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) 2024 Antoine Chambert-Loir. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir
-/
import Mathlib.Algebra.Pointwise.Stabilizer
import Mathlib.Data.Setoid.Partition
import Mathlib.GroupTheory.GroupAction.Pointwise
import Mathlib.GroupTh... | @[to_additive]
theorem IsBlock.image {H Y : Type*} [SMul H Y] {φ : G → H} (j : X →ₑ[φ] Y)
(hφ : Function.Surjective φ) (hj : Function.Injective j) (hB : IsBlock G B) :
IsBlock H (j '' B) := by
simp only [IsBlock, hφ.forall, ← image_smul_setₛₗ]
exact fun g₁ g₂ hg ↦ disjoint_image_of_injective hj <| hB <| ne_... | Mathlib/GroupTheory/GroupAction/Blocks.lean | 351 | 364 |
/-
Copyright (c) 2023 Yaël Dillies, Chenyi Li. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chenyi Li, Ziyu Wang, Yaël Dillies
-/
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.InnerProductSpace.Basic
/-!
# Uniformly and strongly convex functions
I... |
lemma UniformConcaveOn.concaveOn (hf : UniformConcaveOn s φ f) (hφ : 0 ≤ φ) : ConcaveOn ℝ s f := by
| Mathlib/Analysis/Convex/Strong.lean | 66 | 67 |
/-
Copyright (c) 2014 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Leonardo de Moura, Mario Carneiro, Floris van Doorn, Sabbir Rahman
-/
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Algebra.Order.Ring.Pow
import Mathlib.Algeb... | /-- Bernoulli's inequality reformulated to estimate `(n : α)`. -/
theorem Nat.cast_le_pow_sub_div_sub (H : 1 < a) (n : ℕ) : (n : α) ≤ (a ^ n - 1) / (a - 1) :=
(le_div_iff₀ (sub_pos.2 H)).2 <|
le_sub_left_of_add_le <| one_add_mul_sub_le_pow ((neg_le_self zero_le_one).trans H.le) _
| Mathlib/Algebra/Order/Field/Power.lean | 97 | 100 |
/-
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 map_reflexive {r : α → α → Prop} (hr : Reflexive r) {f : α → β} (hf : f.Surjective) :
Reflexive (Relation.Map r f f) := by
intro x
obtain ⟨y, rfl⟩ := hf x
exact ⟨y, y, hr y, rfl, rfl⟩
lemma map_symmetric {r : α → α → Prop} (hr : Symmetric r) (f : α → β) :
| Mathlib/Logic/Relation.lean | 224 | 230 |
/-
Copyright (c) 2021 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Bhavik Mehta
-/
import Mathlib.Analysis.Calculus.Deriv.Support
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
import Mathlib.MeasureTheory.Function.Jacobian
imp... | apply h_infty.congr'
filter_upwards [eventually_ne_atTop a] with x hx
exact (Function.update_of_ne hx a' (u * v)).symm
simpa using integral_Ioi_of_hasDerivAt_of_tendsto
(continuousWithinAt_update_same.mpr h_zero) hderiv huv htendsto
/-- **Integration by parts on (a, ∞).**
For finite intervals, see: `... | Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean | 1,276 | 1,291 |
/-
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.Polynomial.Degree.Domain
import Mathlib.Algebra.Polynomial.Degree.Support
import Mathlib.Algebra.Poly... | simp [C_mul_X_eq_monomial, derivative_monomial, Nat.cast_one, mul_one]
| Mathlib/Algebra/Polynomial/Derivative.lean | 92 | 93 |
/-
Copyright (c) 2020 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
import Mathlib.Algebra.ContinuedFractions.Computation.Basic
import Mathlib.Algebra.ContinuedFractions.Translations
import Mathlib.Algebra.Order.Floor.Ring
/-!
# ... | IntFractPair.stream v n = some ifp_n ∧
ifp_n.fr ≠ 0 ∧ IntFractPair.of ifp_n.fr⁻¹ = ifp_succ_n := by
simp [IntFractPair.stream, ite_eq_iff, Option.bind_eq_some_iff]
/-- An easier to use version of one direction of
`GenContFract.IntFractPair.succ_nth_stream_eq_some_iff`. -/
| Mathlib/Algebra/ContinuedFractions/Computation/Translations.lean | 87 | 92 |
/-
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... | theorem Chain'.append (h₁ : Chain' R l₁) (h₂ : Chain' R l₂)
(h : ∀ x ∈ l₁.getLast?, ∀ y ∈ l₂.head?, R x y) : Chain' R (l₁ ++ l₂) :=
chain'_append.2 ⟨h₁, h₂, h⟩
| Mathlib/Data/List/Chain.lean | 265 | 267 |
/-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Algebra.Order.Pi
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Data.Real.Pointwise
/... |
@[simp]
| Mathlib/Analysis/Seminorm.lean | 955 | 956 |
/-
Copyright (c) 2024 Josha Dekker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Josha Dekker
-/
import Mathlib.Order.Filter.Cofinite
import Mathlib.Order.Filter.CountableInter
import Mathlib.Order.Filter.CardinalInter
import Mathlib.SetTheory.Cardinal.Arithmetic
imp... |
instance instCardinalInterFilter_cocardinal : CardinalInterFilter (cocardinal (α := α) hreg) c where
cardinal_sInter_mem S hS hSs := by
| Mathlib/Order/Filter/Cocardinal.lean | 44 | 46 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Sébastien Gouëzel, Yury Kudryashov, Dylan MacKenzie, Patrick Massot
-/
import Mathlib.Algebra.BigOperators.Module
import Mathlib.Algebra.Order.Field.Power
import M... | apply summable_of_isBigO_nat (summable_geometric_of_lt_one ((norm_nonneg _).trans hrr'.le) h)
calc
fun n ↦ ‖↑(u n) * r ^ n‖
_ =O[atTop] fun n ↦ u n * ‖r‖ ^ n := by
apply (IsBigOWith.of_bound (c := ‖(1 : R)‖) ?_).isBigO
| Mathlib/Analysis/SpecificLimits/Normed.lean | 362 | 366 |
/-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.Polynomial.Reverse
/-!
