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) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Data.Countable.Small
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Powerset
import Mathlib.Dat... |
theorem mk_sep (s : Set α) (t : α → Prop) : #({ x ∈ s | t x } : Set α) = #{ x : s | t x.1 } :=
mk_congr (Equiv.Set.sep s t)
theorem mk_preimage_of_injective_lift {α : Type u} {β : Type v} (f : α → β) (s : Set β)
(h : Injective f) : lift.{v} #(f ⁻¹' s) ≤ lift.{u} #s := by
rw [lift_mk_le.{0}]
-- Porting note:... | Mathlib/SetTheory/Cardinal/Basic.lean | 833 | 845 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov, David Loeffler
-/
import Mathlib.Analysis.Convex.Slope
import Mathlib.Analysis.Calculus.Deriv.MeanValue
/-!
# Convexity of functions and derivat... | (hfd : DifferentiableAt ℝ f x) :
deriv f x ≤ slope f x y :=
le_slope_of_hasDerivAt hfc hx hy hxy hfd.hasDerivAt
end left
section right
/-!
### Convex functions, derivative at right endpoint of secant
-/
| Mathlib/Analysis/Convex/Deriv.lean | 603 | 612 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Reid Barton
-/
import Mathlib.Topology.Separation.Regular
/-!
# The shrinking lemma
In this file we prove a few versions of the shrinking lemma. The lemma says that... | (ne : c.Nonempty) {i v} (hv : v ∈ c) (hi : i ∈ carrier v) : find c ne i i = v i :=
apply_eq_of_chain hc (find_mem _ _) hv ((mem_find_carrier_iff _).2 <| mem_iUnion₂.2 ⟨v, hv, hi⟩)
hi
/-- Least upper bound of a nonempty chain of partial refinements. -/
def chainSup (c : Set (PartialRefinement u s p)) (hc : Is... | Mathlib/Topology/ShrinkingLemma.lean | 124 | 132 |
/-
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... | /--
Similar to `MvPolynomial.induction_on` but only a yet weaker form of `h_add` is required.
| Mathlib/Algebra/MvPolynomial/Basic.lean | 394 | 395 |
/-
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.Basic
/-!
# Maps between real and extended non-negative real numbers
This file focuses on the functions `ENNReal.toReal... |
@[simp]
lemma one_le_ofReal {p : ℝ} : 1 ≤ ENNReal.ofReal p ↔ 1 ≤ p := by
| Mathlib/Data/ENNReal/Real.lean | 198 | 200 |
/-
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.Algebra.Module.ULift
import Mathlib.RingTheory.TensorProduct.Basic
import Mathlib.Tactic.Ring
/-!
# The characteristic predicate of tensor product
## Main ... | cases this
refine
⟨g.comp e.symm.toLinearMap, by
ext
simp, ?_⟩
rintro y (rfl : _ = _)
ext
simp
variable {T O : Type*} [CommSemiring T] [Algebra R T] [Algebra S T] [IsScalarTower R S T]
variable [AddCommMonoid O] [Module R O] [Module S O] [Module T O] [IsScalarTower S T O]
variable [IsScalarTo... | Mathlib/RingTheory/IsTensorProduct.lean | 319 | 341 |
/-
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.Topology.EMetricSpace.Paracompact
import Mathlib.Topology.Instances.ENNReal.Lemmas
import Mathlib.Analysis.Convex.PartitionOfUnity
/-!
# Lemmas abou... | simpa only [← ENNReal.ofReal_coe_nnreal] using hδ i x hi
/-- Let `X` be an extended metric space. Let `K : ι → Set X` be a locally finite family of closed
sets, let `U : ι → Set X` be a family of open sets such that `K i ⊆ U i` for all `i`. Then there
exists a positive continuous function `δ : C(X, ℝ≥0∞)` such that ... | Mathlib/Topology/MetricSpace/PartitionOfUnity.lean | 100 | 106 |
/-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.Order.GameAdd
import Mathlib.Order.RelIso.Set
import Mathlib.SetTheory.ZFC.Basic
/-!
# Von Neumann ordinals
This file works t... |
protected theorem IsTransitive.powerset (h : x.IsTransitive) : (powerset x).IsTransitive :=
fun y hy z hz => by
rw [mem_powerset] at hy ⊢
exact h.subset_of_mem (hy hz)
theorem isTransitive_iff_sUnion_subset : x.IsTransitive ↔ (⋃₀ x : ZFSet) ⊆ x := by
constructor <;>
| Mathlib/SetTheory/ZFC/Ordinal.lean | 77 | 84 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker, Johan Commelin
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Div
import Mathlib.RingTheory... | Mathlib/Algebra/Polynomial/RingDivision.lean | 542 | 547 | |
/-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.LinearAlgebra.Matrix.ZPow
import Mathlib.Data.Matrix.ConjTranspose
/-! # Hermitian matrices
Th... |
variable [RCLike α]
| Mathlib/LinearAlgebra/Matrix/Hermitian.lean | 257 | 259 |
/-
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.Group.Nat.Defs
import Mathlib.CategoryTheory.Category.Preorder
import Mathlib.CategoryTheory.EqToHom
import Mathlib.CategoryTheory.Functor.Const
import M... | fin_cases i
exact h
/-- Constructor for isomorphisms in `ComposableArrows C 0`. -/
@[simps!]
def isoMk₀ {F G : ComposableArrows C 0} (e : F.obj' 0 ≅ G.obj' 0) : F ≅ G where
hom := homMk₀ e.hom
| Mathlib/CategoryTheory/ComposableArrows.lean | 221 | 227 |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Batteries.Data.Rat.Lemmas
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Rat.Init
import Mathlib.Order.Basic
import Mathlib.Tactic.Commo... | theorem num_zero : Rat.num 0 = 0 :=
rfl
-- Not `@[simp]` as `den_ofNat` is stronger.
