Context stringlengths 57 85k | file_name stringlengths 21 79 | start int64 14 2.42k | end int64 18 2.43k | theorem stringlengths 25 2.71k | proof stringlengths 5 10.6k |
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import Mathlib.NumberTheory.FLT.Basic
import Mathlib.NumberTheory.PythagoreanTriples
import Mathlib.RingTheory.Coprime.Lemmas
import Mathlib.Tactic.LinearCombination
#align_import number_theory.fermat4 from "leanprover-community/mathlib"@"10b4e499f43088dd3bb7b5796184ad5216648ab1"
noncomputable section
open scope... | Mathlib/NumberTheory/FLT/Four.lean | 58 | 62 | theorem ne_zero {a b c : ℤ} (h : Fermat42 a b c) : c ≠ 0 := by |
apply ne_zero_pow two_ne_zero _; apply ne_of_gt
rw [← h.2.2, (by ring : a ^ 4 + b ^ 4 = (a ^ 2) ^ 2 + (b ^ 2) ^ 2)]
exact
add_pos (sq_pos_of_ne_zero (pow_ne_zero 2 h.1)) (sq_pos_of_ne_zero (pow_ne_zero 2 h.2.1))
|
import Mathlib.Data.Set.Image
#align_import data.nat.set from "leanprover-community/mathlib"@"cf9386b56953fb40904843af98b7a80757bbe7f9"
namespace Nat
section Set
open Set
| Mathlib/Data/Nat/Set.lean | 21 | 23 | theorem zero_union_range_succ : {0} ∪ range succ = univ := by |
ext n
cases n <;> simp
|
import Mathlib.Algebra.Order.Floor
import Mathlib.Algebra.Order.Field.Power
import Mathlib.Data.Nat.Log
#align_import data.int.log from "leanprover-community/mathlib"@"1f0096e6caa61e9c849ec2adbd227e960e9dff58"
variable {R : Type*} [LinearOrderedSemifield R] [FloorSemiring R]
namespace Int
def log (b : ℕ) (r : ... | Mathlib/Data/Int/Log.lean | 235 | 238 | theorem clog_natCast (b : ℕ) (n : ℕ) : clog b (n : R) = Nat.clog b n := by |
cases' n with n
· simp [clog_of_right_le_one]
· rw [clog_of_one_le_right, (Nat.ceil_eq_iff (Nat.succ_ne_zero n)).mpr] <;> simp
|
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator
import Mathlib.MeasureTheory.Function.UniformIntegrable
import Mathlib.MeasureTheory.Decomposition.RadonNikodym
#align_import measure_theory.function.conditional_expectation.real from "leanprover-community/mathlib"@"b2ff9a3d7a15fd5b0f060b135421d6a... | Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean | 355 | 363 | theorem condexp_stronglyMeasurable_mul₀ {f g : α → ℝ} (hf : AEStronglyMeasurable' m f μ)
(hfg : Integrable (f * g) μ) (hg : Integrable g μ) : μ[f * g|m] =ᵐ[μ] f * μ[g|m] := by |
have : μ[f * g|m] =ᵐ[μ] μ[hf.mk f * g|m] :=
condexp_congr_ae (hf.ae_eq_mk.mul EventuallyEq.rfl)
refine this.trans ?_
have : f * μ[g|m] =ᵐ[μ] hf.mk f * μ[g|m] := hf.ae_eq_mk.mul EventuallyEq.rfl
refine (condexp_stronglyMeasurable_mul hf.stronglyMeasurable_mk ?_ hg).trans this.symm
refine (integrable_congr... |
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 | 125 | 140 | theorem single_order_mul_powerSeriesPart (x : LaurentSeries R) :
(single x.order 1 : LaurentSeries R) * x.powerSeriesPart = x := by |
ext n
rw [← sub_add_cancel n x.order, single_mul_coeff_add, sub_add_cancel, one_mul]
by_cases h : x.order ≤ n
· rw [Int.eq_natAbs_of_zero_le (sub_nonneg_of_le h), coeff_coe_powerSeries,
powerSeriesPart_coeff, ← Int.eq_natAbs_of_zero_le (sub_nonneg_of_le h),
add_sub_cancel]
· rw [ofPowerSeries_app... |
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.MinMax
#align_import data.set.pointwise.interval from "leanprover-community/mathlib"@"2196ab363eb097c... | Mathlib/Data/Set/Pointwise/Interval.lean | 847 | 852 | theorem inv_Ioo_0_left {a : α} (ha : 0 < a) : (Ioo 0 a)⁻¹ = Ioi a⁻¹ := by |
ext x
exact
⟨fun h => inv_inv x ▸ (inv_lt_inv ha h.1).2 h.2, fun h =>
⟨inv_pos.2 <| (inv_pos.2 ha).trans h,
inv_inv a ▸ (inv_lt_inv ((inv_pos.2 ha).trans h) (inv_pos.2 ha)).2 h⟩⟩
|
import Mathlib.Data.Real.Basic
import Mathlib.Data.ENNReal.Real
import Mathlib.Data.Sign
#align_import data.real.ereal from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
open Function ENNReal NNReal Set
noncomputable section
def EReal := WithBot (WithTop ℝ)
deriving Bot, Zero, One,... | Mathlib/Data/Real/EReal.lean | 422 | 425 | theorem coe_toReal_le {x : EReal} (h : x ≠ ⊥) : ↑x.toReal ≤ x := by |
by_cases h' : x = ⊤
· simp only [h', le_top]
· simp only [le_refl, coe_toReal h' h]
|
import Mathlib.Data.Fintype.List
#align_import data.list.cycle from "leanprover-community/mathlib"@"7413128c3bcb3b0818e3e18720abc9ea3100fb49"
assert_not_exists MonoidWithZero
namespace List
variable {α : Type*} [DecidableEq α]
def nextOr : ∀ (_ : List α) (_ _ : α), α
| [], _, default => default
| [_], _, d... | Mathlib/Data/List/Cycle.lean | 365 | 370 | theorem pmap_prev_eq_rotate_length_sub_one (h : Nodup l) :
(l.pmap l.prev fun _ h => h) = l.rotate (l.length - 1) := by |
apply List.ext_nthLe
· simp
· intro n hn hn'
rw [nthLe_rotate, nthLe_pmap, prev_nthLe _ h]
|
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 | 110 | 113 | theorem restrict_apply₀' (hs : NullMeasurableSet s μ) : μ.restrict s t = μ (t ∩ s) := by |
rw [← restrict_congr_set hs.toMeasurable_ae_eq,
restrict_apply' (measurableSet_toMeasurable _ _),
measure_congr ((ae_eq_refl t).inter hs.toMeasurable_ae_eq)]
|
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 | 59 | 61 | theorem birthday_moveRight_lt {x : PGame} (i : x.RightMoves) :
(x.moveRight i).birthday < x.birthday := by |
cases x; rw [birthday]; exact lt_max_of_lt_right (lt_lsub _ i)
|
import Mathlib.Algebra.Bounds
import Mathlib.Algebra.Order.Field.Basic -- Porting note: `LinearOrderedField`, etc
import Mathlib.Data.Set.Pointwise.SMul
#align_import algebra.order.pointwise from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function Set
open Pointwise
variable ... | Mathlib/Algebra/Order/Pointwise.lean | 252 | 262 | theorem smul_Iio : r • Iio a = Iio (r • a) := by |
ext x
simp only [mem_smul_set, smul_eq_mul, mem_Iio]
constructor
· rintro ⟨a_w, a_h_left, rfl⟩
exact (mul_lt_mul_left hr).mpr a_h_left
· rintro h
use x / r
constructor
· exact (div_lt_iff' hr).mpr h
· exact mul_div_cancel₀ _ (ne_of_gt hr)
|
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "leanprover-community/mathlib"@"4698e35ca56a0d4fa53aa5639c3364e0a77f4eba"
section
open CategoryTheory Opposite
namespace CategoryTheory.Limits
-- attribute [local tid... | Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean | 101 | 103 | theorem WalkingParallelPairHom.comp_id
{X Y : WalkingParallelPair} (f : WalkingParallelPairHom X Y) : comp f (id Y) = f := by |
cases f <;> rfl
|
import Mathlib.Algebra.TrivSqZeroExt
import Mathlib.Topology.Algebra.InfiniteSum.Basic
import Mathlib.Topology.Algebra.Module.Basic
#align_import topology.instances.triv_sq_zero_ext from "leanprover-community/mathlib"@"b8d2eaa69d69ce8f03179a5cda774fc0cde984e4"
open scoped Topology
variable {α S R M : Type*}
local... | Mathlib/Topology/Instances/TrivSqZeroExt.lean | 46 | 48 | theorem nhds_def (x : tsze R M) : 𝓝 x = (𝓝 x.fst).prod (𝓝 x.snd) := by |
cases x using Prod.rec
exact nhds_prod_eq
|
import Mathlib.LinearAlgebra.Basis.VectorSpace
import Mathlib.LinearAlgebra.Dimension.Finite
import Mathlib.SetTheory.Cardinal.Subfield
import Mathlib.LinearAlgebra.Dimension.RankNullity
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomput... | Mathlib/LinearAlgebra/Dimension/DivisionRing.lean | 123 | 166 | theorem linearIndependent_of_top_le_span_of_card_eq_finrank {ι : Type*} [Fintype ι] {b : ι → V}
(spans : ⊤ ≤ span K (Set.range b)) (card_eq : Fintype.card ι = finrank K V) :
LinearIndependent K b :=
linearIndependent_iff'.mpr fun s g dependent i i_mem_s => by
classical
by_contra gx_ne_zero
-- We'l... |
rw [Set.toFinset_card, Fintype.card_ofFinset]
_ ≤ (Set.univ \ {i}).toFinset.card := Finset.card_image_le
