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 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Reid Barton
-/
import Mathlib.Topology.Separation.Regular
/-!
# The shrinking lemma
In this file we prove a few versions of the shrinking lemma. The lemma says that... | rcases normal_exists_closure_subset C (v.isOpen i) I with ⟨vi, ovi, hvi, cvi⟩
classical
refine ⟨⟨update v i vi, insert i v.carrier, ?_, ?_, ?_, ?_, ?_⟩, ?_, ?_⟩
· intro j
rcases eq_or_ne j i with (rfl| hne) <;> simp [*, v.isOpen]
· refine fun x hx => mem_iUnion.2 ?_
rcases em (∃ j ≠ i, x ∈ v j) with (... | Mathlib/Topology/ShrinkingLemma.lean | 174 | 204 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, 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 | 1,063 | 1,065 | |
/-
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.Bilinear
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Group.Pointwise.Finset.Basic
import Mathlib.Algebra.Group.Pointwise.Set.B... | theorem pow_eq_npowRec {n : ℕ} : M ^ n = npowRec n M := rfl
protected theorem pow_zero : M ^ 0 = 1 := rfl
protected theorem pow_succ {n : ℕ} : M ^ (n + 1) = M ^ n * M := rfl
protected theorem pow_add {m n : ℕ} (h : n ≠ 0) : M ^ (m + n) = M ^ m * M ^ n :=
npowRec_add m n h _ M.one_mul
| Mathlib/Algebra/Algebra/Operations.lean | 298 | 306 |
/-
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... | Mathlib/GroupTheory/GroupAction/Basic.lean | 850 | 853 | |
/-
Copyright (c) 2022 Richard M. Hill. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Richard M. Hill
-/
import Mathlib.Algebra.Module.Submodule.Invariant
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.LinearAlgebra.DFinsupp
import Mathlib.RingTheory.Finit... | p ≃ₗ[R] mapSubmodule R M a ⟨p, hp⟩ where
toFun x := ⟨of R M a x, by simp⟩
invFun x := ⟨((of R M _).symm (x : AEval R M a)), by obtain ⟨x, hx⟩ := x; simpa using hx⟩
left_inv x := rfl
| Mathlib/Algebra/Polynomial/Module/AEval.lean | 157 | 160 |
/-
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... | Mathlib/Analysis/InnerProductSpace/Basic.lean | 1,678 | 1,687 | |
/-
Copyright (c) 2021 Julian Kuelshammer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Julian Kuelshammer
-/
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.Algebra.GCDMonoid.Finset
import Mathlib.Algebra.GCDMonoid.Nat
import Mathlib.Data.Nat.Factorization.B... | Mathlib/GroupTheory/Exponent.lean | 693 | 698 | |
/-
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.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Analysis.Convex.Hull
import Mathlib.LinearAlgebra.AffineSpace.Basis
/-!
# Convex combinations
... | corresponding barycentric coordinates. -/
theorem AffineBasis.convexHull_eq_nonneg_coord {ι : Type*} (b : AffineBasis ι R E) :
convexHull R (range b) = { x | ∀ i, 0 ≤ b.coord i x } := by
rw [convexHull_range_eq_exists_affineCombination]
ext x
refine ⟨?_, fun hx => ?_⟩
· rintro ⟨s, w, hw₀, hw₁, rfl⟩ i
by... | Mathlib/Analysis/Convex/Combination.lean | 502 | 510 |
/-
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 ... |
/-- `Complex.arg ∘ (↑)` and `expMapCircle` define a partial equivalence between `circle` and `ℝ`
| Mathlib/Analysis/SpecialFunctions/Complex/Circle.lean | 37 | 38 |
/-
Copyright (c) 2023 Scott Carnahan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Carnahan
-/
import Mathlib.Data.PNat.Basic
import Mathlib.Algebra.Notation.Prod
/-!
# Typeclasses for power-associative structures
In this file we define power-associativity f... |
theorem ppow_eq_pow [Monoid M] [Pow M ℕ+] [PNatPowAssoc M] (x : M) (n : ℕ+) :
x ^ n = x ^ (n : ℕ) := by
induction n with
| one => rw [ppow_one, PNat.one_coe, pow_one]
| Mathlib/Algebra/Group/PNatPowAssoc.lean | 88 | 92 |
/-
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.MeasureTheory.MeasurableSpace.EventuallyMeasurable
import Mathlib.MeasureTheory.MeasurableSpace.Basic
import Mathlib.M... | protected theorem union (hs : NullMeasurableSet s μ) (ht : NullMeasurableSet t μ) :
NullMeasurableSet (s ∪ t) μ :=
MeasurableSet.union hs ht
| Mathlib/MeasureTheory/Measure/NullMeasurable.lean | 152 | 155 |
/-
Copyright (c) 2024 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn,
Mario Carneiro
-/
import Mathlib.Data.List.Defs
import Mathlib.Data.Option.Basic
import Mathlib.Util... | | inr h => rw [getD_eq_default _ _ h, getElem?_eq_none_iff.mpr h, Option.getD_none]
@[deprecated (since := "2025-02-14")] alias getD_eq_getD_get? := getD_eq_getD_getElem?
