Context stringlengths 227 76.5k | target stringlengths 0 11.6k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 16 3.69k |
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/-
Copyright (c) 2024 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.Data.EReal.Basic
import Mathlib.NumberTheory.LSeries.Basic
/-!
# Convergence of L-series
We define `LSeries.abscissaOfAbsConv f` (as an `EReal`) to be ... | abscissaOfAbsConv f ≤ s.re :=
sInf_le <| by simpa using h.of_re_le_re (by simp)
lemma LSeries.abscissaOfAbsConv_le_of_forall_lt_LSeriesSummable {f : ℕ → ℂ} {x : ℝ}
(h : ∀ y : ℝ, x < y → LSeriesSummable f y) :
| Mathlib/NumberTheory/LSeries/Convergence.lean | 55 | 59 |
/-
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.Nat.ModEq
import Mathlib.Data.Nat.Prime.Basic
import Mathlib.NumberTheory.Zsqrtd.Basic
/-!
# Pell's equation and Matiyasevic's theorem
This file... |
end
| Mathlib/NumberTheory/PellMatiyasevic.lean | 77 | 79 |
/-
Copyright (c) 2021 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Logic.Function.Basic
import Mathlib.Logic.Relator
/-!
# Types that are empty
In this file we define a typeclass `IsEmpty`, which expresses that a... | variable {α β : Type*} (R : α → β → Prop)
| Mathlib/Logic/IsEmpty.lean | 211 | 212 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Sébastien Gouëzel,
Rémy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
/-!
# Limits and a... | lemma tendsto_rpow_atBot_of_base_gt_one (b : ℝ) (hb : 1 < b) :
Tendsto (b ^ · : ℝ → ℝ) atBot (𝓝 0) := by
simp_rw [Real.rpow_def_of_pos (by positivity : 0 < b)]
refine tendsto_exp_atBot.comp <| (tendsto_const_mul_atBot_iff_pos <| tendsto_id (α := ℝ)).mpr ?_
exact (log_pos_iff (by positivity)).mpr <| by aesop
| Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean | 81 | 85 |
/-
Copyright (c) 2023 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Algebra.NonUnitalHom
import Mathlib.Data.Set.UnionLift
import Mathlib.LinearAlgebra.Span.Basic
import Mathlib.RingTheory.NonUnitalSubring.Basic
... | ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=
(NonUnitalSubalgebra.gc_map_comap f).l_sup
| Mathlib/Algebra/Algebra/NonUnitalSubalgebra.lean | 750 | 752 |
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.Order.LeftRight
import Mathlib.Topology.Order.Monotone
/-!
# Left and right limits
We define the (strict) left and right limits of a f... | rw [leftLim_eq_sSup hf h']
exact hf.tendsto_nhdsLT x
theorem tendsto_leftLim_within (x : α) : Tendsto f (𝓝[<] x) (𝓝[≤] leftLim f x) := by
apply tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within f (hf.tendsto_leftLim x)
filter_upwards [@self_mem_nhdsWithin _ _ x (Iio x)] with y hy using hf.le_leftLim hy... | Mathlib/Topology/Order/LeftRightLim.lean | 163 | 174 |
/-
Copyright (c) 2018 Michael Jendrusch. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Jendrusch, Kim Morrison, Bhavik Mehta, Jakob von Raumer
-/
import Mathlib.CategoryTheory.EqToHom
import Mathlib.CategoryTheory.Functor.Trifunctor
import Mathlib.CategoryTheo... | Mathlib/CategoryTheory/Monoidal/Category.lean | 796 | 796 | |
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kenny Lau
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.RingTheory.Multiplicity
import Mathlib.RingTheory.PowerSeries.Basic
/-! # Formal power series (in one va... |
theorem coe_toNat_order {φ : R⟦X⟧} (hf : φ ≠ 0) : φ.order.toNat = φ.order := by
rw [ENat.coe_toNat_eq_self.mpr (order_eq_top.not.mpr hf)]
/-- If the order of a formal power series is finite,
then the coefficient indexed by the order is nonzero. -/
theorem coeff_order (h : φ ≠ 0) : coeff R φ.order.toNat φ ≠ 0 := by
... | Mathlib/RingTheory/PowerSeries/Order.lean | 68 | 75 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.Complex.Asymptotics
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Data.Complex.Trig... | refine ⟨1, one_pos, fun x hx ↦ ?_⟩
convert exp_bound hx.out.le hn using 1
field_simp [mul_comm]
lemma exp_sub_sum_range_succ_isLittleO_pow (n : ℕ) :
(fun x ↦ exp x - ∑ i ∈ Finset.range (n + 1), x ^ i / i !) =o[𝓝 0] (· ^ n) :=
(exp_sub_sum_range_isBigO_pow (n + 1)).trans_isLittleO <| isLittleO_pow_po... | Mathlib/Analysis/SpecialFunctions/Exp.lean | 80 | 88 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Joey van Langen, Casper Putz
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Group.Fin.Basic
import Mathlib.Algebra.Group.ULift
import Mathlib.Data.Int.ModEq
import M... |
namespace CharP
section AddMonoidWithOne
variable [AddMonoidWithOne R] (p : ℕ)
variable [CharP R p] {a b : ℕ}
lemma natCast_eq_natCast' (h : a ≡ b [MOD p]) : (a : R) = b := by
wlog hle : a ≤ b
· exact (this R p h.symm (le_of_not_le hle)).symm
rw [Nat.modEq_iff_dvd' hle] at h
rw [← Nat.sub_add_cancel hle, Nat... | Mathlib/Algebra/CharP/Basic.lean | 28 | 40 |
/-
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.Data.Nat.PrimeFin
import Mathlib.Data.Nat.Factorization.Defs
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Order.Interval.Finset.Nat
import... | 0 < factorizationLCMRight a b := by
apply Nat.pos_of_ne_zero
rw [factorizationLCMRight, Finsupp.prod_ne_zero_iff]
intro p _ H
by_cases h : b.factorization p ≤ a.factorization p
| Mathlib/Data/Nat/Factorization/Basic.lean | 430 | 434 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Data.Bundle
import Mathlib.Data.Set.Image
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.PartialHomeomorph
import Mathlib.Topology.O... | base sets. -/
noncomputable def disjointUnion (e e' : Trivialization F proj) (H : Disjoint e.baseSet e'.baseSet) :
Trivialization F proj where
toPartialHomeomorph :=
e.toPartialHomeomorph.disjointUnion e'.toPartialHomeomorph
| Mathlib/Topology/FiberBundle/Trivialization.lean | 718 | 722 |
/-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Mario Carneiro, Johan Commelin
-/
import Mathlib.NumberTheory.Padics.PadicNumbers
import Mathlib.RingTheory.DiscreteValuationRing.Basic
/-!