theorem den_zero : Rat.den 0 = 1 :=
rfl
| Mathlib/Data/Rat/Defs.lean | 355 | 360 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Polynomial.Degree.Domain
import Mathlib.Algebra.Polynomial.Degree.Support
import Mathlib.Algebra.Poly... |
theorem derivative_comp (p q : R[X]) :
derivative (p.comp q) = derivative q * p.derivative.comp q := by
induction p using Polynomial.induction_on'
· simp [*, mul_add]
· simp only [derivative_pow, derivative_mul, monomial_comp, derivative_monomial, derivative_C,
zero_mul, C_eq_natCast, zero_add, RingHom... | Mathlib/Algebra/Polynomial/Derivative.lean | 521 | 529 |
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.Group.Action.End
import Mathlib.Algebra.Group.Pointwise.Set.Lattice
import Mathlib.Algebra.Group.Subgroup.MulOppositeLemmas
import Mathlib.Algebra.Gr... | /-- An induction principle for closure membership. If `p` holds for `1` and all elements of
`k` and their inverse, and is preserved under multiplication, then `p` holds for all elements of
the closure of `k`. -/
@[to_additive (attr := elab_as_elim)
"An induction principle for additive closure membership. If `p` holds... | Mathlib/Algebra/Group/Subgroup/Pointwise.lean | 154 | 165 |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.ModEq
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Algebra.Ring.Periodic
import Mathlib.Data.Int.SuccPred
import Mathlib.Order.Cir... | theorem toIocMod_add_toIocDiv_zsmul (a b : α) : toIocMod hp a b + toIocDiv hp a b • p = b := by
rw [toIocMod, sub_add_cancel]
| Mathlib/Algebra/Order/ToIntervalMod.lean | 133 | 134 |
/-
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... | · rw [ascFactorial_zero, choose_zero_right, factorial_zero, Nat.mul_one]
· simp only [zero_ascFactorial, zero_eq, Nat.zero_add, succ_sub_succ_eq_sub,
Nat.le_zero_eq, Nat.sub_zero, choose_succ_self, Nat.mul_zero]
rw [ascFactorial_eq_factorial_mul_choose]
simp only [succ_add_sub_one]
| Mathlib/Data/Nat/Choose/Basic.lean | 235 | 240 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.FreeAlgebra
import Mathlib.RingTheory.Adjoin.Polynomial
import Mathlib.RingTheory.Adjoin.Tower
import Mathlib.RingTheory.Ideal.Quotient.Operati... | simp_rw [← hrw, hs]
exact Set.mem_univ x
· rintro ⟨s, ⟨f, hsur⟩⟩
exact FiniteType.of_surjective (FiniteType.mvPolynomial R { x // x ∈ s }) f hsur
/-- An algebra is finitely generated if and only if it is a quotient
of a polynomial ring whose variables are indexed by a fintype. -/
theorem iff_quotient_fre... | Mathlib/RingTheory/FiniteType.lean | 147 | 158 |
/-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.Sheaf
/-!
# The plus construction for presheaves.
This file contains the construction of `P⁺`, for a presheaf `P : Cᵒᵖ ⥤ D`
where `C` i... |
@[simp]
theorem diagramNatTrans_zero [Preadditive D] (X : C) (P Q : Cᵒᵖ ⥤ D) :
J.diagramNatTrans (0 : P ⟶ Q) X = 0 := by
ext : 2
refine Multiequalizer.hom_ext _ _ _ (fun i => ?_)
simp
| Mathlib/CategoryTheory/Sites/Plus.lean | 71 | 77 |
/-
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... | (Eventually.of_forall fun x => ?_) hf (Eventually.of_forall fun x => ?_)
· exact truncation_nonneg _ (h'f x)
· calc
| Mathlib/Probability/StrongLaw.lean | 178 | 180 |
/-
Copyright (c) 2020 Aaron Anderson, Jalex Stark. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark
-/
import Mathlib.Algebra.Polynomial.Expand
import Mathlib.Algebra.Polynomial.Laurent
import Mathlib.Algebra.Polynomial.Eval.SMul
import Mathli... | M.charpoly.degree = Fintype.card n := by
by_cases h : Fintype.card n = 0
· rw [h]
unfold charpoly
rw [det_of_card_zero]
| Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean | 89 | 93 |
/-
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.Group.Submonoid.Operations
import Mathlib.Algebra.MonoidAlgebra.Defs
import Mathlib.Algebra.Order.Mon... | theorem ofFinsupp_pow (a) (n : ℕ) : (⟨a ^ n⟩ : R[X]) = ⟨a⟩ ^ n := by
change _ = npowRec n _
| Mathlib/Algebra/Polynomial/Basic.lean | 184 | 185 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Geometry.Manifold.ChartedSpace
/-!
# Local properties invariant under a groupoid
We study properties of a triple `(g, s, x)` ... | simp only [liftPropWithinAt_iff']
exact and_congr_right <|
hG.liftPropWithinAt_indep_chart_aux (chart_mem_maximalAtlas _ _) (mem_chart_source _ _) he xe
(chart_mem_maximalAtlas _ _) (mem_chart_source _ _) hf xf
/-- A version of `liftPropWithinAt_indep_chart`, only for the source. -/
theorem liftPropWithi... | Mathlib/Geometry/Manifold/LocalInvariantProperties.lean | 283 | 289 |
/-
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 ... | .fst
theorem right : Primrec₂ fun (_ : α) (b : β) => b :=
.snd
theorem natPair : Primrec₂ Nat.pair := by simp [Primrec₂, Primrec]; constructor
| Mathlib/Computability/Primrec.lean | 363 | 369 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.FinMeasAdditive
/-!
