Context stringlengths 57 92.3k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 18 3.69k | theorem stringlengths 25 2.71k | proof stringlengths 5 10.6k |
|---|---|---|---|---|---|
import Mathlib.Algebra.Group.Hom.Defs
#align_import data.matrix.dmatrix from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
universe u u' v w z
def DMatrix (m : Type u) (n : Type u') (Ξ± : m β n β Type v) : Type max u u' v :=
β i j, Ξ± i j
#align dmatrix DMatrix
variable {l m n o : T... | Mathlib/Data/Matrix/DMatrix.lean | 57 | 59 | theorem map_map {M : DMatrix m n Ξ±} {Ξ² : m β n β Type w} {Ξ³ : m β n β Type z}
{f : β β¦i jβ¦, Ξ± i j β Ξ² i j} {g : β β¦i jβ¦, Ξ² i j β Ξ³ i j} :
(M.map f).map g = M.map fun i j x => g (f x) := by | ext; simp
|
import Mathlib.Data.PFunctor.Multivariate.W
import Mathlib.Data.QPF.Multivariate.Basic
#align_import data.qpf.multivariate.constructions.fix from "leanprover-community/mathlib"@"28aa996fc6fb4317f0083c4e6daf79878d81be33"
universe u v
namespace MvQPF
open TypeVec
open MvFunctor (LiftP LiftR)
open MvFunctor
var... | Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean | 139 | 143 | theorem wrepr_wMk {Ξ± : TypeVec n} (a : q.P.A) (f' : q.P.drop.B a βΉ Ξ±)
(f : q.P.last.B a β q.P.W Ξ±) :
wrepr (q.P.wMk a f' f) =
q.P.wMk' (repr (abs (appendFun id wrepr <$$> β¨a, q.P.appendContents f' fβ©))) := by |
rw [wrepr, recF_eq', q.P.wDest'_wMk]; rfl
|
import Mathlib.Data.Setoid.Partition
import Mathlib.GroupTheory.GroupAction.Basic
import Mathlib.GroupTheory.GroupAction.Pointwise
import Mathlib.GroupTheory.GroupAction.SubMulAction
open scoped BigOperators Pointwise
namespace MulAction
section Group
variable {G : Type*} [Group G] {X : Type*} [MulAction G X]
... | Mathlib/GroupTheory/GroupAction/Blocks.lean | 264 | 277 | theorem IsBlock.inter {Bβ Bβ : Set X} (hβ : IsBlock G Bβ) (hβ : IsBlock G Bβ) :
IsBlock G (Bβ β© Bβ) := by |
rw [IsBlock.def_one]
intro g
rw [Set.smul_set_inter]
cases' hβ.smul_eq_or_disjoint g with hβ hβ
Β· cases' hβ.smul_eq_or_disjoint g with hβ hβ
Β· left; rw [hβ, hβ]
right
apply Disjoint.inter_left'; apply Disjoint.inter_right'
exact hβ
Β· right
apply Disjoint.inter_left; apply Disjoint.inter... |
import Mathlib.Data.List.Range
import Mathlib.Data.List.Perm
#align_import data.list.sigma from "leanprover-community/mathlib"@"f808feb6c18afddb25e66a71d317643cf7fb5fbb"
universe u v
namespace List
variable {Ξ± : Type u} {Ξ² : Ξ± β Type v} {l lβ lβ : List (Sigma Ξ²)}
def keys : List (Sigma Ξ²) β List Ξ± :=
map ... | Mathlib/Data/List/Sigma.lean | 212 | 216 | theorem mem_dlookup {a} {b : Ξ² a} {l : List (Sigma Ξ²)} (nd : l.NodupKeys) (h : Sigma.mk a b β l) :
b β dlookup a l := by |
cases' Option.isSome_iff_exists.mp (dlookup_isSome.mpr (mem_keys_of_mem h)) with b' h'
cases nd.eq_of_mk_mem h (of_mem_dlookup h')
exact h'
|
import Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
#align_import category_theory.limits.shapes.pullbacks from "leanprover-community/mathlib"@"7316286ff2942aa14e540add9058c6b0aa1c8070"
noncomputable section
open CategoryTheory
universe w vβ vβ v u... | Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean | 429 | 430 | theorem cospanExt_inv_app_left :
(cospanExt iX iY iZ wf wg).inv.app WalkingCospan.left = iX.inv := by | dsimp [cospanExt]
|
import Mathlib.Computability.Halting
import Mathlib.Computability.TuringMachine
import Mathlib.Data.Num.Lemmas
import Mathlib.Tactic.DeriveFintype
#align_import computability.tm_to_partrec from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
open Function (update)
open Relation
namespa... | Mathlib/Computability/TMToPartrec.lean | 201 | 201 | theorem zero_eval (v) : zero.eval v = pure [0] := by | simp [zero]
|
import Mathlib.Algebra.Module.Hom
import Mathlib.Algebra.Module.LinearMap.End
#align_import algebra.module.equiv from "leanprover-community/mathlib"@"ea94d7cd54ad9ca6b7710032868abb7c6a104c9c"
open Function
universe u u' v w x y z
variable {R : Type*} {Rβ : Type*} {Rβ : Type*} {Rβ : Type*}
variable {k : Type*} {K... | Mathlib/Algebra/Module/Equiv.lean | 454 | 458 | theorem toLinearMap_symm_comp_eq (f : Mβ βββ[Οββ] Mβ) (g : Mβ βββ[Οββ] Mβ) :
eββ.symm.toLinearMap.comp g = f β g = eββ.toLinearMap.comp f := by |
constructor <;> intro H <;> ext
Β· simp [β H, β eββ.toEquiv.symm_comp_eq f g]
Β· simp [H, eββ.toEquiv.symm_comp_eq f g]
|
import Mathlib.Topology.Instances.Irrational
import Mathlib.Topology.Instances.Rat
import Mathlib.Topology.Compactification.OnePoint
#align_import topology.instances.rat_lemmas from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
open Set Metric Filter TopologicalSpace
open Topology One... | Mathlib/Topology/Instances/RatLemmas.lean | 56 | 62 | theorem not_countably_generated_cocompact : Β¬IsCountablyGenerated (cocompact β) := by |
intro H
rcases exists_seq_tendsto (cocompact β β π 0) with β¨x, hxβ©
rw [tendsto_inf] at hx; rcases hx with β¨hxc, hx0β©
obtain β¨n, hnβ© : β n : β, x n β insert (0 : β) (range x) :=
(hxc.eventually hx0.isCompact_insert_range.compl_mem_cocompact).exists
exact hn (Or.inr β¨n, rflβ©)
|
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Analysis.Convex.Star
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
#align_import analysis.convex.basic from "leanprover-community/mathlib"@"92bd7b1ffeb306a89f450bee126ddd8a284c259d"
variable {π E F Ξ² : Type*}
open LinearMap Set
open scope... | Mathlib/Analysis/Convex/Basic.lean | 513 | 515 | theorem Convex.mapsTo_lineMap (h : Convex π s) {x y : E} (hx : x β s) (hy : y β s) :
MapsTo (AffineMap.lineMap x y) (Icc (0 : π) 1) s := by |
simpa only [mapsTo', segment_eq_image_lineMap] using h.segment_subset hx hy
|
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.Real
#align_import analysis.normed_space.pointwise from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
open Metric Set
open Pointwise Topology
variable {π E : Type*}
variable [NormedField π]
sectio... | Mathlib/Analysis/NormedSpace/Pointwise.lean | 217 | 223 | theorem disjoint_ball_ball_iff (hΞ΄ : 0 < Ξ΄) (hΞ΅ : 0 < Ξ΅) :
Disjoint (ball x Ξ΄) (ball y Ξ΅) β Ξ΄ + Ξ΅ β€ dist x y := by |
refine β¨fun h => le_of_not_lt fun hxy => ?_, ball_disjoint_ballβ©
rw [add_comm] at hxy
obtain β¨z, hxz, hzyβ© := exists_dist_lt_lt hΞ΄ hΞ΅ hxy
rw [dist_comm] at hxz
exact h.le_bot β¨hxz, hzyβ©
|
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Data.Sym.Card
open Finset Function
namespace SimpleGraph
variable {V : Type*} (G : SimpleGraph V) {e : Sym2 V}
theorem edgeFinset_deleteEdges [DecidableEq V] [Fintype G.edgeSet] (s : Finset (Sym2 V))
[Fintyp... | Mathlib/Combinatorics/SimpleGraph/Finite.lean | 455 | 457 | theorem card_commonNeighbors_le_degree_right [DecidableRel G.Adj] (v w : V) :
Fintype.card (G.commonNeighbors v w) β€ G.degree w := by |
simp_rw [commonNeighbors_symm _ v w, card_commonNeighbors_le_degree_left]
|
import Mathlib.Algebra.Group.Conj
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.Group.Subsemigroup.Operations
import Mathlib.Algebra.Group.Submonoid.Operations
import Mathlib.Algebra.Order.Group.Abs
import Mathlib.Data.Set.Image
import Mathlib.Order.Atoms
import Mathlib.Tactic.ApplyFun
#align_import g... | Mathlib/Algebra/Group/Subgroup/Basic.lean | 2,509 | 2,509 | theorem range_eq_map (f : G β* N) : f.range = (β€ : Subgroup G).map f := by | ext; simp
|
import Aesop
import Mathlib.Order.BoundedOrder
#align_import order.disjoint from "leanprover-community/mathlib"@"22c4d2ff43714b6ff724b2745ccfdc0f236a4a76"
open Function
variable {Ξ± : Type*}
section Disjoint
section Codisjoint
section IsCompl
structure IsCompl [PartialOrder Ξ±] [BoundedOrder Ξ±] (x y : Ξ±) : Pro... | Mathlib/Order/Disjoint.lean | 819 | 820 | theorem isCompl_coe : IsCompl (a : Ξ±) b β IsCompl a b := by |
simp_rw [isCompl_iff, disjoint_coe, codisjoint_coe]
|
import Mathlib.Topology.Constructions
#align_import topology.continuous_on from "leanprover-community/mathlib"@"d4f691b9e5f94cfc64639973f3544c95f8d5d494"
open Set Filter Function Topology Filter
variable {Ξ± : Type*} {Ξ² : Type*} {Ξ³ : Type*} {Ξ΄ : Type*}
variable [TopologicalSpace Ξ±]
@[simp]
theorem nhds_bind_nhdsW... | Mathlib/Topology/ContinuousOn.lean | 585 | 587 | theorem continuousOn_prod_of_discrete_right [DiscreteTopology Ξ²] {f : Ξ± Γ Ξ² β Ξ³} {s : Set (Ξ± Γ Ξ²)} :
ContinuousOn f s β β b, ContinuousOn (f β¨Β·, bβ©) {a | (a, b) β s} := by |
simp_rw [ContinuousOn, Prod.forall, continuousWithinAt_prod_of_discrete_right]; apply forall_swap
|
import Mathlib.RingTheory.WittVector.InitTail
#align_import ring_theory.witt_vector.truncated from "leanprover-community/mathlib"@"acbe099ced8be9c9754d62860110295cde0d7181"
open Function (Injective Surjective)
noncomputable section
variable {p : β} [hp : Fact p.Prime] (n : β) (R : Type*)
local notation "π" =>... | Mathlib/RingTheory/WittVector/Truncated.lean | 346 | 350 | theorem mem_ker_truncate (x : π R) :
x β RingHom.ker (@truncate p _ n R _) β β i < n, x.coeff i = 0 := by |
simp only [RingHom.mem_ker, truncate, truncateFun, RingHom.coe_mk, TruncatedWittVector.ext_iff,
TruncatedWittVector.coeff_mk, coeff_zero]
exact Fin.forall_iff
|
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Regular.Pow
import Mathlib.Algebra.MonoidAlgebra.Support
import Mathlib.Data.Finsupp.Antidiagonal
import Mathlib.Order.SymmDiff
import Mathlib.RingTheory.Adjoin.Basic
#align_import data.mv_polynomial.basic fr... | Mathlib/Algebra/MvPolynomial/Basic.lean | 1,574 | 1,575 | theorem evalβHom_zero (f : R β+* Sβ) : evalβHom f (0 : Ο β Sβ) = f.comp constantCoeff := by |
ext <;> simp
|
import Mathlib.CategoryTheory.Sites.Whiskering
import Mathlib.CategoryTheory.Sites.Plus
#align_import category_theory.sites.compatible_plus from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
noncomputable section
namespace CategoryTheory.GrothendieckTopology
open CategoryTheory Limits... | Mathlib/CategoryTheory/Sites/CompatiblePlus.lean | 221 | 224 | theorem plusCompIso_inv_eq_plusLift (hP : Presheaf.IsSheaf J (J.plusObj P β F)) :
(J.plusCompIso F P).inv = J.plusLift (whiskerRight (J.toPlus _) _) hP := by |
apply J.plusLift_unique
simp [Iso.comp_inv_eq]
|
import Mathlib.CategoryTheory.Equivalence
#align_import category_theory.opposites from "leanprover-community/mathlib"@"dde670c9a3f503647fd5bfdf1037bad526d3397a"
universe vβ vβ uβ uβ
-- morphism levels before object levels. See note [CategoryTheory universes].
