Context stringlengths 57 6.04k | file_name stringlengths 21 79 | start int64 14 1.49k | end int64 18 1.5k | theorem stringlengths 25 1.55k | proof stringlengths 5 7.36k |
|---|---|---|---|---|---|
import Mathlib.RingTheory.FiniteType
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Localization.Integer
import Mathlib.RingTheory.Localization.Submodule
import Mathlib.RingTheory.Nilpotent.Lemmas
import Mathlib.RingTheory.RingHomProperties
im... | Mathlib/RingTheory/LocalProperties.lean | 290 | 300 | theorem eq_zero_of_localization (r : R)
(h : ∀ (J : Ideal R) (hJ : J.IsMaximal), algebraMap R (Localization.AtPrime J) r = 0) :
r = 0 := by |
rw [← Ideal.span_singleton_eq_bot]
apply ideal_eq_bot_of_localization
intro J hJ
delta IsLocalization.coeSubmodule
erw [Submodule.map_span, Submodule.span_eq_bot]
rintro _ ⟨_, h', rfl⟩
cases Set.mem_singleton_iff.mpr h'
exact h J hJ
|
import Mathlib.Algebra.Module.Equiv
import Mathlib.Algebra.Module.Submodule.Basic
import Mathlib.Algebra.PUnitInstances
import Mathlib.Data.Set.Subsingleton
#align_import algebra.module.submodule.lattice from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
universe v
variable {R S M : Ty... | Mathlib/Algebra/Module/Submodule/Lattice.lean | 122 | 125 | theorem subsingleton_iff_eq_bot : Subsingleton p ↔ p = ⊥ := by |
rw [subsingleton_iff, Submodule.eq_bot_iff]
refine ⟨fun h x hx ↦ by simpa using h ⟨x, hx⟩ ⟨0, p.zero_mem⟩,
fun h ⟨x, hx⟩ ⟨y, hy⟩ ↦ by simp [h x hx, h y hy]⟩
|
import Mathlib.Data.Vector.Basic
import Mathlib.Data.Vector.Snoc
set_option autoImplicit true
namespace Vector
section Fold
section Binary
variable (xs : Vector α n) (ys : Vector β n)
@[simp]
theorem mapAccumr₂_mapAccumr_left (f₁ : γ → β → σ₁ → σ₁ × ζ) (f₂ : α → σ₂ → σ₂ × γ) :
(mapAccumr₂ f₁ (mapAccumr f₂... | Mathlib/Data/Vector/MapLemmas.lean | 92 | 100 | theorem mapAccumr_mapAccumr₂ (f₁ : γ → σ₁ → σ₁ × ζ) (f₂ : α → β → σ₂ → σ₂ × γ) :
(mapAccumr f₁ (mapAccumr₂ f₂ xs ys s₂).snd s₁)
= let m := mapAccumr₂ (fun x y s =>
let r₂ := f₂ x y s.snd
let r₁ := f₁ r₂.snd s.fst
((r₁.fst, r₂.fst), r₁.snd)
) xs ys (s₁, s₂)
(m.fst.fst,... |
induction xs, ys using Vector.revInductionOn₂ generalizing s₁ s₂ <;> simp_all
|
import Mathlib.NumberTheory.ZetaValues
import Mathlib.NumberTheory.LSeries.RiemannZeta
open Complex Real Set
open scoped Nat
namespace HurwitzZeta
variable {k : ℕ} {x : ℝ}
theorem cosZeta_two_mul_nat (hk : k ≠ 0) (hx : x ∈ Icc 0 1) :
cosZeta x (2 * k) = (-1) ^ (k + 1) * (2 * π) ^ (2 * k) / 2 / (2 * k)! *
... | Mathlib/NumberTheory/LSeries/HurwitzZetaValues.lean | 100 | 110 | theorem cosZeta_two_mul_nat' (hk : k ≠ 0) (hx : x ∈ Icc (0 : ℝ) 1) :
cosZeta x (2 * k) = (-1) ^ (k + 1) / (2 * k) / Gammaℂ (2 * k) *
((Polynomial.bernoulli (2 * k)).map (algebraMap ℚ ℂ)).eval (x : ℂ) := by |
rw [cosZeta_two_mul_nat hk hx]
congr 1
have : (2 * k)! = (2 * k) * Complex.Gamma (2 * k) := by
rw [(by { norm_cast; omega } : 2 * (k : ℂ) = ↑(2 * k - 1) + 1), Complex.Gamma_nat_eq_factorial,
← Nat.cast_add_one, ← Nat.cast_mul, ← Nat.factorial_succ, Nat.sub_add_cancel (by omega)]
simp_rw [this, Gammaℂ... |
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.Adjugate
import Mathlib.LinearAlgebra.Matrix.Transvection
import Mathlib.RingTheory.RootsOfUnity.Basic
#align_import linear_algebra.matrix.special_linear_group from "leanprover-community/mathlib"@"f06058e64b7e8397234455038f3f8aec83aab... | Mathlib/LinearAlgebra/Matrix/SpecialLinearGroup.lean | 181 | 183 | theorem det_ne_zero [Nontrivial R] (g : SpecialLinearGroup n R) : det ↑ₘg ≠ 0 := by |
rw [g.det_coe]
norm_num
|
import Mathlib.Topology.Connected.Basic
import Mathlib.Topology.Separation
open scoped Topology
variable {X Y A} [TopologicalSpace X] [TopologicalSpace A]
theorem embedding_toPullbackDiag (f : X → Y) : Embedding (toPullbackDiag f) :=
Embedding.mk' _ (injective_toPullbackDiag f) fun x ↦ by
rw [toPullbackDiag,... | Mathlib/Topology/SeparatedMap.lean | 79 | 87 | theorem isSeparatedMap_iff_isClosed_diagonal {f : X → Y} :
IsSeparatedMap f ↔ IsClosed f.pullbackDiagonal := by |
simp_rw [isSeparatedMap_iff_nhds, ← isOpen_compl_iff, isOpen_iff_mem_nhds,
Subtype.forall, Prod.forall, nhds_induced, nhds_prod_eq]
refine forall₄_congr fun x₁ x₂ _ _ ↦ ⟨fun h ↦ ?_, fun ⟨t, ht, t_sub⟩ ↦ ?_⟩
· simp_rw [← Filter.disjoint_iff, ← compl_diagonal_mem_prod] at h
exact ⟨_, h, subset_rfl⟩
· obt... |
import Mathlib.Data.Finset.Lattice
import Mathlib.Data.Set.Sigma
#align_import data.finset.sigma from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function Multiset
variable {ι : Type*}
namespace Finset
section SigmaLift
variable {α β γ : ι → Type*} [DecidableEq ι]
def sigm... | Mathlib/Data/Finset/Sigma.lean | 204 | 208 | theorem sigmaLift_eq_empty : sigmaLift f a b = ∅ ↔ ∀ h : a.1 = b.1, f (h ▸ a.2) b.2 = ∅ := by |
simp_rw [sigmaLift]
split_ifs with h
· simp [h, forall_prop_of_true h]
· simp [h, forall_prop_of_false h]
|
import Mathlib.Data.DFinsupp.Basic
import Mathlib.Data.Finset.Pointwise
import Mathlib.LinearAlgebra.Basis.VectorSpace
#align_import algebra.group.unique_prods from "leanprover-community/mathlib"@"d6fad0e5bf2d6f48da9175d25c3dc5706b3834ce"
@[to_additive
"Let `G` be a Type with addition, let `A B : Finset G` ... | Mathlib/Algebra/Group/UniqueProds.lean | 71 | 75 | theorem of_card_le_one (hA : A.Nonempty) (hB : B.Nonempty) (hA1 : A.card ≤ 1) (hB1 : B.card ≤ 1) :
∃ a ∈ A, ∃ b ∈ B, UniqueMul A B a b := by |
rw [Finset.card_le_one_iff] at hA1 hB1
obtain ⟨a, ha⟩ := hA; obtain ⟨b, hb⟩ := hB
exact ⟨a, ha, b, hb, fun _ _ ha' hb' _ ↦ ⟨hA1 ha' ha, hB1 hb' hb⟩⟩
|
import Mathlib.Data.Vector.Basic
import Mathlib.Data.Vector.Snoc
set_option autoImplicit true
namespace Vector
section Fold
section Binary
variable (xs : Vector α n) (ys : Vector β n)
@[simp]
theorem mapAccumr₂_mapAccumr_left (f₁ : γ → β → σ₁ → σ₁ × ζ) (f₂ : α → σ₂ → σ₂ × γ) :
(mapAccumr₂ f₁ (mapAccumr f₂... | Mathlib/Data/Vector/MapLemmas.lean | 108 | 117 | theorem mapAccumr₂_mapAccumr₂_left_left (f₁ : γ → α → σ₁ → σ₁ × φ) (f₂ : α → β → σ₂ → σ₂ × γ) :
(mapAccumr₂ f₁ (mapAccumr₂ f₂ xs ys s₂).snd xs s₁)
= let m := mapAccumr₂ (fun x y (s₁, s₂) =>
let r₂ := f₂ x y s₂
let r₁ := f₁ r₂.snd x s₁
((r₁.fst, r₂.fst), r₁.snd)
... |
induction xs, ys using Vector.revInductionOn₂ generalizing s₁ s₂ <;> simp_all
|
import Mathlib.MeasureTheory.Integral.Lebesgue
open Set hiding restrict restrict_apply
open Filter ENNReal NNReal MeasureTheory.Measure
namespace MeasureTheory
variable {α : Type*} {m0 : MeasurableSpace α} {μ : Measure α}
noncomputable
def Measure.withDensity {m : MeasurableSpace α} (μ : Measure α) (f : α → ℝ≥... | Mathlib/MeasureTheory/Measure/WithDensity.lean | 105 | 107 | theorem withDensity_add_right (f : α → ℝ≥0∞) {g : α → ℝ≥0∞} (hg : Measurable g) :
μ.withDensity (f + g) = μ.withDensity f + μ.withDensity g := by |
simpa only [add_comm] using withDensity_add_left hg f
|
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.DominatedConvergence
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
noncomputable s... | Mathlib/MeasureTheory/Constructions/Prod/Integral.lean | 77 | 122 | theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : StronglyMeasurable fun x => ∫ y, f x y ∂ν := by |
by_cases hE : CompleteSpace E; swap; · simp [integral, hE, stronglyMeasurable_const]
borelize E
haveI : SeparableSpace (range (uncurry f) ∪ {0} : Set E) :=
hf.separableSpace_range_union_singleton
let s : ℕ → SimpleFunc (α × β) E :=
SimpleFunc.approxOn _ hf.measurable (range (uncurry f) ∪ {0}) 0 (by sim... |
import Mathlib.Data.Set.Image
import Mathlib.Data.List.GetD
#align_import data.set.list from "leanprover-community/mathlib"@"2ec920d35348cb2d13ac0e1a2ad9df0fdf1a76b4"
open List
variable {α β : Type*} (l : List α)
namespace Set
theorem range_list_map (f : α → β) : range (map f) = { l | ∀ x ∈ l, x ∈ range f } :=... | Mathlib/Data/Set/List.lean | 52 | 57 | theorem range_list_getD (d : α) : (range fun n => l.getD n d) = insert d { x | x ∈ l } :=
calc
(range fun n => l.getD n d) = (fun o : Option α => o.getD d) '' range l.get? := by |
simp only [← range_comp, (· ∘ ·), getD_eq_getD_get?]
