Context stringlengths 227 76.5k | target stringlengths 0 11.6k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 16 3.69k |
|---|---|---|---|---|
/-
Copyright (c) 2020 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 [h, properDivisors_prime_pow]
@[to_additive (attr := simp) sum_divisors_prime_pow]
theorem prod_divisors_prime_pow {α : Type*} [CommMonoid α] {k p : ℕ} {f : ℕ → α} (h : p.Prime) :
| Mathlib/NumberTheory/Divisors.lean | 522 | 525 |
/-
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... | ∑ i ∈ s, w i * z i = ∑ i ∈ s, w i * x := by
refine sum_congr rfl fun i hi => ?_
rcases eq_or_ne (w i) 0 with hwi | hwi
· rw [hwi, zero_mul, zero_mul]
· rw [hx i hi hwi]
_ = x := by rw [← sum_mul, hw', one_mul]
theorem geom_mean_eq_arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (h... | Mathlib/Analysis/MeanInequalities.lean | 183 | 191 |
/-
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 (U) (hU : U ∈ p.basisSets) :
∃ V ∈ 𝓝 (0 : 𝕜), ∃ W ∈ p.addGroupFilterBasis.sets, V • W ⊆ U := by
rcases p.basisSets_iff.mp hU with ⟨s, r, hr, hU⟩
refine ⟨Metric.ball 0 √r, Metric.ball_mem_nhds 0 (Real.sqrt_pos.mpr hr), ?_⟩
refine ⟨(s.sup p).ball 0 √r, p.basisSets_mem s (Real.sqrt_pos.... | Mathlib/Analysis/LocallyConvex/WithSeminorms.lean | 145 | 155 |
/-
Copyright (c) 2021 Hunter Monroe. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hunter Monroe, Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Dart
import Mathlib.Data.FunLike.Fintype
import Mathlib.Logic.Embedding.Set
/-!
# Maps between graphs
This file ... | RelHom G.Adj G'.Adj
| Mathlib/Combinatorics/SimpleGraph/Maps.lean | 203 | 204 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Order.Filter.Tendsto
import Mathlib.Data.Set.Accumulate
import Mathlib.Topology.Bornology.Basic
import Mathlib.Topolog... | refine ⟨Uᶜ, Set.compl_subset_comm.mp Uc, Ucne, Uo.isClosed_compl, ?_⟩
intro V' V'sub V'ne V'cls
have : V'ᶜ = U := by
| Mathlib/Topology/Compactness/Compact.lean | 1,065 | 1,067 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Geometry.Manifold.MFDeriv.Defs
import Mathlib.Geometry.Manifold.ContMDiff.Defs
/-!
# Basic properties of the manifold Fréchet ... | MDifferentiableWithinAt I I' f s x ↔
ContinuousWithinAt f s x ∧
DifferentiableWithinAt 𝕜 (e'.extend I' ∘ f ∘ (e.extend I).symm) (e.extend I '' s)
(e.extend I x) := by
| Mathlib/Geometry/Manifold/MFDeriv/Basic.lean | 303 | 306 |
/-
Copyright (c) 2024 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.DerivedCategory.Basic
/-!
# The homology sequence
In this file, we construct `homologyFunctor C n : DerivedCategory C ⥤ C` for all `n : ℤ`,
s... | noncomputable def δ (T : Triangle (DerivedCategory C))
(n₀ n₁ : ℤ) (h : n₀ + 1 = n₁) :
(homologyFunctor C n₀).obj T.obj₃ ⟶ (homologyFunctor C n₁).obj T.obj₁ :=
(homologyFunctor C 0).shiftMap T.mor₃ n₀ n₁ (by rw [add_comm 1, h])
| Mathlib/Algebra/Homology/DerivedCategory/HomologySequence.lean | 73 | 77 |
/-
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.Data.Finset.Lattice.Fold
import Mathlib.Data.Fintype.Vector
import Mathlib.Data.Multiset.Sym
/-!
# Symmetric powers of a finset
This file defines the sym... | val_injective <| s.val.sym2_map _
theorem sym2_image [DecidableEq β] (f : α → β) (s : Finset α) :
(s.image f).sym2 = s.sym2.image (Sym2.map f) := by
| Mathlib/Data/Finset/Sym.lean | 62 | 65 |
/-
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.Algebra.Ring.Pointwise.Set
import Mathlib.Order.Filter.AtTopBot.CompleteLattice
import Mathlib.Order.Filter.AtTopBot.G... |
open List in
/-- The following statements are equivalent:
| Mathlib/Topology/Order/LeftRightNhds.lean | 216 | 218 |
/-
Copyright (c) 2023 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.ModelTheory.Algebra.Ring.Basic
import Mathlib.RingTheory.FreeCommRing
/-!
# Making a term in the language of rings from an element of the FreeCommRing
T... | let _ := compatibleRingOfRing (FreeCommRing α)
rw [termOfFreeCommRing]
conv_rhs => rw [← Classical.choose_spec (exists_term_realize_eq_freeCommRing p)]
induction Classical.choose (exists_term_realize_eq_freeCommRing p) with
| var _ => simp
| func f a ih =>
cases f <;>
simp [ih]
end Ring
| Mathlib/ModelTheory/Algebra/Ring/FreeCommRing.lean | 54 | 63 |
/-
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.Tactic.CategoryTheory.Reassoc
/-!
# Isomorphisms
This file defines isomorphisms between objects of a categ... | @[reassoc]
theorem map_hom_inv (F : C ⥤ D) {X Y : C} (f : X ⟶ Y) [IsIso f] :
F.map f ≫ F.map (inv f) = 𝟙 (F.obj X) := by simp
| Mathlib/CategoryTheory/Iso.lean | 554 | 556 |
/-
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
-/
import Mathlib.Tactic.Attr.Register
import Mathlib.Tactic.Basic
import Batteries.Logic
import Batteries.Tactic.Trans
import Batteries.Util.LibraryNot... | Mathlib/Logic/Basic.lean | 1,216 | 1,217 | |
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.CategoryTheory.GlueData
import Mathlib.Topology.Category.TopCat.Limits.Pullbacks
import Mathlib.Topology.Category.TopCat.Opens
import Mathlib.Tactic.Generali... | (colim.map (diagramIsoParallelPair diagram).hom ≫
(colimit.isoColimitCocone (Types.coequalizerColimit _ _)).hom)
this :
_)
-- Porting note: was
-- simp only [eqToHom_refl, types_comp_apply, colimit.ι_map_assoc,
-- diagramIsoParallelPair_hom_app, colimit.isoColimitCocone_ι_hom, ty... | Mathlib/Topology/Gluing.lean | 165 | 202 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.FieldTheory.Finite.Polynomial
import Mathlib.NumberTheory.Basic
import Mathlib.RingTheory.WittVector.WittPolynomial
/-!
