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.Polynomial.Monic
#align_import algebra.polynomial.big_operators from "leanprover-community/mathlib"@"47adfab39a11a072db552f47594bf8ed2cf8a722"
open Finset
open Multiset
open Polynomial
universe u w
variable {R : Type u} {ι : Type w}
namespace Polynomial
variable (s : Finset ι)
sectio... | Mathlib/Algebra/Polynomial/BigOperators.lean | 57 | 59 | theorem natDegree_sum_le (f : ι → S[X]) :
natDegree (∑ i ∈ s, f i) ≤ s.fold max 0 (natDegree ∘ f) := by |
simpa using natDegree_multiset_sum_le (s.val.map f)
| 1 | 2.718282 | 0 | 1.142857 | 7 | 1,216 |
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 | 65 | 69 | theorem isRoot_of_hasEigenvalue {f : End K V} {μ : K} (h : f.HasEigenvalue μ) :
(minpoly K f).IsRoot μ := by |
rcases (Submodule.ne_bot_iff _).1 h with ⟨w, ⟨H, ne0⟩⟩
refine Or.resolve_right (smul_eq_zero.1 ?_) ne0
simp [← aeval_apply_of_hasEigenvector ⟨H, ne0⟩, minpoly.aeval K f]
| 3 | 20.085537 | 1 | 1.8 | 5 | 1,893 |
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
import Mathlib.RingTheory.Localization.AsSubring
#align_import algebraic_geometry.prime_spectrum.maximal from "leanprover-community/mathlib"@"052f6013363326d50cb99c6939814a4b8eb7b301"
noncomputable section
open scoped Classical
universe u v
variable (R : Typ... | Mathlib/AlgebraicGeometry/PrimeSpectrum/Maximal.lean | 65 | 69 | theorem toPrimeSpectrum_range :
Set.range (@toPrimeSpectrum R _) = { x | IsClosed ({x} : Set <| PrimeSpectrum R) } := by |
simp only [isClosed_singleton_iff_isMaximal]
ext ⟨x, _⟩
exact ⟨fun ⟨y, hy⟩ => hy ▸ y.IsMaximal, fun hx => ⟨⟨x, hx⟩, rfl⟩⟩
| 3 | 20.085537 | 1 | 1.5 | 2 | 1,576 |
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Analysis.NormedSpace.OperatorNorm.Basic
import Mathlib.Analysis.NormedSpace.Pointwise
#align_import analysis.normed_space.is_R_or_C from "leanprover-community/mathlib"@"3f655f5297b030a87d641ad4e825af8d9679eb0b"
open Metric
variable {𝕜 : Type*} [RCLike 𝕜] {E :... | Mathlib/Analysis/NormedSpace/RCLike.lean | 49 | 52 | theorem norm_smul_inv_norm' {r : ℝ} (r_nonneg : 0 ≤ r) {x : E} (hx : x ≠ 0) :
‖((r : 𝕜) * (‖x‖ : 𝕜)⁻¹) • x‖ = r := by |
have : ‖x‖ ≠ 0 := by simp [hx]
field_simp [norm_smul, r_nonneg, rclike_simps]
| 2 | 7.389056 | 1 | 1.2 | 5 | 1,253 |
import Mathlib.Data.List.Basic
namespace List
variable {α β : Type*}
@[simp]
theorem reduceOption_cons_of_some (x : α) (l : List (Option α)) :
reduceOption (some x :: l) = x :: l.reduceOption := by
simp only [reduceOption, filterMap, id, eq_self_iff_true, and_self_iff]
#align list.reduce_option_cons_of_some... | Mathlib/Data/List/ReduceOption.lean | 55 | 57 | theorem length_eq_reduceOption_length_add_filter_none {l : List (Option α)} :
l.length = l.reduceOption.length + (l.filter Option.isNone).length := by |
simp_rw [reduceOption_length_eq, l.length_eq_length_filter_add Option.isSome, Option.bnot_isSome]
| 1 | 2.718282 | 0 | 0.615385 | 13 | 540 |
import Mathlib.Data.List.Lattice
import Mathlib.Data.List.Range
import Mathlib.Data.Bool.Basic
#align_import data.list.intervals from "leanprover-community/mathlib"@"7b78d1776212a91ecc94cf601f83bdcc46b04213"
open Nat
namespace List
def Ico (n m : ℕ) : List ℕ :=
range' n (m - n)
#align list.Ico List.Ico
names... | Mathlib/Data/List/Intervals.lean | 51 | 53 | theorem pairwise_lt (n m : ℕ) : Pairwise (· < ·) (Ico n m) := by |
dsimp [Ico]
simp [pairwise_lt_range', autoParam]
| 2 | 7.389056 | 1 | 0.9375 | 16 | 794 |
import Mathlib.Order.BooleanAlgebra
import Mathlib.Tactic.Common
#align_import order.heyting.boundary from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
variable {α : Type*}
namespace Coheyting
variable [CoheytingAlgebra α] {a b : α}
def boundary (a : α) : α :=
a ⊓ ￢a
#align cohe... | Mathlib/Order/Heyting/Boundary.lean | 89 | 93 | theorem boundary_sup_le : ∂ (a ⊔ b) ≤ ∂ a ⊔ ∂ b := by |
rw [boundary, inf_sup_right]
exact
sup_le_sup (inf_le_inf_left _ <| hnot_anti le_sup_left)
(inf_le_inf_left _ <| hnot_anti le_sup_right)
| 4 | 54.59815 | 2 | 0.777778 | 9 | 687 |
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 | 48 | 58 | theorem independent_iff : Independent f ↔ LinearIndependent K (Projectivization.rep ∘ f) := by |
refine ⟨?_, fun h => ?_⟩
· rintro ⟨ff, hff, hh⟩
choose a ha using fun i : ι => exists_smul_eq_mk_rep K (ff i) (hff i)
convert hh.units_smul a
ext i
exact (ha i).symm
· convert Independent.mk _ _ h
· simp only [mk_rep, Function.comp_apply]
· intro i
apply rep_nonzero
| 10 | 22,026.465795 | 2 | 1.142857 | 7 | 1,223 |
import Mathlib.MeasureTheory.Measure.MeasureSpace
open scoped ENNReal NNReal Topology
open Set MeasureTheory Measure Filter MeasurableSpace ENNReal Function
variable {R α β δ γ ι : Type*}
namespace MeasureTheory
variable {m0 : MeasurableSpace α} [MeasurableSpace β] [MeasurableSpace γ]
variable {μ μ₁ μ₂ μ₃ ν ν' ν... | Mathlib/MeasureTheory/Measure/Restrict.lean | 110 | 113 | theorem restrict_apply₀' (hs : NullMeasurableSet s μ) : μ.restrict s t = μ (t ∩ s) := by |
rw [← restrict_congr_set hs.toMeasurable_ae_eq,
restrict_apply' (measurableSet_toMeasurable _ _),
measure_congr ((ae_eq_refl t).inter hs.toMeasurable_ae_eq)]
| 3 | 20.085537 | 1 | 0.666667 | 6 | 583 |
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Sum
import Mathlib.SetTheory.Cardinal.Finite
#align_import data.fintype.units from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
variable {α : Type*}
instance UnitsInt.fintype : Fintype ℤˣ :=
⟨{1, -1}, fun x ↦ by cases Int... | Mathlib/Data/Fintype/Units.lean | 42 | 46 | theorem Nat.card_eq_card_units_add_one [GroupWithZero α] [Finite α] :
Nat.card α = Nat.card αˣ + 1 := by |
have : Fintype α := Fintype.ofFinite α
classical
rw [Nat.card_eq_fintype_card, Nat.card_eq_fintype_card, Fintype.card_eq_card_units_add_one]
| 3 | 20.085537 | 1 | 0.666667 | 3 | 602 |
import Mathlib.Analysis.SpecialFunctions.Complex.Log
import Mathlib.RingTheory.RootsOfUnity.Basic
#align_import ring_theory.roots_of_unity.complex from "leanprover-community/mathlib"@"7fdeecc0d03cd40f7a165e6cf00a4d2286db599f"
namespace Complex
open Polynomial Real
open scoped Nat Real
theorem isPrimitiveRoot_e... | Mathlib/RingTheory/RootsOfUnity/Complex.lean | 96 | 99 | theorem card_primitiveRoots (k : ℕ) : (primitiveRoots k ℂ).card = φ k := by |
by_cases h : k = 0
· simp [h]
exact (isPrimitiveRoot_exp k h).card_primitiveRoots
| 3 | 20.085537 | 1 | 1.5 | 4 | 1,580 |
import Mathlib.Data.Complex.Module
import Mathlib.Data.Complex.Order
import Mathlib.Data.Complex.Exponential
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Topology.Algebra.InfiniteSum.Module
import Mathlib.Topology.Instances.RealVectorSpace
#align_import analysis.complex.basic from "leanprover-community/mathlib... | Mathlib/Analysis/Complex/Basic.lean | 58 | 59 | theorem norm_exp_ofReal_mul_I (t : ℝ) : ‖exp (t * I)‖ = 1 := by |
simp only [norm_eq_abs, abs_exp_ofReal_mul_I]
| 1 | 2.718282 | 0 | 0.222222 | 9 | 282 |
import Batteries.Classes.Order
import Batteries.Control.ForInStep.Basic
namespace Batteries
namespace BinomialHeap
namespace Imp
inductive HeapNode (α : Type u) where
| nil : HeapNode α
| node (a : α) (child sibling : HeapNode α) : HeapNode α
deriving Repr
@[simp] def HeapNode.realSize : HeapNode α → ... | .lake/packages/batteries/Batteries/Data/BinomialHeap/Basic.lean | 205 | 221 | theorem Heap.realSize_merge (le) (s₁ s₂ : Heap α) :
(s₁.merge le s₂).realSize = s₁.realSize + s₂.realSize := by |
unfold merge; split
· simp
· simp
· next r₁ a₁ n₁ t₁ r₂ a₂ n₂ t₂ =>
have IH₁ r a n := realSize_merge le t₁ (cons r a n t₂)
have IH₂ r a n := realSize_merge le (cons r a n t₁) t₂
have IH₃ := realSize_merge le t₁ t₂
split; · simp [IH₁, Nat.add_assoc]
split; · simp [IH₂, Nat.add_assoc, Nat.add... | 15 | 3,269,017.372472 | 2 | 2 | 1 | 2,174 |
import Mathlib.Analysis.RCLike.Lemmas
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.function.l2_space from "leanprover-community/mathlib"@"83a66c8775fa14ee5180c85cab98e970956401ad"
set_option linter.uppercaseLean3 false... | Mathlib/MeasureTheory/Function/L2Space.lean | 42 | 43 | theorem Memℒp.integrable_sq {f : α → ℝ} (h : Memℒp f 2 μ) : Integrable (fun x => f x ^ 2) μ := by |
simpa [← memℒp_one_iff_integrable] using h.norm_rpow two_ne_zero ENNReal.two_ne_top
| 1 | 2.718282 | 0 | 0.888889 | 9 | 770 |
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)
| Mathlib/Data/Nat/Factorization/PrimePow.lean | 20 | 24 | theorem IsPrimePow.minFac_pow_factorization_eq {n : ℕ} (hn : IsPrimePow n) :
n.minFac ^ n.factorization n.minFac = n := by |
obtain ⟨p, k, hp, hk, rfl⟩ := hn
rw [← Nat.prime_iff] at hp
rw [hp.pow_minFac hk.ne', hp.factorization_pow, Finsupp.single_eq_same]
| 3 | 20.085537 | 1 | 1.714286 | 7 | 1,840 |
import Mathlib.Analysis.SpecialFunctions.Complex.Log
import Mathlib.RingTheory.RootsOfUnity.Basic
#align_import ring_theory.roots_of_unity.complex from "leanprover-community/mathlib"@"7fdeecc0d03cd40f7a165e6cf00a4d2286db599f"
namespace Complex
open Polynomial Real
open scoped Nat Real
theorem isPrimitiveRoot_e... | Mathlib/RingTheory/RootsOfUnity/Complex.lean | 53 | 55 | theorem isPrimitiveRoot_exp (n : ℕ) (h0 : n ≠ 0) : IsPrimitiveRoot (exp (2 * π * I / n)) n := by |
simpa only [Nat.cast_one, one_div] using
isPrimitiveRoot_exp_of_coprime 1 n h0 n.coprime_one_left
| 2 | 7.389056 | 1 | 1.5 | 4 | 1,580 |
import Mathlib.Algebra.MvPolynomial.Derivation
import Mathlib.Algebra.MvPolynomial.Variables
#align_import data.mv_polynomial.pderiv from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
universe u v
namespace MvPolynomial
open Set Function Finsupp
variable {R : ... | Mathlib/Algebra/MvPolynomial/PDeriv.lean | 64 | 65 | theorem pderiv_def [DecidableEq σ] (i : σ) : pderiv i = mkDerivation R (Pi.single i 1) := by |
unfold pderiv; congr!
