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.PNat.Prime
import Mathlib.Algebra.IsPrimePow
import Mathlib.NumberTheory.Cyclotomic.Basic
import Mathlib.RingTheory.Adjoin.PowerBasis
import Mathlib.RingTheory.Polynomial.Cyclotomic.Eval
import Mathlib.RingTheory.Norm
import Mathlib.RingTheory.Polynomial.Cyclotomic.Expand
#align_import number_theo... | Mathlib/NumberTheory/Cyclotomic/PrimitiveRoots.lean | 128 | 131 | theorem powerBasis_gen_mem_adjoin_zeta_sub_one :
(hζ.powerBasis K).gen ∈ adjoin K ({ζ - 1} : Set L) := by |
rw [powerBasis_gen, adjoin_singleton_eq_range_aeval, AlgHom.mem_range]
exact ⟨X + 1, by simp⟩
| 2 | 7.389056 | 1 | 0.75 | 4 | 670 |
import Mathlib.Probability.ProbabilityMassFunction.Basic
#align_import probability.probability_mass_function.monad from "leanprover-community/mathlib"@"4ac69b290818724c159de091daa3acd31da0ee6d"
noncomputable section
variable {α β γ : Type*}
open scoped Classical
open NNReal ENNReal
open MeasureTheory
namespac... | Mathlib/Probability/ProbabilityMassFunction/Monad.lean | 74 | 80 | theorem toOuterMeasure_pure_apply : (pure a).toOuterMeasure s = if a ∈ s then 1 else 0 := by |
refine (toOuterMeasure_apply (pure a) s).trans ?_
split_ifs with ha
· refine (tsum_congr fun b => ?_).trans (tsum_ite_eq a 1)
exact ite_eq_left_iff.2 fun hb => symm (ite_eq_right_iff.2 fun h => (hb <| h.symm ▸ ha).elim)
· refine (tsum_congr fun b => ?_).trans tsum_zero
exact ite_eq_right_iff.2 fun hb =... | 6 | 403.428793 | 2 | 1 | 6 | 1,147 |
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.Fin.Tuple.Reflection
#align_import data.matrix.reflection from "leanprover-community/mathlib"@"820b22968a2bc4a47ce5cf1d2f36a9ebe52510aa"
open Matrix
namespace Matrix
variable {l m n : ℕ} {α β : Type*}
def Forall : ∀ {m n}... | Mathlib/Data/Matrix/Reflection.lean | 185 | 188 | theorem mulVecᵣ_eq [NonUnitalNonAssocSemiring α] (A : Matrix (Fin l) (Fin m) α) (v : Fin m → α) :
mulVecᵣ A v = A *ᵥ v := by |
simp [mulVecᵣ, Function.comp]
rfl
| 2 | 7.389056 | 1 | 1 | 3 | 944 |
import Mathlib.Data.Set.Finite
import Mathlib.GroupTheory.GroupAction.FixedPoints
import Mathlib.GroupTheory.Perm.Support
open Equiv List MulAction Pointwise Set Subgroup
variable {G α : Type*} [Group G] [MulAction G α] [DecidableEq α]
theorem finite_compl_fixedBy_closure_iff {S : Set G} :
(∀ g ∈ closure S, ... | Mathlib/GroupTheory/Perm/ClosureSwap.lean | 41 | 44 | theorem Equiv.Perm.IsSwap.finite_compl_fixedBy {σ : Perm α} (h : σ.IsSwap) :
(fixedBy α σ)ᶜ.Finite := by |
obtain ⟨x, y, -, rfl⟩ := h
exact finite_compl_fixedBy_swap
| 2 | 7.389056 | 1 | 1.8 | 5 | 1,882 |
import Mathlib.CategoryTheory.FinCategory.Basic
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Limits.Shapes.FiniteLimits
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryTheory.Category.Preorder
import Mathlib.CategoryTheory.Category.ULift
import Mathlib.CategoryTheory.P... | Mathlib/CategoryTheory/Filtered/Basic.lean | 372 | 388 | theorem of_cocone_nonempty (h : ∀ {J : Type w} [SmallCategory J] [FinCategory J] (F : J ⥤ C),
Nonempty (Cocone F)) : IsFiltered C := by |
have : Nonempty C := by
obtain ⟨c⟩ := h (Functor.empty _)
exact ⟨c.pt⟩
have : IsFilteredOrEmpty C := by
refine ⟨?_, ?_⟩
· intros X Y
obtain ⟨c⟩ := h (ULiftHom.down ⋙ ULift.downFunctor ⋙ pair X Y)
exact ⟨c.pt, c.ι.app ⟨⟨WalkingPair.left⟩⟩, c.ι.app ⟨⟨WalkingPair.right⟩⟩, trivial⟩
· in... | 15 | 3,269,017.372472 | 2 | 2 | 1 | 2,277 |
import Mathlib.Combinatorics.Quiver.Basic
import Mathlib.Combinatorics.Quiver.Path
#align_import combinatorics.quiver.cast from "leanprover-community/mathlib"@"fc2ed6f838ce7c9b7c7171e58d78eaf7b438fb0e"
universe v v₁ v₂ u u₁ u₂
variable {U : Type*} [Quiver.{u + 1} U]
namespace Quiver
def Hom.cast {u v u' v... | Mathlib/Combinatorics/Quiver/Cast.lean | 99 | 103 | theorem Path.cast_cast {u v u' v' u'' v'' : U} (p : Path u v) (hu : u = u') (hv : v = v')
(hu' : u' = u'') (hv' : v' = v'') :
(p.cast hu hv).cast hu' hv' = p.cast (hu.trans hu') (hv.trans hv') := by |
subst_vars
rfl
| 2 | 7.389056 | 1 | 1 | 12 | 1,049 |
import Mathlib.MeasureTheory.MeasurableSpace.Defs
import Mathlib.Order.Filter.CountableInter
open Filter Set MeasurableSpace
variable {α : Type*} (m : MeasurableSpace α) (l : Filter α) [CountableInterFilter l] {s t : Set α}
def EventuallyMeasurableSpace : MeasurableSpace α where
MeasurableSet' s := ∃ t, Measu... | Mathlib/MeasureTheory/Constructions/EventuallyMeasurable.lean | 67 | 70 | theorem EventuallyMeasurableSet.congr
(ht : EventuallyMeasurableSet m l t) (hst : s =ᶠ[l] t) : EventuallyMeasurableSet m l s := by |
rcases ht with ⟨t', ht', htt'⟩
exact ⟨t', ht', hst.trans htt'⟩
| 2 | 7.389056 | 1 | 1 | 1 | 1,033 |
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 | 95 | 98 | theorem DirectedOn.strictConvex_sUnion {S : Set (Set E)} (hdir : DirectedOn (· ⊆ ·) S)
(hS : ∀ s ∈ S, StrictConvex 𝕜 s) : StrictConvex 𝕜 (⋃₀ S) := by |
rw [sUnion_eq_iUnion]
exact (directedOn_iff_directed.1 hdir).strictConvex_iUnion fun s => hS _ s.2
| 2 | 7.389056 | 1 | 1.333333 | 3 | 1,438 |
import Mathlib.Data.Finsupp.Defs
#align_import data.finsupp.indicator from "leanprover-community/mathlib"@"842328d9df7e96fd90fc424e115679c15fb23a71"
noncomputable section
open Finset Function
variable {ι α : Type*}
namespace Finsupp
variable [Zero α] {s : Finset ι} (f : ∀ i ∈ s, α) {i : ι}
def indicator (s ... | Mathlib/Data/Finsupp/Indicator.lean | 54 | 56 | theorem indicator_apply [DecidableEq ι] : indicator s f i = if hi : i ∈ s then f i hi else 0 := by |
simp only [indicator, ne_eq, coe_mk]
congr
| 2 | 7.389056 | 1 | 1.666667 | 3 | 1,791 |
import Mathlib.Data.Set.Prod
import Mathlib.Logic.Function.Conjugate
#align_import data.set.function from "leanprover-community/mathlib"@"996b0ff959da753a555053a480f36e5f264d4207"
variable {α β γ : Type*} {ι : Sort*} {π : α → Type*}
open Equiv Equiv.Perm Function
namespace Set
section Order
variable {s : Se... | Mathlib/Data/Set/Function.lean | 264 | 267 | theorem _root_.MonotoneOn.congr (h₁ : MonotoneOn f₁ s) (h : s.EqOn f₁ f₂) : MonotoneOn f₂ s := by |
intro a ha b hb hab
rw [← h ha, ← h hb]
exact h₁ ha hb hab
| 3 | 20.085537 | 1 | 0.8 | 10 | 704 |
import Mathlib.Topology.Bases
import Mathlib.Topology.DenseEmbedding
#align_import topology.stone_cech from "leanprover-community/mathlib"@"0a0ec35061ed9960bf0e7ffb0335f44447b58977"
noncomputable section
open Filter Set
open Topology
universe u v
section Ultrafilter
def ultrafilterBasis (α : Type u) : Set ... | Mathlib/Topology/StoneCech.lean | 122 | 126 | theorem ultrafilter_pure_injective : Function.Injective (pure : α → Ultrafilter α) := by |
intro x y h
have : {x} ∈ (pure x : Ultrafilter α) := singleton_mem_pure
rw [h] at this
exact (mem_singleton_iff.mp (mem_pure.mp this)).symm
| 4 | 54.59815 | 2 | 2 | 5 | 2,361 |
import Mathlib.Analysis.Asymptotics.Asymptotics
import Mathlib.Analysis.Asymptotics.Theta
import Mathlib.Analysis.Normed.Order.Basic
#align_import analysis.asymptotics.asymptotic_equivalent from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
namespace Asymptotics
open Filter Function
... | Mathlib/Analysis/Asymptotics/AsymptoticEquivalent.lean | 102 | 104 | theorem IsEquivalent.refl : u ~[l] u := by |
rw [IsEquivalent, sub_self]
exact isLittleO_zero _ _
| 2 | 7.389056 | 1 | 1.285714 | 7 | 1,354 |
import Mathlib.Data.Set.Prod
import Mathlib.Logic.Function.Conjugate
#align_import data.set.function from "leanprover-community/mathlib"@"996b0ff959da753a555053a480f36e5f264d4207"
variable {α β γ : Type*} {ι : Sort*} {π : α → Type*}
open Equiv Equiv.Perm Function
namespace Set
section restrict
def restrict (... | Mathlib/Data/Set/Function.lean | 117 | 120 | theorem restrict_extend_range (f : α → β) (g : α → γ) (g' : β → γ) :
(range f).restrict (extend f g g') = fun x => g x.coe_prop.choose := by |
classical
exact restrict_dite _ _
| 2 | 7.389056 | 1 | 0.8 | 10 | 704 |
import Mathlib.Computability.Encoding
import Mathlib.Logic.Small.List
import Mathlib.ModelTheory.Syntax
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import model_theory.encoding from "leanprover-community/mathlib"@"91288e351d51b3f0748f0a38faa7613fb0ae2ada"
universe u v w u' v'
namespace FirstOrder
namespace... | Mathlib/ModelTheory/Encoding.lean | 235 | 287 | theorem listDecode_encode_list (l : List (Σn, L.BoundedFormula α n)) :
