Context stringlengths 227 76.5k | target stringlengths 0 11.6k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 16 3.69k |
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/-
Copyright (c) 2018 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Limits.HasLimits
import Mathlib.CategoryTheory.Discrete.Basic
/-!
# Categorical (co)products
This file defines (co)products ... | theorem Pi.lift_π {β : Type w} {f : β → C} [HasProduct f] {P : C} (p : ∀ b, P ⟶ f b) (b : β) :
Pi.lift p ≫ Pi.π f b = p b := by
simp only [limit.lift_π, Fan.mk_pt, Fan.mk_π_app]
| Mathlib/CategoryTheory/Limits/Shapes/Products.lean | 245 | 248 |
/-
Copyright (c) 2021 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.Algebra.Group.Defs
import Mathlib.Topology.Separation.Regular
/-!
# Idempotents in topological semigroups
This file provides a sufficient condition for a sem... | (continuous_mul_left : ∀ r : M, Continuous (· * r)) : ∃ m : M, m * m = m := by
/- We apply Zorn's lemma to the poset of nonempty closed subsemigroups of `M`.
It will turn out that any minimal element is `{m}` for an idempotent `m : M`. -/
let S : Set (Set M) :=
{ N | IsClosed N ∧ N.Nonempty ∧ ∀ (m) (_ ... | Mathlib/Topology/Algebra/Semigroup.lean | 27 | 72 |
/-
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.Multiset.Dedup
import Mathlib.Data.List.Infix
/-!
# Preparations for defining operations on `Finset`.
The operations here ignore multiplicities,... |
theorem Disjoint.ndunion_eq {s t : Multiset α} (h : Disjoint s t) :
s.ndunion t = s.dedup + t := by
induction s, t using Quot.induction_on₂
exact congr_arg ((↑) : List α → Multiset α) <| List.Disjoint.union_eq <| by simpa using h
| Mathlib/Data/Multiset/FinsetOps.lean | 176 | 180 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl, Sander Dahmen, Kim Morrison
-/
import Mathlib.Algebra.Algebra.Tower
import Mathlib.LinearAlgebra.LinearIndependent.Basic
import Mathlib.Data.Set.Card
/... | theorem rank_eq_of_equiv_equiv (i : R → R') (j : M ≃+ M₁)
(hi : Bijective i) (hc : ∀ (r : R) (m : M), j (r • m) = i r • j m) :
Module.rank R M = Module.rank R' M₁ := by
simpa only [lift_id] using lift_rank_eq_of_equiv_equiv i j hi hc
| Mathlib/LinearAlgebra/Dimension/Basic.lean | 170 | 173 |
/-
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
... | bi-invariant with respect to `f`. -/
def IsOrthogonal : Prop :=
∀ x y, B (f x) (f y) = B x y
| Mathlib/LinearAlgebra/SesquilinearForm.lean | 473 | 475 |
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel, Joël Riou
-/
import Mathlib.Algebra.Homology.ExactSequence
import Mathlib.CategoryTheory.Abelian.Refinements
/-!
# The four and five lemmas
Consider the following commu... | exact hR₁'
theorem mono_of_epi_of_epi_of_mono (hR₁ : R₁.Exact) (hR₂ : R₂.Exact)
(hR₁' : Epi (R₁.map' 1 2)) (h₀ : Epi (app' φ 0)) (h₁ : Mono (app' φ 1)) :
Mono (app' φ 2) :=
mono_of_epi_of_epi_mono' φ (by simpa only [map'_comp R₁ 0 1 2] using hR₁.toIsComplex.zero 0)
hR₁' hR₂ h₀ h₁
theorem epi_of_mono... | Mathlib/CategoryTheory/Abelian/DiagramLemmas/Four.lean | 167 | 179 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Aesop
import Mathlib.Order.BoundedOrder.Lattice
/-!
# Disjointness and complements
This file defines `Disjoint`, `Codisjoint`, and the `IsCompl` predicate.
... |
section BoundedPartialOrder
| Mathlib/Order/Disjoint.lean | 411 | 412 |
/-
Copyright (c) 2018 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Kim Morrison
-/
import Mathlib.CategoryTheory.Opposites
/-!
# Morphisms from equations between objects.
When working categorically, sometimes one encounters an equation `h ... | HEq g (eqToHom h ≫ f) ↔ HEq g f :=
⟨(·.trans (eqToHom_comp_heq ..)), (·.trans (eqToHom_comp_heq ..).symm)⟩
theorem comp_eqToHom_heq {C} [Category C] {X Y Z : C}
| Mathlib/CategoryTheory/EqToHom.lean | 95 | 98 |
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel, Johan Commelin, Kim Morrison
-/
import Mathlib.CategoryTheory.Limits.Constructions.Pullbacks
import Mathlib.CategoryTheory.Preadditive.Biproducts
import Mathlib.CategoryT... | e := coimage.π f
e_strong_epi := strongEpi_of_epi _
| Mathlib/CategoryTheory/Abelian/Basic.lean | 348 | 349 |
/-
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.Probability.IdentDistrib
import Mathlib.Probability.Independence.Integrable
import Mathlib.MeasureTheory.Integral.DominatedConvergence
import Mat... | refine sum_nbij' (fun p ↦ ⟨p.2, p.1⟩) (fun p ↦ ⟨p.2, p.1⟩) ?_ ?_ ?_ ?_ ?_ <;>
aesop (add unsafe lt_trans)
_ ≤ ∑ k ∈ range K, 2 / (k + 1 : ℝ) * ∫ x in k..(k + 1 : ℕ), x ^ 2 ∂ρ := by
apply sum_le_sum fun k _ => ?_
refine mul_le_mul_of_nonneg_right (sum_Ioo_inv_sq_le _ _) ?_
refine inte... | Mathlib/Probability/StrongLaw.lean | 329 | 383 |
/-
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.Finite.Defs
import Mathlib.Data.Finset.BooleanAlgebra
import Mathlib.Data.Finset.Image
import Mathlib.Data.Fintype.Defs
import Mathlib.Data.Fintyp... | | [], _ => False.elim (by tauto)
| a :: _, _ => Trunc.mk a
| Mathlib/Data/Fintype/Basic.lean | 191 | 192 |
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.DominatedConvergence
import Mathlib.Probability.Kernel.MeasurableLIntegral
/-!
