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) 2023 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Integral.IntegralEqImproper
/-!
# Integrals against peak functions
A sequence of peak functions is a sequence of functions with a... | (hlφ : ∀ u : Set α, IsOpen u → x₀ ∈ u → TendstoUniformlyOn φ 0 l (s \ u))
(hiφ : Tendsto (fun i ↦ ∫ x in t, φ i x ∂μ) l (𝓝 1))
(h'iφ : ∀ᶠ i in l, AEStronglyMeasurable (φ i) (μ.restrict s))
(hmg : IntegrableOn g s μ) (hcg : Tendsto g (𝓝[s] x₀) (𝓝 a)) :
∀ᶠ i in l, IntegrableOn (fun x => φ i x • g x... | Mathlib/MeasureTheory/Integral/PeakFunction.lean | 54 | 86 |
/-
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... | apply not_iff_not.1
simpa only [not_forall, not_not] using Option.ne_none_iff_exists
| Mathlib/Data/Option/Basic.lean | 326 | 327 |
/-
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... | Mathlib/Data/Fintype/Card.lean | 528 | 534 | |
/-
Copyright (c) 2020 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Polynomial.Degree.Support
import Mathlib.Data.ENat.Basic
/-!
# Trailing degree of univariate polynomials
## Main definitions
* `trailingDegree... |
lemma trailingDegree_ne_zero : trailingDegree p ≠ 0 ↔ coeff p 0 = 0 :=
trailingDegree_eq_zero.not_left
@[simp] theorem trailingDegree_le_trailingDegree (h : coeff q (natTrailingDegree p) ≠ 0) :
trailingDegree q ≤ trailingDegree p :=
(trailingDegree_le_of_ne_zero h).trans natTrailingDegree_le_trailingDegree
t... | Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean | 142 | 150 |
/-
Copyright (c) 2022 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Heather Macbeth
-/
import Mathlib.Data.Nat.Cast.WithTop
import Mathlib.FieldTheory.IsAlgClosed.Basic
import Mathlib.RingTheory.WittVector.DiscreteValuationRing
/-!
#... |
end RecursionBase
open RecursionMain RecursionBase
section FrobeniusRotation
section IsAlgClosed
variable {k : Type*} [Field k] [CharP k p] [IsAlgClosed k]
/-- Recursively defines the sequence of coefficients for `WittVector.frobeniusRotation`.
| Mathlib/RingTheory/WittVector/FrobeniusFractionField.lean | 171 | 182 |
/-
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.Data.Set.Lattice
import Mathlib.Data.SetLike.Basic
import Mathlib.Order.ModularLattice
import Mathlib.Order.SuccPred.Basic
import Mathlib.Order.WellFou... | refine ⟨fun hla => ?_, fun ha => gi.isAtom_of_u_bot hbot ((h_atom a ha).symm ▸ ha)⟩
obtain ⟨a', ha', hab'⟩ :=
(eq_bot_or_exists_atom_le (u (l a))).resolve_left (hbot ▸ fun h => hla.1 (gi.u_injective h))
have :=
| Mathlib/Order/Atoms.lean | 917 | 920 |
/-
Copyright (c) 2020 Aaron Anderson, Jalex Stark, Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark, Kyle Miller, Alena Gusakov, Hunter Monroe
-/
import Mathlib.Combinatorics.SimpleGraph.Init
import Mathlib.Data.Finite.Prod
import... |
theorem mk'_mem_incidenceSet_iff : s(b, c) ∈ G.incidenceSet a ↔ G.Adj b c ∧ (a = b ∨ a = c) :=
and_congr_right' Sym2.mem_iff
| Mathlib/Combinatorics/SimpleGraph/Basic.lean | 638 | 640 |
/-
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... | for `A^ι` with any finite type `ι`. -/
protected def mkPiAlgebraFin : MultilinearMap R (fun _ : Fin n => A) A :=
MultilinearMap.mk' (fun m ↦ (List.ofFn m).prod)
| Mathlib/LinearAlgebra/Multilinear/Basic.lean | 1,151 | 1,153 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Notation.Pi
import Mathlib.Data.Set.Lattice
import Mathlib.Order.Filter.Defs
/-!
# Theory of... | Mathlib/Order/Filter/Basic.lean | 3,171 | 3,173 | |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Comma.Over.Basic
import Mathlib.CategoryTheory.Discrete.Basic
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryThe... | ext <;> simp
theorem coprod.diag_comp {X Y : C} [HasBinaryCoproduct X X] (f : X ⟶ Y) :
| Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean | 740 | 742 |
/-
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.Support
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.Data.Nat.Choose.Sum
impo... | simpa using h
· rintro rfl
| Mathlib/Algebra/Polynomial/Coeff.lean | 346 | 347 |
/-
Copyright (c) 2022 Praneeth Kolichala. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Praneeth Kolichala
-/
import Mathlib.Topology.Homotopy.Path
import Mathlib.Topology.Homotopy.Equiv
/-!