(listDecode (l.bind fun φ => φ.2.listEncode)).1 = l.headI := by |
suffices h : ∀ (φ : Σn, L.BoundedFormula α n) (l),
(listDecode (listEncode φ.2 ++ l)).1 = φ ∧ (listDecode (listEncode φ.2 ++ l)).2.1 = l by
induction' l with φ l _
· rw [List.nil_bind]
simp [listDecode]
· rw [cons_bind, (h φ _).1, headI_cons]
rintro ⟨n, φ⟩
induction' φ with _ _ _ _ φ_n φ_... | 51 | 14,093,490,824,269,389,000,000 | 2 | 2 | 3 | 2,074 |
import Mathlib.Data.Multiset.Nodup
#align_import data.multiset.dedup from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace Multiset
open List
variable {α β : Type*} [DecidableEq α]
def dedup (s : Multiset α) : Multiset α :=
Quot.liftOn s (fun l => (l.dedup : Multiset α)... | Mathlib/Data/Multiset/Dedup.lean | 126 | 128 | theorem dedup_nsmul {s : Multiset α} {n : ℕ} (h0 : n ≠ 0) : (n • s).dedup = s.dedup := by |
ext a
by_cases h : a ∈ s <;> simp [h, h0]
| 2 | 7.389056 | 1 | 0.25 | 4 | 296 |
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calculus.FDeriv.Equiv
#align_import analysis.calculus.deriv.inverse from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
open scoped Classical
open Topology Filter ENNReal
open Filter Asymptotics Set
va... | Mathlib/Analysis/Calculus/Deriv/Inverse.lean | 112 | 117 | theorem not_differentiableWithinAt_of_local_left_inverse_hasDerivWithinAt_zero {f g : 𝕜 → 𝕜} {a : 𝕜}
{s t : Set 𝕜} (ha : a ∈ s) (hsu : UniqueDiffWithinAt 𝕜 s a) (hf : HasDerivWithinAt f 0 t (g a))
(hst : MapsTo g s t) (hfg : f ∘ g =ᶠ[𝓝[s] a] id) : ¬DifferentiableWithinAt 𝕜 g s a := by |
intro hg
have := (hf.comp a hg.hasDerivWithinAt hst).congr_of_eventuallyEq_of_mem hfg.symm ha
simpa using hsu.eq_deriv _ this (hasDerivWithinAt_id _ _)
| 3 | 20.085537 | 1 | 1 | 2 | 1,005 |
import Mathlib.Algebra.Algebra.Quasispectrum
import Mathlib.Topology.ContinuousFunction.Compact
import Mathlib.Topology.ContinuousFunction.ContinuousMapZero
import Mathlib.Topology.ContinuousFunction.FunctionalCalculus
import Mathlib.Topology.UniformSpace.CompactConvergence
local notation "σₙ" => quasispectrum
open... | Mathlib/Topology/ContinuousFunction/NonUnitalFunctionalCalculus.lean | 147 | 167 | theorem cfcₙHom_comp [UniqueNonUnitalContinuousFunctionalCalculus R A] (f : C(σₙ R a, R)₀)
(f' : C(σₙ R a, σₙ R (cfcₙHom ha f))₀)
(hff' : ∀ x, f x = f' x) (g : C(σₙ R (cfcₙHom ha f), R)₀) :
cfcₙHom ha (g.comp f') = cfcₙHom (cfcₙHom_predicate ha f) g := by |
let ψ : C(σₙ R (cfcₙHom ha f), R)₀ →⋆ₙₐ[R] C(σₙ R a, R)₀ :=
{ toFun := (ContinuousMapZero.comp · f')
map_smul' := fun _ _ ↦ rfl
map_add' := fun _ _ ↦ rfl
map_mul' := fun _ _ ↦ rfl
map_zero' := rfl
map_star' := fun _ ↦ rfl }
let φ : C(σₙ R (cfcₙHom ha f), R)₀ →⋆ₙₐ[R] A := (cfcₙHom ... | 17 | 24,154,952.753575 | 2 | 2 | 1 | 2,069 |
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Data.Real.ConjExponents
#align_import analysis.mean_inequalities from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
universe u... | Mathlib/Analysis/MeanInequalities.lean | 113 | 134 | theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i ∈ s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) :
∏ i ∈ s, z i ^ w i ≤ ∑ i ∈ s, w i * z i := by |
-- If some number `z i` equals zero and has non-zero weight, then LHS is 0 and RHS is nonnegative.
by_cases A : ∃ i ∈ s, z i = 0 ∧ w i ≠ 0
· rcases A with ⟨i, his, hzi, hwi⟩
rw [prod_eq_zero his]
· exact sum_nonneg fun j hj => mul_nonneg (hw j hj) (hz j hj)
· rw [hzi]
exact zero_rpow hwi
-- I... | 19 | 178,482,300.963187 | 2 | 1.6 | 5 | 1,716 |
import Mathlib.Order.Interval.Finset.Fin
#align_import data.fintype.fin from "leanprover-community/mathlib"@"759575657f189ccb424b990164c8b1fa9f55cdfe"
open Finset
open Fintype
namespace Fin
variable {α β : Type*} {n : ℕ}
theorem map_valEmbedding_univ : (Finset.univ : Finset (Fin n)).map Fin.valEmbedding = Iio ... | Mathlib/Data/Fintype/Fin.lean | 61 | 64 | theorem card_filter_univ_succ' (p : Fin (n + 1) → Prop) [DecidablePred p] :
(univ.filter p).card = ite (p 0) 1 0 + (univ.filter (p ∘ Fin.succ)).card := by |
rw [Fin.univ_succ, filter_cons, card_disjUnion, filter_map, card_map]
split_ifs <;> simp
| 2 | 7.389056 | 1 | 1.4 | 5 | 1,485 |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Regular.Basic
#align_import algebra.regular.pow from "leanprover-community/mathlib"@"46a64b5b4268c594af770c44d9e502afc6a515cb"
variable {R : Type*} {a b : R}
section Monoid
variable [Monoid R]
... | Mathlib/Algebra/Regular/Pow.lean | 54 | 58 | theorem IsRightRegular.pow_iff {n : ℕ} (n0 : 0 < n) :
IsRightRegular (a ^ n) ↔ IsRightRegular a := by |
refine ⟨?_, IsRightRegular.pow n⟩
rw [← Nat.succ_pred_eq_of_pos n0, pow_succ']
exact IsRightRegular.of_mul
| 3 | 20.085537 | 1 | 0.75 | 4 | 675 |
import Mathlib.Algebra.EuclideanDomain.Defs
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Regular
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Ring.Basic
#align_import algebra.euclidean_domain.basic from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d0... | Mathlib/Algebra/EuclideanDomain/Basic.lean | 96 | 101 | theorem mul_div_assoc (x : R) {y z : R} (h : z ∣ y) : x * y / z = x * (y / z) := by |
by_cases hz : z = 0
· subst hz
rw [div_zero, div_zero, mul_zero]
rcases h with ⟨p, rfl⟩
rw [mul_div_cancel_left₀ _ hz, mul_left_comm, mul_div_cancel_left₀ _ hz]
| 5 | 148.413159 | 2 | 0.888889 | 9 | 769 |
import Mathlib.Data.List.Count
import Mathlib.Data.List.Dedup
import Mathlib.Data.List.InsertNth
import Mathlib.Data.List.Lattice
import Mathlib.Data.List.Permutation
import Mathlib.Data.Nat.Factorial.Basic
#align_import data.list.perm from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
... | Mathlib/Data/List/Perm.lean | 149 | 164 | theorem perm_comp_forall₂ {l u v} (hlu : Perm l u) (huv : Forall₂ r u v) :
(Forall₂ r ∘r Perm) l v := by |
induction hlu generalizing v with
| nil => cases huv; exact ⟨[], Forall₂.nil, Perm.nil⟩
| cons u _hlu ih =>
cases' huv with _ b _ v hab huv'
rcases ih huv' with ⟨l₂, h₁₂, h₂₃⟩
exact ⟨b :: l₂, Forall₂.cons hab h₁₂, h₂₃.cons _⟩
| swap a₁ a₂ h₂₃ =>
cases' huv with _ b₁ _ l₂ h₁ hr₂₃
cases' hr₂₃... | 14 | 1,202,604.284165 | 2 | 2 | 3 | 2,228 |
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.Topology.MetricSpace.ThickenedIndicator
open MeasureTheory Topology Metric Filter Set ENNReal NNReal
open scoped Topology ENNReal NNReal BoundedContinuousFunction
section auxiliary
namespace MeasureTheory
variable {Ω : Type*} [TopologicalSpace Ω] [Mea... | Mathlib/MeasureTheory/Measure/HasOuterApproxClosed.lean | 75 | 85 | theorem measure_of_cont_bdd_of_tendsto_filter_indicator {ι : Type*} {L : Filter ι}
[L.IsCountablyGenerated] [TopologicalSpace Ω] [OpensMeasurableSpace Ω] (μ : Measure Ω)
[IsFiniteMeasure μ] {c : ℝ≥0} {E : Set Ω} (E_mble : MeasurableSet E) (fs : ι → Ω →ᵇ ℝ≥0)
(fs_bdd : ∀ᶠ i in L, ∀ᵐ ω : Ω ∂μ, fs i ω ≤ c)
... |
convert tendsto_lintegral_nn_filter_of_le_const μ fs_bdd fs_lim
have aux : ∀ ω, indicator E (fun _ ↦ (1 : ℝ≥0∞)) ω = ↑(indicator E (fun _ ↦ (1 : ℝ≥0)) ω) :=
fun ω ↦ by simp only [ENNReal.coe_indicator, ENNReal.coe_one]
simp_rw [← aux, lintegral_indicator _ E_mble]
simp only [lintegral_one, Measure.restrict... | 5 | 148.413159 | 2 | 2 | 4 | 2,283 |
import Mathlib.Algebra.IsPrimePow
import Mathlib.Data.Nat.Factorization.Basic
#align_import data.nat.factorization.prime_pow from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf97388c9e3b441fc4ccf3f"
variable {R : Type*} [CommMonoidWithZero R] (n p : R) (k : ℕ)
theorem IsPrimePow.minFac_pow_factorization_eq ... | Mathlib/Data/Nat/Factorization/PrimePow.lean | 63 | 73 | theorem IsPrimePow.exists_ord_compl_eq_one {n : ℕ} (h : IsPrimePow n) :
∃ p : ℕ, p.Prime ∧ ord_compl[p] n = 1 := by |
rcases eq_or_ne n 0 with (rfl | hn0); · cases not_isPrimePow_zero h
rcases isPrimePow_iff_factorization_eq_single.mp h with ⟨p, k, hk0, h1⟩
rcases em' p.Prime with (pp | pp)
· refine absurd ?_ hk0.ne'
simp [← Nat.factorization_eq_zero_of_non_prime n pp, h1]
refine ⟨p, pp, ?_⟩
refine Nat.eq_of_factoriza... | 9 | 8,103.083928 | 2 | 1.714286 | 7 | 1,840 |
import Mathlib.Analysis.NormedSpace.Basic
#align_import analysis.normed_space.enorm from "leanprover-community/mathlib"@"57ac39bd365c2f80589a700f9fbb664d3a1a30c2"
noncomputable section
attribute [local instance] Classical.propDecidable
