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) 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.Order.Filter.SmallSets
import Mathlib.Topology.UniformSpace.Defs
import Mathlib.Topology.ContinuousOn
/-!
# Basic resu... | Mathlib/Topology/UniformSpace/Basic.lean | 1,723 | 1,725 | |
/-
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.SetTheory.Cardinal.Finite
import Mathlib.Topology.Algebra.InfiniteSum.Basic
import Mathlib.Topology.UniformSpace.Cauchy
import Mathlib.Topology.Algebra... | @[to_additive]
theorem HasProd.hasProd_iff_compl {s : Set β} (hf : HasProd (f ∘ (↑) : s → α) a₁) :
HasProd f a₂ ↔ HasProd (f ∘ (↑) : ↑sᶜ → α) (a₂ / a₁) :=
Iff.symm <| hf.hasProd_compl_iff.trans <| by rw [mul_div_cancel]
@[to_additive]
| Mathlib/Topology/Algebra/InfiniteSum/Group.lean | 91 | 96 |
/-
Copyright (c) 2024 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Lie.Weights.Killing
import Mathlib.LinearAlgebra.RootSystem.Basic
import Mathlib.LinearAlgebra.RootSystem.Reduced
import Mathlib.LinearAlgebra.RootSy... | Nat.cast_inj] at this
rwa [this, Nat.succ_le, two_mul, add_lt_add_iff_left]
| Mathlib/Algebra/Lie/Weights/RootSystem.lean | 280 | 282 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Group.Units.Equiv
import Mathlib.Algebra.Order.Group.End
import Mathlib.Logic.Function.Conjugate
import Mathlib.Order.Bounds.OrderIso
import ... |
theorem semiconj_of_isLUB [PartialOrder α] [Group G] (f₁ f₂ : G →* α ≃o α) {h : α → α}
(H : ∀ x, IsLUB (range fun g' => (f₁ g')⁻¹ (f₂ g' x)) (h x)) (g : G) :
Function.Semiconj h (f₂ g) (f₁ g) := by
refine fun y => (H _).unique ?_
have := (f₁ g).leftOrdContinuous (H y)
| Mathlib/Order/SemiconjSup.lean | 91 | 96 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.MeasurableSpace.Basic
import Mathlib.MeasureTheory.Measure.MeasureSpaceDef
/-!
# Sequence of measurable functions associated to a sequence o... | rw [funext fun n => mk_eq_fun_of_mem_aeSeqSet hf hx n]
have h_ss : aeSeqSet hf p ⊆ { x | p x fun n => f n x } := by
rw [← compl_compl { x | p x fun n => f n x }, aeSeqSet, Set.compl_subset_compl]
| Mathlib/MeasureTheory/Function/AEMeasurableSequence.lean | 64 | 66 |
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.AlgebraicTopology.SimplicialObject.Split
import Mathlib.AlgebraicTopology.DoldKan.PInfty
/-!
# Construction of the inverse functor of the Dold-Kan equivalence
... | simp only [id_comp, eqToHom_refl, comp_id]
· rw [HomologicalComplex.Hom.comm]
· rw [zero_comp, comp_zero]
variable (K)
| Mathlib/AlgebraicTopology/DoldKan/FunctorGamma.lean | 125 | 129 |
/-
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.Topology.Algebra.ConstMulAction
/-!
# Minimal action of a group
In this file we define an action of a monoid `M` on a topological space `α` to be *... | Mathlib/Dynamics/Minimal.lean | 119 | 126 | |
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.AlgebraicTopology.SimplicialObject.Split
import Mathlib.AlgebraicTopology.DoldKan.PInfty
/-!
# Construction of the inverse functor of the Dold-Kan equivalence
... | Mathlib/AlgebraicTopology/DoldKan/FunctorGamma.lean | 352 | 359 | |
/-
Copyright (c) 2022 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed
import Mathlib.RingTheory.PowerBasis
/-!
# A predicate on adjoining root... | theorem aeval_eq (h : IsAdjoinRoot S f) (p : R[X]) : aeval h.root p = h.map p :=
Polynomial.induction_on p (fun x => by rw [aeval_C, h.algebraMap_apply])
| Mathlib/RingTheory/IsAdjoinRoot.lean | 136 | 137 |
/-
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
import Mathlib.Algebra.Ring.NegOnePow
import Mathlib.CategoryTheory.Shift.Quotient
import Mathlib.CategoryTheory.Linear.LinearF... |
lemma shiftFunctorAdd_eq (a b : ℤ) :
CategoryTheory.shiftFunctorAdd (CochainComplex C ℤ) a b = shiftFunctorAdd' C a b _ rfl := by
rw [← CategoryTheory.shiftFunctorAdd'_eq_shiftFunctorAdd, shiftFunctorAdd'_eq]
| Mathlib/Algebra/Homology/HomotopyCategory/Shift.lean | 178 | 181 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Topology.MetricSpace.IsometricSMul
/-!
# Hausdorff distance
The Hausdorff distance on subsets of a... | theorem continuous_infEdist : Continuous fun x => infEdist x s :=
continuous_of_le_add_edist 1 (by simp) <| by
simp only [one_mul, infEdist_le_infEdist_add_edist, forall₂_true_iff]
| Mathlib/Topology/MetricSpace/HausdorffDistance.lean | 133 | 135 |
/-
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, Kim Morrison
-/
import Mathlib.Algebra.Algebra.Equiv
import Mathlib.Algebra.Algebra.NonUnitalHom
import Mathlib.Algebra.Module.BigOperators
import Math... | Mathlib/Algebra/MonoidAlgebra/Basic.lean | 786 | 793 | |
/-
Copyright (c) 2017 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Kim Morrison, Mario Carneiro, Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.Limits.Products
/-!