# p-adic integers
This f... | let u : ℚ_[p] := x * (p : ℚ_[p]) ^ (-x.valuation : ℤ)
have hu : ‖u‖ = 1 := by
simp [u, hx, pow_ne_zero _ (NeZero.ne _), norm_eq_zpow_neg_valuation]
mkUnits hu
| Mathlib/NumberTheory/Padics/PadicIntegers.lean | 355 | 359 |
/-
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.Analysis.Convex.StrictConvexBetween
import Mathlib.Analysis.InnerProductSpace.Convex
import Mathlib.Analysis.Normed.Affine.Convex
import Mathlib.Geometry.E... | point. In two dimensions, this is the same thing as being
concyclic. -/
def Cospherical (ps : Set P) : Prop :=
| Mathlib/Geometry/Euclidean/Sphere/Basic.lean | 141 | 143 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Data.Nat.ModEq
/-!
# Congruences modulo an integer
This file defines the equivalence relation `a ≡ b [ZMOD n]` on the integers, similarly to how
`Data.N... | @[gcongr] protected theorem pow (m : ℕ) (h : a ≡ b [ZMOD n]) : a ^ m ≡ b ^ m [ZMOD n] := by
induction' m with d hd; · rfl
rw [pow_succ, pow_succ]
exact hd.mul h
| Mathlib/Data/Int/ModEq.lean | 164 | 167 |
/-
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... | { TrivSqZeroExt.monoid with
mul_comm := fun x₁ x₂ =>
ext (mul_comm x₁.1 x₂.1) <|
show x₁.1 •> x₂.2 + x₁.2 <• x₂.1 = x₂.1 •> x₁.2 + x₂.2 <• x₁.1 by
| Mathlib/Algebra/TrivSqZeroExt.lean | 660 | 663 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Andrew Zipperer, Haitao Zhang, Minchao Wu, Yury Kudryashov
-/
import Mathlib.Data.Set.Prod
import Mathlib.Data.Set.Restrict
/-!
# Functions over sets
This file contains... | MapsTo f (fa '' s) (fb '' t) :=
mapsTo_image_iff.2 fun x hx ↦ ⟨f x, hst hx, (h x).symm⟩
| Mathlib/Data/Set/Function.lean | 1,094 | 1,096 |
/-
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.RingTheory.Ideal.BigOperators
import Mathlib.RingTheory.FiniteType
/-!
# Rees algebra
The Rees algebra of an ideal `I` is the subalgebra `R[It]` of `R[t]`... | apply Subalgebra.sum_mem _ _
rintro i -
exact monomial_mem_adjoin_monomial (hp i)
variable {I}
theorem reesAlgebra.fg (hI : I.FG) : (reesAlgebra I).FG := by
classical
obtain ⟨s, hs⟩ := hI
rw [← adjoin_monomial_eq_reesAlgebra, ← hs]
use s.image (monomial 1)
| Mathlib/RingTheory/ReesAlgebra.lean | 98 | 108 |
/-
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
-/
import Mathlib.Tactic.Attr.Register
import Mathlib.Tactic.Basic
import Batteries.Logic
import Batteries.Tactic.Trans
import Batteries.Util.LibraryNot... |
On the other hand, making `Nat.prime` a class would require a major refactoring of the library,
| Mathlib/Logic/Basic.lean | 73 | 74 |
/-
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, Yaël Dillies
-/
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Int.Order.Basic
import Mathlib.Logic.Function.Iterate
import Mathlib.Order.Compare
impor... | Mathlib/Order/Monotone/Basic.lean | 949 | 953 | |
/-
Copyright (c) 2021 Hunter Monroe. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hunter Monroe, Kyle Miller, Alena Gusakov
-/
import Mathlib.Combinatorics.SimpleGraph.DeleteEdges
import Mathlib.Data.Fintype.Powerset
/-!
# Subgraphs of a simple graph
A subgraph of ... | theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by
ext e
| Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | 533 | 534 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Kexing Ying, Moritz Doll
-/
import Mathlib.Algebra.GroupWithZero.Action.Opposite
import Mathlib.LinearAlgebra.Finsupp.VectorSpace
import Mathlib.LinearAlgebra.Matrix.Basis
im... | simp_rw [LinearMap.toMatrix₂_compl₁₂ b₁ b₂, toMatrix_toLin, transpose_transpose]
theorem LinearMap.mul_toMatrix₂ (B : M₁ →ₗ[R] M₂ →ₗ[R] R) (M : Matrix n' n R) :
M * LinearMap.toMatrix₂ b₁ b₂ B =
| Mathlib/LinearAlgebra/Matrix/SesquilinearForm.lean | 449 | 452 |
/-
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.Order.Group.Nat
import Mathlib.Data.List.Rotate
import Mathlib.GroupTheory.Perm.Support
/-!
# Permutations from a list
A list `l : List α` ... | · simpa [Nat.succ_lt_succ_iff] using hn
simp only [swap_apply_eq_iff, coe_mul, formPerm_cons_cons, Function.comp]
simp only [getElem_cons_succ] at *
| Mathlib/GroupTheory/Perm/List.lean | 166 | 168 |
/-
Copyright (c) 2015 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis
-/
import Mathlib.Algebra.Order.Monoid.Unbundled.Pow
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Ring.Parity
import Mathlib.Tactic.Bound... |
end OrderedSemiring
section StrictOrderedSemiring
| Mathlib/Algebra/Order/Ring/Basic.lean | 75 | 79 |
/-
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... | Mathlib/Data/Nat/Choose/Basic.lean | 409 | 419 | |
/-
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.Analysis.Calculus.InverseFunctionTheorem.FDeriv
import Mathlib.Analysis.Calculus.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Prod
import Mathl... | fun ⟨_, _⟩ h => congr_arg Prod.fst h
/-- Any point in some neighborhood of `a` can be represented as
`HasStrictFDerivAt.implicitFunctionOfComplemented` of some point. -/
| Mathlib/Analysis/Calculus/Implicit.lean | 301 | 304 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.FieldTheory.Minpoly.Field
import Mathlib.LinearAlgebra.SModEq
import Mathlib.RingTheory.Ideal.BigOperators
/-!
