Context stringlengths 227 76.5k | target stringlengths 0 11.6k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 16 3.69k |
|---|---|---|---|---|
/-
Copyright (c) 2022 Eric Rodriguez. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Rodriguez
-/
import Mathlib.Analysis.InnerProductSpace.Convex
import Mathlib.Analysis.SpecialFunctions.Complex.Arg
/-!
# Rays in the complex numbers
This file links the definiti... | by_cases hy : y = 0; · simp [hy]
simp [hx, hy, arg_div_coe_angle, sub_eq_zero]
theorem norm_add_eq_iff : ‖x + y‖ = ‖x‖ + ‖y‖ ↔ x = 0 ∨ y = 0 ∨ x.arg = y.arg :=
sameRay_iff_norm_add.symm.trans sameRay_iff
| Mathlib/Analysis/Complex/Arg.lean | 41 | 45 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.GroupWithZero.Divisibili... |
@[simp] lemma isRegular_prod_X (s : Finset σ) :
IsRegular (∏ n ∈ s, X n : MvPolynomial σ R) :=
IsRegular.prod fun _ _ ↦ isRegular_X
/-- The finset of nonzero coefficients of a multivariate polynomial. -/
def coeffs (p : MvPolynomial σ R) : Finset R :=
| Mathlib/Algebra/MvPolynomial/Basic.lean | 786 | 792 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Linear.Basic
import Mathlib.Algebra.Homology.ComplexShapeSigns
import Mathlib.Algebra.Homology.HomologicalBicomplex
import Mathlib.Algebra.Module.... | · simp [d₁_eq_zero _ c₁₂ i₁ i₂ i₁₂ h]
@[reassoc (attr := simp)]
lemma d₂_mapMap (i₁ : I₁) (i₂ : I₂) (i₁₂ : I₁₂) :
K.d₂ c₁₂ i₁ i₂ i₁₂ ≫ GradedObject.mapMap (toGradedObjectMap φ) _ i₁₂ =
(φ.f i₁).f i₂ ≫ L.d₂ c₁₂ i₁ i₂ i₁₂ := by
by_cases h : c₂.Rel i₂ (c₂.next i₂)
| Mathlib/Algebra/Homology/TotalComplex.lean | 365 | 371 |
/-
Copyright (c) 2021 Aaron Anderson, Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Kevin Buzzard, Yaël Dillies, Eric Wieser
-/
import Mathlib.Data.Finset.Lattice.Union
import Mathlib.Data.Finset.Pairwise
import Mathlib.Data.Finset.Prod
i... | @[deprecated (since := "2024-11-24")]
alias CompleteLattice.independent_ne_bot_iff_independent := iSupIndep_ne_bot
theorem iSupIndep.injOn (ht : iSupIndep t) : InjOn t {i | t i ≠ ⊥} := by
| Mathlib/Order/SupIndep.lean | 408 | 411 |
/-
Copyright (c) 2020 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard
-/
import Mathlib.RingTheory.AdicCompletion.Basic
import Mathlib.RingTheory.LocalRing.MaximalIdeal.Basic
import Mathlib.RingTheory.LocalRing.RingHom.Basic
import Mathlib.R... | apply le_antisymm
· intro r hr
by_cases hr0 : r = 0
· simp only [hr0, Submodule.zero_mem]
obtain ⟨n, u, rfl⟩ := H hr0
simp only [mem_span_singleton, Units.isUnit, IsUnit.dvd_mul_right]
apply pow_dvd_pow
apply Nat.find_min'
simpa only [Units.mul_inv_cancel_right] using I.mul_mem_right (↑u... | Mathlib/RingTheory/DiscreteValuationRing/Basic.lean | 250 | 279 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Group.Subgroup.Ker
import Mathlib.Algebra.BigOperators.Group.List.Basic
/-!
# Free groups
This file defines free groups over a type. Furthermore, it is... | Mathlib/GroupTheory/FreeGroup/Basic.lean | 1,416 | 1,417 | |
/-
Copyright (c) 2023 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.NoZeroSMulDivisors.Basic
import Mathlib.Data.Int.ModEq
import Mathlib.GroupTheory.QuotientGroup.Defs
import Math... | (add_zsmul_modEq _).trans
| Mathlib/Algebra/ModEq.lean | 115 | 116 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.Module.End
import Mathlib.Algebra.Ring.Prod
import Mathlib.Data.Fintype.Units
import Mathlib.GroupTheory.GroupAc... | theorem intCast_comp_cast : ((↑) : ℤ → R) ∘ (cast : ZMod n → ℤ) = cast := by
cases n
· exact congr_arg (Int.cast ∘ ·) ZMod.cast_id'
· ext
simp [ZMod, ZMod.cast]
| Mathlib/Data/ZMod/Basic.lean | 235 | 239 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
import Mathlib.Analysis.Normed.Module.Basic
import Mathlib.MeasureTheory.Function.SimpleFuncDense
/-!
# Strongly measurable and finitely strongly meas... | theorem _root_.List.stronglyMeasurable_prod' (l : List (α → M))
(hl : ∀ f ∈ l, StronglyMeasurable f) : StronglyMeasurable l.prod := by
induction' l with f l ihl; · exact stronglyMeasurable_one
| Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean | 540 | 542 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.RingTheory.FractionalIdeal.Basic
import Mathlib.RingTheory.IntegralClosure.IsIntegral.Basi... | IsLocalization.mk'_self, spanSingleton_one]
let y' : (FractionalIdeal R₁⁰ K)ˣ := Units.mkOfMulEqOne _ _ this
have coe_y' : ↑y' = spanSingleton R₁⁰ (IsLocalization.mk' K (1 : R₁) ⟨y, hy⟩) := rfl
refine Iff.trans ?_ (y'.mul_right_inj.trans coeIdeal_inj)
rw [coe_y', coeIdeal_mul, coeIdeal_span_singleton, coe... | Mathlib/RingTheory/FractionalIdeal/Operations.lean | 700 | 704 |
/-
Copyright (c) 2022 Jon Eugster. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jon Eugster
-/
import Mathlib.Algebra.CharP.LocalRing
import Mathlib.RingTheory.Ideal.Quotient.Basic
import Mathlib.Tactic.FieldSimp
/-!
