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 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.Topology.Algebra.Nonarchimedean.Bases
import Mathlib.Topology.Algebra.UniformRing
/-!
# Adic topology
Given a co... | constructor
· rintro ⟨-, ⟨i, rfl⟩, h⟩
replace h : ↑(I ^ i) ⊆ U := by simpa using h
exact ⟨i, trivial, h⟩
· rintro ⟨i, -, h⟩
exact ⟨(I ^ i : Ideal R), ⟨i, by simp⟩, h⟩⟩
theorem hasBasis_nhds_adic (I : Ideal R) (x : R) :
HasBasis (@nhds R I.adicTopology x) (fun _n : ℕ => True) fun n =>
... | Mathlib/Topology/Algebra/Nonarchimedean/AdicTopology.lean | 92 | 103 |
/-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Homeomorph.Lemmas
import Mathlib.Topology.Sets.Opens
/-!
# The OnePoint Compactification
We ... |
theorem le_nhds_infty {f : Filter (OnePoint X)} :
f ≤ 𝓝 ∞ ↔ ∀ s : Set X, IsClosed s → IsCompact s → (↑) '' sᶜ ∪ {∞} ∈ f := by
simp only [hasBasis_nhds_infty.ge_iff, and_imp]
| Mathlib/Topology/Compactification/OnePoint.lean | 342 | 346 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.Deriv.ZPow
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
impo... | obtain ⟨a, ha, h⟩ := h
rw [sub_eq_zero.1 h]
exact ⟨Int.ofNat_zero_le _, Int.ofNat_lt.2 <| Finset.mem_range.1 ha⟩
/-- `x^m`, `m : ℤ` is convex on `(0, +∞)` for all `m` except `0` and `1`. -/
theorem strictConvexOn_zpow {m : ℤ} (hm₀ : m ≠ 0) (hm₁ : m ≠ 1) :
StrictConvexOn ℝ (Ioi 0) fun x : ℝ => x ^ m := by
| Mathlib/Analysis/Convex/SpecificFunctions/Deriv.lean | 88 | 94 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Wrenna Robson
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Pi
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.LinearAlgebra.Vandermonde
import Mathlib.RingT... | def nodalWeight (s : Finset ι) (v : ι → F) (i : ι) :=
∏ j ∈ s.erase i, (v i - v j)⁻¹
theorem nodalWeight_eq_eval_nodal_erase_inv :
| Mathlib/LinearAlgebra/Lagrange.lean | 553 | 556 |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Kim Morrison
-/
import Mathlib.CategoryTheory.Subobject.MonoOver
import Mathlib.CategoryTheory.Skeletal
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathlib.... | intro a
exact h a.arrow
protected theorem ind₂ {X : C} (p : Subobject X → Subobject X → Prop)
(h : ∀ ⦃A B : C⦄ (f : A ⟶ X) (g : B ⟶ X) [Mono f] [Mono g],
| Mathlib/CategoryTheory/Subobject/Basic.lean | 118 | 122 |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Geometry.Euclidean.Altitude
import Mathlib.Geometry.Euclidean.Circumcenter
/-!
# Monge point and orthocenter
This file defines the orthocenter of a trian... | Mathlib/Geometry/Euclidean/MongePoint.lean | 697 | 723 | |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Piecewise
import Mathlib.Algebra.BigOperators.Group.Finset.Sigma
import Mathlib.Algebra.BigOperators.Option
import Ma... | @[simp]
theorem card_vector [Fintype α] (n : ℕ) :
Fintype.card (List.Vector α n) = Fintype.card α ^ n := by
rw [Fintype.ofEquiv_card]; simp
| Mathlib/Data/Fintype/BigOperators.lean | 201 | 205 |
/-
Copyright (c) 2020 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.Group.End
import Mathlib.Data.ZMod.Defs
import Mathlib.Tactic.Ring
/-!
# Racks and Quandles
This file defines racks and quandles, algebraic structu... |
lemma act_idem (x : S) : (x ◃ x) = x := by rw [← act_one x, ← Shelf.self_distrib, act_one]
lemma act_self_act_eq (x y : S) : x ◃ (x ◃ y) = x ◃ y := by
have h : x ◃ (x ◃ y) = (x ◃ 1) ◃ (x ◃ y) := by rw [act_one]
rw [h, ← Shelf.self_distrib, one_act]
/--
| Mathlib/Algebra/Quandle.lean | 156 | 163 |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.Heyting.Basic
import Mathlib.Order.Hom.Basic
import Mathlib.Order.WithBot
/-!
# Orders on a sum type
This file defines the disjoint sum and the lin... | · exact (not_inl_lt_inr h).elim⟩,
⟨fun a => by
obtain ⟨b | b, h⟩ := exists_gt (inr a : α ⊕ β)
· exact (not_inr_lt_inl h).elim
· exact ⟨b, inr_lt_inr_iff.1 h⟩⟩⟩,
fun h => @Sum.noMaxOrder _ _ _ _ h.1 h.2⟩
instance denselyOrdered [LT α] [LT β] [DenselyOrdered α] [DenselyOrdered β... | Mathlib/Data/Sum/Order.lean | 226 | 236 |
/-
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.... | map_rel_iff' := map_le_map_iff_of_injective hf.injective _ _
/-- A linear isomorphism induces an order isomorphism of submodules. -/
@[simps! apply]
| Mathlib/Algebra/Module/Submodule/Map.lean | 416 | 419 |
/-
Copyright (c) 2021 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.SetFamily.Shadow
/-!
# UV-compressions
This file defines UV-compression. It is an operation on a set family that reduces its ... | s ∉ ∂ 𝒜 → u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \ u ∈ ∂ 𝒜 ∧ (s ∪ v) \ u ∉ ∂ 𝒜' by
rintro s hs'
rw [mem_compression]
by_cases hs : s ∈ 𝒜.shadow
swap
· obtain ⟨hus, hvs, h, _⟩ := H _ hs' hs
| Mathlib/Combinatorics/SetFamily/Compression/UV.lean | 297 | 302 |
/-
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.Analysis.Convex.Topology
import Mathlib.Topology.Connected.LocPathConnected
/-!
