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) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kim Morrison
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Algebra.Group.InjSurj
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Tactic.FastInstance
impo... |
@[simp]
theorem embDomain_inj {f : α ↪ β} {l₁ l₂ : α →₀ M} : embDomain f l₁ = embDomain f l₂ ↔ l₁ = l₂ :=
| Mathlib/Data/Finsupp/Defs.lean | 412 | 414 |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Finset.Card
import Mathlib.Data.Finset.Union
import Mathlib.Data.Multiset.Pi
import Mathlib.Logic.Function.DependsOn
/-!
# The cartesian product ... |
@[simp]
lemma pi_eq_empty : s.pi t = ∅ ↔ ∃ a ∈ s, t a = ∅ := by
simp [← not_nonempty_iff_eq_empty]
@[simp]
theorem pi_insert [∀ a, DecidableEq (β a)] {s : Finset α} {t : ∀ a : α, Finset (β a)} {a : α}
(ha : a ∉ s) : pi (insert a s) t = (t a).biUnion fun b => (pi s t).image (Pi.cons s a b) := by
apply eq_of_ve... | Mathlib/Data/Finset/Pi.lean | 96 | 112 |
/-
Copyright (c) 2018 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Mario Carneiro, Kim Morrison, Floris van Doorn
-/
import Mathlib.CategoryTheory.Limits.IsLimit
import Mathlib.CategoryTheory.Category.ULift
import Mathlib.CategoryTheory.Ess... | we obtain a formula for `limit.pre F E`.
-/
theorem limit.pre_eq (s : LimitCone (E ⋙ F)) (t : LimitCone F) :
limit.pre F E = (limit.isoLimitCone t).hom ≫ s.isLimit.lift (t.cone.whisker E) ≫
(limit.isoLimitCone s).inv := by aesop_cat
end Pre
| Mathlib/CategoryTheory/Limits/HasLimits.lean | 384 | 390 |
/-
Copyright (c) 2024 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Alex J. Best
-/
import Mathlib.Algebra.Polynomial.Roots
import Mathlib.Tactic.IntervalCases
import Mathlib.Algebra.Polynomial.FieldDivision
/-!
# Polynomials of specific deg... | /-- A polynomial of degree 2 or 3 is irreducible iff it doesn't have roots. -/
theorem Monic.irreducible_iff_roots_eq_zero_of_degree_le_three {p : R[X]} (hp : p.Monic)
(hp2 : 2 ≤ p.natDegree) (hp3 : p.natDegree ≤ 3) : Irreducible p ↔ p.roots = 0 := by
have hp0 : p ≠ 0 := hp.ne_zero
have hp1 : p ≠ 1 := by rintro... | Mathlib/Algebra/Polynomial/SpecificDegree.lean | 22 | 34 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Morenikeji Neri
-/
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.Algebra.EuclideanDomain.Field
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.RingTheor... |
theorem mem_iff_generator_dvd (S : Ideal R) [S.IsPrincipal] {x : R} : x ∈ S ↔ generator S ∣ x :=
(mem_iff_eq_smul_generator S).trans (exists_congr fun a => by simp only [mul_comm, smul_eq_mul])
theorem prime_generator_of_isPrime (S : Ideal R) [S.IsPrincipal] [is_prime : S.IsPrime]
| Mathlib/RingTheory/PrincipalIdealDomain.lean | 133 | 137 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kenny Lau, Robert Y. Lewis
-/
import Mathlib.Algebra.Group.Defs
/-!
# Eckmann-Hilton argument
The Eckmann-Hilton argument says that if a type carries two monoid structures... |
In fact, they give a commutative monoid structure, see `eckmann_hilton.CommMonoid`. -/
| Mathlib/GroupTheory/EckmannHilton.lean | 76 | 77 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Minchao Wu, Mario Carneiro
-/
import Mathlib.Data.Finset.Attach
import Mathlib.Data.Finset.Disjoint
import Mathlib.Data.Finset.Erase
import Mat... | Mathlib/Data/Finset/Basic.lean | 2,077 | 2,083 | |
/-
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.Logic.Equiv.PartialEquiv
import Mathlib.Topology.Homeomorph.Lemmas
import Mathlib.Topology.Sets.Opens
/-!
# Partial homeomorphisms
This file de... | PartialEquiv.IsImage.iff_preimage_eq
alias ⟨preimage_eq, of_preimage_eq⟩ := iff_preimage_eq
theorem iff_symm_preimage_eq : e.IsImage s t ↔ e.target ∩ e.symm ⁻¹' s = e.target ∩ t :=
symm_iff.symm.trans iff_preimage_eq
alias ⟨symm_preimage_eq, of_symm_preimage_eq⟩ := iff_symm_preimage_eq
| Mathlib/Topology/PartialHomeomorph.lean | 483 | 490 |
/-
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, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Pairwise
import Mathlib.Data.Set.BooleanAlgebra
/-!
# The set lattice
This file is a collectio... | Mathlib/Data/Set/Lattice.lean | 1,498 | 1,503 | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Group.Action.End
import Mathlib.Algebra.Group.Action.Pointwise.Set.Basic
import Mathlib.Algebra.Group.Action.Prod
import Mathlib.Algebra.Group.Subg... | Mathlib/GroupTheory/GroupAction/Basic.lean | 634 | 647 | |
/-
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 ... | lemma integrable_pow_mul_norm_one_add_mul_inv (n : ℕ) {z : ℂ} (hz : ‖z‖ < 1) :
IntervalIntegrable (fun t : ℝ ↦ t ^ n * ‖(1 + t * z)⁻¹‖) MeasureTheory.volume 0 1 := by
have := continuousOn_one_add_mul_inv <| mem_slitPlane_of_norm_lt_one hz
rw [← Set.uIcc_of_le zero_le_one] at this
exact ContinuousOn.intervalIn... | Mathlib/Analysis/SpecialFunctions/Complex/LogBounds.lean | 124 | 128 |
/-
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.Kernel.Disintegration.Unique
import Mathlib.Probability.Notation
/-!
