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/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.RingTheory.Localization.Integer
import Mathlib.RingTheory.Localization.Submodule
/-!
# Fractional ideals
This file defines fractional... | end Semiring
| Mathlib/RingTheory/FractionalIdeal/Basic.lean | 584 | 585 |
/-
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 | 1,728 | 1,730 | |
/-
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 Mathlib.Data.Set.Operations
import Mathlib.Order.Basic
import Mathlib.Order.BooleanAlgebra
import Mathlib.Tactic.Tauto
import Mathlib.Tactic.B... | Mathlib/Data/Set/Basic.lean | 2,073 | 2,075 | |
/-
Copyright (c) 2019 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.List.FinRange
import Mathlib.Data.List.Perm.Basic
import Mathlib.Data.List.Lex
import Mathlib.Data.List.Induc... | theorem length_sublists (l : List α) : length (sublists l) = 2 ^ length l := by
simp only [sublists_eq_sublists', length_map, length_sublists', length_reverse]
theorem map_pure_sublist_sublists (l : List α) : map pure l <+ sublists l := by
induction' l using reverseRecOn with l a ih <;> simp only [map, map_append,... | Mathlib/Data/List/Sublists.lean | 169 | 173 |
/-
Copyright (c) 2021 Aaron Anderson, Jesse Michael Han, Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jesse Michael Han, Floris van Doorn
-/
import Mathlib.Data.Finset.Basic
import Mathlib.ModelTheory.Syntax
import Mathlib.Data.List.... | Mathlib/ModelTheory/Semantics.lean | 1,137 | 1,149 | |
/-
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.Data.Sum.Order
import Mathlib.Order.RelIso.Set
import Mathlib.Order.UpperLower.Basic
import Mathlib.Order... | noncomputable def InitialSeg.total (r s) [IsWellOrder α r] [IsWellOrder β s] :
(r ≼i s) ⊕ (s ≼i r) :=
match (leAdd r s).principalSumRelIso,
(RelEmbedding.sumLexInr r s).collapse.principalSumRelIso with
| Sum.inl f, Sum.inr g => Sum.inl <| f.transRelIso g.symm
| Sum.inr f, Sum.inl g => Sum.inr <| g.transRe... | Mathlib/Order/InitialSeg.lean | 535 | 541 |
/-
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... | Equiv.swap (fromCoset ⟨↑f.hom.range, ⟨1, one_leftCoset _⟩⟩) ∞
local notation "τ" => tau f
theorem τ_apply_infinity : τ ∞ = fromCoset ⟨f.hom.range, 1, one_leftCoset _⟩ :=
Equiv.swap_apply_right _ _
theorem τ_apply_fromCoset : τ (fromCoset ⟨f.hom.range, 1, one_leftCoset _⟩) = ∞ :=
| Mathlib/Algebra/Category/Grp/EpiMono.lean | 147 | 154 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.RingTheory.FractionalIdeal.Basic
import Mathlib.RingTheory.IntegralClosure.IsIntegral.Basi... | spanFinset R₁ s f ≠ 0 ↔ ∃ j ∈ s, f j ≠ 0 := by simp
open Submodule.IsPrincipal
variable [IsLocalization S P]
theorem isFractional_span_singleton (x : P) : IsFractional S (span R {x} : Submodule R P) :=
let ⟨a, ha⟩ := exists_integer_multiple S x
isFractional_span_iff.mpr ⟨a, a.2, fun _ hx' => (Set.mem_singlet... | Mathlib/RingTheory/FractionalIdeal/Operations.lean | 545 | 558 |
/-
Copyright (c) 2024 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Data.Set.Functor
import Mathlib.Order.Sublattice
import Mathlib.Order.Hom.CompleteLattice
/-!
# Complete Sublattices
This file defines complete sublattices... | rw [← map_sInf]
exact mem_image_of_mem f (sInfClosed ht)
@[simp] theorem mem_map {b : β} : b ∈ L.map f ↔ ∃ a ∈ L, f a = b := Iff.rfl
| Mathlib/Order/CompleteSublattice.lean | 125 | 128 |
/-
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, Yury Kudryashov
-/
import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
import Mathlib.Algebra.Order.Monoid.Unbundled.OrderDual
import Mathlib.Tactic.Lift... | Mathlib/Algebra/Order/Monoid/Unbundled/Pow.lean | 352 | 355 | |
/-
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
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Analysis.Special... | convert ((hasStrictFDerivAt_fst.log hp.ne').mul hasStrictFDerivAt_snd).exp using 1
rw [rpow_sub_one hp.ne', ← rpow_def_of_pos hp, smul_add, smul_smul, mul_div_left_comm,
div_eq_mul_inv, smul_smul, smul_smul, mul_assoc, add_comm]
/-- `(x, y) ↦ x ^ y` is strictly differentiable at `p : ℝ × ℝ` such that `p.fst < ... | Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean | 321 | 326 |
/-
Copyright (c) 2023 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Geißer, Michael Stoll
-/
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.DiophantineApproximation.Basic
import Mathlib.NumberTheory.Zsqrtd.Basic
import Mathlib.Tactic... | rw [H, sq (0 : ℤ), mul_zero, mul_zero, sub_zero, sq_eq_one_iff] at prop
exact prop.imp (fun h => ext h H) fun h => ext h H
/-- The set of solutions with `x > 0` is closed under multiplication. -/
theorem x_mul_pos {a b : Solution₁ d} (ha : 0 < a.x) (hb : 0 < b.x) : 0 < (a * b).x := by
| Mathlib/NumberTheory/Pell.lean | 226 | 230 |
/-
Copyright (c) 2021 Bryan Gin-ge Chen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Bryan Gin-ge Chen, Yaël Dillies
-/
import Mathlib.Order.BooleanAlgebra
import Mathlib.Logic.Equiv.Basic
/-!
