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.Convex.Cone.Extension
import Mathlib.Analysis.NormedSpace.RCLike
import Mathlib.Analysis.NormedSpace.Extend
import Mathlib.Analysis.RCLike.Lemmas
#align_import analysis.normed_space.hahn_banach.extension from "leanprover-community/mathlib"@"915591b2bb3ea303648db07284a161a7f2a9e3d4"
univers... | Mathlib/Analysis/NormedSpace/HahnBanach/Extension.lean | 44 | 59 | theorem exists_extension_norm_eq (p : Subspace ℝ E) (f : p →L[ℝ] ℝ) :
∃ g : E →L[ℝ] ℝ, (∀ x : p, g x = f x) ∧ ‖g‖ = ‖f‖ := by |
rcases exists_extension_of_le_sublinear ⟨p, f⟩ (fun x => ‖f‖ * ‖x‖)
(fun c hc x => by simp only [norm_smul c x, Real.norm_eq_abs, abs_of_pos hc, mul_left_comm])
(fun x y => by -- Porting note: placeholder filled here
rw [← left_distrib]
exact mul_le_mul_of_nonneg_left (norm_add_le x y) (@... | 14 | 1,202,604.284165 | 2 | 2 | 2 | 2,379 |
import Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho
import Mathlib.LinearAlgebra.Orientation
#align_import analysis.inner_product_space.orientation from "leanprover-community/mathlib"@"bd65478311e4dfd41f48bf38c7e3b02fb75d0163"
noncomputable section
variable {E : Type*} [NormedAddCommGroup E] [InnerProduct... | Mathlib/Analysis/InnerProductSpace/Orientation.lean | 54 | 60 | theorem det_to_matrix_orthonormalBasis_of_same_orientation
(h : e.toBasis.orientation = f.toBasis.orientation) : e.toBasis.det f = 1 := by |
apply (e.det_to_matrix_orthonormalBasis_real f).resolve_right
have : 0 < e.toBasis.det f := by
rw [e.toBasis.orientation_eq_iff_det_pos] at h
simpa using h
linarith
| 5 | 148.413159 | 2 | 1.111111 | 9 | 1,197 |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Data.Finset.Preimage
import Mathlib.Order.Interval.Set.Disjoint
import Mathlib.Order.Int... | Mathlib/Order/Filter/AtTopBot.lean | 118 | 125 | theorem disjoint_atBot_atTop [PartialOrder α] [Nontrivial α] :
Disjoint (atBot : Filter α) atTop := by |
rcases exists_pair_ne α with ⟨x, y, hne⟩
by_cases hle : x ≤ y
· refine disjoint_of_disjoint_of_mem ?_ (Iic_mem_atBot x) (Ici_mem_atTop y)
exact Iic_disjoint_Ici.2 (hle.lt_of_ne hne).not_le
· refine disjoint_of_disjoint_of_mem ?_ (Iic_mem_atBot y) (Ici_mem_atTop x)
exact Iic_disjoint_Ici.2 hle
| 6 | 403.428793 | 2 | 2 | 1 | 2,192 |
import Mathlib.Analysis.Convex.Body
import Mathlib.Analysis.Convex.Measure
import Mathlib.MeasureTheory.Group.FundamentalDomain
#align_import measure_theory.group.geometry_of_numbers from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
namespace MeasureTheory
open ENNReal FiniteDimensio... | Mathlib/MeasureTheory/Group/GeometryOfNumbers.lean | 50 | 58 | theorem exists_pair_mem_lattice_not_disjoint_vadd [AddCommGroup L] [Countable L] [AddAction L E]
[MeasurableSpace L] [MeasurableVAdd L E] [VAddInvariantMeasure L E μ]
(fund : IsAddFundamentalDomain L F μ) (hS : NullMeasurableSet s μ) (h : μ F < μ s) :
∃ x y : L, x ≠ y ∧ ¬Disjoint (x +ᵥ s) (y +ᵥ s) := by |
contrapose! h
exact ((fund.measure_eq_tsum _).trans (measure_iUnion₀
(Pairwise.mono h fun i j hij => (hij.mono inf_le_left inf_le_left).aedisjoint)
fun _ => (hS.vadd _).inter fund.nullMeasurableSet).symm).trans_le
(measure_mono <| Set.iUnion_subset fun _ => Set.inter_subset_right)
| 5 | 148.413159 | 2 | 2 | 3 | 2,272 |
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 | 74 | 79 | theorem sign_intCast (z : ℤ) : sign (z : ℝ) = ↑(Int.sign z) := by |
obtain hn | rfl | hp := lt_trichotomy z (0 : ℤ)
· rw [sign_of_neg (Int.cast_lt_zero.mpr hn), Int.sign_eq_neg_one_of_neg hn, Int.cast_neg,
Int.cast_one]
· rw [Int.cast_zero, sign_zero, Int.sign_zero, Int.cast_zero]
· rw [sign_of_pos (Int.cast_pos.mpr hp), Int.sign_eq_one_of_pos hp, Int.cast_one]
| 5 | 148.413159 | 2 | 1.363636 | 11 | 1,467 |
import Mathlib.Tactic.Qify
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.DiophantineApproximation
import Mathlib.NumberTheory.Zsqrtd.Basic
#align_import number_theory.pell from "leanprover-community/mathlib"@"7ad820c4997738e2f542f8a20f32911f52020e26"
namespace Pell
open Zsqrtd
theorem is_pell_s... | Mathlib/NumberTheory/Pell.lean | 209 | 214 | theorem eq_zero_of_d_neg (h₀ : d < 0) (a : Solution₁ d) : a.x = 0 ∨ a.y = 0 := by |
have h := a.prop
contrapose! h
have h1 := sq_pos_of_ne_zero h.1
have h2 := sq_pos_of_ne_zero h.2
nlinarith
| 5 | 148.413159 | 2 | 1 | 7 | 981 |
import Mathlib.Data.Set.Lattice
import Mathlib.Data.Set.Pairwise.Basic
#align_import data.set.pairwise.lattice from "leanprover-community/mathlib"@"c4c2ed622f43768eff32608d4a0f8a6cec1c047d"
open Function Set Order
variable {α β γ ι ι' : Type*} {κ : Sort*} {r p q : α → α → Prop}
section Pairwise
variable {f g : ... | Mathlib/Data/Set/Pairwise/Lattice.lean | 124 | 130 | theorem biUnion_diff_biUnion_eq {s t : Set ι} {f : ι → Set α} (h : (s ∪ t).PairwiseDisjoint f) :
((⋃ i ∈ s, f i) \ ⋃ i ∈ t, f i) = ⋃ i ∈ s \ t, f i := by |
refine
(biUnion_diff_biUnion_subset f s t).antisymm
(iUnion₂_subset fun i hi a ha => (mem_diff _).2 ⟨mem_biUnion hi.1 ha, ?_⟩)
rw [mem_iUnion₂]; rintro ⟨j, hj, haj⟩
exact (h (Or.inl hi.1) (Or.inr hj) (ne_of_mem_of_not_mem hj hi.2).symm).le_bot ⟨ha, haj⟩
| 5 | 148.413159 | 2 | 1.666667 | 6 | 1,759 |
import Mathlib.Data.PFunctor.Univariate.Basic
#align_import data.pfunctor.univariate.M from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
universe u v w
open Nat Function
open List
variable (F : PFunctor.{u})
-- Porting note: the ♯ tactic is never used
-- local prefix:0 "♯" => cast... | Mathlib/Data/PFunctor/Univariate/M.lean | 101 | 115 | theorem truncate_eq_of_agree {n : ℕ} (x : CofixA F n) (y : CofixA F (succ n)) (h : Agree x y) :
truncate y = x := by |
induction n <;> cases x <;> cases y
· rfl
· -- cases' h with _ _ _ _ _ h₀ h₁
cases h
simp only [truncate, Function.comp, true_and_iff, eq_self_iff_true, heq_iff_eq]
-- Porting note: used to be `ext y`
rename_i n_ih a f y h₁
suffices (fun x => truncate (y x)) = f
by simp [this]
funex... | 12 | 162,754.791419 | 2 | 1.166667 | 6 | 1,239 |
import Mathlib.MeasureTheory.Integral.FundThmCalculus
import Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv
import Mathlib.Analysis.SpecialFunctions.NonIntegrable
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
#align_import analysis.special_functions.integrals from "leanprover-community/mathlib"@"011... | Mathlib/Analysis/SpecialFunctions/Integrals.lean | 73 | 95 | theorem intervalIntegrable_rpow' {r : ℝ} (h : -1 < r) :
IntervalIntegrable (fun x => x ^ r) volume a b := by |
suffices ∀ c : ℝ, IntervalIntegrable (fun x => x ^ r) volume 0 c by
exact IntervalIntegrable.trans (this a).symm (this b)
have : ∀ c : ℝ, 0 ≤ c → IntervalIntegrable (fun x => x ^ r) volume 0 c := by
intro c hc
rw [intervalIntegrable_iff, uIoc_of_le hc]
have hderiv : ∀ x ∈ Ioo 0 c, HasDerivAt (fun x... | 21 | 1,318,815,734.483215 | 2 | 2 | 2 | 2,258 |
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 | 730 | 736 | theorem exists_frequently_lt_of_liminf_ne_top {ι : Type*} {l : Filter ι} {x : ι → ℝ}
(hx : liminf (fun n => (Real.nnabs (x n) : ℝ≥0∞)) l ≠ ∞) : ∃ R, ∃ᶠ n in l, x n < R := by |
by_contra h
simp_rw [not_exists, not_frequently, not_lt] at h
refine hx (ENNReal.eq_top_of_forall_nnreal_le fun r => le_limsInf_of_le (by isBoundedDefault) ?_)
simp only [eventually_map, ENNReal.coe_le_coe]
filter_upwards [h r] with i hi using hi.trans (le_abs_self (x i))
| 5 | 148.413159 | 2 | 1.285714 | 7 | 1,362 |
import Mathlib.AlgebraicTopology.DoldKan.EquivalenceAdditive
import Mathlib.AlgebraicTopology.DoldKan.Compatibility
import Mathlib.CategoryTheory.Idempotents.SimplicialObject
#align_import algebraic_topology.dold_kan.equivalence_pseudoabelian from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190f0b5... | Mathlib/AlgebraicTopology/DoldKan/EquivalencePseudoabelian.lean | 129 | 144 | theorem hε :
Compatibility.υ (isoN₁) =
(Γ₂N₁ : (toKaroubiEquivalence _).functor ≅
(N₁ : SimplicialObject C ⥤ _) ⋙ Preadditive.DoldKan.equivalence.inverse) := by |
dsimp only [isoN₁]
ext1
rw [← cancel_epi Γ₂N₁.inv, Iso.inv_hom_id]
ext X : 2
rw [NatTrans.comp_app]
erw [compatibility_Γ₂N₁_Γ₂N₂_natTrans X]
rw [Compatibility.υ_hom_app, Preadditive.DoldKan.equivalence_unitIso, Iso.app_inv, assoc]
erw [← NatTrans.comp_app_assoc, IsIso.hom_inv_id]
rw [NatTrans.id_app,... | 12 | 162,754.791419 | 2 | 1.5 | 2 | 1,592 |
import Mathlib.Data.Finsupp.Basic
import Mathlib.Data.Finsupp.Order
#align_import data.finsupp.multiset from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
open Finset
variable {α β ι : Type*}
namespace Finsupp
def toMultiset : (α →₀ ℕ) →+ Multiset α where
toFun f := Finsupp.sum f... | Mathlib/Data/Finsupp/Multiset.lean | 71 | 79 | theorem toMultiset_map (f : α →₀ ℕ) (g : α → β) :
f.toMultiset.map g = toMultiset (f.mapDomain g) := by |
refine f.induction ?_ ?_
· rw [toMultiset_zero, Multiset.map_zero, mapDomain_zero, toMultiset_zero]
· intro a n f _ _ ih
rw [toMultiset_add, Multiset.map_add, ih, mapDomain_add, mapDomain_single,
toMultiset_single, toMultiset_add, toMultiset_single, ← Multiset.coe_mapAddMonoidHom,
(Multiset.mapAd... | 7 | 1,096.633158 | 2 | 1.111111 | 9 | 1,194 |
import Mathlib.Topology.Bornology.Basic
#align_import topology.bornology.constructions from "leanprover-community/mathlib"@"e3d9ab8faa9dea8f78155c6c27d62a621f4c152d"
open Set Filter Bornology Function
open Filter
variable {α β ι : Type*} {π : ι → Type*} [Bornology α] [Bornology β]
[∀ i, Bornology (π i)]
