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 |
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/-
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... | Mathlib/LinearAlgebra/Ray.lean | 708 | 714 | |
/-
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 cast_sub' (a b : ZMod n) : (cast (a - b : ZMod n) : R) = cast a - cast b :=
cast_sub dvd_rfl a b
@[simp]
theorem cast_pow' (a : ZMod n) (k : ℕ) : (cast (a ^ k : ZMod n) : R) = (cast a : R) ^ k :=
cast_pow dvd_rfl a k
| Mathlib/Data/ZMod/Basic.lean | 359 | 366 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot, Yury Kudryashov, Rémy Degenne
-/
import Mathlib.Algebra.Order.Group.Abs
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.... |
end Set
| Mathlib/Algebra/Order/Interval/Set/Group.lean | 238 | 241 |
/-
Copyright (c) 2022 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Sébastien Gouëzel, Heather Macbeth, Floris van Doorn
-/
import Mathlib.Topology.FiberBundle.Basic
/-!
# Standard constructions on fiber bundles
This file contains... | open_target := ((map_continuous f).isOpen_preimage _ e.open_baseSet).prod isOpen_univ
open_baseSet := (map_continuous f).isOpen_preimage _ e.open_baseSet
continuousOn_toFun :=
(Pullback.continuous_proj F E f).continuousOn.prodMk
(continuous_snd.comp_continuousOn <|
| Mathlib/Topology/FiberBundle/Constructions.lean | 322 | 326 |
/-
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... | -- Increased priority so this fires before `image_subset_iff`
@[simp high]
| Mathlib/Logic/Equiv/Set.lean | 65 | 66 |
/-
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, Wen Yang
-/
import Mathlib.LinearAlgebra.Matrix.Transvection
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mat... |
theorem det_of_lowerTriangular [LinearOrder m] (M : Matrix m m R) (h : M.BlockTriangular toDual) :
M.det = ∏ i : m, M i i := by
rw [← det_transpose]
exact det_of_upperTriangular h.transpose
open Polynomial
| Mathlib/LinearAlgebra/Matrix/Block.lean | 280 | 286 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Oliver Nash
-/
import Mathlib.Topology.PartialHomeomorph
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.D... | theorem univBall_symm_apply_center (c : P) (r : ℝ) : (univBall c r).symm c = 0 := by
have : 0 ∈ (univBall c r).source := by simp
| Mathlib/Analysis/NormedSpace/HomeomorphBall.lean | 140 | 141 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Simon Hudon, Mario Carneiro
-/
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Init
import Mathlib.Data.Int.Init
import Mathlib.... | Mathlib/Algebra/Group/Basic.lean | 490 | 490 | |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Pairwise
import Mathlib.Data.Set.BooleanAlgebra
/-!
# The set lattice
This file is a collectio... | Mathlib/Data/Set/Lattice.lean | 2,151 | 2,151 | |
/-
Copyright (c) 2024 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.MvPolynomial.Monad
import Mathlib.LinearAlgebra.Charpoly.ToMatrix
import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition
import Mathlib.Li... | lemma isNilRegular_iff_coeff_polyCharpoly_nilRank_ne_zero :
IsNilRegular φ x ↔
MvPolynomial.eval (b.repr x)
((polyCharpoly φ b).coeff (nilRank φ)) ≠ 0 := by
rw [IsNilRegular, polyCharpoly_coeff_eval]
| Mathlib/Algebra/Module/LinearMap/Polynomial.lean | 514 | 518 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Logic.Function.Conjugate
/-!
# Iterations of a function
In this file we prove simple properties of `Nat.iterate f n` a.k.a. `f^[n]`:
* `iterate_ze... | Semiconj.iterate_right h n
theorem iterate_left (h : Commute f g) (n : ℕ) : Commute f^[n] g :=
(h.symm.iterate_right n).symm
theorem iterate_iterate (h : Commute f g) (m n : ℕ) : Commute f^[m] g^[n] :=
(h.iterate_left m).iterate_right n
theorem iterate_eq_of_map_eq (h : Commute f g) (n : ℕ) {x} (hx : f x = g x... | Mathlib/Logic/Function/Iterate.lean | 121 | 129 |
/-
Copyright (c) 2019 Abhimanyu Pallavi Sudhir. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Abhimanyu Pallavi Sudhir
-/
import Mathlib.Order.Filter.FilterProduct
import Mathlib.Analysis.SpecificLimits.Basic
/-!
# Construction of the hyperreal numbers as an ultrapro... | Mathlib/Data/Real/Hyperreal.lean | 781 | 782 | |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.CharP.Reduced
import Mathlib.RingTheory.IntegralDomain
-- TODO: remove Mathlib.Algebra.CharP.Reduced and move the last two lemmas to Lemmas
/-... | exact map_one σ⟩
map_one' := by ext1; simp only [OneMemClass.coe_one, map_one]
map_mul' := fun ξ₁ ξ₂ ↦ by
| Mathlib/RingTheory/RootsOfUnity/Basic.lean | 125 | 127 |
/-
Copyright (c) 2014 Microsoft Corporation. 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.Algebra.Ring.Int.Defs
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.Cast.Order.Basic
import Math... | Mathlib/Data/Num/Lemmas.lean | 983 | 997 | |
/-
Copyright (c) 2022 Floris van Doorn. 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.VectorBundle.Basic
/-! # Tangent bundles
This file defines the tangent bundle as a `C^n` vector bundle.
Let `... | lemma contMDiffOn_vectorSpace_iff_contDiffOn
{V : Π (x : E), TangentSpace 𝓘(𝕜, E) x} {s : Set E} :
ContMDiffOn 𝓘(𝕜, E) 𝓘(𝕜, E).tangent n (fun x ↦ (V x : TangentBundle 𝓘(𝕜, E) E)) s ↔
ContDiffOn 𝕜 n V s := by
| Mathlib/Geometry/Manifold/VectorBundle/Tangent.lean | 471 | 474 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Amelia Livingston, Yury Kudryashov,
Neil Strickland, Aaron Anderson
-/
import Mathlib.Algebra.Divisibility.Basic
import Mathlib.Algeb... | theorem IsRelPrime.dvd_of_dvd_mul_right_of_isPrimal (H1 : IsRelPrime x z) (H2 : x ∣ y * z)
(h : IsPrimal x) : x ∣ y := by
| Mathlib/Algebra/Divisibility/Units.lean | 205 | 206 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl, Patrick Massot
-/
import Mathlib.Data.Set.Image
import Mathlib.Data.SProd
/-!
