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.Topology.Instances.ENNReal
import Mathlib.MeasureTheory.Measure.Dirac
#align_import probability.probability_mass_function.basic from "leanprover-community/mathlib"@"4ac69b290818724c159de091daa3acd31da0ee6d"
noncomputable section
variable {α β γ : Type*}
open scoped Classical
open NNReal ENNReal M... | Mathlib/Probability/ProbabilityMassFunction/Basic.lean | 107 | 108 | theorem apply_eq_zero_iff (p : PMF α) (a : α) : p a = 0 ↔ a ∉ p.support := by |
rw [mem_support_iff, Classical.not_not]
| 1 | 2.718282 | 0 | 1 | 3 | 1,003 |
import Mathlib.Algebra.Group.NatPowAssoc
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Induction
import Mathlib.Algebra.Polynomial.Eval
namespace Polynomial
section MulActionWithZero
variable {R : Type*} [Semiring R] (r : R) (p : R[X]) {S : Type*} [AddCommMonoid S] [Pow S ℕ]
[Mu... | Mathlib/Algebra/Polynomial/Smeval.lean | 57 | 58 | theorem smeval_C : (C r).smeval x = r • x ^ 0 := by |
simp only [smeval_eq_sum, smul_pow, zero_smul, sum_C_index]
| 1 | 2.718282 | 0 | 0.333333 | 9 | 368 |
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Algebra.Ring.Int
import Mathlib.NumberTheory.Padics.PadicVal
import Mathlib.RingTheory.Ideal.Quotient
#align_import number_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
open I... | Mathlib/NumberTheory/Multiplicity.lean | 39 | 43 | theorem dvd_geom_sum₂_iff_of_dvd_sub {x y p : R} (h : p ∣ x - y) :
(p ∣ ∑ i ∈ range n, x ^ i * y ^ (n - 1 - i)) ↔ p ∣ n * y ^ (n - 1) := by |
rw [← mem_span_singleton, ← Ideal.Quotient.eq] at h
simp only [← mem_span_singleton, ← eq_zero_iff_mem, RingHom.map_geom_sum₂, h, geom_sum₂_self,
_root_.map_mul, map_pow, map_natCast]
| 3 | 20.085537 | 1 | 1.25 | 4 | 1,300 |
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Inverse
#align_import analysis.calculus.cont_diff from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputab... | Mathlib/Analysis/Calculus/ContDiff/Basic.lean | 122 | 123 | theorem contDiffOn_of_subsingleton [Subsingleton F] : ContDiffOn 𝕜 n f s := by |
rw [Subsingleton.elim f fun _ => 0]; exact contDiffOn_const
| 1 | 2.718282 | 0 | 0.875 | 8 | 760 |
import Mathlib.Tactic.Ring
import Mathlib.Data.PNat.Prime
#align_import data.pnat.xgcd from "leanprover-community/mathlib"@"6afc9b06856ad973f6a2619e3e8a0a8d537a58f2"
open Nat
namespace PNat
structure XgcdType where
wp : ℕ
x : ℕ
y : ℕ
zp : ℕ
ap : ℕ
bp : ℕ
deriving Inhabited
#alig... | Mathlib/Data/PNat/Xgcd.lean | 136 | 137 | theorem v_eq_succ_vp : u.v = succ₂ u.vp := by |
ext <;> dsimp [v, vp, w, z, a, b, succ₂] <;> ring_nf
| 1 | 2.718282 | 0 | 1 | 4 | 924 |
import Mathlib.CategoryTheory.Balanced
import Mathlib.CategoryTheory.Limits.EssentiallySmall
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
import Mathlib.CategoryTheory.Subobject.Lattice
import Mathlib.CategoryTheory.Subobject.WellPowered
import Mathlib.Data.S... | Mathlib/CategoryTheory/Generator.lean | 109 | 110 | theorem isCoseparating_unop_iff (𝒢 : Set Cᵒᵖ) : IsCoseparating 𝒢.unop ↔ IsSeparating 𝒢 := by |