inst... | Mathlib/Topology/Bornology/Constructions.lean | 126 | 131 | theorem isBounded_pi : IsBounded (pi univ S) ↔ (∃ i, S i = ∅) ∨ ∀ i, IsBounded (S i) := by |
by_cases hne : ∃ i, S i = ∅
· simp [hne, univ_pi_eq_empty_iff.2 hne]
· simp only [hne, false_or_iff]
simp only [not_exists, ← Ne.eq_def, ← nonempty_iff_ne_empty, ← univ_pi_nonempty_iff] at hne
exact isBounded_pi_of_nonempty hne
| 5 | 148.413159 | 2 | 1.333333 | 3 | 1,370 |
import Mathlib.Analysis.Normed.Field.Basic
import Mathlib.Analysis.Normed.Group.InfiniteSum
import Mathlib.Topology.Algebra.InfiniteSum.Real
#align_import analysis.normed.field.infinite_sum from "leanprover-community/mathlib"@"008205aa645b3f194c1da47025c5f110c8406eab"
variable {R : Type*} {ι : Type*} {ι' : Type*}... | Mathlib/Analysis/Normed/Field/InfiniteSum.lean | 73 | 83 | theorem summable_norm_sum_mul_antidiagonal_of_summable_norm {f g : ℕ → R}
(hf : Summable fun x => ‖f x‖) (hg : Summable fun x => ‖g x‖) :
Summable fun n => ‖∑ kl ∈ antidiagonal n, f kl.1 * g kl.2‖ := by |
have :=
summable_sum_mul_antidiagonal_of_summable_mul
(Summable.mul_of_nonneg hf hg (fun _ => norm_nonneg _) fun _ => norm_nonneg _)
refine this.of_nonneg_of_le (fun _ => norm_nonneg _) (fun n ↦ ?_)
calc
‖∑ kl ∈ antidiagonal n, f kl.1 * g kl.2‖ ≤ ∑ kl ∈ antidiagonal n, ‖f kl.1 * g kl.2‖ :=
no... | 8 | 2,980.957987 | 2 | 1.333333 | 3 | 1,381 |
import Mathlib.SetTheory.Ordinal.Arithmetic
namespace Cardinal
universe u
variable {α : Type u}
variable (g : Ordinal → α)
open Cardinal Ordinal SuccOrder Function Set
| Mathlib/SetTheory/Ordinal/FixedPointApproximants.lean | 49 | 56 | theorem not_injective_limitation_set : ¬ InjOn g (Iio (ord <| succ #α)) := by |
intro h_inj
have h := lift_mk_le_lift_mk_of_injective <| injOn_iff_injective.1 h_inj
have mk_initialSeg_subtype :
#(Iio (ord <| succ #α)) = lift.{u + 1} (succ #α) := by
simpa only [coe_setOf, card_typein, card_ord] using mk_initialSeg (ord <| succ #α)
rw [mk_initialSeg_subtype, lift_lift, lift_le] at... | 7 | 1,096.633158 | 2 | 1.8 | 5 | 1,896 |
import Mathlib.Logic.Pairwise
import Mathlib.Logic.Relation
import Mathlib.Data.List.Basic
#align_import data.list.pairwise from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
open Nat Function
namespace List
variable {α β : Type*} {R S T : α → α → Prop} {a : α} {l : List α}
mk_iff_o... | Mathlib/Data/List/Pairwise.lean | 124 | 133 | theorem pairwise_pmap {p : β → Prop} {f : ∀ b, p b → α} {l : List β} (h : ∀ x ∈ l, p x) :
Pairwise R (l.pmap f h) ↔
Pairwise (fun b₁ b₂ => ∀ (h₁ : p b₁) (h₂ : p b₂), R (f b₁ h₁) (f b₂ h₂)) l := by |
induction' l with a l ihl
· simp
obtain ⟨_, hl⟩ : p a ∧ ∀ b, b ∈ l → p b := by simpa using h
simp only [ihl hl, pairwise_cons, exists₂_imp, pmap, and_congr_left_iff, mem_pmap]
refine fun _ => ⟨fun H b hb _ hpb => H _ _ hb rfl, ?_⟩
rintro H _ b hb rfl
exact H b hb _ _
| 7 | 1,096.633158 | 2 | 1.5 | 4 | 1,638 |
import Mathlib.CategoryTheory.Sites.Coherent.Comparison
import Mathlib.CategoryTheory.Sites.Coherent.ExtensiveSheaves
import Mathlib.CategoryTheory.Sites.Coherent.ReflectsPrecoherent
import Mathlib.CategoryTheory.Sites.Coherent.ReflectsPreregular
import Mathlib.CategoryTheory.Sites.InducedTopology
import Mathlib.Categ... | Mathlib/CategoryTheory/Sites/Coherent/SheafComparison.lean | 55 | 76 | theorem exists_effectiveEpiFamily_iff_mem_induced (X : C) (S : Sieve X) :
(∃ (α : Type) (_ : Finite α) (Y : α → C) (π : (a : α) → (Y a ⟶ X)),
EffectiveEpiFamily Y π ∧ (∀ a : α, (S.arrows) (π a)) ) ↔
(S ∈ F.inducedTopologyOfIsCoverDense (coherentTopology _) X) := by |
refine ⟨fun ⟨α, _, Y, π, ⟨H₁, H₂⟩⟩ ↦ ?_, fun hS ↦ ?_⟩
· apply (mem_sieves_iff_hasEffectiveEpiFamily (Sieve.functorPushforward _ S)).mpr
refine ⟨α, inferInstance, fun i => F.obj (Y i),
fun i => F.map (π i), ⟨?_,
fun a => Sieve.image_mem_functorPushforward F S (H₂ a)⟩⟩
exact F.map_finite_effectiv... | 18 | 65,659,969.137331 | 2 | 2 | 2 | 1,991 |
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 | 71 | 77 | theorem diagramNatTrans_id (X : C) (P : Cᵒᵖ ⥤ D) :
J.diagramNatTrans (𝟙 P) X = 𝟙 (J.diagram P X) := by |
ext : 2
refine Multiequalizer.hom_ext _ _ _ (fun i => ?_)
dsimp
simp only [limit.lift_π, Multifork.ofι_pt, Multifork.ofι_π_app, Category.id_comp]
erw [Category.comp_id]
| 5 | 148.413159 | 2 | 2 | 3 | 2,407 |
import Mathlib.Probability.Process.Filtration
import Mathlib.Topology.Instances.Discrete
#align_import probability.process.adapted from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Filter Order TopologicalSpace
open scoped Classical MeasureTheory NNReal ENNReal Topology
namespa... | Mathlib/Probability/Process/Adapted.lean | 99 | 105 | theorem Filtration.adapted_natural [MetrizableSpace β] [mβ : MeasurableSpace β] [BorelSpace β]
{u : ι → Ω → β} (hum : ∀ i, StronglyMeasurable[m] (u i)) :
Adapted (Filtration.natural u hum) u := by |
intro i
refine StronglyMeasurable.mono ?_ (le_iSup₂_of_le i (le_refl i) le_rfl)
rw [stronglyMeasurable_iff_measurable_separable]
exact ⟨measurable_iff_comap_le.2 le_rfl, (hum i).isSeparable_range⟩
| 4 | 54.59815 | 2 | 2 | 2 | 2,288 |
import Mathlib.Analysis.LocallyConvex.Bounded
import Mathlib.Analysis.RCLike.Basic
#align_import analysis.locally_convex.continuous_of_bounded from "leanprover-community/mathlib"@"3f655f5297b030a87d641ad4e825af8d9679eb0b"
open TopologicalSpace Bornology Filter Topology Pointwise
variable {𝕜 𝕜' E F : Type*}
var... | Mathlib/Analysis/LocallyConvex/ContinuousOfBounded.lean | 96 | 166 | theorem LinearMap.continuousAt_zero_of_locally_bounded (f : E →ₛₗ[σ] F)
(hf : ∀ s, IsVonNBounded 𝕜 s → IsVonNBounded 𝕜' (f '' s)) : ContinuousAt f 0 := by |
-- Assume that f is not continuous at 0
by_contra h
-- We use a decreasing balanced basis for 0 : E and a balanced basis for 0 : F
-- and reformulate non-continuity in terms of these bases
rcases (nhds_basis_balanced 𝕜 E).exists_antitone_subbasis with ⟨b, bE1, bE⟩
simp only [_root_.id] at bE
have bE' : ... | 69 | 925,378,172,558,778,900,000,000,000,000 | 2 | 2 | 1 | 2,371 |
import Mathlib.Probability.Kernel.Disintegration.Unique
import Mathlib.Probability.Notation
#align_import probability.kernel.cond_distrib from "leanprover-community/mathlib"@"00abe0695d8767201e6d008afa22393978bb324d"
open MeasureTheory Set Filter TopologicalSpace
open scoped ENNReal MeasureTheory ProbabilityTheo... | Mathlib/Probability/Kernel/CondDistrib.lean | 120 | 130 | theorem condDistrib_ae_eq_of_measure_eq_compProd (hX : Measurable X) (hY : Measurable Y)
(κ : kernel β Ω) [IsFiniteKernel κ] (hκ : μ.map (fun x => (X x, Y x)) = μ.map X ⊗ₘ κ) :
∀ᵐ x ∂μ.map X, κ x = condDistrib Y X μ x := by |
have heq : μ.map X = (μ.map (fun x ↦ (X x, Y x))).fst := by
ext s hs
rw [Measure.map_apply hX hs, Measure.fst_apply hs, Measure.map_apply]
exacts [rfl, Measurable.prod hX hY, measurable_fst hs]
rw [heq, condDistrib]
refine eq_condKernel_of_measure_eq_compProd _ ?_
convert hκ
exact heq.symm
| 8 | 2,980.957987 | 2 | 0.777778 | 9 | 692 |
import Mathlib.Analysis.SpecialFunctions.Exponential
#align_import analysis.special_functions.trigonometric.series from "leanprover-community/mathlib"@"ccf84e0d918668460a34aa19d02fe2e0e2286da0"
open NormedSpace
open scoped Nat
section SinCos
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Series.lean | 32 | 46 | theorem Complex.hasSum_cos' (z : ℂ) :
HasSum (fun n : ℕ => (z * Complex.I) ^ (2 * n) / ↑(2 * n)!) (Complex.cos z) := by |
rw [Complex.cos, Complex.exp_eq_exp_ℂ]
have := ((expSeries_div_hasSum_exp ℂ (z * Complex.I)).add
(expSeries_div_hasSum_exp ℂ (-z * Complex.I))).div_const 2
replace := (Nat.divModEquiv 2).symm.hasSum_iff.mpr this
dsimp [Function.comp_def] at this
simp_rw [← mul_comm 2 _] at this
refine this.prod_fiberwi... | 13 | 442,413.392009 | 2 | 1.5 | 4 | 1,652 |
import Mathlib.FieldTheory.Galois
#align_import field_theory.polynomial_galois_group from "leanprover-community/mathlib"@"e3f4be1fcb5376c4948d7f095bec45350bfb9d1a"
noncomputable section
open scoped Polynomial
open FiniteDimensional
namespace Polynomial
variable {F : Type*} [Field F] (p q : F[X]) (E : Type*) [... | Mathlib/FieldTheory/PolynomialGaloisGroup.lean | 155 | 168 | theorem mapRoots_bijective [h : Fact (p.Splits (algebraMap F E))] :
Function.Bijective (mapRoots p E) := by |
constructor
· exact fun _ _ h => Subtype.ext (RingHom.injective _ (Subtype.ext_iff.mp h))
· intro y
-- this is just an equality of two different ways to write the roots of `p` as an `E`-polynomial
have key :=
roots_map (IsScalarTower.toAlgHom F p.SplittingField E : p.SplittingField →+* E)
(... | 12 | 162,754.791419 | 2 | 1.75 | 4 | 1,873 |
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 | 63 | 75 | theorem continuousWithinAt_right_of_monotoneOn_of_exists_between {f : α → β} {s : Set α} {a : α}
(h_mono : MonotoneOn f s) (hs : s ∈ 𝓝[≥] a) (hfs : ∀ b > f a, ∃ c ∈ s, f c ∈ Ioo (f a) b) :
ContinuousWithinAt f (Ici a) a := by |
have ha : a ∈ Ici a := left_mem_Ici
have has : a ∈ s := mem_of_mem_nhdsWithin ha hs
refine tendsto_order.2 ⟨fun b hb => ?_, fun b hb => ?_⟩
· filter_upwards [hs, @self_mem_nhdsWithin _ _ a (Ici a)] with _ hxs hxa using hb.trans_le
(h_mono has hxs hxa)
· rcases hfs b hb with ⟨c, hcs, hac, hcb⟩
have ... | 10 | 22,026.465795 | 2 | 2 | 3 | 2,171 |
import Mathlib.LinearAlgebra.Matrix.Gershgorin
import Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody
import Mathlib.NumberTheory.NumberField.Units.Basic
import Mathlib.RingTheory.RootsOfUnity.Basic