# Equal and mixed characteristic
In commutative ... | (h_mixed : ∀ p : ℕ, Nat.Prime p → MixedCharZero R p → P) : P := by
cases CharP.exists R with
| intro p p_charP =>
by_cases h : p = 0
· rw [h] at p_charP
haveI h0 : CharZero R := CharP.charP_to_charZero R
exact split_equalCharZero_mixedCharZero R h_equal h_mixed
· exact h_pos p h p_charP
... | Mathlib/Algebra/CharP/MixedCharZero.lean | 321 | 329 |
/-
Copyright (c) 2018 Louis Carlin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Louis Carlin, Mario Carneiro
-/
import Mathlib.Algebra.EuclideanDomain.Defs
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Regular
import Mathlib.Algebra.Grou... | rw [gcd_zero_left]
exact ⟨dvd_zero _, dvd_rfl⟩)
fun a b _ ⟨IH₁, IH₂⟩ => by
| Mathlib/Algebra/EuclideanDomain/Basic.lean | 136 | 138 |
/-
Copyright (c) 2023 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll, Ralf Stephan
-/
import Mathlib.Data.Nat.Factorization.Defs
import Mathlib.Data.Nat.Squarefree
/-!
# Smooth numbers
For `s : Finset ℕ` we define the set `Nat.factoredNum... | rwa [Finset.filter_congr (s := Finset.range (succ N)) fun n _ ↦ hn' n]
rw [Finset.filter_ne', Finset.card_erase_of_mem <| Finset.mem_range_succ_iff.mpr <| zero_le N]
simp only [Finset.card_range, succ_sub_succ_eq_sub, tsub_zero]
/-- A `k`-smooth number can be written as a square times a product of distinct pri... | Mathlib/NumberTheory/SmoothNumbers.lean | 443 | 456 |
/-
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.Topology.Homeomorph.Lemmas
import Mathlib.Topology.Sets.OpenCover
import Mathlib.Topology.LocallyClosed
/-!
# Properties of maps that are local at the targe... | simpa [isClosed_induced_iff]
exact fun u hu e => ⟨f '' u, H u hu, by simp [← e, image_restrictPreimage]⟩
theorem IsOpenMap.restrictPreimage (H : IsOpenMap f) (s : Set β) :
IsOpenMap (s.restrictPreimage f) := by
intro t
suffices ∀ u, IsOpen u → Subtype.val ⁻¹' u = t →
| Mathlib/Topology/LocalAtTarget.lean | 78 | 84 |
/-
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.Sigma.Basic
import Mathlib.Algebra.Order.Ring.Nat
/-!
# A computable model of ZFA without infinity
In this file we define finite hereditary list... | end Lists'
namespace Lists
/-- Sends `a : α` to the corresponding atom in `Lists α`. -/
| Mathlib/SetTheory/Lists.lean | 184 | 188 |
/-
Copyright (c) 2021 Luke Kershaw. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Luke Kershaw, Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.Basic
import Mathlib.CategoryTheory.Limits.Constructions.FiniteProductsOfBinaryProducts
import Mathlib.CategoryThe... | (contractible_distinguished X) hT f 0 (by aesop_cat)
exact ⟨a, by simpa using ha₁⟩
lemma coyoneda_exact₁ {X : C} (f : X ⟶ T.obj₁⟦(1 : ℤ)⟧) (hf : f ≫ T.mor₁⟦1⟧' = 0) :
∃ (g : X ⟶ T.obj₃), f = g ≫ T.mor₃ :=
| Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean | 245 | 249 |
/-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Data.Set.Finite.Lemmas
import Mathlib.ModelTheory.Substructures
/-!
# Finitely Generated First-Order Structures
This file defines what it means for a... | Countable (M ↪[L] N) :=
Function.Embedding.countable ⟨Embedding.toHom, Embedding.toHom_injective⟩
| Mathlib/ModelTheory/FinitelyGenerated.lean | 235 | 237 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Bhavik Mehta, Stuart Presnell
-/
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Order.Monotone.Defs
/-!
# Binomial coefficients
This file defines binomial coeffic... | i.e. ways to select `k` items (up to permutation) from `n` items with replacement.
| Mathlib/Data/Nat/Choose/Basic.lean | 332 | 333 |
/-
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.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Ring.Rat
import Mathlib.Data.NNRat.Defs
/-! # Casting lemm... | Mathlib/Data/NNRat/BigOperators.lean | 52 | 55 | |
/-
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... | Mathlib/Topology/Bases.lean | 1,008 | 1,013 | |
/-
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.TensorAlgebra.Basic
import Mathlib.LinearAlgebra.TensorPower.Basic
/-!
# Tensor algebras as direct sums of tensor powers
In this file we show... | refine Fin.consInduction ?_ ?_ x <;> clear x
· rw [List.finRange_zero, List.map_nil, List.prod_nil]
rfl
· intro n x₀ x ih
| Mathlib/LinearAlgebra/TensorAlgebra/ToTensorPower.lean | 128 | 131 |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho
import Mathlib.LinearAlgebra.Orientation
/-!
# Orientations of real inner product spaces.
Th... | e.toBasis.adjustToOrientation_apply_eq_or_eq_neg x i
theorem det_adjustToOrientation :
| Mathlib/Analysis/InnerProductSpace/Orientation.lean | 122 | 124 |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Kim Morrison
-/
import Mathlib.CategoryTheory.Subobject.MonoOver
import Mathlib.CategoryTheory.Skeletal
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathlib.... | @[simps]
def isoOfEqMk {B A : C} (X : Subobject B) (f : A ⟶ B) [Mono f] (h : X = mk f) : (X : C) ≅ A where
hom := ofLEMk X f h.le
inv := ofMkLE f X h.ge
| Mathlib/CategoryTheory/Subobject/Basic.lean | 423 | 427 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Johan Commelin, Patrick Massot
-/
import Mathlib.Algebra.Order.Hom.Monoid
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.Tac... | theorem ne_zero (h : v₁.IsEquiv v₂) {r : R} : v₁ r ≠ 0 ↔ v₂ r ≠ 0 := by
have : v₁ r ≠ v₁ 0 ↔ v₂ r ≠ v₂ 0 := not_congr h.val_eq
rwa [v₁.map_zero, v₂.map_zero] at this
end IsEquiv
-- end of namespace
section
theorem isEquiv_of_map_strictMono [LinearOrderedCommMonoidWithZero Γ₀]
[LinearOrderedCommMonoidWithZero... | Mathlib/RingTheory/Valuation/Basic.lean | 447 | 480 |
/-
Copyright (c) 2023 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Algebra.Group.Defs
import Batteries.Data.List.Basic
/-!