open ENNReal
structure ENorm (𝕜 : Type*) (V : Type*) [NormedField 𝕜] [Ad... | Mathlib/Analysis/NormedSpace/ENorm.lean | 120 | 124 | theorem map_sub_le (x y : V) : e (x - y) ≤ e x + e y :=
calc
e (x - y) = e (x + -y) := by | rw [sub_eq_add_neg]
_ ≤ e x + e (-y) := e.map_add_le x (-y)
_ = e x + e y := by rw [e.map_neg]
| 3 | 20.085537 | 1 | 1 | 5 | 1,154 |
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
universe u v w
namespace Ideal
open Set
variabl... | Mathlib/RingTheory/Ideal/Quotient.lean | 152 | 154 | theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by |
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
| 2 | 7.389056 | 1 | 0.333333 | 3 | 363 |
import Mathlib.MeasureTheory.Function.SimpleFunc
import Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable
#align_import measure_theory.function.simple_func_dense from "leanprover-community/mathlib"@"7317149f12f55affbc900fc873d0d422485122b9"
open Set Function Filter TopologicalSpace ENNReal EMetric Finset
... | Mathlib/MeasureTheory/Function/SimpleFuncDense.lean | 87 | 92 | theorem nearestPtInd_succ (e : ℕ → α) (N : ℕ) (x : α) :
nearestPtInd e (N + 1) x =
if ∀ k ≤ N, edist (e (N + 1)) x < edist (e k) x then N + 1 else nearestPtInd e N x := by |
simp only [nearestPtInd, coe_piecewise, Set.piecewise]
congr
simp
| 3 | 20.085537 | 1 | 1.8 | 5 | 1,885 |
import Mathlib.Algebra.Lie.OfAssociative
import Mathlib.LinearAlgebra.Matrix.Reindex
import Mathlib.LinearAlgebra.Matrix.ToLinearEquiv
#align_import algebra.lie.matrix from "leanprover-community/mathlib"@"55e2dfde0cff928ce5c70926a3f2c7dee3e2dd99"
universe u v w w₁ w₂
section Matrices
open scoped Matrix
variabl... | Mathlib/Algebra/Lie/Matrix.lean | 69 | 72 | theorem Matrix.lieConj_apply (P A : Matrix n n R) (h : Invertible P) :
P.lieConj h A = P * A * P⁻¹ := by |
simp [LinearEquiv.conj_apply, Matrix.lieConj, LinearMap.toMatrix'_comp,
LinearMap.toMatrix'_toLin']
| 2 | 7.389056 | 1 | 1 | 2 | 1,083 |
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 | 93 | 98 | theorem stoppedValue_stoppedValue_leastGE (f : ℕ → Ω → ℝ) (π : Ω → ℕ) (r : ℝ) {n : ℕ}
(hπn : ∀ ω, π ω ≤ n) : stoppedValue (fun i => stoppedValue f (leastGE f r i)) π =
stoppedValue (stoppedProcess f (leastGE f r n)) π := by |
ext1 ω
simp (config := { unfoldPartialApp := true }) only [stoppedProcess, stoppedValue]
rw [leastGE_eq_min _ _ _ hπn]
| 3 | 20.085537 | 1 | 1.75 | 4 | 1,870 |
import Mathlib.Algebra.Regular.Basic
import Mathlib.GroupTheory.GroupAction.Hom
#align_import algebra.regular.smul from "leanprover-community/mathlib"@"550b58538991c8977703fdeb7c9d51a5aa27df11"
variable {R S : Type*} (M : Type*) {a b : R} {s : S}
def IsSMulRegular [SMul R M] (c : R) :=
Function.Injective ((c ... | Mathlib/Algebra/Regular/SMul.lean | 102 | 105 | theorem of_mul [Mul R] [IsScalarTower R R M] (ab : IsSMulRegular M (a * b)) :
IsSMulRegular M b := by |
rw [← smul_eq_mul] at ab
exact ab.of_smul _
| 2 | 7.389056 | 1 | 1.25 | 4 | 1,318 |
import Mathlib.Data.ZMod.Basic
import Mathlib.RingTheory.Int.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
#align_import data.zmod.coprime from "leanprover-community/mathlib"@"4b4975cf92a1ffe2ddfeff6ff91b0c46a9162bf5"
namespace ZMod
| Mathlib/Data/ZMod/Coprime.lean | 24 | 28 | theorem eq_zero_iff_gcd_ne_one {a : ℤ} {p : ℕ} [pp : Fact p.Prime] :
(a : ZMod p) = 0 ↔ a.gcd p ≠ 1 := by |
rw [Ne, Int.gcd_comm, Int.gcd_eq_one_iff_coprime,
(Nat.prime_iff_prime_int.1 pp.1).coprime_iff_not_dvd, Classical.not_not,
intCast_zmod_eq_zero_iff_dvd]
| 3 | 20.085537 | 1 | 1 | 1 | 828 |
import Mathlib.FieldTheory.PrimitiveElement
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly
import Mathlib.LinearAlgebra.Matrix.ToLinearEquiv
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.G... | Mathlib/RingTheory/Norm.lean | 197 | 207 | theorem norm_eq_norm_adjoin [FiniteDimensional K L] [IsSeparable K L] (x : L) :
norm K x = norm K (AdjoinSimple.gen K x) ^ finrank K⟮x⟯ L := by |
letI := isSeparable_tower_top_of_isSeparable K K⟮x⟯ L
let pbL := Field.powerBasisOfFiniteOfSeparable K⟮x⟯ L
let pbx := IntermediateField.adjoin.powerBasis (IsSeparable.isIntegral K x)
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← AdjoinSimple.algebraMap_gen K x, norm_eq_matrix... | 9 | 8,103.083928 | 2 | 1.272727 | 11 | 1,347 |
import Mathlib.LinearAlgebra.Basis.VectorSpace
import Mathlib.LinearAlgebra.Dimension.Finite
import Mathlib.SetTheory.Cardinal.Subfield
import Mathlib.LinearAlgebra.Dimension.RankNullity
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomput... | Mathlib/LinearAlgebra/Dimension/DivisionRing.lean | 171 | 193 | theorem linearIndependent_iff_card_eq_finrank_span {ι : Type*} [Fintype ι] {b : ι → V} :
LinearIndependent K b ↔ Fintype.card ι = (Set.range b).finrank K := by |
constructor
· intro h
exact (finrank_span_eq_card h).symm
· intro hc
let f := Submodule.subtype (span K (Set.range b))
let b' : ι → span K (Set.range b) := fun i =>
⟨b i, mem_span.2 fun p hp => hp (Set.mem_range_self _)⟩
have hs : ⊤ ≤ span K (Set.range b') := by
intro x
have h :... | 21 | 1,318,815,734.483215 | 2 | 1.625 | 8 | 1,748 |
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Ideal.Operations
namespace Submodule
open Pointwise
variable {R M M' F G : Type*} [CommRing R] [AddCommGroup M] [Module R M]
variable {N N₁ N₂ P P₁ P₂ : Submodule R M}
def colon (N P : Submodule R M) : Ideal R :=
annihilator (P.map N.mkQ)
#align ... | Mathlib/RingTheory/Ideal/Colon.lean | 40 | 42 | theorem colon_top {I : Ideal R} : I.colon ⊤ = I := by |
simp_rw [SetLike.ext_iff, mem_colon, smul_eq_mul]
exact fun x ↦ ⟨fun h ↦ mul_one x ▸ h 1 trivial, fun h _ _ ↦ I.mul_mem_right _ h⟩
| 2 | 7.389056 | 1 | 0.5 | 6 | 471 |
import Batteries.Classes.Order
namespace Batteries.PairingHeapImp
inductive Heap (α : Type u) where
| nil : Heap α
| node (a : α) (child sibling : Heap α) : Heap α
deriving Repr
def Heap.size : Heap α → Nat
| .nil => 0
| .node _ c s => c.size + 1 + s.size
def Heap.singleton (a : α) : Heap α := .... | .lake/packages/batteries/Batteries/Data/PairingHeap.lean | 148 | 152 | theorem Heap.size_tail (le) {s : Heap α} (h : s.NoSibling) : (s.tail le).size = s.size - 1 := by |
simp only [Heap.tail]
match eq : s.tail? le with
| none => cases s with cases eq | nil => rfl
| some tl => simp [Heap.size_tail? h eq]
| 4 | 54.59815 | 2 | 1 | 13 | 899 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.iterated_deriv from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputable section
open scoped Classical Topology
open Filter Asymptotics Set
variable {𝕜... | Mathlib/Analysis/Calculus/IteratedDeriv/Defs.lean | 91 | 95 | theorem iteratedFDerivWithin_eq_equiv_comp :
iteratedFDerivWithin 𝕜 n f s =
ContinuousMultilinearMap.piFieldEquiv 𝕜 (Fin n) F ∘ iteratedDerivWithin n f s := by |
rw [iteratedDerivWithin_eq_equiv_comp, ← Function.comp.assoc, LinearIsometryEquiv.self_comp_symm,
Function.id_comp]
| 2 | 7.389056 | 1 | 0.727273 | 11 | 649 |
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.AddTorsor
#align_import analysis.convex.normed from "leanprover-community/mathlib"@"a63928c34ec358b5edcda2bf7513c50052... | Mathlib/Analysis/Convex/Normed.lean | 133 | 136 | theorem Wbtw.dist_add_dist {x y z : P} (h : Wbtw ℝ x y z) :
dist x y + dist y z = dist x z := by |
obtain ⟨a, ⟨ha₀, ha₁⟩, rfl⟩ := h
simp [abs_of_nonneg, ha₀, ha₁, sub_mul]
| 2 | 7.389056 | 1 | 0.818182 | 11 | 721 |
import Mathlib.Analysis.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
#align_import analysis.calculus.cont_diff_def from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
noncomputable section
open scoped Classical
open NNReal Topology Filter
local notatio... | Mathlib/Analysis/Calculus/ContDiff/Defs.lean | 223 | 226 | theorem HasFTaylorSeriesUpToOn.continuousOn (h : HasFTaylorSeriesUpToOn n f p s) :
ContinuousOn f s := by |
have := (h.cont 0 bot_le).congr fun x hx => (h.zero_eq' hx).symm
rwa [← (continuousMultilinearCurryFin0 𝕜 E F).symm.comp_continuousOn_iff]
| 2 | 7.389056 | 1 | 1.4 | 5 | 1,481 |
import Mathlib.Analysis.InnerProductSpace.Adjoint
#align_import analysis.inner_product_space.positive from "leanprover-community/mathlib"@"caa58cbf5bfb7f81ccbaca4e8b8ac4bc2b39cc1c"
open InnerProductSpace RCLike ContinuousLinearMap
open scoped InnerProduct ComplexConjugate