# "Mirror" of a univariate polynomial
In this file we define `Po... | repeat' rw [mul_assoc]
theorem mirror_smul (a : R) : (a • p).mirror = a • p.mirror := by
rw [← C_mul', ← C_mul', mirror_mul_of_domain, mirror_C]
| Mathlib/Algebra/Polynomial/Mirror.lean | 180 | 184 |
/-
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 | 2,024 | 2,027 | |
/-
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 | 1,527 | 1,539 | |
/-
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 | 175 | 175 | |
/-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson
-/
import Mathlib.Computability.Language
import Mathlib.Tactic.AdaptationNote
/-!
# Regular Expressions
This file contains the formal definition for regular expressions and ... | have IH := fun t (_h : List.length t < List.length x) => star_rmatch_iff P t
clear star_rmatch_iff
constructor
· rcases x with - | ⟨a, x⟩
· intro _h
use []; dsimp; tauto
· rw [rmatch, deriv, mul_rmatch_iff]
rintro ⟨t, u, hs, ht, hu⟩
have hwf : u.length < (List.cons a ... | Mathlib/Computability/RegularExpressions.lean | 254 | 297 |
/-
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... | · obtain rfl | hs := s.eq_empty_or_nonempty
· simp
simp [zero_smul_set hs, ← Set.singleton_zero]
· have := (lipschitzWith_smul c⁻¹).ediam_image_le (c • s)
rwa [← smul_eq_mul, ← ENNReal.smul_def, Set.image_smul, inv_smul_smul₀ hc s, nnnorm_inv,
le_inv_smul_iff_of_pos (nnnorm_pos.2 hc)] at this
the... | Mathlib/Analysis/NormedSpace/Pointwise.lean | 42 | 50 |
/-
Copyright (c) 2023 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Data.Matrix.Kronecker
import Mathlib.LinearAlgebra.Matrix.ToLin
import Mathlib.LinearAlgebra.TensorProduct.Basis
/-!
# Connections between `TensorProduct` a... | TensorProduct.map (toLin bM bM' A) (toLin bN bN' B) := by
rw [← LinearEquiv.eq_symm_apply, toLin_symm, TensorProduct.toMatrix_map, toMatrix_toLin,
toMatrix_toLin]
/-- `TensorProduct.comm` corresponds to a permutation of the identity matrix. -/
| Mathlib/LinearAlgebra/TensorProduct/Matrix.lean | 49 | 53 |
/-
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.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.GeneralLinearG... | theorem three_lt_four_mul_im_sq_of_mem_fdo (h : z ∈ 𝒟ᵒ) : 3 < 4 * z.im ^ 2 := by
have : 1 < z.re * z.re + z.im * z.im := by simpa [Complex.normSq_apply] using h.1
have := h.2
cases abs_cases z.re <;> nlinarith
/-- non-strict variant of `ModularGroup.three_le_four_mul_im_sq_of_mem_fdo` -/
theorem three_le_four_m... | Mathlib/NumberTheory/Modular.lean | 381 | 388 |
/-
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 ... |
end Functor
| Mathlib/CategoryTheory/EqToHom.lean | 311 | 312 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Johan Commelin, Patrick Massot
-/
import Mathlib.Algebra.Order.Hom.Monoid
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.Tac... | (v.map_add_of_distinct_val h.ne).trans (max_eq_right_iff.mpr h.le)
theorem map_add_eq_of_lt_left (h : v y < v x) : v (x + y) = v x := by
rw [add_comm]; exact map_add_eq_of_lt_right _ h
| Mathlib/RingTheory/Valuation/Basic.lean | 296 | 300 |
/-
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 | 504 | 508 | |
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Equivalence
/-! Tools for compatibilities between Dold-Kan equivalences
The purpose of this file is to introduce tools which will enable the
con... | _ ≅ F ⋙ 𝟭 B' ⋙ e'.inverse ⋙ eA.inverse := Iso.refl _
_ ≅ F ⋙ (eB.inverse ⋙ eB.functor) ⋙ e'.inverse ⋙ eA.inverse :=
isoWhiskerLeft _ (isoWhiskerRight eB.counitIso.symm _)
_ ≅ (F ⋙ eB.inverse) ⋙ eB.functor ⋙ e'.inverse ⋙ eA.inverse := Iso.refl _
theorem equivalence₂UnitIso_eq : (equivalence₂ eB hF).u... | Mathlib/AlgebraicTopology/DoldKan/Compatibility.lean | 133 | 138 |
/-
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.LinearAlgebra.Quotient.Basic
import Mathlib.LinearAlgebra.Prod
/-!
# Projection to a subspace
In this file we define
* `Submodule.linearProjOfIsCom... | Mathlib/LinearAlgebra/Projection.lean | 468 | 471 | |
/-
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.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calculus.Deriv.Add
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calcul... | simp only [g, s', Function.comp_def, smul_eq_mul, mul_comm c, ← smul_smul] at Z
convert Z
ext t
simp [← smul_smul]
| Mathlib/Analysis/Calculus/LineDeriv/Basic.lean | 508 | 511 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Moritz Doll
-/
import Mathlib.LinearAlgebra.Prod
/-!