end getD
section getI
variable [Inhabited α]
@[simp]
theorem getI_nil : getI ([] : List α) n = default :=
| Mathlib/Data/List/GetD.lean | 89 | 100 |
/-
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]
theorem norm_mul_cos_add_sin_mul_I (x : ℂ) : (‖x‖ * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by
rw [← exp_mul_I, norm_mul_exp_arg_mul_I]
@[simp]
| Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean | 54 | 58 |
/-
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.FieldTheory.SplittingField.Construction
import Mathlib.RingTheory.Localization.Integral
import Mathlib.RingTheory.IntegralClosure.IntegrallyClosed
impo... | rcases con with (con | con)
· apply h0 con
· apply Units.ne_zero _ con
/-- **Gauss's Lemma** for GCD domains states that a primitive polynomial is irreducible iff it is
irreducible in the fraction field. -/
theorem IsPrimitive.irreducible_iff_irreducible_map_fraction_map {p : R[X]} (hp : p.IsPrimitive) :
I... | Mathlib/RingTheory/Polynomial/GaussLemma.lean | 229 | 244 |
/-
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... | /-- The monad `map` function, pointwise `Part.map` -/
def map (f : β → γ) (g : α →. β) : α →. γ := fun a => (g a).map f
| Mathlib/Data/PFun.lean | 189 | 190 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Function.L1Space.Integrable
import Mathlib.MeasureTheory.Function.LpSpace.Indicator
/-! # Functions integrable on a set an... | Mathlib/MeasureTheory/Integral/IntegrableOn.lean | 772 | 774 | |
/-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Data.Complex.Basic
import Mathlib.MeasureTheory.Integral.CircleIntegral
/-!
# Circle integral transform
In this file we define the circle integral tr... | intro x
simp_rw [circleTransformDeriv, this]
congr 2
simp [this]
theorem circleTransformDeriv_eq (f : ℂ → E) : circleTransformDeriv R z w f =
fun θ => (circleMap z R θ - w)⁻¹ • circleTransform R z w f θ := by
ext
| Mathlib/MeasureTheory/Integral/CircleTransform.lean | 48 | 55 |
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Data.Set.BooleanAlgebra
import Mathlib.Tactic.AdaptationNote
/-!
# Relations
This file defines bundled relations. A relation between `α` and `β` is a f... | Mathlib/Data/Rel.lean | 420 | 421 | |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Moritz Doll
-/
import Mathlib.LinearAlgebra.Prod
/-!
# Partially defined linear maps
A `LinearPMap R E F` or `E →ₗ.[R] F` is a linear map from a submodule of `E` to... | noncomputable def valFromGraph {g : Submodule R (E × F)}
(hg : ∀ (x : E × F) (_hx : x ∈ g) (_hx' : x.fst = 0), x.snd = 0) {a : E}
(ha : a ∈ g.map (LinearMap.fst R E F)) : F :=
(ExistsUnique.exists (existsUnique_from_graph @hg ha)).choose
theorem valFromGraph_mem {g : Submodule R (E × F)}
(hg : ∀ (x : E ×... | Mathlib/LinearAlgebra/LinearPMap.lean | 871 | 880 |
/-
Copyright (c) 2017 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Tim Baumann, Stephen Morgan, Kim Morrison, Floris van Doorn
-/
import Mathlib.Tactic.CategoryTheory.Reassoc
/-!
# Isomorphisms
This file defines isomorphisms between objects of a categ... |
open IsIso
| Mathlib/CategoryTheory/Iso.lean | 411 | 413 |
/-
Copyright (c) 2021 David Wärn,. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn, Kim Morrison
-/
import Mathlib.Combinatorics.Quiver.Prefunctor
import Mathlib.Logic.Lemmas
import Batteries.Data.List.Basic
/-!
# Paths in quivers
Given a quiver `V`, we def... | fun _ _ h => ((comp_inj' rfl).1 h).2
@[simp]
theorem comp_inj_left {p₁ p₂ : Path a b} {q : Path b c} : p₁.comp q = p₂.comp q ↔ p₁ = p₂ :=
| Mathlib/Combinatorics/Quiver/Path.lean | 137 | 140 |
/-
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.Analysis.InnerProductSpace.Subspace
import Mathlib.LinearAlgebra.SesquilinearForm
/-!
# Orthogonal complements of su... | rw [inner_sub_right, sub_eq_zero]
exact h ⟨u, hu⟩
variable (K)
| Mathlib/Analysis/InnerProductSpace/Orthogonal.lean | 86 | 90 |
/-
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.Bounds.Basic
import Mathlib.Order.Preorder.Chain
/-!
# Antichains
This file defines antichains. An antichain is a set where any two distinct elemen... |
theorem IsGreatest.antichain_iff (hs : IsGreatest s a) : IsAntichain (· ≤ ·) s ↔ s = {a} :=
(and_iff_left hs).symm.trans isAntichain_and_greatest_iff
| Mathlib/Order/Antichain.lean | 221 | 224 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Sébastien Gouëzel, Yury Kudryashov, Dylan MacKenzie, Patrick Massot
-/
import Mathlib.Algebra.BigOperators.Module
import Mathlib.Algebra.Order.Field.Power
import M... | (div_nonneg h₁ (h₁.trans h₂.le)) ((div_lt_one H).2 h₂)).congr fun _ ↦ div_pow _ _ _
theorem isBigO_pow_pow_of_le_left {r₁ r₂ : ℝ} (h₁ : 0 ≤ r₁) (h₂ : r₁ ≤ r₂) :
(fun n : ℕ ↦ r₁ ^ n) =O[atTop] fun n ↦ r₂ ^ n :=
h₂.eq_or_lt.elim (fun h ↦ h ▸ isBigO_refl _ _) fun h ↦ (isLittleO_pow_pow_of_lt_left h₁ h).isBigO... | Mathlib/Analysis/SpecificLimits/Normed.lean | 39 | 44 |
/-
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.BigOperators.Ring.Finset
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
/-... | theorem pow (H : IsRelPrime x y) : IsRelPrime (x ^ m) (y ^ n) :=
H.pow_left.pow_right
theorem pow_left_iff (hm : 0 < m) : IsRelPrime (x ^ m) y ↔ IsRelPrime x y := by
refine ⟨fun h ↦ ?_, IsRelPrime.pow_left⟩
rw [← Finset.card_range m, ← Finset.prod_const] at h
exact h.of_prod_left 0 (Finset.mem_range.mpr hm)
t... | Mathlib/RingTheory/Coprime/Lemmas.lean | 281 | 289 |
/-
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.Algebra.Field.NegOnePow
import Mathlib.Algebra.Field.Periodic
import Mathlib.Algebra.Qua... | open Lean.Meta Qq
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | 169 | 170 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Moritz Doll
-/
import Mathlib.LinearAlgebra.Prod
/-!