# Regular conditional probability distribution
We define the regular con... | @[deprecated (since := "2025-03-05")]
alias _root_.MeasureTheory.AEStronglyMeasurable.comp_snd_map_prod_mk :=
MeasureTheory.AEStronglyMeasurable.comp_snd_map_prodMk
theorem _root_.MeasureTheory.Integrable.comp_snd_map_prodMk
{Ω} {mΩ : MeasurableSpace Ω} (X : Ω → β) {μ : Measure Ω} {f : Ω → F} (hf_int : Integrabl... | Mathlib/Probability/Kernel/CondDistrib.lean | 325 | 335 |
/-
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.Algebra.NoZeroSMulDivisors.Basic
import Mathlib.Algebra.Order.GroupWithZero.Action.Synonym
import Mathlib.Tactic.GCongr
import Mathlib.Tactic.Positivity.Co... | simp only [← lt_iff_lt_of_le_iff_le' hf hf, zero, smul] at *
exact smul_lt_smul_of_pos_right ha hb
| Mathlib/Algebra/Order/Module/Defs.lean | 1,104 | 1,105 |
/-
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... | Mathlib/Probability/Process/Stopping.lean | 1,144 | 1,147 | |
/-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Gamma.Deriv
import Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral
/-! # Convexity properties of the Gamma funct... | (tendsto_add_atTop_nat 1)
have :
((fun x_1 : ℕ =>
(fun n : ℕ =>
logGammaSeq x (n - 1) + x * (log (↑(n - 1) + 1) - log ↑(n - 1)) - log (x - 1))
x_1 +
(fun b : ℕ => log (x - 1)) x_1) ∘
| Mathlib/Analysis/SpecialFunctions/Gamma/BohrMollerup.lean | 269 | 275 |
/-
Copyright (c) 2023 Yaël Dillies, Vladimir Ivanov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Vladimir Ivanov
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.BigOperators.... | positivity
variable [Nonempty α]
@[simp] lemma supSum_singleton (hs : s ≠ univ) :
supSum ({s} : Finset (Finset α)) = card α * ∑ k ∈ range (card α), (k : ℚ)⁻¹ := by
have : ∀ t : Finset α,
(card α - #(truncatedSup {s} t) : ℚ) / ((card α - #t) * (card α).choose #t) =
if t ⊆ s then (card α - #s : ℚ) / ((c... | Mathlib/Combinatorics/SetFamily/AhlswedeZhang.lean | 364 | 378 |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Covering.VitaliFamily
import Mathlib.MeasureTheory.Function.AEMeasurableOrder
import Mathlib.MeasureTheory.Integral.Average
import ... | theorem ae_eventually_measure_zero_of_singular (hρ : ρ ⟂ₘ μ) :
∀ᵐ x ∂μ, Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 0) := by
have A : ∀ ε > (0 : ℝ≥0), ∀ᵐ x ∂μ, ∀ᶠ a in v.filterAt x, ρ a < ε * μ a := by
intro ε εpos
set s := {x | ¬∀ᶠ a in v.filterAt x, ρ a < ε * μ a} with hs
change μ s = 0
obta... | Mathlib/MeasureTheory/Covering/Differentiation.lean | 160 | 201 |
/-
Copyright (c) 2022 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.Algebra.IsPrimePow
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.Nat.Prime.Pow
import Mathlib.NumberTheory.Divisors
/-!
# Prime powers a... | obtain ⟨p, k, hp, hk, rfl⟩ := hn
rw [← Nat.prime_iff] at hp
rw [hp.pow_minFac hk.ne', hp.factorization_pow, Finsupp.single_eq_same]
theorem isPrimePow_of_minFac_pow_factorization_eq {n : ℕ}
| Mathlib/Data/Nat/Factorization/PrimePow.lean | 20 | 24 |
/-
Copyright (c) 2022 Yaël Dillies, Ella Yu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Ella Yu
-/
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Fintype.Prod
import Mathlib.Algebra.Group.Poin... | _ ≤ u.card * ∑ c ∈ u, ((s ×ˢ t).filter fun (a, b) ↦ a * b = c).card ^ 2 := by
simpa using sum_mul_sq_le_sq_mul_sq (R := ℕ) _ 1 _
_ ≤ u.card * ∑ c ∈ s * t, ((s ×ˢ t).filter fun (a, b) ↦ a * b = c).card ^ 2 := by
refine mul_le_mul_left' (sum_le_sum_of_ne_zero ?_) _
aesop (add simp [filter_... | Mathlib/Combinatorics/Additive/Energy.lean | 140 | 152 |
/-
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... | Mathlib/RingTheory/FractionalIdeal/Norm.lean | 88 | 88 | |
/-
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... | · rw [ofReal_im, eq_comm, ← Complex.conj_eq_iff_im]
exact RingHom.congr_fun hφ x
| Mathlib/NumberTheory/NumberField/Embeddings.lean | 203 | 204 |
/-
Copyright (c) 2014 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Leonardo de Moura, Johannes Hölzl, Mario Carneiro, Yaël Dillies
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Data.Rat.Init
/-!
# Division (semi)rings and (semi... | Mathlib/Algebra/Field/Defs.lean | 226 | 227 | |
/-
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.Topology.Compactness.Compact
/-!
