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) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Kexing Ying, Eric Wieser
-/
import Mathlib.Data.Finset.Sym
import Mathlib.LinearAlgebra.BilinearMap
import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
import Mathlib.Linea... | fun _ _ β¦ by
simp only [LinearMap.mkβ_apply]
| Mathlib/LinearAlgebra/QuadraticForm/Basic.lean | 331 | 332 |
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes HΓΆlzl, YaΓ«l Dillies
-/
import Mathlib.Analysis.Normed.Group.Seminorm
import Mathlib.Data.NNReal.Basic
import Mathlib.Topology.Algebra.Support
import Mathlib.To... |
@[to_additive norm_nsmul_le]
lemma norm_pow_le_mul_norm : β {n : β}, βa ^ nβ β€ n * βaβ
| 0 => by simp
| Mathlib/Analysis/Normed/Group/Basic.lean | 744 | 747 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Logic.Encodable.Pi
import Mathlib.Logic.Function.Iterate
/-!
# The primitive recursive functions
The primitive ... | Mathlib/Computability/Primrec.lean | 436 | 436 | |
/-
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.Data.Set.Prod
/-!
# N-ary images of sets
This file defines `Set.image2`, the binary image of sets.
This is mostly useful to define pointwise oper... | Mathlib/Data/Set/NAry.lean | 360 | 364 | |
/-
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.Deriv.Pow
import Mathlib.Analysis.Calculus.Deriv.Inv
/-!
# Derivatives of `x ^ m`, `m : β€`
In this file we p... | funext <| deriv_zpow m
theorem derivWithin_zpow (hxs : UniqueDiffWithinAt π s x) (h : x β 0 β¨ 0 β€ m) :
derivWithin (fun x => x ^ m) s x = (m : π) * x ^ (m - 1) :=
(hasDerivWithinAt_zpow m x h s).derivWithin hxs
@[simp]
| Mathlib/Analysis/Calculus/Deriv/ZPow.lean | 86 | 92 |
/-
Copyright (c) 2020 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Halting
import Mathlib.Computability.TuringMachine
import Mathlib.Data.Num.Lemmas
import Mathlib.Tactic.DeriveFintype
import Mathlib.Comp... | Mathlib/Computability/TMToPartrec.lean | 1,728 | 1,746 | |
/-
Copyright (c) 2017 Johannes HΓΆlzl. 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.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.Nat.SuccPred
import Mathlib.Order.SuccPred.Initial... | le_iff_le_iff_lt_iff_lt.2 <| succ_lt_of_isLimit h
theorem limit_le {o} (h : IsLimit o) {a} : o β€ a β β x < o, x β€ a :=
β¨fun h _x l => l.le.trans h, fun H =>
(le_succ_of_isLimit h).1 <| le_of_not_lt fun hn => not_lt_of_le (H _ hn) (lt_succ a)β©
| Mathlib/SetTheory/Ordinal/Arithmetic.lean | 225 | 229 |
/-
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, Peter Pfaffelhuber
-/
import Mathlib.MeasureTheory.SetSemiring
import Mathlib.MeasureTheory.OuterMeasure.Induced
/-!
# Additive Contents
An additive content `m` on a set ... | Β· rw [sum_insert, sum_singleton]
simp only [Finset.mem_singleton]
refine fun hs_eq_t => hs_empty ?_
rw [β hs_eq_t] at h_dis
exact Disjoint.eq_bot_of_self h_dis
/-- An additive content is said to be sigma-sub-additive if for any sequence of sets `f` in `C` such
that `β i, f i β C`, we have `m (β i, f ... | Mathlib/MeasureTheory/Measure/AddContent.lean | 111 | 124 |
/-
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.Data.Fintype.List
import Mathlib.Data.Fintype.OfMap
/-!
# Cycles of a list
Lists have an equivalence relation of whether they are rotational permut... |
instance [DecidableEq Ξ±] (x : Ξ±) (s : Cycle Ξ±) : Decidable (x β s) :=
Quotient.recOnSubsingleton' s fun l => show Decidable (x β l) from inferInstance
| Mathlib/Data/List/Cycle.lean | 477 | 479 |
/-
Copyright (c) 2021 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.Data.Matrix.Basis
import Mathlib.Data.Matrix.DMatrix
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.Re... | i := inl t.i
j := inl t.j
hij := by simp [t.hij]
c := t.c
theorem toMatrix_sumInl (t : TransvectionStruct n R) :
| Mathlib/LinearAlgebra/Matrix/Transvection.lean | 239 | 244 |
/-
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.TotalComplex
/-! The symmetry of the total complex of a bicomplex
Let `K : HomologicalComplexβ C cβ cβ` be a bicomplex. If we assume both
`[To... |
@[reassoc (attr := simp)]
lemma ΞΉTotal_totalFlipIso_f_inv
(iβ : Iβ) (iβ : Iβ) (j : J) (h : ComplexShape.Ο cβ cβ c (iβ, iβ) = j) :
K.ΞΉTotal c iβ iβ j h β« (K.totalFlipIso c).inv.f j =
| Mathlib/Algebra/Homology/TotalComplexSymmetry.lean | 129 | 133 |
/-
Copyright (c) 2022 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.CategoryTheory.Comma.Arrow
/-!
# Commutative squares
This file provide an API for commutative squares in categories.
