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 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Algebra.Group.Conj
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.Group.Subgroup.Ker
/-!
# Basic results on subgroups
We prove basic results... |
namespace IsConj
| Mathlib/Algebra/Group/Subgroup/Basic.lean | 923 | 925 |
/-
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... | Measurable fun x => (f x : EReal) :=
measurable_coe_ennreal_ereal.comp hf
@[measurability, fun_prop]
theorem AEMeasurable.coe_ereal_ennreal {f : α → ℝ≥0∞} {μ : Measure α} (hf : AEMeasurable f μ) :
| Mathlib/MeasureTheory/Constructions/BorelSpace/Real.lean | 425 | 429 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Kexing Ying, Moritz Doll
-/
import Mathlib.Algebra.GroupWithZero.Action.Opposite
import Mathlib.LinearAlgebra.Finsupp.VectorSpace
import Mathlib.LinearAlgebra.Matrix.Basis
im... | theorem _root_.Matrix.separatingLeft_toLinearMap₂'_iff {M : Matrix ι ι R₁} :
(Matrix.toLinearMap₂' R₁ M).SeparatingLeft (R := R₁) ↔ M.Nondegenerate :=
⟨fun h v hv => h v fun w => (M.toLinearMap₂'_apply' _ _).trans <| hv w,
Matrix.Nondegenerate.toLinearMap₂'⟩
| Mathlib/LinearAlgebra/Matrix/SesquilinearForm.lean | 631 | 634 |
/-
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... | refine ⟨fun h𝒜 s hs => ?_, Set.Sized.shadow⟩
obtain ⟨a, ha⟩ := nonempty_iff_ne_empty.2 (ne_of_mem_of_not_mem hs h)
rw [← h𝒜 (erase_mem_shadow hs ha), card_erase_add_one ha]
/-- Being in the shadow of `𝒜` means we have a superset in `𝒜`. -/
lemma exists_subset_of_mem_shadow (hs : t ∈ ∂ 𝒜) : ∃ s ∈ 𝒜, t ⊆ s :... | Mathlib/Combinatorics/SetFamily/Shadow.lean | 163 | 168 |
/-
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, Amelia Livingston, Yury Kudryashov,
Neil Strickland, Aaron Anderson
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Tactic.Commo... | Mathlib/Algebra/Divisibility/Basic.lean | 221 | 222 | |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Yury Kudryashov
-/
import Mathlib.Algebra.Algebra.Rat
import Mathlib.Data.Nat.Prime.Int
import Mathlib.Data.Rat.Sqrt
imp... | Mathlib/Data/Real/Irrational.lean | 692 | 693 | |
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Paul Lezeau
-/
import Mathlib.RingTheory.DedekindDomain.Ideal
import Mathlib.RingTheory.IsAdjoinRoot
/-!
# Kummer-Dedekind theorem
This file proves the monogenic version of... |
open IsLocalization in
| Mathlib/NumberTheory/KummerDedekind.lean | 85 | 86 |
/-
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.Algebra.Order.Field.Pi
import Mathlib.Algebra.Order.Pi
import Mathlib.Analysis.Normed.Field.Basic
import Mathlib.Analysis.Normed.Group.Pointwise
import Mat... | /-!
#### Note
The closure and frontier of an antichain might not be antichains. Take for example the union
of the open segments from `(0, 2)` to `(1, 1)` and from `(2, 1)` to `(3, 0)`. `(1, 1)` and `(2, 1)`
are comparable and both in the closure/frontier.
-/
protected lemma IsClosed.upperClosure_pi (hs : IsClosed s) ... | Mathlib/Analysis/Normed/Order/UpperLower.lean | 189 | 205 |
/-
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.Real
/-!
# Power function... | lift x to ℝ≥0 using ne_of_lt (lt_of_le_of_lt hx coe_lt_top)
simp only [coe_le_one_iff] at hx
simp [← coe_rpow_of_nonneg _ hz, NNReal.rpow_le_one hx hz]
theorem rpow_lt_one_of_one_lt_of_neg {x : ℝ≥0∞} {z : ℝ} (hx : 1 < x) (hz : z < 0) : x ^ z < 1 := by
cases x
| Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | 827 | 832 |
/-
Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Algebra.Order.Chebyshev
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Math... | (Odd.pow_pos_iff (by decide)).mp (eps_pow_five_pos hPε)
| Mathlib/Combinatorics/SimpleGraph/Regularity/Bound.lean | 115 | 115 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Function.L1Space.Integrable
import Mathlib.MeasureTheory.Function.LpSpace.Indicator
/-! # Functions integrable on a set an... | Mathlib/MeasureTheory/Integral/IntegrableOn.lean | 738 | 740 | |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël 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 ... | variable {f : α → β}
theorem minimal_mem_image_monotone (hf : ∀ ⦃x y⦄, x ∈ s → y ∈ s → (f x ≤ f y ↔ x ≤ y))
(hx : Minimal (· ∈ s) x) : Minimal (· ∈ f '' s) (f x) := by
refine ⟨mem_image_of_mem f hx.prop, ?_⟩
| Mathlib/Order/Minimal.lean | 425 | 429 |
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Data.Set.BooleanAlgebra
import Mathlib.Tactic.AdaptationNote
/-!
# Relations
This file defines bundled relations. A relation between `α` and `β` is a f... | theorem dom_mono {r s : Rel α β} (h : r ≤ s) : dom r ⊆ dom s := fun a ⟨b, hx⟩ => ⟨b, h a b hx⟩
/-- Codomain aka range of a relation -/
| Mathlib/Data/Rel.lean | 70 | 72 |
/-
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.MeasureTheory.Measure.Content
import Mathlib.MeasureTheory.Group.Prod
import Mathlib.Topology.Algebra.Group.Compact
/-!