| 1 | 2.718282 | 0 | 0.222222 | 9 | 283 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.NormedSpace.FiniteDimension
import Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic
#align_import analysis.calculus.fderiv_... | Mathlib/Analysis/Calculus/FDeriv/Measurable.lean | 473 | 483 | theorem A_mem_nhdsWithin_Ioi {L : F} {r ε x : ℝ} (hx : x ∈ A f L r ε) : A f L r ε ∈ 𝓝[>] x := by |
rcases hx with ⟨r', rr', hr'⟩
rw [mem_nhdsWithin_Ioi_iff_exists_Ioo_subset]
obtain ⟨s, s_gt, s_lt⟩ : ∃ s : ℝ, r / 2 < s ∧ s < r' := exists_between rr'.1
have : s ∈ Ioc (r / 2) r := ⟨s_gt, le_of_lt (s_lt.trans_le rr'.2)⟩
refine ⟨x + r' - s, by simp only [mem_Ioi]; linarith, fun x' hx' => ⟨s, this, ?_⟩⟩
have... | 10 | 22,026.465795 | 2 | 1.4 | 10 | 1,489 |
import Mathlib.Algebra.Group.Subgroup.Finite
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.GroupTheory.Congruence.Basic
import Mathlib.GroupTheory.Coset
#align_import group_theory.quotient_group from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
open Function
open scope... | Mathlib/GroupTheory/QuotientGroup.lean | 129 | 131 | theorem eq_one_iff {N : Subgroup G} [nN : N.Normal] (x : G) : (x : G ⧸ N) = 1 ↔ x ∈ N := by |
refine QuotientGroup.eq.trans ?_
rw [mul_one, Subgroup.inv_mem_iff]
| 2 | 7.389056 | 1 | 1.333333 | 3 | 1,425 |
import Mathlib.Algebra.Group.Subgroup.Actions
import Mathlib.Algebra.Order.Module.Algebra
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.Algebra.Ring.Subring.Units
#align_import linear_algebra.ray from "leanprover-community/mathlib"@"0f6670b8af2dff699de1c0b4b49039b31bc13c46"
noncomputable section
... | Mathlib/LinearAlgebra/Ray.lean | 74 | 76 | theorem refl (x : M) : SameRay R x x := by |
nontriviality R
exact Or.inr (Or.inr <| ⟨1, 1, zero_lt_one, zero_lt_one, rfl⟩)
| 2 | 7.389056 | 1 | 1.333333 | 3 | 1,453 |
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.Sym.Sym2
namespace List
variable {α : Type*}
section Sym2
protected def sym2 : List α → List (Sym2 α)
| [] => []
| x :: xs => (x :: xs).map (fun y => s(x, y)) ++ xs.sym2
theorem mem_sym2_cons_iff {x : α} {xs : List α} {z : Sym2 α} :
z ∈ (x :: xs)... | Mathlib/Data/List/Sym.lean | 81 | 87 | theorem mk_mem_sym2_iff {xs : List α} {a b : α} :
s(a, b) ∈ xs.sym2 ↔ a ∈ xs ∧ b ∈ xs := by |
constructor
· intro h
exact ⟨left_mem_of_mk_mem_sym2 h, right_mem_of_mk_mem_sym2 h⟩
· rintro ⟨ha, hb⟩
exact mk_mem_sym2 ha hb
| 5 | 148.413159 | 2 | 1.444444 | 9 | 1,529 |
import Mathlib.Data.DFinsupp.Lex
import Mathlib.Order.GameAdd
import Mathlib.Order.Antisymmetrization
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Tactic.AdaptationNote
#align_import data.dfinsupp.well_founded from "leanprover-community/mathlib"@"e9b8651eb1ad354f4de6be35a38ef31efcd2cfaa"
variable {ι : Ty... | Mathlib/Data/DFinsupp/WellFounded.lean | 103 | 109 | theorem Lex.acc_of_single_erase [DecidableEq ι] {x : Π₀ i, α i} (i : ι)
(hs : Acc (DFinsupp.Lex r s) <| single i (x i)) (hu : Acc (DFinsupp.Lex r s) <| x.erase i) :
Acc (DFinsupp.Lex r s) x := by |
classical
convert ← @Acc.of_fibration _ _ _ _ _ (lex_fibration r s) ⟨{i}, _⟩
(InvImage.accessible snd <| hs.prod_gameAdd hu)
convert piecewise_single_erase x i
| 4 | 54.59815 | 2 | 2 | 4 | 2,437 |
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Functor.EpiMono
#align_import category_theory.adjunction.evaluation from "leanprover-community/mathlib"@"937c692d73f5130c7fecd3fd32e81419f4e04eb7"
namespace CategoryTheory
open CategoryTheory.Limits
universe v₁ v₂ u₁ u₂
variable... | Mathlib/CategoryTheory/Adjunction/Evaluation.lean | 140 | 145 | theorem NatTrans.epi_iff_epi_app {F G : C ⥤ D} (η : F ⟶ G) : Epi η ↔ ∀ c, Epi (η.app c) := by |
constructor
· intro h c
exact (inferInstance : Epi (((evaluation _ _).obj c).map η))
· intros
apply NatTrans.epi_of_epi_app
| 5 | 148.413159 | 2 | 2 | 2 | 2,483 |
import Mathlib.AlgebraicGeometry.Morphisms.Basic
import Mathlib.Topology.Spectral.Hom
import Mathlib.AlgebraicGeometry.Limits
#align_import algebraic_geometry.morphisms.quasi_compact from "leanprover-community/mathlib"@"5dc6092d09e5e489106865241986f7f2ad28d4c8"
noncomputable section
open CategoryTheory CategoryT... | Mathlib/AlgebraicGeometry/Morphisms/QuasiCompact.lean | 123 | 126 | theorem quasiCompact_eq_affineProperty :
@QuasiCompact = targetAffineLocally QuasiCompact.affineProperty := by |
ext
exact quasiCompact_iff_affineProperty _
| 2 | 7.389056 | 1 | 1.5 | 6 | 1,632 |
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.LinearAlgebra.Matrix.ToLin
#align_import linear_algebra.matrix.charpoly.linear_map from "leanprover-community/mathlib"@"62c0a4ef1441edb463095ea02a06e87f3dfe135c"
variable {ι : Type*} [Fintype ι]
variable {M : Type*} [AddCommGroup M] (R : Type*) [Co... | Mathlib/LinearAlgebra/Matrix/Charpoly/LinearMap.lean | 136 | 138 | theorem Matrix.Represents.zero : (0 : Matrix ι ι R).Represents b 0 := by |
delta Matrix.Represents
rw [map_zero, map_zero]
| 2 | 7.389056 | 1 | 1.444444 | 9 | 1,527 |
import Mathlib.CategoryTheory.Subobject.MonoOver
import Mathlib.CategoryTheory.Skeletal
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathlib.Tactic.ApplyFun
import Mathlib.Tactic.CategoryTheory.Elementwise
#align_import category_theory.subobject.basic from "leanprover-community/mathlib"@"70fd9563a21e7b... | Mathlib/CategoryTheory/Subobject/Basic.lean | 580 | 582 | theorem map_id (x : Subobject X) : (map (𝟙 X)).obj x = x := by |
induction' x using Quotient.inductionOn' with f
exact Quotient.sound ⟨(MonoOver.mapId _).app f⟩
| 2 | 7.389056 | 1 | 1 | 5 | 901 |
import Mathlib.Algebra.Algebra.Subalgebra.Pointwise
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Maximal
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Noetherian
import Mathlib.RingTheory.ChainOfDivisors
import Mathlib.RingTheory.DedekindDomain.Basic
import Mathlib.RingTheory.FractionalIdeal.Operations
#align_impo... | Mathlib/RingTheory/DedekindDomain/Ideal.lean | 87 | 92 | theorem inv_anti_mono (hI : I ≠ 0) (hJ : J ≠ 0) (hIJ : I ≤ J) : J⁻¹ ≤ I⁻¹ := by |
-- Porting note: in Lean3, introducing `x` would just give `x ∈ J⁻¹ → x ∈ I⁻¹`, but
-- in Lean4, it goes all the way down to the subtypes
intro x
simp only [val_eq_coe, mem_coe, mem_inv_iff hJ, mem_inv_iff hI]
exact fun h y hy => h y (hIJ hy)
| 5 | 148.413159 | 2 | 0.666667 | 6 | 564 |
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.RingTheory.Int.Basic
import Mathlib.Tactic.Ring
import Mathlib.Tactic.FieldSimp
import Mathlib.Data.Int.NatPrime
import Mathlib.Data.ZMod.Basic
#align_import number_theory.pythagorean_tri... | Mathlib/NumberTheory/PythagoreanTriples.lean | 78 | 82 | theorem mul (k : ℤ) : PythagoreanTriple (k * x) (k * y) (k * z) :=
calc
k * x * (k * x) + k * y * (k * y) = k ^ 2 * (x * x + y * y) := by | ring
_ = k ^ 2 * (z * z) := by rw [h.eq]
_ = k * z * (k * z) := by ring
| 3 | 20.085537 | 1 | 1.2 | 10 | 1,251 |
import Mathlib.RingTheory.Polynomial.Basic
import Mathlib.RingTheory.Ideal.LocalRing
#align_import data.polynomial.expand from "leanprover-community/mathlib"@"bbeb185db4ccee8ed07dc48449414ebfa39cb821"
universe u v w
open Polynomial
open Finset
namespace Polynomial
section CommSemiring
variable (R : Type u) [... | Mathlib/Algebra/Polynomial/Expand.lean | 121 | 123 | theorem coeff_expand_mul {p : ℕ} (hp : 0 < p) (f : R[X]) (n : ℕ) :
(expand R p f).coeff (n * p) = f.coeff n := by |
