Context stringlengths 227 76.5k | target stringlengths 0 11.6k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 16 3.69k |
|---|---|---|---|---|
/-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Data.Fintype.Card
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
/-!
# Multiset coercion to type
This module defines a `CoeSort` instance for multi... | imp_not_comm, not_le, Nat.lt_sub_iff_add_lt] using h
have : card (s.1.filter fun x ↦ a = x.1) ≤ card (s.1.filter fun x ↦ a = x.1) - 1 := by
simpa [Finset.card, eq_comm] using Finset.card_mono h
omega
exact Nat.le_of_pred_lt (han.trans_lt <| by simpa using hsm hn)
| Mathlib/Data/Multiset/Fintype.lean | 122 | 126 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Order.CauSeq.BigOperators
import Mathlib.Algebra.Order.Star.Basic
import Mathlib.Data.C... | Mathlib/Data/Complex/Exponential.lean | 984 | 985 | |
/-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Devon Tuma
-/
import Mathlib.Probability.ProbabilityMassFunction.Basic
/-!
# Monad Operations for Probability Mass Functions
This file constructs two operations on `PMF` ... | (p.bindOnSupport fun a _ => f a) = p.bind f := by
ext b
have : ∀ a, ite (p a = 0) 0 (p a * f a b) = p a * f a b :=
fun a => ite_eq_right_iff.2 fun h => h.symm ▸ symm (zero_mul <| f a b)
simp only [bindOnSupport_apply fun a _ => f a, p.bind_apply f, dite_eq_ite, mul_ite,
mul_zero, this]
theorem bindOn... | Mathlib/Probability/ProbabilityMassFunction/Monad.lean | 225 | 236 |
/-
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, Kim Morrison
-/
import Mathlib.Algebra.Algebra.Equiv
import Mathlib.Algebra.Algebra.NonUnitalHom
import Mathlib.Algebra.Module.BigOperators
import Math... | Mathlib/Algebra/MonoidAlgebra/Basic.lean | 609 | 619 | |
/-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth, Eric Wieser
-/
import Mathlib.Analysis.Normed.Lp.PiLp
import Mathlib.Analysis.InnerProductSpace.PiL2
/-!
# Matrices as a normed space
In this file we provide the fo... | @[deprecated (since := "2025-03-20")] alias linfty_opNorm_row := linfty_opNNNorm_replicateRow
@[simp]
| Mathlib/Analysis/Matrix.lean | 288 | 290 |
/-
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.Logic.Encodable.Pi
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Topology.MetricSpace.Closeds
import Mathlib.Topology.MetricSpace.Comple... | have : hausdorffDist (range Φ) (range Ψ) = hausdorffDist (range Φ') (range Ψ') := by
rw [ΦΦ', ΨΨ', range_comp, range_comp]
exact hausdorffDist_image isometry_subtype_coe
rw [this]
-- Embed `s` in `ℓ^∞(ℝ)` through its Kuratowski embedding
let F := kuratowskiEmbedding (Subtype s)
have : hausdorffDist (F... | Mathlib/Topology/MetricSpace/GromovHausdorff.lean | 196 | 248 |
/-
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, Yuyang Zhao
-/
import Mathlib.Algebra.Group.Units.Basic
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra... | section PartialOrder
variable [PartialOrder α] [CanonicallyOrderedMul α] {a b c : α}
@[to_additive] instance CanonicallyOrderedCommMonoid.toUniqueUnits : Unique αˣ where
| Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean | 257 | 260 |
/-
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, Yaël Dillies, Yuyang Zhao
-/
import Mathlib.Algebra.Order.Ring.Unbundled.Basic
import Mathlib.Algebra.CharZero.Defs
import Mathlib.Alge... | /-- The product of non-negative elements is non-negative. -/
protected mul_nonneg : ∀ a b : R, 0 ≤ a → 0 ≤ b → 0 ≤ a * b
set_option linter.deprecated false in
/-- An `OrderedCommRing` is a commutative ring with a partial order such that addition is monotone
| Mathlib/Algebra/Order/Ring/Defs.lean | 265 | 269 |
/-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.JapaneseBracket
import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.MeasureTheory.Group.Integral
import Mathlib... | H.mono (Set.Ioi_subset_Ioi (le_max_right _ _)) le_rfl
have : IntegrableOn (fun x ↦ x⁻¹) (Ioi (max 1 t)) := by
apply H'.mono' measurable_inv.aestronglyMeasurable
filter_upwards [ae_restrict_mem measurableSet_Ioi] with x hx
have x_one : 1 ≤ x := ((le_max_left _ _).trans_lt (mem_Ioi.1 hx)).le
simp on... | Mathlib/Analysis/SpecialFunctions/ImproperIntegrals.lean | 130 | 138 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Sébastien Gouëzel,
Rémy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
/-!
# Power function... | by_cases hx0 : x = 0
· rw [← Ne] at hp0
have hp_pos : 0 < p := lt_of_le_of_ne hp_nonneg hp0.symm
simp [hx0, hp_pos, hp_pos.ne.symm]
rw [← Ne] at hx0
exact ofReal_rpow_of_pos (hx_nonneg.lt_of_ne hx0.symm)
@[simp] lemma rpow_rpow_inv {y : ℝ} (hy : y ≠ 0) (x : ℝ≥0∞) : (x ^ y) ^ y⁻¹ = x := by
rw [← rpow_... | Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | 896 | 905 |
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Mario Carneiro, Simon Hudon
-/
import Mathlib.Data.Fin.Fin2
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.Common
/-!
# Tuples of types, and their categorica... | instance Arrow.inhabited (α β : TypeVec n) [∀ i, Inhabited (β i)] : Inhabited (α ⟹ β) :=
⟨fun _ _ => default⟩
| Mathlib/Data/TypeVec.lean | 60 | 62 |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
/-!
