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) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov, Kim Morrison
-/
import Mathlib.Algebra.Algebra.Equiv
import Mathlib.Algebra.Algebra.NonUnitalHom
import Mathlib.Algebra.Module.BigOperators
import Math... | lemma lift_of' (F : Multiplicative G →* A) (x : G) :
lift k G A F (of' k G x) = F (Multiplicative.ofAdd x) :=
lift_of F x
| Mathlib/Algebra/MonoidAlgebra/Basic.lean | 489 | 492 |
/-
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... | local prefix:100 "π" => cs.wordProd
/-! ### Length -/
| Mathlib/GroupTheory/Coxeter/Length.lean | 61 | 63 |
/-
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.MeasureTheory.Measure.Decomposition.RadonNikodym
/-!
# Exponentially tilted measures
The exponential tilting of a measure `μ` on `α` by a function `f : α... | ∫⁻ x in s, g x ∂(μ.tilted f)
= ∫⁻ x in s, ENNReal.ofReal (exp (f x) / ∫ x, exp (f x) ∂μ) * g x ∂μ := by
by_cases hf : AEMeasurable f μ
· rw [Measure.tilted, setLIntegral_withDensity_eq_setLIntegral_mul_non_measurable₀']
· simp only [Pi.mul_apply]
· refine AEMeasurable.restrict ?_
exact ((mea... | Mathlib/MeasureTheory/Measure/Tilted.lean | 165 | 180 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.Nat.SuccPred
import Mathlib.Order.SuccPred.Initial... | lt_iff_lt_of_le_iff_le <| le_div b0
theorem div_le_of_le_mul {a b c : Ordinal} (h : a ≤ b * c) : a / b ≤ c :=
| Mathlib/SetTheory/Ordinal/Arithmetic.lean | 853 | 855 |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic
import Mathlib.Algebra.Ring.Pointwise.Set
import Mathlib.Topology.MetricSpace.Isometry
import Mathlib.Topol... | ⟨fun c => .piMap (fun _ => (c • ·)) fun i => isometry_smul (X i) c⟩
@[to_additive]
| Mathlib/Topology/MetricSpace/IsometricSMul.lean | 413 | 415 |
/-
Copyright (c) 2024 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.PurelyInseparable.Basic
import Mathlib.FieldTheory.PerfectClosure
/-!
# `IsPerfectClosure` predicate
This file contains `IsPerfectClosure` which asserts... | PerfectRing.liftAux_apply i j p _ _ _ h
theorem equiv_symm_apply (x : M) (n : ℕ) (y : K) (h : j y = x ^ p ^ n) :
| Mathlib/FieldTheory/IsPerfectClosure.lean | 457 | 459 |
/-
Copyright (c) 2023 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.LinearAlgebra.Matrix.Gershgorin
import Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody
import Mathlib.NumberTheory.NumberField.Units.Basic... | (basisModTorsion K).map (logEmbeddingEquiv K)
/-- A fundamental system of units of `K`. The units of `fundSystem` are arbitrary lifts of the
| Mathlib/NumberTheory/NumberField/Units/DirichletTheorem.lean | 452 | 454 |
/-
Copyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Anne Baanen
-/
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Group.Action.Pi
import Mathlib.Data.Fintype.BigOperators
import Mat... | simp [prod_univ_castSucc]
| Mathlib/Algebra/BigOperators/Fin.lean | 113 | 113 |
/-
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.HomologicalComplex
import Mathlib.Algebra.Homology.ShortComplex.SnakeLemma
import Mathlib.Algebra.Homology.ShortComplex.ShortExact
... | @[reassoc (attr := simp)]
lemma opcyclesToCycles_homologyπ :
K.opcyclesToCycles i j ≫ K.homologyπ j = 0 := by
simp only [← cancel_epi (K.pOpcycles i),
pOpcycles_opcyclesToCycles_assoc, toCycles_comp_homologyπ, comp_zero]
| Mathlib/Algebra/Homology/HomologySequence.lean | 69 | 73 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Antoine Chambert-Loir
-/
import Mathlib.Algebra.Group.Hom.CompTypeclasses
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Notation.Prod
import Mathlib.Algebra.Ring.Act... |
@[simp]
theorem id_comp (f : A →ₑ+[φ] B) : comp (DistribMulActionHom.id N) f = f :=
| Mathlib/GroupTheory/GroupAction/Hom.lean | 599 | 601 |
/-
Copyright (c) 2020 Johan Commelin, Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.RingTheory.WittVector.Basic
import Mathlib.RingTheory.WittVector.IsPoly
/-!
# `init` and `tail`
Given a Witt vecto... | IsPoly₂.diag (hf := IsPoly₂.comp)
ghost_calc x
intro n
simp only [RingHom.map_add]
suffices
(bind₁ (selectPoly P)) (wittPolynomial p ℤ n) +
(bind₁ (selectPoly fun i => ¬P i)) (wittPolynomial p ℤ n) =
wittPolynomial p ℤ n by
apply_fun aeval x.coeff at this
simpa only [map_add, aeval... | Mathlib/RingTheory/WittVector/InitTail.lean | 88 | 109 |
/-
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... | @[simp] theorem logb_zero_left_eq_zero : logb 0 = 0 := by ext; rw [logb_zero_left, Pi.zero_apply]
| Mathlib/Analysis/SpecialFunctions/Log/Base.lean | 53 | 53 |
/-
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.CategoryTheory.Shift.Basic
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
/-! Sequences of functors from a category equipped with a shift
Let `F : C... | (shiftIso F n a a' ha').hom.app ((shiftFunctor C m).obj X) ≫
(shiftIso F m a' a'' ha'').hom.app X := by
simp [F.shiftIso_add' n m mn hnm a a' a'' ha' ha'']
lemma shiftIso_add'_inv_app (n m mn : M) (hnm : m + n = mn) (a a' a'' : M)
(ha' : n + a = a') (ha'' : m + a' = a'') (X : C) :
(F.shiftI... | Mathlib/CategoryTheory/Shift/ShiftSequence.lean | 170 | 176 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Notation.Pi
import Mathlib.Data.Set.Lattice
import Mathlib.Order.Filter.Defs
/-!
