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 |
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import Mathlib.NumberTheory.Liouville.Basic
#align_import number_theory.liouville.liouville_number from "leanprover-community/mathlib"@"04e80bb7e8510958cd9aacd32fe2dc147af0b9f1"
noncomputable section
open scoped Nat
open Real Finset
def liouvilleNumber (m : β) : β :=
β' i : β, 1 / m ^ i !
#align liouville_n... | Mathlib/NumberTheory/Liouville/LiouvilleNumber.lean | 110 | 134 | theorem remainder_lt' (n : β) {m : β} (m1 : 1 < m) :
remainder m n < (1 - 1 / m)β»ΒΉ * (1 / m ^ (n + 1)!) :=
-- two useful inequalities
have m0 : 0 < m := zero_lt_one.trans m1
have mi : 1 / m < 1 := (div_lt_one m0).mpr m1
-- to show the strict inequality between these series, we prove that:
calc
(β' i, ... |
simp only [pow_add, one_div, mul_inv, inv_pow]
-- factor the constant `(1 / m ^ (n + 1)!)` out of the series
_ = (β' i, (1 / m) ^ i) * (1 / m ^ (n + 1)!) := tsum_mul_right
-- the series is the geometric series
_ = (1 - 1 / m)β»ΒΉ * (1 / m ^ (n + 1)!) := by rw [tsum_geometric_of_lt_one (by positivit... | 5 | 148.413159 | 2 | 1.333333 | 3 | 1,416 |
import Mathlib.NumberTheory.Liouville.Basic
#align_import number_theory.liouville.liouville_number from "leanprover-community/mathlib"@"04e80bb7e8510958cd9aacd32fe2dc147af0b9f1"
noncomputable section
open scoped Nat
open Real Finset
def liouvilleNumber (m : β) : β :=
β' i : β, 1 / m ^ i !
#align liouville_n... | Mathlib/NumberTheory/Liouville/LiouvilleNumber.lean | 137 | 160 | theorem aux_calc (n : β) {m : β} (hm : 2 β€ m) :
(1 - 1 / m)β»ΒΉ * (1 / m ^ (n + 1)!) β€ 1 / (m ^ n !) ^ n :=
calc
(1 - 1 / m)β»ΒΉ * (1 / m ^ (n + 1)!) β€ 2 * (1 / m ^ (n + 1)!) :=
-- the second factors coincide (and are non-negative),
-- the first factors satisfy the inequality `sub_one_div_inv_le_two`
... |
-- [NB: in this block, I do not follow the brace convention for subgoals -- I wait until
-- I solve all extraneous goals at once with `exact pow_pos (zero_lt_two.trans_le hm) _`.]
-- Clear denominators and massage*
apply (div_le_div_iff _ _).mpr
focus
conv_rhs => rw [one_mul, mul_... | 14 | 1,202,604.284165 | 2 | 1.333333 | 3 | 1,416 |
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.CategoryTheory.Groupoid.VertexGroup
import Mathlib.CategoryTheory.Groupoid.Basic
import Mathlib.CategoryTheory.Groupoid
import Mathlib.Data.Set.Lattice
import Mathlib.Order.GaloisConnection
#align_import category_theory.groupoid.subgroupoid from "leanprover-c... | Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean | 82 | 87 | theorem inv_mem_iff {c d : C} (f : c βΆ d) :
Groupoid.inv f β S.arrows d c β f β S.arrows c d := by |
constructor
Β· intro h
simpa only [inv_eq_inv, IsIso.inv_inv] using S.inv h
Β· apply S.inv
| 4 | 54.59815 | 2 | 1.333333 | 6 | 1,417 |
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.CategoryTheory.Groupoid.VertexGroup
import Mathlib.CategoryTheory.Groupoid.Basic
import Mathlib.CategoryTheory.Groupoid
import Mathlib.Data.Set.Lattice
import Mathlib.Order.GaloisConnection
#align_import category_theory.groupoid.subgroupoid from "leanprover-c... | Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean | 90 | 97 | theorem mul_mem_cancel_left {c d e : C} {f : c βΆ d} {g : d βΆ e} (hf : f β S.arrows c d) :
f β« g β S.arrows c e β g β S.arrows d e := by |
constructor
Β· rintro h
suffices Groupoid.inv f β« f β« g β S.arrows d e by
simpa only [inv_eq_inv, IsIso.inv_hom_id_assoc] using this
apply S.mul (S.inv hf) h
Β· apply S.mul hf
| 6 | 403.428793 | 2 | 1.333333 | 6 | 1,417 |
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.CategoryTheory.Groupoid.VertexGroup
import Mathlib.CategoryTheory.Groupoid.Basic
import Mathlib.CategoryTheory.Groupoid
import Mathlib.Data.Set.Lattice
import Mathlib.Order.GaloisConnection
#align_import category_theory.groupoid.subgroupoid from "leanprover-c... | Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean | 100 | 107 | theorem mul_mem_cancel_right {c d e : C} {f : c βΆ d} {g : d βΆ e} (hg : g β S.arrows d e) :
f β« g β S.arrows c e β f β S.arrows c d := by |
constructor
Β· rintro h
suffices (f β« g) β« Groupoid.inv g β S.arrows c d by
simpa only [inv_eq_inv, IsIso.hom_inv_id, Category.comp_id, Category.assoc] using this
apply S.mul h (S.inv hg)
Β· exact fun hf => S.mul hf hg
| 6 | 403.428793 | 2 | 1.333333 | 6 | 1,417 |
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.CategoryTheory.Groupoid.VertexGroup
import Mathlib.CategoryTheory.Groupoid.Basic
import Mathlib.CategoryTheory.Groupoid
import Mathlib.Data.Set.Lattice
import Mathlib.Order.GaloisConnection
#align_import category_theory.groupoid.subgroupoid from "leanprover-c... | Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean | 123 | 126 | theorem id_mem_of_nonempty_isotropy (c : C) : c β objs S β π c β S.arrows c c := by |
rintro β¨Ξ³, hΞ³β©
convert S.mul hΞ³ (S.inv hΞ³)
simp only [inv_eq_inv, IsIso.hom_inv_id]
| 3 | 20.085537 | 1 | 1.333333 | 6 | 1,417 |
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.CategoryTheory.Groupoid.VertexGroup
import Mathlib.CategoryTheory.Groupoid.Basic
import Mathlib.CategoryTheory.Groupoid
import Mathlib.Data.Set.Lattice
import Mathlib.Order.GaloisConnection
#align_import category_theory.groupoid.subgroupoid from "leanprover-c... | Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean | 152 | 154 | theorem coe_inv_coe' {c d : S.objs} (p : c βΆ d) :
(CategoryTheory.inv p).val = CategoryTheory.inv p.val := by |
simp only [β inv_eq_inv, coe_inv_coe]
| 1 | 2.718282 | 0 | 1.333333 | 6 | 1,417 |
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.CategoryTheory.Groupoid.VertexGroup
import Mathlib.CategoryTheory.Groupoid.Basic
import Mathlib.CategoryTheory.Groupoid
import Mathlib.Data.Set.Lattice
import Mathlib.Order.GaloisConnection
#align_import category_theory.groupoid.subgroupoid from "leanprover-c... | Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean | 165 | 167 | theorem hom.inj_on_objects : Function.Injective (hom S).obj := by |
rintro β¨c, hcβ© β¨d, hdβ© hcd
simp only [Subtype.mk_eq_mk]; exact hcd
| 2 | 7.389056 | 1 | 1.333333 | 6 | 1,417 |
import Mathlib.LinearAlgebra.DFinsupp
import Mathlib.LinearAlgebra.StdBasis
#align_import linear_algebra.finsupp_vector_space from "leanprover-community/mathlib"@"59628387770d82eb6f6dd7b7107308aa2509ec95"
noncomputable section
open Set LinearMap Submodule
open scoped Cardinal
universe u v w
namespace Finsupp
... | Mathlib/LinearAlgebra/FinsuppVectorSpace.lean | 34 | 51 | theorem linearIndependent_single {Ο : ΞΉ β Type*} {f : β ΞΉ, Ο ΞΉ β M}
(hf : β i, LinearIndependent R (f i)) :
LinearIndependent R fun ix : Ξ£i, Ο i => single ix.1 (f ix.1 ix.2) := by |
apply @linearIndependent_iUnion_finite R _ _ _ _ ΞΉ Ο fun i x => single i (f i x)
Β· intro i
have h_disjoint : Disjoint (span R (range (f i))) (ker (lsingle i)) := by
rw [ker_lsingle]
exact disjoint_bot_right
apply (hf i).map h_disjoint
Β· intro i t _ hit
refine (disjoint_lsingle_lsingle {i}... | 15 | 3,269,017.372472 | 2 | 1.333333 | 3 | 1,418 |
import Mathlib.LinearAlgebra.DFinsupp
import Mathlib.LinearAlgebra.StdBasis
#align_import linear_algebra.finsupp_vector_space from "leanprover-community/mathlib"@"59628387770d82eb6f6dd7b7107308aa2509ec95"
noncomputable section
open Set LinearMap Submodule
open scoped Cardinal
universe u v w
namespace Finsupp
... | Mathlib/LinearAlgebra/FinsuppVectorSpace.lean | 161 | 164 | theorem _root_.Finset.sum_single_ite [Fintype n] (a : R) (i : n) :
(β x : n, Finsupp.single x (if i = x then a else 0)) = Finsupp.single i a := by |
simp only [apply_ite (Finsupp.single _), Finsupp.single_zero, Finset.sum_ite_eq,
if_pos (Finset.mem_univ _)]
| 2 | 7.389056 | 1 | 1.333333 | 3 | 1,418 |
import Mathlib.LinearAlgebra.DFinsupp
import Mathlib.LinearAlgebra.StdBasis
#align_import linear_algebra.finsupp_vector_space from "leanprover-community/mathlib"@"59628387770d82eb6f6dd7b7107308aa2509ec95"
noncomputable section
open Set LinearMap Submodule
open scoped Cardinal
universe u v w
namespace Finsupp
... | Mathlib/LinearAlgebra/FinsuppVectorSpace.lean | 167 | 170 | theorem equivFun_symm_stdBasis [Finite n] (b : Basis n R M) (i : n) :
b.equivFun.symm (LinearMap.stdBasis R (fun _ => R) i 1) = b i := by |
cases nonempty_fintype n
simp
| 2 | 7.389056 | 1 | 1.333333 | 3 | 1,418 |
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.Data.ZMod.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
#align_import ring_theory.zmod from "leanprover-community/mathlib"@"00d163e35035c3577c1c79fa53b68de17781ffc1"
| Mathlib/RingTheory/ZMod.lean | 25 | 29 | theorem ZMod.ker_intCastRingHom (n : β) :
RingHom.ker (Int.castRingHom (ZMod n)) = Ideal.span ({(n : β€)} : Set β€) := by |
ext
rw [Ideal.mem_span_singleton, RingHom.mem_ker, Int.coe_castRingHom,
ZMod.intCast_zmod_eq_zero_iff_dvd]
| 3 | 20.085537 | 1 | 1.333333 | 3 | 1,419 |
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 : β) :
RingHom.ker (Int.castRingHom (ZMod n)) =... | Mathlib/RingTheory/ZMod.lean | 33 | 37 | theorem ZMod.ringHom_eq_of_ker_eq {n : β} {R : Type*} [CommRing R] (f g : R β+* ZMod n)
(h : RingHom.ker f = RingHom.ker g) : f = g := by |
