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 |
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/-
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... | · rintro ⟨c, hc, rfl⟩
exact Or.inr (Or.inr <| Or.inl ⟨a, b, c, rfl, rfl, rfl, hc⟩)
· exact fun h ↦ (not_mem_empty _ h).elim
· rw [sumLexLift, mem_map]
rintro ⟨c, hc, rfl⟩
exact Or.inr (Or.inr <| Or.inr <| ⟨a, b, c, rfl, rfl, rfl, hc⟩)
· rintro (⟨a, b, c, rfl, rfl, rfl, hc⟩ | ⟨a, b, c... | Mathlib/Data/Sum/Interval.lean | 143 | 168 |
/-
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 | 1,695 | 1,695 | |
/-
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... | theorem eLpNorm_congr_nnnorm_ae {f : α → F} {g : α → G} (hfg : ∀ᵐ x ∂μ, ‖f x‖₊ = ‖g x‖₊) :
eLpNorm f p μ = eLpNorm g p μ :=
le_antisymm (eLpNorm_mono_nnnorm_ae <| EventuallyEq.le hfg)
(eLpNorm_mono_nnnorm_ae <| (EventuallyEq.symm hfg).le)
theorem eLpNorm_congr_norm_ae {f : α → F} {g : α → G} (hfg : ∀ᵐ x ∂μ, ... | Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean | 466 | 474 |
/-
Copyright (c) 2024 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Group.Action.Pi
import Mathlib.Algebra.Group.End
import Mathlib.Algebra.Module.NatInt
import Mathlib.Algebra.Order.Archimedean.Basic
/-!
# M... | @[scoped simp]
theorem map_sub_zsmul [AddGroup G] [AddGroup H] [AddConstMapClass F G H a b]
(f : F) (x : G) (n : ℤ) : f (x - n • a) = f x - n • b := by
| Mathlib/Algebra/AddConstMap/Basic.lean | 205 | 207 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Order.Filter.Tendsto
import Mathlib.Data.Set.Accumulate
import Mathlib.Topology.Bornology.Basic
import Mathlib.Topolog... | instance ULift.compactSpace [CompactSpace X] : CompactSpace (ULift.{v} X) :=
IsClosedEmbedding.uliftDown.compactSpace
| Mathlib/Topology/Compactness/Compact.lean | 929 | 931 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Michael Stoll
-/
import Mathlib.Data.Nat.Squarefree
import Mathlib.NumberTheory.Zsqrtd.QuadraticReciprocity
import Mathlib.NumberTheory.Padics.PadicVal.Basic
/-!
# Sums of... | conv_rhs => tactic => norm_cast
rw [(by norm_cast : (-1 : ZMod n.natAbs) = (-1 : ℤ))]
exact (ZMod.intCast_eq_intCast_iff_dvd_sub _ _ _).mpr (Int.natAbs_dvd.mpr ⟨_, H⟩)
/-- If the natural number `n` is a sum of two squares of coprime natural numbers, then
`-1` is a square modulo `n`. -/
theorem ZMod.isSquare_neg_... | Mathlib/NumberTheory/SumTwoSquares.lean | 161 | 172 |
/-
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.LinearAlgebra.DFinsupp
import Mathlib.LinearAlgebra.Dual.Basis
import Mathlib.LinearAlgebra.Matrix.ToLin
/-!... | LinearMap.toMatrix B₂.dualBasis B₁.dualBasis (Module.Dual.transpose (R := K) u) =
(LinearMap.toMatrix B₁ B₂ u)ᵀ := by
ext i j
simp only [LinearMap.toMatrix_apply, Module.Dual.transpose_apply, B₁.dualBasis_repr,
B₂.dualBasis_apply, Matrix.transpose_apply, LinearMap.comp_apply]
| Mathlib/LinearAlgebra/Matrix/Dual.lean | 32 | 37 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
/-!
# Kernels and cokernels
In a category with zero morphisms, the kernel of a morphism `f : X... | Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean | 1,198 | 1,202 | |
/-
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 | 2,207 | 2,207 | |
/-
Copyright (c) 2024 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Lie.CartanSubalgebra
import Mathlib.Algebra.Lie.Rank
/-!
# Existence of Cartan subalgebras
In this file we prove existence of Cartan subalge... | lemma lieCharpoly_map_eval (r : R) :
(lieCharpoly R M x y).map (evalRingHom r) = (φ (r • y + x)).charpoly := by
rw [lieCharpoly, map_map]
set b := chooseBasis R L
have aux : (fun i ↦ (b.repr y) i * r + (b.repr x) i) = b.repr (r • y + x) := by
ext i; simp [mul_comm r]
simp_rw [← coe_aeval_eq_evalRingHom,... | Mathlib/Algebra/Lie/CartanExists.lean | 90 | 98 |
/-
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.ModelTheory.Ultraproducts
import Mathlib.ModelTheory.Bundled
import Mathlib.ModelTheory.Skolem
import Mathlib.Order.Filter.AtTopBot.Basic
/-!
# First-... | end Theory
variable (L)
/-- A version of The Downward Löwenheim–Skolem theorem where the structure `N` elementarily embeds
into `M`, but is not by type a substructure of `M`, and thus can be chosen to belong to the universe
of the cardinal `κ`.
