Context stringlengths 227 76.5k | target stringlengths 0 11.6k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 16 3.69k |
|---|---|---|---|---|
/-
Copyright (c) 2022 YaΓ«l Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: YaΓ«l Dillies, Bhavik Mehta
-/
import Mathlib.Algebra.Order.Group.Unbundled.Int
import Mathlib.Algebra.Order.Nonneg.Basic
import Mathlib.Algebra.Order.Ring.Unbundled.Rat
imp... | Mathlib/Data/NNRat/Defs.lean | 432 | 432 | |
/-
Copyright (c) 2020 SΓ©bastien GouΓ«zel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: SΓ©bastien GouΓ«zel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Normed.Operator.BoundedLinear... | Β· exact (measurable_deriv_with_param hf).stronglyMeasurable
theorem aemeasurable_deriv_with_param [LocallyCompactSpace π] [MeasurableSpace π]
[OpensMeasurableSpace π] [MeasurableSpace F]
[BorelSpace F] {f : Ξ± β π β F} (hf : Continuous f.uncurry) (ΞΌ : Measure (Ξ± Γ π)) :
| Mathlib/Analysis/Calculus/FDeriv/Measurable.lean | 946 | 950 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes HΓΆlzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.Polynomial.Basi... | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 1,366 | 1,367 | |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Control.Combinators
import Mathlib.Data.Option.Defs
import Mathlib.Logic.IsEmpty
import Mathlib.Logic.Relator
import Mathlib.Util.CompileInductive
impo... |
theorem getD_default_eq_iget [Inhabited Ξ±] (o : Option Ξ±) :
o.getD default = o.iget := by cases o <;> rfl
| Mathlib/Data/Option/Basic.lean | 218 | 221 |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, Mario Carneiro, Oliver Nash
-/
import Mathlib.Data.Finset.Card
import Mathlib.Data.Finset.Union
/-!
# Finsets in product types
This file defines finset constru... | simp only [or_and_right, mem_union, mem_product]
@[simp]
theorem product_union [DecidableEq Ξ±] [DecidableEq Ξ²] : s ΓΛ’ (t βͺ t') = s ΓΛ’ t βͺ s ΓΛ’ t' := by
| Mathlib/Data/Finset/Prod.lean | 226 | 229 |
/-
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... | /-- This cannot be made an instance because of the `[Module (ZMod n) G]` argument and the fact that
`n` only appears in the second argument of `SMulMemClass`, which is an `OutParam`. -/
lemma smulMemClass : SMulMemClass S (ZMod n) G where smul_mem _ _ {_x} hx := zmod_smul_mem hx _
namespace AddSubgroupClass
instance ... | Mathlib/Data/ZMod/Basic.lean | 1,172 | 1,183 |
/-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Logic.Relation
import Mathlib.Logic.Unique
import Mathlib.Util.Notation3
/-!
# Quotient types
This module extends the core library's treatment of quo... | Lean.Meta.registerCoercion ``Setoid.r
(some { numArgs := 2, coercee := 1, type := .coeFun })
/-- When writing a lemma about `someSetoid x y` (which uses this instance),
call it `someSetoid_apply` not `someSetoid_r`. -/
instance : CoeFun (Setoid Ξ±) (fun _ β¦ Ξ± β Ξ± β Prop) where
| Mathlib/Data/Quot.lean | 26 | 31 |
/-
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, Sabbir Rahman
-/
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Algebra.Order.Ring.Pow
import Mathlib.Algeb... | let some n := (b'.getRevArg! 1).rawNatLit? | throwError "not a ^ -n where n is a literal"
guard (n % 2 = 0)
| Mathlib/Algebra/Order/Field/Power.lean | 133 | 134 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Jakob von Raumer
-/
import Mathlib.CategoryTheory.Category.Cat.AsSmall
import Mathlib.CategoryTheory.Comma.StructuredArrow.Basic
import Mathlib.CategoryTheory.IsConnected
i... | @zigzag_isConnected _ _ β¨uβ© fun f g =>
Relation.ReflTransGen.trans
(Relation.ReflTransGen.single
(show Zag f u from
Or.inl β¨CostructuredArrow.homMk (adj.homEquiv f.left d f.hom) (by simp [u])β©))
(Relation.ReflTransGen.single
(show Zag u g from
... | Mathlib/CategoryTheory/Limits/Final.lean | 142 | 152 |
/-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.OuterMeasure.OfFunction
import Mathlib.MeasureTheory.PiSystem
/-!
# The Caratheodory Ο-algebra of an outer measure
Give... | theorem measure_inter_union (h : sβ β© sβ β β
) (hβ : IsCaratheodory m sβ) {t : Set Ξ±} :
m (t β© (sβ βͺ sβ)) = m (t β© sβ) + m (t β© sβ) := by
rw [hβ, Set.inter_assoc, Set.union_inter_cancel_left, inter_diff_assoc, union_diff_cancel_left h]
theorem isCaratheodory_iUnion_lt {s : β β Set Ξ±} :
β {n : β}, (β i < n, Is... | Mathlib/MeasureTheory/OuterMeasure/Caratheodory.lean | 87 | 94 |
/-
Copyright (c) 2023 Josha Dekker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Josha Dekker
-/
import Mathlib.Topology.Bases
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.Compactness.SigmaCompact
/-!
# LindelΓΆf sets and LindelΓΆf spaces
## Mai... | instance Quot.LindelofSpace {r : X β X β Prop} [LindelofSpace X] : LindelofSpace (Quot r) where
isLindelof_univ := by
rw [β range_quot_mk]
exact isLindelof_range continuous_quot_mk
| Mathlib/Topology/Compactness/Lindelof.lean | 684 | 688 |
/-
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 | 1,893 | 1,901 | |
/-
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 HΓΆlzl, Yuyang Zhao
-/
import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
import Mathlib.Algebra.Order.ZeroLEOne
import Mathli... | (1 : Ξ±) β€ 2 :=
calc (1 : Ξ±) = 1 + 0 := (add_zero 1).symm
_ β€ 1 + 1 := add_le_add_left zero_le_one _
_ = 2 := one_add_one_eq_two
| Mathlib/Algebra/Order/Monoid/NatCast.lean | 44 | 47 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Batteries.Tactic.Congr
import Mathlib.Data.Option.Basic
import Mathlib.Data.Prod.Basic
import Mathlib.Data.Set.Subsingleton
import Mathlib.Dat... | theorem preimage_image_preimage {f : Ξ± β Ξ²} {s : Set Ξ²} : f β»ΒΉ' (f '' (f β»ΒΉ' s)) = f β»ΒΉ' s := by
rw [image_preimage_eq_range_inter, preimage_range_inter]
@[simp, mfld_simps]
| Mathlib/Data/Set/Image.lean | 742 | 745 |
/-
Copyright (c) 2019 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Yury Kudryashov, SΓ©bastien GouΓ«zel, Chris Hughes
-/
import Mathlib.Data.Fin.Rev
import Mathlib.Data.Nat.Find
/-!
