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 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yaël Dillies
-/
import Mathlib.GroupTheory.Perm.Cycle.Basic
/-!
# Closure results for permutation groups
* This file contains several closure results:
* `closure_isCycle... | obtain ⟨i, hi⟩ :=
h1.exists_pow_eq (mem_support.mp ((Finset.ext_iff.mp h2 x).mpr (Finset.mem_univ x)))
(mem_support.mp ((Finset.ext_iff.mp h2 y).mpr (Finset.mem_univ y)))
rw [h5, ← hi]
refine closure_cycle_coprime_swap
(Nat.Coprime.symm (h0.coprime_iff_not_dvd.mpr fun h => h4 ?_)) h1 h2 x
obtain ⟨... | Mathlib/GroupTheory/Perm/Closure.lean | 111 | 122 |
/-
Copyright (c) 2023 Jake Levinson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jake Levinson
-/
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Positivity.Core
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
/-!
# D... | | k + 1 => by
| Mathlib/Data/Nat/Factorial/DoubleFactorial.lean | 48 | 48 |
/-
Copyright (c) 2017 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Tim Baumann, Stephen Morgan, Kim Morrison, Floris van Doorn
-/
import Mathlib.Tactic.CategoryTheory.Reassoc
/-!
# Isomorphisms
This file defines isomorphisms between objects of a categ... | inv := 𝟙 X
| Mathlib/CategoryTheory/Iso.lean | 117 | 117 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Simon Hudon
-/
import Mathlib.CategoryTheory.Monoidal.Braided.Basic
import Mathlib.CategoryTheory.Monoidal.OfChosenFiniteProducts.Basic
/-!
# The symmetric monoidal struct... | apply (ℬ _ _).isLimit.hom_ext
rintro ⟨⟨⟩⟩ <;> · dsimp [Limits.IsLimit.conePointUniqueUpToIso]; simp
end MonoidalOfChosenFiniteProducts
open MonoidalOfChosenFiniteProducts
| Mathlib/CategoryTheory/Monoidal/OfChosenFiniteProducts/Symmetric.lean | 77 | 83 |
/-
Copyright (c) 2021 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Riccardo Brasca
-/
import Mathlib.Analysis.Normed.Module.Basic
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.RingTheory.Ideal.Quotient.Operations
import Mathl... |
/-- The quotient in the category of normed groups. -/
@[deprecated QuotientAddGroup.instNormedAddCommGroup (since := "2025-02-02")]
noncomputable instance AddSubgroup.normedAddCommGroupQuotient (S : AddSubgroup M)
[IsClosed (S : Set M)] : NormedAddCommGroup (M ⧸ S) := inferInstance
namespace AddSubgroup
| Mathlib/Analysis/Normed/Group/Quotient.lean | 328 | 335 |
/-
Copyright (c) 2015 Joseph Hua. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Hua
-/
import Mathlib.Data.W.Basic
/-!
# Examples of W-types
We take the view of W types as inductive types.
Given `α : Type` and `β : α → Type`, the W type determined by this dat... | simp only [toNat, ofNat, leftInverse_nat (f ()), mk.injEq, heq_eq_eq, true_and]
rfl
theorem rightInverse_nat : Function.RightInverse ofNat toNat
| Nat.zero => rfl
| Nat.succ n => by rw [ofNat, toNat, rightInverse_nat n]
/-- The naturals are equivalent to their associated `WType` -/
def equivNat : WType Na... | Mathlib/Data/W/Constructions.lean | 66 | 74 |
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Group.Invertible.Basic
import Mathlib.Logic.Basic
import Mathlib.Data.Set.Basic
/-!
# Cent... | -- cf. `Commute.left_comm`
@[to_additive]
| Mathlib/Algebra/Group/Center.lean | 73 | 74 |
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Data.Prod.Lex
import Mathlib.Data.Sigma.Lex
import Mathlib.Order.RelIso.Set
import Mathlib.Order.WellQuasiOrder
import Mathlib.Tactic.TFAE
/-!
# Well-... | refine Nat.find_min h ?_ ⟨g, fun m mn => (h1 m mn).trans (hg1 m mn), con, rfl⟩
rwa [← h3]
theorem exists_min_bad_of_exists_bad (r : α → α → Prop) (rk : α → ℕ) (s : Set α) :
(∃ f, IsBadSeq r s f) → ∃ f, IsBadSeq r s f ∧ ∀ n, IsMinBadSeq r rk s n f := by
rintro ⟨f0, hf0 : IsBadSeq r s f0⟩
let fs : ∀ n : ... | Mathlib/Order/WellFoundedSet.lean | 711 | 717 |
/-
Copyright (c) 2023 Alex Keizer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Keizer
-/
import Mathlib.Data.Vector.Basic
/-!
