Context stringlengths 227 76.5k | target stringlengths 0 11.6k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 16 3.69k |
|---|---|---|---|---|
/-
Copyright (c) 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.SpecialFunctions.Complex.Circle
import Mathlib.Geometry.Euclidean.Angle.Oriented.Basic
/-!
# Rotations by oriented angles.
This... | @[simp]
theorem inner_smul_rotation_pi_div_two_smul_right (x : V) (r₁ r₂ : ℝ) :
| Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean | 418 | 419 |
/-
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.Operations
import Mathlib.Analysis.SpecificLimits.Basic
/-!
# Outer measures from functions
Given an arbit... | sets that covers that set, where a different measure can be used for each set in the cover. -/
theorem sInf_apply' {m : Set (OuterMeasure α)} {s : Set α} (h : s.Nonempty) :
sInf m s =
⨅ (t : ℕ → Set α) (_ : s ⊆ iUnion t), ∑' n, ⨅ (μ : OuterMeasure α) (_ : μ ∈ m), μ (t n) :=
| Mathlib/MeasureTheory/OuterMeasure/OfFunction.lean | 366 | 369 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Data.Nat.Log
import Mathlib.Data.Nat.Pri... |
/-- A prime power divides `n!` iff it is at most the sum of the quotients `n / p ^ i`.
This sum is expressed over the set `Ico 1 b` where `b` is any bound greater than `log p n` -/
theorem pow_dvd_factorial_iff {p : ℕ} {n r b : ℕ} (hp : p.Prime) (hbn : log p n < b) :
p ^ r ∣ n ! ↔ r ≤ ∑ i ∈ Ico 1 b, n / p ^ i :=... | Mathlib/Data/Nat/Multiplicity.lean | 162 | 169 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Data.Int.Cast.Pi
import Mathlib.Data.Nat.Cast.Basic
import Mathlib.MeasureTheory.MeasurableSpace... | Mathlib/MeasureTheory/MeasurableSpace/Basic.lean | 1,193 | 1,195 | |
/-
Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.SimpleGraph.Path
import Mathlib.Combinatorics.SimpleGraph.Operations
import Mathlib.Data.Finset.Pairwise
import M... |
theorem isNClique_map_iff (hn : 1 < n) {t : Finset β} {f : α ↪ β} :
(G.map f).IsNClique n t ↔ ∃ s : Finset α, G.IsNClique n s ∧ s.map f = t := by
rw [isNClique_iff, isClique_map_finset_iff, or_and_right,
or_iff_right (by rintro ⟨h', rfl⟩; exact h'.not_lt hn)]
constructor
· rintro ⟨⟨s, hs, rfl⟩, rfl⟩
... | Mathlib/Combinatorics/SimpleGraph/Clique.lean | 216 | 224 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Matrix.RowCol
import Mathli... | calc
det (of fun i j => v j * A i j) = det (A * diagonal v) :=
congr_arg det <| by
| Mathlib/LinearAlgebra/Matrix/Determinant/Basic.lean | 287 | 289 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Michael Stoll
-/
import Mathlib.NumberTheory.LegendreSymbol.Basic
import Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum
/-!
# Quadratic reciprocity.
## Main r... | legendreSym q p = -legendreSym p q := by
let nop := @neg_one_pow_div_two_of_three_mod_four
rw [quadratic_reciprocity', pow_mul, nop hp, nop hq, neg_one_mul] <;>
rwa [← Prime.mod_two_eq_one_iff_ne_two, odd_of_mod_four_eq_three]
end legendreSym
namespace ZMod
| Mathlib/NumberTheory/LegendreSymbol/QuadraticReciprocity.lean | 138 | 145 |
/-
Copyright (c) 2021 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Group.Measure
import Mathlib.MeasureTheory.Measure.Prod
/-!
# Measure theory in the product of groups
In this file we show propertie... | This should not be confused with `(measurePreserving_add_left μ g).quasiMeasurePreserving`. "]
theorem quasiMeasurePreserving_mul_left [IsMulRightInvariant μ] (g : G) :
QuasiMeasurePreserving (fun h : G => g * h) μ μ := by
have :=
(quasiMeasurePreserving_mul_right μ.inv g⁻¹).mono (inv_absolutelyContinuous μ.i... | Mathlib/MeasureTheory/Group/Prod.lean | 450 | 454 |
/-
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.FunctorN
/-!
# Comparison with the normalized Moore complex functor
In this file, we show that when the category `A` is abelian,
the... | theorem inclusionOfMooreComplexMap_comp_PInfty (X : SimplicialObject A) :
inclusionOfMooreComplexMap X ≫ PInfty = inclusionOfMooreComplexMap X := by
ext (_|n)
· dsimp
| Mathlib/AlgebraicTopology/DoldKan/Normalized.lean | 83 | 86 |
/-
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, Damiano Testa,
Yuyang Zhao
-/
import Mathlib.Algebra.Order.Monoid.Unbundled.Defs
import Mathlib.Data.Ordering.Basic
imp... |
@[to_additive]
lemma min_mul [CovariantClass α α (swap (· * ·)) (· ≤ ·)] (a b c : α) :
min a b * c = min (a * c) (b * c) := mul_right_mono.map_min
| Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean | 340 | 343 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.Polynomial.Basi... | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 1,360 | 1,361 | |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Casper Putz, Anne Baanen
-/
import Mathlib.Algebra.Algebra.Subalgebra.Tower
import Mathlib.Data.Finite.Sum
import Mathlib.Data.Matrix.Block
import Mathl... | lemma end_apply_apply (ij : ι × ι) (k : ι) : (b.end ij) (b k) = if ij.2 = k then b ij.1 else 0 :=
linearMap_apply_apply b b ij k
| Mathlib/LinearAlgebra/Matrix/ToLin.lean | 979 | 981 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Devon Tuma
-/
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.RingTheory.Coprime.Basic
import Mathlib.Tactic.... |
end Polynomial
| Mathlib/RingTheory/Polynomial/ScaleRoots.lean | 258 | 261 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Data.ENNReal.Basic
/-!
