Context stringlengths 57 6.04k | file_name stringlengths 21 79 | start int64 14 1.49k | end int64 18 1.5k | theorem stringlengths 25 1.57k | proof stringlengths 5 7.36k | hint bool 2
classes |
|---|---|---|---|---|---|---|
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Data.Set.Function
import Mathlib.Order.Interval.Set.Basic
#align_import data.set.intervals.monoid from "leanprover-community/mathlib"@"aba57d4d3dae35460225919dcd82fe91355162f9"
namespace Set
variable {M : Type*} ... | Mathlib/Algebra/Order/Interval/Set/Monoid.lean | 118 | 119 | theorem image_const_add_Ioi : (fun x => a + x) '' Ioi b = Ioi (a + b) := by |
simp only [add_comm a, image_add_const_Ioi]
| true |
import Mathlib.Algebra.Homology.Homotopy
import Mathlib.Algebra.Category.ModuleCat.Abelian
import Mathlib.Algebra.Category.ModuleCat.Subobject
import Mathlib.CategoryTheory.Limits.Shapes.ConcreteCategory
#align_import algebra.homology.Module from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225... | Mathlib/Algebra/Homology/ModuleCat.lean | 72 | 79 | theorem cycles'Map_toCycles' (f : C ⟶ D) {i : ι} (x : LinearMap.ker (C.dFrom i)) :
(cycles'Map f i) (toCycles' x) = toCycles' ⟨f.f i x.1, by
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
rw [LinearMap.mem_ker]; erw [Hom.comm_from_apply, x.2, map_zero]⟩ := by
ext |
ext
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [cycles'Map_arrow_apply, toKernelSubobject_arrow, toKernelSubobject_arrow]
rfl
| true |
import Mathlib.FieldTheory.Separable
import Mathlib.FieldTheory.SplittingField.Construction
import Mathlib.Algebra.CharP.Reduced
open Function Polynomial
class PerfectRing (R : Type*) (p : ℕ) [CommSemiring R] [ExpChar R p] : Prop where
bijective_frobenius : Bijective <| frobenius R p
section PerfectRing
va... | Mathlib/FieldTheory/Perfect.lean | 113 | 114 | theorem iterateFrobeniusEquiv_zero_apply (x : R) : iterateFrobeniusEquiv R p 0 x = x := by |
rw [iterateFrobeniusEquiv_def, pow_zero, pow_one]
| true |
import Mathlib.Data.PFunctor.Univariate.M
#align_import data.qpf.univariate.basic from "leanprover-community/mathlib"@"14b69e9f3c16630440a2cbd46f1ddad0d561dee7"
universe u
class QPF (F : Type u → Type u) [Functor F] where
P : PFunctor.{u}
abs : ∀ {α}, P α → F α
repr : ∀ {α}, F α → P α
abs_repr : ∀ {α} (... | Mathlib/Data/QPF/Univariate/Basic.lean | 134 | 153 | theorem liftr_iff {α : Type u} (r : α → α → Prop) (x y : F α) :
Liftr r x y ↔ ∃ a f₀ f₁, x = abs ⟨a, f₀⟩ ∧ y = abs ⟨a, f₁⟩ ∧ ∀ i, r (f₀ i) (f₁ i) := by
constructor |
constructor
· rintro ⟨u, xeq, yeq⟩
cases' h : repr u with a f
use a, fun i => (f i).val.fst, fun i => (f i).val.snd
constructor
· rw [← xeq, ← abs_repr u, h, ← abs_map]
rfl
constructor
· rw [← yeq, ← abs_repr u, h, ← abs_map]
rfl
intro i
exact (f i).property
rintro ⟨a,... | true |
import Mathlib.Logic.Equiv.Nat
import Mathlib.Logic.Equiv.Fin
import Mathlib.Data.Countable.Defs
#align_import data.countable.basic from "leanprover-community/mathlib"@"1f0096e6caa61e9c849ec2adbd227e960e9dff58"
universe u v w
open Function
instance : Countable ℤ :=
Countable.of_equiv ℕ Equiv.intEquivNat.symm
... | Mathlib/Data/Countable/Basic.lean | 38 | 39 | theorem uncountable_iff_isEmpty_embedding : Uncountable α ↔ IsEmpty (α ↪ ℕ) := by |
rw [← not_countable_iff, countable_iff_nonempty_embedding, not_nonempty_iff]
| true |
import Mathlib.Data.Finset.Basic
import Mathlib.ModelTheory.Syntax
import Mathlib.Data.List.ProdSigma
#align_import model_theory.semantics from "leanprover-community/mathlib"@"d565b3df44619c1498326936be16f1a935df0728"
universe u v w u' v'
namespace FirstOrder
namespace Language
variable {L : Language.{u, v}} {... | Mathlib/ModelTheory/Semantics.lean | 117 | 122 | theorem realize_functions_apply₂ {f : L.Functions 2} {t₁ t₂ : L.Term α} {v : α → M} :
(f.apply₂ t₁ t₂).realize v = funMap f ![t₁.realize v, t₂.realize v] := by
rw [Functions.apply₂, Term.realize] |
rw [Functions.apply₂, Term.realize]
refine congr rfl (funext (Fin.cases ?_ ?_))
· simp only [Matrix.cons_val_zero]
· simp only [Matrix.cons_val_succ, Matrix.cons_val_fin_one, forall_const]
| true |
import Mathlib.Analysis.Analytic.Linear
import Mathlib.Analysis.Analytic.Composition
import Mathlib.Analysis.NormedSpace.Completion
#align_import analysis.analytic.uniqueness from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜] {E : Type... | Mathlib/Analysis/Analytic/Uniqueness.lean | 96 | 101 | theorem eqOn_of_preconnected_of_eventuallyEq {f g : E → F} {U : Set E} (hf : AnalyticOn 𝕜 f U)
(hg : AnalyticOn 𝕜 g U) (hU : IsPreconnected U) {z₀ : E} (h₀ : z₀ ∈ U) (hfg : f =ᶠ[𝓝 z₀] g) :
EqOn f g U := by
have hfg' : f - g =ᶠ[𝓝 z₀] 0 := hfg.mono fun z h => by simp [h] |
have hfg' : f - g =ᶠ[𝓝 z₀] 0 := hfg.mono fun z h => by simp [h]
simpa [sub_eq_zero] using fun z hz =>
(hf.sub hg).eqOn_zero_of_preconnected_of_eventuallyEq_zero hU h₀ hfg' hz
| true |
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.GroupTheory.GroupAction.Ring
#align_import data.polynomial.derivative from "leanprover-community/mathlib"@"bbeb185db4ccee8ed07dc48449414ebfa39cb821"
noncomputable section
open Finset
open Polynomial
namespace Pol... | Mathlib/Algebra/Polynomial/Derivative.lean | 149 | 150 | theorem derivative_X_add_C (c : R) : derivative (X + C c) = 1 := by |
rw [derivative_add, derivative_X, derivative_C, add_zero]
| true |
import Mathlib.Algebra.Field.ULift
import Mathlib.Algebra.MvPolynomial.Cardinal
import Mathlib.Data.Nat.Factorization.PrimePow
import Mathlib.Data.Rat.Denumerable
import Mathlib.FieldTheory.Finite.GaloisField
import Mathlib.Logic.Equiv.TransferInstance
import Mathlib.RingTheory.Localization.Cardinality
import Mathlib.... | Mathlib/FieldTheory/Cardinality.lean | 66 | 76 | theorem Infinite.nonempty_field {α : Type u} [Infinite α] : Nonempty (Field α) := by
letI K := FractionRing (MvPolynomial α <| ULift.{u} ℚ) |
letI K := FractionRing (MvPolynomial α <| ULift.{u} ℚ)
suffices #α = #K by
obtain ⟨e⟩ := Cardinal.eq.1 this
exact ⟨e.field⟩
rw [← IsLocalization.card (MvPolynomial α <| ULift.{u} ℚ)⁰ K le_rfl]
apply le_antisymm
· refine
⟨⟨fun a => MvPolynomial.monomial (Finsupp.single a 1) (1 : ULift.{u} ℚ), fu... | true |
import Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic
#align_import number_theory.legendre_symbol.basic from "leanprover-community/mathlib"@"5b2fe80501ff327b9109fb09b7cc8c325cd0d7d9"
open Nat
section Euler
namespace ZMod
variable (p : ℕ) [Fact p.Prime]
theorem euler_criterion_units (x : (ZMod p)ˣ) :... | Mathlib/NumberTheory/LegendreSymbol/Basic.lean | 61 | 70 | theorem euler_criterion {a : ZMod p} (ha : a ≠ 0) : IsSquare (a : ZMod p) ↔ a ^ (p / 2) = 1 := by
apply (iff_congr _ (by simp [Units.ext_iff])).mp (euler_criterion_units p (Units.mk0 a ha)) |
apply (iff_congr _ (by simp [Units.ext_iff])).mp (euler_criterion_units p (Units.mk0 a ha))
simp only [Units.ext_iff, sq, Units.val_mk0, Units.val_mul]
constructor
· rintro ⟨y, hy⟩; exact ⟨y, hy.symm⟩
· rintro ⟨y, rfl⟩
have hy : y ≠ 0 := by
rintro rfl
simp [zero_pow, mul_zero, ne_eq, not_true... | true |
import Mathlib.Logic.Equiv.Defs
import Mathlib.Tactic.Convert
#align_import control.equiv_functor from "leanprover-community/mathlib"@"d6aae1bcbd04b8de2022b9b83a5b5b10e10c777d"
universe u₀ u₁ u₂ v₀ v₁ v₂
open Function
class EquivFunctor (f : Type u₀ → Type u₁) where
map : ∀ {α β}, α ≃ β → f α → f β
m... | Mathlib/Control/EquivFunctor.lean | 70 | 71 | theorem mapEquiv_refl (α) : mapEquiv f (Equiv.refl α) = Equiv.refl (f α) := by |
simp only [mapEquiv, map_refl', Equiv.refl_symm]; rfl
| true |
import Mathlib.AlgebraicTopology.DoldKan.Faces
import Mathlib.CategoryTheory.Idempotents.Basic
#align_import algebraic_topology.dold_kan.projections from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190f0b504"
open CategoryTheory CategoryTheory.Category CategoryTheory.Limits CategoryTheory.Pread... | Mathlib/AlgebraicTopology/DoldKan/Projections.lean | 118 | 134 | theorem comp_P_eq_self {Y : C} {n q : ℕ} {φ : Y ⟶ X _[n + 1]} (v : HigherFacesVanish q φ) :
φ ≫ (P q).f (n + 1) = φ := by
induction' q with q hq |
induction' q with q hq