# Measurability of the integral against a kernel
Th... | Mathlib/Probability/Kernel/MeasurableIntegral.lean | 253 | 256 | |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jan-David Salchow, Sébastien Gouëzel, Jean Lo, Yury Kudryashov, Frédéric Dupuis,
Heather Macbeth
-/
import Mathlib.Algebra.Module.Opposite
import Mathlib.Topology.Algebra.Group.Qu... | Mathlib/Topology/Algebra/Module/Basic.lean | 2,688 | 2,692 | |
/-
Copyright (c) 2023 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Data.ZMod.Basic
import Mathlib.RingTheory.Polynomial.Basic
import Mathlib.Tactic.NormNum.DivMod
import Mathlib.Tactic.NormNum.PowMod
import Mathlib.Tactic.Re... | _ = ↑(k % p) := by simp [CharP.cast_eq_zero R]
/-- Given an integral expression `e : t` such that `t` is a ring of characteristic `n`,
reduce `e` modulo `n`. -/
partial def normIntNumeral {α : Q(Type u)} (n : Q(ℕ)) (e : Q($α)) (_ : Q(Ring $α))
| Mathlib/Tactic/ReduceModChar.lean | 97 | 101 |
/-
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
... | ∀ x y, B' (f x) y = B x (g y)
variable {B B' f g}
| Mathlib/LinearAlgebra/SesquilinearForm.lean | 411 | 414 |
/-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Data.Finsupp.Lex
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.GameAdd
/-!
# Termination of a hydra game
This file deals with the following version... | tauto
lemma cutExpand_pair_left {a' a b} (hr : r a' a) : CutExpand r {a', b} {a, b} :=
(cutExpand_add_right {b}).2 (cutExpand_singleton_singleton hr)
| Mathlib/Logic/Hydra.lean | 140 | 143 |
/-
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.Order.Filter.Germ.OrderedMonoid
import Mathlib.Algebra.Order.Ring.Defs
/-!
# Lemmas about filters and ordered rings.
-/
namespace Filte... |
theorem eventually_sub_nonneg [Ring β] [PartialOrder β] [IsOrderedRing β]
| Mathlib/Order/Filter/Ring.lean | 34 | 35 |
/-
Copyright (c) 2021 Bryan Gin-ge Chen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bryan Gin-ge Chen, Yaël Dillies
-/
import Mathlib.Algebra.Group.Idempotent
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Algebra.Ring.PUnit
import Mathlib.Order.Hom.BoundedLattic... |
-- Porting note (https://github.com/leanprover-community/mathlib4/issues/11215): TODO: add priority 100. lower instance priority
scoped[BooleanAlgebraOfBooleanRing] attribute [instance] BooleanRing.toBooleanAlgebra
| Mathlib/Algebra/Ring/BooleanRing.lean | 231 | 233 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Cardinal.Arithmetic
import Mathlib.SetTheory.Ordinal.FixedPoint
/-!
# Cofinality
This file co... |
/-! ### Cofinality of ordinals -/
namespace Ordinal
/-- Cofinality of an ordinal. This is the smallest cardinal of a subset `S` of the ordinal which is
unbounded, in the sense `∀ a, ∃ b ∈ S, a ≤ b`.
In particular, `cof 0 = 0` and `cof (succ o) = 1`. -/
def cof (o : Ordinal.{u}) : Cardinal.{u} :=
o.liftOn (fun a ↦... | Mathlib/SetTheory/Cardinal/Cofinality.lean | 90 | 102 |
/-
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.NumberTheory.LegendreSymbol.JacobiSymbol
/-!
# A `norm_num` extension for Jacobi and Legendre symbols
We extend the `norm_num` tactic so that it can be... |
theorem jacobiSymNat.qr₃_mod (a b ab : ℕ) (r : ℤ) (ha : a % 4 = 3) (hb : b % 4 = 3)
(hab : b % a = ab) (hr : jacobiSymNat ab a = r) : jacobiSymNat a b = -r :=
| Mathlib/Tactic/NormNum/LegendreSymbol.lean | 173 | 175 |
/-
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.Logic.Function.Basic
import Mathlib.Logic.Relator
/-!
