t, f x ∂μ] := by
by_cases hse : μ s = 0
| Mathlib/MeasureTheory/Integral/Average.lean | 388 | 391 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
import Mathlib.Analysis.Normed.Module.Basic
import Mathlib.MeasureTheory.Function.SimpleFuncDense
/-!
# Strongly measurable and finitely strongly meas... | Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean | 1,489 | 1,492 | |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Eric Wieser
-/
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.RingTheory.Ideal.Nonunits
import Mathlib.RingTheory.Ideal.Quotient.De... | exact ⟨h x, fun h' => h'.symm ▸ I.zero_mem⟩⟩
/-- `CharP.quotient'` as an `Iff`. -/
theorem quotient_iff {R : Type*} [CommRing R] (n : ℕ) [CharP R n] (I : Ideal R) :
CharP (R ⧸ I) n ↔ ∀ x : ℕ, ↑x ∈ I → (x : R) = 0 := by
| Mathlib/Algebra/CharP/Quotient.lean | 47 | 51 |
/-
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... | rw [← empty_mem_iff_bot, mem_ae_iff, compl_empty, measure_univ_eq_zero]
| Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 1,306 | 1,307 |
/-
Copyright (c) 2020 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap
import Mathlib.MeasureTheory.Integral.Bochner.FundThmCalculus
import Mathlib.MeasureT... | Mathlib/MeasureTheory/Integral/SetIntegral.lean | 288 | 293 |
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