Context stringlengths 227 76.5k | target stringlengths 0 11.6k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 16 3.69k |
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/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Ordmap.Invariants
/-!
# Verification of `Ordnode`
This file uses the invariants defined in `Mathlib.Data.Ordmap.Invariants` to construct `Ordset... | Mathlib/Data/Ordmap/Ordset.lean | 1,542 | 1,546 | |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.GroupTheory.MonoidLocalization.Away
import Mathlib.RingTheory.Ideal.Quotient.Operations
import Ma... | refine fun h => Ideal.mem_sInf.1 h fun z hz => ?_
obtain ⟨y, hy⟩ := hz
let Z : { x // x ∈ I } := ⟨y, hy.left⟩
use ⟨Z, 1⟩
simp [Z, hy.right]
· rintro ⟨⟨a, s⟩, h⟩
rw [← Ideal.unit_mul_mem_iff_mem _ (map_units S s), mul_comm]
exact h.symm ▸ Ideal.mem_map_of_mem _ a.2
| Mathlib/RingTheory/Localization/Ideal.lean | 67 | 75 |
/-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Patrick Stevens, Thomas Browning
-/
import Mathlib.Data.Nat.Choose.Central
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.Nat.Multiplicity
/-!
# Factori... |
theorem factorization_choose_eq_zero_of_lt (h : n < p) : (choose n k).factorization p = 0 := by
by_cases hnk : n < k; · simp [choose_eq_zero_of_lt hnk]
rw [choose_eq_factorial_div_factorial (le_of_not_lt hnk),
| Mathlib/Data/Nat/Choose/Factorization.lean | 100 | 103 |
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Data.List.Nodup
import Mathlib.Data.Set.Prod
/-!
# Finite products of types
This file defines the product of types over a list. For `l : List ι` ... | have h2 : { i | i ∈ l } = univ := by
ext i
simp [h]
rw [← h2, ← mk_preimage_tprod, preimage_preimage]
simp only [TProd.mk_elim hnd h]
dsimp
end Set
| Mathlib/Data/Prod/TProd.lean | 157 | 170 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Morenikeji Neri
-/
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.Algebra.EuclideanDomain.Field
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.RingTheor... | Mathlib/RingTheory/PrincipalIdealDomain.lean | 546 | 590 | |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Data.ENat.Lattice
import Mathlib.Order.OrderIsoNat
import Mathlib.Tactic.TFAE
/-!
# Maximal length of chains
This file contains lemmas to work with the ma... | exact ⟨fun y hy ↦ hx _ (hl.2 _ (mem_of_mem_head? hy)), hl.1⟩
· rintro x (_ | _)
exacts [Or.inl (Set.mem_singleton a), Or.inr (hl.2 x ‹x ∈ l›)]
| Mathlib/Order/Height.lean | 273 | 276 |
/-
Copyright (c) 2024 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.MvPolynomial.Equiv
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.RingTheory.MvPolynomial.Homogeneous
/-!
# The universal c... | rw [MvPolynomial.map, univ_map_eval₂Hom]; rfl
@[simp]
| Mathlib/LinearAlgebra/Matrix/Charpoly/Univ.lean | 60 | 62 |
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Measure.MeasureSpace
import Mathlib.MeasureTheory.Measure.Regular
import Mathlib.Topology.Sets.Compacts
/-!
# Contents
In this file... |
@[deprecated toFun_eq_toNNReal_apply (since := "2025-02-11")]
| Mathlib/MeasureTheory/Measure/Content.lean | 98 | 99 |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Analysis.Normed.Module.Convex
/-!
# Sides of affine subspaces
This ... |
theorem _root_.Function.Injective.sOppSide_map_iff {s : AffineSubspace R P} {x y : P}
{f : P →ᵃ[R] P'} (hf : Function.Injective f) :
(s.map f).SOppSide (f x) (f y) ↔ s.SOppSide x y := by
simp_rw [SOppSide, hf.wOppSide_map_iff, mem_map_iff_mem_of_injective hf]
@[simp]
theorem _root_.AffineEquiv.wOppSide_map_... | Mathlib/Analysis/Convex/Side.lean | 109 | 119 |
/-
Copyright (c) 2023 Ashvni Narayanan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ashvni Narayanan, Moritz Firsching, Michael Stoll
-/
import Mathlib.Algebra.Group.EvenFunction
import Mathlib.Data.ZMod.Units
import Mathlib.NumberTheory.MulChar.Basic
/-!
# Dirichl... | changeLevel hm χ = 1 ↔ χ = 1 :=
map_eq_one_iff _ (changeLevel_injective hm)
| Mathlib/NumberTheory/DirichletCharacter/Basic.lean | 80 | 82 |
/-
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, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Pairwise
import Mathlib.Data.Set.BooleanAlgebra
/-!
# The set lattice
This file is a collectio... | Mathlib/Data/Set/Lattice.lean | 2,268 | 2,269 | |
/-
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.HomologicalComplexBiprod
import Mathlib.Algebra.Homology.Homotopy
import Mathlib.CategoryTheory.MorphismProperty.IsInvertedBy
/-! The homotopy ... | @[reassoc (attr := simp)]
lemma ι₀_π : ι₀ K ≫ π K = 𝟙 K := by simp [π]
| Mathlib/Algebra/Homology/HomotopyCofiber.lean | 416 | 417 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Topology.Compactness.Compact
/-!