# Extension of a linear function from indicators to L1
Give... | Mathlib/MeasureTheory/Integral/SetToL1.lean | 1,693 | 1,695 | |
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel, Alex Keizer
-/
import Mathlib.Algebra.Group.Nat.Even
import Mathlib.Algebra.NeZero
import Mathlib.Algebra.Ring.Nat
import Mathlib.Data.List.GetD
import Mathlib.Data.Nat.B... | | z => exact False.elim (Bool.false_ne_true ((zero_testBit i).symm.trans hm))
| f b' m hm' =>
| Mathlib/Data/Nat/Bitwise.lean | 212 | 213 |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Manuel Candales
-/
import Mathlib.Analysis.InnerProductSpace.Subspace
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
/-!
# Angles between vectors
This fil... |
/-- The inner product of two non-zero vectors equals the negative product of their norms
if and only if the angle between the two vectors is π. -/
| Mathlib/Geometry/Euclidean/Angle/Unoriented/Basic.lean | 219 | 221 |
/-
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 | 1,429 | 1,431 | |
/-
Copyright (c) 2022 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.MeasureTheory.Integral.ExpDecay
/-!
# The Gamma function
This file defines the `Γ` functio... | theorem Gamma_ofNat_eq_factorial (n : ℕ) [(n + 1).AtLeastTwo] :
Gamma (ofNat(n + 1) : ℂ) = n ! :=
mod_cast Gamma_nat_eq_factorial (n : ℕ)
/-- At `0` the Gamma function is undefined; by convention we assign it the value `0`. -/
@[simp]
| Mathlib/Analysis/SpecialFunctions/Gamma/Basic.lean | 332 | 337 |
/-
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 | 193 | 197 | |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Joël Riou
-/
import Mathlib.CategoryTheory.Sites.Subsheaf
import Mathlib.CategoryTheory.Sites.CompatibleSheafification
import Mathlib.CategoryTheory.Sites.LocallyInjective
/-... | exact ⟨F.map i.op s, NatTrans.naturality_apply f i.op s⟩
/-- If a morphism `g : V ⟶ U.unop` belongs to the sieve `imageSieve f s g`, then
this is choice of a preimage of `G.map g.op s` in `F.obj (op V)`, see
`app_localPreimage`. -/
noncomputable def localPreimage {F G : Cᵒᵖ ⥤ A} (f : F ⟶ G) {U : Cᵒᵖ} (s : ToType (G.... | Mathlib/CategoryTheory/Sites/LocallySurjective.lean | 65 | 70 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Set.Image
import Mathlib.Data.Set.BooleanAlgebra
/-!
# Sets in sigma types
This file defines `Set.sigma`, the indexed sum of sets.
-/
namespace Set... | · rintro _ ⟨b, rfl⟩
simp
· rintro ⟨x, y⟩ (rfl | _)
exact mem_range_self y
theorem preimage_image_sigmaMk_of_ne (h : i ≠ j) (s : Set (α j)) :
| Mathlib/Data/Set/Sigma.lean | 23 | 28 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Option.NAry
import Mathlib.Data.Seq.Computation
import Mathlib.Tactic.ApplyFun
import Mathlib.Data.List.Basic
/-!
# Possibly infinite lists
This... | rw [Option.map_id, Stream'.map_id]
| Mathlib/Data/Seq/Seq.lean | 881 | 882 |
/-
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.AddCharacter
import Mathlib.NumberTheory.LegendreSymbol.ZModChar
import Mathlib.Algebra.CharP.CharAndCard
/-!
# Gauss sums
... | have := Fact.mk (hchar.subst hp')
rw [Ne, ← Nat.prime_dvd_prime_iff_eq hp' hp, ← isUnit_iff_not_dvd_char, hchar] at hch₁
exact Char.card_pow_char_pow (hχ₂.comp _) ψ.char (ringChar FF') n' hch₁ (hchar ▸ hch₂)
(gaussSum_sq ((ringHomComp_ne_one_iff (RingHom.injective _)).mpr hχ₁) (hχ₂.comp _) ψ.prim)
end Gau... | Mathlib/NumberTheory/GaussSum.lean | 253 | 318 |
/-
Copyright (c) 2022 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu, Anne Baanen
-/
import Mathlib.Algebra.Module.LocalizedModule.IsLocalization
import Mathlib.LinearAlgebra.Basis.Basic
import Mathlib.RingTheory.Localization.FractionRing
import ... | eq_comm, ← inv_smul_eq_iff] at h
refine h ▸ smul_mem _ _ (span_subset_span R Rₛ _ ?_)
rw [← LinearMap.coe_restrictScalars R, ← LinearMap.map_span, hv]
exact mem_map_of_mem mem_top
theorem LinearIndependent.of_isLocalizedModule {ι : Type*} {v : ι → M}
(hv : LinearIndependent R v) : LinearIndependent Rₛ (... | Mathlib/RingTheory/Localization/Module.lean | 47 | 54 |
/-
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... | theorem _root_.Set.Countable.isSeparable {s : Set α} (hs : s.Countable) : IsSeparable s :=
⟨s, hs, subset_closure⟩
| Mathlib/Topology/Bases.lean | 454 | 456 |
/-
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.Basic
import Mathlib.Topology.MetricSpace.Pseudo.Le... |
/-- Metric space structure pulled back by an injective function. Injectivity is necessary to
| Mathlib/Topology/MetricSpace/Basic.lean | 87 | 88 |
/-
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.CategoryTheory.Abelian.Basic
/-!
# Idempotent complete categories
In this file, we define the notion of idempotent complete categories
(also known as Karoubian... | constructor
· simp only [← unop_comp, h₁]
rfl
· simp only [← unop_comp, h₂]
rfl
| Mathlib/CategoryTheory/Idempotents/Basic.lean | 177 | 181 |
/-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap
import Mathlib.MeasureTheory.Measure.HasOuterApproxClosed
import Mathlib.MeasureTheory.Measure.Prod
impo... | section FiniteMeasureBoundedConvergence
/-! ### Bounded convergence results for finite measures
This section is about bounded convergence theorems for finite measures.
