Context stringlengths 227 76.5k | target stringlengths 0 11.6k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 16 3.69k |
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/-
Copyright (c) 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 ... | unfold subNatNat
cases e : n - m
· simp only [ofNat_eq_coe]
simp [e, Nat.le_of_sub_eq_zero e]
| Mathlib/Data/Int/Cast/Basic.lean | 85 | 88 |
/-
Copyright (c) 2024 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Category.ModuleCat.Presheaf
import Mathlib.Algebra.Category.ModuleCat.Differentials.Basic
/-!
# The presheaf of differentials of a presheaf of modules
... | CommRingCat.KaehlerDifferential.map_d (φ'.naturality f) _
/-- The universal derivation. -/
noncomputable def derivation' : (relativeDifferentials' φ').Derivation' φ' :=
Derivation'.mk (fun X ↦ CommRingCat.KaehlerDifferential.D (φ'.app X))
| Mathlib/Algebra/Category/ModuleCat/Differentials/Presheaf.lean | 200 | 204 |
/-
Copyright (c) 2022 Moritz Firsching. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Firsching, Fabian Kruse, Nikolas Kuhn
-/
import Mathlib.Analysis.PSeries
import Mathlib.Data.Real.Pi.Wallis
import Mathlib.Tactic.AdaptationNote
/-!
# Stirling's formula
Thi... | rw [one_div, inv_pow]
exact inv_lt_one_of_one_lt₀ (one_lt_pow₀ (lt_add_of_pos_left _ <| by positivity) two_ne_zero)
have hab (k : ℕ) : (1 : ℝ) / (2 * ↑(k + 1) + 1) * ((1 / (2 * ↑(n + 1) + 1)) ^ 2) ^ ↑(k + 1) ≤
(((1 : ℝ) / (2 * ↑(n + 1) + 1)) ^ 2) ^ ↑(k + 1) := by
refine mul_le_of_le_one_left (pow_no... | Mathlib/Analysis/SpecialFunctions/Stirling.lean | 104 | 120 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Order.SuccPred
import Mathlib.Data.Sum.Order
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Tactic.PPWithUniv
/-!
# ... | theorem omega0_le_ord {a : Cardinal} : ω ≤ a.ord ↔ ℵ₀ ≤ a := by
rw [← ord_aleph0, ord_le_ord]
@[simp]
theorem ord_le_omega0 {a : Cardinal} : a.ord ≤ ω ↔ a ≤ ℵ₀ := by
rw [← ord_aleph0, ord_le_ord]
@[simp]
| Mathlib/SetTheory/Ordinal/Basic.lean | 1,164 | 1,171 |
/-
Copyright (c) 2024 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.Data.Matroid.Constructions
import Mathlib.Data.Set.Notation
/-!
# Maps between matroids
This file defines maps and comaps, which move a matroid on one ty... | exact ⟨B, hB, rfl⟩
lemma IsBasis.mapEmbedding {X : Set α} (hIX : M.IsBasis I X) (f : α ↪ β) :
| Mathlib/Data/Matroid/Map.lean | 543 | 545 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Yury Kudryashov
-/
import Mathlib.Algebra.CharP.Lemmas
import Mathlib.FieldTheory.Perfect
/-!
# The perfect closure of a characteristic `p` ring
## Main definitions
- `Perfec... |
instance instReduced : IsReduced (PerfectClosure K p) where
eq_zero x := induction_on x fun x ⟨n, h⟩ ↦ by
replace h : mk K p x ^ p ^ n = 0 := by
rw [← Nat.sub_add_cancel ((n.lt_pow_self (Fact.out : p.Prime).one_lt).le),
pow_add, h, mul_zero]
| Mathlib/FieldTheory/PerfectClosure.lean | 388 | 393 |
/-
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.Basic
import Mathlib.Algebra.GroupWithZero.NeZero
import Mathlib.Logic.Unique
import Mathlib.Tactic.Conv
/-!
# Groups with an adjoined z... | rw [zero_zpow_eq, one_ne_zero.ite_eq_left_iff]
| Mathlib/Algebra/GroupWithZero/Basic.lean | 406 | 407 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker, Johan Commelin
-/
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib... | if h : p = 0 then ∅ else Classical.choose (exists_multiset_roots h)
theorem roots_def [DecidableEq R] (p : R[X]) [Decidable (p = 0)] :
p.roots = if h : p = 0 then ∅ else Classical.choose (exists_multiset_roots h) := by
rename_i iR ip0
obtain rfl := Subsingleton.elim iR (Classical.decEq R)
obtain rfl := Sub... | Mathlib/Algebra/Polynomial/Roots.lean | 55 | 61 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yury Kudryashov
-/
import Mathlib.Algebra.Group.Commute.Defs
import Mathlib.Algebra.Opposites
import Mathlib.Tactic.Spread
/-!
# Definitions of group actions
This file de... | /-- `SMul` version of `comp_mul_left` -/
@[to_additive "`VAdd` version of `comp_add_left`"]
lemma comp_smul_left (a₁ a₂ : M) : (a₁ • ·) ∘ (a₂ • ·) = (((a₁ * a₂) • ·) : α → α) :=
funext fun _ ↦ (mul_smul _ _ _).symm
variable {M}
@[to_additive (attr := simp)]
theorem smul_iterate (a : M) : ∀ n : ℕ, (a • · : α → α)^[n... | Mathlib/Algebra/Group/Action/Defs.lean | 349 | 365 |
/-
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, Floris van Doorn, Yury Kudryashov, Neil Strickland
-/
import Mathlib.Algebra.Group.Defs
import Mathlib.Algebra.GroupWithZero.Defs
import Mathlib.Data.I... |
/-- The additive inverse of one multiplied by an element of a ring is the element's additive
| Mathlib/Algebra/Ring/Defs.lean | 283 | 284 |
/-
Copyright (c) 2020 Jalex Stark. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jalex Stark, Kim Morrison, Eric Wieser, Oliver Nash, Wen Yang
-/
import Mathlib.Data.Matrix.Basic
/-!
