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/-
Copyright (c) 2024 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Seminorm
import Mathlib.GroupTheory.GroupAction.Pointwise
/-!
# The Minkowski functional, normed field version
In this file we define `(eg... | Mathlib/Analysis/Convex/EGauge.lean | 51 | 51 | |
/-
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... | Mathlib/Combinatorics/SimpleGraph/Basic.lean | 967 | 968 | |
/-
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... | open Cardinal in
lemma eq_zero_of_forall_eval_zero_of_natDegree_lt_card
(f : R[X]) (hf : ∀ r, f.eval r = 0) (hfR : f.natDegree < #R) : f = 0 := by
obtain hR|hR := finite_or_infinite R
· have := Fintype.ofFinite R
apply eq_zero_of_natDegree_lt_card_of_eval_eq_zero f Function.injective_id hf
simpa only [m... | Mathlib/Algebra/Polynomial/Roots.lean | 606 | 623 |
/-
Copyright (c) 2014 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Leonardo de Moura, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Ord... | Mathlib/Algebra/Order/Field/Basic.lean | 787 | 789 | |
/-
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... | Mathlib/Algebra/Polynomial/Basic.lean | 1,224 | 1,224 | |
/-
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.AlgebraMap
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.Ring.Action.Basic
import Mathlib.GroupTheory.GroupAction.Hom
import ... | variable (G : Type*) [Group G] [Fintype G]
variable (R : Type*) [CommRing R] [MulSemiringAction G R]
| Mathlib/Algebra/Polynomial/GroupRingAction.lean | 76 | 78 |
/-
Copyright (c) 2019 Calle Sönne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 944 | 962 | |
/-
Copyright (c) 2020 Ashvni Narayanan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ashvni Narayanan
-/
import Mathlib.Algebra.Field.Defs
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Algebra.Ring.Subring.Defs
import Mathlib.Algebra.Ring.Subsemiring.Bas... | theorem closure_eq (s : Subring R) : closure (s : Set R) = s :=
(Subring.gi R).l_u_eq s
| Mathlib/Algebra/Ring/Subring/Basic.lean | 614 | 615 |
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.Polynomial.Degree.CardPowDegree
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.NumberTheory.ClassNumber.AdmissibleAbsoluteValue
imp... | -- there must be two elements of A with the same coefficients at
-- `0`, ... `degree b - 1` ≤ `d - 1`.
-- In other words, the following map is not injective:
set f : Fin m.succ → Fin d → Fq := fun i j => (A i).coeff j
have : Fintype.card (Fin d → Fq) < Fintype.card (Fin m.succ) := by
simpa using lt_of_le_... | Mathlib/NumberTheory/ClassNumber/AdmissibleCardPowDegree.lean | 36 | 57 |
/-
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.Analysis.SpecialFunctions.Trigonometric.Arctan
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
/-!
# Right-angled triangles
This file proves ba... | rw [angle_add_eq_arcsin_of_inner_eq_zero h h0,
Real.sin_arcsin (le_trans (by norm_num) (div_nonneg (norm_nonneg _) (norm_nonneg _)))
(div_le_one_of_le₀ _ (norm_nonneg _))]
rw [mul_self_le_mul_self_iff (norm_nonneg _) (norm_nonneg _),
norm_add_sq_eq_norm_sq_add_norm_sq_real h]
exact le_add_of_nonneg_... | Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean | 132 | 139 |
/-
Copyright (c) 2022 Tian Chen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Tian Chen, Mantas Bakšys
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Algebra.Ring.Int.Parity
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.D... | · exact Nat.sub_pos_of_lt hyx
· exact Nat.sub_pos_of_lt (Nat.pow_lt_pow_left hyx hn)
theorem pow_add_pow (hxy : p ∣ x + y) (hx : ¬p ∣ x) {n : ℕ} (hn : Odd n) :
padicValNat p (x ^ n + y ^ n) = padicValNat p (x + y) + padicValNat p n := by
rcases y with - | y
· contradiction
rw [← Nat.cast_inj (R := ℕ∞), N... | Mathlib/NumberTheory/Multiplicity.lean | 405 | 414 |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Complex.Basic
import Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle
/-!
# Closure, interior, and frontier of preimages under `re`... | simpa only [closure_Ioi] using closure_preimage_im (Ioi a)
| Mathlib/Analysis/Complex/ReImTopology.lean | 114 | 115 |
/-
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... |
/-- The `p`-adic numbers `ℚ_[p]` are the Cauchy completion of `ℚ` with respect to the `p`-adic norm.
-/
def Padic (p : ℕ) [Fact p.Prime] :=
CauSeq.Completion.Cauchy (padicNorm p)
| Mathlib/NumberTheory/Padics/PadicNumbers.lean | 426 | 430 |
/-
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.Tactic.CategoryTheory.Elementwise
import Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer
import Mathlib.CategoryTheory.Limits.Constructions.EpiMono
impor... | · exact IsIso.eq_inv_of_hom_inv_id (D.cocycle _ _ _)
· rw [← cancel_mono (pullback.fst (D.f i j) (D.f i k))]
simp [t_fac, t_fac_assoc]
/-- (Implementation) The disjoint union of `U i`. -/
def sigmaOpens [HasCoproduct D.U] : C :=
∐ D.U
| Mathlib/CategoryTheory/GlueData.lean | 123 | 129 |
/-
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... | (encard_mono subset_union_left).trans_eq (encard_diff_add_encard _ _).symm
| Mathlib/Data/Set/Card.lean | 237 | 238 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Data.Nat.Log
import Mathlib.Data.Nat.Pri... | le_antisymm
(by
have hdisj :
Disjoint {i ∈ Ico 1 n.succ | p ^ i ≤ k % p ^ i + (p ^ n - k) % p ^ i}
{i ∈ Ico 1 n.succ | p ^ i ∣ k} := by
simp +contextual [disjoint_right, *, dvd_iff_mod_eq_zero,
Nat.mod_lt _ (pow_pos hp.pos _)]
rw [emultiplicity_choose hp hkn (lt_suc... | Mathlib/Data/Nat/Multiplicity.lean | 232 | 240 |
/-
Copyright (c) 2020 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp
-/
import Mathlib.Algebra.Algebra.Spectrum.Basic
import Mathlib.Algebra.Module.LinearMap.Basic
import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
import Mathl... | f.genEigenspace μ 1 = LinearMap.ker (f - μ • 1) := by
rw [← Nat.cast_one, genEigenspace_nat, pow_one]
| Mathlib/LinearAlgebra/Eigenspace/Basic.lean | 114 | 116 |
/-
Copyright (c) 2020 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Mario Carneiro
-/
import Mathlib.Data.Set.Function
import Mathlib.Order.Bounds.Defs
/-!