open Opposite
variable {C : Type uβ}
namespace Categ... | Mathlib/CategoryTheory/Opposites.lean | 168 | 170 | theorem unop_inv {X Y : Cα΅α΅} (f : X βΆ Y) [IsIso f] : (inv f).unop = inv f.unop := by |
apply IsIso.eq_inv_of_hom_inv_id
rw [β unop_comp, IsIso.inv_hom_id, unop_id]
|
import Mathlib.Data.Real.Basic
import Mathlib.Data.ENNReal.Real
import Mathlib.Data.Sign
#align_import data.real.ereal from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
open Function ENNReal NNReal Set
noncomputable section
def EReal := WithBot (WithTop β)
deriving Bot, Zero, One,... | Mathlib/Data/Real/EReal.lean | 1,269 | 1,269 | theorem coe_coe_sign (x : SignType) : ((x : β) : EReal) = x := by | cases x <;> rfl
|
import Mathlib.Algebra.Ring.Int
import Mathlib.RingTheory.DedekindDomain.IntegralClosure
#align_import number_theory.number_field.basic from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
class NumberField (K : Type*) [Field K] : Prop where
[to_charZero : CharZero K]
[to_finiteDime... | Mathlib/NumberTheory/NumberField/Basic.lean | 225 | 229 | theorem not_isField : Β¬IsField (π K) := by |
have h_inj : Function.Injective (algebraMap β€ (π K)) := RingHom.injective_int (algebraMap β€ (π K))
intro hf
exact Int.not_isField
(((IsIntegralClosure.isIntegral_algebra β€ K).isField_iff_isField h_inj).mpr hf)
|
import Mathlib.Algebra.Category.GroupCat.EquivalenceGroupAddGroup
import Mathlib.GroupTheory.QuotientGroup
#align_import algebra.category.Group.epi_mono from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
noncomputable section
open scoped Pointwise
universe u v
section
open Category... | Mathlib/Algebra/Category/GroupCat/EpiMono.lean | 444 | 445 | theorem epi_iff_surjective : Epi f β Function.Surjective f := by |
rw [epi_iff_range_eq_top, MonoidHom.range_top_iff_surjective]
|
import Mathlib.MeasureTheory.Constructions.Prod.Integral
import Mathlib.MeasureTheory.Function.LocallyIntegrable
import Mathlib.MeasureTheory.Group.Integral
import Mathlib.Topology.Metrizable.Urysohn
import Mathlib.Topology.UrysohnsLemma
import Mathlib.MeasureTheory.Measure.Haar.Basic
import Mathlib.MeasureTheory.Meas... | Mathlib/MeasureTheory/Measure/Haar/Unique.lean | 615 | 638 | theorem measure_isMulInvariant_eq_smul_of_isCompact_closure [LocallyCompactSpace G]
(ΞΌ' ΞΌ : Measure G) [IsHaarMeasure ΞΌ] [IsFiniteMeasureOnCompacts ΞΌ'] [IsMulLeftInvariant ΞΌ']
{s : Set G} (h's : IsCompact (closure s)) :
ΞΌ' s = haarScalarFactor ΞΌ' ΞΌ β’ ΞΌ s := by |
let Ξ½ := haarScalarFactor ΞΌ' ΞΌ β’ ΞΌ
apply le_antisymm
Β· calc
ΞΌ' s β€ ΞΌ' ((toMeasurable Ξ½ s) β© (closure s)) :=
measure_mono <| subset_inter (subset_toMeasurable Ξ½ s) subset_closure
_ = Ξ½ ((toMeasurable Ξ½ s) β© (closure s)) := by
apply measure_isMulInvariant_eq_smul_of_isCompact_closure_of_measura... |
import Mathlib.Order.Cover
import Mathlib.Order.Interval.Finset.Defs
#align_import data.finset.locally_finite from "leanprover-community/mathlib"@"442a83d738cb208d3600056c489be16900ba701d"
assert_not_exists MonoidWithZero
assert_not_exists Finset.sum
open Function OrderDual
open FinsetInterval
variable {ΞΉ Ξ± : T... | Mathlib/Order/Interval/Finset/Basic.lean | 746 | 747 | theorem card_Iio_eq_card_Iic_sub_one (a : Ξ±) : (Iio a).card = (Iic a).card - 1 := by |
rw [Iic_eq_cons_Iio, card_cons, Nat.add_sub_cancel_right]
|
import Mathlib.Init.Data.Sigma.Lex
import Mathlib.Data.Prod.Lex
import Mathlib.Data.Sigma.Lex
import Mathlib.Order.Antichain
import Mathlib.Order.OrderIsoNat
import Mathlib.Order.WellFounded
import Mathlib.Tactic.TFAE
#align_import order.well_founded_set from "leanprover-community/mathlib"@"2c84c2c5496117349007d97104... | Mathlib/Order/WellFoundedSet.lean | 690 | 697 | theorem IsWF.min_union (hs : s.IsWF) (hsn : s.Nonempty) (ht : t.IsWF) (htn : t.Nonempty) :
(hs.union ht).min (union_nonempty.2 (Or.intro_left _ hsn)) =
Min.min (hs.min hsn) (ht.min htn) := by |
refine le_antisymm (le_min (IsWF.min_le_min_of_subset subset_union_left)
(IsWF.min_le_min_of_subset subset_union_right)) ?_
rw [min_le_iff]
exact ((mem_union _ _ _).1 ((hs.union ht).min_mem (union_nonempty.2 (.inl hsn)))).imp
(hs.min_le _) (ht.min_le _)
|
import Mathlib.CategoryTheory.MorphismProperty.Composition
import Mathlib.CategoryTheory.MorphismProperty.IsInvertedBy
import Mathlib.CategoryTheory.Category.Quiv
#align_import category_theory.localization.construction from "leanprover-community/mathlib"@"1a5e56f2166e4e9d0964c71f4273b1d39227678d"
noncomputable se... | Mathlib/CategoryTheory/Localization/Construction.lean | 310 | 315 | theorem natTrans_hcomp_injective {F G : W.Localization β₯€ D} {Οβ Οβ : F βΆ G}
(h : π W.Q β« Οβ = π W.Q β« Οβ) : Οβ = Οβ := by |
ext X
have eq := (objEquiv W).right_inv X
simp only [objEquiv] at eq
rw [β eq, β NatTrans.id_hcomp_app, β NatTrans.id_hcomp_app, h]
|
import Mathlib.Algebra.Group.Nat
import Mathlib.Algebra.Order.Sub.Canonical
import Mathlib.Data.List.Perm
import Mathlib.Data.Set.List
import Mathlib.Init.Quot
import Mathlib.Order.Hom.Basic
#align_import data.multiset.basic from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
universe v
... | Mathlib/Data/Multiset/Basic.lean | 726 | 731 | theorem mem_of_mem_nsmul {a : Ξ±} {s : Multiset Ξ±} {n : β} (h : a β n β’ s) : a β s := by |
induction' n with n ih
Β· rw [zero_nsmul] at h
exact absurd h (not_mem_zero _)
Β· rw [succ_nsmul, mem_add] at h
exact h.elim ih id
|
import Mathlib.SetTheory.Cardinal.Finite
#align_import data.set.ncard from "leanprover-community/mathlib"@"74c2af38a828107941029b03839882c5c6f87a04"
namespace Set
variable {Ξ± Ξ² : Type*} {s t : Set Ξ±}
noncomputable def encard (s : Set Ξ±) : ββ := PartENat.withTopEquiv (PartENat.card s)
@[simp] theorem encard_uni... | Mathlib/Data/Set/Card.lean | 628 | 635 | theorem pred_ncard_le_ncard_diff_singleton (s : Set Ξ±) (a : Ξ±) : s.ncard - 1 β€ (s \ {a}).ncard := by |
cases' s.finite_or_infinite with hs hs
Β· by_cases h : a β s
