Context stringlengths 57 6.04k | file_name stringlengths 21 79 | start int64 14 1.49k | end int64 18 1.5k | theorem stringlengths 25 1.55k | proof stringlengths 5 7.36k | num_lines int64 1 150 | complexity_score float64 2.72 139,370,958,066,637,970,000,000,000,000,000,000,000,000,000,000,000,000,000B | diff_level int64 0 2 | file_diff_level float64 0 2 | theorem_same_file int64 1 32 | rank_file int64 0 2.51k |
|---|---|---|---|---|---|---|---|---|---|---|---|
import Mathlib.Data.Nat.Count
import Mathlib.Data.Nat.SuccPred
import Mathlib.Order.Interval.Set.Monotone
import Mathlib.Order.OrderIsoNat
#align_import data.nat.nth from "leanprover-community/mathlib"@"7fdd4f3746cb059edfdb5d52cba98f66fce418c0"
open Finset
namespace Nat
variable (p : ℕ → Prop)
noncomputable d... | Mathlib/Data/Nat/Nth.lean | 113 | 119 | theorem image_nth_Iio_card (hf : (setOf p).Finite) : nth p '' Set.Iio hf.toFinset.card = setOf p :=
calc
nth p '' Set.Iio hf.toFinset.card = Set.range (hf.toFinset.orderEmbOfFin rfl) := by |
ext x
simp only [Set.mem_image, Set.mem_range, Fin.exists_iff, ← nth_eq_orderEmbOfFin hf,
Set.mem_Iio, exists_prop]
_ = setOf p := by rw [range_orderEmbOfFin, Set.Finite.coe_toFinset]
| 4 | 54.59815 | 2 | 0.5 | 6 | 435 |
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 | 143 | 153 | theorem basisSets_smul_right (v : E) (U : Set E) (hU : U ∈ p.basisSets) :
∀ᶠ x : 𝕜 in 𝓝 0, x • v ∈ U := by |
rcases p.basisSets_iff.mp hU with ⟨s, r, hr, hU⟩
rw [hU, Filter.eventually_iff]
simp_rw [(s.sup p).mem_ball_zero, map_smul_eq_mul]
by_cases h : 0 < (s.sup p) v
· simp_rw [(lt_div_iff h).symm]
rw [← _root_.ball_zero_eq]
exact Metric.ball_mem_nhds 0 (div_pos hr h)
simp_rw [le_antisymm (not_lt.mp h) (... | 9 | 8,103.083928 | 2 | 1.272727 | 11 | 1,349 |
import Mathlib.Data.Int.AbsoluteValue
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
#align_import linear_algebra.matrix.absolute_value from "leanprover-community/mathlib"@"ab0a2959c83b06280ef576bc830d4aa5fe8c8e61"
open Matrix
namespace Matrix
open Equiv Finset
variable {R S : Type*} [CommRing R] [Nontr... | Mathlib/LinearAlgebra/Matrix/AbsoluteValue.lean | 64 | 73 | theorem det_sum_smul_le {ι : Type*} (s : Finset ι) {c : ι → R} {A : ι → Matrix n n R}
{abv : AbsoluteValue R S} {x : S} (hx : ∀ k i j, abv (A k i j) ≤ x) {y : S}
(hy : ∀ k, abv (c k) ≤ y) :
abv (det (∑ k ∈ s, c k • A k)) ≤
Nat.factorial (Fintype.card n) • (Finset.card s • y * x) ^ Fintype.card n := by |
simpa only [smul_mul_assoc] using
det_sum_le s fun k i j =>
calc
abv (c k * A k i j) = abv (c k) * abv (A k i j) := abv.map_mul _ _
_ ≤ y * x := mul_le_mul (hy k) (hx k i j) (abv.nonneg _) ((abv.nonneg _).trans (hy k))
| 5 | 148.413159 | 2 | 1.333333 | 3 | 1,399 |
import Mathlib.Analysis.Calculus.FDeriv.Prod
import Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv
import Mathlib.LinearAlgebra.Dual
#align_import analysis.calculus.lagrange_multipliers from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Filter Set
open scoped Topology Fi... | Mathlib/Analysis/Calculus/LagrangeMultipliers.lean | 44 | 53 | theorem IsLocalExtrOn.range_ne_top_of_hasStrictFDerivAt
(hextr : IsLocalExtrOn φ {x | f x = f x₀} x₀) (hf' : HasStrictFDerivAt f f' x₀)
(hφ' : HasStrictFDerivAt φ φ' x₀) : LinearMap.range (f'.prod φ') ≠ ⊤ := by |
intro htop
set fφ := fun x => (f x, φ x)
have A : map φ (𝓝[f ⁻¹' {f x₀}] x₀) = 𝓝 (φ x₀) := by
change map (Prod.snd ∘ fφ) (𝓝[fφ ⁻¹' {p | p.1 = f x₀}] x₀) = 𝓝 (φ x₀)
rw [← map_map, nhdsWithin, map_inf_principal_preimage, (hf'.prod hφ').map_nhds_eq_of_surj htop]
exact map_snd_nhdsWithin _
exact he... | 7 | 1,096.633158 | 2 | 2 | 4 | 2,176 |
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
| Mathlib/Algebra/CharP/Quotient.lean | 60 | 66 | theorem Ideal.Quotient.index_eq_zero {R : Type*} [CommRing R] (I : Ideal R) :
(↑I.toAddSubgroup.index : R ⧸ I) = 0 := by |
rw [AddSubgroup.index, Nat.card_eq]
split_ifs with hq; swap
· simp
letI : Fintype (R ⧸ I) := @Fintype.ofFinite _ hq
exact Nat.cast_card_eq_zero (R ⧸ I)
| 5 | 148.413159 | 2 | 1 | 3 | 970 |
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 | 46 | 53 | theorem IsSubfield.pow_mem {a : F} {n : ℤ} {s : Set F} (hs : IsSubfield s) (h : a ∈ s) :
a ^ n ∈ s := by |
cases' n with n n
· suffices a ^ (n : ℤ) ∈ s by exact this
rw [zpow_natCast]
exact hs.toIsSubring.toIsSubmonoid.pow_mem h
· rw [zpow_negSucc]
exact hs.inv_mem (hs.toIsSubring.toIsSubmonoid.pow_mem h)
| 6 | 403.428793 | 2 | 1.5 | 6 | 1,579 |
import Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.measure.haar.normed_space from "leanprover-community/mathlib"@"b84aee748341da06a6d78491367e2c0e9f15e8a5"
noncomputable sect... | Mathlib/MeasureTheory/Measure/Haar/NormedSpace.lean | 64 | 83 | theorem integral_comp_smul (f : E → F) (R : ℝ) :
∫ x, f (R • x) ∂μ = |(R ^ finrank ℝ E)⁻¹| • ∫ x, f x ∂μ := by |
by_cases hF : CompleteSpace F; swap
· simp [integral, hF]
rcases eq_or_ne R 0 with (rfl | hR)
· simp only [zero_smul, integral_const]
rcases Nat.eq_zero_or_pos (finrank ℝ E) with (hE | hE)
· have : Subsingleton E := finrank_zero_iff.1 hE
have : f = fun _ => f 0 := by ext x; rw [Subsingleton.elim ... | 18 | 65,659,969.137331 | 2 | 0.75 | 8 | 658 |
import Mathlib.CategoryTheory.Balanced
import Mathlib.CategoryTheory.Limits.EssentiallySmall
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
import Mathlib.CategoryTheory.Subobject.Lattice
import Mathlib.CategoryTheory.Subobject.WellPowered
import Mathlib.Data.S... | Mathlib/CategoryTheory/Generator.lean | 101 | 106 | theorem isCoseparating_op_iff (𝒢 : Set C) : IsCoseparating 𝒢.op ↔ IsSeparating 𝒢 := by |
refine ⟨fun h𝒢 X Y f g hfg => ?_, fun h𝒢 X Y f g hfg => ?_⟩
· refine Quiver.Hom.op_inj (h𝒢 _ _ fun G hG h => Quiver.Hom.unop_inj ?_)
simpa only [unop_comp, Quiver.Hom.unop_op] using hfg _ (Set.mem_op.1 hG) _
· refine Quiver.Hom.unop_inj (h𝒢 _ _ fun G hG h => Quiver.Hom.op_inj ?_)
simpa only [op_comp,... | 5 | 148.413159 | 2 | 1.333333 | 6 | 1,424 |
import Mathlib.Data.Finset.Lattice
#align_import combinatorics.set_family.compression.down from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
variable {α : Type*} [DecidableEq α] {𝒜 ℬ : Finset (Finset α)} {s : Finset α} {a : α}
namespace Finset
def nonMemberSubfamily (a : α) (𝒜 : ... | Mathlib/Combinatorics/SetFamily/Compression/Down.lean | 99 | 110 | theorem memberSubfamily_union_nonMemberSubfamily (a : α) (𝒜 : Finset (Finset α)) :
𝒜.memberSubfamily a ∪ 𝒜.nonMemberSubfamily a = 𝒜.image fun s => s.erase a := by |
ext s
simp only [mem_union, mem_memberSubfamily, mem_nonMemberSubfamily, mem_image, exists_prop]
constructor
· rintro (h | h)
· exact ⟨_, h.1, erase_insert h.2⟩
· exact ⟨_, h.1, erase_eq_of_not_mem h.2⟩
· rintro ⟨s, hs, rfl⟩
by_cases ha : a ∈ s
· exact Or.inl ⟨by rwa [insert_erase ha], not_me... | 10 | 22,026.465795 | 2 | 1.3125 | 16 | 1,367 |
import Mathlib.Algebra.Polynomial.Reverse
import Mathlib.Algebra.Regular.SMul
#align_import data.polynomial.monic from "leanprover-community/mathlib"@"cbdf7b565832144d024caa5a550117c6df0204a5"
noncomputable section
open Finset
open Polynomial
namespace Polynomial
universe u v y
variable {R : Type u} {S : Typ... | Mathlib/Algebra/Polynomial/Monic.lean | 65 | 73 | theorem Monic.map [Semiring S] (f : R →+* S) (hp : Monic p) : Monic (p.map f) := by |
unfold Monic
nontriviality
have : f p.leadingCoeff ≠ 0 := by
rw [show _ = _ from hp, f.map_one]
exact one_ne_zero
rw [Polynomial.leadingCoeff, coeff_map]
suffices p.coeff (p.map f).natDegree = 1 by simp [this]
rwa [natDegree_eq_of_degree_eq (degree_map_eq_of_leadingCoeff_ne_zero f this)]
| 8 | 2,980.957987 | 2 | 1.285714 | 7 | 1,353 |
import Mathlib.Analysis.Calculus.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.FDeriv.Prod
import Mathlib.Analysis.Calculus.Monotone
import Mathlib.Data.Set.Function
import Mathlib.Algebra.Group.Basic
import Mathlib.Tactic.WLOG
#align_import analysis.bounded_variation from ... | Mathlib/Analysis/BoundedVariation.lean | 107 | 124 | theorem sum_le_of_monotoneOn_Icc (f : α → E) {s : Set α} {m n : ℕ} {u : ℕ → α}
(hu : MonotoneOn u (Icc m n)) (us : ∀ i ∈ Icc m n, u i ∈ s) :
(∑ i ∈ Finset.Ico m n, edist (f (u (i + 1))) (f (u i))) ≤ eVariationOn f s := by |
rcases le_total n m with hnm | hmn
· simp [Finset.Ico_eq_empty_of_le hnm]
let π := projIcc m n hmn
let v i := u (π i)
calc
∑ i ∈ Finset.Ico m n, edist (f (u (i + 1))) (f (u i))
= ∑ i ∈ Finset.Ico m n, edist (f (v (i + 1))) (f (v i)) :=
Finset.sum_congr rfl fun i hi ↦ by
rw [Finset.m... | 15 | 3,269,017.372472 | 2 | 1.2 | 5 | 1,250 |
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Data.Real.Sqrt
import Mathlib.Tactic.Polyrith
#align_import algebra.star.chsh from "leanprover-community/mathlib"@"31c24aa72e7b3e5ed97a8412470e904f82b81004"
universe u
--@[nolint has_nonempty_instance] Porting note(#5171): linter not ported yet
structure Is... | Mathlib/Algebra/Star/CHSH.lean | 104 | 112 | theorem CHSH_id [CommRing R] {A₀ A₁ B₀ B₁ : R} (A₀_inv : A₀ ^ 2 = 1) (A₁_inv : A₁ ^ 2 = 1)
(B₀_inv : B₀ ^ 2 = 1) (B₁_inv : B₁ ^ 2 = 1) :
(2 - A₀ * B₀ - A₀ * B₁ - A₁ * B₀ + A₁ * B₁) * (2 - A₀ * B₀ - A₀ * B₁ - A₁ * B₀ + A₁ * B₁) =
4 * (2 - A₀ * B₀ - A₀ * B₁ - A₁ * B₀ + A₁ * B₁) := by |
-- polyrith suggests:
linear_combination
(2 * B₀ * B₁ + 2) * A₀_inv + (B₀ ^ 2 - 2 * B₀ * B₁ + B₁ ^ 2) * A₁_inv +
(A₀ ^ 2 + 2 * A₀ * A₁ + 1) * B₀_inv +
(A₀ ^ 2 - 2 * A₀ * A₁ + 1) * B₁_inv
| 5 | 148.413159 | 2 | 1.75 | 4 | 1,872 |
import Mathlib.FieldTheory.Separable
import Mathlib.RingTheory.IntegralDomain
import Mathlib.Algebra.CharP.Reduced
import Mathlib.Tactic.ApplyFun
#align_import field_theory.finite.basic from "leanprover-community/mathlib"@"12a85fac627bea918960da036049d611b1a3ee43"
variable {K : Type*} {R : Type*}
local notation ... | Mathlib/FieldTheory/Finite/Basic.lean | 104 | 111 | theorem prod_univ_units_id_eq_neg_one [CommRing K] [IsDomain K] [Fintype Kˣ] :