# Matrices with a single non-zero element.
This file provides `Matrix.stdBasisMatri... | @[simp]
theorem diag_zero (h : j ≠ i) : diag (stdBasisMatrix i j c) = 0 :=
| Mathlib/Data/Matrix/Basis.lean | 195 | 196 |
/-
Copyright (c) 2019 Abhimanyu Pallavi Sudhir. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Abhimanyu Pallavi Sudhir
-/
import Mathlib.Order.Filter.FilterProduct
import Mathlib.Analysis.SpecificLimits.Basic
/-!
# Construction of the hyperreal numbers as an ultrapro... |
theorem IsSt.le {x y : ℝ*} {r s : ℝ} (hrx : IsSt x r) (hsy : IsSt y s) (hxy : x ≤ y) : r ≤ s :=
| Mathlib/Data/Real/Hyperreal.lean | 345 | 346 |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.ModEq
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Algebra.Ring.Periodic
import Mathlib.Data.Int.SuccPred
import Mathlib.Order.Cir... |
theorem toIcoMod_eq_toIcoMod : toIcoMod hp a b = toIcoMod hp a c ↔ ∃ n : ℤ, c - b = n • p := by
refine ⟨fun h => ⟨toIcoDiv hp a c - toIcoDiv hp a b, ?_⟩, fun h => ?_⟩
| Mathlib/Algebra/Order/ToIntervalMod.lean | 469 | 471 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov, Kexing Ying
-/
import Mathlib.Topology.Semicontinuous
import Mathlib.MeasureTheory.Function.AEMeasurableSequence
import Mathlib.MeasureTheory.Order.Lat... | theorem borel_eq_generateFrom_Iic :
borel α = MeasurableSpace.generateFrom (range Iic) := by
| Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean | 77 | 78 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Topology.Algebra.Constructions
import Mathlib.Topology.Bases
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Topology.UniformSpac... | refine ⟨Finset.image xs (Finset.range (u i + 1)), Finset.finite_toSet _, fun x hx ↦ ?_⟩
simp only [Finset.coe_image, Finset.coe_range, mem_image, mem_Iio, iUnion_exists, biUnion_and',
iUnion_iUnion_eq_right, Nat.lt_succ_iff]
exact h_subset hx
/-- The construction `interUnionBalls` is used to have a relativel... | Mathlib/Topology/UniformSpace/Cauchy.lean | 663 | 672 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Set.Image
import Mathlib.Data.Set.BooleanAlgebra
/-!
# Sets in sigma types
This file defines `Set.sigma`, the indexed sum of sets.
-/
namespace Set... | Mathlib/Data/Set/Sigma.lean | 274 | 276 | |
/-
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.Order.Atoms
import Mathlib.Order.OrderIsoNat
import Mathlib.Order.RelIso.Set
import Mathlib.Order.SupClosed
import Mathlib.Order.SupIndep
import Mathlib.Orde... | obtain hxk := hbelow x hxs
exact hxk.ne (hxk.le.antisymm hkx)
theorem isCompactElement_finsetSup {α β : Type*} [CompleteLattice α] {f : β → α} (s : Finset β)
(h : ∀ x ∈ s, IsCompactElement (f x)) : IsCompactElement (s.sup f) := by
classical
rw [isCompactElement_iff_le_of_directed_sSup_le]
intro d hem... | Mathlib/Order/CompactlyGenerated/Basic.lean | 174 | 183 |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Covering.Differentiation
/-!
# Besicovitch covering theorems
The topological Besicovitch covering theorem ensures that, in a nice... | (Besicovitch.vitaliFamily μ) hs]
intro x hx
exact hx.comp (tendsto_filterAt μ x)
/-- Given an arbitrary set `s`, then `μ (s ∩ closedBall x r) / μ (closedBall x r)` converges
to `1` when `r` tends to `0`, for almost every `x` in `s`.
This shows that almost every point of `s` is a Lebesgue density point for `s... | Mathlib/MeasureTheory/Covering/Besicovitch.lean | 1,099 | 1,111 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.Group.Defs
/-!
# Invertible elements
This file defines a typeclass `Invertible a` for elements `a` with a two-sided
multiplicative inverse.
The in... | rw [← mul_right_inj_of_invertible (c := ⅟c), invOf_mul_cancel_left]
theorem mul_invOf_eq_iff_eq_mul_right : a * ⅟c = b ↔ a = b * c := by
| Mathlib/Algebra/Group/Invertible/Defs.lean | 231 | 233 |
/-
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.Complement
import Mathlib.GroupTheory.Sylow
/-!
# The Transfer Homomorphism
In this file we construct the transfer homomorphism.
## Ma... | calc
transfer ϕ g = ∏ q : G ⧸ H, _ := transfer_def ϕ (transferTransversal H g) g
_ = _ := ((quotientEquivSigmaZMod H g).symm.prod_comp _).symm
_ = _ := Finset.prod_sigma _ _ _
_ = _ := by
refine Fintype.prod_congr _ _ (fun q => ?_)
simp only [quotientEquivSigmaZMod_symm_apply... | Mathlib/GroupTheory/Transfer.lean | 165 | 181 |
/-
Copyright (c) 2023 Josha Dekker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Josha Dekker
-/
import Mathlib.Topology.Bases
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.Compactness.SigmaCompact
/-!
# Lindelöf sets and Lindelöf spaces
## Mai... | theorem Tendsto.isLindelof_insert_range_of_coLindelof {f : X → Y} {y}
(hf : Tendsto f (coLindelof X) (𝓝 y)) (hfc : Continuous f) :
IsLindelof (insert y (range f)) := by
intro l hne _ hle
by_cases hy : ClusterPt y l
· exact ⟨y, Or.inl rfl, hy⟩
simp only [clusterPt_iff_nonempty, not_forall, ← not_disjoin... | Mathlib/Topology/Compactness/Lindelof.lean | 432 | 447 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Linear.Basic
import Mathlib.Algebra.Homology.ComplexShapeSigns
import Mathlib.Algebra.Homology.HomologicalBicomplex
import Mathlib.Algebra.Module.... | obtain rfl := c₁.next_eq' h
rfl
lemma d₁_eq {i₁ i₁' : I₁} (h : c₁.Rel i₁ i₁') (i₂ : I₂) (i₁₂ : I₁₂)
(h' : ComplexShape.π c₁ c₂ c₁₂ ⟨i₁', i₂⟩ = i₁₂) :
| Mathlib/Algebra/Homology/TotalComplex.lean | 90 | 94 |
/-
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.ModularForms.JacobiTheta.TwoVariable
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
/-! # Jacobi's theta function
This file define... | Mathlib/NumberTheory/ModularForms/JacobiTheta/OneVariable.lean | 135 | 138 | |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Geometry.Manifold.IsManifold.ExtChartAt
import Mathlib.Geometry.Manifold.LocalInvariantProperties
/-!
