Context stringlengths 57 6.04k | file_name stringlengths 21 79 | start int64 14 1.49k | end int64 18 1.5k | theorem stringlengths 25 1.55k | proof stringlengths 5 7.36k | num_lines int64 1 150 | complexity_score float64 2.72 139,370,958,066,637,970,000,000,000,000,000,000,000,000,000,000,000,000,000B | diff_level int64 0 2 | file_diff_level float64 0 2 | theorem_same_file int64 1 32 | rank_file int64 0 2.51k |
|---|---|---|---|---|---|---|---|---|---|---|---|
import Mathlib.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
#align_import data.set.intervals.with_bot_top from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
variable {Ξ± : Type*}
namespace WithTop
@[simp]
theorem preimage_coe_top : (some : Ξ± β WithTop Ξ±) β»ΒΉ' {β€} =... | Mathlib/Order/Interval/Set/WithBotTop.lean | 75 | 76 | theorem preimage_coe_Iio_top : (some : Ξ± β WithTop Ξ±) β»ΒΉ' Iio β€ = univ := by |
rw [β range_coe, preimage_range]
| 1 | 2.718282 | 0 | 0.4 | 15 | 401 |
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 | 119 | 123 | theorem Finset.untrop_sum' [LinearOrder R] [OrderTop R] (s : Finset S) (f : S β Tropical R) :
untrop (β i β s, f i) = s.inf (untrop β f) := by |
convert Multiset.untrop_sum (s.val.map f)
simp only [Multiset.map_map, Function.comp_apply]
rfl
| 3 | 20.085537 | 1 | 0.928571 | 14 | 793 |
import Mathlib.Algebra.Algebra.Subalgebra.Pointwise
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Maximal
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Noetherian
import Mathlib.RingTheory.ChainOfDivisors
import Mathlib.RingTheory.DedekindDomain.Basic
import Mathlib.RingTheory.FractionalIdeal.Operations
#align_impo... | Mathlib/RingTheory/DedekindDomain/Ideal.lean | 136 | 137 | theorem map_inv (I : FractionalIdeal Rββ° K) (h : K ββ[Rβ] K') :
Iβ»ΒΉ.map (h : K ββ[Rβ] K') = (I.map h)β»ΒΉ := by | rw [inv_eq, map_div, map_one, inv_eq]
| 1 | 2.718282 | 0 | 0.666667 | 6 | 564 |
import Mathlib.Init.Algebra.Classes
import Mathlib.Init.Data.Ordering.Basic
#align_import init.data.ordering.lemmas from "leanprover-community/lean"@"4bd314f7bd5e0c9e813fc201f1279a23f13f9f1d"
universe u
namespace Ordering
@[simp]
theorem ite_eq_lt_distrib (c : Prop) [Decidable c] (a b : Ordering) :
((if c th... | Mathlib/Init/Data/Ordering/Lemmas.lean | 26 | 28 | theorem ite_eq_eq_distrib (c : Prop) [Decidable c] (a b : Ordering) :
((if c then a else b) = Ordering.eq) = if c then a = Ordering.eq else b = Ordering.eq := by |
by_cases c <;> simp [*]
| 1 | 2.718282 | 0 | 0 | 3 | 20 |
import Mathlib.Algebra.Module.Submodule.Lattice
import Mathlib.Order.Hom.CompleteLattice
namespace Submodule
variable (S : Type*) {R M : Type*} [Semiring R] [AddCommMonoid M] [Semiring S]
[Module S M] [Module R M] [SMul S R] [IsScalarTower S R M]
def restrictScalars (V : Submodule R M) : Submodule S M where
... | Mathlib/Algebra/Module/Submodule/RestrictScalars.lean | 116 | 117 | theorem restrictScalars_eq_top_iff {p : Submodule R M} : restrictScalars S p = β€ β p = β€ := by |
simp [SetLike.ext_iff]
| 1 | 2.718282 | 0 | 0 | 2 | 128 |
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Monoidal.Functor
#align_import category_theory.monoidal.preadditive from "leanprover-community/mathlib"@"986c4d5761f938b2e1c43c01f001b6d9d88c2055"
noncomputable section
open scoped Classical
namespace CategoryTheory
open Cat... | Mathlib/CategoryTheory/Monoidal/Preadditive.lean | 63 | 64 | theorem add_tensor {W X Y Z : C} (f g : W βΆ X) (h : Y βΆ Z) : (f + g) β h = f β h + g β h := by |
simp [tensorHom_def]
| 1 | 2.718282 | 0 | 0.5 | 8 | 481 |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Combinatorics.SimpleGraph.Dart
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Data.ZMod.Parity
#align_import combinatorics.simple_graph.degree_sum from "leanprover-community/mathlib"@"90659cbe25e59ec302e2fb92b00e9732160cc620"
open Finset
nam... | Mathlib/Combinatorics/SimpleGraph/DegreeSum.lean | 88 | 95 | theorem dart_edge_fiber_card [DecidableEq V] (e : Sym2 V) (h : e β G.edgeSet) :
(univ.filter fun d : G.Dart => d.edge = e).card = 2 := by |
refine Sym2.ind (fun v w h => ?_) e h
let d : G.Dart := β¨(v, w), hβ©
convert congr_arg card d.edge_fiber
rw [card_insert_of_not_mem, card_singleton]
rw [mem_singleton]
exact d.symm_ne.symm
| 6 | 403.428793 | 2 | 1.6 | 5 | 1,746 |
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 | 113 | 115 | theorem Οβ_int_three_mod_four {n : β€} (hn : n % 4 = 3) : Οβ n = -1 := by |
rw [Οβ_int_mod_four, hn]
rfl
| 2 | 7.389056 | 1 | 1.25 | 12 | 1,332 |
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
#align_import linear_algebra.clifford_algebra.star from "leanprover-community/mathlib"@"4d66277cfec381260ba05c68f9ae6ce2a118031d"
variable {R : Type*} [CommRing R]
variable {M : Type*} [AddCommGroup M] [Module R M]
variable {Q : QuadraticForm R M}
namespac... | Mathlib/LinearAlgebra/CliffordAlgebra/Star.lean | 62 | 64 | theorem star_algebraMap (r : R) :
star (algebraMap R (CliffordAlgebra Q) r) = algebraMap R (CliffordAlgebra Q) r := by |
rw [star_def, involute.commutes, reverse.commutes]
| 1 | 2.718282 | 0 | 0 | 3 | 167 |
import Mathlib.Algebra.Module.Card
import Mathlib.SetTheory.Cardinal.CountableCover
import Mathlib.SetTheory.Cardinal.Continuum
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Topology.MetricSpace.Perfect
universe u v
open Filter Pointwise Set Function Cardinal
open scoped Cardinal Topology
theorem c... | Mathlib/Topology/Algebra/Module/Cardinality.lean | 119 | 123 | theorem continuum_le_cardinal_of_isOpen
{E : Type*} (π : Type*) [NontriviallyNormedField π] [CompleteSpace π] [AddCommGroup E]
[Module π E] [Nontrivial E] [TopologicalSpace E] [ContinuousAdd E] [ContinuousSMul π E]
{s : Set E} (hs : IsOpen s) (h's : s.Nonempty) : π β€ #s := by |
simpa [cardinal_eq_of_isOpen π hs h's] using continuum_le_cardinal_of_module π E
| 1 | 2.718282 | 0 | 1.2 | 5 | 1,284 |
import Mathlib.Init.Data.Nat.Notation
import Mathlib.Init.Order.Defs
set_option autoImplicit true
structure UFModel (n) where
parent : Fin n β Fin n
rank : Nat β Nat
rank_lt : β i, (parent i).1 β i β rank i < rank (parent i)
structure UFNode (Ξ± : Type*) where
parent : Nat
value : Ξ±
rank : Nat
inductive... | Mathlib/Data/UnionFind.lean | 79 | 80 | theorem size_eq {arr : Array Ξ±} {m : Fin n β Ξ²} (H : Agrees arr f m) : n = arr.size := by |
cases H; rfl
| 1 | 2.718282 | 0 | 1 | 5 | 1,140 |
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 | 88 | 89 | theorem Ioc_eq_empty_iff : Ioc a b = β
β Β¬a < b := by |
rw [β coe_eq_empty, coe_Ioc, Set.Ioc_eq_empty_iff]
| 1 | 2.718282 | 0 | 0 | 12 | 11 |
import Mathlib.Data.Fin.Tuple.Basic
import Mathlib.Data.List.Join
#align_import data.list.of_fn from "leanprover-community/mathlib"@"bf27744463e9620ca4e4ebe951fe83530ae6949b"
universe u
variable {Ξ± : Type u}
open Nat
namespace List
#noalign list.length_of_fn_aux
@[simp]
theorem length_ofFn_go {n} (f : Fin n ... | Mathlib/Data/List/OfFn.lean | 44 | 45 | theorem length_ofFn {n} (f : Fin n β Ξ±) : length (ofFn f) = n := by |
simp [ofFn, length_ofFn_go]
| 1 | 2.718282 | 0 | 0.6 | 10 | 535 |
import Mathlib.LinearAlgebra.ExteriorAlgebra.Basic
import Mathlib.LinearAlgebra.CliffordAlgebra.Fold
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.Dual