# Levenshtein distances
We define the Levenshtein edit distance `levenshtein C xy ys` between two... | -- Not sure if this belongs in the main `List` API, or can stay local.
theorem List.eq_of_length_one (x : List α) (w : x.length = 1) :
have : 0 < x.length := lt_of_lt_of_eq Nat.zero_lt_one w.symm
x = [x[0]] := by
match x, w with
| Mathlib/Data/List/EditDistance/Defs.lean | 207 | 211 |
/-
Copyright (c) 2024 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.Data.Matroid.Minor.Restrict
/-!
# Some constructions of matroids
This file defines some very elementary examples of matroids, namely those with at most o... |
@[simp] theorem loopyOn_dual_eq : (loopyOn E)✶ = freeOn E := rfl
| Mathlib/Data/Matroid/Constructions.lean | 146 | 148 |
/-
Copyright (c) 2021 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.Topology.Algebra.Module.Basic
/-!
# Group and ring filter bases
A `GroupFilterBasis` is a `FilterBasis` on a group with some properties relating
the ... | rintro _ ⟨x, hx, rfl⟩
calc
| Mathlib/Topology/Algebra/FilterBasis.lean | 149 | 150 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Andrew Yang
-/
import Mathlib.CategoryTheory.Monoidal.Functor
/-!
# Endofunctors as a monoidal category.
We give the monoidal category structure on `C ⥤ C`,
and show that... |
-- This is a simp lemma in the reverse direction via `NatTrans.naturality`.
@[reassoc]
theorem δ_naturality {m n : M} {X Y : C} (f : X ⟶ Y) [F.OplaxMonoidal]:
(δ F m n).app X ≫ (F.obj n).map ((F.obj m).map f) =
(F.obj _).map f ≫ (δ F m n).app Y := by simp
| Mathlib/CategoryTheory/Monoidal/End.lean | 150 | 155 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yaël Dillies, David Loeffler
-/
import Mathlib.Order.PartialSups
import Mathlib.Order.Interval.Finset.Fin
/-!
# Making a sequence disjoint
This file defines the way t... | simp only [partialSups_apply, Iic_eq_cons_Iio, hi, disjointed_apply, sup'_eq_sup, sup_cons,
sup_empty, sdiff_bot]
| succ n ih =>
-- Induction step: first WLOG arrange that `#(Iio i) = r + 1`
rcases lt_or_eq_of_le hi with hn | hn
· exact ih _ <| Nat.le_of_lt_succ hn
simp only [partialSups_app... | Mathlib/Order/Disjointed.lean | 123 | 136 |
/-
Copyright (c) 2022 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Adjunction.Limits
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.AbelianImages
import Mat... | instance hasFiniteLimits [HasFiniteLimits C] :
HasFiniteLimits.{w} (ShrinkHoms C) := ⟨fun _ => inferInstance⟩
end Preadditive
variable (C) in
noncomputable instance abelian [Abelian C] :
Abelian.{w} (ShrinkHoms C) := abelianOfEquivalence (inverse C)
end ShrinkHoms
| Mathlib/CategoryTheory/Abelian/Transfer.lean | 137 | 148 |
/-
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.Deriv.ZPow
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
impo... | | zero => simp
| succ n ihn =>
rw [← two_mul] at ihn
rw [← two_mul, mul_add, mul_one, ← one_add_one_eq_two, ← add_assoc,
Finset.prod_range_succ, Finset.prod_range_succ, mul_assoc]
refine mul_nonneg ihn ?_; generalize (1 + 1) * n = k
rcases le_or_lt m k with hmk | hmk
· have : m ≤ k + 1 := ... | Mathlib/Analysis/Convex/SpecificFunctions/Deriv.lean | 72 | 85 |
/-
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.Topology.MetricSpace.Pseudo.Constructions
import Mathlib.Topology.Order.Densely... | /-- If `u` is a neighborhood of `x`, then for small enough `r`, the closed ball
`Metric.closedBall x r` is contained in `u`. -/
| Mathlib/Topology/MetricSpace/Pseudo/Lemmas.lean | 38 | 39 |
/-
Copyright (c) 2021 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Data.Int.LeastGreatest
/-!
## `ℤ` forms a conditionally complete linear order
The integer... | theorem csSup_empty : sSup (∅ : Set ℤ) = 0 :=
dif_neg (by simp)
theorem csSup_of_not_bdd_above {s : Set ℤ} (h : ¬BddAbove s) : sSup s = 0 :=
dif_neg (by simp [h])
| Mathlib/Data/Int/ConditionallyCompleteOrder.lean | 61 | 65 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Yury Kudryashov, Neil Strickland
-/
import Mathlib.Algebra.Group.Defs
import Mathlib.Algebra.GroupWithZero.Defs
import Mathlib.Data.I... | CommMonoidWithZero α :=
{ inferInstanceAs (CommMonoid α), inferInstanceAs (CommSemiring α) with }
section CommSemiring
| Mathlib/Algebra/Ring/Defs.lean | 218 | 221 |
/-
Copyright (c) 2022 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.Data.Nat.Choose.Dvd
import Mathlib.RingTheory.IntegralClosure.IntegrallyClosed
import Mathlib.RingTheory.Norm.Basic
import Mathlib.RingTheory.Polynom... | exact
hn (mem_adjoin_of_smul_prime_smul_of_minpoly_isEisensteinAt hp hBint (hzint.smul _) hz hei)
end IsIntegral
| Mathlib/RingTheory/Polynomial/Eisenstein/IsIntegral.lean | 378 | 386 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Yury Kudryashov
-/
import Mathlib.Algebra.Algebra.Rat
import Mathlib.Data.Nat.Prime.Int
import Mathlib.Data.Rat.Sqrt
imp... | end
| Mathlib/Data/Real/Irrational.lean | 657 | 658 |
/-
Copyright (c) 2022 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.Algebra.Module.ZLattice.Basic
import Mathlib.Analysis.InnerProductSpace.ProdL2
import Mathlib.MeasureTheory.Measure.Haar.Unique
import Mathlib.NumberTheo... | (LinearMap.ker (commMap K)) := by
refine LinearMap.disjoint_ker.mpr (fun x h_mem h_zero => ?_)
replace h_mem : x ∈ Submodule.span ℝ (Set.range (canonicalEmbedding K)) := by
refine (Submodule.span_mono ?_) h_mem
| Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean | 288 | 291 |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.LinearAlgebra.Finsupp.Span
/-!
