Context stringlengths 57 85k | file_name stringlengths 21 79 | start int64 14 2.42k | end int64 18 2.43k | theorem stringlengths 25 2.71k | proof stringlengths 5 10.6k |
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import Batteries.Data.RBMap.Alter
import Batteries.Data.List.Lemmas
namespace Batteries
namespace RBNode
open RBColor
attribute [simp] fold foldl foldr Any forM foldlM Ordered
@[simp] theorem min?_reverse (t : RBNode α) : t.reverse.min? = t.max? := by
unfold RBNode.max?; split <;> simp [RBNode.min?]
unfold RB... | .lake/packages/batteries/Batteries/Data/RBMap/Lemmas.lean | 32 | 33 | theorem All_def {t : RBNode α} : t.All p ↔ ∀ x ∈ t, p x := by |
induction t <;> simp [or_imp, forall_and, *]
|
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 | 606 | 618 | theorem sub_one_mul_sum_log_div_pow_eq_sub_sum_digits {p : ℕ} (n : ℕ) :
(p - 1) * ∑ i ∈ range (log p n).succ, n / p ^ i.succ = n - (p.digits n).sum := by |
obtain h | rfl | h : 1 < p ∨ 1 = p ∨ p < 1 := trichotomous 1 p
· rcases eq_or_ne n 0 with rfl | hn
· simp
· convert sub_one_mul_sum_div_pow_eq_sub_sum_digits (p.digits n) (getLast_digit_ne_zero p hn) <|
(fun l a ↦ digits_lt_base h a)
· refine (digits_len p n h hn).symm
all_goals exact... |
import Mathlib.Algebra.CharZero.Defs
import Mathlib.Algebra.Group.Hom.Defs
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Algebra.Order.Monoid.OrderDual
import Mathlib.Algebra.Order.ZeroLEOne
import Mathlib.Data.Nat.Cast.Defs
import Mathlib.Order.WithBot
#align_import algebra.order.monoid.with_top ... | Mathlib/Algebra/Order/Monoid/WithTop.lean | 128 | 128 | theorem add_top (a : WithTop α) : a + ⊤ = ⊤ := by | cases a <;> rfl
|
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 {ι α : T... | Mathlib/Order/Interval/Finset/Basic.lean | 381 | 384 | theorem filter_le_lt_eq_Ico [DecidablePred fun j => a ≤ j ∧ j < b] :
(univ.filter fun j => a ≤ j ∧ j < b) = Ico a b := by |
ext
simp
|
import Mathlib.RingTheory.RootsOfUnity.Basic
import Mathlib.RingTheory.AdjoinRoot
import Mathlib.FieldTheory.Galois
import Mathlib.LinearAlgebra.Eigenspace.Minpoly
import Mathlib.RingTheory.Norm
universe u
variable {K : Type u} [Field K]
open Polynomial IntermediateField AdjoinRoot
section Splits
lemma root_X_pow... | Mathlib/FieldTheory/KummerExtension.lean | 74 | 82 | theorem X_pow_sub_C_splits_of_isPrimitiveRoot
{n : ℕ} {ζ : K} (hζ : IsPrimitiveRoot ζ n) {α a : K} (e : α ^ n = a) :
(X ^ n - C a).Splits (RingHom.id _) := by |
cases n.eq_zero_or_pos with
| inl hn =>
rw [hn, pow_zero, ← C.map_one, ← map_sub]
exact splits_C _ _
| inr hn =>
rw [splits_iff_card_roots, ← nthRoots, hζ.card_nthRoots, natDegree_X_pow_sub_C, if_pos ⟨α, e⟩]
|
import Mathlib.ModelTheory.Ultraproducts
import Mathlib.ModelTheory.Bundled
import Mathlib.ModelTheory.Skolem
#align_import model_theory.satisfiability from "leanprover-community/mathlib"@"d565b3df44619c1498326936be16f1a935df0728"
set_option linter.uppercaseLean3 false
universe u v w w'
open Cardinal CategoryTh... | Mathlib/ModelTheory/Satisfiability.lean | 129 | 135 | theorem isSatisfiable_directed_union_iff {ι : Type*} [Nonempty ι] {T : ι → L.Theory}
(h : Directed (· ⊆ ·) T) : Theory.IsSatisfiable (⋃ i, T i) ↔ ∀ i, (T i).IsSatisfiable := by |
refine ⟨fun h' i => h'.mono (Set.subset_iUnion _ _), fun h' => ?_⟩
rw [isSatisfiable_iff_isFinitelySatisfiable, IsFinitelySatisfiable]
intro T0 hT0
obtain ⟨i, hi⟩ := h.exists_mem_subset_of_finset_subset_biUnion hT0
exact (h' i).mono hi
|
import Mathlib.Combinatorics.SetFamily.HarrisKleitman
import Mathlib.Combinatorics.SetFamily.Intersecting
#align_import combinatorics.set_family.kleitman from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
open Finset
open Fintype (card)
variable {ι α : Type*} [Fintype α] [DecidableEq... | Mathlib/Combinatorics/SetFamily/Kleitman.lean | 37 | 85 | theorem Finset.card_biUnion_le_of_intersecting (s : Finset ι) (f : ι → Finset (Finset α))
(hf : ∀ i ∈ s, (f i : Set (Finset α)).Intersecting) :
(s.biUnion f).card ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - s.card) := by |
have : DecidableEq ι := by
classical
infer_instance
obtain hs | hs := le_total (Fintype.card α) s.card
· rw [tsub_eq_zero_of_le hs, pow_zero]
refine (card_le_card <| biUnion_subset.2 fun i hi a ha ↦
mem_compl.2 <| not_mem_singleton.2 <| (hf _ hi).ne_bot ha).trans_eq ?_
rw [card_compl, Finty... |
import Mathlib.Logic.Equiv.Option
import Mathlib.Order.RelIso.Basic
import Mathlib.Order.Disjoint
import Mathlib.Order.WithBot
import Mathlib.Tactic.Monotonicity.Attr
import Mathlib.Util.AssertExists
#align_import order.hom.basic from "leanprover-community/mathlib"@"62a5626868683c104774de8d85b9855234ac807c"
open ... | Mathlib/Order/Hom/Basic.lean | 911 | 913 | theorem symm_symm (e : α ≃o β) : e.symm.symm = e := by |
ext
rfl
|
import Mathlib.Order.Lattice
#align_import order.min_max from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
universe u v
variable {α : Type u} {β : Type v}
attribute [simp] max_eq_left max_eq_right min_eq_left min_eq_right
section