If `top`, `left`, `right`... | β¨by rw [Iso.comp_inv_eq, Category.assoc, Iso.eq_inv_comp, p.w]β©
theorem horiz_inv {f : W β
X} {i : Y β
Z} (p : CommSq f.hom g h i.hom) :
| Mathlib/CategoryTheory/CommSq.lean | 69 | 71 |
/-
Copyright (c) 2020 Aaron Anderson, Jalex Stark, Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark, Kyle Miller, Alena Gusakov
-/
import Mathlib.Combinatorics.SimpleGraph.Maps
import Mathlib.Data.Finset.Max
import Mathlib.Data.Sy... | degree. Note the assumption that the graph is nonempty is necessary as long as `G.minDegree` is
defined to be a natural. -/
theorem le_minDegree_of_forall_le_degree [DecidableRel G.Adj] [Nonempty V] (k : β)
(h : β v, k β€ G.degree v) : k β€ G.minDegree := by
| Mathlib/Combinatorics/SimpleGraph/Finite.lean | 308 | 311 |
/-
Copyright (c) 2023 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck, Ruben Van de Velde
-/
import Mathlib.Analysis.Calculus.ContDiff.Operations
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Shift
impor... | exact IH
variable (f) in
theorem iteratedDerivWithin_neg' :
iteratedDerivWithin n (fun z => -f z) s x = -iteratedDerivWithin n f s x :=
iteratedDerivWithin_neg f
include h hx
theorem iteratedDerivWithin_sub
(hf : ContDiffWithinAt π n f s x) (hg : ContDiffWithinAt π n g s x) :
iteratedDerivWithin ... | Mathlib/Analysis/Calculus/IteratedDeriv/Lemmas.lean | 85 | 100 |
/-
Copyright (c) 2022 Pierre-Alexandre Bazin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Pierre-Alexandre Bazin
-/
import Mathlib.Algebra.Module.DedekindDomain
import Mathlib.LinearAlgebra.FreeModule.PID
import Mathlib.Algebra.Module.Projective
import Mathlib.Algeb... | have : p ^ k β’ f1.choose β R β z := by
rw [β Quotient.mk_eq_zero, mk_smul, f1.choose_spec, β f.map_smul]
convert f.map_zero; change _ β’ Submodule.Quotient.mk _ = _
rw [β mk_smul, Quotient.mk_eq_zero, Algebra.id.smul_eq_mul, mul_one]
exact Submodule.mem_span_singleton_self _
obtain β¨a, haβ© := p_pow_s... | Mathlib/Algebra/Module/PID.lean | 153 | 165 |
/-
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.SpecialFunctions.Gamma.Basic
import Mathlib.Analysis.SpecialFunctions.PolarCoord
import Mathlib.Analysis.Complex.Convex
import Mathlib.D... | have A : β x : β, HasDerivAt (fun x => -(2 * b)β»ΒΉ * cexp (-b * x ^ 2))
(x * cexp (-b * x ^ 2)) x := by
intro x
convert ((hasDerivAt_pow 2 x).const_mul (-b)).cexp.const_mul (-(2 * b)β»ΒΉ) using 1
field_simp [hb']
ring
have B : Tendsto (fun y : β β¦ -(2 * b)β»ΒΉ * cexp (-b * (y : β) ^ 2))
atTop (π... | Mathlib/Analysis/SpecialFunctions/Gaussian/GaussianIntegral.lean | 172 | 180 |
/-
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, Floris van Doorn
-/
import Mathlib.Data.Countable.Small
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Powerset
import Mathlib.Dat... | Mathlib/SetTheory/Cardinal/Basic.lean | 1,483 | 1,484 | |
/-
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, SΓ©bastien GouΓ«zel
-/
import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
import Mathlib.MeasureTheory.Constructions.BorelSpace.Metric
import Mathlib.MeasureTheory... | Β· classical convert parallelepiped_single (ΞΉ := ΞΉ) 1
Β· exact zero_le_one
/-- A parallelepiped can be expressed on the standard basis. -/
theorem Basis.parallelepiped_eq_map {ΞΉ E : Type*} [Fintype ΞΉ] [NormedAddCommGroup E]
[NormedSpace β E] (b : Basis ΞΉ β E) :
| Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean | 75 | 80 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Group.Units.Basic
import Mathlib.Algebra.GroupWithZero.Basic
import Mathlib.Data.Int.Basic
import Mathlib.Lean.Meta.CongrTheorems
import Mathli... | Mathlib/Algebra/GroupWithZero/Units/Basic.lean | 524 | 526 | |
/-
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.Countable.Defs
import Mathlib.Data.Fin.Basic
import Mathlib.Data.Nat.Find
import Mathlib.Data.PNat.Equiv
import Mathlib.... | conv =>
rhs
rw [β hx]
rw [encode_injective.eq_iff, β Option.some_inj, Option.some_get, β hx, encodekβ]
| Mathlib/Logic/Encodable/Basic.lean | 211 | 214 |
/-
Copyright (c) 2020 SΓ©bastien GouΓ«zel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: SΓ©bastien GouΓ«zel, Floris van Doorn
-/
import Mathlib.Geometry.Manifold.MFDeriv.Defs
import Mathlib.Geometry.Manifold.ContMDiff.Defs
/-!
# Basic properties of the manifold FrΓ©chet ... | ### Relating differentiability in a manifold and differentiability in the model space
through extended charts
| Mathlib/Geometry/Manifold/MFDeriv/Basic.lean | 166 | 167 |
/-
Copyright (c) 2016 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, Mario Carneiro
-/
import Mathlib.Data.Set.Defs
import Mathlib.Logic.Basic
import Mathlib.Logic.ExistsUnique
import Mathlib.Logic.Nonempty
import Mathlib.Logic.Nontrivia... |
This definition is mathematically meaningful only when `f aβ = f aβ β g aβ = g aβ` (spelled
`g.FactorsThrough f`). In particular this holds if `f` is injective.
A typical use case is extending a function from a subtype to the entire type. If you wish to extend
`g : {b : Ξ² // p b} β Ξ³` to a function `Ξ² β Ξ³`, you shoul... | Mathlib/Logic/Function/Basic.lean | 660 | 666 |
/-
Copyright (c) 2018 Andreas Swerdlow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andreas Swerdlow, Kexing Ying
-/
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.LinearAlgebra.BilinearForm.Properties
/-!
# Bilinear form
This file defines ort... | rw [smul_right]
convert mul_zero (vi j) using 2
obtain rfl | hij := eq_or_ne i j
Β· exact ho
Β· exact h hij
/-- Given an orthogonal basis with respect to a bilinear form, the bilinear form is nondegenerate
iff the basis has no elements which are self-orthogonal. -/
| Mathlib/LinearAlgebra/BilinearForm/Orthogonal.lean | 224 | 231 |
/-
Copyright (c) 2018 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton
-/
import Mathlib.Topology.Bases
import Mathlib.Topology.DenseEmbedding
import Mathlib.Topology.Connected.TotallyDisconnected
/-! # Stone-Δech compactification
Construction ... | intro s hs
rw [β le_principal_iff]
refine iInf_le_of_le { u | s β u } ?_
refine iInf_le_of_le β¨hs, β¨s, rflβ©β© ?_
exact principal_mono.2 fun _ β¦ id
| Mathlib/Topology/StoneCech.lean | 122 | 126 |
/-
Copyright (c) 2021 YaΓ«l Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: YaΓ«l Dillies
-/
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Data.PNat.Defs
/-!