-/
theorem exists_elementaryEmbedding_card_eq_of_le (M : Type w') [L.Stru... | Mathlib/ModelTheory/Satisfiability.lean | 189 | 201 |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.AlgebraicGeometry.Gluing
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.AlgebraicGeometry.AffineScheme
import Mathlib.CategoryTheory.Limits.Sh... | · simp_rw [t_fst_fst, pullback.lift_fst, pullbackFstιToV_snd, GlueData.openCover_map]
· simp_rw [t_fst_snd, pullback.lift_snd, pullbackFstιToV_fst_assoc, pullback.condition_assoc,
GlueData.openCover_map, p2]
| Mathlib/AlgebraicGeometry/Pullbacks.lean | 345 | 347 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Yury Kudryashov
-/
import Mathlib.Algebra.Algebra.Rat
import Mathlib.Data.Nat.Prime.Int
import Mathlib.Data.Rat.Sqrt
imp... | theorem irrational_sqrt_ofNat_iff {n : ℕ} [n.AtLeastTwo] :
Irrational √(ofNat(n)) ↔ ¬IsSquare ofNat(n) :=
irrational_sqrt_natCast_iff
| Mathlib/Data/Real/Irrational.lean | 125 | 127 |
/-
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.Data.Matroid.IndepAxioms
/-!
# Matroid Duality
For a matroid `M` on ground set `E`, the collection of complements of the bases of `M` is the
collection o... | refine ⟨fun h ↦ ⟨_, h.1, diff_diff_cancel_left h.2⟩,
fun ⟨B', hB', h⟩ ↦ ⟨?_,h.symm.trans_subset diff_subset⟩⟩
rwa [← h, diff_diff_cancel_left hB'.subset_ground]
@[simp] theorem dual_dual (M : Matroid α) : M✶✶ = M :=
ext_isBase rfl (fun B (h : B ⊆ M.E) ↦
| Mathlib/Data/Matroid/Dual.lean | 151 | 156 |
/-
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... | intro z
simp [congrArg (· ≫ h.unitInv.app y) (h.unit.naturality z)]
apply Subsingleton.intro
intro a b
rw [this a, this b]
simp only [Functor.comp_map]
congr 3
apply ULift.ext
simp [eq_iff_true_of_subsingleton]
· rintro ⟨⟨p⟩, h⟩
haveI := fun x y => (h x y).some
refine
... | Mathlib/CategoryTheory/PUnit.lean | 73 | 102 |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Yury Kudryashov
-/
import Mathlib.Data.Finset.Fin
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Order.Interval.Set.Fin
/-!
# Finite intervals in `Fin n`
This fi... |
/-!
### Image under `Fin.castLE`
-/
@[simp]
theorem finsetImage_castLE_Icc (h : n ≤ m) :
| Mathlib/Order/Interval/Finset/Fin.lean | 215 | 221 |
/-
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... | ∀ (n : ℕ) (l : List α) (f : List α → β) (g : β → γ) (r : List β) (s : List γ),
sublistsLenAux n l (g ∘ f) (r.map g ++ s) = (sublistsLenAux n l f r).map g ++ s
| Mathlib/Data/List/Sublists.lean | 196 | 197 |
/-
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... | split_ifs with hs
· refine eq_of_forall_le_iff fun _ => ?_
simp_rw [hs.le_infsep_iff, imp_forall_iff, Finset.le_inf'_iff, Finite.mem_toFinset,
mem_offDiag, Prod.forall, uncurry_apply_pair, and_imp]
· rw [not_nontrivial_iff] at hs
exact hs.infsep_zero
| Mathlib/Topology/MetricSpace/Infsep.lean | 405 | 411 |
/-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
import Mathlib.Topology.Category.Profinite.Nobeling.Basic
import Mathlib.Topology.Category.Profinite.Nobeling.Induction
import Mathlib.Topology.Category.Profinite... | Mathlib/Topology/Category/Profinite/Nobeling.lean | 719 | 725 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Data.ENNReal.Action
import Mathlib.MeasureTheory.MeasurableSpace.Constructions
import Mathlib.MeasureTheory.OuterMeasure.Caratheodory
... | theorem trim_zero : (0 : OuterMeasure α).trim = 0 :=
ext fun s =>
le_antisymm
| Mathlib/MeasureTheory/OuterMeasure/Induced.lean | 358 | 360 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Bhavik Mehta, Stuart Presnell
-/
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Order.Monotone.Defs
/-!
# Binomial coefficients
This file defines binomial coeffic... | Mathlib/Data/Nat/Choose/Basic.lean | 404 | 406 | |
/-
Copyright (c) 2024 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Functor.KanExtension.Basic
import Mathlib.CategoryTheory.Localization.Predicate
/-!
# Right derived functors
In this file, given a functor `F : ... |
variable {F L W}
lemma HasRightDerivedFunctor.mk' [RF.IsRightDerivedFunctor α W] :
| Mathlib/CategoryTheory/Functor/Derived/RightDerived.lean | 181 | 184 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Notation.Pi
import Mathlib.Data.Set.Lattice
import Mathlib.Order.Filter.Defs
/-!