_ = insert d { x | x ∈ l } := by
simp only [range_list_get?, image_insert_eq, Option.getD, image_image, image_id']
|
import Mathlib.Analysis.Complex.RemovableSingularity
import Mathlib.Analysis.Calculus.UniformLimitsDeriv
import Mathlib.Analysis.NormedSpace.FunctionSeries
#align_import analysis.complex.locally_uniform_limit from "leanprover-community/mathlib"@"fe44cd36149e675eb5dec87acc7e8f1d6568e081"
open Set Metric MeasureThe... | Mathlib/Analysis/Complex/LocallyUniformLimit.lean | 95 | 110 | theorem _root_.TendstoUniformlyOn.cderiv (hF : TendstoUniformlyOn F f φ (cthickening δ K))
(hδ : 0 < δ) (hFn : ∀ᶠ n in φ, ContinuousOn (F n) (cthickening δ K)) :
TendstoUniformlyOn (cderiv δ ∘ F) (cderiv δ f) φ K := by |
rcases φ.eq_or_neBot with rfl | hne
· simp only [TendstoUniformlyOn, eventually_bot, imp_true_iff]
have e1 : ContinuousOn f (cthickening δ K) := TendstoUniformlyOn.continuousOn hF hFn
rw [tendstoUniformlyOn_iff] at hF ⊢
rintro ε hε
filter_upwards [hF (ε * δ) (mul_pos hε hδ), hFn] with n h h' z hz
simp_rw... |
import Mathlib.Order.BooleanAlgebra
import Mathlib.Logic.Equiv.Basic
#align_import order.symm_diff from "leanprover-community/mathlib"@"6eb334bd8f3433d5b08ba156b8ec3e6af47e1904"
open Function OrderDual
variable {ι α β : Type*} {π : ι → Type*}
def symmDiff [Sup α] [SDiff α] (a b : α) : α :=
a \ b ⊔ b \ a
#ali... | Mathlib/Order/SymmDiff.lean | 347 | 347 | theorem top_symmDiff' : ⊤ ∆ a = ￢a := by | simp [symmDiff]
|
import Mathlib.Algebra.DirectSum.Finsupp
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.DirectSum.TensorProduct
#align_import linear_algebra.direct_sum.finsupp from "leanprover-community/mathlib"@"9b9d125b7be0930f564a68f1d73ace10cf46064d"
noncomputable section
open DirectSum TensorProduct
ope... | Mathlib/LinearAlgebra/DirectSum/Finsupp.lean | 256 | 259 | theorem finsuppTensorFinsupp_single (i : ι) (m : M) (k : κ) (n : N) :
finsuppTensorFinsupp R S M N ι κ (Finsupp.single i m ⊗ₜ Finsupp.single k n) =
Finsupp.single (i, k) (m ⊗ₜ n) := by |
simp [finsuppTensorFinsupp]
|
import Mathlib.Analysis.BoxIntegral.Partition.Basic
#align_import analysis.box_integral.partition.split from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf97388c9e3b441fc4ccf3f"
noncomputable section
open scoped Classical
open Filter
open Function Set Filter
namespace BoxIntegral
variable {ι M : Type*} {... | Mathlib/Analysis/BoxIntegral/Partition/Split.lean | 88 | 94 | theorem splitLower_def [DecidableEq ι] {i x} (h : x ∈ Ioo (I.lower i) (I.upper i))
(h' : ∀ j, I.lower j < update I.upper i x j :=
(forall_update_iff I.upper fun j y => I.lower j < y).2
⟨h.1, fun j _ => I.lower_lt_upper _⟩) :
I.splitLower i x = (⟨I.lower, update I.upper i x, h'⟩ : Box ι) := by |
simp (config := { unfoldPartialApp := true }) only [splitLower, mk'_eq_coe, min_eq_left h.2.le,
update, and_self]
|
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Div
#align_import data.polynomial.ring_division from "leanprover-community/mathlib"@"8efcf8022aac8e01df8d302dcebdbc25d6a886c8"
noncomputable ... | Mathlib/Algebra/Polynomial/RingDivision.lean | 103 | 107 | theorem aeval_modByMonic_eq_self_of_root [Algebra R S] {p q : R[X]} (hq : q.Monic) {x : S}
(hx : aeval x q = 0) : aeval x (p %ₘ q) = aeval x p := by |
--`eval₂_modByMonic_eq_self_of_root` doesn't work here as it needs commutativity
rw [modByMonic_eq_sub_mul_div p hq, _root_.map_sub, _root_.map_mul, hx, zero_mul,
sub_zero]
|
import Mathlib.Algebra.Lie.BaseChange
import Mathlib.Algebra.Lie.Solvable
import Mathlib.Algebra.Lie.Quotient
import Mathlib.Algebra.Lie.Normalizer
import Mathlib.LinearAlgebra.Eigenspace.Basic
import Mathlib.Order.Filter.AtTopBot
import Mathlib.RingTheory.Artinian
import Mathlib.RingTheory.Nilpotent.Lemmas
import Mat... | Mathlib/Algebra/Lie/Nilpotent.lean | 504 | 508 | theorem ucs_le_of_normalizer_eq_self (h : N₁.normalizer = N₁) (k : ℕ) :
(⊥ : LieSubmodule R L M).ucs k ≤ N₁ := by |
rw [← ucs_eq_self_of_normalizer_eq_self h k]
mono
simp
|
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 | 92 | 92 | theorem volume_Ioc {a b : ℝ} : volume (Ioc a b) = ofReal (b - a) := by | simp [volume_val]
|
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Data.Finset.Sym
import Mathlib.Data.Nat.Choose.Cast
import Mathlib.Data.Nat.Choose.Multinomial
#align_import analysis.calculus.cont_diff from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputable section
open scoped C... | Mathlib/Analysis/Calculus/ContDiff/Bounds.lean | 40 | 122 | theorem ContinuousLinearMap.norm_iteratedFDerivWithin_le_of_bilinear_aux {Du Eu Fu Gu : Type u}
[NormedAddCommGroup Du] [NormedSpace 𝕜 Du] [NormedAddCommGroup Eu] [NormedSpace 𝕜 Eu]
[NormedAddCommGroup Fu] [NormedSpace 𝕜 Fu] [NormedAddCommGroup Gu] [NormedSpace 𝕜 Gu]
(B : Eu →L[𝕜] Fu →L[𝕜] Gu) {f : Du... |
/- We argue by induction on `n`. The bound is trivial for `n = 0`. For `n + 1`, we write
the `(n+1)`-th derivative as the `n`-th derivative of the derivative `B f g' + B f' g`,
and apply the inductive assumption to each of those two terms. For this induction to make sense,
the spaces of linear maps that ... |
import Mathlib.Data.Finset.Sigma
import Mathlib.Data.Finset.Pairwise
import Mathlib.Data.Finset.Powerset
import Mathlib.Data.Fintype.Basic
import Mathlib.Order.CompleteLatticeIntervals
#align_import order.sup_indep from "leanprover-community/mathlib"@"c4c2ed622f43768eff32608d4a0f8a6cec1c047d"
variable {α β ι ι' :... | Mathlib/Order/SupIndep.lean | 92 | 96 | theorem SupIndep.le_sup_iff (hs : s.SupIndep f) (hts : t ⊆ s) (hi : i ∈ s) (hf : ∀ i, f i ≠ ⊥) :
f i ≤ t.sup f ↔ i ∈ t := by |
refine ⟨fun h => ?_, le_sup⟩
by_contra hit
exact hf i (disjoint_self.1 <| (hs hts hi hit).mono_right h)
|
import Mathlib.Algebra.Order.Monoid.OrderDual
import Mathlib.Tactic.Lift
import Mathlib.Tactic.Monotonicity.Attr
open Function
variable {β G M : Type*}
section Monoid
variable [Monoid M]
section Preorder
variable [Preorder M]
section Left
variable [CovariantClass M M (· * ·) (· ≤ ·)] {x : M}
@[to_additive (... | Mathlib/Algebra/Order/Monoid/Unbundled/Pow.lean | 71 | 77 | theorem one_lt_pow' {a : M} (ha : 1 < a) {k : ℕ} (hk : k ≠ 0) : 1 < a ^ k := by |
rcases Nat.exists_eq_succ_of_ne_zero hk with ⟨l, rfl⟩
clear hk
induction' l with l IH
· rw [pow_succ]; simpa using ha
· rw [pow_succ]
exact one_lt_mul'' IH ha
|
import Mathlib.Algebra.Polynomial.Basic
#align_import data.polynomial.monomial from "leanprover-community/mathlib"@"220f71ba506c8958c9b41bd82226b3d06b0991e8"
noncomputable section
namespace Polynomial
open Polynomial
universe u
variable {R : Type u} {a b : R} {m n : ℕ}
variable [Semiring R] {p q r : R[X]}
th... | Mathlib/Algebra/Polynomial/Monomial.lean | 39 | 56 | theorem card_support_le_one_iff_monomial {f : R[X]} :
Finset.card f.support ≤ 1 ↔ ∃ n a, f = monomial n a := by |
constructor
· intro H
rw [Finset.card_le_one_iff_subset_singleton] at H
rcases H with ⟨n, hn⟩
refine ⟨n, f.coeff n, ?_⟩
ext i
by_cases hi : i = n
· simp [hi, coeff_monomial]
· have : f.coeff i = 0 := by
rw [← not_mem_support_iff]
exact fun hi' => hi (Finset.mem_singleton... |
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 | 144 | 145 | theorem div_mem {x y : M} (hx : x ∈ H) (hy : y ∈ H) : x / y ∈ H := by |
rw [div_eq_mul_inv]; exact mul_mem hx (inv_mem hy)
|
import Mathlib.SetTheory.Ordinal.Arithmetic
import Mathlib.Tactic.Abel
#align_import set_theory.ordinal.natural_ops from "leanprover-community/mathlib"@"31b269b60935483943542d547a6dd83a66b37dc7"
set_option autoImplicit true
universe u v
open Function Order
noncomputable section
def NatOrdinal : Type _ :=
