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.Algebra.MvPolynomial.Funext
import Mathlib.Algebra.Ring.ULift
import Mathlib.RingTheory.WittVector.Basic
#align_import ring_theory.witt_vector.is_poly from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc9524185ae994553ebf4"
namespace WittVector
universe u
variable {p : ℕ} {R S : Type u} {σ id... | Mathlib/RingTheory/WittVector/IsPoly.lean | 172 | 195 | theorem ext [Fact p.Prime] {f g} (hf : IsPoly p f) (hg : IsPoly p g)
(h : ∀ (R : Type u) [_Rcr : CommRing R] (x : 𝕎 R) (n : ℕ),
ghostComponent n (f x) = ghostComponent n (g x)) :
∀ (R : Type u) [_Rcr : CommRing R] (x : 𝕎 R), f x = g x := by |
obtain ⟨φ, hf⟩ := hf
obtain ⟨ψ, hg⟩ := hg
intros
ext n
rw [hf, hg, poly_eq_of_wittPolynomial_bind_eq p φ ψ]
intro k
apply MvPolynomial.funext
intro x
simp only [hom_bind₁]
specialize h (ULift ℤ) (mk p fun i => ⟨x i⟩) k
simp only [ghostComponent_apply, aeval_eq_eval₂Hom] at h
apply (ULift.ringEq... | 20 | 485,165,195.40979 | 2 | 2 | 3 | 2,029 |
import Mathlib.MeasureTheory.Measure.Dirac
set_option autoImplicit true
open Set
open scoped ENNReal Classical
variable [MeasurableSpace α] [MeasurableSpace β] {s : Set α}
noncomputable section
namespace MeasureTheory.Measure
def count : Measure α :=
sum dirac
#align measure_theory.measure.count MeasureTheo... | Mathlib/MeasureTheory/Measure/Count.lean | 122 | 126 | theorem empty_of_count_eq_zero [MeasurableSingletonClass α] (hsc : count s = 0) : s = ∅ := by |
have hs : s.Finite := by
rw [← count_apply_lt_top, hsc]
exact WithTop.zero_lt_top
simpa [count_apply_finite _ hs] using hsc
| 4 | 54.59815 | 2 | 1.1 | 10 | 1,189 |
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Algebra.Category.Ring.Colimits
import Mathlib.CategoryTheory.Iso
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.IsTensorProduct
#align_import ring_theory.ring_hom_properties from "leanprover-community/mathlib"@"a7c017d75051... | Mathlib/RingTheory/RingHomProperties.lean | 65 | 91 | theorem RespectsIso.is_localization_away_iff (hP : RingHom.RespectsIso @P) {R S : Type u}
(R' S' : Type u) [CommRing R] [CommRing S] [CommRing R'] [CommRing S'] [Algebra R R']
[Algebra S S'] (f : R →+* S) (r : R) [IsLocalization.Away r R'] [IsLocalization.Away (f r) S'] :
P (Localization.awayMap f r) ↔ P (I... |
let e₁ : R' ≃+* Localization.Away r :=
(IsLocalization.algEquiv (Submonoid.powers r) _ _).toRingEquiv
let e₂ : Localization.Away (f r) ≃+* S' :=
(IsLocalization.algEquiv (Submonoid.powers (f r)) _ _).toRingEquiv
refine (hP.cancel_left_isIso e₁.toCommRingCatIso.hom (CommRingCat.ofHom _)).symm.trans ?_
r... | 23 | 9,744,803,446.248903 | 2 | 2 | 1 | 2,024 |
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
import Mathlib.RingTheory.Polynomial.Basic
#align_import algebraic_geometry.prime_spectrum.is_open_comap_C from "leanprover-community/mathlib"@"052f6013363326d50cb99c6939814a4b8eb7b301"
open Ideal Polynomial PrimeSpectrum Set
namespace AlgebraicGeometry
names... | Mathlib/AlgebraicGeometry/PrimeSpectrum/IsOpenComapC.lean | 74 | 79 | theorem isOpenMap_comap_C : IsOpenMap (PrimeSpectrum.comap (C : R →+* R[X])) := by |
rintro U ⟨s, z⟩
rw [← compl_compl U, ← z, ← iUnion_of_singleton_coe s, zeroLocus_iUnion, compl_iInter,
image_iUnion]
simp_rw [← imageOfDf_eq_comap_C_compl_zeroLocus]
exact isOpen_iUnion fun f => isOpen_imageOfDf
| 5 | 148.413159 | 2 | 1.666667 | 3 | 1,814 |
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryTheory.Conj
#align_import category_theory.adjunction.mates from "leanprover-community/mathlib"@"cea27692b3fdeb328a2ddba6aabf181754543184"
universe v₁ v₂ v₃ v₄ u₁ u₂ u₃ u₄
namespace CategoryTheory
open Category
variable {C : Type u₁} {D : Typ... | Mathlib/CategoryTheory/Adjunction/Mates.lean | 118 | 124 | theorem unit_transferNatTrans (f : G ⋙ L₂ ⟶ L₁ ⋙ H) (X : C) :
G.map (adj₁.unit.app X) ≫ (transferNatTrans adj₁ adj₂ f).app _ =
adj₂.unit.app _ ≫ R₂.map (f.app _) := by |
dsimp [transferNatTrans]
rw [← adj₂.unit_naturality_assoc, ← R₂.map_comp, ← Functor.comp_map G L₂, f.naturality_assoc,
Functor.comp_map, ← H.map_comp]
dsimp; simp
| 4 | 54.59815 | 2 | 1.5 | 2 | 1,636 |
import Mathlib.Data.Set.Lattice
#align_import data.set.intervals.disjoint from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
universe u v w
variable {ι : Sort u} {α : Type v} {β : Type w}
open Set
open OrderDual (toDual)
namespace Set
section UnionIxx
variable [LinearOrder α] {s ... | Mathlib/Order/Interval/Set/Disjoint.lean | 201 | 205 | theorem IsGLB.biUnion_Ioi_eq (h : IsGLB s a) : ⋃ x ∈ s, Ioi x = Ioi a := by |
refine (iUnion₂_subset fun x hx => ?_).antisymm fun x hx => ?_
· exact Ioi_subset_Ioi (h.1 hx)
· rcases h.exists_between hx with ⟨y, hys, _, hyx⟩
exact mem_biUnion hys hyx
| 4 | 54.59815 | 2 | 0.333333 | 18 | 364 |
import Mathlib.AlgebraicGeometry.Pullbacks
import Mathlib.AlgebraicGeometry.AffineScheme
#align_import algebraic_geometry.limits from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
suppress_compilation
set_option linter.uppercaseLean3 false
universe u
open CategoryTheory CategoryTheor... | Mathlib/AlgebraicGeometry/Limits.lean | 133 | 139 | theorem bot_isAffineOpen (X : Scheme) : IsAffineOpen (⊥ : Opens X.carrier) := by |
convert rangeIsAffineOpenOfOpenImmersion (initial.to X)
ext
-- Porting note: added this `erw` to turn LHS to `False`
erw [Set.mem_empty_iff_false]
rw [false_iff_iff]
exact fun x => isEmptyElim (show (⊥_ Scheme).carrier from x.choose)
| 6 | 403.428793 | 2 | 2 | 1 | 2,083 |
import Mathlib.FieldTheory.Minpoly.Field
#align_import ring_theory.power_basis from "leanprover-community/mathlib"@"d1d69e99ed34c95266668af4e288fc1c598b9a7f"
open Polynomial
open Polynomial
variable {R S T : Type*} [CommRing R] [Ring S] [Algebra R S]
variable {A B : Type*} [CommRing A] [CommRing B] [IsDomain B]... | Mathlib/RingTheory/PowerBasis.lean | 105 | 116 | theorem mem_span_pow {x y : S} {d : ℕ} (hd : d ≠ 0) :
y ∈ Submodule.span R (Set.range fun i : Fin d => x ^ (i : ℕ)) ↔
∃ f : R[X], f.natDegree < d ∧ y = aeval x f := by |
rw [mem_span_pow']
constructor <;>
· rintro ⟨f, h, hy⟩
refine ⟨f, ?_, hy⟩
by_cases hf : f = 0
· simp only [hf, natDegree_zero, degree_zero] at h ⊢
first | exact lt_of_le_of_ne (Nat.zero_le d) hd.symm | exact WithBot.bot_lt_coe d
simp_all only [degree_eq_natDegree hf]
· fir... | 9 | 8,103.083928 | 2 | 1.428571 | 7 | 1,518 |
import Batteries.Data.List.Lemmas
import Batteries.Data.Array.Basic
import Batteries.Tactic.SeqFocus
import Batteries.Util.ProofWanted
namespace Array
theorem forIn_eq_data_forIn [Monad m]
(as : Array α) (b : β) (f : α → β → m (ForInStep β)) :
forIn as b f = forIn as.data b f := by
let rec loop : ∀ {i h b ... | .lake/packages/batteries/Batteries/Data/Array/Lemmas.lean | 33 | 73 | theorem zipWith_eq_zipWith_data (f : α → β → γ) (as : Array α) (bs : Array β) :
(as.zipWith bs f).data = as.data.zipWith f bs.data := by |
let rec loop : ∀ (i : Nat) cs, i ≤ as.size → i ≤ bs.size →
(zipWithAux f as bs i cs).data = cs.data ++ (as.data.drop i).zipWith f (bs.data.drop i) := by
intro i cs hia hib
unfold zipWithAux
by_cases h : i = as.size ∨ i = bs.size
case pos =>
have : ¬(i < as.size) ∨ ¬(i < bs.size) := by
... | 39 | 86,593,400,423,993,740 | 2 | 1.5 | 6 | 1,680 |
import Mathlib.NumberTheory.ModularForms.SlashInvariantForms
import Mathlib.NumberTheory.ModularForms.CongruenceSubgroups
noncomputable section
open ModularForm UpperHalfPlane Complex Matrix
open scoped MatrixGroups
namespace EisensteinSeries
variable (N : ℕ) (a : Fin 2 → ZMod N)
variable {N a}
section eisSum... | Mathlib/NumberTheory/ModularForms/EisensteinSeries/Basic.lean | 92 | 100 | theorem eisSummand_SL2_apply (k : ℤ) (i : (Fin 2 → ℤ)) (A : SL(2, ℤ)) (z : ℍ) :
eisSummand k i (A • z) = (z.denom A) ^ k * eisSummand k (i ᵥ* A) z := by |
simp only [eisSummand, specialLinearGroup_apply, algebraMap_int_eq, eq_intCast, ofReal_intCast,
one_div, vecMul, vec2_dotProduct, Int.cast_add, Int.cast_mul]
have h (a b c d u v : ℂ) (hc : c * z + d ≠ 0) : ((u * ((a * z + b) / (c * z + d)) + v) ^ k)⁻¹ =
(c * z + d) ^ k * (((u * a + v * c) * z + (u * b + ... | 7 | 1,096.633158 | 2 | 2 | 1 | 2,015 |
import Mathlib.Algebra.ContinuedFractions.Computation.ApproximationCorollaries
import Mathlib.Algebra.ContinuedFractions.Computation.Translations
import Mathlib.Data.Real.Irrational
import Mathlib.RingTheory.Coprime.Lemmas
import Mathlib.Tactic.Basic
#align_import number_theory.diophantine_approximation from "leanpro... | Mathlib/NumberTheory/DiophantineApproximation.lean | 93 | 132 | theorem exists_int_int_abs_mul_sub_le (ξ : ℝ) {n : ℕ} (n_pos : 0 < n) :
∃ j k : ℤ, 0 < k ∧ k ≤ n ∧ |↑k * ξ - j| ≤ 1 / (n + 1) := by |
let f : ℤ → ℤ := fun m => ⌊fract (ξ * m) * (n + 1)⌋
have hn : 0 < (n : ℝ) + 1 := mod_cast Nat.succ_pos _
have hfu := fun m : ℤ => mul_lt_of_lt_one_left hn <| fract_lt_one (ξ * ↑m)
conv in |_| ≤ _ => rw [mul_comm, le_div_iff hn, ← abs_of_pos hn, ← abs_mul]
let D := Icc (0 : ℤ) n
by_cases H : ∃ m ∈ D, f m = ... | 38 | 31,855,931,757,113,756 | 2 | 2 | 3 | 2,140 |
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Data.Nat.Factorial.BigOperators
import Mathlib.Data.ZMod.Basic
open Finset Nat
namespace ZMod
| Mathlib/Data/ZMod/Factorial.lean | 31 | 42 | theorem cast_descFactorial {n p : ℕ} (h : n ≤ p) :
(descFactorial (p - 1) n : ZMod p) = (-1) ^ n * n ! := by |
rw [descFactorial_eq_prod_range, ← prod_range_add_one_eq_factorial]
simp only [cast_prod]
nth_rw 2 [← card_range n]
rw [pow_card_mul_prod]
refine prod_congr rfl ?_
intro x hx
rw [← tsub_add_eq_tsub_tsub_swap,
Nat.cast_sub <| Nat.le_trans (Nat.add_one_le_iff.mpr (List.mem_range.mp hx)) h,
CharP.ca... | 10 | 22,026.465795 | 2 | 2 | 1 | 1,949 |
import Mathlib.Data.Finsupp.Basic
import Mathlib.Data.Finsupp.Order
#align_import data.finsupp.multiset from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
open Finset
variable {α β ι : Type*}
namespace Finsupp
def toMultiset : (α →₀ ℕ) →+ Multiset α where
