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 |
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import Mathlib.NumberTheory.SmoothNumbers
import Mathlib.Analysis.PSeries
open Set Nat
open scoped Topology
-- This needs `Mathlib.Analysis.RCLike.Basic`, so we put it here
-- instead of in `Mathlib.NumberTheory.SmoothNumbers`.
lemma Nat.roughNumbersUpTo_card_le' (N k : ℕ) :
(roughNumbersUpTo N k).card ≤
... | Mathlib/NumberTheory/SumPrimeReciprocals.lean | 86 | 97 | theorem Nat.Primes.summable_rpow {r : ℝ} :
Summable (fun p : Nat.Primes ↦ (p : ℝ) ^ r) ↔ r < -1 := by |
by_cases h : r < -1
· -- case `r < -1`
simp only [h, iff_true]
exact (Real.summable_nat_rpow.mpr h).subtype _
· -- case `-1 ≤ r`
simp only [h, iff_false]
refine fun H ↦ Nat.Primes.not_summable_one_div <| H.of_nonneg_of_le (fun _ ↦ by positivity) ?_
intro p
rw [one_div, ← Real.rpow_neg_one... |
import Mathlib.Analysis.NormedSpace.Exponential
#align_import analysis.normed_space.star.exponential from "leanprover-community/mathlib"@"1e3201306d4d9eb1fd54c60d7c4510ad5126f6f9"
open NormedSpace -- For `NormedSpace.exp`.
section Star
variable {A : Type*} [NormedRing A] [NormedAlgebra ℂ A] [StarRing A] [Continu... | Mathlib/Analysis/NormedSpace/Star/Exponential.lean | 42 | 48 | theorem Commute.expUnitary_add {a b : selfAdjoint A} (h : Commute (a : A) (b : A)) :
expUnitary (a + b) = expUnitary a * expUnitary b := by |
ext
have hcomm : Commute (I • (a : A)) (I • (b : A)) := by
unfold Commute SemiconjBy
simp only [h.eq, Algebra.smul_mul_assoc, Algebra.mul_smul_comm]
simpa only [expUnitary_coe, AddSubgroup.coe_add, smul_add] using exp_add_of_commute hcomm
|
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 | 57 | 58 | theorem forall_mem_map {f : α → β} {o : Option α} {p : β → Prop} :
(∀ y ∈ o.map f, p y) ↔ ∀ x ∈ o, p (f x) := by | simp
|
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.RingTheory.PowerSeries.Basic
#align_import ring_theory.power_series.well_known from "leanprover-community/mathlib"@"8199f6717c150a7fe91c4534175f4cf99725978f"
namespace PowerS... | Mathlib/RingTheory/PowerSeries/WellKnown.lean | 174 | 176 | theorem constantCoeff_exp : constantCoeff A (exp A) = 1 := by |
rw [← coeff_zero_eq_constantCoeff_apply, coeff_exp]
simp
|
import Mathlib.Analysis.Normed.Field.Basic
import Mathlib.Analysis.Normed.Group.InfiniteSum
import Mathlib.Topology.Algebra.InfiniteSum.Real
#align_import analysis.normed.field.infinite_sum from "leanprover-community/mathlib"@"008205aa645b3f194c1da47025c5f110c8406eab"
variable {R : Type*} {ι : Type*} {ι' : Type*}... | Mathlib/Analysis/Normed/Field/InfiniteSum.lean | 73 | 83 | theorem summable_norm_sum_mul_antidiagonal_of_summable_norm {f g : ℕ → R}
(hf : Summable fun x => ‖f x‖) (hg : Summable fun x => ‖g x‖) :
Summable fun n => ‖∑ kl ∈ antidiagonal n, f kl.1 * g kl.2‖ := by |
have :=
summable_sum_mul_antidiagonal_of_summable_mul
(Summable.mul_of_nonneg hf hg (fun _ => norm_nonneg _) fun _ => norm_nonneg _)
refine this.of_nonneg_of_le (fun _ => norm_nonneg _) (fun n ↦ ?_)
calc
‖∑ kl ∈ antidiagonal n, f kl.1 * g kl.2‖ ≤ ∑ kl ∈ antidiagonal n, ‖f kl.1 * g kl.2‖ :=
no... |
import Batteries.Classes.Order
namespace Batteries.PairingHeapImp
inductive Heap (α : Type u) where
| nil : Heap α
| node (a : α) (child sibling : Heap α) : Heap α
deriving Repr
def Heap.size : Heap α → Nat
| .nil => 0
| .node _ c s => c.size + 1 + s.size
def Heap.singleton (a : α) : Heap α := .... | .lake/packages/batteries/Batteries/Data/PairingHeap.lean | 138 | 140 | theorem Heap.size_deleteMin {s : Heap α} (h : s.NoSibling) (eq : s.deleteMin le = some (a, s')) :
s.size = s'.size + 1 := by |
cases h with cases eq | node a c => rw [size_combine, size, size]
|
import Mathlib.Topology.Algebra.Constructions
import Mathlib.Topology.Bases
import Mathlib.Topology.UniformSpace.Basic
#align_import topology.uniform_space.cauchy from "leanprover-community/mathlib"@"22131150f88a2d125713ffa0f4693e3355b1eb49"
universe u v
open scoped Classical
open Filter TopologicalSpace Set Uni... | Mathlib/Topology/UniformSpace/Cauchy.lean | 63 | 67 | theorem Cauchy.ultrafilter_of {l : Filter α} (h : Cauchy l) :
Cauchy (@Ultrafilter.of _ l h.1 : Filter α) := by |
haveI := h.1
have := Ultrafilter.of_le l
exact ⟨Ultrafilter.neBot _, (Filter.prod_mono this this).trans h.2⟩
|
import Mathlib.Algebra.Group.Units.Equiv
import Mathlib.Logic.Function.Conjugate
import Mathlib.Order.Bounds.OrderIso
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.OrdContinuous
import Mathlib.Order.RelIso.Group
#align_import order.semiconj_Sup from "leanprover-community/mathlib"@"422e7... | Mathlib/Order/SemiconjSup.lean | 101 | 107 | theorem semiconj_of_isLUB [PartialOrder α] [Group G] (f₁ f₂ : G →* α ≃o α) {h : α → α}
(H : ∀ x, IsLUB (range fun g' => (f₁ g')⁻¹ (f₂ g' x)) (h x)) (g : G) :
Function.Semiconj h (f₂ g) (f₁ g) := by |
refine fun y => (H _).unique ?_
have := (f₁ g).leftOrdContinuous (H y)
rw [← range_comp, ← (Equiv.mulRight g).surjective.range_comp _] at this
simpa [(· ∘ ·)] using this
|
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Algebra.Module.Submodule.Basic
#align_import algebra.direct_sum.decomposition from "leanprover-community/mathlib"@"4e861f25ba5ceef42ba0712d8ffeb32f38ad6441"
variable {ι R M σ : Type*}
open DirectSum
namespace DirectSum
section AddCommMonoid
variable [Deci... | Mathlib/Algebra/DirectSum/Decomposition.lean | 136 | 137 | theorem decompose_of_mem_same {x : M} {i : ι} (hx : x ∈ ℳ i) : (decompose ℳ x i : M) = x := by |
rw [decompose_of_mem _ hx, DirectSum.of_eq_same, Subtype.coe_mk]
|
import Mathlib.CategoryTheory.Limits.Types
import Mathlib.CategoryTheory.Functor.Currying
import Mathlib.CategoryTheory.Limits.FunctorCategory
#align_import category_theory.limits.colimit_limit from "leanprover-community/mathlib"@"59382264386afdbaf1727e617f5fdda511992eb9"
universe v₁ v₂ v u₁ u₂ u
open CategoryTh... | Mathlib/CategoryTheory/Limits/ColimitLimit.lean | 89 | 93 | theorem ι_colimitLimitToLimitColimit_π (j) (k) :
colimit.ι _ k ≫ colimitLimitToLimitColimit F ≫ limit.π _ j =
limit.π ((curry.obj (Prod.swap K J ⋙ F)).obj k) j ≫ colimit.ι ((curry.obj F).obj j) k := by |
dsimp [colimitLimitToLimitColimit]
simp
|
import Mathlib.Order.Filter.Basic
import Mathlib.Data.Set.Countable
#align_import order.filter.countable_Inter from "leanprover-community/mathlib"@"b9e46fe101fc897fb2e7edaf0bf1f09ea49eb81a"
open Set Filter
open Filter
variable {ι : Sort*} {α β : Type*}
class CountableInterFilter (l : Filter α) : Prop where
... | Mathlib/Order/Filter/CountableInter.lean | 116 | 121 | theorem EventuallyLE.countable_bInter {ι : Type*} {S : Set ι} (hS : S.Countable)
{s t : ∀ i ∈ S, Set α} (h : ∀ i hi, s i hi ≤ᶠ[l] t i hi) :
⋂ i ∈ S, s i ‹_› ≤ᶠ[l] ⋂ i ∈ S, t i ‹_› := by |
simp only [biInter_eq_iInter]
haveI := hS.toEncodable
exact EventuallyLE.countable_iInter fun i => h i i.2
|
import Mathlib.Data.Fintype.Card
import Mathlib.GroupTheory.Perm.Basic
import Mathlib.Tactic.Ring
#align_import data.fintype.perm from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
open Function
open Nat
universe u v
variable {α β γ : Type*}
open Finset Function List Equiv Equiv.Per... | Mathlib/Data/Fintype/Perm.lean | 102 | 128 | theorem nodup_permsOfList : ∀ {l : List α}, l.Nodup → (permsOfList l).Nodup
| [], _ => by simp [permsOfList]
| a :: l, hl => by
have hl' : l.Nodup := hl.of_cons
have hln' : (permsOfList l).Nodup := nodup_permsOfList hl'
have hmeml : ∀ {f : Perm α}, f ∈ permsOfList l → f a = a := fun {f} hf =>
not_... |
rw [← hf.2, mul_apply, hmeml hf.1, swap_apply_left]
have hiy : x a = List.get l j := by
rw [← hg.2, mul_apply, hmeml hg.1, swap_apply_left]
have hieqj : i = j := nodup_iff_injective_get.1 hl' (hix.symm.trans hiy)
exact absurd hieqj (_root_.ne_of_lt hij)
· intros f hf₁ hf₂
le... |
import Mathlib.Analysis.MeanInequalities
import Mathlib.Data.Fintype.Order
import Mathlib.LinearAlgebra.Matrix.Basis
import Mathlib.Analysis.NormedSpace.WithLp
#align_import analysis.normed_space.pi_Lp from "leanprover-community/mathlib"@"9d013ad8430ddddd350cff5c3db830278ded3c79"
set_option linter.uppercaseLean3 f... | Mathlib/Analysis/NormedSpace/PiLp.lean | 185 | 187 | theorem edist_eq_iSup (f g : PiLp ∞ β) : edist f g = ⨆ i, edist (f i) (g i) := by |
dsimp [edist]
exact if_neg ENNReal.top_ne_zero
|
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 Sigma
variable {α : ι → Type*} {β : Type*} (s s₁ s₂ : Finset ι) (... | Mathlib/Data/Finset/Sigma.lean | 91 | 94 | theorem sigma_eq_biUnion [DecidableEq (Σi, α i)] (s : Finset ι) (t : ∀ i, Finset (α i)) :
s.sigma t = s.biUnion fun i => (t i).map <| Embedding.sigmaMk i := by |
ext ⟨x, y⟩
simp [and_left_comm]
|
import Mathlib.Algebra.GCDMonoid.Finset
import Mathlib.Algebra.Polynomial.CancelLeads
import Mathlib.Algebra.Polynomial.EraseLead
import Mathlib.Algebra.Polynomial.FieldDivision
#align_import ring_theory.polynomial.content from "leanprover-community/mathlib"@"7a030ab8eb5d99f05a891dccc49c5b5b90c947d3"
namespace Po... | Mathlib/RingTheory/Polynomial/Content.lean | 184 | 195 | theorem content_eq_gcd_range_of_lt (p : R[X]) (n : ℕ) (h : p.natDegree < n) :
p.content = (Finset.range n).gcd p.coeff := by |
apply dvd_antisymm_of_normalize_eq normalize_content Finset.normalize_gcd
· rw [Finset.dvd_gcd_iff]
intro i _
apply content_dvd_coeff _
· apply Finset.gcd_mono
intro i
simp only [Nat.lt_succ_iff, mem_support_iff, Ne, Finset.mem_range]
contrapose!