... | Mathlib/SetTheory/Ordinal/NaturalOps.lean | 255 | 258 | theorem nadd_le_nadd_left (h : b ≤ c) (a) : a ♯ b ≤ a ♯ c := by |
rcases lt_or_eq_of_le h with (h | rfl)
· exact (nadd_lt_nadd_left h a).le
· exact le_rfl
|
import Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear
import Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm
import Mathlib.Analysis.NormedSpace.Span
suppress_compilation
open Bornology
open Filter hiding map_smul
open scoped Classical NNReal Topology Uniformity
-- the `ₗ` subscript variables are for special... | Mathlib/Analysis/NormedSpace/OperatorNorm/NormedSpace.lean | 99 | 107 | theorem opNorm_zero_iff [RingHomIsometric σ₁₂] : ‖f‖ = 0 ↔ f = 0 :=
Iff.intro
(fun hn => ContinuousLinearMap.ext fun x => norm_le_zero_iff.1
(calc
_ ≤ ‖f‖ * ‖x‖ := le_opNorm _ _
_ = _ := by | rw [hn, zero_mul]))
(by
rintro rfl
exact opNorm_zero)
|
import Mathlib.CategoryTheory.Filtered.Basic
import Mathlib.Topology.Category.TopCat.Limits.Basic
#align_import topology.category.Top.limits.konig from "leanprover-community/mathlib"@"dbdf71cee7bb20367cb7e37279c08b0c218cf967"
-- Porting note: every ML3 decl has an uppercase letter
set_option linter.uppercaseLean3 ... | Mathlib/Topology/Category/TopCat/Limits/Konig.lean | 107 | 124 | theorem partialSections.closed [∀ j : J, T2Space (F.obj j)] {G : Finset J}
(H : Finset (FiniteDiagramArrow G)) : IsClosed (partialSections F H) := by |
have :
partialSections F H =
⋂ (f : FiniteDiagramArrow G) (_ : f ∈ H), {u | F.map f.2.2.2.2 (u f.1) = u f.2.1} := by
ext1
simp only [Set.mem_iInter, Set.mem_setOf_eq]
rfl
rw [this]
apply isClosed_biInter
intro f _
-- Porting note: can't see through forget
have : T2Space ((forget TopCa... |
import Mathlib.Algebra.Polynomial.Mirror
import Mathlib.Analysis.Complex.Polynomial
#align_import data.polynomial.unit_trinomial from "leanprover-community/mathlib"@"302eab4f46abb63de520828de78c04cb0f9b5836"
namespace Polynomial
open scoped Polynomial
open Finset
section Semiring
variable {R : Type*} [Semirin... | Mathlib/Algebra/Polynomial/UnitTrinomial.lean | 100 | 102 | theorem trinomial_trailingCoeff (hkm : k < m) (hmn : m < n) (hu : u ≠ 0) :
(trinomial k m n u v w).trailingCoeff = u := by |
rw [trailingCoeff, trinomial_natTrailingDegree hkm hmn hu, trinomial_trailing_coeff' hkm hmn]
|
import Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv
#align_import linear_algebra.quadratic_form.prod from "leanprover-community/mathlib"@"9b2755b951bc323c962bd072cd447b375cf58101"
universe u v w
variable {ι : Type*} {R : Type*} {M₁ M₂ N₁ N₂ : Type*} {Mᵢ Nᵢ : ι → Type*}
namespace QuadraticForm
section Pro... | Mathlib/LinearAlgebra/QuadraticForm/Prod.lean | 328 | 339 | theorem nonneg_pi_iff [Fintype ι] {R} [OrderedCommRing R] [∀ i, Module R (Mᵢ i)]
{Q : ∀ i, QuadraticForm R (Mᵢ i)} : (∀ x, 0 ≤ pi Q x) ↔ ∀ i x, 0 ≤ Q i x := by |
simp_rw [pi, sum_apply, comp_apply, LinearMap.proj_apply]
constructor
-- TODO: does this generalize to a useful lemma independent of `QuadraticForm`?
· intro h i x
classical
convert h (Pi.single i x) using 1
rw [Finset.sum_eq_single_of_mem i (Finset.mem_univ _) fun j _ hji => ?_, Pi.single_eq_same]... |
import Mathlib.Analysis.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calculus.FDeriv.Comp
#align_import analysis.calculus.fderiv.add from "leanprover-community/mathlib"@"e3fb84046afd187b710170887195d50bada934ee"
open Filter Asymptotics ContinuousLinearMap Set Metric
open scoped Classical
open Topology NNReal F... | Mathlib/Analysis/Calculus/FDeriv/Add.lean | 353 | 357 | theorem HasFDerivAtFilter.sum (h : ∀ i ∈ u, HasFDerivAtFilter (A i) (A' i) x L) :
HasFDerivAtFilter (fun y => ∑ i ∈ u, A i y) (∑ i ∈ u, A' i) x L := by |
simp only [hasFDerivAtFilter_iff_isLittleO] at *
convert IsLittleO.sum h
simp [ContinuousLinearMap.sum_apply]
|
import Mathlib.Geometry.Manifold.Sheaf.Smooth
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
noncomputable section
universe u
variable {𝕜 : Type u} [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜]
{EM : Type*} [NormedAddCommGroup EM] [NormedSpace 𝕜 EM]
{HM : Type*} [TopologicalSpace HM] (IM : ModelWit... | Mathlib/Geometry/Manifold/Sheaf/LocallyRingedSpace.lean | 43 | 98 | theorem smoothSheafCommRing.isUnit_stalk_iff {x : M}
(f : (smoothSheafCommRing IM 𝓘(𝕜) M 𝕜).presheaf.stalk x) :
IsUnit f ↔ f ∉ RingHom.ker (smoothSheafCommRing.eval IM 𝓘(𝕜) M 𝕜 x) := by |
constructor
· rintro ⟨⟨f, g, hf, hg⟩, rfl⟩ (h' : smoothSheafCommRing.eval IM 𝓘(𝕜) M 𝕜 x f = 0)
simpa [h'] using congr_arg (smoothSheafCommRing.eval IM 𝓘(𝕜) M 𝕜 x) hf
· let S := (smoothSheafCommRing IM 𝓘(𝕜) M 𝕜).presheaf
-- Suppose that `f`, in the stalk at `x`, is nonzero at `x`
rintro (hf :... |
import Mathlib.Algebra.Order.ToIntervalMod
import Mathlib.Algebra.Ring.AddAut
import Mathlib.Data.Nat.Totient
import Mathlib.GroupTheory.Divisible
import Mathlib.Topology.Connected.PathConnected
import Mathlib.Topology.IsLocalHomeomorph
#align_import topology.instances.add_circle from "leanprover-community/mathlib"@"... | Mathlib/Topology/Instances/AddCircle.lean | 64 | 79 | theorem continuous_right_toIcoMod : ContinuousWithinAt (toIcoMod hp a) (Ici x) x := by |
intro s h
rw [Filter.mem_map, mem_nhdsWithin_iff_exists_mem_nhds_inter]
haveI : Nontrivial 𝕜 := ⟨⟨0, p, hp.ne⟩⟩
simp_rw [mem_nhds_iff_exists_Ioo_subset] at h ⊢
obtain ⟨l, u, hxI, hIs⟩ := h
let d := toIcoDiv hp a x • p
have hd := toIcoMod_mem_Ico hp a x
simp_rw [subset_def, mem_inter_iff]
refine ⟨_, ... |
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Fintype.BigOperators
#align_import data.sign from "leanprover-community/mathlib"@"2445c98ae4b87eabebdde552593519b9b6dc350c"
-- Porting note (#11081): cannot automatically derive Fintype, adde... | Mathlib/Data/Sign.lean | 165 | 165 | theorem nonneg_iff_ne_neg_one {a : SignType} : 0 ≤ a ↔ a ≠ -1 := by | cases a <;> decide
|
import Mathlib.Algebra.Polynomial.Degree.Definitions
#align_import ring_theory.polynomial.opposites from "leanprover-community/mathlib"@"63417e01fbc711beaf25fa73b6edb395c0cfddd0"
open Polynomial
open Polynomial MulOpposite
variable {R : Type*} [Semiring R]
noncomputable section
namespace Polynomial
def opRi... | Mathlib/RingTheory/Polynomial/Opposites.lean | 110 | 114 | theorem natDegree_opRingEquiv (p : R[X]ᵐᵒᵖ) : (opRingEquiv R p).natDegree = (unop p).natDegree := by |
by_cases p0 : p = 0
· simp only [p0, _root_.map_zero, natDegree_zero, unop_zero]
· simp only [p0, natDegree_eq_support_max', Ne, AddEquivClass.map_eq_zero_iff, not_false_iff,
support_opRingEquiv, unop_eq_zero_iff]
|
import Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension
import Mathlib.Geometry.Manifold.ContMDiff.Atlas
import Mathlib.Geometry.Manifold.ContMDiff.NormedSpace
#align_import geometry.manifold.bump_function from "leanprover-community/mathlib"@"b018406ad2f2a73223a3a9e198ccae61e6f05318"
universe uE uF uH uM
va... | Mathlib/Geometry/Manifold/BumpFunction.lean | 124 | 128 | theorem support_eq_symm_image :
support f = (extChartAt I c).symm '' (ball (extChartAt I c c) f.rOut ∩ range I) := by |
rw [f.support_eq_inter_preimage, ← extChartAt_source I,
← (extChartAt I c).symm_image_target_inter_eq', inter_comm,
ball_inter_range_eq_ball_inter_target]
|
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.CharP.Algebra
import Mathlib.Data.Nat.Prime
#align_import algebra.char_p.exp_char from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u
variable (R : Type u)
section Semiring
variable [Semiring R]
class inductive Ex... | Mathlib/Algebra/CharP/ExpChar.lean | 105 | 108 | theorem char_zero_of_expChar_one (p : ℕ) [hp : CharP R p] [hq : ExpChar R 1] : p = 0 := by |
