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/-
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.CategoryTheory.Limits.Shapes.FiniteProducts
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Shapes.NormalMono.Eq... |
section
| Mathlib/CategoryTheory/Abelian/NonPreadditive.lean | 293 | 294 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Patrick Massot, Yury Kudryashov
-/
import Mathlib.Topology.Connected.Clopen
/-!
# Totally disconnected and totally separated topological spaces
## Main definitions
We define th... | [TopologicalSpace α] [TotallySeparatedSpace α] : TotallyDisconnectedSpace α :=
⟨TotallySeparatedSpace.isTotallySeparated_univ.isTotallyDisconnected⟩
-- see Note [lower instance priority]
instance (priority := 100) TotallySeparatedSpace.of_discrete (α : Type*) [TopologicalSpace α]
[DiscreteTopology α] : Total... | Mathlib/Topology/Connected/TotallyDisconnected.lean | 202 | 211 |
/-
Copyright (c) 2024 Judith Ludwig, Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Judith Ludwig, Christian Merten
-/
import Mathlib.RingTheory.AdicCompletion.Basic
import Mathlib.RingTheory.AdicCompletion.Algebra
import Mathlib.Algebra.DirectSum.Bas... | simp only [LinearMap.coe_comp, Function.comp_apply, LinearMap.id_coe, id_eq, mk_apply_coe,
Submodule.mkQ_apply]
rw [← DirectSum.sum_univ_of (((sumInv I M) ((AdicCompletion.mk I (⨁ (j : ι), M j)) f)))]
simp only [sumInv_apply, map_mk, map_sum, sum_of, val_sum_apply, mk_apply_coe,
AdicCauchySequence.map_app... | Mathlib/RingTheory/AdicCompletion/Functoriality.lean | 302 | 310 |
/-
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.CategoryTheory.Limits.Shapes.FiniteProducts
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Shapes.NormalMono.Eq... | Mathlib/CategoryTheory/Abelian/NonPreadditive.lean | 439 | 440 | |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.GroupTheory.Submonoid.Inverses
import Mathlib.RingTheory.FiniteType
import Mathlib.RingTheory.Loc... | exact Submodule.smul_mem _ _ (Submodule.subset_span (toInvSubmonoid M S m).prop)
theorem finiteType_of_monoid_fg [Monoid.FG M] : Algebra.FiniteType R S := by
have := Monoid.fg_of_surjective _ (toInvSubmonoid_surjective M S)
rw [Monoid.fg_iff_submonoid_fg] at this
| Mathlib/RingTheory/Localization/InvSubmonoid.lean | 99 | 103 |
/-
Copyright (c) 2024 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Peter Pfaffelhuber
-/
import Mathlib.MeasureTheory.SetSemiring
import Mathlib.MeasureTheory.OuterMeasure.Induced
/-!
# Additive Contents
An additive content `m` on a set ... | (h_dis : PairwiseDisjoint (I : Set (Set α)) id) :
m s = ∑ i ∈ I, m i + ∑ i ∈ hC.disjointOfDiffUnion hs hI, m i := by
classical
conv_lhs => rw [← hC.sUnion_union_disjointOfDiffUnion_of_subset hs hI hI_ss]
rw [addContent_sUnion]
· rw [sum_union]
exact hC.disjoint_disjointOfDiffUnion hs hI
| Mathlib/MeasureTheory/Measure/AddContent.lean | 126 | 132 |
/-
Copyright (c) 2018 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Mario Carneiro, Kim Morrison, Floris van Doorn
-/
import Mathlib.CategoryTheory.Limits.IsLimit
import Mathlib.CategoryTheory.Category.ULift
import Mathlib.CategoryTheory.Ess... | -/
def HasColimit.isoOfNatIso {F G : J ⥤ C} [HasColimit F] [HasColimit G] (w : F ≅ G) :
| Mathlib/CategoryTheory/Limits/HasLimits.lean | 821 | 822 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
/-!
# Kernels and cokernels
In a category with zero morphisms, the kernel of a morphism `f : X... | cokernelComparison _ G ≫ G.map (cokernel.map f g p q hpq) :=
cokernel.map_desc _ _ (by rw [← G.map_comp, cokernel.condition, G.map_zero]) _ _
(by rw [← G.map_comp, cokernel.condition, G.map_zero]) _ _ _ _
(by simp only [← G.map_comp]; exact G.congr_map (cokernel.π_desc _ _ _))
| Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean | 1,165 | 1,169 |
/-
Copyright (c) 2019 Jan-David Salchow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jan-David Salchow, Sébastien Gouëzel, Jean Lo
-/
import Mathlib.Analysis.NormedSpace.OperatorNorm.Basic
/-!
# Operator norm as an `NNNorm`
Operator norm as an `NNNorm`, i.e. takin... | mul_assoc, ← NNReal.lt_inv_iff_mul_lt hy'] at hy
obtain ⟨k, hk₁, hk₂⟩ := NormedField.exists_lt_nnnorm_lt 𝕜 hy
refine ⟨k • y, (nnnorm_smul k y).symm ▸ (NNReal.lt_inv_iff_mul_lt hy').1 hk₂, ?_⟩
have : ‖σ₁₂ k‖₊ = ‖k‖₊ := Subtype.ext RingHomIsometric.is_iso
rwa [map_smulₛₗ f, nnnorm_smul, ← div_lt_iff₀ hfy.bot... | Mathlib/Analysis/NormedSpace/OperatorNorm/NNNorm.lean | 139 | 143 |
/-
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... |
theorem χ₄_nat_eq_if_mod_four (n : ℕ) :
χ₄ n = if n % 2 = 0 then 0 else if n % 4 = 1 then 1 else -1 :=
mod_cast χ₄_int_eq_if_mod_four n
/-- Alternative description of `χ₄ n` for odd `n : ℕ` in terms of powers of `-1` -/
| Mathlib/NumberTheory/LegendreSymbol/ZModChar.lean | 68 | 73 |
/-
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 | 1,970 | 1,971 | |
/-
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.Data.Set.Image
/-!