# Symmetric difference and bi-implication
This file defines... | @[simp]
theorem sdiff_symmDiff_left : a \ a ∆ b = a ⊓ b := by simp [sdiff_symmDiff]
| Mathlib/Order/SymmDiff.lean | 360 | 361 |
/-
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.IsManifold.ExtChartAt
import Mathlib.Geometry.Manifold.LocalInvariantProperties
/-!
# `C^n` functions betwee... | refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩
· apply h.comp_of_eq
· exact (continuousAt_extChartAt_symm x).continuousWithinAt
· exact (mapsTo_preimage _ _).mono_left inter_subset_left
· exact extChartAt_to_inv x
· rw [← continuousWithinAt_inter (extChartAt_source_mem_nhds (I := I) x)]
| Mathlib/Geometry/Manifold/ContMDiff/Defs.lean | 319 | 324 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Control.Combinators
import Mathlib.Data.Option.Defs
import Mathlib.Logic.IsEmpty
import Mathlib.Logic.Relator
import Mathlib.Util.CompileInductive
impo... |
@[simp]
theorem orElse_none' (x : Option α) : x.orElse (fun _ ↦ none) = x := by cases x <;> rfl
| Mathlib/Data/Option/Basic.lean | 206 | 208 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.FreeAlgebra
import Mathlib.RingTheory.Adjoin.Polynomial
import Mathlib.RingTheory.Adjoin.Tower
import Mathlib.RingTheory.Ideal.Quotient.Operati... | end RingHom
namespace AlgHom
| Mathlib/RingTheory/FiniteType.lean | 262 | 264 |
/-
Copyright (c) 2021 Vladimir Goryachev. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Vladimir Goryachev, Kyle Miller, Kim Morrison, Eric Rodriguez
-/
import Mathlib.Data.List.GetD
import Mathlib.Data.Nat.Count
import Mathlib.Data.Nat.SuccPred
import M... | nth p m < nth p n :=
(setOf p).finite_or_infinite.elim (fun hf => nth_lt_nth_of_lt_card hf hlt (h _)) fun hf =>
(nth_lt_nth hf).2 hlt
theorem nth_le_nth' {m n : ℕ} (hle : m ≤ n) (h : ∀ hf : (setOf p).Finite, n < #hf.toFinset) :
nth p m ≤ nth p n :=
(setOf p).finite_or_infinite.elim (fun hf => nth_le_nt... | Mathlib/Data/Nat/Nth.lean | 183 | 190 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Mario Carneiro
-/
import Mathlib.Data.Subtype
import Mathlib.Order.Defs.LinearOrder
import Mathlib.Order.Notation
import Mathlib.Tactic.GCongr.Core
import Mathlib.Tactic.... | Mathlib/Order/Basic.lean | 1,325 | 1,332 | |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Kexing Ying, Moritz Doll
-/
import Mathlib.Algebra.GroupWithZero.Action.Opposite
import Mathlib.LinearAlgebra.Finsupp.VectorSpace
import Mathlib.LinearAlgebra.Matrix.Basis
im... | Mathlib/LinearAlgebra/Matrix/SesquilinearForm.lean | 712 | 714 | |
/-
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... | rw [or_comm]
exact isEmpty_prod
| Mathlib/SetTheory/Ordinal/Arithmetic.lean | 626 | 627 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Algebra.Field.NegOnePow
import Mathlib.Algebra.Field.Periodic
import Mathlib.Algebra.Qua... | @[simp]
noncomputable def sqrtTwoAddSeries (x : ℝ) : ℕ → ℝ
| 0 => x
| n + 1 => √(2 + sqrtTwoAddSeries x n)
theorem sqrtTwoAddSeries_zero : sqrtTwoAddSeries x 0 = x := by simp
theorem sqrtTwoAddSeries_one : sqrtTwoAddSeries 0 1 = √2 := by simp
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | 616 | 624 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yakov Pechersky
-/
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Infix
import Mathlib.Data.Quot
/-!
# List rotation
This file proves basic results about `List.r... |
protected theorem IsRotated.cyclicPermutations {l l' : List α} (h : l ~r l') :
l.cyclicPermutations ~r l'.cyclicPermutations := by
obtain ⟨k, rfl⟩ := h
exact ⟨k, by simp⟩
@[simp]
theorem isRotated_cyclicPermutations_iff {l l' : List α} :
| Mathlib/Data/List/Rotate.lean | 597 | 604 |
/-
Copyright (c) 2024 Colva Roney-Dougal. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Colva Roney-Dougal, Inna Capdeboscq, Susanna Fishel, Kim Morrison
-/
import Mathlib.Order.Atoms
/-!
# The radical of a lattice
This file contains results on the order radical of ... | unfold Order.radical
simp only [OrderIso.map_iInf]
fapply Equiv.iInf_congr
· exact f.toEquiv
· simp
theorem Order.radical_nongenerating [IsCoatomic α] {a : α} (h : a ⊔ radical α = ⊤) : a = ⊤ := by
| Mathlib/Order/Radical.lean | 30 | 36 |
/-
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.Algebra.ModEq
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Algebra.Ring.Periodic
import Mathlib.Data.Int.SuccPred
import Mathlib.Order.Cir... | @[simp]
theorem toIocMod_toIocMod (a₁ a₂ b : α) : toIocMod hp a₁ (toIocMod hp a₂ b) = toIocMod hp a₁ b :=
(toIocMod_eq_toIocMod _).2 ⟨toIocDiv hp a₂ b, self_sub_toIocMod hp a₂ b⟩
@[simp]
theorem toIocMod_toIcoMod (a₁ a₂ b : α) : toIocMod hp a₁ (toIcoMod hp a₂ b) = toIocMod hp a₁ b :=
(toIocMod_eq_toIocMod _).2 ⟨to... | Mathlib/Algebra/Order/ToIntervalMod.lean | 620 | 644 |
/-
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.Basic
import Mathlib.Analysis.Normed.Operator.BoundedLinearMaps
/-!