# Sets in product and pi types
This file proves basic properties of prod... | open Function.Pullback in
/-- The projection $(X \times_Y Z) \times_X (X \times_Y Z) \to Z \times_Y Z$. -/
| Mathlib/Data/Set/Prod.lean | 507 | 508 |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Algebra.Group.Embedding
import Mathlib.Order.Interval.Multiset
/-!
# Finite intervals of naturals
This file proves that `ℕ` is a `LocallyFiniteOrder` and... |
lemma Icc_insert_succ_right (h : a ≤ b + 1) : insert (b + 1) (Icc a b) = Icc a (b + 1) := by
ext x
| Mathlib/Order/Interval/Finset/Nat.lean | 153 | 155 |
/-
Copyright (c) 2022 Praneeth Kolichala. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Praneeth Kolichala
-/
import Mathlib.Topology.Homotopy.Equiv
import Mathlib.CategoryTheory.Equivalence
import Mathlib.AlgebraicTopology.FundamentalGroupoid.Product
/-!
# Homotopic... | hcast (H.apply_one x₁) :=
Quotient.inductionOn p fun p' => by
apply @eq_path_of_eq_image _ _ _ _ H.uliftMap _ _ _ _ _ ((Path.refl (ULift.up _)).prod p')
intros
rw [Path.prod_coe, ulift_apply H]
simp
| Mathlib/AlgebraicTopology/FundamentalGroupoid/InducedMaps.lean | 173 | 179 |
/-
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... | theorem card_sym2 (s : Finset α) : s.sym2.card = Nat.choose (s.card + 1) 2 := by
rw [card_def, sym2_val, Multiset.card_sym2, ← card_def]
end
variable {s t : Finset α} {a b : α}
section
variable [DecidableEq α]
theorem sym2_eq_image : s.sym2 = (s ×ˢ s).image Sym2.mk := by
ext z
| Mathlib/Data/Finset/Sym.lean | 122 | 133 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Yury Kudryashov, Neil Strickland
-/
import Mathlib.Algebra.Group.Defs
import Mathlib.Algebra.GroupWithZero.Defs
import Mathlib.Data.I... |
section Mul
| Mathlib/Algebra/Ring/Defs.lean | 249 | 250 |
/-
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... | theorem tendstoLocallyUniformly_iff {ι : Type*} [TopologicalSpace β] {F : ι → β → α} {f : β → α}
{p : Filter ι} :
TendstoLocallyUniformly F f p ↔
| Mathlib/Topology/EMetricSpace/Basic.lean | 127 | 129 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jan-David Salchow, Sébastien Gouëzel, Jean Lo, Yury Kudryashov, Frédéric Dupuis,
Heather Macbeth
-/
import Mathlib.Algebra.Module.Opposite
import Mathlib.Topology.Algebra.Group.Qu... | Mathlib/Topology/Algebra/Module/Basic.lean | 1,909 | 1,912 | |
/-
Copyright (c) 2023 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Computability.AkraBazzi.GrowsPolynomially
import Mathlib.Analysis.Calculus.Deriv.Inv
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
/-!
# Divid... | DifferentiableAt ℝ (fun z => 1 + ε z) x :=
DifferentiableAt.add (differentiableAt_const _) <| differentiableAt_smoothingFn hx
lemma differentiableOn_one_add_smoothingFn : DifferentiableOn ℝ (fun z => 1 + ε z) (Set.Ioi 1) :=
| Mathlib/Computability/AkraBazzi/AkraBazzi.lean | 378 | 381 |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Julian Kuelshammer
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Algebra.Group.Pointwise.Set.Finite
import Mathlib.Alge... | Mathlib/GroupTheory/OrderOfElement.lean | 1,322 | 1,323 | |
/-
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 | 1,563 | 1,564 | |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Floris van Doorn
-/
import Mathlib.CategoryTheory.Limits.Filtered
import Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts
import Mathlib.CategoryTheory.Limits.Shapes.Ker... | colimit.ι F.leftOp j ≫ (colimitLeftOpIsoUnopLimit F).hom = (limit.π F j.unop).unop := by
simp [colimitLeftOpIsoUnopLimit]
@[reassoc (attr := simp)]
lemma π_comp_colimitLeftOpIsoUnopLimit_inv (F : J ⥤ Cᵒᵖ) [HasLimit F] (j : J) :
(limit.π F j).unop ≫ (colimitLeftOpIsoUnopLimit F).inv = colimit.ι F.leftOp (op j... | Mathlib/CategoryTheory/Limits/Opposites.lean | 437 | 442 |
/-
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... | trans ⨅ (y) (_ : y ∈ s), ‖c‖₊ • edist x y
· refine (this.iInf_congr _ fun y => ?_).symm
simp_rw [smul_mem_smul_set_iff₀ hc, edist_smul₀]
· have : (‖c‖₊ : ENNReal) ≠ 0 := by simp [hc]
simp_rw [ENNReal.smul_def, smul_eq_mul, ENNReal.mul_iInf_of_ne this ENNReal.coe_ne_top]
theorem infDist_smul₀ {c : 𝕜} (hc... | Mathlib/Analysis/NormedSpace/Pointwise.lean | 57 | 66 |
/-
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` ... |
@[simp]
theorem formPerm_apply_getLast (x : α) (xs : List α) :
| Mathlib/GroupTheory/Perm/List.lean | 131 | 133 |
/-
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.Data.NNReal.Basic
import Mathlib.Order.Fin.Tuple
import Mathlib.Order.Interval.Set.Monotone
import Mathlib.Topology.MetricSpace.Basic
import Mathlib.... | le_iff_bounds.2 ⟨fun i ↦ (ha_anti i).antitone hkl, fun i ↦ (hb_mono i).monotone hkl⟩⟩,
fun n x hx i _ ↦ ⟨(ha_mem _ _).1.trans_le (hx.1 _), (hx.2 _).trans_lt (hb_mem _ _).2⟩,
tendsto_pi_nhds.2 ha_tendsto, tendsto_pi_nhds.2 hb_tendsto⟩
section Distortion
variable [Fintype ι]
| Mathlib/Analysis/BoxIntegral/Box/Basic.lean | 428 | 434 |
/-
Copyright (c) 2018 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.GeomSum
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Algebra.Order.CauSeq.Basic
/-!