# Booleans and set operations
This file contains two trivial lemmas about `Bool`, `Set.univ`, and `Set.range`.
-/
open Set
na... | @[simp] theorem compl_singleton (b : Bool) : ({b}ᶜ : Set Bool) = {!b} :=
Set.ext fun _ => eq_not_iff.symm
| Mathlib/Data/Bool/Set.lean | 27 | 28 |
/-
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.Analysis.InnerProductSpace.Orientation
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
/-!
# Volume forms and measures on inner product spa... |
variable [Fintype ι]
variable [FiniteDimensional ℝ E] [FiniteDimensional ℝ F]
section
variable {m n : ℕ} [_i : Fact (finrank ℝ F = n)]
| Mathlib/MeasureTheory/Measure/Haar/InnerProductSpace.lean | 48 | 54 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Jeremy Avigad, Simon Hudon
-/
import Batteries.WF
import Mathlib.Data.Part
import Mathlib.Data.Rel
import Mathlib.Tactic.GeneralizeProofs
/-!
# Partial functions
This... | `a` given that `f a` inherits `P` from `a` and `P` holds for preimages of `b`.
-/
@[elab_as_elim]
| Mathlib/Data/PFun.lean | 306 | 308 |
/-
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
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Order.CauSeq.BigOperators
import Mathlib.Algebra.Order.Star.Basic
import Mathlib.Data.C... | simp_rw [Complex.norm_mul, Complex.norm_pow, Complex.norm_div, norm_natCast]
| Mathlib/Data/Complex/Exponential.lean | 486 | 486 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Bhavik Mehta, Stuart Presnell
-/
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Order.Monotone.Defs
/-!
# Binomial coefficients
This file defines binomial coeffic... | exact factorial_mul_factorial_dvd_factorial (Nat.le_add_right _ _)
@[simp]
theorem choose_symm {n k : ℕ} (hk : k ≤ n) : choose n (n - k) = choose n k := by
| Mathlib/Data/Nat/Choose/Basic.lean | 185 | 188 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.SymmDiff
import Mathlib.Order.SuccPred.Relation
import Mathlib.Topology.Irreducible
/-!
# Connected subsets ... | Mathlib/Topology/Connected/Basic.lean | 1,112 | 1,124 | |
/-
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.Analysis.Normed.Group.Continuity
import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
import Mathlib.Topology.MetricSpace.Thick... | (hg : AEMeasurable g μ) : AEMeasurable (fun a => edist (f a) (g a)) μ :=
(@continuous_edist α _).aemeasurable2 hf hg
end PseudoEMetricSpace
| Mathlib/MeasureTheory/Constructions/BorelSpace/Metric.lean | 178 | 182 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Topology.MetricSpace.IsometricSMul
/-!
# Hausdorff distance
The Hausdorff distance on subsets of a... | simp [mem_closure_iff_infEdist_zero, infDist, ENNReal.toReal_eq_zero_iff, infEdist_ne_top h]
theorem infDist_pos_iff_not_mem_closure (hs : s.Nonempty) :
x ∉ closure s ↔ 0 < infDist x s :=
| Mathlib/Topology/MetricSpace/HausdorffDistance.lean | 543 | 546 |
/-
Copyright (c) 2024 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Basic
import Mathlib.Tactic.ComputeDegree
/-!
# Division polynomials of Weierst... |
lemma coeff_preΨ_ne_zero {n : ℤ} (h : (n : R) ≠ 0) :
(W.preΨ n).coeff ((n.natAbs ^ 2 - if Even n then 4 else 1) / 2) ≠ 0 := by
induction n using Int.negInduction with
| nat n => simpa only [preΨ_ofNat, Int.even_coe_nat]
using W.coeff_preΨ'_ne_zero <| by exact_mod_cast h
| neg ih n => simpa only [preΨ_n... | Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Degree.lean | 286 | 292 |
/-
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.Group.Action.Pi
import Mathlib.Algebra.Order.AbsoluteValue.Basic
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.Mi... |
/-- The constant Cauchy sequence -/
local notation "const" => const abs
/-- The entries of a positive Cauchy sequence eventually have a positive lower bound. -/
def Pos (f : CauSeq α abs) : Prop :=
∃ K > 0, ∃ i, ∀ j ≥ i, K ≤ f j
| Mathlib/Algebra/Order/CauSeq/Basic.lean | 569 | 575 |
/-
Copyright (c) 2019 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Yury Kudryashov
-/
import Mathlib.Analysis.Convex.Combination
import Mathlib.Analysis.Convex.Function
import Mathlib.Tactic.FieldSimp
/-!
# Jensen's inequality... | obtain ⟨y, hy, hxy⟩ := hf.exists_ge_of_mem_convexHull hst hx
exact hxy.trans <| hb <| mem_image_of_mem _ hy
lemma ConcaveOn.bddBelow_convexHull {s t : Set E} (hst : s ⊆ t) (hf : ConcaveOn 𝕜 t f) :
BddBelow (f '' s) → BddBelow (f '' convexHull 𝕜 s) := hf.dual.bddAbove_convexHull hst
| Mathlib/Analysis/Convex/Jensen.lean | 325 | 329 |
/-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Yury Kudryashov
-/
import Mathlib.Algebra.BigOperators.Associated
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Ring.Abs
import M... | calc
(m + n)# = m# * ∏ p ∈ Ico (m + 1) (m + n + 1) with p.Prime, p := primorial_add _ _
_ ∣ m# * choose (m + n) m :=
mul_dvd_mul_left _ <|
prod_primes_dvd _ (fun _ hk ↦ (mem_filter.1 hk).2.prime) fun p hp ↦ by
| Mathlib/NumberTheory/Primorial.lean | 51 | 55 |
/-
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
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Order.CauSeq.BigOperators
import Mathlib.Algebra.Order.Star.Basic
import Mathlib.Data.C... | Mathlib/Data/Complex/Exponential.lean | 1,149 | 1,150 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.Measure.Comap
import Mathlib.MeasureTheory.Measure.QuasiMeasurePreserving
/-!
