Context stringlengths 57 6.04k | file_name stringlengths 21 79 | start int64 14 1.49k | end int64 18 1.5k | theorem stringlengths 25 1.55k | proof stringlengths 5 7.36k | num_lines int64 1 150 | complexity_score float64 2.72 139,370,958,066,637,970,000,000,000,000,000,000,000,000,000,000,000,000,000B | diff_level int64 0 2 | file_diff_level float64 0 2 | theorem_same_file int64 1 32 | rank_file int64 0 2.51k |
|---|---|---|---|---|---|---|---|---|---|---|---|
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
import Qq
#align_import analysis.special_functions.pow.real from "leanprover-community/mathlib"@"4fa54b337f7d52805480306db1b1439c741848c8"
noncomputable section
open scoped Classical
open Real ComplexConjugate
open Finset Set
namespace Real
variable {x y z... | Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 128 | 128 | theorem rpow_zero_pos (x : β) : 0 < x ^ (0 : β) := by | simp
| 1 | 2.718282 | 0 | 0.384615 | 13 | 383 |
import Mathlib.Data.Multiset.Bind
#align_import data.multiset.sections from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
assert_not_exists Ring
namespace Multiset
variable {Ξ± : Type*}
section Sections
def Sections (s : Multiset (Multiset Ξ±)) : Multiset (Multiset Ξ±) :=
Multiset.... | Mathlib/Data/Multiset/Sections.lean | 60 | 64 | theorem mem_sections {s : Multiset (Multiset Ξ±)} :
β {a}, a β Sections s β s.Rel (fun s a => a β s) a := by |
induction s using Multiset.induction_on with
| empty => simp
| cons _ _ ih => simp [ih, rel_cons_left, eq_comm]
| 3 | 20.085537 | 1 | 1 | 1 | 897 |
import Mathlib.Data.Sigma.Basic
import Mathlib.Algebra.Order.Ring.Nat
#align_import set_theory.lists from "leanprover-community/mathlib"@"497d1e06409995dd8ec95301fa8d8f3480187f4c"
variable {Ξ± : Type*}
inductive Lists'.{u} (Ξ± : Type u) : Bool β Type u
| atom : Ξ± β Lists' Ξ± false
| nil : Lists' Ξ± true
| con... | Mathlib/SetTheory/Lists.lean | 88 | 88 | theorem toList_cons (a : Lists Ξ±) (l) : toList (cons a l) = a :: l.toList := by | simp
| 1 | 2.718282 | 0 | 0.666667 | 3 | 567 |
import Mathlib.Combinatorics.Quiver.Basic
import Mathlib.Combinatorics.Quiver.Path
#align_import combinatorics.quiver.cast from "leanprover-community/mathlib"@"fc2ed6f838ce7c9b7c7171e58d78eaf7b438fb0e"
universe v vβ vβ u uβ uβ
variable {U : Type*} [Quiver.{u + 1} U]
namespace Quiver
def Hom.cast {u v u' v... | Mathlib/Combinatorics/Quiver/Cast.lean | 87 | 90 | theorem Path.cast_eq_cast {u v u' v' : U} (hu : u = u') (hv : v = v') (p : Path u v) :
p.cast hu hv = _root_.cast (by rw [hu, hv]) p := by |
subst_vars
rfl
| 2 | 7.389056 | 1 | 1 | 12 | 1,049 |
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Topology.Algebra.Module.Basic
#align_import analysis.normed_space.basic from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
open Metric Set Function Filter
open scoped NNReal Topology
instance Real.punctured_nhds_module_neBot {E ... | Mathlib/Analysis/NormedSpace/Real.lean | 46 | 47 | theorem norm_smul_of_nonneg {t : β} (ht : 0 β€ t) (x : E) : βt β’ xβ = t * βxβ := by |
rw [norm_smul, Real.norm_eq_abs, abs_of_nonneg ht]
| 1 | 2.718282 | 0 | 0.9 | 10 | 783 |
import Mathlib.Data.Int.Range
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.MulChar.Basic
#align_import number_theory.legendre_symbol.zmod_char from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
namespace ZMod
section QuadCharModP
@[simps]
def Οβ : MulChar (ZMod 4) β€... | Mathlib/NumberTheory/LegendreSymbol/ZModChar.lean | 101 | 103 | theorem Οβ_nat_three_mod_four {n : β} (hn : n % 4 = 3) : Οβ n = -1 := by |
rw [Οβ_nat_mod_four, hn]
rfl
| 2 | 7.389056 | 1 | 1.25 | 12 | 1,332 |
import Mathlib.Data.Finset.Pointwise
#align_import combinatorics.additive.e_transform from "leanprover-community/mathlib"@"207c92594599a06e7c134f8d00a030a83e6c7259"
open MulOpposite
open Pointwise
variable {Ξ± : Type*} [DecidableEq Ξ±]
namespace Finset
section Group
variable [Group Ξ±] (e : Ξ±) (x : Finset... | Mathlib/Combinatorics/Additive/ETransform.lean | 150 | 153 | theorem mulETransformRight.fst_mul_snd_subset :
(mulETransformRight e x).1 * (mulETransformRight e x).2 β x.1 * x.2 := by |
refine union_mul_inter_subset_union.trans (union_subset Subset.rfl ?_)
rw [op_smul_finset_mul_eq_mul_smul_finset, smul_inv_smul]
| 2 | 7.389056 | 1 | 0.875 | 8 | 758 |
import Mathlib.Tactic.CategoryTheory.Reassoc
#align_import category_theory.natural_transformation from "leanprover-community/mathlib"@"8350c34a64b9bc3fc64335df8006bffcadc7baa6"
namespace CategoryTheory
-- declare the `v`'s first; see note [CategoryTheory universes].
universe vβ vβ vβ vβ uβ uβ uβ uβ
variable {C :... | Mathlib/CategoryTheory/NatTrans.lean | 63 | 64 | theorem congr_app {F G : C β₯€ D} {Ξ± Ξ² : NatTrans F G} (h : Ξ± = Ξ²) (X : C) : Ξ±.app X = Ξ².app X := by |
aesop_cat
| 1 | 2.718282 | 0 | 0 | 1 | 64 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Comp
import Mathlib.Analysis.Calculus.FDeriv.RestrictScalars
#align_import analysis.calculus.deriv.comp from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
open scoped Classical
open Top... | Mathlib/Analysis/Calculus/Deriv/Comp.lean | 133 | 136 | theorem HasDerivAt.scomp_hasDerivWithinAt_of_eq (hg : HasDerivAt gβ gβ' y)
(hh : HasDerivWithinAt h h' s x) (hy : y = h x) :
HasDerivWithinAt (gβ β h) (h' β’ gβ') s x := by |
rw [hy] at hg; exact hg.scomp_hasDerivWithinAt x hh
| 1 | 2.718282 | 0 | 0 | 14 | 81 |
import Mathlib.Algebra.Group.Defs
#align_import group_theory.eckmann_hilton from "leanprover-community/mathlib"@"41cf0cc2f528dd40a8f2db167ea4fb37b8fde7f3"
universe u
namespace EckmannHilton
variable {X : Type u}
local notation a " <" m:51 "> " b => m a b
structure IsUnital (m : X β X β X) (e : X) extends Std... | Mathlib/GroupTheory/EckmannHilton.lean | 64 | 69 | theorem mul : mβ = mβ := by |
funext a b
calc
mβ a b = mβ (mβ a eβ) (mβ eβ b) := by
{ simp only [one hβ hβ distrib, hβ.left_id, hβ.right_id, hβ.left_id, hβ.right_id] }
_ = mβ a b := by simp only [distrib, hβ.left_id, hβ.right_id, hβ.left_id, hβ.right_id]
| 5 | 148.413159 | 2 | 1 | 2 | 856 |
import Mathlib.RingTheory.Ideal.Operations
import Mathlib.Algebra.Module.Torsion
import Mathlib.Algebra.Ring.Idempotents
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Filtration
import Mathlib.RingTheory.Nakayama
#align_import ring_theory.ideal.cota... | Mathlib/RingTheory/Ideal/Cotangent.lean | 63 | 65 | theorem map_toCotangent_ker : I.toCotangent.ker.map I.subtype = I ^ 2 := by |
rw [Ideal.toCotangent, Submodule.ker_mkQ, pow_two, Submodule.map_smul'' I β€ (Submodule.subtype I),
Algebra.id.smul_eq_mul, Submodule.map_subtype_top]
| 2 | 7.389056 | 1 | 1.333333 | 6 | 1,400 |
import Mathlib.Data.Set.Image
import Mathlib.Data.List.GetD
#align_import data.set.list from "leanprover-community/mathlib"@"2ec920d35348cb2d13ac0e1a2ad9df0fdf1a76b4"
open List
variable {Ξ± Ξ² : Type*} (l : List Ξ±)
namespace Set
theorem range_list_map (f : Ξ± β Ξ²) : range (map f) = { l | β x β l, x β range f } :=... | Mathlib/Data/Set/List.lean | 33 | 34 | theorem range_list_map_coe (s : Set Ξ±) : range (map ((β) : s β Ξ±)) = { l | β x β l, x β s } := by |
rw [range_list_map, Subtype.range_coe]
| 1 | 2.718282 | 0 | 1.2 | 5 | 1,263 |
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 RelPrime
