Context stringlengths 227 76.5k | target stringlengths 0 11.6k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 16 3.69k |
|---|---|---|---|---|
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Asymptotics.SpecificAsympt... | Mathlib/Analysis/SpecialFunctions/CompareExp.lean | 207 | 209 | |
/-
Copyright (c) 2019 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Yury Kudryashov, YaΓ«l Dillies
-/
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Or... | Β· rw [segment_symm]
exact (segment_subset_Icc hyx).trans (hs.out hy hx)
theorem Set.OrdConnected.convex [Semiring π] [PartialOrder π]
[AddCommMonoid E] [LinearOrder E] [IsOrderedAddMonoid E] [Module π E]
[OrderedSMul π E] {s : Set E} (hs : s.OrdConnected) : Convex π s :=
hs.convex_of_chain <| isCh... | Mathlib/Analysis/Convex/Basic.lean | 560 | 569 |
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: SΓ©bastien GouΓ«zel
-/
import Mathlib.Topology.Constructions
/-!
# Neighborhoods and continuity relative to a subset
This file develops API on the relative versions
* `nhdsWithin` ... |
theorem ContinuousWithinAt.congr_set (hf : ContinuousWithinAt f s x) (h : s =αΆ [π x] t) :
ContinuousWithinAt f t x :=
(continuousWithinAt_congr_set h).1 hf
| Mathlib/Topology/ContinuousOn.lean | 726 | 729 |
/-
Copyright (c) 2021 Kalle KytΓΆlΓ€. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle KytΓΆlΓ€
-/
import Mathlib.Topology.MetricSpace.HausdorffDistance
/-!
# Thickenings in pseudo-metric spaces
## Main definitions
* `Metric.thickening Ξ΄ s`, the open thickening by ra... | obtain β¨y, hy, hxyβ© := h
exact infEdist_le_edist_add_infEdist.trans_lt
((ENNReal.add_lt_add_of_lt_of_le (hy.trans_lt ENNReal.ofReal_lt_top).ne hxy hy).trans_eq
(tsub_add_cancel_of_le <| le_self_add.trans (lt_tsub_iff_left.1 hxy).le))
| Mathlib/Topology/MetricSpace/Thickening.lean | 572 | 575 |
/-
Copyright (c) 2019 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, Mario Carneiro
-/
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Ring.Rat
import Mathlib.Algebra.Ring.Int.Parity
import Mathlib.Data.PNat.Defs... |
theorem den_div_eq_of_coprime {a b : β€} (hb0 : 0 < b) (h : Nat.Coprime a.natAbs b.natAbs) :
((a / b : β).den : β€) = b := by
lift b to β using hb0.le
| Mathlib/Data/Rat/Lemmas.lean | 189 | 192 |
/-
Copyright (c) 2022 YaΓ«l Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: YaΓ«l Dillies
-/
import Mathlib.Algebra.Order.Group.Unbundled.Basic
import Mathlib.Algebra.Order.GroupWithZero.Canonical
import Mathlib.Algebra.Order.Monoid.Units
/-!
# Ordered monoid an... | theorem strictAnti_iff_map_pos : StrictAnti (f : Ξ± β Ξ²) β β a < 0, 0 < f a :=
strictMono_comp_ofDual_iff.symm.trans <| strictMono_iff_map_pos (Ξ± := Ξ±α΅α΅) (iamhc := iamhc) _
| Mathlib/Algebra/Order/Hom/Monoid.lean | 282 | 284 |
/-
Copyright (c) 2024 Miyahara KΕ. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Miyahara KΕ
-/
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Data.List.Defs
import Mathlib.Data.Set.Function
/-!
# iterate
Proves various lemmas about `List.iterate`.
-/
variab... | theorem length_iterate (f : Ξ± β Ξ±) (a : Ξ±) (n : β) : length (iterate f a n) = n := by
induction n generalizing a <;> simp [*]
| Mathlib/Data/List/Iterate.lean | 21 | 22 |
/-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, Kim Morrison
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Algebra.Group.InjSurj
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Tactic.FastInstance
impo... | Mathlib/Data/Finsupp/Defs.lean | 1,100 | 1,106 | |
/-
Copyright (c) 2023 RΓ©my Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: RΓ©my Degenne
-/
import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap
import Mathlib.Probability.Kernel.MeasurableLIntegral
/-!
# With Density
For an s-finite kernel `ΞΊ : K... |
theorem withDensity_kernel_sum [Countable ΞΉ] (ΞΊ : ΞΉ β Kernel Ξ± Ξ²) (hΞΊ : β i, IsSFiniteKernel (ΞΊ i))
(f : Ξ± β Ξ² β ββ₯0β) :
withDensity (Kernel.sum ΞΊ) f = Kernel.sum fun i => withDensity (ΞΊ i) f := by
by_cases hf : Measurable (Function.uncurry f)
Β· ext1 a
simp_rw [sum_apply, Kernel.withDensity_apply _ hf,... | Mathlib/Probability/Kernel/WithDensity.lean | 125 | 132 |
/-
Copyright (c) 2021 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Bhavik Mehta
-/
import Mathlib.Analysis.Calculus.Deriv.Support
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
import Mathlib.MeasureTheory.Function.Jacobian
imp... | (hu : β x β Ioi a, HasDerivAt u (u' x) x) (hv : β x β Ioi a, HasDerivAt v (v' x) x)
(huv : IntegrableOn (u' * v + u * v') (Ioi a))
(h_zero : Tendsto (u * v) (π[>] a) (π a')) (h_infty : Tendsto (u * v) atTop (π b')) :
β« (x : β) in Ioi a, u' x * v x + u x * v' x = b' - a' := by
rw [β Ici_diff_left] a... | Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean | 1,264 | 1,271 |
/-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.InnerProductSpace.Adjoint
import Mathlib.Topology.Algebra.Module.Equiv
/-!
# Partially defined linear operators on Hilbert spaces
We will develop... | β¨-- Trivially, every `x : S.domain` is in `T.adjoint.domain`
fun x hx =>
mem_adjoint_domain_of_exists _
| Mathlib/Analysis/InnerProductSpace/LinearPMap.lean | 185 | 187 |
/-
Copyright (c) 2020 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.FieldTheory.RatFunc.AsPolynomial
import Mathlib.NumberTheory.ArithmeticFunction
import Mathlib.RingTheory.Ro... | induction' n using Nat.strong_induction_on with k ihk generalizing ΞΆ
rcases k.eq_zero_or_pos with (rfl | hpos)
Β· use 1
simp only [cyclotomic'_zero, coe_mapRingHom, Polynomial.map_one]
let B : K[X] := β i β Nat.properDivisors k, cyclotomic' i K
have Bmo : B.Monic := by
apply monic_prod_of_monic
int... | Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean | 180 | 194 |
/-
Copyright (c) 2024 Violeta HernΓ‘ndez Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta HernΓ‘ndez Palacios
-/
import Mathlib.SetTheory.Cardinal.Arithmetic
import Mathlib.SetTheory.Ordinal.Principal
/-!
