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/-
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.Homotopy
import Mathlib.Algebra.Ring.NegOnePow
import Mathlib.Algebra.Category.Grp.Preadditive
import Mathlib.Tactic.Linarith
import Mathlib.Cat... | @[simp]
lemma map_ofHom :
(Cochain.ofHom f).map Φ = Cochain.ofHom ((Φ.mapHomologicalComplex _).map f) := by aesop_cat
end Cochain
variable (n)
| Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean | 816 | 822 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Analytic.Within
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries
/-!
# Higher d... | intro y hyt
refine (h_fderiv y (htu hyt)).hasFDerivAt ?_
exact mem_nhds_iff.mpr ⟨t, htu, ht, hyt⟩
· rintro ⟨f', ⟨u, H, h_fderiv⟩, h_cont_diff⟩
refine ⟨u, H, by simp, f', fun x hxu ↦ ?_, h_cont_diff.contDiffWithinAt⟩
exact (h_fderiv x hxu).hasFDerivWithinAt
protected theorem ContDiffAt.eventually ... | Mathlib/Analysis/Calculus/ContDiff/Defs.lean | 996 | 1,003 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Kevin Buzzard, Yury Kudryashov, Eric Wieser
-/
import Mathlib.Algebra.Algebra.Prod
import Mathlib.Algebra.Group.Graph
import Mathlib.LinearAlgebra.Span.... | @[simp]
theorem prod_map_fst : (prod p q).map (fst R M M₂) = p := by
ext x; simp [(⟨0, zero_mem _⟩ : ∃ x, x ∈ q)]
@[simp]
| Mathlib/LinearAlgebra/Prod.lean | 501 | 505 |
/-
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.HomologicalComplex
import Mathlib.Algebra.Homology.ShortComplex.SnakeLemma
import Mathlib.Algebra.Homology.ShortComplex.ShortExact
... | (snakeInput hS i j hij).L₂'_exact
include hS in
/-- Exactness of `S.X₁.homology i ⟶ S.X₂.homology i ⟶ S.X₃.homology i`. -/
lemma homology_exact₂ : (ShortComplex.mk (HomologicalComplex.homologyMap S.f i)
(HomologicalComplex.homologyMap S.g i) (by rw [← HomologicalComplex.homologyMap_comp,
S.zero, Homologica... | Mathlib/Algebra/Homology/HomologySequence.lean | 299 | 312 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.CharP.Frobenius
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.RingTheory.Polynomial.Basic
/... | theorem contract_one {f : R[X]} : contract 1 f = f :=
ext fun n ↦ by rw [coeff_contract one_ne_zero, mul_one]
@[simp] theorem contract_C (r : R) : contract p (C r) = C r := by simp [contract]
| Mathlib/Algebra/Polynomial/Expand.lean | 201 | 204 |
/-
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... | rfl
/-- A single `List.map` of a composition of functions is equal to
composing a `List.map` with another `List.map`, fully applied.
This is the reverse direction of `List.map_map`.
-/
| Mathlib/Data/List/Basic.lean | 720 | 725 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Operations
import Mathlib.Order.Basic
import Mathlib.Order.BooleanAlgebra
import Mathlib.Tactic.Tauto
import Mathlib.Tactic.B... | Mathlib/Data/Set/Basic.lean | 1,937 | 1,939 | |
/-
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.Set.Piecewise
import Mathlib.Logic.Equiv.Defs
import Mathlib.Tactic.Core
import Mathlib.Tactic.Attr.Core
/-!
# Partial equivalences
This f... | simp only [mem_inter_iff, restr_source] at hx
exact he.2 hx.1
| Mathlib/Logic/Equiv/PartialEquiv.lean | 720 | 721 |
/-
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.Basic
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
/-!
# Left Homology of short complexes
Given a short complex `S : Shor... | LeftHomologyMapData.comp_φH]
@[reassoc]
lemma cyclesMap'_comp (φ₁ : S₁ ⟶ S₂) (φ₂ : S₂ ⟶ S₃)
(h₁ : S₁.LeftHomologyData) (h₂ : S₂.LeftHomologyData) (h₃ : S₃.LeftHomologyData) :
cyclesMap' (φ₁ ≫ φ₂) h₁ h₃ = cyclesMap' φ₁ h₁ h₂ ≫ cyclesMap' φ₂ h₂ h₃ := by
let γ₁ := leftHomologyMapData φ₁ h₁ h₂
let γ₂ := le... | Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean | 607 | 614 |
/-
Copyright (c) 2023 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston
-/
import Mathlib.FieldTheory.Fixed
import Mathlib.RepresentationTheory.GroupCohomology.LowDegree
/-!
# Hilbert's Theorem 90
Let `L/K` be a finite extension of f... | /- the set `Aut_K(L)` is linearly independent in the `L`-vector space `L → L`, by Dedekind's
linear independence of characters -/
have : LinearIndependent L (fun (f : L ≃ₐ[K] L) => (f : L → L)) :=
LinearIndependent.comp (ι' := L ≃ₐ[K] L)
(linearIndependent_monoidHom L L) (fun f => f)
(fun x y h => by ... | Mathlib/RepresentationTheory/GroupCohomology/Hilbert90.lean | 61 | 70 |
/-
Copyright (c) 2021 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Algebra.Homology.Linear
import Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex
import Mathlib.Tactic.Abel
/-!