# The derivative of bou... | e.hasFDerivAtFilter
| Mathlib/Analysis/Calculus/FDeriv/Linear.lean | 54 | 55 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.NaturalTransformation
import Mathlib.CategoryTheory.Monoidal.Opposite
import Mathli... | rw [Iso.inv_ext]
rw [braiding_tensorUnit_left]
| Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean | 327 | 328 |
/-
Copyright (c) 2021 Justus Springer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Justus Springer
-/
import Mathlib.CategoryTheory.Sites.Spaces
import Mathlib.Topology.Sheaves.Sheaf
import Mathlib.CategoryTheory.Sites.DenseSubsite.Basic
/-!
# Coverings and sieves... | intro x hxU
rw [Opens.coe_iSup, Set.mem_iUnion]
obtain ⟨V, iVU, ⟨W, iVW, iWU, hiWU, -⟩, hxV⟩ := hR x hxU
exact ⟨⟨W, ⟨iWU, hiWU⟩⟩, iVW.le hxV⟩
end coveringOfPresieve
/-- Given a family of opens `U : ι → Opens X` and any open `Y : Opens X`, we obtain a presieve
on `Y` by declaring that a morphism `f : V ⟶ Y` is... | Mathlib/Topology/Sheaves/SheafCondition/Sites.lean | 58 | 67 |
/-
Copyright (c) 2023 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Yaël Dillies
-/
import Mathlib.Analysis.SpecialFunctions.Exponential
import Mathlib.Data.Complex.Trigonometric
/-!
# Trigonometric functions as sums of infinite series
In t... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Series.lean | 161 | 167 | |
/-
Copyright (c) 2023 Shogo Saito. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Shogo Saito. Adapted for mathlib by Hunter Monroe
-/
import Mathlib.Data.Nat.ModEq
import Mathlib.Data.Nat.ChineseRemainder
import Mathlib.Data.Nat.Prime.Defs
import Mathlib.Data.Nat.Pai... | le_trans (self_le_factorial _) (le_trans (Nat.le_mul_of_pos_left (supOfSeq a)! (succ_pos i))
(le_add_right _ _))
simpa only [coprimes] using lt_of_lt_of_le h₁ h₂
open scoped Function in -- required for scoped `on` notation
private lemma pairwise_coprime_coprimes (a : Fin m → ℕ) : Pairwise (Coprime on copri... | Mathlib/Logic/Godel/GodelBetaFunction.lean | 71 | 81 |
/-
Copyright (c) 2023 Lean FRO LLC. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Comon_
import Mathlib.CategoryTheory.Monoidal.OfHasFiniteProducts
/-!
# Comonoid objects in a cartesian monoidal category.
The ca... | @[simp] theorem comul_eq_diag (A : Comon_ C) : A.comul = diag A.X := by
ext
· simpa using A.comul_counit =≫ prod.fst
· simpa using A.counit_comul =≫ prod.snd
| Mathlib/CategoryTheory/Monoidal/Cartesian/Comon_.lean | 43 | 46 |
/-
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, Jeremy Avigad
-/
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Data.Set.Finite.Range
import Mathlib.Data.Set.Lattice
import Mathlib.Topology.Defs.... | Mathlib/Topology/Basic.lean | 1,826 | 1,829 | |
/-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | ofReal_cpow zero_le_two, show (2 : ℝ) = (2 : ℂ) by norm_cast, ← cpow_neg, ofReal_sub,
ofReal_one, neg_sub, ← div_eq_mul_inv]
end Complex
namespace Real
open Complex
theorem Gamma_mul_Gamma_add_half (s : ℝ) :
Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * (2 : ℝ) ^ (1 - 2 * s) * √π := by
rw [← ofReal_in... | Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean | 571 | 601 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.UniformSpace.Cauchy
/-!
# Uniform convergence
A sequence of functions `Fₙ` (with values in a metric space) converges uniformly on a se... | Mathlib/Topology/UniformSpace/UniformConvergence.lean | 645 | 646 | |
/-
Copyright (c) 2022 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.NumberTheory.LegendreSymbol.AddCharacter
import Mathlib.NumberTheory.LegendreSymbol.ZModChar
import Mathlib.Algebra.CharP.CharAndCard
/-!
# Gauss sums
... | (hψ : IsPrimitive ψ) :
gaussSum χ ψ * gaussSum χ⁻¹ ψ⁻¹ = Fintype.card R := by
simp only [gaussSum, AddChar.inv_apply, Finset.sum_mul, Finset.mul_sum, MulChar.inv_apply']
conv =>
enter [1, 2, x, 2, y]
rw [mul_mul_mul_comm, ← map_mul, ← map_add_eq_mul, ← sub_eq_add_neg]
-- conv in _ * _ * (_ * _) => ... | Mathlib/NumberTheory/GaussSum.lean | 126 | 132 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Patrick Massot, Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic
import Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalcul... | Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | 129 | 131 | |
/-
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.MeasureTheory.Constructions.BorelSpace.Order
import Mathlib.MeasureTheory.MeasurableSpace.Prod
import Mathlib.MeasureTheory.Measure.Ty... | convert isPiSystem_image_Iic (((↑) : ℚ → ℝ) '' univ)
ext x
simp only [iUnion_singleton_eq_range, mem_range, image_univ, mem_image, exists_exists_eq_and]
| Mathlib/MeasureTheory/Constructions/BorelSpace/Real.lean | 101 | 104 |
/-
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 Mathlib.Algebra.Notation.Defs
import Mathlib.Data.Set.Subsingleton
import Mathlib.Logic.Equiv.Defs
/-!