# Ordinal arithmetic with cardinals
This file co... | Mathlib/SetTheory/Cardinal/Ordinal.lean | 211 | 213 | |
/-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Logic.Function.Defs
import Mathlib.Logic.Function.Basic
/-!
# Sigma types
This file proves basic results about sigma types.
A sigma type is a depend... | β {xβ xβ : Ξ£a, Subtype (p a)}, xβ.fst = xβ.fst β (xβ.snd : Ξ²) = xβ.snd β xβ = xβ
| β¨_, _, _β©, β¨_, _, _β©, rfl, rfl => rfl
-- This is not a good simp lemma, as its discrimination tree key is just an arrow.
theorem Β«forallΒ» {p : (Ξ£a, Ξ² a) β Prop} : (β x, p x) β β a b, p β¨a, bβ© :=
β¨fun h a b β¦ h β¨a, bβ©, fun h β¨a, ... | Mathlib/Data/Sigma/Basic.lean | 75 | 80 |
/-
Copyright (c) 2024 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.Algebra.Module.ZLattice.Covolume
import Mathlib.LinearAlgebra.Matrix.Determinant.Misc
import Mathlib.NumberTheory.NumberField.Units.DirichletTheorem
/-!... | |(Matrix.of fun i w : {w // w β w'} β¦ (mult w.val : β) *
Real.log (w.val (fundSystem K (e i) : K))).det| := by
let e' : {w : InfinitePlace K // w β wβ} β Fin (rank K) := Fintype.equivOfCardEq (by
rw [Fintype.card_subtype_compl, Fintype.card_ofSubsingleton, Fintype.card_fin, rank])
simp_rw [regulat... | Mathlib/NumberTheory/NumberField/Units/Regulator.lean | 103 | 110 |
/-
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.Analysis.InnerProductSpace.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Tactic.AdaptationNote
/-!
# Inversion in an affine sp... | div_mul_div_comm, div_mul_cancelβ _ (dist_ne_zero.2 hne), β sq, div_self, one_pow, one_smul,
vsub_vadd]
exact pow_ne_zero _ hR
theorem inversion_involutive (c : P) {R : β} (hR : R β 0) : Involutive (inversion c R) :=
inversion_inversion c hR
theorem inversion_surjective (c : P) {R : β} (hR : R β 0) ... | Mathlib/Geometry/Euclidean/Inversion/Basic.lean | 112 | 119 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Multiset.ZeroCons
/-!
# Basic results on multisets
-/
-- No algebra should be required
assert_not_exists Monoid
universe v
open List S... | Mathlib/Data/Multiset/Basic.lean | 2,979 | 2,981 | |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Convex.Between
import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathli... | exact β¨x, hxβ©
theorem le_mkMetric (m : ββ₯0β β ββ₯0β) (ΞΌ : Measure X) (Ξ΅ : ββ₯0β) (hβ : 0 < Ξ΅)
| Mathlib/MeasureTheory/Measure/Hausdorff.lean | 487 | 489 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Finset.Card
import Mathlib.Data.Finset.Lattice.Union
import Mathlib.Data.Multiset.Powerset
import Mathlib.Data.Set.Pairwise.Lattice
/-!
# The pow... | simpa only [disjiUnion_eq_biUnion] using powerset_card_disjiUnion s
theorem powersetCard_sup [DecidableEq Ξ±] (u : Finset Ξ±) (n : β) (hn : n < u.card) :
(powersetCard n.succ u).sup id = u := by
apply le_antisymm
Β· simp_rw [Finset.sup_le_iff, mem_powersetCard]
rintro x β¨h, -β©
| Mathlib/Data/Finset/Powerset.lean | 280 | 286 |
/-
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.FieldTheory.Finiteness
import Mathlib.LinearAlgebra.AffineSpace.Basis
import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
/-!
# Finite-dimensional subsp... | (hβ : pβ β line[k, pβ, pβ]) : Collinear k ({pβ, pβ, pβ, pβ} : Set P) := by
rw [collinear_insert_iff_of_mem_affineSpan
((AffineSubspace.le_def' _ _).1 (affineSpan_mono k (Set.subset_insert _ _)) _ hβ),
collinear_insert_iff_of_mem_affineSpan hβ]
exact collinear_pair _ _ _
/-- If three points lie in the... | Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean | 578 | 589 |
/-
Copyright (c) 2024 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn,
Mario Carneiro
-/
import Mathlib.Data.List.Defs
import Mathlib.Data.Option.Basic
import Mathlib.Util... | Mathlib/Data/List/GetD.lean | 66 | 66 | |
/-
Copyright (c) 2023 JoΓ«l Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: JoΓ«l Riou
-/
import Mathlib.Algebra.Homology.HomologicalComplexBiprod
import Mathlib.Algebra.Homology.Homotopy
import Mathlib.CategoryTheory.MorphismProperty.IsInvertedBy
/-! The homotopy ... | end cylinder
/-- If a functor inverts homotopy equivalences, it sends homotopic maps to the same map. -/
lemma _root_.Homotopy.map_eq_of_inverts_homotopyEquivalences
{Οβ Οβ : F βΆ G} (h : Homotopy Οβ Οβ)(hc : β j, β i, c.Rel i j)
[β i, HasBinaryBiproduct (F.X i) (F.X i)]
[HasHomotopyCofiber (biprod.lift (π... | Mathlib/Algebra/Homology/HomotopyCofiber.lean | 544 | 553 |
/-
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.Sum
import Mathlib.Data.Fintype.EquivFin
import Mathlib.Logic.Embedding.Set
/-!
## Instances
We provide the `Fintype` instance for the su... |
end right
variable [Fintype Ξ±] [Fintype Ξ²]
@[simp] lemma univ_disjSum_univ : univ.disjSum univ = (univ : Finset (Ξ± β Ξ²)) := rfl
@[simp] lemma toLeft_univ : (univ : Finset (Ξ± β Ξ²)).toLeft = univ := by ext; simp
@[simp] lemma toRight_univ : (univ : Finset (Ξ± β Ξ²)).toRight = univ := by ext; simp
end Finset
| Mathlib/Data/Fintype/Sum.lean | 47 | 57 |
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Joseph Myers
-/
import Mathlib.Analysis.InnerProductSpace.Orthogonal
import Mathlib.Analysis.Normed.Group.AddTorsor
/-!