#align_import number_theory.number_field.units from "leanprover-community/mathlib"@"00f91228655eecdcd3ac... | Mathlib/NumberTheory/NumberField/Units/DirichletTheorem.lean | 100 | 106 | theorem mult_log_place_eq_zero {x : (𝓞 K)ˣ} {w : InfinitePlace K} :
mult w * Real.log (w x) = 0 ↔ w x = 1 := by |
rw [mul_eq_zero, or_iff_right, Real.log_eq_zero, or_iff_right, or_iff_left]
· linarith [(apply_nonneg _ _ : 0 ≤ w x)]
· simp only [ne_eq, map_eq_zero, coe_ne_zero x, not_false_eq_true]
· refine (ne_of_gt ?_)
rw [mult]; split_ifs <;> norm_num
| 5 | 148.413159 | 2 | 1.833333 | 6 | 1,909 |
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 | 739 | 745 | theorem exists_frequently_lt_of_liminf_ne_top' {ι : Type*} {l : Filter ι} {x : ι → ℝ}
(hx : liminf (fun n => (Real.nnabs (x n) : ℝ≥0∞)) l ≠ ∞) : ∃ R, ∃ᶠ n in l, R < x n := by |
by_contra h
simp_rw [not_exists, not_frequently, not_lt] at h
refine hx (ENNReal.eq_top_of_forall_nnreal_le fun r => le_limsInf_of_le (by isBoundedDefault) ?_)
simp only [eventually_map, ENNReal.coe_le_coe]
filter_upwards [h (-r)] with i hi using(le_neg.1 hi).trans (neg_le_abs _)
| 5 | 148.413159 | 2 | 1.285714 | 7 | 1,362 |
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 | 66 | 71 | theorem continuousAt_const_cpow {a b : ℂ} (ha : a ≠ 0) : ContinuousAt (fun x : ℂ => a ^ x) b := by |
have cpow_eq : (fun x : ℂ => a ^ x) = fun x => exp (log a * x) := by
ext1 b
rw [cpow_def_of_ne_zero ha]
rw [cpow_eq]
exact continuous_exp.continuousAt.comp (ContinuousAt.mul continuousAt_const continuousAt_id)
| 5 | 148.413159 | 2 | 1.857143 | 7 | 1,926 |
import Mathlib.AlgebraicGeometry.Morphisms.Basic
import Mathlib.RingTheory.LocalProperties
#align_import algebraic_geometry.morphisms.ring_hom_properties from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
-- Explicit universe annotations were used in this file to improve perfomance #127... | Mathlib/AlgebraicGeometry/Morphisms/RingHomProperties.lean | 163 | 205 | theorem affineLocally_iff_affineOpens_le
(hP : RingHom.RespectsIso @P) {X Y : Scheme.{u}} (f : X ⟶ Y) :
affineLocally.{u} (@P) f ↔
∀ (U : Y.affineOpens) (V : X.affineOpens) (e : V.1 ≤ (Opens.map f.1.base).obj U.1),
P (Scheme.Hom.appLe f e) := by |
apply forall_congr'
intro U
delta sourceAffineLocally
simp_rw [op_comp, Scheme.Γ.map_comp, Γ_map_morphismRestrict, Category.assoc, Scheme.Γ_map_op,
hP.cancel_left_isIso (Y.presheaf.map (eqToHom _).op)]
constructor
· intro H V e
let U' := (Opens.map f.val.base).obj U.1
have e'' : (Scheme.Hom.ope... | 38 | 31,855,931,757,113,756 | 2 | 2 | 6 | 1,934 |
import Mathlib.Data.Countable.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Order.Disjointed
import Mathlib.MeasureTheory.OuterMeasure.Defs
#align_import measure_theory.measure.outer_measure from "leanprover-community/mathlib"@"343e80208d29d2d15f8050b929aa50fe4ce71b55"
noncomputable section
open Set F... | Mathlib/MeasureTheory/OuterMeasure/Basic.lean | 63 | 69 | theorem measure_iUnion_le [Countable ι] (s : ι → Set α) : μ (⋃ i, s i) ≤ ∑' i, μ (s i) := by |
refine rel_iSup_tsum μ measure_empty (· ≤ ·) (fun t ↦ ?_) _
calc
μ (⋃ i, t i) = μ (⋃ i, disjointed t i) := by rw [iUnion_disjointed]
_ ≤ ∑' i, μ (disjointed t i) :=
OuterMeasureClass.measure_iUnion_nat_le _ _ (disjoint_disjointed _)
_ ≤ ∑' i, μ (t i) := by gcongr; apply disjointed_subset
| 6 | 403.428793 | 2 | 0.5 | 10 | 470 |
import Mathlib.CategoryTheory.Balanced
import Mathlib.CategoryTheory.LiftingProperties.Basic
#align_import category_theory.limits.shapes.strong_epi from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
universe v u
namespace CategoryTheory
variable {C : Type u} [Category.{v} C]
variable... | Mathlib/CategoryTheory/Limits/Shapes/StrongEpi.lean | 150 | 158 | theorem StrongEpi.of_arrow_iso {A B A' B' : C} {f : A ⟶ B} {g : A' ⟶ B'}
(e : Arrow.mk f ≅ Arrow.mk g) [h : StrongEpi f] : StrongEpi g :=
{ epi := by |
rw [Arrow.iso_w' e]
haveI := epi_comp f e.hom.right
apply epi_comp
llp := fun {X Y} z => by
intro
apply HasLiftingProperty.of_arrow_iso_left e z }
| 6 | 403.428793 | 2 | 1.428571 | 7 | 1,511 |
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Combinatorics.SimpleGraph.Coloring
import Mathlib.Combinatorics.SimpleGraph.Hasse
import Mathlib.Order.OmegaCompletePartialOrder
namespace SimpleGraph
def pathGraph.bicoloring (n : ℕ) :
Coloring (pathGraph n) Bool :=
Coloring.mk (fun u ↦ u.val % 2 = 0) <|... | Mathlib/Combinatorics/SimpleGraph/ConcreteColorings.lean | 43 | 49 | theorem chromaticNumber_pathGraph (n : ℕ) (h : 2 ≤ n) :
(pathGraph n).chromaticNumber = 2 := by |
have hc := (pathGraph.bicoloring n).colorable
apply le_antisymm
· exact hc.chromaticNumber_le
· simpa only [pathGraph_two_eq_top, chromaticNumber_top] using
chromaticNumber_mono_of_embedding (pathGraph_two_embedding n h)
| 5 | 148.413159 | 2 | 2 | 1 | 2,310 |
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 | 57 | 64 | theorem IsCompact.compl_mem_sets_of_nhdsWithin (hs : IsCompact s) {f : Filter X}
(hf : ∀ x ∈ s, ∃ t ∈ 𝓝[s] x, tᶜ ∈ f) : sᶜ ∈ f := by |
refine hs.compl_mem_sets fun x hx => ?_
rcases hf x hx with ⟨t, ht, hst⟩
replace ht := mem_inf_principal.1 ht
apply mem_inf_of_inter ht hst
rintro x ⟨h₁, h₂⟩ hs
exact h₂ (h₁ hs)
| 6 | 403.428793 | 2 | 1.6 | 5 | 1,733 |
import Mathlib.GroupTheory.Archimedean
import Mathlib.Topology.Order.Basic
#align_import topology.algebra.order.archimedean from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
open Set
theorem Rat.denseRange_cast {𝕜} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜]
... | Mathlib/Topology/Algebra/Order/Archimedean.lean | 67 | 71 | theorem dense_or_cyclic (S : AddSubgroup G) : Dense (S : Set G) ∨ ∃ a : G, S = closure {a} := by |
refine (em _).imp (dense_of_not_isolated_zero S) fun h => ?_
push_neg at h
rcases h with ⟨ε, ε0, hε⟩
exact cyclic_of_isolated_zero ε0 (disjoint_left.2 hε)
| 4 | 54.59815 | 2 | 1.666667 | 3 | 1,825 |
import Mathlib.Probability.Kernel.MeasurableIntegral
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import probability.kernel.with_density from "leanprover-community/mathlib"@"c0d694db494dd4f9aa57f2714b6e4c82b4ebc113"
open MeasureTheory ProbabilityTheory
open scoped MeasureTheory ENNReal NNReal
namesp... | Mathlib/Probability/Kernel/WithDensity.lean | 135 | 144 | theorem withDensity_kernel_sum [Countable ι] (κ : ι → kernel α β) (hκ : ∀ i, IsSFiniteKernel (κ i))
(f : α → β → ℝ≥0∞) :
@withDensity _ _ _ _ (kernel.sum κ) (isSFiniteKernel_sum hκ) f =
kernel.sum fun i => withDensity (κ i) f := by |
by_cases hf : Measurable (Function.uncurry f)
· ext1 a
simp_rw [sum_apply, kernel.withDensity_apply _ hf, sum_apply,
withDensity_sum (fun n => κ n a) (f a)]
· simp_rw [withDensity_of_not_measurable _ hf]
exact sum_zero.symm
| 6 | 403.428793 | 2 | 1.2 | 5 | 1,277 |
import Mathlib.RingTheory.WittVector.Frobenius
import Mathlib.RingTheory.WittVector.Verschiebung
import Mathlib.RingTheory.WittVector.MulP
#align_import ring_theory.witt_vector.identities from "leanprover-community/mathlib"@"0798037604b2d91748f9b43925fb7570a5f3256c"
namespace WittVector
variable {p : ℕ} {R : Typ... | Mathlib/RingTheory/WittVector/Identities.lean | 64 | 71 | theorem coeff_p_pow_eq_zero [CharP R p] {i j : ℕ} (hj : j ≠ i) : ((p : 𝕎 R) ^ i).coeff j = 0 := by |
induction' i with i hi generalizing j
· rw [pow_zero, one_coeff_eq_of_pos]
exact Nat.pos_of_ne_zero hj
· rw [pow_succ, ← frobenius_verschiebung, coeff_frobenius_charP]
cases j
· rw [verschiebung_coeff_zero, zero_pow hp.out.ne_zero]
· rw [verschiebung_coeff_succ, hi (ne_of_apply_ne _ hj), zero_pow... | 7 | 1,096.633158 | 2 | 1 | 13 | 1,103 |
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.MvPolynomial.Variables
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Algebra.MvPolynomial.Expand
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.ZMod.Basic
#align_import ring_theory.witt_vector.witt_polynomial from "leanprover-c... | Mathlib/RingTheory/WittVector/WittPolynomial.lean | 125 | 132 | theorem constantCoeff_wittPolynomial [hp : Fact p.Prime] (n : ℕ) :
constantCoeff (wittPolynomial p R n) = 0 := by |
simp only [wittPolynomial, map_sum, constantCoeff_monomial]
rw [sum_eq_zero]
rintro i _
rw [if_neg]
rw [Finsupp.single_eq_zero]
exact ne_of_gt (pow_pos hp.1.pos _)
| 6 | 403.428793 | 2 | 1.181818 | 11 | 1,247 |
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 | 65 | 70 | theorem coe_splitLower : (splitLower I i x : Set (ι → ℝ)) = ↑I ∩ { y | y i ≤ x } := by |
rw [splitLower, coe_mk']
ext y
simp only [mem_univ_pi, mem_Ioc, mem_inter_iff, mem_coe, mem_setOf_eq, forall_and, ← Pi.le_def,
le_update_iff, le_min_iff, and_assoc, and_forall_ne (p := fun j => y j ≤ upper I j) i, mem_def]
rw [and_comm (a := y i ≤ x)]
| 5 | 148.413159 | 2 | 1.2 | 10 | 1,269 |
import Mathlib.LinearAlgebra.Basis.VectorSpace
import Mathlib.LinearAlgebra.Dimension.Finite
import Mathlib.SetTheory.Cardinal.Subfield
import Mathlib.LinearAlgebra.Dimension.RankNullity
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomput... | Mathlib/LinearAlgebra/Dimension/DivisionRing.lean | 123 | 166 | theorem linearIndependent_of_top_le_span_of_card_eq_finrank {ι : Type*} [Fintype ι] {b : ι → V}
(spans : ⊤ ≤ span K (Set.range b)) (card_eq : Fintype.card ι = finrank K V) :
LinearIndependent K b :=
linearIndependent_iff'.mpr fun s g dependent i i_mem_s => by
classical
by_contra gx_ne_zero
-- We'l... |
rw [Set.toFinset_card, Fintype.card_ofFinset]