# Witt struct... | rcases n with - | n
· simp only [isUnit_one, Int.ofNat_zero, Int.natCast_succ, zero_add, pow_zero, C_1, IsUnit.dvd,
Nat.cast_one]
-- prepare a useful equation for rewriting
have key := bind₁_rename_expand_wittPolynomial Φ n IH
apply_fun map (Int.castRingHom (ZMod (p ^ (n + 1)))) at key
conv_lhs at key... | Mathlib/RingTheory/WittVector/StructurePolynomial.lean | 230 | 268 |
/-
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.Independence.Kernel
import Mathlib.Probability.Kernel.Condexp
/-!
# Conditional Independence
We define conditional independence of sets/σ-alg... | rw [condIndep_iff_condIndepSets, condIndepSets_iff]
· rfl
· exact hm₁
· exact hm₂
end CondIndep
variable (m' : MeasurableSpace Ω) {mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω]
(hm' : m' ≤ mΩ)
| Mathlib/Probability/Independence/Conditional.lean | 275 | 284 |
/-
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... | subst er
refine exp_approx_succ _ en _ _ ?_ h
| Mathlib/Data/Complex/Exponential.lean | 602 | 603 |
/-
Copyright (c) 2022 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots
import Mathlib.FieldTheory.Finite.Trace
import Mathlib.Algebra.Group.AddChar
import Mathlib.Data.ZMod.Units
import... | (φ.val_isUnit a).unit ∈ rootsOfUnity (ringChar R).toPNat' R' := by
simp only [mem_rootsOfUnity', IsUnit.unit_spec, Nat.toPNat'_coe, h, ↓reduceIte,
← map_nsmul_eq_pow, nsmul_eq_mul, CharP.cast_eq_zero, zero_mul, map_zero_eq_one]
/-- An additive character is *primitive* iff all its multiplicative shifts by non... | Mathlib/NumberTheory/LegendreSymbol/AddCharacter.lean | 51 | 55 |
/-
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 ... | iSupIndep N ↔ iSupIndep fun i ↦ (N i : Submodule R M) := iSupIndep_toSubmodule.symm
@[deprecated (since := "2024-12-30")]
| Mathlib/Algebra/Lie/Submodule.lean | 509 | 511 |
/-
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.Analysis.Convex.Side
import Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
/-!
# Oriented an... | theorem left_ne_right_of_oangle_sign_eq_neg_one {p₁ p₂ p₃ : P} (h : (∡ p₁ p₂ p₃).sign = -1) :
p₁ ≠ p₃ :=
| Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean | 158 | 159 |
/-
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.BigOperators.Expect
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Field.Canonical
import Mathlib.Algebra.O... | Mathlib/Data/NNReal/Basic.lean | 813 | 814 | |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.MeasureTheory.Integral.Lebesgue.Countable
/-!
# The Giry monad
Let X be a measurable space. The collection of all measures on X again
forms a measura... | ext1 s hs
simp [hs]
@[fun_prop]
theorem measurable_join : Measurable (join : Measure (Measure α) → Measure α) :=
| Mathlib/MeasureTheory/Measure/GiryMonad.lean | 163 | 167 |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Emily Riehl
-/
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryTheory.Functor.TwoSquare
import Mathlib.CategoryTheory.HomCongr
import Mathlib.Tactic.... | G.map (adj₁.unit.app c) ≫ (mateEquiv adj₁ adj₂ α).app _ =
adj₂.unit.app _ ≫ R₂.map (α.app _) := by
dsimp [mateEquiv]
rw [← adj₂.unit_naturality_assoc]
slice_lhs 2 3 =>
rw [← R₂.map_comp, ← Functor.comp_map G L₂, α.naturality]
rw [R₂.map_comp]
| Mathlib/CategoryTheory/Adjunction/Mates.lean | 118 | 124 |
/-
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, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Field.IsField
import Mathlib.Algebra.GroupWithZero.N... | Mathlib/RingTheory/Localization/Basic.lean | 1,273 | 1,285 | |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Sébastien Gouëzel, Yury Kudryashov, Dylan MacKenzie, Patrick Massot
-/
import Mathlib.Algebra.BigOperators.Module
import Mathlib.Algebra.Order.Field.Power
import M... |
NB: For backwards compatibility, if you add more items to the list, please append them at the end of
the list. -/
theorem TFAE_exists_lt_isLittleO_pow (f : ℕ → ℝ) (R : ℝ) :
TFAE
[∃ a ∈ Ioo (-R) R, f =o[atTop] (a ^ ·), ∃ a ∈ Ioo 0 R, f =o[atTop] (a ^ ·),
∃ a ∈ Ioo (-R) R, f =O[atTop] (a ^ ·), ∃ a ∈ Io... | Mathlib/Analysis/SpecificLimits/Normed.lean | 62 | 68 |
/-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Gabin Kolly
-/
import Mathlib.Data.Finite.Sum
import Mathlib.Data.Fintype.Order
import Mathlib.ModelTheory.FinitelyGenerated
import Mathlib.ModelTheory.Quotients
import... | that respect the directed system structure (i.e. make some diagram commute) give rise
to a unique map out of the direct limit. -/
def lift (g : ∀ i, G i ↪[L] P) (Hg : ∀ i j hij x, g j (f i j hij x) = g i x) :
DirectLimit G f ↪[L] P where
| Mathlib/ModelTheory/DirectLimit.lean | 326 | 329 |
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.MeasureTheory.I... | theorem setIntegral_comp_smul (f : E → F) {R : ℝ} (s : Set E) (hR : R ≠ 0) :
∫ x in s, f (R • x) ∂μ = |(R ^ finrank ℝ E)⁻¹| • ∫ x in R • s, f x ∂μ := by
let e : E ≃ᵐ E := (Homeomorph.smul (Units.mk0 R hR)).toMeasurableEquiv
| Mathlib/MeasureTheory/Measure/Haar/NormedSpace.lean | 128 | 130 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Notation.Pi
import Mathlib.Data.Set.Lattice
import Mathlib.Order.Filter.Defs
/-!