# Partially defined linear maps
A `LinearPMap R E F` or `E →ₗ.[R] F` is a linear map from a submodule of `E` to... | rw [dif_pos]
· rfl
· exact hg
theorem mem_graph_toLinearPMap {g : Submodule R (E × F)}
(hg : ∀ (x : E × F) (_hx : x ∈ g) (_hx' : x.fst = 0), x.snd = 0)
(x : g.map (LinearMap.fst R E F)) : (x.val, g.toLinearPMap x) ∈ g := by
rw [toLinearPMap_apply_aux hg]
exact valFromGraph_mem hg x.2
@[simp]
theorem... | Mathlib/LinearAlgebra/LinearPMap.lean | 920 | 933 |
/-
Copyright (c) 2020 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Eric Wieser
-/
import Mathlib.Data.ENNReal.Holder
import Mathlib.MeasureTheory.Function.LpSeminorm.Basic
import Mathlib.MeasureTheory.Integral.MeanInequalities
import Mathl... | theorem eLpNorm'_le_eLpNorm'_of_exponent_le {p q : ℝ} (hp0_lt : 0 < p)
(hpq : p ≤ q) (μ : Measure α) [IsProbabilityMeasure μ] (hf : AEStronglyMeasurable f μ) :
eLpNorm' f p μ ≤ eLpNorm' f q μ := by
have h_le_μ := eLpNorm'_le_eLpNorm'_mul_rpow_measure_univ hp0_lt hpq hf
rwa [measure_univ, ENNReal.one_rpow, m... | Mathlib/MeasureTheory/Function/LpSeminorm/CompareExp.lean | 88 | 92 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Johan Commelin, Patrick Massot
-/
import Mathlib.Algebra.Order.Hom.Monoid
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.Tac... |
end Monoid
| Mathlib/RingTheory/Valuation/Basic.lean | 337 | 339 |
/-
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.CharP.Lemmas
import Mathlib.GroupTheory.OrderOfElement
/-!
# Lemmas about rings of characteristic two
This file contains results about `CharP R 2`,... | Mathlib/Algebra/CharP/Two.lean | 55 | 55 | |
/-
Copyright (c) 2021 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Order.Antichain
import Mathlib.Topology.ContinuousOn
/-!
# Left and right continuity
In this file we prove a few lemmas about left and right cont... | 𝓝[>] a ⊔ 𝓝[{a}] a = 𝓝[≥] a := by
simp only [union_singleton, Ioi_insert, ← nhdsWithin_union]
| Mathlib/Topology/Order/LeftRight.lean | 127 | 129 |
/-
Copyright (c) 2022 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Group.Nat.Even
import Mathlib.Data.Nat.Cast.Basic
import Mathlib.Data.Nat.Cast.Commute
import Mathlib.Data.Set.Operations
import Mathlib.Logic.Fu... | Mathlib/Algebra/Ring/Parity.lean | 450 | 452 | |
/-
Copyright (c) 2020 Kenji Nakagawa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenji Nakagawa, Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.Algebra.Subalgebra.Pointwise
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.RingTheory.Spect... | rw [← h]
apply mul_left_mono I
apply (le_div_iff_of_nonzero hI).mpr _
intro y hy x hx
rw [mul_comm]
exact mul_mem_mul hy hx
theorem mul_inv_cancel_iff {I : FractionalIdeal R₁⁰ K} : I * I⁻¹ = 1 ↔ ∃ J, I * J = 1 :=
⟨fun h => ⟨I⁻¹, h⟩, fun ⟨J, hJ⟩ => by rwa [← right_inverse_eq K I J hJ]⟩
theorem mul_inv_ca... | Mathlib/RingTheory/DedekindDomain/Ideal.lean | 108 | 122 |
/-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Applicative
import Mathlib.Control.Traversable.Basic
/-!
# Traversing collections
This file proves basic properties of traversable and applicative ... | rw [comp_apply, map_traverse]
theorem naturality_pf (η : ApplicativeTransformation F G) (f : α → F β) :
traverse (@η _ ∘ f) = @η _ ∘ (traverse f : t α → F (t β)) := by
| Mathlib/Control/Traversable/Lemmas.lean | 116 | 119 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Ring.List
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Data.Nat.Prime.Basic
import Ma... | theorem perm_primeFactorsList_mul_of_coprime {a b : ℕ} (hab : Coprime a b) :
(a * b).primeFactorsList ~ a.primeFactorsList ++ b.primeFactorsList := by
rcases a.eq_zero_or_pos with (rfl | ha)
· simp [(coprime_zero_left _).mp hab]
rcases b.eq_zero_or_pos with (rfl | hb)
| Mathlib/Data/Nat/Factors.lean | 197 | 201 |
/-
Copyright (c) 2022 Michael Blyth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Blyth
-/
import Mathlib.LinearAlgebra.Projectivization.Basic
/-!
# Subspaces of Projective Space
In this file we define subspaces of a projective space, and show that the subs... |
theorem subset_span_trans {S T U : Set (ℙ K V)} (hST : S ⊆ span T) (hTU : T ⊆ span U) :
S ⊆ span U :=
gi.gc.le_u_l_trans hST hTU
| Mathlib/LinearAlgebra/Projectivization/Subspace.lean | 155 | 158 |
/-
Copyright (c) 2023 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Data.Int.Order.Units
import Mathlib.Data.ZMod.IntUnitsPower
import Mathlib.RingTheory.TensorProduct.Basic
import Mathlib.LinearAlgebra.DirectSum.TensorProduc... | refine TensorProduct.curry ?_
refine map (LinearMap.mul' R (⨁ i, 𝒜 i)) (LinearMap.mul' R (⨁ i, ℬ i)) ∘ₗ ?_
refine (assoc R _ _ _).symm.toLinearMap ∘ₗ .lTensor _ ?_ ∘ₗ (assoc R _ _ _).toLinearMap
refine (assoc R _ _ _).toLinearMap ∘ₗ .rTensor _ ?_ ∘ₗ (assoc R _ _ _).symm.toLinearMap
exact (gradedComm _ _ _).t... | Mathlib/LinearAlgebra/TensorProduct/Graded/External.lean | 183 | 196 |
/-
Copyright (c) 2014 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Leonardo de Moura, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Ord... | theorem inv_strictAntiOn : StrictAntiOn (fun x : α => x⁻¹) (Set.Ioi 0) := fun _ hx _ hy xy =>
(inv_lt_inv₀ hy hx).2 xy
theorem inv_pow_le_inv_pow_of_le (a1 : 1 ≤ a) {m n : ℕ} (mn : m ≤ n) : (a ^ n)⁻¹ ≤ (a ^ m)⁻¹ := by
| Mathlib/Algebra/Order/Field/Basic.lean | 257 | 260 |
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel, Adam Topaz, Johan Commelin, Jakob von Raumer
-/
import Mathlib.Algebra.Homology.ImageToKernel
import Mathlib.Algebra.Homology.ShortComplex.Exact
import Mathlib.CategoryTh... | Mathlib/CategoryTheory/Abelian/Exact.lean | 387 | 392 | |
/-
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.Algebra.Algebra.Equiv
import Mathlib.Algebra.Algebra.NonUnitalHom
import Mathlib.Algebra.Algebra.Prod
import Mathlib.Algebra.Algebra.Pi
import Mathlib.Al... | end InlInr
| Mathlib/Algebra/Star/StarAlgHom.lean | 579 | 580 |
/-
Copyright (c) 2022 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.MeasureTheory.Measure.Doubling
import Mathlib.MeasureTheory.Covering.Vitali
import Mathlib.MeasureTheory.Covering.Differentiation
/-!