# Haar measure
In this fi... | simp only [mem_iUnion]; use g₀; constructor; swap
· simp only [Finset.mem_filter, h1g₀, true_and]; use g
simp [hg, h2g₀]
exact h2g₀
| Mathlib/MeasureTheory/Measure/Haar/Basic.lean | 214 | 217 |
/-
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.Algebra.Group.Basic
import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
import Mathlib.Order.Lattice
/-!
# Ordered Subtraction
This file proves l... | variable [AddCommMonoid α] [Sub α] [OrderedSub α] {a b : α}
| Mathlib/Algebra/Order/Sub/Defs.lean | 213 | 214 |
/-
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... | choose! t t_open xt ht using A
refine ⟨⋃ x ∈ s, t x, isOpen_biUnion t_open, fun x hx ↦ mem_biUnion hx (xt x hx), ?_⟩
| Mathlib/Topology/UniformSpace/Basic.lean | 978 | 979 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Shapes.Equalizers
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.Mono
import Mathlib.CategoryTheory.Limits.Shape... | (m : ImageMap sq) : HasImageMap sq :=
⟨Nonempty.intro m⟩
theorem HasImageMap.transport {f g : Arrow C} [HasImage f.hom] [HasImage g.hom] (sq : f ⟶ g)
| Mathlib/CategoryTheory/Limits/Shapes/Images.lean | 641 | 644 |
/-
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.Order.Interval.Set.OrderEmbedding
import Mathlib.Order.Antichain
import Mathlib.Order.SetNotation
/-!
# Order-connected sets
We say that a set `s :... |
lemma not_ordConnected_inter_Icc_iff (hx : x ∈ s) (hy : y ∈ s) :
¬ OrdConnected (s ∩ Icc x y) ↔ ∃ z ∉ s, z ∈ Ioo x y := by
simp_rw [ordConnected_inter_Icc_iff hx hy, subset_def, not_forall, exists_prop, and_comm]
| Mathlib/Order/Interval/Set/OrdConnected.lean | 257 | 260 |
/-
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 ... |
variable {G' : G.Subgraph} {s : Set V}
| Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | 1,212 | 1,213 |
/-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.Topology.MetricSpace.HausdorffDistance
/-!
# Thickenings in pseudo-metric spaces
## Main definitions
* `Metric.thickening δ s`, the open thickening by ra... | theorem cthickening_singleton {α : Type*} [PseudoMetricSpace α] (x : α) {δ : ℝ} (hδ : 0 ≤ δ) :
cthickening δ ({x} : Set α) = closedBall x δ := by
| Mathlib/Topology/MetricSpace/Thickening.lean | 237 | 238 |
/-
Copyright (c) 2022 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer, Kevin Klinge, Andrew Yang
-/
import Mathlib.Algebra.Group.Submonoid.DistribMulAction
import Mathlib.GroupTheory.OreLocalization.Basic
import Mathlib.Algebra.GroupWi... | Mathlib/RingTheory/OreLocalization/Basic.lean | 371 | 373 | |
/-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory.Basic
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib... | slice_lhs 1 2 => rw [ihf]
simp
-- p ≠ nil required! See the docstring of `normalizeAux`.
| whisker_left _ _ ih =>
dsimp
rw [associator_inv_naturality_right_assoc, whisker_exchange_assoc, ih]
simp
| whisker_right h η' ih =>
dsimp
rw [associator_inv_naturality_middle_assoc, ← comp_whiske... | Mathlib/CategoryTheory/Bicategory/Coherence.lean | 161 | 183 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
import Mathlib.Probability.Notation
import Mathlib.Probability.Process.Stopping
/-!
# Martingales
A family of functions `f : ι → Ω → E` is a martingale wit... | Submartingale f ℱ μ := by
refine ⟨hadp, fun i j hij => ?_, hint⟩
suffices f i ≤ᵐ[μ.trim (ℱ.le i)] μ[f j|ℱ i] by exact ae_le_of_ae_le_trim this
suffices 0 ≤ᵐ[μ.trim (ℱ.le i)] μ[f j|ℱ i] - f i by
filter_upwards [this] with x hx
| Mathlib/Probability/Martingale/Basic.lean | 256 | 260 |
/-
Copyright (c) 2022 María Inés de Frutos-Fernández. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: María Inés de Frutos-Fernández
-/
import Mathlib.Order.Filter.Cofinite
import Mathlib.RingTheory.DedekindDomain.Ideal
import Mathlib.RingTheory.UniqueFactorizationDomai... |
open Classical in
/-- `val_v(I)` does not depend on the choice of `a` and `J` used to represent `I`. -/
theorem count_well_defined {I : FractionalIdeal R⁰ K} (hI : I ≠ 0) {a : R}
{J : Ideal R} (h_aJ : I = spanSingleton R⁰ ((algebraMap R K) a)⁻¹ * ↑J) :
count K v I = ((Associates.mk v.asIdeal).count (Associates... | Mathlib/RingTheory/DedekindDomain/Factorization.lean | 293 | 325 |
/-
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.Interval.Set.Basic
import Mathlib.Data.Set.Lattice.Image
import Mathlib.Data.SetLike.Basic
/-!
# Order intervals
This file defines (nonempty) close... | Mathlib/Order/Interval/Basic.lean | 765 | 766 | |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Piecewise
import Mathlib.Algebra.Group.Ext
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.BinaryProducts
import Ma... |
end
| Mathlib/CategoryTheory/Preadditive/Biproducts.lean | 647 | 648 |
/-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.MeasureTheory.Measure.Regular
import Mathlib.MeasureTheory.Function.SimpleFuncDenseLp
import Mathlib.Topology.UrysohnsLemma
import Mathlib.MeasureThe... | ∃ g : α → E, eLpNorm (f - g) p μ ≤ ε ∧ Continuous g ∧ MemLp g p μ ∧ HasCompactSupport g by
rcases H with ⟨g, hg, g_cont, g_mem, g_support⟩
exact ⟨g, g_support, hg, g_cont, g_mem⟩
-- It suffices to check that the set of functions we consider approximates characteristic
-- functions, is stable under add... | Mathlib/MeasureTheory/Function/ContinuousMapDense.lean | 139 | 188 |
/-
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.Combinatorics.SimpleGraph.Acyclic
import Mathlib.Data.ENat.Lattice
/-!