# Chain homotopies
We define chain... | { hom := hom
zero := zero
comm := by
intro i
rw [HomologicalComplex.zero_f_apply, add_zero]
rfl }
open Classical in
/-- Homotopy to zero for maps constructed with `nullHomotopicMap'` -/
| Mathlib/Algebra/Homology/Homotopy.lean | 321 | 329 |
/-
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.Localization.CalculusOfFractions.Fractions
import Mathlib.CategoryTheory.Localization.HasLocalization
import Mathlib.CategoryTheory.Preadditive.Ad... | @[reassoc (attr := simp)]
lemma comp_add' (f : L.obj X ⟶ L.obj Y) (g₁ g₂ : L.obj Y ⟶ L.obj Z) :
f ≫ add' W g₁ g₂ = add' W (f ≫ g₁) (f ≫ g₂) := by
obtain ⟨α, hα⟩ := exists_leftFraction L W f
obtain ⟨β, hβ₁, hβ₂⟩ := exists_leftFraction₂ L W g₁ g₂
obtain ⟨γ, hγ₁, hγ₂⟩ := (RightFraction₂.mk _ α.hs β.f β.f').exist... | Mathlib/CategoryTheory/Localization/CalculusOfFractions/Preadditive.lean | 196 | 210 |
/-
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 ... | right_inv := fun N ↦ by ext; simp [e.apply_eq_iff_eq_symm_apply]
map_rel_iff' := fun {_ _} ↦ Set.image_subset_image_iff e.injective
| Mathlib/Algebra/Lie/Submodule.lean | 916 | 918 |
/-
Copyright (c) 2019 Michael Howes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Howes, Newell Jensen
-/
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.GroupTheory.FreeGroup.Basic
import Mathlib.GroupTheory.QuotientGroup.Defs
/-!
# Defining a gr... | Presented groups satisfy a universal property. If `G` is a group and `f : α → G` is a map such that
the images of `f` satisfy all the given relations, then `f` extends uniquely to a group homomorphism
from `PresentedGroup rels` to `G`.
-/
| Mathlib/GroupTheory/PresentedGroup.lean | 101 | 104 |
/-
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_eq_self [DecidableEq n] (A : Matrix m n α) (i : n) :
(A.updateCol i fun j => A j i) = A :=
funext fun j => Function.update_eq_self i (A j)
| Mathlib/Data/Matrix/RowCol.lean | 334 | 337 |
/-
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
/-!
# ... | intro n
induction n with
| zero => rfl
| succ n hn =>
rw [pow_succ', mul_apply, cycleOf_apply, hn, if_pos, pow_succ', mul_apply]
| Mathlib/GroupTheory/Perm/Cycle/Factors.lean | 58 | 62 |
/-
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.Topology.Algebra.Order.LiminfLimsup
import Mathlib.Topology.Metrizable.Real
/-!
# Limsup and liminf of reals
This file compiles filter-related results ab... | exact ⟨b.toNNReal, by simp, fun x _ ↦ by simpa [*] using hb _⟩
· rintro ⟨b, hb₀, hb⟩
exact ⟨b, fun x hx ↦ hb _ (hx.exists.choose_spec.trans' (by simp)) hx⟩
@[simp, norm_cast] lemma isCoboundedUnder_ge_toReal :
IsCoboundedUnder (· ≥ ·) f (fun i ↦ (u i : ℝ)) ↔ IsCoboundedUnder (· ≥ ·) f u := by
simp only... | Mathlib/Order/Filter/ENNReal.lean | 86 | 93 |
/-
Copyright (c) 2024 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable
/-!
# Asymptotic bounds for Jacobi theta functions
The goal of this file is to establish some tech... | lemma summable_f_int (k : ℕ) (a : ℝ) {t : ℝ} (ht : 0 < t) : Summable (f_int k a t) := by
apply Summable.of_norm
suffices ∀ n, ‖f_int k a t n‖ = ‖(Int.rec (f_nat k a t) (f_nat k (1 - a) t) : ℤ → ℝ) n‖ from
funext this ▸ (HasSum.int_rec (summable_f_nat k a ht).hasSum
(summable_f_nat k (1 - a) ht).hasSum).su... | Mathlib/NumberTheory/ModularForms/JacobiTheta/Bounds.lean | 218 | 228 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Kevin Buzzard, Yury Kudryashov, Frédéric Dupuis,
Heather Macbeth
-/
import Mathlib.Algebra.Module.Submodule.Ker
import Mathlib.Algebra.Module.Submodul... | [AddCommGroup P] [Module R P] {f : M →ₗ[R] P} {g : P →ₗ[R] M} (h : f ∘ₗ g = .id) :
ker f = range (LinearMap.id - g ∘ₗ f) :=
le_antisymm (fun x hx ↦ ⟨x, show x - g (f x) = x by rw [hx, map_zero, sub_zero]⟩) <|
range_le_ker_iff.mpr <| by rw [comp_sub, comp_id, ← comp_assoc, h, id_comp, sub_self]
| Mathlib/Algebra/Module/Submodule/Range.lean | 200 | 203 |
/-
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 | 777 | 793 | |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Jeremy Avigad, Simon Hudon
-/
import Batteries.WF
import Mathlib.Data.Part
import Mathlib.Data.Rel
import Mathlib.Tactic.GeneralizeProofs
/-!
# Partial functions
This... | theorem prodMap_id_id : (PFun.id α).prodMap (PFun.id β) = PFun.id _ := by
aesop
| Mathlib/Data/PFun.lean | 585 | 586 |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Batteries.Data.Rat.Lemmas
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Rat.Init
import Mathlib.Order.Basic
import Mathlib.Tactic.Commo... | Mathlib/Data/Rat/Defs.lean | 490 | 491 | |
/-
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 | 698 | 699 | |
/-
Copyright (c) 2022 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.MeasureTheory.Integral.Prod
import Mathlib.Meas... | * `φ` is a sequence of nonnegative functions with integral `1` as `i` tends to `l`;
* The support of `φ` tends to small neighborhoods around `(0 : G)` as `i` tends to `l`;
* `g i` is `mu`-a.e. strongly measurable as `i` tends to `l`;
* `g i x` tends to `z₀` as `(i, x)` tends to `l ×ˢ 𝓝 x₀`;
* `k i` tends to `x₀`.
