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 FrΓ©dΓ©ric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: FrΓ©dΓ©ric Dupuis
-/
import Mathlib.Topology.Algebra.Module.WeakDual
import Mathlib.Algebra.Algebra.Spectrum.Basic
import Mathlib.Topology.ContinuousMap.Algebra
import Mathlib.Data.Set.... | theorem apply_mem_spectrum [Nontrivial π] (Ο : characterSpace π A) (a : A) : Ο a β spectrum π a :=
AlgHom.apply_mem_spectrum Ο a
theorem ext_ker {Ο Ο : characterSpace π A} (h : RingHom.ker Ο = RingHom.ker Ο) : Ο = Ο := by
ext x
| Mathlib/Topology/Algebra/Module/CharacterSpace.lean | 181 | 185 |
/-
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
-/
import Mathlib.Algebra.Order.Monoid.Unbundled.WithTop
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
/-! # Adjoi... | Mathlib/Algebra/Order/Monoid/WithTop.lean | 242 | 248 | |
/-
Copyright (c) 2024 RΓ©my Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: RΓ©my Degenne
-/
import Mathlib.LinearAlgebra.FreeModule.Basic
import Mathlib.MeasureTheory.Measure.Decomposition.Exhaustion
import Mathlib.Probability.ConditionalProbability
/-!
# s-fin... | Mathlib/MeasureTheory/Measure/WithDensityFinite.lean | 158 | 168 | |
/-
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, Johan Commelin
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Combinatorics.Enumerative.Composition
/-!
# Composition of analytic functions
In this fi... | (FormalMultilinearSeries.id π E x) 0 v = x := rfl
/-- The first coefficient of `id π E` is the identity. -/
@[simp]
| Mathlib/Analysis/Analytic/Composition.lean | 337 | 340 |
/-
Copyright (c) 2019 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Module.Submodule.Equiv
import Mathlib.Algebra.Module.Equiv.Basic
import Mathlib.Algebra.Module.Rat
im... | { LinearMap.inverse f.toLinearMap g hβ hβ with
map_lie' := by
intros x n
calc
| Mathlib/Algebra/Lie/Basic.lean | 813 | 816 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Jeremy Avigad, Simon Hudon
-/
import Mathlib.Algebra.Notation.Defs
import Mathlib.Data.Set.Subsingleton
import Mathlib.Logic.Equiv.Defs
/-!
# Partial values of a type
... | theorem mem_coe {a : Ξ±} {o : Option Ξ±} : a β (o : Part Ξ±) β a β o :=
mem_ofOption
| Mathlib/Data/Part.lean | 300 | 302 |
/-
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.... | /-- We define `fourierBasis` to be a `β€`-indexed Hilbert basis for `Lp β 2 haarAddCircle`,
which by definition is an isometric isomorphism from `Lp β 2 haarAddCircle` to `βΒ²(β€, β)`. -/
def fourierBasis : HilbertBasis β€ β (Lp β 2 <| @haarAddCircle T hT) :=
HilbertBasis.mk orthonormal_fourier (span_fourierLp_closure_eq... | Mathlib/Analysis/Fourier/AddCircle.lean | 356 | 359 |
/-
Copyright (c) 2020 David WΓ€rn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David WΓ€rn
-/
import Mathlib.Order.Ideal
import Mathlib.Data.Finset.Max
/-!
# The back and forth method and countable dense linear orders
## Results
Suppose `Ξ± Ξ²` are linear orders, wit... |
-- variable (Ξ± Ξ²)
/-- Any countable linear order embeds in any nontrivial dense linear order. -/
theorem embedding_from_countable_to_dense [Countable Ξ±] [DenselyOrdered Ξ²] [Nontrivial Ξ²] :
Nonempty (Ξ± βͺo Ξ²) := by
cases nonempty_encodable Ξ±
rcases exists_pair_lt Ξ² with β¨x, y, hxyβ©
obtain β¨a, haβ© := exists_be... | Mathlib/Order/CountableDenseLinearOrder.lean | 221 | 235 |
/-
Copyright (c) 2024 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.MvPolynomial.Monad
import Mathlib.LinearAlgebra.Charpoly.ToMatrix
import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition
import Mathlib.Li... | apply Polynomial.natTrailingDegree_le_of_ne_zero
intro h
apply_fun (MvPolynomial.eval ((chooseBasis R L).repr x)) at h
rw [polyCharpoly_coeff_eval, map_zero] at h
apply Polynomial.trailingCoeff_nonzero_iff_nonzero.mpr _ h
apply (LinearMap.charpoly_monic _).ne_zero
end
| Mathlib/Algebra/Module/LinearMap/Polynomial.lean | 491 | 498 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Gabriel Ebner
-/
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Algebra.Group.Basic
import Mathlib.Data.Nat.Basic
/-!
# Cast of integers (additional theorems)
This ... | | (n + 1 : β) => by rw [cast_natCast, neg_ofNat_succ]; simp
| -[n+1] => by rw [Int.neg_negSucc, cast_natCast]; simp
| Mathlib/Data/Int/Cast/Basic.lean | 79 | 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
-/
import Mathlib.Data.Countable.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Order.Disjointed
import Mathlib.MeasureTheory.OuterMeasure.Defs... | [SecondCountableTopology Ξ±] {s : Set Ξ±} (hs : ΞΌ s β 0) :
β x β s, β t β π[s] x, 0 < ΞΌ t := by
contrapose! hs
simp only [nonpos_iff_eq_zero] at hs
exact measure_null_of_locally_null s hs
end OuterMeasureClass
| Mathlib/MeasureTheory/OuterMeasure/Basic.lean | 152 | 158 |
/-
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.MeasureTheory.Integral.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Lebesgue.Countable
import Mathlib.MeasureTheory.Integral.Le... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 150 | 150 | |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Eric Wieser
-/
import Mathlib.Algebra.BigOperators.GroupWithZero.Action
import Mathlib.Algebra.GroupWithZero.Invertible
import Mathlib.LinearAlgebra.Prod
/-!