# Girth of a simple graph
This file defines the girth and the extended girth of a ... | Mathlib/Combinatorics/SimpleGraph/Girth.lean | 27 | 27 | |
/-
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.Algebra.Order.Ring.Canonical
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Combinatorics.SetFamily.Compression.Down
import Mathlib.Data.Fintype.Powe... |
/-- **Harris-Kleitman inequality**: Any two upper sets of finsets correlate. -/
theorem IsUpperSet.le_card_inter_finset (h𝒜 : IsUpperSet (𝒜 : Set (Finset α)))
(hℬ : IsUpperSet (ℬ : Set (Finset α))) :
#𝒜 * #ℬ ≤ 2 ^ Fintype.card α * #(𝒜 ∩ ℬ) := by
| Mathlib/Combinatorics/SetFamily/HarrisKleitman.lean | 114 | 118 |
/-
Copyright (c) 2024 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Module.LocalizedModule.Submodule
import Mathlib.LinearAlgebra.Dimension.DivisionRing
import Mathlib.RingTheory.IsTensorProduct
import Mathlib.RingThe... | universe w in
instance IsDomain.hasRankNullity [IsDomain R] : HasRankNullity.{w} R where
rank_quotient_add_rank := rank_quotient_add_rank_of_isDomain
exists_set_linearIndependent M := exists_set_linearIndependent_of_isDomain R M
namespace IsBaseChange
| Mathlib/LinearAlgebra/Dimension/Localization.lean | 96 | 102 |
/-
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.Complex
import Qq
/-! # P... | MonotoneOn (fun (x : ℝ) => x ^ r) (Set.Ici 0) :=
fun _ ha _ _ hab => rpow_le_rpow ha hab hr
| Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 531 | 532 |
/-
Copyright (c) 2023 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.ModuleCat
import Mathlib.RepresentationTheory.GroupCohomology.Basic
import Mathlib.RepresentationTheory.... | dsimp
rw [Fin.sum_univ_three]
simp only [sub_eq_add_neg, add_assoc, Fin.val_zero, zero_add, pow_one, neg_smul,
one_smul, Fin.val_one, Fin.val_two, pow_succ' (-1 : k) 2, neg_sq, Nat.one_add, one_pow, mul_one]
rcongr i <;> fin_cases i <;> rfl
| Mathlib/RepresentationTheory/GroupCohomology/LowDegree.lean | 198 | 202 |
/-
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... | Mathlib/Algebra/Ring/Defs.lean | 419 | 419 | |
/-
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.Order.SupClosed
/-!
# Sublattices
This file defines sublattices.
## TODO
Subsemilattices, if people care about them.
## Tags
sublattice
-/
open Func... | comap f L ⊔ comap f M ≤ comap f (L ⊔ M) := comap_mono.le_map_sup _ _
lemma le_comap_iSup (f : LatticeHom α β) (L : ι → Sublattice β) :
⨆ i, (L i).comap f ≤ (⨆ i, L i).comap f := comap_mono.le_map_iSup
| Mathlib/Order/Sublattice.lean | 283 | 286 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Topology.MetricSpace.Basic
/-!
# Metrizability of a T₃ topological space with second countable topology
In this file we define metrizable topologic... | Mathlib/Topology/Metrizable/Basic.lean | 133 | 137 | |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Normed.Operator.BoundedLinear... | · rw [← dist_eq_norm]
change dist (f.uncurry (a', z)) (f.uncurry (a, z)) ≤ ε
apply (hu _ _ _).le
· exact ha'x'.1
· simp [dzx]
· rw [← dist_eq_norm']
change dist (f.uncurry (a', y)) (f.uncurry (a, y)) ≤ ε
apply (hu _ _ _).le
· exact ha'x'.1
· si... | Mathlib/Analysis/Calculus/FDeriv/Measurable.lean | 846 | 910 |
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.Perm.Option
import Mathlib.Logic.Equiv.Fin.Rotate
import Mathlib.Logic.Equiv.Fintype
/-!
# Permutatio... | · simp [Equiv.Perm.decomposeFin, EquivFunctor.map]
· intro i
| Mathlib/GroupTheory/Perm/Fin.lean | 44 | 45 |
/-
Copyright (c) 2020 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.Algebra.GeomSum
import Mathlib.Data.Finset.Slice
import Mathlib.Data.Nat.BitIndices
import Mathlib.Order.SupClosed
import Math... | have hau : a ∈ u := mem_u.2 ⟨hat, has, fun b hbs hbt ↦ (hm _ hbs hbt).trans hma⟩
exact hma.not_le <| le_max' _ _ hau
| Mathlib/Combinatorics/Colex.lean | 354 | 355 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Michael Stoll
-/
import Mathlib.Data.Nat.Squarefree
import Mathlib.NumberTheory.Zsqrtd.QuadraticReciprocity
import Mathlib.NumberTheory.Padics.PadicVal.Basic
/-!
