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 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel, Johan Commelin, Kim Morrison
-/
import Mathlib.CategoryTheory.Limits.Constructions.Pullbacks
import Mathlib.CategoryTheory.Preadditive.Biproducts
import Mathlib.CategoryT... | Mathlib/CategoryTheory/Abelian/Basic.lean | 808 | 811 | |
/-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
import Mathlib.SetTheory.Nimber.Basic
/-!
# Nim and the Sprague-Grundy theore... | exact IH _ (typein_lt_self i)
@[simp]
| Mathlib/SetTheory/Game/Nim.lean | 182 | 184 |
/-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Algebra.Order.Pi
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Data.Real.Pointwise
/... | section SMul
variable [SMul ℝ E] [IsScalarTower ℝ 𝕜 E] (p : Seminorm 𝕜 E)
/-- A seminorm is convex. Also see `convexOn_norm`. -/
protected theorem convexOn : ConvexOn ℝ univ p := by
refine ⟨convex_univ, fun x _ y _ a b ha hb _ => ?_⟩
calc
| Mathlib/Analysis/Seminorm.lean | 976 | 983 |
/-
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.Algebra.Defs
import Mathlib.Algebra.Field.Defs
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Math... | variable {R : Type*} [Ring R] {S : Submonoid R} [OreSet S]
variable {X : Type*} [AddCommGroup X] [Module R X]
| Mathlib/RingTheory/OreLocalization/Ring.lean | 182 | 184 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Dia... | instance isFiniteMeasure_restrict_Ioc (x y : ℝ) : IsFiniteMeasure (volume.restrict (Ioc x y)) :=
⟨by simp⟩
instance isFiniteMeasure_restrict_Ioo (x y : ℝ) : IsFiniteMeasure (volume.restrict (Ioo x y)) :=
⟨by simp⟩
| Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | 200 | 204 |
/-
Copyright (c) 2022 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Heather Macbeth
-/
import Mathlib.Algebra.MvPolynomial.Supported
import Mathlib.RingTheory.WittVector.Truncated
/-!
# Leading terms of Witt vector multiplication
Th... | coefficients of the inputs. -/
def nthRemainder (n : ℕ) : (Fin (n + 1) → k) → (Fin (n + 1) → k) → k :=
Classical.choose (nth_mul_coeff p k n)
theorem nthRemainder_spec (n : ℕ) (x y : 𝕎 k) : (x * y).coeff (n + 1) =
x.coeff (n + 1) * y.coeff 0 ^ p ^ (n + 1) + y.coeff (n + 1) * x.coeff 0 ^ p ^ (n + 1) +
nthRem... | Mathlib/RingTheory/WittVector/MulCoeff.lean | 279 | 288 |
/-
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... | lemma rpow_ne_zero (hx : 0 ≤ x) (hy : y ≠ 0) : x ^ y ≠ 0 ↔ x ≠ 0 :=
| Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 73 | 73 |
/-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad
-/
import Mathlib.Logic.Basic
import Mathlib.Logic.Function.Defs
import Mathlib.Order.Defs.LinearOrder
/-!
# Booleans
This file proves various... | lemma self_ne_not : ∀ b : Bool, b ≠ !b := by decide
| Mathlib/Data/Bool/Basic.lean | 125 | 125 |
/-
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.Data.Set.Finite.Lattice
/-!
# Partitions based on membership of a sequence of sets
Let `f : ℕ → Set α` be a sequence of sets. For `n : ℕ`, we can form th... | @[simp]
lemma sUnion_memPartition (f : ℕ → Set α) (n : ℕ) : ⋃₀ memPartition f n = univ := by
induction n with
| zero => simp
| succ n ih =>
rw [memPartition_succ]
ext x
have : x ∈ ⋃₀ memPartition f n := by simp [ih]
simp only [mem_sUnion, mem_iUnion, mem_insert_iff, mem_singleton_iff, exists_prop,... | Mathlib/Data/Set/MemPartition.lean | 73 | 86 |
/-
Copyright (c) 2020 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Analysis.MeanInequalities
import Mathlib.Analysis.MeanInequalitiesPow
import Mathlib.MeasureTheory.Function.SpecialFunctions.Basic
import Mathlib.MeasureTh... | refine lintegral_congr_ae ?_
suffices h_mul_zero : f * g =ᵐ[μ] 0 * g by rwa [zero_mul] at h_mul_zero
have hf_eq_zero : f =ᵐ[μ] 0 := ae_eq_zero_of_lintegral_rpow_eq_zero hp0 hf hf_zero
exact hf_eq_zero.mul (ae_eq_refl g)
theorem lintegral_mul_le_Lp_mul_Lq_of_ne_zero_of_eq_top {p q : ℝ} (hp0_lt : 0 < p) (hq0 : 0... | Mathlib/MeasureTheory/Integral/MeanInequalities.lean | 133 | 138 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Notation.Pi
import Mathlib.Data.Set.Lattice
import Mathlib.Order.Filter.Defs
/-!
# Theory of... | Mathlib/Order/Filter/Basic.lean | 1,664 | 1,666 | |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Algebra.Order.Group.Defs
import Mathlib.Algebra.Order.Group.Unbundled.Abs
import Mathlib.Algebra.Order... | Mathlib/Algebra/Order/Group/Abs.lean | 433 | 434 | |
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Frédéric Dupuis
-/
import Mathlib.Algebra.Star.SelfAdjoint
import Mathlib.Algebra.Module.Basic
import Mathlib.Algebra.Module.Equiv.Defs
import Mathlib.Algebra.Module.LinearMa... | theorem selfAdjointPart_comp_subtype_selfAdjoint :
(selfAdjointPart R).comp (selfAdjoint.submodule R A).subtype = .id :=
LinearMap.ext fun x ↦ x.2.selfAdjointPart_apply R
| Mathlib/Algebra/Star/Module.lean | 169 | 171 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
import Mathlib.Data.Set.Function
import Mathlib.Logic.Equiv.Defs
import Mathlib.Tactic.Says
/-!