# Theory of... | -/
section EventuallyEq
| Mathlib/Order/Filter/Basic.lean | 857 | 859 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.MkIffOfInductiveProp
/-!
# Additional lemmas about sum types
Most of the former contents ... | Mathlib/Data/Sum/Basic.lean | 210 | 210 | |
/-
Copyright (c) 2020 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa, Alex Meiburg
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Degree.Monomial
/-!
# Erase the... | · rw [eraseLead, natDegree_monomial, if_neg hr, erase_monomial]
@[simp]
| Mathlib/Algebra/Polynomial/EraseLead.lean | 132 | 134 |
/-
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.Nilpotent
import Mathlib.Algebra.Lie.Normalizer
/-!
# Cartan subalgebras
Cartan subalgebras are one of the most important concepts in Lie theor... |
@[simp]
theorem ucs_eq_self_of_isCartanSubalgebra (H : LieSubalgebra R L) [H.IsCartanSubalgebra] (k : ℕ) :
H.toLieSubmodule.ucs k = H.toLieSubmodule := by
| Mathlib/Algebra/Lie/CartanSubalgebra.lean | 58 | 61 |
/-
Copyright (c) 2021 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Order.Module.Algebra
import Mathlib.Algebra.Ring.Subring.Units
import Mathlib.LinearAlgebra.LinearIndepende... | Mathlib/LinearAlgebra/Ray.lean | 724 | 728 | |
/-
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.Limits.Shapes.BinaryBiproducts
import Mathlib.GroupTheory.EckmannHilton
import Mathlib.Tactic.CategoryTheory.Reassoc
/-!
# Constructing a ... | end
end CategoryTheory.SemiadditiveOfBinaryBiproducts
| Mathlib/CategoryTheory/Preadditive/OfBiproducts.lean | 129 | 132 |
/-
Copyright (c) 2022 Joseph Hua. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Bhavik Mehta, Johan Commelin, Reid Barton, Robert Y. Lewis, Joseph Hua
-/
import Mathlib.CategoryTheory.Limits.Shapes.IsTerminal
import Mathlib.CategoryTheory.Functor.EpiMono... | { a := V.1
str := (adj.homEquiv V.1 V.1).invFun V.2 }
map f :=
| Mathlib/CategoryTheory/Endofunctor/Algebra.lean | 446 | 448 |
/-
Copyright (c) 2021 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.PiL2
/-!
# Adjoint of operators on Hilbert spaces
Given ... | (U.orthogonalProjection : E →L[𝕜] U)† = U.subtypeL := by
rw [← U.adjoint_subtypeL, adjoint_adjoint]
/-- `E →L[𝕜] E` is a star algebra with the adjoint as the star operation. -/
instance : Star (E →L[𝕜] E) :=
| Mathlib/Analysis/InnerProductSpace/Adjoint.lean | 175 | 179 |
/-
Copyright (c) 2020 Kexing Ying and Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Kevin Buzzard, Yury Kudryashov
-/
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.Algebra.Gro... | /-- If `f a = 1` when `a ∉ s`, then the product of `f i` over `i ∈ insert a s` equals the product of
`f i` over `i ∈ s`. -/
@[to_additive
| Mathlib/Algebra/BigOperators/Finprod.lean | 754 | 756 |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.MeasureTheory.Integral.Lebesgue.Countable
/-!
# The Giry monad
Let X be a measurable space. The collection of all measures on X again
forms a measura... | intro i; exact (measurable_coe (hf i)).aemeasurable)
@[simp]
| Mathlib/MeasureTheory/Measure/GiryMonad.lean | 118 | 120 |
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Group.Center
import Mathlib.Data.Int.Cast.Lemmas
/-!
# Centers of rings
-/
assert_not_exists RelIso Finset Subsemigroup Field
vari... | comm _ := by rw [Nat.commute_cast]
left_assoc _ _ := by
induction n with
| zero => rw [Nat.cast_zero, zero_mul, zero_mul, zero_mul]
| succ n ihn => rw [Nat.cast_succ, add_mul, one_mul, ihn, add_mul, add_mul, one_mul]
mid_assoc _ _ := by
induction n with
| zero => rw [Nat.cast_zero, zero_mul, m... | Mathlib/Algebra/Ring/Center.lean | 24 | 37 |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle
import Mathlib.Geometry.Euclidean.Circumcenter
/-!
# Angles in circles and sphere.
This file proves results ... | have hy : y ≠ 0 := by
rintro rfl
rw [norm_zero, norm_eq_zero] at hxy
exact hxyne hxy
have hx : x ≠ 0 := norm_ne_zero_iff.1 (hxy.symm ▸ norm_ne_zero_iff.2 hy)
have hz : z ≠ 0 := norm_ne_zero_iff.1 (hxz ▸ norm_ne_zero_iff.2 hx)
calc
o.oangle y z = o.oangle x z - o.oangle x y := (o.oangle_sub_left ... | Mathlib/Geometry/Euclidean/Angle/Sphere.lean | 32 | 48 |
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Mathlib.Control.Basic
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Option.Basic
im... |
-- foldl and foldr coincide when f is commutative and associative
variable {f : α → α → α}
theorem foldl1_eq_foldr1 [hassoc : Std.Associative f] :
∀ a b l, foldl f a (l ++ [b]) = foldr f b (a :: l)
| _, _, nil => rfl
| a, b, c :: l => by
simp only [cons_append, foldl_cons, foldr_cons, foldl1_eq_foldr1 _ _... | Mathlib/Data/List/Basic.lean | 891 | 900 |
/-
Copyright (c) 2020 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo
-/
import Mathlib.Logic.Function.Iterate
import Mathlib.Topology.Algebra.Monoid
import Mathlib.Topology.Algebra.Group.Defs
/-!