-/
variable {Ω : Type*} [MeasurableSpace Ω] [TopologicalSpace Ω] [OpensMeasurableSpace Ω]
/-- A bounded convergence theorem for a finite measure:
I... | Mathlib/MeasureTheory/Measure/FiniteMeasure.lean | 567 | 578 |
/-
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.Order.Interval.Set.OrdConnected
import Mathlib.Data.Set.Lattice.Image
/-!
# Order connected components of a set
In this file we define `Set.ordConn... | (hy <| Icc_subset_uIcc ⟨hxb, hby.le⟩)
exact ⟨sol1, sol2⟩
end Set
| Mathlib/Order/Interval/Set/OrdConnectedComponent.lean | 200 | 234 |
/-
Copyright (c) 2019 Abhimanyu Pallavi Sudhir. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Abhimanyu Pallavi Sudhir
-/
import Mathlib.Order.Filter.FilterProduct
import Mathlib.Analysis.SpecificLimits.Basic
/-!
# Construction of the hyperreal numbers as an ultrapro... | or_comm.trans <| infiniteNeg_neg.or infinitePos_neg
nonrec theorem Infinitesimal.not_infinite {x : ℝ*} (h : Infinitesimal x) : ¬Infinite x :=
h.not_infinite
| Mathlib/Data/Real/Hyperreal.lean | 391 | 394 |
/-
Copyright (c) 2022 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.MeasureTheory.Integral.ExpDecay
/-!
# The Gamma function
This file defines the `Γ` functio... | rw [Gamma, Complex.ofReal_add, Complex.ofReal_natCast, Complex.ofReal_one,
Complex.Gamma_nat_eq_factorial, ← Complex.ofReal_natCast, Complex.ofReal_re]
@[simp]
theorem Gamma_ofNat_eq_factorial (n : ℕ) [(n + 1).AtLeastTwo] :
Gamma (ofNat(n + 1) : ℝ) = n ! :=
mod_cast Gamma_nat_eq_factorial (n : ℕ)
/-- At `... | Mathlib/Analysis/SpecialFunctions/Gamma/Basic.lean | 432 | 444 |
/-
Copyright (c) 2020 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang, Johan Commelin
-/
import Mathlib.RingTheory.GradedAlgebra.Homogeneous.Ideal
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.Topology.Sets.Opens
import Mathlib.... | Iff.rfl
| Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Topology.lean | 417 | 418 |
/-
Copyright (c) 2022 Kevin H. Wilson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin H. Wilson
-/
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Analysis.NormedSpace.RCLike
import Mathlib.Order.Filter.Curry
/-!
# Swapping limits and derivatives via u... | (Metric.ball x r) := by
rw [uniformCauchySeqOn_iff_uniformCauchySeqOnFilter]
exact hf'.one_smulRight
exact uniformCauchySeqOn_ball_of_fderiv hf' hf hfg
theorem hasDerivAt_of_tendstoUniformlyOnFilter [NeBot l]
(hf' : TendstoUniformlyOnFilter f' g' l (𝓝 x))
(hf : ∀ᶠ n : ι × 𝕜 in l ×ˢ 𝓝 x, HasD... | Mathlib/Analysis/Calculus/UniformLimitsDeriv.lean | 483 | 493 |
/-
Copyright (c) 2022 Rishikesh Vaishnav. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rishikesh Vaishnav
-/
import Mathlib.MeasureTheory.Measure.Typeclasses.Probability
/-!
# Conditional Probability
This file defines conditional probability and includes basic resu... | scoped macro:max μ:term noWs "[" s:term " | " X:term " ← " x:term "]" : term =>
`($μ[$s | $X in {$x:term}])
| Mathlib/Probability/ConditionalProbability.lean | 143 | 144 |
/-
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 ... | (precomposing _ _ d).obj (f ≫ g) ≅ (precomposing ..).obj g ⋙ (precomposing ..).obj f :=
NatIso.ofComponents (α_ f g ·)
| Mathlib/CategoryTheory/Bicategory/Basic.lean | 464 | 465 |
/-
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.Analysis.Calculus.Deriv.Basic
import Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap
import Mathlib.MeasureTheory.Covering.Bes... | obtain ⟨δ', δ'pos, hδ'⟩ : ∃ (δ' : ℝ), 0 < δ' ∧ ∀ B, dist B A < δ' → dist B.det A.det < ↑ε := by
refine continuousAt_iff.1 ?_ ε εpos
exact ContinuousLinearMap.continuous_det.continuousAt
let δ'' : ℝ≥0 := ⟨δ' / 2, (half_pos δ'pos).le⟩
have I'' : ∀ B : E →L[ℝ] E, ‖B - A‖ ≤ ↑δ'' → |B.det - A.det| ≤ ... | Mathlib/MeasureTheory/Function/Jacobian.lean | 924 | 1,026 |
/-
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.Localization.Predicate
import Mathlib.CategoryTheory.CatCommSq
/-!