# Uniformly locally do... | · filter_upwards [xmem, δlim self_mem_nhdsWithin] with j hj h'j
exact closedBall_mem_vitaliFamily_of_dist_le_mul μ hj h'j
· rcases l.eq_or_neBot with rfl | h
· simp
have hK : 0 ≤ K := by
rcases (xmem.and (δlim self_mem_nhdsWithin)).exists with ⟨j, hj, h'j⟩
have : 0 ≤ K * δ j := nonempty_clos... | Mathlib/MeasureTheory/Covering/DensityTheorem.lean | 112 | 132 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Shapes.Equalizers
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.Mono
import Mathlib.CategoryTheory.Limits.Shape... | /-- A category has strong epi images if it has all images and `factorThruImage f` is a strong
epimorphism for all `f`. -/
class HasStrongEpiImages : Prop where
| Mathlib/CategoryTheory/Limits/Shapes/Images.lean | 834 | 836 |
/-
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.SetTheory.Ordinal.Exponential
import Mathlib.SetTheory.Ordinal.Family
/-!
# Cantor Normal Form
The Cantor normal form of an ordinal is generally defi... | Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | 150 | 158 | |
/-
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.Topology.ContinuousMap.Bounded.ArzelaAscoli
import Mathlib.Topology.ContinuousMap.Bounded.Normed
import Mathlib.Topology.MetricSpace.Gluing
impor... | /- To check that `HD` is continuous, we check that it is Lipschitz. As `HD` is a max, we
prove separately inequalities controlling the two terms (relying too heavily on copy-paste...) -/
private theorem HD_lipschitz_aux1 (f g : Cb X Y) :
(⨆ x, ⨅ y, f (inl x, inr y)) ≤ (⨆ x, ⨅ y, g (inl x, inr y)) + dist f g := by
... | Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean | 268 | 279 |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark, Kyle Miller, Lu-Ming Zhang
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Combinatorics.SimpleGraph.Connectivity.WalkCounting
import Math... | theorem apply_ne_zero_iff [MulZeroOneClass α] [Nontrivial α] (h : IsAdjMatrix A) (i j : V) :
¬A i j = 0 ↔ A i j = 1 := by rw [← apply_ne_one_iff h, Classical.not_not]
| Mathlib/Combinatorics/SimpleGraph/AdjMatrix.lean | 69 | 70 |
/-
Copyright (c) 2021 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Logic.Equiv.Defs
import Mathlib.Tactic.MkIffOfInductiveProp
import Mathlib.Tactic.PPWithUniv
/-!
# Small types
A type is `w`-small if there exists an equ... | -- is resolved.
@[induction_eliminator]
protected noncomputable def Shrink.rec {α : Type*} [Small.{w} α] {F : Shrink α → Sort v}
| Mathlib/Logic/Small/Defs.lean | 56 | 58 |
/-
Copyright (c) 2023 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Analysis.LocallyConvex.WithSeminorms
import Mathlib.Topology.Semicontinuous
import Mathlib.Topology.Baire.Lemmas
/-!
# Barrelled spaces and the Ba... | the seminorms from equicontinuity. -/
protected theorem banach_steinhaus (H : ∀ k x, BddAbove (range fun i ↦ q k (𝓕 i x))) :
UniformEquicontinuous ((↑) ∘ 𝓕) := by
-- We just have to prove that `⊔ i, (q k) ∘ (𝓕 i)` is a (well-defined) continuous seminorm
-- for all `k`.
refine (hq.uniformEquicontinuous_iff_... | Mathlib/Analysis/LocallyConvex/Barrelled.lean | 156 | 171 |
/-
Copyright (c) 2020 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Algebra.GroupWithZero.Basic
/-!
# Indicator functions and support of a function in groups with z... | lemma indicator_mul (s : Set ι) (f g : ι → M₀) :
indicator s (fun i ↦ f i * g i) = fun i ↦ indicator s f i * indicator s g i := by
funext
simp only [indicator]
split_ifs
· rfl
rw [mul_zero]
| Mathlib/Algebra/GroupWithZero/Indicator.lean | 21 | 27 |
/-
Copyright (c) 2018 Violeta Hernández Palacios, Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios, Mario Carneiro
-/
import Mathlib.Logic.Small.List
import Mathlib.SetTheory.Ordinal.Enum
import Mathlib.SetTheory.Ordinal.Exponen... | rw [add_zero, mul_lt_mul_iff_left (opow_pos ω ha)] at this
rwa [succ_le_iff]
| Mathlib/SetTheory/Ordinal/FixedPoint.lean | 487 | 488 |
/-
Copyright (c) 2018 Michael Jendrusch. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Jendrusch, Kim Morrison, Bhavik Mehta, Jakob von Raumer
-/
import Mathlib.CategoryTheory.EqToHom
import Mathlib.CategoryTheory.Functor.Trifunctor
import Mathlib.CategoryTheo... | rw [← comp_whiskerRight, IsIso.hom_inv_id, id_whiskerRight]
@[reassoc (attr := simp)]
| Mathlib/CategoryTheory/Monoidal/Category.lean | 307 | 309 |
/-
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.Nat.ModEq
import Mathlib.Data.Nat.Prime.Basic
import Mathlib.NumberTheory.Zsqrtd.Basic
/-!