namespace ContinuousLinearMap
variable... | Mathlib/Analysis/InnerProductSpace/Positive.lean | 119 | 123 | theorem isPositive_iff_complex (T : E' →L[ℂ] E') :
IsPositive T ↔ ∀ x, (re ⟪T x, x⟫_ℂ : ℂ) = ⟪T x, x⟫_ℂ ∧ 0 ≤ re ⟪T x, x⟫_ℂ := by |
simp_rw [IsPositive, forall_and, isSelfAdjoint_iff_isSymmetric,
LinearMap.isSymmetric_iff_inner_map_self_real, conj_eq_iff_re]
rfl
| 3 | 20.085537 | 1 | 0.875 | 8 | 762 |
import Mathlib.LinearAlgebra.Matrix.Transvection
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.Tactic.FinCases
#align_import linear_algebra.matrix.block from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf97388c9e3b441fc4ccf3f"
open Finset Function OrderDual
open Matrix
universe v
v... | Mathlib/LinearAlgebra/Matrix/Block.lean | 63 | 69 | theorem blockTriangular_reindex_iff {b : n → α} {e : m ≃ n} :
(reindex e e M).BlockTriangular b ↔ M.BlockTriangular (b ∘ e) := by |
refine ⟨fun h => ?_, fun h => ?_⟩
· convert h.submatrix
simp only [reindex_apply, submatrix_submatrix, submatrix_id_id, Equiv.symm_comp_self]
· convert h.submatrix
simp only [comp.assoc b e e.symm, Equiv.self_comp_symm, comp_id]
| 5 | 148.413159 | 2 | 2 | 1 | 2,180 |
import Mathlib.Analysis.Normed.Group.Basic
#align_import information_theory.hamming from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
section HammingDistNorm
open Finset Function
variable {α ι : Type*} {β : ι → Type*} [Fintype ι] [∀ i, DecidableEq (β i)]
variable {γ : ι → Type*} [∀ ... | Mathlib/InformationTheory/Hamming.lean | 45 | 47 | theorem hammingDist_self (x : ∀ i, β i) : hammingDist x x = 0 := by |
rw [hammingDist, card_eq_zero, filter_eq_empty_iff]
exact fun _ _ H => H rfl
| 2 | 7.389056 | 1 | 0.7 | 10 | 642 |
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.Nat.Digits
import Mathlib.Data.Nat.MaxPowDiv
import Mathlib.Data.Nat.Multiplicity
import Mathlib.Tactic.IntervalCases
#align_import number_theory.padics.padic_val from "leanprover-community/mathlib"@"60fa54e778c9e85d930efae172435f42fb0d71f7"
universe u
ope... | Mathlib/NumberTheory/Padics/PadicVal.lean | 126 | 129 | theorem maxPowDiv_eq_multiplicity_get {p n : ℕ} (hp : 1 < p) (hn : 0 < n) (h : Finite p n) :
p.maxPowDiv n = (multiplicity p n).get h := by |
rw [PartENat.get_eq_iff_eq_coe.mpr]
apply maxPowDiv_eq_multiplicity hp hn|>.symm
| 2 | 7.389056 | 1 | 1.333333 | 6 | 1,377 |
import Mathlib.Algebra.Module.Torsion
import Mathlib.SetTheory.Cardinal.Cofinality
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
... | Mathlib/LinearAlgebra/Dimension/Finite.lean | 34 | 40 | theorem rank_le {n : ℕ}
(H : ∀ s : Finset M, (LinearIndependent R fun i : s => (i : M)) → s.card ≤ n) :
Module.rank R M ≤ n := by |
rw [Module.rank_def]
apply ciSup_le'
rintro ⟨s, li⟩
exact linearIndependent_bounded_of_finset_linearIndependent_bounded H _ li
| 4 | 54.59815 | 2 | 1.5 | 4 | 1,598 |
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.Seminorm
import Mathlib.Analysis.LocallyConvex.Bounded
import Mathlib.Analysis.RCLike.Basic
#align_import analysis.convex.gauge from "leanprover-community/mathlib"@"373b03b5b9d0486534edbe94747f23cb3712f93d"
... | Mathlib/Analysis/Convex/Gauge.lean | 119 | 121 | theorem gauge_of_subset_zero (h : s ⊆ 0) : gauge s = 0 := by |
obtain rfl | rfl := subset_singleton_iff_eq.1 h
exacts [gauge_empty, gauge_zero']
| 2 | 7.389056 | 1 | 1.090909 | 11 | 1,185 |
import Mathlib.Analysis.SpecialFunctions.ImproperIntegrals
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.MeasureTheory.Measure.Haar.NormedSpace
#align_import analysis.mellin_transform from "leanprover-community/mathlib"@"917c3c072e487b3cccdbfeff17e75b40e45f66cb"
open MeasureTheory Set Filter A... | Mathlib/Analysis/MellinTransform.lean | 210 | 231 | theorem mellin_convergent_top_of_isBigO {f : ℝ → ℝ}
(hfc : AEStronglyMeasurable f <| volume.restrict (Ioi 0)) {a s : ℝ}
(hf : f =O[atTop] (· ^ (-a))) (hs : s < a) :
∃ c : ℝ, 0 < c ∧ IntegrableOn (fun t : ℝ => t ^ (s - 1) * f t) (Ioi c) := by |
obtain ⟨d, hd'⟩ := hf.isBigOWith
simp_rw [IsBigOWith, eventually_atTop] at hd'
obtain ⟨e, he⟩ := hd'
have he' : 0 < max e 1 := zero_lt_one.trans_le (le_max_right _ _)
refine ⟨max e 1, he', ?_, ?_⟩
· refine AEStronglyMeasurable.mul ?_ (hfc.mono_set (Ioi_subset_Ioi he'.le))
refine (ContinuousAt.continuou... | 18 | 65,659,969.137331 | 2 | 1.333333 | 12 | 1,415 |
import Mathlib.Algebra.Algebra.Unitization
import Mathlib.Analysis.NormedSpace.OperatorNorm.Mul
suppress_compilation
variable (𝕜 A : Type*) [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A]
variable [NormedSpace 𝕜 A] [IsScalarTower 𝕜 A A] [SMulCommClass 𝕜 A A]
open ContinuousLinearMap
namespace Unitizati... | Mathlib/Analysis/NormedSpace/Unitization.lean | 167 | 180 | theorem antilipschitzWith_addEquiv :
AntilipschitzWith 2 (addEquiv 𝕜 A) := by |
refine AddMonoidHomClass.antilipschitz_of_bound (addEquiv 𝕜 A) fun x => ?_
rw [norm_eq_sup, Prod.norm_def, NNReal.coe_two]
refine max_le ?_ ?_
· rw [mul_max_of_nonneg _ _ (zero_le_two : (0 : ℝ) ≤ 2)]
exact le_max_of_le_left ((le_add_of_nonneg_left (norm_nonneg _)).trans_eq (two_mul _).symm)
· nontrivial... | 12 | 162,754.791419 | 2 | 1.6 | 5 | 1,728 |
import Mathlib.Algebra.Module.Equiv
import Mathlib.Data.DFinsupp.Basic
import Mathlib.Data.Finsupp.Basic
#align_import data.finsupp.to_dfinsupp from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
variable {ι : Type*} {R : Type*} {M : Type*}
section Defs
def Finsupp.toDFinsupp [Zer... | Mathlib/Data/Finsupp/ToDFinsupp.lean | 123 | 126 | theorem DFinsupp.toFinsupp_single (i : ι) (m : M) :
(DFinsupp.single i m : Π₀ _ : ι, M).toFinsupp = Finsupp.single i m := by |
ext
simp [Finsupp.single_apply, DFinsupp.single_apply]
| 2 | 7.389056 | 1 | 1 | 4 | 1,019 |
import Mathlib.LinearAlgebra.CliffordAlgebra.Contraction
variable {R M : Type*}
variable [CommRing R] [AddCommGroup M] [Module R M] {Q : QuadraticForm R M}
namespace CliffordAlgebra
variable (Q)
def invertibleιOfInvertible (m : M) [Invertible (Q m)] : Invertible (ι Q m) where
invOf := ι Q (⅟ (Q m) • m)
invO... | Mathlib/LinearAlgebra/CliffordAlgebra/Inversion.lean | 31 | 34 | theorem invOf_ι (m : M) [Invertible (Q m)] [Invertible (ι Q m)] :
⅟ (ι Q m) = ι Q (⅟ (Q m) • m) := by |
letI := invertibleιOfInvertible Q m
convert (rfl : ⅟ (ι Q m) = _)
| 2 | 7.389056 | 1 | 1 | 5 | 870 |
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.Calculus.FDeriv.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
open Topology InnerProductSpace Set
noncomputable section
variable {𝕜 F : Type*} [RCLike 𝕜]
variable [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [CompleteSpace F]
variabl... | Mathlib/Analysis/Calculus/Gradient/Basic.lean | 138 | 140 | theorem hasGradientWithinAt_univ : HasGradientWithinAt f f' univ x ↔ HasGradientAt f f' x := by |
rw [hasGradientWithinAt_iff_hasFDerivWithinAt, hasGradientAt_iff_hasFDerivAt]
exact hasFDerivWithinAt_univ
| 2 | 7.389056 | 1 | 0.538462 | 13 | 511 |
import Mathlib.CategoryTheory.Category.Grpd
import Mathlib.CategoryTheory.Groupoid
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.Topology.Homotopy.Path
import Mathlib.Data.Set.Subsingleton
#align_import algebraic_topology.fundamental_groupoid.basic from "leanprover-community/mathlib"@"3d7987cda72abc473... | Mathlib/AlgebraicTopology/FundamentalGroupoid/Basic.lean | 138 | 140 | theorem transReflReparamAux_mem_I (t : I) : transReflReparamAux t ∈ I := by |
unfold transReflReparamAux
split_ifs <;> constructor <;> linarith [unitInterval.le_one t, unitInterval.nonneg t]
| 2 | 7.389056 | 1 | 1.166667 | 12 | 1,229 |
import Mathlib.Algebra.Associated
import Mathlib.Algebra.Order.Monoid.Unbundled.Pow
import Mathlib.Algebra.Ring.Int
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Order.Bounds.Basic
#align_import data.nat.prime from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82... | Mathlib/Data/Nat/Prime.lean | 147 | 153 | theorem prime_of_coprime (n : ℕ) (h1 : 1 < n) (h : ∀ m < n, m ≠ 0 → n.Coprime m) : Prime n := by |
refine prime_def_lt.mpr ⟨h1, fun m mlt mdvd => ?_⟩
have hm : m ≠ 0 := by
rintro rfl
rw [zero_dvd_iff] at mdvd
exact mlt.ne' mdvd
exact (h m mlt hm).symm.eq_one_of_dvd mdvd
| 6 | 403.428793 | 2 | 2 | 3 | 2,433 |
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts
import Mathlib.Logic.Equiv.Fin