# `C^n` functions betwee... | (e.continuousOn_writtenInExtend_iff hs h2s).1 h.continuousOn
/-- If the set where you want `f` to be `C^n` lies entirely in a single chart, and `f` maps it
into a single chart, the fact that `f` is `C^n` on that set can be expressed by purely looking in
these charts.
Note: this lemma uses `extChartAt I x '' s` ins... | Mathlib/Geometry/Manifold/ContMDiff/Defs.lean | 473 | 484 |
/-
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.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.FreeModule.PID
import Mathlib.LinearAlgebra.Eigenspace.Basic
import Mathlib.LinearAlgebra.... | lemma diag_toMatrix_directSum_collectedBasis_eq_zero_of_mapsTo_ne
{κ : ι → Type*} [∀ i, Fintype (κ i)] [∀ i, DecidableEq (κ i)]
{s : Finset ι} (h : IsInternal fun i : s ↦ N i)
(b : (i : s) → Basis (κ i) R (N i)) (σ : ι → ι) (hσ : ∀ i, σ i ≠ i)
{f : Module.End R M} (hf : ∀ i, MapsTo f (N i) (N <| σ i)) (... | Mathlib/Algebra/DirectSum/LinearMap.lean | 45 | 58 |
/-
Copyright (c) 2019 Calle Sönne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology... | induction θ using Real.Angle.induction_on
convert toIocMod_le_right two_pi_pos _ _
ring
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 464 | 466 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Data.ENNReal.Basic
/-!
# Maps between real and extended non-negative real numbers
This file focuses on the functions `ENNReal.toReal... | Mathlib/Data/ENNReal/Real.lean | 506 | 507 | |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.GroupTheory.Archimedean
import Mathlib.Topology.Algebra.Order.Group
import Mathlib.Algebra.Group.Subgroup.ZPowers.Basic
import Mathlib.Topology.Order... | refine S.dense_of_not_isolated_one fun ε ε1 => ?_
contrapose! H
exact exists_isLeast_one_lt hbot ε1 (disjoint_left.2 H)
/-- A subgroup of an archimedean linear ordered multiplicative commutative group `G` with order
| Mathlib/Topology/Algebra/Order/Archimedean.lean | 67 | 71 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.Ab
import Mathlib.Algebra.Homology.ShortComplex.ExactFunctor
import Mathlib.Algebra.Homology.ShortComplex.SnakeLemma
import Mathlib... |
@[simp]
lemma ShortComplex.zero_apply
[Limits.HasZeroMorphisms C] [(forget₂ C Ab).PreservesZeroMorphisms]
(S : ShortComplex C) (x : (forget₂ C Ab).obj S.X₁) :
((forget₂ C Ab).map S.g) (((forget₂ C Ab).map S.f) x) = 0 := by
rw [← ConcreteCategory.comp_apply, ← Functor.map_comp, S.zero, Functor.map_zero]
| Mathlib/Algebra/Homology/ShortComplex/ConcreteCategory.lean | 30 | 36 |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Multiset.Bind
/-!
# The cartesian product of multisets
## Main definitions
* `Multiset.pi`: Cartesian product of multisets indexed by a multise... | Pi.cons _ _ (φ b) (fun a' ha' ↦ φ (f a' ha')) = (fun a' ha' ↦ φ ((cons _ _ b f) a' ha')) := by
ext a' ha'
refine (congrArg₂ _ ?_ rfl).trans (apply_dite (@φ _) (a' = a) _ _).symm
ext rfl
rfl
theorem forall_rel_cons_ext {r : ∀ ⦃a⦄, δ a → δ a → Prop} {b₁ b₂ : δ a} {f₁ f₂ : ∀ a' ∈ m, δ a'}
(hb : r b₁ b₂) (... | Mathlib/Data/Multiset/Pi.lean | 71 | 80 |
/-
Copyright (c) 2020 Kenji Nakagawa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenji Nakagawa, Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.Algebra.Subalgebra.Pointwise
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.RingTheory.Spect... | namespace Ideal
/-! ### GCD and LCM of ideals in a Dedekind domain
| Mathlib/RingTheory/DedekindDomain/Ideal.lean | 791 | 794 |
/-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Analysis.Convex.Hull
import Mathlib.LinearAlgebra.AffineSpace.Basis
/-!
# Convex combinations
... |
variable [LinearOrder R] [IsStrictOrderedRing R] [IsOrderedAddMonoid α] [OrderedSMul R α]
/-- The center of mass of a finite subset of a convex set belongs to the set
| Mathlib/Analysis/Convex/Combination.lean | 151 | 154 |
/-
Copyright (c) 2024 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Localization.CalculusOfFractions.ComposableArrows
import Mathlib.CategoryTheory.Localization.CalculusOfFractions.Preadditive
import Mathlib.Catego... | complete_distinguished_triangle_morphism := complete_distinguished_triangle_morphism L W
instance isTriangulated_functor :
letI : Pretriangulated D := pretriangulated L W; L.IsTriangulated :=
| Mathlib/CategoryTheory/Localization/Triangulated.lean | 196 | 199 |
/-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Jireh Loreaux
-/
import Mathlib.Algebra.Divisibility.Hom
import Mathlib.Algebra.GroupWithZero.InjSurj
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.Data.Set... | Mathlib/Algebra/Ring/Hom/Basic.lean | 61 | 66 | |
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel, Joël Riou
-/
import Mathlib.Algebra.Homology.ExactSequence
import Mathlib.CategoryTheory.Abelian.Refinements
/-!
