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) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
/-!
# Kernels and cokernels
In a category with zero morphisms, the kernel of a morphism `f : X... |
/- Porting note: induction on Eq is trying instantiate another g... -/
@[reassoc (attr := simp)]
theorem kernelIsoOfEq_hom_comp_ι {f g : X ⟶ Y} [HasKernel f] [HasKernel g] (h : f = g) :
| Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean | 369 | 372 |
/-
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.ZetaValues
import Mathlib.NumberTheory.LSeries.RiemannZeta
/-!
# Special values of Hurwitz and Riemann zeta functions
This file gives t... | theorem cosZeta_two_mul_nat (hk : k ≠ 0) (hx : x ∈ Icc 0 1) :
cosZeta x (2 * k) = (-1) ^ (k + 1) * (2 * π) ^ (2 * k) / 2 / (2 * k)! *
((Polynomial.bernoulli (2 * k)).map (algebraMap ℚ ℂ)).eval (x : ℂ) := by
rw [← (hasSum_nat_cosZeta x (?_ : 1 < re (2 * k))).tsum_eq]
· refine Eq.trans ?_ <|
(congr_ar... | Mathlib/NumberTheory/LSeries/HurwitzZetaValues.lean | 49 | 67 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Data.List.Basic
/-!
# Double universal quantification on a list
This file provides an API for `List.Forall₂` (definition in `Data.Lis... | @right_unique_forall₂' _ _ _ hr
theorem _root_.Relator.BiUnique.forall₂ (hr : BiUnique R) : BiUnique (Forall₂ R) :=
⟨hr.left.forall₂, hr.right.forall₂⟩
| Mathlib/Data/List/Forall2.lean | 123 | 126 |
/-
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.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Or... | Mathlib/Analysis/Convex/Basic.lean | 696 | 698 | |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Multiset.FinsetOps
import Mathlib.Data.Multiset.Fold
/-!
# Lattice operations on multisets
-/
namespace Multiset
variable {α : Type*}
/-! ###... | Mathlib/Data/Multiset/Lattice.lean | 168 | 169 | |
/-
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.Algebra.BigOperators.Field
import Mathlib.NumberTheory.LSeries.Basic
/-!
# Linearity of the L-series of `f` as a function of `f`
We show that the `LSer... | simpa [LSeriesSummable, term_neg] using Summable.neg hf
@[simp]
| Mathlib/NumberTheory/LSeries/Linearity.lean | 62 | 64 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Yuyang Zhao
-/
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Polynomial.AlgebraMap
/-!
# Algebra towers for polynomial
This file proves some basic results about t... | rw [aeval_def, eval_map]
end Semiring
| Mathlib/RingTheory/Polynomial/Tower.lean | 41 | 44 |
/-
Copyright (c) 2019 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
import Mathlib.Algebra.ContinuedFractions.Basic
import Mathlib.Algebra.GroupWithZero.Basic
/-!
# Basic Translation Lemmas Between Functions Defined for Continued... | Mathlib/Algebra/ContinuedFractions/Translations.lean | 166 | 167 | |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Simon Hudon, Mario Carneiro
-/
import Mathlib.Algebra.Notation.Defs
import Mathlib.Data.Int.Notation
import Mathlib.Data.Nat.BinaryRec
import Mathlib.L... | mul_right_cancel := RightCancelSemigroup.mul_right_cancel }
end CancelMonoid
| Mathlib/Algebra/Group/Defs.lean | 726 | 728 |
/-
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 restrictSubtype_indep_iff_of_subset (hIX : I ⊆ X) :
(M.restrictSubtype X).Indep (X ↓∩ I) ↔ M.Indep I := by
rw [restrictSubtype_indep_iff, image_preimage_eq_iff.2]; simpa
| Mathlib/Data/Matroid/Map.lean | 648 | 651 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Group.Subgroup.Ker
import Mathlib.Algebra.BigOperators.Group.List.Basic
/-!
# Free groups
This file defines free groups over a type. Furthermore, it is... | Mathlib/GroupTheory/FreeGroup/Basic.lean | 1,410 | 1,411 | |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Vector.Defs
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.OfFn
import Mathlib.Data.List.Scan
import Mathlib.Control.Applicative
import M... |
@[simp]
| Mathlib/Data/Vector/Basic.lean | 221 | 222 |
/-
Copyright (c) 2021 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno
-/
import Mathlib.CategoryTheory.NatIso
/-!
# Bicategories
In this file we define typeclass for bicategories.
A bicategory `B` consists of
* objects `a : B`,
* 1-morphisms ... | @[reassoc]
theorem whiskerRight_comp_symm {f f' : a ⟶ b} (η : f ⟶ f') (g : b ⟶ c) (h : c ⟶ d) :
η ▷ g ▷ h = (α_ f g h).hom ≫ η ▷ (g ≫ h) ≫ (α_ f' g h).inv := by simp
| Mathlib/CategoryTheory/Bicategory/Basic.lean | 303 | 306 |
/-
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.Minor.Restrict
/-!
# Some constructions of matroids
This file defines some very elementary examples of matroids, namely those with at most o... | theorem uniqueBaseOn_inter_ground_eq (I E : Set α) :
uniqueBaseOn (I ∩ E) E = uniqueBaseOn I E := by
simp only [uniqueBaseOn, restrict_eq_restrict_iff, freeOn_indep_iff, subset_inter_iff,
iff_self_and]
tauto
| Mathlib/Data/Matroid/Constructions.lean | 211 | 215 |
/-
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, Floris van Doorn
-/
import Mathlib.Data.Countable.Small
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Powerset
import Mathlib.Dat... |
theorem mk_sUnion_le {α : Type u} (A : Set (Set α)) : #(⋃₀ A) ≤ #A * ⨆ s : A, #s := by
rw [sUnion_eq_iUnion]
apply mk_iUnion_le
theorem mk_biUnion_le {ι α : Type u} (A : ι → Set α) (s : Set ι) :
| Mathlib/SetTheory/Cardinal/Basic.lean | 736 | 741 |
/-
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.BigOperators.Finprod
import Mathlib.Order.Filter.AtTopBot.BigOperators
import Mathlib.Topology.Separation.Hausdorff
/-!
