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.Analysis.SpecialFunctions.Integrals
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.MeasureTheory.Integral.Layercake
#align_import analysis.special_functions.japanese_bracket from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
noncomputable section
op... | Mathlib/Analysis/SpecialFunctions/JapaneseBracket.lean | 79 | 95 | theorem finite_integral_rpow_sub_one_pow_aux {r : ℝ} (n : ℕ) (hnr : (n : ℝ) < r) :
(∫⁻ x : ℝ in Ioc 0 1, ENNReal.ofReal ((x ^ (-r⁻¹) - 1) ^ n)) < ∞ := by |
have hr : 0 < r := lt_of_le_of_lt n.cast_nonneg hnr
have h_int : ∀ x : ℝ, x ∈ Ioc (0 : ℝ) 1 →
ENNReal.ofReal ((x ^ (-r⁻¹) - 1) ^ n) ≤ ENNReal.ofReal (x ^ (-(r⁻¹ * n))) := fun x hx ↦ by
apply ENNReal.ofReal_le_ofReal
rw [← neg_mul, rpow_mul hx.1.le, rpow_natCast]
refine pow_le_pow_left ?_ (by simp... | 15 | 3,269,017.372472 | 2 | 1.571429 | 7 | 1,710 |
import Mathlib.Combinatorics.Quiver.Basic
import Mathlib.Logic.Lemmas
#align_import combinatorics.quiver.path from "leanprover-community/mathlib"@"18a5306c091183ac90884daa9373fa3b178e8607"
open Function
universe v v₁ v₂ u u₁ u₂
namespace Quiver
inductive Path {V : Type u} [Quiver.{v} V] (a : V) : V → Sort max ... | Mathlib/Combinatorics/Quiver/Path.lean | 81 | 84 | theorem eq_of_length_zero (p : Path a b) (hzero : p.length = 0) : a = b := by |
cases p
· rfl
· cases Nat.succ_ne_zero _ hzero
| 3 | 20.085537 | 1 | 1.5 | 2 | 1,662 |
import Mathlib.Algebra.GradedMonoid
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Algebra.MvPolynomial.Basic
#align_import ring_theory.mv_polynomial.weighted_homogeneous from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Fins... | Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean | 120 | 122 | theorem weightedTotalDegree_zero (w : σ → M) :
weightedTotalDegree w (0 : MvPolynomial σ R) = ⊥ := by |
simp only [weightedTotalDegree, support_zero, Finset.sup_empty]
| 1 | 2.718282 | 0 | 1.125 | 8 | 1,208 |
import Mathlib.Data.Matrix.Basis
import Mathlib.LinearAlgebra.Basis
import Mathlib.LinearAlgebra.Pi
#align_import linear_algebra.std_basis from "leanprover-community/mathlib"@"13bce9a6b6c44f6b4c91ac1c1d2a816e2533d395"
open Function Set Submodule
namespace LinearMap
variable (R : Type*) {ι : Type*} [Semiring R] ... | Mathlib/LinearAlgebra/StdBasis.lean | 55 | 57 | theorem stdBasis_apply' (i i' : ι) : (stdBasis R (fun _x : ι => R) i) 1 i' = ite (i = i') 1 0 := by |
rw [LinearMap.stdBasis_apply, Function.update_apply, Pi.zero_apply]
congr 1; rw [eq_iff_iff, eq_comm]
| 2 | 7.389056 | 1 | 1.5 | 6 | 1,540 |
import Mathlib.Data.List.Basic
namespace List
variable {α β : Type*}
#align list.length_enum_from List.enumFrom_length
#align list.length_enum List.enum_length
@[simp]
theorem get?_enumFrom :
∀ n (l : List α) m, get? (enumFrom n l) m = (get? l m).map fun a => (n + m, a)
| n, [], m => rfl
| n, a :: l, 0 =... | Mathlib/Data/List/Enum.lean | 132 | 133 | theorem enum_append (xs ys : List α) : enum (xs ++ ys) = enum xs ++ enumFrom xs.length ys := by |
simp [enum, enumFrom_append]
| 1 | 2.718282 | 0 | 0.5 | 10 | 472 |
import Mathlib.Data.List.Basic
#align_import data.list.join from "leanprover-community/mathlib"@"18a5306c091183ac90884daa9373fa3b178e8607"
-- Make sure we don't import algebra
assert_not_exists Monoid
variable {α β : Type*}
namespace List
attribute [simp] join
-- Porting note (#10618): simp can prove this
-- @... | Mathlib/Data/List/Join.lean | 65 | 66 | theorem join_join (l : List (List (List α))) : l.join.join = (l.map join).join := by |
induction l <;> simp [*]
| 1 | 2.718282 | 0 | 0.9 | 10 | 782 |
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 | 94 | 106 | theorem nextOr_mem {xs : List α} {x d : α} (hd : d ∈ xs) : nextOr xs x d ∈ xs := by |
revert hd
suffices ∀ xs' : List α, (∀ x ∈ xs, x ∈ xs') → d ∈ xs' → nextOr xs x d ∈ xs' by
exact this xs fun _ => id
intro xs' hxs' hd
induction' xs with y ys ih
· exact hd
cases' ys with z zs
· exact hd
rw [nextOr]
split_ifs with h
· exact hxs' _ (mem_cons_of_mem _ (mem_cons_self _ _))
· exac... | 12 | 162,754.791419 | 2 | 1.4 | 10 | 1,479 |
import Mathlib.Analysis.InnerProductSpace.Calculus
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import analysis.inner_product_space.euclidean_dist from "leanprover-community/mathlib"@"9425b6f8220e53b059f5a4904786c3c4b50fc057"
open scoped Topology
open Set
variable {E : Type*} [AddCommGroup E] [Topologi... | Mathlib/Analysis/InnerProductSpace/EuclideanDist.lean | 82 | 84 | theorem closedBall_eq_image (x : E) (r : ℝ) :
closedBall x r = toEuclidean.symm '' Metric.closedBall (toEuclidean x) r := by |
rw [toEuclidean.image_symm_eq_preimage, closedBall_eq_preimage]
| 1 | 2.718282 | 0 | 0.666667 | 3 | 613 |
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Data.Finset.Antidiagonal
import Mathlib.Data.Finset.Card
import Mathlib.Data.Multiset.NatAntidiagonal
#align_import data.finset.nat_antidiagonal from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function
namespace Finset
name... | Mathlib/Data/Finset/NatAntidiagonal.lean | 78 | 86 | theorem antidiagonal_succ' (n : ℕ) :
antidiagonal (n + 1) =
cons (n + 1, 0)
((antidiagonal n).map
(Embedding.prodMap (Embedding.refl _) ⟨Nat.succ, Nat.succ_injective⟩))
(by simp) := by |
apply eq_of_veq
rw [cons_val, map_val]
exact Multiset.Nat.antidiagonal_succ'
| 3 | 20.085537 | 1 | 0.75 | 4 | 673 |
import Mathlib.Topology.GDelta
#align_import topology.metric_space.baire from "leanprover-community/mathlib"@"b9e46fe101fc897fb2e7edaf0bf1f09ea49eb81a"
noncomputable section
open scoped Topology
open Filter Set TopologicalSpace
variable {X α : Type*} {ι : Sort*}
section BaireTheorem
variable [TopologicalSpace... | Mathlib/Topology/Baire/Lemmas.lean | 132 | 145 | theorem IsGδ.dense_iUnion_interior_of_closed [Countable ι] {s : Set X} (hs : IsGδ s) (hd : Dense s)
{f : ι → Set X} (hc : ∀ i, IsClosed (f i)) (hU : s ⊆ ⋃ i, f i) :
Dense (⋃ i, interior (f i)) := by |
let g i := (frontier (f i))ᶜ
have hgo : ∀ i, IsOpen (g i) := fun i => isClosed_frontier.isOpen_compl
have hgd : Dense (⋂ i, g i) := by
refine dense_iInter_of_isOpen hgo fun i x => ?_
rw [closure_compl, interior_frontier (hc _)]
exact id
refine (hd.inter_of_Gδ hs (.iInter_of_isOpen fun i => (hgo i))... | 11 | 59,874.141715 | 2 | 1.428571 | 7 | 1,523 |
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 | 114 | 116 | theorem memberSubfamily_memberSubfamily : (𝒜.memberSubfamily a).memberSubfamily a = ∅ := by |
ext
simp
| 2 | 7.389056 | 1 | 1.3125 | 16 | 1,367 |
import Batteries.Tactic.Lint.Basic
import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Order.ZeroLEOne
import Mathlib.Data.Nat.Cast.Order
import Mathlib.Init.Data.Int.Order
set_option autoImplicit true
namespace Linarith
theorem lt_irrefl {α : Type u} ... | Mathlib/Tactic/Linarith/Lemmas.lean | 39 | 40 | theorem lt_of_lt_of_eq {α} [OrderedSemiring α] {a b : α} (ha : a < 0) (hb : b = 0) : a + b < 0 := by |
simp [*]
| 1 | 2.718282 | 0 | 0 | 6 | 96 |
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 | 63 | 64 | theorem dvd_mod_iff {a b c : R} (h : c ∣ b) : c ∣ a % b ↔ c ∣ a := by |
rw [← dvd_add_right (h.mul_right _), div_add_mod]
| 1 | 2.718282 | 0 | 0.888889 | 9 | 769 |
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Set.Image
import Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov
import Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp
import Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality... | Mathlib/MeasureTheory/Function/LpSpace.lean | 131 | 132 | theorem toLp_eq_toLp_iff {f g : α → E} (hf : Memℒp f p μ) (hg : Memℒp g p μ) :
hf.toLp f = hg.toLp g ↔ f =ᵐ[μ] g := by | simp [toLp]
| 1 | 2.718282 | 0 | 0.4 | 5 | 396 |
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.CategoryTheory.Groupoid.VertexGroup
import Mathlib.CategoryTheory.Groupoid.Basic
import Mathlib.CategoryTheory.Groupoid
import Mathlib.Data.Set.Lattice
import Mathlib.Order.GaloisConnection