cases hq
· exact CharP.eq R hp inferInstance
· exact False.elim (CharP.char_ne_one R 1 rfl)
|
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
import Mathlib.MeasureTheory.Integral.FundThmCalculus
#align_import analysis.special_functions.non_integrable from "leanprover-community/mathlib"@"55ec6e9af7d3e0043f57e394cb06a72f6275273e"
open scoped MeasureTheory Topology Interval NNReal ENNReal
open MeasureTh... | Mathlib/Analysis/SpecialFunctions/NonIntegrable.lean | 52 | 96 | theorem not_integrableOn_of_tendsto_norm_atTop_of_deriv_isBigO_filter_aux
[CompleteSpace E] {f : ℝ → E} {g : ℝ → F}
{k : Set ℝ} (l : Filter ℝ) [NeBot l] [TendstoIxxClass Icc l l]
(hl : k ∈ l) (hd : ∀ᶠ x in l, DifferentiableAt ℝ f x) (hf : Tendsto (fun x => ‖f x‖) l atTop)
(hfg : deriv f =O[l] g) : ¬Inte... |
intro hgi
obtain ⟨C, hC₀, s, hsl, hsub, hfd, hg⟩ :
∃ (C : ℝ) (_ : 0 ≤ C), ∃ s ∈ l, (∀ x ∈ s, ∀ y ∈ s, [[x, y]] ⊆ k) ∧
(∀ x ∈ s, ∀ y ∈ s, ∀ z ∈ [[x, y]], DifferentiableAt ℝ f z) ∧
∀ x ∈ s, ∀ y ∈ s, ∀ z ∈ [[x, y]], ‖deriv f z‖ ≤ C * ‖g z‖ := by
rcases hfg.exists_nonneg with ⟨C, C₀, hC⟩
have... |
import Mathlib.Algebra.Group.ConjFinite
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Dynamics.PeriodicPts
import Mathlib.GroupTheory.Commutator
import Mathlib.GroupTheory.Coset
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.GroupAction.Hom
#align_import group_theory.group_action.... | Mathlib/GroupTheory/GroupAction/Quotient.lean | 120 | 123 | theorem _root_.QuotientGroup.out'_conj_pow_minimalPeriod_mem (a : α) (q : α ⧸ H) :
q.out'⁻¹ * a ^ Function.minimalPeriod (a • ·) q * q.out' ∈ H := by |
rw [mul_assoc, ← QuotientGroup.eq', QuotientGroup.out_eq', ← smul_eq_mul, Quotient.mk_smul_out',
eq_comm, pow_smul_eq_iff_minimalPeriod_dvd]
|
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.NormedSpace.FiniteDimension
#align_import analysis.calculus.bump_function_inner from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputable section
open Function Set Filter
open scoped Topology Filter
variable... | Mathlib/Analysis/Calculus/BumpFunction/Basic.lean | 172 | 176 | theorem support_eq : Function.support f = Metric.ball c f.rOut := by |
simp only [toFun, support_comp_eq_preimage, ContDiffBumpBase.support _ _ f.one_lt_rOut_div_rIn]
ext x
simp only [mem_ball_iff_norm, sub_zero, norm_smul, mem_preimage, Real.norm_eq_abs, abs_inv,
abs_of_pos f.rIn_pos, ← div_eq_inv_mul, div_lt_div_right f.rIn_pos]
|
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.Order.Filter.IndicatorFunction
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner
import Mathlib.MeasureTheory.Function.LpSeminorm.Trim
#align_import measure_theory.function.conditional... | Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean | 86 | 92 | theorem sub [AddGroup β] [TopologicalAddGroup β] {f g : α → β} (hfm : AEStronglyMeasurable' m f μ)
(hgm : AEStronglyMeasurable' m g μ) : AEStronglyMeasurable' m (f - g) μ := by |
rcases hfm with ⟨f', hf'_meas, hf_ae⟩
rcases hgm with ⟨g', hg'_meas, hg_ae⟩
refine ⟨f' - g', hf'_meas.sub hg'_meas, hf_ae.mp (hg_ae.mono fun x hx1 hx2 => ?_)⟩
simp_rw [Pi.sub_apply]
rw [hx1, hx2]
|
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
#align_import measure_theory.constructions.borel_space.basic from "leanprover-community/mathlib"@"9f55d0d4363ae59948c33864cbc52e0b12e0e8ce"
open Set Filter MeasureTheory MeasurableSpace
open scoped Classical Topology NNReal ENNReal MeasureTheory
univers... | Mathlib/MeasureTheory/Constructions/BorelSpace/Real.lean | 68 | 74 | theorem borel_eq_generateFrom_Iic_rat : borel ℝ = .generateFrom (⋃ a : ℚ, {Iic (a : ℝ)}) := by |
rw [borel_eq_generateFrom_Ioi_rat, iUnion_singleton_eq_range, iUnion_singleton_eq_range]
refine le_antisymm (generateFrom_le ?_) (generateFrom_le ?_) <;>
rintro _ ⟨q, rfl⟩ <;>
dsimp only <;>
[rw [← compl_Iic]; rw [← compl_Ioi]] <;>
exact MeasurableSet.compl (GenerateMeasurable.basic _ (mem_range_self q))
|
import Mathlib.Data.Finset.Lattice
import Mathlib.Data.Fintype.Vector
import Mathlib.Data.Multiset.Sym
#align_import data.finset.sym from "leanprover-community/mathlib"@"02ba8949f486ebecf93fe7460f1ed0564b5e442c"
namespace Finset
variable {α : Type*}
@[simps]
protected def sym2 (s : Finset α) : Finset (Sym2 α) :... | Mathlib/Data/Finset/Sym.lean | 139 | 142 | theorem not_isDiag_mk_of_mem_offDiag {a : α × α} (h : a ∈ s.offDiag) :
¬ (Sym2.mk a).IsDiag := by |
rw [Sym2.isDiag_iff_proj_eq]
exact (mem_offDiag.1 h).2.2
|
import Mathlib.Data.List.OfFn
import Mathlib.Data.List.Range
#align_import data.list.fin_range from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
universe u
open List
| Mathlib/Data/List/FinRange.lean | 79 | 82 | theorem Equiv.Perm.map_finRange_perm {n : ℕ} (σ : Equiv.Perm (Fin n)) :
map σ (finRange n) ~ finRange n := by |
rw [perm_ext_iff_of_nodup ((nodup_finRange n).map σ.injective) <| nodup_finRange n]
simpa [mem_map, mem_finRange, true_and_iff, iff_true_iff] using σ.surjective
|
import Mathlib.Order.Filter.Cofinite
import Mathlib.Order.ZornAtoms
#align_import order.filter.ultrafilter from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
universe u v
variable {α : Type u} {β : Type v} {γ : Type*}
open Set Filter Function
open scoped Classical
open Filter
inst... | Mathlib/Order/Filter/Ultrafilter.lean | 135 | 135 | theorem compl_mem_iff_not_mem : sᶜ ∈ f ↔ s ∉ f := by | rw [← compl_not_mem_iff, compl_compl]
|
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.LinearAlgebra.Matrix.Block
#align_import analysis.inner_product_space.gram_schmidt_ortho from "leanprover-community/mathlib"@"1a4df69ca1a9a0e5e26bfe12e2b92814216016d0"
open Finset Submodule FiniteDimensional
variable (𝕜 : Type*) {E : Type*} [RCLike �... | Mathlib/Analysis/InnerProductSpace/GramSchmidtOrtho.lean | 83 | 108 | theorem gramSchmidt_orthogonal (f : ι → E) {a b : ι} (h₀ : a ≠ b) :
⟪gramSchmidt 𝕜 f a, gramSchmidt 𝕜 f b⟫ = 0 := by |
suffices ∀ a b : ι, a < b → ⟪gramSchmidt 𝕜 f a, gramSchmidt 𝕜 f b⟫ = 0 by
cases' h₀.lt_or_lt with ha hb
· exact this _ _ ha
· rw [inner_eq_zero_symm]
exact this _ _ hb
clear h₀ a b
intro a b h₀
revert a
apply wellFounded_lt.induction b
intro b ih a h₀
simp only [gramSchmidt_def 𝕜 f b... |
import Mathlib.Data.List.Basic
#align_import data.list.lattice from "leanprover-community/mathlib"@"dd71334db81d0bd444af1ee339a29298bef40734"
open Nat
namespace List
variable {α : Type*} {l l₁ l₂ : List α} {p : α → Prop} {a : α}
variable [DecidableEq α]
section Inter
@[simp]
theorem inter_nil (l : L... | Mathlib/Data/List/Lattice.lean | 167 | 169 | theorem inter_eq_nil_iff_disjoint : l₁ ∩ l₂ = [] ↔ Disjoint l₁ l₂ := by |
simp only [eq_nil_iff_forall_not_mem, mem_inter_iff, not_and]
rfl
|
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 | 80 | 81 | theorem le_infEdist {d} : d ≤ infEdist x s ↔ ∀ y ∈ s, d ≤ edist x y := by |
simp only [infEdist, le_iInf_iff]
|
import Mathlib.AlgebraicGeometry.GammaSpecAdjunction
import Mathlib.AlgebraicGeometry.Restrict
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.RingTheory.Localization.InvSubmonoid
#align_import algebraic_geometry.AffineScheme from "leanprover-community/mathlib"@"88474d1b5af6d37c2ab728b757771bced7f5194c"... | Mathlib/AlgebraicGeometry/AffineScheme.lean | 218 | 234 | theorem Scheme.map_PrimeSpectrum_basicOpen_of_affine
(X : Scheme) [IsAffine X] (f : Scheme.Γ.obj (op X)) :
X.isoSpec.hom ⁻¹ᵁ PrimeSpectrum.basicOpen f = X.basicOpen f := by |
rw [← basicOpen_eq_of_affine]
trans
X.isoSpec.hom ⁻¹ᵁ (Scheme.Spec.obj (op (Scheme.Γ.obj (op X)))).basicOpen
((inv (X.isoSpec.hom.1.c.app (op ((Opens.map (inv X.isoSpec.hom).val.base).obj ⊤)))) f)
· congr