# Flows and invariant sets
This file defines a flow on... |
/-- A set `s ⊆ α` is forward-invariant under `ϕ : τ → α → α` if
| Mathlib/Dynamics/Flow.lean | 49 | 50 |
/-
Copyright (c) 2020 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo
-/
import Mathlib.Dynamics.Flow
import Mathlib.Tactic.Monotonicity
/-!
# ω-limits
For a function `ϕ : τ → α → β` where `β` is a topological space, we
define the ω-limit under `ϕ` of... | have hw₃ : k \ n ⊆ (closure (image2 ϕ w s))ᶜ := by
apply Subset.trans hg₃
simp only [j, iUnion_subset_iff, compl_subset_compl]
intros u hu
unfold w
gcongr
refine iInter_subset_of_subset u (iInter_subset_of_subset hu ?_)
all_goals exact Subset.rfl
have hw₄ : kᶜ ⊆ (closure (image2 ϕ w s))ᶜ... | Mathlib/Dynamics/OmegaLimit.lean | 226 | 258 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Yongle Hu
-/
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Group.Subgroup.Actions
import Mathlib.RingTheory.Ideal.Pointwise
import Mathlib.RingTheory.Ideal.Quot... | variable {P} {E : Type*} [EquivLike E B C] [AlgEquivClass E A B C] (σ : E)
/-- An `A ⧸ p`-algebra isomorphism between `B ⧸ P` and `C ⧸ Q` induced by an `A`-algebra
isomorphism between `B` and `C`, where `Q = σ P`. -/
def algEquivOfEqMap (h : Q = P.map σ) : (B ⧸ P) ≃ₐ[A ⧸ p] (C ⧸ Q) where
__ := quotientEquiv P Q σ ... | Mathlib/RingTheory/Ideal/Over.lean | 235 | 240 |
/-
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 | 65 | 72 | |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Minchao Wu, Mario Carneiro
-/
import Mathlib.Algebra.NeZero
import Mathlib.Data.Finset.Attach
import Mathlib.Data.Finset.Disjoint
import Mathli... |
@[simp]
theorem map_ssubset_map {s t : Finset α} : s.map f ⊂ t.map f ↔ s ⊂ t := (mapEmbedding f).lt_iff_lt
| Mathlib/Data/Finset/Image.lean | 167 | 169 |
/-
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.ENNReal.Action
import Mathlib.MeasureTheory.MeasurableSpace.Constructions
import Mathlib.MeasureTheory.OuterMeasure.Caratheodory
... | inducedOuterMeasure m MeasurableSet.empty m0 s = m s hs :=
(inducedOuterMeasure_eq_extend m0 mU hs).trans <| extend_eq _ _
end MeasurableSpace
namespace OuterMeasure
variable {α : Type*} [MeasurableSpace α] (m : OuterMeasure α)
| Mathlib/MeasureTheory/OuterMeasure/Induced.lean | 290 | 297 |
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Algebra.Group.Commute.Hom
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Data.Fintype.B... | rcases eq_or_ne a b with rfl | hab
· rfl
exact comm h (mem_of_mem_erase hb) hab
| Mathlib/Data/Finset/NoncommProd.lean | 215 | 218 |
/-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.LocallyConvex.BalancedCoreHull
import Mathlib.Analysis.Convex.TotallyBounded
/-!
# Absolutely convex sets
A set `s` in an commutative monoid `E` i... | · rintro rfl
exact absConvexHull_empty
@[simp]
theorem absConvexHull_nonempty : (absConvexHull 𝕜 s).Nonempty ↔ s.Nonempty := by
rw [nonempty_iff_ne_empty, nonempty_iff_ne_empty, Ne, Ne]
| Mathlib/Analysis/LocallyConvex/AbsConvex.lean | 141 | 146 |
/-
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.Relation
import Mathlib.Logic.Unique
import Mathlib.Util.Notation3
/-!
# Quotient types
This module extends the core library's treatment of quo... | instance piSetoid {ι : Sort*} {α : ι → Sort*} [∀ i, Setoid (α i)] : Setoid (∀ i, α i) where
r a b := ∀ i, a i ≈ b i
iseqv := ⟨fun _ _ ↦ Setoid.refl _,
fun h _ ↦ Setoid.symm (h _),
| Mathlib/Data/Quot.lean | 392 | 395 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kim Morrison
-/
import Mathlib.Algebra.BigOperators.Finsupp.Basic
import Mathlib.Algebra.BigOperators.Group.Finset.Preimage
import Mathlib.Algebra.Module.Defs
import Ma... | Mathlib/Data/Finsupp/Basic.lean | 1,871 | 1,875 | |
/-
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.Calculus.ContDiff.RCLike
import Mathlib.MeasureTheory.Measure.Hausdorff
/-!
# Hausdorff dimension
The Hausdorff dimension of a set `X` in ... |
variable {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E]
| Mathlib/Topology/MetricSpace/HausdorffDimension.lean | 475 | 476 |
/-
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.CategoryTheory.Linear.Basic
import Mathlib.Algebra.Homology.ComplexShapeSigns
import Mathlib.Algebra.Homology.HomologicalBicomplex
import Mathlib.Algebra.Module.... | · rw [K.D₂_shape c₁₂ _ _ h₂, comp_zero]
· rw [K.D₂_shape c₁₂ _ _ h₁, zero_comp]
@[reassoc (attr := simp)]
lemma D₂_D₁ (i₁₂ i₁₂' i₁₂'' : I₁₂) :
K.D₂ c₁₂ i₁₂ i₁₂' ≫ K.D₁ c₁₂ i₁₂' i₁₂'' = - K.D₁ c₁₂ i₁₂ i₁₂' ≫ K.D₂ c₁₂ i₁₂' i₁₂'' := by
by_cases h₁ : c₁₂.Rel i₁₂ i₁₂'
· by_cases h₂ : c₁₂.Rel i₁₂' i₁₂''
· ... | Mathlib/Algebra/Homology/TotalComplex.lean | 210 | 239 |
/-
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, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Field.Defs
import Mathlib.Algebra.Ring.Commute
import Mathlib.Algebra.Ring.Invertible
impo... | nnqsmul_def := fun _ _ => rfl
qsmul := _
qsmul_def := fun _ _ => rfl
| Mathlib/Algebra/Field/Basic.lean | 193 | 195 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.BoxIntegral.Partition.Split
import Mathlib.Analysis.NormedSpace.OperatorNorm.Mul
/-!