#align_import category_theory.limits.... | Mathlib/CategoryTheory/Limits/Constructions/FiniteProductsOfBinaryProducts.lean | 106 | 110 | theorem hasFiniteProducts_of_has_binary_and_terminal : HasFiniteProducts C := by |
refine ⟨fun n => ⟨fun K => ?_⟩⟩
letI := hasProduct_fin n fun n => K.obj ⟨n⟩
let that : (Discrete.functor fun n => K.obj ⟨n⟩) ≅ K := Discrete.natIso fun ⟨i⟩ => Iso.refl _
apply @hasLimitOfIso _ _ _ _ _ _ this that
| 4 | 54.59815 | 2 | 2 | 1 | 2,045 |
import Mathlib.RingTheory.WittVector.Frobenius
import Mathlib.RingTheory.WittVector.Verschiebung
import Mathlib.RingTheory.WittVector.MulP
#align_import ring_theory.witt_vector.identities from "leanprover-community/mathlib"@"0798037604b2d91748f9b43925fb7570a5f3256c"
namespace WittVector
variable {p : ℕ} {R : Typ... | Mathlib/RingTheory/WittVector/Identities.lean | 74 | 77 | theorem coeff_p [CharP R p] (i : ℕ) : (p : 𝕎 R).coeff i = if i = 1 then 1 else 0 := by |
split_ifs with hi
· simpa only [hi, pow_one] using coeff_p_pow p R 1
· simpa only [pow_one] using coeff_p_pow_eq_zero p R hi
| 3 | 20.085537 | 1 | 1 | 13 | 1,103 |
import Batteries.Data.Fin.Basic
namespace Fin
attribute [norm_cast] val_last
protected theorem le_antisymm_iff {x y : Fin n} : x = y ↔ x ≤ y ∧ y ≤ x :=
Fin.ext_iff.trans Nat.le_antisymm_iff
protected theorem le_antisymm {x y : Fin n} (h1 : x ≤ y) (h2 : y ≤ x) : x = y :=
Fin.le_antisymm_iff.2 ⟨h1, h2⟩
@[simp... | .lake/packages/batteries/Batteries/Data/Fin/Lemmas.lean | 87 | 90 | theorem foldl_eq_foldl_list (f : α → Fin n → α) (x) : foldl n f x = (list n).foldl f x := by |
induction n generalizing x with
| zero => rw [foldl_zero, list_zero, List.foldl_nil]
| succ n ih => rw [foldl_succ, ih, list_succ, List.foldl_cons, List.foldl_map]
| 3 | 20.085537 | 1 | 1.090909 | 11 | 1,186 |
import Mathlib.Algebra.IsPrimePow
import Mathlib.Data.Nat.Factorization.Basic
#align_import data.nat.factorization.prime_pow from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf97388c9e3b441fc4ccf3f"
variable {R : Type*} [CommMonoidWithZero R] (n p : R) (k : ℕ)
theorem IsPrimePow.minFac_pow_factorization_eq ... | Mathlib/Data/Nat/Factorization/PrimePow.lean | 41 | 54 | theorem isPrimePow_iff_factorization_eq_single {n : ℕ} :
IsPrimePow n ↔ ∃ p k : ℕ, 0 < k ∧ n.factorization = Finsupp.single p k := by |
rw [isPrimePow_nat_iff]
refine exists₂_congr fun p k => ?_
constructor
· rintro ⟨hp, hk, hn⟩
exact ⟨hk, by rw [← hn, Nat.Prime.factorization_pow hp]⟩
· rintro ⟨hk, hn⟩
have hn0 : n ≠ 0 := by
rintro rfl
simp_all only [Finsupp.single_eq_zero, eq_comm, Nat.factorization_zero, hk.ne']
rw ... | 12 | 162,754.791419 | 2 | 1.714286 | 7 | 1,840 |
import Mathlib.LinearAlgebra.Matrix.ToLin
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.RingTheory.Nilpotent.Defs
#align_import ring_theory.nilpotent from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
universe u v
open Function Set
variable ... | Mathlib/RingTheory/Nilpotent/Lemmas.lean | 123 | 126 | theorem IsNilpotent.mapQ (hnp : IsNilpotent f) : IsNilpotent (p.mapQ p f hp) := by |
obtain ⟨k, hk⟩ := hnp
use k
simp [← p.mapQ_pow, hk]
| 3 | 20.085537 | 1 | 1 | 3 | 985 |
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
| Mathlib/RingTheory/Int/Basic.lean | 33 | 46 | theorem gcd_eq_one_iff_coprime {a b : ℤ} : Int.gcd a b = 1 ↔ IsCoprime a b := by |
constructor
· intro hg
obtain ⟨ua, -, ha⟩ := exists_unit_of_abs a
obtain ⟨ub, -, hb⟩ := exists_unit_of_abs b
use Nat.gcdA (Int.natAbs a) (Int.natAbs b) * ua, Nat.gcdB (Int.natAbs a) (Int.natAbs b) * ub
rw [mul_assoc, ← ha, mul_assoc, ← hb, mul_comm, mul_comm _ (Int.natAbs b : ℤ), ←
Nat.gcd_eq... | 13 | 442,413.392009 | 2 | 1.153846 | 13 | 1,227 |
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.Data.ZMod.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
#align_import ring_theory.zmod from "leanprover-community/mathlib"@"00d163e35035c3577c1c79fa53b68de17781ffc1"
theorem ZMod.ker_intCastRingHom (n : ℕ) :
RingHom.ker (Int.castRingHom (ZMod n)) =... | Mathlib/RingTheory/ZMod.lean | 42 | 46 | theorem isReduced_zmod {n : ℕ} : IsReduced (ZMod n) ↔ Squarefree n ∨ n = 0 := by |
rw [← RingHom.ker_isRadical_iff_reduced_of_surjective
(ZMod.ringHom_surjective <| Int.castRingHom <| ZMod n),
ZMod.ker_intCastRingHom, ← isRadical_iff_span_singleton, isRadical_iff_squarefree_or_zero,
Int.squarefree_natCast, Nat.cast_eq_zero]
| 4 | 54.59815 | 2 | 1.333333 | 3 | 1,419 |
import Mathlib.ModelTheory.Satisfiability
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import model_theory.graph from "leanprover-community/mathlib"@"e56b8fea84d60fe434632b9d3b829ee685fb0c8f"
set_option linter.uppercaseLean3 false
universe u v w w'
namespace FirstOrder
namespace Language
open FirstOr... | Mathlib/ModelTheory/Graph.lean | 107 | 110 | theorem _root_.SimpleGraph.simpleGraphOfStructure (G : SimpleGraph V) :
@simpleGraphOfStructure V G.structure _ = G := by |
ext
rfl
| 2 | 7.389056 | 1 | 0.5 | 2 | 464 |
import Mathlib.Analysis.NormedSpace.ConformalLinearMap
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.conformal.normed_space from "leanprover-community/mathlib"@"e3fb84046afd187b710170887195d50bada934ee"
noncomputable section
variable {X Y Z : Type*} [NormedAddCommGroup X] [NormedAdd... | Mathlib/Analysis/Calculus/Conformal/NormedSpace.lean | 73 | 82 | theorem conformalAt_iff_isConformalMap_fderiv {f : X → Y} {x : X} :
ConformalAt f x ↔ IsConformalMap (fderiv ℝ f x) := by |
constructor
· rintro ⟨f', hf, hf'⟩
rwa [hf.fderiv]
· intro H
by_cases h : DifferentiableAt ℝ f x
· exact ⟨fderiv ℝ f x, h.hasFDerivAt, H⟩
· nontriviality X
exact absurd (fderiv_zero_of_not_differentiableAt h) H.ne_zero
| 8 | 2,980.957987 | 2 | 1.5 | 2 | 1,611 |
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex
#align_import analysis.special_functions.trigonometric.complex_deriv from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
noncomputable section
namespace Complex
open Set Filter
open scoped Real
| Mathlib/Analysis/SpecialFunctions/Trigonometric/ComplexDeriv.lean | 25 | 28 | theorem hasStrictDerivAt_tan {x : ℂ} (h : cos x ≠ 0) : HasStrictDerivAt tan (1 / cos x ^ 2) x := by |
convert (hasStrictDerivAt_sin x).div (hasStrictDerivAt_cos x) h using 1
rw_mod_cast [← sin_sq_add_cos_sq x]
ring
| 3 | 20.085537 | 1 | 1.333333 | 3 | 1,406 |
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.Deriv.Inv
#align_import analysis.calculus.deriv.zpow from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
open scoped Classical
open Topology Filter
open Filter Asymptotics Set
variable {𝕜 : Typ... | Mathlib/Analysis/Calculus/Deriv/ZPow.lean | 106 | 113 | theorem iter_deriv_zpow' (m : ℤ) (k : ℕ) :
(deriv^[k] fun x : 𝕜 => x ^ m) =
fun x => (∏ i ∈ Finset.range k, ((m : 𝕜) - i)) * x ^ (m - k) := by |
induction' k with k ihk
· simp only [Nat.zero_eq, one_mul, Int.ofNat_zero, id, sub_zero, Finset.prod_range_zero,
Function.iterate_zero]
· simp only [Function.iterate_succ_apply', ihk, deriv_const_mul_field', deriv_zpow',
Finset.prod_range_succ, Int.ofNat_succ, ← sub_sub, Int.cast_sub, Int.cast_natCas... | 5 | 148.413159 | 2 | 2 | 4 | 1,973 |
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Analytic.CPolynomial
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.fderiv_analytic from "leanprover-community/mathlib"@"3bce8d800a6f2... | Mathlib/Analysis/Calculus/FDeriv/Analytic.lean | 449 | 458 | theorem derivSeries_apply_diag (n : ℕ) (x : E) :
derivSeries p n (fun _ ↦ x) x = (n + 1) • p (n + 1) fun _ ↦ x := by |
simp only [derivSeries, compFormalMultilinearSeries_apply, changeOriginSeries,
compContinuousMultilinearMap_coe, ContinuousLinearEquiv.coe_coe, LinearIsometryEquiv.coe_coe,
Function.comp_apply, ContinuousMultilinearMap.sum_apply, map_sum, coe_sum', Finset.sum_apply,
continuousMultilinearCurryFin1_apply, ... | 8 | 2,980.957987 | 2 | 1.857143 | 7 | 1,925 |
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Order.Monotone.Basic
#align_import data.nat.choose.basic from "leanprover-community/mathlib"@"2f3994e1b117b1e1da49bcfb67334f33460c3ce4"
open Nat
namespace Nat
def choose : ℕ → ℕ → ℕ
| _, 0 => 1
| 0, _ + 1 => 0
| n + 1, k + 1 => choose n k + choose n ... | Mathlib/Data/Nat/Choose/Basic.lean | 93 | 95 | theorem triangle_succ (n : ℕ) : (n + 1) * (n + 1 - 1) / 2 = n * (n - 1) / 2 + n := by |