See... | Mathlib/Analysis/Convolution.lean | 794 | 829 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.AlgebraicGeometry.Spec
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.CategoryTheory.Elementwise
/-!
# The category of schemes
A schem... |
@[simp]
theorem id.base (X : Scheme) : (𝟙 X :).base = 𝟙 _ :=
| Mathlib/AlgebraicGeometry/Scheme.lean | 251 | 253 |
/-
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 | 1,369 | 1,373 | |
/-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Homeomorph.Lemmas
import Mathlib.Topology.Sets.Opens
/-!
# The OnePoint Compactification
We ... | IsOpen ((↑) '' s : Set (OnePoint X))ᶜ ↔ IsClosed s ∧ IsCompact s := by
rw [isOpen_iff_of_mem, ← preimage_compl, compl_compl, preimage_image_eq _ coe_injective]
| Mathlib/Topology/Compactification/OnePoint.lean | 244 | 245 |
/-
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... | (g₁ + g₂).support = g₁.support ∪ g₂.support :=
le_antisymm support_zipWith fun a ha =>
(Finset.mem_union.1 ha).elim
| Mathlib/Data/Finsupp/Defs.lean | 481 | 483 |
/-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Eric Wieser
-/
import Mathlib.LinearAlgebra.Span.Basic
import Mathlib.LinearAlgebra.BilinearMap
/-!
# Images of pairs of submodules under bilinear maps
This file provides `Subm... | Mathlib/Algebra/Module/Submodule/Bilinear.lean | 166 | 167 | |
/-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin, Lu-Ming Zhang
-/
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Algebra.Pi
import Mathlib.Algebra.BigOp... | Mathlib/Data/Matrix/Basic.lean | 1,805 | 1,809 | |
/-
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
/... | def closedBall (x : E) (r : ℝ) :=
{ y : E | p (y - x) ≤ r }
| Mathlib/Analysis/Seminorm.lean | 608 | 610 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Simon Hudon, Mario Carneiro
-/
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Init
import Mathlib.Data.Int.Init
import Mathlib.... | Mathlib/Algebra/Group/Basic.lean | 1,220 | 1,221 | |
/-
Copyright (c) 2023 Xavier Généreux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Généreux
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
import Mathlib.Analysis.Complex.PhragmenLindelof
/-!
# Hadamard three-lines Theorem
In this file we present a ... | simp only [norm_cpow_eq_rpow_re_of_pos (sSupNormIm_eps_pos f hε 1), sub_re,
one_re, neg_re, Real.rpow_le_one_of_one_le_of_nonpos
hM1_one (Right.neg_nonpos_iff.mpr hset.1)]
-- `0 < sSupNormIm f 1 < 1`
· rw [not_le] at hM1_one; apply le_trans _ (le_max_right _ _)
simp only [norm_cpow_e... | Mathlib/Analysis/Complex/Hadamard.lean | 180 | 200 |
/-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.Add
/-!
# Local extrema of differentiable functions
## Main definitions
In a real normed space `E` we define `posTangentCo... | else by rw [fderivWithin_zero_of_not_differentiableWithinAt hf]; rfl
/-- If `f` has a local max on `s` at `a`, `f'` is a derivative of `f` at `a` within `s`, and
both `y` and `-y` belong to the positive tangent cone of `s` at `a`, then `f' y ≤ 0`. -/
| Mathlib/Analysis/Calculus/LocalExtr/Basic.lean | 170 | 173 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.CoreAttrs
/-!
# (Semi-)lattices
Semilatti... | @[simp]
theorem inf_apply [∀ i, Min (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊓ g) i = f i ⊓ g i :=
rfl
theorem inf_def [∀ i, Min (α' i)] (f g : ∀ i, α' i) : f ⊓ g = fun i => f i ⊓ g i :=
rfl
| Mathlib/Order/Lattice.lean | 808 | 813 |
/-
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.Dynamics.Ergodic.AddCircle
import Mathlib.MeasureTheory.Covering.LiminfLimsup
/-!
# Well-approximable numbers and Gallagher's ergodic theorem
Gallagher's e... | @[to_additive]
theorem image_pow_subset (n : ℕ) (hm : 0 < m) :
(fun (y : A) => y ^ m) '' approxOrderOf A (n * m) δ ⊆ approxOrderOf A n (m * δ) := by
rintro - ⟨a, ha, rfl⟩
obtain ⟨b, hb : orderOf b = n * m, hab : a ∈ ball b δ⟩ := mem_approxOrderOf_iff.mp ha
replace hb : b ^ m ∈ {y : A | orderOf y = n} := by
... | Mathlib/NumberTheory/WellApproximable.lean | 108 | 116 |
/-
Copyright (c) 2024 Mitchell Lee. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mitchell Lee, Óscar Álvarez
-/
import Mathlib.GroupTheory.Coxeter.Length
import Mathlib.Data.List.GetD
import Mathlib.Tactic.Group
/-!
# Reflections, inversions, and inversion sequences... | simpa [lengthParity_eq_ofAdd_length, ZMod.eq_one_iff_odd]
rcases ht with ⟨w, i, rfl⟩
simp [lengthParity_simple]
theorem length_mul_left_ne (w : W) : ℓ (w * t) ≠ ℓ w := by
suffices cs.lengthParity (w * t) ≠ cs.lengthParity w by
| Mathlib/GroupTheory/Coxeter/Inversion.lean | 88 | 93 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.AtTopBot.BigOperators
import Mathlib.Topology.Separation.Hausdorff
/-!