rw [coeff_expand hp, if_pos (dvd_mul_left _ _), Nat.mul_div_cancel _ hp]
| 1 | 2.718282 | 0 | 0.285714 | 7 | 311 |
import Mathlib.Analysis.BoxIntegral.Partition.Basic
#align_import analysis.box_integral.partition.split from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf97388c9e3b441fc4ccf3f"
noncomputable section
open scoped Classical
open Filter
open Function Set Filter
namespace BoxIntegral
variable {ι M : Type*} {... | Mathlib/Analysis/BoxIntegral/Partition/Split.lean | 106 | 112 | theorem coe_splitUpper : (splitUpper I i x : Set (ι → ℝ)) = ↑I ∩ { y | x < y i } := by |
rw [splitUpper, coe_mk']
ext y
simp only [mem_univ_pi, mem_Ioc, mem_inter_iff, mem_coe, mem_setOf_eq, forall_and,
forall_update_iff I.lower fun j z => z < y j, max_lt_iff, and_assoc (a := x < y i),
and_forall_ne (p := fun j => lower I j < y j) i, mem_def]
exact and_comm
| 6 | 403.428793 | 2 | 1.2 | 10 | 1,269 |
import Batteries.Data.RBMap.Alter
import Batteries.Data.List.Lemmas
namespace Batteries
namespace RBNode
open RBColor
attribute [simp] fold foldl foldr Any forM foldlM Ordered
@[simp] theorem min?_reverse (t : RBNode α) : t.reverse.min? = t.max? := by
unfold RBNode.max?; split <;> simp [RBNode.min?]
unfold RB... | .lake/packages/batteries/Batteries/Data/RBMap/Lemmas.lean | 45 | 65 | theorem isOrdered_iff' [@TransCmp α cmp] {t : RBNode α} :
isOrdered cmp t L R ↔
(∀ a ∈ L, t.All (cmpLT cmp a ·)) ∧
(∀ a ∈ R, t.All (cmpLT cmp · a)) ∧
(∀ a ∈ L, ∀ b ∈ R, cmpLT cmp a b) ∧
Ordered cmp t := by |
induction t generalizing L R with
| nil =>
simp [isOrdered]; split <;> simp [cmpLT_iff]
next h => intro _ ha _ hb; cases h _ _ ha hb
| node _ l v r =>
simp [isOrdered, *]
exact ⟨
fun ⟨⟨Ll, lv, Lv, ol⟩, ⟨vr, rR, vR, or⟩⟩ => ⟨
fun _ h => ⟨Lv _ h, Ll _ h, (Lv _ h).trans_l vr⟩,
... | 15 | 3,269,017.372472 | 2 | 0.666667 | 6 | 556 |
import Mathlib.Analysis.Convex.Cone.Basic
import Mathlib.Data.Real.Archimedean
import Mathlib.LinearAlgebra.LinearPMap
#align_import analysis.convex.cone.basic from "leanprover-community/mathlib"@"915591b2bb3ea303648db07284a161a7f2a9e3d4"
open Set LinearMap
variable {𝕜 E F G : Type*}
variable [AddCommGroup E... | Mathlib/Analysis/Convex/Cone/Extension.lean | 64 | 112 | theorem step (nonneg : ∀ x : f.domain, (x : E) ∈ s → 0 ≤ f x)
(dense : ∀ y, ∃ x : f.domain, (x : E) + y ∈ s) (hdom : f.domain ≠ ⊤) :
∃ g, f < g ∧ ∀ x : g.domain, (x : E) ∈ s → 0 ≤ g x := by |
obtain ⟨y, -, hy⟩ : ∃ y ∈ ⊤, y ∉ f.domain := SetLike.exists_of_lt (lt_top_iff_ne_top.2 hdom)
obtain ⟨c, le_c, c_le⟩ :
∃ c, (∀ x : f.domain, -(x : E) - y ∈ s → f x ≤ c) ∧
∀ x : f.domain, (x : E) + y ∈ s → c ≤ f x := by
set Sp := f '' { x : f.domain | (x : E) + y ∈ s }
set Sn := f '' { x : f.do... | 46 | 94,961,194,206,024,480,000 | 2 | 2 | 2 | 2,033 |
import Mathlib.Topology.Order
#align_import topology.maps from "leanprover-community/mathlib"@"d91e7f7a7f1c7e9f0e18fdb6bde4f652004c735d"
open Set Filter Function
open TopologicalSpace Topology Filter
variable {X : Type*} {Y : Type*} {Z : Type*} {ι : Type*} {f : X → Y} {g : Y → Z}
section OpenMap
variable [Topo... | Mathlib/Topology/Maps.lean | 478 | 482 | theorem of_nonempty (h : ∀ s, IsClosed s → s.Nonempty → IsClosed (f '' s)) :
IsClosedMap f := by |
intro s hs; rcases eq_empty_or_nonempty s with h2s | h2s
· simp_rw [h2s, image_empty, isClosed_empty]
· exact h s hs h2s
| 3 | 20.085537 | 1 | 0.5 | 12 | 442 |
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Data.Nat.Totient
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.GroupTheory.Subgroup.Simple
import Mathlib.Tactic.Group
import Mathlib.GroupTheory.Exponent
#align_import group_theory.specific_groups.cyclic from "leanprover-community/mathli... | Mathlib/GroupTheory/SpecificGroups/Cyclic.lean | 136 | 141 | theorem Subgroup.eq_bot_or_eq_top_of_prime_card {G : Type*} [Group G] {_ : Fintype G}
(H : Subgroup G) [hp : Fact (Fintype.card G).Prime] : H = ⊥ ∨ H = ⊤ := by |
classical
have := card_subgroup_dvd_card H
rwa [Nat.card_eq_fintype_card (α := G), Nat.dvd_prime hp.1, ← Nat.card_eq_fintype_card,
← eq_bot_iff_card, card_eq_iff_eq_top] at this
| 4 | 54.59815 | 2 | 1.333333 | 6 | 1,386 |
import Mathlib.Analysis.InnerProductSpace.Calculus
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.Adjoint
import Mathlib.Analysis.Calculus.LagrangeMultipliers
import Mathlib.LinearAlgebra.Eigenspace.Basic
#align_import analysis.inner_product_space.rayleigh from "leanprover-co... | Mathlib/Analysis/InnerProductSpace/Rayleigh.lean | 67 | 80 | theorem image_rayleigh_eq_image_rayleigh_sphere {r : ℝ} (hr : 0 < r) :
rayleighQuotient T '' {0}ᶜ = rayleighQuotient T '' sphere 0 r := by |
ext a
constructor
· rintro ⟨x, hx : x ≠ 0, hxT⟩
have : ‖x‖ ≠ 0 := by simp [hx]
let c : 𝕜 := ↑‖x‖⁻¹ * r
have : c ≠ 0 := by simp [c, hx, hr.ne']
refine ⟨c • x, ?_, ?_⟩
· field_simp [c, norm_smul, abs_of_pos hr]
· rw [T.rayleigh_smul x this]
exact hxT
· rintro ⟨x, hx, hxT⟩
exact... | 12 | 162,754.791419 | 2 | 2 | 4 | 2,416 |
import Mathlib.Order.Filter.EventuallyConst
import Mathlib.Order.PartialSups
import Mathlib.Algebra.Module.Submodule.IterateMapComap
import Mathlib.RingTheory.OrzechProperty
import Mathlib.RingTheory.Nilpotent.Lemmas
#align_import ring_theory.noetherian from "leanprover-community/mathlib"@"210657c4ea4a4a7b234392f70a3... | Mathlib/RingTheory/Noetherian.lean | 136 | 139 | theorem isNoetherian_top_iff : IsNoetherian R (⊤ : Submodule R M) ↔ IsNoetherian R M := by |
constructor <;> intro h
· exact isNoetherian_of_linearEquiv (LinearEquiv.ofTop (⊤ : Submodule R M) rfl)
· exact isNoetherian_of_linearEquiv (LinearEquiv.ofTop (⊤ : Submodule R M) rfl).symm
| 3 | 20.085537 | 1 | 1.5 | 2 | 1,577 |
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.TryThis
import Mathlib.Util.AtomM
set_option autoImplicit true
namespace Mathlib.Tactic.Abel
open Lean Elab Meta Tactic Qq
initialize registerTraceClass `abel
initialize registerTraceClass `abel.detail
structure Context where
α : Expr
univ :... | Mathlib/Tactic/Abel.lean | 128 | 130 | theorem const_add_term {α} [AddCommMonoid α] (k n x a a') (h : k + a = a') :
k + @term α _ n x a = term n x a' := by |
simp [h.symm, term, add_comm, add_assoc]
| 1 | 2.718282 | 0 | 0.125 | 8 | 249 |
import Mathlib.Order.Filter.Cofinite
#align_import topology.bornology.basic from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
open Set Filter
variable {ι α β : Type*}
class Bornology (α : Type*) where
cobounded' : Filter α
le_cofinite' : cobounded' ≤ cofinite
#align borno... | Mathlib/Topology/Bornology/Basic.lean | 173 | 175 | theorem isBounded_singleton : IsBounded ({x} : Set α) := by |
rw [isBounded_def]
exact le_cofinite _ (finite_singleton x).compl_mem_cofinite
| 2 | 7.389056 | 1 | 0.75 | 4 | 665 |
import Mathlib.RingTheory.TensorProduct.Basic
import Mathlib.Algebra.Module.ULift
#align_import ring_theory.is_tensor_product from "leanprover-community/mathlib"@"c4926d76bb9c5a4a62ed2f03d998081786132105"
universe u v₁ v₂ v₃ v₄
open TensorProduct
section IsTensorProduct
variable {R : Type*} [CommSemiring R]
va... | Mathlib/RingTheory/IsTensorProduct.lean | 109 | 112 | theorem IsTensorProduct.map_eq (hf : IsTensorProduct f) (hg : IsTensorProduct g) (i₁ : M₁ →ₗ[R] N₁)
(i₂ : M₂ →ₗ[R] N₂) (x₁ : M₁) (x₂ : M₂) : hf.map hg i₁ i₂ (f x₁ x₂) = g (i₁ x₁) (i₂ x₂) := by |
delta IsTensorProduct.map
simp
| 2 | 7.389056 | 1 | 1.4 | 5 | 1,506 |