rw [← isSeparating_op_iff, Set.unop_op]
| 1 | 2.718282 | 0 | 1.333333 | 6 | 1,424 |
import Mathlib.Algebra.MvPolynomial.Monad
#align_import data.mv_polynomial.expand from "leanprover-community/mathlib"@"5da451b4c96b4c2e122c0325a7fce17d62ee46c6"
namespace MvPolynomial
variable {σ τ R S : Type*} [CommSemiring R] [CommSemiring S]
noncomputable def expand (p : ℕ) : MvPolynomial σ R →ₐ[R] MvPolyno... | Mathlib/Algebra/MvPolynomial/Expand.lean | 77 | 78 | theorem map_expand (f : R →+* S) (p : ℕ) (φ : MvPolynomial σ R) :
map f (expand p φ) = expand p (map f φ) := by | simp [expand, map_bind₁]
| 1 | 2.718282 | 0 | 0.571429 | 7 | 523 |
import Mathlib.Algebra.Group.Indicator
import Mathlib.Algebra.Group.Submonoid.Basic
import Mathlib.Data.Set.Finite
#align_import data.finsupp.defs from "leanprover-community/mathlib"@"842328d9df7e96fd90fc424e115679c15fb23a71"
noncomputable section
open Finset Function
variable {α β γ ι M M' N P G H R S : Type*}... | Mathlib/Data/Finsupp/Defs.lean | 185 | 185 | theorem coe_eq_zero {f : α →₀ M} : (f : α → M) = 0 ↔ f = 0 := by | rw [← coe_zero, DFunLike.coe_fn_eq]
| 1 | 2.718282 | 0 | 0.2 | 5 | 268 |
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Order.Bounds.OrderIso
import Mathlib.Tactic.Positivity.Core
#align_import algebra.order.field.basic from "leanprover-community/mathlib"@"8477... | Mathlib/Algebra/Order/Field/Basic.lean | 122 | 122 | theorem inv_mul_lt_iff' (h : 0 < b) : b⁻¹ * a < c ↔ a < c * b := by | rw [inv_mul_lt_iff h, mul_comm]
| 1 | 2.718282 | 0 | 0.25 | 16 | 288 |
import Mathlib.Analysis.Complex.Circle
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
#align_import analysis.complex.isometry from "leanprover-community/mathlib"@"ae690b0c236e488a0043f6faa8ce3546e7f2f9c5"
noncomputable section
open Complex
open ComplexConjugate
... | Mathlib/Analysis/Complex/Isometry.lean | 90 | 93 | theorem LinearIsometry.re_apply_eq_re_of_add_conj_eq (f : ℂ →ₗᵢ[ℝ] ℂ)
(h₃ : ∀ z, z + conj z = f z + conj (f z)) (z : ℂ) : (f z).re = z.re := by |
simpa [ext_iff, add_re, add_im, conj_re, conj_im, ← two_mul,
show (2 : ℝ) ≠ 0 by simp [two_ne_zero]] using (h₃ z).symm
| 2 | 7.389056 | 1 | 1.571429 | 7 | 1,705 |
import Mathlib.Deprecated.Group
#align_import deprecated.ring from "leanprover-community/mathlib"@"5a3e819569b0f12cbec59d740a2613018e7b8eec"
universe u v w
variable {α : Type u}
structure IsSemiringHom {α : Type u} {β : Type v} [Semiring α] [Semiring β] (f : α → β) : Prop where
map_zero : f 0 = 0
map... | Mathlib/Deprecated/Ring.lean | 58 | 63 | theorem comp (hf : IsSemiringHom f) {γ} [Semiring γ] {g : β → γ} (hg : IsSemiringHom g) :
IsSemiringHom (g ∘ f) :=
{ map_zero := by | simpa [map_zero hf] using map_zero hg
map_one := by simpa [map_one hf] using map_one hg
map_add := fun {x y} => by simp [map_add hf, map_add hg]
map_mul := fun {x y} => by simp [map_mul hf, map_mul hg] }
| 4 | 54.59815 | 2 | 0.5 | 8 | 496 |
import Mathlib.Algebra.Polynomial.Mirror