· simp only [P_zero]
apply comp_id
· simp only [P_succ, comp_add, HomologicalComplex.comp_f, HomologicalComplex.add_f_apply,
comp_id, ← assoc, hq v.of_succ, add_right_eq_self]
by_cases hqn : n < q
· exact v.of_succ.comp_Hσ_eq_zero hqn
· obtain ⟨a, ha⟩ := Nat.le.dest (... | true |
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Analysis.NormedSpace.Banach
import Mathlib.LinearAlgebra.SesquilinearForm
#align_import analysis.inner_product_space.symmetric from "leanprover-community/mathlib"@"3f655f5297b030a87d641ad4e825af8d9679eb0b"
open RCLike
open ComplexConjugate
variable ... | Mathlib/Analysis/InnerProductSpace/Symmetric.lean | 88 | 92 | theorem IsSymmetric.add {T S : E →ₗ[𝕜] E} (hT : T.IsSymmetric) (hS : S.IsSymmetric) :
(T + S).IsSymmetric := by
intro x y |
intro x y
rw [LinearMap.add_apply, inner_add_left, hT x y, hS x y, ← inner_add_right]
rfl
| true |
import Mathlib.Data.List.Lattice
import Mathlib.Data.List.Range
import Mathlib.Data.Bool.Basic
#align_import data.list.intervals from "leanprover-community/mathlib"@"7b78d1776212a91ecc94cf601f83bdcc46b04213"
open Nat
namespace List
def Ico (n m : ℕ) : List ℕ :=
range' n (m - n)
#align list.Ico List.Ico
names... | Mathlib/Data/List/Intervals.lean | 125 | 127 | theorem succ_top {n m : ℕ} (h : n ≤ m) : Ico n (m + 1) = Ico n m ++ [m] := by
rwa [← succ_singleton, append_consecutive] |
rwa [← succ_singleton, append_consecutive]
exact Nat.le_succ _
| true |
import Mathlib.MeasureTheory.Measure.MeasureSpace
open scoped ENNReal NNReal Topology
open Set MeasureTheory Measure Filter MeasurableSpace ENNReal Function
variable {R α β δ γ ι : Type*}
namespace MeasureTheory
variable {m0 : MeasurableSpace α} [MeasurableSpace β] [MeasurableSpace γ]
variable {μ μ₁ μ₂ μ₃ ν ν' ν... | Mathlib/MeasureTheory/Measure/Restrict.lean | 140 | 141 | theorem restrict_apply_univ (s : Set α) : μ.restrict s univ = μ s := by |
rw [restrict_apply MeasurableSet.univ, Set.univ_inter]
| true |
import Mathlib.CategoryTheory.Limits.Shapes.CommSq
import Mathlib.CategoryTheory.Limits.Shapes.StrictInitial
import Mathlib.CategoryTheory.Limits.Shapes.Types
import Mathlib.Topology.Category.TopCat.Limits.Pullbacks
import Mathlib.CategoryTheory.Limits.FunctorCategory
import Mathlib.CategoryTheory.Limits.Constructions... | Mathlib/CategoryTheory/Extensive.lean | 203 | 216 | theorem finitaryExtensive_iff_of_isTerminal (C : Type u) [Category.{v} C] [HasFiniteCoproducts C]
[HasPullbacksOfInclusions C]
(T : C) (HT : IsTerminal T) (c₀ : BinaryCofan T T) (hc₀ : IsColimit c₀) :
FinitaryExtensive C ↔ IsVanKampenColimit c₀ := by
refine ⟨fun H => H.van_kampen' c₀ hc₀, fun H => ?_⟩ |
refine ⟨fun H => H.van_kampen' c₀ hc₀, fun H => ?_⟩
constructor
simp_rw [BinaryCofan.isVanKampen_iff] at H ⊢
intro X Y c hc X' Y' c' αX αY f hX hY
obtain ⟨d, hd, hd'⟩ :=
Limits.BinaryCofan.IsColimit.desc' hc (HT.from _ ≫ c₀.inl) (HT.from _ ≫ c₀.inr)
rw [H c' (αX ≫ HT.from _) (αY ≫ HT.from _) (f ≫ d) (b... | true |
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Group.Invertible.Defs
import Mathlib.Algebra.Group.Hom.Defs
import Mathlib.Logic.Equiv.Defs
#align_import algebra.invertible from "leanprover-community/mathlib"@"722b3b152ddd5e0cf21c0a29787c76596cb6b422"
assert_not_exists MonoidWithZero
assert_not_ex... | Mathlib/Algebra/Group/Invertible/Basic.lean | 69 | 74 | theorem Commute.invOf_right [Monoid α] {a b : α} [Invertible b] (h : Commute a b) :
Commute a (⅟ b) :=
calc
a * ⅟ b = ⅟ b * (b * a * ⅟ b) := by simp [mul_assoc] | simp [mul_assoc]
_ = ⅟ b * (a * b * ⅟ b) := by rw [h.eq]
_ = ⅟ b * a := by simp [mul_assoc]
| true |
import Mathlib.Algebra.CharZero.Defs
import Mathlib.Algebra.Group.Hom.Defs
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Algebra.Order.Monoid.OrderDual
import Mathlib.Algebra.Order.ZeroLEOne
import Mathlib.Data.Nat.Cast.Defs
import Mathlib.Order.WithBot
#align_import algebra.order.monoid.with_top ... | Mathlib/Algebra/Order/Monoid/WithTop.lean | 175 | 181 | theorem add_left_cancel_iff [IsLeftCancelAdd α] (ha : a ≠ ⊤) : a + b = a + c ↔ b = c := by
lift a to α using ha |
lift a to α using ha
obtain rfl | hb := eq_or_ne b ⊤
· rw [add_top, eq_comm, WithTop.coe_add_eq_top_iff, eq_comm]
lift b to α using hb
simp_rw [← WithTop.coe_add, eq_comm, WithTop.add_eq_coe, eq_comm, coe_eq_coe,
exists_and_left, exists_eq_left', add_right_inj, exists_eq_right']
| true |
import Mathlib.Analysis.Seminorm
import Mathlib.Topology.Algebra.Equicontinuity
import Mathlib.Topology.MetricSpace.Equicontinuity
import Mathlib.Topology.Algebra.FilterBasis
import Mathlib.Topology.Algebra.Module.LocallyConvex
#align_import analysis.locally_convex.with_seminorms from "leanprover-community/mathlib"@"... | Mathlib/Analysis/LocallyConvex/WithSeminorms.lean | 98 | 112 | theorem basisSets_intersect (U V : Set E) (hU : U ∈ p.basisSets) (hV : V ∈ p.basisSets) :
∃ z ∈ p.basisSets, z ⊆ U ∩ V := by
classical |
classical
rcases p.basisSets_iff.mp hU with ⟨s, r₁, hr₁, hU⟩
rcases p.basisSets_iff.mp hV with ⟨t, r₂, hr₂, hV⟩
use ((s ∪ t).sup p).ball 0 (min r₁ r₂)
refine ⟨p.basisSets_mem (s ∪ t) (lt_min_iff.mpr ⟨hr₁, hr₂⟩), ?_⟩
rw [hU, hV, ball_finset_sup_eq_iInter _ _ _ (lt_min_iff.mpr ⟨hr₁, hr₂⟩),
ba... | true |
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 | 141 | 142 | theorem toComplex_inj {x y : ℤ[i]} : (x : ℂ) = y ↔ x = y := by |
cases x; cases y; simp [toComplex_def₂]
| true |
import Mathlib.Algebra.ContinuedFractions.Translations
#align_import algebra.continued_fractions.terminated_stable from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b348ce40ad"
namespace GeneralizedContinuedFraction
variable {K : Type*} {g : GeneralizedContinuedFraction K} {n m : ℕ}
theorem te... | Mathlib/Algebra/ContinuedFractions/TerminatedStable.lean | 80 | 82 | theorem denominators_stable_of_terminated (n_le_m : n ≤ m) (terminated_at_n : g.TerminatedAt n) :
g.denominators m = g.denominators n := by |
simp only [denom_eq_conts_b, continuants_stable_of_terminated n_le_m terminated_at_n]
| true |
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 | 213 | 219 | theorem coe_eq_coe_iff_of_mem_Ico {x y : 𝕜} (hx : x ∈ Ico a (a + p)) (hy : y ∈ Ico a (a + p)) :
(x : AddCircle p) = y ↔ x = y := by
refine ⟨fun h => ?_, by tauto⟩ |
refine ⟨fun h => ?_, by tauto⟩
suffices (⟨x, hx⟩ : Ico a (a + p)) = ⟨y, hy⟩ by exact Subtype.mk.inj this
apply_fun equivIco p a at h
rw [← (equivIco p a).right_inv ⟨x, hx⟩, ← (equivIco p a).right_inv ⟨y, hy⟩]
exact h
| true |
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Algebra.Order.Monoid.WithTop
#align_import data.nat.with_bot from "leanprover-community/mathlib"@"966e0cf0685c9cedf8a3283ac69eef4d5f2eaca2"
namespace Nat
namespace WithBot
instance : WellFoundedRelation (WithBot ℕ) where
rel := (· < ·)
wf := IsWellFounde... | Mathlib/Data/Nat/WithBot.lean | 61 | 63 | theorem coe_nonneg {n : ℕ} : 0 ≤ (n : WithBot ℕ) := by
rw [← WithBot.coe_zero] |
rw [← WithBot.coe_zero]
exact WithBot.coe_le_coe.mpr (Nat.zero_le n)
| true |
import Mathlib.Control.Monad.Basic
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.List.ProdSigma
#align_import data.fin_enum from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
universe u v
open Finset
class FinEnum (α : Sort*) where
card : ℕ
equiv : α ≃ Fin card
[... | Mathlib/Data/FinEnum.lean | 69 | 70 | theorem mem_toList [FinEnum α] (x : α) : x ∈ toList α := by |
simp [toList]; exists equiv x; simp
| true |
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.LinearAlgebra.Matrix.Trace
import Mathlib.LinearAlgebra.Matrix.Symmetric
#align_import combinatorics.simple_graph.adj_matrix from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa1... | Mathlib/Combinatorics/SimpleGraph/AdjMatrix.lean | 64 | 65 | theorem apply_diag_ne [MulZeroOneClass α] [Nontrivial α] (h : IsAdjMatrix A) (i : V) :
¬A i i = 1 := by | simp [h.apply_diag i]
| true |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.Ring.Action.Basic
import Mathlib.GroupTheory.GroupAction.Hom
import Mathlib.GroupTheory.GroupAction.Quotient