variable [LinearOrder α] [LinearOrder β] {f : α → β... | Mathlib/Order/MinMax.lean | 279 | 279 | theorem min_choice (a b : α) : min a b = a ∨ min a b = b := by | cases le_total a b <;> simp [*]
|
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Algebra.Ring.Nat
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.RingTheory.Fintype
import Mathlib.Tactic.IntervalCases
#align_import number_the... | Mathlib/NumberTheory/LucasLehmer.lean | 390 | 390 | theorem coe_natCast (n : ℕ) : ((n : ℤ) : X q) = (n : X q) := by | ext <;> simp
|
import Mathlib.Data.Set.Lattice
#align_import data.set.intervals.disjoint from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
universe u v w
variable {ι : Sort u} {α : Type v} {β : Type w}
open Set
open OrderDual (toDual)
namespace Set
section LinearOrder
variable [LinearOrder α] ... | Mathlib/Order/Interval/Set/Disjoint.lean | 176 | 178 | theorem iUnion_Ioc_eq_Ioi_self_iff {f : ι → α} {a : α} :
⋃ i, Ioc a (f i) = Ioi a ↔ ∀ x, a < x → ∃ i, x ≤ f i := by |
simp [← Ioi_inter_Iic, ← inter_iUnion, subset_def]
|
import Mathlib.Data.Set.Function
import Mathlib.Logic.Equiv.Defs
import Mathlib.Tactic.Says
#align_import logic.equiv.set from "leanprover-community/mathlib"@"aba57d4d3dae35460225919dcd82fe91355162f9"
open Function Set
universe u v w z
variable {α : Sort u} {β : Sort v} {γ : Sort w}
namespace Equiv
@[simp]
th... | Mathlib/Logic/Equiv/Set.lean | 338 | 342 | theorem sumCompl_symm_apply_of_mem {α : Type u} {s : Set α} [DecidablePred (· ∈ s)] {x : α}
(hx : x ∈ s) : (Equiv.Set.sumCompl s).symm x = Sum.inl ⟨x, hx⟩ := by |
have : ((⟨x, Or.inl hx⟩ : (s ∪ sᶜ : Set α)) : α) ∈ s := hx
rw [Equiv.Set.sumCompl]
simpa using Set.union_apply_left (by simp) this
|
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 {ι α : T... | Mathlib/Order/Interval/Finset/Basic.lean | 176 | 177 | theorem Ico_subset_Ico (ha : a₂ ≤ a₁) (hb : b₁ ≤ b₂) : Ico a₁ b₁ ⊆ Ico a₂ b₂ := by |
simpa [← coe_subset] using Set.Ico_subset_Ico ha hb
|
import Mathlib.MeasureTheory.Integral.IntegralEqImproper
#align_import measure_theory.integral.peak_function from "leanprover-community/mathlib"@"13b0d72fd8533ba459ac66e9a885e35ffabb32b2"
open Set Filter MeasureTheory MeasureTheory.Measure TopologicalSpace Metric
open scoped Topology ENNReal
open Set
variable... | Mathlib/MeasureTheory/Integral/PeakFunction.lean | 192 | 225 | theorem tendsto_setIntegral_peak_smul_of_integrableOn_of_tendsto
(hs : MeasurableSet s) {t : Set α} (ht : MeasurableSet t) (hts : t ⊆ s) (h'ts : t ∈ 𝓝[s] x₀)
(h't : μ t ≠ ∞) (hnφ : ∀ᶠ i in l, ∀ x ∈ s, 0 ≤ φ i x)
(hlφ : ∀ u : Set α, IsOpen u → x₀ ∈ u → TendstoUniformlyOn φ 0 l (s \ u))
(hiφ : Tendsto (f... |
let h := g - t.indicator (fun _ ↦ a)
have A : Tendsto (fun i : ι => (∫ x in s, φ i x • h x ∂μ) + (∫ x in t, φ i x ∂μ) • a) l
(𝓝 (0 + (1 : ℝ) • a)) := by
refine Tendsto.add ?_ (Tendsto.smul hiφ tendsto_const_nhds)
apply tendsto_setIntegral_peak_smul_of_integrableOn_of_tendsto_aux hs ht hts h'ts
... |
import Mathlib.Data.Finset.Basic
variable {ι : Sort _} {π : ι → Sort _} {x : ∀ i, π i} [DecidableEq ι]
namespace Function
def updateFinset (x : ∀ i, π i) (s : Finset ι) (y : ∀ i : ↥s, π i) (i : ι) : π i :=
if hi : i ∈ s then y ⟨i, hi⟩ else x i
open Finset Equiv
theorem updateFinset_def {s : Finset ι} {y} :
... | Mathlib/Data/Finset/Update.lean | 43 | 50 | theorem update_eq_updateFinset {i y} :
Function.update x i y = updateFinset x {i} (uniqueElim y) := by |
congr with j
by_cases hj : j = i
· cases hj
simp only [dif_pos, Finset.mem_singleton, update_same, updateFinset]
exact uniqueElim_default (α := fun j : ({i} : Finset ι) => π j) y
· simp [hj, updateFinset]
|
import Mathlib.RingTheory.Localization.Module
import Mathlib.RingTheory.Norm
import Mathlib.RingTheory.Discriminant
#align_import ring_theory.localization.norm from "leanprover-community/mathlib"@"2e59a6de168f95d16b16d217b808a36290398c0a"
open scoped nonZeroDivisors
variable (R : Type*) {S : Type*} [CommRing R] ... | Mathlib/RingTheory/Localization/NormTrace.lean | 101 | 109 | theorem Algebra.traceMatrix_localizationLocalization (b : Basis ι R S) :
Algebra.traceMatrix Rₘ (b.localizationLocalization Rₘ M Sₘ) =
(algebraMap R Rₘ).mapMatrix (Algebra.traceMatrix R b) := by |
have : Module.Finite R S := Module.Finite.of_basis b
have : Module.Free R S := Module.Free.of_basis b
ext i j : 2
simp_rw [RingHom.mapMatrix_apply, Matrix.map_apply, traceMatrix_apply, traceForm_apply,
Basis.localizationLocalization_apply, ← map_mul]
exact Algebra.trace_localization R M _
|
import Mathlib.Algebra.Lie.OfAssociative
import Mathlib.Algebra.Lie.IdealOperations
#align_import algebra.lie.abelian from "leanprover-community/mathlib"@"8983bec7cdf6cb2dd1f21315c8a34ab00d7b2f6d"
universe u v w w₁ w₂
class LieModule.IsTrivial (L : Type v) (M : Type w) [Bracket L M] [Zero M] : Prop where
triv... | Mathlib/Algebra/Lie/Abelian.lean | 136 | 141 | theorem ideal_oper_maxTrivSubmodule_eq_bot (I : LieIdeal R L) :
⁅I, maxTrivSubmodule R L M⁆ = ⊥ := by |
rw [← LieSubmodule.coe_toSubmodule_eq_iff, LieSubmodule.lieIdeal_oper_eq_linear_span,
LieSubmodule.bot_coeSubmodule, Submodule.span_eq_bot]
rintro m ⟨⟨x, hx⟩, ⟨⟨m, hm⟩, rfl⟩⟩
exact hm x
|
import Mathlib.Analysis.SpecialFunctions.Complex.Arg