# Pullbacks and pushouts in the category of topological spaces
-... | rw [← Iso.eq_inv_comp, pullbackIsoProdSubtype_inv_fst]
theorem pullbackIsoProdSubtype_hom_snd (f : X ⟶ Z) (g : Y ⟶ Z) :
| Mathlib/Topology/Category/TopCat/Limits/Pullbacks.lean | 102 | 104 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Jeremy Avigad, Simon Hudon
-/
import Mathlib.Algebra.Notation.Defs
import Mathlib.Data.Set.Subsingleton
import Mathlib.Logic.Equiv.Defs
/-!
# Partial values of a type
... | Mathlib/Data/Part.lean | 781 | 781 | |
/-
Copyright (c) 2020 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
import Mathlib.Analysis.NormedSpace.IndicatorFunction
import Mathlib.Data.Fintype.Order
import Mathlib.MeasureTheory.Function.AEEqFun
import Mathlib.Me... | rfl
theorem _root_.MeasurableEmbedding.memLp_map_measure_iff (hf : MeasurableEmbedding f) :
MemLp g p (Measure.map f μ) ↔ MemLp (g ∘ f) p μ := by
simp_rw [MemLp, hf.aestronglyMeasurable_map_iff, hf.eLpNorm_map_measure]
@[deprecated (since := "2025-02-21")]
alias _root_.MeasurableEmbedding.memℒp_map_measure_... | Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean | 1,117 | 1,143 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Module.Basic
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
/-!
# Midpoint of a segment
## Main definitions
* `midpoint R x y`: midp... | ⇑(ofMapMidpoint R R' f h0 hm) = f :=
rfl
| Mathlib/LinearAlgebra/AffineSpace/Midpoint.lean | 240 | 241 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Kappelmann
-/
import Mathlib.Algebra.Order.Floor.Defs
import Mathlib.Algebra.Order.Floor.Ring
import Mathlib.Algebra.Order.Floor.Semiring
deprecated_module (sinc... | Mathlib/Algebra/Order/Floor.lean | 305 | 307 | |
/-
Copyright (c) 2022 Stuart Presnell. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stuart Presnell
-/
import Mathlib.Algebra.Ring.Parity
import Mathlib.Data.Nat.BinaryRec
/-! # A recursion principle based on even and odd numbers. -/
namespace Nat
/-- Recursion pr... | theorem evenOddRec_odd {P : ℕ → Sort*} (h0 : P 0) (h_even : ∀ i, P i → P (2 * i))
(h_odd : ∀ i, P i → P (2 * i + 1)) (H : h_even 0 h0 = h0) (n : ℕ) :
(2 * n + 1).evenOddRec h0 h_even h_odd = h_odd n (evenOddRec h0 h_even h_odd n) := by
apply binaryRec_eq true n
simp [H]
/-- Strong recursion principle on ev... | Mathlib/Data/Nat/EvenOddRec.lean | 37 | 46 |
/-
Copyright (c) 2024 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Basic
import Mathlib.Tactic.ComputeDegree
/-!
# Division polynomials of Weierst... | (W.ΨSq n).natDegree = n.natAbs ^ 2 - 1 :=
natDegree_eq_of_le_of_coeff_ne_zero (W.natDegree_ΨSq_le n) <| W.coeff_ΨSq_ne_zero h
| Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Degree.lean | 360 | 362 |
/-
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.Tape
import Mathlib.Data.Fintype.Option
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.PFun
import M... | Mathlib/Computability/TuringMachine.lean | 1,185 | 1,194 | |
/-
Copyright (c) 2022 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Group.Subsemigroup.Basic
/-!
# Subsemigroups: membership criteria
In this file we prove various facts about membership in a subsemigroup.
The i... | @[to_additive]
theorem mem_sSup_of_directed_on {S : Set (Subsemigroup M)} (hS : DirectedOn (· ≤ ·) S) {x : M} :
x ∈ sSup S ↔ ∃ s ∈ S, x ∈ s := by
| Mathlib/Algebra/Group/Subsemigroup/Membership.lean | 60 | 62 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.Polynomial.Basi... | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 1,363 | 1,364 | |
/-
Copyright (c) 2022 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston
-/
import Mathlib.Algebra.Category.ModuleCat.Projective
import Mathlib.AlgebraicTopology.ExtraDegeneracy
import Mathlib.CategoryTheory.Abelian.Ext
import Mathlib.G... | Mathlib/RepresentationTheory/GroupCohomology/Resolution.lean | 654 | 662 | |
/-
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.Geometry.RingedSpace.PresheafedSpace
import Mathlib.Topology.Category.TopCat.Limits.Basic
import Mathlib.Topology.Sheaves.Limits
import Mathlib.CategoryThe... | refine (limitObjIsoLimitCompEvaluation _ _).trans (Limits.lim.mapIso ?_)
fapply NatIso.ofComponents
· intro X
refine (F.obj (unop X)).presheaf.mapIso (eqToIso ?_)
simp only [Functor.op_obj, unop_op, op_inj_iff, Opens.map_coe, SetLike.ext'_iff,
Set.preimage_preimage]
refine congr_arg (Set.preimag... | Mathlib/Geometry/RingedSpace/PresheafedSpace/HasColimits.lean | 300 | 310 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Topology.Order.LeftRight
import Mathlib.Topology.Separation.Hausdorff
/-!