# Types that are empty
In this file we define a typeclass `IsEmpty`, which expresses that a... | not_iff_not.mp <| by rw [not_or, not_nonempty_iff, not_nonempty_iff, isEmpty_fun, not_isEmpty_iff]
| Mathlib/Logic/IsEmpty.lean | 154 | 155 |
/-
Copyright (c) 2024 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Kontorovich, David Loeffler, Heather Macbeth, Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.Analysis.Calculus.ContDiff.CPolynomial
import... |
lemma norm_fourierPowSMulRight_le (f : V → E) (v : V) (n : ℕ) :
‖fourierPowSMulRight L f v n‖ ≤ (2 * π * ‖L‖) ^ n * ‖v‖ ^ n * ‖f v‖ := by
apply ContinuousMultilinearMap.opNorm_le_bound (by positivity) (fun m ↦ ?_)
calc
‖fourierPowSMulRight L f v n m‖
= (2 * π) ^ n * ((∏ x : Fin n, |(L v) (m x)|) * ‖f v‖)... | Mathlib/Analysis/Fourier/FourierTransformDeriv.lean | 321 | 333 |
/-
Copyright (c) 2024 Calle Sönne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Paul Lezeau, Calle Sönne
-/
import Mathlib.CategoryTheory.FiberedCategory.HomLift
import Mathlib.CategoryTheory.Bicategory.Strict
import Mathlib.CategoryTheory.Functor.Category
import Ma... | @[ext]
lemma ext (β : BasedNatTrans F G) (h : α.toNatTrans = β.toNatTrans) : α = β := by
cases α; subst h; rfl
| Mathlib/CategoryTheory/FiberedCategory/BasedCategory.lean | 150 | 152 |
/-
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.Option.Basic
import Mathlib.Data.Prod.Basic
import Mathlib.Data.Prod.PProd
import Mathlib.Logic.Equiv.Basic
/-!
# Injective funct... | @[simp]
theorem subtype_apply {α} {p : α → Prop} (x : Subtype p) : subtype p x = x :=
rfl
| Mathlib/Logic/Embedding/Basic.lean | 209 | 211 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Kappelmann
-/
import Mathlib.Algebra.Order.Floor.Defs
import Mathlib.Algebra.Order.Floor.Ring
import Mathlib.Algebra.Order.Floor.Semiring
deprecated_module (sinc... | Mathlib/Algebra/Order/Floor.lean | 1,663 | 1,664 | |
/-
Copyright (c) 2021 Hunter Monroe. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hunter Monroe, Kyle Miller, Alena Gusakov
-/
import Mathlib.Combinatorics.SimpleGraph.DeleteEdges
import Mathlib.Data.Fintype.Powerset
/-!
# Subgraphs of a simple graph
A subgraph of ... | (H.map f).edgeSet = Sym2.map f '' H.edgeSet := Sym2.fromRel_relationMap ..
end map
/-- Graph homomorphisms induce a contravariant function on subgraphs. -/
@[simps]
protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where
verts := f ⁻¹' H.verts
Adj u v := G.Adj u v ∧ H.Adj (... | Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | 648 | 656 |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.WSeq.Basic
import Mathlib.Data.WSeq.Defs
import Mathlib.Data.WSeq.Productive
import Mathlib.Data.WSeq.Relation
deprecated_module (since :=... | Mathlib/Data/Seq/WSeq.lean | 1,804 | 1,807 | |
/-
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.Data.Finset.Sort
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Sign
import Mathlib.LinearAlgebra.AffineSpace.Combination
import Mathlib.LinearAlg... | @[simp]
theorem reindex_trans {n₁ n₂ n₃ : ℕ} (e₁₂ : Fin (n₁ + 1) ≃ Fin (n₂ + 1))
(e₂₃ : Fin (n₂ + 1) ≃ Fin (n₃ + 1)) (s : Simplex k P n₁) :
s.reindex (e₁₂.trans e₂₃) = (s.reindex e₁₂).reindex e₂₃ :=
rfl
| Mathlib/LinearAlgebra/AffineSpace/Independent.lean | 870 | 875 |
/-
Copyright (c) 2020 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca, Johan Commelin
-/
import Mathlib.Algebra.GCDMonoid.IntegrallyClosed
import Mathlib.FieldTheory.Finite.Basic
import Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed
impo... | refine minpoly.isIntegrallyClosed_dvd (h.isIntegral hpos) ?_
rw [aeval_def, coe_expand, ← comp, eval₂_eq_eval_map, map_comp, Polynomial.map_pow, map_X,
eval_comp, eval_pow, eval_X, ← eval₂_eq_eval_map, ← aeval_def]
exact minpoly.aeval _ _
/-- Let `P` be the minimal polynomial of a root of unity `μ` and `Q` b... | Mathlib/RingTheory/RootsOfUnity/Minpoly.lean | 82 | 90 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Algebra.Rat
import Mathlib.Data.Nat.Cast.Field
import Mathlib.RingTheory.PowerSeries.Basic
/-!
# Definition of well-known power series
In t... | simp only [one_div, coeff_mk, cast_pow, coe_mk, MonoidHom.coe_mk, OneHom.coe_mk,
coeff_exp, factorial, map_sum]
end PowerSeries
| Mathlib/RingTheory/PowerSeries/WellKnown.lean | 275 | 282 |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.IsPrimePow
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Ring.Int
import Mathlib.Algebra.Ring.CharZero
im... | simp
@[simp]
lemma nonempty_divisors : (divisors n).Nonempty ↔ n ≠ 0 :=
| Mathlib/NumberTheory/Divisors.lean | 234 | 237 |
/-
Copyright (c) 2024 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Kontorovich, David Loeffler, Heather Macbeth, Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.Analysis.Calculus.ContDiff.CPolynomial
import... | {f : ℝ → E} (hf : Integrable f) (hf' : Integrable (fun x : ℝ ↦ x • f x)) :
deriv (𝓕 f) = 𝓕 (fun x : ℝ ↦ (-2 * π * I * x) • f x) := by
ext x
exact (hasDerivAt_fourierIntegral hf hf' x).deriv
/-- The Fourier integral of the Fréchet derivative of a function is obtained by multiplying the
Fourier integral of... | Mathlib/Analysis/Fourier/FourierTransformDeriv.lean | 781 | 797 |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Algebra.Group.Int.Defs
import Mathlib.Algebra.Order.Monoid.Defs
/-!