#align_import linear_algebra.clifford_algebra.contraction from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2... | Mathlib/LinearAlgebra/CliffordAlgebra/Contraction.lean | 144 | 146 | theorem contractLeft_algebraMap_mul (r : R) (b : CliffordAlgebra Q) :
dβ(algebraMap _ _ r * b) = algebraMap _ _ r * (dβb) := by |
rw [β Algebra.smul_def, map_smul, Algebra.smul_def]
| 1 | 2.718282 | 0 | 0.625 | 8 | 549 |
import Mathlib.Data.Int.Interval
import Mathlib.RingTheory.Binomial
import Mathlib.RingTheory.HahnSeries.PowerSeries
import Mathlib.RingTheory.HahnSeries.Summable
import Mathlib.FieldTheory.RatFunc.AsPolynomial
import Mathlib.RingTheory.Localization.FractionRing
#align_import ring_theory.laurent_series from "leanprov... | Mathlib/RingTheory/LaurentSeries.lean | 87 | 89 | theorem coeff_coe_powerSeries (x : PowerSeries R) (n : β) :
HahnSeries.coeff (x : LaurentSeries R) n = PowerSeries.coeff R n x := by |
rw [ofPowerSeries_apply_coeff]
| 1 | 2.718282 | 0 | 1.2 | 5 | 1,285 |
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 | 60 | 62 | theorem hasMFDerivAt_iff_hasFDerivAt {f'} :
HasMFDerivAt π(π, E) π(π, E') f x f' β HasFDerivAt f f' x := by |
rw [β hasMFDerivWithinAt_univ, hasMFDerivWithinAt_iff_hasFDerivWithinAt, hasFDerivWithinAt_univ]
| 1 | 2.718282 | 0 | 0.5 | 8 | 455 |
import Mathlib.Topology.MetricSpace.Antilipschitz
#align_import topology.metric_space.isometry from "leanprover-community/mathlib"@"b1859b6d4636fdbb78c5d5cefd24530653cfd3eb"
noncomputable section
universe u v w
variable {ΞΉ : Type*} {Ξ± : Type u} {Ξ² : Type v} {Ξ³ : Type w}
open Function Set
open scoped Topology ... | Mathlib/Topology/MetricSpace/Isometry.lean | 46 | 48 | theorem isometry_iff_dist_eq [PseudoMetricSpace Ξ±] [PseudoMetricSpace Ξ²] {f : Ξ± β Ξ²} :
Isometry f β β x y, dist (f x) (f y) = dist x y := by |
simp only [isometry_iff_nndist_eq, β coe_nndist, NNReal.coe_inj]
| 1 | 2.718282 | 0 | 0.6 | 5 | 529 |
import Mathlib.Order.Sublattice
import Mathlib.Order.Hom.CompleteLattice
open Function Set
variable (Ξ± Ξ² : Type*) [CompleteLattice Ξ±] [CompleteLattice Ξ²] (f : CompleteLatticeHom Ξ± Ξ²)
structure CompleteSublattice extends Sublattice Ξ± where
sSupClosed' : β β¦s : Set Ξ±β¦, s β carrier β sSup s β carrier
sInfClosed... | Mathlib/Order/CompleteSublattice.lean | 84 | 85 | theorem coe_sSup' (S : Set L) : (β(sSup S) : Ξ±) = β¨ N β S, (N : Ξ±) := by |
rw [coe_sSup, β Set.image, sSup_image]
| 1 | 2.718282 | 0 | 0 | 2 | 44 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Data.Complex.Exponential
import Mathlib.Data.Complex.Module
import Mathlib.RingTheory.Polynomial.Chebyshev
#align_import analysis.special_functions.trigonometric.chebyshev from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
set_... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Chebyshev.lean | 45 | 47 | theorem algebraMap_eval_U (x : R) (n : β€) :
algebraMap R A ((U R n).eval x) = (U A n).eval (algebraMap R A x) := by |
rw [β aeval_algebraMap_apply_eq_algebraMap_eval, aeval_U]
| 1 | 2.718282 | 0 | 0.666667 | 6 | 568 |
import Mathlib.Algebra.NeZero
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Lifts
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.RingTheory.RootsOfUnity.Complex
import Mathlib.NumberTheory.ArithmeticFunction
import Mathlib.RingTheory.RootsOfUnity.Basic
import Mathlib.FieldTh... | Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean | 131 | 141 | theorem X_pow_sub_one_eq_prod {ΞΆ : R} {n : β} (hpos : 0 < n) (h : IsPrimitiveRoot ΞΆ n) :
X ^ n - 1 = β ΞΆ β nthRootsFinset n R, (X - C ΞΆ) := by |
classical
rw [nthRootsFinset, β Multiset.toFinset_eq (IsPrimitiveRoot.nthRoots_one_nodup h)]
simp only [Finset.prod_mk, RingHom.map_one]
rw [nthRoots]
have hmonic : (X ^ n - C (1 : R)).Monic := monic_X_pow_sub_C (1 : R) (ne_of_lt hpos).symm
symm
apply prod_multiset_X_sub_C_of_monic_of_roots_card_eq hmoni... | 9 | 8,103.083928 | 2 | 1 | 7 | 1,027 |
import Mathlib.Data.List.Lattice
import Mathlib.Data.List.Range
import Mathlib.Data.Bool.Basic
#align_import data.list.intervals from "leanprover-community/mathlib"@"7b78d1776212a91ecc94cf601f83bdcc46b04213"
open Nat
namespace List
def Ico (n m : β) : List β :=
range' n (m - n)
#align list.Ico List.Ico
names... | Mathlib/Data/List/Intervals.lean | 56 | 58 | theorem nodup (n m : β) : Nodup (Ico n m) := by |
dsimp [Ico]
simp [nodup_range', autoParam]
| 2 | 7.389056 | 1 | 0.9375 | 16 | 794 |
import Mathlib.MeasureTheory.Measure.Content
import Mathlib.MeasureTheory.Group.Prod
import Mathlib.Topology.Algebra.Group.Compact
#align_import measure_theory.measure.haar.basic from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
noncomputable section
open Set Inv Function Topological... | Mathlib/MeasureTheory/Measure/Haar/Basic.lean | 185 | 196 | theorem le_index_mul (Kβ : PositiveCompacts G) (K : Compacts G) {V : Set G}
(hV : (interior V).Nonempty) :
index (K : Set G) V β€ index (K : Set G) Kβ * index (Kβ : Set G) V := by |
obtain β¨s, h1s, h2sβ© := index_elim K.isCompact Kβ.interior_nonempty
obtain β¨t, h1t, h2tβ© := index_elim Kβ.isCompact hV
rw [β h2s, β h2t, mul_comm]
refine le_trans ?_ Finset.card_mul_le
apply Nat.sInf_le; refine β¨_, ?_, rflβ©; rw [mem_setOf_eq]; refine Subset.trans h1s ?_
apply iUnionβ_subset; intro gβ hgβ; ... | 9 | 8,103.083928 | 2 | 0.428571 | 7 | 407 |
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 | 61 | 63 | theorem Iio_eq_range : Iio = range := by |
ext b x
rw [mem_Iio, mem_range]
| 2 | 7.389056 | 1 | 0.125 | 8 | 253 |
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.LinearAlgebra.Matrix.Block
#align_import analysis.inner_product_space.gram_schmidt_ortho from "leanprover-community/mathlib"@"1a4df69ca1a9a0e5e26bfe12e2b92814216016d0"
open Finset Submodule FiniteDimensional
variable (π : Type*) {E : Type*} [RCLike οΏ½... | Mathlib/Analysis/InnerProductSpace/GramSchmidtOrtho.lean | 63 | 65 | theorem gramSchmidt_def' (f : ΞΉ β E) (n : ΞΉ) :
f n = gramSchmidt π f n + β i β Iio n, orthogonalProjection (π β gramSchmidt π f i) (f n) := by |
rw [gramSchmidt_def, sub_add_cancel]
| 1 | 2.718282 | 0 | 1.125 | 8 | 1,201 |
import Mathlib.Geometry.Euclidean.Sphere.Basic
#align_import geometry.euclidean.sphere.second_inter from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open RealInnerProductSpace
namespace EuclideanGeometry
variable {V : Type*} {P : Type*} [NormedAddCommGroup V]... | Mathlib/Geometry/Euclidean/Sphere/SecondInter.lean | 62 | 63 | theorem Sphere.secondInter_zero (s : Sphere P) (p : P) : s.secondInter p (0 : V) = p := by |
simp [Sphere.secondInter]
| 1 | 2.718282 | 0 | 1.25 | 8 | 1,314 |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Positivity.Core
#align_import data.nat.factorial.double_factorial from "leanprover-community/mathlib"@"7daeaf3072304c498b653628add84a88d0e78767"
open Nat
namespace Nat
@[sim... | Mathlib/Data/Nat/Factorial/DoubleFactorial.lean | 48 | 48 | theorem doubleFactorial_add_one (n : β) : (n + 1)βΌ = (n + 1) * (n - 1)βΌ := by | cases n <;> rfl
| 1 | 2.718282 | 0 | 0 | 1 | 46 |
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Topology.Algebra.Field
import Mathlib.Topology.Algebra.Order.Group
#align_import topology.algebra.order.field from "leanprover-community/mathlib"@"9a59dcb7a2d06bf55da57b9030169219980660cd"
open Set Filter TopologicalSpace Function
open scoped Pointwise Top... | Mathlib/Topology/Algebra/Order/Field.lean | 63 | 67 | theorem Filter.Tendsto.atTop_mul {C : π} (hC : 0 < C) (hf : Tendsto f l atTop)
