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) 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... | lemma in₀₂_iff' :
(graph t).Adj (in₀ a) (in₂ c) ↔ ∃ x : α × β × γ, x ∈ t ∧ x.1 = a ∧ x.2.2 = c where
mp := by rintro ⟨⟩; exact ⟨_, ‹_›, by simp⟩
mpr := by rintro ⟨⟨a, b, c⟩, h, rfl, rfl⟩; constructor; assumption
| Mathlib/Combinatorics/SimpleGraph/Triangle/Tripartite.lean | 93 | 96 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Matrix.RowCol
import Mathli... | (M.updateCol j (fun k ↦ M k i)).det = 0 := det_zero_of_column_eq h (by simp [h])
@[deprecated (since := "2024-12-11")] alias det_updateColumn_eq_zero := det_updateCol_eq_zero
| Mathlib/LinearAlgebra/Matrix/Determinant/Basic.lean | 374 | 376 |
/-
Copyright (c) 2021 Martin Zinkevich. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Martin Zinkevich, Rémy Degenne
-/
import Mathlib.Logic.Encodable.Lattice
import Mathlib.MeasureTheory.MeasurableSpace.Defs
import Mathlib.Order.Disjointed
/-!
# Indu... | ∃ (T : Finset β) (f : β → Set α), (t = ⋂ b ∈ T, f b) ∧ ∀ b ∈ T, f b ∈ g b := by
classical
induction h_t with
| @base s h_s =>
rcases h_s with ⟨t', ⟨⟨b, rfl⟩, h_s_in_t'⟩⟩
refine ⟨{b}, fun _ => s, ?_⟩
| Mathlib/MeasureTheory/PiSystem.lean | 256 | 261 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Alex Kontorovich, Heather Macbeth
-/
import Mathlib.MeasureTheory.Group.Action
import Mathlib.MeasureTheory.Group.Pointwise
import Mathlib.MeasureTheory.Integral.Lebe... | refine sInf_le_sInf ?_
rintro a (ha : (μ.restrict s) {x : α | a < f x} = 0)
rw [Measure.restrict_apply₀' hs.nullMeasurableSet] at ha
refine measure_zero_of_invariant hs _ ?_ ha
intro γ
ext x
rw [mem_smul_set_iff_inv_smul_mem]
| Mathlib/MeasureTheory/Group/FundamentalDomain.lean | 466 | 472 |
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes Hölzl
-/
import Mathlib.Algebra.Field.Subfield.Defs
import Mathlib.Algebra.Order.Group.Pointwise.Interval
import Mathlib.Analysis.Normed.Ring.Basic
/-!
# Norm... | rintro ⟨-, x, rfl⟩ ⟨-, y, rfl⟩ hxy
let ⟨z, h⟩ := exists_lt_norm_lt α (norm_nonneg _) hxy
exact ⟨⟨‖z‖, z, rfl⟩, h⟩
| Mathlib/Analysis/Normed/Field/Basic.lean | 270 | 272 |
/-
Copyright (c) 2019 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Yury Kudryashov, Yaël Dillies
-/
import Mathlib.Algebra.Order.Invertible
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.LinearAlgebra.AffineSpac... |
variable [Semiring 𝕜] [PartialOrder 𝕜] [AddCommMonoid E] [AddCommMonoid F] [Module 𝕜 E] [Module 𝕜 F]
theorem segment_subset (x y : E × F) : segment 𝕜 x y ⊆ segment 𝕜 x.1 y.1 ×ˢ segment 𝕜 x.2 y.2 := by
| Mathlib/Analysis/Convex/Segment.lean | 561 | 564 |
/-
Copyright (c) 2022 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang, Kim Morrison, Joël Riou
-/
import Mathlib.Algebra.Homology.Additive
import Mathlib.CategoryTheory.Abelian.Injective.Resolution
/-!
# Right-derived functors
We define the ... | (F : C ⥤ D) [F.Additive] (n : ℕ) :
(F.rightDerived n).map f ≫ (J.isoRightDerivedObj F n).hom =
(I.isoRightDerivedObj F n).hom ≫
(F.mapHomologicalComplex _ ⋙ HomologicalComplex.homologyFunctor _ _ n).map φ := by
dsimp [isoRightDerivedObj, Functor.rightDerived]
rw [assoc, ← Functor.map_comp_asso... | Mathlib/CategoryTheory/Abelian/RightDerived.lean | 120 | 133 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Andrew Zipperer, Haitao Zhang, Minchao Wu, Yury Kudryashov
-/
import Mathlib.Data.Set.Prod
import Mathlib.Data.Set.Restrict
/-!
# Functions over sets
This file contains... | @[simp]
theorem mapsTo_union : MapsTo f (s₁ ∪ s₂) t ↔ MapsTo f s₁ t ∧ MapsTo f s₂ t :=
| Mathlib/Data/Set/Function.lean | 185 | 186 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Minchao Wu, Mario Carneiro
-/
import Mathlib.Data.Finset.Attach
import Mathlib.Data.Finset.Disjoint
import Mathlib.Data.Finset.Erase
import Mat... | Mathlib/Data/Finset/Basic.lean | 1,527 | 1,528 | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Algebra.Order.GroupWithZero.Canonical
import Mathlib.Data.Nat.Cast.WithTop
/-!