# Theory of... | Mathlib/Order/Filter/Basic.lean | 2,794 | 2,794 | |
/-
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.Fin.VecNotation
import Mathlib.Logic.Small.Basic
import Mathlib.SetTheory.ZFC.PSet
/-!
# A model of ZFC
In this file, we model Zermelo-Fraenkel ... | let c : (A a).Type := c
let ⟨d, hd⟩ := γδ (by rwa [ea] at c)
use ⟨b, Eq.ndrec d (Eq.symm eb)⟩
change PSet.Equiv ((A a).Func c) ((B b).Func (Eq.ndrec d eb.symm))
match A a, B b, ea, eb, c, d, hd with
| _, _, rfl, rfl, _, _, hd => exact hd
/-- The union operator, the collection of elements of ele... | Mathlib/SetTheory/ZFC/Basic.lean | 436 | 446 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.Bochner.Basic
import Mathlib.MeasureTheory.Integral.Bochner.L1
import Mathlib.Me... | Mathlib/MeasureTheory/Integral/Bochner.lean | 1,448 | 1,458 | |
/-
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.Unbundled.Basic
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Or... | Mathlib/Algebra/Order/Group/Defs.lean | 751 | 752 | |
/-
Copyright (c) 2022 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.ImproperIntegrals
import Mathlib.MeasureTheory.Integral.ExpDecay
/-!
# The Gamma function
This file defines the `Γ` functio... | rw [mul_cpow_ofReal_nonneg hr.le hx.le, ← mul_assoc, one_div, ← ofReal_inv,
← mul_cpow_ofReal_nonneg (inv_pos.mpr hr).le hr.le, ← ofReal_mul r⁻¹,
inv_mul_cancel₀ hr.ne', ofReal_one, one_cpow, one_mul]
_ = 1 / r * ∫ (t : ℝ) in Ioi 0, (1 / r) ^ (a - 1) * t ^ (a - 1) * exp (-t) := by
simp_r... | Mathlib/Analysis/SpecialFunctions/Gamma/Basic.lean | 379 | 390 |
/-
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... | def isLimitConeOfAdj (F : J ⥤ C) :
IsLimit (coneOfAdj adj F) where
lift s := adj.homEquiv _ _ s.π
fac s j := by
have eq := NatTrans.congr_app (adj.counit.naturality s.π) j
have eq' := NatTrans.congr_app (adj.left_triangle_components s.pt) j
| Mathlib/CategoryTheory/Limits/HasLimits.lean | 545 | 550 |
/-
Copyright (c) 2022 Wrenna Robson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Wrenna Robson
-/
import Mathlib.Analysis.Normed.Group.Basic
/-!
# Hamming spaces
The Hamming metric counts the number of places two members of a (finite) Pi type
differ. The Hamming n... |
/-- Corresponds to `dist_smul` with the discrete norm on `α`. -/
| Mathlib/InformationTheory/Hamming.lean | 123 | 124 |
/-
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... | inst₁ = inst₂ := by
have h_monoid : inst₁.toAddMonoid = inst₂.toAddMonoid := by ext : 1; exact h_add
have h_zero' : inst₁.toZero = inst₂.toZero := congrArg (·.toZero) h_monoid
have h_one' : inst₁.toOne = inst₂.toOne :=
congrArg One.mk h_one
| Mathlib/Algebra/Ring/Ext.lean | 101 | 105 |
/-
Copyright (c) 2022 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.List.Induction
import Mathlib.Data.List.TakeWhile
/-!
# Dropping or taking from lists on the right
Taking or removing element from the tail e... | simp only [rtakeWhile, takeWhile, reverse_append, reverse_singleton, singleton_append]
split_ifs with h <;> simp [h]
@[simp]
theorem rtakeWhile_concat_pos (x : α) (h : p x) :
rtakeWhile p (l ++ [x]) = rtakeWhile p l ++ [x] := by rw [rtakeWhile_concat, if_pos h]
@[simp]
theorem rtakeWhile_concat_neg (x : α) (h... | Mathlib/Data/List/DropRight.lean | 166 | 174 |
/-
Copyright (c) 2023 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.Filter.Map
/-!
# Kernel of a filter
In this file we define the *kernel* `Filter.ker f` of a filter `f`
to be the intersection of all its sets.
We ... | gi_principal_ker.gc.u_iInf
| Mathlib/Order/Filter/Ker.lean | 42 | 42 |
/-
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.Game.Basic
import Mathlib.SetTheory.Ordinal.NaturalOps
/-!
# Ordinals as games
We define the canonical map `Ordinal... | /-- Converts an ordinal into the corresponding game. -/
noncomputable def toGame : Ordinal.{u} ↪o Game.{u} where
| Mathlib/SetTheory/Game/Ordinal.lean | 154 | 155 |
/-
Copyright (c) 2021 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.SpecificLimits.Basic
/-!
# A collection of specific asymptotic results
This fil... | have : (fun _ : ℕ => ∑ i ∈ range N, f i) =o[atTop] fun n : ℕ => ∑ i ∈ range n, g i := by
apply isLittleO_const_left.2
exact Or.inr (h'g.congr fun n => (B n).symm)
filter_upwards [isLittleO_iff.1 this (half_pos εpos), Ici_mem_atTop N] with n hn Nn
calc
‖∑ i ∈ range n, f i‖ = ‖(∑ i ∈ range N, f i) + ∑ i... | Mathlib/Analysis/Asymptotics/SpecificAsymptotics.lean | 98 | 128 |
/-
Copyright (c) 2022 Yaël Dillies, Sara Rousta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Sara Rousta
-/
import Mathlib.Logic.Equiv.Set
import Mathlib.Order.Interval.Set.OrderEmbedding
import Mathlib.Order.SetNotation
/-!
