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) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Set.Lattice
/-! # Semiquotients
A data type for semiquotients, which are classically equivalent to
nonempty sets, but are useful for programming... | @Set.mem_univ α
| Mathlib/Data/Semiquot.lean | 215 | 216 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Aaron Anderson, Yakov Pechersky
-/
import Mathlib.Data.Fintype.Card
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Algebra.Group.End
import Mathlib.Data.Finset.N... | theorem disjoint_prod_right (l : List (Perm α)) (h : ∀ g ∈ l, Disjoint f g) :
Disjoint f l.prod := by
| Mathlib/GroupTheory/Perm/Support.lean | 114 | 115 |
/-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Group.Equiv.Defs
import Mathlib.Control.Applicative
import Mathlib.Control.Traversable.Basic
import Mathlib.Logic.Equi... | Mathlib/Algebra/Free.lean | 565 | 565 | |
/-
Copyright (c) 2019 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
import Mathlib.Algebra.ContinuedFractions.Basic
import Mathlib.Algebra.GroupWithZero.Basic
/-!
# Basic Translation Lemmas Between Functions Defined for Continued... | Here we give some basic translations that hold by definition for the computational methods of a
continued fraction.
-/
| Mathlib/Algebra/ContinuedFractions/Translations.lean | 71 | 74 |
/-
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.IsPrimePow
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Ring.Int
import Mathlib.Algebra.Ring.CharZero
im... | @[simp]
theorem image_fst_divisorsAntidiagonal : (divisorsAntidiagonal n).image Prod.fst = divisors n := by
ext
simp [Dvd.dvd, @eq_comm _ n (_ * _)]
| Mathlib/NumberTheory/Divisors.lean | 342 | 346 |
/-
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.CharP.Basic
import Mathlib.Algebra.Module.End
import Mathlib.Algebra.Ring.Prod
import Mathlib.Data.Fintype.Units
import Mathlib.GroupTheory.GroupAc... |
@[simp]
theorem lift_coe (x : ℤ) : lift n f (x : ZMod n) = f.val x :=
AddMonoidHom.liftOfRightInverse_comp_apply _ _ (fun _ => intCast_zmod_cast _) _ _
| Mathlib/Data/ZMod/Basic.lean | 1,123 | 1,126 |
/-
Copyright (c) 2017 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Tim Baumann, Stephen Morgan, Kim Morrison, Floris van Doorn
-/
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.ObjectProperty.FullSubcategory
import Mat... | simp only [← Category.assoc, cancel_mono]
@[simp]
| Mathlib/CategoryTheory/Equivalence.lean | 405 | 407 |
/-
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.CharP.Basic
import Mathlib.Algebra.Module.End
import Mathlib.Algebra.Ring.Prod
import Mathlib.Data.Fintype.Units
import Mathlib.GroupTheory.GroupAc... | theorem natCast_rightInverse [NeZero n] : Function.RightInverse val ((↑) : ℕ → ZMod n) :=
natCast_zmod_val
theorem natCast_zmod_surjective [NeZero n] : Function.Surjective ((↑) : ℕ → ZMod n) :=
| Mathlib/Data/ZMod/Basic.lean | 192 | 195 |
/-
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.RCLike.Basic
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Complex.Module
import Mathlib.Data.Complex.Order
import Mathli... | theorem restrictScalars_one_smulRight' (x : E) :
ContinuousLinearMap.restrictScalars ℝ ((1 : ℂ →L[ℂ] ℂ).smulRight x : ℂ →L[ℂ] E) =
| Mathlib/Analysis/Complex/Basic.lean | 184 | 185 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Devon Tuma
-/
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.RingTheory.Coprime.Basic
import Mathlib.Tactic.... | ext; simp [mul_pow, mul_assoc]
/-- Multiplication and `scaleRoots` commute up to a power of `r`. The factor disappears if we
assume that the product of the leading coeffs does not vanish. See `Polynomial.mul_scaleRoots'`. -/
lemma mul_scaleRoots (p q : R[X]) (r : R) :
r ^ (natDegree p + natDegree q - natDegree (... | Mathlib/RingTheory/Polynomial/ScaleRoots.lean | 170 | 178 |
/-
Copyright (c) 2021 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Order.Module.Algebra
import Mathlib.Algebra.Ring.Subring.Units
import Mathlib.LinearAlgebra.LinearIndepende... | · rcases h with ((hx0 | hy0 | ⟨r₁, r₂, hr₁, _, h⟩) | (hx0 | hy0 | ⟨r₁, r₂, hr₁, _, h⟩))
· exact False.elim (hx hx0)
· exact False.elim (hy hy0)
· refine ⟨![r₁, -r₂], ?_⟩
rw [Fin.sum_univ_two, Fin.exists_fin_two]
| Mathlib/LinearAlgebra/Ray.lean | 513 | 517 |
/-
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 | 167 | 167 | |
/-
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... | Nat.card_eq_fintype_card (α := K ≃ₐ[k] K) ▸ even_nat_card_aut_of_not_isUnramified hw
lemma even_finrank_of_not_isUnramified [IsGalois k K]
(hw : ¬ IsUnramified k w) : Even (finrank k K) := by
by_cases FiniteDimensional k K
· exact IsGalois.card_aut_eq_finrank k K ▸ even_card_aut_of_not_isUnramified hw
· ex... | Mathlib/NumberTheory/NumberField/Embeddings.lean | 958 | 977 |
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kenny Lau
-/
import Mathlib.Algebra.Order.Antidiag.Finsupp
import Mathlib.Data.Finsupp.Weight
import Mathlib.Tactic.Linarith
import Mathlib.LinearAlgebra.Pi
import Mat... | ((φ ^ n : MvPolynomial σ R) : MvPowerSeries σ R) = (φ : MvPowerSeries σ R) ^ n :=
| Mathlib/RingTheory/MvPowerSeries/Basic.lean | 867 | 867 |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.LinearAlgebra.Finsupp.Span
/-!