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Group.Invertible.Defs
import Mathlib.Algebra.Group.Hom.Defs
import Mathlib.Logic.Equiv.Defs
#align_import algebra.invertible from "leanprover-community/mathlib"@"722b3b152ddd5e0cf21c0a29787c76596cb6b422"
assert_not_exists MonoidWithZero
assert_not_ex... | Mathlib/Algebra/Group/Invertible/Basic.lean | 77 | 82 | theorem Commute.invOf_left [Monoid α] {a b : α} [Invertible b] (h : Commute b a) :
Commute (⅟ b) a :=
calc
⅟ b * a = ⅟ b * (a * b * ⅟ b) := by | simp [mul_assoc]
_ = ⅟ b * (b * a * ⅟ b) := by rw [h.eq]
_ = a * ⅟ b := by simp [mul_assoc]
| 3 | 20.085537 | 1 | 1 | 2 | 1,017 |
import Mathlib.Analysis.NormedSpace.PiLp
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import analysis.matrix from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped NNReal Matrix
namespace Matrix
variable {R l m n α β : Type*} [Fintype l] [Fintyp... | Mathlib/Analysis/Matrix.lean | 574 | 575 | theorem frobenius_nnnorm_map_eq (A : Matrix m n α) (f : α → β) (hf : ∀ a, ‖f a‖₊ = ‖a‖₊) :
‖A.map f‖₊ = ‖A‖₊ := by | simp_rw [frobenius_nnnorm_def, Matrix.map_apply, hf]
| 1 | 2.718282 | 0 | 0.533333 | 15 | 509 |
import Mathlib.Order.Filter.Bases
#align_import order.filter.pi from "leanprover-community/mathlib"@"ce64cd319bb6b3e82f31c2d38e79080d377be451"
open Set Function
open scoped Classical
open Filter
namespace Filter
variable {ι : Type*} {α : ι → Type*} {f f₁ f₂ : (i : ι) → Filter (α i)} {s : (i : ι) → Set (α i)}
... | Mathlib/Order/Filter/Pi.lean | 244 | 245 | theorem coprodᵢ_neBot_iff [∀ i, Nonempty (α i)] : NeBot (Filter.coprodᵢ f) ↔ ∃ d, NeBot (f d) := by |
simp [coprodᵢ_neBot_iff', *]
| 1 | 2.718282 | 0 | 0.666667 | 12 | 565 |
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Zip
import Mathlib.Data.Nat.Defs
import Mathlib.Data.List.Infix
#align_import data.list.rotate from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
universe u
variable {α : Type u}
open Nat Function
namespace List
theorem rotate... | Mathlib/Data/List/Rotate.lean | 41 | 41 | theorem rotate_nil (n : ℕ) : ([] : List α).rotate n = [] := by | simp [rotate]
| 1 | 2.718282 | 0 | 0.153846 | 13 | 257 |
import Mathlib.Combinatorics.Enumerative.DoubleCounting
import Mathlib.Combinatorics.SimpleGraph.AdjMatrix
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Data.Set.Finite
#align_import combinatorics.simple_graph.strongly_regular from "leanprover-community/mathlib"@"2b35fc7bea4640cb75e477e83f32fbd5389208... | Mathlib/Combinatorics/SimpleGraph/StronglyRegular.lean | 125 | 134 | theorem sdiff_compl_neighborFinset_inter_eq {v w : V} (h : G.Adj v w) :
((G.neighborFinset v)ᶜ ∩ (G.neighborFinset w)ᶜ) \ ({w} ∪ {v}) =
(G.neighborFinset v)ᶜ ∩ (G.neighborFinset w)ᶜ := by |
ext
simp only [and_imp, mem_union, mem_sdiff, mem_compl, and_iff_left_iff_imp, mem_neighborFinset,
mem_inter, mem_singleton]
rintro hnv hnw (rfl | rfl)
· exact hnv h
· apply hnw
rwa [adj_comm]
| 7 | 1,096.633158 | 2 | 1.428571 | 7 | 1,524 |
import Mathlib.Analysis.MeanInequalities
import Mathlib.Analysis.NormedSpace.WithLp
open Real Set Filter RCLike Bornology Uniformity Topology NNReal ENNReal
noncomputable section
variable (p : ℝ≥0∞) (𝕜 α β : Type*)
namespace WithLp
section DistNorm
section EDist
variable [EDist α] [EDist β]
open scope... | Mathlib/Analysis/NormedSpace/ProdLp.lean | 171 | 174 | theorem prod_edist_eq_sup (f g : WithLp ∞ (α × β)) :
edist f g = edist f.fst g.fst ⊔ edist f.snd g.snd := by |
dsimp [edist]
exact if_neg ENNReal.top_ne_zero
| 2 | 7.389056 | 1 | 0.5 | 6 | 431 |
import Mathlib.Topology.Separation
#align_import topology.shrinking_lemma from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
open Set Function
open scoped Classical
noncomputable section
variable {ι X : Type*} [TopologicalSpace X] [NormalSpace X]
namespace ShrinkingLemma
-- the tr... | Mathlib/Topology/ShrinkingLemma.lean | 118 | 121 | theorem find_mem {c : Set (PartialRefinement u s)} (i : ι) (ne : c.Nonempty) : find c ne i ∈ c := by |
rw [find]
split_ifs with h
exacts [h.choose_spec.1, ne.some_mem]
| 3 | 20.085537 | 1 | 1.5 | 2 | 1,663 |
import Mathlib.Geometry.Euclidean.Sphere.Basic
import Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional
import Mathlib.Tactic.DeriveFintype
#align_import geometry.euclidean.circumcenter from "leanprover-community/mathlib"@"2de9c37fa71dde2f1c6feff19876dd6a7b1519f0"
noncomputable section
open scoped Classical
o... | Mathlib/Geometry/Euclidean/Circumcenter.lean | 76 | 81 | theorem exists_dist_eq_iff_exists_dist_orthogonalProjection_eq {s : AffineSubspace ℝ P} [Nonempty s]
[HasOrthogonalProjection s.direction] {ps : Set P} (hps : ps ⊆ s) (p : P) :
(∃ r, ∀ p1 ∈ ps, dist p1 p = r) ↔ ∃ r, ∀ p1 ∈ ps, dist p1 ↑(orthogonalProjection s p) = r := by |
have h := dist_set_eq_iff_dist_orthogonalProjection_eq hps p
simp_rw [Set.pairwise_eq_iff_exists_eq] at h
exact h
| 3 | 20.085537 | 1 | 1.666667 | 3 | 1,756 |
import Mathlib.Topology.MetricSpace.Basic
#align_import topology.metric_space.infsep from "leanprover-community/mathlib"@"5316314b553dcf8c6716541851517c1a9715e22b"
variable {α β : Type*}
namespace Set
section Einfsep
open ENNReal
open Function
noncomputable def einfsep [EDist α] (s : Set α) : ℝ≥0∞ :=
⨅ (x... | Mathlib/Topology/MetricSpace/Infsep.lean | 79 | 81 | theorem einfsep_lt_iff {d} :
s.einfsep < d ↔ ∃ x ∈ s, ∃ y ∈ s, x ≠ y ∧ edist x y < d := by |
simp_rw [einfsep, iInf_lt_iff, exists_prop]
| 1 | 2.718282 | 0 | 0.25 | 12 | 302 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Monic
#align_import data.polynomial.lifts from "leanprover-community/mathlib"@"63417e01fbc711beaf25fa73b6edb395c0cfddd0"
open Polynomial
noncomputable section
namespace Polynomial
universe u v w
section Semiring
variable {R : Type... | Mathlib/Algebra/Polynomial/Lifts.lean | 257 | 258 | theorem lifts_iff_liftsRing (p : S[X]) : p ∈ lifts f ↔ p ∈ liftsRing f := by |
simp only [lifts, liftsRing, RingHom.mem_range, RingHom.mem_rangeS]
| 1 | 2.718282 | 0 | 1 | 13 | 1,139 |
import Mathlib.RingTheory.EisensteinCriterion
import Mathlib.RingTheory.Polynomial.ScaleRoots
#align_import ring_theory.polynomial.eisenstein.basic from "leanprover-community/mathlib"@"2032a878972d5672e7c27c957e7a6e297b044973"
universe u v w z
variable {R : Type u}
open Ideal Algebra Finset
open Polynomial
na... | Mathlib/RingTheory/Polynomial/Eisenstein/Basic.lean | 111 | 121 | theorem exists_mem_adjoin_mul_eq_pow_natDegree_le {x : S} (hx : aeval x f = 0) (hmo : f.Monic)
(hf : f.IsWeaklyEisensteinAt (Submodule.span R {p})) :
∀ i, (f.map (algebraMap R S)).natDegree ≤ i →
∃ y ∈ adjoin R ({x} : Set S), (algebraMap R S) p * y = x ^ i := by |
intro i hi
obtain ⟨k, hk⟩ := exists_add_of_le hi
rw [hk, pow_add]
obtain ⟨y, hy, H⟩ := exists_mem_adjoin_mul_eq_pow_natDegree hx hmo hf
refine ⟨y * x ^ k, ?_, ?_⟩
· exact Subalgebra.mul_mem _ hy (Subalgebra.pow_mem _ (subset_adjoin (Set.mem_singleton x)) _)
· rw [← mul_assoc _ y, H]
| 7 | 1,096.633158 | 2 | 1.8 | 5 | 1,889 |
import Mathlib.Analysis.NormedSpace.IndicatorFunction
import Mathlib.MeasureTheory.Function.EssSup
import Mathlib.MeasureTheory.Function.AEEqFun
import Mathlib.MeasureTheory.Function.SpecialFunctions.Basic
#align_import measure_theory.function.lp_seminorm from "leanprover-community/mathlib"@"c4015acc0a223449d44061e27... | Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean | 117 | 120 | theorem lintegral_rpow_nnnorm_eq_rpow_snorm' {f : α → F} (hq0_lt : 0 < q) :
