Context stringlengths 57 6.04k | file_name stringlengths 21 79 | start int64 14 1.49k | end int64 18 1.5k | theorem stringlengths 25 1.55k | proof stringlengths 5 7.36k | num_lines int64 1 150 | complexity_score float64 2.72 139,370,958,066,637,970,000,000,000,000,000,000,000,000,000,000,000,000,000B | diff_level int64 0 2 | file_diff_level float64 0 2 | theorem_same_file int64 1 32 | rank_file int64 0 2.51k |
|---|---|---|---|---|---|---|---|---|---|---|---|
import Mathlib.Data.Set.Image
import Mathlib.Data.SProd
#align_import data.set.prod from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc9524185ae994553ebf4"
open Function
namespace Set
section Prod
variable {α β γ δ : Type*} {s s₁ s₂ : Set α} {t t₁ t₂ : Set β} {a : α} {b : β}
theorem Subsingleton.pro... | Mathlib/Data/Set/Prod.lean | 132 | 134 | theorem prod_union : s ×ˢ (t₁ ∪ t₂) = s ×ˢ t₁ ∪ s ×ˢ t₂ := by |
ext ⟨x, y⟩
simp [and_or_left]
| 2 | 7.389056 | 1 | 0.692308 | 13 | 638 |
import Mathlib.Analysis.Complex.Circle
import Mathlib.MeasureTheory.Group.Integral
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Haar.OfBasis
import Mathlib.MeasureTheory.Constructions.Prod.Integral
import Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace
import Mathlib.Alge... | Mathlib/Analysis/Fourier/FourierTransform.lean | 132 | 147 | theorem fourierIntegral_convergent_iff (he : Continuous e)
(hL : Continuous fun p : V × W ↦ L p.1 p.2) {f : V → E} (w : W) :
Integrable (fun v : V ↦ e (-L v w) • f v) μ ↔ Integrable f μ := by |
-- first prove one-way implication
have aux {g : V → E} (hg : Integrable g μ) (x : W) :
Integrable (fun v : V ↦ e (-L v x) • g v) μ := by
have c : Continuous fun v ↦ e (-L v x) :=
he.comp (hL.comp (continuous_prod_mk.mpr ⟨continuous_id, continuous_const⟩)).neg
simp_rw [← integrable_norm_iff (c.... | 13 | 442,413.392009 | 2 | 1.75 | 4 | 1,844 |
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 | 105 | 109 | theorem take_sum_join' (L : List (List α)) (i : ℕ) :
L.join.take (Nat.sum ((L.map length).take i)) = (L.take i).join := by |
induction L generalizing i
· simp
· cases i <;> simp [take_append, *]
| 3 | 20.085537 | 1 | 0.9 | 10 | 782 |
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.Data.Nat.Lattice
#align_import combinatorics.simple_graph.metric from "leanprover-community/mathlib"@"352ecfe114946c903338006dd3287cb5a9955ff2"
namespace SimpleGraph
variable {V : Type*} (G : SimpleGraph V)
noncomputable def dist (u v : V)... | Mathlib/Combinatorics/SimpleGraph/Metric.lean | 137 | 142 | theorem dist_eq_one_iff_adj {u v : V} : G.dist u v = 1 ↔ G.Adj u v := by |
refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩
· let ⟨w, hw⟩ := exists_walk_of_dist_ne_zero <| ne_zero_of_eq_one h
exact w.adj_of_length_eq_one <| h ▸ hw
· have : h.toWalk.length = 1 := Walk.length_cons _ _
exact ge_antisymm (h.reachable.pos_dist_of_ne h.ne) (this ▸ dist_le _)
| 5 | 148.413159 | 2 | 1 | 7 | 1,136 |
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Analysis.NormedSpace.OperatorNorm.Basic
import Mathlib.Analysis.NormedSpace.Pointwise
#align_import analysis.normed_space.is_R_or_C from "leanprover-community/mathlib"@"3f655f5297b030a87d641ad4e825af8d9679eb0b"
open Metric
variable {𝕜 : Type*} [RCLike 𝕜] {E :... | Mathlib/Analysis/NormedSpace/RCLike.lean | 43 | 45 | theorem norm_smul_inv_norm {x : E} (hx : x ≠ 0) : ‖(‖x‖⁻¹ : 𝕜) • x‖ = 1 := by |
have : ‖x‖ ≠ 0 := by simp [hx]
field_simp [norm_smul]
| 2 | 7.389056 | 1 | 1.2 | 5 | 1,253 |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Fintype.Card
#align_import data.multiset.fintype from "leanprover-community/mathlib"@"e3d9ab8faa9dea8f78155c6c27d62a621f4c152d"
variable {α : Type*} [DecidableEq α] {m : Multiset α}
def Multiset.ToType (m : Multiset α) : Type _ := (x : α) × Fi... | Mathlib/Data/Multiset/Fintype.lean | 130 | 141 | theorem Multiset.toEnumFinset_subset_iff {m₁ m₂ : Multiset α} :
m₁.toEnumFinset ⊆ m₂.toEnumFinset ↔ m₁ ≤ m₂ := by |
refine ⟨fun h ↦ ?_, Multiset.toEnumFinset_mono⟩
rw [Multiset.le_iff_count]
intro x
by_cases hx : x ∈ m₁
· apply Nat.le_of_pred_lt
have : (x, m₁.count x - 1) ∈ m₁.toEnumFinset := by
rw [Multiset.mem_toEnumFinset]
exact Nat.pred_lt (ne_of_gt (Multiset.count_pos.mpr hx))
simpa only [Multiset... | 10 | 22,026.465795 | 2 | 1.5 | 2 | 1,610 |
import Mathlib.Analysis.RCLike.Lemmas
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.function.l2_space from "leanprover-community/mathlib"@"83a66c8775fa14ee5180c85cab98e970956401ad"
set_option linter.uppercaseLean3 false... | Mathlib/MeasureTheory/Function/L2Space.lean | 124 | 140 | theorem snorm_inner_lt_top (f g : α →₂[μ] E) : snorm (fun x : α => ⟪f x, g x⟫) 1 μ < ∞ := by |
have h : ∀ x, ‖⟪f x, g x⟫‖ ≤ ‖‖f x‖ ^ (2 : ℝ) + ‖g x‖ ^ (2 : ℝ)‖ := by
intro x
rw [← @Nat.cast_two ℝ, Real.rpow_natCast, Real.rpow_natCast]
calc
‖⟪f x, g x⟫‖ ≤ ‖f x‖ * ‖g x‖ := norm_inner_le_norm _ _
_ ≤ 2 * ‖f x‖ * ‖g x‖ :=
(mul_le_mul_of_nonneg_right (le_mul_of_one_le_left (norm_non... | 15 | 3,269,017.372472 | 2 | 0.888889 | 9 | 770 |
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
#align_import analysis.special_functions.pow.asymptotics from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8"
set_option linter.uppercaseLean3 false
noncomputable section
open scoped Classical
open Real Topology NNReal ENNReal Filter C... | Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean | 259 | 266 | theorem IsBigOWith.rpow (h : IsBigOWith c l f g) (hc : 0 ≤ c) (hr : 0 ≤ r) (hg : 0 ≤ᶠ[l] g) :
IsBigOWith (c ^ r) l (fun x => f x ^ r) fun x => g x ^ r := by |
apply IsBigOWith.of_bound
filter_upwards [hg, h.bound] with x hgx hx
calc
|f x ^ r| ≤ |f x| ^ r := abs_rpow_le_abs_rpow _ _
_ ≤ (c * |g x|) ^ r := rpow_le_rpow (abs_nonneg _) hx hr
_ = c ^ r * |g x ^ r| := by rw [mul_rpow hc (abs_nonneg _), abs_rpow_of_nonneg hgx]
| 6 | 403.428793 | 2 | 1.5 | 10 | 1,605 |
import Mathlib.Analysis.SpecialFunctions.ImproperIntegrals
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.MeasureTheory.Measure.Haar.NormedSpace
#align_import analysis.mellin_transform from "leanprover-community/mathlib"@"917c3c072e487b3cccdbfeff17e75b40e45f66cb"
open MeasureTheory Set Filter A... | Mathlib/Analysis/MellinTransform.lean | 237 | 264 | theorem mellin_convergent_zero_of_isBigO {b : ℝ} {f : ℝ → ℝ}
(hfc : AEStronglyMeasurable f <| volume.restrict (Ioi 0))
(hf : f =O[𝓝[>] 0] (· ^ (-b))) {s : ℝ} (hs : b < s) :
∃ c : ℝ, 0 < c ∧ IntegrableOn (fun t : ℝ => t ^ (s - 1) * f t) (Ioc 0 c) := by |
obtain ⟨d, _, hd'⟩ := hf.exists_pos
simp_rw [IsBigOWith, eventually_nhdsWithin_iff, Metric.eventually_nhds_iff, gt_iff_lt] at hd'
obtain ⟨ε, hε, hε'⟩ := hd'
refine ⟨ε, hε, integrableOn_Ioc_iff_integrableOn_Ioo.mpr ⟨?_, ?_⟩⟩
· refine AEStronglyMeasurable.mul ?_ (hfc.mono_set Ioo_subset_Ioi_self)
refine (C... | 24 | 26,489,122,129.84347 | 2 | 1.333333 | 12 | 1,415 |
import Mathlib.CategoryTheory.Sites.Spaces
import Mathlib.Topology.Sheaves.Sheaf
import Mathlib.CategoryTheory.Sites.DenseSubsite
#align_import topology.sheaves.sheaf_condition.sites from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
noncomputable section
set_option linter.uppercaseLe... | Mathlib/Topology/Sheaves/SheafCondition/Sites.lean | 161 | 168 | theorem OpenEmbedding.compatiblePreserving (hf : OpenEmbedding f) :
CompatiblePreserving (Opens.grothendieckTopology Y) hf.isOpenMap.functor := by |
haveI : Mono f := (TopCat.mono_iff_injective f).mpr hf.inj
apply compatiblePreservingOfDownwardsClosed
intro U V i
refine ⟨(Opens.map f).obj V, eqToIso <| Opens.ext <| Set.image_preimage_eq_of_subset fun x h ↦ ?_⟩
obtain ⟨_, _, rfl⟩ := i.le h
exact ⟨_, rfl⟩
| 6 | 403.428793 | 2 | 1.6 | 5 | 1,729 |
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.Calculus.FDeriv.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
open Topology InnerProductSpace Set
noncomputable section
variable {𝕜 F : Type*} [RCLike 𝕜]
variable [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [CompleteSpace F]
variabl... | Mathlib/Analysis/Calculus/Gradient/Basic.lean | 118 | 121 | theorem DifferentiableAt.hasGradientAt (h : DifferentiableAt 𝕜 f x) :
HasGradientAt f (∇ f x) x := by |
rw [hasGradientAt_iff_hasFDerivAt, gradient, (toDual 𝕜 F).apply_symm_apply (fderiv 𝕜 f x)]
exact h.hasFDerivAt
| 2 | 7.389056 | 1 | 0.538462 | 13 | 511 |
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanprover-community/mathlib"@"00f91228655eecdcd3ac97a7fd8dbcb139fe990a"
universe u v w
open scoped Classical Polynomial
open Polynomial
vari... | Mathlib/FieldTheory/IsAlgClosed/Basic.lean | 104 | 111 | theorem roots_eq_zero_iff [IsAlgClosed k] {p : k[X]} :
p.roots = 0 ↔ p = Polynomial.C (p.coeff 0) := by |
refine ⟨fun h => ?_, fun hp => by rw [hp, roots_C]⟩
rcases le_or_lt (degree p) 0 with hd | hd
· exact eq_C_of_degree_le_zero hd
· obtain ⟨z, hz⟩ := IsAlgClosed.exists_root p hd.ne'