# Infinite sum and pro... | simp only [HasProd, Tendsto, comp_apply, hg.map_atTop_finset_prod_eq hf]
@[to_additive]
| Mathlib/Topology/Algebra/InfiniteSum/Defs.lean | 129 | 131 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Yury Kudryashov, Neil Strickland
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Group.Hom.Defs
import Mathlib.Algebra.G... | mul_left_cancel_of_ne_zero ha h := by
rw [← sub_eq_zero, ← mul_sub] at h
exact sub_eq_zero.1 ((eq_zero_or_eq_zero_of_mul_eq_zero h).resolve_left ha)
mul_right_cancel_of_ne_zero hb h := by
rw [← sub_eq_zero, ← sub_mul] at h
| Mathlib/Algebra/Ring/Basic.lean | 130 | 134 |
/-
Copyright (c) 2020 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard
-/
import Mathlib.RingTheory.AdicCompletion.Basic
import Mathlib.RingTheory.LocalRing.MaximalIdeal.Basic
import Mathlib.RingTheory.LocalRing.RingHom.Basic
import Mathlib.R... | rw [← Associates.mk_eq_mk_iff_associated] at H ⊢
rw [← H, ← Associates.prod_mk, Associates.mk_pow, ← Multiset.prod_replicate]
congr 1
rw [Multiset.eq_replicate]
simp only [true_and, and_imp, Multiset.card_map, eq_self_iff_true, Multiset.mem_map, exists_imp]
rintro _ _ _ rfl
rw [Associates.mk_eq_mk_iff_ass... | Mathlib/RingTheory/DiscreteValuationRing/Basic.lean | 323 | 339 |
/-
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... | { liftRingHom φ.toRingHom hφ with
commutes' := fun r => by
simp_rw [RingHom.toFun_eq_coe, AlgHom.toRingHom_eq_coe, algebraMap_apply r,
liftRingHom_apply_div, AlgHom.coe_toRingHom, map_one, div_one, AlgHom.commutes] }
theorem liftAlgHom_apply_ofFractionRing_mk (n : K[X]) (d : K[X]⁰) :
liftAlgHom... | Mathlib/FieldTheory/RatFunc/Basic.lean | 625 | 632 |
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calculus.FDeriv.Comp
/-!
# Additive operations on derivative... |
@[simp]
theorem differentiableOn_const_add_iff (c : F) :
| Mathlib/Analysis/Calculus/FDeriv/Add.lean | 288 | 290 |
/-
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 | 1,430 | 1,431 | |
/-
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, Kevin Buzzard, Yury Kudryashov, Frédéric Dupuis,
Heather Macbeth
-/
import Mathlib.Algebra.Module.Submodule.Ker
import Mathlib.Algebra.Module.Submodul... | lemma comap_subtype_le_iff {p q r : Submodule R M} :
q.comap p.subtype ≤ r.comap p.subtype ↔ p ⊓ q ≤ p ⊓ r :=
| Mathlib/Algebra/Module/Submodule/Range.lean | 297 | 298 |
/-
Copyright (c) 2022 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang, Eric Wieser
-/
import Mathlib.RingTheory.GradedAlgebra.Homogeneous.Ideal
/-!
This file contains a proof that the radical of any homogeneous ideal is a homogeneous ideal
... | rintro x y hx hy hxy
have H := h.mem_or_mem (Ideal.toIdeal_homogeneousCore_le 𝒜 I hxy)
refine H.imp ?_ ?_
· exact Ideal.mem_homogeneousCore_of_homogeneous_of_mem hx
· exact Ideal.mem_homogeneousCore_of_homogeneous_of_mem hy
theorem Ideal.IsHomogeneous.radical_eq {I : Ideal A} (hI : I.IsHomogeneous 𝒜) :
... | Mathlib/RingTheory/GradedAlgebra/Radical.lean | 149 | 157 |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Module.BigOperators
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.Nat.Squarefree
imp... | · exact hp.ne_one
· exact hn.ne'
· rw [ZeroHom.map_zero, ZeroHom.map_zero]
· simp
· intro a b _ha _hb hab ha' hb'
| Mathlib/NumberTheory/ArithmeticFunction.lean | 1,073 | 1,077 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker, Johan Commelin
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Div
import Mathlib.RingTheory... | Mathlib/Algebra/Polynomial/RingDivision.lean | 718 | 719 | |
/-
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.SetTheory.Game.State
/-!
# Domineering as a combinatorial game.
We define the game of Domineering, played on a chessboard of arbitrary shape
(possibly ev... |
theorem card_of_mem_left {b : Board} {m : ℤ × ℤ} (h : m ∈ left b) : 2 ≤ Finset.card b := by
have w₁ : m ∈ b := (Finset.mem_inter.1 h).1
have w₂ : (m.1, m.2 - 1) ∈ b.erase m := snd_pred_mem_erase_of_mem_left h
have i₁ := Finset.card_erase_lt_of_mem w₁
| Mathlib/SetTheory/Game/Domineering.lean | 79 | 83 |
/-
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, Floris van Doorn
-/
import Mathlib.Data.Countable.Small
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Powerset
import Mathlib.Dat... | Mathlib/SetTheory/Cardinal/Basic.lean | 1,211 | 1,214 | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Polynomial.Reverse
import Mathlib.Algebra.Regular.SMul
/-!