# Locally compact spaces
This file contains basic results about locally compact spac... | have (x : X) : (𝓝 x).HasBasis (fun s ↦ (s ∈ 𝓝 (f x) ∧ IsCompact s) ∧ s ⊆ U)
(fun s ↦ f ⁻¹' (s ∩ Z)) := by
have H : U ∈ 𝓝 (f x) := hU.mem_nhds (hUZ.subset <| mem_range_self _).1
rw [hf.nhds_eq_comap, ← comap_nhdsWithin_range, hUZ,
nhdsWithin_inter_of_mem (nhdsWithin_le_nhds H)]
exact (nhdsWi... | Mathlib/Topology/Compactness/LocallyCompact.lean | 185 | 194 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Algebra.Group.TypeTags.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Finset.Piecewise
import Mathlib.Or... | cases nonempty_fintype ι
rw [isOpen_iff_nhds]
simp_rw [le_principal_iff, nhds_pi, Filter.mem_pi', mem_nhds_iff]
| Mathlib/Topology/Constructions.lean | 889 | 891 |
/-
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.Analysis.SpecialFunctions.Complex.Log
import Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots
/-!
# Complex roots of unity
In this file we show that th... | constructor
· intro h
obtain ⟨i, hi, H⟩ : ∃ i < (n : ℕ), exp (2 * π * I / n) ^ i = x := by
simpa only using (isPrimitiveRoot_exp n NeZero.out).eq_pow_of_pow_eq_one h
refine ⟨i, hi, ?_⟩
rw [← H, ← exp_nat_mul]
congr 1
field_simp [hn0, mul_comm (i : ℂ)]
· rintro ⟨i, _, H⟩
rw [← H, ← ex... | Mathlib/RingTheory/RootsOfUnity/Complex.lean | 74 | 89 |
/-
Copyright (c) 2019 Neil Strickland. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Neil Strickland
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Group.NatPowAssoc
import Mathlib.Algebra.Order.... |
theorem Odd.add_dvd_pow_add_pow [CommRing R] (x y : R) {n : ℕ} (h : Odd n) :
x + y ∣ x ^ n + y ^ n := by
have h₁ := geom_sum₂_mul x (-y) n
rw [Odd.neg_pow h y, sub_neg_eq_add, sub_neg_eq_add] at h₁
exact Dvd.intro_left _ h₁
| Mathlib/Algebra/GeomSum.lean | 211 | 216 |
/-
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... | rw [degree_eq_natDegree h]
exact WithBot.succ_coe p.natDegree
| Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 246 | 248 |
/-
Copyright (c) 2021 Eric Weiser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
import Mathlib.Algebra.Ring.Subring.Pointwise
/-!
# Pointwise actions on subalgebras.
If `R'` acts on an `R`-algebra `A`... | · intro x hx1
rw [← mul_one x]
exact Submodule.mul_mem_mul hx1 (show (1 : A) ∈ S from one_mem S)
@[deprecated (since := "2025-01-12")] alias mul_self := isIdempotentElem_toSubmodule
/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/
theorem mul_toSubmodule {R : Type*} {A : Type*} [Co... | Mathlib/Algebra/Algebra/Subalgebra/Pointwise.lean | 37 | 44 |
/-
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... | hβ.tendsto_iff (UniformFun.hasBasis_uniformity ι α)]
simp only [Prod.forall]
rfl
theorem Filter.HasBasis.uniformEquicontinuousOn_iff_left {p : κ → Prop}
{s : κ → Set (β × β)} {F : ι → β → α} {S : Set β} (hβ : (𝓤 β ⊓ 𝓟 (S ×ˢ S)).HasBasis p s) :
| Mathlib/Topology/UniformSpace/Equicontinuity.lean | 656 | 661 |
/-
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 ... | then it is a constant on this set. -/
theorem is_const_of_fderivWithin_eq_zero (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s)
(hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = 0) (hx : x ∈ s) (hy : y ∈ s) : f x = f y := by
have bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ 0 := fun x hx => by
simp only [hf' x hx, norm_z... | Mathlib/Analysis/Calculus/MeanValue.lean | 554 | 560 |
/-
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.Order.Group.Finset
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Algebra.Polynomial.Eva... | obtain ⟨x, hf⟩ := natDegree_eq_zero.1 H
by_cases hx : x = 0
· rw [← hf, hx, map_zero] at h; exact not_irreducible_zero h
exact h.1 (hf ▸ isUnit_C.2 (Ne.isUnit hx))
| Mathlib/Algebra/Polynomial/FieldDivision.lean | 673 | 682 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Multiset.ZeroCons
/-!
# Basic results on multisets
-/
-- No algebra should be required
assert_not_exists Monoid
universe v
open List S... | Mathlib/Data/Multiset/Basic.lean | 2,347 | 2,348 | |
/-
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 only [Prod.snd_add, add_apply, coprod_apply, Prod.fst_add, add_add_add_comm]
| Mathlib/LinearAlgebra/Prod.lean | 255 | 255 |
/-
Copyright (c) 2021 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Devon Tuma
-/
import Mathlib.Algebra.Polynomial.Eval.Defs
import Mathlib.Analysis.Asymptotics.Lemmas
/-!
# Super-Polynomial Function Decay
This file defines a predicate `Asymptotics.Supe... |
theorem SuperpolynomialDecay.trans_eventuallyLE (hk : 0 ≤ᶠ[l] k) (hg : SuperpolynomialDecay l k g)
(hg' : SuperpolynomialDecay l k g') (hfg : g ≤ᶠ[l] f) (hfg' : f ≤ᶠ[l] g') :
SuperpolynomialDecay l k f := fun z =>
tendsto_of_tendsto_of_tendsto_of_le_of_le' (hg z) (hg' z)
| Mathlib/Analysis/Asymptotics/SuperpolynomialDecay.lean | 128 | 132 |
/-
Copyright (c) 2022 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Order.Filter.Cofinite
/-!