# Maps between real and extended non-negative real numbers
This file focuses on the functions `ENNReal.toReal... | Mathlib/Data/ENNReal/Real.lean | 215 | 215 | |
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes Hölzl, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Seminorm
import Mathlib.Data.NNReal.Basic
import Mathlib.Topology.Algebra.Support
import Mathlib.To... | · simpa [h] using norm_mul_le' x y
· simpa [h] using norm_le_mul_norm_add' x y
| Mathlib/Analysis/Normed/Group/Basic.lean | 552 | 553 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Topology.Bases
import Mathlib.Topology.Compactness.LocallyCompact
import Mathlib.Topology.Compactness.LocallyFinite
/... | refine
⟨⋃ n, (t n : Set X), iUnion_subset fun n x hx => (ht n x hx).2,
countable_iUnion fun n => (t n).countable_toSet, fun x hx => mem_iUnion₂.2 ?_⟩
rcases exists_mem_compactCovering x with ⟨n, hn⟩
rcases mem_iUnion₂.1 (hsub n ⟨hn, hx⟩) with ⟨y, hyt : y ∈ t n, hyf : x ∈ s → x ∈ f y⟩
exact ⟨y, mem_iUn... | Mathlib/Topology/Compactness/SigmaCompact.lean | 306 | 311 |
/-
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.Category.GaloisConnection
import Mathlib.CategoryTheory.EqToHom
import Mathlib.Topology.Category.TopCat.EpiMono
import Mathlib.Topology.Sets... |
theorem map_id_eq : map (𝟙 X) = 𝟭 (Opens X) := by
rfl
| Mathlib/Topology/Category/TopCat/Opens.lean | 226 | 229 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yaël Dillies
-/
import Mathlib.Data.List.Iterate
import Mathlib.GroupTheory.Perm.Cycle.Basic
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Tactic.Group
/-!
# ... | p ∈ cycleFactorsFinset f ↔ p.IsCycle ∧ ∀ a ∈ p.support, p a = f a := by
obtain ⟨l, hl, hl'⟩ := f.cycleFactorsFinset.exists_list_nodup_eq
| Mathlib/GroupTheory/Perm/Cycle/Factors.lean | 551 | 552 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Johannes Hölzl, Yury Kudryashov, Patrick Massot
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Order.Filter.AtTopBot.Archimedean
import Mathlib.Order.Iterate
impor... | · funext n
simp only [one_div, inv_pow]
rfl
· norm_num
| Mathlib/Analysis/SpecificLimits/Basic.lean | 347 | 351 |
/-
Copyright (c) 2021 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.Normed.Module.Basic
import Mathlib.MeasureTheory.Function.SimpleFuncDense
/-!
# Strongly measurable and finitely strongly meas... | protected theorem snd [mα : MeasurableSpace α] {_ : MeasurableSpace β} [TopologicalSpace γ]
{f : β → γ} (hf : StronglyMeasurable f) :
StronglyMeasurable (fun z : α × β => f z.2) :=
hf.comp_measurable measurable_snd
section Arithmetic
variable {mα : MeasurableSpace α} [TopologicalSpace β]
| Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean | 374 | 381 |
/-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Joël Riou
-/
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.Equalizers
import Mathlib.CategoryTheory.Limits.ConeCategory
/-!
... |
@[simp]
theorem fst_app_right (a) : K.ι.app (WalkingMultispan.left a) = I.fst a ≫ K.π _ := by
| Mathlib/CategoryTheory/Limits/Shapes/Multiequalizer.lean | 547 | 549 |
/-
Copyright (c) 2024 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Star.Basic
/-!
# Morphisms of star rings
This file defines a new type of morphism between (non-unita... | Function.bijective_iff_has_inverse.mpr ⟨_, symm_symm, symm_symm⟩
theorem coe_mk (e h₁) : ⇑(⟨e, h₁⟩ : A ≃⋆+* B) = e := rfl
| Mathlib/Algebra/Star/StarRingHom.lean | 360 | 362 |
/-
Copyright (c) 2021 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Logic.Function.Basic
import Mathlib.Logic.Relator
/-!
# Types that are empty
In this file we define a typeclass `IsEmpty`, which expresses that a... | theorem wellFounded_of_isEmpty {α} [IsEmpty α] (r : α → α → Prop) : WellFounded r :=
⟨isEmptyElim⟩
| Mathlib/Logic/IsEmpty.lean | 191 | 192 |
/-
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... | Mathlib/SetTheory/Ordinal/Arithmetic.lean | 2,444 | 2,446 | |
/-
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 | 98 | 100 | |
/-
Copyright (c) 2023 Yaël Dillies, Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Data.Finset.Lattice.Prod
import Mathlib.Data.Finset.Powerset
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Or... | Mathlib/Order/SupClosed.lean | 352 | 352 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.MeasurableSpace.MeasurablyGenerated
import Mathlib.MeasureTheory.Measure.NullMeasurable
import Mathlib.Order.Interval.Set... | rw [← measure_univ_eq_zero, sum_apply _ MeasurableSet.univ, tsum_empty]
theorem sum_add_sum_compl (s : Set ι) (μ : ι → Measure α) :
((sum fun i : s => μ i) + sum fun i : ↥sᶜ => μ i) = sum μ := by
ext1 t ht
simp only [add_apply, sum_apply _ ht]
exact ENNReal.summable.tsum_add_tsum_compl (f := fun i => μ i t... | Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 1,241 | 1,248 |
/-
Copyright (c) 2024 Josha Dekker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Josha Dekker
-/
import Mathlib.Probability.Notation
import Mathlib.Probability.CDF
import Mathlib.Analysis.SpecialFunctions.Gamma.Basic
/-! # Gamma distributions over ℝ
Define the gamm... |
/-- The gamma pdf is positive for all positive reals -/
lemma gammaPDFReal_pos {x a r : ℝ} (ha : 0 < a) (hr : 0 < r) (hx : 0 < x) :
0 < gammaPDFReal a r x := by
simp only [gammaPDFReal, if_pos hx.le]
| Mathlib/Probability/Distributions/Gamma.lean | 83 | 87 |
/-
Copyright (c) 2018 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Functor.Const
import Mathlib.CategoryTheory.Discrete.Basic
/-!