# Partial values of a type
... |
@[simp] lemma toOption_eq_none (o : Part α) [Decidable o.Dom] : o.toOption = none ↔ ¬o.Dom := by
| Mathlib/Data/Part.lean | 68 | 69 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Wrenna Robson
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Pi
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.LinearAlgebra.Vandermonde
import Mathlib.RingT... | theorem nodalWeight_ne_zero (hvs : Set.InjOn v s) (hi : i ∈ s) : nodalWeight s v i ≠ 0 := by
rw [nodalWeight, prod_ne_zero_iff]
intro j hj
| Mathlib/LinearAlgebra/Lagrange.lean | 569 | 571 |
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel, Joël Riou
-/
import Mathlib.Algebra.Homology.ExactSequence
import Mathlib.CategoryTheory.Abelian.Refinements
/-!
# The four and five lemmas
Consider the following commu... |
end Four
section Five
variable {R₁ R₂ : ComposableArrows C 4} (hR₁ : R₁.Exact) (hR₂ : R₂.Exact) (φ : R₁ ⟶ R₂)
| Mathlib/CategoryTheory/Abelian/DiagramLemmas/Four.lean | 123 | 128 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Topology.Algebra.InfiniteSum.Order
import Mathlib.Topology.Instances.ENNReal.Lemma... | simpa using mod_cast NNReal.not_summable_iff_tendsto_nat_atTop
theorem summable_iff_not_tendsto_nat_atTop_of_nonneg {f : ℕ → ℝ} (hf : ∀ n, 0 ≤ f n) :
| Mathlib/Topology/Algebra/InfiniteSum/Real.lean | 60 | 62 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Kevin Buzzard, Yury Kudryashov, Eric Wieser
-/
import Mathlib.Algebra.Algebra.Prod
import Mathlib.Algebra.Group.Graph
import Mathlib.LinearAlgebra.Span.... | (M →ₗ[R] M₃) × (M₂ →ₗ[R] M₄) →ₗ[S] M × M₂ →ₗ[R] M₃ × M₄ where
toFun f := prodMap f.1 f.2
map_add' _ _ := rfl
map_smul' _ _ := rfl
| Mathlib/LinearAlgebra/Prod.lean | 332 | 335 |
/-
Copyright (c) 2021 Jordan Brown, Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jordan Brown, Thomas Browning, Patrick Lutz
-/
import Mathlib.GroupTheory.Abelianization
import Mathlib.GroupTheory.Perm.ViaEmbedding
import Mathlib.GroupT... | end Solvable
section IsSimpleGroup
variable [IsSimpleGroup G]
theorem IsSimpleGroup.derivedSeries_succ {n : ℕ} : derivedSeries G n.succ = commutator G := by
| Mathlib/GroupTheory/Solvable.lean | 178 | 184 |
/-
Copyright (c) 2024 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.Data.Matroid.Constructions
import Mathlib.Data.Set.Notation
/-!
# Maps between matroids
This file defines maps and comaps, which move a matroid on one ty... | def restrictSubtype (M : Matroid α) (X : Set α) : Matroid X := (M ↾ X).comap (↑)
| Mathlib/Data/Matroid/Map.lean | 640 | 641 |
/-
Copyright (c) 2020 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Analysis.Asymptotics.Theta
/-!
# Asymptotic equivalence
In this file, we define the relation `IsEquivalent l u v`, which means that `u-v` is litt... | theorem IsEquivalent.tendsto_const {c : β} (hu : u ~[l] const _ c) : Tendsto u l (𝓝 c) := by
rcases em <| c = 0 with rfl | h
· exact (tendsto_congr' <| isEquivalent_zero_iff_eventually_zero.mp hu).mpr tendsto_const_nhds
· exact (isEquivalent_const_iff_tendsto h).mp hu
| Mathlib/Analysis/Asymptotics/AsymptoticEquivalent.lean | 133 | 136 |
/-
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... | rw [← IsGalois.card_aut_eq_finrank,
Finset.card_eq_sum_card_fiberwise (f := (comap · (algebraMap k K)))
(t := ({w : InfinitePlace k | w.IsUnramifiedIn K}: Finset _)ᶜ), ← smul_eq_mul, ← sum_const]
· refine sum_congr rfl (fun w hw ↦ ?_)
obtain ⟨w, rfl⟩ := comap_surjective (K := K) w
simp only [mem_uni... | Mathlib/NumberTheory/NumberField/Embeddings.lean | 1,029 | 1,034 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Set.NAry
import Mathlib.Order.SupClosed
import Mathlib.Order.UpperLower.Closure
/-!
# Set family operations
This file defines a few binary operation... | UpperSet.coe_sup, Set.mem_inter_iff]
constructor
| Mathlib/Data/Set/Sups.lean | 346 | 347 |
/-
Copyright (c) 2020 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Algebra.Group.Conj
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.Group.Subgroup.Ker
/-!
# Basic results on subgroups
We prove basic results... | ⟨fun _ => by simp +contextual [Subgroup.mem_comap, hH.conj_mem]⟩
@[to_additive]
| Mathlib/Algebra/Group/Subgroup/Basic.lean | 785 | 787 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot, Yury Kudryashov, Rémy Degenne
-/
import Mathlib.Data.Set.Subsingleton
import Mathlib.Order.Interval.Set.Defs
/-!
# Intervals
In any pr... | Mathlib/Order/Interval/Set/Basic.lean | 1,882 | 1,884 | |
/-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.Ring.Parity
import Mathlib.Combinatorics.SimpleGraph.Path
/-!