# `WithBot ℕ`
Lemmas about the type... | · simp [WithBot.add_bot]
simp [← WithBot.coe_add, Nat.add_eq_one_iff]
theorem add_eq_two_iff {n m : WithBot ℕ} :
n + m = 2 ↔ n = 0 ∧ m = 2 ∨ n = 1 ∧ m = 1 ∨ n = 2 ∧ m = 0 := by
cases n
| Mathlib/Data/Nat/WithBot.lean | 36 | 41 |
/-
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... |
/-- For any natural numbers n, a, and b, one of the following holds:
1. n < a
2. n ≥ b
3. n ∈ Ico a b
| Mathlib/Data/List/Intervals.lean | 185 | 189 |
/-
Copyright (c) 2017 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Tim Baumann, Stephen Morgan, Kim Morrison, Floris van Doorn
-/
import Mathlib.Tactic.CategoryTheory.Reassoc
/-!
# Isomorphisms
This file defines isomorphisms between objects of a categ... | namespace Functor
universe u₁ v₁ u₂ v₂
| Mathlib/CategoryTheory/Iso.lean | 520 | 522 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
/-... | simpa only [map_natCast] using IsCoprime.map (Nat.Coprime.isCoprime h) (Int.castRingHom A)
theorem IsCoprime.prod_left : (∀ i ∈ t, IsCoprime (s i) x) → IsCoprime (∏ i ∈ t, s i) x := by
classical
| Mathlib/RingTheory/Coprime/Lemmas.lean | 56 | 59 |
/-
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 | 857 | 857 | |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Yaël Dillies
-/
import Mathlib.Order.Cover
import Mathlib.Order.Interval.Finset.Defs
/-!
# Intervals as finsets
This file provides basic results about all the `Finset.Ixx... |
@[simp]
| Mathlib/Order/Interval/Finset/Basic.lean | 404 | 405 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yaël Dillies
-/
import Mathlib.Algebra.Module.BigOperators
import Mathlib.GroupTheory.Perm.Basic
import Mathlib.GroupTheory.Perm.Finite
import Mathlib.GroupTheory.Perm.Lis... |
variable [DecidableEq α]
theorem isCycle_swap (hxy : x ≠ y) : IsCycle (swap x y) :=
⟨y, by rwa [swap_apply_right], fun a (ha : ite (a = x) y (ite (a = y) x a) ≠ a) =>
if hya : y = a then ⟨0, hya⟩
| Mathlib/GroupTheory/Perm/Cycle/Basic.lean | 291 | 296 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Group.Equiv.Basic
import Mathlib.Data.ENat.Lattice
import Mathlib.Data.Part
import Mathlib.Tactic.NormNum
/-!
# Natural numbers with infinity
The... | Mathlib/Data/Nat/PartENat.lean | 753 | 754 | |
/-
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... | @[simp]
theorem bagInter_consecutive (n m l : Nat) :
@List.bagInter ℕ instBEqOfDecidableEq (Ico n m) (Ico m l) = [] :=
(bagInter_nil_iff_inter_nil _ _).2 (by convert inter_consecutive n m l)
@[simp]
| Mathlib/Data/List/Intervals.lean | 95 | 100 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kim Morrison
-/
import Mathlib.Algebra.BigOperators.Finsupp.Basic
import Mathlib.Algebra.BigOperators.Group.Finset.Preimage
import Mathlib.Algebra.Module.Defs
import Ma... |
@[to_additive]
theorem prod_mapDomain_index_inj [CommMonoid N] {f : α → β} {s : α →₀ M} {h : β → M → N}
(hf : Function.Injective f) : (s.mapDomain f).prod h = s.prod fun a b => h (f a) b := by
rw [← Function.Embedding.coeFn_mk f hf, ← embDomain_eq_mapDomain, prod_embDomain]
| Mathlib/Data/Finsupp/Basic.lean | 537 | 541 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Yury Kudryashov
-/
import Mathlib.Analysis.Normed.Module.Convex
import Mathlib.Analysis.Normed.Module.Ray
import Mathlib.Analysis.NormedSpace.Pointwise
/-!
# Strictly conv... | Mathlib/Analysis/Convex/StrictConvexSpace.lean | 230 | 233 | |
/-
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.... | rw [descFactorial_succ, Nat.pow_succ, Nat.pow_one, descFactorial_one]
exact Nat.mul_lt_mul_of_pos_left (by omega) (Nat.sub_pos_of_lt h)
| k + 1, h => by
rw [descFactorial_succ, Nat.pow_succ, Nat.mul_comm]
refine Nat.mul_lt_mul_of_pos_left ?_ (Nat.sub_pos_of_lt h)
refine Nat.lt_of_le_of_lt (Nat.pow... | Mathlib/Data/Nat/Factorial/Basic.lean | 414 | 420 |
/-
Copyright (c) 2024 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Kernel.Composition.IntegralCompProd
import Mathlib.Probability.Kernel.Disintegration.StandardBorel
/-!
# Lebesgue and Bochner integrals of con... |
lemma setLIntegral_condKernel_univ_left (hf : Measurable f) (a : α) {t : Set Ω}
(ht : MeasurableSet t) :
∫⁻ b, ∫⁻ ω in t, f (b, ω) ∂(Kernel.condKernel κ (a, b)) ∂(Kernel.fst κ a)
= ∫⁻ x in Set.univ ×ˢ t, f x ∂(κ a) := by
| Mathlib/Probability/Kernel/Disintegration/Integral.lean | 76 | 80 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Prime.Basic
import Mathlib.Data.Real.Archimedean
import Mathlib.NumberTheory.Zsqrtd.Basic
/-!