This file establishes a `snoc : Vector α n → α → Vector α (n+1)` operation, that appends a single
element to the back of a vector.... | = let q := f x y s
let r := mapAccumr₂ f xs ys q.1
(r.1, r.2.snoc q.2) := by
induction xs, ys using Vector.inductionOn₂ <;> simp_all
end Simp
| Mathlib/Data/Vector/Snoc.lean | 154 | 159 |
/-
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 | 1,765 | 1,767 | |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Covering.VitaliFamily
import Mathlib.MeasureTheory.Function.AEMeasurableOrder
import Mathlib.MeasureTheory.Integral.Average
import ... | simp -zeta only [not_lt, not_eventually, nonpos_iff_eq_zero] at hs
change μ s = 0
let f : α → Set (Set α) := fun _ => {a | μ a = 0}
have h : v.FineSubfamilyOn f s := by
intro x hx ε εpos
rw [hs] at hx
simp only [frequently_filterAt_iff, exists_prop, gt_iff_lt, mem_setOf_eq] at hx
rcases hx ε εpo... | Mathlib/MeasureTheory/Covering/Differentiation.lean | 97 | 113 |
/-
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.Data.Set.Order
import Mathlib.Order.Bounds.Basic
import Mathlib.Order.Interval.Set.Image
import Mathlib.Order.Interval.Set.LinearOrder
import Mathlib.Tac... | theorem Ioo_min_max : Ioo (min a b) (max a b) = uIoo a b := rfl
| Mathlib/Order/Interval/Set/UnorderedInterval.lean | 364 | 365 |
/-
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.Order.Atoms
import Mathlib.Order.OrderIsoNat
import Mathlib.Order.RelIso.Set
import Mathlib.Order.SupClosed
import Mathlib.Order.SupIndep
import Mathlib.Orde... |
theorem sSupIndep_iff_finite {s : Set α} :
sSupIndep s ↔
∀ t : Finset α, ↑t ⊆ s → sSupIndep (↑t : Set α) :=
⟨fun hs _ ht => hs.mono ht, fun h a ha => by
rw [disjoint_iff, inf_sSup_eq_iSup_inf_sup_finset, iSup_eq_bot]
intro t
rw [iSup_eq_bot, Finset.sup_id_eq_sSup]
intro ht
classical
... | Mathlib/Order/CompactlyGenerated/Basic.lean | 434 | 447 |
/-
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.Algebra.ModEq
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Algebra.Ring.Periodic
import Mathlib.Data.Int.SuccPred
import Mathlib.Order.Cir... | TFAE
[a ≡ b [PMOD p], ∀ z : ℤ, b - z • p ∉ Set.Ioo a (a + p), toIcoMod hp a b ≠ toIocMod hp a b,
| Mathlib/Algebra/Order/ToIntervalMod.lean | 515 | 516 |
/-
Copyright (c) 2020 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Andrew Yang
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
/-!
# Derivations
This file defines derivation. A derivat... | /-- If adjoin of a set is the whole algebra, then any two derivations equal on this set are equal
on the whole algebra. -/
theorem ext_of_adjoin_eq_top (s : Set A) (hs : adjoin R s = ⊤) (h : Set.EqOn D1 D2 s) : D1 = D2 :=
ext fun _ => eqOn_adjoin h <| hs.symm ▸ trivial
-- Data typeclasses
instance : Zero (Derivation... | Mathlib/RingTheory/Derivation/Basic.lean | 158 | 165 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.BoxIntegral.Partition.Basic
/-!
# Split a box along one or more hyperplanes
## Main definitions
A hyperplane `{x : ι → ℝ | x i = a}` spli... | haveI := fun π (_ : π ∈ s) => eventually_splitMany_inf_eq_filter π
(hs.eventually_all.2 this).exists
/-- If `π` is a partition of `I`, then there exists a finite set `s` of hyperplanes such that
`splitMany I s ≤ π`. -/
theorem IsPartition.exists_splitMany_le {I : Box ι} {π : Prepartition I} (h : IsPartition π) :
... | Mathlib/Analysis/BoxIntegral/Partition/Split.lean | 310 | 318 |
/-
Copyright (c) 2020 Kenji Nakagawa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenji Nakagawa, Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.Algebra.Subalgebra.Pointwise
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.RingTheory.Spect... | · simpa only [prime_span_singleton_iff]
theorem count_span_normalizedFactors_eq_of_normUnit {r X : R}
(hr : r ≠ 0) (hX₁ : normUnit X = 1) (hX : Prime X) :
Multiset.count (Ideal.span {X} : Ideal R) (normalizedFactors (Ideal.span {r})) =
Multiset.count X (normalizedFactors r) := by
simpa [hX₁, norm... | Mathlib/RingTheory/DedekindDomain/Ideal.lean | 1,442 | 1,449 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Finset.Card
import Mathlib.Data.Fintype.Basic
/-!
# Cardinalities of finite types
This file defines the cardinality `Fintype.card α` as the numb... | Mathlib/Data/Fintype/Card.lean | 659 | 660 | |
/-
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... | (hA : MeasurableSet A) {r : ℝ≥0∞} (hr : r < μ A) :
∃ K, K ⊆ A ∧ IsCompact K ∧ r < μ K :=
InnerRegular.innerRegular hA _ hr
protected theorem map_of_continuous [BorelSpace α] [MeasurableSpace β] [TopologicalSpace β]
[BorelSpace β] [h : InnerRegular μ] {f : α → β} (hf : Continuous f) :
| Mathlib/MeasureTheory/Measure/Regular.lean | 676 | 681 |
/-
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 | 1,731 | 1,732 | |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.Basic
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
/-!
# Left Homology of short complexes
Given a short complex `S : Shor... | lemma isIso_i (hg : S.g = 0) : IsIso h.i :=
⟨h.liftK (𝟙 S.X₂) (by rw [hg, id_comp]),
by simp only [← cancel_mono h.i, id_comp, assoc, liftK_i, comp_id], liftK_i _ _ _⟩
| Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean | 128 | 130 |
/-
Copyright (c) 2024 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.GroupTheory.Perm.Cycle.Type
/-!
# Fixed-point-free automorphisms
This file defines fixed-point-free automorphisms and proves some basic properties.... | /-- If a finite group admits a fixed-point-free involution, then it is commutative. -/
def commGroupOfInvolutive (hφ : FixedPointFree φ) (h2 : Function.Involutive φ):
CommGroup G := .mk (hφ.commute_all_of_involutive h2)
| Mathlib/GroupTheory/FixedPointFree.lean | 69 | 71 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl, Patrick Massot
-/
import Mathlib.Data.Set.Image
import Mathlib.Data.SProd
/-!