intro h1
apply coeff_eq_zero_of_natDeg... |
import Batteries.Data.Rat.Basic
import Batteries.Tactic.SeqFocus
namespace Rat
theorem ext : {p q : Rat} → p.num = q.num → p.den = q.den → p = q
| ⟨_,_,_,_⟩, ⟨_,_,_,_⟩, rfl, rfl => rfl
@[simp] theorem mk_den_one {r : Int} :
⟨r, 1, Nat.one_ne_zero, (Nat.coprime_one_right _)⟩ = (r : Rat) := rfl
@[simp] theor... | .lake/packages/batteries/Batteries/Data/Rat/Lemmas.lean | 36 | 42 | theorem normalize_eq {num den} (den_nz) : normalize num den den_nz =
{ num := num / num.natAbs.gcd den
den := den / num.natAbs.gcd den
den_nz := normalize.den_nz den_nz rfl
reduced := normalize.reduced' den_nz rfl } := by |
simp only [normalize, maybeNormalize_eq,
Int.div_eq_ediv_of_dvd (Int.ofNat_dvd_left.2 (Nat.gcd_dvd_left ..))]
|
import Mathlib.Data.Fin.Fin2
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.Common
#align_import data.typevec from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc9524185ae994553ebf4"
universe u v w
@[pp_with_univ]
def TypeVec (n : ℕ) :=
Fin2 n → Type*
#align typevec TypeVec
instance {n} : Inh... | Mathlib/Data/TypeVec.lean | 60 | 62 | theorem Arrow.ext {α β : TypeVec n} (f g : α ⟹ β) :
(∀ i, f i = g i) → f = g := by |
intro h; funext i; apply h
|
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open ENNReal MeasureTheory MeasureTheory.Measure Metric Set Filter TopologicalSpace Function
open scoped Topology ENNReal Convex
variable... | Mathlib/MeasureTheory/Integral/Average.lean | 158 | 162 | theorem laverage_lt_top (hf : ∫⁻ x, f x ∂μ ≠ ∞) : ⨍⁻ x, f x ∂μ < ∞ := by |
obtain rfl | hμ := eq_or_ne μ 0
· simp
· rw [laverage_eq]
exact div_lt_top hf (measure_univ_ne_zero.2 hμ)
|
import Mathlib.Data.Real.Cardinality
import Mathlib.Topology.Separation
import Mathlib.Topology.TietzeExtension
open Set Function Cardinal Topology TopologicalSpace
universe u
variable {X : Type u} [TopologicalSpace X] [SeparableSpace X]
| Mathlib/Topology/Separation/NotNormal.lean | 26 | 53 | theorem IsClosed.mk_lt_continuum [NormalSpace X] {s : Set X} (hs : IsClosed s)
[DiscreteTopology s] : #s < 𝔠 := by |
-- Proof by contradiction: assume `𝔠 ≤ #s`
by_contra! h
-- Choose a countable dense set `t : Set X`
rcases exists_countable_dense X with ⟨t, htc, htd⟩
haveI := htc.to_subtype
-- To obtain a contradiction, we will prove `2 ^ 𝔠 ≤ 𝔠`.
refine (Cardinal.cantor 𝔠).not_le ?_
calc
-- Any function `s → ... |
import Mathlib.RingTheory.WittVector.Identities
#align_import ring_theory.witt_vector.domain from "leanprover-community/mathlib"@"b1d911acd60ab198808e853292106ee352b648ea"
noncomputable section
open scoped Classical
namespace WittVector
open Function
variable {p : ℕ} {R : Type*}
local notation "𝕎" => WittVe... | Mathlib/RingTheory/WittVector/Domain.lean | 79 | 85 | theorem eq_iterate_verschiebung {x : 𝕎 R} {n : ℕ} (h : ∀ i < n, x.coeff i = 0) :
x = verschiebung^[n] (x.shift n) := by |
induction' n with k ih
· cases x; simp [shift]
· dsimp; rw [verschiebung_shift]
· exact ih fun i hi => h _ (hi.trans (Nat.lt_succ_self _))
· exact h
|
import Mathlib.Geometry.Manifold.MFDeriv.UniqueDifferential
import Mathlib.Geometry.Manifold.ContMDiffMap
#align_import geometry.manifold.cont_mdiff_mfderiv from "leanprover-community/mathlib"@"e473c3198bb41f68560cab68a0529c854b618833"
open Set Function Filter ChartedSpace SmoothManifoldWithCorners Bundle
open sc... | Mathlib/Geometry/Manifold/ContMDiffMFDeriv.lean | 287 | 339 | theorem ContMDiffOn.contMDiffOn_tangentMapWithin_aux {f : H → H'} {s : Set H}
(hf : ContMDiffOn I I' n f s) (hmn : m + 1 ≤ n) (hs : UniqueMDiffOn I s) :
ContMDiffOn I.tangent I'.tangent m (tangentMapWithin I I' f s)
(π E (TangentSpace I) ⁻¹' s) := by |
have m_le_n : m ≤ n := (le_add_right le_rfl).trans hmn
have one_le_n : 1 ≤ n := (le_add_left le_rfl).trans hmn
have U' : UniqueDiffOn 𝕜 (range I ∩ I.symm ⁻¹' s) := fun y hy ↦ by
simpa only [UniqueMDiffOn, UniqueMDiffWithinAt, hy.1, inter_comm, mfld_simps]
using hs (I.symm y) hy.2
rw [contMDiffOn_iff... |
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.MeasureTheory.Measure.Haar.Quotient
import Mathlib.MeasureTheory.Constructions.Polish
import Mathlib.MeasureTheory.Integral.IntervalIntegral
import Mathlib.Topology.Algebra.Order.Floor
#align_import measure_theory.integral.periodic from "leanprover-c... | Mathlib/MeasureTheory/Integral/Periodic.lean | 39 | 46 | theorem isAddFundamentalDomain_Ioc {T : ℝ} (hT : 0 < T) (t : ℝ)
(μ : Measure ℝ := by | volume_tac) :
IsAddFundamentalDomain (AddSubgroup.zmultiples T) (Ioc t (t + T)) μ := by
refine IsAddFundamentalDomain.mk' measurableSet_Ioc.nullMeasurableSet fun x => ?_
have : Bijective (codRestrict (fun n : ℤ => n • T) (AddSubgroup.zmultiples T) _) :=
(Equiv.ofInjective (fun n : ℤ => n • T) (zsmul_strict... |
import Mathlib.Topology.Algebra.GroupWithZero
import Mathlib.Topology.Order.OrderClosed
#align_import topology.algebra.with_zero_topology from "leanprover-community/mathlib"@"3e0c4d76b6ebe9dfafb67d16f7286d2731ed6064"
open Topology Filter TopologicalSpace Filter Set Function
namespace WithZeroTopology
variable {α... | Mathlib/Topology/Algebra/WithZeroTopology.lean | 78 | 79 | theorem tendsto_zero : Tendsto f l (𝓝 (0 : Γ₀)) ↔ ∀ (γ₀) (_ : γ₀ ≠ 0), ∀ᶠ x in l, f x < γ₀ := by |
simp [nhds_zero]
|
import Mathlib.Order.Interval.Set.Image
import Mathlib.Order.CompleteLatticeIntervals
import Mathlib.Topology.Order.DenselyOrdered
import Mathlib.Topology.Order.Monotone
#align_import topology.algebra.order.intermediate_value from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
open Filt... | Mathlib/Topology/Order/IntermediateValue.lean | 105 | 112 | theorem IsPreconnected.intermediate_value₂_eventually₁ {s : Set X} (hs : IsPreconnected s) {a : X}
{l : Filter X} (ha : a ∈ s) [NeBot l] (hl : l ≤ 𝓟 s) {f g : X → α} (hf : ContinuousOn f s)
(hg : ContinuousOn g s) (ha' : f a ≤ g a) (he : g ≤ᶠ[l] f) : ∃ x ∈ s, f x = g x := by |
rw [continuousOn_iff_continuous_restrict] at hf hg
obtain ⟨b, h⟩ :=
@intermediate_value_univ₂_eventually₁ _ _ _ _ _ _ (Subtype.preconnectedSpace hs) ⟨a, ha⟩ _
(comap_coe_neBot_of_le_principal hl) _ _ hf hg ha' (he.comap _)
exact ⟨b, b.prop, h⟩
|
import Mathlib.Algebra.Order.Floor
import Mathlib.Data.Rat.Cast.Order
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Ring
#align_import data.rat.floor from "leanprover-community/mathlib"@"e1bccd6e40ae78370f01659715d3c948716e3b7e"
open Int
namespace Rat
variable {α : Type*} [LinearOrderedField α] [FloorRi... | Mathlib/Data/Rat/Floor.lean | 56 | 66 | theorem floor_int_div_nat_eq_div {n : ℤ} {d : ℕ} : ⌊(↑n : ℚ) / (↑d : ℚ)⌋ = n / (↑d : ℤ) := by |
rw [Rat.floor_def]
obtain rfl | hd := @eq_zero_or_pos _ _ d
· simp
set q := (n : ℚ) / d with q_eq
obtain ⟨c, n_eq_c_mul_num, d_eq_c_mul_denom⟩ : ∃ c, n = c * q.num ∧ (d : ℤ) = c * q.den := by
rw [q_eq]
exact mod_cast @Rat.exists_eq_mul_div_num_and_eq_mul_div_den n d (mod_cast hd.ne')
rw [n_eq_c_mul... |
import Mathlib.RingTheory.WittVector.Basic
import Mathlib.RingTheory.WittVector.IsPoly
#align_import ring_theory.witt_vector.verschiebung from "leanprover-community/mathlib"@"32b08ef840dd25ca2e47e035c5da03ce16d2dc3c"
namespace WittVector
open MvPolynomial
variable {p : ℕ} {R S : Type*} [hp : Fact p.Prime] [Comm... | Mathlib/RingTheory/WittVector/Verschiebung.lean | 65 | 71 | theorem ghostComponent_verschiebungFun (x : 𝕎 R) (n : ℕ) :
ghostComponent (n + 1) (verschiebungFun x) = p * ghostComponent n x := by |
simp only [ghostComponent_apply, aeval_wittPolynomial]
rw [Finset.sum_range_succ', verschiebungFun_coeff, if_pos rfl,
zero_pow (pow_ne_zero _ hp.1.ne_zero), mul_zero, add_zero, Finset.mul_sum, Finset.sum_congr rfl]
rintro i -
simp only [pow_succ', verschiebungFun_coeff_succ, Nat.succ_sub_succ_eq_sub, mul_a... |