# Properties of unbundled ... | Mathlib/Order/UpperLower/Basic.lean | 1,363 | 1,364 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Topology.Algebra.Constructions
import Mathlib.Topology.Bases
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Topology.UniformSpac... | Cauchy (l.map f) ↔ Tendsto (fun p : β × β => (f p.1, f p.2)) (l ×ˢ l) (𝓤 α) :=
cauchy_map_iff.trans <| and_iff_right hl
theorem Cauchy.mono {f g : Filter α} [hg : NeBot g] (h_c : Cauchy f) (h_le : g ≤ f) : Cauchy g :=
⟨hg, le_trans (Filter.prod_mono h_le h_le) h_c.right⟩
| Mathlib/Topology/UniformSpace/Cauchy.lean | 63 | 67 |
/-
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.Small.Module
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.Isomorphisms
import Mathlib.LinearAlgebra.TensorProduc... | · simp only [f, LinearMap.comp_apply, b.repr.apply_symm_apply,
LinearEquiv.coe_toLinearMap, Finsupp.lmapDomain_apply]
rw [← Finsupp.mapDomain_comp, hσ₁, Finsupp.mapDomain_id]
· rintro ⟨y, hy, rfl⟩
simp [f, hπ, ← Finsupp.apply_linearCombination, hy]
-- Ideally this should be an instance but it m... | Mathlib/Algebra/Module/FinitePresentation.lean | 142 | 157 |
/-
Copyright (c) 2020 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Algebra.Algebra.Field
import Mathlib.Algebra.BigOperators.Balance
import Mathlib.Algebra.Order.BigOperators.Expect
import Mathlib.Algebra.Order.Star.... | rw [lt_iff_le_and_ne, Ne, eq_comm]; simp [normSq_nonneg]
| Mathlib/Analysis/RCLike/Basic.lean | 407 | 408 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.Group.Defs
/-!
# Invertible elements
This file defines a typeclass `Invertible a` for elements `a` with a two-sided
multiplicative inverse.
The in... | Mathlib/Algebra/Group/Invertible/Defs.lean | 253 | 254 | |
/-
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... | variable [PosSMulStrictMono α β] [PosSMulReflectLT α β]
| Mathlib/Algebra/Order/Module/Defs.lean | 893 | 894 |
/-
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.Partrec
import Mathlib.Data.Option.Basic
/-!
# Gödel Numbering for Partial Recursive Functions.
This file defines `Nat.Partrec.Code`, a... | Primrec₂.encode_iff.1 <|
nat_add.comp
(nat_double.comp <|
nat_double.comp <|
| Mathlib/Computability/PartrecCode.lean | 212 | 215 |
/-
Copyright (c) 2020 Paul van Wamelen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Paul van Wamelen
-/
import Mathlib.Data.Int.NatPrime
import Mathlib.Data.ZMod.Basic
import Mathlib.RingTheory.Int.Basic
import Mathlib.Tactic.FieldSimp
/-!
# Pythagorean Triples
Th... | end PythagoreanTriple
section circleEquivGen
/-!
### A parametrization of the unit circle
For the classification of Pythagorean triples, we will use a parametrization of the unit circle.
-/
variable {K : Type*} [Field K]
| Mathlib/NumberTheory/PythagoreanTriples.lean | 241 | 252 |
/-
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.Ideal
/-!
# Ideal operations for Lie algebras
Given a Lie module `M` over a Lie algebra `L`, there is a natural action of the Lie ideals of `L`... | rw [← map_sup_ker_eq_map, ← comap_bracket_eq h, map_comap_eq h, inf_eq_right]
exact le_trans (LieSubmodule.lie_le_left _ _) inf_le_left
theorem comap_bracket_incl {I₁ I₂ : LieIdeal R L} :
⁅comap I.incl I₁, comap I.incl I₂⁆ = comap I.incl ⁅I ⊓ I₁, I ⊓ I₂⁆ := by
conv_rhs =>
congr
next => skip
rw [←... | Mathlib/Algebra/Lie/IdealOperations.lean | 297 | 316 |
/-
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, Yaël Dillies, Yuyang Zhao
-/
import Mathlib.Algebra.Order.Ring.Unbundled.Basic
import Mathlib.Algebra.CharZero.Defs
import Mathlib.Alge... | by a non-negative element `0 ≤ c` to obtain `c * a ≤ c * b`. -/
protected mul_le_mul_of_nonneg_left : ∀ a b c : R, a ≤ b → 0 ≤ c → c * a ≤ c * b
/-- In an ordered semiring, we can multiply an inequality `a ≤ b` on the right
by a non-negative element `0 ≤ c` to obtain `a * c ≤ b * c`. -/
protected mul_le_mul_o... | Mathlib/Algebra/Order/Ring/Defs.lean | 241 | 247 |
/-
Copyright (c) 2022 Cuma Kökmen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Cuma Kökmen, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Integral.CircleIntegral
import Mathlib.MeasureTheory.Integral.Prod
import Mathlib.Order.Fin.Tuple
import Mathlib.Util.Superscr... | have heπ : (e ⁻¹' Icc 0 fun _ => 2 * π) = Icc 0 (2 * π) ×ˢ Icc (0 : ℝⁿ) fun _ => 2 * π :=
((Fin.insertNthOrderIso (fun _ => ℝ) i).preimage_Icc _ _).trans (Icc_prod_eq _ _)
rw [torusIntegral, ← hem.map_eq, setIntegral_map_equiv, heπ, Measure.volume_eq_prod,
setIntegral_prod, circleIntegral_def_Icc]
· refin... | Mathlib/MeasureTheory/Integral/TorusIntegral.lean | 217 | 228 |
/-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Inductions
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.RingTheory.Polynom... | exact continuous_finset_sum _ fun c _ => continuous_const.mul (continuous_pow _)
@[continuity, fun_prop]
protected theorem continuous : Continuous fun x => p.eval x :=
| Mathlib/Topology/Algebra/Polynomial.lean | 49 | 52 |
/-
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.Order.Interval.Finset.Basic
/-!