toFun f := Finsupp.sum f... | Mathlib/Data/Finsupp/Multiset.lean | 94 | 101 | theorem toFinset_toMultiset [DecidableEq α] (f : α →₀ ℕ) : f.toMultiset.toFinset = f.support := by |
refine f.induction ?_ ?_
· rw [toMultiset_zero, Multiset.toFinset_zero, support_zero]
· intro a n f ha hn ih
rw [toMultiset_add, Multiset.toFinset_add, ih, toMultiset_single, support_add_eq,
support_single_ne_zero _ hn, Multiset.toFinset_nsmul _ _ hn, Multiset.toFinset_singleton]
refine Disjoint.mo... | 7 | 1,096.633158 | 2 | 1.111111 | 9 | 1,194 |
import Mathlib.Analysis.Calculus.FDeriv.Measurable
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calculus.Deriv.Add
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.NormedSpace.Dual
import Mathlib.MeasureTheory.Integral.DominatedConve... | Mathlib/MeasureTheory/Integral/FundThmCalculus.lean | 273 | 288 | theorem measure_integral_sub_linear_isLittleO_of_tendsto_ae' [IsMeasurablyGenerated l']
[TendstoIxxClass Ioc l l'] (hfm : StronglyMeasurableAtFilter f l' μ)
(hf : Tendsto f (l' ⊓ ae μ) (𝓝 c)) (hl : μ.FiniteAtFilter l') (hu : Tendsto u lt l)
(hv : Tendsto v lt l) :
(fun t => (∫ x in u t..v t, f x ∂μ) - ... |
by_cases hE : CompleteSpace E; swap
· simp [intervalIntegral, integral, hE]
have A := hf.integral_sub_linear_isLittleO_ae hfm hl (hu.Ioc hv)
have B := hf.integral_sub_linear_isLittleO_ae hfm hl (hv.Ioc hu)
simp_rw [integral_const', sub_smul]
refine ((A.trans_le fun t ↦ ?_).sub (B.trans_le fun t ↦ ?_)).cong... | 10 | 22,026.465795 | 2 | 2 | 2 | 2,292 |
import Mathlib.SetTheory.Game.Short
#align_import set_theory.game.state from "leanprover-community/mathlib"@"b134b2f5cf6dd25d4bbfd3c498b6e36c11a17225"
universe u
namespace SetTheory
namespace PGame
class State (S : Type u) where
turnBound : S → ℕ
l : S → Finset S
r : S → Finset S
left_bound : ∀ {s t : S... | Mathlib/SetTheory/Game/State.lean | 50 | 54 | theorem turnBound_ne_zero_of_left_move {s t : S} (m : t ∈ l s) : turnBound s ≠ 0 := by |
intro h
have t := left_bound m
rw [h] at t
exact Nat.not_succ_le_zero _ t
| 4 | 54.59815 | 2 | 2 | 2 | 1,955 |
import Mathlib.Analysis.NormedSpace.Multilinear.Basic
import Mathlib.Analysis.NormedSpace.Units
import Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness
import Mathlib.Analysis.NormedSpace.OperatorNorm.Mul
#align_import analysis.normed_space.bounded_linear_maps from "leanprover-community/mathlib"@"ce11c3c2a285b... | Mathlib/Analysis/NormedSpace/BoundedLinearMaps.lean | 236 | 254 | theorem isBoundedLinearMap_continuousMultilinearMap_comp_linear (g : G →L[𝕜] E) :
IsBoundedLinearMap 𝕜 fun f : ContinuousMultilinearMap 𝕜 (fun _ : ι => E) F =>
f.compContinuousLinearMap fun _ => g := by |
refine
IsLinearMap.with_bound
⟨fun f₁ f₂ => by ext; rfl,
fun c f => by ext; rfl⟩
(‖g‖ ^ Fintype.card ι) fun f => ?_
apply ContinuousMultilinearMap.opNorm_le_bound _ _ _
· apply_rules [mul_nonneg, pow_nonneg, norm_nonneg]
intro m
calc
‖f (g ∘ m)‖ ≤ ‖f‖ * ∏ i, ‖g (m i)‖ := f.le_opNo... | 16 | 8,886,110.520508 | 2 | 0.538462 | 13 | 510 |
import Mathlib.LinearAlgebra.Matrix.DotProduct
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Diagonal
#align_import data.matrix.rank from "leanprover-community/mathlib"@"17219820a8aa8abe85adf5dfde19af1dd1bd8ae7"
open Matrix
namespace Matrix
open FiniteDimensional
variable {l m n ... | Mathlib/Data/Matrix/Rank.lean | 255 | 264 | theorem ker_mulVecLin_transpose_mul_self (A : Matrix m n R) :
LinearMap.ker (Aᵀ * A).mulVecLin = LinearMap.ker (mulVecLin A) := by |
ext x
simp only [LinearMap.mem_ker, mulVecLin_apply, ← mulVec_mulVec]
constructor
· intro h
replace h := congr_arg (dotProduct x) h
rwa [dotProduct_mulVec, dotProduct_zero, vecMul_transpose, dotProduct_self_eq_zero] at h
· intro h
rw [h, mulVec_zero]
| 8 | 2,980.957987 | 2 | 0.916667 | 12 | 792 |
import Mathlib.Algebra.GradedMonoid
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Algebra.MvPolynomial.Basic
#align_import ring_theory.mv_polynomial.weighted_homogeneous from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Fins... | Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean | 196 | 204 | theorem isWeightedHomogeneous_monomial (w : σ → M) (d : σ →₀ ℕ) (r : R) {m : M}
(hm : weightedDegree w d = m) : IsWeightedHomogeneous w (monomial d r) m := by |
classical
intro c hc
rw [coeff_monomial] at hc
split_ifs at hc with h
· subst c
exact hm
· contradiction
| 7 | 1,096.633158 | 2 | 1.125 | 8 | 1,208 |
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Data.Finite.Card
#align_import group_theory.subgroup.finite from "leanprover-community/mathlib"@"f93c11933efbc3c2f0299e47b8ff83e9b539cbf6"
variable {G : Type*} [Group G]
variable {A : Type*} [AddGroup A]
... | Mathlib/Algebra/Group/Subgroup/Finite.lean | 127 | 137 | theorem eq_top_of_card_eq [Finite H] (h : Nat.card H = Nat.card G) :
H = ⊤ := by |
have : Nonempty H := ⟨1, one_mem H⟩
have h' : Nat.card H ≠ 0 := Nat.card_pos.ne'
have : Finite G := (Nat.finite_of_card_ne_zero (h ▸ h'))
have : Fintype G := Fintype.ofFinite G
have : Fintype H := Fintype.ofFinite H
rw [Nat.card_eq_fintype_card, Nat.card_eq_fintype_card] at h
rw [SetLike.ext'_iff, coe_to... | 9 | 8,103.083928 | 2 | 1.8 | 5 | 1,897 |
import Mathlib.Algebra.DualNumber
import Mathlib.Algebra.QuaternionBasis
import Mathlib.Data.Complex.Module
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Star
import Mathlib.LinearAlgebra.QuadraticForm.Prod
#align_import linear_algebra.clifford_algebra.equivs fr... | Mathlib/LinearAlgebra/CliffordAlgebra/Equivs.lean | 223 | 228 | theorem reverse_apply (x : CliffordAlgebra Q) : reverse (R := ℝ) x = x := by |
induction x using CliffordAlgebra.induction with
| algebraMap r => exact reverse.commutes _
| ι x => rw [reverse_ι]
| mul x₁ x₂ hx₁ hx₂ => rw [reverse.map_mul, mul_comm, hx₁, hx₂]
| add x₁ x₂ hx₁ hx₂ => rw [reverse.map_add, hx₁, hx₂]
| 5 | 148.413159 | 2 | 1.4 | 10 | 1,508 |
import Mathlib.AlgebraicTopology.SimplicialObject
import Mathlib.CategoryTheory.Limits.Shapes.Products
#align_import algebraic_topology.split_simplicial_object from "leanprover-community/mathlib"@"dd1f8496baa505636a82748e6b652165ea888733"
noncomputable section
open CategoryTheory CategoryTheory.Category Category... | Mathlib/AlgebraicTopology/SplitSimplicialObject.lean | 77 | 84 | theorem ext (A₁ A₂ : IndexSet Δ) (h₁ : A₁.1 = A₂.1) (h₂ : A₁.e ≫ eqToHom (by rw [h₁]) = A₂.e) :
A₁ = A₂ := by |
rcases A₁ with ⟨Δ₁, ⟨α₁, hα₁⟩⟩
rcases A₂ with ⟨Δ₂, ⟨α₂, hα₂⟩⟩
simp only at h₁
subst h₁
simp only [eqToHom_refl, comp_id, IndexSet.e] at h₂
simp only [h₂]
| 6 | 403.428793 | 2 | 2 | 5 | 2,188 |
import Mathlib.Geometry.Manifold.ChartedSpace
#align_import geometry.manifold.local_invariant_properties from "leanprover-community/mathlib"@"431589bce478b2229eba14b14a283250428217db"
noncomputable section
open scoped Classical
open Manifold Topology
open Set Filter TopologicalSpace
variable {H M H' M' X : Typ... | Mathlib/Geometry/Manifold/LocalInvariantProperties.lean | 121 | 136 | theorem left_invariance {s : Set H} {x : H} {f : H → H'} {e' : PartialHomeomorph H' H'}
(he' : e' ∈ G') (hfs : ContinuousWithinAt f s x) (hxe' : f x ∈ e'.source) :
P (e' ∘ f) s x ↔ P f s x := by |
have h2f := hfs.preimage_mem_nhdsWithin (e'.open_source.mem_nhds hxe')
have h3f :=
((e'.continuousAt hxe').comp_continuousWithinAt hfs).preimage_mem_nhdsWithin <|
e'.symm.open_source.mem_nhds <| e'.mapsTo hxe'
constructor
· intro h
rw [hG.is_local_nhds h3f] at h
have h2 := hG.left_invariance'... | 13 | 442,413.392009 | 2 | 1.6 | 5 | 1,739 |
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Tactic.Ring
#align_import data.nat.count from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
open Finset
namespace Nat
variable (p : ℕ → Prop)
section Count
variable [DecidablePred p]
def count (n : ℕ) : ℕ :=
(List.range n).... | Mathlib/Data/Nat/Count.lean | 133 | 137 | theorem count_injective {m n : ℕ} (hm : p m) (hn : p n) (heq : count p m = count p n) : m = n := by |
by_contra! h : m ≠ n
wlog hmn : m < n
· exact this hn hm heq.symm h.symm (h.lt_or_lt.resolve_left hmn)
· simpa [heq] using count_strict_mono hm hmn
| 4 | 54.59815 | 2 | 0.642857 | 14 | 554 |
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.Orientation
import Mathlib.Data.Complex.Orientation
import Mathlib.Tactic.LinearCombination
#align_import analysis.inner_product_space.two_dim from "leanprover-community/mathlib"@"cd8fafa2fac98e1a67097e8a91ad9901cfde48af"
non... | Mathlib/Analysis/InnerProductSpace/TwoDim.lean | 172 | 180 | theorem areaForm_comp_linearIsometryEquiv (φ : E ≃ₗᵢ[ℝ] E)
(hφ : 0 < LinearMap.det (φ.toLinearEquiv : E →ₗ[ℝ] E)) (x y : E) :
o.areaForm (φ x) (φ y) = o.areaForm x y := by |
convert o.areaForm_map φ (φ x) (φ y)
· symm
rwa [← o.map_eq_iff_det_pos φ.toLinearEquiv] at hφ
rw [@Fact.out (finrank ℝ E = 2), Fintype.card_fin]
· simp
· simp
| 6 | 403.428793 | 2 | 1.222222 | 9 | 1,295 |
import Mathlib.Topology.Algebra.UniformConvergence
#align_import topology.algebra.equicontinuity from "leanprover-community/mathlib"@"01ad394a11bf06b950232720cf7e8fc6b22f0d6a"
open Function
open UniformConvergence
@[to_additive]
theorem equicontinuous_of_equicontinuousAt_one {ι G M hom : Type*} [TopologicalSpac... | Mathlib/Topology/Algebra/Equicontinuity.lean | 36 | 47 | theorem uniformEquicontinuous_of_equicontinuousAt_one {ι G M hom : Type*} [UniformSpace G]
[UniformSpace M] [Group G] [Group M] [UniformGroup G] [UniformGroup M]
[FunLike hom G M] [MonoidHomClass hom G M]
(F : ι → hom) (hf : EquicontinuousAt ((↑) ∘ F) (1 : G)) :
UniformEquicontinuous ((↑) ∘ F) := by |
rw [uniformEquicontinuous_iff_uniformContinuous]
rw [equicontinuousAt_iff_continuousAt] at hf
let φ : G →* (ι →ᵤ M) :=
{ toFun := swap ((↑) ∘ F)
map_one' := by dsimp [UniformFun]; ext; exact map_one _
map_mul' := fun a b => by dsimp [UniformFun]; ext; exact map_mul _ _ _ }
exact uniformContinuo... | 7 | 1,096.633158 | 2 | 2 | 2 | 2,011 |
import Mathlib.Analysis.Convex.Combination