import Mathlib.Analysis.SpecialFunctions.Log.Basic
#align_import analysis.special_functions.complex.log from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
namespace Complex
open Set Filter Bornology
open scop... | Mathlib/Analysis/SpecialFunctions/Complex/Log.lean | 106 | 106 | theorem log_zero : log 0 = 0 := by | simp [log]
|
import Mathlib.Topology.Algebra.Module.Basic
import Mathlib.Analysis.Normed.MulAction
#align_import analysis.normed_space.continuous_linear_map from "leanprover-community/mathlib"@"fe18deda804e30c594e75a6e5fe0f7d14695289f"
open Metric ContinuousLinearMap
open Set Real
open NNReal
variable {𝕜 𝕜₂ E F G : Type*}... | Mathlib/Analysis/NormedSpace/ContinuousLinearMap.lean | 198 | 205 | theorem ContinuousLinearEquiv.homothety_inverse (a : ℝ) (ha : 0 < a) (f : E ≃ₛₗ[σ] F) :
(∀ x : E, ‖f x‖ = a * ‖x‖) → ∀ y : F, ‖f.symm y‖ = a⁻¹ * ‖y‖ := by |
intro hf y
calc
‖f.symm y‖ = a⁻¹ * (a * ‖f.symm y‖) := by
rw [← mul_assoc, inv_mul_cancel (ne_of_lt ha).symm, one_mul]
_ = a⁻¹ * ‖f (f.symm y)‖ := by rw [hf]
_ = a⁻¹ * ‖y‖ := by simp
|
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 | 723 | 724 | theorem lintegral_mul_const (r : ℝ≥0∞) {f : α → ℝ≥0∞} (hf : Measurable f) :
∫⁻ a, f a * r ∂μ = (∫⁻ a, f a ∂μ) * r := by | simp_rw [mul_comm, lintegral_const_mul r hf]
|
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.HasseDeriv
#align_import data.polynomial.taylor from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
noncomputable section
namespace Polynomial
open Polynomial... | Mathlib/Algebra/Polynomial/Taylor.lean | 146 | 149 | theorem sum_taylor_eq {R} [CommRing R] (f : R[X]) (r : R) :
((taylor r f).sum fun i a => C a * (X - C r) ^ i) = f := by |
rw [← comp_eq_sum_left, sub_eq_add_neg, ← C_neg, ← taylor_apply, taylor_taylor, neg_add_self,
taylor_zero]
|
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 {α : TypeVec n} (a : q.P.A) (f' : q.P.drop.B a ⟹ α)
(f : q.P.last.B a → q.P.W α) :
wrepr (q.P.wMk a f' f) =
q.P.wMk' (repr (abs (appendFun id wrepr <$$> ⟨a, q.P.appendContents f' f⟩))) := by |
rw [wrepr, recF_eq', q.P.wDest'_wMk]; rfl
|
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 {α : Type u} {β : α → Type v} {l l₁ l₂ : List (Sigma β)}
def keys : List (Sigma β) → List α :=
map ... | Mathlib/Data/List/Sigma.lean | 212 | 216 | theorem mem_dlookup {a} {b : β a} {l : List (Sigma β)} (nd : l.NodupKeys) (h : Sigma.mk a b ∈ l) :
b ∈ 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.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.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 : ¬IsCountablyGenerated (cocompact ℚ) := by |
intro H
rcases exists_seq_tendsto (cocompact ℚ ⊓ 𝓝 0) with ⟨x, hx⟩
rw [tendsto_inf] at hx; rcases hx with ⟨hxc, hx0⟩
obtain ⟨n, hn⟩ : ∃ n : ℕ, x n ∉ insert (0 : ℚ) (range x) :=
(hxc.eventually hx0.isCompact_insert_range.compl_mem_cocompact).exists
exact hn (Or.inr ⟨n, rfl⟩)
|
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 {𝕜 E : Type*}
variable [NormedField 𝕜]
sectio... | Mathlib/Analysis/NormedSpace/Pointwise.lean | 217 | 223 | theorem disjoint_ball_ball_iff (hδ : 0 < δ) (hε : 0 < ε) :
Disjoint (ball x δ) (ball y ε) ↔ δ + ε ≤ dist x y := by |
refine ⟨fun h => le_of_not_lt fun hxy => ?_, ball_disjoint_ball⟩
rw [add_comm] at hxy
obtain ⟨z, hxz, hzy⟩ := exists_dist_lt_lt hδ hε hxy
rw [dist_comm] at hxz
exact h.le_bot ⟨hxz, hzy⟩
|
import Mathlib.Topology.Constructions
#align_import topology.continuous_on from "leanprover-community/mathlib"@"d4f691b9e5f94cfc64639973f3544c95f8d5d494"
open Set Filter Function Topology Filter
variable {α : Type*} {β : Type*} {γ : Type*} {δ : Type*}
variable [TopologicalSpace α]
@[simp]
theorem nhds_bind_nhdsW... | Mathlib/Topology/ContinuousOn.lean | 585 | 587 | theorem continuousOn_prod_of_discrete_right [DiscreteTopology β] {f : α × β → γ} {s : Set (α × β)} :
ContinuousOn f s ↔ ∀ b, ContinuousOn (f ⟨·, b⟩) {a | (a, b) ∈ s} := by |
simp_rw [ContinuousOn, Prod.forall, continuousWithinAt_prod_of_discrete_right]; apply forall_swap
|
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 ⋙ F)) :
(J.plusCompIso F P).inv = J.plusLift (whiskerRight (J.toPlus _) _) hP := by |
apply J.plusLift_unique
simp [Iso.comp_inv_eq]
|
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 ℝ)
deriving Bot, Zero, One,... | Mathlib/Data/Real/EReal.lean | 1,269 | 1,269 | theorem coe_coe_sign (x : SignType) : ((x : ℝ) : EReal) = x := by | cases x <;> rfl
|
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]
(μ' μ : Measure G) [IsHaarMeasure μ] [IsFiniteMeasureOnCompacts μ'] [IsMulLeftInvariant μ']
{s : Set G} (h's : IsCompact (closure s)) :
μ' s = haarScalarFactor μ' μ • μ s := by |
let ν := haarScalarFactor μ' μ • μ
apply le_antisymm
· calc
μ' s ≤ μ' ((toMeasurable ν s) ∩ (closure s)) :=
measure_mono <| subset_inter (subset_toMeasurable ν s) subset_closure
_ = ν ((toMeasurable ν s) ∩ (closure s)) := by
apply measure_isMulInvariant_eq_smul_of_isCompact_closure_of_measura... |
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 ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1CLM F' hm μ (hf.toL1 f) := by |