# Infinite sum and pro... |
@[to_additive]
theorem hasProd_fintype [Fintype β] (f : β → α) : HasProd f (∏ b, f b) :=
| Mathlib/Topology/Algebra/InfiniteSum/Defs.lean | 135 | 137 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Yury Kudryashov, Neil Strickland
-/
import Mathlib.Algebra.Ring.Semiconj
import Mathlib.Algebra.Ring.Units
import Mathlib.Algebra.Gro... | Mathlib/Algebra/Ring/Commute.lean | 265 | 266 | |
/-
Copyright (c) 2023 Yaël Dillies, Vladimir Ivanov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Vladimir Ivanov
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.BigOperators.... | congr with h
simp only [filter_map, Function.comp_def, Equiv.coe_toEmbedding, RelIso.coe_fn_toEquiv,
| Mathlib/Combinatorics/SetFamily/AhlswedeZhang.lean | 214 | 215 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Data.Nat.Log
import Mathlib.Data.Nat.Pri... | exact Nat.geom_sum_Ico_le hp.two_le _ _
theorem multiplicity_choose_aux {p n b k : ℕ} (hp : p.Prime) (hkn : k ≤ n) :
| Mathlib/Data/Nat/Multiplicity.lean | 174 | 176 |
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Comma.Basic
/-!
# The category of arrows
The category of arrows, with morphisms commutative squares.
We set this up as a specialization ... | IsIso g.hom := by
rw [iso_w' (asIso sq)]
| Mathlib/CategoryTheory/Comma/Arrow.lean | 208 | 209 |
/-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
import Mathlib.CategoryTheory.Sites.DenseSubsite.InducedTopology
import Mathlib.CategoryTheory.Sites.LocallyBijective
import Mathlib.CategoryTheory.Sites.Preserve... | instance (priority := 900) [G.IsEquivalence] : IsCoverDense G J where
is_cover U := by
let e := (asEquivalence G).symm
convert J.top_mem U
ext Y f
simp only [Sieve.functorPushforward_apply, Presieve.functorPushforward, exists_and_left,
Sieve.top_apply, iff_true]
let g : e.inverse.obj _ ⟶ U :... | Mathlib/CategoryTheory/Sites/Equivalence.lean | 51 | 65 |
/-
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 ... | (leftInv p i x (c : Composition (n + 2)).length).compAlongComposition
(p.compContinuousLinearMap i.symm) c
| Mathlib/Analysis/Analytic/Inverse.lean | 68 | 69 |
/-
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.Algebra.Ring.Pointwise.Set
import Mathlib.Order.Filter.AtTopBot.CompleteLattice
import Mathlib.Order.Filter.AtTopBot.G... | variable [TopologicalSpace α] [CommGroup α] [LinearOrder α] [IsOrderedMonoid α]
[OrderTopology α]
variable {l : Filter β} {f g : β → α}
@[to_additive]
theorem nhds_eq_iInf_mabs_div (a : α) : 𝓝 a = ⨅ r > 1, 𝓟 { b | |a / b|ₘ < r } := by
| Mathlib/Topology/Order/LeftRightNhds.lean | 368 | 373 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Data.List.Basic
/-!
# Double universal quantification on a list
This file provides an API for `List.Forall₂` (definition in `Data.Lis... | ⟨fun H => by cases H; rfl, by rintro rfl; exact Forall₂.nil⟩
@[simp]
theorem forall₂_nil_right_iff {l} : Forall₂ R l nil ↔ l = nil :=
⟨fun H => by cases H; rfl, by rintro rfl; exact Forall₂.nil⟩
theorem forall₂_cons_left_iff {a l u} :
Forall₂ R (a :: l) u ↔ ∃ b u', R a b ∧ Forall₂ R l u' ∧ u = b :: u' :=
If... | Mathlib/Data/List/Forall2.lean | 61 | 69 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Sébastien Gouëzel,
Rémy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
import Qq
/-! # P... | · rw [zero_rpow h, zero_eq_mul]
have : y ≠ 0 ∨ z ≠ 0 := not_and_or.1 fun ⟨hy, hz⟩ => h <| hy.symm ▸ hz.symm ▸ zero_add 0
exact this.imp zero_rpow zero_rpow
· exact rpow_add pos _ _
| Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 187 | 191 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
import Mathlib.Data.Sum.Basic
import Mathlib.Logic.Equiv.Option
import Mathlib.Logic.Equiv.Sum
import Mathlib.Logic.Function.Conjugate
impor... | Mathlib/Logic/Equiv/Basic.lean | 1,685 | 1,687 | |
/-
Copyright (c) 2020 Kenji Nakagawa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenji Nakagawa, Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.Algebra.Subalgebra.Pointwise
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.RingTheory.Spect... | (e : ι → ℕ) (prime : ∀ i ∈ s, Prime (P i))
(coprime : ∀ᵉ (i ∈ s) (j ∈ s), i ≠ j → P i ≠ P j) (x : s → R) :
∃ y, ∀ (i) (hi : i ∈ s), y - x ⟨i, hi⟩ ∈ P i ^ e i := by
obtain ⟨y, hy⟩ :=
IsDedekindDomain.exists_representative_mod_finset P e prime coprime fun i =>
Ideal.Quotient.mk _ (x i)
exact ⟨y,... | Mathlib/RingTheory/DedekindDomain/Ideal.lean | 1,308 | 1,333 |
/-
Copyright (c) 2020 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.NormedSpace.IndicatorFunction
import Mathlib.Data.Fintype.Order
import Mathlib.MeasureTheory.Function.AEEqFun
import Mathlib.Me... | eLpNorm (f - g) p μ = eLpNorm (g - f) p μ := by simp [← eLpNorm_neg (f := f - g)]
theorem MemLp.neg {f : α → E} (hf : MemLp f p μ) : MemLp (-f) p μ :=
⟨AEStronglyMeasurable.neg hf.1, by simp [hf.right]⟩
| Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean | 192 | 195 |
/-
Copyright (c) 2023 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Set.Image
import Mathlib.Topology.Bases
import Mathlib.Topology.Inseparable
import Mathlib.Topology.Compactness.Exterior
/-!