# Pell's equation and Matiyasevic's theorem
This file... | rw [Nat.mod_eq_of_lt (strictMono_x _ (Nat.lt_of_succ_lt jn)),
Nat.mod_eq_of_lt (strictMono_x _ jn)]
exact strictMono_x _ (Nat.lt_succ_self _)
| Mathlib/NumberTheory/PellMatiyasevic.lean | 563 | 566 |
/-
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
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Order.CauSeq.BigOperators
import Mathlib.Algebra.Order.Star.Basic
import Mathlib.Data.C... | |exp x - expNear n x a₁| ≤ |x| ^ n / n.factorial * b₁ := by
refine (abs_sub_le _ _ _).trans ((add_le_add_right h _).trans ?_)
subst e₁; rw [expNear_succ, expNear_sub, abs_mul]
convert mul_le_mul_of_nonneg_left (a := |x| ^ n / ↑(Nat.factorial n))
(le_sub_iff_add_le'.1 e) ?_ using 1
· simp [mul_add, pow... | Mathlib/Data/Complex/Exponential.lean | 584 | 590 |
/-
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... | @[simp]
| Mathlib/Data/Multiset/Bind.lean | 323 | 323 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.CharP.Frobenius
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.RingTheory.Polynomial.Basic
/... |
theorem expand_contract [CharP R p] [NoZeroDivisors R] {f : R[X]} (hf : Polynomial.derivative f = 0)
| Mathlib/Algebra/Polynomial/Expand.lean | 233 | 234 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Finset.Lattice.Prod
import Mathlib.Data.Finite.Prod
import Mathlib.Data.Set.Lattice.Image
/-!
# N-ary images of finsets
This file defines `Finset.im... | (h_distrib : ∀ a b, g (f a b) = f' a (g' b)) :
(image₂ f s t).image g = image₂ f' s (t.image g') :=
| Mathlib/Data/Finset/NAry.lean | 356 | 357 |
/-
Copyright (c) 2019 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Mario Carneiro, Isabel Longbottom, Kim Morrison, Apurva Nakade, Yuyang Zhao
-/
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.SetTheory.PGame.Algebra
import Mathl... | def Fuzzy : Game → Game → Prop :=
Quotient.lift₂ PGame.Fuzzy fun _ _ _ _ hx hy => propext (fuzzy_congr hx hy)
| Mathlib/SetTheory/Game/Basic.lean | 119 | 121 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Wrenna Robson
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Pi
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.LinearAlgebra.Vandermonde
import Mathlib.RingT... | (hlc : f.leadingCoeff = g.leadingCoeff)
(h_eval : ∀ x ∈ s, f.eval x = g.eval x) :
f = g := by
rcases eq_or_ne f 0 with rfl | hf
· rwa [degree_zero, eq_comm, degree_eq_bot, eq_comm] at h_deg_eq
· exact eq_of_degree_sub_lt_of_eval_finset_eq s
(lt_of_lt_of_le (degree_sub_lt h_deg_eq hf hlc) h_deg_l... | Mathlib/LinearAlgebra/Lagrange.lean | 74 | 83 |
/-
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.Geometry.Euclidean.Projection
import Mathlib.Geometry.Euclidean.Sphere.Basic
import Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional
import Mathlib.Tact... | finite-dimensional, that contains a set of points, those points are
cospherical if and only if they are equidistant from some point in
that subspace. -/
theorem cospherical_iff_exists_mem_of_finiteDimensional {s : AffineSubspace ℝ P} {ps : Set P}
(h : ps ⊆ s) [Nonempty s] [FiniteDimensional ℝ s.direction] :
Cos... | Mathlib/Geometry/Euclidean/Circumcenter.lean | 623 | 635 |
/-
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.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
rw [-- Manually apply `elementwise_of%` to generate a `ConcreteCategory` lemma
elementwise_of% PresheafedSpace.stalkMap_germ
(coequalizer.π (C := SheafedSpace _) f.toShHom g.toShHom) U _ hU] at ha
let V := imageBasicOpen f g U s
... | Mathlib/Geometry/RingedSpace/LocallyRingedSpace/HasColimits.lean | 214 | 223 |
/-
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 ... | (x y : E) : f x = f y := by
letI : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E
exact convex_univ.is_const_of_fderivWithin_eq_zero hf.differentiableOn
(fun x _ => by rw [fderivWithin_univ]; exact hf' x) trivial trivial
/-- If two functions have ... | Mathlib/Analysis/Calculus/MeanValue.lean | 565 | 578 |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Casper Putz, Anne Baanen
-/
import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.L... | set v' := Basis.mk hli hspan.ge
| Mathlib/LinearAlgebra/Determinant.lean | 530 | 530 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad
-/
import Mathlib.Data.Finset.Basic
import Mathlib.Data.Finset.Image
/-!
# Cardinality of a finite set
This defines the cardinality of a `Fins... | Mathlib/Data/Finset/Card.lean | 917 | 921 | |
/-
Copyright (c) 2022 Vincent Beffara. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Vincent Beffara, Stefan Kebekus
-/
import Mathlib.Analysis.Analytic.Constructions
import Mathlib.Analysis.Calculus.DSlope
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathl... |
theorem frequently_zero_iff_eventually_zero {f : 𝕜 → E} {w : 𝕜} (hf : AnalyticAt 𝕜 f w) :
(∃ᶠ z in 𝓝[≠] w, f z = 0) ↔ ∀ᶠ z in 𝓝 w, f z = 0 :=
| Mathlib/Analysis/Analytic/IsolatedZeros.lean | 135 | 137 |
/-
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 | 577 | 589 | |
/-
Copyright (c) 2022 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex J. Best, Xavier Roblot
-/
import Mathlib.Algebra.Algebra.Hom.Rat
import Mathlib.Analysis.Complex.Polynomial.Basic
import Mathlib.NumberTheory.NumberField.Norm
import Mathlib.RingTh... | ← h.ext_iff, eq_comm (a := σ)]
variable (k w)
lemma nat_card_stabilizer_eq_one_or_two :
Nat.card (Stab w) = 1 ∨ Nat.card (Stab w) = 2 := by
classical
rw [← SetLike.coe_sort_coe, ← mk_embedding w]
by_cases h : ∃ σ, ComplexEmbedding.IsConj (k := k) (embedding w) σ
| Mathlib/NumberTheory/NumberField/Embeddings.lean | 900 | 908 |
/-
Copyright (c) 2023 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Topology.Connected.Basic
import Mathlib.Topology.Separation.Hausdorff
import Mathlib.Topology.Connected.Clopen
/-!
