Context stringlengths 227 76.5k | target stringlengths 0 11.6k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 16 3.69k |
|---|---|---|---|---|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.List.Sublists
import Mathlib.Data.List.Zip
import Mathlib.Data.Multiset.Bind
import Mathlib.Data.Multiset.Range
/-!
# The powerset of a multiset
... | congr_arg ((↑) : List (Multiset α) → Multiset (Multiset α)) powersetCardAux_eq_map_coe
@[simp]
theorem powersetCard_zero_left (s : Multiset α) : powersetCard 0 s = {0} :=
Quotient.inductionOn s fun l => by simp [powersetCard_coe']
theorem powersetCard_zero_right (n : ℕ) : @powersetCard α (n + 1) 0 = 0 :=
rfl
@... | Mathlib/Data/Multiset/Powerset.lean | 211 | 227 |
/-
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.... | end AddCommGroup
end Submodule
| Mathlib/Algebra/Module/Submodule/Map.lean | 494 | 496 |
/-
Copyright (c) 2022 Tian Chen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Tian Chen, Mantas Bakšys
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Algebra.Ring.Int.Parity
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.D... | rw [← geom_sum₂_mul, emultiplicity_mul hp, emultiplicity_geom_sum₂_eq_one hp hp1 hxy hx, add_comm]
@[deprecated (since := "2024-11-30")]
| Mathlib/NumberTheory/Multiplicity.lean | 176 | 178 |
/-
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... | theorem sdiff_lt_sdiff_right (h : x < y) (hz : z ≤ x) : x \ z < y \ z :=
(sdiff_le_sdiff_right h.le).lt_of_not_le
fun h' => h.not_le <| le_sdiff_sup.trans <| sup_le_of_le_sdiff_right h' hz
theorem sup_inf_inf_sdiff : x ⊓ y ⊓ z ⊔ y \ z = x ⊓ y ⊔ y \ z :=
| Mathlib/Order/BooleanAlgebra.lean | 308 | 312 |
/-
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.RingTheory.WittVector.Basic
import Mathlib.RingTheory.WittVector.IsPoly
/-!
## The Verschiebung operator
## References
* [Hazewinkel, *Witt Vectors*... | theorem verschiebung_coeff_zero (x : 𝕎 R) : (verschiebung x).coeff 0 = 0 :=
rfl
-- simp_nf complains if this is simp
theorem verschiebung_coeff_add_one (x : 𝕎 R) (n : ℕ) :
| Mathlib/RingTheory/WittVector/Verschiebung.lean | 139 | 143 |
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.FieldTheory.RatFunc.AsPolynomial
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Poly... | @[simp]
theorem intDegree_polynomial {p : K[X]} :
intDegree (algebraMap K[X] (RatFunc K) p) = natDegree p := by
| Mathlib/FieldTheory/RatFunc/Degree.lean | 59 | 61 |
/-
Copyright (c) 2023 Sidharth Hariharan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Sidharth Hariharan
-/
import Mathlib.Algebra.Polynomial.Div
import Mathlib.Logic.Function.Basic
import Mathlib.RingTheory.Coprime.Lemmas
import Mathlib.RingTheory.Lo... | ∃ q r₁ r₂ : R[X],
r₁.degree < g₁.degree ∧
r₂.degree < g₂.degree ∧ (f : K) / (↑g₁ * ↑g₂) = ↑q + ↑r₁ / ↑g₁ + ↑r₂ / ↑g₂ := by
rcases hcoprime with ⟨c, d, hcd⟩
refine
⟨f * d /ₘ g₁ + f * c /ₘ g₂, f * d %ₘ g₁, f * c %ₘ g₂, degree_modByMonic_lt _ hg₁,
degree_modByMonic_lt _ hg₂, ?_⟩
have hg₁'... | Mathlib/Algebra/Polynomial/PartialFractions.lean | 60 | 79 |
/-
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.Analysis.Convex.Hull
/-!
# Convex join
This file defines the convex join of two sets. The convex join of `s` and `t` is the union of the
segments with on... | @[simp]
theorem convexJoin_union_left (s₁ s₂ t : Set E) :
| Mathlib/Analysis/Convex/Join.lean | 65 | 66 |
/-
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
-/
import Mathlib.Algebra.Order.Group.Pointwise.Interval
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.Rat.Cardinal
import Mathlib.SetTheory.Cardi... | cases h' : f n
· simp [h']
| Mathlib/Data/Real/Cardinality.lean | 69 | 70 |
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.AlgebraicTopology.DoldKan.Decomposition
import Mathlib.Tactic.FinCases
/-!
# Behaviour of P_infty with respect to degeneracies
For any `X : SimplicialObject C... | theorem σ_comp_PInfty (X : SimplicialObject C) {n : ℕ} (i : Fin (n + 1)) :
X.σ i ≫ PInfty.f (n + 1) = 0 := by
rw [PInfty_f, σ_comp_P_eq_zero X i]
simp only [le_add_iff_nonneg_left, zero_le]
| Mathlib/AlgebraicTopology/DoldKan/Degeneracies.lean | 120 | 123 |
/-
Copyright (c) 2023 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Algebra.Order.ToIntervalMod
import Mathli... | protected lemma GrowsPolynomially.add {f g : ℝ → ℝ} (hf : GrowsPolynomially f)
(hg : GrowsPolynomially g) (hf' : 0 ≤ᶠ[atTop] f) (hg' : 0 ≤ᶠ[atTop] g) :
GrowsPolynomially fun x => f x + g x := by
intro b hb
have hf := hf b hb
have hg := hg b hb
obtain ⟨c₁, hc₁_mem, c₂, hc₂_mem, hf⟩ := hf
obtain ⟨c₃, hc... | Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean | 361 | 399 |
/-
Copyright (c) 2021 Luke Kershaw. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Luke Kershaw
-/
import Mathlib.CategoryTheory.Adjunction.Limits
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproduct... | lemma _root_.CategoryTheory.Iso.inv_hom_id_triangle_hom₁ {A B : Triangle C} (e : A ≅ B) :
e.inv.hom₁ ≫ e.hom.hom₁ = 𝟙 _ := by rw [← comp_hom₁, e.inv_hom_id, id_hom₁]
@[reassoc (attr := simp)]
| Mathlib/CategoryTheory/Triangulated/Basic.lean | 217 | 219 |
/-
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.Data.Set.Constructions
import Mathlib.Order.Filter.AtTopBot.CountablyGenerated
import Mathlib.Topology.Constructions
import Mathlib.Top... | (hb.isOpen hs).mem_nhds ha
theorem IsTopologicalBasis.exists_subset_of_mem_open {b : Set (Set α)} (hb : IsTopologicalBasis b)
{a : α} {u : Set α} (au : a ∈ u) (ou : IsOpen u) : ∃ v ∈ b, a ∈ v ∧ v ⊆ u :=
| Mathlib/Topology/Bases.lean | 165 | 168 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou, Johan Commelin
-/
import Mathlib.Algebra.Ring.Int.Parity
import Mathlib.Algebra.Ring.Int.Units
import Mathlib.Data.ZMod.IntUnitsPower
/-!