# Sets in product and pi types
This file proves basic properties of prod... | exact prod_mono H.1 H.2
theorem prod_eq_prod_iff_of_nonempty (h : (s ×ˢ t).Nonempty) :
| Mathlib/Data/Set/Prod.lean | 345 | 347 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
import Mathlib.Probability.Notation
import Mathlib.Probability.Process.Stopping
/-!
# Martingales
A family of functions `f : ι → Ω → E` is a martingale wit... | simpa
end Supermartingale
namespace Submartingale
| Mathlib/Probability/Martingale/Basic.lean | 179 | 184 |
/-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Game.Basic
import Mathlib.SetTheory.Ordinal.NaturalOps
/-!
# Ordinals as games
We define the canonical map `Ordinal... | -- Porting note: was `rw [PGame.Equiv]`
change _ ≤_ ∧ _ ≤ _ ↔ _
rw [le_antisymm_iff, toPGame_le_iff, toPGame_le_iff]
| Mathlib/SetTheory/Game/Ordinal.lean | 134 | 137 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩
rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN
| Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean | 135 | 136 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Fin.Fin2
import Mathlib.Data.PFun
import Mathlib.Data.Vector3
import Mathlib.NumberTheory.PellMatiyasevic
/-!
# Diophantine functions and Matiyas... |
@[inherit_doc]
scoped infixl:80 " D- " => Dioph.sub_dioph
| Mathlib/NumberTheory/Dioph.lean | 594 | 597 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro
-/
import Mathlib.Algebra.Module.Submodule.Bilinear
import Mathlib.Algebra.Module.Equiv.Basic
import Mathlib.GroupTheory.Congruence.Hom
import Mathlib.Tactic.Abel
... | Mathlib/LinearAlgebra/TensorProduct/Basic.lean | 1,469 | 1,469 | |
/-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Fold
import Mathlib.LinearAlgebra.ExteriorAlgebra.Basic
import Mathlib... |
include h h' in
/-- Auxiliary lemma used as an argument to `CliffordAlgebra.changeForm` -/
theorem changeForm.add_proof : (B + B').toQuadraticMap = Q'' - Q :=
(congr_arg₂ (· + ·) h h').trans <| sub_add_sub_cancel' _ _ _
| Mathlib/LinearAlgebra/CliffordAlgebra/Contraction.lean | 249 | 253 |
/-
Copyright (c) 2024 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Kernel.Composition.MapComap
import Mathlib.Probability.Martingale.Convergence
import Mathlib.Probability.Process.PartitionFiltration
/-!
# Ker... | · exact ⟨0, measure_ne_top _ _⟩
simpa [← prod_iInter, hseq_iInter] using h
lemma tendsto_density_atTop_ae_of_antitone (hκν : fst κ ≤ ν) [IsFiniteKernel ν] (a : α)
(seq : ℕ → Set β) (hseq : Antitone seq) (hseq_iInter : ⋂ i, seq i = ∅)
| Mathlib/Probability/Kernel/Disintegration/Density.lean | 623 | 627 |
/-
Copyright (c) 2021 Aaron Anderson, Jesse Michael Han, Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jesse Michael Han, Floris van Doorn
-/
import Mathlib.ModelTheory.Basic
/-!
# Language Maps
Maps between first-order languages in... | LHom.funext (funext fun _n => Sum.comp_elim _ _ _) (funext fun _n => Sum.comp_elim _ _ _)
end SumElim
| Mathlib/ModelTheory/LanguageMap.lean | 153 | 155 |
/-
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... | csInf_le := (WithTop.conditionallyCompleteLattice (α := αᵒᵈ)).le_csSup
le_csInf := (WithTop.conditionallyCompleteLattice (α := αᵒᵈ)).csSup_le }
open Classical in
noncomputable instance WithTop.WithBot.completeLattice {α : Type*}
[ConditionallyCompleteLattice α] : CompleteLattice (WithTop (WithBot α)) :=
... | Mathlib/Order/ConditionallyCompleteLattice/Basic.lean | 902 | 911 |
/-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.JapaneseBracket
import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.MeasureTheory.Group.Integral
import Mathlib... |
theorem integrableOn_exp_Iic (c : ℝ) : IntegrableOn exp (Iic c) := by
refine
integrableOn_Iic_of_intervalIntegral_norm_bounded (exp c) c
(fun y => intervalIntegrable_exp.1) tendsto_id
(eventually_of_mem (Iic_mem_atBot 0) fun y _ => ?_)
simp_rw [norm_of_nonneg (exp_pos _).le, integral_exp, sub_le_se... | Mathlib/Analysis/SpecialFunctions/ImproperIntegrals.lean | 32 | 38 |
/-
Copyright (c) 2023 Matthew Robert Ballard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matthew Robert Ballard
-/
import Mathlib.Algebra.Divisibility.Units
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Tactic.Common
/-!
# The maximal power of one natural n... | rw [maxPowDiv.go]
simp
theorem base_mul_eq_succ {p n : ℕ} (hp : 1 < p) (hn : 0 < n) :
| Mathlib/Data/Nat/MaxPowDiv.lean | 63 | 66 |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Mario Carneiro
-/
import Mathlib.Algebra.Order.Group.OrderIso
import Mathlib.Algebra.Ring.Int.Defs
import Mathlib.Data.Nat.Find
import Mathlib.Order.Bounds.Defs
/-! # Le... | let ⟨b, Hb⟩ := Hbdd
let ⟨lb, H⟩ := leastOfBdd b Hb Hinh
exact ⟨lb, H⟩
theorem coe_leastOfBdd_eq {P : ℤ → Prop} [DecidablePred P] {b b' : ℤ} (Hb : ∀ z : ℤ, P z → b ≤ z)
(Hb' : ∀ z : ℤ, P z → b' ≤ z) (Hinh : ∃ z : ℤ, P z) :
| Mathlib/Data/Int/LeastGreatest.lean | 71 | 76 |
/-
Copyright (c) 2021 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.CategoryTheory.Comma.StructuredArrow.Small
import Mathlib.CategoryTheory.Generator.Basic
import Mathlib.CategoryTheory.Limits.ConeCategory
import Mathlib.C... | if `G` satisfies the solution set condition then `G` is a right adjoint.