# Theory of... | Mathlib/Order/Filter/Basic.lean | 2,499 | 2,501 | |
/-
Copyright (c) 2014 Microsoft Corporation. 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.Algebra.Ring.Int.Defs
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.Cast.Order.Basic
import Math... | rfl
| Mathlib/Data/Num/Lemmas.lean | 238 | 238 |
/-
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.Ring.Divisibility.Lemmas
import Mathlib.Algebra.Lie.Nilpotent
import Mathlib.Algebra.Lie.Engel
import Mathlib.LinearAlgebra.Eigenspace.Pi
import Math... | (posFittingComp R L M).map e = posFittingComp R L M₂ := by
refine le_antisymm (map_posFittingComp_le _) ?_
suffices posFittingComp R L M₂ = ((posFittingComp R L M₂).map (e.symm : M₂ →ₗ⁅R,L⁆ M)).map e by
rw [this]
exact LieSubmodule.map_mono (map_posFittingComp_le _)
| Mathlib/Algebra/Lie/Weights/Basic.lean | 556 | 560 |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Algebra.Order.Monoid.Unbundled.WithTop
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
/-! # Adjoi... | Mathlib/Algebra/Order/Monoid/WithTop.lean | 294 | 302 | |
/-
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... | lemma cast_ncard {s : Set α} (hs : s.Finite) :
(s.ncard : Cardinal) = Cardinal.mk s := @Nat.cast_card _ hs
| Mathlib/Data/Set/Card.lean | 524 | 525 |
/-
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... | theorem ncard_singleton_inter (a : α) (s : Set α) : ({a} ∩ s).ncard ≤ 1 := by
rw [← Nat.cast_le (α := ℕ∞), (toFinite _).cast_ncard_eq, Nat.cast_one]
apply encard_singleton_inter
@[simp]
| Mathlib/Data/Set/Card.lean | 580 | 584 |
/-
Copyright (c) 2022 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Data.Nat.Cast.Field
import Mathlib.NumberTheory.ArithmeticFunction
/-!
# The von Mangoldt Function
In ... | exact (Real.log_pos (one_lt_cast.2 (minFac_prime hn).one_lt)).ne'.ite_ne_right_iff
| Mathlib/NumberTheory/VonMangoldt.lean | 90 | 91 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Batteries.Data.String.Lemmas
import Mathlib.Data.List.Lex
import Mathlib.Data.Char
import Mathlib.Algebra.Order.Group.Nat
/-!
# Strings
Supplementary theorem... | | ⟨s₁⟩, ⟨s₂⟩ => show ltb ⟨⟨s₁⟩, 0⟩ ⟨⟨s₂⟩, 0⟩ ↔ s₁ < s₂ by
induction s₁ generalizing s₂ <;> cases s₂
· unfold ltb; decide
· rename_i c₂ cs₂; apply iff_of_true
· unfold ltb
simp [Iterator.hasNext, Char.utf8Size_pos]
· apply List.nil_lt_cons
· rename_i c₁ cs₁ ih; apply iff_of_false
... | Mathlib/Data/String/Basic.lean | 77 | 103 |
/-
Copyright (c) 2024 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.Cardinal.Arithmetic
import Mathlib.SetTheory.Ordinal.Principal
/-!
# Ordinal arithmetic with cardinals
This file co... | Mathlib/SetTheory/Cardinal/Ordinal.lean | 659 | 660 | |
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Equivalence
/-! Tools for compatibilities between Dold-Kan equivalences
The purpose of this file is to introduce tools which will enable the
con... | @[simps!]
def υ : eA.functor ≅ F ⋙ e'.inverse :=
calc
eA.functor ≅ eA.functor ⋙ 𝟭 A' := (Functor.leftUnitor _).symm
_ ≅ eA.functor ⋙ e'.functor ⋙ e'.inverse := isoWhiskerLeft _ e'.unitIso
_ ≅ (eA.functor ⋙ e'.functor) ⋙ e'.inverse := Iso.refl _
_ ≅ F ⋙ e'.inverse := isoWhiskerRight hF _
variable (ε ... | Mathlib/AlgebraicTopology/DoldKan/Compatibility.lean | 215 | 227 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Data.Countable.Small
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Powerset
import Mathlib.Dat... | Mathlib/SetTheory/Cardinal/Basic.lean | 1,289 | 1,290 | |
/-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad
-/
import Mathlib.Logic.Basic
import Mathlib.Logic.Function.Defs
import Mathlib.Order.Defs.LinearOrder
/-!
# Booleans
This file proves various... | Mathlib/Data/Bool/Basic.lean | 249 | 249 | |
/-
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... | theorem isOpen_iSup_iff {s : Set α} : IsOpen[⨆ i, t i] s ↔ ∀ i, IsOpen[t i] s :=
show s ∈ {s | IsOpen[iSup t] s} ↔ s ∈ { x : Set α | ∀ i : ι, IsOpen[t i] x } by
simp [setOf_isOpen_iSup]
| Mathlib/Topology/Order.lean | 879 | 882 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Group.Equiv.Basic
import Mathlib.Data.ENat.Lattice
import Mathlib.Data.Part
import Mathlib.Tactic.NormNum
/-!