# Trails and Eulerian trails
This module contains additional theory about trails, in... | Mathlib/Combinatorics/SimpleGraph/Trails.lean | 163 | 169 | |
/-
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.Algebra.Group.Pi.Basic
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.Images
import Mathlib.CategoryTheor... | Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean | 308 | 308 | |
/-
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.Data.List.Lemmas
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Data.Li... | theorem permutations_append (is ts : List α) :
permutations (is ++ ts) = (permutations is).map (· ++ ts) ++ permutationsAux ts is.reverse := by
simp [permutations, permutationsAux_append]
| Mathlib/Data/List/Permutation.lean | 244 | 246 |
/-
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... | theorem ContDiff.continuous_fderiv_apply (h : ContDiff 𝕜 n f) (hn : 1 ≤ n) :
Continuous fun p : E × E => (fderiv 𝕜 f p.1 : E → F) p.2 :=
have A : Continuous fun q : (E →L[𝕜] F) × E => q.1 q.2 := isBoundedBilinearMap_apply.continuous
have B : Continuous fun p : E × E => (fderiv 𝕜 f p.1, p.2) :=
((h.conti... | Mathlib/Analysis/Calculus/ContDiff/Defs.lean | 1,223 | 1,231 |
/-
Copyright (c) 2023 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Order.ConditionallyCompleteLattice.Indexed
import Mathlib.Order.SuccPred.Limit
/-!
# Relation between `IsSuccPrelimit` and `iSup` in (conditionally) comple... | lemma csSup_mem_of_not_isSuccPrelimit
(hne : s.Nonempty) (hbdd : BddAbove s) (hlim : ¬ IsSuccPrelimit (sSup s)) : sSup s ∈ s := by
obtain ⟨y, hy⟩ := not_forall_not.mp hlim
obtain ⟨i, his, hi⟩ := exists_lt_of_lt_csSup hne hy.lt
exact eq_of_le_of_not_lt (le_csSup hbdd his) (hy.2 hi) ▸ his
| Mathlib/Order/SuccPred/CompleteLinearOrder.lean | 22 | 27 |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Batteries.Data.List.Perm
import Mathlib.Data.List.OfFn
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.TakeWhile
import Mathlib.Order.Fin.Basic
/-!
# So... | Pairwise.nodup h
protected theorem Sorted.filter {l : List α} (f : α → Bool) (h : Sorted r l) :
Sorted r (filter f l) :=
h.sublist filter_sublist
theorem eq_of_perm_of_sorted [IsAntisymm α r] {l₁ l₂ : List α} (hp : l₁ ~ l₂) (hs₁ : Sorted r l₁)
(hs₂ : Sorted r l₂) : l₁ = l₂ := by
induction hs₁ generalizi... | Mathlib/Data/List/Sort.lean | 104 | 120 |
/-
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.Algebra.Exact
import Mathlib.RingTheory.AdicCompletion.Functoriality
import Mathlib.RingTheory.Filtration
/-!
# Exac... | Function.Injective (map I f) := by
obtain ⟨k, hk⟩ := Ideal.exists_pow_inf_eq_pow_smul I (range f)
rw [← LinearMap.ker_eq_bot, LinearMap.ker_eq_bot']
intro x
apply AdicCompletion.induction_on I M x (fun a ↦ ?_)
intro hx
refine AdicCompletion.mk_zero_of _ _ _ ⟨42, fun n _ ↦ ⟨n + k, by omega, n, by omega, ... | Mathlib/RingTheory/AdicCompletion/Exactness.lean | 91 | 104 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Sébastien Gouëzel, Patrick Massot
-/
import Mathlib.Topology.UniformSpace.Cauchy
import Mathlib.Topology.UniformSpace.Separation
import Mathlib.Topology.DenseEmbedding
... | IsComplete s ↔ IsComplete ((↑) '' s : Set α) :=
isUniformEmbedding_subtype_val.isComplete_iff.symm
alias ⟨isComplete_of_complete_image, _⟩ := isComplete_image_iff
theorem completeSpace_iff_isComplete_range {f : α → β} (hf : IsUniformInducing f) :
| Mathlib/Topology/UniformSpace/UniformEmbedding.lean | 297 | 302 |
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Logic.Encodable.Pi
import Mathlib.MeasureTheory.Group.Measure
import Mathlib.MeasureTheory.MeasurableSpace.... | variable (μ)
@[to_additive]
instance pi.isMulLeftInvariant [∀ i, Group (α i)] [∀ i, MeasurableMul (α i)]
[∀ i, IsMulLeftInvariant (μ i)] : IsMulLeftInvariant (Measure.pi μ) := by
refine ⟨fun v => (pi_eq fun s hs => ?_).symm⟩
rw [map_apply (measurable_const_mul _) (MeasurableSet.univ_pi hs),
show (v * ·) ⁻¹... | Mathlib/MeasureTheory/Constructions/Pi.lean | 511 | 531 |
/-
Copyright (c) 2022 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.Data.ENNReal.Lemmas
import Mathlib.Topology.MetricSpace.Thickening
import Mathlib.Topology.ContinuousMap.Bounded.Basic
/-!
# Thickened indicators
This fi... |
variable [TopologicalSpace β]
/-- The multiplicative indicators of δ-thickenings of a set tend pointwise to the multiplicative
indicator of the set, as δ>0 tends to zero. -/
@[to_additive "The indicators of δ-thickenings of a set tend pointwise to the indicator of the
set, as δ>0 tends to zero."]
lemma tendsto_mulInd... | Mathlib/Topology/MetricSpace/ThickenedIndicator.lean | 266 | 279 |
/-
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.Constructions
import Mathlib.Tactic.FunProp
/-!