# Intervals as multisets
This file defines intervals as multisets.
## Main declarations
In a `LocallyFiniteOrder`,
* `M... | Ico a₁ b₁ ⊆ Ico a₂ b₂ ↔ a₂ ≤ a₁ ∧ b₁ ≤ b₂ :=
Finset.Ico_subset_Ico_iff h
theorem Ico_add_Ico_eq_Ico {a b c : α} (hab : a ≤ b) (hbc : b ≤ c) :
| Mathlib/Order/Interval/Multiset.lean | 271 | 274 |
/-
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.Countable
import Mathlib.MeasureTheory.Measure.Decomposition.Exhaustion
import Mathlib.MeasureTheory.Me... | apply lintegral_withDensity_eq_lintegral_mul_non_measurable _ hf.measurable_mk
filter_upwards [h'f, hf.ae_eq_mk]
intro x hx h'x
rwa [← h'x]
_ = ∫⁻ a, (f * g) a ∂μ := by
apply lintegral_congr_ae
filter_upwards [hf.ae_eq_mk]
intro x hx
simp only [f', hx, Pi.mul_apply]
... | Mathlib/MeasureTheory/Measure/WithDensity.lean | 465 | 488 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.MvPolynomial.Rename
/-!
# Degrees of polynomials
This file establ... | _ ≤ d.sum fun _ e => e := by
rw [Finsupp.sum_fintype]
intros
rfl
_ ≤ f.totalDegree := le_totalDegree hd
| Mathlib/Algebra/MvPolynomial/Degrees.lean | 489 | 493 |
/-
Copyright (c) 2021 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Bhavik Mehta
-/
import Mathlib.Analysis.Calculus.Deriv.Support
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
import Mathlib.MeasureTheory.Function.Jacobian
imp... | derivative is zero. -/
theorem integral_eq_zero_of_hasDerivAt_of_integrable
(hderiv : ∀ x, HasDerivAt f (f' x) x) (hf' : Integrable f') (hf : Integrable f) :
∫ x, f' x = 0 := by
by_cases hE : CompleteSpace E; swap
· simp [integral, hE]
| Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean | 1,008 | 1,013 |
/-
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.HomotopyCategory.HomComplex
import Mathlib.Algebra.Homology.HomotopyCofiber
/-! # The mapping cone of a morphism of cochain complexes
In this ... | rw [← Cochain.comp_assoc_of_first_is_zero_cochain, inr_fst, Cochain.zero_comp]
@[simp]
lemma inr_snd_assoc {K : CochainComplex C ℤ} {d e : ℤ} (γ : Cochain G K d) (he : 0 + d = e) :
(Cochain.ofHom (inr φ)).comp ((snd φ).comp γ he) (by simp only [← he, zero_add]) = γ := by
| Mathlib/Algebra/Homology/HomotopyCategory/MappingCone.lean | 141 | 145 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.PropInstances
import Mathlib.Order.GaloisConnection.Defs
/-!
# Heyting algebras
This file defines Heyting, co-Heyting and bi-Heyting algebras.
A H... | sup_le le_himp compl_le_himp
-- `p → ¬ p ↔ ¬ p`
| Mathlib/Order/Heyting/Basic.lean | 632 | 634 |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.Real
/-!
# Properties of pointwise scalar multiplication of se... | nonnegative. -/
@[simp]
theorem NormedSpace.sphere_nonempty [Nontrivial E] {x : E} {r : ℝ} :
(sphere x r).Nonempty ↔ 0 ≤ r := by
| Mathlib/Analysis/NormedSpace/Pointwise.lean | 383 | 386 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
import Mathlib.Data.Set.Function
import Mathlib.Logic.Equiv.Defs
import Mathlib.Tactic.Says
/-!
# Equivalences and sets
In this file we pr... | invFun x := ⟨(x.1, x.2.1), x.2.2⟩
left_inv := fun ⟨⟨_, _⟩, _⟩ => rfl
right_inv := fun ⟨_, _, _⟩ => rfl
| Mathlib/Logic/Equiv/Set.lean | 155 | 158 |
/-
Copyright (c) 2018 Andreas Swerdlow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andreas Swerdlow
-/
import Mathlib.LinearAlgebra.Basis.Basic
import Mathlib.LinearAlgebra.BilinearMap
import Mathlib.LinearAlgebra.LinearIndependent.Lemmas
/-!