# Restricting a measure to a subset or a s... | Mathlib/MeasureTheory/Measure/Restrict.lean | 1,093 | 1,101 | |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Algebra.Order.Group.Unbundled.Basic
import Mathlib.Algebra.Order.GroupWithZero.Canonical
import Mathlib.Algebra.Order.Monoid.Units
/-!
# Ordered monoid an... | @[to_additive]
theorem symm_apply_eq (e : α ≃*o β) {x y} : e.symm x = y ↔ x = e y :=
| Mathlib/Algebra/Order/Hom/Monoid.lean | 714 | 715 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Yury Kudryashov
-/
import Mathlib.Algebra.CharP.Lemmas
import Mathlib.FieldTheory.Perfect
/-!
# The perfect closure of a characteristic `p` ring
## Main definitions
- `Perfec... | toFun x := mk _ _ (0, x)
map_one' := rfl
map_mul' _ _ := rfl
| Mathlib/FieldTheory/PerfectClosure.lean | 380 | 382 |
/-
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.BigOperators.Ring.Finset
import Mathlib.Combinatorics.SimpleGraph.Dart
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Data.ZMod.Basic... | rcases G.odd_card_odd_degree_vertices_ne v h with ⟨k, hg⟩
have hg' : 0 < #{w | w ≠ v ∧ Odd (G.degree w)} := by
rw [hg]
apply Nat.succ_pos
rcases card_pos.mp hg' with ⟨w, hw⟩
simp only [true_and, mem_filter, mem_univ, Ne] at hw
exact ⟨w, hw⟩
end SimpleGraph
| Mathlib/Combinatorics/SimpleGraph/DegreeSum.lean | 163 | 172 |
/-
Copyright (c) 2021 Ashvni Narayanan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ashvni Narayanan, Anne Baanen
-/
import Mathlib.Algebra.Algebra.Rat
import Mathlib.Algebra.Ring.Int.Parity
import Mathlib.Algebra.Ring.Int.Units
import Mathlib.RingTheory.DedekindDom... | an isomorphism of algebras `e : K ≃ₐ[k] L` to `𝓞 K`. -/
def mapAlgEquiv {k K L E : Type*} [Field k] [Field K] [Field L] [Algebra k K]
[Algebra k L] [EquivLike E K L] [AlgEquivClass E k K L] (e : E) : (𝓞 K) ≃ₐ[𝓞 k] (𝓞 L) :=
AlgEquiv.ofAlgHom (mapAlgHom e) (mapAlgHom (e : K ≃ₐ[k] L).symm)
| Mathlib/NumberTheory/NumberField/Basic.lean | 193 | 196 |
/-
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.Order.Field.Rat
import Mathlib.Data.Rat.Cast.CharZero
import Mathlib.Tactic.Positivity.Core
/-!
# Casts of rational numbers in... | rw [← cast_le (K := K), cast_natCast]
| Mathlib/Data/Rat/Cast/Order.lean | 65 | 65 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Multiset.ZeroCons
/-!
# Basic results on multisets
-/
-- No algebra should be required
assert_not_exists Monoid
universe v
open List S... | Mathlib/Data/Multiset/Basic.lean | 1,912 | 1,913 | |
/-
Copyright (c) 2022 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.NumberTheory.BernoulliPolynomials
import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic
import Mathlib.Analysis.Calculus.Deriv.Polynomial
import... | apply (congr_arg₂ HasSum ?_ ?_).to_iff.mp <|
hasSum_one_div_nat_pow_mul_sin one_ne_zero (?_ : 1 / 4 ∈ Icc (0 : ℝ) 1)
· ext1 n
norm_num
| Mathlib/NumberTheory/ZetaValues.lean | 355 | 358 |
/-
Copyright (c) 2024 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Order.Ring.Int
import Mathlib.Data.Nat.Cast.Order.Basic
import Mathlib.Order.Interval.Set.OrdConnected
import Mathlib.Order.Nat
import Mathli... | (castOrderEmbedding (α := ℤ)).image_Ioc (by simp [ordConnected_Ici]) a b
| Mathlib/Data/Nat/Cast/SetInterval.lean | 34 | 35 |
/-
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.Algebra.Ring.Action.Basic
import Mathlib.GroupTheory.Congruence.Basic
import Mathlib.RingTheory.Congruence.Defs
/-!
# Congruence relations on rings
This fi... | Mathlib/RingTheory/Congruence/Basic.lean | 576 | 577 | |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Joey van Langen, Casper Putz
-/
import Mathlib.Data.Nat.Cast.Basic
import Mathlib.Data.Nat.Find
import Mathlib.Data.Nat.Prime.Defs
import Mathlib.Order.Lattice
/-!