# Box additive functions
We say that a function `f : ... | exacts [(WithTop.coe_le_coe.2 <| π₁.le_of_mem hJ).trans hI,
(WithTop.coe_le_coe.2 <| π₂.le_of_mem hJ).trans hI]
section ToSMul
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
/-- If `f` is a box-additive map, then so is the map sending `I` to the scalar multiplication
by `f I` as a continuous lin... | Mathlib/Analysis/BoxIntegral/Partition/Additive.lean | 159 | 175 |
/-
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.Ordmap.Invariants
/-!
# Verification of `Ordnode`
This file uses the invariants defined in `Mathlib.Data.Ordmap.Invariants` to construct `Ordset... | Mathlib/Data/Ordmap/Ordset.lean | 881 | 886 | |
/-
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.Analysis.Calculus.MeanValue
import Mathlib.MeasureTheory.Integral.DominatedConvergence
import Mathlib.MeasureTheory.Integral.Bochner.Set
import Mathlib... | (hF'_meas : AEStronglyMeasurable (F' x₀) μ)
(h_bound : ∀ᵐ a ∂μ, ∀ x ∈ ball x₀ ε, ‖F' x a‖ ≤ bound a)
(bound_integrable : Integrable (bound : α → ℝ) μ)
(h_diff : ∀ᵐ a ∂μ, ∀ x ∈ ball x₀ ε, HasFDerivAt (F · a) (F' x a) x) :
HasFDerivAt (fun x ↦ ∫ a, F x a ∂μ) (∫ a, F' x₀ a ∂μ) x₀ := by
letI : NormedS... | Mathlib/Analysis/Calculus/ParametricIntegral.lean | 207 | 226 |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.GroupWithZero.Units.Basic
import Mathlib.Algebra.Ring.Basic
import Mathlib.Algebra.Ring.Int.Defs
import Mathlib.Data.Rat.Defs
i... |
lemma natCast_eq_divInt (n : ℕ) : ↑n = n /. 1 := by rw [← Int.cast_natCast, intCast_eq_divInt]
| Mathlib/Algebra/Ring/Rat.lean | 85 | 86 |
/-
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 ... | induction x using QuotientAddGroup.induction_on
induction y using QuotientAddGroup.induction_on
simp_rw [← coe_add, toCircle_apply_mk, mul_add, Circle.exp_add]
@[simp] lemma toCircle_zero : toCircle (0 : AddCircle T) = 1 := by
| Mathlib/Analysis/SpecialFunctions/Complex/Circle.lean | 142 | 146 |
/-
Copyright (c) 2020 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.Order.Group.Multiset
import Mathlib.Data.Setoid.Basic
import Mathlib.Data.Vector.Basic
import Mathlib.Logic.Nontrivial.Basic
import Mathlib.Tactic.Ap... | theorem mem_attach (s : Sym α n) (x : { x // x ∈ s }) : x ∈ s.attach :=
Multiset.mem_attach _ _
| Mathlib/Data/Sym/Basic.lean | 414 | 416 |
/-
Copyright (c) 2024 Jeremy Tan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Tan
-/
import Mathlib.Analysis.SpecialFunctions.Complex.LogBounds
/-!
# Complex arctangent
This file defines the complex arctangent `Complex.arctan` as
$$\arctan z = -\frac i2 \lo... | theorem Real.hasSum_arctan {x : ℝ} (hx : ‖x‖ < 1) :
HasSum (fun n : ℕ => (-1) ^ n * x ^ (2 * n + 1) / ↑(2 * n + 1)) (arctan x) :=
mod_cast Complex.hasSum_arctan (z := x) (by simpa)
| Mathlib/Analysis/SpecialFunctions/Complex/Arctan.lean | 137 | 139 |
/-
Copyright (c) 2024 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.Analysis.Complex.LocallyUniformLimit
import Mathlib.NumberTheory.LSeries.Convergence
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
import Mathlib.An... | @[simp]
lemma LSeries.absicssaOfAbsConv_logPowMul {f : ℕ → ℂ} {m : ℕ} :
abscissaOfAbsConv (logMul^[m] f) = abscissaOfAbsConv f := by
induction m with
| zero => simp
| succ n ih => simp [ih, Function.iterate_succ', Function.comp_def,
-Function.comp_apply, -Function.iterate_succ]
| Mathlib/NumberTheory/LSeries/Deriv.lean | 119 | 126 |
/-
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.BigOperators.Group.Finset.Indicator
import Mathlib.Algebra.Module.BigOperators
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic
import... | theorem centroid_eq_affineCombination_fintype [Fintype ι] (p : ι → P) :
s.centroid k p = univ.affineCombination k p (s.centroidWeightsIndicator k) :=
affineCombination_indicator_subset _ _ (subset_univ _)
/-- An indexed family of points that is injective on the given
`Finset` has the same centroid as the image o... | Mathlib/LinearAlgebra/AffineSpace/Combination.lean | 853 | 864 |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kenny Lau, Johan Commelin, Mario Carneiro, Kevin Buzzard,
Amelia Livingston, Yury Kudryashov
-/
import Mathlib.Algebra.Group.Action.Faithful
import Mathlib.Algebra.Grou... | Mathlib/Algebra/Group/Submonoid/Operations.lean | 1,132 | 1,135 | |
/-
Copyright (c) 2024 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.CommShift
/-!
