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 |
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/-
Copyright (c) 2022 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Dynamics.Ergodic.AddCircle
import Mathlib.MeasureTheory.Covering.LiminfLimsup
/-!
# Well-approximable numbers and Gallagher's ergodic theorem
Gallagher's e... | a • approxOrderOf A n δ ⊆ approxOrderOf A (orderOf a * n) δ := by
simp_rw [approxOrderOf, thickening_eq_biUnion_ball, ← image_smul, image_iUnion₂, image_smul,
smul_ball'', smul_eq_mul, mem_setOf_eq]
refine iUnion₂_subset_iff.mpr fun b hb c hc => ?_
simp only [mem_iUnion, exists_prop]
refine ⟨a * b, ?_, ... | Mathlib/NumberTheory/WellApproximable.lean | 121 | 129 |
/-
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... | theorem one_le_iff_zero_lt {x : WithBot ℕ} : 1 ≤ x ↔ 0 < x := by
refine ⟨zero_lt_one.trans_le, fun h => ?_⟩
cases x
| Mathlib/Data/Nat/WithBot.lean | 62 | 64 |
/-
Copyright (c) 2024 Theodore Hwa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kim Morrison, Violeta Hernández Palacios, Junyan Xu, Theodore Hwa
-/
import Mathlib.Logic.Hydra
import Mathlib.SetTheory.Surreal.Basic
/-!
### Surreal multiplication
In... | lemma ih4_neg : IH4 x₁ x₂ y → IH4 (-x₂) (-x₁) y ∧ IH4 x₁ x₂ (-y) := by
simp_rw [IH4, isOption_neg]
refine fun h ↦ ⟨fun z w h' ↦ ?_, fun z w h' ↦ ?_⟩
· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]
· convert h h' using 2 <;> rw [P2_neg_right]
| Mathlib/SetTheory/Surreal/Multiplication.lean | 333 | 338 |
/-
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.Group.Pi.Basic
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.Images
import Mathlib.CategoryTheor... |
@[simp]
theorem image.ι_zero {X Y : C} [HasImage (0 : X ⟶ Y)] : image.ι (0 : X ⟶ Y) = 0 := by
rw [← image.lift_fac (monoFactorisationZero X Y)]
| Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean | 530 | 533 |
/-
Copyright (c) 2023 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.MonoidAlgebra.Ideal
import Mathlib.Algebra.MvPolynomial.Division
/-!
# Lemmas about ideals of `MvPolynomial`
Notably this contains results about mo... | refine mem_ideal_span_monomial_image.trans (forall₂_congr fun xi hxi => ?_)
simp_rw [monomial_dvd_monomial, one_dvd, and_true, mem_support_iff.mp hxi, false_or]
/-- `x` is in a monomial ideal generated by variables `X` iff every element of its support
has a component in `s`. -/
| Mathlib/RingTheory/MvPolynomial/Ideal.lean | 39 | 43 |
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv
/-! # Quadratic form on product and pi types
## Main definitions
* `QuadraticForm.prod Q₁ Q₂`: the quadratic for... | rfl
end Ring
end Prod
section Pi
| Mathlib/LinearAlgebra/QuadraticForm/Prod.lean | 226 | 233 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Analysis.InnerProductSpace.LinearMap
import Mathlib.MeasureTheory.Function.LpSpace.ContinuousFunctions
import Mathlib.MeasureTheory.Function.StronglyMeasur... | theorem eLpNorm_inner_lt_top (f g : α →₂[μ] E) : eLpNorm (fun x : α => ⟪f x, g x⟫) 1 μ < ∞ := by
have h : ∀ x, ‖⟪f x, g x⟫‖ ≤ ‖‖f x‖ ^ (2 : ℝ) + ‖g x‖ ^ (2 : ℝ)‖ := by
intro x
rw [← @Nat.cast_two ℝ, Real.rpow_natCast, Real.rpow_natCast]
calc
‖⟪f x, g x⟫‖ ≤ ‖f x‖ * ‖g x‖ := norm_inner_le_norm _ _
... | Mathlib/MeasureTheory/Function/L2Space.lean | 124 | 140 |
/-
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 | 3,011 | 3,013 | |
/-
Copyright (c) 2020 Jean Lo, Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yury Kudryashov
-/
import Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set
import Mathlib.Algebra.Ring.Action.Pointwise.Set
import Mathlib.Topology.Bornology.Basic... | protected lemma empty : Absorbs M s ∅ := by simp [Absorbs]
| Mathlib/Topology/Bornology/Absorbs.lean | 66 | 66 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Data.Set.Constructions
import Mathlib.Order.Filter.AtTopBot.CountablyGenerated
import Mathlib.Topology.Constructions
import Mathlib.Top... | exact congr_arg _ (Set.ext fun U ↦ and_congr_right <| h _)
/-- A point `a` is in the closure of `s` iff all basis sets containing `a` intersect `s`. -/
theorem IsTopologicalBasis.mem_closure_iff {b : Set (Set α)} (hb : IsTopologicalBasis b) {s : Set α}
{a : α} : a ∈ closure s ↔ ∀ o ∈ b, a ∈ o → (o ∩ s).Nonempty ... | Mathlib/Topology/Bases.lean | 201 | 205 |
/-
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... | iIndepFun g μ :=
Kernel.iIndepFun.congr' hf (by simp [h])
nonrec lemma iIndepFun.comp {β γ : ι → Type*} {mβ : ∀ i, MeasurableSpace (β i)}
| Mathlib/Probability/Independence/Basic.lean | 582 | 585 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Johan Commelin, Andrew Yang, Joël Riou
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
import Mathlib.CategoryTheory.Monoid... |
variable (X Y : C) (f : X ⟶ Y)
/-- Shifting by `i` is an equivalence. -/
| Mathlib/CategoryTheory/Shift/Basic.lean | 408 | 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.Fin.Tuple.Basic
/-!