# The category `Discrete PUnit`
We define `star : C ⥤ Discret... | @[simps]
def equiv : Discrete PUnit.{w + 1} ⥤ C ≌ C where
| Mathlib/CategoryTheory/PUnit.lean | 48 | 49 |
/-
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.Group.Basic
import Mathlib.Algebra.GroupWithZero.NeZero
import Mathlib.Logic.Unique
import Mathlib.Tactic.Conv
/-!
# Groups with an adjoined z... | @[field_simps]
lemma pow_ne_zero (n : ℕ) (h : a ≠ 0) : a ^ n ≠ 0 := mt pow_eq_zero h
| Mathlib/Algebra/GroupWithZero/Basic.lean | 179 | 181 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Frédéric Dupuis, Heather Macbeth
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Analysis.InnerProductSpace.Orthogonal
import Mathlib.Analysis.InnerProductSpace.Sy... | /-- A point equals its orthogonal projection if and only if it lies in the subspace. -/
theorem orthogonalProjection_eq_self_iff {v : E} : (K.orthogonalProjection v : E) = v ↔ v ∈ K := by
refine ⟨fun h => ?_, fun h => eq_orthogonalProjection_of_mem_of_inner_eq_zero h ?_⟩
· rw [← h]
| Mathlib/Analysis/InnerProductSpace/Projection.lean | 536 | 539 |
/-
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.Convex.Topology
import Mathlib.Analysis.Normed.Module.Basic
import Mathlib.Analysis.Seminorm
import Mathlib.Analysis.SpecificLimits.Basi... | section TVS
variable [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E]
variable {x y : E} {s t : Set E}
| Mathlib/Analysis/Calculus/TangentCone.lean | 415 | 418 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Finset.NAry
import Mathlib.Data.Finset.Slice
import Mathlib.Data.Set.Sups
/-!
# Set family operations
This file defines a few binary operations on `... | @[simp]
| Mathlib/Data/Finset/Sups.lean | 546 | 546 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Minchao Wu, Mario Carneiro
-/
import Mathlib.Data.Finset.Attach
import Mathlib.Data.Finset.Disjoint
import Mathlib.Data.Finset.Erase
import Mat... | Mathlib/Data/Finset/Basic.lean | 2,467 | 2,468 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Data.ENNReal.Operations
/-!
# Results about division in extended non-negative reals
This file establishes basic properties related t... | @[simp 900]
theorem le_inv_iff_mul_le : a ≤ b⁻¹ ↔ a * b ≤ 1 := by
rw [← one_div, ENNReal.le_div_iff_mul_le] <;>
| Mathlib/Data/ENNReal/Inv.lean | 405 | 407 |
/-
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.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.Order.Sub.WithTop
import Mathlib.Data.NNReal.Defs
import Mathlib.Order.Interval.Set.... |
theorem coe_lt_natCast {n : ℕ} : (r : ℝ≥0∞) < n ↔ r < n := ENNReal.coe_natCast n ▸ coe_lt_coe
| Mathlib/Data/ENNReal/Basic.lean | 601 | 603 |
/-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Regular.Basic
import Mathlib.GroupTheory.GroupAction.Hom
/-!
# Action of regular elements on a module
We introduce `M`-regular elements, in the... | IsSMulRegular M (a * b) ∧ IsSMulRegular M (b * a) ↔ IsSMulRegular M a ∧ IsSMulRegular M b := by
refine ⟨?_, ?_⟩
· rintro ⟨ab, ba⟩
exact ⟨ba.of_mul, ab.of_mul⟩
| Mathlib/Algebra/Regular/SMul.lean | 102 | 105 |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.LinearAlgebra.Finsupp.Span
/-!
# Lie submodules of a Lie algebra
In this file we define Lie submodules, we construct ... | replace hne : Nonempty s := Set.nonempty_coe_sort.mpr hne
have := Submodule.coe_iSup_of_directed _ hdir.directed_val
simp_rw [← iSup_toSubmodule, Set.iUnion_coe_set, coe_toSubmodule] at this
rw [← this, SetLike.coe_set_eq, sSup_eq_iSup, iSup_subtype]
| Mathlib/Algebra/Lie/Submodule.lean | 752 | 756 |
/-
Copyright (c) 2022 Wrenna Robson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Wrenna Robson
-/
import Mathlib.Topology.MetricSpace.Basic
/-!
# Infimum separation
This file defines the extended infimum separation of a set. This is approximately dual to the
diame... | Mathlib/Topology/MetricSpace/Infsep.lean | 532 | 534 | |
/-
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 ... | induction n with
| zero => rfl
| succ n IH => simp [IH, H, List.range_succ]
theorem listLookup [DecidableEq α] : Primrec₂ (List.lookup : α → List (α × β) → Option β) :=
(to₂ <| list_rec snd (const none) <|
to₂ <|
| Mathlib/Computability/Primrec.lean | 997 | 1,003 |
/-
Copyright (c) 2017 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Keeley Hoek
-/
import Mathlib.Algebra.NeZero
import Mathlib.Data.Int.DivMod
import Mathlib.Logic.Embedding.Basic
import Mathlib.Logic.Equiv.Set
import Mathlib.Tactic.... | rcases hi with ⟨y, rfl⟩
exact predAbove_last_castSucc
| Mathlib/Data/Fin/Basic.lean | 1,269 | 1,271 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.Polynomial.Basi... | WithBot.coe_zero]
theorem degree_C_le : degree (C a) ≤ 0 := by
by_cases h : a = 0
· rw [h, C_0]
| Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 150 | 154 |
/-
Copyright (c) 2018 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton
-/
import Mathlib.Topology.Bases
import Mathlib.Topology.DenseEmbedding
import Mathlib.Topology.Connected.TotallyDisconnected
/-! # Stone-Čech compactification
Construction ... |
/-- Every ultrafilter `u` on `Ultrafilter α` converges to a unique
point of `Ultrafilter α`, namely `joinM u`. -/
theorem ultrafilter_converges_iff {u : Ultrafilter (Ultrafilter α)} {x : Ultrafilter α} :
↑u ≤ 𝓝 x ↔ x = joinM u := by
| Mathlib/Topology/StoneCech.lean | 76 | 80 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Star.SelfAdjoint
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
/-!