Β· rw [ncard_diff_singleton_of_mem h hs]
rw [diff_singleton_eq_self h]
apply Nat.pred_le
convert Nat.zero_le _
rw [hs.ncard]
|
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology MeasureTheory
names... | Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean | 152 | 155 | theorem condexp_ae_eq_condexpL1CLM (hm : m β€ m0) [SigmaFinite (ΞΌ.trim hm)] (hf : Integrable f ΞΌ) :
ΞΌ[f|m] =α΅[ΞΌ] condexpL1CLM F' hm ΞΌ (hf.toL1 f) := by |
refine (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => ?_)
rw [condexpL1_eq hf]
|
import Mathlib.Algebra.Group.Indicator
import Mathlib.Data.Finset.Piecewise
import Mathlib.Data.Finset.Preimage
#align_import algebra.big_operators.basic from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
-- TODO
-- assert_not_exists AddCommMonoidWithOne
assert_not_exists MonoidWithZero... | Mathlib/Algebra/BigOperators/Group/Finset.lean | 2,140 | 2,142 | theorem prod_sdiff_div_prod_sdiff :
(β x β sβ \ sβ, f x) / β x β sβ \ sβ, f x = (β x β sβ, f x) / β x β sβ, f x := by |
simp [β Finset.prod_sdiff (@inf_le_left _ _ sβ sβ), β Finset.prod_sdiff (@inf_le_right _ _ sβ sβ)]
|
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.Group.FiniteSupport
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Set.Subsingleton
#align_import algebra.big_operators.finprod from "leanprover-community/mathlib"@"d6fad0e5bf... | Mathlib/Algebra/BigOperators/Finprod.lean | 716 | 720 | theorem MonoidHom.map_finprod_mem' {f : Ξ± β M} (g : M β* N) (hβ : (s β© mulSupport f).Finite) :
g (βαΆ j β s, f j) = βαΆ i β s, g (f i) := by |
rw [g.map_finprod]
Β· simp only [g.map_finprod_Prop]
Β· simpa only [finprod_eq_mulIndicator_apply, mulSupport_mulIndicator]
|
import Mathlib.Order.MinMax
import Mathlib.Data.Set.Subsingleton
import Mathlib.Tactic.Says
#align_import data.set.intervals.basic from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
open Function
open OrderDual (toDual ofDual)
variable {Ξ± Ξ² : Type*}
namespace Set
theorem Icc_bot_top... | Mathlib/Order/Interval/Set/Basic.lean | 1,700 | 1,709 | theorem Icc_union_Icc' (hβ : c β€ b) (hβ : a β€ d) : Icc a b βͺ Icc c d = Icc (min a c) (max b d) := by |
ext1 x
simp_rw [mem_union, mem_Icc, min_le_iff, le_max_iff]
by_cases hc : c β€ x <;> by_cases hd : x β€ d
Β· simp only [hc, hd, and_self, or_true] -- Porting note: restore `tauto`
Β· have hax : a β€ x := hβ.trans (le_of_not_ge hd)
simp only [hax, true_and, hc, or_self] -- Porting note: restore `tauto`
Β· hav... |
import Mathlib.FieldTheory.Separable
import Mathlib.RingTheory.IntegralDomain
import Mathlib.Algebra.CharP.Reduced
import Mathlib.Tactic.ApplyFun
#align_import field_theory.finite.basic from "leanprover-community/mathlib"@"12a85fac627bea918960da036049d611b1a3ee43"
variable {K : Type*} {R : Type*}
local notation ... | Mathlib/FieldTheory/Finite/Basic.lean | 608 | 613 | theorem pow_dichotomy (hF : ringChar F β 2) {a : F} (ha : a β 0) :
a ^ (Fintype.card F / 2) = 1 β¨ a ^ (Fintype.card F / 2) = -1 := by |
have hβ := FiniteField.pow_card_sub_one_eq_one a ha
rw [β Nat.two_mul_odd_div_two (FiniteField.odd_card_of_char_ne_two hF), mul_comm, pow_mul,
pow_two] at hβ
exact mul_self_eq_one_iff.mp hβ
|
import Mathlib.Init.Function
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
universe u
open Function
namespace Option
variable {Ξ± Ξ² Ξ³ Ξ΄ : Type*} {f : Ξ± β Ξ² β Ξ³} {a : Option Ξ±} {b : Option Ξ²} {c : Option Ξ³}
def mapβ (f : Ξ± β Ξ² β Ξ³) (a : Option Ξ±) ... | Mathlib/Data/Option/NAry.lean | 146 | 149 | theorem map_mapβ_distrib {g : Ξ³ β Ξ΄} {f' : Ξ±' β Ξ²' β Ξ΄} {gβ : Ξ± β Ξ±'} {gβ : Ξ² β Ξ²'}
(h_distrib : β a b, g (f a b) = f' (gβ a) (gβ b)) :
(mapβ f a b).map g = mapβ f' (a.map gβ) (b.map gβ) := by |
cases a <;> cases b <;> simp [h_distrib]
|
import Mathlib.Analysis.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
#align_import analysis.calculus.cont_diff_def from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
noncomputable section
open scoped Classical
open NNReal Topology Filter
local notatio... | Mathlib/Analysis/Calculus/ContDiff/Defs.lean | 901 | 905 | theorem iteratedFDerivWithin_one_apply (h : UniqueDiffWithinAt π s x) (m : Fin 1 β E) :
iteratedFDerivWithin π 1 f s x m = fderivWithin π f s x (m 0) := by |
simp only [iteratedFDerivWithin_succ_apply_left, iteratedFDerivWithin_zero_eq_comp,
(continuousMultilinearCurryFin0 π E F).symm.comp_fderivWithin h]
rfl
|
import Mathlib.Control.Functor.Multivariate
import Mathlib.Data.PFunctor.Univariate.Basic
#align_import data.pfunctor.multivariate.basic from "leanprover-community/mathlib"@"e3d9ab8faa9dea8f78155c6c27d62a621f4c152d"
universe u v
open MvFunctor
@[pp_with_univ]
structure MvPFunctor (n : β) where
A : Type u
... | Mathlib/Data/PFunctor/Multivariate/Basic.lean | 116 | 119 | theorem const.mk_get (x : const n A Ξ±) : const.mk n (const.get x) = x := by |
cases x
dsimp [const.get, const.mk]
congr with (_β¨β©)
|
import Mathlib.RingTheory.Ideal.IsPrimary
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
universe u v
namespace Ideal
variable {R : Type u} {S : Type v}... | Mathlib/RingTheory/JacobsonIdeal.lean | 322 | 340 | theorem jacobson_bot_polynomial_le_sInf_map_maximal :
jacobson (β₯ : Ideal R[X]) β€ sInf (map (C : R β+* R[X]) '' { J : Ideal R | J.IsMaximal }) := by |
refine le_sInf fun J => exists_imp.2 fun j hj => ?_
haveI : j.IsMaximal := hj.1
refine Trans.trans (jacobson_mono bot_le) (le_of_eq ?_ : J.jacobson β€ J)
suffices t : (β₯ : Ideal (Polynomial (R β§Έ j))).jacobson = β₯ by
rw [β hj.2, jacobson_eq_iff_jacobson_quotient_eq_bot]
replace t := congr_arg (map (polyn... |
import Mathlib.Data.Set.Lattice
import Mathlib.Logic.Small.Basic
import Mathlib.Logic.Function.OfArity
import Mathlib.Order.WellFounded
#align_import set_theory.zfc.basic from "leanprover-community/mathlib"@"f0b3759a8ef0bd8239ecdaa5e1089add5feebe1a"
-- Porting note: Lean 3 uses `Set` for `ZFSet`.