# Order-closed topologies
In this file we ... | end Preorder
section NoBotOrder
variable [Preorder α] [NoBotOrder α] [TopologicalSpace α] [ClosedIicTopology α] {a : α}
{l : Filter β} [NeBot l] {f : β → α}
theorem disjoint_nhds_atBot (a : α) : Disjoint (𝓝 a) atBot := by simp
| Mathlib/Topology/Order/OrderClosed.lean | 154 | 162 |
/-
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 | 814 | 816 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.Nat.SuccPred
import Mathlib.Order.SuccPred.Initial... | obtain rfl | hx := Ordinal.eq_zero_or_pos x
| Mathlib/SetTheory/Ordinal/Arithmetic.lean | 1,031 | 1,031 |
/-
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.Algebra.Group.Indicator
import Mathlib.Data.Int.Cast.Pi
import Mathlib.Data.Nat.Cast.Basic
import Mathlib.MeasureTheory.MeasurableSpace... | Mathlib/MeasureTheory/MeasurableSpace/Basic.lean | 1,285 | 1,291 | |
/-
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, Kim Morrison
-/
import Mathlib.Algebra.Algebra.Equiv
import Mathlib.Algebra.Algebra.NonUnitalHom
import Mathlib.Algebra.Module.BigOperators
import Math... | Mathlib/Algebra/MonoidAlgebra/Basic.lean | 1,938 | 1,939 | |
/-
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... | simp_rw [mem_union, mem_preimage]
by_contra! ha
refine s.ne_insert_of_not_mem _ ha.1 (h _ ?_ <| s.subset_insert _)
rw [coe_insert]
refine hs.insert ?_ fun b hb hab => ha.2 <| (hs.isUpperSet' h) hab.le_compl_left hb
| Mathlib/Combinatorics/SetFamily/Intersecting.lean | 163 | 167 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.InnerProductSpace.Defs
impor... | simp only [norm_div, norm_mul, norm_ofReal, abs_norm]
exact norm_inner_div_norm_mul_norm_eq_one_of_ne_zero_of_ne_zero_mul hx hr
/-- The inner product of two vectors, divided by the product of their
norms, has absolute value 1 if and only if they are nonzero and one is
a multiple of the other. One form of equal... | Mathlib/Analysis/InnerProductSpace/Basic.lean | 713 | 725 |
/-
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.Order.LeftRight
import Mathlib.Topology.Order.Monotone
/-!
# Left and right limits
We define the (strict) left and right limits of a f... | ContinuousWithinAt f (Iio x) x ↔ leftLim f x = f x := by
rcases eq_or_ne (𝓝[<] x) ⊥ with (h' | h')
· simp [leftLim_eq_of_eq_bot f h', ContinuousWithinAt, h']
haveI : (𝓝[Iio x] x).NeBot := neBot_iff.2 h'
refine ⟨fun h => tendsto_nhds_unique (hf.tendsto_leftLim x) h.tendsto, fun h => ?_⟩
| Mathlib/Topology/Order/LeftRightLim.lean | 179 | 183 |
/-
Copyright (c) 2021 Ashwin Iyengar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Johan Commelin, Ashwin Iyengar, Patrick Massot
-/
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Topology.Algebra.OpenSubgroup
import Mathlib.Topology.Algebr... | Mathlib/Topology/Algebra/Nonarchimedean/Basic.lean | 140 | 148 | |
/-
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... | refine fun a => imp_congr_right fun _ => (sum_congr rfl fun x _hx => ?_).congr_left
rw [zsmul_eq_mul]
/-- Möbius inversion for functions to a `CommGroup`. -/
theorem prod_eq_iff_prod_pow_moebius_eq [CommGroup R] {f g : ℕ → R} :
(∀ n > 0, ∏ i ∈ n.divisors, f i = g n) ↔
∀ n > 0, ∏ x ∈ n.divisorsAntidiagona... | Mathlib/NumberTheory/ArithmeticFunction.lean | 1,156 | 1,165 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Andrew Zipperer, Haitao Zhang, Minchao Wu, Yury Kudryashov
-/
import Mathlib.Data.Set.Prod
import Mathlib.Data.Set.Restrict
/-!
# Functions over sets
This file contains... | Mathlib/Data/Set/Function.lean | 1,845 | 1,848 | |
/-
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... | /-- The index type `J` -/
{J : Type u}
/-- For each `i : J`, a bundled topological space `U i` -/
U : J → TopCat.{u}
/-- For each `i j : J`, an open set `V i j ⊆ U i` -/
V : ∀ i, J → Opens (U i)
/-- For each `i j : ι`, a transition map `t i j : V i j ⟶ V j i` -/
t : ∀ i j, (Opens.toTopCat _).obj (V i j)... | Mathlib/Topology/Gluing.lean | 286 | 300 |
/-
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... | rw [Module.finrank, cast_toNat_of_lt_aleph0 (rank_lt_aleph0 R M)]
/-- If `M` is finite, then `finrank N = rank N` for all `N : Submodule M`. Note that
| Mathlib/LinearAlgebra/Dimension/StrongRankCondition.lean | 478 | 480 |
/-
Copyright (c) 2023 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.Algebra.Polynomial.Bivariate
import Mathlib.AlgebraicGeometry.EllipticCurve.Weierstrass
import Mathlib.AlgebraicGeometry.EllipticCu... | rw [Nonsingular, and_iff_right hx', negAddY, polynomialX, polynomialY]
eval_simp
contrapose! hx
rw [addPolynomial, linePolynomial, polynomial]
| Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean | 430 | 433 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Topology.MetricSpace.HausdorffDistance
import Mathlib.Topology.Sets.Compacts
/-!
# Closed subsets
... | Mathlib/Topology/MetricSpace/Closeds.lean | 413 | 416 | |
/-
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.Vector.Defs
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.OfFn
import Mathlib.Data.List.Scan
import Mathlib.Control.Applicative
import M... | obtain ⟨a, v', h⟩ := Vector.exists_eq_cons v
rw [h, map_cons, tail_cons, tail_cons]
| Mathlib/Data/Vector/Basic.lean | 101 | 102 |
/-
Copyright (c) 2019 Calle Sönne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology... | (Set.mem_image_of_mem _ hx)
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 880 | 880 |
/-
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 | 3,242 | 3,244 | |
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Abelian.Basic
/-!
