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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.Algebra.Order.Hom.Ring
import Mathlib.Data.ENat.Basic
import Mathlib.SetTheory.Cardinal.Basic
/-!
# Conversion between `Cardinal` and `ℕ∞`
In this ... | lemma ofENat_toENat_le (a : Cardinal) : ↑(toENat a) ≤ a := enat_gc.l_u_le _
@[simp]
lemma ofENat_toENat_eq_self {a : Cardinal} : toENat a = a ↔ a ≤ ℵ₀ := by
| Mathlib/SetTheory/Cardinal/ENat.lean | 234 | 237 |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Module.BigOperators
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.Nat.Squarefree
imp... | theorem map_lcm [CommGroupWithZero R] {f : ArithmeticFunction R}
(hf : f.IsMultiplicative) {x y : ℕ} (hf_gcd : f (x.gcd y) ≠ 0) :
f (x.lcm y) = f x * f y / f (x.gcd y) := by
rw [← hf.lcm_apply_mul_gcd_apply, mul_div_cancel_right₀ _ hf_gcd]
theorem eq_zero_of_squarefree_of_dvd_eq_zero [MonoidWithZero R] {f : ... | Mathlib/NumberTheory/ArithmeticFunction.lean | 760 | 768 |
/-
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.HomotopyCategory.HomComplex
import Mathlib.Algebra.Homology.HomotopyCofiber
/-! # The mapping cone of a morphism of cochain complexes
In this ... |
variable (α : Cochain K F m) (β : Cochain K G n) (h : n + 1 = m)
@[simp]
| Mathlib/Algebra/Homology/HomotopyCategory/MappingCone.lean | 415 | 418 |
/-
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... | rw [mul_smul]
exact im_lt_im_S_smul hg₀'
· show |(g • z).re| ≤ 1 / 2
-- if not, then either `T` or `T'` decrease |Re|.
rw [abs_le]
constructor
· contrapose! hg'
refine ⟨T * g, (T_mul_apply_one _).symm, ?_⟩
rw [mul_smul, re_T_smul]
cases abs_cases ((g • z).re + 1) <;> cases ab... | Mathlib/NumberTheory/Modular.lean | 428 | 438 |
/-
Copyright (c) 2024 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Order.CompactlyGenerated.Basic
/-!
# Generators for boolean algebras
In this file, we provide an alternative constructor for boolean algebras.
A set... | intro h a ha
apply (hS.mono hY).mem_of_isAtom_of_le_sSup_atoms _ _ ((le_sSup ha).trans h)
exact (hS.mono hX).isAtom a ha
lemma eq_atoms_of_sSup_eq_top (hS : BooleanGenerators S) (h : sSup S = ⊤) :
S = {a : α | IsAtom a} := by
| Mathlib/Order/BooleanGenerators.lean | 167 | 172 |
/-
Copyright (c) 2023 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.CharP.Reduced
import Mathlib.FieldTheory.KummerPolynomial
import Mathlib.FieldTheory.Separable
/-!
# Perfect fie... | (X ^ p ^ n - C y).roots = p ^ n • {(iterateFrobeniusEquiv R p n).symm y} := by
have H := roots_expand_pow (p := p) (n := n) (f := X - C y)
rwa [roots_X_sub_C, Multiset.map_singleton, map_sub, expand_X, expand_C] at H
theorem roots_X_pow_char_pow_sub_C_pow {y : R} {m : ℕ} :
| Mathlib/FieldTheory/Perfect.lean | 291 | 295 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Jeremy Avigad, Simon Hudon
-/
import Mathlib.Algebra.Notation.Defs
import Mathlib.Data.Set.Subsingleton
import Mathlib.Logic.Equiv.Defs
/-!
# Partial values of a type
... | some_injective.eq_iff
@[simp]
| Mathlib/Data/Part.lean | 194 | 196 |
/-
Copyright (c) 2023 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Integral.Bochner.Basic
import Mathlib.Topology.ContinuousMap.Bounded.Normed
/-!
# Integration of bounded continuous functions
In this file,... |
lemma lintegral_le_edist_mul (f : X →ᵇ ℝ≥0) (μ : Measure X) :
(∫⁻ x, f x ∂μ) ≤ edist 0 f * (μ Set.univ) :=
| Mathlib/MeasureTheory/Integral/BoundedContinuousFunction.lean | 31 | 33 |
/-
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.OuterMeasure.Operations
import Mathlib.Analysis.SpecificLimits.Basic
/-!
# Outer measures from functions
Given an arbit... | that don't satisfy `P`.
This is similar to `ofFunction_apply`, except that the sets `tᵢ` satisfy `P`.
The hypothesis `m_top` applies in particular to a function of the form `extend m'`. -/
theorem ofFunction_eq_iInf_mem {P : Set α → Prop} (m_top : ∀ s, ¬ P s → m s = ∞) (s : Set α) :
OuterMeasure.ofFunction m m_empt... | Mathlib/MeasureTheory/OuterMeasure/OfFunction.lean | 99 | 106 |
/-
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.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Order.WithBot
/-!
# Intervals in `WithTop α` and `WithBot α`
In this file we ... | Mathlib/Order/Interval/Set/WithBotTop.lean | 226 | 229 | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.Polynomial.Basi... | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 1,431 | 1,432 | |
/-
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... | · rw [nodal_monic, leadingCoeff_sub_of_degree_lt h, monic_X_pow]
· intros i hi
| Mathlib/LinearAlgebra/Lagrange.lean | 536 | 537 |
/-
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
/... | theorem coe_restrictScalars (p : Seminorm 𝕜' E) : (p.restrictScalars 𝕜 : E → ℝ) = p :=
rfl
@[simp]
theorem restrictScalars_ball (p : Seminorm 𝕜' E) : (p.restrictScalars 𝕜).ball = p.ball :=
rfl
| Mathlib/Analysis/Seminorm.lean | 1,025 | 1,031 |
/-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Topology.UniformSpace.UniformConvergenceTopology
/-!