# Integer powers of (-1)
This file def... |
lemma negOnePow_succ (n : ℤ) : (n + 1).negOnePow = - n.negOnePow := by
rw [negOnePow_add, negOnePow_one, mul_neg, mul_one]
| Mathlib/Algebra/Ring/NegOnePow.lean | 39 | 41 |
/-
Copyright (c) 2021 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Logic.Function.Basic
import Mathlib.Logic.Relator
/-!
# Types that are empty
In this file we define a typeclass `IsEmpty`, which expresses that a... |
theorem isEmpty_subtype (p : α → Prop) : IsEmpty (Subtype p) ↔ ∀ x, ¬p x := by
| Mathlib/Logic/IsEmpty.lean | 163 | 164 |
/-
Copyright (c) 2019 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.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno
import Mathlib.Analysis.Calculus.FDeriv.Ad... | /-- `x ↦ fderiv 𝕜 (f x) (g x)` is smooth at `x₀`. -/
protected theorem ContDiffAt.fderiv {f : E → F → G} {g : E → F}
(hf : ContDiffAt 𝕜 n (Function.uncurry f) (x₀, g x₀)) (hg : ContDiffAt 𝕜 m g x₀)
(hmn : m + 1 ≤ n) : ContDiffAt 𝕜 m (fun x => fderiv 𝕜 (f x) (g x)) x₀ := by
simp_rw [← fderivWithin_univ]
| Mathlib/Analysis/Calculus/ContDiff/Basic.lean | 1,163 | 1,167 |
/-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Bryan Gin-ge Chen
-/
import Mathlib.Logic.Relation
import Mathlib.Order.CompleteLattice.Basic
import Mathlib.Order.GaloisConnection.Defs
/-!
# Equivalence relati... |
/-- The supremum of a set S of equivalence relations is the equivalence closure of the binary
relation `there exists r ∈ S relating x and y`. -/
theorem sSup_eq_eqvGen (S : Set (Setoid α)) :
sSup S = EqvGen.setoid fun x y => ∃ r : Setoid α, r ∈ S ∧ r x y := by
rw [eqvGen_eq]
apply congr_arg sInf
| Mathlib/Data/Setoid/Basic.lean | 246 | 252 |
/-
Copyright (c) 2019 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Yury Kudryashov, Sébastien Gouëzel, Chris Hughes
-/
import Mathlib.Data.Fin.Rev
import Mathlib.Data.Nat.Find
/-!
# Operation on tuples
We interpret maps `∀ i : Fi... | dif_pos _
@[simp]
theorem snoc_comp_cast_add {n m : ℕ} {α : Sort*} (f : Fin (n + m) → α) (a : α) :
| Mathlib/Data/Fin/Tuple/Basic.lean | 537 | 540 |
/-
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.Filter.SmallSets
import Mathlib.Topology.ContinuousOn
/-!
### Locally finite families of sets
We say that a family of sets in a topological s... | /-- Let `f : ℕ → Π a, β a` be a sequence of (dependent) functions on a topological space. Suppose
that the family of sets `s n = {x | f (n + 1) x ≠ f n x}` is locally finite. Then there exists a
function `F : Π a, β a` such that for any `x`, we have `f n x = F x` on the product of an infinite
| Mathlib/Topology/LocallyFinite.lean | 129 | 131 |
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.Limits.Pullbacks
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
/-!
# Open immersions of structured spaces
We say that a m... | @[simp, reassoc]
theorem app_invApp (U : Opens Y) :
f.c.app (op U) ≫ H.invApp _ ((Opens.map f.base).obj U) =
Y.presheaf.map
((homOfLE (Set.image_preimage_subset f.base U.1)).op :
op U ⟶ op (opensFunctor f |>.obj ((Opens.map f.base).obj U))) := by
erw [← Category.assoc]; rw [IsIso.comp_inv_... | Mathlib/Geometry/RingedSpace/OpenImmersion.lean | 196 | 203 |
/-
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.... | @[to_additive]
theorem div_mul_div_comm : a / b * (c / d) = a * c / (b * d) := by simp
| Mathlib/Algebra/Group/Basic.lean | 604 | 606 |
/-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Fold
import Mathlib.LinearAlgebra.ExteriorAlgebra.Basic
import Mathlib... |
This is $\bar \lambda_B$ from [bourbaki2007][] $9 Proposition 3. -/
@[simps apply]
def changeFormEquiv : CliffordAlgebra Q ≃ₗ[R] CliffordAlgebra Q' :=
{ changeForm h with
toFun := changeForm h
invFun := changeForm (changeForm.neg_proof h)
left_inv := fun x => by
exact (changeForm_changeForm _ _ x).... | Mathlib/LinearAlgebra/CliffordAlgebra/Contraction.lean | 323 | 331 |
/-
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, Johan Commelin
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Div
import Mathlib.RingTheory... | Mathlib/Algebra/Polynomial/RingDivision.lean | 654 | 667 | |
/-
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... | | ⟨-[x+1], (y + 1 : Nat)⟩, (xy : SqLe _ _ _ _), (yx : SqLe _ _ _ _) => by
let t := le_antisymm xy yx
rw [one_mul] at t
exact absurd t (not_divides_sq _ _)
theorem le_antisymm {a b : ℤ√d} (ab : a ≤ b) (ba : b ≤ a) : a = b :=
eq_of_sub_eq_zero <| nonneg_antisymm ba (by rwa [neg_sub])
instance linearOrde... | Mathlib/NumberTheory/Zsqrtd/Basic.lean | 793 | 803 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Order.Filter.Tendsto
import Mathlib.Data.Set.Accumulate
import Mathlib.Topology.Bornology.Basic
import Mathlib.Topolog... | ∃ b', b' ⊆ b ∧ Set.Finite b' ∧ s ⊆ ⋃ i ∈ b', c i := by
simp only [Subtype.forall', biUnion_eq_iUnion] at hc₁ hc₂
rcases hs.elim_finite_subcover (fun i => c i : b → Set X) hc₁ hc₂ with ⟨d, hd⟩
refine ⟨Subtype.val '' d.toSet, ?_, d.finite_toSet.image _, ?_⟩
· simp
· rwa [biUnion_image]
| Mathlib/Topology/Compactness/Compact.lean | 312 | 317 |
/-
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.Homology
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Preadditive.Opposite
/-!