# Locally convex topological modules
A `LocallyConvexSpace` is a ... | (fun x => ?_) fun x If hif => convex_iInter fun i => convex_iInter fun hi => (hif.2 i hi).2
rw [nhds_sInf, ← iInf_subtype'']
exact hasBasis_iInf' fun i : ts => (@locallyConvexSpace_iff 𝕜 E _ _ _ _ _ ↑i).mp (h (↑i) i.2) x
theorem locallyConvexSpace_iInf {ts' : ι → TopologicalSpace E}
| Mathlib/Topology/Algebra/Module/LocallyConvex.lean | 164 | 168 |
/-
Copyright (c) 2018 Violeta Hernández Palacios, Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios, Mario Carneiro
-/
import Mathlib.Logic.Small.List
import Mathlib.SetTheory.Ordinal.Enum
import Mathlib.SetTheory.Ordinal.Exponen... | Mathlib/SetTheory/Ordinal/FixedPoint.lean | 624 | 628 | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Polynomial.Degree.Domain
import Mathlib.Algebra.Polynomial.Degree.Support
import Mathlib.Algebra.Poly... | Mathlib/Algebra/Polynomial/Derivative.lean | 669 | 671 | |
/-
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.Data.Set.Lattice
import Mathlib.Data.SetLike.Basic
import Mathlib.Order.ModularLattice
import Mathlib.Order.SuccPred.Basic
import Mathlib.Order.WellFou... | instance _root_.OrderDual.instIsCoatomistic [OrderBot α] [h : IsAtomistic α] : IsCoatomistic αᵒᵈ :=
isCoatomistic_dual_iff_isAtomistic.2 h
variable [OrderBot α] [IsAtomistic α]
instance (priority := 100) : IsAtomic α :=
⟨fun b => by
rcases isLUB_atoms b with ⟨s, hsb, hs⟩
rcases s.eq_empty_or_nonempty with... | Mathlib/Order/Atoms.lean | 533 | 541 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.MvPolynomial.Rename
/-!
# Degrees of polynomials
This file establ... |
theorem degreeOf_X [DecidableEq σ] (i j : σ) [Nontrivial R] :
| Mathlib/Algebra/MvPolynomial/Degrees.lean | 241 | 242 |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Abelian
import Mathlib.LinearAlgebra.TensorProduct.Tower
/-!
# Tensor products of Lie modules
Tensor products of Lie modules carry natural Lie ... |
theorem liftLie_apply (f : M →ₗ⁅R,L⁆ N →ₗ[R] P) (m : M) (n : N) :
liftLie R L M N P f (m ⊗ₜ n) = f m n := by
simp only [coe_liftLie_eq_lift_coe, LieModuleHom.coe_toLinearMap, lift_apply]
variable {R L M N P Q}
/-- A pair of Lie module morphisms `f : M → P` and `g : N → Q`, induce a Lie module morphism:
| Mathlib/Algebra/Lie/TensorProduct.lean | 115 | 122 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.RingTheory.WittVector.InitTail
/-!
# Truncated Witt vectors
The ring of truncated Witt vectors (of length `n`) is a quotient of the... |
theorem out_injective : Injective (@out p n R _) := by
| Mathlib/RingTheory/WittVector/Truncated.lean | 100 | 101 |
/-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.Shapes.StrongEpi
import Mathlib.CategoryTheory.LiftingProperties.Adjunction
/-!
# Preservati... | · exact F.mono_of_mono_map
· intro h
exact F.map_mono f
/-- If `F : C ⥤ D` is an equivalence of categories and `C` is a `split_epi_category`,
then `D` also is. -/
| Mathlib/CategoryTheory/Functor/EpiMono.lean | 235 | 240 |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Analysis.Normed.Module.Convex
/-!
# Sides of affine subspaces
This ... | rcases hfp₁ with ⟨p₁, hp₁, rfl⟩
rcases hfp₂ with ⟨p₂, hp₂, rfl⟩
refine ⟨p₁, hp₁, p₂, hp₂, ?_⟩
simp_rw [← linearMap_vsub, (f.linear_injective_iff.2 hf).sameRay_map_iff] at h
exact h
theorem _root_.Function.Injective.sSameSide_map_iff {s : AffineSubspace R P} {x y : P}
{f : P →ᵃ[R] P'} (hf : Function.Injec... | Mathlib/Analysis/Convex/Side.lean | 70 | 80 |
/-
Copyright (c) 2022 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang, Kim Morrison, Joël Riou
-/
import Mathlib.Algebra.Homology.Additive
import Mathlib.CategoryTheory.Abelian.Injective.Resolution
/-!
# Right-derived functors
We define the ... | @[simp, reassoc]
theorem NatTrans.rightDerived_comp {F G H : C ⥤ D} [F.Additive] [G.Additive] [H.Additive]
(α : F ⟶ G) (β : G ⟶ H) (n : ℕ) :
NatTrans.rightDerived (α ≫ β) n = NatTrans.rightDerived α n ≫ NatTrans.rightDerived β n := by
simp [NatTrans.rightDerived]
| Mathlib/CategoryTheory/Abelian/RightDerived.lean | 219 | 223 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.Complex.Asymptotics
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Data.Complex.Trig... | simp [nhdsWithin, comap_exp_nhds_zero, this]
| Mathlib/Analysis/SpecialFunctions/Exp.lean | 463 | 464 |
/-
Copyright (c) 2019 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Mario Carneiro, Isabel Longbottom, Kim Morrison, Apurva Nakade, Yuyang Zhao
-/
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.SetTheory.PGame.Algebra
import Mathl... | theorem quot_mul_comm (x y : PGame.{u}) : (⟦x * y⟧ : Game) = ⟦y * x⟧ :=
game_eq (x.mul_comm y).equiv
/-- `x * y` is equivalent to `y * x`. -/
theorem mul_comm_equiv (x y : PGame) : x * y ≈ y * x :=
Quotient.exact <| quot_mul_comm _ _
| Mathlib/SetTheory/Game/Basic.lean | 405 | 411 |
/-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Homeomorph.Lemmas
import Mathlib.Topology.Sets.Opens
/-!