refine (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => ?_)
rw [condexpL1_eq hf]
|
import Mathlib.Init.Function
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
universe u
open Function
namespace Option
variable {α β γ δ : Type*} {f : α → β → γ} {a : Option α} {b : Option β} {c : Option γ}
def map₂ (f : α → β → γ) (a : Option α) ... | Mathlib/Data/Option/NAry.lean | 146 | 149 | theorem map_map₂_distrib {g : γ → δ} {f' : α' → β' → δ} {g₁ : α → α'} {g₂ : β → β'}
(h_distrib : ∀ a b, g (f a b) = f' (g₁ a) (g₂ b)) :
(map₂ f a b).map g = map₂ f' (a.map g₁) (b.map g₂) := by |
cases a <;> cases b <;> simp [h_distrib]
|
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 : ℕ) where
A : Type u
... | Mathlib/Data/PFunctor/Multivariate/Basic.lean | 116 | 119 | theorem const.mk_get (x : const n A α) : const.mk n (const.get x) = x := by |
cases x
dsimp [const.get, const.mk]
congr with (_⟨⟩)
|
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 (⊤ : Submodule R M).annihilator).toFinset =>
torsionBySet R M (p ^ (factors (⊤ : Submodule R M).annihilator).count ↑p : 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 [← Set.nonempty_iff_ne_empty] at hI; rw [Submodule.ne_bot_iff]
obtain ⟨x, H, hx⟩ := hI; exact ⟨x, H, nonZeroDivisors.ne_ze... |
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 (θ : Real.Angle) : cos θ ^ 2 + sin θ ^ 2 = 1 := by |
induction θ 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 ∈ s ↔ infEdist x s = 0 := by |
rw [← 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 {α G A S... | Mathlib/Algebra/Group/Subgroup/Pointwise.lean | 366 | 372 | theorem conj_smul_subgroupOf {P H : Subgroup G} (hP : P ≤ H) (h : H) :
MulAut.conj h • P.subgroupOf H = (MulAut.conj (h : G) • P).subgroupOf H := by |
refine le_antisymm ?_ ?_
· rintro - ⟨g, hg, rfl⟩
exact ⟨g, hg, rfl⟩
· rintro p ⟨g, hg, hp⟩
exact ⟨⟨g, hP hg⟩, hg, Subtype.ext hp⟩
|
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 γ] (a : α) : Tendsto f (comap e (𝓝 a)) (𝓝 (ψ a)) := by |
simpa only [DenseInducing.extend] using
tendsto_nhds_limUnder (uniformly_extend_exists h_e ‹_› h_f _)
|
import Mathlib.Topology.Constructions
#align_import topology.continuous_on from "leanprover-community/mathlib"@"d4f691b9e5f94cfc64639973f3544c95f8d5d494"
open Set Filter Function Topology Filter
variable {α : Type*} {β : Type*} {γ : Type*} {δ : Type*}
variable [TopologicalSpace α]
@[simp]
theorem nhds_bind_nhdsW... | Mathlib/Topology/ContinuousOn.lean | 276 | 277 | theorem nhdsWithin_singleton (a : α) : 𝓝[{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 : ι → k) (p : ι → P) {s₁ s₂ : Finset ι}
(h : s₁ ⊆ s₂) :
s₁.affineCombination k p w = s₂.affineCombination k p (Set.indicator (↑s₁) 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 - ∏ i : n, (X - C (M i i))).degree < ↑(Fintype.card n - 1) := by |
rw [charpoly, det_apply', ← 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 [← mem_degreeLT]
apply Submodule.sum_mem... |
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 {σ τ : p.Gal} (h : ∀ x ∈ p.rootSet p.SplittingField, σ x = τ x) : σ = τ := by |
refine
AlgEquiv.ext fun x =>
(AlgHom.mem_equalizer σ.toAlgHom τ.toAlgHom x).mp
((SetLike.ext_iff.mp ?_ x).mpr Algebra.mem_top)
rwa [eq_top_iff, ← 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 : ℝ≥0∞) (f : α → ℝ≥0∞) (hr : r ≠ ∞) :
∫⁻ a, r * f a ∂μ = r * ∫⁻ a, f a ∂μ := by |
by_cases h : r = 0
· simp [h]
apply le_antisymm _ (lintegral_const_mul_le r f)
have rinv : r * r⁻¹ = 1 := ENNReal.mul_inv_cancel h hr
have rinv' : r⁻¹ * r = 1 := by
rw [mul_comm]
exact rinv
have := lintegral_const_mul_le (μ := μ) r⁻¹ fun x => r * f x
simp? [(mul_assoc _ _ _).symm, rinv'] at this ... |
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 (γ : Path x y) : γ.map continuous_id = γ := by |
ext
rfl
|
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 {β} (s : Finset β) (f : n → β → α) (i : n) :
(∑ x ∈ s, cramer A (fun j => f j x) i) = cramer A (fun j : n => ∑ x ∈ s, f j x) i :=
calc
(∑ x ∈ s, cramer A (fun j => f j x) i) = (∑ x ∈ s, cramer A fun j => f j x) i :=
(Finset.sum_apply i s _).symm
_ = cramer A (fun j : n => ∑ ... |
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} (θ : Real.Angle) :
o.oangle x y = θ ↔
(x ≠ 0 ∧ y ≠ 0 ∧ ∃ r : ℝ, 0 < r ∧ y = r • o.rotation θ x) ∨ θ = 0 ∧ (x = 0 ∨ y = 0) := by |
by_cases hx : x = 0
· simp [hx, eq_comm]
· by_cases hy : y = 0
· simp [hy, eq_comm]
· rw [o.oangle_eq_iff_eq_pos_smul_rotation_of_ne_zero hx hy]
simp [hx, hy]
|
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 𝕜 l t y)
(hf : DifferentiableWithinAt 𝕜 f s x) (hs : MapsTo f s t) (hxs : UniqueDiffWithinAt 𝕜 s x)
(hy : y = f x) :
derivWithin (l ∘ f) s x = (fderivWithin 𝕜 l t (f x) : F → 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 ⥤ C} {s : Cone F} {t : Cone G} (P : IsLimit s)
(Q : IsLimit t) (w : F ≅ G) (j : J) :
(conePointsIsoOfNatIso P Q w).hom ≫ t.π.app j = s.π.app j ≫ 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 {α β γ : Sort*} {f : α → β}
@[reducible, simp] de... | Mathlib/Logic/Function/Basic.lean | 691 | 694 | theorem update_idem {α} [DecidableEq α] {β : α → Sort*} {a : α} (v w : β a) (f : ∀ a, β 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 : α → δ} {g : δ → β}