# Homolo... | @[simp]
lemma leftHomologyMap'_sub :
leftHomologyMap' (φ - φ') h₁ h₂ = leftHomologyMap' φ h₁ h₂ -
leftHomologyMap' φ' h₁ h₂ := by
simp only [sub_eq_add_neg, leftHomologyMap'_add, leftHomologyMap'_neg]
| Mathlib/Algebra/Homology/ShortComplex/Preadditive.lean | 126 | 130 |
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.Polynomial.Degree.Domain
import Mathlib.Algebra.Ring.NonZeroDivisors
import Mathlib.RingTheory.Localization.FractionRing
/-!
# The field of rational... | rw [mk_def_of_ne _ hq, Localization.mk_eq_mk']
theorem mk_one' (p : K[X]) :
RatFunc.mk p 1 = ofFractionRing (algebraMap _ _ p) := by
| Mathlib/FieldTheory/RatFunc/Defs.lean | 162 | 165 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Polynomial.Expand
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.FieldTheory.IntermediateField.Basic
imp... |
theorem separable_C (r : R) : (C r).Separable ↔ IsUnit r := by
rw [separable_def, derivative_C, isCoprime_zero_right, isUnit_C]
| Mathlib/FieldTheory/Separable.lean | 76 | 78 |
/-
Copyright (c) 2018 . All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.SpecificGroups.Cyclic
/-!
# p-groups
This file contains a proof that if `G` is a `p`-group ac... |
/-- A group of order `p ^ 2` is commutative. See also `IsPGroup.commGroupOfCardEqPrimeSq`
for the `CommGroup` instance. -/
theorem commutative_of_card_eq_prime_sq (hG : Nat.card G = p ^ 2) : ∀ a b : G, a * b = b * a :=
(commGroupOfCardEqPrimeSq hG).mul_comm
| Mathlib/GroupTheory/PGroup.lean | 368 | 373 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.Polynomial.Basi... | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 1,378 | 1,381 | |
/-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
import Mathlib.Topology.Category.Profinite.Nobeling.Basic
import Mathlib.Topology.Category.Profinite.Nobeling.Induction
import Mathlib.Topology.Category.Profinite... | Mathlib/Topology/Category/Profinite/Nobeling.lean | 1,396 | 1,400 | |
/-
Copyright (c) 2017 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Keeley Hoek
-/
import Mathlib.Algebra.NeZero
import Mathlib.Data.Int.DivMod
import Mathlib.Logic.Embedding.Basic
import Mathlib.Logic.Equiv.Set
import Mathlib.Tactic.... | Mathlib/Data/Fin/Basic.lean | 1,528 | 1,529 | |
/-
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.Algebra.Order.Group.Pointwise.Interval
import Mathlib.LinearAlgebra.BilinearMap
import Mathlib.LinearAlgebra.Pi
import Mathlib.LinearAlgebra.Prod
import Ma... |
/-- Decomposition of an affine map in the special case when the point space and vector space
| Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean | 564 | 565 |
/-
Copyright (c) 2020 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Data.Stream.Init
import Mathlib.Tactic.ApplyFun
import Mathlib.Control.Fix
import Mathlib.Order.OmegaCompletePartialOrder
/-!
# Lawful fixed point operators... | exists 0
intro b' h'
simp only [mem_iff f] at h'
| Mathlib/Control/LawfulFix.lean | 94 | 96 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Independence.Kernel
import Mathlib.MeasureTheory.Constructions.Pi
/-!
# Independence of sets of sets and measure spaces (σ-algebras)
* A fami... | have h₀ {s : Set β} {t : Set β'} (hs : MeasurableSet s) (ht : MeasurableSet t) :
μ (f ⁻¹' s) * μ (g ⁻¹' t) = μ.map f s * μ.map g t ∧
μ (f ⁻¹' s ∩ g ⁻¹' t) = μ.map (fun ω ↦ (f ω, g ω)) (s ×ˢ t) :=
⟨by rw [Measure.map_apply_of_aemeasurable hf hs, Measure.map_apply_of_aemeasurable hg ht],
(Measure.... | Mathlib/Probability/Independence/Basic.lean | 607 | 613 |
/-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Ordinal.Family
import Mathlib.Tactic.Abel
/-!
