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) 2018 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Kim Morrison
-/
import Mathlib.CategoryTheory.Opposites
/-!
# Morphisms from equations between objects.
When working categorically, sometimes one encounters an equation `h ... | F.mapIso (eqToIso p) ≪≫ F.mapIso (eqToIso q) = F.mapIso (eqToIso <| p.trans q) := by aesop_cat
@[simp]
theorem eqToHom_app {F G : C ⥤ D} (h : F = G) (X : C) :
| Mathlib/CategoryTheory/EqToHom.lean | 334 | 337 |
/-
Copyright (c) 2022 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
import Mathlib.LinearAlgebra.LinearPMap
import Mathlib.Algebra.Equiv.TransferInstance
import Mathlib.Logic.Small.Basic
import Mathlib.RingTheory.Ideal.Defs
/-!
# Injecti... | end lifting_property
| Mathlib/Algebra/Module/Injective.lean | 439 | 447 |
/-
Copyright (c) 2021 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Order.Module.Algebra
import Mathlib.Algebra.Ring.Subring.Units
import Mathlib.LinearAlgebra.LinearIndepende... | theorem exists_pos_right (h : SameRay R x y) (hx : x ≠ 0) (hy : y ≠ 0) :
∃ r : R, 0 < r ∧ x = r • y :=
(h.symm.exists_pos_left hy hx).imp fun _ => And.imp_right Eq.symm
| Mathlib/LinearAlgebra/Ray.lean | 571 | 573 |
/-
Copyright (c) 2022 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Patrick Massot
-/
import Mathlib.Topology.Neighborhoods
/-!
# Neighborhoods of a set
In this file we define the filter `𝓝ˢ s` or `nhdsSet s` consisting of all ne... | mem_nhdsSet_iff_exists
| Mathlib/Topology/NhdsSet.lean | 63 | 64 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Joey van Langen, Casper Putz
-/
import Mathlib.Data.Nat.Cast.Basic
import Mathlib.Data.Nat.Find
import Mathlib.Data.Nat.Prime.Defs
import Mathlib.Order.Lattice
/-!
# Characteris... | _ = (1 : ℕ) * r := by rw [Nat.cast_one]
_ = 0 * r := by rw [CharP.cast_eq_zero]
_ = 0 := by rw [zero_mul]
lemma false_of_nontrivial_of_char_one [Nontrivial R] [CharP R 1] : False := by
have : Subsingleton R := CharOne.subsingleton
exact false_of_nontrivial_of_subsingleton R
lemma ringChar_ne_one... | Mathlib/Algebra/CharP/Defs.lean | 262 | 278 |
/-
Copyright (c) 2017 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Tim Baumann, Stephen Morgan, Kim Morrison, Floris van Doorn
-/
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.ObjectProperty.FullSubcategory
import Mat... | dsimp [adjointifyη,Trans.trans]
simp only [comp_id, assoc, map_comp]
have := ε.hom.naturality (F.map (η.inv.app X)); dsimp at this; rw [this]; clear this
rw [← assoc _ _ (F.map _)]
have := ε.hom.naturality (ε.inv.app <| F.obj X); dsimp at this; rw [this]; clear this
have := (ε.app <| F.obj X).hom_inv_id; ds... | Mathlib/CategoryTheory/Equivalence.lean | 259 | 267 |
/-
Copyright (c) 2014 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Leonardo de Moura, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Ord... | simpa using lt_inv_comm₀ ha hb
@[bound] lemma Bound.one_lt_div_of_pos_of_lt (b0 : 0 < b) : b < a → 1 < a / b := (one_lt_div b0).mpr
| Mathlib/Algebra/Order/Field/Basic.lean | 99 | 101 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.Hom.Basic
/-!
# Bounded order homomorphisms
This file defines (bounded) order homomorphisms.
We use the `DFunLike` design, so each type of morphis... | instance : Min (TopHom α β) :=
⟨fun f g => ⟨f ⊓ g, by rw [Pi.inf_apply, map_top, map_top, inf_top_eq]⟩⟩
instance : SemilatticeInf (TopHom α β) :=
| Mathlib/Order/Hom/Bounded.lean | 302 | 305 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Yaël Dillies
-/
import Mathlib.Order.Cover
import Mathlib.Order.Interval.Finset.Defs
/-!
# Intervals as finsets
This file provides basic results about all the `Finset.Ixx... | theorem Ico_inter_Ico {a b c d : α} : Ico a b ∩ Ico c d = Ico (max a c) (min b d) := by
rw [← coe_inj, coe_inter, coe_Ico, coe_Ico, coe_Ico, Set.Ico_inter_Ico]
theorem Ioc_inter_Ioc {a b c d : α} : Ioc a b ∩ Ioc c d = Ioc (max a c) (min b d) := by
| Mathlib/Order/Interval/Finset/Basic.lean | 842 | 845 |
/-
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.Basic
import Mathlib.Algebra.Module.End
import Mathlib.Algebra.Ring.Prod
import Mathlib.Data.Fintype.Units
import Mathlib.GroupTheory.GroupAc... | theorem cast_intCast (h : m ∣ n) (k : ℤ) : (cast (k : ZMod n) : R) = k :=
map_intCast (castHom h R) k
end CharDvd
section CharEq
/-! Some specialised simp lemmas which apply when `R` has characteristic `n`. -/
variable [CharP R n]
| Mathlib/Data/ZMod/Basic.lean | 335 | 346 |
/-
Copyright (c) 2020 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard
-/
import Mathlib.RingTheory.AdicCompletion.Basic
import Mathlib.RingTheory.LocalRing.MaximalIdeal.Basic
import Mathlib.RingTheory.LocalRing.RingHom.Basic
import Mathlib.R... | by_cases hm : m = 0
· simp only [hm, one_mul, pow_zero] at h ⊢
right
exact h
left
obtain ⟨m, rfl⟩ := Nat.exists_eq_succ_of_ne_zero hm
rw [pow_succ']
apply dvd_mul_of_dvd_left dvd_rfl _
· rw [Multiset.prod_replicate]
exact Classical.choose_spec (spec.2 hx)
t... | Mathlib/RingTheory/DiscreteValuationRing/Basic.lean | 198 | 224 |
/-
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.Order.Filter.Curry
import Mathlib.Data.Set.Countable
/-!