variable {Ξ± I} [Comm... | Mathlib/RingTheory/Coprime/Lemmas.lean | 235 | 240 | theorem IsRelPrime.prod_left : (β i β t, IsRelPrime (s i) x) β IsRelPrime (β i β t, s i) x := by |
classical
refine Finset.induction_on t (fun _ β¦ isRelPrime_one_left) fun b t hbt ih H β¦ ?_
rw [Finset.prod_insert hbt]
rw [Finset.forall_mem_insert] at H
exact H.1.mul_left (ih H.2)
| 5 | 148.413159 | 2 | 1.111111 | 18 | 1,195 |
import Mathlib.RingTheory.Algebraic
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.RingTheory.Localization.Integral
#align_import ring_theory.ideal.over from "leanprover-community/mathlib"@"198cb64d5c961e1a8d0d3e219feb7058d5353861"
variable {R : Type*} [CommRing R]
namespace Ideal
open Polynomial... | Mathlib/RingTheory/Ideal/Over.lean | 116 | 126 | theorem exists_nonzero_mem_of_ne_bot {P : Ideal R[X]} (Pb : P β β₯) (hP : β x : R, C x β P β x = 0) :
β p : R[X], p β P β§ Polynomial.map (Quotient.mk (P.comap (C : R β+* R[X]))) p β 0 := by |
obtain β¨m, hmβ© := Submodule.nonzero_mem_of_bot_lt (bot_lt_iff_ne_bot.mpr Pb)
refine β¨m, Submodule.coe_mem m, fun pp0 => hm (Submodule.coe_eq_zero.mp ?_)β©
refine
(injective_iff_map_eq_zero (Polynomial.mapRingHom (Ideal.Quotient.mk
(P.comap (C : R β+* R[X]))))).mp
?_ _ pp0
refine map_injective _ ... | 9 | 8,103.083928 | 2 | 1.666667 | 6 | 1,826 |
import Mathlib.Data.Vector.Basic
set_option autoImplicit true
namespace Vector
def snoc : Vector Ξ± n β Ξ± β Vector Ξ± (n+1) :=
fun xs x => append xs (x ::α΅₯ Vector.nil)
section Simp
variable (xs : Vector Ξ± n)
@[simp]
theorem snoc_cons : (x ::α΅₯ xs).snoc y = x ::α΅₯ (xs.snoc y) :=
rfl
@[simp]
theorem snoc_nil... | Mathlib/Data/Vector/Snoc.lean | 42 | 45 | theorem reverse_cons : reverse (x ::α΅₯ xs) = (reverse xs).snoc x := by |
cases xs
simp only [reverse, cons, toList_mk, List.reverse_cons, snoc]
congr
| 3 | 20.085537 | 1 | 1.666667 | 3 | 1,811 |
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 | 106 | 108 | theorem integral_normed_smul {X} [NormedAddCommGroup X] [NormedSpace β X]
[CompleteSpace X] (z : X) : β« x, f.normed ΞΌ x β’ z βΞΌ = z := by |
simp_rw [integral_smul_const, f.integral_normed (ΞΌ := ΞΌ), one_smul]
| 1 | 2.718282 | 0 | 0.818182 | 11 | 722 |
import Mathlib.Data.Complex.Module
import Mathlib.Data.Complex.Order
import Mathlib.Data.Complex.Exponential
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Topology.Algebra.InfiniteSum.Module
import Mathlib.Topology.Instances.RealVectorSpace
#align_import analysis.complex.basic from "leanprover-community/mathlib... | Mathlib/Analysis/Complex/Basic.lean | 121 | 122 | theorem edist_of_re_eq {z w : β} (h : z.re = w.re) : edist z w = edist z.im w.im := by |
rw [edist_nndist, edist_nndist, nndist_of_re_eq h]
| 1 | 2.718282 | 0 | 0.222222 | 9 | 282 |
import Mathlib.Data.Matrix.Basis
import Mathlib.Data.Matrix.DMatrix
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.Reindex
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.matrix.transvection from "leanprover-community/mathlib"@"0e2aab2b0d521f060f62a14d2cf2e2c54e... | Mathlib/LinearAlgebra/Matrix/Transvection.lean | 113 | 116 | theorem transvection_mul_transvection_same (h : i β j) (c d : R) :
transvection i j c * transvection i j d = transvection i j (c + d) := by |
simp [transvection, Matrix.add_mul, Matrix.mul_add, h, h.symm, add_smul, add_assoc,
stdBasisMatrix_add]
| 2 | 7.389056 | 1 | 0.666667 | 12 | 572 |
import Mathlib.Topology.Basic
#align_import topology.nhds_set from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Filter Topology
variable {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y] {f : Filter X}
{s t sβ sβ tβ tβ : Set X} {x : X}
theorem nhdsSet_diagonal (X) [T... | Mathlib/Topology/NhdsSet.lean | 41 | 42 | theorem mem_nhdsSet_iff_forall : s β πΛ’ t β β x : X, x β t β s β π x := by |
simp_rw [nhdsSet, Filter.mem_sSup, forall_mem_image]
| 1 | 2.718282 | 0 | 0.333333 | 9 | 339 |
import Mathlib.Order.Filter.Ultrafilter
import Mathlib.Order.Filter.Germ
#align_import order.filter.filter_product from "leanprover-community/mathlib"@"2738d2ca56cbc63be80c3bd48e9ed90ad94e947d"
universe u v
variable {Ξ± : Type u} {Ξ² : Type v} {Ο : Ultrafilter Ξ±}
open scoped Classical
namespace Filter
local not... | Mathlib/Order/Filter/FilterProduct.lean | 161 | 162 | theorem const_max [LinearOrder Ξ²] (x y : Ξ²) : (β(max x y : Ξ²) : Ξ²*) = max βx βy := by |
rw [max_def, mapβ_const]
| 1 | 2.718282 | 0 | 0.333333 | 3 | 340 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.iterated_deriv from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputable section
open scoped Classical Topology
open Filter Asymptotics Set
variable {π... | Mathlib/Analysis/Calculus/IteratedDeriv/Defs.lean | 84 | 86 | theorem iteratedDerivWithin_eq_equiv_comp : iteratedDerivWithin n f s =
(ContinuousMultilinearMap.piFieldEquiv π (Fin n) F).symm β iteratedFDerivWithin π n f s := by |
ext x; rfl
| 1 | 2.718282 | 0 | 0.727273 | 11 | 649 |
import Mathlib.Algebra.Ring.InjSurj
import Mathlib.Algebra.Group.Units.Hom
import Mathlib.Algebra.Ring.Hom.Defs
#align_import algebra.ring.units from "leanprover-community/mathlib"@"2ed7e4aec72395b6a7c3ac4ac7873a7a43ead17c"
universe u v w x
variable {Ξ± : Type u} {Ξ² : Type v} {Ξ³ : Type w} {R : Type x}
open Funct... | Mathlib/Algebra/Ring/Units.lean | 50 | 50 | theorem neg_divp (a : Ξ±) (u : Ξ±Λ£) : -(a /β u) = -a /β u := by | simp only [divp, neg_mul]
| 1 | 2.718282 | 0 | 0 | 2 | 68 |
import Mathlib.Algebra.Algebra.Pi
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.RingTheory.Adjoin.Basic
#align_import data.polynomial.algebra_map from "leanprover-community/mathlib"@"e064a7bf82ad94c3c17b5128bbd860d1ec34874e"
noncomputable section
open Finset
open Polynomial
namespace Polynomial
univer... | Mathlib/Algebra/Polynomial/AlgebraMap.lean | 123 | 127 | theorem algHom_evalβ_algebraMap {R A B : Type*} [CommSemiring R] [Semiring A] [Semiring B]
[Algebra R A] [Algebra R B] (p : R[X]) (f : A ββ[R] B) (a : A) :
f (evalβ (algebraMap R A) a p) = evalβ (algebraMap R B) (f a) p := by |
simp only [evalβ_eq_sum, sum_def]
simp only [f.map_sum, f.map_mul, f.map_pow, eq_intCast, map_intCast, AlgHom.commutes]
| 2 | 7.389056 | 1 | 1.5 | 2 | 1,679 |
import Mathlib.Algebra.Field.Defs
import Mathlib.Tactic.Common
#align_import algebra.field.defs from "leanprover-community/mathlib"@"2651125b48fc5c170ab1111afd0817c903b1fc6c"
universe u
section IsField
structure IsField (R : Type u) [Semiring R] : Prop where
exists_pair_ne : β x y : R, x β y
mul_comm ... | Mathlib/Algebra/Field/IsField.lean | 84 | 93 | theorem uniq_inv_of_isField (R : Type u) [Ring R] (hf : IsField R) :
β x : R, x β 0 β β! y : R, x * y = 1 := by |
intro x hx
apply exists_unique_of_exists_of_unique
Β· exact hf.mul_inv_cancel hx
Β· intro y z hxy hxz
calc
y = y * (x * z) := by rw [hxz, mul_one]
_ = x * y * z := by rw [β mul_assoc, hf.mul_comm y x]
_ = z := by rw [hxy, one_mul]
| 8 | 2,980.957987 | 2 | 2 | 1 | 2,321 |
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
import Mathlib.Topology.Instances.RealVectorSpace
#align_import analysis.normed_space.add_torsor from "leanprover-community/mathlib"@... | Mathlib/Analysis/NormedSpace/AddTorsor.lean | 57 | 58 | theorem dist_homothety_center (pβ pβ : P) (c : π) :
dist (homothety pβ c pβ) pβ = βcβ * dist pβ pβ := by | rw [dist_comm, dist_center_homothety]