# Alexandrov-discrete to... | alexandrovDiscrete_coinduced.sup alexandrovDiscrete_coinduced
instance Sigma.instAlexandrovDiscrete {ι : Type*} {π : ι → Type*} [∀ i, TopologicalSpace (π i)]
[∀ i, AlexandrovDiscrete (π i)] : AlexandrovDiscrete (Σ i, π i) :=
| Mathlib/Topology/AlexandrovDiscrete.lean | 197 | 200 |
/-
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... |
@[simp]
theorem add_ofNat' (m n) : Num.ofNat' (m + n) = Num.ofNat' m + Num.ofNat' n := by
induction n
| Mathlib/Data/Num/Lemmas.lean | 221 | 224 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad
-/
import Mathlib.Data.Finset.Basic
import Mathlib.Data.Finset.Image
/-!
# Cardinality of a finite set
This defines the cardinality of a `Fins... |
theorem inj_on_of_surj_on_of_card_le (f : ∀ a ∈ s, β) (hf : ∀ a ha, f a ha ∈ t)
(hsurj : ∀ b ∈ t, ∃ a ha, f a ha = b) (hst : #s ≤ #t) ⦃a₁⦄ (ha₁ : a₁ ∈ s) ⦃a₂⦄
(ha₂ : a₂ ∈ s) (ha₁a₂ : f a₁ ha₁ = f a₂ ha₂) : a₁ = a₂ := by
let f' : s → β := fun a ↦ f a a.2
have hsurj' : Set.SurjOn f' s.attach t := fun x hx ↦ ... | Mathlib/Data/Finset/Card.lean | 458 | 465 |
/-
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.Measure.Lebesgue.EqHaar
import Mathlib.MeasureTheory.Covering.Besicovitch
import Mathlib.Tactic.AdaptationNote
import Mathlib.Algeb... | Finset.mem_image, imp_true_iff, true_and]
have h's : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 - goodδ E ≤ ‖c - d‖ := by
simp only [s, forall_apply_eq_imp_iff, forall_exists_index, Finset.mem_univ, Finset.mem_image,
Ne, exists_true_left, forall_apply_eq_imp_iff, forall_true_left, true_and]
intro i j hij
hav... | Mathlib/MeasureTheory/Covering/BesicovitchVectorSpace.lean | 277 | 298 |
/-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Dagur Asgeirsson
-/
import Mathlib.Topology.Category.CompHaus.Basic
import Mathlib.Topology.Category.CompHausLike.Limits
/-!
# Explicit limits and colimits
This file applies ... | Mathlib/Topology/Category/CompHaus/Limits.lean | 178 | 183 | |
/-
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.RingTheory.Regular.IsSMulRegular
import Mathlib.RingTheory.Artinian.Module
import Mathlib.RingTheory.Nakayama
import Mathlib.Algebra.Equiv.TransferInst... |
section Perm
open LinearMap in
private lemma IsWeaklyRegular.swap {a b : R} (h1 : IsWeaklyRegular M [a, b])
(h2 : torsionBy R M b = a • torsionBy R M b → torsionBy R M b = ⊥) :
IsWeaklyRegular M [b, a] := by
rw [isWeaklyRegular_cons_iff, isWeaklyRegular_singleton_iff] at h1 ⊢
obtain ⟨ha, hb⟩ := h1
rw [←... | Mathlib/RingTheory/Regular/RegularSequence.lean | 575 | 587 |
/-
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.Topology.Bornology.Basic
/-!
# Locally bounded maps
This file defines locally bounded maps between bornologies.
We use the `DFunLike` design, so each ty... | Mathlib/Topology/Bornology/Hom.lean | 195 | 197 | |
/-
Copyright (c) 2023 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Algebra.Unitization
import Mathlib.Algebra.Star.Subalgebra
import Mathlib.GroupTheory.GroupAction.Ring
/-!
# Relating unital and non-unital subs... | (unitization_range s).ge x.property
exact ⟨a, Subtype.ext ha⟩
end Field
end NonUnitalSubalgebra
/-! ## Subsemirings -/
namespace NonUnitalSubsemiring
variable {R S : Type*} [Semiring R] [SetLike S R] [hSR : NonUnitalSubsemiringClass S R] (s : S)
| Mathlib/Algebra/Algebra/Subalgebra/Unitization.lean | 145 | 157 |
/-
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, Violeta Hernández Palacios
-/
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.Nat.SuccPred
import Mathlib.Order.SuccPred.Initial... | rcases a with ⟨a, ⟨⟨⟩⟩⟩; rcases b with ⟨b, ⟨⟨⟩⟩⟩
simp only [Prod.lex_def, EmptyRelation, and_false, or_false]
rfl⟩⟩
@[simp]
theorem type_prod_lex {α β : Type u} (r : α → α → Prop) (s : β → β → Prop) [IsWellOrder α r]
[IsWellOrder β s] : type (Prod.Lex s r) = type r * type s :=
... | Mathlib/SetTheory/Ordinal/Arithmetic.lean | 613 | 622 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Data.FunLike.Equiv
import Mathlib.Data.Quot
import Mathlib.Data.Subtype
import Mathlib.Logic.U... | Mathlib/Logic/Equiv/Defs.lean | 365 | 365 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.MvPolynomial.Rename
/-!
# Degrees of polynomials
This file establ... | specialize hi j ⟨x, hx, hj⟩
rw [Finsupp.single_apply, if_neg hi]
theorem degrees_map_of_injective [CommSemiring S] (p : MvPolynomial σ R) {f : R →+* S}
(hf : Injective f) : (map f p).degrees = p.degrees := by
simp only [degrees, MvPolynomial.support_map_of_injective _ hf]
| Mathlib/Algebra/MvPolynomial/Degrees.lean | 188 | 194 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Order.CauSeq.Completion
import Mathlib.Algebra.Order.Ring.Rat
import Mathlib.Data.Rat.Cast.Defs
/-!