# Filters with countable intersection property
In this file we define `CountableInterFilte... | Mathlib/Order/Filter/CountableInter.lean | 302 | 306 | |
/-
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
-/
import Mathlib.Algebra.Order.SuccPred
import Mathlib.Data.Sum.Order
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Tactic.PPWithUniv
/-!
# ... | enum r ⟨typein r a, typein_lt_type r a⟩ = a :=
enum_type (PrincipalSeg.ofElement r a)
theorem enum_lt_enum {r : α → α → Prop} [IsWellOrder α r] {o₁ o₂ : Iio (type r)} :
r (enum r o₁) (enum r o₂) ↔ o₁ < o₂ :=
(enum _).map_rel_iff
theorem enum_le_enum (r : α → α → Prop) [IsWellOrder α r] {o₁ o₂ : Iio (type ... | Mathlib/SetTheory/Ordinal/Basic.lean | 461 | 471 |
/-
Copyright (c) 2022 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.Tactic.IntervalCases
/-!
# Cubics and discriminants
This file defines cubic polynomials ... |
theorem disc_ne_zero_iff_roots_nodup (ha : P.a ≠ 0) (h3 : (map φ P).roots = {x, y, z}) :
P.disc ≠ 0 ↔ (map φ P).roots.Nodup := by
rw [disc_ne_zero_iff_roots_ne ha h3, h3]
| Mathlib/Algebra/CubicDiscriminant.lean | 486 | 489 |
/-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Johan Commelin
-/
import Mathlib.Algebra.MvPolynomial.Equiv
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Logic.Equiv.Functor
import Mathlib.RingTheory.FreeRing
/-... | rcases this with ⟨w, H⟩
rw [← Polynomial.C_eq_intCast] at H
have : Polynomial.X.coeff 1 = (Polynomial.C ↑w).coeff 1 := by rw [H]; rfl
rwa [Polynomial.coeff_C, if_neg (one_ne_zero : 1 ≠ 0), Polynomial.coeff_X, if_pos rfl] at this
-- Porting note: Changed `(Subtype.val : s → α)` to `(↑)` in the type
theorem map_... | Mathlib/RingTheory/FreeCommRing.lean | 269 | 279 |
/-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson
-/
import Mathlib.Computability.Language
import Mathlib.Tactic.AdaptationNote
/-!
# Regular Expressions
This file contains the formal definition for regular expressions and ... | Mathlib/Computability/RegularExpressions.lean | 412 | 418 | |
/-
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.BigOperators.Group.Finset.Piecewise
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Pi
import Mathlib.Algebra.O... | /-- The `n = 1` case of the Ahlswede-Daykin inequality. Note that we can't just expand everything
out and bound termwise since `c₀ * d₁` appears twice on the RHS of the assumptions while `c₁ * d₀`
does not appear. -/
private lemma ineq [ExistsAddOfLE β] {a₀ a₁ b₀ b₁ c₀ c₁ d₀ d₁ : β}
(ha₀ : 0 ≤ a₀) (ha₁ : 0 ≤ a₁) (h... | Mathlib/Combinatorics/SetFamily/FourFunctions.lean | 65 | 88 |
/-
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 | 1,166 | 1,170 | |
/-
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.... | @[simp]
| Mathlib/Data/Fin/Basic.lean | 632 | 632 |
/-
Copyright (c) 2020 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Algebra.Group.Conj
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.Group.Subgroup.Ker
/-!
# Basic results on subgroups
We prove basic results... | Mathlib/Algebra/Group/Subgroup/Basic.lean | 1,686 | 1,687 | |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.RingTheory.FractionalIdeal.Basic
import Mathlib.RingTheory.IntegralClosure.IsIntegral.Basi... | rcases exist_integer_multiples_of_finset S I with ⟨⟨s, hs1⟩, hs⟩
rw [isFractional_span_iff]
exact ⟨s, hs1, hs⟩
theorem mem_span_mul_finite_of_mem_mul {I J : FractionalIdeal S P} {x : P} (hx : x ∈ I * J) :
∃ T T' : Finset P, (T : Set P) ⊆ I ∧ (T' : Set P) ⊆ J ∧ x ∈ span R (T * T' : Set P) :=
Submodule.mem_s... | Mathlib/RingTheory/FractionalIdeal/Operations.lean | 175 | 188 |
/-
Copyright (c) 2022 Pierre-Alexandre Bazin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Pierre-Alexandre Bazin
-/
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Algebra.Module.ZMod
import Mathlib.GroupTheory.Torsion
import Mathlib.LinearAlgebra.Isomorphism... |
/-- Torsion by a set is torsion by the ideal generated by it. -/
theorem torsionBySet_eq_torsionBySet_span :
| Mathlib/Algebra/Module/Torsion.lean | 263 | 265 |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Batteries.Data.List.Perm
import Mathlib.Data.List.OfFn
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.TakeWhile
import Mathlib.Order.Fin.Basic
/-!