# Natural numbers with infinity
The... | section WithTopEquiv
| Mathlib/Data/Nat/PartENat.lean | 605 | 606 |
/-
Copyright (c) 2022 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu, Anne Baanen
-/
import Mathlib.Algebra.Module.LocalizedModule.IsLocalization
import Mathlib.LinearAlgebra.Basis.Basic
import Mathlib.RingTheory.Localization.FractionRing
import ... | rw [← Finsupp.range_linearCombination]
rintro _ ⟨_, r, rfl⟩ _ ⟨_, r', rfl⟩ eq
congr; ext i
have ⟨s, eq⟩ := IsLocalizedModule.exists_of_eq (S := S) eq
| Mathlib/RingTheory/Localization/Module.lean | 73 | 76 |
/-
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... |
theorem setOf_set {s : Set α} : setOf s = s :=
| Mathlib/Data/Set/Basic.lean | 224 | 225 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.Bochner.Basic
import Mathlib.MeasureTheory.Integral.Bochner.L1
import Mathlib.Me... | Mathlib/MeasureTheory/Integral/Bochner.lean | 850 | 854 | |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl, Sander Dahmen, Kim Morrison, Chris Hughes, Anne Baanen
-/
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
import Mathlib.LinearAlgebra.Basis.Prod
impo... | /-- If `m` and `n` are finite and lie in the same universe, the rank of `m × n` matrices over a
module `M` is `(#m * #n).lift * rank R M`. -/
| Mathlib/LinearAlgebra/Dimension/Constructions.lean | 219 | 220 |
/-
Copyright (c) 2021 Aaron Anderson, Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Kevin Buzzard, Yaël Dillies, Eric Wieser
-/
import Mathlib.Data.Finset.Lattice.Union
import Mathlib.Data.Finset.Pairwise
import Mathlib.Data.Finset.Prod
i... | @[simp]
theorem iSupIndep_ne_bot :
iSupIndep (fun i : {i // t i ≠ ⊥} ↦ t i) ↔ iSupIndep t := by
refine ⟨fun h ↦ ?_, fun h ↦ h.comp Subtype.val_injective⟩
simp only [iSupIndep_def] at h ⊢
| Mathlib/Order/SupIndep.lean | 393 | 397 |
/-
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.Solvable
import Mathlib.Algebra.Lie.Quotient
import Mathlib.Algebra.Lie.Normalizer
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.... | rw [← LieSubmodule.toSubmodule_inj] at hk ⊢ <;>
simpa using hk
| Mathlib/Algebra/Lie/Nilpotent.lean | 527 | 528 |
/-
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.Data.List.Cycle
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.Perm.List
/-!
# Properties of cyclic permutations constructed... | rw [formPerm_eq_one_iff _ hn]
exact Nat.le_of_lt_succ hx
· rw [← mem_toFinset]
| Mathlib/GroupTheory/Perm/Cycle/Concrete.lean | 428 | 430 |
/-
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 | 2,318 | 2,322 | |
/-
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 | 2,322 | 2,323 | |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Analysis.Convex.Side
import Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
/-!
# Oriented an... |
/-- An oriented angle is zero if and only if the angle with the order of the points reversed is
| Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean | 170 | 171 |
/-
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.SetTheory.Cardinal.Arithmetic
/-!
# Cardinality of continuum
In this file we define `Cardinal.continuum` (notation: `𝔠`, localized in `Cardinal`) ... | rw [← lift_continuum.{v, u}, lift_le]
| Mathlib/SetTheory/Cardinal/Continuum.lean | 46 | 47 |
/-
Copyright (c) 2018 Michael Jendrusch. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Jendrusch, Kim Morrison, Bhavik Mehta, Jakob von Raumer
-/
import Mathlib.CategoryTheory.EqToHom
import Mathlib.CategoryTheory.Functor.Trifunctor
import Mathlib.CategoryTheo... | Mathlib/CategoryTheory/Monoidal/Category.lean | 1,012 | 1,015 | |
/-
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.Algebra.Module.BigOperators
import Mathlib.GroupTheory.Perm.Basic
import Mathlib.GroupTheory.Perm.Finite
import Mathlib.GroupTheory.Perm.Lis... | def IsCycle (f : Perm α) : Prop :=
∃ x, f x ≠ x ∧ ∀ ⦃y⦄, f y ≠ y → SameCycle f x y
theorem IsCycle.ne_one (h : IsCycle f) : f ≠ 1 := fun hf => by simp [hf, IsCycle] at h
@[simp]
theorem not_isCycle_one : ¬(1 : Perm α).IsCycle := fun H => H.ne_one rfl
protected theorem IsCycle.sameCycle (hf : IsCycle f) (hx : f x ≠... | Mathlib/GroupTheory/Perm/Cycle/Basic.lean | 225 | 234 |
/-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin, Lu-Ming Zhang
-/
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Algebra.Pi
import Mathlib.Algebra.BigOp... | Mathlib/Data/Matrix/Basic.lean | 1,100 | 1,103 | |
/-
Copyright (c) 2022 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll, Thomas Zhu, Mario Carneiro
-/
import Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity
/-!
# The Jacobi Symbol
We define the Jacobi symbol and prove its main pro... | (by
rw [List.mem_pmap, Int.gcd_eq_natAbs, Ne, Prime.not_coprime_iff_dvd]
simp_rw [legendreSym.eq_zero_iff _ _, intCast_zmod_eq_zero_iff_dvd,
mem_primeFactorsList (NeZero.ne b), ← Int.natCast_dvd, Int.natCast_dvd_natCast, exists_prop,
and_assoc, _root_.and_comm])
| Mathlib/NumberTheory/LegendreSymbol/JacobiSymbol.lean | 159 | 163 |
/-
Copyright (c) 2020 Paul van Wamelen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Paul van Wamelen
-/
import Mathlib.Data.Nat.Factors
import Mathlib.NumberTheory.FLT.Basic
import Mathlib.NumberTheory.PythagoreanTriples
import Mathlib.RingTheory.Coprime.Lemmas
impo... | decide
· exact ⟨b0, ⟨a0, ⟨c0, minimal_comm hf, hbp⟩⟩⟩
exact ⟨a0, ⟨b0, ⟨c0, hf, hap⟩⟩⟩
/-- We can assume that a minimal solution to `a ^ 4 + b ^ 4 = c ^ 2` has
`a` odd and `c` positive. -/
theorem exists_pos_odd_minimal {a b c : ℤ} (h : Fermat42 a b c) :
∃ a0 b0 c0, Minimal a0 b0 c0 ∧ a0 % 2 = 1 ∧ 0 < c... | Mathlib/NumberTheory/FLT/Four.lean | 124 | 136 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.CharP.Frobenius
import Mathlib.Algebra.CharP.Pi
import Mathlib.Algebra.CharP.Quotient
import Mathlib.Algebra.CharP.Subring
import Mathlib.Analysis.Specia... | open NNReal
variable [hp : Fact p.Prime]
theorem mul_ne_zero_of_pow_p_ne_zero {x y : ModP O p} (hx : x ^ p ≠ 0) (hy : y ^ p ≠ 0) :
x * y ≠ 0 := by
obtain ⟨r, rfl⟩ := Ideal.Quotient.mk_surjective x
obtain ⟨s, rfl⟩ := Ideal.Quotient.mk_surjective y
| Mathlib/RingTheory/Perfection.lean | 420 | 427 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Pairwise
import Mathlib.Data.Set.BooleanAlgebra
/-!