| 1 | 2.718282 | 0 | 0.5 | 6 | 418 |
import Mathlib.Algebra.GCDMonoid.Finset
import Mathlib.Algebra.Polynomial.CancelLeads
import Mathlib.Algebra.Polynomial.EraseLead
import Mathlib.Algebra.Polynomial.FieldDivision
#align_import ring_theory.polynomial.content from "leanprover-community/mathlib"@"7a030ab8eb5d99f05a891dccc49c5b5b90c947d3"
namespace Po... | Mathlib/RingTheory/Polynomial/Content.lean | 154 | 155 | theorem content_monomial {r : R} {k : β} : content (monomial k r) = normalize r := by |
rw [β C_mul_X_pow_eq_monomial, content_C_mul, content_X_pow, mul_one]
| 1 | 2.718282 | 0 | 1.2 | 15 | 1,288 |
import Mathlib.Order.Cover
import Mathlib.Order.Interval.Finset.Defs
#align_import data.finset.locally_finite from "leanprover-community/mathlib"@"442a83d738cb208d3600056c489be16900ba701d"
assert_not_exists MonoidWithZero
assert_not_exists Finset.sum
open Function OrderDual
open FinsetInterval
variable {ΞΉ Ξ± : T... | Mathlib/Order/Interval/Finset/Basic.lean | 144 | 144 | theorem right_mem_Icc : b β Icc a b β a β€ b := by | simp only [mem_Icc, and_true_iff, le_rfl]
| 1 | 2.718282 | 0 | 0 | 12 | 11 |
import Mathlib.Algebra.Quaternion
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Topology.Algebra.Algebra
#align_import analysis.quaternion from "leanprover-community/mathlib"@"07992a1d1f7a4176c6d3f160209608be4e198566"
@[inherit_doc] scoped[Quaternion... | Mathlib/Analysis/Quaternion.lean | 65 | 66 | theorem normSq_eq_norm_mul_self (a : β) : normSq a = βaβ * βaβ := by |
rw [β inner_self, real_inner_self_eq_norm_mul_norm]
| 1 | 2.718282 | 0 | 0 | 6 | 50 |
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.MkIffOfInductiveProp
#align_import data.sum.basic from "leanprover-community/mathlib"@"bd9851ca476957ea4549eb19b40e7b5ade9428cc"
universe u v w x
variable {Ξ± : Type u} {Ξ±' : Type w} {Ξ² : Type v} {Ξ²' : Type x} {Ξ³ Ξ΄ : Type*}
namespace Sum
#align sum.foral... | Mathlib/Data/Sum/Basic.lean | 57 | 58 | theorem eq_right_iff_getRight_eq {b : Ξ²} : x = inr b β β h, x.getRight h = b := by |
cases x <;> simp
| 1 | 2.718282 | 0 | 0.142857 | 7 | 254 |
import Mathlib.MeasureTheory.Constructions.Cylinders
import Mathlib.MeasureTheory.Measure.Typeclasses
open Set
namespace MeasureTheory
variable {ΞΉ : Type*} {Ξ± : ΞΉ β Type*} [β i, MeasurableSpace (Ξ± i)]
{P : β J : Finset ΞΉ, Measure (β j : J, Ξ± j)}
def IsProjectiveMeasureFamily (P : β J : Finset ΞΉ, Measure (β j ... | Mathlib/MeasureTheory/Constructions/Projective.lean | 143 | 150 | theorem unique [β i, IsFiniteMeasure (P i)]
(hΞΌ : IsProjectiveLimit ΞΌ P) (hΞ½ : IsProjectiveLimit Ξ½ P) :
ΞΌ = Ξ½ := by |
haveI : IsFiniteMeasure ΞΌ := hΞΌ.isFiniteMeasure
refine ext_of_generate_finite (measurableCylinders Ξ±) generateFrom_measurableCylinders.symm
isPiSystem_measurableCylinders (fun s hs β¦ ?_) (hΞΌ.measure_univ_unique hΞ½)
obtain β¨I, S, hS, rflβ© := (mem_measurableCylinders _).mp hs
rw [hΞΌ.measure_cylinder _ hS, hΞ½... | 5 | 148.413159 | 2 | 2 | 1 | 2,390 |
import Mathlib.MeasureTheory.Constructions.Prod.Integral
import Mathlib.MeasureTheory.Integral.CircleIntegral
#align_import measure_theory.integral.torus_integral from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
variable {n : β}
variable {E : Type*} [NormedAddCommGroup E]
noncomputa... | Mathlib/MeasureTheory/Integral/TorusIntegral.lean | 113 | 114 | theorem torusIntegrable_const (a : E) (c : ββΏ) (R : ββΏ) : TorusIntegrable (fun _ => a) c R := by |
simp [TorusIntegrable, measure_Icc_lt_top]
| 1 | 2.718282 | 0 | 0.428571 | 7 | 406 |
import Mathlib.Data.Finset.Lattice
import Mathlib.Data.Fintype.Vector
import Mathlib.Data.Multiset.Sym
#align_import data.finset.sym from "leanprover-community/mathlib"@"02ba8949f486ebecf93fe7460f1ed0564b5e442c"
namespace Finset
variable {Ξ± : Type*}
@[simps]
protected def sym2 (s : Finset Ξ±) : Finset (Sym2 Ξ±) :... | Mathlib/Data/Finset/Sym.lean | 46 | 47 | theorem mk_mem_sym2_iff : s(a, b) β s.sym2 β a β s β§ b β s := by |
rw [mem_mk, sym2_val, Multiset.mk_mem_sym2_iff, mem_mk, mem_mk]
| 1 | 2.718282 | 0 | 0.769231 | 13 | 684 |
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.PEquiv
#align_import data.matrix.pequiv from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa1"
namespace PEquiv
open Matrix
universe u v
variable {k l m n : Type*}
variable {Ξ± : Type v}
open Matrix
def toMatrix [DecidableEq n] [Zer... | Mathlib/Data/Matrix/PEquiv.lean | 142 | 148 | theorem toMatrix_swap [DecidableEq n] [Ring Ξ±] (i j : n) :
(Equiv.swap i j).toPEquiv.toMatrix =
(1 : Matrix n n Ξ±) - (single i i).toMatrix - (single j j).toMatrix + (single i j).toMatrix +
(single j i).toMatrix := by |
ext
dsimp [toMatrix, single, Equiv.swap_apply_def, Equiv.toPEquiv, one_apply]
split_ifs <;> simp_all
| 3 | 20.085537 | 1 | 1.2 | 10 | 1,267 |
import Mathlib.RingTheory.MvPowerSeries.Basic
import Mathlib.RingTheory.Ideal.LocalRing
#align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60"
noncomputable section
open Finset (antidiagonal mem_antidiagonal)
namespace MvPowerSeries
open Fi... | Mathlib/RingTheory/MvPowerSeries/Inverse.lean | 90 | 97 | theorem coeff_invOfUnit [DecidableEq Ο] (n : Ο ββ β) (Ο : MvPowerSeries Ο R) (u : RΛ£) :
coeff R n (invOfUnit Ο u) =
if n = 0 then βuβ»ΒΉ
else
-βuβ»ΒΉ *
β x β antidiagonal n,
if x.2 < n then coeff R x.1 Ο * coeff R x.2 (invOfUnit Ο u) else 0 := by |
convert coeff_inv_aux n (βuβ»ΒΉ) Ο
| 1 | 2.718282 | 0 | 1 | 3 | 920 |
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.PEquiv
#align_import data.matrix.pequiv from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa1"
namespace PEquiv
open Matrix
universe u v
variable {k l m n : Type*}
variable {Ξ± : Type v}
open Matrix
def toMatrix [DecidableEq n] [Zer... | Mathlib/Data/Matrix/PEquiv.lean | 109 | 114 | theorem toMatrix_trans [Fintype m] [DecidableEq m] [DecidableEq n] [Semiring Ξ±] (f : l β. m)
(g : m β. n) : ((f.trans g).toMatrix : Matrix l n Ξ±) = f.toMatrix * g.toMatrix := by |
ext i j
rw [mul_matrix_apply]
dsimp [toMatrix, PEquiv.trans]
cases f i <;> simp
| 4 | 54.59815 | 2 | 1.2 | 10 | 1,267 |
import Mathlib.Algebra.Module.Submodule.Ker
#align_import linear_algebra.basic from "leanprover-community/mathlib"@"9d684a893c52e1d6692a504a118bfccbae04feeb"
variable {R : Type*} {Rβ : Type*}
variable {M : Type*} {Mβ : Type*}
namespace LinearMap
section AddCommMonoid
variable [Semiring R] [Semiring Rβ]
varia... | Mathlib/Algebra/Module/Submodule/EqLocus.lean | 64 | 65 | theorem eqLocus_eq_top {f g : F} : eqLocus f g = β€ β f = g := by |
simp [SetLike.ext_iff, DFunLike.ext_iff]
| 1 | 2.718282 | 0 | 0.5 | 2 | 433 |
import Mathlib.Order.Interval.Multiset
#align_import data.nat.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
-- TODO
-- assert_not_exists Ring
open Finset Nat
variable (a b c : β)
namespace Nat
instance instLocallyFiniteOrder : LocallyFiniteOrder β where
finsetIcc a b... | Mathlib/Order/Interval/Finset/Nat.lean | 126 | 127 | theorem card_fintypeIoc : Fintype.card (Set.Ioc a b) = b - a := by |
rw [Fintype.card_ofFinset, card_Ioc]
| 1 | 2.718282 | 0 | 0.125 | 8 | 253 |
import Mathlib.AlgebraicTopology.DoldKan.FunctorN
#align_import algebraic_topology.dold_kan.normalized from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190f0b504"