(∫⁻ a, (‖f a‖₊ : ℝ≥0∞) ^ q ∂μ) = snorm' f q μ ^ q := by |
rw [snorm', ← ENNReal.rpow_mul, one_div, inv_mul_cancel, ENNReal.rpow_one]
exact (ne_of_lt hq0_lt).symm
| 2 | 7.389056 | 1 | 0.4 | 5 | 402 |
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Polynomial.Degree.Lemmas
#align_import data.polynomial.erase_lead from "leanprover-community/mathlib"@"fa256f00ce018e7b40e1dc756e403c86680bf448"
noncomputable section
open Polynomial
open Polynomial Finset
namespace Polynomial
variable {R : Type*}... | Mathlib/Algebra/Polynomial/EraseLead.lean | 83 | 85 | theorem self_sub_C_mul_X_pow {R : Type*} [Ring R] (f : R[X]) :
f - C f.leadingCoeff * X ^ f.natDegree = f.eraseLead := by |
rw [C_mul_X_pow_eq_monomial, self_sub_monomial_natDegree_leadingCoeff]
| 1 | 2.718282 | 0 | 0.5 | 14 | 465 |
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring.List
import Mathlib.Data.Int.ModEq
import Mathlib.Data.Nat.Bits
import Mathlib.Data.Nat.Log
import Mathlib.Data.List.Indexes
import Mathlib.Data.List.Palindrome
import Mathlib.Tactic.IntervalCases
import Mathlib.Tactic.Linarith
impo... | Mathlib/Data/Nat/Digits.lean | 143 | 153 | theorem digits_add (b : ℕ) (h : 1 < b) (x y : ℕ) (hxb : x < b) (hxy : x ≠ 0 ∨ y ≠ 0) :
digits b (x + b * y) = x :: digits b y := by |
rcases Nat.exists_eq_add_of_le' h with ⟨b, rfl : _ = _ + 2⟩
cases y
· simp [hxb, hxy.resolve_right (absurd rfl)]
dsimp [digits]
rw [digitsAux_def]
· congr
· simp [Nat.add_mod, mod_eq_of_lt hxb]
· simp [add_mul_div_left, div_eq_of_lt hxb]
· apply Nat.succ_pos
| 9 | 8,103.083928 | 2 | 0.857143 | 7 | 752 |
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv
import Mathlib.Analysis.Calculus.FDeriv.Extend
import Mathlib.Analysis.Calculus.Deriv.Prod
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv
#... | Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean | 52 | 60 | theorem hasStrictDerivAt_const_cpow {x y : ℂ} (h : x ≠ 0 ∨ y ≠ 0) :
HasStrictDerivAt (fun y => x ^ y) (x ^ y * log x) y := by |
rcases em (x = 0) with (rfl | hx)
· replace h := h.neg_resolve_left rfl
rw [log_zero, mul_zero]
refine (hasStrictDerivAt_const _ 0).congr_of_eventuallyEq ?_
exact (isOpen_ne.eventually_mem h).mono fun y hy => (zero_cpow hy).symm
· simpa only [cpow_def_of_ne_zero hx, mul_one] using
((hasStrictDe... | 7 | 1,096.633158 | 2 | 1.666667 | 9 | 1,769 |
import Mathlib.Order.Lattice
import Mathlib.Data.List.Sort
import Mathlib.Logic.Equiv.Fin
import Mathlib.Logic.Equiv.Functor
import Mathlib.Data.Fintype.Card
import Mathlib.Order.RelSeries
#align_import order.jordan_holder from "leanprover-community/mathlib"@"91288e351d51b3f0748f0a38faa7613fb0ae2ada"
universe u
... | Mathlib/Order/JordanHolder.lean | 102 | 106 | theorem isMaximal_inf_right_of_isMaximal_sup {x y : X} (hxz : IsMaximal x (x ⊔ y))
(hyz : IsMaximal y (x ⊔ y)) : IsMaximal (x ⊓ y) y := by |
rw [inf_comm]
rw [sup_comm] at hxz hyz
exact isMaximal_inf_left_of_isMaximal_sup hyz hxz
| 3 | 20.085537 | 1 | 1 | 4 | 952 |
import Mathlib.Analysis.Complex.AbsMax
import Mathlib.Analysis.Complex.RemovableSingularity
#align_import analysis.complex.schwarz from "leanprover-community/mathlib"@"3f655f5297b030a87d641ad4e825af8d9679eb0b"
open Metric Set Function Filter TopologicalSpace
open scoped Topology
namespace Complex
section Space... | Mathlib/Analysis/Complex/Schwarz.lean | 92 | 108 | theorem norm_dslope_le_div_of_mapsTo_ball (hd : DifferentiableOn ℂ f (ball c R₁))
(h_maps : MapsTo f (ball c R₁) (ball (f c) R₂)) (hz : z ∈ ball c R₁) :
‖dslope f c z‖ ≤ R₂ / R₁ := by |
have hR₁ : 0 < R₁ := nonempty_ball.1 ⟨z, hz⟩
have hR₂ : 0 < R₂ := nonempty_ball.1 ⟨f z, h_maps hz⟩
rcases eq_or_ne (dslope f c z) 0 with hc | hc
· rw [hc, norm_zero]; exact div_nonneg hR₂.le hR₁.le
rcases exists_dual_vector ℂ _ hc with ⟨g, hg, hgf⟩
have hg' : ‖g‖₊ = 1 := NNReal.eq hg
have hg₀ : ‖g‖₊ ≠ 0 ... | 14 | 1,202,604.284165 | 2 | 2 | 3 | 2,213 |
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
#align_import linear_algebra.affine_space.midpoint from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
open AffineMap AffineEquiv
section
variable (R : Type*) {V V' P P' : Type*} [Ring R] [Invertible (2 : R)] [AddCommGroup V]
[Modu... | Mathlib/LinearAlgebra/AffineSpace/Midpoint.lean | 133 | 137 | theorem midpoint_vsub (p₁ p₂ p : P) :
midpoint R p₁ p₂ -ᵥ p = (⅟ 2 : R) • (p₁ -ᵥ p) + (⅟ 2 : R) • (p₂ -ᵥ p) := by |
rw [← vsub_sub_vsub_cancel_right p₁ p p₂, smul_sub, sub_eq_add_neg, ← smul_neg,
neg_vsub_eq_vsub_rev, add_assoc, invOf_two_smul_add_invOf_two_smul, ← vadd_vsub_assoc,
midpoint_comm, midpoint, lineMap_apply]
| 3 | 20.085537 | 1 | 0.444444 | 9 | 412 |
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
open Filter Metric Set
open scoped ComplexConjugate Real To... | Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean | 76 | 83 | theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by |
refine ⟨fun hz => ⟨arg z, ?_⟩, ?_⟩
· calc
exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul]
_ = z := abs_mul_exp_arg_mul_I z
· rintro ⟨θ, rfl⟩
exact Complex.abs_exp_ofReal_mul_I θ
| 6 | 403.428793 | 2 | 1.571429 | 7 | 1,706 |
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 | 25 | 29 | theorem RingHom.ker_isRadical_iff_reduced_of_surjective {S F} [CommSemiring R] [CommRing S]
[FunLike F R S] [RingHomClass F R S] {f : F} (hf : Function.Surjective f) :
(RingHom.ker f).IsRadical ↔ IsReduced S := by |
simp_rw [isReduced_iff, hf.forall, IsNilpotent, ← map_pow, ← RingHom.mem_ker]
rfl
| 2 | 7.389056 | 1 | 1 | 3 | 985 |
import Mathlib.Analysis.Calculus.ContDiff.RCLike
import Mathlib.MeasureTheory.Measure.Hausdorff
#align_import topology.metric_space.hausdorff_dimension from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
open scoped MeasureTheory ENNReal NNReal Topology
open MeasureTheory MeasureTheory... | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | 133 | 135 | theorem le_dimH_of_hausdorffMeasure_eq_top {s : Set X} {d : ℝ≥0} (h : μH[d] s = ∞) :
↑d ≤ dimH s := by |
rw [dimH_def]; exact le_iSup₂ (α := ℝ≥0∞) d h
| 1 | 2.718282 | 0 | 1 | 4 | 902 |
import Mathlib.CategoryTheory.Monoidal.Functor
#align_import category_theory.monoidal.End from "leanprover-community/mathlib"@"85075bccb68ab7fa49fb05db816233fb790e4fe9"
universe v u
namespace CategoryTheory
variable (C : Type u) [Category.{v} C]
def endofunctorMonoidalCategory : MonoidalCategory (C ⥤ C) where... | Mathlib/CategoryTheory/Monoidal/End.lean | 129 | 131 | theorem ε_inv_naturality {X Y : C} (f : X ⟶ Y) :
(MonoidalFunctor.εIso F).inv.app X ≫ (𝟙_ (C ⥤ C)).map f = F.εIso.inv.app X ≫ f := by |
aesop_cat
| 1 | 2.718282 | 0 | 0 | 1 | 153 |
import Mathlib.MeasureTheory.Measure.Typeclasses
import Mathlib.MeasureTheory.Measure.MutuallySingular
import Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated
open Function Set
open scoped ENNReal Classical
noncomputable section
variable {α β δ : Type*} [MeasurableSpace α] [MeasurableSpace β] {s : Set α} ... | Mathlib/MeasureTheory/Measure/Dirac.lean | 53 | 59 | theorem dirac_apply [MeasurableSingletonClass α] (a : α) (s : Set α) :
dirac a s = s.indicator 1 a := by |
by_cases h : a ∈ s; · rw [dirac_apply_of_mem h, indicator_of_mem h, Pi.one_apply]
rw [indicator_of_not_mem h, ← nonpos_iff_eq_zero]
calc