# Separated maps and locally injective maps ou... | then two lifts `g₁, g₂ : A → E` of a map `f : A → X` are equal if they agree at one point. -/
theorem eq_of_comp_eq
[PreconnectedSpace A] (h₁ : Continuous g₁) (h₂ : Continuous g₂)
(he : p ∘ g₁ = p ∘ g₂) (a : A) (ha : g₁ a = g₂ a) : g₁ = g₂ := funext fun a' ↦ by
apply (IsClopen.eq_univ ⟨sep.isClosed_eqLocus ... | Mathlib/Topology/SeparatedMap.lean | 201 | 206 |
/-
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... | @[simp] lemma map₂_bot_left (f : α → β → γ) (b) : map₂ f ⊥ b = ⊥ := rfl
@[simp] lemma map₂_bot_right (f : α → β → γ) (a) : map₂ f a ⊥ = ⊥ := by cases a <;> rfl
@[simp] lemma map₂_coe_left (f : α → β → γ) (a : α) (b) : map₂ f a b = b.map fun b ↦ f a b := rfl
| Mathlib/Order/WithBot.lean | 143 | 145 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Data.ENNReal.Holder
/-!
# Real conjugate exponents
This file defines Hölder triple and Hölder conjugate exponents in `ℝ` and `... | positive and `p⁻¹ + q⁻¹ = r⁻¹`. -/
@[mk_iff]
structure HolderTriple (p q r : ℝ≥0) : Prop where
inv_add_inv_eq_inv : p⁻¹ + q⁻¹ = r⁻¹
| Mathlib/Data/Real/ConjExponents.lean | 208 | 211 |
/-
Copyright (c) 2023 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.Analysis.Complex.Convex
import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.Analysis.Calculus.Deriv.Shift
/-!
# Estimates for the complex ... | lemma logTaylor_at_zero (n : ℕ) : logTaylor n 0 = 0 := by
induction n with
| zero => simp [logTaylor_zero]
| succ n ih => simpa [logTaylor_succ, ih] using ne_or_eq n 0
| Mathlib/Analysis/SpecialFunctions/Complex/LogBounds.lean | 71 | 74 |
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, María Inés de Frutos-Fernández, Filippo A. E. Nuccio
-/
import Mathlib.Data.Int.Interval
import Mathlib.FieldTheory.RatFunc.AsPolynomial
import Mathlib.RingTheory.Binom... |
/-- The coercion `RatFunc F → F⸨X⸩` as bundled alg hom. -/
def coeAlgHom (F : Type u) [Field F] : RatFunc F →ₐ[F[X]] F⸨X⸩ :=
liftAlgHom (Algebra.ofId _ _) <|
nonZeroDivisors_le_comap_nonZeroDivisors_of_injective _ <|
| Mathlib/RingTheory/LaurentSeries.lean | 367 | 371 |
/-
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... | Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean | 687 | 689 | |
/-
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, Yury Kudryashov
-/
import Mathlib.Data.Finite.Prod
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.Set.Finite.Lemmas
import Mathlib.Order.Conditionall... | Filter.coext fun s => by
simp only [compl_mem_coprodᵢ, mem_cofinite, compl_compl, forall_finite_image_eval_iff]
| Mathlib/Order/Filter/Cofinite.lean | 122 | 124 |
/-
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... |
theorem mem_nhds_iff : s ∈ 𝓝 x ↔ ∃ ε > 0, ball x ε ⊆ s :=
nhds_basis_ball.mem_iff
| Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | 716 | 719 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Polynomial.GroupRingAction
import Mathlib.Algebra.Ring.Action.Field
import Mathlib.Algebra.Ring.Action.Invariant
import Mathlib.FieldTheory.Finiteness
im... | theorem coe_algebraMap :
algebraMap (FixedPoints.subfield M F) F = Subfield.subtype (FixedPoints.subfield M F) :=
rfl
| Mathlib/FieldTheory/Fixed.lean | 110 | 113 |
/-
Copyright (c) 2020 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
import Mathlib.Analysis.NormedSpace.IndicatorFunction
import Mathlib.Data.Fintype.Order
import Mathlib.MeasureTheory.Function.AEEqFun
import Mathlib.Me... | · have := Fact.mk hμ.lt_top
apply memLp_const
@[deprecated (since := "2025-02-21")]
alias memℒp_indicator_const := memLp_indicator_const
lemma eLpNormEssSup_piecewise (f g : α → ε) [DecidablePred (· ∈ s)] (hs : MeasurableSet s) :
eLpNormEssSup (Set.piecewise s f g) μ
= max (eLpNormEssSup f (μ.restrict... | Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean | 754 | 771 |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Manuel Candales
-/
import Mathlib.Geometry.Euclidean.PerpBisector
import Mathlib.Algebra.QuadraticDiscriminant
/-!
# Euclidean spaces
This file makes some definitions and... | Mathlib/Geometry/Euclidean/Basic.lean | 543 | 550 | |
/-
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.Additive
import Mathlib.Algebra.Homology.ShortComplex.Exact
import Mathlib.Algebra.Homology.ShortComplex.Preadditive
import Mathlib.Tactic.Linar... | @[reassoc (attr := simp)]
lemma fromOpcycles_d :
K.fromOpcycles i j ≫ K.d j k = 0 := by
simp only [← cancel_epi (K.pOpcycles i), p_fromOpcycles_assoc, d_comp_d, comp_zero]
variable {i j} in
lemma fromOpcycles_eq_zero (hij : ¬ c.Rel i j) :
K.fromOpcycles i j = 0 := by
rw [← cancel_epi (K.pOpcycles i), p_fro... | Mathlib/Algebra/Homology/ShortComplex/HomologicalComplex.lean | 263 | 273 |
/-
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.CompactConvergence
import Mathlib.Topology.UniformSpace.Equicontinuity
import Mathlib.Topology.UniformSpace.Equiv
/-!