import Mathlib.Analysis.Convex.Extreme
#align_import analysis.convex.independent from "leanprover-community/mathlib"@"fefd8a38be7811574cd2ec2f77d3a393a407f112"
open scoped Classical
open Affine
open Finset Function
variable {𝕜 E ι : Type*}
section OrderedSemiring
va... | Mathlib/Analysis/Convex/Independent.lean | 133 | 141 | theorem convexIndependent_iff_not_mem_convexHull_diff {p : ι → E} :
ConvexIndependent 𝕜 p ↔ ∀ i s, p i ∉ convexHull 𝕜 (p '' (s \ {i})) := by |
refine ⟨fun hc i s h => ?_, fun h s i hi => ?_⟩
· rw [hc.mem_convexHull_iff] at h
exact h.2 (Set.mem_singleton _)
· by_contra H
refine h i s ?_
rw [Set.diff_singleton_eq_self H]
exact hi
| 7 | 1,096.633158 | 2 | 1.8 | 5 | 1,891 |
import Mathlib.Algebra.Category.GroupCat.Basic
import Mathlib.CategoryTheory.Limits.Shapes.ZeroObjects
#align_import algebra.category.Group.zero from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open CategoryTheory
open CategoryTheory.Limits
universe u
namespace CommGroupCat
@[to_... | Mathlib/Algebra/Category/GroupCat/Zero.lean | 49 | 55 | theorem isZero_of_subsingleton (G : CommGroupCat) [Subsingleton G] : IsZero G := by |
refine ⟨fun X => ⟨⟨⟨1⟩, fun f => ?_⟩⟩, fun X => ⟨⟨⟨1⟩, fun f => ?_⟩⟩⟩
· ext x
have : x = 1 := Subsingleton.elim _ _
rw [this, map_one, map_one]
· ext
apply Subsingleton.elim
| 6 | 403.428793 | 2 | 2 | 2 | 2,492 |
import Mathlib.Data.List.Join
#align_import data.list.permutation from "leanprover-community/mathlib"@"dd71334db81d0bd444af1ee339a29298bef40734"
-- Make sure we don't import algebra
assert_not_exists Monoid
open Nat
variable {α β : Type*}
namespace List
theorem permutationsAux2_fst (t : α) (ts : List α) (r : L... | Mathlib/Data/List/Permutation.lean | 140 | 146 | theorem permutations'Aux_eq_permutationsAux2 (t : α) (ts : List α) :
permutations'Aux t ts = (permutationsAux2 t [] [ts ++ [t]] ts id).2 := by |
induction' ts with a ts ih; · rfl
simp only [permutations'Aux, ih, cons_append, permutationsAux2_snd_cons, append_nil, id_eq,
cons.injEq, true_and]
simp (config := { singlePass := true }) only [← permutationsAux2_append]
simp [map_permutationsAux2]
| 5 | 148.413159 | 2 | 1 | 9 | 903 |
import Mathlib.Analysis.Convex.Basic
import Mathlib.Order.Closure
#align_import analysis.convex.hull from "leanprover-community/mathlib"@"92bd7b1ffeb306a89f450bee126ddd8a284c259d"
open Set
open Pointwise
variable {𝕜 E F : Type*}
section convexHull
section OrderedSemiring
variable [OrderedSemiring 𝕜]
secti... | Mathlib/Analysis/Convex/Hull.lean | 127 | 131 | theorem convexHull_pair (x y : E) : convexHull 𝕜 {x, y} = segment 𝕜 x y := by |
refine (convexHull_min ?_ <| convex_segment _ _).antisymm
(segment_subset_convexHull (mem_insert _ _) <| subset_insert _ _ <| mem_singleton _)
rw [insert_subset_iff, singleton_subset_iff]
exact ⟨left_mem_segment _ _ _, right_mem_segment _ _ _⟩
| 4 | 54.59815 | 2 | 1.166667 | 6 | 1,236 |
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 ⟶ d) :
Groupoid.inv f ∈ S.arrows d c ↔ f ∈ S.arrows c d := by |
constructor
· intro h
simpa only [inv_eq_inv, IsIso.inv_inv] using S.inv h
· apply S.inv
| 4 | 54.59815 | 2 | 1.333333 | 6 | 1,417 |
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 u₁ u₂ u₃ u₄
variable {ι : Type u₁} {k : Type... | Mathlib/LinearAlgebra/AffineSpace/Matrix.lean | 137 | 146 | theorem isUnit_toMatrix_iff [Nontrivial k] (p : ι → P) :
IsUnit (b.toMatrix p) ↔ AffineIndependent k p ∧ affineSpan k (range p) = ⊤ := by |
constructor
· rintro ⟨⟨B, A, hA, hA'⟩, rfl : B = b.toMatrix p⟩
exact ⟨b.affineIndependent_of_toMatrix_right_inv p hA,
b.affineSpan_eq_top_of_toMatrix_left_inv p hA'⟩
· rintro ⟨h_tot, h_ind⟩
let b' : AffineBasis ι k P := ⟨p, h_tot, h_ind⟩
change IsUnit (b.toMatrix b')
exact b.isUnit_toMatrix... | 8 | 2,980.957987 | 2 | 1.428571 | 7 | 1,512 |
import Mathlib.Geometry.Manifold.Diffeomorph
import Mathlib.Geometry.Manifold.Instances.Real
import Mathlib.Geometry.Manifold.PartitionOfUnity
#align_import geometry.manifold.whitney_embedding from "leanprover-community/mathlib"@"86c29aefdba50b3f33e86e52e3b2f51a0d8f0282"
universe uι uE uH uM
variable {ι : Type u... | Mathlib/Geometry/Manifold/WhitneyEmbedding.lean | 68 | 75 | theorem embeddingPiTangent_injOn : InjOn f.embeddingPiTangent s := by |
intro x hx y _ h
simp only [embeddingPiTangent_coe, funext_iff] at h
obtain ⟨h₁, h₂⟩ := Prod.mk.inj_iff.1 (h (f.ind x hx))
rw [f.apply_ind x hx] at h₂
rw [← h₂, f.apply_ind x hx, one_smul, one_smul] at h₁
have := f.mem_extChartAt_source_of_eq_one h₂.symm
exact (extChartAt I (f.c _)).injOn (f.mem_extChart... | 7 | 1,096.633158 | 2 | 2 | 4 | 2,356 |
import Batteries.Data.DList
import Mathlib.Mathport.Rename
import Mathlib.Tactic.Cases
#align_import data.dlist from "leanprover-community/lean"@"855e5b74e3a52a40552e8f067169d747d48743fd"
universe u
#align dlist Batteries.DList
namespace Batteries.DList
open Function
variable {α : Type u}
#align dlist.of_list... | Mathlib/Data/DList/Defs.lean | 62 | 66 | theorem ofList_toList (l : DList α) : DList.ofList (DList.toList l) = l := by |
cases' l with app inv
simp only [ofList, toList, mk.injEq]
funext x
rw [(inv x)]
| 4 | 54.59815 | 2 | 0.333333 | 6 | 333 |
import Mathlib.Data.List.Lex
import Mathlib.Data.Char
import Mathlib.Tactic.AdaptationNote
import Mathlib.Algebra.Order.Group.Nat
#align_import data.string.basic from "leanprover-community/mathlib"@"d13b3a4a392ea7273dfa4727dbd1892e26cfd518"
namespace String
def ltb (s₁ s₂ : Iterator) : Bool :=
if s₂.hasNext th... | Mathlib/Data/String/Basic.lean | 60 | 74 | theorem ltb_cons_addChar (c : Char) (cs₁ cs₂ : List Char) (i₁ i₂ : Pos) :
ltb ⟨⟨c :: cs₁⟩, i₁ + c⟩ ⟨⟨c :: cs₂⟩, i₂ + c⟩ = ltb ⟨⟨cs₁⟩, i₁⟩ ⟨⟨cs₂⟩, i₂⟩ := by |
apply ltb.inductionOn ⟨⟨cs₁⟩, i₁⟩ ⟨⟨cs₂⟩, i₂⟩ (motive := fun ⟨⟨cs₁⟩, i₁⟩ ⟨⟨cs₂⟩, i₂⟩ ↦
ltb ⟨⟨c :: cs₁⟩, i₁ + c⟩ ⟨⟨c :: cs₂⟩, i₂ + c⟩ =
ltb ⟨⟨cs₁⟩, i₁⟩ ⟨⟨cs₂⟩, i₂⟩) <;> simp only <;>
intro ⟨cs₁⟩ ⟨cs₂⟩ i₁ i₂ <;>
intros <;>
(conv => lhs; unfold ltb) <;> (conv => rhs; unfold ltb) <;>
simp only [Iterator.... | 13 | 442,413.392009 | 2 | 2 | 1 | 1,947 |
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 | 278 | 292 | theorem stream_succ_nth_fr_num_lt_nth_fr_num_rat {ifp_n ifp_succ_n : IntFractPair ℚ}
(stream_nth_eq : IntFractPair.stream q n = some ifp_n)
(stream_succ_nth_eq : IntFractPair.stream q (n + 1) = some ifp_succ_n) :
ifp_succ_n.fr.num < ifp_n.fr.num := by |
obtain ⟨ifp_n', stream_nth_eq', ifp_n_fract_ne_zero, IntFractPair.of_eq_ifp_succ_n⟩ :
∃ ifp_n',
IntFractPair.stream q n = some ifp_n' ∧
ifp_n'.fr ≠ 0 ∧ IntFractPair.of ifp_n'.fr⁻¹ = ifp_succ_n :=
succ_nth_stream_eq_some_iff.mp stream_succ_nth_eq
have : ifp_n = ifp_n' := by injection Eq.trans ... | 11 | 59,874.141715 | 2 | 1.272727 | 11 | 1,350 |
import Mathlib.Dynamics.Ergodic.AddCircle
import Mathlib.MeasureTheory.Covering.LiminfLimsup
#align_import number_theory.well_approximable from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open Set Filter Function Metric MeasureTheory
open scoped MeasureTheory Topology Pointwise
@[... | Mathlib/NumberTheory/WellApproximable.lean | 183 | 191 | theorem mem_addWellApproximable_iff (δ : ℕ → ℝ) (x : UnitAddCircle) :
x ∈ addWellApproximable UnitAddCircle δ ↔
{n : ℕ | ∃ m < n, gcd m n = 1 ∧ ‖x - ↑((m : ℝ) / n)‖ < δ n}.Infinite := by |
simp only [mem_add_wellApproximable_iff, ← Nat.cofinite_eq_atTop, cofinite.blimsup_set_eq,
mem_setOf_eq]
refine iff_of_eq (congr_arg Set.Infinite <| ext fun n => ⟨fun hn => ?_, fun hn => ?_⟩)
· exact (mem_approxAddOrderOf_iff hn.1).mp hn.2
· have h : 0 < n := by obtain ⟨m, hm₁, _, _⟩ := hn; exact pos_of_gt... | 6 | 403.428793 | 2 | 1.714286 | 7 | 1,835 |
import Mathlib.Analysis.Calculus.Deriv.ZPow
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv
import Mathlib.Analysis.Convex.Deriv
#align_import analysis.convex.specific_functions.deriv from "leanprover-communi... | Mathlib/Analysis/Convex/SpecificFunctions/Deriv.lean | 88 | 94 | theorem int_prod_range_pos {m : ℤ} {n : ℕ} (hn : Even n) (hm : m ∉ Ico (0 : ℤ) n) :
0 < ∏ k ∈ Finset.range n, (m - k) := by |
refine (int_prod_range_nonneg m n hn).lt_of_ne fun h => hm ?_
rw [eq_comm, Finset.prod_eq_zero_iff] at h
obtain ⟨a, ha, h⟩ := h
rw [sub_eq_zero.1 h]
exact ⟨Int.ofNat_zero_le _, Int.ofNat_lt.2 <| Finset.mem_range.1 ha⟩
| 5 | 148.413159 | 2 | 1.6 | 10 | 1,744 |
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.MvPolynomial.Variables
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Algebra.MvPolynomial.Expand
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.ZMod.Basic
#align_import ring_theory.witt_vector.witt_polynomial from "leanprover-c... | Mathlib/RingTheory/WittVector/WittPolynomial.lean | 81 | 86 | theorem wittPolynomial_eq_sum_C_mul_X_pow (n : ℕ) :
wittPolynomial p R n = ∑ i ∈ range (n + 1), C ((p : R) ^ i) * X i ^ p ^ (n - i) := by |
apply sum_congr rfl
rintro i -
rw [monomial_eq, Finsupp.prod_single_index]
rw [pow_zero]
| 4 | 54.59815 | 2 | 1.181818 | 11 | 1,247 |
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 | 98 | 106 | theorem dart_card_eq_twice_card_edges : Fintype.card G.Dart = 2 * G.edgeFinset.card := by |
classical
rw [← card_univ]
rw [@card_eq_sum_card_fiberwise _ _ _ Dart.edge _ G.edgeFinset fun d _h =>
by rw [mem_edgeFinset]; apply Dart.edge_mem]
rw [← mul_comm, sum_const_nat]
intro e h
apply G.dart_edge_fiber_card e
rwa [← mem_edgeFinset]
| 8 | 2,980.957987 | 2 | 1.6 | 5 | 1,746 |
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 {α : 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 | 125 | 131 | theorem ofFn_succ' {n} (f : Fin (succ n) → α) :
ofFn f = (ofFn fun i => f (Fin.castSucc i)).concat (f (Fin.last _)) := by |
induction' n with n IH
· rw [ofFn_zero, concat_nil, ofFn_succ, ofFn_zero]
rfl
· rw [ofFn_succ, IH, ofFn_succ, concat_cons, Fin.castSucc_zero]
congr
| 5 | 148.413159 | 2 | 0.6 | 10 | 535 |
import Mathlib.CategoryTheory.Filtered.Basic
import Mathlib.Topology.Category.TopCat.Limits.Basic