_ ≤ (Set.univ \ {i}).toFinset.card := Finset.card_image_le
_ = (Finset.univ.erase i).card := (congr_arg Finset.card (Finset.ext (by simp [and_comm])))
_ < Finset.univ.card := Finset.card_erase_lt_of_mem (Finset.mem_univ i)
_ = finr... | 32 | 78,962,960,182,680.7 | 2 | 1.625 | 8 | 1,748 |
import Mathlib.SetTheory.Ordinal.Arithmetic
import Mathlib.Tactic.TFAE
import Mathlib.Topology.Order.Monotone
#align_import set_theory.ordinal.topology from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
noncomputable section
universe u v
open Cardinal Order Topology
namespace Ordina... | Mathlib/SetTheory/Ordinal/Topology.lean | 86 | 124 | theorem mem_closure_tfae (a : Ordinal.{u}) (s : Set Ordinal) :
TFAE [a ∈ closure s,
a ∈ closure (s ∩ Iic a),
(s ∩ Iic a).Nonempty ∧ sSup (s ∩ Iic a) = a,
∃ t, t ⊆ s ∧ t.Nonempty ∧ BddAbove t ∧ sSup t = a,
∃ (o : Ordinal.{u}), o ≠ 0 ∧ ∃ (f : ∀ x < o, Ordinal),
(∀ x hx, f x hx ∈ s) ∧ b... |
tfae_have 1 → 2
· simp only [mem_closure_iff_nhdsWithin_neBot, inter_comm s, nhdsWithin_inter', nhds_left_eq_nhds]
exact id
tfae_have 2 → 3
· intro h
rcases (s ∩ Iic a).eq_empty_or_nonempty with he | hne
· simp [he] at h
· refine ⟨hne, (isLUB_of_mem_closure ?_ h).csSup_eq hne⟩
exact fun x... | 31 | 29,048,849,665,247.426 | 2 | 1.2 | 5 | 1,259 |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Finset.NatAntidiagonal
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Logic.Function.Iterate
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Zify
#align_import data.nat.fib from "leanprover-community/mathlib"@"... | Mathlib/Data/Nat/Fib/Basic.lean | 187 | 194 | theorem fib_two_mul_add_two (n : ℕ) :
fib (2 * n + 2) = fib (n + 1) * (2 * fib n + fib (n + 1)) := by |
rw [fib_add_two, fib_two_mul, fib_two_mul_add_one]
-- Porting note: A bunch of issues similar to [this zulip thread](https://github.com/leanprover-community/mathlib4/pull/1576) with `zify`
have : fib n ≤ 2 * fib (n + 1) :=
le_trans fib_le_fib_succ (mul_comm 2 _ ▸ Nat.le_mul_of_pos_right _ two_pos)
zify [th... | 6 | 403.428793 | 2 | 1.181818 | 11 | 1,246 |
import Mathlib.Data.List.Nodup
#align_import data.list.duplicate from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
variable {α : Type*}
namespace List
inductive Duplicate (x : α) : List α → Prop
| cons_mem {l : List α} : x ∈ l → Duplicate x (x :: l)
| cons_duplicate {y : α} {l ... | Mathlib/Data/List/Duplicate.lean | 88 | 95 | theorem duplicate_cons_iff {y : α} : x ∈+ y :: l ↔ y = x ∧ x ∈ l ∨ x ∈+ l := by |
refine ⟨fun h => ?_, fun h => ?_⟩
· cases' h with _ hm _ _ hm
· exact Or.inl ⟨rfl, hm⟩
· exact Or.inr hm
· rcases h with (⟨rfl | h⟩ | h)
· simpa
· exact h.cons_duplicate
| 7 | 1,096.633158 | 2 | 0.818182 | 11 | 719 |
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Algebra.Order.Group.Indicator
import Mathlib.Order.LiminfLimsup
import Mathlib.Order.Filter.Archimedean
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.Algebra.Group.Basic
import Mathlib.Data.Set.La... | Mathlib/Topology/Algebra/Order/LiminfLimsup.lean | 568 | 578 | theorem limsup_eq_tendsto_sum_indicator_atTop (R : Type*) [StrictOrderedSemiring R] [Archimedean R]
(s : ℕ → Set α) : limsup s atTop = { ω | Tendsto
(fun n ↦ ∑ k ∈ Finset.range n, (s (k + 1)).indicator (1 : α → R) ω) atTop atTop } := by |
rw [limsup_eq_tendsto_sum_indicator_nat_atTop s]
ext ω
simp only [Set.mem_setOf_eq]
rw [(_ : (fun n ↦ ∑ k ∈ Finset.range n, (s (k + 1)).indicator (1 : α → R) ω) = fun n ↦
↑(∑ k ∈ Finset.range n, (s (k + 1)).indicator (1 : α → ℕ) ω))]
· exact tendsto_natCast_atTop_iff.symm
· ext n
simp only [Set.ind... | 8 | 2,980.957987 | 2 | 1.8 | 5 | 1,883 |
import Mathlib.Algebra.Lie.Nilpotent
import Mathlib.Algebra.Lie.Normalizer
#align_import algebra.lie.cartan_subalgebra from "leanprover-community/mathlib"@"938fead7abdc0cbbca8eba7a1052865a169dc102"
universe u v w w₁ w₂
variable {R : Type u} {L : Type v}
variable [CommRing R] [LieRing L] [LieAlgebra R L] (H : Lie... | Mathlib/Algebra/Lie/CartanSubalgebra.lean | 72 | 94 | theorem isCartanSubalgebra_iff_isUcsLimit : H.IsCartanSubalgebra ↔ H.toLieSubmodule.IsUcsLimit := by |
constructor
· intro h
have h₁ : LieAlgebra.IsNilpotent R H := by infer_instance
obtain ⟨k, hk⟩ := H.toLieSubmodule.isNilpotent_iff_exists_self_le_ucs.mp h₁
replace hk : H.toLieSubmodule = LieSubmodule.ucs k ⊥ :=
le_antisymm hk
(LieSubmodule.ucs_le_of_normalizer_eq_self H.normalizer_eq_sel... | 22 | 3,584,912,846.131591 | 2 | 1.25 | 4 | 1,305 |
import Mathlib.RingTheory.Ideal.IsPrimary
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.Order.Minimal
#align_import ring_theory.ideal.minimal_prime from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
section
variable {R S : Type*} [CommSemiring R] [CommSemiring S] (I J ... | Mathlib/RingTheory/Ideal/MinimalPrime.lean | 134 | 164 | theorem Ideal.exists_comap_eq_of_mem_minimalPrimes {I : Ideal S} (f : R →+* S) (p)
(H : p ∈ (I.comap f).minimalPrimes) : ∃ p' : Ideal S, p'.IsPrime ∧ I ≤ p' ∧ p'.comap f = p := by |
have := H.1.1
let f' := (Ideal.Quotient.mk I).comp f
have e : RingHom.ker f' = I.comap f := by
ext1
exact Submodule.Quotient.mk_eq_zero _
have : RingHom.ker (Ideal.Quotient.mk <| RingHom.ker f') ≤ p := by
rw [Ideal.mk_ker, e]
exact H.1.2
suffices _ by
have ⟨p', hp₁, hp₂⟩ := Ideal.exists_c... | 29 | 3,931,334,297,144.042 | 2 | 2 | 5 | 2,263 |
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.Galois
universe u v w
open scoped Classical Polynomial
open Polynomial
variable (k : Type u) [Field k] (K : Type v) [Field K]
class IsSepClosed : Prop where
splits_of_separable : ∀ p : k[X], p.Separable → (p.Splits <| RingHom.... | Mathlib/FieldTheory/IsSepClosed.lean | 104 | 116 | theorem exists_pow_nat_eq [IsSepClosed k] (x : k) (n : ℕ) [hn : NeZero (n : k)] :
∃ z, z ^ n = x := by |
have hn' : 0 < n := Nat.pos_of_ne_zero fun h => by
rw [h, Nat.cast_zero] at hn
exact hn.out rfl
have : degree (X ^ n - C x) ≠ 0 := by
rw [degree_X_pow_sub_C hn' x]
exact (WithBot.coe_lt_coe.2 hn').ne'
by_cases hx : x = 0
· exact ⟨0, by rw [hx, pow_eq_zero_iff hn'.ne']⟩
· obtain ⟨z, hz⟩ := exi... | 11 | 59,874.141715 | 2 | 1.5 | 6 | 1,639 |
import Mathlib.Topology.Algebra.GroupCompletion
import Mathlib.Topology.Algebra.InfiniteSum.Group
open UniformSpace.Completion
variable {α β : Type*} [AddCommGroup α] [UniformSpace α] [UniformAddGroup α]
theorem hasSum_iff_hasSum_compl (f : β → α) (a : α):
HasSum (toCompl ∘ f) a ↔ HasSum f a := (denseInducin... | Mathlib/Topology/Algebra/InfiniteSum/GroupCompletion.lean | 32 | 45 | theorem summable_iff_cauchySeq_finset_and_tsum_mem (f : β → α) :
Summable f ↔ CauchySeq (fun s : Finset β ↦ ∑ b in s, f b) ∧
∑' i, toCompl (f i) ∈ Set.range toCompl := by |
classical
constructor
· rintro ⟨a, ha⟩
exact ⟨ha.cauchySeq, ((summable_iff_summable_compl_and_tsum_mem f).mp ⟨a, ha⟩).2⟩
· rintro ⟨h_cauchy, h_tsum⟩
apply (summable_iff_summable_compl_and_tsum_mem f).mpr
constructor
· apply summable_iff_cauchySeq_finset.mpr
simp_rw [Function.comp_apply, ←... | 11 | 59,874.141715 | 2 | 2 | 1 | 2,491 |
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Algebra.Polynomial.Roots
import Mathlib.RingTheory.EuclideanDomain
#align_import data.polynomial.field_division from "leanprover-community/mathlib"@"bbeb185db4ccee8ed07dc48449414ebfa39cb821"
noncomputable section
open Polynomial
namespace Polynomial
u... | Mathlib/Algebra/Polynomial/FieldDivision.lean | 65 | 76 | theorem eval_iterate_derivative_rootMultiplicity {p : R[X]} {t : R} :
(derivative^[p.rootMultiplicity t] p).eval t =
(p.rootMultiplicity t).factorial • (p /ₘ (X - C t) ^ p.rootMultiplicity t).eval t := by |
set m := p.rootMultiplicity t with hm
conv_lhs => rw [← p.pow_mul_divByMonic_rootMultiplicity_eq t, ← hm]
rw [iterate_derivative_mul, eval_finset_sum, sum_eq_single_of_mem _ (mem_range.mpr m.succ_pos)]
· rw [m.choose_zero_right, one_smul, eval_mul, m.sub_zero, iterate_derivative_X_sub_pow_self,
eval_natC... | 9 | 8,103.083928 | 2 | 2 | 4 | 2,297 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Monic
#align_import data.polynomial.lifts from "leanprover-community/mathlib"@"63417e01fbc711beaf25fa73b6edb395c0cfddd0"
open Polynomial
noncomputable section
namespace Polynomial
universe u v w
section Semiring
variable {R : Type... | Mathlib/Algebra/Polynomial/Lifts.lean | 120 | 124 | theorem monomial_mem_lifts {s : S} (n : ℕ) (h : s ∈ Set.range f) : monomial n s ∈ lifts f := by |
obtain ⟨r, rfl⟩ := Set.mem_range.1 h
use monomial n r
simp only [coe_mapRingHom, Set.mem_univ, map_monomial, Subsemiring.coe_top, eq_self_iff_true,
and_self_iff]
| 4 | 54.59815 | 2 | 1 | 13 | 1,139 |
import Mathlib.MeasureTheory.MeasurableSpace.Defs
import Mathlib.SetTheory.Cardinal.Cofinality
import Mathlib.SetTheory.Cardinal.Continuum
#align_import measure_theory.card_measurable_space from "leanprover-community/mathlib"@"f2b108e8e97ba393f22bf794989984ddcc1da89b"
universe u
variable {α : Type u}
open Cardi... | Mathlib/MeasureTheory/MeasurableSpace/Card.lean | 117 | 151 | theorem generateMeasurable_eq_rec (s : Set (Set α)) :
{ t | GenerateMeasurable s t } =
⋃ (i : (Quotient.out (aleph 1).ord).α), generateMeasurableRec s i := by |
ext t; refine ⟨fun ht => ?_, fun ht => ?_⟩
· inhabit ω₁
induction' ht with u hu u _ IH f _ IH
· exact mem_iUnion.2 ⟨default, self_subset_generateMeasurableRec s _ hu⟩
· exact mem_iUnion.2 ⟨default, empty_mem_generateMeasurableRec s _⟩
· rcases mem_iUnion.1 IH with ⟨i, hi⟩