# Sums of... | have ha := mt sq_eq_zero_iff.mpr ha₂
have hb := right_ne_zero_of_mul hab
haveI hqi : Fact q.Prime := ⟨hq⟩
simp_rw [h₁, padicValNat.mul ha₂ hb, padicValNat.pow 2 ha, hqb, add_zero]
exact even_two_mul _
· obtain ⟨b, a, hb₀, ha₀, hab, hb⟩ := Nat.sq_mul_squarefree_of_pos hn₀
refine ⟨a, b, hab.symm... | Mathlib/NumberTheory/SumTwoSquares.lean | 214 | 240 |
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Mathlib.Control.Basic
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Option.Basic
im... | Mathlib/Data/List/Basic.lean | 3,108 | 3,110 | |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.MeasureTheory.Integral.Lebesgue.Countable
/-!
# The Giry monad
Let X be a measurable space. The collection of all measures on X again
forms a measura... |
@[simp]
theorem join_smul {R : Type*} [SMul R ℝ≥0∞] [IsScalarTower R ℝ≥0∞ ℝ≥0∞] (c : R)
(m : Measure (Measure α)) : (c • m).join = c • m.join := by
ext s hs
simp [hs]
@[simp]
theorem join_dirac (μ : Measure α) : join (dirac μ) = μ := by
ext s hs
simp [hs, lintegral_dirac', measurable_coe]
theorem le_ae_j... | Mathlib/MeasureTheory/Measure/GiryMonad.lean | 128 | 149 |
/-
Copyright (c) 2021 Aaron Anderson, Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Kevin Buzzard, Yaël Dillies, Eric Wieser
-/
import Mathlib.Data.Finset.Lattice.Union
import Mathlib.Data.Finset.Pairwise
import Mathlib.Data.Finset.Prod
i... | suffices t j ≤ ⨆ (k) (_ : k ≠ i), t k by
replace ht := (ht i).mono_right this
rwa [h, disjoint_self] at ht
replace contra : j ≠ i := Ne.symm contra
-- Porting note: needs explicit `f`
exact le_iSup₂ (f := fun x _ ↦ t x) j contra
@[deprecated (since := "2024-11-24")] alias CompleteLattice.Independent.in... | Mathlib/Order/SupIndep.lean | 415 | 427 |
/-
Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Johannes Hölzl, Rémy Degenne
-/
import Mathlib.Order.ConditionallyCompleteLattice.Indexed
import Mathlib.Order.Filter.IsBounded
import Mathlib.Order.Hom.CompleteL... | cofinite.blimsup_set_eq]
rfl
/-- In other words, `limsup cofinite s` is the set of elements lying inside the family `s`
| Mathlib/Order/LiminfLimsup.lean | 714 | 717 |
/-
Copyright (c) 2021 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Ines Wright, Joachim Breitner
-/
import Mathlib.GroupTheory.Solvable
import Mathlib.GroupTheory.Sylow
import Mathlib.Algebra.Group.Subgroup.Order
import Mathlib.GroupTheo... | Mathlib/GroupTheory/Nilpotent.lean | 890 | 906 | |
/-
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.GroupTheory.Index
/-!
# Complements
In this file we define the complement of a subgroup.
## Main definitions
- `Subgroup.IsComplement S T` where ... | obtain ⟨S, cmem, -⟩ := H''.exists_isComplement_right 1
refine ⟨H.subtype '' S, ?_, ?_⟩
· have : H.subtype '' (H'' * S) = H''.map H.subtype * H.subtype '' S := image_mul H.subtype
rw [← this, cmem.mul_eq]
| Mathlib/GroupTheory/Complement.lean | 431 | 434 |
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Algebra.Group.Commute.Hom
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Data.Fintype.B... | ih (comm_ff.mono fun _ => mem_cons_of_mem) (comm_gg.mono fun _ => mem_cons_of_mem)
(comm_gf.mono fun _ => mem_cons_of_mem),
(noncommProd_commute _ _ _ _ fun y hy => ?_).mul_mul_mul_comm]
exact comm_gf (mem_cons_self x s) (mem_cons_of_mem hy) (ne_of_mem_of_not_mem hy hnmem).symm
section FinitePi
| Mathlib/Data/Finset/NoncommProd.lean | 393 | 398 |
/-
Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
import Mathlib.Topology.Bornology.Constructions
import Mathlib.Topology.MetricSpace.Pseudo.Def... | lemma dist_eq (x y : ULift β) : dist x y = dist x.down y.down := rfl
| Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean | 133 | 134 |
/-
Copyright (c) 2024 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.LinearAlgebra.Dimension.Finite
import Mathlib.LinearAlgebra.Dimension.Constructions
/-!
# Some results on free modules over rings satisfying strong rank condition
T... | constructor
· rw [rank_submodule_le_one_iff]
rintro ⟨v₀, _, h⟩
exact ⟨v₀, h⟩
· rintro ⟨v₀, h⟩
obtain ⟨κ, b⟩ := Module.Free.exists_basis (R := K) (M := s)
simpa [b.mk_eq_rank''] using b.linearIndependent.map' _ (ker_inclusion _ _ h)
|>.cardinal_le_rank.trans (rank_span_le {v₀})
theorem Submo... | Mathlib/LinearAlgebra/Dimension/FreeAndStrongRankCondition.lean | 158 | 168 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
import Mathlib.Analysis.Normed.Module.Basic
import Mathlib.MeasureTheory.Function.SimpleFuncDense
/-!