# The set lattice
This file is a collectio... | ⋂ (x) (h : x = b), s x h = s b rfl :=
iInf_iInf_eq_left
| Mathlib/Data/Set/Lattice.lean | 478 | 479 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique
import Mathlib.MeasureTheory.Function.L2Space
/-... | rw [lintegral_eq_zero_iff] at h_nnnorm_eq_zero
· refine h_nnnorm_eq_zero.mono fun x hx => ?_
dsimp only at hx
rw [Pi.zero_apply] at hx ⊢
· rwa [ENNReal.coe_eq_zero, nnnorm_eq_zero] at hx
· refine Measurable.coe_nnreal_ennreal (Measurable.nnnorm ?_)
exact (Lp.stronglyMeasurable _).mea... | Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean | 206 | 221 |
/-
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.Algebra.Field.Rat
import Mathlib.Algebra.Group.Indicator
import Mathlib.Algebra.GroupWithZero.Action.End
import Mathlib.Algebra... |
/-! ### Numerator and denominator -/
| Mathlib/Data/NNRat/Lemmas.lean | 58 | 59 |
/-
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... | cases nonempty_fintype ι
refine eventually_atTop.2
⟨s.biUnion fun J => Finset.univ.biUnion fun i => {(i, J.lower i), (i, J.upper i)},
fun t ht I J hJ J' hJ' => not_disjoint_imp_le_of_subset_of_mem_splitMany (fun i => ?_) hJ'⟩
exact fun p hp =>
ht (Finset.mem_biUnion.2 ⟨J, hJ, Finset.mem_biUnion.2 ⟨i... | Mathlib/Analysis/BoxIntegral/Partition/Split.lean | 285 | 298 |
/-
Copyright (c) 2024 Lawrence Wu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lawrence Wu
-/
import Mathlib.Analysis.Fourier.Inversion
import Mathlib.Analysis.MellinTransform
/-!
# Mellin inversion formula
We derive the Mellin inversion formula as a consequence o... |
theorem mellin_eq_fourierIntegral (f : ℝ → E) {s : ℂ} :
mellin f s = 𝓕 (fun (u : ℝ) ↦ (Real.exp (-s.re * u) • f (Real.exp (-u)))) (s.im / (2 * π)) :=
calc
mellin f s
= ∫ (u : ℝ), Complex.exp (-s * u) • f (Real.exp (-u)) := by
rw [mellin, ← rexp_neg_image_aux, integral_image_eq_integral_abs_deriv... | Mathlib/Analysis/MellinInversion.lean | 44 | 67 |
/-
Copyright (c) 2022 Yaël Dillies, Sara Rousta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Sara Rousta
-/
import Mathlib.Logic.Equiv.Set
import Mathlib.Order.Interval.Set.OrderEmbedding
import Mathlib.Order.SetNotation
/-!
# Properties of unbundled ... | Mathlib/Order/UpperLower/Basic.lean | 1,716 | 1,717 | |
/-
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 [braiding_tensorUnit_right]
monoidal
| Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean | 347 | 349 |
/-
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... | instance colimMap_epi' {F G : J ⥤ C} [HasColimitsOfShape J C] (α : F ⟶ G) [Epi α] :
Epi (colimMap α) :=
(colim : (J ⥤ C) ⥤ C).map_epi α
instance colimMap_epi {F G : J ⥤ C} [HasColimit F] [HasColimit G] (α : F ⟶ G) [∀ j, Epi (α.app j)] :
Epi (colimMap α) :=
⟨fun {Z} u v h =>
colimit.hom_ext fun j => (ca... | Mathlib/CategoryTheory/Limits/HasLimits.lean | 1,064 | 1,072 |
/-
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 `... | forall_mem_image₂
| Mathlib/Data/Finset/Sups.lean | 542 | 542 |
/-
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.Finite.Defs
import Mathlib.Data.Finset.BooleanAlgebra
import Mathlib.Data.Finset.Image
import Mathlib.Data.Fintype.Defs
import Mathlib.Data.Fintyp... | Mathlib/Data/Fintype/Basic.lean | 727 | 729 | |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark
-/
import Mathlib.Algebra.Polynomial.Monic
/-!