# The OnePoint Compactification
We ... | toFun := fun ⟨f, hf⟩ ↦ ⟨f, by rwa [← Nat.cofinite_eq_atTop]⟩
invFun := fun ⟨f, hf⟩ ↦ ⟨f, by rwa [Nat.cofinite_eq_atTop]⟩
left_inv := fun _ ↦ rfl
| Mathlib/Topology/Compactification/OnePoint.lean | 450 | 452 |
/-
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 ... | exact fun j _ => this j trivial
have hconn : IsPreconnected S :=
(hs i).image _ (continuous_const.update i continuous_id).continuousOn
| Mathlib/Topology/Connected/Basic.lean | 443 | 445 |
/-
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.MeasureTheory.Measure.AEMeasurable
/-!
# Typeclasses for measurability of operations
In this file we define classes `MeasurableMul` etc and prove d... | variable [MeasurableSpace α] [Mul α] [Div α] [Inv α]
@[to_additive] -- See note [lower instance priority]
instance (priority := 100) DiscreteMeasurableSpace.toMeasurableMul [DiscreteMeasurableSpace α] :
MeasurableMul α where
measurable_const_mul _ := .of_discrete
| Mathlib/MeasureTheory/Group/Arithmetic.lean | 905 | 910 |
/-
Copyright (c) 2022 Yaël Dillies, Sara Rousta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Sara Rousta
-/
import Mathlib.Logic.Equiv.Set
import Mathlib.Order.Interval.Set.OrderEmbedding
import Mathlib.Order.SetNotation
/-!
# Properties of unbundled ... | Mathlib/Order/UpperLower/Basic.lean | 1,682 | 1,683 | |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Cardinal.Arithmetic
import Mathlib.SetTheory.Ordinal.FixedPoint
/-!
# Cofinality
This file co... | hf.1.antisymm' <| by
rw [← hf.2.2]
exact (ord_le_ord.2 (cof_blsub_le f)).trans (ord_card_le o)
| Mathlib/SetTheory/Cardinal/Cofinality.lean | 440 | 443 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yakov Pechersky
-/
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Infix
import Mathlib.Data.Quot
/-!
# List rotation
This file proves basic results about `List.r... | theorem rotate'_length_mul (l : List α) : ∀ n : ℕ, l.rotate' (l.length * n) = l
| 0 => by simp
| n + 1 =>
calc
l.rotate' (l.length * (n + 1)) =
(l.rotate' (l.length * n)).rotate' (l.rotate' (l.length * n)).length := by
| Mathlib/Data/List/Rotate.lean | 79 | 84 |
/-
Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Data.Rat.Cast.Order
import Mathlib.Orde... | simp_rw [← P.biUnion_parts, interedges_biUnion_left, id]
rw [card_biUnion]
exact fun x hx y hy h ↦ interedges_disjoint_left r (P.disjoint hx hy h) _
theorem card_interedges_finpartition_right [DecidableEq β] (s : Finset α) (P : Finpartition t) :
| Mathlib/Combinatorics/SimpleGraph/Density.lean | 146 | 150 |
/-
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.MeasurableSpace.MeasurablyGenerated
import Mathlib.MeasureTheory.Measure.NullMeasurable
import Mathlib.Order.Interval.Set... | Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 2,224 | 2,227 | |
/-
Copyright (c) 2024 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.NumberTheory.LSeries.RiemannZeta
import Mathlib.NumberTheory.Harmonic.GammaDeriv
/-!
# Asymptotics of `ζ s` as `s → 1`
The goal of this file is to ev... | induction N with
| zero =>
simp_rw [term_sum, Finset.sum_range_zero, harmonic_succ, harmonic_zero,
Nat.cast_zero, zero_add, Nat.cast_one, inv_one, Rat.cast_one, log_one, sub_add_cancel]
| succ N hN =>
unfold term_sum at hN ⊢
rw [Finset.sum_range_succ, hN, harmonic_succ (N + 1),
term_one (b... | Mathlib/NumberTheory/Harmonic/ZetaAsymp.lean | 112 | 120 |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Yaël Dillies, Yuyang Zhao
-/
import Mathlib.Algebra.Order.Ring.Unbundled.Basic
import Mathlib.Algebra.CharZero.Defs
import Mathlib.Alge... | Mathlib/Algebra/Order/Ring/Defs.lean | 1,126 | 1,133 | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Group.Action.End
import Mathlib.Algebra.Group.Action.Pointwise.Set.Basic
import Mathlib.Algebra.Group.Action.Prod
import Mathlib.Algebra.Group.Subg... | rw [smul_sub, h', sub_self]
namespace MulAction
variable {G α β : Type*} [Group G] [MulAction G α] [MulAction G β]
| Mathlib/GroupTheory/GroupAction/Basic.lean | 110 | 113 |
/-
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... | theorem orderOf_dvd_of_pow_eq_one (h : x ^ n = 1) : orderOf x ∣ n :=
IsPeriodicPt.minimalPeriod_dvd ((isPeriodicPt_mul_iff_pow_eq_one _).mpr h)
| Mathlib/GroupTheory/OrderOfElement.lean | 239 | 240 |
/-
Copyright (c) 2023 Mantas Bakšys, Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mantas Bakšys, Yaël Dillies
-/
import Mathlib.Algebra.Order.Monovary
import Mathlib.Algebra.Order.Rearrangement
import Mathlib.GroupTheory.Perm.Cycle.Basic
import Mathlib.... | end Mul
variable [Semifield α] [LinearOrder α] [IsStrictOrderedRing α] [ExistsAddOfLE α]
{s : Finset ι} {f : ι → α}
/-- Special case of **Jensen's inequality** for sums of powers. -/
lemma pow_sum_div_card_le_sum_pow (hf : ∀ i ∈ s, 0 ≤ f i) (n : ℕ) :
(∑ i ∈ s, f i) ^ (n + 1) / #s ^ n ≤ ∑ i ∈ s, f i ^ (n + 1) :=... | Mathlib/Algebra/Order/Chebyshev.lean | 153 | 160 |
/-
Copyright (c) 2018 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Mario Carneiro, Kim Morrison, Floris van Doorn
-/
import Mathlib.CategoryTheory.Limits.IsLimit
import Mathlib.CategoryTheory.Category.ULift
import Mathlib.CategoryTheory.Ess... | theorem HasLimit.isoOfNatIso_inv_π {F G : J ⥤ C} [HasLimit F] [HasLimit G] (w : F ≅ G) (j : J) :
(HasLimit.isoOfNatIso w).inv ≫ limit.π F j = limit.π G j ≫ w.inv.app j :=
| Mathlib/CategoryTheory/Limits/HasLimits.lean | 309 | 310 |
/-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Joël Riou
-/
import Mathlib.CategoryTheory.Limits.Final
/-!