# Strongly measurable and finitely strongly meas... | Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean | 1,761 | 1,774 | |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kenny Lau
-/
import Mathlib.Data.DFinsupp.Submonoid
import Mathlib.Data.Finsupp.ToDFinsupp
import Mathlib.LinearAlgebra.Finsupp.SumProd
import Mathlib.LinearAlgebra.LinearIn... | iSupIndep. -/
theorem iSupIndep_of_dfinsupp_lsum_injective (p : ι → Submodule R N)
(h : Function.Injective (lsum ℕ fun i => (p i).subtype)) :
iSupIndep p := by
rw [iSupIndep_iff_forall_dfinsupp]
intro i x v hv
replace hv : lsum ℕ (fun i => (p i).subtype) (erase i v) =
lsum ℕ (fun i => (p i).subtype)... | Mathlib/LinearAlgebra/DFinsupp.lean | 463 | 471 |
/-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Group.Commute.Defs
import Mathlib.Algebra.Group.Units.Defs
import Mathlib.Algebra.GroupWithZero.Defs
import Mathlib.Algebra.Order.Monoid.Unbundle... | end MulZeroClass
section MulOneClass
variable [MulOneClass R]
| Mathlib/Algebra/Regular/Basic.lean | 254 | 258 |
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Mathlib.Control.Basic
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Option.Basic
im... | Mathlib/Data/List/Basic.lean | 2,134 | 2,137 | |
/-
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.Data.Set.Image
/-!
### Recursion on the natural numbers and `Set.range`
-/
namespace Nat
section Set
open Set
theorem zero_union_range_succ : {... | rename_i n
induction n with
| zero => rfl
| succ n ihn => dsimp at ihn ⊢; rw [ihn]
theorem range_casesOn {α : Type*} (x : α) (f : ℕ → α) :
(Set.range fun n => Nat.casesOn n x f : Set α) = {x} ∪ Set.range f :=
(range_of_succ _).symm
end Set
| Mathlib/Data/Nat/Set.lean | 37 | 46 |
/-
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.Analysis.Normed.Ring.InfiniteSum
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.NumberTheory.ArithmeticFunction
import Mathlib.NumberTheory... |
include hf₁ hmul in
/-- A version of `EulerProduct.summable_and_hasSum_smoothNumbers_prod_primesBelow_tsum`
in terms of the value of the series. -/
lemma prod_primesBelow_tsum_eq_tsum_smoothNumbers (hsum : Summable (‖f ·‖)) (N : ℕ) :
∏ p ∈ N.primesBelow, ∑' n : ℕ, f (p ^ n) = ∑' m : N.smoothNumbers, f m :=
(summ... | Mathlib/NumberTheory/EulerProduct/Basic.lean | 143 | 152 |
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Mario Carneiro, Simon Hudon
-/
import Mathlib.Data.PFunctor.Multivariate.Basic
import Mathlib.Data.PFunctor.Univariate.M
/-!
# The M construction as a multivariate polyn... | rw [MvPFunctor.map_eq]; congr
conv_rhs => rw [← split_dropFun_lastFun f, appendFun_comp_splitFun]
rfl
| Mathlib/Data/PFunctor/Multivariate/M.lean | 195 | 198 |
/-
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.Abelian.LeftDerived
import Mathlib.CategoryTheory.Monoidal.Preadditive
/-!
# Tor, the left-derived functor of tensor product
We define `To... | Mathlib/CategoryTheory/Monoidal/Tor.lean | 68 | 71 | |
/-
Copyright (c) 2019 Gabriel Ebner. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Gabriel Ebner, Sébastien Gouëzel, Yury Kudryashov, Yuyang Zhao
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Comp
import Mathlib.Analysis.Calcu... | nonrec theorem HasDerivAt.comp (hh₂ : HasDerivAt h₂ h₂' (h x)) (hh : HasDerivAt h h' x) :
HasDerivAt (h₂ ∘ h) (h₂' * h') x :=
hh₂.comp x hh hh.continuousAt
/-- The chain rule.
| Mathlib/Analysis/Calculus/Deriv/Comp.lean | 237 | 241 |
/-
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.Algebra.Order.Pi
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
/-!
# Simple functions
A function `f` from a measurable ... |
protected theorem mul {β} [MulZeroClass β] {f g : α →ₛ β} (hf : f.FinMeasSupp μ)
(hg : g.FinMeasSupp μ) : (f * g).FinMeasSupp μ := by
rw [mul_eq_map₂]
exact hf.map₂ hg (zero_mul 0)
theorem lintegral_lt_top {f : α →ₛ ℝ≥0∞} (hm : f.FinMeasSupp μ) (hf : ∀ᵐ a ∂μ, f a ≠ ∞) :
f.lintegral μ < ∞ := by
refine su... | Mathlib/MeasureTheory/Function/SimpleFunc.lean | 1,103 | 1,113 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Order.CauSeq.BigOperators
import Mathlib.Algebra.Order.Star.Basic
import Mathlib.Data.C... | Mathlib/Data/Complex/Exponential.lean | 936 | 937 | |
/-
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.Operations
/-!
# Results about division in extended non-negative reals
This file establishes basic properties related t... | induction' a with a
· replace hb : b ≠ 0 := coe_ne_zero.1 (hb.neg_resolve_left rfl)
| Mathlib/Data/ENNReal/Inv.lean | 229 | 230 |
/-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Ordinal.FixedPoint
/-!
# Principal ordinals
We define principal or indecomposable ordinals, and we prove the standa... | constructor <;> exact fun h a b ha hb => h hb ha
theorem not_principal_iff : ¬ Principal op o ↔ ∃ a < o, ∃ b < o, o ≤ op a b := by
| Mathlib/SetTheory/Ordinal/Principal.lean | 52 | 54 |
/-
Copyright (c) 2018 Mario Carneiro. 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.Archimedean.Basic
import Mathlib.Algebra.Order.Group.Pointwise.Bounds
import Mathlib.Data.Real.Basic
import Mathlib.Ord... | theorem sInf_le_sSup (s : Set ℝ) (h₁ : BddBelow s) (h₂ : BddAbove s) : sInf s ≤ sSup s := by
rcases s.eq_empty_or_nonempty with (rfl | hne)
· rw [sInf_empty, sSup_empty]
· exact csInf_le_csSup h₁ h₂ hne
| Mathlib/Data/Real/Archimedean.lean | 296 | 299 |
/-
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.Combinatorics.Hall.Basic
import Mathlib.Data.Matrix.Rank
import Mathlib.LinearAlgebra.Projectivization.Constructions
/-!