# The four and five lemmas
Consider the following commu... | end Three
end Abelian
namespace ShortComplex
variable {C : Type*} [Category C] [Abelian C]
variable {R₁ R₂ : ShortComplex C} (φ : R₁ ⟶ R₂)
attribute [local simp] ComposableArrows.Precomp.map
| Mathlib/CategoryTheory/Abelian/DiagramLemmas/Four.lean | 221 | 231 |
/-
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... | rw [RatFunc.liftOn', liftOn_div _ _ _ f0]
apply liftOn_condition_of_liftOn'_condition H
| Mathlib/FieldTheory/RatFunc/Basic.lean | 683 | 685 |
/-
Copyright (c) 2021 Vladimir Goryachev. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Vladimir Goryachev, Kyle Miller, Kim Morrison, Eric Rodriguez
-/
import Mathlib.Data.List.GetD
import Mathlib.Data.Nat.Count
import Mathlib.Data.Nat.SuccPred
import M... | (hlt : k + 1 < #hf.toFinset) :
{n ∈ range (nth p (k + 1)) | p n} = insert (nth p k) {n ∈ range (nth p k) | p n} :=
| Mathlib/Data/Nat/Nth.lean | 354 | 355 |
/-
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.Algebra.Lie.Weights.Basic
import Mathlib.LinearAlgebra.Trace
import Mathlib.LinearAlgebra.FreeModule.PID
/-!
# Lie modules with linear weights
Given a Lie ... | Mathlib/Algebra/Lie/Weights/Linear.lean | 87 | 87 | |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou, Johan Commelin
-/
import Mathlib.Algebra.Ring.Int.Parity
import Mathlib.Algebra.Ring.Int.Units
import Mathlib.Data.ZMod.IntUnitsPower
/-!
# Integer powers of (-1)
This file def... |
@[simp]
| Mathlib/Algebra/Ring/NegOnePow.lean | 36 | 37 |
/-
Copyright (c) 2023 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Algebra.Unitization
import Mathlib.Algebra.Star.Subalgebra
import Mathlib.GroupTheory.GroupAction.Ring
/-!
# Relating unital and non-unital subs... | Unitization.lift (NonUnitalSubalgebraClass.subtype s)
@[simp]
theorem unitization_apply (x : Unitization R s) :
| Mathlib/Algebra/Algebra/Subalgebra/Unitization.lean | 78 | 81 |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Oliver Nash
-/
import Mathlib.Data.Finset.Card
import Mathlib.Data.Finset.Union
/-!
# Finsets in product types
This file defines finset constru... | classical
rw [← card_product, ← card_product, ← filter_product, ← filter_product, ← card_union_of_disjoint]
· apply congr_arg
| Mathlib/Data/Finset/Prod.lean | 164 | 166 |
/-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.Plus
import Mathlib.CategoryTheory.Limits.Shapes.ConcreteCategory
/-!
# Sheafification
We construct the sheafification of a presheaf ov... | exact h
@[simp]
theorem isoSheafify_inv {P : Cᵒᵖ ⥤ D} (hP : Presheaf.IsSheaf J P) :
(J.isoSheafify hP).inv = J.sheafifyLift (𝟙 _) hP := by
| Mathlib/CategoryTheory/Sites/ConcreteSheafification.lean | 523 | 527 |
/-
Copyright (c) 2019 mathlib community. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Wojciech Nawrocki
-/
import Mathlib.Data.Nat.Notation
import Mathlib.Tactic.TypeStar
import Mathlib.Util.CompileInductive
/-!
# Binary tree
Provides binary tree st... | theorem numLeaves_eq_numNodes_succ (x : Tree α) : x.numLeaves = x.numNodes + 1 := by
induction x <;> simp [*, Nat.add_comm, Nat.add_assoc, Nat.add_left_comm]
| Mathlib/Data/Tree/Basic.lean | 94 | 96 |
/-
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... | integrable on every compact subset contained in `s`. -/
theorem locallyIntegrableOn_iff [LocallyCompactSpace X] (hs : IsLocallyClosed s) :
LocallyIntegrableOn f s μ ↔ ∀ (k : Set X), k ⊆ s → IsCompact k → IntegrableOn f k μ := by
refine ⟨fun hf k hk ↦ hf.integrableOn_compact_subset hk, fun hf x hx ↦ ?_⟩
rcases h... | Mathlib/MeasureTheory/Function/LocallyIntegrable.lean | 131 | 136 |
/-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth, Yury Kudryashov
-/
import Mathlib.Topology.Order.Basic
/-!
# Bounded monotone sequences converge
In this file we prove a few theorems of the form “if the range of a... | section IsGLB
variable [Preorder α] [InfConvergenceClass α] {f : ι → α} {a : α}
theorem tendsto_atBot_isGLB (h_mono : Monotone f) (ha : IsGLB (Set.range f) a) :
| Mathlib/Topology/Order/MonotoneConvergence.lean | 96 | 100 |
/-
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.Order.Disjoint
import Mathlib.Order.RelIso.Basic
import Mathlib.Tactic.Monotonicity.Attr
/-!
# Order homomorphisms
This file defines order homomorphi... | Mathlib/Order/Hom/Basic.lean | 1,452 | 1,454 | |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Eric Wieser
-/
import Mathlib.Data.Fin.Tuple.Basic
/-!
# Matrix and vector notation
This file defines notation for vectors and matrices. Given `a b c d : α`,
the notation a... | @[simp]
theorem vecHead_vecAlt1 (hm : m + 2 = n + 1 + (n + 1)) (v : Fin (m + 2) → α) :
| Mathlib/Data/Fin/VecNotation.lean | 374 | 375 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Algebra.Pi
import Mathlib.LinearAlgebra.Finsupp.SumProd
import Mathlib.LinearAlgebra.FreeModule.Basic
import Mathlib.LinearAlgebra.LinearIndepe... | [Finite G] (φ : (G → k) →ₐ[k] k) : ∃ (s : G), φ = Pi.evalAlgHom _ _ s := by
have h1 := map_one φ
classical
have := Fintype.ofFinite G
simp only [← Finset.univ_sum_single (1 : G → k), Pi.one_apply, map_sum] at h1
obtain ⟨s, hs⟩ : ∃ (s : G), φ (Pi.single s 1) ≠ 0 := by
by_contra
simp_all
have h2 :... | Mathlib/LinearAlgebra/StdBasis.lean | 175 | 200 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Group.Nat.Defs
import Mathlib.CategoryTheory.Category.Preorder
import Mathlib.CategoryTheory.EqToHom
import Mathlib.CategoryTheory.Functor.Const
import M... | obtain _ | i := i
· simp only [← Fin.mk_zero]
simp
· exact h i (by valid)
/-- Given `obj : Fin (n + 1) → C` and `mapSucc i : obj i.castSucc ⟶ obj i.succ`
for all `i : Fin n`, this is `F : ComposableArrows C n` such that `F.obj i` is
definitionally equal to `obj i` and such that `F.map' i (i + 1) = ma... | Mathlib/CategoryTheory/ComposableArrows.lean | 849 | 866 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Sébastien Gouëzel,
Rémy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
/-!