# Lists from functions
Theorems and lemmas for dealing with `List.ofFn`, which converts a function on `Fin n` to a list
of l... | Mathlib/Data/List/OfFn.lean | 286 | 294 | |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jan-David Salchow, Sébastien Gouëzel, Jean Lo, Yury Kudryashov, Frédéric Dupuis,
Heather Macbeth
-/
import Mathlib.Algebra.Module.Opposite
import Mathlib.Topology.Algebra.Group.Qu... | Mathlib/Topology/Algebra/Module/Basic.lean | 2,701 | 2,702 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Jeremy Avigad
-/
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Data.Set.Finite.Range
import Mathlib.Data.Set.Lattice
import Mathlib.Topology.Defs.... | simpa only [imp_iff_not_or] using hp.isClosed_compl.union hq
| Mathlib/Topology/Basic.lean | 183 | 184 |
/-
Copyright (c) 2020 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Analysis.Asymptotics.Theta
/-!
# Asymptotic equivalence
In this file, we define the relation `IsEquivalent l u v`, which means that `u-v` is litt... | theorem IsEquivalent.trans {l : Filter α} {u v w : α → β} (huv : u ~[l] v) (hvw : v ~[l] w) :
u ~[l] w :=
(huv.isLittleO.trans_isBigO hvw.isBigO).triangle hvw.isLittleO
| Mathlib/Analysis/Asymptotics/AsymptoticEquivalent.lean | 102 | 104 |
/-
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... | @[to_additive]
theorem discreteTopology_of_isOpen_singleton_one (h : IsOpen ({1} : Set G)) :
DiscreteTopology G := by
rw [← singletons_open_iff_discrete]
| Mathlib/Topology/Algebra/Group/Basic.lean | 114 | 117 |
/-
Copyright (c) 2020 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Basic
import Mathlib.RingTheory.RootsOfUnity.Minpoly
/-!
# Roots of cyclotomic polynomials.
We gather results abou... | (cyclotomic.irreducible_rat hmzero) hdiv
end minpoly
| Mathlib/RingTheory/Polynomial/Cyclotomic/Roots.lean | 206 | 209 |
/-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.Complex.AbsMax
import Mathlib.Analysis.LocallyConvex.WithSeminorms
import Mathlib.Geometry.Manifold.MFDeriv.Basic
import Mathlib.Topology.Lo... | (ho : IsOpen U) (ha : a ∈ U) (hb : b ∈ U) : f a = f b := by
refine ?_
-- Subtract `f b` to avoid the assumption `[StrictConvexSpace ℝ F]`
wlog hb₀ : f b = 0 generalizing f
· have hd' : MDifferentiableOn I 𝓘(ℂ, F) (f · - f b) U := fun x hx ↦
⟨(hd x hx).1.sub continuousWithinAt_const, (hd x hx).2.sub_c... | Mathlib/Geometry/Manifold/Complex.lean | 123 | 135 |
/-
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.Topology.Order.LeftRightNhds
/-!
# Properties of LUB and GLB in an order topology
-/
open Set Filter TopologicalSpa... | by_cases h : x ∈ t
· exact ⟨fun _ => x, monotone_const, fun n => le_rfl, tendsto_const_nhds, fun _ => h⟩
· rcases htx.exists_seq_strictMono_tendsto_of_not_mem h ht with ⟨u, hu⟩
exact ⟨u, hu.1.monotone, fun n => (hu.2.1 n).le, hu.2.2⟩
theorem exists_seq_strictMono_tendsto' {α : Type*} [LinearOrder α] [Topolog... | Mathlib/Topology/Order/IsLUB.lean | 186 | 192 |
/-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Eric Wieser, Jeremy Avigad, Johan Commelin
-/
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.Linear... |
/-- An expression for the inverse of a lower block-triangular matrix, when either both elements of
diagonal are invertible, or both are not. -/
theorem inv_fromBlocks_zero₁₂_of_isUnit_iff (A : Matrix m m α) (C : Matrix n m α) (D : Matrix n n α)
(hAD : IsUnit A ↔ IsUnit D) :
(fromBlocks A 0 C D)⁻¹ = fromBlocks ... | Mathlib/LinearAlgebra/Matrix/SchurComplement.lean | 202 | 215 |
/-
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.Eval.Degree
import Mathlib.Algebra.Prime.Lemmas
/-!
# Theory of degrees of polynomials
S... | have h4 : map f p ≠ 0 := by
rwa [ne_eq, ← degree_eq_bot, h2, degree_eq_bot]
rwa [← coeff_natDegree, ← coeff_map, ← h3, coeff_natDegree, ne_eq, leadingCoeff_eq_zero]
@[simp]
theorem natDegree_map_eq_iff {f : R →+* S} {p : Polynomial R} :
natDegree (map f p) = natDegree p ↔ f (p.leadingCoeff) ≠ 0 ∨ natDegree... | Mathlib/Algebra/Polynomial/Degree/Lemmas.lean | 270 | 276 |
/-
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... | lemma inf'_Ici (a : α) : (Ici a).inf' nonempty_Ici id = a :=
ge_antisymm (le_inf' _ _ fun _ ↦ mem_Ici.1) <| inf'_le (f := id) <| mem_Ici.2 <| le_refl a
| Mathlib/Order/Interval/Finset/Basic.lean | 801 | 803 |
/-
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 | 1,464 | 1,475 | |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.InnerProductSpace.Defs
impor... | Mathlib/Analysis/InnerProductSpace/Basic.lean | 1,525 | 1,527 | |
/-
Copyright (c) 2024 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Cardinal.Arithmetic
import Mathlib.SetTheory.Ordinal.Principal
/-!