∏ x : Kˣ, x = (-1 : Kˣ) := by |
classical
have : (∏ x ∈ (@univ Kˣ _).erase (-1), x) = 1 :=
prod_involution (fun x _ => x⁻¹) (by simp)
(fun a => by simp (config := { contextual := true }) [Units.inv_eq_self_iff])
(fun a => by simp [@inv_eq_iff_eq_inv _ _ a]) (by simp)
rw [← insert_erase (mem_univ (-1 : Kˣ)), prod_inser... | 6 | 403.428793 | 2 | 1.833333 | 6 | 1,921 |
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.RingTheory.Ideal.Maps
#align_import algebra.algebra.subalgebra.basic from "leanprover-community/mathlib"@"b915e9392ecb2a861e1e766f0e1df6ac481188ca"
namespace Subalgebra
open Algebra
variable {R S : Type*} [CommSemiring R] [CommRing S] [Algebra R S]
... | Mathlib/Algebra/Algebra/Subalgebra/Operations.lean | 40 | 68 | theorem mem_of_finset_sum_eq_one_of_pow_smul_mem
{ι : Type*} (ι' : Finset ι) (s : ι → S) (l : ι → S)
(e : ∑ i ∈ ι', l i * s i = 1) (hs : ∀ i, s i ∈ S') (hl : ∀ i, l i ∈ S') (x : S)
(H : ∀ i, ∃ n : ℕ, (s i ^ n : S) • x ∈ S') : x ∈ S' := by |
-- Porting note: needed to add this instance
let _i : Algebra { x // x ∈ S' } { x // x ∈ S' } := Algebra.id _
suffices x ∈ Subalgebra.toSubmodule (Algebra.ofId S' S).range by
obtain ⟨x, rfl⟩ := this
exact x.2
choose n hn using H
let s' : ι → S' := fun x => ⟨s x, hs x⟩
let l' : ι → S' := fun x => ⟨l... | 25 | 72,004,899,337.38586 | 2 | 2 | 1 | 2,420 |
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial In... | Mathlib/FieldTheory/PrimitiveElement.lean | 282 | 292 | theorem FiniteDimensional.of_finite_intermediateField
[Finite (IntermediateField F E)] : FiniteDimensional F E := by |
let IF := { K : IntermediateField F E // ∃ x, K = F⟮x⟯ }
have := isAlgebraic_of_finite_intermediateField F E
haveI : ∀ K : IF, FiniteDimensional F K.1 := fun ⟨_, x, rfl⟩ ↦ adjoin.finiteDimensional
(Algebra.IsIntegral.isIntegral _)
have hfin := finiteDimensional_iSup_of_finite (t := fun K : IF ↦ K.1)
have... | 9 | 8,103.083928 | 2 | 2 | 5 | 2,432 |
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.RingTheory.Localization.Basic
import Mathlib.RingTheory.Localization.FractionRing
#align_import ring_theory.localization.localization_localization from "leanprover-community/mathlib"@"831c494092374cfe9f50591ed0ac81a25efc5b86"
open Function
namespace ... | Mathlib/RingTheory/Localization/LocalizationLocalization.lean | 53 | 61 | theorem mem_localizationLocalizationSubmodule {x : R} :
x ∈ localizationLocalizationSubmodule M N ↔
∃ (y : N) (z : M), algebraMap R S x = y * algebraMap R S z := by |
rw [localizationLocalizationSubmodule, Submonoid.mem_comap, Submonoid.mem_sup]
constructor
· rintro ⟨y, hy, _, ⟨z, hz, rfl⟩, e⟩
exact ⟨⟨y, hy⟩, ⟨z, hz⟩, e.symm⟩
· rintro ⟨y, z, e⟩
exact ⟨y, y.prop, _, ⟨z, z.prop, rfl⟩, e.symm⟩
| 6 | 403.428793 | 2 | 1.8 | 5 | 1,899 |
import Mathlib.MeasureTheory.Function.L1Space
import Mathlib.MeasureTheory.Function.SimpleFuncDense
#align_import measure_theory.function.simple_func_dense_lp from "leanprover-community/mathlib"@"5a2df4cd59cb31e97a516d4603a14bed5c2f9425"
noncomputable section
set_option linter.uppercaseLean3 false
open Set Func... | Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean | 68 | 74 | theorem nnnorm_approxOn_le [OpensMeasurableSpace E] {f : β → E} (hf : Measurable f) {s : Set E}
{y₀ : E} (h₀ : y₀ ∈ s) [SeparableSpace s] (x : β) (n : ℕ) :
‖approxOn f hf s y₀ h₀ n x - f x‖₊ ≤ ‖f x - y₀‖₊ := by |
have := edist_approxOn_le hf h₀ x n
rw [edist_comm y₀] at this
simp only [edist_nndist, nndist_eq_nnnorm] at this
exact mod_cast this
| 4 | 54.59815 | 2 | 1.666667 | 6 | 1,781 |
import Mathlib.Algebra.ContinuedFractions.Computation.Basic
import Mathlib.Algebra.ContinuedFractions.Translations
#align_import algebra.continued_fractions.computation.translations from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b348ce40ad"
namespace GeneralizedContinuedFraction
open Generali... | Mathlib/Algebra/ContinuedFractions/Computation/Translations.lean | 226 | 234 | theorem IntFractPair.exists_succ_get?_stream_of_gcf_of_get?_eq_some {gp_n : Pair K}
(s_nth_eq : (of v).s.get? n = some gp_n) :
∃ ifp : IntFractPair K, IntFractPair.stream v (n + 1) = some ifp ∧ (ifp.b : K) = gp_n.b := by |
obtain ⟨ifp, stream_succ_nth_eq, gp_n_eq⟩ :
∃ ifp, IntFractPair.stream v (n + 1) = some ifp ∧ Pair.mk 1 (ifp.b : K) = gp_n := by
unfold of IntFractPair.seq1 at s_nth_eq
simpa [Stream'.Seq.get?_tail, Stream'.Seq.map_get?] using s_nth_eq
cases gp_n_eq
simp_all only [Option.some.injEq, exists_eq_left']
| 6 | 403.428793 | 2 | 1.307692 | 13 | 1,366 |
import Mathlib.NumberTheory.Liouville.Basic
#align_import number_theory.liouville.liouville_number from "leanprover-community/mathlib"@"04e80bb7e8510958cd9aacd32fe2dc147af0b9f1"
noncomputable section
open scoped Nat
open Real Finset
def liouvilleNumber (m : ℝ) : ℝ :=
∑' i : ℕ, 1 / m ^ i !
#align liouville_n... | Mathlib/NumberTheory/Liouville/LiouvilleNumber.lean | 137 | 160 | theorem aux_calc (n : ℕ) {m : ℝ} (hm : 2 ≤ m) :
(1 - 1 / m)⁻¹ * (1 / m ^ (n + 1)!) ≤ 1 / (m ^ n !) ^ n :=
calc
(1 - 1 / m)⁻¹ * (1 / m ^ (n + 1)!) ≤ 2 * (1 / m ^ (n + 1)!) :=
-- the second factors coincide (and are non-negative),
-- the first factors satisfy the inequality `sub_one_div_inv_le_two`
... |
-- [NB: in this block, I do not follow the brace convention for subgoals -- I wait until
-- I solve all extraneous goals at once with `exact pow_pos (zero_lt_two.trans_le hm) _`.]
-- Clear denominators and massage*
apply (div_le_div_iff _ _).mpr
focus
conv_rhs => rw [one_mul, mul_... | 14 | 1,202,604.284165 | 2 | 1.333333 | 3 | 1,416 |
import Mathlib.Analysis.Complex.CauchyIntegral
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Analysis.NormedSpace.Completion
#align_import analysis.complex.liouville from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open TopologicalSpace Metric Set Filter Asymptotics ... | Mathlib/Analysis/Complex/Liouville.lean | 71 | 84 | theorem norm_deriv_le_of_forall_mem_sphere_norm_le {c : ℂ} {R C : ℝ} {f : ℂ → F} (hR : 0 < R)
(hd : DiffContOnCl ℂ f (ball c R)) (hC : ∀ z ∈ sphere c R, ‖f z‖ ≤ C) :
‖deriv f c‖ ≤ C / R := by |
set e : F →L[ℂ] F̂ := UniformSpace.Completion.toComplL
have : HasDerivAt (e ∘ f) (e (deriv f c)) c :=
e.hasFDerivAt.comp_hasDerivAt c
(hd.differentiableAt isOpen_ball <| mem_ball_self hR).hasDerivAt
calc
‖deriv f c‖ = ‖deriv (e ∘ f) c‖ := by
rw [this.deriv]
exact (UniformSpace.Completio... | 11 | 59,874.141715 | 2 | 1.75 | 4 | 1,851 |
import Mathlib.Analysis.InnerProductSpace.Calculus
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.Adjoint
import Mathlib.Analysis.Calculus.LagrangeMultipliers
import Mathlib.LinearAlgebra.Eigenspace.Basic
#align_import analysis.inner_product_space.rayleigh from "leanprover-co... | Mathlib/Analysis/InnerProductSpace/Rayleigh.lean | 107 | 114 | theorem _root_.LinearMap.IsSymmetric.hasStrictFDerivAt_reApplyInnerSelf {T : F →L[ℝ] F}
(hT : (T : F →ₗ[ℝ] F).IsSymmetric) (x₀ : F) :
HasStrictFDerivAt T.reApplyInnerSelf (2 • (innerSL ℝ (T x₀))) x₀ := by |
convert T.hasStrictFDerivAt.inner ℝ (hasStrictFDerivAt_id x₀) using 1
ext y
rw [ContinuousLinearMap.smul_apply, ContinuousLinearMap.comp_apply, fderivInnerCLM_apply,
ContinuousLinearMap.prod_apply, innerSL_apply, id, ContinuousLinearMap.id_apply,
hT.apply_clm x₀ y, real_inner_comm _ x₀, two_smul]
| 5 | 148.413159 | 2 | 2 | 4 | 2,416 |
import Mathlib.CategoryTheory.Monoidal.Category
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Mathlib.CategoryTheory.PEmpty
#align_import category_theory.monoidal.of_chosen_finite_products.basic from "leanprover-community/mathlib"@"95a87616d63b3cb49d3fe678d416fbe9c4217bf4"
universe v u
names... | Mathlib/CategoryTheory/Monoidal/OfChosenFiniteProducts/Basic.lean | 249 | 254 | theorem tensor_comp {X₁ Y₁ Z₁ X₂ Y₂ Z₂ : C} (f₁ : X₁ ⟶ Y₁) (f₂ : X₂ ⟶ Y₂) (g₁ : Y₁ ⟶ Z₁)
(g₂ : Y₂ ⟶ Z₂) : tensorHom ℬ (f₁ ≫ g₁) (f₂ ≫ g₂) = tensorHom ℬ f₁ f₂ ≫ tensorHom ℬ g₁ g₂ := by |
apply IsLimit.hom_ext (ℬ _ _).isLimit;
rintro ⟨⟨⟩⟩ <;>
· dsimp [tensorHom]
simp
| 4 | 54.59815 | 2 | 2 | 2 | 1,981 |
import Mathlib.Algebra.BigOperators.WithTop
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.ENNReal.Basic
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
variable {a b c d : ℝ≥0∞} {r p q... | Mathlib/Data/ENNReal/Operations.lean | 246 | 252 | theorem mul_lt_top_iff {a b : ℝ≥0∞} : a * b < ∞ ↔ a < ∞ ∧ b < ∞ ∨ a = 0 ∨ b = 0 := by |
constructor
· intro h
rw [← or_assoc, or_iff_not_imp_right, or_iff_not_imp_right]
intro hb ha
exact ⟨lt_top_of_mul_ne_top_left h.ne hb, lt_top_of_mul_ne_top_right h.ne ha⟩
· rintro (⟨ha, hb⟩ | rfl | rfl) <;> [exact mul_lt_top ha.ne hb.ne; simp; simp]
| 6 | 403.428793 | 2 | 0.666667 | 12 | 570 |
import Mathlib.Algebra.EuclideanDomain.Defs
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Regular
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Ring.Basic
#align_import algebra.euclidean_domain.basic from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d0... | Mathlib/Algebra/EuclideanDomain/Basic.lean | 123 | 128 | theorem dvd_div_of_mul_dvd {a b c : R} (h : a * b ∣ c) : b ∣ c / a := by |
rcases eq_or_ne a 0 with (rfl | ha)
· simp only [div_zero, dvd_zero]
rcases h with ⟨d, rfl⟩
refine ⟨d, ?_⟩
rw [mul_assoc, mul_div_cancel_left₀ _ ha]
| 5 | 148.413159 | 2 | 0.888889 | 9 | 769 |
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Range
#align_import data.list.nat_antidiagonal from "leanprover-community/mathlib"@"7b78d1776212a91ecc94cf601f83bdcc46b04213"
open List Function Nat
namespace List
namespace Nat
def antidiagonal (n : ℕ) : List (ℕ × ℕ) :=
(range (n + 1)).map fun i ↦ (i,... | Mathlib/Data/List/NatAntidiagonal.lean | 76 | 82 | theorem antidiagonal_succ' {n : ℕ} :
antidiagonal (n + 1) = (antidiagonal n).map (Prod.map id Nat.succ) ++ [(n + 1, 0)] := by |
simp only [antidiagonal, range_succ, Nat.add_sub_cancel_left, map_append, append_assoc,
Nat.sub_self, singleton_append, map_map, map]