-/
lemma isRightAdjoint_of_preservesLimits_of_solutionSetCondition [HasLimits D]
[PreservesLimits G] (hG : SolutionSetCondition G) : G.IsRightAdjoint := by
refine @isRightAdjointOfStructuredArrowInitials _ _ _ _ G ?_
intro A
specialize h... | Mathlib/CategoryTheory/Adjunction/AdjointFunctorTheorems.lean | 78 | 93 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Option.NAry
import Mathlib.Data.Seq.Computation
import Mathlib.Tactic.ApplyFun
import Mathlib.Data.List.Basic
/-!
# Possibly infinite lists
This... |
@[simp]
theorem terminatedAt_nil {n : ℕ} : TerminatedAt (nil : Seq α) n := rfl
| Mathlib/Data/Seq/Seq.lean | 697 | 700 |
/-
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... | lemma eq_mul_of_mul_inv_eq₀ (hb : b ≠ 0) (h : a * c⁻¹ = b) : a = b * c := by
| Mathlib/Algebra/GroupWithZero/Units/Basic.lean | 288 | 288 |
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.AlgebraicGeometry.Cover.Open
import Mathlib.AlgebraicGeometry.Over
/-!
# Restriction of Schemes and Morphisms
## Main definition
- `AlgebraicGeometry.Schem... | /-- The global sections of the restriction is isomorphic to the sections on the open set. -/
@[simps!]
def topIso : Γ(U, ⊤) ≅ Γ(X, U) :=
X.presheaf.mapIso (eqToIso U.ι_image_top.symm).op
/-- The stalks of an open subscheme are isomorphic to the stalks of the original scheme. -/
| Mathlib/AlgebraicGeometry/Restrict.lean | 129 | 134 |
/-
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
-/
import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.L... | apply isUnit_det_of_left_inverse
simpa using (LinearMap.toMatrix_comp v v' v f.symm f).symm
/-- Specialization of `LinearEquiv.isUnit_det` -/
| Mathlib/LinearAlgebra/Determinant.lean | 424 | 427 |
/-
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... | Mathlib/Algebra/Group/Defs.lean | 1,324 | 1,325 | |
/-
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... |
@[simp]
theorem fderiv_natCast [NatCast F] (n : ℕ) : fderiv 𝕜 (n : E → F) = 0 := fderiv_const _
| Mathlib/Analysis/Calculus/FDeriv/Basic.lean | 1,254 | 1,257 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Dia... | simp only [volume_pi_Ico, ENNReal.toReal_prod, ENNReal.toReal_ofReal (sub_nonneg.2 (h _))]
@[simp]
nonrec theorem volume_pi_ball (a : ι → ℝ) {r : ℝ} (hr : 0 < r) :
| Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | 257 | 260 |
/-
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... | by_cases h : a < b
· rw [Ico_eq_cons_Ioo h, card_cons]
exact (Nat.add_sub_cancel _ _).symm
· rw [Ioo_eq_empty h, Ico_eq_empty h, card_empty, Nat.zero_sub]
theorem card_Ioo_eq_card_Ioc_sub_one (a b : α) : #(Ioo a b) = #(Ioc a b) - 1 :=
| Mathlib/Order/Interval/Finset/Basic.lean | 651 | 656 |
/-
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 | 779 | 780 | |
/-
Copyright (c) 2019 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.Adjunction.Reflective
import Mathlib.CategoryTheory.Monad.Algebra
/-!
# Adjunctions and (co)monads
We develop the basic rela... | (comonadicAdjunction L).isLeftAdjoint
noncomputable instance (G : Comonad C) : ComonadicLeftAdjoint G.forget where
| Mathlib/CategoryTheory/Monad/Adjunction.lean | 290 | 292 |
/-
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... | def isoZeroOfMonoEqZero {X Y : C} {f : X ⟶ Y} [Mono f] (h : f = 0) : X ≅ 0 := by
subst h
apply isoZeroOfMonoZero ‹_›
| Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean | 390 | 392 |
/-
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.ContMDiff.Defs
/-!
## Basic properties of `C^n` functions between manifolds
In this file, we show that stan... | @[deprecated (since := "2024-11-20")] alias Smooth.extend_one := ContMDiff.extend_one
@[deprecated (since := "2024-11-20")] alias Smooth.extend_zero := ContMDiff.extend_zero
@[deprecated (since := "2024-11-20")] alias smooth_inclusion := contMDiff_inclusion
end Inclusion
| Mathlib/Geometry/Manifold/ContMDiff/Basic.lean | 332 | 338 |
/-
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... | rfl
| Mathlib/Data/List/OfFn.lean | 65 | 66 |
/-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Kernel.Defs
/-!
# Basic kernels
This file contains basic results about kernels in general and definitions of some particular
kernels.
## Mai... | Mathlib/Probability/Kernel/Basic.lean | 592 | 595 | |
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.Constructions
/-!
# Neighborhoods and continuity relative to a subset
This file develops API on the relative versions
* `nhdsWithin` ... | continuousOn_of_forall_continuousAt <| fun _ hx => hx.elim
(fun h => ContinuousWithinAt.continuousAt (continuousWithinAt hs h) <| IsOpen.mem_nhds s_op h)
(ht _)
| Mathlib/Topology/ContinuousOn.lean | 1,347 | 1,350 |
/-
Copyright (c) 2022 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.Data.Int.Range
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.MulChar.Basic
/-!