# Operation on tuples
We interpret maps `β i : Fi... | variable {Ξ± : Sort*}
/-- Append a tuple of length `m` to a tuple of length `n` to get a tuple of length `m + n`.
| Mathlib/Data/Fin/Tuple/Basic.lean | 312 | 314 |
/-
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, Floris van Doorn, Amelia Livingston, Yury Kudryashov,
Neil Strickland, Aaron Anderson
-/
import Mathlib.Algebra.GroupWithZero.Units.Basic
import Mathli... |
section GroupWithZero
variable [GroupWithZero Ξ±]
/-- `β£` is not a useful definition if an inverse is available. -/
@[simp]
lemma GroupWithZero.dvd_iff {m n : Ξ±} : m β£ n β (m = 0 β n = 0) := by
refine β¨fun β¨a, haβ© hm => ?_, fun h => ?_β©
Β· simp [hm, ha]
Β· refine β¨mβ»ΒΉ * n, ?_β©
obtain rfl | hn := eq_or_ne n 0
... | Mathlib/Algebra/GroupWithZero/Divisibility.lean | 172 | 183 |
/-
Copyright (c) 2019 SΓ©bastien GouΓ«zel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: SΓ©bastien GouΓ«zel
-/
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Comp
/-!
# Derivative of `(f x) ^ n`, `n : β`
In this file we prove that `(x... | (hasDerivAt_pow n x).hasDerivWithinAt
theorem differentiableAt_pow : DifferentiableAt π (fun x : π => x ^ n) x :=
(hasDerivAt_pow n x).differentiableAt
theorem differentiableWithinAt_pow :
DifferentiableWithinAt π (fun x : π => x ^ n) s x :=
| Mathlib/Analysis/Calculus/Deriv/Pow.lean | 45 | 51 |
/-
Copyright (c) 2018 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp
-/
import Mathlib.Data.Nat.Find
import Mathlib.Algebra.Module.Pi
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
/-!
# Basic properties of holors
Holors... | nonrec theorem zero_mul {Ξ± : Type} [MulZeroClass Ξ±] (x : Holor Ξ± dsβ) : (0 : Holor Ξ± dsβ) β x = 0 :=
funext fun t => zero_mul (x (HolorIndex.drop t))
| Mathlib/Data/Holor.lean | 156 | 157 |
/-
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... | (Classical.choose_spec (exist_integer_multiples_of_finset M (p.support.image p.coeff))
(p.coeff i) (Finset.mem_image.mpr β¨i, hi, rflβ©))
Β· rw [RingHom.map_zero, not_mem_support_iff.mp hi, smul_zero]
-- Porting note: was `convert (smul_zero _).symm, ...`
theorem integerNormalization_map_to_map (p... | Mathlib/RingTheory/Localization/Integral.lean | 80 | 90 |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Algebra.Ring.Int.Defs
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.Size
import Batteries.Data.Int
/-!
# Bitwise operations on integers
Possi... |
@[simp]
theorem shiftLeft_natCast (m n : β) : (m : β€) <<< (n : β€) = β(m <<< n) := by
unfold_projs; simp
@[simp]
theorem shiftRight_natCast (m n : β) : (m : β€) >>> (n : β€) = m >>> n := by cases n <;> rfl
@[deprecated (since := "2025-03-10")] alias shiftLeft_coe_nat := shiftLeft_natCast
@[deprecated (since := "2025-... | Mathlib/Data/Int/Bitwise.lean | 352 | 367 |
/-
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.Data.ENat.Lattice
import Mathlib.Order.OrderIsoNat
import Mathlib.Tactic.TFAE
/-!
# Maximal length of chains
This file contains lemmas to work with the ma... | exact β¨l.take n, β¨hβ.take _, fun x h β¦ hβ _ <| take_subset _ _ hβ©,
(l.length_take).trans <| min_eq_left <| le_of_not_ge hββ©
Β· rw [ENat.iSup_coe_lt_top] at ha
obtain β¨β¨l, hβ, hββ©, e : l.length = _β© := Nat.sSup_mem (Set.range_nonempty _) ha
refine
β¨l.take n, β¨hβ.take _, fun x h β¦ hβ _ <| take_su... | Mathlib/Order/Height.lean | 93 | 106 |
/-
Copyright (c) 2018 Violeta HernΓ‘ndez Palacios, Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta HernΓ‘ndez Palacios, Mario Carneiro
-/
import Mathlib.Logic.Small.List
import Mathlib.SetTheory.Ordinal.Enum
import Mathlib.SetTheory.Ordinal.Exponen... | /-- `Ordinal.deriv` enumerates the fixed points of a normal function. -/
theorem deriv_eq_enumOrd (H : IsNormal f) : deriv f = enumOrd (Function.fixedPoints f) := by
convert derivFamily_eq_enumOrd fun _ : Unit => H
exact (Set.iInter_const _).symm
| Mathlib/SetTheory/Ordinal/FixedPoint.lean | 342 | 346 |
/-
Copyright (c) 2021 David WΓ€rn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David WΓ€rn, Joachim Breitner
-/
import Mathlib.Algebra.Group.Action.End
import Mathlib.Algebra.Group.Action.Pointwise.Set.Basic
import Mathlib.Algebra.Group.Submonoid.Membership
import Mat... | /-- A `NeWord M i j` is a representation of a non-empty reduced words where the first letter comes
from `M i` and the last letter comes from `M j`. It can be constructed from singletons and via
concatenation, and thus provides a useful induction principle. -/
inductive NeWord : ΞΉ β ΞΉ β Type _
| singleton : β {i : ΞΉ} ... | Mathlib/GroupTheory/CoprodI.lean | 619 | 630 |
/-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Order.Filter.Bases.Finit... | β V β π (1 : G), β v β V, β w β V, v / w β s := by
have : (fun p : G Γ G => p.1 * p.2β»ΒΉ) β»ΒΉ' s β π ((1, 1) : G Γ G) :=
| Mathlib/Topology/Algebra/Group/Basic.lean | 689 | 690 |
/-
Copyright (c) 2023 JoΓ«l Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: JoΓ«l Riou
-/
import Mathlib.CategoryTheory.Shift.Basic
/-!