# So... | @[simp]
theorem sorted_lt_listMap (e : α ↪o β) {l : List α} :
(l.map e).Sorted (· < ·) ↔ l.Sorted (· < ·) :=
| Mathlib/Data/List/Sort.lean | 273 | 275 |
/-
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.Regularity.Bound
import Mathlib.Combinatorics.SimpleGraph.Regularity.Equitabilise
import Mathlib.Comb... | refine mul_le_mul ?_ ?_ (cast_nonneg _) (by positivity)
· norm_cast
refine card_biUnion_le_card_mul _ _ _ fun x hx => ?_
apply card_le_m_add_one_of_mem_chunk_parts (h𝒜 hx)
· exact mod_cast m_le_card_of_mem_chunk_parts (h𝒜 hs)
private theorem one_sub_le_m_div_m_add_one_sq [Nonempty α]
| Mathlib/Combinatorics/SimpleGraph/Regularity/Chunk.lean | 208 | 214 |
/-
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.Algebra.BigOperators.Group.Finset.Piecewise
import Mathlib.Algebra.Group.Ext
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.BinaryProducts
import Ma... | (n : ∀ j k, g j ⟶ h k) :
biproduct.map m ≫ biproduct.matrix n = biproduct.matrix fun j k => m j ≫ n j k := by
ext
simp
| Mathlib/CategoryTheory/Preadditive/Biproducts.lean | 265 | 269 |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Algebra.GroupWithZero.Indicator
import Mathlib.Topology.Piecewise
import Mathlib.Topology.Instances.ENNReal.Lemmas
/-!
# Semicontinuous maps
A ... | variable {δ : Type*} [LinearOrder δ] [TopologicalSpace δ] [OrderTopology δ]
variable {ι : Type*} [TopologicalSpace ι]
| Mathlib/Topology/Semicontinuous.lean | 339 | 341 |
/-
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.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.Perm.Option
import Mathlib.Logic.Equiv.Fin.Rotate
import Mathlib.Logic.Equiv.Fintype
/-!
# Permutatio... | end CycleRange
section Sign
variable {n : ℕ}
| Mathlib/GroupTheory/Perm/Fin.lean | 280 | 284 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Anne Baanen
-/
import Mathlib.Algebra.Algebra.Equiv
import Mathlib.LinearAlgebra.Span.Basic
/-!
# Towers of algebras
In this file we prove basic facts about towers of algebra.
... | theorem of_algebraMap_eq' [Algebra R A]
(h : algebraMap R A = (algebraMap S A).comp (algebraMap R S)) : IsScalarTower R S A :=
of_algebraMap_eq <| RingHom.ext_iff.1 h
| Mathlib/Algebra/Algebra/Tower.lean | 110 | 112 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.SetTheory.Cardinal.Finite
import Mathlib.Topology.Algebra.InfiniteSum.Basic
import Mathlib.Topology.UniformSpace.Cauchy
import Mathlib.Topology.Algebra... | · refine tendsto_const_nhds.congr fun _ ↦ (tprod_eq_one_of_not_multipliable ?_).symm
rwa [Finset.multipliable_compl_iff]
/-- Product divergence test: if `f` is unconditionally multipliable, then `f x` tends to one along
`cofinite`. -/
@[to_additive "Series divergence test: if `f` is unconditionally summable, the... | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | 377 | 386 |
/-
Copyright (c) 2022 Praneeth Kolichala. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Praneeth Kolichala
-/
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Nat.BinaryRec
import Mathlib.Data.List.Defs
import Mathlib.Tactic.Convert
import Mathlib.Tactic.GeneralizePr... | Mathlib/Data/Nat/Bits.lean | 497 | 500 | |
/-
Copyright (c) 2022 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Tactic.NormNum.Eq
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Algebra.Order.Invertible
import Mathlib.Algebra.Order.Monoid.WithTop
import Ma... | /-- Helper function to synthesize typed `Field α` `LinearOrder α` `IsStrictOrderedRing α`
expressions. -/
def inferLinearOrderedField (α : Q(Type u)) : MetaM <|
(_ : Q(Field $α)) × (_ : Q(LinearOrder $α)) × Q(IsStrictOrderedRing $α) :=
let go := do
let field ← synthInstanceQ q(Field $α)
let linearOrder ← ... | Mathlib/Tactic/NormNum/Ineq.lean | 44 | 53 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.Homology
/-!
# Quasi-isomorphisms of short complexes
This file introduces the typeclass `QuasiIso φ` for a morphism `φ : S₁ ⟶ S₂... | lemma quasiIso_iff_comp_right (φ : S₁ ⟶ S₂) (φ' : S₂ ⟶ S₃) [hφ' : QuasiIso φ'] :
QuasiIso (φ ≫ φ') ↔ QuasiIso φ := by
constructor
· intro
exact quasiIso_of_comp_right φ φ'
· intro
exact quasiIso_comp φ φ'
| Mathlib/Algebra/Homology/ShortComplex/QuasiIso.lean | 75 | 81 |
/-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Algebra.Order.Pi
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Data.Real.Pointwise
/... | p.closedBall x r₁ ⊆ p.closedBall x r₂ := fun _ (hx : _ ≤ _) => hx.trans h
| Mathlib/Analysis/Seminorm.lean | 692 | 692 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Order.CauSeq.BigOperators
import Mathlib.Algebra.Order.Star.Basic
import Mathlib.Data.C... | Mathlib/Data/Complex/Exponential.lean | 1,194 | 1,194 | |
/-
Copyright (c) 2021 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.GroupWithZero.Action.Defs
import Mathlib.Algebra.Order.Interval.Finset.Basic
import Mathlib.Combinatorics.Additive.Frei... | · exact ha hc hd h
· exact mul_right_cancel h
· exact ha' hb hd h
· obtain rfl := ha hc hb ((mul_comm _ _).trans h)
exact ha' hb hc h
| Mathlib/Combinatorics/Additive/AP/Three/Defs.lean | 178 | 182 |
/-
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... | theorem setOf_exists (p : ι → β → Prop) : { x | ∃ i, p i x } = ⋃ i, { x | p i x } :=
ext fun _ => mem_iUnion.symm
| Mathlib/Data/Set/Lattice.lean | 132 | 133 |
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Mario Carneiro, Simon Hudon
-/
import Mathlib.Data.Fin.Fin2
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.Common
/-!