# Partially defined linear maps
A `LinearPMap R E F` or `E →ₗ.[R] F` is a linear map from a submodule of `E` to... | theorem mem_domain_iff {f : E →ₗ.[R] F} {x : E} : x ∈ f.domain ↔ ∃ y : F, (x, y) ∈ f.graph := by
constructor <;> intro h
· use f ⟨x, h⟩
exact f.mem_graph ⟨x, h⟩
| Mathlib/LinearAlgebra/LinearPMap.lean | 772 | 775 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
/-!
# (Generalized) Boolean algebras
A Boolean algebra is a bounded distributive lattice with a complement ope... |
@[simp] lemma sdiff_eq_right : x \ y = y ↔ x = ⊥ ∧ y = ⊥ := by
rw [disjoint_sdiff_self_left.eq_iff]; aesop
| Mathlib/Order/BooleanAlgebra.lean | 302 | 305 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Data.Finset.Lattice.Fold
import Mathlib.Data.Set.Sigma
import Mathlib.Order.CompleteLattice.Finset
/-!
# Finite sets in a ... | simp_rw [sigmaLift]
split_ifs with h
· simp [h, forall_prop_of_true h]
· simp [h, forall_prop_of_false h]
| Mathlib/Data/Finset/Sigma.lean | 190 | 193 |
/-
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.Analysis.InnerProductSpace.Projection
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique
import Mathlib.MeasureTheory.Function.L2Space
/-... | rw [indicatorConstLp_eq_toSpanSingleton_compLp hs hμs x]
have h_comp :=
condExpL2_comp_continuousLinearMap ℝ 𝕜 hm (toSpanSingleton ℝ x)
(indicatorConstLp 2 hs hμs (1 : ℝ))
refine h_comp.trans ?_
exact (toSpanSingleton ℝ x).coeFn_compLp _
@[deprecated (since := "2025-01-21")]
alias condexpL2_indicato... | Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean | 337 | 351 |
/-
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... | ⟨fun _ => preservesEpimorphisms.of_iso α, fun _ => preservesEpimorphisms.of_iso α.symm⟩
theorem reflectsMonomorphisms.of_iso {F G : C ⥤ D} [ReflectsMonomorphisms F] (α : F ≅ G) :
ReflectsMonomorphisms G :=
{ reflects := fun {X} {Y} f h => by
apply F.mono_of_mono_map
| Mathlib/CategoryTheory/Functor/EpiMono.lean | 121 | 126 |
/-
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.MeasureTheory.MeasurableSpace.CountablyGenerated
import Mathlib.MeasureTheory.Measure.MutuallySingular
import Mathlib.MeasureTheory.Measure.Typeclasses... | end Measure
open Measure
| Mathlib/MeasureTheory/Measure/Dirac.lean | 127 | 129 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
-/
import Mathlib.MeasureTheory.Integral.Lebesgue.Countable
import Mathlib.MeasureTheory.Measure.Decomposition.Exhaustion
import Mathlib.MeasureTheory.Me... | theorem prod_withDensity_right₀ {g : β → ℝ≥0∞} (hg : AEMeasurable g ν) :
μ.prod (ν.withDensity g) = (μ.prod ν).withDensity (fun z ↦ g z.2) := by
refine ext_of_lintegral _ fun φ hφ ↦ ?_
rw [lintegral_prod _ hφ.aemeasurable, lintegral_withDensity_eq_lintegral_mul₀ _ hφ.aemeasurable,
lintegral_prod]
· refine... | Mathlib/MeasureTheory/Measure/WithDensity.lean | 651 | 658 |
/-
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... | unfold One.one
unfold instOnePGame
change mk _ _ _ _ * mk _ _ _ _ ≡r _
| Mathlib/SetTheory/Game/Basic.lean | 629 | 631 |
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.Topology.Algebra.Ring.Basic
import Mathlib.Topology.Algebra.Group.Quotient
import Mathlib.RingTheory.Ideal.Quotient.Defs
/-!
# Ideals and quotients of... | instance topologicalRing_quotient : IsTopologicalRing (R ⧸ N) where
__ := QuotientAddGroup.instIsTopologicalAddGroup _
continuous_mul := (QuotientRing.isQuotientMap_coe_coe N).continuous_iff.2 <|
continuous_quot_mk.comp continuous_mul
| Mathlib/Topology/Algebra/Ring/Ideal.lean | 68 | 71 |
/-
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.Order.Bounds.Defs
import Mathlib.Order.Directed
import Mathlib.Order.BoundedOrder.Monotone
import Mathlib.Order.Interval.Set.Basic
/-... |
variable [LinearOrder α] {s : Set α} {a b : α}
| Mathlib/Order/Bounds/Basic.lean | 892 | 893 |
/-
Copyright (c) 2018 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton
-/
import Mathlib.Topology.Bases
import Mathlib.Topology.DenseEmbedding
import Mathlib.Topology.Connected.TotallyDisconnected
/-! # Stone-Čech compactification
Construction ... | -- The following refined version will never be used
/-- `pure : α → Ultrafilter α` defines a dense embedding of `α` in `Ultrafilter α`. -/
theorem isDenseEmbedding_pure : @IsDenseEmbedding _ _ ⊥ _ (pure : α → Ultrafilter α) :=
letI : TopologicalSpace α := ⊥
{ isDenseInducing_pure with injective := ultrafilter_pure_... | Mathlib/Topology/StoneCech.lean | 156 | 161 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Sébastien Gouëzel, Yury Kudryashov, Dylan MacKenzie, Patrick Massot
-/
import Mathlib.Algebra.BigOperators.Module
import Mathlib.Algebra.Order.Field.Power
import M... | (fun n ↦ ?_)
rw [norm_mul, norm_pow, div_pow, ← mul_comm_div]
conv at hC => enter [n]; rw [norm_mul, norm_pow, ← _root_.le_div_iff₀ (by positivity)]
exact mul_le_mul_of_nonneg_right (hC n) (pow_nonneg (norm_nonneg z) n)
| Mathlib/Analysis/SpecificLimits/Normed.lean | 657 | 660 |
/-
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.Ordering.Basic
import Mathlib.Order.Synonym
/-!