# Limits and a... | · refine (tendsto_exp_mul_div_rpow_atTop (-b) a ha).inv_tendsto_atTop.congr' ?_
refine (eventually_ge_atTop 0).mono fun t ht => ?_
| Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean | 323 | 324 |
/-
Copyright (c) 2023 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Order.Atoms
import Mathlib.Order.Grade
import Mathlib.Order.Nat
/-!
# Finsets and multisets form a graded order
This... | lemma _root_.CovBy.exists_finset_insert (h : s ⋖ t) : ∃ a ∉ s, insert a s = t := by
simpa using h.exists_finset_cons
| Mathlib/Data/Finset/Grade.lean | 107 | 108 |
/-
Copyright (c) 2021 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Monoidal.Functor
/-!
# Preadditive monoidal categories
A monoidal category is `M... | (leftDistributor X f).inv ≫ (X ◁ biproduct.π _ j) = biproduct.π _ j := by
classical
cases nonempty_fintype J
simp [leftDistributor_inv, Preadditive.sum_comp, ← MonoidalCategory.whiskerLeft_comp,
| Mathlib/CategoryTheory/Monoidal/Preadditive.lean | 176 | 179 |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq
import Mathlib.CategoryTheory.Limits.Shapes.StrictInitial
import Mathlib.CategoryTheory.Limits.Types.Shapes
impo... | · simp
· simp only [coprod.desc_comp, colimit.ι_desc, Cofan.mk_pt, Cofan.mk_ι_app,
eqToHom_refl, Category.id_comp, dite_true, BinaryCofan.mk_pt, BinaryCofan.ι_app_right,
BinaryCofan.mk_inr, colimit.ι_desc_assoc, Discrete.functor_obj, Category.comp_id]
exact dif_neg j.prop }
let... | Mathlib/CategoryTheory/Extensive.lean | 488 | 520 |
/-
Copyright (c) 2021 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Alena Gusakov, Yaël Dillies
-/
import Mathlib.Data.Finset.Grade
import Mathlib.Data.Finset.Sups
import Mathlib.Logic.Function.Iterate
/-!
# Shadows
This file defines shad... | lemma mem_shadow_iff : t ∈ ∂ 𝒜 ↔ ∃ s ∈ 𝒜, ∃ a ∈ s, erase s a = t := by
simp only [shadow, mem_sup, mem_image]
theorem erase_mem_shadow (hs : s ∈ 𝒜) (ha : a ∈ s) : erase s a ∈ ∂ 𝒜 :=
| Mathlib/Combinatorics/SetFamily/Shadow.lean | 90 | 93 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Analysis.Convex.Hull
/-!
# Convex join
This file defines the convex join of two sets. The convex join of `s` and `t` is the union of the
segments with on... |
theorem convexJoin_comm (s t : Set E) : convexJoin 𝕜 s t = convexJoin 𝕜 t s :=
| Mathlib/Analysis/Convex/Join.lean | 36 | 37 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
-/
import Mathlib.MeasureTheory.Integral.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Lebesgue.Countable
import Mathlib.MeasureTheory.Integral.Le... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 1,491 | 1,494 | |
/-
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.Algebra.Field.Subfield.Basic
import Mathlib.Data.W.Cardinal
import Mathlib.Tactic.FinCases
/-!
# Cardinality of the division ring generated by a set
`Subfield.... | private lemma rangeOfWType_eq_top : rangeOfWType s = ⊤ := top_le_iff.mp fun a _ ↦ by
rw [← SetLike.mem_coe, ← Subtype.val_injective.mem_set_image]
change ↑a ∈ map (closure s).subtype _
refine closure_le.mpr (fun a ha ↦ ?_) a.prop
exact ⟨⟨a, subset_closure ha⟩, ⟨WType.mk (.inr ⟨a, ha⟩) Empty.rec, rfl⟩, rfl⟩
| Mathlib/SetTheory/Cardinal/Subfield.lean | 57 | 61 |
/-
Copyright (c) 2024 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Derivation.Killing
import Mathlib.Algebra.Lie.Killing
import Mathlib.Algebra.Lie.Sl2
import Mathlib.Algebra.Lie.Weights.Chain
import Mathlib.Line... |
section
variable [IsTriangularizable K H L]
/-- For any `α` and `β`, the corresponding root spaces are orthogonal with respect to the Killing
form, provided `α + β ≠ 0`. -/
lemma killingForm_apply_eq_zero_of_mem_rootSpace_of_add_ne_zero {α β : H → K} {x y : L}
(hx : x ∈ rootSpace H α) (hy : y ∈ rootSpace H β) (hα... | Mathlib/Algebra/Lie/Weights/Killing.lean | 92 | 114 |
/-
Copyright (c) 2023 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calculus.Deriv.Add
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calcul... | theorem hasLineDerivWithinAt_smul_iff {c : 𝕜} (hc : c ≠ 0) :
HasLineDerivWithinAt 𝕜 f (c • f') s x (c • v) ↔ HasLineDerivWithinAt 𝕜 f f' s x v :=
⟨fun h ↦ by simpa [smul_smul, inv_mul_cancel₀ hc] using h.smul (c ⁻¹), fun h ↦ h.smul c⟩
| Mathlib/Analysis/Calculus/LineDeriv/Basic.lean | 513 | 515 |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Complex.Basic
import Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle
/-!