have := f.liftOfRightInverse_comp _ (ZMod.ringHom_rightInverse f) β¨g, le_of_eq hβ©
rw [Subtype.coe_mk] at this
rw [β this, RingHom.ext_zmod (f.liftOfRightInverse _ _ β¨g, _β©) _, RingHom.id_comp]
| 3 | 20.085537 | 1 | 1.333333 | 3 | 1,419 |
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 : β) :
RingHom.ker (Int.castRingHom (ZMod n)) =... | Mathlib/RingTheory/ZMod.lean | 42 | 46 | theorem isReduced_zmod {n : β} : IsReduced (ZMod n) β Squarefree n β¨ n = 0 := by |
rw [β RingHom.ker_isRadical_iff_reduced_of_surjective
(ZMod.ringHom_surjective <| Int.castRingHom <| ZMod n),
ZMod.ker_intCastRingHom, β isRadical_iff_span_singleton, isRadical_iff_squarefree_or_zero,
Int.squarefree_natCast, Nat.cast_eq_zero]
| 4 | 54.59815 | 2 | 1.333333 | 3 | 1,419 |
import Mathlib.Data.Finset.Lattice
#align_import data.finset.pairwise from "leanprover-community/mathlib"@"c4c2ed622f43768eff32608d4a0f8a6cec1c047d"
open Finset
variable {Ξ± ΞΉ ΞΉ' : Type*}
instance [DecidableEq Ξ±] {r : Ξ± β Ξ± β Prop} [DecidableRel r] {s : Finset Ξ±} :
Decidable ((s : Set Ξ±).Pairwise r) :=
dec... | Mathlib/Data/Finset/Pairwise.lean | 27 | 30 | theorem Finset.pairwiseDisjoint_range_singleton :
(Set.range (singleton : Ξ± β Finset Ξ±)).PairwiseDisjoint id := by |
rintro _ β¨a, rflβ© _ β¨b, rflβ© h
exact disjoint_singleton.2 (ne_of_apply_ne _ h)
| 2 | 7.389056 | 1 | 1.333333 | 3 | 1,420 |
import Mathlib.Data.Finset.Lattice
#align_import data.finset.pairwise from "leanprover-community/mathlib"@"c4c2ed622f43768eff32608d4a0f8a6cec1c047d"
open Finset
variable {Ξ± ΞΉ ΞΉ' : Type*}
instance [DecidableEq Ξ±] {r : Ξ± β Ξ± β Prop} [DecidableRel r] {s : Finset Ξ±} :
Decidable ((s : Set Ξ±).Pairwise r) :=
dec... | Mathlib/Data/Finset/Pairwise.lean | 44 | 48 | theorem PairwiseDisjoint.image_finset_of_le [DecidableEq ΞΉ] {s : Finset ΞΉ} {f : ΞΉ β Ξ±}
(hs : (s : Set ΞΉ).PairwiseDisjoint f) {g : ΞΉ β ΞΉ} (hf : β a, f (g a) β€ f a) :
(s.image g : Set ΞΉ).PairwiseDisjoint f := by |
rw [coe_image]
exact hs.image_of_le hf
| 2 | 7.389056 | 1 | 1.333333 | 3 | 1,420 |
import Mathlib.Data.Finset.Lattice
#align_import data.finset.pairwise from "leanprover-community/mathlib"@"c4c2ed622f43768eff32608d4a0f8a6cec1c047d"
open Finset
variable {Ξ± ΞΉ ΞΉ' : Type*}
instance [DecidableEq Ξ±] {r : Ξ± β Ξ± β Prop} [DecidableRel r] {s : Finset Ξ±} :
Decidable ((s : Set Ξ±).Pairwise r) :=
dec... | Mathlib/Data/Finset/Pairwise.lean | 62 | 71 | theorem PairwiseDisjoint.biUnion_finset {s : Set ΞΉ'} {g : ΞΉ' β Finset ΞΉ} {f : ΞΉ β Ξ±}
(hs : s.PairwiseDisjoint fun i' : ΞΉ' => (g i').sup f)
(hg : β i β s, (g i : Set ΞΉ).PairwiseDisjoint f) : (β i β s, β(g i)).PairwiseDisjoint f := by |
rintro a ha b hb hab
simp_rw [Set.mem_iUnion] at ha hb
obtain β¨c, hc, haβ© := ha
obtain β¨d, hd, hbβ© := hb
obtain hcd | hcd := eq_or_ne (g c) (g d)
Β· exact hg d hd (by rwa [hcd] at ha) hb hab
Β· exact (hs hc hd (ne_of_apply_ne _ hcd)).mono (Finset.le_sup ha) (Finset.le_sup hb)
| 7 | 1,096.633158 | 2 | 1.333333 | 3 | 1,420 |
import Mathlib.RingTheory.DedekindDomain.Ideal
import Mathlib.RingTheory.IsAdjoinRoot
#align_import number_theory.kummer_dedekind from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
variable (R : Type*) {S : Type*} [CommRing R] [CommRing S] [Algebra R S]
open Ideal Polynomial DoubleQuo... | Mathlib/NumberTheory/KummerDedekind.lean | 85 | 86 | theorem conductor_eq_top_of_adjoin_eq_top (h : R<x> = β€) : conductor R x = β€ := by |
simp only [Ideal.eq_top_iff_one, mem_conductor_iff, h, mem_top, forall_const]
| 1 | 2.718282 | 0 | 1.333333 | 3 | 1,421 |
import Mathlib.RingTheory.DedekindDomain.Ideal
import Mathlib.RingTheory.IsAdjoinRoot
#align_import number_theory.kummer_dedekind from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
variable (R : Type*) {S : Type*} [CommRing R] [CommRing S] [Algebra R S]
open Ideal Polynomial DoubleQuo... | Mathlib/NumberTheory/KummerDedekind.lean | 119 | 148 | theorem prod_mem_ideal_map_of_mem_conductor {p : R} {z : S}
(hp : p β Ideal.comap (algebraMap R S) (conductor R x)) (hz' : z β I.map (algebraMap R S)) :
algebraMap R S p * z β algebraMap R<x> S '' β(I.map (algebraMap R R<x>)) := by |
rw [Ideal.map, Ideal.span, Finsupp.mem_span_image_iff_total] at hz'
obtain β¨l, H, H'β© := hz'
rw [Finsupp.total_apply] at H'
rw [β H', mul_comm, Finsupp.sum_mul]
have lem : β {a : R}, a β I β l a β’ algebraMap R S a * algebraMap R S p β
algebraMap R<x> S '' I.map (algebraMap R R<x>) := by
intro a ha
... | 27 | 532,048,240,601.79865 | 2 | 1.333333 | 3 | 1,421 |
import Mathlib.RingTheory.DedekindDomain.Ideal
import Mathlib.RingTheory.IsAdjoinRoot
#align_import number_theory.kummer_dedekind from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
variable (R : Type*) {S : Type*} [CommRing R] [CommRing S] [Algebra R S]
open Ideal Polynomial DoubleQuo... | Mathlib/NumberTheory/KummerDedekind.lean | 152 | 186 | theorem comap_map_eq_map_adjoin_of_coprime_conductor
(hx : (conductor R x).comap (algebraMap R S) β I = β€)
(h_alg : Function.Injective (algebraMap R<x> S)) :
(I.map (algebraMap R S)).comap (algebraMap R<x> S) = I.map (algebraMap R R<x>) := by |
apply le_antisymm
Β· -- This is adapted from [Neukirch1992]. Let `C = (conductor R x)`. The idea of the proof
-- is that since `I` and `C β© R` are coprime, we have
-- `(I * S) β© R<x> β (I + C) * ((I * S) β© R<x>) β I * R<x> + I * C * S β I * R<x>`.
intro y hy
obtain β¨z, hzβ© := y
obtain β¨p, hp, q,... | 31 | 29,048,849,665,247.426 | 2 | 1.333333 | 3 | 1,421 |
import Mathlib.Algebra.Group.Subgroup.MulOpposite
import Mathlib.Algebra.Group.Submonoid.Pointwise
import Mathlib.GroupTheory.GroupAction.ConjAct
#align_import group_theory.subgroup.pointwise from "leanprover-community/mathlib"@"e655e4ea5c6d02854696f97494997ba4c31be802"
open Set
open Pointwise
variable {Ξ± G A S... | Mathlib/Algebra/Group/Subgroup/Pointwise.lean | 73 | 81 | theorem closure_toSubmonoid (S : Set G) :
(closure S).toSubmonoid = Submonoid.closure (S βͺ Sβ»ΒΉ) := by |
refine le_antisymm (fun x hx => ?_) (Submonoid.closure_le.2 ?_)
Β· refine
closure_induction hx
(fun x hx => Submonoid.closure_mono subset_union_left (Submonoid.subset_closure hx))
(Submonoid.one_mem _) (fun x y hx hy => Submonoid.mul_mem _ hx hy) fun x hx => ?_
rwa [β Submonoid.mem_closure... | 7 | 1,096.633158 | 2 | 1.333333 | 3 | 1,422 |
import Mathlib.Algebra.Group.Subgroup.MulOpposite
import Mathlib.Algebra.Group.Submonoid.Pointwise
import Mathlib.GroupTheory.GroupAction.ConjAct
#align_import group_theory.subgroup.pointwise from "leanprover-community/mathlib"@"e655e4ea5c6d02854696f97494997ba4c31be802"
open Set
open Pointwise
variable {Ξ± G A S... | Mathlib/Algebra/Group/Subgroup/Pointwise.lean | 89 | 102 | theorem closure_induction_left {p : (x : G) β x β closure s β Prop} (one : p 1 (one_mem _))
(mul_left : β x (hx : x β s), β (y) hy, p y hy β p (x * y) (mul_mem (subset_closure hx) hy))
(mul_left_inv : β x (hx : x β s), β (y) hy, p y hy β
p (xβ»ΒΉ * y) (mul_mem (inv_mem (subset_closure hx)) hy))
{x : G} ... |
revert h
simp_rw [β mem_toSubmonoid, closure_toSubmonoid] at *
intro h
induction h using Submonoid.closure_induction_left with
| one => exact one
| mul_left x hx y hy ih =>
cases hx with
| inl hx => exact mul_left _ hx _ hy ih
| inr hx => simpa only [inv_inv] using mul_left_inv _ hx _ hy ih
| 9 | 8,103.083928 | 2 | 1.333333 | 3 | 1,422 |
import Mathlib.Algebra.Group.Subgroup.MulOpposite
import Mathlib.Algebra.Group.Submonoid.Pointwise
import Mathlib.GroupTheory.GroupAction.ConjAct
#align_import group_theory.subgroup.pointwise from "leanprover-community/mathlib"@"e655e4ea5c6d02854696f97494997ba4c31be802"
open Set
open Pointwise
variable {Ξ± G A S... | Mathlib/Algebra/Group/Subgroup/Pointwise.lean | 125 | 126 | theorem closure_inv (s : Set G) : closure sβ»ΒΉ = closure s := by |
simp only [β toSubmonoid_eq, closure_toSubmonoid, inv_inv, union_comm]
| 1 | 2.718282 | 0 | 1.333333 | 3 | 1,422 |
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.List.Sublists
import Mathlib.Data.List.InsertNth
#align_import group_theory.free_group from "leanprover-community/mathlib"@"f93c11933efbc3c2f0299e47b8ff83e9b539cbf6"
open Relation
universe u v w
variable {Ξ± : Type u... | Mathlib/GroupTheory/FreeGroup/Basic.lean | 115 | 116 | theorem Step.not_rev {x b} : Step (Lβ ++ (x, !b) :: (x, b) :: Lβ) (Lβ ++ Lβ) := by |
cases b <;> exact Step.not
| 1 | 2.718282 | 0 | 1.333333 | 3 | 1,423 |
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.List.Sublists
import Mathlib.Data.List.InsertNth
#align_import group_theory.free_group from "leanprover-community/mathlib"@"f93c11933efbc3c2f0299e47b8ff83e9b539cbf6"
open Relation
universe u v w
variable {Ξ± : Type u... | Mathlib/GroupTheory/FreeGroup/Basic.lean | 151 | 155 | theorem not_step_nil : Β¬Step [] L := by |
generalize h' : [] = L'
intro h
cases' h with Lβ Lβ
simp [List.nil_eq_append] at h'
| 4 | 54.59815 | 2 | 1.333333 | 3 | 1,423 |
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.List.Sublists
import Mathlib.Data.List.InsertNth
#align_import group_theory.free_group from "leanprover-community/mathlib"@"f93c11933efbc3c2f0299e47b8ff83e9b539cbf6"
open Relation