# Sesquilinear maps
... |
protected theorem eq (H : B.IsSymm) (x y) : I (B x y) = B y x :=
H x y
| Mathlib/LinearAlgebra/SesquilinearForm.lean | 196 | 199 |
/-
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... | Mathlib/Data/Nat/ModEq.lean | 485 | 486 | |
/-
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, Mitchell Lee
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.Algebra.Group.Submonoid.Basic
import M... | mul_mem' ha hb := s.top_closure_mul_self_subset ⟨_, ha, _, hb, rfl⟩
@[to_additive]
| Mathlib/Topology/Algebra/Monoid.lean | 546 | 548 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Algebra.Group.Action.Opposite
import Mathlib.Algebra.Group.Action.Units
import Mathlib.Algebra.Group.Invertible.Defs
import Mathlib.Algebra.GroupWithZero.U... | Mathlib/Algebra/Star/Basic.lean | 577 | 579 | |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Batteries.Tactic.Congr
import Mathlib.Data.Option.Basic
import Mathlib.Data.Prod.Basic
import Mathlib.Data.Set.Subsingleton
import Mathlib.Dat... | fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
| Mathlib/Data/Set/Image.lean | 266 | 266 |
/-
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.Data.Sum.Order
import Mathlib.Order.RelIso.Set
import Mathlib.Order.UpperLower.Basic
import Mathlib.Order... | (fun a => _root_.trans (f.lt_top a) h) h).transRelIso g.subrelIso⟩
· let f := f.subrelIso
rw [h] at f
exact ⟨Sum.inl <| (f.symm.trans g.subrelIso).toInitialSeg⟩
· exact ⟨Sum.inr <| (g.codRestrict {x | Sum.Lex r s x f.top}
| Mathlib/Order/InitialSeg.lean | 545 | 549 |
/-
Copyright (c) 2024 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.NumberTheory.LSeries.AbstractFuncEq
import Mathlib.NumberTheory.ModularForms.JacobiTheta.Bounds
import Mathlib.NumberTheory.LSeries.MellinEqDirichlet
i... | /-- Reformulation of `hasSum_nat_sinZeta` using `LSeriesHasSum`. -/
lemma LSeriesHasSum_sin (a : ℝ) {s : ℂ} (hs : 1 < re s) :
LSeriesHasSum (Real.sin <| 2 * π * a * ·) s (sinZeta a s) :=
(hasSum_nat_sinZeta a hs).congr_fun (LSeries.term_of_ne_zero' (ne_zero_of_one_lt_re hs) _)
/-- The trivial zeroes of the odd H... | Mathlib/NumberTheory/LSeries/HurwitzZetaOdd.lean | 533 | 546 |
/-
Copyright (c) 2019 Neil Strickland. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Neil Strickland
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Group.NatPowAssoc
import Mathlib.Algebra.Order.... | rw [← h.geom_sum₂_Ico_mul hmn, mul_div_cancel_right₀ _ this]
theorem geom_sum₂_Ico [Field K] {x y : K} (hxy : x ≠ y) {m n : ℕ} (hmn : m ≤ n) :
(∑ i ∈ Finset.Ico m n, x ^ i * y ^ (n - 1 - i)) = (x ^ n - y ^ (n - m) * x ^ m) / (x - y) :=
| Mathlib/Algebra/GeomSum.lean | 370 | 373 |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Casper Putz, Anne Baanen
-/
import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.L... |
/-- If we fix a background basis `e`, then for any other basis `v`, we can characterise the
coordinates provided by `v` in terms of determinants relative to `e`. -/
| Mathlib/LinearAlgebra/Determinant.lean | 625 | 627 |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Heather Macbeth, Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Analysis.Normed.Group.Uniform
import Mathlib.Topo... | theorem cauchySeq_finset_of_norm_bounded_eventually {f : ι → E} {g : ι → ℝ} (hg : Summable g)
(h : ∀ᶠ i in cofinite, ‖f i‖ ≤ g i) : CauchySeq fun s => ∑ i ∈ s, f i := by
refine cauchySeq_finset_iff_vanishing_norm.2 fun ε hε => ?_
| Mathlib/Analysis/Normed/Group/InfiniteSum.lean | 49 | 51 |
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Michael Stoll
-/
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Normed.Module.FiniteDimension
import Mathlib.Data.Complex.FiniteDimensional
/-!
# L-series
Gi... |
Let `R` be a ring with a coercion to `ℂ`. Then we can write `↗χ` when `χ : DirichletCharacter R`
or `↗f` when `f : ArithmeticFunction R` or simply `f : N → R` with a coercion from `ℕ` to `N`
as an argument to `LSeries`, `LSeriesHasSum`, `LSeriesSummable` etc. -/
| Mathlib/NumberTheory/LSeries/Basic.lean | 253 | 256 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.PropInstances
import Mathlib.Order.GaloisConnection.Defs
/-!
# Heyting algebras
This file defines Heyting, co-Heyting and bi-Heyting algebras.
A H... | theorem compl_sup_le_himp : aᶜ ⊔ b ≤ a ⇨ b :=
sup_le compl_le_himp le_himp
| Mathlib/Order/Heyting/Basic.lean | 628 | 629 |
/-
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.Topology.MetricSpace.HausdorffDistance
/-!
# Topological study of spaces `Π (n : ℕ), E n`
When `E n` are topological spaces, the space `Π (n : ... | intro x y h
| Mathlib/Topology/MetricSpace/PiNat.lean | 219 | 219 |
/-
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.Solvable
import Mathlib.Algebra.Lie.Quotient
import Mathlib.Algebra.Lie.Normalizer
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.... |
/-- The upper (aka ascending) central series.
See also `LieSubmodule.lcs`. -/
def ucs (k : ℕ) : LieSubmodule R L M → LieSubmodule R L M :=
| Mathlib/Algebra/Lie/Nilpotent.lean | 536 | 540 |
/-
Copyright (c) 2021 Filippo A. E. Nuccio. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Filippo A. E. Nuccio, Eric Wieser
-/
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.Matrix.Block
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.Linear... | Prefer the notation `⊗ₖₜ` rather than this definition. -/
@[simp]
def kroneckerTMul : Matrix l m α → Matrix n p β → Matrix (l × n) (m × p) (α ⊗[R] β) :=
kroneckerMap (· ⊗ₜ ·)
@[inherit_doc kroneckerTMul]
scoped[Kronecker] infixl:100 " ⊗ₖₜ " => Matrix.kroneckerMap (· ⊗ₜ ·)
@[inherit_doc kroneckerTMul] scoped[Kroneck... | Mathlib/Data/Matrix/Kronecker.lean | 410 | 418 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Yaël Dillies
-/
import Mathlib.Order.Cover
import Mathlib.Order.Interval.Finset.Defs
/-!
# Intervals as finsets
This file provides basic results about all the `Finset.Ixx... |
@[simp, aesop safe apply (rule_sets := [finsetNonempty])]
| Mathlib/Order/Interval/Finset/Basic.lean | 408 | 409 |
/-
Copyright (c) 2023 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta, Doga Can Sertbas
-/
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Data.Nat.ModEq
import Mathlib.Data.Nat.Prime.Defs
import Mathlib.Data.Real.Archimedea... | /-- The Schnirelmann density of any finset is `0`. -/
lemma schnirelmannDensity_finset (A : Finset ℕ) : schnirelmannDensity A = 0 := by
refine le_antisymm ?_ schnirelmannDensity_nonneg
simp only [schnirelmannDensity_le_iff_forall, zero_add]
intro ε hε
wlog hε₁ : ε ≤ 1 generalizing ε
· obtain ⟨n, hn, hn'⟩ := t... | Mathlib/Combinatorics/Schnirelmann.lean | 188 | 200 |
/-
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
/-!