# Well-founded relations
A relation is well-founded if it can be used for induct... | set_option linter.deprecated false in
@[deprecated "`WellFounded.succ` is deprecated" (since := "2024-10-25")]
protected theorem lt_succ {r : α → α → Prop} (wf : WellFounded r) {x : α} (h : ∃ y, r x y) :
r x (wf.succ x) := by
rw [WellFounded.succ, dif_pos h]
apply min_mem
set_option linter.deprecated false in
... | Mathlib/Order/WellFounded.lean | 123 | 138 |
/-
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... | {F | F.2.2 ∈ S.arrows F.1 F.2.1}
instance : SetLike (Subgroupoid C) (Σ c d : C, c ⟶ d) where
| Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean | 165 | 167 |
/-
Copyright (c) 2023 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Algebra.NonUnitalHom
import Mathlib.Data.Set.UnionLift
import Mathlib.LinearAlgebra.Span.Basic
import Mathlib.RingTheory.NonUnitalSubring.Basic
... | exact
{ toFun :=
Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)
(fun i j x hxi hxj => by
let ⟨k, hik, hjk⟩ := dir i j
| Mathlib/Algebra/Algebra/NonUnitalSubalgebra.lean | 991 | 995 |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.Data.Nat.Factorization.PrimePow
import Mathlib.RingTheory.UniqueFactorizationDomain.Nat
/-!
# Lemmas about squ... | (ih : ∀ m, Prime m → m ∣ n → k ≤ m) : MinSqFacProp n (minSqFacAux n k) := by
rw [minSqFacAux]
by_cases h : n < k * k <;> simp only [h, ↓reduceDIte]
· refine squarefree_iff_prime_squarefree.2 fun p pp d => ?_
have := ih p pp (dvd_trans ⟨_, rfl⟩ d)
have := Nat.mul_le_mul this this
exact not_le_of_lt... | Mathlib/Data/Nat/Squarefree.lean | 148 | 161 |
/-
Copyright (c) 2022 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic
/-!
# The layer cake formula / Cavalieri's principle / tail probability formula
In this file we prove the f... | intervalIntegrable_const
have key :=
lintegral_comp_eq_lintegral_meas_le_mul μ f_nn f_mble cst_intble
(Eventually.of_forall fun _ => zero_le_one)
simp_rw [cst, ENNReal.ofReal_one, mul_one] at key
rw [← key]
congr with ω
simp only [intervalIntegral.integral_const, sub_zero, Algebra.id.smul_eq_mul... | Mathlib/MeasureTheory/Integral/Layercake.lean | 447 | 459 |
/-
Copyright (c) 2020 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.Geometry.Manifold.IsManifold.ExtChartAt
import Mathlib.Geometry.Manifold.LocalInvariantProperties
/-!
# `C^n` functions betwee... | (continuousWithinAt_univ _ _).1 <| ContMDiffWithinAt.continuousWithinAt hf
theorem ContMDiffOn.continuousOn (hf : ContMDiffOn I I' n f s) : ContinuousOn f s := fun x hx =>
(hf x hx).continuousWithinAt
theorem ContMDiff.continuous (hf : ContMDiff I I' n f) : Continuous f :=
| Mathlib/Geometry/Manifold/ContMDiff/Defs.lean | 621 | 626 |
/-
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, Mitchell Lee
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.Algebra.Group.Submonoid.Basic
import M... | variable {M₁ M₂} [MulOneClass M₁] [MulOneClass M₂] [ContinuousMul M₂]
{F : Type*} [FunLike F M₁ M₂] [MonoidHomClass F M₁ M₂] {l : Filter α}
/-- Construct a bundled monoid homomorphism `M₁ →* M₂` from a function `f` and a proof that it
belongs to the closure of the range of the coercion from `M₁ →* M₂` (or another ty... | Mathlib/Topology/Algebra/Monoid.lean | 293 | 321 |
/-
Copyright (c) 2021 Arthur Paulino. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Arthur Paulino, Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Clique
import Mathlib.Data.ENat.Lattice
import Mathlib.Data.Nat.Lattice
import Mathlib.Data.Setoid.Partition
imp... | refine ⟨G.recolorOfCardLE (by simpa using h.le) C, fun hC ↦ ?_⟩
dsimp at hC
simpa [h.not_le] using Fintype.card_le_of_surjective _ hC.of_comp
lemma le_chromaticNumber_iff_forall_surjective :
| Mathlib/Combinatorics/SimpleGraph/Coloring.lean | 353 | 357 |
/-
Copyright (c) 2020 Aaron Anderson, Jalex Stark. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark
-/
import Mathlib.Algebra.Polynomial.Expand
import Mathlib.Algebra.Polynomial.Laurent
import Mathlib.Algebra.Polynomial.Eval.SMul
import Mathli... |
variable {p : ℕ} [Fact p.Prime]
theorem matPolyEquiv_eq_X_pow_sub_C {K : Type*} (k : ℕ) [CommRing K] (M : Matrix n n K) :
matPolyEquiv ((expand K k : K[X] →+* K[X]).mapMatrix (charmatrix (M ^ k))) =
X ^ k - C (M ^ k) := by
ext m i j
rw [coeff_sub, coeff_C, matPolyEquiv_coeff_apply, RingHom.mapMatrix_app... | Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean | 241 | 248 |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.LinearAlgebra.Finsupp.Span
/-!