# Measurability criterion for ennreal-valued functions
Consider a function `f : α → ℝ≥0∞`. If ... | [BorelSpace β] (s : Set β) (s_count : s.Countable) (s_dense : Dense s) (f : α → β)
(h : ∀ p ∈ s, ∀ q ∈ s, p < q → ∃ u v, MeasurableSet u ∧ MeasurableSet v ∧
{ x | f x < p } ⊆ u ∧ { x | q < f x } ⊆ v ∧ μ (u ∩ v) = 0) :
AEMeasurable f μ := by
classical
haveI : Encodable s := s_count.toEncodable
ha... | Mathlib/MeasureTheory/Function/AEMeasurableOrder.lean | 34 | 107 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Johan Commelin, Andrew Yang, Joël Riou
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
import Mathlib.CategoryTheory.Monoid... | Category.assoc, ← Functor.map_comp_assoc, Iso.inv_hom_id_app, Functor.map_id,
Category.id_comp, eqToHom_trans, eqToHom_refl]
lemma add_zero_inv_app (h : ShiftMkCore C A) (n : A) (X : C) :
(h.add n 0).inv.app X = h.zero.hom.app ((h.F n).obj X) ≫
eqToHom (by dsimp; rw [add_zero]) := by
| Mathlib/CategoryTheory/Shift/Basic.lean | 105 | 110 |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Sign
import Mathlib.LinearAlgebra.AffineSpace.Combination
import Mathlib.LinearAlg... | two subsets of the points of a simplex are equal if and only if those
faces are given by the same subset of points. -/
| Mathlib/LinearAlgebra/AffineSpace/Independent.lean | 913 | 914 |
/-
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
... |
/-- The restriction of a reflexive bilinear map `B` onto a submodule `W` is
nondegenerate if `W` has trivial intersection with its orthogonal complement,
that is `Disjoint W (W.orthogonalBilin B)`. -/
theorem nondegenerate_restrict_of_disjoint_orthogonal {B : M →ₗ[R] M →ₗ[R] M₁} (hB : B.IsRefl)
| Mathlib/LinearAlgebra/SesquilinearForm.lean | 749 | 753 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.RingTheory.Localization.Integer
import Mathlib.RingTheory.Localization.Submodule
/-!
# Fractional ideals
This file defines fractional... | · rintro ⟨y, hy, eq_y⟩
rwa [← eq_y]
· intro hx
| Mathlib/RingTheory/FractionalIdeal/Basic.lean | 636 | 638 |
/-
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... |
theorem Valid'.balance' {l} {x : α} {r o₁ o₂} (hl : Valid' o₁ l x) (hr : Valid' x r o₂)
(H : ∃ l' r', BalancedSz l' r' ∧
| Mathlib/Data/Ordmap/Ordset.lean | 326 | 328 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kenny Lau
-/
import Mathlib.Data.List.Forall2
/-!
# zip & unzip
This file provides results about `List.zipWith`, `List.zip` and `List.unzip` (definitions are in
core ... | | a :: l₁, b :: l₂, h => by
simp only [length_cons, succ_inj] at h
simp [forall_zipWith h]
theorem unzip_swap (l : List (α × β)) : unzip (l.map Prod.swap) = (unzip l).swap := by
| Mathlib/Data/List/Zip.lean | 43 | 47 |
/-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.SimpleGraph.Triangle.Basic
/-!
# Construct a tripartite graph from its triangles
This file contains the constru... | ext; exact is3Clique_iff
| Mathlib/Combinatorics/SimpleGraph/Triangle/Tripartite.lean | 208 | 209 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Yury Kudryashov
-/
import Mathlib.Analysis.Convex.Combination
import Mathlib.Analysis.Convex.Strict
import Mathlib.Topology.Algebra.Affine
import Mathlib.Topology.... |
variable [Ring 𝕜] [LinearOrder 𝕜] [IsStrictOrderedRing 𝕜] [DenselyOrdered 𝕜]
[PseudoMetricSpace 𝕜] [OrderTopology 𝕜]
[ProperSpace 𝕜] [CompactIccSpace 𝕜] [AddCommGroup E] [TopologicalSpace E] [T2Space E]
| Mathlib/Analysis/Convex/Topology.lean | 109 | 112 |
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.Group.Support
import Mathlib.Algebra.Order.Monoid.Unbundled.WithTop
import Mathlib.Order.WellFoundedSet
/-!
# Hahn Series
If `Γ` is ordered an... | def leadingCoeff (x : HahnSeries Γ R) : R :=
if h : x = 0 then 0 else x.coeff (x.isWF_support.min (support_nonempty_iff.2 h))
| Mathlib/RingTheory/HahnSeries/Basic.lean | 313 | 315 |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.Real
/-!
# Properties of pointwise scalar multiplication of se... | Mathlib/Analysis/NormedSpace/Pointwise.lean | 423 | 432 | |
/-
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 | 548 | 548 | |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou, Johan Commelin
-/
import Mathlib.Algebra.Ring.Int.Parity
import Mathlib.Algebra.Ring.Int.Units
import Mathlib.Data.ZMod.IntUnitsPower
/-!
# Integer powers of (-1)
This file def... | lemma negOnePow_neg (n : ℤ) : (-n).negOnePow = n.negOnePow := by
dsimp [negOnePow]
simp only [zpow_neg, ← inv_zpow, inv_neg, inv_one]
| Mathlib/Algebra/Ring/NegOnePow.lean | 82 | 84 |
/-
Copyright (c) 2023 Ali Ramsey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ali Ramsey, Eric Wieser
-/
import Mathlib.Algebra.Algebra.Bilinear
import Mathlib.LinearAlgebra.DFinsupp
import Mathlib.LinearAlgebra.Prod
import Mathlib.LinearAlgebra.TensorProduct.Finite... | ∑ i ∈ repr.index, counit (R := R) (repr.left i) ⊗ₜ f (repr.right i) = 1 ⊗ₜ[R] f a := by
have := sum_counit_tmul_eq repr
apply_fun LinearMap.lTensor R (f : A →ₗ[R] B) at this
simp_all only [map_sum, LinearMap.lTensor_tmul, LinearMap.coe_coe]
@[simp]
theorem sum_map_tmul_counit_eq {B : Type*} [AddCommMonoid B]... | Mathlib/RingTheory/Coalgebra/Basic.lean | 137 | 143 |
/-
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... | · exact ⟨n, Or.inr (by simp [hn])⟩
· rw [Set.mem_singleton_iff] at hb
rw [hb]
exact ⟨-n, Or.inr (by simp [hn])⟩
end IsFundamental
open Solution₁ IsFundamental
| Mathlib/NumberTheory/Pell.lean | 626 | 634 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Johan Commelin
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Combinatorics.Enumerative.Composition
/-!