# Quate... | rw [← star_comm_self']
| Mathlib/Algebra/Quaternion.lean | 709 | 709 |
/-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.Topology.MetricSpace.HausdorffDistance
/-!
# Thickenings in pseudo-metric spaces
## Main definitions
* `Metric.thickening δ s`, the open thickening by ra... | apply cthickening_eq_iInter_thickening' δ_nn (Ioi δ) rfl.subset
simp_rw [inter_eq_right.mpr Ioc_subset_Ioi_self]
exact fun _ hε => nonempty_Ioc.mpr hε
theorem cthickening_eq_iInter_thickening'' (δ : ℝ) (E : Set α) :
cthickening δ E = ⋂ (ε : ℝ) (_ : max 0 δ < ε), thickening ε E := by
| Mathlib/Topology/MetricSpace/Thickening.lean | 469 | 474 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kim Morrison
-/
import Mathlib.Algebra.BigOperators.Finsupp.Basic
import Mathlib.Algebra.BigOperators.Group.Finset.Preimage
import Mathlib.Algebra.Module.Defs
import Ma... |
@[to_additive]
lemma prod_sumElim {ι₁ ι₂ α M : Type*} [Zero α] [CommMonoid M]
(f₁ : ι₁ →₀ α) (f₂ : ι₂ →₀ α) (g : ι₁ ⊕ ι₂ → α → M) :
(f₁.sumElim f₂).prod g = f₁.prod (g ∘ Sum.inl) * f₂.prod (g ∘ Sum.inr) := by
simp [Finsupp.prod, Finset.prod_disj_sum]
/-- The equivalence between `(α ⊕ β) →₀ γ` and `(α →₀ γ) ... | Mathlib/Data/Finsupp/Basic.lean | 1,169 | 1,179 |
/-
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.ShortComplex.PreservesHomology
import Mathlib.Algebra.Homology.ShortComplex.Abelian
import Mathlib.Algebra.Homology.ShortComplex.QuasiIso
import... |
lemma mono_f (s : S.Splitting) : Mono S.f := by
| Mathlib/Algebra/Homology/ShortComplex/Exact.lean | 498 | 499 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kim Morrison
-/
import Mathlib.Algebra.BigOperators.Finsupp.Basic
import Mathlib.Algebra.BigOperators.Group.Finset.Preimage
import Mathlib.Algebra.Module.Defs
import Ma... | conv_lhs => rw [← f.apply_symm_apply a]
exact mapDomain_apply f.injective _ _
/-- `Finsupp.mapDomain` is an `AddMonoidHom`. -/
@[simps]
def mapDomain.addMonoidHom (f : α → β) : (α →₀ M) →+ β →₀ M where
toFun := mapDomain f
| Mathlib/Data/Finsupp/Basic.lean | 451 | 457 |
/-
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, Mario Carneiro
-/
import Mathlib.Tactic.NormNum.Basic
import Mathlib.Data.Nat.Prime.Basic
/-!
# `norm_num` extensions on natural numbers
This... | | n, _, k, ⟨rfl⟩, h1, h2 => by
rw [eq_comm, ← Nat.dvd_iff_mod_eq_zero] at h2
exact ⟨le_antisymm (minFac_le_of_dvd h1.1.le h2) h1.2.2⟩
theorem isNat_minFac_4 : {n n' k : ℕ} →
| Mathlib/Tactic/NormNum/Prime.lean | 103 | 107 |
/-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
import Mathlib.CategoryTheory.Limits.Shapes.Countable
import Mathlib.Topology.Category.Profinite.AsLimit
import Mathlib.Topology.Category.Profinite.CofilteredLimi... | end LightProfinite
/-- The functor part of the equivalence `LightProfinite ≌ LightDiagram` -/
@[simps]
noncomputable def lightProfiniteToLightDiagram : LightProfinite.{u} ⥤ LightDiagram.{u} where
obj X := X.toLightDiagram
map f := f
open scoped Classical in
instance (S : LightDiagram.{u}) : SecondCountableTopolog... | Mathlib/Topology/Category/LightProfinite/Basic.lean | 276 | 312 |
/-
Copyright (c) 2019 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, François Dupuis
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Order.Filter.Extr
import Mathlib.Tactic.NormNum
/-!
# Convex and concave functions
This... | theorem ConcaveOn.congr (hf : ConcaveOn 𝕜 s f) (hfg : EqOn f g s) : ConcaveOn 𝕜 s g :=
⟨hf.1, fun x hx y hy a b ha hb hab => by
simpa only [← hfg hx, ← hfg hy, ← hfg (hf.1 hx hy ha hb hab)] using hf.2 hx hy ha hb hab⟩
| Mathlib/Analysis/Convex/Function.lean | 99 | 102 |
/-
Copyright (c) 2019 Rohan Mitta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rohan Mitta, Kevin Buzzard, Alistair Tucker, Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.Setoid.Basic
import Mathlib.Dynamics.Fixed... | theorem aposteriori_dist_iterate_fixedPoint_le (x n) :
dist (f^[n] x) (fixedPoint f hf) ≤ dist (f^[n] x) (f^[n + 1] x) / (1 - K) := by
| Mathlib/Topology/MetricSpace/Contracting.lean | 283 | 284 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Data.Countable.Small
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Powerset
import Mathlib.Dat... |
theorem mul_lt_aleph0 {a b : Cardinal} (ha : a < ℵ₀) (hb : b < ℵ₀) : a * b < ℵ₀ :=
| Mathlib/SetTheory/Cardinal/Basic.lean | 461 | 462 |
/-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.Topology.MetricSpace.HausdorffDistance
/-!