# Power basis
This file defines a structure ... | variable [Algebra A S]
/-- `pb.minpolyGen` is the minimal polynomial for `pb.gen`. -/
noncomputable def minpolyGen (pb : PowerBasis A S) : A[X] :=
X ^ pb.dim - ∑ i : Fin pb.dim, C (pb.basis.repr (pb.gen ^ pb.dim) i) * X ^ (i : ℕ)
| Mathlib/RingTheory/PowerBasis.lean | 155 | 160 |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.Order.Group.Pointwise.Interval
import Mathlib.LinearAlgebra.BilinearMap
import Mathlib.LinearAlgebra.Pi
import Mathlib.LinearAlgebra.Prod
import Ma... |
This is the affine version of `LinearMap.pi`.
-/
def pi (f : (i : ι) → (P1 →ᵃ[k] φp i)) : P1 →ᵃ[k] ((i : ι) → φp i) where
toFun m a := f a m
linear := LinearMap.pi (fun a ↦ (f a).linear)
map_vadd' _ _ := funext fun _ ↦ map_vadd _ _ _
--fp for when the image is a dependent AffineSpace φp i, fv for when the
| Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean | 676 | 684 |
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kenny Lau
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.RingTheory.Multiplicity
import Mathlib.RingTheory.PowerSeries.Basic
/-! # Formal power series (in one va... | · simpa only [add_comm, inf_comm] using order_add_of_order_eq.aux _ _ ψ_lt_φ
/-- The order of the product of two formal power series
is at least the sum of their orders. -/
theorem le_order_mul (φ ψ : R⟦X⟧) : order φ + order ψ ≤ order (φ * ψ) := by
apply le_order
intro n hn; rw [coeff_mul, Finset.sum_eq_zero]
... | Mathlib/RingTheory/PowerSeries/Order.lean | 167 | 180 |
/-
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.Analysis.Convex.Topology
import Mathlib.Analysis.Normed.Module.Basic
import Mathlib.Analysis.Seminorm
import Mathlib.Analysis.SpecificLimits.Basi... |
/-!
### Properties of `UniqueDiffWithinAt` and `UniqueDiffOn`
| Mathlib/Analysis/Calculus/TangentCone.lean | 409 | 412 |
/-
Copyright (c) 2019 Neil Strickland. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Neil Strickland
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Group.NatPowAssoc
import Mathlib.Algebra.Order.... | (Commute.all x y).sub_dvd_pow_sub_pow n
theorem nat_sub_dvd_pow_sub_pow (x y n : ℕ) : x - y ∣ x ^ n - y ^ n := by
| Mathlib/Algebra/GeomSum.lean | 183 | 185 |
/-
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... | theorem imageToKernel_comp_hom_inv_comp [HasEqualizers V] [HasImages V] {Z : V} {i : B ≅ Z} (w) :
imageToKernel (f ≫ i.hom) (i.inv ≫ g) w =
(imageSubobjectCompIso _ _).hom ≫
imageToKernel f g (by simpa using w) ≫ (kernelSubobjectIsoComp i.inv g).inv := by
ext
simp
| Mathlib/Algebra/Homology/ImageToKernel.lean | 127 | 132 |
/-
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... | (hB : M.IsBase B) (h : B.Finite) (hB' : M.IsBase B') : B'.Finite :=
(finite_iff_finite_of_encard_eq_encard (hB.encard_eq_encard_of_isBase hB')).mp h
| Mathlib/Data/Matroid/Basic.lean | 430 | 431 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Data.Nat.ModEq
/-!
# Congruences modulo an integer
This file defines the equivalence relation `a ≡ b [ZMOD n]` on the integers, similarly to how
`Data.N... | ⟨z.natAbs, by
| Mathlib/Data/Int/ModEq.lean | 256 | 256 |
/-
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.AbsolutelyContinuous
/-!
# Vitali families
On a metric space `X` with a measure `μ`, consider for each `x : X` a family o... | simp only [filterAt, tendsto_inf, nhds_basis_closedBall.smallSets.tendsto_right_iff,
tendsto_principal, and_comm, mem_powerset_iff]
| Mathlib/MeasureTheory/Covering/VitaliFamily.lean | 226 | 228 |
/-
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... | have hps : p ∈ s := h'ε (mem_ball_iff_norm.2 hp)
by_cases hx : x ∈ k
· have H : (p, x) ∈ t := by
apply hε
refine mem_thickening_iff.2 ⟨(q₀.1, x), ?_, ?_⟩
· simp only [hx, singleton_prod, mem_image, Prod.mk_inj, eq_self_iff_true, true_and,
| Mathlib/Analysis/Convolution.lean | 1,078 | 1,083 |
/-
Copyright (c) 2018 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, Winston Yin
-/
import Mathlib.Algebra.Group.End
import Mathlib.Topology.EMetricSpace.Diam
/-!