# Equicontinuity of a family of functions
Let `X` be a topological space and `α` a `UniformS... | (hu : IsUniformInducing u) :
UniformEquicontinuousOn F S ↔ UniformEquicontinuousOn ((u ∘ ·) ∘ F) S := by
have := UniformFun.postcomp_isUniformInducing (α := ι) hu
simp only [uniformEquicontinuousOn_iff_uniformContinuousOn, this.uniformContinuousOn_iff]
rfl
| Mathlib/Topology/UniformSpace/Equicontinuity.lean | 751 | 756 |
/-
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.Finite.Defs
import Mathlib.Data.Finset.BooleanAlgebra
import Mathlib.Data.Finset.Image
import Mathlib.Data.Fintype.Defs
import Mathlib.Data.Fintyp... | Mathlib/Data/Fintype/Basic.lean | 462 | 463 | |
/-
Copyright (c) 2021 Jordan Brown, Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jordan Brown, Thomas Browning, Patrick Lutz
-/
import Mathlib.GroupTheory.Abelianization
import Mathlib.GroupTheory.Perm.ViaEmbedding
import Mathlib.GroupT... | | succ n ih =>
rw [_root_.derivedSeries_succ, ih, _root_.commutator]
rcases (commutator_normal (⊤ : Subgroup G) (⊤ : Subgroup G)).eq_bot_or_eq_top with h | h
· rw [h, commutator_bot_left]
· rwa [h]
theorem IsSimpleGroup.comm_iff_isSolvable : (∀ a b : G, a * b = b * a) ↔ IsSolvable G :=
⟨isSolvable_... | Mathlib/GroupTheory/Solvable.lean | 187 | 197 |
/-
Copyright (c) 2024 Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Merten
-/
import Mathlib.CategoryTheory.Galois.GaloisObjects
import Mathlib.CategoryTheory.Limits.Shapes.CombinedProducts
import Mathlib.Data.Finite.Sum
/-!
# Decompositio... | /-- The trivial case if `X` is connected. -/
private lemma has_decomp_connected_components_aux_conn (X : C) [IsConnected X] :
∃ (ι : Type) (f : ι → C) (g : (i : ι) → (f i) ⟶ X) (_ : IsColimit (Cofan.mk X g)),
(∀ i, IsConnected (f i)) ∧ Finite ι := by
refine ⟨Unit, fun _ ↦ X, fun _ ↦ 𝟙 X, mkCofanColimit _ (fu... | Mathlib/CategoryTheory/Galois/Decomposition.lean | 57 | 62 |
/-
Copyright (c) 2022 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll, Thomas Zhu, Mario Carneiro
-/
import Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity
/-!
# The Jacobi Symbol
We define the Jacobi symbol and prove its main pro... |
/-- The **Law of Quadratic Reciprocity for the Jacobi symbol**, version with `qrSign` -/
theorem quadratic_reciprocity' {a b : ℕ} (ha : Odd a) (hb : Odd b) :
J(a | b) = qrSign b a * J(b | a) := by
-- define the right hand side for fixed `a` as a `ℕ →* ℤ`
let rhs : ℕ → ℕ →* ℤ := fun a =>
| Mathlib/NumberTheory/LegendreSymbol/JacobiSymbol.lean | 397 | 402 |
/-
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 [sub_eq_add_neg, angle_add_eq_arccos_of_inner_eq_zero h]
/-- An angle in a right-angled triangle expressed using `arcsin`, version subtracting vectors. -/
theorem angle_sub_eq_arcsin_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) (h0 : x ≠ 0 ∨ y ≠ 0) :
angle x (x - y) = Real.arcsin (‖y‖ / ‖x - y‖) := by
| Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean | 207 | 211 |
/-
Copyright (c) 2020 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Topology.Algebra.Group.Defs
/-!
# Topological propertie... | ext y
simp [homothety_apply]
end CommRing
section Field
| Mathlib/Topology/Algebra/Affine.lean | 61 | 67 |
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Probability.Variance
import Mathlib.MeasureTheory.Function.UniformIntegrable
/-!
# Identically distributed random variables
Two random variable... | apply (ae_map_iff h.aemeasurable_snd pmeas).1
rw [← h.map_eq]
exact (ae_map_iff h.aemeasurable_fst pmeas).2 hp
| Mathlib/Probability/IdentDistrib.lean | 132 | 135 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Data.List.Lattice
import Mathlib.Data.Bool.Basic
import Mathlib.Order.Lattice
/-!
# Intervals in ℕ
This file defines intervals of naturals. `List.Ico m n... | by_cases h₁ : n < a
· left
exact h₁
· right
| Mathlib/Data/List/Intervals.lean | 192 | 195 |
/-
Copyright (c) 2022 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Data.Nat.Cast.Field
import Mathlib.GroupTheory.Abelianization
import Mathlib.GroupTheory.GroupAction.CardComm... | · exact pow_pos (Nat.cast_pos.mpr Finite.card_pos) 2
theorem Subgroup.commProb_quotient_le [H.Normal] : commProb (G ⧸ H) ≤ commProb G * Nat.card H := by
/- After rewriting with `commProb_def'`, we reduce to showing that `G` has at least as many
conjugacy classes as `G ⧸ H`. -/
rw [commProb_def', commProb_d... | Mathlib/GroupTheory/CommutingProbability.lean | 108 | 116 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Kappelmann
-/
import Mathlib.Algebra.Order.Floor.Defs
import Mathlib.Algebra.Order.Floor.Ring
import Mathlib.Algebra.Order.Floor.Semiring
deprecated_module (sinc... | Mathlib/Algebra/Order/Floor.lean | 1,423 | 1,425 | |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Alex Kontorovich
-/
import Mathlib.Data.Set.Piecewise
import Mathlib.Order.Filter.Tendsto
import Mathlib.Order.Filter.Bases.Finite
/-!