# Finite intervals of positive naturals
This file proves that `β+` is a `LocallyFiniteOrder` ... | theorem card_Ioo : #(Ioo a b) = b - a - 1 := by
rw [β Nat.card_Ioo, β map_subtype_embedding_Ioo, card_map]
@[simp]
theorem card_uIcc : #(uIcc a b) = (b - a : β€).natAbs + 1 := by
rw [β Nat.card_uIcc, β map_subtype_embedding_uIcc, card_map]
| Mathlib/Data/PNat/Interval.lean | 67 | 72 |
/-
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.Probability.Kernel.Disintegration.Density
import Mathlib.Probability.Kernel.WithDensity
/-!
# Radon-Nikodym derivative and Lebesgue decomposition for kern... |
lemma withDensity_rnDeriv_mutuallySingularSetSlice (ΞΊ Ξ· : Kernel Ξ± Ξ³) [IsFiniteKernel ΞΊ]
[IsFiniteKernel Ξ·] (a : Ξ±) :
withDensity Ξ· (rnDeriv ΞΊ Ξ·) a (mutuallySingularSetSlice ΞΊ Ξ· a) = 0 := by
rw [Kernel.withDensity_apply']
Β· exact setLIntegral_measure_zero _ _ (measure_mutuallySingularSetSlice ΞΊ Ξ· a)
Β· ex... | Mathlib/Probability/Kernel/RadonNikodym.lean | 321 | 341 |
/-
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
-/
import Mathlib.Data.Finset.Basic
import Mathlib.Data.Finset.Image
/-!
# Cardinality of a finite set
This defines the cardinality of a `Fins... | Β· intro aβ haβ aβ haβ haa
rw [mem_coe] at haβ haβ
simp only [f', haβ, haβ, β Subtype.ext_iff] at haa
exact Subtype.ext_iff.mp (hf haa)
lemma card_le_card_of_surjOn (f : Ξ± β Ξ²) (hf : Set.SurjOn f s t) : #t β€ #s := by
classical unfold Set.SurjOn at hf; exact (card_le_card (mod_cast hf)).trans car... | Mathlib/Data/Finset/Card.lean | 408 | 417 |
/-
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.Fin.Fin2
import Mathlib.Util.Notation3
import Mathlib.Tactic.TypeStar
/-!
# Alternate definition of `Vector` in terms of `Fin2`
This file provid... |
section Vector3
| Mathlib/Data/Vector3.lean | 189 | 190 |
/-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
import Mathlib.CategoryTheory.Limits.Opposites
/-!
# Constructing colimits from finite... | map := fun Ξ² => { app := fun _ => Sigma.map (fun x => Ξ².app x.val) }
/-- The converse of the construction in `liftToFinsetColimitCocone`: we can form a cocone on the
| Mathlib/CategoryTheory/Limits/Constructions/Filtered.lean | 79 | 81 |
/-
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.BigOperators.Expect
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Field.Canonical
import Mathlib.Algebra.O... | Mathlib/Data/NNReal/Basic.lean | 206 | 207 | |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Multiset.Powerset
/-!
# The antidiagonal on a multiset.
The antidiagonal of a multiset `s` consists of all pairs `(tβ, tβ)`
such that `tβ + tβ =... | antidiagonal (a ::β s) =
map (Prod.map id (cons a)) (antidiagonal s) + map (Prod.map (cons a) id) (antidiagonal s) :=
| Mathlib/Data/Multiset/Antidiagonal.lean | 66 | 67 |
/-
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.GroupTheory.Abelianization
import Mathlib.GroupTheory.Commutator.Finite
import Mathlib.GroupTheory.Transfer
import Mathlib.Algebra.Group.Pointwise.Fi... | rw [β hS1, β Fintype.card_coe, β Finset.card_univ, β Finset.prod_const]
let f : (β g : S, zpowers (g : G)) β* G := noncommPiCoprod fun s t _ x y _ _ => mul_comm x _
have hf : Function.Surjective f := by
rw [β MonoidHom.range_eq_top, eq_top_iff, β hS2, closure_le]
exact fun g hg => β¨Pi.mulSingle β¨g, hgβ© β¨g... | Mathlib/GroupTheory/Schreier.lean | 40 | 51 |
/-
Copyright (c) 2023 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.NumberTheory.EulerProduct.ExpLog
import Mathlib.NumberTheory.LSeries.Dirichlet
/-!
# The Euler Product for the Riemann Zeta Function and Dirichlet L-Ser... | Summable (fun n β¦ βdirichletSummandHom Ο (ne_zero_of_one_lt_re hs) nβ) := by
simp only [dirichletSummandHom, MonoidWithZeroHom.coe_mk, ZeroHom.coe_mk, norm_mul]
exact (summable_riemannZetaSummand hs).of_nonneg_of_le (fun _ β¦ by positivity)
(fun n β¦ mul_le_of_le_one_left (norm_nonneg _) <| Ο.norm_le_one n)
... | Mathlib/NumberTheory/EulerProduct/DirichletLSeries.lean | 70 | 75 |
/-
Copyright (c) 2024 Theodore Hwa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kim Morrison, Violeta HernΓ‘ndez Palacios, Junyan Xu, Theodore Hwa
-/
import Mathlib.Logic.Hydra
import Mathlib.SetTheory.Surreal.Basic
/-!