# Ordinal arithmetic with cardinals
This file co... | Mathlib/SetTheory/Cardinal/Ordinal.lean | 655 | 656 | |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Ring.List
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Data.Nat.Prime.Basic
import Ma... | rw [mem_primeFactorsList (mul_ne_zero ha hb), mem_primeFactorsList ha, mem_primeFactorsList hb,
← and_or_left]
simpa only [and_congr_right_iff] using Prime.dvd_mul
/-- The sets of factors of coprime `a` and `b` are disjoint -/
theorem coprime_primeFactorsList_disjoint {a b : ℕ} (hab : a.Coprime b) :
List.D... | Mathlib/Data/Nat/Factors.lean | 260 | 274 |
/-
Copyright (c) 2023 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Algebra.NonUnitalHom
import Mathlib.Data.Set.UnionLift
import Mathlib.LinearAlgebra.Span.Basic
import Mathlib.RingTheory.NonUnitalSubring.Basic
... | apply span_induction₂ ?Hs (by simp) (by simp) ?Hadd_l ?Hadd_r ?Hsmul_l ?Hsmul_r hpx hpy
case Hs => exact fun x y hx hy ↦ subset_span <| mul_mem hx hy
case Hadd_l => exact fun x y z _ _ _ hxz hyz ↦ by simpa [add_mul] using add_mem hxz hyz
case Hadd_r => exact fun x y z _ _ _ hxz hyz ↦ by simpa [m... | Mathlib/Algebra/Algebra/NonUnitalSubalgebra.lean | 665 | 668 |
/-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Applicative
import Mathlib.Control.Traversable.Basic
import Mathlib.Data.List.Forall2
import Mathlib.Data.Set.Functor
/-!
# LawfulTraversable instan... |
end Option
instance : LawfulTraversable Option :=
{ show LawfulMonad Option from inferInstance with
| Mathlib/Control/Traversable/Instances.lean | 47 | 51 |
/-
Copyright (c) 2021 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.Group.End
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Fintype.Sum
import Mathlib.Data.Prod.Lex
import Mathlib.Order.Interval.Finset.Fin
/-!
... | theorem comp_perm_comp_sort_eq_comp_sort : (f ∘ σ) ∘ sort (f ∘ σ) = f ∘ sort f := by
rw [Function.comp_assoc, ← Equiv.Perm.coe_mul]
exact unique_monotone (monotone_sort (f ∘ σ)) (monotone_sort f)
| Mathlib/Data/Fin/Tuple/Sort.lean | 186 | 189 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn
/-!
# Inverse functio... | simp [@Subsingleton.elim E hE x y]
have := hf.isLittleO.def hc
rw [nhds_prod_eq, Filter.Eventually, mem_prod_same_iff] at this
rcases this with ⟨s, has, hs⟩
exact ⟨s, has, fun x hx y hy => hs (mk_mem_prod hx hy)⟩
theorem map_nhds_eq_of_surj [CompleteSpace E] [CompleteSpace F] {f : E → F} {f' : E →L[𝕜] F} ... | Mathlib/Analysis/Calculus/InverseFunctionTheorem/FDeriv.lean | 74 | 83 |
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.SetTheory.Cardinal.Finite
import Mathlib.Data.Set.Finite.Powerset
/-!
# Noncomputable Set Cardinality
We define the cardinality of set `s` as a term `Set... | rw [ht.cast_ncard_eq, hs.cast_ncard_eq, hs.diff.cast_ncard_eq]
apply encard_le_encard_diff_add_encard
convert Nat.zero_le _
| Mathlib/Data/Set/Card.lean | 888 | 890 |
/-
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 | 1,471 | 1,476 | |
/-
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... | ∃ f : Perm α, f.IsCycleOn s ∧ f.support ⊆ s := by
refine ⟨s.toList.formPerm, ?_, fun x hx => by
simpa using List.mem_of_formPerm_apply_ne (Perm.mem_support.1 hx)⟩
convert s.nodup_toList.isCycleOn_formPerm
simp
end Finset
| Mathlib/GroupTheory/Perm/Cycle/Basic.lean | 869 | 876 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Sophie Morel, Yury Kudryashov
-/
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
import Mathlib.Logic.Embedding.Basic
import Mathlib.Data.Fintype.Car... |
private lemma uniformity_eq_seminorm :
𝓤 (ContinuousMultilinearMap 𝕜 E G) = ⨅ r > 0, 𝓟 {f | ‖f.1 - f.2‖ < r} := by
refine (ContinuousMultilinearMap.seminorm 𝕜 E G).uniformity_eq_of_hasBasis
(ContinuousMultilinearMap.hasBasis_nhds_zero_of_basis Metric.nhds_basis_closedBall)
| Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean | 453 | 457 |
/-
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.Topology.EMetricSpace.Paracompact
import Mathlib.Topology.MetricSpace.Basic
import Mathlib.Topology.MetricSpace.ProperSpace.Lemmas
import Mathlib.Top... | ∃ r' : ι → ℝ, (s ⊆ ⋃ i, ball (c i) (r' i)) ∧ ∀ i, r' i < r i := by
rcases exists_subset_iUnion_closed_subset hs (fun i => @isOpen_ball _ _ (c i) (r i)) uf us with
⟨v, hsv, hvc, hcv⟩
have := fun i => exists_lt_subset_ball (hvc i) (hcv i)
choose r' hlt hsub using this
exact ⟨r', hsv.trans <| iUnion_mono <... | Mathlib/Topology/MetricSpace/ShrinkingLemma.lean | 39 | 46 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Data.Finset.Pi
import Mathlib.Data.Finset.Sigma
import Mathlib.Data.Finset.Sum
import Mathlib.Data.Set.Finite.Basic
/-!