congr 1
apply map_congr
simp (config := { contextual := true }) [le_of_lt, Nat.succ_eq_add_one, Nat.sub_add_comm]
| 5 | 148.413159 | 2 | 1.666667 | 6 | 1,790 |
import Mathlib.Geometry.Manifold.MFDeriv.FDeriv
noncomputable section
open scoped Manifold
open Bundle Set Topology
section SpecificFunctions
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜] {E : Type*} [NormedAddCommGroup E]
[NormedSpace 𝕜 E] {H : Type*} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H)... | Mathlib/Geometry/Manifold/MFDeriv/SpecificFunctions.lean | 291 | 297 | theorem tangentMapWithin_prod_fst {s : Set (M × M')} {p : TangentBundle (I.prod I') (M × M')}
(hs : UniqueMDiffWithinAt (I.prod I') s p.proj) :
tangentMapWithin (I.prod I') I Prod.fst s p = ⟨p.proj.1, p.2.1⟩ := by |
simp only [tangentMapWithin]
rw [mfderivWithin_fst]
· rcases p with ⟨⟩; rfl
· exact hs
| 4 | 54.59815 | 2 | 1.3 | 10 | 1,364 |
import Mathlib.ModelTheory.Satisfiability
#align_import model_theory.types from "leanprover-community/mathlib"@"98bd247d933fb581ff37244a5998bd33d81dd46d"
set_option linter.uppercaseLean3 false
universe u v w w'
open Cardinal Set
open scoped Classical
open Cardinal FirstOrder
namespace FirstOrder
namespace La... | Mathlib/ModelTheory/Types.lean | 147 | 151 | theorem nonempty_iff : Nonempty (T.CompleteType α) ↔ T.IsSatisfiable := by |
rw [← isSatisfiable_onTheory_iff (lhomWithConstants_injective L α)]
rw [nonempty_iff_univ_nonempty, nonempty_iff_ne_empty, Ne, not_iff_comm,
← union_empty ((L.lhomWithConstants α).onTheory T), ← setOf_subset_eq_empty_iff]
simp
| 4 | 54.59815 | 2 | 1.8 | 5 | 1,884 |
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Analysis.SpecialFunctions.Complex.Circle
import Mathlib.Analysis.InnerProductSpace.l2Space
import Mathlib.MeasureTheory.Function.ContinuousMapDense
import Mathlib.MeasureTheory.Function.L2Space
import Mathlib.MeasureTheory.Group.Integral
import Mathlib.M... | Mathlib/Analysis/Fourier/AddCircle.lean | 184 | 193 | theorem fourier_add_half_inv_index {n : ℤ} (hn : n ≠ 0) (hT : 0 < T) (x : AddCircle T) :
@fourier T n (x + ↑(T / 2 / n)) = -fourier n x := by |
rw [fourier_apply, zsmul_add, ← QuotientAddGroup.mk_zsmul, toCircle_add, coe_mul_unitSphere]
have : (n : ℂ) ≠ 0 := by simpa using hn
have : (@toCircle T (n • (T / 2 / n) : ℝ) : ℂ) = -1 := by
rw [zsmul_eq_mul, toCircle, Function.Periodic.lift_coe, expMapCircle_apply]
replace hT := Complex.ofReal_ne_zero.m... | 8 | 2,980.957987 | 2 | 0.916667 | 12 | 791 |
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calculus.LocalExtr.Rolle
import Mathlib.Analysis.Convex.Normed
import Mathlib.Analysis.RCLike.Basic
#align_import... | Mathlib/Analysis/Calculus/MeanValue.lean | 156 | 175 | theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ}
(hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b))
(hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x)
-- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x`
(bound ... |
have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by
apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound
· rwa [sub_self, mul_zero, add_zero]
· exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const))
· intro x hx
exact (hB' x hx).a... | 14 | 1,202,604.284165 | 2 | 2 | 2 | 2,354 |
import Mathlib.CategoryTheory.Sites.SheafOfTypes
import Mathlib.Order.Closure
#align_import category_theory.sites.closed from "leanprover-community/mathlib"@"4cfc30e317caad46858393f1a7a33f609296cc30"
universe v u
namespace CategoryTheory
variable {C : Type u} [Category.{v} C]
variable (J₁ J₂ : GrothendieckTopol... | Mathlib/CategoryTheory/Sites/Closed.lean | 149 | 159 | theorem close_eq_top_iff_mem {X : C} (S : Sieve X) : J₁.close S = ⊤ ↔ S ∈ J₁ X := by |
constructor
· intro h
apply J₁.transitive (J₁.top_mem X)
intro Y f hf
change J₁.close S f
rwa [h]
· intro hS
rw [eq_top_iff]
intro Y f _
apply J₁.pullback_stable _ hS
| 10 | 22,026.465795 | 2 | 2 | 2 | 2,090 |
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Topology.Algebra.OpenSubgroup
import Mathlib.Topology.Algebra.Ring.Basic
#align_import topology.algebra.nonarchimedean.basic from "leanprover-community/mathlib"@"83f81aea33931a1edb94ce0f32b9a5d484de6978"
open scoped Pointwise Topology
class Nonarchimede... | Mathlib/Topology/Algebra/Nonarchimedean/Basic.lean | 69 | 75 | theorem nonarchimedean_of_emb (f : G →* H) (emb : OpenEmbedding f) : NonarchimedeanGroup H :=
{ is_nonarchimedean := fun U hU =>
have h₁ : f ⁻¹' U ∈ 𝓝 (1 : G) := by |
apply emb.continuous.tendsto
rwa [f.map_one]
let ⟨V, hV⟩ := is_nonarchimedean (f ⁻¹' U) h₁
⟨{ Subgroup.map f V with isOpen' := emb.isOpenMap _ V.isOpen }, Set.image_subset_iff.2 hV⟩ }
| 4 | 54.59815 | 2 | 2 | 2 | 2,165 |
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Analysis.Convex.Hull
import Mathlib.LinearAlgebra.AffineSpace.Basis
#align_import analysis.convex.combination from "leanprover-community/mathlib"@"92bd7b1ffeb306a89f450bee126ddd8a284c259d"
open Set Function
open scoped Classical
open Pointwise
... | Mathlib/Analysis/Convex/Combination.lean | 93 | 100 | theorem Finset.centerMass_segment' (s : Finset ι) (t : Finset ι') (ws : ι → R) (zs : ι → E)
(wt : ι' → R) (zt : ι' → E) (hws : ∑ i ∈ s, ws i = 1) (hwt : ∑ i ∈ t, wt i = 1) (a b : R)
(hab : a + b = 1) : a • s.centerMass ws zs + b • t.centerMass wt zt = (s.disjSum t).centerMass
(Sum.elim (fun i => a * ws i) f... |
rw [s.centerMass_eq_of_sum_1 _ hws, t.centerMass_eq_of_sum_1 _ hwt, smul_sum, smul_sum, ←
Finset.sum_sum_elim, Finset.centerMass_eq_of_sum_1]
· congr with ⟨⟩ <;> simp only [Sum.elim_inl, Sum.elim_inr, mul_smul]
· rw [sum_sum_elim, ← mul_sum, ← mul_sum, hws, hwt, mul_one, mul_one, hab]
| 4 | 54.59815 | 2 | 0.777778 | 9 | 690 |
import Mathlib.Analysis.LocallyConvex.BalancedCoreHull
import Mathlib.Analysis.LocallyConvex.WithSeminorms
import Mathlib.Analysis.Convex.Gauge
#align_import analysis.locally_convex.abs_convex from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open NormedField Set
open NNReal Pointwis... | Mathlib/Analysis/LocallyConvex/AbsConvex.lean | 52 | 60 | theorem nhds_basis_abs_convex :
(𝓝 (0 : E)).HasBasis (fun s : Set E => s ∈ 𝓝 (0 : E) ∧ Balanced 𝕜 s ∧ Convex ℝ s) id := by |
refine
(LocallyConvexSpace.convex_basis_zero ℝ E).to_hasBasis (fun s hs => ?_) fun s hs =>
⟨s, ⟨hs.1, hs.2.2⟩, rfl.subset⟩
refine ⟨convexHull ℝ (balancedCore 𝕜 s), ?_, convexHull_min (balancedCore_subset s) hs.2⟩
refine ⟨Filter.mem_of_superset (balancedCore_mem_nhds_zero hs.1) (subset_convexHull ℝ _),... | 7 | 1,096.633158 | 2 | 2 | 2 | 1,994 |
import Mathlib.Data.Num.Lemmas
import Mathlib.Data.Nat.Prime
import Mathlib.Tactic.Ring
#align_import data.num.prime from "leanprover-community/mathlib"@"58581d0fe523063f5651df0619be2bf65012a94a"
namespace PosNum
def minFacAux (n : PosNum) : ℕ → PosNum → PosNum
| 0, _ => n
| fuel + 1, k =>
if n < k.bit1... | Mathlib/Data/Num/Prime.lean | 44 | 54 | theorem minFacAux_to_nat {fuel : ℕ} {n k : PosNum} (h : Nat.sqrt n < fuel + k.bit1) :
(minFacAux n fuel k : ℕ) = Nat.minFacAux n k.bit1 := by |
induction' fuel with fuel ih generalizing k <;> rw [minFacAux, Nat.minFacAux]
· rw [Nat.zero_add, Nat.sqrt_lt] at h
simp only [h, ite_true]
simp_rw [← mul_to_nat]
simp only [cast_lt, dvd_to_nat]
split_ifs <;> try rfl
rw [ih] <;> [congr; convert Nat.lt_succ_of_lt h using 1] <;>
simp only [_root_.bit... | 9 | 8,103.083928 | 2 | 2 | 2 | 2,405 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.HasseDeriv
#align_import data.polynomial.taylor from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
noncomputable section
namespace Polynomial
open Polynomial... | Mathlib/Algebra/Polynomial/Taylor.lean | 98 | 102 | theorem natDegree_taylor (p : R[X]) (r : R) : natDegree (taylor r p) = natDegree p := by |
refine map_natDegree_eq_natDegree _ ?_
nontriviality R
intro n c c0
simp [taylor_monomial, natDegree_C_mul_eq_of_mul_ne_zero, natDegree_pow_X_add_C, c0]
| 4 | 54.59815 | 2 | 0.466667 | 15 | 415 |
import Mathlib.Algebra.Polynomial.UnitTrinomial
import Mathlib.RingTheory.Polynomial.GaussLemma
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.selmer from "leanprover-community/mathlib"@"3e00d81bdcbf77c8188bbd18f5524ddc3ed8cac6"
namespace Polynomial
open scoped Polynomial
variable ... | Mathlib/RingTheory/Polynomial/Selmer.lean | 31 | 45 | theorem X_pow_sub_X_sub_one_irreducible_aux (z : ℂ) : ¬(z ^ n = z + 1 ∧ z ^ n + z ^ 2 = 0) := by |
rintro ⟨h1, h2⟩
replace h3 : z ^ 3 = 1 := by
linear_combination (1 - z - z ^ 2 - z ^ n) * h1 + (z ^ n - 2) * h2
have key : z ^ n = 1 ∨ z ^ n = z ∨ z ^ n = z ^ 2 := by
rw [← Nat.mod_add_div n 3, pow_add, pow_mul, h3, one_pow, mul_one]
have : n % 3 < 3 := Nat.mod_lt n zero_lt_three
interval_cases n... | 14 | 1,202,604.284165 | 2 | 2 | 3 | 2,319 |
import Mathlib.Analysis.Calculus.LineDeriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Measurable
open MeasureTheory
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜] [LocallyCompactSpace 𝕜]
{E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [MeasurableSpace E] [OpensMeasurableSpace E]
{F : Type*} [Norm... | Mathlib/Analysis/Calculus/LineDeriv/Measurable.lean | 33 | 38 | theorem measurableSet_lineDifferentiableAt (hf : Continuous f) :
MeasurableSet {x : E | LineDifferentiableAt 𝕜 f x v} := by |
borelize 𝕜
let g : E → 𝕜 → F := fun x t ↦ f (x + t • v)
have hg : Continuous g.uncurry := by apply hf.comp; continuity
exact measurable_prod_mk_right (measurableSet_of_differentiableAt_with_param 𝕜 hg)
| 4 | 54.59815 | 2 | 2 | 6 | 2,214 |
import Mathlib.RingTheory.RootsOfUnity.Basic
import Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed
import Mathlib.Algebra.GCDMonoid.IntegrallyClosed
import Mathlib.FieldTheory.Finite.Basic
#align_import ring_theory.roots_of_unity.minpoly from "leanprover-community/mathlib"@"7fdeecc0d03cd40f7a165e6cf00a4d2286db599f"