#align_import category_theory.groupoid.subgroupoid from "leanprover-c... | Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean | 165 | 167 | theorem hom.inj_on_objects : Function.Injective (hom S).obj := by |
rintro ⟨c, hc⟩ ⟨d, hd⟩ hcd
simp only [Subtype.mk_eq_mk]; exact hcd
| 2 | 7.389056 | 1 | 1.333333 | 6 | 1,417 |
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 | 86 | 90 | theorem log_ofReal_mul {r : ℝ} (hr : 0 < r) {x : ℂ} (hx : x ≠ 0) :
log (r * x) = Real.log r + log x := by |
replace hx := Complex.abs.ne_zero_iff.mpr hx
simp_rw [log, map_mul, abs_ofReal, arg_real_mul _ hr, abs_of_pos hr, Real.log_mul hr.ne' hx,
ofReal_add, add_assoc]
| 3 | 20.085537 | 1 | 0.375 | 16 | 378 |
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 OpenMap
variable [Topo... | Mathlib/Topology/Maps.lean | 371 | 378 | theorem of_sections
(h : ∀ x, ∃ g : Y → X, ContinuousAt g (f x) ∧ g (f x) = x ∧ RightInverse g f) : IsOpenMap f :=
of_nhds_le fun x =>
let ⟨g, hgc, hgx, hgf⟩ := h x
calc
𝓝 (f x) = map f (map g (𝓝 (f x))) := by | rw [map_map, hgf.comp_eq_id, map_id]
_ ≤ map f (𝓝 (g (f x))) := map_mono hgc
_ = map f (𝓝 x) := by rw [hgx]
| 3 | 20.085537 | 1 | 0.5 | 12 | 442 |
import Mathlib.MeasureTheory.Group.GeometryOfNumbers
import Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls
import Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic
#align_import number_theory.number_field.canonical_embedding from "leanprover-community/mathlib"@"60da01b41bbe4206f05d34fd70c8dd7498717a30"
... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean | 316 | 324 | theorem convexBodySumFun_eq_zero_iff (x : E K) :
convexBodySumFun x = 0 ↔ x = 0 := by |
rw [← normAtPlace_eq_zero, convexBodySumFun, Finset.sum_eq_zero_iff_of_nonneg fun _ _ =>
mul_nonneg (Nat.cast_pos.mpr mult_pos).le (normAtPlace_nonneg _ _)]
conv =>
enter [1, w, hw]
rw [mul_left_mem_nonZeroDivisors_eq_zero_iff
(mem_nonZeroDivisors_iff_ne_zero.mpr <| Nat.cast_ne_zero.mpr mult_ne_z... | 7 | 1,096.633158 | 2 | 1.666667 | 15 | 1,803 |
import Mathlib.LinearAlgebra.Ray
import Mathlib.Analysis.NormedSpace.Real
#align_import analysis.normed_space.ray from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
open Real
variable {E : Type*} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*}
[NormedAddCommGroup F] [NormedSp... | Mathlib/Analysis/NormedSpace/Ray.lean | 68 | 69 | theorem norm_injOn_ray_right (hy : y ≠ 0) : { x | SameRay ℝ x y }.InjOn norm := by |
simpa only [SameRay.sameRay_comm] using norm_injOn_ray_left hy
| 1 | 2.718282 | 0 | 1.2 | 5 | 1,273 |
import Mathlib.Algebra.Module.Submodule.Localization
import Mathlib.LinearAlgebra.Dimension.DivisionRing
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.OreLocalization.OreSet
open Cardinal nonZeroDivisors
section CommRing
universe u u' v v'
variable {R : Type u} (S : Type u') {M : T... | Mathlib/LinearAlgebra/Dimension/Localization.lean | 96 | 102 | theorem rank_quotient_add_rank_of_isDomain [IsDomain R] (M' : Submodule R M) :
Module.rank R (M ⧸ M') + Module.rank R M' = Module.rank R M := by |
apply lift_injective.{max u v}
rw [lift_add, ← IsLocalizedModule.lift_rank_eq (FractionRing R) R⁰ (M'.toLocalized R⁰) le_rfl,
← IsLocalizedModule.lift_rank_eq (FractionRing R) R⁰ (LocalizedModule.mkLinearMap R⁰ M) le_rfl,
← IsLocalizedModule.lift_rank_eq (FractionRing R) R⁰ (M'.toLocalizedQuotient R⁰) le_r... | 5 | 148.413159 | 2 | 2 | 2 | 2,141 |
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Data.Nat.Cast.Order
import Mathlib.Tactic.Common
#align_import data.nat.cast.field from "leanprover-community/mathlib"@"acee671f47b8e7972a1eb6f4eed74b4b3abce829"
namespace Nat
variable {α : Type*}
@[simp]
theorem cast_div [DivisionSemiring α] {m n : ℕ} (n_... | Mathlib/Data/Nat/Cast/Field.lean | 36 | 41 | theorem cast_div_div_div_cancel_right [DivisionSemiring α] [CharZero α] {m n d : ℕ}
(hn : d ∣ n) (hm : d ∣ m) :
(↑(m / d) : α) / (↑(n / d) : α) = (m : α) / n := by |
rcases eq_or_ne d 0 with (rfl | hd); · simp [Nat.zero_dvd.1 hm]
replace hd : (d : α) ≠ 0 := by norm_cast
rw [cast_div hm, cast_div hn, div_div_div_cancel_right _ hd] <;> exact hd
| 3 | 20.085537 | 1 | 1.166667 | 6 | 1,240 |
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.NumberTheory.Liouville.Residual
import Mathlib.NumberTheory.Liouville.LiouvilleWith
import Mathlib.Analysis.PSeries
#align_import number_theory.liouville.measure from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
open sc... | Mathlib/NumberTheory/Liouville/Measure.lean | 77 | 106 | theorem volume_iUnion_setOf_liouvilleWith :
volume (⋃ (p : ℝ) (_hp : 2 < p), { x : ℝ | LiouvilleWith p x }) = 0 := by |
simp only [← setOf_exists, exists_prop]
refine measure_mono_null setOf_liouvilleWith_subset_aux ?_
rw [measure_iUnion_null_iff]; intro m; rw [measure_preimage_add_right]; clear m
refine (measure_biUnion_null_iff <| to_countable _).2 fun n (hn : 1 ≤ n) => ?_
generalize hr : (2 + 1 / n : ℝ) = r
replace hr : ... | 28 | 1,446,257,064,291.475 | 2 | 2 | 2 | 2,265 |
import Mathlib.Algebra.Group.Center
#align_import group_theory.subsemigroup.centralizer from "leanprover-community/mathlib"@"cc67cd75b4e54191e13c2e8d722289a89e67e4fa"
variable {M : Type*} {S T : Set M}
namespace Set
variable (S)
@[to_additive addCentralizer " The centralizer of a subset of an additive magma. ... | Mathlib/Algebra/Group/Centralizer.lean | 94 | 97 | theorem div_mem_centralizer [Group M] (ha : a ∈ centralizer S) (hb : b ∈ centralizer S) :
a / b ∈ centralizer S := by |
rw [div_eq_mul_inv]
exact mul_mem_centralizer ha (inv_mem_centralizer hb)
| 2 | 7.389056 | 1 | 0.5 | 4 | 427 |
import Mathlib.Tactic.Ring
#align_import algebra.group_power.identities from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
variable {R : Type*} [CommRing R] {a b x₁ x₂ x₃ x₄ x₅ x₆ x₇ x₈ y₁ y₂ y₃ y₄ y₅ y₆ y₇ y₈ n : R}
theorem sq_add_sq_mul_sq_add_sq :
(x₁ ^ 2 + x₂ ^ 2) * (y₁ ^ 2 +... | Mathlib/Algebra/Ring/Identities.lean | 55 | 60 | theorem sum_four_sq_mul_sum_four_sq :
(x₁ ^ 2 + x₂ ^ 2 + x₃ ^ 2 + x₄ ^ 2) * (y₁ ^ 2 + y₂ ^ 2 + y₃ ^ 2 + y₄ ^ 2) =
(x₁ * y₁ - x₂ * y₂ - x₃ * y₃ - x₄ * y₄) ^ 2 + (x₁ * y₂ + x₂ * y₁ + x₃ * y₄ - x₄ * y₃) ^ 2 +
(x₁ * y₃ - x₂ * y₄ + x₃ * y₁ + x₄ * y₂) ^ 2 +
(x₁ * y₄ + x₂ * y₃ - x₃ * y₂ + x₄ * y₁) ... |
ring
| 1 | 2.718282 | 0 | 0 | 6 | 55 |
import Mathlib.Logic.Function.Basic
import Mathlib.Logic.Relator
import Mathlib.Init.Data.Quot
import Mathlib.Tactic.Cases
import Mathlib.Tactic.Use
import Mathlib.Tactic.MkIffOfInductiveProp
import Mathlib.Tactic.SimpRw
#align_import logic.relation from "leanprover-community/mathlib"@"3365b20c2ffa7c35e47e5209b89ba9a... | Mathlib/Logic/Relation.lean | 159 | 164 | theorem comp_assoc : (r ∘r p) ∘r q = r ∘r p ∘r q := by |
funext a d
apply propext
constructor
· exact fun ⟨c, ⟨b, hab, hbc⟩, hcd⟩ ↦ ⟨b, hab, c, hbc, hcd⟩
· exact fun ⟨b, hab, c, hbc, hcd⟩ ↦ ⟨c, ⟨b, hab, hbc⟩, hcd⟩
| 5 | 148.413159 | 2 | 1.6 | 15 | 1,743 |
import Mathlib.Topology.Homeomorph
import Mathlib.Topology.Order.LeftRightNhds
#align_import topology.algebra.order.monotone_continuity from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
open Set Filter
open Topology
section LinearOrder
variable {α β : Type*} [LinearOrder α] [Topolo... | Mathlib/Topology/Order/MonotoneContinuity.lean | 81 | 89 | theorem continuousWithinAt_right_of_monotoneOn_of_closure_image_mem_nhdsWithin [DenselyOrdered β]
{f : α → β} {s : Set α} {a : α} (h_mono : MonotoneOn f s) (hs : s ∈ 𝓝[≥] a)
(hfs : closure (f '' s) ∈ 𝓝[≥] f a) : ContinuousWithinAt f (Ici a) a := by |
refine continuousWithinAt_right_of_monotoneOn_of_exists_between h_mono hs fun b hb => ?_
rcases (mem_nhdsWithin_Ici_iff_exists_mem_Ioc_Ico_subset hb).1 hfs with ⟨b', ⟨hab', hbb'⟩, hb'⟩
rcases exists_between hab' with ⟨c', hc'⟩
rcases mem_closure_iff.1 (hb' ⟨hc'.1.le, hc'.2⟩) (Ioo (f a) b') isOpen_Ioo hc' with
... | 6 | 403.428793 | 2 | 2 | 3 | 2,171 |
import Mathlib.Analysis.NormedSpace.Real
import Mathlib.Analysis.Seminorm