# Lemmas for the interaction between polynomials and `∑` and `∏`.
Recall that `∑` and `∏` are notation for ... | · simp
· simp
· simpa using h
theorem coeff_zero_multiset_prod : t.prod.coeff 0 = (t.map fun f => coeff f 0).prod := by
refine Multiset.induction_on t ?_ fun a t ht => ?_; · simp
rw [Multiset.prod_cons, Multiset.map_cons, Multiset.prod_cons, Polynomial.mul_coeff_zero, ht]
| Mathlib/Algebra/Polynomial/BigOperators.lean | 225 | 231 |
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.FDeriv.Prod
import Mathlib.Analysis.C... | Mathlib/Analysis/BoundedVariation.lean | 133 | 136 | |
/-
Copyright (c) 2021 Benjamin Davidson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Log.NegMulLog
import Mathlib.Analysis.SpecialFunctions.NonIntegrable
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv... | Mathlib/Analysis/SpecialFunctions/Integrals.lean | 826 | 829 | |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Jakob von Raumer
-/
import Mathlib.Data.List.Chain
import Mathlib.CategoryTheory.PUnit
import Mathlib.CategoryTheory.Groupoid
import Mathlib.CategoryTheory.Category.ULift
... | IsPreconnected J :=
IsPreconnected.of_any_functor_const_on_obj fun F =>
h F.obj fun f => by ext; exact Discrete.eq_of_hom (F.map f)
/-- `J` is connected if: given any function `F : J → α` which is constant for any
`j₁, j₂` for which there is a morphism `j₁ ⟶ j₂`, then `F` is constant.
| Mathlib/CategoryTheory/IsConnected.lean | 157 | 162 |
/-
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... | Mathlib/Data/List/Basic.lean | 3,421 | 3,422 | |
/-
Copyright (c) 2023 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Algebra.Group.Action.Pointwise.Finset
import Mathlib.GroupTheory.QuotientGroup.Defs
import Mathlib.Order.ConditionallyCompleteLattice.Basic
/-!
# Stabiliz... | lemma stabilizer_inf_stabilizer_le_stabilizer_inter :
stabilizer G s ⊓ stabilizer G t ≤ stabilizer G (s ∩ t) :=
stabilizer_inf_stabilizer_le_stabilizer_apply₂ fun _ ↦ smul_set_inter
@[to_additive]
lemma stabilizer_inf_stabilizer_le_stabilizer_sdiff :
| Mathlib/Algebra/Pointwise/Stabilizer.lean | 74 | 79 |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Thomas Read, Andrew Yang, Dagur Asgeirsson, Joël Riou
-/
import Mathlib.CategoryTheory.Adjunction.Mates
/-!
# Uniqueness of adjoints
This file shows that adjoints are uni... | (x : D) : (rightAdjointUniq adj1 adj2).inv.app x = (rightAdjointUniq adj2 adj1).hom.app x :=
rfl
@[reassoc (attr := simp)]
| Mathlib/CategoryTheory/Adjunction/Unique.lean | 131 | 134 |
/-
Copyright (c) 2017 Mario Carneiro. 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.SetTheory.Ordinal.Family
/-! # Ordinal exponential
In this file we define the power function and the lo... | theorem log_zero_left : ∀ b, log 0 b = 0 :=
log_of_left_le_one zero_le_one
@[simp]
theorem log_zero_right (b : Ordinal) : log b 0 = 0 := by
obtain hb | hb := lt_or_le 1 b
· rw [log_def hb, ← Ordinal.le_zero, pred_le, succ_zero]
apply csInf_le'
| Mathlib/SetTheory/Ordinal/Exponential.lean | 284 | 291 |
/-
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... | -/
def IsOpenMap.adjunction {X Y : TopCat} {f : X ⟶ Y} (hf : IsOpenMap f) :
hf.functor ⊣ Opens.map f where
| Mathlib/Topology/Category/TopCat/Opens.lean | 290 | 292 |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark, Kyle Miller, Lu-Ming Zhang
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Combinatorics.SimpleGraph.Connectivity.WalkCounting
import Math... | Mathlib/Combinatorics/SimpleGraph/AdjMatrix.lean | 281 | 291 | |
/-
Copyright (c) 2022 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu, Anne Baanen
-/
import Mathlib.Algebra.Module.LocalizedModule.IsLocalization
import Mathlib.LinearAlgebra.Basis.Basic
import Mathlib.RingTheory.Localization.FractionRing
import ... | hf t (fun i ↦ _ * (sec (v i)).2) (fun i ↦ _ * (sec (v i)).2) ?_ i hit
simp_rw [mul_smul, ← Submonoid.smul_def, hsec, ← map_smul, ← map_sum, eq]
| Mathlib/RingTheory/Localization/Module.lean | 97 | 98 |
/-
Copyright (c) 2021 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Riccardo Brasca, Adam Topaz, Jujian Zhang, Joël Riou
-/
import Mathlib.Algebra.Homology.Additive
import Mathlib.CategoryTheory.Abelian.Projective.Resolution
/-!
# Left-der... | have h₂ : (P.isoLeftDerivedToHomotopyCategoryObj F).inv =
(F.mapHomologicalComplex _ ⋙ HomotopyCategory.quotient _ _).map (lift (𝟙 X) _ _) :=
id_comp _
simp only [Functor.map_id, comp_id] at h₁
rw [assoc, ← cancel_epi ((HomotopyCategory.homologyFunctor _ _ 0).map
(P.isoLeftDerivedToHomotopyCatego... | Mathlib/CategoryTheory/Abelian/LeftDerived.lean | 302 | 319 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Group.Action.Pi
import Mathlib.Algebra.Order.AbsoluteValue.Basic
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.Mi... | mul_equiv_mul (const_equiv.mpr <| Eq.refl <| c • (1 : β)) hf
theorem pow_equiv_pow {f1 f2 : CauSeq β abv} (hf : f1 ≈ f2) (n : ℕ) : f1 ^ n ≈ f2 ^ n := by
induction n with
| zero => simp only [pow_zero, Setoid.refl]
| succ n ih => simpa only [pow_succ'] using mul_equiv_mul hf ih
end Ring
section IsDomain
va... | Mathlib/Algebra/Order/CauSeq/Basic.lean | 502 | 514 |
/-
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.... | ext
simp
| Mathlib/RingTheory/Polynomial/ScaleRoots.lean | 42 | 44 |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle
/-!