# Trivial Square-Ze... | theorem snd_natCast [AddMonoidWithOne R] [AddMonoid M] (n : β) : (n : tsze R M).snd = 0 :=
rfl
@[simp]
| Mathlib/Algebra/TrivSqZeroExt.lean | 494 | 497 |
/-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.NumberTheory.LSeries.HurwitzZeta
import Mathlib.Analysis.PSeriesComplex
/-!
# Definition of the Riemann zeta function
## Main definitions:
* `rieman... | riemannZeta k = β' n : β, 1 / (n : β) ^ k := by
simp only [zeta_eq_tsum_one_div_nat_cpow
(by rwa [β ofReal_natCast, ofReal_re, β Nat.cast_one, Nat.cast_lt] : 1 < re k),
cpow_natCast]
| Mathlib/NumberTheory/LSeries/RiemannZeta.lean | 195 | 198 |
/-
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.Analysis.InnerProductSpace.Convex
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Combinatorics.Additive.AP.Three... | have hβ := ceil_lt_add_one hx.le
have hβ : 1 - x β€ 2 - βxββ := by linarith
calc
_ β€ exp (1 - x) / (x + 1) := ?_
_ β€ exp (2 - βxββ) / (x + 1) := by gcongr
_ < _ := by gcongr
rw [le_div_iffβ (add_pos hx zero_lt_one), β le_div_iffβ' (exp_pos _), β exp_sub, neg_mul,
sub_neg_eq_add, two_mul, sub_add... | Mathlib/Combinatorics/Additive/AP/Three/Behrend.lean | 312 | 320 |
/-
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... |
theorem DifferentiableWithinAt.congr_of_eventuallyEq_insert (h : DifferentiableWithinAt π f s x)
(hβ : fβ =αΆ [π[insert x s] x] f) : DifferentiableWithinAt π fβ s x :=
(h.insert.congr_of_eventuallyEq_of_mem hβ (mem_insert _ _)).of_insert
| Mathlib/Analysis/Calculus/FDeriv/Basic.lean | 923 | 926 |
/-
Copyright (c) 2022 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Group.Nat.Even
import Mathlib.Data.Nat.Cast.Basic
import Mathlib.Data.Nat.Cast.Commute
import Mathlib.Data.Set.Operations
import Mathlib.Logic.Fu... | Mathlib/Algebra/Ring/Parity.lean | 405 | 412 | |
/-
Copyright (c) 2023 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael GeiΓer, Michael Stoll
-/
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.DiophantineApproximation.Basic
import Mathlib.NumberTheory.Zsqrtd.Basic
import Mathlib.Tactic... | refine β¨fun H => ?_, fun H => h.y_strictMono Hβ©
contrapose! H
exact h.y_strictMono.monotone H
/-- The `n`th power of a fundamental solution is trivial if and only if `n = 0`. -/
| Mathlib/NumberTheory/Pell.lean | 508 | 512 |
/-
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.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.C... | Mathlib/FieldTheory/RatFunc/Basic.lean | 1,048 | 1,049 | |
/-
Copyright (c) 2020 Paul van Wamelen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Paul van Wamelen
-/
import Mathlib.Data.Nat.Factors
import Mathlib.NumberTheory.FLT.Basic
import Mathlib.NumberTheory.PythagoreanTriples
import Mathlib.RingTheory.Coprime.Lemmas
impo... |
theorem Int.isCoprime_of_sq_sum {r s : β€} (h2 : IsCoprime s r) : IsCoprime (r ^ 2 + s ^ 2) r := by
rw [sq, sq]
exact (IsCoprime.mul_left h2 h2).mul_add_left_left r
@[deprecated (since := "2025-01-23")] alias Int.coprime_of_sq_sum := Int.isCoprime_of_sq_sum
theorem Int.isCoprime_of_sq_sum' {r s : β€} (h : IsCoprim... | Mathlib/NumberTheory/FLT/Four.lean | 141 | 149 |
/-
Copyright (c) 2019 SΓ©bastien GouΓ«zel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jan-David Salchow, SΓ©bastien GouΓ«zel, Jean Lo, Yury Kudryashov, FrΓ©dΓ©ric Dupuis,
Heather Macbeth
-/
import Mathlib.Algebra.Module.Opposite
import Mathlib.Topology.Algebra.Group.Qu... | Mathlib/Topology/Algebra/Module/Basic.lean | 820 | 822 | |
/-
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 | 3,139 | 3,141 | |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Group.Multiset.Basic
/-!
# Bind operation for multisets
This file defines a few basic operations on `Multiset`, notably the mona... |
open scoped Function in -- required for scoped `on` notation
| Mathlib/Data/Multiset/Bind.lean | 202 | 203 |
/-
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, Sander Dahmen,
Kim Morrison, Chris Hughes, Anne Baanen, Junyan Xu
-/
import Mathlib.LinearAlgebra.Basis.VectorSpace
import Mathlib.LinearAlgebra.Dimensi... | Mathlib/LinearAlgebra/Dimension/DivisionRing.lean | 288 | 300 | |
/-
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.FDeriv.Linear
import Mathlib.Analysis.Calculus.FDeriv.Comp
/-!
# Additive operations on derivative... |
@[fun_prop]
| Mathlib/Analysis/Calculus/FDeriv/Add.lean | 637 | 638 |
/-
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.Localization.Opposite
/-!
# Calculus of fractions
Following the definitions by [Gabriel and Zisman][gabriel-zisman-1967],
given a morphism prope... | obtain β¨Ο, eqβ© := Localization.exists_leftFraction L.op W.op f.op
refine β¨Ο.unop, Quiver.Hom.op_inj ?_β©
rw [eq, MorphismProperty.RightFraction.op_map]
rfl
lemma MorphismProperty.RightFraction.map_eq_iff
| Mathlib/CategoryTheory/Localization/CalculusOfFractions.lean | 943 | 948 |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, YaΓ«l Dillies
-/
import Mathlib.Order.Cover
import Mathlib.Order.LatticeIntervals
import Mathlib.Order.GaloisConnection.Defs
/-!