import Mathlib.Topology.MetricSpace.HausdorffDistance
#align_import analysis.normed_space.riesz_lemma from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Metric
open Topology
variable {𝕜 : Type*} [Norm... | Mathlib/Analysis/NormedSpace/RieszLemma.lean | 41 | 70 | theorem riesz_lemma {F : Subspace 𝕜 E} (hFc : IsClosed (F : Set E)) (hF : ∃ x : E, x ∉ F) {r : ℝ}
(hr : r < 1) : ∃ x₀ : E, x₀ ∉ F ∧ ∀ y ∈ F, r * ‖x₀‖ ≤ ‖x₀ - y‖ := by |
classical
obtain ⟨x, hx⟩ : ∃ x : E, x ∉ F := hF
let d := Metric.infDist x F
have hFn : (F : Set E).Nonempty := ⟨_, F.zero_mem⟩
have hdp : 0 < d :=
lt_of_le_of_ne Metric.infDist_nonneg fun heq =>
hx ((hFc.mem_iff_infDist_zero hFn).2 heq.symm)
let r' := max r 2⁻¹
have hr' : r' < 1... | 28 | 1,446,257,064,291.475 | 2 | 2 | 3 | 2,077 |
import Mathlib.Probability.ProbabilityMassFunction.Basic
#align_import probability.probability_mass_function.monad from "leanprover-community/mathlib"@"4ac69b290818724c159de091daa3acd31da0ee6d"
noncomputable section
variable {α β γ : Type*}
open scoped Classical
open NNReal ENNReal
open MeasureTheory
namespac... | Mathlib/Probability/ProbabilityMassFunction/Monad.lean | 54 | 54 | theorem mem_support_pure_iff : a' ∈ (pure a).support ↔ a' = a := by | simp
| 1 | 2.718282 | 0 | 1 | 6 | 1,147 |
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.GroupTheory.GroupAction.Ring
#align_import data.polynomial.derivative from "leanprover-community/mathlib"@"bbeb185db4ccee8ed07dc48449414ebfa39cb821"
noncomputable section
open Finset
open Polynomial
namespace Pol... | Mathlib/Algebra/Polynomial/Derivative.lean | 121 | 121 | theorem derivative_C {a : R} : derivative (C a) = 0 := by | simp [derivative_apply]
| 1 | 2.718282 | 0 | 0.3 | 10 | 317 |
import Mathlib.Algebra.BigOperators.Finsupp
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Data.Fintype.BigOperators
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.SetTheory.Cardinal.Cofinality
#align_import linear_algebra.basis from "leanprover-communit... | Mathlib/LinearAlgebra/Basis.lean | 149 | 150 | theorem repr_self_apply (j) [Decidable (i = j)] : b.repr (b i) j = if i = j then 1 else 0 := by |
rw [repr_self, Finsupp.single_apply]
| 1 | 2.718282 | 0 | 0.9 | 10 | 779 |
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.SesquilinearForm
import Mathlib.LinearAlgebra.Matrix.Symmetric
#align_import linear_algebra.quadratic_form.basic from "leanprover-community/mathlib"@"d11f435d4e34a6cea0a1797d6b625b0c170be845"
universe u v w
variable {S T : ... | Mathlib/LinearAlgebra/QuadraticForm/Basic.lean | 103 | 104 | theorem polar_neg (f : M → R) (x y : M) : polar (-f) x y = -polar f x y := by |
simp only [polar, Pi.neg_apply, sub_eq_add_neg, neg_add]
| 1 | 2.718282 | 0 | 0.5 | 6 | 482 |
import Mathlib.Algebra.MvPolynomial.Rename
#align_import data.mv_polynomial.comap from "leanprover-community/mathlib"@"aba31c938d3243cc671be7091b28a1e0814647ee"
namespace MvPolynomial
variable {σ : Type*} {τ : Type*} {υ : Type*} {R : Type*} [CommSemiring R]
noncomputable def comap (f : MvPolynomial σ R →ₐ[R] M... | Mathlib/Algebra/MvPolynomial/Comap.lean | 83 | 87 | theorem comap_eq_id_of_eq_id (f : MvPolynomial σ R →ₐ[R] MvPolynomial σ R) (hf : ∀ φ, f φ = φ)
(x : σ → R) : comap f x = x := by |
convert comap_id_apply x
ext1 φ
simp [hf, AlgHom.id_apply]
| 3 | 20.085537 | 1 | 1.166667 | 6 | 1,243 |
import Mathlib.RepresentationTheory.Action.Limits
import Mathlib.RepresentationTheory.Action.Concrete
import Mathlib.CategoryTheory.Monoidal.FunctorCategory
import Mathlib.CategoryTheory.Monoidal.Transport
import Mathlib.CategoryTheory.Monoidal.Rigid.OfEquivalence
import Mathlib.CategoryTheory.Monoidal.Rigid.FunctorCa... | Mathlib/RepresentationTheory/Action/Monoidal.lean | 112 | 114 | theorem rightUnitor_hom_hom {X : Action V G} : Hom.hom (ρ_ X).hom = (ρ_ X.V).hom := by |
dsimp
simp
| 2 | 7.389056 | 1 | 1 | 6 | 1,092 |
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.NormedSpace.FiniteDimension
#align_import analysis.calculus.cont_diff from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputable section
universe uD uE uF uG
variable {𝕜 : Type*} [NontriviallyNormedField ... | Mathlib/Analysis/Calculus/ContDiff/FiniteDimension.lean | 46 | 48 | theorem contDiff_clm_apply_iff {n : ℕ∞} {f : E → F →L[𝕜] G} [FiniteDimensional 𝕜 F] :
ContDiff 𝕜 n f ↔ ∀ y, ContDiff 𝕜 n fun x => f x y := by |
simp_rw [← contDiffOn_univ, contDiffOn_clm_apply]
| 1 | 2.718282 | 0 | 0.5 | 4 | 438 |
import Mathlib.LinearAlgebra.Matrix.Reindex
import Mathlib.LinearAlgebra.Matrix.ToLin
#align_import linear_algebra.matrix.basis from "leanprover-community/mathlib"@"6c263e4bfc2e6714de30f22178b4d0ca4d149a76"
noncomputable section
open LinearMap Matrix Set Submodule
open Matrix
section BasisToMatrix
variable {ι... | Mathlib/LinearAlgebra/Matrix/Basis.lean | 73 | 76 | theorem coePiBasisFun.toMatrix_eq_transpose [Finite ι] :
((Pi.basisFun R ι).toMatrix : Matrix ι ι R → Matrix ι ι R) = Matrix.transpose := by |
ext M i j
rfl
| 2 | 7.389056 | 1 | 1.125 | 8 | 1,207 |
import Mathlib.Geometry.Manifold.ContMDiff.NormedSpace
#align_import geometry.manifold.vector_bundle.fiberwise_linear from "leanprover-community/mathlib"@"be2c24f56783935652cefffb4bfca7e4b25d167e"
noncomputable section
open Set TopologicalSpace
open scoped Manifold Topology
variable {𝕜 B F : Type*} [Topolog... | Mathlib/Geometry/Manifold/VectorBundle/FiberwiseLinear.lean | 87 | 95 | theorem target_trans_partialHomeomorph (hU : IsOpen U)
(hφ : ContinuousOn (fun x => φ x : B → F →L[𝕜] F) U)
(h2φ : ContinuousOn (fun x => (φ x).symm : B → F →L[𝕜] F) U) (hU' : IsOpen U')
(hφ' : ContinuousOn (fun x => φ' x : B → F →L[𝕜] F) U')
(h2φ' : ContinuousOn (fun x => (φ' x).symm : B → F →L[𝕜] ... |
dsimp only [FiberwiseLinear.partialHomeomorph]; mfld_set_tac
| 1 | 2.718282 | 0 | 0 | 2 | 187 |
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
import Mathlib.Analysis.Complex.CauchyIntegral
#align_import analysis.special_functions.gamma.beta from "l... | Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean | 530 | 534 | theorem one_div_Gamma_eq_self_mul_one_div_Gamma_add_one (s : ℂ) :
(Gamma s)⁻¹ = s * (Gamma (s + 1))⁻¹ := by |
rcases ne_or_eq s 0 with (h | rfl)
· rw [Gamma_add_one s h, mul_inv, mul_inv_cancel_left₀ h]
· rw [zero_add, Gamma_zero, inv_zero, zero_mul]
| 3 | 20.085537 | 1 | 1.857143 | 7 | 1,923 |
import Mathlib.Analysis.Complex.Basic
import Mathlib.FieldTheory.IntermediateField
import Mathlib.Topology.Algebra.Field
import Mathlib.Topology.Algebra.UniformRing
#align_import topology.instances.complex from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
section ComplexSubfield
open... | Mathlib/Topology/Instances/Complex.lean | 25 | 44 | theorem Complex.subfield_eq_of_closed {K : Subfield ℂ} (hc : IsClosed (K : Set ℂ)) :
K = ofReal.fieldRange ∨ K = ⊤ := by |
suffices range (ofReal' : ℝ → ℂ) ⊆ K by
rw [range_subset_iff, ← coe_algebraMap] at this
have :=
(Subalgebra.isSimpleOrder_of_finrank finrank_real_complex).eq_bot_or_eq_top
(Subfield.toIntermediateField K this).toSubalgebra
simp_rw [← SetLike.coe_set_eq, IntermediateField.coe_toSubalgebra] a... | 18 | 65,659,969.137331 | 2 | 2 | 2 | 2,026 |
import Mathlib.Data.Real.Irrational
import Mathlib.Data.Nat.Fib.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Algebra.LinearRecurrence
import Mathlib.Tactic.NormNum.NatFib
import Mathlib.Tactic.NormNum.Prime
#align_import data.real.golden_ratio from "leanprover-community/mathlib"@"2196ab363eb097c008d449712... | Mathlib/Data/Real/GoldenRatio.lean | 117 | 119 | theorem gold_lt_two : φ < 2 := by | calc
(1 + sqrt 5) / 2 < (1 + 3) / 2 := by gcongr; rw [sqrt_lt'] <;> norm_num
_ = 2 := by norm_num
| 3 | 20.085537 | 1 | 0.894737 | 19 | 776 |
import Mathlib.Geometry.Manifold.ContMDiff.Basic
open Set Function Filter ChartedSpace SmoothManifoldWithCorners
open scoped Topology Manifold
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜]
-- declare a smooth manifold `M` over the pair `(E, H)`.