# Lie submodules of a Lie algebra
In this file we define Lie submodules, we construct ... | theorem mem_iInf {ι} (p : ι → LieSubmodule R L M) {x} : (x ∈ ⨅ i, p i) ↔ ∀ i, x ∈ p i := by
rw [← SetLike.mem_coe, iInf_coe, Set.mem_iInter]; rfl
| Mathlib/Algebra/Lie/Submodule.lean | 375 | 376 |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Complex.AbsMax
import Mathlib.Analysis.Asymptotics.SuperpolynomialDecay
/-!
# Phragmen-Lindelöf principle
In this file we prove several ve... | have : ∀ {c₁ c₂ B₁ B₂ : ℝ}, c₁ ≤ c₂ → 0 ≤ B₂ → B₁ ≤ B₂ →
(fun z : ℂ => expR (B₁ * ‖z‖ ^ c₁)) =O[cobounded ℂ ⊓ l]
fun z => expR (B₂ * ‖z‖ ^ c₂) := fun hc hB₀ hB ↦ .of_norm_eventuallyLE <| by
filter_upwards [(eventually_cobounded_le_norm 1).filter_mono inf_le_left] with z hz
simp only [Real.norm_e... | Mathlib/Analysis/Complex/PhragmenLindelof.lean | 80 | 94 |
/-
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.Data.Nat.Lattice
import Mathlib.Data.NNReal.Basic
import Mathlib.Topology.Metrizable.Basic
/-!
# Metrizable uniform spaces
In this file we prove th... | exact ⟨t, htc, hts⟩
open TopologicalSpace in
instance (priority := 100) DiscreteTopology.metrizableSpace
{α} [TopologicalSpace α] [DiscreteTopology α] :
MetrizableSpace α := by
obtain rfl := DiscreteTopology.eq_bot (α := α)
exact @UniformSpace.metrizableSpace α ⊥ (isCountablyGenerated_principal _) _
| Mathlib/Topology/Metrizable/Uniformity.lean | 282 | 295 |
/-
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, Damiano Testa,
Yuyang Zhao
-/
import Mathlib.Algebra.Order.Monoid.Unbundled.Defs
import Mathlib.Data.Ordering.Basic
imp... | @[to_additive (attr := simp) lt_add_iff_pos_left]
theorem lt_mul_iff_one_lt_left' [MulRightStrictMono α]
[MulRightReflectLT α] (a : α) {b : α} : a < b * a ↔ 1 < b :=
| Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean | 541 | 543 |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.TwoDim
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
/-!
# Oriented angles.
This file defines orie... | Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | 1,039 | 1,042 | |
/-
Copyright (c) 2018 Andreas Swerdlow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andreas Swerdlow, Kexing Ying
-/
import Mathlib.Algebra.Algebra.Bilinear
import Mathlib.LinearAlgebra.Basis.Defs
import Mathlib.LinearAlgebra.BilinearForm.Basic
import Mathlib.Linear... | Mathlib/LinearAlgebra/BilinearForm/Hom.lean | 393 | 397 | |
/-
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.Group.Units.Equiv
import Mathlib.Algebra.Order.Group.Unbundled.Basic
import Mathlib.Order.Hom.... | Mathlib/Algebra/Order/Group/OrderIso.lean | 137 | 139 | |
/-
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.Minimal
import Mathlib.Order.Zorn
import Mathlib.Topology.ContinuousOn
/-!
# Irreducibility in topological space... | exact ⟨x, h₁ hx'.1, hx'.2⟩
theorem IsPreirreducible.open_subset {U : Set X} (ht : IsPreirreducible t) (hU : IsOpen U)
(hU' : U ⊆ t) : IsPreirreducible U :=
U.eq_empty_or_nonempty.elim (fun h => h.symm ▸ isPreirreducible_empty) fun h =>
(ht.subset_irreducible h hU (fun _ => id) hU').2
| Mathlib/Topology/Irreducible.lean | 295 | 300 |
/-
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.Algebra.Order.Group.Unbundled.Int
import Mathlib.Algebra.Ring.Nat
import Mathlib.Data.Int.GCD
/-!
# Congruences modulo a natural number
This file def... | simp_all [dvd_iff_mod_eq_zero]
theorem odd_mul_odd {n m : ℕ} : n % 2 = 1 → m % 2 = 1 → n * m % 2 = 1 := by
simpa [Nat.ModEq] using @ModEq.mul 2 n 1 m 1
theorem odd_mul_odd_div_two {m n : ℕ} (hm1 : m % 2 = 1) (hn1 : n % 2 = 1) :
m * n / 2 = m * (n / 2) + m / 2 :=
| Mathlib/Data/Nat/ModEq.lean | 412 | 418 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
import Mathlib.MeasureTheory.Measure.Count
import Mathlib.Order.Filter.ENNReal
import Mathlib.Probability.Unif... | eq_of_forall_le_iff fun r => by
simp [essSup, limsup, limsSup, eventually_map, ENNReal.forall_ennreal]; rfl
lemma essSup_restrict_eq_of_support_subset {s : Set α} {f : α → ℝ≥0∞} (hsf : f.support ⊆ s) :
essSup f (μ.restrict s) = essSup f μ := by
| Mathlib/MeasureTheory/Function/EssSup.lean | 300 | 304 |
/-
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.ExactSequence
import Mathlib.Algebra.Homology.ShortComplex.Limits
import Mathlib.CategoryTheory.Abelian.Refinements
/-!