# The integers form a linear ordered group
This file contains the instance necessar... | Mathlib/Algebra/Order/Group/Int.lean | 81 | 82 | |
/-
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 | 559 | 571 | |
/-
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.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Order.WithBot
/-!
# Intervals in `WithTop α` and `WithBot α`
In this file we ... |
theorem range_coe : range (some : α → WithBot α) = Ioi ⊥ :=
@WithTop.range_coe αᵒᵈ _
| Mathlib/Order/Interval/Set/WithBotTop.lean | 118 | 121 |
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Eric Wieser
-/
import Mathlib.Data.Real.Basic
import Mathlib.Tactic.NormNum.Inv
/-!
# Real sign function
This file introduces and contains some results about `Real.sign` wh... | obtain hn | hz | hp := sign_apply_eq r
· rw [hn]
norm_num
· rw [hz]
exact inv_zero
· rw [hp]
exact inv_one
| Mathlib/Data/Real/Sign.lean | 93 | 99 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Data.Finset.BooleanAlgebra
import Mathlib.Data.Set.Piecewise
import Mathlib.Order.Interval.Set.Basic
/-!
# Functions defined piecewise on a fins... |
lemma piecewise_same : s.piecewise f f = f := by
ext i
by_cases h : i ∈ s <;> simp [h]
| Mathlib/Data/Finset/Piecewise.lean | 117 | 120 |
/-
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.Two
import Mathlib.Data.Nat.Cast.Field
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.Nat.Factorization.Induction
import Mat... | #{x ∈ Ico k (k + n) | a.Coprime x} ≤ totient a * (n / a + 1) := by
conv_lhs => rw [← Nat.mod_add_div n a]
induction' n / a with i ih
· rw [← filter_coprime_Ico_eq_totient a k]
| Mathlib/Data/Nat/Totient.lean | 77 | 80 |
/-
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
-/
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Data.Nat.Factorization.Defs
import Mathlib.Analysis.NormedSpac... |
/-- The real logarithm function tends to `+∞` at `+∞`. -/
theorem tendsto_log_atTop : Tendsto log atTop atTop :=
tendsto_comp_exp_atTop.1 <| by simpa only [log_exp] using tendsto_id
| Mathlib/Analysis/SpecialFunctions/Log/Basic.lean | 320 | 323 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Group.Unbundled.Int
import Mathlib.Algebra.Ring.Nat
import Mathlib.Data.Int.GCD
/-!
# Congruences modulo a natural number
This file def... | Mathlib/Data/Nat/ModEq.lean | 448 | 449 | |
/-
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... | Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean | 1,706 | 1,720 | |
/-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
import Mathlib.SetTheory.Nimber.Basic
/-!
# Nim and the Sprague-Grundy theore... | theorem grundyValue_le_of_forall_moveRight {G : PGame} [G.Impartial] {o : Nimber}
(h : ∀ i, grundyValue (G.moveRight i) ≠ o) : G.grundyValue ≤ o := by
| Mathlib/SetTheory/Game/Nim.lean | 332 | 333 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Batteries.Tactic.Init
import Mathlib.Logic.Function.Defs
/-!
# Binary map of options
This file defines the binary map of `Option`. This is mostly useful to defin... | Mathlib/Data/Option/NAry.lean | 207 | 209 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.MeasureTheory.MeasurableSpace.EventuallyMeasurable
import Mathlib.MeasureTheory.MeasurableSpace.Basic
import Mathlib.M... | theorem measure_union₀' (hs : NullMeasurableSet s μ) (hd : AEDisjoint μ s t) :
μ (s ∪ t) = μ s + μ t := by rw [union_comm, measure_union₀ hs (AEDisjoint.symm hd), add_comm]
theorem measure_add_measure_compl₀ {s : Set α} (hs : NullMeasurableSet s μ) :
μ s + μ sᶜ = μ univ := by rw [← measure_union₀' hs aedisjoin... | Mathlib/MeasureTheory/Measure/NullMeasurable.lean | 290 | 301 |
/-
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 | 1,410 | 1,411 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Data.Countable.Small
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Powerset
import Mathlib.Dat... | Mathlib/SetTheory/Cardinal/Basic.lean | 1,705 | 1,707 | |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Ring.Associated
import Mathlib.Algebra.Star.Unitary
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.Tactic.Ring
import Mathlib.Al... | Mathlib/NumberTheory/Zsqrtd/Basic.lean | 280 | 280 | |
/-
Copyright (c) 2022 Antoine Chambert-Loir. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir
-/
import Mathlib.Algebra.Group.Subgroup.Lattice
import Mathlib.GroupTheory.GroupAction.FixedPoints
/-!
# Fixing submonoid, fixing subgroup of an action
... |
/-- The Galois connection between fixing subgroups and fixed points of a group action -/
@[to_additive]
| Mathlib/GroupTheory/GroupAction/FixingSubgroup.lean | 129 | 131 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Star.SelfAdjoint
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
/-!