# Preimage of ... |
@[simp]
| Mathlib/Data/Finset/Preimage.lean | 44 | 45 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.Polynomial.Basi... | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 849 | 855 | |
/-
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.Sites.CoverLifting
import Mathlib.CategoryTheory.Sites.CoverPreserving
/-! Localization
In this file, given a Grothendieck topology `J` on a cat... | (over_forget_compatiblePreserving J X)
(over_forget_coverPreserving J X)
/-- The pullback functor `Sheaf J A ⥤ Sheaf (J.over X) A` -/
abbrev overPullback (A : Type u') [Category.{v'} A] (X : C) :
Sheaf J A ⥤ Sheaf (J.over X) A :=
(Over.forget X).sheafPushforwardContinuous _ _ _
| Mathlib/CategoryTheory/Sites/Over.lean | 160 | 166 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Finset.Sum
import Mathlib.Data.Sum.Order
import Mathlib.Order.Interval.Finset.Defs
/-!
# Finite intervals in a disjoint union
This file provides the... | Mathlib/Data/Sum/Interval.lean | 369 | 371 | |
/-
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... | rw [inverse_unit, Units.mul_inv]
| Mathlib/Algebra/GroupWithZero/Units/Basic.lean | 98 | 99 |
/-
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... | /- Porting note: Commented out `Tactic` namespace.
namespace Tactic
open Positivity
/-- Extension for the `positivity` tactic: `Rel.edgeDensity` and `SimpleGraph.edgeDensity` are
always nonnegative. -/
@[positivity]
unsafe def positivity_edge_density : expr → tactic strictness
| Mathlib/Combinatorics/SimpleGraph/Density.lean | 376 | 384 |
/-
Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.SimpleGraph.Path
import Mathlib.Combinatorics.SimpleGraph.Operations
import Mathlib.Data.Finset.Pairwise
import M... | | zero => exact False.elim <| not_mem_empty _ (isNClique_zero.1 hs ▸ hb)
| succ _ => exact (hs.erase_of_mem hb).insert fun w h ↦ by aesop
theorem is3Clique_triple_iff : G.IsNClique 3 {a, b, c} ↔ G.Adj a b ∧ G.Adj a c ∧ G.Adj b c := by
simp only [isNClique_iff, isClique_iff, Set.pairwise_insert_of_symmetric G.sym... | Mathlib/Combinatorics/SimpleGraph/Clique.lean | 262 | 269 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.PropInstances
import Mathlib.Order.GaloisConnection.Defs
/-!
# Heyting algebras
This file defines Heyting, co-Heyting and bi-Heyting algebras.
A H... |
theorem le_sup_sdiff : a ≤ b ⊔ a \ b :=
sdiff_le_iff.1 le_rfl
| Mathlib/Order/Heyting/Basic.lean | 404 | 406 |
/-
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.Insert
/-!
# Subsingleton
Defines the predicate `Subsingleton s : Prop`, saying that `s` has at most one element.
Also def... | /-- The `coe_sort` of a set `s` in a subsingleton type is a subsingleton.
For the corresponding result for `Subtype`, see `subtype.subsingleton`. -/
instance subsingleton_coe_of_subsingleton [Subsingleton α] {s : Set α} : Subsingleton s := by
rw [s.subsingleton_coe]
exact subsingleton_of_subsingleton
| Mathlib/Data/Set/Subsingleton.lean | 109 | 113 |
/-
Copyright (c) 2024 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.NumberTheory.DirichletCharacter.Bounds
import Mathlib.NumberTheory.LSeries.Convolution
import Mathlib.NumberTheory.LSeries.Deriv
import Mathlib.NumberThe... |
/-- The L-series of a Dirichlet character mod `N > 0` converges absolutely at `s` if and only if
`re s > 1`. -/
lemma LSeriesSummable_iff {N : ℕ} (hN : N ≠ 0) (χ : DirichletCharacter ℂ N) {s : ℂ} :
LSeriesSummable ↗χ s ↔ 1 < s.re := by
refine ⟨fun H ↦ ?_, LSeriesSummable_of_one_lt_re χ⟩
| Mathlib/NumberTheory/LSeries/Dirichlet.lean | 180 | 185 |
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform
import Mathlib.Probability.Moments.ComplexMGF
/-!
# Gaussian dis... | lemma integrable_gaussianPDFReal (μ : ℝ) (v : ℝ≥0) :
Integrable (gaussianPDFReal μ v) := by
rw [gaussianPDFReal_def]
by_cases hv : v = 0
· simp [hv]
let g : ℝ → ℝ := fun x ↦ (√(2 * π * v))⁻¹ * rexp (- x ^ 2 / (2 * v))
have hg : Integrable g := by
suffices g = fun x ↦ (√(2 * π * v))⁻¹ * rexp (- (2 * v)... | Mathlib/Probability/Distributions/Gaussian.lean | 75 | 94 |
/-
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.TangentCone
import Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics
import Mathlib.Analysis.As... | variable {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E]
variable {F : Type*} [NormedAddCommGroup F] [NormedSpace 𝕜 F]
variable {f f₀ f₁ g : E → F}
| Mathlib/Analysis/Calculus/FDeriv/Basic.lean | 225 | 228 |
/-
Copyright (c) 2023 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Computability.AkraBazzi.GrowsPolynomially
import Mathlib.Analysis.Calculus.Deriv.Inv
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
/-!
# Divid... | lemma asympBound_def' {α} [Fintype α] (a b : α → ℝ) {n : ℕ} :
asympBound g a b n = n ^ p a b * (1 + (∑ u ∈ range n, g u / u ^ (p a b + 1))) := by
simp [asympBound_def, sumTransform, mul_add, mul_one, Finset.sum_Ico_eq_sum_range]
| Mathlib/Computability/AkraBazzi/AkraBazzi.lean | 595 | 597 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johan Commelin
-/
import Mathlib.Algebra.Group.Equiv.Defs
import Mathlib.Algebra.Group.WithOne.Defs
import Mathlib.Algebra.GroupWithZero.Equiv
import Mathlib.Algebra.Gr... | MonoidWithZeroHom.ext fun x => (map'_map' f g x).symm
| Mathlib/Algebra/GroupWithZero/WithZero.lean | 135 | 136 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Geometry.Manifold.IsManifold.ExtChartAt
import Mathlib.Geometry.Manifold.LocalInvariantProperties
/-!