# Functors which commute with shifts
Let `C` and `D` be two categories equipped with shifts by an additive monoid `A`. In this ... |
@[reassoc]
lemma app_shift (a : A) (X : C) :
Ο.app (Xβ¦aβ§) = (Fβ.commShiftIso a).hom.app X β« (Ο.app X)β¦aβ§' β«
(Fβ.commShiftIso a).inv.app X := by
simp [shift_app_comm_assoc Ο a X]
| Mathlib/CategoryTheory/Shift/CommShift.lean | 292 | 298 |
/-
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.Nat.Lattice
import Mathlib.Logic.Denumerable
import Mathlib.Logic.Function.Iterate
import Mathlib.Order.Hom.Basic
import Mathlib.Data.Set.Subsing... |
theorem orderEmbeddingOfSet_range [DecidablePred (Β· β s)] :
Set.range (Nat.orderEmbeddingOfSet s) = s :=
| Mathlib/Order/OrderIsoNat.lean | 142 | 144 |
/-
Copyright (c) 2018 SΓ©bastien GouΓ«zel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: SΓ©bastien GouΓ«zel, Johannes HΓΆlzl, RΓ©my Degenne
-/
import Mathlib.Order.ConditionallyCompleteLattice.Indexed
import Mathlib.Order.Filter.IsBounded
import Mathlib.Order.Hom.CompleteL... | ext x
simp only [mem_setOf_eq, iInter_coe_set, mem_iUnion, mem_iInter, mem_Iic, Subtype.exists,
exists_prop]
| Mathlib/Order/LiminfLimsup.lean | 279 | 281 |
/-
Copyright (c) 2019 SΓ©bastien GouΓ«zel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: SΓ©bastien GouΓ«zel
-/
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Complex.Module
import Mathlib.Data.Complex.Order
import Mathli... | rfl
@[simp]
| Mathlib/Analysis/Complex/Basic.lean | 413 | 415 |
/-
Copyright (c) 2022 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.Finsupp.Defs
import Mathlib.Data.Finset.Pairwise
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
/-!
# Sums of collections of Finsupp, ... | theorem List.mem_foldr_sup_support_iff [Zero M] {l : List (ΞΉ ββ M)} {x : ΞΉ} :
x β l.foldr (Finsupp.support Β· β Β·) β
β β f β l, x β f.support := by
simp only [Finset.sup_eq_union, List.foldr_map, Finsupp.mem_support_iff, exists_prop]
| Mathlib/Data/Finsupp/BigOperators.lean | 55 | 57 |
/-
Copyright (c) 2022 MarΓa InΓ©s de Frutos-FernΓ‘ndez. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir, MarΓa InΓ©s de Frutos-FernΓ‘ndez
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Algebra.DirectSum.Decomposition
import Mathlib.Algeb... | theorem isWeightedHomogeneous_zero_iff_weightedTotalDegree_eq_zero {p : MvPolynomial Ο R} :
IsWeightedHomogeneous w p 0 β p.weightedTotalDegree w = 0 := by
rw [weightedTotalDegree, β bot_eq_zero, Finset.sup_eq_bot_iff, bot_eq_zero, IsWeightedHomogeneous]
apply forall_congr'
intro m
rw [mem_support_iff]
/--... | Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean | 555 | 563 |
/-
Copyright (c) 2024 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
import Mathlib.CategoryTheory.Sites.Coherent.ExtensiveTopology
import Mathlib.CategoryTheory.Sites.Coherent.SheafComparison
import Mathlib.CategoryTheory.Sites.Lo... | attribute [local instance] Types.instFunLike Types.instConcreteCategory in
lemma extensiveTopology.surjective_of_isLocallySurjective_sheaf_of_types [FinitaryPreExtensive C]
{F G : Cα΅α΅ β₯€ Type w} (f : F βΆ G) [PreservesFiniteProducts F] [PreservesFiniteProducts G]
(h : Presheaf.IsLocallySurjective (extensiveTopo... | Mathlib/CategoryTheory/Sites/Coherent/LocallySurjective.lean | 56 | 85 |
/-
Copyright (c) 2021 RΓ©my Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: RΓ©my Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` ... | | _ => throw ()
| Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean | 109 | 110 |
/-
Copyright (c) 2018 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Algebra.Order.Group.Finset
import Mathlib.Data.Finsupp.Order
import Mathlib.Data.Sym.Basic
/-!
# Equivalence between `Multiset` and `β`-valued finitel... |
theorem toMultiset_sup [DecidableEq Ξ±] (f g : Ξ± ββ β) :
toMultiset (f β g) = toMultiset f βͺ toMultiset g := by
ext
simp_rw [Multiset.count_union, Finsupp.count_toMultiset, Finsupp.sup_apply]
theorem toMultiset_inf [DecidableEq Ξ±] (f g : Ξ± ββ β) :
toMultiset (f β g) = toMultiset f β© toMultiset g := by
ex... | Mathlib/Data/Finsupp/Multiset.lean | 105 | 114 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle SΓΆnne, SΓ©bastien GouΓ«zel,
RΓ©my Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
import Qq
/-! # P... | rw [β log_le_log_iff (rpow_pos_of_pos x_pos y) (rpow_pos_of_pos x_pos z), log_rpow x_pos,
log_rpow x_pos, mul_le_mul_right (log_pos hx)]
| Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 625 | 627 |
/-
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 ... | simp
| Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | 401 | 402 |
/-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.NumberTheory.Padics.PadicVal.Basic
/-!
# p-adic norm
This file defines the `p`-adic norm on `β`.