# Oriented angles in right-angled triangles.
T... |
/-- The sine of an angle in a right-angled triangle as a ratio of sides, version subtracting
vectors. -/
theorem sin_oangle_sub_right_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
Real.Angle.sin (o.oangle y (y - x)) = ‖x‖ / ‖y - x‖ := by
have hs : (o.oangle y (y - x)).sign = 1 := by
rw [o... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 302 | 308 |
/-
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 | 1,638 | 1,639 | |
/-
Copyright (c) 2023 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Algebra.Order.ToIntervalMod
import Mathli... | lemma GrowsPolynomially.add_isLittleO {f g : ℝ → ℝ} (hf : GrowsPolynomially f)
(hfg : g =o[atTop] f) : GrowsPolynomially fun x => f x + g x := by
intro b hb
have hb_ub := hb.2
rw [isLittleO_iff] at hfg
cases hf.eventually_atTop_nonneg_or_nonpos with
| inl hf' => -- f is eventually non-negative
have hf... | Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean | 401 | 505 |
/-
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 | 879 | 880 | |
/-
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.CharP.Invertible
import Mathlib.Algebra.Order.Interval.Set.Group
import Mathlib.Analysis.Convex.Basic
import Mathlib.Analysis.Convex.Segment
import... | theorem affineSegment_vsub_const_image (x y p : P) :
(· -ᵥ p) '' affineSegment R x y = affineSegment R (x -ᵥ p) (y -ᵥ p) :=
affineSegment_image ((AffineEquiv.vaddConst R p).symm : P →ᵃ[R] V) x y
| Mathlib/Analysis/Convex/Between.lean | 67 | 69 |
/-
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.TangentCone
import Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics
import Mathlib.Analysis.As... | hasFDerivAtFilter_iff_tendsto
theorem hasFDerivAt_iff_tendsto :
HasFDerivAt f f' x ↔ Tendsto (fun x' => ‖x' - x‖⁻¹ * ‖f x' - f x - f' (x' - x)‖) (𝓝 x) (𝓝 0) :=
| Mathlib/Analysis/Calculus/FDeriv/Basic.lean | 327 | 330 |
/-
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... |
end Cardinal
| Mathlib/Algebra/Quaternion.lean | 1,318 | 1,319 |
/-
Copyright (c) 2021 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Bhavik Mehta
-/
import Mathlib.Analysis.Calculus.Deriv.Support
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
import Mathlib.MeasureTheory.Function.Jacobian
imp... | integral_Ioi_of_hasDerivAt_of_tendsto' hderiv (integrableOn_Ioi_deriv_of_nonpos' hderiv g'neg hg)
hg
end IoiFTC
| Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean | 858 | 861 |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jens Wagemaker, Anne Baanen
-/
import Mathlib.Algebra.BigOperators.Finsupp.Basic
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.GroupWithZero.... | theorem divisor_closure_eq_closure [CancelCommMonoidWithZero α]
(x y : α) (hxy : x * y ∈ closure { r : α | IsUnit r ∨ Prime r}) :
x ∈ closure { r : α | IsUnit r ∨ Prime r} := by
obtain ⟨m, hm, hprod⟩ := exists_multiset_of_mem_closure hxy
induction m using Multiset.induction generalizing x y with
| empty =... | Mathlib/Algebra/BigOperators/Associated.lean | 82 | 100 |
/-
Copyright (c) 2019 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Yury Kudryashov
-/
import Mathlib.Analysis.Convex.Combination
import Mathlib.Analysis.Convex.Function
import Mathlib.Tactic.FieldSimp
/-!
# Jensen's inequality... | ∀ ⦃j⦄, j ∈ t → ∀ ⦃k⦄, k ∈ t → p j = p k := by
by_contra!; exact h_eq.not_lt <| hf.map_sum_lt h₀ h₁ hmem this
/-- A form of the **equality case of Jensen's equality**.
For a strictly concave function `f` and positive weights `w`, if
`f (∑ i ∈ t, w i • p i) = ∑ i ∈ t, w i • f (p i)`, then the points `p` are all e... | Mathlib/Analysis/Convex/Jensen.lean | 158 | 168 |
/-
Copyright (c) 2020 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
import Mathlib.Algebra.ContinuedFractions.ContinuantsRecurrence
import Mathlib.Algebra.ContinuedFractions.TerminatedStable
import Mathlib.Tactic.FieldSimp
import ... | simp only [succ_succ_nth_conv'Aux_eq_succ_nth_conv'Aux_squashSeq, convs',
squashGCF]
| Mathlib/Algebra/ContinuedFractions/ConvergentsEquiv.lean | 217 | 219 |
/-
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... |
@[simp] theorem ncard_empty (α : Type*) : (∅ : Set α).ncard = 0 := by
| Mathlib/Data/Set/Card.lean | 556 | 557 |
/-
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, David Loeffler
-/
import Mathlib.Analysis.Convex.Slope
import Mathlib.Analysis.Calculus.Deriv.MeanValue
/-!