# (Co)product of a family of f... | @[simp]
theorem pi_principal [Finite ι] (s : (i : ι) → Set (α i)) :
pi (fun i ↦ 𝓟 (s i)) = 𝓟 (univ.pi s) := by
| Mathlib/Order/Filter/Pi.lean | 133 | 135 |
/-
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... | C_injective.eq_iff' (map_zero C)
theorem C_ne_zero : C a ≠ 0 ↔ a ≠ 0 :=
| Mathlib/Algebra/Polynomial/Basic.lean | 693 | 695 |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Data.Finset.Fold
import Mathlib.Algebra.GCDMonoid.Multiset
/-!
# GCD and LCM operations on finsets
## Main definitions
- `Finset.gcd` - the greatest... |
theorem lcm_eq_lcm_image [DecidableEq α] : s.lcm f = (s.image f).lcm id :=
Eq.symm <| lcm_image _
| Mathlib/Algebra/GCDMonoid/Finset.lean | 100 | 103 |
/-
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... | left_inv := fun ⟨_, _, _⟩ => rfl
right_inv := fun ⟨_, _⟩ => rfl
variable (R)
/-- A Dedekind domain is equal to the intersection of its localizations at all its height one
non-zero prime ideals viewed as subalgebras of its field of fractions. -/
theorem iInf_localization_eq_bot [Algebra R K] [hK : IsFractionRing R... | Mathlib/RingTheory/DedekindDomain/Ideal.lean | 984 | 991 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Alex J. Best
-/
import Mathlib.MeasureTheory.Group.Arithmetic
/-!
# Pointwise set operations on `MeasurableSet`s
In this file we prove several versions of the follo... | MeasurableSet (a • s) := by
rw [← preimage_smul_inv]
exact measurable_const_smul _ hs
theorem MeasurableSet.const_smul_of_ne_zero {G₀ α : Type*} [GroupWithZero G₀] [MulAction G₀ α]
| Mathlib/MeasureTheory/Group/Pointwise.lean | 24 | 28 |
/-
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 | 1,025 | 1,029 | |
/-
Copyright (c) 2021 Benjamin Davidson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Log.NegMulLog
import Mathlib.Analysis.SpecialFunctions.NonIntegrable
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv... | simpa only [mul_one] using ((hasDerivAt_id (x : ℂ)).const_mul _).comp_ofReal
rw [integral_deriv_eq_sub' _ (funext fun x => (D x).deriv) fun x _ => (D x).differentiableAt]
| Mathlib/Analysis/SpecialFunctions/Integrals.lean | 477 | 478 |
/-
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 | 886 | 948 | |
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Data.Set.Prod
/-!
# N-ary images of sets
This file defines `Set.image2`, the binary image of sets.
This is mostly useful to define pointwise oper... |
@[simp]
theorem image2_nonempty_iff : (image2 f s t).Nonempty ↔ s.Nonempty ∧ t.Nonempty :=
| Mathlib/Data/Set/NAry.lean | 116 | 118 |
/-
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... | rw [h.ncard]
apply zero_le
theorem ncard_union_eq (h : Disjoint s t) (hs : s.Finite := by toFinite_tac)
| Mathlib/Data/Set/Card.lean | 860 | 863 |
/-
Copyright (c) 2024 Emilie Burgun. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Emilie Burgun
-/
import Mathlib.Algebra.Group.Action.Pointwise.Set.Basic
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Dynamics.PeriodicPts.Defs
import Mathlib.GroupTheory.G... | theorem fixedBy_inv (g : G) : fixedBy α g⁻¹ = fixedBy α g := by
ext
rw [mem_fixedBy, mem_fixedBy, inv_smul_eq_iff, eq_comm]
| Mathlib/GroupTheory/GroupAction/FixedPoints.lean | 60 | 62 |
/-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Math... | -- yielding an inequality which we have seen is false for large enough n.
have H1 : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) ≤ 4 ^ n := bertrand_main_inequality n_large
have H2 : 4 ^ n < n * n.centralBinom :=
Nat.four_pow_lt_mul_centralBinom n (le_trans (by norm_num1) n_large)
have H3 : n.centralBinom ≤... | Mathlib/NumberTheory/Bertrand.lean | 189 | 201 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Topology.Algebra.Constructions
import Mathlib.Topology.Bases
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Topology.UniformSpac... | hu.map hf
theorem CauchySeq.subseq_mem {V : ℕ → Set (α × α)} (hV : ∀ n, V n ∈ 𝓤 α) {u : ℕ → α}
(hu : CauchySeq u) : ∃ φ : ℕ → ℕ, StrictMono φ ∧ ∀ n, (u <| φ (n + 1), u <| φ n) ∈ V n := by
have : ∀ n, ∃ N, ∀ k ≥ N, ∀ l ≥ k, (u l, u k) ∈ V n := fun n => by
| Mathlib/Topology/UniformSpace/Cauchy.lean | 249 | 253 |
/-
Copyright (c) 2021 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp
-/
import Mathlib.Analysis.Convex.Cone.Basic
import Mathlib.Analysis.InnerProductSpace.Projection
/-!
# Convex cones in inner product spaces
We define `Set.inn... |
variable [CompleteSpace H]
open scoped InnerProductSpace in
/-- This is a stronger version of the Hahn-Banach separation theorem for closed convex cones. This
is also the geometric interpretation of Farkas' lemma. -/
theorem ConvexCone.hyperplane_separation_of_nonempty_of_isClosed_of_nmem (K : ConvexCone ℝ H)
(ne... | Mathlib/Analysis/Convex/Cone/InnerDual.lean | 144 | 161 |
/-
Copyright (c) 2023 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Algebra.GroupWithZero.Action.Pi
import Mathlib.Algebra.Order.Module.Defs
import Mathlib.Algebra.Order.Pi
import Mathlib.Data.Finsupp.Order
/-!