# Natural operations on ordinals
The goal of this file is to define n... | · exact le_max_of_le_right ((h hi).trans <| Ordinal.le_iSup (fun x : Iio b ↦ _) ⟨x, hx⟩)
all_goals
apply csSup_le_csSup' (bddAbove_of_small _)
rintro _ ⟨⟨c, hc⟩, rfl⟩
| Mathlib/SetTheory/Ordinal/NaturalOps.lean | 259 | 262 |
/-
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, Yaël Dillies
-/
import Mathlib.Algebra.Order.Group.DenselyOrdered
import Mathlib.Data.Real.Archimedean
import Mathlib.Order.LiminfLimsu... | /-- A continuous antitone function between (conditionally) complete linear ordered spaces sends a
`Filter.limsup` to the `Filter.liminf` of the images (if the filter is bounded from above and
frequently bounded from below). -/
theorem Antitone.map_limsup_of_continuousAt {f : R → S} (f_decr : Antitone f) (a : ι → R)
... | Mathlib/Topology/Algebra/Order/LiminfLimsup.lean | 339 | 385 |
/-
Copyright (c) 2024 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
/-!
# Angle between complex numbers
This file rel... | simpa using angle_exp_exp x 0
| Mathlib/Analysis/Complex/Angle.lean | 69 | 70 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Order.Filter.Interval
import Mathlib.Order.Interval.Set.Pi
import Mathlib.Tactic.TFAE
import Mathlib.Tactic.NormNum
im... | ⟨c, hac, hc_nhds, hcs⟩
refine ⟨b, c, ⟨hba, hac⟩, ?_⟩
rw [← Icc_union_Icc_eq_Icc hba hac, ← nhdsLE_sup_nhdsGE]
exact ⟨union_mem_sup hb_nhds hc_nhds, union_subset hbs hcs⟩
theorem IsOpen.exists_Ioo_subset [OrderTopology α] [Nontrivial α] {s : Set α} (hs : IsOpen s)
(h : s.Nonempty) : ∃ a b, a < b ∧ Ioo a b... | Mathlib/Topology/Order/Basic.lean | 420 | 430 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot, Yury Kudryashov, Rémy Degenne
-/
import Mathlib.Data.Set.Subsingleton
import Mathlib.Order.Interval.Set.Defs
/-!
# Intervals
In any pr... | section Sup
| Mathlib/Order/Interval/Set/Basic.lean | 917 | 918 |
/-
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.Data.Int.Range
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.MulChar.Basic
/-!
# Quadratic characters on ℤ/nℤ
This file defines some quadr... | exact help _ (Nat.mod_lt n (by omega)) <| (Nat.mod_mod_of_dvd n (by omega : 2 ∣ 4)).trans hn
/-- If `n % 4 = 1`, then `χ₄ n = 1`. -/
theorem χ₄_nat_one_mod_four {n : ℕ} (hn : n % 4 = 1) : χ₄ n = 1 := by
rw [χ₄_nat_mod_four, hn]
rfl
/-- If `n % 4 = 3`, then `χ₄ n = -1`. -/
theorem χ₄_nat_three_mod_four {n : ℕ} (... | Mathlib/NumberTheory/LegendreSymbol/ZModChar.lean | 82 | 90 |
/-
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.Finset.Card
import Mathlib.Data.Fintype.Basic
/-!
# Cardinalities of finite types
This file defines the cardinality `Fintype.card α` as the numb... | Mathlib/Data/Fintype/Card.lean | 575 | 587 | |
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky, Chris Hughes
-/
import Mathlib.Data.List.Nodup
/-!
# List duplicates
## Main definitions
* `List.Duplicate x l : Prop` is an inductive property that holds when `x`... |
theorem Duplicate.elim_singleton {y : α} (h : x ∈+ [y]) : False :=
not_duplicate_singleton x y h
| Mathlib/Data/List/Duplicate.lean | 70 | 73 |
/-
Copyright (c) 2023 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Data.List.Sym
/-! # Unordered tuples of elements of a multiset
Defines `Multiset.sym` and the specialized `Multiset.sym2` for computing multisets of all
un... | m.inductionOn (fun _ h => List.Nodup.sym2 h) h
open scoped List in
@[simp, mono]
| Mathlib/Data/Multiset/Sym.lean | 70 | 73 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.AdaptationNote
import Mathlib.Tactic.Simps.Basic
/-!
# Subtypes
This file provides basic API for subtypes,... | Setoid.trans h₁ h₂
| Mathlib/Data/Subtype.lean | 197 | 198 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.CharP.Frobenius
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.RingTheory.Polynomial.Basic
/... | noncomputable def contract (p : ℕ) (f : R[X]) : R[X] :=
∑ n ∈ range (f.natDegree + 1), monomial n (f.coeff (n * p))
| Mathlib/Algebra/Polynomial/Expand.lean | 179 | 180 |
/-
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.Hom.Ring
import Mathlib.Data.ENat.Basic
import Mathlib.SetTheory.Cardinal.Basic
/-!
# Conversion between `Cardinal` and `ℕ∞`
In this ... | @[simp]
theorem toENat_lift {a : Cardinal.{v}} : toENat (lift.{u} a) = toENat a := by
| Mathlib/SetTheory/Cardinal/ENat.lean | 263 | 264 |
/-
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... | @[simp]
theorem interior_Ioc [NoMaxOrder α] {a b : α} : interior (Ioc a b) = Ioo a b := by
rw [← Ioi_inter_Iic, interior_inter, interior_Ioi, interior_Iic, Ioi_inter_Iio]
| Mathlib/Topology/Order/DenselyOrdered.lean | 106 | 108 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Data.List.Lattice
import Mathlib.Data.Bool.Basic
import Mathlib.Order.Lattice
/-!
# Intervals in ℕ
This file defines intervals of naturals. `List.Ico m n... | exact h₂
· left
simp only [Ico.mem, not_and, not_lt] at *
exact h₂ h₁
end Ico
| Mathlib/Data/List/Intervals.lean | 198 | 203 |
/-
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
/-... |
lemma sup_iInf₂_eq {f : ∀ i, κ i → α} (a : α) : (a ⊔ ⨅ i, ⨅ j, f i j) = ⨅ i, ⨅ j, a ⊔ f i j := by
| Mathlib/Order/CompleteBooleanAlgebra.lean | 189 | 190 |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Module.ULift
import Mathlib.RingTheory.TensorProduct.Basic
import Mathlib.Tactic.Ring
/-!