rw [← IsIso.inv_eq_inv, IsIso.inv_inv, IsIso.Iso.inv_inv, NatIso.app_hom]
-- Porting not... |
import Mathlib.Probability.Variance
import Mathlib.MeasureTheory.Function.UniformIntegrable
#align_import probability.ident_distrib from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open MeasureTheory Filter Finset
noncomputable section
open scoped Topology MeasureTheory ENNReal NNR... | Mathlib/Probability/IdentDistrib.lean | 141 | 145 | theorem ae_snd (h : IdentDistrib f g μ ν) {p : γ → Prop} (pmeas : MeasurableSet {x | p x})
(hp : ∀ᵐ x ∂μ, p (f x)) : ∀ᵐ x ∂ν, p (g x) := by |
apply (ae_map_iff h.aemeasurable_snd pmeas).1
rw [← h.map_eq]
exact (ae_map_iff h.aemeasurable_fst pmeas).2 hp
|
import Mathlib.FieldTheory.Finite.Basic
#align_import field_theory.chevalley_warning from "leanprover-community/mathlib"@"e001509c11c4d0f549d91d89da95b4a0b43c714f"
universe u v
section FiniteField
open MvPolynomial
open Function hiding eval
open Finset FiniteField
variable {K σ ι : Type*} [Fintype K] [Field ... | Mathlib/FieldTheory/ChevalleyWarning.lean | 53 | 97 | theorem MvPolynomial.sum_eval_eq_zero (f : MvPolynomial σ K)
(h : f.totalDegree < (q - 1) * Fintype.card σ) : ∑ x, eval x f = 0 := by |
haveI : DecidableEq K := Classical.decEq K
calc
∑ x, eval x f = ∑ x : σ → K, ∑ d ∈ f.support, f.coeff d * ∏ i, x i ^ d i := by
simp only [eval_eq']
_ = ∑ d ∈ f.support, ∑ x : σ → K, f.coeff d * ∏ i, x i ^ d i := sum_comm
_ = 0 := sum_eq_zero ?_
intro d hd
obtain ⟨i, hi⟩ : ∃ i, d i < q - 1 := ... |
import Mathlib.AlgebraicTopology.SimplexCategory
import Mathlib.CategoryTheory.Comma.Arrow
import Mathlib.CategoryTheory.Limits.FunctorCategory
import Mathlib.CategoryTheory.Opposites
#align_import algebraic_topology.simplicial_object from "leanprover-community/mathlib"@"5ed51dc37c6b891b79314ee11a50adc2b1df6fd6"
o... | Mathlib/AlgebraicTopology/SimplicialObject.lean | 146 | 149 | theorem δ_comp_σ_of_le {n} {i : Fin (n + 2)} {j : Fin (n + 1)} (H : i ≤ Fin.castSucc j) :
X.σ j.succ ≫ X.δ (Fin.castSucc i) = X.δ i ≫ X.σ j := by |
dsimp [δ, σ]
simp only [← X.map_comp, ← op_comp, SimplexCategory.δ_comp_σ_of_le H]
|
import Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.measure.haar.normed_space from "leanprover-community/mathlib"@"b84aee748341da06a6d78491367e2c0e9f15e8a5"
noncomputable sect... | Mathlib/MeasureTheory/Measure/Haar/NormedSpace.lean | 64 | 83 | theorem integral_comp_smul (f : E → F) (R : ℝ) :
∫ x, f (R • x) ∂μ = |(R ^ finrank ℝ E)⁻¹| • ∫ x, f x ∂μ := by |
by_cases hF : CompleteSpace F; swap
· simp [integral, hF]
rcases eq_or_ne R 0 with (rfl | hR)
· simp only [zero_smul, integral_const]
rcases Nat.eq_zero_or_pos (finrank ℝ E) with (hE | hE)
· have : Subsingleton E := finrank_zero_iff.1 hE
have : f = fun _ => f 0 := by ext x; rw [Subsingleton.elim ... |
import Mathlib.Analysis.NormedSpace.PiLp
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import analysis.matrix from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped NNReal Matrix
namespace Matrix
variable {R l m n α β : Type*} [Fintype l] [Fintyp... | Mathlib/Analysis/Matrix.lean | 98 | 99 | theorem norm_lt_iff {r : ℝ} (hr : 0 < r) {A : Matrix m n α} : ‖A‖ < r ↔ ∀ i j, ‖A i j‖ < r := by |
simp_rw [norm_def, pi_norm_lt_iff hr]
|
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 | 229 | 237 | theorem hasFTaylorSeriesUpToOn_zero_iff :
HasFTaylorSeriesUpToOn 0 f p s ↔ ContinuousOn f s ∧ ∀ x ∈ s, (p x 0).uncurry0 = f x := by |
refine ⟨fun H => ⟨H.continuousOn, H.zero_eq⟩, fun H =>
⟨H.2, fun m hm => False.elim (not_le.2 hm bot_le), fun m hm ↦ ?_⟩⟩
obtain rfl : m = 0 := mod_cast hm.antisymm (zero_le _)
have : EqOn (p · 0) ((continuousMultilinearCurryFin0 𝕜 E F).symm ∘ f) s := fun x hx ↦
(continuousMultilinearCurryFin0 𝕜 E F)... |
import Mathlib.CategoryTheory.Limits.Shapes.Images
import Mathlib.CategoryTheory.Limits.Constructions.EpiMono
#align_import category_theory.limits.preserves.shapes.images from "leanprover-community/mathlib"@"fc78e3c190c72a109699385da6be2725e88df841"
noncomputable section
namespace CategoryTheory
namespace Prese... | Mathlib/CategoryTheory/Limits/Preserves/Shapes/Images.lean | 57 | 58 | theorem hom_comp_map_image_ι {X Y : A} (f : X ⟶ Y) :
(iso L f).hom ≫ L.map (image.ι f) = image.ι (L.map f) := by | rw [iso_hom, image.lift_fac]
|
import Mathlib.Analysis.Complex.CauchyIntegral
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Analysis.NormedSpace.Completion
#align_import analysis.complex.liouville from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open TopologicalSpace Metric Set Filter Asymptotics ... | Mathlib/Analysis/Complex/Liouville.lean | 71 | 84 | theorem norm_deriv_le_of_forall_mem_sphere_norm_le {c : ℂ} {R C : ℝ} {f : ℂ → F} (hR : 0 < R)
(hd : DiffContOnCl ℂ f (ball c R)) (hC : ∀ z ∈ sphere c R, ‖f z‖ ≤ C) :
‖deriv f c‖ ≤ C / R := by |
set e : F →L[ℂ] F̂ := UniformSpace.Completion.toComplL
have : HasDerivAt (e ∘ f) (e (deriv f c)) c :=
e.hasFDerivAt.comp_hasDerivAt c
(hd.differentiableAt isOpen_ball <| mem_ball_self hR).hasDerivAt
calc
‖deriv f c‖ = ‖deriv (e ∘ f) c‖ := by
rw [this.deriv]
exact (UniformSpace.Completio... |
import Mathlib.Control.Bitraversable.Basic
#align_import control.bitraversable.lemmas from "leanprover-community/mathlib"@"58581d0fe523063f5651df0619be2bf65012a94a"
universe u
variable {t : Type u → Type u → Type u} [Bitraversable t]
variable {β : Type u}
namespace Bitraversable
open Functor LawfulApplicative
... | Mathlib/Control/Bitraversable/Lemmas.lean | 95 | 99 | theorem comp_tsnd {α β₀ β₁ β₂} (g : β₀ → F β₁) (g' : β₁ → G β₂) (x : t α β₀) :
Comp.mk (tsnd g' <$> tsnd g x) = tsnd (Comp.mk ∘ map g' ∘ g) x := by |
rw [← comp_bitraverse]
simp only [Function.comp, map_pure]
rfl
|
import Mathlib.Analysis.Calculus.FDeriv.Bilinear
#align_import analysis.calculus.fderiv.mul from "leanprover-community/mathlib"@"d608fc5d4e69d4cc21885913fb573a88b0deb521"
open scoped Classical
open Filter Asymptotics ContinuousLinearMap Set Metric Topology NNReal ENNReal
noncomputable section
section
variable ... | Mathlib/Analysis/Calculus/FDeriv/Mul.lean | 391 | 395 | theorem HasFDerivWithinAt.mul (hc : HasFDerivWithinAt c c' s x) (hd : HasFDerivWithinAt d d' s x) :
HasFDerivWithinAt (fun y => c y * d y) (c x • d' + d x • c') s x := by |
convert hc.mul' hd
ext z
apply mul_comm
|
import Mathlib.Algebra.EuclideanDomain.Defs
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Regular
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Ring.Basic
#align_import algebra.euclidean_domain.basic from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d0... | Mathlib/Algebra/EuclideanDomain/Basic.lean | 96 | 101 | theorem mul_div_assoc (x : R) {y z : R} (h : z ∣ y) : x * y / z = x * (y / z) := by |
by_cases hz : z = 0
· subst hz
rw [div_zero, div_zero, mul_zero]
rcases h with ⟨p, rfl⟩
rw [mul_div_cancel_left₀ _ hz, mul_left_comm, mul_div_cancel_left₀ _ hz]
|
import Mathlib.Analysis.Convex.Topology
import Mathlib.LinearAlgebra.Dimension.DivisionRing
import Mathlib.Topology.Algebra.Module.Cardinality
open Convex Set Metric
section TopologicalVectorSpace
variable {E : Type*} [AddCommGroup E] [Module ℝ E]
[TopologicalSpace E] [ContinuousAdd E] [ContinuousSMul ℝ E]
| Mathlib/Analysis/NormedSpace/Connected.lean | 34 | 103 | theorem Set.Countable.isPathConnected_compl_of_one_lt_rank
(h : 1 < Module.rank ℝ E) {s : Set E} (hs : s.Countable) :
IsPathConnected sᶜ := by |
have : Nontrivial E := (rank_pos_iff_nontrivial (R := ℝ)).1 (zero_lt_one.trans h)
-- the set `sᶜ` is dense, therefore nonempty. Pick `a ∈ sᶜ`. We have to show that any
-- `b ∈ sᶜ` can be joined to `a`.