# Basic theory of bornology
We develop the basic theory of bornologies. Instead of axiomatizing bounded sets and defining
bor... | lemma eventually_ne_cobounded (a : α) : ∀ᶠ x in cobounded α, x ≠ a :=
le_cofinite_iff_eventually_ne.1 (le_cofinite _) a
| Mathlib/Topology/Bornology/Basic.lean | 255 | 257 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Algebra.Algebra.Hom
import Mathlib.RingTheory.Congruence.Basic
import Mathlib.RingTheory.Ideal.Quotient.Defs
import Mathlib.RingTheory.Ideal.Span
/-!
# Qu... | Mathlib/Algebra/RingQuot.lean | 656 | 659 | |
/-
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.End
import Mathlib.Data.ZMod.Defs
import Mathlib.Tactic.Ring
/-!
# Racks and Quandles
This file defines racks and quandles, algebraic structu... | -/
instance oppositeRack : Rack Rᵐᵒᵖ where
act x y := op (invAct (unop x) (unop y))
self_distrib := by
intro x y z
| Mathlib/Algebra/Quandle.lean | 232 | 236 |
/-
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.Extensive
import Mathlib.CategoryTheory.Limits.Shapes.KernelPair
import Mathlib.CategoryTheory.Limits.Constructions.EpiMono
/-!
# Adhesive c... | · rintro ⟨hc₃, hc₄⟩
exact ⟨(IsPullback.of_hasPullback αY c.inr).paste_horiz hc₄, hc₃⟩
theorem IsPushout.IsVanKampen.isPullback_of_mono_left [Mono f] {H : IsPushout f g h i}
(H' : H.IsVanKampen) : IsPullback f g h i :=
| Mathlib/CategoryTheory/Adhesive.lean | 187 | 191 |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Logic.Relator
import Mathlib.Tactic.Use
import Mathlib.Tactic.MkIffOfInductiveProp
import Mathlib.Tactic.SimpRw
import Mathlib.Logic.Basic
import Mathl... |
lemma reflGen_minimal {r' : α → α → Prop} (hr' : Reflexive r') (h : ∀ x y, r x y → r' x y) {x y : α}
(hxy : ReflGen r x y) : r' x y := by
simpa [reflGen_eq_self hr'] using ReflGen.mono h hxy
end reflGen
| Mathlib/Logic/Relation.lean | 452 | 458 |
/-
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.Operations
/-!
# Results about division in extended non-negative reals
This file establishes basic properties related t... | by_cases h0 : c = 0
· have : a = 0 := by simpa [h0] using h
simp [*]
| Mathlib/Data/ENNReal/Inv.lean | 359 | 361 |
/-
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.Ring.Associated
import Mathlib.Algebra.Star.Unitary
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.Tactic.Ring
import Mathlib.Al... | a.re * b.re + (a.re * b.im + a.im * b.re) * r + a.im * b.im * (r * r) := by
ring
simp only [mul_re, Int.cast_add, Int.cast_mul, mul_im, this, r.prop]
ring }
invFun f := ⟨f sqrtd, by rw [← f.map_mul, dmuld, map_intCast]⟩
| Mathlib/NumberTheory/Zsqrtd/Basic.lean | 900 | 904 |
/-
Copyright (c) 2022 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler, Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
import Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv
/-!
# Polynomia... | rcases lt_or_le 1 x with h' | h'
· exact (sin_le_one x).trans_lt h'
have hx : |x| = x := abs_of_nonneg h.le
have := le_of_abs_le (sin_bound <| show |x| ≤ 1 by rwa [hx])
rw [sub_le_iff_le_add', hx] at this
apply this.trans_lt
rw [sub_add, sub_lt_self_iff, sub_pos, div_eq_mul_inv (x ^ 3)]
refine mul_lt_mu... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Bounds.lean | 39 | 49 |
/-
Copyright (c) 2024 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.TotalComplex
/-! The symmetry of the total complex of a bicomplex
Let `K : HomologicalComplex₂ C c₁ c₂` be a bicomplex. If we assume both
`[To... |
/-- The symmetry isomorphism `K.flip.total c ≅ K.total c` of the total complex of a
bicomplex when we have `[TotalComplexShapeSymmetry c₁ c₂ c]`. -/
noncomputable def totalFlipIso : K.flip.total c ≅ K.total c :=
HomologicalComplex.Hom.isoOfComponents (K.totalFlipIsoX c) (fun j j' _ => by
| Mathlib/Algebra/Homology/TotalComplexSymmetry.lean | 101 | 105 |
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.SetTheory.Cardinal.Finite
import Mathlib.Data.Set.Finite.Powerset
/-!
# Noncomputable Set Cardinality
We define the cardinality of set `s` as a term `Set... | (s ∪ t).encard + (s ∩ t).encard = s.encard + t.encard := by
rw [← diff_union_self, encard_union_eq disjoint_sdiff_left, add_right_comm,
encard_diff_add_encard_inter]
| Mathlib/Data/Set/Card.lean | 184 | 187 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Localization.Basic
import Mathlib.RingTheory.Localization.Ideal
import Mathlib.RingThe... | rw [← hry] at hy
obtain ⟨t, ht⟩ := IsLocalization.eq.1 hxyz
simp only [mul_one, one_mul, Submonoid.coe_mul, Subtype.coe_mk] at ht
suffices (t : R) * (sx * sy * sz) ∈ P from
not_or_intro (mt hp.mem_or_mem <| not_or_intro sx.2 sy.2) sz.2
(hp.mem_or_mem <| (hp.mem_or_mem this).res... | Mathlib/RingTheory/Localization/AtPrime.lean | 79 | 103 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Notation.Pi
import Mathlib.Data.Set.Lattice
import Mathlib.Order.Filter.Defs
/-!