The `p`-adic valuation on `β` is the difference o... | Β· norm_cast at hz
simp [hz]
Β· rw [zpow_le_zpow_iff_rightβ, neg_le_neg_iff, padicValRat.of_int,
padicValInt.of_ne_one_ne_zero hp.1.ne_one _]
Β· norm_cast
rw [β FiniteMultiplicity.pow_dvd_iff_le_multiplicity]
Β· norm_cast
Β· apply Int.finiteMultiplicity_iff.2 β¨by simp [hp.out.ne_one], mod... | Mathlib/NumberTheory/Padics/PadicNorm.lean | 225 | 243 |
/-
Copyright (c) 2022 SΓ©bastien GouΓ«zel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: SΓ©bastien GouΓ«zel
-/
import Mathlib.MeasureTheory.Function.Jacobian
import Mathlib.MeasureTheory.Measure.Lebesgue.Complex
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deri... | protected noncomputable def polarCoord : PartialHomeomorph β (β Γ β) :=
equivRealProdCLM.toHomeomorph.transPartialHomeomorph polarCoord
| Mathlib/Analysis/SpecialFunctions/PolarCoord.lean | 178 | 180 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Shing Tak Lam, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring.List
import Mathlib.Data.Int.ModEq
import Mathlib.Da... | Β· simp only [ofDigits, add_sub_add_left_eq_sub]
exact dvd_mul_sub_mul h ih
theorem ofDigits_modEq' (b b' : β) (k : β) (h : b β‘ b' [MOD k]) (L : List β) :
ofDigits b L β‘ ofDigits b' L [MOD k] := by
induction' L with d L ih
Β· rfl
Β· dsimp [ofDigits]
dsimp [Nat.ModEq] at *
conv_lhs => rw [Nat.add_m... | Mathlib/Data/Nat/Digits.lean | 623 | 632 |
/-
Copyright (c) 2023 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.Logic.Small.Set
import Mathlib.CategoryTheory.Filtered.Final
/-!
# Finally small categories
A category given by `(J : Type u) [Category.{v} J]` is `w`-... | refine Functor.final_of_exists_of_isFiltered_of_fullyFaithful _ (fun i => ?_)
obtain β¨j, hjβ, hjββ© := hs i
exact β¨β¨j, hjββ©, hjββ©
theorem finallySmall_iff_exists_small_weakly_terminal_set [IsFilteredOrEmpty J] :
| Mathlib/CategoryTheory/Limits/FinallySmall.lean | 147 | 151 |
/-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.Measure.Typeclasses.Finite
import Mathlib.MeasureTheory.Measure.Typeclasses.NoAtoms
import Mathlib.MeasureTheory.Measure.... | Mathlib/MeasureTheory/Measure/Typeclasses.lean | 975 | 980 | |
/-
Copyright (c) 2022 Violeta HernΓ‘ndez Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta HernΓ‘ndez Palacios
-/
import Mathlib.SetTheory.Ordinal.Family
import Mathlib.Tactic.Abel
/-!
# Natural operations on ordinals
The goal of this file is to define n... | instance : AddLeftStrictMono NatOrdinal.{u} :=
| Mathlib/SetTheory/Ordinal/NaturalOps.lean | 338 | 338 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Batteries.Tactic.Congr
import Mathlib.Data.Option.Basic
import Mathlib.Data.Prod.Basic
import Mathlib.Data.Set.Subsingleton
import Mathlib.Dat... |
theorem forall_mem_range {p : Ξ± β Prop} : (β a β range f, p a) β β i, p (f i) := by simp
theorem forall_subtype_range_iff {p : range f β Prop} :
| Mathlib/Data/Set/Image.lean | 573 | 577 |
/-
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, YaΓ«l Dillies
-/
import Mathlib.Algebra.Order.Invertible
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.LinearAlgebra.AffineSpac... | simp only [smul_sub, sub_smul, one_smul]
abel
theorem openSegment_eq_image' (x y : E) :
openSegment π x y = (fun ΞΈ : π => x + ΞΈ β’ (y - x)) '' Ioo (0 : π) 1 := by
convert openSegment_eq_image π x y using 2
| Mathlib/Analysis/Convex/Segment.lean | 191 | 196 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle SΓΆnne, Benjamin Davidson
-/
import Mathlib.Algebra.QuadraticDiscriminant
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
/-!
... |
theorem tan_int_mul_pi_div_two (n : β€) : tan (n * Ο / 2) = 0 :=
tan_eq_zero_iff.mpr (by use n)
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Complex.lean | 69 | 72 |
/-
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 HΓΆlzl, Yury Kudryashov
-/
import Mathlib.Order.Interval.Set.ProjIcc
import Mathlib.Topology.Algebra.Order.Field
import Mathlib.Topolo... | Mathlib/Topology/MetricSpace/Lipschitz.lean | 420 | 431 | |
/-
Copyright (c) 2022 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Jireh Loreaux
-/
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Star.Module
import Mathlib.Algebra.Star.NonUn... | variable [StarModule R A]
| Mathlib/Algebra/Star/Subalgebra.lean | 715 | 716 |
/-
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.Calculus.ParametricIntegral
import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic
/-!
# Derivatives of interval integrals depending ... | (h_bound : βα΅ t βΞΌ, t β Ξ a b β β x β ball xβ Ξ΅, βF' x tβ β€ bound t)
(bound_integrable : IntervalIntegrable bound ΞΌ a b)
(h_diff : βα΅ t βΞΌ, t β Ξ a b β β x β ball xβ Ξ΅, HasDerivAt (fun x => F x t) (F' x t) x) :
IntervalIntegrable (F' xβ) ΞΌ a b β§
HasDerivAt (fun x => β« t in a..b, F x t βΞΌ) (β« t in ... | Mathlib/Analysis/Calculus/ParametricIntervalIntegral.lean | 97 | 112 |
/-
Copyright (c) 2023 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.TensorAlgebra.Basic
import Mathlib.LinearAlgebra.FreeAlgebra
/-!
# A basis for `TensorAlgebra R M`
## Main definitions
* `TensorAlgebra.equi... | lemma rank_eq [Nontrivial R] [Module.Free R M] :
Module.rank R (TensorAlgebra R M) = Cardinal.lift.{uR} (sum fun n β¦ Module.rank R M ^ n) := by
let β¨β¨ΞΊ, bβ©β© := Module.Free.exists_basis (R := R) (M := M)
rw [(equivFreeAlgebra b).toLinearEquiv.rank_eq, FreeAlgebra.rank_eq, mk_list_eq_sum_pow,
Basis.mk_eq_rank... | Mathlib/LinearAlgebra/TensorAlgebra/Basis.lean | 83 | 87 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Sean Leather
-/
import Batteries.Data.List.Perm
import Mathlib.Data.List.Pairwise
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Lookmap
import Mathlib.Data.Si... |
/-! ### `kinsert` -/
/-- Insert the pair `β¨a, bβ©` and erase the first pair with the key `a`. -/
def kinsert (a : Ξ±) (b : Ξ² a) (l : List (Sigma Ξ²)) : List (Sigma Ξ²) :=
| Mathlib/Data/List/Sigma.lean | 526 | 531 |
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.Order.AbsoluteValue.Euclidean
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Algebra.Polynomial.FieldDivision
/-!