# Localization functors are preserved through equivalences
In `Localization/P... |
/-- If `L₁ : C₁ ⥤ D₁` is a localization functor for `W₁ : MorphismProperty C₁`, then if we
transport this functor `L₁` via equivalences `C₁ ≌ C₂` and `D₁ ≌ D₂` to get a functor
`L₂ : C₂ ⥤ D₂`, then `L₂` is also a localization functor for
a suitable `W₂ : MorphismProperty C₂`. -/
lemma of_equivalences (L₁ : C₁ ⥤ D₁) (W... | Mathlib/CategoryTheory/Localization/Equivalence.lean | 95 | 106 |
/-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Algebra.Order.Pi
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Data.Real.Pointwise
/... | refine fun hx => ⟨k⁻¹ • x, ?_, ?_⟩
· rwa [Seminorm.mem_ball_zero, map_smul_eq_mul, norm_inv, ←
mul_lt_mul_left <| norm_pos_iff.mpr hk, ← mul_assoc, ← div_eq_mul_inv ‖k‖ ‖k‖,
div_self (ne_of_gt <| norm_pos_iff.mpr hk), one_mul]
| Mathlib/Analysis/Seminorm.lean | 897 | 900 |
/-
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 ... |
@[reassoc (attr := simp)]
lemma Pi.map'_comp_π {f : α → C} {g : β → C} [HasProduct f] [HasProduct g] (p : β → α)
(q : ∀ (b : β), f (p b) ⟶ g b) (b : β) : Pi.map' p q ≫ Pi.π g b = Pi.π f (p b) ≫ q b :=
| Mathlib/CategoryTheory/Limits/Shapes/Products.lean | 336 | 339 |
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Mathlib.Control.Basic
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Option.Basic
im... | theorem cons_head!_tail [Inhabited α] {l : List α} (h : l ≠ []) : head! l :: tail l = l :=
cons_head?_tail (head!_mem_head? h)
theorem head!_mem_self [Inhabited α] {l : List α} (h : l ≠ nil) : l.head! ∈ l := by
| Mathlib/Data/List/Basic.lean | 460 | 463 |
/-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.GCDMonoid.Multiset
import Mathlib.Algebra.GCDMonoid.Nat
import Mathlib.Algebra.Group.TypeTags.Finite
import Mathlib.Combinatorics.Enumerative... | intro d h
simp only [Function.comp_apply, eq_comm, Finset.mem_val.mp h, exists_const]
| Mathlib/GroupTheory/Perm/Cycle/Type.lean | 87 | 88 |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Measure.Trim
import Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated
/-!
# Almost everywhere measurable functions
A funct... | Mathlib/MeasureTheory/Measure/AEMeasurable.lean | 449 | 454 | |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kenny Lau
-/
import Mathlib.Data.List.Forall2
/-!
# Lists with no duplicates
`List.Nodup` is defined in `Data/List/Basic`. In this file we prove various properties of... | ⟨fun h => attach_map_subtype_val l ▸ h.map fun _ _ => Subtype.eq, fun h =>
Nodup.of_map Subtype.val ((attach_map_subtype_val l).symm ▸ h)⟩
| Mathlib/Data/List/Nodup.lean | 227 | 229 |
/-
Copyright (c) 2021 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.CategoryTheory.Subobject.Limits
/-!
# Image-to-kernel comparison maps
Whenever `f : A ⟶ B` and `g : B ⟶ C` satisfy `w : f ≫ g = 0`,
we have `image_le_ker... | Mathlib/Algebra/Homology/ImageToKernel.lean | 295 | 298 | |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Basic
import Mathlib.RingTheory.Artinian.Module
/-!
# Lie subalgebras
This file defines Lie subalgebras of a Lie algebra and provides basic rel... | @[simp]
theorem coe_inclusion (x : K) : (inclusion h x : L) = x :=
rfl
| Mathlib/Algebra/Lie/Subalgebra.lean | 553 | 555 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.Bochner.Basic
import Mathlib.MeasureTheory.Integral.Bochner.L1
import Mathlib.Me... | Mathlib/MeasureTheory/Integral/Bochner.lean | 1,270 | 1,274 | |
/-
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.Topology.Algebra.GroupCompletion
import Mathlib.Topology.Algebra.Ring.Real
import Mathlib.Topology.MetricSpace.Algebra
import Mathlib.Topology.Me... | exact Completion.continuous_dist continuous_id continuous_id
· intro a
rw [Completion.dist_eq, dist_self]
protected theorem dist_comm (x y : Completion α) : dist x y = dist y x := by
refine induction_on₂ x y ?_ ?_
| Mathlib/Topology/MetricSpace/Completion.lean | 60 | 65 |
/-
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.FunctorN
/-!
# Comparison with the normalized Moore complex functor
In this file, we show that when the category `A` is abelian,
the... | ⟨fun _ _ hf => by
ext n
| Mathlib/AlgebraicTopology/DoldKan/Normalized.lean | 91 | 92 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Peter Nelson
-/
import Mathlib.Order.Antichain
/-!
# Minimality and Maximality
This file proves basic facts about minimality and maximality
of an element with respect to ... |
theorem minimal_iff_forall_lt : Minimal P x ↔ P x ∧ ∀ ⦃y⦄, y < x → ¬ P y := by
simp [Minimal, lt_iff_le_not_le, not_imp_not, imp.swap]
theorem maximal_iff_forall_gt : Maximal P x ↔ P x ∧ ∀ ⦃y⦄, x < y → ¬ P y :=
minimal_iff_forall_lt (α := αᵒᵈ)
| Mathlib/Order/Minimal.lean | 166 | 171 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.MvPolynomial.Variables
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Alg... | one_mul, pow_one, C_1, pow_zero, tsub_self, tsub_zero]
| Mathlib/RingTheory/WittVector/WittPolynomial.lean | 136 | 137 |
/-
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.Exact
import Mathlib.CategoryTheory.ComposableArrows
/-!