# Theory of... | theorem EventuallyLE.refl (l : Filter α) (f : α → β) : f ≤ᶠ[l] f :=
EventuallyEq.rfl.le
theorem EventuallyLE.rfl : f ≤ᶠ[l] f :=
| Mathlib/Order/Filter/Basic.lean | 1,063 | 1,066 |
/-
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.Topology.Algebra.Constructions
import Mathlib.Topology.Bases
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Topology.UniformSpac... | let ⟨x2, hx2⟩ := CompleteSpace.complete <| hf.map uniformContinuous_snd
⟨(x1, x2), by rw [nhds_prod_eq, le_prod]; constructor <;> assumption⟩
| Mathlib/Topology/UniformSpace/Cauchy.lean | 374 | 376 |
/-
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 | 2,347 | 2,348 | |
/-
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 | 1,072 | 1,072 | |
/-
Copyright (c) 2022 Apurva Nakade. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Apurva Nakade
-/
import Mathlib.Analysis.Convex.Cone.Closure
import Mathlib.Analysis.InnerProductSpace.Adjoint
/-!
# Proper cones
We define a *proper cone* as a closed, pointed cone. ... | toSubmodule := PointedCone.comap (f : E →ₗ[ℝ] F) S
isClosed' := by
| Mathlib/Analysis/Convex/Cone/Proper.lean | 177 | 178 |
/-
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... | end NNReal
namespace ULift
variable [PseudoMetricSpace β]
instance : PseudoMetricSpace (ULift β) := PseudoMetricSpace.induced ULift.down ‹_›
| Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean | 126 | 131 |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Sébastien Gouëzel, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Analysis.Normed.Lp.PiLp
import Mathlib.LinearAlgebra.FiniteDimensi... | variable [DecidableEq ι]
section
open scoped Matrix
| Mathlib/Analysis/InnerProductSpace/PiL2.lean | 733 | 737 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Minchao Wu, Mario Carneiro
-/
import Mathlib.Algebra.NeZero
import Mathlib.Data.Finset.Attach
import Mathlib.Data.Finset.Disjoint
import Mathli... |
instance canLift (c) (p) [CanLift β α c p] :
| Mathlib/Data/Finset/Image.lean | 304 | 305 |
/-
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,... | Tendsto (fun i ↦ ∫ x, f x ∂ (μs i)) L (𝓝 (∫ x, f x ∂μ)) := by
rcases eq_or_neBot L with rfl|hL
· simp only [tendsto_bot]
have obs := BoundedContinuousFunction.isBounded_range_integral μs f
have bdd_above := BddAbove.isBoundedUnder L.univ_mem (by simpa using obs.bddAbove)
have bdd_below := BddBelow.isBoun... | Mathlib/MeasureTheory/Integral/BoundedContinuousFunction.lean | 164 | 186 |
/-
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 | 1,452 | 1,454 | |
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.Group.Action.End
import Mathlib.Algebra.Group.Pointwise.Set.Lattice
import Mathlib.Algebra.Group.Subgroup.MulOppositeLemmas
import Mathlib.Algebra.Gr... | lemma coe_div_coe [SetLike S G] [DivisionMonoid G] [SubgroupClass S G] (H : S) :
H / H = (H : Set G) := by simp [div_eq_mul_inv]
| Mathlib/Algebra/Group/Subgroup/Pointwise.lean | 57 | 59 |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Neil Strickland
-/
import Mathlib.Data.Nat.Prime.Defs
import Mathlib.Data.PNat.Basic
/-!
# Primality and GCD on pnat
This file extends the theory of `ℕ+` with ... | exists (⟨p, Nat.Prime.pos hp.left⟩ : ℕ+); rw [dvd_iff]; apply hp
end Prime
| Mathlib/Data/PNat/Prime.lean | 144 | 147 |
/-
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 | 1,178 | 1,187 | |
/-
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.CategoryTheory.Subobject.Limits
/-!
# Image-to-kernel comparison maps
Whenever `f : A ⟶ B` and `g : B ⟶ C` satisfy `w : f ≫ g = 0`,
we have `image_le_ker... |
end
section
variable {A B C : V} (f : A ⟶ B) (g : B ⟶ C)
| Mathlib/Algebra/Homology/ImageToKernel.lean | 73 | 78 |
/-
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.List.Cycle
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.Perm.List
/-!
# Properties of cyclic permutations constructed... | cycleOf_pow_apply_self p x m]
rcases n with - | n <;> rcases m with - | m
| Mathlib/GroupTheory/Perm/Cycle/Concrete.lean | 253 | 254 |
/-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Ordinal.Family
import Mathlib.Tactic.Abel
/-!