(hg : Tendsto g l (π C)) : Tendsto (fun x => f x * g x) l atTop := by |
refine tendsto_atTop_mono' _ ?_ (hf.atTop_mul_const (half_pos hC))
filter_upwards [hg.eventually (lt_mem_nhds (half_lt_self hC)), hf.eventually_ge_atTop 0]
with x hg hf using mul_le_mul_of_nonneg_left hg.le hf
| 3 | 20.085537 | 1 | 0.666667 | 9 | 577 |
import Mathlib.Algebra.ContinuedFractions.Basic
import Mathlib.Algebra.GroupWithZero.Basic
#align_import algebra.continued_fractions.translations from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b348ce40ad"
namespace GeneralizedContinuedFraction
section General
variable {Ξ± : Type*} {g : Gen... | Mathlib/Algebra/ContinuedFractions/Translations.lean | 41 | 42 | theorem part_num_none_iff_s_none : g.partialNumerators.get? n = none β g.s.get? n = none := by |
cases s_nth_eq : g.s.get? n <;> simp [partialNumerators, s_nth_eq]
| 1 | 2.718282 | 0 | 0.052632 | 19 | 240 |
import Mathlib.Data.Fintype.List
#align_import data.list.cycle from "leanprover-community/mathlib"@"7413128c3bcb3b0818e3e18720abc9ea3100fb49"
assert_not_exists MonoidWithZero
open List
def Cycle (Ξ± : Type*) : Type _ :=
Quotient (IsRotated.setoid Ξ±)
#align cycle Cycle
namespace Cycle
variable {Ξ± : Type*}
--... | Mathlib/Data/List/Cycle.lean | 605 | 610 | theorem Subsingleton.congr {s : Cycle Ξ±} (h : Subsingleton s) :
β β¦xβ¦ (_hx : x β s) β¦yβ¦ (_hy : y β s), x = y := by |
induction' s using Quot.inductionOn with l
simp only [length_subsingleton_iff, length_coe, mk_eq_coe, le_iff_lt_or_eq, Nat.lt_add_one_iff,
length_eq_zero, length_eq_one, Nat.not_lt_zero, false_or_iff] at h
rcases h with (rfl | β¨z, rflβ©) <;> simp
| 4 | 54.59815 | 2 | 1.4 | 10 | 1,479 |
import Mathlib.Algebra.Group.Defs
import Mathlib.Algebra.GroupWithZero.Defs
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Tactic.Spread
import Mathlib.Util.AssertExists
#align_import algebra.ring.defs from "leanprover-community/mathlib"@"76de8ae01554c3b37d66544866659ff174e66e1f"
universe u v w x
variable {Ξ± : ... | Mathlib/Algebra/Ring/Defs.lean | 218 | 221 | theorem ite_sub_ite {Ξ±} [Sub Ξ±] (P : Prop) [Decidable P] (a b c d : Ξ±) :
((if P then a else b) - if P then c else d) = if P then a - c else b - d := by |
split
repeat rfl
| 2 | 7.389056 | 1 | 0.181818 | 11 | 267 |
import Mathlib.Order.Lattice
import Mathlib.Data.List.Sort
import Mathlib.Logic.Equiv.Fin
import Mathlib.Logic.Equiv.Functor
import Mathlib.Data.Fintype.Card
import Mathlib.Order.RelSeries
#align_import order.jordan_holder from "leanprover-community/mathlib"@"91288e351d51b3f0748f0a38faa7613fb0ae2ada"
universe u
... | Mathlib/Order/JordanHolder.lean | 109 | 113 | theorem isMaximal_of_eq_inf (x b : X) {a y : X} (ha : x β y = a) (hxy : x β y) (hxb : IsMaximal x b)
(hyb : IsMaximal y b) : IsMaximal a y := by |
have hb : x β y = b := sup_eq_of_isMaximal hxb hyb hxy
substs a b
exact isMaximal_inf_right_of_isMaximal_sup hxb hyb
| 3 | 20.085537 | 1 | 1 | 4 | 952 |
import Mathlib.Data.Set.Lattice
import Mathlib.Order.Hom.Lattice
#align_import order.hom.complete_lattice from "leanprover-community/mathlib"@"9d684a893c52e1d6692a504a118bfccbae04feeb"
open Function OrderDual Set
variable {F Ξ± Ξ² Ξ³ Ξ΄ : Type*} {ΞΉ : Sort*} {ΞΊ : ΞΉ β Sort*}
-- Porting note: mathport made this & sInf... | Mathlib/Order/Hom/CompleteLattice.lean | 142 | 143 | theorem map_iInfβ [InfSet Ξ±] [InfSet Ξ²] [sInfHomClass F Ξ± Ξ²] (f : F) (g : β i, ΞΊ i β Ξ±) :
f (β¨
(i) (j), g i j) = β¨
(i) (j), f (g i j) := by | simp_rw [map_iInf]
| 1 | 2.718282 | 0 | 0 | 2 | 5 |
import Mathlib.MeasureTheory.Group.GeometryOfNumbers
import Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls
import Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic
#align_import number_theory.number_field.canonical_embedding from "leanprover-community/mathlib"@"60da01b41bbe4206f05d34fd70c8dd7498717a30"
... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean | 108 | 130 | theorem convexBodyLT_volume :
volume (convexBodyLT K f) = (convexBodyLTFactor K) * β w, (f w) ^ (mult w) := by |
calc
_ = (β x : {w // InfinitePlace.IsReal w}, ENNReal.ofReal (2 * (f x.val))) *
β x : {w // InfinitePlace.IsComplex w}, ENNReal.ofReal (f x.val) ^ 2 * NNReal.pi := by
simp_rw [volume_eq_prod, prod_prod, volume_pi, pi_pi, Real.volume_ball, Complex.volume_ball]
_ = ((2:ββ₯0) ^ NrRealPlaces K * ... | 21 | 1,318,815,734.483215 | 2 | 1.666667 | 15 | 1,803 |
import Mathlib.Algebra.NeZero
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Lifts
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.RingTheory.RootsOfUnity.Complex
import Mathlib.NumberTheory.ArithmeticFunction
import Mathlib.RingTheory.RootsOfUnity.Basic
import Mathlib.FieldTh... | Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean | 118 | 120 | theorem degree_cyclotomic' {ΞΆ : R} {n : β} (h : IsPrimitiveRoot ΞΆ n) :
(cyclotomic' n R).degree = Nat.totient n := by |
simp only [degree_eq_natDegree (cyclotomic'_ne_zero n R), natDegree_cyclotomic' h]
| 1 | 2.718282 | 0 | 1 | 7 | 1,027 |
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Inverse
#align_import analysis.calculus.cont_diff from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputab... | Mathlib/Analysis/Calculus/ContDiff/Basic.lean | 107 | 108 | theorem contDiff_of_subsingleton [Subsingleton F] : ContDiff π n f := by |
rw [Subsingleton.elim f fun _ => 0]; exact contDiff_const
| 1 | 2.718282 | 0 | 0.875 | 8 | 760 |
import Mathlib.LinearAlgebra.AffineSpace.Basis
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
#align_import linear_algebra.affine_space.matrix from "leanprover-community/mathlib"@"2de9c37fa71dde2f1c6feff19876dd6a7b1519f0"
open Affine Matrix
open Set
universe uβ uβ uβ uβ
variable {ΞΉ : Type uβ} {k : Type... | Mathlib/LinearAlgebra/AffineSpace/Matrix.lean | 48 | 50 | theorem toMatrix_self [DecidableEq ΞΉ] : b.toMatrix b = (1 : Matrix ΞΉ ΞΉ k) := by |
ext i j
rw [toMatrix_apply, coord_apply, Matrix.one_eq_pi_single, Pi.single_apply]
| 2 | 7.389056 | 1 | 1.428571 | 7 | 1,512 |
import Mathlib.Algebra.Group.Defs
import Mathlib.Algebra.GroupWithZero.Defs
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Tactic.Spread
import Mathlib.Util.AssertExists
#align_import algebra.ring.defs from "leanprover-community/mathlib"@"76de8ae01554c3b37d66544866659ff174e66e1f"
universe u v w x
variable {Ξ± : ... | Mathlib/Algebra/Ring/Defs.lean | 164 | 165 | theorem one_add_mul [RightDistribClass Ξ±] (a b : Ξ±) : (1 + a) * b = b + a * b := by |
rw [add_mul, one_mul]
| 1 | 2.718282 | 0 | 0.181818 | 11 | 267 |
import Mathlib.Order.BooleanAlgebra
import Mathlib.Logic.Equiv.Basic
#align_import order.symm_diff from "leanprover-community/mathlib"@"6eb334bd8f3433d5b08ba156b8ec3e6af47e1904"
open Function OrderDual
variable {ΞΉ Ξ± Ξ² : Type*} {Ο : ΞΉ β Type*}
def symmDiff [Sup Ξ±] [SDiff Ξ±] (a b : Ξ±) : Ξ± :=
a \ b β b \ a
#ali... | Mathlib/Order/SymmDiff.lean | 137 | 138 | theorem symmDiff_of_le {a b : Ξ±} (h : a β€ b) : a β b = b \ a := by |
rw [symmDiff, sdiff_eq_bot_iff.2 h, bot_sup_eq]
| 1 | 2.718282 | 0 | 0.181818 | 22 | 266 |
import Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing
import Mathlib.AlgebraicGeometry.OpenImmersion
#align_import algebraic_geometry.gluing from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