# Comparison
This file provides basic results about orderings and comparison in linear orders.
... | | eq, _, _, h₁, h₂ => h₁.eq_eq.2 h₂
| gt, _, _, h₁, h₂ => h₁.eq_gt.2 h₂
theorem compares_iff_of_compares_impl [LinearOrder α] [Preorder β] {a b : α} {a' b' : β}
| Mathlib/Order/Compare.lean | 88 | 91 |
/-
Copyright (c) 2022 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Tactic.NormNum.Inv
/-!
# `norm_num` extension for equalities
-/
variable {α : Type*}
open Lean Meta Qq
namespace Mathlib.Meta.NormNum
| theorem isNat_eq_false [AddMonoidWithOne α] [CharZero α] : {a b : α} → {a' b' : ℕ} →
IsNat a a' → IsNat b b' → Nat.beq a' b' = false → ¬a = b
| _, _, _, _, ⟨rfl⟩, ⟨rfl⟩, h => by simpa using Nat.ne_of_beq_eq_false h
| Mathlib/Tactic/NormNum/Eq.lean | 18 | 20 |
/-
Copyright (c) 2024 Mitchell Lee. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mitchell Lee
-/
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Coxeter.Basic
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.Zify
/-!
# The length function, reduced word... | cs.IsRightDescent w i ↔ ℓ (w * s i) + 1 = ℓ w := by
unfold IsRightDescent
constructor
· intro _
exact (cs.length_mul_simple w i).resolve_left (by omega)
· omega
theorem not_isRightDescent_iff {w : W} {i : B} :
| Mathlib/GroupTheory/Coxeter/Length.lean | 323 | 330 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
import Mathlib.Data.Sum.Basic
import Mathlib.Logic.Equiv.Option
import Mathlib.Logic.Equiv.Sum
import Mathlib.Logic.Function.Conjugate
impor... | Mathlib/Logic/Equiv/Basic.lean | 1,651 | 1,652 | |
/-
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... | Mathlib/GroupTheory/GroupAction/Basic.lean | 347 | 351 | |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yaël Dillies
-/
import Mathlib.Data.List.Iterate
import Mathlib.GroupTheory.Perm.Cycle.Basic
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Tactic.Group
/-!
# ... | split_ifs at hy <;> tauto))
(by
rw [hfg hx]
intro y hy
simp [inv_eq_iff_eq, cycleOf_apply, eq_comm (a := g y)] at hy
rw [hfg (Ne.symm hy.right), ← mul_inv_eq_one (a := g.cycleOf y), cycleOf_inv]
simp_rw [mul_inv_rev]
rw [i... | Mathlib/GroupTheory/Perm/Cycle/Factors.lean | 386 | 411 |
/-
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... | suffices ∀ i j : ℕ, ∃ k, I ^ k ≤ I ^ i ∧ I ^ k ≤ I ^ j by
simpa only [smul_eq_mul, mul_top, Algebra.id.map_eq_id, map_id, le_inf_iff] using this
intro i j
exact ⟨max i j, pow_le_pow_right (le_max_left i j), pow_le_pow_right (le_max_right i j)⟩
leftMul := by
suffices ∀ (a : R) (i : ℕ)... | Mathlib/Topology/Algebra/Nonarchimedean/AdicTopology.lean | 54 | 73 |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.CategoryTheory.Limits.Preserves.Finite
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Limits.Preserves.... | haveI := hasProduct_fin n
HasLimit.mk ⟨_, extendFanIsLimit f (limit.isLimit _) (limit.isLimit _)⟩
/-- If `C` has a terminal object and binary products, then it has finite products. -/
theorem hasFiniteProducts_of_has_binary_and_terminal : HasFiniteProducts C :=
⟨fun n => ⟨fun K => by
let that : (Discrete... | Mathlib/CategoryTheory/Limits/Constructions/FiniteProductsOfBinaryProducts.lean | 96 | 103 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Geometry.Manifold.MFDeriv.FDeriv
/-!
# Differentiability of specific functions
In this file, we establish differentiability r... |
theorem mdifferentiableWithinAt_prod_module_iff (f : M → F₁ × F₂) :
MDifferentiableWithinAt I 𝓘(𝕜, F₁ × F₂) f s x ↔
MDifferentiableWithinAt I 𝓘(𝕜, F₁) (Prod.fst ∘ f) s x ∧
| Mathlib/Geometry/Manifold/MFDeriv/SpecificFunctions.lean | 357 | 360 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yaël Dillies
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Algebra.Order.CauSeq.Basic
/-!