# Characteris... | | 0, _ => Or.inr rfl
| 1, hc => absurd (Eq.refl (1 : ℕ)) (@char_ne_one R _ _ (1 : ℕ) hc)
| m + 2, hc => Or.inl (@char_is_prime_of_two_le R _ _ (m + 2) hc (Nat.le_add_left 2 m))
| Mathlib/Algebra/CharP/Defs.lean | 227 | 229 |
/-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator
import Mathlib.MeasureTheory.Function.UniformIntegrable
import Mathlib.MeasureTheory.V... | ∀ᵐ x ∂μ, |(μ[f|m]) x| ≤ R := by
by_cases hnm : m ≤ m0
swap
· simp_rw [condExp_of_not_le hnm, Pi.zero_apply, abs_zero]
exact Eventually.of_forall fun _ => R.coe_nonneg
by_cases hfint : Integrable f μ
swap
· simp_rw [condExp_of_not_integrable hfint]
filter_upwards [hbdd] with x hx
rw [Pi.zero_... | Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean | 146 | 179 |
/-
Copyright (c) 2021 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn, Joachim Breitner
-/
import Mathlib.Algebra.Group.Action.End
import Mathlib.Algebra.Group.Action.Pointwise.Set.Basic
import Mathlib.Algebra.Group.Submonoid.Membership
import Mat... | variable {α : Type*} [MulAction G α]
variable (X : ι → Set α)
variable (hXnonempty : ∀ i, (X i).Nonempty)
variable (hXdisj : Pairwise (Disjoint on X))
variable (hpp : Pairwise fun i j => ∀ h : H i, h ≠ 1 → f i h • X j ⊆ X i)
| Mathlib/GroupTheory/CoprodI.lean | 820 | 824 |
/-
Copyright (c) 2022 Eric Rodriguez. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Rodriguez
-/
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Ring.Cast
import Mathlib.Data.Fi... | top_le_iff
| Mathlib/Data/Sign.lean | 174 | 174 |
/-
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 Batteries.Data.Rat.Lemmas
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Rat.Init
import Mathlib.Order.Basic
import Mathlib.Tactic.Commo... | simp [-divInt_ofNat, mkRat_eq_divInt, Int.mul_comm, Int.mul_ne_zero hn (Int.ofNat_ne_zero.2 hd)]
protected theorem inv_mul_cancel (h : a ≠ 0) : a⁻¹ * a = 1 :=
| Mathlib/Data/Rat/Defs.lean | 277 | 279 |
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Limits.Shapes.RegularMon... | show Mono c'.inl
apply MonoCoprod.binaryCofan_inl
apply mapIsColimitOfPreservesOfIsColimit F
apply IsColimit.ofIsoColimit h
refine Cocones.ext (φ := eqToIso rfl) ?_
rintro ⟨(j₁|j₂)⟩ <;> simp only [const_obj_obj, eqToIso_refl, Iso.refl_hom,
Category.comp_id, BinaryCofan.mk_inl, BinaryCofan.... | Mathlib/CategoryTheory/Limits/MonoCoprod.lean | 230 | 242 |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Measure.Trim
import Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated
/-!
# Almost everywhere measurable functions
A funct... | apply measurable_const
· rw [restrict_piecewise_compl, restrict_eq]
refine measurable_generateFrom fun s hs ↦ .of_subtype_image ?_
rw [preimage_comp, Subtype.image_preimage_coe]
convert (hTm s hs).diff hvm using 1
rw [inter_comm]
refine Set.ext fun x ↦ and_congr_left fun hxv ↦ ⟨fun hx ↦ ?_, fu... | Mathlib/MeasureTheory/Measure/AEMeasurable.lean | 380 | 394 |
/-
Copyright (c) 2018 Violeta Hernández Palacios, Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios, Mario Carneiro
-/
import Mathlib.Logic.Small.List
import Mathlib.SetTheory.Ordinal.Enum
import Mathlib.SetTheory.Ordinal.Exponen... | rintro _ ⟨i, rfl⟩
exact nfpFamily_fp (H i) _
/-- The derivative of a family of normal functions is the sequence of their common fixed points.
This is defined for all functions such that `Ordinal.derivFamily_zero`,
`Ordinal.derivFamily_succ`, and `Ordinal.derivFamily_limit` are satisfied. -/
| Mathlib/SetTheory/Ordinal/FixedPoint.lean | 128 | 134 |
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calculus.FDeriv.Comp
/-!
# Additive operations on derivative... | theorem DifferentiableAt.const_sub (hf : DifferentiableAt 𝕜 f x) (c : F) :
DifferentiableAt 𝕜 (fun y => c - f y) x :=
| Mathlib/Analysis/Calculus/FDeriv/Add.lean | 683 | 684 |
/-
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... | theorem factors_self {X : C} (P : Subobject X) : P.Factors P.arrow :=
(factors_iff _ _).mpr ⟨𝟙 (P : C), by simp⟩
| Mathlib/CategoryTheory/Subobject/FactorThru.lean | 79 | 80 |
/-
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.Algebra.MvPolynomial.Rename
import Mathlib.Algebra.MvPolynomial.Variables
/-!
# Monad operations on `MvPolynomial`
... | @[simp]
theorem join₂_map (f : R →+* MvPolynomial σ S) (φ : MvPolynomial σ R) :
join₂ (map f φ) = bind₂ f φ := by simp only [join₂, bind₂, eval₂Hom_map_hom, RingHom.id_comp]
| Mathlib/Algebra/MvPolynomial/Monad.lean | 160 | 162 |
/-
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... | it is sufficient to show that for all `y₀ ∈ K` there `P x y` holds for `(x, y)` close enough
to `(x₀, y₀)`.
| Mathlib/Topology/Compactness/Compact.lean | 418 | 420 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Heather Macbeth
-/
import Mathlib.Topology.Homeomorph.Defs
import Mathlib.Topology.Order.LeftRightNhds
/-!
# Continuity of monotone functions
In this file we prove ... | Mathlib/Topology/Order/MonotoneContinuity.lean | 304 | 310 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.MeasureTheory.MeasurableSpace.EventuallyMeasurable
import Mathlib.MeasureTheory.MeasurableSpace.Basic
import Mathlib.M... | /-- If `sᵢ` is a countable family of (null) measurable pairwise `μ`-a.e. disjoint sets, then there
exists a subordinate family `tᵢ ⊆ sᵢ` of measurable pairwise disjoint sets such that
`tᵢ =ᵐ[μ] sᵢ`. -/
theorem exists_subordinate_pairwise_disjoint [Countable ι] {s : ι → Set α}
(h : ∀ i, NullMeasurableSet (s i) μ) (h... | Mathlib/MeasureTheory/Measure/NullMeasurable.lean | 217 | 223 |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Algebra.Order.Group.Unbundled.Basic
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Or... | Mathlib/Algebra/Order/Group/Defs.lean | 807 | 807 | |
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Data.Set.BooleanAlgebra
import Mathlib.Tactic.AdaptationNote
/-!