# Lipschitz co... | protected lemma comp {f : β → γ} {g : α → β}
(hf : LocallyLipschitz f) (hg : LocallyLipschitz g) : LocallyLipschitz (f ∘ g) := by
intro x
-- g is Lipschitz on t ∋ x, f is Lipschitz on u ∋ g(x)
rcases hg x with ⟨Kg, t, ht, hgL⟩
| Mathlib/Topology/EMetricSpace/Lipschitz.lean | 366 | 370 |
/-
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.... | map_zero' := by simp only [add_zero, mul_zero, map_zero]
map_one' := by simp only [one_im, add_zero, mul_one, one_re, mul_zero]
map_mul' z w := by
simp only [mul_im, mul_re]
| Mathlib/Analysis/RCLike/Basic.lean | 375 | 378 |
/-
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.Data.Nat.PrimeFin
import Mathlib.Data.Nat.Factorization.Defs
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Order.Interval.Finset.Nat
import... | theorem coprime_ordCompl {n p : ℕ} (hp : Prime p) (hn : n ≠ 0) : Coprime p (ordCompl[p] n) :=
(or_iff_left (not_dvd_ordCompl hp hn)).mp <| coprime_or_dvd_of_prime hp _
@[deprecated (since := "2024-10-24")] alias coprime_ord_compl := coprime_ordCompl
theorem factorization_ordCompl (n p : ℕ) :
| Mathlib/Data/Nat/Factorization/Basic.lean | 232 | 237 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.InnerProductSpace.Defs
impor... | /-- If the inner product of two unit vectors is `1`, then the two vectors are equal. One form of
the equality case for Cauchy-Schwarz. -/
theorem inner_eq_one_iff_of_norm_one {x y : E} (hx : ‖x‖ = 1) (hy : ‖y‖ = 1) :
⟪x, y⟫ = 1 ↔ x = y := by
| Mathlib/Analysis/InnerProductSpace/Basic.lean | 777 | 780 |
/-
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.Calculus.ContDiff.Bounds
import Mathlib.Analysis.Calculus.IteratedDeriv.Defs
import Mathlib.Analysis.Calculus.LineDeriv.Basic
import Mathlib.Analysi... | rw [Real.div_rpow (by linarith) zero_le_two]
simp [div_eq_inv_mul, ← Real.rpow_neg_one, ← Real.rpow_mul]
ring
rw [this]
rcases le_total x 1 with h'x|h'x
· gcongr
· apply (pow_le_one₀ hx h'x).trans
apply Real.one_le_rpow_of_pos_of_le_one_of_nonpos
· linarith
· linarith
· sim... | Mathlib/Analysis/Distribution/SchwartzSpace.lean | 613 | 629 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
/-!
# Kernels and cokernels
In a category with zero morphisms, the kernel of a morphism `f : X... | (by rw [iff, Category.assoc, Iso.inv_hom_id_assoc, hs])))
(fun s hs ↦ by simp [← cancel_mono e.inv])
(fun s hs m hm ↦ Fork.IsLimit.hom_ext hc (by simpa [← cancel_mono e.hom] using hm))
/-- If `c` is a limit kernel fork for `g : X ⟶ Y`, and `g' : X ⟶ Y'` is a another morphism,
then there is a limit kernel... | Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean | 212 | 220 |
/-
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.MFDeriv.Defs
import Mathlib.Geometry.Manifold.ContMDiff.Defs
/-!
# Basic properties of the manifold Fréchet ... | simp only [mfderivWithin, mfderiv, mfld_simps]
rw [mdifferentiableWithinAt_univ]
theorem mfderivWithin_zero_of_not_mdifferentiableWithinAt
(h : ¬MDifferentiableWithinAt I I' f s x) : mfderivWithin I I' f s x = 0 := by
simp only [mfderivWithin, h, if_neg, not_false_iff]
| Mathlib/Geometry/Manifold/MFDeriv/Basic.lean | 613 | 619 |
/-
Copyright (c) 2022 Cuma Kökmen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Cuma Kökmen, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Integral.CircleIntegral
import Mathlib.MeasureTheory.Integral.Prod
import Mathlib.Order.Fin.Tuple
import Mathlib.Util.Superscr... | Mathlib/MeasureTheory/Integral/TorusIntegral.lean | 257 | 260 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Order.SuccPred
import Mathlib.Data.Sum.Order
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Tactic.PPWithUniv
/-!
# ... | Mathlib/SetTheory/Ordinal/Basic.lean | 1,425 | 1,425 | |
/-
Copyright (c) 2024 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Calculus
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
/-!
# Properties about the powers of the n... | rw [hf.fderiv_norm_rpow hp, norm_smul, norm_mul]
simp_rw [norm_rpow_of_nonneg (norm_nonneg _), norm_norm, norm_eq_abs,
abs_eq_self.mpr <| zero_le_one.trans hp.le, mul_assoc]
gcongr _ * ?_
refine mul_le_mul_of_nonneg_left (ContinuousLinearMap.opNorm_comp_le ..) (by positivity)
|>.trans_eq ?_
rw [innerS... | Mathlib/Analysis/InnerProductSpace/NormPow.lean | 76 | 86 |
/-
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.Piecewise
import Mathlib.Order.FixedPoints
import Mathlib.Order.Zorn
/-!
# Schröder-Bernstein theorem, well-ordering of cardi... | · have : IsEmpty α := Function.isEmpty f
exact ⟨_, ((Equiv.equivEmpty α).trans (Equiv.equivEmpty β).symm).bijective⟩
set F : Set α →o Set α :=
{ toFun := fun s => (g '' (f '' s)ᶜ)ᶜ
monotone' := fun s t hst =>
compl_subset_compl.mpr <| image_subset _ <| compl_subset_compl.mpr <| image_subset _ ... | Mathlib/SetTheory/Cardinal/SchroederBernstein.lean | 47 | 76 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johan Commelin
-/
import Mathlib.RingTheory.IntegralClosure.IsIntegral.Basic
/-!
# Minimal polynomials
This file defines the minimal polynomial of an element `x` of an `A... | then the minimal polynomial of `algebraMap A B a` is `X - C a`. -/
theorem eq_X_sub_C_of_algebraMap_inj (a : A) (hf : Function.Injective (algebraMap A B)) :
minpoly A (algebraMap A B a) = X - C a := by
nontriviality A
| Mathlib/FieldTheory/Minpoly/Basic.lean | 223 | 226 |
/-
Copyright (c) 2024 Frédéric Marbach. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Marbach
-/
import Mathlib.Algebra.Lie.NonUnitalNonAssocAlgebra
import Mathlib.Algebra.Lie.OfAssociative
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.RingTheory.Nilpo... | /-- The natural map from a Lie module to the derivations taking values in it. -/
@[simps!]
def inner : M →ₗ[R] LieDerivation R L M where
toFun m :=
| Mathlib/Algebra/Lie/Derivation/Basic.lean | 352 | 355 |
/-
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.Calculus.ContDiff.RCLike
import Mathlib.MeasureTheory.Measure.Hausdorff
/-!