# Locally compact spaces
This file contains basic results about locally compact spac... |
/-- A reformulation of the definition of locally compact space: In a locally compact space,
every open set containing `x` has a compact subset containing `x` in its interior. -/
theorem exists_compact_subset [LocallyCompactSpace X] {x : X} {U : Set X} (hU : IsOpen U)
| Mathlib/Topology/Compactness/LocallyCompact.lean | 141 | 144 |
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Probability.Variance
import Mathlib.MeasureTheory.Function.UniformIntegrable
/-!
# Identically distributed random variables
Two random variable... | have I : ∀ a, μ {x : α | a < f x} = ν {x : β | a < g x} := fun a =>
h.measure_mem_eq measurableSet_Ioi
simp_rw [essSup_eq_sInf, I]
theorem lintegral_eq {f : α → ℝ≥0∞} {g : β → ℝ≥0∞} (h : IdentDistrib f g μ ν) :
∫⁻ x, f x ∂μ = ∫⁻ x, g x ∂ν := by
change ∫⁻ x, id (f x) ∂μ = ∫⁻ x, id (g x) ∂ν
| Mathlib/Probability/IdentDistrib.lean | 162 | 168 |
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Kexing Ying
-/
import Mathlib.Probability.Notation
import Mathlib.Probability.Integration
import Mathlib.MeasureTheory.Function.L2Space
/-!
# Variance of random ... | This is defined as the Lebesgue integral of `(X - 𝔼[X])^2`. -/
scoped notation "eVar[" X "; " μ "]" => ProbabilityTheory.evariance X μ
/-- The `ℝ≥0∞`-valued variance of the real-valued random variable `X` according to the volume
measure.
This is defined as the Lebesgue integral of `(X - 𝔼[X])^2`. -/
scoped notation... | Mathlib/Probability/Variance.lean | 62 | 69 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Finset.Finsupp
import Mathlib.Data.Finsupp.Order
import Mathlib.Order.Interval.Finset.Basic
/-!
# Finite intervals of finitely supported functions
T... |
theorem card_Iio : #(Iio f) = ∏ i ∈ f.support, #(Iic (f i)) - 1 := by
| Mathlib/Data/Finsupp/Interval.lean | 123 | 124 |
/-
Copyright (c) 2023 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calculus.Deriv.Add
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calcul... | exact B.preimage_mem_nhds this
theorem lineDifferentiableWithinAt_congr_set (h : s =ᶠ[𝓝 x] t) :
LineDifferentiableWithinAt 𝕜 f s x v ↔ LineDifferentiableWithinAt 𝕜 f t x v :=
⟨fun h' ↦ ((hasLineDerivWithinAt_congr_set h).1
h'.hasLineDerivWithinAt).lineDifferentiableWithinAt,
fun h' ↦ ((hasLineDerivWit... | Mathlib/Analysis/Calculus/LineDeriv/Basic.lean | 300 | 306 |
/-
Copyright (c) 2022 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Star.Basic
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Data.Set.Lattice.Image
import Mathlib.Algebra.Group.Pointwise.Set.Basic
/-!
# Poi... | Mathlib/Algebra/Star/Pointwise.lean | 137 | 139 | |
/-
Copyright (c) 2022 Sebastian Monnet. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sebastian Monnet
-/
import Mathlib.FieldTheory.Galois.Basic
import Mathlib.Topology.Algebra.FilterBasis
import Mathlib.Topology.Algebra.OpenSubgroup
import Mathlib.Tactic.ByContra
/... | def galBasis (K L : Type*) [Field K] [Field L] [Algebra K L] : FilterBasis (L ≃ₐ[K] L) where
sets := (fun g => g.carrier) '' fixedByFinite K L
nonempty := ⟨⊤, ⊤, top_fixedByFinite, rfl⟩
inter_sets := by
| Mathlib/FieldTheory/KrullTopology.lean | 124 | 127 |
/-
Copyright (c) 2024 Raghuram Sundararajan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Raghuram Sundararajan
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Ext
/-!
# Extensionality lemmas for rings and similar structures
In this file we prove e... | ext : 1 <;> assumption
have h₂ : inst₁.toNonAssocSemiring = inst₂.toNonAssocSemiring := by
ext : 1 <;> assumption
-- Mathematically non-trivial fact: `intCast` is determined by the rest.
have h₃ : inst₁.toAddCommGroupWithOne = inst₂.toAddCommGroupWithOne :=
AddCommGroupWithOne.ext h_add (congrArg (·.t... | Mathlib/Algebra/Ring/Ext.lean | 249 | 264 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kenny Lau, Kim Morrison
-/
import Mathlib.Data.List.Chain
/-!