#align_import topology.category.Top.limits.konig from "leanprover-community/mathlib"@"dbdf71cee7bb20367cb7e37279c08b0c218cf967"
-- Porting note: every ML3 decl has an uppercase letter
set_option linter.uppercaseLean3 ... | Mathlib/Topology/Category/TopCat/Limits/Konig.lean | 130 | 146 | theorem nonempty_limitCone_of_compact_t2_cofiltered_system (F : J ⥤ TopCat.{max v u})
[IsCofilteredOrEmpty J]
[∀ j : J, Nonempty (F.obj j)] [∀ j : J, CompactSpace (F.obj j)] [∀ j : J, T2Space (F.obj j)] :
Nonempty (TopCat.limitCone F).pt := by |
classical
obtain ⟨u, hu⟩ :=
IsCompact.nonempty_iInter_of_directed_nonempty_isCompact_isClosed (fun G => partialSections F _)
(partialSections.directed F) (fun G => partialSections.nonempty F _)
(fun G => IsClosed.isCompact (partialSections.closed F _)) fun G =>
partialSections.closed F _
us... | 13 | 442,413.392009 | 2 | 2 | 4 | 2,008 |
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Interval.Set.IsoIoo
import Mathlib.Topology.Order.MonotoneContinuity
import Mathlib.Topology.UrysohnsBounded
#align_import topology.tietze_extension from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
variable {X Y : T... | Mathlib/Topology/TietzeExtension.lean | 169 | 213 | theorem tietze_extension_step (f : X →ᵇ ℝ) (e : C(X, Y)) (he : ClosedEmbedding e) :
∃ g : Y →ᵇ ℝ, ‖g‖ ≤ ‖f‖ / 3 ∧ dist (g.compContinuous e) f ≤ 2 / 3 * ‖f‖ := by |
have h3 : (0 : ℝ) < 3 := by norm_num1
have h23 : 0 < (2 / 3 : ℝ) := by norm_num1
-- In the trivial case `f = 0`, we take `g = 0`
rcases eq_or_ne f 0 with (rfl | hf)
· use 0
simp
replace hf : 0 < ‖f‖ := norm_pos_iff.2 hf
/- Otherwise, the closed sets `e '' (f ⁻¹' (Iic (-‖f‖ / 3)))` and `e '' (f ⁻¹' (I... | 43 | 4,727,839,468,229,346,000 | 2 | 1.5 | 6 | 1,691 |
import Mathlib.Algebra.Lie.Nilpotent
import Mathlib.Algebra.Lie.Normalizer
#align_import algebra.lie.engel from "leanprover-community/mathlib"@"210657c4ea4a4a7b234392f70a3a2a83346dfa90"
universe u₁ u₂ u₃ u₄
variable {R : Type u₁} {L : Type u₂} {L₂ : Type u₃} {M : Type u₄}
variable [CommRing R] [LieRing L] [LieAl... | Mathlib/Algebra/Lie/Engel.lean | 173 | 183 | theorem Function.Surjective.isEngelian {f : L →ₗ⁅R⁆ L₂} (hf : Function.Surjective f)
(h : LieAlgebra.IsEngelian.{u₁, u₂, u₄} R L) : LieAlgebra.IsEngelian.{u₁, u₃, u₄} R L₂ := by |
intro M _i1 _i2 _i3 _i4 h'
letI : LieRingModule L M := LieRingModule.compLieHom M f
letI : LieModule R L M := compLieHom M f
have hnp : ∀ x, IsNilpotent (toEnd R L M x) := fun x => h' (f x)
have surj_id : Function.Surjective (LinearMap.id : M →ₗ[R] M) := Function.surjective_id
haveI : LieModule.IsNilpotent... | 9 | 8,103.083928 | 2 | 2 | 6 | 2,255 |
import Mathlib.MeasureTheory.Integral.IntegralEqImproper
#align_import measure_theory.integral.peak_function from "leanprover-community/mathlib"@"13b0d72fd8533ba459ac66e9a885e35ffabb32b2"
open Set Filter MeasureTheory MeasureTheory.Measure TopologicalSpace Metric
open scoped Topology ENNReal
open Set
variable... | Mathlib/MeasureTheory/Integral/PeakFunction.lean | 54 | 86 | theorem integrableOn_peak_smul_of_integrableOn_of_tendsto
(hs : MeasurableSet s) (h'st : t ∈ 𝓝[s] x₀)
(hlφ : ∀ u : Set α, IsOpen u → x₀ ∈ u → TendstoUniformlyOn φ 0 l (s \ u))
(hiφ : Tendsto (fun i ↦ ∫ x in t, φ i x ∂μ) l (𝓝 1))
(h'iφ : ∀ᶠ i in l, AEStronglyMeasurable (φ i) (μ.restrict s))
(hmg : ... |
obtain ⟨u, u_open, x₀u, ut, hu⟩ :
∃ u, IsOpen u ∧ x₀ ∈ u ∧ s ∩ u ⊆ t ∧ ∀ x ∈ u ∩ s, g x ∈ ball a 1 := by
rcases mem_nhdsWithin.1 (Filter.inter_mem h'st (hcg (ball_mem_nhds _ zero_lt_one)))
with ⟨u, u_open, x₀u, hu⟩
refine ⟨u, u_open, x₀u, ?_, hu.trans inter_subset_right⟩
rw [inter_comm]
e... | 26 | 195,729,609,428.83878 | 2 | 2 | 2 | 2,209 |
import Mathlib.Data.PFunctor.Univariate.Basic
#align_import data.pfunctor.univariate.M from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
universe u v w
open Nat Function
open List
variable (F : PFunctor.{u})
-- Porting note: the ♯ tactic is never used
-- local prefix:0 "♯" => cast... | Mathlib/Data/PFunctor/Univariate/M.lean | 152 | 174 | theorem head_succ' (n m : ℕ) (x : ∀ n, CofixA F n) (Hconsistent : AllAgree x) :
head' (x (succ n)) = head' (x (succ m)) := by |
suffices ∀ n, head' (x (succ n)) = head' (x 1) by simp [this]
clear m n
intro n
cases' h₀ : x (succ n) with _ i₀ f₀
cases' h₁ : x 1 with _ i₁ f₁
dsimp only [head']
induction' n with n n_ih
· rw [h₁] at h₀
cases h₀
trivial
· have H := Hconsistent (succ n)
cases' h₂ : x (succ n) with _ i₂ f... | 21 | 1,318,815,734.483215 | 2 | 1.166667 | 6 | 1,239 |
import Mathlib.MeasureTheory.Constructions.Pi
import Mathlib.MeasureTheory.Integral.Lebesgue
open scoped Classical ENNReal
open Set Function Equiv Finset
noncomputable section
namespace MeasureTheory
section LMarginal
variable {δ δ' : Type*} {π : δ → Type*} [∀ x, MeasurableSpace (π x)]
variable {μ : ∀ i, Measu... | Mathlib/MeasureTheory/Integral/Marginal.lean | 110 | 116 | theorem lmarginal_update_of_mem {i : δ} (hi : i ∈ s)
(f : (∀ i, π i) → ℝ≥0∞) (x : ∀ i, π i) (y : π i) :
(∫⋯∫⁻_s, f ∂μ) (Function.update x i y) = (∫⋯∫⁻_s, f ∂μ) x := by |
apply lmarginal_congr
intro j hj
have : j ≠ i := by rintro rfl; exact hj hi
apply update_noteq this
| 4 | 54.59815 | 2 | 1.333333 | 6 | 1,378 |
import Mathlib.Analysis.Convex.Combination
import Mathlib.Analysis.Convex.Function
import Mathlib.Tactic.FieldSimp
#align_import analysis.convex.jensen from "leanprover-community/mathlib"@"bfad3f455b388fbcc14c49d0cac884f774f14d20"
open Finset LinearMap Set
open scoped Classical
open Convex Pointwise
variable {�... | Mathlib/Analysis/Convex/Jensen.lean | 52 | 58 | theorem ConvexOn.map_centerMass_le (hf : ConvexOn 𝕜 s f) (h₀ : ∀ i ∈ t, 0 ≤ w i)
(h₁ : 0 < ∑ i ∈ t, w i) (hmem : ∀ i ∈ t, p i ∈ s) :
f (t.centerMass w p) ≤ t.centerMass w (f ∘ p) := by |
have hmem' : ∀ i ∈ t, (p i, (f ∘ p) i) ∈ { p : E × β | p.1 ∈ s ∧ f p.1 ≤ p.2 } := fun i hi =>
⟨hmem i hi, le_rfl⟩
convert (hf.convex_epigraph.centerMass_mem h₀ h₁ hmem').2 <;>
simp only [centerMass, Function.comp, Prod.smul_fst, Prod.fst_sum, Prod.smul_snd, Prod.snd_sum]
| 4 | 54.59815 | 2 | 1.5 | 2 | 1,586 |
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 {α G A S... | Mathlib/Algebra/Group/Subgroup/Pointwise.lean | 73 | 81 | theorem closure_toSubmonoid (S : Set G) :
(closure S).toSubmonoid = Submonoid.closure (S ∪ S⁻¹) := by |
refine le_antisymm (fun x hx => ?_) (Submonoid.closure_le.2 ?_)
· 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 [← Submonoid.mem_closure... | 7 | 1,096.633158 | 2 | 1.333333 | 3 | 1,422 |
import Mathlib.Algebra.BigOperators.Finsupp
import Mathlib.Algebra.Module.Basic
import Mathlib.Algebra.Regular.SMul
import Mathlib.Data.Finset.Preimage
import Mathlib.Data.Rat.BigOperators
import Mathlib.GroupTheory.GroupAction.Hom
import Mathlib.Data.Set.Subsingleton
#align_import data.finsupp.basic from "leanprover... | Mathlib/Data/Finsupp/Basic.lean | 68 | 74 | theorem mk_mem_graph_iff {a : α} {m : M} {f : α →₀ M} : (a, m) ∈ f.graph ↔ f a = m ∧ m ≠ 0 := by |
simp_rw [graph, mem_map, mem_support_iff]
constructor
· rintro ⟨b, ha, rfl, -⟩
exact ⟨rfl, ha⟩
· rintro ⟨rfl, ha⟩
exact ⟨a, ha, rfl⟩
| 6 | 403.428793 | 2 | 1.25 | 4 | 1,325 |
import Mathlib.CategoryTheory.Action
import Mathlib.Combinatorics.Quiver.Arborescence
import Mathlib.Combinatorics.Quiver.ConnectedComponent
import Mathlib.GroupTheory.FreeGroup.IsFreeGroup
#align_import group_theory.nielsen_schreier from "leanprover-community/mathlib"@"1bda4fc53de6ade5ab9da36f2192e24e2084a2ce"
n... | Mathlib/GroupTheory/FreeGroup/NielsenSchreier.lean | 195 | 202 | theorem loopOfHom_eq_id {a b : Generators G} (e) (H : e ∈ wideSubquiverSymmetrify T a b) :
loopOfHom T (of e) = 𝟙 (root' T) := by |
rw [loopOfHom, ← Category.assoc, IsIso.comp_inv_eq, Category.id_comp]
cases' H with H H
· rw [treeHom_eq T (Path.cons default ⟨Sum.inl e, H⟩), homOfPath]
rfl
· rw [treeHom_eq T (Path.cons default ⟨Sum.inr e, H⟩), homOfPath]
simp only [IsIso.inv_hom_id, Category.comp_id, Category.assoc, treeHom]
| 6 | 403.428793 | 2 | 1.333333 | 3 | 1,388 |
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Preadditive.FunctorCategory
import Mathlib.CategoryTheory.Limits.Shapes.FunctorCategory
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels
#align_import category_theory.abelian.functor_category from "leanprover-community/mathlib"@"8a... | Mathlib/CategoryTheory/Abelian/FunctorCategory.lean | 64 | 76 | theorem coimageImageComparison_app :
coimageImageComparison (α.app X) =
(coimageObjIso α X).inv ≫ (coimageImageComparison α).app X ≫ (imageObjIso α X).hom := by |
ext
dsimp
dsimp [imageObjIso, coimageObjIso, cokernel.map]
simp only [coimage_image_factorisation, PreservesKernel.iso_hom, Category.assoc,
kernel.lift_ι, Category.comp_id, PreservesCokernel.iso_inv,
cokernel.π_desc_assoc, Category.id_comp]
erw [kernelComparison_comp_ι _ ((evaluation C D).obj X),
... | 10 | 22,026.465795 | 2 | 1.5 | 2 | 1,572 |
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Tactic.Ring
#align_import data.nat.hyperoperation from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
def hyperoperation : ℕ → ℕ → ℕ → ℕ
| 0, _, k => k + 1
| 1, m, 0 => m
| 2, _, 0 => 0
| _ + 3, _, 0 => 1
| n + 1, m, k + 1 ... | Mathlib/Data/Nat/Hyperoperation.lean | 69 | 78 | theorem hyperoperation_two : hyperoperation 2 = (· * ·) := by |
ext m k
induction' k with bn bih
· rw [hyperoperation]
exact (Nat.mul_zero m).symm
· rw [hyperoperation_recursion, hyperoperation_one, bih]
-- Porting note: was `ring`
dsimp only
nth_rewrite 1 [← mul_one m]
rw [← mul_add, add_comm]
| 9 | 8,103.083928 | 2 | 1.444444 | 9 | 1,532 |
import Mathlib.Data.Matroid.Dual
open Set
namespace Matroid
variable {α : Type*} {M : Matroid α} {R I J X Y : Set α}
section restrict
@[simps] def restrictIndepMatroid (M : Matroid α) (R : Set α) : IndepMatroid α where
E := R
Indep I := M.Indep I ∧ I ⊆ R
indep_empty := ⟨M.empty_indep, empty_subset _⟩