# Conditions for `parallelPair` to be initial
In this file we give sufficient conditions on a category... | lemma parallelPair_initial_mk {X Y : C} (f g : X ⟶ Y)
(h₁ : ∀ Z, Nonempty (X ⟶ Z))
(h₂ : ∀ ⦃Z : C⦄ (i j : X ⟶ Z), ∃ (a : Y ⟶ Z), i = f ≫ a ∧ j = g ≫ a) :
(parallelPair f g).Initial :=
parallelPair_initial_mk' f g h₁ (fun Z i j => by
obtain ⟨a, rfl, rfl⟩ := h₂ i j
let f₁ : (mk (Y := zero) (f ≫ a) :... | Mathlib/CategoryTheory/Limits/Final/ParallelPair.lean | 42 | 52 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Algebra.Order.Field.Pi
import Mathlib.Algebra.Order.Pi
import Mathlib.Analysis.Normed.Field.Basic
import Mathlib.Analysis.Normed.Group.Pointwise
import Mat... | ∃ y, closedBall y (δ / 4) ⊆ closedBall x δ ∧ closedBall y (δ / 4) ⊆ interior s := by
refine ⟨x + const _ (3 / 4 * δ), closedBall_subset_closedBall' ?_, ?_⟩
| Mathlib/Analysis/Normed/Order/UpperLower.lean | 149 | 150 |
/-
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... |
/-- val_v(0) = 0. -/
| Mathlib/RingTheory/DedekindDomain/Factorization.lean | 282 | 283 |
/-
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.LegendreSymbol.AddCharacter
import Mathlib.NumberTheory.LegendreSymbol.ZModChar
import Mathlib.Algebra.CharP.CharAndCard
/-!
# Gauss sums
... | gaussSum χ ψ ^ 2 = χ (-1) * Fintype.card R := by
rw [pow_two, ← gaussSum_mul_gaussSum_eq_card hχ₁ hψ, hχ₂.inv, mul_rotate']
congr
rw [mul_comm, ← gaussSum_mulShift _ _ (-1 : Rˣ), inv_mulShift]
rfl
| Mathlib/NumberTheory/GaussSum.lean | 161 | 166 |
/-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Group.Equiv.Defs
import Mathlib.Control.Applicative
import Mathlib.Control.Traversable.Basic
import Mathlib.Logic.Equi... | (ih2 : ∀ x y, C (of x) → C y → C (of x * y)) : C x :=
FreeSemigroup.recOn x fun f s ↦
| Mathlib/Algebra/Free.lean | 507 | 508 |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Algebra.Order.Group.Unbundled.Basic
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Or... | Mathlib/Algebra/Order/Group/Defs.lean | 974 | 974 | |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.Bochner.Basic
import Mathlib.MeasureTheory.Integral.Bochner.L1
import Mathlib.Me... | Mathlib/MeasureTheory/Integral/Bochner.lean | 2,009 | 2,012 | |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Edward Ayers
-/
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.HasPullback
import Mathlib.Data.Set.BooleanAlgebra
/-!
# Theory of sieves
- For an object `X` of a ca... | (Sieve.functorPullback F) := by
apply (functor_galoisConnection F X).toGaloisCoinsertion
rintro S Y f ⟨Z, g, h, h₁, h₂⟩
rw [← F.map_preimage h, ← F.map_comp] at h₂
| Mathlib/CategoryTheory/Sites/Sieves.lean | 771 | 774 |
/-
Copyright (c) 2024 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Cardinal.Arithmetic
import Mathlib.SetTheory.Ordinal.Principal
/-!
# Ordinal arithmetic with cardinals
This file co... | Mathlib/SetTheory/Cardinal/Ordinal.lean | 1,380 | 1,386 | |
/-
Copyright (c) 2024 Colva Roney-Dougal. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Colva Roney-Dougal, Inna Capdeboscq, Susanna Fishel, Kim Morrison
-/
import Mathlib.Order.Atoms
/-!
# The radical of a lattice
This file contains results on the order radical of ... | -- or there is a coatom above it.
obtain (rfl | w) := eq_top_or_exists_le_coatom a
· -- In the first case, we're done, this was already the goal.
rfl
· obtain ⟨m, c, le⟩ := w
have q : a ⊔ radical α ≤ m := sup_le le (radical_le_coatom c)
-- Now note that `a ⊔ radical α ≤ m` since both `a ≤ m` and `ra... | Mathlib/Order/Radical.lean | 38 | 48 |
/-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno
-/
import Mathlib.CategoryTheory.EqToHom
import Mathlib.CategoryTheory.Bicategory.Basic
/-!