# Real numbers from Cauc... | ⟨Real.zero_lt_one⟩
| Mathlib/Data/Real/Basic.lean | 342 | 343 |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set
import Mathlib.Algebra.Module.LinearMap.Prod
import Mathlib.Algebra.Order.Module.Synonym
import Mathlib.Analysis... |
theorem starConvex_iUnion {ι : Sort*} {s : ι → Set E} (hs : ∀ i, StarConvex 𝕜 x (s i)) :
StarConvex 𝕜 x (⋃ i, s i) := by
rintro y hy a b ha hb hab
rw [mem_iUnion] at hy ⊢
| Mathlib/Analysis/Convex/Star.lean | 122 | 126 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Yaël Dillies
-/
import Mathlib.Order.Cover
import Mathlib.Order.Interval.Finset.Defs
/-!
# Intervals as finsets
This file provides basic results about all the `Finset.Ixx... |
@[simp]
theorem nonempty_uIcc : Finset.Nonempty [[a, b]] :=
nonempty_Icc.2 inf_le_sup
| Mathlib/Order/Interval/Finset/Basic.lean | 966 | 969 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
import Mathlib.Analy... |
variable {c : E} {r : ℝ}
theorem contDiff_unitBallBall (hr : 0 < r) : ContDiff ℝ n (unitBallBall c r hr) :=
| Mathlib/Analysis/InnerProductSpace/Calculus.lean | 347 | 350 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Batteries.Tactic.Congr
import Mathlib.Data.Option.Basic
import Mathlib.Data.Prod.Basic
import Mathlib.Data.Set.Subsingleton
import Mathlib.Dat... | (h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
| Mathlib/Data/Set/Image.lean | 348 | 350 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro
-/
import Mathlib.Algebra.Field.IsField
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Finsupp.L... | Mathlib/RingTheory/Ideal/Basic.lean | 679 | 701 | |
/-
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.ConditionalProbability
import Mathlib.Probability.Kernel.Basic
import Mathlib.Probability.Kernel.Composition.MeasureComp
import Mathlib.Tactic.... | (κ : Kernel α Ω) (μ : Measure α) [IsZeroOrMarkovKernel κ]
(h_indep : IndepSets S T κ μ) :
IndepSet s t κ μ :=
| Mathlib/Probability/Independence/Kernel.lean | 831 | 833 |
/-
Copyright (c) 2024 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.Algebra.Module.Submodule.Ker
/-!
# Iterate maps and comaps of submodules
Some preliminary work for establishing the strong rank condition for noetherian rings.
Giv... | theorem iterateMapComap_le_succ (K : Submodule R N) (h : K.map f ≤ K.map i) (n : ℕ) :
f.iterateMapComap i n K ≤ f.iterateMapComap i (n + 1) K := by
nth_rw 2 [iterateMapComap]
rw [iterate_succ', Function.comp_apply, ← iterateMapComap, ← map_le_iff_le_comap]
induction n with
| zero => exact h
| succ n ih =>... | Mathlib/Algebra/Module/Submodule/IterateMapComap.lean | 45 | 56 |
/-
Copyright (c) 2014 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Divisibility.Hom
import Mathlib.Algebra.Group.Even
import Mathlib.Algebra.Group.Nat.Hom
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.Alg... | Mathlib/Data/Nat/Cast/Basic.lean | 276 | 279 | |
/-
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.Algebra.Group.Indicator
import Mathlib.Data.Int.Cast.Pi
import Mathlib.Data.Nat.Cast.Basic
import Mathlib.MeasureTheory.MeasurableSpace... | Mathlib/MeasureTheory/MeasurableSpace/Basic.lean | 1,235 | 1,241 | |
/-
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.Adjunction.FullyFaithful
import Mathlib.CategoryTheory.Functor.EpiMono
import Mathlib.CategoryTheory.HomCongr
/-!
# Reflective functors
Ba... | Mathlib/CategoryTheory/Adjunction/Reflective.lean | 251 | 256 | |
/-
Copyright (c) 2023 Ali Ramsey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ali Ramsey, Eric Wieser
-/
import Mathlib.Algebra.Algebra.Bilinear
import Mathlib.LinearAlgebra.DFinsupp
import Mathlib.LinearAlgebra.Prod
import Mathlib.LinearAlgebra.TensorProduct.Finite... | f (repr.left i) ⊗ₜ (g ((a₂ i).left j) ⊗ₜ h ((a₂ i).right j)) =
| Mathlib/RingTheory/Coalgebra/Basic.lean | 155 | 155 |
/-
Copyright (c) 2020 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Yury Kudryashov
-/
import Mathlib.Topology.Instances.NNReal.Lemmas
import Mathlib.Topology.Order.MonotoneContinuity
/-!
# Square root of a real numbe... | Mathlib/Data/Real/Sqrt.lean | 463 | 465 | |
/-
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... | theorem ae_restrict_iff'₀ {p : α → Prop} (hs : NullMeasurableSet s μ) :
(∀ᵐ x ∂μ.restrict s, p x) ↔ ∀ᵐ x ∂μ, x ∈ s → p x := by
| Mathlib/MeasureTheory/Measure/Restrict.lean | 568 | 569 |
/-
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... |
@[simp]
theorem cardFactors_apply_prime_pow {p k : ℕ} (hp : p.Prime) : Ω (p ^ k) = k := by
rw [cardFactors_apply, hp.primeFactorsList_pow, List.length_replicate]
/-- `ω n` is the number of distinct prime factors of `n`. -/
def cardDistinctFactors : ArithmeticFunction ℕ :=
⟨fun n => n.primeFactorsList.dedup.length... | Mathlib/NumberTheory/ArithmeticFunction.lean | 909 | 917 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Finset.Card
import Mathlib.Data.Fintype.Basic
/-!