{E : Type*}
[NormedAddCommGroup E] [NormedSpace 𝕜 E] {H ... | Mathlib/Geometry/Manifold/ContMDiff/Product.lean | 218 | 231 | theorem contMDiffWithinAt_snd {s : Set (M × N)} {p : M × N} :
ContMDiffWithinAt (I.prod J) J n Prod.snd s p := by |
/- porting note: `simp` fails to apply lemmas to `ModelProd`. Was
rw [contMDiffWithinAt_iff']
refine' ⟨continuousWithinAt_snd, _⟩
refine' contDiffWithinAt_snd.congr (fun y hy => _) _
· simp only [mfld_simps] at hy
simp only [hy, mfld_simps]
· simp only [mfld_simps]
-/
rw [contMDiffWithinAt_iff']
... | 12 | 162,754.791419 | 2 | 1.5 | 4 | 1,657 |
import Mathlib.Topology.Order.MonotoneContinuity
import Mathlib.Topology.Algebra.Order.LiminfLimsup
import Mathlib.Topology.Instances.NNReal
import Mathlib.Topology.EMetricSpace.Lipschitz
import Mathlib.Topology.Metrizable.Basic
import Mathlib.Topology.Order.T5
#align_import topology.instances.ennreal from "leanprove... | Mathlib/Topology/Instances/ENNReal.lean | 123 | 127 | theorem eventuallyEq_of_toReal_eventuallyEq {l : Filter α} {f g : α → ℝ≥0∞}
(hfi : ∀ᶠ x in l, f x ≠ ∞) (hgi : ∀ᶠ x in l, g x ≠ ∞)
(hfg : (fun x => (f x).toReal) =ᶠ[l] fun x => (g x).toReal) : f =ᶠ[l] g := by |
filter_upwards [hfi, hgi, hfg] with _ hfx hgx _
rwa [← ENNReal.toReal_eq_toReal hfx hgx]
| 2 | 7.389056 | 1 | 1.285714 | 7 | 1,362 |
import Mathlib.Analysis.Calculus.LineDeriv.Measurable
import Mathlib.Analysis.NormedSpace.FiniteDimension
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.Analysis.BoundedVariation
import Mathlib.MeasureTheory.Group.Integral
import Mathlib.Analysis.Distribution.AEEqOfIntegralContDiff
import Mathlib.... | Mathlib/Analysis/Calculus/Rademacher.lean | 119 | 160 | theorem integral_inv_smul_sub_mul_tendsto_integral_lineDeriv_mul'
(hf : LipschitzWith C f) (h'f : HasCompactSupport f) (hg : Continuous g) (v : E) :
Tendsto (fun (t : ℝ) ↦ ∫ x, (t⁻¹ • (f (x + t • v) - f x)) * g x ∂μ) (𝓝[>] 0)
(𝓝 (∫ x, lineDeriv ℝ f x v * g x ∂μ)) := by |
let K := cthickening (‖v‖) (tsupport f)
have K_compact : IsCompact K := IsCompact.cthickening h'f
apply tendsto_integral_filter_of_dominated_convergence
(K.indicator (fun x ↦ (C * ‖v‖) * ‖g x‖))
· filter_upwards with t
apply AEStronglyMeasurable.mul ?_ hg.aestronglyMeasurable
apply aestronglyMeas... | 38 | 31,855,931,757,113,756 | 2 | 2 | 3 | 2,148 |
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Valuation.PrimeMultiplicity
import Mathlib.RingTheory.AdicCompletion.Basic
#align_import ring_theory.discrete_valuation_ring.basic from "leanprover-community/mathlib"@"c163ec99dfc664628ca15d215fce0a5b9c2... | Mathlib/RingTheory/DiscreteValuationRing/Basic.lean | 118 | 145 | theorem iff_pid_with_one_nonzero_prime (R : Type u) [CommRing R] [IsDomain R] :
DiscreteValuationRing R ↔ IsPrincipalIdealRing R ∧ ∃! P : Ideal R, P ≠ ⊥ ∧ IsPrime P := by |
constructor
· intro RDVR
rcases id RDVR with ⟨Rlocal⟩
constructor
· assumption
use LocalRing.maximalIdeal R
constructor
· exact ⟨Rlocal, inferInstance⟩
· rintro Q ⟨hQ1, hQ2⟩
obtain ⟨q, rfl⟩ := (IsPrincipalIdealRing.principal Q).1
have hq : q ≠ 0 := by
rintro rfl
... | 26 | 195,729,609,428.83878 | 2 | 1.666667 | 6 | 1,772 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Data.Complex.Exponential
import Mathlib.Data.Complex.Module
import Mathlib.RingTheory.Polynomial.Chebyshev
#align_import analysis.special_functions.trigonometric.chebyshev from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
set_... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Chebyshev.lean | 39 | 41 | theorem algebraMap_eval_T (x : R) (n : ℤ) :
algebraMap R A ((T R n).eval x) = (T A n).eval (algebraMap R A x) := by |
rw [← aeval_algebraMap_apply_eq_algebraMap_eval, aeval_T]
| 1 | 2.718282 | 0 | 0.666667 | 6 | 568 |
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 | 68 | 70 | theorem pow_two : t ^ 2 = 1 := by |
rcases ht with ⟨w, i, rfl⟩
simp
| 2 | 7.389056 | 1 | 1.111111 | 9 | 1,198 |
import Mathlib.Algebra.Algebra.Bilinear
import Mathlib.RingTheory.Localization.Basic
#align_import algebra.module.localized_module from "leanprover-community/mathlib"@"831c494092374cfe9f50591ed0ac81a25efc5b86"
namespace LocalizedModule
universe u v
variable {R : Type u} [CommSemiring R] (S : Submonoid R)
variab... | Mathlib/Algebra/Module/LocalizedModule.lean | 99 | 102 | theorem induction_on {β : LocalizedModule S M → Prop} (h : ∀ (m : M) (s : S), β (mk m s)) :
∀ x : LocalizedModule S M, β x := by |
rintro ⟨⟨m, s⟩⟩
exact h m s
| 2 | 7.389056 | 1 | 1 | 7 | 797 |
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.List.Sublists
import Mathlib.Data.List.InsertNth
#align_import group_theory.free_group from "leanprover-community/mathlib"@"f93c11933efbc3c2f0299e47b8ff83e9b539cbf6"
open Relation
universe u v w
variable {α : Type u... | Mathlib/GroupTheory/FreeGroup/Basic.lean | 115 | 116 | theorem Step.not_rev {x b} : Step (L₁ ++ (x, !b) :: (x, b) :: L₂) (L₁ ++ L₂) := by |
cases b <;> exact Step.not
| 1 | 2.718282 | 0 | 1.333333 | 3 | 1,423 |
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Data.Nat.Totient
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.GroupTheory.Subgroup.Simple
import Mathlib.Tactic.Group
import Mathlib.GroupTheory.Exponent
#align_import group_theory.specific_groups.cyclic from "leanprover-community/mathli... | Mathlib/GroupTheory/SpecificGroups/Cyclic.lean | 145 | 149 | theorem zpowers_eq_top_of_prime_card {G : Type*} [Group G] {_ : Fintype G} {p : ℕ}
[hp : Fact p.Prime] (h : Fintype.card G = p) {g : G} (hg : g ≠ 1) : zpowers g = ⊤ := by |
subst h
have := (zpowers g).eq_bot_or_eq_top_of_prime_card
rwa [zpowers_eq_bot, or_iff_right hg] at this
| 3 | 20.085537 | 1 | 1.333333 | 6 | 1,386 |
import Mathlib.Algebra.Polynomial.Splits
#align_import algebra.cubic_discriminant from "leanprover-community/mathlib"@"930133160e24036d5242039fe4972407cd4f1222"
noncomputable section
@[ext]
structure Cubic (R : Type*) where
(a b c d : R)
#align cubic Cubic
namespace Cubic
open Cubic Polynomial
open Polynom... | Mathlib/Algebra/CubicDiscriminant.lean | 458 | 459 | theorem map_toPoly : (map φ P).toPoly = Polynomial.map φ P.toPoly := by |
simp only [map, toPoly, map_C, map_X, Polynomial.map_add, Polynomial.map_mul, Polynomial.map_pow]
| 1 | 2.718282 | 0 | 0.1 | 10 | 246 |
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 | 189 | 197 | theorem continuous_transAssocReparamAux : Continuous transAssocReparamAux := by |
refine continuous_if_le ?_ ?_ (Continuous.continuousOn ?_)
(continuous_if_le ?_ ?_
(Continuous.continuousOn ?_) (Continuous.continuousOn ?_) ?_).continuousOn
?_ <;>
[continuity; continuity; continuity; continuity; continuity; continuity; continuity; skip;
skip] <;>
· intro x hx
se... | 8 | 2,980.957987 | 2 | 1.166667 | 12 | 1,229 |
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Data.Real.ConjExponents
#align_import analysis.mean_inequalities from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
universe u... | Mathlib/Analysis/MeanInequalities.lean | 180 | 183 | theorem geom_mean_eq_arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i ∈ s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i ∈ s, z i ^ w i = ∑ i ∈ s, w i * z i := by |
rw [geom_mean_weighted_of_constant, arith_mean_weighted_of_constant] <;> assumption
| 1 | 2.718282 | 0 | 1.6 | 5 | 1,716 |
import Mathlib.Topology.Algebra.UniformConvergence
#align_import topology.algebra.module.strong_topology from "leanprover-community/mathlib"@"8905e5ed90859939681a725b00f6063e65096d95"
open scoped Topology UniformConvergence
section General
variable {𝕜₁ 𝕜₂ : Type*} [NormedField 𝕜₁] [NormedField 𝕜₂] (σ : 𝕜... | Mathlib/Topology/Algebra/Module/StrongTopology.lean | 152 | 157 | theorem t2Space [TopologicalSpace F] [TopologicalAddGroup F] [T2Space F]