# `C^n` functions betwee... | /-- One can reformulate being `Cⁿ` within a set at a point as being `Cⁿ` in the source space when
composing with the extended chart. -/
| Mathlib/Geometry/Manifold/ContMDiff/Defs.lean | 309 | 310 |
/-
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.GroupTheory.GroupAction.Pointwise
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.LocallyConvex.BalancedCoreHull
import Mathlib.Analysis.... | theorem isVonNBounded_insert (x : E) {s : Set E} :
IsVonNBounded 𝕜 (insert x s) ↔ IsVonNBounded 𝕜 s := by
simp only [← singleton_union, isVonNBounded_union, isVonNBounded_singleton, true_and]
protected alias ⟨_, IsVonNBounded.insert⟩ := isVonNBounded_insert
| Mathlib/Analysis/LocallyConvex/Bounded.lean | 273 | 277 |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Batteries.Data.Rat.Lemmas
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Rat.Init
import Mathlib.Order.Basic
import Mathlib.Tactic.Commo... | @[simp] alias mkRat_num_den' := mkRat_self
| Mathlib/Data/Rat/Defs.lean | 105 | 105 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Data.Complex.Trigonometric
import Mathlib.Data.Complex.Module
import Mathlib.RingTheory.Polynomial.Chebyshev
/-!
# Multiple angle formulas in terms of... | | one => simp [one_add_one_eq_two, sin_two_mul]; ring
| add_two n ih1 ih2 =>
simp only [U_add_two, add_sub_cancel_right, eval_sub, eval_mul, eval_X, eval_ofNat, sub_mul,
mul_assoc, ih1, ih2, sub_eq_iff_eq_add, sin_add_sin]
push_cast
ring_nf
| neg_add_one n ih1 ih2 =>
simp only [U_sub_one, ad... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Chebyshev.lean | 73 | 86 |
/-
Copyright (c) 2022 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Field.Defs
import Mathlib.Algebra.GroupWithZero.Invertible
import Mathlib.Data.Sigma.Basic
import Mathlib.Algebra.Ring.Nat
import Mathlib.Data.... | theorem IsInt.of_raw (α) [Ring α] (n : ℤ) : IsInt (n.rawCast : α) n := ⟨rfl⟩
theorem IsInt.neg_to_eq {α} [Ring α] {n} :
| Mathlib/Tactic/NormNum/Result.lean | 176 | 178 |
/-
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... | lemma mem_liminf_iff_eventually_mem : (a ∈ liminf s 𝓕) ↔ (∀ᶠ i in 𝓕, a ∈ s i) := by
simpa only [liminf_eq_iSup_iInf, iSup_eq_iUnion, iInf_eq_iInter, mem_iUnion, mem_iInter]
using ⟨fun ⟨S, hS, hS'⟩ ↦ mem_of_superset hS (by tauto), fun h ↦ ⟨{i | a ∈ s i}, h, by tauto⟩⟩
| Mathlib/Order/LiminfLimsup.lean | 694 | 696 |
/-
Copyright (c) 2022 Kyle Miller, Adam Topaz, Rémi Bottinelli, Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller, Adam Topaz, Rémi Bottinelli, Junyan Xu
-/
import Mathlib.Topology.Category.TopCat.Limits.Konig
/-!
# Cofiltered systems
This file de... | refine ⟨⟨y, mem_iInter.2 fun g₂ => ?_⟩, Subtype.ext ?_⟩
· obtain ⟨j₄, f₄, h₄⟩ := IsCofilteredOrEmpty.cone_maps g₂ ((f₃ ≫ f₂) ≫ g₁)
obtain ⟨y, rfl⟩ := F.mem_eventualRange_iff.1 hy f₄
rw [← map_comp_apply] at h₃
rw [mem_preimage, ← map_comp_apply, h₄, ← Category.assoc, map_comp_apply, h₃,
... | Mathlib/CategoryTheory/CofilteredSystem.lean | 222 | 230 |
/-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Algebra.Order.Pi
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Data.Real.Pointwise
/... | Metric.vadd_ball x y r
/-- The image of a closed ball under addition with a singleton is another closed ball. -/
theorem vadd_closedBall (p : Seminorm 𝕜 E) : x +ᵥ p.closedBall y r = p.closedBall (x +ᵥ y) r :=
letI := AddGroupSeminorm.toSeminormedAddCommGroup p.toAddGroupSeminorm
| Mathlib/Analysis/Seminorm.lean | 722 | 726 |
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.AlgebraicTopology.DoldKan.GammaCompN
import Mathlib.AlgebraicTopology.DoldKan.NReflectsIso
/-! The unit isomorphism of the Dold-Kan equivalence
In order to con... |
end Γ₂N₂
theorem compatibility_Γ₂N₁_Γ₂N₂_natTrans (X : SimplicialObject C) :
| Mathlib/AlgebraicTopology/DoldKan/NCompGamma.lean | 180 | 183 |
/-
Copyright (c) 2020 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Analysis.Asymptotics.Theta
/-!