# Hausdorff dimension
The Hausdorff dimension of a set `X` in ... | {s t : Set E} (h : ContDiffOn ℝ 1 f s) (hc : Convex ℝ s) (ht : t ⊆ s)
(htF : dimH t < finrank ℝ F) : Dense (f '' t)ᶜ :=
dense_compl_of_dimH_lt_finrank <| (h.dimH_image_le hc ht).trans_lt htF
/-- A particular case of Sard's Theorem. If `f` is a `C¹` smooth map from a real vector space to a
real vector space `... | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | 516 | 526 |
/-
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.... | Mathlib/Algebra/Group/Basic.lean | 1,321 | 1,321 | |
/-
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.Analytic.CPolynomial
import Mathlib.Analysis.Analytic.Inverse
import Mathlib.Analysis.Analytic.Within
import Mathlib.Analysis.Calculus.Deriv... | apply HasFDerivWithinAt.fderivWithin _ (hu _ h'y)
exact a.hasFDerivAt.comp_hasFDerivWithinAt (x + y) hf'
rw [this] at Z
convert Z with n
ext v
simp only [FormalMultilinearSeries.derivSeries,
ContinuousLinearMap.compFormalMultilinearSeries_apply,
FormalMultilinearSeries.changeOriginSeries,
Co... | Mathlib/Analysis/Calculus/FDeriv/Analytic.lean | 314 | 346 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Fin.Tuple.Basic
/-!
# Lists from functions
Theorems and lemmas for dealing with `List.ofFn`, which converts a function on `Fin n` to a list
of l... | theorem last_ofFn {n : ℕ} (f : Fin n → α) (h : ofFn f ≠ [])
(hn : n - 1 < n := Nat.pred_lt <| ofFn_eq_nil_iff.not.mp h) :
| Mathlib/Data/List/OfFn.lean | 146 | 147 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Algebra.Field.NegOnePow
import Mathlib.Algebra.Field.Periodic
import Mathlib.Algebra.Qua... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | 1,373 | 1,374 | |
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Sébastien Gouëzel, Yury Kudryashov, Eric Wieser
-/
import Mathlib.Analysis.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calculus.FDeriv.Comp
/-!
# Derivative of the ca... | have h' : comp (proj i) id' = proj i := by rfl
rw [← h']; apply h; apply hasFDerivWithinAt_id
theorem hasFDerivWithinAt_pi :
| Mathlib/Analysis/Calculus/FDeriv/Prod.lean | 451 | 454 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Joey van Langen, Casper Putz
-/
import Mathlib.Data.Nat.Cast.Basic
import Mathlib.Data.Nat.Find
import Mathlib.Data.Nat.Prime.Defs
import Mathlib.Order.Lattice
/-!
# Characteris... | Nat.pos_of_ne_zero <| not_of_not_imp <| Nat.find_spec (not_forall.1 H))
(not_imp_of_and_not
| Mathlib/Algebra/CharP/Defs.lean | 112 | 113 |
/-
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... | /-- Closed balls defined using the distance or the edistance coincide -/
theorem Metric.emetric_closedBall {x : α} {ε : ℝ} (h : 0 ≤ ε) :
EMetric.closedBall x (ENNReal.ofReal ε) = closedBall x ε := by
ext y; simp [edist_le_ofReal h]
| Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | 934 | 937 |
/-
Copyright (c) 2019 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Mario Carneiro, Isabel Longbottom, Kim Morrison, Apurva Nakade, Yuyang Zhao
-/
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.SetTheory.PGame.Algebra
import Mathl... | (Equiv.sumComm _ _).trans (Equiv.sumCongr (Equiv.prodComm _ _) (Equiv.prodComm _ _)), ?_, ?_⟩
<;>
rintro (⟨i, j⟩ | ⟨i, j⟩) <;>
{ dsimp
exact ((addCommRelabelling _ _).trans <|
(mulCommRelabelling _ _).addCongr (mulCommRelabelling _ _)).subCongr
(mulCommRelabelling _ _) }
| Mathlib/SetTheory/Game/Basic.lean | 396 | 402 |
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel, Joël Riou
-/
import Mathlib.Algebra.Homology.ExactSequence
import Mathlib.CategoryTheory.Abelian.Refinements
/-!
# The four and five lemmas
Consider the following commu... | refine epi_of_epi_of_epi_of_mono' ψ (hR₁.exact 0).exact_toComposableArrows
(exact₂_mk _ (by simp) ?_) ?_ (by dsimp [ψ]; infer_instance) h₀ h₁
· rw [ShortComplex.exact_iff_mono _ (by simp)]
exact hR₂'
· dsimp
| Mathlib/CategoryTheory/Abelian/DiagramLemmas/Four.lean | 181 | 185 |
/-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Analysis.SpecialFunctions.Complex.Circle
import Mathlib.Analysis.InnerProductSpace.... | convert hasDerivAt_mul_const (2 * ↑π * I * ↑n / T) using 1
ext1 y; ring
theorem hasDerivAt_fourier_neg (n : ℤ) (x : ℝ) :
| Mathlib/Analysis/Fourier/AddCircle.lean | 448 | 451 |
/-
Copyright (c) 2020 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard
-/
import Mathlib.RingTheory.AdicCompletion.Basic
import Mathlib.RingTheory.LocalRing.MaximalIdeal.Basic
import Mathlib.RingTheory.LocalRing.RingHom.Basic
import Mathlib.R... | constructor
· contrapose
intro h
| Mathlib/RingTheory/DiscreteValuationRing/Basic.lean | 426 | 428 |
/-
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
-/
import Mathlib.Tactic.Attr.Register
import Mathlib.Tactic.Basic
import Batteries.Logic
import Batteries.Tactic.Trans
import Batteries.Util.LibraryNot... | Mathlib/Logic/Basic.lean | 323 | 323 | |
/-
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.HomologicalComplexBiprod
import Mathlib.Algebra.Homology.Homotopy
import Mathlib.CategoryTheory.MorphismProperty.IsInvertedBy
/-! The homotopy ... | @[reassoc]
lemma d_sndX (i j : ι) (hij : c.Rel i j) :
d φ i j ≫ sndX φ j = fstX φ i j hij ≫ φ.f j + sndX φ i ≫ G.d i j := by
dsimp [d]
split_ifs with hij <;> simp
| Mathlib/Algebra/Homology/HomotopyCofiber.lean | 179 | 183 |
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Logic.Encodable.Pi
import Mathlib.MeasureTheory.Group.Measure
import Mathlib.MeasureTheory.MeasurableSpace.... | instance pi.instIsProbabilityMeasure [∀ i, IsProbabilityMeasure (μ i)] :
IsProbabilityMeasure (Measure.pi μ) :=
⟨by simp only [Measure.pi_univ, measure_univ, Finset.prod_const_one]⟩
instance {α : ι → Type*} [∀ i, MeasureSpace (α i)]
[∀ i, IsProbabilityMeasure (volume : Measure (α i))] :
IsProbabilityMeas... | Mathlib/MeasureTheory/Constructions/Pi.lean | 317 | 329 |
/-
Copyright (c) 2021 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Order.Antichain
import Mathlib.Topology.ContinuousOn
/-!