# The snake lemma
The ... | lemma L₀'_exact : S.L₀'.Exact := by
rw [ShortComplex.exact_iff_exact_up_to_refinements]
intro A x₂ hx₂
dsimp [L₀'] at x₂ hx₂
obtain ⟨A', π, hπ, x₁, fac⟩ := S.L₁_exact.exact_up_to_refinements (x₂ ≫ pullback.fst _ _)
(by rw [assoc, pullback.condition, reassoc_of% hx₂, zero_comp])
exact ⟨A', π, hπ, x₁, pullb... | Mathlib/Algebra/Homology/ShortComplex/SnakeLemma.lean | 257 | 263 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Johan Commelin, Andrew Yang, Joël Riou
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
import Mathlib.CategoryTheory.Monoid... | apply NatIso.naturality_1
variable (A)
/-- Shifting by zero is the identity functor. -/
abbrev shiftZero : X⟦(0 : A)⟧ ≅ X :=
(shiftFunctorZero C A).app _
| Mathlib/CategoryTheory/Shift/Basic.lean | 360 | 366 |
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Function.ConvergenceInMeasure
import Mathlib.MeasureTheory.Function.L1Space.Integrable
/-!
# Uniform integrability
This file contains the def... | refine (lt_of_le_of_lt (le_trans
(hM <| ℐ <| 2 * max M 1 * δ⁻¹ ^ (1 / p.toReal)) (le_max_left (M : ℝ≥0∞) 1))
(lt_of_lt_of_le ?_ this)).ne rfl
rw [← ENNReal.coe_one, ← ENNReal.coe_max, ← ENNReal.coe_mul, ENNReal.coe_lt_coe]
exact lt_two_mul_self (lt_max_of_lt_right one_pos)
exact ⟨C, fun i =>... | Mathlib/MeasureTheory/Function/UniformIntegrable.lean | 855 | 888 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Algebra.Group.TypeTags.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Finset.Piecewise
import Mathlib.Or... | (hf : ContinuousAt f x) : ContinuousAt (fun a ↦ Fin.tail (f a)) x :=
hf.tendsto.finTail
| Mathlib/Topology/Constructions.lean | 845 | 847 |
/-
Copyright (c) 2022 Floris van Doorn, Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Heather Macbeth
-/
import Mathlib.Geometry.Manifold.ContMDiff.Atlas
import Mathlib.Geometry.Manifold.VectorBundle.FiberwiseLinear
import Mathlib.To... | refine (contMDiffWithinAt_prod_iff _).trans (and_congr ?_ Iff.rfl)
have h1 : (fun x => (f x).proj) ⁻¹' (trivializationAt F E (f x₀).proj).baseSet ∈ 𝓝[s] x₀ :=
((FiberBundle.continuous_proj F E).continuousWithinAt.comp hf (mapsTo_image f s))
((Trivialization.open_baseSet _).mem_nhds (mem_baseSet_trivializ... | Mathlib/Geometry/Manifold/VectorBundle/Basic.lean | 178 | 196 |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Yury Kudryashov
-/
import Mathlib.Data.Finset.Fin
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Order.Interval.Set.Fin
/-!
# Finite intervals in `Fin n`
This fi... | theorem map_valEmbedding_Iic : (Iic a).map Fin.valEmbedding = Iic (a : ℕ) := by
rw [← attachFin_Iic, map_valEmbedding_attachFin]
| Mathlib/Order/Interval/Finset/Fin.lean | 205 | 206 |
/-
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... | iSup₂_disjoint_iff
| Mathlib/Data/Set/Lattice.lean | 1,211 | 1,212 |
/-
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.Data.Complex.Trigonometric
import Mathlib.Data.Complex.Module
import Mathlib.RingTheory.Polynomial.Chebyshev
/-!
# Multiple angle formulas in terms of... | @[simp, norm_cast]
theorem complex_ofReal_eval_U : ∀ (x : ℝ) n, (((U ℝ n).eval x : ℝ) : ℂ) = (U ℂ n).eval (x : ℂ) :=
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Chebyshev.lean | 29 | 30 |
/-
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 | 2,789 | 2,792 | |
/-
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.Fin
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Finset.Sort
/-!
# Compositions
A compositio... | exact ones_length n
· contrapose
intro H length_n
apply lt_irrefl n
| Mathlib/Combinatorics/Enumerative/Composition.lean | 499 | 502 |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Eric Wieser
-/
import Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Order.Module.Pointwise
impor... | Mathlib/Data/Real/Pointwise.lean | 140 | 141 | |
/-
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.Nat.ModEq
import Mathlib.Data.Nat.Prime.Basic
import Mathlib.NumberTheory.Zsqrtd.Basic
/-!
# Pell's equation and Matiyasevic's theorem
This file... | apply Int.lt_of_ofNat_lt_ofNat
rw [lem2 (n + 1) (Nat.lt_succ_self _) j2n,
show 2 * n - (n + 1) = n - 1 by
rw [two_mul, tsub_add_eq_tsub_tsub, add_tsub_cancel_right]]
refine lt_sub_left_of_add_lt (Int.ofNat_lt_ofNat_of_lt ?_)
rcases lt_or_eq_of_le <| Nat.... | Mathlib/NumberTheory/PellMatiyasevic.lean | 604 | 617 |
/-
Copyright (c) 2021 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno
-/
import Mathlib.CategoryTheory.NatIso
/-!
# Bicategories
In this file we define typeclass for bicategories.
A bicategory `B` consists of
* objects `a : B`,
* 1-morphisms ... | /-- The right whiskering of a 2-isomorphism is a 2-isomorphism. -/
@[simps!]
| Mathlib/CategoryTheory/Bicategory/Basic.lean | 206 | 207 |
/-
Copyright (c) 2022 Antoine Labelle, Rémi Bottinelli. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Labelle, Rémi Bottinelli
-/
import Mathlib.Combinatorics.Quiver.Basic
import Mathlib.Combinatorics.Quiver.Path
/-!