(hg : AEStronglyMeasurable g (Measure.map f μ)) (hf : AEMeasurable f μ) :
Integrable g (Measure.map f μ) ↔ Integrable (g ∘ f) μ := by |
simp_rw [← memℒp_one_iff_integrable]
exact memℒp_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 : α × ℝ | 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 {α : Type u}
open Nat Function
namespace List
theorem rotate... | Mathlib/Data/List/Rotate.lean | 572 | 573 | theorem length_cyclicPermutations_cons (x : α) (l : List α) :
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
{𝕜 : 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 ∘ f) s x (g'.comp f') := by |
rw [← hasMFDerivWithinAt_univ] at *
exact HasMFDerivWithinAt.comp x (hg.mono (subset_univ _)) hf subset_preimage_univ
|
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 : β → β → Prop) [IsTrans β r] {f : ℕ → β} {a : ℕ}
(h : ∀ n, a ≤ n → r (f n) (f (n + 1))) ⦃b c : ℕ⦄ (hab : a ≤ 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.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 :
∃ S : Set α,
MeasurableSet S ∧
j.toSignedMeasure ≤[S] 0 ∧
0 ≤[Sᶜ] j.toSignedMeasure ∧ j.posPart S = 0 ∧ j.negPart Sᶜ = 0 := by |
obtain ⟨S, hS₁, hS₂, hS₃⟩ := j.mutuallySingular
refine ⟨S, hS₁, ?_, ?_, hS₂, hS₃⟩
· refine restrict_le_restrict_of_subset_le _ _ fun A hA hA₁ => ?_
rw [toSignedMeasure, toSignedMeasure_sub_apply hA,
show j.posPart A = 0 from nonpos_iff_eq_zero.1 (hS₂ ▸ measure_mono hA₁), ENNReal.zero_toReal,
zero... |
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 = ⊥ := 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 ⟶ G) {x y : X} (h : x ⤳ y) :
F.stalkSpecializes h ≫ (stalkFunctor C x).map f =
(stalkFunctor C y).map f ≫ G.stalkSpecializes h := by |
change (_ : colimit _ ⟶ _) = (_ : colimit _ ⟶ _)
ext; delta stalkFunctor; simpa [stalkSpecializes] using by rfl
|
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 : ℝ} (hp0 : 0 ≤ p) {f g : α → ℝ≥0∞}
(hf : AEMeasurable f μ) (hf_zero : ∫⁻ a, f a ^ p ∂μ = 0) : (∫⁻ a, (f * g) a ∂μ) = 0 := by |
rw [← @lintegral_zero_fun α _ μ]
refine lintegral_congr_ae ?_
suffices h_mul_zero : f * g =ᵐ[μ] 0 * g by rwa [zero_mul] at h_mul_zero
have hf_eq_zero : f =ᵐ[μ] 0 := ae_eq_zero_of_lintegral_rpow_eq_zero hp0 hf hf_zero
exact hf_eq_zero.mul (ae_eq_refl g)
|
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 α} {b c : α} (h : a = ⊥ → b ≤ c) :
a.unbot' b ≤ c ↔ a ≤ c := by |
induction a
· simpa using h rfl
· 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 ⟶ Z) (g : Y ⟶ Z) (U : Set X) :
(pullback.snd : pullback f g ⟶ _) '' ((pullback.fst : pullback f g ⟶ _) ⁻¹' U) =
g ⁻¹' (f '' U) := by |
ext x
constructor
· rintro ⟨(y : (forget TopCat).obj _), hy, rfl⟩
exact
⟨(pullback.fst : pullback f g ⟶ _) y, hy, ConcreteCategory.congr_hom pullback.condition y⟩
· rintro ⟨y, hy, eq⟩
-- next 5 lines were
-- `exact ⟨(TopCat.pullbackIsoProdSubtype f g).inv ⟨⟨_, _⟩, eq⟩, by simpa, by simp⟩` before ... |
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 ⟶ 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 ≫ (imageUnopOp g).inv = (image.ι g.unop).op := by |
rw [Iso.comp_inv_eq, image_ι_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 • mk m s' = mk (r • m) (s * s') := by |
rw [smul_def, mk_eq]
obtain ⟨c, hc⟩ := IsLocalization.eq.mp <| IsLocalization.mk'_sec T (IsLocalization.mk' T r s)
use c
simp_rw [← mul_smul, Submonoid.smul_def, Submonoid.coe_mul, ← mul_smul, ← 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 {α : Type*} [LinearOrder α] {E : Type*} [PseudoEMetr... | Mathlib/Analysis/ConstantSpeed.lean | 176 | 190 | theorem HasConstantSpeedOnWith.ratio {l' : ℝ≥0} (hl' : l' ≠ 0) {φ : ℝ → ℝ} (φm : MonotoneOn φ s)
(hfφ : HasConstantSpeedOnWith (f ∘ φ) s l) (hf : HasConstantSpeedOnWith f (φ '' s) l') ⦃x : ℝ⦄
(xs : x ∈ s) : EqOn φ (fun y => l / l' * (y - x) + φ x) s := by |
rintro y ys
rw [← sub_eq_iff_eq_add, mul_comm, ← 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 hfφ
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 ≤ m0) (f : @SimpleFunc α m ℝ) {g : α → ℝ}
(hg : Integrable g μ) : μ[(f * g : α → ℝ)|m] =ᵐ[μ] f * μ[g|m] := by |
have : ∀ (s c) (f : α → ℝ), Set.indicator s (Function.const α c) * f = s.indicator (c • f) := by
intro s c f
ext1 x
by_cases hx : x ∈ s
· simp only [hx, Pi.mul_apply, Set.indicator_of_mem, Pi.smul_apply, Algebra.id.smul_eq_mul,
Function.const_apply]
· 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 ℤ) = atBot ⊔ atTop := by |
rw [← 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_φ : (id X n).φ = 𝟙 _ := by |
simp only [← P_add_Q_f (n + 1) (n + 1), φ]
congr 1
· simp only [id, PInfty_f, P_f_idem]
· exact Eq.trans (by congr; simp) (decomposition_Q n (n + 1)).symm
|
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
{α : Type} [Finite α] {B : Stonean.{u}}
(X : α → Stonean.{u}) (π : (a : α) → (X a ⟶ B)) :
TFAE
[ EffectiveEpiFamily X π
, Epi (Sigma.desc π)
, ∀ b : B, ∃ (a : α) (x : X a), π a x = b
] := by |
tfae_have 2 → 1
· intro
simpa [← effectiveEpi_desc_iff_effectiveEpiFamily, (effectiveEpi_tfae (Sigma.desc π)).out 0 1]
tfae_have 1 → 2
· intro; infer_instance