rw [← mem_roots (ne_zero_of_degree_gt hd), h] at hz
simp at hz
| 6 | 403.428793 | 2 | 1.5 | 6 | 1,683 |
import Mathlib.Data.Set.Image
import Mathlib.Data.List.InsertNth
import Mathlib.Init.Data.List.Lemmas
#align_import data.list.lemmas from "leanprover-community/mathlib"@"2ec920d35348cb2d13ac0e1a2ad9df0fdf1a76b4"
open List
variable {α β γ : Type*}
namespace List
theorem injOn_insertNth_index_of_not_mem (l : List... | Mathlib/Data/List/Lemmas.lean | 55 | 66 | theorem foldl_range_subset_of_range_subset {f : α → β → α} {g : α → γ → α}
(hfg : (Set.range fun a c => f c a) ⊆ Set.range fun b c => g c b) (a : α) :
Set.range (foldl f a) ⊆ Set.range (foldl g a) := by |
change (Set.range fun l => _) ⊆ Set.range fun l => _
-- Porting note: This was simply `simp_rw [← foldr_reverse]`
simp_rw [← foldr_reverse _ (fun z w => g w z), ← foldr_reverse _ (fun z w => f w z)]
-- Porting note: This `change` was not necessary in mathlib3
change (Set.range (foldr (fun z w => f w z) a ∘ r... | 9 | 8,103.083928 | 2 | 2 | 3 | 2,041 |
import Mathlib.Algebra.Module.Submodule.Basic
import Mathlib.Topology.Algebra.Monoid
import Mathlib.Analysis.Asymptotics.Asymptotics
import Mathlib.Algebra.Algebra.Pi
#align_import order.filter.zero_and_bounded_at_filter from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
namespace Filt... | Mathlib/Order/Filter/ZeroAndBoundedAtFilter.lean | 84 | 87 | theorem ZeroAtFilter.boundedAtFilter [NormedAddCommGroup β] {l : Filter α} {f : α → β}
(hf : ZeroAtFilter l f) : BoundedAtFilter l f := by |
rw [ZeroAtFilter, ← Asymptotics.isLittleO_const_iff (one_ne_zero' ℝ)] at hf
exact hf.isBigO
| 2 | 7.389056 | 1 | 0.5 | 2 | 476 |
import Mathlib.RepresentationTheory.Basic
import Mathlib.RepresentationTheory.FdRep
#align_import representation_theory.invariants from "leanprover-community/mathlib"@"55b3f8206b8596db8bb1804d8a92814a0b6670c9"
suppress_compilation
open MonoidAlgebra
open Representation
namespace GroupAlgebra
variable (k G : Ty... | Mathlib/RepresentationTheory/Invariants.lean | 43 | 48 | theorem mul_average_left (g : G) : ↑(Finsupp.single g 1) * average k G = average k G := by |
simp only [mul_one, Finset.mul_sum, Algebra.mul_smul_comm, average, MonoidAlgebra.of_apply,
Finset.sum_congr, MonoidAlgebra.single_mul_single]
set f : G → MonoidAlgebra k G := fun x => Finsupp.single x 1
show ⅟ (Fintype.card G : k) • ∑ x : G, f (g * x) = ⅟ (Fintype.card G : k) • ∑ x : G, f x
rw [Function.B... | 5 | 148.413159 | 2 | 2 | 3 | 2,112 |
import Mathlib.CategoryTheory.Monad.Types
import Mathlib.CategoryTheory.Monad.Limits
import Mathlib.CategoryTheory.Equivalence
import Mathlib.Topology.Category.CompHaus.Basic
import Mathlib.Topology.Category.Profinite.Basic
import Mathlib.Data.Set.Constructions
#align_import topology.category.Compactum from "leanprov... | Mathlib/Topology/Category/Compactum.lean | 158 | 162 | theorem join_distrib (X : Compactum) (uux : Ultrafilter (Ultrafilter X)) :
X.str (X.join uux) = X.str (map X.str uux) := by |
change ((β ).μ.app _ ≫ X.a) _ = _
rw [Monad.Algebra.assoc]
rfl
| 3 | 20.085537 | 1 | 1.25 | 4 | 1,307 |
import Mathlib.Data.Finset.Lattice
import Mathlib.Data.Set.Sigma
#align_import data.finset.sigma from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function Multiset
variable {ι : Type*}
namespace Finset
section Sigma
variable {α : ι → Type*} {β : Type*} (s s₁ s₂ : Finset ι) (... | Mathlib/Data/Finset/Sigma.lean | 91 | 94 | theorem sigma_eq_biUnion [DecidableEq (Σi, α i)] (s : Finset ι) (t : ∀ i, Finset (α i)) :
s.sigma t = s.biUnion fun i => (t i).map <| Embedding.sigmaMk i := by |
ext ⟨x, y⟩
simp [and_left_comm]
| 2 | 7.389056 | 1 | 1.214286 | 14 | 1,292 |
import Mathlib.NumberTheory.Cyclotomic.Discriminant
import Mathlib.RingTheory.Polynomial.Eisenstein.IsIntegral
import Mathlib.RingTheory.Ideal.Norm
#align_import number_theory.cyclotomic.rat from "leanprover-community/mathlib"@"b353176c24d96c23f0ce1cc63efc3f55019702d9"
universe u
open Algebra IsCyclotomicExtensio... | Mathlib/NumberTheory/Cyclotomic/Rat.lean | 38 | 43 | theorem discr_prime_pow_ne_two' [IsCyclotomicExtension {p ^ (k + 1)} ℚ K]
(hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))) (hk : p ^ (k + 1) ≠ 2) :
discr ℚ (hζ.subOnePowerBasis ℚ).basis =
(-1) ^ ((p ^ (k + 1) : ℕ).totient / 2) * p ^ ((p : ℕ) ^ k * ((p - 1) * (k + 1) - 1)) := by |
rw [← discr_prime_pow_ne_two hζ (cyclotomic.irreducible_rat (p ^ (k + 1)).pos) hk]
exact hζ.discr_zeta_eq_discr_zeta_sub_one.symm
| 2 | 7.389056 | 1 | 1.2 | 5 | 1,265 |
import Mathlib.Data.ZMod.Quotient
#align_import group_theory.complement from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf97388c9e3b441fc4ccf3f"
open Set
open scoped Pointwise
namespace Subgroup
variable {G : Type*} [Group G] (H K : Subgroup G) (S T : Set G)
@[to_additive "`S` and `T` are complements if ... | Mathlib/GroupTheory/Complement.lean | 124 | 128 | theorem isComplement_singleton_left {g : G} : IsComplement {g} S ↔ S = univ := by |
refine
⟨fun h => top_le_iff.mp fun x _ => ?_, fun h => (congr_arg _ h).mpr isComplement_singleton_univ⟩
obtain ⟨⟨⟨z, rfl : z = g⟩, y, _⟩, hy⟩ := h.2 (g * x)
rwa [← mul_left_cancel hy]
| 4 | 54.59815 | 2 | 2 | 3 | 2,365 |
import Mathlib.AlgebraicTopology.DoldKan.FunctorGamma
import Mathlib.AlgebraicTopology.DoldKan.SplitSimplicialObject
import Mathlib.CategoryTheory.Idempotents.HomologicalComplex
#align_import algebraic_topology.dold_kan.gamma_comp_n from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190f0b504"
no... | Mathlib/AlgebraicTopology/DoldKan/GammaCompN.lean | 105 | 108 | theorem N₁Γ₀_hom_app_f_f (K : ChainComplex C ℕ) (n : ℕ) :
(N₁Γ₀.hom.app K).f.f n = (Γ₀.splitting K).toKaroubiNondegComplexIsoN₁.inv.f.f n := by |
rw [N₁Γ₀_hom_app]
apply comp_id
| 2 | 7.389056 | 1 | 1.2 | 5 | 1,264 |
import Mathlib.Data.Nat.Cast.Basic
import Mathlib.Algebra.CharZero.Defs
import Mathlib.Algebra.Order.Group.Abs
import Mathlib.Data.Nat.Cast.NeZero
import Mathlib.Algebra.Order.Ring.Nat
#align_import data.nat.cast.basic from "leanprover-community/mathlib"@"acebd8d49928f6ed8920e502a6c90674e75bd441"
variable {α β : T... | Mathlib/Data/Nat/Cast/Order.lean | 80 | 83 | theorem cast_add_one_pos (n : ℕ) : 0 < (n : α) + 1 := by |
apply zero_lt_one.trans_le
convert (@mono_cast α _).imp (?_ : 1 ≤ n + 1)
<;> simp
| 3 | 20.085537 | 1 | 0.166667 | 6 | 261 |
import Mathlib.CategoryTheory.GlueData
import Mathlib.Topology.Category.TopCat.Limits.Pullbacks
import Mathlib.Topology.Category.TopCat.Opens
import Mathlib.Tactic.Generalize
import Mathlib.CategoryTheory.Elementwise
#align_import topology.gluing from "leanprover-community/mathlib"@"178a32653e369dce2da68dc6b2694e385d... | Mathlib/Topology/Gluing.lean | 132 | 158 | theorem rel_equiv : Equivalence D.Rel :=
⟨fun x => Or.inl (refl x), by
rintro a b (⟨⟨⟩⟩ | ⟨x, e₁, e₂⟩)
exacts [Or.inl rfl, Or.inr ⟨D.t _ _ x, e₂, by erw [← e₁, D.t_inv_apply]⟩], by
-- previous line now `erw` after #13170
rintro ⟨i, a⟩ ⟨j, b⟩ ⟨k, c⟩ (⟨⟨⟩⟩ | ⟨x, e₁, e₂⟩)
· exact id
rintro (⟨⟨⟩⟩... |
dsimp only [coe_of, z]
erw [pullbackIsoProdSubtype_inv_fst_apply, D.t_inv_apply]-- now `erw` after #13170
have eq₂ : (pullback.snd : _ ⟶ D.V _) z = y := pullbackIsoProdSubtype_inv_snd_apply _ _ _
clear_value z
right
use (pullback.fst : _ ⟶ D.V (i, k)) (D.t' _ _ _ z)
dsimp only at *
... | 16 | 8,886,110.520508 | 2 | 2 | 3 | 2,317 |
import Mathlib.Data.Complex.Module
import Mathlib.RingTheory.Norm
import Mathlib.RingTheory.Trace
#align_import ring_theory.complex from "leanprover-community/mathlib"@"9015c511549dc77a0f8d6eba021d8ac4bba20c82"
open Complex
| Mathlib/RingTheory/Complex.lean | 17 | 28 | theorem Algebra.leftMulMatrix_complex (z : ℂ) :
Algebra.leftMulMatrix Complex.basisOneI z = !![z.re, -z.im; z.im, z.re] := by |
ext i j
rw [Algebra.leftMulMatrix_eq_repr_mul, Complex.coe_basisOneI_repr, Complex.coe_basisOneI, mul_re,
mul_im, Matrix.of_apply]
fin_cases j
· simp only [Fin.mk_zero, Matrix.cons_val_zero, one_re, mul_one, one_im, mul_zero, sub_zero,
zero_add]
fin_cases i <;> rfl
· simp only [Fin.mk_one, Matr... | 10 | 22,026.465795 | 2 | 1.333333 | 3 | 1,455 |
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.MeasureTheory.Group.Measure
#align_import measure_theory.group.integration from "leanprover-community/mathlib"@"ec247d43814751ffceb33b758e8820df2372bf6f"
namespace MeasureTheory
open Measure TopologicalSpace
open scoped ENNReal
variable {𝕜 M α G E F ... | Mathlib/MeasureTheory/Group/Integral.lean | 79 | 83 | theorem integral_div_right_eq_self [IsMulRightInvariant μ] (f : G → E) (g : G) :
(∫ x, f (x / g) ∂μ) = ∫ x, f x ∂μ := by |
simp_rw [div_eq_mul_inv]
-- Porting note: was `simp_rw`
rw [integral_mul_right_eq_self f g⁻¹]
| 3 | 20.085537 | 1 | 0.75 | 8 | 660 |
import Mathlib.Algebra.IsPrimePow
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.Order.Hom.Bounded
import Mathlib.Algebra.GCDMonoid.Basic