# Measurable embeddings and equivalences
A measurable e... | rw [← coe_toEquiv, Equiv.image_preimage]
@[simp]
| Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean | 308 | 310 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Fintype.EquivFin
import Mathlib.Data.List.MinMax
import Mathlib.Data.Nat.Order.Lemmas
import Mathlib.Logic.Encodable.Basic
/-!
# Denumerable type... | ofEquiv _ Equiv.ulift
/-- The lift of a denumerable type is denumerable. -/
| Mathlib/Logic/Denumerable.lean | 161 | 163 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yaël Dillies
-/
import Mathlib.Logic.Equiv.Set
import Mathlib.Order.CompleteLattice.Lemmas
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection.Basic
/-... |
/-- Turn minimal axioms for `CompleteDistribLattice` into minimal axioms for `Order.Coframe`. -/
| Mathlib/Order/CompleteBooleanAlgebra.lean | 226 | 227 |
/-
Copyright (c) 2022 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Eisenstein.Criterion
import Mathlib.RingTheory.Polynomial.ScaleRoots
/-!
# Eisenstein polynomials
Given an ideal `𝓟` of a com... | rw [IsRoot.def, eval_eq_sum_range, Finset.range_add_one,
Finset.sum_insert Finset.not_mem_range_self, Finset.sum_range, hmo.coeff_natDegree, one_mul] at
*
rw [eq_neg_of_add_eq_zero_left hroot, Ideal.neg_mem_iff]
exact Submodule.sum_mem _ fun i _ => mul_mem_right _ _ (hf.mem (Fin.is_lt i))
theorem pow_nat... | Mathlib/RingTheory/Polynomial/Eisenstein/Basic.lean | 128 | 138 |
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Logic.Encodable.Pi
import Mathlib.MeasureTheory.Group.Measure
import Mathlib.MeasureTheory.MeasurableSpace.... | (Measure.pi fun i ↦ μ (e i)).map (MeasurableEquiv.piCongrLeft (fun i ↦ β i) e)
= Measure.pi μ := by
let e_meas : ((b : ι) → β (e b)) ≃ᵐ ((a : ι') → β a) :=
| Mathlib/MeasureTheory/Constructions/Pi.lean | 397 | 399 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Sébastien Gouëzel,
Rémy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
/-!
# Power function... |
theorem rpow_left_surjective {x : ℝ} (hx : x ≠ 0) : Function.Surjective fun y : ℝ≥0∞ => y ^ x :=
| Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | 929 | 930 |
/-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Topology.Coherent
import Mathlib.Topology.UniformSpace.Equiv
import Mathlib.Topology.UniformSpace.Pi
import Mathlib.Topology.UniformSpace.UniformAp... | _ = _
simp_rw [UniformFun]
rw [← uniformity_comap]
congr
unfold instUniformSpaceProd
| Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | 468 | 472 |
/-
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... | @[simp]
theorem head!_append [Inhabited α] (t : List α) {s : List α} (h : s ≠ []) :
| Mathlib/Data/List/Basic.lean | 427 | 428 |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Algebra.Ring.Int.Defs
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.Size
import Batteries.Data.Int
/-!
# Bitwise operations on integers
Possi... | Mathlib/Data/Int/Bitwise.lean | 469 | 469 | |
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Topology.Bases
import Mathlib.Topology.Separation.Regular
/-!
# Dense embeddings
This file defines three properties of f... | exact le_trans lim1 lim2
protected theorem nhdsWithin_neBot (di : IsDenseInducing i) (b : β) : NeBot (𝓝[range i] b) :=
di.dense.nhdsWithin_neBot b
theorem comap_nhds_neBot (di : IsDenseInducing i) (b : β) : NeBot (comap i (𝓝 b)) :=
comap_neBot fun s hs => by
rcases mem_closure_iff_nhds.1 (di.dense b) s hs... | Mathlib/Topology/DenseEmbedding.lean | 117 | 124 |
/-
Copyright (c) 2021 Arthur Paulino. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Arthur Paulino, Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Clique
import Mathlib.Data.ENat.Lattice
import Mathlib.Data.Nat.Lattice
import Mathlib.Data.Setoid.Partition
imp... | by_contra h'
simp only [not_le] at h'
obtain ⟨i, hi⟩ := hm.some (Classical.arbitrary V)
have h₁ : i < 0 := lt_of_lt_of_le hi (Nat.le_of_lt_succ h')
exact Nat.not_lt_zero _ h₁
| Mathlib/Combinatorics/SimpleGraph/Coloring.lean | 306 | 311 |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Kim Morrison
-/
import Mathlib.CategoryTheory.Subobject.MonoOver
import Mathlib.CategoryTheory.Skeletal
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathlib.... | simp only [ofMkLE, ofLEMk, ofLE, ofMkLEMk, ← Functor.map_comp_assoc underlying, assoc]
congr 1
@[reassoc (attr := simp)]
| Mathlib/CategoryTheory/Subobject/Basic.lean | 385 | 388 |
/-
Copyright (c) 2021 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Topology.Constructions
import Mathlib.Topology.Order.OrderClosed
/-!