# Ranges of naturals as lists
This file shows basic results about `List.iota`, `List.range`, `List.range... | Mathlib/Data/List/Range.lean | 273 | 276 | |
/-
Copyright (c) 2021 Paul Lezeau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Paul Lezeau
-/
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Algebra.IsPrimePow
import Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity
import Mathlib.Data.Z... | rcases n with - | n; · contradiction
refine (Associates.isAtom_iff (ne_zero_of_dvd_ne_zero hq (h₂.2 ⟨1, rfl⟩))).mp ⟨?_, fun b hb => ?_⟩
· exact ne_bot_of_gt (h₁ (show (0 : Fin (n + 2)) < 1 from Fin.one_pos))
obtain ⟨⟨i, hi⟩, rfl⟩ := h₂.1 (hb.le.trans (h₂.2 ⟨1, rfl⟩))
cases i
· exact (Associates.isUnit_iff_e... | Mathlib/RingTheory/ChainOfDivisors.lean | 99 | 108 |
/-
Copyright (c) 2024 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Kontorovich, David Loeffler, Heather Macbeth, Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.Analysis.Calculus.ContDiff.CPolynomial
import... |
/-- The `w`-derivative of the Fourier transform integrand. -/
lemma hasFDerivAt_fourierChar_smul (v : V) (w : W) :
HasFDerivAt (fun w' ↦ 𝐞 (-L v w') • f v) (𝐞 (-L v w) • fourierSMulRight L f v) w := by
have ha : HasFDerivAt (fun w' : W ↦ L v w') (L v) w := ContinuousLinearMap.hasFDerivAt (L v)
convert ((hasD... | Mathlib/Analysis/Fourier/FourierTransformDeriv.lean | 165 | 176 |
/-
Copyright (c) 2022 Hanting Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hanting Zhang
-/
import Mathlib.Topology.MetricSpace.Antilipschitz
import Mathlib.Topology.MetricSpace.Isometry
import Mathlib.Topology.MetricSpace.Lipschitz
import Mathlib.Data.FunLike... | h.imp fun r hr =>
⟨hr.1, fun x y => NNReal.eq <| by rw [coe_nndist, hr.2, NNReal.coe_mul, coe_nndist]⟩
@[simp]
theorem coe_mkOfDistEq {α β} [PseudoMetricSpace α] [PseudoMetricSpace β] (f : α → β) (h) :
| Mathlib/Topology/MetricSpace/Dilation.lean | 214 | 218 |
/-
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.W.Basic
/-!
# Polynomial functors
This file defines polynomial functors and the W-type construction as a
polynomial functor. (For the M-type cons... | intro
exact ⟨_, rfl⟩
· simp only [liftp_iff']
cases h
subst x
tauto
end PFunctor
| Mathlib/Data/PFunctor/Univariate/Basic.lean | 220 | 239 |
/-
Copyright (c) 2014 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Leonardo de Moura, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Ord... |
section LinearOrderedSemifield
| Mathlib/Algebra/Order/Field/Basic.lean | 677 | 678 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Logic.Nontrivial.Basic
import Mathlib.Order.TypeTags
import Mathlib.Data.Option.NAry
import Mathlib.Tactic.Contrapose
import Mathlib.Tactic.Lift
import... | a.unbot h = b ↔ a = b := by
induction a
| Mathlib/Order/WithBot.lean | 179 | 180 |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Order.AddGroupWithTop
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.Order.Sub.WithTop
import Mathlib.Data.ENat.Defs
import... | rfl
| Mathlib/Data/ENat/Basic.lean | 246 | 246 |
/-
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, Kevin Buzzard, Yury Kudryashov, Eric Wieser
-/
import Mathlib.Algebra.Group.Fin.Tuple
import Mathlib.Algebra.BigOperators.GroupWithZero.Action
import Ma... | Mathlib/LinearAlgebra/Pi.lean | 69 | 69 | |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle
import Mathlib.Geometry.Euclidean.Circumcenter
/-!
# Angles in circles and sphere.
This file proves results ... |
/-- Given three points on a circle, twice the radius of that circle may be expressed explicitly as
the distance between two of those points divided by the absolute value of the sine of the angle
at the third point (a version of the law of sines or sine rule). -/
| Mathlib/Geometry/Euclidean/Angle/Sphere.lean | 219 | 222 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPi... | apply IsPGroup.disjoint_of_ne p₁ p₂ hne' _ _ (P p₁).isPGroup' (P p₂).isPGroup'
refine MulEquiv.trans (N := ∀ p : ps, P p) ?_ ?_
-- There is only one Sylow subgroup for each p, so the inner product is trivial
· -- here we need to help the elaborator with an explicit instantiation
apply @MulEquiv.piCongrRig... | Mathlib/GroupTheory/Sylow.lean | 791 | 801 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Filippo A. E. Nuccio, Sam van Gool
-/
import Mathlib.Data.Fintype.Order
import Mathlib.Order.Interval.Finset.Basic
import Mathlib.Order.Irreducible
import Mathlib.Order.Upp... | @[simp] lemma infIrred_Ici (a : α) : InfIrred (Ici a) := by
refine ⟨fun h ↦ Ici_ne_top h.eq_top, fun s t hst ↦ ?_⟩
have := mem_Ici_iff.2 (le_refl a)
rw [← hst] at this
exact this.imp (fun ha ↦ le_antisymm (le_Ici.2 ha) <| hst.ge.trans inf_le_left) fun ha ↦
le_antisymm (le_Ici.2 ha) <| hst.ge.trans inf_le_... | Mathlib/Order/Birkhoff.lean | 61 | 66 |
/-
Copyright (c) 2022 Rishikesh Vaishnav. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rishikesh Vaishnav
-/
import Mathlib.MeasureTheory.Measure.Typeclasses.Probability
/-!
# Conditional Probability
This file defines conditional probability and includes basic resu... | Mathlib/Probability/ConditionalProbability.lean | 138 | 138 | |
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky, Chris Hughes
-/
import Mathlib.Data.List.Nodup
/-!