# Natural operations on ordinals
The goal of this file is to define n... |
@[simp]
theorem nadd_le_nadd_iff_right : ∀ (a) {b c}, b ♯ a ≤ c ♯ a ↔ b ≤ c :=
| Mathlib/SetTheory/Ordinal/NaturalOps.lean | 416 | 418 |
/-
Copyright (c) 2014 Floris van Doorn (c) 2016 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Leonardo de Moura, Jeremy Avigad, Mario Carneiro
-/
import Mathlib.Algebra.Group.Nat.Defs
import Mathlib.Algebra.Group.Basic
import M... | Mathlib/Data/Nat/Size.lean | 137 | 138 | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Topology.Order.ProjIcc
/-!... | (lt_arcsin_iff_sin_lt' <| left_mem_Ico.2 <| neg_lt_self pi_div_two_pos).trans <| by
rw [sin_neg, sin_pi_div_two]
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean | 202 | 204 |
/-
Copyright (c) 2022 Eric Rodriguez. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Rodriguez
-/
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Ring.Cast
import Mathlib.Data.Fi... | rcases lt_trichotomy 0 a with (h | h | h)
· simp [h, h.not_le]
· simp [← h]
· simp [h, h.le]
| Mathlib/Data/Sign.lean | 364 | 368 |
/-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
-- Some proofs and docs came from mathlib3 `src/algebra/commute.lean` (c) Neil Strickland
import Mathlib.Algebra.Group.Defs
import Mathlib.Order.Defs.Unbundled
/-... | Mathlib/Algebra/Group/Semiconj/Defs.lean | 147 | 152 | |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Patrick Massot
-/
import Mathlib.Data.Set.Function
import Mathlib.Order.Interval.Set.OrdConnected
/-!
# Projection of a line onto a closed interval
Given a linearly... | @[simp]
theorem projIcc_val (x : Icc a b) : projIcc a b h x = x := by
| Mathlib/Order/Interval/Set/ProjIcc.lean | 105 | 106 |
/-
Copyright (c) 2015 Nathaniel Thomas. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nathaniel Thomas, Jeremy Avigad, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.GroupWithZero.Action.Defs
import Mathlib.Algebra.Ring.Defs
/-!
# Modules over a ring
In th... | Mathlib/Algebra/Module/Defs.lean | 269 | 269 | |
/-
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.BigOperators.Ring.Finset
import Mathlib.Combinatorics.SimpleGraph.Density
import Mathlib.Data.Nat.Cast.Order.Field
impo... | G.edgeDensity s t| := by
rw [nonuniformWitnesses, dif_pos h]
exact (not_isUniform_iff.1 h).choose_spec.2.choose_spec.2.2.2
| Mathlib/Combinatorics/SimpleGraph/Regularity/Uniform.lean | 148 | 151 |
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Order.Interval.Set.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
import Mathlib.Tactic.AdaptationNote
... | theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
(hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N :=
lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ
(upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn)
theorem le_sub_of_le_upcrossingsBef... | Mathlib/Probability/Martingale/Upcrossing.lean | 522 | 534 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kenny Lau, Yury Kudryashov
-/
import Mathlib.Data.List.Forall2
import Mathlib.Data.List.Lex
import Mathlib.Logic.Function.Iterate
import Mathlib.Logic.Relation
/-!
# R... | rw [← cons_head?_tail hy] at h
exact h.rel_head
| Mathlib/Data/List/Chain.lean | 246 | 248 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Devon Tuma
-/
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.RingTheory.Coprime.Basic
import Mathlib.Tactic.... |
end Semiring
section CommSemiring
variable [Semiring S] [CommSemiring R] [Semiring A] [Field K]
theorem scaleRoots_eval₂_mul_of_commute {p : S[X]} (f : S →+* A) (a : A) (s : S)
(hsa : Commute (f s) a) (hf : ∀ s₁ s₂, Commute (f s₁) (f s₂)) :
eval₂ f (f s * a) (scaleRoots p s) = f s ^ p.natDegree * eval₂ f a ... | Mathlib/RingTheory/Polynomial/ScaleRoots.lean | 112 | 122 |
/-
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.Types.Colimits
import Mathlib.CategoryTheory.Limits.Types.Limits
import Mathlib.CategoryTheory.SingleObj
import Mathlib.Data.... | section Colimits
variable {G : Type v} [Group G] (J : SingleObj G ⥤ Type u)
/-- The relation used to construct colimits in types for `J : SingleObj G ⥤ Type u` is
equivalent to the `MulAction.orbitRel` equivalence relation on `J.obj (SingleObj.star G)`. -/
lemma Types.Quot.Rel.iff_orbitRel (x y : J.obj (SingleObj.sta... | Mathlib/CategoryTheory/Limits/Shapes/SingleObj.lean | 74 | 82 |
/-
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... |
/-- A commutative algebra is finitely generated if and only if it is a quotient of a polynomial ring
in `n` variables. -/
theorem iff_quotient_mvPolynomial'' :
FiniteType R S ↔ ∃ (n : ℕ) (f : MvPolynomial (Fin n) R →ₐ[R] S), Surjective f := by
constructor
· rw [iff_quotient_mvPolynomial']
rintro ⟨ι, hfinty... | Mathlib/RingTheory/FiniteType.lean | 175 | 183 |
/-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.RingTheory.AlgebraTower
import Mathli... | Cardinal.lift.{w} (Module.rank F K) * Cardinal.lift.{v} (Module.rank K A) =
Cardinal.lift.{v} (Module.rank F A) := by
let b := Module.Free.chooseBasis F K
let c := Module.Free.chooseBasis K A
rw [← (Module.rank F K).lift_id, ← b.mk_eq_rank, ← (Module.rank K A).lift_id, ← c.mk_eq_rank,
← lift_umax.{w... | Mathlib/LinearAlgebra/Dimension/Free.lean | 41 | 48 |
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.Limits.Pullbacks
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
/-!