# Rewriting arrows and paths along vertex... | Mathlib/Combinatorics/Quiver/Cast.lean | 142 | 145 | |
/-
Copyright (c) 2020 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Ira Fesefeldt
-/
import Mathlib.Control.Monad.Basic
import Mathlib.Dynamics.FixedPoints.Basic
import Mathlib.Order.CompleteLattice.Basic
import Mathlib.Order.Iterate
import M... | /-- When proving that a chain of applications is below a bound `z`, it suffices to consider the
functions and values being selected from the same index in the chains.
This lemma is more specific than necessary, i.e. `c₀` only needs to be a
chain of monotone functions, but it is only used with continuous functions. -/
... | Mathlib/Order/OmegaCompletePartialOrder.lean | 675 | 700 |
/-
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.Lattice.Prod
import Mathlib.Data.Finite.Prod
import Mathlib.Data.Set.Lattice.Image
/-!
# N-ary images of finsets
This file defines `Finset.im... | theorem image₂_singleton : image₂ f {a} {b} = {f a b} := by simp
| Mathlib/Data/Finset/NAry.lean | 145 | 146 |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Analysis.Convex.Function
/-!
# Quasiconvex and quasiconcave functions
This file defines quasiconvexity, quasiconcavity and quasilinearity of functions, w... | Mathlib/Analysis/Convex/Quasiconvex.lean | 238 | 242 | |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
-/
import Mathlib.MeasureTheory.Integral.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Lebesgue.Countable
import Mathlib.MeasureTheory.Integral.Le... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 744 | 752 | |
/-
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... | Mathlib/Data/Ordmap/Ordset.lean | 1,779 | 1,782 | |
/-
Copyright (c) 2021 Martin Zinkevich. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Martin Zinkevich, Vincent Beffara
-/
import Mathlib.MeasureTheory.Integral.Bochner.Set
import Mathlib.Probability.Independence.Basic
/-!
# Integration in Probability Theory
Integra... | let pos : ℝ → ℝ := fun x => max x 0
let neg : ℝ → ℝ := fun x => max (-x) 0
have posm : Measurable pos := measurable_id'.max measurable_const
have negm : Measurable neg := measurable_id'.neg.max measurable_const
let Xp := pos ∘ X
-- `X⁺` would look better but it makes `simp_rw` below fail
let Xm := neg ∘ X... | Mathlib/Probability/Integration.lean | 225 | 265 |
/-
Copyright (c) 2021 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Data.Fintype.Shrink
import Mathlib.Data.Fintype.Sum
import Mathlib.Data.Finite.Prod
import Mathlib.Algebra.BigOperators.Grou... | /-- Consider a line in `ι → η → α` as a `η`-dimensional subspace in `ι × η → α`. -/
def toSubspace (l : Line (η → α) ι) : Subspace η α (ι × η) where
idxFun ie := (l.idxFun ie.1).elim (.inr ie.2) (fun f ↦ .inl <| f ie.2)
proper e := let ⟨i, hi⟩ := l.proper; ⟨(i, e), by simp [hi]⟩
@[simp] lemma toSubspace_apply (l :... | Mathlib/Combinatorics/HalesJewett.lean | 218 | 322 |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.AlgebraicGeometry.Morphisms.Constructors
import Mathlib.AlgebraicGeometry.Morphisms.QuasiCompact
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.Equaliz... | let g : pullback U.1.ι V.1.ι ⟶ X := pullback.fst _ _ ≫ U.1.ι
have : IsOpenImmersion g := inferInstance
have e := this.base_open.isEmbedding.toHomeomorph
rw [IsOpenImmersion.range_pullback_to_base_of_left] at e
erw [Subtype.range_coe, Subtype.range_coe] at e
rw [isCompact_iff_compactSpace]
ex... | Mathlib/AlgebraicGeometry/Morphisms/QuasiSeparated.lean | 87 | 115 |
/-
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... | have hpq := Real.HolderConjugate.conjExponent hp1_lt
by_cases h0 : (∫⁻ a, (f + g) a ^ p ∂μ) = 0
· rw [h0, @ENNReal.zero_rpow_of_pos (1 / p) (by simp [lt_of_lt_of_le zero_lt_one hp1])]
exact zero_le _
have htop : (∫⁻ a, (f + g) a ^ p ∂μ) ≠ ⊤ := by
rw [← Ne] at hf_top hg_top
rw [← lt_top_iff_ne_top] a... | Mathlib/MeasureTheory/Integral/MeanInequalities.lean | 437 | 462 |
/-
Copyright (c) 2022 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed
import Mathlib.RingTheory.PowerBasis
/-!
# A predicate on adjoining root... |
section lift
| Mathlib/RingTheory/IsAdjoinRoot.lean | 174 | 175 |
/-
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 | 1,635 | 1,636 | |
/-
Copyright (c) 2024 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.NumberTheory.ModularForms.EisensteinSeries.UniformConvergence
import Mathlib.Analysis.Complex.UpperHalfPlane.Manifold
import Mathlib.Analysis.Complex.... | lemma div_linear_zpow_differentiableOn (k : ℤ) (a : Fin 2 → ℤ) :
DifferentiableOn ℂ (fun z : ℂ => (a 0 * z + a 1) ^ (-k)) {z : ℂ | 0 < z.im} := by
rcases ne_or_eq a 0 with ha | rfl
· apply DifferentiableOn.zpow
· fun_prop
· left
exact fun z hz ↦ linear_ne_zero _ ⟨z, hz⟩
((comp_ne_zero_iff ... | Mathlib/NumberTheory/ModularForms/EisensteinSeries/MDifferentiable.lean | 28 | 42 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Order.SuccPred
import Mathlib.Data.Sum.Order
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Tactic.PPWithUniv
/-!