# Strict bicategories
A bicategory is called `Strict` if the left unitors, the ri... | end CategoryTheory
| Mathlib/CategoryTheory/Bicategory/Strict.lean | 78 | 81 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Piecewise
import Mathlib.Algebra.Group.Ext
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.BinaryProducts
import Ma... | ext
simp [biproduct.ι_π, biproduct.ι_π_assoc, comp_sum, sum_comp, comp_dite, dite_comp]
@[reassoc]
| Mathlib/CategoryTheory/Preadditive/Biproducts.lean | 231 | 234 |
/-
Copyright (c) 2024 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Order.Ring.Int
import Mathlib.Data.Nat.Cast.Order.Basic
import Mathlib.Order.Interval.Set.OrdConnected
import Mathlib.Order.Nat
import Mathli... | rw [← Icc_bot, image_cast_int_Icc]; rfl
| Mathlib/Data/Nat/Cast/SetInterval.lean | 40 | 41 |
/-
Copyright (c) 2024 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.PurelyInseparable.Basic
import Mathlib.FieldTheory.PerfectClosure
/-!
# `IsPerfectClosure` predicate
This file contains `IsPerfectClosure` which asserts... | RingEquiv.ext (equiv_comp_equiv_apply i j k p)
theorem equiv_comp_equiv_apply_eq_self (x : L) :
| Mathlib/FieldTheory/IsPerfectClosure.lean | 489 | 491 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Field.IsField
import Mathlib.Algebra.GroupWithZero.N... | namespace Localization
| Mathlib/RingTheory/Localization/Basic.lean | 352 | 353 |
/-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.EuclideanDomain.Int
import Mathlib.Algebra.MvPolynomial.Eval
import Mathlib.RingTheory.Adjoin.Basic
import Mathlib.RingTheory.Polynomial.Basic
import Mat... | rw [← S.range_val, ← Algebra.map_top]
exact FG.map _ h, fun h ↦
fg_of_fg_map _ S.val Subtype.val_injective <| by
rw [Algebra.map_top, range_val]
exact h⟩
theorem induction_on_adjoin [IsNoetherian R A] (P : Subalgebra R A → Prop) (base : P ⊥)
(ih : ∀ (S : Subalgebra R A) (x : A), P S → P (Al... | Mathlib/RingTheory/Adjoin/FG.lean | 151 | 158 |
/-
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, Sander Dahmen, Kim Morrison
-/
import Mathlib.SetTheory.Cardinal.Cofinality
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAl... | exact ⟨t, ht', hs.mono ht⟩
lemma exists_finset_linearIndependent_of_le_rank {n : ℕ} (hn : n ≤ Module.rank R M) :
∃ s : Finset M, s.card = n ∧ LinearIndepOn R id (s : Set M) := by
have := nonempty_linearIndependent_set
| Mathlib/LinearAlgebra/Dimension/Finite.lean | 214 | 218 |
/-
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.Exact
import Mathlib.CategoryTheory.Preadditive.Injective.Basic
/-!
# Short exact short complexes
A short complex `S : ShortCompl... | lemma shortExact_iff_of_iso (e : S₁ ≅ S₂) : S₁.ShortExact ↔ S₂.ShortExact := by
constructor
· exact shortExact_of_iso e
· exact shortExact_of_iso e.symm
| Mathlib/Algebra/Homology/ShortComplex/ShortExact.lean | 56 | 59 |
/-
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... | variable [LinearOrder α]
section LocallyFiniteOrder
| Mathlib/Order/Interval/Finset/Basic.lean | 811 | 813 |
/-
Copyright (c) 2020 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap
import Mathlib.MeasureTheory.Integral.Bochner.FundThmCalculus
import Mathlib.MeasureT... | Mathlib/MeasureTheory/Integral/SetIntegral.lean | 1,295 | 1,311 | |
/-
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.IsLUB
/-!
# Order topology on a densely ordered set
-/
open Set Filter TopologicalSpace Topology Func... | rw [filter_eq_bot_of_isEmpty atTop, Filter.map_bot, hb', nhdsWithin_empty]
· rw [← comap_coe_nhdsLT_of_Ioo_subset hb fun _ => hs a ha, map_comap_of_mem]
rw [Subtype.range_val]
exact (mem_nhdsLT_iff_exists_Ioo_subset' ha).2 (hs a ha)
theorem map_coe_atBot_of_Ioo_subset (ha : s ⊆ Ioi a) (hs : ∀ b' > a, ∃ b... | Mathlib/Topology/Order/DenselyOrdered.lean | 256 | 271 |
/-
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, Chris Hughes, Daniel Weber
-/
import Batteries.Data.Nat.Gcd
import Mathlib.Algebra.GroupWithZero.Associated
import Mathlib.Algebra.Ring.Divisibility.Basic
import Math... | FiniteMultiplicity p a ∨ FiniteMultiplicity p b := by
by_contra! nh
obtain ⟨c, hc⟩ := hf
simp_all [dvd_add]
@[deprecated (since := "2024-11-30")]
alias multiplicity.Finite.or_of_add := FiniteMultiplicity.or_of_add
| Mathlib/RingTheory/Multiplicity.lean | 602 | 609 |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Analysis.Convex.Side
import Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
/-!
# Oriented an... | convert Iff.rfl
ext i
fin_cases i <;> rfl
| Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean | 188 | 190 |
/-
Copyright (c) 2023 Koundinya Vajjha. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Koundinya Vajjha, Thomas Browning
-/
import Mathlib.NumberTheory.Harmonic.Defs
import Mathlib.NumberTheory.Padics.PadicNumbers
import Mathlib.Tactic.Positivity
/-!