_ = (Finset.univ.erase i).card := (congr_arg Finset.card (Finset.ext (by simp [and_comm])))
_ < Finset.univ.card := Finset.card_erase_lt_of_mem (Finset.mem_univ i)
_ = finr... |
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.Order.Sub.WithTop
import Mathlib.Data.Real.NNReal
import Mathlib.Order.Interval.Set.WithBotTop
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Function Set NNReal
variable {α : Typ... | Mathlib/Data/ENNReal/Basic.lean | 713 | 714 | theorem iUnion_Ico_coe_nat : ⋃ n : ℕ, Ico a n = Ici a \ {∞} := by |
simp only [← Ici_inter_Iio, ← inter_iUnion, iUnion_Iio_coe_nat, diff_eq]
|
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.RingTheory.Int.Basic
import Mathlib.Tactic.Ring
import Mathlib.Tactic.FieldSimp
import Mathlib.Data.Int.NatPrime
import Mathlib.Data.ZMod.Basic
#align_import number_theory.pythagorean_tri... | Mathlib/NumberTheory/PythagoreanTriples.lean | 241 | 252 | theorem isPrimitiveClassified_of_coprime_of_zero_left (hc : Int.gcd x y = 1) (hx : x = 0) :
h.IsPrimitiveClassified := by |
subst x
change Nat.gcd 0 (Int.natAbs y) = 1 at hc
rw [Nat.gcd_zero_left (Int.natAbs y)] at hc
cases' Int.natAbs_eq y with hy hy
· use 1, 0
rw [hy, hc, Int.gcd_zero_right]
decide
· use 0, 1
rw [hy, hc, Int.gcd_zero_left]
decide
|
import Mathlib.NumberTheory.LegendreSymbol.Basic
import Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum
#align_import number_theory.legendre_symbol.quadratic_reciprocity from "leanprover-community/mathlib"@"5b2fe80501ff327b9109fb09b7cc8c325cd0d7d9"
open Nat
section Values
variable {p : ℕ} [Fact p.Pri... | Mathlib/NumberTheory/LegendreSymbol/QuadraticReciprocity.lean | 60 | 62 | theorem at_two : legendreSym p 2 = χ₈ p := by |
have : (2 : ZMod p) = (2 : ℤ) := by norm_cast
rw [legendreSym, ← this, quadraticChar_two ((ringChar_zmod_n p).substr hp), card p]
|
import Mathlib.Data.Real.Sqrt
import Mathlib.Analysis.NormedSpace.Star.Basic
import Mathlib.Analysis.NormedSpace.ContinuousLinearMap
import Mathlib.Analysis.NormedSpace.Basic
#align_import data.is_R_or_C.basic from "leanprover-community/mathlib"@"baa88307f3e699fa7054ef04ec79fa4f056169cb"
section
local notation "�... | Mathlib/Analysis/RCLike/Basic.lean | 621 | 621 | theorem norm_conj {z : K} : ‖conj z‖ = ‖z‖ := by | simp only [← sqrt_normSq_eq_norm, normSq_conj]
|
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 | 57 | 58 | theorem nonempty_Icc : (Icc a b).Nonempty ↔ a ≤ b := by |
rw [← coe_nonempty, coe_Icc, Set.nonempty_Icc]
|
import Mathlib.Algebra.Algebra.Subalgebra.Pointwise
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Maximal
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Noetherian
import Mathlib.RingTheory.ChainOfDivisors
import Mathlib.RingTheory.DedekindDomain.Basic
import Mathlib.RingTheory.FractionalIdeal.Operations
#align_impo... | Mathlib/RingTheory/DedekindDomain/Ideal.lean | 170 | 175 | theorem coe_ideal_span_singleton_mul_inv {x : R₁} (hx : x ≠ 0) :
(Ideal.span ({x} : Set R₁) : FractionalIdeal R₁⁰ K) *
(Ideal.span ({x} : Set R₁) : FractionalIdeal R₁⁰ K)⁻¹ = 1 := by |
rw [coeIdeal_span_singleton,
spanSingleton_mul_inv K <|
(map_ne_zero_iff _ <| NoZeroSMulDivisors.algebraMap_injective R₁ K).mpr hx]
|
import Mathlib.Algebra.Associated
import Mathlib.Algebra.Star.Unitary
import Mathlib.RingTheory.Int.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.Tactic.Ring
#align_import number_theory.zsqrtd.basic from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
@[ext]
struct... | Mathlib/NumberTheory/Zsqrtd/Basic.lean | 378 | 392 | theorem coprime_of_dvd_coprime {a b : ℤ√d} (hcoprime : IsCoprime a.re a.im) (hdvd : b ∣ a) :
IsCoprime b.re b.im := by |
apply isCoprime_of_dvd
· rintro ⟨hre, him⟩
obtain rfl : b = 0 := Zsqrtd.ext b 0 hre him
rw [zero_dvd_iff] at hdvd
simp [hdvd, zero_im, zero_re, not_isCoprime_zero_zero] at hcoprime
· rintro z hz - hzdvdu hzdvdv
apply hz
obtain ⟨ha, hb⟩ : z ∣ a.re ∧ z ∣ a.im := by
rw [← intCast_dvd]
... |
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.MinMax
#align_import data.set.pointwise.interval from "leanprover-community/mathlib"@"2196ab363eb097c... | Mathlib/Data/Set/Pointwise/Interval.lean | 629 | 630 | theorem preimage_mul_const_Ico (a b : α) {c : α} (h : 0 < c) :
(fun x => x * c) ⁻¹' Ico a b = Ico (a / c) (b / c) := by | simp [← Ici_inter_Iio, h]
|
import Mathlib.Tactic.Ring
import Mathlib.Data.PNat.Prime
#align_import data.pnat.xgcd from "leanprover-community/mathlib"@"6afc9b06856ad973f6a2619e3e8a0a8d537a58f2"
open Nat
namespace PNat
structure XgcdType where
wp : ℕ
x : ℕ
y : ℕ
zp : ℕ
ap : ℕ
bp : ℕ
deriving Inhabited
#alig... | Mathlib/Data/PNat/Xgcd.lean | 348 | 350 | theorem reduce_a {u : XgcdType} (h : u.r = 0) : u.reduce = u.finish := by |
rw [reduce]
exact if_pos h
|
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.MeasureTheory.Covering.OneDim
import Mathlib.Order.Monotone.Extension
#align_import analysis.calculus.monotone from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
open Set Filter Function Metric MeasureTheory MeasureTheory.Meas... | Mathlib/Analysis/Calculus/Monotone.lean | 136 | 214 | theorem Monotone.ae_hasDerivAt {f : ℝ → ℝ} (hf : Monotone f) :
∀ᵐ x, HasDerivAt f (rnDeriv hf.stieltjesFunction.measure volume x).toReal x := by |
/- We already know that the Stieltjes function associated to `f` (i.e., `g : x ↦ f (x^+)`) is
differentiable almost everywhere. We reduce to this statement by sandwiching values of `f` with
values of `g`, by shifting with `(y - x)^2` (which has no influence on the relevant
scale `y - x`.)-/
filter_upwa... |
import Mathlib.Analysis.Complex.AbsMax
import Mathlib.Analysis.Asymptotics.SuperpolynomialDecay
#align_import analysis.complex.phragmen_lindelof from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Function Filter Asymptotics Metric Complex Bornology
open scoped Topology Filter R... | Mathlib/Analysis/Complex/PhragmenLindelof.lean | 80 | 94 | theorem isBigO_sub_exp_rpow {a : ℝ} {f g : ℂ → E} {l : Filter ℂ}
(hBf : ∃ c < a, ∃ B, f =O[cobounded ℂ ⊓ l] fun z => expR (B * abs z ^ c))
(hBg : ∃ c < a, ∃ B, g =O[cobounded ℂ ⊓ l] fun z => expR (B * abs z ^ c)) :
∃ c < a, ∃ B, (f - g) =O[cobounded ℂ ⊓ l] fun z => expR (B * abs z ^ c) := by |
have : ∀ {c₁ c₂ B₁ B₂ : ℝ}, c₁ ≤ c₂ → 0 ≤ B₂ → B₁ ≤ B₂ →
(fun z : ℂ => expR (B₁ * abs z ^ c₁)) =O[cobounded ℂ ⊓ l]
fun z => expR (B₂ * abs z ^ c₂) := fun hc hB₀ hB ↦ .of_bound 1 <| by
filter_upwards [(eventually_cobounded_le_norm 1).filter_mono inf_le_left] with z hz
simp only [one_mul, Real.no... |
import Mathlib.Combinatorics.SimpleGraph.Clique
import Mathlib.Data.ENat.Lattice
import Mathlib.Data.Nat.Lattice
import Mathlib.Data.Setoid.Partition
import Mathlib.Order.Antichain
#align_import combinatorics.simple_graph.coloring from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open ... | Mathlib/Combinatorics/SimpleGraph/Coloring.lean | 352 | 356 | theorem colorable_of_chromaticNumber_ne_top (h : G.chromaticNumber ≠ ⊤) :
G.Colorable (ENat.toNat G.chromaticNumber) := by |
rw [chromaticNumber_ne_top_iff_exists] at h
obtain ⟨n, hn⟩ := h
exact colorable_chromaticNumber hn
|
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Data.ENat.Basic
#align_import data.polynomial.degree.trailing_degree from "leanprover-community/mathlib"@"302eab4f46abb63de520828de78c04cb0f9b5836"
noncomputable section
open Function Polynomial Finsupp Finset
open scoped Polynomial
namespace ... | Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean | 233 | 235 | theorem trailingDegree_monomial (ha : a ≠ 0) : trailingDegree (monomial n a) = n := by |