universe u v w
variable {Ξ± : Type u... | Mathlib/GroupTheory/FreeGroup/Basic.lean | 160 | 173 | theorem Step.cons_left_iff {a : Ξ±} {b : Bool} :
Step ((a, b) :: Lβ) Lβ β (β L, Step Lβ L β§ Lβ = (a, b) :: L) β¨ Lβ = (a, ! b) :: Lβ := by |
constructor
Β· generalize hL : ((a, b) :: Lβ : List _) = L
rintro @β¨_ | β¨p, s'β©, e, a', b'β©
Β· simp at hL
simp [*]
Β· simp at hL
rcases hL with β¨rfl, rflβ©
refine Or.inl β¨s' ++ e, Step.not, ?_β©
simp
Β· rintro (β¨L, h, rflβ© | rfl)
Β· exact Step.cons h
Β· exact Step.cons_not
| 12 | 162,754.791419 | 2 | 1.333333 | 3 | 1,423 |
import Mathlib.CategoryTheory.Balanced
import Mathlib.CategoryTheory.Limits.EssentiallySmall
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
import Mathlib.CategoryTheory.Subobject.Lattice
import Mathlib.CategoryTheory.Subobject.WellPowered
import Mathlib.Data.S... | Mathlib/CategoryTheory/Generator.lean | 93 | 98 | theorem isSeparating_op_iff (π’ : Set C) : IsSeparating π’.op β IsCoseparating π’ := by |
refine β¨fun hπ’ X Y f g hfg => ?_, fun hπ’ X Y f g hfg => ?_β©
Β· refine Quiver.Hom.op_inj (hπ’ _ _ fun G hG h => Quiver.Hom.unop_inj ?_)
simpa only [unop_comp, Quiver.Hom.unop_op] using hfg _ (Set.mem_op.1 hG) _
Β· refine Quiver.Hom.unop_inj (hπ’ _ _ fun G hG h => Quiver.Hom.op_inj ?_)
simpa only [op_comp,... | 5 | 148.413159 | 2 | 1.333333 | 6 | 1,424 |
import Mathlib.CategoryTheory.Balanced
import Mathlib.CategoryTheory.Limits.EssentiallySmall
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
import Mathlib.CategoryTheory.Subobject.Lattice
import Mathlib.CategoryTheory.Subobject.WellPowered
import Mathlib.Data.S... | Mathlib/CategoryTheory/Generator.lean | 101 | 106 | theorem isCoseparating_op_iff (π’ : Set C) : IsCoseparating π’.op β IsSeparating π’ := by |
refine β¨fun hπ’ X Y f g hfg => ?_, fun hπ’ X Y f g hfg => ?_β©
Β· refine Quiver.Hom.op_inj (hπ’ _ _ fun G hG h => Quiver.Hom.unop_inj ?_)
simpa only [unop_comp, Quiver.Hom.unop_op] using hfg _ (Set.mem_op.1 hG) _
Β· refine Quiver.Hom.unop_inj (hπ’ _ _ fun G hG h => Quiver.Hom.op_inj ?_)
simpa only [op_comp,... | 5 | 148.413159 | 2 | 1.333333 | 6 | 1,424 |
import Mathlib.CategoryTheory.Balanced
import Mathlib.CategoryTheory.Limits.EssentiallySmall
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
import Mathlib.CategoryTheory.Subobject.Lattice
import Mathlib.CategoryTheory.Subobject.WellPowered
import Mathlib.Data.S... | Mathlib/CategoryTheory/Generator.lean | 109 | 110 | theorem isCoseparating_unop_iff (π’ : Set Cα΅α΅) : IsCoseparating π’.unop β IsSeparating π’ := by |
rw [β isSeparating_op_iff, Set.unop_op]
| 1 | 2.718282 | 0 | 1.333333 | 6 | 1,424 |
import Mathlib.CategoryTheory.Balanced
import Mathlib.CategoryTheory.Limits.EssentiallySmall
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
import Mathlib.CategoryTheory.Subobject.Lattice
import Mathlib.CategoryTheory.Subobject.WellPowered
import Mathlib.Data.S... | Mathlib/CategoryTheory/Generator.lean | 113 | 114 | theorem isSeparating_unop_iff (π’ : Set Cα΅α΅) : IsSeparating π’.unop β IsCoseparating π’ := by |
rw [β isCoseparating_op_iff, Set.unop_op]
| 1 | 2.718282 | 0 | 1.333333 | 6 | 1,424 |
import Mathlib.CategoryTheory.Balanced
import Mathlib.CategoryTheory.Limits.EssentiallySmall
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
import Mathlib.CategoryTheory.Subobject.Lattice
import Mathlib.CategoryTheory.Subobject.WellPowered
import Mathlib.Data.S... | Mathlib/CategoryTheory/Generator.lean | 117 | 126 | theorem isDetecting_op_iff (π’ : Set C) : IsDetecting π’.op β IsCodetecting π’ := by |
refine β¨fun hπ’ X Y f hf => ?_, fun hπ’ X Y f hf => ?_β©
Β· refine (isIso_op_iff _).1 (hπ’ _ fun G hG h => ?_)
obtain β¨t, ht, ht'β© := hf (unop G) (Set.mem_op.1 hG) h.unop
exact
β¨t.op, Quiver.Hom.unop_inj ht, fun y hy => Quiver.Hom.unop_inj (ht' _ (Quiver.Hom.op_inj hy))β©
Β· refine (isIso_unop_iff _).1... | 9 | 8,103.083928 | 2 | 1.333333 | 6 | 1,424 |
import Mathlib.CategoryTheory.Balanced
import Mathlib.CategoryTheory.Limits.EssentiallySmall
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
import Mathlib.CategoryTheory.Subobject.Lattice
import Mathlib.CategoryTheory.Subobject.WellPowered
import Mathlib.Data.S... | Mathlib/CategoryTheory/Generator.lean | 129 | 138 | theorem isCodetecting_op_iff (π’ : Set C) : IsCodetecting π’.op β IsDetecting π’ := by |
refine β¨fun hπ’ X Y f hf => ?_, fun hπ’ X Y f hf => ?_β©
Β· refine (isIso_op_iff _).1 (hπ’ _ fun G hG h => ?_)
obtain β¨t, ht, ht'β© := hf (unop G) (Set.mem_op.1 hG) h.unop
exact
β¨t.op, Quiver.Hom.unop_inj ht, fun y hy => Quiver.Hom.unop_inj (ht' _ (Quiver.Hom.op_inj hy))β©
Β· refine (isIso_unop_iff _).1... | 9 | 8,103.083928 | 2 | 1.333333 | 6 | 1,424 |
import Mathlib.Algebra.Group.Subgroup.Finite
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.GroupTheory.Congruence.Basic
import Mathlib.GroupTheory.Coset
#align_import group_theory.quotient_group from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
open Function
open scope... | Mathlib/GroupTheory/QuotientGroup.lean | 108 | 113 | theorem sound (U : Set (G β§Έ N)) (g : N.op) :
g β’ (mk' N) β»ΒΉ' U = (mk' N) β»ΒΉ' U := by |
ext x
simp only [Set.mem_preimage, Set.mem_smul_set_iff_inv_smul_mem]
congr! 1
exact Quotient.sound β¨gβ»ΒΉ, rflβ©
| 4 | 54.59815 | 2 | 1.333333 | 3 | 1,425 |
import Mathlib.Algebra.Group.Subgroup.Finite
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.GroupTheory.Congruence.Basic
import Mathlib.GroupTheory.Coset
#align_import group_theory.quotient_group from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
open Function
open scope... | Mathlib/GroupTheory/QuotientGroup.lean | 129 | 131 | theorem eq_one_iff {N : Subgroup G} [nN : N.Normal] (x : G) : (x : G β§Έ N) = 1 β x β N := by |
refine QuotientGroup.eq.trans ?_
rw [mul_one, Subgroup.inv_mem_iff]
| 2 | 7.389056 | 1 | 1.333333 | 3 | 1,425 |
import Mathlib.Algebra.Group.Subgroup.Finite
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.GroupTheory.Congruence.Basic
import Mathlib.GroupTheory.Coset
#align_import group_theory.quotient_group from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
open Function
open scope... | Mathlib/GroupTheory/QuotientGroup.lean | 149 | 152 | theorem eq_iff_div_mem {N : Subgroup G} [nN : N.Normal] {x y : G} :
(x : G β§Έ N) = y β x / y β N := by |
refine eq_comm.trans (QuotientGroup.eq.trans ?_)
rw [nN.mem_comm_iff, div_eq_mul_inv]
| 2 | 7.389056 | 1 | 1.333333 | 3 | 1,425 |
import Mathlib.Combinatorics.Quiver.Cast
import Mathlib.Combinatorics.Quiver.Symmetric
import Mathlib.Data.Sigma.Basic
import Mathlib.Logic.Equiv.Basic
import Mathlib.Tactic.Common
#align_import combinatorics.quiver.covering from "leanprover-community/mathlib"@"188a411e916e1119e502dbe35b8b475716362401"
open Funct... | Mathlib/Combinatorics/Quiver/Covering.lean | 114 | 118 | theorem Prefunctor.IsCovering.map_injective (hΟ : Ο.IsCovering) {u v : U} :
Injective fun f : u βΆ v => Ο.map f := by |
rintro f g he
have : Ο.star u (Quiver.Star.mk f) = Ο.star u (Quiver.Star.mk g) := by simpa using he
simpa using (hΟ.star_bijective u).left this
| 3 | 20.085537 | 1 | 1.333333 | 3 | 1,426 |
import Mathlib.Combinatorics.Quiver.Cast
import Mathlib.Combinatorics.Quiver.Symmetric
import Mathlib.Data.Sigma.Basic
import Mathlib.Logic.Equiv.Basic
import Mathlib.Tactic.Common
#align_import combinatorics.quiver.covering from "leanprover-community/mathlib"@"188a411e916e1119e502dbe35b8b475716362401"
open Funct... | Mathlib/Combinatorics/Quiver/Covering.lean | 132 | 136 | theorem Prefunctor.IsCovering.of_comp_left (hΟ : Ο.IsCovering) (hΟΟ : (Ο βq Ο).IsCovering)
(Οsur : Surjective Ο.obj) : Ο.IsCovering := by |
refine β¨fun v => ?_, fun v => ?_β© <;> obtain β¨u, rflβ© := Οsur v
exacts [(Bijective.of_comp_iff _ (hΟ.star_bijective u)).mp (hΟΟ.star_bijective u),
(Bijective.of_comp_iff _ (hΟ.costar_bijective u)).mp (hΟΟ.costar_bijective u)]
| 3 | 20.085537 | 1 | 1.333333 | 3 | 1,426 |
import Mathlib.Combinatorics.Quiver.Cast
import Mathlib.Combinatorics.Quiver.Symmetric
import Mathlib.Data.Sigma.Basic
import Mathlib.Logic.Equiv.Basic
import Mathlib.Tactic.Common
#align_import combinatorics.quiver.covering from "leanprover-community/mathlib"@"188a411e916e1119e502dbe35b8b475716362401"
open Funct... | Mathlib/Combinatorics/Quiver/Covering.lean | 153 | 163 | theorem Prefunctor.symmetrifyStar (u : U) :
Ο.symmetrify.star u =
(Quiver.symmetrifyStar _).symm β Sum.map (Ο.star u) (Ο.costar u) β
Quiver.symmetrifyStar u := by |
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [Equiv.eq_symm_comp]
ext β¨v, f | gβ© <;>
-- porting note (#10745): was `simp [Quiver.symmetrifyStar]`
simp only [Quiver.symmetrifyStar, Function.comp_apply] <;>
erw [Equiv.sigmaSumDistrib_apply, Equiv.sigmaSumDistrib_apply] <;>... | 7 | 1,096.633158 | 2 | 1.333333 | 3 | 1,426 |
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import linear_algebra.projective_space.basic from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
variable (K V : Type*) [DivisionRing K] [AddCommGroup V] [Module K V]
def projectivizationSetoid : Setoid { v : V // v β 0 } :=