# Lie submodules of a Lie algebra
In this file we define Lie submodules, we construct ... | h.trans <| Submodule.subsingleton_iff R
| Mathlib/Algebra/Lie/Submodule.lean | 591 | 592 |
/-
Copyright (c) 2021 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.GroupTheory.Index
/-!
# Complements
In this file we define the complement of a subgroup.
## Main definitions
- `Subgroup.IsComplement S T` where ... | map_mul' := fun _ _ ↦ rfl }
theorem IsComplement.card_mul (h : IsComplement S T) :
| Mathlib/GroupTheory/Complement.lean | 794 | 796 |
/-
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... | theorem Ioo_subset_Ici_self : Ioo a b ⊆ Ici a :=
Ioo_subset_Ico_self.trans Ico_subset_Ici_self
end LocallyFiniteOrderTop
| Mathlib/Order/Interval/Finset/Basic.lean | 387 | 390 |
/-
Copyright (c) 2020 Jannis Limperg. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jannis Limperg
-/
import Mathlib.Data.List.Induction
/-!
# Lemmas about List.*Idx functions.
Some specification lemmas for `List.mapIdx`, `List.mapIdxM`, `List.foldlIdx` and `List.fo... | Mathlib/Data/List/Indexes.lean | 364 | 367 | |
/-
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... |
@[simp]
theorem support_inv (σ : Perm α) : support σ⁻¹ = σ.support := by
simp_rw [Finset.ext_iff, mem_support, not_iff_not, inv_eq_iff_eq.trans eq_comm, imp_true_iff]
| Mathlib/GroupTheory/Perm/Support.lean | 332 | 336 |
/-
Copyright (c) 2017 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Kim Morrison, Mario Carneiro, Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.Limits.Products
/-!
# Pullbacks and pushouts in the category of topological spaces
-... | -- `exact ⟨(TopCat.pullbackIsoProdSubtype f g).inv ⟨⟨_, _⟩, eq.symm⟩, by simpa, by simp⟩`
-- before https://github.com/leanprover-community/mathlib4/pull/13170
refine ⟨(TopCat.pullbackIsoProdSubtype f g).inv ⟨⟨_, _⟩, eq.symm⟩, ?_, ?_⟩
· simp only [coe_of, Set.mem_preimage]
convert hy
rw [pul... | Mathlib/Topology/Category/TopCat/Limits/Pullbacks.lean | 391 | 402 |
/-
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
/-!
# Traversing collections
This file proves basic properties of traversable and applicative ... | Mathlib/Control/Traversable/Lemmas.lean | 135 | 138 | |
/-
Copyright (c) 2021 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn, Joachim Breitner
-/
import Mathlib.Algebra.Group.Action.End
import Mathlib.Algebra.Group.Action.Pointwise.Set.Basic
import Mathlib.Algebra.Group.Submonoid.Membership
import Mat... | exact mul m y ihy
@[elab_as_elim]
| Mathlib/GroupTheory/CoprodI.lean | 210 | 212 |
/-
Copyright (c) 2021 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno
-/
import Mathlib.CategoryTheory.NatIso
/-!
# Bicategories
In this file we define typeclass for bicategories.
A bicategory `B` consists of
* objects `a : B`,
* 1-morphisms ... | /-- Precomposition of a 1-morphism as a functor. -/
@[simps]
def precomp (c : B) (f : a ⟶ b) : (b ⟶ c) ⥤ (a ⟶ c) where
obj := (f ≫ ·)
map := (f ◁ ·)
| Mathlib/CategoryTheory/Bicategory/Basic.lean | 422 | 426 |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Data.List.Chain
/-!
# List of booleans
In this file we prove lemmas about the number of `false`s and `true`s ... | Mathlib/Data/Bool/Count.lean | 120 | 123 | |
/-
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 | 957 | 961 | |
/-
Copyright (c) 2020 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.Topology.Path
/-!
# Path connectedness
Continuing from `Mathlib.Topology.Path`, this file defines path components and path-connected
spaces.
## Main... | Mathlib/Topology/Connected/PathConnected.lean | 930 | 932 | |
/-
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-... |
nonrec theorem IsWF.not_lt_min (hs : IsWF s) (hn : s.Nonempty) (ha : a ∈ s) : ¬a < hs.min hn :=
hs.not_lt_min univ (nonempty_iff_univ_nonempty.1 hn.to_subtype) (mem_univ (⟨a, ha⟩ : s))
| Mathlib/Order/WellFoundedSet.lean | 597 | 599 |
/-
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... | rw [← norm_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul,
arg_mul_cos_add_sin_mul_I (mul_pos hr (norm_pos_iff.mpr hx)) x.arg_mem_Ioc]
theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x :=
mul_comm x r ▸ arg_real_mul x hr
theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) ... | Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean | 173 | 181 |
/-
Copyright (c) 2021 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.GroupWithZero.Subgroup
import Mathlib.Data.Finite.Card
import Mathlib.Data.Finite.Pr... | theorem relindex_eq_zero_of_le_left (hHK : H ≤ K) (hKL : K.relindex L = 0) : H.relindex L = 0 :=
eq_zero_of_zero_dvd (hKL ▸ relindex_dvd_of_le_left L hHK)
| Mathlib/GroupTheory/Index.lean | 290 | 292 |
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.MetricSpace.HausdorffDistance
/-!