... | Mathlib/RingTheory/RootsOfUnity/Minpoly.lean | 40 | 45 | theorem isIntegral (hpos : 0 < n) : IsIntegral ℤ μ := by |
use X ^ n - 1
constructor
· exact monic_X_pow_sub_C 1 (ne_of_lt hpos).symm
· simp only [((IsPrimitiveRoot.iff_def μ n).mp h).left, eval₂_one, eval₂_X_pow, eval₂_sub,
sub_self]
| 5 | 148.413159 | 2 | 2 | 6 | 2,240 |
import Mathlib.Analysis.SpecificLimits.Basic
#align_import analysis.hofer from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open scoped Classical
open Topology
open Filter Finset
local notation "d" => dist
#noalign pos_div_pow_pos
| Mathlib/Analysis/Hofer.lean | 33 | 104 | theorem hofer {X : Type*} [MetricSpace X] [CompleteSpace X] (x : X) (ε : ℝ) (ε_pos : 0 < ε)
{ϕ : X → ℝ} (cont : Continuous ϕ) (nonneg : ∀ y, 0 ≤ ϕ y) : ∃ ε' > 0, ∃ x' : X,
ε' ≤ ε ∧ d x' x ≤ 2 * ε ∧ ε * ϕ x ≤ ε' * ϕ x' ∧ ∀ y, d x' y ≤ ε' → ϕ y ≤ 2 * ϕ x' := by |
by_contra H
have reformulation : ∀ (x') (k : ℕ), ε * ϕ x ≤ ε / 2 ^ k * ϕ x' ↔ 2 ^ k * ϕ x ≤ ϕ x' := by
intro x' k
rw [div_mul_eq_mul_div, le_div_iff, mul_assoc, mul_le_mul_left ε_pos, mul_comm]
positivity
-- Now let's specialize to `ε/2^k`
replace H : ∀ k : ℕ, ∀ x', d x' x ≤ 2 * ε ∧ 2 ^ k * ϕ x ≤ ϕ... | 69 | 925,378,172,558,778,900,000,000,000,000 | 2 | 2 | 1 | 2,401 |
import Mathlib.Algebra.Polynomial.Smeval
import Mathlib.GroupTheory.GroupAction.Ring
import Mathlib.RingTheory.Polynomial.Pochhammer
section Multichoose
open Function Polynomial
class BinomialRing (R : Type*) [AddCommMonoid R] [Pow R ℕ] where
nsmul_right_injective (n : ℕ) (h : n ≠ 0) : Injective (n • · : R →... | Mathlib/RingTheory/Binomial.lean | 129 | 138 | theorem ascPochhammer_smeval_neg_eq_descPochhammer (r : R) (k : ℕ) :
(ascPochhammer ℕ k).smeval (-r) = (-1)^k * (descPochhammer ℤ k).smeval r := by |
induction k with
| zero => simp only [ascPochhammer_zero, descPochhammer_zero, smeval_one, npow_zero, one_mul]
| succ k ih =>
simp only [ascPochhammer_succ_right, smeval_mul, ih, descPochhammer_succ_right, sub_eq_add_neg]
have h : (X + (k : ℕ[X])).smeval (-r) = - (X + (-k : ℤ[X])).smeval r := by
si... | 8 | 2,980.957987 | 2 | 1.6 | 5 | 1,723 |
import Mathlib.Analysis.NormedSpace.Multilinear.Basic
import Mathlib.Analysis.NormedSpace.Units
import Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness
import Mathlib.Analysis.NormedSpace.OperatorNorm.Mul
#align_import analysis.normed_space.bounded_linear_maps from "leanprover-community/mathlib"@"ce11c3c2a285b... | Mathlib/Analysis/NormedSpace/BoundedLinearMaps.lean | 217 | 231 | theorem isBoundedLinearMap_prod_multilinear {E : ι → Type*} [∀ i, NormedAddCommGroup (E i)]
[∀ i, NormedSpace 𝕜 (E i)] :
IsBoundedLinearMap 𝕜 fun p : ContinuousMultilinearMap 𝕜 E F × ContinuousMultilinearMap 𝕜 E G =>
p.1.prod p.2 where
map_add p₁ p₂ := by | ext : 1; rfl
map_smul c p := by ext : 1; rfl
bound := by
refine ⟨1, zero_lt_one, fun p ↦ ?_⟩
rw [one_mul]
apply ContinuousMultilinearMap.opNorm_le_bound _ (norm_nonneg _) _
intro m
rw [ContinuousMultilinearMap.prod_apply, norm_prod_le_iff]
constructor
· exact (p.1.le_opNorm m).trans (mu... | 11 | 59,874.141715 | 2 | 0.538462 | 13 | 510 |
import Mathlib.LinearAlgebra.Matrix.DotProduct
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Diagonal
#align_import data.matrix.rank from "leanprover-community/mathlib"@"17219820a8aa8abe85adf5dfde19af1dd1bd8ae7"
open Matrix
namespace Matrix
open FiniteDimensional
variable {l m n ... | Mathlib/Data/Matrix/Rank.lean | 125 | 130 | theorem rank_submatrix_le [StrongRankCondition R] [Fintype m] (f : n → m) (e : n ≃ m)
(A : Matrix m m R) : rank (A.submatrix f e) ≤ rank A := by |
rw [rank, rank, mulVecLin_submatrix, LinearMap.range_comp, LinearMap.range_comp,
show LinearMap.funLeft R R e.symm = LinearEquiv.funCongrLeft R R e.symm from rfl,
LinearEquiv.range, Submodule.map_top]
exact Submodule.finrank_map_le _ _
| 4 | 54.59815 | 2 | 0.916667 | 12 | 792 |
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Analysis.NormedSpace.Dual
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.function.ae_eq_of_integral from "leanprover-community/mathlib"@"915591b2bb3ea303648db07284... | Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean | 164 | 221 | theorem ae_le_of_forall_set_lintegral_le_of_sigmaFinite [SigmaFinite μ] {f g : α → ℝ≥0∞}
(hf : Measurable f) (hg : Measurable g)
(h : ∀ s, MeasurableSet s → μ s < ∞ → (∫⁻ x in s, f x ∂μ) ≤ ∫⁻ x in s, g x ∂μ) : f ≤ᵐ[μ] g := by |
have A :
∀ (ε N : ℝ≥0) (p : ℕ), 0 < ε → μ ({x | g x + ε ≤ f x ∧ g x ≤ N} ∩ spanningSets μ p) = 0 := by
intro ε N p εpos
let s := {x | g x + ε ≤ f x ∧ g x ≤ N} ∩ spanningSets μ p
have s_meas : MeasurableSet s := by
have A : MeasurableSet {x | g x + ε ≤ f x} := measurableSet_le (hg.add measurable... | 55 | 769,478,526,514,201,800,000,000 | 2 | 2 | 6 | 1,968 |
import Mathlib.Algebra.CharP.Two
import Mathlib.Algebra.CharP.Reduced
import Mathlib.Algebra.NeZero
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.GroupTheory.SpecificGroups.Cyclic
import Mathlib.NumberTheory.Divisors
import Mathlib.RingTheory.IntegralDomain
import Mathlib.Tactic.Zify
#align_import rin... | Mathlib/RingTheory/RootsOfUnity/Basic.lean | 335 | 342 | theorem mk_of_lt (ζ : M) (hk : 0 < k) (h1 : ζ ^ k = 1) (h : ∀ l : ℕ, 0 < l → l < k → ζ ^ l ≠ 1) :
IsPrimitiveRoot ζ k := by |
refine ⟨h1, fun l hl => ?_⟩
suffices k.gcd l = k by exact this ▸ k.gcd_dvd_right l
rw [eq_iff_le_not_lt]
refine ⟨Nat.le_of_dvd hk (k.gcd_dvd_left l), ?_⟩
intro h'; apply h _ (Nat.gcd_pos_of_pos_left _ hk) h'
exact pow_gcd_eq_one _ h1 hl
| 6 | 403.428793 | 2 | 0.727273 | 11 | 648 |
import Mathlib.GroupTheory.Coxeter.Length
import Mathlib.Data.ZMod.Parity
namespace CoxeterSystem
open List Matrix Function
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
local prefi... | Mathlib/GroupTheory/Coxeter/Inversion.lean | 88 | 93 | theorem length_mul_left_ne (w : W) : ℓ (w * t) ≠ ℓ w := by |
suffices cs.lengthParity (w * t) ≠ cs.lengthParity w by
contrapose! this
simp only [lengthParity_eq_ofAdd_length, this]
rcases ht with ⟨w, i, rfl⟩
simp [lengthParity_simple]
| 5 | 148.413159 | 2 | 1.111111 | 9 | 1,198 |
import Mathlib.FieldTheory.Extension
import Mathlib.FieldTheory.SplittingField.Construction
import Mathlib.GroupTheory.Solvable
#align_import field_theory.normal from "leanprover-community/mathlib"@"9fb8964792b4237dac6200193a0d533f1b3f7423"
noncomputable section
open scoped Classical Polynomial
open Polynomial ... | Mathlib/FieldTheory/Normal.lean | 66 | 81 | theorem Normal.exists_isSplittingField [h : Normal F K] [FiniteDimensional F K] :
∃ p : F[X], IsSplittingField F K p := by |
let s := Basis.ofVectorSpace F K
refine
⟨∏ x, minpoly F (s x), splits_prod _ fun x _ => h.splits (s x),
Subalgebra.toSubmodule.injective ?_⟩
rw [Algebra.top_toSubmodule, eq_top_iff, ← s.span_eq, Submodule.span_le, Set.range_subset_iff]
refine fun x =>
Algebra.subset_adjoin
(Multiset.mem_toF... | 14 | 1,202,604.284165 | 2 | 2 | 3 | 2,047 |
import Mathlib.Algebra.Module.PID
import Mathlib.Data.ZMod.Quotient
#align_import group_theory.finite_abelian from "leanprover-community/mathlib"@"879155bff5af618b9062cbb2915347dafd749ad6"
open scoped DirectSum
private def directSumNeZeroMulHom {ι : Type} [DecidableEq ι] (p : ι → ℕ) (n : ι → ℕ) :
(⨁ i : {i ... | Mathlib/GroupTheory/FiniteAbelian.lean | 91 | 100 | theorem finite_of_fg_torsion [AddCommGroup M] [Module ℤ M] [Module.Finite ℤ M]
(hM : Module.IsTorsion ℤ M) : _root_.Finite M := by |
rcases Module.equiv_directSum_of_isTorsion hM with ⟨ι, _, p, h, e, ⟨l⟩⟩
haveI : ∀ i : ι, NeZero (p i ^ e i).natAbs := fun i =>
⟨Int.natAbs_ne_zero.mpr <| pow_ne_zero (e i) (h i).ne_zero⟩
haveI : ∀ i : ι, _root_.Finite <| ℤ ⧸ Submodule.span ℤ {p i ^ e i} := fun i =>
Finite.of_equiv _ (p i ^ e i).quotientS... | 8 | 2,980.957987 | 2 | 2 | 3 | 2,160 |
import Mathlib.Data.List.Infix
#align_import data.list.rdrop from "leanprover-community/mathlib"@"26f081a2fb920140ed5bc5cc5344e84bcc7cb2b2"
-- Make sure we don't import algebra
assert_not_exists Monoid
variable {α : Type*} (p : α → Bool) (l : List α) (n : ℕ)
namespace List
def rdrop : List α :=
l.take (l.leng... | Mathlib/Data/List/DropRight.lean | 81 | 87 | theorem rtake_eq_reverse_take_reverse : l.rtake n = reverse (l.reverse.take n) := by |
rw [rtake]
induction' l using List.reverseRecOn with xs x IH generalizing n
· simp
· cases n
· exact drop_length _
· simp [drop_append_eq_append_drop, IH]
| 6 | 403.428793 | 2 | 0.631579 | 19 | 550 |
import Mathlib.Analysis.Complex.Isometry
import Mathlib.Analysis.NormedSpace.ConformalLinearMap
import Mathlib.Analysis.NormedSpace.FiniteDimension
#align_import analysis.complex.conformal from "leanprover-community/mathlib"@"468b141b14016d54b479eb7a0fff1e360b7e3cf6"
noncomputable section
open Complex Continuous... | Mathlib/Analysis/Complex/Conformal.lean | 49 | 62 | theorem isConformalMap_complex_linear {map : ℂ →L[ℂ] E} (nonzero : map ≠ 0) :
IsConformalMap (map.restrictScalars ℝ) := by |
have minor₁ : ‖map 1‖ ≠ 0 := by simpa only [ext_ring_iff, Ne, norm_eq_zero] using nonzero
refine ⟨‖map 1‖, minor₁, ⟨‖map 1‖⁻¹ • ((map : ℂ →ₗ[ℂ] E) : ℂ →ₗ[ℝ] E), ?_⟩, ?_⟩
· intro x
simp only [LinearMap.smul_apply]
have : x = x • (1 : ℂ) := by rw [smul_eq_mul, mul_one]
nth_rw 1 [this]
rw [LinearMap... | 12 | 162,754.791419 | 2 | 2 | 2 | 2,054 |
import Mathlib.RingTheory.FiniteType
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Localization.Integer
import Mathlib.RingTheory.Localization.Submodule
import Mathlib.RingTheory.Nilpotent.Lemmas
import Mathlib.RingTheory.RingHomProperties
im... | Mathlib/RingTheory/LocalProperties.lean | 166 | 176 | theorem RingHom.ofLocalizationSpanTarget_iff_finite :
RingHom.OfLocalizationSpanTarget @P ↔ RingHom.OfLocalizationFiniteSpanTarget @P := by |
delta RingHom.OfLocalizationSpanTarget RingHom.OfLocalizationFiniteSpanTarget
apply forall₅_congr
-- TODO: Using `refine` here breaks `resetI`.