set_option linter.uppercaseLean3 false
noncomputable section
universe u
open Topologica... | Mathlib/AlgebraicGeometry/Gluing.lean | 319 | 322 | theorem glued_cover_cocycle_fst (x y z : π°.J) :
gluedCoverT' π° x y z β« gluedCoverT' π° y z x β« gluedCoverT' π° z x y β« pullback.fst =
pullback.fst := by |
apply pullback.hom_ext <;> simp
| 1 | 2.718282 | 0 | 0.142857 | 7 | 256 |
import Mathlib.Analysis.Calculus.ContDiff.Bounds
import Mathlib.Analysis.Calculus.IteratedDeriv.Defs
import Mathlib.Analysis.Calculus.LineDeriv.Basic
import Mathlib.Analysis.LocallyConvex.WithSeminorms
import Mathlib.Analysis.Normed.Group.ZeroAtInfty
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Ana... | Mathlib/Analysis/Distribution/SchwartzSpace.lean | 203 | 205 | theorem decay_neg_aux (k n : β) (f : π’(E, F)) (x : E) :
βxβ ^ k * βiteratedFDeriv β n (-f : E β F) xβ = βxβ ^ k * βiteratedFDeriv β n f xβ := by |
rw [iteratedFDeriv_neg_apply, norm_neg]
| 1 | 2.718282 | 0 | 1.25 | 8 | 1,308 |
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.Algebra.Module.Torsion
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v v' uβ' w w'
variable {R S : Type u} {M : Type v} {M' : Type v'} {Mβ : Type v}... | Mathlib/LinearAlgebra/Dimension/Constructions.lean | 211 | 213 | theorem rank_matrix' (m n : Type v) [Finite m] [Finite n] :
Module.rank R (Matrix m n R) = Cardinal.lift.{u} (#m * #n) := by |
rw [rank_matrix, lift_mul, lift_umax.{v, u}]
| 1 | 2.718282 | 0 | 0.75 | 24 | 667 |
import Mathlib.MeasureTheory.MeasurableSpace.Basic
import Mathlib.Data.Set.MemPartition
import Mathlib.Order.Filter.CountableSeparatingOn
open Set MeasureTheory
namespace MeasurableSpace
variable {Ξ± Ξ² : Type*}
class CountablyGenerated (Ξ± : Type*) [m : MeasurableSpace Ξ±] : Prop where
isCountablyGenerated : β b... | Mathlib/MeasureTheory/MeasurableSpace/CountablyGenerated.lean | 157 | 163 | theorem separating_of_generateFrom (S : Set (Set Ξ±))
[h : @SeparatesPoints Ξ± (generateFrom S)] :
β x y : Ξ±, (β s β S, x β s β y β s) β x = y := by |
letI := generateFrom S
intros x y hxy
rw [β forall_generateFrom_mem_iff_mem_iff] at hxy
exact separatesPoints_def $ fun _ hs β¦ (hxy _ hs).mp
| 4 | 54.59815 | 2 | 1.5 | 4 | 1,686 |
import Mathlib.Data.Finset.Lattice
import Mathlib.Data.Set.Sigma
#align_import data.finset.sigma from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function Multiset
variable {ΞΉ : Type*}
namespace Finset
section Sigma
variable {Ξ± : ΞΉ β Type*} {Ξ² : Type*} (s sβ sβ : Finset ΞΉ) (... | Mathlib/Data/Finset/Sigma.lean | 112 | 114 | theorem _root_.biSup_finsetSigma [CompleteLattice Ξ²] (s : Finset ΞΉ) (t : β i, Finset (Ξ± i))
(f : Sigma Ξ± β Ξ²) : β¨ ij β s.sigma t, f ij = β¨ (i β s) (j β t i), f β¨i, jβ© := by |
simp_rw [β Finset.iSup_coe, Finset.coe_sigma, biSup_sigma]
| 1 | 2.718282 | 0 | 1.214286 | 14 | 1,292 |
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Data.Finset.Sym
import Mathlib.Data.Matrix.Basic
#align_import combinatorics.simple_graph.inc_matrix from "leanprover-community/mathlib"@"bb168510ef455e9280a152e7f31673cabd3d7496"
open Finset Matrix SimpleGraph Sym2
open Matrix
namespace SimpleGraph... | Mathlib/Combinatorics/SimpleGraph/IncMatrix.lean | 92 | 93 | theorem incMatrix_of_not_mem_incidenceSet (h : e β G.incidenceSet a) : G.incMatrix R a e = 0 := by |
rw [incMatrix_apply, Set.indicator_of_not_mem h]
| 1 | 2.718282 | 0 | 0.9 | 10 | 784 |
import Mathlib.Algebra.Group.Units
import Mathlib.Algebra.GroupWithZero.Basic
import Mathlib.Logic.Equiv.Defs
import Mathlib.Tactic.Contrapose
import Mathlib.Tactic.Nontriviality
import Mathlib.Tactic.Spread
import Mathlib.Util.AssertExists
#align_import algebra.group_with_zero.units.basic from "leanprover-community/... | Mathlib/Algebra/GroupWithZero/Units/Basic.lean | 130 | 131 | theorem inverse_mul_cancel_left (x y : Mβ) (h : IsUnit x) : inverse x * (x * y) = y := by |
rw [β mul_assoc, inverse_mul_cancel x h, one_mul]
| 1 | 2.718282 | 0 | 0.375 | 8 | 377 |
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 | 83 | 84 | theorem isBounded_principal (s : Set Ξ±) : IsBounded r (π s) β β t, β x β s, r x t := by |
simp [IsBounded, subset_def]
| 1 | 2.718282 | 0 | 0.25 | 4 | 306 |
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Order.Group.Int
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Algebra.Ring.Rat
import Mathlib.Data.PNat.Defs
#align_import data.rat.lemmas from "leanprover-community/mathlib"@"550b58538991c8977703fdeb7c9d51a5aa27df11"
namespace Rat
o... | Mathlib/Data/Rat/Lemmas.lean | 114 | 119 | theorem add_num_den (q r : β) :
q + r = (q.num * r.den + q.den * r.num : β€) /. (βq.den * βr.den : β€) := by |
have hqd : (q.den : β€) β 0 := Int.natCast_ne_zero_iff_pos.2 q.den_pos
have hrd : (r.den : β€) β 0 := Int.natCast_ne_zero_iff_pos.2 r.den_pos
conv_lhs => rw [β num_divInt_den q, β num_divInt_den r, divInt_add_divInt _ _ hqd hrd]
rw [mul_comm r.num q.den]
| 4 | 54.59815 | 2 | 1.333333 | 12 | 1,389 |
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 | 467 | 478 | theorem coequalizer_isOpen_iff (F : WalkingParallelPair β₯€ TopCat.{u})
(U : Set ((colimit F : _) : Type u)) :
IsOpen U β IsOpen (colimit.ΞΉ F WalkingParallelPair.one β»ΒΉ' U) := by |
rw [colimit_isOpen_iff]
constructor
Β· intro H
exact H _
Β· intro H j
cases j
Β· rw [β colimit.w F WalkingParallelPairHom.left]
exact (F.map WalkingParallelPairHom.left).continuous_toFun.isOpen_preimage _ H
Β· exact H
| 9 | 8,103.083928 | 2 | 0.714286 | 7 | 647 |
import Mathlib.Data.Stream.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Init.Data.List.Basic
import Mathlib.Data.List.Basic
#align_import data.stream.init from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
set_option autoImplicit true
open Nat Function Option
namespace Stre... | Mathlib/Data/Stream/Init.lean | 76 | 76 | theorem tail_drop (n : Nat) (s : Stream' Ξ±) : tail (drop n s) = drop n (tail s) := by | simp
| 1 | 2.718282 | 0 | 0.2 | 5 | 275 |
import Mathlib.Order.Bounds.Basic
import Mathlib.Order.WellFounded
import Mathlib.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Data.Set.Lattice
#align_import order.conditionally_complete_lattice.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
open Func... | Mathlib/Order/ConditionallyCompleteLattice/Basic.lean | 91 | 92 | theorem WithTop.iInf_empty [IsEmpty ΞΉ] [InfSet Ξ±] (f : ΞΉ β WithTop Ξ±) :
β¨
i, f i = β€ := by | rw [iInf, range_eq_empty, WithTop.sInf_empty]
| 1 | 2.718282 | 0 | 1 | 5 | 871 |
import Mathlib.Algebra.MvPolynomial.Derivation
import Mathlib.Algebra.MvPolynomial.Variables
#align_import data.mv_polynomial.pderiv from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
universe u v
namespace MvPolynomial
open Set Function Finsupp
variable {R : ... | Mathlib/Algebra/MvPolynomial/PDeriv.lean | 115 | 117 | theorem pderiv_mul {i : Ο} {f g : MvPolynomial Ο R} :
pderiv i (f * g) = pderiv i f * g + f * pderiv i g := by |
simp only [(pderiv i).leibniz f g, smul_eq_mul, mul_comm, add_comm]
| 1 | 2.718282 | 0 | 0.222222 | 9 | 283 |
import Mathlib.Data.List.Duplicate
import Mathlib.Data.List.Sort
#align_import data.list.nodup_equiv_fin from "leanprover-community/mathlib"@"008205aa645b3f194c1da47025c5f110c8406eab"
namespace List
variable {Ξ± : Type*}
section Sublist
theorem sublist_of_orderEmbedding_get?_eq {l l' : List Ξ±} (f : β βͺo β)