# Gaussian intege... | Mathlib/NumberTheory/Zsqrtd/GaussianInt.lean | 97 | 97 | |
/-
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.FreeMonoid.Basic
import Mathlib.Algebra.Group.TypeTags.Basic
/-!
# `List.count` as a bundled homomorphism
In this file we define `FreeMonoi... |
namespace FreeAddMonoid
/-- `List.countP` as a bundled additive monoid homomorphism. -/
| Mathlib/Algebra/FreeMonoid/Count.lean | 61 | 64 |
/-
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.Data.Set.Lattice
import Mathlib.Data.Set.Pairwise.Basic
/-!
# Relations holding pairwise
In this file we prove many facts about `Pairwise` and the se... | noncomputable def biUnionEqSigmaOfDisjoint {s : Set ι} {f : ι → Set α} (h : s.PairwiseDisjoint f) :
(⋃ i ∈ s, f i) ≃ Σi : s, f i :=
(Equiv.setCongr (biUnion_eq_iUnion _ _)).trans <|
unionEqSigmaOfDisjoint fun ⟨_i, hi⟩ ⟨_j, hj⟩ ne => h hi hj fun eq => ne <| Subtype.eq eq
end Set
| Mathlib/Data/Set/Pairwise/Lattice.lean | 124 | 130 |
/-
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.Algebra.Category.Ring.Basic
import Mathlib.CategoryTheory.Limits.HasLimits
/-!
# The category of commutative rings has all colimits.
This file uses a "pr... | apply Relation.map
@[simp]
theorem cocone_naturality_components (j j' : J) (f : j ⟶ j') (x : F.obj j) :
(coconeMorphism F j') (F.map f x) = (coconeMorphism F j) x := by
| Mathlib/Algebra/Category/Ring/Colimits.lean | 199 | 203 |
/-
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.Vector.Basic
/-!
# The `zipWith` operation on vectors.
-/
namespace List
namespace Vector
section ZipWith
variable {α β γ : Type*} {n : ℕ} (f : α... | @[to_additive]
theorem prod_mul_prod_eq_prod_zipWith [CommMonoid α] (x y : Vector α n) :
x.toList.prod * y.toList.prod = (Vector.zipWith (· * ·) x y).toList.prod :=
List.prod_mul_prod_eq_prod_zipWith_of_length_eq x.toList y.toList
| Mathlib/Data/Vector/Zip.lean | 40 | 43 |
/-
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.Basic
/-!
# Split a box along one or more hyperplanes
## Main definitions
A hyperplane `{x : ι → ℝ | x i = a}` spli... | Mathlib/Analysis/BoxIntegral/Partition/Split.lean | 381 | 382 | |
/-
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
-/
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.FieldTheory.Minpoly.Basic
import Mathlib.RingTheory.A... | `h₂ : aeval pb.gen (minpoly R (root g)) = 0`; that minimal polynomial is not
guaranteed to be identical to `g`. -/
@[simps -fullyApplied]
| Mathlib/RingTheory/AdjoinRoot.lean | 595 | 597 |
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kenny Lau
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.RingTheory.MvPowerSer... | theorem constantCoeff_C (a : R) : constantCoeff R (C R a) = a :=
rfl
| Mathlib/RingTheory/PowerSeries/Basic.lean | 361 | 363 |
/-
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... |
protected theorem mul_div_cancel' {a b : R} (hb : b ≠ 0) (hab : b ∣ a) : b * (a / b) = a := by
rw [← mul_div_assoc _ hab, mul_div_cancel_left₀ _ hb]
-- This generalizes `Int.div_one`, see note [simp-normal form]
@[simp]
| Mathlib/Algebra/EuclideanDomain/Basic.lean | 96 | 101 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Order.Filter.Bases.Finit... | Mathlib/Topology/Algebra/Group/Basic.lean | 1,507 | 1,508 | |
/-
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.Lie.CartanSubalgebra
import Mathlib.Algebra.Lie.Weights.Basic
/-!
# Weights and roots of Lie modules and Lie algebras with respect to Cartan subalge... | induction n with
| zero => simpa using hm
| succ n IH =>
simp only [pow_succ', Module.End.mul_apply, toEnd_apply_apply,
Nat.cast_add, Nat.cast_one, rootSpace]
convert lie_mem_genWeightSpace_of_mem_genWeightSpace hx IH using 2
rw [succ_nsmul, ← add_assoc, add_comm (n • _)]
variable (R L H M)
| Mathlib/Algebra/Lie/Weights/Cartan.lean | 72 | 81 |
/-
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 ... | Mathlib/Data/List/Zip.lean | 366 | 373 | |
/-
Copyright (c) 2018 Michael Jendrusch. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Jendrusch, Kim Morrison, Bhavik Mehta, Jakob von Raumer
-/
import Mathlib.CategoryTheory.EqToHom
import Mathlib.CategoryTheory.Functor.Trifunctor
import Mathlib.CategoryTheo... | Mathlib/CategoryTheory/Monoidal/Category.lean | 1,026 | 1,029 | |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Andrew Zipperer, Haitao Zhang, Minchao Wu, Yury Kudryashov
-/
import Mathlib.Data.Set.Prod
import Mathlib.Data.Set.Restrict
/-!