#align_import algebra.polynomial.group_ring_action from "leanprover-community/mathlib"@"afad8e438d03f... | Mathlib/Algebra/Polynomial/GroupRingAction.lean | 31 | 39 | theorem smul_eq_map [MulSemiringAction M R] (m : M) :
HSMul.hSMul m = map (MulSemiringAction.toRingHom M R m) := by
suffices DistribMulAction.toAddMonoidHom R[X] m = |
suffices DistribMulAction.toAddMonoidHom R[X] m =
(mapRingHom (MulSemiringAction.toRingHom M R m)).toAddMonoidHom by
ext1 r
exact DFunLike.congr_fun this r
ext n r : 2
change m • monomial n r = map (MulSemiringAction.toRingHom M R m) (monomial n r)
rw [Polynomial.map_monomial, Polynomial.smul_mon... | true |
import Mathlib.Algebra.MonoidAlgebra.Support
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.Algebra.Regular.Basic
import Mathlib.Data.Nat.Choose.Sum
#align_import data.polynomial.coeff from "leanprover-community/mathlib"@"2651125b48fc5c170ab1111afd0817c903b1fc6c"
set_option linter.uppercaseLean3 false
no... | Mathlib/Algebra/Polynomial/Coeff.lean | 130 | 134 | theorem coeff_mul (p q : R[X]) (n : ℕ) :
coeff (p * q) n = ∑ x ∈ antidiagonal n, coeff p x.1 * coeff q x.2 := by
rcases p with ⟨p⟩; rcases q with ⟨q⟩ |
rcases p with ⟨p⟩; rcases q with ⟨q⟩
simp_rw [← ofFinsupp_mul, coeff]
exact AddMonoidAlgebra.mul_apply_antidiagonal p q n _ Finset.mem_antidiagonal
| true |
import Mathlib.Init.Order.Defs
import Mathlib.Logic.Nontrivial.Defs
import Mathlib.Tactic.Attr.Register
import Mathlib.Data.Prod.Basic
import Mathlib.Data.Subtype
import Mathlib.Logic.Function.Basic
import Mathlib.Logic.Unique
#align_import logic.nontrivial from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc95... | Mathlib/Logic/Nontrivial/Basic.lean | 32 | 34 | theorem exists_pair_lt (α : Type*) [Nontrivial α] [LinearOrder α] : ∃ x y : α, x < y := by
rcases exists_pair_ne α with ⟨x, y, hxy⟩ |
rcases exists_pair_ne α with ⟨x, y, hxy⟩
cases lt_or_gt_of_ne hxy <;> exact ⟨_, _, ‹_›⟩
| true |
import Mathlib.Data.Bracket
import Mathlib.LinearAlgebra.Basic
#align_import algebra.lie.basic from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
universe u v w w₁ w₂
open Function
class LieRing (L : Type v) extends AddCommGroup L, Bracket L L where
protected add_lie : ∀ x y z ... | Mathlib/Algebra/Lie/Basic.lean | 163 | 165 | theorem neg_lie : ⁅-x, m⁆ = -⁅x, m⁆ := by
rw [← sub_eq_zero, sub_neg_eq_add, ← add_lie] |
rw [← sub_eq_zero, sub_neg_eq_add, ← add_lie]
simp
| true |
import Mathlib.AlgebraicGeometry.Gluing
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.AlgebraicGeometry.AffineScheme
import Mathlib.CategoryTheory.Limits.Shapes.Diagonal
#align_import algebraic_geometry.pullbacks from "leanprover-community/mathlib"@"7316286ff2942aa14e540add9058c6b0aa1c8070"
set_opt... | Mathlib/AlgebraicGeometry/Pullbacks.lean | 78 | 81 | theorem t_snd (i j : 𝒰.J) : t 𝒰 f g i j ≫ pullback.snd = pullback.fst ≫ pullback.fst := by
simp only [t, Category.assoc, pullbackSymmetry_hom_comp_snd, pullbackAssoc_hom_fst, |
simp only [t, Category.assoc, pullbackSymmetry_hom_comp_snd, pullbackAssoc_hom_fst,
pullback.lift_fst_assoc, pullbackSymmetry_hom_comp_fst, pullbackAssoc_inv_fst_snd,
pullbackSymmetry_hom_comp_snd_assoc]
| true |
import Mathlib.Analysis.InnerProductSpace.Orientation
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
#align_import measure_theory.measure.haar.inner_product_space from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
open FiniteDimensional MeasureTheory MeasureTheory.Measure Set
var... | Mathlib/MeasureTheory/Measure/Haar/InnerProductSpace.lean | 34 | 43 | theorem Orientation.measure_orthonormalBasis (o : Orientation ℝ F (Fin n))
(b : OrthonormalBasis ι ℝ F) : o.volumeForm.measure (parallelepiped b) = 1 := by
have e : ι ≃ Fin n := by |
have e : ι ≃ Fin n := by
refine Fintype.equivFinOfCardEq ?_
rw [← _i.out, finrank_eq_card_basis b.toBasis]
have A : ⇑b = b.reindex e ∘ e := by
ext x
simp only [OrthonormalBasis.coe_reindex, Function.comp_apply, Equiv.symm_apply_apply]
rw [A, parallelepiped_comp_equiv, AlternatingMap.measure_paral... | true |
import Mathlib.Algebra.Category.Ring.FilteredColimits
import Mathlib.Geometry.RingedSpace.SheafedSpace
import Mathlib.Topology.Sheaves.Stalks
import Mathlib.Algebra.Category.Ring.Colimits
import Mathlib.Algebra.Category.Ring.Limits
#align_import algebraic_geometry.ringed_space from "leanprover-community/mathlib"@"5dc... | Mathlib/Geometry/RingedSpace/Basic.lean | 84 | 125 | theorem isUnit_of_isUnit_germ (U : Opens X) (f : X.presheaf.obj (op U))
(h : ∀ x : U, IsUnit (X.presheaf.germ x f)) : IsUnit f := by
-- We pick a cover of `U` by open sets `V x`, such that `f` is a unit on each `V x`. |
-- We pick a cover of `U` by open sets `V x`, such that `f` is a unit on each `V x`.
choose V iVU m h_unit using fun x : U => X.isUnit_res_of_isUnit_germ U f x (h x)
have hcover : U ≤ iSup V := by
intro x hxU
-- Porting note: in Lean3 `rw` is sufficient
erw [Opens.mem_iSup]
exact ⟨⟨x, hxU⟩, m ⟨x,... | true |
import Mathlib.AlgebraicGeometry.Morphisms.Basic
import Mathlib.Topology.LocalAtTarget
#align_import algebraic_geometry.morphisms.universally_closed from "leanprover-community/mathlib"@"a8ae1b3f7979249a0af6bc7cf20c1f6bf656ca73"
noncomputable section
open CategoryTheory CategoryTheory.Limits Opposite TopologicalS... | Mathlib/AlgebraicGeometry/Morphisms/UniversallyClosed.lean | 72 | 76 | theorem topologically_isClosedMap_respectsIso : RespectsIso (topologically @IsClosedMap) := by
apply MorphismProperty.respectsIso_of_isStableUnderComposition |
apply MorphismProperty.respectsIso_of_isStableUnderComposition
intro _ _ f hf
have : IsIso f := hf
exact (TopCat.homeoOfIso (Scheme.forgetToTop.mapIso (asIso f))).isClosedMap
| true |
import Mathlib.Order.Filter.Bases
#align_import order.filter.pi from "leanprover-community/mathlib"@"ce64cd319bb6b3e82f31c2d38e79080d377be451"
open Set Function
open scoped Classical
open Filter
namespace Filter
variable {ι : Type*} {α : ι → Type*} {f f₁ f₂ : (i : ι) → Filter (α i)} {s : (i : ι) → Set (α i)}
... | Mathlib/Order/Filter/Pi.lean | 121 | 125 | theorem hasBasis_pi {ι' : ι → Type} {s : ∀ i, ι' i → Set (α i)} {p : ∀ i, ι' i → Prop}
(h : ∀ i, (f i).HasBasis (p i) (s i)) :
(pi f).HasBasis (fun If : Set ι × ∀ i, ι' i => If.1.Finite ∧ ∀ i ∈ If.1, p i (If.2 i))
fun If : Set ι × ∀ i, ι' i => If.1.pi fun i => s i <| If.2 i := by |
simpa [Set.pi_def] using hasBasis_iInf' fun i => (h i).comap (eval i : (∀ j, α j) → α i)
| true |
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Normed.Group.Lemmas
import Mathlib.Analysis.NormedSpace.AddTorsor
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
import Mathlib.Analysis.NormedSpace.RieszLemma
import Mathli... | Mathlib/Analysis/NormedSpace/FiniteDimension.lean | 246 | 255 | theorem ContinuousLinearMap.isOpen_injective [FiniteDimensional 𝕜 E] :
IsOpen { L : E →L[𝕜] F | Injective L } := by
rw [isOpen_iff_eventually] |
rw [isOpen_iff_eventually]
rintro φ₀ hφ₀
rcases φ₀.injective_iff_antilipschitz.mp hφ₀ with ⟨K, K_pos, H⟩
have : ∀ᶠ φ in 𝓝 φ₀, ‖φ - φ₀‖₊ < K⁻¹ := eventually_nnnorm_sub_lt _ <| inv_pos_of_pos K_pos
filter_upwards [this] with φ hφ
apply φ.injective_iff_antilipschitz.mpr
exact ⟨(K⁻¹ - ‖φ - φ₀‖₊)⁻¹, inv_pos_... | true |
import Mathlib.RingTheory.WittVector.Frobenius
import Mathlib.RingTheory.WittVector.Verschiebung
import Mathlib.RingTheory.WittVector.MulP
#align_import ring_theory.witt_vector.identities from "leanprover-community/mathlib"@"0798037604b2d91748f9b43925fb7570a5f3256c"
namespace WittVector
variable {p : ℕ} {R : Typ... | Mathlib/RingTheory/WittVector/Identities.lean | 64 | 71 | theorem coeff_p_pow_eq_zero [CharP R p] {i j : ℕ} (hj : j ≠ i) : ((p : 𝕎 R) ^ i).coeff j = 0 := by
induction' i with i hi generalizing j |
induction' i with i hi generalizing j
· rw [pow_zero, one_coeff_eq_of_pos]
exact Nat.pos_of_ne_zero hj
· rw [pow_succ, ← frobenius_verschiebung, coeff_frobenius_charP]
cases j
· rw [verschiebung_coeff_zero, zero_pow hp.out.ne_zero]
· rw [verschiebung_coeff_succ, hi (ne_of_apply_ne _ hj), zero_pow... | true |
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.Perm
import Mathlib.GroupTheory.Perm.Finite
import Mathlib.GroupTheory.Perm.List