set_option linter... | Mathlib/SetTheory/ZFC/Basic.lean | 1,492 | 1,500 | theorem mem_wf : @WellFounded Class.{u} (Β· β Β·) :=
β¨by
have H : β x : ZFSet.{u}, @Acc Class.{u} (Β· β Β·) βx := by |
refine fun a => ZFSet.inductionOn a fun x IH => β¨_, ?_β©
rintro A β¨z, rfl, hzβ©
exact IH z hz
refine fun A => β¨A, ?_β©
rintro B β¨x, rfl, _β©
exact H xβ©
|
import Mathlib.Analysis.Normed.Group.Seminorm
import Mathlib.Order.LiminfLimsup
import Mathlib.Topology.Instances.Rat
import Mathlib.Topology.MetricSpace.Algebra
import Mathlib.Topology.MetricSpace.IsometricSMul
import Mathlib.Topology.Sequences
#align_import analysis.normed.group.basic from "leanprover-community/mat... | Mathlib/Analysis/Normed/Group/Basic.lean | 2,040 | 2,051 | theorem mul_lipschitzWith (hf : AntilipschitzWith Kf f) (hg : LipschitzWith Kg g) (hK : Kg < Kfβ»ΒΉ) :
AntilipschitzWith (Kfβ»ΒΉ - Kg)β»ΒΉ fun x => f x * g x := by |
letI : PseudoMetricSpace Ξ± := PseudoEMetricSpace.toPseudoMetricSpace hf.edist_ne_top
refine AntilipschitzWith.of_le_mul_dist fun x y => ?_
rw [NNReal.coe_inv, β _root_.div_eq_inv_mul]
rw [le_div_iff (NNReal.coe_pos.2 <| tsub_pos_iff_lt.2 hK)]
rw [mul_comm, NNReal.coe_sub hK.le, _root_.sub_mul]
-- Porting n... |
import Mathlib.Algebra.Module.Torsion
import Mathlib.RingTheory.DedekindDomain.Ideal
#align_import algebra.module.dedekind_domain from "leanprover-community/mathlib"@"cdc34484a07418af43daf8198beaf5c00324bca8"
universe u v
variable {R : Type u} [CommRing R] [IsDomain R] {M : Type v} [AddCommGroup M] [Module R M]
... | Mathlib/Algebra/Module/DedekindDomain.lean | 65 | 72 | theorem isInternal_prime_power_torsion [Module.Finite R M] (hM : Module.IsTorsion R M) :
DirectSum.IsInternal fun p : (factors (β€ : Submodule R M).annihilator).toFinset =>
torsionBySet R M (p ^ (factors (β€ : Submodule R M).annihilator).count βp : Ideal R) := by |
have hM' := Module.isTorsionBySet_annihilator_top R M
have hI := Submodule.annihilator_top_inter_nonZeroDivisors hM
refine isInternal_prime_power_torsion_of_is_torsion_by_ideal ?_ hM'
rw [β Set.nonempty_iff_ne_empty] at hI; rw [Submodule.ne_bot_iff]
obtain β¨x, H, hxβ© := hI; exact β¨x, H, nonZeroDivisors.ne_ze... |
import Mathlib.Logic.Function.Basic
import Mathlib.Logic.Relator
import Mathlib.Init.Data.Quot
import Mathlib.Tactic.Cases
import Mathlib.Tactic.Use
import Mathlib.Tactic.MkIffOfInductiveProp
import Mathlib.Tactic.SimpRw
#align_import logic.relation from "leanprover-community/mathlib"@"3365b20c2ffa7c35e47e5209b89ba9a... | Mathlib/Logic/Relation.lean | 436 | 439 | theorem trans_right (hab : ReflTransGen r a b) (hbc : TransGen r b c) : TransGen r a c := by |
induction hbc with
| single hbc => exact tail' hab hbc
| tail _ hcd hac => exact hac.tail hcd
|
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology.Instances.Sign
#align_import analysis.special_functions.trigonometric.angle from "leanprover-community/mathlib"@"213b0cff7bc5ab6696ee07cceec80829... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 444 | 446 | theorem cos_sq_add_sin_sq (ΞΈ : Real.Angle) : cos ΞΈ ^ 2 + sin ΞΈ ^ 2 = 1 := by |
induction ΞΈ using Real.Angle.induction_on
exact Real.cos_sq_add_sin_sq _
|
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Topology.MetricSpace.IsometricSMul
#align_import topology.metric_space.hausdorff_distance from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
noncomputable section
open NNReal ENNReal Topology Set Filter Pointwise Bornology
u... | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | 180 | 181 | theorem mem_iff_infEdist_zero_of_closed (h : IsClosed s) : x β s β infEdist x s = 0 := by |
rw [β mem_closure_iff_infEdist_zero, h.closure_eq]
|
import Mathlib.Algebra.Group.Subgroup.MulOpposite
import Mathlib.Algebra.Group.Submonoid.Pointwise
import Mathlib.GroupTheory.GroupAction.ConjAct
#align_import group_theory.subgroup.pointwise from "leanprover-community/mathlib"@"e655e4ea5c6d02854696f97494997ba4c31be802"
open Set
open Pointwise
variable {Ξ± G A S... | Mathlib/Algebra/Group/Subgroup/Pointwise.lean | 366 | 372 | theorem conj_smul_subgroupOf {P H : Subgroup G} (hP : P β€ H) (h : H) :
MulAut.conj h β’ P.subgroupOf H = (MulAut.conj (h : G) β’ P).subgroupOf H := by |
refine le_antisymm ?_ ?_
Β· rintro - β¨g, hg, rflβ©
exact β¨g, hg, rflβ©
Β· rintro p β¨g, hg, hpβ©
exact β¨β¨g, hP hgβ©, hg, Subtype.ext hpβ©
|
import Mathlib.Topology.UniformSpace.Cauchy
import Mathlib.Topology.UniformSpace.Separation
import Mathlib.Topology.DenseEmbedding
#align_import topology.uniform_space.uniform_embedding from "leanprover-community/mathlib"@"195fcd60ff2bfe392543bceb0ec2adcdb472db4c"
open Filter Function Set Uniformity Topology
sec... | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | 473 | 475 | theorem uniformly_extend_spec [CompleteSpace Ξ³] (a : Ξ±) : Tendsto f (comap e (π a)) (π (Ο a)) := by |
simpa only [DenseInducing.extend] using
tendsto_nhds_limUnder (uniformly_extend_exists h_e βΉ_βΊ h_f _)
|
import Mathlib.Topology.Constructions
#align_import topology.continuous_on from "leanprover-community/mathlib"@"d4f691b9e5f94cfc64639973f3544c95f8d5d494"
open Set Filter Function Topology Filter
variable {Ξ± : Type*} {Ξ² : Type*} {Ξ³ : Type*} {Ξ΄ : Type*}
variable [TopologicalSpace Ξ±]
@[simp]
theorem nhds_bind_nhdsW... | Mathlib/Topology/ContinuousOn.lean | 276 | 277 | theorem nhdsWithin_singleton (a : Ξ±) : π[{a}] a = pure a := by |
rw [nhdsWithin, principal_singleton, inf_eq_right.2 (pure_le_nhds a)]
|
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.BigOperators
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.Tactic.FinCases
#align_import linear_algebra.affine_space.combination from ... | Mathlib/LinearAlgebra/AffineSpace/Combination.lean | 487 | 491 | theorem affineCombination_indicator_subset (w : ΞΉ β k) (p : ΞΉ β P) {sβ sβ : Finset ΞΉ}
(h : sβ β sβ) :
sβ.affineCombination k p w = sβ.affineCombination k p (Set.indicator (βsβ) w) := by |
rw [affineCombination_apply, affineCombination_apply,
weightedVSubOfPoint_indicator_subset _ _ _ h]
|
import Mathlib.Algebra.Polynomial.Expand
import Mathlib.Algebra.Polynomial.Laurent
import Mathlib.LinearAlgebra.Matrix.Charpoly.Basic
import Mathlib.LinearAlgebra.Matrix.Reindex
import Mathlib.RingTheory.Polynomial.Nilpotent
#align_import linear_algebra.matrix.charpoly.coeff from "leanprover-community/mathlib"@"9745b... | Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean | 61 | 78 | theorem charpoly_sub_diagonal_degree_lt :
(M.charpoly - β i : n, (X - C (M i i))).degree < β(Fintype.card n - 1) := by |
rw [charpoly, det_apply', β insert_erase (mem_univ (Equiv.refl n)),
sum_insert (not_mem_erase (Equiv.refl n) univ), add_comm]
simp only [charmatrix_apply_eq, one_mul, Equiv.Perm.sign_refl, id, Int.cast_one,
Units.val_one, add_sub_cancel_right, Equiv.coe_refl]
rw [β mem_degreeLT]
apply Submodule.sum_mem... |
import Mathlib.Order.Interval.Set.Disjoint
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
#align_import measure_theory.integral.interval_integral from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
noncomputable section
open scoped... | Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | 742 | 744 | theorem integral_comp_mul_left (hc : c β 0) :
(β« x in a..b, f (c * x)) = cβ»ΒΉ β’ β« x in c * a..c * b, f x := by |
simpa only [mul_comm c] using integral_comp_mul_right f hc
|
import Mathlib.Data.Finset.Attr
import Mathlib.Data.Multiset.FinsetOps
import Mathlib.Logic.Equiv.Set
import Mathlib.Order.Directed
import Mathlib.Order.Interval.Set.Basic
#align_import data.finset.basic from "leanprover-community/mathlib"@"442a83d738cb208d3600056c489be16900ba701d"
-- Assert that we define `Finset... | Mathlib/Data/Finset/Basic.lean | 1,581 | 1,590 | theorem induction_on_union (P : Finset Ξ± β Finset Ξ± β Prop) (symm : β {a b}, P a b β P b a)
(empty_right : β {a}, P a β
) (singletons : β {a b}, P {a} {b})
(union_of : β {a b c}, P a c β P b c β P (a βͺ b) c) : β a b, P a b := by |
intro a b
refine Finset.induction_on b empty_right fun x s _xs hi => symm ?_
rw [Finset.insert_eq]
apply union_of _ (symm hi)
refine Finset.induction_on a empty_right fun a t _ta hi => symm ?_
rw [Finset.insert_eq]
exact union_of singletons (symm hi)
|
import Mathlib.FieldTheory.Galois
#align_import field_theory.polynomial_galois_group from "leanprover-community/mathlib"@"e3f4be1fcb5376c4948d7f095bec45350bfb9d1a"