open CategoryTheory CategoryTheory.Category CategoryTheory.Limits
CategoryTheory.Subobject CategoryTheory.Idempotents DoldKan
non... | Mathlib/AlgebraicTopology/DoldKan/Normalized.lean | 83 | 86 | theorem PInftyToNormalizedMooreComplex_naturality {X Y : SimplicialObject A} (f : X βΆ Y) :
AlternatingFaceMapComplex.map f β« PInftyToNormalizedMooreComplex Y =
PInftyToNormalizedMooreComplex X β« NormalizedMooreComplex.map f := by |
aesop_cat
| 1 | 2.718282 | 0 | 0.8 | 5 | 699 |
import Mathlib.Topology.Order.IsLUB
open Set Filter TopologicalSpace Topology Function
open OrderDual (toDual ofDual)
variable {Ξ± Ξ² Ξ³ : Type*}
section ConditionallyCompleteLinearOrder
variable [ConditionallyCompleteLinearOrder Ξ±] [TopologicalSpace Ξ±] [OrderTopology Ξ±]
[ConditionallyCompleteLinearOrder Ξ²] [Top... | Mathlib/Topology/Order/Monotone.lean | 230 | 232 | theorem Monotone.map_ciSup_of_continuousAt {f : Ξ± β Ξ²} {g : Ξ³ β Ξ±} (Cf : ContinuousAt f (β¨ i, g i))
(Mf : Monotone f) (H : BddAbove (range g)) : f (β¨ i, g i) = β¨ i, f (g i) := by |
rw [iSup, Mf.map_csSup_of_continuousAt Cf (range_nonempty _) H, β range_comp, iSup]; rfl
| 1 | 2.718282 | 0 | 1 | 7 | 829 |
import Mathlib.SetTheory.Game.Ordinal
import Mathlib.SetTheory.Ordinal.NaturalOps
#align_import set_theory.game.birthday from "leanprover-community/mathlib"@"a347076985674932c0e91da09b9961ed0a79508c"
universe u
open Ordinal
namespace SetTheory
open scoped NaturalOps PGame
namespace PGame
noncomputable def b... | Mathlib/SetTheory/Game/Birthday.lean | 64 | 78 | theorem lt_birthday_iff {x : PGame} {o : Ordinal} :
o < x.birthday β
(β i : x.LeftMoves, o β€ (x.moveLeft i).birthday) β¨
β i : x.RightMoves, o β€ (x.moveRight i).birthday := by |
constructor
Β· rw [birthday_def]
intro h
cases' lt_max_iff.1 h with h' h'
Β· left
rwa [lt_lsub_iff] at h'
Β· right
rwa [lt_lsub_iff] at h'
Β· rintro (β¨i, hiβ© | β¨i, hiβ©)
Β· exact hi.trans_lt (birthday_moveLeft_lt i)
Β· exact hi.trans_lt (birthday_moveRight_lt i)
| 11 | 59,874.141715 | 2 | 0.4 | 10 | 387 |
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Diagonal
import Mathlib.LinearAlgebra.Matrix.Transvection
import Mathlib.MeasureTheory.Group.LIntegral
import Mathlib.MeasureTheory.Integral.Marginal
import Mathlib.MeasureTheory.Measure.Stiel... | Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | 96 | 96 | theorem volume_singleton {a : β} : volume ({a} : Set β) = 0 := by | simp [volume_val]
| 1 | 2.718282 | 0 | 0.909091 | 22 | 790 |
import Mathlib.Order.Filter.Cofinite
import Mathlib.Order.Hom.CompleteLattice
#align_import order.liminf_limsup from "leanprover-community/mathlib"@"ffde2d8a6e689149e44fd95fa862c23a57f8c780"
set_option autoImplicit true
open Filter Set Function
variable {Ξ± Ξ² Ξ³ ΞΉ ΞΉ' : Type*}
namespace Filter
section Relation
... | Mathlib/Order/LiminfLimsup.lean | 77 | 77 | theorem isBounded_bot : IsBounded r β₯ β Nonempty Ξ± := by | simp [IsBounded, exists_true_iff_nonempty]
| 1 | 2.718282 | 0 | 0.25 | 4 | 306 |
import Mathlib.Algebra.Group.Submonoid.Operations
import Mathlib.Algebra.Star.SelfAdjoint
#align_import algebra.star.order from "leanprover-community/mathlib"@"31c24aa72e7b3e5ed97a8412470e904f82b81004"
open Set
open scoped NNRat
universe u
variable {R : Type u}
class StarOrderedRing (R : Type u) [NonUnitalSemi... | Mathlib/Algebra/Star/Order.lean | 137 | 139 | theorem nonneg_iff [NonUnitalSemiring R] [PartialOrder R] [StarRing R] [StarOrderedRing R] {x : R} :
0 β€ x β x β AddSubmonoid.closure (Set.range fun s : R => star s * s) := by |
simp only [le_iff, zero_add, exists_eq_right']
| 1 | 2.718282 | 0 | 0 | 1 | 151 |
import Mathlib.Algebra.GroupWithZero.Indicator
import Mathlib.Topology.ContinuousOn
import Mathlib.Topology.Instances.ENNReal
#align_import topology.semicontinuous from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Topology ENNReal
open Set Function Filter
variable {Ξ± : Type*} [... | Mathlib/Topology/Semicontinuous.lean | 213 | 220 | theorem IsOpen.lowerSemicontinuous_indicator (hs : IsOpen s) (hy : 0 β€ y) :
LowerSemicontinuous (indicator s fun _x => y) := by |
intro x z hz
by_cases h : x β s <;> simp [h] at hz
Β· filter_upwards [hs.mem_nhds h]
simp (config := { contextual := true }) [hz]
Β· refine Filter.eventually_of_forall fun x' => ?_
by_cases h' : x' β s <;> simp [h', hz.trans_le hy, hz]
| 6 | 403.428793 | 2 | 0.666667 | 3 | 607 |
import Mathlib.Data.List.Sublists
import Mathlib.Data.Multiset.Bind
#align_import data.multiset.powerset from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace Multiset
open List
variable {Ξ± : Type*}
-- Porting note (#11215): TODO: Write a more efficient version
def powerset... | Mathlib/Data/Multiset/Powerset.lean | 132 | 137 | theorem revzip_powersetAux' {l : List Ξ±} β¦xβ¦ (h : x β revzip (powersetAux' l)) :
x.1 + x.2 = βl := by |
rw [revzip, powersetAux', β map_reverse, zip_map, β revzip, List.mem_map] at h
simp only [Prod.map_apply, Prod.exists] at h
rcases h with β¨lβ, lβ, h, rfl, rflβ©
exact Quot.sound (revzip_sublists' _ _ _ h)
| 4 | 54.59815 | 2 | 1.2 | 5 | 1,258 |
import Mathlib.Algebra.BigOperators.Group.Multiset
import Mathlib.Data.Multiset.Dedup
#align_import data.multiset.bind from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
assert_not_exists MonoidWithZero
assert_not_exists MulAction
universe v
variable {Ξ± : Type*} {Ξ² : Type v} {Ξ³ Ξ΄ : Ty... | Mathlib/Data/Multiset/Bind.lean | 158 | 159 | theorem mem_bind {b s} {f : Ξ± β Multiset Ξ²} : b β bind s f β β a β s, b β f a := by |
simp [bind]
| 1 | 2.718282 | 0 | 0.384615 | 13 | 382 |
import Mathlib.Algebra.Divisibility.Basic
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Ring.Defs
#align_import algebra.euclidean_domain.defs from "leanprover-community/mathlib"@"ee7b9f9a9ac2a8d9f04ea39bbfe6b1a3be053b38"
universe u
class EuclideanDomain (R : Type u) extends CommRing R, Nontrivial R ... | Mathlib/Algebra/EuclideanDomain/Defs.lean | 157 | 157 | theorem mod_zero (a : R) : a % 0 = a := by | simpa only [zero_mul, zero_add] using div_add_mod a 0
| 1 | 2.718282 | 0 | 0.666667 | 6 | 589 |
import Mathlib.RingTheory.DedekindDomain.Dvr
import Mathlib.RingTheory.DedekindDomain.Ideal
#align_import ring_theory.dedekind_domain.pid from "leanprover-community/mathlib"@"6010cf523816335f7bae7f8584cb2edaace73940"
variable {R : Type*} [CommRing R]
open Ideal
open UniqueFactorizationMonoid
open scoped nonZer... | Mathlib/RingTheory/DedekindDomain/PID.lean | 109 | 168 | theorem FractionalIdeal.isPrincipal.of_finite_maximals_of_inv {A : Type*} [CommRing A]
[Algebra R A] {S : Submonoid R} [IsLocalization S A] (hS : S β€ Rβ°)
(hf : {I : Ideal R | I.IsMaximal}.Finite) (I I' : FractionalIdeal S A) (hinv : I * I' = 1) :
Submodule.IsPrincipal (I : Submodule R A) := by |
have hinv' := hinv
rw [Subtype.ext_iff, val_eq_coe, coe_mul] at hinv
let s := hf.toFinset
haveI := Classical.decEq (Ideal R)
have coprime : β M β s, β M' β s.erase M, M β M' = β€ := by
simp_rw [Finset.mem_erase, hf.mem_toFinset]
rintro M hM M' β¨hne, hM'β©
exact Ideal.IsMaximal.coprime_of_ne hM hM' ... | 56 | 2,091,659,496,012,996,000,000,000 | 2 | 2 | 3 | 2,278 |
import Mathlib.Algebra.Category.ModuleCat.EpiMono
import Mathlib.CategoryTheory.ConcreteCategory.Elementwise
#align_import algebra.category.Module.kernels from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