# Functors from a category to a category with a shift
Given a category `C`, and a category `D` equipped with a shift by a mo... | lemma shiftIso_add'_hom_app (n m mn : A) (hnm : m + n = mn) (a a' a'' : A)
(ha' : n + a = a') (ha'' : m + a' = a'') (X : C) :
(F.shiftIso mn a a'' (by rw [← hnm, ← ha'', ← ha', add_assoc])).hom.app X =
(shiftFunctorAdd' D m n mn hnm).hom.app ((F.functor a'').obj X) ≫
((F.shiftIso m a' a'' ha'').ho... | Mathlib/CategoryTheory/Shift/SingleFunctors.lean | 78 | 83 |
/-
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.Convex.Between
import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathli... | end IsMetric
/-!
### Constructors of metric outer measures
| Mathlib/MeasureTheory/Measure/Hausdorff.lean | 229 | 232 |
/-
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... | (sum fun n => sum (μ n)) = sum (fun (p : ι × ι') ↦ μ p.1 p.2) := by
ext1 s hs
simp [sum_apply _ hs, ENNReal.tsum_prod']
theorem sum_comm {ι' : Type*} (μ : ι → ι' → Measure α) :
| Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 1,203 | 1,207 |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark
-/
import Mathlib.Algebra.Polynomial.Monic
/-!
# Lemmas for the interaction between polynomials and `∑` and `∏`.
Recall that `∑` and `∏` are notation for ... | natDegree l.sum ≤ (l.map natDegree).foldr max 0 :=
Quotient.inductionOn l (by simpa using natDegree_list_sum_le)
| Mathlib/Algebra/Polynomial/BigOperators.lean | 52 | 54 |
/-
Copyright (c) 2022 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.Algebra.IsPrimePow
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.Nat.Prime.Pow
import Mathlib.NumberTheory.Divisors
/-!
# Prime powers a... | ∃ p : ℕ, p.Prime ∧ ordCompl[p] n = 1 := by
rcases eq_or_ne n 0 with (rfl | hn0); · cases not_isPrimePow_zero h
rcases isPrimePow_iff_factorization_eq_single.mp h with ⟨p, k, hk0, h1⟩
rcases em' p.Prime with (pp | pp)
| Mathlib/Data/Nat/Factorization/PrimePow.lean | 57 | 60 |
/-
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.Convex.Between
import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathli... |
/-- If `f` is a `K`-Lipschitz map, then it increases the Hausdorff `d`-measures of sets at most
by the factor of `K ^ d`. -/
theorem hausdorffMeasure_image_le (h : LipschitzWith K f) {d : ℝ} (hd : 0 ≤ d) (s : Set X) :
μH[d] (f '' s) ≤ (K : ℝ≥0∞) ^ d * μH[d] s :=
h.lipschitzOnWith.hausdorffMeasure_image_le hd
en... | Mathlib/MeasureTheory/Measure/Hausdorff.lean | 728 | 764 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.MvPolynomial.Rename
/-!
# Degrees of polynomials
This file establ... |
theorem totalDegree_add (a b : MvPolynomial σ R) :
| Mathlib/Algebra/MvPolynomial/Degrees.lean | 387 | 388 |
/-
Copyright (c) 2024 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.Algebra.BigOperators.Field
import Mathlib.NumberTheory.LSeries.Basic
/-!
# Linearity of the L-series of `f` as a function of `f`
We show that the `LSer... |
lemma LSeriesHasSum.neg {f : ℕ → ℂ} {s a : ℂ} (hf : LSeriesHasSum f s a) :
| Mathlib/NumberTheory/LSeries/Linearity.lean | 55 | 56 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Group.Units.Equiv
import Mathlib.Algebra.Order.Group.End
import Mathlib.Logic.Function.Conjugate
import Mathlib.Order.Bounds.OrderIso
import ... | isomorphisms. Then the map `x ↦ ⨆ g : G, (f₁ g)⁻¹ (f₂ g x)` semiconjugates each `f₁ g'` to `f₂ g'`.
This is a version of Proposition 5.4 from [Étienne Ghys, Groupes d'homéomorphismes du cercle et
cohomologie bornée][ghys87:groupes]. -/
theorem sSup_div_semiconj [CompleteLattice α] [Group G] (f₁ f₂ : G →* α ≃o α) (g : ... | Mathlib/Order/SemiconjSup.lean | 101 | 107 |
/-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.Sheaf
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "covering pres... | lemma isSheafFor_of_factorsThru
{X : C} {S T : Presieve X}
(P : Cᵒᵖ ⥤ Type*)
(H : S.FactorsThru T) (hS : S.IsSheafFor P)
(h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ (R : Presieve Y),
R.IsSeparatedFor P ∧ R.FactorsThruAlong S f) :
T.IsSheafFor P := by
simp only [← Presieve.isSeparatedFor_and_exists_i... | Mathlib/CategoryTheory/Sites/Coverage.lean | 99 | 130 |
/-
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... | @[simp]
| Mathlib/SetTheory/Cardinal/Basic.lean | 570 | 570 |
/-
Copyright (c) 2022 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa, Yuyang Zhao
-/
import Mathlib.Algebra.Order.GroupWithZero.Unbundled.Basic
import Mathlib.Algebra.Order.GroupWithZero.Unbundled.Defs
import Mathlib.Tactic.Linter.Deprecate... | Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean | 713 | 715 | |
/-
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.Order.Hom.Basic
/-!