# ... |
instance wellFoundedLT : WellFoundedLT Ordinal :=
⟨lt_wf⟩
instance : ConditionallyCompleteLinearOrderBot Ordinal :=
| Mathlib/SetTheory/Ordinal/Basic.lean | 547 | 551 |
/-
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.Computability.Primrec
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Linarith
/-!
# Ackermann function
In this file, we def... | ack_add_one_sq_lt_ack_add_three _ _⟩
rw [max_ack_left]
exact max_le_max (ha n).le (hb n).le
| comp hf hg IHf IHg =>
obtain ⟨a, ha⟩ := IHf; obtain ⟨b, hb⟩ := IHg
exact
⟨max a b + 2, fun n =>
(ha _).trans <| (ack_strictMono_right a <| hb n).trans <| ack_ack_lt_ack_max_add_two a... | Mathlib/Computability/Ackermann.lean | 295 | 302 |
/-
Copyright (c) 2024 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Lie.Weights.Killing
import Mathlib.LinearAlgebra.RootSystem.Basic
import Mathlib.LinearAlgebra.RootSystem.Reduced
import Mathlib.LinearAlgebra.RootSy... | (by convert span_weight_isNonZero_eq_top K L H; ext; simp)
(fun α β ↦
⟨chainBotCoeff β.1 α.1 - chainTopCoeff β.1 α.1, by simp [apply_coroot_eq_cast β.1 α.1]⟩)
| Mathlib/Algebra/Lie/Weights/RootSystem.lean | 394 | 396 |
/-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.Polynomial.Reverse
/-!
# "Mirror" of a univariate polynomial
In this file we define `Po... | (p * p.mirror).natTrailingDegree = 2 * p.natTrailingDegree := by
by_cases hp : p = 0
| Mathlib/Algebra/Polynomial/Mirror.lean | 157 | 158 |
/-
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.Algebra.GroupWithZero.Indicator
import Mathlib.Topology.Piecewise
import Mathlib.Topology.Instances.ENNReal.Lemmas
/-!
# Semicontinuous maps
A ... | calc f x ≤ liminf f (𝓝 x) := hf x
_ ≤ liminf f (map Prod.fst F) := liminf_le_liminf_of_le h'.1
_ = liminf (f ∘ Prod.fst) F := (Filter.liminf_comp _ _ _).symm
_ ≤ liminf Prod.snd F := liminf_le_liminf <| by
simpa using (eventually_principal.2 fun (_ : α × γ) ↦ id).filter_mono h
_ = y := h'... | Mathlib/Topology/Semicontinuous.lean | 320 | 328 |
/-
Copyright (c) 2020 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.FieldTheory.RatFunc.AsPolynomial
import Mathlib.NumberTheory.ArithmeticFunction
import Mathlib.RingTheory.Ro... | theorem cyclotomic.eval_apply {R S : Type*} (q : R) (n : ℕ) [Ring R] [Ring S] (f : R →+* S) :
eval (f q) (cyclotomic n S) = f (eval q (cyclotomic n R)) := by
rw [← map_cyclotomic n f, eval_map, eval₂_at_apply]
@[simp] theorem cyclotomic.eval_apply_ofReal (q : ℝ) (n : ℕ) :
| Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean | 266 | 270 |
/-
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... | have : d ≫ (-g) = 0 := calc
d ≫ (-g) = d ≫ biprod.lift f (-g) ≫ biprod.snd := by rw [biprod.lift_snd]
_ = biprod.lift e (0 : R ⟶ Z) ≫ biprod.snd := by rw [← Category.assoc, hd]
_ = 0 := biprod.lift_snd _ _
| Mathlib/CategoryTheory/Abelian/Basic.lean | 724 | 727 |
/-
Copyright (c) 2024 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Topology.Bornology.Constructions
/-!
# Bornology of order-bounded sets
This file relates the notion of bornology-boundedness (sets that lie in a bornolog... | protected isBounded_iff_bddBelow_bddAbove (s : Set α) : IsBounded s ↔ BddBelow s ∧ BddAbove s
lemma isOrderBornology_iff_eq_orderBornology [Lattice α] [Nonempty α] :
IsOrderBornology α ↔ ‹Bornology α› = orderBornology := by
refine ⟨fun h ↦ ?_, fun h ↦ ⟨fun s ↦ by rw [h, orderBornology_isBounded]⟩⟩
| Mathlib/Topology/Order/Bornology.lean | 47 | 51 |
/-
Copyright (c) 2020 Zhangir Azerbayev. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Zhangir Azerbayev
-/
import Mathlib.GroupTheory.Perm.Sign
import Mathlib.LinearAlgebra.LinearIndependent.Defs
import Mathlib.LinearAlgebra.Multilinear.Basis
/-!
# Alte... |
/-- `AlternatingMap.domDomCongr` as an equivalence.