# Thickenings in pseudo-metric spaces
## Main definitions
* `Metric.thickening δ s`, the open thickening by ra... | obtain ⟨δ, hδ, hthick⟩ := (isCompact_singleton (x := f x)).exists_thickening_subset_open ho this
let V := f ⁻¹' (thickening (δ / 2) {f x})
have : V ∈ 𝓝 x := by
apply hf x hx
apply isOpen_thickening.mem_nhds
exact (self_subset_thickening (by positivity) _) rfl
refine ⟨K ∩ (interior V),... | Mathlib/Topology/MetricSpace/Thickening.lean | 660 | 669 |
/-
Copyright (c) 2022 Matej Penciak. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matej Penciak, Moritz Doll, Fabien Clery
-/
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
/-!
# The Symplectic Group
This file defines the symplectic group and proves element... | _ = -(J l R)⁻¹ := by
rw [mul_nonsing_inv_cancel_left _ _ huAT, nonsing_inv_mul_cancel_right _ _ huA]
_ = J l R := by simp [J_inv]
@[simp]
theorem transpose_mem_iff : Aᵀ ∈ symplecticGroup l R ↔ A ∈ symplecticGroup l R :=
⟨fun hA => by simpa using transpose_mem hA, transpose_mem⟩
theorem mem_iff' : A ∈ ... | Mathlib/LinearAlgebra/SymplecticGroup.lean | 142 | 151 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.CharP.Two
import Mathlib.Data.Nat.Cast.Field
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.Nat.Factorization.Induction
import Mat... | suffices ∃ x < n + 1, (n + 1).gcd x = 1 by simpa [totient, filter_eq_empty_iff]
⟨1 % (n + 1), mod_lt _ n.succ_pos, by rw [gcd_comm, ← gcd_rec, gcd_one_right]⟩
| Mathlib/Data/Nat/Totient.lean | 63 | 64 |
/-
Copyright (c) 2020. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Massot
-/
import Mathlib.Tactic.Ring
import Mathlib.Tactic.FailIfNoProgress
import Mathlib.Algebra.Group.Commutator
/-!
# `group` tactic
Normalizes expressions in the langu... | macro_rules
| `(tactic| aux_group₁ $[at $location]?) =>
| Mathlib/Tactic/Group.lean | 49 | 50 |
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kenny Lau
-/
import Mathlib.Algebra.Polynomial.Coeff
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.RingTheory.PowerSeries.Basic
/-!
# Formal power se... | @[simp]
theorem trunc_one (n) : trunc (n + 1) (1 : R⟦X⟧) = 1 :=
Polynomial.ext fun m => by
rw [coeff_trunc, coeff_one, Polynomial.coeff_one]
| Mathlib/RingTheory/PowerSeries/Trunc.lean | 50 | 53 |
/-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.Orientation
import Mathlib.Data.Complex.FiniteDimensional
import Mathlib.Da... | · rcases eq_or_lt_of_le (norm_nonneg (o.rightAngleRotationAux₁ x)) with h | h
· rw [← h]
positivity
refine le_of_mul_le_mul_right ?_ h
rw [← real_inner_self_eq_norm_mul_norm, o.inner_rightAngleRotationAux₁_left]
exact o.areaForm_le x (o.rightAngleRotationAux₁ x)
· l... | Mathlib/Analysis/InnerProductSpace/TwoDim.lean | 193 | 199 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Joey van Langen, Casper Putz
-/
import Mathlib.Algebra.CharP.Algebra
import Mathlib.Algebra.CharP.Reduced
import Mathlib.Algebra.Field.ZMod
import Mathlib.Data.Nat.Prime.In... | rw [← ZMod.eq_iff_modEq_nat]
let x' : Units (ZMod n) := ZMod.unitOfCoprime _ h
have := ZMod.pow_totient x'
apply_fun ((fun (x : Units (ZMod n)) => (x : ZMod n)) : Units (ZMod n) → ZMod n) at this
simpa only [Nat.succ_eq_add_one, Nat.cast_pow, Units.val_one, Nat.cast_one,
| Mathlib/FieldTheory/Finite/Basic.lean | 535 | 539 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kim Morrison
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Algebra.Group.InjSurj
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Tactic.FastInstance
impo... | Mathlib/Data/Finsupp/Defs.lean | 1,127 | 1,130 | |
/-
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... | · rfl
· dsimp
intro j
rw [← w]
rfl
/-- If `t.op : Cocone F.op` is a colimit cocone, then `t : Cone F` is a limit cone. -/
def isLimitOfOp {t : Cone F} (P : IsColimit t.op) : IsLimit t :=
P.unop
/-- If `t.op : Cone F.op` is a limit cone, then `t : Cocone F` is a colimit cocone. -/
| Mathlib/CategoryTheory/Limits/HasLimits.lean | 1,153 | 1,163 |
/-
Copyright (c) 2023 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Prod
/-!
# Products of finite measures and probability measures
This file i... | lemma prod_prod (s : Set α) (t : Set β) : μ.prod ν (s ×ˢ t) = μ s * ν t := by simp [coeFn_def]
| Mathlib/MeasureTheory/Measure/FiniteMeasureProd.lean | 111 | 111 |
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Data.ULift
import Mathlib.Data.ZMod.Defs
import Mathlib.SetTheory.Cardinal.ToNat
import Mathlib.SetTheory.Cardinal.ENat
/-!