# Topological lattices
In this file we define mixin classes `ContinuousI... | lemma Continuous.finset_sup (hs : ∀ i ∈ s, Continuous (f i)) : Continuous (s.sup f) :=
continuous_iff_continuousAt.2 fun _ ↦ ContinuousAt.finset_sup fun i hi ↦ (hs i hi).continuousAt
| Mathlib/Topology/Order/Lattice.lean | 311 | 313 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
import Mathlib.Analy... | end
theorem hasStrictFDerivAt_norm_sq (x : F) :
| Mathlib/Analysis/InnerProductSpace/Calculus.lean | 190 | 192 |
/-
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... |
end DivisionSemiring
| Mathlib/Algebra/Field/Basic.lean | 80 | 82 |
/-
Copyright (c) 2020 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Bhavik Mehta, Dagur Asgeirsson
-/
import Mathlib.CategoryTheory.Monad.Limits
import Mathlib.Topology.StoneCech
import Mathlib.Topology.UrysohnsLemma
import Mathlib.Topology.Cat... | instance CompHaus.hasColimits : Limits.HasColimits CompHaus :=
hasColimits_of_reflective compHausToTop
namespace CompHaus
/-- An explicit limit cone for a functor `F : J ⥤ CompHaus`, defined in terms of
`TopCat.limitCone`. -/
def limitCone {J : Type v} [SmallCategory J] (F : J ⥤ CompHaus.{max v u}) : Limits.Cone F ... | Mathlib/Topology/Category/CompHaus/Basic.lean | 123 | 135 |
/-
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.CategoryTheory.Limits.Preserves.Shapes.Zero
/-!
# Preservation of biproducts
We define the... |
end Limits
end HasZeroMorphisms
end CategoryTheory
| Mathlib/CategoryTheory/Limits/Preserves/Shapes/Biproducts.lean | 409 | 416 |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Algebra.Order.Module.Defs
import Mathlib.Data.DFinsupp.Module
/-!
# Pointwise order on finitely supported dependent functions
This file lifts order struc... | end DFinsupp
| Mathlib/Data/DFinsupp/Order.lean | 297 | 299 |
/-
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... | · rw [dropWhile]
refine ⟨fun h => ?_, fun h => ?_⟩
· intro _ H
| Mathlib/Data/List/DropRight.lean | 131 | 133 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Finset.Card
import Mathlib.Data.Fintype.Basic
/-!
# Cardinalities of finite types
This file defines the cardinality `Fintype.card α` as the numb... | Mathlib/Data/Fintype/Card.lean | 627 | 628 | |
/-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Damiano Testa, Jens Wagemaker
-/
import Mathlib.Algebra.MonoidAlgebra.Division
import Mathlib.Algebra.Polynomial.Degree.Operations
import Mat... | ext
simpa only [coeff_divX, coeff_zero] using coeff_one
| Mathlib/Algebra/Polynomial/Inductions.lean | 65 | 66 |
/-
Copyright (c) 2018 Andreas Swerdlow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andreas Swerdlow
-/
import Mathlib.LinearAlgebra.Basis.Basic
import Mathlib.LinearAlgebra.BilinearMap
import Mathlib.LinearAlgebra.LinearIndependent.Lemmas
/-!
# Sesquilinear maps
... |
theorem isOrtho_zero_right (B : M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] M) (x) : IsOrtho B x (0 : M₂) :=
map_zero (B x)
| Mathlib/LinearAlgebra/SesquilinearForm.lean | 62 | 64 |
/-
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.Topology.PartialHomeomorph
import Mathlib.Topology.Connected.LocPathConnected
/-!
# Charted spaces
A smooth manifold is a topological space `M`... | id_mem := trivial
locality _ _ := trivial
congr _ _ _ := trivial
instance (H : Type*) [TopologicalSpace H] : Inhabited (Pregroupoid H) :=
| Mathlib/Geometry/Manifold/ChartedSpace.lean | 419 | 423 |
/-
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, Jeremy Avigad
-/
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Data.Set.Finite.Range
import Mathlib.Data.Set.Lattice
import Mathlib.Topology.Defs.... | Mathlib/Topology/Basic.lean | 1,722 | 1,724 | |
/-
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 | 782 | 790 | |
/-
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.Analysis.SpecialFunctions.ImproperIntegrals
import Mathlib.MeasureTheory.Integral.ExpDecay
/-!
# The Gamma function
This file defines the `Γ` functio... | rw [this, intervalIntegrable_iff_integrableOn_Ioc_of_le hY]
constructor
· refine (continuousOn_const.mul ?_).aestronglyMeasurable measurableSet_Ioc
apply (continuous_ofReal.comp continuous_neg.rexp).continuousOn.mul
| Mathlib/Analysis/SpecialFunctions/Gamma/Basic.lean | 169 | 172 |
/-
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.InnerProductSpace.Calculus
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Topology.MetricSpace.ProperSpace.Lemmas
/-!
# Eucl... |
variable {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] {G : Type*} [NormedAddCommGroup G]
[NormedSpace ℝ G] [FiniteDimensional ℝ G] {f g : F → G} {n : ℕ∞}
| Mathlib/Analysis/InnerProductSpace/EuclideanDist.lean | 108 | 110 |
/-
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.Covering.VitaliFamily
import Mathlib.MeasureTheory.Function.AEMeasurableOrder
import Mathlib.MeasureTheory.Integral.Average
import ... | above sketch. -/
let A := MeasureTheory.Measure.finiteSpanningSetsInOpen' μ
rcases h'f.isSeparable_range with ⟨t, t_count, ht⟩
have main :
∀ᵐ x ∂μ,
∀ᵉ (n : ℕ) (c ∈ t),
Tendsto (fun a => (∫⁻ y in a, ‖f y - (A.set n).indicator (fun _ => c) y‖ₑ ∂μ) / μ a)
(v.filterAt x) (𝓝 ‖f x - (A.... | Mathlib/MeasureTheory/Covering/Differentiation.lean | 780 | 792 |
/-
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.BigOperators.Expect
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Field.Canonical
import Mathlib.Algebra.O... | Mathlib/Data/NNReal/Basic.lean | 749 | 751 | |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Aurélien Saue, Anne Baanen
-/
import Mathlib.Tactic.NormNum.Inv
import Mathlib.Tactic.NormNum.Pow
import Mathlib.Util.AtomM
/-!