(MulA... | Mathlib/LinearAlgebra/Projectivization/Basic.lean | 108 | 116 | theorem mk_eq_mk_iff' (v w : V) (hv : v β 0) (hw : w β 0) :
mk K v hv = mk K w hw β β a : K, a β’ w = v := by |
rw [mk_eq_mk_iff K v w hv hw]
constructor
Β· rintro β¨a, haβ©
exact β¨a, haβ©
Β· rintro β¨a, haβ©
refine β¨Units.mk0 a fun c => hv.symm ?_, haβ©
rwa [c, zero_smul] at ha
| 7 | 1,096.633158 | 2 | 1.333333 | 3 | 1,427 |
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import linear_algebra.projective_space.basic from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
variable (K V : Type*) [DivisionRing K] [AddCommGroup V] [Module K V]
def projectivizationSetoid : Setoid { v : V // v β 0 } :=
(MulA... | Mathlib/LinearAlgebra/Projectivization/Basic.lean | 137 | 139 | theorem submodule_eq (v : β K V) : v.submodule = K β v.rep := by |
conv_lhs => rw [β v.mk_rep]
rfl
| 2 | 7.389056 | 1 | 1.333333 | 3 | 1,427 |
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import linear_algebra.projective_space.basic from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
variable (K V : Type*) [DivisionRing K] [AddCommGroup V] [Module K V]
def projectivizationSetoid : Setoid { v : V // v β 0 } :=
(MulA... | Mathlib/LinearAlgebra/Projectivization/Basic.lean | 142 | 144 | theorem finrank_submodule (v : β K V) : finrank K v.submodule = 1 := by |
rw [submodule_eq]
exact finrank_span_singleton v.rep_nonzero
| 2 | 7.389056 | 1 | 1.333333 | 3 | 1,427 |
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.GroupAction.Quotient
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.Topology.Algebra.Monoid
import Mathlib.Topology.Algebra.Constructions
#align_import topology.algebra.group.basic from "leanprover-community/mathlib"@"3b1890e71632be9e3... | Mathlib/Topology/Algebra/Group/Basic.lean | 71 | 73 | theorem Homeomorph.mulLeft_symm (a : G) : (Homeomorph.mulLeft a).symm = Homeomorph.mulLeft aβ»ΒΉ := by |
ext
rfl
| 2 | 7.389056 | 1 | 1.333333 | 3 | 1,428 |
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.GroupAction.Quotient
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.Topology.Algebra.Monoid
import Mathlib.Topology.Algebra.Constructions
#align_import topology.algebra.group.basic from "leanprover-community/mathlib"@"3b1890e71632be9e3... | Mathlib/Topology/Algebra/Group/Basic.lean | 114 | 117 | theorem Homeomorph.mulRight_symm (a : G) :
(Homeomorph.mulRight a).symm = Homeomorph.mulRight aβ»ΒΉ := by |
ext
rfl
| 2 | 7.389056 | 1 | 1.333333 | 3 | 1,428 |
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.GroupAction.Quotient
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.Topology.Algebra.Monoid
import Mathlib.Topology.Algebra.Constructions
#align_import topology.algebra.group.basic from "leanprover-community/mathlib"@"3b1890e71632be9e3... | Mathlib/Topology/Algebra/Group/Basic.lean | 146 | 154 | theorem discreteTopology_of_isOpen_singleton_one (h : IsOpen ({1} : Set G)) :
DiscreteTopology G := by |
rw [β singletons_open_iff_discrete]
intro g
suffices {g} = (gβ»ΒΉ * Β·) β»ΒΉ' {1} by
rw [this]
exact (continuous_mul_left gβ»ΒΉ).isOpen_preimage _ h
simp only [mul_one, Set.preimage_mul_left_singleton, eq_self_iff_true, inv_inv,
Set.singleton_eq_singleton_iff]
| 7 | 1,096.633158 | 2 | 1.333333 | 3 | 1,428 |
import Mathlib.RingTheory.JacobsonIdeal
#align_import ring_theory.nakayama from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
variable {R M : Type*} [CommRing R] [AddCommGroup M] [Module R M]
open Ideal
namespace Submodule
| Mathlib/RingTheory/Nakayama.lean | 52 | 61 | theorem eq_smul_of_le_smul_of_le_jacobson {I J : Ideal R} {N : Submodule R M} (hN : N.FG)
(hIN : N β€ I β’ N) (hIjac : I β€ jacobson J) : N = J β’ N := by |
refine le_antisymm ?_ (Submodule.smul_le.2 fun _ _ _ => Submodule.smul_mem _ _)
intro n hn
cases' Submodule.exists_sub_one_mem_and_smul_eq_zero_of_fg_of_le_smul I N hN hIN with r hr
cases' exists_mul_sub_mem_of_sub_one_mem_jacobson r (hIjac hr.1) with s hs
have : n = -(s * r - 1) β’ n := by
rw [neg_sub, s... | 8 | 2,980.957987 | 2 | 1.333333 | 3 | 1,429 |
import Mathlib.RingTheory.JacobsonIdeal
#align_import ring_theory.nakayama from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
variable {R M : Type*} [CommRing R] [AddCommGroup M] [Module R M]
open Ideal
namespace Submodule
theorem eq_smul_of_le_smul_of_le_jacobson {I J : Ideal R} {... | Mathlib/RingTheory/Nakayama.lean | 109 | 111 | theorem eq_bot_of_le_smul_of_le_jacobson_bot (I : Ideal R) (N : Submodule R M) (hN : N.FG)
(hIN : N β€ I β’ N) (hIjac : I β€ jacobson β₯) : N = β₯ := by |
rw [eq_smul_of_le_smul_of_le_jacobson hN hIN hIjac, Submodule.bot_smul]
| 1 | 2.718282 | 0 | 1.333333 | 3 | 1,429 |
import Mathlib.RingTheory.JacobsonIdeal
#align_import ring_theory.nakayama from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
variable {R M : Type*} [CommRing R] [AddCommGroup M] [Module R M]
open Ideal
namespace Submodule
theorem eq_smul_of_le_smul_of_le_jacobson {I J : Ideal R} {... | Mathlib/RingTheory/Nakayama.lean | 114 | 126 | theorem sup_eq_sup_smul_of_le_smul_of_le_jacobson {I J : Ideal R} {N N' : Submodule R M}
(hN' : N'.FG) (hIJ : I β€ jacobson J) (hNN : N' β€ N β I β’ N') : N β N' = N β J β’ N' := by |
have hNN' : N β N' = N β I β’ N' :=
le_antisymm (sup_le le_sup_left hNN)
(sup_le_sup_left (Submodule.smul_le.2 fun _ _ _ => Submodule.smul_mem _ _) _)
have h_comap := Submodule.comap_injective_of_surjective (LinearMap.range_eq_top.1 N.range_mkQ)
have : (I β’ N').map N.mkQ = N'.map N.mkQ := by
simpa onl... | 11 | 59,874.141715 | 2 | 1.333333 | 3 | 1,429 |
import Mathlib.LinearAlgebra.Matrix.Adjugate
import Mathlib.RingTheory.PolynomialAlgebra
#align_import linear_algebra.matrix.charpoly.basic from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
noncomputable section
universe u v w
namespace Matrix
open Finset Matrix Polynomial
variable... | Mathlib/LinearAlgebra/Matrix/Charpoly/Basic.lean | 55 | 57 | theorem charmatrix_apply_eq : charmatrix M i i = (X : R[X]) - C (M i i) := by |
simp only [charmatrix, RingHom.mapMatrix_apply, sub_apply, scalar_apply, map_apply,
diagonal_apply_eq]
| 2 | 7.389056 | 1 | 1.333333 | 6 | 1,430 |
import Mathlib.LinearAlgebra.Matrix.Adjugate
import Mathlib.RingTheory.PolynomialAlgebra
#align_import linear_algebra.matrix.charpoly.basic from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
noncomputable section
universe u v w
namespace Matrix
open Finset Matrix Polynomial
variable... | Mathlib/LinearAlgebra/Matrix/Charpoly/Basic.lean | 62 | 64 | theorem charmatrix_apply_ne (h : i β j) : charmatrix M i j = -C (M i j) := by |
simp only [charmatrix, RingHom.mapMatrix_apply, sub_apply, scalar_apply, diagonal_apply_ne _ h,
map_apply, sub_eq_neg_self]
| 2 | 7.389056 | 1 | 1.333333 | 6 | 1,430 |
import Mathlib.LinearAlgebra.Matrix.Adjugate
import Mathlib.RingTheory.PolynomialAlgebra
#align_import linear_algebra.matrix.charpoly.basic from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
noncomputable section
universe u v w
namespace Matrix
open Finset Matrix Polynomial
variable... | Mathlib/LinearAlgebra/Matrix/Charpoly/Basic.lean | 67 | 76 | theorem matPolyEquiv_charmatrix : matPolyEquiv (charmatrix M) = X - C M := by |
ext k i j
simp only [matPolyEquiv_coeff_apply, coeff_sub, Pi.sub_apply]
by_cases h : i = j
Β· subst h
rw [charmatrix_apply_eq, coeff_sub]
simp only [coeff_X, coeff_C]
split_ifs <;> simp
Β· rw [charmatrix_apply_ne _ _ _ h, coeff_X, coeff_neg, coeff_C, coeff_C]
split_ifs <;> simp [h]
| 9 | 8,103.083928 | 2 | 1.333333 | 6 | 1,430 |
import Mathlib.LinearAlgebra.Matrix.Adjugate
import Mathlib.RingTheory.PolynomialAlgebra
#align_import linear_algebra.matrix.charpoly.basic from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
noncomputable section
universe u v w
namespace Matrix
open Finset Matrix Polynomial
variable... | Mathlib/LinearAlgebra/Matrix/Charpoly/Basic.lean | 79 | 83 | theorem charmatrix_reindex (e : n β m) :
charmatrix (reindex e e M) = reindex e e (charmatrix M) := by |
ext i j x
by_cases h : i = j
all_goals simp [h]
| 3 | 20.085537 | 1 | 1.333333 | 6 | 1,430 |
import Mathlib.LinearAlgebra.Matrix.Adjugate
import Mathlib.RingTheory.PolynomialAlgebra
#align_import linear_algebra.matrix.charpoly.basic from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
noncomputable section
universe u v w
namespace Matrix
open Finset Matrix Polynomial
variable... | Mathlib/LinearAlgebra/Matrix/Charpoly/Basic.lean | 103 | 106 | theorem charpoly_reindex (e : n β m)
(M : Matrix n n R) : (reindex e e M).charpoly = M.charpoly := by |
unfold Matrix.charpoly
rw [charmatrix_reindex, Matrix.det_reindex_self]
| 2 | 7.389056 | 1 | 1.333333 | 6 | 1,430 |
import Mathlib.LinearAlgebra.Matrix.Adjugate
import Mathlib.RingTheory.PolynomialAlgebra
#align_import linear_algebra.matrix.charpoly.basic from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
noncomputable section
universe u v w
namespace Matrix
open Finset Matrix Polynomial
variable... | Mathlib/LinearAlgebra/Matrix/Charpoly/Basic.lean | 134 | 154 | theorem aeval_self_charpoly (M : Matrix n n R) : aeval M M.charpoly = 0 := by |
-- We begin with the fact $Ο_M(t) I = adjugate (t I - M) * (t I - M)$,
-- as an identity in `Matrix n n R[X]`.