# Finite Cardinality Funct... | card (f '' s) = card s :=
card_congr (Equiv.Set.imageOfInjOn f s h).symm
theorem card_image_of_injective {α β : Type*} (f : α → β) (s : Set α)
(h : Function.Injective f) : card (f '' s) = card s := card_image_of_injOn h.injOn
| Mathlib/SetTheory/Cardinal/Finite.lean | 296 | 300 |
/-
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]
lemma rank_mul_eq_left_of_isUnit_det [DecidableEq n]
| Mathlib/Data/Matrix/Rank.lean | 181 | 182 |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.FieldTheory.Finiteness
import Mathlib.LinearAlgebra.AffineSpace.Basis
import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
/-!
# Finite-dimensional subsp... | obtain rfl | ht' := t.eq_empty_or_nonempty
· simp [Set.subset_empty_iff] at hst
have := hs'.to_subtype
have := ht'.to_set.to_subtype
have dir_lt := AffineSubspace.direction_lt_of_nonempty (k := k) hst <| hs'.to_set.affineSpan k
rw [direction_affineSpan, direction_affineSpan,
← @Subtype.range_coe _ (s : ... | Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean | 253 | 270 |
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Function.ConvergenceInMeasure
import Mathlib.MeasureTheory.Function.L1Space.Integrable
/-!
# Uniform integrability
This file contains the def... | theorem tendsto_indicator_ge (f : α → β) (x : α) :
Tendsto (fun M : ℕ => { x | (M : ℝ) ≤ ‖f x‖₊ }.indicator f x) atTop (𝓝 0) := by
refine tendsto_atTop_of_eventually_const (i₀ := Nat.ceil (‖f x‖₊ : ℝ) + 1) fun n hn => ?_
rw [Set.indicator_of_not_mem]
simp only [not_le, Set.mem_setOf_eq]
refine lt_of_le_of_... | Mathlib/MeasureTheory/Function/UniformIntegrable.lean | 176 | 208 |
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.BigOperators.Group.List.Lemmas
import Mathlib.Algebra.Group.Action.Hom
import Mathlib.Algebra.Group.Submonoid.Defs
import Mathlib.Data.List.FinRange
... | -- { Mul.gMul ι, One.gOne ι with
{ One.gOne ι with
mul := fun x y => x * y
one_mul := fun _ => Sigma.ext (zero_add _) (heq_of_eq (one_mul _))
mul_one := fun _ => Sigma.ext (add_zero _) (heq_of_eq (mul_one _))
mul_assoc := fun _ _ _ => Sigma.ext (add_assoc _ _ _) (heq_of_eq (mul_assoc _ _ _))
| Mathlib/Algebra/GradedMonoid.lean | 447 | 452 |
/-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston
-/
import Mathlib.Algebra.Group.Submonoid.Operations
import Mathlib.Data.Setoid.Basic
import Mathlib.GroupTheory.Congruence.Hom
/-!
# Congruence relations
This f... | Mathlib/GroupTheory/Congruence/Basic.lean | 884 | 885 | |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.BoxIntegral.Partition.Basic
/-!
# Split a box along one or more hyperplanes
## Main definitions
A hyperplane `{x : ι → ℝ | x i = a}` spli... | variable [Finite ι]
/-- Let `s` be a finite set of boxes in `ℝⁿ = ι → ℝ`. Then there exists a finite set `t₀` of
hyperplanes (namely, the set of all hyperfaces of boxes in `s`) such that for any `t ⊇ t₀`
and any box `I` in `ℝⁿ` the following holds. The hyperplanes from `t` split `I` into subboxes.
| Mathlib/Analysis/BoxIntegral/Partition/Split.lean | 274 | 278 |
/-
Copyright (c) 2021 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.AlgebraicGeometry.Restrict
import Mathlib.CategoryTheory.Adjunction.Limits
import Mathlib.CategoryTheory.Adjunction.Reflective
/-!
# Adjunction between `Γ` and ... |
/-- The functor `Spec : CommRingCatᵒᵖ ⥤ Scheme` is fully faithful. -/
def Spec.fullyFaithful : Scheme.Spec.FullyFaithful :=
ΓSpec.adjunction.fullyFaithfulROfIsIsoCounit
| Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | 502 | 505 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Minchao Wu, Mario Carneiro
-/
import Mathlib.Algebra.NeZero
import Mathlib.Data.Finset.Attach
import Mathlib.Data.Finset.Disjoint
import Mathli... | · simp
conv_rhs => rw [range_succ]
rw [range_succ, image_insert, ← hk, insert_sdiff_of_not_mem]
simp
| Mathlib/Data/Finset/Image.lean | 654 | 657 |
/-
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
/-!
# The derivative of a composition (chain rule)
For detailed documentation of the... | exact hx.symm
@[fun_prop]
protected theorem HasFDerivWithinAt.iterate {f : E → E} {f' : E →L[𝕜] E}
(hf : HasFDerivWithinAt f f' s x) (hx : f x = x) (hs : MapsTo f s s) (n : ℕ) :
HasFDerivWithinAt f^[n] (f' ^ n) s x := by
refine HasFDerivAtFilter.iterate hf ?_ hx n
| Mathlib/Analysis/Calculus/FDeriv/Comp.lean | 233 | 239 |
/-
Copyright (c) 2021 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.GroupTheory.Index
/-!
# Complements
In this file we define the complement of a subgroup.
## Main definitions
- `Subgroup.IsComplement S T` where ... | · have : Nat.card H'' * Nat.card S = Nat.card H := cmem.card_mul_card
rw [← this]
refine congr_arg₂ (· * ·) ?_ ?_ <;>
exact Nat.card_congr (Equiv.Set.image _ _ <| subtype_injective H).symm
| Mathlib/GroupTheory/Complement.lean | 436 | 439 |
/-
Copyright (c) 2022 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler, Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
import Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv
/-!