# Composition of analytic functions
In this fi... | theorem compAlongComposition_nnnorm {n : ℕ} (q : FormalMultilinearSeries 𝕜 F G)
(p : FormalMultilinearSeries 𝕜 E F) (c : Composition n) :
‖q.compAlongComposition p c‖₊ ≤ ‖q c.length‖₊ * ∏ i, ‖p (c.blocksFun i)‖₊ := by
rw [← NNReal.coe_le_coe]; push_cast; exact q.compAlongComposition_norm p c
/-!
### The id... | Mathlib/Analysis/Analytic/Composition.lean | 311 | 326 |
/-
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.Exact
import Mathlib.CategoryTheory.Preadditive.Injective.Basic
/-!
# Short exact short complexes
A short complex `S : ShortCompl... | lemma ShortExact.map_of_exact (hS : S.ShortExact)
(F : C ⥤ D) [F.PreservesZeroMorphisms] [PreservesFiniteLimits F]
[PreservesFiniteColimits F] : (S.map F).ShortExact := by
have := hS.mono_f
have := hS.epi_g
have := preserves_mono_of_preservesLimit F S.f
have := preserves_epi_of_preservesColimit F S.g
... | Mathlib/Algebra/Homology/ShortComplex/ShortExact.lean | 101 | 108 |
/-
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.... | def ascFactorial (n : ℕ) : ℕ → ℕ
| 0 => 1
| k + 1 => (n + k) * ascFactorial n k
@[simp]
theorem ascFactorial_zero (n : ℕ) : n.ascFactorial 0 = 1 :=
rfl
| Mathlib/Data/Nat/Factorial/Basic.lean | 194 | 200 |
/-
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
... | chirality; in addition to this "left" orthogonal complement one could define a "right" orthogonal
complement for which, for all `y` in `N`, `B y x = 0`. This variant definition is not currently
provided in mathlib. -/
def orthogonalBilin (N : Submodule R₁ M₁) (B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] M) : Submodule R₁ M₁ where
car... | Mathlib/LinearAlgebra/SesquilinearForm.lean | 317 | 326 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.Bochner.Basic
import Mathlib.MeasureTheory.Integral.Bochner.L1
import Mathlib.Me... | Mathlib/MeasureTheory/Integral/Bochner.lean | 881 | 886 | |
/-
Copyright (c) 2019 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard
-/
import Mathlib.Data.EReal.Basic
deprecated_module (since := "2025-04-13")
| Mathlib/Data/Real/EReal.lean | 1,513 | 1,521 | |
/-
Copyright (c) 2024 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
/-!
# Angle between complex numbers
This file rel... | rw [angle_comm, angle_div_left_eq_angle_mul_right, angle_comm]
| Mathlib/Analysis/Complex/Angle.lean | 61 | 62 |
/-
Copyright (c) 2020 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.Topology.Path
/-!
# Path connectedness
Continuing from `Mathlib.Topology.Path`, this file defines path components and path-connected
spaces.
## Main... | Mathlib/Topology/Connected/PathConnected.lean | 689 | 691 | |
/-
Copyright (c) 2024 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.RingTheory.FractionalIdeal.Norm
import Mathlib.RingTheory.FractionalIdeal.Operations
/-!
# Fractional ide... | theorem fractionalIdeal_rank (I : (FractionalIdeal (𝓞 K)⁰ K)ˣ) :
finrank ℤ I = finrank ℤ (𝓞 K) := by
rw [finrank_eq_card_chooseBasisIndex, RingOfIntegers.rank,
finrank_eq_card_basis (basisOfFractionalIdeal K I)]
| Mathlib/NumberTheory/NumberField/FractionalIdeal.lean | 93 | 96 |
/-
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.GradedObject.Bifunctor
import Mathlib.CategoryTheory.Functor.Trifunctor
/-!
# The action of trifunctors on graded objects
Given a trifunctor `F. ... | bifunctors can be computed as a composition of the actions of these two bifunctors. -/
noncomputable def mapBifunctorComp₁₂MapObjIso :
mapTrifunctorMapObj (bifunctorComp₁₂ F₁₂ G) r X₁ X₂ X₃ ≅
mapBifunctorMapObj G ρ₁₂.q (mapBifunctorMapObj F₁₂ ρ₁₂.p X₁ X₂) X₃ :=
isoMk _ _ (fun j => (CofanMapObjFun.iso
(isC... | Mathlib/CategoryTheory/GradedObject/Trifunctor.lean | 363 | 370 |
/-
Copyright (c) 2020 Kexing Ying and Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Kevin Buzzard, Yury Kudryashov
-/
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.Algebra.Gro... | @[to_additive
| Mathlib/Algebra/BigOperators/Finprod.lean | 767 | 767 |
/-
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.Basic
import Mathlib.RingTheory.Artinian.Module
/-!
# Lie subalgebras
This file defines Lie subalgebras of a Lie algebra and provides basic rel... | rw [lieSpan, sInf_coe]
exact Set.mem_iInter₂
theorem subset_lieSpan : s ⊆ lieSpan R L s := by
intro m hm
rw [SetLike.mem_coe, mem_lieSpan]
intro K hK
| Mathlib/Algebra/Lie/Subalgebra.lean | 617 | 623 |
/-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois.Basic
/-!