# Perpendicular bisector of a segment
We def... |
theorem perpBisector_nonempty : (perpBisector pβ pβ : Set P).Nonempty :=
β¨_, midpoint_mem_perpBisector _ _β©
| Mathlib/Geometry/Euclidean/PerpBisector.lean | 65 | 67 |
/-
Copyright (c) 2019 Calle SΓΆnne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Calle SΓΆnne
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology... |
theorem sign_two_nsmul_eq_sign_iff {ΞΈ : Angle} :
((2 : β) β’ ΞΈ).sign = ΞΈ.sign β ΞΈ = Ο β¨ |ΞΈ.toReal| < Ο / 2 := by
by_cases hpi : ΞΈ = Ο; Β· simp [hpi]
rw [or_iff_right hpi]
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 818 | 822 |
/-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, Mario Carneiro
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Data.Int.Cast.Pi
import Mathlib.Data.Nat.Cast.Basic
import Mathlib.MeasureTheory.MeasurableSpace... | Mathlib/MeasureTheory/MeasurableSpace/Basic.lean | 982 | 985 | |
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Data.Matrix.Basic
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.Adjugate
/-!
# Matrices associated with non-dege... | Mathlib/LinearAlgebra/Matrix/Nondegenerate.lean | 72 | 74 | |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kevin Buzzard
-/
import Mathlib.Algebra.BigOperators.Field
import Mathlib.RingTheory.PowerSeries.Inverse
import Mathlib.RingTheory.PowerSeries.WellKnown
/-!
# Bernoull... | end Faulhaber
| Mathlib/NumberTheory/Bernoulli.lean | 375 | 414 |
/-
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... |
instance hasColimits : HasColimits AffineScheme.{u} :=
haveI := Adjunction.has_limits_of_equivalence.{u} Ξ.{u}
Adjunction.has_colimits_of_equivalence.{u} (opOpEquivalence AffineScheme.{u}).inverse
| Mathlib/AlgebraicGeometry/AffineScheme.lean | 195 | 198 |
/-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Topology.UniformSpace.CompactConvergence
import Mathlib.Topology.UniformSpace.Equicontinuity
import Mathlib.Topology.UniformSpace.Equiv
/-!
# Asco... | /-- Let `X` be a topological space, `π` a family of compact subsets of `X` and
`Ξ±` a uniform space. An equicontinuous subset of `X β Ξ±` is closed in the topology of uniform
convergence on all `K β π` iff it is closed in the topology of pointwise convergence on `ββ π`. -/
theorem EquicontinuousOn.isClosed_range_pi_of... | Mathlib/Topology/UniformSpace/Ascoli.lean | 369 | 382 |
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.Data.Finite.Prod
import Mathlib.Data.Matroid.Init
import Mathlib.Data.Set.Card
import Mathlib.Data.Set.Finite.Powerset
import Mathlib.Order.UpperLower.Clos... |
theorem Dep.not_indep (hD : M.Dep D) : Β¬ M.Indep D :=
hD.1
| Mathlib/Data/Matroid/Basic.lean | 555 | 557 |
/-
Copyright (c) 2024 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.AlgebraicGeometry.EllipticCurve.Group
import Mathlib.NumberTheory.EllipticDivisibilitySequence
/-!
# Division polynomials of Weier... | rw [β Nat.cast_one, Ξ¨Sq_ofNat, preΞ¨'_one, one_pow, one_mul, if_neg Nat.not_even_one]
@[simp]
| Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Basic.lean | 267 | 269 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Eric Wieser
-/
import Mathlib.Algebra.DirectSum.Internal
import Mathlib.Algebra.GradedMonoid
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Algebra.MvPolyn... | ((finSuccEquiv _ _ Ο).coeff i).IsHomogeneous j := by
intro d hd
rw [finSuccEquiv_coeff_coeff] at hd
have h' : (weight 1) (Finsupp.cons i d) = i + j := by
simpa [Finset.sum_subset_zero_on_sdiff (g := d.cons i)
(d.cons_support (y := i)) (by simp) (fun _ _ β¦ rfl), β h] using hΟ hd
simp only [weight_ap... | Mathlib/RingTheory/MvPolynomial/Homogeneous.lean | 281 | 288 |
/-
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.Limits.Constructions.LimitsOfProductsAndEqualizers
import Mathlib.CategoryTheory.Limits.FintypeCat
import Mathlib.CategoryTheory.Lim... | /-- An object is initial if and only if its fiber is empty. -/
lemma initial_iff_fiber_empty (X : C) : Nonempty (IsInitial X) β IsEmpty (F.obj X) := by
rw [(IsInitial.isInitialIffObj F X).nonempty_congr]
haveI : PreservesFiniteColimits (forget FintypeCat) := by
show PreservesFiniteColimits FintypeCat.incl
i... | Mathlib/CategoryTheory/Galois/Basic.lean | 212 | 218 |
/-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Topology.Order.IsLUB
/-!