# Modular Lattices
This file defines (... | Β· simp [disjoint_iff, Subtype.ext_iff, inf_comm a c, inf_assoc a, β inf_assoc b, hβ.inf_eq_bot]
Β· simp only [Iic.codisjoint_iff, inf_comm a, IsModularLattice.inf_sup_inf_assoc]
simp [inf_of_le_left hβ, hβ.sup_eq_top]
| Mathlib/Order/ModularLattice.lean | 376 | 379 |
/-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, Mitchell Lee
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Indicator
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Topology.Algebra.InfiniteSum.Defs
imp... | @[to_additive]
theorem Function.Surjective.tprod_eq_tprod_of_hasProd_iff_hasProd {Ξ±' : Type*} [CommMonoid Ξ±']
[TopologicalSpace Ξ±'] {e : Ξ±' β Ξ±} (hes : Function.Surjective e) (h1 : e 1 = 1) {f : Ξ² β Ξ±}
{g : Ξ³ β Ξ±'} (h : β {a}, HasProd f (e a) β HasProd g a) : β' b, f b = e (β' c, g c) :=
| Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | 544 | 547 |
/-
Copyright (c) 2023 Michael Rothgang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Rothgang
-/
import Mathlib.Geometry.Manifold.Diffeomorph
import Mathlib.Topology.IsLocalHomeomorph
/-!
# Local diffeomorphisms between manifolds
In this file, we define `C... | hf.localInverse_contMDiffAt.mdifferentiableAt hn
end IsLocalDiffeomorphAt
/-- `f : M β N` is called a **`C^n` local diffeomorphism on *s*** iff it is a local diffeomorphism
| Mathlib/Geometry/Manifold/LocalDiffeomorph.lean | 218 | 222 |
/-
Copyright (c) 2021 RΓ©my Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, SΓ©bastien GouΓ«zel, RΓ©my Degenne
-/
import Mathlib.MeasureTheory.Integral.FinMeasAdditive
/-!
# Extension of a linear function from indicators to L1
Give... | exact (setToL1SCLM_congr_left' hT hT' h f).symm
theorem setToL1_add_left (hT : DominatedFinMeasAdditive ΞΌ T C)
(hT' : DominatedFinMeasAdditive ΞΌ T' C') (f : Ξ± ββ[ΞΌ] E) :
setToL1 (hT.add hT') f = setToL1 hT f + setToL1 hT' f := by
suffices setToL1 (hT.add hT') = setToL1 hT + setToL1 hT' by
rw [this, Con... | Mathlib/MeasureTheory/Integral/SetToL1.lean | 448 | 463 |
/-
Copyright (c) 2024 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
import Mathlib.CategoryTheory.EffectiveEpi.Enough
import Mathlib.CategoryTheory.EffectiveEpi.Preserves
import Mathlib.CategoryTheory.Sites.Coherent.RegularTopolog... | include F in
lemma Functor.reflects_preregular : Preregular C where
exists_fac f g _ := by
obtain β¨W, f', _, i, wβ© := Preregular.exists_fac (F.map f) (F.map g)
refine β¨_, F.preimage (F.effectiveEpiOver W β« f'),
β¨F.effectiveEpi_of_map _ ?_, F.preimage (F.effectiveEpiOver W β« i), ?_β©β©
Β· simp only [Fun... | Mathlib/CategoryTheory/Sites/Coherent/ReflectsPreregular.lean | 25 | 33 |
/-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.ImproperIntegrals
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.MeasureTheory.Measure.Haar.NormedSpace
... | Mathlib/Analysis/MellinTransform.lean | 464 | 476 | |
/-
Copyright (c) 2022 Wrenna Robson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Wrenna Robson
-/
import Mathlib.Topology.MetricSpace.Basic
/-!
# Infimum separation
This file defines the extended infimum separation of a set. This is approximately dual to the
diame... | theorem einfsep_lt_top_iff : s.einfsep < β β s.Nontrivial :=
β¨nontrivial_of_einfsep_lt_top, Nontrivial.einfsep_lt_topβ©
theorem einfsep_ne_top_iff : s.einfsep β β β s.Nontrivial :=
| Mathlib/Topology/MetricSpace/Infsep.lean | 227 | 230 |
/-
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... | protected theorem MeasurableSet.preimage {t : Set Ξ²} (ht : MeasurableSet t) (hf : Measurable f) :
MeasurableSet (f β»ΒΉ' t) :=
hf ht
@[measurability, fun_prop]
| Mathlib/MeasureTheory/MeasurableSpace/Basic.lean | 274 | 278 |
/-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Bilinear
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Group.Pointwise.Finset.Basic
import Mathlib.Algebra.Group.Pointwise.Set.B... | exact smul_induction_on hx (fun a ha x hx β¦ β¨_, smul _ ha _ hxβ©)
fun x y β¨_, hxβ© β¨_, hyβ© β¦ β¨_, add _ _ _ _ hx hyβ©
theorem smul_mono (hij : I β€ J) (hnp : N β€ P) : I β’ N β€ J β’ P :=
| Mathlib/Algebra/Algebra/Operations.lean | 136 | 139 |
/-
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, Peter Pfaffelhuber
-/
import Mathlib.MeasureTheory.Constructions.Cylinders
import Mathlib.MeasureTheory.Measure.Typeclasses.Probability
/-!
# Projective measure families a... | obtain β¨i, hi_emptyβ© := h
rw [measure_univ_eq hP I {i}]
have : (univ : Set ((j : {x // x β ({i} : Finset ΞΉ)}) β Ξ± j)) = β
:= by simp [hi_empty]
simp [this]
| inr h =>
have : S = Finset.restrictβ hJI β»ΒΉ' T :=
eq_of_cylinder_eq_of_subset h_eq hJI
rw [hP I J hJI, Measure.map_apply _ hT, thi... | Mathlib/MeasureTheory/Constructions/Projective.lean | 87 | 102 |
/-
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... | arrows c d := {p | Discrete.Arrows c d p}
inv := by rintro _ _ _ β¨β©; simp only [inv_eq_inv, IsIso.inv_id]; constructor
mul := by rintro _ _ _ _ β¨β© _ β¨β©; rw [Category.comp_id]; constructor
| Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean | 267 | 269 |
/-
Copyright (c) 2024 Ira Fesefeldt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ira Fesefeldt
-/
import Mathlib.SetTheory.Ordinal.Arithmetic
/-!