# Left and right continuity
In this file we prove a few lemmas about left and right cont... | simp only [ContinuousWithinAt, ContinuousAt, ← tendsto_sup, nhdsLE_sup_nhdsGE]
theorem continuousAt_iff_continuous_left'_right' {a : α} {f : α → β} :
ContinuousAt f a ↔ ContinuousWithinAt f (Iio a) a ∧ ContinuousWithinAt f (Ioi a) a := by
| Mathlib/Topology/Order/LeftRight.lean | 132 | 135 |
/-
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.HomologicalComplexLimits
import Mathlib.Algebra.Homology.Additive
/-! Binary biproducts of homological complexes
In this file, it is shown tha... | @[reassoc (attr := simp)]
lemma inr_biprodXIso_inv (i : ι) :
biprod.inr ≫ (biprodXIso K L i).inv = (biprod.inr : L ⟶ K ⊞ L).f i := by
simp [biprodXIso]
| Mathlib/Algebra/Homology/HomologicalComplexBiprod.lean | 50 | 53 |
/-
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... | Mathlib/Data/Rat/Defs.lean | 502 | 504 | |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Patrick Massot, Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus
deprecated_module (since := "2025-04-06")
| Mathlib/MeasureTheory/Integral/FundThmCalculus.lean | 525 | 536 | |
/-
Copyright (c) 2020 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.Group.Action.Pi
import Mathlib.Data.Finset.Prod
import Mathlib.Data.SetLike.Basic
import Mathlib.Data.Sym.Basic
import Mathlib.Data.Sym.Sym2.Init
/-... |
@[simp]
| Mathlib/Data/Sym/Sym2.lean | 377 | 378 |
/-
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.ContinuousMap.Bounded.ArzelaAscoli
import Mathlib.Topology.ContinuousMap.Bounded.Normed
import Mathlib.Topology.MetricSpace.Gluing
impor... | refine Monotone.map_ciInf_of_continuousAt (continuousAt_id.add continuousAt_const) ?_ ?_
· intro x y hx
simpa
· show BddBelow (range fun x : X => g (inl x, inr y))
exact ⟨cg, forall_mem_range.2 fun i => Hcg _⟩
have E2 : (⨆ y, ⨅ x, g (inl x, inr y)) + dist f g = ⨆ y, (⨅ x, g (inl x, inr y)) + d... | Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean | 313 | 319 |
/-
Copyright (c) 2022 Justin Thomas. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Justin Thomas
-/
import Mathlib.FieldTheory.Minpoly.Field
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.Algebra.Polynomial.Module.AEval
/-!
# Annihilating Ideal
Given ... | end Polynomial
| Mathlib/LinearAlgebra/AnnihilatingPolynomial.lean | 169 | 176 |
/-
Copyright (c) 2023 Michael Lee. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Lee, Geoffrey Irving
-/
import Mathlib.Deprecated.AnalyticManifold
deprecated_module
"Analytic manifolds are deprecated, use `IsManifold I ω M`" (since := "2025-04-13")
set_op... | Mathlib/Geometry/Manifold/AnalyticManifold.lean | 176 | 190 | |
/-
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.Basic
import Mathlib.Data.Finset.Image
import Mathlib.Data.Multiset.Fold
/-!
# The fold operation for a commutative associative operation ... |
@[simp]
theorem fold_insert_idem [DecidableEq α] [hi : Std.IdempotentOp op] :
(insert a s).fold op b f = f a * s.fold op b f := by
by_cases h : a ∈ s
| Mathlib/Data/Finset/Fold.lean | 99 | 103 |
/-
Copyright (c) 2024 Sophie Morel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sophie Morel
-/
import Mathlib.Analysis.NormedSpace.PiTensorProduct.ProjectiveSeminorm
import Mathlib.LinearAlgebra.Isomorphisms
/-!