obtain ⟨a, ha⟩ : sᶜ.Nonempty := (hs.dense_compl ℝ).nonempty
refine ⟨a, ha, ?_⟩
intro b hb
rcases eq_or_... |
import Mathlib.Order.PartialSups
#align_import order.disjointed from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
variable {α β : Type*}
section GeneralizedBooleanAlgebra
variable [GeneralizedBooleanAlgebra α]
def disjointed (f : ℕ → α) : ℕ → α
| 0 => f 0
| n + 1 => f (n + 1) ... | Mathlib/Order/Disjointed.lean | 123 | 136 | theorem disjointed_unique {f d : ℕ → α} (hdisj : Pairwise (Disjoint on d))
(hsups : partialSups d = partialSups f) : d = disjointed f := by |
ext n
cases' n with n
· rw [← partialSups_zero d, hsups, partialSups_zero, disjointed_zero]
suffices h : d n.succ = partialSups d n.succ \ partialSups d n by
rw [h, hsups, partialSups_succ, disjointed_succ, sup_sdiff, sdiff_self, bot_sup_eq]
rw [partialSups_succ, sup_sdiff, sdiff_self, bot_sup_eq, eq_com... |
import Mathlib.MeasureTheory.Covering.Differentiation
import Mathlib.MeasureTheory.Covering.VitaliFamily
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.MeasureTheory.Measure.Regular
import Mathlib.SetTheory.Ordinal.Arithmetic
import Mathlib.Topology.MetricSpace.Basic
import Mathlib.Data.Set.Pairwise.Lat... | Mathlib/MeasureTheory/Covering/Besicovitch.lean | 278 | 281 | theorem monotone_iUnionUpTo : Monotone p.iUnionUpTo := by |
intro i j hij
simp only [iUnionUpTo]
exact iUnion_mono' fun r => ⟨⟨r, r.2.trans_le hij⟩, Subset.rfl⟩
|
import Mathlib.Order.Filter.Bases
import Mathlib.Order.ConditionallyCompleteLattice.Basic
#align_import order.filter.lift from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
open Set Classical Filter Function
namespace Filter
variable {α β γ : Type*} {ι : Sort*}
section lift
protect... | Mathlib/Order/Filter/Lift.lean | 117 | 118 | theorem comap_lift_eq {m : γ → β} : comap m (f.lift g) = f.lift (comap m ∘ g) := by |
simp only [Filter.lift, comap_iInf]; rfl
|
import Mathlib.ModelTheory.Satisfiability
#align_import model_theory.types from "leanprover-community/mathlib"@"98bd247d933fb581ff37244a5998bd33d81dd46d"
set_option linter.uppercaseLean3 false
universe u v w w'
open Cardinal Set
open scoped Classical
open Cardinal FirstOrder
namespace FirstOrder
namespace La... | Mathlib/ModelTheory/Types.lean | 147 | 151 | theorem nonempty_iff : Nonempty (T.CompleteType α) ↔ T.IsSatisfiable := by |
rw [← isSatisfiable_onTheory_iff (lhomWithConstants_injective L α)]
rw [nonempty_iff_univ_nonempty, nonempty_iff_ne_empty, Ne, not_iff_comm,
← union_empty ((L.lhomWithConstants α).onTheory T), ← setOf_subset_eq_empty_iff]
simp
|
import Mathlib.Data.Finsupp.Encodable
import Mathlib.LinearAlgebra.Pi
import Mathlib.LinearAlgebra.Span
import Mathlib.Data.Set.Countable
#align_import linear_algebra.finsupp from "leanprover-community/mathlib"@"9d684a893c52e1d6692a504a118bfccbae04feeb"
noncomputable section
open Set LinearMap Submodule
namespa... | Mathlib/LinearAlgebra/Finsupp.lean | 260 | 263 | theorem iSup_lsingle_range : ⨆ a, LinearMap.range (lsingle a : M →ₗ[R] α →₀ M) = ⊤ := by |
refine eq_top_iff.2 <| SetLike.le_def.2 fun f _ => ?_
rw [← sum_single f]
exact sum_mem fun a _ => Submodule.mem_iSup_of_mem a ⟨_, rfl⟩
|
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 | 624 | 653 | theorem ae_restrict_of_ae_restrict_inter_Ioo {μ : Measure ℝ} [NoAtoms μ] {s : Set ℝ} {p : ℝ → Prop}
(h : ∀ a b, a ∈ s → b ∈ s → a < b → ∀ᵐ x ∂μ.restrict (s ∩ Ioo a b), p x) :
∀ᵐ x ∂μ.restrict s, p x := by |
/- By second-countability, we cover `s` by countably many intervals `(a, b)` (except maybe for
two endpoints, which don't matter since `μ` does not have any atom). -/
let T : s × s → Set ℝ := fun p => Ioo p.1 p.2
let u := ⋃ i : ↥s × ↥s, T i
have hfinite : (s \ u).Finite := s.finite_diff_iUnion_Ioo'
obtai... |
import Mathlib.Data.Fintype.Quotient
import Mathlib.ModelTheory.Semantics
#align_import model_theory.quotients from "leanprover-community/mathlib"@"d78597269638367c3863d40d45108f52207e03cf"
namespace FirstOrder
namespace Language
variable (L : Language) {M : Type*}
open FirstOrder
open Structure
class Prest... | Mathlib/ModelTheory/Quotients.lean | 57 | 62 | theorem funMap_quotient_mk' {n : ℕ} (f : L.Functions n) (x : Fin n → M) :
(funMap f fun i => (⟦x i⟧ : Quotient s)) = ⟦@funMap _ _ ps.toStructure _ f x⟧ := by |
change
Quotient.map (@funMap L M ps.toStructure n f) Prestructure.fun_equiv (Quotient.finChoice _) =
_
rw [Quotient.finChoice_eq, Quotient.map_mk]
|
import Mathlib.Order.SuccPred.LinearLocallyFinite
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.optional_sampling from "leanprover-community/mathlib"@"ba074af83b6cf54c3104e59402b39410ddbd6dca"
open scoped MeasureTheory ENNReal
open TopologicalSpace
namespace MeasureTheory
nam... | Mathlib/Probability/Martingale/OptionalSampling.lean | 157 | 183 | theorem condexp_stoppedValue_stopping_time_ae_eq_restrict_le (h : Martingale f ℱ μ)
(hτ : IsStoppingTime ℱ τ) (hσ : IsStoppingTime ℱ σ) [SigmaFinite (μ.trim hσ.measurableSpace_le)]
(hτ_le : ∀ x, τ x ≤ n) :
μ[stoppedValue f τ|hσ.measurableSpace] =ᵐ[μ.restrict {x : Ω | τ x ≤ σ x}] stoppedValue f τ := by |
rw [ae_eq_restrict_iff_indicator_ae_eq
(hτ.measurableSpace_le _ (hτ.measurableSet_le_stopping_time hσ))]
refine (condexp_indicator (integrable_stoppedValue ι hτ h.integrable hτ_le)
(hτ.measurableSet_stopping_time_le hσ)).symm.trans ?_
have h_int :
Integrable ({ω : Ω | τ ω ≤ σ ω}.indicator (stoppedV... |
import Mathlib.Algebra.ContinuedFractions.Basic
import Mathlib.Algebra.GroupWithZero.Basic
#align_import algebra.continued_fractions.translations from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b348ce40ad"
namespace GeneralizedContinuedFraction
section General
variable {α : Type*} {g : Gen... | Mathlib/Algebra/ContinuedFractions/Translations.lean | 62 | 63 | theorem part_denom_eq_s_b {gp : Pair α} (s_nth_eq : g.s.get? n = some gp) :
g.partialDenominators.get? n = some gp.b := by | simp [partialDenominators, s_nth_eq]
|
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.Int.ModEq
import Mathlib.Data.Set.Pointwise.Basic
import Mathlib.Dynamics.PeriodicPts
import Mathlib.GroupTheory.Index
import Mathlib.Order.Interval.Finset.Nat
import Mat... | Mathlib/GroupTheory/OrderOfElement.lean | 48 | 49 | theorem isPeriodicPt_mul_iff_pow_eq_one (x : G) : IsPeriodicPt (x * ·) n 1 ↔ x ^ n = 1 := by |
rw [IsPeriodicPt, IsFixedPt, mul_left_iterate]; beta_reduce; rw [mul_one]
|
import Mathlib.Mathport.Rename
import Mathlib.Tactic.Lemma
import Mathlib.Tactic.TypeStar
#align_import data.option.defs from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
namespace Option
#align option.lift_or_get Option.liftOrGet
protected def traverse.{u, v}
{F : Type u → Type... | Mathlib/Data/Option/Defs.lean | 61 | 61 | theorem mem_some_iff {α : Type*} {a b : α} : a ∈ some b ↔ b = a := by | simp
|
import Mathlib.Algebra.MvPolynomial.Basic
import Mathlib.Data.Finset.PiAntidiagonal
import Mathlib.LinearAlgebra.StdBasis
import Mathlib.Tactic.Linarith
#align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60"
noncomputable section
open Finset (... | Mathlib/RingTheory/MvPowerSeries/Basic.lean | 134 | 140 | theorem coeff_monomial [DecidableEq σ] (m n : σ →₀ ℕ) (a : R) :
coeff R m (monomial R n a) = if m = n then a else 0 := by |
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [coeff, monomial_def, LinearMap.proj_apply (i := m)]
dsimp only
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [LinearMap.stdBasis_apply, Function.update_apply, Pi.zero_apply]
|
import Mathlib.Probability.ProbabilityMassFunction.Constructions
import Mathlib.Tactic.FinCases
namespace PMF
open ENNReal
noncomputable
def binomial (p : ℝ≥0∞) (h : p ≤ 1) (n : ℕ) : PMF (Fin (n + 1)) :=
.ofFintype (fun i => p^(i : ℕ) * (1-p)^((Fin.last n - i) : ℕ) * (n.choose i : ℕ)) (by
convert (add_pow ... | Mathlib/Probability/ProbabilityMassFunction/Binomial.lean | 53 | 55 | theorem binomial_one_eq_bernoulli (p : ℝ≥0∞) (h : p ≤ 1) :
binomial p h 1 = (bernoulli p h).map (cond · 1 0) := by |
ext i; fin_cases i <;> simp [tsum_bool, binomial_apply]
|
import Mathlib.Init.Order.Defs