# Ordinal Approximants for the Fixed points on complete lattices
This file sets up the ordinal-indexed approximation t... | theorem le_lfpApprox {a : Ordinal} : x β€ lfpApprox f x a := by
rw [lfpApprox]
apply le_sSup
simp only [exists_prop, Set.union_singleton, Set.mem_insert_iff, Set.mem_setOf_eq, true_or]
| Mathlib/SetTheory/Ordinal/FixedPointApproximants.lean | 87 | 90 |
/-
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.Combinatorics.SimpleGraph.Path
import Mathlib.Combinatorics.SimpleGraph.Operations
import Mathlib.Data.Finset.Pairwise
import M... | (β€ : SimpleGraph (Fin n)) βͺg G := by
simp only [CliqueFree, isNClique_iff, isClique_iff_induce_eq, not_forall, Classical.not_not] at h
| Mathlib/Combinatorics/SimpleGraph/Clique.lean | 335 | 336 |
/-
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.MeasureTheory.Integral.IntegrableOn
/-!
# Locally integrable functions
A function is called *locally integrable* (`MeasureTheory.LocallyIntegrabl... | choose u u_open xu hu using this
obtain β¨T, T_count, hTβ© : β T : Set s, T.Countable β§ s β β i β T, u i := by
have : s β β x : s, u x := fun y hy => mem_iUnion_of_mem β¨y, hyβ© (xu β¨y, hyβ©)
obtain β¨T, hT_count, hT_unβ© := isOpen_iUnion_countable u u_open
exact β¨T, hT_count, by rwa [hT_un]β©
refine β¨u '' T,... | Mathlib/MeasureTheory/Function/LocallyIntegrable.lean | 83 | 100 |
/-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: SΓ©bastien GouΓ«zel, Johannes HΓΆlzl, Yury Kudryashov, Patrick Massot
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Order.Filter.AtTopBot.Archimedean
import Mathlib.Order.Iterate
impor... | theorem ENNReal.tendsto_pow_atTop_nhds_zero_of_lt_one {r : ββ₯0β} (hr : r < 1) :
Tendsto (fun n : β β¦ r ^ n) atTop (π 0) := by
rcases ENNReal.lt_iff_exists_coe.1 hr with β¨r, rfl, hr'β©
rw [β ENNReal.coe_zero]
norm_cast at *
apply NNReal.tendsto_pow_atTop_nhds_zero_of_lt_one hr
@[simp]
protected theorem ENNR... | Mathlib/Analysis/SpecificLimits/Basic.lean | 248 | 256 |
/-
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.Localization.Opposite
/-!
# Calculus of fractions
Following the definitions by [Gabriel and Zisman][gabriel-zisman-1967],
given a morphism prope... | @[reassoc (attr := simp)]
lemma map_s_comp_map (Ο : W.RightFraction X Y) (L : C β₯€ D) (hL : W.IsInvertedBy L) :
L.map Ο.s β« Ο.map L hL = L.map Ο.f := by
letI := hL _ Ο.hs
simp [map]
| Mathlib/CategoryTheory/Localization/CalculusOfFractions.lean | 140 | 144 |
/-
Copyright (c) 2022 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.CharP.Lemmas
import Mathlib.Data.Fintype.Units
import Mathlib.GroupTheory.OrderOfElement
/-!
# Multiplicative... | Mathlib/NumberTheory/MulChar/Basic.lean | 274 | 274 | |
/-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.MeasureTheory.Integral.PeakFunction
/-! # Euler's infinite product for the sine function
This file... |
end EulerSine
| Mathlib/Analysis/SpecialFunctions/Trigonometric/EulerSineProd.lean | 325 | 338 |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, JoΓ«l Riou
-/
import Mathlib.CategoryTheory.Comma.Presheaf.Basic
import Mathlib.CategoryTheory.Elements
import Mathlib.CategoryTheory.Functor.KanExtension.Adjunction
import ... |
/-- `F β yoneda` is naturally isomorphic to `yoneda β F.op.lan`. -/
noncomputable def compYonedaIsoYonedaCompLan :
F β yoneda β
yoneda β F.op.lan :=
NatIso.ofComponents (fun X => Functor.leftKanExtensionUnique _
| Mathlib/CategoryTheory/Limits/Presheaf.lean | 346 | 350 |
/-
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
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Powerset
import Mathlib.Algebra.NoZeroSMulDivisors.Pi
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Finty... | @[simp]
theorem map_update_zero [DecidableEq ΞΉ] (m : β i, Mβ i) (i : ΞΉ) : f (update m i 0) = 0 :=
f.map_coord_zero i (update_self i 0 m)
| Mathlib/LinearAlgebra/Multilinear/Basic.lean | 172 | 175 |
/-
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.Data.Set.Countable
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Tactic.FunProp.Attr
import Mathlib.Tactic.Mea... |
@[simp]
lemma MeasurableSet.singleton [MeasurableSpace Ξ±] [MeasurableSingletonClass Ξ±] (a : Ξ±) :
MeasurableSet {a} :=
| Mathlib/MeasureTheory/MeasurableSpace/Defs.lean | 228 | 231 |
/-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
import Mathlib.SetTheory.Nimber.Basic
/-!