intros
constructor
· intro h s; exact h s
· intro h s hs hs'
obtain ⟨s', h₁, h₂⟩ := (Ideal.span_eq_top_iff_finite s).mp hs
exact h s' h₂ fun x => hs' ⟨_,... | 9 | 8,103.083928 | 2 | 1.833333 | 6 | 1,920 |
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 | 177 | 182 | theorem normUnit_content {p : R[X]} : normUnit (content p) = 1 := by |
by_cases hp0 : p.content = 0
· simp [hp0]
· ext
apply mul_left_cancel₀ hp0
erw [← normalize_apply, normalize_content, mul_one]
| 5 | 148.413159 | 2 | 1.2 | 15 | 1,288 |
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.GroupTheory.EckmannHilton
import Mathlib.Tactic.CategoryTheory.Reassoc
#align_import category_theory.preadditive.of_biproducts from "leanprover-community/mathlib"@"061ea99a5610cfc72c286aa930d3c1f47f74f3d0"
noncomputable section
universe v u
op... | Mathlib/CategoryTheory/Preadditive/OfBiproducts.lean | 71 | 85 | theorem isUnital_rightAdd : EckmannHilton.IsUnital (· +ᵣ ·) 0 := by |
have h₂ : ∀ f : X ⟶ Y, biprod.desc (0 : X ⟶ Y) f = biprod.snd ≫ f := by
intro f
ext
· aesop_cat
· simp only [biprod.inr_desc, BinaryBicone.inr_snd_assoc]
have h₁ : ∀ f : X ⟶ Y, biprod.desc f (0 : X ⟶ Y) = biprod.fst ≫ f := by
intro f
ext
· aesop_cat
· simp only [biprod.inr_desc, Bin... | 14 | 1,202,604.284165 | 2 | 2 | 3 | 2,018 |
import Mathlib.Algebra.Algebra.Unitization
import Mathlib.Algebra.Star.NonUnitalSubalgebra
import Mathlib.Algebra.Star.Subalgebra
import Mathlib.GroupTheory.GroupAction.Ring
namespace NonUnitalSubalgebra
| Mathlib/Algebra/Algebra/Subalgebra/Unitization.lean | 145 | 157 | theorem _root_.AlgHomClass.unitization_injective' {F R S A : Type*} [CommRing R] [Ring A]
[Algebra R A] [SetLike S A] [hSA : NonUnitalSubringClass S A] [hSRA : SMulMemClass S R A]
(s : S) (h : ∀ r, r ≠ 0 → algebraMap R A r ∉ s)
[FunLike F (Unitization R s) A] [AlgHomClass F R (Unitization R s) A]
(f : F... |
refine (injective_iff_map_eq_zero f).mpr fun x hx => ?_
induction' x with r a
simp_rw [map_add, hf, ← Unitization.algebraMap_eq_inl, AlgHomClass.commutes] at hx
rw [add_eq_zero_iff_eq_neg] at hx ⊢
by_cases hr : r = 0
· ext <;> simp [hr] at hx ⊢
exact hx
· exact (h r hr <| hx ▸ (neg_mem a.property)).e... | 8 | 2,980.957987 | 2 | 0.75 | 4 | 668 |
import Mathlib.CategoryTheory.EffectiveEpi.Preserves
import Mathlib.CategoryTheory.Limits.Final.ParallelPair
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.Sites.Canonical
import Mathlib.CategoryTheory.Sites.Coherent.Basic
import Mathlib.CategoryTheory.Sites.EffectiveEpimorphic
na... | Mathlib/CategoryTheory/Sites/Coherent/RegularSheaves.lean | 87 | 100 | theorem EqualizerCondition.bijective_mapToEqualizer_pullback (P : Cᵒᵖ ⥤ Type*)
(hP : EqualizerCondition P) : ∀ (X B : C) (π : X ⟶ B) [EffectiveEpi π] [HasPullback π π],
Function.Bijective
(MapToEqualizer P π (pullback.fst (f := π) (g := π)) (pullback.snd (f := π) (g := π))
pullback.condition) := b... |
intro X B π _ _
specialize hP π _ (pullbackIsPullback π π)
rw [Types.type_equalizer_iff_unique] at hP
rw [Function.bijective_iff_existsUnique]
intro ⟨b, hb⟩
obtain ⟨a, ha₁, ha₂⟩ := hP b hb
refine ⟨a, ?_, ?_⟩
· simpa [MapToEqualizer] using ha₁
· simpa [MapToEqualizer] using ha₂
| 9 | 8,103.083928 | 2 | 2 | 3 | 1,987 |
import Mathlib.Geometry.Manifold.Diffeomorph
import Mathlib.Geometry.Manifold.Instances.Real
import Mathlib.Geometry.Manifold.PartitionOfUnity
#align_import geometry.manifold.whitney_embedding from "leanprover-community/mathlib"@"86c29aefdba50b3f33e86e52e3b2f51a0d8f0282"
universe uι uE uH uM
variable {ι : Type u... | Mathlib/Geometry/Manifold/WhitneyEmbedding.lean | 83 | 98 | theorem comp_embeddingPiTangent_mfderiv (x : M) (hx : x ∈ s) :
((ContinuousLinearMap.fst ℝ E ℝ).comp
(@ContinuousLinearMap.proj ℝ _ ι (fun _ => E × ℝ) _ _ (fun _ => inferInstance)
(f.ind x hx))).comp
(mfderiv I 𝓘(ℝ, ι → E × ℝ) f.embeddingPiTangent x) =
mfderiv I I (chartAt H... |
set L :=
(ContinuousLinearMap.fst ℝ E ℝ).comp
(@ContinuousLinearMap.proj ℝ _ ι (fun _ => E × ℝ) _ _ (fun _ => inferInstance) (f.ind x hx))
have := L.hasMFDerivAt.comp x f.embeddingPiTangent.smooth.mdifferentiableAt.hasMFDerivAt
convert hasMFDerivAt_unique this _
refine (hasMFDerivAt_extChartAt I (f.m... | 10 | 22,026.465795 | 2 | 2 | 4 | 2,356 |
import Mathlib.Topology.Category.TopCat.Adjunctions
#align_import topology.category.Top.epi_mono from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u
open CategoryTheory
open TopCat
namespace TopCat
| Mathlib/Topology/Category/TopCat/EpiMono.lean | 27 | 34 | theorem epi_iff_surjective {X Y : TopCat.{u}} (f : X ⟶ Y) : Epi f ↔ Function.Surjective f := by |
suffices Epi f ↔ Epi ((forget TopCat).map f) by
rw [this, CategoryTheory.epi_iff_surjective]
rfl
constructor
· intro
infer_instance
· apply Functor.epi_of_epi_map
| 7 | 1,096.633158 | 2 | 2 | 2 | 2,340 |
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 | 71 | 76 | theorem d_comp_d (C : HomologicalComplex V c) (i j k : ι) : C.d i j ≫ C.d j k = 0 := by |
by_cases hij : c.Rel i j
· by_cases hjk : c.Rel j k
· exact C.d_comp_d' i j k hij hjk
· rw [C.shape j k hjk, comp_zero]
· rw [C.shape i j hij, zero_comp]
| 5 | 148.413159 | 2 | 1.4 | 10 | 1,494 |
import Batteries.Data.UnionFind.Basic
namespace Batteries.UnionFind
@[simp] theorem arr_empty : empty.arr = #[] := rfl
@[simp] theorem parent_empty : empty.parent a = a := rfl
@[simp] theorem rank_empty : empty.rank a = 0 := rfl
@[simp] theorem rootD_empty : empty.rootD a = a := rfl
@[simp] theorem arr_push {m : Un... | .lake/packages/batteries/Batteries/Data/UnionFind/Lemmas.lean | 64 | 97 | theorem root_link {self : UnionFind} {x y : Fin self.size}
(xroot : self.parent x = x) (yroot : self.parent y = y) :
∃ r, (r = x ∨ r = y) ∧ ∀ i,
(link self x y yroot).rootD i =
if self.rootD i = x ∨ self.rootD i = y then r.1 else self.rootD i := by |
if h : x.1 = y then
refine ⟨x, .inl rfl, fun i => ?_⟩
rw [rootD_ext (m2 := self) (fun _ => by rw [parent_link, if_pos h])]
split <;> [obtain _ | _ := ‹_› <;> simp [*]; rfl]
else
have {x y : Fin self.size}
(xroot : self.parent x = x) (yroot : self.parent y = y) {m : UnionFind}
(hm : ∀ i, m... | 29 | 3,931,334,297,144.042 | 2 | 1 | 5 | 1,144 |
import Mathlib.Init.Data.Ordering.Basic
import Mathlib.Order.Synonym
#align_import order.compare from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
variable {α β : Type*}
def cmpLE {α} [LE α] [@DecidableRel α (· ≤ ·)] (x y : α) : Ordering :=
if x ≤ y then if y ≤ x then Ordering.eq ... | Mathlib/Order/Compare.lean | 67 | 71 | theorem compares_swap [LT α] {a b : α} {o : Ordering} : o.swap.Compares a b ↔ o.Compares b a := by |
cases o
· exact Iff.rfl
· exact eq_comm
· exact Iff.rfl
| 4 | 54.59815 | 2 | 1 | 4 | 1,096 |
import Mathlib.Algebra.DualNumber
import Mathlib.Algebra.QuaternionBasis
import Mathlib.Data.Complex.Module
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Star
import Mathlib.LinearAlgebra.QuadraticForm.Prod
#align_import linear_algebra.clifford_algebra.equivs fr... | Mathlib/LinearAlgebra/CliffordAlgebra/Equivs.lean | 311 | 322 | theorem toQuaternion_star (c : CliffordAlgebra (Q c₁ c₂)) :
toQuaternion (star c) = star (toQuaternion c) := by |
simp only [CliffordAlgebra.star_def']
induction c using CliffordAlgebra.induction with
| algebraMap r =>
simp only [reverse.commutes, AlgHom.commutes, QuaternionAlgebra.coe_algebraMap,
QuaternionAlgebra.star_coe]
| ι x =>
rw [reverse_ι, involute_ι, toQuaternion_ι, AlgHom.map_neg, toQuaternion_ι,
... | 10 | 22,026.465795 | 2 | 1.4 | 10 | 1,508 |
import Mathlib.CategoryTheory.Sites.CompatiblePlus
import Mathlib.CategoryTheory.Sites.ConcreteSheafification
#align_import category_theory.sites.compatible_sheafification from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
namespace CategoryTheory.GrothendieckTopology
open CategoryThe... | Mathlib/CategoryTheory/Sites/CompatibleSheafification.lean | 118 | 125 | theorem whiskerRight_toSheafify_sheafifyCompIso_hom :
whiskerRight (J.toSheafify _) _ ≫ (J.sheafifyCompIso F P).hom = J.toSheafify _ := by |
dsimp [sheafifyCompIso]
erw [whiskerRight_comp, Category.assoc]
slice_lhs 2 3 => rw [plusCompIso_whiskerRight]
rw [Category.assoc, ← J.plusMap_comp, whiskerRight_toPlus_comp_plusCompIso_hom, ←
Category.assoc, whiskerRight_toPlus_comp_plusCompIso_hom]
rfl
| 6 | 403.428793 | 2 | 1.2 | 5 | 1,281 |
import Mathlib.Algebra.Homology.Linear
import Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex
import Mathlib.Tactic.Abel
#align_import algebra.homology.homotopy from "leanprover-community/mathlib"@"618ea3d5c99240cd7000d8376924906a148bf9ff"
universe v u
open scoped Classical
noncomputable section
open ... | Mathlib/Algebra/Homology/Homotopy.lean | 115 | 121 | theorem dNext_nat (C D : ChainComplex V ℕ) (i : ℕ) (f : ∀ i j, C.X i ⟶ D.X j) :
dNext i f = C.d i (i - 1) ≫ f (i - 1) i := by |
dsimp [dNext]
cases i
· simp only [shape, ChainComplex.next_nat_zero, ComplexShape.down_Rel, Nat.one_ne_zero,
not_false_iff, zero_comp]
· congr <;> simp
| 5 | 148.413159 | 2 | 1.4 | 5 | 1,477 |
import Mathlib.Algebra.MonoidAlgebra.Basic
import Mathlib.RingTheory.Ideal.Basic
#align_import algebra.monoid_algebra.ideal from "leanprover-community/mathlib"@"72c366d0475675f1309d3027d3d7d47ee4423951"
variable {k A G : Type*}