# Flooring,... | @[norm_cast] lemma coe_floorDiv (f : ι →₀ β) (a : α) : f ⌊/⌋ a = fun i ↦ f i ⌊/⌋ a := rfl
@[simp] lemma floorDiv_apply (f : ι →₀ β) (a : α) (i : ι) : (f ⌊/⌋ a) i = f i ⌊/⌋ a := rfl
| Mathlib/Algebra/Order/Floor/Div.lean | 244 | 245 |
/-
Copyright (c) 2024 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.HomotopyCategory.Shift
import Mathlib.Algebra.Homology.TotalComplex
/-!
# Behaviour of the total complex with respect to shifts
There are two ... |
/-- The compatibility isomorphisms of the total complex with the shifts
in both variables "commute" only up to a sign `(x * y).negOnePow`. -/
lemma totalShift₁Iso_trans_totalShift₂Iso :
((shiftFunctor₂ C y).obj K).totalShift₁Iso x ≪≫
(shiftFunctor (CochainComplex C ℤ) x).mapIso (K.totalShift₂Iso y) =
(x ... | Mathlib/Algebra/Homology/TotalComplexShift.lean | 351 | 374 |
/-
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.Group.Defs
import Mathlib.Algebra.GroupWithZero.Defs
import Mathlib.Data.I... | alias add_pow_two := add_sq
| Mathlib/Algebra/Ring/Defs.lean | 234 | 234 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Topology.MetricSpace.IsometricSMul
/-!
# Hausdorff distance
The Hausdorff distance on subsets of a... | theorem infEdist_closure_pos_iff_not_mem_closure {x : α} {E : Set α} :
0 < infEdist x (closure E) ↔ x ∉ closure E := by
rw [infEdist_closure, infEdist_pos_iff_not_mem_closure]
theorem exists_real_pos_lt_infEdist_of_not_mem_closure {x : α} {E : Set α} (h : x ∉ closure E) :
∃ ε : ℝ, 0 < ε ∧ ENNReal.ofReal ε < ... | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | 171 | 176 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Ordmap.Invariants
/-!
# Verification of `Ordnode`
This file uses the invariants defined in `Mathlib.Data.Ordmap.Invariants` to construct `Ordset... | Mathlib/Data/Ordmap/Ordset.lean | 1,317 | 1,333 | |
/-
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.Preserves.Shapes.Zero
/-!
# Kernels and cokernels
In a category with zero morphisms, the kernel of a morphism `f : X... | instance hasCokernel_epi_comp {X Y : C} (f : X ⟶ Y) [HasCokernel f] {W} (g : W ⟶ X) [Epi g] :
HasCokernel (g ≫ f) :=
⟨⟨{ cocone := _
isColimit := isCokernelEpiComp (colimit.isColimit _) g rfl }⟩⟩
| Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean | 909 | 912 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Localization.Opposite
/-!
# Calculus of fractions
Following the definitions by [Gabriel and Zisman][gabriel-zisman-1967],
given a morphism prope... | z₁.map L (Localization.inverts L W) ≫ z₂.map L (Localization.inverts L W) =
(z₁.comp₀ z₂ z₃).map L (Localization.inverts L W) := by
have := Localization.inverts L W _ z₂.hs
have := Localization.inverts L W _ z₃.hs
have : IsIso (L.map (z₂.s ≫ z₃.s)) := by
rw [L.map_comp]
infer_instance
dsimp [L... | Mathlib/CategoryTheory/Localization/CalculusOfFractions.lean | 696 | 709 |
/-
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.NAry
import Mathlib.Data.Finset.Slice
import Mathlib.Data.Set.Sups
/-!
# Set family operations
This file defines a few binary operations on `... | simp_rw [← disjSups_assoc, disjSups_comm s]
theorem disjSups_right_comm : s ○ t ○ u = s ○ u ○ t := by simp_rw [disjSups_assoc, disjSups_comm]
| Mathlib/Data/Finset/Sups.lean | 493 | 495 |
/-
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, Wen Yang
-/
import Mathlib.LinearAlgebra.Matrix.Transvection
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mat... | exact det_isEmpty
theorem det_of_upperTriangular [LinearOrder m] (h : M.BlockTriangular id) :
M.det = ∏ i : m, M i i := by
haveI : DecidableEq R := Classical.decEq _
| Mathlib/LinearAlgebra/Matrix/Block.lean | 274 | 278 |
/-
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.Equiv.Basic
import Mathlib.Algebra.Group.Equiv.Opposite
import Mathlib.Algebra.Group.TypeTags.Basic
/-!
# Squares and even elements
This ... | @[to_additive Even.zsmul_right] lemma IsSquare.zpow (n : ℤ) : IsSquare a → IsSquare (a ^ n) := by
rintro ⟨r, rfl⟩; exact ⟨r ^ n, (Commute.refl _).mul_zpow _⟩
end DivisionMonoid
| Mathlib/Algebra/Group/Even.lean | 150 | 154 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.RingTheory.IntegralClosure.IntegrallyClosed
import Mathlib.RingTheory.Localization.NumDen
import Mathlib.RingTheory.Polynomial.ScaleRoots
/-!
# Rational roo... | (hr : aeval (mk' S r s) p = 0) : aeval (algebraMap A S r) (scaleRoots p s) = 0 := by
convert scaleRoots_eval₂_eq_zero (algebraMap A S) hr
-- Porting note: added
funext
rw [aeval_def, mk'_spec' _ r s]
| Mathlib/RingTheory/Polynomial/RationalRoot.lean | 39 | 44 |
/-
Copyright (c) 2021 Aaron Anderson, Jesse Michael Han, Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jesse Michael Han, Floris van Doorn
-/
import Mathlib.Data.Finset.Basic
import Mathlib.ModelTheory.Syntax
import Mathlib.Data.List.... | def ElementarilyEquivalent : Prop :=
L.completeTheory M = L.completeTheory N
@[inherit_doc FirstOrder.Language.ElementarilyEquivalent]
scoped[FirstOrder]
notation:25 A " ≅[" L "] " B:50 => FirstOrder.Language.ElementarilyEquivalent L A B
| Mathlib/ModelTheory/Semantics.lean | 652 | 657 |
/-
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.Order.Filter.Cofinite
/-!