obtain ⟨j, hj⟩ := ex... | 32 | 78,962,960,182,680.7 | 2 | 1.333333 | 6 | 1,448 |
import Mathlib.Geometry.RingedSpace.PresheafedSpace
import Mathlib.CategoryTheory.Limits.Final
import Mathlib.Topology.Sheaves.Stalks
#align_import algebraic_geometry.stalks from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
noncomputable section
universe v u v' u'
open Opposite Cate... | Mathlib/Geometry/RingedSpace/Stalks.lean | 108 | 121 | theorem restrictStalkIso_inv_eq_ofRestrict {U : TopCat} (X : PresheafedSpace.{_, _, v} C)
{f : U ⟶ (X : TopCat.{v})} (h : OpenEmbedding f) (x : U) :
(X.restrictStalkIso h x).inv = stalkMap (X.ofRestrict h) x := by |
-- We can't use `ext` here due to https://github.com/leanprover/std4/pull/159
refine colimit.hom_ext fun V => ?_
induction V with | h V => ?_
let i : (h.isOpenMap.functorNhds x).obj ((OpenNhds.map f x).obj V) ⟶ V :=
homOfLE (Set.image_preimage_subset f _)
erw [Iso.comp_inv_eq, colimit.ι_map_assoc, colimi... | 11 | 59,874.141715 | 2 | 0.875 | 8 | 765 |
import Mathlib.Data.Set.Basic
open Function
universe u v
namespace Set
section Nontrivial
variable {α : Type u} {a : α} {s t : Set α}
protected def Nontrivial (s : Set α) : Prop :=
∃ x ∈ s, ∃ y ∈ s, x ≠ y
#align set.nontrivial Set.Nontrivial
theorem nontrivial_of_mem_mem_ne {x y} (hx : x ∈ s) (hy : y ∈... | Mathlib/Data/Set/Subsingleton.lean | 194 | 198 | theorem Nontrivial.exists_ne (hs : s.Nontrivial) (z) : ∃ x ∈ s, x ≠ z := by |
by_contra! H
rcases hs with ⟨x, hx, y, hy, hxy⟩
rw [H x hx, H y hy] at hxy
exact hxy rfl
| 4 | 54.59815 | 2 | 1.75 | 4 | 1,876 |
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Data.Nat.ModEq
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Zify
#align_import number_theory.frobenius_number from "leanprover-community/mathlib"@"1126441d6bccf98c81214a0780c73d499f6721fe"
open Nat
def FrobeniusNumber (n : ℕ) (s : Set ℕ) : Pro... | Mathlib/NumberTheory/FrobeniusNumber.lean | 55 | 82 | theorem frobeniusNumber_pair (cop : Coprime m n) (hm : 1 < m) (hn : 1 < n) :
FrobeniusNumber (m * n - m - n) {m, n} := by |
simp_rw [FrobeniusNumber, AddSubmonoid.mem_closure_pair]
have hmn : m + n ≤ m * n := add_le_mul hm hn
constructor
· push_neg
intro a b h
apply cop.mul_add_mul_ne_mul (add_one_ne_zero a) (add_one_ne_zero b)
simp only [Nat.sub_sub, smul_eq_mul] at h
zify [hmn] at h ⊢
rw [← sub_eq_zero] at h ⊢... | 26 | 195,729,609,428.83878 | 2 | 2 | 1 | 2,040 |
import Mathlib.Analysis.Convex.Combination
import Mathlib.Analysis.Convex.Extreme
#align_import analysis.convex.independent from "leanprover-community/mathlib"@"fefd8a38be7811574cd2ec2f77d3a393a407f112"
open scoped Classical
open Affine
open Finset Function
variable {𝕜 E ι : Type*}
section OrderedSemiring
va... | Mathlib/Analysis/Convex/Independent.lean | 158 | 166 | theorem convexIndependent_set_iff_not_mem_convexHull_diff {s : Set E} :
ConvexIndependent 𝕜 ((↑) : s → E) ↔ ∀ x ∈ s, x ∉ convexHull 𝕜 (s \ {x}) := by |
rw [convexIndependent_set_iff_inter_convexHull_subset]
constructor
· rintro hs x hxs hx
exact (hs _ Set.diff_subset ⟨hxs, hx⟩).2 (Set.mem_singleton _)
· rintro hs t ht x ⟨hxs, hxt⟩
by_contra h
exact hs _ hxs (convexHull_mono (Set.subset_diff_singleton ht h) hxt)
| 7 | 1,096.633158 | 2 | 1.8 | 5 | 1,891 |
import Mathlib.Data.PFunctor.Multivariate.Basic
#align_import data.qpf.multivariate.basic from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
universe u
open MvFunctor
class MvQPF {n : ℕ} (F : TypeVec.{u} n → Type*) [MvFunctor F] where
P : MvPFunctor.{u} n
abs : ∀ {α}, P α → F α
... | Mathlib/Data/QPF/Multivariate/Basic.lean | 164 | 177 | theorem mem_supp {α : TypeVec n} (x : F α) (i) (u : α i) :
u ∈ supp x i ↔ ∀ a f, abs ⟨a, f⟩ = x → u ∈ f i '' univ := by |
rw [supp]; dsimp; constructor
· intro h a f haf
have : LiftP (fun i u => u ∈ f i '' univ) x := by
rw [liftP_iff]
refine ⟨a, f, haf.symm, ?_⟩
intro i u
exact mem_image_of_mem _ (mem_univ _)
exact h this
intro h p; rw [liftP_iff]
rintro ⟨a, f, xeq, h'⟩
rcases h a f xeq.symm with... | 12 | 162,754.791419 | 2 | 1.666667 | 6 | 1,775 |
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
namespace Con... | Mathlib/Analysis/NormedSpace/ContinuousAffineMap.lean | 102 | 114 | theorem contLinear_eq_zero_iff_exists_const (f : P →ᴬ[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by |
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine ⟨fun h => ?_, fun h => ?_⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
intro q
refine ⟨fun ... | 11 | 59,874.141715 | 2 | 1 | 4 | 935 |
import Mathlib.Topology.EMetricSpace.Paracompact
import Mathlib.Topology.Instances.ENNReal
import Mathlib.Analysis.Convex.PartitionOfUnity
#align_import topology.metric_space.partition_of_unity from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Topology ENNReal NNReal Filter Set Fu... | Mathlib/Topology/MetricSpace/PartitionOfUnity.lean | 42 | 61 | theorem eventually_nhds_zero_forall_closedBall_subset (hK : ∀ i, IsClosed (K i))
(hU : ∀ i, IsOpen (U i)) (hKU : ∀ i, K i ⊆ U i) (hfin : LocallyFinite K) (x : X) :
∀ᶠ p : ℝ≥0∞ × X in 𝓝 0 ×ˢ 𝓝 x, ∀ i, p.2 ∈ K i → closedBall p.2 p.1 ⊆ U i := by |
suffices ∀ i, x ∈ K i → ∀ᶠ p : ℝ≥0∞ × X in 𝓝 0 ×ˢ 𝓝 x, closedBall p.2 p.1 ⊆ U i by
apply mp_mem ((eventually_all_finite (hfin.point_finite x)).2 this)
(mp_mem (@tendsto_snd ℝ≥0∞ _ (𝓝 0) _ _ (hfin.iInter_compl_mem_nhds hK x)) _)
apply univ_mem'
rintro ⟨r, y⟩ hxy hyU i hi
simp only [mem_iInter... | 17 | 24,154,952.753575 | 2 | 1.75 | 4 | 1,871 |
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Set.Finite
#align_import combinatorics.hall.finite from "leanprover-community/mathlib"@"d6fad0e5bf2d6f48da9175d25c3dc5706b3834ce"
open Finset
universe u v
namespace HallMarriageTheorem
variable {ι : Type u} {α : Type v} [DecidableEq α] {t : ι → Finset α}
s... | Mathlib/Combinatorics/Hall/Finite.lean | 136 | 158 | theorem hall_cond_of_compl {ι : Type u} {t : ι → Finset α} {s : Finset ι}
(hus : s.card = (s.biUnion t).card) (ht : ∀ s : Finset ι, s.card ≤ (s.biUnion t).card)
(s' : Finset (sᶜ : Set ι)) : s'.card ≤ (s'.biUnion fun x' => t x' \ s.biUnion t).card := by |
haveI := Classical.decEq ι
have disj : Disjoint s (s'.image fun z => z.1) := by
simp only [disjoint_left, not_exists, mem_image, exists_prop, SetCoe.exists, exists_and_right,
exists_eq_right, Subtype.coe_mk]
intro x hx hc _
exact absurd hx hc
have : s'.card = (s ∪ s'.image fun z => z.1).card - ... | 20 | 485,165,195.40979 | 2 | 2 | 4 | 2,298 |
import Mathlib.Order.Interval.Set.Disjoint
import Mathlib.Order.SuccPred.Basic
#align_import data.set.intervals.monotone from "leanprover-community/mathlib"@"4d06b17aea8cf2e220f0b0aa46cd0231593c5c97"
open Set
section SuccOrder
open Order
variable {α β : Type*} [PartialOrder α]
theorem StrictMonoOn.Iic_id_le [... | Mathlib/Order/Interval/Set/Monotone.lean | 230 | 253 | theorem strictMonoOn_Iic_of_lt_succ [SuccOrder α] [IsSuccArchimedean α] {n : α}
(hψ : ∀ m, m < n → ψ m < ψ (succ m)) : StrictMonoOn ψ (Set.Iic n) := by |
intro x hx y hy hxy
obtain ⟨i, rfl⟩ := hxy.le.exists_succ_iterate
induction' i with k ih
· simp at hxy
cases' k with k
· exact hψ _ (lt_of_lt_of_le hxy hy)
rw [Set.mem_Iic] at *
simp only [Function.iterate_succ', Function.comp_apply] at ih hxy hy ⊢
by_cases hmax : IsMax (succ^[k] x)
· rw [succ_eq_i... | 22 | 3,584,912,846.131591 | 2 | 2 | 2 | 2,012 |
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 | 134 | 142 | theorem integrable_toReal_condDistrib (hX : AEMeasurable X μ) (hs : MeasurableSet s) :
Integrable (fun a => (condDistrib Y X μ (X a) s).toReal) μ := by |
refine integrable_toReal_of_lintegral_ne_top ?_ ?_
· exact Measurable.comp_aemeasurable (kernel.measurable_coe _ hs) hX
· refine ne_of_lt ?_
calc
∫⁻ a, condDistrib Y X μ (X a) s ∂μ ≤ ∫⁻ _, 1 ∂μ := lintegral_mono fun a => prob_le_one
_ = μ univ := lintegral_one
_ < ∞ := measure_lt_top _ _
| 7 | 1,096.633158 | 2 | 0.777778 | 9 | 692 |
import Mathlib.Algebra.MvPolynomial.Supported
import Mathlib.RingTheory.WittVector.Truncated
#align_import ring_theory.witt_vector.mul_coeff from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
namespace WittVector
variable (p : ℕ) [hp : Fact p.Prime]
variable {k ... | Mathlib/RingTheory/WittVector/MulCoeff.lean | 120 | 135 | theorem mul_polyOfInterest_aux1 (n : ℕ) :
∑ i ∈ range (n + 1), (p : 𝕄) ^ i * wittMul p i ^ p ^ (n - i) = wittPolyProd p n := by |
simp only [wittPolyProd]
convert wittStructureInt_prop p (X (0 : Fin 2) * X 1) n using 1
· simp only [wittPolynomial, wittMul]
rw [AlgHom.map_sum]
congr 1 with i
congr 1
have hsupp : (Finsupp.single i (p ^ (n - i))).support = {i} := by
rw [Finsupp.support_eq_singleton]
simp only [and_... | 14 | 1,202,604.284165 | 2 | 1.833333 | 6 | 1,912 |
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.Mul
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
#align_import analysis.mean_inequalities_pow from "leanprover-community/mathlib"@"ccdbfb6e5614667af5aa3ab2d50885e0ef44a46f"
universe u... | Mathlib/Analysis/MeanInequalitiesPow.lean | 101 | 110 | theorem arith_mean_le_rpow_mean (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i) (hw' : ∑ i ∈ s, w i = 1)
(hz : ∀ i ∈ s, 0 ≤ z i) {p : ℝ} (hp : 1 ≤ p) :
∑ i ∈ s, w i * z i ≤ (∑ i ∈ s, w i * z i ^ p) ^ (1 / p) := by |
have : 0 < p := by positivity
rw [← rpow_le_rpow_iff _ _ this, ← rpow_mul, one_div_mul_cancel (ne_of_gt this), rpow_one]
· exact rpow_arith_mean_le_arith_mean_rpow s w z hw hw' hz hp
all_goals
apply_rules [sum_nonneg, rpow_nonneg]