# Unbounded lattice homomorphisms
This file defines unbounded lattice homomorphisms. _Bounded_ lattice homomorphisms are defined in
`... | Mathlib/Order/Hom/Lattice.lean | 1,864 | 1,866 | |
/-
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... | let (a', m) := split a
(oadd e n a', m)
/-- `scale x o` is the ordinal notation for `ω ^ x * o`. -/
def scale (x : ONote) : ONote → ONote
| 0 => 0
| oadd e n a => oadd (x + e) n (scale x a)
/-- `mulNat o n` is the ordinal notation for `o * n`. -/
def mulNat : ONote → ℕ → ONote
| 0, _ => 0
| _, 0 =... | Mathlib/SetTheory/Ordinal/Notation.lean | 591 | 605 |
/-
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... | induction θ using Real.Angle.induction_on
exact Real.cos_sq_add_sin_sq _
theorem sin_add_pi_div_two (θ : Angle) : sin (θ + ↑(π / 2)) = cos θ := by
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 365 | 368 |
/-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa, Junyan Xu
-/
import Mathlib.Data.DFinsupp.Defs
/-!
# Locus of unequal values of finitely supported dependent functions
Let `N : α → Type*` be a type family, assume that `N ... | theorem neLocus_self_add_left : neLocus (f + g) f = g.support := by
| Mathlib/Data/DFinsupp/NeLocus.lean | 142 | 142 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Alex Kontorovich, Heather Macbeth
-/
import Mathlib.MeasureTheory.Group.Action
import Mathlib.MeasureTheory.Group.Pointwise
import Mathlib.MeasureTheory.Integral.Lebe... | (h : IsFundamentalDomain G s μ) (g : G') : IsFundamentalDomain G (g • s) μ :=
h.image_of_equiv (MulAction.toPerm g) (measurePreserving_smul _ _).quasiMeasurePreserving
(Equiv.refl _) <| smul_comm g
variable [MeasurableSpace G] [MeasurableSMul G α] [SMulInvariantMeasure G α μ]
@[to_additive]
| Mathlib/MeasureTheory/Group/FundamentalDomain.lean | 182 | 188 |
/-
Copyright (c) 2022 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex J. Best, Xavier Roblot
-/
import Mathlib.Algebra.Algebra.Hom.Rat
import Mathlib.Analysis.Complex.Polynomial.Basic
import Mathlib.NumberTheory.NumberField.Norm
import Mathlib.RingTh... | lemma comap_surjective [Algebra k K] [Algebra.IsAlgebraic k K] :
Function.Surjective (comap · (algebraMap k K)) := fun w ↦
| Mathlib/NumberTheory/NumberField/Embeddings.lean | 704 | 705 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Logic.Function.Basic
/-!
# Semiconjugate and commuting maps
We define the following predicates:
* `Function.Semiconj`: `f : α → β` semiconjugates ... | Mathlib/Logic/Function/Conjugate.lean | 190 | 191 | |
/-
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.Order.PropInstances
import Mathlib.Order.GaloisConnection.Defs
/-!
# Heyting algebras
This file defines Heyting, co-Heyting and bi-Heyting algebras.
A H... |
@[simp]
theorem sup_hnot_self (a : α) : a ⊔ ￢a = ⊤ :=
Codisjoint.eq_top codisjoint_hnot_right
| Mathlib/Order/Heyting/Basic.lean | 839 | 842 |
/-
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... | /-- If `‖r‖ < 1`, then `∑' n : ℕ, n * r ^ n = r / (1 - r) ^ 2`, `HasSum` version in a general ring
with summable geometric series. For a version in a field, using division instead of `Ring.inverse`,
see `hasSum_coe_mul_geometric_of_norm_lt_one`. -/
| Mathlib/Analysis/SpecificLimits/Normed.lean | 467 | 469 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Mathlib.Data.Stream.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Data.List.Defs
import Mathlib.Data.Nat.Basic
import Mathlib.Tactic.Common... | · have h₁ : some (get s₁ (succ n)) = some (get s₂ (succ n)) := by
rw [← getElem?_take_succ, ← getElem?_take_succ, h (succ (succ n))]
injection h₁
protected theorem cycle_g_cons (a : α) (a₁ : α) (l₁ : List α) (a₀ : α) (l₀ : List α) :
| Mathlib/Data/Stream/Init.lean | 608 | 612 |
/-
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.ContMDiff.Defs
/-!
## Basic properties of `C^n` functions between manifolds
In this file, we show that stan... |
/-- If the `ChartedSpace` structure on a manifold `M` is given by an open embedding `e : M → H`,
then the inverse of `e` is `C^n`. -/
lemma contMDiffOn_isOpenEmbedding_symm [Nonempty M] :
haveI := h.singletonChartedSpace; ContMDiffOn I I
n (IsOpenEmbedding.toPartialHomeomorph e h).symm (range e) := by
have... | Mathlib/Geometry/Manifold/ContMDiff/Basic.lean | 371 | 379 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad
-/
import Mathlib.Data.Finset.Basic
import Mathlib.Data.Finset.Image
/-!
# Cardinality of a finite set
This defines the cardinality of a `Fins... | rw [← card_union_add_card_inter, Nat.add_sub_cancel]
lemma card_inter (s t : Finset α) : #(s ∩ t) = #s + #t - #(s ∪ t) := by
rw [← card_inter_add_card_union, Nat.add_sub_cancel]
theorem card_union_le (s t : Finset α) : #(s ∪ t) ≤ #s + #t :=
card_union_add_card_inter s t ▸ Nat.le_add_right _ _
lemma card_union_... | Mathlib/Data/Finset/Card.lean | 489 | 507 |
/-
Copyright (c) 2021 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.GroupTheory.Index
/-!
# Complements
In this file we define the complement of a subgroup.
## Main definitions
- `Subgroup.IsComplement S T` where ... |
@[to_additive]
theorem isComplement'_bot_top : IsComplement' (⊥ : Subgroup G) ⊤ :=
| Mathlib/GroupTheory/Complement.lean | 171 | 173 |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Analysis.Normed.Module.Convex
/-!