# Cardinalities of finite types
This file defines the cardinality `Fintype.card α` as the numb... | Eq.symm <| Fintype.card_congr <| Equiv.ofInjective _ <| EmbeddingLike.injective f
theorem card_eq_zero_iff : card α = 0 ↔ IsEmpty α := by
| Mathlib/Data/Fintype/Card.lean | 254 | 256 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Powerset
import Mathlib.Algebra.NoZeroSMulDivisors.Pi
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Finty... | eq_neg_of_add_eq_zero_left <| by
rw [← MultilinearMap.map_update_add, neg_add_cancel, f.map_coord_zero i (update_self i 0 m)]
@[deprecated (since := "2024-11-03")] protected alias map_neg := MultilinearMap.map_update_neg
@[simp]
| Mathlib/LinearAlgebra/Multilinear/Basic.lean | 1,269 | 1,275 |
/-
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
/-!
# ... | Mathlib/SetTheory/Ordinal/Basic.lean | 1,637 | 1,638 | |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno
import Mathlib.Analysis.Calculus.FDeriv.Ad... | Mathlib/Analysis/Calculus/ContDiff/Basic.lean | 1,386 | 1,389 | |
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton
-/
import Mathlib.Topology.Category.TopCat.Adjunctions
import Mathlib.CategoryTheory.Functor.EpiMono
/-!
# Epi- and monomorphisms in `Top`
This file shows that a continuous ... | constructor
· intro
infer_instance
· apply Functor.mono_of_mono_map
end TopCat
| Mathlib/Topology/Category/TopCat/EpiMono.lean | 39 | 46 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.List.Sublists
import Mathlib.Data.List.Zip
import Mathlib.Data.Multiset.Bind
import Mathlib.Data.Multiset.Range
/-!
# The powerset of a multiset
... | rw [← List.map_map]
exact ((map_pure_sublist_sublists _).map _).subperm
@[simp]
| Mathlib/Data/Multiset/Powerset.lean | 101 | 104 |
/-
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... | (∫⁻ x, ENNReal.ofReal (f x) ∂μ) < ∞ :=
calc
(∫⁻ x, ENNReal.ofReal (f x) ∂μ) ≤ ∫⁻ x, ↑‖f x‖₊ ∂μ := lintegral_ofReal_le_lintegral_enorm f
_ < ∞ := hf.2
| Mathlib/MeasureTheory/Integral/IntegrableOn.lean | 351 | 355 |
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.SmoothSeries
import Mathlib.Analysis.Calculus.BumpFunction.InnerProduct
import Mathlib.Analysis.Convolution
import Mathlib.Anal... | tsupport f ⊆ s ∧ HasCompactSupport f ∧ ContDiff ℝ ∞ f ∧ range f ⊆ Icc 0 1 ∧ f x = 1 := by
obtain ⟨d : ℝ, d_pos : 0 < d, hd : Euclidean.closedBall x d ⊆ s⟩ :=
Euclidean.nhds_basis_closedBall.mem_iff.1 hs
let c : ContDiffBump (toEuclidean x) :=
{ rIn := d / 2
rOut := d
rIn_pos := half_pos d_... | Mathlib/Analysis/Calculus/BumpFunction/FiniteDimension.lean | 43 | 73 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Data.Nat.Prime.Factorial
import Mathlib.NumberTheory.LegendreSymbol.Basic
import Mathlib.Analysis.Normed.Ring.Lemmas
/-!
# Lemmas of Gauss and Eisenstein
... | rw [Ne, CharP.cast_eq_zero_iff (ZMod p) p, hp.1.dvd_factorial, not_le]
exact Nat.div_lt_self hp.1.pos (by decide))).1 <| by
simpa using gauss_lemma_aux₁ p hap
/-- **Gauss' lemma**. The Legendre symbol can be computed by considering the number of naturals less
than `p/2` such that `(a * x) % p > p / 2`.... | Mathlib/NumberTheory/LegendreSymbol/GaussEisensteinLemmas.lean | 94 | 100 |
/-
Copyright (c) 2024 Antoine Chambert-Loir. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir
-/
import Mathlib.Logic.Function.Defs
/-!
# Propositional typeclasses on several maps
This file contains typeclasses that are used in the definition of
... | lemma comp_apply {M N P : Type*}
{φ : M → N} {ψ : N → P} {χ : M → P} (h : CompTriple φ ψ χ) (x : M) :
ψ (φ x) = χ x := by
rw [← h.comp_eq, Function.comp_apply]
| Mathlib/Logic/Function/CompTypeclasses.lean | 61 | 64 |
/-
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.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... |
We show that the reciprocal Gamma function `1 / Γ(s)` is entire. These lemmas show that (in this
case at least) mathlib's conventions for division by zero do actually give a mathematically useful
answer! (These results are useful in the theory of zeta and L-functions.) -/
/-- A reformulation of the Gamma recurrence ... | Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean | 495 | 502 |
/-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.SplittingField.Construction
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.Separable
import Mathlib.FieldTheory.Normal.... | `X ^ (q ^ n) - C y` for some `n : ℕ` and `y : F`. -/
theorem natSepDegree_eq_one_iff_eq_X_pow_sub_C : (minpoly F x).natSepDegree = 1 ↔
∃ (n : ℕ) (y : F), minpoly F x = X ^ q ^ n - C y := by
simp only [minpoly.natSepDegree_eq_one_iff_eq_expand_X_sub_C q, map_sub, expand_X, expand_C]
/-- The minimal polynomial of ... | Mathlib/FieldTheory/SeparableDegree.lean | 639 | 645 |
/-
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.Analysis.InnerProductSpace.Continuous
import Mathlib.Analysis.Normed.Module.Dual
import Mathlib.MeasureTheory.Function.AEEqOfLIntegral
import Mathlib.Measu... | refine measure_eq_zero_of_trim_eq_zero hm ?_
refine ae_eq_zero_of_forall_setIntegral_eq_of_sigmaFinite ?_ ?_
· intro s hs hμs
unfold IntegrableOn
rw [restrict_trim hm (μ.restrict t) hs, Measure.restrict_restrict (hm s hs)]
rw [← restrict_trim hm μ ht_meas, Measure.restrict_apply hs,
trim_measura... | Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean | 338 | 345 |
/-
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.IsPrimePow
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Ring.Int
import Mathlib.Algebra.Ring.CharZero
im... | apply card_le_card
| Mathlib/NumberTheory/Divisors.lean | 206 | 206 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.MvPolynomial.Rename
/-!