# Polynomia... | simpa only [tan_zero, sub_zero, sub_pos] using mono zero_in_U x_in_U h1
theorem le_tan {x : ℝ} (h1 : 0 ≤ x) (h2 : x < π / 2) : x ≤ tan x := by
rcases eq_or_lt_of_le h1 with (rfl | h1')
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Bounds.lean | 196 | 199 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.LeftHomology
import Mathlib.CategoryTheory.Limits.Opposites
/-!
# Right Homology of short complexes
In this file, we define the ... | LeftHomologyData.cyclesIso_inv_comp_iCycles, RightHomologyData.op_i,
← op_comp, RightHomologyData.p_g', op_f]
@[reassoc (attr := simp)]
lemma op_pOpcycles_opcyclesOpIso_hom [S.HasLeftHomology] :
S.op.pOpcycles ≫ S.opcyclesOpIso.hom = S.iCycles.op := by
dsimp [opcyclesOpIso]
| Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean | 983 | 989 |
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.SetTheory.Cardinal.Finite
import Mathlib.Data.Set.Finite.Powerset
/-!
# Noncomputable Set Cardinality
We define the cardinality of set `s` as a term `Set... | hf.injOn.encard_image
theorem _root_.Function.Embedding.encard_le (e : s ↪ t) : s.encard ≤ t.encard := by
| Mathlib/Data/Set/Card.lean | 410 | 412 |
/-
Copyright (c) 2021 Ivan Sadofschi Costa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ivan Sadofschi Costa
-/
import Mathlib.Data.Finsupp.Single
/-!
# `cons` and `tail` for maps `Fin n →₀ M`
We interpret maps `Fin n →₀ M` as `n`-tuples of elements of `M`,
We def... | theorem cons_ne_zero_of_right (h : s ≠ 0) : cons y s ≠ 0 := by
contrapose! h with c
ext a
| Mathlib/Data/Finsupp/Fin.lean | 78 | 80 |
/-
Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Data.Set.Lattice
import Mathlib.Order.ConditionallyCompleteLattice.Defs
/-!
# Theory of conditionally complete lattices
A conditionally complet... | Mathlib/Order/ConditionallyCompleteLattice/Basic.lean | 1,728 | 1,728 | |
/-
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... | ⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨_, hi⟩ => ⟨_, hi⟩⟩
| Mathlib/Data/Set/Image.lean | 587 | 590 |
/-
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, Mario Carneiro
-/
import Mathlib.Tactic.NormNum.Basic
import Mathlib.Data.Nat.Prime.Basic
/-!
# `norm_num` extensions on natural numbers
This... | refine minFacHelper_1 e h fun h2 ↦ ?_
have nk := Nat.ne_of_beq_eq_false nk
rw [← Nat.dvd_iff_mod_eq_zero, ← h2] at nk
exact nk <| minFac_dvd n
theorem isNat_minFac_1 : {a : ℕ} → IsNat a (nat_lit 1) → IsNat a.minFac 1
| Mathlib/Tactic/NormNum/Prime.lean | 90 | 95 |
/-
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... | simp_rw [restrict, restrictₗ, liftLinear, LinearMap.coe_mk, AddHom.coe_mk,
toMeasure_toOuterMeasure, OuterMeasure.restrict_trim h, μ.trimmed]
theorem restrict_apply₀ (ht : NullMeasurableSet t (μ.restrict s)) : μ.restrict s t = μ (t ∩ s) := by
| Mathlib/MeasureTheory/Measure/Restrict.lean | 56 | 59 |
/-
Copyright (c) 2024 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.Algebra.BigOperators.Field
import Mathlib.NumberTheory.LSeries.Basic
/-!
# Linearity of the L-series of `f` as a function of `f`
We show that the `LSer... |
lemma LSeriesSummable.sub {f g : ℕ → ℂ} {s : ℂ} (hf : LSeriesSummable f s)
(hg : LSeriesSummable g s) :
LSeriesSummable (f - g) s := by
| Mathlib/NumberTheory/LSeries/Linearity.lean | 88 | 91 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.InnerProductSpace.Defs
impor... | Mathlib/Analysis/InnerProductSpace/Basic.lean | 1,731 | 1,741 | |
/-
Copyright (c) 2021 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.LinearAlgebra.Ray
import Mathlib.LinearAlgebra.Determinant
/-!
# Orientations of modules
This file defines orientations of modules.
## Main definitions
... | variable {M : Type*} [AddCommGroup M] [Module R M]
variable {ι : Type*}
namespace Orientation
variable [Fintype ι]
| Mathlib/LinearAlgebra/Orientation.lean | 322 | 328 |
/-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Ordinal.Family
import Mathlib.Tactic.Abel
/-!
# Natural operations on ordinals
The goal of this file is to define n... | a < b + c ↔ (∃ b' < b, a ≤ b' + c) ∨ ∃ c' < c, a ≤ b + c' :=
Ordinal.lt_nadd_iff
theorem add_le_iff {a b c : NatOrdinal} :
| Mathlib/SetTheory/Ordinal/NaturalOps.lean | 331 | 334 |
/-
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.Algebra.Order.Group.OrderIso
import Mathlib.SetTheory.Game.Ordinal
import Mathlib.SetTheory.Ordinal.NaturalOps
/-!
# Birthdays... | unfold birthday
congr 1
all_goals
apply lsub_eq_of_range_eq.{u, u, u}
ext i; constructor
all_goals rintro ⟨j, rfl⟩
· exact ⟨_, (r.moveLeft j).birthday_congr.symm⟩
· exact ⟨_, (r.moveLeftSymm j).birthday_congr⟩
· exact ⟨_, (r.moveRight j).birthday_congr.symm⟩
· exact ⟨_, (r.mo... | Mathlib/SetTheory/Game/Birthday.lean | 81 | 93 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Andrew Zipperer, Haitao Zhang, Minchao Wu, Yury Kudryashov
-/
import Mathlib.Data.Set.Prod
import Mathlib.Data.Set.Restrict
/-!