This is declared separately because it does not work with dot notation. -/
@[simps apply symm_apply]
def domDomCongrEquiv (σ : ι ≃ ι') : M [⋀^ι]→ₗ[R] N ≃+ M [⋀^ι']→ₗ[R] N where
| Mathlib/LinearAlgebra/Alternating/Basic.lean | 678 | 683 |
/-
Copyright (c) 2019 Calle Sönne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology... |
/-- The sign of a `Real.Angle` is `0` if the angle is `0` or `π`, `1` if the angle is strictly
between `0` and `π` and `-1` is the angle is strictly between `-π` and `0`. It is defined as the
sign of the sine of the angle. -/
def sign (θ : Angle) : SignType :=
SignType.sign (sin θ)
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 698 | 704 |
/-
Copyright (c) 2020 Alexander Bentkamp, Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Sébastien Gouëzel, Eric Wieser
-/
import Mathlib.Algebra.Algebra.RestrictScalars
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Data.... | Mathlib/Data/Complex/Module.lean | 554 | 558 | |
/-
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 | 1,748 | 1,749 | |
/-
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.Algebra.BigOperators.Group.Finset.Piecewise
import Mathlib.Algebra.BigOperators.Group.Finset.Sigma
import Mathlib.Algebra.BigOperators.Option
import Ma... | · rw [filter_piFinset_of_not_mem _ _ _ h, Finset.card_empty]
| Mathlib/Data/Fintype/BigOperators.lean | 185 | 186 |
/-
Copyright (c) 2022 Alex Kontorovich and Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Kontorovich, Heather Macbeth
-/
import Mathlib.Algebra.Group.Opposite
import Mathlib.MeasureTheory.Constructions.Polish.Basic
import Mathlib.MeasureTheory.Gr... | MeasurePreserving (@QuotientGroup.mk G _ Γ) (ν.restrict 𝓕) μ :=
h𝓕.measurePreserving_quotient_mk μ
local notation "π" => @QuotientGroup.mk G _ Γ
variable [TopologicalSpace G] [IsTopologicalGroup G] [BorelSpace G] [PolishSpace G]
[T2Space (G ⧸ Γ)] [SecondCountableTopology (G ⧸ Γ)]
/-- If `μ` satisfies `Quot... | Mathlib/MeasureTheory/Measure/Haar/Quotient.lean | 77 | 101 |
/-
Copyright (c) 2021 Arthur Paulino. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Arthur Paulino, Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Coloring
/-!
# Graph partitions
This module provides an interface for dealing with partitions on simple graphs... | obtain ⟨⟨_, h⟩, _⟩ := (P.isPartition.2 v).choose_spec
exact h
| Mathlib/Combinatorics/SimpleGraph/Partition.lean | 89 | 91 |
/-
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... | rw [add_zero]
| Mathlib/SetTheory/Ordinal/Arithmetic.lean | 1,034 | 1,035 |
/-
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... | have := ENNReal.lintegral_mul_le_Lp_mul_Lq μ (Real.holderConjugate_iff.mpr ⟨h2p, h2pq⟩)
(hf.pow_const p) (hg.pow_const q)
simpa [← ENNReal.rpow_mul, hp.ne', hq.ne'] using this
/-- A version of Hölder with multiple arguments -/
theorem lintegral_prod_norm_pow_le {α ι : Type*} [MeasurableSpace α] {μ : Measure α}... | Mathlib/MeasureTheory/Integral/MeanInequalities.lean | 183 | 199 |
/-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll, Anatole Dedecker
-/
import Mathlib.Analysis.LocallyConvex.Bounded
import Mathlib.Analysis.Seminorm
import Mathlib.Data.Real.Sqrt
import Mathlib.Topology.Algebra.Equicontinuit... | theorem basisSets_smul_right (v : E) (U : Set E) (hU : U ∈ p.basisSets) :
∀ᶠ x : 𝕜 in 𝓝 0, x • v ∈ U := by
rcases p.basisSets_iff.mp hU with ⟨s, r, hr, hU⟩
rw [hU, Filter.eventually_iff]
simp_rw [(s.sup p).mem_ball_zero, map_smul_eq_mul]
| Mathlib/Analysis/LocallyConvex/WithSeminorms.lean | 132 | 136 |
/-
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, Jeremy Avigad
-/
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Data.Set.Finite.Range
import Mathlib.Data.Set.Lattice
import Mathlib.Topology.Defs.... | Mathlib/Topology/Basic.lean | 1,190 | 1,193 | |
/-
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.Analysis.Calculus.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.FDeriv.Prod
import Mathlib.Analysis.C... | Mathlib/Analysis/BoundedVariation.lean | 164 | 174 | |
/-
Copyright (c) 2020 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
import Mathlib.Analysis.NormedSpace.IndicatorFunction
import Mathlib.Data.Fintype.Order
import Mathlib.MeasureTheory.Function.AEEqFun
import Mathlib.Me... | meas_essSup_lt
lemma eLpNorm_lt_top_of_finite [Finite α] [IsFiniteMeasure μ] : eLpNorm f p μ < ∞ := by
| Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean | 1,029 | 1,031 |
/-
Copyright (c) 2019 Gabriel Ebner. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Gabriel Ebner, Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.FDeriv.Basic
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
/-!
# One-dimensional derivatives
This ... | theorem HasDerivWithinAt.congr (h : HasDerivWithinAt f f' s x) (hs : ∀ x ∈ s, f₁ x = f x)
(hx : f₁ x = f x) : HasDerivWithinAt f₁ f' s x :=
h.congr_mono hs hx (Subset.refl _)
| Mathlib/Analysis/Calculus/Deriv/Basic.lean | 536 | 538 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.QuasiIso
import Mathlib.CategoryTheory.Limits.Preserves.Finite
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels
/-!