# Closure, interior, and frontier of preimages under `re`... | protected lemma TendstoUniformlyOn.re {f : ι → α → ℂ} {p : Filter ι} {g : α → ℂ} {K : Set α}
(hf : TendstoUniformlyOn f g p K) :
TendstoUniformlyOn (fun n x => (f n x).re) (fun y => (g y).re) p K := by
| Mathlib/Analysis/Complex/ReImTopology.lean | 188 | 190 |
/-
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.Data.Fintype.Parity
import Mathlib.NumberTheory.LegendreSymbol.ZModChar
import Mathlib.FieldTheory.Finite.Basic
/-!
# Quadratic characters of finite fie... | Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/Basic.lean | 311 | 323 | |
/-
Copyright (c) 2023 Adrian Wüthrich. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adrian Wüthrich
-/
import Mathlib.Combinatorics.SimpleGraph.AdjMatrix
import Mathlib.LinearAlgebra.Matrix.PosDef
/-!
# Laplacian Matrix
This module defines the Laplacian matrix of a... |
theorem isSymm_degMatrix [AddMonoidWithOne R] : (G.degMatrix R).IsSymm :=
isSymm_diagonal _
| Mathlib/Combinatorics/SimpleGraph/LapMatrix.lean | 49 | 51 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.MeasurableSpace.Basic
import Mathlib.MeasureTheory.Measure.MeasureSpaceDef
/-!
# Sequence of measurable functions associated to a sequence o... | Mathlib/MeasureTheory/Function/AEMeasurableSequence.lean | 127 | 134 | |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.MeasureTheory.Covering.Besicovitch
import Mathlib.Tactic.AdaptationNote
import Mathlib.Algeb... | -/
by_contra! h
set N := multiplicity E + 1 with hN
have :
∀ δ : ℝ, 0 < δ → ∃ f : Fin N → E, (∀ i : Fin N, ‖f i‖ ≤ 2) ∧
Pairwise fun i j => 1 - δ ≤ ‖f i - f j‖ := by
intro δ hδ
rcases lt_or_le δ 1 with (hδ' | hδ')
· rcases h δ hδ hδ' with ⟨s, hs, h's, s_card⟩
obtain ⟨f, f_inj, hfs... | Mathlib/MeasureTheory/Covering/BesicovitchVectorSpace.lean | 174 | 246 |
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Analysis.Normed.Group.Int
import Mathlib.Analysis.Normed.Group.Subgroup
import Mathlib.Analysis.Normed.Group.Uniform
/-!
# Normed groups homomorphisms... | `AddMonoidHom.mkNormedAddGroupHom`, then its norm is bounded by the bound given to the constructor
if it is nonnegative. -/
| Mathlib/Analysis/Normed/Group/Hom.lean | 253 | 254 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Chris Hughes, Floris van Doorn, Yaël Dillies
-/
import Mathlib.Data.Nat.Basic
import Mathlib.Tactic.GCongr.CoreAttrs
import Mathlib.Tactic.Common
import Mathlib.Tactic.... | assumption
· match i with
| 0 => simp
| 1 =>
rw [← Nat.add_assoc, factorial_succ (1 + n), Nat.add_mul, Nat.one_mul, Nat.add_comm 1 n,
Nat.add_le_add_iff_right]
| Mathlib/Data/Nat/Factorial/Basic.lean | 158 | 163 |
/-
Copyright (c) 2020 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap
import Mathlib.MeasureTheory.Integral.Bochner.FundThmCalculus
import Mathlib.MeasureT... | Mathlib/MeasureTheory/Integral/SetIntegral.lean | 555 | 562 | |
/-
Copyright (c) 2021 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey
-/
import Mathlib.Algebra.Ring.Periodic
import Mathlib.Data.Nat.Count
/-!
# Periodic Functions on ℕ
This file identifies a few functions on `ℕ` which are periodic, and a... |
end Multiset
section Finset
open Finset
| Mathlib/Data/Nat/Periodic.lean | 48 | 54 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.MeasurableSpace.Constructions
import Mathlib.Tactic.FunProp
/-!
# Measurable embeddings and equivalences
A measurable e... | · rw [restrict_extend_compl_range]
exact hg'.comp measurable_subtype_coe
theorem exists_measurable_extend (hf : MeasurableEmbedding f) {g : α → γ} (hg : Measurable g)
(hne : β → Nonempty γ) : ∃ g' : β → γ, Measurable g' ∧ g' ∘ f = g :=
⟨extend f g fun x => Classical.choice (hne x),
hf.measurable_extend... | Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean | 107 | 113 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2
import Mathlib.MeasureTheory.Measure.Real
/-! # Conditional expectation in L1
This file contains ... |
variable {m m0 : MeasurableSpace α} {μ : Measure α} {hm : m ≤ m0} [SigmaFinite (μ.trim hm)]
{f g : α → F'} {s : Set α}
variable (F')
/-- Conditional expectation of a function as a linear map from `α →₁[μ] F'` to itself. -/
def condExpL1CLM (hm : m ≤ m0) (μ : Measure α) [SigmaFinite (μ.trim hm)] :
(α →₁[μ] F') ... | Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean | 405 | 425 |
/-
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.RingTheory.WittVector.Frobenius
import Mathlib.RingTheory.WittVector.Verschiebung
import Mathlib.RingTheory.WittVector.MulP
/-!
## Identities between ... | · simp
· rw [pow_succ, ← mul_assoc, ← add_assoc,mul_charP_coeff_succ, pow_succ, pow_mul]
congr
exact ih
| Mathlib/RingTheory/WittVector/Identities.lean | 124 | 127 |
/-
Copyright (c) 2024 Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Merten
-/
import Mathlib.CategoryTheory.Galois.GaloisObjects
import Mathlib.CategoryTheory.Limits.Shapes.CombinedProducts
import Mathlib.Data.Finite.Sum
/-!