#align_import init.algebra.functions from "leanprover-community/lean"@"c2bcdbcbe741ed37c361a30d38e179182b989f76"
universe u
section
open Decidable
variable {α : Type u} [LinearOrder α]
theorem min_def (a b : α) : min a b = if a ≤ b then a else b := by
rw [LinearOrder.min_def a]... | Mathlib/Init/Order/LinearOrder.lean | 33 | 37 | theorem min_le_left (a b : α) : min a b ≤ a := by |
-- Porting note: no `min_tac` tactic
if h : a ≤ b
then simp [min_def, if_pos h, le_refl]
else simp [min_def, if_neg h]; exact le_of_not_le h
|
import Mathlib.SetTheory.Ordinal.Arithmetic
import Mathlib.Tactic.TFAE
import Mathlib.Topology.Order.Monotone
#align_import set_theory.ordinal.topology from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
noncomputable section
universe u v
open Cardinal Order Topology
namespace Ordina... | Mathlib/SetTheory/Ordinal/Topology.lean | 60 | 61 | theorem nhds_left'_eq_nhds_ne (a : Ordinal) : 𝓝[<] a = 𝓝[≠] a := by |
rw [← nhds_left'_sup_nhds_right', nhds_right', sup_bot_eq]
|
import Mathlib.Algebra.Module.Zlattice.Basic
import Mathlib.NumberTheory.NumberField.Embeddings
import Mathlib.NumberTheory.NumberField.FractionalIdeal
#align_import number_theory.number_field.canonical_embedding from "leanprover-community/mathlib"@"60da01b41bbe4206f05d34fd70c8dd7498717a30"
variable (K : Type*) [F... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean | 93 | 105 | theorem integerLattice.inter_ball_finite [NumberField K] (r : ℝ) :
((integerLattice K : Set ((K →+* ℂ) → ℂ)) ∩ Metric.closedBall 0 r).Finite := by |
obtain hr | _ := lt_or_le r 0
· simp [Metric.closedBall_eq_empty.2 hr]
· have heq : ∀ x, canonicalEmbedding K x ∈ Metric.closedBall 0 r ↔
∀ φ : K →+* ℂ, ‖φ x‖ ≤ r := by
intro x; rw [← norm_le_iff, mem_closedBall_zero_iff]
convert (Embeddings.finite_of_norm_le K ℂ r).image (canonicalEmbedding K)... |
import Mathlib.Data.Vector.Basic
#align_import data.vector.mem from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
namespace Vector
variable {α β : Type*} {n : ℕ} (a a' : α)
@[simp]
theorem get_mem (i : Fin n) (v : Vector α n) : v.get i ∈ v.toList := by
rw [get_eq_get]
exact List.... | Mathlib/Data/Vector/Mem.lean | 70 | 73 | theorem mem_of_mem_tail (v : Vector α n) (ha : a ∈ v.tail.toList) : a ∈ v.toList := by |
induction' n with n _
· exact False.elim (Vector.not_mem_zero a v.tail ha)
· exact (mem_succ_iff a v).2 (Or.inr ha)
|
import Mathlib.Control.Bitraversable.Basic
#align_import control.bitraversable.lemmas from "leanprover-community/mathlib"@"58581d0fe523063f5651df0619be2bf65012a94a"
universe u
variable {t : Type u → Type u → Type u} [Bitraversable t]
variable {β : Type u}
namespace Bitraversable
open Functor LawfulApplicative
... | Mathlib/Control/Bitraversable/Lemmas.lean | 87 | 91 | theorem tsnd_tfst {α₀ α₁ β₀ β₁} (f : α₀ → F α₁) (f' : β₀ → G β₁) (x : t α₀ β₀) :
Comp.mk (tsnd f' <$> tfst f x)
= bitraverse (Comp.mk ∘ map pure ∘ f) (Comp.mk ∘ pure ∘ f') x := by |
rw [← comp_bitraverse]
simp only [Function.comp, map_pure]
|
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Dynamics.BirkhoffSum.Average
open Function Set Filter
open scoped Topology ENNReal Uniformity
section
variable {α E : Type*}
theorem Function.IsFixedPt.tendsto_birkhoffAverage
(R : Type*) [DivisionSemiring R] [CharZero R]
[AddCommMonoid E] [Topological... | Mathlib/Dynamics/BirkhoffSum/NormedSpace.lean | 64 | 67 | theorem dist_birkhoffAverage_apply_birkhoffAverage (f : α → α) (g : α → E) (n : ℕ) (x : α) :
dist (birkhoffAverage 𝕜 f g n (f x)) (birkhoffAverage 𝕜 f g n x) =
dist (g (f^[n] x)) (g x) / n := by |
simp [dist_birkhoffAverage_birkhoffAverage, dist_birkhoffSum_apply_birkhoffSum]
|
import Mathlib.Algebra.Homology.Homotopy
import Mathlib.Algebra.Homology.Linear
import Mathlib.CategoryTheory.MorphismProperty.IsInvertedBy
import Mathlib.CategoryTheory.Quotient.Linear
import Mathlib.CategoryTheory.Quotient.Preadditive
#align_import algebra.homology.homotopy_category from "leanprover-community/mathl... | Mathlib/Algebra/Homology/HomotopyCategory.lean | 138 | 139 | theorem quotient_map_out_comp_out {C D E : HomotopyCategory V c} (f : C ⟶ D) (g : D ⟶ E) :
(quotient V c).map (Quot.out f ≫ Quot.out g) = f ≫ g := by | simp
|
import Mathlib.Data.Fin.Tuple.Basic
import Mathlib.Data.List.Join
#align_import data.list.of_fn from "leanprover-community/mathlib"@"bf27744463e9620ca4e4ebe951fe83530ae6949b"
universe u
variable {α : Type u}
open Nat
namespace List
#noalign list.length_of_fn_aux
@[simp]
theorem length_ofFn_go {n} (f : Fin n ... | Mathlib/Data/List/OfFn.lean | 44 | 45 | theorem length_ofFn {n} (f : Fin n → α) : length (ofFn f) = n := by |
simp [ofFn, length_ofFn_go]
|
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 | 94 | 95 | theorem Ioo_eq_empty_iff [DenselyOrdered α] : Ioo a b = ∅ ↔ ¬a < b := by |
rw [← coe_eq_empty, coe_Ioo, Set.Ioo_eq_empty_iff]
|
import Mathlib.Data.Vector.Basic
import Mathlib.Data.Vector.Snoc
set_option autoImplicit true
namespace Vector
section Fold
section Comm
variable (xs ys : Vector α n)
| Mathlib/Data/Vector/MapLemmas.lean | 369 | 371 | theorem map₂_comm (f : α → α → β) (comm : ∀ a₁ a₂, f a₁ a₂ = f a₂ a₁) :
map₂ f xs ys = map₂ f ys xs := by |
induction xs, ys using Vector.inductionOn₂ <;> simp_all
|
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 Preorder
variable [Preorder α] {a b c... | Mathlib/Order/Interval/Set/Disjoint.lean | 92 | 93 | theorem iUnion_Ioc_right (a : α) : ⋃ b, Ioc a b = Ioi a := by |
simp only [← Ioi_inter_Iic, ← inter_iUnion, iUnion_Iic, inter_univ]
|
import Mathlib.Topology.Order
#align_import topology.maps from "leanprover-community/mathlib"@"d91e7f7a7f1c7e9f0e18fdb6bde4f652004c735d"
open Set Filter Function
open TopologicalSpace Topology Filter
variable {X : Type*} {Y : Type*} {Z : Type*} {ι : Type*} {f : X → Y} {g : Y → Z}
section Inducing
variable [To... | Mathlib/Topology/Maps.lean | 69 | 72 | theorem Inducing.of_comp_iff (hg : Inducing g) :
Inducing (g ∘ f) ↔ Inducing f := by |
refine ⟨fun h ↦ ?_, hg.comp⟩
rw [inducing_iff, hg.induced, induced_compose, h.induced]
|
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.Orientation
import Mathlib.Data.Complex.Orientation
import Mathlib.Tactic.LinearCombination
#align_import analysis.inner_product_space.two_dim from "leanprover-community/mathlib"@"cd8fafa2fac98e1a67097e8a91ad9901cfde48af"
non... | Mathlib/Analysis/InnerProductSpace/TwoDim.lean | 116 | 121 | theorem areaForm_swap (x y : E) : ω x y = -ω y x := by |
simp only [areaForm_to_volumeForm]
convert o.volumeForm.map_swap ![y, x] (_ : (0 : Fin 2) ≠ 1)
· ext i
fin_cases i <;> rfl
· norm_num
|
import Mathlib.Analysis.SpecialFunctions.JapaneseBracket
import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.MeasureTheory.Group.Integral
import Mathlib.MeasureTheory.Integral.IntegralEqImproper
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.special_functions.improper_inte... | Mathlib/Analysis/SpecialFunctions/ImproperIntegrals.lean | 62 | 73 | theorem integrableOn_Ioi_rpow_of_lt {a : ℝ} (ha : a < -1) {c : ℝ} (hc : 0 < c) :
IntegrableOn (fun t : ℝ => t ^ a) (Ioi c) := by |
have hd : ∀ x ∈ Ici c, HasDerivAt (fun t => t ^ (a + 1) / (a + 1)) (x ^ a) x := by
intro x hx
-- Porting note: helped `convert` with explicit arguments
convert (hasDerivAt_rpow_const (p := a + 1) (Or.inl (hc.trans_le hx).ne')).div_const _ using 1
field_simp [show a + 1 ≠ 0 from ne_of_lt (by linarith)... |
import Mathlib.Analysis.NormedSpace.Star.Spectrum
import Mathlib.Analysis.Normed.Group.Quotient
import Mathlib.Analysis.NormedSpace.Algebra
import Mathlib.Topology.ContinuousFunction.Units
import Mathlib.Topology.ContinuousFunction.Compact
import Mathlib.Topology.Algebra.Algebra
import Mathlib.Topology.ContinuousFunct... | Mathlib/Analysis/NormedSpace/Star/GelfandDuality.lean | 99 | 105 | theorem WeakDual.CharacterSpace.exists_apply_eq_zero {a : A} (ha : ¬IsUnit a) :
∃ f : characterSpace ℂ A, f a = 0 := by |
obtain ⟨M, hM, haM⟩ := (span {a}).exists_le_maximal (span_singleton_ne_top ha)
exact
⟨M.toCharacterSpace,
M.toCharacterSpace_apply_eq_zero_of_mem
(haM (mem_span_singleton.mpr ⟨1, (mul_one a).symm⟩))⟩
|