# Contractible spaces
In this file, we define `ContractibleSpace`, a space ... | theorem Nullhomotopic.comp_right {f : C(X, Y)} (hf : f.Nullhomotopic) (g : C(Y, Z)) :
(g.comp f).Nullhomotopic := by
| Mathlib/Topology/Homotopy/Contractible.lean | 28 | 29 |
/-
Copyright (c) 2022 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa, Yuyang Zhao
-/
import Mathlib.Algebra.Order.GroupWithZero.Unbundled.Basic
import Mathlib.Algebra.Order.GroupWithZero.Unbundled.Defs
import Mathlib.Tactic.Linter.Deprecate... | Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean | 1,022 | 1,025 | |
/-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.RingTheory.Valuation.Basic
import Mathlib.NumberTheory.Padics.PadicNorm
import Mathlib.Analysis.Normed.Field.Lemmas
import Mathlib.Tactic.Peel
import... | push_cast
rw [padicNormE.mul]
| Mathlib/NumberTheory/Padics/PadicNumbers.lean | 852 | 853 |
/-
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 | 917 | 918 | |
/-
Copyright (c) 2021 Aaron Anderson, Jesse Michael Han, Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jesse Michael Han, Floris van Doorn
-/
import Mathlib.Data.Finset.Basic
import Mathlib.ModelTheory.Syntax
import Mathlib.Data.List.... | @[simp]
| Mathlib/ModelTheory/Semantics.lean | 853 | 853 |
/-
Copyright (c) 2019 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Yury Kudryashov, Yaël Dillies
-/
import Mathlib.Algebra.Order.Invertible
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.LinearAlgebra.AffineSpac... | section DenselyOrdered
variable [ZeroLEOneClass 𝕜] [Nontrivial 𝕜] [DenselyOrdered 𝕜]
| Mathlib/Analysis/Convex/Segment.lean | 158 | 160 |
/-
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.... | {f : Ω → β} {g : Ω → γ} (hfg : IndepFun f g κ μ)
{s t : Set Ω} (hs : MeasurableSet[mβ.comap f] s) (ht : MeasurableSet[mγ.comap g] t) :
∀ᵐ a ∂μ, κ a (s ∩ t) = κ a s * κ a t := hfg _ _ hs ht
end ByDefinition
| Mathlib/Probability/Independence/Kernel.lean | 268 | 273 |
/-
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 | 366 | 367 | |
/-
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.Support
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.Data.Nat.Choose.Sum
impo... | subst h1
rfl
· exact fun h1 => (h1 (mem_antidiagonal.2 rfl)).elim
@[simp]
theorem coeff_X_pow_mul (p : R[X]) (n d : ℕ) :
coeff (Polynomial.X ^ n * p) (d + n) = coeff p d := by
rw [(commute_X_pow p n).eq, coeff_mul_X_pow]
| Mathlib/Algebra/Polynomial/Coeff.lean | 230 | 237 |
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Data.PFunctor.Univariate.M
/-!
# Quotients of Polynomial Functors
We assume the following:
* `P`: a polynomial functor
* `W`: its W-type
* `M`: its M... | Cofix.bisim R
(fun x y ⟨x', Qx', xeq, yeq⟩ => by
rcases h x' Qx' with ⟨a, f, f', ux'eq, vx'eq, h'⟩
rw [liftr_iff]
exact ⟨a, f, f', xeq.symm ▸ ux'eq, yeq.symm ▸ vx'eq, h'⟩)
_ _ ⟨x, Qx, rfl, rfl⟩
end QPF
/-
Composition of qpfs.
-/
namespace QPF
variable {F₂ : Type u → Type u} [q₂ : QPF F₂]
... | Mathlib/Data/QPF/Univariate/Basic.lean | 433 | 467 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Group.Invertible.Defs
import Mathlib.Algebra.Group.Hom.Defs
import Mathlib.Logic.Equiv.Defs
/-!
# Theorems... | Mathlib/Algebra/Group/Invertible/Basic.lean | 164 | 169 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Logic.Nontrivial.Basic
import Mathlib.Order.TypeTags
import Mathlib.Data.Option.NAry
import Mathlib.Tactic.Contrapose
import Mathlib.Tactic.Lift
import... | Mathlib/Order/WithBot.lean | 1,163 | 1,165 | |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Mathlib.Data.Stream.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Data.List.Defs
import Mathlib.Data.Nat.Basic
import Mathlib.Tactic.Common... | rfl
theorem homomorphism (f : α → β) (a : α) : pure f ⊛ pure a = pure (f a) :=
rfl
theorem interchange (fs : Stream' (α → β)) (a : α) :
fs ⊛ pure a = (pure fun f : α → β => f a) ⊛ fs :=
rfl
theorem map_eq_apply (f : α → β) (s : Stream' α) : map f s = pure f ⊛ s :=
| Mathlib/Data/Stream/Init.lean | 702 | 711 |
/-
Copyright (c) 2022 Mantas Bakšys. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mantas Bakšys
-/
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Order.Module.Synonym
import Mathlib.Data.Prod.Lex
import Mathlib.Data.Set.Image
import Mathlib.Da... | (hfg.monovaryOn _).sum_comp_perm_smul_le_sum_smul fun _ _ ↦ mem_univ _
/-- **Rearrangement Inequality**: Pointwise scalar multiplication of `f` and `g` is minimized when
`f` and `g` antivary together. Stated by permuting the entries of `f`. -/
theorem Antivary.sum_smul_le_sum_comp_perm_smul (hfg : Antivary f g) :
| Mathlib/Algebra/Order/Rearrangement.lean | 152 | 156 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.Complex.Asymptotics
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Data.Complex.Trig... | /-- The function `(b * exp x + c) / (x ^ n)` tends to `+∞` at `+∞`, for any natural number
`n` and any real numbers `b` and `c` such that `b` is positive. -/
theorem tendsto_mul_exp_add_div_pow_atTop (b c : ℝ) (n : ℕ) (hb : 0 < b) :
Tendsto (fun x => (b * exp x + c) / x ^ n) atTop atTop := by
| Mathlib/Analysis/SpecialFunctions/Exp.lean | 275 | 278 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Cardinal.Arithmetic
import Mathlib.SetTheory.Ordinal.FixedPoint
/-!