noncomputable section
open scoped Polynomial
open FiniteDimensional
namespace Polynomial
variable {F : Type*} [Field F] (p q : F[X]) (E : Type*) [... | Mathlib/FieldTheory/PolynomialGaloisGroup.lean | 74 | 79 | theorem ext {Ο Ο : p.Gal} (h : β x β p.rootSet p.SplittingField, Ο x = Ο x) : Ο = Ο := by |
refine
AlgEquiv.ext fun x =>
(AlgHom.mem_equalizer Ο.toAlgHom Ο.toAlgHom x).mp
((SetLike.ext_iff.mp ?_ x).mpr Algebra.mem_top)
rwa [eq_top_iff, β SplittingField.adjoin_rootSet, Algebra.adjoin_le_iff]
|
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Function.SimpleFunc
import Mathlib.MeasureTheory.Measure.MutuallySingular
import Mathlib.MeasureTheory.Measure.Count
import Mathlib.Topology.IndicatorConstPointwise
import Mathlib.MeasureTheory.Constructions.BorelSpace.Real
#align_import m... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 708 | 720 | theorem lintegral_const_mul' (r : ββ₯0β) (f : Ξ± β ββ₯0β) (hr : r β β) :
β«β» a, r * f a βΞΌ = r * β«β» a, f a βΞΌ := by |
by_cases h : r = 0
Β· simp [h]
apply le_antisymm _ (lintegral_const_mul_le r f)
have rinv : r * rβ»ΒΉ = 1 := ENNReal.mul_inv_cancel h hr
have rinv' : rβ»ΒΉ * r = 1 := by
rw [mul_comm]
exact rinv
have := lintegral_const_mul_le (ΞΌ := ΞΌ) rβ»ΒΉ fun x => r * f x
simp? [(mul_assoc _ _ _).symm, rinv'] at this ... |
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.MeasureTheory.Constructions.Prod.Integral
import Mathlib.MeasureTheory.Function.LocallyIntegrable
import Mathlib.MeasureTheory.Group.Integral
import Mathlib.MeasureTheory.Group.Prod
import Mathlib.Measure... | Mathlib/Analysis/Convolution.lean | 1,380 | 1,385 | theorem contDiffOn_convolution_left_with_param [ΞΌ.IsAddLeftInvariant] [ΞΌ.IsNegInvariant]
(L : E' βL[π] E βL[π] F) {f : G β E} {n : ββ} {g : P β G β E'} {s : Set P} {k : Set G}
(hs : IsOpen s) (hk : IsCompact k) (hgs : β p, β x, p β s β x β k β g p x = 0)
(hf : LocallyIntegrable f ΞΌ) (hg : ContDiffOn π n ... |
simpa only [convolution_flip] using contDiffOn_convolution_right_with_param L.flip hs hk hgs hf hg
|
import Mathlib.Topology.Order.ProjIcc
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.UnitInterval
#align_import topology.path_connected from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open scoped Classical
open Topology Filter unitInterval Set Fun... | Mathlib/Topology/Connected/PathConnected.lean | 436 | 438 | theorem map_id (Ξ³ : Path x y) : Ξ³.map continuous_id = Ξ³ := by |
ext
rfl
|
import Mathlib.Data.Set.Card
import Mathlib.Order.Minimal
import Mathlib.Data.Matroid.Init
set_option autoImplicit true
open Set
def Matroid.ExchangeProperty {Ξ± : Type _} (P : Set Ξ± β Prop) : Prop :=
β X Y, P X β P Y β β a β X \ Y, β b β Y \ X, P (insert b (X \ {a}))
def Matroid.ExistsMaximalSubsetProperty {... | Mathlib/Data/Matroid/Basic.lean | 427 | 428 | theorem not_finiteRk (M : Matroid Ξ±) [InfiniteRk M] : Β¬ FiniteRk M := by |
intro h; obtain β¨B,hBβ© := M.exists_base; exact hB.infinite hB.finite
|
import Mathlib.Algebra.Regular.Basic
import Mathlib.LinearAlgebra.Matrix.MvPolynomial
import Mathlib.LinearAlgebra.Matrix.Polynomial
import Mathlib.RingTheory.Polynomial.Basic
#align_import linear_algebra.matrix.adjugate from "leanprover-community/mathlib"@"a99f85220eaf38f14f94e04699943e185a5e1d1a"
namespace Matr... | Mathlib/LinearAlgebra/Matrix/Adjugate.lean | 160 | 170 | theorem sum_cramer_apply {Ξ²} (s : Finset Ξ²) (f : n β Ξ² β Ξ±) (i : n) :
(β x β s, cramer A (fun j => f j x) i) = cramer A (fun j : n => β x β s, f j x) i :=
calc
(β x β s, cramer A (fun j => f j x) i) = (β x β s, cramer A fun j => f j x) i :=
(Finset.sum_apply i s _).symm
_ = cramer A (fun j : n => β ... |
rw [sum_cramer, cramer_apply, cramer_apply]
simp only [updateColumn]
congr with j
congr
apply Finset.sum_apply
|
import Mathlib.Analysis.SpecialFunctions.Complex.Circle
import Mathlib.Geometry.Euclidean.Angle.Oriented.Basic
#align_import geometry.euclidean.angle.oriented.rotation from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
noncomputable section
open FiniteDimensional Complex
open scoped ... | Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean | 365 | 373 | theorem oangle_eq_iff_eq_pos_smul_rotation_or_eq_zero {x y : V} (ΞΈ : Real.Angle) :
o.oangle x y = ΞΈ β
(x β 0 β§ y β 0 β§ β r : β, 0 < r β§ y = r β’ o.rotation ΞΈ x) β¨ ΞΈ = 0 β§ (x = 0 β¨ y = 0) := by |
by_cases hx : x = 0
Β· simp [hx, eq_comm]
Β· by_cases hy : y = 0
Β· simp [hy, eq_comm]
Β· rw [o.oangle_eq_iff_eq_pos_smul_rotation_of_ne_zero hx hy]
simp [hx, hy]
|
import Mathlib.Algebra.Order.Invertible
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.LinearAlgebra.AffineSpace.Midpoint
import Mathlib.LinearAlgebra.Ray
import Mathlib.Tactic.GCongr
#align_import analysis.convex.segment from "leanprover-community/mathlib"@"c5773405394e073885e2a144c9ca14637e8eb963"
... | Mathlib/Analysis/Convex/Segment.lean | 595 | 608 | theorem Convex.mem_Ico (h : x < y) :
z β Ico x y β β a b, 0 < a β§ 0 β€ b β§ a + b = 1 β§ a * x + b * y = z := by |
refine β¨fun hz => ?_, ?_β©
Β· obtain β¨a, b, ha, hb, hab, rflβ© := (Convex.mem_Icc h.le).1 (Ico_subset_Icc_self hz)
obtain rfl | ha' := ha.eq_or_lt
Β· rw [zero_add] at hab
rw [hab, one_mul, zero_mul, zero_add] at hz
exact (hz.2.ne rfl).elim
Β· exact β¨a, b, ha', hb, hab, rflβ©
Β· rintro β¨a, b, ha,... |
import Mathlib.LinearAlgebra.FreeModule.PID
import Mathlib.MeasureTheory.Group.FundamentalDomain
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.RingTheory.Localization.Module
#align_import algebra.module.zlattice from "leanprover-community/mathlib"@"a3e83f0fa4391c8740f7d773a7a9b74e311ae2a3"
n... | Mathlib/Algebra/Module/Zlattice/Basic.lean | 419 | 452 | theorem Zlattice.FG [hs : IsZlattice K L] : AddSubgroup.FG L := by |
suffices (AddSubgroup.toIntSubmodule L).FG by exact (fg_iff_add_subgroup_fg _).mp this
obtain β¨s, β¨h_incl, β¨h_span, h_lindβ©β©β© := exists_linearIndependent K (L : Set E)
-- Let `s` be a maximal `K`-linear independent family of elements of `L`. We show that
-- `L` is finitely generated (as a β€-module) because it ... |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Comp
import Mathlib.Analysis.Calculus.FDeriv.RestrictScalars
#align_import analysis.calculus.deriv.comp from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
open scoped Classical
open Top... | Mathlib/Analysis/Calculus/Deriv/Comp.lean | 404 | 408 | theorem fderivWithin.comp_derivWithin_of_eq {t : Set F} (hl : DifferentiableWithinAt π l t y)
(hf : DifferentiableWithinAt π f s x) (hs : MapsTo f s t) (hxs : UniqueDiffWithinAt π s x)
(hy : y = f x) :
derivWithin (l β f) s x = (fderivWithin π l t (f x) : F β E) (derivWithin f s x) := by |
rw [hy] at hl; exact fderivWithin.comp_derivWithin x hl hf hs hxs
|
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryTheory.Limits.Cones
#align_import category_theory.limits.is_limit from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
noncomputable section
open CategoryTheory CategoryTheory.Category CategoryTheory.Functor Opposite
... | Mathlib/CategoryTheory/Limits/IsLimit.lean | 309 | 311 | theorem conePointsIsoOfNatIso_hom_comp {F G : J β₯€ C} {s : Cone F} {t : Cone G} (P : IsLimit s)
(Q : IsLimit t) (w : F β
G) (j : J) :
(conePointsIsoOfNatIso P Q w).hom β« t.Ο.app j = s.Ο.app j β« w.hom.app j := by | simp
|
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
import Mathlib.Logic.Basic
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
open Function
universe u v w
namespace Function
section
variable {Ξ± Ξ² Ξ³ : Sort*} {f : Ξ± β Ξ²}
@[reducible, simp] de... | Mathlib/Logic/Function/Basic.lean | 691 | 694 | theorem update_idem {Ξ±} [DecidableEq Ξ±] {Ξ² : Ξ± β Sort*} {a : Ξ±} (v w : Ξ² a) (f : β a, Ξ² a) :
update (update f a v) a w = update f a w := by |
funext b
by_cases h : b = a <;> simp [update, h]
|
import Mathlib.MeasureTheory.Function.LpOrder
#align_import measure_theory.function.l1_space from "leanprover-community/mathlib"@"ccdbfb6e5614667af5aa3ab2d50885e0ef44a46f"
noncomputable section
open scoped Classical
open Topology ENNReal MeasureTheory NNReal
open Set Filter TopologicalSpace ENNReal EMetric Meas... | Mathlib/MeasureTheory/Function/L1Space.lean | 604 | 608 | theorem integrable_map_measure {f : Ξ± β Ξ΄} {g : Ξ΄ β Ξ²}