# Nim and the Sprague-Grundy theore... | iterate 3 apply congr_arg
ext i
exact @grundyValue_neg _ (@Impartial.moveRight_impartial β¨l, r, L, Rβ© _ _)
| Mathlib/SetTheory/Game/Nim.lean | 311 | 314 |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.WSeq.Basic
import Mathlib.Data.WSeq.Defs
import Mathlib.Data.WSeq.Productive
import Mathlib.Data.WSeq.Relation
deprecated_module (since :=... | Mathlib/Data/Seq/WSeq.lean | 705 | 705 | |
/-
Copyright (c) 2019 Gabriel Ebner. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Gabriel Ebner, SΓ©bastien GouΓ«zel, Yury Kudryashov, Yuyang Zhao
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Comp
import Mathlib.Analysis.Calcu... | protected nonrec theorem HasDerivAt.iterate {f : π β π} {f' : π} (hf : HasDerivAt f f' x)
(hx : f x = x) (n : β) : HasDerivAt f^[n] (f' ^ n) x :=
hf.iterate _ (have := hf.tendsto_nhds le_rfl; by rwa [hx] at this) hx n
| Mathlib/Analysis/Calculus/Deriv/Comp.lean | 305 | 308 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Kim Morrison
-/
import Mathlib.CategoryTheory.Subobject.Lattice
/-!
# Specific subobjects
We define `equalizerSubobject`, `kernelSubobject` and `imageSubobject`, which ar... |
section
| Mathlib/CategoryTheory/Subobject/Limits.lean | 314 | 315 |
/-
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.SetTheory.Cardinal.Finite
import Mathlib.Data.Set.Finite.Powerset
/-!
# Noncomputable Set Cardinality
We define the cardinality of set `s` as a term `Set... | refine β¨a, t, hat, ?_, ?_β©
Β· simp [Set.ext_iff, hts]
Β· simp
| Mathlib/Data/Set/Card.lean | 1,088 | 1,091 |
/-
Copyright (c) 2018 Jan-David Salchow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jan-David Salchow, Patrick Massot, Yury Kudryashov
-/
import Mathlib.Topology.Defs.Sequences
import Mathlib.Topology.UniformSpace.Cauchy
/-!
# Sequences in topological spaces
In t... | Mathlib/Topology/Sequences.lean | 387 | 388 | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Group.Conj
import Mathlib.Algebra.Group.Subgroup.Lattice
import Mathlib.Algebra.Group.Submonoid.BigOperators
import Mathlib.Data.Finset.Fin
import ... | wlog lt : i < j generalizing i j
Β· rw [swap_comm]; exact this _ _ ne.symm (ne.lt_or_lt.resolve_left lt)
induction' j using Fin.induction with j ih
Β· cases lt
have mem : swap j.castSucc j.succ β Submonoid.closure
(Set.range fun (i : Fin n) β¦ swap i.castSucc i.succ) := Submonoid.subset_closure β¨_, rflβ©
... | Mathlib/GroupTheory/Perm/Sign.lean | 126 | 140 |
/-
Copyright (c) 2017 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stephen Morgan, Kim Morrison, Johannes HΓΆlzl
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.Data.Set.CoeSort
import Mathlib.T... | Mathlib/CategoryTheory/Types.lean | 156 | 156 | |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.GroupWithZero.Action.Defs
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.NoZeroSMulDiviso... | theorem add_pow_eq_zero_of_add_le_succ_of_pow_eq_zero (h_comm : Commute x y) {m n k : β}
(hx : x ^ m = 0) (hy : y ^ n = 0) (h : m + n β€ k + 1) :
(x + y) ^ k = 0 := by
rw [h_comm.add_pow']
| Mathlib/RingTheory/Nilpotent/Basic.lean | 129 | 132 |
/-
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 | 379 | 379 | |
/-
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, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Logic.Equiv.Fin.Basic
imp... | refine β¨a, ha, C, hC, fun y hy n ys => (hp y hy n ys).trans ?_β©
have yr' : βyβ < r' := by rwa [ball_zero_eq] at hy
have := ha.1.le -- needed to discharge a side goal on the next line
gcongr
exact mul_le_of_le_one_right ha.1.le (div_le_one_of_leβ yr'.le r'.coe_nonneg)
/-- If a function admits a power series e... | Mathlib/Analysis/Analytic/Basic.lean | 1,010 | 1,021 |
/-
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 | 1,713 | 1,721 | |
/-
Copyright (c) 2024 Kalle KytΓΆlΓ€. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle KytΓΆlΓ€
-/
import Mathlib.Analysis.Normed.Group.Basic
import Mathlib.Topology.MetricSpace.ProperSpace.Real
import Mathlib.Analysis.Normed.Ring.Lemmas
/-!
# Bounded operations
This... |
lemma SeminormedAddCommGroup.lipschitzWith_sub :
LipschitzWith 2 (fun (p : R Γ R) β¦ p.1 - p.2) := by
convert LipschitzWith.prod_fst.sub LipschitzWith.prod_snd
norm_num
instance : BoundedSub R := boundedSub_of_lipschitzWith_sub SeminormedAddCommGroup.lipschitzWith_sub
end SeminormedAddCommGroup
| Mathlib/Topology/Bornology/BoundedOperation.lean | 159 | 168 |
/-
Copyright (c) 2023 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, JoΓ«l Riou
-/
import Mathlib.Algebra.Category.ModuleCat.ChangeOfRings
import Mathlib.Algebra.Category.Ring.Basic
/-!
# Presheaves of modules over a presheaf of rings.
Give... | { app := fun X β¦ AddCommGrp.ofHom <| AddMonoidHom.mk' (Hom.app f X) (by simp)
naturality := fun X Y g β¦ by ext x; exact naturality_apply f g x }
@[simp]
lemma toPresheaf_obj_coe (X : Cα΅α΅) :
| Mathlib/Algebra/Category/ModuleCat/Presheaf.lean | 137 | 141 |
/-
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.VecNotation
import Mathlib.Logic.Small.Basic
import Mathlib.SetTheory.ZFC.PSet
/-!