# Sides of affine subspaces
This ... | theorem _root_.Wbtw.wSameSide₃₂ {s : AffineSubspace R P} {x y z : P} (h : Wbtw R x y z)
(hx : x ∈ s) : s.WSameSide z y :=
(h.wSameSide₂₃ hx).symm
| Mathlib/Analysis/Convex/Side.lean | 316 | 318 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Bhavik Mehta, Eric Wieser
-/
import Mathlib.Algebra.BigOperators.Group.Multiset.Basic
import Mathlib.Algebra.BigOperators.Ring.List
import Mathlib.Data.Multiset.Antidiagona... | variable [NoZeroDivisors α] [Nontrivial α] {s : Multiset α}
| Mathlib/Algebra/BigOperators/Ring/Multiset.lean | 31 | 32 |
/-
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.NNReal
/-!
# Limits and a... |
theorem IsBigOWith.rpow (h : IsBigOWith c l f g) (hc : 0 ≤ c) (hr : 0 ≤ r) (hg : 0 ≤ᶠ[l] g) :
IsBigOWith (c ^ r) l (fun x => f x ^ r) fun x => g x ^ r := by
apply IsBigOWith.of_bound
filter_upwards [hg, h.bound] with x hgx hx
calc
|f x ^ r| ≤ |f x| ^ r := abs_rpow_le_abs_rpow _ _
_ ≤ (c * |g x|) ^ r ... | Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean | 223 | 234 |
/-
Copyright (c) 2019 Gabriel Ebner. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Gabriel Ebner, Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.FDeriv.Basic
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
/-!
# One-dimensional derivatives
This ... | rw [fderivWithin_inter ht]
| Mathlib/Analysis/Calculus/Deriv/Basic.lean | 481 | 481 |
/-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll, Anatole Dedecker
-/
import Mathlib.Analysis.LocallyConvex.Bounded
import Mathlib.Analysis.Seminorm
import Mathlib.Data.Real.Sqrt
import Mathlib.Topology.Algebra.Equicontinuit... | theorem SeminormFamily.withSeminorms_iff_topologicalSpace_eq_iInf [IsTopologicalAddGroup E]
(p : SeminormFamily 𝕜 E ι) :
WithSeminorms p ↔
t = ⨅ i, (p i).toSeminormedAddCommGroup.toUniformSpace.toTopologicalSpace := by
rw [p.withSeminorms_iff_nhds_eq_iInf,
IsTopologicalAddGroup.ext_iff inferInstanc... | Mathlib/Analysis/LocallyConvex/WithSeminorms.lean | 416 | 421 |
/-
Copyright (c) 2020 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.Topology.Path
/-!
# Path connectedness
Continuing from `Mathlib.Topology.Path`, this file defines path components and path-connected
spaces.
## Main... |
theorem Topology.IsInducing.joinedIn_image {f : X → Y} (hf : IsInducing f) (hx : x ∈ F)
(hy : y ∈ F) : JoinedIn (f '' F) (f x) (f y) ↔ JoinedIn F x y := by
refine ⟨?_, (.map · hf.continuous)⟩
| Mathlib/Topology/Connected/PathConnected.lean | 178 | 181 |
/-
Copyright (c) 2022 Joanna Choules. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joanna Choules
-/
import Mathlib.CategoryTheory.CofilteredSystem
import Mathlib.Combinatorics.SimpleGraph.Subgraph
/-!
# Homomorphisms from finite subgraphs
This file defines the typ... | /-- The finite subgraph of G generated by a single vertex. -/
def singletonFinsubgraph (v : V) : G.Finsubgraph :=
⟨SimpleGraph.singletonSubgraph _ v, by simp⟩
/-- The finite subgraph of G generated by a single edge. -/
def finsubgraphOfAdj {u v : V} (e : G.Adj u v) : G.Finsubgraph :=
⟨SimpleGraph.subgraphOfAdj _ e... | Mathlib/Combinatorics/SimpleGraph/Finsubgraph.lean | 119 | 153 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Minchao Wu, Mario Carneiro
-/
import Mathlib.Data.Finset.Attach
import Mathlib.Data.Finset.Disjoint
import Mathlib.Data.Finset.Erase
import Mat... | Mathlib/Data/Finset/Basic.lean | 2,382 | 2,383 | |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2
import Mathlib.MeasureTheory.Measure.Real
/-! # Conditional expectation in L1
This file contains ... | theorem condExpL1_measure_zero (hm : m ≤ m0) : condExpL1 hm (0 : Measure α) f = 0 :=
setToFun_measure_zero _ rfl
| Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean | 585 | 587 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Yury Kudryashov
-/
import Mathlib.Algebra.Algebra.Rat
import Mathlib.Data.Nat.Prime.Int
import Mathlib.Data.Rat.Sqrt
imp... | /-!
### Simplification lemmas about operations
-/
@[simp]
theorem irrational_ratCast_add_iff : Irrational (q + x) ↔ Irrational x :=
| Mathlib/Data/Real/Irrational.lean | 518 | 524 |
/-
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 Mathlib.Algebra.Notation.Defs
import Mathlib.Data.Int.Notation
import Mathlib.Data.Nat.BinaryRec
import Mathlib.L... |
/-- Commutative version of `CancelMonoid`. -/
| Mathlib/Algebra/Group/Defs.lean | 708 | 709 |
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.ConditionalProbability
import Mathlib.Probability.Kernel.Basic
import Mathlib.Probability.Kernel.Composition.MeasureComp
import Mathlib.Tactic.... | have h_f_mem : ∀ n ∈ insert a s, f n ∈ π n := by
intro n hn_mem_insert
dsimp only [f]
rcases Finset.mem_insert.mp hn_mem_insert with hn_mem | hn_mem
· simp [hn_mem, ht2_mem_pia]
· have hn_ne_a : n ≠ a := by rintro rfl; exact haS (hs_mem hn_mem)
simp [hn_ne_a, hn_mem, hft1_mem n hn_mem]
hav... | Mathlib/Probability/Independence/Kernel.lean | 704 | 719 |
/-
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... | tfae_have 1 → 2 := by
refine fun h => or_iff_not_imp_left.2 fun hx₀ => ?_
have : ‖x‖ ^ 2 ≠ 0 := pow_ne_zero _ (norm_ne_zero_iff.2 hx₀)
rw [← sq_eq_sq₀, mul_pow, ← mul_right_inj' this, eq_comm, ← sub_eq_zero, ← mul_sub] at h <;>
try positivity
| Mathlib/Analysis/InnerProductSpace/Basic.lean | 669 | 673 |
/-
Copyright (c) 2019 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard
-/
import Mathlib.Data.EReal.Basic
deprecated_module (since := "2025-04-13")
| Mathlib/Data/Real/EReal.lean | 825 | 826 | |
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes Hölzl
-/
import Mathlib.Algebra.Field.Subfield.Defs
import Mathlib.Algebra.Order.Group.Pointwise.Interval
import Mathlib.Analysis.Normed.Ring.Basic
/-!