import Mathlib.Analysis.Complex.Polynomial
#align_import data.polynomial.unit_trinomial from "leanprover-community/mathlib"@"302eab4f46abb63de520828de78c04cb0f9b5836"
namespace Polynomial
open scoped Polynomial
open Finset
section Semiring
variable {R : Type*} [Semirin... | Mathlib/Algebra/Polynomial/UnitTrinomial.lean | 100 | 102 | theorem trinomial_trailingCoeff (hkm : k < m) (hmn : m < n) (hu : u ≠ 0) :
(trinomial k m n u v w).trailingCoeff = u := by |
rw [trailingCoeff, trinomial_natTrailingDegree hkm hmn hu, trinomial_trailing_coeff' hkm hmn]
| 1 | 2.718282 | 0 | 1.111111 | 9 | 1,193 |
import Mathlib.Order.MinMax
import Mathlib.Data.Set.Subsingleton
import Mathlib.Tactic.Says
#align_import data.set.intervals.basic from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
open Function
open OrderDual (toDual ofDual)
variable {α β : Type*}
namespace Set
section Preorder
v... | Mathlib/Order/Interval/Set/Basic.lean | 186 | 186 | theorem left_mem_Ico : a ∈ Ico a b ↔ a < b := by | simp [le_refl]
| 1 | 2.718282 | 0 | 0 | 4 | 54 |
import Mathlib.Algebra.Group.Aut
import Mathlib.Algebra.Group.Invertible.Basic
import Mathlib.Algebra.GroupWithZero.Units.Basic
import Mathlib.GroupTheory.GroupAction.Units
#align_import group_theory.group_action.group from "leanprover-community/mathlib"@"3b52265189f3fb43aa631edffce5d060fafaf82f"
universe u v w
... | Mathlib/GroupTheory/GroupAction/Group.lean | 30 | 30 | theorem inv_smul_smul (c : α) (x : β) : c⁻¹ • c • x = x := by | rw [smul_smul, mul_left_inv, one_smul]
| 1 | 2.718282 | 0 | 0 | 2 | 60 |
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.Probability.Independence.Basic
#align_import probability.integration from "leanprover-community/mathlib"@"2f8347015b12b0864dfaf366ec4909eb70c78740"
noncomputable section
open Set MeasureTheory
open scoped ENNReal MeasureTheory
variable {Ω : Type*... | Mathlib/Probability/Integration.lean | 45 | 73 | theorem lintegral_mul_indicator_eq_lintegral_mul_lintegral_indicator {Mf mΩ : MeasurableSpace Ω}
{μ : Measure Ω} (hMf : Mf ≤ mΩ) (c : ℝ≥0∞) {T : Set Ω} (h_meas_T : MeasurableSet T)
(h_ind : IndepSets {s | MeasurableSet[Mf] s} {T} μ) (h_meas_f : Measurable[Mf] f) :
(∫⁻ ω, f ω * T.indicator (fun _ => c) ω ∂μ)... |
revert f
have h_mul_indicator : ∀ g, Measurable g → Measurable fun a => g a * T.indicator (fun _ => c) a :=
fun g h_mg => h_mg.mul (measurable_const.indicator h_meas_T)
apply @Measurable.ennreal_induction _ Mf
· intro c' s' h_meas_s'
simp_rw [← inter_indicator_mul]
rw [lintegral_indicator _ (Measur... | 24 | 26,489,122,129.84347 | 2 | 2 | 2 | 2,223 |
import Mathlib.CategoryTheory.Subobject.Lattice
#align_import category_theory.subobject.limits from "leanprover-community/mathlib"@"956af7c76589f444f2e1313911bad16366ea476d"
universe v u
noncomputable section
open CategoryTheory CategoryTheory.Category CategoryTheory.Limits CategoryTheory.Subobject Opposite
var... | Mathlib/CategoryTheory/Subobject/Limits.lean | 350 | 353 | theorem factorThruImageSubobject_comp_self_assoc {W W' : C} (k : W ⟶ W') (k' : W' ⟶ X) (h) :