# Quate... | Mathlib/Algebra/Quaternion.lean | 1,551 | 1,552 | |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.Real
/-!
# Properties of pointwise scalar multiplication of se... | use a • x + b • z
nth_rw 1 [← one_smul ℝ x]
| Mathlib/Analysis/NormedSpace/Pointwise.lean | 150 | 151 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
-/
import Mathlib.MeasureTheory.Integral.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Lebesgue.Countable
import Mathlib.MeasureTheory.Integral.Le... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 1,708 | 1,710 | |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jan-David Salchow, Sébastien Gouëzel, Jean Lo, Yury Kudryashov, Frédéric Dupuis,
Heather Macbeth
-/
import Mathlib.Algebra.Module.Opposite
import Mathlib.Topology.Algebra.Group.Qu... | Mathlib/Topology/Algebra/Module/Basic.lean | 1,559 | 1,563 | |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Robert Y. Lewis
-/
import Mathlib.Algebra.Order.CauSeq.Basic
import Mathlib.Algebra.Ring.Action.Rat
import Mathlib.Tactic.FastInstance
/-!
# Cauchy completion
This fi... | { Cauchy.divisionRing, Cauchy.commRing with }
| Mathlib/Algebra/Order/CauSeq/Completion.lean | 269 | 270 |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Equalizers
import Mathlib.CategoryTheory.Limits.Shapes.Reflexive
import Mathlib.CategoryTheory.Monad.Coequalizer
imp... | /-- The cofork which describes the counit of the adjunction: the morphism from the coequalizer of
this pair to this morphism is the counit.
-/
@[simps!]
def counitCofork (B : D) :
Cofork (F.map (G.map (adj.counit.app B)))
(adj.counit.app (F.obj (G.obj B))) :=
Cofork.ofπ (adj.counit.app B) (adj.counit_natura... | Mathlib/CategoryTheory/Monad/Monadicity.lean | 178 | 188 |
/-
Copyright (c) 2020 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Yaël Dillies
-/
import Mathlib.Order.Interval.Set.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Tactic.Bound.Attribute
import Mathlib.Tactic.Contrapose
import Mathl... |
theorem log_lt_of_lt_pow {b x y : ℕ} (hy : y ≠ 0) : y < b ^ x → log b y < x :=
lt_imp_lt_of_le_imp_le (pow_le_of_le_log hy)
| Mathlib/Data/Nat/Log.lean | 108 | 111 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.SymmDiff
import Mathlib.Order.SuccPred.Relation
import Mathlib.Topology.Irreducible
/-!
# Connected subsets ... | exact this.imp fun _ h => ⟨ts h.1, h.2.2, h.2.1⟩
/-- If any two points of a set are contained in a preconnected subset,
then the original set is preconnected as well. -/
theorem isPreconnected_of_forall_pair {s : Set α}
(H : ∀ x ∈ s, ∀ y ∈ s, ∃ t, t ⊆ s ∧ x ∈ t ∧ y ∈ t ∧ IsPreconnected t) :
IsPreconnected ... | Mathlib/Topology/Connected/Basic.lean | 96 | 111 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Group.Action.End
import Mathlib.Algebra.Group.Action.Pointwise.Set.Basic
import Mathlib.Algebra.Group.Action.Prod
import Mathlib.Algebra.Group.Subg... |
end Orbit
| Mathlib/GroupTheory/GroupAction/Basic.lean | 69 | 70 |
/-
Copyright (c) 2021 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.Analysis.Convex.Topology
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.Seminorm
import Mathlib.Analysis... | Mathlib/Analysis/Convex/Gauge.lean | 614 | 618 | |
/-
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.Field.Basic
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Ord... | 1 / a ^ n < 1 / a ^ m := by
refine (one_div_lt_one_div ?_ ?_).2 (pow_lt_pow_right₀ a1 mn) <;>
exact pow_pos (zero_lt_one.trans a1) _
| Mathlib/Algebra/Order/Field/Basic.lean | 247 | 250 |
/-
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.... |
theorem inv_inj (h : A⁻¹ = B⁻¹) (h' : IsUnit A.det) : A = B := by
| Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean | 476 | 477 |
/-
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.Order.Group.Pointwise.Interval
import Mathlib.LinearAlgebra.BilinearMap
import Mathlib.LinearAlgebra.Pi
import Mathlib.LinearAlgebra.Prod
import Ma... |
/-- Two affine maps are equal if they have equal linear maps and are equal at some point. -/
| Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean | 136 | 137 |
/-
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... | rwa [LinearMap.det_def, dif_neg H]
@[simp]
| Mathlib/LinearAlgebra/Determinant.lean | 219 | 221 |
/-
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.BigOperators.Expect
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Field.Canonical
import Mathlib.Algebra.O... | Mathlib/Data/NNReal/Basic.lean | 928 | 929 | |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.AddTorsor.Defs
import Mathlib.GroupTheory.GroupAction.SubMulAction
import Mathlib.Topology.Algebra.Constructions
import Mathlib.Topology.Alge... | IsInducing.id.continuousSMul continuous_subtype_val rfl
end Monoid
section Group
| Mathlib/Topology/Algebra/MulAction.lean | 211 | 216 |
/-
Copyright (c) 2022 Wrenna Robson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Wrenna Robson
-/
import Mathlib.Topology.MetricSpace.Basic
/-!