# List duplicates
## Main definitions
* `List.Duplicate x l : Prop` is an inductive property that holds when `x`... | theorem Duplicate.mem_cons_self (h : x ∈+ x :: l) : x ∈ l := by
obtain h | h := h
· exact h
· exact h.mem
| Mathlib/Data/List/Duplicate.lean | 46 | 49 |
/-
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 | 73 | 76 | |
/-
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... | rw [invFunOn_apply_eq (f := f) hx']
theorem injOn_iff_invFunOn_image_image_eq_self [Nonempty α] :
| Mathlib/Data/Set/Function.lean | 937 | 939 |
/-
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.Topology.Algebra.GroupCompletion
import Mathlib.Topology.Algebra.Ring.Real
import Mathlib.Topology.MetricSpace.Algebra
import Mathlib.Topology.Me... | closed properties pass to the completion. -/
intro hs
rcases mem_uniformity_isClosed hs with ⟨t, ht, ⟨tclosed, ts⟩⟩
have A : { x : α × α | (↑x.1, ↑x.2) ∈ t } ∈ uniformity α :=
uniformContinuous_def.1 (uniformContinuous_coe α) t ht
rcases mem_uniformity_dist.1 A with ⟨ε, εpos, hε⟩
refine ... | Mathlib/Topology/MetricSpace/Completion.lean | 87 | 142 |
/-
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
... | one_smul := TensorProduct.one_smul
smul_zero := TensorProduct.smul_zero }
instance : DistribMulAction R (M ⊗[R] N) :=
TensorProduct.leftDistribMulAction
| Mathlib/LinearAlgebra/TensorProduct/Basic.lean | 332 | 337 |
/-
Copyright (c) 2020 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap
import Mathlib.MeasureTheory.Integral.Bochner.FundThmCalculus
import Mathlib.MeasureT... | Mathlib/MeasureTheory/Integral/SetIntegral.lean | 542 | 548 | |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno
import Mathlib.Analysis.Calculus.FDeriv.Ad... | /-- Composition by continuous linear equivs on the right respects higher differentiability. -/
theorem ContinuousLinearEquiv.contDiff_comp_iff (e : G ≃L[𝕜] E) :
ContDiff 𝕜 n (f ∘ e) ↔ ContDiff 𝕜 n f := by
rw [← contDiffOn_univ, ← contDiffOn_univ, ← preimage_univ]
exact e.contDiffOn_comp_iff
end linear
| Mathlib/Analysis/Calculus/ContDiff/Basic.lean | 462 | 468 |
/-
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.Group.Action.Units
import Mathlib.Algebra.Group.Nat.Units
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra... | rw [add_comm] at h
exact h.of_add_mul_left_right
theorem IsCoprime.of_mul_add_right_right (h : IsCoprime x (z * x + y)) : IsCoprime x y := by
rw [add_comm] at h
| Mathlib/RingTheory/Coprime/Basic.lean | 195 | 199 |
/-
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.Topology.UniformSpace.UniformEmbedding
/-!
# Theory of complete separated uniform spaces.
This file is for elementary lemmas that depend on bot... | isClosed_iff_clusterPt.2 fun a ha => by
let f := 𝓝[s] a
have : Cauchy f := cauchy_nhds.mono' ha inf_le_left
rcases h f this inf_le_right with ⟨y, ys, fy⟩
rwa [(tendsto_nhds_unique' ha inf_le_left fy : a = y)]
theorem IsUniformEmbedding.isClosedEmbedding [UniformSpace α] [UniformSpace β] [CompleteSpa... | Mathlib/Topology/UniformSpace/CompleteSeparated.lean | 24 | 30 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Sina Hazratpour
-/
import Mathlib.CategoryTheory.Category.Cat.AsSmall
import Mathlib.CategoryTheory.Elements
import Mathlib.CategoryTheory.Comma.Over.Basic
/-!
# The Groth... | rw [← NatIso.naturality_2 (eqToIso (F.map_comp _ _)) f.fiber]
simp
| Mathlib/CategoryTheory/Grothendieck.lean | 127 | 129 |
/-
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.Logic.Function.Conjugate
/-!
# Iterations of a function
In this file we prove simple properties of `Nat.iterate f n` a.k.a. `f^[n]`:
* `iterate_ze... | (refl f).iterate_iterate m n
end Commute
| Mathlib/Logic/Function/Iterate.lean | 150 | 153 |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.Group.Finset
import Mathlib.Data.Finsupp.Order
import Mathlib.Data.Sym.Basic
/-!
# Equivalence between `Multiset` and `ℕ`-valued finitel... | the multiplicities of the elements of `s`. -/
@[simps symm_apply]
def toFinsupp : Multiset α ≃+ (α →₀ ℕ) where
| Mathlib/Data/Finsupp/Multiset.lean | 128 | 130 |
/-
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... | ⟨invFun f, leftInverse_invFun hf⟩
theorem injective_iff_hasLeftInverse : Injective f ↔ HasLeftInverse f :=
⟨Injective.hasLeftInverse, HasLeftInverse.injective⟩
end InvFun
section SurjInv
variable {α : Sort u} {β : Sort v} {γ : Sort w} {f : α → β}
/-- The inverse of a surjective function. (Unlike `invFun`, this... | Mathlib/Logic/Function/Basic.lean | 407 | 419 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.Homotopy
import Mathlib.Algebra.Ring.NegOnePow
import Mathlib.Algebra.Category.Grp.Preadditive
import Mathlib.Tactic.Linarith
import Mathlib.Cat... | Int.negOnePow_one, Units.neg_smul, one_smul, sub_eq_add_neg]
@[simp]
lemma δ_ofHom {p : ℤ} (φ : F ⟶ G) : δ 0 p (Cochain.ofHom φ) = 0 := by
by_cases h : p = 1
· subst h
ext
simp
· rw [δ_shape]
| Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean | 525 | 533 |
/-
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.Cardinal.Arithmetic
import Mathlib.SetTheory.Ordinal.FixedPoint
/-!