### Surreal multiplication
In... | (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y := by
have he' := neg_equiv_neg_iff.2 he
apply PGame.le_of_forall_lt <;> simp_rw [lt_iff_game_lt]
Β· rw [leftMoves_mul_iff (_ > Β·)]
refine β¨mulOption_lt_mul_of_equiv hβ hββ he, ?_β©
rw [β quot_neg_mul_neg]
exact mulOption_lt_... | Mathlib/SetTheory/Surreal/Multiplication.lean | 349 | 362 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro
-/
import Mathlib.Algebra.Field.IsField
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Finsupp.L... | exact β¨x, hx, not_unitβ©
theorem not_isField_iff_exists_ideal_bot_lt_and_lt_top [Nontrivial R] :
Β¬IsField R β β I : Ideal R, β₯ < I β§ I < β€ := by
constructor
Β· intro h
obtain β¨x, nz, nuβ© := exists_not_isUnit_of_not_isField h
use Ideal.span {x}
rw [bot_lt_iff_ne_bot, lt_top_iff_ne_top]
| Mathlib/RingTheory/Ideal/Basic.lean | 224 | 232 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes HΓΆlzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.Polynomial.Basi... | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 1,607 | 1,608 | |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Sean Leather
-/
import Batteries.Data.List.Perm
import Mathlib.Data.List.Pairwise
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Lookmap
import Mathlib.Data.Si... | lookmap_of_forall_not _ <| by
rintro β¨a', b'β© h; dsimp; split_ifs
Β· subst a'
| Mathlib/Data/List/Sigma.lean | 330 | 332 |
/-
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.CharP.Defs
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Algebra.MvPolynomial.Equiv
import Mathlib.Algebra.P... |
theorem exists_C_coeff_not_mem : f β I β β i : β, C (coeff f i) β I :=
Not.imp_symm fun cf => span_le_of_C_coeff_mem (not_exists_not.mp cf) mem_span_C_coeff
end Ideal
| Mathlib/RingTheory/Polynomial/Basic.lean | 736 | 741 |
/-
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.Sphere.Basic
/-!
# Second intersection of a sphere and a line
This file defines and proves basic results about the second intersection... | rw [β neg_one_smul β v, s.secondInter_smul p v (by norm_num : (-1 : β) β 0)]
/-- Applying `secondInter` twice returns the original point. -/
@[simp]
theorem Sphere.secondInter_secondInter (s : Sphere P) (p : P) (v : V) :
s.secondInter (s.secondInter p v) v = p := by
| Mathlib/Geometry/Euclidean/Sphere/SecondInter.lean | 103 | 108 |
/-
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.Pi.Lemmas
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Topology.Algebra.Monoid
/-!
# Topological group with zero
I... | end Div
/-! ### Left and right multiplication as homeomorphisms -/
namespace Homeomorph
variable [TopologicalSpace Ξ±] [GroupWithZero Ξ±] [ContinuousMul Ξ±]
/-- Left multiplication by a nonzero element in a `GroupWithZero` with continuous multiplication
| Mathlib/Topology/Algebra/GroupWithZero.lean | 236 | 245 |
/-
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 `Ξ` functio... | Mathlib/Analysis/SpecialFunctions/Gamma/Basic.lean | 539 | 541 | |
/-
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.MeasureTheory.Function.ConditionalExpectation.CondexpL1
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` ... |
@[deprecated (since := "2025-01-21")] alias integral_condexp := integral_condExp
/-- **Law of total probability** using `condExp` as conditional probability. -/
theorem integral_condExp_indicator [mΞ² : MeasurableSpace Ξ²] {Y : Ξ± β Ξ²} (hY : Measurable Y)
[SigmaFinite (ΞΌ.trim hY.comap_le)] {A : Set Ξ±} (hA : Measurab... | Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean | 261 | 277 |
/-
Copyright (c) 2019 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Yury Kudryashov
-/
import Mathlib.Analysis.Convex.Combination
import Mathlib.Analysis.Convex.Function
import Mathlib.Tactic.FieldSimp
/-!
# Jensen's inequality... | β β¦jβ¦, j β t β β β¦kβ¦, k β t β p j = p k := by
by_contra!; exact h_eq.not_lt <| hf.lt_map_sum hβ hβ hmem this
/-- Canonical form of the **equality case of Jensen's equality**.
For a strictly convex function `f` and positive weights `w`, we have
`f (β i β t, w i β’ p i) = β i β t, w i β’ f (p i)` if and only if the... | Mathlib/Analysis/Convex/Jensen.lean | 170 | 180 |
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, RΓ©my Degenne
-/
import Mathlib.Probability.Process.Adapted
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
/-!
# Stopping times, stopped processes and stopped va... | Β· simp_rw [eq_comm, β Nat.sub_eq_iff_eq_add hij, eq_comm]
exact f.mono (j.sub_le i) _ (hΟ.measurableSet_eq (j - i))
Β· rw [not_le] at hij
convert @MeasurableSet.empty _ (f.1 j)
ext Ο
| Mathlib/Probability/Process/Stopping.lean | 260 | 264 |
/-
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.Geometry.Euclidean.Altitude
import Mathlib.Geometry.Euclidean.Circumcenter
/-!
# Monge point and orthocenter
This file defines the orthocenter of a trian... | /-- `mongePointWeightsWithCircumcenter` sums to 1. -/
@[simp]
theorem sum_mongePointWeightsWithCircumcenter (n : β) :
β i, mongePointWeightsWithCircumcenter n i = 1 := by
| Mathlib/Geometry/Euclidean/MongePoint.lean | 103 | 106 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Polynomial.Expand
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.FieldTheory.IntermediateField.Basic
imp... | (ha : Associated f g) : f.Separable β g.Separable := β¨ha.separable, ha.symm.separableβ©
theorem Separable.mul_unit {f g : R[X]} (hf : f.Separable) (hg : IsUnit g) : (f * g).Separable :=
(associated_mul_unit_right f g hg).separable hf
theorem Separable.unit_mul {f g : R[X]} (hf : IsUnit f) (hg : g.Separable) : (f... | Mathlib/FieldTheory/Separable.lean | 138 | 149 |
/-
Copyright (c) 2022 YaΓ«l Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: YaΓ«l Dillies, Peter Nelson
-/
import Mathlib.Order.Antichain
/-!