set_option linter.uppercaseLean3 false
open CategoryTheory CategoryTheory.Limits... | Mathlib/Algebra/Category/ModuleCat/Kernels.lean | 137 | 140 | theorem cokernel_Ο_ext {M N : ModuleCat.{u} R} (f : M βΆ N) {x y : N} (m : M) (w : x = y + f m) :
cokernel.Ο f x = cokernel.Ο f y := by |
subst w
simpa only [map_add, add_right_eq_self] using cokernel.condition_apply f m
| 2 | 7.389056 | 1 | 1 | 1 | 1,115 |
import Mathlib.Algebra.Group.Units.Equiv
import Mathlib.CategoryTheory.Endomorphism
#align_import category_theory.conj from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
universe v u
namespace CategoryTheory
namespace Iso
variable {C : Type u} [Category.{v} C]
def homCongr {X Y Xβ... | Mathlib/CategoryTheory/Conj.lean | 114 | 115 | theorem refl_conj (f : End X) : (Iso.refl X).conj f = f := by |
rw [conj_apply, Iso.refl_inv, Iso.refl_hom, Category.id_comp, Category.comp_id]
| 1 | 2.718282 | 0 | 0 | 6 | 130 |
import Mathlib.Algebra.Order.Group.Defs
import Mathlib.Algebra.Order.Monoid.WithTop
#align_import algebra.order.group.with_top from "leanprover-community/mathlib"@"f178c0e25af359f6cbc72a96a243efd3b12423a3"
namespace WithTop
variable {Ξ± : Type*}
namespace LinearOrderedAddCommGroup
variable [LinearOrderedAddCommG... | Mathlib/Algebra/Order/Group/WithTop.lean | 65 | 65 | theorem sub_top {a : WithTop Ξ±} : a - β€ = β€ := by | cases a <;> rfl
| 1 | 2.718282 | 0 | 0 | 2 | 139 |
import Mathlib.Data.PNat.Prime
import Mathlib.Algebra.IsPrimePow
import Mathlib.NumberTheory.Cyclotomic.Basic
import Mathlib.RingTheory.Adjoin.PowerBasis
import Mathlib.RingTheory.Polynomial.Cyclotomic.Eval
import Mathlib.RingTheory.Norm
import Mathlib.RingTheory.Polynomial.Cyclotomic.Expand
#align_import number_theo... | Mathlib/NumberTheory/Cyclotomic/PrimitiveRoots.lean | 289 | 291 | theorem norm_eq_neg_one_pow (hΞΆ : IsPrimitiveRoot ΞΆ 2) [IsDomain L] :
norm K ΞΆ = (-1 : K) ^ finrank K L := by |
rw [hΞΆ.eq_neg_one_of_two_right, show -1 = algebraMap K L (-1) by simp, Algebra.norm_algebraMap]
| 1 | 2.718282 | 0 | 0.75 | 4 | 670 |
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 | 126 | 130 | theorem untrop_sum_eq_sInf_image [ConditionallyCompleteLinearOrder R] (s : Finset S)
(f : S β Tropical (WithTop R)) : untrop (β i β s, f i) = sInf (untrop β f '' s) := by |
rcases s.eq_empty_or_nonempty with (rfl | h)
Β· simp only [Set.image_empty, coe_empty, sum_empty, WithTop.sInf_empty, untrop_zero]
Β· rw [β inf'_eq_csInf_image _ h, inf'_eq_inf, Finset.untrop_sum']
| 3 | 20.085537 | 1 | 0.928571 | 14 | 793 |
import Mathlib.Data.Nat.Count
import Mathlib.Data.Nat.SuccPred
import Mathlib.Order.Interval.Set.Monotone
import Mathlib.Order.OrderIsoNat
#align_import data.nat.nth from "leanprover-community/mathlib"@"7fdd4f3746cb059edfdb5d52cba98f66fce418c0"
open Finset
namespace Nat
variable (p : β β Prop)
noncomputable d... | Mathlib/Data/Nat/Nth.lean | 127 | 129 | theorem exists_lt_card_finite_nth_eq (hf : (setOf p).Finite) {x} (h : p x) :
β n, n < hf.toFinset.card β§ nth p n = x := by |
rwa [β @Set.mem_setOf_eq _ _ p, β image_nth_Iio_card hf] at h
| 1 | 2.718282 | 0 | 0.5 | 6 | 435 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
#align_import ring_theory.polynomial.opposites from "leanprover-community/mathlib"@"63417e01fbc711beaf25fa73b6edb395c0cfddd0"
open Polynomial
open Polynomial MulOpposite
variable {R : Type*} [Semiring R]
noncomputable section
namespace Polynomial
def opRi... | Mathlib/RingTheory/Polynomial/Opposites.lean | 118 | 120 | theorem leadingCoeff_opRingEquiv (p : R[X]α΅α΅α΅) :
(opRingEquiv R p).leadingCoeff = op (unop p).leadingCoeff := by |
rw [leadingCoeff, coeff_opRingEquiv, natDegree_opRingEquiv, leadingCoeff]
| 1 | 2.718282 | 0 | 0.714286 | 7 | 643 |
import Mathlib.Algebra.Module.Submodule.Map
#align_import linear_algebra.basic from "leanprover-community/mathlib"@"9d684a893c52e1d6692a504a118bfccbae04feeb"
open Function
open Pointwise
variable {R : Type*} {Rβ : Type*} {Rβ : Type*} {Rβ : Type*}
variable {K : Type*}
variable {M : Type*} {Mβ : Type*} {Mβ : Type*... | Mathlib/Algebra/Module/Submodule/Ker.lean | 129 | 132 | theorem ker_restrict [AddCommMonoid Mβ] [Module R Mβ] {p : Submodule R M} {q : Submodule R Mβ}
{f : M ββ[R] Mβ} (hf : β x : M, x β p β f x β q) :
ker (f.restrict hf) = LinearMap.ker (f.domRestrict p) := by |
rw [restrict_eq_codRestrict_domRestrict, ker_codRestrict]
| 1 | 2.718282 | 0 | 0.142857 | 7 | 255 |
import Mathlib.Algebra.CharP.ExpChar
import Mathlib.GroupTheory.OrderOfElement
#align_import algebra.char_p.two from "leanprover-community/mathlib"@"7f1ba1a333d66eed531ecb4092493cd1b6715450"
variable {R ΞΉ : Type*}
namespace CharTwo
section CommSemiring
variable [CommSemiring R] [CharP R 2]
theorem add_sq (x y... | Mathlib/Algebra/CharP/Two.lean | 115 | 116 | theorem sum_mul_self (s : Finset ΞΉ) (f : ΞΉ β R) :
((β i β s, f i) * β i β s, f i) = β i β s, f i * f i := by | simp_rw [β pow_two, sum_sq]
| 1 | 2.718282 | 0 | 0.2 | 10 | 273 |
import Batteries.Classes.SatisfiesM
namespace Array
| .lake/packages/batteries/Batteries/Data/Array/Monadic.lean | 18 | 30 | theorem SatisfiesM_foldlM [Monad m] [LawfulMonad m]
{as : Array Ξ±} (motive : Nat β Ξ² β Prop) {init : Ξ²} (h0 : motive 0 init) {f : Ξ² β Ξ± β m Ξ²}
(hf : β i : Fin as.size, β b, motive i.1 b β SatisfiesM (motive (i.1 + 1)) (f b as[i])) :
SatisfiesM (motive as.size) (as.foldlM f init) := by |
let rec go {i j b} (hβ : j β€ as.size) (hβ : as.size β€ i + j) (H : motive j b) :
SatisfiesM (motive as.size) (foldlM.loop f as as.size (Nat.le_refl _) i j b) := by
unfold foldlM.loop; split
Β· next hj =>
split
Β· cases Nat.not_le_of_gt (by simp [hj]) hβ
Β· exact (hf β¨j, hjβ© b H).bind fun _ ... | 9 | 8,103.083928 | 2 | 2 | 4 | 2,443 |
import Mathlib.Data.List.Basic
#align_import data.bool.all_any from "leanprover-community/mathlib"@"5a3e819569b0f12cbec59d740a2613018e7b8eec"
variable {Ξ± : Type*} {p : Ξ± β Prop} [DecidablePred p] {l : List Ξ±} {a : Ξ±}
namespace List
-- Porting note: in Batteries
#align list.all_nil List.all_nil
#align list.all_... | Mathlib/Data/Bool/AllAny.lean | 33 | 34 | theorem all_iff_forall_prop : (all l fun a => p a) β β a β l, p a := by |
simp only [all_iff_forall, decide_eq_true_iff]
| 1 | 2.718282 | 0 | 0.5 | 4 | 499 |
import Mathlib.Algebra.Associated
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.Data.Nat.Lattice
import Mathlib.RingTheory.Nilpotent.Defs
#align_import ring_th... | Mathlib/RingTheory/Nilpotent/Basic.lean | 64 | 66 | theorem IsNilpotent.isUnit_one_sub [Ring R] {r : R} (hnil : IsNilpotent r) : IsUnit (1 - r) := by |
rw [β IsUnit.neg_iff, neg_sub]
exact isUnit_sub_one hnil
| 2 | 7.389056 | 1 | 1.25 | 8 | 1,320 |
import Mathlib.Data.List.Infix
#align_import data.list.rdrop from "leanprover-community/mathlib"@"26f081a2fb920140ed5bc5cc5344e84bcc7cb2b2"
-- Make sure we don't import algebra
assert_not_exists Monoid
variable {Ξ± : Type*} (p : Ξ± β Bool) (l : List Ξ±) (n : β)
namespace List
def rdrop : List Ξ± :=
l.take (l.leng... | Mathlib/Data/List/DropRight.lean | 112 | 113 | theorem rdropWhile_concat_pos (x : Ξ±) (h : p x) : rdropWhile p (l ++ [x]) = rdropWhile p l := by |
rw [rdropWhile_concat, if_pos h]
| 1 | 2.718282 | 0 | 0.631579 | 19 | 550 |
import Mathlib.Topology.UniformSpace.CompleteSeparated