# Quadratic characters on ℤ/nℤ
This file defines some quadr... | theorem χ₄_int_mod_four (n : ℤ) : χ₄ n = χ₄ (n % 4 : ℤ) := by
rw [← ZMod.intCast_mod n 4, Nat.cast_ofNat]
| Mathlib/NumberTheory/LegendreSymbol/ZModChar.lean | 58 | 59 |
/-
Copyright (c) 2018 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Mario Carneiro, Kim Morrison, Floris van Doorn
-/
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryTheory.Limits.Cones
import Batteries.Tactic.Congr
/-... | attribute [reassoc (attr := simp)] IsColimit.fac
namespace IsColimit
instance subsingleton {t : Cocone F} : Subsingleton (IsColimit t) :=
⟨by intro P Q; cases P; cases Q; congr; aesop_cat⟩
/-- Given a natural transformation `α : F ⟶ G`, we give a morphism from the cocone point
| Mathlib/CategoryTheory/Limits/IsLimit.lean | 513 | 520 |
/-
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... | Finset.le_sup (α := ℕ) (f := fun s => sum s fun _ e => e) h
theorem totalDegree_le_degrees_card (p : MvPolynomial σ R) :
p.totalDegree ≤ Multiset.card p.degrees := by
classical
| Mathlib/Algebra/MvPolynomial/Degrees.lean | 358 | 362 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Aurélien Saue, Anne Baanen
-/
import Mathlib.Tactic.NormNum.Inv
import Mathlib.Tactic.NormNum.Pow
import Mathlib.Util.AtomM
/-!
# `ring` tactic
A tactic for solving e... | * `(a * b) ^ e = a ^ e * b ^ e`
| Mathlib/Tactic/Ring/Basic.lean | 766 | 766 |
/-
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 | 1,457 | 1,460 | |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Aurélien Saue, Anne Baanen
-/
import Mathlib.Tactic.NormNum.Inv
import Mathlib.Tactic.NormNum.Pow
import Mathlib.Util.AtomM
/-!
# `ring` tactic
A tactic for solving e... | ve' : ExProd sℕ e'
| Mathlib/Tactic/Ring/Basic.lean | 803 | 803 |
/-
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.Algebra.Order.Ring.Nat
import Mathlib.Logic.Encodable.Pi
import Mathlib.Logic.Function.Iterate
/-!
# The primitive recursive functions
The primitive ... | Mathlib/Computability/Primrec.lean | 1,510 | 1,511 | |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Jakob von Raumer
-/
import Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
/-!
# Biproducts and binary biproducts
... | Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean | 2,234 | 2,239 | |
/-
Copyright (c) 2020 Kexing Ying and Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Kevin Buzzard, Yury Kudryashov
-/
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.Algebra.Gro... | rw [← Set.ext_iff.mp h x, mem_inter_iff]
exact ⟨hx, hxf⟩
| Mathlib/Algebra/BigOperators/Finprod.lean | 422 | 423 |
/-
Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
import Mathlib.Data.ENNReal.Real
import Mathlib.Tactic.Bound.Attribute
import Mathlib.Topology... | ⟨n, trivial, hn.le⟩
theorem mem_uniformity_dist {s : Set (α × α)} :
s ∈ 𝓤 α ↔ ∃ ε > 0, ∀ ⦃a b : α⦄, dist a b < ε → (a, b) ∈ s :=
uniformity_basis_dist.mem_uniformity_iff
| Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | 690 | 694 |
/-
Copyright (c) 2024 Judith Ludwig, Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Judith Ludwig, Christian Merten
-/
import Mathlib.Algebra.Exact
import Mathlib.RingTheory.AdicCompletion.Functoriality
import Mathlib.RingTheory.Filtration
/-!
# Exac... | apply AdicCompletion.induction_on I N y (fun b ↦ ?_)
let a := mapPreimage hf b
refine ⟨AdicCompletion.mk I M (AdicCauchySequence.mk I M (fun n ↦ (a n : M)) ?_), ?_⟩
· refine fun n ↦ SModEq.symm ?_
simp only [SModEq.symm, SModEq, mapPreimage, Submodule.Quotient.mk_sub,
sub_eq_self, Submodule.Quotient.m... | Mathlib/RingTheory/AdicCompletion/Exactness.lean | 69 | 76 |
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Algebra.Group.Commute.Hom
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Data.Fintype.B... | @[to_additive (attr := simp)]
theorem noncommProd_insert_of_not_mem [DecidableEq α] (s : Finset α) (a : α) (f : α → β) (comm)
(ha : a ∉ s) :
noncommProd (insert a s) f comm =
f a * noncommProd s f (comm.mono fun _ => mem_insert_of_mem) := by
simp only [← cons_eq_insert _ _ ha, noncommProd_cons]
@[to_ad... | Mathlib/Data/Finset/NoncommProd.lean | 288 | 296 |
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Choose.Cast
import Mathlib.Data.Nat.Choose.Vanderm... | @[simp]
theorem hasseDeriv_zero : @hasseDeriv R _ 0 = LinearMap.id :=
LinearMap.ext <| hasseDeriv_zero'
| Mathlib/Algebra/Polynomial/HasseDeriv.lean | 83 | 85 |
/-
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.Subalgebra.Prod
import Mathlib.Algebra.Algebra.Subalgebra.Tower
import Mathlib.LinearAlgebra.Basis.Basic
import Mathlib.LinearAlgebra.Prod
/-!