#align_import ring_theory.chain_of_divisors from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
variable {M : Type*} [CancelCommMonoidWithZero... | Mathlib/RingTheory/ChainOfDivisors.lean | 91 | 95 | theorem first_of_chain_isUnit {q : Associates M} {n : ℕ} {c : Fin (n + 1) → Associates M}
(h₁ : StrictMono c) (h₂ : ∀ {r}, r ≤ q ↔ ∃ i, r = c i) : IsUnit (c 0) := by |
obtain ⟨i, hr⟩ := h₂.mp Associates.one_le
rw [Associates.isUnit_iff_eq_one, ← Associates.le_one_iff, hr]
exact h₁.monotone (Fin.zero_le i)
| 3 | 20.085537 | 1 | 1.8 | 5 | 1,895 |
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Topology.ContinuousFunction.Algebra
import Mathlib.Topology.Compactness.Paracompact
import Mathlib.Topology.ShrinkingLemma
import Mathlib.Topology.UrysohnsLemma
#align_import topology.partition_of_unity from "leanprover-... | Mathlib/Topology/PartitionOfUnity.lean | 193 | 196 | theorem coe_finsupport (x₀ : X) :
(ρ.finsupport x₀ : Set ι) = support fun i ↦ ρ i x₀ := by |
ext
rw [Finset.mem_coe, mem_finsupport]
| 2 | 7.389056 | 1 | 1.3 | 10 | 1,365 |
import Mathlib.Analysis.SpecialFunctions.PolarCoord
import Mathlib.Analysis.SpecialFunctions.Gamma.Basic
open Real Set MeasureTheory MeasureTheory.Measure
section real
theorem integral_rpow_mul_exp_neg_rpow {p q : ℝ} (hp : 0 < p) (hq : - 1 < q) :
∫ x in Ioi (0:ℝ), x ^ q * exp (- x ^ p) = (1 / p) * Gamma ((q +... | Mathlib/MeasureTheory/Integral/Gamma.lean | 59 | 63 | theorem integral_exp_neg_rpow {p : ℝ} (hp : 0 < p) :
∫ x in Ioi (0:ℝ), exp (- x ^ p) = Gamma (1 / p + 1) := by |
convert (integral_rpow_mul_exp_neg_rpow hp neg_one_lt_zero) using 1
· simp_rw [rpow_zero, one_mul]
· rw [zero_add, Gamma_add_one (one_div_ne_zero (ne_of_gt hp))]
| 3 | 20.085537 | 1 | 1.5 | 4 | 1,616 |
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Inverse
#align_import analysis.calculus.cont_diff from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputab... | Mathlib/Analysis/Calculus/ContDiff/Basic.lean | 140 | 145 | theorem iteratedFDerivWithin_const_of_ne {n : ℕ} (hn : n ≠ 0) (c : F)
(hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) :
iteratedFDerivWithin 𝕜 n (fun _ : E ↦ c) s x = 0 := by |
cases n with
| zero => contradiction
| succ n => exact iteratedFDerivWithin_succ_const n c hs hx
| 3 | 20.085537 | 1 | 0.875 | 8 | 760 |
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 | 84 | 89 | theorem isCompactOperator_iff_exists_mem_nhds_isCompact_closure_image [T2Space M₂] (f : M₁ → M₂) :
IsCompactOperator f ↔ ∃ V ∈ (𝓝 0 : Filter M₁), IsCompact (closure <| f '' V) := by |
rw [isCompactOperator_iff_exists_mem_nhds_image_subset_compact]
exact
⟨fun ⟨V, hV, K, hK, hKV⟩ => ⟨V, hV, hK.closure_of_subset hKV⟩,
fun ⟨V, hV, hVc⟩ => ⟨V, hV, closure (f '' V), hVc, subset_closure⟩⟩
| 4 | 54.59815 | 2 | 1.6 | 5 | 1,713 |
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.Polynomial.Roots
import Mathlib.GroupTheory.SpecificGroups.Cyclic
#align_import ring_theory.integral_domain from "leanprover-community/mathlib"@"6e70e0d419bf686784937d64ed4bfde866ff229e"
section
open Finset Polynomial Function Nat
variable {R : Type*} {G : Ty... | Mathlib/RingTheory/IntegralDomain.lean | 137 | 142 | theorem isCyclic_of_subgroup_isDomain [Finite G] (f : G →* R) (hf : Injective f) : IsCyclic G := by |
classical
cases nonempty_fintype G
apply isCyclic_of_card_pow_eq_one_le
intro n hn
exact le_trans (card_nthRoots_subgroup_units f hf hn 1) (card_nthRoots n (f 1))
| 5 | 148.413159 | 2 | 2 | 6 | 2,481 |
import Mathlib.Analysis.Analytic.Constructions
import Mathlib.Analysis.Calculus.Dslope
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Analysis.Analytic.Uniqueness
#align_import analysis.analytic.isolated_zeros from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
open sco... | Mathlib/Analysis/Analytic/IsolatedZeros.lean | 90 | 93 | theorem iterate_dslope_fslope_ne_zero (hp : HasFPowerSeriesAt f p z₀) (h : p ≠ 0) :
(swap dslope z₀)^[p.order] f z₀ ≠ 0 := by |
rw [← coeff_zero (has_fpower_series_iterate_dslope_fslope p.order hp) 1]
simpa [coeff_eq_zero] using apply_order_ne_zero h
| 2 | 7.389056 | 1 | 1.2 | 5 | 1,274 |
import Mathlib.MeasureTheory.Decomposition.RadonNikodym
import Mathlib.MeasureTheory.Measure.Haar.OfBasis
import Mathlib.Probability.Independence.Basic
#align_import probability.density from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open scoped Classical MeasureTheory NNReal ENNRea... | Mathlib/Probability/Density.lean | 158 | 164 | theorem hasPDF_of_pdf_ne_zero {m : MeasurableSpace Ω} {ℙ : Measure Ω} {μ : Measure E} {X : Ω → E}
(hac : map X ℙ ≪ μ) (hpdf : ¬pdf X ℙ μ =ᵐ[μ] 0) : HasPDF X ℙ μ := by |
refine ⟨?_, ?_, hac⟩
· exact aemeasurable_of_pdf_ne_zero X hpdf
· contrapose! hpdf
have := pdf_of_not_haveLebesgueDecomposition hpdf
filter_upwards using congrFun this
| 5 | 148.413159 | 2 | 1.428571 | 7 | 1,513 |
import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.Analysis.SumIntegralComparisons
import Mathlib.NumberTheory.Harmonic.Defs
| Mathlib/NumberTheory/Harmonic/Bounds.lean | 17 | 24 | theorem log_add_one_le_harmonic (n : ℕ) :
Real.log ↑(n+1) ≤ harmonic n := by |
calc _ = ∫ x in (1:ℕ)..↑(n+1), x⁻¹ := ?_
_ ≤ ∑ d ∈ Finset.Icc 1 n, (d:ℝ)⁻¹ := ?_
_ = harmonic n := ?_
· rw [Nat.cast_one, integral_inv (by simp [(show ¬ (1 : ℝ) ≤ 0 by norm_num)]), div_one]
· exact (inv_antitoneOn_Icc_right <| by norm_num).integral_le_sum_Ico (Nat.le_add_left 1 n)
· simp only [ha... | 6 | 403.428793 | 2 | 2 | 4 | 2,289 |
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 | 96 | 103 | theorem iSup_range_stdBasis_le_iInf_ker_proj (I J : Set ι) (h : Disjoint I J) :
⨆ i ∈ I, range (stdBasis R φ i) ≤ ⨅ i ∈ J, ker (proj i : (∀ i, φ i) →ₗ[R] φ i) := by |
refine iSup_le fun i => iSup_le fun hi => range_le_iff_comap.2 ?_
simp only [← ker_comp, eq_top_iff, SetLike.le_def, mem_ker, comap_iInf, mem_iInf]
rintro b - j hj
rw [proj_stdBasis_ne R φ j i, zero_apply]
rintro rfl
exact h.le_bot ⟨hi, hj⟩
| 6 | 403.428793 | 2 | 1.5 | 6 | 1,540 |
import Mathlib.RingTheory.AdjoinRoot
import Mathlib.FieldTheory.Minpoly.Field
import Mathlib.RingTheory.Polynomial.GaussLemma
#align_import field_theory.minpoly.is_integrally_closed from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open scoped Classical Polynomial
open Polynomial Set... | Mathlib/FieldTheory/Minpoly/IsIntegrallyClosed.lean | 75 | 92 | theorem isIntegrallyClosed_dvd {s : S} (hs : IsIntegral R s) {p : R[X]}
(hp : Polynomial.aeval s p = 0) : minpoly R s ∣ p := by |
let K := FractionRing R
let L := FractionRing S
let _ : Algebra K L := FractionRing.liftAlgebra R L
have := FractionRing.isScalarTower_liftAlgebra R L
have : minpoly K (algebraMap S L s) ∣ map (algebraMap R K) (p %ₘ minpoly R s) := by
rw [map_modByMonic _ (minpoly.monic hs), modByMonic_eq_sub_mul_div]
... | 16 | 8,886,110.520508 | 2 | 1.571429 | 7 | 1,708 |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.List.MinMax
import Mathlib.Algebra.Tropical.Basic
import Mathlib.Order.ConditionallyCompleteLattice.Finset
#align_import algebra.tropical.big_operators from "leanprover-community/mathlib"@"d6fad0e5bf2d6f48da9175d25c3dc5706b3834ce"
variable {R S :... | Mathlib/Algebra/Tropical/BigOperators.lean | 133 | 136 | theorem untrop_sum [ConditionallyCompleteLinearOrder R] [Fintype S] (f : S → Tropical (WithTop R)) :
untrop (∑ i : S, f i) = ⨅ i : S, untrop (f i) := by |
rw [iInf,← Set.image_univ,← coe_univ, untrop_sum_eq_sInf_image]
rfl
| 2 | 7.389056 | 1 | 0.928571 | 14 | 793 |
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Data.Finset.Sym
import Mathlib.Data.Nat.Choose.Cast
import Mathlib.Data.Nat.Choose.Multinomial
#align_import analysis.calculus.cont_diff from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputable section
open scoped C... | Mathlib/Analysis/Calculus/ContDiff/Bounds.lean | 40 | 122 | theorem ContinuousLinearMap.norm_iteratedFDerivWithin_le_of_bilinear_aux {Du Eu Fu Gu : Type u}
[NormedAddCommGroup Du] [NormedSpace 𝕜 Du] [NormedAddCommGroup Eu] [NormedSpace 𝕜 Eu]
[NormedAddCommGroup Fu] [NormedSpace 𝕜 Fu] [NormedAddCommGroup Gu] [NormedSpace 𝕜 Gu]
(B : Eu →L[𝕜] Fu →L[𝕜] Gu) {f : Du... |
/- We argue by induction on `n`. The bound is trivial for `n = 0`. For `n + 1`, we write
the `(n+1)`-th derivative as the `n`-th derivative of the derivative `B f g' + B f' g`,
and apply the inductive assumption to each of those two terms. For this induction to make sense,
the spaces of linear maps that ... | 75 | 373,324,199,679,900,150,000,000,000,000,000 | 2 | 2 | 1 | 2,007 |
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Data.Set.Function
import Mathlib.Order.Interval.Set.Basic
#align_import data.set.intervals.monoid from "leanprover-community/mathlib"@"aba57d4d3dae35460225919dcd82fe91355162f9"
namespace Set
variable {M : Type*} ... | Mathlib/Algebra/Order/Interval/Set/Monoid.lean | 35 | 41 | theorem Ioi_add_bij : BijOn (· + d) (Ioi a) (Ioi (a + d)) := by |