have h : M.charpoly β’ (1 : Matrix n n R[X]) = adjugate (charmatrix M) * charmatrix M :=
(adjugate_mul _).symm
-- Using the algebra isomorphism `Matrix n n R[X] ββ[R] Polynomial (Matrix n n R)`,
... | 20 | 485,165,195.40979 | 2 | 1.333333 | 6 | 1,430 |
import Mathlib.RingTheory.Valuation.Basic
import Mathlib.RingTheory.Ideal.QuotientOperations
#align_import ring_theory.valuation.quotient from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
namespace Valuation
variable {R Ξβ : Type*} [CommRing R] [LinearOrderedCommMonoidWithZero Ξβ]
va... | Mathlib/RingTheory/Valuation/Quotient.lean | 51 | 54 | theorem self_le_supp_comap (J : Ideal R) (v : Valuation (R β§Έ J) Ξβ) :
J β€ (v.comap (Ideal.Quotient.mk J)).supp := by |
rw [comap_supp, β Ideal.map_le_iff_le_comap]
simp
| 2 | 7.389056 | 1 | 1.333333 | 3 | 1,431 |
import Mathlib.RingTheory.Valuation.Basic
import Mathlib.RingTheory.Ideal.QuotientOperations
#align_import ring_theory.valuation.quotient from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
namespace Valuation
variable {R Ξβ : Type*} [CommRing R] [LinearOrderedCommMonoidWithZero Ξβ]
va... | Mathlib/RingTheory/Valuation/Quotient.lean | 66 | 74 | theorem supp_quot {J : Ideal R} (hJ : J β€ supp v) :
supp (v.onQuot hJ) = (supp v).map (Ideal.Quotient.mk J) := by |
apply le_antisymm
Β· rintro β¨xβ© hx
apply Ideal.subset_span
exact β¨x, hx, rflβ©
Β· rw [Ideal.map_le_iff_le_comap]
intro x hx
exact hx
| 7 | 1,096.633158 | 2 | 1.333333 | 3 | 1,431 |
import Mathlib.RingTheory.Valuation.Basic
import Mathlib.RingTheory.Ideal.QuotientOperations
#align_import ring_theory.valuation.quotient from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
namespace Valuation
variable {R Ξβ : Type*} [CommRing R] [LinearOrderedCommMonoidWithZero Ξβ]
va... | Mathlib/RingTheory/Valuation/Quotient.lean | 77 | 79 | theorem supp_quot_supp : supp (v.onQuot le_rfl) = 0 := by |
rw [supp_quot]
exact Ideal.map_quotient_self _
| 2 | 7.389056 | 1 | 1.333333 | 3 | 1,431 |
import Mathlib.CategoryTheory.Limits.Shapes.Terminal
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
#align_import category_theory.limits.shapes.strict_initial from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe v u
namespace CategoryTheory
namespace Limits
open C... | Mathlib/CategoryTheory/Limits/Shapes/StrictInitial.lean | 74 | 77 | theorem IsInitial.strict_hom_ext (hI : IsInitial I) {A : C} (f g : A βΆ I) : f = g := by |
haveI := hI.isIso_to f
haveI := hI.isIso_to g
exact eq_of_inv_eq_inv (hI.hom_ext (inv f) (inv g))
| 3 | 20.085537 | 1 | 1.333333 | 3 | 1,432 |
import Mathlib.CategoryTheory.Limits.Shapes.Terminal
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
#align_import category_theory.limits.shapes.strict_initial from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe v u
namespace CategoryTheory
namespace Limits
open C... | Mathlib/CategoryTheory/Limits/Shapes/StrictInitial.lean | 192 | 195 | theorem IsTerminal.strict_hom_ext (hI : IsTerminal I) {A : C} (f g : I βΆ A) : f = g := by |
haveI := hI.isIso_from f
haveI := hI.isIso_from g
exact eq_of_inv_eq_inv (hI.hom_ext (inv f) (inv g))
| 3 | 20.085537 | 1 | 1.333333 | 3 | 1,432 |
import Mathlib.CategoryTheory.Limits.Shapes.Terminal
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
#align_import category_theory.limits.shapes.strict_initial from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe v u
namespace CategoryTheory
namespace Limits
open C... | Mathlib/CategoryTheory/Limits/Shapes/StrictInitial.lean | 206 | 237 | theorem limit_Ο_isIso_of_is_strict_terminal (F : J β₯€ C) [HasLimit F] (i : J)
(H : β (j) (_ : j β i), IsTerminal (F.obj j)) [Subsingleton (i βΆ i)] : IsIso (limit.Ο F i) := by |
classical
refine β¨β¨limit.lift _ β¨_, β¨?_, ?_β©β©, ?_, ?_β©β©
Β· exact fun j =>
dite (j = i)
(fun h => eqToHom (by cases h; rfl))
fun h => (H _ h).from _
Β· intro j k f
split_ifs with h h_1 h_1
Β· cases h
cases h_1
obtain rfl : f = π _ := Subsingleton.elim ... | 30 | 10,686,474,581,524.463 | 2 | 1.333333 | 3 | 1,432 |
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Asymptotics.SpecificAsymptotics
#align_import analysis.special_functions.compare_exp from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8"
open Asympto... | Mathlib/Analysis/SpecialFunctions/CompareExp.lean | 107 | 116 | theorem isLittleO_im_pow_exp_re (hl : IsExpCmpFilter l) (n : β) :
(fun z : β => z.im ^ n) =o[l] fun z => Real.exp z.re :=
flip IsLittleO.of_pow two_ne_zero <|
calc
(fun z : β β¦ (z.im ^ n) ^ 2) = (fun z β¦ z.im ^ (2 * n)) := by | simp only [pow_mul']
_ =O[l] fun z β¦ Real.exp z.re := hl.isBigO_im_pow_re _
_ = fun z β¦ (Real.exp z.re) ^ 1 := by simp only [pow_one]
_ =o[l] fun z β¦ (Real.exp z.re) ^ 2 :=
(isLittleO_pow_pow_atTop_of_lt one_lt_two).comp_tendsto <|
Real.tendsto_exp_atTop.comp hl.tendsto_re
| 6 | 403.428793 | 2 | 1.333333 | 3 | 1,433 |
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Asymptotics.SpecificAsymptotics
#align_import analysis.special_functions.compare_exp from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8"
open Asympto... | Mathlib/Analysis/SpecialFunctions/CompareExp.lean | 119 | 121 | theorem abs_im_pow_eventuallyLE_exp_re (hl : IsExpCmpFilter l) (n : β) :
(fun z : β => |z.im| ^ n) β€αΆ [l] fun z => Real.exp z.re := by |
simpa using (hl.isLittleO_im_pow_exp_re n).bound zero_lt_one
| 1 | 2.718282 | 0 | 1.333333 | 3 | 1,433 |
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Asymptotics.SpecificAsymptotics
#align_import analysis.special_functions.compare_exp from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8"
open Asympto... | Mathlib/Analysis/SpecialFunctions/CompareExp.lean | 127 | 151 | theorem isLittleO_log_abs_re (hl : IsExpCmpFilter l) : (fun z => Real.log (abs z)) =o[l] re :=
calc
(fun z => Real.log (abs z)) =O[l] fun z => Real.log (β2) + Real.log (max z.re |z.im|) :=
IsBigO.of_bound 1 <|
(hl.tendsto_re.eventually_ge_atTop 1).mono fun z hz => by
have h2 : 0 < β2 := by | simp
have hz' : 1 β€ abs z := hz.trans (re_le_abs z)
have hmβ : 0 < max z.re |z.im| := lt_max_iff.2 (Or.inl <| one_pos.trans_le hz)
rw [one_mul, Real.norm_eq_abs, _root_.abs_of_nonneg (Real.log_nonneg hz')]
refine le_trans ?_ (le_abs_self _)
rw [β Real.log_mul, Real.log... | 20 | 485,165,195.40979 | 2 | 1.333333 | 3 | 1,433 |
import Mathlib.Data.Nat.Defs
import Mathlib.Tactic.GCongr.Core
import Mathlib.Tactic.Common
import Mathlib.Tactic.Monotonicity.Attr
#align_import data.nat.factorial.basic from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
namespace Nat
def factorial : β β β
| 0 => 1
| succ n => s... | Mathlib/Data/Nat/Factorial/Basic.lean | 73 | 76 | theorem factorial_dvd_factorial {m n} (h : m β€ n) : m ! β£ n ! := by |
induction' h with n _ ih
Β· exact Nat.dvd_refl _
Β· exact Nat.dvd_trans ih (Nat.dvd_mul_left _ _)
| 3 | 20.085537 | 1 | 1.333333 | 9 | 1,434 |
import Mathlib.Data.Nat.Defs
import Mathlib.Tactic.GCongr.Core
import Mathlib.Tactic.Common
import Mathlib.Tactic.Monotonicity.Attr
#align_import data.nat.factorial.basic from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
namespace Nat
def factorial : β β β
| 0 => 1
| succ n => s... | Mathlib/Data/Nat/Factorial/Basic.lean | 95 | 103 | theorem factorial_lt (hn : 0 < n) : n ! < m ! β n < m := by |
refine β¨fun h => not_le.mp fun hmn => Nat.not_le_of_lt h (factorial_le hmn), fun h => ?_β©
have : β {n}, 0 < n β n ! < (n + 1)! := by
intro k hk
rw [factorial_succ, succ_mul, Nat.lt_add_left_iff_pos]
exact Nat.mul_pos hk k.factorial_pos
induction' h with k hnk ih generalizing hn
Β· exact this hn
Β· ... | 8 | 2,980.957987 | 2 | 1.333333 | 9 | 1,434 |
import Mathlib.Data.Nat.Defs
import Mathlib.Tactic.GCongr.Core
import Mathlib.Tactic.Common
import Mathlib.Tactic.Monotonicity.Attr
#align_import data.nat.factorial.basic from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
namespace Nat
def factorial : β β β
| 0 => 1
| succ n => s... | Mathlib/Data/Nat/Factorial/Basic.lean | 113 | 118 | theorem factorial_eq_one : n ! = 1 β n β€ 1 := by |