rw [trailingDegree, support_monomial n ha, min_singleton]
rfl
|
import Mathlib.Topology.UniformSpace.UniformConvergenceTopology
#align_import topology.uniform_space.equicontinuity from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
section
open UniformSpace Filter Set Uniformity Topology UniformConvergence Function
variable {ι κ X X' Y Z α α' β β'... | Mathlib/Topology/UniformSpace/Equicontinuity.lean | 751 | 756 | theorem UniformInducing.equicontinuousWithinAt_iff {F : ι → X → α} {S : Set X} {x₀ : X} {u : α → β}
(hu : UniformInducing u) : EquicontinuousWithinAt F S x₀ ↔
EquicontinuousWithinAt ((u ∘ ·) ∘ F) S x₀ := by |
have := (UniformFun.postcomp_uniformInducing (α := ι) hu).inducing
simp only [equicontinuousWithinAt_iff_continuousWithinAt, this.continuousWithinAt_iff]
rfl
|
import Mathlib.Data.Finset.Lattice
import Mathlib.Data.Fintype.Vector
import Mathlib.Data.Multiset.Sym
#align_import data.finset.sym from "leanprover-community/mathlib"@"02ba8949f486ebecf93fe7460f1ed0564b5e442c"
namespace Finset
variable {α : Type*}
@[simps]
protected def sym2 (s : Finset α) : Finset (Sym2 α) :... | Mathlib/Data/Finset/Sym.lean | 114 | 115 | theorem card_sym2 (s : Finset α) : s.sym2.card = Nat.choose (s.card + 1) 2 := by |
rw [card_def, sym2_val, Multiset.card_sym2, ← card_def]
|
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.TensorProduct.Tower
import Mathlib.RingTheory.Adjoin.Basic
import Mathlib.LinearAlgebra.DirectSum.Finsupp
#align_import ring_theory.tensor_product from "leanprover-community/mathlib"@"88fcdc3da43943f5b01925deddaa5bf0c0e85e4e"
suppress_comp... | Mathlib/RingTheory/TensorProduct/Basic.lean | 96 | 98 | theorem baseChange_smul : (r • f).baseChange A = r • f.baseChange A := by |
ext
simp [baseChange_tmul]
|
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
#align_import measure_theory.function.simple_func from "leanprover-community/mathlib"@"bf6a01357ff5684b1ebcd0f1a13be314fc82c0bf"
noncomputable section
open Set hiding restrict restrict_apply
open Filter ENNReal
open Function (support)
open scoped Cla... | Mathlib/MeasureTheory/Function/SimpleFunc.lean | 125 | 126 | theorem forall_mem_range {f : α →ₛ β} {p : β → Prop} : (∀ y ∈ f.range, p y) ↔ ∀ x, p (f x) := by |
simp only [mem_range, Set.forall_mem_range]
|
import Mathlib.Algebra.Group.Equiv.Basic
import Mathlib.Algebra.Group.Aut
import Mathlib.Data.ZMod.Defs
import Mathlib.Tactic.Ring
#align_import algebra.quandle from "leanprover-community/mathlib"@"28aa996fc6fb4317f0083c4e6daf79878d81be33"
open MulOpposite
universe u v
class Shelf (α : Type u) where
act : ... | Mathlib/Algebra/Quandle.lean | 315 | 317 | theorem self_invAct_eq_iff_eq {x y : R} : x ◃⁻¹ x = y ◃⁻¹ y ↔ x = y := by |
have h := @self_act_eq_iff_eq _ _ (op x) (op y)
simpa using h
|
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 | 38 | 41 | theorem Hom.cast_eq_cast {u v u' v' : U} (hu : u = u') (hv : v = v') (e : u ⟶ v) :
e.cast hu hv = _root_.cast (by {rw [hu, hv]}) e := by |
subst_vars
rfl
|
import Mathlib.Order.Filter.Lift
import Mathlib.Topology.Defs.Filter
#align_import topology.basic from "leanprover-community/mathlib"@"e354e865255654389cc46e6032160238df2e0f40"
noncomputable section
open Set Filter
universe u v w x
def TopologicalSpace.ofClosed {X : Type u} (T : Set (Set X)) (empty_mem : ∅ ∈... | Mathlib/Topology/Basic.lean | 672 | 674 | theorem self_diff_frontier (s : Set X) : s \ frontier s = interior s := by |
rw [frontier, diff_diff_right, diff_eq_empty.2 subset_closure,
inter_eq_self_of_subset_right interior_subset, empty_union]
|
import Mathlib.CategoryTheory.NatIso
#align_import category_theory.bicategory.basic from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
namespace CategoryTheory
universe w v u
open Category Iso
-- intended to be used with explicit universe parameters
@[nolint checkUnivs]
class Bicate... | Mathlib/CategoryTheory/Bicategory/Basic.lean | 206 | 207 | theorem whiskerLeft_inv_hom (f : a ⟶ b) {g h : b ⟶ c} (η : g ≅ h) :
f ◁ η.inv ≫ f ◁ η.hom = 𝟙 (f ≫ h) := by | rw [← whiskerLeft_comp, inv_hom_id, whiskerLeft_id]
|
import Mathlib.Algebra.MvPolynomial.Basic
import Mathlib.Topology.Algebra.Ring.Basic
variable {X σ : Type*} [TopologicalSpace X] [CommSemiring X] [TopologicalSemiring X]
(p : MvPolynomial σ X)
| Mathlib/Topology/Algebra/MvPolynomial.lean | 25 | 26 | theorem MvPolynomial.continuous_eval : Continuous fun x ↦ eval x p := by |
continuity
|
import Mathlib.Algebra.MvPolynomial.Variables
#align_import data.mv_polynomial.comm_ring from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Finsupp AddMonoidAlgebra
universe u v
variable {R : Type u} {S : Type v}
namespace MvPolynomial
varia... | Mathlib/Algebra/MvPolynomial/CommRing.lean | 207 | 212 | theorem totalDegree_sub (a b : MvPolynomial σ R) :
(a - b).totalDegree ≤ max a.totalDegree b.totalDegree :=
calc
(a - b).totalDegree = (a + -b).totalDegree := by | rw [sub_eq_add_neg]
_ ≤ max a.totalDegree (-b).totalDegree := totalDegree_add a (-b)
_ = max a.totalDegree b.totalDegree := by rw [totalDegree_neg]
|
import Mathlib.Init.Function
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.Inhabit
#align_import data.prod.basic from "leanprover-community/mathlib"@"d07245fd37786daa997af4f1a73a49fa3b748408"
variable {α : Type*} {β : Type*} {γ : Type*} {δ : Type*}
@[simp]
theorem Prod.map_apply (f : α → γ) (g : β → δ... | Mathlib/Data/Prod/Basic.lean | 110 | 113 | theorem mk.inj_right {α β : Type*} (b : β) :
Function.Injective (fun a ↦ Prod.mk a b : α → α × β) := by |
intro b₁ b₂ h
simpa only [and_true, eq_self_iff_true, mk.inj_iff] using h
|
import Mathlib.Analysis.SpecialFunctions.Complex.Arg
import Mathlib.Analysis.SpecialFunctions.Log.Basic
#align_import analysis.special_functions.complex.log from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
namespace Complex
open Set Filter Bornology
open scop... | Mathlib/Analysis/SpecialFunctions/Complex/Log.lean | 110 | 110 | theorem log_one : log 1 = 0 := by | simp [log]
|
import Mathlib.Data.Part
import Mathlib.Data.Rel
#align_import data.pfun from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
open Function
def PFun (α β : Type*) :=
α → Part β
#align pfun PFun
infixr:25 " →. " => PFun
namespace PFun
variable {α β γ δ ε ι : Type*}
instance inhab... | Mathlib/Data/PFun.lean | 550 | 552 | theorem mem_toSubtype_iff {p : β → Prop} {f : α → β} {a : α} {b : Subtype p} :
b ∈ toSubtype p f a ↔ ↑b = f a := by |
rw [toSubtype_apply, Part.mem_mk_iff, exists_subtype_mk_eq_iff, eq_comm]
|
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Order.Bounds.OrderIso
import Mathlib.Tactic.Positivity.Core
#align_import algebra.order.field.basic from "leanprover-community/mathlib"@"8477... | Mathlib/Algebra/Order/Field/Basic.lean | 76 | 76 | theorem div_le_iff' (hb : 0 < b) : a / b ≤ c ↔ a ≤ b * c := by | rw [mul_comm, div_le_iff hb]
|
import Mathlib.Algebra.Algebra.Subalgebra.Unitization
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Topology.Algebra.StarSubalgebra
import Mathlib.Topology.ContinuousFunction.ContinuousMapZero
import Mathlib.Topology.ContinuousFunction.Weierstrass
#align_import topology.continuous_function.stone_weierstrass fro... | Mathlib/Topology/ContinuousFunction/StoneWeierstrass.lean | 275 | 289 | theorem subalgebra_topologicalClosure_eq_top_of_separatesPoints (A : Subalgebra ℝ C(X, ℝ))
(w : A.SeparatesPoints) : A.topologicalClosure = ⊤ := by |
-- The closure of `A` is closed under taking `sup` and `inf`,
-- and separates points strongly (since `A` does),
-- so we can apply `sublattice_closure_eq_top`.