rw [← add_mul_div_left, Nat.mul_comm 2 n, ← Nat.mul_add, Nat.add_sub_cancel, Nat.mul_comm]
cases n <;> rfl; apply zero_lt_succ
| 2 | 7.389056 | 1 | 1 | 5 | 972 |
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.CharP.Algebra
import Mathlib.Data.Nat.Prime
#align_import algebra.char_p.exp_char from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u
variable (R : Type u)
section Semiring
variable [Semiring R]
class inductive Ex... | Mathlib/Algebra/CharP/ExpChar.lean | 133 | 136 | theorem char_prime_of_ne_zero {p : ℕ} [hp : CharP R p] (p_ne_zero : p ≠ 0) : Nat.Prime p := by |
cases' CharP.char_is_prime_or_zero R p with h h
· exact h
· contradiction
| 3 | 20.085537 | 1 | 1.272727 | 11 | 1,345 |
import Mathlib.Topology.Algebra.Module.WeakDual
import Mathlib.Analysis.Normed.Field.Basic
import Mathlib.Analysis.LocallyConvex.WithSeminorms
#align_import analysis.locally_convex.weak_dual from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
variable {𝕜 E F ι : Type*}
open Topology
... | Mathlib/Analysis/LocallyConvex/WeakDual.lean | 73 | 76 | theorem toSeminorm_comp (f : F →ₗ[𝕜] 𝕜) (g : E →ₗ[𝕜] F) :
f.toSeminorm.comp g = (f.comp g).toSeminorm := by |
ext
simp only [Seminorm.comp_apply, toSeminorm_apply, coe_comp, Function.comp_apply]
| 2 | 7.389056 | 1 | 0.5 | 2 | 443 |
import Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.measure.haar.normed_space from "leanprover-community/mathlib"@"b84aee748341da06a6d78491367e2c0e9f15e8a5"
noncomputable sect... | Mathlib/MeasureTheory/Measure/Haar/NormedSpace.lean | 163 | 177 | theorem integrable_comp_smul_iff {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
[MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : Measure E) [IsAddHaarMeasure μ]
(f : E → F) {R : ℝ} (hR : R ≠ 0) : Integrable (fun x => f (R • x)) μ ↔ Integrable f μ := by |
-- reduce to one-way implication
suffices
∀ {g : E → F} (_ : Integrable g μ) {S : ℝ} (_ : S ≠ 0), Integrable (fun x => g (S • x)) μ by
refine ⟨fun hf => ?_, fun hf => this hf hR⟩
convert this hf (inv_ne_zero hR)
rw [← mul_smul, mul_inv_cancel hR, one_smul]
-- now prove
intro g hg S hS
let t :... | 12 | 162,754.791419 | 2 | 0.75 | 8 | 658 |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
#align_import ring_theory.coprime.lemmas from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
universe u v
section RelPrime
variable {α I} [Comm... | Mathlib/RingTheory/Coprime/Lemmas.lean | 281 | 289 | theorem pairwise_isRelPrime_iff_isRelPrime_prod [DecidableEq I] :
Pairwise (IsRelPrime on fun i : t ↦ s i) ↔ ∀ i ∈ t, IsRelPrime (s i) (∏ j ∈ t \ {i}, s j) := by |
refine ⟨fun hp i hi ↦ IsRelPrime.prod_right_iff.mpr fun j hj ↦ ?_, fun hp ↦ ?_⟩
· rw [Finset.mem_sdiff, Finset.mem_singleton] at hj
obtain ⟨hj, ji⟩ := hj
exact @hp ⟨i, hi⟩ ⟨j, hj⟩ fun h ↦ ji (congrArg Subtype.val h).symm
· rintro ⟨i, hi⟩ ⟨j, hj⟩ h
apply IsRelPrime.prod_right_iff.mp (hp i hi)
exac... | 7 | 1,096.633158 | 2 | 1.111111 | 18 | 1,195 |
import Mathlib.Algebra.Polynomial.Module.AEval
#align_import data.polynomial.module from "leanprover-community/mathlib"@"63417e01fbc711beaf25fa73b6edb395c0cfddd0"
universe u v
open Polynomial BigOperators
@[nolint unusedArguments]
def PolynomialModule (R M : Type*) [CommRing R] [AddCommGroup M] [Module R M] := ℕ ... | Mathlib/Algebra/Polynomial/Module/Basic.lean | 123 | 135 | theorem monomial_smul_single (i : ℕ) (r : R) (j : ℕ) (m : M) :
monomial i r • single R j m = single R (i + j) (r • m) := by |
simp only [LinearMap.mul_apply, Polynomial.aeval_monomial, LinearMap.pow_apply,
Module.algebraMap_end_apply, smul_def]
induction i generalizing r j m with
| zero =>
rw [Function.iterate_zero, zero_add]
exact Finsupp.smul_single r j m
| succ n hn =>
rw [Function.iterate_succ, Function.comp_apply... | 11 | 59,874.141715 | 2 | 2 | 3 | 2,085 |
import Mathlib.MeasureTheory.Integral.IntervalIntegral
import Mathlib.Analysis.Calculus.Deriv.ZPow
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.SpecialFunctions.NonIntegrable
import Mathlib.Analysis.Analytic.Basic
#align_import measure_theory.integral.circle_integral from "leanprover-communit... | Mathlib/MeasureTheory/Integral/CircleIntegral.lean | 141 | 148 | theorem range_circleMap (c : ℂ) (R : ℝ) : range (circleMap c R) = sphere c |R| :=
calc
range (circleMap c R) = c +ᵥ R • range fun θ : ℝ => exp (θ * I) := by |
simp (config := { unfoldPartialApp := true }) only [← image_vadd, ← image_smul, ← range_comp,
vadd_eq_add, circleMap, Function.comp_def, real_smul]
_ = sphere c |R| := by
rw [Complex.range_exp_mul_I, smul_sphere R 0 zero_le_one]
simp
| 5 | 148.413159 | 2 | 0.333333 | 9 | 352 |
import Mathlib.Data.Finset.Sort
import Mathlib.Data.List.FinRange
import Mathlib.Data.Prod.Lex
import Mathlib.GroupTheory.Perm.Basic
import Mathlib.Order.Interval.Finset.Fin
#align_import data.fin.tuple.sort from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
namespace Tuple
variable {... | Mathlib/Data/Fin/Tuple/Sort.lean | 99 | 102 | theorem monotone_proj (f : Fin n → α) : Monotone (graph.proj : graph f → α) := by |
rintro ⟨⟨x, i⟩, hx⟩ ⟨⟨y, j⟩, hy⟩ (_ | h)
· exact le_of_lt ‹_›
· simp [graph.proj]
| 3 | 20.085537 | 1 | 1.5 | 4 | 1,661 |
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Order.Module.Pointwise
import Mathlib.Data.Real.Archimedean
#align_import data.real.pointwise from "leanprover-community/mathlib"@"dde670c9a3f503647fd5bfdf1037bad526d3397a"
open Set
open Pointwise
variable {ι : Sort*} {α : Type*} [LinearOrde... | Mathlib/Data/Real/Pointwise.lean | 37 | 46 | theorem Real.sInf_smul_of_nonneg (ha : 0 ≤ a) (s : Set ℝ) : sInf (a • s) = a • sInf s := by |
obtain rfl | hs := s.eq_empty_or_nonempty
· rw [smul_set_empty, Real.sInf_empty, smul_zero]
obtain rfl | ha' := ha.eq_or_lt
· rw [zero_smul_set hs, zero_smul]
exact csInf_singleton 0
by_cases h : BddBelow s
· exact ((OrderIso.smulRight ha').map_csInf' hs h).symm
· rw [Real.sInf_of_not_bddBelow (mt (b... | 9 | 8,103.083928 | 2 | 2 | 4 | 2,412 |
import Mathlib.Data.Nat.Count
import Mathlib.Data.Nat.SuccPred
import Mathlib.Order.Interval.Set.Monotone
import Mathlib.Order.OrderIsoNat
#align_import data.nat.nth from "leanprover-community/mathlib"@"7fdd4f3746cb059edfdb5d52cba98f66fce418c0"
open Finset
namespace Nat
variable (p : ℕ → Prop)
noncomputable d... | Mathlib/Data/Nat/Nth.lean | 76 | 80 | theorem nth_strictMonoOn (hf : (setOf p).Finite) :
StrictMonoOn (nth p) (Set.Iio hf.toFinset.card) := by |
rintro m (hm : m < _) n (hn : n < _) h
simp only [nth_eq_orderEmbOfFin, *]
exact OrderEmbedding.strictMono _ h
| 3 | 20.085537 | 1 | 0.5 | 6 | 435 |
import Mathlib.Data.Set.Image
import Mathlib.Order.SuccPred.Relation
import Mathlib.Topology.Clopen
import Mathlib.Topology.Irreducible
#align_import topology.connected from "leanprover-community/mathlib"@"d101e93197bb5f6ea89bd7ba386b7f7dff1f3903"
open Set Function Topology TopologicalSpace Relation
open scoped C... | Mathlib/Topology/Connected/Basic.lean | 124 | 128 | theorem isPreconnected_sUnion (x : α) (c : Set (Set α)) (H1 : ∀ s ∈ c, x ∈ s)
(H2 : ∀ s ∈ c, IsPreconnected s) : IsPreconnected (⋃₀ c) := by |
apply isPreconnected_of_forall x
rintro y ⟨s, sc, ys⟩
exact ⟨s, subset_sUnion_of_mem sc, H1 s sc, ys, H2 s sc⟩
| 3 | 20.085537 | 1 | 1.4 | 5 | 1,507 |
import Mathlib.Topology.Sets.Closeds
import Mathlib.Topology.QuasiSeparated
#align_import topology.sets.compacts from "leanprover-community/mathlib"@"8c1b484d6a214e059531e22f1be9898ed6c1fd47"
open Set
variable {α β γ : Type*} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ]
namespace TopologicalSp... | Mathlib/Topology/Sets/Compacts.lean | 125 | 129 | theorem coe_finset_sup {ι : Type*} {s : Finset ι} {f : ι → Compacts α} :
(↑(s.sup f) : Set α) = s.sup fun i => ↑(f i) := by |
refine Finset.cons_induction_on s rfl fun a s _ h => ?_
simp_rw [Finset.sup_cons, coe_sup, sup_eq_union]
congr
| 3 | 20.085537 | 1 | 1 | 1 | 853 |
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 | 58 | 62 | theorem List.untrop_prod [AddMonoid R] (l : List (Tropical R)) :
untrop l.prod = List.sum (l.map untrop) := by |
induction' l with hd tl IH
· simp
· simp [← IH]
| 3 | 20.085537 | 1 | 0.928571 | 14 | 793 |
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Dynamics.BirkhoffSum.NormedSpace
open Filter Finset Function Bornology