# The characteristic predicate of tensor product
## Main ... | variable [Module A N] [IsScalarTower S A N] [IsScalarTower R A N]
/-- If `N` is the base change of `M` to `A`, then `N ⊗[R] P` is the base change
of `M ⊗[R] P` to `A`. This is simply the isomorphism
`A ⊗[S] (M ⊗[R] P) ≃ₗ[A] (A ⊗[S] M) ⊗[R] P`. -/
lemma isBaseChange_tensorProduct_map {f : M →ₗ[S] N} (hf : IsBaseChange ... | Mathlib/RingTheory/IsTensorProduct.lean | 249 | 280 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Amelia Livingston, Yury Kudryashov,
Neil Strickland, Aaron Anderson
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Tactic.Commo... | Mathlib/Algebra/Divisibility/Basic.lean | 225 | 229 | |
/-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Contraction
/-! # Results about inverses in Clifford algebras
This contains some basic results about the inversion of vectors... | letI := invertibleιOfInvertible Q m
exact isUnit_of_invertible (ι Q m)
/-- $aba^{-1}$ is a vector. -/
| Mathlib/LinearAlgebra/CliffordAlgebra/Inversion.lean | 37 | 40 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Data.ENNReal.Basic
/-!
# Maps between real and extended non-negative real numbers
This file focuses on the functions `ENNReal.toReal... | theorem toReal_mul_top (a : ℝ≥0∞) : ENNReal.toReal (a * ∞) = 0 := by
rw [toReal_mul, toReal_top, mul_zero]
| Mathlib/Data/ENNReal/Real.lean | 329 | 330 |
/-
Copyright (c) 2021 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
import Mathlib.Order.Lattice
/-!
# Ordered Subtraction
This file proves l... | rw [tsub_le_iff_left, add_assoc]
exact add_le_add_left le_add_tsub _
| Mathlib/Algebra/Order/Sub/Defs.lean | 160 | 162 |
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.RingTheory.Ideal.Maps
/-!
# Ideals in product rings
For commutative rings `R` and `S` and ideals `I ≤ R`, `J ≤ S`, we define `Ideal.prod I J` as the
pr... | rw [← map_prodComm_prod]
-- Note: couldn't synthesize the right instances without the `R` and `S` hints
exact map_isPrime_of_equiv (RingEquiv.prodComm (R := R) (S := S))
theorem isPrime_ideal_prod_top {I : Ideal R} [h : I.IsPrime] : (prod I (⊤ : Ideal S)).IsPrime := by
constructor
· rcases h with ⟨h, -⟩
... | Mathlib/RingTheory/Ideal/Prod.lean | 108 | 118 |
/-
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.Basic
import Mathlib.CategoryTheory.Preadditive.Basic
/-!
# Factoring through subobjects
The predicate `h : P.Fact... | ⟨u ≫ ((MonoOver.forget _).map e.inv).left, by simp [hu]⟩⟩
/-- `P.factorThru f h` provides a factorisation of `f : X ⟶ Y` through some `P : MonoOver Y`,
given the evidence `h : P.Factors f` that such a factorisation exists. -/
| Mathlib/CategoryTheory/Subobject/FactorThru.lean | 41 | 44 |
/-
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, Sébastien Gouëzel,
Rémy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
/-!
# Power function... | wlog hxy : x ≤ y
· convert this y x z hz (le_of_not_le hxy) using 2 <;> simp only [mul_comm, and_comm, or_comm]
rcases eq_or_ne x 0 with (rfl | hx0)
· induction y <;> rcases hz with hz | hz <;> simp [*, hz.not_lt]
rcases eq_or_ne y 0 with (rfl | hy0)
| Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | 642 | 646 |
/-
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, Heather Macbeth
-/
import Mathlib.Analysis.Convex.Slope
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Tactic.LinearCombination
... | _ < log (a • -x + b • -y) := strictConcaveOn_log_Ioi.2 hx' hy' hxy' ha hb hab
_ = log (-(a • x + b • y)) := by congr 1; simp only [Algebra.id.smul_eq_mul]; ring
_ = _ := by rw [log_neg_eq_log]
namespace Real
lemma exp_mul_le_cosh_add_mul_sinh {t : ℝ} (ht : |t| ≤ 1) (x : ℝ) :
exp (t * x) ≤ cosh x + t *... | Mathlib/Analysis/Convex/SpecificFunctions/Basic.lean | 212 | 222 |
/-
Copyright (c) 2023 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.DirectSum.LinearMap
import Mathlib.Algebra.Lie.InvariantForm
import Mathlib.Algebra.Lie.Weights.Cartan
import Mathlib.Algebra.Lie.Weights.Linear
impo... | LieAlgebra.InvariantForm.orthogonal (killingForm R L) (LieModule.traceForm_lieInvariant R L L) I
@[simp] lemma toSubmodule_killingCompl :
LieSubmodule.toSubmodule I.killingCompl = (killingForm R L).orthogonal I.toSubmodule :=
| Mathlib/Algebra/Lie/TraceForm.lean | 361 | 364 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Yury Kudryashov, Neil Strickland
-/
import Mathlib.Algebra.Group.Defs
import Mathlib.Algebra.GroupWithZero.Defs
import Mathlib.Data.I... | Mathlib/Algebra/Ring/Defs.lean | 327 | 327 | |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Frédéric Dupuis, Heather Macbeth
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Analysis.InnerProductSpace.Orthogonal
import Mathlib.Analysis.InnerProductSpace.Sy... | @[simp]
theorem ker_orthogonalProjection : LinearMap.ker K.orthogonalProjection = Kᗮ := by
ext; exact orthogonalProjection_eq_zero_iff
| Mathlib/Analysis/InnerProductSpace/Projection.lean | 550 | 552 |
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Eric Wieser
-/
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Dual.Lemmas
import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
import Mathlib.Li... | variable [Fintype n] [Fintype o]
section CommRing
variable [CommRing R]
/-- The rank of a matrix is the rank of its image. -/
noncomputable def rank (A : Matrix m n R) : ℕ :=
finrank R <| LinearMap.range A.mulVecLin
| Mathlib/Data/Matrix/Rank.lean | 102 | 111 |
/-
Copyright (c) 2024 Yaël Dillies, Kin Yau James Wong. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Kin Yau James Wong, Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.AEEqOfLIntegral
import Mathlib.Probability.Kernel.Composition.MeasureCompProd
... | Mathlib/Probability/Kernel/Disintegration/Basic.lean | 303 | 324 | |
/-
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.Option.NAry
import Mathlib.Data.Seq.Computation
import Mathlib.Tactic.ApplyFun
import Mathlib.Data.List.Basic
/-!