# Equivalences and sets
In this file we pr... | rfl
@[simp]
| Mathlib/Logic/Equiv/Set.lean | 264 | 266 |
/-
Copyright (c) 2022 Benjamin Davidson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Benjamin Davidson, Devon Tuma, Eric Rodriguez, Oliver Nash
-/
import Mathlib.Algebra.Order.Group.Pointwise.Interval
import Mathlib.Order.Filter.AtTopBot.Field
import Mathlib.Topolog... | tendsto_inv_atTop_nhdsGT_zero.mono_right inf_le_left
/-- The function `x ↦ x⁻¹` tends to `-∞` on the left of `0`. -/
theorem tendsto_inv_zero_atBot : Tendsto (fun x : 𝕜 => x⁻¹) (𝓝[<] (0 : 𝕜)) atBot :=
inv_nhdsLT_zero.le
| Mathlib/Topology/Algebra/Order/Field.lean | 165 | 169 |
/-
Copyright (c) 2023 Jason Yuen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jason Yuen
-/
import Mathlib.Data.Real.ConjExponents
import Mathlib.Data.Real.Irrational
/-!
# Rayleigh's theorem on Beatty sequences
This file proves Rayleigh's theorem on Beatty sequen... | `1/r + 1/s = 1`. Then `B⁺_r` and `B⁺_s` partition the positive integers. -/
theorem Irrational.beattySeq_symmDiff_beattySeq_pos {r s : ℝ}
(hrs : r.HolderConjugate s) (hr : Irrational r) :
{beattySeq r k | k > 0} ∆ {beattySeq s k | k > 0} = {n | 0 < n} := by
| Mathlib/NumberTheory/Rayleigh.lean | 172 | 175 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro
-/
import Mathlib.Algebra.Module.Submodule.Bilinear
import Mathlib.Algebra.Module.Equiv.Basic
import Mathlib.GroupTheory.Congruence.Hom
import Mathlib.Tactic.Abel
... | def lift.equiv : (M →ₗ[R] N →ₗ[R] P) ≃ₗ[R] M ⊗[R] N →ₗ[R] P :=
{ uncurry R M N P with
invFun := fun f => (mk R M N).compr₂ f
left_inv := fun _ => LinearMap.ext₂ fun _ _ => lift.tmul _ _
| Mathlib/LinearAlgebra/TensorProduct/Basic.lean | 565 | 568 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Topology.Algebra.Monoid
/-!
# Topological group with zero
I... | theorem continuousAt_zpow₀ (x : G₀) (m : ℤ) (h : x ≠ 0 ∨ 0 ≤ m) :
ContinuousAt (fun x => x ^ m) x := by
| Mathlib/Topology/Algebra/GroupWithZero.lean | 314 | 315 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
/-... | Mathlib/RingTheory/Coprime/Lemmas.lean | 306 | 309 | |
/-
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, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.FinMeasAdditive
/-!
# Extension of a linear function from indicators to L1
Give... | rw [← Integrable.toL1_sub]
refine ((fs_int n).sub f_int).coeFn_toL1.mono fun x hx => ?_
dsimp only
rw [hx, ofReal_norm_eq_enorm, Pi.sub_apply]
/-- Lebesgue dominated convergence theorem for filters with a countable basis -/
theorem tendsto_setToFun_filter_of_dominated_convergence (hT : DominatedFinMeasAdditive... | Mathlib/MeasureTheory/Integral/SetToL1.lean | 1,054 | 1,062 |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.WSeq.Relation
/-!
# Parallel computation
Parallel computation of a computable sequence of computations by
a diagonal enumeration.
The imp... | intro n; induction' n with n IH <;> intro l S c o T
· rcases o with a | a
· exact terminates_parallel.aux a T
have H : Seq.destruct S = some (some c, Seq.tail S) := by simp [Seq.destruct, (· <$> ·), ← a]
induction' h : parallel.aux2 l with a l'
· have C : corec parallel.aux1 (l, S) = pure a := by
... | Mathlib/Data/Seq/Parallel.lean | 122 | 186 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Analysis.SumOverResidueClass
/-!
# Conve... | (hu : Monotone u) (n : ℕ) :
(∑ k ∈ Ico (u 0) (u n), f k) ≤ ∑ k ∈ range n, (u (k + 1) - u k) • f (u k) := by
induction n with
| zero => simp
| succ n ihn =>
suffices (∑ k ∈ Ico (u n) (u (n + 1)), f k) ≤ (u (n + 1) - u n) • f (u n) by
rw [sum_range_succ, ← sum_Ico_consecutive]
· exact add_le... | Mathlib/Analysis/PSeries.lean | 50 | 62 |
/-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson
-/
import Mathlib.Algebra.Order.Group.Defs
import Mathlib.SetTheory.Game.Basic
import Mathlib.Tactic.NthRewrite
/-!
# Basic definitions about impartial (pre-)games
We will d... | ⟨(equiv_or_fuzzy_zero G).resolve_right, Equiv.not_fuzzy⟩
theorem add_self : G + G ≈ 0 :=
Equiv.trans (add_congr_left (neg_equiv_self G)) (neg_add_cancel_equiv G)
| Mathlib/SetTheory/Game/Impartial.lean | 130 | 133 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Andrew Zipperer, Haitao Zhang, Minchao Wu, Yury Kudryashov
-/
import Mathlib.Data.Set.Prod
import Mathlib.Data.Set.Restrict
/-!
# Functions over sets
This file contains... | ⟨h.1.extendDomain, h.2.extendDomain⟩
end Set
| Mathlib/Data/Set/Function.lean | 1,192 | 1,195 |
/-
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.Combinatorics.SimpleGraph.Path
import Mathlib.Combinatorics.SimpleGraph.Operations
import Mathlib.Data.Finset.Pairwise
import M... | theorem isNClique_zero : G.IsNClique 0 s ↔ s = ∅ := by
simp only [isNClique_iff, Finset.card_eq_zero, and_iff_right_iff_imp]; rintro rfl; simp
| Mathlib/Combinatorics/SimpleGraph/Clique.lean | 235 | 236 |
/-
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.MonoidAlgebra.Degree
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.Polynomial.Basi... | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 821 | 828 | |
/-
Copyright (c) 2022 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.Stopping
import Mathlib.Tactic.AdaptationNote
/-!