# Idempotent complete categories
In this file, we define the notion of idempotent complete categories
(also known as Karoubian... | rw [hpp', ← assoc, φ.inv_hom_id, id_comp]
theorem split_iff_of_iso {X X' : C} (φ : X ≅ X') (p : X ⟶ X) (p' : X' ⟶ X')
(hpp' : p ≫ φ.hom = φ.hom ≫ p') :
(∃ (Y : C) (i : Y ⟶ X) (e : X ⟶ Y), i ≫ e = 𝟙 Y ∧ e ≫ i = p) ↔
∃ (Y' : C) (i' : Y' ⟶ X') (e' : X' ⟶ Y'), i' ≫ e' = 𝟙 Y' ∧ e' ≫ i' = p' := by
cons... | Mathlib/CategoryTheory/Idempotents/Basic.lean | 130 | 140 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johan Commelin
-/
import Mathlib.RingTheory.IntegralClosure.IsIntegral.Basic
/-!
# Minimal polynomials
This file defines the minimal polynomial of an element `x` of an `A... | section IsDomain
variable [Ring B] [Algebra A B]
variable {x : B}
/-- If `a` strictly divides the minimal polynomial of `x`, then `x` cannot be a root for `a`. -/
theorem aeval_ne_zero_of_dvdNotUnit_minpoly {a : A[X]} (hx : IsIntegral A x) (hamonic : a.Monic)
(hdvd : DvdNotUnit a (minpoly A x)) : Polynomial.aeval... | Mathlib/FieldTheory/Minpoly/Basic.lean | 237 | 246 |
/-
Copyright (c) 2020 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Kim Morrison
-/
import Mathlib.Algebra.Order.Interval.Set.Instances
import Mathlib.Order.Interval.Set.ProjIcc
import Mathlib.Topology.Algebra.Ring.Real
/-!
# The unit ... | have hδ := half_pos δ_pos
simp_rw [Subtype.ext_iff]
| Mathlib/Topology/UnitInterval.lean | 323 | 324 |
/-
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.Group.Units.Basic
import Mathlib.Algebra.GroupWithZero.Basic
import Mathlib.Data.Int.Basic
import Mathlib.Lean.Meta.CongrTheorems
import Mathli... | fun h1 => by rw [h1, inverse_mul_cancel_left _ _ h]⟩
| Mathlib/Algebra/GroupWithZero/Units/Basic.lean | 118 | 119 |
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.FDeriv.Basic
/-!
# The derivative of the scalar restriction of a linear map
For detailed document... | (h.hasFDerivAt.restrictScalars 𝕜).fderiv
theorem differentiableWithinAt_iff_restrictScalars (hf : DifferentiableWithinAt 𝕜 f s x)
(hs : UniqueDiffWithinAt 𝕜 s x) : DifferentiableWithinAt 𝕜' f s x ↔
| Mathlib/Analysis/Calculus/FDeriv/RestrictScalars.lean | 92 | 95 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Logic.Function.Defs
import Mathlib.Logic.Function.Iterate
import Aesop
import Mathlib.Tactic.Inhabit
/-!
# Extra facts about `Prod`
This file proves ... | Mathlib/Data/Prod/Basic.lean | 388 | 392 | |
/-
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.Algebra.Group.Action.Defs
import Mathlib.Algebra.Group.End
import Mathlib.Logic.Equiv.Set
import Mathlib.Tactic.Common
/-!
#... | Mathlib/GroupTheory/Perm/Basic.lean | 630 | 631 | |
/-
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.Algebra.CharP.Invertible
import Mathlib.Algebra.Order.Interval.Set.Group
import Mathlib.Analysis.Convex.Basic
import Mathlib.Analysis.Convex.Segment
import... |
theorem Sbtw.trans_wbtw_right_ne [NoZeroSMulDivisors R V] {w x y z : P} (h₁ : Sbtw R w x z)
(h₂ : Wbtw R x y z) : w ≠ y :=
h₁.wbtw.trans_right_ne h₂ h₁.left_ne
| Mathlib/Analysis/Convex/Between.lean | 462 | 466 |
/-
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 | 753 | 754 | |
/-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.FiniteMeasure
import Mathlib.MeasureTheory.Integral.Average
import Mathlib.MeasureTheory.Measure.Prod
/-!
# Probability measures
Th... | {f : Ω → Ω'} (f_cont : Continuous f) :
Tendsto (fun i ↦ (νs i).map f_cont.measurable.aemeasurable) L
(𝓝 (ν.map f_cont.measurable.aemeasurable)) := by
rw [ProbabilityMeasure.tendsto_iff_forall_lintegral_tendsto] at lim ⊢
intro g
convert lim (g.compContinuous ⟨f, f_cont⟩) <;>
· simp only [map, comp... | Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean | 571 | 584 |
/-
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.Algebra.Module.Torsion
import Mathlib.Algebra.Ring.Idempotent
import Mathlib.LinearAlgebra.Dimension.FreeAndStrongRankCondition
import Mathlib.LinearAlgebra.... |
theorem mem_toCotangent_ker {x : I} : x ∈ LinearMap.ker I.toCotangent ↔ (x : R) ∈ I ^ 2 := by
rw [← I.map_toCotangent_ker]
| Mathlib/RingTheory/Ideal/Cotangent.lean | 63 | 65 |
/-
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.Polynomial.Expand
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.FieldTheory.IntermediateField.Basic
imp... | insert_erase (mem_erase_of_ne_of_mem (Ne.symm hxy) hy), prod_insert (not_mem_erase _ _), ←
mul_assoc, hfxy, ← sq] at hfs
cases (hfs.of_mul_left.of_pow (not_isUnit_X_sub_C _) two_ne_zero).2
theorem Separable.injective_of_prod_X_sub_C [Nontrivial R] {ι : Sort _} [Fintype ι] {f : ι → R}
(hfs : (∏ i, (X - C ... | Mathlib/FieldTheory/Separable.lean | 227 | 234 |
/-
Copyright (c) 2022 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca, Eric Rodriguez
-/
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.RingTheory.Localization.NormTrace
import Mathlib.RingTheory.Norm.Transitivity
/-!