# Relations
This file defines bundled relations. A relation between `α` and `β` is a f... | Iff.intro (fun h x xs _y rxy => h ⟨x, xs, rxy⟩) fun h y ⟨_x, xs, rxy⟩ => h xs y rxy
theorem image_core_gc : GaloisConnection r.image r.core :=
| Mathlib/Data/Rel.lean | 311 | 313 |
/-
Copyright (c) 2020 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Yaël Dillies
-/
import Mathlib.Order.Interval.Set.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Tactic.Bound.Attribute
import Mathlib.Tactic.Contrapose
import Mathl... |
@[mono]
theorem log_anti_left {b c n : ℕ} (hc : 1 < c) (hb : c ≤ b) : log b n ≤ log c n := by
| Mathlib/Data/Nat/Log.lean | 178 | 180 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Analytic.Within
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries
/-!
# Higher d... | Mathlib/Analysis/Calculus/ContDiff/Defs.lean | 1,298 | 1,300 | |
/-
Copyright (c) 2024 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.Algebra.Module.ZLattice.Covolume
import Mathlib.LinearAlgebra.Matrix.Determinant.Misc
import Mathlib.NumberTheory.NumberField.Units.DirichletTheorem
/-!... | -- We construct an equiv `Fin (rank K + 1) ≃ InfinitePlace K` from `e₂.symm`
let f : Fin (rank K + 1) ≃ InfinitePlace K :=
(finSuccEquiv _).trans ((Equiv.optionSubtype _).symm e₁.symm).val
-- And `g` corresponds to the restriction of `f⁻¹` to `{w // w ≠ w₂}`
let g : {w // w ≠ w₂} ≃ Fin (rank K) :=
(Equi... | Mathlib/NumberTheory/NumberField/Units/Regulator.lean | 69 | 99 |
/-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth, Eric Wieser
-/
import Mathlib.Analysis.Normed.Lp.PiLp
import Mathlib.Analysis.InnerProductSpace.PiL2
/-!
# Matrices as a normed space
In this file we provide the fo... | rw [linfty_opNNNorm_eq_opNNNorm]
simp only [← toLin'_apply', toLin'_toMatrix']
@[simp] lemma linfty_opNorm_toMatrix (f : (n → α) →L[α] (m → α)) :
‖LinearMap.toMatrix' (↑f : (n → α) →ₗ[α] (m → α))‖ = ‖f‖ :=
congr_arg NNReal.toReal (linfty_opNNNorm_toMatrix f)
| Mathlib/Analysis/Matrix.lean | 441 | 446 |
/-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Bhavik Mehta, Yaël Dillies
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Analysis.Convex.Hull
import Mathlib.Analysis.Normed.Module.Basic
import Mathlib.Topology.Bornology.A... | theorem balanced_univ : Balanced 𝕜 (univ : Set E) := fun _a _ha => subset_univ _
| Mathlib/Analysis/LocallyConvex/Basic.lean | 81 | 82 |
/-
Copyright (c) 2015 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis
-/
import Mathlib.Algebra.Order.Monoid.Unbundled.Pow
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Ring.Parity
import Mathlib.Tactic.Bound... |
namespace MonoidHom
| Mathlib/Algebra/Order/Ring/Basic.lean | 27 | 28 |
/-
Copyright (c) 2022 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.List.Induction
import Mathlib.Data.List.TakeWhile
/-!
# Dropping or taking from lists on the right
Taking or removing element from the tail e... |
@[simp]
lemma rdrop_append_length_add {l₁ l₂ : List α} (k : ℕ) :
| Mathlib/Data/List/DropRight.lean | 214 | 216 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Eric Wieser
-/
import Mathlib.Data.Matrix.Basis
import Mathlib.Data.Matrix.Composition
import Mathlib.Data.Matrix.Kronecker
import Mathlib.RingTheory.TensorProduct.Basic
... | simp only [invFun, map_sum, toFunAlgHom_apply]
convert Finset.sum_product (β := Matrix n n A) ..
conv_lhs => rw [matrix_eq_sum_stdBasisMatrix M]
refine Finset.sum_congr rfl fun i _ => Finset.sum_congr rfl fun j _ => Matrix.ext fun a b => ?_
| Mathlib/RingTheory/MatrixAlgebra.lean | 164 | 167 |
/-
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.AlgebraicGeometry.EllipticCurve.Affine
import Mathlib.LinearAlgebra.FreeModule.Norm
import Mathlib.RingTheory.ClassGroup
import Mat... | have h := Fintype.linearIndependent_iff.mp (CoordinateRing.basis W).linearIndependent ![p, q]
rw [Fin.sum_univ_succ, basis_zero, Fin.sum_univ_one, Fin.succ_zero_eq_one, basis_one] at h
exact ⟨h hpq 0, h hpq 1⟩
| Mathlib/AlgebraicGeometry/EllipticCurve/Group.lean | 146 | 148 |
/-
Copyright (c) 2021 Alex J. Best. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex J. Best
-/
import Mathlib.Analysis.Convex.Body
import Mathlib.Analysis.Convex.Measure
import Mathlib.MeasureTheory.Group.FundamentalDomain
/-!
# Geometry of numbers
In this file w... | (h : μ F * 2 ^ finrank ℝ E ≤ μ s) :
∃ x ≠ 0, ((x : L) : E) ∈ s := by
have h_mes : μ s ≠ 0 := by
intro hμ
suffices μ F = 0 from fund.measure_ne_zero (NeZero.ne μ) this
rw [hμ, le_zero_iff, mul_eq_zero] at h
exact h.resolve_right <| pow_ne_zero _ two_ne_zero
have h_nemp : s.Nonempty := nonempt... | Mathlib/MeasureTheory/Group/GeometryOfNumbers.lean | 92 | 142 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.IsolatedZeros
import Mathlib.Analysis.SpecialFunctions.Complex.CircleMap
import Mathlib.Analysis.SpecialFunctions.NonIntegrable
/-... | ((2 * π)⁻¹ * ∫ θ : ℝ in (0)..2 * π, ‖f (circleMap c R θ)‖) fun n => ?_
refine (mul_le_mul_of_nonneg_right (norm_cauchyPowerSeries_le _ _ _ _)
(pow_nonneg R.coe_nonneg _)).trans ?_
rw [abs_of_nonneg R.coe_nonneg]
rcases eq_or_ne (R ^ n : ℝ) 0 with hR | hR
· rw_mod_cast [hR, mul_zero]
exact mul_nonn... | Mathlib/MeasureTheory/Integral/CircleIntegral.lean | 501 | 516 |
/-
Copyright (c) 2019 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Mario Carneiro, Isabel Longbottom, Kim Morrison, Apurva Nakade, Yuyang Zhao
-/
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.SetTheory.PGame.Algebra
import Mathl... | · fconstructor
| Mathlib/SetTheory/Game/Basic.lean | 523 | 523 |
/-
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
-/
import Batteries.Tactic.Congr
import Mathlib.Data.Option.Basic
import Mathlib.Data.Prod.Basic
import Mathlib.Data.Set.Subsingleton
import Mathlib.Dat... | Mathlib/Data/Set/Image.lean | 183 | 183 | |
/-
Copyright (c) 2024 Lawrence Wu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lawrence Wu
-/
import Mathlib.MeasureTheory.Group.Measure
import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap
/-!