# Configurations of Points ... | theorem HasLines.pointCount_le_lineCount [HasLines P L] {p : P} {l : L} (h : p ∉ l)
[Finite { l : L // p ∈ l }] : pointCount P l ≤ lineCount L p := by
by_cases hf : Infinite { p : P // p ∈ l }
· exact (le_of_eq Nat.card_eq_zero_of_infinite).trans (zero_le (lineCount L p))
haveI := fintypeOfNotInfinite hf
ca... | Mathlib/Combinatorics/Configuration.lean | 186 | 195 |
/-
Copyright (c) 2021 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.MvPolynomial.Variables
/-!
# Polynomials supported by a set of variables
This file contains the definition and lemmas about `MvPolynomial.support... | Mathlib/Algebra/MvPolynomial/Supported.lean | 135 | 141 | |
/-
Copyright (c) 2014 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Jeremy Avigad
-/
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Algebra.Order.Ring.Canonical
/-!
# Distance function on ℕ
This file defines a simple dista... | Mathlib/Data/Nat/Dist.lean | 103 | 104 | |
/-
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... | trop l.minimum = List.sum (l.map (trop ∘ WithTop.some)) := by
induction' l with hd tl IH
· simp
· simp [List.minimum_cons, ← IH]
| Mathlib/Algebra/Tropical/BigOperators.lean | 70 | 74 |
/-
Copyright (c) 2024 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.PurelyInseparable.Basic
import Mathlib.FieldTheory.PerfectClosure
/-!
# `IsPerfectClosure` predicate
This file contains `IsPerfectClosure` which asserts... |
theorem lift_comp_lift_apply_eq_self [PerfectRing L p] (x : L) :
lift j i p (lift i j p x) = x := by
| Mathlib/FieldTheory/IsPerfectClosure.lean | 398 | 400 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Algebra.Field.NegOnePow
import Mathlib.Algebra.Field.Periodic
import Mathlib.Algebra.Qua... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | 1,287 | 1,288 | |
/-
Copyright (c) 2021 Aaron Anderson, Jesse Michael Han, Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jesse Michael Han, Floris van Doorn
-/
import Mathlib.Data.Fin.VecNotation
import Mathlib.SetTheory.Cardinal.Basic
/-!
# Basics on... |
/-- The cardinality of a language is the cardinality of its type of symbols. -/
def card : Cardinal :=
#L.Symbols
variable {L} {L' : Language.{u', v'}}
| Mathlib/ModelTheory/Basic.lean | 90 | 95 |
/-
Copyright (c) 2022 Julian Berman. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Julian Berman
-/
import Mathlib.GroupTheory.PGroup
import Mathlib.LinearAlgebra.Quotient.Defs
/-!
# Torsion groups
This file defines torsion groups, i.e. groups where all elements hav... | /-- Elements of the `p`-primary component have order `p^n` for some `n`. -/
@[to_additive primaryComponent.exists_orderOf_eq_prime_nsmul
"Elements of the `p`-primary component have additive order `p^n` for some `n`"]
theorem primaryComponent.exists_orderOf_eq_prime_pow (g : CommMonoid.primaryComponent G p) :
∃ n ... | Mathlib/GroupTheory/Torsion.lean | 201 | 208 |
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Rémy Degenne
-/
import Mathlib.Probability.Process.Adapted
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
/-!
# Stopping times, stopped processes and stopped va... | section MeasurableSet
| Mathlib/Probability/Process/Stopping.lean | 57 | 58 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Order.Monotone.Odd
import Mathlib.Analysis.Calculus.LogDeriv
import Mathlib.Analysis.Spe... | theorem differentiable_cosh : Differentiable ℂ cosh := fun x => (hasDerivAt_cosh x).differentiableAt
@[simp]
theorem differentiableAt_cosh {x : ℂ} : DifferentiableAt ℂ cosh x :=
differentiable_cosh x
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean | 134 | 138 |
/-
Copyright (c) 2024 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.Topology.Separation.CompletelyRegular
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
/-!
# Dirac deltas as probability measures and embedding of ... | funext μ; simp [diracProbaEquiv]
lemma tendsto_diracProbaEquivSymm_iff_tendsto [T0Space X] [CompletelyRegularSpace X]
{μ : range (diracProba (X := X))} (F : Filter (range (diracProba (X := X)))) :
Tendsto diracProbaEquiv.symm F (𝓝 (diracProbaEquiv.symm μ)) ↔ Tendsto id F (𝓝 μ) := by
have key :=
tends... | Mathlib/MeasureTheory/Measure/DiracProba.lean | 148 | 157 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Mathlib.Data.Stream.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Data.List.Defs
import Mathlib.Data.Nat.Basic
import Mathlib.Tactic.Common... | -- Take theorem reduces a proof of equality of infinite streams to an
-- induction over all their finite approximations.
theorem take_theorem (s₁ s₂ : Stream' α) : (∀ n : ℕ, take n s₁ = take n s₂) → s₁ = s₂ := by
| Mathlib/Data/Stream/Init.lean | 599 | 601 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mitchell Lee
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Indicator
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Topology.Algebra.InfiniteSum.Defs
imp... | @[to_additive]
theorem tprod_bool (f : Bool → α) : ∏' i : Bool, f i = f false * f true := by
rw [tprod_fintype, Fintype.prod_bool, mul_comm]
| Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | 405 | 407 |
/-
Copyright (c) 2022 Yaël Dillies, Sara Rousta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Sara Rousta
-/
import Mathlib.Logic.Equiv.Set
import Mathlib.Order.Interval.Set.OrderEmbedding
import Mathlib.Order.SetNotation
/-!