# Cofinality
This file co... | Mathlib/SetTheory/Cardinal/Cofinality.lean | 1,104 | 1,106 | |
/-
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... |
@[simp]
theorem getElem_set_of_ne {l : List α} {i j : ℕ} (h : i ≠ j) (a : α)
(hj : j < (l.set i a).length) :
(l.set i a)[j] = l[j]'(by simpa using hj) := by
rw [← Option.some_inj, ← List.getElem?_eq_getElem, List.getElem?_set_ne h,
List.getElem?_eq_getElem]
/-! ### map -/
| Mathlib/Data/List/Basic.lean | 692 | 701 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Jakob von Raumer
-/
import Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
/-!
# Biproducts and binary biproducts
... | Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean | 1,848 | 1,856 | |
/-
Copyright (c) 2023 Geoffrey Irving. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler, Geoffrey Irving, Stefan Kebekus
-/
import Mathlib.Analysis.Analytic.Composition
import Mathlib.Analysis.Analytic.Linear
import Mathlib.Analysis.NormedSpace.OperatorNor... | @[fun_prop]
theorem AnalyticAt.sub (hf : AnalyticAt 𝕜 f x) (hg : AnalyticAt 𝕜 g x) :
AnalyticAt 𝕜 (f - g) x :=
hf.fun_sub hg
@[deprecated (since := "2025-03-11")] alias AnalyticAt.sub' := AnalyticAt.fun_sub
theorem HasFPowerSeriesWithinOnBall.const_smul (hf : HasFPowerSeriesWithinOnBall f pf s x r) :
Has... | Mathlib/Analysis/Analytic/Constructions.lean | 181 | 190 |
/-
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.Data.ENNReal.Real
import Mathlib.Tactic.Bound.Attribute
import Mathlib.Topology... | -- ensure that the bornology is unchanged when replacing the uniformity.
example {α} [U : UniformSpace α] (m : PseudoMetricSpace α)
(H : 𝓤[U] = 𝓤[PseudoEMetricSpace.toUniformSpace]) :
(PseudoMetricSpace.replaceUniformity m H).toBornology = m.toBornology := by
with_reducible_and_instances rfl
| Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | 965 | 970 |
/-
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.LSeries.RiemannZeta
import Mathlib.NumberTheory.Harmonic.GammaDeriv
/-!
# Asymptotics of `ζ s` as `s → 1`
The goal of this file is to ev... | rw [Pi.div_apply, ← sub_div, div_right_comm, sub_div' (Gammaℝ_ne_zero_of_re_pos ha), one_mul]
lemma tendsto_riemannZeta_sub_one_div_Gammaℝ :
Tendsto (fun s ↦ riemannZeta s - 1 / Gammaℝ s / (s - 1)) (𝓝[≠] 1)
(𝓝 ((γ - Complex.log (4 * ↑π)) / 2)) := by
have := tendsto_riemannZeta_sub_one_div.add tendsto_Gam... | Mathlib/NumberTheory/Harmonic/ZetaAsymp.lean | 375 | 381 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.RingTheory.Ideal.Operations
/-!
# Maps on modules and ideals
Main definitions include `Ideal.map`, `Ideal.comap`, `RingHom.ker`, `Module.annihilator`
and `Subm... | simp_rw [show map e I = _ from map_comap_of_equiv (e : R ≃+* S)] at h
exact ⟨(e : R ≃+* S).symm y, h, (e : R ≃+* S).apply_symm_apply y⟩
· rintro ⟨x, hx, rfl⟩
exact mem_map_of_mem e hx
section Bijective
variable (hf : Function.Bijective f) {I : Ideal R} {K : Ideal S}
| Mathlib/RingTheory/Ideal/Maps.lean | 445 | 452 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Data.Set.Constructions
import Mathlib.Order.Filter.AtTopBot.CountablyGenerated
import Mathlib.Topology.Constructions
import Mathlib.Top... | h₁.inf_induced h₂ Prod.fst Prod.snd
theorem isTopologicalBasis_of_cover {ι} {U : ι → Set α} (Uo : ∀ i, IsOpen (U i))
(Uc : ⋃ i, U i = univ) {b : ∀ i, Set (Set (U i))} (hb : ∀ i, IsTopologicalBasis (b i)) :
IsTopologicalBasis (⋃ i : ι, image ((↑) : U i → α) '' b i) := by
refine isTopologicalBasis_of_isOpen_... | Mathlib/Topology/Bases.lean | 260 | 266 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker, Johan Commelin
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Div
import Mathlib.RingTheory... | simp_rw [rootMultiplicity_eq_natTrailingDegree, mul_comp, natTrailingDegree_mul' hpq]
theorem Monic.neg_one_pow_natDegree_mul_comp_neg_X {p : R[X]} (hp : p.Monic) :
((-1) ^ p.natDegree * p.comp (-X)).Monic := by
| Mathlib/Algebra/Polynomial/RingDivision.lean | 172 | 175 |
/-
Copyright (c) 2021 Hunter Monroe. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hunter Monroe, Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Dart
import Mathlib.Data.FunLike.Fintype
import Mathlib.Logic.Embedding.Set
/-!
# Maps between graphs
This file ... |
theorem map_mem_edgeSet_iff {e : Sym2 V} : e.map f ∈ G'.edgeSet ↔ e ∈ G.edgeSet :=
Sym2.ind (fun _ _ => f.map_adj_iff) e
| Mathlib/Combinatorics/SimpleGraph/Maps.lean | 489 | 492 |
/-
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.AlgebraicGeometry.Cover.Open
import Mathlib.AlgebraicGeometry.Over
/-!