refine
⟨fun x h => add_lt_add_right (mem_Ioi.mp h) _, fun _ _ _ _ h => add_right_cancel h, fun _ h =>
?_⟩
obtain ⟨c, rfl⟩ := exists_add_of_le (mem_Ioi.mp h).le
rw [mem_Ioi, add_right_comm, add_lt_add_iff_right] at h
exact ⟨a + c, h, by rw [add_right_comm]⟩
| 6 | 403.428793 | 2 | 1.090909 | 11 | 1,187 |
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.Tactic.Linarith
#align_import combinatorics.simple_graph.acyclic from "leanprover-community/mathlib"@"b07688016d62f81d14508ff339ea3415558d6353"
universe u v
namespace SimpleGraph
open Walk
variable {V : Type u} (G : SimpleGraph V)
def IsAcy... | Mathlib/Combinatorics/SimpleGraph/Acyclic.lean | 68 | 80 | theorem isAcyclic_iff_forall_adj_isBridge :
G.IsAcyclic ↔ ∀ ⦃v w : V⦄, G.Adj v w → G.IsBridge s(v, w) := by |
simp_rw [isBridge_iff_adj_and_forall_cycle_not_mem]
constructor
· intro ha v w hvw
apply And.intro hvw
intro u p hp
cases ha p hp
· rintro hb v (_ | ⟨ha, p⟩) hp
· exact hp.not_of_nil
· apply (hb ha).2 _ hp
rw [Walk.edges_cons]
apply List.mem_cons_self
| 11 | 59,874.141715 | 2 | 1.6 | 5 | 1,737 |
import Mathlib.Data.Finset.Fin
import Mathlib.Data.Int.Order.Units
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.GroupTheory.Perm.Support
import Mathlib.Logic.Equiv.Fintype
#align_import group_theory.perm.sign from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
universe u v
o... | Mathlib/GroupTheory/Perm/Finite.lean | 132 | 163 | theorem mem_sumCongrHom_range_of_perm_mapsTo_inl {m n : Type*} [Finite m] [Finite n]
{σ : Perm (Sum m n)} (h : Set.MapsTo σ (Set.range Sum.inl) (Set.range Sum.inl)) :
σ ∈ (sumCongrHom m n).range := by |
classical
have h1 : ∀ x : Sum m n, (∃ a : m, Sum.inl a = x) → ∃ a : m, Sum.inl a = σ x := by
rintro x ⟨a, ha⟩
apply h
rw [← ha]
exact ⟨a, rfl⟩
have h3 : ∀ x : Sum m n, (∃ b : n, Sum.inr b = x) → ∃ b : n, Sum.inr b = σ x := by
rintro x ⟨b, hb⟩
apply (perm_mapsTo_inl_iff_map... | 29 | 3,931,334,297,144.042 | 2 | 2 | 5 | 2,072 |
import Mathlib.CategoryTheory.Sites.Spaces
import Mathlib.Topology.Sheaves.Sheaf
import Mathlib.CategoryTheory.Sites.DenseSubsite
#align_import topology.sheaves.sheaf_condition.sites from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
noncomputable section
set_option linter.uppercaseLe... | Mathlib/Topology/Sheaves/SheafCondition/Sites.lean | 137 | 144 | theorem coverDense_iff_isBasis [Category ι] (B : ι ⥤ Opens X) :
B.IsCoverDense (Opens.grothendieckTopology X) ↔ Opens.IsBasis (Set.range B.obj) := by |
rw [Opens.isBasis_iff_nbhd]
constructor
· intro hd U x hx; rcases hd.1 U x hx with ⟨V, f, ⟨i, f₁, f₂, _⟩, hV⟩
exact ⟨B.obj i, ⟨i, rfl⟩, f₁.le hV, f₂.le⟩
intro hb; constructor; intro U x hx; rcases hb hx with ⟨_, ⟨i, rfl⟩, hx, hi⟩
exact ⟨B.obj i, ⟨⟨hi⟩⟩, ⟨⟨i, 𝟙 _, ⟨⟨hi⟩⟩, rfl⟩⟩, hx⟩
| 6 | 403.428793 | 2 | 1.6 | 5 | 1,729 |
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Topology.Algebra.Field
import Mathlib.Topology.Algebra.Order.Group
#align_import topology.algebra.order.field from "leanprover-community/mathlib"@"9a59dcb7a2d06bf55da57b9030169219980660cd"
open Set Filter TopologicalSpace Function
open scoped Pointwise Top... | Mathlib/Topology/Algebra/Order/Field.lean | 102 | 105 | theorem Filter.Tendsto.atBot_mul_neg {C : 𝕜} (hC : C < 0) (hf : Tendsto f l atBot)
(hg : Tendsto g l (𝓝 C)) : Tendsto (fun x => f x * g x) l atTop := by |
have := (tendsto_neg_atBot_atTop.comp hf).atTop_mul_neg hC hg
simpa [(· ∘ ·)] using tendsto_neg_atBot_atTop.comp this
| 2 | 7.389056 | 1 | 0.666667 | 9 | 577 |
import Mathlib.Algebra.Associated
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Group.Abs
import Mathlib.Algebra.Ring.Divisibility.Basic
#align_import ring_theory.prime from "leanprover-community/mathlib"@"008205aa645b3f194c1da47025c5f110c8406eab"
section CancelCommMonoidWithZero
... | Mathlib/RingTheory/Prime.lean | 28 | 46 | theorem mul_eq_mul_prime_prod {α : Type*} [DecidableEq α] {x y a : R} {s : Finset α} {p : α → R}
(hp : ∀ i ∈ s, Prime (p i)) (hx : x * y = a * ∏ i ∈ s, p i) :
∃ (t u : Finset α) (b c : R),
t ∪ u = s ∧ Disjoint t u ∧ a = b * c ∧ (x = b * ∏ i ∈ t, p i) ∧ y = c * ∏ i ∈ u, p i := by |
induction' s using Finset.induction with i s his ih generalizing x y a
· exact ⟨∅, ∅, x, y, by simp [hx]⟩
· rw [prod_insert his, ← mul_assoc] at hx
have hpi : Prime (p i) := hp i (mem_insert_self _ _)
rcases ih (fun i hi ↦ hp i (mem_insert_of_mem hi)) hx with
⟨t, u, b, c, htus, htu, hbc, rfl, rfl⟩
... | 15 | 3,269,017.372472 | 2 | 1.5 | 4 | 1,602 |
import Mathlib.Analysis.Fourier.Inversion
open Real Complex Set MeasureTheory
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℂ E]
open scoped FourierTransform
private theorem rexp_neg_deriv_aux :
∀ x ∈ univ, HasDerivWithinAt (rexp ∘ Neg.neg) (-rexp (-x)) univ x :=
fun x _ ↦ mul_neg_one (rexp (-x)... | Mathlib/Analysis/MellinInversion.lean | 44 | 67 | theorem mellin_eq_fourierIntegral (f : ℝ → E) {s : ℂ} :
mellin f s = 𝓕 (fun (u : ℝ) ↦ (Real.exp (-s.re * u) • f (Real.exp (-u)))) (s.im / (2 * π)) :=
calc
mellin f s
= ∫ (u : ℝ), Complex.exp (-s * u) • f (Real.exp (-u)) := by |
rw [mellin, ← rexp_neg_image_aux, integral_image_eq_integral_abs_deriv_smul
MeasurableSet.univ rexp_neg_deriv_aux rexp_neg_injOn_aux]
simp [rexp_cexp_aux]
_ = ∫ (u : ℝ), Complex.exp (↑(-2 * π * (u * (s.im / (2 * π)))) * I) •
(Real.exp (-s.re * u) • f (Real.exp (-u))) := by
congr
... | 19 | 178,482,300.963187 | 2 | 2 | 3 | 1,966 |
import Mathlib.Data.List.Cycle
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.Perm.List
#align_import group_theory.perm.cycle.concrete from "leanprover-community/mathlib"@"00638177efd1b2534fc5269363ebf42a7871df9a"
open Equiv Equiv.Perm List
variable {α : Type*}
namespace List
variable [D... | Mathlib/GroupTheory/Perm/Cycle/Concrete.lean | 97 | 102 | theorem cycleOf_formPerm (hl : Nodup l) (hn : 2 ≤ l.length) (x) :
cycleOf l.attach.formPerm x = l.attach.formPerm :=
have hn : 2 ≤ l.attach.length := by | rwa [← length_attach] at hn
have hl : l.attach.Nodup := by rwa [← nodup_attach] at hl
(isCycle_formPerm hl hn).cycleOf_eq
((formPerm_apply_mem_ne_self_iff _ hl _ (mem_attach _ _)).mpr hn)
| 4 | 54.59815 | 2 | 1 | 18 | 1,030 |
import Mathlib.Analysis.Calculus.BumpFunction.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
#align_import analysis.calculus.bump_function_inner from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputable section
open F... | Mathlib/Analysis/Calculus/BumpFunction/Normed.lean | 75 | 77 | theorem integral_normed : ∫ x, f.normed μ x ∂μ = 1 := by |
simp_rw [ContDiffBump.normed, div_eq_mul_inv, mul_comm (f _), ← smul_eq_mul, integral_smul]
exact inv_mul_cancel f.integral_pos.ne'
| 2 | 7.389056 | 1 | 0.818182 | 11 | 722 |
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.Data.ZMod.Algebra
#align_import ring_theory.polynomial.cyclotomic.expand from "leanprover-community/mathlib"@"0723536a0522d24fc2f159a096fb3304bef77472"
namespace Polynomial
@[simp]
| Mathlib/RingTheory/Polynomial/Cyclotomic/Expand.lean | 36 | 72 | theorem cyclotomic_expand_eq_cyclotomic_mul {p n : ℕ} (hp : Nat.Prime p) (hdiv : ¬p ∣ n)
(R : Type*) [CommRing R] :
expand R p (cyclotomic n R) = cyclotomic (n * p) R * cyclotomic n R := by |
rcases Nat.eq_zero_or_pos n with (rfl | hnpos)
· simp
haveI := NeZero.of_pos hnpos
suffices expand ℤ p (cyclotomic n ℤ) = cyclotomic (n * p) ℤ * cyclotomic n ℤ by
rw [← map_cyclotomic_int, ← map_expand, this, Polynomial.map_mul, map_cyclotomic_int,
map_cyclotomic]
refine eq_of_monic_of_dvd_of_natDe... | 34 | 583,461,742,527,454.9 | 2 | 2 | 3 | 2,215 |
import Mathlib.Algebra.Lie.Submodule
#align_import algebra.lie.ideal_operations from "leanprover-community/mathlib"@"8983bec7cdf6cb2dd1f21315c8a34ab00d7b2f6d"
universe u v w w₁ w₂
namespace LieSubmodule
variable {R : Type u} {L : Type v} {M : Type w} {M₂ : Type w₁}
variable [CommRing R] [LieRing L] [LieAlgebra ... | Mathlib/Algebra/Lie/IdealOperations.lean | 96 | 100 | theorem lie_le_iff : ⁅I, N⁆ ≤ N' ↔ ∀ x ∈ I, ∀ m ∈ N, ⁅x, m⁆ ∈ N' := by |
rw [lieIdeal_oper_eq_span, LieSubmodule.lieSpan_le]
refine ⟨fun h x hx m hm => h ⟨⟨x, hx⟩, ⟨m, hm⟩, rfl⟩, ?_⟩
rintro h _ ⟨⟨x, hx⟩, ⟨m, hm⟩, rfl⟩
exact h x hx m hm
| 4 | 54.59815 | 2 | 1.285714 | 7 | 1,363 |
import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
#align_import algebra.order.monoid.min_max from "leanprover-community/mathlib"@"de87d5053a9fe5cbde723172c0fb7e27e7436473"
open Function
variable {α β : Type*}
section CovariantClassMulLe
variable [LinearOrder α]
section Mul
variable [Mul α]
@[to_additive... | Mathlib/Algebra/Order/Monoid/Unbundled/MinMax.lean | 90 | 94 | theorem lt_or_lt_of_mul_lt_mul [CovariantClass α α (· * ·) (· ≤ ·)]
[CovariantClass α α (Function.swap (· * ·)) (· ≤ ·)] {a₁ a₂ b₁ b₂ : α} :
a₁ * b₁ < a₂ * b₂ → a₁ < a₂ ∨ b₁ < b₂ := by |
contrapose!