# Asco... | `α` a uniform space, `F : ι → (X → α)` a family equicontinuous on each `K ∈ 𝔖`, and `ℱ` a filter
on `ι`. Then, `F` tends to `f : X → α` along `ℱ` *uniformly on each `K ∈ 𝔖`* iff it tends to `f`
*pointwise on `⋃₀ 𝔖`* along `ℱ`. -/
theorem EquicontinuousOn.tendsto_uniformOnFun_iff_pi'
{𝔖 : Set (Set X)} (𝔖_compac... | Mathlib/Topology/UniformSpace/Ascoli.lean | 327 | 345 |
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.SetTheory.Cardinal.ENat
/-!
# Projection from cardinal numbers to natural numbers
In this file we define `Cardinal.toNat` to be the natural projectio... | · exact fun h ↦ (Cardinal.cast_toNat_of_lt_aleph0 h)
theorem toNat_strictMonoOn : StrictMonoOn toNat (Iio ℵ₀) := by
| Mathlib/SetTheory/Cardinal/ToNat.lean | 64 | 66 |
/-
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 ... | Mathlib/Analysis/Calculus/MeanValue.lean | 937 | 941 | |
/-
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.Option.NAry
import Mathlib.Data.Seq.Computation
import Mathlib.Tactic.ApplyFun
import Mathlib.Data.List.Basic
/-!
# Possibly infinite lists
This... | theorem cons_append (a : α) (s t) : append (cons a s) t = cons a (append s t) :=
destruct_eq_cons <| by
| Mathlib/Data/Seq/Seq.lean | 838 | 839 |
/-
Copyright (c) 2020 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard
-/
import Mathlib.RingTheory.AdicCompletion.Basic
import Mathlib.RingTheory.LocalRing.MaximalIdeal.Basic
import Mathlib.RingTheory.LocalRing.RingHom.Basic
import Mathlib.R... | such that every nonzero element is associated to a power of `p`
is a discrete valuation ring.
-/
theorem ofHasUnitMulPowIrreducibleFactorization {R : Type u} [CommRing R] [IsDomain R]
(hR : HasUnitMulPowIrreducibleFactorization R) : IsDiscreteValuationRing R := by
letI : UniqueFactorizationMonoid R := hR.toUnique... | Mathlib/RingTheory/DiscreteValuationRing/Basic.lean | 286 | 303 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker, Johan Commelin
-/
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib... | theorem aroots_X_pow [CommRing S] [IsDomain S] [Algebra T S] (n : ℕ) :
(X ^ n : T[X]).aroots S = n • ({0} : Multiset S) := by
rw [aroots_pow, aroots_X]
| Mathlib/Algebra/Polynomial/Roots.lean | 460 | 462 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker, Johan Commelin
-/
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib... | simp only [Finset.mem_image, Finset.mem_univ, true_and, Multiset.mem_toFinset, mem_roots', ne_eq,
IsRoot.def, forall_exists_index, hp, not_false_eq_true]
rintro x rfl
exact heval _
lemma eq_zero_of_natDegree_lt_card_of_eval_eq_zero' {R} [CommRing R] [IsDomain R]
(p : R[X]) (s : Finset R) (heval : ∀ i ∈ s... | Mathlib/Algebra/Polynomial/Roots.lean | 595 | 601 |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark, Kyle Miller, Lu-Ming Zhang
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Combinatorics.SimpleGraph.Connectivity.WalkCounting
import Math... | theorem isSymm_compl [Zero α] [One α] (h : A.IsSymm) : A.compl.IsSymm := by
| Mathlib/Combinatorics/SimpleGraph/AdjMatrix.lean | 105 | 105 |
/-
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
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Order.CauSeq.BigOperators
import Mathlib.Algebra.Order.Star.Basic
import Mathlib.Data.C... | Mathlib/Data/Complex/Exponential.lean | 762 | 768 | |
/-
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.EuclideanDomain.Int
import Mathlib.Algebra.MvPolynomial.Eval
import Mathlib.RingTheory.Adjoin.Basic
import Mathlib.RingTheory.Polynomial.Basic
import Mat... | theorem fg_trans (h1 : (adjoin R s).toSubmodule.FG) (h2 : (adjoin (adjoin R s) t).toSubmodule.FG) :
(adjoin R (s ∪ t)).toSubmodule.FG := by
rcases fg_def.1 h1 with ⟨p, hp, hp'⟩
rcases fg_def.1 h2 with ⟨q, hq, hq'⟩
refine fg_def.2 ⟨p * q, hp.mul hq, le_antisymm ?_ ?_⟩
· rw [span_le, Set.mul_subset_iff]
i... | Mathlib/RingTheory/Adjoin/FG.lean | 40 | 80 |
/-
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.Topology.PartialHomeomorph
import Mathlib.Topology.Connected.LocPathConnected
/-!
# Charted spaces
A smooth manifold is a topological space `M`... | def ChartedSpace.sum_of_nonempty [Nonempty H] : ChartedSpace H (M ⊕ M') where
atlas := ((fun e ↦ e.lift_openEmbedding IsOpenEmbedding.inl) '' cm.atlas) ∪
((fun e ↦ e.lift_openEmbedding IsOpenEmbedding.inr) '' cm'.atlas)
-- At `x : M`, the chart is the chart in `M`; at `x' ∈ M'`, it is the chart in `M'`.
chart... | Mathlib/Geometry/Manifold/ChartedSpace.lean | 905 | 909 |
/-
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.Analysis.MellinTransform
/-!