# Theory of monic polynomials
We give se... | rw [degree_mul', hp.degree_mul]
· exact add_comm _ _
| Mathlib/Algebra/Polynomial/Monic.lean | 171 | 172 |
/-
Copyright (c) 2022 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer, Kevin Klinge, Andrew Yang
-/
import Mathlib.Algebra.Group.Submonoid.DistribMulAction
import Mathlib.GroupTheory.OreLocalization.Basic
import Mathlib.Algebra.GroupWi... | rw [← Subtype.coe_eq_of_eq_mk ha]
repeat rw [oreDiv_smul_oreDiv]
simp only [smul_add, add_oreDiv]
instance : DistribMulAction R[S⁻¹] X[S⁻¹] where
smul_zero := OreLocalization.smul_zero
smul_add := OreLocalization.smul_add
instance {R₀} [Monoid R₀] [MulAction R₀ X] [MulAction R₀ R]
[IsScalarTower R₀ R X]... | Mathlib/RingTheory/OreLocalization/Basic.lean | 214 | 234 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.MeasurableSpace.MeasurablyGenerated
import Mathlib.MeasureTheory.Measure.NullMeasurable
import Mathlib.Order.Interval.Set... | ⟨fun μ₁ μ₂ =>
{ toOuterMeasure := μ₁.toOuterMeasure + μ₂.toOuterMeasure
m_iUnion := fun s hs hd =>
show μ₁ (⋃ i, s i) + μ₂ (⋃ i, s i) = ∑' i, (μ₁ (s i) + μ₂ (s i)) by
rw [ENNReal.tsum_add, measure_iUnion hd hs, measure_iUnion hd hs]
trim_le := by rw [OuterMeasure.trim_add, μ₁.trimmed... | Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 782 | 793 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Amelia Livingston, Yury Kudryashov,
Neil Strickland, Aaron Anderson
-/
import Mathlib.Algebra.GroupWithZero.Units.Basic
import Mathli... | obtain rfl | h0 := eq_or_ne m 0; · rwa [zero_mul]
intro b c h
obtain ⟨a₁, a₂, ⟨b, rfl⟩, ⟨c, rfl⟩, rfl⟩ := hm (dvd_of_mul_right_dvd h)
| Mathlib/Algebra/GroupWithZero/Divisibility.lean | 122 | 124 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Peter Nelson
-/
import Mathlib.Order.Antichain
/-!
# Minimality and Maximality
This file proves basic facts about minimality and maximality
of an element with respect to ... | theorem Set.maximal_iff_forall_ssuperset : Maximal P s ↔ P s ∧ ∀ ⦃t⦄, s ⊂ t → ¬ P t :=
maximal_iff_forall_gt
| Mathlib/Order/Minimal.lean | 327 | 328 |
/-
Copyright (c) 2023 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Data.List.Infix
import Mathlib.Data.List.MinMax
import Mathlib.Data.List.EditDistance.Defs
/-!
# Lower ... | (suffixLevenshtein C xs ys₂).1.minimum ≤ (suffixLevenshtein C xs (ys₁ ++ ys₂)).1.minimum := by
induction ys₁ with
| nil => exact le_refl _
| cons y ys₁ ih => exact ih.trans (le_suffixLevenshtein_cons_minimum _ _ _)
| Mathlib/Data/List/EditDistance/Bounds.lean | 75 | 79 |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Batteries.Data.List.Perm
import Mathlib.Data.List.OfFn
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.TakeWhile
import Mathlib.Order.Fin.Basic
/-!
# So... | simp [Sorted, pairwise_map, e.map_rel_iff]
end RelEmbedding
namespace OrderEmbedding
| Mathlib/Data/List/Sort.lean | 265 | 270 |
/-
Copyright (c) 2020 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Yaël Dillies
-/
import Mathlib.Order.Interval.Set.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Tactic.Bound.Attribute
import Mathlib.Tactic.Contrapose
import Mathl... | ```
#eval Nat.logTR 2 (2 ^ 1000000)
#eval Nat.logC 2 (2 ^ 1000000)
```
The definition `Nat.logC` is not tail-recursive, however, but the stack limit will only be reached
if the output size is around 2^10000, meaning the input will be around 2^(2^10000), which will
take far too long to compute in the first place.
Adap... | Mathlib/Data/Nat/Log.lean | 349 | 359 |
/-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll, Anatole Dedecker
-/
import Mathlib.Analysis.LocallyConvex.Bounded
import Mathlib.Analysis.Seminorm
import Mathlib.Data.Real.Sqrt
import Mathlib.Topology.Algebra.Equicontinuit... | variable [NontriviallyNormedField 𝕜] [AddCommGroup E] [Module 𝕜 E]
[SeminormedAddCommGroup F] [NormedSpace 𝕜 F]
{p : SeminormFamily 𝕜 E ι}
/-- In a semi-`NormedSpace`, a continuous seminorm is zero on elements of norm `0`. -/
lemma map_eq_zero_of_norm_zero (q : Seminorm 𝕜 F)
(hq : Continuous q) {x : F} (h... | Mathlib/Analysis/LocallyConvex/WithSeminorms.lean | 734 | 743 |
/-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Bilinear
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Group.Pointwise.Finset.Basic
import Mathlib.Algebra.Group.Pointwise.Set.B... | (∏ i ∈ s, Submodule.span R (M i)) = Submodule.span R (∏ i ∈ s, M i) := by
letI := Classical.decEq ι
| Mathlib/Algebra/Algebra/Operations.lean | 733 | 734 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Data.List.Basic
/-!