# Infimum separation
This file defines the extended infimum separation of a set. This is approximately dual to the
diame... | · rw [not_nontrivial_iff] at hs
exact hs.einfsep.symm ▸ WithTop.top_pos
| Mathlib/Topology/MetricSpace/Infsep.lean | 248 | 250 |
/-
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.Fintype.Lattice
import Mathlib.Data.Fintype.Sum
import Mathlib.Topology.Homeomorph.Lemmas
import Mathlib.Topology.MetricSpace.Antilipschitz
... | @[simp] theorem inv_apply_self (e : α ≃ᵢ α) (x : α) : e⁻¹ (e x) = x := e.symm_apply_apply x
@[simp] theorem apply_inv_self (e : α ≃ᵢ α) (x : α) : e (e⁻¹ x) = x := e.apply_symm_apply x
| Mathlib/Topology/MetricSpace/Isometry.lean | 483 | 485 |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Nat.Find
import Mathlib.Data.Stream.Init
import Mathlib.Tactic.Common
/-!
# Coinductive formalization of unbounded computations.
This fil... | Mathlib/Data/Seq/Computation.lean | 460 | 460 | |
/-
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... | exact (sum_mapDomain_index (fun _ => rfl) (fun _ _ _ => rfl)).trans_le (le_totalDegree hs)
lemma totalDegree_renameEquiv (f : σ ≃ τ) (p : MvPolynomial σ R) :
(renameEquiv R f p).totalDegree = p.totalDegree :=
(totalDegree_rename_le f p).antisymm (le_trans (by simp) (totalDegree_rename_le f.symm _))
end Tota... | Mathlib/Algebra/MvPolynomial/Degrees.lean | 523 | 531 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.Polynomial.Basi... | (degree_sub_le _ _).trans (max_le degree_X_le (degree_C_le.trans zero_le_one))
| Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 563 | 563 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.MeasurableSpace.MeasurablyGenerated
import Mathlib.MeasureTheory.Measure.NullMeasurable
import Mathlib.Order.Interval.Set... | suffices h : ∀ u v, μ (u ∆ v) ≠ ∞ → μ u = ∞ → μ v = ∞
from ⟨h s t hμst, h t s (symmDiff_comm s t ▸ hμst)⟩
intro u v hμuv hμu
| Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 243 | 245 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2
import Mathlib.MeasureTheory.Measure.Real
/-! # Conditional expectation in L1
This file contains ... |
theorem condExpInd_ae_eq_condExpIndSMul (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
(hs : MeasurableSet s) (hμs : μ s ≠ ∞) (x : G) :
condExpInd G hm μ s x =ᵐ[μ] condExpIndSMul hm hs hμs x := by
refine EventuallyEq.trans ?_ (condExpIndL1Fin_ae_eq_condExpIndSMul hm hs hμs x)
simp [condExpInd, condExpIndL1, hs, h... | Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean | 290 | 297 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Andrew Zipperer, Haitao Zhang, Minchao Wu, Yury Kudryashov
-/
import Mathlib.Data.Set.Prod
import Mathlib.Data.Set.Restrict
/-!
# Functions over sets
This file contains... | Mathlib/Data/Set/Function.lean | 1,684 | 1,686 | |
/-
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.Projections
import Mathlib.CategoryTheory.Idempotents.FunctorCategories
import Mathlib.CategoryTheory.Idempotents.FunctorExtension
/-!... | variable (C)
/-- `PInfty` induces a natural transformation, i.e. an endomorphism of
the functor `alternatingFaceMapComplex C`. -/
| Mathlib/AlgebraicTopology/DoldKan/PInfty.lean | 130 | 133 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.Ring.Action.Basic
import Mathlib.GroupTheory.GroupAction.Hom
import ... | g • (prodXSubSMul G R x).coeff n = (prodXSubSMul G R x).coeff n := by
rw [← Polynomial.coeff_smul, prodXSubSMul.smul]
end CommRing
| Mathlib/Algebra/Polynomial/GroupRingAction.lean | 102 | 105 |
/-
Copyright (c) 2024 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.Cotangent
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.MvPolynomial.Tower
import Mathlib.RingTheory.TensorPro... | rw [map_add, ha, hb]
/-- Given generators `P` and an equivalence `ι ≃ P.vars`, these
are the induced generators indexed by `ι`. -/
@[simps -isSimp vars]
noncomputable def reindex (P : Generators.{w} R S) {ι : Type*} (e : ι ≃ P.vars) :
Generators R S where
vars := ι
val := P.val ∘ e
| Mathlib/RingTheory/Generators.lean | 248 | 256 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Prime.Basic
import Mathlib.Data.Real.Archimedean
import Mathlib.NumberTheory.Zsqrtd.Basic
/-!
# Gaussian intege... | quotient := (· / ·)
remainder := (· % ·)
quotient_zero := by simp [div_def]; rfl
quotient_mul_add_remainder_eq := fun _ _ => by simp [mod_def]
r := _
r_wellFounded := (measure (Int.natAbs ∘ norm)).wf
remainder_lt := natAbs_norm_mod_lt
mul_left_not_lt := fun a _ hb0 => not_lt_of_ge <| nor... | Mathlib/NumberTheory/Zsqrtd/GaussianInt.lean | 230 | 243 |
/-
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
... | refine ⟨hx, ?_⟩
ext y
obtain ⟨u, hu, v, hv, rfl⟩ : ∃ u ∈ W, ∃ v ∈ ker B, u + v = y := by
rw [← Submodule.mem_sup, hW.sup_eq_top]; exact Submodule.mem_top
suffices B x u = 0 by rw [mem_ker] at hv; simpa [← hB.eq v, hv]
| Mathlib/LinearAlgebra/SesquilinearForm.lean | 742 | 746 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot, Yury Kudryashov, Rémy Degenne
-/
import Mathlib.Data.Set.Subsingleton
import Mathlib.Order.Interval.Set.Defs
/-!