# Cauchy sequences and big oper... | abv ((∑ k ∈ range j, f k) * ∑ k ∈ range j, g k -
∑ n ∈ range j, ∑ m ∈ range (n + 1), f m * g (n - m)) < ε := by
let ⟨P, hP⟩ := ha.bounded
let ⟨Q, hQ⟩ := hb.bounded
have hP0 : 0 < P := lt_of_le_of_lt (abs_nonneg _) (hP 0)
have hPε0 : 0 < ε / (2 * P) := div_pos ε0 (mul_pos (show (2 : α) > 0 by norm_... | Mathlib/Algebra/Order/CauSeq/BigOperators.lean | 57 | 141 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Comma.Over.Basic
import Mathlib.CategoryTheory.Discrete.Basic
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryThe... | Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean | 1,387 | 1,388 | |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.AddTorsor.Defs
import Mathlib.GroupTheory.GroupAction.SubMulAction
import Mathlib.Topology.Algebra.Constructions
import Mathlib.Topology.Alge... | end AddTorsor
| Mathlib/Topology/Algebra/MulAction.lean | 293 | 299 |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Algebra.Order.Group.Defs
import Mathlib.Algebra.Order.Group.Unbundled.Abs
import Mathlib.Algebra.Order... | Mathlib/Algebra/Order/Group/Abs.lean | 382 | 382 | |
/-
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 | 791 | 793 | |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Data.Nat.Log
import Mathlib.Data.Nat.Pri... | (add_comm n k ▸ hnb)), multiplicity_choose_aux hp (le_add_left k n)]
simp [add_comm]
refine WithTop.add_right_cancel ?_ h₁
apply finiteMultiplicity_iff_emultiplicity_ne_top.1
exact Nat.finiteMultiplicity_iff.2 ⟨hp.ne_one, mul_pos (factorial_pos k) (factorial_pos n)⟩
/-- The multiplicity of `p` in `choo... | Mathlib/Data/Nat/Multiplicity.lean | 201 | 217 |
/-
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... | variable {T : Type*} [CommRing T] [Algebra R T]
/-- Adjoining a root gives a unique ring up to algebra isomorphism.
This is the converse of `IsAdjoinRoot.ofEquiv`: this turns an `IsAdjoinRoot` into an
`AlgEquiv`, and `IsAdjoinRoot.ofEquiv` turns an `AlgEquiv` into an `IsAdjoinRoot`.
-/
def aequiv (h : IsAdjoinRoot S ... | Mathlib/RingTheory/IsAdjoinRoot.lean | 547 | 562 |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.IsPrimePow
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Ring.Int
import Mathlib.Algebra.Ring.CharZero
im... |
/-- `Nat.swap_mem_divisorsAntidiagonal` with the LHS in simp normal form. -/
@[deprecated swap_mem_divisorsAntidiagonal (since := "2025-02-17")]
| Mathlib/NumberTheory/Divisors.lean | 314 | 316 |
/-
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.Analytic.Within
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries
/-!
# Higher d... | DifferentiableWithinAt 𝕜 f (insert x s) x := by
rcases contDiffWithinAt_nat.1 (h.of_le hn) with ⟨u, hu, p, H⟩
rcases mem_nhdsWithin.1 hu with ⟨t, t_open, xt, tu⟩
rw [inter_comm] at tu
exact (differentiableWithinAt_inter (IsOpen.mem_nhds t_open xt)).1 <|
((H.mono tu).differentiableOn le_rfl) x ⟨mem_inse... | Mathlib/Analysis/Calculus/ContDiff/Defs.lean | 344 | 367 |
/-
Copyright (c) 2017 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Mario Carneiro
-/
import Mathlib.Algebra.Ring.CharZero
import Mathlib.Algebra.Star.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Order.Interval.Set.UnorderedInterva... | simp only [equivRealProd_apply, singleton_prod, mem_image, Prod.mk.injEq,
exists_eq_right_right, mem_preimage] at hx
obtain ⟨x₁, hx₁, hx₁', hx₁''⟩ := hx
| Mathlib/Data/Complex/Basic.lean | 835 | 837 |
/-
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 | 870 | 871 | |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Johan Commelin, Patrick Massot
-/
import Mathlib.Algebra.GroupWithZero.InjSurj
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.GroupWithZero.WithZero
impo... | set_option linter.deprecated false in
@[deprecated OrderIso.mulLeft₀_symm (since := "2024-11-18")]
theorem OrderIso.mulLeft₀'_symm {a : α} (ha : a ≠ 0) :
| Mathlib/Algebra/Order/GroupWithZero/Canonical.lean | 186 | 188 |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Finset.Grade
import Mathlib.Data.Finset.Powerset
import Mathlib.Order.Interval.Finset.Basic
/-!
# Intervals of finsets as finsets
This file provides... |
/-- Cardinality of a non-empty `Icc` of finsets. -/
theorem card_Icc_finset (h : s ⊆ t) : (Icc s t).card = 2 ^ (t.card - s.card) := by
rw [← card_sdiff h, ← card_powerset, Icc_eq_image_powerset h, Finset.card_image_iff]
rintro u hu v hv (huv : s ⊔ u = s ⊔ v)
rw [mem_coe, mem_powerset] at hu hv
rw [← (disjoint_... | Mathlib/Data/Finset/Interval.lean | 90 | 97 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Justus Springer
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.AlgebraicGeometry.StructureSheaf
import Mathlib.RingTheory.Localization.Localizatio... | lemma Spec.locallyRingedSpaceObj_sheaf' (R : Type u) [CommRing R] :
(Spec.locallyRingedSpaceObj <| CommRingCat.of R).sheaf = structureSheaf R := rfl
lemma Spec.locallyRingedSpaceObj_presheaf_map (R : CommRingCat.{u}) {U V} (i : U ⟶ V) :
(Spec.locallyRingedSpaceObj R).presheaf.map i =
(structureSheaf R).1.m... | Mathlib/AlgebraicGeometry/Spec.lean | 184 | 197 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Finset.Finsupp
import Mathlib.Data.Finsupp.Order
import Mathlib.Order.Interval.Finset.Basic
/-!
# Finite intervals of finitely supported functions
T... | Mathlib/Data/Finsupp/Interval.lean | 145 | 147 | |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Simon Hudon, Mario Carneiro
-/
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Init
import Mathlib.Data.Int.Init
import Mathlib.... | Mathlib/Algebra/Group/Basic.lean | 1,135 | 1,136 | |
/-
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.Primrec
import Mathlib.Data.Nat.PSub
import Mathlib.Data.PFun
/-!