# Double universal quantification on a list
This file provides an API for `List.Forall₂` (definition in `Data.Lis... | Iff.intro
(fun h => by
rw [← reverse_reverse l₁, ← reverse_reverse l₂]
exact rel_reverse h)
| Mathlib/Data/List/Forall2.lean | 219 | 222 |
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Data.Set.Prod
/-!
# N-ary images of sets
This file defines `Set.image2`, the binary image of sets.
This is mostly useful to define pointwise oper... | Mathlib/Data/Set/NAry.lean | 382 | 387 | |
/-
Copyright (c) 2023 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta, Doga Can Sertbas
-/
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Data.Nat.ModEq
import Mathlib.Data.Nat.Prime.Defs
import Mathlib.Data.Real.Archimedea... | lemma schnirelmannDensity_setOf_modeq_one {m : ℕ} :
schnirelmannDensity {n | n ≡ 1 [MOD m]} = (m⁻¹ : ℝ) := by
rcases eq_or_ne m 1 with rfl | hm
· simp [Nat.modEq_one]
rw [← schnirelmannDensity_setOf_mod_eq_one hm]
apply schnirelmannDensity_congr
ext n
simp only [Set.mem_setOf_eq, Nat.ModEq, Nat.one_mod_... | Mathlib/Combinatorics/Schnirelmann.lean | 255 | 262 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Independence.Kernel
import Mathlib.MeasureTheory.Constructions.Pi
/-!
# Independence of sets of sets and measure spaces (σ-algebras)
* A fami... |
lemma iIndepSets.meas_iInter [Fintype ι] (h : iIndepSets π μ) (hs : ∀ i, s i ∈ π i) :
μ (⋂ i, s i) = ∏ i, μ (s i) := by simp [← h.meas_biInter _ fun _i _ ↦ hs _]
| Mathlib/Probability/Independence/Basic.lean | 152 | 155 |
/-
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... | Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean | 777 | 781 | |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.GroupWithZero.Invertible
import Mathlib.Algebra.Ring.Defs
/-!
# Theorems about invertible elements in rings
-/
universe u
variable {R : Type u}
/... | theorem Ring.inverse_add_inverse [Semiring R] {a b : R} (h : IsUnit a ↔ IsUnit b) :
Ring.inverse a + Ring.inverse b = Ring.inverse a * (a + b) * Ring.inverse b := by
by_cases ha : IsUnit a
| Mathlib/Algebra/Ring/Invertible.lean | 49 | 51 |
/-
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 | 780 | 781 | |
/-
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]
theorem Ioo_insert_left (h : a < b) : insert a (Ioo a b) = Ico a b := by
| Mathlib/Order/Interval/Finset/Basic.lean | 586 | 587 |
/-
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.AlgebraicGeometry.EllipticCurve.Affine
import Mathlib.LinearAlgebra.FreeModule.Norm
import Mathlib.RingTheory.ClassGroup
import Mat... | @[simps]
noncomputable def toClass : W.Point →+ Additive (ClassGroup W.CoordinateRing) where
toFun P := match P with
| 0 => 0
| some h => Additive.ofMul <| ClassGroup.mk <| CoordinateRing.XYIdeal' h
| Mathlib/AlgebraicGeometry/EllipticCurve/Group.lean | 490 | 494 |
/-
Copyright (c) 2020 Alexander Bentkamp, Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Sébastien Gouëzel, Eric Wieser
-/
import Mathlib.Algebra.Algebra.RestrictScalars
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Data.... | lemma realPart_imaginaryPart {x : A} : ℜ (ℑ x : A) = ℑ x :=
Subtype.ext <| (ℑ x).property.coe_realPart
lemma realPart_surjective : Function.Surjective (realPart (A := A)) :=
fun x ↦ ⟨(x : A), Subtype.ext x.property.coe_realPart⟩
| Mathlib/Data/Complex/Module.lean | 445 | 450 |
/-
Copyright (c) 2020 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Yaël Dillies
-/
import Mathlib.Order.Interval.Set.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Tactic.Bound.Attribute
import Mathlib.Tactic.Contrapose
import Mathl... | section computation
private lemma logC_aux {m b : ℕ} (hb : 1 < b) (hbm : b ≤ m) : m / (b * b) < m / b := by
have hb' : 0 < b := zero_lt_of_lt hb
rw [div_lt_iff_lt_mul (Nat.mul_pos hb' hb'), ← Nat.mul_assoc, ← div_lt_iff_lt_mul hb']
exact (Nat.lt_mul_iff_one_lt_right (Nat.div_pos hbm hb')).2 hb
| Mathlib/Data/Nat/Log.lean | 326 | 331 |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Casper Putz, Anne Baanen, Wen Yang
-/
import Mathlib.LinearAlgebra.Matrix.Transvection
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mat... | simpa using hM (lt_of_lt_of_le j.2 <| le_of_not_lt i.2)
have h_mul_eq_zero : M⁻¹.toBlock q p * M.toBlock p p = 0 := by simpa [h_zero] using h_sum
haveI : Invertible (M.toBlock p p) := hM.invertibleToBlock k
have : (fun i => k ≤ b i) = q := by
ext
exact not_lt.symm
rw [this, ← Matrix.zero_mul (M.toBl... | Mathlib/LinearAlgebra/Matrix/Block.lean | 347 | 367 |
/-
Copyright (c) 2021 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.AlgebraicGeometry.Restrict
import Mathlib.CategoryTheory.Adjunction.Limits
import Mathlib.CategoryTheory.Adjunction.Reflective
/-!