open scoped Topology
variable {𝕜 E : Type*} [RCLike 𝕜] [NormedAddCommGroup E]
theorem LinearMap.tendsto_birkhoffAverage_of_ker_subset_closure [NormedSpace 𝕜 E]
(f : E ... | Mathlib/Analysis/InnerProductSpace/MeanErgodic.lean | 84 | 103 | theorem ContinuousLinearMap.tendsto_birkhoffAverage_orthogonalProjection (f : E →L[𝕜] E)
(hf : ‖f‖ ≤ 1) (x : E) :
Tendsto (birkhoffAverage 𝕜 f _root_.id · x) atTop
(𝓝 <| orthogonalProjection (LinearMap.eqLocus f 1) x) := by |
/- Due to the previous theorem, it suffices to verify
that the range of `f - 1` is dense in the orthogonal complement
to the submodule of fixed points of `f`. -/
apply (f : E →ₗ[𝕜] E).tendsto_birkhoffAverage_of_ker_subset_closure (f.lipschitz.weaken hf)
· exact orthogonalProjection_mem_subspace_eq_self (K :... | 16 | 8,886,110.520508 | 2 | 2 | 2 | 2,172 |
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.Algebra.Module.Basic
#align_import topology.algebra.continuous_affine_map from "leanprover-community/mathlib"@"bd1fc183335ea95a9519a1630bcf901fe9326d83"
structure ContinuousAffineMap (R : T... | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | 127 | 129 | theorem mk_coe (f : P →ᴬ[R] Q) (h) : (⟨(f : P →ᵃ[R] Q), h⟩ : P →ᴬ[R] Q) = f := by |
ext
rfl
| 2 | 7.389056 | 1 | 1 | 3 | 1,146 |
import Mathlib.Algebra.Group.Equiv.Basic
import Mathlib.Algebra.Group.Aut
import Mathlib.Data.ZMod.Defs
import Mathlib.Tactic.Ring
#align_import algebra.quandle from "leanprover-community/mathlib"@"28aa996fc6fb4317f0083c4e6daf79878d81be33"
open MulOpposite
universe u v
class Shelf (α : Type u) where
act : ... | Mathlib/Algebra/Quandle.lean | 287 | 289 | theorem self_invAct_invAct_eq {x y : R} : (x ◃⁻¹ x) ◃⁻¹ y = x ◃⁻¹ y := by |
have h := @self_act_act_eq _ _ (op x) (op y)
simpa using h
| 2 | 7.389056 | 1 | 1.285714 | 7 | 1,357 |
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 {α E G: Type*}
[NormedAddCommGroup E] [NormedSpace ℝ E] [C... | Mathlib/MeasureTheory/Integral/DominatedConvergence.lean | 107 | 137 | theorem integral_tsum {ι} [Countable ι] {f : ι → α → G} (hf : ∀ i, AEStronglyMeasurable (f i) μ)
(hf' : ∑' i, ∫⁻ a : α, ‖f i a‖₊ ∂μ ≠ ∞) :
∫ a : α, ∑' i, f i a ∂μ = ∑' i, ∫ a : α, f i a ∂μ := by |
by_cases hG : CompleteSpace G; swap
· simp [integral, hG]
have hf'' : ∀ i, AEMeasurable (fun x => (‖f i x‖₊ : ℝ≥0∞)) μ := fun i => (hf i).ennnorm
have hhh : ∀ᵐ a : α ∂μ, Summable fun n => (‖f n a‖₊ : ℝ) := by
rw [← lintegral_tsum hf''] at hf'
refine (ae_lt_top' (AEMeasurable.ennreal_tsum hf'') hf').mon... | 28 | 1,446,257,064,291.475 | 2 | 2 | 4 | 2,302 |
import Mathlib.MeasureTheory.Constructions.Pi
import Mathlib.MeasureTheory.Constructions.Prod.Integral
open Fintype MeasureTheory MeasureTheory.Measure
variable {𝕜 : Type*} [RCLike 𝕜]
namespace MeasureTheory
theorem Integrable.fin_nat_prod {n : ℕ} {E : Fin n → Type*}
[∀ i, MeasureSpace (E i)] [∀ i, SigmaF... | Mathlib/MeasureTheory/Integral/Pi.lean | 65 | 84 | theorem integral_fin_nat_prod_eq_prod {n : ℕ} {E : Fin n → Type*}
[∀ i, MeasureSpace (E i)] [∀ i, SigmaFinite (volume : Measure (E i))]
(f : (i : Fin n) → E i → 𝕜) :
∫ x : (i : Fin n) → E i, ∏ i, f i (x i) = ∏ i, ∫ x, f i x := by |
induction n with
| zero =>
simp only [Nat.zero_eq, volume_pi, Finset.univ_eq_empty, Finset.prod_empty, integral_const,
pi_empty_univ, ENNReal.one_toReal, smul_eq_mul, mul_one, pow_zero, one_smul]
| succ n n_ih =>
calc
_ = ∫ x : E 0 × ((i : Fin n) → E (Fin.succ i)),
f 0 x.1... | 16 | 8,886,110.520508 | 2 | 1.6 | 5 | 1,745 |
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 | 122 | 126 | theorem logEmbedding_component_le {r : ℝ} {x : (𝓞 K)ˣ} (hr : 0 ≤ r) (h : ‖logEmbedding K x‖ ≤ r)
(w : {w : InfinitePlace K // w ≠ w₀}) : |logEmbedding K x w| ≤ r := by |
lift r to NNReal using hr
simp_rw [Pi.norm_def, NNReal.coe_le_coe, Finset.sup_le_iff, ← NNReal.coe_le_coe] at h
exact h w (mem_univ _)
| 3 | 20.085537 | 1 | 1.833333 | 6 | 1,909 |
import Mathlib.Data.Nat.Cast.WithTop
import Mathlib.FieldTheory.IsAlgClosed.Basic
import Mathlib.RingTheory.WittVector.DiscreteValuationRing
#align_import ring_theory.witt_vector.frobenius_fraction_field from "leanprover-community/mathlib"@"cead93130da7100f8a9fe22ee210f7636a91168f"
noncomputable section
namespac... | Mathlib/RingTheory/WittVector/FrobeniusFractionField.lean | 79 | 95 | theorem succNthDefiningPoly_degree [IsDomain k] (n : ℕ) (a₁ a₂ : 𝕎 k) (bs : Fin (n + 1) → k)
(ha₁ : a₁.coeff 0 ≠ 0) (ha₂ : a₂.coeff 0 ≠ 0) :
(succNthDefiningPoly p n a₁ a₂ bs).degree = p := by |
have : (X ^ p * C (a₁.coeff 0 ^ p ^ (n + 1))).degree = (p : WithBot ℕ) := by
rw [degree_mul, degree_C]
· simp only [Nat.cast_withBot, add_zero, degree_X, degree_pow, Nat.smul_one_eq_cast]
· exact pow_ne_zero _ ha₁
have : (X ^ p * C (a₁.coeff 0 ^ p ^ (n + 1)) - X * C (a₂.coeff 0 ^ p ^ (n + 1))).degree =... | 14 | 1,202,604.284165 | 2 | 2 | 1 | 2,261 |
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Fintype.BigOperators
import Mathlib.RingTheory.PowerSeries.Inverse
import Mathlib.RingTheory.PowerSeries.WellKnown
import Mathlib.Tactic.FieldSimp
#align_import number_theory.bernoulli from "leanprover-community/mat... | Mathlib/NumberTheory/Bernoulli.lean | 91 | 95 | theorem bernoulli'_spec' (n : ℕ) :
(∑ k ∈ antidiagonal n, ((k.1 + k.2).choose k.2 : ℚ) / (k.2 + 1) * bernoulli' k.1) = 1 := by |
refine ((sum_antidiagonal_eq_sum_range_succ_mk _ n).trans ?_).trans (bernoulli'_spec n)
refine sum_congr rfl fun x hx => ?_
simp only [add_tsub_cancel_of_le, mem_range_succ_iff.mp hx, cast_sub]
| 3 | 20.085537 | 1 | 1.272727 | 11 | 1,346 |
import Mathlib.LinearAlgebra.Dimension.Finite
import Mathlib.LinearAlgebra.Dimension.Constructions
open Cardinal Submodule Set FiniteDimensional
universe u v
section Module
variable {K : Type u} {V : Type v} [Ring K] [StrongRankCondition K] [AddCommGroup V] [Module K V]
noncomputable def Basis.ofRankEqZero [Mo... | Mathlib/LinearAlgebra/Dimension/FreeAndStrongRankCondition.lean | 46 | 60 | theorem le_rank_iff_exists_linearIndependent [Module.Free K V] {c : Cardinal} :
c ≤ Module.rank K V ↔ ∃ s : Set V, #s = c ∧ LinearIndependent K ((↑) : s → V) := by |
haveI := nontrivial_of_invariantBasisNumber K
constructor
· intro h
obtain ⟨κ, t'⟩ := Module.Free.exists_basis (R := K) (M := V)
let t := t'.reindexRange
have : LinearIndependent K ((↑) : Set.range t' → V) := by
convert t.linearIndependent
ext; exact (Basis.reindexRange_apply _ _).symm
... | 13 | 442,413.392009 | 2 | 1.636364 | 11 | 1,751 |
import Mathlib.Algebra.BigOperators.Group.Multiset
import Mathlib.Data.PNat.Prime
import Mathlib.Data.Nat.Factors
import Mathlib.Data.Multiset.Sort
#align_import data.pnat.factors from "leanprover-community/mathlib"@"e3d9ab8faa9dea8f78155c6c27d62a621f4c152d"
-- Porting note: `deriving` contained Inhabited, Canonic... | Mathlib/Data/PNat/Factors.lean | 130 | 133 | theorem coePNat_nat (v : PrimeMultiset) : ((v : Multiset ℕ+) : Multiset ℕ) = (v : Multiset ℕ) := by |
change (v.map (Coe.coe : Nat.Primes → ℕ+)).map Subtype.val = v.map Subtype.val
rw [Multiset.map_map]
congr
| 3 | 20.085537 | 1 | 1.25 | 4 | 1,335 |
import Mathlib.NumberTheory.Padics.PadicIntegers
import Mathlib.RingTheory.ZMod
#align_import number_theory.padics.ring_homs from "leanprover-community/mathlib"@"565eb991e264d0db702722b4bde52ee5173c9950"
noncomputable section
open scoped Classical
open Nat LocalRing Padic
namespace PadicInt
variable {p : ℕ} [h... | Mathlib/NumberTheory/Padics/RingHoms.lean | 124 | 134 | theorem norm_sub_modPart (h : ‖(r : ℚ_[p])‖ ≤ 1) : ‖(⟨r, h⟩ - modPart p r : ℤ_[p])‖ < 1 := by |
let n := modPart p r
rw [norm_lt_one_iff_dvd, ← (isUnit_den r h).dvd_mul_right]
suffices ↑p ∣ r.num - n * r.den by
convert (Int.castRingHom ℤ_[p]).map_dvd this
simp only [sub_mul, Int.cast_natCast, eq_intCast, Int.cast_mul, sub_left_inj, Int.cast_sub]
apply Subtype.coe_injective
simp only [coe_mu... | 10 | 22,026.465795 | 2 | 1.833333 | 12 | 1,916 |
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Shift
import Mathlib.Analysis.Calculus.IteratedDeriv.Defs
variable
{𝕜 : Type*} [NontriviallyNormedField 𝕜]
{F : Type*} [NormedAddCommGroup F] [NormedSpace 𝕜 F]
{R : Type*} [Semi... | Mathlib/Analysis/Calculus/IteratedDeriv/Lemmas.lean | 24 | 28 | theorem iteratedDerivWithin_add (hf : ContDiffOn 𝕜 n f s) (hg : ContDiffOn 𝕜 n g s) :