# Functions over sets
This file contains... | Mathlib/Data/Set/Function.lean | 1,497 | 1,499 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Topology.Continuous
import Mathlib.Topology.Defs.Induced
/-!
# Ordering on topologies and (co)induced topologies
Topologies on a fixe... | (hs : s ∈ @nhds β (TopologicalSpace.coinduced π ‹_›) (π a)) : π ⁻¹' s ∈ 𝓝 a := by
letI := TopologicalSpace.coinduced π ‹_›
rcases mem_nhds_iff.mp hs with ⟨V, hVs, V_op, mem_V⟩
| Mathlib/Topology/Order.lean | 360 | 362 |
/-
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... |
@[simp]
theorem coe_comp_rangeFactorization (f : ι → β) : (↑) ∘ rangeFactorization f = f := rfl
theorem surjective_onto_range : Surjective (rangeFactorization f) := fun ⟨_, ⟨i, rfl⟩⟩ => ⟨i, rfl⟩
| Mathlib/Data/Set/Image.lean | 893 | 898 |
/-
Copyright (c) 2022 Richard M. Hill. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Richard M. Hill
-/
import Mathlib.Algebra.Module.Submodule.Invariant
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.LinearAlgebra.DFinsupp
import Mathlib.RingTheory.Finit... |
@[simp] lemma C_smul (t : R) (m : AEval R M a) : C t • m = t • m :=
| Mathlib/Algebra/Polynomial/Module/AEval.lean | 70 | 71 |
/-
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.Calculus.Deriv.Inv
import Mathlib.Analysis.Calculus.Deriv.MeanValue
/-!
# L'Hôpital's rule for 0/0 indeterminate forms
In this file, we ... | (hftop : Tendsto f atTop (𝓝 0)) (hgtop : Tendsto g atTop (𝓝 0))
(hdiv : Tendsto (fun x => f' x / g' x) atTop l) : Tendsto (fun x => f x / g x) atTop l := by
rw [eventually_iff_exists_mem] at *
rcases hff' with ⟨s₁, hs₁, hff'⟩
rcases hgg' with ⟨s₂, hs₂, hgg'⟩
rcases hg' with ⟨s₃, hs₃, hg'⟩
let s := s... | Mathlib/Analysis/Calculus/LHopital.lean | 330 | 336 |
/-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.EuclideanDomain.Int
import Mathlib.Algebra.MvPolynomial.Eval
import Mathlib.RingTheory.Adjoin.Basic
import Mathlib.RingTheory.Polynomial.Basic
import Mat... | (S.prod T).FG := by
obtain ⟨s, hs⟩ := fg_def.1 hS
obtain ⟨t, ht⟩ := fg_def.1 hT
rw [← hs.2, ← ht.2]
exact fg_def.2 ⟨LinearMap.inl R A B '' (s ∪ {1}) ∪ LinearMap.inr R A B '' (t ∪ {1}),
| Mathlib/RingTheory/Adjoin/FG.lean | 122 | 126 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Finset.Sum
import Mathlib.Data.Sum.Order
import Mathlib.Order.Interval.Finset.Defs
/-!
# Finite intervals in a disjoint union
This file provides the... | cases a <;> cases b
exacts [map_subset_map.2 (hf₁ _ _), disjSum_mono (hg₁ _ _) (hg₂ _ _), Subset.rfl,
map_subset_map.2 (hf₂ _ _)]
lemma sumLexLift_eq_empty :
| Mathlib/Data/Sum/Interval.lean | 171 | 175 |
/-
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.Analysis.SpecialFunctions.Trigonometric.Arctan
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
/-!
# Right-angled triangles
This file proves ba... | the opposite side, version subtracting vectors. -/
theorem tan_angle_sub_mul_norm_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) (h0 : x ≠ 0 ∨ y = 0) :
Real.tan (angle x (x - y)) * ‖x‖ = ‖y‖ := by
rw [← neg_eq_zero, ← inner_neg_right] at h
rw [← neg_eq_zero] at h0
| Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean | 283 | 287 |
/-
Copyright (c) 2021 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Sébastien Gouëzel
-/
import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
import Mathlib.MeasureTheory.Constructions.BorelSpace.Metric
import Mathlib.MeasureTheory... |
theorem addHaar_real_closedBall (x : E) {r : ℝ} (hr : 0 ≤ r) :
μ.real (closedBall x r) = r ^ finrank ℝ E * μ.real (ball 0 1) := by
simp [addHaar_real_closedBall' μ x hr, measureReal_def,
addHaar_unitClosedBall_eq_addHaar_unitBall]
theorem addHaar_closedBall_eq_addHaar_ball [Nontrivial E] (x : E) (r : ℝ) :
... | Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean | 501 | 514 |
/-
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.CharP.Reduced
import Mathlib.RingTheory.IntegralDomain
-- TODO: remove Mathlib.Algebra.CharP.Reduced and move the last two lemmas to Lemmas
/-... | Mathlib/RingTheory/RootsOfUnity/Basic.lean | 709 | 712 | |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Group.Multiset.Basic
/-!
# Bind operation for multisets
This file defines a few basic operations on `Multiset`, notably the mona... | (s.bind t).prod = (s.map fun a => (t a).prod).prod := by simp [bind]
open scoped Relator in
theorem rel_bind {r : α → β → Prop} {p : γ → δ → Prop} {s t} {f : α → Multiset γ}
{g : β → Multiset δ} (h : (r ⇒ Rel p) f g) (hst : Rel r s t) :
| Mathlib/Data/Multiset/Bind.lean | 170 | 174 |
/-
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... |
/-- If `t` is a measurable set, then the measure of `t` with respect to the restriction of
the measure to `s` equals the outer measure of `t ∩ s`. An alternate version requiring that `s`
| Mathlib/MeasureTheory/Measure/Restrict.lean | 62 | 64 |
/-
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... |
/-! ### Division on ordinals -/
| Mathlib/SetTheory/Ordinal/Arithmetic.lean | 808 | 810 |
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