import Mathlib.NumberTheory.Zsqrtd.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Real.Archimedean
#align_import number_theory.zsqrtd.gaussian_int from "leanprover-community/mathlib"@"5b2fe80501ff327b9109fb09b7cc8c325cd0d7d9"
open Zsqrtd Complex
open sc... | Mathlib/NumberTheory/Zsqrtd/GaussianInt.lean | 162 | 163 | theorem intCast_complex_norm (x : ℤ[i]) : (x.norm : ℂ) = Complex.normSq (x : ℂ) := by |
cases x; rw [Zsqrtd.norm, normSq]; simp
|
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.LinearAlgebra.AffineSpace.Basic
import Mathlib.LinearAlgebra.BilinearMap
import Mathlib.LinearAlgebra.Pi
import Mathlib.LinearAlgebra.Prod
#align_import linear_algebra.affine_space.affine_map from "leanprover-community/mathlib"@"bd1fc183335ea95a9519a1630bcf901... | Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean | 135 | 136 | theorem linearMap_vsub (f : P1 →ᵃ[k] P2) (p1 p2 : P1) : f.linear (p1 -ᵥ p2) = f p1 -ᵥ f p2 := by |
conv_rhs => rw [← vsub_vadd p1 p2, map_vadd, vadd_vsub]
|
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open ENNReal MeasureTheory MeasureTheory.Measure Metric Set Filter TopologicalSpace Function
open scoped Topology ENNReal Convex
variable... | Mathlib/MeasureTheory/Integral/Average.lean | 108 | 108 | theorem laverage_zero : ⨍⁻ _x, (0 : ℝ≥0∞) ∂μ = 0 := by | rw [laverage, lintegral_zero]
|
import Mathlib.Algebra.Polynomial.Eval
#align_import data.polynomial.degree.lemmas from "leanprover-community/mathlib"@"728baa2f54e6062c5879a3e397ac6bac323e506f"
noncomputable section
open Polynomial
open Finsupp Finset
namespace Polynomial
universe u v w
variable {R : Type u} {S : Type v} {ι : Type w} {a b ... | Mathlib/Algebra/Polynomial/Degree/Lemmas.lean | 111 | 117 | theorem natDegree_C_mul_eq_of_mul_eq_one {ai : R} (au : ai * a = 1) :
(C a * p).natDegree = p.natDegree :=
le_antisymm (natDegree_C_mul_le a p)
(calc
p.natDegree = (1 * p).natDegree := by | nth_rw 1 [← one_mul p]
_ = (C ai * (C a * p)).natDegree := by rw [← C_1, ← au, RingHom.map_mul, ← mul_assoc]
_ ≤ (C a * p).natDegree := natDegree_C_mul_le ai (C a * p))
|
import Mathlib.Analysis.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calculus.FDeriv.Comp
#align_import analysis.calculus.fderiv.prod from "leanprover-community/mathlib"@"e354e865255654389cc46e6032160238df2e0f40"
open Filter Asymptotics ContinuousLinearMap Set Metric
open scoped Classical
open Topology NNReal ... | Mathlib/Analysis/Calculus/FDeriv/Prod.lean | 451 | 454 | theorem hasFDerivAt_apply (i : ι) (f : ∀ i, F' i) :
HasFDerivAt (𝕜:=𝕜) (fun f : ∀ i, F' i => f i) (proj i) f := by |
apply HasStrictFDerivAt.hasFDerivAt
apply hasStrictFDerivAt_apply
|
import Mathlib.Algebra.MvPolynomial.Supported
import Mathlib.RingTheory.WittVector.Truncated
#align_import ring_theory.witt_vector.mul_coeff from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
namespace WittVector
variable (p : ℕ) [hp : Fact p.Prime]
variable {k ... | Mathlib/RingTheory/WittVector/MulCoeff.lean | 120 | 135 | theorem mul_polyOfInterest_aux1 (n : ℕ) :
∑ i ∈ range (n + 1), (p : 𝕄) ^ i * wittMul p i ^ p ^ (n - i) = wittPolyProd p n := by |
simp only [wittPolyProd]
convert wittStructureInt_prop p (X (0 : Fin 2) * X 1) n using 1
· simp only [wittPolynomial, wittMul]
rw [AlgHom.map_sum]
congr 1 with i
congr 1
have hsupp : (Finsupp.single i (p ^ (n - i))).support = {i} := by
rw [Finsupp.support_eq_singleton]
simp only [and_... |
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#align_import algebra.order.group.defs from "leanprover-community/mathlib"@"b599f4e4e5cf1fbcb4194503671d3d9e569c1fce"
open Function
universe u
variable {α : Type u}
class OrderedAddCommGroup (α : Ty... | Mathlib/Algebra/Order/Group/Defs.lean | 318 | 319 | theorem inv_mul_lt_one_iff_lt : a * b⁻¹ < 1 ↔ a < b := by |
rw [← mul_lt_mul_iff_right b, inv_mul_cancel_right, one_mul]
|
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Ext
local macro:max "local_hAdd[" type:term ", " inst:term "]" : term =>
`(term| (letI := $inst; HAdd.hAdd : $type → $type → $type))
local macro:max "local_hMul[" type:term ", " inst:term "]" : term =>
`(term| (letI := $inst; HMul.hMul : $type → $typ... | Mathlib/Algebra/Ring/Ext.lean | 497 | 499 | theorem toSemiring_injective :
Function.Injective (@toSemiring R) := by |
rintro ⟨⟩ ⟨⟩ _; congr
|
import Mathlib.Analysis.Normed.Group.Basic
#align_import information_theory.hamming from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
section HammingDistNorm
open Finset Function
variable {α ι : Type*} {β : ι → Type*} [Fintype ι] [∀ i, DecidableEq (β i)]
variable {γ : ι → Type*} [∀ ... | Mathlib/InformationTheory/Hamming.lean | 61 | 67 | theorem hammingDist_triangle (x y z : ∀ i, β i) :
hammingDist x z ≤ hammingDist x y + hammingDist y z := by |
classical
unfold hammingDist
refine le_trans (card_mono ?_) (card_union_le _ _)
rw [← filter_or]
exact monotone_filter_right _ fun i h ↦ (h.ne_or_ne _).imp_right Ne.symm
|
import Mathlib.Topology.MetricSpace.Antilipschitz
#align_import topology.metric_space.isometry from "leanprover-community/mathlib"@"b1859b6d4636fdbb78c5d5cefd24530653cfd3eb"
noncomputable section
universe u v w
variable {ι : Type*} {α : Type u} {β : Type v} {γ : Type w}
open Function Set
open scoped Topology ... | Mathlib/Topology/MetricSpace/Isometry.lean | 138 | 141 | theorem preimage_emetric_closedBall (h : Isometry f) (x : α) (r : ℝ≥0∞) :
f ⁻¹' EMetric.closedBall (f x) r = EMetric.closedBall x r := by |
ext y
simp [h.edist_eq]
|
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
import Mathlib.Topology.Algebra.Module.Basic
open Function
structure ContinuousAffineEquiv (k P₁ P₂ : Type*) {V₁ V₂ : Type*} [Ring k]
[AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [TopologicalSpace P₁]
[AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P... | Mathlib/LinearAlgebra/AffineSpace/ContinuousAffineEquiv.lean | 84 | 87 | theorem coe_injective : Function.Injective ((↑) : (P₁ ≃ᵃL[k] P₂) → P₁ ≃ᵃ[k] P₂) := by |
intro e e' H
cases e
congr
|
import Mathlib.AlgebraicTopology.SplitSimplicialObject
import Mathlib.AlgebraicTopology.DoldKan.Degeneracies
import Mathlib.AlgebraicTopology.DoldKan.FunctorN
#align_import algebraic_topology.dold_kan.split_simplicial_object from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190f0b504"
open Categ... | Mathlib/AlgebraicTopology/DoldKan/SplitSimplicialObject.lean | 73 | 85 | theorem σ_comp_πSummand_id_eq_zero {n : ℕ} (i : Fin (n + 1)) :
X.σ i ≫ s.πSummand (IndexSet.id (op [n + 1])) = 0 := by |
apply s.hom_ext'
intro A
dsimp only [SimplicialObject.σ]
rw [comp_zero, s.cofan_inj_epi_naturality_assoc A (SimplexCategory.σ i).op,
cofan_inj_πSummand_eq_zero]
rw [ne_comm]
change ¬(A.epiComp (SimplexCategory.σ i).op).EqId
rw [IndexSet.eqId_iff_len_eq]
have h := SimplexCategory.len_le_of_epi (infe... |
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
import Mathlib.CategoryTheory.ConcreteCategory.BundledHom
import Mathlib.CategoryTheory.Elementwise
#align_import analysis.normed.group.SemiNormedGroup from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11... | Mathlib/Analysis/Normed/Group/SemiNormedGroupCat.lean | 111 | 114 | theorem isZero_of_subsingleton (V : SemiNormedGroupCat) [Subsingleton V] : Limits.IsZero V := by |
refine ⟨fun X => ⟨⟨⟨0⟩, fun f => ?_⟩⟩, fun X => ⟨⟨⟨0⟩, fun f => ?_⟩⟩⟩
· ext x; have : x = 0 := Subsingleton.elim _ _; simp only [this, map_zero]
· ext; apply Subsingleton.elim
|
import Batteries.Classes.Order
namespace Batteries.PairingHeapImp
inductive Heap (α : Type u) where
| nil : Heap α
| node (a : α) (child sibling : Heap α) : Heap α
deriving Repr
def Heap.size : Heap α → Nat
| .nil => 0