i... | Mathlib/Data/Matroid/Restrict.lean | 159 | 163 | theorem Basis.restrict_base (h : M.Basis I X) : (M ↾ X).Base I := by |
rw [basis_iff'] at h
simp_rw [base_iff_maximal_indep, restrict_indep_iff, and_imp, and_assoc, and_iff_right h.1.1,
and_iff_right h.1.2.1]
exact fun J hJ hJX hIJ ↦ h.1.2.2 _ hJ hIJ hJX
| 4 | 54.59815 | 2 | 1 | 3 | 810 |
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.RingTheory.GradedAlgebra.Basic
#align_import ring_theory.graded_algebra.homogeneous_ideal from "leanprover-community/mathlib"@"4e861f25ba5ceef42ba0712d8ffeb32f38ad6441"
open SetLike Direc... | Mathlib/RingTheory/GradedAlgebra/HomogeneousIdeal.lean | 64 | 69 | theorem Ideal.IsHomogeneous.mem_iff {I} (hI : Ideal.IsHomogeneous 𝒜 I) {x} :
x ∈ I ↔ ∀ i, (decompose 𝒜 x i : A) ∈ I := by |
classical
refine ⟨fun hx i ↦ hI i hx, fun hx ↦ ?_⟩
rw [← DirectSum.sum_support_decompose 𝒜 x]
exact Ideal.sum_mem _ (fun i _ ↦ hx i)
| 4 | 54.59815 | 2 | 2 | 2 | 2,435 |
import Mathlib.SetTheory.Ordinal.Arithmetic
import Mathlib.SetTheory.Ordinal.Exponential
#align_import set_theory.ordinal.cantor_normal_form from "leanprover-community/mathlib"@"991ff3b5269848f6dd942ae8e9dd3c946035dc8b"
noncomputable section
universe u
open List
namespace Ordinal
@[elab_as_elim]
noncomputabl... | Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | 121 | 129 | theorem CNF_fst_le_log {b o : Ordinal.{u}} {x : Ordinal × Ordinal} :
x ∈ CNF b o → x.1 ≤ log b o := by |
refine CNFRec b ?_ (fun o ho H ↦ ?_) o
· rw [CNF_zero]
intro contra; contradiction
· rw [CNF_ne_zero ho, mem_cons]
rintro (rfl | h)
· exact le_rfl
· exact (H h).trans (log_mono_right _ (mod_opow_log_lt_self b ho).le)
| 7 | 1,096.633158 | 2 | 0.888889 | 9 | 775 |
import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Subobject.Lattice
import Mathlib.Order.Atoms
#align_import category_theory.simple from "leanprover-community/mathlib"@"4ed0bcaef698011... | Mathlib/CategoryTheory/Simple.lean | 61 | 77 | theorem Simple.of_iso {X Y : C} [Simple Y] (i : X ≅ Y) : Simple X :=
{ mono_isIso_iff_nonzero := fun f m => by
haveI : Mono (f ≫ i.hom) := mono_comp _ _
constructor
· intro h w
have j : IsIso (f ≫ i.hom) := by | infer_instance
rw [Simple.mono_isIso_iff_nonzero] at j
subst w
simp at j
· intro h
have j : IsIso (f ≫ i.hom) := by
apply isIso_of_mono_of_nonzero
intro w
apply h
simpa using (cancel_mono i.inv).2 w
rw [← Category.comp_id f, ← i.hom_... | 12 | 162,754.791419 | 2 | 1.5 | 8 | 1,594 |
import Mathlib.Algebra.Order.Ring.Abs
#align_import data.int.order.units from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
namespace Int
theorem isUnit_iff_abs_eq {x : ℤ} : IsUnit x ↔ abs x = 1 := by
rw [isUnit_iff_natAbs_eq, abs_eq_natAbs, ← Int.ofNat_one, natCast_inj]
#align int.... | Mathlib/Data/Int/Order/Units.lean | 63 | 67 | theorem units_pow_eq_pow_mod_two (u : ℤˣ) (n : ℕ) : u ^ n = u ^ (n % 2) := by |
conv =>
lhs
rw [← Nat.mod_add_div n 2];
rw [pow_add, pow_mul, units_sq, one_pow, mul_one]
| 4 | 54.59815 | 2 | 0.222222 | 9 | 285 |
import Mathlib.Data.ENNReal.Inv
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
section iInf
variable {ι : Sort*} {f g : ι → ℝ≥0∞}
variable {a b c d : ℝ≥0∞} {r p q : ℝ≥0}
theorem toNNReal_iInf (hf : ∀ i, f ... | Mathlib/Data/ENNReal/Real.lean | 556 | 561 | theorem toNNReal_iSup (hf : ∀ i, f i ≠ ∞) : (iSup f).toNNReal = ⨆ i, (f i).toNNReal := by |
lift f to ι → ℝ≥0 using hf
simp_rw [toNNReal_coe]
by_cases h : BddAbove (range f)
· rw [← coe_iSup h, toNNReal_coe]
· rw [NNReal.iSup_of_not_bddAbove h, iSup_coe_eq_top.2 h, top_toNNReal]
| 5 | 148.413159 | 2 | 0.857143 | 21 | 755 |
import Mathlib.Order.SuccPred.Basic
import Mathlib.Topology.Order.Basic
import Mathlib.Topology.Metrizable.Uniformity
#align_import topology.instances.discrete from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78ad6527b7e4"
open Order Set TopologicalSpace Filter
variable {α : Type*} [TopologicalSp... | Mathlib/Topology/Instances/Discrete.lean | 111 | 117 | theorem discreteTopology_iff_orderTopology_of_pred_succ [LinearOrder α] [PredOrder α]
[SuccOrder α] : DiscreteTopology α ↔ OrderTopology α := by |
refine ⟨fun h => ⟨?_⟩, fun h => ⟨?_⟩⟩
· rw [h.eq_bot]
exact LinearOrder.bot_topologicalSpace_eq_generateFrom
· rw [h.topology_eq_generate_intervals]
exact LinearOrder.bot_topologicalSpace_eq_generateFrom.symm
| 5 | 148.413159 | 2 | 2 | 4 | 2,225 |
import Mathlib.Data.Real.Basic
#align_import data.real.sign from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace Real
noncomputable def sign (r : ℝ) : ℝ :=
if r < 0 then -1 else if 0 < r then 1 else 0
#align real.sign Real.sign
theorem sign_of_neg {r : ℝ} (hr : r < 0) : si... | Mathlib/Data/Real/Sign.lean | 119 | 123 | theorem sign_inv (r : ℝ) : sign r⁻¹ = sign r := by |
obtain hn | rfl | hp := lt_trichotomy r (0 : ℝ)
· rw [sign_of_neg hn, sign_of_neg (inv_lt_zero.mpr hn)]
· rw [sign_zero, inv_zero, sign_zero]
· rw [sign_of_pos hp, sign_of_pos (inv_pos.mpr hp)]
| 4 | 54.59815 | 2 | 1.363636 | 11 | 1,467 |
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle
#align_import geometry.euclidean.angle.oriented.right_angle from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped EuclideanGeometry
ope... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 54 | 61 | theorem oangle_add_right_eq_arcsin_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
o.oangle x (x + y) = Real.arcsin (‖y‖ / ‖x + y‖) := by |
have hs : (o.oangle x (x + y)).sign = 1 := by
rw [oangle_sign_add_right, h, Real.Angle.sign_coe_pi_div_two]
rw [o.oangle_eq_angle_of_sign_eq_one hs,
InnerProductGeometry.angle_add_eq_arcsin_of_inner_eq_zero
(o.inner_eq_zero_of_oangle_eq_pi_div_two h)
(Or.inl (o.left_ne_zero_of_oangle_eq_pi_div_... | 6 | 403.428793 | 2 | 1.642857 | 14 | 1,752 |
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 {𝕜 𝕝 E F β : Type*}
open Function Se... | Mathlib/Analysis/Convex/Strict.lean | 85 | 92 | theorem Directed.strictConvex_iUnion {ι : Sort*} {s : ι → Set E} (hdir : Directed (· ⊆ ·) s)
(hs : ∀ ⦃i : ι⦄, StrictConvex 𝕜 (s i)) : StrictConvex 𝕜 (⋃ i, s i) := by |
rintro x hx y hy hxy a b ha hb hab
rw [mem_iUnion] at hx hy
obtain ⟨i, hx⟩ := hx
obtain ⟨j, hy⟩ := hy
obtain ⟨k, hik, hjk⟩ := 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.Data.List.Chain
#align_import data.list.destutter from "leanprover-community/mathlib"@"7b78d1776212a91ecc94cf601f83bdcc46b04213"
variable {α : Type*} (l : List α) (R : α → α → Prop) [DecidableRel R] {a b : α}
namespace List
@[simp]
theorem destutter'_nil : destutter' R a [] = [a] :=
rfl
#align ... | Mathlib/Data/List/Destutter.lean | 92 | 98 | theorem destutter'_is_chain' (a) : (l.destutter' R a).Chain' R := by |
induction' l with b l hl generalizing a
· simp
rw [destutter']
split_ifs with h
· exact destutter'_is_chain R l h
· exact hl a
| 6 | 403.428793 | 2 | 1.142857 | 7 | 1,214 |
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathlib.CategoryTheory.FullSubcategory
import Mathlib.CategoryTheory.Skeletal
import Mathlib.Data.Fintype.Card
#align_import category_theory.Fintype from "leanprover-community/mathlib"@"c3019c79074b0619edb4b27553a91b2e82242395"
open scoped Classical
ope... | Mathlib/CategoryTheory/FintypeCat.lean | 160 | 179 | theorem is_skeletal : Skeletal Skeleton.{u} := fun X Y ⟨h⟩ =>
ext _ _ <|
Fin.equiv_iff_eq.mp <|
Nonempty.intro <|
{ toFun := fun x => (h.hom ⟨x⟩).down
invFun := fun x => (h.inv ⟨x⟩).down
left_inv := by |
intro a
change ULift.down _ = _
rw [ULift.up_down]
change ((h.hom ≫ h.inv) _).down = _
simp
rfl
right_inv := by
intro a
change ULift.down _ = _
rw [ULift.up_down]
change ((h.inv ≫ h.hom) _)... | 13 | 442,413.392009 | 2 | 1.5 | 2 | 1,559 |
import Mathlib.Algebra.MvPolynomial.Supported
import Mathlib.RingTheory.WittVector.Truncated
#align_import ring_theory.witt_vector.mul_coeff from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
namespace WittVector
variable (p : ℕ) [hp : Fact p.Prime]
variable {k ... | Mathlib/RingTheory/WittVector/MulCoeff.lean | 145 | 176 | theorem mul_polyOfInterest_aux3 (n : ℕ) : wittPolyProd p (n + 1) =
-((p : 𝕄) ^ (n + 1) * X (0, n + 1)) * ((p : 𝕄) ^ (n + 1) * X (1, n + 1)) +
(p : 𝕄) ^ (n + 1) * X (0, n + 1) * rename (Prod.mk (1 : Fin 2)) (wittPolynomial p ℤ (n + 1)) +
(p : 𝕄) ^ (n + 1) * X (1, n + 1) * rename (Prod.mk (0 : Fin 2)) (wi... |
-- a useful auxiliary fact
have mvpz : (p : 𝕄) ^ (n + 1) = MvPolynomial.C ((p : ℤ) ^ (n + 1)) := by norm_cast
-- Porting note: the original proof applies `sum_range_succ` through a non-`conv` rewrite,
-- but this does not work in Lean 4; the whole proof also times out very badly. The proof has been
-- nearl... | 27 | 532,048,240,601.79865 | 2 | 1.833333 | 6 | 1,912 |
import Mathlib.Analysis.NormedSpace.lpSpace
import Mathlib.Topology.Sets.Compacts
#align_import topology.metric_space.kuratowski from "leanprover-community/mathlib"@"95d4f6586d313c8c28e00f36621d2a6a66893aa6"
noncomputable section
set_option linter.uppercaseLean3 false
open Set Metric TopologicalSpace NNReal ENNR... | Mathlib/Topology/MetricSpace/Kuratowski.lean | 61 | 87 | theorem embeddingOfSubset_isometry (H : DenseRange x) : Isometry (embeddingOfSubset x) := by |
refine Isometry.of_dist_eq fun a b => ?_
refine (embeddingOfSubset_dist_le x a b).antisymm (le_of_forall_pos_le_add fun e epos => ?_)
-- First step: find n with dist a (x n) < e
rcases Metric.mem_closure_range_iff.1 (H a) (e / 2) (half_pos epos) with ⟨n, hn⟩
-- Second step: use the norm control at index n to... | 26 | 195,729,609,428.83878 | 2 | 2 | 3 | 2,145 |
import Mathlib.MeasureTheory.Measure.Restrict
open scoped ENNReal NNReal Topology
open Set MeasureTheory Measure Filter Function MeasurableSpace ENNReal
variable {α β δ ι : Type*}
namespace MeasureTheory
variable {m0 : MeasurableSpace α} [MeasurableSpace β] {μ ν ν₁ ν₂: Measure α}
{s t : Set α}
section IsFinit... | Mathlib/MeasureTheory/Measure/Typeclasses.lean | 132 | 139 | theorem Measure.isFiniteMeasure_map {m : MeasurableSpace α} (μ : Measure α) [IsFiniteMeasure μ]