#align_import group_theory.perm.cycle.basic from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
open Equiv Function Finset
variable {... | Mathlib/GroupTheory/Perm/Cycle/Basic.lean | 90 | 90 | theorem sameCycle_one : SameCycle 1 x y ↔ x = y := by | simp [SameCycle]
| true |
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Data.Nat.Cast.Order
import Mathlib.Tactic.Common
#align_import data.nat.cast.field from "leanprover-community/mathlib"@"acee671f47b8e7972a1eb6f4eed74b4b3abce829"
namespace Nat
variable {α : Type*}
@[simp]
theorem cast_div [DivisionSemiring α] {m n : ℕ} (n_... | Mathlib/Data/Nat/Cast/Field.lean | 36 | 41 | theorem cast_div_div_div_cancel_right [DivisionSemiring α] [CharZero α] {m n d : ℕ}
(hn : d ∣ n) (hm : d ∣ m) :
(↑(m / d) : α) / (↑(n / d) : α) = (m : α) / n := by
rcases eq_or_ne d 0 with (rfl | hd); · simp [Nat.zero_dvd.1 hm] |
rcases eq_or_ne d 0 with (rfl | hd); · simp [Nat.zero_dvd.1 hm]
replace hd : (d : α) ≠ 0 := by norm_cast
rw [cast_div hm, cast_div hn, div_div_div_cancel_right _ hd] <;> exact hd
| true |
import Mathlib.Data.Bracket
import Mathlib.LinearAlgebra.Basic
#align_import algebra.lie.basic from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
universe u v w w₁ w₂
open Function
class LieRing (L : Type v) extends AddCommGroup L, Bracket L L where
protected add_lie : ∀ x y z ... | Mathlib/Algebra/Lie/Basic.lean | 179 | 179 | theorem lie_sub : ⁅x, m - n⁆ = ⁅x, m⁆ - ⁅x, n⁆ := by | simp [sub_eq_add_neg]
| true |
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
#align_import geometry.euclidean.angle.unoriented.right_angle from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped EuclideanGeometry
... | Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean | 77 | 91 | theorem angle_add_eq_arcsin_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) (h0 : x ≠ 0 ∨ y ≠ 0) :
angle x (x + y) = Real.arcsin (‖y‖ / ‖x + y‖) := by
have hxy : ‖x + y‖ ^ 2 ≠ 0 := by |
have hxy : ‖x + y‖ ^ 2 ≠ 0 := by
rw [pow_two, norm_add_sq_eq_norm_sq_add_norm_sq_real h, ne_comm]
refine ne_of_lt ?_
rcases h0 with (h0 | h0)
· exact
Left.add_pos_of_pos_of_nonneg (mul_self_pos.2 (norm_ne_zero_iff.2 h0)) (mul_self_nonneg _)
· exact
Left.add_pos_of_nonneg_of_pos (m... | true |
import Mathlib.Algebra.Algebra.Quasispectrum
import Mathlib.Topology.ContinuousFunction.Compact
import Mathlib.Topology.ContinuousFunction.ContinuousMapZero
import Mathlib.Topology.ContinuousFunction.FunctionalCalculus
import Mathlib.Topology.UniformSpace.CompactConvergence
local notation "σₙ" => quasispectrum
open... | Mathlib/Topology/ContinuousFunction/NonUnitalFunctionalCalculus.lean | 147 | 167 | theorem cfcₙHom_comp [UniqueNonUnitalContinuousFunctionalCalculus R A] (f : C(σₙ R a, R)₀)
(f' : C(σₙ R a, σₙ R (cfcₙHom ha f))₀)
(hff' : ∀ x, f x = f' x) (g : C(σₙ R (cfcₙHom ha f), R)₀) :
cfcₙHom ha (g.comp f') = cfcₙHom (cfcₙHom_predicate ha f) g := by
let ψ : C(σₙ R (cfcₙHom ha f), R)₀ →⋆ₙₐ[R] C(σₙ R ... |
let ψ : C(σₙ R (cfcₙHom ha f), R)₀ →⋆ₙₐ[R] C(σₙ R a, R)₀ :=
{ toFun := (ContinuousMapZero.comp · f')
map_smul' := fun _ _ ↦ rfl
map_add' := fun _ _ ↦ rfl
map_mul' := fun _ _ ↦ rfl
map_zero' := rfl
map_star' := fun _ ↦ rfl }
let φ : C(σₙ R (cfcₙHom ha f), R)₀ →⋆ₙₐ[R] A := (cfcₙHom ... | true |
import Mathlib.Algebra.Module.Defs
import Mathlib.Data.Fintype.BigOperators
import Mathlib.GroupTheory.GroupAction.BigOperators
#align_import algebra.module.big_operators from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
variable {ι κ α β R M : Type*}
section AddCommMonoid
variable [... | Mathlib/Algebra/Module/BigOperators.lean | 30 | 34 | theorem Multiset.sum_smul_sum {s : Multiset R} {t : Multiset M} :
s.sum • t.sum = ((s ×ˢ t).map fun p : R × M ↦ p.fst • p.snd).sum := by
induction' s using Multiset.induction with a s ih |
induction' s using Multiset.induction with a s ih
· simp
· simp [add_smul, ih, ← Multiset.smul_sum]
| true |
import Mathlib.Algebra.CharP.ExpChar
import Mathlib.GroupTheory.OrderOfElement
#align_import algebra.char_p.two from "leanprover-community/mathlib"@"7f1ba1a333d66eed531ecb4092493cd1b6715450"
variable {R ι : Type*}
namespace CharTwo
section CommSemiring
variable [CommSemiring R] [CharP R 2]
theorem add_sq (x y... | Mathlib/Algebra/CharP/Two.lean | 91 | 92 | theorem add_mul_self (x y : R) : (x + y) * (x + y) = x * x + y * y := by |
rw [← pow_two, ← pow_two, ← pow_two, add_sq]
| true |
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 | 90 | 94 | theorem natDegree_C_mul_le (a : R) (f : R[X]) : (C a * f).natDegree ≤ f.natDegree :=
calc
(C a * f).natDegree ≤ (C a).natDegree + f.natDegree := natDegree_mul_le
_ = 0 + f.natDegree := by | rw [natDegree_C a]
_ = f.natDegree := zero_add _
| true |
import Mathlib.SetTheory.Game.Ordinal
import Mathlib.SetTheory.Ordinal.NaturalOps
#align_import set_theory.game.birthday from "leanprover-community/mathlib"@"a347076985674932c0e91da09b9961ed0a79508c"
universe u
open Ordinal
namespace SetTheory
open scoped NaturalOps PGame
namespace PGame
noncomputable def b... | Mathlib/SetTheory/Game/Birthday.lean | 142 | 143 | theorem neg_birthday_le : -x.birthday.toPGame ≤ x := by |
simpa only [neg_birthday, ← neg_le_iff] using le_birthday (-x)
| true |
import Mathlib.Combinatorics.SimpleGraph.DegreeSum
import Mathlib.Combinatorics.SimpleGraph.Subgraph
#align_import combinatorics.simple_graph.matching from "leanprover-community/mathlib"@"138448ae98f529ef34eeb61114191975ee2ca508"
universe u
namespace SimpleGraph
variable {V : Type u} {G : SimpleGraph V} (M : Su... | Mathlib/Combinatorics/SimpleGraph/Matching.lean | 96 | 98 | theorem isMatching_iff_forall_degree {M : Subgraph G} [∀ v : V, Fintype (M.neighborSet v)] :
M.IsMatching ↔ ∀ v : V, v ∈ M.verts → M.degree v = 1 := by |
simp only [degree_eq_one_iff_unique_adj, IsMatching]
| true |
import Mathlib.RingTheory.Ideal.Operations
import Mathlib.Algebra.Module.Torsion
import Mathlib.Algebra.Ring.Idempotents
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Filtration
import Mathlib.RingTheory.Nakayama
#align_import ring_theory.ideal.cota... | Mathlib/RingTheory/Ideal/Cotangent.lean | 69 | 71 | theorem mem_toCotangent_ker {x : I} : x ∈ LinearMap.ker I.toCotangent ↔ (x : R) ∈ I ^ 2 := by
rw [← I.map_toCotangent_ker] |
rw [← I.map_toCotangent_ker]
simp
| true |
import Mathlib.Analysis.LocallyConvex.Bounded
import Mathlib.Topology.Algebra.Module.StrongTopology
#align_import analysis.normed_space.compact_operator from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open Function Set Filter Bornology Metric Pointwise Topology
def IsCompactOperat... | Mathlib/Analysis/NormedSpace/CompactOperator.lean | 252 | 257 | theorem IsCompactOperator.comp_clm [AddCommMonoid M₂] [Module R₂ M₂] {f : M₂ → M₃}
(hf : IsCompactOperator f) (g : M₁ →SL[σ₁₂] M₂) : IsCompactOperator (f ∘ g) := by
have := g.continuous.tendsto 0 |
have := g.continuous.tendsto 0
rw [map_zero] at this
rcases hf with ⟨K, hK, hKf⟩
exact ⟨K, hK, this hKf⟩
| true |
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.RingTheory.Ideal.Quotient
#align_import algebra.char_p.quotient from "leanprover-community/mathlib"@"85e3c05a94b27c84dc6f234cf88326d5e0096ec3"
universe u v
namespace CharP
theorem quotient (R : Type u) [CommRing R] (p ... | Mathlib/Algebra/CharP/Quotient.lean | 54 | 56 | theorem quotient_iff_le_ker_natCast {R : Type*} [CommRing R] (n : ℕ) [CharP R n] (I : Ideal R) :
CharP (R ⧸ I) n ↔ I.comap (Nat.castRingHom R) ≤ RingHom.ker (Nat.castRingHom R) := by |
rw [CharP.quotient_iff, RingHom.ker_eq_comap_bot]; rfl
| true |
import Mathlib.Topology.MetricSpace.HausdorffDistance
#align_import topology.metric_space.pi_nat from "leanprover-community/mathlib"@"49b7f94aab3a3bdca1f9f34c5d818afb253b3993"
noncomputable section
open scoped Classical
open Topology Filter
open TopologicalSpace Set Metric Filter Function
attribute [local simp... | Mathlib/Topology/MetricSpace/PiNat.lean | 154 | 161 | theorem mem_cylinder_iff_le_firstDiff {x y : ∀ n, E n} (hne : x ≠ y) (i : ℕ) :
x ∈ cylinder y i ↔ i ≤ firstDiff x y := by
constructor |
constructor
· intro h
by_contra!