The nth Harmonic... | unfold harmonic
rw [← Finset.nonempty_range_iff] at Hn
positivity
/-- The 2-adic valuation of the n-th harmonic number is the negative of the logarithm
of n. -/
theorem padicValRat_two_harmonic (n : ℕ) : padicValRat 2 (harmonic n) = -Nat.log 2 n := by
induction n with
| zero => simp
| succ n ih =>
... | Mathlib/NumberTheory/Harmonic/Int.lean | 21 | 33 |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jens Wagemaker
-/
import Mathlib.Algebra.Ring.Associated
import Mathlib.Algebra.Ring.Regular
/-!
# Monoids with normalization functions, `gcd`, and `lcm`
This file de... | theorem gcd_mul_dvd_mul_gcd [GCDMonoid α] (k m n : α) : gcd k (m * n) ∣ gcd k m * gcd k n := by
obtain ⟨m', n', hm', hn', h⟩ := exists_dvd_and_dvd_of_dvd_mul (gcd_dvd_right k (m * n))
replace h : gcd k (m * n) = m' * n' := h
| Mathlib/Algebra/GCDMonoid/Basic.lean | 488 | 490 |
/-
Copyright (c) 2021 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.AlgebraicGeometry.Restrict
import Mathlib.CategoryTheory.Adjunction.Limits
import Mathlib.CategoryTheory.Adjunction.Reflective
/-!
# Adjunction between `Γ` and ... | dsimp [toΓSpecFun]
rw [← IsLocalRing.comap_closedPoint (f.stalkMap x).hom, ←
PrimeSpectrum.comap_comp_apply, ← PrimeSpectrum.comap_comp_apply,
← CommRingCat.hom_comp, ← CommRingCat.hom_comp]
congr 3
exact (PresheafedSpace.stalkMap_germ f.1 ⊤ x trivial).symm
· intro r
| Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | 278 | 284 |
/-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.Add
/-!
# Local extrema of differentiable functions
## Main definitions
In a real normed space `E` we define `posTangentCo... | ### Fermat's Theorem (vector space)
-/
/-- If `f` has a local max on `s` at `a`, `f'` is the derivative of `f` at `a` within `s`, and
`y` belongs to the positive tangent cone of `s` at `a`, then `f' y ≤ 0`. -/
theorem IsLocalMaxOn.hasFDerivWithinAt_nonpos (h : IsLocalMaxOn f s a)
(hf : HasFDerivWithinAt f f' s a) ... | Mathlib/Analysis/Calculus/LocalExtr/Basic.lean | 114 | 124 |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic
import Mathlib.Algebra.Ring.Pointwise.Set
import Mathlib.Topology.MetricSpace.Isometry
import Mathlib.Topol... | Mathlib/Topology/MetricSpace/IsometricSMul.lean | 470 | 471 | |
/-
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... | (∀ᵐ x ∂μ.restrict (⋃ i ∈ t, s i), p x) ↔ ∀ i ∈ t, ∀ᵐ x ∂μ.restrict (s i), p x := by
simp_rw [Filter.Eventually, ae_restrict_biUnion_eq s ht, mem_iSup]
@[simp]
| Mathlib/MeasureTheory/Measure/Restrict.lean | 522 | 525 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Yury Kudryashov
-/
import Mathlib.Analysis.Convex.Combination
import Mathlib.Analysis.Convex.Strict
import Mathlib.Topology.Algebra.Affine
import Mathlib.Topology.... | alias ⟨_, IsPreconnected.convex⟩ := Real.convex_iff_isPreconnected
/-! ### Standard simplex -/
| Mathlib/Analysis/Convex/Topology.lean | 46 | 50 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mitchell Lee
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Indicator
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Topology.Algebra.InfiniteSum.Defs
imp... | exact ha.mul hb
@[to_additive]
theorem hasProd_prod_disjoint {ι} (s : Finset ι) {t : ι → Set β} {a : ι → α}
(hs : (s : Set ι).Pairwise (Disjoint on t)) (hf : ∀ i ∈ s, HasProd (f ∘ (↑) : t i → α) (a i)) :
HasProd (f ∘ (↑) : (⋃ i ∈ s, t i) → α) (∏ i ∈ s, a i) := by
simp_rw [hasProd_subtype_iff_mulIndicator] ... | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | 287 | 294 |
/-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller, Vincent Beffara, Rida Hamadani
-/
import Mathlib.Combinatorics.SimpleGraph.Path
import Mathlib.Data.ENat.Lattice
/-!
# Graph metric
This module defines the `SimpleGraph.edi... | by_cases h : u = v <;> simp [h, edist_bot_of_ne]
lemma edist_top_of_ne (h : u ≠ v) : (⊤ : SimpleGraph V).edist u v = 1 := by
simp [h]
lemma edist_top [DecidableEq V] : (⊤ : SimpleGraph V).edist u v = (if u = v then 0 else 1) := by
| Mathlib/Combinatorics/SimpleGraph/Metric.lean | 137 | 142 |
/-
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 isIntegral_root (h : IsAdjoinRootMonic S f) : IsIntegral R h.root :=
⟨f, h.Monic, h.aeval_root⟩
end IsAdjoinRootMonic
| Mathlib/RingTheory/IsAdjoinRoot.lean | 519 | 523 |
/-
Copyright (c) 2023 Josha Dekker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Josha Dekker
-/
import Mathlib.Topology.Bases
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.Compactness.SigmaCompact
/-!
# Lindelöf sets and Lindelöf spaces
## Mai... | theorem Set.Finite.isLindelof_sUnion {S : Set (Set X)} (hf : S.Finite)
(hc : ∀ s ∈ S, IsLindelof s) : IsLindelof (⋃₀ S) := by
rw [sUnion_eq_biUnion]; exact hf.isLindelof_biUnion hc
| Mathlib/Topology/Compactness/Lindelof.lean | 351 | 353 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Malo Jaffré
-/
import Mathlib.Analysis.Convex.Function
import Mathlib.Tactic.AdaptationNote
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Linarith
/-!