| Mathlib/Algebra/MonoidAlgebra/Ideal.lean | 23 | 58 | theorem MonoidAlgebra.mem_ideal_span_of_image [Monoid G] [Semiring k] {s : Set G}
{x : MonoidAlgebra k G} :
x ∈ Ideal.span (MonoidAlgebra.of k G '' s) ↔ ∀ m ∈ x.support, ∃ m' ∈ s, ∃ d, m = d * m' := by |
let RHS : Ideal (MonoidAlgebra k G) :=
{ carrier := { p | ∀ m : G, m ∈ p.support → ∃ m' ∈ s, ∃ d, m = d * m' }
add_mem' := fun {x y} hx hy m hm => by
classical exact (Finset.mem_union.1 <| Finsupp.support_add hm).elim (hx m) (hy m)
zero_mem' := fun m hm => by cases hm
smul_mem' := fun x... | 33 | 214,643,579,785,916.06 | 2 | 2 | 1 | 2,118 |
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.LinearAlgebra.Dimension.Finite
import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition
open FiniteDimensional
namespace Subalgebra
variable {R S : Type*} [CommRing R] [CommRing S] [Algebra R S]
(A B : Subalgebra R S) [Module.Free R A] [Module.Free R... | Mathlib/Algebra/Algebra/Subalgebra/Rank.lean | 30 | 41 | theorem rank_sup_eq_rank_left_mul_rank_of_free :
Module.rank R ↥(A ⊔ B) = Module.rank R A * Module.rank A (Algebra.adjoin A (B : Set S)) := by |
rcases subsingleton_or_nontrivial R with _ | _
· haveI := Module.subsingleton R S; simp
nontriviality S using rank_subsingleton'
letI : Algebra A (Algebra.adjoin A (B : Set S)) := Subalgebra.algebra _
letI : SMul A (Algebra.adjoin A (B : Set S)) := Algebra.toSMul
haveI : IsScalarTower R A (Algebra.adjoin A... | 10 | 22,026.465795 | 2 | 0.5 | 4 | 495 |
import Mathlib.Data.Set.Pointwise.SMul
import Mathlib.GroupTheory.GroupAction.Hom
open Set Pointwise
| Mathlib/GroupTheory/GroupAction/Pointwise.lean | 33 | 41 | theorem MulAction.smul_bijective_of_is_unit
{M : Type*} [Monoid M] {α : Type*} [MulAction M α] {m : M} (hm : IsUnit m) :
Function.Bijective (fun (a : α) ↦ m • a) := by |
lift m to Mˣ using hm
rw [Function.bijective_iff_has_inverse]
use fun a ↦ m⁻¹ • a
constructor
· intro x; simp [← Units.smul_def]
· intro x; simp [← Units.smul_def]
| 6 | 403.428793 | 2 | 1.2 | 5 | 1,282 |
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 | 48 | 58 | theorem snorm'_le_snormEssSup_mul_rpow_measure_univ {q : ℝ} (hq_pos : 0 < q) :
snorm' f q μ ≤ snormEssSup f μ * μ Set.univ ^ (1 / q) := by |
have h_le : (∫⁻ a : α, (‖f a‖₊ : ℝ≥0∞) ^ q ∂μ) ≤ ∫⁻ _ : α, snormEssSup f μ ^ q ∂μ := by
refine lintegral_mono_ae ?_
have h_nnnorm_le_snorm_ess_sup := coe_nnnorm_ae_le_snormEssSup f μ
exact h_nnnorm_le_snorm_ess_sup.mono fun x hx => by gcongr
rw [snorm', ← ENNReal.rpow_one (snormEssSup f μ)]
nth_rw 2 ... | 9 | 8,103.083928 | 2 | 1.833333 | 6 | 1,918 |
import Mathlib.LinearAlgebra.AffineSpace.Independent
import Mathlib.LinearAlgebra.Basis
#align_import linear_algebra.affine_space.basis from "leanprover-community/mathlib"@"2de9c37fa71dde2f1c6feff19876dd6a7b1519f0"
open Affine
open Set
universe u₁ u₂ u₃ u₄
structure AffineBasis (ι : Type u₁) (k : Type u₂) {V ... | Mathlib/LinearAlgebra/AffineSpace/Basis.lean | 174 | 179 | theorem coord_apply_ne (h : i ≠ j) : b.coord i (b j) = 0 := by |
-- Porting note:
-- in mathlib3 we didn't need to given the `fun j => j ≠ i` argument to `Subtype.coe_mk`,
-- but I don't think we can complain: this proof was over-golfed.
rw [coord, AffineMap.coe_mk, ← @Subtype.coe_mk _ (fun j => j ≠ i) j h.symm, ← b.basisOf_apply,
Basis.sumCoords_self_apply, sub_self]
| 5 | 148.413159 | 2 | 0.833333 | 6 | 735 |
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 | 97 | 110 | theorem IsSymmetric.continuous [CompleteSpace E] {T : E →ₗ[𝕜] E} (hT : IsSymmetric T) :
Continuous T := by |
-- We prove it by using the closed graph theorem
refine T.continuous_of_seq_closed_graph fun u x y hu hTu => ?_
rw [← sub_eq_zero, ← @inner_self_eq_zero 𝕜]
have hlhs : ∀ k : ℕ, ⟪T (u k) - T x, y - T x⟫ = ⟪u k - x, T (y - T x)⟫ := by
intro k
rw [← T.map_sub, hT]
refine tendsto_nhds_unique ((hTu.sub_c... | 12 | 162,754.791419 | 2 | 1.5 | 6 | 1,614 |
import Mathlib.Topology.EMetricSpace.Basic
import Mathlib.Topology.Bornology.Constructions
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Topology.Order.DenselyOrdered
open Set Filter TopologicalSpace Bornology
open scoped ENNReal NNReal Uniformity Topology
universe u v w
variable {α : Type u} {β : Typ... | Mathlib/Topology/MetricSpace/PseudoMetric.lean | 130 | 140 | theorem PseudoMetricSpace.ext {α : Type*} {m m' : PseudoMetricSpace α}
(h : m.toDist = m'.toDist) : m = m' := by |
cases' m with d _ _ _ ed hed U hU B hB
cases' m' with d' _ _ _ ed' hed' U' hU' B' hB'
obtain rfl : d = d' := h
congr
· ext x y : 2
rw [hed, hed']
· exact UniformSpace.ext (hU.trans hU'.symm)
· ext : 2
rw [← Filter.mem_sets, ← Filter.mem_sets, hB, hB']
| 9 | 8,103.083928 | 2 | 2 | 1 | 2,262 |
import Mathlib.Data.TypeMax
import Mathlib.Logic.UnivLE
import Mathlib.CategoryTheory.Limits.Shapes.Images
#align_import category_theory.limits.types from "leanprover-community/mathlib"@"4aa2a2e17940311e47007f087c9df229e7f12942"
open CategoryTheory CategoryTheory.Limits
universe v u w
namespace CategoryTheory.L... | Mathlib/CategoryTheory/Limits/Types.lean | 52 | 60 | theorem isLimit_iff (c : Cone F) :
Nonempty (IsLimit c) ↔ ∀ s ∈ F.sections, ∃! x : c.pt, ∀ j, c.π.app j x = s j := by |
refine ⟨fun ⟨t⟩ s hs ↦ ?_, fun h ↦ ⟨?_⟩⟩
· let cs := coneOfSection hs
exact ⟨t.lift cs ⟨⟩, fun j ↦ congr_fun (t.fac cs j) ⟨⟩,
fun x hx ↦ congr_fun (t.uniq cs (fun _ ↦ x) fun j ↦ funext fun _ ↦ hx j) ⟨⟩⟩
· choose x hx using fun c y ↦ h _ (sectionOfCone c y).2
exact ⟨x, fun c j ↦ funext fun y ↦ (hx c... | 7 | 1,096.633158 | 2 | 1.333333 | 3 | 1,452 |
import Mathlib.Probability.Kernel.Disintegration.Unique
import Mathlib.Probability.Notation
#align_import probability.kernel.cond_distrib from "leanprover-community/mathlib"@"00abe0695d8767201e6d008afa22393978bb324d"
open MeasureTheory Set Filter TopologicalSpace
open scoped ENNReal MeasureTheory ProbabilityTheo... | Mathlib/Probability/Kernel/CondDistrib.lean | 198 | 206 | theorem set_lintegral_preimage_condDistrib (hX : Measurable X) (hY : AEMeasurable Y μ)
(hs : MeasurableSet s) (ht : MeasurableSet t) :
∫⁻ a in X ⁻¹' t, condDistrib Y X μ (X a) s ∂μ = μ (X ⁻¹' t ∩ Y ⁻¹' s) := by |
-- Porting note: need to massage the LHS integrand into the form accepted by `lintegral_comp`
-- (`rw` does not see that the two forms are defeq)
conv_lhs => arg 2; change (fun a => ((condDistrib Y X μ) a) s) ∘ X
rw [lintegral_comp (kernel.measurable_coe _ hs) hX, condDistrib, ← Measure.restrict_map hX ht, ←
... | 6 | 403.428793 | 2 | 0.777778 | 9 | 692 |
import Mathlib.CategoryTheory.Comma.Over
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Yoneda
import Mathlib.Data.Set.Lattice
import Mathlib.Order.CompleteLattice
#align_import category_theory.sites.sieves from "leanprover-community/mathlib"@"239d882c4fb58361ee8b3b39fb2091320edef... | Mathlib/CategoryTheory/Sites/Sieves.lean | 151 | 161 | theorem ofArrows_pullback [HasPullbacks C] {ι : Type*} (Z : ι → C) (g : ∀ i : ι, Z i ⟶ X) :
(ofArrows (fun i => pullback (g i) f) fun i => pullback.snd) =
pullbackArrows f (ofArrows Z g) := by |
funext T
ext h
constructor
· rintro ⟨hk⟩
exact pullbackArrows.mk _ _ (ofArrows.mk hk)
· rintro ⟨W, k, hk₁⟩
cases' hk₁ with i hi
apply ofArrows.mk
| 8 | 2,980.957987 | 2 | 2 | 5 | 2,346 |
import Mathlib.Order.PartialSups
#align_import order.disjointed from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
variable {α β : Type*}
section GeneralizedBooleanAlgebra
variable [GeneralizedBooleanAlgebra α]
def disjointed (f : ℕ → α) : ℕ → α
| 0 => f 0
| n + 1 => f (n + 1) ... | Mathlib/Order/Disjointed.lean | 123 | 136 | theorem disjointed_unique {f d : ℕ → α} (hdisj : Pairwise (Disjoint on d))
(hsups : partialSups d = partialSups f) : d = disjointed f := by |
ext n
cases' n with n
· rw [← partialSups_zero d, hsups, partialSups_zero, disjointed_zero]
suffices h : d n.succ = partialSups d n.succ \ partialSups d n by
rw [h, hsups, partialSups_succ, disjointed_succ, sup_sdiff, sdiff_self, bot_sup_eq]
rw [partialSups_succ, sup_sdiff, sdiff_self, bot_sup_eq, eq_com... | 12 | 162,754.791419 | 2 | 2 | 4 | 2,431 |
import Mathlib.Data.Finset.Lattice
#align_import combinatorics.set_family.compression.down from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
variable {α : Type*} [DecidableEq α] {𝒜 ℬ : Finset (Finset α)} {s : Finset α} {a : α}
namespace Finset
def nonMemberSubfamily (a : α) (𝒜 : ... | Mathlib/Combinatorics/SetFamily/Compression/Down.lean | 91 | 96 | theorem card_memberSubfamily_add_card_nonMemberSubfamily (a : α) (𝒜 : Finset (Finset α)) :
(𝒜.memberSubfamily a).card + (𝒜.nonMemberSubfamily a).card = 𝒜.card := by |
rw [memberSubfamily, nonMemberSubfamily, card_image_of_injOn]
· conv_rhs => rw [← filter_card_add_filter_neg_card_eq_card (fun s => (a ∈ s))]
· apply (erase_injOn' _).mono
simp
| 4 | 54.59815 | 2 | 1.3125 | 16 | 1,367 |
import Mathlib.Algebra.MonoidAlgebra.Basic
import Mathlib.Data.Finset.Pointwise
#align_import algebra.monoid_algebra.support from "leanprover-community/mathlib"@"16749fc4661828cba18cd0f4e3c5eb66a8e80598"
open scoped Pointwise
universe u₁ u₂ u₃
namespace MonoidAlgebra
open Finset Finsupp
variable {k : Type u₁} ... | Mathlib/Algebra/MonoidAlgebra/Support.lean | 45 | 52 | theorem support_single_mul_eq_image [DecidableEq G] [Mul G] (f : MonoidAlgebra k G) {r : k}