# Computational realization of filters (experimental)
This file provides infrastructure to compute with filters.
## Main d... | Mathlib/Data/Analysis/Filter.lean | 347 | 352 | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Order.Group.Finset
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Algebra.Polynomial.Eva... | (by rw [degree_mul_leadingCoeff_inv _ hq0]; exact hq)
theorem isUnit_map [Field k] (f : R →+* k) : IsUnit (p.map f) ↔ IsUnit p := by
simp_rw [isUnit_iff_degree_eq_zero, degree_map]
theorem map_div [Field k] (f : R →+* k) : (p / q).map f = p.map f / q.map f := by
| Mathlib/Algebra/Polynomial/FieldDivision.lean | 391 | 396 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Analysis.InnerProductSpace.Continuous
import Mathlib.Analysis.Normed.Module.Dual
import Mathlib.MeasureTheory.Function.AEEqOfLIntegral
import Mathlib.Measu... | Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean | 667 | 687 | |
/-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.Mul
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
imp... | · simp
have h := rpow_add_rpow_le a b hp_pos hp1
rw [one_div_one, one_div] at h
repeat' rw [NNReal.rpow_one] at h
exact (NNReal.le_rpow_inv_iff hp_pos).mp h
end NNReal
namespace Real
lemma add_rpow_le_rpow_add {p : ℝ} {a b : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b) (hp1 : 1 ≤ p) :
a ^ p + b ^ p ≤ (a + b) ^ p := b... | Mathlib/Analysis/MeanInequalitiesPow.lean | 179 | 193 |
/-
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... |
theorem smul_X_eq_monomial {n} : a • X ^ n = monomial n (a : R) := by
rw [X_pow_eq_monomial, smul_monomial, smul_eq_mul, mul_one]
| Mathlib/Algebra/Polynomial/Basic.lean | 808 | 811 |
/-
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... |
theorem Option.comp_traverse {α β γ} (f : β → F γ) (g : α → G β) (x : Option α) :
| Mathlib/Control/Traversable/Instances.lean | 31 | 32 |
/-
Copyright (c) 2022 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Field.Defs
import Mathlib.Algebra.GroupWithZero.Invertible
import Mathlib.Data.Sigma.Basic
import Mathlib.Algebra.Ring.Nat
import Mathlib.Data.... | | _, _, ⟨inv, rfl⟩ => have := @invertibleOne α _; ⟨by simp⟩
| Mathlib/Tactic/NormNum/Result.lean | 218 | 219 |
/-
Copyright (c) 2018 Andreas Swerdlow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andreas Swerdlow
-/
import Mathlib.LinearAlgebra.Basis.Basic
import Mathlib.LinearAlgebra.BilinearMap
import Mathlib.LinearAlgebra.LinearIndependent.Lemmas
/-!
# Sesquilinear maps
... |
/-! ### Self-adjoint pairs -/
| Mathlib/LinearAlgebra/SesquilinearForm.lean | 499 | 501 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl, Sander Dahmen, Kim Morrison, Chris Hughes, Anne Baanen
-/
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
import Mathlib.LinearAlgebra.Basis.Prod
impo... | section SubmoduleRank
section
| Mathlib/LinearAlgebra/Dimension/Constructions.lean | 390 | 392 |
/-
Copyright (c) 2021 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Sébastien Gouëzel
-/
import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
import Mathlib.MeasureTheory.Constructions.BorelSpace.Metric
import Mathlib.MeasureTheory... | have : closedBall (0 : E) r = (x + ·) ⁻¹' closedBall x r := by simp [preimage_add_closedBall]
rw [this, measure_preimage_add]
theorem addHaar_real_closedBall_center {E : Type*} [NormedAddCommGroup E] [MeasurableSpace E]
[BorelSpace E] (μ : Measure E) [IsAddHaarMeasure μ] (x : E) (r : ℝ) :
μ.real (closedBal... | Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean | 426 | 432 |
/-
Copyright (c) 2019 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Algebra.Order.Group.Pointwise.Interval
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.Rat.Cardinal
import Mathlib.SetTheory.Cardi... | ∑' n, cantorFunctionAux c f n
theorem cantorFunction_le (h1 : 0 ≤ c) (h2 : c < 1) (h3 : ∀ n, f n → g n) :
cantorFunction c f ≤ cantorFunction c g := by
| Mathlib/Data/Real/Cardinality.lean | 93 | 96 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Simon Hudon, Mario Carneiro
-/
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Init
import Mathlib.Data.Int.Init
import Mathlib.... |
@[to_additive (attr := simp) neg_zsmul]
| Mathlib/Algebra/Group/Basic.lean | 450 | 451 |
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Mathlib.Control.Basic
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Option.Basic
im... | Mathlib/Data/List/Basic.lean | 2,552 | 2,554 | |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Oliver Nash
-/
import Mathlib.Topology.PartialHomeomorph
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.D... | exact subset_univ _
@[simp]
theorem univBall_apply_zero (c : P) (r : ℝ) : univBall c r 0 = c := by
unfold univBall; split_ifs <;> simp
| Mathlib/Analysis/NormedSpace/HomeomorphBall.lean | 133 | 137 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.Order.Sub.WithTop
import Mathlib.Data.NNReal.Defs
import Mathlib.Order.Interval.Set.... | theorem le_of_top_imp_top_of_toNNReal_le {a b : ℝ≥0∞} (h : a = ⊤ → b = ⊤)
(h_nnreal : a ≠ ⊤ → b ≠ ⊤ → a.toNNReal ≤ b.toNNReal) : a ≤ b := by
by_contra! hlt
lift b to ℝ≥0 using hlt.ne_top
lift a to ℝ≥0 using mt h coe_ne_top
refine hlt.not_le ?_
simpa using h_nnreal
| Mathlib/Data/ENNReal/Basic.lean | 650 | 656 |
/-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Data.Set.Finite.Lemmas
import Mathlib.ModelTheory.Substructures
/-!