import Mathlib.Topology.EMetricSpace.Lipschitz
import Mathlib.Topology.MetricSpace.Basic
import Mathlib.Topology.MetricSpace.Bounded
#align_import topology.metric_space.antilipschitz from "leanprover-community/mathlib"@"c8f305514e0d47dfaa710f5a52f0d21b588e6328"
... | Mathlib/Topology/MetricSpace/Antilipschitz.lean | 110 | 113 | theorem mul_le_edist (hf : AntilipschitzWith K f) (x y : Ξ±) :
(K : ββ₯0β)β»ΒΉ * edist x y β€ edist (f x) (f y) := by |
rw [mul_comm, β div_eq_mul_inv]
exact ENNReal.div_le_of_le_mul' (hf x y)
| 2 | 7.389056 | 1 | 0.6 | 5 | 537 |
import Mathlib.ModelTheory.Basic
#align_import model_theory.language_map from "leanprover-community/mathlib"@"b3951c65c6e797ff162ae8b69eab0063bcfb3d73"
universe u v u' v' w w'
namespace FirstOrder
set_option linter.uppercaseLean3 false
namespace Language
open Structure Cardinal
open Cardinal
variable (L : L... | Mathlib/ModelTheory/LanguageMap.lean | 159 | 161 | theorem comp_id (F : L βα΄Έ L') : F βα΄Έ LHom.id L = F := by |
cases F
rfl
| 2 | 7.389056 | 1 | 1 | 2 | 992 |
import Mathlib.Order.SuccPred.Basic
#align_import order.succ_pred.relation from "leanprover-community/mathlib"@"9aba7801eeecebb61f58a5763c2b6dd1b47dc6ef"
open Function Order Relation Set
section PartialSucc
variable {Ξ± : Type*} [PartialOrder Ξ±] [SuccOrder Ξ±] [IsSuccArchimedean Ξ±]
theorem reflTransGen_of_succ_... | Mathlib/Order/SuccPred/Relation.lean | 40 | 43 | theorem reflTransGen_of_succ_of_ge (r : Ξ± β Ξ± β Prop) {n m : Ξ±} (h : β i β Ico m n, r (succ i) i)
(hmn : m β€ n) : ReflTransGen r n m := by |
rw [β reflTransGen_swap]
exact reflTransGen_of_succ_of_le (swap r) h hmn
| 2 | 7.389056 | 1 | 1.5 | 2 | 1,646 |
import Mathlib.Order.Interval.Set.OrdConnected
import Mathlib.Data.Set.Lattice
#align_import data.set.intervals.ord_connected_component from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
open Interval Function OrderDual
namespace Set
variable {Ξ± : Type*} [LinearOrder Ξ±] {s t : Set Ξ±}... | Mathlib/Order/Interval/Set/OrdConnectedComponent.lean | 73 | 74 | theorem ordConnectedComponent_univ : ordConnectedComponent univ x = univ := by |
simp [ordConnectedComponent]
| 1 | 2.718282 | 0 | 0.571429 | 7 | 519 |
import Mathlib.LinearAlgebra.Matrix.Symmetric
import Mathlib.LinearAlgebra.Matrix.Orthogonal
import Mathlib.Data.Matrix.Kronecker
#align_import linear_algebra.matrix.is_diag from "leanprover-community/mathlib"@"55e2dfde0cff928ce5c70926a3f2c7dee3e2dd99"
namespace Matrix
variable {Ξ± Ξ² R n m : Type*}
open Function... | Mathlib/LinearAlgebra/Matrix/IsDiag.lean | 98 | 101 | theorem IsDiag.sub [AddGroup Ξ±] {A B : Matrix n n Ξ±} (ha : A.IsDiag) (hb : B.IsDiag) :
(A - B).IsDiag := by |
intro i j h
simp [ha h, hb h]
| 2 | 7.389056 | 1 | 1.25 | 8 | 1,302 |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Finset.Sym
import Mathlib.Data.Fintype.Sum
import Mathlib.Data.Fintype.Prod
#align_import data.sym.card from "leanprover-community/mathlib"@"0bd2ea37bcba5769e14866170f251c9bc64e35d7"
open Finset Fintype Function Sum Nat
variable {Ξ± Ξ² : Type*}
... | Mathlib/Data/Sym/Card.lean | 120 | 122 | theorem card_sym_eq_choose {Ξ± : Type*} [Fintype Ξ±] (k : β) [Fintype (Sym Ξ± k)] :
card (Sym Ξ± k) = (card Ξ± + k - 1).choose k := by |
rw [card_sym_eq_multichoose, Nat.multichoose_eq]
| 1 | 2.718282 | 0 | 1 | 2 | 1,025 |
import Mathlib.LinearAlgebra.Contraction
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
#align_import linear_algebra.trace from "leanprover-community/mathlib"@"4cf7ca0e69e048b006674cf4499e5c7d296a89e0"
noncomputable section
universe u v w
namespace LinearMap
open Matrix
open FiniteDimensional
open Tensor... | Mathlib/LinearAlgebra/Trace.lean | 116 | 119 | theorem trace_conj (g : M ββ[R] M) (f : (M ββ[R] M)Λ£) :
trace R M (βf * g * βfβ»ΒΉ) = trace R M g := by |
rw [trace_mul_comm]
simp
| 2 | 7.389056 | 1 | 1.333333 | 6 | 1,414 |
import Mathlib.Algebra.MvPolynomial.Counit
import Mathlib.Algebra.MvPolynomial.Invertible
import Mathlib.RingTheory.WittVector.Defs
#align_import ring_theory.witt_vector.basic from "leanprover-community/mathlib"@"9556784a5b84697562e9c6acb40500d4a82e675a"
noncomputable section
open MvPolynomial Function
variable... | Mathlib/RingTheory/WittVector/Basic.lean | 102 | 102 | theorem zero : mapFun f (0 : π R) = 0 := by | map_fun_tac
| 1 | 2.718282 | 0 | 0.090909 | 11 | 242 |
import Mathlib.Topology.Order
#align_import topology.maps from "leanprover-community/mathlib"@"d91e7f7a7f1c7e9f0e18fdb6bde4f652004c735d"
open Set Filter Function
open TopologicalSpace Topology Filter
variable {X : Type*} {Y : Type*} {Z : Type*} {ΞΉ : Type*} {f : X β Y} {g : Y β Z}
section Inducing
variable [To... | Mathlib/Topology/Maps.lean | 156 | 157 | theorem isClosed_iff' (hf : Inducing f) {s : Set X} :
IsClosed s β β x, f x β closure (f '' s) β x β s := by | rw [hf.induced, isClosed_induced_iff']
| 1 | 2.718282 | 0 | 0.5 | 12 | 442 |
import Mathlib.Topology.Category.TopCat.Limits.Pullbacks
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
#align_import algebraic_geometry.open_immersion.basic from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
-- Porting note: due to `PresheafedSpace`, `SheafedSpace` and `Locally... | Mathlib/Geometry/RingedSpace/OpenImmersion.lean | 145 | 146 | theorem isoRestrict_inv_ofRestrict : H.isoRestrict.inv β« f = Y.ofRestrict _ := by |
rw [Iso.inv_comp_eq, isoRestrict_hom_ofRestrict]
| 1 | 2.718282 | 0 | 1 | 2 | 889 |
import Mathlib.LinearAlgebra.Contraction
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
#align_import linear_algebra.trace from "leanprover-community/mathlib"@"4cf7ca0e69e048b006674cf4499e5c7d296a89e0"
noncomputable section
universe u v w
namespace LinearMap
open Matrix
open FiniteDimensional
open Tensor... | Mathlib/LinearAlgebra/Trace.lean | 155 | 157 | theorem trace_eq_contract_of_basis' [Fintype ΞΉ] [DecidableEq ΞΉ] (b : Basis ΞΉ R M) :
LinearMap.trace R M = contractLeft R M ββ (dualTensorHomEquivOfBasis b).symm.toLinearMap := by |
simp [LinearEquiv.eq_comp_toLinearMap_symm, trace_eq_contract_of_basis b]
| 1 | 2.718282 | 0 | 1.333333 | 6 | 1,414 |
import Mathlib.Order.Interval.Set.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
import Mathlib.Tactic.AdaptationNote
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
open Topological... | Mathlib/Probability/Martingale/Upcrossing.lean | 206 | 209 | theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n Ο β€ upperCrossingTime a b f N (n + 1) Ο := by |
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le Ο
| 2 | 7.389056 | 1 | 0.625 | 8 | 548 |
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import analysis.inner_product_space.adjoint from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open RCLike
open scoped ComplexConjugate
variable {π E F G : Type... | Mathlib/Analysis/InnerProductSpace/Adjoint.lean | 150 | 152 | theorem apply_norm_eq_sqrt_inner_adjoint_left (A : E βL[π] F) (x : E) :
βA xβ = β(re βͺ(Aβ βL A) x, xβ«) := by |
rw [β apply_norm_sq_eq_inner_adjoint_left, Real.sqrt_sq (norm_nonneg _)]
| 1 | 2.718282 | 0 | 0.875 | 8 | 763 |
import Mathlib.Order.SuccPred.Basic
import Mathlib.Topology.Order.Basic
import Mathlib.Topology.Metrizable.Uniformity
#align_import topology.instances.discrete from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78ad6527b7e4"