# Cauchy sequences and big oper... | conv in _ / _ => rw [← neg_div_neg_eq, neg_sub, neg_sub]
have : 0 < 1 - abv x := sub_pos.2 hx1
refine of_mono_bounded _ (a := (1 : α) / (1 - abv x)) (m := 0) ?_ ?_
· intro n _
rw [abs_of_nonneg]
· gcongr
exact sub_le_self _ (abv_pow abv x n ▸ abv_nonneg _ _)
refine div_nonneg (sub_nonneg.2 ?_)... | Mathlib/Algebra/Order/CauSeq/BigOperators.lean | 184 | 200 |
/-
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.Fintype.Card
import Mathlib.Order.UpperLower.Basic
/-!
# Intersecting families
This file defines intersecting families and proves their basic proper... |
theorem Intersecting.exists_card_eq (hs : (s : Set α).Intersecting) :
∃ t, s ⊆ t ∧ 2 * #t = Fintype.card α ∧ (t : Set α).Intersecting := by
have := hs.card_le
rw [Nat.mul_comm, ← Nat.le_div_iff_mul_le Nat.two_pos] at this
revert hs
refine s.strongDownwardInductionOn ?_ this
rintro s ih _hcard hs
by_cas... | Mathlib/Combinatorics/SetFamily/Intersecting.lean | 173 | 192 |
/-
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.MeasureTheory.Measure.Comap
import Mathlib.MeasureTheory.Measure.QuasiMeasurePreserving
/-!
# Restricting a measure to a subset or a s... | restrict_restrict₀' ht.nullMeasurableSet
theorem restrict_comm (hs : MeasurableSet s) :
(μ.restrict t).restrict s = (μ.restrict s).restrict t := by
| Mathlib/MeasureTheory/Measure/Restrict.lean | 187 | 190 |
/-
Copyright (c) 2023 Richard M. Hill. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Richard M. Hill
-/
import Mathlib.RingTheory.PowerSeries.Trunc
import Mathlib.RingTheory.PowerSeries.Inverse
import Mathlib.RingTheory.Derivation.Basic
/-!
# Definitions
In this fil... | theorem derivativeFun_smul (r : R) (f : R⟦X⟧) : derivativeFun (r • f) = r • derivativeFun f := by
rw [smul_eq_C_mul, smul_eq_C_mul, derivativeFun_mul, derivativeFun_C, smul_zero, add_zero,
smul_eq_mul]
| Mathlib/RingTheory/PowerSeries/Derivative.lean | 90 | 92 |
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Eric Wieser
-/
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Dual.Lemmas
import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
import Mathlib.Li... |
@[simp]
theorem cRank_zero {m n : Type*} [Nontrivial R] : cRank (0 : Matrix m n R) = 0 := by
obtain hn | hn := isEmpty_or_nonempty n
| Mathlib/Data/Matrix/Rank.lean | 133 | 136 |
/-
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.LinearAlgebra.Basis.Basic
import Mathlib.LinearAlgebra.Basis.Submodule
import Mathlib.LinearAlgebra.Dimension.Finrank
import Mathlib.LinearAlgebra.Invarian... | (finrank R M : Cardinal.{w}) = #ι := by
cases @nonempty_fintype _ (Module.Finite.finite_basis h)
| Mathlib/LinearAlgebra/Dimension/StrongRankCondition.lean | 431 | 432 |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Module.BigOperators
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.Nat.Squarefree
imp... | end AddMonoid
instance instAddMonoidWithOne [AddMonoidWithOne R] : AddMonoidWithOne (ArithmeticFunction R) :=
{ ArithmeticFunction.instAddMonoid,
| Mathlib/NumberTheory/ArithmeticFunction.lean | 216 | 219 |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Nat.Find
import Mathlib.Data.Stream.Init
import Mathlib.Tactic.Common
/-!
# Coinductive formalization of unbounded computations.
This fil... | theorem ret_orElse (a : α) (c₂ : Computation α) : (pure a <|> c₂) = pure a :=
destruct_eq_pure <| by
unfold_projs
simp [orElse]
@[simp]
theorem orElse_pure (c₁ : Computation α) (a : α) : (think c₁ <|> pure a) = pure a :=
destruct_eq_pure <| by
unfold_projs
simp [orElse]
@[simp]
theorem orElse_thin... | Mathlib/Data/Seq/Computation.lean | 776 | 789 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Dia... | MeasurableSet { p : α × ℝ | p.fst ∈ s ∧ p.snd ∈ Ico (f p.fst) (g p.fst) } := by
dsimp only [regionBetween, Ico, mem_setOf_eq, setOf_and]
refine
MeasurableSet.inter ?_
((measurableSet_le (hf.comp measurable_fst) measurable_snd).inter
(measurableSet_lt measurable_snd (hg.comp measurable_fst)))
... | Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | 468 | 476 |
/-
Copyright (c) 2018 Louis Carlin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Louis Carlin, Mario Carneiro
-/
import Mathlib.Algebra.EuclideanDomain.Defs
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Regular
import Mathlib.Algebra.Grou... |
theorem mul_sub_div_left (x y z : R) (h1 : z ≠ 0) (h2 : z ∣ y) : (z * x - y) / z = x - y / z := by
rw [eq_comm]
apply eq_div_of_mul_eq_right h1
rw [mul_sub, EuclideanDomain.mul_div_cancel' h1 h2]
theorem mul_sub_div_right (x y z : R) (h1 : z ≠ 0) (h2 : z ∣ y) : (x * z - y) / z = x - y / z := by
rw [mul_comm x... | Mathlib/Algebra/EuclideanDomain/Basic.lean | 347 | 354 |
/-
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.Topology.Homeomorph.Lemmas
import Mathlib.Topology.Sets.Closeds
/-!