... | Mathlib/Data/List/NodupEquivFin.lean | 144 | 161 | theorem sublist_iff_exists_orderEmbedding_get?_eq {l l' : List Ξ±} :
l <+ l' β β f : β βͺo β, β ix : β, l.get? ix = l'.get? (f ix) := by |
constructor
Β· intro H
induction' H with xs ys y _H IH xs ys x _H IH
Β· simp
Β· obtain β¨f, hfβ© := IH
refine β¨f.trans (OrderEmbedding.ofStrictMono (Β· + 1) fun _ => by simp), ?_β©
simpa using hf
Β· obtain β¨f, hfβ© := IH
refine
β¨OrderEmbedding.ofMapLEIff (fun ix : β => if ix = 0 th... | 16 | 8,886,110.520508 | 2 | 2 | 4 | 1,962 |
import Mathlib.Order.Filter.Lift
import Mathlib.Topology.Separation
import Mathlib.Order.Interval.Set.Monotone
#align_import topology.filter from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
open Set Filter TopologicalSpace
open Filter Topology
variable {ΞΉ : Sort*} {Ξ± Ξ² X Y : Type*}... | Mathlib/Topology/Filter.lean | 105 | 106 | theorem HasBasis.nhds' {l : Filter Ξ±} {p : ΞΉ β Prop} {s : ΞΉ β Set Ξ±} (h : HasBasis l p s) :
HasBasis (π l) p fun i => { l' | s i β l' } := by | simpa only [Iic_principal] using h.nhds
| 1 | 2.718282 | 0 | 0.5 | 6 | 489 |
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 | 60 | 60 | theorem exp_mul (x y : β) : exp (x * y) = exp x ^ y := by | rw [rpow_def_of_pos (exp_pos _), log_exp]
| 1 | 2.718282 | 0 | 0.384615 | 13 | 383 |
import Mathlib.Algebra.CharP.Defs
import Mathlib.RingTheory.Multiplicity
import Mathlib.RingTheory.PowerSeries.Basic
#align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60"
noncomputable section
open Polynomial
open Finset (antidiagonal mem_anti... | Mathlib/RingTheory/PowerSeries/Order.lean | 162 | 164 | theorem le_order_add (Ο Ο : Rβ¦Xβ§) : min (order Ο) (order Ο) β€ order (Ο + Ο) := by |
refine le_order _ _ ?_
simp (config := { contextual := true }) [coeff_of_lt_order]
| 2 | 7.389056 | 1 | 1.8 | 10 | 1,890 |
import Mathlib.CategoryTheory.Monoidal.Mon_
import Mathlib.CategoryTheory.Monoidal.Braided.Opposite
import Mathlib.CategoryTheory.Monoidal.Transport
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.CategoryTheory.Limits.Shapes.Terminal
universe vβ vβ uβ uβ u
open CategoryTheory MonoidalCategor... | Mathlib/CategoryTheory/Monoidal/Comon_.lean | 73 | 74 | theorem counit_comul_hom {Z : C} (f : M.X βΆ Z) : M.comul β« (M.counit β f) = f β« (Ξ»_ Z).inv := by |
rw [leftUnitor_inv_naturality, tensorHom_def, counit_comul_assoc]
| 1 | 2.718282 | 0 | 0 | 2 | 134 |
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.Data.Nat.Lattice
#align_import combinatorics.simple_graph.metric from "leanprover-community/mathlib"@"352ecfe114946c903338006dd3287cb5a9955ff2"
namespace SimpleGraph
variable {V : Type*} (G : SimpleGraph V)
noncomputable def dist (u v : V)... | Mathlib/Combinatorics/SimpleGraph/Metric.lean | 74 | 74 | theorem dist_self {v : V} : dist G v v = 0 := by | simp
| 1 | 2.718282 | 0 | 1 | 7 | 1,136 |
import Batteries.Data.List.Count
import Batteries.Data.Fin.Lemmas
open Nat Function
namespace List
theorem rel_of_pairwise_cons (p : (a :: l).Pairwise R) : β {a'}, a' β l β R a a' :=
(pairwise_cons.1 p).1 _
theorem Pairwise.of_cons (p : (a :: l).Pairwise R) : Pairwise R l :=
(pairwise_cons.1 p).2
theorem... | .lake/packages/batteries/Batteries/Data/List/Pairwise.lean | 104 | 104 | theorem pairwise_singleton (R) (a : Ξ±) : Pairwise R [a] := by | simp
| 1 | 2.718282 | 0 | 1 | 8 | 814 |
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 | 84 | 89 | theorem trace_eq_matrix_trace_of_finset {s : Finset M} (b : Basis s R M) (f : M ββ[R] M) :
trace R M f = Matrix.trace (LinearMap.toMatrix b b f) := by |
have : β s : Finset M, Nonempty (Basis s R M) := β¨s, β¨bβ©β©
rw [trace, dif_pos this, β traceAux_def]
congr 1
apply traceAux_eq
| 4 | 54.59815 | 2 | 1.333333 | 6 | 1,414 |
import Mathlib.Control.Applicative
import Mathlib.Control.Traversable.Basic
import Mathlib.Data.List.Forall2
import Mathlib.Data.Set.Functor
#align_import control.traversable.instances from "leanprover-community/mathlib"@"18a5306c091183ac90884daa9373fa3b178e8607"
universe u v
section Option
open Functor
variab... | Mathlib/Control/Traversable/Instances.lean | 41 | 42 | theorem Option.traverse_eq_map_id {Ξ± Ξ²} (f : Ξ± β Ξ²) (x : Option Ξ±) :
Option.traverse ((pure : _ β Id _) β f) x = (pure : _ β Id _) (f <$> x) := by | cases x <;> rfl
| 1 | 2.718282 | 0 | 0.25 | 4 | 301 |
import Mathlib.AlgebraicGeometry.Restrict
import Mathlib.CategoryTheory.Adjunction.Limits
import Mathlib.CategoryTheory.Adjunction.Reflective
#align_import algebraic_geometry.Gamma_Spec_adjunction from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
-- Explicit universe annotations were u... | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | 146 | 160 | theorem toΞSpecCApp_iff
(f :
(structureSheaf <| Ξ.obj <| op X).val.obj (op <| basicOpen r) βΆ
X.presheaf.obj (op <| X.toΞSpecMapBasicOpen r)) :
toOpen _ (basicOpen r) β« f = X.toToΞSpecMapBasicOpen r β f = X.toΞSpecCApp r := by |
-- Porting Note: Type class problem got stuck in `IsLocalization.Away.AwayMap.lift_comp`
-- created instance manually. This replaces the `pick_goal` tactics
have loc_inst := IsLocalization.to_basicOpen (Ξ.obj (op X)) r
rw [β @IsLocalization.Away.AwayMap.lift_comp _ _ _ _ _ _ _ r loc_inst _
(X.isUnit_res_... | 10 | 22,026.465795 | 2 | 1.4 | 5 | 1,475 |
import Mathlib.Order.BooleanAlgebra
import Mathlib.Logic.Equiv.Basic
#align_import order.symm_diff from "leanprover-community/mathlib"@"6eb334bd8f3433d5b08ba156b8ec3e6af47e1904"
open Function OrderDual
variable {ΞΉ Ξ± Ξ² : Type*} {Ο : ΞΉ β Type*}
def symmDiff [Sup Ξ±] [SDiff Ξ±] (a b : Ξ±) : Ξ± :=
a \ b β b \ a
#ali... | Mathlib/Order/SymmDiff.lean | 125 | 125 | theorem symmDiff_bot : a β β₯ = a := by | rw [symmDiff, sdiff_bot, bot_sdiff, sup_bot_eq]
| 1 | 2.718282 | 0 | 0.181818 | 22 | 266 |
import Mathlib.AlgebraicGeometry.Restrict
import Mathlib.CategoryTheory.Adjunction.Limits
import Mathlib.CategoryTheory.Adjunction.Reflective
#align_import algebraic_geometry.Gamma_Spec_adjunction from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
-- Explicit universe annotations were u... | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | 77 | 79 | theorem not_mem_prime_iff_unit_in_stalk (r : Ξ.obj (op X)) (x : X) :
r β (X.toΞSpecFun x).asIdeal β IsUnit (X.ΞToStalk x r) := by |
erw [LocalRing.mem_maximalIdeal, Classical.not_not]
| 1 | 2.718282 | 0 | 1.4 | 5 | 1,475 |
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Tactic.NthRewrite
#align_import data.nat.gcd.basic from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
namespace Nat
theorem gcd_greatest {a b d : β} (hda : d β£ a) (hdb : d β£ b) (hd ... | Mathlib/Data/Nat/GCD/Basic.lean | 40 | 41 | theorem gcd_add_mul_left_right (m n k : β) : gcd m (n + m * k) = gcd m n := by |
simp [gcd_rec m (n + m * k), gcd_rec m n]
| 1 | 2.718282 | 0 | 0.352941 | 17 | 375 |
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 | 61 | 62 | theorem top_sub {a : WithTop Ξ±} : (β€ : WithTop Ξ±) - a = β€ := by |
cases a <;> rfl
| 1 | 2.718282 | 0 | 0 | 2 | 139 |
import Mathlib.Algebra.Order.Ring.Nat
#align_import data.nat.dist from "leanprover-community/mathlib"@"d50b12ae8e2bd910d08a94823976adae9825718b"
namespace Nat
def dist (n m : β) :=
n - m + (m - n)
#align nat.dist Nat.dist