# Degrees of polynomials
This file establ... |
-- TODO we can prove equality with `NoZeroDivisors R`
theorem degreeOf_prod_le {ι : Type*} (i : σ) (s : Finset ι) (f : ι → MvPolynomial σ R) :
degreeOf i (∏ j ∈ s, f j) ≤ ∑ j ∈ s, (f j).degreeOf i := by
simp_rw [degreeOf_eq_sup]
exact supDegree_prod_le (by simp only [coe_zero, Pi.zero_apply])
| Mathlib/Algebra/MvPolynomial/Degrees.lean | 274 | 279 |
/-
Copyright (c) 2024 Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Merten
-/
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.CategoryTheory.Limits.Preserves.Limits
import Mathlib.CategoryTheory.Limits.Yoneda
/-!
# Ind- and ... | variable [HasColimit (D.unop ⋙ yoneda)] [HasLimitsOfShape Iᵒᵖ (Type (max u₁ u₂))]
/-- Ind-Yoneda lemma: morphisms from colimit of yoneda of diagram `D` to `F` is limit of `F`
evaluated at `D`. This version is for covariant diagrams, see `colimitYonedaHomIsoLimit'` for a
contravariant version. -/
noncomputable def coli... | Mathlib/CategoryTheory/Limits/IndYoneda.lean | 191 | 202 |
/-
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 ... | theorem rightInv_coeff_one (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F) (x : E) :
p.rightInv i x 1 = (continuousMultilinearCurryFin1 𝕜 F E).symm i.symm := by rw [rightInv]
| Mathlib/Analysis/Analytic/Inverse.lean | 182 | 183 |
/-
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.Limits.Shapes.Pullback.CommSq
import Mathlib.CategoryTheory.Limits.Shapes.StrictInitial
import Mathlib.CategoryTheory.Limits.Types.Shapes
impo... | end FiniteCoproducts
end Extensive
end CategoryTheory
| Mathlib/CategoryTheory/Extensive.lean | 549 | 565 |
/-
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.Analysis.Normed.Group.Continuity
import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
import Mathlib.Topology.MetricSpace.Thick... | measurable_enorm.comp_aemeasurable hf
@[deprecated (since := "2025-01-21")] alias measurable_ennnorm := measurable_enorm
@[deprecated (since := "2025-01-21")] alias Measurable.ennnorm := Measurable.enorm
@[deprecated (since := "2025-01-21")] alias AEMeasurable.ennnorm := AEMeasurable.enorm
| Mathlib/MeasureTheory/Constructions/BorelSpace/Metric.lean | 229 | 234 |
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes Hölzl, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Continuity
import Mathlib.Topology.Algebra.IsUniformGroup.Basic
import Mathlib.Topology.MetricSpace... |
@[to_additive (attr := simp)]
| Mathlib/Analysis/Normed/Group/Uniform.lean | 250 | 251 |
/-
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.Shift.CommShift
import Mathlib.CategoryTheory.Shift.Induced
import Mathlib.CategoryTheory.Quotient
/-!
# The shift on a quotient category
Let `C... | @[simp]
lemma iso_inv_app (a : A) (X : C) :
(iso F r hF a).inv.app ((functor r).obj X) =
(F.commShiftIso a).inv.app X ≫
(lift r F hF).map (((functor r).commShiftIso a).hom.app X) := by
dsimp only [iso, natIsoLift]
rw [natTransLift_app]
dsimp
erw [id_comp, comp_id, comp_id, comp_id, Functor.map_i... | Mathlib/CategoryTheory/Shift/Quotient.lean | 87 | 95 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Data.ENNReal.Real
/-!
# Properties of addition, multiplication and subtraction on extended non-negative real numbers
In this file we... | Mathlib/Data/ENNReal/Operations.lean | 609 | 612 | |
/-
Copyright (c) 2023 Ziyu Wang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ziyu Wang, Chenyi Li, Sébastien Gouëzel, Penghao Yu, Zhipeng Cao
-/
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.Calculus.FDeriv.Basic
import Mathlib.Analysis.Calc... |
theorem HasGradientWithinAt.congr (h : HasGradientWithinAt f f' s x) (hs : ∀ x ∈ s, f₁ x = f x)
(hx : f₁ x = f x) : HasGradientWithinAt f₁ f' s x :=
| Mathlib/Analysis/Calculus/Gradient/Basic.lean | 269 | 271 |
/-
Copyright (c) 2021 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Data.Set.Finite.Lattice
import Mathlib.Order.ConditionallyCompleteLattice.Indexed
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Order.SuccPred.Ba... | (f : ι → α) (i : ι) :
partialSups f (Order.succ i) = partialSups f i ⊔ f (Order.succ i) := by
suffices Iic (Order.succ i) = Iic i ∪ {Order.succ i} by simp only [partialSups_apply, this,
| Mathlib/Order/PartialSups.lean | 132 | 134 |
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