# ... | Mathlib/RingTheory/Adjoin/Basic.lean | 209 | 211 | |
/-
Copyright (c) 2019 Calle Sönne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology... | theorem sign_sub_pi (θ : Angle) : (θ - π).sign = -θ.sign :=
sign_antiperiodic.sub_eq θ
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 727 | 728 |
/-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Jireh Loreaux
-/
import Mathlib.Algebra.GroupWithZero.Hom
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Ring.Basic
/-!
# Homomorphisms of semirings and... | /-- Turn an element of a type `F` satisfying `RingHomClass F α β` into an actual
`RingHom`. This is declared as the default coercion from `F` to `α →+* β`. -/
@[coe]
| Mathlib/Algebra/Ring/Hom/Defs.lean | 329 | 331 |
/-
Copyright (c) 2022 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Integral.IntegrableOn
/-!
# Locally integrable functions
A function is called *locally integrable* (`MeasureTheory.LocallyIntegrabl... | let K := tsupport g
have hK : IsCompact K := h'g
have : K.indicator (fun x ↦ f x • g x) = (fun x ↦ f x • g x) := by
| Mathlib/MeasureTheory/Function/LocallyIntegrable.lean | 341 | 343 |
/-
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... | namespace ArithmeticFunction
/-- A translation of the relation `Λ * ↑ζ = log` of (real-valued) arithmetic functions
to an equality of complex sequences. -/
lemma convolution_vonMangoldt_zeta : ↗Λ ⍟ ↗ζ = ↗Complex.log := by
ext n
simpa [apply_ite, LSeries.convolution_def, -vonMangoldt_mul_zeta]
| Mathlib/NumberTheory/LSeries/Dirichlet.lean | 327 | 333 |
/-
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.Ring.Semiconj
import Mathlib.Algebra.Ring.Units
import Mathlib.Algebra.Gro... | lemma sub_sq_comm (a b : R) : (a - b) ^ 2 = (b - a) ^ 2 := by
| Mathlib/Algebra/Ring/Commute.lean | 196 | 196 |
/-
Copyright (c) 2020 Kenji Nakagawa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenji Nakagawa, Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.Algebra.Subalgebra.Pointwise
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.RingTheory.Spect... | exact
idealFactorsEquivOfQuotEquiv_mem_normalizedFactors_of_mem_normalizedFactors f.symm hI
j.prop⟩
left_inv := fun ⟨j, hj⟩ => by simp
right_inv := fun ⟨j, hj⟩ => by simp [-Set.coe_setOf]
@[simp]
theorem normalizedFactorsEquivOfQuotEquiv_symm (hI : I ≠ ⊥) (hJ : J ≠ ⊥) :
(normalizedFacto... | Mathlib/RingTheory/DedekindDomain/Ideal.lean | 1,116 | 1,124 |
/-
Copyright (c) 2023 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Probability.Kernel.Composition.Comp
/-!
# Invariance of measures along a kernel
We say that a measure `μ` is invariant with respect to a kernel `κ` if its ... |
/-- A measure `μ` is invariant with respect to the kernel `κ` if the push-forward measure of `μ`
along `κ` equals `μ`. -/
| Mathlib/Probability/Kernel/Invariance.lean | 63 | 65 |
/-
Copyright (c) 2022 Kevin H. Wilson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin H. Wilson
-/
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Analysis.NormedSpace.RCLike
import Mathlib.Order.Filter.Curry
/-!
# Swapping limits and derivatives via u... | (hf : ∀ n : ι, ∀ y : E, y ∈ Metric.ball x r → HasFDerivAt (f n) (f' n y) y)
(hfg : Cauchy (map (fun n => f n x) l)) : UniformCauchySeqOn f l (Metric.ball x r) := by
letI : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
letI : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 _
have : NeBot l := (cauchy_map_... | Mathlib/Analysis/Calculus/UniformLimitsDeriv.lean | 176 | 220 |
/-
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.Abelian
import Mathlib.Algebra.Lie.BaseChange
import Mathlib.Algebra.Lie.IdealOperations
import Mathlib.Order.Hom.Basic
import Mathlib.RingTheory... | @[gcongr, mono]
theorem derivedSeriesOfIdeal_le {I J : LieIdeal R L} {k l : ℕ} (h₁ : I ≤ J) (h₂ : l ≤ k) :
D k I ≤ D l J := by
revert l; induction' k with k ih <;> intro l h₂
| Mathlib/Algebra/Lie/Solvable.lean | 82 | 85 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Alex Kontorovich, Heather Macbeth
-/
import Mathlib.MeasureTheory.Group.Action
import Mathlib.MeasureTheory.Group.Pointwise
import Mathlib.MeasureTheory.Integral.Lebe... | variable {G s}
@[to_additive (attr := simp) MeasureTheory.mem_addFundamentalFrontier]
theorem mem_fundamentalFrontier :
x ∈ fundamentalFrontier G s ↔ x ∈ s ∧ ∃ g : G, g ≠ 1 ∧ x ∈ g • s := by
simp [fundamentalFrontier]
@[to_additive (attr := simp) MeasureTheory.mem_addFundamentalInterior]
theorem mem_fundamental... | Mathlib/MeasureTheory/Group/FundamentalDomain.lean | 496 | 505 |
/-
Copyright (c) 2021 Filippo A. E. Nuccio. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Filippo A. E. Nuccio, Eric Wieser
-/
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.Matrix.Block
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.Linear... | · intro; exact map_zero _
_ = _ := by simp_rw [det_blockDiagonal, Finset.prod_const, Finset.card_univ]
end KroneckerMap
/-! ### Specialization to `Matrix.kroneckerMap (*)` -/
section Kronecker
open Matrix
/-- The Kronecker product. This is just a shorthand for `kroneckerMap (*)`. Prefer the notation
`⊗ₖ... | Mathlib/Data/Matrix/Kronecker.lean | 236 | 253 |
/-
Copyright (c) 2014 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Leonardo de Moura, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Field.Defs
/-!