# Tuples of types, and their categorica... | Mathlib/Data/TypeVec.lean | 761 | 766 | |
/-
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... | (hV_closed i).inter W_op.isClosed_compl
rcases hV i j with ⟨k, hki, hkj⟩
refine ⟨k, ⟨fun x => ?_, fun x => ?_⟩⟩ <;> simp only [and_imp, mem_inter_iff, mem_compl_iff] <;>
| Mathlib/Topology/Compactness/Compact.lean | 497 | 499 |
/-
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 ... | exact
⟨x.2, (congr_arg (· ∈ T) (eq_mul_inv_of_mul_eq h1)).mp x.1.2, fun y hy =>
(Prod.ext_iff.mp (h2 ⟨⟨g * (↑y)⁻¹, hy⟩, y⟩ (inv_mul_cancel_right g y))).2⟩
· obtain ⟨x, h1, h2⟩ := h g
refine ⟨⟨⟨g * (↑x)⁻¹, h1⟩, x⟩, inv_mul_cancel_right g x, fun y hy => ?_⟩
have hf := h2 y.2 ((congr_arg (· ∈ T... | Mathlib/GroupTheory/Complement.lean | 232 | 243 |
/-
Copyright (c) 2023 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed
import Mathlib.FieldTheory.PrimitiveElement
import Mathlib.FieldTheory.IsAlgClosed.Basic
/-!
# Results about `minpoly... | · rw [← eval₂_map, ← minpolyDiv_spec] at hy
simpa [sub_eq_zero, Ne.symm h] using hy
lemma eval₂_minpolyDiv_self {T} [CommRing T] [Algebra R T] [IsDomain T] [DecidableEq T] (x : S)
(σ₁ σ₂ : S →ₐ[R] T) :
| Mathlib/FieldTheory/Minpoly/MinpolyDiv.lean | 62 | 66 |
/-
Copyright (c) 2024 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
import Mathlib.LinearAlgebra.PiTensorProduct
import Mathlib.Algebra.Algebra.Bilinear
import Mathlib.Algebra.Algebra.Equiv
import Mathlib.Data.Finset.NoncommProd
/-!
# T... | (ha : SemiconjBy a₁ a₂ a₃) :
SemiconjBy (tprod R a₁) (tprod R a₂) (tprod R a₃) := by
rw [SemiconjBy, tprod_mul_tprod, tprod_mul_tprod, ha]
| Mathlib/RingTheory/PiTensorProduct.lean | 68 | 71 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Order.Filter.SmallSets
import Mathlib.Topology.UniformSpace.Defs
import Mathlib.Topology.ContinuousOn
/-!
# Basic resu... | Mathlib/Topology/UniformSpace/Basic.lean | 1,934 | 1,936 | |
/-
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.Computability.Tape
import Mathlib.Data.Fintype.Option
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.PFun
import M... | structure Cfg where
/-- The current label to run (or `none` for the halting state) -/
l : Option Λ
/-- The internal state -/
var : σ
/-- The (finite) collection of internal stacks -/
stk : ∀ k, List (Γ k)
instance Cfg.inhabited [Inhabited σ] : Inhabited (Cfg Γ Λ σ) :=
⟨⟨default, default, default⟩⟩
| Mathlib/Computability/TuringMachine.lean | 140 | 150 |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Data.Finset.Fold
import Mathlib.Algebra.GCDMonoid.Multiset
/-!
# GCD and LCM operations on finsets
## Main definitions
- `Finset.gcd` - the greatest... | lcm_dvd fun b hb ↦ (h b hb).trans (dvd_lcm hb)
| Mathlib/Algebra/GCDMonoid/Finset.lean | 92 | 92 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` ... | lemma condExp_congr_ae_trim (hm : m ≤ m₀) (hfg : f =ᵐ[μ] g) :
μ[f|m] =ᵐ[μ.trim hm] μ[g|m] :=
StronglyMeasurable.ae_eq_trim_of_stronglyMeasurable hm
stronglyMeasurable_condExp stronglyMeasurable_condExp (condExp_congr_ae hfg)
| Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean | 221 | 224 |
/-
Copyright (c) 2022 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston
-/
import Mathlib.Algebra.Category.ModuleCat.Projective
import Mathlib.AlgebraicTopology.ExtraDegeneracy
import Mathlib.CategoryTheory.Abelian.Ext
import Mathlib.G... | actionDiagonalSucc_inv_apply]
rfl
theorem diagonalSucc_inv_single_left (g : G) (f : Gⁿ →₀ k) (r : k) :
(diagonalSucc k G n).inv.hom (Finsupp.single g r ⊗ₜ f) =
Finsupp.lift (Gⁿ⁺¹ →₀ k) k Gⁿ (fun f => single (g • partialProd f) r) f := by
refine f.induction ?_ ?_
· simp only [TensorProduct.tmul_zero... | Mathlib/RepresentationTheory/GroupCohomology/Resolution.lean | 203 | 221 |
/-
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.Analysis.Convex.Combination
import Mathlib.Analysis.Convex.Join
/-!