# Lie submodules of a Lie algebra
In this file we define Lie submodules, we construct ... | Mathlib/Algebra/Lie/Submodule.lean | 1,208 | 1,214 | |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kim Morrison
-/
import Mathlib.Tactic.NormNum.Basic
import Mathlib.Tactic.TryThis
import Mathlib.Util.AtomM
/-!
# The `abel` tactic
Evaluate expressions in the langua... | theorem zero_smul {α} [AddCommMonoid α] (c) : smul c (0 : α) = 0 := by
simp [smul, nsmul_zero]
| Mathlib/Tactic/Abel.lean | 244 | 245 |
/-
Copyright (c) 2023 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Geißer, Michael Stoll
-/
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.DiophantineApproximation.Basic
import Mathlib.NumberTheory.Zsqrtd.Basic
import Mathlib.Tactic... | rw [div_le_one h0, one_le_sq_iff_one_le_abs, Nat.abs_cast, Nat.one_le_cast]
exact q.pos
obtain ⟨m, hm⟩ : ∃ m : ℤ, {q : ℚ | q.1 ^ 2 - d * (q.den : ℤ) ^ 2 = m}.Infinite := by
contrapose! hM
simp only [not_infinite] at hM ⊢
refine (congr_arg _ (ext fun x => ?_)).mp (Finite.biUnion (finite_Ioo (-M) M)... | Mathlib/NumberTheory/Pell.lean | 341 | 349 |
/-
Copyright (c) 2021 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Data.Complex.Norm
/-!
# The partial order on the complex numbers
This order is defined by `z ≤ w ↔ z.re ≤ w.re ∧ z.im = w.im`.
This is a natural order o... | not_le_iff
| Mathlib/Data/Complex/Order.lean | 91 | 92 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Aurélien Saue, Anne Baanen
-/
import Mathlib.Tactic.NormNum.Inv
import Mathlib.Tactic.NormNum.Pow
import Mathlib.Util.AtomM
/-!
# `ring` tactic
A tactic for solving e... | Given monomials `va, vb`, attempts to add them together to get another monomial.
If the monomials are not compatible, returns `none`.
For example, `xy + 2xy = 3xy` is a `.nonzero` overlap, while `xy + xz` returns `none`
| Mathlib/Tactic/Ring/Basic.lean | 320 | 322 |
/-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.MeasureTheory.Integral.Layercake
/-!
# Japanese Brac... | have hr : 0 < r := lt_of_le_of_lt (finrank ℝ E).cast_nonneg hnr
refine ((integrable_one_add_norm hnr).const_mul <| (2 : ℝ) ^ (r / 2)).mono'
?_ (Eventually.of_forall fun x => ?_)
· apply Measurable.aestronglyMeasurable (by fun_prop)
refine (abs_of_pos ?_).trans_le (rpow_neg_one_add_norm_sq_le x hr)
positiv... | Mathlib/Analysis/SpecialFunctions/JapaneseBracket.lean | 142 | 150 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.Basic
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
/-!
# Left Homology of short complexes
Given a short complex `S : Shor... | lemma isIso_π (hf : S.f = 0) : IsIso h.π := by
have ⟨φ, hφ⟩ := CokernelCofork.IsColimit.desc' h.hπ' (𝟙 _)
(by rw [← cancel_mono h.i, comp_id, f'_i, zero_comp, hf])
dsimp at hφ
exact ⟨φ, hφ, by rw [← cancel_epi h.π, reassoc_of% hφ, comp_id]⟩
| Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean | 132 | 136 |
/-
Copyright (c) 2017 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Kim Morrison, Mario Carneiro, Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.EpiMono
import Mathlib.Topology.Category.TopCat.Limits.Basic
import Mathlib.CategoryT... |
/-- The binary coproduct cofan in `TopCat`. -/
| Mathlib/Topology/Category/TopCat/Limits/Products.lean | 233 | 234 |
/-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Kernel.Defs
/-!
# Basic kernels
This file contains basic results about kernels in general and definitions of some particular
kernels.
## Mai... | Mathlib/Probability/Kernel/Basic.lean | 457 | 461 | |
/-
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... | Function.apply_update (fun k => f k) _ _ _ _
simp [this]
map_update_smul' m i c x := by
| Mathlib/LinearAlgebra/Multilinear/Basic.lean | 351 | 353 |
/-
Copyright (c) 2024 Calle Sönne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Paul Lezeau, Calle Sönne
-/
import Mathlib.CategoryTheory.Functor.Category
import Mathlib.CategoryTheory.CommSq
/-!
# HomLift
Given a functor `p : 𝒳 ⥤ 𝒮`, this file provides API for... | subst ha hb h; simp
lemma of_fac' {R S : 𝒮} {a b : 𝒳} (f : R ⟶ S) (φ : a ⟶ b) (ha : p.obj a = R) (hb : p.obj b = S)
(h : p.map φ = eqToHom ha ≫ f ≫ eqToHom hb.symm) : p.IsHomLift f φ := by
subst ha hb
| Mathlib/CategoryTheory/FiberedCategory/HomLift.lean | 101 | 105 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mitchell Lee
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Indicator
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Topology.Algebra.InfiniteSum.Defs
imp... | theorem tprod_subtype_mulSupport (f : β → α) : ∏' x : mulSupport f, f x = ∏' x, f x :=
tprod_subtype_eq_of_mulSupport_subset Set.Subset.rfl
| Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | 473 | 475 |
/-
Copyright (c) 2020 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Kim Morrison
-/
import Mathlib.Algebra.Order.Interval.Set.Instances
import Mathlib.Order.Interval.Set.ProjIcc
import Mathlib.Topology.Algebra.Ring.Real
/-!