# Injective seminorm on the tensor of a finite famil... | private theorem mapL_add_smul_aux {ι : Type uι}
{E : ι → Type uE} [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)]
{E' : ι → Type u_1} [(i : ι) → SeminormedAddCommGroup (E' i)] [(i : ι) → NormedSpace 𝕜 (E' i)]
(f : (i : ι) → E i →L[𝕜] E' i)
[DecidableEq ι] (i : ι) (u : E i →L[... | Mathlib/Analysis/NormedSpace/PiTensorProduct/InjectiveSeminorm.lean | 401 | 408 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Normed.Operator.BoundedLinear... | rintro ⟨a, x⟩ ⟨r', ⟨Irr', Ir'r⟩, hr⟩
have ha : Continuous (f a) := hf.uncurry_left a
rcases exists_between Irr' with ⟨t, hrt, htr'⟩
| Mathlib/Analysis/Calculus/FDeriv/Measurable.lean | 798 | 800 |
/-
Copyright (c) 2020 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
import Mathlib.Algebra.ContinuedFractions.Computation.Translations
import Mathlib.Algebra.ContinuedFractions.TerminatedStable
import Mathlib.Algebra.ContinuedFrac... |
open GenContFract
(compExactValue compExactValue_correctness_of_stream_eq_some_aux_comp)
/-- Shows the correctness of `compExactValue` in case the continued fraction
`GenContFract.of v` did not terminate at position `n`. That is, we obtain the
| Mathlib/Algebra/ContinuedFractions/Computation/CorrectnessTerminating.lean | 78 | 83 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Kevin Buzzard, Yury Kudryashov, Frédéric Dupuis,
Heather Macbeth
-/
import Mathlib.Algebra.Group.Subgroup.Map
import Mathlib.Algebra.Module.Submodule.... | @[gcongr]
theorem map_mono {f : F} {p p' : Submodule R M} : p ≤ p' → map f p ≤ map f p' :=
image_subset _
| Mathlib/Algebra/Module/Submodule/Map.lean | 106 | 108 |
/-
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 Hölzl
-/
import Mathlib.Algebra.Order.Group.Defs
import Mathlib.Algebra.Order.Group.Unbundled.Abs
import Mathlib.Algebra.Order... | Mathlib/Algebra/Order/Group/Abs.lean | 385 | 385 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Logic.Nontrivial.Basic
import Mathlib.Order.TypeTags
import Mathlib.Data.Option.NAry
import Mathlib.Tactic.Contrapose
import Mathlib.Tactic.Lift
import... | Mathlib/Order/WithBot.lean | 1,226 | 1,227 | |
/-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Ordinal.Family
import Mathlib.Tactic.Abel
/-!
# Natural operations on ordinals
The goal of this file is to define n... | rintro ⟨i, hi⟩
rwa [nadd_one, succ_le_succ_iff, succ_le_iff]
termination_by a
@[simp]
theorem one_nadd : 1 ♯ a = succ a := by rw [nadd_comm, nadd_one]
| Mathlib/SetTheory/Ordinal/NaturalOps.lean | 291 | 296 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yaël Dillies
-/
import Mathlib.Data.Set.Image
/-!
# Directed indexed families and sets
This file defines directed indexed families and directed sets. An indexed famil... | theorem directedOn_range {f : ι → α} : Directed r f ↔ DirectedOn r (Set.range f) := by
simp_rw [Directed, DirectedOn, Set.forall_mem_range, Set.exists_range_iff]
| Mathlib/Order/Directed.lean | 58 | 60 |
/-
Copyright (c) 2022 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Algebra.Ring.Idempotent
import Mathlib.Analysis.Normed.Group.Basic
import Mathlib.Order.Basic
import Mathlib.Tactic.NoncommRing
/-!
# M-struct... |
instance Subtype.inf [FaithfulSMul M X] : Min { P : M // IsLprojection X P } :=
⟨fun P Q => ⟨P * Q, P.prop.mul Q.prop⟩⟩
| Mathlib/Analysis/NormedSpace/MStructure.lean | 164 | 167 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Order.Minimal
import Mathlib.Order.Zorn
import Mathlib.Topology.ContinuousOn
/-!
# Irreducibility in topological space... | isIrreducible_iff_sUnion_closed := isIrreducible_iff_sUnion_isClosed
/-- A nonempty open subset of a preirreducible subspace is dense in the subspace. -/
theorem subset_closure_inter_of_isPreirreducible_of_isOpen {S U : Set X} (hS : IsPreirreducible S)
(hU : IsOpen U) (h : (S ∩ U).Nonempty) : S ⊆ closure (S ∩ U) :... | Mathlib/Topology/Irreducible.lean | 269 | 276 |
/-
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
... | theorem lookup_union_right {a} {s₁ s₂ : Finmap β} : a ∉ s₁ → lookup a (s₁ ∪ s₂) = lookup a s₂ :=
induction_on₂ s₁ s₂ fun _ _ => AList.lookup_union_right
theorem lookup_union_left_of_not_in {a} {s₁ s₂ : Finmap β} (h : a ∉ s₂) :
| Mathlib/Data/Finmap.lean | 513 | 516 |
/-
Copyright (c) 2018 Guy Leroy. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sangwoo Jo (aka Jason), Guy Leroy, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.GroupWithZero.Semiconj
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Data.Nat.GCD.Bas... | ∀ {s t s' t'}, P x y (r, s, t) → P x y (r', s', t') → P x y (xgcdAux r s t r' s' t') := by
induction r, r' using gcd.induction with
| Mathlib/Data/Int/GCD.lean | 112 | 113 |
/-
Copyright (c) 2017 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Keeley Hoek
-/
import Mathlib.Algebra.NeZero
import Mathlib.Data.Int.DivMod
import Mathlib.Logic.Embedding.Basic
import Mathlib.Logic.Equiv.Set
import Mathlib.Tactic.... |
-- to match `Fin.eq_zero_or_eq_succ`
| Mathlib/Data/Fin/Basic.lean | 627 | 628 |
/-
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.Algebra.BigOperators.Fin
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Finset.Sort
/-!
# Compositions
A compositio... | let (l₁, l₂) := l.splitAt n
l₁::splitWrtCompositionAux l₂ ns
/-- Given a list of length `n` and a composition `[i₁, ..., iₖ]` of `n`, split `l` into a list of
| Mathlib/Combinatorics/Enumerative/Composition.lean | 694 | 697 |
/-
Copyright (c) 2023 Josha Dekker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Josha Dekker
-/
import Mathlib.Topology.Bases
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.Compactness.SigmaCompact
/-!
# Lindelöf sets and Lindelöf spaces
## Mai... | theorem isLindelof_open_iff_eq_countable_iUnion_of_isTopologicalBasis (b : ι → Set X)
(hb : IsTopologicalBasis (Set.range b)) (hb' : ∀ i, IsLindelof (b i)) (U : Set X) :
IsLindelof U ∧ IsOpen U ↔ ∃ s : Set ι, s.Countable ∧ U = ⋃ i ∈ s, b i := by
constructor
· rintro ⟨h₁, h₂⟩
obtain ⟨Y, f, rfl, hf⟩ := hb... | Mathlib/Topology/Compactness/Lindelof.lean | 382 | 406 |
/-
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, Kexing Ying
-/
import Mathlib.Topology.Semicontinuous
import Mathlib.MeasureTheory.Function.AEMeasurableSequence
import Mathlib.MeasureTheory.Order.Lat... | have mc_meas : MeasurableSet {x | ∀ (j : Subtype p), x ∉ m j} := by
rw [setOf_forall]
exact MeasurableSet.iInter (fun j ↦ (m_meas j).compl)
refine measurable_const.piecewise mc_meas <| .iSup fun j ↦ ?_
let reparam : δ → Subtype p → Subtype p := fun x ↦ liminf_reparam (fun i ↦ f i x) s p
let F0 : Subtype... | Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean | 864 | 874 |
/-
Copyright (c) 2021 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Jireh Loreaux
-/
import Mathlib.Algebra.Group.Center
import Mathlib.Algebra.Ring.Defs
/-!