# Adjunction between `Γ` and ... |
/-- Spec preserves limits. -/
instance : Limits.PreservesLimits Spec.toLocallyRingedSpace :=
ΓSpec.locallyRingedSpaceAdjunction.rightAdjoint_preservesLimits
instance Spec.preservesLimits : Limits.PreservesLimits Scheme.Spec :=
| Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | 481 | 487 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.HomotopyCategory.HomComplex
import Mathlib.Algebra.Homology.HomotopyCofiber
/-! # The mapping cone of a morphism of cochain complexes
In this ... | lemma inr_descCochain :
(Cochain.ofHom (inr φ)).comp (descCochain φ α β h) (zero_add n) = β := by
simp [descCochain]
@[reassoc (attr := simp)]
lemma inl_v_descCochain_v (p₁ p₂ p₃ : ℤ) (h₁₂ : p₁ + (-1) = p₂) (h₂₃ : p₂ + n = p₃) :
(inl φ).v p₁ p₂ h₁₂ ≫ (descCochain φ α β h).v p₂ p₃ h₂₃ =
| Mathlib/Algebra/Homology/HomotopyCategory/MappingCone.lean | 307 | 313 |
/-
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.TypeTags.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Finset.Piecewise
import Mathlib.Or... | Mathlib/Topology/Constructions.lean | 1,537 | 1,552 | |
/-
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.ImproperIntegrals
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.MeasureTheory.Measure.Haar.NormedSpace
... | congr 1
abel
/-- Suppose `f` is locally integrable on `(0, ∞)`, is `O(x ^ (-a))` as `x → ∞`, and is
`O(x ^ (-b))` as `x → 0`. Then its Mellin transform is differentiable on the domain `b < re s < a`,
with derivative equal to the Mellin transform of `log • f`. -/
theorem mellin_hasDerivAt_of_isBigO_rpow [NormedSpac... | Mathlib/Analysis/MellinTransform.lean | 304 | 312 |
/-
Copyright (c) 2024 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.AlgebraicGeometry.EllipticCurve.Group
import Mathlib.NumberTheory.EllipticDivisibilitySequence
/-!
# Division polynomials of Weier... | rw [φ, ψ_neg, neg_sq (R := R[X][Y]), neg_add_eq_sub, ← neg_sub n, ψ_neg, ← neg_add', ψ_neg,
neg_mul_neg (α := R[X][Y]), mul_comm <| W.ψ _, φ]
| Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Basic.lean | 530 | 532 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.MeasureTheory.MeasurableSpace.EventuallyMeasurable
import Mathlib.MeasureTheory.MeasurableSpace.Basic
import Mathlib.M... | (h : ∀ b ∈ s, NullMeasurableSet (f b) μ) : NullMeasurableSet (⋃ b ∈ s, f b) μ :=
Finite.measurableSet_biUnion hs h
| Mathlib/MeasureTheory/Measure/NullMeasurable.lean | 323 | 324 |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Callum Sutton, Yury Kudryashov
-/
import Mathlib.Algebra.Group.Equiv.Opposite
import Mathlib.Algebra.GroupWithZero.Equiv
import Mathlib.Algebra.GroupWithZero.InjSurj
im... | rfl
| Mathlib/Algebra/Ring/Equiv.lean | 611 | 612 |
/-
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... | instance : Add (LieSubalgebra R L) where add := max
instance : Zero (LieSubalgebra R L) where zero := ⊥
| Mathlib/Algebra/Lie/Subalgebra.lean | 490 | 493 |
/-
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.Group.Units.Basic
import Mathlib.Algebra.GroupWithZero.Basic
import Mathlib.Data.Int.Basic
import Mathlib.Lean.Meta.CongrTheorems
import Mathli... | theorem inverse_mul_cancel_left (x y : M₀) (h : IsUnit x) : inverse x * (x * y) = y := by
rw [← mul_assoc, inverse_mul_cancel x h, one_mul]
| Mathlib/Algebra/GroupWithZero/Units/Basic.lean | 113 | 115 |
/-
Copyright (c) 2023 Moritz Firsching. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Firsching
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.LinearAlgebra.Vanderm... | | 0 => rfl
| (n + 1) => by
simp only [prod_range_succ, mul_pow, mul_add, mul_one, superFactorial_succ,
superFactorial_two_mul n, factorial_succ]
ring
theorem superFactorial_four_mul (n : ℕ) :
| Mathlib/Data/Nat/Factorial/SuperFactorial.lean | 88 | 94 |
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Yury Kudryashov
-/
import Mathlib.Order.UpperLower.Closure
import Mathlib.Order.UpperLower.Fibration
import Mathlib.Tactic.TFAE
import Mathlib.Topology.ContinuousOn
import Ma... | e ▸ refl x
@[symm]
nonrec theorem symm (h : x ~ᵢ y) : y ~ᵢ x := h.symm
@[trans]
nonrec theorem trans (h₁ : x ~ᵢ y) (h₂ : y ~ᵢ z) : x ~ᵢ z := h₁.trans h₂
| Mathlib/Topology/Inseparable.lean | 492 | 499 |
/-
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.Support
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.Data.Nat.Choose.Sum
impo... | rcases p with ⟨p⟩
simp_rw [← monomial_zero_left, ← ofFinsupp_single, ← ofFinsupp_mul, coeff]
exact AddMonoidAlgebra.mul_single_zero_apply p a n
@[simp] lemma coeff_mul_natCast {a k : ℕ} :
| Mathlib/Algebra/Polynomial/Coeff.lean | 160 | 164 |
/-
Copyright (c) 2022 Antoine Labelle, Rémi Bottinelli. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Labelle, Rémi Bottinelli
-/
import Mathlib.Combinatorics.Quiver.Cast
import Mathlib.Combinatorics.Quiver.Symmetric
import Mathlib.Data.Sigma.Basic
import Math... | Quiver.Costar (Symmetrify.of.obj u) ≃ Quiver.Costar u ⊕ Quiver.Star u :=
Equiv.sigmaSumDistrib _ _
theorem Prefunctor.symmetrifyStar (u : U) :
φ.symmetrify.star u =
| Mathlib/Combinatorics/Quiver/Covering.lean | 132 | 136 |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Nat.Totient
import Mathlib.Data.ZMod.Aut
import Mathlib.Data.ZMod.QuotientGroup
import Mathlib.GroupTheory.Exponent
import Mathlib.GroupTheory.Sub... |
/-- Two group homomorphisms `G →* G'` are equal if and only if they agree on a generator of `G`. -/
@[to_additive
"Two homomorphisms `G →+ G'` of additive groups are equal if and only if they agree
on a generator of `G`."]