# Possibly infinite lists
This... |
@[simp]
| Mathlib/Data/Seq/Seq.lean | 452 | 453 |
/-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin, Lu-Ming Zhang
-/
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Algebra.Pi
import Mathlib.Algebra.BigOp... | Mathlib/Data/Matrix/Basic.lean | 2,130 | 2,132 | |
/-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib... |
theorem logb_neg_of_base_lt_one (h1 : 1 < x) : logb b x < 0 :=
| Mathlib/Analysis/SpecialFunctions/Log/Base.lean | 308 | 309 |
/-
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.Adjunction.FullyFaithful
import Mathlib.CategoryTheory.Adjunction.Limits
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq
import Ma... | subst this
refine PullbackCone.mono_of_isLimitMkIdId _ (IsPullback.isLimit ?_)
nth_rw 1 [← Category.id_comp (c.ι.app i)]
convert IsPullback.paste_vert _ (isPullback_of_cofan_isVanKampen hc i.as i.as)
swap
· exact (eqToHom (if_pos rfl).symm)
· simp
· exact IsPullback.of_vert_isIso ⟨by simp⟩
end FiniteCo... | Mathlib/CategoryTheory/Limits/VanKampen.lean | 757 | 768 |
/-
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.Algebra.Ring.Divisibility.Basic
import Mathlib.Data.Ordering.Lemmas
import Mathlib.Data.PNat.Basic
import Mathlib.SetTheory.Ordinal.Principal
import Ma... | theorem repr_mul (a b) : repr (a * b) = repr a * repr b :=
ONote.repr_mul a.1 b.1
| Mathlib/SetTheory/Ordinal/Notation.lean | 1,241 | 1,243 |
/-
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... | (n : ℕ) : (f ^ n) v = μ ^ n • v := by
induction n <;> simp [*, pow_succ f, hv.apply_eq_smul, smul_smul, pow_succ' μ]
| Mathlib/LinearAlgebra/Eigenspace/Basic.lean | 185 | 187 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
import Mathlib.Analy... | refine (contDiffAt_sqrt h.ne').comp y ?_
exact contDiffAt_const.sub (contDiff_norm_sq ℝ).contDiffAt
theorem Homeomorph.contDiff_unitBall : ContDiff ℝ n fun x : E => (unitBall x : E) :=
PartialHomeomorph.contDiff_univUnitBall
| Mathlib/Analysis/InnerProductSpace/Calculus.lean | 340 | 344 |
/-
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Chris Hughes, Kevin Buzzard
-/
import Mathlib.Algebra.Group.Hom.Defs
import Mathlib.Algebra.Group.Units.Basic
/-!
# Monoid homomorphisms and units
This file allows to... | @[to_additive (attr := simp)]
theorem mul_liftRight_inv (f : M →* N) (h : ∀ x, IsUnit (f x)) (x) :
f x * ↑(IsUnit.liftRight f h x)⁻¹ = 1 := Units.mul_liftRight_inv (by intro; rfl) x
| Mathlib/Algebra/Group/Units/Hom.lean | 204 | 206 |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash, Eric Wieser
-/
import Mathlib.Topology.Algebra.InfiniteSum.Basic
import Mathlib.Topology.Algebra.Ring.Basic
import Mathlib.Topology.Algebra.Star
import Mathlib.LinearAlgebra.... | theorem Continuous.matrix_blockDiagonal' [Zero R] [DecidableEq l]
{A : X → ∀ i, Matrix (m' i) (n' i) R} (hA : Continuous A) :
Continuous fun x => blockDiagonal' (A x) :=
continuous_matrix fun ⟨i₁, i₂⟩ ⟨j₁, j₂⟩ => by
dsimp only [blockDiagonal'_apply']
split_ifs with h
| Mathlib/Topology/Instances/Matrix.lean | 243 | 248 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Algebra.Group.TypeTags.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Finset.Piecewise
import Mathlib.Or... | Mathlib/Topology/Constructions.lean | 1,351 | 1,354 | |
/-
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.Sum
import Mathlib.Data.Sum.Order
import Mathlib.Order.Interval.Finset.Defs
/-!
# Finite intervals in a disjoint union
This file provides the... | Mathlib/Data/Sum/Interval.lean | 426 | 428 | |
/-
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.Combinatorics.SimpleGraph.Clique
import Mathlib.Combinatorics.SimpleGraph.Regularity.Uniform
import Mathlib.Data.Real.Basic
imp... | private lemma card_badVertices_le (dst : 2 * ε ≤ G.edgeDensity s t) (hst : G.IsUniform ε s t) :
#(badVertices G ε s t) ≤ #s * ε := by
have hε : ε ≤ 1 := (le_rfl.trans <| le_mul_of_one_le_left hst.pos.le (by norm_num)).trans
(dst.trans <| by exact_mod_cast edgeDensity_le_one _ _ _)
by_contra! h
have : |(G.... | Mathlib/Combinatorics/SimpleGraph/Triangle/Counting.lean | 51 | 59 |
/-
Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Data.Fintype.Lattice
import Mathlib.Data.Fintype.Sum
import Mathlib.Topology.Homeomorph.Lemmas
import Mathlib.Topology.MetricSpace.Antilipschitz
... | isComplete_image_iff e.isometry.isUniformInducing]
protected theorem completeSpace [CompleteSpace β] (e : α ≃ᵢ β) : CompleteSpace α :=
| Mathlib/Topology/MetricSpace/Isometry.lean | 489 | 491 |
/-
Copyright (c) 2014 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Leonardo de Moura, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Ord... | Mathlib/Algebra/Order/Field/Basic.lean | 809 | 809 | |
/-
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.Logic.Function.Defs
import Mathlib.Order.Defs.Unbundled
/-!