# Topological study of spaces `Π (n : ℕ), E n`
When `E n` are topological spaces, the space `Π (n : ... | * if `x ∈ s` but `y ∉ s`, then the longest prefix `w` of `y` shared by an element of `s` is of
length at least `n` (because of `x`), and then `f y` starts with `w` and therefore stays in the
same length `n` cylinder.
* if `x ∉ s`, `y ∉ s`, let `w` be the longest prefix of `x` shared by an element of `s`... | Mathlib/Topology/MetricSpace/PiNat.lean | 557 | 563 |
/-
Copyright (c) 2018 Andreas Swerdlow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andreas Swerdlow
-/
import Mathlib.LinearAlgebra.Basis.Basic
import Mathlib.LinearAlgebra.BilinearMap
import Mathlib.LinearAlgebra.LinearIndependent.Lemmas
/-!
# Sesquilinear maps
... | IsAdjointPair B B' (c • f) (c • g) := fun _ _ ↦ by
| Mathlib/LinearAlgebra/SesquilinearForm.lean | 462 | 462 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.MeasurableSpace.Constructions
import Mathlib.Tactic.FunProp
/-!
# Measurable embeddings and equivalences
A measurable e... | coe_comp_rangeFactorization, hf.measurableSet_image]
theorem measurable_extend (hf : MeasurableEmbedding f) {g : α → γ} {g' : β → γ} (hg : Measurable g)
(hg' : Measurable g') : Measurable (extend f g g') := by
refine measurable_of_restrict_of_restrict_compl hf.measurableSet_range ?_ ?_
| Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean | 100 | 104 |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.LinearAlgebra.Finsupp.Span
/-!
# Lie submodules of a Lie algebra
In this file we define Lie submodules, we construct ... | Mathlib/Algebra/Lie/Submodule.lean | 1,312 | 1,314 | |
/-
Copyright (c) 2019 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Lu-Ming Zhang
-/
import Mathlib.Data.Matrix.Invertible
import Mathlib.Data.Matrix.Kronecker
import Mathlib.LinearAlgebra.FiniteDimensional.Basic
import Mathlib.LinearAlgebra.... |
@[simp]
theorem inv_submatrix_equiv (A : Matrix m m α) (e₁ e₂ : n ≃ m) :
(A.submatrix e₁ e₂)⁻¹ = A⁻¹.submatrix e₂ e₁ := by
| Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean | 698 | 701 |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Topology.Algebra.Group.Pointwise
import Mathlib.Topology.Order.Basic
/-!
# Strictly convex sets
This file defines st... | rw [h]
exact hs hx hxy (fun h => hy <| add_left_cancel (a := x) (by rw [← h, add_zero]))
(sub_pos_of_lt ht₁) ht₀ (sub_add_cancel 1 t)
theorem StrictConvex.smul_mem_of_zero_mem [AddRightStrictMono 𝕜]
(hs : StrictConvex 𝕜 s) (zero_mem : (0 : E) ∈ s)
(hx : x ∈ s) (hx₀ : x ≠ 0) {t : 𝕜} (ht₀ : 0 < t) (ht... | Mathlib/Analysis/Convex/Strict.lean | 308 | 318 |
/-
Copyright (c) 2023 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.Algebra.Polynomial.Bivariate
import Mathlib.AlgebraicGeometry.EllipticCurve.Weierstrass
import Mathlib.AlgebraicGeometry.EllipticCu... | lemma Y_sub_polynomialY : Y - W'.polynomialY = W'.negPolynomial := by
rw [polynomialY, negPolynomial]
C_simp
ring1
| Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean | 317 | 321 |
/-
Copyright (c) 2022 Eric Rodriguez. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Rodriguez
-/
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Ring.Cast
import Mathlib.Data.Fi... | Mathlib/Data/Sign.lean | 534 | 545 | |
/-
Copyright (c) 2019 Neil Strickland. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Neil Strickland
-/
import Mathlib.Algebra.BigOperators.Group.Multiset.Basic
import Mathlib.Data.PNat.Prime
import Mathlib.Data.Nat.Factors
import Mathlib.Data.Multiset.OrderedMonoid
i... | · exact dvd_lcm_right m n
/-- The number of occurrences of p in the factor multiset of m
is the same as the p-adic valuation of m. -/
theorem count_factorMultiset (m : ℕ+) (p : Nat.Primes) (k : ℕ) :
| Mathlib/Data/PNat/Factors.lean | 333 | 337 |
/-
Copyright (c) 2024 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.CommShift
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
/-! Shifted morphisms
Given a category `C` endowed with a shift by an ... | lemma comp_mk₀ {a : M} (f : ShiftedHom X Y a) (m₀ : M) (hm₀ : m₀ = 0) (g : Y ⟶ Z) :
f.comp (mk₀ m₀ hm₀ g) (by rw [hm₀, zero_add]) = f ≫ g⟦a⟧' := by
subst hm₀
simp only [comp, shiftFunctorAdd'_zero_add_inv_app, mk₀, shiftFunctorZero',
eqToIso_refl, Iso.refl_trans, ← Functor.map_comp, assoc, Iso.inv_hom_id_ap... | Mathlib/CategoryTheory/Shift/ShiftedHom.lean | 83 | 88 |
/-
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.Convex.Between
import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathli... | simpa only [Set.image_image]
borelize E
rw [IsometryEquiv.hausdorffMeasure_image, Set.image_smul, Measure.hausdorffMeasure_smul₀ hd hc,
IsometryEquiv.hausdorffMeasure_image]
| Mathlib/MeasureTheory/Measure/Hausdorff.lean | 1,046 | 1,049 |
/-
Copyright (c) 2023 Mark Andrew Gerads. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mark Andrew Gerads, Junyan Xu, Eric Wieser
-/
import Mathlib.Tactic.Ring
/-!
# Hyperoperation sequence
This file defines the Hyperoperation sequence.