# Minimality and Maximality
This file proves basic facts about minimality and maximality
of an element with respect to ... | simp [Minimal]
@[simp] theorem maximal_false : Β¬ Maximal (fun _ β¦ False) x := by
simp [Maximal]
@[simp] theorem minimal_true : Minimal (fun _ β¦ True) x β IsMin x := by
| Mathlib/Order/Minimal.lean | 61 | 66 |
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Choose.Cast
import Mathlib.Data.Nat.Choose.Vanderm... |
theorem hasseDeriv_mul (f g : R[X]) :
hasseDeriv k (f * g) = β ij β antidiagonal k, hasseDeriv ij.1 f * hasseDeriv ij.2 g := by
let D k := (@hasseDeriv R _ k).toAddMonoidHom
let Ξ¦ := @AddMonoidHom.mul R[X] _
show
(compHom (D k)).comp Ξ¦ f g =
β ij β antidiagonal k, ((compHom.comp ((compHom Ξ¦) (D ij.... | Mathlib/Algebra/Polynomial/HasseDeriv.lean | 210 | 221 |
/-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | have hn' : (n : β) β 0 := Nat.cast_ne_zero.mpr hn
have A : (n : β) ^ s = (n : β) ^ (s - 1) * n := by
conv_lhs => rw [(by ring : s = s - 1 + 1), cpow_add _ _ hn']
simp
have B : ((x : β) * βn) ^ (s - 1) = (x : β) ^ (s - 1) * (n : β) ^ (s - 1) := by
rw [β ofReal_natCast,
mul_cpow_ofReal_nonneg hx.1... | Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean | 265 | 286 |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, Mario Carneiro, Oliver Nash
-/
import Mathlib.Data.Finset.Card
import Mathlib.Data.Finset.Union
/-!
# Finsets in product types
This file defines finset constru... |
theorem offDiag_filter_lt_eq_filter_le {ΞΉ} [PartialOrder ΞΉ]
[DecidableEq ΞΉ] [DecidableLE ΞΉ] [DecidableLT ΞΉ] (s : Finset ΞΉ) :
| Mathlib/Data/Finset/Prod.lean | 379 | 381 |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Thomas Read, Andrew Yang, Dagur Asgeirsson, JoΓ«l Riou
-/
import Mathlib.CategoryTheory.Adjunction.Mates
/-!
# Uniqueness of adjoints
This file shows that adjoints are uni... | theorem leftAdjointUniq_hom_app_counit {F F' : C β₯€ D} {G : D β₯€ C} (adj1 : F β£ G) (adj2 : F' β£ G)
(x : D) :
(leftAdjointUniq adj1 adj2).hom.app (G.obj x) β« adj2.counit.app x = adj1.counit.app x := by
| Mathlib/CategoryTheory/Adjunction/Unique.lean | 64 | 66 |
/-
Copyright (c) 2018 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated
import Mathlib.MeasureTheory.Measure.MutuallySingular
import Mathlib.MeasureTheory.Measure.Typeclasses... |
@[simp]
| Mathlib/MeasureTheory/Measure/Dirac.lean | 137 | 138 |
/-
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 Mathlib.Algebra.Notation.Defs
import Mathlib.Data.Int.Notation
import Mathlib.Data.Nat.BinaryRec
import Mathlib.L... | /-- Any `CancelMonoid G` satisfies `IsCancelMul G`. -/
@[to_additive toIsCancelAdd "Any `AddCancelMonoid G` satisfies `IsCancelAdd G`."]
| Mathlib/Algebra/Group/Defs.lean | 721 | 722 |
/-
Copyright (c) 2021 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Topology.PartialHomeomorph
import Mathlib.Topology.SeparatedMap
/-!
# Local homeomorphisms
This file defines local homeomorphisms.
## Main definit... | end IsLocalHomeomorphOn
/-- A function `f : X β Y` satisfies `IsLocalHomeomorph f` if each `x : x` is contained in
the source of some `e : PartialHomeomorph X Y` with `f = e`. -/
def IsLocalHomeomorph :=
β x : X, β e : PartialHomeomorph X Y, x β e.source β§ f = e
| Mathlib/Topology/IsLocalHomeomorph.lean | 125 | 130 |
/-
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... | dist x y < Ξ΅β + Ξ΅β :=
let β¨z, hzβ© := h
| Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | 539 | 540 |
/-
Copyright (c) 2021 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.Convex.Hull
/-!
# Extreme sets
This file defines extreme sets and extreme points for sets in a module.
An extreme s... |
theorem Convex.mem_extremePoints_iff_convex_diff (hA : Convex π A) :
x β A.extremePoints π β x β A β§ Convex π (A \ {x}) := by
use fun hx β¦ β¨hx.1, (isExtreme_singleton.2 hx).convex_diff hAβ©
rintro β¨hxA, hAxβ©
refine mem_extremePoints_iff_forall_segment.2 β¨hxA, fun xβ hxβ xβ hxβ hx β¦ ?_β©
rw [convex_iff_seg... | Mathlib/Analysis/Convex/Extreme.lean | 258 | 267 |
/-
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
-/
import Batteries.Tactic.Congr
import Mathlib.Data.Option.Basic
import Mathlib.Data.Prod.Basic
import Mathlib.Data.Set.Subsingleton
import Mathlib.Dat... | range_subset_iff.2 fun _ => rfl
| Mathlib/Data/Set/Image.lean | 858 | 859 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kim Morrison
-/
import Mathlib.Tactic.NormNum.Basic
import Mathlib.Tactic.TryThis
import Mathlib.Util.AtomM
/-!
# The `abel` tactic
Evaluate expressions in the langua... | else do
let (a', h) β evalAdd heβ aβ
| Mathlib/Tactic/Abel.lean | 218 | 219 |
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: SΓ©bastien GouΓ«zel
-/
import Mathlib.Topology.Constructions
/-!
# Neighborhoods and continuity relative to a subset
This file develops API on the relative versions
* `nhdsWithin` ... | (hβ : β y β s, g y = f y) (hx : g x = f x) : ContinuousWithinAt g s x :=
h.congr_of_eventuallyEq (mem_of_superset self_mem_nhdsWithin hβ) hx
theorem continuousWithinAt_congr (hβ : β y β s, g y = f y) (hx : g x = f x) :
ContinuousWithinAt g s x β ContinuousWithinAt f s x :=
β¨fun h' β¦ h'.congr (fun x hx β¦ (h... | Mathlib/Topology/ContinuousOn.lean | 877 | 884 |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.FieldTheory.Finiteness
import Mathlib.LinearAlgebra.AffineSpace.Basis
import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
/-!