import Mathlib.Data.Matroid.Restrict
variable {α : Type*} {M : Matroid α} {E B I X R J : Set α}
namespace Matroid
open Set
section EmptyOn
def emptyOn (α : Type*) : Matroid α where
E := ∅
Base := (· = ∅)
Indep := (· = ∅)
indep_iff' := by simp [subset_empty_iff]
exists_base := ⟨∅, rfl⟩
base_exchange... | Mathlib/Data/Matroid/Constructions.lean | 71 | 73 | theorem eq_emptyOn [IsEmpty α] (M : Matroid α) : M = emptyOn α := by |
rw [← ground_eq_empty_iff]
exact M.E.eq_empty_of_isEmpty
|
import Mathlib.CategoryTheory.Limits.Shapes.WideEqualizers
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.Terminal
#align_import category_theory.limits.constructions.weakly_initial from "leanprover-community/mathlib"@"239d882c4fb58361ee8b3b39fb2091320edef10a"
univ... | Mathlib/CategoryTheory/Limits/Constructions/WeaklyInitial.lean | 46 | 64 | theorem hasInitial_of_weakly_initial_and_hasWideEqualizers [HasWideEqualizers.{v} C] {T : C}
(hT : ∀ X, Nonempty (T ⟶ X)) : HasInitial C := by |
let endos := T ⟶ T
let i := wideEqualizer.ι (id : endos → endos)
haveI : Nonempty endos := ⟨𝟙 _⟩
have : ∀ X : C, Unique (wideEqualizer (id : endos → endos) ⟶ X) := by
intro X
refine ⟨⟨i ≫ Classical.choice (hT X)⟩, fun a => ?_⟩
let E := equalizer a (i ≫ Classical.choice (hT _))
let e : E ⟶ wide... |
import Mathlib.Data.Set.Defs
import Mathlib.Order.Heyting.Basic
import Mathlib.Order.RelClasses
import Mathlib.Order.Hom.Basic
import Mathlib.Lean.Thunk
set_option autoImplicit true
class EstimatorData (a : Thunk α) (ε : Type*) where
bound : ε → α
improve : ε → Option ε
class Estimator [Preorder α] (a... | Mathlib/Order/Estimator.lean | 126 | 142 | theorem Estimator.improveUntilAux_spec (a : Thunk α) (p : α → Bool)
[Estimator a ε] [WellFoundedGT (range (bound a : ε → α))] (e : ε) (r : Bool) :
match Estimator.improveUntilAux a p e r with
| .error _ => ¬ p a.get
| .ok e' => p (bound a e') := by |
rw [Estimator.improveUntilAux]
by_cases h : p (bound a e)
· simp only [h]; exact h
· simp only [h]
match improve a e, improve_spec e with
| none, eq =>
simp only [Bool.not_eq_true]
rw [eq] at h
exact Bool.bool_eq_false h
| some e', _ =>
exact Estimator.improveUntilAux_spec a... |
import Mathlib.Data.Nat.Prime
#align_import data.int.nat_prime from "leanprover-community/mathlib"@"422e70f7ce183d2900c586a8cda8381e788a0c62"
open Nat
namespace Int
theorem not_prime_of_int_mul {a b : ℤ} {c : ℕ} (ha : a.natAbs ≠ 1) (hb : b.natAbs ≠ 1)
(hc : a * b = (c : ℤ)) : ¬Nat.Prime c :=
not_prime_mul... | Mathlib/Data/Int/NatPrime.lean | 24 | 33 | theorem succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul {p : ℕ} (p_prime : Nat.Prime p) {m n : ℤ}
{k l : ℕ} (hpm : ↑(p ^ k) ∣ m) (hpn : ↑(p ^ l) ∣ n) (hpmn : ↑(p ^ (k + l + 1)) ∣ m * n) :
↑(p ^ (k + 1)) ∣ m ∨ ↑(p ^ (l + 1)) ∣ n :=
have hpm' : p ^ k ∣ m.natAbs := Int.natCast_dvd_natCast.1 <| Int.dvd_natAbs.2 hpm
ha... |
rw [← Int.natAbs_mul]; apply Int.natCast_dvd_natCast.1 <| Int.dvd_natAbs.2 hpmn
let hsd := Nat.succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul p_prime hpm' hpn' hpmn'
hsd.elim (fun hsd1 => Or.inl (by apply Int.dvd_natAbs.1; apply Int.natCast_dvd_natCast.2 hsd1))
fun hsd2 => Or.inr (by apply Int.dvd_natAbs.1; appl... |
import Mathlib.CategoryTheory.Bicategory.Functor.Oplax
#align_import category_theory.bicategory.natural_transformation from "leanprover-community/mathlib"@"4ff75f5b8502275a4c2eb2d2f02bdf84d7fb8993"
namespace CategoryTheory
open Category Bicategory
open scoped Bicategory
universe w₁ w₂ v₁ v₂ u₁ u₂
variable {B :... | Mathlib/CategoryTheory/Bicategory/NaturalTransformation.lean | 104 | 107 | theorem whiskerLeft_naturality_naturality (f : a' ⟶ G.obj a) {g h : a ⟶ b} (β : g ⟶ h) :
f ◁ G.map₂ β ▷ θ.app b ≫ f ◁ θ.naturality h =
f ◁ θ.naturality g ≫ f ◁ θ.app a ◁ H.map₂ β := by |
simp_rw [← whiskerLeft_comp, naturality_naturality]
|
import Mathlib.MeasureTheory.Integral.IntervalIntegral
import Mathlib.Order.Filter.IndicatorFunction
open MeasureTheory
section DominatedConvergenceTheorem
open Set Filter TopologicalSpace ENNReal
open scoped Topology
namespace MeasureTheory
variable {α E G: Type*}
[NormedAddCommGroup E] [NormedSpace ℝ E] [C... | Mathlib/MeasureTheory/Integral/DominatedConvergence.lean | 66 | 75 | theorem tendsto_integral_filter_of_dominated_convergence {ι} {l : Filter ι} [l.IsCountablyGenerated]
{F : ι → α → G} {f : α → G} (bound : α → ℝ) (hF_meas : ∀ᶠ n in l, AEStronglyMeasurable (F n) μ)
(h_bound : ∀ᶠ n in l, ∀ᵐ a ∂μ, ‖F n a‖ ≤ bound a) (bound_integrable : Integrable bound μ)
(h_lim : ∀ᵐ a ∂μ, Ten... |
by_cases hG : CompleteSpace G
· simp only [integral, hG, L1.integral]
exact tendsto_setToFun_filter_of_dominated_convergence (dominatedFinMeasAdditive_weightedSMul μ)
bound hF_meas h_bound bound_integrable h_lim
· simp [integral, hG, tendsto_const_nhds]
|
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.Algebra.GCDMonoid.Nat
#align_import ring_theory.int.basic from "leanprover-community/mathlib"@"e655e4ea5c6d02854696f97494997ba4c31be802"
namespace Int
theorem gcd_eq_one_iff_coprime {a b : ℤ} : Int.gcd a b ... | Mathlib/RingTheory/Int/Basic.lean | 54 | 56 | theorem gcd_ne_one_iff_gcd_mul_right_ne_one {a : ℤ} {m n : ℕ} :
a.gcd (m * n) ≠ 1 ↔ a.gcd m ≠ 1 ∨ a.gcd n ≠ 1 := by |
simp only [gcd_eq_one_iff_coprime, ← not_and_or, not_iff_not, IsCoprime.mul_right_iff]
|
import Mathlib.Order.Filter.Basic
#align_import order.filter.prod from "leanprover-community/mathlib"@"d6fad0e5bf2d6f48da9175d25c3dc5706b3834ce"
open Set
open Filter
namespace Filter
variable {α β γ δ : Type*} {ι : Sort*}
section Prod
variable {s : Set α} {t : Set β} {f : Filter α} {g : Filter β}
protected ... | Mathlib/Order/Filter/Prod.lean | 131 | 135 | theorem eventually_prod_iff {p : α × β → Prop} :
(∀ᶠ x in f ×ˢ g, p x) ↔
∃ pa : α → Prop, (∀ᶠ x in f, pa x) ∧ ∃ pb : β → Prop, (∀ᶠ y in g, pb y) ∧
∀ {x}, pa x → ∀ {y}, pb y → p (x, y) := by |
simpa only [Set.prod_subset_iff] using @mem_prod_iff α β p f g
|
import Mathlib.Algebra.Module.Zlattice.Basic
import Mathlib.NumberTheory.NumberField.Embeddings
import Mathlib.NumberTheory.NumberField.FractionalIdeal
#align_import number_theory.number_field.canonical_embedding from "leanprover-community/mathlib"@"60da01b41bbe4206f05d34fd70c8dd7498717a30"
variable (K : Type*) [F... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean | 259 | 262 | theorem normAtPlace_nonneg (w : InfinitePlace K) (x : E K) :
0 ≤ normAtPlace w x := by |
rw [normAtPlace, MonoidWithZeroHom.coe_mk, ZeroHom.coe_mk]
split_ifs <;> exact norm_nonneg _
|
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.LinearAlgebra.Matrix.ToLin
#align_import linear_algebra.matrix.charpoly.linear_map from "leanprover-community/mathlib"@"62c0a4ef1441edb463095ea02a06e87f3dfe135c"
variable {ι : Type*} [Fintype ι]
variable {M : Type*} [AddCommGroup M] (R : Type*) [Co... | Mathlib/LinearAlgebra/Matrix/Charpoly/LinearMap.lean | 141 | 144 | theorem Matrix.Represents.smul {A : Matrix ι ι R} {f : Module.End R M} (h : A.Represents b f)
(r : R) : (r • A).Represents b (r • f) := by |
delta Matrix.Represents at h ⊢
rw [_root_.map_smul, _root_.map_smul, h]
|
import Mathlib.Topology.MetricSpace.Basic
#align_import topology.metric_space.infsep from "leanprover-community/mathlib"@"5316314b553dcf8c6716541851517c1a9715e22b"
variable {α β : Type*}
namespace Set
section Einfsep
open ENNReal
open Function
noncomputable def einfsep [EDist α] (s : Set α) : ℝ≥0∞ :=
⨅ (x... | Mathlib/Topology/MetricSpace/Infsep.lean | 340 | 341 | theorem infsep_pos : 0 < s.infsep ↔ 0 < s.einfsep ∧ s.einfsep < ∞ := by |
simp_rw [infsep, ENNReal.toReal_pos_iff]
|
import Mathlib.Algebra.Group.Semiconj.Defs
import Mathlib.Init.Algebra.Classes
#align_import algebra.group.commute from "leanprover-community/mathlib"@"05101c3df9d9cfe9430edc205860c79b6d660102"
assert_not_exists MonoidWithZero
assert_not_exists DenselyOrdered
variable {G M S : Type*}
@[to_additive "Two elements... | Mathlib/Algebra/Group/Commute/Defs.lean | 262 | 263 | theorem mul_inv_cancel_assoc (h : Commute a b) : a * (b * a⁻¹) = b := by |
rw [← mul_assoc, h.mul_inv_cancel]
|
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