# Functions over sets
This file contains... | Mathlib/Data/Set/Function.lean | 1,973 | 1,978 | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Topology.Order.ProjIcc
/-!... | rw [arccos, cos_pi_div_two_sub, sin_arcsin hx₁ hx₂]
theorem arccos_cos {x : ℝ} (hx₁ : 0 ≤ x) (hx₂ : x ≤ π) : arccos (cos x) = x := by
rw [arccos, ← sin_pi_div_two_sub, arcsin_sin] <;> simp [sub_eq_add_neg] <;> linarith
lemma arccos_eq_of_eq_cos (hy₀ : 0 ≤ y) (hy₁ : y ≤ π) (hxy : x = cos y) : arccos x = y := by
... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean | 306 | 318 |
/-
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.Complex
import Qq
/-! # P... | Antitone (b ^ · : ℝ → ℝ) := by
rcases lt_or_eq_of_le hb₁ with hb₁ | rfl
case inl => exact (strictAnti_rpow_of_base_lt_one hb₀ hb₁).antitone
case inr => intro _ _ _; simp
lemma rpow_right_inj (hx₀ : 0 < x) (hx₁ : x ≠ 1) : x ^ y = x ^ z ↔ y = z := by
refine ⟨fun H ↦ ?_, fun H ↦ by rw [H]⟩
| Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 908 | 914 |
/-
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... | Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean | 1,066 | 1,069 | |
/-
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... | have hlt :=
sum_lt_sum_of_nonempty h fun x hx => pos_of_mem_properDivisors (sdiff_subset hx)
| Mathlib/NumberTheory/Divisors.lean | 434 | 435 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Eric Wieser
-/
import Mathlib.Algebra.DirectSum.Internal
import Mathlib.Algebra.GradedMonoid
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Algebra.MvPolyn... | exact Finset.le_sup (f := fun s ↦ ∑ x ∈ s.support, s x) hd
theorem rename_isHomogeneous {f : σ → τ} (h : φ.IsHomogeneous n) :
(rename f φ).IsHomogeneous n := by
rw [← φ.support_sum_monomial_coeff, map_sum]; simp_rw [rename_monomial]
apply IsHomogeneous.sum _ _ _ fun d hd ↦ isHomogeneous_monomial _ _
intro ... | Mathlib/RingTheory/MvPolynomial/Homogeneous.lean | 260 | 269 |
/-
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... | to `X → α` satisfying the right continuity conditions. See also `Set.EquicontinuousWithinAt.closure`
for a more familiar (but weaker) statement.
Note: This could *technically* be called `EquicontinuousWithinAt.closure` without name clashes
| Mathlib/Topology/UniformSpace/Equicontinuity.lean | 760 | 763 |
/-
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 | 2,390 | 2,392 | |
/-
Copyright (c) 2019 Gabriel Ebner. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Gabriel Ebner, Sébastien Gouëzel, Yury Kudryashov, Yuyang Zhao
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Comp
import Mathlib.Analysis.Calcu... | variable {l : F → E} {l' : F →L[𝕜] E} {y : F}
variable (x)
/-- The composition `l ∘ f` where `l : F → E` and `f : 𝕜 → F`, has a derivative within a set
equal to the Fréchet derivative of `l` applied to the derivative of `f`. -/
| Mathlib/Analysis/Calculus/Deriv/Comp.lean | 328 | 332 |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Algebra.Order.Module.Defs
import Mathlib.Data.DFinsupp.Module
/-!
# Pointwise order on finitely supported dependent functions
This file lifts order struc... |
lemma support_monotone : Monotone (support (ι := ι) (β := α)) :=
| Mathlib/Data/DFinsupp/Order.lean | 205 | 206 |
/-
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.Homotopy
import Mathlib.Algebra.Ring.NegOnePow
import Mathlib.Algebra.Category.Grp.Preadditive
import Mathlib.Tactic.Linarith
import Mathlib.Cat... | simp only [ofHoms, mk_v, eqToHom_refl, comp_id]
| Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean | 138 | 139 |
/-
Copyright (c) 2021 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, David Kurniadi Angdinata
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.CubicDiscriminant
import Mathlib.RingTheory.Nilpotent.Defs
import Mathlib.Tactic.Fiel... |
variable [CharP R 2]
| Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean | 137 | 139 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Nat.ModEq
import Mathlib.Data.Nat.Prime.Basic
import Mathlib.NumberTheory.Zsqrtd.Basic
/-!
# Pell's equation and Matiyasevic's theorem
This file... | rw [tsub_add_cancel_of_le h, two_mul]; exact Nat.add_le_add_left jn _
let t := xn_modEq_x2n_sub_lem a1 (Nat.le_of_add_le_add_right this)
| Mathlib/NumberTheory/PellMatiyasevic.lean | 539 | 540 |
/-
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.Algebra.Algebra.Pi
import Mathlib.LinearAlgebra.Finsupp.SumProd
import Mathlib.LinearAlgebra.FreeModule.Basic
import Mathlib.LinearAlgebra.LinearIndepe... | (Pi.basis s).repr x ji = (s ji.1).repr (x ji.1) ji.2 :=
rfl
end Fintype
section
variable [Finite η]
variable (R η)
/-- The basis on `η → R` where the `i`th basis vector is `Function.update 0 i 1`. -/
noncomputable def basisFun : Basis η R (η → R) :=
Basis.ofEquivFun (LinearEquiv.refl _ _)
| Mathlib/LinearAlgebra/StdBasis.lean | 140 | 153 |
/-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston
-/
import Mathlib.Algebra.Group.Submonoid.Operations
import Mathlib.Data.Setoid.Basic
import Mathlib.GroupTheory.Congruence.Hom
/-!