`hyperoperation 0 m k = k + ... | @[simp]
theorem hyperoperation_three : hyperoperation 3 = (· ^ ·) := by
ext m k
induction' k with bn bih
· rw [hyperoperation_ge_three_eq_one]
exact (pow_zero m).symm
· rw [hyperoperation_recursion, hyperoperation_two, bih]
exact (pow_succ' m bn).symm
theorem hyperoperation_ge_two_eq_self (n m : ℕ) : h... | Mathlib/Data/Nat/Hyperoperation.lean | 69 | 78 |
/-
Copyright (c) 2024 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Lie.Engel
import Mathlib.Algebra.Lie.Normalizer
import Mathlib.Algebra.Lie.OfAssociative
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.D... | /-- Engel subalgebras are self-normalizing.
See `LieSubalgebra.normalizer_eq_self_of_engel_le` for a proof that Lie-subalgebras
containing an Engel subalgebra are also self-normalizing,
provided that the ambient Lie algebra is artinina. -/
@[simp]
lemma normalizer_engel (x : L) : normalizer (engel R x) = engel R x := b... | Mathlib/Algebra/Lie/EngelSubalgebra.lean | 78 | 93 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Wrenna Robson
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Pi
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.LinearAlgebra.Vandermonde
import Mathlib.RingT... | rw [eval_finset_sum, eval_one, ← add_sum_erase _ _ hi, eval_basis_self hvs hi,
add_eq_left]
refine sum_eq_zero fun j hj => ?_
rcases mem_erase.mp hj with ⟨hij, _⟩
rw [eval_basis_of_ne hij hi]
theorem basisDivisor_add_symm {x y : F} (hxy : x ≠ y) :
basisDivisor x y + basisDivisor y x = 1 := by... | Mathlib/LinearAlgebra/Lagrange.lean | 262 | 271 |
/-
Copyright (c) 2020 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Grading
import Mathlib.Algebra.Module.Opposite
/-!
# Conjugations
This file defines the grade reversal and grade involution f... | @[simp]
theorem involute_mem_evenOdd_iff {x : CliffordAlgebra Q} {n : ZMod 2} :
involute x ∈ evenOdd Q n ↔ x ∈ evenOdd Q n :=
SetLike.ext_iff.mp (evenOdd_comap_involute Q n) x
| Mathlib/LinearAlgebra/CliffordAlgebra/Conjugation.lean | 281 | 285 |
/-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Rayleigh
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Algebra.DirectSum.Decomposition
import Mathlib.Line... | · simp [hv']
have H := hT.conj_eigenvalue_eq_self (hasEigenvalue_of_hasEigenvector ⟨hv, hv'⟩)
rw [mem_eigenspace_iff] at hv hw
refine Or.resolve_left ?_ hμν.symm
simpa [inner_smul_left, inner_smul_right, hv, hw, H] using (hT v w).symm
theorem orthogonalFamily_eigenspaces' (hT : T.IsSymmetric) :
Orthogona... | Mathlib/Analysis/InnerProductSpace/Spectrum.lean | 83 | 91 |
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Algebra.Order.Archimedean.IndicatorCard
import Mathlib.Probability.Martingale.Centering
import Mathlib.Probability.Martingale.Convergence
import Mathlib.Prob... | have hg : Submartingale g ℱ μ :=
hf.sub_martingale (martingale_const_fun _ _ (hf.adapted 0) (hf.integrable 0))
have hg0 : g 0 = 0 := by
ext ω
simp only [g, sub_self, Pi.zero_apply]
| Mathlib/Probability/Martingale/BorelCantelli.lean | 178 | 182 |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kenny Lau, Johan Commelin, Mario Carneiro, Kevin Buzzard,
Amelia Livingston, Yury Kudryashov
-/
import Mathlib.Algebra.BigOperators.Group.Multiset.Defs
import Mathlib.A... | c ∈ Submonoid.closure ({a, b} : Set A) ↔ ∃ m n : ℕ, a ^ m * b ^ n = c := by
| Mathlib/Algebra/Group/Submonoid/Membership.lean | 495 | 495 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.GroupWithZero.Divisibili... | @[simp] lemma isRegular_X : IsRegular (X n : MvPolynomial σ R) := by
suffices IsLeftRegular (X n : MvPolynomial σ R) from
⟨this, this.right_of_commute <| Commute.all _⟩
intro P Q (hPQ : (X n) * P = (X n) * Q)
ext i
| Mathlib/Algebra/MvPolynomial/Basic.lean | 778 | 782 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Moritz Doll
-/
import Mathlib.LinearAlgebra.Prod
/-!
# Partially defined linear maps
A `LinearPMap R E F` or `E →ₗ.[R] F` is a linear map from a submodule of `E` to... | Mathlib/LinearAlgebra/LinearPMap.lean | 1,107 | 1,110 | |
/-
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.CharP.Lemmas
import Mathlib.FieldTheory.Perfect
/-!
# The perfect closure of a characteristic `p` ring
## Main definitions
- `Perfec... |
private theorem mul_aux_right (x y1 y2 : ℕ × K) (H : R K p y1 y2) :
mk K p (x.1 + y1.1, (frobenius K p)^[y1.1] x.2 * (frobenius K p)^[x.1] y1.2) =
mk K p (x.1 + y2.1, (frobenius K p)^[y2.1] x.2 * (frobenius K p)^[x.1] y2.2) :=
match y1, y2, H with
| _, _, R.intro n y =>
Quot.sound <| by
rw [← i... | Mathlib/FieldTheory/PerfectClosure.lean | 121 | 129 |
/-
Copyright (c) 2020 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.GroupTheory.Complement
/-!
# Semidirect product
This file defines semidirect products of groups, and the canonical maps in and out of the
semidirect prod... | H ⋊[(H.normalizerMonoidHom).comp (inclusion h)] K →* G :=
| Mathlib/GroupTheory/SemidirectProduct.lean | 232 | 232 |
/-
Copyright (c) 2021 Martin Zinkevich. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Martin Zinkevich, Rémy Degenne
-/
import Mathlib.Logic.Encodable.Lattice
import Mathlib.MeasureTheory.MeasurableSpace.Defs
import Mathlib.Order.Disjointed
/-!