# A model of ZFC
In this file, we model Zermelo-Fraenkel ... | Mathlib/SetTheory/ZFC/Basic.lean | 1,271 | 1,290 | |
/-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, SΓ©bastien GouΓ«zel, RΓ©my Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.Mul
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
imp... | rwa [one_div_div] at h_rpow_add_rpow_le_add
theorem rpow_add_le_add_rpow {p : β} (a b : ββ₯0β) (hp : 0 β€ p) (hp1 : p β€ 1) :
(a + b) ^ p β€ a ^ p + b ^ p := by
rcases hp.eq_or_lt with (rfl | hp_pos)
| Mathlib/Analysis/MeanInequalitiesPow.lean | 312 | 316 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Data.Ordering.Lemmas
import Mathlib.Data.PNat.Basic
import Mathlib.SetTheory.Ordinal.Principal
import Ma... | (k β 0 β R < ((Ο ^ repr a0) ^ succ (k : Ordinal))) β§
((Ο ^ repr a0) ^ (k : Ordinal)) * ((Ο ^ repr a0) * (n : β) + repr a') + R =
((Ο ^ repr a0) * (n : β) + repr a' + m) ^ succ (k : Ordinal) := by
intro R'
haveI No : NF (oadd a0 n a') :=
N0.oadd n (Na'.below_of_lt' <| lt_of_le_of_lt (le_add_rig... | Mathlib/SetTheory/Ordinal/Notation.lean | 802 | 807 |
/-
Copyright (c) 2018 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp
-/
import Mathlib.Data.Nat.Find
import Mathlib.Algebra.Module.Pi
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
/-!
# Basic properties of holors
Holors... | theorem cprankMax_sum [NonUnitalNonAssocSemiring Ξ±] {Ξ²} {n : β} (s : Finset Ξ²)
(f : Ξ² β Holor Ξ± ds) : (β x β s, CPRankMax n (f x)) β CPRankMax (s.card * n) (β x β s, f x) :=
letI := Classical.decEq Ξ²
Finset.induction_on s (by simp [CPRankMax.zero])
| Mathlib/Data/Holor.lean | 282 | 285 |
/-
Copyright (c) 2022 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Data.Nat.Cast.Field
import Mathlib.GroupTheory.Abelianization
import Mathlib.GroupTheory.GroupAction.CardComm... | commProb_odd (by decide)]
field_simp [h0, h2.two_dvd]
norm_num
Β· have := div_four_lt h0 h1
rw [reciprocalFactors_odd h1 h2, commProb_cons, commProb_reciprocal (n / 4 + 1)]
have key : n % 4 = 1 β¨ n % 4 = 3 := Nat.odd_mod_four_iff.mp (Nat.odd_iff.mp h2)
have hn : Odd (n % 4) := by rcases key... | Mathlib/GroupTheory/CommutingProbability.lean | 197 | 220 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Minchao Wu, Mario Carneiro
-/
import Mathlib.Data.Finset.Attach
import Mathlib.Data.Finset.Disjoint
import Mathlib.Data.Finset.Erase
import Mat... | theorem erase_insert_subset (a : Ξ±) (s : Finset Ξ±) : erase (insert a s) a β s :=
| Mathlib/Data/Finset/Basic.lean | 194 | 194 |
/-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson
-/
import Mathlib.Computability.DFA
import Mathlib.Data.Fintype.Powerset
/-!
# Nondeterministic Finite Automata
This file contains the definition of a Nondeterministic Finite... |
end DFA
| Mathlib/Computability/NFA.lean | 147 | 154 |
/-
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
/-... | Mathlib/Data/Sym/Sym2.lean | 524 | 525 | |
/-
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, Jeremy Avigad
-/
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Data.Set.Finite.Range
import Mathlib.Data.Set.Lattice
import Mathlib.Topology.Defs.... | Mathlib/Topology/Basic.lean | 808 | 814 | |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta HernΓ‘ndez Palacios
-/
import Mathlib.SetTheory.Cardinal.Arithmetic
import Mathlib.SetTheory.Ordinal.FixedPoint
/-!
# Cofinality
This file co... | obtain β¨n, eβ© := Cardinal.lt_aleph0.1 h
have := cof_cof o
rw [e, ord_nat] at this
cases n
Β· simp at e
simp [e, not_zero_isLimit] at l
| Mathlib/SetTheory/Cardinal/Cofinality.lean | 584 | 589 |
/-
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.MeasureTheory.Measure.Typeclasses.Finite
import Mathlib.MeasureTheory.Measure.Typeclasses.NoAtoms
import Mathlib.MeasureTheory.Measure.... | Mathlib/MeasureTheory/Measure/Typeclasses.lean | 41 | 44 | |
/-
Copyright (c) 2023 Geoffrey Irving. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler, Geoffrey Irving, Stefan Kebekus
-/
import Mathlib.Analysis.Analytic.Composition
import Mathlib.Analysis.Analytic.Linear
import Mathlib.Analysis.NormedSpace.OperatorNor... | (q.isLittleO_one_of_lt_radius hr.2)).isBigO
refine (p.prod q).le_radius_of_isBigO ((isBigO_of_le _ fun n β¦ ?_).trans this)
rw [norm_mul, norm_norm, β add_mul, norm_mul]
| Mathlib/Analysis/Analytic/Constructions.lean | 274 | 276 |
/-
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... | end BinaryOp
section ULift
| Mathlib/Logic/Equiv/Basic.lean | 917 | 920 |
/-
Copyright (c) 2021 Aaron Anderson, Jesse Michael Han, Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jesse Michael Han, Floris van Doorn
-/
import Mathlib.Data.Finset.Basic
import Mathlib.ModelTheory.Syntax
import Mathlib.Data.List.... | have h := ih3 (Sum.elim (v β Sum.inl) (snoc (v β Sum.inr) a))
simp only [Sum.elim_comp_inl, Sum.elim_comp_inr] at h
| Mathlib/ModelTheory/Semantics.lean | 822 | 823 |
/-
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.Data.ENNReal.Real
import Mathlib.Tactic.Bound.Attribute
import Mathlib.Topology... | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | 1,557 | 1,559 | |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta HernΓ‘ndez Palacios
-/
import Mathlib.SetTheory.Cardinal.Arithmetic
import Mathlib.SetTheory.Ordinal.FixedPoint
/-!