# Cyclotomic extens... | · exact ⟨fun h => h.elim, fun x => by convert (mem_top : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic ex... | Mathlib/NumberTheory/Cyclotomic/Basic.lean | 154 | 168 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.Complex.Asymptotics
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Data.Complex.Trig... | /-- The real exponential function tends to `1` at `0`. -/
theorem tendsto_exp_nhds_zero_nhds_one : Tendsto exp (𝓝 0) (𝓝 1) := by
convert continuous_exp.tendsto 0
| Mathlib/Analysis/SpecialFunctions/Exp.lean | 224 | 226 |
/-
Copyright (c) 2020 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Polynomial.Degree.TrailingDegree
import Mathlib.Algebra.Polynomial.EraseLead
/-!
# Reverse of a univariate polynomial
The main definition is `r... | g.natDegree ≤ O → reflect (N + O) (f * g) = reflect N f * reflect O g := by
induction' cf with cf hcf
| Mathlib/Algebra/Polynomial/Reverse.lean | 160 | 161 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Data.ENNReal.Operations
/-!
# Results about division in extended non-negative reals
This file establishes basic properties related t... | /-- See `ENNReal.mul_div_cancel` for a simpler version assuming `a ≠ 0`, `a ≠ ∞`. -/
protected lemma mul_div_cancel' (ha₀ : a = 0 → b = 0) (ha : a = ∞ → b = 0) : a * (b / a) = b := by
rw [mul_comm, ENNReal.div_mul_cancel' ha₀ ha]
| Mathlib/Data/ENNReal/Inv.lean | 165 | 167 |
/-
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 | 1,319 | 1,319 | |
/-
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.Shift.Adjunction
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
/-!
# The pullback of a shift by a monoid morphism
Given a shift by a... | isomorphism `shiftFunctor (PullbackShift C φ) a ≅ shiftFunctor C b`. -/
noncomputable def pullbackShiftIso (a : A) (b : B) (h : b = φ a) :
shiftFunctor (PullbackShift C φ) a ≅ shiftFunctor C b := eqToIso (by subst h; rfl)
variable {C}
variable (X : PullbackShift C φ) (a₁ a₂ a₃ : A) (h : a₁ + a₂ = a₃) (b₁ b₂ b₃ : B... | Mathlib/CategoryTheory/Shift/Pullback.lean | 70 | 75 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Iterate
import Mathlib.Order.SemiconjSup
import Mathlib.Topology.Order.MonotoneContinuity
import M... | theorem le_iterate_pos_iff {x : ℝ} {m : ℤ} {n : ℕ} (hn : 0 < n) :
x + n * m ≤ f^[n] x ↔ x + m ≤ f x := by
simpa only [not_lt] using not_congr (f.iterate_pos_lt_iff hn)
theorem lt_iterate_pos_iff {x : ℝ} {m : ℤ} {n : ℕ} (hn : 0 < n) :
x + n * m < f^[n] x ↔ x + m < f x := by
simpa only [not_le] using not_con... | Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean | 523 | 531 |
/-
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... | exact logb_lt_logb_iff hb h zero_lt_one
| Mathlib/Analysis/SpecialFunctions/Log/Base.lean | 219 | 220 |
/-
Copyright (c) 2018 Michael Jendrusch. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Jendrusch, Kim Morrison, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Monoidal.Category
import Mathlib.CategoryTheory.Adjunction.FullyFaithful
import Mathlib.CategoryTheory... | ∀ X : C, (λ_ (F.obj X)).hom = (ε' ⊗ 𝟙 (F.obj X)) ≫ μ' (𝟙_ C) X ≫ F.map (λ_ X).hom := by
aesop_cat)
(right_unitality' :
∀ X : C, (ρ_ (F.obj X)).hom = (𝟙 (F.obj X) ⊗ ε') ≫ μ' X (𝟙_ C) ≫ F.map (ρ_ X).hom := by
| Mathlib/CategoryTheory/Monoidal/Functor.lean | 171 | 174 |
/-
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 | 2,605 | 2,608 | |
/-
Copyright (c) 2024 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Derivation.Killing
import Mathlib.Algebra.Lie.Killing
import Mathlib.Algebra.Lie.Sl2
import Mathlib.Algebra.Lie.Weights.Chain
import Mathlib.Line... | (LieSubmodule.iSupIndep_toSubmodule.mpr <| iSupIndep_genWeightSpace K H L)
(LieSubmodule.iSup_toSubmodule_eq_top.mpr <| iSup_genWeightSpace_eq_top K H L)
exact LinearMap.trace_eq_zero_of_mapsTo_ne hds σ hσ hf
/-- Elements of the `α` root space which are Killing-orthogonal to the `-α` root space are
Killing-o... | Mathlib/Algebra/Lie/Weights/Killing.lean | 116 | 131 |
/-
Copyright (c) 2021 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Order.Lattice
import Mathlib.Data.List.Sort
import Mathlib.Logic.Equiv.Fin.Basic
import Mathlib.Logic.Equiv.Functor
import Mathlib.Data.Fintype.Pigeonhole
... | (hb : s₁.head = s₂.head) (ht : s₁.last = s₂.last) (hs₁0 : s₁.length = 0) : s₁ = s₂ := by
have : ∀ x, x ∈ s₁ ↔ x = s₁.last := fun x =>
| Mathlib/Order/JordanHolder.lean | 349 | 350 |
/-
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.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.C... | (f : RatFunc K) : liftRingHom φ hφ f = φ f.num / φ f.denom :=
liftMonoidWithZeroHom_apply _ hφ _
theorem liftAlgHom_apply {L S : Type*} [Field L] [CommSemiring S] [Algebra S K[X]] [Algebra S L]
(φ : K[X] →ₐ[S] L) (hφ : K[X]⁰ ≤ L⁰.comap φ) (f : RatFunc K) :
liftAlgHom φ hφ f = φ f.num / φ f.denom :=
| Mathlib/FieldTheory/RatFunc/Basic.lean | 989 | 994 |
/-
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.Topology.UniformSpace.Cauchy
/-!