# Linear ordered (semi)fields
A ... | Mathlib/Algebra/Order/Field/Defs.lean | 64 | 64 | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Aaron Anderson, Yakov Pechersky
-/
import Mathlib.Data.Fintype.Card
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Algebra.Group.End
import Mathlib.Data.Finset.N... |
theorem Disjoint.conj (H : Disjoint f g) (h : Perm α) : Disjoint (h * f * h⁻¹) (h * g * h⁻¹) :=
| Mathlib/GroupTheory/Perm/Support.lean | 110 | 111 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Yury Kudryashov
-/
import Mathlib.Algebra.Algebra.Hom
import Mathlib.Algebra.Ring.Action.Group
/-!
# Isomorphisms of `R`-algebras
This file defines bundled isomorphisms of `R`-... | @[simp]
theorem equivCongr_refl : equivCongr AlgEquiv.refl AlgEquiv.refl = Equiv.refl (A₁ ≃ₐ[R] A₁') :=
rfl
| Mathlib/Algebra/Algebra/Equiv.lean | 433 | 435 |
/-
Copyright (c) 2023 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.NumberTheory.EulerProduct.ExpLog
import Mathlib.NumberTheory.LSeries.Dirichlet
/-!
# The Euler Product for the Riemann Zeta Function and Dirichlet L-Ser... | simp only [dirichletSummandHom, cpow_neg, MonoidWithZeroHom.coe_mk, ZeroHom.coe_mk, LSeries,
LSeries.term_of_ne_zero' (ne_zero_of_one_lt_re hs), div_eq_mul_inv]
open Filter Nat Topology EulerProduct
/-- The Euler product for the Riemann ζ function, valid for `s.re > 1`.
This version is stated in terms of `HasPr... | Mathlib/NumberTheory/EulerProduct/DirichletLSeries.lean | 78 | 85 |
/-
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.HomologySequence
import Mathlib.Algebra.Homology.ShortComplex.ConcreteCategory
/-!
# Homology of complexes in concrete categories
The homology... |
@[simp]
lemma i_cyclesMk {i : ι} (x : (forget₂ C Ab).obj (K.X i)) (j : ι) (hj : c.next i = j)
(hx : ((forget₂ C Ab).map (K.d i j)) x = 0) :
((forget₂ C Ab).map (K.iCycles i)) (K.cyclesMk x j hj hx) = x := by
subst hj
| Mathlib/Algebra/Homology/ConcreteCategory.lean | 39 | 44 |
/-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Kernel.Defs
/-!
# Basic kernels
This file contains basic results about kernels in general and definitions of some particular
kernels.
## Mai... | Mathlib/Probability/Kernel/Basic.lean | 566 | 568 | |
/-
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... | fun _ j h => by
exact coeff_eq_zero_of_natDegree_lt <| Nat.lt_of_le_of_lt (h_deg j) h
theorem det_matrixOfPolynomials {n : ℕ} (p : Fin n → R[X])
(h_deg : ∀ i, (p i).natDegree = i) (h_monic : ∀ i, Monic <| p i) :
(Matrix.of (fun (i j : Fin n) => (p j).coeff i)).det = 1 := by
rw [Matrix.det_of_upperTrian... | Mathlib/LinearAlgebra/Matrix/Block.lean | 291 | 305 |
/-
Copyright (c) 2024 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.NumberTheory.LSeries.AbstractFuncEq
import Mathlib.NumberTheory.ModularForms.JacobiTheta.Bounds
import Mathlib.NumberTheory.LSeries.MellinEqDirichlet
i... | simp only [← mul_div_assoc, push_cast, mul_assoc (Gammaℝ _), ← mul_add]
congr 3
rw [mul_one, mul_neg_one, neg_neg, neg_mul I, ← sub_eq_neg_add, ← mul_sub, mul_comm,
mul_neg, neg_mul]
congr 3 <;> ring
/-- Formula for `completedHurwitzZetaOdd` as a Dirichlet series in the convergence range. -/
lemma hasSum_i... | Mathlib/NumberTheory/LSeries/HurwitzZetaOdd.lean | 420 | 438 |
/-
Copyright (c) 2018 Violeta Hernández Palacios, Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios, Mario Carneiro
-/
import Mathlib.Logic.Small.List
import Mathlib.SetTheory.Ordinal.Enum
import Mathlib.SetTheory.Ordinal.Exponen... | theorem nfp_eq_self {a} (h : f a = a) : nfp f a = a :=
nfpFamily_eq_self fun _ => h
/-- The fixed point lemma for normal functions: any normal function has an unbounded set of
| Mathlib/SetTheory/Ordinal/FixedPoint.lean | 293 | 296 |
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Tactic.FinCases
import Mathlib.Topology.Connected.LocallyConnected
import Mathlib.Topology.Sets.Closeds
/-!
# L... | funext fun y => hf.apply_eq_of_preconnectedSpace y x
theorem exists_eq_const [PreconnectedSpace X] [Nonempty Y] {f : X → Y} (hf : IsLocallyConstant f) :
∃ y, f = Function.const X y := by
| Mathlib/Topology/LocallyConstant/Basic.lean | 143 | 146 |
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.Galois.Basic
/-!