# Congruence relations
This f... | /-- Given a monoid homomorphism `f`, the induced homomorphism on the quotient by `f`'s kernel has
the same image as `f`. -/
| Mathlib/GroupTheory/Congruence/Basic.lean | 186 | 187 |
/-
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.... | | refl => exact Nat.dvd_refl _
| step _ ih => exact Nat.dvd_trans ih (Nat.dvd_mul_left _ _)
theorem dvd_factorial : ∀ {m n}, 0 < m → m ≤ n → m ∣ n !
| Mathlib/Data/Nat/Factorial/Basic.lean | 73 | 76 |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.Cover
import Mathlib.Order.Iterate
/-!
# Successor and predecessor
This file defines succes... | theorem not_isMin_succ [Nontrivial α] (a : α) : ¬ IsMin (succ a) := by
obtain ha | ha := (le_succ a).eq_or_lt
· exact (ha ▸ succ_eq_iff_isMax.1 ha.symm).not_isMin
· exact not_isMin_of_lt ha
| Mathlib/Order/SuccPred/Basic.lean | 431 | 434 |
/-
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.List.Nodup
import Mathlib.Data.Set.Prod
/-!
# Finite products of types
This file defines the product of types over a list. For `l : List ι` ... | Mathlib/Data/Prod/TProd.lean | 173 | 180 | |
/-
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.Algebra.BigOperators.Group.Finset.Indicator
import Mathlib.Algebra.Module.BigOperators
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic
import... | `weightedVSubOfPoint` using a `Finset` lying within that subset and
with a given sum of weights if and only if it can be expressed as
`weightedVSubOfPoint` with that sum of weights for the
corresponding indexed family whose index type is the subtype
corresponding to that subset. -/
| Mathlib/LinearAlgebra/AffineSpace/Combination.lean | 526 | 530 |
/-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin
-/
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.Matrix.Composition
import Mathlib.Data.Matrix.ConjTranspose
/-!... |
end
@[simp]
theorem blockDiagonal'_pow [∀ i, DecidableEq (m' i)] [Fintype o] [∀ i, Fintype (m' i)] [Semiring α]
| Mathlib/Data/Matrix/Block.lean | 693 | 697 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Johannes Hölzl, Yury Kudryashov, Patrick Massot
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Order.Filter.AtTopBot.Archimedean
import Mathlib.Order.Iterate
impor... | (tendsto_add_atTop_iff_nat 1).2 (_root_.tendsto_const_div_atTop_nhds_zero_nat 1)
theorem NNReal.tendsto_algebraMap_inverse_atTop_nhds_zero_nat (𝕜 : Type*) [Semiring 𝕜]
| Mathlib/Analysis/SpecificLimits/Basic.lean | 59 | 61 |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Neil Strickland
-/
import Mathlib.Data.Nat.Prime.Defs
import Mathlib.Data.PNat.Basic
/-!
# Primality and GCD on pnat
This file extends the theory of `ℕ+` with ... | Mathlib/Data/PNat/Prime.lean | 279 | 285 | |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Antoine Chambert-Loir
-/
import Mathlib.Algebra.DirectSum.Finsupp
import Mathlib.LinearAlgebra.DirectSum.TensorProduct
import Mathlib.LinearAlgebra.Finsupp.SumProd
/-!... | finsuppLeft R M N ι (p ⊗ₜ[R] n) i = p i ⊗ₜ[R] n := by
rw [finsuppLeft_apply_tmul, Finsupp.sum_apply,
Finsupp.sum_eq_single i (fun _ _ ↦ Finsupp.single_eq_of_ne) (by simp), Finsupp.single_eq_same]
theorem finsuppLeft_apply (t : (ι →₀ M) ⊗[R] N) (i : ι) :
| Mathlib/LinearAlgebra/DirectSum/Finsupp.lean | 96 | 100 |
/-
Copyright (c) 2022 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Group.Nat.Even
import Mathlib.Data.Nat.Cast.Basic
import Mathlib.Data.Nat.Cast.Commute
import Mathlib.Data.Set.Operations
import Mathlib.Logic.Fu... | lemma Odd.neg (hp : Odd a) : Odd (-a) := by
obtain ⟨k, hk⟩ := hp
use -(k + 1)
rw [mul_neg, mul_add, neg_add, add_assoc, two_mul (1 : α), neg_add, neg_add_cancel_right,
← neg_add, hk]
| Mathlib/Algebra/Ring/Parity.lean | 174 | 178 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.Module.End
import Mathlib.Algebra.Ring.Prod
import Mathlib.Data.Fintype.Units
import Mathlib.GroupTheory.GroupAc... | rw [← Nat.mod_zero n, ← val_natCast, val_unit'.mp h]
| Mathlib/Data/ZMod/Basic.lean | 101 | 102 |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.LinearAlgebra.Finsupp.Span
/-!
# Lie submodules of a Lie algebra
In this file we define Lie submodules, we construct ... | rw [lie_add]
refine add_mem ?_ (ih (Subset.trans (by simp) hs) hu)
obtain ⟨p, hp, rfl⟩ : ∃ p ∈ S, ↑p = q := hs (Finset.mem_insert_self q t)
| Mathlib/Algebra/Lie/Submodule.lean | 404 | 406 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.GDelta.Basic
/-!
# Baire spaces
A topological space is called a *Baire space*
if a countable intersection of dense open subsets is den... |
/-- Baire theorem: if countably many closed sets cover the whole space, then their interiors
are dense. Formulated here with `⋃₀`. -/
theorem dense_sUnion_interior_of_closed {S : Set (Set X)} (hc : ∀ s ∈ S, IsClosed s)
(hS : S.Countable) (hU : ⋃₀ S = univ) : Dense (⋃ s ∈ S, interior s) :=
IsGδ.univ.dense_sUnion_... | Mathlib/Topology/Baire/Lemmas.lean | 151 | 156 |
/-
Copyright (c) 2024 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.Data.Matroid.Constructions
import Mathlib.Data.Set.Notation
/-!