# Restriction of Schemes and Morphisms
## Main definition
- `AlgebraicGeometry.Schem... | @[simp]
lemma ι_appLE (V W e) :
U.ι.appLE V W e =
| Mathlib/AlgebraicGeometry/Restrict.lean | 80 | 82 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.Group.Action.Units
import Mathlib.Algebra.Group.Nat.Units
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra... | (add_comm _ y ▸ h).of_add_mul_left_right
theorem IsRelPrime.of_mul_add_right_right (h : IsRelPrime x (z * x + y)) : IsRelPrime x y :=
| Mathlib/RingTheory/Coprime/Basic.lean | 222 | 224 |
/-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Topology.Algebra.Algebra
import Mathlib.Analysis.InnerProductSpace.Convex
import Mathlib.Algebra.Module.LinearMap.Rat
import Mathlib.Tactic.Module
/... | private theorem add_left_aux2 (x y z : E) : ‖2 • x + y‖ * ‖2 • x + y‖ + ‖y - 2 • z‖ * ‖y - 2 • z‖
= 2 * (‖x + y - z‖ * ‖x + y - z‖ + ‖x + z‖ * ‖x + z‖) := by
convert parallelogram_identity (x + y - z) (x + z) using 4 <;> abel
private theorem add_left_aux3 (y z : E) :
‖2 • z + y‖ * ‖2 • z + y‖ + ‖y‖ * ‖y‖ = 2... | Mathlib/Analysis/InnerProductSpace/OfNorm.lean | 138 | 160 |
/-
Copyright (c) 2022 Mario Carneiro, Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Heather Macbeth, Yaël Dillies
-/
import Mathlib.Tactic.NormNum.Core
import Mathlib.Tactic.HaveI
import Mathlib.Algebra.Order.Invertible
import Mathlib.... | | ⟨inv, eq⟩, h => by rw [eq, of_decide_eq_true h]; simp
lemma nz_of_isRat {n : ℤ} {d : ℕ} [Ring A] [LinearOrder A] [IsStrictOrderedRing A] :
| Mathlib/Tactic/Positivity/Core.lean | 151 | 153 |
/-
Copyright (c) 2022 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Oleksandr Manzyuk
-/
import Mathlib.CategoryTheory.Bicategory.Basic
import Mathlib.CategoryTheory.Monoidal.Mon_
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Equali... | slice_rhs 3 4 => rw [ι_colimMap, parallelPairHom_app_one]
slice_rhs 2 3 => rw [← MonoidalCategory.whiskerLeft_comp, ι_colimMap, parallelPairHom_app_one]
dsimp [AssociatorBimod.inv]
slice_rhs 3 4 => rw [coequalizer.π_desc]
dsimp [AssociatorBimod.invAux]
slice_rhs 2 2 => rw [MonoidalCategory.whiskerLeft_comp]... | Mathlib/CategoryTheory/Monoidal/Bimod.lean | 869 | 894 |
/-
Copyright (c) 2022 Rémi Bottinelli. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémi Bottinelli, Junyan Xu
-/
import Mathlib.Algebra.Group.Subgroup.Defs
import Mathlib.CategoryTheory.Groupoid.VertexGroup
import Mathlib.CategoryTheory.Groupoid.Basic
import Mathlib... | @ST ⟨_, _, _⟩
theorem isNormal_comap {S : Subgroupoid D} (Sn : IsNormal S) : IsNormal (comap φ S) where
wide c := by rw [comap, mem_setOf, Functor.map_id]; apply Sn.wide
| Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean | 384 | 387 |
/-
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.Connected.Basic
/-!
# Locally connected topological spaces
A topological space is **locally connected** if each neighborhood filter admi... |
theorem locallyConnectedSpace_iff_subsets_isOpen_isConnected :
LocallyConnectedSpace α ↔
∀ x, ∀ U ∈ 𝓝 x, ∃ V : Set α, V ⊆ U ∧ IsOpen V ∧ x ∈ V ∧ IsConnected V := by
simp_rw [locallyConnectedSpace_iff_hasBasis_isOpen_isConnected]
refine forall_congr' fun _ => ?_
constructor
· intro h U hU
rcases ... | Mathlib/Topology/Connected/LocallyConnected.lean | 41 | 52 |
/-
Copyright (c) 2022 Yaël Dillies, Sara Rousta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Sara Rousta
-/
import Mathlib.Logic.Equiv.Set
import Mathlib.Order.Interval.Set.OrderEmbedding
import Mathlib.Order.SetNotation
/-!
# Properties of unbundled ... | Mathlib/Order/UpperLower/Basic.lean | 1,713 | 1,714 | |
/-
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.Group.Nat.Defs
import Mathlib.CategoryTheory.Category.Preorder
import Mathlib.CategoryTheory.EqToHom
import Mathlib.CategoryTheory.Functor.Const
import M... | variable (obj : Fin (n + 1) → C) (mapSucc : ∀ (i : Fin n), obj i.castSucc ⟶ obj i.succ)
lemma mkOfObjOfMapSucc_exists : ∃ (F : ComposableArrows C n) (e : ∀ i, F.obj i ≅ obj i),
∀ (i : ℕ) (hi : i < n), mapSucc ⟨i, hi⟩ =
(e ⟨i, _⟩).inv ≫ F.map' i (i + 1) ≫ (e ⟨i + 1, _⟩).hom := by
| Mathlib/CategoryTheory/ComposableArrows.lean | 835 | 839 |
/-
Copyright (c) 2019 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.Topology.UniformSpace.UniformEmbedding
import Mathlib.Topology.UniformSpace.Equiv
/-!