# Finitely Generated First-Order Structures
This file defines what it means for a... | theorem CG.sup {N₁ N₂ : L.Substructure M} (hN₁ : N₁.CG) (hN₂ : N₂.CG) : (N₁ ⊔ N₂).CG :=
let ⟨t₁, ht₁⟩ := cg_def.1 hN₁
let ⟨t₂, ht₂⟩ := cg_def.1 hN₂
cg_def.2 ⟨t₁ ∪ t₂, ht₁.1.union ht₂.1, by rw [closure_union, ht₁.2, ht₂.2]⟩
| Mathlib/ModelTheory/FinitelyGenerated.lean | 150 | 153 |
/-
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... |
This is a specialization of `EquicontinuousOn.isClosed_range_pi_of_uniformOnFun'` to the case where
`𝔖` covers `X`. -/
theorem EquicontinuousOn.isClosed_range_uniformOnFun_iff_pi
{𝔖 : Set (Set X)} (𝔖_compact : ∀ K ∈ 𝔖, IsCompact K) (𝔖_covers : ⋃₀ 𝔖 = univ)
(F_eqcont : ∀ K ∈ 𝔖, EquicontinuousOn F K) :
... | Mathlib/Topology/UniformSpace/Ascoli.lean | 390 | 399 |
/-
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 | 362 | 366 | |
/-
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.LinearAlgebra.Matrix.ToLin
import Mathlib.LinearAlgebra.Quotient.Basic
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.RingTheory.Nilpotent.Defs
/-!
# N... | lemma isNilpotent.restrict
{f : M →ₗ[R] M} {p : Submodule R M} (hf : MapsTo f p p) (hnil : IsNilpotent f) :
IsNilpotent (f.restrict hf) := by
obtain ⟨n, hn⟩ := hnil
exact ⟨n, LinearMap.ext fun m ↦ by simp only [Module.End.pow_restrict n, hn,
| Mathlib/RingTheory/Nilpotent/Lemmas.lean | 114 | 118 |
/-
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.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.LinearAlgebra.Matrix.ToLin
/-!
# Cayley-Hamilton theorem for f.g. modules.
Given a fixed finite spannin... | variable (hb : Submodule.span R (Set.range b) = ⊤)
include hb
| Mathlib/LinearAlgebra/Matrix/Charpoly/LinearMap.lean | 147 | 149 |
/-
Copyright (c) 2022 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll, Thomas Zhu, Mario Carneiro
-/
import Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity
/-!
# The Jacobi Symbol
We define the Jacobi symbol and prove its main pro... | · simp_rw [qrSign]
rw [χ₄_nat_mod_four, χ₄_nat_mod_four (b % (4 * a)), mod_mod_of_dvd b (dvd_mul_right 4 a)]
· rw [mod_left ↑(b % _), mod_left b, Int.natCast_mod, Int.emod_emod_of_dvd b]
| Mathlib/NumberTheory/LegendreSymbol/JacobiSymbol.lean | 464 | 466 |
/-
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.Algebra.BigOperators.Fin
import Mathlib.Algebra.Group.Fin.Tuple
import Mathlib.Data.Finset.NatAntidiagonal
import Mathlib.Order.Fin.Tuple
/-!
# Collections ... | | 0, 0 => List.pairwise_singleton _ _
| 0, _ + 1 => List.Pairwise.nil
| k + 1, n => by
simp_rw [antidiagonalTuple, List.pairwise_flatMap, List.pairwise_map, List.mem_map,
forall_exists_index, and_imp, forall_apply_eq_imp_iff₂]
simp only [mem_antidiagonal, Prod.forall, and_imp, forall_apply_eq_imp_if... | Mathlib/Data/Fin/Tuple/NatAntidiagonal.lean | 140 | 145 |
/-
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.Data.Countable.Defs
import Mathlib.Data.Nat.Factors
import Mathlib.Data.Nat.Prime.Infinite
import Mathlib.Data... |
@[simp] lemma disjoint_primeFactors (ha : a ≠ 0) (hb : b ≠ 0) :
Disjoint a.primeFactors b.primeFactors ↔ Coprime a b := by
| Mathlib/Data/Nat/PrimeFin.lean | 93 | 95 |
/-
Copyright (c) 2023 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.Algebra.Polynomial.Bivariate
import Mathlib.AlgebraicGeometry.EllipticCurve.Weierstrass
import Mathlib.AlgebraicGeometry.EllipticCu... | simp only [zero_add, add_zero, sub_zero, zero_mul, mul_one]
eval_simp
linear_combination (norm := (norm_num1; ring1)) hx.left + ℓ * hx.right
| Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean | 436 | 439 |
/-
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... |
theorem cocompact_le_coclosedCompact : cocompact X ≤ coclosedCompact X :=
iInf_mono fun _ => le_iInf fun _ => le_rfl
end Filter
theorem IsCompact.compl_mem_coclosedCompact_of_isClosed (hs : IsCompact s) (hs' : IsClosed s) :
sᶜ ∈ Filter.coclosedCompact X :=
hasBasis_coclosedCompact.mem_of_mem ⟨hs', hs⟩
| Mathlib/Topology/Compactness/Compact.lean | 587 | 595 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.BoxIntegral.Partition.Filter
import Mathlib.Analysis.BoxIntegral.Partition.Measure
import Mathlib.Analysis.Oscillation
import Mathlib.Data.B... | ((l.toFilter I ×ˢ l.toFilter I) ⊓ 𝓟 {π | π.1.iUnion = π.2.iUnion}) (𝓤 F) := by
refine (((l.hasBasis_toFilter I).prod_self.inf_principal _).tendsto_iff
uniformity_basis_dist_le).2 fun ε ε0 => ?_
replace ε0 := half_pos ε0
use h.convergenceR (ε / 2), h.convergenceR_cond (ε / 2); rintro ⟨π₁, π₂⟩ ⟨⟨h₁, h... | Mathlib/Analysis/BoxIntegral/Basic.lean | 455 | 459 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Sophie Morel, Yury Kudryashov
-/
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
import Mathlib.Logic.Embedding.Basic
import Mathlib.Data.Fintype.Car... | simpa only [div_self hc₀.ne'] using hcx i
_ = 1 * ∏ i, ‖x i‖ := (one_mul _).symm
· rcases (NormedSpace.isVonNBounded_iff' _).1 hs with ⟨ε, hε⟩
rcases exists_pos_mul_lt hr (ε ^ Fintype.card ι) with ⟨δ, hδ₀, hδ⟩
refine ⟨δ, hδ₀, fun f hf x hx ↦ ?_⟩
simp only [Seminorm.mem_ball_zero, mem_closedB... | Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean | 472 | 501 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Piecewise
import Mathlib.Algebra.Group.Ext
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.BinaryProducts
import Ma... | section Finite
variable {J : Type} [Finite J]
/-- A functor between preadditive categories that preserves (zero morphisms and) finite biproducts
preserves finite products. -/
lemma preservesProduct_of_preservesBiproduct {f : J → C} [PreservesBiproduct f F] :
PreservesLimit (Discrete.functor f) F where
prese... | Mathlib/CategoryTheory/Preadditive/Biproducts.lean | 852 | 875 |
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.Constructions
/-!