# Monotone functions on an order topology
This file contains lemmas about limits and contin... | comp_def]
/-- An antitone function continuous at the infimum of a nonempty set sends this infimum to
the supremum of the image of this set. -/
theorem AntitoneOn.map_csInf_of_continuousWithinAt {f : Ξ± β Ξ²} {A : Set Ξ±}
| Mathlib/Topology/Order/Monotone.lean | 75 | 79 |
/-
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... |
/-- If a bounded group homomorphism map is constructed from a group homomorphism via the constructor
`NormedAddGroupHom.ofLipschitz`, then its norm is bounded by the bound given to the constructor. -/
theorem ofLipschitz_norm_le (f : Vβ β+ Vβ) {K : ββ₯0} (h : LipschitzWith K f) :
| Mathlib/Analysis/Normed/Group/Hom.lean | 258 | 261 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
import Mathlib.Data.Sum.Basic
import Mathlib.Logic.Equiv.Option
import Mathlib.Logic.Equiv.Sum
import Mathlib.Logic.Function.Conjugate
impor... | Mathlib/Logic/Equiv/Basic.lean | 1,535 | 1,537 | |
/-
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... | ((hg.comp_sub_right t).norm.const_mul _) (Eventually.of_forall fun t => L.le_opNormβ _ _)
theorem Integrable.ae_convolution_exists (hf : Integrable f Ξ½) (hg : Integrable g ΞΌ) :
βα΅ x βΞΌ, ConvolutionExistsAt f g x L Ξ½ :=
((integrable_prod_iff <|
hf.aestronglyMeasurable.convolution_integrand L hg.ae... | Mathlib/Analysis/Convolution.lean | 294 | 309 |
/-
Copyright (c) 2022 RΓ©mi Bottinelli. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: RΓ©mi Bottinelli, Junyan Xu
-/
import Mathlib.Algebra.Group.Subgroup.Defs
import Mathlib.CategoryTheory.Groupoid.VertexGroup
import Mathlib.CategoryTheory.Groupoid.Basic
import Mathlib... | constructor; exact Sn.wide c
conj := fun {d d'} g Ξ΄ hΞ΄ => by
rw [mem_map_iff] at hΞ΄
obtain β¨c, c', Ξ³, cd, cd', Ξ³S, hΞ³β© := hΞ΄; subst_vars; cases hΟ cd'
have : d' β (im Ο hΟ).objs := by rw [hΟ']; apply mem_top_objs
rw [mem_im_objs_iff] at this
| Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean | 504 | 509 |
/-
Copyright (c) 2019 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, Mario Carneiro
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.Field.Rat
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Algebra.GroupWithZero.Un... | β(p * q) = (p * q : Ξ±) := by
rw [mul_eq_mkRat, cast_mkRat_of_ne_zero, cast_def, cast_def,
(Nat.commute_cast _ _).div_mul_div_comm (Int.commute_cast _ _)]
Β· push_cast
rfl
Β· push_cast
exact mul_ne_zero hp hq
@[norm_cast]
| Mathlib/Data/Rat/Cast/Defs.lean | 179 | 187 |
/-
Copyright (c) 2019 Gabriel Ebner. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Gabriel Ebner, Anatole Dedecker, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.FDeriv.Add
... | HasStrictDerivAt (fun x => c x / d) (c' / d) x := by
simpa only [div_eq_mul_inv] using hc.mul_const dβ»ΒΉ
| Mathlib/Analysis/Calculus/Deriv/Mul.lean | 394 | 396 |
/-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, Kim Morrison
-/
import Mathlib.Algebra.BigOperators.Finsupp.Basic
import Mathlib.Algebra.BigOperators.Group.Finset.Preimage
import Mathlib.Algebra.Module.Defs
import Ma... | Β· exact β¨f, Multiset.mem_cons_self _ _, hβ©
Β· simp only [Multiset.sum_cons, mem_support_iff, add_apply, not_mem_support_iff.1 h,
zero_add] at ha
rcases ih (mem_support_iff.2 ha) with β¨f', hβ, hββ©
exact β¨f', Multiset.mem_cons_of_mem hβ, hββ©)
theorem mem_support_finset_sum [AddCommMo... | Mathlib/Data/Finsupp/Basic.lean | 1,040 | 1,046 |
/-
Copyright (c) 2022 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, Kenny Lau, Johan Commelin, Mario Carneiro, Kevin Buzzard,
Amelia Livingston, Yury Kudryashov, Yakov Pechersky, Jireh Loreaux
-/
import Mathlib.Algebra.Group.Prod
impor... | end MulHom
| Mathlib/Algebra/Group/Subsemigroup/Operations.lean | 666 | 667 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Vector.Defs
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.OfFn
import Mathlib.Data.List.Scan
import Mathlib.Control.Applicative
import M... | (v.map f)[i] = f v[i] := by
simp only [getElem_def, toList_map, List.getElem_map]
| Mathlib/Data/Vector/Basic.lean | 106 | 108 |
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, SΓ©bastien GouΓ«zel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.TangentCone
import Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics
import Mathlib.Analysis.As... | simp only [fderivWithin, dif_pos h]
split_ifs with hβ
exacts [hβ, Classical.choose_spec h]
theorem DifferentiableAt.hasFDerivAt (h : DifferentiableAt π f x) :
| Mathlib/Analysis/Calculus/FDeriv/Basic.lean | 552 | 556 |
/-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, Yury Kudryashov
-/
import Mathlib.Data.ENNReal.Real
/-!
# Properties of addition, multiplication and subtraction on extended non-negative real numbers
In this file we... | protected lemma pow_right_strictMono {n : β} (hn : n β 0) : StrictMono fun a : ββ₯0β β¦ a ^ n :=
WithTop.pow_right_strictMono hn
@[gcongr] protected lemma pow_lt_pow_left (hab : a < b) {n : β} (hn : n β 0) : a ^ n < b ^ n :=
WithTop.pow_lt_pow_left hab hn
-- TODO: generalize to `WithTop`
theorem mul_left_strictMono... | Mathlib/Data/ENNReal/Operations.lean | 33 | 41 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Fintype.EquivFin
import Mathlib.Data.List.MinMax
import Mathlib.Data.Nat.Order.Lemmas
import Mathlib.Logic.Encodable.Basic
/-!
# Denumerable type... |
/-- A denumerable type is equivalent to `β`. -/
def eqv (Ξ±) [Denumerable Ξ±] : Ξ± β β :=
| Mathlib/Logic/Denumerable.lean | 66 | 68 |
/-
Copyright (c) 2020 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Data.Part
import Mathlib.Data.Nat.Find
import Mathlib.Data.Nat.Upto
import Mathlib.Data.Stream.Defs
import Mathlib.Tactic.Common
/-!
# Fixed point
This mod... | rw [hk] at this
revert hk
dsimp [Part.fix]; rw [assert_pos h']; revert this
generalize Upto.zero = z; intro _this hk
suffices β x' hwf,
WellFounded.fix hwf (fixAux f) z x' = Fix.approx f (succ k) x'
from this _ _
induction k generalizing z with
| zero =>
intro x' _
rw [Fix.approx, WellFoun... | Mathlib/Control/Fix.lean | 74 | 108 |
/-
Copyright (c) 2020 FrΓ©dΓ©ric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: FrΓ©dΓ©ric Dupuis, Eric Wieser
-/
import Mathlib.LinearAlgebra.Multilinear.TensorProduct
import Mathlib.Tactic.AdaptationNote
import Mathlib.LinearAlgebra.Multilinear.Curry
/-!
# Tenso... |
Note that `PiTensorProduct.piTensorHomMap (tprod R f)` is equal to `PiTensorProduct.map f`.