# Bounding of integrals by asymptotics
We establish integ... | obtain ⟨w, hwl, hw⟩ := hfm.exists_mem
refine eventually_iff_exists_mem.mpr ⟨w ∩ v, inter_mem hwl hv, fun x hx ↦ ?_⟩
haveI : IsFiniteMeasure (μ.restrict s) := ⟨Measure.restrict_apply_univ s ▸ hμ⟩
refine Integrable.mono' (integrable_const (C * ‖g x‖)) (hw x hx.1) ?_
filter_upwards [MeasureTheory.self_mem_ae_res... | Mathlib/MeasureTheory/Integral/Asymptotics.lean | 70 | 77 |
/-
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.Algebra.BigOperators.Group.Finset.Sigma
import Mathlib.Algebra.Order.Interval.Finset.Basic
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Tact... | rw [mul_comm, mul_prod_Ioi_eq_prod_Ici]
end LocallyFiniteOrderTop
| Mathlib/Algebra/BigOperators/Intervals.lean | 55 | 57 |
/-
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... |
@[simp]
| Mathlib/Data/Matrix/Rank.lean | 285 | 286 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.Group.Defs
/-!
# Invertible elements
This file defines a typeclass `Invertible a` for elements `a` with a two-sided
multiplicative inverse.
The in... | @[congr]
theorem Invertible.congr [Monoid α] (a b : α) [Invertible a] [Invertible b] (h : a = b) :
⅟a = ⅟b := by subst h; congr; apply Subsingleton.allEq
| Mathlib/Algebra/Group/Invertible/Defs.lean | 156 | 159 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Yury Kudryashov
-/
import Mathlib.Algebra.Algebra.Basic
/-!
# Homomorphisms of `R`-algebras
This file defines bundled homomorphisms of `R`-algebras.
## Main definitions
* `Al... | def ofId : R →ₐ[R] A :=
{ algebraMap R A with commutes' := fun _ => rfl }
variable {R}
| Mathlib/Algebra/Algebra/Hom.lean | 388 | 391 |
/-
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, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Pairwise
import Mathlib.Data.Set.BooleanAlgebra
/-!
# The set lattice
This file is a collectio... |
/-- Equivalence from the disjoint union of a family of sets forming a partition of `β`, to `β`
| Mathlib/Data/Set/Lattice.lean | 1,316 | 1,317 |
/-
Copyright (c) 2021 Hunter Monroe. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hunter Monroe, Kyle Miller, Alena Gusakov
-/
import Mathlib.Combinatorics.SimpleGraph.DeleteEdges
import Mathlib.Data.Fintype.Powerset
/-!
# Subgraphs of a simple graph
A subgraph of ... | lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) :
Subgraph.coeSubgraph H' ≤ H := by
constructor
· simp
· rintro v w ⟨_, _, h, rfl, rfl⟩
exact H'.adj_sub h
| Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | 1,008 | 1,013 |
/-
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... |
variable {k : Type*} {V : Type*} {P : Type*} [Ring k] [AddCommGroup V] [Module k V]
variable [S : AffineSpace V P]
| Mathlib/LinearAlgebra/AffineSpace/Combination.lean | 51 | 53 |
/-
Copyright (c) 2018 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Mario Carneiro, Reid Barton, Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.Opens
import Mathlib.CategoryTheory.Adjunction.Unique
import Mathlib.CategoryTheory.Func... | Mathlib/Topology/Sheaves/Presheaf.lean | 413 | 420 | |
/-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Dagur Asgeirsson
-/
import Mathlib.Topology.Category.CompHaus.Basic
import Mathlib.Topology.Category.CompHausLike.Limits
/-!
# Explicit limits and colimits
This file applies ... | Mathlib/Topology/Category/CompHaus/Limits.lean | 66 | 68 | |
/-
Copyright (c) 2018 Guy Leroy. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sangwoo Jo (aka Jason), Guy Leroy, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.GroupWithZero.Semiconj
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Data.Nat.GCD.Bas... |
theorem gcd_mul_lcm (i j : ℤ) : gcd i j * lcm i j = natAbs (i * j) := by
rw [Int.gcd, Int.lcm, Nat.gcd_mul_lcm, natAbs_mul]
theorem gcd_comm (i j : ℤ) : gcd i j = gcd j i :=
Nat.gcd_comm _ _
theorem gcd_assoc (i j k : ℤ) : gcd (gcd i j) k = gcd i (gcd j k) :=
Nat.gcd_assoc _ _ _
| Mathlib/Data/Int/GCD.lean | 190 | 199 |
/-
Copyright (c) 2022 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
import Mathlib.Algebra.Category.Grp.EquivalenceGroupAddGroup
import Mathlib.CategoryTheory.ConcreteCategory.EpiMono
import Mathlib.CategoryTheory.Limits.Constructions.Epi... | f.ker = ⊥ := by simpa using congr_arg range (h f.ker.subtype 1 (by aesop_cat))
| Mathlib/Algebra/Category/Grp/EpiMono.lean | 37 | 38 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Function.L1Space.Integrable
import Mathlib.MeasureTheory.Function.LpSpace.Indicator
/-! # Functions integrable on a set an... | rw [← s.toSet.biUnion_of_singleton]
simp [integrableOn_finset_iUnion, measure_lt_top]
lemma IntegrableOn.of_finite [MeasurableSingletonClass α] {μ : Measure α} [IsFiniteMeasure μ]
{s : Set α} (hs : s.Finite) {f : α → E} : IntegrableOn f s μ := by
simpa using IntegrableOn.finset (s := hs.toFinset)
| Mathlib/MeasureTheory/Integral/IntegrableOn.lean | 196 | 202 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Order.Filter.SmallSets
import Mathlib.Topology.UniformSpace.Defs
import Mathlib.Topology.ContinuousOn
/-!