exact apply_firstDiff_ne hne (h _ this)
· intro hi j hj
exact apply_eq_of_lt_firstDiff (hj.trans_le hi)
| true |
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 | 117 | 128 | theorem gramSchmidt_inv_triangular (v : ι → E) {i j : ι} (hij : i < j) :
⟪gramSchmidt 𝕜 v j, v i⟫ = 0 := by
rw [gramSchmidt_def'' 𝕜 v] |
rw [gramSchmidt_def'' 𝕜 v]
simp only [inner_add_right, inner_sum, inner_smul_right]
set b : ι → E := gramSchmidt 𝕜 v
convert zero_add (0 : 𝕜)
· exact gramSchmidt_orthogonal 𝕜 v hij.ne'
apply Finset.sum_eq_zero
rintro k hki'
have hki : k < i := by simpa using hki'
have : ⟪b j, b k⟫ = 0 := gramSchm... | true |
import Mathlib.Combinatorics.Quiver.Basic
import Mathlib.Combinatorics.Quiver.Path
#align_import combinatorics.quiver.cast from "leanprover-community/mathlib"@"fc2ed6f838ce7c9b7c7171e58d78eaf7b438fb0e"
universe v v₁ v₂ u u₁ u₂
variable {U : Type*} [Quiver.{u + 1} U]
namespace Quiver
def Hom.cast {u v u' v... | Mathlib/Combinatorics/Quiver/Cast.lean | 107 | 109 | theorem Path.cast_nil {u u' : U} (hu : u = u') : (Path.nil : Path u u).cast hu hu = Path.nil := by
subst_vars |
subst_vars
rfl
| true |
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
universe u v w
namespace Ideal
open Set
variabl... | Mathlib/RingTheory/Ideal/Quotient.lean | 129 | 130 | theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by |
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
| true |
import Mathlib.Algebra.ContinuedFractions.Translations
#align_import algebra.continued_fractions.terminated_stable from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b348ce40ad"
namespace GeneralizedContinuedFraction
variable {K : Type*} {g : GeneralizedContinuedFraction K} {n m : ℕ}
theorem te... | Mathlib/Algebra/ContinuedFractions/TerminatedStable.lean | 91 | 93 | theorem convergents'_stable_of_terminated (n_le_m : n ≤ m) (terminated_at_n : g.TerminatedAt n) :
g.convergents' m = g.convergents' n := by |
simp only [convergents', convergents'Aux_stable_of_terminated n_le_m terminated_at_n]
| true |
import Mathlib.Data.Complex.Module
import Mathlib.Data.Complex.Order
import Mathlib.Data.Complex.Exponential
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Topology.Algebra.InfiniteSum.Module
import Mathlib.Topology.Instances.RealVectorSpace
#align_import analysis.complex.basic from "leanprover-community/mathlib... | Mathlib/Analysis/Complex/Basic.lean | 149 | 150 | theorem nndist_self_conj (z : ℂ) : nndist z (conj z) = 2 * Real.nnabs z.im := by |
rw [nndist_comm, nndist_conj_self]
| true |
import Mathlib.Algebra.Order.Floor
import Mathlib.Topology.Algebra.Order.Group
import Mathlib.Topology.Order.Basic
#align_import topology.algebra.order.floor from "leanprover-community/mathlib"@"84dc0bd6619acaea625086d6f53cb35cdd554219"
open Filter Function Int Set Topology
variable {α β γ : Type*} [LinearOrdere... | Mathlib/Topology/Algebra/Order/Floor.lean | 88 | 93 | theorem tendsto_floor_left_pure_ceil_sub_one (x : α) :
Tendsto (floor : α → ℤ) (𝓝[<] x) (pure (⌈x⌉ - 1)) :=
have h₁ : ↑(⌈x⌉ - 1) < x := by rw [cast_sub, cast_one, sub_lt_iff_lt_add]; exact ceil_lt_add_one _ | rw [cast_sub, cast_one, sub_lt_iff_lt_add]; exact ceil_lt_add_one _
have h₂ : x ≤ ↑(⌈x⌉ - 1) + 1 := by rw [cast_sub, cast_one, sub_add_cancel]; exact le_ceil _
tendsto_pure.2 <| mem_of_superset (Ico_mem_nhdsWithin_Iio' h₁) fun _y hy =>
floor_eq_on_Ico _ _ ⟨hy.1, hy.2.trans_le h₂⟩
| true |
import Mathlib.Algebra.MvPolynomial.Counit
import Mathlib.Algebra.MvPolynomial.Invertible
import Mathlib.RingTheory.WittVector.Defs
#align_import ring_theory.witt_vector.basic from "leanprover-community/mathlib"@"9556784a5b84697562e9c6acb40500d4a82e675a"
noncomputable section
open MvPolynomial Function
variable... | Mathlib/RingTheory/WittVector/Basic.lean | 114 | 114 | theorem mul : mapFun f (x * y) = mapFun f x * mapFun f y := by | map_fun_tac
| true |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Finset.NatAntidiagonal
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Logic.Function.Iterate
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Zify
#align_import data.nat.fib from "leanprover-community/mathlib"@"... | Mathlib/Data/Nat/Fib/Basic.lean | 110 | 111 | theorem fib_add_two_sub_fib_add_one {n : ℕ} : fib (n + 2) - fib (n + 1) = fib n := by |
rw [fib_add_two, add_tsub_cancel_right]
| true |
import Mathlib.Tactic.Basic
import Mathlib.Init.Data.Int.Basic
class CanLift (α β : Sort*) (coe : outParam <| β → α) (cond : outParam <| α → Prop) : Prop where
prf : ∀ x : α, cond x → ∃ y : β, coe y = x
#align can_lift CanLift
instance : CanLift ℤ ℕ (fun n : ℕ ↦ n) (0 ≤ ·) :=
⟨fun n hn ↦ ⟨n.natAbs, Int.nat... | Mathlib/Tactic/Lift.lean | 38 | 43 | theorem Subtype.exists_pi_extension {ι : Sort*} {α : ι → Sort*} [ne : ∀ i, Nonempty (α i)]
{p : ι → Prop} (f : ∀ i : Subtype p, α i) :
∃ g : ∀ i : ι, α i, (fun i : Subtype p => g i) = f := by
haveI : DecidablePred p := fun i ↦ Classical.propDecidable (p i) |
haveI : DecidablePred p := fun i ↦ Classical.propDecidable (p i)
exact ⟨fun i => if hi : p i then f ⟨i, hi⟩ else Classical.choice (ne i),
funext fun i ↦ dif_pos i.2⟩
| true |
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.Seminorm
import Mathlib.Analysis.LocallyConvex.Bounded
import Mathlib.Analysis.RCLike.Basic
#align_import analysis.convex.gauge from "leanprover-community/mathlib"@"373b03b5b9d0486534edbe94747f23cb3712f93d"
... | Mathlib/Analysis/Convex/Gauge.lean | 114 | 116 | theorem gauge_empty : gauge (∅ : Set E) = 0 := by
ext |
ext
simp only [gauge_def', Real.sInf_empty, mem_empty_iff_false, Pi.zero_apply, sep_false]
| true |
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.List.Perm
import Mathlib.Data.List.Range
#align_import data.list.sublists from "leanprover-community/mathlib"@"ccad6d5093bd2f5c6ca621fc74674cce51355af6"
universe u v w
variable {α : Type u} {β : Type v} {γ : Type w}
open Nat
namespace List
@[simp]
theo... | Mathlib/Data/List/Sublists.lean | 61 | 66 | theorem sublists'_eq_sublists'Aux (l : List α) :
sublists' l = l.foldr (fun a r => sublists'Aux a r r) [[]] := by
simp only [sublists', sublists'Aux_eq_array_foldl] |
simp only [sublists', sublists'Aux_eq_array_foldl]
rw [← List.foldr_hom Array.toList]
· rfl
· intros _ _; congr <;> simp
| true |
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Algebra.Order.Monoid.WithTop
#align_import data.nat.with_bot from "leanprover-community/mathlib"@"966e0cf0685c9cedf8a3283ac69eef4d5f2eaca2"
namespace Nat
namespace WithBot
instance : WellFoundedRelation (WithBot ℕ) where
rel := (· < ·)
wf := IsWellFounde... | Mathlib/Data/Nat/WithBot.lean | 52 | 58 | theorem add_eq_three_iff {n m : WithBot ℕ} :
n + m = 3 ↔ n = 0 ∧ m = 3 ∨ n = 1 ∧ m = 2 ∨ n = 2 ∧ m = 1 ∨ n = 3 ∧ m = 0 := by
rcases n, m with ⟨_ | _, _ | _⟩ |
rcases n, m with ⟨_ | _, _ | _⟩
repeat refine ⟨fun h => Option.noConfusion h, fun h => ?_⟩;
aesop (simp_config := { decide := true })
repeat erw [WithBot.coe_eq_coe]
exact Nat.add_eq_three_iff
| true |
import Mathlib.SetTheory.Ordinal.Arithmetic
#align_import set_theory.ordinal.exponential from "leanprover-community/mathlib"@"b67044ba53af18680e1dd246861d9584e968495d"
noncomputable section
open Function Cardinal Set Equiv Order
open scoped Classical
open Cardinal Ordinal
universe u v w
namespace Ordinal
in... | Mathlib/SetTheory/Ordinal/Exponential.lean | 68 | 69 | theorem opow_le_of_limit {a b c : Ordinal} (a0 : a ≠ 0) (h : IsLimit b) :
a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ c := by | rw [opow_limit a0 h, bsup_le_iff]
| true |
import Mathlib.CategoryTheory.Sites.Coherent.Basic
import Mathlib.CategoryTheory.EffectiveEpi.Comp
import Mathlib.CategoryTheory.EffectiveEpi.Extensive
namespace CategoryTheory
open Limits GrothendieckTopology Sieve
variable (C : Type*) [Category C]
instance [Precoherent C] [HasFiniteCoproducts C] : Preregular C ... | Mathlib/CategoryTheory/Sites/Coherent/Comparison.lean | 57 | 94 | theorem extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) := by
ext B S |
ext B S
refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩
· induction h with
| of Y T hT =>
apply Coverage.saturate.of
simp only [Coverage.sup_covering, Set.mem_union] at hT