# Lexicographic order on a sigma type
This defines the lexicographical order of two arbitrary... | · exact Lex.right _ _ (hs _ _ _ hab)
theorem Lex.mono_left (hr : ∀ a b, r₁ a b → r₂ a b) {a b : Σ i, α i} (h : Lex r₁ s a b) :
Lex r₂ s a b :=
h.mono hr fun _ _ _ => id
| Mathlib/Data/Sigma/Lex.lean | 63 | 67 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
import Mathlib.Probability.Notation
import Mathlib.Probability.Process.Stopping
/-!
# Martingales
A family of functions `f : ι → Ω → E` is a martingale wit... |
@[deprecated (since := "2025-01-21")]
alias submartingale_iff_condexp_sub_nonneg := submartingale_iff_condExp_sub_nonneg
end Submartingale
namespace Supermartingale
theorem sub_submartingale [Preorder E] [AddLeftMono E]
(hf : Supermartingale f ℱ μ) (hg : Submartingale g ℱ μ) : Supermartingale (f - g) ℱ μ := by
... | Mathlib/Probability/Martingale/Basic.lean | 297 | 312 |
/-
Copyright (c) 2022 Daniel Roca González. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Daniel Roca González
-/
import Mathlib.Analysis.InnerProductSpace.Dual
/-!
# The Lax-Milgram Theorem
We consider a Hilbert space `V` over `ℝ`
equipped with a bounded bilinear f... | refine ⟨C, C_ge_0, ?_⟩
intro v
by_cases h : 0 < ‖v‖
· refine (mul_le_mul_right h).mp ?_
calc
C * ‖v‖ * ‖v‖ ≤ B v v := coercivity v
_ = ⟪B♯ v, v⟫_ℝ := (continuousLinearMapOfBilin_apply B v v).symm
_ ≤ ‖B♯ v‖ * ‖v‖ := real_inner_le_norm (B♯ v) v
· have : v = 0 := by simpa using h
simp ... | Mathlib/Analysis/InnerProductSpace/LaxMilgram.lean | 51 | 62 |
/-
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.Multiset.Basic
/-!
# Bind operation for multisets
This file defines a few basic operations on `Multiset`, notably the mona... | Mathlib/Data/Multiset/Bind.lean | 163 | 163 | |
/-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Topology.UniformSpace.Equicontinuity
import Mathlib.Topology.MetricSpace.Pseudo.Lemmas
/-!
# Equicontinuity in metric spaces
This files contains ... | rw [equicontinuousAt_iff_pair]
constructor <;> intro H
· intro ε hε
exact H _ (dist_mem_uniformity hε)
· intro U hU
rcases mem_uniformity_dist.mp hU with ⟨ε, hε, hεU⟩
refine Exists.imp (fun V => And.imp_right fun h => ?_) (H _ hε)
exact fun x hx x' hx' i => hεU (h _ hx _ hx' i)
/-- Characteriza... | Mathlib/Topology/MetricSpace/Equicontinuity.lean | 58 | 69 |
/-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Devon Tuma
-/
import Mathlib.Probability.ProbabilityMassFunction.Basic
/-!
# Monad Operations for Probability Mass Functions
This file constructs two operations on `PMF` ... |
end BindOnSupport
end PMF
| Mathlib/Probability/ProbabilityMassFunction/Monad.lean | 310 | 323 |
/-
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.Conj
import Mathlib.Algebra.Group.Subgroup.Lattice
import Mathlib.Algebra.Group.Submonoid.BigOperators
import Mathlib.Data.Finset.Fin
import ... | have lt : 1 < a₁ := le.trans_lt ha₁
have h01 : Equiv.swap (0 : Fin (n + 2)) 1 1 = 0 := by simp only [swap_apply_right]
rcases le.eq_or_lt with (rfl | lt')
· rw [swap_apply_of_ne_of_ne H.ne' lt.ne', h01, if_neg H.not_le]
· rw [swap_apply_of_ne_of_ne (ne_of_gt H) (ne_of_gt lt),
swa... | Mathlib/GroupTheory/Perm/Sign.lean | 267 | 289 |
/-
Copyright (c) 2020 James Arthur. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: James Arthur, Chris Hughes, Shing Tak Lam
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv
import Mathlib.Analysis.SpecialFunctions.Log.Basic
/-!
# Inverse of the sinh fu... |
/-- `sinh` is surjective, `∀ b, ∃ a, sinh a = b`. In this case, we use `a = arsinh b`. -/
| Mathlib/Analysis/SpecialFunctions/Arsinh.lean | 78 | 79 |
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.Perm.Option
import Mathlib.Logic.Equiv.Fin.Rotate
import Mathlib.Logic.Equiv.Fintype
/-!
# Permutatio... | _ ≤ n := Nat.lt_succ_iff.mp i.2)
theorem cycleRange_of_lt {n : ℕ} [NeZero n] {i j : Fin n} (h : j < i) : cycleRange i j = j + 1 := by
rw [cycleRange_of_le h.le, if_neg h.ne]
theorem coe_cycleRange_of_lt {n : ℕ} {i j : Fin n} (h : j < i) :
(cycleRange i j : ℕ) = j + 1 := by rw [coe_cycleRange_of_le h.le,... | Mathlib/GroupTheory/Perm/Fin.lean | 175 | 190 |
/-
Copyright (c) 2022 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Yaël Dillies
-/
import Mathlib.Analysis.Convex.Cone.Extension
import Mathlib.Analysis.Convex.Gauge
import Mathlib.Topology.Algebra.Module.FiniteDimension
import Mathlib.Top... | (ht₁ : Convex ℝ t) (ht₂ : IsClosed t) (disj : Disjoint s t) :
∃ (f : E →L[ℝ] ℝ) (u v : ℝ), (∀ a ∈ s, f a < u) ∧ u < v ∧ ∀ b ∈ t, v < f b := by
obtain rfl | hs := s.eq_empty_or_nonempty
· exact ⟨0, -2, -1, by simp, by norm_num, fun b _hb => by norm_num⟩
obtain rfl | _ht := t.eq_empty_or_nonempty
· exact ... | Mathlib/Analysis/NormedSpace/HahnBanach/Separation.lean | 153 | 168 |
/-
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 [← Real.cos_neg]
refine cos_neg_of_pi_div_two_lt_of_lt (by linarith) ?_
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 613 | 614 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.BigOperators.Option
import Mathlib.Analysis.BoxIntegral.Box.Basic
import Mathlib.Data.Set.Pairwise.Lattice
/-!