# No... | RingOfIntegers.coe_eq_algebraMap, map_pow]
theorem isUnit_norm_of_isGalois [IsGalois K L] {x : 𝓞 L} : IsUnit (norm K x) ↔ IsUnit x := by
classical
refine ⟨fun hx => ?_, IsUnit.map _⟩
| Mathlib/NumberTheory/NumberField/Norm.lean | 65 | 69 |
/-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Topology.Coherent
import Mathlib.Topology.UniformSpace.Equiv
import Mathlib.Topology.UniformSpace.Pi
import Mathlib.Topology.UniformSpace.UniformAp... | /-- The natural bijection between `α → Π i, δ i` and `Π i, α → δ i`, upgraded to a uniform
isomorphism between `α →ᵤ (Π i, δ i)` and `Π i, α →ᵤ δ i`. -/
protected def uniformEquivPiComm : UniformEquiv (α →ᵤ ∀ i, δ i) (∀ i, α →ᵤ δ i) :=
-- Denote `φ` this bijection. We want to show that
| Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | 481 | 484 |
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Mario Carneiro, Simon Hudon
-/
import Mathlib.Data.Fin.Fin2
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.Common
/-!
# Tuples of types, and their categorica... | induction i <;> simp only [id, toSubtype', comp, subtypeVal, prod.mk] at *
simp [*]
end TypeVec
| Mathlib/Data/TypeVec.lean | 692 | 695 |
/-
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 | 873 | 878 | |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Eric Wieser
-/
import Mathlib.Algebra.Quaternion
import Mathlib.Analysis.InnerProductSpace.Continuous
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.To... | Subtype.ext <| norm_coe a
| Mathlib/Analysis/Quaternion.lean | 73 | 74 |
/-
Copyright (c) 2019 Calle Sönne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology... |
theorem two_zsmul_toReal_eq_two_mul {θ : Angle} :
((2 : ℤ) • θ).toReal = 2 * θ.toReal ↔ θ.toReal ∈ Set.Ioc (-π / 2) (π / 2) := by
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 554 | 556 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Module.Basic
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
/-!
# Midpoint of a segment
## Main definitions
* `midpoint R x y`: midp... |
end
| Mathlib/LinearAlgebra/AffineSpace/Midpoint.lean | 216 | 217 |
/-
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.... |
theorem isNilpotent_iff_le_maxNilpotentSubmodule [IsNoetherian R M] (N : LieSubmodule R L M) :
IsNilpotent L N ↔ N ≤ maxNilpotentSubmodule R L M :=
⟨fun h ↦ le_sSup h, fun h ↦ isNilpotent_of_le R L M N (maxNilpotentSubmodule R L M) h⟩
@[simp] lemma maxNilpotentSubmodule_eq_top_of_isNilpotent [LieModule.IsNilpot... | Mathlib/Algebra/Lie/Nilpotent.lean | 730 | 736 |
/-
Copyright (c) 2021 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.CategoryTheory.Limits.Shapes.WideEqualizers
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.Terminal
/-!
... | let i := wideEqualizer.ι (id : endos → endos)
haveI : Nonempty endos := ⟨𝟙 _⟩
have : ∀ X : C, Unique (wideEqualizer (id : endos → endos) ⟶ X) := by
intro X
refine ⟨⟨i ≫ Classical.choice (hT X)⟩, fun a => ?_⟩
let E := equalizer a (i ≫ Classical.choice (hT _))
let e : E ⟶ wideEqualizer id := equali... | Mathlib/CategoryTheory/Limits/Constructions/WeaklyInitial.lean | 46 | 64 |
/-
Copyright (c) 2017 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Tim Baumann, Stephen Morgan, Kim Morrison, Floris van Doorn
-/
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.ObjectProperty.FullSubcategory
import Mat... |
@[simp]
| Mathlib/CategoryTheory/Equivalence.lean | 400 | 401 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Kevin Buzzard, Yury Kudryashov, Frédéric Dupuis,
Heather Macbeth
-/
import Mathlib.Algebra.Group.Subgroup.Map
import Mathlib.Algebra.Module.Submodule.... | variable [RingHomInvPair τ₁₂ τ₂₁] [RingHomInvPair τ₂₁ τ₁₂]
variable (p : Submodule R M) (q : Submodule R₂ M₂)
variable (pₗ : Submodule R N) (qₗ : Submodule R N₂)
theorem comap_le_comap_smul (fₗ : N →ₗ[R] N₂) (c : R) : comap fₗ qₗ ≤ comap (c • fₗ) qₗ := by
rw [SetLike.le_def]
| Mathlib/Algebra/Module/Submodule/Map.lean | 607 | 612 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.ExactSequence
import Mathlib.Algebra.Homology.ShortComplex.Limits
import Mathlib.CategoryTheory.Abelian.Refinements
/-!