# Galois Groups of Polynomials
In this file, we introduce the Galois group of a polynomial... | have qx_int := Normal.isIntegral h_normal (aeval x q)
exact
splits_of_splits_of_dvd _ (minpoly.ne_zero qx_int) (Normal.splits h_normal _)
((minpoly.irreducible qx_int).dvd_symm hr (minpoly.dvd F _ hx))
have key2 : ∀ {p₁ p₂ : F[X]}, P p₁ → P p₂ → P (p₁ * p₂) := by
intro p₁ p₂ hp₁ hp₂
by_c... | Mathlib/FieldTheory/PolynomialGaloisGroup.lean | 321 | 335 |
/-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Analysis.SpecialFunctions.Complex.Circle
import Mathlib.Analysis.InnerProductSpace.... | Mathlib/Analysis/Fourier/AddCircle.lean | 150 | 150 | |
/-
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.FieldTheory.Minpoly.Field
import Mathlib.LinearAlgebra.SModEq
import Mathlib.RingTheory.Ideal.BigOperators
/-!
# Power basis
This file defines a structure ... | by_cases hf : f %ₘ minpoly A pb.gen = 0
· simp only [hf, map_zero]
have : (f %ₘ minpoly A pb.gen).natDegree < pb.dim := by
rw [← pb.natDegree_minpoly]
apply natDegree_lt_natDegree hf
exact degree_modByMonic_lt _ (minpoly.monic pb.isIntegral_gen)
rw [aeval_eq_sum_range' this, aeval_eq_sum_range' this... | Mathlib/RingTheory/PowerBasis.lean | 246 | 263 |
/-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Ordinal.Family
import Mathlib.Tactic.Abel
/-!
# Natural operations on ordinals
The goal of this file is to define n... | @[simp]
theorem nmul_one (a : Ordinal) : a ⨳ 1 = a := by
rw [nmul]
convert csInf_Ici
| Mathlib/SetTheory/Ordinal/NaturalOps.lean | 521 | 524 |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro
-/
import Mathlib.Algebra.Order.Group.Abs
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Algebra.Order.Ring.Int
import Mathlib.A... | rcases abs_choice a with h | h <;> simp only [h, neg_dvd]
| Mathlib/Algebra/Order/Ring/Abs.lean | 192 | 193 |
/-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
import Mathlib.Topology.Category.Profinite.Nobeling.Basic
import Mathlib.Topology.Category.Profinite.Nobeling.Induction
import Mathlib.Topology.Category.Profinite... | Mathlib/Topology/Category/Profinite/Nobeling.lean | 860 | 862 | |
/-
Copyright (c) 2022 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.MeasureTheory.Integral.Prod
import Mathlib.Meas... | intro t; by_cases ht : t ∈ support f
· have h2t := hf ht
rw [mem_ball_zero_iff] at h2t
specialize hg (x₀ - t)
rw [sub_eq_add_neg, add_mem_ball_iff_norm, norm_neg, ← sub_eq_add_neg] at hg
refine ((L (f t)).dist_le_opNorm _ _).trans ?_
| Mathlib/Analysis/Convolution.lean | 753 | 758 |
/-
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.Group.Subgroup.Map
import Mathlib.Algebra.Module.Submodule.... | def comap [SemilinearMapClass F σ₁₂ M M₂] (f : F) (p : Submodule R₂ M₂) : Submodule R M :=
{ p.toAddSubmonoid.comap f with
carrier := f ⁻¹' p
-- Note: https://github.com/leanprover-community/mathlib4/pull/8386 added `map_smulₛₗ _`
| Mathlib/Algebra/Module/Submodule/Map.lean | 170 | 173 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.Homotopy
import Mathlib.Algebra.Ring.NegOnePow
import Mathlib.Algebra.Category.Grp.Preadditive
import Mathlib.Tactic.Linarith
import Mathlib.Cat... | simp only [ofHom, d_comp_ofHoms_v]
@[simp]
| Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean | 180 | 182 |
/-
Copyright (c) 2020 Kenji Nakagawa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenji Nakagawa, Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.Algebra.Subalgebra.Pointwise
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.RingTheory.Spect... | -/
protected theorem div_eq_mul_inv [IsDedekindDomain A] (I J : FractionalIdeal A⁰ K) :
I / J = I * J⁻¹ := by
by_cases hJ : J = 0
· rw [hJ, div_zero, inv_zero', mul_zero]
refine le_antisymm ((mul_right_le_iff hJ).mp ?_) ((le_div_iff_mul_le hJ).mpr ?_)
· rw [mul_assoc, mul_comm J⁻¹, FractionalIdeal.mul_inv_c... | Mathlib/RingTheory/DedekindDomain/Ideal.lean | 533 | 545 |
/-
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... | sin_add_pi_div_two x ▸ sin_pos_of_mem_Ioo ⟨by linarith [hx.1], by linarith [hx.2]⟩
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | 442 | 443 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Data.ENNReal.Operations
/-!
# Results about division in extended non-negative reals
This file establishes basic properties related t... | simp_rw [div_eq_mul_inv]
| Mathlib/Data/ENNReal/Inv.lean | 259 | 259 |
/-
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.Basic
import Mathlib.Algebra.GroupWithZero.NeZero
import Mathlib.Logic.Unique
import Mathlib.Tactic.Conv
/-!
# Groups with an adjoined z... | `a` (whether or not `a` is zero). -/
@[simp]
theorem div_div_self (a : G₀) : a / (a / a) = a := by
rw [div_div_eq_mul_div]
| Mathlib/Algebra/GroupWithZero/Basic.lean | 377 | 380 |
/-
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.Algebra.CharP.Invertible
import Mathlib.Algebra.Order.Interval.Set.Group
import Mathlib.Analysis.Convex.Basic
import Mathlib.Analysis.Convex.Segment
import... |
lemma AffineSubspace.mem_of_wbtw {s : AffineSubspace R P} {x y z : P} (hxyz : Wbtw R x y z)
(hx : x ∈ s) (hz : z ∈ s) : y ∈ s := by obtain ⟨ε, -, rfl⟩ := hxyz; exact lineMap_mem _ hx hz
| Mathlib/Analysis/Convex/Between.lean | 155 | 158 |
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