intro i hi
apply_rules [mul_nonneg, rpow_nonneg, hw i hi, hz i hi]
| 7 | 1,096.633158 | 2 | 2 | 2 | 2,374 |
import Mathlib.Analysis.Calculus.Deriv.Inv
import Mathlib.Analysis.Calculus.Deriv.Polynomial
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Analysis.SpecialFunctions.PolynomialExp
#align_import analysis.calculus.bump_function_inner from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9... | Mathlib/Analysis/SpecialFunctions/SmoothTransition.lean | 127 | 135 | theorem contDiff_polynomial_eval_inv_mul {n : ℕ∞} (p : ℝ[X]) :
ContDiff ℝ n (fun x ↦ p.eval x⁻¹ * expNegInvGlue x) := by |
apply contDiff_all_iff_nat.2 (fun m => ?_) n
induction m generalizing p with
| zero => exact contDiff_zero.2 <| continuous_polynomial_eval_inv_mul _
| succ m ihm =>
refine contDiff_succ_iff_deriv.2 ⟨differentiable_polynomial_eval_inv_mul _, ?_⟩
convert ihm (X ^ 2 * (p - derivative (R := ℝ) p)) using 2
... | 7 | 1,096.633158 | 2 | 1.166667 | 6 | 1,237 |
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 | 152 | 163 | theorem trans_refl_reparam (p : Path x₀ x₁) :
p.trans (Path.refl x₁) =
p.reparam (fun t => ⟨transReflReparamAux t, transReflReparamAux_mem_I t⟩) (by continuity)
(Subtype.ext transReflReparamAux_zero) (Subtype.ext transReflReparamAux_one) := by |
ext
unfold transReflReparamAux
simp only [Path.trans_apply, not_le, coe_reparam, Function.comp_apply, one_div, Path.refl_apply]
split_ifs
· rfl
· rfl
· simp
· simp
| 8 | 2,980.957987 | 2 | 1.166667 | 12 | 1,229 |
import Mathlib.RingTheory.OrzechProperty
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.PrincipalIdealDomain
#align_import linear_algebra.invariant_basis_number from "leanprover-community/mathlib"@"5fd3186f1ec30a75d5f65732e3ce5e623382556f"
noncomputable section
open Function
universe u v w
... | Mathlib/LinearAlgebra/InvariantBasisNumber.lean | 197 | 203 | theorem card_le_of_surjective' [RankCondition R] {α β : Type*} [Fintype α] [Fintype β]
(f : (α →₀ R) →ₗ[R] β →₀ R) (i : Surjective f) : Fintype.card β ≤ Fintype.card α := by |
let P := Finsupp.linearEquivFunOnFinite R R β
let Q := (Finsupp.linearEquivFunOnFinite R R α).symm
exact
card_le_of_surjective R ((P.toLinearMap.comp f).comp Q.toLinearMap)
((P.surjective.comp i).comp Q.surjective)
| 5 | 148.413159 | 2 | 2 | 5 | 2,178 |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
#align_import ring_theory.coprime.lemmas from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
universe u v
section IsCoprime
variable {R : Type ... | Mathlib/RingTheory/Coprime/Lemmas.lean | 94 | 108 | theorem Finset.prod_dvd_of_coprime :
(t : Set I).Pairwise (IsCoprime on s) → (∀ i ∈ t, s i ∣ z) → (∏ x ∈ t, s x) ∣ z := by |
classical
exact Finset.induction_on t (fun _ _ ↦ one_dvd z)
(by
intro a r har ih Hs Hs1
rw [Finset.prod_insert har]
have aux1 : a ∈ (↑(insert a r) : Set I) := Finset.mem_insert_self a r
refine
(IsCoprime.prod_right fun i hir ↦
Hs aux1 (Finset.mem_insert_of_mem hir)... | 13 | 442,413.392009 | 2 | 1.111111 | 18 | 1,195 |
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 | 118 | 153 | theorem gal_X_pow_sub_C_isSolvable_aux (n : ℕ) (a : F)
(h : (X ^ n - 1 : F[X]).Splits (RingHom.id F)) : IsSolvable (X ^ n - C a).Gal := by |
by_cases ha : a = 0
· rw [ha, C_0, sub_zero]
exact gal_X_pow_isSolvable n
have ha' : algebraMap F (X ^ n - C a).SplittingField a ≠ 0 :=
mt ((injective_iff_map_eq_zero _).mp (RingHom.injective _) a) ha
by_cases hn : n = 0
· rw [hn, pow_zero, ← C_1, ← C_sub]
exact gal_C_isSolvable (1 - a)
have hn... | 34 | 583,461,742,527,454.9 | 2 | 0.909091 | 11 | 788 |
import Batteries.Data.List.Lemmas
import Batteries.Tactic.Classical
import Mathlib.Tactic.TypeStar
import Mathlib.Mathport.Rename
#align_import data.list.tfae from "leanprover-community/mathlib"@"5a3e819569b0f12cbec59d740a2613018e7b8eec"
namespace List
def TFAE (l : List Prop) : Prop :=
∀ x ∈ l, ∀ y ∈ l, x ↔ ... | Mathlib/Data/List/TFAE.lean | 91 | 96 | theorem forall_tfae {α : Type*} (l : List (α → Prop)) (H : ∀ a : α, (l.map (fun p ↦ p a)).TFAE) :
(l.map (fun p ↦ ∀ a, p a)).TFAE := by |
simp only [TFAE, List.forall_mem_map_iff]
intros p₁ hp₁ p₂ hp₂
exact forall_congr' fun a ↦ H a (p₁ a) (mem_map_of_mem (fun p ↦ p a) hp₁)
(p₂ a) (mem_map_of_mem (fun p ↦ p a) hp₂)
| 4 | 54.59815 | 2 | 1.166667 | 6 | 1,233 |
import Mathlib.GroupTheory.Abelianization
import Mathlib.GroupTheory.Exponent
import Mathlib.GroupTheory.Transfer
#align_import group_theory.schreier from "leanprover-community/mathlib"@"8350c34a64b9bc3fc64335df8006bffcadc7baa6"
open scoped Pointwise
namespace Subgroup
open MemRightTransversals
variable {G : T... | Mathlib/GroupTheory/Schreier.lean | 37 | 58 | theorem closure_mul_image_mul_eq_top
(hR : R ∈ rightTransversals (H : Set G)) (hR1 : (1 : G) ∈ R) (hS : closure S = ⊤) :
(closure ((R * S).image fun g => g * (toFun hR g : G)⁻¹)) * R = ⊤ := by |
let f : G → R := fun g => toFun hR g
let U : Set G := (R * S).image fun g => g * (f g : G)⁻¹
change (closure U : Set G) * R = ⊤
refine top_le_iff.mp fun g _ => ?_
refine closure_induction_right ?_ ?_ ?_ (eq_top_iff.mp hS (mem_top g))
· exact ⟨1, (closure U).one_mem, 1, hR1, one_mul 1⟩
· rintro - - s hs ⟨... | 19 | 178,482,300.963187 | 2 | 1.6 | 5 | 1,726 |
import Mathlib.Analysis.NormedSpace.Multilinear.Basic
import Mathlib.LinearAlgebra.PiTensorProduct
universe uι u𝕜 uE uF
variable {ι : Type uι} [Fintype ι]
variable {𝕜 : Type u𝕜} [NontriviallyNormedField 𝕜]
variable {E : ι → Type uE} [∀ i, SeminormedAddCommGroup (E i)] [∀ i, NormedSpace 𝕜 (E i)]
variable {F : ... | Mathlib/Analysis/NormedSpace/PiTensorProduct/ProjectiveSeminorm.lean | 84 | 90 | theorem bddBelow_projectiveSemiNormAux (x : ⨂[𝕜] i, E i) :
BddBelow (Set.range (fun (p : lifts x) ↦ projectiveSeminormAux p.1)) := by |
existsi 0
rw [mem_lowerBounds]
simp only [Set.mem_range, Subtype.exists, exists_prop, forall_exists_index, and_imp,
forall_apply_eq_imp_iff₂]
exact fun p _ ↦ projectiveSeminormAux_nonneg p
| 5 | 148.413159 | 2 | 2 | 6 | 2,058 |
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Int.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.Cases
import Mathlib.Tactic.SimpRw
import Mathlib.Tactic.SplitIfs
#align_import algebra.group.basic from "leanprover-community/mathlib"@"a07d750983b94c530a... | Mathlib/Algebra/Group/Basic.lean | 1,426 | 1,432 | theorem multiplicative_of_isTotal (p : α → Prop) (hswap : ∀ {a b}, p a → p b → f a b * f b a = 1)
(hmul : ∀ {a b c}, r a b → r b c → p a → p b → p c → f a c = f a b * f b c) {a b c : α}
(pa : p a) (pb : p b) (pc : p c) : f a c = f a b * f b c := by |
apply multiplicative_of_symmetric_of_isTotal (fun a b => p a ∧ p b) r f fun _ _ => And.symm
· simp_rw [and_imp]; exact @hswap
· exact fun rab rbc pab _pbc pac => hmul rab rbc pab.1 pab.2 pac.2
exacts [⟨pa, pb⟩, ⟨pb, pc⟩, ⟨pa, pc⟩]
| 4 | 54.59815 | 2 | 0.333333 | 18 | 367 |
import Mathlib.CategoryTheory.Filtered.Basic
import Mathlib.Data.Set.Finite
import Mathlib.Data.Set.Subsingleton
import Mathlib.Topology.Category.TopCat.Limits.Konig
import Mathlib.Tactic.AdaptationNote
#align_import category_theory.cofiltered_system from "leanprover-community/mathlib"@"178a32653e369dce2da68dc6b2694e... | Mathlib/CategoryTheory/CofilteredSystem.lean | 158 | 163 | theorem IsMittagLeffler.subset_image_eventualRange (h : F.IsMittagLeffler) (f : j ⟶ i) :
F.eventualRange i ⊆ F.map f '' F.eventualRange j := by |
obtain ⟨k, g, hg⟩ := F.isMittagLeffler_iff_eventualRange.1 h j
rw [hg]; intro x hx
obtain ⟨x, rfl⟩ := F.mem_eventualRange_iff.1 hx (g ≫ f)
exact ⟨_, ⟨x, rfl⟩, by rw [map_comp_apply]⟩
| 4 | 54.59815 | 2 | 2 | 5 | 2,164 |
import Mathlib.CategoryTheory.Closed.Cartesian
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Adjunction.FullyFaithful
#align_import category_theory.closed.functor from "leanprover-community/mathlib"@"cea27692b3fdeb328a2ddba6aabf181754543184"
noncomputable secti... | Mathlib/CategoryTheory/Closed/Functor.lean | 107 | 116 | theorem expComparison_whiskerLeft {A A' : C} (f : A' ⟶ A) :
expComparison F A ≫ whiskerLeft _ (pre (F.map f)) =
whiskerRight (pre f) _ ≫ expComparison F A' := by |
ext B
dsimp
apply uncurry_injective
rw [uncurry_natural_left, uncurry_natural_left, uncurry_expComparison, uncurry_pre,
prod.map_swap_assoc, ← F.map_id, expComparison_ev, ← F.map_id, ←
prodComparison_inv_natural_assoc, ← prodComparison_inv_natural_assoc, ← F.map_comp, ←
F.map_comp, prod_map_pre_app... | 7 | 1,096.633158 | 2 | 1.142857 | 7 | 1,210 |
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.SchwartzSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.fourier.poisson_summation from "leanprover-community/mathlib"@"fd5... | Mathlib/Analysis/Fourier/PoissonSummation.lean | 56 | 103 | theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)}
(hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖)
(m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 f m := by |
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩
have neK : ∀ (K : Comp... | 45 | 34,934,271,057,485,095,000 | 2 | 2 | 3 | 2,059 |
import Mathlib.Algebra.Group.Hom.Defs
#align_import algebra.group.ext from "leanprover-community/mathlib"@"e574b1a4e891376b0ef974b926da39e05da12a06"
assert_not_exists MonoidWithZero
assert_not_exists DenselyOrdered
open Function
universe u
@[to_additive (attr := ext)]
theorem Monoid.ext {M : Type u} ⦃m₁ m₂ : Mo... | Mathlib/Algebra/Group/Ext.lean | 119 | 124 | theorem CancelCommMonoid.toCommMonoid_injective {M : Type u} :