# Maps between matroids
This file defines maps and comaps, which move a matroid on one ty... | lemma comapOn_dual_eq_of_bijOn (h : BijOn f E N.E) :
(N.comapOn E f)✶ = N✶.comapOn E f := by
refine ext_isBase (by simp) (fun B hB ↦ ?_)
rw [comapOn_isBase_iff_of_bijOn (by simpa), dual_isBase_iff, comapOn_isBase_iff_of_bijOn h,
dual_isBase_iff _, comapOn_ground_eq, and_iff_left diff_subset, and_iff_left (b... | Mathlib/Data/Matroid/Map.lean | 280 | 286 |
/-
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... | ext a
simp only [mem_interedges_iff, mem_biUnion]
exact ⟨fun ⟨x₁, ⟨x₂, x₃, x₄⟩, x₅⟩ ↦ ⟨x₂, x₃, x₁, x₄, x₅⟩,
| Mathlib/Combinatorics/SimpleGraph/Density.lean | 105 | 107 |
/-
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.Homology
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Preadditive.Opposite
/-!
# Homolo... | lemma homologyMap'_congr (h : Homotopy φ₁ φ₂) (h₁ : S₁.HomologyData)
(h₂ : S₂.HomologyData) : homologyMap' φ₁ h₁ h₂ = homologyMap' φ₂ h₁ h₂ := by
rw [h.eq_add_nullHomotopic, homologyMap'_add, homologyMap'_nullHomotopic, add_zero]
| Mathlib/Algebra/Homology/ShortComplex/Preadditive.lean | 655 | 657 |
/-
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 | 3,164 | 3,173 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Notation.Pi
import Mathlib.Data.Set.Lattice
import Mathlib.Order.Filter.Defs
/-!
# Theory of... | Mathlib/Order/Filter/Basic.lean | 3,237 | 3,239 | |
/-
Copyright (c) 2022 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.Algebra.Ring.Regular
import Mathlib.Algebra.Equiv.TransferInstance
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.Algebra.BigOperators.Ring.Finset... | def compAddMonoidHom (φ : AddChar B M) (f : A →+ B) : AddChar A M :=
toAddMonoidHomEquiv.symm (φ.toAddMonoidHom.comp f)
@[simp, norm_cast]
| Mathlib/Algebra/Group/AddChar.lean | 229 | 232 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Algebra.Field.NegOnePow
import Mathlib.Algebra.Field.Periodic
import Mathlib.Algebra.Qua... |
theorem sin_add_pi (x : ℂ) : sin (x + π) = -sin x :=
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | 1,019 | 1,020 |
/-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Joël Riou
-/
import Mathlib.CategoryTheory.Adjunction.Restrict
import Mathlib.CategoryTheory.Adjunction.Whiskering
import Mathlib.CategoryTheory.Sites.PreservesSheafification
... | Mathlib/CategoryTheory/Sites/Adjunction.lean | 136 | 143 | |
/-
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... | Mathlib/Analysis/SpecialFunctions/Integrals.lean | 833 | 836 | |
/-
Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Johannes Hölzl, Rémy Degenne
-/
import Mathlib.Order.ConditionallyCompleteLattice.Indexed
import Mathlib.Order.Filter.IsBounded
import Mathlib.Order.Hom.CompleteL... | Mathlib/Order/LiminfLimsup.lean | 1,217 | 1,221 | |
/-
Copyright (c) 2020 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.Fin.Tuple.Basic
/-!
# Matrix and vector notation
This file defines notation for vectors and matrices. Given `a b c d : α`,
the notation a... | let _ ← synthInstanceQ q(NeZero $etailn)
have i_lit : Q(ℕ) := mkRawNatLit (wrapped_i - xs.length)
| Mathlib/Data/Fin/VecNotation.lean | 183 | 184 |
/-
Copyright (c) 2023 Xavier Généreux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Généreux
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
import Mathlib.Analysis.Complex.PhragmenLindelof
/-!
# Hadamard three-lines Theorem
In this file we present a ... |
Note that if `sSupNormIm f 0 = 0 ∨ sSupNormIm f 1 = 0` then the power is not continuous
since `0 ^ 0 = 1`. Hence the use of `ite`. -/
noncomputable def interpStrip (z : ℂ) : ℂ :=
| Mathlib/Analysis/Complex/Hadamard.lean | 244 | 247 |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jens Wagemaker
-/
import Mathlib.Algebra.Ring.Associated
import Mathlib.Algebra.Ring.Regular
/-!