exact fun h => mul_le_mul' h.1 h.2
| 2 | 7.389056 | 1 | 1.2 | 5 | 1,252 |
import Mathlib.RingTheory.WittVector.Identities
#align_import ring_theory.witt_vector.domain from "leanprover-community/mathlib"@"b1d911acd60ab198808e853292106ee352b648ea"
noncomputable section
open scoped Classical
namespace WittVector
open Function
variable {p : ℕ} {R : Type*}
local notation "𝕎" => WittVe... | Mathlib/RingTheory/WittVector/Domain.lean | 88 | 98 | theorem verschiebung_nonzero {x : 𝕎 R} (hx : x ≠ 0) :
∃ n : ℕ, ∃ x' : 𝕎 R, x'.coeff 0 ≠ 0 ∧ x = verschiebung^[n] x' := by |
have hex : ∃ k : ℕ, x.coeff k ≠ 0 := by
by_contra! hall
apply hx
ext i
simp only [hall, zero_coeff]
let n := Nat.find hex
use n, x.shift n
refine ⟨Nat.find_spec hex, eq_iterate_verschiebung fun i hi => not_not.mp ?_⟩
exact Nat.find_min hex hi
| 9 | 8,103.083928 | 2 | 2 | 3 | 2,323 |
import Mathlib.Algebra.ContinuedFractions.Computation.Basic
import Mathlib.Algebra.ContinuedFractions.Translations
#align_import algebra.continued_fractions.computation.translations from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b348ce40ad"
namespace GeneralizedContinuedFraction
open Generali... | Mathlib/Algebra/ContinuedFractions/Computation/Translations.lean | 163 | 165 | theorem of_h_eq_intFractPair_seq1_fst_b : (of v).h = (IntFractPair.seq1 v).fst.b := by |
cases aux_seq_eq : IntFractPair.seq1 v
simp [of, aux_seq_eq]
| 2 | 7.389056 | 1 | 1.307692 | 13 | 1,366 |
import Mathlib.Combinatorics.Enumerative.DoubleCounting
import Mathlib.Combinatorics.SimpleGraph.AdjMatrix
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Data.Set.Finite
#align_import combinatorics.simple_graph.strongly_regular from "leanprover-community/mathlib"@"2b35fc7bea4640cb75e477e83f32fbd5389208... | Mathlib/Combinatorics/SimpleGraph/StronglyRegular.lean | 84 | 95 | theorem IsSRGWith.card_neighborFinset_union_eq {v w : V} (h : G.IsSRGWith n k ℓ μ) :
(G.neighborFinset v ∪ G.neighborFinset w).card =
2 * k - Fintype.card (G.commonNeighbors v w) := by |
apply Nat.add_right_cancel (m := Fintype.card (G.commonNeighbors v w))
rw [Nat.sub_add_cancel, ← Set.toFinset_card]
-- Porting note: Set.toFinset_inter needs workaround to use unification to solve for one of the
-- instance arguments:
· simp [commonNeighbors, @Set.toFinset_inter _ _ _ _ _ _ (_),
← neig... | 9 | 8,103.083928 | 2 | 1.428571 | 7 | 1,524 |
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.Data.ZMod.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
#align_import ring_theory.zmod from "leanprover-community/mathlib"@"00d163e35035c3577c1c79fa53b68de17781ffc1"
theorem ZMod.ker_intCastRingHom (n : ℕ) :
RingHom.ker (Int.castRingHom (ZMod n)) =... | Mathlib/RingTheory/ZMod.lean | 33 | 37 | theorem ZMod.ringHom_eq_of_ker_eq {n : ℕ} {R : Type*} [CommRing R] (f g : R →+* ZMod n)
(h : RingHom.ker f = RingHom.ker g) : f = g := by |
have := f.liftOfRightInverse_comp _ (ZMod.ringHom_rightInverse f) ⟨g, le_of_eq h⟩
rw [Subtype.coe_mk] at this
rw [← this, RingHom.ext_zmod (f.liftOfRightInverse _ _ ⟨g, _⟩) _, RingHom.id_comp]
| 3 | 20.085537 | 1 | 1.333333 | 3 | 1,419 |
import Mathlib.NumberTheory.SmoothNumbers
import Mathlib.Analysis.PSeries
open Set Nat
open scoped Topology
-- This needs `Mathlib.Analysis.RCLike.Basic`, so we put it here
-- instead of in `Mathlib.NumberTheory.SmoothNumbers`.
lemma Nat.roughNumbersUpTo_card_le' (N k : ℕ) :
(roughNumbersUpTo N k).card ≤
... | Mathlib/NumberTheory/SumPrimeReciprocals.lean | 64 | 79 | theorem not_summable_one_div_on_primes :
¬ Summable (indicator {p | p.Prime} (fun n : ℕ ↦ (1 : ℝ) / n)) := by |
intro h
obtain ⟨k, hk⟩ := h.nat_tsum_vanishing (Iio_mem_nhds one_half_pos : Iio (1 / 2 : ℝ) ∈ 𝓝 0)
specialize hk ({p | Nat.Prime p} ∩ {p | k ≤ p}) inter_subset_right
rw [tsum_subtype, indicator_indicator, inter_eq_left.mpr fun n hn ↦ hn.1, mem_Iio] at hk
have h' : Summable (indicator ({p | Nat.Prime p} ∩ {p... | 14 | 1,202,604.284165 | 2 | 1.333333 | 3 | 1,446 |
import Mathlib.Geometry.Manifold.MFDeriv.Defs
#align_import geometry.manifold.mfderiv from "leanprover-community/mathlib"@"e473c3198bb41f68560cab68a0529c854b618833"
noncomputable section
open scoped Topology Manifold
open Set Bundle
section DerivativesProperties
variable
{𝕜 : Type*} [NontriviallyNormedFiel... | Mathlib/Geometry/Manifold/MFDeriv/Basic.lean | 46 | 49 | theorem uniqueMDiffWithinAt_univ : UniqueMDiffWithinAt I univ x := by |
unfold UniqueMDiffWithinAt
simp only [preimage_univ, univ_inter]
exact I.unique_diff _ (mem_range_self _)
| 3 | 20.085537 | 1 | 1 | 2 | 890 |
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 | 167 | 170 | theorem card_sum [Finite α] [Finite β] : Nat.card (α ⊕ β) = Nat.card α + Nat.card β := by |
have := Fintype.ofFinite α
have := Fintype.ofFinite β
simp_rw [Nat.card_eq_fintype_card, Fintype.card_sum]
| 3 | 20.085537 | 1 | 0.666667 | 3 | 595 |
import Mathlib.Topology.MetricSpace.HausdorffDistance
#align_import topology.metric_space.pi_nat from "leanprover-community/mathlib"@"49b7f94aab3a3bdca1f9f34c5d818afb253b3993"
noncomputable section
open scoped Classical
open Topology Filter
open TopologicalSpace Set Metric Filter Function
attribute [local simp... | Mathlib/Topology/MetricSpace/PiNat.lean | 168 | 172 | theorem cylinder_eq_cylinder_of_le_firstDiff (x y : ∀ n, E n) {n : ℕ} (hn : n ≤ firstDiff x y) :
cylinder x n = cylinder y n := by |
rw [← mem_cylinder_iff_eq]
intro i hi
exact apply_eq_of_lt_firstDiff (hi.trans_le hn)
| 3 | 20.085537 | 1 | 1.083333 | 12 | 1,184 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.Tactic.Abel
#align_import ring_theory.polynomial.pochhammer from "leanprover-community/mathlib"@"53b216bcc1146df1c4a0a868778... | Mathlib/RingTheory/Polynomial/Pochhammer.lean | 284 | 287 | theorem descPochhammer_eval_cast (n : ℕ) (k : ℤ) :
(((descPochhammer ℤ n).eval k : ℤ) : R) = ((descPochhammer R n).eval k : R) := by |
rw [← descPochhammer_map (algebraMap ℤ R), eval_map, ← eq_intCast (algebraMap ℤ R)]
simp only [algebraMap_int_eq, eq_intCast, eval₂_at_intCast, Nat.cast_id, eq_natCast, Int.cast_id]
| 2 | 7.389056 | 1 | 0.96 | 25 | 796 |
import Mathlib.RingTheory.FiniteType
#align_import ring_theory.rees_algebra from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open Polynomial
def reesAlgebra : Subalgebra... | Mathlib/RingTheory/ReesAlgebra.lean | 98 | 108 | theorem adjoin_monomial_eq_reesAlgebra :
Algebra.adjoin R (Submodule.map (monomial 1 : R →ₗ[R] R[X]) I : Set R[X]) = reesAlgebra I := by |
apply le_antisymm
· apply Algebra.adjoin_le _
rintro _ ⟨r, hr, rfl⟩
exact reesAlgebra.monomial_mem.mpr (by rwa [pow_one])
· intro p hp
rw [p.as_sum_support]
apply Subalgebra.sum_mem _ _
rintro i -
exact monomial_mem_adjoin_monomial (hp i)
| 9 | 8,103.083928 | 2 | 1.8 | 5 | 1,902 |
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.Order.Interval.Set.Group
import Mathlib.Analysis.Convex.Segment
import Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional
import Mathlib.Tactic.FieldSimp
#align_import analysis.convex.between from "leanprover-community/mathlib"@"571e13cacbed7bf042fd3058c... | Mathlib/Analysis/Convex/Between.lean | 67 | 74 | theorem affineSegment_same (x : P) : affineSegment R x x = {x} := by |
-- Porting note: added as this doesn't do anything in `simp_rw` any more
rw [affineSegment]
-- Note: when adding "simp made no progress" in lean4#2336,
-- had to change `lineMap_same` to `lineMap_same _`. Not sure why?