# Basic resu... | instance instUniformSpaceSubtype {p : α → Prop} [t : UniformSpace α] : UniformSpace (Subtype p) :=
UniformSpace.comap Subtype.val t
theorem uniformity_subtype {p : α → Prop} [UniformSpace α] :
𝓤 (Subtype p) = comap (fun q : Subtype p × Subtype p => (q.1.1, q.2.1)) (𝓤 α) :=
rfl
| Mathlib/Topology/UniformSpace/Basic.lean | 575 | 580 |
/-
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.Nontrivial.Basic
import Mathlib.Order.TypeTags
import Mathlib.Data.Option.NAry
import Mathlib.Tactic.Contrapose
import Mathlib.Tactic.Lift
import... |
@[simp, norm_cast] lemma coe_lt_coe : (a : WithBot α) < b ↔ a < b := by simp [lt_def]
@[simp] lemma bot_lt_coe (a : α) : ⊥ < (a : WithBot α) := by simp [lt_def]
| Mathlib/Order/WithBot.lean | 249 | 251 |
/-
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... | rintro ⟨g, hg, rfl⟩
choose t ht hg using @hg
refine ⟨@t, @ht, ?_⟩
simp_rw [hg]
theorem convex_parallelepiped (v : ι → E) : Convex ℝ (parallelepiped v) := by
rw [parallelepiped_eq_sum_segment]
exact convex_sum _ fun _i _hi => convex_segment _ _
/-- A `parallelepiped` is the convex hull of its vertices -/
t... | Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean | 128 | 138 |
/-
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... | refine ⟨‖c‖ / ε, this, fun x ↦ ?_⟩
by_cases hx : ‖x‖ = 0
· rw [hx, mul_zero]
exact le_of_eq (map_eq_zero_of_norm_zero q hq hx)
· refine (normSeminorm 𝕜 F).bound_of_shell q ε_pos hc (fun x hle hlt ↦ ?_) hx
refine (le_of_lt <| show q x < _ from hε hlt).trans ?_
rwa [← div_le_iff₀' this, one_div_div]
... | Mathlib/Analysis/LocallyConvex/WithSeminorms.lean | 757 | 767 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro
-/
import Mathlib.Algebra.Module.Submodule.Bilinear
import Mathlib.Algebra.Module.Equiv.Basic
import Mathlib.GroupTheory.Congruence.Hom
import Mathlib.Tactic.Abel
... | left_inv x := x.induction_on (map_zero _) (fun _ _ ↦ rfl)
fun _ _ h h' ↦ by simpa using congr($h + $h')
right_inv x := x.induction_on (map_zero _) (fun _ _ ↦ rfl)
fun _ _ h h' ↦ by simpa using congr($h + $h')
| Mathlib/LinearAlgebra/TensorProduct/Basic.lean | 684 | 687 |
/-
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 | 552 | 555 | |
/-
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.GammaCompN
import Mathlib.AlgebraicTopology.DoldKan.NReflectsIso
/-! The unit isomorphism of the Dold-Kan equivalence
In order to con... | rw [hom_comp_eq_id] at h
rw [h]
infer_instance
exact isIso_of_reflects_iso _ N₂
apply NatIso.isIso_of_isIso_app
instance : IsIso (Γ₂N₁.natTrans : (N₁ : SimplicialObject C ⥤ _) ⋙ _ ⟶ _) := by
have : ∀ X : SimplicialObject C, IsIso (Γ₂N₁.natTrans.app X) := by
intro X
rw [compatibility_Γ... | Mathlib/AlgebraicTopology/DoldKan/NCompGamma.lean | 228 | 249 |
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel, Kim Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
/-!
# The abelian image and coimage.
In an abelian category we usually want the image of a morphism ... | Mathlib/CategoryTheory/Abelian/Images.lean | 122 | 123 | |
/-
Copyright (c) 2024 Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Merten
-/
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.CategoryTheory.Limits.Preserves.Limits
import Mathlib.CategoryTheory.Limits.Yoneda
/-!
# Ind- and ... | rw [coyonedaOpColimitIsoLimitCoyoneda_hom_comp_π]
@[reassoc (attr := simp)]
lemma colimitHomIsoLimitYoneda_inv_comp_π [HasLimitsOfShape Iᵒᵖ (Type u₂)] (A : C) (i : I) :
| Mathlib/CategoryTheory/Limits/IndYoneda.lean | 75 | 78 |
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Sébastien Gouëzel, Yury Kudryashov, Eric Wieser
-/
import Mathlib.Analysis.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calculus.FDeriv.Comp
/-!
# Derivative of the ca... | ∀ i, HasStrictFDerivAt (φ i) (φ' i) x :=
hasStrictFDerivAt_pi'
@[simp]
| Mathlib/Analysis/Calculus/FDeriv/Prod.lean | 400 | 403 |
/-
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]
theorem ofStream_append (l : List α) (s : Stream' α) :
| Mathlib/Data/Seq/Seq.lean | 1,000 | 1,001 |
/-
Copyright (c) 2022 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.Algebra.Module.ZLattice.Basic
import Mathlib.Analysis.InnerProductSpace.ProdL2
import Mathlib.MeasureTheory.Measure.Haar.Unique
import Mathlib.NumberTheo... | variable {K}
open scoped Classical in
/-- The norm at the infinite place `w` of an element of the mixed space -/
| Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean | 314 | 317 |
/-
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.Data.Finsupp.Basic
import Mathlib.Data.List.AList
/-!