# The unit ... | /-- like `unitInterval.le_one`, but with the inequality in `I`. -/
theorem le_one' {t : I} : t ≤ 1 :=
t.2.2
protected lemma pos_iff_ne_zero {x : I} : 0 < x ↔ x ≠ 0 := bot_lt_iff_ne_bot
| Mathlib/Topology/UnitInterval.lean | 193 | 198 |
/-
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
-/
import Mathlib.Algebra.Order.Group.Abs
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Algebra.Order.Ring.Int
import Mathlib.A... |
private def geomSum : ℕ → α
| Mathlib/Algebra/Order/Ring/Abs.lean | 158 | 159 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.Measure.Typeclasses.Finite
import Mathlib.MeasureTheory.Measure.Typeclasses.NoAtoms
import Mathlib.MeasureTheory.Measure.... | Mathlib/MeasureTheory/Measure/Typeclasses.lean | 1,515 | 1,517 | |
/-
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.GramSchmidtOrtho
import Mathlib.LinearAlgebra.Orientation
/-!
# Orientations of real inner product spaces.
Th... | vector. -/
def adjustToOrientation : OrthonormalBasis ι ℝ E :=
(e.toBasis.adjustToOrientation x).toOrthonormalBasis (e.orthonormal_adjustToOrientation x)
| Mathlib/Analysis/InnerProductSpace/Orientation.lean | 103 | 105 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Order.Group.Indicator
import Mathlib.Analysis.Normed.Affine.AddTorsor
import Mathlib.Analysis.NormedSpace.FunctionSeries
import Mathlib.Analy... | theorem lim_of_nmem_U (c : CU P) (x : X) (h : x ∉ c.U) : c.lim x = 1 := by
simp only [CU.lim, approx_of_nmem_U c _ h, ciSup_const]
| Mathlib/Topology/UrysohnsLemma.lean | 246 | 247 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.MeasurableSpace.MeasurablyGenerated
import Mathlib.MeasureTheory.Measure.NullMeasurable
import Mathlib.Order.Interval.Set... |
theorem ae_sum_iff' {μ : ι → Measure α} {p : α → Prop} (h : MeasurableSet { x | p x }) :
| Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 1,216 | 1,217 |
/-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Contraction
/-! # Results about inverses in Clifford algebras
This contains some basic results about the inversion of vectors... | ⅟ (ι Q m) = ι Q (⅟ (Q m) • m) := by
letI := invertibleιOfInvertible Q m
convert (rfl : ⅟ (ι Q m) = _)
| Mathlib/LinearAlgebra/CliffordAlgebra/Inversion.lean | 31 | 34 |
/-
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.SpecialFunctions.Complex.Circle
import Mathlib.Geometry.Euclidean.Angle.Oriented.Basic
/-!
# Rotations by oriented angles.
This... | Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean | 140 | 140 | |
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Simon Hudon
-/
import Mathlib.Data.PFunctor.Multivariate.W
import Mathlib.Data.QPF.Multivariate.Basic
/-!
# The initial algebra of a multivariate qpf is again a qpf.
Fo... | /-- Equivalence relation on W-types that represent the same `Fix F`
value -/
inductive WEquiv {α : TypeVec n} : q.P.W α → q.P.W α → Prop
| ind (a : q.P.A) (f' : q.P.drop.B a ⟹ α) (f₀ f₁ : q.P.last.B a → q.P.W α) :
(∀ x, WEquiv (f₀ x) (f₁ x)) → WEquiv (q.P.wMk a f' f₀) (q.P.wMk a f' f₁)
| Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean | 71 | 75 |
/-
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.FieldTheory.Finite.Polynomial
import Mathlib.NumberTheory.Basic
import Mathlib.RingTheory.WittVector.WittPolynomial
/-!
# Witt struct... | · intro n; apply wittStructureRat_prop
· intro φ H
funext n
rw [show φ n = bind₁ φ (bind₁ (W_ ℚ) (xInTermsOfW p ℚ n)) by
rw [bind₁_wittPolynomial_xInTermsOfW p, bind₁_X_right]]
rw [bind₁_bind₁]
exact eval₂Hom_congr (RingHom.ext_rat _ _) (funext H) rfl
theorem wittStructureRat_rec_aux (Φ : M... | Mathlib/RingTheory/WittVector/StructurePolynomial.lean | 152 | 162 |
/-
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.Analysis.NormedSpace.Multilinear.Basic
import Mathlib.LinearAlgebra.Multilinear.Curry
/-!
# Currying and uncurrying continuous multilinear maps
... | continuousMultilinearCurryFin0 𝕜 G G' f = f 0 :=
rfl
| Mathlib/Analysis/NormedSpace/Multilinear/Curry.lean | 434 | 435 |
/-
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, Mario Carneiro, Johan Commelin
-/
import Mathlib.NumberTheory.Padics.PadicNumbers
import Mathlib.RingTheory.DiscreteValuationRing.Basic
/-!