# The snake lemma
The ... | simp only [← cancel_mono S₂.L₂.f, assoc, φ₁_L₂_f, ← naturality_φ₂, f.f₂.comm₁₂, φ₁_L₂_f_assoc]
@[reassoc]
lemma naturality_δ (f : S₁ ⟶ S₂) : S₁.δ ≫ f.f₃.τ₁ = f.f₀.τ₃ ≫ S₂.δ := by
| Mathlib/Algebra/Homology/ShortComplex/SnakeLemma.lean | 490 | 493 |
/-
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.Complex.Circle
import Mathlib.Analysis.SpecialFunctions.Complex.Log
/-!
# Maps on the unit circle
In this file we prove some basic lemmas ... | · intro x; simp_rw [mul_add]; rw [div_mul_cancel₀ _ hT, Circle.periodic_exp]
| Mathlib/Analysis/SpecialFunctions/Complex/Circle.lean | 130 | 131 |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Alena Gusakov, Yaël Dillies
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Finset.Slice
import Mathlib.Data.Nat.BitIndices
import Mathlib.Order.SupClosed
import Math... | ⟨fun h hst ↦ h <| ofColex_ne_ofColex.2 hst, fun h hst ↦ h <| ofColex_ne_ofColex.1 hst⟩
lemma toColex_lt_toColex_iff_max'_mem :
toColex s < toColex t ↔ ∃ hst : s ≠ t, (s ∆ t).max' (symmDiff_nonempty.2 hst) ∈ t := by
rw [lt_iff_le_and_ne, toColex_le_toColex_iff_max'_mem]; aesop
lemma lt_iff_max'_mem {s t : Co... | Mathlib/Combinatorics/Colex.lean | 333 | 345 |
/-
Copyright (c) 2022 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa, Yuyang Zhao
-/
import Mathlib.Algebra.Order.GroupWithZero.Unbundled.Basic
import Mathlib.Algebra.Order.GroupWithZero.Unbundled.Defs
import Mathlib.Tactic.Linter.Deprecate... | Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean | 681 | 683 | |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.RightHomology
/-!
# Homology of short complexes
In this file, we shall define the homology of short complexes `S`, i.e. diagrams... |
@[reassoc (attr := simp)]
lemma LeftHomologyData.leftHomologyIso_hom_comp_homologyIso_inv (h : S.LeftHomologyData) :
h.leftHomologyIso.hom ≫ h.homologyIso.inv = S.leftHomologyIso.hom := by
dsimp only [homologyIso]
simp only [Iso.trans_inv, Iso.symm_inv, Iso.hom_inv_id_assoc]
| Mathlib/Algebra/Homology/ShortComplex/Homology.lean | 969 | 974 |
/-
Copyright (c) 2022 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex J. Best, Xavier Roblot
-/
import Mathlib.Algebra.Algebra.Hom.Rat
import Mathlib.Analysis.Complex.Polynomial.Basic
import Mathlib.NumberTheory.NumberField.Norm
import Mathlib.RingTh... |
lemma mult_comap_le (f : k →+* K) (w : InfinitePlace K) : mult (w.comap f) ≤ mult w := by
rw [mult, mult]
split_ifs with h₁ h₂ h₂
pick_goal 3
· exact (h₁ (h₂.comap _)).elim
all_goals decide
variable [Algebra k K] (σ : K ≃ₐ[k] K) (w : InfinitePlace K)
variable (k K)
| Mathlib/NumberTheory/NumberField/Embeddings.lean | 707 | 716 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Patrick Massot, Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic
import Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalcul... | Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | 829 | 831 | |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.ContinuousMap.Bounded.ArzelaAscoli
import Mathlib.Topology.ContinuousMap.Bounded.Normed
import Mathlib.Topology.MetricSpace.Gluing
impor... | tendsto_const_nhds.mul tendsto_id
simpa using this
· rintro x y ⟨f, hf⟩
exact (candidates_lipschitz hf).dist_le_mul _ _
/-- candidates give rise to elements of `BoundedContinuousFunction`s -/
def candidatesBOfCandidates (f : ProdSpaceFun X Y) (fA : f ∈ candidates X Y) : Cb X Y :=
BoundedConti... | Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean | 348 | 374 |
/-
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.FieldTheory.Minpoly.Field
import Mathlib.LinearAlgebra.SModEq
import Mathlib.RingTheory.Ideal.BigOperators
/-!
# Power basis
This file defines a structure ... |
theorem minpolyGen_map (pb : PowerBasis A S) (e : S ≃ₐ[A] S') :
(pb.map e).minpolyGen = pb.minpolyGen := by
dsimp only [minpolyGen, map_dim]
-- Turn `Fin (pb.map e).dim` into `Fin pb.dim`
simp only [LinearEquiv.trans_apply, map_basis, Basis.map_repr, map_gen,
AlgEquiv.toLinearEquiv_apply, e.toLinearEquiv... | Mathlib/RingTheory/PowerBasis.lean | 442 | 450 |
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Localization.Construction
/-!
# Predicate for localized categories
In this file, a predicate `L.IsLocalization W` is introduced for a functor `... |
/-- The functor of equivalence of localized categories given by `Localization.uniq` is
compatible with the localization functors. -/
def compUniqFunctor : L₁ ⋙ (uniq L₁ L₂ W').functor ≅ L₂ :=
calc
| Mathlib/CategoryTheory/Localization/Predicate.lean | 438 | 442 |
/-
Copyright (c) 2019 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Yury Kudryashov, Sébastien Gouëzel, Chris Hughes
-/
import Mathlib.Data.Fin.Rev
import Mathlib.Data.Nat.Find
/-!