# Cofinality
This file co... | Quotient.sound
⟨RelIso.ofSurjective (RelEmbedding.ofMonotone ?_ fun x y => ?_) fun x => ?_⟩⟩
· apply Sum.rec <;> [exact Subtype.val; exact fun _ => a]
| Mathlib/SetTheory/Cardinal/Cofinality.lean | 406 | 408 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Polynomial.Degree.Domain
import Mathlib.Algebra.Polynomial.Degree.Support
import Mathlib.Algebra.Poly... | derivative (p ^ (n + 1)) = C (n + 1 : R) * p ^ n * derivative p :=
Nat.recOn n (by simp) fun n ih => by
rw [pow_succ, derivative_mul, ih, Nat.add_one, mul_right_comm, C_add,
add_mul, add_mul, pow_succ, ← mul_assoc, C_1, one_mul]; simp [add_mul]
| Mathlib/Algebra/Polynomial/Derivative.lean | 450 | 453 |
/-
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... | have (k : ℕ) : (u (k + 1) - (u k : ℝ)) = ((u (k + 1) : ℝ≥0) - (u k : ℝ≥0) : ℝ≥0) := by
have := Nat.cast_le (α := ℝ≥0).mpr <| (hu_strict k.lt_succ_self).le
simp [NNReal.coe_sub this]
simp_rw [this]
exact_mod_cast NNReal.summable_schlomilch_iff hf h_pos hu_strict hC_nonzero h_succ_diff
/-- Cauchy condensat... | Mathlib/Analysis/PSeries.lean | 225 | 235 |
/-
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, Kevin Buzzard, Yury Kudryashov, Eric Wieser
-/
import Mathlib.Algebra.Algebra.Prod
import Mathlib.Algebra.Group.Graph
import Mathlib.LinearAlgebra.Span.... | Mathlib/LinearAlgebra/Prod.lean | 1,011 | 1,012 | |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Wrenna Robson
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Pi
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.LinearAlgebra.Vandermonde
import Mathlib.RingT... |
theorem natDegree_basisDivisor_self : (basisDivisor x x).natDegree = 0 := by
| Mathlib/LinearAlgebra/Lagrange.lean | 168 | 169 |
/-
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... | · intro h
by_contra! h_contra
rw [(geom_mean_eq_arith_mean_weighted_iff' s w z hw hw' hz).mpr ?_] at h
· exact (lt_self_iff_false _).mp h
· intro j hjs
rw [← arith_mean_weighted_of_constant s w (fun _ => z j) (z j) hw' fun _ _ => congrFun rfl]
apply sum_congr rfl (fun x a => congrArg (HMul... | Mathlib/Analysis/MeanInequalities.lean | 257 | 263 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Countable.Basic
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Data.Set.Subsingleton
import Mathlib.Logic.Equiv.List
/-!
# Countable sets
I... |
alias ⟨Countable.exists_surjective, _⟩ := Set.countable_iff_exists_surjective
theorem countable_univ_iff : (univ : Set α).Countable ↔ Countable α :=
countable_coe_iff.symm.trans (Equiv.Set.univ _).countable_iff
| Mathlib/Data/Set/Countable.lean | 129 | 134 |
/-
Copyright (c) 2022 Anand Rao, Rémi Bottinelli. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anand Rao, Rémi Bottinelli
-/
import Mathlib.CategoryTheory.CofilteredSystem
import Mathlib.Combinatorics.SimpleGraph.Path
import Mathlib.Data.Finite.Set
/-!
# Ends
This ... | exact ((h v).mp ⟨hv, Reachable.refl _⟩).choose_spec
theorem ComponentCompl.supp_inj {C D : G.ComponentCompl K} : C.supp = D.supp ↔ C = D :=
ComponentCompl.supp_injective.eq_iff
instance ComponentCompl.setLike : SetLike (G.ComponentCompl K) V where
| Mathlib/Combinatorics/SimpleGraph/Ends/Defs.lean | 44 | 49 |
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.FieldTheory.RatFunc.AsPolynomial
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Poly... | theorem intDegree_mul {x y : RatFunc K} (hx : x ≠ 0) (hy : y ≠ 0) :
intDegree (x * y) = intDegree x + intDegree y := by
simp only [intDegree, add_sub, sub_add, sub_sub_eq_add_sub, sub_sub, sub_eq_sub_iff_add_eq_add]
norm_cast
| Mathlib/FieldTheory/RatFunc/Degree.lean | 65 | 68 |
/-
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.Enum
import Mathlib.Tactic.TFAE
import Mathlib.Topology.Order.Monotone
/-!
### Topology of ordinals
We prov... | (bddAbove_range.{u, u} f)
tfae_finish
theorem mem_closure_iff_iSup :
| Mathlib/SetTheory/Ordinal/Topology.lean | 126 | 129 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.PartrecCode
import Mathlib.Data.Set.Subsingleton
/-!
# Computability theory and the halting problem
A universal partial recursive funct... | theorem of_part : ∀ {n f}, _root_.Partrec f → @Partrec' n f :=
@(suffices ∀ f, Nat.Partrec f → @Partrec' 1 fun v => f v.head from fun {n f} hf => by
let g := fun n₁ =>
(Part.ofOption (decode (α := List.Vector ℕ n) n₁)).bind (fun a => Part.map encode (f a))
exact
(comp₁ g (this g hf) (prim ... | Mathlib/Computability/Halting.lean | 375 | 395 |
/-
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 | 1,043 | 1,043 | |
/-
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 | 1,685 | 1,691 | |
/-
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 inner_sub_right (x y z : E) : ⟪x, y - z⟫ = ⟪x, y⟫ - ⟪x, z⟫ := by
simp [sub_eq_add_neg, inner_add_right]
| Mathlib/Analysis/InnerProductSpace/Basic.lean | 221 | 222 |
/-
Copyright (c) 2023 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Mario Carneiro
-/
import Mathlib.Tactic.NormNum.Basic
import Mathlib.Tactic.NormNum.Ineq
/-!