(hr : ∀ y, r * y = 0 ↔ y = 0) {x : G} (lx : IsLeftRegular x) :
(single x r * f : MonoidAlgebra k G).support = Finset.image (x * ·) f.support := by |
refine subset_antisymm (support_single_mul_subset f _ _) fun y hy => ?_
obtain ⟨y, yf, rfl⟩ : ∃ a : G, a ∈ f.support ∧ x * a = y := by
simpa only [Finset.mem_image, exists_prop] using hy
simp only [mul_apply, mem_support_iff.mp yf, hr, mem_support_iff, sum_single_index,
Finsupp.sum_ite_eq', Ne, not_false... | 5 | 148.413159 | 2 | 1.666667 | 6 | 1,758 |
import Mathlib.Combinatorics.SimpleGraph.Clique
open Finset
namespace SimpleGraph
variable {V : Type*} [Fintype V] [DecidableEq V] (G H : SimpleGraph V) [DecidableRel G.Adj]
{n r : ℕ}
def IsTuranMaximal (r : ℕ) : Prop :=
G.CliqueFree (r + 1) ∧ ∀ (H : SimpleGraph V) [DecidableRel H.Adj],
H.CliqueFree (r +... | Mathlib/Combinatorics/SimpleGraph/Turan.lean | 66 | 75 | theorem turanGraph_cliqueFree : (turanGraph n r).CliqueFree (r + 1) := by |
rw [cliqueFree_iff]
by_contra h
rw [not_isEmpty_iff] at h
obtain ⟨f, ha⟩ := h
simp only [turanGraph, top_adj] at ha
obtain ⟨x, y, d, c⟩ := Fintype.exists_ne_map_eq_of_card_lt (fun x ↦
(⟨(f x).1 % r, Nat.mod_lt _ hr⟩ : Fin r)) (by simp)
simp only [Fin.mk.injEq] at c
exact absurd c ((@ha x y).mpr d)
| 9 | 8,103.083928 | 2 | 2 | 3 | 2,273 |
import Mathlib.CategoryTheory.Limits.Shapes.Equalizers
import Mathlib.CategoryTheory.Limits.Shapes.CommSq
import Mathlib.CategoryTheory.Limits.Shapes.RegularMono
#align_import category_theory.limits.shapes.kernel_pair from "leanprover-community/mathlib"@"f6bab67886fb92c3e2f539cc90a83815f69a189d"
universe v u u₂
... | Mathlib/CategoryTheory/Limits/Shapes/KernelPair.lean | 139 | 150 | theorem comp_of_mono {f₁ : X ⟶ Y} {f₂ : Y ⟶ Z} [Mono f₂] (small_k : IsKernelPair f₁ a b) :
IsKernelPair (f₁ ≫ f₂) a b :=
{ w := by | rw [small_k.w_assoc]
isLimit' := ⟨by
refine PullbackCone.isLimitAux _
(fun s => small_k.lift s.fst s.snd (by rw [← cancel_mono f₂, assoc, s.condition, assoc]))
(by simp) (by simp) ?_
intro s m hm
apply small_k.isLimit.hom_ext
apply PullbackCone.equalizer_ext small_k.cone _ _... | 10 | 22,026.465795 | 2 | 2 | 1 | 2,294 |
import Mathlib.Analysis.BoxIntegral.Partition.Basic
#align_import analysis.box_integral.partition.split from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf97388c9e3b441fc4ccf3f"
noncomputable section
open scoped Classical
open Filter
open Function Set Filter
namespace BoxIntegral
variable {ι M : Type*} {... | Mathlib/Analysis/BoxIntegral/Partition/Split.lean | 147 | 153 | theorem splitLower_ne_splitUpper (I : Box ι) (i : ι) (x : ℝ) :
I.splitLower i x ≠ I.splitUpper i x := by |
cases' le_or_lt x (I.lower i) with h
· rw [splitUpper_eq_self.2 h, splitLower_eq_bot.2 h]
exact WithBot.bot_ne_coe
· refine (disjoint_splitLower_splitUpper I i x).ne ?_
rwa [Ne, splitLower_eq_bot, not_le]
| 5 | 148.413159 | 2 | 1.2 | 10 | 1,269 |
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.PEquiv
#align_import data.matrix.pequiv from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa1"
namespace PEquiv
open Matrix
universe u v
variable {k l m n : Type*}
variable {α : Type v}
open Matrix
def toMatrix [DecidableEq n] [Zer... | Mathlib/Data/Matrix/PEquiv.lean | 123 | 139 | theorem toMatrix_injective [DecidableEq n] [MonoidWithZero α] [Nontrivial α] :
Function.Injective (@toMatrix m n α _ _ _) := by |
classical
intro f g
refine not_imp_not.1 ?_
simp only [Matrix.ext_iff.symm, toMatrix_apply, PEquiv.ext_iff, not_forall, exists_imp]
intro i hi
use i
cases' hf : f i with fi
· cases' hg : g i with gi
-- Porting note: was `cc`
· rw [hf, hg] at hi
exact (hi rfl).elim
... | 15 | 3,269,017.372472 | 2 | 1.2 | 10 | 1,267 |
import Mathlib.Data.Multiset.Bind
#align_import data.multiset.pi from "leanprover-community/mathlib"@"b2c89893177f66a48daf993b7ba5ef7cddeff8c9"
namespace Multiset
section Pi
variable {α : Type*}
open Function
def Pi.empty (δ : α → Sort*) : ∀ a ∈ (0 : Multiset α), δ a :=
nofun
#align multiset.pi.empty Multi... | Mathlib/Data/Multiset/Pi.lean | 49 | 58 | theorem Pi.cons_swap {a a' : α} {b : δ a} {b' : δ a'} {m : Multiset α} {f : ∀ a ∈ m, δ a}
(h : a ≠ a') : HEq (Pi.cons (a' ::ₘ m) a b (Pi.cons m a' b' f))
(Pi.cons (a ::ₘ m) a' b' (Pi.cons m a b f)) := by |
apply hfunext rfl
simp only [heq_iff_eq]
rintro a'' _ rfl
refine hfunext (by rw [Multiset.cons_swap]) fun ha₁ ha₂ _ => ?_
rcases ne_or_eq a'' a with (h₁ | rfl)
on_goal 1 => rcases eq_or_ne a'' a' with (rfl | h₂)
all_goals simp [*, Pi.cons_same, Pi.cons_ne]
| 7 | 1,096.633158 | 2 | 1.666667 | 3 | 1,801 |
import Mathlib.Order.Filter.Basic
import Mathlib.Topology.Bases
import Mathlib.Data.Set.Accumulate
import Mathlib.Topology.Bornology.Basic
import Mathlib.Topology.LocallyFinite
open Set Filter Topology TopologicalSpace Classical Function
universe u v
variable {X : Type u} {Y : Type v} {ι : Type*}
variable [Topolog... | Mathlib/Topology/Compactness/Compact.lean | 104 | 116 | theorem IsCompact.image_of_continuousOn {f : X → Y} (hs : IsCompact s) (hf : ContinuousOn f s) :
IsCompact (f '' s) := by |
intro l lne ls
have : NeBot (l.comap f ⊓ 𝓟 s) :=
comap_inf_principal_neBot_of_image_mem lne (le_principal_iff.1 ls)
obtain ⟨x, hxs, hx⟩ : ∃ x ∈ s, ClusterPt x (l.comap f ⊓ 𝓟 s) := @hs _ this inf_le_right
haveI := hx.neBot
use f x, mem_image_of_mem f hxs
have : Tendsto f (𝓝 x ⊓ (comap f l ⊓ 𝓟 s)) (�... | 11 | 59,874.141715 | 2 | 1.6 | 5 | 1,733 |
import Mathlib.RingTheory.Ideal.Cotangent
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.TensorProduct.Basic
import Mathlib.RingTheory.FinitePresentation
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Localization.Away.AdjoinRoot
#align_import ring_theory.etale from ... | Mathlib/RingTheory/Smooth/Basic.lean | 148 | 153 | theorem of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by |
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
| 5 | 148.413159 | 2 | 2 | 4 | 2,341 |
import Mathlib.MeasureTheory.Function.SimpleFunc
import Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable
#align_import measure_theory.function.simple_func_dense from "leanprover-community/mathlib"@"7317149f12f55affbc900fc873d0d422485122b9"
open Set Function Filter TopologicalSpace ENNReal EMetric Finset
... | Mathlib/MeasureTheory/Function/SimpleFuncDense.lean | 95 | 99 | theorem nearestPtInd_le (e : ℕ → α) (N : ℕ) (x : α) : nearestPtInd e N x ≤ N := by |
induction' N with N ihN; · simp
simp only [nearestPtInd_succ]
split_ifs
exacts [le_rfl, ihN.trans N.le_succ]
| 4 | 54.59815 | 2 | 1.8 | 5 | 1,885 |
import Mathlib.Data.Real.Basic
#align_import data.real.sign from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace Real
noncomputable def sign (r : ℝ) : ℝ :=
if r < 0 then -1 else if 0 < r then 1 else 0
#align real.sign Real.sign
theorem sign_of_neg {r : ℝ} (hr : r < 0) : si... | Mathlib/Data/Real/Sign.lean | 92 | 98 | theorem sign_mul_nonneg (r : ℝ) : 0 ≤ sign r * r := by |
obtain hn | rfl | hp := lt_trichotomy r (0 : ℝ)
· rw [sign_of_neg hn]
exact mul_nonneg_of_nonpos_of_nonpos (by norm_num) hn.le
· rw [mul_zero]
· rw [sign_of_pos hp, one_mul]
exact hp.le
| 6 | 403.428793 | 2 | 1.363636 | 11 | 1,467 |
import Mathlib.Topology.Algebra.InfiniteSum.Basic
import Mathlib.Topology.Algebra.UniformGroup
noncomputable section
open Filter Finset Function
open scoped Topology
variable {α β γ δ : Type*}
section TopologicalGroup
variable [CommGroup α] [TopologicalSpace α] [TopologicalGroup α]
variable {f g : β → α} {a a₁... | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | 75 | 81 | theorem HasProd.update (hf : HasProd f a₁) (b : β) [DecidableEq β] (a : α) :
HasProd (update f b a) (a / f b * a₁) := by |
convert (hasProd_ite_eq b (a / f b)).mul hf with b'
by_cases h : b' = b
· rw [h, update_same]
simp [eq_self_iff_true, if_true, sub_add_cancel]
· simp only [h, update_noteq, if_false, Ne, one_mul, not_false_iff]
| 5 | 148.413159 | 2 | 0.833333 | 6 | 738 |
import Mathlib.Probability.Notation
import Mathlib.Probability.Integration
import Mathlib.MeasureTheory.Function.L2Space
#align_import probability.variance from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open MeasureTheory Filter Finset
noncomputable section
open scoped MeasureThe... | Mathlib/Probability/Variance.lean | 75 | 89 | theorem evariance_eq_top [IsFiniteMeasure μ] (hXm : AEStronglyMeasurable X μ) (hX : ¬Memℒp X 2 μ) :
evariance X μ = ∞ := by |
by_contra h
rw [← Ne, ← lt_top_iff_ne_top] at h
have : Memℒp (fun ω => X ω - μ[X]) 2 μ := by
refine ⟨hXm.sub aestronglyMeasurable_const, ?_⟩
rw [snorm_eq_lintegral_rpow_nnnorm two_ne_zero ENNReal.two_ne_top]
simp only [coe_two, ENNReal.one_toReal, ENNReal.rpow_two, Ne]
exact ENNReal.rpow_lt_top_o... | 13 | 442,413.392009 | 2 | 1.857143 | 7 | 1,927 |
import Mathlib.Algebra.Order.Group.Abs
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Group.OrderIso
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Logic.Pairwise
#align_import data.set.intervals.group from "lean... | Mathlib/Algebra/Order/Interval/Set/Group.lean | 171 | 183 | theorem pairwise_disjoint_Ioc_mul_zpow :
Pairwise (Disjoint on fun n : ℤ => Ioc (a * b ^ n) (a * b ^ (n + 1))) := by |
simp (config := { unfoldPartialApp := true }) only [Function.onFun]
simp_rw [Set.disjoint_iff]
intro m n hmn x hx
apply hmn
have hb : 1 < b := by