dirac a s ≤ dirac a {a}ᶜ := measure_mono (subset_compl_comm.1 <| singleton_subset_iff.2 h)
_ = 0 := by simp [dirac_apply' _ (measurableSet_singleton _).compl]
| 5 | 148.413159 | 2 | 1.5 | 6 | 1,557 |
import Mathlib.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
#align_import data.set.intervals.with_bot_top from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
variable {α : Type*}
namespace WithTop
@[simp]
theorem preimage_coe_top : (some : α → WithTop α) ⁻¹' {⊤} =... | Mathlib/Order/Interval/Set/WithBotTop.lean | 67 | 67 | theorem preimage_coe_Ioc : (some : α → WithTop α) ⁻¹' Ioc a b = Ioc a b := by | simp [← Ioi_inter_Iic]
| 1 | 2.718282 | 0 | 0.4 | 15 | 401 |
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 | 76 | 78 | theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} :
r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by |
simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul]
| 1 | 2.718282 | 0 | 0.5 | 6 | 471 |
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Order.Hom.Set
#align_import data.set.intervals.order_iso from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
namespace OrderIso
section Preorder
variable {α β : Type*} [Preorder α] [Preorder β]
@[simp]
theorem preimage_I... | Mathlib/Order/Interval/Set/OrderIso.lean | 88 | 89 | theorem image_Ioo (e : α ≃o β) (a b : α) : e '' Ioo a b = Ioo (e a) (e b) := by |
rw [e.image_eq_preimage, e.symm.preimage_Ioo, e.symm_symm]
| 1 | 2.718282 | 0 | 0.285714 | 14 | 310 |
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Data.Finset.Sym
import Mathlib.Data.Matrix.Basic
#align_import combinatorics.simple_graph.inc_matrix from "leanprover-community/mathlib"@"bb168510ef455e9280a152e7f31673cabd3d7496"
open Finset Matrix SimpleGraph Sym2
open Matrix
namespace SimpleGraph... | Mathlib/Combinatorics/SimpleGraph/IncMatrix.lean | 69 | 72 | theorem incMatrix_apply' [Zero R] [One R] [DecidableEq α] [DecidableRel G.Adj] {a : α}
{e : Sym2 α} : G.incMatrix R a e = if e ∈ G.incidenceSet a then 1 else 0 := by |
unfold incMatrix Set.indicator
convert rfl
| 2 | 7.389056 | 1 | 0.9 | 10 | 784 |
import Mathlib.RingTheory.DedekindDomain.Ideal
#align_import ring_theory.dedekind_domain.factorization from "leanprover-community/mathlib"@"2f588be38bb5bec02f218ba14f82fc82eb663f87"
noncomputable section
open scoped Classical nonZeroDivisors
open Set Function UniqueFactorizationMonoid IsDedekindDomain IsDedekind... | Mathlib/RingTheory/DedekindDomain/Factorization.lean | 131 | 144 | theorem finprod_not_dvd (I : Ideal R) (hI : I ≠ 0) :
¬v.asIdeal ^ ((Associates.mk v.asIdeal).count (Associates.mk I).factors + 1) ∣
∏ᶠ v : HeightOneSpectrum R, v.maxPowDividing I := by |
have hf := finite_mulSupport hI
have h_ne_zero : v.maxPowDividing I ≠ 0 := pow_ne_zero _ v.ne_bot
rw [← mul_finprod_cond_ne v hf, pow_add, pow_one, finprod_cond_ne _ _ hf]
intro h_contr
have hv_prime : Prime v.asIdeal := Ideal.prime_of_isPrime v.ne_bot v.isPrime
obtain ⟨w, hw, hvw'⟩ :=
Prime.exists_mem... | 11 | 59,874.141715 | 2 | 1.714286 | 7 | 1,839 |
import Mathlib.Combinatorics.Enumerative.DoubleCounting
import Mathlib.Combinatorics.SimpleGraph.AdjMatrix
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Data.Set.Finite
#align_import combinatorics.simple_graph.strongly_regular from "leanprover-community/mathlib"@"2b35fc7bea4640cb75e477e83f32fbd5389208... | Mathlib/Combinatorics/SimpleGraph/StronglyRegular.lean | 102 | 106 | theorem IsSRGWith.card_neighborFinset_union_of_not_adj {v w : V} (h : G.IsSRGWith n k ℓ μ)
(hne : v ≠ w) (ha : ¬G.Adj v w) :
(G.neighborFinset v ∪ G.neighborFinset w).card = 2 * k - μ := by |
rw [← h.of_not_adj hne ha]
apply h.card_neighborFinset_union_eq
| 2 | 7.389056 | 1 | 1.428571 | 7 | 1,524 |
import Mathlib.Topology.Algebra.Module.StrongTopology
import Mathlib.Topology.Algebra.Module.LocallyConvex
#align_import analysis.locally_convex.strong_topology from "leanprover-community/mathlib"@"47b12e7f2502f14001f891ca87fbae2b4acaed3f"
open Topology UniformConvergence
variable {R 𝕜₁ 𝕜₂ E F : Type*}
variab... | Mathlib/Analysis/LocallyConvex/StrongTopology.lean | 47 | 54 | theorem locallyConvexSpace (𝔖 : Set (Set E)) (h𝔖₁ : 𝔖.Nonempty)
(h𝔖₂ : DirectedOn (· ⊆ ·) 𝔖) :
LocallyConvexSpace R (UniformConvergenceCLM σ F 𝔖) := by |
apply LocallyConvexSpace.ofBasisZero _ _ _ _
(UniformConvergenceCLM.hasBasis_nhds_zero_of_basis _ _ _ h𝔖₁ h𝔖₂
(LocallyConvexSpace.convex_basis_zero R F)) _
rintro ⟨S, V⟩ ⟨_, _, hVconvex⟩ f hf g hg a b ha hb hab x hx
exact hVconvex (hf x hx) (hg x hx) ha hb hab
| 5 | 148.413159 | 2 | 2 | 1 | 2,485 |
import Mathlib.Data.DFinsupp.Lex
import Mathlib.Order.GameAdd
import Mathlib.Order.Antisymmetrization
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Tactic.AdaptationNote
#align_import data.dfinsupp.well_founded from "leanprover-community/mathlib"@"e9b8651eb1ad354f4de6be35a38ef31efcd2cfaa"
variable {ι : Ty... | Mathlib/Data/DFinsupp/WellFounded.lean | 69 | 98 | theorem lex_fibration [∀ (i) (s : Set ι), Decidable (i ∈ s)] :
Fibration (InvImage (GameAdd (DFinsupp.Lex r s) (DFinsupp.Lex r s)) snd) (DFinsupp.Lex r s)
fun x => piecewise x.2.1 x.2.2 x.1 := by |
rintro ⟨p, x₁, x₂⟩ x ⟨i, hr, hs⟩
simp_rw [piecewise_apply] at hs hr
split_ifs at hs with hp
· refine ⟨⟨{ j | r j i → j ∈ p }, piecewise x₁ x { j | r j i }, x₂⟩,
.fst ⟨i, fun j hj ↦ ?_, ?_⟩, ?_⟩ <;> simp only [piecewise_apply, Set.mem_setOf_eq]
· simp only [if_pos hj]
· split_ifs with hi
· r... | 27 | 532,048,240,601.79865 | 2 | 2 | 4 | 2,437 |
import Mathlib.MeasureTheory.Constructions.BorelSpace.Metric
import Mathlib.Topology.Metrizable.Basic
import Mathlib.Topology.IndicatorConstPointwise
#align_import measure_theory.constructions.borel_space.metrizable from "leanprover-community/mathlib"@"bf6a01357ff5684b1ebcd0f1a13be314fc82c0bf"
open Filter MeasureT... | Mathlib/MeasureTheory/Constructions/BorelSpace/Metrizable.lean | 87 | 101 | theorem aemeasurable_of_unif_approx {β} [MeasurableSpace β] [PseudoMetricSpace β] [BorelSpace β]
{μ : Measure α} {g : α → β}
(hf : ∀ ε > (0 : ℝ), ∃ f : α → β, AEMeasurable f μ ∧ ∀ᵐ x ∂μ, dist (f x) (g x) ≤ ε) :
AEMeasurable g μ := by |
obtain ⟨u, -, u_pos, u_lim⟩ :
∃ u : ℕ → ℝ, StrictAnti u ∧ (∀ n : ℕ, 0 < u n) ∧ Tendsto u atTop (𝓝 0) :=
exists_seq_strictAnti_tendsto (0 : ℝ)
choose f Hf using fun n : ℕ => hf (u n) (u_pos n)
have : ∀ᵐ x ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x)) := by
have : ∀ᵐ x ∂μ, ∀ n, dist (f n x) (g x) ≤ u ... | 11 | 59,874.141715 | 2 | 2 | 3 | 2,089 |
import Mathlib.Algebra.Polynomial.Expand
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.FieldTheory.Minpoly.Field
import Mathlib.RingTheory.PowerBasis
#align_import field_theory.separable from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
... | Mathlib/FieldTheory/Separable.lean | 66 | 67 | theorem separable_of_subsingleton [Subsingleton R] (f : R[X]) : f.Separable := by |
simp [Separable, IsCoprime, eq_iff_true_of_subsingleton]
| 1 | 2.718282 | 0 | 0.9 | 10 | 781 |
import Mathlib.Algebra.ModEq
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Algebra.Periodic
import Mathlib.Data.Int.SuccPred
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.Order.Circular
import Mathlib.Data.List.TFAE
import Mathlib.Data.Set.Lattice
#align_import a... | Mathlib/Algebra/Order/ToIntervalMod.lean | 87 | 89 | theorem toIcoMod_mem_Ico' (b : α) : toIcoMod hp 0 b ∈ Set.Ico 0 p := by |