# `norm_num` extension for integer div/mod and divides
This file adds support f... |
lemma isNat_neg_of_isNegNat {a : ℤ} {b : ℕ} (h : IsInt a (.negOfNat b)) : IsNat (-a) b :=
| Mathlib/Tactic/NormNum/DivMod.lean | 37 | 38 |
/-
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... | simp only [sdiff_eq, inf_comm _ aᶜ, inf_liminf]
theorem sdiff_limsup [NeBot f] (a : α) : a \ limsup u f = liminf (fun b => a \ u b) f := by
| Mathlib/Order/LiminfLimsup.lean | 678 | 680 |
/-
Copyright (c) 2021 Alena Gusakov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alena Gusakov, Jeremy Tan
-/
import Mathlib.Combinatorics.Enumerative.DoubleCounting
import Mathlib.Combinatorics.SimpleGraph.AdjMatrix
/-!
# Strongly regular graphs
## Main definitio... | rw [Nat.sub_add_cancel, ← Set.toFinset_card]
· simp [commonNeighbors, ← neighborFinset_def, Finset.card_union_add_card_inter,
h.regular.degree_eq, two_mul]
· apply le_trans (card_commonNeighbors_le_degree_left _ _ _)
simp [h.regular.degree_eq, two_mul]
/-- Assuming `G` is strongly regular, `2*(k + 1) -... | Mathlib/Combinatorics/SimpleGraph/StronglyRegular.lean | 84 | 95 |
/-
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.Order.Group.Nat
import Mathlib.Data.List.Rotate
import Mathlib.GroupTheory.Perm.Support
/-!
# Permutations from a list
A list `l : List α` ... | rw [← hx, IH] at h
simp [getElem_mem] at h
| Mathlib/GroupTheory/Perm/List.lean | 171 | 172 |
/-
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... | of `1 / d n`. Then `c n • d n` belongs to a fixed annulus. By compactness, one can extract
a subsequence converging to a limit `l`. Then `l` is nonzero, and by definition it belongs to
the tangent cone. -/
obtain ⟨u, -, u_pos, u_lim⟩ :
∃ u, StrictAnti u ∧ (∀ (n : ℕ), 0 < u n) ∧ Tendsto u atTop (𝓝 (0 : ℝ)... | Mathlib/Analysis/Calculus/TangentCone.lean | 321 | 330 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth, Mitchell Lee
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Algebra.Ri... | simpa only [Int.negSucc_eq, neg_add] using neg_add_one n hn hnm
@[simp]
| Mathlib/RingTheory/Polynomial/Chebyshev.lean | 85 | 87 |
/-
Copyright (c) 2020 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Algebra.Group.Conj
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.Group.Subgroup.Ker
/-!
# Basic results on subgroups
We prove basic results... | Mathlib/Algebra/Group/Subgroup/Basic.lean | 3,057 | 3,059 | |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Eric Wieser
-/
import Mathlib.Algebra.DirectSum.Internal
import Mathlib.Algebra.GradedMonoid
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Algebra.MvPolyn... |
theorem add (hφ : IsHomogeneous φ n) (hψ : IsHomogeneous ψ n) : IsHomogeneous (φ + ψ) n :=
(homogeneousSubmodule σ R n).add_mem hφ hψ
| Mathlib/RingTheory/MvPolynomial/Homogeneous.lean | 163 | 166 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Simon Hudon, Mario Carneiro
-/
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Init
import Mathlib.Data.Int.Init
import Mathlib.... | section DivisionCommMonoid
| Mathlib/Algebra/Group/Basic.lean | 533 | 533 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Jeremy Avigad, Simon Hudon
-/
import Mathlib.Algebra.Notation.Defs
import Mathlib.Data.Set.Subsingleton
import Mathlib.Logic.Equiv.Defs
/-!
# Partial values of a type
... | have h := h b
simp only [and_imp, exists_prop, bind_eq_bind, mem_bind_iff, exists_imp] at h
apply h _ h'
· intro b h'
simp only [exists_prop, bind_eq_bind, mem_bind_iff] at h'
rcases h' with ⟨a, h₀, h₁⟩
apply h _ h₀ _ h₁
| Mathlib/Data/Part.lean | 529 | 536 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Ordmap.Invariants
/-!
# Verification of `Ordnode`
This file uses the invariants defined in `Mathlib.Data.Ordmap.Invariants` to construct `Ordset... | apply Raised.add_right
apply Raised.add_left
exact t_r_size
right; exists t_r.size; exact And.intro t_r_size h.bal.1
| Mathlib/Data/Ordmap/Ordset.lean | 599 | 603 |
/-
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... |
theorem mem_range_of_degree_eq_one (hx : (minpoly A x).degree = 1) :
x ∈ (algebraMap A B).range := by
have h : IsIntegral A x := by
by_contra h
rw [eq_zero h, degree_zero, ← WithBot.coe_one] at hx
| Mathlib/FieldTheory/Minpoly/Basic.lean | 115 | 120 |
/-
Copyright (c) 2024 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca, Sanyam Gupta, Omar Haddad, David Lowry-Duda,
Lorenzo Luccioli, Pietro Monticone, Alexis Saurin, Florent Schaffhauser
-/
import Mathlib.NumberTheory.FLT.Basic
import... | congruences modulo `9`.