# Partitions of rectangular b... | · exact π.le_of_mem (π.biUnionIndex_mem hJ)
· rw [biUnionIndex, dif_neg hJ]
| Mathlib/Analysis/BoxIntegral/Partition/Basic.lean | 326 | 328 |
/-
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.Probability.Independence.Kernel
import Mathlib.MeasureTheory.Constructions.Pi
/-!
# Independence of sets of sets and measure spaces (σ-algebras)
* A fami... |
lemma iIndepFun_iff_iIndep {β : ι → Type*}
(m : ∀ x : ι, MeasurableSpace (β x)) (f : ∀ x : ι, Ω → β x) (μ : Measure Ω) :
iIndepFun f μ ↔ iIndep (fun x ↦ (m x).comap (f x)) μ := by
| Mathlib/Probability/Independence/Basic.lean | 212 | 215 |
/-
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.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.Basic
import Mathlib.Geometry.RingedSpace.Stalks
import Mathlib.RingTheory.Nilp... | obtain ⟨n, hn⟩ := hf
cases n.eq_zero_or_pos with
| inr h =>
rw [← X.toRingedSpace.basicOpen_pow f n h, hn]
simp [basicOpen_zero]
| inl h =>
rw [h, pow_zero] at hn
simp [eq_zero_of_zero_eq_one hn.symm f, basicOpen_zero]
instance component_nontrivial (X : LocallyRingedSpace.{u}) (U : Opens X.car... | Mathlib/Geometry/RingedSpace/LocallyRingedSpace.lean | 302 | 315 |
/-
Copyright (c) 2023 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.DedekindDomain.Ideal
import Mathlib.RingTheory.Discriminant
import Mathlib.RingTheory.DedekindDomain.IntegralClosure
import Mathlib.NumberTheory.K... |
end FractionalIdeal
variable (B)
variable [IsIntegrallyClosed A] [IsDedekindDomain B]
/-- The different ideal of an extension of integral domains `B/A` is the inverse of the dual of `A`
as an ideal of `B`. See `coeIdeal_differentIdeal` and `coeSubmodule_differentIdeal`. -/
def differentIdeal [NoZeroSMulDivisors A B]... | Mathlib/RingTheory/DedekindDomain/Different.lean | 388 | 412 |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.LinearAlgebra.Finsupp.Span
/-!
# Lie submodules of a Lie algebra
In this file we define Lie submodules, we construct ... | Mathlib/Algebra/Lie/Submodule.lean | 1,169 | 1,171 | |
/-
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.Group.Action.Pi
import Mathlib.Algebra.Group.End
import Mathlib.Algebra.Module.NatInt
import Mathlib.Algebra.Order.Archimedean.Basic
/-!
# M... | @[scoped simp]
theorem map_zsmul_add [AddCommGroup G] [AddGroup H] [AddConstMapClass F G H a b]
(f : F) (n : ℤ) (x : G) : f (n • a + x) = f x + n • b := by
| Mathlib/Algebra/AddConstMap/Basic.lean | 218 | 220 |
/-
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.ModularForms.JacobiTheta.TwoVariable
/-!
# Asymptotic bounds for Jacobi theta functions
The goal of this file is to establish some tech... | lemma isBigO_atTop_F_nat_zero_sub {a : ℝ} (ha : 0 ≤ a) : ∃ p, 0 < p ∧
(fun t ↦ F_nat 0 a t - (if a = 0 then 1 else 0)) =O[atTop] fun t ↦ exp (-p * t) := by
split_ifs with h
· rw [h]
have : (fun t ↦ F_nat 0 0 t - 1) =O[atTop] fun t ↦ rexp (-π * t) / (1 - rexp (-π * t)) := by
apply Eventually.isBigO
... | Mathlib/NumberTheory/ModularForms/JacobiTheta/Bounds.lean | 130 | 146 |
/-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth, Frédéric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Calculus
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.Adjoint... | simpa [← norm_eq_zero, Ne]
exact hasEigenvalue_of_hasEigenvector (T'.prop.hasEigenvector_of_isMaxOn hx₀_ne this)
/-- The infimum of the Rayleigh quotient of a symmetric operator `T` on a nontrivial
finite-dimensional vector space is an eigenvalue for that operator. -/
theorem hasEigenvalue_iInf_of_finiteDimensio... | Mathlib/Analysis/InnerProductSpace/Rayleigh.lean | 242 | 257 |
/-
Copyright (c) 2020 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Alexey Soloyev, Junyan Xu, Kamila Szewczyk
-/
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.Algebra.LinearRecurrence
import Mathlib.Data.Fin.VecNotati... |
end Fibrec
/-- Binet's formula as a function equality. -/
| Mathlib/Data/Real/GoldenRatio.lean | 178 | 181 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.Measure.Typeclasses.Finite
import Mathlib.MeasureTheory.Measure.Typeclasses.NoAtoms
import Mathlib.MeasureTheory.Measure.... | Mathlib/MeasureTheory/Measure/Typeclasses.lean | 1,320 | 1,322 | |
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.Perm.Option
import Mathlib.Logic.Equiv.Fin.Rotate
import Mathlib.Logic.Equiv.Fintype
/-!
# Permutatio... |
@[deprecated (since := "2025-01-28")]
| Mathlib/GroupTheory/Perm/Fin.lean | 213 | 214 |
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