# Noetherian space
A Noetherian space is a topological space that satisfies any of the ... | rcases NoetherianSpace.exists_finite_set_closeds_irreducible s with ⟨S, hSf, hS, rfl⟩
refine ⟨(↑) '' S, hSf.image _, Set.forall_mem_image.2 fun S _ ↦ S.2, Set.forall_mem_image.2 hS,
?_⟩
lift S to Finset (Closeds α) using hSf
simp [← Finset.sup_id_eq_sSup, Closeds.coe_finset_sup]
/-- In a Noetherian space, ... | Mathlib/Topology/NoetherianSpace.lean | 175 | 191 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.RingTheory.Ideal.Operations
/-!
# Maps on modules and ideals
Main definitions include `Ideal.map`, `Ideal.comap`, `RingHom.ker`, `Module.annihilator`
and `Subm... | theorem LinearEquiv.annihilator_eq (e : M ≃ₗ[R] M') :
Module.annihilator R M = Module.annihilator R M' :=
(e.annihilator_le_of_surjective e.surjective).antisymm (e.annihilator_le_of_injective e.injective)
| Mathlib/RingTheory/Ideal/Maps.lean | 785 | 788 |
/-
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.Order.Interval.Set.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
import Mathlib.Tactic.AdaptationNote
... | · obtain ⟨m, hm⟩ := exists_nat_ge k
refine ⟨m, fun N => Nat.cast_le.1 ((hk N).trans ?_)⟩
rwa [← ENNReal.coe_natCast, ENNReal.coe_le_coe]
· refine ⟨k, fun N => ?_⟩
simp only [ENNReal.coe_natCast, Nat.cast_le, hk N]
/-- A variant of Doob's upcrossing estimate obtained by taking the supremum on both sides... | Mathlib/Probability/Martingale/Upcrossing.lean | 778 | 801 |
/-
Copyright (c) 2022 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Group.Nat.Even
import Mathlib.Data.Nat.Cast.Basic
import Mathlib.Data.Nat.Cast.Commute
import Mathlib.Data.Set.Operations
import Mathlib.Logic.Fu... | lemma even_or_odd' (n : ℕ) : ∃ k, n = 2 * k ∨ n = 2 * k + 1 := by
| Mathlib/Algebra/Ring/Parity.lean | 218 | 218 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kim Morrison
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Algebra.Group.InjSurj
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Tactic.FastInstance
impo... | Mathlib/Data/Finsupp/Defs.lean | 1,060 | 1,063 | |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.InnerProductSpace.Defs
impor... | theorem inner_sub_sub_self (x y : E) : ⟪x - y, x - y⟫ = ⟪x, x⟫ - ⟪x, y⟫ - ⟪y, x⟫ + ⟪y, y⟫ := by
simp only [inner_sub_left, inner_sub_right]; ring
| Mathlib/Analysis/InnerProductSpace/Basic.lean | 240 | 242 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.BoxIntegral.Partition.Filter
import Mathlib.Analysis.BoxIntegral.Partition.Measure
import Mathlib.Analysis.Oscillation
import Mathlib.Data.B... | have HJi : Integrable J l f vol := h.to_subbox Hle
set r := fun x => min (h.convergenceR δ' C x) (HJi.convergenceR δ' C x)
have hJd : J.distortion ≤ C := le_trans (Finset.le_sup hJ) (le_max_left _ _)
rcases l.exists_memBaseSet_isPartition J hJd r with ⟨πJ, hC, hp⟩
have hC₁ : l.MemBaseSet J C (HJi.co... | Mathlib/Analysis/BoxIntegral/Basic.lean | 527 | 532 |
/-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Algebra.Order.Pi
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Data.Real.Pointwise
/... | inf := (· ⊓ ·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; rfl
inf_le_right := fun p q x =>
| Mathlib/Analysis/Seminorm.lean | 451 | 455 |
/-
Copyright (c) 2014 Microsoft Corporation. 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.Algebra.Ring.Int.Defs
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.Cast.Order.Basic
import Math... | lt_iff_le_not_le := by
intro a b
transfer_rw
| Mathlib/Data/Num/Lemmas.lean | 545 | 547 |
/-
Copyright (c) 2022 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.Polynomial.Mirror
import Mathlib.Algebra.Ring.Regular
import Mathlib.Data.Int.Order.Units
import Mathlib.RingTheory.Coprime.Basic
/-!
# Unit... | rw [trailingCoeff, trinomial_natTrailingDegree hkm hmn hu, trinomial_trailing_coeff' hkm hmn]
theorem trinomial_monic (hkm : k < m) (hmn : m < n) : (trinomial k m n u v 1).Monic := by
| Mathlib/Algebra/Polynomial/UnitTrinomial.lean | 95 | 97 |
/-
Copyright (c) 2019 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard
-/
import Mathlib.Data.EReal.Basic
deprecated_module (since := "2025-04-13")
| Mathlib/Data/Real/EReal.lean | 554 | 557 | |
/-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.Galois.Basic
/-!