(hg : AEStronglyMeasurable g (Measure.map f ΞΌ)) (hf : AEMeasurable f ΞΌ) :
Integrable g (Measure.map f ΞΌ) β Integrable (g β f) ΞΌ := by |
simp_rw [β memβp_one_iff_integrable]
exact memβp_map_measure_iff hg hf
|
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Diagonal
import Mathlib.LinearAlgebra.Matrix.Transvection
import Mathlib.MeasureTheory.Group.LIntegral
import Mathlib.MeasureTheory.Integral.Marginal
import Mathlib.MeasureTheory.Measure.Stiel... | Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | 506 | 508 | theorem measurableSet_graph (hf : Measurable f) :
MeasurableSet { p : Ξ± Γ β | p.snd = f p.fst } := by |
simpa using measurableSet_region_between_cc hf hf MeasurableSet.univ
|
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Zip
import Mathlib.Data.Nat.Defs
import Mathlib.Data.List.Infix
#align_import data.list.rotate from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
universe u
variable {Ξ± : Type u}
open Nat Function
namespace List
theorem rotate... | Mathlib/Data/List/Rotate.lean | 572 | 573 | theorem length_cyclicPermutations_cons (x : Ξ±) (l : List Ξ±) :
length (cyclicPermutations (x :: l)) = length l + 1 := by | simp [cyclicPermutations_cons]
|
import Mathlib.Geometry.Manifold.MFDeriv.Defs
#align_import geometry.manifold.mfderiv from "leanprover-community/mathlib"@"e473c3198bb41f68560cab68a0529c854b618833"
noncomputable section
open scoped Topology Manifold
open Set Bundle
section DerivativesProperties
variable
{π : Type*} [NontriviallyNormedFiel... | Mathlib/Geometry/Manifold/MFDeriv/Basic.lean | 704 | 708 | theorem HasMFDerivAt.comp_hasMFDerivWithinAt (hg : HasMFDerivAt I' I'' g (f x) g')
(hf : HasMFDerivWithinAt I I' f s x f') :
HasMFDerivWithinAt I I'' (g β f) s x (g'.comp f') := by |
rw [β hasMFDerivWithinAt_univ] at *
exact HasMFDerivWithinAt.comp x (hg.mono (subset_univ _)) hf subset_preimage_univ
|
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Data.Nat.Totient
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.GroupTheory.Subgroup.Simple
import Mathlib.Tactic.Group
import Mathlib.GroupTheory.Exponent
#align_import group_theory.specific_groups.cyclic from "leanprover-community/mathli... | Mathlib/GroupTheory/SpecificGroups/Cyclic.lean | 606 | 626 | theorem prime_card [Fintype Ξ±] : (Fintype.card Ξ±).Prime := by |
have h0 : 0 < Fintype.card Ξ± := Fintype.card_pos_iff.2 (by infer_instance)
obtain β¨g, hgβ© := IsCyclic.exists_generator (Ξ± := Ξ±)
rw [Nat.prime_def_lt'']
refine β¨Fintype.one_lt_card_iff_nontrivial.2 inferInstance, fun n hn => ?_β©
refine (IsSimpleOrder.eq_bot_or_eq_top (Subgroup.zpowers (g ^ n))).symm.imp ?_ ?_... |
import Mathlib.Logic.Function.Iterate
import Mathlib.Init.Data.Int.Order
import Mathlib.Order.Compare
import Mathlib.Order.Max
import Mathlib.Order.RelClasses
import Mathlib.Tactic.Choose
#align_import order.monotone.basic from "leanprover-community/mathlib"@"554bb38de8ded0dafe93b7f18f0bfee6ef77dc5d"
open Functio... | Mathlib/Order/Monotone/Basic.lean | 1,014 | 1,018 | theorem Nat.rel_of_forall_rel_succ_of_le_of_lt (r : Ξ² β Ξ² β Prop) [IsTrans Ξ² r] {f : β β Ξ²} {a : β}
(h : β n, a β€ n β r (f n) (f (n + 1))) β¦b c : ββ¦ (hab : a β€ b) (hbc : b < c) :
r (f b) (f c) := by |
induction' hbc with k b_lt_k r_b_k
exacts [h _ hab, _root_.trans r_b_k (h _ (hab.trans_lt b_lt_k).le)]
|
import Mathlib.NumberTheory.Cyclotomic.Discriminant
import Mathlib.RingTheory.Polynomial.Eisenstein.IsIntegral
import Mathlib.RingTheory.Ideal.Norm
#align_import number_theory.cyclotomic.rat from "leanprover-community/mathlib"@"b353176c24d96c23f0ce1cc63efc3f55019702d9"
universe u
open Algebra IsCyclotomicExtensio... | Mathlib/NumberTheory/Cyclotomic/Rat.lean | 429 | 464 | theorem not_exists_int_prime_dvd_sub_of_prime_pow_ne_two
[hcycl : IsCyclotomicExtension {p ^ (k + 1)} β K]
(hΞΆ : IsPrimitiveRoot ΞΆ β(p ^ (k + 1))) (htwo : p ^ (k + 1) β 2) :
Β¬(β n : β€, (p : π K) β£ (hΞΆ.toInteger - n : π K)) := by |
intro β¨n, x, hβ©
-- Let `pB` be the power basis of `π K` given by powers of `ΞΆ`.
let pB := hΞΆ.integralPowerBasis
have hdim : pB.dim = βp ^ k * (βp - 1) := by
simp [integralPowerBasis_dim, pB, Nat.totient_prime_pow hp.1 (Nat.zero_lt_succ k)]
replace hdim : 1 < pB.dim := by
rw [Nat.one_lt_iff_ne_zero_a... |
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition
import Mathlib.LinearAlgebra.Projection
import Mathlib.LinearAlgebra.SesquilinearForm
import Mathlib.RingTheory.TensorProduct.Basic
import Mathlib.RingTheory.I... | Mathlib/LinearAlgebra/Dual.lean | 1,844 | 1,846 | theorem dualDistribInvOfBasis_apply (b : Basis ΞΉ R M) (c : Basis ΞΊ R N) (f : Dual R (M β[R] N)) :
dualDistribInvOfBasis b c f = β i, β j, f (b i ββ c j) β’ b.dualBasis i ββ c.dualBasis j := by |
simp [dualDistribInvOfBasis]
|
import Mathlib.MeasureTheory.Decomposition.SignedHahn
import Mathlib.MeasureTheory.Measure.MutuallySingular
#align_import measure_theory.decomposition.jordan from "leanprover-community/mathlib"@"70a4f2197832bceab57d7f41379b2592d1110570"
noncomputable section
open scoped Classical MeasureTheory ENNReal NNReal
va... | Mathlib/MeasureTheory/Decomposition/Jordan.lean | 196 | 212 | theorem exists_compl_positive_negative :
β S : Set Ξ±,
MeasurableSet S β§
j.toSignedMeasure β€[S] 0 β§
0 β€[SαΆ] j.toSignedMeasure β§ j.posPart S = 0 β§ j.negPart SαΆ = 0 := by |
obtain β¨S, hSβ, hSβ, hSββ© := j.mutuallySingular
refine β¨S, hSβ, ?_, ?_, hSβ, hSββ©
Β· refine restrict_le_restrict_of_subset_le _ _ fun A hA hAβ => ?_
rw [toSignedMeasure, toSignedMeasure_sub_apply hA,
show j.posPart A = 0 from nonpos_iff_eq_zero.1 (hSβ βΈ measure_mono hAβ), ENNReal.zero_toReal,
zero... |
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.GroupTheory.GroupAction.Ring
#align_import data.polynomial.derivative from "leanprover-community/mathlib"@"bbeb185db4ccee8ed07dc48449414ebfa39cb821"
noncomputable section
open Finset
open Polynomial
namespace Pol... | Mathlib/Algebra/Polynomial/Derivative.lean | 659 | 661 | theorem derivative_X_sub_C_pow (c : R) (m : β) :
derivative ((X - C c) ^ m) = C (m : R) * (X - C c) ^ (m - 1) := by |
rw [derivative_pow, derivative_X_sub_C, mul_one]
|
import Mathlib.LinearAlgebra.Dimension.Finite
import Mathlib.LinearAlgebra.Dimension.Constructions
open Cardinal Submodule Set FiniteDimensional
universe u v
namespace Subalgebra
variable {F E : Type*} [CommRing F] [StrongRankCondition F] [Ring E] [Algebra F E]
{S : Subalgebra F E}
theorem eq_bot_of_rank_le_o... | Mathlib/LinearAlgebra/Dimension/FreeAndStrongRankCondition.lean | 277 | 280 | theorem eq_bot_of_finrank_one (h : finrank F S = 1) [Module.Free F S] : S = β₯ := by |
refine Subalgebra.eq_bot_of_rank_le_one ?_
rw [finrank, toNat_eq_one] at h
rw [h]
|
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing
import Mathlib.CategoryTheory.Adjunction.Evaluation
import Mathlib.CategoryTheory.Limits.Types
import Mathlib.CategoryTheory.Limits.Preserves.Filtered
import Mathlib.Cat... | Mathlib/Topology/Sheaves/Stalks.lean | 354 | 358 | theorem stalkSpecializes_stalkFunctor_map {F G : X.Presheaf C} (f : F βΆ G) {x y : X} (h : x β€³ y) :
F.stalkSpecializes h β« (stalkFunctor C x).map f =
(stalkFunctor C y).map f β« G.stalkSpecializes h := by |
change (_ : colimit _ βΆ _) = (_ : colimit _ βΆ _)
ext; delta stalkFunctor; simpa [stalkSpecializes] using by rfl
|
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.Combinatorics.SimpleGraph.Operations
import Mathlib.Data.Finset.Pairwise
#align_import combinatorics.simple_graph.clique from "leanprover-community/mathlib"@"3365b20c2ffa7c35e47e5209b89ba9abdddf3ffe"
open Finset Fintype Function SimpleGraph.Walk... | Mathlib/Combinatorics/SimpleGraph/Clique.lean | 523 | 528 | theorem cliqueSet_map_of_equiv (G : SimpleGraph Ξ±) (e : Ξ± β Ξ²) (n : β) :
(G.map e.toEmbedding).cliqueSet n = map e.toEmbedding '' G.cliqueSet n := by |
obtain rfl | hn := eq_or_ne n 1
Β· ext
simp [e.exists_congr_left]
Β· exact cliqueSet_map hn _ _
|
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.Analysis.MeanInequalities
import Mathlib.Analysis.MeanInequalitiesPow
import Mathlib.MeasureTheory.Function.SpecialFunctions.Basic
#align_import measure_theory.integral.mean_inequalities from "leanprover-community/mathlib"@"13bf7613c96a9fd66a81b9020a82cad9... | Mathlib/MeasureTheory/Integral/MeanInequalities.lean | 141 | 147 | theorem lintegral_mul_eq_zero_of_lintegral_rpow_eq_zero {p : β} (hp0 : 0 β€ p) {f g : Ξ± β ββ₯0β}