open Order Set TopologicalSpace Filter
variable {Ξ± : Type*} [TopologicalSp... | Mathlib/Topology/Instances/Discrete.lean | 66 | 72 | theorem discreteTopology_iff_orderTopology_of_pred_succ' [PartialOrder Ξ±] [PredOrder Ξ±]
[SuccOrder Ξ±] [NoMinOrder Ξ±] [NoMaxOrder Ξ±] : DiscreteTopology Ξ± β OrderTopology Ξ± := by |
refine β¨fun h => β¨?_β©, fun h => β¨?_β©β©
Β· rw [h.eq_bot]
exact bot_topologicalSpace_eq_generateFrom_of_pred_succOrder
Β· rw [h.topology_eq_generate_intervals]
exact bot_topologicalSpace_eq_generateFrom_of_pred_succOrder.symm
| 5 | 148.413159 | 2 | 2 | 4 | 2,225 |
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.centering from "leanprover-community/mathlib"@"bea6c853b6edbd15e9d0941825abd04d77933ed0"
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory
namespace MeasureTheory
variable {Ξ© E : Type*} {m0 : ... | Mathlib/Probability/Martingale/Centering.lean | 75 | 79 | theorem martingalePart_eq_sum : martingalePart f β± ΞΌ = fun n =>
f 0 + β i β Finset.range n, (f (i + 1) - f i - ΞΌ[f (i + 1) - f i|β± i]) := by |
unfold martingalePart predictablePart
ext1 n
rw [Finset.eq_sum_range_sub f n, β add_sub, β Finset.sum_sub_distrib]
| 3 | 20.085537 | 1 | 1 | 4 | 951 |
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Analysis.NormedSpace.Real
#align_import analysis.special_functions.log.basic from "leanprover-community/mathlib"@"f23a09ce6d3f367220dc3cecad6b7eb69eb01690"
open Set Filter Function
open Topology
noncomputable ... | Mathlib/Analysis/SpecialFunctions/Log/Basic.lean | 142 | 143 | theorem log_le_log_iff (h : 0 < x) (hβ : 0 < y) : log x β€ log y β x β€ y := by |
rw [β exp_le_exp, exp_log h, exp_log hβ]
| 1 | 2.718282 | 0 | 0.583333 | 12 | 525 |
import Mathlib.Analysis.MeanInequalities
import Mathlib.Analysis.NormedSpace.WithLp
open Real Set Filter RCLike Bornology Uniformity Topology NNReal ENNReal
noncomputable section
variable (p : ββ₯0β) (π Ξ± Ξ² : Type*)
namespace WithLp
section DistNorm
section Dist
variable [Dist Ξ±] [Dist Ξ²]
open scoped C... | Mathlib/Analysis/NormedSpace/ProdLp.lean | 240 | 243 | theorem prod_dist_eq_sup (f g : WithLp β (Ξ± Γ Ξ²)) :
dist f g = dist f.fst g.fst β dist f.snd g.snd := by |
dsimp [dist]
exact if_neg ENNReal.top_ne_zero
| 2 | 7.389056 | 1 | 0.5 | 6 | 431 |
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Iterate
import Mathlib.Order.SemiconjSup
import Mathlib.Tactic.Monotonicity
import Mathlib.Topology.Order.MonotoneContinuity
#align_import dynamics.circle.rotation_number.translation_number from "leanprover-... | Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean | 213 | 214 | theorem units_inv_apply_apply (f : CircleDeg1LiftΛ£) (x : β) :
(fβ»ΒΉ : CircleDeg1LiftΛ£) (f x) = x := by | simp only [β mul_apply, f.inv_mul, coe_one, id]
| 1 | 2.718282 | 0 | 0 | 3 | 35 |
import Mathlib.Data.W.Basic
#align_import data.pfunctor.univariate.basic from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
-- "W", "Idx"
set_option linter.uppercaseLean3 false
universe u v vβ vβ vβ
@[pp_with_univ]
structure PFunctor where
A : Type u
B : A β Type u
#align p... | Mathlib/Data/PFunctor/Univariate/Basic.lean | 125 | 125 | theorem W.dest_mk (p : P (W P)) : W.dest (W.mk p) = p := by | cases p; rfl
| 1 | 2.718282 | 0 | 0.25 | 4 | 305 |
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 {Ξ± : Type*}
namespace Equiv.Perm
secti... | Mathlib/GroupTheory/Perm/Cycle/Concrete.lean | 237 | 238 | theorem two_le_length_toList_iff_mem_support {p : Perm Ξ±} {x : Ξ±} :
2 β€ length (toList p x) β x β p.support := by | simp
| 1 | 2.718282 | 0 | 1 | 18 | 1,030 |
import Mathlib.RingTheory.IntegrallyClosed
import Mathlib.RingTheory.Trace
import Mathlib.RingTheory.Norm
#align_import ring_theory.discriminant from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa1"
universe u v w z
open scoped Matrix
open Matrix FiniteDimensional Fintype Polynomial Fin... | Mathlib/RingTheory/Discriminant.lean | 76 | 79 | theorem discr_eq_discr_of_algEquiv [Fintype ΞΉ] (b : ΞΉ β B) (f : B ββ[A] C) :
Algebra.discr A b = Algebra.discr A (f β b) := by |
rw [discr_def]; congr; ext
simp_rw [traceMatrix_apply, traceForm_apply, Function.comp, β map_mul f, trace_eq_of_algEquiv]
| 2 | 7.389056 | 1 | 1.2 | 10 | 1,266 |
import Mathlib.Topology.Separation
#align_import topology.sober from "leanprover-community/mathlib"@"0a0ec35061ed9960bf0e7ffb0335f44447b58977"
open Set
variable {Ξ± Ξ² : Type*} [TopologicalSpace Ξ±] [TopologicalSpace Ξ²]
section genericPoint
def IsGenericPoint (x : Ξ±) (S : Set Ξ±) : Prop :=
closure ({x} : Set Ξ±)... | Mathlib/Topology/Sober.lean | 107 | 111 | theorem isGenericPoint_iff_forall_closed (hS : IsClosed S) (hxS : x β S) :
IsGenericPoint x S β β Z : Set Ξ±, IsClosed Z β x β Z β S β Z := by |
have : closure {x} β S := closure_minimal (singleton_subset_iff.2 hxS) hS
simp_rw [IsGenericPoint, subset_antisymm_iff, this, true_and_iff, closure, subset_sInter_iff,
mem_setOf_eq, and_imp, singleton_subset_iff]
| 3 | 20.085537 | 1 | 0.2 | 5 | 278 |
import Mathlib.Topology.Constructions
#align_import topology.continuous_on from "leanprover-community/mathlib"@"d4f691b9e5f94cfc64639973f3544c95f8d5d494"
open Set Filter Function Topology Filter
variable {Ξ± : Type*} {Ξ² : Type*} {Ξ³ : Type*} {Ξ΄ : Type*}
variable [TopologicalSpace Ξ±]
@[simp]
theorem nhds_bind_nhdsW... | Mathlib/Topology/ContinuousOn.lean | 57 | 59 | theorem mem_closure_ne_iff_frequently_within {z : Ξ±} {s : Set Ξ±} :
z β closure (s \ {z}) β βαΆ x in π[β ] z, x β s := by |
simp [mem_closure_iff_frequently, frequently_nhdsWithin_iff]
| 1 | 2.718282 | 0 | 0.5 | 6 | 483 |
import Mathlib.Topology.Category.TopCat.Limits.Products
#align_import topology.category.Top.limits.pullbacks from "leanprover-community/mathlib"@"178a32653e369dce2da68dc6b2694e385d484ef1"
-- Porting note: every ML3 decl has an uppercase letter
set_option linter.uppercaseLean3 false
open TopologicalSpace
open Cat... | Mathlib/Topology/Category/TopCat/Limits/Pullbacks.lean | 131 | 133 | theorem pullbackIsoProdSubtype_hom_snd (f : X βΆ Z) (g : Y βΆ Z) :
(pullbackIsoProdSubtype f g).hom β« pullbackSnd f g = pullback.snd := by |
rw [β Iso.eq_inv_comp, pullbackIsoProdSubtype_inv_snd]
| 1 | 2.718282 | 0 | 0.714286 | 7 | 647 |
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 | 85 | 90 | theorem interedges_disjoint_left {s s' : Finset Ξ±} (hs : Disjoint s s') (t : Finset Ξ²) :
Disjoint (interedges r s t) (interedges r s' t) := by |
rw [Finset.disjoint_left] at hs β’
intro _ hx hy
rw [mem_interedges_iff] at hx hy
exact hs hx.1 hy.1
| 4 | 54.59815 | 2 | 0.785714 | 14 | 695 |
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.RingTheory.Coprime.Basic
import Mathlib.Tactic.AdaptationNote
#align_import ring_theory.polynomial.scale_roots from "leanprover-community/mathlib"@"40ac1b258344e0c2b4568dc37bfad937ec35a727"
variable {R... | Mathlib/RingTheory/Polynomial/ScaleRoots.lean | 94 | 95 | theorem monic_scaleRoots_iff {p : R[X]} (s : R) : Monic (scaleRoots p s) β Monic p := by |
simp only [Monic, leadingCoeff, natDegree_scaleRoots, coeff_scaleRoots_natDegree]
| 1 | 2.718282 | 0 | 0.777778 | 9 | 688 |
import Mathlib.Geometry.Manifold.MFDeriv.Basic
noncomputable section
open scoped Manifold