# Slop... | have hxz' := hxz.ne
rw [← sub_pos] at hxy hxz hyz
suffices f y / (y - x) + f y / (z - y) < f x / (y - x) + f z / (z - y) by
ring_nf at this ⊢
linarith
set a := (z - y) / (z - x)
set b := (y - x) / (z - x)
have hy : a • x + b • z = y := by field_simp [a, b]; ring
have key :=
hf.2 hx hz hxz' (di... | Mathlib/Analysis/Convex/Slope.lean | 60 | 80 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.Group.Action.Units
import Mathlib.Algebra.Group.Nat.Units
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra... |
theorem IsCoprime.of_isCoprime_of_dvd_left (h : IsCoprime y z) (hdvd : x ∣ y) : IsCoprime x z := by
obtain ⟨d, rfl⟩ := hdvd
| Mathlib/RingTheory/Coprime/Basic.lean | 144 | 146 |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Order.Interval.Set.OrdConnected
import Mathlib.Data.Set.Lattice.Image
/-!
# Order connected components of a set
In this file we define `Set.ordConn... |
theorem dual_ordSeparatingSet :
ordSeparatingSet (ofDual ⁻¹' s) (ofDual ⁻¹' t) = ofDual ⁻¹' ordSeparatingSet s t := by
simp only [ordSeparatingSet, mem_preimage, ← toDual.surjective.iUnion_comp, ofDual_toDual,
dual_ordConnectedComponent, ← preimage_compl, preimage_inter, preimage_iUnion]
/-- An auxiliary ne... | Mathlib/Order/Interval/Set/OrdConnectedComponent.lean | 156 | 163 |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Nat.Totient
import Mathlib.Data.ZMod.Aut
import Mathlib.Data.ZMod.QuotientGroup
import Mathlib.GroupTheory.Exponent
import Mathlib.GroupTheory.Sub... | rcases Finset.card_pos.1 hpos with ⟨a, ha'⟩
have ha : orderOf a = d := (mem_filter.1 ha').2
have h1 :
(∑ m ∈ d.properDivisors, #{a : α | orderOf a = m}) =
∑ m ∈ d.properDivisors, φ m := by
refine Finset.sum_congr rfl fun m hm => ?_
simp only [mem_filter, mem_range, mem_properDivisors] at hm
... | Mathlib/GroupTheory/SpecificGroups/Cyclic.lean | 465 | 497 |
/-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Subalgebra.Prod
import Mathlib.Algebra.Algebra.Subalgebra.Tower
import Mathlib.LinearAlgebra.Basis.Basic
import Mathlib.LinearAlgebra.Prod
/-!
# ... | Mathlib/RingTheory/Adjoin/Basic.lean | 467 | 469 | |
/-
Copyright (c) 2021 Jordan Brown, Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jordan Brown, Thomas Browning, Patrick Lutz
-/
import Mathlib.GroupTheory.Abelianization
import Mathlib.GroupTheory.Perm.ViaEmbedding
import Mathlib.GroupT... | clear hm
induction' m with m hm
· exact f.range_eq_map ▸ ((derivedSeries G n).map_eq_bot_iff.mp
(le_bot_iff.mp ((map_derivedSeries_le_derivedSeries g n).trans hn.le))).trans hfg
· exact commutator_le_map_commutator hm hm
| Mathlib/GroupTheory/Solvable.lean | 118 | 122 |
/-
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... | def _root_.CategoryTheory.freeGroupoidFunctor (φ : V ⥤q V') : FreeGroupoid V ⥤ FreeGroupoid V' :=
lift (φ ⋙q of V')
theorem freeGroupoidFunctor_id :
freeGroupoidFunctor (Prefunctor.id V) = Functor.id (FreeGroupoid V) := by
dsimp only [freeGroupoidFunctor]; symm
apply lift_unique; rfl
theorem freeGroupoidFun... | Mathlib/CategoryTheory/Groupoid/FreeGroupoid.lean | 174 | 186 |
/-
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.MeasurableSpace.MeasurablyGenerated
import Mathlib.MeasureTheory.Measure.NullMeasurable
import Mathlib.Order.Interval.Set... | Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 1,513 | 1,514 | |
/-
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, Sander Dahmen, Kim Morrison, Chris Hughes, Anne Baanen
-/
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
import Mathlib.LinearAlgebra.Basis.Prod
impo... |
namespace Submodule
theorem lt_of_le_of_finrank_lt_finrank {s t : Submodule R M} (le : s ≤ t)
| Mathlib/LinearAlgebra/Dimension/Constructions.lean | 395 | 398 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Complex.Module
import Mathlib.Data.Complex.Order
import Mathli... | theorem lipschitz_equivRealProd : LipschitzWith 1 equivRealProd := by
simpa using AddMonoidHomClass.lipschitz_of_bound equivRealProdLm 1 equivRealProd_apply_le'
| Mathlib/Analysis/Complex/Basic.lean | 105 | 107 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Alex Kontorovich, Heather Macbeth
-/
import Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic
deprecated_module (since := "2025-04-13")
| Mathlib/MeasureTheory/Integral/Periodic.lean | 209 | 209 | |
/-
Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Johannes Hölzl, Rémy Degenne
-/
import Mathlib.Order.ConditionallyCompleteLattice.Indexed
import Mathlib.Order.Filter.IsBounded
import Mathlib.Order.Hom.CompleteL... |
theorem eventually_lt_of_lt_liminf {f : Filter α} [ConditionallyCompleteLinearOrder β] {u : α → β}
| Mathlib/Order/LiminfLimsup.lean | 758 | 759 |
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Kim Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.FiniteLimits
/-!
# Filtered categories
A category is filtered if every finite diagram admits a cocone.
We give a... | -- This should be an instance but it causes significant slowdown
[nonempty : Nonempty C]
instance (priority := 100) isCofilteredOrEmpty_of_semilatticeInf (α : Type u) [SemilatticeInf α] :
IsCofilteredOrEmpty α where
cone_objs X Y := ⟨X ⊓ Y, homOfLE inf_le_left, homOfLE inf_le_right, trivial⟩
| Mathlib/CategoryTheory/Filtered/Basic.lean | 501 | 506 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... |
@[simp]
| Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean | 118 | 119 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Batteries.Tactic.Init
import Mathlib.Logic.Function.Defs
/-!