# Stone's separation theorem
This file proves Stone's separation theorem. This tells ... | ⟨hc.directedOn.convex_sUnion fun s hs => (hcS hs).1,
disjoint_sUnion_left.2 fun c hc => (hcS hc).2⟩,
fun s => subset_sUnion_of_mem⟩)
s ⟨hs, hst⟩
obtain hC : _ ∧ _ := hmax.prop
refine
⟨C, hC.1, convex_iff_segment_subset.2 fun x hx y hy z hz hzC => ?_, hsC, hC.2.subset_compl_... | Mathlib/Analysis/Convex/StoneSeparation.lean | 81 | 109 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker, Johan Commelin
-/
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib... | noncomputable abbrev aroots (p : T[X]) (S) [CommRing S] [IsDomain S] [Algebra T S] : Multiset S :=
(p.map (algebraMap T S)).roots
| Mathlib/Algebra/Polynomial/Roots.lean | 389 | 390 |
/-
Copyright (c) 2017 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Mario Carneiro
-/
import Mathlib.Algebra.Ring.CharZero
import Mathlib.Algebra.Star.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Order.Interval.Set.UnorderedInterva... | Mathlib/Data/Complex/Basic.lean | 264 | 264 | |
/-
Copyright (c) 2024 Lean FRO. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Data.List.InsertIdx
/-!
This is a stub file for importing `Mathlib.Data.List.InsertNth`,
which has been renamed to `Mathlib.Data.List.InsertIdx`.
This file c... | Mathlib/Data/List/InsertNth.lean | 52 | 54 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Dia... | simp [hab]
| Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | 92 | 92 |
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Data.Set.BooleanAlgebra
import Mathlib.Tactic.AdaptationNote
/-!
# Relations
This file defines bundled relations. A relation between `α` and `β` is a f... | rcases hx with ⟨hx, ⟨y, rxy⟩⟩
use y
| Mathlib/Data/Rel.lean | 268 | 269 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.SetTheory.Cardinal.Finite
import Mathlib.Topology.Algebra.InfiniteSum.Basic
import Mathlib.Topology.UniformSpace.Cauchy
import Mathlib.Topology.Algebra... | @[to_additive existing, deprecated (since := "2025-04-12")] alias tprod_eq_mul_tprod_ite :=
Multipliable.tprod_eq_mul_tprod_ite
end tprod
end IsTopologicalGroup
section IsUniformGroup
variable [CommGroup α] [UniformSpace α]
/-- The **Cauchy criterion** for infinite products, also known as the **Cauchy convergenc... | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | 200 | 222 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yaël Dillies
-/
import Mathlib.Logic.Equiv.Set
import Mathlib.Order.CompleteLattice.Lemmas
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection.Basic
/-... | end MinimalAxioms
/-- Construct a complete distrib lattice instance using the minimal amount of work needed.
| Mathlib/Order/CompleteBooleanAlgebra.lean | 230 | 232 |
/-
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 | 724 | 728 | |
/-
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 cosine of an angle in a right-angled triangle multiplied by the hypotenuse equals the
adjacent side. -/
theorem cos_oangle_right_mul_dist_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P}
(h : ∡ p₁ p₂ p₃ = ↑(π / 2)) : Real.Angle.cos (∡ p₂ p₃ p₁) * dist p₁ p₃ = dist p₃ p₂ := by
have hs : (∡ p₂ p₃ p₁).sign = 1 := by ... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 620 | 625 |
/-
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... | theorem angle_sub_le_pi_div_two_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) :
angle x (x - y) ≤ π / 2 := by
rw [← neg_eq_zero, ← inner_neg_right] at h
rw [sub_eq_add_neg]
exact angle_add_le_pi_div_two_of_inner_eq_zero h
| Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean | 232 | 236 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Order.CauSeq.BigOperators
import Mathlib.Algebra.Order.Star.Basic
import Mathlib.Data.C... | Mathlib/Data/Complex/Exponential.lean | 1,087 | 1,088 | |
/-
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 ... | Mathlib/Algebra/Lie/Submodule.lean | 1,441 | 1,442 | |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Alena Gusakov, Yaël Dillies
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Finset.Slice
import Mathlib.Data.Nat.BitIndices
import Mathlib.Order.SupClosed
import Math... | toColex (insert a s) < toColex (insert b s) ↔ a < b := by
rw [← cons_eq_insert _ _ ha, ← cons_eq_insert _ _ hb, cons_lt_cons]
| Mathlib/Combinatorics/Colex.lean | 270 | 271 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.FieldTheory.Minpoly.Field
import Mathlib.LinearAlgebra.SModEq
import Mathlib.RingTheory.Ideal.BigOperators
/-!