# Operation on tuples
We interpret maps `∀ i : Fi... | rw [← cons_succ x p]; rfl
| Mathlib/Data/Fin/Tuple/Basic.lean | 122 | 122 |
/-
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.Analysis.Convex.Deriv
import Mathlib.Analysis.SpecialFunctions.Gamma.Deligne
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.NumberTheory.Harmo... | lemma hasDerivAt_Gamma_nat (n : ℕ) :
HasDerivAt Gamma (n ! * (-γ + harmonic n)) (n + 1) := by
exact_mod_cast HasDerivAt.complex_of_real
(by exact_mod_cast differentiable_at_Gamma_nat_add_one n)
(Real.hasDerivAt_Gamma_nat n) Gamma_ofReal
| Mathlib/NumberTheory/Harmonic/GammaDeriv.lean | 168 | 172 |
/-
Copyright (c) 2018 Andreas Swerdlow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andreas Swerdlow, Kexing Ying
-/
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.LinearAlgebra.BilinearForm.Properties
/-!
# Bilinear form
This file defines ort... | Mathlib/LinearAlgebra/BilinearForm/Orthogonal.lean | 437 | 446 | |
/-
Copyright (c) 2022 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.Algebra.Module.ZLattice.Basic
import Mathlib.Analysis.InnerProductSpace.ProdL2
import Mathlib.MeasureTheory.Measure.Haar.Unique
import Mathlib.NumberTheo... | obtain hr | _ := lt_or_le r 0
· simp [Metric.closedBall_eq_empty.2 hr]
· have heq : ∀ x, canonicalEmbedding K x ∈ Metric.closedBall 0 r ↔
∀ φ : K →+* ℂ, ‖φ x‖ ≤ r := by
intro x; rw [← norm_le_iff, mem_closedBall_zero_iff]
convert (Embeddings.finite_of_norm_le K ℂ r).image (canonicalEmbedding K)
... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean | 93 | 105 |
/-
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.Analytic.IsolatedZeros
import Mathlib.Analysis.SpecialFunctions.Complex.CircleMap
import Mathlib.Analysis.SpecialFunctions.NonIntegrable
/-... | theorem integral_radius_zero (f : ℂ → E) (c : ℂ) : (∮ z in C(c, 0), f z) = 0 := by
simp +unfoldPartialApp [circleIntegral, const]
theorem integral_congr {f g : ℂ → E} {c : ℂ} {R : ℝ} (hR : 0 ≤ R) (h : EqOn f g (sphere c R)) :
(∮ z in C(c, R), f z) = ∮ z in C(c, R), g z :=
intervalIntegral.integral_congr fun θ ... | Mathlib/MeasureTheory/Integral/CircleIntegral.lean | 301 | 331 |
/-
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.Group.Basic
import Mathlib.Tactic.Commo... |
theorem dvd_of_mul_left_dvd (h : a * b ∣ c) : b ∣ c :=
Dvd.elim h fun d ceq => Dvd.intro (a * d) (by simp [ceq])
| Mathlib/Algebra/Divisibility/Basic.lean | 171 | 174 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kenny Lau
-/
import Mathlib.Data.DFinsupp.Submonoid
import Mathlib.Data.Finsupp.ToDFinsupp
import Mathlib.LinearAlgebra.Finsupp.SumProd
import Mathlib.LinearAlgebra.LinearIn... | @[deprecated (since := "2025-04-06")]
alias iSupIndep_of_dfinsupp_sumAddHom_injective' := iSupIndep_of_dfinsuppSumAddHom_injective'
@[deprecated (since := "2024-11-24")]
| Mathlib/LinearAlgebra/DFinsupp.lean | 516 | 519 |
/-
Copyright (c) 2023 Ziyu Wang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ziyu Wang, Chenyi Li, Sébastien Gouëzel, Penghao Yu, Zhipeng Cao
-/
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.Calculus.FDeriv.Basic
import Mathlib.Analysis.Calc... |
theorem DifferentiableAt.hasGradientAt (h : DifferentiableAt 𝕜 f x) :
HasGradientAt f (∇ f x) x := by
rw [hasGradientAt_iff_hasFDerivAt, gradient, (toDual 𝕜 F).apply_symm_apply (fderiv 𝕜 f x)]
| Mathlib/Analysis/Calculus/Gradient/Basic.lean | 118 | 121 |
/-
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.RingTheory.Valuation.Basic
import Mathlib.NumberTheory.Padics.PadicNorm
import Mathlib.Analysis.Normed.Field.Lemmas
import Mathlib.Tactic.Peel
import... |
variable {p : ℕ} [hp : Fact p.Prime]
-- Porting note: Linter thinks this is a duplicate simp lemma, so `priority` is assigned
@[simp (high)]
protected theorem mul (q r : ℚ_[p]) : ‖q * r‖ = ‖q‖ * ‖r‖ := by simp [Norm.norm, map_mul]
protected theorem is_norm (q : ℚ_[p]) : ↑(padicNormE q) = ‖q‖ := rfl
| Mathlib/NumberTheory/Padics/PadicNumbers.lean | 748 | 754 |
/-
Copyright (c) 2022 Rémi Bottinelli. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémi Bottinelli
-/
import Mathlib.CategoryTheory.Groupoid
import Mathlib.CategoryTheory.PathCategory.Basic
/-!
# Free groupoid on a quiver
This file defines the free groupoid on a q... | Mathlib/CategoryTheory/Groupoid/FreeGroupoid.lean | 200 | 203 | |
/-
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.Algebra.Order.Pi
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
/-!