# Hitting time
Given a stochastic process, the hitting time provides th... | theorem hitting_of_lt {m : ι} (h : m < n) : hitting u s n m ω = m := by
simp_rw [hitting]
have h_not : ¬∃ (j : ι) (_ : j ∈ Set.Icc n m), u j ω ∈ s := by
push_neg
intro j
rw [Set.Icc_eq_empty_of_lt h]
simp only [Set.mem_empty_iff_false, IsEmpty.forall_iff]
simp only [exists_prop] at h_not
simp on... | Mathlib/Probability/Process/HittingTime.lean | 67 | 75 |
/-
Copyright (c) 2020 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
import Mathlib.Algebra.ContinuedFractions.Determinant
import Mathlib.Algebra.ContinuedFractions.Computation.CorrectnessTerminating
import Mathlib.Algebra.Order.Ri... | GenContFract.contsAux_recurrence nth_s_eq rfl rfl]
/-- Shows that `bₙ * Bₙ ≤ Bₙ₊₁`, where `bₙ` is the `n`th partial denominator and `Bₙ₊₁` and `Bₙ` are
the `n + 1`th and `n`th denominator of the continued fraction. -/
theorem le_of_succ_get?_den {b : K}
| Mathlib/Algebra/ContinuedFractions/Computation/Approximations.lean | 286 | 290 |
/-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth, Floris van Doorn
-/
import Mathlib.Topology.VectorBundle.Basic
/-!
# The vector bundle of continuous (semi)linear maps
We define the (topological) vector bundle of ... |
/-- The continuous `σ`-semilinear maps between two topological vector bundles form a
`VectorPrebundle` (this is an auxiliary construction for the
`VectorBundle` instance, in which the pretrivializations are collated but no topology
on the total space is yet provided). -/
def Bundle.ContinuousLinearMap.vectorPrebundle ... | Mathlib/Topology/VectorBundle/Hom.lean | 186 | 198 |
/-
Copyright (c) 2019 Minchao Wu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Minchao Wu, Chris Hughes, Mantas Bakšys
-/
import Mathlib.Data.List.Basic
import Mathlib.Order.BoundedOrder.Lattice
import Mathlib.Data.List.Induction
import Mathlib.Order.MinMax
import Ma... |
theorem not_of_mem_foldl_argAux (hr₀ : Irreflexive r) (hr₁ : Transitive r) :
∀ {a m : α} {o : Option α}, a ∈ l → m ∈ foldl (argAux r) o l → ¬r a m := by
induction' l using List.reverseRecOn with tl a ih
· simp
intro b m o hb ho
rw [foldl_append, foldl_cons, foldl_nil, argAux] at ho
rcases hf : foldl (arg... | Mathlib/Data/List/MinMax.lean | 69 | 86 |
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.AlgebraicTopology.DoldKan.Projections
import Mathlib.CategoryTheory.Idempotents.FunctorCategories
import Mathlib.CategoryTheory.Idempotents.FunctorExtension
/-!... |
/-- The endomorphism `PInfty : K[X] ⟶ K[X]` obtained from the `P q` by passing to the limit. -/
noncomputable def PInfty : K[X] ⟶ K[X] :=
| Mathlib/AlgebraicTopology/DoldKan/PInfty.lean | 46 | 48 |
/-
Copyright (c) 2020 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp
-/
import Mathlib.Algebra.Algebra.Spectrum.Basic
import Mathlib.Algebra.Module.LinearMap.Basic
import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
import Mathl... | genEigenspace_div f a b hb
@[deprecated genEigenspace_nat (since := "2024-10-28")]
lemma genEigenspace_def (f : End R M) (μ : R) (k : ℕ) :
f.genEigenspace μ k = LinearMap.ker ((f - μ • 1) ^ k) :=
genEigenspace_nat
/-- A nonzero element of a generalized eigenspace is a generalized eigenvector.
(Def 8.9 of [axl... | Mathlib/LinearAlgebra/Eigenspace/Basic.lean | 469 | 484 |
/-
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.NumberTheory.LegendreSymbol.QuadraticChar.Basic
import Mathlib.NumberTheory.GaussSum
/-!
# Quadratic characters of finite fields
Further facts relying ... | simp only [FiniteField.isSquare_of_char_two hF, true_iff]
omega
· have h := FiniteField.odd_card_of_char_ne_two hF
rw [← quadraticChar_one_iff_isSquare (neg_ne_zero.mpr (Ring.two_ne_zero hF)),
quadraticChar_neg_two hF, χ₈'_nat_eq_if_mod_eight]
omega
/-- The relation between the values of the qu... | Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/GaussSum.lean | 63 | 73 |
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.Constructions
/-!
# Neighborhoods and continuity relative to a subset
This file develops API on the relative versions
* `nhdsWithin` ... | {f : α → ∀ i, π i} {s : Set α} : ContinuousOn f s ↔ ∀ i, ContinuousOn (fun y => f y i) s :=
⟨fun h i x hx => tendsto_pi_nhds.1 (h x hx) i, fun h x hx => tendsto_pi_nhds.2 fun i => h i x hx⟩
| Mathlib/Topology/ContinuousOn.lean | 1,174 | 1,176 |
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes Hölzl
-/
import Mathlib.Algebra.Field.Subfield.Defs
import Mathlib.Algebra.Order.Group.Pointwise.Interval
import Mathlib.Analysis.Normed.Ring.Basic
/-!
# Norm... | Mathlib/Analysis/Normed/Field/Basic.lean | 1,106 | 1,107 | |
/-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Algebra.Order.Pi
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Data.Real.Pointwise
/... | theorem ball_norm_mul_subset {p : Seminorm 𝕜 E} {k : 𝕜} {r : ℝ} :
p.ball 0 (‖k‖ * r) ⊆ k • p.ball 0 r := by
rcases eq_or_ne k 0 with (rfl | hk)
· rw [norm_zero, zero_mul, ball_eq_emptyset _ le_rfl]
exact empty_subset _
| Mathlib/Analysis/Seminorm.lean | 890 | 894 |
/-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator
import Mathlib.MeasureTheory.Function.UniformIntegrable
import Mathlib.MeasureTheory.V... | @[deprecated (since := "2025-01-21")]
alias condexp_stronglyMeasurable_mul_of_bound₀ := condExp_stronglyMeasurable_mul_of_bound₀
/-- Pull-out property of the conditional expectation. -/
theorem condExp_mul_of_stronglyMeasurable_left {f g : α → ℝ} (hf : StronglyMeasurable[m] f)
(hfg : Integrable (f * g) μ) (hg : In... | Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean | 317 | 351 |
/-
Copyright (c) 2021 Praneeth Kolichala. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Praneeth Kolichala
-/
import Mathlib.Topology.Constructions
import Mathlib.Topology.Homotopy.Path
/-!