# Open immersions of structured spaces
We say that a m... | variable (f : X ⟶ Y.toPresheafedSpace) [H : IsOpenImmersion f]
@[simp]
theorem toSheafedSpace_toPresheafedSpace : (toSheafedSpace Y f).toPresheafedSpace = X :=
rfl
/-- If `X ⟶ Y` is an open immersion of PresheafedSpaces, and `Y` is a SheafedSpace, we can
upgrade it into a morphism of SheafedSpaces.
-/
| Mathlib/Geometry/RingedSpace/OpenImmersion.lean | 522 | 530 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Localization.Integer
import Mathli... |
@[simp]
lemma num_zero : IsFractionRing.num A (0 : K) = 0 := by
have := mk'_num_den' A (0 : K)
| Mathlib/RingTheory/Localization/NumDen.lean | 82 | 85 |
/-
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.Order.SuccPred.Archimedean
import Mathlib.Order.BoundedOrder.Lattice
/-!
# Successor and predecessor limits
We define the pre... | theorem IsSuccLimit.succ_ne (h : IsSuccLimit a) (b : α) : succ b ≠ a :=
h.isSuccPrelimit.succ_ne b
| Mathlib/Order/SuccPred/Limit.lean | 135 | 137 |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.Order.Interval.Set.Group
import Mathlib.Analysis.Convex.Basic
import Mathlib.Analysis.Convex.Segment
import... | Wbtw R x (r * (y - x) + x) y ↔ x = y ∨ r ∈ Set.Icc (0 : R) 1 :=
wbtw_lineMap_iff
| Mathlib/Analysis/Convex/Between.lean | 388 | 389 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Patrick Massot, Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic
import Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalcul... | Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | 303 | 305 | |
/-
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.Basic
import Mathlib.Topology.MetricSpace.Pseudo.Le... | instance Real.metricSpace : MetricSpace ℝ := .ofT0PseudoMetricSpace ℝ
end Real
section NNReal
| Mathlib/Topology/MetricSpace/Basic.lean | 121 | 126 |
/-
Copyright (c) 2024 Thomas Browning, Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Junyan Xu
-/
import Mathlib.Algebra.Group.Subgroup.Ker
import Mathlib.GroupTheory.GroupAction.Basic
import Mathlib.GroupTheory.GroupAction.FixedPoints
impo... | /-- If the support of each element in a generating set of a permutation group is finite,
then the support of every element in the group is finite. -/
theorem finite_compl_fixedBy_closure_iff {S : Set G} :
(∀ g ∈ closure S, (fixedBy α g)ᶜ.Finite) ↔ ∀ g ∈ S, (fixedBy α g)ᶜ.Finite :=
⟨fun h g hg ↦ h g (subset_closur... | Mathlib/GroupTheory/Perm/ClosureSwap.lean | 31 | 35 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yaël Dillies
-/
import Mathlib.Logic.Equiv.Set
import Mathlib.Order.CompleteLattice.Lemmas
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection.Basic
/-... | ⨅ i, ⨆ j, f i j = ⨆ g : ∀ i, κ i, ⨅ i, f i (g i) := by
let _ := minAx.toCompleteLattice
refine le_antisymm ?_ le_iInf_iSup
calc
_ = ⨅ a : range (range <| f ·), ⨆ b : a.1, b.1 := by
simp_rw [iInf_subtype, iInf_range, iSup_subtype, iSup_range]
_ = _ := minAx.iInf_iSup_eq _
| Mathlib/Order/CompleteBooleanAlgebra.lean | 248 | 254 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Order.Filter.Interval
import Mathlib.Order.Interval.Set.Pi
import Mathlib.Tactic.TFAE
import Mathlib.Tactic.NormNum
im... | theorem isOpen_lt' [OrderTopology α] (a : α) : IsOpen { b : α | a < b } :=
isOpen_iff_generate_intervals.2 <| .basic _ ⟨a, .inl rfl⟩
| Mathlib/Topology/Order/Basic.lean | 91 | 93 |
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Choose.Cast
import Mathlib.Data.Nat.Choose.Vanderm... | push_neg at hikl
apply @cast_injective ℚ
have h1 : l ≤ i := le_of_add_le_right hikl
have h2 : k ≤ i - l := le_tsub_of_add_le_right hikl
have h3 : k ≤ k + l := le_self_add
push_cast
rw [cast_choose ℚ h1, cast_choose ℚ h2, cast_choose ℚ h3, cast_choose ℚ hikl]
rw [show i - (k + l) = i - l - k by rw [add_c... | Mathlib/Algebra/Polynomial/HasseDeriv.lean | 164 | 189 |
/-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Ordinal.FixedPoint
/-!
# Principal ordinals
We define principal or indecomposable ordinals, and we prove the standa... | _ ≤ ω ^ b :=
opow_le_opow_right omega0_pos <| (max_lt xb yb).le
theorem principal_add_omega0_opow (o : Ordinal) : Principal (· + ·) (ω ^ o) :=
principal_add_iff_add_left_eq_self.2 fun _ => add_omega0_opow
/-- The main characterization theorem for additive principal ordinals. -/
theorem principal_add_iff_z... | Mathlib/SetTheory/Ordinal/Principal.lean | 186 | 201 |
/-
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 ... | have B : ∀ᶠ a in 𝓝 0, r * (I + 1) * a < 1 / 2 := by
apply (tendsto_order.1 this).2; simp [zero_lt_one]
have C : ∀ᶠ a in 𝓝[>] (0 : ℝ), (0 : ℝ) < a := by
filter_upwards [self_mem_nhdsWithin] with _ ha using ha
rcases (C.and ((A.and B).filter_mono inf_le_left)).exists with ⟨a, ha⟩
exact ⟨a, h... | Mathlib/Analysis/Analytic/Inverse.lean | 501 | 571 |
/-
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.Analysis.SpecialFunctions.Log.Deriv
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics
import Mathlib.Analysis.Convex.Deriv
/-!