(imageSubobject f).factorThru (k ≫ k' ≫ f) h = k ≫ k' ≫ factorThruImageSubobject f := by |
ext
simp
| 2 | 7.389056 | 1 | 0.263158 | 19 | 308 |
import Mathlib.Algebra.CharP.ExpChar
import Mathlib.GroupTheory.OrderOfElement
#align_import algebra.char_p.two from "leanprover-community/mathlib"@"7f1ba1a333d66eed531ecb4092493cd1b6715450"
variable {R ι : Type*}
namespace CharTwo
section Semiring
variable [Semiring R] [CharP R 2]
theorem two_eq_zero : (2 : ... | Mathlib/Algebra/CharP/Two.lean | 55 | 55 | theorem bit1_apply_eq_one (x : R) : (bit1 x : R) = 1 := by | simp
| 1 | 2.718282 | 0 | 0.2 | 10 | 273 |
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 | 73 | 78 | theorem card_filter_univ_eq_vector_get_eq_count [DecidableEq α] (a : α) (v : Vector α n) :
(univ.filter fun i => a = v.get i).card = v.toList.count a := by |
induction' v with n x xs hxs
· simp
· simp_rw [card_filter_univ_succ', Vector.get_cons_zero, Vector.toList_cons, Function.comp,
Vector.get_cons_succ, hxs, List.count_cons, add_comm (ite (a = x) 1 0)]
| 4 | 54.59815 | 2 | 1.4 | 5 | 1,485 |
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv
import Mathlib.Analysis.SpecialFunctions.Log.Basic
#align_import analysis.special_functions.arsinh from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open Function Filter Set
open scoped Topology
name... | Mathlib/Analysis/SpecialFunctions/Arsinh.lean | 83 | 84 | theorem cosh_arsinh (x : ℝ) : cosh (arsinh x) = √(1 + x ^ 2) := by |
rw [← sqrt_sq (cosh_pos _).le, cosh_sq', sinh_arsinh]
| 1 | 2.718282 | 0 | 0.625 | 8 | 547 |
import Mathlib.Algebra.Category.MonCat.Basic
import Mathlib.CategoryTheory.Limits.HasLimits
import Mathlib.CategoryTheory.ConcreteCategory.Elementwise
#align_import algebra.category.Mon.colimits from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe v
open CategoryTheory
open Ca... | Mathlib/Algebra/Category/MonCat/Colimits.lean | 188 | 191 | theorem cocone_naturality_components (j j' : J) (f : j ⟶ j') (x : F.obj j) :
(coconeMorphism F j') (F.map f x) = (coconeMorphism F j) x := by |
rw [← cocone_naturality F f]
rfl
| 2 | 7.389056 | 1 | 1 | 2 | 966 |
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 | 468 | 476 | theorem measurableSet_region_between_oc (hf : Measurable f) (hg : Measurable g)
(hs : MeasurableSet s) :
MeasurableSet { p : α × ℝ | p.fst ∈ s ∧ p.snd ∈ Ioc (f p.fst) (g p.fst) } := by |
dsimp only [regionBetween, Ioc, mem_setOf_eq, setOf_and]
refine
MeasurableSet.inter ?_
((measurableSet_lt (hf.comp measurable_fst) measurable_snd).inter
(measurableSet_le measurable_snd (hg.comp measurable_fst)))
exact measurable_fst hs
| 6 | 403.428793 | 2 | 0.909091 | 22 | 790 |
import Mathlib.Algebra.DirectLimit
import Mathlib.Algebra.CharP.Algebra
import Mathlib.FieldTheory.IsAlgClosed.Basic
import Mathlib.FieldTheory.SplittingField.Construction
#align_import field_theory.is_alg_closed.algebraic_closure from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