# Centralizers of rings
-/
assert_not_exists RelIso
variable {M : Type*} ... | a + b ∈ centralizer S := fun c hc => by rw [add_mul, mul_add, ha c hc, hb c hc]
| Mathlib/Algebra/Ring/Centralizer.lean | 23 | 24 |
/-
Copyright (c) 2019 Gabriel Ebner. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Gabriel Ebner, Sébastien Gouëzel, Yury Kudryashov, Yuyang Zhao
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Comp
import Mathlib.Analysis.Calcu... | HasStrictFDerivAt (h₂ ∘ f) (h₂' • f') x := by
rw [hy] at hh; exact hh.comp_hasStrictFDerivAt x hf
theorem HasDerivAt.comp_hasFDerivAt {f : E → 𝕜'} {f' : E →L[𝕜] 𝕜'} (x)
(hh : HasDerivAt h₂ h₂' (f x)) (hf : HasFDerivAt f f' x) : HasFDerivAt (h₂ ∘ f) (h₂' • f') x :=
hh.comp_hasFDerivAtFilter x hf hf.conti... | Mathlib/Analysis/Calculus/Deriv/Comp.lean | 176 | 182 |
/-
Copyright (c) 2021 Bryan Gin-ge Chen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bryan Gin-ge Chen, Yaël Dillies
-/
import Mathlib.Algebra.Group.Idempotent
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Algebra.Ring.PUnit
import Mathlib.Order.Hom.BoundedLattic... | * `a ⊓ b` unfolds to `a * b`
* `a ≤ b` unfolds to `a + b + a * b = b`
* `⊥` unfolds to `0`
| Mathlib/Algebra/Ring/BooleanRing.lean | 207 | 209 |
/-
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... |
/-- Expresses the integral over `Ioi 0` of `t ^ (a - 1) * exp (-(r * t))` in terms of the Gamma
function, for complex `a`. -/
lemma integral_cpow_mul_exp_neg_mul_Ioi {a : ℂ} {r : ℝ} (ha : 0 < a.re) (hr : 0 < r) :
∫ (t : ℝ) in Ioi 0, t ^ (a - 1) * exp (-(r * t)) = (1 / r) ^ a * Gamma a := by
have aux : (1 / r : ℂ... | Mathlib/Analysis/SpecialFunctions/Gamma/Basic.lean | 367 | 376 |
/-
Copyright (c) 2020 Kexing Ying and Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Kevin Buzzard, Yury Kudryashov
-/
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.Algebra.Gro... | apply Finset.prod_erase _ ha
/-- If `s : Set α` and `t : Set β` are finite sets, then taking the product over `s` commutes with
taking the product over `t`. -/
| Mathlib/Algebra/BigOperators/Finprod.lean | 966 | 969 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Mathlib.Data.Stream.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Data.List.Defs
import Mathlib.Data.Nat.Basic
import Mathlib.Tactic.Common... | · rfl
· simp [odd_eq, odd_eq, tail_interleave, tail_even])
| Mathlib/Data/Stream/Init.lean | 422 | 423 |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.IsPrimePow
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Ring.Int
import Mathlib.Algebra.Ring.CharZero
im... | have : IsAsymm (ℕ × ℕ) (·.snd < ·.snd) := ⟨fun _ _ ↦ lt_asymm⟩
refine List.eq_of_perm_of_sorted ?_ sorted_divisorsAntidiagonalList_snd.reverse <|
sorted_divisorsAntidiagonalList_fst.map _ fun _ _ ↦ id
| Mathlib/NumberTheory/Divisors.lean | 176 | 178 |
/-
Copyright (c) 2022 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.LinearAlgebra.Dimension.FreeAndStrongRankCondition
import Mathlib.LinearAlgebra.FiniteDimensional.Basic
/-!
# Projective Spaces
This file contains the defin... |
/-- Two nonzero vectors go to the same point in projective space if and only if one is
a scalar multiple of the other. -/
theorem mk_eq_mk_iff' (v w : V) (hv : v ≠ 0) (hw : w ≠ 0) :
mk K v hv = mk K w hw ↔ ∃ a : K, a • w = v := by
rw [mk_eq_mk_iff K v w hv hw]
constructor
· rintro ⟨a, ha⟩
exact ⟨a, ha⟩
| Mathlib/LinearAlgebra/Projectivization/Basic.lean | 108 | 116 |
/-
Copyright (c) 2023 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Finset.Lattice.Fold
/-!
# Irreducible and prime elements in an order
This file defines irreducible and prime elements in an order and shows that in ... | Mathlib/Order/Irreducible.lean | 348 | 349 | |
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes Hölzl, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Seminorm
import Mathlib.Data.NNReal.Basic
import Mathlib.Topology.Algebra.Support
import Mathlib.To... | @[to_additive]
theorem norm_le_norm_add_norm_div' (u v : E) : ‖u‖ ≤ ‖v‖ + ‖u / v‖ := by
| Mathlib/Analysis/Normed/Group/Basic.lean | 521 | 522 |
/-
Copyright (c) 2024 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convex.Deriv
import Mathlib.Analysis.SpecialFunctions.Gamma.Deligne
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.NumberTheory.Harmo... | lemma hasDerivAt_Gamma_one : HasDerivAt Gamma (-γ) 1 := by
simpa only [factorial_zero, cast_one, harmonic_zero, Rat.cast_zero, add_zero, mul_neg, one_mul,
cast_zero, zero_add] using hasDerivAt_Gamma_nat 0
| Mathlib/NumberTheory/Harmonic/GammaDeriv.lean | 180 | 182 |
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