tfae_have 3 ↔ 1
· erw [((CompHaus.effectiveEpiFamily_tfae
(fun a ↦ Stonean.toCompHaus.obj (X a)) (fun a ↦ Stonean.toCompHaus.map (π 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 : α} {l : List α} : a ∈ l.toArray ↔ a ∈ l := by
simp [Array.mem_def]
@[simp]
theorem drop_one : ∀ l : List α, 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.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 {Ω ι ... | Mathlib/Probability/Moments.lean | 73 | 77 | theorem centralMoment_one' [IsFiniteMeasure μ] (h_int : Integrable X μ) :
centralMoment X 1 μ = (1 - (μ Set.univ).toReal) * μ[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 v₂ u u₂
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 ≫ g) h := by |
refine ⟨by simp [hgh.w], ?_⟩
rw [imageToKernel_comp_left]
· haveI := hgh.epi
infer_instance
|
import Mathlib.CategoryTheory.Sites.Sieves
#align_import category_theory.sites.sheaf_of_types from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe w v₁ v₂ u₁ u₂
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.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 : ℕ) (h : 2 ≤ b) (n : ℕ) (w : 0 < n) :
digitsAux b h n = (n % b) :: digitsAux b h (n / b) := by |
cases n
· cases w
· rw [digitsAux]
|
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 : β → α} (hs : s₁ = s₂) (hfg : ∀ a ∈ s₂, f a = g a) :
s₁.lcm f = s₂.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 {α : Type u} {β : Type v} {γ : Type w}
open Nat
namespace List
@[simp]
theo... | Mathlib/Data/List/Sublists.lean | 422 | 428 | theorem sublists_perm_sublists' (l : List α) : sublists l ~ sublists' l := by |
rw [← finRange_map_get l, sublists_map, sublists'_map]
apply Perm.map
apply (perm_ext_iff_of_nodup _ _).mpr
· simp
· exact nodup_sublists.mpr (nodup_finRange _)
· exact (nodup_sublists'.mpr (nodup_finRange _))
|
import Mathlib.Algebra.QuadraticDiscriminant
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
#align_import analysis.special_functions.trigonometric.complex from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
noncomputable secti... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Complex.lean | 166 | 173 | theorem tan_eq {z : ℂ}
(h :
((∀ k : ℤ, (z.re : ℂ) ≠ (2 * k + 1) * π / 2) ∧
∀ l : ℤ, (z.im : ℂ) * I ≠ (2 * l + 1) * π / 2) ∨
(∃ k : ℤ, (z.re : ℂ) = (2 * k + 1) * π / 2) ∧
∃ l : ℤ, (z.im : ℂ) * I = (2 * l + 1) * π / 2) :
tan z = (tan z.re + tanh z.im * I) / (1 - tan z.re * tanh z... |
convert tan_add_mul_I h; exact (re_add_im z).symm
|
import Mathlib.Algebra.ModEq
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Algebra.Periodic
import Mathlib.Data.Int.SuccPred
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.Order.Circular
import Mathlib.Data.List.TFAE
import Mathlib.Data.Set.Lattice
#align_import a... | Mathlib/Algebra/Order/ToIntervalMod.lean | 464 | 465 | theorem toIocMod_sub_zsmul (a b : α) (m : ℤ) : toIocMod hp a (b - m • p) = toIocMod hp a b := by |
rw [sub_eq_add_neg, ← neg_smul, toIocMod_add_zsmul]
|
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 {α β : Type*}
namespace Set
section Preorder
v... | Mathlib/Order/Interval/Set/Basic.lean | 181 | 181 | theorem left_mem_Ioo : a ∈ Ioo a b ↔ False := by | simp [lt_irrefl]
|
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Tactic.NthRewrite
#align_import data.nat.gcd.basic from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
namespace Nat
theorem gcd_greatest {a b d : ℕ} (hda : d ∣ a) (hdb : d ∣ b) (hd ... | Mathlib/Data/Nat/GCD/Basic.lean | 139 | 140 | theorem lcm_mul_right {m n k : ℕ} : (m * n).lcm (k * n) = m.lcm k * n := by |
rw [mul_comm, mul_comm k n, lcm_mul_left, mul_comm]
|
import Mathlib.CategoryTheory.Comma.StructuredArrow
import Mathlib.CategoryTheory.PUnit
import Mathlib.CategoryTheory.Functor.ReflectsIso
import Mathlib.CategoryTheory.Functor.EpiMono
#align_import category_theory.over from "leanprover-community/mathlib"@"8a318021995877a44630c898d0b2bc376fceef3b"
namespace Catego... | Mathlib/CategoryTheory/Comma/Over.lean | 59 | 63 | theorem OverMorphism.ext {X : T} {U V : Over X} {f g : U ⟶ V} (h : f.left = g.left) : f = g := by |
let ⟨_,b,_⟩ := f
let ⟨_,e,_⟩ := g
congr
simp only [eq_iff_true_of_subsingleton]
|
import Mathlib.Data.Fin.VecNotation
import Mathlib.Logic.Embedding.Set
#align_import logic.equiv.fin from "leanprover-community/mathlib"@"bd835ef554f37ef9b804f0903089211f89cb370b"
assert_not_exists MonoidWithZero
universe u
variable {m n : ℕ}
def finZeroEquiv : Fin 0 ≃ Empty :=
Equiv.equivEmpty _
#align fin_... | Mathlib/Logic/Equiv/Fin.lean | 394 | 398 | theorem finRotate_of_lt {k : ℕ} (h : k < n) :
finRotate (n + 1) ⟨k, h.trans_le n.le_succ⟩ = ⟨k + 1, Nat.succ_lt_succ h⟩ := by |
ext
dsimp [finRotate_succ]
simp [finAddFlip_apply_mk_left h, Nat.add_comm]
|
import Mathlib.Logic.Relation
import Mathlib.Data.Option.Basic
import Mathlib.Data.Seq.Seq
#align_import data.seq.wseq from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b348ce40ad"
namespace Stream'
open Function
universe u v w
def WSeq (α) :=
Seq (Option α)
#align stream.wseq Stream'.WSeq
... | Mathlib/Data/Seq/WSeq.lean | 717 | 717 | theorem dropn_nil (n) : drop (nil : WSeq α) n = nil := by | induction n <;> simp [*, drop]