# Exact sequences
A sequence of `n` composable arrows `S : ComposableArr... | S.Exact ↔ S.toComposableArrows.Exact :=
(S.toComposableArrows.exact₂_iff S.isComplex_toComposableArrows).symm
lemma exact_iff_δ₀ (S : ComposableArrows C (n + 2)) :
S.Exact ↔ (mk₂ (S.map' 0 1) (S.map' 1 2)).Exact ∧ S.δ₀.Exact := by
constructor
· intro h
constructor
| Mathlib/Algebra/Homology/ExactSequence.lean | 241 | 248 |
/-
Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
import Mathlib.Data.ENNReal.Real
import Mathlib.Tactic.Bound.Attribute
import Mathlib.Topology... | class NNDist (α : Type*) where
| Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | 251 | 251 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.HomotopyCategory.HomComplex
import Mathlib.Algebra.Homology.HomotopyCategory.Shift
/-! Shifting cochains
Let `C` be a preadditive category. Gi... | add_right_inj, smul_smul]
congr 1
simp only [← hm', add_comm m', Int.negOnePow_add, ← mul_assoc,
Int.units_mul_self, one_mul]
· have hnm' : ¬ n' + 1 = m' := fun _ => hnm (by omega)
rw [δ_shape _ _ hnm', δ_shape _ _ hnm, rightShift_zero, smul_zero]
| Mathlib/Algebra/Homology/HomotopyCategory/HomComplexShift.lean | 408 | 414 |
/-
Copyright (c) 2023 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Geißer, Michael Stoll
-/
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.DiophantineApproximation.Basic
import Mathlib.NumberTheory.Zsqrtd.Basic
import Mathlib.Tactic... |
/-- If there is a fundamental solution, it is unique. -/
theorem subsingleton {a b : Solution₁ d} (ha : IsFundamental a) (hb : IsFundamental b) : a = b := by
have hx := le_antisymm (ha.2.2 hb.1) (hb.2.2 ha.1)
refine Solution₁.ext hx ?_
| Mathlib/NumberTheory/Pell.lean | 451 | 455 |
/-
Copyright (c) 2024 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.Data.Matroid.Constructions
import Mathlib.Data.Set.Notation
/-!
# Maps between matroids
This file defines maps and comaps, which move a matroid on one ty... |
@[simp] lemma map_id : M.map id (injOn_id M.E) = M := by
| Mathlib/Data/Matroid/Map.lean | 460 | 461 |
/-
Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
import Mathlib.Data.ENNReal.Real
import Mathlib.Tactic.Bound.Attribute
import Mathlib.Topology... | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | 1,364 | 1,367 | |
/-
Copyright (c) 2024 Emilie Burgun. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Emilie Burgun
-/
import Mathlib.Dynamics.PeriodicPts.Lemmas
import Mathlib.GroupTheory.Exponent
import Mathlib.GroupTheory.GroupAction.Basic
/-!
# Period of a group action
This modul... | theorem period_inv (g : G) (a : α) : period g⁻¹ a = period g a := by
simp only [period_eq_minimalPeriod, Function.minimalPeriod_eq_minimalPeriod_iff,
isPeriodicPt_smul_iff]
intro n
rw [smul_eq_iff_eq_inv_smul, eq_comm, ← zpow_natCast, inv_zpow, inv_inv, zpow_natCast]
| Mathlib/GroupTheory/GroupAction/Period.lean | 71 | 75 |
/-
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.Univariate.Basic
/-!
# Multivariate polynomial functors.
Multivariate polynomial... | theorem supp_eq {α : TypeVec n} (a : P.A) (f : P.B a ⟹ α) (i) :
@supp.{u} _ P.Obj _ α (⟨a, f⟩ : P α) i = f i '' univ := by
ext x; simp only [supp, image_univ, mem_range, mem_setOf_eq]
constructor <;> intro h
· apply @h fun i x => ∃ y : P.B a i, f i y = x
rw [liftP_iff']
intros
exact ⟨_, rfl⟩
· s... | Mathlib/Data/PFunctor/Multivariate/Basic.lean | 182 | 201 |
/-
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.Algebra.Homology.Embedding.IsSupported
import Mathlib.Algebra.Homology.Additive
import Mathlib.Algebra.Homology.Opposite
/-!
# The extension of a homological co... | | some _ => Iso.refl _
| none => IsZero.iso (isZero_X _ rfl) (isZero_X K rfl).op
| Mathlib/Algebra/Homology/Embedding/Extend.lean | 53 | 54 |
/-
Copyright (c) 2018 Sean Leather. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sean Leather, Mario Carneiro
-/
import Mathlib.Data.List.AList
import Mathlib.Data.Finset.Sigma
import Mathlib.Data.Part
/-!
# Finite maps over `Multiset`
-/
universe u v w
open List
... | induction_on s fun s => by simp [extract]
/-! ### union -/
/-- `s₁ ∪ s₂` is the key-based union of two finite maps. It is left-biased: if
| Mathlib/Data/Finmap.lean | 484 | 488 |
/-
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.Algebra.Ring.Regular
import Mathlib.Algebra.Equiv.TransferInstance
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.Algebra.BigOperators.Ring.Finset... |
lemma div_apply' (ψ χ : AddChar A M) (a : A) : (ψ / χ) a = ψ a / χ a := by
rw [div_apply, map_neg_eq_inv, div_eq_mul_inv]
lemma sub_apply' (ψ χ : AddChar A M) (a : A) : (ψ - χ) a = ψ a / χ a := by
rw [sub_apply, map_neg_eq_inv, div_eq_mul_inv]
@[simp] lemma zsmul_apply (n : ℤ) (ψ : AddChar A M) (a : A) : (n • ψ)... | Mathlib/Algebra/Group/AddChar.lean | 396 | 404 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Group.Defs
import Mathlib.Logic.Nontrivial.Defs
import Mathlib.Tactic.SplitIfs
import Mathlib.Logic.Basic
/-!
# Typeclasses for groups with an... | Mathlib/Algebra/GroupWithZero/Defs.lean | 309 | 309 | |
/-
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... |
theorem exists_blsub_cof (o : Ordinal) :
∃ f : ∀ a < (cof o).ord, Ordinal, blsub.{u, u} _ f = o := by
rcases exists_lsub_cof o with ⟨ι, f, hf, hι⟩
rcases Cardinal.ord_eq ι with ⟨r, hr, hι'⟩
rw [← @blsub_eq_lsub' ι r hr] at hf
rw [← hι, hι']
| Mathlib/SetTheory/Cardinal/Cofinality.lean | 308 | 314 |
/-
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.TwoDim
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
/-!
# Oriented angles.
This file defines orie... | rw [Real.Angle.sign_two_zsmul_eq_sign_iff] at hs
rcases hs with (hs | hs)
· rw [oangle_eq_pi_iff_oangle_rev_eq_pi, oangle_eq_pi_iff_sameRay_neg, neg_sub] at hs
| Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | 932 | 934 |
/-
Copyright (c) 2023 Moritz Firsching. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Firsching, Ashvni Narayanan, Michael Stoll
-/
import Mathlib.Algebra.BigOperators.Associated
import Mathlib.Data.ZMod.Basic
import Mathlib.RingTheory.Coprime.Lemmas
/-!