constructor
Β· intro h
rw [β not_lt, β one_lt_factorial, h]
apply lt_irrefl
Β· rintro (_|_|_) <;> rfl
| 5 | 148.413159 | 2 | 1.333333 | 9 | 1,434 |
import Mathlib.Data.Nat.Defs
import Mathlib.Tactic.GCongr.Core
import Mathlib.Tactic.Common
import Mathlib.Tactic.Monotonicity.Attr
#align_import data.nat.factorial.basic from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
namespace Nat
def factorial : β β β
| 0 => 1
| succ n => s... | Mathlib/Data/Nat/Factorial/Basic.lean | 121 | 129 | theorem factorial_inj (hn : 1 < n) : n ! = m ! β n = m := by |
refine β¨fun h => ?_, congr_arg _β©
obtain hnm | rfl | hnm := lt_trichotomy n m
Β· rw [β factorial_lt <| lt_of_succ_lt hn, h] at hnm
cases lt_irrefl _ hnm
Β· rfl
rw [β one_lt_factorial, h, one_lt_factorial] at hn
rw [β factorial_lt <| lt_of_succ_lt hn, h] at hnm
cases lt_irrefl _ hnm
| 8 | 2,980.957987 | 2 | 1.333333 | 9 | 1,434 |
import Mathlib.Data.Nat.Defs
import Mathlib.Tactic.GCongr.Core
import Mathlib.Tactic.Common
import Mathlib.Tactic.Monotonicity.Attr
#align_import data.nat.factorial.basic from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
namespace Nat
def factorial : β β β
| 0 => 1
| succ n => s... | Mathlib/Data/Nat/Factorial/Basic.lean | 132 | 135 | theorem factorial_inj' (h : 1 < n β¨ 1 < m) : n ! = m ! β n = m := by |
obtain hn|hm := h
Β· exact factorial_inj hn
Β· rw [eq_comm, factorial_inj hm, eq_comm]
| 3 | 20.085537 | 1 | 1.333333 | 9 | 1,434 |
import Mathlib.Data.Nat.Defs
import Mathlib.Tactic.GCongr.Core
import Mathlib.Tactic.Common
import Mathlib.Tactic.Monotonicity.Attr
#align_import data.nat.factorial.basic from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
namespace Nat
def factorial : β β β
| 0 => 1
| succ n => s... | Mathlib/Data/Nat/Factorial/Basic.lean | 142 | 147 | theorem lt_factorial_self {n : β} (hi : 3 β€ n) : n < n ! := by |
have : 0 < n := by omega
have hn : 1 < pred n := le_pred_of_lt (succ_le_iff.mp hi)
rw [β succ_pred_eq_of_pos βΉ0 < nβΊ, factorial_succ]
exact (Nat.lt_mul_iff_one_lt_right (pred n).succ_pos).2
((Nat.lt_of_lt_of_le hn (self_le_factorial _)))
| 5 | 148.413159 | 2 | 1.333333 | 9 | 1,434 |
import Mathlib.Data.Nat.Defs
import Mathlib.Tactic.GCongr.Core
import Mathlib.Tactic.Common
import Mathlib.Tactic.Monotonicity.Attr
#align_import data.nat.factorial.basic from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
namespace Nat
def factorial : β β β
| 0 => 1
| succ n => s... | Mathlib/Data/Nat/Factorial/Basic.lean | 150 | 155 | theorem add_factorial_succ_lt_factorial_add_succ {i : β} (n : β) (hi : 2 β€ i) :
i + (n + 1)! < (i + n + 1)! := by |
rw [factorial_succ (i + _), Nat.add_mul, Nat.one_mul]
have := (i + n).self_le_factorial
refine Nat.add_lt_add_of_lt_of_le (Nat.lt_of_le_of_lt ?_ ((Nat.lt_mul_iff_one_lt_right ?_).2 ?_))
(factorial_le ?_) <;> omega
| 4 | 54.59815 | 2 | 1.333333 | 9 | 1,434 |
import Mathlib.Data.Nat.Defs
import Mathlib.Tactic.GCongr.Core
import Mathlib.Tactic.Common
import Mathlib.Tactic.Monotonicity.Attr
#align_import data.nat.factorial.basic from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
namespace Nat
def factorial : β β β
| 0 => 1
| succ n => s... | Mathlib/Data/Nat/Factorial/Basic.lean | 340 | 341 | theorem zero_descFactorial_succ (k : β) : (0 : β).descFactorial (k + 1) = 0 := by |
rw [descFactorial_succ, Nat.zero_sub, Nat.zero_mul]
| 1 | 2.718282 | 0 | 1.333333 | 9 | 1,434 |
import Mathlib.Data.Nat.Defs
import Mathlib.Tactic.GCongr.Core
import Mathlib.Tactic.Common
import Mathlib.Tactic.Monotonicity.Attr
#align_import data.nat.factorial.basic from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
namespace Nat
def factorial : β β β
| 0 => 1
| succ n => s... | Mathlib/Data/Nat/Factorial/Basic.lean | 344 | 344 | theorem descFactorial_one (n : β) : n.descFactorial 1 = n := by | simp
| 1 | 2.718282 | 0 | 1.333333 | 9 | 1,434 |
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.MeasureTheory.Covering.Besicovitch
import Mathlib.Tactic.AdaptationNote
#align_import measure_theory.covering.besicovitch_vector_space from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
universe u
open Metric Set Fini... | Mathlib/MeasureTheory/Covering/BesicovitchVectorSpace.lean | 83 | 84 | theorem centerAndRescale_center : a.centerAndRescale.c (last N) = 0 := by |
simp [SatelliteConfig.centerAndRescale]
| 1 | 2.718282 | 0 | 1.333333 | 6 | 1,435 |
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.MeasureTheory.Covering.Besicovitch
import Mathlib.Tactic.AdaptationNote
#align_import measure_theory.covering.besicovitch_vector_space from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
universe u
open Metric Set Fini... | Mathlib/MeasureTheory/Covering/BesicovitchVectorSpace.lean | 87 | 89 | theorem centerAndRescale_radius {N : β} {Ο : β} (a : SatelliteConfig E N Ο) :
a.centerAndRescale.r (last N) = 1 := by |
simp [SatelliteConfig.centerAndRescale, inv_mul_cancel (a.rpos _).ne']
| 1 | 2.718282 | 0 | 1.333333 | 6 | 1,435 |
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.MeasureTheory.Covering.Besicovitch
import Mathlib.Tactic.AdaptationNote
#align_import measure_theory.covering.besicovitch_vector_space from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
universe u
open Metric Set Fini... | Mathlib/MeasureTheory/Covering/BesicovitchVectorSpace.lean | 110 | 150 | theorem card_le_of_separated (s : Finset E) (hs : β c β s, βcβ β€ 2)
(h : β c β s, β d β s, c β d β 1 β€ βc - dβ) : s.card β€ 5 ^ finrank β E := by |
/- We consider balls of radius `1/2` around the points in `s`. They are disjoint, and all
contained in the ball of radius `5/2`. A volume argument gives `s.card * (1/2)^dim β€ (5/2)^dim`,
i.e., `s.card β€ 5^dim`. -/
borelize E
let ΞΌ : Measure E := Measure.addHaar
let Ξ΄ : β := (1 : β) / 2
let Ο : β := (... | 39 | 86,593,400,423,993,740 | 2 | 1.333333 | 6 | 1,435 |
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.MeasureTheory.Covering.Besicovitch
import Mathlib.Tactic.AdaptationNote
#align_import measure_theory.covering.besicovitch_vector_space from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
universe u
open Metric Set Fini... | Mathlib/MeasureTheory/Covering/BesicovitchVectorSpace.lean | 153 | 157 | theorem multiplicity_le : multiplicity E β€ 5 ^ finrank β E := by |
apply csSup_le
Β· refine β¨0, β¨β
, by simpβ©β©
Β· rintro _ β¨s, β¨rfl, hβ©β©
exact Besicovitch.card_le_of_separated s h.1 h.2
| 4 | 54.59815 | 2 | 1.333333 | 6 | 1,435 |
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.MeasureTheory.Covering.Besicovitch
import Mathlib.Tactic.AdaptationNote
#align_import measure_theory.covering.besicovitch_vector_space from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
universe u
open Metric Set Fini... | Mathlib/MeasureTheory/Covering/BesicovitchVectorSpace.lean | 160 | 167 | theorem card_le_multiplicity {s : Finset E} (hs : β c β s, βcβ β€ 2)
(h's : β c β s, β d β s, c β d β 1 β€ βc - dβ) : s.card β€ multiplicity E := by |
apply le_csSup
Β· refine β¨5 ^ finrank β E, ?_β©
rintro _ β¨s, β¨rfl, hβ©β©
exact Besicovitch.card_le_of_separated s h.1 h.2
Β· simp only [mem_setOf_eq, Ne]
exact β¨s, rfl, hs, h'sβ©
| 6 | 403.428793 | 2 | 1.333333 | 6 | 1,435 |
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.MeasureTheory.Covering.Besicovitch
import Mathlib.Tactic.AdaptationNote
#align_import measure_theory.covering.besicovitch_vector_space from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
universe u
open Metric Set Fini... | Mathlib/MeasureTheory/Covering/BesicovitchVectorSpace.lean | 174 | 246 | theorem exists_goodΞ΄ :
β Ξ΄ : β, 0 < Ξ΄ β§ Ξ΄ < 1 β§ β s : Finset E, (β c β s, βcβ β€ 2) β
(β c β s, β d β s, c β d β 1 - Ξ΄ β€ βc - dβ) β s.card β€ multiplicity E := by |
classical
/- This follows from a compactness argument: otherwise, one could extract a converging
subsequence, to obtain a `1`-separated set in the ball of radius `2` with cardinality
`N = multiplicity E + 1`. To formalize this, we work with functions `Fin N β E`.