exact Or.elim hT
(fun ⟨α, x, X, π, ⟨h, _⟩⟩ ↦ ⟨α, x, X, π, ⟨h, inferInstance⟩⟩)
(fun ⟨Z, f, ⟨h, _⟩⟩ ↦ ⟨Unit, inferInstan... | true |
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Analysis.NormedSpace.Banach
import Mathlib.LinearAlgebra.SesquilinearForm
#align_import analysis.inner_product_space.symmetric from "leanprover-community/mathlib"@"3f655f5297b030a87d641ad4e825af8d9679eb0b"
open RCLike
open ComplexConjugate
variable ... | Mathlib/Analysis/InnerProductSpace/Symmetric.lean | 163 | 180 | theorem IsSymmetric.inner_map_polarization {T : E →ₗ[𝕜] E} (hT : T.IsSymmetric) (x y : E) :
⟪T x, y⟫ =
(⟪T (x + y), x + y⟫ - ⟪T (x - y), x - y⟫ - I * ⟪T (x + (I : 𝕜) • y), x + (I : 𝕜) • y⟫ +
I * ⟪T (x - (I : 𝕜) • y), x - (I : 𝕜) • y⟫) /
4 := by
rcases@I_mul_I_ax 𝕜 _ with (h | h) |
rcases@I_mul_I_ax 𝕜 _ with (h | h)
· simp_rw [h, zero_mul, sub_zero, add_zero, map_add, map_sub, inner_add_left,
inner_add_right, inner_sub_left, inner_sub_right, hT x, ← inner_conj_symm x (T y)]
suffices (re ⟪T y, x⟫ : 𝕜) = ⟪T y, x⟫ by
rw [conj_eq_iff_re.mpr this]
ring
rw [← re_add_im ... | true |
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.Algebra.Module.Torsion
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v v' u₁' w w'
variable {R S : Type u} {M : Type v} {M' : Type v'} {M₁ : Type v}... | Mathlib/LinearAlgebra/Dimension/Constructions.lean | 211 | 213 | theorem rank_matrix' (m n : Type v) [Finite m] [Finite n] :
Module.rank R (Matrix m n R) = Cardinal.lift.{u} (#m * #n) := by |
rw [rank_matrix, lift_mul, lift_umax.{v, u}]
| true |
import Mathlib.LinearAlgebra.Dimension.LinearMap
import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition
#align_import linear_algebra.free_module.finite.matrix from "leanprover-community/mathlib"@"b1c23399f01266afe392a0d8f71f599a0dad4f7b"
universe u u' v w
variable (R : Type u) (S : Type u') (M : Type v) (N ... | Mathlib/LinearAlgebra/FreeModule/Finite/Matrix.lean | 113 | 119 | theorem Matrix.rank_vecMulVec {K m n : Type u} [CommRing K] [Fintype n]
[DecidableEq n] (w : m → K) (v : n → K) : (Matrix.vecMulVec w v).toLin'.rank ≤ 1 := by
nontriviality K |
nontriviality K
rw [Matrix.vecMulVec_eq, Matrix.toLin'_mul]
refine le_trans (LinearMap.rank_comp_le_left _ _) ?_
refine (LinearMap.rank_le_domain _).trans_eq ?_
rw [rank_fun', Fintype.card_unit, Nat.cast_one]
| true |
import Mathlib.GroupTheory.Solvable
import Mathlib.FieldTheory.PolynomialGaloisGroup
import Mathlib.RingTheory.RootsOfUnity.Basic
#align_import field_theory.abel_ruffini from "leanprover-community/mathlib"@"e3f4be1fcb5376c4948d7f095bec45350bfb9d1a"
noncomputable section
open scoped Classical Polynomial Intermedi... | Mathlib/FieldTheory/AbelRuffini.lean | 98 | 114 | theorem gal_X_pow_sub_one_isSolvable (n : ℕ) : IsSolvable (X ^ n - 1 : F[X]).Gal := by
by_cases hn : n = 0 |
by_cases hn : n = 0
· rw [hn, pow_zero, sub_self]
exact gal_zero_isSolvable
have hn' : 0 < n := pos_iff_ne_zero.mpr hn
have hn'' : (X ^ n - 1 : F[X]) ≠ 0 := X_pow_sub_C_ne_zero hn' 1
apply isSolvable_of_comm
intro σ τ
ext a ha
simp only [mem_rootSet_of_ne hn'', map_sub, aeval_X_pow, aeval_one, sub_... | true |
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 | 45 | 50 | theorem deriv_eq_smul_circleIntegral [CompleteSpace F] {R : ℝ} {c : ℂ} {f : ℂ → F} (hR : 0 < R)
(hf : DiffContOnCl ℂ f (ball c R)) :
deriv f c = (2 * π * I : ℂ)⁻¹ • ∮ z in C(c, R), (z - c) ^ (-2 : ℤ) • f z := by
lift R to ℝ≥0 using hR.le |
lift R to ℝ≥0 using hR.le
refine (hf.hasFPowerSeriesOnBall hR).hasFPowerSeriesAt.deriv.trans ?_
simp only [cauchyPowerSeries_apply, one_div, zpow_neg, pow_one, smul_smul, zpow_two, mul_inv]
| true |
import Mathlib.Order.Filter.Germ
import Mathlib.Topology.NhdsSet
import Mathlib.Topology.LocallyConstant.Basic
import Mathlib.Analysis.NormedSpace.Basic
variable {F G : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F]
[NormedAddCommGroup G] [NormedSpace ℝ G]
open scoped Topology
open Filter Set
variable {X Y Z ... | Mathlib/Topology/Germ.lean | 112 | 115 | theorem forall_restrictGermPredicate_iff {P : ∀ x : X, Germ (𝓝 x) Y → Prop} :
(∀ x, RestrictGermPredicate P A x f) ↔ ∀ᶠ x in 𝓝ˢ A, P x f := by
rw [eventually_nhdsSet_iff_forall] |
rw [eventually_nhdsSet_iff_forall]
rfl
| true |
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev from "leanprover-community/mathlib"@"d774451114d6045faeb6751c396bea1eb9058946"
namespace Polynomial.Chebyshev
set_option linter.uppercaseLean3 false -- `T` `U` `X`
open Polynomial
v... | Mathlib/RingTheory/Polynomial/Chebyshev.lean | 159 | 160 | theorem U_sub_one (n : ℤ) : U R (n - 1) = 2 * X * U R n - U R (n + 1) := by |
linear_combination (norm := ring_nf) U_add_two R (n - 1)
| true |
import Mathlib.SetTheory.Cardinal.ToNat
import Mathlib.Data.Nat.PartENat
#align_import set_theory.cardinal.basic from "leanprover-community/mathlib"@"3ff3f2d6a3118b8711063de7111a0d77a53219a8"
universe u v
open Function
variable {α : Type u}
namespace Cardinal
noncomputable def toPartENat : Cardinal →+o PartEN... | Mathlib/SetTheory/Cardinal/PartENat.lean | 47 | 51 | theorem toPartENat_eq_top {c : Cardinal} :
toPartENat c = ⊤ ↔ ℵ₀ ≤ c := by
rw [← partENatOfENat_toENat, ← PartENat.withTopEquiv_symm_top, ← toENat_eq_top, |
rw [← partENatOfENat_toENat, ← PartENat.withTopEquiv_symm_top, ← toENat_eq_top,
← PartENat.withTopEquiv.symm.injective.eq_iff]
simp
| true |
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 | 276 | 278 | theorem norm_eq_ciSup (f : PiLp ∞ β) : ‖f‖ = ⨆ i, ‖f i‖ := by
dsimp [Norm.norm] |
dsimp [Norm.norm]
exact if_neg ENNReal.top_ne_zero
| true |
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 | 113 | 117 | theorem Heap.noSibling_tail (le) (s : Heap α) : (s.tail le).NoSibling := by
simp only [Heap.tail] |
simp only [Heap.tail]
match eq : s.tail? le with
| none => cases s with cases eq | nil => constructor
| some tl => exact Heap.noSibling_tail? eq
| true |
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.SesquilinearForm
import Mathlib.LinearAlgebra.Matrix.Symmetric
#align_import linear_algebra.quadratic_form.basic from "leanprover-community/mathlib"@"d11f435d4e34a6cea0a1797d6b625b0c170be845"
universe u v w
variable {S T : ... | Mathlib/LinearAlgebra/QuadraticForm/Basic.lean | 107 | 108 | theorem polar_smul [Monoid S] [DistribMulAction S R] (f : M → R) (s : S) (x y : M) :
polar (s • f) x y = s • polar f x y := by | simp only [polar, Pi.smul_apply, smul_sub]
| true |
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Group.Measure
import Mathlib.Topology.Constructions
#align_import measure_theory.constructions.pi from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
noncomputable section
open Function Set MeasureTheory... | Mathlib/MeasureTheory/Constructions/Pi.lean | 69 | 73 | theorem IsPiSystem.pi {C : ∀ i, Set (Set (α i))} (hC : ∀ i, IsPiSystem (C i)) :
IsPiSystem (pi univ '' pi univ C) := by
rintro _ ⟨s₁, hs₁, rfl⟩ _ ⟨s₂, hs₂, rfl⟩ hst |
rintro _ ⟨s₁, hs₁, rfl⟩ _ ⟨s₂, hs₂, rfl⟩ hst
rw [← pi_inter_distrib] at hst ⊢; rw [univ_pi_nonempty_iff] at hst
exact mem_image_of_mem _ fun i _ => hC i _ (hs₁ i (mem_univ i)) _ (hs₂ i (mem_univ i)) (hst i)
| true |
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
import Mathlib.Topology.Order.LeftRightLim
#align_import measure_theory.measure.stieltjes from "leanprover-community/mathlib"@"20d5763051978e9bc6428578ed070445df6a18b3"
noncomputable section
open scoped Classical
open Set Filter Function ENNReal NNReal T... | Mathlib/MeasureTheory/Measure/Stieltjes.lean | 138 | 142 | theorem countable_leftLim_ne (f : StieltjesFunction) : Set.Countable { x | leftLim f x ≠ f x } := by
refine Countable.mono ?_ f.mono.countable_not_continuousAt |
refine Countable.mono ?_ f.mono.countable_not_continuousAt
intro x hx h'x
apply hx
exact tendsto_nhds_unique (f.mono.tendsto_leftLim x) (h'x.tendsto.mono_left nhdsWithin_le_nhds)
| true |
import Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho
import Mathlib.LinearAlgebra.Matrix.PosDef
#align_import linear_algebra.matrix.ldl from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
variable {𝕜 : Type*} [RCLike 𝕜]