(hf : AEMeasurable f ΞΌ) (hf_zero : β«β» a, f a ^ p βΞΌ = 0) : (β«β» a, (f * g) a βΞΌ) = 0 := by |
rw [β @lintegral_zero_fun Ξ± _ ΞΌ]
refine lintegral_congr_ae ?_
suffices h_mul_zero : f * g =α΅[ΞΌ] 0 * g by rwa [zero_mul] at h_mul_zero
have hf_eq_zero : f =α΅[ΞΌ] 0 := ae_eq_zero_of_lintegral_rpow_eq_zero hp0 hf hf_zero
exact hf_eq_zero.mul (ae_eq_refl g)
|
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calculus.LocalExtr.Rolle
import Mathlib.Analysis.Convex.Normed
import Mathlib.Analysis.RCLike.Basic
#align_import... | Mathlib/Analysis/Calculus/MeanValue.lean | 1,100 | 1,118 | theorem hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt
(hder : βαΆ y in π x, HasFDerivAt f (f' y) y) (hcont : ContinuousAt f' x) :
HasStrictFDerivAt f (f' x) x := by |
-- turn little-o definition of strict_fderiv into an epsilon-delta statement
refine isLittleO_iff.mpr fun c hc => Metric.eventually_nhds_iff_ball.mpr ?_
-- the correct Ξ΅ is the modulus of continuity of f'
rcases Metric.mem_nhds_iff.mp (inter_mem hder (hcont <| ball_mem_nhds _ hc)) with β¨Ξ΅, Ξ΅0, hΞ΅β©
refine β¨Ξ΅,... |
import Mathlib.Init.Algebra.Classes
import Mathlib.Logic.Nontrivial.Basic
import Mathlib.Order.BoundedOrder
import Mathlib.Data.Option.NAry
import Mathlib.Tactic.Lift
import Mathlib.Data.Option.Basic
#align_import order.with_bot from "leanprover-community/mathlib"@"0111834459f5d7400215223ea95ae38a1265a907"
variabl... | Mathlib/Order/WithBot.lean | 276 | 280 | theorem unbot'_le_iff {a : WithBot Ξ±} {b c : Ξ±} (h : a = β₯ β b β€ c) :
a.unbot' b β€ c β a β€ c := by |
induction a
Β· simpa using h rfl
Β· simp
|
import Mathlib.Topology.Category.TopCat.Limits.Products
#align_import topology.category.Top.limits.pullbacks from "leanprover-community/mathlib"@"178a32653e369dce2da68dc6b2694e385d484ef1"
-- Porting note: every ML3 decl has an uppercase letter
set_option linter.uppercaseLean3 false
open TopologicalSpace
open Cat... | Mathlib/Topology/Category/TopCat/Limits/Pullbacks.lean | 406 | 421 | theorem pullback_snd_image_fst_preimage (f : X βΆ Z) (g : Y βΆ Z) (U : Set X) :
(pullback.snd : pullback f g βΆ _) '' ((pullback.fst : pullback f g βΆ _) β»ΒΉ' U) =
g β»ΒΉ' (f '' U) := by |
ext x
constructor
Β· rintro β¨(y : (forget TopCat).obj _), hy, rflβ©
exact
β¨(pullback.fst : pullback f g βΆ _) y, hy, ConcreteCategory.congr_hom pullback.condition yβ©
Β· rintro β¨y, hy, eqβ©
-- next 5 lines were
-- `exact β¨(TopCat.pullbackIsoProdSubtype f g).inv β¨β¨_, _β©, eqβ©, by simpa, by simpβ©` before ... |
import Mathlib.Order.Filter.Basic
import Mathlib.Topology.Bases
import Mathlib.Data.Set.Accumulate
import Mathlib.Topology.Bornology.Basic
import Mathlib.Topology.LocallyFinite
open Set Filter Topology TopologicalSpace Classical Function
universe u v
variable {X : Type u} {Y : Type v} {ΞΉ : Type*}
variable [Topolog... | Mathlib/Topology/Compactness/Compact.lean | 1,010 | 1,013 | theorem Inducing.isCompact_preimage {f : X β Y} (hf : Inducing f) (hf' : IsClosed (range f))
{K : Set Y} (hK : IsCompact K) : IsCompact (f β»ΒΉ' K) := by |
replace hK := hK.inter_right hf'
rwa [hf.isCompact_iff, image_preimage_eq_inter_range]
|
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks
#align_import category_theory.limits.constructions.epi_mono from "leanprover-community/mathlib"@"f7baecbb54bd0f24f228576f97b1752fc3c9b318"
... | Mathlib/CategoryTheory/Limits/Constructions/EpiMono.lean | 58 | 62 | theorem preserves_epi_of_preservesColimit {X Y : C} (f : X βΆ Y) [PreservesColimit (span f f) F]
[Epi f] : Epi (F.map f) := by |
have := isColimitPushoutCoconeMapOfIsColimit F _ (PushoutCocone.isColimitMkIdId f)
simp_rw [F.map_id] at this
apply PushoutCocone.epi_of_isColimitMkIdId _ this
|
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Preadditive.Opposite
import Mathlib.CategoryTheory.Limits.Opposites
#align_import category_theory.abelian.opposite from "leanprover-community/mathlib"@"a5ff45a1c92c278b03b52459a620cfd9c49ebc80"
noncomputable section
namespace CategoryTheor... | Mathlib/CategoryTheory/Abelian/Opposite.lean | 181 | 183 | theorem factorThruImage_comp_imageUnopOp_inv :
factorThruImage g β« (imageUnopOp g).inv = (image.ΞΉ g.unop).op := by |
rw [Iso.comp_inv_eq, image_ΞΉ_op_comp_imageUnopOp_hom]
|
import Mathlib.Algebra.Algebra.Bilinear
import Mathlib.RingTheory.Localization.Basic
#align_import algebra.module.localized_module from "leanprover-community/mathlib"@"831c494092374cfe9f50591ed0ac81a25efc5b86"
namespace LocalizedModule
universe u v
variable {R : Type u} [CommSemiring R] (S : Submonoid R)
variab... | Mathlib/Algebra/Module/LocalizedModule.lean | 339 | 345 | theorem mk'_smul_mk (r : R) (m : M) (s s' : S) :
IsLocalization.mk' T r s β’ mk m s' = mk (r β’ m) (s * s') := by |
rw [smul_def, mk_eq]
obtain β¨c, hcβ© := IsLocalization.eq.mp <| IsLocalization.mk'_sec T (IsLocalization.mk' T r s)
use c
simp_rw [β mul_smul, Submonoid.smul_def, Submonoid.coe_mul, β mul_smul, β mul_assoc,
mul_comm _ (s':R), mul_assoc, hc]
|
import Mathlib.Data.Set.Function
import Mathlib.Analysis.BoundedVariation
#align_import analysis.constant_speed from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open scoped NNReal ENNReal
open Set MeasureTheory Classical
variable {Ξ± : Type*} [LinearOrder Ξ±] {E : Type*} [PseudoEMetr... | Mathlib/Analysis/ConstantSpeed.lean | 176 | 190 | theorem HasConstantSpeedOnWith.ratio {l' : ββ₯0} (hl' : l' β 0) {Ο : β β β} (Οm : MonotoneOn Ο s)
(hfΟ : HasConstantSpeedOnWith (f β Ο) s l) (hf : HasConstantSpeedOnWith f (Ο '' s) l') β¦x : ββ¦
(xs : x β s) : EqOn Ο (fun y => l / l' * (y - x) + Ο x) s := by |
rintro y ys
rw [β sub_eq_iff_eq_add, mul_comm, β mul_div_assoc, eq_div_iff (NNReal.coe_ne_zero.mpr hl')]
rw [hasConstantSpeedOnWith_iff_variationOnFromTo_eq] at hf
rw [hasConstantSpeedOnWith_iff_variationOnFromTo_eq] at hfΟ
symm
calc
(y - x) * l = l * (y - x) := by rw [mul_comm]
_ = variationOnFrom... |
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator
import Mathlib.MeasureTheory.Function.UniformIntegrable
import Mathlib.MeasureTheory.Decomposition.RadonNikodym
#align_import measure_theory.function.conditional_expectation.real from "leanprover-community/mathlib"@"b2ff9a3d7a15fd5b0f060b135421d6a... | Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean | 230 | 256 | theorem condexp_stronglyMeasurable_simpleFunc_mul (hm : m β€ m0) (f : @SimpleFunc Ξ± m β) {g : Ξ± β β}
(hg : Integrable g ΞΌ) : ΞΌ[(f * g : Ξ± β β)|m] =α΅[ΞΌ] f * ΞΌ[g|m] := by |
have : β (s c) (f : Ξ± β β), Set.indicator s (Function.const Ξ± c) * f = s.indicator (c β’ f) := by
intro s c f
ext1 x
by_cases hx : x β s
Β· simp only [hx, Pi.mul_apply, Set.indicator_of_mem, Pi.smul_apply, Algebra.id.smul_eq_mul,
Function.const_apply]
Β· simp only [hx, Pi.mul_apply, Set.indi... |
import Mathlib.Data.Int.Interval
import Mathlib.Data.Int.SuccPred
import Mathlib.Data.Int.ConditionallyCompleteOrder
import Mathlib.Topology.Instances.Discrete
import Mathlib.Topology.MetricSpace.Bounded
import Mathlib.Order.Filter.Archimedean
#align_import topology.instances.int from "leanprover-community/mathlib"@"... | Mathlib/Topology/Instances/Int.lean | 84 | 85 | theorem cofinite_eq : (cofinite : Filter β€) = atBot β atTop := by |
rw [β cocompact_eq_cofinite, cocompact_eq_atBot_atTop]
|
import Mathlib.AlgebraicTopology.DoldKan.PInfty
#align_import algebraic_topology.dold_kan.decomposition from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190f0b504"
open CategoryTheory CategoryTheory.Category CategoryTheory.Preadditive
Opposite Simplicial
noncomputable section
namespace Alge... | Mathlib/AlgebraicTopology/DoldKan/Decomposition.lean | 120 | 124 | theorem id_Ο : (id X n).Ο = π _ := by |
simp only [β P_add_Q_f (n + 1) (n + 1), Ο]
congr 1
Β· simp only [id, PInfty_f, P_f_idem]
Β· exact Eq.trans (by congr; simp) (decomposition_Q n (n + 1)).symm
|
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.Perm
import Mathlib.GroupTheory.Perm.Finite
import Mathlib.GroupTheory.Perm.List
#align_import group_theory.perm.cycle.basic from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
open Equiv Function Finset
variable {... | Mathlib/GroupTheory/Perm/Cycle/Basic.lean | 1,047 | 1,061 | theorem product_self_eq_disjiUnion_perm_aux (hf : f.IsCycleOn s) :