iteratedDerivWithin n (f + g) s x =
iteratedDerivWithin n f s x + iteratedDerivWithin n g s x := by |
simp_rw [iteratedDerivWithin, iteratedFDerivWithin_add_apply hf hg h hx,
ContinuousMultilinearMap.add_apply]
| 2 | 7.389056 | 1 | 1.2 | 10 | 1,290 |
import Mathlib.MeasureTheory.Measure.Haar.Basic
import Mathlib.Analysis.NormedSpace.FiniteDimension
import Mathlib.MeasureTheory.Measure.Haar.Unique
open MeasureTheory Measure Set
open scoped ENNReal
variable {𝕜 E F : Type*}
[NontriviallyNormedField 𝕜] [CompleteSpace 𝕜]
[NormedAddCommGroup E] [MeasurableSp... | Mathlib/MeasureTheory/Measure/Haar/Disintegration.lean | 106 | 109 | theorem LinearMap.exists_map_addHaar_eq_smul_addHaar (h : Function.Surjective L) :
∃ (c : ℝ≥0∞), 0 < c ∧ μ.map L = c • ν := by |
rcases L.exists_map_addHaar_eq_smul_addHaar' μ ν h with ⟨c, c_pos, -, hc⟩
exact ⟨_, by simp [c_pos, NeZero.ne addHaar], hc⟩
| 2 | 7.389056 | 1 | 1.5 | 2 | 1,596 |
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Logic.Equiv.Fin
import Mathlib.Topology.Algebra.InfiniteSum.Module
#align_import analysis.analytic.basic from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
noncomputable... | Mathlib/Analysis/Analytic/Basic.lean | 102 | 105 | theorem partialSum_continuous (p : FormalMultilinearSeries 𝕜 E F) (n : ℕ) :
Continuous (p.partialSum n) := by |
unfold partialSum -- Porting note: added
continuity
| 2 | 7.389056 | 1 | 1.5 | 2 | 1,600 |
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 | 174 | 177 | theorem strictConcaveOn_cos_Icc : StrictConcaveOn ℝ (Icc (-(π / 2)) (π / 2)) cos := by |
apply strictConcaveOn_of_deriv2_neg (convex_Icc _ _) continuousOn_cos fun x hx => ?_
rw [interior_Icc] at hx
simp [cos_pos_of_mem_Ioo hx]
| 3 | 20.085537 | 1 | 1.6 | 10 | 1,744 |
import Mathlib.Algebra.Homology.ExactSequence
import Mathlib.CategoryTheory.Abelian.Refinements
#align_import category_theory.abelian.diagram_lemmas.four from "leanprover-community/mathlib"@"d34cbcf6c94953e965448c933cd9cc485115ebbd"
namespace CategoryTheory
open Category Limits Preadditive
namespace Abelian
va... | Mathlib/CategoryTheory/Abelian/DiagramLemmas/Four.lean | 62 | 83 | theorem mono_of_epi_of_mono_of_mono' (hR₁ : R₁.map' 0 2 = 0)
(hR₁' : (mk₂ (R₁.map' 1 2) (R₁.map' 2 3)).Exact)
(hR₂ : (mk₂ (R₂.map' 0 1) (R₂.map' 1 2)).Exact)
(h₀ : Epi (app' φ 0)) (h₁ : Mono (app' φ 1)) (h₃ : Mono (app' φ 3)) :
Mono (app' φ 2) := by |
apply mono_of_cancel_zero
intro A f₂ h₁
have h₂ : f₂ ≫ R₁.map' 2 3 = 0 := by
rw [← cancel_mono (app' φ 3 _), assoc, NatTrans.naturality, reassoc_of% h₁,
zero_comp, zero_comp]
obtain ⟨A₁, π₁, _, f₁, hf₁⟩ := (hR₁'.exact 0).exact_up_to_refinements f₂ h₂
dsimp at hf₁
have h₃ : (f₁ ≫ app' φ 1) ≫ R₂.ma... | 17 | 24,154,952.753575 | 2 | 2 | 2 | 2,128 |
import Mathlib.Topology.Order.Basic
#align_import topology.algebra.order.monotone_convergence from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
open Filter Set Function
open scoped Classical
open Filter Topology
variable {α β : Type*}
class SupConvergenceClass (α : Type*) [Preorde... | Mathlib/Topology/Order/MonotoneConvergence.lean | 96 | 100 | theorem tendsto_atTop_isLUB (h_mono : Monotone f) (ha : IsLUB (Set.range f) a) :
Tendsto f atTop (𝓝 a) := by |
suffices Tendsto (rangeFactorization f) atTop atTop from
(SupConvergenceClass.tendsto_coe_atTop_isLUB _ _ ha).comp this
exact h_mono.rangeFactorization.tendsto_atTop_atTop fun b => b.2.imp fun a ha => ha.ge
| 3 | 20.085537 | 1 | 0.5 | 2 | 462 |
import Mathlib.CategoryTheory.Balanced
import Mathlib.CategoryTheory.Limits.EssentiallySmall
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
import Mathlib.CategoryTheory.Subobject.Lattice
import Mathlib.CategoryTheory.Subobject.WellPowered
import Mathlib.Data.S... | Mathlib/CategoryTheory/Generator.lean | 93 | 98 | theorem isSeparating_op_iff (𝒢 : Set C) : IsSeparating 𝒢.op ↔ IsCoseparating 𝒢 := by |
refine ⟨fun h𝒢 X Y f g hfg => ?_, fun h𝒢 X Y f g hfg => ?_⟩
· refine Quiver.Hom.op_inj (h𝒢 _ _ fun G hG h => Quiver.Hom.unop_inj ?_)
simpa only [unop_comp, Quiver.Hom.unop_op] using hfg _ (Set.mem_op.1 hG) _
· refine Quiver.Hom.unop_inj (h𝒢 _ _ fun G hG h => Quiver.Hom.op_inj ?_)
simpa only [op_comp,... | 5 | 148.413159 | 2 | 1.333333 | 6 | 1,424 |
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Factorial.DoubleFactorial
#align_import ring_theory.polynomial.hermite.basic from "leanprover-community/mathlib"@"938d3db9c278f8a52c0f964a405806f0f2b09b74"
noncomputable section
open Polynomial
namespace P... | Mathlib/RingTheory/Polynomial/Hermite/Basic.lean | 86 | 89 | theorem coeff_hermite_succ_succ (n k : ℕ) : coeff (hermite (n + 1)) (k + 1) =
coeff (hermite n) k - (k + 2) * coeff (hermite n) (k + 2) := by |
rw [hermite_succ, coeff_sub, coeff_X_mul, coeff_derivative, mul_comm]
norm_cast
| 2 | 7.389056 | 1 | 1.1 | 10 | 1,191 |
import Mathlib.CategoryTheory.Limits.KanExtension
import Mathlib.Topology.Category.TopCat.Opens
import Mathlib.CategoryTheory.Adjunction.Unique
import Mathlib.Topology.Sheaves.Init
import Mathlib.Data.Set.Subsingleton
#align_import topology.sheaves.presheaf from "leanprover-community/mathlib"@"5dc6092d09e5e4891068652... | Mathlib/Topology/Sheaves/Presheaf.lean | 154 | 158 | theorem map_restrict {X : TopCat} {C : Type*} [Category C] [ConcreteCategory C]
{F G : X.Presheaf C} (e : F ⟶ G) {U V : Opens X} (h : U ≤ V) (x : F.obj (op V)) :
e.app _ (x |_ U) = e.app _ x |_ U := by |
delta restrictOpen restrict
rw [← comp_apply, NatTrans.naturality, comp_apply]
| 2 | 7.389056 | 1 | 1 | 2 | 831 |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Group.Submonoid.Basic
import Mathlib.Deprecated.Group
#align_import deprecated.submonoid from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
variable {M : Type*} [Monoid M] {s : Set M}
variable {A : Type*} [AddMonoi... | Mathlib/Deprecated/Submonoid.lean | 232 | 237 | theorem list_prod_mem (hs : IsSubmonoid s) : ∀ {l : List M}, (∀ x ∈ l, x ∈ s) → l.prod ∈ s
| [], _ => hs.one_mem
| a :: l, h =>
suffices a * l.prod ∈ s by simpa
have : a ∈ s ∧ ∀ x ∈ l, x ∈ s := by | simpa using h
hs.mul_mem this.1 (list_prod_mem hs this.2)
| 2 | 7.389056 | 1 | 0.666667 | 3 | 571 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.Tactic.Abel
#align_import ring_theory.polynomial.pochhammer from "leanprover-community/mathlib"@"53b216bcc1146df1c4a0a868778... | Mathlib/RingTheory/Polynomial/Pochhammer.lean | 137 | 140 | theorem ascPochhammer_succ_eval {S : Type*} [Semiring S] (n : ℕ) (k : S) :
(ascPochhammer S (n + 1)).eval k = (ascPochhammer S n).eval k * (k + n) := by |
rw [ascPochhammer_succ_right, mul_add, eval_add, eval_mul_X, ← Nat.cast_comm, ← C_eq_natCast,
eval_C_mul, Nat.cast_comm, ← mul_add]
| 2 | 7.389056 | 1 | 0.96 | 25 | 796 |
import Mathlib.Analysis.Calculus.LocalExtr.Basic
import Mathlib.Topology.Algebra.Order.Rolle
#align_import analysis.calculus.local_extr from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
open Set Filter Topology
variable {f f' : ℝ → ℝ} {a b l : ℝ}
theorem exists_hasDerivAt_eq_zero (h... | Mathlib/Analysis/Calculus/LocalExtr/Rolle.lean | 78 | 84 | theorem exists_deriv_eq_zero' (hab : a < b) (hfa : Tendsto f (𝓝[>] a) (𝓝 l))
(hfb : Tendsto f (𝓝[<] b) (𝓝 l)) : ∃ c ∈ Ioo a b, deriv f c = 0 := by |
by_cases h : ∀ x ∈ Ioo a b, DifferentiableAt ℝ f x
· exact exists_hasDerivAt_eq_zero' hab hfa hfb fun x hx => (h x hx).hasDerivAt
· obtain ⟨c, hc, hcdiff⟩ : ∃ x ∈ Ioo a b, ¬DifferentiableAt ℝ f x := by
push_neg at h; exact h
exact ⟨c, hc, deriv_zero_of_not_differentiableAt hcdiff⟩
| 5 | 148.413159 | 2 | 2 | 1 | 2,468 |
import Mathlib.Topology.Sets.Closeds
#align_import topology.noetherian_space from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
variable (α β : Type*) [TopologicalSpace α] [TopologicalSpace β]
namespace TopologicalSpace
@[mk_iff]
class NoetherianSpace : Prop where
wellFounded_open... | Mathlib/Topology/NoetherianSpace.lean | 53 | 56 | theorem noetherianSpace_iff_opens : NoetherianSpace α ↔ ∀ s : Opens α, IsCompact (s : Set α) := by |
rw [noetherianSpace_iff, CompleteLattice.wellFounded_iff_isSupFiniteCompact,
CompleteLattice.isSupFiniteCompact_iff_all_elements_compact]
exact forall_congr' Opens.isCompactElement_iff
| 3 | 20.085537 | 1 | 1.5 | 4 | 1,545 |
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