variable {π : Type*} [NontriviallyNormedField π] {E : Type*} [NormedAddCommGroup E]
[NormedSpace π E] {E' : Type*} [NormedAddCommGroup E'] [NormedSpace π E'] {f : E β E'}
{s : Set E} {x : E}
section MFDerivFderiv
t... | Mathlib/Geometry/Manifold/MFDeriv/FDeriv.lean | 84 | 87 | theorem mdifferentiableAt_iff_differentiableAt :
MDifferentiableAt π(π, E) π(π, E') f x β DifferentiableAt π f x := by |
simp only [mdifferentiableAt_iff, differentiableWithinAt_univ, mfld_simps]
exact β¨fun H => H.2, fun H => β¨H.continuousAt, Hβ©β©
| 2 | 7.389056 | 1 | 0.5 | 8 | 455 |
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open ENNReal MeasureTheory MeasureTheory.Measure Metric Set Filter TopologicalSpace Function
open scoped Topology ENNReal Convex
variable... | Mathlib/MeasureTheory/Integral/Average.lean | 315 | 315 | theorem average_zero : β¨ _, (0 : E) βΞΌ = 0 := by | rw [average, integral_zero]
| 1 | 2.718282 | 0 | 0.347826 | 23 | 374 |
import Mathlib.Analysis.BoxIntegral.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.Tactic.Generalize
#align_import analysis.box_integral.integrability from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
open scoped Classical NNReal ENNReal Topology
universe u v
... | Mathlib/Analysis/BoxIntegral/Integrability.lean | 104 | 155 | theorem HasIntegral.of_aeEq_zero {l : IntegrationParams} {I : Box ΞΉ} {f : (ΞΉ β β) β E}
{ΞΌ : Measure (ΞΉ β β)} [IsLocallyFiniteMeasure ΞΌ] (hf : f =α΅[ΞΌ.restrict I] 0)
(hl : l.bRiemann = false) : HasIntegral.{u, v, v} I l f ΞΌ.toBoxAdditive.toSMul 0 := by |
/- Each set `{x | n < βf xβ β€ n + 1}`, `n : β`, has measure zero. We cover it by an open set of
measure less than `Ξ΅ / 2 ^ n / (n + 1)`. Then the norm of the integral sum is less than `Ξ΅`. -/
refine hasIntegral_iff.2 fun Ξ΅ Ξ΅0 => ?_
lift Ξ΅ to ββ₯0 using Ξ΅0.lt.le; rw [gt_iff_lt, NNReal.coe_pos] at Ξ΅0
rcases N... | 49 | 1,907,346,572,495,099,800,000 | 2 | 2 | 2 | 1,943 |
import Mathlib.Topology.Order
import Mathlib.Topology.Sets.Opens
import Mathlib.Topology.ContinuousFunction.Basic
#align_import topology.continuous_function.t0_sierpinski from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
noncomputable section
namespace TopologicalSpace
theorem eq_in... | Mathlib/Topology/ContinuousFunction/T0Sierpinski.lean | 50 | 52 | theorem productOfMemOpens_inducing : Inducing (productOfMemOpens X) := by |
convert inducing_iInf_to_pi fun (u : Opens X) (x : X) => x β u
apply eq_induced_by_maps_to_sierpinski
| 2 | 7.389056 | 1 | 1.333333 | 3 | 1,382 |
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 | 73 | 73 | theorem oangle_zero_right (x : V) : o.oangle x 0 = 0 := by | simp [oangle]
| 1 | 2.718282 | 0 | 0.571429 | 7 | 521 |
import Mathlib.Combinatorics.SimpleGraph.Coloring
#align_import combinatorics.simple_graph.partition from "leanprover-community/mathlib"@"2303b3e299f1c75b07bceaaac130ce23044d1386"
universe u v
namespace SimpleGraph
variable {V : Type u} (G : SimpleGraph V)
structure Partition where
parts : Set (Set V)
... | Mathlib/Combinatorics/SimpleGraph/Partition.lean | 98 | 102 | theorem partOfVertex_ne_of_adj {v w : V} (h : G.Adj v w) : P.partOfVertex v β P.partOfVertex w := by |
intro hn
have hw := P.mem_partOfVertex w
rw [β hn] at hw
exact P.independent _ (P.partOfVertex_mem v) (P.mem_partOfVertex v) hw (G.ne_of_adj h) h
| 4 | 54.59815 | 2 | 1.333333 | 3 | 1,402 |
import Mathlib.RingTheory.GradedAlgebra.HomogeneousIdeal
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.Topology.Sets.Opens
import Mathlib.Data.Set.Subsingleton
#align_import algebraic_geometry.projective_spectrum.topology from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
... | Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Topology.lean | 109 | 111 | theorem mem_vanishingIdeal (t : Set (ProjectiveSpectrum π)) (f : A) :
f β vanishingIdeal t β β x : ProjectiveSpectrum π, x β t β f β x.asHomogeneousIdeal := by |
rw [β SetLike.mem_coe, coe_vanishingIdeal, Set.mem_setOf_eq]
| 1 | 2.718282 | 0 | 0.8 | 5 | 697 |
import Mathlib.Analysis.Convex.Slope
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Tactic.LinearCombination
#align_import analysis.convex.specific_functions.basic from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
open Real Set NNReal
theorem strictConvexOn_exp : St... | Mathlib/Analysis/Convex/SpecificFunctions/Basic.lean | 127 | 133 | theorem one_add_mul_self_le_rpow_one_add {s : β} (hs : -1 β€ s) {p : β} (hp : 1 β€ p) :
1 + p * s β€ (1 + s) ^ p := by |
rcases eq_or_lt_of_le hp with (rfl | hp)
Β· simp
by_cases hs' : s = 0
Β· simp [hs']
exact (one_add_mul_self_lt_rpow_one_add hs hs' hp).le
| 5 | 148.413159 | 2 | 2 | 5 | 2,149 |
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 | 117 | 122 | theorem compl_neighborFinset_sdiff_inter_eq {v w : V} :
(G.neighborFinset v)αΆ \ {v} β© ((G.neighborFinset w)αΆ \ {w}) =
((G.neighborFinset v)αΆ β© (G.neighborFinset w)αΆ) \ ({w} βͺ {v}) := by |
ext
rw [β not_iff_not]
simp [imp_iff_not_or, or_assoc, or_comm, or_left_comm]
| 3 | 20.085537 | 1 | 1.428571 | 7 | 1,524 |
import Mathlib.Algebra.GeomSum
import Mathlib.Order.Filter.Archimedean
import Mathlib.Order.Iterate
import Mathlib.Topology.Algebra.Algebra
import Mathlib.Topology.Algebra.InfiniteSum.Real
#align_import analysis.specific_limits.basic from "leanprover-community/mathlib"@"57ac39bd365c2f80589a700f9fbb664d3a1a30c2"
n... | Mathlib/Analysis/SpecificLimits/Basic.lean | 39 | 41 | theorem tendsto_const_div_atTop_nhds_zero_nat (C : β) :
Tendsto (fun n : β β¦ C / n) atTop (π 0) := by |
simpa only [mul_zero] using tendsto_const_nhds.mul tendsto_inverse_atTop_nhds_zero_nat
| 1 | 2.718282 | 0 | 0.8 | 5 | 710 |
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.Reindex
import Mathlib.Tactic.FieldSimp
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.Basis
#align_import linear_algebra.determinant from "lea... | Mathlib/LinearAlgebra/Determinant.lean | 96 | 99 | theorem det_conj_of_mul_eq_one [DecidableEq m] [DecidableEq n] {M : Matrix m n A}
{M' : Matrix n m A} {N : Matrix n n A} (hMM' : M * M' = 1) (hM'M : M' * M = 1) :
det (M * N * M') = det N := by |
rw [β det_comm' hM'M hMM', β Matrix.mul_assoc, hM'M, Matrix.one_mul]
| 1 | 2.718282 | 0 | 0.666667 | 3 | 584 |
import Mathlib.Data.List.Range
import Mathlib.Algebra.Order.Ring.Nat
variable {Ξ± : Type*}
namespace List
@[simp]
theorem length_iterate (f : Ξ± β Ξ±) (a : Ξ±) (n : β) : length (iterate f a n) = n := by
induction n generalizing a <;> simp [*]
@[simp]
theorem iterate_eq_nil {f : Ξ± β Ξ±} {a : Ξ±} {n : β} : iterate f ... | Mathlib/Data/List/Iterate.lean | 39 | 41 | theorem mem_iterate {f : Ξ± β Ξ±} {a : Ξ±} {n : β} {b : Ξ±} :
b β iterate f a n β β m < n, b = f^[m] a := by |
simp [List.mem_iff_get, Fin.exists_iff, eq_comm (b := b)]
| 1 | 2.718282 | 0 | 0.166667 | 6 | 264 |
import Mathlib.Algebra.BigOperators.Group.Finset
#align_import data.nat.gcd.big_operators from "leanprover-community/mathlib"@"008205aa645b3f194c1da47025c5f110c8406eab"
namespace Nat
variable {ΞΉ : Type*}
| Mathlib/Data/Nat/GCD/BigOperators.lean | 20 | 22 | theorem coprime_list_prod_left_iff {l : List β} {k : β} :
Coprime l.prod k β β n β l, Coprime n k := by |
induction l <;> simp [Nat.coprime_mul_iff_left, *]
| 1 | 2.718282 | 0 | 0 | 8 | 159 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.