# Product of homotopies
In this file, we introduce definitions for the produ... | Quotient.map₂ Path.prod (fun _ _ h₁ _ _ h₂ => Nonempty.map2 prodHomotopy h₁ h₂) q₁ q₂
variable (p₁ p₁' p₂ p₂')
theorem prod_lift : prod ⟦p₁⟧ ⟦p₂⟧ = ⟦p₁.prod p₂⟧ :=
rfl
| Mathlib/Topology/Homotopy/Product.lean | 168 | 173 |
/-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Data.Set.Defs
import Mathlib.Logic.Basic
import Mathlib.Logic.ExistsUnique
import Mathlib.Logic.Nonempty
import Mathlib.Logic.Nontrivia... | protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
theorem not_injective_iff : ¬ Injective f ↔ ∃ a b, f a = f b ∧ a ≠ b := by
simp only [Injective, not_forall, exists_prop]
| Mathlib/Logic/Function/Basic.lean | 74 | 78 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Topology.MetricSpace.IsometricSMul
/-!
# Hausdorff distance
The Hausdorff distance on subsets of a... | theorem infEdist_le_edist_add_infEdist : infEdist x s ≤ edist x y + infEdist y s := by
rw [add_comm]
| Mathlib/Topology/MetricSpace/HausdorffDistance.lean | 120 | 121 |
/-
Copyright (c) 2021 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Riccardo Brasca, Adam Topaz, Jujian Zhang, Joël Riou
-/
import Mathlib.Algebra.Homology.Additive
import Mathlib.CategoryTheory.Abelian.Projective.Resolution
/-!
# Left-der... | (F : C ⥤ D) [F.Additive] (n : ℕ) :
(F.leftDerived n).map f ≫ (Q.isoLeftDerivedObj F n).hom =
(P.isoLeftDerivedObj F n).hom ≫
(F.mapHomologicalComplex _ ⋙ HomologicalComplex.homologyFunctor _ _ n).map φ := by
dsimp [isoLeftDerivedObj, Functor.leftDerived]
rw [assoc, ← Functor.map_comp_assoc,
... | Mathlib/CategoryTheory/Abelian/LeftDerived.lean | 119 | 132 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.Normed.Module.Basic
import Mathlib.Analysis.Seminorm
import Mathlib.Analysis.SpecificLimits.Basi... |
end TVS
| Mathlib/Analysis/Calculus/TangentCone.lean | 435 | 436 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Kim Morrison
-/
import Mathlib.CategoryTheory.Functor.Currying
import Mathlib.CategoryTheory.Subobject.FactorThru
import Mathlib.CategoryTheory.Subobject.WellPowered
import... | simp only [Finset.sup_insert]
simp only [Finset.mem_insert] at m
rcases m with (rfl | m)
· exact sup_factors_of_factors_left h
· exact sup_factors_of_factors_right (ih ⟨j, ⟨m, h⟩⟩)
end SemilatticeSup
section Lattice
| Mathlib/CategoryTheory/Subobject/Lattice.lean | 486 | 494 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johan Commelin
-/
import Mathlib.RingTheory.IntegralClosure.IsIntegral.Basic
/-!
# Minimal polynomials
This file defines the minimal polynomial of an element `x` of an `A... | rw [dif_pos hx]
exact (degree_lt_wf.min_mem _ hx).1
/-- A minimal polynomial is nonzero. -/
| Mathlib/FieldTheory/Minpoly/Basic.lean | 52 | 55 |
/-
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 | 3,523 | 3,528 | |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.Convex.StrictConvexSpace
import Mathlib.MeasureTheory.Function.AEEqOfIntegral
import Mathlib.Measure... | have : ∀ {t}, μ t ≠ 0 → (⨍ x in t, f x ∂μ) ∈ s ∧ g (⨍ x in t, f x ∂μ) ≤ ⨍ x in t, g (f x) ∂μ :=
fun ht =>
hg.convexOn.set_average_mem_epigraph hgc hsc ht (measure_ne_top _ _) (ae_restrict_of_ae hfs)
hfi.integrableOn hgi.integrableOn
refine (ae_eq_const_or_exists_average_ne_compl hfi).imp_right ?_
ri... | Mathlib/Analysis/Convex/Integral.lean | 271 | 280 |
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.AlgebraicTopology.DoldKan.FunctorGamma
import Mathlib.AlgebraicTopology.DoldKan.SplitSimplicialObject
import Mathlib.CategoryTheory.Idempotents.HomologicalComple... |
@[simp]
theorem N₁Γ₀_hom_app_f_f (K : ChainComplex C ℕ) (n : ℕ) :
(N₁Γ₀.hom.app K).f.f n = (Γ₀.splitting K).toKaroubiNondegComplexIsoN₁.inv.f.f n := by
rw [N₁Γ₀_hom_app]
apply comp_id
| Mathlib/AlgebraicTopology/DoldKan/GammaCompN.lean | 95 | 100 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.InnerProductSpace.Defs
impor... | theorem dist_div_norm_sq_smul {x y : F} (hx : x ≠ 0) (hy : y ≠ 0) (R : ℝ) :
| Mathlib/Analysis/InnerProductSpace/Basic.lean | 616 | 616 |
/-
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.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Order.Interval.Finset.Na... | Mathlib/Topology/EMetricSpace/Basic.lean | 881 | 888 | |
/-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad
-/
import Mathlib.Logic.Basic
import Mathlib.Logic.Function.Defs
import Mathlib.Order.Defs.LinearOrder
/-!
# Booleans
This file proves various... | Mathlib/Data/Bool/Basic.lean | 247 | 247 | |
/-
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.NumberTheory.LSeries.AbstractFuncEq
import Mathlib.NumberTheory.ModularForms.JacobiTheta.Bounds
import Mathlib.Analysis.SpecialFunctions.Gamma.Deligne
... | Tendsto (fun s ↦ (s - 1) * completedHurwitzZetaEven a s) (𝓝[≠] 1) (𝓝 1) := by
have h1 : Tendsto (fun s : ℂ ↦ (s - ↑(1 / 2 : ℝ)) * _) (𝓝[≠] ↑(1 / 2 : ℝ))
(𝓝 ((1 : ℂ) * (1 : ℂ))) := (hurwitzEvenFEPair a).Λ_residue_k
simp only [push_cast, one_mul] at h1
refine (h1.comp <| tendsto_div_two_punctured_nhds... | Mathlib/NumberTheory/LSeries/HurwitzZetaEven.lean | 452 | 461 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Yakov Pechersky, Eric Wieser
-/
import Mathlib.Data.List.Basic
/-!