|
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 | 147 | 155 | theorem pullback_topology {X Y Z : TopCat.{u}} (f : X ⟶ Z) (g : Y ⟶ Z) :
(pullback f g).str =
induced (pullback.fst : pullback f g ⟶ _) X.str ⊓
induced (pullback.snd : pullback f g ⟶ _) Y.str := by |
let homeo := homeoOfIso (pullbackIsoProdSubtype f g)
refine homeo.inducing.induced.trans ?_
change induced homeo (induced _ ( (induced Prod.fst X.str) ⊓ (induced Prod.snd Y.str))) = _
simp only [induced_compose, induced_inf]
congr
|
import Mathlib.Logic.Relation
import Mathlib.Data.Option.Basic
import Mathlib.Data.Seq.Seq
#align_import data.seq.wseq from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b348ce40ad"
namespace Stream'
open Function
universe u v w
def WSeq (α) :=
Seq (Option α)
#align stream.wseq Stream'.WSeq
... | Mathlib/Data/Seq/WSeq.lean | 1,485 | 1,497 | theorem destruct_map (f : α → β) (s : WSeq α) :
destruct (map f s) = Computation.map (Option.map (Prod.map f (map f))) (destruct s) := by |
apply
Computation.eq_of_bisim fun c1 c2 =>
∃ s,
c1 = destruct (map f s) ∧
c2 = Computation.map (Option.map (Prod.map f (map f))) (destruct s)
· intro c1 c2 h
cases' h with s h
rw [h.left, h.right]
induction' s using WSeq.recOn with a s s <;> simp
exact ⟨s, rfl, rfl⟩
· ... |
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 | 206 | 209 | theorem HasDerivAt.comp_hasFDerivWithinAt_of_eq {f : E → 𝕜'} {f' : E →L[𝕜] 𝕜'} {s} (x)
(hh : HasDerivAt h₂ h₂' y) (hf : HasFDerivWithinAt f f' s x) (hy : y = f x) :
HasFDerivWithinAt (h₂ ∘ f) (h₂' • f') s x := by |
rw [hy] at hh; exact hh.comp_hasFDerivWithinAt x hf
|
import Mathlib.Algebra.Module.Submodule.EqLocus
import Mathlib.Algebra.Module.Submodule.RestrictScalars
import Mathlib.Algebra.Ring.Idempotents
import Mathlib.Data.Set.Pointwise.SMul
import Mathlib.LinearAlgebra.Basic
import Mathlib.Order.CompactlyGenerated.Basic
import Mathlib.Order.OmegaCompletePartialOrder
#align_... | Mathlib/LinearAlgebra/Span.lean | 266 | 272 | theorem span_nat_eq_addSubmonoid_closure (s : Set M) :
(span ℕ s).toAddSubmonoid = AddSubmonoid.closure s := by |
refine Eq.symm (AddSubmonoid.closure_eq_of_le subset_span ?_)
apply (OrderIso.to_galoisConnection (AddSubmonoid.toNatSubmodule (M := M)).symm).l_le
(a := span ℕ s) (b := AddSubmonoid.closure s)
rw [span_le]
exact AddSubmonoid.subset_closure
|
import Mathlib.NumberTheory.ZetaValues
import Mathlib.NumberTheory.LSeries.RiemannZeta
open Complex Real Set
open scoped Nat
open HurwitzZeta
theorem riemannZeta_two_mul_nat {k : ℕ} (hk : k ≠ 0) :
riemannZeta (2 * k) = (-1) ^ (k + 1) * (2 : ℂ) ^ (2 * k - 1)
* (π : ℂ) ^ (2 * k) * bernoulli (2 * k) / (... | Mathlib/NumberTheory/LSeries/HurwitzZetaValues.lean | 220 | 224 | theorem riemannZeta_two : riemannZeta 2 = (π : ℂ) ^ 2 / 6 := by |
convert congr_arg ((↑) : ℝ → ℂ) hasSum_zeta_two.tsum_eq
· rw [← Nat.cast_two, zeta_nat_eq_tsum_of_gt_one one_lt_two]
simp only [push_cast]
· norm_cast
|
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Data.Nat.PrimeFin
import Mathlib.NumberTheory.Padics.PadicVal
import Mathlib.Order.Interval.Finset.Nat
#align_import data.nat.factorization.basic from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
... | Mathlib/Data/Nat/Factorization/Basic.lean | 558 | 565 | theorem ord_proj_dvd_ord_proj_iff_dvd {a b : ℕ} (ha0 : a ≠ 0) (hb0 : b ≠ 0) :
(∀ p : ℕ, ord_proj[p] a ∣ ord_proj[p] b) ↔ a ∣ b := by |
refine ⟨fun h => ?_, fun hab p => ord_proj_dvd_ord_proj_of_dvd hb0 hab p⟩
rw [← factorization_le_iff_dvd ha0 hb0]
intro q
rcases le_or_lt q 1 with (hq_le | hq1)
· interval_cases q <;> simp
exact (pow_dvd_pow_iff_le_right hq1).1 (h q)
|
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Ideal.Over
import Mathlib.RingTheory.JacobsonIdeal
#align_import ring_theory.jacobson from "leanprover-community/mathlib"@"a7c017d750512a352b623b1824d75da5998457d0"
set_option autoImplicit true
universe u
namespace Ideal
open Polynomial
... | Mathlib/RingTheory/Jacobson.lean | 108 | 117 | theorem isJacobson_of_surjective [H : IsJacobson R] :
(∃ f : R →+* S, Function.Surjective ↑f) → IsJacobson S := by |
rintro ⟨f, hf⟩
rw [isJacobson_iff_sInf_maximal]
intro p hp
use map f '' { J : Ideal R | comap f p ≤ J ∧ J.IsMaximal }
use fun j ⟨J, hJ, hmap⟩ => hmap ▸ (map_eq_top_or_isMaximal_of_surjective f hf hJ.right).symm
have : p = map f (comap f p).jacobson :=
(IsJacobson.out' _ <| hp.isRadical.comap f).symm ▸ ... |
import Mathlib.Analysis.NormedSpace.AddTorsorBases
#align_import analysis.convex.intrinsic from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open AffineSubspace Set
open scoped Pointwise
variable {𝕜 V W Q P : Type*}
section AddTorsor
variable (𝕜) [Ring 𝕜] [AddCommGroup V] [Modu... | Mathlib/Analysis/Convex/Intrinsic.lean | 231 | 238 | theorem intrinsicClosure_idem (s : Set P) :
intrinsicClosure 𝕜 (intrinsicClosure 𝕜 s) = intrinsicClosure 𝕜 s := by |
refine IsClosed.intrinsicClosure ?_
set t := affineSpan 𝕜 (intrinsicClosure 𝕜 s) with ht
clear_value t
obtain rfl := ht.trans (affineSpan_intrinsicClosure _)
rw [intrinsicClosure, preimage_image_eq _ Subtype.coe_injective]