lemma MonoidHom.eq_iff_eq_on_generator (f₁ f₂ : G →* G') : f₁ = f₂ ↔ f₁ g = f₂ g := by
r... | Mathlib/GroupTheory/SpecificGroups/Cyclic.lean | 834 | 842 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Ring.Associated
import Mathlib.Algebra.Star.Unitary
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.Tactic.Ring
import Mathlib.Al... | · apply Nat.le_add_right
· dsimp
rw [add_comm, add_comm (y : ℤ)]
exact nonneg_add_lem hb ha
· have : Nonneg ⟨_, _⟩ :=
nonnegg_neg_pos.2 (sqLe_add (nonnegg_neg_pos.1 ha) (nonnegg_neg_pos.1 hb))
rw [Nat.cast_add, Nat.cast_add, neg_add] at this
rwa [add_def]
theorem nonneg_iff_zero_le {a : ℤ... | Mathlib/NumberTheory/Zsqrtd/Basic.lean | 597 | 609 |
/-
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.MonoidAlgebra.Degree
import Mathlib.Algebra.MvPolynomial.Rename
/-!
# Degrees of polynomials
This file establ... | intro i
rw [mem_degrees, Multiset.mem_map]
rintro ⟨d, hd, hi⟩
obtain ⟨x, rfl, hx⟩ := coeff_rename_ne_zero _ _ _ hd
simp only [Finsupp.mapDomain, Finsupp.mem_support_iff] at hi
rw [sum_apply, Finsupp.sum] at hi
contrapose! hi
rw [Finset.sum_eq_zero]
| Mathlib/Algebra/MvPolynomial/Degrees.lean | 178 | 185 |
/-
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... | real eigenvalue, with eigenvector `v`. Then a manipulation using the identity
`A ^ 2 - B ^ 2 = A * (A - B) + (A - B) * B` leads to the conclusion that
`⟨v, A v⟩ + ⟨v, B v⟩ = 0`. Since `A, B` are positive semidefinite, both terms must be zero. Thus
| Mathlib/LinearAlgebra/Matrix/PosDef.lean | 224 | 226 |
/-
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... | rw [coe_lt_iff] at h
specialize h H
exact lt_of_le_of_lt hm h
| Mathlib/Data/Nat/PartENat.lean | 721 | 723 |
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Constructions.Polish.Basic
import Mathlib.MeasureTheory.Function.UniformIntegrable
import Mathlib.Probability.Martingale.Upcrossing
/-!
# Mar... |
theorem Submartingale.ae_tendsto_limitProcess_of_uniformIntegrable (hf : Submartingale f ℱ μ)
(hunif : UniformIntegrable f 1 μ) :
∀ᵐ ω ∂μ, Tendsto (fun n => f n ω) atTop (𝓝 (ℱ.limitProcess f μ ω)) :=
let ⟨_, hR⟩ := hunif.2.2
hf.ae_tendsto_limitProcess hR
/-- If a martingale `f` adapted to `ℱ` converges i... | Mathlib/Probability/Martingale/Convergence.lean | 319 | 327 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker, Johan Commelin
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Div
import Mathlib.RingTheory... | Mathlib/Algebra/Polynomial/RingDivision.lean | 738 | 750 | |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Multiset.ZeroCons
/-!