# Properties of `List.enum`
## Deprecation note
Many lemmas in this file have been replaced by the... | Mathlib/Data/List/Enum.lean | 57 | 59 | |
/-
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.Shift.Adjunction
import Mathlib.CategoryTheory.Preadditive.Opposite
/-!
# The (naive) shift on the opposite category
If `C` is a category equipp... | def OppositeShift (A : Type*) [AddMonoid A] [HasShift C A] := Cᵒᵖ
instance : Category (OppositeShift C A) := by
dsimp only [OppositeShift]
infer_instance
noncomputable instance : HasShift (OppositeShift C A) A :=
| Mathlib/CategoryTheory/Shift/Opposite.lean | 78 | 84 |
/-
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.Order.Ring.Defs
import Mathlib.Algebra.Ring.Invertible
import Mathlib.Data.Nat.Cast.Order.Ring
/-!
# Lemmas about `invOf` in ordered (semi)r... | ⟨fun h => pos_of_mul_pos_left this h.le, fun h => pos_of_mul_pos_right this h.le⟩
@[simp]
| Mathlib/Algebra/Order/Invertible.lean | 19 | 21 |
/-
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, Patrick Massot
-/
import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic
import Mathlib.MeasureTheory.Measure.Real
import Mathlib.Order.Filter.Indi... | simp only [integral_symm b]
exact (continuousOn_primitive_interval h_int).neg
theorem continuous_primitive (h_int : ∀ a b, IntervalIntegrable f μ a b) (a : ℝ) :
Continuous fun b => ∫ x in a..b, f x ∂μ := by
| Mathlib/MeasureTheory/Integral/DominatedConvergence.lean | 499 | 503 |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Module.BigOperators
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.Nat.Squarefree
imp... | simp [y2ne]
mul_assoc := mul_smul' }
instance instSemiring : Semiring (ArithmeticFunction R) :=
{ ArithmeticFunction.instAddMonoidWithOne,
ArithmeticFunction.instMonoid,
ArithmeticFunction.instAddCommMonoid with
zero_mul := fun f => by
ext
simp
mul_zero := fun f => by
ext
... | Mathlib/NumberTheory/ArithmeticFunction.lean | 325 | 344 |
/-
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.UniformSpace.UniformConvergenceTopology
/-!
# Equicontinuity of a family of functions
Let `X` be a topological space and `α` a `UniformS... | have := (UniformFun.postcomp_isUniformInducing (α := ι) hu).isInducing
rw [equicontinuousAt_iff_continuousAt, equicontinuousAt_iff_continuousAt, this.continuousAt_iff]
rfl
/-- Given `u : α → β` a uniform inducing map, a family `𝓕 : ι → X → α` is equicontinuous at a point
`x₀ : X` within a subset `S : Set X` iff... | Mathlib/Topology/UniformSpace/Equicontinuity.lean | 709 | 715 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Powerset
import Mathlib.Algebra.NoZeroSMulDivisors.Pi
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Finty... | /-- The map is always nonempty. This lemma is needed to apply `SubMulAction.zero_mem`. -/
theorem map_nonempty [Nonempty ι] (f : MultilinearMap R M₁ M₂) (p : ∀ i, Submodule R (M₁ i)) :
(map f p : Set M₂).Nonempty :=
⟨f 0, 0, fun i => (p i).zero_mem, rfl⟩
/-- The range of a multilinear map, closed under scalar mu... | Mathlib/LinearAlgebra/Multilinear/Basic.lean | 1,390 | 1,398 |
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.ConditionalProbability
import Mathlib.Probability.Kernel.Basic
import Mathlib.Probability.Kernel.Composition.MeasureComp
import Mathlib.Tactic.... | have hπS_pi : IsPiSystem πS := by exact IsPiSystem.comap (@isPiSystem_pi _ _ ?_) _
have hπS_gen : (MeasurableSpace.pi.comap fun a (i : S) => f i a) = generateFrom πS := by
rw [generateFrom_pi.symm, comap_generateFrom]
congr
let πTβ := Set.pi (Set.univ : Set T) ''
Set.pi (Set.univ : Set T) fun i => {... | Mathlib/Probability/Independence/Kernel.lean | 1,038 | 1,057 |
/-
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.CharP.Reduced
import Mathlib.RingTheory.IntegralDomain
-- TODO: remove Mathlib.Algebra.CharP.Reduced and move the last two lemmas to Lemmas
/-... | Mathlib/RingTheory/RootsOfUnity/Basic.lean | 528 | 535 | |
/-
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.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Order.Interval.Set.OrdConnected
/-! # Subtypes of cond... | · simp
| Mathlib/Order/CompleteLatticeIntervals.lean | 266 | 266 |
/-
Copyright (c) 2020 Kevin Buzzard, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Equalizers
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Products
import Mathl... | erw [(homEquivAmalgamation (eqv π).2).uniqueCongr.nonempty_congr]
rw [unique_subtype_iff_existsUnique]
exact h _ _ (eqv π).2
/-- Given sieve `S` and presheaf `P : Cᵒᵖ ⥤ A`, their natural associated cone admits at most one
morphism from every cone in the same category (i.e. over the same diagram),
i... | Mathlib/CategoryTheory/Sites/Sheaf.lean | 149 | 164 |
/-
Copyright (c) 2019 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.Regular.Basic
import Mathlib.GroupTheory.MonoidLocalization.Basic
import Mathlib.LinearAlgebra.Matrix.MvPolynomial
import Mathlib.LinearAlgebra.Matri... | AlgHom.mapMatrix_apply, AlgHom.mapMatrix_apply, Matrix.map_smul' _ _ _ (map_mul _)]
have h_card' : Fintype.card n - 2 + 1 = Fintype.card n - 1 := by simp [h_card]
have is_reg : IsSMulRegular (MvPolynomial (n × n) ℤ) (det A') := fun x y =>
mul_left_cancel₀ (det_mvPolynomialX_ne_zero n ℤ)
apply is_reg.mat... | Mathlib/LinearAlgebra/Matrix/Adjugate.lean | 496 | 516 |
/-
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.... | @[simp]
lemma succAbove_ne_zero_zero [NeZero n] {a : Fin (n + 1)} (ha : a ≠ 0) : a.succAbove 0 = 0 := by
rw [Fin.succAbove_of_castSucc_lt]
· exact castSucc_zero'
| Mathlib/Data/Fin/Basic.lean | 1,023 | 1,026 |