# Finite-dimensional subsp... | rw [hs, bot_coe, span_empty, bot_coe, direction_affineSpan]
convert finiteDimensional_bot k V <;> simp
Β· rw [affineSpan_coe, direction_affineSpan_insert hpβ]
infer_instance
| Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean | 340 | 343 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Sean Leather
-/
import Batteries.Data.List.Perm
import Mathlib.Data.List.Pairwise
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Lookmap
import Mathlib.Data.Si... |
theorem lookupAll_length_le_one (a : Ξ±) {l : List (Sigma Ξ²)} (h : l.NodupKeys) :
length (lookupAll a l) β€ 1 := by
have := Nodup.sublist ((lookupAll_sublist a l).map _) h
rw [map_map] at this
rwa [β nodup_replicate, β map_const]
theorem lookupAll_eq_dlookup (a : Ξ±) {l : List (Sigma Ξ²)} (h : l.NodupKeys) :
| Mathlib/Data/List/Sigma.lean | 290 | 297 |
/-
Copyright (c) 2023 YaΓ«l Dillies, Chenyi Li. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chenyi Li, Ziyu Wang, YaΓ«l Dillies
-/
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.InnerProductSpace.Basic
/-!
# Uniformly and strongly convex functions
I... |
lemma UniformConcaveOn.sub (hf : UniformConcaveOn s Ο f) (hg : UniformConvexOn s Ο g) :
| Mathlib/Analysis/Convex/Strong.lean | 108 | 109 |
/-
Copyright (c) 2022 YaΓ«l Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: YaΓ«l Dillies
-/
import Mathlib.Order.PropInstances
import Mathlib.Order.GaloisConnection.Defs
/-!
# Heyting algebras
This file defines Heyting, co-Heyting and bi-Heyting algebras.
A H... | rw [compl_sup_distrib, le_compl_iff_disjoint_right, disjoint_right_comm, β
| Mathlib/Order/Heyting/Basic.lean | 750 | 750 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Johan Commelin, Andrew Yang, JoΓ«l Riou
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
import Mathlib.CategoryTheory.Monoid... | (h : m + n = p)
lemma shiftFunctorCompIsoId_add'_inv_app :
(shiftFunctorCompIsoId C p' p hp).inv.app X =
(shiftFunctorCompIsoId C n' n hn).inv.app X β«
(shiftFunctorCompIsoId C m' m hm).inv.app (Xβ¦n'β§)β¦nβ§' β«
(shiftFunctorAdd' C m n p h).inv.app (Xβ¦n'β§β¦m'β§) β«
((shiftFunctorAdd' C n' m' p'
| Mathlib/CategoryTheory/Shift/Basic.lean | 504 | 511 |
/-
Copyright (c) 2021 YaΓ«l Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: YaΓ«l Dillies
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Topology.Algebra.Group.Pointwise
import Mathlib.Topology.Order.Basic
/-!
# Strictly convex sets
This file defines st... |
variable [CommSemiring π] [PartialOrder π] [TopologicalSpace E]
| Mathlib/Analysis/Convex/Strict.lean | 261 | 263 |
/-
Copyright (c) 2018 SΓ©bastien GouΓ«zel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: SΓ©bastien GouΓ«zel
-/
import Mathlib.Data.Set.Lattice
import Mathlib.Order.ConditionallyCompleteLattice.Defs
/-!
# Theory of conditionally complete lattices
A conditionally complet... | Mathlib/Order/ConditionallyCompleteLattice/Basic.lean | 1,400 | 1,401 | |
/-
Copyright (c) 2021 YaΓ«l Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: YaΓ«l Dillies
-/
import Mathlib.Algebra.Group.Embedding
import Mathlib.Order.Interval.Multiset
/-!
# Finite intervals of naturals
This file proves that `β` is a `LocallyFiniteOrder` and... | convert (Nat.mul_lt_mul_right <| seed.succ_pos.trans_le hl).2 hk
rw [one_mul]
theorem Nat.cauchy_induction_two_mul (h : β n, P (n + 1) β P n) (seed : β) (hs : P seed.succ)
(hm : β x, seed < x β P x β P (2 * x)) (n : β) : P n :=
Nat.cauchy_induction_mul h 2 seed Nat.one_lt_two hs hm n
theorem Nat.pow_imp_sel... | Mathlib/Order/Interval/Finset/Nat.lean | 316 | 327 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Aaron Anderson, Yakov Pechersky
-/
import Mathlib.Data.Fintype.Card
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Algebra.Group.End
import Mathlib.Data.Finset.N... | simp [h]
Β· rintro β¨a, ha, hbβ©
| Mathlib/GroupTheory/Perm/Support.lean | 562 | 563 |
/-
Copyright (c) 2023 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.LinearAlgebra.Trace
import Mathlib.LinearAlgebra.FreeModule.PID
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
/-!
# Linear maps of modules with coeff... | /-- If a linear endomorphism of a (finite, free) module `M` takes values in a submodule `p β M`,
then the trace of its restriction to `p` is equal to its trace on `M`. -/
lemma trace_restrict_eq_of_forall_mem [IsDomain R] [IsPrincipalIdealRing R]
(p : Submodule R M) (f : M ββ[R] M)
(hf : β x, f x β p) (hf' : β ... | Mathlib/LinearAlgebra/PID.lean | 29 | 44 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kenny Lau
-/
import Mathlib.Data.List.Forall2
/-!
# zip & unzip
This file provides results about `List.zipWith`, `List.zip` and `List.unzip` (definitions are in
core ... | Mathlib/Data/List/Zip.lean | 349 | 363 | |
/-
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.Equiv.Basic
import Mathlib.Data.ENat.Lattice
import Mathlib.Data.Part
import Mathlib.Tactic.NormNum
/-!
# Natural numbers with infinity
The... | PartENat.casesOn b (le_top.trans_eq (add_top _).symm) fun _ =>
PartENat.casesOn a (top_add _).ge fun _ =>
| Mathlib/Data/Nat/PartENat.lean | 405 | 406 |
/-
Copyright (c) 2021 Aaron Anderson, YaΓ«l Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Kevin Buzzard, YaΓ«l Dillies, Eric Wieser
-/
import Mathlib.Data.Finset.Lattice.Union
import Mathlib.Data.Finset.Pairwise
import Mathlib.Data.Finset.Prod
i... |
section DistribLattice
variable [DistribLattice Ξ±] [OrderBot Ξ±] {s : Finset ΞΉ} {f : ΞΉ β Ξ±}
| Mathlib/Order/SupIndep.lean | 158 | 161 |
/-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, Mario Carneiro, Patrick Massot, Yury Kudryashov, RΓ©my Degenne
-/
import Mathlib.Data.Set.Subsingleton
import Mathlib.Order.Interval.Set.Defs
/-!