# The partial recursive functions
The partial recursive functions are... | (∃ n : ℕ,
((∃ b' : σ, Sum.inl b' ∈ F a n) ∧ ∀ {m : ℕ}, m < n → ∃ b : α, Sum.inr b ∈ F a m) ∧
Sum.inl b ∈ F a n) ↔
b ∈ PFun.fix f a := by
intro F; refine ⟨fun h => ?_, fun h => ?_⟩
· rcases h with ⟨n, ⟨_x, h₁⟩, h₂⟩
have : ∀ m a', Sum.inr a' ∈ F a m → b ∈ PFun.fix f a' → b ∈ PFun.fix f... | Mathlib/Computability/Partrec.lean | 723 | 738 |
/-
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.Combinatorics.SetFamily.Compression.Down
import Mathlib.Data.Fintype.Powerset
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Algebra.BigOperators.... | Mathlib/Combinatorics/SetFamily/Shatter.lean | 98 | 98 | |
/-
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... | ring_nf
/-- Formula for `ζ 1`. Note that mathematically `ζ 1` is undefined, but our construction ascribes
this particular value to it. -/
lemma _root_.riemannZeta_one : riemannZeta 1 = (γ - Complex.log (4 * ↑π)) / 2 := by
have := (HurwitzZeta.tendsto_hurwitzZetaEven_sub_one_div_nhds_one 0).mono_left
<| nhdsWit... | Mathlib/NumberTheory/Harmonic/ZetaAsymp.lean | 383 | 389 |
/-
Copyright (c) 2018 Andreas Swerdlow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andreas Swerdlow, Kexing Ying
-/
import Mathlib.Algebra.Algebra.Bilinear
import Mathlib.LinearAlgebra.Basis.Defs
import Mathlib.LinearAlgebra.BilinearForm.Basic
import Mathlib.Linear... | LinearEquiv.congrRight₂ (.refl R N₁) = .refl R (BilinMap R M N₁) := rfl
@[simp]
| Mathlib/LinearAlgebra/BilinearForm/Hom.lean | 254 | 256 |
/-
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.Ring.Associated
import Mathlib.Algebra.Star.Unitary
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.Tactic.Ring
import Mathlib.Al... | Mathlib/NumberTheory/Zsqrtd/Basic.lean | 996 | 1,011 | |
/-
Copyright (c) 2020 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.FieldTheory.RatFunc.AsPolynomial
import Mathlib.NumberTheory.ArithmeticFunction
import Mathlib.RingTheory.Ro... | rw [(div_modByMonic_unique (cyclotomic' n K) 0 prod_monic _).1]
simp only [degree_zero, zero_add]
refine ⟨by rw [mul_comm], ?_⟩
rw [bot_lt_iff_ne_bot]
intro h
exact Monic.ne_zero prod_monic (degree_eq_bot.1 h)
/-- If there is a primitive `n`-th root of unity in `K`, then `cyclotomic' n K` comes from a
| Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean | 167 | 174 |
/-
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.FieldTheory.Finiteness
import Mathlib.LinearAlgebra.AffineSpace.Basis
import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
/-!
# Finite-dimensional subsp... | obtain ⟨r, hr⟩ := hv (p -ᵥ p₀) (vsub_mem_vectorSpan k hp h)
use r
rw [eq_vadd_iff_vsub_eq]
| Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean | 426 | 428 |
/-
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
-/
import Mathlib.Data.Set.Lattice.Image
import Mathlib.Order.Interval.Set.LinearOrder
/-!
# Extra lemmas about intervals
This file contains lemma... | theorem eq_of_Ico_disjoint {x₁ x₂ y₁ y₂ : α} (h : Disjoint (Ico x₁ x₂) (Ico y₁ y₂)) (hx : x₁ < x₂)
(h2 : x₂ ∈ Ico y₁ y₂) : y₁ = x₂ := by
rw [Ico_disjoint_Ico, min_eq_left (le_of_lt h2.2), le_max_iff] at h
| Mathlib/Order/Interval/Set/Disjoint.lean | 143 | 145 |
/-
Copyright (c) 2018 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.CategoryTheory.Iso
import Mathlib.CategoryTheory.Functor.Category
import Mathlib.CategoryTheory.Functor.FullyFaithful
/-!
# Whiskering
Given a functor `F... | @[simp]
theorem isoWhiskerLeft_twice (F : B ⥤ C) (G : C ⥤ D) {H K : D ⥤ E} (α : H ≅ K) :
isoWhiskerLeft F (isoWhiskerLeft G α) =
(Functor.associator _ _ _).symm ≪≫ isoWhiskerLeft (F ⋙ G) α ≪≫ Functor.associator _ _ _ := by
| Mathlib/CategoryTheory/Whiskering.lean | 301 | 304 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.Exact
import Mathlib.CategoryTheory.ComposableArrows
/-!
# Exact sequences
A sequence of `n` composable arrows `S : ComposableArr... |
/-- The isomorphism `S₁.sc _ i ≅ S₂.sc _ i` induced by an isomorphism `S₁ ≅ S₂`
in `ComposableArrows C n`. -/
| Mathlib/Algebra/Homology/ExactSequence.lean | 164 | 166 |
/-
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.Data.Bundle
import Mathlib.Data.Set.Image
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.PartialHomeomorph
import Mathlib.Topology.O... | theorem proj_symm_apply {x : B × F} (hx : x ∈ e.target) :
proj (e.toPartialHomeomorph.symm x) = x.1 :=
e.toPretrivialization.proj_symm_apply hx
theorem proj_symm_apply' {b : B} {x : F} (hx : b ∈ e.baseSet) :
| Mathlib/Topology/FiberBundle/Trivialization.lean | 339 | 343 |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.WSeq.Basic
import Mathlib.Data.WSeq.Defs
import Mathlib.Data.WSeq.Productive
import Mathlib.Data.WSeq.Relation
deprecated_module (since :=... | Mathlib/Data/Seq/WSeq.lean | 1,529 | 1,540 | |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Kim Morrison
-/
import Mathlib.CategoryTheory.Subobject.Lattice
/-!