apply SetLike.ext'
let L := A.topologicalClosure
have n : Set.Nonempty (L : Set C(X, ℝ)) := ⟨(1 : C(X, ℝ)), A.le_topologicalClosure A.one_mem⟩
c... |
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex
#align_import analysis.special_functions.trigonometric.arctan from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
namespace Real
open Set Filter
open scoped Topology Real
theorem tan_add {x y : ℝ}
... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Arctan.lean | 218 | 223 | theorem arctan_inv_of_pos {x : ℝ} (h : 0 < x) : arctan x⁻¹ = π / 2 - arctan x := by |
rw [← arctan_tan (x := _ - _), tan_pi_div_two_sub, tan_arctan]
· norm_num
exact (arctan_lt_pi_div_two x).trans (half_lt_self_iff.mpr pi_pos)
· rw [sub_lt_self_iff, ← arctan_zero]
exact tanOrderIso.symm.strictMono h
|
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 | 142 | 143 | theorem log_le_log_iff (h : 0 < x) (h₁ : 0 < y) : log x ≤ log y ↔ x ≤ y := by |
rw [← exp_le_exp, exp_log h, exp_log h₁]
|
import Mathlib.Algebra.Group.Fin
import Mathlib.LinearAlgebra.Matrix.Symmetric
#align_import linear_algebra.matrix.circulant from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa1"
variable {α β m n R : Type*}
namespace Matrix
open Function
open Matrix
def circulant [Sub n] (v : n → α)... | Mathlib/LinearAlgebra/Matrix/Circulant.lean | 191 | 193 | theorem circulant_isSymm_iff [AddGroup n] {v : n → α} :
(circulant v).IsSymm ↔ ∀ i, v (-i) = v i := by |
rw [IsSymm, transpose_circulant, circulant_inj, funext_iff]
|
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Option.Basic
import Mathlib.Data.List.Defs
import Mathlib.Init.Data.List.Basic
import Mathlib.Init.Data.List.Instances
import Mathlib.Init.Data.List.Lemmas
import Mathlib.Logic.Unique
import Mathlib.Order.Basic
import Mathlib.Tactic.Common
#align_import data.list.basic... | Mathlib/Data/List/Basic.lean | 695 | 704 | theorem mem_getLast?_eq_getLast : ∀ {l : List α} {x : α}, x ∈ l.getLast? → ∃ h, x = getLast l h
| [], x, hx => False.elim <| by simp at hx
| [a], x, hx =>
have : a = x := by | simpa using hx
this ▸ ⟨cons_ne_nil a [], rfl⟩
| a :: b :: l, x, hx => by
rw [getLast?_cons_cons] at hx
rcases mem_getLast?_eq_getLast hx with ⟨_, h₂⟩
use cons_ne_nil _ _
assumption
|
import Mathlib.NumberTheory.Cyclotomic.Discriminant
import Mathlib.RingTheory.Polynomial.Eisenstein.IsIntegral
import Mathlib.RingTheory.Ideal.Norm
#align_import number_theory.cyclotomic.rat from "leanprover-community/mathlib"@"b353176c24d96c23f0ce1cc63efc3f55019702d9"
universe u
open Algebra IsCyclotomicExtensio... | Mathlib/NumberTheory/Cyclotomic/Rat.lean | 46 | 49 | theorem discr_odd_prime' [IsCyclotomicExtension {p} ℚ K] (hζ : IsPrimitiveRoot ζ p) (hodd : p ≠ 2) :
discr ℚ (hζ.subOnePowerBasis ℚ).basis = (-1) ^ (((p : ℕ) - 1) / 2) * p ^ ((p : ℕ) - 2) := by |
rw [← discr_odd_prime hζ (cyclotomic.irreducible_rat hp.out.pos) hodd]
exact hζ.discr_zeta_eq_discr_zeta_sub_one.symm
|
import Mathlib.Topology.Order
#align_import topology.maps from "leanprover-community/mathlib"@"d91e7f7a7f1c7e9f0e18fdb6bde4f652004c735d"
open Set Filter Function
open TopologicalSpace Topology Filter
variable {X : Type*} {Y : Type*} {Z : Type*} {ι : Type*} {f : X → Y} {g : Y → Z}
section Inducing
variable [To... | Mathlib/Topology/Maps.lean | 137 | 139 | theorem continuousAt_iff' (hf : Inducing f) {x : X} (h : range f ∈ 𝓝 (f x)) :
ContinuousAt (g ∘ f) x ↔ ContinuousAt g (f x) := by |
simp_rw [ContinuousAt, Filter.Tendsto, ← hf.map_nhds_of_mem _ h, Filter.map_map, comp]
|
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 | 138 | 139 | theorem le_inv_mul_iff_le : 1 ≤ b⁻¹ * a ↔ b ≤ a := by |
rw [← mul_le_mul_iff_left b, mul_one, mul_inv_cancel_left]
|
import Mathlib.Algebra.Group.Prod
import Mathlib.Order.Cover
#align_import algebra.support from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
assert_not_exists MonoidWithZero
open Set
namespace Function
variable {α β A B M N P G : Type*}
section One
variable [One M] [One N] [One P]
... | Mathlib/Algebra/Group/Support.lean | 134 | 138 | theorem mulSupport_eq_empty_iff {f : α → M} : mulSupport f = ∅ ↔ f = 1 := by |
#adaptation_note /-- This used to be `simp_rw` rather than `rw`,
but this broke `to_additive` as of `nightly-2024-03-07` -/
rw [← subset_empty_iff, mulSupport_subset_iff', funext_iff]
simp
|
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Polynomial.AlgebraMap
#align_import ring_theory.polynomial.tower from "leanprover-community/mathlib"@"bb168510ef455e9280a152e7f31673cabd3d7496"
open Polynomial
variable (R A B : Type*)
namespace Polynomial
section CommSemiring
variable [CommSemiring ... | Mathlib/RingTheory/Polynomial/Tower.lean | 60 | 63 | theorem aeval_algebraMap_eq_zero_iff [NoZeroSMulDivisors A B] [Nontrivial B] (x : A) (p : R[X]) :
aeval (algebraMap A B x) p = 0 ↔ aeval x p = 0 := by |
rw [aeval_algebraMap_apply, Algebra.algebraMap_eq_smul_one, smul_eq_zero,
iff_false_intro (one_ne_zero' B), or_false_iff]
|
import Mathlib.Data.List.Basic
#align_import data.list.lattice from "leanprover-community/mathlib"@"dd71334db81d0bd444af1ee339a29298bef40734"
open Nat
namespace List
variable {α : Type*} {l l₁ l₂ : List α} {p : α → Prop} {a : α}
variable [DecidableEq α]
section BagInter
@[simp]
theorem nil_bagInt... | Mathlib/Data/List/Lattice.lean | 199 | 199 | theorem bagInter_nil (l : List α) : l.bagInter [] = [] := by | cases l <;> rfl
|
import Mathlib.Topology.Basic
#align_import topology.nhds_set from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Filter Topology
variable {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y] {f : Filter X}
{s t s₁ s₂ t₁ t₂ : Set X} {x : X}
theorem nhdsSet_diagonal (X) [T... | Mathlib/Topology/NhdsSet.lean | 213 | 214 | theorem nhdsSet_iUnion {ι : Sort*} (s : ι → Set X) : 𝓝ˢ (⋃ i, s i) = ⨆ i, 𝓝ˢ (s i) := by |
simp only [nhdsSet, image_iUnion, sSup_iUnion (β := Filter X)]
|
import Mathlib.Order.CompleteLattice
import Mathlib.Order.Cover
import Mathlib.Order.Iterate
import Mathlib.Order.WellFounded
#align_import order.succ_pred.basic from "leanprover-community/mathlib"@"0111834459f5d7400215223ea95ae38a1265a907"
open Function OrderDual Set
variable {α β : Type*}
@[ext]
class SuccOr... | Mathlib/Order/SuccPred/Basic.lean | 301 | 303 | theorem le_succ_iterate (k : ℕ) (x : α) : x ≤ succ^[k] x := by |
conv_lhs => rw [(by simp only [Function.iterate_id, id] : x = id^[k] x)]
exact Monotone.le_iterate_of_le succ_mono le_succ k x
|
import Mathlib.SetTheory.Ordinal.Arithmetic
import Mathlib.Tactic.TFAE
import Mathlib.Topology.Order.Monotone
#align_import set_theory.ordinal.topology from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
noncomputable section
universe u v
open Cardinal Order Topology
namespace Ordina... | Mathlib/SetTheory/Ordinal/Topology.lean | 86 | 124 | theorem mem_closure_tfae (a : Ordinal.{u}) (s : Set Ordinal) :
TFAE [a ∈ closure s,
a ∈ closure (s ∩ Iic a),
(s ∩ Iic a).Nonempty ∧ sSup (s ∩ Iic a) = a,
∃ t, t ⊆ s ∧ t.Nonempty ∧ BddAbove t ∧ sSup t = a,
∃ (o : Ordinal.{u}), o ≠ 0 ∧ ∃ (f : ∀ x < o, Ordinal),