# Abstract functional equations for Mellin transforms
This file formalises a general version of an argument used to pro... | field_simp
· simp_rw [norm_mul, norm_real, one_div, inv_rpow hx.le, rpow_neg hx.le, inv_inv, norm_inv,
norm_of_nonneg (rpow_pos_of_pos hx _).le, rpow_add hx]
field_simp
ring
/-- Power asymptotic for `f - f₀` as `x → 0`. -/
lemma hf_zero' (P : WeakFEPair E) :
(fun x : ℝ ↦ P.f x - P.f₀) =O[𝓝[>] ... | Mathlib/NumberTheory/LSeries/AbstractFuncEq.lean | 158 | 170 |
/-
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.Geometry.RingedSpace.PresheafedSpace.Gluing
import Mathlib.AlgebraicGeometry.Cover.Open
/-!
# Gluing Schemes
Given a family of gluing data of schemes, we m... | Mathlib/AlgebraicGeometry/Gluing.lean | 466 | 470 | |
/-
Copyright (c) 2021 Filippo A. E. Nuccio. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Filippo A. E. Nuccio, Eric Wieser
-/
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.Matrix.Block
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.Linear... | kroneckerMap f (Matrix.reindex el em M) N =
reindex (el.prodCongr (Equiv.refl _)) (em.prodCongr (Equiv.refl _)) (kroneckerMap f M N) :=
kroneckerMap_reindex _ _ _ (Equiv.refl _) (Equiv.refl _) _ _
theorem kroneckerMap_reindex_right (f : α → β → γ) (em : m ≃ m') (en : n ≃ n') (M : Matrix l l' α)
(N : Ma... | Mathlib/Data/Matrix/Kronecker.lean | 155 | 160 |
/-
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 | 1,222 | 1,223 | |
/-
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 | 868 | 870 | |
/-
Copyright (c) 2017 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Algebra.Group.Defs
import Mathlib.Control.Functor
import Mathlib.Control.Basic
/-!
# `applicative` instances
This file provides `Applicative` instances for... | (∀ {α β : Type u} (f : F (α → β)) (x : F α),
@Seq.seq _ A1.toSeq _ _ f (fun _ => x) = @Seq.seq _ A2.toSeq _ _ f (fun _ => x)) →
A1 = A2
| { toFunctor := F1, seq := s1, pure := p1, seqLeft := sl1, seqRight := sr1 },
{ toFunctor := F2, seq := s2, pure := p2, seqLeft := sl2, seqRight := sr2 },
... | Mathlib/Control/Applicative.lean | 40 | 63 |
/-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
/-! # Jacobi's theta function
This file define... | theorem jacobiTheta_two_add (τ : ℂ) : jacobiTheta (2 + τ) = jacobiTheta τ := by
simp_rw [jacobiTheta_eq_jacobiTheta₂, add_comm, jacobiTheta₂_add_right]
| Mathlib/NumberTheory/ModularForms/JacobiTheta/OneVariable.lean | 30 | 31 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Yury Kudryashov, Neil Strickland
-/
import Mathlib.Algebra.Ring.Semiconj
import Mathlib.Algebra.Ring.Units
import Mathlib.Algebra.Gro... | Mathlib/Algebra/Ring/Commute.lean | 251 | 253 | |
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.ModelTheory.Ultraproducts
import Mathlib.ModelTheory.Bundled
import Mathlib.ModelTheory.Skolem
import Mathlib.Order.Filter.AtTopBot.Basic
/-!
# First-... |
theorem isSatisfiable_iUnion_iff_isSatisfiable_iUnion_finset {ι : Type*} (T : ι → L.Theory) :
IsSatisfiable (⋃ i, T i) ↔ ∀ s : Finset ι, IsSatisfiable (⋃ i ∈ s, T i) := by
classical
refine
⟨fun h s => h.mono (Set.iUnion_mono fun _ => Set.iUnion_subset_iff.2 fun _ => refl _),
fun h => ?_⟩
rw... | Mathlib/ModelTheory/Satisfiability.lean | 174 | 186 |
/-
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.Algebra.Group.Hom.End
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Group.Subsemigroup.Membership
import Mathlib.Algebra.Group... | K.comap (f : R →ₙ+* S) = K.map f.symm :=
(map_equiv_eq_comap_symm f.symm K).symm
| Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | 381 | 382 |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Abelian
import Mathlib.Algebra.Lie.BaseChange
import Mathlib.Algebra.Lie.IdealOperations
import Mathlib.Order.Hom.Basic
import Mathlib.RingTheory... | rw [← smul_lie]
refine Submodule.subset_span ⟨c • a, ?_, b, hb, rfl⟩
rw [← ih] at ha ⊢
exact Submodule.smul_mem _ _ ha
theorem coe_derivedSeries_eq_int (k : ℕ) :
| Mathlib/Algebra/Lie/Solvable.lean | 222 | 227 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Subspace
import Mathlib.LinearAlgebra.SesquilinearForm
/-!
# Orthogonal complements of su... | theorem orthogonal_eq_inter : Kᗮ = ⨅ v : K, LinearMap.ker (innerSL 𝕜 (v : E)) := by
apply le_antisymm
· rw [le_iInf_iff]
rintro ⟨v, hv⟩ w hw
simpa using hw _ hv
| Mathlib/Analysis/InnerProductSpace/Orthogonal.lean | 103 | 107 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
/-!
# Grönwall's inequality
The main technical result of this file is the Grönwall-like inequality
`norm_le_gron... | · have hss : Icc t₀ t' ⊆ Ioo a b :=
fun _ ht'' ↦ ⟨lt_of_lt_of_le ht.1 ht''.1, lt_of_le_of_lt ht''.2 ht'.2⟩
exact ODE_solution_unique_of_mem_Icc_right
(fun t'' ht'' ↦ hv t'' ((Ico_subset_Icc_self.trans hss) ht''))
(HasDerivAt.continuousOn fun _ ht'' ↦ (hf _ <| hss ht'').1)
(fun _ ht'' ↦ (hf... | Mathlib/Analysis/ODE/Gronwall.lean | 325 | 352 |
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