# Convexity of functions and derivat... | -/
/-- If `f : ℝ → ℝ` is convex on `S` and right-differentiable at `x ∈ S`, then the slope of any
secant line with left endpoint at `x` is bounded below by the right derivative of `f` at `x`. -/
lemma le_slope_of_hasDerivWithinAt_Ioi (hfc : ConvexOn ℝ S f)
(hx : x ∈ S) (hy : y ∈ S) (hxy : x < y) (hf' : HasDerivWit... | Mathlib/Analysis/Convex/Deriv.lean | 549 | 559 |
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.ConditionalProbability
import Mathlib.Probability.Kernel.Basic
import Mathlib.Probability.Kernel.Composition.MeasureComp
import Mathlib.Tactic.... | rw [← ae_all_iff] at ihf
filter_upwards [ihf] with a ha
rw [inter_iUnion, measure_iUnion, measure_iUnion hfd fun i ↦ h2 _ (hfm i)]
· simp only [ENNReal.tsum_mul_left, ha]
· exact hfd.mono fun i j h ↦ (h.inter_left' _).inter_right' _
· exact fun i ↦ .inter ht1m (h2 _ <| hfm i)
/-- The measurable... | Mathlib/Probability/Independence/Kernel.lean | 510 | 519 |
/-
Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
import Mathlib.Data.ENNReal.Real
import Mathlib.Tactic.Bound.Attribute
import Mathlib.Topology... | of the uniformity filter. -/
theorem uniformity_basis_dist_le :
(𝓤 α).HasBasis ((0 : ℝ) < ·) fun ε => { p : α × α | dist p.1 p.2 ≤ ε } :=
Metric.mk_uniformity_basis_le (fun _ => id) fun ε ε₀ => ⟨ε, ε₀, le_refl ε⟩
| Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | 681 | 685 |
/-
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... | Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean | 498 | 504 | |
/-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Data.Fintype.Quotient
import Mathlib.ModelTheory.Semantics
/-!
# Quotients of First-Order Structures
This file defines prestructures and quotients of... | rw [Quotient.finChoice_eq, Quotient.map_mk]
theorem relMap_quotient_mk' {n : ℕ} (r : L.Relations n) (x : Fin n → M) :
(RelMap r fun i => (⟦x i⟧ : Quotient s)) ↔ @RelMap _ _ ps.toStructure _ r x := by
change
Quotient.lift (@RelMap L M ps.toStructure n r) Prestructure.rel_equiv (Quotient.finChoice _) ↔
| Mathlib/ModelTheory/Quotients.lean | 57 | 62 |
/-
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.Field.IsField
import Mathlib.Algebra.Polynomial.Inductions
import Mathlib.Algebra.Polynomial.Monic
im... | theorem modByMonic_one (p : R[X]) : p %ₘ 1 = 0 :=
(modByMonic_eq_zero_iff_dvd (by convert monic_one (R := R))).2 (one_dvd _)
@[simp]
theorem divByMonic_one (p : R[X]) : p /ₘ 1 = p := by
conv_rhs => rw [← modByMonic_add_div p monic_one]; simp
| Mathlib/Algebra/Polynomial/Div.lean | 417 | 422 |
/-
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 interpolate_eq_iff_values_eq_on (hvs : Set.InjOn v s) :
interpolate s v r = interpolate s v r' ↔ ∀ i ∈ s, r i = r' i :=
⟨values_eq_on_of_interpolate_eq _ _ hvs, interpolate_eq_of_values_eq_on _ _⟩
theorem eq_interpolate {f : F[X]} (hvs : Set.InjOn v s) (degree_f_lt : f.degree < #s) :
| Mathlib/LinearAlgebra/Lagrange.lean | 346 | 351 |
/-
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.Limits.HasLimits
import Mathlib.CategoryTheory.Discrete.Basic
/-!
# Categorical (co)products
This file defines (co)products ... | /-- Construct a morphism between categorical coproducts from a family of morphisms between the
factors. -/
def Sigma.map' {f : α → C} {g : β → C} [HasCoproduct f] [HasCoproduct g] (p : α → β)
| Mathlib/CategoryTheory/Limits/Shapes/Products.lean | 446 | 448 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kim Morrison
-/
import Mathlib.Algebra.BigOperators.Finsupp.Basic
import Mathlib.Algebra.BigOperators.Group.Finset.Preimage
import Mathlib.Algebra.Module.Defs
import Ma... | Mathlib/Data/Finsupp/Basic.lean | 1,753 | 1,764 | |
/-
Copyright (c) 2018 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, Winston Yin
-/
import Mathlib.Algebra.Group.End
import Mathlib.Topology.EMetricSpace.Diam
/-!
# Lipschitz co... | | i::l => by
simp only [List.map_cons, List.prod_cons]
exact (h i).mul_end (LipschitzWith.list_prod f K h l)
protected theorem pow_end {f : Function.End α} {K} (h : LipschitzWith K f) :
∀ n : ℕ, LipschitzWith (K ^ n) (f ^ n : Function.End α)
| Mathlib/Topology/EMetricSpace/Lipschitz.lean | 281 | 286 |
/-
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.Projection
import Mathlib.Analysis.Normed.Lp.lpSpace
import Mathlib.Analysis.InnerProductSpace.PiL2
/-!
# Hilbert sum of ... | protected theorem hasSum_inner_mul_inner (b : HilbertBasis ι 𝕜 E) (x y : E) :
HasSum (fun i => ⟪x, b i⟫ * ⟪b i, y⟫) ⟪x, y⟫ := by
| Mathlib/Analysis/InnerProductSpace/l2Space.lean | 457 | 458 |
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