# p-adic integers
This f... | Mathlib/NumberTheory/Padics/PadicIntegers.lean | 302 | 302 | |
/-
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 ... | lt_of_le_of_lt (le_max_right a 0) (lt_one_add _)⟩⟩
have fact1 : ∀ x : ℝ, x ∈ Ioo 0 a'⁻¹ → x ≠ 0 := fun _ hx => (ne_of_lt hx.1).symm
have fact2 (x) (hx : x ∈ Ioo 0 a'⁻¹) : a < x⁻¹ := lt_trans haa' ((lt_inv_comm₀ ha' hx.1).mpr hx.2)
have hdnf : ∀ x ∈ Ioo 0 a'⁻¹, HasDerivAt (f ∘ Inv.inv) (f' x⁻¹ * -(x ^ 2)⁻¹) ... | Mathlib/Analysis/Calculus/LHopital.lean | 144 | 174 |
/-
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.Bounded
import Mathlib.Analysis.Normed.Group.Uniform
import Mathlib.Topology.MetricSpace.Thickening
/-!
# P... | Mathlib/Analysis/Normed/Group/Pointwise.lean | 232 | 232 | |
/-
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
import Mathlib.Data.List.Forall2
import Mathlib.Data.Set.Functor
/-!
# LawfulTraversable instan... |
instance {σ : Type u} : LawfulTraversable.{u} (Sum σ) :=
{ show LawfulMonad (Sum σ) from inferInstance with
| Mathlib/Control/Traversable/Instances.lean | 170 | 172 |
/-
Copyright (c) 2021 David Wärn,. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn, Kim Morrison
-/
import Mathlib.Combinatorics.Quiver.Prefunctor
import Mathlib.Logic.Lemmas
import Batteries.Data.List.Basic
/-!
# Paths in quivers
Given a quiver `V`, we def... | lemma cons_ne_nil (p : Path a b) (e : b ⟶ a) : p.cons e ≠ Path.nil :=
fun h => by injection h
| Mathlib/Combinatorics/Quiver/Path.lean | 42 | 43 |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.GCDMonoid.Finset
import Mathlib.Algebra.Polynomial.CancelLeads
import Mathlib.Algebra.Polynomial.EraseLead
import Mathlib.Algebra.Polynomial.Fi... |
@[simp]
theorem content_C {r : R} : (C r).content = normalize r := by
rw [content]
by_cases h0 : r = 0
· simp [h0]
| Mathlib/RingTheory/Polynomial/Content.lean | 92 | 97 |
/-
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.Subsemigroup.Basic
/-!
# Subsemigroups: membership criteria
In this file we prove various facts about membership in a subsemigroup.
The i... |
@[to_additive]
theorem mem_sSup_of_mem {S : Set (Subsemigroup M)} {s : Subsemigroup M} (hs : s ∈ S) :
| Mathlib/Algebra/Group/Subsemigroup/Membership.lean | 89 | 91 |
/-
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
-/
import Mathlib.Tactic.Attr.Register
import Mathlib.Tactic.Basic
import Batteries.Logic
import Batteries.Tactic.Trans
import Batteries.Util.LibraryNot... |
/-- Construct a function from a default value `H0`, and a function to use if there exists a value
satisfying the predicate. -/
noncomputable def existsCases {α C : Sort*} {p : α → Prop} (H0 : C) (H : ∀ a, p a → C) : C :=
if h : ∃ a, p a then H (Classical.choose h) (Classical.choose_spec h) else H0
| Mathlib/Logic/Basic.lean | 703 | 707 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Kim Morrison
-/
import Mathlib.CategoryTheory.Subobject.Lattice
/-!
# Specific subobjects
We define `equalizerSubobject`, `kernelSubobject` and `imageSubobject`, which ar... | ⟨equalizer.lift h w, by simp⟩
theorem equalizerSubobject_factors_iff {W : C} (h : W ⟶ X) :
| Mathlib/CategoryTheory/Subobject/Limits.lean | 62 | 64 |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.CategoryTheory.Comma.Arrow
import Mathlib.Order.CompleteBooleanAlgebra
/-!
# Properties of morphisms
We provide the basic framework for talking about prope... | precomp e (_ : IsIso e) f hf := hprecomp (asIso e) f hf
postcomp e (_ : IsIso e) f hf := hpostcomp (asIso e) f hf
lemma RespectsIso.precomp (P : MorphismProperty C) [P.RespectsIso] {X Y Z : C} (e : X ⟶ Y)
[IsIso e] (f : Y ⟶ Z) (hf : P f) : P (e ≫ f) :=
RespectsLeft.precomp (Q := isomorphisms C) e ‹IsIso e› f... | Mathlib/CategoryTheory/MorphismProperty/Basic.lean | 229 | 238 |
/-
Copyright (c) 2023 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash, Deepro Choudhury
-/
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.LinearAlgebra.Span.Defs
import Mathlib.Algebra.Module.Equiv.Basic
/-!
# Additional results abou... |
/-- A linear equivalence which preserves a finite spanning set must have finite order. -/
lemma LinearEquiv.isOfFinOrder_of_finite_of_span_eq_top_of_mapsTo
{R M : Type*} [Semiring R] [AddCommMonoid M] [Module R M]
{Φ : Set M} (hΦ₁ : Φ.Finite) (hΦ₂ : span R Φ = ⊤) {e : M ≃ₗ[R] M} (he : MapsTo e Φ Φ) :
IsOfF... | Mathlib/LinearAlgebra/FiniteSpan.lean | 18 | 36 |
/-
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
/... | fun _ (hx : _ < _) => hx.trans_le h
| Mathlib/Analysis/Seminorm.lean | 689 | 689 |
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Data.Finset.Lattice.Fold
import Mathlib.Logic.Encodable.Pi
/-!
# W types
Given `α : Type` and `β : α → Type`, the W type determined by this data, `WTyp... | refine
not_injective_infinite_finite
(fun n : ℕ =>
show WType β from Nat.recOn n ⟨b, IsEmpty.elim' he⟩ fun _ ih => ⟨a, fun _ => ih⟩)
?_
intro n m h
| Mathlib/Data/W/Basic.lean | 92 | 97 |
/-
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.Subsemigroup.Operations
import Mathlib.Algebra.MonoidAlgebra.Support
import Mathlib.Order.Filter.Extr
/-!