# Separably Closed Field
In this file we define the typeclass for separably closed fields and separable closures,
and prove some of thei... | obtain ⟨x, hx⟩ := H (q * C (leadingCoeff q)⁻¹) (monic_mul_leadingCoeff_inv hq.ne_zero) hirr' hsep'
exact degree_mul_leadingCoeff_inv q hq.ne_zero ▸ degree_eq_one_of_irreducible_of_root hirr' hx
theorem degree_eq_one_of_irreducible [IsSepClosed k] {p : k[X]}
| Mathlib/FieldTheory/IsSepClosed.lean | 185 | 188 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Bhavik Mehta, Stuart Presnell
-/
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Order.Monotone.Defs
/-!
# Binomial coefficients
This file defines binomial coeffic... |
TODO: Prove that `choose (-n) k = (-1)^k * multichoose n k`,
where `choose` is the generalized binomial coefficient.
<https://github.com/leanprover-community/mathlib/pull/15072#issuecomment-1171415738>
| Mathlib/Data/Nat/Choose/Basic.lean | 336 | 339 |
/-
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... | norm_setIntegral_le_of_norm_le_const measure_Icc_lt_top fun θ _ =>
calc
‖(∏ i : Fin n, R i * exp (θ i * I) * I : ℂ) • f (torusMap c R θ)‖ =
| Mathlib/MeasureTheory/Integral/TorusIntegral.lean | 176 | 178 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.RingTheory.Ideal.Operations
/-!
# Maps on modules and ideals
Main definitions include `Ideal.map`, `Ideal.comap`, `RingHom.ker`, `Module.annihilator`
and `Subm... | theorem map_comap_of_equiv {I : Ideal R} (f : R ≃+* S) : I.map (f : R →+* S) = I.comap f.symm :=
le_antisymm (Ideal.map_le_comap_of_inverse _ _ _ (Equiv.left_inv' _))
(Ideal.comap_le_map_of_inverse _ _ _ (Equiv.right_inv' _))
/-- If `f : R ≃+* S` is a ring isomorphism and `I : Ideal R`, then `comap f.symm I = ma... | Mathlib/RingTheory/Ideal/Maps.lean | 416 | 423 |
/-
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... | (completeSpace_congr isUniformEmbedding_equivRealProd).mpr inferInstance
| Mathlib/Analysis/Complex/Basic.lean | 116 | 117 |
/-
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.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.Polynomial.Lifts
import Mathlib.Grou... |
/-- A ring is algebraic over the ring `A` iff it is algebraic over the field of fractions of `A`.
-/
theorem comap_isAlgebraic_iff [Algebra A C] [Algebra K C] [IsScalarTower A K C] :
Algebra.IsAlgebraic A C ↔ Algebra.IsAlgebraic K C :=
⟨fun h => ⟨fun x => (isAlgebraic_iff A K C).mp (h.isAlgebraic x)⟩,
fun h =... | Mathlib/RingTheory/Localization/Integral.lean | 149 | 159 |
/-
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... | ⟨∅, isCompact_empty⟩
theorem mem_cocompact : s ∈ cocompact X ↔ ∃ t, IsCompact t ∧ tᶜ ⊆ s :=
hasBasis_cocompact.mem_iff
| Mathlib/Topology/Compactness/Compact.lean | 511 | 515 |
/-
Copyright (c) 2017 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Keeley Hoek
-/
import Mathlib.Algebra.NeZero
import Mathlib.Data.Int.DivMod
import Mathlib.Logic.Embedding.Basic
import Mathlib.Logic.Equiv.Set
import Mathlib.Tactic.... | end CastPred
| Mathlib/Data/Fin/Basic.lean | 923 | 924 |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Yaël Dillies
-/
import Mathlib.Order.Cover
import Mathlib.Order.LatticeIntervals
import Mathlib.Order.GaloisConnection.Defs
/-!
# Modular Lattices
This file defines (... | instance : IsWeakLowerModularLattice (OrderDual α) :=
⟨fun ha hb => (ha.ofDual.sup_of_inf_of_inf_left hb.ofDual).toDual⟩
| Mathlib/Order/ModularLattice.lean | 103 | 105 |
/-
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... | (- a).val = n - a.val := by simp_all [neg_val a, na.out]
theorem val_sub {n : ℕ} [NeZero n] {a b : ZMod n} (h : b.val ≤ a.val) :
(a - b).val = a.val - b.val := by
by_cases hb : b = 0
· cases hb; simp
· have : NeZero b := ⟨hb⟩
rw [sub_eq_add_neg, val_add, val_neg_of_ne_zero, ← Nat.add_sub_assoc (le_of... | Mathlib/Data/ZMod/Basic.lean | 996 | 1,003 |
/-
Copyright (c) 2020 Benjamin Davidson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Benjamin Davidson, Jeremy Tan
-/
import Mathlib.Analysis.Complex.AbelLimit
import Mathlib.Analysis.SpecialFunctions.Complex.Arctan
/-! ### Leibniz's series for `π` -/
namespace Re... | -- The series is alternating with terms of decreasing magnitude, so it converges to some limit
obtain ⟨l, h⟩ :
∃ l, Tendsto (fun n ↦ ∑ i ∈ range n, (-1 : ℝ) ^ i / (2 * i + 1)) atTop (𝓝 l) := by
apply Antitone.tendsto_alternating_series_of_tendsto_zero
· exact antitone_iff_forall_lt.mpr fun _ _ _ ↦ by... | Mathlib/Data/Real/Pi/Leibniz.lean | 21 | 57 |
/-
Copyright (c) 2022 María Inés de Frutos-Fernández. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: María Inés de Frutos-Fernández
-/
import Mathlib.Order.Filter.Cofinite
import Mathlib.RingTheory.DedekindDomain.Ideal
import Mathlib.RingTheory.UniqueFactorizationDomai... | · rw [hI, zero_zpow n hn, zero_zpow (-n) (neg_ne_zero.mpr hn), count_zero, neg_zero]
· rw [eq_neg_iff_add_eq_zero, ← count_mul K v (zpow_ne_zero _ hI) (zpow_ne_zero _ hI),
← zpow_add₀ hI, neg_add_cancel, zpow_zero]
| Mathlib/RingTheory/DedekindDomain/Factorization.lean | 422 | 424 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Yury Kudryashov, Neil Strickland
-/
import Mathlib.Algebra.Group.Semiconj.Defs
import Mathlib.Algebra.Ring.Defs
/-!