# Power basis
This file defines a structure ... | mem_supported', linearCombination, Finsupp.sum, Algebra.smul_def, eval₂_zero, exists_prop,
LinearMap.id_coe, eval₂_one, id, not_lt, Finsupp.coe_lsum, LinearMap.coe_smulRight,
Finset.mem_range, AlgHom.coe_mks, Finset.mem_coe]
simp_rw [@eq_comm _ y]
exact Iff.rfl
theorem mem_span_pow {x y : S} {d : ℕ} (h... | Mathlib/RingTheory/PowerBasis.lean | 90 | 103 |
/-
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.Algebra.Bilinear
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Group.Pointwise.Finset.Basic
import Mathlib.Algebra.Group.Pointwise.Set.B... |
protected theorem mul_one : M * 1 = M := by
conv_lhs => rw [one_eq_span, ← span_eq M]
rw [span_mul_span]
simp
protected theorem map_mul {A'} [Semiring A'] [Algebra R A'] (f : A →ₐ[R] A') :
map f.toLinearMap (M * N) = map f.toLinearMap M * map f.toLinearMap N :=
calc
map f.toLinearMap (M * N) = ⨆ i : M... | Mathlib/Algebra/Algebra/Operations.lean | 405 | 417 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Mario Carneiro, Yaël Dillies
-/
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Int.Order.Basic
import Mathlib.Logic.Function.Iterate
import Mathlib.Order.Compare
impor... | Mathlib/Order/Monotone/Basic.lean | 1,236 | 1,239 | |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.Polynomial.Lifts
import Mathlib.Grou... | Mathlib/RingTheory/Localization/Integral.lean | 463 | 478 | |
/-
Copyright (c) 2022 John Nicol. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: John Nicol
-/
import Mathlib.FieldTheory.Finite.Basic
/-!
# Wilson's theorem.
This file contains a proof of Wilson's theorem.
The heavy lifting is mostly done by the previous `wilsons_l... | Mathlib/NumberTheory/Wilson.lean | 101 | 104 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot, Yury Kudryashov, Rémy Degenne
-/
import Mathlib.Data.Set.Subsingleton
import Mathlib.Order.Interval.Set.Defs
/-!
# Intervals
In any pr... | Mathlib/Order/Interval/Set/Basic.lean | 1,351 | 1,354 | |
/-
Copyright (c) 2021 Hunter Monroe. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hunter Monroe, Kyle Miller, Alena Gusakov
-/
import Mathlib.Combinatorics.SimpleGraph.DeleteEdges
import Mathlib.Data.Fintype.Powerset
/-!
# Subgraphs of a simple graph
A subgraph of ... | end Subgraph
section MkProperties
| Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | 825 | 828 |
/-
Copyright (c) 2022 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Data.Nat.Cast.Field
import Mathlib.GroupTheory.Abelianization
import Mathlib.GroupTheory.GroupAction.CardComm... | by_cases h0 : n = 0
· rw [h0, reciprocalFactors_zero, commProb_cons, commProb_nil, mul_one, Nat.cast_zero, div_zero]
| Mathlib/GroupTheory/CommutingProbability.lean | 189 | 190 |
/-
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... | /-- The `vectorSpan` of a finite subset of an affinely independent
family has dimension one less than its cardinality. -/
theorem AffineIndependent.finrank_vectorSpan_image_finset [DecidableEq P]
{p : ι → P} (hi : AffineIndependent k p) {s : Finset ι} {n : ℕ} (hc : #s = n + 1) :
finrank k (vectorSpan k (s.image... | Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean | 100 | 115 |
/-
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_cthickening' (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} (δ_nn : 0 ≤ δ) (s : Set ℝ) (hsδ : s ⊆ Ioi δ)
(hs : ∀ ε, δ < ε → (s ∩ Ioc δ ε).Nonempty) (E : Set α) :
| Mathlib/Topology/MetricSpace/Thickening.lean | 453 | 458 |
/-
Copyright (c) 2020 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.FieldTheory.RatFunc.AsPolynomial
import Mathlib.NumberTheory.ArithmeticFunction
import Mathlib.RingTheory.Ro... | apply monic_prod_of_monic
intro i _
exact cyclotomic.monic i R
rw [@cyclotomic_eq_X_pow_sub_one_div R _ _ hpos, (div_modByMonic_unique P 0 prod_monic _).1]
refine ⟨by rwa [zero_add, mul_comm], ?_⟩
rw [degree_zero, bot_lt_iff_ne_bot]
intro h
exact Monic.ne_zero prod_monic (degree_eq... | Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean | 487 | 502 |
/-
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.Prod.Lex
import Mathlib.Data.Sigma.Lex
import Mathlib.Order.RelIso.Set
import Mathlib.Order.WellQuasiOrder
import Mathlib.Tactic.TFAE
/-!
# Well-... | (fun m hm => (if_pos hm).symm) ?_
swap
· simp only [if_neg (lt_irrefl (g 0)), Nat.sub_self]
rw [List.length_tail, ← Nat.pred_eq_sub_one]
exact Nat.pred_lt fun con => hnil _ (List.length_eq_zero_iff.1 con)
rw [IsBadSeq] at hf'
push_neg at hf'
obtain ⟨m, n, mn, hmn⟩ := hf' fun n x hx => by
spl... | Mathlib/Order/WellFoundedSet.lean | 761 | 781 |
/-
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.Analysis.InnerProductSpace.Convex
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Combinatorics.Additive.AP.Three... | quadrant. -/
| Mathlib/Combinatorics/Additive/AP/Three/Behrend.lean | 105 | 105 |
/-
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... | simp [(I.lower_lt_upper _).not_le]
| Mathlib/Analysis/BoxIntegral/Partition/Split.lean | 73 | 74 |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Complex.Basic
import Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle
/-!