Function.Injective (@CancelCommMonoid.toCommMonoid M) := by |
rintro @⟨@⟨@⟨⟩⟩⟩ @⟨@⟨@⟨⟩⟩⟩ h
congr <;> {
injection h with h'
injection h' }
| 4 | 54.59815 | 2 | 1.333333 | 6 | 1,372 |
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.Algebra.MonoidAlgebra.Support
import Mathlib.Algebra.DirectSum.Internal
import Mathlib.RingTheory.GradedAlgebra.Basic
#align_import algebra.monoid_algebra.grading from "leanprover-community/mathlib"@"feb99064803fd3108e37c18b0f77d0a8344677a3"
noncomputable sectio... | Mathlib/Algebra/MonoidAlgebra/Grading.lean | 72 | 78 | theorem mem_grade_iff' (m : M) (a : R[M]) :
a ∈ grade R m ↔ a ∈ (LinearMap.range (Finsupp.lsingle m : R →ₗ[R] M →₀ R) :
Submodule R R[M]) := by |
rw [mem_grade_iff, Finsupp.support_subset_singleton']
apply exists_congr
intro r
constructor <;> exact Eq.symm
| 4 | 54.59815 | 2 | 1.2 | 5 | 1,256 |
import Mathlib.Algebra.Algebra.Bilinear
import Mathlib.RingTheory.Localization.Basic
#align_import algebra.module.localized_module from "leanprover-community/mathlib"@"831c494092374cfe9f50591ed0ac81a25efc5b86"
section IsLocalizedModule
universe u v
variable {R : Type*} [CommSemiring R] (S : Submonoid R)
variabl... | Mathlib/Algebra/Module/LocalizedModule.lean | 574 | 588 | theorem IsLocalizedModule.of_linearEquiv (e : M' ≃ₗ[R] M'') [hf : IsLocalizedModule S f] :
IsLocalizedModule S (e ∘ₗ f : M →ₗ[R] M'') where
map_units s := by |
rw [show algebraMap R (Module.End R M'') s = e ∘ₗ (algebraMap R (Module.End R M') s) ∘ₗ e.symm
by ext; simp, Module.End_isUnit_iff, LinearMap.coe_comp, LinearMap.coe_comp,
LinearEquiv.coe_coe, LinearEquiv.coe_coe, EquivLike.comp_bijective, EquivLike.bijective_comp]
exact (Module.End_isUnit_iff _).m... | 12 | 162,754.791419 | 2 | 1 | 7 | 797 |
import Mathlib.Algebra.Quaternion
import Mathlib.Tactic.Ring
#align_import algebra.quaternion_basis from "leanprover-community/mathlib"@"3aa5b8a9ed7a7cabd36e6e1d022c9858ab8a8c2d"
open Quaternion
namespace QuaternionAlgebra
structure Basis {R : Type*} (A : Type*) [CommRing R] [Ring A] [Algebra R A] (c₁ c₂ : R) ... | Mathlib/Algebra/QuaternionBasis.lean | 125 | 135 | theorem lift_mul (x y : ℍ[R,c₁,c₂]) : q.lift (x * y) = q.lift x * q.lift y := by |
simp only [lift, Algebra.algebraMap_eq_smul_one]
simp_rw [add_mul, mul_add, smul_mul_assoc, mul_smul_comm, one_mul, mul_one, smul_smul]
simp only [i_mul_i, j_mul_j, i_mul_j, j_mul_i, i_mul_k, k_mul_i, k_mul_j, j_mul_k, k_mul_k]
simp only [smul_smul, smul_neg, sub_eq_add_neg, add_smul, ← add_assoc, mul_neg, neg... | 10 | 22,026.465795 | 2 | 0.4 | 10 | 394 |
import Mathlib.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.Content
#align_import field_theory.ratfunc from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
universe u v
noncompu... | Mathlib/FieldTheory/RatFunc/Basic.lean | 235 | 239 | theorem mk_smul (c : R) (p q : K[X]) : RatFunc.mk (c • p) q = c • RatFunc.mk p q := by |
by_cases hq : q = 0
· rw [hq, mk_zero, mk_zero, ← ofFractionRing_smul, smul_zero]
· rw [mk_eq_localization_mk _ hq, mk_eq_localization_mk _ hq, ← Localization.smul_mk, ←
ofFractionRing_smul]
| 4 | 54.59815 | 2 | 0.416667 | 12 | 404 |
import Mathlib.CategoryTheory.Abelian.Opposite
import Mathlib.CategoryTheory.Abelian.Homology
import Mathlib.Algebra.Homology.Additive
import Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex
#align_import algebra.homology.opposite from "leanprover-community/mathlib"@"8c75ef3517d4106e89fe524e6281d0b0545f47fc"
... | Mathlib/Algebra/Homology/Opposite.lean | 40 | 50 | theorem imageToKernel_op {X Y Z : V} (f : X ⟶ Y) (g : Y ⟶ Z) (w : f ≫ g = 0) :
imageToKernel g.op f.op (by rw [← op_comp, w, op_zero]) =
(imageSubobjectIso _ ≪≫ (imageOpOp _).symm).hom ≫
(cokernel.desc f (factorThruImage g)
(by rw [← cancel_mono (image.ι g), Category.assoc, image.fac, w,... |
ext
simp only [Iso.trans_hom, Iso.symm_hom, Iso.trans_inv, kernelOpOp_inv, Category.assoc,
imageToKernel_arrow, kernelSubobject_arrow', kernel.lift_ι, ← op_comp, cokernel.π_desc,
← imageSubobject_arrow, ← imageUnopOp_inv_comp_op_factorThruImage g.op]
rfl
| 5 | 148.413159 | 2 | 2 | 2 | 2,303 |
import Mathlib.AlgebraicGeometry.Morphisms.RingHomProperties
import Mathlib.RingTheory.RingHom.FiniteType
#align_import algebraic_geometry.morphisms.finite_type from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
noncomputable section
open CategoryTheory CategoryTheory.Limits Opposite ... | Mathlib/AlgebraicGeometry/Morphisms/FiniteType.lean | 65 | 71 | theorem locallyOfFiniteTypeOfComp {X Y Z : Scheme} (f : X ⟶ Y) (g : Y ⟶ Z)
[hf : LocallyOfFiniteType (f ≫ g)] : LocallyOfFiniteType f := by |
revert hf
rw [locallyOfFiniteType_eq]
apply RingHom.finiteType_is_local.affineLocally_of_comp
introv H
exact RingHom.FiniteType.of_comp_finiteType H
| 5 | 148.413159 | 2 | 1.5 | 2 | 1,631 |
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.Analysis.MeanInequalities
import Mathlib.Analysis.MeanInequalitiesPow
import Mathlib.MeasureTheory.Function.SpecialFunctions.Basic
#align_import measure_theory.integral.mean_inequalities from "leanprover-community/mathlib"@"13bf7613c96a9fd66a81b9020a82cad9... | Mathlib/MeasureTheory/Integral/MeanInequalities.lean | 93 | 98 | theorem funMulInvSnorm_rpow {p : ℝ} (hp0 : 0 < p) {f : α → ℝ≥0∞} {a : α} :
funMulInvSnorm f p μ a ^ p = f a ^ p * (∫⁻ c, f c ^ p ∂μ)⁻¹ := by |
rw [funMulInvSnorm, mul_rpow_of_nonneg _ _ (le_of_lt hp0)]
suffices h_inv_rpow : ((∫⁻ c : α, f c ^ p ∂μ) ^ (1 / p))⁻¹ ^ p = (∫⁻ c : α, f c ^ p ∂μ)⁻¹ by
rw [h_inv_rpow]
rw [inv_rpow, ← rpow_mul, one_div_mul_cancel hp0.ne', rpow_one]
| 4 | 54.59815 | 2 | 1.4 | 5 | 1,491 |
import Mathlib.Data.Finset.Lattice
import Mathlib.Order.Hom.Basic
import Mathlib.Data.Set.Finite
import Mathlib.Order.ConditionallyCompleteLattice.Basic
#align_import order.partial_sups from "leanprover-community/mathlib"@"d6fad0e5bf2d6f48da9175d25c3dc5706b3834ce"
variable {α : Type*}
section SemilatticeSup
var... | Mathlib/Order/PartialSups.lean | 97 | 101 | theorem Monotone.partialSups_eq {f : ℕ → α} (hf : Monotone f) : (partialSups f : ℕ → α) = f := by |
ext n
induction' n with n ih
· rfl
· rw [partialSups_succ, ih, sup_eq_right.2 (hf (Nat.le_succ _))]
| 4 | 54.59815 | 2 | 2 | 1 | 1,993 |
import Mathlib.Data.List.Count
import Mathlib.Data.List.Dedup
import Mathlib.Data.List.InsertNth
import Mathlib.Data.List.Lattice
import Mathlib.Data.List.Permutation
import Mathlib.Data.Nat.Factorial.Basic
#align_import data.list.perm from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
... | Mathlib/Data/List/Perm.lean | 167 | 175 | theorem forall₂_comp_perm_eq_perm_comp_forall₂ : Forall₂ r ∘r Perm = Perm ∘r Forall₂ r := by |
funext l₁ l₃; apply propext
constructor
· intro h
rcases h with ⟨l₂, h₁₂, h₂₃⟩
have : Forall₂ (flip r) l₂ l₁ := h₁₂.flip
rcases perm_comp_forall₂ h₂₃.symm this with ⟨l', h₁, h₂⟩
exact ⟨l', h₂.symm, h₁.flip⟩
· exact fun ⟨l₂, h₁₂, h₂₃⟩ => perm_comp_forall₂ h₁₂ h₂₃
| 8 | 2,980.957987 | 2 | 2 | 3 | 2,228 |
import Mathlib.Algebra.Module.DedekindDomain
import Mathlib.LinearAlgebra.FreeModule.PID
import Mathlib.Algebra.Module.Projective
import Mathlib.Algebra.Category.ModuleCat.Biproducts
import Mathlib.RingTheory.SimpleModule
#align_import algebra.module.pid from "leanprover-community/mathlib"@"cdc34484a07418af43daf8198b... | Mathlib/Algebra/Module/PID.lean | 75 | 84 | theorem Submodule.isInternal_prime_power_torsion_of_pid [Module.Finite R M]
(hM : Module.IsTorsion R M) :
DirectSum.IsInternal fun p : (factors (⊤ : Submodule R M).annihilator).toFinset =>
torsionBy R M
(IsPrincipal.generator (p : Ideal R) ^
(factors (⊤ : Submodule R M).annihilator).coun... |
convert isInternal_prime_power_torsion hM
ext p : 1
rw [← torsionBySet_span_singleton_eq, Ideal.submodule_span_eq, ← Ideal.span_singleton_pow,
Ideal.span_singleton_generator]
| 4 | 54.59815 | 2 | 2 | 5 | 2,030 |
import Mathlib.Algebra.Polynomial.Module.Basic
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv.Defs
import Mathlib.Analysis.Calculus.MeanValue
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
open scoped... | Mathlib/Analysis/Calculus/Taylor.lean | 141 | 146 | theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by |
simp_rw [sub_eq_neg_add]
rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
| 4 | 54.59815 | 2 | 1.714286 | 7 | 1,841 |
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 | 118 | 169 | theorem minpoly_eq_pow {p : ℕ} [hprime : Fact p.Prime] (hdiv : ¬p ∣ n) :
minpoly ℤ μ = minpoly ℤ (μ ^ p) := by |
classical
by_cases hn : n = 0
· simp_all
have hpos := Nat.pos_of_ne_zero hn
by_contra hdiff
set P := minpoly ℤ μ
set Q := minpoly ℤ (μ ^ p)
have Pmonic : P.Monic := minpoly.monic (h.isIntegral hpos)
have Qmonic : Q.Monic := minpoly.monic ((h.pow_of_prime hprime.1 hdiv).isIntegral hpos)
have Pirr : ... | 50 | 5,184,705,528,587,073,000,000 | 2 | 2 | 6 | 2,240 |
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Ext
local macro:max "local_hAdd[" type:term ", " inst:term "]" : term =>
`(term| (letI := $inst; HAdd.hAdd : $type → $type → $type))
local macro:max "local_hMul[" type:term ", " inst:term "]" : term =>
`(term| (letI := $inst; HMul.hMul : $type → $typ... | Mathlib/Algebra/Ring/Ext.lean | 195 | 201 | theorem toNonUnitalNonAssocSemiring_injective :
Function.Injective (@toNonUnitalNonAssocSemiring R) := by |
intro _ _ h
-- Use above extensionality lemma to prove injectivity by showing that `h_add` and `h_mul` hold.