-- Porting note: added `_ _` and `Function.const`
simp_rw [lineMap_same _, AffineMap.coe... | 7 | 1,096.633158 | 2 | 0.625 | 8 | 545 |
import Mathlib.Algebra.Group.ConjFinite
import Mathlib.GroupTheory.Abelianization
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.GroupAction.Quotient
import Mathlib.GroupTheory.Index
import Mathlib.GroupTheory.SpecificGroups.Dihedral
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Line... | Mathlib/GroupTheory/CommutingProbability.lean | 47 | 52 | theorem commProb_prod (M' : Type*) [Mul M'] : commProb (M × M') = commProb M * commProb M' := by |
simp_rw [commProb_def, div_mul_div_comm, Nat.card_prod, Nat.cast_mul, mul_pow, ← Nat.cast_mul,
← Nat.card_prod, Commute, SemiconjBy, Prod.ext_iff]
congr 2
exact Nat.card_congr ⟨fun x => ⟨⟨⟨x.1.1.1, x.1.2.1⟩, x.2.1⟩, ⟨⟨x.1.1.2, x.1.2.2⟩, x.2.2⟩⟩,
fun x => ⟨⟨⟨x.1.1.1, x.2.1.1⟩, ⟨x.1.1.2, x.2.1.2⟩⟩, ⟨x.1.2,... | 5 | 148.413159 | 2 | 1.625 | 8 | 1,749 |
import Mathlib.Algebra.Order.Ring.Nat
#align_import data.nat.dist from "leanprover-community/mathlib"@"d50b12ae8e2bd910d08a94823976adae9825718b"
namespace Nat
def dist (n m : ℕ) :=
n - m + (m - n)
#align nat.dist Nat.dist
-- Should be aligned to `Nat.dist.eq_def`, but that is generated on demand and isn't pr... | Mathlib/Data/Nat/Dist.lean | 74 | 78 | theorem dist_add_add_right (n k m : ℕ) : dist (n + k) (m + k) = dist n m :=
calc
dist (n + k) (m + k) = n + k - (m + k) + (m + k - (n + k)) := rfl
_ = n - m + (m + k - (n + k)) := by | rw [@add_tsub_add_eq_tsub_right]
_ = n - m + (m - n) := by rw [@add_tsub_add_eq_tsub_right]
| 2 | 7.389056 | 1 | 0.266667 | 15 | 309 |
import Mathlib.Algebra.BigOperators.Finsupp
import Mathlib.Algebra.Module.Basic
import Mathlib.Algebra.Regular.SMul
import Mathlib.Data.Finset.Preimage
import Mathlib.Data.Rat.BigOperators
import Mathlib.GroupTheory.GroupAction.Hom
import Mathlib.Data.Set.Subsingleton
#align_import data.finsupp.basic from "leanprover... | Mathlib/Data/Finsupp/Basic.lean | 101 | 106 | theorem graph_injective (α M) [Zero M] : Injective (@graph α M _) := by |
intro f g h
classical
have hsup : f.support = g.support := by rw [← image_fst_graph, h, image_fst_graph]
refine ext_iff'.2 ⟨hsup, fun x hx => apply_eq_of_mem_graph <| h.symm ▸ ?_⟩
exact mk_mem_graph _ (hsup ▸ hx)
| 5 | 148.413159 | 2 | 1.25 | 4 | 1,325 |
import Batteries.Data.UnionFind.Basic
namespace Batteries.UnionFind
@[simp] theorem arr_empty : empty.arr = #[] := rfl
@[simp] theorem parent_empty : empty.parent a = a := rfl
@[simp] theorem rank_empty : empty.rank a = 0 := rfl
@[simp] theorem rootD_empty : empty.rootD a = a := rfl
@[simp] theorem arr_push {m : Un... | .lake/packages/batteries/Batteries/Data/UnionFind/Lemmas.lean | 41 | 51 | theorem parentD_linkAux {self} {x y : Fin self.size} :
parentD (linkAux self x y) i =
if x.1 = y then
parentD self i
else
if (self.get y).rank < (self.get x).rank then
if y = i then x else parentD self i
else
if x = i then y else parentD self i := by |
dsimp only [linkAux]; split <;> [rfl; split] <;> [rw [parentD_set]; split] <;> rw [parentD_set]
split <;> [(subst i; rwa [if_neg, parentD_eq]); rw [parentD_set]]
| 2 | 7.389056 | 1 | 1 | 5 | 1,144 |
import Mathlib.Combinatorics.Quiver.Path
import Mathlib.Combinatorics.Quiver.Push
#align_import combinatorics.quiver.symmetric from "leanprover-community/mathlib"@"706d88f2b8fdfeb0b22796433d7a6c1a010af9f2"
universe v u w v'
namespace Quiver
-- Porting note: no hasNonemptyInstance linter yet
def Symmetrify (V : ... | Mathlib/Combinatorics/Quiver/Symmetric.lean | 158 | 163 | theorem Path.reverse_reverse [h : HasInvolutiveReverse V] {a b : V} (p : Path a b) :
p.reverse.reverse = p := by |
induction' p with _ _ _ _ h
· simp
· rw [Path.reverse, Path.reverse_comp, h, Path.reverse_toPath, Quiver.reverse_reverse]
rfl
| 4 | 54.59815 | 2 | 1.375 | 8 | 1,472 |
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 | 126 | 129 | theorem nonMemberSubfamily_memberSubfamily :
(𝒜.memberSubfamily a).nonMemberSubfamily a = 𝒜.memberSubfamily a := by |
ext
simp
| 2 | 7.389056 | 1 | 1.3125 | 16 | 1,367 |
import Mathlib.Analysis.InnerProductSpace.Rayleigh
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Algebra.DirectSum.Decomposition
import Mathlib.LinearAlgebra.Eigenspace.Minpoly
#align_import analysis.inner_product_space.spectrum from "leanprover-community/mathlib"@"6b0169218d01f2837d79ea2784882009a0da... | Mathlib/Analysis/InnerProductSpace/Spectrum.lean | 125 | 131 | theorem orthogonalComplement_iSup_eigenspaces_eq_bot : (⨆ μ, eigenspace T μ)ᗮ = ⊥ := by |
have hT' : IsSymmetric _ :=
hT.restrict_invariant hT.orthogonalComplement_iSup_eigenspaces_invariant
-- a self-adjoint operator on a nontrivial inner product space has an eigenvalue
haveI :=
hT'.subsingleton_of_no_eigenvalue_finiteDimensional hT.orthogonalComplement_iSup_eigenspaces
exact Submodule.eq_... | 6 | 403.428793 | 2 | 1.5 | 6 | 1,660 |
import Mathlib.LinearAlgebra.Dimension.Constructions
import Mathlib.LinearAlgebra.Dimension.Finite
universe u v
open Function Set Cardinal
variable {R} {M M₁ M₂ M₃ : Type u} {M' : Type v} [Ring R]
variable [AddCommGroup M] [AddCommGroup M₁] [AddCommGroup M₂] [AddCommGroup M₃] [AddCommGroup M']
variable [Module R M... | Mathlib/LinearAlgebra/Dimension/RankNullity.lean | 68 | 72 | theorem lift_rank_range_add_rank_ker (f : M →ₗ[R] M') :
lift.{u} (Module.rank R (LinearMap.range f)) + lift.{v} (Module.rank R (LinearMap.ker f)) =
lift.{v} (Module.rank R M) := by |
haveI := fun p : Submodule R M => Classical.decEq (M ⧸ p)
rw [← f.quotKerEquivRange.lift_rank_eq, ← lift_add, rank_quotient_add_rank]
| 2 | 7.389056 | 1 | 1 | 7 | 820 |
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 | 83 | 92 | theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) :
taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x +
(((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by |
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Na... | 7 | 1,096.633158 | 2 | 1.714286 | 7 | 1,841 |
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.GroupTheory.Perm.Basic
#align_import group_theory.perm.support from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Equiv Finset
namespace Equiv.Perm
variable {α : Type*}
section support
s... | Mathlib/GroupTheory/Perm/Support.lean | 264 | 267 | theorem set_support_inv_eq : { x | p⁻¹ x ≠ x } = { x | p x ≠ x } := by |
ext x
simp only [Set.mem_setOf_eq, Ne]
rw [inv_def, symm_apply_eq, eq_comm]
| 3 | 20.085537 | 1 | 0.944444 | 18 | 795 |
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex
#align_import analysis.special_functions.trigonometric.arctan from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
namespace Real
open Set Filter
open scoped Topology Real
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Arctan.lean | 32 | 38 | theorem tan_add {x y : ℝ}
(h : ((∀ k : ℤ, x ≠ (2 * k + 1) * π / 2) ∧ ∀ l : ℤ, y ≠ (2 * l + 1) * π / 2) ∨
(∃ k : ℤ, x = (2 * k + 1) * π / 2) ∧ ∃ l : ℤ, y = (2 * l + 1) * π / 2) :
tan (x + y) = (tan x + tan y) / (1 - tan x * tan y) := by |
simpa only [← Complex.ofReal_inj, Complex.ofReal_sub, Complex.ofReal_add, Complex.ofReal_div,
Complex.ofReal_mul, Complex.ofReal_tan] using
@Complex.tan_add (x : ℂ) (y : ℂ) (by convert h <;> norm_cast)
| 3 | 20.085537 | 1 | 1.5 | 4 | 1,690 |
import Mathlib.Algebra.FreeMonoid.Basic
import Mathlib.Algebra.Group.Submonoid.MulOpposite
import Mathlib.Algebra.Group.Submonoid.Operations
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.Finset.NoncommProd
import Mathlib.Data.Int.Order.Lemmas
#align_import group_theory.submonoid.membership fro... | Mathlib/Algebra/Group/Submonoid/Membership.lean | 262 | 265 | theorem mem_sSup_of_mem {S : Set (Submonoid M)} {s : Submonoid M} (hs : s ∈ S) :
∀ {x : M}, x ∈ s → x ∈ sSup S := by |
rw [← SetLike.le_def]
exact le_sSup hs
| 2 | 7.389056 | 1 | 0.875 | 8 | 759 |
import Mathlib.FieldTheory.Finite.Basic
#align_import number_theory.wilson from "leanprover-community/mathlib"@"c471da714c044131b90c133701e51b877c246677"
open Finset Nat FiniteField ZMod
open scoped Nat
namespace Nat
variable {n : ℕ}
| Mathlib/NumberTheory/Wilson.lean | 89 | 97 | theorem prime_of_fac_equiv_neg_one (h : ((n - 1)! : ZMod n) = -1) (h1 : n ≠ 1) : Prime n := by |
rcases eq_or_ne n 0 with (rfl | h0)
· norm_num at h
replace h1 : 1 < n := n.two_le_iff.mpr ⟨h0, h1⟩
by_contra h2
obtain ⟨m, hm1, hm2 : 1 < m, hm3⟩ := exists_dvd_of_not_prime2 h1 h2
have hm : m ∣ (n - 1)! := Nat.dvd_factorial (pos_of_gt hm2) (le_pred_of_lt hm3)
refine hm2.ne' (Nat.dvd_one.mp ((Nat.dvd_add... | 8 | 2,980.957987 | 2 | 2 | 3 | 2,454 |
import Mathlib.Algebra.Lie.OfAssociative
import Mathlib.LinearAlgebra.Matrix.Reindex
import Mathlib.LinearAlgebra.Matrix.ToLinearEquiv
#align_import algebra.lie.matrix from "leanprover-community/mathlib"@"55e2dfde0cff928ce5c70926a3f2c7dee3e2dd99"
universe u v w w₁ w₂
section Matrices
open scoped Matrix
variabl... | Mathlib/Algebra/Lie/Matrix.lean | 76 | 79 | theorem Matrix.lieConj_symm_apply (P A : Matrix n n R) (h : Invertible P) :
(P.lieConj h).symm A = P⁻¹ * A * P := by |
simp [LinearEquiv.symm_conj_apply, Matrix.lieConj, LinearMap.toMatrix'_comp,
LinearMap.toMatrix'_toLin']
| 2 | 7.389056 | 1 | 1 | 2 | 1,083 |
import Mathlib.SetTheory.Cardinal.Finite
#align_import data.finite.card from "leanprover-community/mathlib"@"3ff3f2d6a3118b8711063de7111a0d77a53219a8"
noncomputable section
open scoped Classical
variable {α β γ : Type*}
def Finite.equivFin (α : Type*) [Finite α] : α ≃ Fin (Nat.card α) := by
have := (Finite.... | Mathlib/Data/Finite/Card.lean | 83 | 85 | theorem one_lt_card_iff_nontrivial [Finite α] : 1 < Nat.card α ↔ Nontrivial α := by |
haveI := Fintype.ofFinite α
simp only [Nat.card_eq_fintype_card, Fintype.one_lt_card_iff_nontrivial]
| 2 | 7.389056 | 1 | 1.2 | 10 | 1,279 |
import Mathlib.Algebra.Category.Ring.FilteredColimits
import Mathlib.Geometry.RingedSpace.SheafedSpace
import Mathlib.Topology.Sheaves.Stalks
import Mathlib.Algebra.Category.Ring.Colimits
import Mathlib.Algebra.Category.Ring.Limits
#align_import algebraic_geometry.ringed_space from "leanprover-community/mathlib"@"5dc... | Mathlib/Geometry/RingedSpace/Basic.lean | 84 | 125 | theorem isUnit_of_isUnit_germ (U : Opens X) (f : X.presheaf.obj (op U))
(h : ∀ x : U, IsUnit (X.presheaf.germ x f)) : IsUnit f := by |
-- We pick a cover of `U` by open sets `V x`, such that `f` is a unit on each `V x`.