# Closure, interior, and frontier of preimages under `re`... | @[simp]
theorem closure_setOf_im_lt (a : ℝ) : closure { z : ℂ | z.im < a } = { z | z.im ≤ a } := by
| Mathlib/Analysis/Complex/ReImTopology.lean | 104 | 105 |
/-
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
-/
import Mathlib.MeasureTheory.Integral.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Lebesgue.Countable
import Mathlib.MeasureTheory.Integral.Le... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 541 | 546 | |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Operations
import Mathlib.Order.Basic
import Mathlib.Order.BooleanAlgebra
import Mathlib.Tactic.Tauto
import Mathlib.Tactic.B... | Mathlib/Data/Set/Basic.lean | 2,401 | 2,406 | |
/-
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 two_lt_ncard_iff (hs : s.Finite := by toFinite_tac) :
2 < s.ncard ↔ ∃ a b c, a ∈ s ∧ b ∈ s ∧ c ∈ s ∧ a ≠ b ∧ a ≠ c ∧ b ≠ c := by
| Mathlib/Data/Set/Card.lean | 1,067 | 1,069 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Kevin Buzzard, Yury Kudryashov, Frédéric Dupuis,
Heather Macbeth
-/
import Mathlib.Algebra.Group.Subgroup.Map
import Mathlib.Algebra.Module.Submodule.... | rintro x ⟨m, hm, rfl⟩
exact add_mem_sup (mem_map_of_mem hm) (mem_map_of_mem hm)
| Mathlib/Algebra/Module/Submodule/Map.lean | 116 | 118 |
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Eval.SMul
import Mathlib.Algebra.Polyno... | apply taylor_injective r
rw [LinearMap.map_zero]
ext k
| Mathlib/Algebra/Polynomial/Taylor.lean | 116 | 118 |
/-
Copyright (c) 2021 Stuart Presnell. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stuart Presnell
-/
import Mathlib.Data.Nat.PrimeFin
import Mathlib.Data.Nat.Factorization.Defs
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Order.Interval.Finset.Nat
import... | theorem Prime.dvd_iff_one_le_factorization {p n : ℕ} (pp : Prime p) (hn : n ≠ 0) :
p ∣ n ↔ 1 ≤ n.factorization p :=
Iff.trans (by simp) (pp.pow_dvd_iff_le_factorization hn)
theorem exists_factorization_lt_of_lt {a b : ℕ} (ha : a ≠ 0) (hab : a < b) :
| Mathlib/Data/Nat/Factorization/Basic.lean | 198 | 202 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Order.Filter.Bases.Finit... | Mathlib/Topology/Algebra/Group/Basic.lean | 1,656 | 1,666 | |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Tactic.AdaptationNote
/-!
# Inversion in an affine sp... | theorem inversion_mul (c : P) (a R : ℝ) (x : P) :
| Mathlib/Geometry/Euclidean/Inversion/Basic.lean | 69 | 69 |
/-
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.Algebra.Subalgebra.Prod
import Mathlib.Algebra.Algebra.Subalgebra.Tower
import Mathlib.LinearAlgebra.Basis.Basic
import Mathlib.LinearAlgebra.Prod
/-!
# ... | Mathlib/RingTheory/Adjoin/Basic.lean | 390 | 411 | |
/-
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_repr (b : HilbertBasis ι 𝕜 E) (x : E) :
HasSum (fun i => b.repr x i • b i) x := by simpa using b.hasSum_repr_symm (b.repr x)
@[simp]
protected theorem dense_span (b : HilbertBasis ι 𝕜 E) :
(span 𝕜 (Set.range b)).topologicalClosure = ⊤ := by
classical
rw [eq_top_iff]
rintr... | Mathlib/Analysis/InnerProductSpace/l2Space.lean | 439 | 454 |
/-
Copyright (c) 2021 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Order.Module.Algebra
import Mathlib.Algebra.Ring.Subring.Units
import Mathlib.LinearAlgebra.LinearIndepende... | /-- Two vectors are in the same ray, or the first is in the same ray as the negation of the
second, if and only if they are not linearly independent. -/
theorem sameRay_or_sameRay_neg_iff_not_linearIndependent {x y : M} :
SameRay R x y ∨ SameRay R x (-y) ↔ ¬LinearIndependent R ![x, y] := by
| Mathlib/LinearAlgebra/Ray.lean | 505 | 508 |
/-
Copyright (c) 2022 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.RingTheory.Bezout
import Mathlib.RingTheory.LocalRing.Basic
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Localization.Integer
... | obtain ⟨c, h | h⟩ := PreValuationRing.cond a b
· rw [← h]
exact Ideal.mul_mem_right _ _ h₁
· exfalso; apply h₂; rw [← h]
apply Ideal.mul_mem_right _ _ hb
| Mathlib/RingTheory/Valuation/ValuationRing.lean | 282 | 286 |
/-
Copyright (c) 2022 Jake Levinson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jake Levinson
-/
import Mathlib.Data.Finset.Preimage
import Mathlib.Data.Finset.Prod
import Mathlib.Data.SetLike.Basic
import Mathlib.Order.UpperLower.Basic
/-!
# Young diagrams
A You... |
/-! ### The list of row lengths of a Young diagram
| Mathlib/Combinatorics/Young/YoungDiagram.lean | 347 | 348 |
/-
Copyright (c) 2024 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Kernel.Disintegration.Density
import Mathlib.Probability.Kernel.WithDensity
/-!
# Radon-Nikodym derivative and Lebesgue decomposition for kern... | (hf : Measurable (Function.uncurry f)) (a : α) (hξ : ξ a ⟂ₘ η a) :
f a =ᵐ[η a] ∂(κ a)/∂(η a) := by
have : κ a = ξ a + (η a).withDensity (f a) := by
rw [h, coe_add, Pi.add_apply, η.withDensity_apply hf, add_comm]
exact (κ a).eq_rnDeriv₀ (hf.comp measurable_prodMk_left).aemeasurable hξ this
omit hαγ in
l... | Mathlib/Probability/Kernel/RadonNikodym.lean | 489 | 496 |
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.AlgebraicGeometry.Cover.Open
import Mathlib.AlgebraicGeometry.Over
/-!