# Properties of unbundled ... | Mathlib/Order/UpperLower/Basic.lean | 688 | 689 | |
/-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth, Frédéric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Calculus
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.Adjoint... | (hextr : IsMaxOn T.reApplyInnerSelf (sphere (0 : E) ‖x₀‖) x₀) :
HasEigenvector (T : E →ₗ[𝕜] E) (↑(⨆ x : { x : E // x ≠ 0 }, T.rayleighQuotient x)) x₀ := by
convert hT.hasEigenvector_of_isLocalExtrOn hx₀ (Or.inr hextr.localize)
have hx₀' : 0 < ‖x₀‖ := by simp [hx₀]
have hx₀'' : x₀ ∈ sphere (0 : E) ‖x₀‖ :=... | Mathlib/Analysis/InnerProductSpace/Rayleigh.lean | 180 | 185 |
/-
Copyright (c) 2023 Antoine Chambert-Loir. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir
-/
import Mathlib.Algebra.Exact
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.RingTheory.Ideal.Quotient.Defs
import Mathlib.RingTheory.TensorProduct... | {f : M →ₗ[R] N} (g : N →ₗ[R] P)
lemma le_comap_range_lTensor (q : Q) :
LinearMap.range g ≤ (LinearMap.range (lTensor Q g)).comap (TensorProduct.mk R Q P q) := by
| Mathlib/LinearAlgebra/TensorProduct/RightExactness.lean | 96 | 99 |
/-
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... | subtypePerm c fun _ : α => apply_mem_support.symm
/-- Restrict a permutation to a Finset containing its support -/
def subtypePerm_of_support_le (c : Perm α) {s : Finset α}
(hcs : c.support ⊆ s) : Equiv.Perm s :=
subtypePerm c (isInvariant_of_support_le hcs)
/-- Support of a cycle is nonempty -/
theorem IsCyc... | Mathlib/GroupTheory/Perm/Cycle/Basic.lean | 989 | 997 |
/-
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... | theorem range_inl_union_range_inr : range (Sum.inl : α → α ⊕ β) ∪ range Sum.inr = univ :=
isCompl_range_inl_range_inr.sup_eq_top
| Mathlib/Data/Set/Image.lean | 777 | 778 |
/-
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, Yury Kudryashov, Rémy Degenne
-/
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Order.Hom.Set
/-!
# Lemmas about images of inte... | Mathlib/Order/Interval/Set/OrderIso.lean | 103 | 104 | |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Mario Carneiro, Yaël Dillies
-/
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Int.Order.Basic
import Mathlib.Logic.Function.Iterate
import Mathlib.Order.Compare
impor... | Mathlib/Order/Monotone/Basic.lean | 1,167 | 1,170 | |
/-
Copyright (c) 2024 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Basic
import Mathlib.Tactic.ComputeDegree
/-!
# Division polynomials of Weierst... |
private lemma expCoeff_rec (m : ℕ) :
(expCoeff (2 * (m + 3)) =
expCoeff (m + 2) ^ 2 * expCoeff (m + 3) * expCoeff (m + 5) -
expCoeff (m + 1) * expCoeff (m + 3) * expCoeff (m + 4) ^ 2) ∧
(expCoeff (2 * (m + 2) + 1) =
expCoeff (m + 4) * expCoeff (m + 2) ^ 3 * (if Even m then 4 ^ 2 else 1) -
... | Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Degree.lean | 183 | 192 |
/-
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.Operations
/-!
# Results about division in extended non-negative reals
This file establishes basic properties related t... |
@[simp]
| Mathlib/Data/ENNReal/Inv.lean | 304 | 305 |
/-
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.Continuous
import Mathlib.Topology.Defs.Induced
/-!
# Ordering on topologies and (co)induced topologies
Topologies on a fixe... | generateFrom (⋂ i, { s | IsOpen[f i] s }) = ⨆ i, f i :=
(gciGenerateFrom α).u_iSup_l _
| Mathlib/Topology/Order.lean | 869 | 870 |
/-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Bilinear
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Group.Pointwise.Finset.Basic
import Mathlib.Algebra.Group.Pointwise.Set.B... | @[simp]
theorem map_op_one :
map (↑(opLinearEquiv R : A ≃ₗ[R] Aᵐᵒᵖ) : A →ₗ[R] Aᵐᵒᵖ) (1 : Submodule R A) = 1 := by
ext x
| Mathlib/Algebra/Algebra/Operations.lean | 370 | 373 |
/-
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.Algebra.Polynomial.Module.AEval
/-!
# Polynomial module
In this file, we define the polynomial module for an `R`-module `M`, i.e. the `R[X]`-module `M[X]`.... | theorem comp_smul (p p' : R[X]) (q : PolynomialModule R M) :
comp p (p' • q) = p'.comp p • comp p q := by
rw [comp_apply, map_smul, eval_smul, Polynomial.comp, Polynomial.eval_map, comp_apply]
rfl
end PolynomialModule
| Mathlib/Algebra/Polynomial/Module/Basic.lean | 324 | 333 |
/-
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.Algebra.NonUnitalHom
import Mathlib.LinearAlgebra.TensorProduct.Basic
/-!
# Facts about algebras involving bilinear maps and tensor pro... |
end one_side
| Mathlib/Algebra/Algebra/Bilinear.lean | 134 | 136 |
/-
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, Violeta Hernández Palacios
-/
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.Nat.SuccPred
import Mathlib.Order.SuccPred.Initial... | Mathlib/SetTheory/Ordinal/Arithmetic.lean | 2,249 | 2,254 | |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.SetTheory.Game.State
/-!