# Binary map of options
This file defines the binary map of `Option`. This is mostly useful to defin... | map₂ f (some a) o = o := by
cases o; exacts [rfl, congr_arg some (h _)]
/-- If `b` is a right identity for a binary operation `f`, then `some b` is a right identity for
| Mathlib/Data/Option/NAry.lean | 188 | 191 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Multiset.Powerset
/-!
# The antidiagonal on a multiset.
The antidiagonal of a multiset `s` consists of all pairs `(t₁, t₂)`
such that `t₁ + t₂ =... | theorem card_antidiagonal (s : Multiset α) : card (antidiagonal s) = 2 ^ card s := by
have := card_powerset s
rwa [← antidiagonal_map_fst, card_map] at this
end Multiset
| Mathlib/Data/Multiset/Antidiagonal.lean | 90 | 99 |
/-
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... | rw [← inner_conj_symm, inner_neg_left]; simp only [RingHom.map_neg, inner_conj_symm]
theorem inner_neg_neg (x y : E) : ⟪-x, -y⟫ = ⟪x, y⟫ := by simp
| Mathlib/Analysis/InnerProductSpace/Basic.lean | 212 | 214 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro
-/
import Mathlib.Algebra.Field.IsField
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Finsupp.L... | Mathlib/RingTheory/Ideal/Basic.lean | 468 | 481 | |
/-
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.Algebra.ZMod
import Mathlib.Data.Nat.Multiplicity
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.WittVector.Basic
import Mathlib.... | /-- The polynomials that give the coefficients of `frobenius x`,
in terms of the coefficients of `x`. -/
def frobeniusPoly (n : ℕ) : MvPolynomial ℕ ℤ :=
X n ^ p + C (p : ℤ) * frobeniusPolyAux p n
/-
Our next goal is to prove
```
| Mathlib/RingTheory/WittVector/Frobenius.lean | 97 | 104 |
/-
Copyright (c) 2019 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.Topology.UniformSpace.UniformEmbedding
/-!
# Indexed product of uniform spaces
-/
noncomputable section
open scoped Uniformity Topology
open Filter... | Cauchy l ↔ ∀ i, Cauchy (map (eval i) l) := by
simp_rw [Pi.uniformSpace_eq, cauchy_iInf_uniformSpace, cauchy_comap_uniformSpace]
lemma cauchy_pi_iff' {l : Filter (∀ i, α i)} [l.NeBot] :
Cauchy l ↔ ∀ i, Cauchy (map (eval i) l) := by
| Mathlib/Topology/UniformSpace/Pi.lean | 89 | 93 |
/-
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 | 511 | 514 | |
/-
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... |
@[simp]
| Mathlib/RingTheory/IsAdjoinRoot.lean | 565 | 566 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro
-/
import Mathlib.Algebra.Module.Submodule.Bilinear
import Mathlib.Algebra.Module.Equiv.Basic
import Mathlib.GroupTheory.Congruence.Hom
import Mathlib.Tactic.Abel
... | Mathlib/LinearAlgebra/TensorProduct/Basic.lean | 1,398 | 1,400 | |
/-
Copyright (c) 2024 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.PurelyInseparable.Basic
import Mathlib.FieldTheory.PerfectClosure
/-!
# `IsPerfectClosure` predicate
This file contains `IsPerfectClosure` which asserts... | rwa [map_pow, ← iterateFrobenius_def, ← iterateFrobeniusEquiv_apply, RingEquiv.symm_apply_apply,
map_zero, map_zero] at h
variable [ExpChar L p] in
theorem IsPerfectClosure.ker_eq [PerfectRing L p] [IsPerfectClosure i p] :
RingHom.ker i = pNilradical K p :=
| Mathlib/FieldTheory/IsPerfectClosure.lean | 196 | 201 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Fin.VecNotation
import Mathlib.Logic.Small.Basic
import Mathlib.SetTheory.ZFC.PSet
/-!
# A model of ZFC
In this file, we model Zermelo-Fraenkel ... | Quotient.sound h
| Mathlib/SetTheory/ZFC/Basic.lean | 158 | 159 |
/-
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, Antoine Labelle
-/
import Mathlib.LinearAlgebra.Contraction
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.... | trace R P ((g ∘ₗ f) ∘ₗ h) = trace R M ((h ∘ₗ g) ∘ₗ f) := by
rw [trace_comp_comm', ← comp_assoc]
@[simp]
theorem trace_conj' (f : M →ₗ[R] M) (e : M ≃ₗ[R] N) : trace R N (e.conj f) = trace R M f := by
classical
| Mathlib/LinearAlgebra/Trace.lean | 270 | 275 |
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Data.Prod.Lex
import Mathlib.Data.Sigma.Lex
import Mathlib.Order.RelIso.Set
import Mathlib.Order.WellQuasiOrder
import Mathlib.Tactic.TFAE
/-!
# Well-... | Mathlib/Order/WellFoundedSet.lean | 928 | 937 | |
/-
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 | 1,035 | 1,041 | |
/-
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 Batteries.WF
import Mathlib.Data.Part
import Mathlib.Data.Rel
import Mathlib.Tactic.GeneralizeProofs
/-!
# Partial functions
This... |
theorem image_union (s t : Set α) : f.image (s ∪ t) = f.image s ∪ f.image t :=
Rel.image_union _ s t
/-- Preimage of a set under a partial function. -/
def preimage (s : Set β) : Set α :=
Rel.image (fun x y => x ∈ f y) s
theorem Preimage_def (s : Set β) : f.preimage s = { x | ∃ y ∈ s, y ∈ f x } :=
rfl
| Mathlib/Data/PFun.lean | 362 | 372 |
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