# `ring` tactic
A tactic for solving e... | OptionT Lean.Core.CoreM (Overlap sα q($a + $b)) := do
Lean.Core.checkSystem decl_name%.toString
| Mathlib/Tactic/Ring/Basic.lean | 326 | 327 |
/-
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 | 1,048 | 1,049 | |
/-
Copyright (c) 2023 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Algebra.Module.Card
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.SetTheory.Cardinal.Continuum
import Mathlib.SetTheory.Cardinal.C... | /-- A complete nontrivially normed field has cardinality at least continuum. -/
theorem continuum_le_cardinal_of_nontriviallyNormedField
(𝕜 : Type*) [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜] : 𝔠 ≤ #𝕜 := by
suffices ∃ f : (ℕ → Bool) → 𝕜, range f ⊆ univ ∧ Continuous f ∧ Injective f by
rcases this with... | Mathlib/Topology/Algebra/Module/Cardinality.lean | 29 | 45 |
/-
Copyright (c) 2022 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Jireh Loreaux
-/
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Star.Module
import Mathlib.Algebra.Star.NonUn... | (adjoin R (s : Set A)).toSubalgebra.toSubmodule = span R {1} ⊔ s.toSubmodule := by
rw [adjoin_eq_span, Submonoid.closure_eq_one_union, span_union,
← NonUnitalStarAlgebra.adjoin_eq_span, NonUnitalStarAlgebra.adjoin_eq]
theorem _root_.Subalgebra.starClosure_eq_adjoin (S : Subalgebra R A) :
S.starClosure = ... | Mathlib/Algebra/Star/Subalgebra.lean | 468 | 473 |
/-
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.Order.Ring.Nat
import Mathlib.Logic.Encodable.Pi
import Mathlib.Logic.Function.Iterate
/-!
# The primitive recursive functions
The primitive ... |
theorem list_rec {f : α → List β} {g : α → σ} {h : α → β × List β × σ → σ} (hf : Primrec f)
(hg : Primrec g) (hh : Primrec₂ h) :
@Primrec _ σ _ _ fun a => List.recOn (f a) (g a) fun b l IH => h a (b, l, IH) :=
let F (a : α) := (f a).foldr (fun (b : β) (s : List β × σ) => (b :: s.1, h a (b, s))) ([], g a)
h... | Mathlib/Computability/Primrec.lean | 890 | 914 |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.TwoDim
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
/-!
# Oriented angles.
This file defines orie... | /-- Subtracting the second vector passed to `oangle` from the first vector does not change the
sign of the angle. -/
@[simp]
| Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | 848 | 850 |
/-
Copyright (c) 2021 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Kim Morrison
-/
import Mathlib.Algebra.Field.Subfield.Defs
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Order.Group.Pointwise.Interval
impor... | section Preconnected
/-! Some results about functions on preconnected sets valued in a ring or field with a topology. -/
open Set
variable {α 𝕜 : Type*} {f g : α → 𝕜} {S : Set α} [TopologicalSpace α] [TopologicalSpace 𝕜]
[T1Space 𝕜]
| Mathlib/Topology/Algebra/Field.lean | 142 | 149 |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.CategoryTheory.Sites.Sieves
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.Mono
/-!
# The sheaf condition for a presieve
We define what it means fo... | simp only [op_id, FunctorToTypes.map_id_apply] at q₁ q₂
rw [q₁, q₂]
/-- We define `P` to be a sheaf for the presieve `R` if every compatible family has a unique
amalgamation.
| Mathlib/CategoryTheory/Sites/IsSheafFor.lean | 399 | 403 |
/-
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.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.Symmetric
/-!
# Integer powers of square matrices
In this file, we defi... | | -[n+1] => by rw [zpow_negSucc, zpow_negSucc, inv_pow']
@[simp]
theorem zpow_neg_one (A : M) : A ^ (-1 : ℤ) = A⁻¹ := by
| Mathlib/LinearAlgebra/Matrix/ZPow.lean | 92 | 95 |
/-
Copyright (c) 2020 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
import Mathlib.Analysis.NormedSpace.IndicatorFunction
import Mathlib.Data.Fintype.Order
import Mathlib.MeasureTheory.Function.AEEqFun
import Mathlib.Me... | @[gcongr, mono]
theorem eLpNorm_mono_measure (f : α → ε) (hμν : ν ≤ μ) : eLpNorm f p ν ≤ eLpNorm f p μ := by
by_cases hp0 : p = 0
· simp [hp0]
by_cases hp_top : p = ∞
· simp [hp_top, eLpNormEssSup_mono_measure f (Measure.absolutelyContinuous_of_le hμν)]
simp_rw [eLpNorm_eq_eLpNorm' hp0 hp_top]
| Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean | 615 | 621 |
/-
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 | 177 | 179 | |
/-
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.Additive
import Mathlib.Algebra.Homology.ShortComplex.Exact
import Mathlib.Algebra.Homology.ShortComplex.Preadditive
import Mathlib.Tactic.Linar... | namespace ChainComplex
variable {C : Type*} [Category C] [Abelian C]
lemma isIso_descOpcycles_iff (K : ChainComplex C ℕ) {X : C} (φ : K.X 0 ⟶ X)
[K.HasHomology 0] (hφ : K.d 1 0 ≫ φ = 0) :
| Mathlib/Algebra/Homology/ShortComplex/HomologicalComplex.lean | 741 | 746 |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.Group.Finset
import Mathlib.Data.Finsupp.Order
import Mathlib.Data.Sym.Basic
/-!
# Equivalence between `Multiset` and `ℕ`-valued finitel... |
/-- The `n`th symmetric power of a type `α` is naturally equivalent to the subtype of
finitely-supported maps `α →₀ ℕ` with total mass `n`.
| Mathlib/Data/Finsupp/Multiset.lean | 234 | 237 |
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