# ... | refine le_antisymm (le_of_forall_lt fun a ha => ?_) ?_
· rcases Ordinal.lt_lift_iff.1 ha with ⟨a, _, rfl⟩
| Mathlib/SetTheory/Ordinal/Basic.lean | 1,127 | 1,128 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Chris Hughes
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.Polynomial.Roots
import Mathlib.Data.Fintype.Inv
import Mathlib.GroupTheory.SpecificGroups.Cyclic
... | Mathlib/RingTheory/IntegralDomain.lean | 255 | 260 | |
/-
Copyright (c) 2019 Neil Strickland. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Neil Strickland
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Group.NatPowAssoc
import Mathlib.Algebra.Order.... | @[simp]
theorem op_geom_sum₂ (x y : R) (n : ℕ) : ∑ i ∈ range n, op y ^ (n - 1 - i) * op x ^ i =
∑ i ∈ range n, op y ^ i * op x ^ (n - 1 - i) := by
rw [← sum_range_reflect]
refine sum_congr rfl fun j j_in => ?_
rw [mem_range, Nat.lt_iff_add_one_le] at j_in
| Mathlib/Algebra/GeomSum.lean | 69 | 74 |
/-
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.Fintype.Card
import Mathlib.Algebra.Order.BigOperators.Group.Multiset
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Data.Multiset.OrderedM... | Mathlib/Algebra/Order/BigOperators/Group/Finset.lean | 632 | 634 | |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
import Mathlib.Order.Closure
/-!
# Closed sieves
A natural closure operator on sieves is a closure operator on `Sieve X... | -/
theorem le_close_of_isClosed {X : C} {S T : Sieve X} (h : S ≤ T) (hT : J₁.IsClosed T) :
J₁.close S ≤ T :=
| Mathlib/CategoryTheory/Sites/Closed.lean | 88 | 90 |
/-
Copyright (c) 2021 Aaron Anderson, Jesse Michael Han, Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jesse Michael Han, Floris van Doorn
-/
import Mathlib.Data.Finset.Basic
import Mathlib.ModelTheory.Syntax
import Mathlib.Data.List.... | rw [Sentence.Realize, Sentence.Realize, ← realize_formula g,
Unique.eq_default (g ∘ default)]
theorem theory_model [M ⊨ T] : N ⊨ T :=
⟨fun φ hφ => (realize_sentence g φ).1 (Theory.realize_sentence_of_mem T hφ)⟩
theorem elementarilyEquivalent : M ≅[L] N :=
elementarilyEquivalent_iff.2 (realize_sentence g)
e... | Mathlib/ModelTheory/Semantics.lean | 911 | 927 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Reid Barton
-/
import Mathlib.Topology.Separation.Regular
/-!
# The shrinking lemma
In this file we prove a few versions of the shrinking lemma. The lemma says that... | IsChain (· ≤ ·) c → c.Nonempty → ∃ ub, ∀ v ∈ c, v ≤ ub :=
fun c hc ne => ⟨.chainSup c hc ne uf us, fun v hv => PartialRefinement.le_chainSup _ _ _ _ hv⟩
rcases zorn_le_nonempty this with ⟨v, hv⟩
suffices ∀ i, i ∈ v.carrier from
⟨v, v.subset_iUnion, fun i => v.isOpen _, fun i => v.closure_subset (this ... | Mathlib/Topology/ShrinkingLemma.lean | 218 | 231 |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Algebra.Ring.Int.Defs
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.Size
import Batteries.Data.Int
/-!
# Bitwise operations on integers
Possi... | | -[_+1] => congr_arg ((↑) : ℕ → ℤ) (by simp)
| Mathlib/Data/Int/Bitwise.lean | 417 | 418 |
/-
Copyright (c) 2023 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.RingTheory.DedekindDomain.Dvr
import Mathlib.RingTheory.DedekindDomain.Ideal
/-!
# Criteria under which a Dedekind domain is a PID
This file contains some ... | · exact isUnit_of_mul_eq_one _ _ (FractionalIdeal.coe_ideal_mul_inv I hI)⟩
variable [IsDedekindDomain R]
variable (S : Type*) [CommRing S]
variable [Algebra R S] [NoZeroSMulDivisors R S] [Module.Finite R S]
variable (p : Ideal R) (hp0 : p ≠ ⊥) [IsPrime p]
variable {Sₚ : Type*} [CommRing Sₚ] [Algebra S Sₚ]
variable... | Mathlib/RingTheory/DedekindDomain/PID.lean | 182 | 189 |
/-
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.UnderlyingMap
import Mathlib.Topology.Spectral.Hom
import Mathlib.AlgebraicGeometry.Limits
/-!
# Quasi-compact morphisms
A morp... | intro i
show _ = (X.sheaf.val.map _) 0
rw [map_zero]
apply this
intro i
replace hn :=
congr_arg (fun x => X.presheaf.map (homOfLE (h₁ i)).op (f ^ (Finset.univ.sup n - n i)) * x)
(hn i)
dsimp at hn
simp only [← map_mul, ← map_pow] at hn
rwa [mul_zero, ← mul_assoc, ← pow_add, tsub_add_... | Mathlib/AlgebraicGeometry/Morphisms/QuasiCompact.lean | 300 | 341 |
/-
Copyright (c) 2024 Arend Mellendijk. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Arend Mellendijk
-/
import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.Analysis.SumIntegralComparisons
import Mathlib.NumberTheory.Harmonic.Defs
/-!