# Decompositio... | rw [map_comp, FintypeCat.comp_apply, h]
convert_to F.map (selfProdProj u t) b =
(F.map (Pi.map' (fiberPerm h b) fun _ ↦ 𝟙 X) ≫
F.map (Pi.π (fun _ ↦ X) t)) (mkSelfProdFib F X)
rw [← map_comp, Pi.map'_comp_π, Category.comp_id, mkSelfProdFib_map_π F X (fiberPerm h b t)]
rfl
| Mathlib/CategoryTheory/Galois/Decomposition.lean | 255 | 261 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Logic.Equiv.Fin.Basic
imp... | apply Z.hasSum ?_ h'y
simp only [mem_insert_iff, add_eq_left] at hy
rcases hy with rfl | hy
· simp
apply mem_insert_of_mem _ (hr' ?_)
simp only [EMetric.mem_ball, edist_eq_enorm_sub, sub_zero, lt_min_iff, mem_inter_iff,
add_sub_cancel_left, hy, and_true] at h'y ⊢
exact h'y.2
@[deprecated (since := "2... | Mathlib/Analysis/Analytic/Basic.lean | 670 | 699 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
import Mathlib.MeasureTheory.MeasurableSpace.Prod
import Mathlib.MeasureTheory.Measure.Ty... | theorem AEMeasurable.ennreal_toNNReal {f : α → ℝ≥0∞} {μ : Measure α} (hf : AEMeasurable f μ) :
AEMeasurable (fun x => (f x).toNNReal) μ :=
ENNReal.measurable_toNNReal.comp_aemeasurable hf
@[simp, norm_cast]
theorem measurable_coe_nnreal_ennreal_iff {f : α → ℝ≥0} :
(Measurable fun x => (f x : ℝ≥0∞)) ↔ Measura... | Mathlib/MeasureTheory/Constructions/BorelSpace/Real.lean | 326 | 347 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Polynomial.Degree.Domain
import Mathlib.Algebra.Polynomial.Degree.Support
import Mathlib.Algebra.Poly... | exact (natDegree_derivative_le _).trans <| Nat.sub_le_sub_right hd 1
@[simp]
| Mathlib/Algebra/Polynomial/Derivative.lean | 193 | 195 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.BooleanAlgebra
import Mathlib.Tactic.Common
/-!
# Co-Heyting boundary
The boundary of an element of a co-Heyting algebra is the intersection of its... | exact boundary_le_boundary_sup_sup_boundary_inf_left
theorem boundary_sup_sup_boundary_inf (a b : α) : ∂ (a ⊔ b) ⊔ ∂ (a ⊓ b) = ∂ a ⊔ ∂ b :=
le_antisymm (sup_le boundary_sup_le boundary_inf_le) <|
sup_le boundary_le_boundary_sup_sup_boundary_inf_left
boundary_le_boundary_sup_sup_boundary_inf_right
@[simp... | Mathlib/Order/Heyting/Boundary.lean | 105 | 117 |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.TwoDim
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
/-!
# Oriented angles.
This file defines orie... | have h' : ∀ r' : ℝ, o.oangle x (r' • x + y) ≠ 0 ∧ o.oangle x (r' • x + y) ≠ π := by
intro r'
rwa [← o.oangle_smul_add_right_eq_zero_or_eq_pi_iff r', not_or] at h
let s : Set (V × V) := (fun r' : ℝ => (x, r' • x + y)) '' Set.univ
have hc : IsConnected s := isConnected_univ.image _ (by fun_prop)
have hf :... | Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | 784 | 792 |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Yaël Dillies
-/
import Mathlib.Data.Set.BooleanAlgebra
import Mathlib.Data.SetLike.Basic
import Mathlib.Order.Hom.Basic
/-!
# Closure operators between preorders
We defin... | Mathlib/Order/Closure.lean | 604 | 607 | |
/-
Copyright (c) 2024 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Measure.GiryMonad
import Mathlib.MeasureTheory.Measure.Stieltjes
import Mathlib.Analysis.Normed.Order.Lattice
import Mathlib.MeasureTheory.Fu... | exact IsRatStieltjesPoint.ite (IsRatStieltjesPoint f) id
(fun _ ↦ isRatStieltjesPoint_defaultRatCDF a)
measurable := measurable_toRatCDF hf
| Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean | 262 | 265 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.Lattice
import Mathlib.RingTheory.Coprime.Lemmas
import Mathlib... | · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩
theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι}
| Mathlib/RingTheory/Ideal/Operations.lean | 681 | 683 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.RingTheory.Ideal.Operations
/-!
# Maps on modules and ideals
Main definitions include `Ideal.map`, `Ideal.comap`, `RingHom.ker`, `Module.annihilator`
and `Subm... |
theorem LinearMap.annihilator_le_of_injective (f : M →ₗ[R] M') (hf : Function.Injective f) :
Module.annihilator R M' ≤ Module.annihilator R M := fun x h ↦ by
| Mathlib/RingTheory/Ideal/Maps.lean | 774 | 776 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.MvPolynomial.Rename
/-!
# Degrees of polynomials
This file establ... | degreeOf i (f * X j) = degreeOf i f := by
classical
simp only [degreeOf_eq_sup i, support_mul_X, Finset.sup_map]
congr
| Mathlib/Algebra/MvPolynomial/Degrees.lean | 288 | 291 |
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Topology.Bases
import Mathlib.Topology.Separation.Regular
/-!
# Dense embeddings
This file defines three properties of f... | trivial
/-- If `i : α → β` is a dense embedding with dense complement of the range, then any compact set in
`α` has empty interior. -/
| Mathlib/Topology/DenseEmbedding.lean | 75 | 78 |
/-
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.Algebra.BigOperators.Finprod
import Mathlib.Topology.ContinuousMap.Algebra
import Mathlib.Topology.Compactness.Paracompact
import Mathlib.Topology.Sh... | apply Finset.sum_congr rfl
rintro x hx
simp only [Finset.mem_sdiff, ρ.mem_finsupport, mem_support, Classical.not_not] at hx
exact hx.2
theorem sum_finsupport_smul_eq_finsum {M : Type*} [AddCommMonoid M] [Module ℝ M] (φ : ι → X → M) :
∑ i ∈ ρ.finsupport x₀, ρ i x₀ • φ i x₀ = ∑ᶠ i, ρ i x₀ • φ i x₀ := by
ap... | Mathlib/Topology/PartitionOfUnity.lean | 203 | 212 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.Bochner.Basic
import Mathlib.MeasureTheory.Integral.Bochner.L1
import Mathlib.Me... | Mathlib/MeasureTheory/Integral/Bochner.lean | 1,113 | 1,140 | |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.RingTheory.Ideal.Operations
/-!