(∀ x hx, f x hx ∈ s) ∧ b... |
tfae_have 1 → 2
· simp only [mem_closure_iff_nhdsWithin_neBot, inter_comm s, nhdsWithin_inter', nhds_left_eq_nhds]
exact id
tfae_have 2 → 3
· intro h
rcases (s ∩ Iic a).eq_empty_or_nonempty with he | hne
· simp [he] at h
· refine ⟨hne, (isLUB_of_mem_closure ?_ h).csSup_eq hne⟩
exact fun x... |
import Mathlib.Algebra.IsPrimePow
import Mathlib.NumberTheory.ArithmeticFunction
import Mathlib.Analysis.SpecialFunctions.Log.Basic
#align_import number_theory.von_mangoldt from "leanprover-community/mathlib"@"c946d6097a6925ad16d7ec55677bbc977f9846de"
namespace ArithmeticFunction
open Finset Nat
open scoped Arit... | Mathlib/NumberTheory/VonMangoldt.lean | 90 | 91 | theorem vonMangoldt_apply_pow {n k : ℕ} (hk : k ≠ 0) : Λ (n ^ k) = Λ n := by |
simp only [vonMangoldt_apply, isPrimePow_pow_iff hk, pow_minFac hk]
|
import Mathlib.Topology.Constructions
#align_import topology.continuous_on from "leanprover-community/mathlib"@"d4f691b9e5f94cfc64639973f3544c95f8d5d494"
open Set Filter Function Topology Filter
variable {α : Type*} {β : Type*} {γ : Type*} {δ : Type*}
variable [TopologicalSpace α]
@[simp]
theorem nhds_bind_nhdsW... | Mathlib/Topology/ContinuousOn.lean | 567 | 571 | theorem continuousWithinAt_prod_of_discrete_right [DiscreteTopology β]
{f : α × β → γ} {s : Set (α × β)} {x : α × β} :
ContinuousWithinAt f s x ↔ ContinuousWithinAt (f ⟨·, x.2⟩) {a | (a, x.2) ∈ s} x.1 := by |
rw [← x.eta]; simp_rw [ContinuousWithinAt, nhdsWithin, nhds_prod_eq, nhds_discrete, prod_pure,
← map_inf_principal_preimage]; rfl
|
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.GroupTheory.GroupAction.Pi
open Function Set
structure AddConstMap (G H : Type*) [Add G] [Add H] (a : G) (b : H) where
protected... | Mathlib/Algebra/AddConstMap/Basic.lean | 121 | 122 | theorem map_nat [AddMonoidWithOne G] [AddMonoidWithOne H] [AddConstMapClass F G H 1 1]
(f : F) (n : ℕ) : f n = f 0 + n := by | simp
|
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "leanprover-community/mathlib"@"4698e35ca56a0d4fa53aa5639c3364e0a77f4eba"
section
open CategoryTheory Opposite
namespace CategoryTheory.Limits
-- attribute [local tid... | Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean | 105 | 108 | theorem WalkingParallelPairHom.assoc {X Y Z W : WalkingParallelPair}
(f : WalkingParallelPairHom X Y) (g: WalkingParallelPairHom Y Z)
(h : WalkingParallelPairHom Z W) : comp (comp f g) h = comp f (comp g h) := by |
cases f <;> cases g <;> cases h <;> rfl
|
import Mathlib.CategoryTheory.Monoidal.Free.Basic
import Mathlib.CategoryTheory.Groupoid
import Mathlib.CategoryTheory.DiscreteCategory
#align_import category_theory.monoidal.free.coherence from "leanprover-community/mathlib"@"f187f1074fa1857c94589cc653c786cadc4c35ff"
universe u
namespace CategoryTheory
open Mo... | Mathlib/CategoryTheory/Monoidal/Free/Coherence.lean | 285 | 305 | theorem normalize_naturality (n : NormalMonoidalObject C) {X Y : F C} (f : X ⟶ Y) :
inclusionObj n ◁ f ≫ (normalizeIsoApp' C Y n).hom =
(normalizeIsoApp' C X n).hom ≫
inclusion.map (eqToHom (Discrete.ext _ _ (normalizeObj_congr n f))) := by |
revert n
induction f using Hom.inductionOn
case comp f g ihf ihg => simp [ihg, reassoc_of% (ihf _)]
case whiskerLeft X' X Y f ih =>
intro n
dsimp only [normalizeObj_tensor, normalizeIsoApp', tensor_eq_tensor, Iso.trans_hom,
Iso.symm_hom, whiskerRightIso_hom, Function.comp_apply, inclusion_obj]
... |
import Mathlib.Algebra.Field.Basic
import Mathlib.Deprecated.Subring
#align_import deprecated.subfield from "leanprover-community/mathlib"@"bd9851ca476957ea4549eb19b40e7b5ade9428cc"
variable {F : Type*} [Field F] (S : Set F)
structure IsSubfield extends IsSubring S : Prop where
inv_mem : ∀ {x : F}, x ∈ S → x⁻... | Mathlib/Deprecated/Subfield.lean | 40 | 43 | theorem IsSubfield.div_mem {S : Set F} (hS : IsSubfield S) {x y : F} (hx : x ∈ S) (hy : y ∈ S) :
x / y ∈ S := by |
rw [div_eq_mul_inv]
exact hS.toIsSubring.toIsSubmonoid.mul_mem hx (hS.inv_mem hy)
|
import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Subobject.Lattice
import Mathlib.Order.Atoms
#align_import category_theory.simple from "leanprover-community/mathlib"@"4ed0bcaef698011... | Mathlib/CategoryTheory/Simple.lean | 84 | 89 | theorem kernel_zero_of_nonzero_from_simple {X Y : C} [Simple X] {f : X ⟶ Y} [HasKernel f]
(w : f ≠ 0) : kernel.ι f = 0 := by |
classical
by_contra h
haveI := isIso_of_mono_of_nonzero h
exact w (eq_zero_of_epi_kernel f)
|
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.Probability.Independence.Basic
#align_import probability.integration from "leanprover-community/mathlib"@"2f8347015b12b0864dfaf366ec4909eb70c78740"
noncomputable section
open Set MeasureTheory
open scoped ENNReal MeasureTheory
variable {Ω : Type*... | Mathlib/Probability/Integration.lean | 270 | 293 | theorem IndepFun.integral_mul (hXY : IndepFun X Y μ) (hX : AEStronglyMeasurable X μ)
(hY : AEStronglyMeasurable Y μ) : integral μ (X * Y) = integral μ X * integral μ Y := by |
by_cases h'X : X =ᵐ[μ] 0
· have h' : X * Y =ᵐ[μ] 0 := by
filter_upwards [h'X] with ω hω
simp [hω]
simp only [integral_congr_ae h'X, integral_congr_ae h', Pi.zero_apply, integral_const,
Algebra.id.smul_eq_mul, mul_zero, zero_mul]
by_cases h'Y : Y =ᵐ[μ] 0
· have h' : X * Y =ᵐ[μ] 0 := by
... |
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Ring.Subring.Pointwise
import Mathlib.RingTheory.Adjoin.Basic
#align_import algebra.algebra.subalgebra.pointwise from "leanprover-community/mathlib"@"b2c707cd190a58ea0565c86695a19e99ccecc215"
namespace... | Mathlib/Algebra/Algebra/Subalgebra/Pointwise.lean | 37 | 44 | theorem mul_self (S : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule S)
= (Subalgebra.toSubmodule S) := by |
apply le_antisymm
· refine (mul_toSubmodule_le _ _).trans_eq ?_
rw [sup_idem]
· intro x hx1
rw [← mul_one x]
exact Submodule.mul_mem_mul hx1 (show (1 : A) ∈ S from one_mem S)
|
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.MinMax
#align_import data.set.pointwise.interval from "leanprover-community/mathlib"@"2196ab363eb097c... | Mathlib/Data/Set/Pointwise/Interval.lean | 92 | 95 | theorem Iic_mul_Iio_subset' (a b : α) : Iic a * Iio b ⊆ Iio (a * b) := by |
haveI := covariantClass_le_of_lt
rintro x ⟨y, hya, z, hzb, rfl⟩
exact mul_lt_mul_of_le_of_lt hya hzb
|
import Mathlib.RingTheory.WittVector.InitTail
#align_import ring_theory.witt_vector.truncated from "leanprover-community/mathlib"@"acbe099ced8be9c9754d62860110295cde0d7181"
open Function (Injective Surjective)
noncomputable section
variable {p : ℕ} [hp : Fact p.Prime] (n : ℕ) (R : Type*)
local notation "𝕎" =>... | Mathlib/RingTheory/WittVector/Truncated.lean | 114 | 115 | theorem coeff_out (x : TruncatedWittVector p n R) (i : Fin n) : x.out.coeff i = x.coeff i := by |
rw [out]; dsimp only; rw [dif_pos i.is_lt, Fin.eta]
|