# Specific subobjects
We define `equalizerSubobject`, `kernelSubobject` and `imageSubobject`, which ar... | (kernelSubobject g).arrow ≫ inv f := by
simp [kernelSubobjectIsoComp]
| Mathlib/CategoryTheory/Subobject/Limits.lean | 190 | 191 |
/-
Copyright (c) 2015 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis
-/
import Mathlib.Algebra.Order.Monoid.Unbundled.Pow
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Ring.Parity
import Mathlib.Tactic.Bound... | rw [pow_succ]
calc
_ ≤ 2 ^ n * (a ^ (n + 1) + b ^ (n + 1)) * (a + b) :=
mul_le_mul_of_nonneg_right (add_pow_le ha hb (n + 1)) <| add_nonneg ha hb
| Mathlib/Algebra/Order/Ring/Basic.lean | 208 | 211 |
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Simon Hudon
-/
import Mathlib.Control.Functor.Multivariate
import Mathlib.Data.PFunctor.Univariate.Basic
/-!
# Multivariate polynomial functors.
Multivariate polynomial... |
end Const
| Mathlib/Data/PFunctor/Multivariate/Basic.lean | 106 | 108 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Amelia Livingston, Yury Kudryashov,
Neil Strickland, Aaron Anderson
-/
import Mathlib.Algebra.Divisibility.Basic
import Mathlib.Algeb... | apply Units.mul_left_dvd
end CommMonoid
| Mathlib/Algebra/Divisibility/Units.lean | 110 | 112 |
/-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Algebra.Ring.Int.Defs
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.Cast.Order.Basic
import Math... | Mathlib/Data/Num/Lemmas.lean | 1,193 | 1,194 | |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Aurélien Saue, Anne Baanen
-/
import Mathlib.Tactic.NormNum.Inv
import Mathlib.Tactic.NormNum.Pow
import Mathlib.Util.AtomM
/-!
# `ring` tactic
A tactic for solving e... | private local instance {m} [Pure m] : MonadLift Option (OptionT m) where
monadLift f := .mk <| pure f
| Mathlib/Tactic/Ring/Basic.lean | 316 | 318 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yaël Dillies
-/
import Mathlib.Logic.Equiv.Set
import Mathlib.Order.CompleteLattice.Lemmas
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection.Basic
/-... |
/-- A complete distributive lattice is a complete lattice whose `⊔` and `⊓` respectively
distribute over `⨅` and `⨆`. -/
class CompleteDistribLattice (α : Type*) extends Frame α, Coframe α, BiheytingAlgebra α
/-- Structure containing the minimal axioms required to check that an order is a completely
distributive. Do ... | Mathlib/Order/CompleteBooleanAlgebra.lean | 107 | 120 |
/-
Copyright (c) 2021 Roberto Alvarez. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Roberto Alvarez
-/
import Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup
import Mathlib.GroupTheory.EckmannHilton
import Mathlib.Algebra.Equiv.TransferInstance
import ... |
theorem fromLoop_apply (i : N) {p : Ω (Ω^ { j // j ≠ i } X x) const} {t : I^N} :
fromLoop i p t = p (t i) (Cube.splitAt i t).snd :=
rfl
| Mathlib/Topology/Homotopy/HomotopyGroup.lean | 242 | 245 |
/-
Copyright (c) 2018 Andreas Swerdlow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andreas Swerdlow
-/
import Mathlib.LinearAlgebra.Basis.Basic
import Mathlib.LinearAlgebra.BilinearMap
import Mathlib.LinearAlgebra.LinearIndependent.Lemmas
/-!
# Sesquilinear maps
... | simpa only [← two_mul] using this
end OrthogonalMap
| Mathlib/LinearAlgebra/SesquilinearForm.lean | 494 | 496 |
/-
Copyright (c) 2021 Vladimir Goryachev. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Vladimir Goryachev, Kyle Miller, Kim Morrison, Eric Rodriguez
-/
import Mathlib.Algebra.Group.Nat.Range
import Mathlib.Data.Set.Finite.Basic
/-!
# Counting on ℕ
Thi... | theorem count_le_card (hp : (setOf p).Finite) (n : ℕ) : count p n ≤ #hp.toFinset := by
rw [count_eq_card_filter_range]
exact Finset.card_mono fun x hx ↦ hp.mem_toFinset.2 (mem_filter.1 hx).2
| Mathlib/Data/Nat/Count.lean | 120 | 122 |
/-
Copyright (c) 2022 Julian Kuelshammer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Julian Kuelshammer
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Nat.Choose.Central
import Mathlib.Tactic.Field... | Mathlib/Combinatorics/Enumerative/Catalan.lean | 202 | 204 | |
/-
Copyright (c) 2020 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp
-/
import Mathlib.Algebra.Algebra.Spectrum.Basic
import Mathlib.Algebra.Module.LinearMap.Basic
import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
import Mathl... | `(f - μ • id) ^ k` for some `k`. -/
lemma genEigenspace_top_eq_maxUnifEigenspaceIndex [IsNoetherian R M] (f : End R M) (μ : R) :
genEigenspace f μ ⊤ = f.genEigenspace μ (maxUnifEigenspaceIndex f μ) := by
have := WellFoundedGT.iSup_eq_monotonicSequenceLimit <|
(f.genEigenspace μ).comp <| WithTop.coeOrderHom.to... | Mathlib/LinearAlgebra/Eigenspace/Basic.lean | 265 | 269 |
/-
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... | section
| Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 710 | 711 |
/-
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.Linear
import Mathlib.Analysis.Calculus.FDeriv.Comp
/-!