(f : α → β) : IsFiniteMeasure (μ.map f) := by |
by_cases hf : AEMeasurable f μ
· constructor
rw [map_apply_of_aemeasurable hf MeasurableSet.univ]
exact measure_lt_top μ _
· rw [map_of_not_aemeasurable hf]
exact MeasureTheory.isFiniteMeasureZero
| 6 | 403.428793 | 2 | 1.25 | 8 | 1,315 |
import Mathlib.LinearAlgebra.Projectivization.Basic
#align_import linear_algebra.projective_space.independence from "leanprover-community/mathlib"@"1e82f5ec4645f6a92bb9e02fce51e44e3bc3e1fe"
open scoped LinearAlgebra.Projectivization
variable {ι K V : Type*} [DivisionRing K] [AddCommGroup V] [Module K V] {f : ι → ... | Mathlib/LinearAlgebra/Projectivization/Independence.lean | 84 | 94 | theorem dependent_iff : Dependent f ↔ ¬LinearIndependent K (Projectivization.rep ∘ f) := by |
refine ⟨?_, fun h => ?_⟩
· rintro ⟨ff, hff, hh1⟩
contrapose! hh1
choose a ha using fun i : ι => exists_smul_eq_mk_rep K (ff i) (hff i)
convert hh1.units_smul a⁻¹
ext i
simp only [← ha, inv_smul_smul, Pi.smul_apply', Pi.inv_apply, Function.comp_apply]
· convert Dependent.mk _ _ h
· simp on... | 10 | 22,026.465795 | 2 | 1.142857 | 7 | 1,223 |
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Polynomial.IntegralNormalization
#align_import ring_theory.algebraic from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
universe u v w
open scoped Classical
open Polynomi... | Mathlib/RingTheory/Algebraic.lean | 330 | 338 | theorem algHom_bijective [Algebra.IsAlgebraic K L] (f : L →ₐ[K] L) :
Function.Bijective f := by |
refine ⟨f.injective, fun b ↦ ?_⟩
obtain ⟨p, hp, he⟩ := Algebra.IsAlgebraic.isAlgebraic (R := K) b
let f' : p.rootSet L → p.rootSet L := (rootSet_maps_to' (fun x ↦ x) f).restrict f _ _
have : f'.Surjective := Finite.injective_iff_surjective.1
fun _ _ h ↦ Subtype.eq <| f.injective <| Subtype.ext_iff.1 h
ob... | 7 | 1,096.633158 | 2 | 1.125 | 8 | 1,205 |
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 | 111 | 116 | theorem Multiset.untrop_sum [LinearOrder R] [OrderTop R] (s : Multiset (Tropical R)) :
untrop s.sum = Multiset.inf (s.map untrop) := by |
induction' s using Multiset.induction with s x IH
· simp
· simp only [sum_cons, ge_iff_le, untrop_add, untrop_le_iff, map_cons, inf_cons, ← IH]
rfl
| 4 | 54.59815 | 2 | 0.928571 | 14 | 793 |
import Mathlib.Probability.Martingale.Convergence
import Mathlib.Probability.Martingale.OptionalStopping
import Mathlib.Probability.Martingale.Centering
#align_import probability.martingale.borel_cantelli from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
open Filter
open scoped NNRea... | Mathlib/Probability/Martingale/BorelCantelli.lean | 101 | 115 | theorem Submartingale.stoppedValue_leastGE [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) (r : ℝ) :
Submartingale (fun i => stoppedValue f (leastGE f r i)) ℱ μ := by |
rw [submartingale_iff_expected_stoppedValue_mono]
· intro σ π hσ hπ hσ_le_π hπ_bdd
obtain ⟨n, hπ_le_n⟩ := hπ_bdd
simp_rw [stoppedValue_stoppedValue_leastGE f σ r fun i => (hσ_le_π i).trans (hπ_le_n i)]
simp_rw [stoppedValue_stoppedValue_leastGE f π r hπ_le_n]
refine hf.expected_stoppedValue_mono ?_... | 13 | 442,413.392009 | 2 | 1.75 | 4 | 1,870 |
import Mathlib.Topology.GDelta
#align_import topology.metric_space.baire from "leanprover-community/mathlib"@"b9e46fe101fc897fb2e7edaf0bf1f09ea49eb81a"
noncomputable section
open scoped Topology
open Filter Set TopologicalSpace
variable {X α : Type*} {ι : Sort*}
section BaireTheorem
variable [TopologicalSpace... | Mathlib/Topology/Baire/Lemmas.lean | 74 | 81 | theorem mem_residual {s : Set X} : s ∈ residual X ↔ ∃ t ⊆ s, IsGδ t ∧ Dense t := by |
constructor
· rw [mem_residual_iff]
rintro ⟨S, hSo, hSd, Sct, Ss⟩
refine ⟨_, Ss, ⟨_, fun t ht => hSo _ ht, Sct, rfl⟩, ?_⟩
exact dense_sInter_of_isOpen hSo Sct hSd
rintro ⟨t, ts, ho, hd⟩
exact mem_of_superset (residual_of_dense_Gδ ho hd) ts
| 7 | 1,096.633158 | 2 | 1.428571 | 7 | 1,523 |
import Mathlib.LinearAlgebra.Matrix.BilinearForm
import Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.Vandermonde
import Mathlib.LinearAlgebra.Trace
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosu... | Mathlib/RingTheory/Trace.lean | 115 | 119 | theorem trace_algebraMap_of_basis (x : R) : trace R S (algebraMap R S x) = Fintype.card ι • x := by |
haveI := Classical.decEq ι
rw [trace_apply, LinearMap.trace_eq_matrix_trace R b, Matrix.trace]
convert Finset.sum_const x
simp [-coe_lmul_eq_mul]
| 4 | 54.59815 | 2 | 1 | 8 | 843 |
import Mathlib.RingTheory.RootsOfUnity.Basic
import Mathlib.RingTheory.AdjoinRoot
import Mathlib.FieldTheory.Galois
import Mathlib.LinearAlgebra.Eigenspace.Minpoly
import Mathlib.RingTheory.Norm
universe u
variable {K : Type u} [Field K]
open Polynomial IntermediateField AdjoinRoot
section Splits
lemma root_X_pow... | Mathlib/FieldTheory/KummerExtension.lean | 74 | 82 | theorem X_pow_sub_C_splits_of_isPrimitiveRoot
{n : ℕ} {ζ : K} (hζ : IsPrimitiveRoot ζ n) {α a : K} (e : α ^ n = a) :
(X ^ n - C a).Splits (RingHom.id _) := by |
cases n.eq_zero_or_pos with
| inl hn =>
rw [hn, pow_zero, ← C.map_one, ← map_sub]
exact splits_C _ _
| inr hn =>
rw [splits_iff_card_roots, ← nthRoots, hζ.card_nthRoots, natDegree_X_pow_sub_C, if_pos ⟨α, e⟩]
| 6 | 403.428793 | 2 | 1.5 | 2 | 1,633 |
import Mathlib.Topology.Constructions
import Mathlib.Topology.ContinuousOn
#align_import topology.bases from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78ad6527b7e4"
open Set Filter Function Topology
noncomputable section
namespace TopologicalSpace
universe u
variable {α : Type u} {β : Type*} ... | Mathlib/Topology/Bases.lean | 122 | 129 | theorem IsTopologicalBasis.of_hasBasis_nhds {s : Set (Set α)}
(h_nhds : ∀ a, (𝓝 a).HasBasis (fun t ↦ t ∈ s ∧ a ∈ t) id) : IsTopologicalBasis s where
exists_subset_inter t₁ ht₁ t₂ ht₂ x hx := by |
simpa only [and_assoc, (h_nhds x).mem_iff]
using (inter_mem ((h_nhds _).mem_of_mem ⟨ht₁, hx.1⟩) ((h_nhds _).mem_of_mem ⟨ht₂, hx.2⟩))
sUnion_eq := sUnion_eq_univ_iff.2 fun x ↦ (h_nhds x).ex_mem
eq_generateFrom := ext_nhds fun x ↦ by
simpa only [nhds_generateFrom, and_comm] using (h_nhds x).eq_biInf
| 5 | 148.413159 | 2 | 2 | 5 | 2,210 |
import Mathlib.Data.Fintype.Basic
#align_import data.fintype.quotient from "leanprover-community/mathlib"@"d78597269638367c3863d40d45108f52207e03cf"
def Quotient.finChoiceAux {ι : Type*} [DecidableEq ι] {α : ι → Type*} [S : ∀ i, Setoid (α i)] :
∀ l : List ι, (∀ i ∈ l, Quotient (S i)) → @Quotient (∀ i ∈ l, α ... | Mathlib/Data/Fintype/Quotient.lean | 76 | 84 | theorem Quotient.finChoice_eq {ι : Type*} [DecidableEq ι] [Fintype ι] {α : ι → Type*}
[∀ i, Setoid (α i)] (f : ∀ i, α i) : (Quotient.finChoice fun i => ⟦f i⟧) = ⟦f⟧ := by |
dsimp only [Quotient.finChoice]
conv_lhs =>
enter [1]
tactic =>
change _ = ⟦fun i _ => f i⟧
exact Quotient.inductionOn (@Finset.univ ι _).1 fun l => Quotient.finChoiceAux_eq _ _
rfl
| 7 | 1,096.633158 | 2 | 2 | 1 | 2,203 |
import Mathlib.LinearAlgebra.Matrix.Gershgorin
import Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody
import Mathlib.NumberTheory.NumberField.Units.Basic
import Mathlib.RingTheory.RootsOfUnity.Basic
#align_import number_theory.number_field.units from "leanprover-community/mathlib"@"00f91228655eecdcd3ac... | Mathlib/NumberTheory/NumberField/Units/DirichletTheorem.lean | 108 | 120 | theorem logEmbedding_eq_zero_iff {x : (𝓞 K)ˣ} :
logEmbedding K x = 0 ↔ x ∈ torsion K := by |
rw [mem_torsion]
refine ⟨fun h w => ?_, fun h => ?_⟩
· by_cases hw : w = w₀
· suffices -mult w₀ * Real.log (w₀ (x : K)) = 0 by
rw [neg_mul, neg_eq_zero, ← hw] at this
exact mult_log_place_eq_zero.mp this
rw [← sum_logEmbedding_component, sum_eq_zero]
exact fun w _ => congrFun h w
... | 11 | 59,874.141715 | 2 | 1.833333 | 6 | 1,909 |
import Mathlib.Data.Finset.Lattice
import Mathlib.Data.Fintype.Vector
import Mathlib.Data.Multiset.Sym
#align_import data.finset.sym from "leanprover-community/mathlib"@"02ba8949f486ebecf93fe7460f1ed0564b5e442c"
namespace Finset
variable {α : Type*}
@[simps]
protected def sym2 (s : Finset α) : Finset (Sym2 α) :... | Mathlib/Data/Finset/Sym.lean | 122 | 133 | theorem sym2_eq_image : s.sym2 = (s ×ˢ s).image Sym2.mk := by |
ext z
refine z.ind fun x y ↦ ?_
rw [mk_mem_sym2_iff, mem_image]
constructor
· intro h
use (x, y)
simp only [mem_product, h, and_self, true_and]
· rintro ⟨⟨a, b⟩, h⟩
simp only [mem_product, Sym2.eq_iff] at h
obtain ⟨h, (⟨rfl, rfl⟩ | ⟨rfl, rfl⟩)⟩ := h
<;> simp [h]
| 11 | 59,874.141715 | 2 | 0.769231 | 13 | 684 |
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2
#align_import measure_theory.function.conditional_expectation.condexp_L1 from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
noncomputable section
open TopologicalSpace MeasureTheory.Lp Filter ContinuousLinearMap
o... | Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean | 128 | 143 | theorem norm_condexpIndL1Fin_le (hs : MeasurableSet s) (hμs : μ s ≠ ∞) (x : G) :
‖condexpIndL1Fin hm hs hμs x‖ ≤ (μ s).toReal * ‖x‖ := by |
have : 0 ≤ ∫ a : α, ‖condexpIndL1Fin hm hs hμs x a‖ ∂μ := by positivity
rw [L1.norm_eq_integral_norm, ← ENNReal.toReal_ofReal (norm_nonneg x), ← ENNReal.toReal_mul, ←
ENNReal.toReal_ofReal this,
ENNReal.toReal_le_toReal ENNReal.ofReal_ne_top (ENNReal.mul_ne_top hμs ENNReal.ofReal_ne_top),
ofReal_integr... | 14 | 1,202,604.284165 | 2 | 2 | 4 | 2,398 |
import Mathlib.Algebra.Category.ModuleCat.EpiMono
import Mathlib.Algebra.Category.ModuleCat.Kernels
import Mathlib.CategoryTheory.Subobject.WellPowered
import Mathlib.CategoryTheory.Subobject.Limits
#align_import algebra.category.Module.subobject from "leanprover-community/mathlib"@"6d584f1709bedbed9175bd9350df46599b... | Mathlib/Algebra/Category/ModuleCat/Subobject.lean | 89 | 96 | theorem toKernelSubobject_arrow {M N : ModuleCat R} {f : M ⟶ N} (x : LinearMap.ker f) :
(kernelSubobject f).arrow (toKernelSubobject x) = x.1 := by |
-- Porting note: The whole proof was just `simp [toKernelSubobject]`.