# Lem... | have h := Nat.dvd_sub hp ((Finset.dvd_prod_of_mem id pps).mul_right n)
rw [Nat.add_sub_cancel] at h
contradiction
| Mathlib/Data/ZMod/Units.lean | 67 | 70 |
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Analysis.Normed.Group.Int
import Mathlib.Analysis.Normed.Group.Subgroup
import Mathlib.Analysis.Normed.Group.Uniform
/-!
# Normed groups homomorphisms... | theorem ι_normNoninc : (ι f g).NormNoninc := fun _v => le_rfl
| Mathlib/Analysis/Normed/Group/Hom.lean | 842 | 843 |
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Mathlib.Control.Basic
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Option.Basic
im... | Mathlib/Data/List/Basic.lean | 1,424 | 1,426 | |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Yury Kudryashov
-/
import Mathlib.Analysis.Convex.Combination
import Mathlib.Analysis.Convex.Strict
import Mathlib.Topology.Algebra.Affine
import Mathlib.Topology.... | intro x hx y hy hne
by_cases hx' : x ∈ interior s
· exact hs.openSegment_interior_self_subset_interior hx' hy
by_cases hy' : y ∈ interior s
· exact hs.openSegment_self_interior_subset_interior hx hy'
rcases h ⟨hx, hx'⟩ ⟨hy, hy'⟩ hne with ⟨c, hc⟩
refine (openSegment_subset_union x y ⟨c, rfl⟩).trans
(in... | Mathlib/Analysis/Convex/Topology.lean | 259 | 272 |
/-
Copyright (c) 2022 Yaël Dillies, George Shakan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, George Shakan
-/
import Mathlib.Algebra.Order.Field.Rat
import Mathlib.Combinatorics.Enumerative.DoubleCounting
import Mathlib.Tactic.FieldSimp
import Mathli... | exact ruzsa_triangle_inequality_mul_div_div _ _ _
-- Auxiliary lemma towards the Plünnecke-Ruzsa inequality
@[to_additive]
private lemma card_mul_pow_le (hAB : ∀ A' ⊆ A, #(A * B) * #A' ≤ #(A' * B) * #A) (n : ℕ) :
#(A * B ^ n) ≤ (#(A * B) / #A : ℚ≥0) ^ n * #A := by
obtain rfl | hA := A.eq_empty_or_nonempty
· ... | Mathlib/Combinatorics/Additive/PluenneckeRuzsa.lean | 207 | 219 |
/-
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.Analysis.Filter
import Mathlib.Topology.Bases
import Mathlib.Topology.LocallyFinite
/-!
# Computational realization of topological spaces (experi... | theorem isClosed_iff [TopologicalSpace α] (F : Realizer α) {s : Set α} :
IsClosed s ↔ ∀ a, (∀ b, a ∈ F.F b → ∃ z, z ∈ F.F b ∩ s) → a ∈ s :=
isOpen_compl_iff.symm.trans <|
| Mathlib/Data/Analysis/Topology.lean | 127 | 129 |
/-
Copyright (c) 2019 Rohan Mitta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rohan Mitta, Kevin Buzzard, Alistair Tucker, Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.Setoid.Basic
import Mathlib.Dynamics.Fixed... |
end
/-- Banach fixed-point theorem for maps contracting on a complete subset. -/
| Mathlib/Topology/MetricSpace/Contracting.lean | 143 | 146 |
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Analysis.Normed.Group.Int
import Mathlib.Analysis.Normed.Group.Subgroup
import Mathlib.Analysis.Normed.Group.Uniform
/-!
# Normed groups homomorphisms... | theorem coe_mkNormedAddGroupHom (f : V₁ →+ V₂) (C) (hC) : ⇑(f.mkNormedAddGroupHom C hC) = f :=
rfl
| Mathlib/Analysis/Normed/Group/Hom.lean | 117 | 118 |
/-
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.Algebra.Pi
import Mathlib.Algebra.Algebra.Prod
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
impo... | theorem aevalTower_comp_algebraMap : (aevalTower g y : R[X] →+* A').comp (algebraMap R R[X]) = g :=
aevalTower_comp_C _ _
theorem aevalTower_toAlgHom (x : R) : aevalTower g y (IsScalarTower.toAlgHom S R R[X] x) = g x :=
aevalTower_algebraMap _ _ _
@[simp]
theorem aevalTower_comp_toAlgHom : (aevalTower g y).comp (... | Mathlib/Algebra/Polynomial/AlgebraMap.lean | 488 | 499 |
/-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.Topology.MetricSpace.HausdorffDistance
/-!
# Thickenings in pseudo-metric spaces
## Main definitions
* `Metric.thickening δ s`, the open thickening by ra... | rcases E.eq_empty_or_nonempty with rfl | ⟨x, hx⟩
· simp
· refine (isBounded_iff_subset_closedBall x).2 ⟨δ + diam E, fun y hy ↦ ?_⟩
| Mathlib/Topology/MetricSpace/Thickening.lean | 162 | 164 |
/-
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.Topology.Algebra.Module.LinearMapPiProd
import Mathlib.LinearAlgebra.Multilinear.Basic
/-!
# Continuous multilinear maps
We define continuous m... |
@[deprecated (since := "2024-11-03")]
| Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean | 113 | 114 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kim Morrison
-/
import Mathlib.Algebra.BigOperators.Finsupp.Basic
import Mathlib.Algebra.BigOperators.Group.Finset.Preimage
import Mathlib.Algebra.Module.Defs
import Ma... | f.subtypeDomain p = g.subtypeDomain p ↔ ∀ x, p x → f x = g x := by
simp_rw [DFunLike.ext_iff, subtypeDomain_apply, Subtype.forall]
theorem subtypeDomain_eq_iff {f g : α →₀ M}
(hf : ∀ x ∈ f.support, p x) (hg : ∀ x ∈ g.support, p x) :
f.subtypeDomain p = g.subtypeDomain p ↔ f = g :=
| Mathlib/Data/Finsupp/Basic.lean | 930 | 935 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.Algebra.MvPolynomial.Monad
/-!