# Absolute value on polyno... |
@[simp]
theorem cardPowDegree_zero : cardPowDegree (0 : Fq[X]) = 0 := rfl
@[simp]
| Mathlib/Algebra/Polynomial/Degree/CardPowDegree.lean | 79 | 83 |
/-
Copyright (c) 2020 SΓ©bastien GouΓ«zel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: SΓ©bastien GouΓ«zel, Floris van Doorn
-/
import Mathlib.Geometry.Manifold.MFDeriv.Defs
import Mathlib.Geometry.Manifold.ContMDiff.Defs
/-!
# Basic properties of the manifold FrΓ©chet ... | DifferentiableWithinAt π (extChartAt I' y β f β (extChartAt I x).symm) (range I)
(extChartAt I x x') :=
(mdifferentiableWithinAt_iff_of_mem_source hx hy).trans <| by
rw [continuousWithinAt_univ, preimage_univ, univ_inter]
| Mathlib/Geometry/Manifold/MFDeriv/Basic.lean | 346 | 349 |
/-
Copyright (c) 2022 YaΓ«l Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: YaΓ«l Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.SimpleGraph.Path
import Mathlib.Combinatorics.SimpleGraph.Operations
import Mathlib.Data.Finset.Pairwise
import M... | theorem cliqueSet_one (G : SimpleGraph Ξ±) : G.cliqueSet 1 = Set.range singleton :=
Set.ext fun s => by simp [eq_comm]
@[simp]
theorem cliqueSet_bot (hn : 1 < n) : (β₯ : SimpleGraph Ξ±).cliqueSet n = β
:=
(cliqueFree_bot hn).cliqueSet
@[simp]
theorem cliqueSet_map (hn : n β 1) (G : SimpleGraph Ξ±) (f : Ξ± βͺ Ξ²) :
(... | Mathlib/Combinatorics/SimpleGraph/Clique.lean | 578 | 594 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kim Morrison
-/
import Mathlib.Tactic.NormNum.Basic
import Mathlib.Tactic.TryThis
import Mathlib.Util.AtomM
/-!
# The `abel` tactic
Evaluate expressions in the langua... | def intToExpr (n : β€) : M Expr := do
Expr.ofInt (mkConst (if (β read).isGroup then ``Int else ``Nat) []) n
| Mathlib/Tactic/Abel.lean | 133 | 135 |
/-
Copyright (c) 2014 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Divisibility.Hom
import Mathlib.Algebra.Group.Even
import Mathlib.Algebra.Group.Nat.Hom
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.Alg... |
-- these versions are primed so that the `RingHomClass` versions aren't
theorem eq_natCast' [AddMonoidHomClass F β A] (f : F) (h1 : f 1 = 1) : β n : β, f n = n
| 0 => by simp
| Mathlib/Data/Nat/Cast/Basic.lean | 92 | 95 |
/-
Copyright (c) 2019 Neil Strickland. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Neil Strickland
-/
import Mathlib.Algebra.BigOperators.Group.Multiset.Basic
import Mathlib.Data.PNat.Prime
import Mathlib.Data.Nat.Factors
import Mathlib.Data.Multiset.OrderedMonoid
i... | theorem factorMultiset_pow (n : β+) (m : β) :
factorMultiset (n ^ m) = m β’ factorMultiset n := by
let u := factorMultiset n
have : n = u.prod := (prod_factorMultiset n).symm
rw [this, β PrimeMultiset.prod_smul]
repeat' rw [PrimeMultiset.factorMultiset_prod]
/-- Factoring a prime gives the corresponding one... | Mathlib/Data/PNat/Factors.lean | 259 | 271 |
/-
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.Logic.Function.Conjugate
/-!
# Iterations of a function
In this file we prove simple properties of `Nat.iterate f n` a.k.a. `f^[n]`:
* `iterate_ze... | namespace Semiconj
| Mathlib/Logic/Function/Iterate.lean | 98 | 99 |
/-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.MeasureTheory.Measure.Regular
import Mathlib.MeasureTheory.Function.SimpleFuncDenseLp
import Mathlib.Topology.UrysohnsLemma
import Mathlib.MeasureThe... | @[deprecated (since := "2025-02-21")]
alias Memβp.exists_boundedContinuous_integral_rpow_sub_le :=
MemLp.exists_boundedContinuous_integral_rpow_sub_le
/-- Any integrable function can be approximated by bounded continuous functions,
| Mathlib/MeasureTheory/Function/ContinuousMapDense.lean | 308 | 312 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Batteries.Data.Nat.Gcd
import Mathlib.Algebra.Group.Nat.Units
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.GroupWi... | Mathlib/Data/Nat/GCD/Basic.lean | 309 | 318 | |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Multiset.ZeroCons
/-!
# Basic results on multisets
-/
-- No algebra should be required
assert_not_exists Monoid
universe v
open List S... | Mathlib/Data/Multiset/Basic.lean | 2,789 | 2,795 | |
/-
Copyright (c) 2019 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, Mario Carneiro
-/
import Mathlib.Algebra.Field.Rat
import Mathlib.Algebra.Order.Nonneg.Field
import Mathlib.Algebra.Order.Ring.Rat
/-!
# The rational numbers form a li... | Mathlib/Algebra/Order/Field/Rat.lean | 47 | 47 | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes HΓΆlzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.Polynomial.Basi... |
@[simp]
theorem nextCoeff_C_eq_zero (c : R) : nextCoeff (C c) = 0 := by
| Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 311 | 313 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Fin.VecNotation
import Mathlib.Logic.Small.Basic
import Mathlib.SetTheory.ZFC.PSet
/-!
# A model of ZFC
In this file, we model Zermelo-Fraenkel ... | Mathlib/SetTheory/ZFC/Basic.lean | 1,028 | 1,030 | |
/-
Copyright (c) 2024 JoΓ«l Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: JoΓ«l Riou
-/
import Mathlib.CategoryTheory.Localization.LocalizerMorphism
import Mathlib.CategoryTheory.HomCongr
/-!