exact isClosed_closure
|
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 | 186 | 188 | theorem imageUnopOp_inv_comp_op_factorThruImage :
(imageUnopOp g).inv ≫ (factorThruImage g.unop).op = image.ι g := by |
rw [Iso.inv_comp_eq, imageUnopOp_hom_comp_image_ι]
|
import Mathlib.GroupTheory.Subgroup.Center
import Mathlib.GroupTheory.Submonoid.Centralizer
#align_import group_theory.subgroup.basic from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
open Function
open Int
variable {G : Type*} [Group G]
namespace Subgroup
variable {H K : Subgroup ... | Mathlib/GroupTheory/Subgroup/Centralizer.lean | 42 | 44 | theorem mem_centralizer_iff_commutator_eq_one {g : G} {s : Set G} :
g ∈ centralizer s ↔ ∀ h ∈ s, h * g * h⁻¹ * g⁻¹ = 1 := by |
simp only [mem_centralizer_iff, mul_inv_eq_iff_eq_mul, one_mul]
|
import Mathlib.AlgebraicTopology.DoldKan.FunctorGamma
import Mathlib.AlgebraicTopology.DoldKan.SplitSimplicialObject
import Mathlib.CategoryTheory.Idempotents.HomologicalComplex
#align_import algebraic_topology.dold_kan.gamma_comp_n from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190f0b504"
no... | Mathlib/AlgebraicTopology/DoldKan/GammaCompN.lean | 105 | 108 | theorem N₁Γ₀_hom_app_f_f (K : ChainComplex C ℕ) (n : ℕ) :
(N₁Γ₀.hom.app K).f.f n = (Γ₀.splitting K).toKaroubiNondegComplexIsoN₁.inv.f.f n := by |
rw [N₁Γ₀_hom_app]
apply comp_id
|
import Mathlib.Data.Bundle
import Mathlib.Data.Set.Image
import Mathlib.Topology.PartialHomeomorph
import Mathlib.Topology.Order.Basic
#align_import topology.fiber_bundle.trivialization from "leanprover-community/mathlib"@"e473c3198bb41f68560cab68a0529c854b618833"
open TopologicalSpace Filter Set Bundle Function
... | Mathlib/Topology/FiberBundle/Trivialization.lean | 273 | 275 | theorem mk_symm (e : Pretrivialization F (π F E)) {b : B} (hb : b ∈ e.baseSet) (y : F) :
TotalSpace.mk b (e.symm b y) = e.toPartialEquiv.symm (b, y) := by |
simp only [e.symm_apply hb, TotalSpace.mk_cast (e.proj_symm_apply' hb), TotalSpace.eta]
|
import Mathlib.Topology.Constructions
#align_import topology.continuous_on from "leanprover-community/mathlib"@"d4f691b9e5f94cfc64639973f3544c95f8d5d494"
open Set Filter Function Topology Filter
variable {α : Type*} {β : Type*} {γ : Type*} {δ : Type*}
variable [TopologicalSpace α]
@[simp]
theorem nhds_bind_nhdsW... | Mathlib/Topology/ContinuousOn.lean | 334 | 336 | theorem nhdsWithin_pi_eq_bot {ι : Type*} {α : ι → Type*} [∀ i, TopologicalSpace (α i)] {I : Set ι}
{s : ∀ i, Set (α i)} {x : ∀ i, α i} : 𝓝[pi I s] x = ⊥ ↔ ∃ i ∈ I, 𝓝[s i] x i = ⊥ := by |
simp only [nhdsWithin, nhds_pi, pi_inf_principal_pi_eq_bot]
|
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Preserves.Basic
#align_import category_theory.limits.preserves.shapes.pullbacks from "leanprover-community/mathlib"@"f11e306adb9f2a393539d2bb4293bf1b42caa7ac"
noncomputable section
universe v₁ v₂ u₁ u₂
-- Porting note: ne... | Mathlib/CategoryTheory/Limits/Preserves/Shapes/Pullbacks.lean | 132 | 134 | theorem PreservesPullback.iso_inv_fst :
(PreservesPullback.iso G f g).inv ≫ G.map pullback.fst = pullback.fst := by |
simp [PreservesPullback.iso, Iso.inv_comp_eq]
|
import Mathlib.Init.Control.Combinators
import Mathlib.Data.Option.Defs
import Mathlib.Logic.IsEmpty
import Mathlib.Logic.Relator
import Mathlib.Util.CompileInductive
import Aesop
#align_import data.option.basic from "leanprover-community/mathlib"@"f340f229b1f461aa1c8ee11e0a172d0a3b301a4a"
universe u
namespace Op... | Mathlib/Data/Option/Basic.lean | 342 | 343 | theorem getD_default_eq_iget [Inhabited α] (o : Option α) :
o.getD default = o.iget := by | cases o <;> rfl
|
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Combinatorics.Additive.AP.Three.Defs
import Mathlib.Combinatorics.Pigeonhole
import Mathlib.Data.Complex.ExponentialBounds
#align_import combinatorics.additive.behrend from "leanprover-community/mathlib"@"4fa54b337f7d52805480306db1b1439c741848c8"
open N... | Mathlib/Combinatorics/Additive/AP/Three/Behrend.lean | 312 | 320 | theorem log_two_mul_two_le_sqrt_log_eight : log 2 * 2 ≤ √(log 8) := by |
have : (8 : ℝ) = 2 ^ ((3 : ℕ) : ℝ) := by rw [rpow_natCast]; norm_num
rw [this, log_rpow zero_lt_two (3 : ℕ)]
apply le_sqrt_of_sq_le
rw [mul_pow, sq (log 2), mul_assoc, mul_comm]
refine mul_le_mul_of_nonneg_right ?_ (log_nonneg one_le_two)
rw [← le_div_iff]
on_goal 1 => apply log_two_lt_d9.le.trans
all_... |
import Mathlib.MeasureTheory.Measure.NullMeasurable
import Mathlib.MeasureTheory.MeasurableSpace.Basic
import Mathlib.Topology.Algebra.Order.LiminfLimsup
#align_import measure_theory.measure.measure_space from "leanprover-community/mathlib"@"343e80208d29d2d15f8050b929aa50fe4ce71b55"
noncomputable section
open Set... | Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 416 | 420 | theorem tsum_measure_le_measure_univ {s : ι → Set α} (hs : ∀ i, MeasurableSet (s i))
(H : Pairwise (Disjoint on s)) : (∑' i, μ (s i)) ≤ μ (univ : Set α) := by |
rw [ENNReal.tsum_eq_iSup_sum]
exact iSup_le fun s =>
sum_measure_le_measure_univ (fun i _hi => hs i) fun i _hi j _hj hij => H hij
|
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