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.SchwartzSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.fourier.poisson_summation from "leanprover-community/mathlib"@"fd5... | Mathlib/Analysis/Fourier/PoissonSummation.lean | 161 | 179 | theorem isBigO_norm_Icc_restrict_atBot {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : f =O[atBot] fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
(fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) =O[atBot] fun x : ℝ => |x| ^ (-b) := by |
have h1 : (f.comp (ContinuousMap.mk _ continuous_neg)) =O[atTop] fun x : ℝ => |x| ^ (-b) := by
convert hf.comp_tendsto tendsto_neg_atTop_atBot using 1
ext1 x; simp only [Function.comp_apply, abs_neg]
have h2 := (isBigO_norm_Icc_restrict_atTop hb h1 (-S) (-R)).comp_tendsto tendsto_neg_atBot_atTop
have : (... |
import Mathlib.Data.Set.Lattice
#align_import data.semiquot from "leanprover-community/mathlib"@"09597669f02422ed388036273d8848119699c22f"
-- Porting note: removed universe parameter
structure Semiquot (α : Type*) where mk' ::
s : Set α
val : Trunc s
#align semiquot Semiquot
namespace Semiquot
vari... | Mathlib/Data/Semiquot.lean | 90 | 91 | theorem blur_eq_blur' (q : Semiquot α) (s : Set α) (h : q.s ⊆ s) : blur s q = blur' q h := by |
unfold blur; congr; exact Set.union_eq_self_of_subset_right h
|
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.MeasureTheory.Group.Pointwise
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Measure.Haar.Basic
import Mathlib.MeasureTheory.Measure.Doubling
import Mathlib.MeasureTheory.Constructions.BorelSpace.Metric
#align_import measu... | Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean | 426 | 432 | theorem addHaar_image_homothety (x : E) (r : ℝ) (s : Set E) :
μ (AffineMap.homothety x r '' s) = ENNReal.ofReal (abs (r ^ finrank ℝ E)) * μ s :=
calc
μ (AffineMap.homothety x r '' s) = μ ((fun y => y + x) '' (r • (fun y => y + -x) '' s)) := by |
simp only [← image_smul, image_image, ← sub_eq_add_neg]; rfl
_ = ENNReal.ofReal (abs (r ^ finrank ℝ E)) * μ s := by
simp only [image_add_right, measure_preimage_add_right, addHaar_smul]
|
import Mathlib.MeasureTheory.Function.LpSeminorm.Basic
import Mathlib.MeasureTheory.Integral.MeanInequalities
#align_import measure_theory.function.lp_seminorm from "leanprover-community/mathlib"@"c4015acc0a223449d44061e27ddac1835a3852b9"
open Filter
open scoped ENNReal Topology
namespace MeasureTheory
section S... | Mathlib/MeasureTheory/Function/LpSeminorm/CompareExp.lean | 61 | 85 | theorem snorm_le_snorm_mul_rpow_measure_univ {p q : ℝ≥0∞} (hpq : p ≤ q)
(hf : AEStronglyMeasurable f μ) :
snorm f p μ ≤ snorm f q μ * μ Set.univ ^ (1 / p.toReal - 1 / q.toReal) := by |
by_cases hp0 : p = 0
· simp [hp0, zero_le]
rw [← Ne] at hp0
have hp0_lt : 0 < p := lt_of_le_of_ne (zero_le _) hp0.symm
have hq0_lt : 0 < q := lt_of_lt_of_le hp0_lt hpq
by_cases hq_top : q = ∞
· simp only [hq_top, _root_.div_zero, one_div, ENNReal.top_toReal, sub_zero, snorm_exponent_top,
GroupWithZ... |
import Mathlib.CategoryTheory.Generator
import Mathlib.CategoryTheory.Preadditive.Yoneda.Basic
#align_import category_theory.preadditive.generator from "leanprover-community/mathlib"@"09f981f72d43749f1fa072deade828d9c1e185bb"
universe v u
open CategoryTheory Opposite
namespace CategoryTheory
variable {C : Type... | Mathlib/CategoryTheory/Preadditive/Generator.lean | 77 | 81 | theorem isCoseparator_iff_faithful_preadditiveYonedaObj (G : C) :
IsCoseparator G ↔ (preadditiveYonedaObj G).Faithful := by |
rw [isCoseparator_iff_faithful_preadditiveYoneda, preadditiveYoneda_obj]
exact ⟨fun h => Functor.Faithful.of_comp _ (forget₂ _ AddCommGroupCat.{v}),
fun h => Functor.Faithful.comp _ _⟩
|
import Mathlib.Algebra.CharP.ExpChar
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Algebra.MvPolynomial.Equiv
import Mathlib.RingTheory.Polynomial.Content
import Mathlib.RingTheory.UniqueFactorizationDomain
#align_import ring_theory.polynomial.basic from "leanprover-commun... | Mathlib/RingTheory/Polynomial/Basic.lean | 223 | 230 | theorem span_le_degreeLE_of_finite {s : Set R[X]} (s_fin : s.Finite) :
∃ n : ℕ, Submodule.span R s ≤ degreeLE R n := by |
by_cases s_emp : s.Nonempty
· rcases exists_degree_le_of_mem_span_of_finite s_fin s_emp with ⟨p', _, hp'max⟩
exact ⟨natDegree p', fun p hp => mem_degreeLE.mpr ((hp'max _ hp).trans degree_le_natDegree)⟩
· rw [Set.not_nonempty_iff_eq_empty] at s_emp
rw [s_emp, Submodule.span_empty]
exact ⟨0, bot_le⟩
|
import Mathlib.Data.Nat.Defs
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Tactic.Monotonicity.Attr
#align_import data.nat.log from "leanprover-community/mathlib"@"3e00d81bdcbf77c8188bbd18f5524ddc3ed8cac6"
namespace Nat
--@[pp_nodot] porting note: unknown attribute
def log (b : ℕ) : ℕ → ℕ
| n => i... | Mathlib/Data/Nat/Log.lean | 114 | 116 | theorem le_log_of_pow_le {b x y : ℕ} (hb : 1 < b) (h : b ^ x ≤ y) : x ≤ log b y := by |
rcases ne_or_eq y 0 with (hy | rfl)
exacts [(pow_le_iff_le_log hb hy).1 h, (h.not_lt (Nat.pow_pos (Nat.zero_lt_one.trans hb))).elim]
|
import Mathlib.Data.Set.Lattice
import Mathlib.Data.Set.Pairwise.Basic
#align_import data.set.pairwise.lattice from "leanprover-community/mathlib"@"c4c2ed622f43768eff32608d4a0f8a6cec1c047d"
open Function Set Order
variable {α β γ ι ι' : Type*} {κ : Sort*} {r p q : α → α → Prop}
section Pairwise
variable {f g : ... | Mathlib/Data/Set/Pairwise/Lattice.lean | 124 | 130 | theorem biUnion_diff_biUnion_eq {s t : Set ι} {f : ι → Set α} (h : (s ∪ t).PairwiseDisjoint f) :
((⋃ i ∈ s, f i) \ ⋃ i ∈ t, f i) = ⋃ i ∈ s \ t, f i := by |
refine
(biUnion_diff_biUnion_subset f s t).antisymm
(iUnion₂_subset fun i hi a ha => (mem_diff _).2 ⟨mem_biUnion hi.1 ha, ?_⟩)
rw [mem_iUnion₂]; rintro ⟨j, hj, haj⟩
exact (h (Or.inl hi.1) (Or.inr hj) (ne_of_mem_of_not_mem hj hi.2).symm).le_bot ⟨ha, haj⟩
|
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Coxeter.Basic
namespace CoxeterSystem
open List Matrix Function Classical
variable {B : Type*}
variable {W : Type*} [Group W]
variable {M : CoxeterMatrix B} (cs : CoxeterSystem M W)
local prefix:100 "s" => cs.simple
local prefix:100 "π" => cs.wordProd
... | Mathlib/GroupTheory/Coxeter/Length.lean | 277 | 279 | theorem isRightDescent_inv_iff {w : W} {i : B} :
cs.IsRightDescent w⁻¹ i ↔ cs.IsLeftDescent w i := by |
simpa using (cs.isLeftDescent_inv_iff (w := w⁻¹)).symm
|
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 | 83 | 88 | theorem content_dvd_coeff {p : R[X]} (n : ℕ) : p.content ∣ p.coeff n := by |
by_cases h : n ∈ p.support
· apply Finset.gcd_dvd h
rw [mem_support_iff, Classical.not_not] at h
rw [h]
apply dvd_zero
|
import Mathlib.Topology.UniformSpace.Basic
import Mathlib.Topology.Separation
import Mathlib.Order.Filter.CountableInter
#align_import topology.G_delta from "leanprover-community/mathlib"@"b9e46fe101fc897fb2e7edaf0bf1f09ea49eb81a"
noncomputable section
open Topology TopologicalSpace Filter Encodable Set
open sco... | Mathlib/Topology/GDelta.lean | 160 | 163 | theorem IsGδ.biUnion {s : Set ι} (hs : s.Finite) {f : ι → Set X} (h : ∀ i ∈ s, IsGδ (f i)) :
IsGδ (⋃ i ∈ s, f i) := by |
rw [← sUnion_image]
exact .sUnion (hs.image _) (forall_mem_image.2 h)
|