-/
by_contra! h
set N := multiplic... | 70 | 2,515,438,670,919,167,200,000,000,000,000 | 2 | 1.333333 | 6 | 1,435 |
import Mathlib.Data.Rat.Cast.Defs
import Mathlib.Algebra.Field.Basic
#align_import data.rat.cast from "leanprover-community/mathlib"@"acebd8d49928f6ed8920e502a6c90674e75bd441"
namespace Rat
variable {Ξ± : Type*} [DivisionRing Ξ±]
-- Porting note: rewrote proof
@[simp]
| Mathlib/Data/Rat/Cast/Lemmas.lean | 28 | 32 | theorem cast_inv_nat (n : β) : ((nβ»ΒΉ : β) : Ξ±) = (n : Ξ±)β»ΒΉ := by |
cases' n with n
Β· simp
rw [cast_def, inv_natCast_num, inv_natCast_den, if_neg n.succ_ne_zero,
Int.sign_eq_one_of_pos (Nat.cast_pos.mpr n.succ_pos), Int.cast_one, one_div]
| 4 | 54.59815 | 2 | 1.333333 | 6 | 1,436 |
import Mathlib.Data.Rat.Cast.Defs
import Mathlib.Algebra.Field.Basic
#align_import data.rat.cast from "leanprover-community/mathlib"@"acebd8d49928f6ed8920e502a6c90674e75bd441"
namespace Rat
variable {Ξ± : Type*} [DivisionRing Ξ±]
-- Porting note: rewrote proof
@[simp]
theorem cast_inv_nat (n : β) : ((nβ»ΒΉ : β) : Ξ±... | Mathlib/Data/Rat/Cast/Lemmas.lean | 37 | 40 | theorem cast_inv_int (n : β€) : ((nβ»ΒΉ : β) : Ξ±) = (n : Ξ±)β»ΒΉ := by |
cases' n with n n
Β· simp [ofInt_eq_cast, cast_inv_nat]
Β· simp only [ofInt_eq_cast, Int.cast_negSucc, β Nat.cast_succ, cast_neg, inv_neg, cast_inv_nat]
| 3 | 20.085537 | 1 | 1.333333 | 6 | 1,436 |
import Mathlib.Data.Rat.Cast.Defs
import Mathlib.Algebra.Field.Basic
#align_import data.rat.cast from "leanprover-community/mathlib"@"acebd8d49928f6ed8920e502a6c90674e75bd441"
namespace Rat
variable {Ξ± : Type*} [DivisionRing Ξ±]
-- Porting note: rewrote proof
@[simp]
theorem cast_inv_nat (n : β) : ((nβ»ΒΉ : β) : Ξ±... | Mathlib/Data/Rat/Cast/Lemmas.lean | 44 | 51 | theorem cast_nnratCast {K} [DivisionRing K] (q : ββ₯0) :
((q : β) : K) = (q : K) := by |
rw [Rat.cast_def, NNRat.cast_def, NNRat.cast_def]
have hn := @num_div_eq_of_coprime q.num q.den ?hdp q.coprime_num_den
on_goal 1 => have hd := @den_div_eq_of_coprime q.num q.den ?hdp q.coprime_num_den
case hdp => simpa only [Nat.cast_pos] using q.den_pos
simp only [Int.cast_natCast, Nat.cast_inj] at hn hd
... | 6 | 403.428793 | 2 | 1.333333 | 6 | 1,436 |
import Mathlib.Data.Rat.Cast.Defs
import Mathlib.Algebra.Field.Basic
#align_import data.rat.cast from "leanprover-community/mathlib"@"acebd8d49928f6ed8920e502a6c90674e75bd441"
namespace Rat
variable {Ξ± : Type*} [DivisionRing Ξ±]
-- Porting note: rewrote proof
@[simp]
theorem cast_inv_nat (n : β) : ((nβ»ΒΉ : β) : Ξ±... | Mathlib/Data/Rat/Cast/Lemmas.lean | 55 | 57 | theorem cast_ofScientific {K} [DivisionRing K] (m : β) (s : Bool) (e : β) :
(OfScientific.ofScientific m s e : β) = (OfScientific.ofScientific m s e : K) := by |
rw [β NNRat.cast_ofScientific (K := K), β NNRat.cast_ofScientific, cast_nnratCast]
| 1 | 2.718282 | 0 | 1.333333 | 6 | 1,436 |
import Mathlib.Data.Rat.Cast.Defs
import Mathlib.Algebra.Field.Basic
#align_import data.rat.cast from "leanprover-community/mathlib"@"acebd8d49928f6ed8920e502a6c90674e75bd441"
namespace NNRat
@[simp, norm_cast]
| Mathlib/Data/Rat/Cast/Lemmas.lean | 64 | 67 | theorem cast_pow {K} [DivisionSemiring K] (q : ββ₯0) (n : β) :
NNRat.cast (q ^ n) = (NNRat.cast q : K) ^ n := by |
rw [cast_def, cast_def, den_pow, num_pow, Nat.cast_pow, Nat.cast_pow, div_eq_mul_inv, β inv_pow,
β (Nat.cast_commute _ _).mul_pow, β div_eq_mul_inv]
| 2 | 7.389056 | 1 | 1.333333 | 6 | 1,436 |
import Mathlib.Data.Rat.Cast.Defs
import Mathlib.Algebra.Field.Basic
#align_import data.rat.cast from "leanprover-community/mathlib"@"acebd8d49928f6ed8920e502a6c90674e75bd441"
namespace NNRat
@[simp, norm_cast]
theorem cast_pow {K} [DivisionSemiring K] (q : ββ₯0) (n : β) :
NNRat.cast (q ^ n) = (NNRat.cast q :... | Mathlib/Data/Rat/Cast/Lemmas.lean | 69 | 75 | theorem cast_zpow_of_ne_zero {K} [DivisionSemiring K] (q : ββ₯0) (z : β€) (hq : (q.num : K) β 0) :
NNRat.cast (q ^ z) = (NNRat.cast q : K) ^ z := by |
obtain β¨n, rfl | rflβ© := z.eq_nat_or_neg
Β· simp
Β· simp_rw [zpow_neg, zpow_natCast, β inv_pow, NNRat.cast_pow]
congr
rw [cast_inv_of_ne_zero hq]
| 5 | 148.413159 | 2 | 1.333333 | 6 | 1,436 |
import Mathlib.Geometry.Manifold.ContMDiff.Atlas
import Mathlib.Geometry.Manifold.VectorBundle.FiberwiseLinear
import Mathlib.Topology.VectorBundle.Constructions
#align_import geometry.manifold.vector_bundle.basic from "leanprover-community/mathlib"@"e473c3198bb41f68560cab68a0529c854b618833"
assert_not_exists mfde... | Mathlib/Geometry/Manifold/VectorBundle/Basic.lean | 108 | 114 | theorem FiberBundle.chartedSpace_chartAt (x : TotalSpace F E) :
chartAt (ModelProd HB F) x =
(trivializationAt F E x.proj).toPartialHomeomorph β«β
(chartAt HB x.proj).prod (PartialHomeomorph.refl F) := by |
dsimp only [chartAt_comp, prodChartedSpace_chartAt, FiberBundle.chartedSpace'_chartAt,
chartAt_self_eq]
rw [Trivialization.coe_coe, Trivialization.coe_fst' _ (mem_baseSet_trivializationAt F E x.proj)]
| 3 | 20.085537 | 1 | 1.333333 | 3 | 1,437 |
import Mathlib.Geometry.Manifold.ContMDiff.Atlas
import Mathlib.Geometry.Manifold.VectorBundle.FiberwiseLinear
import Mathlib.Topology.VectorBundle.Constructions
#align_import geometry.manifold.vector_bundle.basic from "leanprover-community/mathlib"@"e473c3198bb41f68560cab68a0529c854b618833"
assert_not_exists mfde... | Mathlib/Geometry/Manifold/VectorBundle/Basic.lean | 117 | 121 | theorem FiberBundle.chartedSpace_chartAt_symm_fst (x : TotalSpace F E) (y : ModelProd HB F)
(hy : y β (chartAt (ModelProd HB F) x).target) :
((chartAt (ModelProd HB F) x).symm y).proj = (chartAt HB x.proj).symm y.1 := by |
simp only [FiberBundle.chartedSpace_chartAt, mfld_simps] at hy β’
exact (trivializationAt F E x.proj).proj_symm_apply hy.2
| 2 | 7.389056 | 1 | 1.333333 | 3 | 1,437 |
import Mathlib.Geometry.Manifold.ContMDiff.Atlas
import Mathlib.Geometry.Manifold.VectorBundle.FiberwiseLinear
import Mathlib.Topology.VectorBundle.Constructions
#align_import geometry.manifold.vector_bundle.basic from "leanprover-community/mathlib"@"e473c3198bb41f68560cab68a0529c854b618833"
assert_not_exists mfde... | Mathlib/Geometry/Manifold/VectorBundle/Basic.lean | 178 | 196 | theorem contMDiffWithinAt_totalSpace (f : M β TotalSpace F E) {s : Set M} {xβ : M} :
ContMDiffWithinAt IM (IB.prod π(π, F)) n f s xβ β
ContMDiffWithinAt IM IB n (fun x => (f x).proj) s xβ β§
ContMDiffWithinAt IM π(π, F) n (fun x β¦ (trivializationAt F E (f xβ).proj (f x)).2) s xβ := by |
simp (config := { singlePass := true }) only [contMDiffWithinAt_iff_target]
rw [and_and_and_comm, β FiberBundle.continuousWithinAt_totalSpace, and_congr_right_iff]
intro hf
simp_rw [modelWithCornersSelf_prod, FiberBundle.extChartAt, Function.comp,
PartialEquiv.trans_apply, PartialEquiv.prod_coe, PartialEqu... | 15 | 3,269,017.372472 | 2 | 1.333333 | 3 | 1,437 |
import Mathlib.Analysis.Convex.Basic
import Mathlib.Topology.Algebra.Group.Basic
import Mathlib.Topology.Order.Basic
#align_import analysis.convex.strict from "leanprover-community/mathlib"@"84dc0bd6619acaea625086d6f53cb35cdd554219"
open Set
open Convex Pointwise
variable {π π E F Ξ² : Type*}
open Function Se... | Mathlib/Analysis/Convex/Strict.lean | 67 | 70 | theorem strictConvex_univ : StrictConvex π (univ : Set E) := by |
intro x _ y _ _ a b _ _ _
rw [interior_univ]
exact mem_univ _
| 3 | 20.085537 | 1 | 1.333333 | 3 | 1,438 |
import Mathlib.Analysis.Convex.Basic
import Mathlib.Topology.Algebra.Group.Basic
import Mathlib.Topology.Order.Basic
#align_import analysis.convex.strict from "leanprover-community/mathlib"@"84dc0bd6619acaea625086d6f53cb35cdd554219"
open Set
open Convex Pointwise
variable {π π E F Ξ² : Type*}
open Function Se... | Mathlib/Analysis/Convex/Strict.lean | 85 | 92 | theorem Directed.strictConvex_iUnion {ΞΉ : Sort*} {s : ΞΉ β Set E} (hdir : Directed (Β· β Β·) s)
(hs : β β¦i : ΞΉβ¦, StrictConvex π (s i)) : StrictConvex π (β i, s i) := by |
rintro x hx y hy hxy a b ha hb hab
rw [mem_iUnion] at hx hy
obtain β¨i, hxβ© := hx
obtain β¨j, hyβ© := hy
obtain β¨k, hik, hjkβ© := hdir i j
exact interior_mono (subset_iUnion s k) (hs (hik hx) (hjk hy) hxy ha hb hab)
| 6 | 403.428793 | 2 | 1.333333 | 3 | 1,438 |
import Mathlib.Analysis.Convex.Basic
import Mathlib.Topology.Algebra.Group.Basic
import Mathlib.Topology.Order.Basic
#align_import analysis.convex.strict from "leanprover-community/mathlib"@"84dc0bd6619acaea625086d6f53cb35cdd554219"
open Set
open Convex Pointwise
variable {π π E F Ξ² : Type*}