# Intervals
In any pr... | Mathlib/Order/Interval/Set/Basic.lean | 1,830 | 1,831 | |
/-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Data.Fintype.Card
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
/-!
# Multiset coercion to type
This module defines a `CoeSort` instance for multi... | theorem prod_eq_prod_toEnumFinset [CommMonoid α] (m : Multiset α) :
m.prod = ∏ x ∈ m.toEnumFinset, x.1 := by
congr
simp
@[to_additive]
| Mathlib/Data/Multiset/Fintype.lean | 219 | 224 |
/-
Copyright (c) 2023 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Order.ScottContinuity
import Mathlib.Topology.Order.UpperLowerSetTopology
/-!
# Scott topology
This file introduces the Scott topology on a p... | lemma topology_eq [IsScottHausdorff α D] : ‹_› = scottHausdorff α D := topology_eq_scottHausdorff
variable {α D}
| Mathlib/Topology/Order/ScottTopology.lean | 174 | 176 |
/-
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.Order.SupClosed
/-!
# Sublattices
This file defines sublattices.
## TODO
Subsemilattices, if people care about them.
## Tags
sublattice
-/
open Func... | @[simp] lemma not_mem_bot (a : α) : a ∉ (⊥ : Sublattice α) := id
@[simp] lemma mem_top (a : α) : a ∈ (⊤ : Sublattice α) := mem_univ _
| Mathlib/Order/Sublattice.lean | 176 | 177 |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.AlgebraicGeometry.Cover.Open
import Mathlib.AlgebraicGeometry.GammaSpecAdjunction
import Mathlib.AlgebraicGeometry.Restrict
import Mathlib.CategoryTheory.Lim... | include hU in
theorem ι_basicOpen_preimage (r : Γ(X, ⊤)) :
IsAffineOpen ((X.basicOpen r).ι ⁻¹ᵁ U) := by
apply (X.basicOpen r).ι.isAffineOpen_iff_of_isOpenImmersion.mp
dsimp [Scheme.Hom.opensFunctor, LocallyRingedSpace.IsOpenImmersion.opensFunctor]
rw [Opens.functor_obj_map_obj, Opens.isOpenEmbedding_obj_top, ... | Mathlib/AlgebraicGeometry/AffineScheme.lean | 553 | 572 |
/-
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.Pi
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Set.Finite.Basic
/-!
# Fintype instances for pi types
-/
assert_not_exists Order... | (ha : a ∈ s i) :
piFinset (Function.update s i {a}) = {f ∈ piFinset s | f i = a} := by
simp [piFinset_update_eq_filter_piFinset_mem, ha]
| Mathlib/Data/Fintype/Pi.lean | 128 | 131 |
/-
Copyright (c) 2023 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.ModuleCat
import Mathlib.RepresentationTheory.GroupCohomology.Basic
import Mathlib.RepresentationTheory.... | v v
Fun(G, A) -dOne-> Fun(G × G, A)
```
where the vertical arrows are `oneCochainsLequiv` and `twoCochainsLequiv` respectively.
-/
theorem dOne_comp_eq : dOne A ∘ₗ oneCochainsLequiv A =
twoCochainsLequiv A ∘ₗ ((inhomogeneousCochains A).d 1 2).hom := by
ext x y
| Mathlib/RepresentationTheory/GroupCohomology/LowDegree.lean | 166 | 173 |
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kenny Lau
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.RingTheory.MvPowerSer... | have := coe_monomial 0 (1 : R)
rwa [PowerSeries.monomial_zero_eq_C_apply] at this
@[simp, norm_cast]
theorem coe_add : ((φ + ψ : R[X]) : PowerSeries R) = φ + ψ := by
ext
simp
@[simp, norm_cast]
| Mathlib/RingTheory/PowerSeries/Basic.lean | 801 | 809 |
/-
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... | Mathlib/Algebra/Group/Submonoid/Membership.lean | 582 | 585 | |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Data.Finset.BooleanAlgebra
import Mathlib.Data.Set.Piecewise
import Mathlib.Order.Interval.Set.Basic
/-!
# Functions defined piecewise on a fins... | p (s.piecewise f g i) := by
by_cases hi : i ∈ s <;> simpa [hi]
lemma piecewise_singleton [DecidableEq ι] (i : ι) : piecewise {i} f g = update g i (f i) := by
rw [← insert_empty_eq, piecewise_insert, piecewise_empty]
| Mathlib/Data/Finset/Piecewise.lean | 71 | 76 |
/-
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, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Localization.Ideal
import Mathlib.... | rw [coeSubmodule, Submodule.map_top, Submodule.one_eq_range]
| Mathlib/RingTheory/Localization/Submodule.lean | 48 | 49 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.Algebra.Notation.Prod
import Mathlib.Data.Set.Image
/-!
# Support of a func... | section DivisionMonoid
variable [DivisionMonoid G] (f g : α → G)
| Mathlib/Algebra/Group/Support.lean | 221 | 224 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Data.Set.Piecewise
import Mathlib.Logic.Equiv.Defs
import Mathlib.Tactic.Core
import Mathlib.Tactic.Attr.Core
/-!