# Uniform convergence
A sequence of functions `Fₙ` (with values in a metric space) converges uniformly on a se... | Mathlib/Topology/UniformSpace/UniformConvergence.lean | 678 | 682 | |
/-
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... | dvd_add rcs (Dvd.intro_left (C (leadingCoeff s) * X ^ (natDegree s - natDegree r)) rfl)
have hC0 := rprim.ne_zero
rw [Ne, ← leadingCoeff_eq_zero, ← C_eq_zero] at hC0
rw [sub_add_cancel, ← rprim.dvd_primPart_iff_dvd (mul_ne_zero hC0 s0)] at h
rcases isUnit_primPart_C r.leadingCoeff with ⟨u, hu⟩
... | Mathlib/RingTheory/Polynomial/Content.lean | 420 | 425 |
/-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
/-! # Positive Definite Matrices
This file defi... | theorem transpose {M : Matrix n n R} (hM : M.PosSemidef) : Mᵀ.PosSemidef := by
refine ⟨IsHermitian.transpose hM.1, fun x => ?_⟩
convert hM.2 (star x) using 1
rw [mulVec_transpose, dotProduct_mulVec, star_star, dotProduct_comm]
| Mathlib/LinearAlgebra/Matrix/PosDef.lean | 90 | 93 |
/-
Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.SimpleGraph.Path
import Mathlib.Combinatorics.SimpleGraph.Operations
import Mathlib.Data.Finset.Pairwise
import M... | lemma isClique_insert : G.IsClique (insert a s) ↔ G.IsClique s ∧ ∀ b ∈ s, a ≠ b → G.Adj a b :=
| Mathlib/Combinatorics/SimpleGraph/Clique.lean | 77 | 77 |
/-
Copyright (c) 2023 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Ring.CharZero
import Mathlib.Algebra.Ring.Int.Units
import Mathlib.GroupTheory.Coprod.Basic
import Mathlib.GroupTheory.Complement
/-!
## HNN Exte... | delta HNNExtension; simp [lift, t]
@[simp]
| Mathlib/GroupTheory/HNNExtension.lean | 102 | 104 |
/-
Copyright (c) 2020 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Algebra.Group.Conj
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.Group.Subgroup.Ker
/-!
# Basic results on subgroups
We prove basic results... | Mathlib/Algebra/Group/Subgroup/Basic.lean | 3,213 | 3,222 | |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Analysis.Convex.Side
import Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
/-!
# Oriented an... |
/-- If the angle between three points is `π`, the first two points are not equal. -/
| Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean | 75 | 76 |
/-
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.AlgebraicGeometry.Morphisms.Constructors
import Mathlib.RingTheory.LocalProperties.Basic
import Mathlib.RingTheory.RingHom.Locally
/-!
# Properties of morp... | simp [Scheme.Hom.appLE]
have heq : f.appLE (Y.basicOpen r') (X.basicOpen s') (hBrr' ▸ hBss' ▸ ers) =
f.appLE (Y.basicOpen r') (X.basicOpen (f.appLE U₂ V₂ e₂ r')) (by simp [Scheme.Hom.appLE]) ≫
CommRingCat.ofHom (algebraMap _ _) := by
simp only [Scheme.Hom.appLE, homOfLE_leOfHom, CommRingCat.comp... | Mathlib/AlgebraicGeometry/Morphisms/RingHomProperties.lean | 207 | 220 |
/-
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.Exact
import Mathlib.CategoryTheory.ComposableArrows
/-!
# Exact sequences
A sequence of `n` composable arrows `S : ComposableArr... |
lemma exact_iff_δlast {n : ℕ} (S : ComposableArrows C (n + 2)) :
S.Exact ↔ S.δlast.Exact ∧ (mk₂ (S.map' n (n + 1)) (S.map' (n + 1) (n + 2))).Exact := by
constructor
· intro h
| Mathlib/Algebra/Homology/ExactSequence.lean | 277 | 281 |
/-
Copyright (c) 2024 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.NumberTheory.LSeries.AbstractFuncEq
import Mathlib.NumberTheory.ModularForms.JacobiTheta.Bounds
import Mathlib.Analysis.SpecialFunctions.Gamma.Deligne
... | lemma completedHurwitzZetaEven₀_one_sub (a : UnitAddCircle) (s : ℂ) :
completedHurwitzZetaEven₀ a (1 - s) = completedCosZeta₀ a s := by
rw [completedHurwitzZetaEven₀, completedCosZeta₀, sub_div,
| Mathlib/NumberTheory/LSeries/HurwitzZetaEven.lean | 372 | 374 |
/-
Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
import Mathlib.Data.ENNReal.Real
import Mathlib.Tactic.Bound.Attribute
import Mathlib.Topology... | obtain ⟨R, hR, h⟩ : ∃ R ≥ dist y x, ∀ z : α, z ∈ closedBall x R → p z :=
| Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | 577 | 577 |
/-
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, Jeremy Avigad
-/
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Data.Set.Finite.Range
import Mathlib.Data.Set.Lattice
import Mathlib.Topology.Defs.... | Mathlib/Topology/Basic.lean | 1,106 | 1,109 | |
/-
Copyright (c) 2018 Violeta Hernández Palacios, Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios, Mario Carneiro
-/
import Mathlib.Logic.Small.List
import Mathlib.SetTheory.Ordinal.Enum
import Mathlib.SetTheory.Ordinal.Exponen... | rcases eq_or_lt_of_le h₁ with h | h
· exact ⟨_, h.symm⟩
rw [derivFamily_limit _ l, ← not_le, Ordinal.iSup_le_iff, not_forall] at h
obtain ⟨o', h⟩ := h
exact IH o' o'.2 (le_of_not_le h),
fun ⟨_, e⟩ i => e ▸ (derivFamily_fp (H i) _).le⟩
theorem fp_iff_derivFamily [Small.{u} ι] (H : ∀ i,... | Mathlib/SetTheory/Ordinal/FixedPoint.lean | 196 | 220 |
/-
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... | [FiniteDimensional K V] (f : End K V) (μ : K) :
f.maxGenEigenspace μ = f.genEigenspace μ (finrank K V) := by
apply le_antisymm _ <| (f.genEigenspace μ).monotone le_top
rw [genEigenspace_top_eq_maxUnifEigenspaceIndex]
apply genEigenspace_le_genEigenspace_finrank f μ
lemma mapsTo_maxGenEigenspace_of_comm {... | Mathlib/LinearAlgebra/Eigenspace/Basic.lean | 565 | 577 |
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Measure.MeasureSpace
import Mathlib.MeasureTheory.Measure.Regular
import Mathlib.Topology.Sets.Compacts
/-!