* To prove case 2, we consider the generalized equation `a ^ 3 + b ^ 3 = u * c ^ 3`, where `a`, `b`,
and `c` are in the cyclotomic ring `ℤ[ζ₃]` (where `ζ₃` is a primitive cube root of unity) and `u`
| Mathlib/NumberTheory/FLT/Three.lean | 25 | 27 |
/-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
import Mathlib.Topology.Category.Profinite.Nobeling.Basic
import Mathlib.Topology.Category.Profinite.Nobeling.Induction
import Mathlib.Topology.Category.Profinite... | Mathlib/Topology/Category/Profinite/Nobeling.lean | 905 | 910 | |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Finite.Defs
import Mathlib.Data.Finset.BooleanAlgebra
import Mathlib.Data.Finset.Image
import Mathlib.Data.Fintype.Defs
import Mathlib.Data.Fintyp... | Mathlib/Data/Fintype/Basic.lean | 661 | 662 | |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.WSeq.Basic
import Mathlib.Data.WSeq.Defs
import Mathlib.Data.WSeq.Productive
import Mathlib.Data.WSeq.Relation
deprecated_module (since :=... | Mathlib/Data/Seq/WSeq.lean | 635 | 636 | |
/-
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 | 1,371 | 1,372 | |
/-
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
-/
import Mathlib.Analysis.Complex.Asymptotics
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Data.Complex.Trig... |
@[simp]
theorem comap_exp_nhds_zero : comap exp (𝓝 0) = comap re atBot :=
| Mathlib/Analysis/SpecialFunctions/Exp.lean | 452 | 454 |
/-
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.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.Order.Sub.WithTop
import Mathlib.Data.NNReal.Defs
import Mathlib.Order.Interval.Set.... | simp [coe_toNNReal (hf x)]
@[simp]
| Mathlib/Data/ENNReal/Basic.lean | 224 | 226 |
/-
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.Algebra.Quaternion
import Mathlib.Tactic.Ring
/-!
# Basis on a quaternion-like algebra
## Main definitions
* `QuaternionAlgebra.Basis A c₁ c₂ c₃`: a basis... | simp only [lift, add_re, map_add, add_imI, add_smul, add_imJ, add_imK]
| Mathlib/Algebra/QuaternionBasis.lean | 117 | 117 |
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calculus.FDeriv.Comp
/-!
# Additive operations on derivative... |
section CompAdd
| Mathlib/Analysis/Calculus/FDeriv/Add.lean | 703 | 704 |
/-
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
/... | map_smul_eq_mul]
intro a ha b hb
rw [mul_comm, mul_comm r₁]
| Mathlib/Analysis/Seminorm.lean | 834 | 836 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kim Morrison
-/
import Mathlib.Topology.Sheaves.PresheafOfFunctions
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing
/-!
# Sheaf conditions for presheaves of ... | -- In the informal comments below, I'll just write `U` to represent the union.
-- Our first goal is to define a function "lifted" to all of `U`.
-- We do this one point at a time. Using the axiom of choice, we can pick for each
-- `x : ↑(iSup U)` an index `i : ι` such that `x` lies in `U i`
choose ind... | Mathlib/Topology/Sheaves/SheafOfFunctions.lean | 50 | 76 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Logic.Nontrivial.Basic
import Mathlib.Order.TypeTags
import Mathlib.Data.Option.NAry
import Mathlib.Tactic.Contrapose
import Mathlib.Tactic.Lift
import... | Mathlib/Order/WithBot.lean | 1,158 | 1,159 | |
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Probability.IdentDistrib
import Mathlib.Probability.Independence.Integrable
import Mathlib.MeasureTheory.Integral.DominatedConvergence
import Mat... | exact ⟨hx.1, hx.2.trans (Nat.cast_le.2 hmn)⟩
apply le_of_tendsto_of_tendsto A tendsto_const_nhds
filter_upwards [Ici_mem_atTop K] with N hN
exact sum_prob_mem_Ioc_le hint hnonneg hN
theorem sum_variance_truncation_le {X : Ω → ℝ} (hint : Integrable X) (hnonneg : 0 ≤ X) (K : ℕ) :
∑ j ∈ range K, ((j : ℝ) ^ ... | Mathlib/Probability/StrongLaw.lean | 304 | 326 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kenny Lau, Kim Morrison
-/
import Mathlib.Data.List.Chain
/-!
# Ranges of naturals as lists
This file shows basic results about `List.iota`, `List.range`, `List.range... | Mathlib/Data/List/Range.lean | 224 | 228 | |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Finite.Defs
import Mathlib.Data.Finset.BooleanAlgebra
import Mathlib.Data.Finset.Image
import Mathlib.Data.Fintype.Defs
import Mathlib.Data.Fintyp... | simp [univ_image_def]
| Mathlib/Data/Fintype/Basic.lean | 96 | 96 |
/-
Copyright (c) 2023 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.LineDeriv.Measurable
import Mathlib.Analysis.Normed.Module.FiniteDimension
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar... | /-- *Rademacher's theorem*: a function between finite-dimensional real vector spaces which is
Lipschitz on a set is differentiable almost everywere in this set. -/
theorem ae_differentiableWithinAt {f : E → F} (hf : LipschitzOnWith C f s)
(hs : MeasurableSet s) :
| Mathlib/Analysis/Calculus/Rademacher.lean | 378 | 381 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Sean Leather
-/
import Batteries.Data.List.Perm
import Mathlib.Data.List.Pairwise
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Lookmap
import Mathlib.Data.Si... | lookmap fun s => if a = s.1 then some ⟨a, b⟩ else none
| Mathlib/Data/List/Sigma.lean | 326 | 327 |
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