# Bijections between morphisms in two localized categories
Given two ... | lemma homEquiv_eq (G : Dβ β₯€ Dβ) (e : Lβ β G β
Lβ) (f : Lβ.obj X βΆ Lβ.obj Y) :
homEquiv W Lβ Lβ f = e.inv.app X β« G.map f β« e.hom.app Y := by
rw [homEquiv_apply, LocalizerMorphism.homMap_apply (LocalizerMorphism.id W) Lβ Lβ G e.symm,
Iso.symm_hom, Iso.symm_inv]
| Mathlib/CategoryTheory/Localization/HomEquiv.lean | 132 | 135 |
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Logic.Encodable.Pi
import Mathlib.MeasureTheory.Group.Measure
import Mathlib.MeasureTheory.MeasurableSpace.... | MeasurePreserving (fun (a : (i : ΞΉ) β Ξ±' i) (i : ΞΉ) β¦ (f i) (a i)) :=
measurePreserving_pi _ _ hf
/-- The measurable equiv `(Ξ±β β Ξ²β) βα΅ (Ξ±β β Ξ²β)` induced by `Ξ±β β Ξ±β` and `Ξ²β βα΅ Ξ²β` is
measure preserving. -/
theorem measurePreserving_arrowCongr' {Ξ±β Ξ²β Ξ±β Ξ²β : Type*} [Fintype Ξ±β] [Fintype Ξ±β]
[MeasurableSp... | Mathlib/MeasureTheory/Constructions/Pi.lean | 862 | 872 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.GroupWithZero.NeZero
import Mathlib.Logic.Unique
import Mathlib.Tactic.Conv
/-!
# Groups with an adjoined z... |
end GroupWithZero
| Mathlib/Algebra/GroupWithZero/Basic.lean | 440 | 441 |
/-
Copyright (c) 2017 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stephen Morgan, Kim Morrison, Johannes HΓΆlzl
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.Data.Set.CoeSort
import Mathlib.T... |
variable (m : Type u β Type v) [_root_.Functor m] [LawfulFunctor m]
@[simp]
theorem ofTypeFunctor_obj : (ofTypeFunctor m).obj = m :=
rfl
| Mathlib/CategoryTheory/Types.lean | 259 | 264 |
/-
Copyright (c) 2023 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.DirectSum.LinearMap
import Mathlib.Algebra.Lie.InvariantForm
import Mathlib.Algebra.Lie.Weights.Cartan
import Mathlib.Algebra.Lie.Weights.Linear
impo... | simp [this]
exact fun m β¦ N.lie_mem (h hy m)
include h in
lemma traceForm_eq_of_le_idealizer :
traceForm R I N = (traceForm R L M).restrict I := by
ext β¨x, hxβ© β¨y, hyβ©
change _ = trace R M (Ο x ββ Ο y)
rw [N.trace_eq_trace_restrict_of_le_idealizer I h x hy]
rfl
| Mathlib/Algebra/Lie/TraceForm.lean | 309 | 319 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes HΓΆlzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Polynomial.Reverse
import Mathlib.Algebra.Regular.SMul
/-!
# Theory of monic polynomials
We give se... | have : (p.comp q).natDegree = p.natDegree * q.natDegree :=
natDegree_comp_eq_of_mul_ne_zero <| by simp [hp.leadingCoeff, hq.leadingCoeff]
| Mathlib/Algebra/Polynomial/Monic.lean | 135 | 136 |
/-
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.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Limits.Shapes.KernelPair
import Mathlib.CategoryTheory.Limits.Constructions.EpiMono
/-!
# Adhesive c... | (IsPushout.of_horiz_isIso β¨by simpβ©)).1.flip
theorem IsPushout.IsVanKampen.isPullback_of_mono_right [Mono g] {H : IsPushout f g h i}
(H' : H.IsVanKampen) : IsPullback f g h i :=
((H' f (π _) (π _) f (π _) (π _) g h (IsPullback.of_vert_isIso β¨by simpβ©)
| Mathlib/CategoryTheory/Adhesive.lean | 194 | 198 |
/-
Copyright (c) 2021 Bryan Gin-ge Chen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bryan Gin-ge Chen, YaΓ«l Dillies
-/
import Mathlib.Algebra.Group.Idempotent
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Algebra.Ring.PUnit
import Mathlib.Order.Hom.BoundedLattic... |
/-- The Boolean algebra structure on a Boolean ring.
| Mathlib/Algebra/Ring/BooleanRing.lean | 202 | 204 |
/-
Copyright (c) 2024 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.NumberTheory.LSeries.RiemannZeta
import Mathlib.NumberTheory.Harmonic.GammaDeriv
/-!
# Asymptotics of `ΞΆ s` as `s β 1`
The goal of this file is to ev... |
lemma term_sum_of_lt (N : β) {s : β} (hs : 1 < s) :
term_sum s N = 1 / (s - 1) * (1 - 1 / (N + 1) ^ (s - 1))
- 1 / s * ((β n β Finset.range N, 1 / (n + 1 : β) ^ s) - N / (N + 1) ^ s) := by
simp only [term_sum]
conv => enter [1, 2, n]; rw [term_of_lt (by simp) hs]
rw [Finset.sum_sub_distrib]
congr 1
Β·... | Mathlib/NumberTheory/Harmonic/ZetaAsymp.lean | 174 | 191 |
/-
Copyright (c) 2019 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Order.Ring.Unbundled.Rat
/-!
# The rational numbers form a linear ordered field
This f... | Mathlib/Algebra/Order/Ring/Rat.lean | 41 | 45 | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPi... | lemma exists_subgroup_le_card_pow_prime_of_le_card {n p : β} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n β€ Nat.card H) : β H' β€ H, Nat.card H' = p ^ n := by
obtain β¨H', H'cardβ© := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine β¨H'.map H.subtype, map_subtype_le _, ?_β©... | Mathlib/GroupTheory/Sylow.lean | 671 | 679 |
/-
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.LinearAlgebra.Quotient.Basic
import Mathlib.LinearAlgebra.Prod
/-!
# Projection to a subspace
In this file we define
* `Submodule.linearProjOfIsCom... | LinearMap.fst R p q ββ β(prodEquivOfIsCompl p q h).symm
variable {p q}
@[simp]
| Mathlib/LinearAlgebra/Projection.lean | 139 | 143 |
/-
Copyright (c) 2024 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.Data.Matroid.Minor.Restrict
/-!