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.Data.Finite.Prod
import Mathlib.Data.Matroid.Init
import Mathlib.Data.Set.Card
import Mathlib.Data.Set.Finite.Powerset
import Mathlib.Order.UpperLower.Clos... |
theorem isBase_iff_maximal_indep : M.IsBase B ↔ Maximal M.Indep B := by
rw [maximal_subset_iff]
| Mathlib/Data/Matroid/Basic.lean | 611 | 613 |
/-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Algebra.BigOperators.Expect
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Analysis.Convex.Jensen
import M... | variable {f g}
/-- **Hölder inequality**: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued nonnegative functions. -/
theorem inner_le_Lp_mul_Lq_of_nonneg (hpq : HolderConjugate p q)... | Mathlib/Analysis/MeanInequalities.lean | 737 | 750 |
/-
Copyright (c) 2022 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex J. Best, Xavier Roblot
-/
import Mathlib.Algebra.Algebra.Hom.Rat
import Mathlib.Analysis.Complex.Polynomial.Basic
import Mathlib.NumberTheory.NumberField.Norm
import Mathlib.RingTh... |
lemma IsUnramifiedAtInfinitePlaces.bot [h₁ : IsUnramifiedAtInfinitePlaces k F]
[Algebra.IsAlgebraic K F] :
IsUnramifiedAtInfinitePlaces k K where
isUnramified w := by
obtain ⟨w, rfl⟩ := InfinitePlace.comap_surjective (K := F) w
exact (h₁.1 w).comap K
variable {K}
lemma NumberField.InfinitePlace.isU... | Mathlib/NumberTheory/NumberField/Embeddings.lean | 1,074 | 1,094 |
/-
Copyright (c) 2023 Mohanad Ahmed. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mohanad Ahmed
-/
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.FieldTheory.IsAlgClosed.Basic
/-!
# Eigenvalues are characteristic polynomial roots.
In fields we show that:
*... | rw [det_eq_sign_charpoly_coeff, ← charpoly_natDegree_eq_dim A,
Polynomial.prod_roots_eq_coeff_zero_of_monic_of_splits A.charpoly_monic hAps, ← mul_assoc,
← pow_two, pow_right_comm, neg_one_sq, one_pow, one_mul]
theorem trace_eq_sum_roots_charpoly_of_splits (hAps : A.charpoly.Splits (RingHom.id R)) :
| Mathlib/LinearAlgebra/Matrix/Charpoly/Eigs.lean | 60 | 64 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Ordinal.Family
/-! # Ordinal exponential
In this file we define the power function and the lo... | theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :
x < b ^ succ (log b x) := by
rcases eq_or_ne x 0 with (rfl | hx)
· apply opow_pos _ (zero_lt_one.trans hb)
· rw [succ_log_def hb hx]
exact csInf_mem (log_nonempty hb)
| Mathlib/SetTheory/Ordinal/Exponential.lean | 311 | 316 |
/-
Copyright (c) 2022 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Jireh Loreaux
-/
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Star.Module
import Mathlib.Algebra.Star.NonUn... | rfl
| Mathlib/Algebra/Star/Subalgebra.lean | 652 | 652 |
/-
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... | @[simp]
theorem roots_pow (p : R[X]) (n : ℕ) : (p ^ n).roots = n • p.roots := by
induction n with
| zero => rw [pow_zero, roots_one, zero_smul, empty_eq_zero]
| succ n ihn =>
rcases eq_or_ne p 0 with (rfl | hp)
| Mathlib/Algebra/Polynomial/Roots.lean | 228 | 233 |
/-
Copyright (c) 2022 Jake Levinson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jake Levinson
-/
import Mathlib.Data.Finset.Preimage
import Mathlib.Data.Finset.Prod
import Mathlib.Data.SetLike.Basic
import Mathlib.Order.UpperLower.Basic
/-!
# Young diagrams
A You... | 1. `YoungDiagram.rowLens_sorted` : It is weakly decreasing (`List.Sorted (· ≥ ·)`).
| Mathlib/Combinatorics/Young/YoungDiagram.lean | 351 | 351 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Data.Finite.Sum
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.GroupTheory.Perm.Support
import Mathlib.Logic.Equiv.Fintype
/-!
# Permutations on... | Finset.surj_on_of_inj_on_of_card_le (fun x hx => (fun i _ => f i) x hx) (fun a ha => h a ha)
(fun a₁ a₂ ha₁ ha₂ heq => (Equiv.apply_eq_iff_eq f).mp heq) rfl.ge
obtain ⟨y2, hy2, heq⟩ := h0 y hy
convert hy2
rw [heq]
simp only [inv_apply_self]
theorem perm_inv_mapsTo_of_mapsTo (f : Perm α) {s : Set α} [... | Mathlib/GroupTheory/Perm/Finite.lean | 57 | 65 |
/-
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.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | theorem arg_mul_cos_add_sin_mul_I_eq_toIocMod {r : ℝ} (hr : 0 < r) (θ : ℝ) :
arg (r * (cos θ + sin θ * I)) = toIocMod Real.two_pi_pos (-π) θ := by
have hi : toIocMod Real.two_pi_pos (-π) θ ∈ Set.Ioc (-π) π := by
| Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean | 448 | 450 |
/-
Copyright (c) 2023 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed
import Mathlib.FieldTheory.PrimitiveElement
import Mathlib.FieldTheory.IsAlgClosed.Basic
/-!
# Results about `minpoly... | rw [e, zero_mul] at this
exact ((minpoly.monic hx).map (algebraMap R S)).ne_zero this.symm
lemma minpolyDiv_monic : Monic (minpolyDiv R x) := by
| Mathlib/FieldTheory/Minpoly/MinpolyDiv.lean | 105 | 108 |
/-
Copyright (c) 2018 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.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
/-!
# Equalizers and coequalizers
This file defines (co)equalizers a... | theorem hasCoequalizer_epi_comp [HasCoequalizer f g] {W : C} (h : W ⟶ X) [Epi h] :
HasCoequalizer (h ≫ f) (h ≫ g) :=
⟨⟨{ cocone := _
isColimit := isCoequalizerEpiComp (colimit.isColimit _) h }⟩⟩
variable (C f g)
| Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean | 1,168 | 1,173 |
/-
Copyright (c) 2024 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.RingTheory.FractionalIdeal.Basic
import Mathlib.RingTheory.Ideal.Norm.AbsNorm
import Mathlib.RingTheory.Localization.NormTrace
/-!