# Basic results on multisets
-/
-- No algebra should be required
assert_not_exists Monoid
universe v
open List S... | Mathlib/Data/Multiset/Basic.lean | 2,964 | 2,972 | |
/-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.ContDiff.Bounds
import Mathlib.Analysis.Calculus.IteratedDeriv.Defs
import Mathlib.Analysis.Calculus.LineDeriv.Basic
import Mathlib.Analysi... | theorem _root_.schwartz_withSeminorms : WithSeminorms (schwartzSeminormFamily 𝕜 E F) := by
have A : WithSeminorms (schwartzSeminormFamily ℝ E F) := ⟨rfl⟩
rw [SeminormFamily.withSeminorms_iff_nhds_eq_iInf] at A ⊢
rw [A]
| Mathlib/Analysis/Distribution/SchwartzSpace.lean | 488 | 491 |
/-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Yury Kudryashov
-/
import Mathlib.Topology.Algebra.Module.Equiv
import Mathlib.Topology.Algebra.Module.UniformConvergence
import Mathlib.Topology.Algebra.Separation... | [TopologicalSpace F] [ContinuousConstSMul R F] [IsTopologicalAddGroup F] (𝔖 : Set (Set E)) :
Module R (UniformConvergenceCLM σ F 𝔖) := ContinuousLinearMap.module
theorem continuousSMul [RingHomSurjective σ] [RingHomIsometric σ]
[TopologicalSpace F] [IsTopologicalAddGroup F] [ContinuousSMul 𝕜₂ F] (𝔖 : S... | Mathlib/Topology/Algebra/Module/StrongTopology.lean | 180 | 189 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
-/
import Mathlib.MeasureTheory.Integral.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Lebesgue.Countable
import Mathlib.MeasureTheory.Integral.Le... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 1,409 | 1,412 | |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.HomotopyCategory.HomComplex
import Mathlib.Algebra.Homology.HomotopyCofiber
/-! # The mapping cone of a morphism of cochain complexes
In this ... | apply d_snd_v
@[simp]
lemma δ_inl :
δ (-1) 0 (inl φ) = Cochain.ofHom (φ ≫ inr φ) := by
ext p
| Mathlib/Algebra/Homology/HomotopyCategory/MappingCone.lean | 274 | 279 |
/-
Copyright (c) 2019 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Algebra.Order.Group.Pointwise.Interval
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.Rat.Cardinal
import Mathlib.SetTheory.Cardi... | not_countable_univ
/-- The cardinality of the interval (a, ∞). -/
theorem mk_Ioi_real (a : ℝ) : #(Ioi a) = 𝔠 := by
refine le_antisymm (mk_real ▸ mk_set_le _) ?_
rw [← not_lt]
intro h
refine _root_.ne_of_lt ?_ mk_univ_real
have hu : Iio a ∪ {a} ∪ Ioi a = Set.univ := by
convert @Iic_union_Ioi ℝ _ _
... | Mathlib/Data/Real/Cardinality.lean | 201 | 211 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Data.Int.Cast.Pi
import Mathlib.Data.Nat.Cast.Basic
import Mathlib.MeasureTheory.MeasurableSpace... | Mathlib/MeasureTheory/MeasurableSpace/Basic.lean | 800 | 804 | |
/-
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.GroupTheory.CoprodI
import Mathlib.GroupTheory.Coprod.Basic
import Mathlib.GroupTheory.Complement
/-!
## Pushouts of Monoids and Groups
This file defin... | let _ := fun i => Classical.decEq (G i)
refine Function.Injective.of_comp
(f := ((· • ·) : PushoutI φ → NormalWord d → NormalWord d)) ?_
intros _ _ h
exact eq_of_smul_eq_smul (fun w : NormalWord d =>
by simp_all [funext_iff, base_smul_eq_smul])
section Reduced
open NormalWord
| Mathlib/GroupTheory/PushoutI.lean | 603 | 612 |
/-
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.Linear.Basic
import Mathlib.Algebra.Homology.ComplexShapeSigns
import Mathlib.Algebra.Homology.HomologicalBicomplex
import Mathlib.Algebra.Module.... | K.ιTotalOrZero c₁₂ i₁ i₂ i₁₂ = 0 := dif_neg h
@[reassoc (attr := simp)]
lemma ι_D₁ (i₁₂ i₁₂' : I₁₂) (i₁ : I₁) (i₂ : I₂) (h : ComplexShape.π c₁ c₂ c₁₂ ⟨i₁, i₂⟩ = i₁₂) :
K.ιTotal c₁₂ i₁ i₂ i₁₂ h ≫ K.D₁ c₁₂ i₁₂ i₁₂' =
| Mathlib/Algebra/Homology/TotalComplex.lean | 289 | 293 |
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Joël Riou
-/
import Mathlib.CategoryTheory.Adjunction.Restrict
import Mathlib.CategoryTheory.Functor.KanExtension.Adjunction
import Mathlib.CategoryTheory.Sites.Continuous
im... | Mathlib/CategoryTheory/Sites/CoverLifting.lean | 399 | 405 | |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Operations
import Mathlib.Order.Basic
import Mathlib.Order.BooleanAlgebra
import Mathlib.Tactic.Tauto
import Mathlib.Tactic.B... | Mathlib/Data/Set/Basic.lean | 2,462 | 2,465 | |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Complex.CauchyIntegral
import Mathlib.Analysis.InnerProductSpace.Convex
import Mathlib.Analysis.NormedSpace.Extr
import Mathlib.Data.Complex... | (hd : DiffContOnCl ℂ f (ball z (dist w z)))
(hz : IsMaxOn (norm ∘ f) (closedBall z (dist w z)) z) : ‖f w‖ = ‖f z‖ := by
-- Consider a circle of radius `r = dist w z`.
set r : ℝ := dist w z
have hw : w ∈ closedBall z r := mem_closedBall.2 le_rfl
-- Assume the converse. Since `‖f w‖ ≤ ‖f z‖`, we have `‖f ... | Mathlib/Analysis/Complex/AbsMax.lean | 106 | 137 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Tactic.FieldSimp
/-!
# Slope of a function
In this file we define the slope of a function `f : k... | theorem slope_sub_smul (f : k → E) {a b : k} (h : a ≠ b) :
slope (fun x => (x - a) • f x) a b = f b := by
| Mathlib/LinearAlgebra/AffineSpace/Slope.lean | 62 | 63 |
/-
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.Localization.CalculusOfFractions.Fractions
import Mathlib.CategoryTheory.Localization.HasLocalization
import Mathlib.CategoryTheory.Preadditive.Ad... | @[reassoc]
lemma add_comp (f₁ f₂ : X' ⟶ Y') (g : Y' ⟶ Z') :
add W eX eY f₁ f₂ ≫ g = add W eX eZ (f₁ ≫ g) (f₂ ≫ g) := by
obtain ⟨f₁, rfl⟩ := (homEquiv eX eY).symm.surjective f₁
obtain ⟨f₂, rfl⟩ := (homEquiv eX eY).symm.surjective f₂
obtain ⟨g, rfl⟩ := (homEquiv eY eZ).symm.surjective g
simp [add]
| Mathlib/CategoryTheory/Localization/CalculusOfFractions/Preadditive.lean | 255 | 261 |
/-
Copyright (c) 2019 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.List.FinRange
import Mathlib.Data.List.Perm.Basic
import Mathlib.Data.List.Lex
import Mathlib.Data.List.Induc... | sublistsLenAux_append]
theorem sublistsLenAux_eq (l : List α) (n) (f : List α → β) (r) :
| Mathlib/Data/List/Sublists.lean | 203 | 205 |
/-
Copyright (c) 2024 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Robin Carlier
-/
import Mathlib.CategoryTheory.Limits.FullSubcategory
import Mathlib.CategoryTheory.Limits.Constructions.FiniteProductsOfBinaryProducts
import Mathlib.CategoryT... |
@[ext 1050]
lemma hom_ext {T X Y : C} (f g : T ⟶ X ⊗ Y)
(h_fst : f ≫ fst _ _ = g ≫ fst _ _)
| Mathlib/CategoryTheory/ChosenFiniteProducts.lean | 136 | 139 |
/-
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... | -- TODO these next two lemmas aren't so fundamental, and perhaps could be removed
-- (replacing their usages by their proofs).