# Neighborhoods and continuity relative to a subset
This file develops API on the relative versions
* `nhdsWithin` ... | 𝓝[pi I s] x = ⨅ i, comap (fun x => x i) (𝓝 (x i) ⊓ ⨅ (_ : i ∈ I), 𝓟 (s i)) := by
simp only [nhdsWithin, nhds_pi, Filter.pi, comap_inf, comap_iInf, pi_def, comap_principal, ←
iInf_principal_finite hI, ← iInf_inf_eq]
| Mathlib/Topology/ContinuousOn.lean | 310 | 313 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Minchao Wu, Mario Carneiro
-/
import Mathlib.Data.Finset.Attach
import Mathlib.Data.Finset.Disjoint
import Mathlib.Data.Finset.Erase
import Mat... | Mathlib/Data/Finset/Basic.lean | 2,785 | 2,786 | |
/-
Copyright (c) 2022 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Yury Kudryashov, Kevin H. Wilson, Heather Macbeth
-/
import Mathlib.Order.Filter.Tendsto
/-!
# Product and coproduct filters
In this file we define `F... | haveI := neBot_of_le hle.1; haveI := neBot_of_le hle.2
exact ⟨hle.1.antisymm <| (prod_le_prod.1 h.ge).1, hle.2.antisymm <| (prod_le_prod.1 h.ge).2⟩
theorem eventually_swap_iff {p : α × β → Prop} :
(∀ᶠ x : α × β in f ×ˢ g, p x) ↔ ∀ᶠ y : β × α in g ×ˢ f, p y.swap := by
| Mathlib/Order/Filter/Prod.lean | 277 | 281 |
/-
Copyright (c) 2020 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Alexey Soloyev, Junyan Xu, Kamila Szewczyk
-/
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.Algebra.LinearRecurrence
import Mathlib.Data.Fin.VecNotati... | theorem one_sub_gold : 1 - ψ = φ := by
linarith [gold_add_goldConj]
| Mathlib/Data/Real/GoldenRatio.lean | 70 | 72 |
/-
Copyright (c) 2023 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Topology.AlexandrovDiscrete
import Mathlib.Topology.ContinuousMap.Basic
import Mathlib.Topology.Order.LowerUpperTopology
/-!
# Upper and lower... | intro s hs
exact (IsUpper.isUpperSet_of_isOpen hs).preimage hf
lemma upperSet_le_upper {t₁ t₂ : TopologicalSpace α} [@Topology.IsUpperSet α t₁ _]
| Mathlib/Topology/Order/UpperLowerSetTopology.lean | 279 | 282 |
/-
Copyright (c) 2022 Benjamin Davidson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Benjamin Davidson, Devon Tuma, Eric Rodriguez, Oliver Nash
-/
import Mathlib.Algebra.Order.Group.Pointwise.Interval
import Mathlib.Order.Filter.AtTopBot.Field
import Mathlib.Topolog... | tendsto_const_nhds.div_atBot hg
theorem Filter.Tendsto.inv_tendsto_atTop (h : Tendsto f l atTop) : Tendsto f⁻¹ l (𝓝 0) :=
tendsto_inv_atTop_zero.comp h
theorem Filter.Tendsto.inv_tendsto_atBot (h : Tendsto f l atBot) : Tendsto f⁻¹ l (𝓝 0) :=
tendsto_inv_atBot_zero.comp h
theorem Filter.Tendsto.inv_tendsto_nh... | Mathlib/Topology/Algebra/Order/Field.lean | 194 | 207 |
/-
Copyright (c) 2023 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.Logic.Small.Set
import Mathlib.CategoryTheory.Filtered.Final
/-!
# Finally small categories
A category given by `(J : Type u) [Category.{v} J]` is `w`-... |
variable (J : Type u) [Category.{v} J]
/-- The converse is true if `J` is cofiltered, see `initiallySmall_of_small_weakly_initial_set`. -/
theorem InitiallySmall.exists_small_weakly_initial_set [InitiallySmall.{w} J] :
| Mathlib/CategoryTheory/Limits/FinallySmall.lean | 160 | 164 |
/-
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... | closed_C := k_closed
open_U := ht.isOpen_compl
subset := kt
hP := by
rintro c u - c_comp u_open cu
rcases exists_compact_closed_between c_comp u_open cu with ⟨k, k_comp, k_closed, ck, ku⟩
have A : closure (interior k) ⊆ k :=
(IsClosed.closure_subset_iff k_closed).2 interior_sub... | Mathlib/Topology/UrysohnsLemma.lean | 354 | 375 |
/-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.Algebra.Order.Group.OrderIso
import Mathlib.SetTheory.Game.Ordinal
import Mathlib.SetTheory.Ordinal.NaturalOps
/-!