## Expand multivariate polynomials
Given a multivariate polynomial `φ`, one may replace every occurre... | simp only [rename_expand, AlgHom.comp_apply]
| Mathlib/Algebra/MvPolynomial/Expand.lean | 77 | 78 |
/-
Copyright (c) 2024 Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Merten
-/
import Mathlib.CategoryTheory.Galois.GaloisObjects
import Mathlib.CategoryTheory.Limits.Shapes.CombinedProducts
import Mathlib.Data.Finite.Sum
/-!
# Decompositio... | @[simp]
private noncomputable def selfProdProj (x : F.obj X) : A ⟶ X := u ≫ Pi.π _ x
variable {u a}
| Mathlib/CategoryTheory/Galois/Decomposition.lean | 218 | 221 |
/-
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.MeasureTheory.Group.Arithmetic
import Mathlib.Topology.GDelta.UniformSpace
import Mathlib.Topology.Instances.EReal.Lemmas
import Mathl... | Measurable f := by
apply measurable_of_isOpen; intro s hs
rw [← MeasurableSet.compl_iff, ← preimage_compl]; apply hf; rw [isClosed_compl_iff]; exact hs
| Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean | 303 | 306 |
/-
Copyright (c) 2023 Yaël Dillies, Vladimir Ivanov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Vladimir Ivanov
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.BigOperators.... |
lemma truncatedInf_of_isAntichain (hs : IsAntichain (· ≤ ·) (s : Set α)) (ha : a ∈ s) :
truncatedInf s a = a := by
refine le_antisymm truncatedInf_le ?_
| Mathlib/Combinatorics/SetFamily/AhlswedeZhang.lean | 217 | 220 |
/-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.Topology.MetricSpace.HausdorffDistance
/-!
# Thickenings in pseudo-metric spaces
## Main definitions
* `Metric.thickening δ s`, the open thickening by ra... | simp_rw [thickening, infEdist_iUnion, iInf_lt_iff, setOf_exists]
lemma thickening_biUnion {ι : Type*} (δ : ℝ) (f : ι → Set α) (I : Set ι) :
| Mathlib/Topology/MetricSpace/Thickening.lean | 333 | 335 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Mario Carneiro
-/
import Mathlib.Data.Countable.Defs
import Mathlib.Data.Nat.Factors
import Mathlib.Data.Nat.Prime.Infinite
import Mathlib.Data... |
lemma prime_of_mem_primeFactors (hp : p ∈ n.primeFactors) : p.Prime := (mem_primeFactors.1 hp).1
| Mathlib/Data/Nat/PrimeFin.lean | 47 | 48 |
/-
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... | @[simp]
| Mathlib/Data/Seq/Computation.lean | 394 | 394 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Polynomial.Eval.Degree
import Mathlib.Algebra.Prime.Lemmas
/-!
# Theory of degrees of polynomials
S... | simp_rw [← Polynomial.toFinsupp_apply, toFinsupp_mul]
refine AddMonoidAlgebra.apply_add_of_supDegree_le ?_ Function.injective_id ?_ ?_
· simp
| Mathlib/Algebra/Polynomial/Degree/Lemmas.lean | 153 | 155 |
/-
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.... | natCast_re n
@[simp, rclike_simps]
theorem ofNat_im (n : ℕ) [n.AtLeastTwo] : im (ofNat(n) : K) = 0 :=
| Mathlib/Analysis/RCLike/Basic.lean | 551 | 553 |
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Scott Carnahan
-/
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Finset.MulAntidiagonal
import Mathlib... | instance [NonUnitalNonAssocRing R] : NonUnitalNonAssocRing (HahnSeries Γ R) :=
{ inferInstanceAs (NonUnitalNonAssocSemiring (HahnSeries Γ R)),
inferInstanceAs (AddGroup (HahnSeries Γ R)) with }
| Mathlib/RingTheory/HahnSeries/Multiplication.lean | 584 | 586 |
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Simon Hudon
-/
import Mathlib.Data.PFunctor.Multivariate.Basic
/-!
# Multivariate quotients of polynomial functors.
Basic definition of multivariate QPF. QPFs form a co... | end MvQPF
| Mathlib/Data/QPF/Multivariate/Basic.lean | 278 | 279 |
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Algebra.Group.Commute.Hom
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Data.Fintype.B... | Mathlib/Data/Finset/NoncommProd.lean | 489 | 494 | |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Eric Wieser
-/
import Mathlib.Algebra.BigOperators.GroupWithZero.Action
import Mathlib.Algebra.GroupWithZero.Invertible
import Mathlib.LinearAlgebra.Prod
/-!
# Trivial Square-Ze... | theorem snd_map (f : M →ₗ[R'] N) (x : TrivSqZeroExt R' M) : snd (map f x) = f (snd x) := by
simp [map, lift_def, Algebra.ofId_apply, algebraMap_eq_inl]
| Mathlib/Algebra/TrivSqZeroExt.lean | 1,066 | 1,067 |
/-
Copyright (c) 2020 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.NormedSpace.IndicatorFunction
import Mathlib.Data.Fintype.Order
import Mathlib.MeasureTheory.Function.AEEqFun
import Mathlib.Me... | refine le_antisymm (eLpNormEssSup_indicator_const_le s c) ?_
by_contra! h
have h' := ae_iff.mp (ae_lt_of_essSup_lt h)
push_neg at h'
refine hμs (measure_mono_null (fun x hx_mem => ?_) h')
| Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean | 685 | 689 |
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