# Restriction of Schemes and Morphisms
## Main definition
- `AlgebraicGeometry.Schem... | def Scheme.restrictFunctor : X.Opens ⥤ Over X where
obj U := Over.mk U.ι
map {U V} i := Over.homMk (X.homOfLE i.le) (by simp)
map_id U := by
| Mathlib/AlgebraicGeometry/Restrict.lean | 275 | 278 |
/-
Copyright (c) 2024 Thomas Browning, Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Junyan Xu
-/
import Mathlib.Algebra.Group.Subgroup.Ker
import Mathlib.GroupTheory.GroupAction.Basic
import Mathlib.GroupTheory.GroupAction.FixedPoints
impo... | theorem closure_of_isSwap_of_isPretransitive [Finite α] {S : Set (Perm α)} (hS : ∀ σ ∈ S, σ.IsSwap)
[MulAction.IsPretransitive (Subgroup.closure S) α] : Subgroup.closure S = ⊤ := by
simp [eq_top_iff', mem_closure_isSwap hS, orbit_eq_univ, Set.toFinite]
/-- A transitive permutation group generated by transpositio... | Mathlib/GroupTheory/Perm/ClosureSwap.lean | 132 | 138 |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.DFinsupp.BigOperators
import Mathlib.Data.DFinsupp.Order
import Mathlib.Order.Interval.Finset.Basic
import Ma... | end Lattice
| Mathlib/Data/DFinsupp/Interval.lean | 181 | 182 |
/-
Copyright (c) 2021 Noam Atar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Noam Atar
-/
import Mathlib.Order.Ideal
import Mathlib.Order.PFilter
/-!
# Prime ideals
## Main definitions
Throughout this file, `P` is at least a preorder, but some sections require mo... | sup_mem (J.lower inf_le_right hyJ)
(hIJ.le <| I.lower inf_le_right <| IsPrime.mem_compl_of_not_mem ‹_› hyI)
end BooleanAlgebra
| Mathlib/Order/PrimeIdeal.lean | 176 | 179 |
/-
Copyright (c) 2018 Jan-David Salchow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jan-David Salchow, Patrick Massot, Yury Kudryashov
-/
import Mathlib.Topology.Defs.Sequences
import Mathlib.Topology.UniformSpace.Cauchy
/-!
# Sequences in topological spaces
In t... | @[deprecated (since := "2024-10-22")]
alias QuotientMap.sequentialSpace := IsQuotientMap.sequentialSpace
/-- The quotient of a sequential space is a sequential space. -/
instance Quotient.instSequentialSpace [SequentialSpace X] {s : Setoid X} :
SequentialSpace (Quotient s) :=
| Mathlib/Topology/Sequences.lean | 216 | 221 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Finset.Max
import Mathlib.Data.Fintype.EquivFin
import Mathlib.Data.Multiset.Sort
import Mathlib.Order.RelIso.Set
/-!
# Construct a sorted list f... | obtain ⟨i, hi⟩ : ∃ i, l.get i = s.min' H := List.mem_iff_get.1 this
rw [← hi]
exact (s.sort_sorted (· ≤ ·)).rel_get_of_le (Nat.zero_le i)
| Mathlib/Data/Finset/Sort.lean | 114 | 116 |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Batteries.Data.List.Perm
import Mathlib.Data.List.OfFn
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.TakeWhile
import Mathlib.Order.Fin.Basic
/-!
# So... | | [] => rfl
| b :: l => by
dsimp only [orderedInsert]
split_ifs with h <;> simp [takeWhile, dropWhile, *, orderedInsert_eq_take_drop a l]
theorem insertionSort_cons_eq_take_drop (a : α) (l : List α) :
| Mathlib/Data/List/Sort.lean | 411 | 416 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Finset.Sort
/-!
# Compositions
A compositio... | ext1
exact c.toCompositionAsSet_blocks
right_inv c := by
ext1
exact c.toComposition_boundaries
instance compositionFintype (n : ℕ) : Fintype (Composition n) :=
Fintype.ofEquiv _ (compositionEquiv n).symm
theorem composition_card (n : ℕ) : Fintype.card (Composition n) = 2 ^ (n - 1) := by
rw [← co... | Mathlib/Combinatorics/Enumerative/Composition.lean | 1,040 | 1,052 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Analytic.CPolynomialDef
/-!
# Linear functions are analytic
In this file we prove that a `Continuou... | HasFiniteFPowerSeriesOnBall f (f.fpowerSeries x) x 2 ∞ where
r_le := by simp
r_pos := ENNReal.coe_lt_top
hasSum := fun _ => (hasSum_nat_add_iff' 2).1 <| by
simp [Finset.sum_range_succ, ← sub_sub, hasSum_zero, fpowerSeries]
finite := by
| Mathlib/Analysis/Analytic/Linear.lean | 35 | 40 |
/-
Copyright (c) 2021 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kim Morrison
-/
import Mathlib.Algebra.Homology.ComplexShape
import Mathlib.CategoryTheory.Subobject.Limits
import Mathlib.CategoryTheory.GradedObject
import Mathlib.Alge... | dsimp [ComplexShape.prev]
rw [dif_neg]
push_neg; intro i hi
have : c.prev j = i := c.prev_eq' hi
| Mathlib/Algebra/Homology/HomologicalComplex.lean | 405 | 408 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Kappelmann
-/
import Mathlib.Algebra.Order.Floor.Defs
import Mathlib.Algebra.Order.Floor.Ring
import Mathlib.Algebra.Order.Floor.Semiring
deprecated_module (sinc... | Mathlib/Algebra/Order/Floor.lean | 1,405 | 1,407 |
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