# Contents
In this file... | exact μ.outerMeasure_opens ⟨U, h1U⟩
variable [S : MeasurableSpace G] [BorelSpace G]
/-- For the outer measure coming from a content, all Borel sets are measurable. -/
theorem borel_le_caratheodory : S ≤ μ.outerMeasure.caratheodory := by
rw [BorelSpace.measurable_eq (α := G)]
refine MeasurableSpace.generateFrom_... | Mathlib/MeasureTheory/Measure/Content.lean | 312 | 320 |
/-
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.... |
/-- A special case of `realize_restrictFreeVar`, included because we can add the `simp` attribute
to it -/
@[simp]
theorem realize_restrictFreeVar' [DecidableEq α] {n : ℕ} {φ : L.BoundedFormula α n} {s : Set α}
| Mathlib/ModelTheory/Semantics.lean | 433 | 437 |
/-
Copyright (c) 2022 Vincent Beffara. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Vincent Beffara
-/
import Mathlib.Analysis.Complex.RemovableSingularity
import Mathlib.Analysis.Calculus.UniformLimitsDeriv
import Mathlib.Analysis.NormedSpace.FunctionSeries
/-!
# L... | have h3 : TendstoUniformlyOn F f φ (cthickening δ K) :=
(tendstoLocallyUniformlyOn_iff_forall_isCompact hU).mp hf (cthickening δ K) hKU h2
apply (h3.cderiv hδ h1).congr
filter_upwards [hF] with n h z hz
exact cderiv_eq_deriv hU h hδ ((closedBall_subset_cthickening hz δ).trans hKU)
theorem exists_cthickenin... | Mathlib/Analysis/Complex/LocallyUniformLimit.lean | 117 | 128 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Analysis.SumOverResidueClass
/-!
# Conve... | rpow_mul zero_lt_two.le, rpow_natCast, ← inv_pow, ← mul_pow,
summable_geometric_iff_norm_lt_one]
nth_rw 1 [← rpow_one 2]
rw [← division_def, ← rpow_sub zero_lt_two, norm_eq_abs,
abs_of_pos (rpow_pos_of_pos zero_lt_two _), rpow_lt_one_iff zero_lt_two.le]
norm_num
· intro n
... | Mathlib/Analysis/PSeries.lean | 267 | 295 |
/-
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.Analysis.Filter
import Mathlib.Topology.Bases
import Mathlib.Topology.LocallyFinite
/-!
# Computational realization of topological spaces (experi... | show (a ∉ s → ∃ b : F.σ, a ∈ F.F b ∧ ∀ z ∈ F.F b, z ∉ s) ↔ _ by
haveI := Classical.propDecidable; rw [not_imp_comm]
simp [not_exists, not_and, not_forall, and_comm]
| Mathlib/Data/Analysis/Topology.lean | 132 | 134 |
/-
Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Data.Set.Equitable
import Mathlib.Logic.Equiv.Fin.Basic
import Mathlib.Order.Partition.Fi... | P.IsEquipartition ↔
∀ a : Finset α, a ∈ P.parts → #a = #s / #P.parts ∨ #a = #s / #P.parts + 1 := by
simp_rw [IsEquipartition, Finset.equitableOn_iff, P.sum_card_parts]
variable {P}
| Mathlib/Order/Partition/Equipartition.lean | 38 | 42 |
/-
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.ChartedSpace
/-!
# Local properties invariant under a groupoid
We study properties of a triple `(g, s, x)` ... |
/-- For a groupoid `G` which is `ClosedUnderRestriction`, being a local structomorphism is a local
invariant property. -/
theorem isLocalStructomorphWithinAt_localInvariantProp [ClosedUnderRestriction G] :
LocalInvariantProp G G (IsLocalStructomorphWithinAt G) :=
{ is_local := by
intro s x u f hu hux
... | Mathlib/Geometry/Manifold/LocalInvariantProperties.lean | 559 | 568 |
/-
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
-/
import Mathlib.LinearAlgebra.BilinearForm.Properties
import Mathlib.LinearAlgebra.Matrix.SesquilinearForm
/-!
# Bilinear form
This file defines the conversio... | (BilinForm.toMatrix b).symm_symm
theorem Matrix.toBilin_basisFun : Matrix.toBilin (Pi.basisFun R₁ n) = Matrix.toBilin' := by
ext M
| Mathlib/LinearAlgebra/Matrix/BilinearForm.lean | 212 | 215 |
/-
Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Data.Rat.Cast.Order
import Mathlib.Orde... | theorem edgeDensity_empty_left (t : Finset β) : edgeDensity r ∅ t = 0 := by
rw [edgeDensity, Finset.card_empty, Nat.cast_zero, zero_mul, div_zero]
| Mathlib/Combinatorics/SimpleGraph/Density.lean | 136 | 137 |
/-
Copyright (c) 2019 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Eric Wieser
-/
import Mathlib.Data.Matrix.ConjTranspose
/-!
# Row and column matrices
This file provides results about row and column matrices.