| .node _ c s => c.size + 1 + s.size
def Heap.singleton (a : α) : Heap α := .... | .lake/packages/batteries/Batteries/Data/PairingHeap.lean | 119 | 121 | theorem Heap.size_merge_node (le) (a₁ : α) (c₁ s₁ : Heap α) (a₂ : α) (c₂ s₂ : Heap α) :
(merge le (.node a₁ c₁ s₁) (.node a₂ c₂ s₂)).size = c₁.size + c₂.size + 2 := by |
unfold merge; dsimp; split <;> simp_arith [size]
|
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.Data.ZMod.Algebra
#align_import ring_theory.polynomial.cyclotomic.expand from "leanprover-community/mathlib"@"0723536a0522d24fc2f159a096fb3304bef77472"
namespace Polynomial
@[simp]
| Mathlib/RingTheory/Polynomial/Cyclotomic/Expand.lean | 36 | 72 | theorem cyclotomic_expand_eq_cyclotomic_mul {p n : ℕ} (hp : Nat.Prime p) (hdiv : ¬p ∣ n)
(R : Type*) [CommRing R] :
expand R p (cyclotomic n R) = cyclotomic (n * p) R * cyclotomic n R := by |
rcases Nat.eq_zero_or_pos n with (rfl | hnpos)
· simp
haveI := NeZero.of_pos hnpos
suffices expand ℤ p (cyclotomic n ℤ) = cyclotomic (n * p) ℤ * cyclotomic n ℤ by
rw [← map_cyclotomic_int, ← map_expand, this, Polynomial.map_mul, map_cyclotomic_int,
map_cyclotomic]
refine eq_of_monic_of_dvd_of_natDe... |
import Mathlib.Analysis.Normed.Order.Basic
import Mathlib.Analysis.Asymptotics.Asymptotics
import Mathlib.Analysis.NormedSpace.Basic
#align_import analysis.asymptotics.specific_asymptotics from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Filter Asymptotics
open Topology
sectio... | Mathlib/Analysis/Asymptotics/SpecificAsymptotics.lean | 42 | 46 | theorem pow_div_pow_eventuallyEq_atTop {p q : ℕ} :
(fun x : 𝕜 => x ^ p / x ^ q) =ᶠ[atTop] fun x => x ^ ((p : ℤ) - q) := by |
apply (eventually_gt_atTop (0 : 𝕜)).mono fun x hx => _
intro x hx
simp [zpow_sub₀ hx.ne']
|
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Data.Rat.Cast.Order
import Mathlib.Order.Partition.Finpartition
import Mathlib.Tactic.GCongr
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.Positivity
import Mathlib.Tactic.Ring
#align_import combinatorics.simp... | Mathlib/Combinatorics/SimpleGraph/Density.lean | 159 | 160 | theorem edgeDensity_empty_right (s : Finset α) : edgeDensity r s ∅ = 0 := by |
rw [edgeDensity, Finset.card_empty, Nat.cast_zero, mul_zero, div_zero]
|
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 | 91 | 96 | theorem commutative_ring_iff_abelian_lie_ring {A : Type v} [Ring A] :
Std.Commutative (α := A) (· * ·) ↔ IsLieAbelian A := by |
have h₁ : Std.Commutative (α := A) (· * ·) ↔ ∀ a b : A, a * b = b * a :=
⟨fun h => h.1, fun h => ⟨h⟩⟩
have h₂ : IsLieAbelian A ↔ ∀ a b : A, ⁅a, b⁆ = 0 := ⟨fun h => h.1, fun h => ⟨h⟩⟩
simp only [h₁, h₂, LieRing.of_associative_ring_bracket, sub_eq_zero]
|
import Mathlib.MeasureTheory.PiSystem
import Mathlib.Order.OmegaCompletePartialOrder
import Mathlib.Topology.Constructions
import Mathlib.MeasureTheory.MeasurableSpace.Basic
open Set
namespace MeasureTheory
variable {ι : Type _} {α : ι → Type _}
section cylinder
def cylinder (s : Finset ι) (S : Set (∀ i : s, α... | Mathlib/MeasureTheory/Constructions/Cylinders.lean | 169 | 183 | theorem cylinder_eq_empty_iff [h_nonempty : Nonempty (∀ i, α i)] (s : Finset ι)
(S : Set (∀ i : s, α i)) :
cylinder s S = ∅ ↔ S = ∅ := by |
refine ⟨fun h ↦ ?_, fun h ↦ by (rw [h]; exact cylinder_empty _)⟩
by_contra hS
rw [← Ne, ← nonempty_iff_ne_empty] at hS
let f := hS.some
have hf : f ∈ S := hS.choose_spec
classical
let f' : ∀ i, α i := fun i ↦ if hi : i ∈ s then f ⟨i, hi⟩ else h_nonempty.some i
have hf' : f' ∈ cylinder s S := by
rw ... |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Reverse
import Mathlib.Algebra.Polynomial.Inductions
import Mathlib.RingTheory.Localization.Basic
#align_import data.polynomial.laurent from "leanprover-community/mathlib"@"831c494092374cfe9f50591ed0ac81a25efc5b86"
open Polynomial Func... | Mathlib/Algebra/Polynomial/Laurent.lean | 203 | 204 | theorem mul_T_assoc (f : R[T;T⁻¹]) (m n : ℤ) : f * T m * T n = f * T (m + n) := by |
simp [← T_add, mul_assoc]
|
import Mathlib.LinearAlgebra.FreeModule.PID
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.BilinearForm.DualLattice
import Mathlib.RingTheory.DedekindDomain.Basic
import Mathlib.RingTheory.Localization.Module
import Mathlib.RingTheory.Trace
#align_import ring_theory.dedekind_domain.... | Mathlib/RingTheory/DedekindDomain/IntegralClosure.lean | 106 | 112 | theorem integralClosure_le_span_dualBasis [IsSeparable K L] {ι : Type*} [Fintype ι] [DecidableEq ι]
(b : Basis ι K L) (hb_int : ∀ i, IsIntegral A (b i)) [IsIntegrallyClosed A] :
Subalgebra.toSubmodule (integralClosure A L) ≤
Submodule.span A (Set.range <| (traceForm K L).dualBasis (traceForm_nondegenerate K... |
refine le_trans ?_ (IsIntegralClosure.range_le_span_dualBasis (integralClosure A L) b hb_int)
intro x hx
exact ⟨⟨x, hx⟩, 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 | 144 | 144 | theorem right_mem_Icc : b ∈ Icc a b ↔ a ≤ b := by | simp only [mem_Icc, and_true_iff, le_rfl]
|
import Mathlib.LinearAlgebra.Dimension.DivisionRing
import Mathlib.LinearAlgebra.Dimension.FreeAndStrongRankCondition
noncomputable section
universe u v v' v''
variable {K : Type u} {V V₁ : Type v} {V' V'₁ : Type v'} {V'' : Type v''}
open Cardinal Basis Submodule Function Set
namespace LinearMap
section Ring
... | Mathlib/LinearAlgebra/Dimension/LinearMap.lean | 58 | 60 | theorem lift_rank_comp_le_right (g : V →ₗ[K] V') (f : V' →ₗ[K] V'') :
Cardinal.lift.{v'} (rank (f.comp g)) ≤ Cardinal.lift.{v''} (rank g) := by |
rw [rank, rank, LinearMap.range_comp]; exact lift_rank_map_le _ _
|
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open ENNReal MeasureTheory MeasureTheory.Measure Metric Set Filter TopologicalSpace Function
open scoped Topology ENNReal Convex
variable... | Mathlib/MeasureTheory/Integral/Average.lean | 112 | 112 | theorem laverage_zero_measure (f : α → ℝ≥0∞) : ⨍⁻ x, f x ∂(0 : Measure α) = 0 := by | simp [laverage]
|
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Analysis.NormedSpace.Real
#align_import analysis.special_functions.log.basic from "leanprover-community/mathlib"@"f23a09ce6d3f367220dc3cecad6b7eb69eb01690"
open Set Filter Function
open Topology
noncomputable ... | Mathlib/Analysis/SpecialFunctions/Log/Basic.lean | 114 | 115 | theorem sinh_log {x : ℝ} (hx : 0 < x) : sinh (log x) = (x - x⁻¹) / 2 := by |
rw [sinh_eq, exp_neg, exp_log hx]
|
import Mathlib.Analysis.SpecialFunctions.Integrals
#align_import data.real.pi.wallis from "leanprover-community/mathlib"@"980755c33b9168bc82f774f665eaa27878140fac"
open scoped Real Topology Nat
open Filter Finset intervalIntegral
namespace Real
namespace Wallis
set_option linter.uppercaseLean3 false
noncomp... | Mathlib/Data/Real/Pi/Wallis.lean | 62 | 75 | theorem W_eq_factorial_ratio (n : ℕ) :
W n = 2 ^ (4 * n) * n ! ^ 4 / ((2 * n)! ^ 2 * (2 * n + 1)) := by |
induction' n with n IH
· simp only [W, prod_range_zero, Nat.factorial_zero, mul_zero, pow_zero,
algebraMap.coe_one, one_pow, mul_one, algebraMap.coe_zero, zero_add, div_self, Ne,
one_ne_zero, not_false_iff]
norm_num
· unfold W at IH ⊢
rw [prod_range_succ, IH, _root_.div_mul_div_comm, _root_.d... |
import Mathlib.Data.ENNReal.Inv
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
section iInf
variable {ι : Sort*} {f g : ι → ℝ≥0∞}
variable {a b c d : ℝ≥0∞} {r p q : ℝ≥0}
theorem toNNReal_iInf (hf : ∀ i, f ... | Mathlib/Data/ENNReal/Real.lean | 548 | 553 | theorem toNNReal_sInf (s : Set ℝ≥0∞) (hs : ∀ r ∈ s, r ≠ ∞) :
(sInf s).toNNReal = sInf (ENNReal.toNNReal '' s) := by |
have hf : ∀ i, ((↑) : s → ℝ≥0∞) i ≠ ∞ := fun ⟨r, rs⟩ => hs r rs
-- Porting note: `← sInf_image'` had to be replaced by `← image_eq_range` as the lemmas are used
-- in a different order.