# Semirings and ... |
end
| Mathlib/Algebra/Ring/Semiconj.lean | 57 | 58 |
/-
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.Matroid.IndepAxioms
/-!
# Matroid Duality
For a matroid `M` on ground set `E`, the collection of complements of the bases of `M` is the
collection o... | theorem IsBase.compl_inter_isBasis_of_inter_isBasis (hB : M.IsBase B) (hBX : M.IsBasis (B ∩ X) X) :
M✶.IsBasis ((M.E \ B) ∩ (M.E \ X)) (M.E \ X) := by
| Mathlib/Data/Matroid/Dual.lean | 179 | 180 |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Covering.VitaliFamily
import Mathlib.MeasureTheory.Function.AEMeasurableOrder
import Mathlib.MeasureTheory.Integral.Average
import ... | filter_upwards [ae_all_iff.2 this] with x hx
obtain ⟨n, hn⟩ : ∃ n, x ∈ u n := by simpa only [← u_univ, mem_iUnion] using mem_univ x
apply Tendsto.congr' _ (hx n)
filter_upwards [v.eventually_filterAt_subset_of_nhds ((u_open n).mem_nhds hn),
v.eventually_filterAt_measurableSet x] with a ha h'a
congr 1
re... | Mathlib/MeasureTheory/Covering/Differentiation.lean | 872 | 883 |
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Data.Set.BooleanAlgebra
import Mathlib.Tactic.AdaptationNote
/-!
# Relations
This file defines bundled relations. A relation between `α` and `β` is a f... | theorem image_inter_dom_eq (s : Set α) : r.image (s ∩ r.dom) = r.image s := by
apply Set.eq_of_subset_of_subset
| Mathlib/Data/Rel.lean | 250 | 251 |
/-
Copyright (c) 2021 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Devon Tuma
-/
import Mathlib.Algebra.Polynomial.Eval.Defs
import Mathlib.Analysis.Asymptotics.Lemmas
/-!
# Super-Polynomial Function Decay
This file defines a predicate `Asymptotics.Supe... | theorem SuperpolynomialDecay.mul_const [ContinuousMul β] (hf : SuperpolynomialDecay l k f) (c : β) :
SuperpolynomialDecay l k fun n => f n * c := fun z => by
simpa only [← mul_assoc, zero_mul] using Tendsto.mul_const c (hf z)
| Mathlib/Analysis/Asymptotics/SuperpolynomialDecay.lean | 84 | 86 |
/-
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... | /-- There is a primitive additive character on `ZMod n` if the characteristic of the target
does not divide `n` -/
noncomputable def primitiveZModChar (n : ℕ+) (F' : Type v) [Field F'] (h : (n : F') ≠ 0) :
PrimitiveAddChar (ZMod n) F' :=
have : NeZero (n : F') := ⟨h⟩
⟨n, zmodChar n (IsCyclotomicExtension.zeta_p... | Mathlib/NumberTheory/LegendreSymbol/AddCharacter.lean | 186 | 192 |
/-
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.MeasureTheory.Measure.Typeclasses.Finite
import Mathlib.MeasureTheory.Measure.Typeclasses.NoAtoms
import Mathlib.MeasureTheory.Measure.... | Mathlib/MeasureTheory/Measure/Typeclasses.lean | 686 | 688 | |
/-
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, Yuyang Zhao
-/
import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
import Mathlib.Algebra.Order.ZeroLEOne
import Mathli... | lemma zero_le_four [Preorder α] [ZeroLEOneClass α] [AddLeftMono α] :
(0 : α) ≤ 4 := by
rw [← three_add_one_eq_four]
exact add_nonneg zero_le_three zero_le_one
| Mathlib/Algebra/Order/Monoid/NatCast.lean | 38 | 41 |
/-
Copyright (c) 2024 Newell Jensen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Newell Jensen, Mitchell Lee, Óscar Álvarez
-/
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.Algebra.Ring.Int.Parity
import Mathlib.GroupTheory.Coxeter.Matrix
import Mat... | theorem length_alternatingWord (i i' : B) (m : ℕ) :
List.length (alternatingWord i i' m) = m := by
induction' m with m ih generalizing i i'
· dsimp [alternatingWord]
· simpa [alternatingWord] using ih i' i
lemma getElem_alternatingWord (i j : B) (p k : ℕ) (hk : k < p) :
(alternatingWord i j p)[k]'(by sim... | Mathlib/GroupTheory/Coxeter/Basic.lean | 408 | 427 |
/-
Copyright (c) 2021 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Joël Riou
-/
import Mathlib.Algebra.Homology.HomologicalComplex
/-!
# Bicomplexes
Given a category `C` with zero morphisms and two complex shapes
`c₁ : ComplexShape I₁` a... | (i₁ i₁' i₁'' : I₁) (i₂ : I₂) :
(K.d i₁ i₁').f i₂ ≫ (K.d i₁' i₁'').f i₂ = 0 := by
rw [← comp_f, d_comp_d, zero_f]
@[reassoc]
| Mathlib/Algebra/Homology/HomologicalBicomplex.lean | 130 | 134 |
/-
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... | @[deprecated (since := "2025-01-02")]
alias Filter.Tendsto.fin_insertNth := Filter.Tendsto.finInsertNth
theorem ContinuousAt.finInsertNth
(i : Fin (n + 1)) {f : X → π i} {g : X → ∀ j : Fin n, π (i.succAbove j)} {x : X}
(hf : ContinuousAt f x) (hg : ContinuousAt g x) :
ContinuousAt (fun a => i.insertNth (f ... | Mathlib/Topology/Constructions.lean | 805 | 821 |
/-
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... | rw [rtake]
| Mathlib/Data/List/DropRight.lean | 74 | 74 |
/-
Copyright (c) 2021 Aaron Anderson, Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Kevin Buzzard, Yaël Dillies, Eric Wieser
-/
import Mathlib.Data.Finset.Lattice.Union
import Mathlib.Data.Finset.Pairwise
import Mathlib.Data.Finset.Prod
i... | · refine (hs.pairwiseDisjoint hi.1 hj.1 hij).mono ?_ ?_
· convert le_sup (α := α) hi.2; simp
· convert le_sup (α := α) hj.2; simp
theorem supIndep_product_iff {s : Finset ι} {t : Finset ι'} {f : ι × ι' → α} :
(s.product t).SupIndep f ↔ (s.SupIndep fun i => t.sup fun i' => f (i, i'))
| Mathlib/Order/SupIndep.lean | 213 | 218 |
/-
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, Mario Carneiro, Johan Commelin
-/
import Mathlib.NumberTheory.Padics.PadicNumbers
import Mathlib.RingTheory.DiscreteValuationRing.Basic
/-!