open Function Se... | Mathlib/Analysis/Convex/Strict.lean | 95 | 98 | theorem DirectedOn.strictConvex_sUnion {S : Set (Set E)} (hdir : DirectedOn (Β· β Β·) S)
(hS : β s β S, StrictConvex π s) : StrictConvex π (ββ S) := by |
rw [sUnion_eq_iUnion]
exact (directedOn_iff_directed.1 hdir).strictConvex_iUnion fun s => hS _ s.2
| 2 | 7.389056 | 1 | 1.333333 | 3 | 1,438 |
import Mathlib.AlgebraicGeometry.Morphisms.Basic
import Mathlib.Topology.LocalAtTarget
#align_import algebraic_geometry.morphisms.universally_closed from "leanprover-community/mathlib"@"a8ae1b3f7979249a0af6bc7cf20c1f6bf656ca73"
noncomputable section
open CategoryTheory CategoryTheory.Limits Opposite TopologicalS... | Mathlib/AlgebraicGeometry/Morphisms/UniversallyClosed.lean | 45 | 46 | theorem universallyClosed_eq : @UniversallyClosed = universally (topologically @IsClosedMap) := by |
ext X Y f; rw [universallyClosed_iff]
| 1 | 2.718282 | 0 | 1.333333 | 3 | 1,439 |
import Mathlib.AlgebraicGeometry.Morphisms.Basic
import Mathlib.Topology.LocalAtTarget
#align_import algebraic_geometry.morphisms.universally_closed from "leanprover-community/mathlib"@"a8ae1b3f7979249a0af6bc7cf20c1f6bf656ca73"
noncomputable section
open CategoryTheory CategoryTheory.Limits Opposite TopologicalS... | Mathlib/AlgebraicGeometry/Morphisms/UniversallyClosed.lean | 72 | 76 | theorem topologically_isClosedMap_respectsIso : RespectsIso (topologically @IsClosedMap) := by |
apply MorphismProperty.respectsIso_of_isStableUnderComposition
intro _ _ f hf
have : IsIso f := hf
exact (TopCat.homeoOfIso (Scheme.forgetToTop.mapIso (asIso f))).isClosedMap
| 4 | 54.59815 | 2 | 1.333333 | 3 | 1,439 |
import Mathlib.AlgebraicGeometry.Morphisms.Basic
import Mathlib.Topology.LocalAtTarget
#align_import algebraic_geometry.morphisms.universally_closed from "leanprover-community/mathlib"@"a8ae1b3f7979249a0af6bc7cf20c1f6bf656ca73"
noncomputable section
open CategoryTheory CategoryTheory.Limits Opposite TopologicalS... | Mathlib/AlgebraicGeometry/Morphisms/UniversallyClosed.lean | 88 | 94 | theorem universallyClosed_is_local_at_target : PropertyIsLocalAtTarget @UniversallyClosed := by |
rw [universallyClosed_eq]
apply universallyIsLocalAtTargetOfMorphismRestrict
Β· exact topologically_isClosedMap_respectsIso
Β· intro X Y f ΞΉ U hU H
simp_rw [topologically, morphismRestrict_base] at H
exact (isClosedMap_iff_isClosedMap_of_iSup_eq_top hU).mpr H
| 6 | 403.428793 | 2 | 1.333333 | 3 | 1,439 |
import Mathlib.Order.Interval.Set.ProjIcc
import Mathlib.Topology.Algebra.Order.Field
import Mathlib.Topology.Bornology.Hom
import Mathlib.Topology.EMetricSpace.Lipschitz
import Mathlib.Topology.MetricSpace.Basic
import Mathlib.Topology.MetricSpace.Bounded
#align_import topology.metric_space.lipschitz from "leanprove... | Mathlib/Topology/MetricSpace/Lipschitz.lean | 41 | 44 | theorem lipschitzWith_iff_dist_le_mul [PseudoMetricSpace Ξ±] [PseudoMetricSpace Ξ²] {K : ββ₯0}
{f : Ξ± β Ξ²} : LipschitzWith K f β β x y, dist (f x) (f y) β€ K * dist x y := by |
simp only [LipschitzWith, edist_nndist, dist_nndist]
norm_cast
| 2 | 7.389056 | 1 | 1.333333 | 3 | 1,440 |
import Mathlib.Order.Interval.Set.ProjIcc
import Mathlib.Topology.Algebra.Order.Field
import Mathlib.Topology.Bornology.Hom
import Mathlib.Topology.EMetricSpace.Lipschitz
import Mathlib.Topology.MetricSpace.Basic
import Mathlib.Topology.MetricSpace.Bounded
#align_import topology.metric_space.lipschitz from "leanprove... | Mathlib/Topology/MetricSpace/Lipschitz.lean | 51 | 55 | theorem lipschitzOnWith_iff_dist_le_mul [PseudoMetricSpace Ξ±] [PseudoMetricSpace Ξ²] {K : ββ₯0}
{s : Set Ξ±} {f : Ξ± β Ξ²} :
LipschitzOnWith K f s β β x β s, β y β s, dist (f x) (f y) β€ K * dist x y := by |
simp only [LipschitzOnWith, edist_nndist, dist_nndist]
norm_cast
| 2 | 7.389056 | 1 | 1.333333 | 3 | 1,440 |
import Mathlib.Order.Interval.Set.ProjIcc
import Mathlib.Topology.Algebra.Order.Field
import Mathlib.Topology.Bornology.Hom
import Mathlib.Topology.EMetricSpace.Lipschitz
import Mathlib.Topology.MetricSpace.Basic
import Mathlib.Topology.MetricSpace.Bounded
#align_import topology.metric_space.lipschitz from "leanprove... | Mathlib/Topology/MetricSpace/Lipschitz.lean | 371 | 378 | theorem continuousAt_of_locally_lipschitz {x : Ξ±} {r : β} (hr : 0 < r) (K : β)
(h : β y, dist y x < r β dist (f y) (f x) β€ K * dist y x) : ContinuousAt f x := by |
-- We use `h` to squeeze `dist (f y) (f x)` between `0` and `K * dist y x`
refine tendsto_iff_dist_tendsto_zero.2 (squeeze_zero' (eventually_of_forall fun _ => dist_nonneg)
(mem_of_superset (ball_mem_nhds _ hr) h) ?_)
-- Then show that `K * dist y x` tends to zero as `y β x`
refine (continuous_const.mul (c... | 6 | 403.428793 | 2 | 1.333333 | 3 | 1,440 |
import Mathlib.ModelTheory.Quotients
import Mathlib.Order.Filter.Germ
import Mathlib.Order.Filter.Ultrafilter
#align_import model_theory.ultraproducts from "leanprover-community/mathlib"@"f1ae620609496a37534c2ab3640b641d5be8b6f0"
universe u v
variable {Ξ± : Type*} (M : Ξ± β Type*) (u : Ultrafilter Ξ±)
open FirstOr... | Mathlib/ModelTheory/Ultraproducts.lean | 77 | 80 | theorem funMap_cast {n : β} (f : L.Functions n) (x : Fin n β β a, M a) :
(funMap f fun i => (x i : (u : Filter Ξ±).Product M)) =
(fun a => funMap f fun i => x i a : (u : Filter Ξ±).Product M) := by |
apply funMap_quotient_mk'
| 1 | 2.718282 | 0 | 1.333333 | 3 | 1,441 |
import Mathlib.ModelTheory.Quotients
import Mathlib.Order.Filter.Germ
import Mathlib.Order.Filter.Ultrafilter
#align_import model_theory.ultraproducts from "leanprover-community/mathlib"@"f1ae620609496a37534c2ab3640b641d5be8b6f0"
universe u v
variable {Ξ± : Type*} (M : Ξ± β Type*) (u : Ultrafilter Ξ±)
open FirstOr... | Mathlib/ModelTheory/Ultraproducts.lean | 83 | 91 | theorem term_realize_cast {Ξ² : Type*} (x : Ξ² β β a, M a) (t : L.Term Ξ²) :
(t.realize fun i => (x i : (u : Filter Ξ±).Product M)) =
(fun a => t.realize fun i => x i a : (u : Filter Ξ±).Product M) := by |
convert @Term.realize_quotient_mk' L _ ((u : Filter Ξ±).productSetoid M)
(Ultraproduct.setoidPrestructure M u) _ t x using 2
ext a
induction t with
| var => rfl
| func _ _ t_ih => simp only [Term.realize, t_ih]; rfl
| 6 | 403.428793 | 2 | 1.333333 | 3 | 1,441 |
import Mathlib.ModelTheory.Quotients
import Mathlib.Order.Filter.Germ
import Mathlib.Order.Filter.Ultrafilter
#align_import model_theory.ultraproducts from "leanprover-community/mathlib"@"f1ae620609496a37534c2ab3640b641d5be8b6f0"
universe u v
variable {Ξ± : Type*} (M : Ξ± β Type*) (u : Ultrafilter Ξ±)
open FirstOr... | Mathlib/ModelTheory/Ultraproducts.lean | 96 | 144 | theorem boundedFormula_realize_cast {Ξ² : Type*} {n : β} (Ο : L.BoundedFormula Ξ² n)
(x : Ξ² β β a, M a) (v : Fin n β β a, M a) :
(Ο.Realize (fun i : Ξ² => (x i : (u : Filter Ξ±).Product M))
(fun i => (v i : (u : Filter Ξ±).Product M))) β
βαΆ a : Ξ± in u, Ο.Realize (fun i : Ξ² => x i a) fun i => v i a := b... |
letI := (u : Filter Ξ±).productSetoid M
induction' Ο with _ _ _ _ _ _ _ _ m _ _ ih ih' k Ο ih
Β· simp only [BoundedFormula.Realize, eventually_const]
Β· have h2 : β a : Ξ±, (Sum.elim (fun i : Ξ² => x i a) fun i => v i a) = fun i => Sum.elim x v i a :=
fun a => funext fun i => Sum.casesOn i (fun i => rfl) fun ... | 44 | 12,851,600,114,359,308,000 | 2 | 1.333333 | 3 | 1,441 |
import Mathlib.MeasureTheory.Measure.Typeclasses
import Mathlib.Analysis.Complex.Basic
#align_import measure_theory.measure.vector_measure from "leanprover-community/mathlib"@"70a4f2197832bceab57d7f41379b2592d1110570"
noncomputable section
open scoped Classical
open NNReal ENNReal MeasureTheory
namespace Measur... | Mathlib/MeasureTheory/Measure/VectorMeasure.lean | 124 | 125 | theorem ext_iff' (v w : VectorMeasure Ξ± M) : v = w β β i : Set Ξ±, v i = w i := by |
rw [β coe_injective.eq_iff, Function.funext_iff]
| 1 | 2.718282 | 0 | 1.333333 | 3 | 1,442 |
import Mathlib.MeasureTheory.Measure.Typeclasses
import Mathlib.Analysis.Complex.Basic
#align_import measure_theory.measure.vector_measure from "leanprover-community/mathlib"@"70a4f2197832bceab57d7f41379b2592d1110570"
noncomputable section
open scoped Classical
open NNReal ENNReal MeasureTheory
namespace Measur... | Mathlib/MeasureTheory/Measure/VectorMeasure.lean | 128 | 136 | theorem ext_iff (v w : VectorMeasure Ξ± M) : v = w β β i : Set Ξ±, MeasurableSet i β v i = w i := by |
constructor
Β· rintro rfl _ _
rfl
Β· rw [ext_iff']
intro h i
by_cases hi : MeasurableSet i
Β· exact h i hi
Β· simp_rw [not_measurable _ hi]
| 8 | 2,980.957987 | 2 | 1.333333 | 3 | 1,442 |
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