# The function `x ↦ ... | /-- Not differentiable, hence `deriv` has junk value zero. -/
lemma deriv_mul_log_zero : deriv (fun x ↦ x * log x) 0 = 0 :=
| Mathlib/Analysis/SpecialFunctions/Log/NegMulLog.lean | 103 | 104 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kim Morrison
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Algebra.Group.InjSurj
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Tactic.FastInstance
impo... | Mathlib/Data/Finsupp/Defs.lean | 923 | 939 | |
/-
Copyright (c) 2023 Hanneke Wiersema. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Hanneke Wiersema, Andrew Yang
-/
import Mathlib.Algebra.Ring.Aut
import Mathlib.NumberTheory.Padics.RingHoms
import Mathlib.RingTheory.RootsOfUnity.EnoughRootsOfUnity
... | -/
variable (n : ℕ) [NeZero n]
| Mathlib/NumberTheory/Cyclotomic/CyclotomicCharacter.lean | 72 | 75 |
/-
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.Complex.UpperHalfPlane.Topology
import Mathlib.Analysis.SpecialFunctions.Arsinh
import Mathlib.Geometry.Euclidean.Inversion.Basic
/-!
# Met... | theorem cosh_half_dist (z w : ℍ) :
cosh (dist z w / 2) = dist (z : ℂ) (conj (w : ℂ)) / (2 * √(z.im * w.im)) := by
rw [← sq_eq_sq₀, cosh_sq', sinh_half_dist, div_pow, div_pow, one_add_div, mul_pow, sq_sqrt]
| Mathlib/Analysis/Complex/UpperHalfPlane/Metric.lean | 45 | 47 |
/-
Copyright (c) 2022 Wrenna Robson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Wrenna Robson
-/
import Mathlib.Topology.MetricSpace.Basic
/-!
# Infimum separation
This file defines the extended infimum separation of a set. This is approximately dual to the
diame... | exact ⟨w.fst, hxy.1, w.snd, hxy.2.1, hxy.2.2, hed⟩
theorem Finite.einfsep_exists_of_nontrivial (hsf : s.Finite) (hs : s.Nontrivial) :
∃ x ∈ s, ∃ y ∈ s, x ≠ y ∧ s.einfsep = edist x y :=
| Mathlib/Topology/MetricSpace/Infsep.lean | 163 | 166 |
/-
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.Control.Combinators
import Mathlib.Data.Option.Defs
import Mathlib.Logic.IsEmpty
import Mathlib.Logic.Relator
import Mathlib.Util.CompileInductive
impo... | Option.some_get h
| Mathlib/Data/Option/Basic.lean | 61 | 62 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Dia... |
/-!
### Images of the Lebesgue measure under translation/linear maps in ℝⁿ
-/
| Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | 335 | 340 |
/-
Copyright (c) 2024 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Lie.Semisimple.Defs
import Mathlib.LinearAlgebra.BilinearForm.Orthogonal
/-!
# Lie algebras with non-degenerate invariant bilinear forms are s... | include hΦ_nondeg hΦ_refl hL
open Module Submodule in
lemma orthogonal_isCompl_toSubmodule (I : LieIdeal K L) (hI : IsAtom I) :
IsCompl I.toSubmodule (orthogonal Φ hΦ_inv I).toSubmodule := by
| Mathlib/Algebra/Lie/InvariantForm.lean | 125 | 129 |
/-
Copyright (c) 2024 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Category.ModuleCat.Presheaf
import Mathlib.Algebra.Category.ModuleCat.Limits
import Mathlib.CategoryTheory.Sites.LocallyBijective
import Mathlib.Categor... | @[simp]
lemma unitHomEquiv_apply_coe (M : SheafOfModules R) (f : unit R ⟶ M) (X : Cᵒᵖ) :
(M.unitHomEquiv f).val X = f.val.app X (1 : R.val.obj X) := rfl
| Mathlib/Algebra/Category/ModuleCat/Sheaf.lean | 166 | 168 |
/-
Copyright (c) 2021 Apurva Nakade. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Apurva Nakade
-/
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Ring.Regular
import Mathlib.GroupTheory.MonoidLocalization.Away
impo... | theorem powHalf_zero_rightMoves : (powHalf 0).RightMoves = PEmpty :=
| Mathlib/SetTheory/Surreal/Dyadic.lean | 52 | 52 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.BigOperators.Option
import Mathlib.Analysis.BoxIntegral.Box.Basic
import Mathlib.Data.Set.Pairwise.Lattice
/-!