# Indu... | /-- The least Dynkin system containing a collection of basic sets. -/
def generate (s : Set (Set α)) : DynkinSystem α where
Has := GenerateHas s
has_empty := GenerateHas.empty
has_compl {_} := GenerateHas.compl
| Mathlib/MeasureTheory/PiSystem.lean | 586 | 590 |
/-
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.Computability.Tape
import Mathlib.Data.Fintype.Option
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.PFun
import M... | Mathlib/Computability/TuringMachine.lean | 1,372 | 1,376 | |
/-
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... | theorem div_lt_of_lt_mul' (h : a < b * c) : a / b < c :=
div_lt_of_lt_mul <| by rwa [mul_comm]
theorem inv_le_iff_le_mul (h₁ : b = ∞ → a ≠ 0) (h₂ : a = ∞ → b ≠ 0) : a⁻¹ ≤ b ↔ 1 ≤ a * b := by
rw [← one_div, ENNReal.div_le_iff_le_mul, mul_comm]
| Mathlib/Data/ENNReal/Inv.lean | 398 | 402 |
/-
Copyright (c) 2019 Jan-David Salchow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jan-David Salchow, Sébastien Gouëzel, Jean Lo
-/
import Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear
import Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm
/-!
# Operators o... | `m, n ≥ N`. -/
rcases cauchySeq_iff_le_tendsto_0.1 hf with ⟨b, hb₀, hfb, hb_lim⟩
-- Since `b → 0`, it suffices to show that `‖f n x - g x‖ ≤ b n * ‖x‖` for all `n` and `x`.
suffices ∀ n x, ‖f n x - g x‖ ≤ b n * ‖x‖ from
tendsto_iff_norm_sub_tendsto_zero.2
(squeeze_zero (fun n => norm_nonneg _) (fun n ... | Mathlib/Analysis/NormedSpace/OperatorNorm/Completeness.lean | 70 | 86 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Sophie Morel
-/
import Mathlib.Algebra.Category.Grp.Preadditive
import Mathlib.Algebra.Equiv.TransferInstance
import Mathlib.CategoryTheory.ConcreteCategory.Elementwise
imp... |
/--
The additive equivalence between `Quot F` and `Quot (F ⋙ uliftFunctor.{u'})`.
-/
@[simp]
| Mathlib/Algebra/Category/Grp/Colimits.lean | 146 | 150 |
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic
/-!
# Egorov theorem
This file contains the Egorov theorem which states that an almost everywhere convergen... | theorem iUnionNotConvergentSeq_subset (hε : 0 < ε) (hf : ∀ n, StronglyMeasurable (f n))
(hg : StronglyMeasurable g) (hsm : MeasurableSet s) (hs : μ s ≠ ∞)
(hfg : ∀ᵐ x ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x))) :
iUnionNotConvergentSeq hε hf hg hsm hs hfg ⊆ s := by
rw [iUnionNotConvergentSeq, ←... | Mathlib/MeasureTheory/Function/Egorov.lean | 146 | 157 |
/-
Copyright (c) 2020 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Algebra.Algebra.Field
import Mathlib.Algebra.BigOperators.Balance
import Mathlib.Algebra.Order.BigOperators.Expect
import Mathlib.Algebra.Order.Star.... | Mathlib/Analysis/RCLike/Basic.lean | 607 | 607 | |
/-
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.Topology.Bornology.Constructions
import Mathlib.Topology.MetricSpace.Pseudo.Def... | Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean | 313 | 316 | |
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Simon Hudon
-/
import Mathlib.Control.Functor.Multivariate
import Mathlib.Data.PFunctor.Multivariate.Basic
import Mathlib.Data.PFunctor.Multivariate.M
import Mathlib.Data... | * holds for `x y`: `R x y`
* for any values `x y` that satisfy `R`, their root has the same shape
and their children can be paired in such a way that they satisfy `R`.
-/
| Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean | 184 | 190 |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Indicator
import Mathlib.Algebra.Module.BigOperators
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic
import... | expressed as a subtraction involving a `weightedVSubOfPoint` expression. -/
theorem sum_smul_vsub_const_eq_weightedVSubOfPoint_sub (w : ι → k) (p₁ : ι → P) (p₂ b : P) :
(∑ i ∈ s, w i • (p₁ i -ᵥ p₂)) = s.weightedVSubOfPoint p₁ b w - (∑ i ∈ s, w i) • (p₂ -ᵥ b) := by
rw [sum_smul_vsub_eq_weightedVSubOfPoint_sub, wei... | Mathlib/LinearAlgebra/AffineSpace/Combination.lean | 179 | 182 |
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes Hölzl, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Continuity
import Mathlib.Topology.Algebra.IsUniformGroup.Basic
import Mathlib.Topology.MetricSpace... | theorem dist_div_eq_dist_mul_right (a b c : E) : dist a (b / c) = dist (a * c) b := by
rw [← dist_mul_right _ _ c, div_mul_cancel]
| Mathlib/Analysis/Normed/Group/Uniform.lean | 47 | 48 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yakov Pechersky
-/
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Infix
import Mathlib.Data.Quot
/-!
# List rotation
This file proves basic results about `List.r... | l.get k = (l.rotate n).get ⟨(l.length - n % l.length + k) % l.length,
(Nat.mod_lt _ (k.1.zero_le.trans_lt k.2)).trans_eq (length_rotate _ _).symm⟩ := by
rw [get_rotate]
refine congr_arg l.get (Fin.eq_of_val_eq ?_)
simp only [mod_add_mod]
| Mathlib/Data/List/Rotate.lean | 248 | 252 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Notation.Pi
import Mathlib.Data.Set.Lattice
import Mathlib.Order.Filter.Defs
/-!
# Theory of... | theorem EventuallyEq.eventually (h : f =ᶠ[l] g) : ∀ᶠ x in l, f x = g x := h
@[simp] lemma eventuallyEq_top : f =ᶠ[⊤] g ↔ f = g := by simp [EventuallyEq, funext_iff]
| Mathlib/Order/Filter/Basic.lean | 862 | 865 |
/-
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... | ‖g (f m)‖ ≤ ‖g‖ * (‖f‖ * ∏ i, ‖m i‖) := g.le_opNorm_of_le <| f.le_opNorm _
_ = _ := (mul_assoc _ _ _).symm
variable (𝕜 E G G')
/-- `ContinuousLinearMap.compContinuousMultilinearMap` as a bundled continuous bilinear map. -/
| Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean | 843 | 848 |
/-
Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Data.Set.Lattice
import Mathlib.Order.ConditionallyCompleteLattice.Defs
/-!