(𝔖 : Set (Set E)) (h𝔖 : ⋃₀ 𝔖 = Set.univ) : T2Space (UniformConvergenceCLM σ F 𝔖) := by |
letI : UniformSpace F := TopologicalAddGroup.toUniformSpace F
haveI : UniformAddGroup F := comm_topologicalAddGroup_is_uniform
haveI : T2Space (E →ᵤ[𝔖] F) := UniformOnFun.t2Space_of_covering h𝔖
exact (embedding_coeFn σ F 𝔖).t2Space
| 4 | 54.59815 | 2 | 1 | 3 | 861 |
import Mathlib.Algebra.Divisibility.Basic
import Mathlib.Algebra.Group.Equiv.Basic
import Mathlib.Algebra.Group.TypeTags
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.Algebra.Ring.Nat
#align_import data.nat.cast.basic from "leanprover-community/mathlib"@"acebd8d49928f6ed8920e502a6c90674e75bd441"
assert_not_e... | Mathlib/Data/Nat/Cast/Basic.lean | 159 | 164 | theorem ext_nat'' [MonoidWithZeroHomClass F ℕ A] (f g : F) (h_pos : ∀ {n : ℕ}, 0 < n → f n = g n) :
f = g := by |
apply DFunLike.ext
rintro (_ | n)
· simp [map_zero f, map_zero g]
· exact h_pos n.succ_pos
| 4 | 54.59815 | 2 | 2 | 1 | 2,300 |
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Tree.Basic
import Mathlib.Logic.Basic
import Mathlib.Tactic.NormNum.Core
import Mathlib.Util.SynthesizeUsing
import Mathlib.Util.Qq
open Lean Parser Tactic Mathlib Meta NormNum Qq
initialize registerTraceClass `CancelDen... | Mathlib/Tactic/CancelDenoms/Core.lean | 81 | 86 | theorem cancel_factors_le {α} [LinearOrderedField α] {a b ad bd a' b' gcd : α}
(ha : ad * a = a') (hb : bd * b = b') (had : 0 < ad) (hbd : 0 < bd) (hgcd : 0 < gcd) :
(a ≤ b) = (1 / gcd * (bd * a') ≤ 1 / gcd * (ad * b')) := by |
rw [mul_le_mul_left, ← ha, ← hb, ← mul_assoc, ← mul_assoc, mul_comm bd, mul_le_mul_left]
· exact mul_pos had hbd
· exact one_div_pos.2 hgcd
| 3 | 20.085537 | 1 | 0.545455 | 11 | 512 |
import Mathlib.CategoryTheory.Linear.Basic
import Mathlib.CategoryTheory.Preadditive.Biproducts
import Mathlib.LinearAlgebra.Matrix.InvariantBasisNumber
import Mathlib.Data.Set.Subsingleton
#align_import category_theory.preadditive.hom_orthogonal from "leanprover-community/mathlib"@"829895f162a1f29d0133f4b3538f4cd1fb... | Mathlib/CategoryTheory/Preadditive/HomOrthogonal.lean | 146 | 166 | theorem matrixDecomposition_comp (o : HomOrthogonal s) {α β γ : Type} [Finite α] [Fintype β]
[Finite γ] {f : α → ι} {g : β → ι} {h : γ → ι} (z : (⨁ fun a => s (f a)) ⟶ ⨁ fun b => s (g b))
(w : (⨁ fun b => s (g b)) ⟶ ⨁ fun c => s (h c)) (i : ι) :
o.matrixDecomposition (z ≫ w) i = o.matrixDecomposition w i * ... |
ext ⟨c, ⟨⟩⟩ ⟨a, j_property⟩
simp only [Set.mem_preimage, Set.mem_singleton_iff] at j_property
simp only [Matrix.mul_apply, Limits.biproduct.components,
HomOrthogonal.matrixDecomposition_apply, Category.comp_id, Category.id_comp, Category.assoc,
End.mul_def, eqToHom_refl, eqToHom_trans_assoc, Finset.sum_c... | 17 | 24,154,952.753575 | 2 | 2 | 2 | 2,391 |
import Mathlib.Order.Interval.Set.OrdConnected
import Mathlib.Data.Set.Lattice
#align_import data.set.intervals.ord_connected_component from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
open Interval Function OrderDual
namespace Set
variable {α : Type*} [LinearOrder α] {s t : Set α}... | Mathlib/Order/Interval/Set/OrdConnectedComponent.lean | 63 | 64 | theorem ordConnectedComponent_eq_empty : ordConnectedComponent s x = ∅ ↔ x ∉ s := by |
rw [← not_nonempty_iff_eq_empty, nonempty_ordConnectedComponent]
| 1 | 2.718282 | 0 | 0.571429 | 7 | 519 |
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
| 2 | 7.389056 | 1 | 0.727273 | 11 | 649 |
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 | 148 | 149 | theorem transReflReparamAux_one : transReflReparamAux 1 = 1 := by |
set_option tactic.skipAssignedInstances false in norm_num [transReflReparamAux]
| 1 | 2.718282 | 0 | 1.166667 | 12 | 1,229 |
import Mathlib.Algebra.CharZero.Lemmas
import Mathlib.Algebra.GroupWithZero.Commute
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Ring.Pow
import Mathlib.Algebra.Ring.Int
#align_import algebra.order.field.power from "leanprover-community/mathlib"@"acb3d204d4ee883eb686f45d486a2a6811a01329"
... | Mathlib/Algebra/Order/Field/Power.lean | 97 | 100 | theorem zpow_injective (h₀ : 0 < a) (h₁ : a ≠ 1) : Injective (a ^ · : ℤ → α) := by |
rcases h₁.lt_or_lt with (H | H)
· exact (zpow_strictAnti h₀ H).injective
· exact (zpow_strictMono H).injective
| 3 | 20.085537 | 1 | 1 | 7 | 1,080 |
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Tactic.Cases
import Mathlib.Algebra.NeZero
import Mathlib.Logic.Function.Basic
#align_import algebra.char_zero.defs from "leanprover-community/mathlib"@"d6aae1bcbd04b8de2022b9b83a5b5b10e10c777d"
class CharZero (R) [AddMonoidWithOne R] : Prop where
cast_injecti... | Mathlib/Algebra/CharZero/Defs.lean | 79 | 79 | theorem cast_eq_zero {n : ℕ} : (n : R) = 0 ↔ n = 0 := by | rw [← cast_zero, cast_inj]
| 1 | 2.718282 | 0 | 0 | 2 | 58 |
import Mathlib.Algebra.MvPolynomial.Variables
#align_import data.mv_polynomial.supported from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
universe u v w
namespace MvPolynomial
variable {σ τ : Type*} {R : Type u} {S : Type v} {r : R} {e : ℕ} {n m : σ}
section CommSemiring
variable... | Mathlib/Algebra/MvPolynomial/Supported.lean | 46 | 48 | theorem supported_eq_range_rename (s : Set σ) : supported R s = (rename ((↑) : s → σ)).range := by |
rw [supported, Set.image_eq_range, adjoin_range_eq_range_aeval, rename]
congr
| 2 | 7.389056 | 1 | 0.888889 | 9 | 773 |
import Mathlib.Data.Finset.Image
#align_import data.finset.card from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
assert_not_exists MonoidWithZero
-- TODO: After a lot more work,
-- assert_not_exists OrderedCommMonoid
open Function Multiset Nat
variable {α β R : Type*}
namespace Fin... | Mathlib/Data/Finset/Card.lean | 91 | 94 | theorem card_singleton_inter [DecidableEq α] : ({a} ∩ s).card ≤ 1 := by |
cases' Finset.decidableMem a s with h h
· simp [Finset.singleton_inter_of_not_mem h]
· simp [Finset.singleton_inter_of_mem h]
| 3 | 20.085537 | 1 | 0.6 | 10 | 527 |
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 | 108 | 110 | theorem monic_X_pow_add_C {n : ℕ} (h : n ≠ 0) : (X ^ n + C a).Monic := by |
obtain ⟨k, rfl⟩ := Nat.exists_eq_succ_of_ne_zero h
exact monic_X_pow_add <| degree_C_le.trans Nat.WithBot.coe_nonneg
| 2 | 7.389056 | 1 | 1.285714 | 7 | 1,353 |
import Mathlib.Algebra.Group.Semiconj.Defs
import Mathlib.Algebra.Ring.Defs
#align_import algebra.ring.semiconj from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
universe u v w x
variable {α : Type u} {β : Type v} {γ : Type w} {R : Type x}
open Function
namespace SemiconjBy
@[simp... | Mathlib/Algebra/Ring/Semiconj.lean | 48 | 49 | theorem neg_right (h : SemiconjBy a x y) : SemiconjBy a (-x) (-y) := by |
simp only [SemiconjBy, h.eq, neg_mul, mul_neg]
| 1 | 2.718282 | 0 | 0 | 6 | 216 |
import Mathlib.Data.Matrix.Basic
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.Adjugate
#align_import linear_algebra.matrix.nondegenerate from "leanprover-community/mathlib"@"2a32c70c78096758af93e997b978a5d461007b4f"
namespace Matrix
variable {m R A : Type*} [Fintype m... | Mathlib/LinearAlgebra/Matrix/Nondegenerate.lean | 50 | 63 | theorem nondegenerate_of_det_ne_zero [DecidableEq m] {M : Matrix m m A} (hM : M.det ≠ 0) :
Nondegenerate M := by |
intro v hv
ext i
specialize hv (M.cramer (Pi.single i 1))
refine (mul_eq_zero.mp ?_).resolve_right hM
convert hv
simp only [mulVec_cramer M (Pi.single i 1), dotProduct, Pi.smul_apply, smul_eq_mul]
rw [Finset.sum_eq_single i, Pi.single_eq_same, mul_one]
· intro j _ hj
simp [hj]
· intros
have :... | 12 | 162,754.791419 | 2 | 2 | 1 | 2,056 |
import Mathlib.Data.Set.Function