# Asymptotic equivalence
In this file, we define the relation `IsEquivalent l u v`, which means that `u-v` is litt... | theorem IsEquivalent.add_isLittleO (huv : u ~[l] v) (hwv : w =o[l] v) : u + w ~[l] v := by
simpa only [IsEquivalent, add_sub_right_comm] using huv.add hwv
theorem IsEquivalent.sub_isLittleO (huv : u ~[l] v) (hwv : w =o[l] v) : u - w ~[l] v := by
| Mathlib/Analysis/Asymptotics/AsymptoticEquivalent.lean | 151 | 154 |
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.Data.Finite.Prod
import Mathlib.Data.Matroid.Init
import Mathlib.Data.Set.Card
import Mathlib.Data.Set.Finite.Powerset
import Mathlib.Order.UpperLower.Clos... | /-- create a copy of `M : Matroid α` with an independence predicate and ground set defeq
to supplied arguments that are provably equal to those of `M`. -/
@[simps!] def copyIndep (M : Matroid α) (E : Set α) (Indep : Set α → Prop)
| Mathlib/Data/Matroid/Basic.lean | 793 | 795 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Powerset
import Mathlib.Algebra.NoZeroSMulDivisors.Pi
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Finty... | end MultilinearMap
| Mathlib/LinearAlgebra/Multilinear/Basic.lean | 1,401 | 1,408 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Order.Module.Synonym
import Mathlib.LinearAlgebra.Aff... | theorem lineMap_mono_endpoints (ha : a ≤ a') (hb : b ≤ b') (h₀ : 0 ≤ r) (h₁ : r ≤ 1) :
lineMap a b r ≤ lineMap a' b' r :=
(lineMap_mono_left ha h₁).trans (lineMap_mono_right hb h₀)
| Mathlib/LinearAlgebra/AffineSpace/Ordered.lean | 67 | 69 |
/-
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 | 2,074 | 2,077 | |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Shing Tak Lam, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring.List
import Mathlib.Data.Int.ModEq
import Mathlib.Da... | · simp
· simpa [List.range_succ_eq_map, List.zipWith_map_right, ofDigits_eq_sum_mapIdx_aux] using
Or.inl hl
@[simp]
theorem ofDigits_nil {b : ℕ} : ofDigits b [] = 0 := rfl
| Mathlib/Data/Nat/Digits.lean | 167 | 172 |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq
import Mathlib.CategoryTheory.Limits.Shapes.StrictInitial
import Mathlib.CategoryTheory.Limits.Types.Shapes
impo... | Limits.BinaryCofan.IsColimit.desc' hc (HT.from _ ≫ c₀.inl) (HT.from _ ≫ c₀.inr)
rw [H c' (αX ≫ HT.from _) (αY ≫ HT.from _) (f ≫ d) (by rw [← reassoc_of% hX, hd, Category.assoc])
(by rw [← reassoc_of% hY, hd', Category.assoc])]
obtain ⟨hl, hr⟩ := (H c (HT.from _) (HT.from _) d hd.symm hd'.symm).mp ⟨hc⟩
r... | Mathlib/CategoryTheory/Extensive.lean | 203 | 216 |
/-
Copyright (c) 2017 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Mario Carneiro
-/
import Mathlib.Algebra.Ring.CharZero
import Mathlib.Algebra.Star.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Order.Interval.Set.UnorderedInterva... | Mathlib/Data/Complex/Basic.lean | 890 | 891 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.Nat.SuccPred
import Mathlib.Order.SuccPred.Initial... | Mathlib/SetTheory/Ordinal/Arithmetic.lean | 1,969 | 1,974 | |
/-
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.Control.Combinators
import Mathlib.Data.Option.Defs
import Mathlib.Logic.IsEmpty
import Mathlib.Logic.Relator
import Mathlib.Util.CompileInductive
impo... | (h : ∀ a ∈ x, f a = g a) : x.bind f = x.bind g := by
cases x <;> simp only [some_bind, none_bind, mem_def, h]
| Mathlib/Data/Option/Basic.lean | 101 | 103 |
/-
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 | 1,722 | 1,723 | |
/-
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.EuclideanDomain.Int
import Mathlib.Algebra.MvPolynomial.Eval
import Mathlib.RingTheory.Adjoin.Basic
import Mathlib.RingTheory.Polynomial.Basic
import Mat... | section Semiring
variable {R : Type u} {A : Type v} {B : Type w}
variable [CommSemiring R] [CommRing A] [CommRing B] [Algebra R A] [Algebra R B]
/-- The image of a Noetherian R-algebra under an R-algebra map is a Noetherian ring. -/
instance AlgHom.isNoetherianRing_range (f : A →ₐ[R] B) [IsNoetherianRing A] :
IsN... | Mathlib/RingTheory/Adjoin/FG.lean | 170 | 179 |
/-
Copyright (c) 2022 Wrenna Robson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Wrenna Robson
-/
import Mathlib.Analysis.Normed.Group.Basic
/-!
# Hamming spaces
The Hamming metric counts the number of places two members of a (finite) Pi type
differ. The Hamming n... | hammingDist_lt_one
| Mathlib/InformationTheory/Hamming.lean | 170 | 171 |
/-
Copyright (c) 2023 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calculus.Deriv.Add
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calcul... | lineDerivWithin 𝕜 f s x v = lineDeriv 𝕜 f x v := by
apply derivWithin_of_mem_nhds
apply (Continuous.continuousAt _).preimage_mem_nhds (by simpa using h)
fun_prop
theorem lineDerivWithin_of_isOpen (hs : IsOpen s) (hx : x ∈ s) :
lineDerivWithin 𝕜 f s x v = lineDeriv 𝕜 f x v :=
| Mathlib/Analysis/Calculus/LineDeriv/Basic.lean | 285 | 291 |
/-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang
-/
import Mathlib.Analysis.Calculus.Conformal.NormedSpace
import Mathlib.Analysis.InnerProductSpace.ConformalLinearMap
/-!
# Conformal maps between inner product spaces
A ... |
/-- The conformal factor of a conformal map at some point `x`. Some authors refer to this function
as the characteristic function of the conformal map. -/
| Mathlib/Analysis/Calculus/Conformal/InnerProduct.lean | 36 | 38 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.PropInstances
import Mathlib.Order.GaloisConnection.Defs
/-!