-- Should be aligned to `Nat.dist.eq_def`, but that is generated on demand and isn't pr... | Mathlib/Data/Nat/Dist.lean | 27 | 27 | theorem dist_comm (n m : β) : dist n m = dist m n := by | simp [dist, add_comm]
| 1 | 2.718282 | 0 | 0.266667 | 15 | 309 |
import Mathlib.Logic.Function.Iterate
import Mathlib.Order.GaloisConnection
import Mathlib.Order.Hom.Basic
#align_import order.hom.order from "leanprover-community/mathlib"@"ba2245edf0c8bb155f1569fd9b9492a9b384cde6"
namespace OrderHom
variable {Ξ± Ξ² : Type*}
section Preorder
variable [Preorder Ξ±]
instance [Sem... | Mathlib/Order/Hom/Order.lean | 117 | 119 | theorem coe_iSup {ΞΉ : Sort*} [CompleteLattice Ξ²] (f : ΞΉ β Ξ± βo Ξ²) :
((β¨ i, f i : Ξ± βo Ξ²) : Ξ± β Ξ²) = β¨ i, (f i : Ξ± β Ξ²) := by |
funext x; simp [iSup_apply]
| 1 | 2.718282 | 0 | 0.666667 | 3 | 566 |
import Mathlib.Algebra.Order.Ring.Nat
#align_import data.nat.dist from "leanprover-community/mathlib"@"d50b12ae8e2bd910d08a94823976adae9825718b"
namespace Nat
def dist (n m : β) :=
n - m + (m - n)
#align nat.dist Nat.dist
-- Should be aligned to `Nat.dist.eq_def`, but that is generated on demand and isn't pr... | Mathlib/Data/Nat/Dist.lean | 112 | 113 | theorem dist_succ_succ {i j : Nat} : dist (succ i) (succ j) = dist i j := by |
simp [dist, succ_sub_succ]
| 1 | 2.718282 | 0 | 0.266667 | 15 | 309 |
import Mathlib.Algebra.Homology.Single
#align_import algebra.homology.augment from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
noncomputable section
open CategoryTheory Limits HomologicalComplex
universe v u
variable {V : Type u} [Category.{v} V]
namespace ChainComplex
@[simps]... | Mathlib/Algebra/Homology/Augment.lean | 92 | 94 | theorem augment_d_succ_succ (C : ChainComplex V β) {X : V} (f : C.X 0 βΆ X) (w : C.d 1 0 β« f = 0)
(i j : β) : (augment C f w).d (i + 1) (j + 1) = C.d i j := by |
cases i <;> rfl
| 1 | 2.718282 | 0 | 0.666667 | 3 | 587 |
import Batteries.Data.Fin.Basic
namespace Fin
attribute [norm_cast] val_last
protected theorem le_antisymm_iff {x y : Fin n} : x = y β x β€ y β§ y β€ x :=
Fin.ext_iff.trans Nat.le_antisymm_iff
protected theorem le_antisymm {x y : Fin n} (h1 : x β€ y) (h2 : y β€ x) : x = y :=
Fin.le_antisymm_iff.2 β¨h1, h2β©
@[simp... | .lake/packages/batteries/Batteries/Data/Fin/Lemmas.lean | 116 | 120 | theorem foldr_succ_last (f : Fin (n+1) β Ξ± β Ξ±) (x) :
foldr (n+1) f x = foldr n (f Β·.castSucc) (f (last n) x) := by |
induction n generalizing x with
| zero => simp [foldr_succ, Fin.last]
| succ n ih => rw [foldr_succ, ih (f Β·.succ), foldr_succ]; simp [succ_castSucc]
| 3 | 20.085537 | 1 | 1.090909 | 11 | 1,186 |
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 | 365 | 366 | theorem setAverage_congr (h : s =α΅[ΞΌ] t) : β¨ x in s, f x βΞΌ = β¨ x in t, f x βΞΌ := by |
simp only [setAverage_eq, setIntegral_congr_set_ae h, measure_congr h]
| 1 | 2.718282 | 0 | 0.347826 | 23 | 374 |
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Maps
open Finset
namespace SimpleGraph
variable {V : Type*} [DecidableEq V] (G : SimpleGraph V) (s t : V)
section ReplaceVertex
def replaceVertex : SimpleGraph V where
Adj v w := if v = t then if w = t then False else G... | Mathlib/Combinatorics/SimpleGraph/Operations.lean | 76 | 80 | theorem edgeSet_replaceVertex_of_not_adj (hn : Β¬G.Adj s t) : (G.replaceVertex s t).edgeSet =
G.edgeSet \ G.incidenceSet t βͺ (s(Β·, t)) '' (G.neighborSet s) := by |
ext e; refine e.inductionOn ?_
simp only [replaceVertex, mem_edgeSet, Set.mem_union, Set.mem_diff, mk'_mem_incidenceSet_iff]
intros; split_ifs; exacts [by simp_all, by aesop, by rw [adj_comm]; aesop, by aesop]
| 3 | 20.085537 | 1 | 1.111111 | 9 | 1,199 |
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.Algebra.GCDMonoid.Nat
#align_import ring_theory.int.basic from "leanprover-community/mathlib"@"e655e4ea5c6d02854696f97494997ba4c31be802"
namespace Int
theorem gcd_eq_one_iff_coprime {a b : β€} : Int.gcd a b ... | Mathlib/RingTheory/Int/Basic.lean | 49 | 50 | theorem coprime_iff_nat_coprime {a b : β€} : IsCoprime a b β Nat.Coprime a.natAbs b.natAbs := by |
rw [β gcd_eq_one_iff_coprime, Nat.coprime_iff_gcd_eq_one, gcd_eq_natAbs]
| 1 | 2.718282 | 0 | 1.153846 | 13 | 1,227 |
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 | 108 | 109 | theorem volume_ball (a r : β) : volume (Metric.ball a r) = ofReal (2 * r) := by |
rw [ball_eq_Ioo, volume_Ioo, β sub_add, add_sub_cancel_left, two_mul]
| 1 | 2.718282 | 0 | 0.909091 | 22 | 790 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Div
#align_import data.polynomial.ring_division from "leanprover-community/mathlib"@"8efcf8022aac8e01df8d302dcebdbc25d6a886c8"
noncomputable ... | Mathlib/Algebra/Polynomial/RingDivision.lean | 190 | 195 | theorem natDegree_sub_eq_of_prod_eq {pβ pβ qβ qβ : R[X]} (hpβ : pβ β 0) (hqβ : qβ β 0)
(hpβ : pβ β 0) (hqβ : qβ β 0) (h_eq : pβ * qβ = pβ * qβ) :
(pβ.natDegree : β€) - qβ.natDegree = (pβ.natDegree : β€) - qβ.natDegree := by |
rw [sub_eq_sub_iff_add_eq_add]
norm_cast
rw [β natDegree_mul hpβ hqβ, β natDegree_mul hpβ hqβ, h_eq]
| 3 | 20.085537 | 1 | 1.5 | 32 | 1,561 |
import Mathlib.Algebra.Polynomial.Degree.TrailingDegree
import Mathlib.Algebra.Polynomial.EraseLead
import Mathlib.Algebra.Polynomial.Eval
#align_import data.polynomial.reverse from "leanprover-community/mathlib"@"44de64f183393284a16016dfb2a48ac97382f2bd"
namespace Polynomial
open Polynomial Finsupp Finset
open... | Mathlib/Algebra/Polynomial/Reverse.lean | 160 | 161 | theorem reflect_C (r : R) (N : β) : reflect N (C r) = C r * X ^ N := by |
conv_lhs => rw [β mul_one (C r), β pow_zero X, reflect_C_mul_X_pow, revAt_zero]
| 1 | 2.718282 | 0 | 1 | 12 | 945 |
import Mathlib.Geometry.Manifold.ContMDiff.Defs
open Set Filter Function
open scoped Topology Manifold
variable {π : Type*} [NontriviallyNormedField π]
-- declare a smooth manifold `M` over the pair `(E, H)`.
{E : Type*}
[NormedAddCommGroup E] [NormedSpace π E] {H : Type*} [TopologicalSpace H]
(I : Mode... | Mathlib/Geometry/Manifold/ContMDiff/Basic.lean | 52 | 77 | theorem ContMDiffWithinAt.comp {t : Set M'} {g : M' β M''} (x : M)
(hg : ContMDiffWithinAt I' I'' n g t (f x)) (hf : ContMDiffWithinAt I I' n f s x)
(st : MapsTo f s t) : ContMDiffWithinAt I I'' n (g β f) s x := by |
rw [contMDiffWithinAt_iff] at hg hf β’
refine β¨hg.1.comp hf.1 st, ?_β©
set e := extChartAt I x
set e' := extChartAt I' (f x)
have : e' (f x) = (writtenInExtChartAt I I' x f) (e x) := by simp only [e, e', mfld_simps]
rw [this] at hg
have A : βαΆ y in π[e.symm β»ΒΉ' s β© range I] e x, f (e.symm y) β t β§ f (e.sy... | 23 | 9,744,803,446.248903 | 2 | 0.833333 | 6 | 725 |
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Set.Finite
#align_import combinatorics.hall.finite from "leanprover-community/mathlib"@"d6fad0e5bf2d6f48da9175d25c3dc5706b3834ce"
open Finset
universe u v
namespace HallMarriageTheorem
variable {ΞΉ : Type u} {Ξ± : Type v} [DecidableEq Ξ±] {t : ΞΉ β Finset Ξ±}
s... | Mathlib/Combinatorics/Hall/Finite.lean | 78 | 121 | theorem hall_hard_inductive_step_A {n : β} (hn : Fintype.card ΞΉ = n + 1)
(ht : β s : Finset ΞΉ, s.card β€ (s.biUnion t).card)
(ih :
β {ΞΉ' : Type u} [Fintype ΞΉ'] (t' : ΞΉ' β Finset Ξ±),
Fintype.card ΞΉ' β€ n β
(β s' : Finset ΞΉ', s'.card β€ (s'.biUnion t').card) β
β f : ΞΉ' β Ξ±, Functi... |
haveI : Nonempty ΞΉ := Fintype.card_pos_iff.mp (hn.symm βΈ Nat.succ_pos _)
haveI := Classical.decEq ΞΉ
-- Choose an arbitrary element `x : ΞΉ` and `y : t x`.