import Mathlib.Order.Interval.Set.OrdConnected
#align_import data.set.intervals.proj_Icc from "leanprover-community/mathlib"@"4e24c4bfcff371c71f7ba22050308aa17815626c"
variable {α β : Type*} [LinearOrder α]
open Function
namespace Set
def projIci (a x : α) : Ici a := ⟨max a x,... | Mathlib/Order/Interval/Set/ProjIcc.lean | 105 | 106 | theorem projIcc_eq_left (h : a < b) : projIcc a b h.le x = ⟨a, left_mem_Icc.mpr h.le⟩ ↔ x ≤ a := by |
simp [projIcc, Subtype.ext_iff, h.not_le]
| 1 | 2.718282 | 0 | 0.083333 | 12 | 241 |
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics
#align_import analysis.special_functions.pow.continuity from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8"
noncomputable section
open scoped Classical
open Real Topology NNReal ENNReal Filter ComplexConjugate
open Filter Finset... | Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean | 36 | 41 | theorem zero_cpow_eq_nhds {b : ℂ} (hb : b ≠ 0) : (fun x : ℂ => (0 : ℂ) ^ x) =ᶠ[𝓝 b] 0 := by |
suffices ∀ᶠ x : ℂ in 𝓝 b, x ≠ 0 from
this.mono fun x hx ↦ by
dsimp only
rw [zero_cpow hx, Pi.zero_apply]
exact IsOpen.eventually_mem isOpen_ne hb
| 5 | 148.413159 | 2 | 1.857143 | 7 | 1,926 |
import Mathlib.LinearAlgebra.TensorProduct.RightExactness
import Mathlib.LinearAlgebra.TensorProduct.Finiteness
universe u
variable (R : Type u) [CommRing R]
variable {M : Type u} [AddCommGroup M] [Module R M]
variable {N : Type u} [AddCommGroup N] [Module R N]
open Classical DirectSum LinearMap Function Submodul... | Mathlib/LinearAlgebra/TensorProduct/Vanishing.lean | 102 | 157 | theorem vanishesTrivially_of_sum_tmul_eq_zero (hm : Submodule.span R (Set.range m) = ⊤)
(hmn : ∑ i, m i ⊗ₜ n i = (0 : M ⊗[R] N)) : VanishesTrivially R m n := by |
-- Define a map $G \colon R^\iota \to M$ whose matrix entries are the $m_i$. It is surjective.
set G : (ι →₀ R) →ₗ[R] M := Finsupp.total ι M R m with hG
have G_basis_eq (i : ι) : G (Finsupp.single i 1) = m i := by simp [hG, toModule_lof]
have G_surjective : Surjective G := by
apply LinearMap.range_eq_top.m... | 54 | 283,075,330,327,469,400,000,000 | 2 | 2 | 3 | 2,251 |
import Mathlib.Algebra.ContinuedFractions.Translations
#align_import algebra.continued_fractions.terminated_stable from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b348ce40ad"
namespace GeneralizedContinuedFraction
variable {K : Type*} {g : GeneralizedContinuedFraction K} {n m : ℕ}
theorem te... | Mathlib/Algebra/ContinuedFractions/TerminatedStable.lean | 31 | 34 | theorem continuantsAux_stable_step_of_terminated (terminated_at_n : g.TerminatedAt n) :
g.continuantsAux (n + 2) = g.continuantsAux (n + 1) := by |
rw [terminatedAt_iff_s_none] at terminated_at_n
simp only [continuantsAux, Nat.add_eq, Nat.add_zero, terminated_at_n]
| 2 | 7.389056 | 1 | 1 | 9 | 1,021 |
import Mathlib.CategoryTheory.Sites.Sheaf
#align_import category_theory.sites.plus from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
namespace CategoryTheory.GrothendieckTopology
open CategoryTheory
open CategoryTheory.Limits
open Opposite
universe w v u
variable {C : Type u} [Ca... | Mathlib/CategoryTheory/Sites/Plus.lean | 81 | 86 | theorem diagramNatTrans_zero [Preadditive D] (X : C) (P Q : Cᵒᵖ ⥤ D) :
J.diagramNatTrans (0 : P ⟶ Q) X = 0 := by |
ext : 2
refine Multiequalizer.hom_ext _ _ _ (fun i => ?_)
dsimp
rw [zero_comp, Multiequalizer.lift_ι, comp_zero]
| 4 | 54.59815 | 2 | 2 | 3 | 2,407 |
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.affine_space.slope from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open AffineMap
variable {k E PE : Type*} [Field k] [AddCommGroup E] [Module k E] [AddTorsor E PE]
def ... | Mathlib/LinearAlgebra/AffineSpace/Slope.lean | 67 | 69 | theorem slope_vadd_const (f : k → E) (c : PE) : (slope fun x => f x +ᵥ c) = slope f := by |
ext a b
simp only [slope, vadd_vsub_vadd_cancel_right, vsub_eq_sub]
| 2 | 7.389056 | 1 | 0.7 | 10 | 639 |
import Batteries.Data.Array.Lemmas
import Batteries.Tactic.Lint.Misc
namespace Batteries
structure UFNode where
parent : Nat
rank : Nat
namespace UnionFind
def panicWith (v : α) (msg : String) : α := @panic α ⟨v⟩ msg
@[simp] theorem panicWith_eq (v : α) (msg) : panicWith v msg = v := rfl
def parentD... | .lake/packages/batteries/Batteries/Data/UnionFind/Basic.lean | 47 | 50 | theorem parentD_set {arr : Array UFNode} {x v i} :
parentD (arr.set x v) i = if x.1 = i then v.parent else parentD arr i := by |
rw [parentD]; simp [Array.get_eq_getElem, parentD]
split <;> [split <;> simp [Array.get_set, *]; split <;> [(subst i; cases ‹¬_› x.2); rfl]]
| 2 | 7.389056 | 1 | 1 | 2 | 1,059 |
import Mathlib.GroupTheory.Solvable
import Mathlib.FieldTheory.PolynomialGaloisGroup
import Mathlib.RingTheory.RootsOfUnity.Basic
#align_import field_theory.abel_ruffini from "leanprover-community/mathlib"@"e3f4be1fcb5376c4948d7f095bec45350bfb9d1a"
noncomputable section
open scoped Classical Polynomial Intermedi... | Mathlib/FieldTheory/AbelRuffini.lean | 248 | 280 | theorem induction (P : solvableByRad F E → Prop)
(base : ∀ α : F, P (algebraMap F (solvableByRad F E) α))
(add : ∀ α β : solvableByRad F E, P α → P β → P (α + β))
(neg : ∀ α : solvableByRad F E, P α → P (-α))
(mul : ∀ α β : solvableByRad F E, P α → P β → P (α * β))
(inv : ∀ α : solvableByRad F E, P ... |
revert α
suffices ∀ α : E, IsSolvableByRad F α → ∃ β : solvableByRad F E, ↑β = α ∧ P β by
intro α
obtain ⟨α₀, hα₀, Pα⟩ := this α (Subtype.mem α)
convert Pα
exact Subtype.ext hα₀.symm
apply IsSolvableByRad.rec
· exact fun α => ⟨algebraMap F (solvableByRad F E) α, rfl, base α⟩
· intro α β _ _ P... | 25 | 72,004,899,337.38586 | 2 | 0.909091 | 11 | 788 |
import Mathlib.CategoryTheory.Sites.Canonical
#align_import category_theory.sites.types from "leanprover-community/mathlib"@"9f9015c645d85695581237cc761981036be8bd37"
universe u
namespace CategoryTheory
--open scoped CategoryTheory.Type -- Porting note: unknown namespace
def typesGrothendieckTopology : Grothe... | Mathlib/CategoryTheory/Sites/Types.lean | 102 | 105 | theorem eval_typesGlue {S hs α} (f) : eval.{u} S α (typesGlue S hs α f) = f := by |
funext x
apply (IsSheafFor.valid_glue _ _ _ <| ⟨PUnit.unit, fun _ => Subsingleton.elim _ _⟩).trans
convert FunctorToTypes.map_id_apply S _
| 3 | 20.085537 | 1 | 1 | 3 | 989 |
import Mathlib.LinearAlgebra.Dual
open Function Module
variable (R M N : Type*) [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N]
structure PerfectPairing :=
toLin : M →ₗ[R] N →ₗ[R] R
bijectiveLeft : Bijective toLin
bijectiveRight : Bijective toLin.flip
attribute [nolint docBlame] P... | Mathlib/LinearAlgebra/PerfectPairing.lean | 102 | 105 | theorem toDualRight_symm_comp_toDualLeft :
p.toDualRight.symm.dualMap ∘ₗ (p.toDualLeft : M →ₗ[R] Dual R N) = Dual.eval R M := by |
ext1 x
exact p.toDualRight_symm_toDualLeft x
| 2 | 7.389056 | 1 | 1 | 6 | 845 |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Finsupp.Defs
import Mathlib.Data.Finset.Pairwise
#align_import data.finsupp.big_operators from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
variable {ι M : Type*} [DecidableEq ι]
theorem List.support_sum_subset [Add... | Mathlib/Data/Finsupp/BigOperators.lean | 55 | 57 | theorem Finset.support_sum_subset [AddCommMonoid M] (s : Finset (ι →₀ M)) :
(s.sum id).support ⊆ Finset.sup s Finsupp.support := by |
classical convert Multiset.support_sum_subset s.1; simp
| 1 | 2.718282 | 0 | 1.571429 | 7 | 1,707 |
import Mathlib.Combinatorics.Quiver.Cast
import Mathlib.Combinatorics.Quiver.Symmetric
#align_import combinatorics.quiver.single_obj from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
namespace Quiver
-- Porting note: Removed `deriving Unique`.