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Convex.Uniform
import Mathlib.Analysis.NormedSpace.Completion
import Mathlib.Analysis.NormedSpace.BoundedLinearMaps
#align_import analysis.inner_product_space.basic from "leanprover-community/mathlib"@"3f655f5297b030... | Mathlib/Analysis/InnerProductSpace/Basic.lean | 313 | 320 | theorem cauchy_schwarz_aux (x y : F) :
normSqF (⟪x, y⟫ • x - ⟪x, x⟫ • y) = normSqF x * (normSqF x * normSqF y - ‖⟪x, y⟫‖ ^ 2) := by |
rw [← @ofReal_inj 𝕜, ofReal_normSq_eq_inner_self]
simp only [inner_sub_sub_self, inner_smul_left, inner_smul_right, conj_ofReal, mul_sub, ←
ofReal_normSq_eq_inner_self x, ← ofReal_normSq_eq_inner_self y]
rw [← mul_assoc, mul_conj, RCLike.conj_mul, mul_left_comm, ← inner_conj_symm y, mul_conj]
push_cast
... |
import Mathlib.Topology.MetricSpace.HausdorffDistance
#align_import topology.metric_space.hausdorff_distance from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
noncomputable section
open NNReal ENNReal Topology Set Filter Bornology
universe u v w
variable {ι : Sort*} {α : Type u} {β :... | Mathlib/Topology/MetricSpace/Thickening.lean | 271 | 275 | theorem closedBall_subset_cthickening_singleton {α : Type*} [PseudoMetricSpace α] (x : α) (δ : ℝ) :
closedBall x δ ⊆ cthickening δ ({x} : Set α) := by |
rcases lt_or_le δ 0 with (hδ | hδ)
· simp only [closedBall_eq_empty.mpr hδ, empty_subset]
· simp only [cthickening_singleton x hδ, Subset.rfl]
|
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Data.Finset.Sym
import Mathlib.Data.Nat.Choose.Cast
import Mathlib.Data.Nat.Choose.Multinomial
#align_import analysis.calculus.cont_diff from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputable section
open scoped C... | Mathlib/Analysis/Calculus/ContDiff/Bounds.lean | 526 | 533 | theorem norm_iteratedFDeriv_comp_le {g : F → G} {f : E → F} {n : ℕ} {N : ℕ∞} (hg : ContDiff 𝕜 N g)
(hf : ContDiff 𝕜 N f) (hn : (n : ℕ∞) ≤ N) (x : E) {C : ℝ} {D : ℝ}
(hC : ∀ i, i ≤ n → ‖iteratedFDeriv 𝕜 i g (f x)‖ ≤ C)
(hD : ∀ i, 1 ≤ i → i ≤ n → ‖iteratedFDeriv 𝕜 i f x‖ ≤ D ^ i) :
‖iteratedFDeriv 𝕜 ... |
simp_rw [← iteratedFDerivWithin_univ] at hC hD ⊢
exact norm_iteratedFDerivWithin_comp_le hg.contDiffOn hf.contDiffOn hn uniqueDiffOn_univ
uniqueDiffOn_univ (mapsTo_univ _ _) (mem_univ x) hC hD
|
import Mathlib.Data.Finset.Sigma
import Mathlib.Data.Finset.Pairwise
import Mathlib.Data.Finset.Powerset
import Mathlib.Data.Fintype.Basic
import Mathlib.Order.CompleteLatticeIntervals
#align_import order.sup_indep from "leanprover-community/mathlib"@"c4c2ed622f43768eff32608d4a0f8a6cec1c047d"
variable {α β ι ι' :... | Mathlib/Order/SupIndep.lean | 106 | 117 | theorem SupIndep.image [DecidableEq ι] {s : Finset ι'} {g : ι' → ι} (hs : s.SupIndep (f ∘ g)) :
(s.image g).SupIndep f := by |
intro t ht i hi hit
rw [mem_image] at hi
obtain ⟨i, hi, rfl⟩ := hi
haveI : DecidableEq ι' := Classical.decEq _
suffices hts : t ⊆ (s.erase i).image g by
refine (supIndep_iff_disjoint_erase.1 hs i hi).mono_right ((sup_mono hts).trans ?_)
rw [sup_image]
rintro j hjt
obtain ⟨j, hj, rfl⟩ := mem_image... |
import Mathlib.Algebra.Associated
import Mathlib.Algebra.Order.Monoid.Unbundled.Pow
import Mathlib.Algebra.Ring.Int
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Order.Bounds.Basic
#align_import data.nat.prime from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82... | Mathlib/Data/Nat/Prime.lean | 279 | 280 | theorem minFac_one : minFac 1 = 1 := by |
simp [minFac, minFacAux]
|
import Mathlib.Data.W.Basic
#align_import data.pfunctor.univariate.basic from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
-- "W", "Idx"
set_option linter.uppercaseLean3 false
universe u v v₁ v₂ v₃
@[pp_with_univ]
structure PFunctor where
A : Type u
B : A → Type u
#align p... | Mathlib/Data/PFunctor/Univariate/Basic.lean | 154 | 154 | theorem fst_map (x : P α) (f : α → β) : (P.map f x).1 = x.1 := by | cases x; rfl
|
import Mathlib.Data.ENNReal.Real
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Topology.UniformSpace.Pi
import Mathlib.Topology.UniformSpace.UniformConvergence
import Mathlib.Topology.UniformSpace.UniformEmbedding
#align_import topology.metric_space.emetric_space from "leanprover-community/mathlib"@"c8f3055... | Mathlib/Topology/EMetricSpace/Basic.lean | 843 | 848 | theorem _root_.TopologicalSpace.IsSeparable.separableSpace {s : Set α} (hs : IsSeparable s) :
SeparableSpace s := by |
rcases hs.exists_countable_dense_subset with ⟨t, hts, htc, hst⟩
lift t to Set s using hts
refine ⟨⟨t, countable_of_injective_of_countable_image Subtype.coe_injective.injOn htc, ?_⟩⟩
rwa [inducing_subtype_val.dense_iff, Subtype.forall]
|
import Mathlib.Tactic.Ring
import Mathlib.Data.PNat.Prime
#align_import data.pnat.xgcd from "leanprover-community/mathlib"@"6afc9b06856ad973f6a2619e3e8a0a8d537a58f2"
open Nat
namespace PNat
structure XgcdType where
wp : ℕ
x : ℕ
y : ℕ
zp : ℕ
ap : ℕ
bp : ℕ
deriving Inhabited
#alig... | Mathlib/Data/PNat/Xgcd.lean | 258 | 259 | theorem start_isSpecial (a b : ℕ+) : (start a b).IsSpecial := by |
dsimp [start, IsSpecial]
|
import Mathlib.Algebra.Homology.ComplexShape
import Mathlib.CategoryTheory.Subobject.Limits
import Mathlib.CategoryTheory.GradedObject
import Mathlib.Algebra.Homology.ShortComplex.Basic
#align_import algebra.homology.homological_complex from "leanprover-community/mathlib"@"88bca0ce5d22ebfd9e73e682e51d60ea13b48347"
... | Mathlib/Algebra/Homology/HomologicalComplex.lean | 79 | 92 | theorem ext {C₁ C₂ : HomologicalComplex V c} (h_X : C₁.X = C₂.X)
(h_d :
∀ i j : ι,
c.Rel i j → C₁.d i j ≫ eqToHom (congr_fun h_X j) = eqToHom (congr_fun h_X i) ≫ C₂.d i j) :
C₁ = C₂ := by |
obtain ⟨X₁, d₁, s₁, h₁⟩ := C₁
obtain ⟨X₂, d₂, s₂, h₂⟩ := C₂
dsimp at h_X
subst h_X
simp only [mk.injEq, heq_eq_eq, true_and]
ext i j
by_cases hij: c.Rel i j
· simpa only [comp_id, id_comp, eqToHom_refl] using h_d i j hij
· rw [s₁ i j hij, s₂ i j hij]
|
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Iterate
import Mathlib.Order.SemiconjSup
import Mathlib.Tactic.Monotonicity
import Mathlib.Topology.Order.MonotoneContinuity
#align_import dynamics.circle.rotation_number.translation_number from "leanprover-... | Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean | 342 | 344 | theorem commute_sub_nat (n : ℕ) : Function.Commute f (· - n) := by |
simpa only [sub_eq_add_neg] using
(f.commute_add_nat n).inverses_right (Equiv.addRight _).right_inv (Equiv.addRight _).left_inv
|
import Mathlib.Init.ZeroOne
import Mathlib.Data.Set.Defs
import Mathlib.Order.Basic
import Mathlib.Order.SymmDiff
import Mathlib.Tactic.Tauto
import Mathlib.Tactic.ByContra
import Mathlib.Util.Delaborators
#align_import data.set.basic from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
... | Mathlib/Data/Set/Basic.lean | 585 | 586 | theorem not_nonempty_iff_eq_empty' : ¬Nonempty s ↔ s = ∅ := by |
rw [nonempty_subtype, not_exists, eq_empty_iff_forall_not_mem]
|
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.iterated_deriv from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputable section
open scoped Classical Topology