# Connections between `Finsupp` and `AList`
## Main definitions
* `Fin... | Mathlib/Data/Finsupp/AList.lean | 124 | 132 | |
/-
Copyright (c) 2023 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Algebra.Group.Defs
import Batteries.Data.List.Basic
/-!
# Levenshtein distances
We define the Levenshtein edit distance `levenshtein C xy ys` between two... |
theorem suffixLevenshtein_length (xs : List α) (ys : List β) :
(suffixLevenshtein C xs ys).1.length = xs.length + 1 := by
induction ys with
| nil =>
dsimp [suffixLevenshtein]
induction xs with
| nil => rfl
| cons _ xs ih =>
simp_all
| cons y ys ih =>
dsimp [suffixLevenshtein]
rw... | Mathlib/Data/List/EditDistance/Defs.lean | 164 | 176 |
/-
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.OuterMeasure.Induced
import Mathlib.MeasureTheory.OuterMeasure.AE
import Mathlib.Order.Filter.CountableInter
/-!
# Measu... | theorem measure_eq_iInf (s : Set α) : μ s = ⨅ (t) (_ : s ⊆ t) (_ : MeasurableSet t), μ t := by
rw [measure_eq_trim, OuterMeasure.trim_eq_iInf, μ.coe_toOuterMeasure]
| Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean | 161 | 162 |
/-
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
-/
import Batteries.Tactic.Congr
import Mathlib.Data.Option.Basic
import Mathlib.Data.Prod.Basic
import Mathlib.Data.Set.Subsingleton
import Mathlib.Dat... | rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
theorem range_eq_empty [IsEmpty ι] (f : ι → α) : range f = ∅ :=
range_eq_empty_iff.2 ‹_›
| Mathlib/Data/Set/Image.lean | 667 | 671 |
/-
Copyright (c) 2023 Xavier Généreux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Généreux
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
import Mathlib.Analysis.Complex.PhragmenLindelof
/-!
# Hadamard three-lines Theorem
In this file we present a ... | rw [← ofReal_add, ofReal_ne_zero]
simp only [ne_eq, ne_of_gt (sSupNormIm_eps_pos f hε 0), not_false_eq_true]
· apply Differentiable.const_cpow (Differentiable.neg differentiable_id')
apply Or.inl
rw [← ofReal_add, ofReal_ne_zero]
exact (ne_of_gt (sSupNormIm_eps_pos f hε 1))
/-- If `f` is bounded ... | Mathlib/Analysis/Complex/Hadamard.lean | 124 | 132 |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Emily Riehl
-/
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryTheory.Functor.TwoSquare
import Mathlib.CategoryTheory.HomCongr
import Mathlib.Tactic.... | end mateEquivHComp
section mateEquivSquareComp
| Mathlib/CategoryTheory/Adjunction/Mates.lean | 199 | 202 |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Callum Sutton, Yury Kudryashov
-/
import Mathlib.Algebra.Group.Equiv.Basic
import Mathlib.Algebra.Group.Units.Hom
/-!
# Multiplicative and additive equivalence acting ... | /-- A multiplicative equivalence of monoids defines a multiplicative equivalence
of their groups of units. -/
def mapEquiv (h : M ≃* N) : Mˣ ≃* Nˣ :=
| Mathlib/Algebra/Group/Units/Equiv.lean | 35 | 37 |
/-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Dagur Asgeirsson
-/
import Mathlib.Topology.ExtremallyDisconnected
import Mathlib.Topology.Category.CompHaus.Projective
import Mathlib.Topology.Category.Profinite.Basic
/-!
# E... | Mathlib/Topology/Category/Stonean/Basic.lean | 310 | 319 | |
/-
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.GammaCompN
import Mathlib.AlgebraicTopology.DoldKan.NReflectsIso
/-! The unit isomorphism of the Dold-Kan equivalence
In order to con... | induction' Δ using SimplexCategory.rec with n
induction' Δ' using SimplexCategory.rec with n'
dsimp
-- We start with the case `i` is an identity
by_cases h : n = n'
· subst h
simp only [SimplexCategory.eq_id_of_mono i, Γ₀.Obj.Termwise.mapMono_id, op_id, X.map_id]
dsimp
simp only [id_comp, comp_i... | Mathlib/AlgebraicTopology/DoldKan/NCompGamma.lean | 83 | 124 |
/-
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.CategoryTheory.Comma.StructuredArrow.Basic
import Mathlib.CategoryTheory.Category.Cat
/-!
# The category of elements
This file defines the category of el... | (fromCostructuredArrow F).obj (op (CostructuredArrow.mk f)) = ⟨op X, yonedaEquiv.1 f⟩ :=
rfl
/-- The equivalence `F.Elementsᵒᵖ ≅ (yoneda, F)` given by yoneda lemma. -/
@[simps]
def costructuredArrowYonedaEquivalence (F : Cᵒᵖ ⥤ Type v) :
F.Elementsᵒᵖ ≌ CostructuredArrow yoneda F where
functor := toCostructu... | Mathlib/CategoryTheory/Elements.lean | 229 | 240 |
/-
Copyright (c) 2022 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Star.Basic
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Data.Set.Lattice.Image
import Mathlib.Algebra.Group.Pointwise.Set.Basic
/-!
# Poi... | simp only [← star_preimage]
| Mathlib/Algebra/Star/Pointwise.lean | 62 | 62 |
/-
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.Basic
/-!
## Smoothness of charts and local structomorphisms
We show that the model with corners,... | (contMDiffWithinAt_extChartAt_symm_target x hy).mono_of_mem_nhdsWithin
(extChartAt_target_mem_nhdsWithin_of_mem hy)
omit [IsManifold I n M] in
| Mathlib/Geometry/Manifold/ContMDiff/Atlas.lean | 113 | 116 |
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