# Some constructions of matroids
This file defines some very elementary examples of matroids, namely those with at most o... | rw [dual_isBase_iff, uniqueBaseOn_isBase_iff inter_subset_right,
uniqueBaseOn_isBase_iff diff_subset, uniqueBaseOn_ground]
exact β¨fun h β¦ by rw [β diff_diff_cancel_left hB, h, diff_inter_self_eq_diff],
fun h β¦ by rw [h, inter_comm I]; simpβ©
| Mathlib/Data/Matroid/Constructions.lean | 236 | 240 |
/-
Copyright (c) 2019 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, Patrick Massot, Casper Putz, Anne Baanen
-/
import Mathlib.Data.Matrix.Basis
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.RowCol
import Mathlib.Data.Matr... |
@[simp]
| Mathlib/LinearAlgebra/Matrix/Trace.lean | 58 | 59 |
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kenny Lau
-/
import Mathlib.Algebra.Polynomial.Coeff
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.RingTheory.PowerSeries.Basic
/-!
# Formal power se... | @[simp] theorem trunc_mul_trunc {n} (f g : Rβ¦Xβ§) :
trunc n (f * (trunc n g) : Rβ¦Xβ§) = trunc n (f * g) := by
rw [mul_comm, trunc_trunc_mul, mul_comm]
| Mathlib/RingTheory/PowerSeries/Trunc.lean | 180 | 182 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Group.Abs
import Mathlib.Algebra.Order.Ring.Int
import Mathlib.Data.Nat.Cast.Order.Ring
/-!
# Order properties of cast of integers
This... | @[simp] lemma cast_nonpos : (n : R) β€ 0 β n β€ 0 := by rw [β cast_zero, cast_le]
| Mathlib/Algebra/Order/Ring/Cast.lean | 60 | 60 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.Images
import Mathlib.CategoryTheor... | hom := 0
| Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean | 378 | 378 |
/-
Copyright (c) 2023 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash, Deepro Choudhury, Mitchell Lee, Johan Commelin
-/
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.Algebra.EuclideanDomain.Int
import Mathlib.Algebra.Module.Linear... |
/-- Given an element `x` in a module `M` and a linear form `f` on `M`, we define the endomorphism
of `M` for which `y β¦ y - (f y) β’ x`.
| Mathlib/LinearAlgebra/Reflection.lean | 61 | 63 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Data.Finite.Sum
import Mathlib.RingTheory.FiniteType
import Mathlib.RingTheory.Finiteness.Ideal
import Mathlib.RingTheory.Ideal.Quotient.Operations
imp... | obtain β¨n, g, hg, hg'β© := hf
let g' := g.toRingHom
change Surjective g' at hg
change (ker g').FG at hg'
have : g'.comp MvPolynomial.C = f := g.comp_algebraMap
clear_value g'
subst this
clear g
induction n generalizing R S with
| zero =>
refine fg_ker _ _ _ (hg.comp (MvPolynomial.C_surjective (Fi... | Mathlib/RingTheory/FinitePresentation.lean | 474 | 484 |
/-
Copyright (c) 2021 Martin Zinkevich. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Martin Zinkevich, Vincent Beffara
-/
import Mathlib.MeasureTheory.Integral.Bochner.Set
import Mathlib.Probability.Independence.Basic
/-!
# Integration in Probability Theory
Integra... | refine β¨fun hfg _ _ hΟ hΟ => IndepFun.integral_mul_of_integrable (hfg.comp hΟ hΟ), ?_β©
rw [IndepFun_iff]
rintro h _ _ β¨A, hA, rflβ© β¨B, hB, rflβ©
specialize
h (measurable_one.indicator hA) (measurable_one.indicator hB)
((integrable_const 1).indicator (hfm.comp measurable_id hA))
((integrable_const... | Mathlib/Probability/Integration.lean | 305 | 320 |
/-
Copyright (c) 2017 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stephen Morgan, Kim Morrison, Johannes HΓΆlzl, Reid Barton
-/
import Mathlib.CategoryTheory.Category.Init
import Mathlib.Combinatorics.Quiver.Basic
import Mathlib.Tactic.PPWithUniv
import... |
theorem ite_comp {P : Prop} [Decidable P] {X Y Z : C} (f f' : X βΆ Y) (g : Y βΆ Z) :
(if P then f else f') β« g = if P then f β« g else f' β« g := by aesop
| Mathlib/CategoryTheory/Category/Basic.lean | 242 | 244 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
/-... | theorem Nat.isCoprime_iff_coprime {m n : β} : IsCoprime (m : β€) n β Nat.Coprime m n := by
rw [Int.isCoprime_iff_gcd_eq_one, Int.gcd_natCast_natCast]
| Mathlib/RingTheory/Coprime/Lemmas.lean | 42 | 43 |
/-
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.Algebra.BigOperators.Finprod
import Mathlib.Topology.ContinuousMap.Algebra
import Mathlib.Topology.Compactness.Paracompact
import Mathlib.Topology.Sh... | rw [Finset.mem_filter, Iff.comm, and_iff_right_iff_imp]
exact flip (ht j) hj
theorem sum_toPOUFun_eq (x : X) : βαΆ i, f.toPOUFun i x = 1 - βαΆ i, (1 - f i x) := by
set s := (f.point_finite x).toFinset
have hs : (s : Set ΞΉ) = { i | f i x β 0 } := Finite.coe_toFinset _
have A : (support fun i => toPOUFun f i x) ... | Mathlib/Topology/PartitionOfUnity.lean | 508 | 516 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Finset.Card
import Mathlib.Data.Fintype.Basic
/-!
# Cardinalities of finite types
This file defines the cardinality `Fintype.card Ξ±` as the numb... | Mathlib/Data/Fintype/Card.lean | 857 | 858 | |
/-
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, Floris van Doorn, Amelia Livingston, Yury Kudryashov,
Neil Strickland, Aaron Anderson
-/
import Mathlib.Algebra.GroupWithZero.Units.Basic
import Mathli... | rw [h]
exact β¨0, by simpβ©β©
@[simp]
| Mathlib/Algebra/GroupWithZero/Divisibility.lean | 34 | 37 |
/-
Copyright (c) 2023 YaΓ«l Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: YaΓ«l Dillies
-/
import Mathlib.Algebra.Order.Monovary
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Analysis.Convex.Function
import Mathlib.Tactic.FieldSimp
/-!