# Lemmas about the `sup` and `in... | · intro h; rw [if_pos rfl, Finsupp.not_mem_support_iff.1 h, mul_zero]
· refine fun a ha hne => Finset.sum_eq_zero (fun a' ha' => if_neg <| fun he => ?_)
apply_fun D at he
simp_rw [hadd] at he
| Mathlib/Algebra/MonoidAlgebra/Degree.lean | 314 | 317 |
/-
Copyright (c) 2020 Johan Commelin, Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.Algebra.Field.ZMod
import Mathlib.NumberTheory.Padics.PadicIntegers
import Mathlib.RingTheory.LocalRing.ResidueField.... | Subtype.ext <| Quot.sound <| nthHomSeq_mul f_compat _ _
-- TODO: generalize this to arbitrary complete discrete valuation rings
/-- `lift f_compat` is the limit of a sequence `f` of compatible ring homs `R →+* ZMod (p^k)`,
with the equality `lift f_compat r = PadicInt.limNthHom f_compat r`.
-/
def lift : R →+* ℤ_[p]... | Mathlib/NumberTheory/Padics/RingHoms.lean | 586 | 597 |
/-
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.Fin.Tuple.Basic
/-!
# Lists from functions
Theorems and lemmas for dealing with `List.ofFn`, which converts a function on `Fin n` to a list
of l... | Mathlib/Data/List/OfFn.lean | 244 | 246 | |
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Joseph Myers
-/
import Mathlib.Analysis.InnerProductSpace.Orthogonal
import Mathlib.Analysis.Normed.Group.AddTorsor
/-!
# Perpendicular bisector of a segment
We def... | MapsTo f (perpBisector p₁ p₂) (perpBisector (f p₁) (f p₂)) :=
(h.preimage_perpBisector p₁ p₂).ge
| Mathlib/Geometry/Euclidean/PerpBisector.lean | 140 | 142 |
/-
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 | 1,216 | 1,217 | |
/-
Copyright (c) 2023 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Topology.Separation.Basic
/-!
# Pointwise convergence of indicator functions
In this file, we prove the equivalenc... |
lemma tendsto_indicator_const_iff_tendsto_pi_pure [T1Space β] (b : β) [NeZero b] :
Tendsto (fun i ↦ (As i).indicator (fun (_ : α) ↦ b)) L (𝓝 (A.indicator (fun (_ : α) ↦ b)))
↔ (Tendsto As L <| Filter.pi (pure <| · ∈ A)) := by
rw [tendsto_indicator_const_iff_forall_eventually _ b, tendsto_pi]
simp_rw [te... | Mathlib/Topology/IndicatorConstPointwise.lean | 117 | 122 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.GroupWithZero.Divisibili... | Mathlib/Algebra/MvPolynomial/Basic.lean | 1,380 | 1,385 | |
/-
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.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.Perm.Option
import Mathlib.Logic.Equiv.Fin.Rotate
import Mathlib.Logic.Equiv.Fintype
/-!
# Permutatio... | | succ n ih =>
rw [finRotate_succ_eq_decomposeFin]
simp [ih, pow_succ]
@[simp]
theorem support_finRotate {n : ℕ} : support (finRotate (n + 2)) = Finset.univ := by
ext
simp
theorem support_finRotate_of_le {n : ℕ} (h : 2 ≤ n) : support (finRotate n) = Finset.univ := by
obtain ⟨m, rfl⟩ := exists_add_of_l... | Mathlib/GroupTheory/Perm/Fin.lean | 95 | 106 |
/-
Copyright (c) 2024 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Kernel.Composition.IntegralCompProd
import Mathlib.Probability.Kernel.Disintegration.StandardBorel
/-!
# Lebesgue and Bochner integrals of con... | Mathlib/Probability/Kernel/Disintegration/Integral.lean | 301 | 305 | |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Complex.CauchyIntegral
import Mathlib.Analysis.InnerProductSpace.Convex
import Mathlib.Analysis.NormedSpace.Extr
import Mathlib.Data.Complex... | rcases eq_or_ne w z with (rfl | hne); · rfl
rw [← dist_ne_zero] at hne
exact norm_max_aux₂ hd (closure_ball z hne ▸ hz.closure hd.continuousOn.norm)
/-!