... | (S.leftHomologyData.map F).cyclesIso
@[reassoc (attr := simp)]
lemma mapCyclesIso_hom_iCycles [S.HasLeftHomology] [F.PreservesLeftHomologyOf S] :
| Mathlib/Algebra/Homology/ShortComplex/PreservesHomology.lean | 403 | 406 |
/-
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 | 2,013 | 2,016 | |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Sign
import Mathlib.LinearAlgebra.AffineSpace.Combination
import Mathlib.LinearAlg... | points : Fin (n + 1) → P
independent : AffineIndependent k points
/-- A `Triangle k P` is a collection of three affinely independent points. -/
abbrev Triangle :=
Simplex k P 2
namespace Simplex
variable {P}
/-- Construct a 0-simplex from a point. -/
def mkOfPoint (p : P) : Simplex k P 0 :=
have : Subsingle... | Mathlib/LinearAlgebra/AffineSpace/Independent.lean | 769 | 786 |
/-
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 | 288 | 289 | |
/-
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... | rintro m hm rfl
exact lt_of_lt_of_le hm (le_hitting (hτbdd _) _)
rw [h₁, h₂, Set.union_empty]
exact MeasurableSet.iUnion fun i => MeasurableSet.iUnion fun hi =>
(f.mono hi _ (hτ.measurableSet_eq i)).inter (hitting_isStoppingTime hf hs n)
section CompleteLattice
variable [CompleteLattice ι] {u : ι → Ω ... | Mathlib/Probability/Process/HittingTime.lean | 260 | 280 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Group.Arithmetic
import Mathlib.Topology.GDelta.UniformSpace
import Mathlib.Topology.Instances.EReal.Lemmas
import Mathl... |
instance Nat.borelSpace : BorelSpace ℕ :=
⟨borel_eq_top_of_discrete.symm⟩
instance Int.borelSpace : BorelSpace ℤ :=
| Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean | 659 | 663 |
/-
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.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Ring.Nat
/-!
# Double countings
This... | ∏ a ∈ s, ∏ b ∈ t.bipartiteAbove r a, f a b = ∏ b ∈ t, ∏ a ∈ s.bipartiteBelow r b, f a b := by
simp_rw [bipartiteAbove, bipartiteBelow, prod_filter]
exact prod_comm
| Mathlib/Combinatorics/Enumerative/DoubleCounting.lean | 79 | 82 |
/-
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... | choose T _ hTc hsubT using fun n =>
subset_countable_closure_of_compact (isCompact_compactCovering α n)
refine ⟨⟨⋃ n, T n, countable_iUnion hTc, fun x => ?_⟩⟩
rcases iUnion_eq_univ_iff.1 (iUnion_compactCovering α) x with ⟨n, hn⟩
exact closure_mono (subset_iUnion _ n) (hsubT _ hn)
theorem secondCountable_of... | Mathlib/Topology/EMetricSpace/Basic.lean | 206 | 214 |
/-
Copyright (c) 2022 Kevin H. Wilson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin H. Wilson
-/
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Analysis.NormedSpace.RCLike
import Mathlib.Order.Filter.Curry
/-!
# Swapping limits and derivatives via u... |
/-- A slight variant of `hasDerivAt_of_tendstoLocallyUniformlyOn` with the assumption stated in
terms of `DifferentiableOn` rather than `HasDerivAt`. This makes a few proofs nicer in complex
analysis where holomorphicity is assumed but the derivative is not known a priori. -/
theorem hasDerivAt_of_tendsto_locally_unif... | Mathlib/Analysis/Calculus/UniformLimitsDeriv.lean | 525 | 533 |
/-
Copyright (c) 2022 Apurva Nakade. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Apurva Nakade
-/
import Mathlib.Analysis.Convex.Cone.Closure
import Mathlib.Analysis.InnerProductSpace.Adjoint
/-!
# Proper cones
We define a *proper cone* as a closed, pointed cone. ... | Mathlib/Analysis/Convex/Cone/Proper.lean | 286 | 288 | |
/-
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... | comp_assoc _ _ _ _ _ (by omega)
@[simp]
lemma comp_assoc_of_second_degree_eq_neg_third_degree {n₁ n₂ n₁₂ : ℤ}
(z₁ : Cochain F G n₁) (z₂ : Cochain G K (-n₂)) (z₃ : Cochain K L n₂) (h₁₂ : n₁ + (-n₂) = n₁₂) :
(z₁.comp z₂ h₁₂).comp z₃
(show n₁₂ + n₂ = n₁ by rw [← h₁₂, add_assoc, neg_add_cancel, add_zero]) ... | Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean | 289 | 295 |
/-
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... | mul_assoc] at this
rwa [pow_two, mul_pow, mul_assoc] at this
rcases eq_or_ne n 0 with rfl | hn
· simp only [pow_zero, one_mul, mul_one, zero_add, Finset.range_one, Finset.product_singleton,
Finset.sum_map, Function.Embedding.coeFn_mk, norm_iteratedFDeriv_zero] at Z ⊢
apply Z.trans
conv_rhs =... | Mathlib/Analysis/Fourier/FourierTransformDeriv.lean | 738 | 759 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Patrick Massot, Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic
import Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalcul... | Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | 867 | 869 | |
/-
Copyright (c) 2024 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Localization.CalculusOfFractions
/-!
# Lemmas on fractions
Let `W : MorphismProperty C`, and objects `X` and `Y` in `C`. In this file,
we introd... | lemma exists_leftFraction₃ {X Y : C} (f f' f'' : L.obj X ⟶ L.obj Y) :
∃ (φ : W.LeftFraction₃ X Y), f = φ.fst.map L (inverts L W) ∧
f' = φ.snd.map L (inverts L W) ∧
f'' = φ.thd.map L (inverts L W) := by
obtain ⟨α, hα, hα'⟩ := exists_leftFraction₂ L W f f'
have ⟨β, hβ⟩ := exists_leftFraction L W f''
... | Mathlib/CategoryTheory/Localization/CalculusOfFractions/Fractions.lean | 287 | 314 |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Abelian
import Mathlib.Algebra.Lie.Solvable
import Mathlib.LinearAlgebra.Dual.Defs
/-!