# Cofinality
This file co... | Β· rw [β Cardinal.succ_zero, succ_le_iff]
simpa [lt_iff_le_and_ne, Cardinal.zero_le] using fun h =>
succ_ne_zero o (cof_eq_zero.1 (Eq.symm h))
| Mathlib/SetTheory/Cardinal/Cofinality.lean | 394 | 397 |
/-
Copyright (c) 2024 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.Category.ModuleCat.Sheaf.ChangeOfRings
import Mathlib.CategoryTheory.Sites.LocallySurjective
/-!
# The associated sheaf of a presheaf of modules
In thi... | smul_add := Sheafify.smul_add Ξ± Ο
add_smul := Sheafify.add_smul Ξ± Ο
mul_smul := Sheafify.mul_smul Ξ± Ο
protected lemma map_smul :
A.val.map Ο (smul Ξ± Ο r m) = smul Ξ± Ο (R.val.map Ο r) (A.val.map Ο m) := by
let S := Presheaf.imageSieve Ξ± (R.val.map Ο r) β Presheaf.imageSieve Ο (A.val.map Ο m)
have hS : S β... | Mathlib/Algebra/Category/ModuleCat/Presheaf/Sheafify.lean | 288 | 299 |
/-
Copyright (c) 2020 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp
-/
import Mathlib.Algebra.Algebra.Spectrum.Basic
import Mathlib.Algebra.Module.LinearMap.Basic
import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
import Mathl... | /-- The exponent of a generalized eigenvalue is never 0. -/
theorem exp_ne_zero_of_hasGenEigenvalue {f : End R M} {ΞΌ : R} {k : β}
(h : f.HasGenEigenvalue ΞΌ k) : k β 0 :=
HasUnifEigenvalue.exp_ne_zero h
/-- The union of the kernels of `(f - ΞΌ β’ id) ^ k` over all `k`. -/
abbrev maxGenEigenspace (f : End R M) (ΞΌ : ... | Mathlib/LinearAlgebra/Eigenspace/Basic.lean | 497 | 504 |
/-
Copyright (c) 2020 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Algebra.Group.Conj
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.Group.Subgroup.Ker
/-!
# Basic results on subgroups
We prove basic results... | Mathlib/Algebra/Group/Subgroup/Basic.lean | 3,373 | 3,379 | |
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Data.SetLike.Basic
import Mathlib.ModelTheory.Semantics
/-!
# Definable Sets
This file defines what it means for a set over a first-order structure t... | (Ο : L βα΄Έ L') [Ο.IsExpansionOn M] : A.Definable L' s := by
obtain β¨Ο, rflβ© := h
refine β¨(Ο.addConstants A).onFormula Ο, ?_β©
ext x
simp only [mem_setOf_eq, LHom.realize_onFormula]
| Mathlib/ModelTheory/Definability.lean | 52 | 57 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.CharP.Two
import Mathlib.Data.Nat.Cast.Field
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.Nat.Factorization.Induction
import Mat... | right_inv := fun m => by simp only [Subtype.coe_mk, Subtype.coe_eta] }
rw [totient_eq_card_coprime, card_congr e, card_eq_fintype_card, Fintype.card_coe]
theorem totient_le (n : β) : Ο n β€ n :=
((range n).card_filter_le _).trans_eq (card_range n)
theorem totient_lt (n : β) (hn : 1 < n) : Ο n < n :=
| Mathlib/Data/Nat/Totient.lean | 50 | 56 |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle
import Mathlib.Geometry.Euclidean.Circumcenter
/-!
# Angles in circles and sphere.
This file proves results ... |
end Sphere
end EuclideanGeometry
namespace Affine
namespace Triangle
| Mathlib/Geometry/Euclidean/Angle/Sphere.lean | 228 | 235 |
/-
Copyright (c) 2022 YaΓ«l Dillies, Sara Rousta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: YaΓ«l Dillies, Sara Rousta
-/
import Mathlib.Logic.Equiv.Set
import Mathlib.Order.Interval.Set.OrderEmbedding
import Mathlib.Order.SetNotation
/-!
# Properties of unbundled ... | Mathlib/Order/UpperLower/Basic.lean | 2,013 | 2,015 | |
/-
Copyright (c) 2024 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.ShortComplex.ShortExact
import Mathlib.CategoryTheory.MorphismProperty.Composition
/-!
# Epimorphisms with an injective kernel
In this file, w... | lemma epiWithInjectiveKernel_of_iso {X Y : C} (f : X βΆ Y) [IsIso f] :
epiWithInjectiveKernel f := by
rw [epiWithInjectiveKernel_iff]
exact β¨0, inferInstance, 0, by simp,
β¨ShortComplex.Splitting.ofIsZeroOfIsIso _ (isZero_zero C) (by dsimp; infer_instance)β©β©
| Mathlib/CategoryTheory/Abelian/EpiWithInjectiveKernel.lean | 53 | 57 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.Algebra.Notation.Prod
import Mathlib.Data.Set.Image
/-!
# Support of a func... | Mathlib/Algebra/Group/Support.lean | 335 | 337 | |
/-
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.Topology.MetricSpace.Lipschitz
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
/-!
# HΓΆlder continuous functions
In this file we define HΓΆl... |
theorem ediam_image_le (hf : HolderOnWith C r f s) :
EMetric.diam (f '' s) β€ (C : ββ₯0β) * EMetric.diam s ^ (r : β) :=
hf.ediam_image_le_of_le le_rfl
theorem ediam_image_le_of_subset (hf : HolderOnWith C r f s) (ht : t β s) :
EMetric.diam (f '' t) β€ (C : ββ₯0β) * EMetric.diam t ^ (r : β) :=
| Mathlib/Topology/MetricSpace/Holder.lean | 136 | 142 |
/-
Copyright (c) 2022 Stuart Presnell. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stuart Presnell, Eric Wieser, YaΓ«l Dillies, Patrick Massot, Kim Morrison
-/
import Mathlib.Algebra.GroupWithZero.InjSurj
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.... | Mathlib/Algebra/Order/Interval/Set/Instances.lean | 385 | 385 | |
/-
Copyright (c) 2021 RΓ©my Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: RΓ©my Degenne
-/
import Mathlib.Probability.Independence.Kernel
import Mathlib.MeasureTheory.Constructions.Pi
/-!