-/
def piTensorHomMap : (β¨[R] i, s i ββ[R] t i) ββ[R] (β¨[R] i, s i) ββ[R] β¨[R] i, t i :=
lift.toLinearMap ββ lift (MultilinearMap.piLinearMap <| tprod R)
| Mathlib/LinearAlgebra/PiTensorProduct.lean | 590 | 594 |
/-
Copyright (c) 2023 JoΓ«l Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: JoΓ«l Riou
-/
import Mathlib.Algebra.Homology.Homotopy
import Mathlib.Algebra.Ring.NegOnePow
import Mathlib.Algebra.Category.Grp.Preadditive
import Mathlib.Tactic.Linarith
import Mathlib.Cat... | Cochain.ofHom (homOf z) = (z : Cochain F G 0) := by
simpa only [Cocycle.ext_iff] using ofHom_homOf_eq_self z
| Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean | 680 | 681 |
/-
Copyright (c) 2016 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, Mario Carneiro
-/
import Mathlib.Data.Set.Defs
import Mathlib.Logic.Basic
import Mathlib.Logic.ExistsUnique
import Mathlib.Logic.Nonempty
import Mathlib.Logic.Nontrivia... | (hf.factorsThrough g).extend_apply e' a
@[simp]
theorem extend_apply' (g : Ξ± β Ξ³) (e' : Ξ² β Ξ³) (b : Ξ²) (hb : Β¬β a, f a = b) :
| Mathlib/Logic/Function/Basic.lean | 691 | 694 |
/-
Copyright (c) 2022 YaΓ«l Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: YaΓ«l Dillies, Bhavik Mehta
-/
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Algebra.Order.Chebyshev
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Math... | scoped macro "sz_positivity" : tactic =>
`(tactic|
{ try have := m_pos βΉ_βΊ
| Mathlib/Combinatorics/SimpleGraph/Regularity/Bound.lean | 78 | 80 |
/-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, Mario Carneiro, Patrick Massot, Yury Kudryashov, RΓ©my Degenne
-/
import Mathlib.Data.Set.Subsingleton
import Mathlib.Order.Interval.Set.Defs
/-!
# Intervals
In any pr... | Mathlib/Order/Interval/Set/Basic.lean | 1,957 | 1,959 | |
/-
Copyright (c) 2020 Kevin Buzzard, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Equalizers
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Products
import Mathl... | Mathlib/CategoryTheory/Sites/Sheaf.lean | 755 | 759 | |
/-
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.Topology.UniformSpace.CompleteSeparated
import Mathlib.Topology.EMetricSpace.Lipschitz
import Mathlib.Topology.MetricSpace.Basic
import Mathlib.Topol... | (hf.restrict univ).to_rightInvOn' (mapsTo_univ g t) h
theorem to_rightInverse (hf : AntilipschitzWith K f) {g : Ξ² β Ξ±} (hg : Function.RightInverse g f) :
LipschitzWith K g := by
intro x y
have := hf (g x) (g y)
| Mathlib/Topology/MetricSpace/Antilipschitz.lean | 129 | 134 |
/-
Copyright (c) 2023 JoΓ«l Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: JoΓ«l Riou
-/
import Mathlib.CategoryTheory.CatCommSq
import Mathlib.CategoryTheory.Localization.Predicate
import Mathlib.CategoryTheory.Adjunction.FullyFaithful
/-!
# Localization of adjun... | @[simp]
lemma localization_counit_app (Xβ : Cβ) :
(adj.localization Lβ Wβ Lβ Wβ G' F').counit.app (Lβ.obj Xβ) =
G'.map ((CatCommSq.iso F Lβ Lβ F').inv.app Xβ) β«
(CatCommSq.iso G Lβ Lβ G').inv.app (F.obj Xβ) β«
Lβ.map (adj.counit.app Xβ) := by
apply Localization.Ξ·_app
| Mathlib/CategoryTheory/Localization/Adjunction.lean | 120 | 126 |
/-
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 SΓΆnne, SΓ©bastien GouΓ«zel,
RΓ©my Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
/-!
# Power function... | theorem rpow_pos {p : β} {x : ββ₯0β} (hx_pos : 0 < x) (hx_ne_top : x β β€) : 0 < x ^ p := by
rcases lt_or_le 0 p with hp_pos | hp_nonpos
Β· exact rpow_pos_of_nonneg hx_pos (le_of_lt hp_pos)
Β· rw [β neg_neg p, rpow_neg, ENNReal.inv_pos]
| Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | 815 | 818 |
/-
Copyright (c) 2021 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kim Morrison
-/
import Mathlib.Algebra.Homology.ComplexShape
import Mathlib.CategoryTheory.Subobject.Limits
import Mathlib.CategoryTheory.GradedObject
import Mathlib.Alge... | Cβ = Cβ := by
obtain β¨Xβ, dβ, sβ, hββ© := Cβ
obtain β¨Xβ, dβ, sβ, hββ© := Cβ
dsimp at h_X
subst h_X
simp only [mk.injEq, heq_eq_eq, true_and]
ext i j
by_cases hij : c.Rel i j
Β· simpa only [comp_id, id_comp, eqToHom_refl] using h_d i j hij
Β· rw [sβ i j hij, sβ i j hij]
/-- The obvious isomorphism `K.... | Mathlib/Algebra/Homology/HomologicalComplex.lean | 79 | 92 |
/-
Copyright (c) 2020 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Data.Stream.Init
import Mathlib.Tactic.ApplyFun
import Mathlib.Control.Fix
import Mathlib.Order.OmegaCompletePartialOrder
/-!
# Lawful fixed point operators... | namespace Pi
instance lawfulFix {Ξ²} : LawfulFix (Ξ± β Part Ξ²) :=
β¨fun {_f} β¦ Part.fix_eq_of_ΟScottContinuousβ©
| Mathlib/Control/LawfulFix.lean | 203 | 207 |
/-
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.Topology.UniformSpace.CompleteSeparated
import Mathlib.Topology.EMetricSpace.Lipschitz
import Mathlib.Topology.MetricSpace.Basic
import Mathlib.Topol... | antilipschitzWith_iff_le_mul_nndist
theorem antilipschitzWith_iff_le_mul_dist :
AntilipschitzWith K f β β x y, dist x y β€ K * dist (f x) (f y) := by
| Mathlib/Topology/MetricSpace/Antilipschitz.lean | 53 | 56 |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Neil Strickland
-/
import Mathlib.Data.Nat.Prime.Defs
import Mathlib.Data.PNat.Basic
/-!