# Heyting algebras
This file defines Heyting, co-Heyting and bi-Heyting algebras.
A H... | · rw [inf_le_iff, or_iff_left h],
| Mathlib/Order/Heyting/Basic.lean | 960 | 960 |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.AlgebraicGeometry.Morphisms.Basic
import Mathlib.RingTheory.RingHomProperties
/-!
# Constructors for properties of morphisms between schemes
This file pro... | rw [← Q.cancel_left_of_respectsIso (pullbackDiagonalMapIso f _ f₁ f₂).hom]
convert HasAffineProperty.of_isPullback (P := P) (.of_hasPullback _ _) H
· apply pullback.hom_ext <;> simp
· infer_instance
· infer_instance
theorem HasAffineProperty.diagonal_iff
(P) {Q} [HasAffineProperty P Q] {X Y} {f : X ⟶ Y} ... | Mathlib/AlgebraicGeometry/Morphisms/Constructors.lean | 104 | 131 |
/-
Copyright (c) 2020 Ashvni Narayanan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ashvni Narayanan
-/
import Mathlib.Algebra.Field.Defs
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Algebra.Ring.Subring.Defs
import Mathlib.Algebra.Ring.Subsemiring.Bas... | Mathlib/Algebra/Ring/Subring/Basic.lean | 1,105 | 1,112 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.OuterMeasure.OfFunction
import Mathlib.MeasureTheory.PiSystem
/-!
# The Caratheodory σ-algebra of an outer measure
Give... | simp [IsCaratheodory, diff_eq, add_comm]
| Mathlib/MeasureTheory/OuterMeasure/Caratheodory.lean | 62 | 62 |
/-
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.Algebra.BigOperators.Fin
import Mathlib.Logic.Encodable.Pi
import Mathlib.MeasureTheory.Group.Measure
import Mathlib.MeasureTheory.MeasurableSpace.... | classical
rw [pi']
rw [← MeasurableEquiv.piMeasurableEquivTProd_symm_apply, MeasurableEquiv.map_apply,
| Mathlib/MeasureTheory/Constructions/Pi.lean | 182 | 184 |
/-
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.Algebra.Group.AddChar
import Mathlib.Analysis.Complex.Circle
import Mathlib.MeasureTheory.Group.Integral
import Mathlib.MeasureTheory.Integral.Prod
imp... | · exact hf.norm
· refine ae_of_all _ fun v ↦ (he.comp ?_).smul continuous_const
exact (hL.comp (.prodMk_right _)).neg
end Continuous
section Fubini
variable [TopologicalSpace 𝕜] [IsTopologicalRing 𝕜] [TopologicalSpace V] [BorelSpace V]
| Mathlib/Analysis/Fourier/FourierTransform.lean | 155 | 163 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Cardinal.Arithmetic
import Mathlib.SetTheory.Ordinal.FixedPoint
/-!
# Cofinality
This file co... | (ho : Cardinal.lift.{v, u} o.card < c.cof) (hf : ∀ i hi, f i hi < c) : blsub.{u, v} o f < c :=
lt_of_le_of_ne (blsub_le hf) fun h =>
ho.not_le (by simpa [← iSup_ord, hf, h] using cof_blsub_le_lift.{u, v} f)
| Mathlib/SetTheory/Cardinal/Cofinality.lean | 335 | 337 |
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Peter Pfaffelhuber, Yaël Dillies, Kin Yau James Wong
-/
import Mathlib.MeasureTheory.MeasurableSpace.Constructions
import Mathlib.MeasureTheory.PiSystem
import Mathlib.Topo... | the projection from `∀ i, α i` to `∀ i : s, α i`.
`measurableCylinders` is the set of all cylinders with measurable base `S`. -/
def measurableCylinders (α : ι → Type*) [∀ i, MeasurableSpace (α i)] : Set (Set (∀ i, α i)) :=
⋃ (s) (S) (_ : MeasurableSet S), {cylinder s S}
| Mathlib/MeasureTheory/Constructions/Cylinders.lean | 265 | 268 |
/-
Copyright (c) 2022 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Tactic.NormNum.Basic
import Mathlib.Data.Rat.Cast.CharZero
import Mathlib.Algebra.Field.Basic
/-!
# `norm_num` plugins for `Rat.cast` and `⁻¹`.
-/
va... |
theorem isRat_inv_neg_one {α} [DivisionRing α] : {a : α} →
IsInt a (.negOfNat (nat_lit 1)) → IsInt a⁻¹ (.negOfNat (nat_lit 1))
| Mathlib/Tactic/NormNum/Inv.lean | 120 | 122 |
/-
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... | noncomputable def zmodCyclicMulEquiv [Group G] (h : IsCyclic G) :
Multiplicative (ZMod (Nat.card G)) ≃* G :=
AddEquiv.toMultiplicative <| zmodAddCyclicAddEquiv <| isAddCyclic_additive_iff.2 h
/-- Two cyclic additive groups of the same cardinality are isomorphic. -/
| Mathlib/GroupTheory/SpecificGroups/Cyclic.lean | 743 | 747 |
/-
Copyright (c) 2020 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp
-/
import Mathlib.LinearAlgebra.Eigenspace.Basic
import Mathlib.FieldTheory.Minpoly.Field
/-!