# Fractional ideal n... | absNorm (spanSingleton R⁰ x) = |(Algebra.norm ℚ x)| := by
have : IsDomain K := IsFractionRing.isDomain R
obtain ⟨d, ⟨r, hr⟩⟩ := IsLocalization.exists_integer_multiple R⁰ x
rw [absNorm_eq' d (Ideal.span {r})]
· rw [Ideal.absNorm_span_singleton]
simp_rw [Int.cast_natAbs, Int.cast_abs, show ((Algebra.norm ... | Mathlib/RingTheory/FractionalIdeal/Norm.lean | 132 | 147 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.BoxIntegral.Box.Basic
import Mathlib.Analysis.SpecificLimits.Basic
/-!
# Induction on subboxes
In this file we prove the following inducti... | refine forall_congr' fun i ↦ ?_
dsimp only [Set.piecewise]
split_ifs with hs <;> simp only [hs, iff_true, iff_false, not_lt]
exacts [⟨fun H ↦ ⟨⟨(left_lt_add_div_two.2 (I.lower_lt_upper i)).trans H.1, H.2⟩, H.1⟩,
fun H ↦ ⟨H.2, H.1.2⟩⟩,
⟨fun H ↦ ⟨⟨H.1, H.2.trans (add_div_two_lt_right.2 (I.lower_lt_upper... | Mathlib/Analysis/BoxIntegral/Box/SubboxInduction.lean | 53 | 62 |
/-
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.Basic
import Mathlib.Order.Basic
import Mathlib.Order.Monotone.Defs
/-!
# Covariants and contravariants
This file contains general lemma... | /-- Typeclass for monotonicity of addition on the left,
namely `b₁ < b₂ → a + b₁ < a + b₂`.
You should usually not use this very granular typeclass directly, but rather a typeclass like
`OrderedAddCommGroup`. -/
abbrev AddLeftStrictMono [Add M] [LT M] : Prop :=
CovariantClass M M (· + ·) (· < ·)
| Mathlib/Algebra/Order/Monoid/Unbundled/Defs.lean | 170 | 176 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yaël Dillies
-/
import Mathlib.Data.Set.Image
/-!
# Directed indexed families and sets
This file defines directed indexed families and directed sets. An indexed famil... |
theorem IsTotal.directed [IsTotal α r] (f : ι → α) : Directed r f := fun i j =>
Or.casesOn (total_of r (f i) (f j)) (fun h => ⟨j, h, refl _⟩) fun h => ⟨i, refl _, h⟩
/-- `IsDirected α r` states that for any elements `a`, `b` there exists an element `c` such that
`r a c` and `r b c`. -/
class IsDirected (α : Type*) ... | Mathlib/Order/Directed.lean | 116 | 128 |
/-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.IntegralClosure.IntegrallyClosed
import Mathlib.RingTheory.Trace.Basic
import Mathlib.RingTheory.Norm.Basic
/-!
# Discriminant of a famil... | (hli : ¬LinearIndependent A b) : discr A b = 0 := by
classical
| Mathlib/RingTheory/Discriminant.lean | 88 | 89 |
/-
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... | | empty =>
simp_rw [subset_empty, ← subset_singleton_iff', subset_singleton_iff] at h𝒜s hℬs
obtain rfl | rfl := h𝒜s
· simp only [card_empty, zero_mul, empty_inter, mul_zero, le_refl]
obtain rfl | rfl := hℬs
· simp only [card_empty, inter_empty, mul_zero, zero_mul, le_refl]
· simp only [card_... | Mathlib/Combinatorics/SetFamily/HarrisKleitman.lean | 55 | 91 |
/-
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... | · rw [h2U.interior_eq]; exact ⟨1, h3U⟩
· refine disjoint_of_subset ?_ ?_ hU
· refine Subset.trans (mul_subset_mul Subset.rfl ?_) h2L₁
exact Subset.trans (inv_subset.mpr h1U) inter_subset_left
· refine Subset.trans (mul_subset_mul Subset.rfl ?_) h2L₂
exact Subset.trans (inv_subset.mpr... | Mathlib/MeasureTheory/Measure/Haar/Basic.lean | 436 | 443 |
/-
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.Connected.Basic
/-!
# Locally connected topological spaces
A topological space is **locally connected** if each neighborhood filter admi... | ∀ F : Set α, IsOpen F → ∀ x ∈ F, IsOpen (connectedComponentIn F x) := by
constructor
· intro h
exact fun F hF x _ => hF.connectedComponentIn
· intro h
rw [locallyConnectedSpace_iff_subsets_isOpen_isConnected]
refine fun x U hU =>
⟨connectedComponentIn (interior U) x,
(connected... | Mathlib/Topology/Connected/LocallyConnected.lean | 89 | 101 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Multiset.ZeroCons
/-!
# Basic results on multisets
-/
-- No algebra should be required
assert_not_exists Monoid
universe v
open List S... | Mathlib/Data/Multiset/Basic.lean | 2,718 | 2,722 | |
/-
Copyright (c) 2022 Antoine Labelle. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Labelle
-/
import Mathlib.LinearAlgebra.Contraction
import Mathlib.Algebra.Group.Equiv.TypeTags
/-!
# Monoid representations
This file introduces monoid representations and ... | Mathlib/RepresentationTheory/Basic.lean | 109 | 109 | |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.InnerProductSpace.Defs
impor... | simp only [inner_add_left, inner_add_right]; ring
| Mathlib/Analysis/InnerProductSpace/Basic.lean | 230 | 230 |
/-
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... | [IsMeasurablyGenerated l] (hfm : StronglyMeasurableAtFilter f l μ) (hμ : μ.FiniteAtFilter l) {b}
(hf : Tendsto f l (𝓝 b)) : IntegrableAtFilter f l μ :=
hμ.integrableAtFilter hfm hf.norm.isBoundedUnder_le
| Mathlib/MeasureTheory/Integral/IntegrableOn.lean | 490 | 493 |
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.Group.Support
import Mathlib.Algebra.Order.Monoid.Unbundled.WithTop
import Mathlib.Order.WellFoundedSet
/-!