@[nolint unusedArguments]
def SingleObj ... | Mathlib/Combinatorics/Quiver/SingleObj.lean | 139 | 142 | theorem pathToList_listToPath (l : List α) : pathToList (listToPath l) = l := by |
induction' l with a l ih
· rfl
· change a :: pathToList (listToPath l) = a :: l; rw [ih]
| 3 | 20.085537 | 1 | 0.666667 | 3 | 561 |
import Mathlib.RingTheory.RingHomProperties
#align_import ring_theory.ring_hom.finite from "leanprover-community/mathlib"@"b5aecf07a179c60b6b37c1ac9da952f3b565c785"
namespace RingHom
open scoped TensorProduct
open TensorProduct Algebra.TensorProduct
| Mathlib/RingTheory/RingHom/Finite.lean | 23 | 25 | theorem finite_stableUnderComposition : StableUnderComposition @Finite := by |
introv R hf hg
exact hg.comp hf
| 2 | 7.389056 | 1 | 1.333333 | 3 | 1,384 |
import Mathlib.SetTheory.Ordinal.Arithmetic
#align_import set_theory.ordinal.exponential from "leanprover-community/mathlib"@"b67044ba53af18680e1dd246861d9584e968495d"
noncomputable section
open Function Cardinal Set Equiv Order
open scoped Classical
open Cardinal Ordinal
universe u v w
namespace Ordinal
in... | Mathlib/SetTheory/Ordinal/Exponential.lean | 94 | 102 | theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b := by |
have h0 : 0 < a ^ (0 : Ordinal) := by simp only [opow_zero, zero_lt_one]
induction b using limitRecOn with
| H₁ => exact h0
| H₂ b IH =>
rw [opow_succ]
exact mul_pos IH a0
| H₃ b l _ =>
exact (lt_opow_of_limit (Ordinal.pos_iff_ne_zero.1 a0) l).2 ⟨0, l.pos, h0⟩
| 8 | 2,980.957987 | 2 | 0.555556 | 9 | 514 |
import Mathlib.Analysis.SpecialFunctions.Gamma.Beta
import Mathlib.NumberTheory.LSeries.HurwitzZeta
import Mathlib.Analysis.Complex.RemovableSingularity
import Mathlib.Analysis.PSeriesComplex
#align_import number_theory.zeta_function from "leanprover-community/mathlib"@"57f9349f2fe19d2de7207e99b0341808d977cdcf"
o... | Mathlib/NumberTheory/LSeries/RiemannZeta.lean | 103 | 105 | theorem completedRiemannZeta₀_one_sub (s : ℂ) :
completedRiemannZeta₀ (1 - s) = completedRiemannZeta₀ s := by |
rw [← completedHurwitzZetaEven₀_zero, ← completedCosZeta₀_zero, completedHurwitzZetaEven₀_one_sub]
| 1 | 2.718282 | 0 | 0.4 | 5 | 389 |
import Mathlib.LinearAlgebra.Ray
import Mathlib.Analysis.NormedSpace.Real
#align_import analysis.normed_space.ray from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
open Real
variable {E : Type*} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*}
[NormedAddCommGroup F] [NormedSp... | Mathlib/Analysis/NormedSpace/Ray.lean | 32 | 35 | theorem norm_add (h : SameRay ℝ x y) : ‖x + y‖ = ‖x‖ + ‖y‖ := by |
rcases h.exists_eq_smul with ⟨u, a, b, ha, hb, -, rfl, rfl⟩
rw [← add_smul, norm_smul_of_nonneg (add_nonneg ha hb), norm_smul_of_nonneg ha,
norm_smul_of_nonneg hb, add_mul]
| 3 | 20.085537 | 1 | 1.2 | 5 | 1,273 |
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Order.Group.Int
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Algebra.Ring.Rat
import Mathlib.Data.PNat.Defs
#align_import data.rat.lemmas from "leanprover-community/mathlib"@"550b58538991c8977703fdeb7c9d51a5aa27df11"
namespace Rat
o... | Mathlib/Data/Rat/Lemmas.lean | 109 | 111 | theorem mul_self_den (q : ℚ) : (q * q).den = q.den * q.den := by |
rw [Rat.mul_den, Int.natAbs_mul, Nat.Coprime.gcd_eq_one, Nat.div_one]
exact (q.reduced.mul_right q.reduced).mul (q.reduced.mul_right q.reduced)
| 2 | 7.389056 | 1 | 1.333333 | 12 | 1,389 |
import Mathlib.Data.Set.Subsingleton
import Mathlib.Order.WithBot
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
theorem powerset_insert (s : Set α) (a : α)... | Mathlib/Data/Set/Image.lean | 666 | 666 | theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by | simp
| 1 | 2.718282 | 0 | 0.666667 | 15 | 590 |
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
variable ... | Mathlib/MeasureTheory/Function/LpSeminorm/TriangleInequality.lean | 98 | 109 | theorem snorm_add_le' {f g : α → E} (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ)
(p : ℝ≥0∞) : snorm (f + g) p μ ≤ LpAddConst p * (snorm f p μ + snorm g p μ) := by |
rcases eq_or_ne p 0 with (rfl | hp)
· simp only [snorm_exponent_zero, add_zero, mul_zero, le_zero_iff]
rcases lt_or_le p 1 with (h'p | h'p)
· simp only [snorm_eq_snorm' hp (h'p.trans ENNReal.one_lt_top).ne]
convert snorm'_add_le_of_le_one hf ENNReal.toReal_nonneg _
· have : p ∈ Set.Ioo (0 : ℝ≥0∞) 1 := ... | 10 | 22,026.465795 | 2 | 1.5 | 8 | 1,656 |
import Mathlib.Data.ULift
import Mathlib.Data.ZMod.Defs
import Mathlib.SetTheory.Cardinal.PartENat
#align_import set_theory.cardinal.finite from "leanprover-community/mathlib"@"3ff3f2d6a3118b8711063de7111a0d77a53219a8"
set_option autoImplicit true
open Cardinal Function
noncomputable section
variable {α β : Typ... | Mathlib/SetTheory/Cardinal/Finite.lean | 144 | 146 | theorem card_of_subsingleton (a : α) [Subsingleton α] : Nat.card α = 1 := by |
letI := Fintype.ofSubsingleton a
rw [card_eq_fintype_card, Fintype.card_ofSubsingleton a]
| 2 | 7.389056 | 1 | 0.666667 | 3 | 595 |
import Mathlib.CategoryTheory.Galois.Basic
import Mathlib.CategoryTheory.Limits.FintypeCat
import Mathlib.CategoryTheory.Limits.Preserves.Limits
import Mathlib.CategoryTheory.Limits.Shapes.SingleObj
import Mathlib.Logic.Equiv.TransferInstance
universe u₁ u₂ v₁ v₂ v w
namespace CategoryTheory
namespace PreGaloisCat... | Mathlib/CategoryTheory/Galois/GaloisObjects.lean | 81 | 84 | theorem isGalois_iff_pretransitive (X : C) [IsConnected X] :
IsGalois X ↔ MulAction.IsPretransitive (Aut X) (F.obj X) := by |
rw [isGalois_iff_aux, Equiv.nonempty_congr <| quotientByAutTerminalEquivUniqueQuotient F X]
exact (MulAction.pretransitive_iff_unique_quotient_of_nonempty (Aut X) (F.obj X)).symm
| 2 | 7.389056 | 1 | 1 | 1 | 1,087 |
import Mathlib.Analysis.LocallyConvex.Bounded
import Mathlib.Topology.Algebra.Module.StrongTopology
#align_import analysis.normed_space.compact_operator from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open Function Set Filter Bornology Metric Pointwise Topology
def IsCompactOperat... | Mathlib/Analysis/NormedSpace/CompactOperator.lean | 228 | 230 | theorem IsCompactOperator.sub [TopologicalAddGroup M₄] {f g : M₁ → M₄} (hf : IsCompactOperator f)
(hg : IsCompactOperator g) : IsCompactOperator (f - g) := by |
rw [sub_eq_add_neg]; exact hf.add hg.neg
| 1 | 2.718282 | 0 | 1.6 | 5 | 1,713 |
import Mathlib.Geometry.Euclidean.Inversion.Basic
import Mathlib.Geometry.Euclidean.PerpBisector
open Metric Function AffineMap Set AffineSubspace
open scoped Topology
variable {V P : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P]
[NormedAddTorsor V P] {c x y : P} {R : ℝ}
namespace Euclid... | Mathlib/Geometry/Euclidean/Inversion/ImageHyperplane.lean | 66 | 71 | theorem preimage_inversion_sphere_dist_center (hR : R ≠ 0) (hy : y ≠ c) :
inversion c R ⁻¹' sphere y (dist y c) =
insert c (perpBisector c (inversion c R y) : Set P) := by |
ext x
rcases eq_or_ne x c with rfl | hx; · simp [dist_comm]