# Abstract theory of Hausdorff completions of uniform spaces
Th... |
local notation "hatβ" => pkg'.space
local notation "ι'" => pkg'.coe
| Mathlib/Topology/UniformSpace/AbstractCompletion.lean | 158 | 161 |
/-
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... | normUnit_mul hp0 hq0 :=
Units.ext
| Mathlib/Algebra/Polynomial/FieldDivision.lean | 201 | 202 |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.Real
/-!
# Properties of pointwise scalar multiplication of se... |
open EMetric ENNReal
@[simp]
theorem infEdist_thickening (hδ : 0 < δ) (s : Set E) (x : E) :
infEdist x (thickening δ s) = infEdist x s - ENNReal.ofReal δ := by
obtain hs | hs := lt_or_le (infEdist x s) (ENNReal.ofReal δ)
| Mathlib/Analysis/NormedSpace/Pointwise.lean | 227 | 233 |
/-
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... | Chain R a (l ++ [b]) ↔ Forall₂ R (a :: l) (l ++ [b]) := by simp [chain_iff_forall₂]
theorem chain_map (f : β → α) {b : β} {l : List β} :
| Mathlib/Data/List/Chain.lean | 69 | 71 |
/-
Copyright (c) 2023 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Algebra.NoZeroSMulDivisors.Basic
import Mathlib.Algebra.Order.GroupWithZero.Action.Synonym
import Mathlib.Tactic.GCongr
import Mathlib.Tactic.Positivity.Co... | b₂ < b₁ := by
rw [← neg_neg a, neg_smul, neg_smul (-a), neg_lt_neg_iff] at h
exact lt_of_smul_lt_smul_of_nonneg_left h (neg_nonneg_of_nonpos ha)
| Mathlib/Algebra/Order/Module/Defs.lean | 879 | 882 |
/-
Copyright (c) 2022 María Inés de Frutos-Fernández. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: María Inés de Frutos-Fernández
-/
import Mathlib.RingTheory.DedekindDomain.Ideal
import Mathlib.RingTheory.Valuation.ExtendToLocalization
import Mathlib.Topology.Algebr... | exact
⟨v.ne_bot, v.asIdeal, (not_congr Ideal.isUnit_iff).mpr (Ideal.IsPrime.ne_top v.isPrime),
sq v.asIdeal⟩
obtain ⟨π, mem, nmem⟩ := SetLike.exists_of_lt hlt
have hπ : Associates.mk (Ideal.span {π}) ≠ 0 := by
rw [Associates.mk_ne_zero']
intro h
rw [h] at nmem
exact nmem (Submodule... | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | 233 | 254 |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Sign
import Mathlib.LinearAlgebra.AffineSpace.Combination
import Mathlib.LinearAlg... | protected theorem AffineIndependent.mem_affineSpan_iff [Nontrivial k] {p : ι → P}
(ha : AffineIndependent k p) (i : ι) (s : Set ι) : p i ∈ affineSpan k (p '' s) ↔ i ∈ s := by
constructor
· intro hs
have h :=
AffineIndependent.exists_mem_inter_of_exists_mem_inter_affineSpan ha hs
| Mathlib/LinearAlgebra/AffineSpace/Independent.lean | 433 | 438 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.Convex.StrictConvexSpace
import Mathlib.MeasureTheory.Function.AEEqOfIntegral
import Mathlib.Measure... | ### Strict Jensen's inequality
-/
| Mathlib/Analysis/Convex/Integral.lean | 210 | 213 |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
/-!
# Lemmas about `min` and `max` in an ordered monoid.
-/
op... | exact fun h => mul_lt_mul_of_lt_of_lt h.1 h.2
@[to_additive]
theorem mul_lt_mul_iff_of_le_of_le [MulLeftMono α]
[MulRightMono α] [MulLeftStrictMono α]
| Mathlib/Algebra/Order/Monoid/Unbundled/MinMax.lean | 99 | 103 |
/-
Copyright (c) 2024 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Cardinal.Arithmetic
import Mathlib.SetTheory.Ordinal.Principal
/-!
# Ordinal arithmetic with cardinals
This file co... | Mathlib/SetTheory/Cardinal/Ordinal.lean | 948 | 953 | |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Shing Tak Lam, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring.List
import Mathlib.Data.Int.ModEq
import Mathlib.Da... |
theorem digits_zero_succ' : ∀ {n : ℕ}, n ≠ 0 → digits 0 n = [n]
| Mathlib/Data/Nat/Digits.lean | 90 | 91 |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Casper Putz, Anne Baanen
-/
import Mathlib.Algebra.Algebra.Subalgebra.Tower
import Mathlib.Data.Finite.Sum
import Mathlib.Data.Matrix.Block
import Mathl... |
/-- `Matrix.toLin'` is a particular case of `Matrix.toLin`, for the standard basis
`Pi.basisFun R n`. -/
| Mathlib/LinearAlgebra/Matrix/ToLin.lean | 497 | 499 |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Polynomial.Module.AEval
/-!
# Polynomial module
In this file, we define the polynomial module for an `R`-module `M`, i.e. the `R[X]`-module `M[X]`.... | omega
theorem smul_apply (f : R[X]) (g : PolynomialModule R M) (n : ℕ) :
(f • g) n = ∑ x ∈ Finset.antidiagonal n, f.coeff x.1 • g x.2 := by
induction f using Polynomial.induction_on' with
| add p q hp hq =>
rw [add_smul, Finsupp.add_apply, hp, hq, ← Finset.sum_add_distrib]
congr
ext
rw [c... | Mathlib/Algebra/Polynomial/Module/Basic.lean | 157 | 169 |
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Mathlib.Control.Basic
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Option.Basic
im... | Mathlib/Data/List/Basic.lean | 2,833 | 2,841 | |
/-
Copyright (c) 2018 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Kim Morrison
-/
import Mathlib.CategoryTheory.Opposites
/-!