open Filter Asymptotics Set
variable {𝕜... | Mathlib/Analysis/Calculus/IteratedDeriv/Defs.lean | 151 | 154 | theorem ContDiffOn.continuousOn_iteratedDerivWithin {n : ℕ∞} {m : ℕ} (h : ContDiffOn 𝕜 n f s)
(hmn : (m : ℕ∞) ≤ n) (hs : UniqueDiffOn 𝕜 s) : ContinuousOn (iteratedDerivWithin m f s) s := by |
simpa only [iteratedDerivWithin_eq_equiv_comp, LinearIsometryEquiv.comp_continuousOn_iff] using
h.continuousOn_iteratedFDerivWithin hmn hs
|
import Mathlib.Analysis.NormedSpace.AddTorsor
import Mathlib.LinearAlgebra.AffineSpace.Ordered
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.GDelta
import Mathlib.Analysis.NormedSpace.FunctionSeries
import Mathlib.Analysis.SpecificLimits.Basic
#align_import topology.urysohns_lemma from "lea... | Mathlib/Topology/UrysohnsLemma.lean | 169 | 175 | theorem approx_of_nmem_U (c : CU P) (n : ℕ) {x : X} (hx : x ∉ c.U) : c.approx n x = 1 := by |
induction' n with n ihn generalizing c
· rw [← mem_compl_iff] at hx
exact indicator_of_mem hx _
· simp only [approx]
rw [ihn, ihn, midpoint_self]
exacts [hx, fun hU => hx <| c.left_U_subset hU]
|
import Mathlib.CategoryTheory.Comma.Arrow
import Mathlib.CategoryTheory.Pi.Basic
import Mathlib.Order.CompleteBooleanAlgebra
#align_import category_theory.morphism_property from "leanprover-community/mathlib"@"7f963633766aaa3ebc8253100a5229dd463040c7"
universe w v v' u u'
open CategoryTheory Opposite
noncomputa... | Mathlib/CategoryTheory/MorphismProperty/Basic.lean | 354 | 359 | theorem RespectsIso.isomorphisms : RespectsIso (isomorphisms C) := by |
constructor <;>
· intro X Y Z e f
simp only [isomorphisms.iff]
intro
infer_instance
|
import Mathlib.Topology.Maps
import Mathlib.Topology.NhdsSet
#align_import topology.constructions from "leanprover-community/mathlib"@"f7ebde7ee0d1505dfccac8644ae12371aa3c1c9f"
noncomputable section
open scoped Classical
open Topology TopologicalSpace Set Filter Function
universe u v
variable {X : Type u} {Y :... | Mathlib/Topology/Constructions.lean | 1,706 | 1,709 | theorem inducing_sigma_map {f₁ : ι → κ} {f₂ : ∀ i, σ i → τ (f₁ i)} (h₁ : Injective f₁) :
Inducing (Sigma.map f₁ f₂) ↔ ∀ i, Inducing (f₂ i) := by |
simp only [inducing_iff_nhds, Sigma.forall, Sigma.nhds_mk, Sigma.map_mk, ← map_sigma_mk_comap h₁,
map_inj sigma_mk_injective]
|
import Mathlib.Algebra.Order.Hom.Ring
import Mathlib.Data.ENat.Basic
import Mathlib.SetTheory.Cardinal.Basic
open Function Set
universe u v
namespace Cardinal
@[coe] def ofENat : ℕ∞ → Cardinal
| (n : ℕ) => n
| ⊤ => ℵ₀
instance : Coe ENat Cardinal := ⟨Cardinal.ofENat⟩
@[simp, norm_cast] lemma ofENat_top : o... | Mathlib/SetTheory/Cardinal/ENat.lean | 258 | 261 | theorem toENat_lift {a : Cardinal.{v}} : toENat (lift.{u} a) = toENat a := by |
cases le_total a ℵ₀ with
| inl ha => lift a to ℕ∞ using ha; simp
| inr ha => simp [toENat_eq_top.2, ha]
|
import Mathlib.LinearAlgebra.QuadraticForm.TensorProduct
import Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv
suppress_compilation
universe uR uM₁ uM₂ uM₃ uM₄
variable {R : Type uR} {M₁ : Type uM₁} {M₂ : Type uM₂} {M₃ : Type uM₃} {M₄ : Type uM₄}
open scoped TensorProduct
namespace QuadraticForm
variable [Co... | Mathlib/LinearAlgebra/QuadraticForm/TensorProduct/Isometries.lean | 186 | 192 | theorem comp_tensorLId_eq (Q₂ : QuadraticForm R M₂) :
Q₂.comp (TensorProduct.lid R M₂) = (sq (R := R)).tmul Q₂ := by |
refine (QuadraticForm.associated_rightInverse R).injective ?_
ext m₂ m₂'
dsimp [-associated_apply]
simp only [associated_tmul, QuadraticForm.associated_comp]
simp [-associated_apply, mul_one]
|
import Mathlib.Data.Bundle
import Mathlib.Data.Set.Image
import Mathlib.Topology.PartialHomeomorph
import Mathlib.Topology.Order.Basic
#align_import topology.fiber_bundle.trivialization from "leanprover-community/mathlib"@"e473c3198bb41f68560cab68a0529c854b618833"
open TopologicalSpace Filter Set Bundle Function
... | Mathlib/Topology/FiberBundle/Trivialization.lean | 278 | 280 | theorem symm_proj_apply (e : Pretrivialization F (π F E)) (z : TotalSpace F E)
(hz : z.proj ∈ e.baseSet) : e.symm z.proj (e z).2 = z.2 := by |
rw [e.symm_apply hz, cast_eq_iff_heq, e.mk_proj_snd' hz, e.symm_apply_apply (e.mem_source.mpr hz)]
|
import Mathlib.MeasureTheory.Function.LpOrder
#align_import measure_theory.function.l1_space from "leanprover-community/mathlib"@"ccdbfb6e5614667af5aa3ab2d50885e0ef44a46f"
noncomputable section
open scoped Classical
open Topology ENNReal MeasureTheory NNReal
open Set Filter TopologicalSpace ENNReal EMetric Meas... | Mathlib/MeasureTheory/Function/L1Space.lean | 538 | 540 | theorem Integrable.left_of_add_measure {f : α → β} (h : Integrable f (μ + ν)) : Integrable f μ := by |
rw [← memℒp_one_iff_integrable] at h ⊢
exact h.left_of_add_measure
|
import Mathlib.LinearAlgebra.Eigenspace.Basic
import Mathlib.FieldTheory.IsAlgClosed.Spectrum
#align_import linear_algebra.eigenspace.is_alg_closed from "leanprover-community/mathlib"@"6b0169218d01f2837d79ea2784882009a0da1aa1"
open Set Function Module FiniteDimensional
variable {K V : Type*} [Field K] [AddCommGro... | Mathlib/LinearAlgebra/Eigenspace/Triangularizable.lean | 132 | 192 | theorem inf_iSup_genEigenspace [FiniteDimensional K V] (h : ∀ x ∈ p, f x ∈ p) :
p ⊓ ⨆ μ, ⨆ k, f.genEigenspace μ k = ⨆ μ, ⨆ k, p ⊓ f.genEigenspace μ k := by |
simp_rw [← (f.genEigenspace _).mono.directed_le.inf_iSup_eq]
refine le_antisymm (fun m hm ↦ ?_)
(le_inf_iff.mpr ⟨iSup_le fun μ ↦ inf_le_left, iSup_mono fun μ ↦ inf_le_right⟩)
classical
obtain ⟨hm₀ : m ∈ p, hm₁ : m ∈ ⨆ μ, ⨆ k, f.genEigenspace μ k⟩ := hm
obtain ⟨m, hm₂, rfl⟩ := (mem_iSup_iff_exists_finsupp... |
import Mathlib.Data.Matrix.Basic
variable {l m n o : Type*}
universe u v w
variable {R : Type*} {α : Type v} {β : Type w}
namespace Matrix
def col (w : m → α) : Matrix m Unit α :=
of fun x _ => w x
#align matrix.col Matrix.col
-- TODO: set as an equation lemma for `col`, see mathlib4#3024
@[simp]
theorem col... | Mathlib/Data/Matrix/RowCol.lean | 88 | 90 | theorem row_smul [SMul R α] (x : R) (v : m → α) : row (x • v) = x • row v := by |
ext
rfl
|
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Int.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.Cases
import Mathlib.Tactic.SimpRw
import Mathlib.Tactic.SplitIfs
#align_import algebra.group.basic from "leanprover-community/mathlib"@"a07d750983b94c530a... | Mathlib/Algebra/Group/Basic.lean | 456 | 457 | theorem mul_div_assoc (a b c : G) : a * b / c = a * (b / c) := by |
rw [div_eq_mul_inv, div_eq_mul_inv, mul_assoc _ _ _]
|
import Mathlib.Algebra.BigOperators.Option
import Mathlib.Analysis.BoxIntegral.Box.Basic
import Mathlib.Data.Set.Pairwise.Lattice
#align_import analysis.box_integral.partition.basic from "leanprover-community/mathlib"@"84dc0bd6619acaea625086d6f53cb35cdd554219"
open Set Finset Function
open scoped Classical
open ... | Mathlib/Analysis/BoxIntegral/Partition/Basic.lean | 236 | 236 | theorem iUnion_top : (⊤ : Prepartition I).iUnion = I := by | simp [Prepartition.iUnion]
|
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