# Monoids with normalization functions, `gcd`, and `lcm`
This file de... | ((mul_dvd_mul_iff_left ha).1 <| eq ▸ gcd_dvd_right _ _)))
(dvd_gcd (mul_dvd_mul_left a <| gcd_dvd_left _ _) (mul_dvd_mul_left a <| gcd_dvd_right _ _))
theorem gcd_mul_left' [GCDMonoid α] (a b c : α) :
Associated (gcd (a * b) (a * c)) (a * gcd b c) := by
obtain rfl | ha := eq_or_ne a 0
· simp ... | Mathlib/Algebra/GCDMonoid/Basic.lean | 403 | 412 |
/-
Copyright (c) 2019 Calle Sönne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology... | exact abs_sin_eq_of_two_nsmul_eq h
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 402 | 402 |
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Simon Hudon
-/
import Mathlib.Control.Functor.Multivariate
import Mathlib.Data.PFunctor.Multivariate.Basic
import Mathlib.Data.PFunctor.Multivariate.M
import Mathlib.Data... | elab_rules : tactic
| `(tactic| mv_bisim $e $[ with $ids:binderIdent*]?) => do
let ids : TSyntaxArray `Lean.binderIdent := ids.getD #[]
let idsn (n : ℕ) : Name :=
| Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean | 422 | 425 |
/-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin, Lu-Ming Zhang
-/
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Algebra.Pi
import Mathlib.Algebra.BigOp... | Mathlib/Data/Matrix/Basic.lean | 2,320 | 2,324 | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Data.Nat.ModEq
/-!
# Congruences modulo an integer
This file defines the equivalence relation `a ≡ b [ZMOD n]` on the integers, similarly to how
`Data.N... | Mathlib/Data/Int/ModEq.lean | 311 | 314 | |
/-
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.Analytic.Uniqueness
import Mathlib.Analysis.Calculus.DiffContOnCl
import Mathlib.Analysis.Calculus.DSlope
import Mathlib.Analysis.Calculus.F... | $\oint_{|z-c|=R}(z-w)^{-1}f(z)\,dz=2πif(w)$. -/
theorem _root_.DiffContOnCl.circleIntegral_sub_inv_smul {R : ℝ} {c w : ℂ} {f : ℂ → E}
(h : DiffContOnCl ℂ f (ball c R)) (hw : w ∈ ball c R) :
(∮ z in C(c, R), (z - w)⁻¹ • f z) = (2 * π * I : ℂ) • f w :=
circleIntegral_sub_inv_smul_of_differentiable_on_off_counta... | Mathlib/Analysis/Complex/CauchyIntegral.lean | 496 | 502 |
/-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Algebra.DirectSum.Algebra
import Mathlib.Analysis.Complex.UpperHalfPlane.FunctionsBoundedAtInfty
import Mathlib.Analysis.Complex.UpperHalfPlane.Manifol... | Mathlib/NumberTheory/ModularForms/Basic.lean | 445 | 450 | |
/-
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, Patrick Massot
-/
import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic
import Mathlib.MeasureTheory.Measure.Real
import Mathlib.Order.Filter.Indi... | theorem tendsto_integral_filter_of_dominated_convergence {ι} {l : Filter ι} [l.IsCountablyGenerated]
{F : ι → α → G} {f : α → G} (bound : α → ℝ) (hF_meas : ∀ᶠ n in l, AEStronglyMeasurable (F n) μ)
(h_bound : ∀ᶠ n in l, ∀ᵐ a ∂μ, ‖F n a‖ ≤ bound a) (bound_integrable : Integrable bound μ)
(h_lim : ∀ᵐ a ∂μ, Ten... | Mathlib/MeasureTheory/Integral/DominatedConvergence.lean | 66 | 75 |
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Rémy Degenne
-/
import Mathlib.Probability.Process.Adapted
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
/-!
# Stopping times, stopped processes and stopped va... | theorem condExp_stopping_time_ae_eq_restrict_eq_of_countable [Countable ι]
[SigmaFiniteFiltration μ ℱ] (hτ : IsStoppingTime ℱ τ)
[SigmaFinite (μ.trim hτ.measurableSpace_le_of_countable)] (i : ι) :
μ[f|hτ.measurableSpace] =ᵐ[μ.restrict {x | τ x = i}] μ[f|ℱ i] :=
condExp_stopping_time_ae_eq_restrict_eq_of_c... | Mathlib/Probability/Process/Stopping.lean | 1,064 | 1,073 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Justus Springer
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.AlgebraicGeometry.StructureSheaf
import Mathlib.RingTheory.Localization.Localizatio... | map f := Spec.sheafedSpaceMap f.unop
map_comp f g := by simp [Spec.sheafedSpaceMap_comp]
/-- Spec, as a contravariant functor from commutative rings to presheafed spaces.
-/
def Spec.toPresheafedSpace : CommRingCat.{u}ᵒᵖ ⥤ PresheafedSpace CommRingCat :=
Spec.toSheafedSpace ⋙ SheafedSpace.forgetToPresheafedSpace
... | Mathlib/AlgebraicGeometry/Spec.lean | 129 | 137 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Analytic.Within
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries
/-!
# Higher d... | Mathlib/Analysis/Calculus/ContDiff/Defs.lean | 1,303 | 1,306 | |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Andrew Zipperer, Haitao Zhang, Minchao Wu, Yury Kudryashov
-/
import Mathlib.Data.Set.Prod
import Mathlib.Data.Set.Restrict
/-!
# Functions over sets
This file contains... | Mathlib/Data/Set/Function.lean | 1,479 | 1,493 | |
/-
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... | rw [mem_divisors, Nat.dvd_prime_pow pp, and_iff_left (ne_of_gt (pow_pos pp.pos k))]
theorem Prime.divisors {p : ℕ} (pp : p.Prime) : divisors p = {1, p} := by
ext
rw [mem_divisors, dvd_prime pp, and_iff_left pp.ne_zero, Finset.mem_insert, Finset.mem_singleton]
theorem Prime.properDivisors {p : ℕ} (pp : p.Prime) ... | Mathlib/NumberTheory/Divisors.lean | 396 | 414 |
/-
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 log_b_ne_zero b_pos b_ne_one
theorem rpow_logb_eq_abs (hx : x ≠ 0) : b ^ logb b x = |x| := by
apply log_injOn_pos
| Mathlib/Analysis/SpecialFunctions/Log/Base.lean | 126 | 129 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Moritz Doll
-/
import Mathlib.LinearAlgebra.Prod
/-!