variable {n : Type*} [LinearOrder n] [IsWellOrder n (· < ·)... | Mathlib/LinearAlgebra/Matrix/LDL.lean | 93 | 97 | theorem LDL.lowerInv_triangular {i j : n} (hij : i < j) : LDL.lowerInv hS i j = 0 := by
rw [← |
rw [←
@gramSchmidt_triangular 𝕜 (n → 𝕜) _ (_ : _) (InnerProductSpace.ofMatrix hS.transpose) n _ _ _
i j hij (Pi.basisFun 𝕜 n),
Pi.basisFun_repr, LDL.lowerInv]
| true |
import Mathlib.Algebra.Polynomial.Coeff
import Mathlib.Data.Nat.Choose.Basic
#align_import data.nat.choose.vandermonde from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"
open Polynomial Finset Finset.Nat
| Mathlib/Data/Nat/Choose/Vandermonde.lean | 27 | 34 | theorem Nat.add_choose_eq (m n k : ℕ) :
(m + n).choose k = ∑ ij ∈ antidiagonal k, m.choose ij.1 * n.choose ij.2 := by
calc |
calc
(m + n).choose k = ((X + 1) ^ (m + n)).coeff k := by rw [coeff_X_add_one_pow, Nat.cast_id]
_ = ((X + 1) ^ m * (X + 1) ^ n).coeff k := by rw [pow_add]
_ = ∑ ij ∈ antidiagonal k, m.choose ij.1 * n.choose ij.2 := by
rw [coeff_mul, Finset.sum_congr rfl]
simp only [coeff_X_add_one_pow, Nat.ca... | true |
import Mathlib.RingTheory.FinitePresentation
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Localization.Away.AdjoinRoot
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.TensorProduct.Basic
-- Porting note: added to make the syntax work below.
open scoped TensorProd... | Mathlib/RingTheory/Unramified/Basic.lean | 139 | 152 | theorem comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor |
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F... | true |
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Inverse
#align_import analysis.calculus.cont_diff from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputab... | Mathlib/Analysis/Calculus/ContDiff/Basic.lean | 126 | 132 | theorem iteratedFDerivWithin_succ_const (n : ℕ) (c : F) (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) :
iteratedFDerivWithin 𝕜 (n + 1) (fun _ : E ↦ c) s x = 0 := by
ext m |
ext m
rw [iteratedFDerivWithin_succ_apply_right hs hx]
rw [iteratedFDerivWithin_congr (fun y hy ↦ fderivWithin_const_apply c (hs y hy)) hx]
rw [iteratedFDerivWithin_zero_fun hs hx]
simp [ContinuousMultilinearMap.zero_apply (R := 𝕜)]
| true |
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Analysis.NormedSpace.LinearIsometry
#align_import analysis.normed_space.conformal_linear_map from "leanprover-community/mathlib"@"d1bd9c5df2867c1cb463bc6364446d57bdd9f7f1"
noncomputable section
open Function LinearIsometry ContinuousLinearMap
def IsConf... | Mathlib/Analysis/NormedSpace/ConformalLinearMap.lean | 84 | 89 | theorem comp (hg : IsConformalMap g) (hf : IsConformalMap f) : IsConformalMap (g.comp f) := by
rcases hf with ⟨cf, hcf, lif, rfl⟩ |
rcases hf with ⟨cf, hcf, lif, rfl⟩
rcases hg with ⟨cg, hcg, lig, rfl⟩
refine ⟨cg * cf, mul_ne_zero hcg hcf, lig.comp lif, ?_⟩
rw [smul_comp, comp_smul, mul_smul]
rfl
| true |
import Mathlib.FieldTheory.SplittingField.Construction
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.Separable
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.Polynomial.SeparableDegree
open scoped Classical Polynomial
open FiniteDimensional Polynomial Interm... | Mathlib/FieldTheory/SeparableDegree.lean | 168 | 172 | theorem finSepDegree_self : finSepDegree F F = 1 := by
have : Cardinal.mk (Emb F F) = 1 := le_antisymm |
have : Cardinal.mk (Emb F F) = 1 := le_antisymm
(Cardinal.le_one_iff_subsingleton.2 AlgHom.subsingleton)
(Cardinal.one_le_iff_ne_zero.2 <| Cardinal.mk_ne_zero _)
rw [finSepDegree, Nat.card, this, Cardinal.one_toNat]
| true |
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
#align_import analysis.special_functions.pow.asymptotics from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8"
set_option linter.uppercaseLean3 false
noncomputable section
open scoped Classical
open Real Topology NNReal ENNReal Filter C... | Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean | 132 | 137 | theorem tendsto_exp_div_rpow_atTop (s : ℝ) : Tendsto (fun x : ℝ => exp x / x ^ s) atTop atTop := by
cases' archimedean_iff_nat_lt.1 Real.instArchimedean s with n hn |
cases' archimedean_iff_nat_lt.1 Real.instArchimedean s with n hn
refine tendsto_atTop_mono' _ ?_ (tendsto_exp_div_pow_atTop n)
filter_upwards [eventually_gt_atTop (0 : ℝ), eventually_ge_atTop (1 : ℝ)] with x hx₀ hx₁
rw [div_le_div_left (exp_pos _) (pow_pos hx₀ _) (rpow_pos_of_pos hx₀ _), ← Real.rpow_natCast]
... | true |
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Nat.Factors
import Mathlib.Order.Interval.Finset.Nat
#align_import number_theory.divisors from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
open scoped Classical
open Finset
namespace Nat
variable (n : ℕ)
d... | Mathlib/NumberTheory/Divisors.lean | 89 | 91 | theorem cons_self_properDivisors (h : n ≠ 0) :
cons n (properDivisors n) properDivisors.not_self_mem = divisors n := by |
rw [cons_eq_insert, insert_self_properDivisors h]
| true |
import Mathlib.Analysis.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn
#align_import analysis.calculus.inverse from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
open Function Set Filter Metric
open scoped Topology Classical NNReal
n... | Mathlib/Analysis/Calculus/InverseFunctionTheorem/FDeriv.lean | 86 | 96 | theorem map_nhds_eq_of_surj [CompleteSpace E] [CompleteSpace F] {f : E → F} {f' : E →L[𝕜] F} {a : E}
(hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) (h : LinearMap.range f' = ⊤) :
map f (𝓝 a) = 𝓝 (f a) := by
let f'symm := f'.nonlinearRightInverseOfSurjective h |
let f'symm := f'.nonlinearRightInverseOfSurjective h
set c : ℝ≥0 := f'symm.nnnorm⁻¹ / 2 with hc
have f'symm_pos : 0 < f'symm.nnnorm := f'.nonlinearRightInverseOfSurjective_nnnorm_pos h
have cpos : 0 < c := by simp [hc, half_pos, inv_pos, f'symm_pos]
obtain ⟨s, s_nhds, hs⟩ : ∃ s ∈ 𝓝 a, ApproximatesLinearOn f... | true |
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 | 493 | 496 | theorem ucs_eq_self_of_normalizer_eq_self (h : N₁.normalizer = N₁) (k : ℕ) : N₁.ucs k = N₁ := by
induction' k with k ih |
induction' k with k ih
· simp
· rwa [ucs_succ, ih]
| true |
import Mathlib.Data.Finset.Grade
import Mathlib.Order.Interval.Finset.Basic
#align_import data.finset.interval from "leanprover-community/mathlib"@"98e83c3d541c77cdb7da20d79611a780ff8e7d90"
variable {α β : Type*}
namespace Finset
section Decidable
variable [DecidableEq α] (s t : Finset α)
instance instLocally... | Mathlib/Data/Finset/Interval.lean | 101 | 106 | theorem card_Icc_finset (h : s ⊆ t) : (Icc s t).card = 2 ^ (t.card - s.card) := by
rw [← card_sdiff h, ← card_powerset, Icc_eq_image_powerset h, Finset.card_image_iff] |
rw [← card_sdiff h, ← card_powerset, Icc_eq_image_powerset h, Finset.card_image_iff]
rintro u hu v hv (huv : s ⊔ u = s ⊔ v)
rw [mem_coe, mem_powerset] at hu hv
rw [← (disjoint_sdiff.mono_right hu : Disjoint s u).sup_sdiff_cancel_left, ←
(disjoint_sdiff.mono_right hv : Disjoint s v).sup_sdiff_cancel_left, h... | true |
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Order.BigOperators.Group.List
import Mathlib.Data.Set.Pointwise.SMul
import Mathlib.Order.WellFoundedSet
#align_import group_theory.submonoid.pointwise from "leanprover-community/mathlib"@"2bbc7e3884ba234309d2a43b19144105a753292e"
open Set P... | Mathlib/Algebra/Group/Submonoid/Pointwise.lean | 700 | 704 | theorem submonoid_closure (hpos : ∀ x : α, x ∈ s → 1 ≤ x) (h : s.IsPWO) :
IsPWO (Submonoid.closure s : Set α) := by
rw [Submonoid.closure_eq_image_prod] |
rw [Submonoid.closure_eq_image_prod]
refine (h.partiallyWellOrderedOn_sublistForall₂ (· ≤ ·)).image_of_monotone_on ?_
exact fun l1 _ l2 hl2 h12 => h12.prod_le_prod' fun x hx => hpos x <| hl2 x hx
| true |
import Mathlib.Data.Int.Interval
import Mathlib.RingTheory.Binomial
import Mathlib.RingTheory.HahnSeries.PowerSeries
import Mathlib.RingTheory.HahnSeries.Summable
import Mathlib.FieldTheory.RatFunc.AsPolynomial
import Mathlib.RingTheory.Localization.FractionRing
#align_import ring_theory.laurent_series from "leanprov... | Mathlib/RingTheory/LaurentSeries.lean | 112 | 121 | theorem powerSeriesPart_eq_zero (x : LaurentSeries R) : x.powerSeriesPart = 0 ↔ x = 0 := by
constructor |
constructor
· contrapose!