# Norm... |
-- see Note [lower instance priority]
instance (priority := 100) NormedField.toNormedDivisionRing : NormedDivisionRing α :=
| Mathlib/Analysis/Normed/Field/Basic.lean | 182 | 184 |
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Measure.Haar.Basic
import Mathlib.Analysis.InnerProductSpace.PiL2
/-!
# Additive Haar measure constructed from a basis
Given a ba... | volume `1` to the parallelepiped spanned by any orthonormal basis. We define the measure using
some arbitrary choice of orthonormal basis. The fact that it works with any orthonormal basis
is proved in `orthonormalBasis.volume_parallelepiped`.
| Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean | 287 | 289 |
/-
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... | ring_nf
have nz : sqrt 5 ≠ 0 := by norm_num
rw [← (mul_inv_cancel₀ nz).symm, one_mul]
/-- Relationship between the Fibonacci Sequence, Golden Ratio and its exponents -/
theorem fib_golden_exp' (n : ℕ) : φ * Nat.fib (n + 1) + Nat.fib n = φ ^ (n + 1) := by
induction n with
| zero => norm_num
| succ n ih =>
... | Mathlib/Data/Real/GoldenRatio.lean | 210 | 229 |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Bounded
import Mathlib.Analysis.Normed.Group.Uniform
import Mathlib.Topology.MetricSpace.Thickening
/-!
# P... | @[to_additive (attr := simp)]
theorem mul_ball : s * ball x δ = x • thickening δ s := by
| Mathlib/Analysis/Normed/Group/Pointwise.lean | 178 | 179 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPi... | /-- The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order. -/
theorem card_eq_multiplicity [Finite G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
Nat.card P = p ^ Nat.factorization (Nat.card G) p := by
obtain ⟨n, heq : Nat.card P = _⟩ := IsPGroup.iff_card.mp P.i... | Mathlib/GroupTheory/Sylow.lean | 712 | 717 |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Ring.Rat
import Mathlib.Algebra.Ring.Int.Parity
import Mathlib.Data.PNat.Defs... | (h2 : Nat.Coprime c.natAbs d.natAbs) (h : (a : ℚ) / b = (c : ℚ) / d) : a = c ∧ b = d := by
apply And.intro
· rw [← num_div_eq_of_coprime hb0 h1, h, num_div_eq_of_coprime hd0 h2]
| Mathlib/Data/Rat/Lemmas.lean | 197 | 199 |
/-
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.Topology.Continuous
import Mathlib.Topology.Defs.Induced
/-!
# Ordering on topologies and (co)induced topologies
Topologies on a fixe... | letI := T.induced f
simp_rw [mem_nhds_iff, isOpen_induced_iff]
constructor
| Mathlib/Topology/Order.lean | 729 | 731 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Topology.Algebra.Monoid
/-!
# Topological group with zero
I... | normed (semi)field has this property. -/
class HasContinuousInv₀ (G₀ : Type*) [Zero G₀] [Inv G₀] [TopologicalSpace G₀] : Prop where
| Mathlib/Topology/Algebra/GroupWithZero.lean | 75 | 76 |
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel, Alex Keizer
-/
import Mathlib.Algebra.Group.Nat.Even
import Mathlib.Algebra.NeZero
import Mathlib.Algebra.Ring.Nat
import Mathlib.Data.List.GetD
import Mathlib.Data.Nat.B... | refine lt_of_testBit i ?_ h fun j hj => ?_
· rw [testBit_xor, h, hi]
rfl
· simp only [testBit_xor, hi' _ hj, Bool.bne_false]
theorem lt_xor_cases {a b c : ℕ} (h : a < b ^^^ c) : a ^^^ c < b ∨ a ^^^ b < c := by
obtain ha | hb | hc := xor_trichotomy <| Nat.xor_assoc _ _ _ ▸ xor_ne_zero.2 h.ne
| Mathlib/Data/Nat/Bitwise.lean | 339 | 345 |
/-
Copyright (c) 2023 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.Algebra.Polynomial.Bivariate
import Mathlib.AlgebraicGeometry.EllipticCurve.Weierstrass
import Mathlib.AlgebraicGeometry.EllipticCu... | simp only [evalEval_polynomial, zero_add, zero_sub, mul_zero, zero_pow <| Nat.succ_ne_zero _]
variable (W') in
| Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean | 183 | 185 |
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Mathlib.Control.Basic
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Option.Basic
im... |
end FoldlMFoldrM
/-! ### intersperse -/
| Mathlib/Data/List/Basic.lean | 989 | 993 |
/-
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.Group.TypeTags.Hom
import Mathlib.Algebra.Ring.Hom.Basic
import Mathlib.Algebra.Ring.Int.Defs
import Mathlib.Algebra.Ring.Parity
/-!
# Cast of... | Mathlib/Data/Int/Cast/Lemmas.lean | 364 | 366 |
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