## Main definitions
* `Mat... | ext
rfl
@[simp]
| Mathlib/Data/Matrix/RowCol.lean | 141 | 144 |
/-
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, Jeremy Avigad
-/
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Data.Set.Finite.Range
import Mathlib.Data.Set.Lattice
import Mathlib.Topology.Defs.... | Mathlib/Topology/Basic.lean | 611 | 612 | |
/-
Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Analysis.InnerProductSpace.Convex
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Combinatorics.Additive.AP.Three... | @[simp]
| Mathlib/Combinatorics/Additive/AP/Three/Behrend.lean | 101 | 101 |
/-
Copyright (c) 2017 Johannes Hölzl. 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.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.Nat.SuccPred
import Mathlib.Order.SuccPred.Initial... | Mathlib/SetTheory/Ordinal/Arithmetic.lean | 1,635 | 1,648 | |
/-
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.Fin.VecNotation
import Mathlib.SetTheory.Cardinal.Basic
/-!
# Basics on... | Mathlib/ModelTheory/Basic.lean | 914 | 914 | |
/-
Copyright (c) 2018 Louis Carlin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Louis Carlin, Mario Carneiro
-/
import Mathlib.Algebra.EuclideanDomain.Defs
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Regular
import Mathlib.Algebra.Grou... | end LCM
| Mathlib/Algebra/EuclideanDomain/Basic.lean | 302 | 302 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.MvPolynomial.Expand
import Mathlib.FieldTheory.Finite.Basic
import Mathlib.LinearAlgebra.Dual.Lemmas
import Mathlib.LinearAlgebra.FiniteDimensi... | degrees (indicator c) ≤ ∑ s : σ, (Fintype.card K - 1) • {s} := by
rw [indicator]
classical
refine degrees_prod_le.trans <| Finset.sum_le_sum fun s _ ↦ degrees_sub_le.trans ?_
rw [degrees_one, Multiset.zero_union]
| Mathlib/FieldTheory/Finite/Polynomial.lean | 72 | 76 |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing
import Mathlib.AlgebraicGeometry.Cover.Open
/-!
# Gluing Schemes
Given a family of gluing data of schemes, we m... | use Set.inter_subset_left
constructor
· rw [← Set.image_preimage_eq_inter_range]
apply (show IsOpenImmersion (𝒰.map (𝒰.f x)) from inferInstance).base_open.isOpenMap
convert hU (𝒰.f x) using 1
simp only [← ι_fromGlued, gluedCover_U, comp_coeBase, TopCat.hom_comp, ContinuousMap.coe_comp,
Set.pr... | Mathlib/AlgebraicGeometry/Gluing.lean | 371 | 388 |
/-
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.Algebra.Order.Ring.Nat
import Mathlib.Data.List.Chain
/-!
# List of booleans
In this file we prove lemmas about the number of `false`s and `true`s ... | aesop (add simp this)
@[simp]
theorem count_add_count_not (l : List Bool) (b : Bool) : count b l + count (!b) l = length l := by
rw [add_comm, count_not_add_count]
| Mathlib/Data/Bool/Count.lean | 24 | 29 |
/-
Copyright (c) 2014 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.Subtype
import Mathlib.Order.Defs.LinearOrder
import Mathlib.Order.Notation
import Mathlib.Tactic.GCongr.Core
import Mathlib.Tactic.... | lemma ltByCases_eq_iff {h₁ : x < y → P} {h₂ : x = y → P} {h₃ : y < x → P} {p : P} :
ltByCases x y h₁ h₂ h₃ = p ↔ (∃ h, h₁ h = p) ∨ (∃ h, h₂ h = p) ∨ (∃ h, h₃ h = p) := by
refine ltByCases x y (fun h => ?_) (fun h => ?_) (fun h => ?_)
· simp only [ltByCases_lt, exists_prop_of_true, h, h.not_lt, not_false_eq_true... | Mathlib/Order/Basic.lean | 447 | 452 |
/-
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... | end Semiring
end MultilinearMap
namespace LinearMap
variable [Semiring R] [∀ i, AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [AddCommMonoid M₃]
[AddCommMonoid M'] [∀ i, Module R (M₁ i)] [Module R M₂] [Module R M₃] [Module R M']
/-- Composing a multilinear map with a linear map gives again a multilinear map. -/
def co... | Mathlib/LinearAlgebra/Multilinear/Basic.lean | 768 | 783 |
/-
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... | theorem Filter.Tendsto.atTop_mul_neg {C : 𝕜} (hC : C < 0) (hf : Tendsto f l atTop)
(hg : Tendsto g l (𝓝 C)) : Tendsto (fun x => f x * g x) l atBot := by
have := hf.atTop_mul_pos (neg_pos.2 hC) hg.neg
| Mathlib/Topology/Algebra/Order/Field.lean | 87 | 89 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yaël Dillies
-/
import Mathlib.Data.Set.Image
/-!
# Directed indexed families and sets
This file defines directed indexed families and directed sets. An indexed famil... | ⟨IsTop.isMax, IsMax.isTop⟩
end Preorder
| Mathlib/Order/Directed.lean | 254 | 257 |
/-
Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Data.Set.Equitable
import Mathlib.Logic.Equiv.Fin.Basic
import Mathlib.Order.Partition.Fi... | exact add_lt_add_of_le_of_lt (mul_le_mul_left' (by omega) _) ((hg (f a).1).mp c)
let z' : s → Fin #s := fun a ↦ ⟨z a, less a⟩
have bij : z'.Bijective := by
refine (bijective_iff_injective_and_card z').mpr ⟨fun a b e ↦ ?_, by simp⟩
simp_rw [z', z, Fin.mk.injEq, mul_comm #P.parts] at e
haveI : NeZer... | Mathlib/Order/Partition/Equipartition.lean | 138 | 165 |
/-
Copyright (c) 2021 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.GroupTheory.Transfer
/-!
# The Schur-Zassenhaus Theorem
In this file we prove the Schur-Zassenhaus theorem.