let x := Classical.arbitrary ΞΉ
have tx_ne : (t x).Nonempty := by
rw [β Finset.card_pos]
calc
0 < 1 := Nat.one_pos
_ β€ (Finset.biUnion {... | 33 | 214,643,579,785,916.06 | 2 | 2 | 4 | 2,298 |
import Mathlib.MeasureTheory.MeasurableSpace.Basic
import Mathlib.MeasureTheory.Measure.MeasureSpaceDef
#align_import measure_theory.function.ae_measurable_sequence from "leanprover-community/mathlib"@"d003c55042c3cd08aefd1ae9a42ef89441cdaaf3"
open MeasureTheory
open scoped Classical
variable {ΞΉ : Sort*} {Ξ± Ξ² Ξ³... | Mathlib/MeasureTheory/Function/AEMeasurableSequence.lean | 81 | 86 | theorem fun_prop_of_mem_aeSeqSet (hf : β i, AEMeasurable (f i) ΞΌ) {x : Ξ±} (hx : x β aeSeqSet hf p) :
p x fun n => f n x := by |
have h_eq : (fun n => f n x) = fun n => aeSeq hf p n x :=
funext fun n => (aeSeq_eq_fun_of_mem_aeSeqSet hf hx n).symm
rw [h_eq]
exact prop_of_mem_aeSeqSet hf hx
| 4 | 54.59815 | 2 | 1.333333 | 6 | 1,404 |
import Mathlib.Algebra.PUnitInstances
import Mathlib.Tactic.Abel
import Mathlib.Tactic.Ring
import Mathlib.Order.Hom.Lattice
#align_import algebra.ring.boolean_ring from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
open scoped symmDiff
variable {Ξ± Ξ² Ξ³ : Type*}
class BooleanRing (Ξ±) ... | Mathlib/Algebra/Ring/BooleanRing.lean | 101 | 101 | theorem sub_eq_add : a - b = a + b := by | rw [sub_eq_add_neg, add_right_inj, neg_eq]
| 1 | 2.718282 | 0 | 0.833333 | 6 | 724 |
import Mathlib.Data.Fin.VecNotation
import Mathlib.Logic.Embedding.Set
#align_import logic.equiv.fin from "leanprover-community/mathlib"@"bd835ef554f37ef9b804f0903089211f89cb370b"
assert_not_exists MonoidWithZero
universe u
variable {m n : β}
def finZeroEquiv : Fin 0 β Empty :=
Equiv.equivEmpty _
#align fin_... | Mathlib/Logic/Equiv/Fin.lean | 121 | 123 | theorem finSuccEquiv'_below {i : Fin (n + 1)} {m : Fin n} (h : Fin.castSucc m < i) :
(finSuccEquiv' i) (Fin.castSucc m) = m := by |
rw [β Fin.succAbove_of_castSucc_lt _ _ h, finSuccEquiv'_succAbove]
| 1 | 2.718282 | 0 | 0.25 | 4 | 298 |
import Mathlib.SetTheory.Cardinal.ToNat
import Mathlib.Data.Nat.PartENat
#align_import set_theory.cardinal.basic from "leanprover-community/mathlib"@"3ff3f2d6a3118b8711063de7111a0d77a53219a8"
universe u v
open Function
variable {Ξ± : Type u}
namespace Cardinal
noncomputable def toPartENat : Cardinal β+o PartEN... | Mathlib/SetTheory/Cardinal/PartENat.lean | 39 | 40 | theorem toPartENat_natCast (n : β) : toPartENat n = n := by |
simp only [β partENatOfENat_toENat, toENat_nat, PartENat.ofENat_coe]
| 1 | 2.718282 | 0 | 0.166667 | 6 | 262 |
import Mathlib.Topology.Category.CompHaus.Basic
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Limits.Preserves.Finite
namespace CompHaus
attribute [local instance] CategoryTheory.ConcreteCategory.instFunLike
universe u w
open Categor... | Mathlib/Topology/Category/CompHaus/Limits.lean | 131 | 134 | theorem pullback_fst_eq :
CompHaus.pullback.fst f g = (pullbackIsoPullback f g).hom β« Limits.pullback.fst := by |
dsimp [pullbackIsoPullback]
simp only [Limits.limit.conePointUniqueUpToIso_hom_comp, pullback.cone_pt, pullback.cone_Ο]
| 2 | 7.389056 | 1 | 0.666667 | 3 | 579 |
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Set.Finite
#align_import data.finset.n_ary from "leanprover-community/mathlib"@"eba7871095e834365616b5e43c8c7bb0b37058d0"
open Function Set
variable {Ξ± Ξ±' Ξ² Ξ²' Ξ³ Ξ³' Ξ΄ Ξ΄' Ξ΅ Ξ΅' ΞΆ ΞΆ' Ξ½ : Type*}
namespace Finset
variable [DecidableEq Ξ±'] [DecidableEq Ξ²'] [Decidabl... | Mathlib/Data/Finset/NAry.lean | 98 | 100 | theorem forall_imageβ_iff {p : Ξ³ β Prop} :
(β z β imageβ f s t, p z) β β x β s, β y β t, p (f x y) := by |
simp_rw [β mem_coe, coe_imageβ, forall_image2_iff]
| 1 | 2.718282 | 0 | 0.375 | 8 | 379 |
import Mathlib.Algebra.ContinuedFractions.Computation.Approximations
import Mathlib.Algebra.ContinuedFractions.Computation.CorrectnessTerminating
import Mathlib.Data.Rat.Floor
#align_import algebra.continued_fractions.computation.terminates_iff_rat from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b3... | Mathlib/Algebra/ContinuedFractions/Computation/TerminatesIffRat.lean | 112 | 115 | theorem exists_rat_eq_nth_denominator : β q : β, (of v).denominators n = (q : K) := by |
rcases exists_gcf_pair_rat_eq_nth_conts v n with β¨β¨_, bβ©, nth_cont_eqβ©
use b
simp [denom_eq_conts_b, nth_cont_eq]
| 3 | 20.085537 | 1 | 1.272727 | 11 | 1,350 |
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Inv
#align_import analysis.calculus.dslope from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
open scoped Classical Topology Filter
open Function Set Filter
variable {π E : Type*} [NontriviallyNormed... | Mathlib/Analysis/Calculus/Dslope.lean | 114 | 115 | theorem continuousAt_dslope_of_ne (h : b β a) : ContinuousAt (dslope f a) b β ContinuousAt f b := by |
simp only [β continuousWithinAt_univ, continuousWithinAt_dslope_of_ne h]
| 1 | 2.718282 | 0 | 0.875 | 8 | 764 |
import Mathlib.RingTheory.Polynomial.Cyclotomic.Eval
#align_import number_theory.primes_congruent_one from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
namespace Nat
open Polynomial Nat Filter
open scoped Nat
theorem exists_prime_gt_modEq_one {k : β} (n : β) (hk0 : k β 0) :
β ... | Mathlib/NumberTheory/PrimesCongruentOne.lean | 60 | 64 | theorem frequently_atTop_modEq_one {k : β} (hk0 : k β 0) :
βαΆ p in atTop, Nat.Prime p β§ p β‘ 1 [MOD k] := by |
refine frequently_atTop.2 fun n => ?_
obtain β¨p, hpβ© := exists_prime_gt_modEq_one n hk0
exact β¨p, β¨hp.2.1.le, hp.1, hp.2.2β©β©
| 3 | 20.085537 | 1 | 1.5 | 2 | 1,593 |
import Mathlib.Geometry.RingedSpace.PresheafedSpace
import Mathlib.CategoryTheory.Limits.Final
import Mathlib.Topology.Sheaves.Stalks
#align_import algebraic_geometry.stalks from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
noncomputable section
universe v u v' u'
open Opposite Cate... | Mathlib/Geometry/RingedSpace/Stalks.lean | 171 | 177 | theorem congr {X Y : PresheafedSpace.{_, _, v} C} (Ξ± Ξ² : X βΆ Y)
(hβ : Ξ± = Ξ²) (x x' : X) (hβ : x = x') :
stalkMap Ξ± x β« eqToHom (show X.stalk x = X.stalk x' by rw [hβ]) =
eqToHom (show Y.stalk (Ξ±.base x) = Y.stalk (Ξ².base x') by rw [hβ, hβ]) β« stalkMap Ξ² x' := by |
ext
substs hβ hβ
simp
| 3 | 20.085537 | 1 | 0.875 | 8 | 765 |
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Normed.Group.Lemmas
import Mathlib.Analysis.NormedSpace.AddTorsor
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
import Mathlib.Analysis.NormedSpace.RieszLemma
import Mathli... | Mathlib/Analysis/NormedSpace/FiniteDimension.lean | 189 | 192 | theorem lipschitzExtensionConstant_pos (E' : Type*) [NormedAddCommGroup E'] [NormedSpace β E']
[FiniteDimensional β E'] : 0 < lipschitzExtensionConstant E' := by |
rw [lipschitzExtensionConstant]
exact zero_lt_one.trans_le (le_max_right _ _)
| 2 | 7.389056 | 1 | 1.833333 | 6 | 1,910 |
import Mathlib.Topology.Compactness.SigmaCompact
import Mathlib.Topology.Connected.TotallyDisconnected
import Mathlib.Topology.Inseparable
#align_import topology.separation from "leanprover-community/mathlib"@"d91e7f7a7f1c7e9f0e18fdb6bde4f652004c735d"