suffices ((arrow ((kernelSubobject f))) ∘ (kernelSubobjectIso f ≪≫ kernelIsoKer f).inv) x = x by
convert this
rw [Iso.trans_inv, ← coe_comp, Category.assoc]
simp only [Category.assoc, kernelSubobject_arrow', kernelIsoKer_inv_kernel_ι]
... | 6 | 403.428793 | 2 | 2 | 2 | 1,985 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Data.ENat.Basic
#align_import data.polynomial.degree.trailing_degree from "leanprover-community/mathlib"@"302eab4f46abb63de520828de78c04cb0f9b5836"
noncomputable section
open Function Polynomial Finsupp Finset
open scoped Polynomial
namespace ... | Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean | 117 | 130 | theorem trailingDegree_eq_iff_natTrailingDegree_eq_of_pos {p : R[X]} {n : ℕ} (hn : 0 < n) :
p.trailingDegree = n ↔ p.natTrailingDegree = n := by |
constructor
· intro H
rwa [← trailingDegree_eq_iff_natTrailingDegree_eq]
rintro rfl
rw [trailingDegree_zero] at H
exact Option.noConfusion H
· intro H
rwa [trailingDegree_eq_iff_natTrailingDegree_eq]
rintro rfl
rw [natTrailingDegree_zero] at H
rw [H] at hn
exact lt_irrefl _ hn... | 12 | 162,754.791419 | 2 | 1.666667 | 6 | 1,754 |
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Analysis.NormedSpace.Dual
import Mathlib.Analysis.NormedSpace.Star.Basic
#align_import analysis.inner_product_space.dual from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped Classical
o... | Mathlib/Analysis/InnerProductSpace/Dual.lean | 82 | 91 | theorem ext_inner_left_basis {ι : Type*} {x y : E} (b : Basis ι 𝕜 E)
(h : ∀ i : ι, ⟪b i, x⟫ = ⟪b i, y⟫) : x = y := by |
apply (toDualMap 𝕜 E).map_eq_iff.mp
refine (Function.Injective.eq_iff ContinuousLinearMap.coe_injective).mp (Basis.ext b ?_)
intro i
simp only [ContinuousLinearMap.coe_coe]
rw [toDualMap_apply, toDualMap_apply]
rw [← inner_conj_symm]
conv_rhs => rw [← inner_conj_symm]
exact congr_arg conj (h i)
| 8 | 2,980.957987 | 2 | 1.666667 | 3 | 1,798 |
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Analysis.SumOverResidueClass
#align_import analysis.p_series from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8"
def SuccDiffBounded (C : ℕ) (u : ℕ → ℕ) : Prop :=... | Mathlib/Analysis/PSeries.lean | 84 | 98 | theorem sum_schlomilch_le' (hf : ∀ ⦃m n⦄, 1 < m → m ≤ n → f n ≤ f m) (h_pos : ∀ n, 0 < u n)
(hu : Monotone u) (n : ℕ) :
(∑ k ∈ range n, (u (k + 1) - u k) • f (u (k + 1))) ≤ ∑ k ∈ Ico (u 0 + 1) (u n + 1), f k := by |
induction' n with n ihn
· simp
suffices (u (n + 1) - u n) • f (u (n + 1)) ≤ ∑ k ∈ Ico (u n + 1) (u (n + 1) + 1), f k by
rw [sum_range_succ, ← sum_Ico_consecutive]
exacts [add_le_add ihn this,
(add_le_add_right (hu n.zero_le) _ : u 0 + 1 ≤ u n + 1),
add_le_add_right (hu n.le_succ) _]
have : ... | 12 | 162,754.791419 | 2 | 1.333333 | 6 | 1,396 |
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
#align_import analysis.special_functions.pow.asymptotics from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8"
set_option linter.uppercaseLean3 false
noncomputable section
open scoped Classical
open Real Topology NNReal ENNReal Filter C... | Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean | 200 | 207 | theorem isTheta_exp_arg_mul_im (hl : IsBoundedUnder (· ≤ ·) l fun x => |(g x).im|) :
(fun x => Real.exp (arg (f x) * im (g x))) =Θ[l] fun _ => (1 : ℝ) := by |
rcases hl with ⟨b, hb⟩
refine Real.isTheta_exp_comp_one.2 ⟨π * b, ?_⟩
rw [eventually_map] at hb ⊢
refine hb.mono fun x hx => ?_
erw [abs_mul]
exact mul_le_mul (abs_arg_le_pi _) hx (abs_nonneg _) Real.pi_pos.le
| 6 | 403.428793 | 2 | 1.5 | 10 | 1,605 |
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Nat.Factors
import Mathlib.Order.Interval.Finset.Nat
#align_import number_theory.divisors from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
open scoped Classical
open Finset
namespace Nat
variable (n : ℕ)
d... | Mathlib/NumberTheory/Divisors.lean | 95 | 99 | theorem mem_divisors {m : ℕ} : n ∈ divisors m ↔ n ∣ m ∧ m ≠ 0 := by |
rcases eq_or_ne m 0 with (rfl | hm); · simp [divisors]
simp only [hm, Ne, not_false_iff, and_true_iff, ← filter_dvd_eq_divisors hm, mem_filter,
mem_range, and_iff_right_iff_imp, Nat.lt_succ_iff]
exact le_of_dvd hm.bot_lt
| 4 | 54.59815 | 2 | 1 | 11 | 1,070 |
import Mathlib.LinearAlgebra.Eigenspace.Basic
import Mathlib.FieldTheory.Minpoly.Field
#align_import linear_algebra.eigenspace.minpoly from "leanprover-community/mathlib"@"c3216069e5f9369e6be586ccbfcde2592b3cec92"
universe u v w
namespace Module
namespace End
open Polynomial FiniteDimensional
open scoped Poly... | Mathlib/LinearAlgebra/Eigenspace/Minpoly.lean | 32 | 43 | theorem eigenspace_aeval_polynomial_degree_1 (f : End K V) (q : K[X]) (hq : degree q = 1) :
eigenspace f (-q.coeff 0 / q.leadingCoeff) = LinearMap.ker (aeval f q) :=
calc
eigenspace f (-q.coeff 0 / q.leadingCoeff)
_ = LinearMap.ker (q.leadingCoeff • f - algebraMap K (End K V) (-q.coeff 0)) := by |
rw [eigenspace_div]
intro h
rw [leadingCoeff_eq_zero_iff_deg_eq_bot.1 h] at hq
cases hq
_ = LinearMap.ker (aeval f (C q.leadingCoeff * X + C (q.coeff 0))) := by
rw [C_mul', aeval_def]; simp [algebraMap, Algebra.toRingHom]
_ = LinearMap.ker (aeval f q) := by rwa... | 7 | 1,096.633158 | 2 | 1.8 | 5 | 1,893 |
import Mathlib.Data.Finset.Lattice
import Mathlib.Data.Set.Sigma
#align_import data.finset.sigma from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function Multiset
variable {ι : Type*}
namespace Finset
section Sigma
variable {α : ι → Type*} {β : Type*} (s s₁ s₂ : Finset ι) (... | Mathlib/Data/Finset/Sigma.lean | 75 | 81 | theorem pairwiseDisjoint_map_sigmaMk :
(s : Set ι).PairwiseDisjoint fun i => (t i).map (Embedding.sigmaMk i) := by |
intro i _ j _ hij
rw [Function.onFun, disjoint_left]
simp_rw [mem_map, Function.Embedding.sigmaMk_apply]
rintro _ ⟨y, _, rfl⟩ ⟨z, _, hz'⟩
exact hij (congr_arg Sigma.fst hz'.symm)
| 5 | 148.413159 | 2 | 1.214286 | 14 | 1,292 |
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.Bounded
import Mathlib.SetTheory.Cardinal.PartENat
import Mathlib.SetTheory.Ordinal.Principal
import Mathlib.Tactic.Linarith
#align_import set_theory.cardinal.ordinal from "leanprover-community/mathlib"@"7c2ce0c2da15516b4e65d0c9e254bb6dc93abd1f"
noncomputa... | Mathlib/SetTheory/Cardinal/Ordinal.lean | 500 | 543 | theorem mul_eq_self {c : Cardinal} (h : ℵ₀ ≤ c) : c * c = c := by |
refine le_antisymm ?_ (by simpa only [mul_one] using mul_le_mul_left' (one_le_aleph0.trans h) c)
-- the only nontrivial part is `c * c ≤ c`. We prove it inductively.