# Morphisms from equations between objects.
When working categorically, sometimes one encounters an equation `h ... | cases w
simp
/-- A variant on `eqToHom_naturality` that helps Lean identify the families `f` and `g`. -/
| Mathlib/CategoryTheory/EqToHom.lean | 138 | 141 |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.CategoryTheory.Sites.IsSheafFor
import Mathlib.CategoryTheory.Limits.Types.Shapes
import Mathlib.Tactic.ApplyFun
/-!
# The equalizer diagram sheaf conditi... | /--
The family of elements given by `x : FirstObj P S` is compatible iff `firstMap` and `secondMap`
map it to the same point.
-/
theorem compatible_iff (x : FirstObj P R) :
((firstObjEqFamily P R).hom x).Compatible ↔ firstMap P R x = secondMap P R x := by
rw [Presieve.pullbackCompatible_iff]
constructor
| Mathlib/CategoryTheory/Sites/EqualizerSheafCondition.lean | 216 | 223 |
/-
Copyright (c) 2021 Julian Kuelshammer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Julian Kuelshammer
-/
import Mathlib.Algebra.CharP.Algebra
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.CharP.Lemmas
import Mathlib.Algebra.EuclideanDomain.Field
... | dickson_two_one_eq_chebyshev_S (n + 1), dickson_two_one_eq_chebyshev_S n]
push_cast
ring
| Mathlib/RingTheory/Polynomial/Dickson.lean | 166 | 169 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Algebra.Field.NegOnePow
import Mathlib.Algebra.Field.Periodic
import Mathlib.Algebra.Qua... | tan_periodic x
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | 1,138 | 1,138 |
/-
Copyright (c) 2021 Alex Kontorovich and Heather Macbeth and Marc Masdeu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Kontorovich, Heather Macbeth, Marc Masdeu
-/
import Mathlib.Analysis.Complex.Basic
import Mathlib.Data.Fintype.Parity
import Mathlib.LinearAl... | rfl
| Mathlib/Analysis/Complex/UpperHalfPlane/Basic.lean | 440 | 441 |
/-
Copyright (c) 2020 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Topology.Algebra.Group.Defs
/-!
# Topological propertie... | end AffineMap
| Mathlib/Topology/Algebra/Affine.lean | 76 | 78 |
/-
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.Projection
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique
import Mathlib.MeasureTheory.Function.L2Space
/-... |
section Real
variable {hm : m ≤ m0}
theorem integral_condExpL2_eq_of_fin_meas_real (f : Lp 𝕜 2 μ) (hs : MeasurableSet[m] s)
(hμs : μ s ≠ ∞) : ∫ x in s, (condExpL2 𝕜 𝕜 hm f : α → 𝕜) x ∂μ = ∫ x in s, f x ∂μ := by
rw [← L2.inner_indicatorConstLp_one (𝕜 := 𝕜) (hm s hs) hμs f]
have h_eq_inner : ∫ x in s, (c... | Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean | 161 | 182 |
/-
Copyright (c) 2022 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Algebra.BigOperators.Group.List.Basic
import Mathlib.Algebra.Group.Embedding
import Mathlib.Algebra.Group.Nat.Defs
import Mathlib.Data.Finsupp.Single... | Mathlib/Data/List/ToFinsupp.lean | 139 | 143 | |
/-
Copyright (c) 2020 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Analysis.Calculus.Deriv.Inv
import Mathlib.Analysis.Calculus.Deriv.MeanValue
/-!
# L'Hôpital's rule for 0/0 indeterminate forms
In this file, we ... | end deriv
| Mathlib/Analysis/Calculus/LHopital.lean | 449 | 460 |
/-
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.Group.Subgroup.Map
import Mathlib.Algebra.Module.Submodule.... | end Submodule
namespace Submodule
variable [Semiring R] [Semiring R₂]
variable [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂]
variable {τ₁₂ : R →+* R₂} {τ₂₁ : R₂ →+* R}
| Mathlib/Algebra/Module/Submodule/Map.lean | 543 | 549 |
/-
Copyright (c) 2021 Bryan Gin-ge Chen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bryan Gin-ge Chen, Yaël Dillies
-/
import Mathlib.Algebra.Group.Idempotent
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Algebra.Ring.PUnit
import Mathlib.Order.Hom.BoundedLattic... | protected def RingHom.asBoolAlg (f : α →+* β) : BoundedLatticeHom (AsBoolAlg α) (AsBoolAlg β) where
toFun := toBoolAlg ∘ f ∘ ofBoolAlg
map_sup' a b := by
dsimp
| Mathlib/Algebra/Ring/BooleanRing.lean | 305 | 308 |
/-
Copyright (c) 2021 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
import Mathlib.Order.Lattice
/-!
# Ordered Subtraction
This file proves l... | theorem lt_tsub_iff_left : a < b - c ↔ c + a < b :=
lt_iff_lt_of_le_iff_le tsub_le_iff_left
theorem lt_tsub_comm : a < b - c ↔ c < b - a :=
lt_tsub_iff_left.trans lt_tsub_iff_right.symm
| Mathlib/Algebra/Order/Sub/Defs.lean | 389 | 393 |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Ring.Divisibility.Lemmas
import Mathlib.Algebra.Lie.Nilpotent
import Mathlib.Algebra.Lie.Engel
import Mathlib.LinearAlgebra.Eigenspace.Pi
import Math... | simp_rw [← LieSubmodule.mem_toSubmodule, posFittingCompOf, hk k (le_refl k)]
apply LinearMap.mem_range_self
suffices (toEnd R L (M ⧸ F) x ^ k) (LieSubmodule.Quotient.mk (N := F) m) =
LieSubmodule.Quotient.mk (N := F) ((toEnd R L M x ^ k) m)
by simpa [Submodule.Quotient.quot_mk_eq_mk, this]
have :=... | Mathlib/Algebra/Lie/Weights/Basic.lean | 481 | 490 |
/-
Copyright (c) 2022 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.GroupTheory.Complement
import Mathlib.GroupTheory.Sylow
/-!