rw [mem_preimage, mem_sphere, ← inversion_mem_perpBisector_inversion_iff hR] <;> simp [*]
| 3 | 20.085537 | 1 | 1.166667 | 6 | 1,232 |
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 | 133 | 134 | theorem log_one_right (b : ℕ) : log b (1 : R) = 0 := by |
rw [log_of_one_le_right _ le_rfl, Nat.floor_one, Nat.log_one_right, Int.ofNat_zero]
| 1 | 2.718282 | 0 | 1.5 | 8 | 1,647 |
import Mathlib.Data.Set.Function
import Mathlib.Logic.Relation
import Mathlib.Logic.Pairwise
#align_import data.set.pairwise.basic from "leanprover-community/mathlib"@"c4c2ed622f43768eff32608d4a0f8a6cec1c047d"
open Function Order Set
variable {α β γ ι ι' : Type*} {r p q : α → α → Prop}
section Pairwise
variabl... | Mathlib/Data/Set/Pairwise/Basic.lean | 137 | 143 | theorem pairwise_union :
(s ∪ t).Pairwise r ↔
s.Pairwise r ∧ t.Pairwise r ∧ ∀ a ∈ s, ∀ b ∈ t, a ≠ b → r a b ∧ r b a := by |
simp only [Set.Pairwise, mem_union, or_imp, forall_and]
exact
⟨fun H => ⟨H.1.1, H.2.2, H.1.2, fun x hx y hy hne => H.2.1 y hy x hx hne.symm⟩,
fun H => ⟨⟨H.1, H.2.2.1⟩, fun x hx y hy hne => H.2.2.2 y hy x hx hne.symm, H.2.1⟩⟩
| 4 | 54.59815 | 2 | 1 | 5 | 1,138 |
import Mathlib.Analysis.Convex.Between
import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.Topology.MetricSpace.Holder
import Mathlib.Topology.MetricSpace.MetricSeparated
#align_import measure_theory.measure.hausdorff from "leanprover-communit... | Mathlib/MeasureTheory/Measure/Hausdorff.lean | 270 | 271 | theorem le_pre : μ ≤ pre m r ↔ ∀ s : Set X, diam s ≤ r → μ s ≤ m s := by |
simp only [pre, le_boundedBy, extend, le_iInf_iff]
| 1 | 2.718282 | 0 | 1.428571 | 7 | 1,520 |
import Mathlib.Topology.PartialHomeomorph
import Mathlib.Topology.SeparatedMap
#align_import topology.is_locally_homeomorph from "leanprover-community/mathlib"@"e97cf15cd1aec9bd5c193b2ffac5a6dc9118912b"
open Topology
variable {X Y Z : Type*} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (g : Y →... | Mathlib/Topology/IsLocalHomeomorph.lean | 90 | 99 | theorem of_comp_left (hgf : IsLocalHomeomorphOn (g ∘ f) s) (hg : IsLocalHomeomorphOn g (f '' s))
(cont : ∀ x ∈ s, ContinuousAt f x) : IsLocalHomeomorphOn f s := mk f s fun x hx ↦ by
obtain ⟨g, hxg, rfl⟩ := hg (f x) ⟨x, hx, rfl⟩
obtain ⟨gf, hgf, he⟩ := hgf x hx
refine ⟨(gf.restr <| f ⁻¹' g.source).trans g.symm... | apply interior_subset hy.1.2
rw [← he, g.eq_symm_apply this (by apply g.map_source this), Function.comp_apply]
| 2 | 7.389056 | 1 | 1.5 | 4 | 1,673 |
import Mathlib.RingTheory.Polynomial.Cyclotomic.Basic
import Mathlib.RingTheory.RootsOfUnity.Minpoly
#align_import ring_theory.polynomial.cyclotomic.roots from "leanprover-community/mathlib"@"7fdeecc0d03cd40f7a165e6cf00a4d2286db599f"
namespace Polynomial
variable {R : Type*} [CommRing R] {n : ℕ}
theorem isRoot_... | Mathlib/RingTheory/Polynomial/Cyclotomic/Roots.lean | 56 | 59 | theorem _root_.isRoot_of_unity_iff (h : 0 < n) (R : Type*) [CommRing R] [IsDomain R] {ζ : R} :
ζ ^ n = 1 ↔ ∃ i ∈ n.divisors, (cyclotomic i R).IsRoot ζ := by |
rw [← mem_nthRoots h, nthRoots, mem_roots <| X_pow_sub_C_ne_zero h _, C_1, ←
prod_cyclotomic_eq_X_pow_sub_one h, isRoot_prod]
| 2 | 7.389056 | 1 | 1.666667 | 6 | 1,797 |
import Mathlib.Topology.MetricSpace.HausdorffDistance
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
#align_import measure_theory.measure.regular from "leanprover-community/mathlib"@"bf6a01357ff5684b1ebcd0f1a13be314fc82c0bf"
open Set Filter ENNReal Topology NNReal TopologicalSpace
namespace MeasureTh... | Mathlib/MeasureTheory/Measure/Regular.lean | 260 | 264 | theorem trans {q' : Set α → Prop} (H : InnerRegularWRT μ p q) (H' : InnerRegularWRT μ q q') :
InnerRegularWRT μ p q' := by |
intro U hU r hr
rcases H' hU r hr with ⟨F, hFU, hqF, hF⟩; rcases H hqF _ hF with ⟨K, hKF, hpK, hrK⟩
exact ⟨K, hKF.trans hFU, hpK, hrK⟩
| 3 | 20.085537 | 1 | 1.444444 | 9 | 1,530 |
import Mathlib.FieldTheory.PrimitiveElement
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly
import Mathlib.LinearAlgebra.Matrix.ToLinearEquiv
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.G... | Mathlib/RingTheory/Norm.lean | 72 | 73 | theorem norm_eq_one_of_not_exists_basis (h : ¬∃ s : Finset S, Nonempty (Basis s R S)) (x : S) :
norm R x = 1 := by | rw [norm_apply, LinearMap.det]; split_ifs <;> trivial
| 1 | 2.718282 | 0 | 1.272727 | 11 | 1,347 |
import Mathlib.Algebra.ContinuedFractions.Basic
import Mathlib.Algebra.GroupWithZero.Basic
#align_import algebra.continued_fractions.translations from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b348ce40ad"
namespace GeneralizedContinuedFraction
section General
variable {α : Type*} {g : Gen... | Mathlib/Algebra/ContinuedFractions/Translations.lean | 53 | 55 | theorem terminatedAt_iff_part_denom_none :
g.TerminatedAt n ↔ g.partialDenominators.get? n = none := by |
rw [terminatedAt_iff_s_none, part_denom_none_iff_s_none]
| 1 | 2.718282 | 0 | 0.052632 | 19 | 240 |
import Mathlib.GroupTheory.Solvable
import Mathlib.FieldTheory.PolynomialGaloisGroup
import Mathlib.RingTheory.RootsOfUnity.Basic
#align_import field_theory.abel_ruffini from "leanprover-community/mathlib"@"e3f4be1fcb5376c4948d7f095bec45350bfb9d1a"
noncomputable section
open scoped Classical Polynomial Intermedi... | Mathlib/FieldTheory/AbelRuffini.lean | 45 | 45 | theorem gal_C_isSolvable (x : F) : IsSolvable (C x).Gal := by | infer_instance
| 1 | 2.718282 | 0 | 0.909091 | 11 | 788 |
import Mathlib.Order.BooleanAlgebra
import Mathlib.Tactic.Common
#align_import order.heyting.boundary from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
variable {α : Type*}
namespace Coheyting
variable [CoheytingAlgebra α] {a b : α}
def boundary (a : α) : α :=
a ⊓ ￢a
#align cohe... | Mathlib/Order/Heyting/Boundary.lean | 76 | 76 | theorem hnot_boundary (a : α) : ￢∂ a = ⊤ := by | rw [boundary, hnot_inf_distrib, sup_hnot_self]
| 1 | 2.718282 | 0 | 0.777778 | 9 | 687 |
import Batteries.Data.Fin.Basic
namespace Fin
attribute [norm_cast] val_last
protected theorem le_antisymm_iff {x y : Fin n} : x = y ↔ x ≤ y ∧ y ≤ x :=
Fin.ext_iff.trans Nat.le_antisymm_iff
protected theorem le_antisymm {x y : Fin n} (h1 : x ≤ y) (h2 : y ≤ x) : x = y :=
Fin.le_antisymm_iff.2 ⟨h1, h2⟩
@[simp... | .lake/packages/batteries/Batteries/Data/Fin/Lemmas.lean | 38 | 39 | theorem list_succ (n) : list (n+1) = 0 :: (list n).map Fin.succ := by |
apply List.ext_get; simp; intro i; cases i <;> simp
| 1 | 2.718282 | 0 | 1.090909 | 11 | 1,186 |
import Batteries.Data.Sum.Basic
import Batteries.Logic
open Function
namespace Sum
@[simp] protected theorem «forall» {p : α ⊕ β → Prop} :
(∀ x, p x) ↔ (∀ a, p (inl a)) ∧ ∀ b, p (inr b) :=
⟨fun h => ⟨fun _ => h _, fun _ => h _⟩, fun ⟨h₁, h₂⟩ => Sum.rec h₁ h₂⟩
@[simp] protected theorem «exists» {p : α ⊕ β ... | .lake/packages/batteries/Batteries/Data/Sum/Lemmas.lean | 83 | 83 | theorem isLeft_iff : x.isLeft ↔ ∃ y, x = Sum.inl y := by | cases x <;> simp
| 1 | 2.718282 | 0 | 0.285714 | 7 | 316 |
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