convert toIcoMod_mem_Ico hp 0 b
exact (zero_add p).symm
| 2 | 7.389056 | 1 | 0.2 | 5 | 281 |
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"
theorem Int.Prime.dvd_mul {m n : ℤ} {p : ℕ} (hp : Nat.Prime p) (h : (p ... | Mathlib/RingTheory/Int/Basic.lean | 139 | 141 | theorem span_natAbs (a : ℤ) : Ideal.span ({(a.natAbs : ℤ)} : Set ℤ) = Ideal.span {a} := by |
rw [Ideal.span_singleton_eq_span_singleton]
exact (associated_natAbs _).symm
| 2 | 7.389056 | 1 | 1.153846 | 13 | 1,227 |
import Mathlib.LinearAlgebra.Matrix.Symmetric
import Mathlib.LinearAlgebra.Matrix.Orthogonal
import Mathlib.Data.Matrix.Kronecker
#align_import linear_algebra.matrix.is_diag from "leanprover-community/mathlib"@"55e2dfde0cff928ce5c70926a3f2c7dee3e2dd99"
namespace Matrix
variable {α β R n m : Type*}
open Function... | Mathlib/LinearAlgebra/Matrix/IsDiag.lean | 159 | 165 | theorem IsDiag.fromBlocks [Zero α] {A : Matrix m m α} {D : Matrix n n α} (ha : A.IsDiag)
(hd : D.IsDiag) : (A.fromBlocks 0 0 D).IsDiag := by |
rintro (i | i) (j | j) hij
· exact ha (ne_of_apply_ne _ hij)
· rfl
· rfl
· exact hd (ne_of_apply_ne _ hij)
| 5 | 148.413159 | 2 | 1.25 | 8 | 1,302 |
import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.Topology.MetricSpace.Contracting
#align_import analysis.ODE.picard_lindelof from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Filter Function Set Metric TopologicalSpace intervalIntegral MeasureTheory
open MeasureTh... | Mathlib/Analysis/ODE/PicardLindelof.lean | 146 | 147 | theorem proj_of_mem {t : ℝ} (ht : t ∈ Icc v.tMin v.tMax) : ↑(v.proj t) = t := by |
simp only [proj, projIcc_of_mem v.tMin_le_tMax ht]
| 1 | 2.718282 | 0 | 1 | 2 | 888 |
import Mathlib.SetTheory.Ordinal.FixedPoint
#align_import set_theory.ordinal.principal from "leanprover-community/mathlib"@"31b269b60935483943542d547a6dd83a66b37dc7"
universe u v w
noncomputable section
open Order
namespace Ordinal
-- Porting note: commented out, doesn't seem necessary
--local infixr:0 "^" => ... | Mathlib/SetTheory/Ordinal/Principal.lean | 52 | 54 | theorem principal_iff_principal_swap {op : Ordinal → Ordinal → Ordinal} {o : Ordinal} :
Principal op o ↔ Principal (Function.swap op) o := by |
constructor <;> exact fun h a b ha hb => h hb ha
| 1 | 2.718282 | 0 | 1.25 | 4 | 1,306 |
import Mathlib.RingTheory.RingHomProperties
#align_import ring_theory.ring_hom.finite from "leanprover-community/mathlib"@"b5aecf07a179c60b6b37c1ac9da952f3b565c785"
namespace RingHom
open scoped TensorProduct
open TensorProduct Algebra.TensorProduct
theorem finite_stableUnderComposition : StableUnderCompositio... | Mathlib/RingTheory/RingHom/Finite.lean | 34 | 42 | theorem finite_stableUnderBaseChange : StableUnderBaseChange @Finite := by |
refine StableUnderBaseChange.mk _ finite_respectsIso ?_
classical
introv h
replace h : Module.Finite R T := by
rw [RingHom.Finite] at h; convert h; ext; simp_rw [Algebra.smul_def]; rfl
suffices Module.Finite S (S ⊗[R] T) by
rw [RingHom.Finite]; convert this; congr; ext; simp_rw [Algebra.smul_def]; rf... | 8 | 2,980.957987 | 2 | 1.333333 | 3 | 1,384 |
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.CharZero.Lemmas
import Mathlib.Data.Finset.NatAntidiagonal
import Mathlib.Data.Nat.Choose.Central
import Mathlib.Data.Tree.Basic
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.GCongr
import Mathlib... | Mathlib/Combinatorics/Enumerative/Catalan.lean | 65 | 65 | theorem catalan_zero : catalan 0 = 1 := by | rw [catalan]
| 1 | 2.718282 | 0 | 0.428571 | 7 | 409 |
import Mathlib.Algebra.MvPolynomial.Derivation
import Mathlib.Algebra.MvPolynomial.Variables
#align_import data.mv_polynomial.pderiv from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
universe u v
namespace MvPolynomial
open Set Function Finsupp
variable {R : ... | Mathlib/Algebra/MvPolynomial/PDeriv.lean | 69 | 77 | theorem pderiv_monomial {i : σ} :
pderiv i (monomial s a) = monomial (s - single i 1) (a * s i) := by |
classical
simp only [pderiv_def, mkDerivation_monomial, Finsupp.smul_sum, smul_eq_mul, ← smul_mul_assoc,
← (monomial _).map_smul]
refine (Finset.sum_eq_single i (fun j _ hne => ?_) fun hi => ?_).trans ?_
· simp [Pi.single_eq_of_ne hne]
· rw [Finsupp.not_mem_support_iff] at hi; simp [hi]
· s... | 7 | 1,096.633158 | 2 | 0.222222 | 9 | 283 |
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 | 358 | 360 | theorem toQuaternion_comp_ofQuaternion :
toQuaternion.comp ofQuaternion = AlgHom.id R ℍ[R,c₁,c₂] := by |
ext : 1 <;> simp
| 1 | 2.718282 | 0 | 1.4 | 10 | 1,508 |
import Mathlib.Algebra.Quaternion
import Mathlib.Tactic.Ring
#align_import algebra.quaternion_basis from "leanprover-community/mathlib"@"3aa5b8a9ed7a7cabd36e6e1d022c9858ab8a8c2d"
open Quaternion
namespace QuaternionAlgebra
structure Basis {R : Type*} (A : Type*) [CommRing R] [Ring A] [Algebra R A] (c₁ c₂ : R) ... | Mathlib/Algebra/QuaternionBasis.lean | 89 | 90 | theorem k_mul_i : q.k * q.i = -c₁ • q.j := by |
rw [← i_mul_j, mul_assoc, j_mul_i, mul_neg, i_mul_k, neg_smul]
| 1 | 2.718282 | 0 | 0.4 | 10 | 394 |
import Mathlib.Algebra.Group.Units.Hom
import Mathlib.Algebra.GroupWithZero.Commute
import Mathlib.Algebra.GroupWithZero.Hom
import Mathlib.GroupTheory.GroupAction.Units
#align_import algebra.group_with_zero.units.lemmas from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
assert_not_exis... | Mathlib/Algebra/GroupWithZero/Units/Lemmas.lean | 49 | 52 | theorem eq_on_inv₀ (f g : F') (h : f a = g a) : f a⁻¹ = g a⁻¹ := by |
rcases eq_or_ne a 0 with (rfl | ha)
· rw [inv_zero, map_zero, map_zero]
· exact (IsUnit.mk0 a ha).eq_on_inv f g h
| 3 | 20.085537 | 1 | 1.5 | 2 | 1,547 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
#align_import ring_theory.polynomial.opposites from "leanprover-community/mathlib"@"63417e01fbc711beaf25fa73b6edb395c0cfddd0"
open Polynomial
open Polynomial MulOpposite
variable {R : Type*} [Semiring R]
noncomputable section
namespace Polynomial
def opRi... | Mathlib/RingTheory/Polynomial/Opposites.lean | 95 | 99 | theorem coeff_opRingEquiv (p : R[X]ᵐᵒᵖ) (n : ℕ) :
(opRingEquiv R p).coeff n = op ((unop p).coeff n) := by |
induction' p using MulOpposite.rec' with p
cases p
rfl
| 3 | 20.085537 | 1 | 0.714286 | 7 | 643 |
import Mathlib.Data.ZMod.Basic
import Mathlib.Algebra.Group.Nat
import Mathlib.Tactic.IntervalCases
import Mathlib.GroupTheory.SpecificGroups.Dihedral
import Mathlib.GroupTheory.SpecificGroups.Cyclic
#align_import group_theory.specific_groups.quaternion from "leanprover-community/mathlib"@"879155bff5af618b9062cbb2915... | Mathlib/GroupTheory/SpecificGroups/Quaternion.lean | 174 | 176 | theorem card [NeZero n] : Fintype.card (QuaternionGroup n) = 4 * n := by |
rw [← Fintype.card_eq.mpr ⟨fintypeHelper⟩, Fintype.card_sum, ZMod.card, two_mul]
ring
| 2 | 7.389056 | 1 | 1.333333 | 6 | 1,410 |
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Asymptotics.SpecificAsymptotics
#align_import analysis.special_functions.compare_exp from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8"
open Asympto... | Mathlib/Analysis/SpecialFunctions/CompareExp.lean | 119 | 121 | theorem abs_im_pow_eventuallyLE_exp_re (hl : IsExpCmpFilter l) (n : ℕ) :
(fun z : ℂ => |z.im| ^ n) ≤ᶠ[l] fun z => Real.exp z.re := by |
simpa using (hl.isLittleO_im_pow_exp_re n).bound zero_lt_one
| 1 | 2.718282 | 0 | 1.333333 | 3 | 1,433 |