# Independence of sets of sets and measure spaces (Ο-algebras)
* A fami... | simp only [iIndepSet, iIndep, Kernel.iIndepSet]
lemma iIndepSet.isProbabilityMeasure (h : iIndepSet s ΞΌ) : IsProbabilityMeasure ΞΌ :=
((iIndepSet_iff_iIndep _ _).1 h).isProbabilityMeasure
| Mathlib/Probability/Independence/Basic.lean | 193 | 196 |
/-
Copyright (c) 2023 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Logic.UnivLE
import Mathlib.SetTheory.Ordinal.Basic
/-!
# UnivLE and cardinals
-/
noncomputable section
universe u v
open Cardinal
| theorem univLE_iff_cardinal_le : UnivLE.{u, v} β univ.{u, v+1} β€ univ.{v, u+1} := by
rw [β not_iff_not, univLE_iff]; simp_rw [small_iff_lift_mk_lt_univ]; push_neg
-- strange: simp_rw [univ_umax.{v,u}] doesn't work
refine β¨fun β¨Ξ±, leβ© β¦ ?_, fun h β¦ ?_β©
Β· rw [univ_umax.{v,u}, β lift_le.{u+1}, lift_univ, lift_lift... | Mathlib/SetTheory/Cardinal/UnivLE.lean | 19 | 27 |
/-
Copyright (c) 2020 Jannis Limperg. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jannis Limperg
-/
import Mathlib.Data.List.Induction
/-!
# Lemmas about List.*Idx functions.
Some specification lemmas for `List.mapIdx`, `List.mapIdxM`, `List.foldlIdx` and `List.fo... | map (uncurry f) (enumFrom n l) = zipWith (fun i β¦ f (i + n)) (range (length l)) l := by
intro l
generalize e : l.length = len
revert l
induction' len with len ih <;> intros l e n f
Β· have : l = [] := by
| Mathlib/Data/List/Indexes.lean | 46 | 51 |
/-
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.Ordering.Basic
import Mathlib.Order.Synonym
/-!
# Comparison
This file provides basic results about orderings and comparison in linear orders.
... | toMin := minOfLe,
toMax := maxOfLe,
toDecidableLE := H,
toDecidableLT := fun a b => decidable_of_iff _ (h a b).eq_lt,
| Mathlib/Order/Compare.lean | 154 | 157 |
/-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.MvPolynomial.Rename
/-!
# Degrees of polynomials
This file establ... | classical
refine (supDegree_single s a).trans_le ?_
| Mathlib/Algebra/MvPolynomial/Degrees.lean | 84 | 85 |
/-
Copyright (c) 2022 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.Calculus.Deriv.Basic
import Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap
import Mathlib.MeasureTheory.Covering.Bes... | {f' : β β β} (hf' : β x β s, HasDerivWithinAt f (f' x) s x) :
(volume.withDensity (fun x β¦ ENNReal.ofReal (g x))).map f.symm s
= ENNReal.ofReal (β« x in s, |f' x| * g (f x)) := by
rw [MeasurableEquiv.withDensity_ofReal_map_symm_apply_eq_integral_abs_det_fderiv_mul volume hs
f hg hg_int hf']
simp ... | Mathlib/MeasureTheory/Function/Jacobian.lean | 1,253 | 1,264 |
/-
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 HΓΆlzl, Yury Kudryashov
-/
import Mathlib.Order.Interval.Set.ProjIcc
import Mathlib.Topology.Algebra.Order.Field
import Mathlib.Topolo... | end Metric
| Mathlib/Topology/MetricSpace/Lipschitz.lean | 221 | 222 |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.GroupWithZero.Action.Defs
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.NoZeroSMulDiviso... |
theorem IsNilpotent.isUnit_add_one [Ring R] {r : R} (hnil : IsNilpotent r) : IsUnit (r + 1) := by
rw [β IsUnit.neg_iff, neg_add']
| Mathlib/RingTheory/Nilpotent/Basic.lean | 64 | 66 |
/-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Calle SΓΆnne, Adam Topaz
-/
import Mathlib.Data.Setoid.Partition
import Mathlib.Topology.LocallyConstant.Basic
import Mathlib.Topology.Separation.Regular
import Mathlib.Topology.Connected.T... | Mathlib/Topology/DiscreteQuotient.lean | 324 | 324 | |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Analysis.Convex.Side
import Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
/-!
# Oriented an... | the angle between the first and the second equals the angle between the first and the third. -/
@[simp]
theorem oangle_add_swap {p pβ pβ pβ : P} (hpβ : pβ β p) (hpβ : pβ β p) (hpβ : pβ β p) :
β‘ pβ p pβ + β‘ pβ p pβ = β‘ pβ p pβ :=
| Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean | 242 | 245 |
/-
Copyright (c) 2024 YaΓ«l Dillies, Kin Yau James Wong. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: YaΓ«l Dillies, Kin Yau James Wong, RΓ©my Degenne
-/
import Mathlib.MeasureTheory.Function.AEEqOfLIntegral
import Mathlib.Probability.Kernel.Composition.MeasureCompProd
... | Mathlib/Probability/Kernel/Disintegration/Basic.lean | 479 | 485 | |
/-
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.... | them for `max` and `min` fields. It also takes `[Ord Ξ±]` as an argument and uses them for `compare`
fields. See `LinearOrder.lift` for a version that autogenerates `compare` fields, and
`LinearOrder.liftWithOrd'` for one that auto-generates `min` and `max` fields.
| Mathlib/Order/Basic.lean | 947 | 949 |
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