| Mathlib/CategoryTheory/Monoidal/Category.lean | 630 | 631 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yakov Pechersky
-/
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Infix
import Mathlib.Data.Quot
/-!
# List rotation
This file proves basic results about `List.r... | theorem rotate'_zero (l : List α) : l.rotate' 0 = l := by cases l <;> rfl
| Mathlib/Data/List/Rotate.lean | 45 | 45 |
/-
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... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | 739 | 739 | |
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Simon Hudon
-/
import Mathlib.Data.PFunctor.Multivariate.Basic
/-!
# Multivariate quotients of polynomial functors.
Basic definition of multivariate QPF. QPFs form a co... |
theorem suppPreservation_iff_liftpPreservation : q.SuppPreservation ↔ q.LiftPPreservation := by
constructor <;> intro h
· rintro α p ⟨a, f⟩
have h' := h
| Mathlib/Data/QPF/Multivariate/Basic.lean | 247 | 251 |
/-
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 | 3,402 | 3,403 | |
/-
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.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Monoidal.Functor
/-!
# Preadditive monoidal categories
A monoidal category is `M... | biproduct.ι_π, Finset.sum_dite_irrel, Finset.sum_dite_eq', Finset.sum_const_zero,
Finset.mem_univ, if_true]
simp_rw [← tensorHom_id]
simp only [← comp_tensor_id, biproduct.ι_π, dite_tensor, comp_dite]
simp
theorem leftDistributor_rightDistributor_assoc {J : Type _} [Finite J]
(X : C) (f : J → C) (Y :... | Mathlib/CategoryTheory/Monoidal/Preadditive.lean | 266 | 281 |
/-
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, Eric Wieser
-/
import Mathlib.Data.Finset.Sym
import Mathlib.LinearAlgebra.BilinearMap
import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
import Mathlib.Linea... | theorem choose_exists_companion : Q.exists_companion.choose = polarBilin Q :=
LinearMap.ext₂ fun x y => by
| Mathlib/LinearAlgebra/QuadraticForm/Basic.lean | 336 | 337 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Pairwise
import Mathlib.Data.Set.BooleanAlgebra
/-!
# The set lattice
This file is a collectio... | Mathlib/Data/Set/Lattice.lean | 2,135 | 2,136 | |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Manuel Candales
-/
import Mathlib.Geometry.Euclidean.PerpBisector
import Mathlib.Algebra.QuadraticDiscriminant
/-!
# Euclidean spaces
This file makes some definitions and... | Mathlib/Geometry/Euclidean/Basic.lean | 228 | 236 | |
/-
Copyright (c) 2020 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou
-/
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.Group.Support
import Mathlib.Data.Set.SymmDiff
/-!
# Indicator function
- `Set.indicator (s : Set α) (f ... | eq_mul_inv_of_mul_eq <| by
rw [Pi.mul_def, ← mulIndicator_union_of_disjoint, diff_union_self,
union_eq_self_of_subset_right h]
| Mathlib/Algebra/Group/Indicator.lean | 415 | 417 |
/-
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... | Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean | 552 | 554 | |
/-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Mario Carneiro, Johan Commelin
-/
import Mathlib.NumberTheory.Padics.PadicNumbers
import Mathlib.RingTheory.DiscreteValuationRing.Basic
/-!
# p-adic integers
This f... | Mathlib/NumberTheory/Padics/PadicIntegers.lean | 604 | 610 | |
/-
Copyright (c) 2024 Raghuram Sundararajan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Raghuram Sundararajan
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Ext
/-!
# Extensionality lemmas for rings and similar structures
In this file we prove e... | @[ext] theorem ext ⦃inst₁ inst₂ : CommRing R⦄
(h_add : local_hAdd[R, inst₁] = local_hAdd[R, inst₂])
(h_mul : local_hMul[R, inst₁] = local_hMul[R, inst₂]) :
inst₁ = inst₂ :=
toRing_injective <| Ring.ext h_add h_mul
| Mathlib/Algebra/Ring/Ext.lean | 438 | 442 |
/-
Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Data.Set.Lattice
import Mathlib.Order.ConditionallyCompleteLattice.Defs
/-!
# Theory of conditionally complete lattices
A conditionally complet... | Mathlib/Order/ConditionallyCompleteLattice/Basic.lean | 1,005 | 1,007 | |
/-
Copyright (c) 2022 Yaël Dillies, George Shakan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, George Shakan
-/
import Mathlib.Algebra.Order.Field.Rat
import Mathlib.Combinatorics.Enumerative.DoubleCounting
import Mathlib.Tactic.FieldSimp
import Mathli... |
end CommGroup
end Finset
| Mathlib/Combinatorics/Additive/PluenneckeRuzsa.lean | 265 | 267 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.