# Hahn Series
If `Γ` is ordered an... | exact x.isWF_support.min_mem (support_nonempty_iff.2 hx)
theorem order_le_of_coeff_ne_zero {Γ} [Zero Γ] [LinearOrder Γ] {x : HahnSeries Γ R}
{g : Γ} (h : x.coeff g ≠ 0) : x.order ≤ g :=
le_trans (le_of_eq (order_of_ne (ne_zero_of_coeff_ne_zero h)))
(Set.IsWF.min_le _ _ ((mem_support _ _).2 h))
@[simp]
the... | Mathlib/RingTheory/HahnSeries/Basic.lean | 357 | 367 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Order.Filter.Tendsto
import Mathlib.Data.Set.Accumulate
import Mathlib.Topology.Bornology.Basic
import Mathlib.Topolog... | ∃ t : Finset X, ⋃ x ∈ t, U x = ⊤ := by
obtain ⟨t, -, s⟩ := IsCompact.elim_nhds_subcover isCompact_univ U fun x _ => hU x
exact ⟨t, top_unique s⟩
theorem compactSpace_of_finite_subfamily_closed
(h : ∀ {ι : Type u} (t : ι → Set X), (∀ i, IsClosed (t i)) → ⋂ i, t i = ∅ →
∃ u : Finset ι, ⋂ i ∈ u, t i = ∅... | Mathlib/Topology/Compactness/Compact.lean | 686 | 692 |
/-
Copyright (c) 2024 Brendan Murphy. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Brendan Murphy
-/
import Mathlib.RingTheory.Regular.IsSMulRegular
import Mathlib.RingTheory.Artinian.Module
import Mathlib.RingTheory.Nakayama
import Mathlib.Algebra.Equiv.TransferInst... | -- TODO: apply the above to localization and completion (Corollary 1.1.3 in B&H)
lemma map_first_exact_on_four_term_right_exact_of_isSMulRegular_last
{rs : List R} {f₁ : M →ₗ[R] M₂} {f₂ : M₂ →ₗ[R] M₃} {f₃ : M₃ →ₗ[R] M₄}
(h₁₂ : Exact f₁ f₂) (h₂₃ : Exact f₂ f₃) (h₃ : Surjective f₃)
(h₄ : IsWeaklyRegular M₄ r... | Mathlib/RingTheory/Regular/RegularSequence.lean | 552 | 568 |
/-
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... | classical
simp [biproduct.lift_eq, biproduct.desc_eq, comp_sum, sum_comp, biproduct.ι_π_assoc, comp_dite,
dite_comp]
theorem biproduct.map_eq [HasFiniteBiproducts C] {f g : J → C} {h : ∀ j, f j ⟶ g j} :
| Mathlib/CategoryTheory/Preadditive/Biproducts.lean | 224 | 228 |
/-
Copyright (c) 2021 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Thomas Murrills
-/
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.Tactic.NormNum.Basic
/-!
## `norm_num` plugin for `^`.
-/
assert_not_exists RelIso
namespace Ma... | let ⟨c2, p2⟩ := go d a hi c1 b q(Nat.pow $a $hi = $c1) ⟨⟩
⟨c2, q(($p1).trans $p2)⟩
| Mathlib/Tactic/NormNum/Pow.lean | 102 | 103 |
/-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... |
namespace Complex
theorem Gamma_mul_Gamma_add_half (s : ℂ) :
Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * (2 : ℂ) ^ (1 - 2 * s) * ↑(√π) := by
suffices (fun z => (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) = fun z =>
(Gamma (2 * z))⁻¹ * (2 : ℂ) ^ (2 * z - 1) / ↑(√π) by
convert congr_arg Inv.inv (congr_fun th... | Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean | 539 | 551 |
/-
Copyright (c) 2024 María Inés de Frutos-Fernández. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata, Fabrizio Barroero, Laura Capuano, Nirvana Coppola,
María Inés de Frutos-Fernández, Sam van Gool, Silvain Rideau-Kikuchi, Amos Turchet,
Frances... | ## Tags
absolute value, Ostrowski's theorem
-/
open Filter Nat Real Topology
| Mathlib/NumberTheory/Ostrowski.lean | 36 | 41 |
/-
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... | theorem exp_one_rpow (x : ℝ) : exp 1 ^ x = exp x := by rw [← exp_mul, one_mul]
| Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 60 | 60 |
/-
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... |
end Ordinal
namespace Cardinal
open Ordinal
| Mathlib/SetTheory/Ordinal/Arithmetic.lean | 1,180 | 1,186 |
/-
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
... | have hW₁ : (s ∪ s').sum W = 1 := by
rw [sum_add_distrib, ← sum_subset subset_union_left,
← sum_subset subset_union_right, sum_ite_of_true,
sum_ite_of_true, ← mul_sum, ← mul_sum, hw₁, hw₁', ← add_mul, hab,
mul_one] <;> intros <;> simp_all
refine ⟨s ∪ s', W, ?_, hW₁, ?_⟩
· rintro... | Mathlib/Analysis/Convex/Combination.lean | 283 | 290 |
/-
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.Complex.Convex
import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.Analysis.Calculus.Deriv.Shift
/-!
# Estimates for the complex ... | · simp
conv => enter [2]; rw [← div_one (‖z‖ ^ n)]
gcongr
norm_cast
· rw [← tendsto_sub_nhds_zero_iff]
conv => enter [1, x]; rw [← div_one (_ - _), ← logTaylor]
rw [← isLittleO_iff_tendsto fun _ h ↦ (one_ne_zero h).elim]
refine IsLittleO.trans_isBigO ?_ <| isBigO_const_one ℂ (1 : ℝ) atTop
... | Mathlib/Analysis/SpecialFunctions/Complex/LogBounds.lean | 232 | 242 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Shing Tak Lam, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring.List
import Mathlib.Data.Int.ModEq
import Mathlib.Da... | exact ofDigits_monotone (digits p.succ n.succ) <| Nat.succ_pos p
theorem pow_length_le_mul_ofDigits {b : ℕ} {l : List ℕ} (hl : l ≠ []) (hl2 : l.getLast hl ≠ 0) :
(b + 2) ^ l.length ≤ (b + 2) * ofDigits (b + 2) l := by
rw [← List.dropLast_append_getLast hl]
| Mathlib/Data/Nat/Digits.lean | 490 | 494 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.