# Partitions of rectangular b... |
@[simp]
theorem distortion_bot (I : Box ι) : distortion (⊥ : Prepartition I) = 0 :=
sup_empty
end Distortion
/-- A prepartition `π` of `I` is a partition if the boxes of `π` cover the whole `I`. -/
def IsPartition (π : Prepartition I) :=
| Mathlib/Analysis/BoxIntegral/Partition/Basic.lean | 622 | 630 |
/-
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... | refine le_iff_count.2 fun a ↦ ?_
obtain ⟨m', hm'⟩ := exists_cons_of_mem <| mem_map_of_mem (fun b ↦ count a (f b)) hx
rw [count_bind, hm', sum_cons]
| Mathlib/Data/Multiset/Bind.lean | 191 | 193 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yaël Dillies
-/
import Mathlib.Algebra.Module.BigOperators
import Mathlib.GroupTheory.Perm.Basic
import Mathlib.GroupTheory.Perm.Finite
import Mathlib.GroupTheory.Perm.Lis... |
theorem IsCycleOn.of_pow {n : ℕ} (hf : (f ^ n).IsCycleOn s) (h : Set.BijOn f s s) : f.IsCycleOn s :=
⟨h, fun _ hx _ hy => (hf.2 hx hy).of_pow⟩
theorem IsCycleOn.of_zpow {n : ℤ} (hf : (f ^ n).IsCycleOn s) (h : Set.BijOn f s s) :
f.IsCycleOn s :=
⟨h, fun _ hx _ hy => (hf.2 hx hy).of_zpow⟩
| Mathlib/GroupTheory/Perm/Cycle/Basic.lean | 820 | 827 |
/-
Copyright (c) 2023 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Ahmad Alkhalawi
-/
import Mathlib.Data.Matrix.ConjTranspose
import Mathlib.Tactic.Abel
/-! # Extra lemmas about invertible matrices
A few of the `Invertible` lemmas general... |
section ConjTranspose
| Mathlib/Data/Matrix/Invertible.lean | 50 | 51 |
/-
Copyright (c) 2023 Mohanad ahmed. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mohanad Ahmed
-/
import Mathlib.Data.Matrix.Block
import Mathlib.LinearAlgebra.Matrix.SemiringInverse
/-! # Block Matrices from Rows and Columns
This file provides the basic definiti... |
end Semiring
section CommRing
variable [CommRing R]
| Mathlib/Data/Matrix/ColumnRowPartitioned.lean | 293 | 298 |
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.Symmetric
/-!
# Integer powers of square matrices
In this file, we defi... | · simp [← inv_pow', hB]
theorem Commute.zpow_left {A B : M} (h : Commute A B) (m : ℤ) : Commute (A ^ m) B :=
(Commute.zpow_right h.symm m).symm
theorem Commute.zpow_zpow {A B : M} (h : Commute A B) (m n : ℤ) : Commute (A ^ m) (B ^ n) :=
Commute.zpow_right (Commute.zpow_left h _) _
theorem Commute.zpow_self (... | Mathlib/LinearAlgebra/Matrix/ZPow.lean | 195 | 211 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Ordmap.Invariants
/-!
# Verification of `Ordnode`
This file uses the invariants defined in `Mathlib.Data.Ordmap.Invariants` to construct `Ordset... | Mathlib/Data/Ordmap/Ordset.lean | 1,416 | 1,417 | |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Geometry.RingedSpace.PresheafedSpace
import Mathlib.CategoryTheory.Limits.Final
import Mathlib.Topology.Sheaves.Stalks
/-!
# Stalks for presheaved spaces
... | {f : U ⟶ (X : TopCat.{v})} (h : IsOpenEmbedding f) (x : U) :
IsIso ((X.ofRestrict h).stalkMap x) := by
rw [← restrictStalkIso_inv_eq_ofRestrict]; infer_instance
end Restrict
| Mathlib/Geometry/RingedSpace/Stalks.lean | 99 | 104 |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Sébastien Gouëzel, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Analysis.Normed.Lp.PiLp
import Mathlib.LinearAlgebra.FiniteDimensi... |
/-- The `Basis ι 𝕜 E` underlying the `OrthonormalBasis` -/
protected def toBasis (b : OrthonormalBasis ι 𝕜 E) : Basis ι 𝕜 E :=
Basis.ofEquivFun b.repr.toLinearEquiv
| Mathlib/Analysis/InnerProductSpace/PiL2.lean | 399 | 403 |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jens Wagemaker
-/
import Mathlib.Algebra.Ring.Associated
import Mathlib.Algebra.Ring.Regular
/-!
# Monoids with normalization functions, `gcd`, and `lcm`
This file de... | theorem gcd_dvd_gcd_mul_left [GCDMonoid α] (m n k : α) : gcd m n ∣ gcd (k * m) n :=
| Mathlib/Algebra/GCDMonoid/Basic.lean | 436 | 436 |
/-
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 | 2,295 | 2,295 | |
/-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Algebra.MvPolynomial.Equiv
import Mathlib.Algebra.P... | apply hp0
ext i
refine (mul_one _).symm.trans ?_
rw [← h, mul_zero]
rfl
haveI : Nontrivial R := ⟨⟨0, 1, this⟩⟩
have : p.leadingCoeff ∈ I.leadingCoeffNth N := by
rw [HN]
exact hm2 k ((I.mem_leadingCoeffNth _ _).2
... | Mathlib/RingTheory/Polynomial/Basic.lean | 862 | 878 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl, Patrick Massot
-/
import Mathlib.Data.Set.Image
import Mathlib.Data.SProd
/-!
# Sets in product and pi types
This file proves basic properties of prod... | def Function.pullbackDiagonal (f : X → Y) : Set (f.Pullback f) := {p | p.fst = p.snd}
| Mathlib/Data/Set/Prod.lean | 489 | 490 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Multiset.ZeroCons
/-!
# Basic results on multisets
-/
-- No algebra should be required
assert_not_exists Monoid
universe v
open List S... | left_inv _ := rfl
right_inv m := Quot.inductionOn m fun _ => rfl
| Mathlib/Data/Multiset/Basic.lean | 163 | 164 |
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