# Domineering as a combinatorial game.
We define the game of Domineering, played on a chessboard of arbitrary shape
(possibly ev... | l s := (left s).image (moveLeft s)
r s := (right s).image (moveRight s)
left_bound m := by
simp only [Finset.mem_image, Prod.exists] at m
rcases m with ⟨_, _, ⟨h, rfl⟩⟩
exact moveLeft_smaller h
| Mathlib/SetTheory/Game/Domineering.lean | 117 | 122 |
/-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Topology.Coherent
import Mathlib.Topology.UniformSpace.Equiv
import Mathlib.Topology.UniformSpace.Pi
import Mathlib.Topology.UniformSpace.UniformAp... | exact he' hs
uniformContinuous_invFun := UniformOnFun.precomp_uniformContinuous he }
/-- If `𝔖` covers `α`, then the topology of `𝔖`-convergence is T₂. -/
theorem t2Space_of_covering [T2Space β] (h : ⋃₀ 𝔖 = univ) : T2Space (α →ᵤ[𝔖] β) where
t2 f g hfg := by
obtain ⟨x, hx⟩ := not_forall.mp (mt funex... | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | 915 | 924 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl, Patrick Massot
-/
import Mathlib.Data.Set.Image
import Mathlib.Data.SProd
/-!
# Sets in product and pi types
This file proves basic properties of prod... | by_cases h₁ : x.1 ∈ s₁ <;> by_cases h₂ : x.2 ∈ t₁ <;> simp [*]
| Mathlib/Data/Set/Prod.lean | 330 | 331 |
/-
Copyright (c) 2022 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Data.Nat.Cast.Field
import Mathlib.GroupTheory.Abelianization
import Mathlib.GroupTheory.GroupAction.CardComm... | fun x => ⟨⟨⟨x.1.1.1, x.2.1.1⟩, ⟨x.1.1.2, x.2.1.2⟩⟩, ⟨x.1.2, x.2.2⟩⟩, fun x => rfl, fun x => rfl⟩
theorem commProb_pi {α : Type*} (i : α → Type*) [Fintype α] [∀ a, Mul (i a)] :
commProb (∀ a, i a) = ∏ a, commProb (i a) := by
simp_rw [commProb_def, Finset.prod_div_distrib, Finset.prod_pow, ← Nat.cast_prod,
... | Mathlib/GroupTheory/CommutingProbability.lean | 47 | 52 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Aurélien Saue, Anne Baanen
-/
import Mathlib.Tactic.NormNum.Inv
import Mathlib.Tactic.NormNum.Pow
import Mathlib.Util.AtomM
/-!
# `ring` tactic
A tactic for solving e... | /-- Embed `ExProd` in `ExSum` by adding 0. -/
def ExProd.toSum {sα : Q(CommSemiring $α)} {e : Q($α)} (v : ExProd sα e) : ExSum sα q($e + 0) :=
.add v .zero
| Mathlib/Tactic/Ring/Basic.lean | 283 | 285 |
/-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Analysis.Normed.Lp.lpSpace
import Mathlib.Analysis.InnerProductSpace.PiL2
/-!
# Hilbert sum of ... | rw [SetLike.mem_coe]
refine sum_mem ?_
intro i _
| Mathlib/Analysis/InnerProductSpace/l2Space.lean | 237 | 239 |
/-
Copyright (c) 2017 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Keeley Hoek
-/
import Mathlib.Algebra.NeZero
import Mathlib.Data.Int.DivMod
import Mathlib.Logic.Embedding.Basic
import Mathlib.Logic.Equiv.Set
import Mathlib.Tactic.... |
theorem lt_castPred_iff {j : Fin n} {i : Fin (n + 1)} (hi : i ≠ last n) :
j < castPred i hi ↔ castSucc j < i := by
| Mathlib/Data/Fin/Basic.lean | 843 | 845 |
/-
Copyright (c) 2024 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.LinearAlgebra.DFinsupp
import Mathlib.RingTheory.Finiteness.Basic
import Mathlib.LinearAlgebra.TensorProduct.Basic
/-!
# Some finiteness results of tensor product
T... | of `M × N`, such that `x` is equal to the sum of `m_i ⊗ₜ[R] n_i`. -/
theorem exists_multiset (x : M ⊗[R] N) :
∃ S : Multiset (M × N), x = (S.map fun i ↦ i.1 ⊗ₜ[R] i.2).sum := by
induction x with
| zero => exact ⟨0, by simp⟩
| tmul x y => exact ⟨{(x, y)}, by simp⟩
| add x y hx hy =>
obtain ⟨Sx, hx⟩ := hx... | Mathlib/LinearAlgebra/TensorProduct/Finiteness.lean | 52 | 60 |
/-
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.Algebra.GroupWithZero.Action.Pointwise.Set
import Mathlib.Algebra.Module.LinearMap.Prod
import Mathlib.Algebra.Order.Module.Synonym
import Mathlib.Analysis... | obtain ⟨y', hy', rfl⟩ := hy
refine ⟨a • x + b • y', hs hy' ha hb hab, ?_⟩
match_scalars <;> simp [hab]
| Mathlib/Analysis/Convex/Star.lean | 228 | 231 |
/-
Copyright (c) 2020 Jannis Limperg. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jannis Limperg
-/
import Mathlib.Data.List.Induction
/-!
# Lemmas about List.*Idx functions.
Some specification lemmas for `List.mapIdx`, `List.mapIdxM`, `List.foldlIdx` and `List.fo... |
-- Note: `traverse` the class method would require a less universe-polymorphic
-- `m : Type u → Type u`.
set_option linter.deprecated false in
@[deprecated "Deprecated without replacement." (since := "2025-01-29")]
| Mathlib/Data/List/Indexes.lean | 190 | 194 |
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Logic.IsEmpty
import Mathlib.Tactic.Inhabit
/-!
# Types with a unique term
In this file we define a typeclass `Unique`,
which expresses that a type h... | inhabit α
exact ⟨Unique.mk' α⟩
| Mathlib/Logic/Unique.lean | 146 | 147 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.