# The Transfer Homomorphism
In this file we construct the transfer homomorphism.
## Ma... | · rw [hH, pow_zero]
exact H.one_mem
letI := fintypeOfIndexNeZero hH
classical
| Mathlib/GroupTheory/Transfer.lean | 184 | 187 |
/-
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
/... | rw [mem_preimage, mem_ball, mem_ball, lt_div_iff₀ (norm_pos_iff.mpr ha), mul_comm, ←
map_smul_eq_mul p, smul_sub, smul_inv_smul₀ ha]
| Mathlib/Analysis/Seminorm.lean | 960 | 962 |
/-
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... | left_distrib := by transfer; simp [mul_add]
right_distrib := by transfer; simp [mul_add, mul_comm]
| Mathlib/Data/Num/Lemmas.lean | 540 | 541 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Aaron Anderson, Yakov Pechersky
-/
import Mathlib.Data.Fintype.Card
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Algebra.Group.End
import Mathlib.Data.Finset.N... | | n + 1 => by rw [pow_succ, mul_apply, hfx, pow_apply_eq_self_of_apply_eq_self hfx n]
theorem zpow_apply_eq_self_of_apply_eq_self {x : α} (hfx : f x = x) : ∀ n : ℤ, (f ^ n) x = x
| (n : ℕ) => pow_apply_eq_self_of_apply_eq_self hfx n
| Int.negSucc n => by rw [zpow_negSucc, inv_eq_iff_eq, pow_apply_eq_self_of_appl... | Mathlib/GroupTheory/Perm/Support.lean | 144 | 152 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Kappelmann
-/
import Mathlib.Algebra.Order.Floor.Defs
import Mathlib.Algebra.Order.Floor.Ring
import Mathlib.Algebra.Order.Floor.Semiring
deprecated_module (sinc... | Mathlib/Algebra/Order/Floor.lean | 1,598 | 1,600 | |
/-
Copyright (c) 2021 Justus Springer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Justus Springer, Andrew Yang
-/
import Mathlib.Algebra.Category.Ring.Colimits
import Mathlib.Algebra.Category.Ring.FilteredColimits
import Mathlib.Algebra.Category.Ring.Limits
import ... |
theorem basicOpen_of_isUnit {U : Opens X} {f : X.presheaf.obj (op U)} (hf : IsUnit f) :
X.basicOpen f = U := by
apply le_antisymm
· exact X.basicOpen_le f
intro x hx
rw [SetLike.mem_coe, X.mem_basicOpen f x hx]
| Mathlib/Geometry/RingedSpace/Basic.lean | 226 | 232 |
/-
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 | 1,370 | 1,383 | |
/-
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... | theorem encard_le_encard (h : s ⊆ t) : s.encard ≤ t.encard := by
rw [← union_diff_cancel h, encard_union_eq disjoint_sdiff_right]; exact le_self_add
@[deprecated (since := "2025-01-05")] alias encard_le_card := encard_le_encard
| Mathlib/Data/Set/Card.lean | 164 | 167 |
/-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Data.Sym.Sym2
import Mathlib.Logic.Relation
/-!
# Game addition relation
This file defines, given relations `rα : α → α → Prop` and `rβ : β → β → Prop`, a rela... | gameAdd_swap_swap rβ rα (b₁, a₁) (b₂, a₂)
| Mathlib/Order/GameAdd.lean | 73 | 74 |
/-
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.Abelian.Exact
import Mathlib.CategoryTheory.Comma.Over.Basic
import Mathlib.Algebra.Category.ModuleCat.EpiMono
/-!
# Pseudoelements in ab... | refine ⟨pullback (Abelian.factorThruImage S.f) c, 𝟙 _,
pullback.snd _ _, inferInstance, inferInstance, ?_⟩
-- Now we can verify that the diagram commutes.
calc
𝟙 (pullback (Abelian.factorThruImage S.f) c) ≫ pullback.fst _ _ ≫ S.f =
pullback.fst _ _ ≫ S.f :=
... | Mathlib/CategoryTheory/Abelian/Pseudoelements.lean | 330 | 336 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Shing Tak Lam, Yury Kudryashov
-/
import Mathlib.Algebra.MvPolynomial.Derivation
import Mathlib.Algebra.MvPolynomial.Variables
/-!
# Partial derivatives of polynomials
This file defi... | (x : σ) (p : MvPolynomial σ R) :
pderiv (f x) (rename f p) = rename f (pderiv x p) := by
| Mathlib/Algebra/MvPolynomial/PDeriv.lean | 125 | 126 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Group.Submonoid.Operations
import Mathlib.Algebra.MonoidAlgebra.Defs
import Mathlib.Algebra.Order.Mon... | instance inhabited : Inhabited R[X] :=
⟨0⟩
| Mathlib/Algebra/Polynomial/Basic.lean | 266 | 267 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Analytic.Within
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries
/-!