(range s.card : Set β).PairwiseDisjoint fun k =>
s.map β¨fun i => (i, (f ^ k) i), fun i j => congr_arg Prod.fstβ© := by |
obtain hs | _ := (s : Set Ξ±).subsingleton_or_nontrivial
Β· refine Set.Subsingleton.pairwise ?_ _
simp_rw [Set.Subsingleton, mem_coe, β card_le_one] at hs β’
rwa [card_range]
classical
rintro m hm n hn hmn
simp only [disjoint_left, Function.onFun, mem_map, Function.Embedding.coeFn_mk, exists_prop,
... |
import Mathlib.CategoryTheory.Sites.Coherent.ReflectsPreregular
import Mathlib.Topology.Category.CompHaus.EffectiveEpi
import Mathlib.Topology.Category.Stonean.Limits
import Mathlib.Topology.Category.CompHaus.EffectiveEpi
universe u
open CategoryTheory Limits
namespace Stonean
noncomputable
def struct {B X : St... | Mathlib/Topology/Category/Stonean/EffectiveEpi.lean | 103 | 121 | theorem effectiveEpiFamily_tfae
{Ξ± : Type} [Finite Ξ±] {B : Stonean.{u}}
(X : Ξ± β Stonean.{u}) (Ο : (a : Ξ±) β (X a βΆ B)) :
TFAE
[ EffectiveEpiFamily X Ο
, Epi (Sigma.desc Ο)
, β b : B, β (a : Ξ±) (x : X a), Ο a x = b
] := by |
tfae_have 2 β 1
Β· intro
simpa [β effectiveEpi_desc_iff_effectiveEpiFamily, (effectiveEpi_tfae (Sigma.desc Ο)).out 0 1]
tfae_have 1 β 2
Β· intro; infer_instance
tfae_have 3 β 1
Β· erw [((CompHaus.effectiveEpiFamily_tfae
(fun a β¦ Stonean.toCompHaus.obj (X a)) (fun a β¦ Stonean.toCompHaus.map (Ο a))).o... |
import Batteries.Control.ForInStep.Lemmas
import Batteries.Data.List.Basic
import Batteries.Tactic.Init
import Batteries.Tactic.Alias
namespace List
open Nat
@[simp] theorem mem_toArray {a : Ξ±} {l : List Ξ±} : a β l.toArray β a β l := by
simp [Array.mem_def]
@[simp]
theorem drop_one : β l : List Ξ±, drop 1 l =... | .lake/packages/batteries/Batteries/Data/List/Lemmas.lean | 1,344 | 1,345 | theorem range'_concat (s n : Nat) : range' s (n + 1) step = range' s n step ++ [s + step * n] := by |
rw [Nat.add_comm n 1]; exact (range'_append s n 1 step).symm
|
import Mathlib.Data.Prod.PProd
import Mathlib.Data.Set.Countable
import Mathlib.Order.Filter.Prod
import Mathlib.Order.Filter.Ker
#align_import order.filter.bases from "leanprover-community/mathlib"@"996b0ff959da753a555053a480f36e5f264d4207"
set_option autoImplicit true
open Set Filter
open scoped Classical
ope... | Mathlib/Order/Filter/Bases.lean | 770 | 771 | theorem HasBasis.eq_iInf (h : l.HasBasis (fun _ => True) s) : l = β¨
i, π (s i) := by |
simpa only [iInf_true] using h.eq_biInf
|
import Mathlib.Probability.Variance
#align_import probability.moments from "leanprover-community/mathlib"@"85453a2a14be8da64caf15ca50930cf4c6e5d8de"
open MeasureTheory Filter Finset Real
noncomputable section
open scoped MeasureTheory ProbabilityTheory ENNReal NNReal
namespace ProbabilityTheory
variable {Ξ© ΞΉ ... | Mathlib/Probability/Moments.lean | 73 | 77 | theorem centralMoment_one' [IsFiniteMeasure ΞΌ] (h_int : Integrable X ΞΌ) :
centralMoment X 1 ΞΌ = (1 - (ΞΌ Set.univ).toReal) * ΞΌ[X] := by |
simp only [centralMoment, Pi.sub_apply, pow_one]
rw [integral_sub h_int (integrable_const _)]
simp only [sub_mul, integral_const, smul_eq_mul, one_mul]
|
import Mathlib.Algebra.Homology.ImageToKernel
#align_import algebra.homology.exact from "leanprover-community/mathlib"@"3feb151caefe53df080ca6ca67a0c6685cfd1b82"
universe v vβ u uβ
open CategoryTheory CategoryTheory.Limits
variable {V : Type u} [Category.{v} V]
variable [HasImages V]
namespace CategoryTheory
... | Mathlib/Algebra/Homology/Exact.lean | 189 | 193 | theorem exact_epi_comp (hgh : Exact g h) [Epi f] : Exact (f β« g) h := by |
refine β¨by simp [hgh.w], ?_β©
rw [imageToKernel_comp_left]
Β· haveI := hgh.epi
infer_instance
|
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
import Mathlib.Tactic.NormNum.Prime
#align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b37837... | Mathlib/NumberTheory/Bertrand.lean | 208 | 212 | theorem exists_prime_lt_and_le_two_mul_succ {n} (q) {p : β} (prime_p : Nat.Prime p)
(covering : p β€ 2 * q) (H : n < q β β p : β, p.Prime β§ n < p β§ p β€ 2 * n) (hn : n < p) :
β p : β, p.Prime β§ n < p β§ p β€ 2 * n := by |
by_cases h : p β€ 2 * n; Β· exact β¨p, prime_p, hn, hβ©
exact H (lt_of_mul_lt_mul_left' (lt_of_lt_of_le (not_le.1 h) covering))
|
import Mathlib.CategoryTheory.Sites.Sieves
#align_import category_theory.sites.sheaf_of_types from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe w vβ vβ uβ uβ
namespace CategoryTheory
open Opposite CategoryTheory Category Limits Sieve
namespace Presieve
variable {C : Type ... | Mathlib/CategoryTheory/Sites/IsSheafFor.lean | 246 | 252 | theorem extend_restrict {x : FamilyOfElements P (generate R)} (t : x.Compatible) :
(x.restrict (le_generate R)).sieveExtend = x := by |
rw [compatible_iff_sieveCompatible] at t
funext _ _ h
apply (t _ _ _).symm.trans
congr
exact h.choose_spec.choose_spec.choose_spec.2
|
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
import Mathlib.Analysis.Complex.CauchyIntegral
#align_import analysis.special_functions.gamma.beta from "l... | Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean | 450 | 468 | theorem Gamma_ne_zero {s : β} (hs : β m : β, s β -m) : Gamma s β 0 := by |
by_cases h_im : s.im = 0
Β· have : s = βs.re := by
conv_lhs => rw [β Complex.re_add_im s]
rw [h_im, ofReal_zero, zero_mul, add_zero]
rw [this, Gamma_ofReal, ofReal_ne_zero]
refine Real.Gamma_ne_zero fun n => ?_
specialize hs n
contrapose! hs
rwa [this, β ofReal_natCast, β ofReal_neg,... |
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring.List
import Mathlib.Data.Int.ModEq
import Mathlib.Data.Nat.Bits
import Mathlib.Data.Nat.Log
import Mathlib.Data.List.Indexes
import Mathlib.Data.List.Palindrome
import Mathlib.Tactic.IntervalCases
import Mathlib.Tactic.Linarith
impo... | Mathlib/Data/Nat/Digits.lean | 63 | 67 | theorem digitsAux_def (b : β) (h : 2 β€ b) (n : β) (w : 0 < n) :
digitsAux b h n = (n % b) :: digitsAux b h (n / b) := by |
cases n
Β· cases w
Β· rw [digitsAux]
|
import Mathlib.Algebra.Module.MinimalAxioms
import Mathlib.Topology.ContinuousFunction.Algebra
import Mathlib.Analysis.Normed.Order.Lattice
import Mathlib.Analysis.NormedSpace.OperatorNorm.Basic
import Mathlib.Analysis.NormedSpace.Star.Basic
import Mathlib.Analysis.NormedSpace.ContinuousLinearMap
import Mathlib.Topolo... | Mathlib/Topology/ContinuousFunction/Bounded.lean | 1,467 | 1,471 | theorem NNReal.upper_bound {Ξ± : Type*} [TopologicalSpace Ξ±] (f : Ξ± βα΅ ββ₯0) (x : Ξ±) :
f x β€ nndist f 0 := by |
have key : nndist (f x) ((0 : Ξ± βα΅ ββ₯0) x) β€ nndist f 0 := @dist_coe_le_dist Ξ± ββ₯0 _ _ f 0 x
simp only [coe_zero, Pi.zero_apply] at key
rwa [NNReal.nndist_zero_eq_val' (f x)] at key
|
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.Bounded
import Mathlib.SetTheory.Cardinal.PartENat
import Mathlib.SetTheory.Ordinal.Principal
import Mathlib.Tactic.Linarith
#align_import set_theory.cardinal.ordinal from "leanprover-community/mathlib"@"7c2ce0c2da15516b4e65d0c9e254bb6dc93abd1f"
noncomputa... | Mathlib/SetTheory/Cardinal/Ordinal.lean | 269 | 270 | theorem aleph_succ {o : Ordinal} : aleph (succ o) = succ (aleph o) := by |
rw [aleph, add_succ, aleph'_succ, aleph]
|
import Mathlib.Data.Finset.Fold
import Mathlib.Algebra.GCDMonoid.Multiset
#align_import algebra.gcd_monoid.finset from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
#align_import algebra.gcd_monoid.div from "leanprover-community/mathlib"@"b537794f8409bc9598febb79cd510b1df5f4539d"
variab... | Mathlib/Algebra/GCDMonoid/Finset.lean | 100 | 103 | theorem lcm_congr {f g : Ξ² β Ξ±} (hs : sβ = sβ) (hfg : β a β sβ, f a = g a) :
sβ.lcm f = sβ.lcm g := by |
subst hs
exact Finset.fold_congr hfg
|
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.List.Perm
import Mathlib.Data.List.Range
#align_import data.list.sublists from "leanprover-community/mathlib"@"ccad6d5093bd2f5c6ca621fc74674cce51355af6"
universe u v w
variable {Ξ± : Type u} {Ξ² : Type v} {Ξ³ : Type w}
open Nat
namespace List
@[simp]
theo... | Mathlib/Data/List/Sublists.lean | 422 | 428 | theorem sublists_perm_sublists' (l : List Ξ±) : sublists l ~ sublists' l := by |
rw [β finRange_map_get l, sublists_map, sublists'_map]
apply Perm.map
apply (perm_ext_iff_of_nodup _ _).mpr
Β· simp
Β· exact nodup_sublists.mpr (nodup_finRange _)
Β· exact (nodup_sublists'.mpr (nodup_finRange _))
|
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