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Option.Basic
import Mathlib.Data.List.Defs
import Mathlib.Init.Data.List.Basic
import Mathlib.Init.Data.List.Instances
import Mathlib.Init.Data.List.Lemmas
import Mathlib.Logic.Unique
import Mathlib.Order.Basic
import Mathlib.Tactic.Common
#align_import data.list.basic... | Mathlib/Data/List/Basic.lean | 137 | 138 | theorem mem_map_of_involutive {f : α → α} (hf : Involutive f) {a : α} {l : List α} :
a ∈ map f l ↔ f a ∈ l := by | rw [mem_map, hf.exists_mem_and_apply_eq_iff]
| 1 | 2.718282 | 0 | 0.5 | 2 | 452 |
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 | 70 | 72 | theorem Convex.thickening (hs : Convex ℝ s) (δ : ℝ) : Convex ℝ (thickening δ s) := by |
rw [← add_ball_zero]
exact hs.add (convex_ball 0 _)
| 2 | 7.389056 | 1 | 0.818182 | 11 | 721 |
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Analysis.NormedSpace.Real
#align_import analysis.special_functions.log.basic from "leanprover-community/mathlib"@"f23a09ce6d3f367220dc3cecad6b7eb69eb01690"
open Set Filter Function
open Topology
noncomputable ... | Mathlib/Analysis/SpecialFunctions/Log/Basic.lean | 137 | 139 | theorem log_inv (x : ℝ) : log x⁻¹ = -log x := by |
by_cases hx : x = 0; · simp [hx]
rw [← exp_eq_exp, exp_log_eq_abs (inv_ne_zero hx), exp_neg, exp_log_eq_abs hx, abs_inv]
| 2 | 7.389056 | 1 | 0.583333 | 12 | 525 |
import Mathlib.Analysis.RCLike.Lemmas
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.function.l2_space from "leanprover-community/mathlib"@"83a66c8775fa14ee5180c85cab98e970956401ad"
set_option linter.uppercaseLean3 false... | Mathlib/MeasureTheory/Function/L2Space.lean | 154 | 167 | theorem integral_inner_eq_sq_snorm (f : α →₂[μ] E) :
∫ a, ⟪f a, f a⟫ ∂μ = ENNReal.toReal (∫⁻ a, (‖f a‖₊ : ℝ≥0∞) ^ (2 : ℝ) ∂μ) := by |
simp_rw [inner_self_eq_norm_sq_to_K]
norm_cast
rw [integral_eq_lintegral_of_nonneg_ae]
rotate_left
· exact Filter.eventually_of_forall fun x => sq_nonneg _
· exact ((Lp.aestronglyMeasurable f).norm.aemeasurable.pow_const _).aestronglyMeasurable
congr
ext1 x
have h_two : (2 : ℝ) = ((2 : ℕ) : ℝ) := by ... | 12 | 162,754.791419 | 2 | 0.888889 | 9 | 770 |
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.GroupAction.Quotient
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.Topology.Algebra.Monoid
import Mathlib.Topology.Algebra.Constructions
#align_import topology.algebra.group.basic from "leanprover-community/mathlib"@"3b1890e71632be9e3... | Mathlib/Topology/Algebra/Group/Basic.lean | 71 | 73 | theorem Homeomorph.mulLeft_symm (a : G) : (Homeomorph.mulLeft a).symm = Homeomorph.mulLeft a⁻¹ := by |
ext
rfl
| 2 | 7.389056 | 1 | 1.333333 | 3 | 1,428 |
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Diagonal
import Mathlib.LinearAlgebra.Matrix.Transvection
import Mathlib.MeasureTheory.Group.LIntegral
import Mathlib.MeasureTheory.Integral.Marginal
import Mathlib.MeasureTheory.Measure.Stiel... | Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | 113 | 114 | theorem volume_closedBall (a r : ℝ) : volume (Metric.closedBall a r) = ofReal (2 * r) := by |
rw [closedBall_eq_Icc, volume_Icc, ← sub_add, add_sub_cancel_left, two_mul]
| 1 | 2.718282 | 0 | 0.909091 | 22 | 790 |
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.Data.Nat.Factorial.BigOperators
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Finset.Sym
import Mathlib.Data.Finsupp.Multiset
#align_import data.nat.choose.multinomial from "leanprover-community/mathlib"@"2738d2ca56cbc... | Mathlib/Data/Nat/Choose/Multinomial.lean | 88 | 92 | theorem multinomial_congr {f g : α → ℕ} (h : ∀ a ∈ s, f a = g a) :
multinomial s f = multinomial s g := by |
simp only [multinomial]; congr 1
· rw [Finset.sum_congr rfl h]
· exact Finset.prod_congr rfl fun a ha => by rw [h a ha]
| 3 | 20.085537 | 1 | 0.777778 | 9 | 694 |
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 | 62 | 63 | theorem nth_of_card_le (hf : (setOf p).Finite) {n : ℕ} (hn : hf.toFinset.card ≤ n) :
nth p n = 0 := by | rw [nth, dif_pos hf, List.getD_eq_default]; rwa [Finset.length_sort]
| 1 | 2.718282 | 0 | 0.5 | 6 | 435 |
import Batteries.Data.Sum.Basic
import Batteries.Logic
open Function
namespace Sum
@[simp] protected theorem «forall» {p : α ⊕ β → Prop} :
(∀ x, p x) ↔ (∀ a, p (inl a)) ∧ ∀ b, p (inr b) :=
⟨fun h => ⟨fun _ => h _, fun _ => h _⟩, fun ⟨h₁, h₂⟩ => Sum.rec h₁ h₂⟩
@[simp] protected theorem «exists» {p : α ⊕ β ... | .lake/packages/batteries/Batteries/Data/Sum/Lemmas.lean | 81 | 81 | theorem not_isRight {x : α ⊕ β} : ¬x.isRight ↔ x.isLeft := by | simp
| 1 | 2.718282 | 0 | 0.285714 | 7 | 316 |
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.Polynomial.Coeff
import Mathlib.Algebra.Polynomial.Monomial
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Nat.WithBot
import Mathlib.Data.Nat.Cast.WithTop
import Mathlib.Data.Nat.SuccPred
#align_import data.polynomial.degree.definitions... | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 128 | 129 | theorem natDegree_of_subsingleton [Subsingleton R] : natDegree p = 0 := by |
rw [Subsingleton.elim p 0, natDegree_zero]
| 1 | 2.718282 | 0 | 0.625 | 8 | 546 |
import Mathlib.Algebra.FreeMonoid.Basic
#align_import algebra.free_monoid.count from "leanprover-community/mathlib"@"a2d2e18906e2b62627646b5d5be856e6a642062f"
variable {α : Type*} (p : α → Prop) [DecidablePred p]
namespace FreeAddMonoid
def countP : FreeAddMonoid α →+ ℕ where
toFun := List.countP p
map_zero... | Mathlib/Algebra/FreeMonoid/Count.lean | 43 | 45 | theorem count_of [DecidableEq α] (x y : α) : count x (of y) = (Pi.single x 1 : α → ℕ) y := by |
simp [Pi.single, Function.update, count, countP, List.countP, List.countP.go,
Bool.beq_eq_decide_eq]
| 2 | 7.389056 | 1 | 0.5 | 2 | 429 |
import Mathlib.Data.Int.Range
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.MulChar.Basic
#align_import number_theory.legendre_symbol.zmod_char from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
namespace ZMod
section QuadCharModP
@[simps]
def χ₄ : MulChar (ZMod 4) ℤ... | Mathlib/NumberTheory/LegendreSymbol/ZModChar.lean | 60 | 62 | theorem χ₄_int_mod_four (n : ℤ) : χ₄ n = χ₄ (n % 4 : ℤ) := by |
rw [← ZMod.intCast_mod n 4]
norm_cast
| 2 | 7.389056 | 1 | 1.25 | 12 | 1,332 |
import Mathlib.Order.Interval.Finset.Nat
#align_import data.fin.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
assert_not_exists MonoidWithZero
open Finset Fin Function
namespace Fin
variable (n : ℕ)
instance instLocallyFiniteOrder : LocallyFiniteOrder (Fin n) :=
Orde... | Mathlib/Order/Interval/Finset/Fin.lean | 84 | 85 | theorem map_valEmbedding_Ico : (Ico a b).map Fin.valEmbedding = Ico ↑a ↑b := by |
simp [Ico_eq_finset_subtype, Finset.fin, Finset.map_map]
| 1 | 2.718282 | 0 | 0.125 | 16 | 251 |
import Mathlib.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
#align_import data.set.intervals.with_bot_top from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
variable {α : Type*}
namespace WithTop
@[simp]
theorem preimage_coe_top : (some : α → WithTop α) ⁻¹' {⊤} =... | Mathlib/Order/Interval/Set/WithBotTop.lean | 93 | 94 | theorem image_coe_Ici : (some : α → WithTop α) '' Ici a = Ico (a : WithTop α) ⊤ := by |
rw [← preimage_coe_Ici, image_preimage_eq_inter_range, range_coe, Ici_inter_Iio]
| 1 | 2.718282 | 0 | 0.4 | 15 | 401 |
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