# Higher d... | Mathlib/Analysis/Calculus/ContDiff/Defs.lean | 1,379 | 1,380 | |
/-
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... | Mathlib/Topology/Constructions.lean | 1,428 | 1,430 | |
/-
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... | (surjective_of_injective (injective_surjInv _))
(rightInverse_surjInv _)⟩⟩
theorem injective_iff_bijective {f : α → α} : Injective f ↔ Bijective f := by
simp [Bijective, injective_iff_surjective]
theorem surjective_iff_bijective {f : α → α} : Surjective f ↔ Bijective f := by
simp [Bijective, injective... | Mathlib/Data/Fintype/Card.lean | 315 | 322 |
/-
Copyright (c) 2021 Aaron Anderson, Jesse Michael Han, Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jesse Michael Han, Floris van Doorn
-/
import Mathlib.Data.Set.Prod
import Mathlib.Logic.Equiv.Fin.Basic
import Mathlib.ModelTheory... | unfold distinctConstantsTheory; gcongr
theorem monotone_distinctConstantsTheory :
Monotone (L.distinctConstantsTheory : Set α → L[[α]].Theory) := fun _s _t st =>
L.distinctConstantsTheory_mono st
theorem directed_distinctConstantsTheory :
| Mathlib/ModelTheory/Syntax.lean | 842 | 848 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Alex Kontorovich, Heather Macbeth
-/
import Mathlib.MeasureTheory.Group.Action
import Mathlib.MeasureTheory.Group.Pointwise
import Mathlib.MeasureTheory.Integral.Lebe... | delicate question which was recently addressed by Misha Kapovich: https://arxiv.org/abs/2301.05325
TODO: Formalize the existence of a Dirichlet domain as in Kapovich's paper.
| Mathlib/MeasureTheory/Group/FundamentalDomain.lean | 658 | 660 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Data.Countable.Small
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Powerset
import Mathlib.Dat... | (add_le_aleph0.2 ⟨le_rfl, (nat_lt_aleph0 n).le⟩).antisymm le_self_add
| Mathlib/SetTheory/Cardinal/Basic.lean | 565 | 565 |
/-
Copyright (c) 2019 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Yury Kudryashov
-/
import Mathlib.Data.Set.Lattice.Image
import Mathlib.Order.Interval.Set.LinearOrder
/-!
# Extra lemmas about intervals
This file contains lemma... | gcongr
exact Iio_subset_Iic_self
end UnionIxx
| Mathlib/Order/Interval/Set/Disjoint.lean | 245 | 248 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Kappelmann
-/
import Mathlib.Algebra.Order.Floor.Defs
import Mathlib.Algebra.Order.Floor.Ring
import Mathlib.Algebra.Order.Floor.Semiring
deprecated_module (sinc... | Mathlib/Algebra/Order/Floor.lean | 522 | 524 | |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.RingTheory.WittVector.InitTail
/-!
# Truncated Witt vectors
The ring of truncated Witt vectors (of length `n`) is a quotient of the... | ⟨fun x y => truncateFun n (x.out + y.out)⟩
| Mathlib/RingTheory/WittVector/Truncated.lean | 166 | 167 |
/-
Copyright (c) 2018 Sean Leather. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sean Leather, Mario Carneiro
-/
import Mathlib.Data.List.AList
import Mathlib.Data.Finset.Sigma
import Mathlib.Data.Part
/-!
# Finite maps over `Multiset`
-/
universe u v w
open List
... | theorem union_toFinmap (s₁ s₂ : AList β) : (toFinmap s₁) ∪ (toFinmap s₂) = toFinmap (s₁ ∪ s₂) := by
simp [(· ∪ ·), union]
| Mathlib/Data/Finmap.lean | 502 | 504 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.BoxIntegral.DivergenceTheorem
import Mathlib.Analysis.BoxIntegral.Integrability
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.... | (((∫ x in a.1..b.1, g (x, b.2)) - ∫ x in a.1..b.1, g (x, a.2)) +
∫ y in a.2..b.2, f (b.1, y)) -
∫ y in a.2..b.2, f (a.1, y) :=
let e : (ℝ × ℝ) ≃L[ℝ] ℝ² := (ContinuousLinearEquiv.finTwoArrow ℝ ℝ).symm
calc
(∫ x in Icc a b, f' x (1, 0) + g' x (0, 1)) =
∑ i : Fin 2,
((∫ x ... | Mathlib/MeasureTheory/Integral/DivergenceTheorem.lean | 441 | 480 |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.TwoDim
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
/-!
# Oriented angles.
This file defines orie... | o.ne_of_oangle_sign_ne_zero (h.symm ▸ by decide : (o.oangle x y).sign ≠ 0)
| Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | 163 | 164 |
/-
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.MeasureTheory.Integral.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Lebesgue.Countable
import Mathlib.MeasureTheory.Integral.Le... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 175 | 176 | |
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Mathlib.Control.Basic
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Option.Basic
im... | Mathlib/Data/List/Basic.lean | 1,386 | 1,391 | |
/-
Copyright (c) 2023 Yaël Dillies, Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Data.Finset.Lattice.Prod
import Mathlib.Data.Finset.Powerset
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Or... | infClosed_sInter <| forall_mem_range.2 hf
lemma InfClosed.codirectedOn (hs : InfClosed s) : DirectedOn (· ≥ ·) s :=
fun _a ha _b hb ↦ ⟨_, hs ha hb, inf_le_left, inf_le_right⟩
| Mathlib/Order/SupClosed.lean | 132 | 136 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Multiset.FinsetOps
import Mathlib.Data.Multiset.Fold
/-!
# Lattice operations on multisets
-/
namespace Multiset
variable {α : Type*}
/-! ###... |
theorem le_sup {s : Multiset α} {a : α} (h : a ∈ s) : a ≤ s.sup :=
| Mathlib/Data/Multiset/Lattice.lean | 53 | 54 |
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