# Maps on modules and ideals
Main definitions include `Ideal.map`, `Ideal.comap`, `RingHom.ker`, `Module.annihilator`
and `Subm... |
section CommRing
| Mathlib/RingTheory/Ideal/Maps.lean | 558 | 560 |
/-
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 | 1,226 | 1,228 | |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
import Mathlib.MeasureTheory.Measure.Typeclasses.NoAtoms
import Mathlib.MeasureTheory.Measure.Typeclasse... | Mathlib/MeasureTheory/Measure/OpenPos.lean | 280 | 289 | |
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Aaron Anderson
-/
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Module.Defs
import Mathlib.Algebra.Order.Pi
import Mathlib.Data.Fi... | ext
simp only [inf_apply, mem_support_iff, Ne, Finset.mem_union, Finset.mem_filter,
Finset.mem_inter]
simp only [← nonpos_iff_eq_zero, min_le_iff, not_or]
| Mathlib/Data/Finsupp/Order.lean | 291 | 295 |
/-
Copyright (c) 2024 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Gamma.Beta
/-!
# Deligne's archimedean Gamma-factors
In the theory of L-series one frequently encounters the following fun... | lemma Gammaℝ_one : Gammaℝ 1 = 1 := by
rw [Gammaℝ_def, Complex.Gamma_one_half_eq]
simp [neg_div, cpow_neg, inv_mul_cancel, pi_ne_zero]
| Mathlib/Analysis/SpecialFunctions/Gamma/Deligne.lean | 76 | 79 |
/-
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 | 964 | 964 | |
/-
Copyright (c) 2021 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Group.Measure
import Mathlib.MeasureTheory.Measure.Prod
/-!
# Measure theory in the product of groups
In this file we show propertie... | @[to_additive]
theorem absolutelyContinuous_inv : μ ≪ μ.inv := by
refine AbsolutelyContinuous.mk fun s _ => ?_
simp_rw [inv_apply μ s, measure_inv_null, imp_self]
| Mathlib/MeasureTheory/Group/Prod.lean | 177 | 180 |
/-
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.List.MinMax
import Mathlib.Algebra.Tropical.Basic
import Mathlib.Order.ConditionallyCompleteLattice.Finset
import Mathlib.Algebra.BigOperators.G... | · simp
· simp [← IH]
theorem Finset.trop_inf [LinearOrder R] [OrderTop R] (s : Finset S) (f : S → R) :
trop (s.inf f) = ∑ i ∈ s, trop (f i) := by
| Mathlib/Algebra/Tropical/BigOperators.lean | 78 | 82 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Order.Filter.SmallSets
import Mathlib.Topology.UniformSpace.Defs
import Mathlib.Topology.ContinuousOn
/-!
# Basic resu... | Mathlib/Topology/UniformSpace/Basic.lean | 1,285 | 1,288 | |
/-
Copyright (c) 2021 Stuart Presnell. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stuart Presnell
-/
import Mathlib.Data.Nat.PrimeFin
import Mathlib.Data.Nat.Factorization.Defs
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Order.Interval.Finset.Nat
import... | rcases eq_or_ne b 0 with rfl | hb
· simp [factorizationLCMLeft]
nth_rewrite 2 [← factorization_prod_pow_eq_self ha]
| Mathlib/Data/Nat/Factorization/Basic.lean | 461 | 463 |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Logic.Function.Defs
import Mathlib.Order.Defs.Unbundled
/-!
# Lexicographic order on a sigma type
This defines the lexicographical order of two arbitrary... | · rintro (⟨a, b, hij⟩ | ⟨a, b, hab⟩)
· exact Or.inl hij
· exact Or.inr ⟨rfl, hab⟩
· obtain ⟨i, a⟩ := a
obtain ⟨j, b⟩ := b
dsimp only
rintro (h | ⟨rfl, h⟩)
· exact Lex.left _ _ h
· exact Lex.right _ _ h
instance Lex.decidable (r : ι → ι → Prop) (s : ∀ i, α i → α i → Prop) [DecidableEq ι]... | Mathlib/Data/Sigma/Lex.lean | 45 | 55 |
/-
Copyright (c) 2022 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Analysis.Normed.Group.Quotient
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Topology.Instances.AddCircle
/-!
# The additive circle as a norm... | Mathlib/Analysis/Normed/Group/AddCircle.lean | 241 | 244 | |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yaël Dillies
-/
import Mathlib.Algebra.Module.BigOperators
import Mathlib.GroupTheory.Perm.Basic
import Mathlib.GroupTheory.Perm.Finite
import Mathlib.GroupTheory.Perm.Lis... | @[simp]
theorem sameCycle_zpow_left {n : ℤ} : SameCycle f ((f ^ n) x) y ↔ SameCycle f x y :=
(Equiv.addRight (n : ℤ)).exists_congr_left.trans <| by simp [SameCycle, zpow_add]
| Mathlib/GroupTheory/Perm/Cycle/Basic.lean | 126 | 128 |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Kim Morrison
-/
import Mathlib.CategoryTheory.Subobject.MonoOver
import Mathlib.CategoryTheory.Skeletal
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathlib.... |
@[reassoc (attr := simp)]
theorem underlying_arrow {X : C} {Y Z : Subobject X} (f : Y ⟶ Z) :
underlying.map f ≫ arrow Z = arrow Y :=
| Mathlib/CategoryTheory/Subobject/Basic.lean | 210 | 213 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Ordmap.Invariants
/-!
# Verification of `Ordnode`
This file uses the invariants defined in `Mathlib.Data.Ordmap.Invariants` to construct `Ordset... | · refine @findMin'_all (P := fun a : α => Bounded nil o₁ (a : WithBot α))
_ rl rx (sep.2.1.1.imp ?_) hr.1.1.to_nil
exact fun y h => hl.1.1.to_nil.mono_right (le_of_lt h)
· exact
@findMin'_all _ (fun a => All (· < a) (.node ls ll lx lr)) rl rx
| Mathlib/Data/Ordmap/Ordset.lean | 423 | 427 |
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