# Additive operations on derivative... | theorem HasStrictFDerivAt.sub (hf : HasStrictFDerivAt f f' x) (hg : HasStrictFDerivAt g g' x) :
HasStrictFDerivAt (fun x => f x - g x) (f' - g') x := by
| Mathlib/Analysis/Calculus/FDeriv/Add.lean | 464 | 465 |
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kenny Lau
-/
import Mathlib.Algebra.Group.Units.Basic
import Mathlib.RingTheory.MvPowerSeries.Basic
import Mathlib.RingTheory.MvPowerSeries.NoZeroDivisors
import Mathl... | φ⁻¹ * φ = 1 := by rw [mul_comm, φ.mul_inv_cancel h]
protected theorem eq_mul_inv_iff_mul_eq {φ₁ φ₂ φ₃ : MvPowerSeries σ k}
(h : constantCoeff σ k φ₃ ≠ 0) : φ₁ = φ₂ * φ₃⁻¹ ↔ φ₁ * φ₃ = φ₂ :=
| Mathlib/RingTheory/MvPowerSeries/Inverse.lean | 247 | 250 |
/-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Ring.Idempotent
import Mathlib.Order.Basic
import Mathlib.RingTheory.Finiteness.Nakayama
/-!
## Lemmas on idempo... | constructor
· intro e
obtain ⟨r, hr, hr'⟩ :=
Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I I h
(by
rw [smul_eq_mul]
exact e.ge)
simp_rw [smul_eq_mul] at hr'
refine ⟨r, hr' r hr, antisymm ?_ ((Submodule.span_singleton_le_iff_mem _ _).mpr hr)⟩
intro x hx
... | Mathlib/RingTheory/Ideal/IdempotentFG.lean | 20 | 35 |
/-
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.Algebra.MvPolynomial.Counit
import Mathlib.Algebra.MvPolynomial.Invertible
import Mathlib.RingTheory.WittVector.Defs
/-!
# Witt vecto... | /-- Auxiliary tactic for showing that `mapFun` respects the ring operations. -/
-- porting note: a very crude port.
macro "map_fun_tac" : tactic => `(tactic| (
| Mathlib/RingTheory/WittVector/Basic.lean | 79 | 81 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kenny Lau
-/
import Mathlib.Data.List.Forall2
/-!
# zip & unzip
This file provides results about `List.zipWith`, `List.zip` and `List.unzip` (definitions are in
core ... | use tl', tail
simp_all
end List
| Mathlib/Data/List/Zip.lean | 137 | 140 |
/-
Copyright (c) 2022 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer, Kevin Klinge, Andrew Yang
-/
import Mathlib.Algebra.Group.Submonoid.DistribMulAction
import Mathlib.GroupTheory.OreLocalization.Basic
import Mathlib.Algebra.GroupWi... | Mathlib/RingTheory/OreLocalization/Basic.lean | 739 | 749 | |
/-
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.CategoryTheory.Comma.Over.Pullback
import Mathlib.CategoryTheory.Limits.Shapes.KernelPair
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq
import ... | @[reassoc (attr := simp)]
theorem pullbackDiagonalMapIdIso_inv_fst :
(pullbackDiagonalMapIdIso f g i).inv ≫ pullback.fst _ _ = pullback.fst _ _ ≫ f := by
rw [Iso.inv_comp_eq, ← Category.comp_id (pullback.fst _ _), ← diagonal_fst i,
| Mathlib/CategoryTheory/Limits/Shapes/Diagonal.lean | 259 | 262 |
/-
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 exists_pow_eq_self_of_coprime (h : n.Coprime (orderOf x)) : ∃ m : ℕ, (x ^ n) ^ m = x := by
by_cases h0 : orderOf x = 0
| Mathlib/GroupTheory/OrderOfElement.lean | 279 | 280 |
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics
import Mathlib.Analysis.Convex.Deriv
/-!
# The function `x ↦ ... | @[simp]
lemma deriv_mul_log {x : ℝ} (hx : x ≠ 0) : deriv (fun x ↦ x * log x) x = log x + 1 := by
rw [deriv_mul differentiableAt_id' (differentiableAt_log hx)]
simp only [deriv_id'', one_mul, deriv_log', ne_eq, add_right_inj]
| Mathlib/Analysis/SpecialFunctions/Log/NegMulLog.lean | 50 | 53 |
/-
Copyright (c) 2023 Scott Carnahan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Carnahan
-/
import Mathlib.Algebra.Group.NatPowAssoc
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Eval.SMul
/-!
# Scalar-multiple polynomial ev... | theorem smeval_one : (1 : R[X]).smeval x = 1 • x ^ 0 := by
rw [← C_1, smeval_C]
simp only [Nat.cast_one, one_smul]
| Mathlib/Algebra/Polynomial/Smeval.lean | 83 | 85 |
/-
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... | rw [coe_nsmul, two_nsmul_eq_iff]
-- Porting note: `congr` didn't simplify the goal of iff of `Or`s
convert Iff.rfl
rw [add_comm, ← coe_add, ← sub_eq_zero, ← coe_sub, neg_div, ← neg_sub, sub_neg_eq_add, add_assoc,
add_halves, ← two_mul, coe_neg, coe_two_pi, neg_zero]
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 186 | 191 |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.CategoryTheory.Adjunction.FullyFaithful
import Mathlib.CategoryTheory.Functor.EpiMono
import Mathlib.CategoryTheory.HomCongr
/-!
# Reflective functors
Ba... | Mathlib/CategoryTheory/Adjunction/Reflective.lean | 263 | 265 |
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