# Characters of Lie algebras
A character of a Lie algebr... | rw [derivedSeries_def, derivedSeriesOfIdeal_succ, derivedSeriesOfIdeal_zero, ←
LieSubmodule.mem_toSubmodule, LieSubmodule.lieIdeal_oper_eq_linear_span] at h
| Mathlib/Algebra/Lie/Character.lean | 49 | 50 |
/-
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.Eval.Degree
import Mathlib.Algebra.Prime.Lemmas
/-!
# Theory of degrees of polynomials
S... | simp only [natDegree, degree_mul_C a0]
| Mathlib/Algebra/Polynomial/Degree/Lemmas.lean | 346 | 347 |
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.MeasureTheory.MeasurableSpace.Defs
/-!
# σ-algebra of sets invariant under a self-map
In this file we define `MeasurableSpace.invariants (f : α → α)... | theorem measurable_invariants_of_semiconj {fa : α → α} {fb : β → β} {g : α → β} (hg : Measurable g)
(hfg : Semiconj g fa fb) : @Measurable _ _ (invariants fa) (invariants fb) g := fun s hs ↦
⟨hg hs.1, by rw [← preimage_comp, hfg.comp_eq, preimage_comp, hs.2]⟩
| Mathlib/MeasureTheory/MeasurableSpace/Invariants.lean | 62 | 64 |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.TwoDim
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
/-!
# Oriented angles.
This file defines orie... | between the second and the third equals the angle between the first and the third. -/
@[simp]
theorem oangle_add {x y z : V} (hx : x ≠ 0) (hy : y ≠ 0) (hz : z ≠ 0) :
o.oangle x y + o.oangle y z = o.oangle x z := by
| Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | 439 | 442 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yaël Dillies
-/
import Mathlib.Algebra.Module.BigOperators
import Mathlib.GroupTheory.Perm.Basic
import Mathlib.GroupTheory.Perm.Finite
import Mathlib.GroupTheory.Perm.Lis... | Mathlib/GroupTheory/Perm/Cycle/Basic.lean | 90 | 90 | |
/-
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, Frédéric Dupuis,
Heather Macbeth
-/
import Mathlib.Algebra.Group.Subgroup.Ker
import Mathlib.Algebra.Module.Submodule.... | theorem ker_eq_bot {f : M →ₛₗ[τ₁₂] M₂} : ker f = ⊥ ↔ Injective f :=
LinearMapClass.ker_eq_bot _
@[simp] lemma injective_domRestrict_iff {f : M →ₛₗ[τ₁₂] M₂} {S : Submodule R M} :
Injective (f.domRestrict S) ↔ S ⊓ LinearMap.ker f = ⊥ := by
| Mathlib/Algebra/Module/Submodule/Ker.lean | 190 | 194 |
/-
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 | 3,607 | 3,621 | |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Logic.Encodable.Pi
import Mathlib.Logic.Function.Iterate
/-!
# The primitive recursive functions
The primitive ... | open Nat.Primrec
instance (priority := 10) ofDenumerable (α) [Denumerable α] : Primcodable α :=
| Mathlib/Computability/Primrec.lean | 137 | 139 |
/-
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.Analysis.Convex.Combination
import Mathlib.LinearAlgebra.AffineSpace.Independent
import Mathlib.Tactic.FieldSimp
/-!
# Carathéodory's conv... | Caratheodory.minCardFinsetOfMemConvexHull_subseteq hx,
Caratheodory.affineIndependent_minCardFinsetOfMemConvexHull hx,
Caratheodory.mem_minCardFinsetOfMemConvexHull hx⟩
· iterate 3 convert Set.iUnion_subset _; intro
exact convexHull_mono ‹_›
/-- A more explicit version of `convexHull_eq_union`.... | Mathlib/Analysis/Convex/Caratheodory.lean | 151 | 161 |
/-
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... | rw [nonempty_iff_ne_empty]; rintro rfl; exact hD.not_indep M.empty_indep
theorem Indep.finite [RankFinite M] (hI : M.Indep I) : I.Finite :=
let ⟨_, hB, hIB⟩ := hI.exists_isBase_superset
hB.finite.subset hIB
theorem Indep.rankPos_of_nonempty (hI : M.Indep I) (hne : I.Nonempty) : M.RankPos := by
obtain ⟨B, hB, ... | Mathlib/Data/Matroid/Basic.lean | 589 | 603 |
/-
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... | of the `a` such that, eventually for `f`, `a ≤ u x` whenever `p x` holds. -/
| Mathlib/Order/LiminfLimsup.lean | 74 | 74 |
/-
Copyright (c) 2021 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.LinearAlgebra.Ray
import Mathlib.LinearAlgebra.Determinant
/-!
# Orientations of modules
This file defines orientations of modules.
## Main definitions
... | Mathlib/LinearAlgebra/Orientation.lean | 414 | 424 | |
/-
Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Data.Fintype.Lattice
import Mathlib.Data.Fintype.Sum
import Mathlib.Topology.Homeomorph.Lemmas
import Mathlib.Topology.MetricSpace.Antilipschitz
... | @[simp] theorem coe_one : ⇑(1 : α ≃ᵢ α) = id := rfl
@[simp] theorem coe_mul (e₁ e₂ : α ≃ᵢ α) : ⇑(e₁ * e₂) = e₁ ∘ e₂ := rfl
| Mathlib/Topology/MetricSpace/Isometry.lean | 477 | 479 |
/-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin
-/
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.Matrix.Composition
import Mathlib.Data.Matrix.ConjTranspose
/-!... | @[simp]
theorem blockDiagonal'_smul {R : Type*} [Zero α] [SMulZeroClass R α] (x : R)
(M : ∀ i, Matrix (m' i) (n' i) α) : blockDiagonal' (x • M) = x • blockDiagonal' M := by
ext
| Mathlib/Data/Matrix/Block.lean | 701 | 704 |
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