# Intervals
In any pr... | Mathlib/Order/Interval/Set/Basic.lean | 1,772 | 1,773 | |
/-
Copyright (c) 2023 RΓ©my Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: RΓ©my Degenne
-/
import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap
import Mathlib.Probability.Kernel.MeasurableLIntegral
/-!
# With Density
For an s-finite kernel `ΞΊ : K... | lemma withDensity_zero' (ΞΊ : Kernel Ξ± Ξ²) [IsSFiniteKernel ΞΊ] :
Kernel.withDensity ΞΊ (fun _ _ β¦ 0) = 0 := Kernel.withDensity_zero _
theorem lintegral_withDensity (ΞΊ : Kernel Ξ± Ξ²) [IsSFiniteKernel ΞΊ]
| Mathlib/Probability/Kernel/WithDensity.lean | 99 | 102 |
/-
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.Card
import Mathlib.Data.Fintype.Basic
/-!
# Cardinalities of finite types
This file defines the cardinality `Fintype.card Ξ±` as the numb... | @[simp] theorem card_eq_zero [IsEmpty Ξ±] : card Ξ± = 0 :=
card_eq_zero_iff.2 βΉ_βΊ
alias card_of_isEmpty := card_eq_zero
| Mathlib/Data/Fintype/Card.lean | 259 | 263 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, Kenny Lau
-/
import Mathlib.Data.DFinsupp.Submonoid
import Mathlib.Data.Finsupp.ToDFinsupp
import Mathlib.LinearAlgebra.Finsupp.SumProd
import Mathlib.LinearAlgebra.LinearIn... | @[simps apply]
def mapRange.linearEquiv (e : β i, Ξ²β i ββ[R] Ξ²β i) : (Ξ β i, Ξ²β i) ββ[R] Ξ β i, Ξ²β i :=
{ mapRange.addEquiv fun i => (e i).toAddEquiv,
mapRange.linearMap fun i => (e i).toLinearMap with
toFun := mapRange (fun i x => e i x) fun i => (e i).map_zero
| Mathlib/LinearAlgebra/DFinsupp.lean | 240 | 244 |
/-
Copyright (c) 2021 YaΓ«l Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: YaΓ«l Dillies, Violeta HernΓ‘ndez Palacios, Grayson Burton, Floris van Doorn
-/
import Mathlib.Order.Antisymmetrization
import Mathlib.Order.Hom.WithTopBot
import Mathlib.Order.Interval.Se... |
end Bool
| Mathlib/Order/Cover.lean | 439 | 440 |
/-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.SetTheory.Cardinal.Finite
import Mathlib.Topology.Algebra.InfiniteSum.Basic
import Mathlib.Topology.UniformSpace.Cauchy
import Mathlib.Topology.Algebra... |
/-- The product over the complement of a finset tends to `1` when the finset grows to cover the
whole space. This does not need a multipliability assumption, as otherwise all such products are
one. -/
@[to_additive "The sum over the complement of a finset tends to `0` when the finset grows to cover
the whole space. Th... | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | 363 | 368 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Multiset.ZeroCons
/-!
# Basic results on multisets
-/
-- No algebra should be required
assert_not_exists Monoid
universe v
open List S... | Mathlib/Data/Multiset/Basic.lean | 2,315 | 2,316 | |
/-
Copyright (c) 2021 YaΓ«l Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: YaΓ«l Dillies, Bhavik Mehta
-/
import Mathlib.Algebra.GroupWithZero.Action.Defs
import Mathlib.Algebra.Order.Interval.Finset.Basic
import Mathlib.Combinatorics.Additive.Frei... | (β β¦bβ¦, b β s β β β¦cβ¦, c β s β a * c = b * b β a = b) β§
β β¦bβ¦, b β s β β β¦cβ¦, c β s β b * c = a * a β b = a := by
refine β¨fun hs β¦ β¨hs.mono (subset_insert _ _),
fun b hb c hc β¦ hs (Or.inl rfl) (Or.inr hb) (Or.inr hc),
| Mathlib/Combinatorics/Additive/AP/Three/Defs.lean | 167 | 170 |
/-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, JoΓ«l Riou
-/
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.Equalizers
import Mathlib.CategoryTheory.Limits.ConeCategory
/-!
... | variable {J : MultispanShape.{w, w'}} (I : MultispanIndex J C) [HasMulticoequalizer I]
/-- The canonical map from the multiequalizer to the objects on the left. -/
| Mathlib/CategoryTheory/Limits/Shapes/Multiequalizer.lean | 826 | 828 |
/-
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.... | zero_div, Complex.exp_zero]
theorem fourier_one {x : AddCircle T} : fourier 1 x = toCircle x := by rw [fourier_apply, one_zsmul]
| Mathlib/Analysis/Fourier/AddCircle.lean | 139 | 141 |
/-
Copyright (c) 2023 Kyle Miller, RΓ©mi Bottinelli. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller, RΓ©mi Bottinelli
-/
import Mathlib.Combinatorics.SimpleGraph.Path
import Mathlib.Data.Set.Card
/-!
# Connectivity of subgraphs and induced graphs
## Main de... | namespace Walk
variable {u v w : V}
/-- The subgraph consisting of the vertices and edges of the walk. -/
@[simp]
protected def toSubgraph {u v : V} : G.Walk u v β G.Subgraph
| Mathlib/Combinatorics/SimpleGraph/Connectivity/Subgraph.lean | 126 | 132 |
/-
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 ... | end DeleteVerts
| Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | 1,268 | 1,269 |
/-
Copyright (c) 2022 YaΓ«l Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: YaΓ«l Dillies
-/
import Mathlib.Order.PropInstances
import Mathlib.Order.GaloisConnection.Defs
/-!
# Heyting algebras
This file defines Heyting, co-Heyting and bi-Heyting algebras.
A H... | theorem le_compl_iff_disjoint_right : a β€ bαΆ β Disjoint a b := by
rw [β himp_bot, le_himp_iff, disjoint_iff_inf_le]
| Mathlib/Order/Heyting/Basic.lean | 641 | 642 |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.FieldTheory.Finiteness
import Mathlib.LinearAlgebra.AffineSpace.Basis
import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
/-!