# Simple functions
A function `f` from a measurable ... | by_cases hx : x ∈ Function.support f
· simpa only [SimpleFunc.coe_add, Pi.add_apply, Function.mem_support, ne_eq, not_not,
Set.piecewise_eq_of_mem _ _ _ hx, _root_.add_right_inj, add_eq_left]
using Set.disjoint_left.1 hff' hx
· simpa only [SimpleFunc.coe_add, Pi.add_apply, Function... | Mathlib/MeasureTheory/Function/SimpleFunc.lean | 1,242 | 1,251 |
/-
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... | s.strongInductionOn H = H s fun t _ => t.strongInductionOn H := by
dsimp only [strongInductionOn]
rw [strongInduction]
@[elab_as_elim]
| Mathlib/Data/Finset/Card.lean | 752 | 756 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Jakob von Raumer
-/
import Mathlib.CategoryTheory.Category.Cat.AsSmall
import Mathlib.CategoryTheory.Comma.StructuredArrow.Basic
import Mathlib.CategoryTheory.IsConnected
i... | CreatesLimit := createsLimitOfComp F
end Initial
| Mathlib/CategoryTheory/Limits/Final.lean | 763 | 766 |
/-
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 | 644 | 646 | |
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.Constructions
/-!
# Neighborhoods and continuity relative to a subset
This file develops API on the relative versions
* `nhdsWithin` ... | theorem continuousOn_apply {ι : Type*} {π : ι → Type*} [∀ i, TopologicalSpace (π i)]
(i : ι) (s) : ContinuousOn (fun p : ∀ i, π i => p i) s :=
Continuous.continuousOn (continuous_apply i)
| Mathlib/Topology/ContinuousOn.lean | 1,184 | 1,187 |
/-
Copyright (c) 2022 Matej Penciak. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matej Penciak, Moritz Doll, Fabien Clery
-/
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
/-!
# The Symplectic Group
This file defines the symplectic group and proves element... | theorem inv_eq_symplectic_inv (A : Matrix (l ⊕ l) (l ⊕ l) R) (hA : A ∈ symplecticGroup l R) :
A⁻¹ = (-J l R) * Aᵀ * J l R :=
| Mathlib/LinearAlgebra/SymplecticGroup.lean | 174 | 175 |
/-
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... | filter_upwards [eventually_deriv_one_sub_smoothingFn, eventually_gt_atTop 1] with x hx hx_pos
have : 0 ≤ x⁻¹ / (log x ^ 2) := by
have hlog : 0 < log x := Real.log_pos hx_pos
positivity
simp only [hp, Real.rpow_zero, one_mul, differentiableAt_const, hx, Real.norm_of_nonneg this]
ref... | Mathlib/Computability/AkraBazzi/AkraBazzi.lean | 900 | 906 |
/-
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.Ring.Divisibility.Lemmas
import Mathlib.Algebra.Lie.Nilpotent
import Mathlib.Algebra.Lie.Engel
import Mathlib.LinearAlgebra.Eigenspace.Pi
import Math... | ← Module.End.mul_eq_comp, ← hq] }
variable {M} in
lemma mem_posFittingCompOf (x : L) (m : M) :
m ∈ posFittingCompOf R M x ↔ ∀ (k : ℕ), ∃ n, (toEnd R L M x ^ k) n = m := by
| Mathlib/Algebra/Lie/Weights/Basic.lean | 419 | 423 |
/-
Copyright (c) 2022 Floris van Doorn, Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Heather Macbeth
-/
import Mathlib.Geometry.Manifold.ContMDiff.Atlas
import Mathlib.Geometry.Manifold.VectorBundle.FiberwiseLinear
import Mathlib.To... | /-- For a `C^n` vector bundle `E` over `B` with fiber modelled on `F`, the change-of-co-ordinates
between two trivializations `e`, `e'` for `E`, considered as charts to `B × F`, is `C^n` and
fiberwise linear. -/
instance ContMDiffFiberwiseLinear.hasGroupoid :
HasGroupoid (TotalSpace F E) (contMDiffFiberwiseLinear B... | Mathlib/Geometry/Manifold/VectorBundle/Basic.lean | 446 | 455 |
/-
Copyright (c) 2024 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.Data.Nat.EvenOddRec
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.LinearCombination
/-!
# Elliptic divisibility sequences
... | have h2 : m + 2 < n + 5 := (lt_add_one _).trans h3
have _ : m + 1 < n + 5 := (lt_add_one _).trans h2
if hn : Even n then
preNormEDS' b c d (m + 4) * preNormEDS' b c d (m + 2) ^ 3 * (if Even m then b else 1) -
preNormEDS' b c d (m + 1) * preNormEDS' b c d (m + 3) ^ 3 * (if Even m then 1 else b)... | Mathlib/NumberTheory/EllipticDivisibilitySequence.lean | 155 | 160 |
/-
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, Floris van Doorn
-/
import Mathlib.Data.Countable.Small
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Powerset
import Mathlib.Dat... | Mathlib/SetTheory/Cardinal/Basic.lean | 1,579 | 1,582 | |
/-
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... | theorem Icc_top [OrderTop α] : Icc a (⊤ : α) = Ici a := rfl
@[simp]
| Mathlib/Order/Interval/Finset/Basic.lean | 519 | 521 |
/-
Copyright (c) 2022 Jake Levinson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jake Levinson
-/
import Mathlib.Data.Finset.Preimage
import Mathlib.Data.Finset.Prod
import Mathlib.Data.SetLike.Basic
import Mathlib.Order.UpperLower.Basic
/-!
# Young diagrams
A You... | @[simp]
theorem length_rowLens {μ : YoungDiagram} : μ.rowLens.length = μ.colLen 0 := by
| Mathlib/Combinatorics/Young/YoungDiagram.lean | 365 | 366 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Minchao Wu, Mario Carneiro
-/
import Mathlib.Data.Finset.Attach
import Mathlib.Data.Finset.Disjoint
import Mathlib.Data.Finset.Erase
import Mat... | Mathlib/Data/Finset/Basic.lean | 2,486 | 2,492 | |
/-
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.Operations
/-!
# Results about division in extended non-negative reals
This file establishes basic properties related t... | rw [← pos_iff_ne_zero, div_pos_iff]
| Mathlib/Data/ENNReal/Inv.lean | 492 | 493 |
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