### Maximum modulus principle for any codomain
| Mathlib/Analysis/Complex/AbsMax.lean | 159 | 164 |
/-
Copyright (c) 2021 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kim Morrison
-/
import Mathlib.Algebra.Homology.ComplexShape
import Mathlib.CategoryTheory.Subobject.Limits
import Mathlib.CategoryTheory.GradedObject
import Mathlib.Alge... | comm' n m := by
rintro (rfl : m + 1 = n)
exact (mkHomAux P Q zero one one_zero_comm succ m).2.2
| Mathlib/Algebra/Homology/HomologicalComplex.lean | 798 | 800 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro
-/
import Mathlib.Algebra.Module.Submodule.Bilinear
import Mathlib.Algebra.Module.Equiv.Basic
import Mathlib.GroupTheory.Congruence.Hom
import Mathlib.Tactic.Abel
... | @[simp] theorem congr_zpow (f : M ≃ₗ[R] M) (g : N ≃ₗ[R] N) (n : ℤ) :
congr f g ^ n = congr (f ^ n) (g ^ n) := by
cases n with
| Mathlib/LinearAlgebra/TensorProduct/Basic.lean | 903 | 905 |
/-
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.BigOperators.Expect
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Field.Canonical
import Mathlib.Algebra.O... | Mathlib/Data/NNReal/Basic.lean | 1,024 | 1,026 | |
/-
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.CategoryTheory.GlueData
import Mathlib.Topology.Category.TopCat.Limits.Pullbacks
import Mathlib.Topology.Category.TopCat.Opens
import Mathlib.Tactic.Generali... | have : D.f _ _ ⁻¹' (𝖣.ι j ⁻¹' (𝖣.ι i '' U)) = (D.t j i ≫ D.f _ _) ⁻¹' U := by
ext x
conv_rhs => rw [← Set.preimage_image_eq U (D.ι_injective _)]
simp
rw [← this, Set.image_preimage_eq_inter_range]
symm
apply Set.inter_eq_self_of_subset_left
rw [← D.preimage_range i j]
| Mathlib/Topology/Gluing.lean | 239 | 246 |
/-
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... |
/-! ### Cofinality of suprema and least strict upper bounds -/
private theorem card_mem_cof {o} : ∃ (ι : _) (f : ι → Ordinal), lsub.{u, u} f = o ∧ #ι = o.card :=
⟨_, _, lsub_typein o, mk_toType o⟩
| Mathlib/SetTheory/Cardinal/Cofinality.lean | 161 | 165 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kim Morrison
-/
import Mathlib.Algebra.BigOperators.Finsupp.Basic
import Mathlib.Algebra.BigOperators.Group.Finset.Preimage
import Mathlib.Algebra.Module.Defs
import Ma... | a ∈ s.sum.support → ∃ f ∈ s, a ∈ (f : α →₀ M).support :=
Multiset.induction_on s (fun h => False.elim (by simp at h))
(by
| Mathlib/Data/Finsupp/Basic.lean | 1,035 | 1,037 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kim Morrison
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Algebra.Group.InjSurj
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Tactic.FastInstance
impo... | Mathlib/Data/Finsupp/Defs.lean | 682 | 683 | |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Andrew Zipperer, Haitao Zhang, Minchao Wu, Yury Kudryashov
-/
import Mathlib.Data.Set.Prod
import Mathlib.Data.Set.Restrict
/-!
# Functions over sets
This file contains... | theorem image_image' (hf : LeftInvOn f' f s) (hs : s₁ ⊆ s) : f' '' (f '' s₁) = s₁ :=
(hf.mono hs).image_image
| Mathlib/Data/Set/Function.lean | 799 | 801 |
/-
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.Order.Filter.Cofinite
import Mathlib.RingTheory.DedekindDomain.Ideal
import Mathlib.RingTheory.UniqueFactorizationDomai... | rw [← associated_iff_eq, ← Associates.mk_eq_mk_iff_associated]
classical
apply Associates.eq_of_eq_counts
· apply Associates.finprod_ne_zero I
· apply Associates.mk_ne_zero.mpr hI
intro v hv
obtain ⟨J, hJv⟩ := Associates.exists_rep v
rw [← hJv, Associates.irreducible_mk] at hv
rw [← hJv]
apply Ideal... | Mathlib/RingTheory/DedekindDomain/Factorization.lean | 178 | 189 |
/-
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.Algebra.Polynomial.Eval.Defs
import Mathlib.LinearAlgebra.Dimension.Constructions
/-!
# Linear recurrence
Informally, a "linear recurrence" is an... | Mathlib/Algebra/LinearRecurrence.lean | 217 | 227 | |
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kenny Lau
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.RingTheory.MvPowerSer... |
@[simp]
| Mathlib/RingTheory/PowerSeries/Basic.lean | 316 | 317 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Chris Hughes, Floris van Doorn, Yaël Dillies
-/
import Mathlib.Data.Nat.Basic
import Mathlib.Tactic.GCongr.CoreAttrs
import Mathlib.Tactic.Common
import Mathlib.Tactic.... | exact (le_trans (Nat.pow_le_pow_left (Nat.sub_le_sub_right n.le_succ _) k)
(pow_sub_le_descFactorial n k))
theorem pow_sub_lt_descFactorial' {n : ℕ} :
| Mathlib/Data/Nat/Factorial/Basic.lean | 408 | 411 |
/-
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.Combinatorics.SimpleGraph.Acyclic
import Mathlib.Data.ENat.Lattice
/-!
# Girth of a simple graph
This file defines the girth and the extended girth of a ... |
@[simp]
lemma le_egirth {n : ℕ∞} : n ≤ G.egirth ↔ ∀ a (w : G.Walk a a), w.IsCycle → n ≤ w.length := by
simp [egirth]
@[simp]
lemma egirth_eq_top : G.egirth = ⊤ ↔ G.IsAcyclic := by simp [egirth, IsAcyclic]
| Mathlib/Combinatorics/SimpleGraph/Girth.lean | 34 | 41 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Sébastien Gouëzel,
Rémy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
/-!
# Power function... |
@[gcongr] theorem rpow_le_rpow_of_exponent_le {x : ℝ≥0∞} {y z : ℝ} (hx : 1 ≤ x) (hyz : y ≤ z) :
x ^ y ≤ x ^ z := by
cases x
· rcases lt_trichotomy y 0 with (Hy | Hy | Hy) <;>
| Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | 769 | 773 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Kexing Ying, Moritz Doll
-/
import Mathlib.Algebra.GroupWithZero.Action.Opposite
import Mathlib.LinearAlgebra.Finsupp.VectorSpace
import Mathlib.LinearAlgebra.Matrix.Basis
im... | theorem LinearMap.toMatrix₂_mul (B : M₁ →ₗ[R] M₂ →ₗ[R] R) (M : Matrix m m' R) :
LinearMap.toMatrix₂ b₁ b₂ B * M =
LinearMap.toMatrix₂ b₁ b₂' (B.compl₂ (toLin b₂' b₂ M)) := by
rw [LinearMap.toMatrix₂_compl₂ b₁ b₂, toMatrix_toLin]
theorem Matrix.toLinearMap₂_compl₁₂ (M : Matrix n m R) (P : Matrix n n' R) (Q ... | Mathlib/LinearAlgebra/Matrix/SesquilinearForm.lean | 456 | 461 |
/-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang
-/
import Mathlib.Analysis.Calculus.Conformal.NormedSpace
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
/-!