simpa only [← sInf_range, ← image_eq_range, Subtype.range_coe_subtype] using (toNNReal_iInf hf)
|
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.TryThis
import Mathlib.Util.AtomM
set_option autoImplicit true
namespace Mathlib.Tactic.Abel
open Lean Elab Meta Tactic Qq
initialize registerTraceClass `abel
initialize registerTraceClass `abel.detail
structure Context where
α : Expr
univ :... | Mathlib/Tactic/Abel.lean | 132 | 134 | theorem const_add_termg {α} [AddCommGroup α] (k n x a a') (h : k + a = a') :
k + @termg α _ n x a = termg n x a' := by |
simp [h.symm, termg, add_comm, add_assoc]
|
import Mathlib.Topology.LocalAtTarget
import Mathlib.AlgebraicGeometry.Morphisms.Basic
#align_import algebraic_geometry.morphisms.open_immersion from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
noncomputable section
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace... | Mathlib/AlgebraicGeometry/Morphisms/OpenImmersion.lean | 34 | 38 | theorem isOpenImmersion_iff_stalk {f : X ⟶ Y} : IsOpenImmersion f ↔
OpenEmbedding f.1.base ∧ ∀ x, IsIso (PresheafedSpace.stalkMap f.1 x) := by |
constructor
· intro h; exact ⟨h.1, inferInstance⟩
· rintro ⟨h₁, h₂⟩; exact IsOpenImmersion.of_stalk_iso f h₁
|
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.Prod
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Tactic.FinCases
import Mathlib.Tactic.LinearCombination
import Mathlib.Lean.Expr.ExtraRecognizers
import Mathlib.Data.Set.Subsingleton
#align_import lin... | Mathlib/LinearAlgebra/LinearIndependent.lean | 192 | 194 | theorem Fintype.not_linearIndependent_iff [Fintype ι] :
¬LinearIndependent R v ↔ ∃ g : ι → R, ∑ i, g i • v i = 0 ∧ ∃ i, g i ≠ 0 := by |
simpa using not_iff_not.2 Fintype.linearIndependent_iff
|
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import set_theory.cardinal.continuum from "leanprover-community/mathlib"@"e08a42b2dd544cf11eba72e5fc7bf199d4349925"
namespace Cardinal
universe u v
open Cardinal
def continuum : Cardinal.{u} :=
2 ^ ℵ₀
#align cardinal.continuum Cardinal.continuum
scoped notat... | Mathlib/SetTheory/Cardinal/Continuum.lean | 90 | 90 | theorem mk_set_nat : #(Set ℕ) = 𝔠 := by | simp
|
import Mathlib.MeasureTheory.Function.SimpleFuncDenseLp
#align_import measure_theory.integral.set_to_l1 from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open scoped Classical Topology NNReal ENNReal MeasureTheory Pointwise
open Set Filter TopologicalSpace ENNR... | Mathlib/MeasureTheory/Integral/SetToL1.lean | 122 | 127 | theorem of_smul_measure (c : ℝ≥0∞) (hc_ne_top : c ≠ ∞) (hT : FinMeasAdditive (c • μ) T) :
FinMeasAdditive μ T := by |
refine of_eq_top_imp_eq_top (fun s _ hμs => ?_) hT
rw [Measure.smul_apply, smul_eq_mul, ENNReal.mul_eq_top] at hμs
simp only [hc_ne_top, or_false_iff, Ne, false_and_iff] at hμs
exact hμs.2
|
namespace Nat
@[reducible] def Coprime (m n : Nat) : Prop := gcd m n = 1
instance (m n : Nat) : Decidable (Coprime m n) := inferInstanceAs (Decidable (_ = 1))
theorem coprime_iff_gcd_eq_one : Coprime m n ↔ gcd m n = 1 := .rfl
theorem Coprime.gcd_eq_one : Coprime m n → gcd m n = 1 := id
theorem Coprime.symm ... | .lake/packages/batteries/Batteries/Data/Nat/Gcd.lean | 65 | 75 | theorem exists_coprime (m n : Nat) :
∃ m' n', Coprime m' n' ∧ m = m' * gcd m n ∧ n = n' * gcd m n := by |
cases eq_zero_or_pos (gcd m n) with
| inl h0 =>
rw [gcd_eq_zero_iff] at h0
refine ⟨1, 1, gcd_one_left 1, ?_⟩
simp [h0]
| inr hpos =>
exact ⟨_, _, coprime_div_gcd_div_gcd hpos,
(Nat.div_mul_cancel (gcd_dvd_left m n)).symm,
(Nat.div_mul_cancel (gcd_dvd_right m n)).symm⟩
|
import Mathlib.Analysis.SpecialFunctions.ImproperIntegrals
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.MeasureTheory.Measure.Haar.NormedSpace
#align_import analysis.mellin_transform from "leanprover-community/mathlib"@"917c3c072e487b3cccdbfeff17e75b40e45f66cb"
open MeasureTheory Set Filter A... | Mathlib/Analysis/MellinTransform.lean | 304 | 312 | theorem isBigO_rpow_top_log_smul [NormedSpace ℝ E] {a b : ℝ} {f : ℝ → E} (hab : b < a)
(hf : f =O[atTop] (· ^ (-a))) :
(fun t : ℝ => log t • f t) =O[atTop] (· ^ (-b)) := by |
refine
((isLittleO_log_rpow_atTop (sub_pos.mpr hab)).isBigO.smul hf).congr'
(eventually_of_forall fun t => by rfl)
((eventually_gt_atTop 0).mp (eventually_of_forall fun t ht => ?_))
simp only
rw [smul_eq_mul, ← rpow_add ht, ← sub_eq_add_neg, sub_eq_add_neg a, add_sub_cancel_left]
|
import Mathlib.Probability.Notation
import Mathlib.Probability.Density
import Mathlib.Probability.ConditionalProbability
import Mathlib.Probability.ProbabilityMassFunction.Constructions
open scoped Classical MeasureTheory NNReal ENNReal
-- TODO: We can't `open ProbabilityTheory` without opening the `ProbabilityThe... | Mathlib/Probability/Distributions/Uniform.lean | 114 | 121 | theorem pdf_eq_zero_of_measure_eq_zero_or_top {X : Ω → E} {s : Set E}
(hu : IsUniform X s ℙ μ) (hμs : μ s = 0 ∨ μ s = ∞) : pdf X ℙ μ =ᵐ[μ] 0 := by |
rcases hμs with H|H
· simp only [IsUniform, ProbabilityTheory.cond, H, ENNReal.inv_zero, restrict_eq_zero.mpr H,
smul_zero] at hu
simp [pdf, hu]
· simp only [IsUniform, ProbabilityTheory.cond, H, ENNReal.inv_top, zero_smul] at hu
simp [pdf, hu]
|
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.Fin.Tuple.Reflection
#align_import data.matrix.reflection from "leanprover-community/mathlib"@"820b22968a2bc4a47ce5cf1d2f36a9ebe52510aa"
open Matrix
namespace Matrix
variable {l m n : ℕ} {α β : Type*}
def Forall : ∀ {m n}... | Mathlib/Data/Matrix/Reflection.lean | 159 | 162 | theorem mulᵣ_eq [Mul α] [AddCommMonoid α] (A : Matrix (Fin l) (Fin m) α)
(B : Matrix (Fin m) (Fin n) α) : mulᵣ A B = A * B := by |
simp [mulᵣ, Function.comp, Matrix.transpose]
rfl
|
import Mathlib.Order.Filter.Ultrafilter
import Mathlib.Order.Filter.Germ
#align_import order.filter.filter_product from "leanprover-community/mathlib"@"2738d2ca56cbc63be80c3bd48e9ed90ad94e947d"
universe u v
variable {α : Type u} {β : Type v} {φ : Ultrafilter α}
open scoped Classical
namespace Filter
local not... | Mathlib/Order/Filter/FilterProduct.lean | 82 | 84 | theorem lt_def [Preorder β] : ((· < ·) : β* → β* → Prop) = LiftRel (· < ·) := by |
ext ⟨f⟩ ⟨g⟩
exact coe_lt
|
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 | 63 | 63 | theorem map₂_none_right (f : α → β → γ) (a : Option α) : map₂ f a none = none := by | cases a <;> rfl
|
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 | 116 | 119 | theorem withDensity_sum {ι : Type*} {m : MeasurableSpace α} (μ : ι → Measure α) (f : α → ℝ≥0∞) :
(sum μ).withDensity f = sum fun n => (μ n).withDensity f := by |
ext1 s hs
simp_rw [sum_apply _ hs, withDensity_apply f hs, restrict_sum μ hs, lintegral_sum_measure]
|
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.UniformLimitsDeriv