refine Acc.recOn (Cardinal.lt_wf.apply c) (fun c _ => Quotient.inductionOn c fun α IH ol => ?_) h
-- consider the minimal well-order `r` on `α` (... | 43 | 4,727,839,468,229,346,000 | 2 | 1 | 8 | 1,056 |
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"
noncomputable section
section coevaluation
open TensorProduct FiniteDimensional
open TensorProduct
universe u v
variable (K : Type u) [Field K]
var... | Mathlib/LinearAlgebra/Coevaluation.lean | 47 | 54 | theorem coevaluation_apply_one :
(coevaluation K V) (1 : K) =
let bV := Basis.ofVectorSpace K V
∑ i : Basis.ofVectorSpaceIndex K V, bV i ⊗ₜ[K] bV.coord i := by |
simp only [coevaluation, id]
rw [(Basis.singleton Unit K).constr_apply_fintype K]
simp only [Fintype.univ_punit, Finset.sum_const, one_smul, Basis.singleton_repr,
Basis.equivFun_apply, Basis.coe_ofVectorSpace, one_nsmul, Finset.card_singleton]
| 4 | 54.59815 | 2 | 2 | 3 | 2,162 |
import Mathlib.MeasureTheory.Constructions.Pi
import Mathlib.MeasureTheory.Constructions.Prod.Integral
open Fintype MeasureTheory MeasureTheory.Measure
variable {𝕜 : Type*} [RCLike 𝕜]
namespace MeasureTheory
| Mathlib/MeasureTheory/Integral/Pi.lean | 26 | 41 | theorem Integrable.fin_nat_prod {n : ℕ} {E : Fin n → Type*}
[∀ i, MeasureSpace (E i)] [∀ i, SigmaFinite (volume : Measure (E i))]
{f : (i : Fin n) → E i → 𝕜} (hf : ∀ i, Integrable (f i)) :
Integrable (fun (x : (i : Fin n) → E i) ↦ ∏ i, f i (x i)) := by |
induction n with
| zero => simp only [Nat.zero_eq, Finset.univ_eq_empty, Finset.prod_empty, volume_pi,
integrable_const_iff, one_ne_zero, pi_empty_univ, ENNReal.one_lt_top, or_true]
| succ n n_ih =>
have := ((measurePreserving_piFinSuccAbove (fun i => (volume : Measure (E i))) 0).symm)
rw [volu... | 12 | 162,754.791419 | 2 | 1.6 | 5 | 1,745 |
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Algebra.Order.Group.Indicator
import Mathlib.Order.LiminfLimsup
import Mathlib.Order.Filter.Archimedean
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.Algebra.Group.Basic
import Mathlib.Data.Set.La... | Mathlib/Topology/Algebra/Order/LiminfLimsup.lean | 487 | 498 | theorem iUnion_Ici_eq_Ioi_of_lt_of_tendsto (x : R) {as : ι → R} (x_lt : ∀ i, x < as i)
{F : Filter ι} [Filter.NeBot F] (as_lim : Filter.Tendsto as F (𝓝 x)) :
⋃ i : ι, Ici (as i) = Ioi x := by |
have obs : x ∉ range as := by
intro maybe_x_is
rcases mem_range.mp maybe_x_is with ⟨i, hi⟩
simpa only [hi, lt_self_iff_false] using x_lt i
-- Porting note: `rw at *` was too destructive. Let's only rewrite `obs` and the goal.
have := iInf_eq_of_forall_le_of_tendsto (fun i ↦ (x_lt i).le) as_lim
rw [... | 9 | 8,103.083928 | 2 | 1.8 | 5 | 1,883 |
import Mathlib.FieldTheory.SplittingField.Construction
import Mathlib.RingTheory.Int.Basic
import Mathlib.RingTheory.Localization.Integral
import Mathlib.RingTheory.IntegrallyClosed
#align_import ring_theory.polynomial.gauss_lemma from "leanprover-community/mathlib"@"e3f4be1fcb5376c4948d7f095bec45350bfb9d1a"
open... | Mathlib/RingTheory/Polynomial/GaussLemma.lean | 124 | 130 | theorem IsPrimitive.irreducible_of_irreducible_map_of_injective (h_irr : Irreducible (map φ f)) :
Irreducible f := by |
refine
⟨fun h => h_irr.not_unit (IsUnit.map (mapRingHom φ) h), fun a b h =>
(h_irr.isUnit_or_isUnit <| by rw [h, Polynomial.map_mul]).imp ?_ ?_⟩
all_goals apply ((isPrimitive_of_dvd hf _).isUnit_iff_isUnit_map_of_injective hinj).mpr
exacts [Dvd.intro _ h.symm, Dvd.intro_left _ h.symm]
| 5 | 148.413159 | 2 | 2 | 4 | 2,151 |
import Mathlib.Analysis.Calculus.LineDeriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Measurable
open MeasureTheory
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜] [LocallyCompactSpace 𝕜]
{E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [MeasurableSpace E] [OpensMeasurableSpace E]
{F : Type*} [Norm... | Mathlib/Analysis/Calculus/LineDeriv/Measurable.lean | 92 | 99 | theorem stronglyMeasurable_lineDeriv_uncurry (hf : Continuous f) :
StronglyMeasurable (fun (p : E × E) ↦ lineDeriv 𝕜 f p.1 p.2) := by |
borelize 𝕜
let g : (E × E) → 𝕜 → F := fun p t ↦ f (p.1 + t • p.2)
have : Continuous g.uncurry :=
hf.comp <| (continuous_fst.comp continuous_fst).add
<| continuous_snd.smul (continuous_snd.comp continuous_fst)
exact (stronglyMeasurable_deriv_with_param this).comp_measurable measurable_prod_mk_right
| 6 | 403.428793 | 2 | 2 | 6 | 2,214 |
import Mathlib.NumberTheory.FLT.Basic
import Mathlib.NumberTheory.PythagoreanTriples
import Mathlib.RingTheory.Coprime.Lemmas
import Mathlib.Tactic.LinearCombination
#align_import number_theory.fermat4 from "leanprover-community/mathlib"@"10b4e499f43088dd3bb7b5796184ad5216648ab1"
noncomputable section
open scope... | Mathlib/NumberTheory/FLT/Four.lean | 89 | 105 | theorem coprime_of_minimal {a b c : ℤ} (h : Minimal a b c) : IsCoprime a b := by |
apply Int.gcd_eq_one_iff_coprime.mp
by_contra hab
obtain ⟨p, hp, hpa, hpb⟩ := Nat.Prime.not_coprime_iff_dvd.mp hab
obtain ⟨a1, rfl⟩ := Int.natCast_dvd.mpr hpa
obtain ⟨b1, rfl⟩ := Int.natCast_dvd.mpr hpb
have hpc : (p : ℤ) ^ 2 ∣ c := by
rw [← Int.pow_dvd_pow_iff two_ne_zero, ← h.1.2.2]
apply Dvd.int... | 16 | 8,886,110.520508 | 2 | 1.666667 | 9 | 1,779 |
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Algebra.Order.Group.Indicator
import Mathlib.Order.LiminfLimsup
import Mathlib.Order.Filter.Archimedean
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.Algebra.Group.Basic
import Mathlib.Data.Set.La... | Mathlib/Topology/Algebra/Order/LiminfLimsup.lean | 511 | 565 | theorem limsup_eq_tendsto_sum_indicator_nat_atTop (s : ℕ → Set α) :
limsup s atTop = { ω | Tendsto
(fun n ↦ ∑ k ∈ Finset.range n, (s (k + 1)).indicator (1 : α → ℕ) ω) atTop atTop } := by |
ext ω
simp only [limsup_eq_iInf_iSup_of_nat, ge_iff_le, Set.iSup_eq_iUnion, Set.iInf_eq_iInter,
Set.mem_iInter, Set.mem_iUnion, exists_prop]
constructor
· intro hω
refine tendsto_atTop_atTop_of_monotone' (fun n m hnm ↦ Finset.sum_mono_set_of_nonneg
(fun i ↦ Set.indicator_nonneg (fun _ _ ↦ zero_le... | 52 | 38,310,080,007,165,770,000,000 | 2 | 1.8 | 5 | 1,883 |
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.optional_stopping from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory
namespace MeasureTheory
vari... | Mathlib/Probability/Martingale/OptionalStopping.lean | 112 | 133 | theorem smul_le_stoppedValue_hitting [IsFiniteMeasure μ] (hsub : Submartingale f 𝒢 μ) {ε : ℝ≥0}
(n : ℕ) : ε • μ {ω | (ε : ℝ) ≤ (range (n + 1)).sup' nonempty_range_succ fun k => f k ω} ≤
ENNReal.ofReal (∫ ω in {ω | (ε : ℝ) ≤ (range (n + 1)).sup' nonempty_range_succ fun k => f k ω},
stoppedValue f (hitting... |
have hn : Set.Icc 0 n = {k | k ≤ n} := by ext x; simp
have : ∀ ω, ((ε : ℝ) ≤ (range (n + 1)).sup' nonempty_range_succ fun k => f k ω) →
(ε : ℝ) ≤ stoppedValue f (hitting f {y : ℝ | ↑ε ≤ y} 0 n) ω := by
intro x hx
simp_rw [le_sup'_iff, mem_range, Nat.lt_succ_iff] at hx
refine stoppedValue_hitting_... | 18 | 65,659,969.137331 | 2 | 2 | 3 | 1,978 |
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.Data.Set.Basic
import Mathlib.Data.Setoid.Basic
import Mathlib.GroupTheory.Coset
#align_import group_theory.double_coset from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
-- Porting note: removed import
-- import Mathlib.Tac... | Mathlib/GroupTheory/DoubleCoset.lean | 52 | 57 | theorem doset_eq_of_mem {H K : Subgroup G} {a b : G} (hb : b ∈ doset a H K) :
doset b H K = doset a H K := by |
obtain ⟨h, hh, k, hk, rfl⟩ := mem_doset.1 hb
rw [doset, doset, ← Set.singleton_mul_singleton, ← Set.singleton_mul_singleton, mul_assoc,
mul_assoc, Subgroup.singleton_mul_subgroup hk, ← mul_assoc, ← mul_assoc,
Subgroup.subgroup_mul_singleton hh]
| 4 | 54.59815 | 2 | 1.428571 | 7 | 1,515 |
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Group.Measure
#align_import measure_theory.group.prod from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
noncomputable section
open Set hiding prod_eq
open Function MeasureTheory
open Filter hiding ma... | Mathlib/MeasureTheory/Group/Prod.lean | 108 | 116 | theorem measurable_measure_mul_right (hs : MeasurableSet s) :
Measurable fun x => μ ((fun y => y * x) ⁻¹' s) := by |
suffices
Measurable fun y =>
μ ((fun x => (x, y)) ⁻¹' ((fun z : G × G => ((1 : G), z.1 * z.2)) ⁻¹' univ ×ˢ s))
by convert this using 1; ext1 x; congr 1 with y : 1; simp
apply measurable_measure_prod_mk_right
apply measurable_const.prod_mk measurable_mul (MeasurableSet.univ.prod hs)
infer_instance... | 7 | 1,096.633158 | 2 | 2 | 4 | 2,269 |
import Mathlib.Topology.VectorBundle.Basic
#align_import topology.vector_bundle.hom from "leanprover-community/mathlib"@"8905e5ed90859939681a725b00f6063e65096d95"
noncomputable section
open scoped Bundle
open Bundle Set ContinuousLinearMap
variable {𝕜₁ : Type*} [NontriviallyNormedField 𝕜₁] {𝕜₂ : Type*} [Non... | Mathlib/Topology/VectorBundle/Hom.lean | 92 | 112 | theorem continuousOn_continuousLinearMapCoordChange [VectorBundle 𝕜₁ F₁ E₁] [VectorBundle 𝕜₂ F₂ E₂]
[MemTrivializationAtlas e₁] [MemTrivializationAtlas e₁'] [MemTrivializationAtlas e₂]
[MemTrivializationAtlas e₂'] :
ContinuousOn (continuousLinearMapCoordChange σ e₁ e₁' e₂ e₂')
(e₁.baseSet ∩ e₂.baseS... |
have h₁ := (compSL F₁ F₂ F₂ σ (RingHom.id 𝕜₂)).continuous
have h₂ := (ContinuousLinearMap.flip (compSL F₁ F₁ F₂ (RingHom.id 𝕜₁) σ)).continuous
have h₃ := continuousOn_coordChange 𝕜₁ e₁' e₁
have h₄ := continuousOn_coordChange 𝕜₂ e₂ e₂'
refine ((h₁.comp_continuousOn (h₄.mono ?_)).clm_comp (h₂.comp_continuo... | 16 | 8,886,110.520508 | 2 | 2 | 1 | 2,453 |
import Mathlib.Algebra.Group.Nat
set_option autoImplicit true
open Lean hiding Literal HashMap
open Batteries
namespace Sat
inductive Literal
| pos : Nat → Literal
| neg : Nat → Literal
def Literal.ofInt (i : Int) : Literal :=
if i < 0 then Literal.neg (-i-1).toNat else Literal.pos (i-1).toNat
def Lit... | Mathlib/Tactic/Sat/FromLRAT.lean | 156 | 166 | theorem Valuation.mk_implies {as ps} (as₁) : as = List.reverseAux as₁ ps →
(Valuation.mk as).implies p ps as₁.length → p := by |
induction ps generalizing as₁ with
| nil => exact fun _ ↦ id
| cons a as ih =>
refine fun e H ↦ @ih (a::as₁) e (H ?_)
subst e; clear ih H
suffices ∀ n n', n' = List.length as₁ + n →
∀ bs, mk (as₁.reverseAux bs) n' ↔ mk bs n from this 0 _ rfl (a::as)
induction as₁ with simp
| cons b as₁ ... | 9 | 8,103.083928 | 2 | 2 | 2 | 2,084 |
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