simp only [ne_eq]
intro h
rw [PowerSeries.ext_iff, not_forall]
refine ⟨0, ?_⟩
simp [coeff_order_ne_zero h]
· rintro rfl
simp
| true |
import Mathlib.MeasureTheory.Measure.MeasureSpace
import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
#align_import measure_theory.measure.open_pos from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Topology ENNReal MeasureTheory
open Set Function Filter
namespace Measur... | Mathlib/MeasureTheory/Measure/OpenPos.lean | 97 | 100 | theorem _root_.IsClosed.ae_eq_univ_iff_eq (hF : IsClosed F) :
F =ᵐ[μ] univ ↔ F = univ := by
refine ⟨fun h ↦ ?_, fun h ↦ by rw [h]⟩ |
refine ⟨fun h ↦ ?_, fun h ↦ by rw [h]⟩
rwa [ae_eq_univ, hF.isOpen_compl.measure_eq_zero_iff μ, compl_empty_iff] at h
| true |
import Mathlib.LinearAlgebra.Basis.VectorSpace
import Mathlib.LinearAlgebra.Dimension.Constructions
import Mathlib.LinearAlgebra.Dimension.Finite
#align_import field_theory.finiteness from "leanprover-community/mathlib"@"039a089d2a4b93c761b234f3e5f5aeb752bac60f"
universe u v
open scoped Classical
open Cardinal
... | Mathlib/FieldTheory/Finiteness.lean | 103 | 112 | theorem iff_fg : IsNoetherian K V ↔ Module.Finite K V := by
constructor |
constructor
· intro h
exact
⟨⟨finsetBasisIndex K V, by
convert (finsetBasis K V).span_eq
simp⟩⟩
· rintro ⟨s, hs⟩
rw [IsNoetherian.iff_rank_lt_aleph0, ← rank_top, ← hs]
exact lt_of_le_of_lt (rank_span_le _) s.finite_toSet.lt_aleph0
| true |
import Mathlib.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
#align_import data.set.intervals.with_bot_top from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
variable {α : Type*}
namespace WithTop
@[simp]
theorem preimage_coe_top : (some : α → WithTop α) ⁻¹' {⊤} =... | Mathlib/Order/Interval/Set/WithBotTop.lean | 89 | 90 | theorem image_coe_Ioi : (some : α → WithTop α) '' Ioi a = Ioo (a : WithTop α) ⊤ := by |
rw [← preimage_coe_Ioi, image_preimage_eq_inter_range, range_coe, Ioi_inter_Iio]
| true |
import Mathlib.Analysis.Convex.Gauge
import Mathlib.Analysis.Convex.Normed
open Metric Bornology Filter Set
open scoped NNReal Topology Pointwise
noncomputable section
section Module
variable {E : Type*} [AddCommGroup E] [Module ℝ E]
def gaugeRescale (s t : Set E) (x : E) : E := (gauge s x / gauge t x) • x
the... | Mathlib/Analysis/Convex/GaugeRescale.lean | 63 | 67 | theorem gauge_gaugeRescale (s : Set E) {t : Set E} (hta : Absorbent ℝ t) (htb : IsVonNBounded ℝ t)
(x : E) : gauge t (gaugeRescale s t x) = gauge s x := by
rcases eq_or_ne x 0 with rfl | hx |
rcases eq_or_ne x 0 with rfl | hx
· simp
· exact gauge_gaugeRescale' s ((gauge_pos hta htb).2 hx).ne'
| true |
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 | 162 | 169 | theorem ext_linear {f g : P1 →ᵃ[k] P2} (h₁ : f.linear = g.linear) {p : P1} (h₂ : f p = g p) :
f = g := by
ext q |
ext q
have hgl : g.linear (q -ᵥ p) = toFun g ((q -ᵥ p) +ᵥ q) -ᵥ toFun g q := by simp
have := f.map_vadd' q (q -ᵥ p)
rw [h₁, hgl, toFun_eq_coe, map_vadd, linearMap_vsub, h₂] at this
simp at this
exact this
| true |
import Mathlib.Data.List.Duplicate
import Mathlib.Data.List.Sort
#align_import data.list.nodup_equiv_fin from "leanprover-community/mathlib"@"008205aa645b3f194c1da47025c5f110c8406eab"
namespace List
variable {α : Type*}
section Sublist
| Mathlib/Data/List/NodupEquivFin.lean | 116 | 137 | theorem sublist_of_orderEmbedding_get?_eq {l l' : List α} (f : ℕ ↪o ℕ)
(hf : ∀ ix : ℕ, l.get? ix = l'.get? (f ix)) : l <+ l' := by
induction' l with hd tl IH generalizing l' f |
induction' l with hd tl IH generalizing l' f
· simp
have : some hd = _ := hf 0
rw [eq_comm, List.get?_eq_some] at this
obtain ⟨w, h⟩ := this
let f' : ℕ ↪o ℕ :=
OrderEmbedding.ofMapLEIff (fun i => f (i + 1) - (f 0 + 1)) fun a b => by
dsimp only
rw [Nat.sub_le_sub_iff_right, OrderEmbedding.le... | true |
import Mathlib.Algebra.FreeMonoid.Basic
import Mathlib.Algebra.Group.Submonoid.MulOpposite
import Mathlib.Algebra.Group.Submonoid.Operations
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.Finset.NoncommProd
import Mathlib.Data.Int.Order.Lemmas
#align_import group_theory.submonoid.membership fro... | Mathlib/Algebra/Group/Submonoid/Membership.lean | 234 | 236 | theorem mem_sup_left {S T : Submonoid M} : ∀ {x : M}, x ∈ S → x ∈ S ⊔ T := by
rw [← SetLike.le_def] |
rw [← SetLike.le_def]
exact le_sup_left
| true |
import Mathlib.MeasureTheory.Measure.Restrict
open scoped ENNReal NNReal Topology
open Set MeasureTheory Measure Filter Function MeasurableSpace ENNReal
variable {α β δ ι : Type*}
namespace MeasureTheory
variable {m0 : MeasurableSpace α} [MeasurableSpace β] {μ ν ν₁ ν₂: Measure α}
{s t : Set α}
section IsFinit... | Mathlib/MeasureTheory/Measure/Typeclasses.lean | 132 | 139 | theorem Measure.isFiniteMeasure_map {m : MeasurableSpace α} (μ : Measure α) [IsFiniteMeasure μ]
(f : α → β) : IsFiniteMeasure (μ.map f) := by
by_cases hf : AEMeasurable f μ |
by_cases hf : AEMeasurable f μ
· constructor
rw [map_apply_of_aemeasurable hf MeasurableSet.univ]
exact measure_lt_top μ _
· rw [map_of_not_aemeasurable hf]
exact MeasureTheory.isFiniteMeasureZero
| true |
import Mathlib.Topology.Constructions
import Mathlib.Topology.ContinuousOn
#align_import topology.bases from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78ad6527b7e4"
open Set Filter Function Topology
noncomputable section
namespace TopologicalSpace
universe u
variable {α : Type u} {β : Type*} ... | Mathlib/Topology/Bases.lean | 77 | 90 | theorem IsTopologicalBasis.insert_empty {s : Set (Set α)} (h : IsTopologicalBasis s) :
IsTopologicalBasis (insert ∅ s) := by
refine ⟨?_, by rw [sUnion_insert, empty_union, h.sUnion_eq], ?_⟩ |
refine ⟨?_, by rw [sUnion_insert, empty_union, h.sUnion_eq], ?_⟩
· rintro t₁ (rfl | h₁) t₂ (rfl | h₂) x ⟨hx₁, hx₂⟩
· cases hx₁
· cases hx₁
· cases hx₂
· obtain ⟨t₃, h₃, hs⟩ := h.exists_subset_inter _ h₁ _ h₂ x ⟨hx₁, hx₂⟩
exact ⟨t₃, .inr h₃, hs⟩
· rw [h.eq_generateFrom]
refine le_antisym... | true |
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.Seminorm
import Mathlib.Analysis.LocallyConvex.Bounded
import Mathlib.Analysis.RCLike.Basic
#align_import analysis.convex.gauge from "leanprover-community/mathlib"@"373b03b5b9d0486534edbe94747f23cb3712f93d"
... | Mathlib/Analysis/Convex/Gauge.lean | 86 | 89 | theorem exists_lt_of_gauge_lt (absorbs : Absorbent ℝ s) (h : gauge s x < a) :
∃ b, 0 < b ∧ b < a ∧ x ∈ b • s := by
obtain ⟨b, ⟨hb, hx⟩, hba⟩ := exists_lt_of_csInf_lt absorbs.gauge_set_nonempty h |
obtain ⟨b, ⟨hb, hx⟩, hba⟩ := exists_lt_of_csInf_lt absorbs.gauge_set_nonempty h
exact ⟨b, hb, hba, hx⟩
| true |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.