# Primality and GCD on pnat
This file extends the theory of `β+` with ... | @[simp]
theorem one_gcd {n : β+} : gcd 1 n = 1 := by
rw [gcd_eq_left_iff_dvd]
apply one_dvd
| Mathlib/Data/PNat/Prime.lean | 204 | 207 |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes HΓΆlzl, Damiano Testa,
Yuyang Zhao
-/
import Mathlib.Algebra.Order.Monoid.Unbundled.Defs
import Mathlib.Data.Ordering.Basic
imp... |
@[to_additive (attr := simp) le_add_iff_nonneg_left]
theorem le_mul_iff_one_le_left' [MulRightMono Ξ±]
[MulRightReflectLE Ξ±] (a : Ξ±) {b : Ξ±} :
| Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean | 458 | 461 |
/-
Copyright (c) 2023 Sophie Morel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sophie Morel
-/
import Mathlib.Analysis.Analytic.Constructions
import Mathlib.Analysis.Analytic.CPolynomialDef
/-! # Properties of continuously polynomial functions
We expand the API a... | Mathlib/Analysis/Analytic/CPolynomial.lean | 239 | 244 | |
/-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Order.Filter.Interval
import Mathlib.Order.Interval.Set.Pi
import Mathlib.Tactic.TFAE
import Mathlib.Tactic.NormNum
im... | instance tendstoIcoClassNhds [OrderTopology Ξ±] (a : Ξ±) : TendstoIxxClass Ico (π a) (π a) :=
tendstoIxxClass_of_subset fun _ _ => Ico_subset_Icc_self
| Mathlib/Topology/Order/Basic.lean | 126 | 128 |
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.RingTheory.HahnSeries.Multiplication
import Mathlib.RingTheory.PowerSeries.Basic
import Mathlib.RingTheory.MvPowerSeries.NoZeroDivisors
import Mathlib.... | Β· rw [embDomain_notin_image_support]
simp only [not_exists, Set.mem_image, toPowerSeries_symm_apply_coeff, mem_support,
PowerSeries.coeff_C]
intro
simp +contextual [Ne.symm hn]
@[simp]
theorem ofPowerSeries_X : ofPowerSeries Ξ R PowerSeries.X = single 1 1 := by
ext n
simp only [coeff_single, of... | Mathlib/RingTheory/HahnSeries/PowerSeries.lean | 117 | 128 |
/-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston
-/
import Mathlib.Algebra.Group.Submonoid.Operations
import Mathlib.Data.Setoid.Basic
import Mathlib.GroupTheory.Congruence.Hom
/-!
# Congruence relations
This f... | Mathlib/GroupTheory/Congruence/Basic.lean | 501 | 511 | |
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Mario Carneiro, Simon Hudon
-/
import Mathlib.Data.PFunctor.Multivariate.Basic
import Mathlib.Data.PFunctor.Univariate.M
/-!
# The M construction as a multivariate polyn... | Mathlib/Data/PFunctor/Multivariate/M.lean | 327 | 332 | |
/-
Copyright (c) 2021 Apurva Nakade. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Apurva Nakade
-/
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Ring.Regular
import Mathlib.GroupTheory.MonoidLocalization.Away
impo... | Mathlib/SetTheory/Surreal/Dyadic.lean | 272 | 276 | |
/-
Copyright (c) 2019 SΓ©bastien GouΓ«zel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: SΓ©bastien GouΓ«zel
-/
import Mathlib.Analysis.Analytic.Within
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries
/-!
# Higher d... | constructor
Β· intro h
obtain β¨u, H, p, hpβ© := h 0 le_rfl
refine β¨u, ?_, ?_β©
Β· simpa [hx] using H
Β· simp only [Nat.cast_zero, hasFTaylorSeriesUpToOn_zero_iff] at hp
exact hp.1.mono inter_subset_right
Β· rintro β¨u, H, huβ©
rw [β contDiffWithinAt_inter' H]
have h' : x β s β© u := β¨hx, mem_... | Mathlib/Analysis/Calculus/ContDiff/Defs.lean | 610 | 627 |
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv
/-! # Quadratic form on product and pi types
## Main definitions
* `QuadraticForm.prod Qβ Qβ`: the quadratic for... | obtain β¨haβ, haββ© := anisotropic_of_prod ha
exact β¨β¨hleβ, haββ©, β¨hleβ, haββ©β©
Β· rintro β¨β¨hleβ, haββ©, β¨hleβ, haββ©β©
refine β¨β¨hleβ, hleββ©, ?_β©
rintro β¨xβ, xββ© (hx : Qβ xβ + Qβ xβ = 0)
rw [add_eq_zero_iff_of_nonneg (hleβ xβ) (hleβ xβ), haβ.eq_zero_iff, haβ.eq_zero_iff] at hx
rwa [Prod.mk_eq_zero]
... | Mathlib/LinearAlgebra/QuadraticForm/Prod.lean | 163 | 175 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Simon Hudon, Mario Carneiro
-/
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Init
import Mathlib.Data.Int.Init
import Mathlib.... | /-- To show a property of all powers of `g` it suffices to show it is closed under multiplication
| Mathlib/Algebra/Group/Basic.lean | 900 | 900 |
/-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Aesop
import Mathlib.Order.BoundedOrder.Lattice
/-!
# Disjointness and complements
This file defines `Disjoint`, `Codisjoint`, and the `IsCompl` predicate.
... | theorem eq_top_of_isCompl_bot (h : IsCompl x β₯) : x = β€ := by rw [β sup_bot_eq x, h.sup_eq_top]
theorem eq_top_of_bot_isCompl (h : IsCompl β₯ x) : x = β€ :=
| Mathlib/Order/Disjoint.lean | 548 | 550 |
/-
Copyright (c) 2024 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.Data.Matroid.Minor.Restrict
/-!
# Some constructions of matroids
This file defines some very elementary examples of matroids, namely those with at most o... | simp [h, eq_freeOn_iff, Subset.rfl]
| Mathlib/Data/Matroid/Constructions.lean | 181 | 182 |
/-
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
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Order.CauSeq.BigOperators
import Mathlib.Algebra.Order.Star.Basic
import Mathlib.Data.C... | open scoped ComplexConjugate
namespace Complex
theorem isCauSeq_norm_exp (z : β) :
IsCauSeq abs fun n => β m β range n, βz ^ m / m.factorialβ :=
let β¨n, hnβ© := exists_nat_gt βzβ
have hn0 : (0 : β) < n := lt_of_le_of_lt (norm_nonneg _) hn
IsCauSeq.series_ratio_test n (βzβ / n) (div_nonneg (norm_nonneg _) (le... | Mathlib/Data/Complex/Exponential.lean | 27 | 36 |
/-
Copyright (c) 2023 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.GroupTheory.CoprodI
import Mathlib.GroupTheory.Coprod.Basic
import Mathlib.GroupTheory.Complement
/-!