open Function Set Filter Topology TopologicalSpace
open scoped... | Mathlib/Topology/Separation.lean | 261 | 264 | theorem t0Space_iff_exists_isOpen_xor'_mem (X : Type u) [TopologicalSpace X] :
T0Space X β Pairwise fun x y => β U : Set X, IsOpen U β§ Xor' (x β U) (y β U) := by |
simp only [t0Space_iff_not_inseparable, xor_iff_not_iff, not_forall, exists_prop,
inseparable_iff_forall_open, Pairwise]
| 2 | 7.389056 | 1 | 0.5 | 4 | 446 |
import Mathlib.Data.Set.Image
import Mathlib.Data.Set.Lattice
#align_import data.set.sigma from "leanprover-community/mathlib"@"2258b40dacd2942571c8ce136215350c702dc78f"
namespace Set
variable {ΞΉ ΞΉ' : Type*} {Ξ± Ξ² : ΞΉ β Type*} {s sβ sβ : Set ΞΉ} {t tβ tβ : β i, Set (Ξ± i)}
{u : Set (Ξ£ i, Ξ± i)} {x : Ξ£ i, Ξ± i} {i j ... | Mathlib/Data/Set/Sigma.lean | 31 | 34 | theorem preimage_image_sigmaMk_of_ne (h : i β j) (s : Set (Ξ± j)) :
Sigma.mk i β»ΒΉ' (Sigma.mk j '' s) = β
:= by |
ext x
simp [h.symm]
| 2 | 7.389056 | 1 | 1.666667 | 3 | 1,824 |
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
noncomputable section
universe u
namespace SetTheory
open scoped PGame
namespace PGame... | Mathlib/SetTheory/Game/Nim.lean | 73 | 75 | theorem moveLeft_nim_hEq (o : Ordinal) :
have : IsWellOrder (Quotient.out o).Ξ± (Β· < Β·) := inferInstance
HEq (nim o).moveLeft fun i : o.out.Ξ± => nim (typein (Β· < Β·) i) := by | rw [nim_def]; rfl
| 1 | 2.718282 | 0 | 0 | 7 | 205 |
import Mathlib.Analysis.NormedSpace.Star.Spectrum
import Mathlib.Analysis.Normed.Group.Quotient
import Mathlib.Analysis.NormedSpace.Algebra
import Mathlib.Topology.ContinuousFunction.Units
import Mathlib.Topology.ContinuousFunction.Compact
import Mathlib.Topology.Algebra.Algebra
import Mathlib.Topology.ContinuousFunct... | Mathlib/Analysis/NormedSpace/Star/GelfandDuality.lean | 99 | 105 | theorem WeakDual.CharacterSpace.exists_apply_eq_zero {a : A} (ha : Β¬IsUnit a) :
β f : characterSpace β A, f a = 0 := by |
obtain β¨M, hM, haMβ© := (span {a}).exists_le_maximal (span_singleton_ne_top ha)
exact
β¨M.toCharacterSpace,
M.toCharacterSpace_apply_eq_zero_of_mem
(haM (mem_span_singleton.mpr β¨1, (mul_one a).symmβ©))β©
| 5 | 148.413159 | 2 | 1.75 | 4 | 1,867 |
import Mathlib.Analysis.Normed.Field.Basic
import Mathlib.RingTheory.Valuation.RankOne
import Mathlib.Topology.Algebra.Valuation
noncomputable section
open Filter Set Valuation
open scoped NNReal
variable {K : Type*} [hK : NormedField K] (h : IsNonarchimedean (norm : K β β))
namespace Valued
variable {L : Typ... | Mathlib/Topology/Algebra/NormedValued.lean | 68 | 68 | theorem norm_nonneg (x : L) : 0 β€ norm x := by | simp only [norm, NNReal.zero_le_coe]
| 1 | 2.718282 | 0 | 0.333333 | 3 | 351 |
import Mathlib.MeasureTheory.Integral.IntervalIntegral
import Mathlib.Order.Filter.IndicatorFunction
open MeasureTheory
section DominatedConvergenceTheorem
open Set Filter TopologicalSpace ENNReal
open scoped Topology
namespace MeasureTheory
variable {Ξ± E G: Type*}
[NormedAddCommGroup E] [NormedSpace β E] [C... | Mathlib/MeasureTheory/Integral/DominatedConvergence.lean | 66 | 75 | theorem tendsto_integral_filter_of_dominated_convergence {ΞΉ} {l : Filter ΞΉ} [l.IsCountablyGenerated]
{F : ΞΉ β Ξ± β G} {f : Ξ± β G} (bound : Ξ± β β) (hF_meas : βαΆ n in l, AEStronglyMeasurable (F n) ΞΌ)
(h_bound : βαΆ n in l, βα΅ a βΞΌ, βF n aβ β€ bound a) (bound_integrable : Integrable bound ΞΌ)
(h_lim : βα΅ a βΞΌ, Ten... |
by_cases hG : CompleteSpace G
Β· simp only [integral, hG, L1.integral]
exact tendsto_setToFun_filter_of_dominated_convergence (dominatedFinMeasAdditive_weightedSMul ΞΌ)
bound hF_meas h_bound bound_integrable h_lim
Β· simp [integral, hG, tendsto_const_nhds]
| 5 | 148.413159 | 2 | 2 | 4 | 2,302 |
import Mathlib.Algebra.Algebra.Defs
import Mathlib.RingTheory.Ideal.Operations
import Mathlib.RingTheory.JacobsonIdeal
import Mathlib.Logic.Equiv.TransferInstance
import Mathlib.Tactic.TFAE
#align_import ring_theory.ideal.local_ring from "leanprover-community/mathlib"@"ec1c7d810034d4202b0dd239112d1792be9f6fdc"
un... | Mathlib/RingTheory/Ideal/LocalRing.lean | 136 | 138 | theorem le_maximalIdeal {J : Ideal R} (hJ : J β β€) : J β€ maximalIdeal R := by |
rcases Ideal.exists_le_maximal J hJ with β¨M, hM1, hM2β©
rwa [β eq_maximalIdeal hM1]
| 2 | 7.389056 | 1 | 1 | 2 | 1,155 |
import Mathlib.Geometry.Manifold.SmoothManifoldWithCorners
import Mathlib.Topology.Compactness.Paracompact
import Mathlib.Topology.Metrizable.Urysohn
#align_import geometry.manifold.metrizable from "leanprover-community/mathlib"@"d1bd9c5df2867c1cb463bc6364446d57bdd9f7f1"
open TopologicalSpace
| Mathlib/Geometry/Manifold/Metrizable.lean | 24 | 31 | theorem ManifoldWithCorners.metrizableSpace {E : Type*} [NormedAddCommGroup E] [NormedSpace β E]
[FiniteDimensional β E] {H : Type*} [TopologicalSpace H] (I : ModelWithCorners β E H)
(M : Type*) [TopologicalSpace M] [ChartedSpace H M] [SigmaCompactSpace M] [T2Space M] :
MetrizableSpace M := by |
haveI := I.locallyCompactSpace; haveI := ChartedSpace.locallyCompactSpace H M
haveI := I.secondCountableTopology
haveI := ChartedSpace.secondCountable_of_sigma_compact H M
exact metrizableSpace_of_t3_second_countable M
| 4 | 54.59815 | 2 | 2 | 1 | 2,233 |
import Mathlib.Topology.Basic
import Mathlib.Order.UpperLower.Basic
import Mathlib.Order.OmegaCompletePartialOrder
#align_import topology.omega_complete_partial_order from "leanprover-community/mathlib"@"2705404e701abc6b3127da906f40bae062a169c9"
open Set OmegaCompletePartialOrder
open scoped Classical
universe ... | Mathlib/Topology/OmegaCompletePartialOrder.lean | 41 | 43 | theorem isΟSup_iff_isLUB {Ξ± : Type u} [Preorder Ξ±] {c : Chain Ξ±} {x : Ξ±} :
IsΟSup c x β IsLUB (range c) x := by |
simp [IsΟSup, IsLUB, IsLeast, upperBounds, lowerBounds]
| 1 | 2.718282 | 0 | 1 | 2 | 1,061 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Monic
#align_import data.polynomial.lifts from "leanprover-community/mathlib"@"63417e01fbc711beaf25fa73b6edb395c0cfddd0"
open Polynomial
noncomputable section
namespace Polynomial
universe u v w
section Semiring
variable {R : Type... | Mathlib/Algebra/Polynomial/Lifts.lean | 73 | 75 | theorem lifts_iff_coeff_lifts (p : S[X]) : p β lifts f β β n : β, p.coeff n β Set.range f := by |
rw [lifts_iff_ringHom_rangeS, mem_map_rangeS f]
rfl
| 2 | 7.389056 | 1 | 1 | 13 | 1,139 |
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Data.Finset.Antidiagonal
import Mathlib.Data.Finset.Card
import Mathlib.Data.Multiset.NatAntidiagonal
#align_import data.finset.nat_antidiagonal from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function
namespace Finset
name... | Mathlib/Data/Finset/NatAntidiagonal.lean | 89 | 99 | theorem antidiagonal_succ_succ' {n : β} :
antidiagonal (n + 2) =
cons (0, n + 2)
(cons (n + 2, 0)
((antidiagonal n).map
(Embedding.prodMap β¨Nat.succ, Nat.succ_injectiveβ©
β¨Nat.succ, Nat.succ_injectiveβ©)) <|
by simp)
(by simp) := by |
simp_rw [antidiagonal_succ (n + 1), antidiagonal_succ', Finset.map_cons, map_map]
rfl
| 2 | 7.389056 | 1 | 0.75 | 4 | 673 |
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