# Finite-dimensional subsp... | have hd := direction_le hle
rw [direction_affineSpan] at hd β’
exact hi.vectorSpan_image_finset_eq_of_le_of_card_eq_finrank_add_one hd hc
/-- If the `affineSpan` of a finite affinely independent family lies
in an affine subspace whose direction has dimension one less than its
cardinality, it equals that subspace.... | Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean | 296 | 307 |
/-
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.Data.Fintype.List
import Mathlib.Data.Fintype.OfMap
/-!
# Cycles of a list
Lists have an equivalence relation of whether they are rotational permut... | @[simp]
theorem coe_nil : β([] : List Ξ±) = @nil Ξ± :=
rfl
| Mathlib/Data/List/Cycle.lean | 433 | 436 |
/-
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... | rcases hf with β¨r, hrβ©
filter_upwards [EMetric.ball_mem_nhds (0 : E) hr.r_pos] with y hy
exact hr.tendsto_partialSum hy
open Finset in
/-- If a function admits a power series expansion within a ball, then the partial sums
`p.partialSum n z` converge to `f (x + y)` as `n β β` and `z β y`. Note that `x + z` doesn'... | Mathlib/Analysis/Analytic/Basic.lean | 873 | 882 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Eric Wieser
-/
import Mathlib.Algebra.Quaternion
import Mathlib.Analysis.InnerProductSpace.Continuous
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.To... | Mathlib/Analysis/Quaternion.lean | 238 | 240 | |
/-
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.GCDMonoid.Finset
import Mathlib.Algebra.Polynomial.CancelLeads
import Mathlib.Algebra.Polynomial.EraseLead
import Mathlib.Algebra.Polynomial.Fi... | Β· apply Finset.gcd_mono
intro i
simp only [Nat.lt_succ_iff, mem_support_iff, Ne, Finset.mem_range]
contrapose!
intro h1
apply coeff_eq_zero_of_natDegree_lt (lt_of_lt_of_le h h1)
| Mathlib/RingTheory/Polynomial/Content.lean | 177 | 182 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Alistair Tucker, Wen Yang
-/
import Mathlib.Order.Interval.Set.Image
import Mathlib.Order.CompleteLatticeIntervals
import Mathlib.Topology.Order.DenselyOrdered
import... | (heβ.comap _) (heβ.comap _)
exact β¨b, b.prop, hβ©
/-- **Intermediate Value Theorem** for continuous functions on connected sets. -/
theorem IsPreconnected.intermediate_value {s : Set X} (hs : IsPreconnected s) {a b : X} (ha : a β s)
(hb : b β s) {f : X β Ξ±} (hf : ContinuousOn f s) : Icc (f a) (f b) β f '' s... | Mathlib/Topology/Order/IntermediateValue.lean | 115 | 124 |
/-
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... | simp
| Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean | 527 | 528 |
/-
Copyright (c) 2022 Pim Otte. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller, Pim Otte
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Order.Antidiag.Pi
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.Data.Nat.Factorial.BigOperators
im... | rw [Finset.sum_insert h, Finset.prod_insert h, hβ, add_comm, β succ_eq_add_one, factorial_succ]
simp only [factorial, succ_eq_add_one, zero_add, mul_one, one_mul]
| Mathlib/Data/Nat/Choose/Multinomial.lean | 75 | 76 |
/-
Copyright (c) 2022 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, Patrick Massot, Yury Kudryashov, Kevin H. Wilson, Heather Macbeth
-/
import Mathlib.Order.Filter.Tendsto
/-!
# Product and coproduct filters
In this file we define `F... | theorem pure_prod {a : Ξ±} {f : Filter Ξ²} : pure a ΓΛ’ f = map (Prod.mk a) f := by
rw [prod_eq, map_pure, pure_seq_eq_map]
| Mathlib/Order/Filter/Prod.lean | 373 | 375 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.RingTheory.FractionalIdeal.Basic
import Mathlib.RingTheory.IntegralClosure.IsIntegral.Basi... | β¨_, isFractional_adjoin_integral S x hxβ©
@[simp]
theorem adjoinIntegral_coe (hx : IsIntegral R x) :
| Mathlib/RingTheory/FractionalIdeal/Operations.lean | 882 | 885 |
/-
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.MeasureTheory.Function.LpSeminorm.Trim
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner
import Mathlib.MeasureTheory.Function.StronglyMeasura... | instance [hm : Fact (m β€ m0)] [CompleteSpace F] [hp : Fact (1 β€ p)] :
CompleteSpace (lpMeas F π m p ΞΌ) := by
rw [(lpMeasSubgroupToLpMeasIso F π p ΞΌ).symm.completeSpace_iff]; infer_instance
theorem isComplete_aestronglyMeasurable [hp : Fact (1 β€ p)] [CompleteSpace F] (hm : m β€ m0) :
| Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean | 426 | 430 |
/-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin, Lu-Ming Zhang
-/
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Algebra.Pi
import Mathlib.Algebra.BigOp... | Mathlib/Data/Matrix/Basic.lean | 2,024 | 2,025 | |
/-
Copyright (c) 2014 Floris van Doorn (c) 2016 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Leonardo de Moura, Jeremy Avigad, Mario Carneiro
-/
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Algebra.Order.GroupWithZero... | Mathlib/Algebra/Order/Ring/Nat.lean | 62 | 63 | |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Winston Yin
-/
import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.Topology.Algebra.Order.Floor
import Mathlib.Topology.MetricSpace.Contracting
/-!
# P... | rcases ((FloorSemiring.tendsto_pow_div_factorial_atTop (v.L * v.tDist)).eventually
(gt_mem_nhds zero_lt_one)).exists with β¨N, hNβ©
have : (0 : β) β€ (v.L * v.tDist) ^ N / N ! :=
div_nonneg (pow_nonneg (mul_nonneg v.L.2 v.tDist_nonneg) _) (Nat.cast_nonneg _)
exact β¨N, β¨_, thisβ©, hN, LipschitzWith.of_dist_le_... | Mathlib/Analysis/ODE/PicardLindelof.lean | 317 | 323 |
/-
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.Fin
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Finset.Sort
/-!
# Compositions
A compositio... | simp
| Mathlib/Combinatorics/Enumerative/Composition.lean | 204 | 204 |
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