# Product of co... | ConvexOn π s (f β’ g) := by
rw [β neg_concaveOn_iff, β smul_neg]
exact hf.smul' hg.neg hfβ (fun x hx β¦ neg_nonneg.2 <| hgβ hx) hfg.neg_right
lemma ConvexOn.smul_concaveOn' [OrderedSMul π E] (hf : ConvexOn π s f) (hg : ConcaveOn π s g)
| Mathlib/Analysis/Convex/Mul.lean | 93 | 97 |
/-
Copyright (c) 2018 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Data.Multiset.Bind
/-!
# The cartesian product of multisets
## Main definitions
* `Multiset.pi`: Cartesian product of multisets indexed by a multise... | apply bind_hcongr
Β· rw [cons_swap a a']
intro b' _
| Mathlib/Data/Multiset/Pi.lean | 115 | 117 |
/-
Copyright (c) 2023 JoΓ«l Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: JoΓ«l Riou
-/
import Mathlib.Algebra.Homology.HomotopyCategory.HomComplex
import Mathlib.Algebra.Homology.HomotopyCofiber
/-! # The mapping cone of a morphism of cochain complexes
In this ... | Cochain.ofHoms (homotopyCofiber.sndX Ο)
@[reassoc (attr := simp)]
lemma inl_v_fst_v (p q : β€) (hpq : q + 1 = p) :
(inl Ο).v p q (by rw [β hpq, add_neg_cancel_right]) β«
| Mathlib/Algebra/Homology/HomotopyCategory/MappingCone.lean | 71 | 75 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes HΓΆlzl
-/
import Mathlib.MeasureTheory.Integral.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Lebesgue.Countable
import Mathlib.MeasureTheory.Integral.Le... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 207 | 228 | |
/-
Copyright (c) 2023 JoΓ«l Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: JoΓ«l Riou
-/
import Mathlib.Algebra.Group.Nat.Defs
import Mathlib.CategoryTheory.Category.Preorder
import Mathlib.CategoryTheory.EqToHom
import Mathlib.CategoryTheory.Functor.Const
import M... |
/-- Inductive construction of isomorphisms in `ComposableArrows C (n + 1)`: in order to
construct an isomorphism `F β
G`, it suffices to provide `Ξ± : F.obj' 0 β
G.obj' 0` and
`Ξ² : F.Ξ΄β β
G.Ξ΄β` such that `F.map' 0 1 β« app' Ξ².hom 0 = Ξ±.hom β« G.map' 0 1`. -/
@[simps]
def isoMkSucc {F G : ComposableArrows C (n + 1)} (Ξ± : ... | Mathlib/CategoryTheory/ComposableArrows.lean | 509 | 514 |
/-
Copyright (c) 2021 Arthur Paulino. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Arthur Paulino, Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Clique
import Mathlib.Data.ENat.Lattice
import Mathlib.Data.Nat.Lattice
import Mathlib.Data.Setoid.Partition
imp... | @[simp] lemma coe_recolorOfEmbedding (f : Ξ± βͺ Ξ²) :
β(G.recolorOfEmbedding f) = (Embedding.completeGraph f).toHom.comp := rfl
/-- Given an equivalence, there is an induced equivalence between colorings. -/
| Mathlib/Combinatorics/SimpleGraph/Coloring.lean | 177 | 180 |
/-
Copyright (c) 2022 YaΓ«l Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: YaΓ«l Dillies
-/
import Mathlib.Algebra.Order.UpperLower
import Mathlib.Topology.Algebra.Group.Pointwise
/-!
# Topological facts about upper/lower/order-connected sets
The topological c... | Mathlib/Topology/Algebra/Order/UpperLower.lean | 109 | 112 | |
/-
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... | theorem coe_algebraMap : β(algebraMap R β[R,cβ,cβ,cβ]) = coe :=
rfl
| Mathlib/Algebra/Quaternion.lean | 614 | 616 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Aaron Anderson, Yakov Pechersky
-/
import Mathlib.Data.Fintype.Card
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Algebra.Group.End
import Mathlib.Data.Finset.N... | apply le_antisymm
Β· convert support_mul_le (swap x y) (swap y z) using 1
rw [support_swap h.left.left, support_swap h.right.left]
simp [Finset.ext_iff]
Β· intro
simp only [mem_insert, mem_singleton]
rintro (rfl | rfl | rfl | _) <;>
simp [swap_apply_of_ne_of_ne, h.left.left, h.left.left.symm, ... | Mathlib/GroupTheory/Perm/Support.lean | 477 | 488 |
/-
Copyright (c) 2022 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Junyan Xu, Jack McKoen
-/
import Mathlib.RingTheory.Valuation.ValuationRing
import Mathlib.RingTheory.Localization.AsSubring
import Mathlib.Algebra.Ring.Subring.Pointwise
impor... | Function.comp_apply]
theorem surjective_unitGroupToResidueFieldUnits :
Function.Surjective A.unitGroupToResidueFieldUnits :=
(IsLocalRing.surjective_units_map_of_local_ringHom _ Ideal.Quotient.mk_surjective
IsLocalRing.isLocalHom_residue).comp
(MulEquiv.surjective _)
/-- The quotient of the unit... | Mathlib/RingTheory/Valuation/ValuationSubring.lean | 655 | 667 |
/-
Copyright (c) 2018 Louis Carlin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Louis Carlin, Mario Carneiro
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Order.RelClasses
/-!
# Euclidean domains
This file introduces Euclidean domains and provides the extende... | Mathlib/Algebra/EuclideanDomain/Defs.lean | 269 | 271 | |
/-
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... |
@[reassoc]
| Mathlib/CategoryTheory/Monoidal/Category.lean | 466 | 467 |
/-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.Plus
import Mathlib.CategoryTheory.Limits.Shapes.ConcreteCategory
/-!
# Sheafification
We construct the sheafification of a presheaf ov... | erw [Meq.equiv_symm_eq_apply]
cases i; rfl
/-- `PβΊ` is always separated. -/
theorem sep {X : C} (P : Cα΅α΅ β₯€ D) (S : J.Cover X) (x y : ToType ((J.plusObj P).obj (op X)))
(h : β I : S.Arrow, (J.plusObj P).map I.f.op x = (J.plusObj P).map I.f.op y) : x = y := by
-- First, we choose representatives for x ... | Mathlib/CategoryTheory/Sites/ConcreteSheafification.lean | 236 | 260 |
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