choose V iVU m h_unit using fun x : U => X.isUnit_res_of_isUnit_germ U f x (h x)
have hcover : U ≤ iSup V := by
intro x hxU
-- Porting note: in Lean3 `rw` is sufficient
erw [Opens.mem_iSup]
exact ⟨⟨x, hxU⟩, m ⟨x,... | 40 | 235,385,266,837,019,970 | 2 | 2 | 2 | 1,958 |
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 | 62 | 65 | theorem empty_mem_generateMeasurableRec (s : Set (Set α)) (i : ω₁) :
∅ ∈ generateMeasurableRec s i := by |
unfold generateMeasurableRec
exact mem_union_left _ (mem_union_left _ (mem_union_right _ (mem_singleton ∅)))
| 2 | 7.389056 | 1 | 1.333333 | 6 | 1,448 |
import Mathlib.NumberTheory.BernoulliPolynomials
import Mathlib.MeasureTheory.Integral.IntervalIntegral
import Mathlib.Analysis.Calculus.Deriv.Polynomial
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.PSeries
#align_import number_theory.zeta_values from "leanprover-community/mathlib"@"f0c8bf9245297... | Mathlib/NumberTheory/ZetaValues.lean | 80 | 87 | theorem integral_bernoulliFun_eq_zero {k : ℕ} (hk : k ≠ 0) :
∫ x : ℝ in (0)..1, bernoulliFun k x = 0 := by |
rw [integral_eq_sub_of_hasDerivAt (fun x _ => antideriv_bernoulliFun k x)
((Polynomial.continuous _).intervalIntegrable _ _)]
rw [bernoulliFun_eval_one]
split_ifs with h
· exfalso; exact hk (Nat.succ_inj'.mp h)
· simp
| 6 | 403.428793 | 2 | 1.166667 | 6 | 1,241 |
import Mathlib.MeasureTheory.Covering.VitaliFamily
import Mathlib.MeasureTheory.Measure.Regular
import Mathlib.MeasureTheory.Function.AEMeasurableOrder
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.MeasureTheory.Integral.Average
import Mathlib.MeasureTheory.Decomposition.Lebesgue
#align_import measure... | Mathlib/MeasureTheory/Covering/Differentiation.lean | 160 | 201 | theorem ae_eventually_measure_zero_of_singular (hρ : ρ ⟂ₘ μ) :
∀ᵐ x ∂μ, Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 0) := by |
have A : ∀ ε > (0 : ℝ≥0), ∀ᵐ x ∂μ, ∀ᶠ a in v.filterAt x, ρ a < ε * μ a := by
intro ε εpos
set s := {x | ¬∀ᶠ a in v.filterAt x, ρ a < ε * μ a} with hs
change μ s = 0
obtain ⟨o, _, ρo, μo⟩ : ∃ o : Set α, MeasurableSet o ∧ ρ o = 0 ∧ μ oᶜ = 0 := hρ
apply le_antisymm _ bot_le
calc
μ s ≤ μ (s... | 40 | 235,385,266,837,019,970 | 2 | 2 | 3 | 2,421 |
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 | 123 | 129 | theorem drop_take_succ_eq_cons_get (L : List α) (i : Fin L.length) :
(L.take (i + 1)).drop i = [get L i] := by |
induction' L with head tail ih
· exact (Nat.not_succ_le_zero i i.isLt).elim
rcases i with ⟨_ | i, hi⟩
· simp
· simpa using ih ⟨i, Nat.lt_of_succ_lt_succ hi⟩
| 5 | 148.413159 | 2 | 0.9 | 10 | 782 |
import Mathlib.Algebra.Lie.OfAssociative
import Mathlib.Algebra.Lie.IdealOperations
#align_import algebra.lie.abelian from "leanprover-community/mathlib"@"8983bec7cdf6cb2dd1f21315c8a34ab00d7b2f6d"
universe u v w w₁ w₂
class LieModule.IsTrivial (L : Type v) (M : Type w) [Bracket L M] [Zero M] : Prop where
triv... | Mathlib/Algebra/Lie/Abelian.lean | 91 | 96 | theorem commutative_ring_iff_abelian_lie_ring {A : Type v} [Ring A] :
Std.Commutative (α := A) (· * ·) ↔ IsLieAbelian A := by |
have h₁ : Std.Commutative (α := A) (· * ·) ↔ ∀ a b : A, a * b = b * a :=
⟨fun h => h.1, fun h => ⟨h⟩⟩
have h₂ : IsLieAbelian A ↔ ∀ a b : A, ⁅a, b⁆ = 0 := ⟨fun h => h.1, fun h => ⟨h⟩⟩
simp only [h₁, h₂, LieRing.of_associative_ring_bracket, sub_eq_zero]
| 4 | 54.59815 | 2 | 1.75 | 4 | 1,869 |
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Combinatorics.Enumerative.DoubleCounting
import Mathlib.Data.Finset.Pointwise
import Mathlib.Tactic.GCongr
#align_import combinatorics.additive.pluennecke_ruzsa from "leanprover-community/mathlib"@"4aab2abced69a9e579b1e6dc... | Mathlib/Combinatorics/Additive/PluenneckeRuzsa.lean | 73 | 76 | theorem card_mul_mul_le_card_div_mul_card_mul (A B C : Finset α) :
(A * C).card * B.card ≤ (A / B).card * (B * C).card := by |
rw [← div_inv_eq_mul, ← div_inv_eq_mul B]
exact card_div_mul_le_card_div_mul_card_div _ _ _
| 2 | 7.389056 | 1 | 1.4 | 5 | 1,480 |
import Mathlib.Algebra.MvPolynomial.Funext
import Mathlib.Algebra.Ring.ULift
import Mathlib.RingTheory.WittVector.Basic
#align_import ring_theory.witt_vector.is_poly from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc9524185ae994553ebf4"
namespace WittVector
universe u
variable {p : ℕ} {R S : Type u} {σ id... | Mathlib/RingTheory/WittVector/IsPoly.lean | 125 | 133 | theorem poly_eq_of_wittPolynomial_bind_eq [Fact p.Prime] (f g : ℕ → MvPolynomial ℕ ℤ)
(h : ∀ n, bind₁ f (wittPolynomial p _ n) = bind₁ g (wittPolynomial p _ n)) : f = g := by |
ext1 n
apply MvPolynomial.map_injective (Int.castRingHom ℚ) Int.cast_injective
rw [← Function.funext_iff] at h
replace h :=
congr_arg (fun fam => bind₁ (MvPolynomial.map (Int.castRingHom ℚ) ∘ fam) (xInTermsOfW p ℚ n)) h
simpa only [Function.comp, map_bind₁, map_wittPolynomial, ← bind₁_bind₁,
bind₁_wi... | 7 | 1,096.633158 | 2 | 2 | 3 | 2,029 |
import Mathlib.CategoryTheory.Limits.Preserves.Finite
import Mathlib.CategoryTheory.Sites.Canonical
import Mathlib.CategoryTheory.Sites.Coherent.Basic
import Mathlib.CategoryTheory.Sites.Preserves
universe v u w
namespace CategoryTheory
open Limits
variable {C : Type u} [Category.{v} C]
variable [FinitaryPreExten... | Mathlib/CategoryTheory/Sites/Coherent/ExtensiveSheaves.lean | 52 | 58 | theorem isSheafFor_extensive_of_preservesFiniteProducts {X : C} (S : Presieve X) [S.Extensive]
(F : Cᵒᵖ ⥤ Type w) [PreservesFiniteProducts F] : S.IsSheafFor F := by |
obtain ⟨α, _, Z, π, rfl, ⟨hc⟩⟩ := Extensive.arrows_nonempty_isColimit (R := S)
have : (ofArrows Z (Cofan.mk X π).inj).hasPullbacks :=
(inferInstance : (ofArrows Z π).hasPullbacks)
cases nonempty_fintype α
exact isSheafFor_of_preservesProduct _ _ hc
| 5 | 148.413159 | 2 | 2 | 4 | 2,378 |
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanprover-community/mathlib"@"00f91228655eecdcd3ac97a7fd8dbcb139fe990a"
universe u v w
open scoped Classical Polynomial
open Polynomial
vari... | Mathlib/FieldTheory/IsAlgClosed/Basic.lean | 149 | 162 | theorem of_ringEquiv (k' : Type u) [Field k'] (e : k ≃+* k')
[IsAlgClosed k] : IsAlgClosed k' := by |
apply IsAlgClosed.of_exists_root
intro p hmp hp
have hpe : degree (p.map e.symm.toRingHom) ≠ 0 := by
rw [degree_map]
exact ne_of_gt (degree_pos_of_irreducible hp)
rcases IsAlgClosed.exists_root (k := k) (p.map e.symm) hpe with ⟨x, hx⟩
use e x
rw [IsRoot] at hx
apply e.symm.injective
rw [map_zer... | 12 | 162,754.791419 | 2 | 1.5 | 6 | 1,683 |
import Mathlib.LinearAlgebra.CliffordAlgebra.Contraction
variable {R M : Type*}
variable [CommRing R] [AddCommGroup M] [Module R M] {Q : QuadraticForm R M}
namespace CliffordAlgebra
variable (Q)
def invertibleιOfInvertible (m : M) [Invertible (Q m)] : Invertible (ι Q m) where
invOf := ι Q (⅟ (Q m) • m)
invO... | Mathlib/LinearAlgebra/CliffordAlgebra/Inversion.lean | 51 | 54 | theorem invOf_ι_mul_ι_mul_ι (a b : M) [Invertible (ι Q a)] [Invertible (Q a)] :
⅟ (ι Q a) * ι Q b * ι Q a = ι Q ((⅟ (Q a) * QuadraticForm.polar Q a b) • a - b) := by |
rw [invOf_ι, map_smul, smul_mul_assoc, smul_mul_assoc, ι_mul_ι_mul_ι, ← map_smul, smul_sub,
smul_smul, smul_smul, invOf_mul_self, one_smul]
| 2 | 7.389056 | 1 | 1 | 5 | 870 |
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.Algebra.Module.Basic
#align_import topology.algebra.continuous_affine_map from "leanprover-community/mathlib"@"bd1fc183335ea95a9519a1630bcf901fe9326d83"
structure ContinuousAffineMap (R : T... | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | 53 | 57 | theorem to_affineMap_injective {f g : P →ᴬ[R] Q} (h : (f : P →ᵃ[R] Q) = (g : P →ᵃ[R] Q)) :
f = g := by |
cases f
cases g
congr
| 3 | 20.085537 | 1 | 1 | 3 | 1,146 |
import Mathlib.Data.Finset.Basic
import Mathlib.ModelTheory.Syntax
import Mathlib.Data.List.ProdSigma
#align_import model_theory.semantics from "leanprover-community/mathlib"@"d565b3df44619c1498326936be16f1a935df0728"
universe u v w u' v'
namespace FirstOrder
namespace Language
variable {L : Language.{u, v}} {... | Mathlib/ModelTheory/Semantics.lean | 158 | 174 | theorem realize_constantsToVars [L[[α]].Structure M] [(lhomWithConstants L α).IsExpansionOn M]