# Angles and conformal maps
This file proves that co... | obtain ⟨c, hc, li, rfl⟩ := h
exact (angle_smul_smul hc _ _).trans (li.angle_map _ _)
/-- If a real differentiable map `f` is conformal at a point `x`,
| Mathlib/Geometry/Euclidean/Angle/Unoriented/Conformal.lean | 25 | 28 |
/-
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]
theorem sigma_add :
| Mathlib/Data/Multiset/Bind.lean | 332 | 333 |
/-
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.Analysis.Calculus.FormalMultilinearSeries
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Logic.Equiv.Fin.Basic
imp... | protected theorem HasFPowerSeriesWithinAt.eventually (hf : HasFPowerSeriesWithinAt f p s x) :
∀ᶠ r : ℝ≥0∞ in 𝓝[>] 0, HasFPowerSeriesWithinOnBall f p s x r :=
let ⟨_, hr⟩ := hf
| Mathlib/Analysis/Analytic/Basic.lean | 591 | 593 |
/-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib... | (logb_surjective b_pos b_ne_one).range_eq
theorem surjOn_logb' : SurjOn (logb b) (Iio 0) univ := by
intro x _
| Mathlib/Analysis/SpecialFunctions/Log/Base.lean | 157 | 160 |
/-
Copyright (c) 2019 Jan-David Salchow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jan-David Salchow, Sébastien Gouëzel, Jean Lo
-/
import Mathlib.Analysis.NormedSpace.OperatorNorm.Basic
/-!
# Operator norm as an `NNNorm`
Operator norm as an `NNNorm`, i.e. takin... | by_cases hr₀ : r < 0
· exact ⟨0, by simpa using hr₀⟩
· lift r to ℝ≥0 using not_lt.1 hr₀
exact f.exists_lt_apply_of_lt_opNNNorm hr
| Mathlib/Analysis/NormedSpace/OperatorNorm/NNNorm.lean | 149 | 152 |
/-
Copyright (c) 2020 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Andrew Yang
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
/-!
# Derivations
This file defines derivation. A derivat... |
variable {M : Type*} [AddCommGroup M] [Module A M] [Module R M]
variable (D : Derivation R A M) {D1 D2 : Derivation R A M} (r : R) (a b : A)
protected theorem map_neg : D (-a) = -D a :=
| Mathlib/RingTheory/Derivation/Basic.lean | 430 | 434 |
/-
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... | rw [hB.eq_generateFrom, continuous_generateFrom_iff]
@[simp] lemma isTopologicalBasis_empty : IsTopologicalBasis (∅ : Set (Set α)) ↔ IsEmpty α where
mp h := by simpa using h.sUnion_eq.symm
mpr h := ⟨by simp, by simp [Set.univ_eq_empty_iff.2], Subsingleton.elim ..⟩
variable (α)
/-- A separable space is one with... | Mathlib/Topology/Bases.lean | 280 | 293 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro
-/
import Mathlib.Algebra.Module.Submodule.Bilinear
import Mathlib.Algebra.Module.Equiv.Basic
import Mathlib.GroupTheory.Congruence.Hom
import Mathlib.Tactic.Abel
... | obtain ⟨m₁, n₁, rfl⟩ := h₁
obtain ⟨m₂, n₂, rfl⟩ := h₂
apply h
end Module
| Mathlib/LinearAlgebra/TensorProduct/Basic.lean | 473 | 478 |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Topology.Algebra.Group.Pointwise
import Mathlib.Topology.Order.Basic
/-!
# Strictly convex sets
This file defines st... | (mem_image_of_mem _ <| ht hw hy (ne_of_apply_ne _ h) ha hb hab)
exact
subset_interior_add_left
(add_mem_add (hs hv hx hvx ha hb hab) <| ht.convex hw hy ha.le hb.le hab)
theorem StrictConvex.add_left (hs : StrictConvex 𝕜 s) (z : E) :
| Mathlib/Analysis/Convex/Strict.lean | 222 | 227 |
/-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Data.Countable.Basic
import Mathlib.Data.Finset.Max
import Mathlib.Data.Fintype.Pigeonhole
import Mathlib.Logic.Enco... | · nth_rw 2 [← hmn]
rw [Int.toNat_eq_max, toZ_of_ge (le_succ_iterate _ _), max_eq_left]
exact Int.natCast_nonneg _
suffices IsMax (succ^[n] i0) from absurd this hn
exact isMax_iterate_succ_of_eq_of_ne h_eq.symm (Ne.symm hmn)
theorem toZ_iterate_pred_of_not_isMin (n : ℕ) (hn : ¬IsMin (pred^[n] i0)) :
t... | Mathlib/Order/SuccPred/LinearLocallyFinite.lean | 263 | 280 |
/-
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... | end Polynomial
| Mathlib/Algebra/Polynomial/Roots.lean | 806 | 809 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Order.Filter.Tendsto
import Mathlib.Data.Set.Accumulate
import Mathlib.Topology.Bornology.Basic
import Mathlib.Topolog... | /-- If `y` is a unique `MapClusterPt` for `f` along `l`
and the codomain of `f` is a compact space,
| Mathlib/Topology/Compactness/Compact.lean | 705 | 706 |
/-
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.Group.Abs
import Mathlib.Algebra.Order.Monoid.Unbundled.MinMax
/-!
# `min` and `max` in... |
alias max_zero_sub_eq_self := max_zero_sub_max_neg_zero_eq_self
| Mathlib/Algebra/Order/Group/MinMax.lean | 22 | 23 |
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