# Partial equivalences
This f... |
@[simp, mfld_simps]
theorem refl_symm : (PartialEquiv.refl α).symm = PartialEquiv.refl α :=
rfl
| Mathlib/Logic/Equiv/PartialEquiv.lean | 519 | 522 |
/-
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.Algebra.Order.Star.Basic
import Mathlib.Data.Matrix.RowCol
/-!
# Dot product of two vectors
This file conta... | lemma self_mul_conjTranspose_eq_zero {m} {A : Matrix m n R} : A * Aᴴ = 0 ↔ A = 0 :=
⟨fun h => Matrix.ext fun i j =>
(congr_fun <| dotProduct_self_star_eq_zero.1 <| Matrix.ext_iff.2 h i i) j,
fun h => h ▸ Matrix.zero_mul _⟩
| Mathlib/LinearAlgebra/Matrix/DotProduct.lean | 150 | 153 |
/-
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.BigOperators.Ring.Finset
import Mathlib.Combinatorics.SimpleGraph.Density
import Mathlib.Data.Nat.Cast.Order.Field
impo... | rcases trichotomous_of WellOrderingRel s t with (lt | rfl | gt)
· rw [if_pos lt, if_neg (asymm lt)]
exact G.nonuniformWitnesses_spec h₂
· cases h₁ rfl
· rw [if_neg (asymm gt), if_pos gt, edgeDensity_comm, edgeDensity_comm _ s]
| Mathlib/Combinatorics/SimpleGraph/Regularity/Uniform.lean | 175 | 179 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Minchao Wu, Mario Carneiro
-/
import Mathlib.Data.Finset.Attach
import Mathlib.Data.Finset.Disjoint
import Mathlib.Data.Finset.Erase
import Mat... | Mathlib/Data/Finset/Basic.lean | 2,355 | 2,356 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Sigma
import Mathlib.Algebra.Order.Interval.Finset.Basic
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Tact... | simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h
@[to_additive]
| Mathlib/Algebra/BigOperators/Intervals.lean | 162 | 164 |
/-
Copyright (c) 2018 Michael Jendrusch. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Jendrusch, Kim Morrison, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Monoidal.Category
import Mathlib.CategoryTheory.Adjunction.FullyFaithful
import Mathlib.CategoryTheory... | (α_ (F.obj X) (F.obj Y) (F.obj Z)).hom ≫ F.obj X ◁ μ F Y Z ≫ μ F X (Y ⊗ Z) := by
apply associativity'
@[simp, reassoc]
| Mathlib/CategoryTheory/Monoidal/Functor.lean | 113 | 116 |
/-
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.HomologicalComplex
import Mathlib.Algebra.Homology.ShortComplex.SnakeLemma
import Mathlib.Algebra.Homology.ShortComplex.ShortExact
... | @[reassoc (attr := simp)]
lemma pOpcycles_opcyclesToCycles :
K.pOpcycles i ≫ K.opcyclesToCycles i j = K.toCycles i j := by
simp only [← cancel_mono (K.iCycles j), assoc, opcyclesToCycles_iCycles,
p_fromOpcycles, toCycles_i]
| Mathlib/Algebra/Homology/HomologySequence.lean | 57 | 61 |
/-
Copyright (c) 2022 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Integral.BoundedContinuousFunction
import Mathlib.Topology.MetricSpace.ThickenedIndicator
/-!
# Spaces where indicators of closed sets have ... | · simp only [hxF, indicator_of_mem, apprSeq_apply_eq_one hF n, le_refl]
· simp only [hxF, not_false_eq_true, indicator_of_not_mem, zero_le]
/-- The measure of a closed set is at most the integral of any function in a decreasing
approximating sequence to the indicator of the set. -/
theorem measure_le_lintegral [Me... | Mathlib/MeasureTheory/Measure/HasOuterApproxClosed.lean | 157 | 162 |
/-
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.Data.Finset.Sort
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Sign
import Mathlib.LinearAlgebra.AffineSpace.Combination
import Mathlib.LinearAlg... | · rcases h with ⟨r, hr⟩
refine ⟨r, ?_⟩
let w' i := r * (w₁ i - w₂ i)
change ∀ i ∈ s, w i = w' i at hr
rw [s.weightedVSub_congr hr fun _ _ => rfl, s.affineCombination_vsub, ←
s.weightedVSub_const_smul]
congr
/-- Given an affinely independent family of points, an affine combination lies in th... | Mathlib/LinearAlgebra/AffineSpace/Independent.lean | 520 | 543 |
/-
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... | convert h.comp_sub_left x
simp_rw [sub_sub_self]
theorem convolutionExistsAt_iff_integrable_swap :
ConvolutionExistsAt f g x L μ ↔ Integrable (fun t => L (f (x - t)) (g t)) μ :=
convolutionExistsAt_flip.symm
end MeasurableGroup
variable [TopologicalSpace G] [IsTopologicalAddGroup G] [BorelSpace G]
| Mathlib/Analysis/Convolution.lean | 379 | 388 |
/-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
/-! # Positive Definite Matrices
This file defi... | rw [mulVec_mulVec, mulVec_mulVec, ← dotProduct_add, ← add_mulVec, mul_sub, sub_mul,
add_sub, sub_add_cancel, pow_two, pow_two]
_ = t * (star v ⬝ᵥ A *ᵥ v) + (star v) ᵥ* (A - B)ᴴ ⬝ᵥ B *ᵥ v := by
| Mathlib/LinearAlgebra/Matrix/PosDef.lean | 233 | 235 |
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