# Theory of conditionally complete lattices
A conditionally complet... | Mathlib/Order/ConditionallyCompleteLattice/Basic.lean | 1,194 | 1,195 | |
/-
Copyright (c) 2024 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.Data.Matroid.Minor.Restrict
/-!
# Some constructions of matroids
This file defines some very elementary examples of matroids, namely those with at most o... |
@[simp] theorem uniqueBaseOn_ground : (uniqueBaseOn I E).E = E :=
| Mathlib/Data/Matroid/Constructions.lean | 204 | 205 |
/-
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.NonUnitalHom
import Mathlib.LinearAlgebra.TensorProduct.Basic
/-!
# Facts about algebras involving bilinear maps and tensor pro... | mul_right_injective₀ hx
end Ring
| Mathlib/Algebra/Algebra/Bilinear.lean | 267 | 270 |
/-
Copyright (c) 2024 Jeremy Tan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Tan
-/
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Tactic.Peel
import Mathlib.Tactic.Positivity
/-!
# Abel's limit theorem
I... | ext z
rw [stolzSet, Set.mem_setOf, Set.mem_empty_iff_false, iff_false, not_and, not_lt, ← sub_pos]
intro zn
calc
_ ≤ 1 * (1 - ‖z‖) := mul_le_mul_of_nonneg_right hM zn.le
_ = ‖(1 : ℂ)‖ - ‖z‖ := by rw [one_mul, norm_one]
_ ≤ _ := norm_sub_norm_le _ _
| Mathlib/Analysis/Complex/AbelLimit.lean | 50 | 57 |
/-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.ImproperIntegrals
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.MeasureTheory.Measure.Haar.NormedSpace
... | mul_zero, ← Complex.coe_smul, ← mul_smul, sub_eq_add_neg,
cpow_add _ _ (ofReal_ne_zero.mpr ha.ne'), cpow_one, ofReal_inv,
mul_assoc, mul_comm, inv_mul_cancel_right₀ (ofReal_ne_zero.mpr ha.ne')]
theorem mellin_comp_mul_right (f : ℝ → E) (s : ℂ) {a : ℝ} (ha : 0 < a) :
mellin (fun t => f (t * a)) s = (a :... | Mathlib/Analysis/MellinTransform.lean | 140 | 155 |
/-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Topology.Algebra.UniformMulAction
import Mathlib.Algebra.Module.Pi
import Mathlib.Topology.UniformSpace.UniformConvergenceTopology
/-!
# Algebraic... | simp [UniformOnFun.gen]
@[to_additive]
protected theorem UniformOnFun.hasBasis_nhds_one (𝔖 : Set <| Set α) (h𝔖₁ : 𝔖.Nonempty)
(h𝔖₂ : DirectedOn (· ⊆ ·) 𝔖) :
(𝓝 1 : Filter (α →ᵤ[𝔖] G)).HasBasis
(fun SV : Set α × Set G => SV.1 ∈ 𝔖 ∧ SV.2 ∈ (𝓝 1 : Filter G)) fun SV =>
{ f : α →ᵤ[𝔖] G | ∀ x... | Mathlib/Topology/Algebra/UniformConvergence.lean | 258 | 267 |
/-
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.Topology.Algebra.Constructions
import Mathlib.Topology.Bases
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Topology.UniformSpac... | obtain ⟨t, htS, htl, htU⟩ : ∃ t, t ⊆ S ∧ t ∈ l ∧ t ×ˢ t ⊆ U := by
rcases hl' U hU with ⟨t, htl, htU⟩
refine ⟨t ∩ S, inter_subset_right, inter_mem htl hls, Subset.trans ?_ htU⟩
gcongr <;> apply inter_subset_left
obtain ⟨i, hi⟩ : ∃ i, t ⊆ s i := by
rcases Filter.nonempty_of_mem htl with ⟨x, hx⟩
rc... | Mathlib/Topology/UniformSpace/Cauchy.lean | 346 | 357 |
/-
Copyright (c) 2024 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nathaniel Thomas, Jeremy Avigad, Johannes Hölzl, Mario Carneiro, Anne Baanen,
Frédéric Dupuis, Heather Macbeth
-/
import Mathlib.Algebra.Module.Equiv.Opposite
import Mathlib.Algebra.... | exact h.of_comp
| Mathlib/Algebra/Module/LinearMap/End.lean | 175 | 175 |
/-
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 ... | rcases n with - | n
· dsimp [F]
clear IH
induction' l with _ l IH <;> simp_all
· simpa using IH ..
@[deprecated (since := "2025-02-14")] alias list_get? := list_getElem?
theorem list_getD (d : α) : Primrec₂ fun l n => List.getD l n d := by
| Mathlib/Computability/Primrec.lean | 920 | 927 |
/-
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.Decomposition
import Mathlib.Tactic.FinCases
/-!
# Behaviour of P_infty with respect to degeneracies
For any `X : SimplicialObject C... | rw [SimplexCategory.mono_iff_injective] at hθ
cases n
· exfalso
apply hθ
intro x y h
fin_cases x
fin_cases y
rfl
· obtain ⟨i, α, h⟩ := SimplexCategory.eq_σ_comp_of_not_injective θ hθ
rw [h, op_comp, X.map_comp, assoc, show X.map (SimplexCategory.σ i).op = X.σ i by rfl,
σ_comp_PInft... | Mathlib/AlgebraicTopology/DoldKan/Degeneracies.lean | 128 | 140 |
/-
Copyright (c) 2020 Hanting Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hanting Zhang
-/
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.RingTheory.MvPolynomial.Symmetric.Defs
/-!
# Vieta's Formula
The main result is `Multiset.prod_X_add_C_eq_sum_... | Mathlib/RingTheory/Polynomial/Vieta.lean | 181 | 191 | |
/-
Copyright (c) 2023 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.NoZeroSMulDivisors.Basic
import Mathlib.Data.Int.ModEq
import Mathlib.GroupTheory.QuotientGroup.Defs
import Math... | Mathlib/Algebra/ModEq.lean | 348 | 349 |
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