import Mathlib.Topology.Algebra.InfiniteSum.Module
import Mathlib.Analysis.NormedSpace.FunctionSeries
#align_import analysis.calculus.series from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982... | Mathlib/Analysis/Calculus/SmoothSeries.lean | 60 | 66 | theorem summable_of_summable_hasDerivAt_of_isPreconnected (hu : Summable u) (ht : IsOpen t)
(h't : IsPreconnected t) (hg : ∀ n y, y ∈ t → HasDerivAt (g n) (g' n y) y)
(hg' : ∀ n y, y ∈ t → ‖g' n y‖ ≤ u n) (hy₀ : y₀ ∈ t) (hg0 : Summable (g · y₀))
(hy : y ∈ t) : Summable fun n => g n y := by |
simp_rw [hasDerivAt_iff_hasFDerivAt] at hg
refine summable_of_summable_hasFDerivAt_of_isPreconnected hu ht h't hg ?_ hy₀ hg0 hy
simpa? says simpa only [ContinuousLinearMap.norm_smulRight_apply, norm_one, one_mul]
|
import Mathlib.Data.Set.Image
#align_import order.directed from "leanprover-community/mathlib"@"ffde2d8a6e689149e44fd95fa862c23a57f8c780"
open Function
universe u v w
variable {α : Type u} {β : Type v} {ι : Sort w} (r r' s : α → α → Prop)
local infixl:50 " ≼ " => r
def Directed (f : ι → α) :=
∀ x y, ∃ z, ... | Mathlib/Order/Directed.lean | 58 | 60 | theorem directedOn_iff_directed {s} : @DirectedOn α r s ↔ Directed r (Subtype.val : s → α) := by |
simp only [DirectedOn, Directed, Subtype.exists, exists_and_left, exists_prop, Subtype.forall]
exact forall₂_congr fun x _ => by simp [And.comm, and_assoc]
|
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
#align_import linear_algebra.affine_space.midpoint from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
open AffineMap AffineEquiv
section
variable (R : Type*) {V V' P P' : Type*} [Ring R] [Invertible (2 : R)] [AddCommGroup V]
[Modu... | Mathlib/LinearAlgebra/AffineSpace/Midpoint.lean | 77 | 79 | theorem AffineEquiv.pointReflection_midpoint_right (x y : P) :
pointReflection R (midpoint R x y) y = x := by |
rw [midpoint_comm, AffineEquiv.pointReflection_midpoint_left]
|
import Mathlib.Tactic.NormNum.Inv
set_option autoImplicit true
open Lean Meta Qq
namespace Mathlib.Meta.NormNum
theorem isNat_eq_false [AddMonoidWithOne α] [CharZero α] : {a b : α} → {a' b' : ℕ} →
IsNat a a' → IsNat b b' → Nat.beq a' b' = false → ¬a = b
| _, _, _, _, ⟨rfl⟩, ⟨rfl⟩, h => by simp; exact Nat.n... | Mathlib/Tactic/NormNum/Eq.lean | 26 | 29 | theorem Rat.invOf_denom_swap [Ring α] (n₁ n₂ : ℤ) (a₁ a₂ : α)
[Invertible a₁] [Invertible a₂] : n₁ * ⅟a₁ = n₂ * ⅟a₂ ↔ n₁ * a₂ = n₂ * a₁ := by |
rw [mul_invOf_eq_iff_eq_mul_right, ← Int.commute_cast, mul_assoc,
← mul_left_eq_iff_eq_invOf_mul, Int.commute_cast]
|
import Mathlib.RingTheory.EisensteinCriterion
import Mathlib.RingTheory.Polynomial.ScaleRoots
#align_import ring_theory.polynomial.eisenstein.basic from "leanprover-community/mathlib"@"2032a878972d5672e7c27c957e7a6e297b044973"
universe u v w z
variable {R : Type u}
open Ideal Algebra Finset
open Polynomial
na... | Mathlib/RingTheory/Polynomial/Eisenstein/Basic.lean | 66 | 69 | theorem map {A : Type v} [CommRing A] (φ : R →+* A) : (f.map φ).IsWeaklyEisensteinAt (𝓟.map φ) := by |
refine (isWeaklyEisensteinAt_iff _ _).2 fun hn => ?_
rw [coeff_map]
exact mem_map_of_mem _ (hf.mem (lt_of_lt_of_le hn (natDegree_map_le _ _)))
|
import Mathlib.Order.UpperLower.Basic
import Mathlib.Data.Finset.Preimage
#align_import combinatorics.young.young_diagram from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
open Function
@[ext]
structure YoungDiagram where
cells : Finset (ℕ × ℕ)
isLowerSet : IsLowerSet (cel... | Mathlib/Combinatorics/Young/YoungDiagram.lean | 174 | 179 | theorem coe_bot : (⊥ : YoungDiagram).cells = (∅ : Set (ℕ × ℕ)) := by |
refine Set.eq_of_subset_of_subset ?_ ?_
· intros x h
simp? [mem_mk, Finset.coe_empty, Set.mem_empty_iff_false] at h says
simp only [cells_bot, Finset.coe_empty, Set.mem_empty_iff_false] at h
· simp only [cells_bot, Finset.coe_empty, Set.empty_subset]
|
import Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform
import Mathlib.Analysis.Fourier.PoissonSummation
open Real Set MeasureTheory Filter Asymptotics intervalIntegral
open scoped Real Topology FourierTransform RealInnerProductSpace
open Complex hiding exp continuous_exp abs_of_nonneg sq_abs
noncomp... | Mathlib/Analysis/SpecialFunctions/Gaussian/PoissonSummation.lean | 88 | 122 | theorem Complex.tsum_exp_neg_quadratic {a : ℂ} (ha : 0 < a.re) (b : ℂ) :
(∑' n : ℤ, cexp (-π * a * n ^ 2 + 2 * π * b * n)) =
1 / a ^ (1 / 2 : ℂ) * ∑' n : ℤ, cexp (-π / a * (n + I * b) ^ 2) := by |
let f : ℝ → ℂ := fun x ↦ cexp (-π * a * x ^ 2 + 2 * π * b * x)
have hCf : Continuous f := by
refine Complex.continuous_exp.comp (Continuous.add ?_ ?_)
· exact continuous_const.mul (Complex.continuous_ofReal.pow 2)
· exact continuous_const.mul Complex.continuous_ofReal
have hFf : 𝓕 f = fun x : ℝ ↦ 1 ... |
import Mathlib.Data.Complex.Exponential
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
#align_import data.complex.exponential_bounds from "leanprover-community/mathlib"@"402f8982dddc1864bd703da2d6e2ee304a866973"
namespace Real
open IsAbsoluteValue Finset CauSeq Complex
| Mathlib/Data/Complex/ExponentialBounds.lean | 20 | 25 | theorem exp_one_near_10 : |exp 1 - 2244083 / 825552| ≤ 1 / 10 ^ 10 := by |
apply exp_approx_start
iterate 13 refine exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) ?_
norm_num1
refine exp_approx_end' _ (by norm_num1; rfl) _ (by norm_cast) (by simp) ?_
rw [_root_.abs_one, abs_of_pos] <;> norm_num1
|
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.Polynomial.Roots
import Mathlib.GroupTheory.SpecificGroups.Cyclic
#align_import ring_theory.integral_domain from "leanprover-community/mathlib"@"6e70e0d419bf686784937d64ed4bfde866ff229e"
section
open Finset Polynomial Function Nat
section CancelMonoidWithZero... | Mathlib/RingTheory/IntegralDomain.lean | 61 | 69 | theorem exists_eq_pow_of_mul_eq_pow_of_coprime {R : Type*} [CommSemiring R] [IsDomain R]
[GCDMonoid R] [Unique Rˣ] {a b c : R} {n : ℕ} (cp : IsCoprime a b) (h : a * b = c ^ n) :
∃ d : R, a = d ^ n := by |
refine exists_eq_pow_of_mul_eq_pow (isUnit_of_dvd_one ?_) h
obtain ⟨x, y, hxy⟩ := cp
rw [← hxy]
exact -- Porting note: added `GCDMonoid.` twice
dvd_add (dvd_mul_of_dvd_right (GCDMonoid.gcd_dvd_left _ _) _)
(dvd_mul_of_dvd_right (GCDMonoid.gcd_dvd_right _ _) _)
|
import Mathlib.Algebra.IsPrimePow
import Mathlib.SetTheory.Cardinal.Ordinal
import Mathlib.Tactic.WLOG
#align_import set_theory.cardinal.divisibility from "leanprover-community/mathlib"@"ea050b44c0f9aba9d16a948c7cc7d2e7c8493567"
namespace Cardinal
open Cardinal
universe u
variable {a b : Cardinal.{u}} {n m : ℕ... | Mathlib/SetTheory/Cardinal/Divisibility.lean | 43 | 58 | theorem isUnit_iff : IsUnit a ↔ a = 1 := by |
refine
⟨fun h => ?_, by
rintro rfl
exact isUnit_one⟩
rcases eq_or_ne a 0 with (rfl | ha)
· exact (not_isUnit_zero h).elim