# p-adic integers
This f... | theorem unitCoeff_spec {x : ℤ_[p]} (hx : x ≠ 0) :
x = (unitCoeff hx : ℤ_[p]) * (p : ℤ_[p]) ^ x.valuation := by
apply Subtype.coe_injective
| Mathlib/NumberTheory/Padics/PadicIntegers.lean | 364 | 366 |
/-
Copyright (c) 2020 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky, Anthony DeRossi
-/
import Mathlib.Data.List.Basic
/-!
# Properties of `List.reduceOption`
In this file we prove basic lemmas about `List.reduceOption`.
-/
namespac... | l.reduceOption = [] ↔ ∃ n, l = replicate n none := by
dsimp [reduceOption]
rw [filterMap_eq_nil_iff]
constructor
· intro h
| Mathlib/Data/List/ReduceOption.lean | 49 | 53 |
/-
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.Group.Multiset.Basic
import Mathlib.Data.PNat.Prime
import Mathlib.Data.Nat.Factors
import Mathlib.Data.Multiset.OrderedMonoid
i... | theorem prod_ofPNatMultiset (v : Multiset ℕ+) (h) : ((ofPNatMultiset v h).prod : ℕ+) = v.prod := by
dsimp [prod]
rw [to_ofPNatMultiset]
/-- Lists can be coerced to multisets; here we have some results
about how this interacts with our constructions on multisets. -/
| Mathlib/Data/PNat/Factors.lean | 158 | 163 |
/-
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.Group.Units.Basic
import Mathlib.RingTheory.MvPowerSeries.Basic
import Mathlib.RingTheory.MvPowerSeries.NoZeroDivisors
import Mathl... | if n = 0 then a
else
-a *
∑ x ∈ antidiagonal n, if x.2 < n then coeff R x.1 φ * coeff R x.2 (inv.aux a φ) else 0 :=
show inv.aux a φ n = _ by
cases Subsingleton.elim ‹DecidableEq σ› (Classical.decEq σ)
rw [inv.aux]
rfl
/-- A multivariate formal power series is invertible if ... | Mathlib/RingTheory/MvPowerSeries/Inverse.lean | 73 | 82 |
/-
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.Geometry.Manifold.Algebra.Structures
import Mathlib.Geometry.Manifold.BumpFunction
import Mathlib.Topology.MetricSpace.PartitionOfUnity
import Mathli... | theorem exists_contMDiffOn_forall_mem_convex_of_local (ht : ∀ x, Convex ℝ (t x))
(Hloc : ∀ x : M, ∃ U ∈ 𝓝 x, ∃ g : M → F, ContMDiffOn I 𝓘(ℝ, F) n g U ∧ ∀ y ∈ U, g y ∈ t y) :
∃ g : C^n⟮I, M; 𝓘(ℝ, F), F⟯, ∀ x, g x ∈ t x := by
choose U hU g hgs hgt using Hloc
obtain ⟨f, hf⟩ :=
SmoothPartitionOfUnity.exi... | Mathlib/Geometry/Manifold/PartitionOfUnity.lean | 593 | 600 |
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Kernel.Disintegration.Unique
import Mathlib.Probability.Notation
/-!
# Regular conditional probability distribution
We define the regular con... | StronglyMeasurable (fun x ↦ ∫ y, f (x, y) ∂condDistrib Y X μ x) := by
rw [condDistrib]; exact hf.integral_kernel_prod_right'
theorem _root_.MeasureTheory.AEStronglyMeasurable.integral_condDistrib_map
| Mathlib/Probability/Kernel/CondDistrib.lean | 98 | 101 |
/-
Copyright (c) 2022 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Dynamics.Ergodic.AddCircle
import Mathlib.MeasureTheory.Covering.LiminfLimsup
/-!
# Well-approximable numbers and Gallagher's ergodic theorem
Gallagher's e... | end UnitAddCircle
namespace AddCircle
variable {T : ℝ} [hT : Fact (0 < T)]
local notation a "∤" b => ¬a ∣ b
| Mathlib/NumberTheory/WellApproximable.lean | 174 | 180 |
/-
Copyright (c) 2024 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn,
Mario Carneiro
-/
import Mathlib.Data.List.Defs
import Mathlib.Data.Option.Basic
import Mathlib.Util... | @[simp]
theorem getI_cons_zero : getI (x :: xs) 0 = x :=
rfl
| Mathlib/Data/List/GetD.lean | 103 | 106 |
/-
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
-/
import Mathlib.Data.Finset.Basic
import Mathlib.Data.Finset.Image
/-!
# Cardinality of a finite set
This defines the cardinality of a `Fins... | Mathlib/Data/Finset/Card.lean | 901 | 905 | |
/-
Copyright (c) 2021 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, David Kurniadi Angdinata
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.CubicDiscriminant
import Mathlib.RingTheory.Nilpotent.Defs
import Mathlib.Tactic.Fiel... | Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean | 603 | 605 | |
/-
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.Data.Set.SymmDiff
import Mathlib.Order.SuccPred.Relation
import Mathlib.Topology.Irreducible
/-!