# Birthdays... | · rintro rfl
| Mathlib/SetTheory/Game/Birthday.lean | 201 | 201 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno
import Mathlib.Analysis.Calculus.FDeriv.Ad... | Mathlib/Analysis/Calculus/ContDiff/Basic.lean | 1,669 | 1,672 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Floris van Doorn, Sébastien Gouëzel, Alex J. Best
-/
import Mathlib.Algebra.GroupWithZero.Commute
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algeb... | | [] => by simp
| a :: l => by rw [prod_cons, mul_eq_zero, prod_eq_zero_iff, mem_cons, eq_comm]
lemma prod_ne_zero (hL : (0 : M₀) ∉ l) : l.prod ≠ 0 := mt prod_eq_zero_iff.1 hL
| Mathlib/Algebra/BigOperators/Ring/List.lean | 67 | 71 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Algebra.Order.Pi
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
/-!
# Simple functions
A function `f` from a measurable ... | Mathlib/MeasureTheory/Function/SimpleFunc.lean | 1,343 | 1,364 | |
/-
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.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Order.Interval.Finset.Na... | Mathlib/Topology/EMetricSpace/Basic.lean | 574 | 575 | |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Data.Complex.Trigonometric
import Mathlib.Data.Complex.Module
import Mathlib.RingTheory.Polynomial.Chebyshev
/-!
# Multiple angle formulas in terms of... | open Complex
variable (θ : ℂ)
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Chebyshev.lean | 45 | 47 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Group.Subgroup.Ker
import Mathlib.Algebra.BigOperators.Group.List.Basic
/-!
# Free groups
This file defines free groups over a type. Furthermore, it is... | theorem church_rosser : Red L₁ L₂ → Red L₁ L₃ → Join Red L₂ L₃ :=
Relation.church_rosser fun _ b c hab hac =>
match b, c, Red.Step.diamond hab hac rfl with
| Mathlib/GroupTheory/FreeGroup/Basic.lean | 190 | 192 |
/-
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.Encodable.Lattice
import Mathlib.Order.Filter.AtTopBot.Finset
import Mathlib.Topology.Algebra.InfiniteSum.Group
/-!
# Infinite sums and product... | /-- If a function is countably sub-multiplicative then it is sub-multiplicative on countable
types -/
@[to_additive "If a function is countably sub-additive then it is sub-additive on countable types"]
theorem rel_iSup_tprod [CompleteLattice α] (m : α → M) (m0 : m ⊥ = 1) (R : M → M → Prop)
(m_iSup : ∀ s : ℕ → α, R ... | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | 155 | 160 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Joey van Langen, Casper Putz
-/
import Mathlib.Data.Nat.Cast.Basic
import Mathlib.Data.Nat.Find
import Mathlib.Data.Nat.Prime.Defs
import Mathlib.Order.Lattice
/-!
# Characteris... | lemma expChar_pos (q : ℕ) [ExpChar R q] : 0 < q := by
rcases expChar_is_prime_or_one R q with h | rfl
exacts [Nat.Prime.pos h, Nat.one_pos]
/-- Any power of the exponential characteristic is positive. -/
| Mathlib/Algebra/CharP/Defs.lean | 356 | 360 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.RingTheory.WittVector.StructurePolynomial
/-!
# Witt vectors
In this file we define the type of `p`-typical Witt vectors and ring op... | theorem one_coeff_eq_of_pos (n : ℕ) (hn : 0 < n) : coeff (1 : 𝕎 R) n = 0 :=
show (aeval _ (wittOne p n) : R) = 0 by simp only [hn, wittOne_pos_eq_zero, map_zero]
| Mathlib/RingTheory/WittVector/Defs.lean | 298 | 300 |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.IsPrimePow
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Ring.Int
import Mathlib.Algebra.Ring.CharZero
im... | theorem properDivisors.not_self_mem : ¬n ∈ properDivisors n := by simp [properDivisors]
@[simp]
| Mathlib/NumberTheory/Divisors.lean | 84 | 86 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.AEEqOfIntegral
import Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable
/-!
# Uniqueness of the conditional expect... | (hf_zero : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, (f : Lp E' p μ) x ∂μ = 0) :
f =ᵐ[μ] (0 : α → E') := by
obtain ⟨g, hg_sm, hfg⟩ := lpMeas.ae_fin_strongly_measurable' hm f hp_ne_zero hp_ne_top
refine hfg.trans ?_
refine ae_eq_zero_of_forall_setIntegral_eq_of_finStronglyMeasurable_trim hm ?_ ... | Mathlib/MeasureTheory/Function/ConditionalExpectation/Unique.lean | 49 | 67 |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.WSeq.Basic
import Mathlib.Data.WSeq.Defs
import Mathlib.Data.WSeq.Productive
import Mathlib.Data.WSeq.Relation
deprecated_module (since :=... | Mathlib/Data/Seq/WSeq.lean | 1,303 | 1,327 | |
/-
Copyright (c) 2024 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Category.ModuleCat.Presheaf.Abelian
import Mathlib.Algebra.Category.ModuleCat.Presheaf.Sheafify
import Mathlib.Algebra.Category.ModuleCat.Presheaf.Limits... | apply (toPresheaf _).map_injective
erw [toPresheaf_map_sheafificationHomEquiv] }
lemma sheafificationAdjunction_homEquiv_apply {P : PresheafOfModules.{v} R₀}
| Mathlib/Algebra/Category/ModuleCat/Presheaf/Sheafification.lean | 125 | 128 |
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