rw [isUnit_iff_forall_dvd] at h
cases' h 1 with t ht
rw [eq_comm, mul_eq_one_iff'] at ht
· exact ht.1
· exact one_le_iff_ne_zero.mpr ha
· apply one_le_iff_ne_zero.... |
import Mathlib.Init.Core
import Mathlib.LinearAlgebra.AffineSpace.Basis
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import linear_algebra.affine_space.finite_dimensional from "leanprover-community/mathlib"@"67e606eaea14c7854bdc556bd53d98aefdf76ec0"
noncomputable section
open Affine
section AffineSpace... | Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean | 81 | 87 | theorem finite_of_fin_dim_affineIndependent [FiniteDimensional k V] {p : ι → P}
(hi : AffineIndependent k p) : Finite ι := by |
nontriviality ι; inhabit ι
rw [affineIndependent_iff_linearIndependent_vsub k p default] at hi
letI : IsNoetherian k V := IsNoetherian.iff_fg.2 inferInstance
exact
(Set.finite_singleton default).finite_of_compl (Set.finite_coe_iff.1 hi.finite_of_isNoetherian)
|
import Mathlib.RingTheory.EisensteinCriterion
import Mathlib.RingTheory.Polynomial.ScaleRoots
#align_import ring_theory.polynomial.eisenstein.basic from "leanprover-community/mathlib"@"2032a878972d5672e7c27c957e7a6e297b044973"
universe u v w z
variable {R : Type u}
open Ideal Algebra Finset
open Polynomial
na... | Mathlib/RingTheory/Polynomial/Eisenstein/Basic.lean | 161 | 166 | theorem scaleRoots.isWeaklyEisensteinAt (p : R[X]) {x : R} {P : Ideal R} (hP : x ∈ P) :
(scaleRoots p x).IsWeaklyEisensteinAt P := by |
refine ⟨fun i => ?_⟩
rw [coeff_scaleRoots]
rw [natDegree_scaleRoots, ← tsub_pos_iff_lt] at i
exact Ideal.mul_mem_left _ _ (Ideal.pow_mem_of_mem P hP _ i)
|
import Mathlib.MeasureTheory.Measure.Typeclasses
#align_import measure_theory.measure.sub from "leanprover-community/mathlib"@"562bbf524c595c153470e53d36c57b6f891cc480"
open Set
namespace MeasureTheory
namespace Measure
noncomputable instance instSub {α : Type*} [MeasurableSpace α] : Sub (Measure α) :=
⟨fun ... | Mathlib/MeasureTheory/Measure/Sub.lean | 105 | 134 | theorem restrict_sub_eq_restrict_sub_restrict (h_meas_s : MeasurableSet s) :
(μ - ν).restrict s = μ.restrict s - ν.restrict s := by |
repeat rw [sub_def]
have h_nonempty : { d | μ ≤ d + ν }.Nonempty := ⟨μ, Measure.le_add_right le_rfl⟩
rw [restrict_sInf_eq_sInf_restrict h_nonempty h_meas_s]
apply le_antisymm
· refine sInf_le_sInf_of_forall_exists_le ?_
intro ν' h_ν'_in
rw [mem_setOf_eq] at h_ν'_in
refine ⟨ν'.restrict s, ?_, rest... |
import Mathlib.Algebra.Order.Group.OrderIso
import Mathlib.Algebra.Order.Monoid.OrderDual
import Mathlib.Data.Set.Pointwise.Basic
import Mathlib.Order.Bounds.OrderIso
import Mathlib.Order.ConditionallyCompleteLattice.Basic
#align_import algebra.bounds from "leanprover-community/mathlib"@"dd71334db81d0bd444af1ee339a29... | Mathlib/Algebra/Bounds.lean | 185 | 186 | theorem ciInf_div (hf : BddBelow (range f)) (a : G) : (⨅ i, f i) / a = ⨅ i, f i / a := by |
simp only [div_eq_mul_inv, ciInf_mul hf]
|
import Mathlib.MeasureTheory.PiSystem
import Mathlib.Order.OmegaCompletePartialOrder
import Mathlib.Topology.Constructions
import Mathlib.MeasureTheory.MeasurableSpace.Basic
open Set
namespace MeasureTheory
variable {ι : Type _} {α : ι → Type _}
section cylinder
def cylinder (s : Finset ι) (S : Set (∀ i : s, α... | Mathlib/MeasureTheory/Constructions/Cylinders.lean | 205 | 207 | theorem union_cylinder_same (s : Finset ι) (S₁ : Set (∀ i : s, α i)) (S₂ : Set (∀ i : s, α i)) :
cylinder s S₁ ∪ cylinder s S₂ = cylinder s (S₁ ∪ S₂) := by |
classical rw [union_cylinder]; rfl
|
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 | 76 | 82 | theorem borel_eq_generateFrom_Ici_rat : borel ℝ = .generateFrom (⋃ a : ℚ, {Ici (a : ℝ)}) := by |
rw [borel_eq_generateFrom_Iio_rat, iUnion_singleton_eq_range, iUnion_singleton_eq_range]
refine le_antisymm (generateFrom_le ?_) (generateFrom_le ?_) <;>
rintro _ ⟨q, rfl⟩ <;>
dsimp only <;>
[rw [← compl_Ici]; rw [← compl_Iio]] <;>
exact MeasurableSet.compl (GenerateMeasurable.basic _ (mem_range_self q))
|
import Mathlib.Topology.Category.TopCat.Limits.Pullbacks
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
#align_import algebraic_geometry.open_immersion.basic from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
-- Porting note: due to `PresheafedSpace`, `SheafedSpace` and `Locally... | Mathlib/Geometry/RingedSpace/OpenImmersion.lean | 145 | 146 | theorem isoRestrict_inv_ofRestrict : H.isoRestrict.inv ≫ f = Y.ofRestrict _ := by |
rw [Iso.inv_comp_eq, isoRestrict_hom_ofRestrict]
|
import Mathlib.Data.Matrix.Basis
import Mathlib.LinearAlgebra.Basis
import Mathlib.LinearAlgebra.Pi
#align_import linear_algebra.std_basis from "leanprover-community/mathlib"@"13bce9a6b6c44f6b4c91ac1c1d2a816e2533d395"
open Function Set Submodule
namespace LinearMap
variable (R : Type*) {ι : Type*} [Semiring R] ... | Mathlib/LinearAlgebra/StdBasis.lean | 123 | 129 | theorem iSup_range_stdBasis_eq_iInf_ker_proj {I J : Set ι} (hd : Disjoint I J)
(hu : Set.univ ⊆ I ∪ J) (hI : Set.Finite I) :
⨆ i ∈ I, range (stdBasis R φ i) = ⨅ i ∈ J, ker (proj i : (∀ i, φ i) →ₗ[R] φ i) := by |
refine le_antisymm (iSup_range_stdBasis_le_iInf_ker_proj _ _ _ _ hd) ?_
have : Set.univ ⊆ ↑hI.toFinset ∪ J := by rwa [hI.coe_toFinset]
refine le_trans (iInf_ker_proj_le_iSup_range_stdBasis R φ this) (iSup_mono fun i => ?_)
rw [Set.Finite.mem_toFinset]
|
import Batteries.Data.List.Lemmas
import Batteries.Data.Array.Basic
import Batteries.Tactic.SeqFocus
import Batteries.Util.ProofWanted
namespace Array
theorem forIn_eq_data_forIn [Monad m]
(as : Array α) (b : β) (f : α → β → m (ForInStep β)) :
forIn as b f = forIn as.data b f := by
let rec loop : ∀ {i h b ... | .lake/packages/batteries/Batteries/Data/Array/Lemmas.lean | 121 | 125 | theorem size_shrink_loop (a : Array α) (n) : (shrink.loop n a).size = a.size - n := by |
induction n generalizing a with simp[shrink.loop]
| succ n ih =>
simp[ih]
omega
|
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.Tactic.Abel
#align_import ring_theory.polynomial.pochhammer from "leanprover-community/mathlib"@"53b216bcc1146df1c4a0a868778... | Mathlib/RingTheory/Polynomial/Pochhammer.lean | 124 | 134 | theorem ascPochhammer_succ_right (n : ℕ) :
ascPochhammer S (n + 1) = ascPochhammer S n * (X + (n : S[X])) := by |
suffices h : ascPochhammer ℕ (n + 1) = ascPochhammer ℕ n * (X + (n : ℕ[X])) by
apply_fun Polynomial.map (algebraMap ℕ S) at h
simpa only [ascPochhammer_map, Polynomial.map_mul, Polynomial.map_add, map_X,
Polynomial.map_natCast] using h
induction' n with n ih
· simp
· conv_lhs =>
rw [ascPoch... |
import Mathlib.SetTheory.Game.State
#align_import set_theory.game.domineering from "leanprover-community/mathlib"@"b134b2f5cf6dd25d4bbfd3c498b6e36c11a17225"
namespace SetTheory
namespace PGame
namespace Domineering
open Function
@[simps!]