## Pushouts of Monoids and Groups
This file defin... | MulAction.ofEndHom <|
lift
(fun _ => MulAction.toEndHom)
MulAction.toEndHom <| by
intro i
| Mathlib/GroupTheory/PushoutI.lean | 453 | 457 |
/-
Copyright (c) 2020 SΓ©bastien GouΓ«zel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: SΓ©bastien GouΓ«zel, Floris van Doorn
-/
import Mathlib.Geometry.Manifold.MFDeriv.FDeriv
/-!
# Differentiability of specific functions
In this file, we establish differentiability r... |
theorem mdifferentiableAt_snd {x : M Γ M'} : MDifferentiableAt (I.prod I') I' Prod.snd x :=
(hasMFDerivAt_snd x).mdifferentiableAt
theorem mdifferentiableWithinAt_snd {s : Set (M Γ M')} {x : M Γ M'} :
MDifferentiableWithinAt (I.prod I') I' Prod.snd s x :=
mdifferentiableAt_snd.mdifferentiableWithinAt
theorem... | Mathlib/Geometry/Manifold/MFDeriv/SpecificFunctions.lean | 300 | 316 |
/-
Copyright (c) 2023 Kalle KytΓΆlΓ€. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle KytΓΆlΓ€
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Topology.Separation.Basic
/-!
# Pointwise convergence of indicator functions
In this file, we prove the equivalenc... |
lemma tendsto_indicator_const_iff_forall_eventually'
(b : Ξ²) (nhd_b : {0}αΆ β π b) (nhd_o : {b}αΆ β π 0) :
Tendsto (fun i β¦ (As i).indicator (fun (_ : Ξ±) β¦ b)) L (π (A.indicator (fun (_ : Ξ±) β¦ b)))
β β x, βαΆ i in L, (x β As i β x β A) := by
simp_rw [tendsto_pi_nhds]
apply forall_congr'
| Mathlib/Topology/IndicatorConstPointwise.lean | 82 | 88 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes HΓΆlzl
-/
import Mathlib.Data.List.Basic
/-!
# Double universal quantification on a list
This file provides an API for `List.Forallβ` (definition in `Data.Lis... | rwa [drop_left] at h'
theorem rel_mem (hr : BiUnique R) : (R β Forallβ R β Iff) (Β· β Β·) (Β· β Β·)
| a, b, _, [], [], Forallβ.nil => by simp only [not_mem_nil]
| a, b, h, a' :: as, b' :: bs, Forallβ.cons hβ hβ => by
simp only [mem_cons]
| Mathlib/Data/List/Forall2.lean | 195 | 200 |
/-
Copyright (c) 2020 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap
import Mathlib.MeasureTheory.Integral.Bochner.FundThmCalculus
import Mathlib.MeasureT... | Mathlib/MeasureTheory/Integral/SetIntegral.lean | 423 | 440 | |
/-
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 HΓΆlzl, Mario Carneiro, SΓ©bastien GouΓ«zel
-/
import Mathlib.Topology.Order.Compact
import Mathlib.Topology.MetricSpace.ProperSpace
import M... | (atTop_basis.comap _).eq_of_same_basis <| by
simpa only [compl_def, mem_ball, not_lt] using hasBasis_cobounded_compl_ball c
| Mathlib/Topology/MetricSpace/Bounded.lean | 119 | 121 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.Lattice
import Mathlib.RingTheory.Coprime.Lemmas
import Mathlib... | le_antisymm (smul_le.mpr fun _m hm _n β¦ Submodule.apply_mem_mapβ _ hm)
(mapβ_le.mpr fun _m hm _n β¦ smul_mem_smul hm)
theorem span_smul_span : Ideal.span S β’ span R T = span R (β (s β S) (t β T), {s β’ t}) := by
rw [smul_eq_mapβ]
exact (mapβ_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _... | Mathlib/RingTheory/Ideal/Operations.lean | 142 | 148 |
/-
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 (f β« g) (f' β« g') := by
cases eq1; cases eq2; cases eq3; cases H1; cases H2; rfl
variable {Ξ² : Sort*}
| Mathlib/CategoryTheory/EqToHom.lean | 116 | 119 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes HΓΆlzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Group.Submonoid.Operations
import Mathlib.Algebra.MonoidAlgebra.Defs
import Mathlib.Algebra.Order.Mon... | exact β¨n, a, Polynomial.ofFinsupp_single _ _β©
theorem addHom_ext {M : Type*} [AddZeroClass M] {f g : R[X] β+ M}
| Mathlib/Algebra/Polynomial/Basic.lean | 726 | 728 |
/-
Copyright (c) 2021 Alena Gusakov, Bhavik Mehta, Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alena Gusakov, Bhavik Mehta, Kyle Miller
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Fintype.Powerset
import Mathlib.Data.Set.Finite.Basic
/-!
#... | (β s' : Finset ΞΉ', #s' β€ #(s'.biUnion t')) β
β f : ΞΉ' β Ξ±, Function.Injective f β§ β x, f x β t' x)
(s : Finset ΞΉ) (hs : s.Nonempty) (hns : s β univ) (hus : #s = #(s.biUnion t)) :
β f : ΞΉ β Ξ±, Function.Injective f β§ β x, f x β t x := by
haveI := Classical.decEq ΞΉ
-- Restrict to `s`
rw... | Mathlib/Combinatorics/Hall/Finite.lean | 166 | 215 |
/-
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 SΓΆnne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | ((continuous_im.tendsto _).eventually (lt_mem_nhds hz)).mono fun _ => arg_of_im_pos
theorem arg_eq_nhds_of_im_neg (hz : im z < 0) : arg =αΆ [π z] fun x => -Real.arccos (x.re / βxβ) :=
| Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean | 563 | 565 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Multiset.ZeroCons
/-!
# Basic results on multisets
-/
-- No algebra should be required
assert_not_exists Monoid
universe v
open List S... | Mathlib/Data/Multiset/Basic.lean | 2,914 | 2,922 | |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.NaturalTransformation
import Mathlib.CategoryTheory.Monoidal.Opposite
import Mathli... | theorem leftUnitor_inv_braiding (X : C) : (Ξ»_ X).inv β« (Ξ²_ (π_ C) X).hom = (Ο_ X).inv := by
simp
| Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean | 332 | 333 |
/-
Copyright (c) 2023 Paul Reichert. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Paul Reichert, YaΓ«l Dillies
-/
import Mathlib.Analysis.Normed.Affine.AddTorsorBases
/-!