# Eigenvalues are the roots of the minimal polynomial.
## Tags
e... | · intro p q hp hq; simp [hp, hq, add_smul]
· intro n a hna
rw [mul_comm, pow_succ', mul_assoc, map_mul, Module.End.mul_apply, mul_comm, hna]
simp only [mem_eigenspace_iff.1 h.1, smul_smul, aeval_X, eval_mul, eval_C, eval_pow, eval_X,
LinearMap.map_smulₛₗ, RingHom.id_apply, mul_comm]
theorem isRoot_of... | Mathlib/LinearAlgebra/Eigenspace/Minpoly.lean | 54 | 62 |
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.DualNumber
import Mathlib.Algebra.QuaternionBasis
import Mathlib.Data.Complex.Module
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import ... | | ι x => rw [ι_eq_zero, LinearMap.zero_apply, reverse.map_zero]
| mul x₁ x₂ hx₁ hx₂ => rw [reverse.map_mul, mul_comm, hx₁, hx₂]
| add x₁ x₂ hx₁ hx₂ => rw [reverse.map_add, hx₁, hx₂]
@[simp]
theorem reverse_eq_id :
(reverse : CliffordAlgebra (0 : QuadraticForm R Unit) →ₗ[R] _) = LinearMap.id :=
| Mathlib/LinearAlgebra/CliffordAlgebra/Equivs.lean | 90 | 96 |
/-
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... | /-- `Function.update f i x` is continuous in `(f, x)`. -/
@[continuity, fun_prop]
theorem continuous_update [DecidableEq ι] (i : ι) :
Continuous fun f : (∀ j, π j) × π i => update f.1 i f.2 :=
| Mathlib/Topology/Constructions.lean | 737 | 740 |
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Mathlib.Control.Basic
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Option.Basic
im... | Mathlib/Data/List/Basic.lean | 2,012 | 2,016 | |
/-
Copyright (c) 2021 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou, Adam Topaz, Johan Commelin
-/
import Mathlib.Algebra.Homology.Additive
import Mathlib.AlgebraicTopology.MooreComplex
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Catego... | /-- The differential on the alternating coface map complex is the alternate
sum of the coface maps -/
@[simp]
def objD (n : ℕ) : X.obj ⦋n⦌ ⟶ X.obj ⦋n + 1⦌ :=
| Mathlib/AlgebraicTopology/AlternatingFaceMapComplex.lean | 287 | 290 |
/-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
/-!
# Preservation of biproducts
We define the... | Mathlib/CategoryTheory/Limits/Preserves/Shapes/Biproducts.lean | 440 | 442 | |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
import Mathlib.MeasureTheory.Group.MeasurableEquiv
import Mathlib... | refine InnerRegularWRT.map h.innerRegular hf.aemeasurable ?_ (fun K hK ↦ hK.image hf) ?_
· rintro s ⟨hs, h's⟩
exact ⟨hf.measurable hs, by rwa [map_apply hf.measurable hs] at h's⟩
· rintro s ⟨hs, -⟩
exact hs
| Mathlib/MeasureTheory/Measure/Regular.lean | 877 | 882 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPi... | MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun _ => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
... | Mathlib/GroupTheory/Sylow.lean | 538 | 543 |
/-
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 | 1,795 | 1,800 | |
/-
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.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGrou... | instance SLAction : SlashAction ℤ SL(2, ℤ) (ℍ → ℂ) ℂ :=
monoidHomSlashAction
| Mathlib/NumberTheory/ModularForms/SlashActions.lean | 130 | 131 |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.GCDMonoid.Finset
import Mathlib.Algebra.Polynomial.CancelLeads
import Mathlib.Algebra.Polynomial.EraseLead
import Mathlib.Algebra.Polynomial.Fi... | Mathlib/RingTheory/Polynomial/Content.lean | 500 | 502 | |
/-
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... | simpa [← coe_subset] using Set.Icc_subset_Iic_self
| Mathlib/Order/Interval/Finset/Basic.lean | 441 | 442 |
/-
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.Group.Basic
import Mathlib.Algebra.Group.Nat.Defs
import Mathlib.Tactic.Common
import Mathlib.Data.Set.Insert
/-!
# Set enumeration
This file ... | | none => simp [enumerate_eq_none_of_sel, h]
| some a' =>
simp only [enumerate, h, Nat.add_eq, add_zero]
exact fun h' : enumerate sel (s \ {a'}) n = some a ↦
have : a ∈ s \ {a'} := enumerate_mem h_sel h'
this.left
theorem enumerate_inj {n₁ n₂ : ℕ} {a : α} {s : Set α} (h_sel : ∀ s a,... | Mathlib/Data/Set/Enumerate.lean | 62 | 72 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Limits.Preserves.Basic
/-!
# Isomorphisms about functors which preserve (co)limits
If `G` preserves limits, and `C` and `D` ... | @[reassoc (attr := simp)]
theorem preservesColimitIso_inv_comp_desc (t : Cocone F) :
(preservesColimitIso G F).inv ≫ G.map (colimit.desc _ t) =
colimit.desc _ (G.mapCocone t) := by
ext
| Mathlib/CategoryTheory/Limits/Preserves/Limits.lean | 142 | 146 |
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.SetTheory.Cardinal.Finite
import Mathlib.Data.Set.Finite.Powerset
/-!
# Noncomputable Set Cardinality
We define the cardinality of set `s` as a term `Set... |
theorem encard_diff_singleton_add_one (h : a ∈ s) :
(s \ {a}).encard + 1 = s.encard := by
| Mathlib/Data/Set/Card.lean | 256 | 258 |
/-
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 [NormedField 𝕝] [Module 𝕝 E]
variable [NontriviallyNormedField 𝕜₂] [AddCommGroup F] [Module 𝕜₂ F]
variable [NormedField 𝕝₂] [Module 𝕝₂ F]
variable {σ₁₂ : 𝕜 →+* 𝕜₂} [RingHomIsometric σ₁₂]
variable {τ₁₂ : 𝕝 →+* 𝕝₂} [RingHomIsometric τ₁₂]
variable [Nonempty ι] [Nonempty ι']
theorem continuous_of_contin... | Mathlib/Analysis/LocallyConvex/WithSeminorms.lean | 542 | 561 |
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