{t : L[[α]].Term β} {v : β → M} :
t.constantsToVars.realize (Sum.elim (fun a => ↑(L.con a)) v) = t.realize v := by |
induction' t with _ n f ts ih
· simp
· cases n
· cases f
· simp only [realize, ih, Nat.zero_eq, constantsOn, mk₂_Functions]
-- Porting note: below lemma does not work with simp for some reason
rw [withConstants_funMap_sum_inl]
· simp only [realize, constantsToVars, Sum.elim_inl, f... | 14 | 1,202,604.284165 | 2 | 1.571429 | 7 | 1,709 |
import Mathlib.RingTheory.EisensteinCriterion
import Mathlib.RingTheory.Polynomial.ScaleRoots
#align_import ring_theory.polynomial.eisenstein.basic from "leanprover-community/mathlib"@"2032a878972d5672e7c27c957e7a6e297b044973"
universe u v w z
variable {R : Type u}
open Ideal Algebra Finset
open Polynomial
na... | Mathlib/RingTheory/Polynomial/Eisenstein/Basic.lean | 161 | 166 | theorem scaleRoots.isWeaklyEisensteinAt (p : R[X]) {x : R} {P : Ideal R} (hP : x ∈ P) :
(scaleRoots p x).IsWeaklyEisensteinAt P := by |
refine ⟨fun i => ?_⟩
rw [coeff_scaleRoots]
rw [natDegree_scaleRoots, ← tsub_pos_iff_lt] at i
exact Ideal.mul_mem_left _ _ (Ideal.pow_mem_of_mem P hP _ i)
| 4 | 54.59815 | 2 | 1.8 | 5 | 1,889 |
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Num.Lemmas
import Mathlib.Data.Option.Basic
import Mathlib.SetTheory.Cardinal.Basic
#align_import computability.encoding from "leanprover-community/mathlib"@"b6395b3a5acd655b16385fa0cdbf1961d6c34b3e"
universe u v
open Cardinal
namespace Computability
struc... | Mathlib/Computability/Encoding.lean | 143 | 149 | theorem decode_encodeNum : ∀ n, decodeNum (encodeNum n) = n := by |
intro n
cases' n with n <;> unfold encodeNum decodeNum
· rfl
rw [decode_encodePosNum n]
rw [PosNum.cast_to_num]
exact if_neg (encodePosNum_nonempty n)
| 6 | 403.428793 | 2 | 1.5 | 4 | 1,566 |
import Mathlib.AlgebraicGeometry.Morphisms.ClosedImmersion
import Mathlib.AlgebraicGeometry.Morphisms.QuasiSeparated
import Mathlib.AlgebraicGeometry.Pullbacks
import Mathlib.CategoryTheory.MorphismProperty.Limits
noncomputable section
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
universe ... | Mathlib/AlgebraicGeometry/Morphisms/Separated.lean | 57 | 60 | theorem respectsIso : MorphismProperty.RespectsIso @IsSeparated := by |
rw [isSeparated_eq_diagonal_isClosedImmersion]
apply MorphismProperty.RespectsIso.diagonal
exact IsClosedImmersion.respectsIso
| 3 | 20.085537 | 1 | 1 | 2 | 879 |
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.LinearAlgebra.SesquilinearForm
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
variable {𝕜 E F : Type*} [RCLike 𝕜]
variable [NormedAddCommGroup E] [InnerProductSpace... | Mathlib/Analysis/InnerProductSpace/Orthogonal.lean | 116 | 123 | theorem orthogonal_eq_inter : Kᗮ = ⨅ v : K, LinearMap.ker (innerSL 𝕜 (v : E)) := by |
apply le_antisymm
· rw [le_iInf_iff]
rintro ⟨v, hv⟩ w hw
simpa using hw _ hv
· intro v hv w hw
simp only [mem_iInf] at hv
exact hv ⟨w, hw⟩
| 7 | 1,096.633158 | 2 | 1.1 | 10 | 1,190 |
import Mathlib.Data.Finsupp.Defs
#align_import data.finsupp.ne_locus from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
variable {α M N P : Type*}
namespace Finsupp
variable [DecidableEq α]
section NHasZero
variable [DecidableEq N] [Zero N] (f g : α →₀ N)
def neLocus (f g : α →₀ ... | Mathlib/Data/Finsupp/NeLocus.lean | 52 | 54 | theorem coe_neLocus : ↑(f.neLocus g) = { x | f x ≠ g x } := by |
ext
exact mem_neLocus
| 2 | 7.389056 | 1 | 0.75 | 4 | 669 |
import Mathlib.Algebra.Group.Prod
import Mathlib.Order.Cover
#align_import algebra.support from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
assert_not_exists MonoidWithZero
open Set
namespace Function
variable {α β A B M N P G : Type*}
section One
variable [One M] [One N] [One P]
... | Mathlib/Algebra/Group/Support.lean | 73 | 76 | theorem mulSupport_eq_iff {f : α → M} {s : Set α} :
mulSupport f = s ↔ (∀ x, x ∈ s → f x ≠ 1) ∧ ∀ x, x ∉ s → f x = 1 := by |
simp (config := { contextual := true }) only [ext_iff, mem_mulSupport, ne_eq, iff_def,
not_imp_comm, and_comm, forall_and]
| 2 | 7.389056 | 1 | 0.333333 | 6 | 332 |
import Mathlib.Data.Set.Function
import Mathlib.Analysis.BoundedVariation
#align_import analysis.constant_speed from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open scoped NNReal ENNReal
open Set MeasureTheory Classical
variable {α : Type*} [LinearOrder α] {E : Type*} [PseudoEMetr... | Mathlib/Analysis/ConstantSpeed.lean | 64 | 68 | theorem hasConstantSpeedOnWith_of_subsingleton (f : ℝ → E) {s : Set ℝ} (hs : s.Subsingleton)
(l : ℝ≥0) : HasConstantSpeedOnWith f s l := by |
rintro x hx y hy; cases hs hx hy
rw [eVariationOn.subsingleton f (fun y hy z hz => hs hy.1 hz.1 : (s ∩ Icc x x).Subsingleton)]
simp only [sub_self, mul_zero, ENNReal.ofReal_zero]
| 3 | 20.085537 | 1 | 1.75 | 4 | 1,875 |
import Mathlib.Algebra.Group.Prod
import Mathlib.Data.Set.Lattice
#align_import data.nat.pairing from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
assert_not_exists MonoidWithZero
open Prod Decidable Function
namespace Nat
-- Porting note: no pp_nodot
--@[pp_nodot]
def pair (a b : ... | Mathlib/Data/Nat/Pairing.lean | 64 | 73 | theorem unpair_pair (a b : ℕ) : unpair (pair a b) = (a, b) := by |
dsimp only [pair]; split_ifs with h
· show unpair (b * b + a) = (a, b)
have be : sqrt (b * b + a) = b := sqrt_add_eq _ (le_trans (le_of_lt h) (Nat.le_add_left _ _))
simp [unpair, be, Nat.add_sub_cancel_left, h]
· show unpair (a * a + a + b) = (a, b)
have ae : sqrt (a * a + (a + b)) = a := by
rw... | 9 | 8,103.083928 | 2 | 1.2 | 10 | 1,270 |
import Mathlib.NumberTheory.Liouville.Basic
import Mathlib.Topology.Baire.Lemmas
import Mathlib.Topology.Baire.LocallyCompactRegular
import Mathlib.Topology.Instances.Irrational
#align_import number_theory.liouville.residual from "leanprover-community/mathlib"@"32b08ef840dd25ca2e47e035c5da03ce16d2dc3c"
open scope... | Mathlib/NumberTheory/Liouville/Residual.lean | 34 | 38 | theorem IsGδ.setOf_liouville : IsGδ { x | Liouville x } := by |
rw [setOf_liouville_eq_iInter_iUnion]
refine .iInter fun n => IsOpen.isGδ ?_
refine isOpen_iUnion fun a => isOpen_iUnion fun b => isOpen_iUnion fun _hb => ?_
exact isOpen_ball.inter isClosed_singleton.isOpen_compl
| 4 | 54.59815 | 2 | 1.75 | 4 | 1,847 |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Fin
import Mathlib.GroupTheory.GroupAction.Pi
import Mathlib.Logic.Equiv.Fin
#align_import algebra.big_operators.fin from "leanprover-community/mathlib"@"cc5dd6244981976cc9da7afc4eee5682b037a013"
open Fins... | Mathlib/Algebra/BigOperators/Fin.lean | 69 | 72 | theorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :
∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by |
rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb]
rfl
| 2 | 7.389056 | 1 | 0.333333 | 12 | 353 |
import Mathlib.Algebra.Group.Support
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Nat.Cast.Field
#align_import algebra.char_zero.lemmas from "leanprover-community/mathlib"@"acee671f47b8e7972a1eb6f4eed74b4b3abce829"
open Function Set
section AddMonoidWithOne
variable {α M : Type*} [AddMonoidWith... | Mathlib/Algebra/CharZero/Lemmas.lean | 127 | 129 | theorem nat_mul_inj {n : ℕ} {a b : R} (h : (n : R) * a = (n : R) * b) : n = 0 ∨ a = b := by |
rw [← sub_eq_zero, ← mul_sub, mul_eq_zero, sub_eq_zero] at h
exact mod_cast h
| 2 | 7.389056 | 1 | 0.5 | 12 | 426 |
import Mathlib.Algebra.Bounds
import Mathlib.Algebra.Order.Field.Basic -- Porting note: `LinearOrderedField`, etc
import Mathlib.Data.Set.Pointwise.SMul
#align_import algebra.order.pointwise from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function Set
open Pointwise
variable ... | Mathlib/Algebra/Order/Pointwise.lean | 265 | 275 | theorem smul_Ici : r • Ici a = Ici (r • a) := by |
ext x
simp only [mem_smul_set, smul_eq_mul, mem_Ioi]
constructor
· rintro ⟨a_w, a_h_left, rfl⟩
exact (mul_le_mul_left hr).mpr a_h_left
· rintro h
use x / r
constructor
· exact (le_div_iff' hr).mpr h
· exact mul_div_cancel₀ _ (ne_of_gt hr)
| 10 | 22,026.465795 | 2 | 1.25 | 16 | 1,340 |
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