def shiftUp : ℤ × ℤ ≃ ℤ × ℤ :=
(Equiv.refl ℤ).prodCongr (Equiv.addRig... | Mathlib/SetTheory/Game/Domineering.lean | 109 | 114 | theorem moveLeft_card {b : Board} {m : ℤ × ℤ} (h : m ∈ left b) :
Finset.card (moveLeft b m) + 2 = Finset.card b := by |
dsimp [moveLeft]
rw [Finset.card_erase_of_mem (snd_pred_mem_erase_of_mem_left h)]
rw [Finset.card_erase_of_mem (Finset.mem_of_mem_inter_left h)]
exact tsub_add_cancel_of_le (card_of_mem_left h)
|
import Mathlib.MeasureTheory.PiSystem
import Mathlib.Order.OmegaCompletePartialOrder
import Mathlib.Topology.Constructions
import Mathlib.MeasureTheory.MeasurableSpace.Basic
open Set
namespace MeasureTheory
variable {ι : Type _} {α : ι → Type _}
section squareCylinders
def squareCylinders (C : ∀ i, Set (Set (α... | Mathlib/MeasureTheory/Constructions/Cylinders.lean | 63 | 105 | theorem isPiSystem_squareCylinders {C : ∀ i, Set (Set (α i))} (hC : ∀ i, IsPiSystem (C i))
(hC_univ : ∀ i, univ ∈ C i) :
IsPiSystem (squareCylinders C) := by |
rintro S₁ ⟨s₁, t₁, h₁, rfl⟩ S₂ ⟨s₂, t₂, h₂, rfl⟩ hst_nonempty
classical
let t₁' := s₁.piecewise t₁ (fun i ↦ univ)
let t₂' := s₂.piecewise t₂ (fun i ↦ univ)
have h1 : ∀ i ∈ (s₁ : Set ι), t₁ i = t₁' i :=
fun i hi ↦ (Finset.piecewise_eq_of_mem _ _ _ hi).symm
have h1' : ∀ i ∉ (s₁ : Set ι), t₁' i = univ :=
... |
import Mathlib.Probability.Kernel.Disintegration.Unique
import Mathlib.Probability.Notation
#align_import probability.kernel.cond_distrib from "leanprover-community/mathlib"@"00abe0695d8767201e6d008afa22393978bb324d"
open MeasureTheory Set Filter TopologicalSpace
open scoped ENNReal MeasureTheory ProbabilityTheo... | Mathlib/Probability/Kernel/CondDistrib.lean | 120 | 130 | theorem condDistrib_ae_eq_of_measure_eq_compProd (hX : Measurable X) (hY : Measurable Y)
(κ : kernel β Ω) [IsFiniteKernel κ] (hκ : μ.map (fun x => (X x, Y x)) = μ.map X ⊗ₘ κ) :
∀ᵐ x ∂μ.map X, κ x = condDistrib Y X μ x := by |
have heq : μ.map X = (μ.map (fun x ↦ (X x, Y x))).fst := by
ext s hs
rw [Measure.map_apply hX hs, Measure.fst_apply hs, Measure.map_apply]
exacts [rfl, Measurable.prod hX hY, measurable_fst hs]
rw [heq, condDistrib]
refine eq_condKernel_of_measure_eq_compProd _ ?_
convert hκ
exact heq.symm
|
import Mathlib.Analysis.NormedSpace.Star.Basic
import Mathlib.Analysis.NormedSpace.Spectrum
import Mathlib.Analysis.SpecialFunctions.Exponential
import Mathlib.Algebra.Star.StarAlgHom
#align_import analysis.normed_space.star.spectrum from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
l... | Mathlib/Analysis/NormedSpace/Star/Spectrum.lean | 60 | 69 | theorem IsSelfAdjoint.spectralRadius_eq_nnnorm {a : A} (ha : IsSelfAdjoint a) :
spectralRadius ℂ a = ‖a‖₊ := by |
have hconst : Tendsto (fun _n : ℕ => (‖a‖₊ : ℝ≥0∞)) atTop _ := tendsto_const_nhds
refine tendsto_nhds_unique ?_ hconst
convert
(spectrum.pow_nnnorm_pow_one_div_tendsto_nhds_spectralRadius (a : A)).comp
(Nat.tendsto_pow_atTop_atTop_of_one_lt one_lt_two) using 1
refine funext fun n => ?_
rw [Function... |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.LinearAlgebra.AffineSpace.Slope
#align_import analysis.calculus.deriv.slope from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
noncomputable section
open Topology Filter TopologicalSpace
open Filter Set
secti... | Mathlib/Analysis/Calculus/Deriv/Slope.lean | 51 | 63 | theorem hasDerivAtFilter_iff_tendsto_slope {x : 𝕜} {L : Filter 𝕜} :
HasDerivAtFilter f f' x L ↔ Tendsto (slope f x) (L ⊓ 𝓟 {x}ᶜ) (𝓝 f') :=
calc HasDerivAtFilter f f' x L
↔ Tendsto (fun y ↦ slope f x y - (y - x)⁻¹ • (y - x) • f') L (𝓝 0) := by |
simp only [hasDerivAtFilter_iff_tendsto, ← norm_inv, ← norm_smul,
← tendsto_zero_iff_norm_tendsto_zero, slope_def_module, smul_sub]
_ ↔ Tendsto (fun y ↦ slope f x y - (y - x)⁻¹ • (y - x) • f') (L ⊓ 𝓟 {x}ᶜ) (𝓝 0) :=
.symm <| tendsto_inf_principal_nhds_iff_of_forall_eq <| by simp
_ ↔ Tend... |
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 | 804 | 809 | theorem sum_centroidWeights_eq_one_of_card_ne_zero [CharZero k] (h : card s ≠ 0) :
∑ i ∈ s, s.centroidWeights k i = 1 := by |
-- Porting note: `simp` cannot find `mul_inv_cancel` and does not use `norm_cast`
simp only [centroidWeights_apply, sum_const, nsmul_eq_mul, ne_eq, Nat.cast_eq_zero, card_eq_zero]
refine mul_inv_cancel ?_
norm_cast
|
import Mathlib.AlgebraicGeometry.Morphisms.ClosedImmersion
import Mathlib.AlgebraicGeometry.Morphisms.QuasiSeparated
import Mathlib.AlgebraicGeometry.Pullbacks
import Mathlib.CategoryTheory.MorphismProperty.Limits
noncomputable section
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
universe ... | Mathlib/AlgebraicGeometry/Morphisms/Separated.lean | 57 | 60 | theorem respectsIso : MorphismProperty.RespectsIso @IsSeparated := by |
rw [isSeparated_eq_diagonal_isClosedImmersion]
apply MorphismProperty.RespectsIso.diagonal
exact IsClosedImmersion.respectsIso
|
import Mathlib.Analysis.Convex.Combination
import Mathlib.Analysis.Convex.Join
#align_import analysis.convex.stone_separation from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf97388c9e3b441fc4ccf3f"
open Set
variable {𝕜 E ι : Type*} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {s t : Set E}
| Mathlib/Analysis/Convex/StoneSeparation.lean | 30 | 77 | theorem not_disjoint_segment_convexHull_triple {p q u v x y z : E} (hz : z ∈ segment 𝕜 x y)
(hu : u ∈ segment 𝕜 x p) (hv : v ∈ segment 𝕜 y q) :
¬Disjoint (segment 𝕜 u v) (convexHull 𝕜 {p, q, z}) := by |
rw [not_disjoint_iff]
obtain ⟨az, bz, haz, hbz, habz, rfl⟩ := hz
obtain rfl | haz' := haz.eq_or_lt
· rw [zero_add] at habz
rw [zero_smul, zero_add, habz, one_smul]
refine ⟨v, by apply right_mem_segment, segment_subset_convexHull ?_ ?_ hv⟩ <;> simp
obtain ⟨av, bv, hav, hbv, habv, rfl⟩ := hv
obtain r... |
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
#align_import analysis.ODE.gronwall from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F]
[NormedSpace ℝ F]
open Metric Set Asymptotics Fil... | Mathlib/Analysis/ODE/Gronwall.lean | 79 | 83 | theorem gronwallBound_x0 (δ K ε : ℝ) : gronwallBound δ K ε 0 = δ := by |
by_cases hK : K = 0
· simp only [gronwallBound, if_pos hK, mul_zero, add_zero]
· simp only [gronwallBound, if_neg hK, mul_zero, exp_zero, sub_self, mul_one,
add_zero]
|
import Mathlib.LinearAlgebra.Basis
import Mathlib.LinearAlgebra.Multilinear.Basic
#align_import linear_algebra.multilinear.basis from "leanprover-community/mathlib"@"ce11c3c2a285bbe6937e26d9792fda4e51f3fe1a"
open MultilinearMap
variable {R : Type*} {ι : Type*} {n : ℕ} {M : Fin n → Type*} {M₂ : Type*} {M₃ : Type*... | Mathlib/LinearAlgebra/Multilinear/Basis.lean | 56 | 61 | theorem Basis.ext_multilinear [Finite ι] {f g : MultilinearMap R (fun _ : ι => M₂) M₃} {ι₁ : Type*}
(e : Basis ι₁ R M₂) (h : ∀ v : ι → ι₁, (f fun i => e (v i)) = g fun i => e (v i)) : f = g := by |
cases nonempty_fintype ι
exact
(domDomCongr_eq_iff (Fintype.equivFin ι) f g).mp
(Basis.ext_multilinear_fin (fun _ => e) fun i => h (i ∘ _))
|
import Mathlib.RingTheory.DedekindDomain.Ideal
#align_import ring_theory.dedekind_domain.factorization from "leanprover-community/mathlib"@"2f588be38bb5bec02f218ba14f82fc82eb663f87"
noncomputable section
open scoped Classical nonZeroDivisors
open Set Function UniqueFactorizationMonoid IsDedekindDomain IsDedekind... | Mathlib/RingTheory/DedekindDomain/Factorization.lean | 81 | 90 | theorem Associates.finite_factors {I : Ideal R} (hI : I ≠ 0) :
∀ᶠ v : HeightOneSpectrum R in Filter.cofinite,
((Associates.mk v.asIdeal).count (Associates.mk I).factors : ℤ) = 0 := by |
have h_supp : {v : HeightOneSpectrum R | ¬((Associates.mk v.asIdeal).count
(Associates.mk I).factors : ℤ) = 0} = {v : HeightOneSpectrum R | v.asIdeal ∣ I} := by
ext v
simp_rw [Int.natCast_eq_zero]
exact Associates.count_ne_zero_iff_dvd hI v.irreducible
rw [Filter.eventually_cofinite, h_supp]
ex... |
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 | 320 | 323 | theorem finsuppTensorFinsuppRid_single_tmul_single (a : ι) (b : κ) (m : M) (r : R) :
finsuppTensorFinsuppRid R M ι κ (Finsupp.single a m ⊗ₜ[R] Finsupp.single b r) =
Finsupp.single (a, b) (r • m) := by |
simp [finsuppTensorFinsuppRid]
|
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 | 121 | 123 | theorem sup_prod (f₁ f₂ : Filter α) (g : Filter β) : (f₁ ⊔ f₂) ×ˢ g = (f₁ ×ˢ g) ⊔ (f₂ ×ˢ g) := by |
dsimp only [SProd.sprod]
rw [Filter.prod, comap_sup, inf_sup_right, ← Filter.prod, ← Filter.prod]
|
import Mathlib.Topology.MetricSpace.PseudoMetric
#align_import topology.metric_space.basic from "leanprover-community/mathlib"@"c8f305514e0d47dfaa710f5a52f0d21b588e6328"
open Set Filter Bornology
open scoped NNReal Uniformity
universe u v w
variable {α : Type u} {β : Type v} {X ι : Type*}
variable [PseudoMetricS... | Mathlib/Topology/MetricSpace/Basic.lean | 140 | 144 | theorem uniformEmbedding_iff' [MetricSpace β] {f : γ → β} :
UniformEmbedding f ↔
(∀ ε > 0, ∃ δ > 0, ∀ {a b : γ}, dist a b < δ → dist (f a) (f b) < ε) ∧
∀ δ > 0, ∃ ε > 0, ∀ {a b : γ}, dist (f a) (f b) < ε → dist a b < δ := by |
rw [uniformEmbedding_iff_uniformInducing, uniformInducing_iff, uniformContinuous_iff]
|
import Mathlib.LinearAlgebra.Dual
open Function Module
variable (R M N : Type*) [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N]
structure PerfectPairing :=
toLin : M →ₗ[R] N →ₗ[R] R
bijectiveLeft : Bijective toLin
bijectiveRight : Bijective toLin.flip
attribute [nolint docBlame] P... | Mathlib/LinearAlgebra/PerfectPairing.lean | 112 | 115 | theorem reflexive_left : IsReflexive R M where
bijective_dual_eval' := by |
rw [← p.toDualRight_symm_comp_toDualLeft]
exact p.bijective_toDualRight_symm_toDualLeft
|
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