have : a * b ^ m < a * b ^ (m + 1) := hx.1.1.trans_le hx.1.2
rwa [mul_lt_mul_iff_left, ← mul_one (b ^ m), zpow_add_one, mul_lt_mul_iff_left] at this
have... | 11 | 59,874.141715 | 2 | 1 | 6 | 939 |
import Mathlib.Algebra.Polynomial.Mirror
import Mathlib.Analysis.Complex.Polynomial
#align_import data.polynomial.unit_trinomial from "leanprover-community/mathlib"@"302eab4f46abb63de520828de78c04cb0f9b5836"
namespace Polynomial
open scoped Polynomial
open Finset
section Semiring
variable {R : Type*} [Semirin... | Mathlib/Algebra/Polynomial/UnitTrinomial.lean | 81 | 92 | theorem trinomial_natTrailingDegree (hkm : k < m) (hmn : m < n) (hu : u ≠ 0) :
(trinomial k m n u v w).natTrailingDegree = k := by |
refine
natTrailingDegree_eq_of_trailingDegree_eq_some
((Finset.le_inf fun i h => ?_).antisymm <|
trailingDegree_le_of_ne_zero <| by rwa [trinomial_trailing_coeff' hkm hmn]).symm
replace h := support_trinomial' k m n u v w h
rw [mem_insert, mem_insert, mem_singleton] at h
rcases h with (rfl ... | 10 | 22,026.465795 | 2 | 1.111111 | 9 | 1,193 |
import Mathlib.Algebra.CharP.Defs
import Mathlib.RingTheory.Multiplicity
import Mathlib.RingTheory.PowerSeries.Basic
#align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60"
noncomputable section
open Polynomial
open Finset (antidiagonal mem_anti... | Mathlib/RingTheory/PowerSeries/Order.lean | 144 | 157 | theorem order_eq {φ : R⟦X⟧} {n : PartENat} :
order φ = n ↔ (∀ i : ℕ, ↑i = n → coeff R i φ ≠ 0) ∧ ∀ i : ℕ, ↑i < n → coeff R i φ = 0 := by |
induction n using PartENat.casesOn
· rw [order_eq_top]
constructor
· rintro rfl
constructor <;> intros
· exfalso
exact PartENat.natCast_ne_top ‹_› ‹_›
· exact (coeff _ _).map_zero
· rintro ⟨_h₁, h₂⟩
ext i
exact h₂ i (PartENat.natCast_lt_top i)
· simpa [PartENat.n... | 12 | 162,754.791419 | 2 | 1.8 | 10 | 1,890 |
import Mathlib.Combinatorics.Hall.Finite
import Mathlib.CategoryTheory.CofilteredSystem
import Mathlib.Data.Rel
#align_import combinatorics.hall.basic from "leanprover-community/mathlib"@"8195826f5c428fc283510bc67303dd4472d78498"
open Finset CategoryTheory
universe u v
def hallMatchingsOn {ι : Type u} {α : Typ... | Mathlib/Combinatorics/Hall/Basic.lean | 123 | 163 | theorem Finset.all_card_le_biUnion_card_iff_exists_injective {ι : Type u} {α : Type v}
[DecidableEq α] (t : ι → Finset α) :
(∀ s : Finset ι, s.card ≤ (s.biUnion t).card) ↔
∃ f : ι → α, Function.Injective f ∧ ∀ x, f x ∈ t x := by |
constructor
· intro h
-- Set up the functor
haveI : ∀ ι' : (Finset ι)ᵒᵖ, Nonempty ((hallMatchingsFunctor t).obj ι') := fun ι' =>
hallMatchingsOn.nonempty t h ι'.unop
classical
haveI : ∀ ι' : (Finset ι)ᵒᵖ, Finite ((hallMatchingsFunctor t).obj ι') := by
intro ι'
rw [hallMatchi... | 37 | 11,719,142,372,802,612 | 2 | 2 | 2 | 2,193 |
import Mathlib.Probability.Notation
import Mathlib.Probability.Density
import Mathlib.Probability.ConditionalProbability
import Mathlib.Probability.ProbabilityMassFunction.Constructions
open scoped Classical MeasureTheory NNReal ENNReal
-- TODO: We can't `open ProbabilityTheory` without opening the `ProbabilityThe... | Mathlib/Probability/Distributions/Uniform.lean | 66 | 75 | theorem aemeasurable {X : Ω → E} {s : Set E} (hns : μ s ≠ 0) (hnt : μ s ≠ ∞)
(hu : IsUniform X s ℙ μ) : AEMeasurable X ℙ := by |
dsimp [IsUniform, ProbabilityTheory.cond] at hu
by_contra h
rw [map_of_not_aemeasurable h] at hu
apply zero_ne_one' ℝ≥0∞
calc
0 = (0 : Measure E) Set.univ := rfl
_ = _ := by rw [hu, smul_apply, restrict_apply MeasurableSet.univ,
Set.univ_inter, smul_eq_mul, ENNReal.inv_mul_cancel hns hnt]
| 8 | 2,980.957987 | 2 | 1.428571 | 7 | 1,510 |
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Tactic.AdaptationNote
#align_import analysis.special_functions.log.deriv from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
ope... | Mathlib/Analysis/SpecialFunctions/Log/Deriv.lean | 42 | 47 | theorem hasStrictDerivAt_log (hx : x ≠ 0) : HasStrictDerivAt log x⁻¹ x := by |
cases' hx.lt_or_lt with hx hx
· convert (hasStrictDerivAt_log_of_pos (neg_pos.mpr hx)).comp x (hasStrictDerivAt_neg x) using 1
· ext y; exact (log_neg_eq_log y).symm
· field_simp [hx.ne]
· exact hasStrictDerivAt_log_of_pos hx
| 5 | 148.413159 | 2 | 1.666667 | 3 | 1,796 |
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.LinearAlgebra.Matrix.ToLin
import Mathlib.RingTheory.PowerBasis
#align_import linear_algebra.matrix.charpoly.minpoly from "leanprover-community/mathlib"@"7ae139f966795f684fc689186f9ccbaedd31bf31"
noncomputable section
universe u v w
open Polynomi... | Mathlib/LinearAlgebra/Matrix/Charpoly/Minpoly.lean | 83 | 92 | theorem charpoly_leftMulMatrix {S : Type*} [Ring S] [Algebra R S] (h : PowerBasis R S) :
(leftMulMatrix h.basis h.gen).charpoly = minpoly R h.gen := by |
cases subsingleton_or_nontrivial R; · apply Subsingleton.elim
apply minpoly.unique' R h.gen (charpoly_monic _)
· apply (injective_iff_map_eq_zero (G := S) (leftMulMatrix _)).mp
(leftMulMatrix_injective h.basis)
rw [← Polynomial.aeval_algHom_apply, aeval_self_charpoly]
refine fun q hq => or_iff_not_im... | 8 | 2,980.957987 | 2 | 2 | 1 | 2,417 |
import Mathlib.Algebra.Order.Ring.Abs
#align_import data.int.order.lemmas from "leanprover-community/mathlib"@"fc2ed6f838ce7c9b7c7171e58d78eaf7b438fb0e"
open Function Nat
namespace Int
variable {a b : ℤ} {n : ℕ}
theorem natAbs_eq_iff_mul_self_eq {a b : ℤ} : a.natAbs = b.natAbs ↔ a * a = b * b := by
rw [← a... | Mathlib/Data/Int/Order/Lemmas.lean | 62 | 68 | theorem eq_zero_of_abs_lt_dvd {m x : ℤ} (h1 : m ∣ x) (h2 : |x| < m) : x = 0 := by |
obtain rfl | hm := eq_or_ne m 0
· exact Int.zero_dvd.1 h1
rcases h1 with ⟨d, rfl⟩
apply mul_eq_zero_of_right
rw [← abs_lt_one_iff, ← mul_lt_iff_lt_one_right (abs_pos.mpr hm), ← abs_mul]
exact lt_of_lt_of_le h2 (le_abs_self m)
| 6 | 403.428793 | 2 | 1.4 | 5 | 1,483 |
import Mathlib.MeasureTheory.Integral.SetToL1
#align_import measure_theory.integral.bochner from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc9524185ae994553ebf4"
assert_not_exists Differentiable
noncomputable section
open scoped Topology NNReal ENNReal MeasureTheory
open Set Filter TopologicalSpace EN... | Mathlib/MeasureTheory/Integral/Bochner.lean | 282 | 286 | theorem posPart_sub_negPart (f : α →ₛ ℝ) : f.posPart - f.negPart = f := by |
simp only [posPart, negPart]
ext a
rw [coe_sub]
exact max_zero_sub_eq_self (f a)
| 4 | 54.59815 | 2 | 0.692308 | 13 | 637 |
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 | 152 | 159 | theorem length_eq_one_iff {w : W} : ℓ w = 1 ↔ ∃ i : B, w = s i := by |
constructor
· intro h
rcases cs.exists_reduced_word w with ⟨ω, hω, rfl⟩
rcases List.length_eq_one.mp (hω.trans h) with ⟨i, rfl⟩
exact ⟨i, cs.wordProd_singleton i⟩
· rintro ⟨i, rfl⟩
exact cs.length_simple i
| 7 | 1,096.633158 | 2 | 1.363636 | 11 | 1,468 |
import Mathlib.RingTheory.WittVector.Truncated
import Mathlib.RingTheory.WittVector.Identities
import Mathlib.NumberTheory.Padics.RingHoms
#align_import ring_theory.witt_vector.compare from "leanprover-community/mathlib"@"168ad7fc5d8173ad38be9767a22d50b8ecf1cd00"
noncomputable section
variable {p : ℕ} [hp : Fact... | Mathlib/RingTheory/WittVector/Compare.lean | 43 | 53 | theorem eq_of_le_of_cast_pow_eq_zero [CharP R p] (i : ℕ) (hin : i ≤ n)
(hpi : (p : TruncatedWittVector p n R) ^ i = 0) : i = n := by |
contrapose! hpi
replace hin := lt_of_le_of_ne hin hpi; clear hpi
have : (p : TruncatedWittVector p n R) ^ i = WittVector.truncate n ((p : 𝕎 R) ^ i) := by
rw [RingHom.map_pow, map_natCast]
rw [this, ne_eq, ext_iff, not_forall]; clear this
use ⟨i, hin⟩
rw [WittVector.coeff_truncate, coeff_zero, Fin.val_... | 9 | 8,103.083928 | 2 | 0.75 | 4 | 656 |
import Mathlib.CategoryTheory.Category.Grpd
import Mathlib.CategoryTheory.Groupoid
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.Topology.Homotopy.Path
import Mathlib.Data.Set.Subsingleton
#align_import algebraic_topology.fundamental_groupoid.basic from "leanprover-community/mathlib"@"3d7987cda72abc473... | Mathlib/AlgebraicTopology/FundamentalGroupoid/Basic.lean | 56 | 79 | theorem reflTransSymmAux_mem_I (x : I × I) : reflTransSymmAux x ∈ I := by |
dsimp only [reflTransSymmAux]
split_ifs
· constructor
· apply mul_nonneg
· apply mul_nonneg
· unit_interval
· norm_num
· unit_interval
· rw [mul_assoc]
apply mul_le_one
· unit_interval
· apply mul_nonneg
· norm_num
· unit_interval
· lina... | 23 | 9,744,803,446.248903 | 2 | 1.166667 | 12 | 1,229 |
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 _
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⟩
| 4 | 54.59815 | 2 | 0.5 | 6 | 492 |
import Mathlib.Data.Finset.Image
import Mathlib.Data.Multiset.Pi
#align_import data.finset.pi from "leanprover-community/mathlib"@"b2c89893177f66a48daf993b7ba5ef7cddeff8c9"
namespace Finset
open Multiset
section Pi
variable {α : Type*}
def Pi.empty (β : α → Sort*) (a : α) (h : a ∈ (∅ : Finset α)) : β a :=... | Mathlib/Data/Finset/Pi.lean | 115 | 123 | theorem pi_singletons {β : Type*} (s : Finset α) (f : α → β) :
(s.pi fun a => ({f a} : Finset β)) = {fun a _ => f a} := by |
rw [eq_singleton_iff_unique_mem]
constructor
· simp
intro a ha
ext i hi
rw [mem_pi] at ha
simpa using ha i hi
| 7 | 1,096.633158 | 2 | 1.666667 | 3 | 1,804 |
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