This file prove... | _ ≤ ∑ d ∈ Finset.Icc 1 n, (d : ℝ)⁻¹ := ?_
_ = harmonic n := ?_
· rw [Nat.cast_one, integral_inv (by simp [(show ¬ (1 : ℝ) ≤ 0 by norm_num)]), div_one]
· exact (inv_antitoneOn_Icc_right <| by norm_num).integral_le_sum_Ico (Nat.le_add_left 1 n)
· simp only [harmonic_eq_sum_Icc, Rat.cast_sum, Rat.cast_... | Mathlib/NumberTheory/Harmonic/Bounds.lean | 26 | 50 |
/-
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.Data.Fintype.BigOperators
import Mathlib.Data.DFinsupp.BigOperators
import Mathlib.Data.DFinsupp.Order
import Mathlib.Order.Interval.Finset.Basic
import Ma... | Mathlib/Data/DFinsupp/Interval.lean | 211 | 213 | |
/-
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` ... | by_contra fun H => h <| mem_of_formPerm_ne_self _ _ H
theorem formPerm_eq_one_iff (hl : Nodup l) : formPerm l = 1 ↔ l.length ≤ 1 := by
rcases l with - | ⟨hd, tl⟩
· simp
· rw [← formPerm_apply_mem_eq_self_iff _ hl hd mem_cons_self]
constructor
| Mathlib/GroupTheory/Perm/List.lean | 322 | 328 |
/-
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.Data.Set.Finite.Lattice
import Mathlib.Order.ConditionallyCompleteLattice.Indexed
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Order.SuccPred.Ba... | simp only [partialSups_apply, Finset.sup'_apply]
end Preorder
| Mathlib/Order/PartialSups.lean | 126 | 128 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.CharP.Frobenius
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.RingTheory.Polynomial.Basic
/... | refine Polynomial.induction_on P (fun a => by simp) (fun f g hf hg => ?_) fun n a _ => by simp
rw [map_add, aeval_add, aeval_add, hf, hg]
| Mathlib/Algebra/Polynomial/Expand.lean | 175 | 177 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Topology.ContinuousMap.Algebra
import Mathlib.Topology.Compactness.Paracompact
import Mathlib.Topology.Sh... | ∀ i, HasCompactSupport (f i) := by
rcases exists_subset_iUnion_closure_subset_t2space hs ho (fun x _ => hf.point_finite x) hU with
⟨V, hsV, hVo, hVU, hcp⟩
have hVU' i : V i ⊆ U i := subset_closure.trans (hVU i)
rcases exists_subset_iUnion_closure_subset_t2space hs hVo
| Mathlib/Topology/PartitionOfUnity.lean | 440 | 444 |
/-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.Algebra.Polynomial.Identities
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.NumberTheory.Padics.PadicIntegers
import Mathlib.Topology.A... | (max_le_max (newton_seq_succ_dist _ _) (newton_seq_dist_aux _ _))
_ = ‖F.derivative.eval a‖ * T ^ 2 ^ n :=
max_eq_right <|
mul_le_mul_of_nonneg_left (pow_le_pow_of_le_one (norm_nonneg _)
(le_of_lt (T_lt_one hnorm)) this) (norm_nonneg _)
private theorem newton_seq_dist {n k :... | Mathlib/NumberTheory/Padics/Hensel.lean | 279 | 297 |
/-
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.Order.SuccPred.Archimedean
import Mathlib.Order.BoundedOrder.Lattice
/-!
# Successor and predecessor limits
We define the pre... | end NoMaxOrder
section IsSuccArchimedean
variable [IsSuccArchimedean α] [NoMaxOrder α]
theorem IsSuccPrelimit.isMin_of_noMax (h : IsSuccPrelimit a) : IsMin a := by
| Mathlib/Order/SuccPred/Limit.lean | 144 | 150 |
/-
Copyright (c) 2023 Ashvni Narayanan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ashvni Narayanan, Moritz Firsching, Michael Stoll
-/
import Mathlib.Algebra.Group.EvenFunction
import Mathlib.Data.ZMod.Units
import Mathlib.NumberTheory.MulChar.Basic
/-!
# Dirichl... | noncomputable def primitiveCharacter : DirichletCharacter R χ.conductor :=
Classical.choose (factorsThrough_conductor χ).choose_spec
lemma primitiveCharacter_isPrimitive : IsPrimitive (χ.primitiveCharacter) := by
by_cases h : χ.conductor = 0
· rw [isPrimitive_def]
| Mathlib/NumberTheory/DirichletCharacter/Basic.lean | 249 | 254 |
/-
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... | refine ⟨fun h _ h' ↦ eventually_lt_of_limsup_lt (h.trans_lt h') h₂, fun h ↦ ?_⟩
--Two cases: Either `x` is a cluster point from above, or it is not.
| Mathlib/Order/LiminfLimsup.lean | 843 | 844 |
/-
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... | Mathlib/GroupTheory/Perm/Support.lean | 316 | 316 | |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Data.FunLike.Equiv
import Mathlib.Data.Quot
import Mathlib.Data.Subtype
import Mathlib.Logic.U... | toFun ac := (ab.symm.trans ac).trans cd
invFun bd := ab.trans <| bd.trans <| cd.symm
left_inv ac := by ext x; simp only [trans_apply, comp_apply, symm_apply_apply]
| Mathlib/Logic/Equiv/Defs.lean | 333 | 335 |
/-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Kernel.Defs
/-!
# Basic kernels
This file contains basic results about kernels in general and definitions of some particular
kernels.
## Mai... | refine ⟨⟨fun n => Kernel.restrict (seq κ n) hs, inferInstance, ?_⟩⟩
ext1 a
simp_rw [sum_apply, restrict_apply, ← Measure.restrict_sum _ hs, ← sum_apply, kernel_sum_seq]
| Mathlib/Probability/Kernel/Basic.lean | 276 | 278 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Data.ENNReal.Real
/-!
# Properties of addition, multiplication and subtraction on extended non-negative real numbers
In this file we... |
@[simp]
theorem image_coe_Ico (x y : ℝ≥0) : (↑) '' Ico x y = Ico (x : ℝ≥0∞) y := WithTop.image_coe_Ico
| Mathlib/Data/ENNReal/Operations.lean | 461 | 464 |
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Yury Kudryashov
-/
import Mathlib.Order.UpperLower.Closure
import Mathlib.Order.UpperLower.Fibration
import Mathlib.Tactic.TFAE
import Mathlib.Topology.ContinuousOn
import Ma... | (specializes_of_nhdsWithin h.le hx).antisymm (specializes_of_nhdsWithin h.ge hy)
| Mathlib/Topology/Inseparable.lean | 460 | 461 |
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