# Partially defined linear maps
A `LinearPMap R E F` or `E →ₗ.[R] F` is a linear map from a submodule of `E` to... |
end Zero
section SMul
variable {M N : Type*} [Monoid M] [DistribMulAction M F] [SMulCommClass R M F]
| Mathlib/LinearAlgebra/LinearPMap.lean | 346 | 351 |
/-
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.BigOperators.Option
import Mathlib.Analysis.BoxIntegral.Box.Basic
import Mathlib.Data.Set.Pairwise.Lattice
/-!
# Partitions of rectangular b... | J ∈ π₁ ⊓ π₂ ↔ ∃ J₁ ∈ π₁, ∃ J₂ ∈ π₂, (J : WithBot (Box ι)) = ↑J₁ ⊓ ↑J₂ := by
simp only [inf_def, mem_biUnion, mem_restrict]
@[simp]
theorem iUnion_inf (π₁ π₂ : Prepartition I) : (π₁ ⊓ π₂).iUnion = π₁.iUnion ∩ π₂.iUnion := by
simp only [inf_def, iUnion_biUnion, iUnion_restrict, ← iUnion_inter, ← iUnion_def]
| Mathlib/Analysis/BoxIntegral/Partition/Basic.lean | 513 | 519 |
/-
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.Ring.Int
import Mathlib.Data.Nat.Cast.Order.Basic
import Mathlib.Order.Interval.Set.OrdConnected
import Mathlib.Order.Nat
import Mathli... | (castOrderEmbedding (α := ℤ)).image_Ici (by simp [isUpperSet_Ici]) a
| Mathlib/Data/Nat/Cast/SetInterval.lean | 46 | 47 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.GroupWithZero.Divisibili... |
theorem C_eq_coe_nat (n : ℕ) : (C ↑n : MvPolynomial σ R) = n := by
induction n <;> simp [*]
theorem C_mul' : MvPolynomial.C a * p = a • p :=
(Algebra.smul_def a p).symm
| Mathlib/Algebra/MvPolynomial/Basic.lean | 243 | 248 |
/-
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.Combinatorics.Hall.Basic
import Mathlib.Data.Matrix.Rank
import Mathlib.LinearAlgebra.Projectivization.Constructions
/-!
# Configurations of Points ... | variable (P) {L}
theorem pointCount_eq [Finite P] [Finite L] (l : L) : pointCount P l = order P L + 1 :=
| Mathlib/Combinatorics/Configuration.lean | 400 | 402 |
/-
Copyright (c) 2023 Sophie Morel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sophie Morel
-/
import Mathlib.Analysis.Analytic.Constructions
import Mathlib.Analysis.Analytic.CPolynomialDef
/-! # Properties of continuously polynomial functions
We expand the API a... | Mathlib/Analysis/Analytic/CPolynomial.lean | 284 | 294 | |
/-
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.Data.Countable.Basic
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Data.Set.Subsingleton
import Mathlib.Logic.Equiv.List
/-!
# Countable sets
I... | rw [image_eq_range]
have := hs.to_subtype
apply countable_range
theorem MapsTo.countable_of_injOn {s : Set α} {t : Set β} {f : α → β} (hf : MapsTo f s t)
| Mathlib/Data/Set/Countable.lean | 157 | 161 |
/-
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, Yaël Dillies
-/
import Mathlib.Topology.Sets.Opens
import Mathlib.Topology.Clopen
/-!
# Closed sets
We define a few types of closed sets in a topological space.
... |
/-! ### Clopen sets -/
/-- The type of clopen sets of a topological space. -/
| Mathlib/Topology/Sets/Closeds.lean | 256 | 260 |
/-
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... | · rw [isComplex₂_iff]
exact h.toIsComplex.zero 0
exact h.exact 0 (by omega)
· exact Exact.mk (IsComplex.mk (fun i hi => h.toIsComplex.zero (i + 1)))
(fun i hi => h.exact (i + 1))
· rintro ⟨h, h₀⟩
refine Exact.mk (IsComplex.mk (fun i hi => ?_)) (fun i hi => ?_)
· obtain _ | i :=... | Mathlib/Algebra/Homology/ExactSequence.lean | 250 | 270 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Eric Wieser
-/
import Mathlib.Algebra.Group.Fin.Tuple
import Mathlib.Data.Matrix.RowCol
import Mathlib.Data.Fin.VecNotation
import Mathlib.Tactic.FinCases
import Mathlib.Alge... | section Submatrix
@[simp]
theorem submatrix_empty (A : Matrix m' n' α) (row : Fin 0 → m') (col : o' → n') :
| Mathlib/Data/Matrix/Notation.lean | 379 | 382 |
/-
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... | | nil => rfl
| node _ l y r => by
unfold Ordnode.insert insertWith; cases cmpLE x y <;> simp [insert_eq_insertWith]
| Mathlib/Data/Ordmap/Ordset.lean | 518 | 520 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Kexing Ying, Moritz Doll
-/
import Mathlib.Algebra.GroupWithZero.Action.Opposite
import Mathlib.LinearAlgebra.Finsupp.VectorSpace
import Mathlib.LinearAlgebra.Matrix.Basis
im... | Mathlib/LinearAlgebra/Matrix/SesquilinearForm.lean | 722 | 724 |
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