# Intrinsic frontier and interior
This file defines the intrinsic frontier, interior and closur... | end AddTorsor
namespace AffineIsometry
| Mathlib/Analysis/Convex/Intrinsic.lean | 202 | 204 |
/-
Copyright (c) 2021 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Alena Gusakov, YaΓ«l Dillies
-/
import Mathlib.Data.Finset.Grade
import Mathlib.Data.Finset.Sups
import Mathlib.Logic.Function.Iterate
/-!
# Shadows
This file defines shad... | Mathlib/Combinatorics/SetFamily/Shadow.lean | 327 | 331 | |
/-
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.Ordmap.Invariants
/-!
# Verification of `Ordnode`
This file uses the invariants defined in `Mathlib.Data.Ordmap.Invariants` to construct `Ordset... | Mathlib/Data/Ordmap/Ordset.lean | 1,566 | 1,594 | |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Mario Carneiro
-/
import Mathlib.Data.Subtype
import Mathlib.Order.Defs.LinearOrder
import Mathlib.Order.Notation
import Mathlib.Tactic.GCongr.Core
import Mathlib.Tactic.... | theorem mk_lt_mk [LT Ξ±] {p : Ξ± β Prop} {x y : Ξ±} {hx : p x} {hy : p y} :
| Mathlib/Order/Basic.lean | 1,009 | 1,009 |
/-
Copyright (c) 2018 SΓ©bastien GouΓ«zel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: SΓ©bastien GouΓ«zel, Johannes HΓΆlzl, RΓ©my Degenne
-/
import Mathlib.Order.ConditionallyCompleteLattice.Indexed
import Mathlib.Order.Filter.IsBounded
import Mathlib.Order.Hom.CompleteL... | Mathlib/Order/LiminfLimsup.lean | 1,203 | 1,205 | |
/-
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.Topology.Homeomorph.Lemmas
import Mathlib.Topology.StoneCech
/-!
# Extremally disconnected spaces
An extremally disconnected topological space is a s... | Β· simpa only [G_empty, image_empty] using empty_subset _
Β· -- let $a \in \rho(G)$
intro a ha
rw [mem_closure_iff]
-- let $N$ be a neighbourhood of $a$
intro N N_open hN
-- get $x \in A$ from nonempty open $G \cap \rho^{-1}(N)$
rcases (G.mem_image Ο a).mp ha with β¨e, he, rflβ©
have nonempt... | Mathlib/Topology/ExtremallyDisconnected.lean | 182 | 208 |
/-
Copyright (c) 2023 Xavier GΓ©nΓ©reux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier GΓ©nΓ©reux
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
import Mathlib.Analysis.Complex.PhragmenLindelof
/-!
# Hadamard three-lines Theorem
In this file we present a ... |
/-- Alternative version of `norm_le_sSupNormIm` with a strict inequality and a positive `Ξ΅`. -/
lemma norm_lt_sSupNormIm_eps (f : β β E) (Ξ΅ : β) (hΞ΅ : Ξ΅ > 0) (z : β)
(hD : z β verticalClosedStrip 0 1) (hB : BddAbove ((norm β f) '' verticalClosedStrip 0 1)) :
βf zβ < Ξ΅ + sSupNormIm f (z.re) :=
lt_add_of_pos_o... | Mathlib/Analysis/Complex/Hadamard.lean | 140 | 178 |
/-
Copyright (c) 2021 Stuart Presnell. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stuart Presnell
-/
import Mathlib.Algebra.Order.Ring.Canonical
import Mathlib.Algebra.BigOperators.Group.List.Defs
/-!
# Big operators on a list in ordered rings
This file contains ... | [CommSemiring Ξ±] [PartialOrder Ξ±] [CanonicallyOrderedAdd Ξ±] [NoZeroDivisors Ξ±] [Nontrivial Ξ±] :
β {l : List Ξ±}, 0 < l.prod β (β x β l, (0 : Ξ±) < x)
| [] => by simp
| (x :: xs) => by simp_rw [List.prod_cons, List.forall_mem_cons, CanonicallyOrderedAdd.mul_pos,
list_prod_pos]
| Mathlib/Algebra/Order/BigOperators/Ring/List.lean | 19 | 26 |
/-
Copyright (c) 2024 Jack McKoen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jack McKoen
-/
import Mathlib.CategoryTheory.Limits.FunctorCategory.Basic
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Limits.Types.Limits
impo... | right_inv _ := by simp [prodMk]
@[ext]
lemma prod_ext' (a : C) (z w : (F β¨― G).obj a)
(h1 : (Limits.prod.fst (X := F)).app a z = (Limits.prod.fst (X := F)).app a w)
(h2 : (Limits.prod.snd (X := F)).app a z = (Limits.prod.snd (X := F)).app a w) :
z = w := by
| Mathlib/CategoryTheory/Limits/Shapes/FunctorToTypes.lean | 150 | 156 |
/-
Copyright (c) 2024 Violeta HernΓ‘ndez Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta HernΓ‘ndez Palacios
-/
import Mathlib.SetTheory.Cardinal.Arithmetic
import Mathlib.SetTheory.Ordinal.Principal
/-!
# Ordinal arithmetic with cardinals
This file co... | Mathlib/SetTheory/Cardinal/Ordinal.lean | 1,227 | 1,234 | |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Jakob von Raumer
-/
import Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
/-!
# Biproducts and binary biproducts
... | theorem biproduct.ΞΉ_toSubtype_subtype (j : Subtype p) :
biproduct.ΞΉ f j β« biproduct.toSubtype f p = biproduct.ΞΉ (Subtype.restrict p f) j := by
classical
ext
rw [biproduct.toSubtype, Category.assoc, biproduct.lift_Ο, biproduct.ΞΉ_Ο, biproduct.ΞΉ_Ο]
split_ifs with hβ hβ hβ
exacts [rfl, False.elim (hβ (Subtype... | Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean | 770 | 779 |
/-
Copyright (c) 2018 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Mario Carneiro, Kim Morrison, Floris van Doorn
-/
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryTheory.Limits.Cones
import Batteries.Tactic.Congr
/-... | congr
rw [coneOfHom_fac]
dsimp [Cone.extend]; cases s; congr with j; exact w j
end
| Mathlib/CategoryTheory/Limits/IsLimit.lean | 492 | 497 |
/-
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.SetTheory.Cardinal.Arithmetic
/-!
# Cardinality of continuum
In this file we define `Cardinal.continuum` (notation: `π `, localized in `Cardinal`) ... | Mathlib/SetTheory/Cardinal/Continuum.lean | 208 | 209 |
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