ext x y
· exact congrArg (·.toAdd.add x y) h
· exact congrArg (·.toMul.mul x y) h
| 5 | 148.413159 | 2 | 0.4 | 10 | 397 |
import Mathlib.MeasureTheory.Integral.IntervalIntegral
import Mathlib.Data.Set.Function
#align_import analysis.sum_integral_comparisons from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Set MeasureTheory.MeasureSpace
variable {x₀ : ℝ} {a b : ℕ} {f : ℝ → ℝ}
| Mathlib/Analysis/SumIntegralComparisons.lean | 47 | 70 | theorem AntitoneOn.integral_le_sum (hf : AntitoneOn f (Icc x₀ (x₀ + a))) :
(∫ x in x₀..x₀ + a, f x) ≤ ∑ i ∈ Finset.range a, f (x₀ + i) := by |
have hint : ∀ k : ℕ, k < a → IntervalIntegrable f volume (x₀ + k) (x₀ + (k + 1 : ℕ)) := by
intro k hk
refine (hf.mono ?_).intervalIntegrable
rw [uIcc_of_le]
· apply Icc_subset_Icc
· simp only [le_add_iff_nonneg_right, Nat.cast_nonneg]
· simp only [add_le_add_iff_left, Nat.cast_le, Nat.suc... | 22 | 3,584,912,846.131591 | 2 | 2 | 4 | 2,208 |
import Mathlib.AlgebraicTopology.DoldKan.Faces
import Mathlib.CategoryTheory.Idempotents.Basic
#align_import algebraic_topology.dold_kan.projections from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190f0b504"
open CategoryTheory CategoryTheory.Category CategoryTheory.Limits CategoryTheory.Pread... | Mathlib/AlgebraicTopology/DoldKan/Projections.lean | 61 | 65 | theorem P_f_0_eq (q : ℕ) : ((P q).f 0 : X _[0] ⟶ X _[0]) = 𝟙 _ := by |
induction' q with q hq
· rfl
· simp only [P_succ, HomologicalComplex.add_f_apply, HomologicalComplex.comp_f,
HomologicalComplex.id_f, id_comp, hq, Hσ_eq_zero, add_zero]
| 4 | 54.59815 | 2 | 1.2 | 5 | 1,271 |
import Mathlib.Analysis.InnerProductSpace.TwoDim
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
#align_import geometry.euclidean.angle.oriented.basic from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
noncomputable section
open FiniteDimensional Complex
open scoped Real Rea... | Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | 78 | 82 | theorem oangle_self (x : V) : o.oangle x x = 0 := by |
rw [oangle, kahler_apply_self, ← ofReal_pow]
convert QuotientAddGroup.mk_zero (AddSubgroup.zmultiples (2 * π))
apply arg_ofReal_of_nonneg
positivity
| 4 | 54.59815 | 2 | 0.571429 | 7 | 521 |
import Mathlib.Data.Int.GCD
import Mathlib.Tactic.NormNum
namespace Tactic
namespace NormNum
| Mathlib/Tactic/NormNum/GCD.lean | 22 | 28 | theorem int_gcd_helper' {d : ℕ} {x y : ℤ} (a b : ℤ) (h₁ : (d : ℤ) ∣ x) (h₂ : (d : ℤ) ∣ y)
(h₃ : x * a + y * b = d) : Int.gcd x y = d := by |
refine Nat.dvd_antisymm ?_ (Int.natCast_dvd_natCast.1 (Int.dvd_gcd h₁ h₂))
rw [← Int.natCast_dvd_natCast, ← h₃]
apply dvd_add
· exact Int.gcd_dvd_left.mul_right _
· exact Int.gcd_dvd_right.mul_right _
| 5 | 148.413159 | 2 | 1 | 4 | 950 |
import Mathlib.RingTheory.Adjoin.FG
#align_import ring_theory.adjoin.tower from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open Pointwise
universe u v w u₁
variable (R : Type u) (S : Type v) (A : Type w) (B : Type u₁)
namespace Algebra
theorem adjoin_restrictScalars (C D E : Typ... | Mathlib/RingTheory/Adjoin/Tower.lean | 49 | 58 | theorem adjoin_res_eq_adjoin_res (C D E F : Type*) [CommSemiring C] [CommSemiring D]
[CommSemiring E] [CommSemiring F] [Algebra C D] [Algebra C E] [Algebra C F] [Algebra D F]
[Algebra E F] [IsScalarTower C D F] [IsScalarTower C E F] {S : Set D} {T : Set E}
(hS : Algebra.adjoin C S = ⊤) (hT : Algebra.adjoin ... |
rw [adjoin_restrictScalars C E, adjoin_restrictScalars C D, ← hS, ← hT, ← Algebra.adjoin_image,
← Algebra.adjoin_image, ← AlgHom.coe_toRingHom, ← AlgHom.coe_toRingHom,
IsScalarTower.coe_toAlgHom, IsScalarTower.coe_toAlgHom, ← adjoin_union_eq_adjoin_adjoin, ←
adjoin_union_eq_adjoin_adjoin, Set.union_comm]... | 4 | 54.59815 | 2 | 2 | 3 | 2,229 |
import Mathlib.LinearAlgebra.CliffordAlgebra.Fold
import Mathlib.LinearAlgebra.CliffordAlgebra.Grading
#align_import linear_algebra.clifford_algebra.even from "leanprover-community/mathlib"@"9264b15ee696b7ca83f13c8ad67c83d6eb70b730"
namespace CliffordAlgebra
-- Porting note: explicit universes
universe uR uM uA ... | Mathlib/LinearAlgebra/CliffordAlgebra/Even.lean | 116 | 128 | theorem even.algHom_ext ⦃f g : even Q →ₐ[R] A⦄ (h : (even.ι Q).compr₂ f = (even.ι Q).compr₂ g) :
f = g := by |
rw [EvenHom.ext_iff] at h
ext ⟨x, hx⟩
induction x, hx using even_induction with
| algebraMap r =>
exact (f.commutes r).trans (g.commutes r).symm
| add x y hx hy ihx ihy =>
have := congr_arg₂ (· + ·) ihx ihy
exact (f.map_add _ _).trans (this.trans <| (g.map_add _ _).symm)
| ι_mul_ι_mul m₁ m₂ x h... | 11 | 59,874.141715 | 2 | 2 | 1 | 1,969 |
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Data.Rat.Cast.Order
import Mathlib.Order.Partition.Finpartition
import Mathlib.Tactic.GCongr
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.Positivity
import Mathlib.Tactic.Ring
#align_import combinatorics.simp... | Mathlib/Combinatorics/SimpleGraph/Density.lean | 115 | 120 | theorem interedges_biUnion_right (s : Finset α) (t : Finset ι) (f : ι → Finset β) :
interedges r s (t.biUnion f) = t.biUnion fun b ↦ interedges r s (f b) := by |
ext a
simp only [mem_interedges_iff, mem_biUnion]
exact ⟨fun ⟨x₁, ⟨x₂, x₃, x₄⟩, x₅⟩ ↦ ⟨x₂, x₃, x₁, x₄, x₅⟩,
fun ⟨x₂, x₃, x₁, x₄, x₅⟩ ↦ ⟨x₁, ⟨x₂, x₃, x₄⟩, x₅⟩⟩
| 4 | 54.59815 | 2 | 0.785714 | 14 | 695 |
import Mathlib.Algebra.IsPrimePow
import Mathlib.Data.Nat.Factorization.Basic
#align_import data.nat.factorization.prime_pow from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf97388c9e3b441fc4ccf3f"
variable {R : Type*} [CommMonoidWithZero R] (n p : R) (k : ℕ)
theorem IsPrimePow.minFac_pow_factorization_eq ... | Mathlib/Data/Nat/Factorization/PrimePow.lean | 89 | 108 | theorem isPrimePow_iff_unique_prime_dvd {n : ℕ} : IsPrimePow n ↔ ∃! p : ℕ, p.Prime ∧ p ∣ n := by |
rw [isPrimePow_nat_iff]
constructor
· rintro ⟨p, k, hp, hk, rfl⟩
refine ⟨p, ⟨hp, dvd_pow_self _ hk.ne'⟩, ?_⟩
rintro q ⟨hq, hq'⟩
exact (Nat.prime_dvd_prime_iff_eq hq hp).1 (hq.dvd_of_dvd_pow hq')
rintro ⟨p, ⟨hp, hn⟩, hq⟩
rcases eq_or_ne n 0 with (rfl | hn₀)
· cases (hq 2 ⟨Nat.prime_two, dvd_zero... | 19 | 178,482,300.963187 | 2 | 1.714286 | 7 | 1,840 |
import Mathlib.Data.Real.Basic
import Mathlib.Combinatorics.Pigeonhole
import Mathlib.Algebra.Order.EuclideanAbsoluteValue
#align_import number_theory.class_number.admissible_absolute_value from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
local infixl:50 " ≺ " => EuclideanDomain.r
na... | Mathlib/NumberTheory/ClassNumber/AdmissibleAbsoluteValue.lean | 117 | 123 | theorem exists_approx {ι : Type*} [Fintype ι] {ε : ℝ} (hε : 0 < ε) {b : R} (hb : b ≠ 0)
(h : abv.IsAdmissible) (A : Fin (h.card ε ^ Fintype.card ι).succ → ι → R) :
∃ i₀ i₁, i₀ ≠ i₁ ∧ ∀ k, (abv (A i₁ k % b - A i₀ k % b) : ℝ) < abv b • ε := by |
let e := Fintype.equivFin ι
obtain ⟨i₀, i₁, ne, h⟩ := h.exists_approx_aux (Fintype.card ι) hε hb fun x y ↦ A x (e.symm y)
refine ⟨i₀, i₁, ne, fun k ↦ ?_⟩
convert h (e k) <;> simp only [e.symm_apply_apply]
| 4 | 54.59815 | 2 | 2 | 3 | 2,197 |
import Mathlib.Init.Data.Sigma.Lex
import Mathlib.Data.Prod.Lex
import Mathlib.Data.Sigma.Lex
import Mathlib.Order.Antichain
import Mathlib.Order.OrderIsoNat
import Mathlib.Order.WellFounded
import Mathlib.Tactic.TFAE
#align_import order.well_founded_set from "leanprover-community/mathlib"@"2c84c2c5496117349007d97104... | Mathlib/Order/WellFoundedSet.lean | 101 | 108 | theorem wellFoundedOn_range : (range f).WellFoundedOn r ↔ WellFounded (r on f) := by |
let f' : β → range f := fun c => ⟨f c, c, rfl⟩
refine ⟨fun h => (InvImage.wf f' h).mono fun c c' => id, fun h => ⟨?_⟩⟩
rintro ⟨_, c, rfl⟩
refine Acc.of_downward_closed f' ?_ _ ?_
· rintro _ ⟨_, c', rfl⟩ -
exact ⟨c', rfl⟩
· exact h.apply _
| 7 | 1,096.633158 | 2 | 1.5 | 10 | 1,649 |
import Mathlib.CategoryTheory.Sites.IsSheafFor
import Mathlib.CategoryTheory.Limits.Shapes.Types
import Mathlib.Tactic.ApplyFun
#align_import category_theory.sites.sheaf_of_types from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe w v u
namespace CategoryTheory
open Opposite ... | Mathlib/CategoryTheory/Sites/EqualizerSheafCondition.lean | 216 | 223 | theorem w : forkMap P R ≫ firstMap P R = forkMap P R ≫ secondMap P R := by |
dsimp
ext fg
simp only [firstMap, secondMap, forkMap]
simp only [limit.lift_π, limit.lift_π_assoc, assoc, Fan.mk_π_app]
haveI := Presieve.hasPullbacks.has_pullbacks fg.1.2.2 fg.2.2.2
rw [← P.map_comp, ← op_comp, pullback.condition]
simp
| 7 | 1,096.633158 | 2 | 1.833333 | 6 | 1,913 |
import Mathlib.Probability.Kernel.Composition
#align_import probability.kernel.invariance from "leanprover-community/mathlib"@"3b92d54a05ee592aa2c6181a4e76b1bb7cc45d0b"
open MeasureTheory
open scoped MeasureTheory ENNReal ProbabilityTheory
namespace ProbabilityTheory
variable {α β γ : Type*} {mα : MeasurableSp... | Mathlib/Probability/Kernel/Invariance.lean | 87 | 92 | theorem Invariant.comp [IsSFiniteKernel κ] (hκ : Invariant κ μ) (hη : Invariant η μ) :
Invariant (κ ∘ₖ η) μ := by |
cases' isEmpty_or_nonempty α with _ hα
· exact Subsingleton.elim _ _
· simp_rw [Invariant, ← comp_const_apply_eq_bind (κ ∘ₖ η) μ hα.some, comp_assoc, hη.comp_const,
hκ.comp_const, const_apply]
| 4 | 54.59815 | 2 | 1 | 6 | 1,165 |
import Mathlib.Data.Multiset.Nodup
#align_import data.multiset.sum from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Sum
namespace Multiset
variable {α β : Type*} (s : Multiset α) (t : Multiset β)
def disjSum : Multiset (Sum α β) :=
s.map inl + t.map inr
#align multiset.dis... | Mathlib/Data/Multiset/Sum.lean | 64 | 69 | theorem inr_mem_disjSum : inr b ∈ s.disjSum t ↔ b ∈ t := by |
rw [mem_disjSum, or_iff_right]
-- Porting note: Previous code for L72 was: simp only [exists_eq_right]
· simp only [inr.injEq, exists_eq_right]
rintro ⟨a, _, ha⟩
exact inl_ne_inr ha
| 5 | 148.413159 | 2 | 1 | 4 | 1,085 |
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