# Connected subsets ... | rcases H with ⟨x, hxs, hxt⟩
exact hs.union x hxs hxt ht
theorem IsConnected.union {s t : Set α} (H : (s ∩ t).Nonempty) (Hs : IsConnected s)
(Ht : IsConnected t) : IsConnected (s ∪ t) := by
| Mathlib/Topology/Connected/Basic.lean | 124 | 128 |
/-
Copyright (c) 2020 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.Topology.Path
/-!
# Path connectedness
Continuing from `Mathlib.Topology.Path`, this file defines path components and path-connected
spaces.
## Main... | Mathlib/Topology/Connected/PathConnected.lean | 674 | 679 | |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set
import Mathlib.Algebra.Module.LinearMap.Prod
import Mathlib.Algebra.Order.Module.Synonym
import Mathlib.Analysis... | rw [add_smul, sub_lt_iff_lt_add']
gcongr
/-- If `x < y`, then `(Set.Ici y)ᶜ` is star convex at `x`. -/
| Mathlib/Analysis/Convex/Star.lean | 373 | 376 |
/-
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, Eric Wieser
-/
import Mathlib.LinearAlgebra.TensorProduct.Tower
import Mathlib.Algebra.DirectSum.Module
/-!
# Tensor products of direct sums
This file shows that... | Mathlib/LinearAlgebra/DirectSum/TensorProduct.lean | 189 | 192 | |
/-
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... | {f : R' →* R''} (hφ : φ.IsPrimitive) (hf : Function.Injective f) :
(f.compAddChar φ).IsPrimitive := fun a ha ↦ by
simpa [DFunLike.ext_iff] using (MonoidHom.compAddChar_injective_right f hf).ne (hφ ha)
/-- The map associating to `a : R` the multiplicative shift of `ψ` by `a`
is injective when `ψ` is primitive... | Mathlib/NumberTheory/LegendreSymbol/AddCharacter.lean | 62 | 67 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Anne Baanen
-/
import Mathlib.Algebra.Algebra.Equiv
import Mathlib.LinearAlgebra.Span.Basic
/-!
# Towers of algebras
In this file we prove basic facts about towers of algebra.
... |
/-- In a tower, the canonical map from the middle element to the top element is an
| Mathlib/Algebra/Algebra/Tower.lean | 130 | 131 |
/-
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.Logic.OpClass
import Mathlib.Order.Lattice
/-!
# `max` and `min`
This file proves basic properties about maxima and minima on a `LinearOrder`.
## Ta... | Mathlib/Order/MinMax.lean | 279 | 279 | |
/-
Copyright (c) 2020 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
import Mathlib.Algebra.ContinuedFractions.Determinant
import Mathlib.Algebra.ContinuedFractions.Computation.CorrectnessTerminating
import Mathlib.Algebra.Order.Ri... | have : IntFractPair.of ifp_n_fr⁻¹ = ifp_succ_n := by
simpa [this, IntFractPair.stream, nth_stream_eq, Option.coe_def] using succ_nth_stream_eq
rwa [← this]
exact floor_le ifp_n.fr⁻¹
end IntFractPair
/-!
Next we translate above results about the stream of `IntFractPair`s to the computed continued
fract... | Mathlib/Algebra/ContinuedFractions/Computation/Approximations.lean | 115 | 127 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kim Morrison
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Algebra.Group.InjSurj
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Tactic.FastInstance
impo... | Mathlib/Data/Finsupp/Defs.lean | 1,367 | 1,370 | |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.FinMeasAdditive
/-!
# Extension of a linear function from indicators to L1
Give... | Mathlib/MeasureTheory/Integral/SetToL1.lean | 1,361 | 1,373 | |
/-
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.SetTheory.Cardinal.Finite
import Mathlib.Data.Set.Finite.Powerset
/-!
# Noncomputable Set Cardinality
We define the cardinality of set `s` as a term `Set... | theorem encard_le_one_iff_eq : s.encard ≤ 1 ↔ s = ∅ ∨ ∃ x, s = {x} := by
rw [le_iff_lt_or_eq, lt_iff_not_le, ENat.one_le_iff_ne_zero, not_not, encard_eq_zero,
encard_eq_one]
theorem encard_le_one_iff : s.encard ≤ 1 ↔ ∀ a b, a ∈ s → b ∈ s → a = b := by
| Mathlib/Data/Set/Card.lean | 309 | 313 |
/-
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... | have finiteCovol : μ univ < ⊤ := measure_lt_top μ univ
rw [fund_dom_s.covolume_eq_volume] at h
by_cases meas_s_ne_zero : ν s = 0
· convert fund_dom_s.quotientMeasureEqMeasurePreimage_of_zero meas_s_ne_zero
rw [← @measure_univ_eq_zero, ← h, meas_s_ne_zero]
apply IsMulLeftInvariant.quotientMeasureEqMeasureP... | Mathlib/MeasureTheory/Measure/Haar/Quotient.lean | 191 | 207 |
/-
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.SetTheory.Cardinal.Finite
import Mathlib.Data.Set.Finite.Powerset
/-!
# Noncomputable Set Cardinality
We define the cardinality of set `s` as a term `Set... | lemma one_lt_ncard_of_nonempty_of_even (hs : Set.Finite s) (hn : Set.Nonempty s := by toFinite_tac)
(he : Even (s.ncard)) : 1 < s.ncard := by
rw [← Set.ncard_pos hs] at hn
| Mathlib/Data/Set/Card.lean | 1,062 | 1,064 |
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.Fintype.List
import Mathlib.Data.Fintype.OfMap
/-!
# Cycles of a list
Lists have an equivalence relation of whether they are rotational permut... | | x :: y :: l =>
if h : x = y then
@decidable_of_iff' _ (Nontrivial (x :: l : Cycle α)) (by simp [h, Nontrivial])
(decidableNontrivialCoe (x :: l))
else isTrue ⟨x, y, h, by simp, by simp⟩
| Mathlib/Data/List/Cycle.lean | 666 | 671 |
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