## Main results
- `Subgroup.exists_ri... | let ϕ : H →* H :=
{ toFun := fun h =>
⟨g.unop⁻¹ * h * g.unop,
hH.mem_comm ((congr_arg (· ∈ H) (mul_inv_cancel_left _ _)).mpr (SetLike.coe_mem _))⟩
map_one' := by rw [Subtype.ext_iff, coe_mk, coe_one, mul_one, inv_mul_cancel]
map_mul' := fun h₁ h₂ => by
simp only [Subtype.ext_... | Mathlib/GroupTheory/SchurZassenhaus.lean | 48 | 62 |
/-
Copyright (c) 2024 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.NumberTheory.DirichletCharacter.Bounds
import Mathlib.NumberTheory.LSeries.Convolution
import Mathlib.NumberTheory.LSeries.Deriv
import Mathlib.NumberThe... | exact mul_mul_mul_comm ..
lemma mul_delta {n : ℕ} (χ : DirichletCharacter ℂ n) : ↗χ * δ = δ :=
LSeries.mul_delta <| by rw [cast_one, map_one]
lemma delta_mul {n : ℕ} (χ : DirichletCharacter ℂ n) : δ * ↗χ = δ :=
mul_comm δ _ ▸ mul_delta ..
| Mathlib/NumberTheory/LSeries/Dirichlet.lean | 127 | 134 |
/-
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 | 1,546 | 1,556 | |
/-
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... | rw [← neg_re, ← norm_neg z, re_eq_norm]
exact neg_nonneg.and <| eq_comm.trans neg_eq_zero
@[simp]
lemma re_eq_neg_norm {z : ℂ} : z.re = -‖z‖ ↔ z ≤ 0 := by rw [← neg_eq_iff_eq_neg, neg_re_eq_norm]
| Mathlib/Data/Complex/Order.lean | 107 | 111 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Module.Basic
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
/-!
# Midpoint of a segment
## Main definitions
* `midpoint R x y`: midp... | @[simp]
theorem left_vsub_midpoint (p₁ p₂ : P) : p₁ -ᵥ midpoint R p₁ p₂ = (⅟ 2 : R) • (p₁ -ᵥ p₂) :=
| Mathlib/LinearAlgebra/AffineSpace/Midpoint.lean | 119 | 120 |
/-
Copyright (c) 2019 Neil Strickland. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Neil Strickland
-/
import Mathlib.Tactic.Ring
import Mathlib.Data.PNat.Prime
/-!
# Euclidean algorithm for ℕ
This file sets up a version of the Euclidean algorithm that only works w... | theorem finish_isSpecial (hs : u.IsSpecial) : u.finish.IsSpecial := by
dsimp [IsSpecial, finish] at hs ⊢
rw [add_mul _ _ u.y, add_comm _ (u.x * u.y), ← hs]
ring
theorem finish_v (hr : u.r = 0) : u.finish.v = u.v := by
| Mathlib/Data/PNat/Xgcd.lean | 241 | 246 |
/-
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... | ⟨fun h =>
Classical.by_contradiction fun hp =>
mt mem_support_iff.1 (Classical.not_not.2 h) (mem_of_max (degree_eq_natDegree hp)),
fun h => h.symm ▸ leadingCoeff_zero⟩
theorem leadingCoeff_ne_zero : leadingCoeff p ≠ 0 ↔ p ≠ 0 := by rw [Ne, leadingCoeff_eq_zero]
| Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 354 | 360 |
/-
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.LinearAlgebra.FreeModule.Basic
import Mathlib.MeasureTheory.Measure.Decomposition.Exhaustion
import Mathlib.Probability.ConditionalProbability
/-!
# s-fin... | Mathlib/MeasureTheory/Measure/WithDensityFinite.lean | 136 | 138 | |
/-
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.CategoryTheory.ObjectProperty.Shift
import Mathlib.CategoryTheory.Triangulated.Pretriangulated
/-!
# t-structures on triangulated categories
This files introdu... | exact t.le_shift _ _ _ hn' X hX
lemma shift_ge (a n n' : ℤ) (hn' : a + n = n') :
(t.ge n).shift a = t.ge n' := by
ext X
constructor
· intro hX
exact ((t.ge n').prop_iff_of_iso ((shiftEquiv C a).unitIso.symm.app X)).1
(t.ge_shift n (-a) n' (by omega) _ hX)
| Mathlib/CategoryTheory/Triangulated/TStructure/Basic.lean | 98 | 106 |
/-
Copyright (c) 2022 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex J. Best, Xavier Roblot
-/
import Mathlib.Algebra.Algebra.Hom.Rat
import Mathlib.Analysis.Complex.Polynomial.Basic
import Mathlib.NumberTheory.NumberField.Norm
import Mathlib.RingTh... | split
· rwa [← isUnramified_iff_card_stabilizer_eq_one]
· rwa [← not_isUnramified_iff_card_stabilizer_eq_two]
lemma even_nat_card_aut_of_not_isUnramified [IsGalois k K] (hw : ¬ IsUnramified k w) :
Even (Nat.card <| K ≃ₐ[k] K) := by
| Mathlib/NumberTheory/NumberField/Embeddings.lean | 941 | 946 |
/-
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.CategoryTheory.ObjectProperty.ClosedUnderIsomorphisms
import Mathlib.CategoryTheory.Localization.CalculusOfFractions
import Mathlib.CategoryTheory.Localization.T... | ⟨_, _, _, hT, h⟩
lemma W.mk' {T : Triangle C} (hT : T ∈ distTriang C) (h : S.P T.obj₁) : S.W T.mor₂ := by
| Mathlib/CategoryTheory/Triangulated/Subcategory.lean | 141 | 143 |
/-
Copyright (c) 2020 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Nat.Prime.Defs
import Mathlib.Data.Num.ZNum
import Mathlib.Tactic.Ring
/-!
# Primality for binary natural numbers
This file defines versions of ... | simp only [h, ite_true]
simp_rw [← mul_to_nat]
simp only [cast_lt, dvd_to_nat]
split_ifs <;> try rfl
rw [ih] <;> [congr; convert Nat.lt_succ_of_lt h using 1] <;>
simp only [cast_bit1, cast_succ, Nat.succ_eq_add_one, add_assoc,
add_left_comm, ← one_add_one_eq_two]
/-- Returns the smallest prime fa... | Mathlib/Data/Num/Prime.lean | 44 | 54 |
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