univ... | Mathlib/FieldTheory/IsAlgClosed/AlgebraicClosure.lean | 77 | 81 | theorem toSplittingField_evalXSelf {s : Finset (MonicIrreducible k)} {f} (hf : f ∈ s) :
toSplittingField k s (evalXSelf k f) = 0 := by |
rw [toSplittingField, evalXSelf, ← AlgHom.coe_toRingHom, hom_eval₂, AlgHom.coe_toRingHom,
MvPolynomial.aeval_X, dif_pos hf, ← MvPolynomial.algebraMap_eq, AlgHom.comp_algebraMap]
exact map_rootOfSplits _ _ _
| 3 | 20.085537 | 1 | 1.666667 | 3 | 1,763 |
import Mathlib.Algebra.MvPolynomial.Basic
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
#align_import linear_algebra.matrix.mv_polynomial from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa1"
set_option linter.uppercaseLean3 false
varia... | Mathlib/LinearAlgebra/Matrix/MvPolynomial.lean | 75 | 80 | theorem det_mvPolynomialX_ne_zero [DecidableEq m] [Fintype m] [CommRing R] [Nontrivial R] :
det (mvPolynomialX m m R) ≠ 0 := by |
intro h_det
have := congr_arg Matrix.det (mvPolynomialX_mapMatrix_eval (1 : Matrix m m R))
rw [det_one, ← RingHom.map_det, h_det, RingHom.map_zero] at this
exact zero_ne_one this
| 4 | 54.59815 | 2 | 2 | 1 | 2,424 |
import Mathlib.Probability.ProbabilityMassFunction.Monad
#align_import probability.probability_mass_function.constructions from "leanprover-community/mathlib"@"4ac69b290818724c159de091daa3acd31da0ee6d"
universe u
namespace PMF
noncomputable section
variable {α β γ : Type*}
open scoped Classical
open NNReal ENN... | Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | 60 | 60 | theorem mem_support_map_iff : b ∈ (map f p).support ↔ ∃ a ∈ p.support, f a = b := by | simp
| 1 | 2.718282 | 0 | 0.2 | 10 | 279 |
import Mathlib.SetTheory.Cardinal.Finite
#align_import data.set.ncard from "leanprover-community/mathlib"@"74c2af38a828107941029b03839882c5c6f87a04"
namespace Set
variable {α β : Type*} {s t : Set α}
noncomputable def encard (s : Set α) : ℕ∞ := PartENat.withTopEquiv (PartENat.card s)
@[simp] theorem encard_uni... | Mathlib/Data/Set/Card.lean | 73 | 76 | theorem Finite.encard_eq_coe_toFinset_card (h : s.Finite) : s.encard = h.toFinset.card := by |
have := h.fintype
rw [encard, PartENat.card_eq_coe_fintype_card,
PartENat.withTopEquiv_natCast, toFinite_toFinset, toFinset_card]
| 3 | 20.085537 | 1 | 0.5 | 14 | 454 |
import Mathlib.Algebra.Group.Indicator
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.Field.Rat
import Mathlib.GroupTheory.GroupAction.Group
import Mathlib.GroupTheory.GroupAction.Pi
#align_import algebra.module.basic from "leanprover-community/mathlib"@"30413fc89f202a090a54d78e540963ed3de0056e"
o... | Mathlib/Algebra/Module/Basic.lean | 61 | 66 | theorem map_ratCast_smul [AddCommGroup M] [AddCommGroup M₂] {F : Type*} [FunLike F M M₂]
[AddMonoidHomClass F M M₂] (f : F) (R S : Type*) [DivisionRing R] [DivisionRing S] [Module R M]
[Module S M₂] (c : ℚ) (x : M) :
f ((c : R) • x) = (c : S) • f x := by |
rw [Rat.cast_def, Rat.cast_def, div_eq_mul_inv, div_eq_mul_inv, mul_smul, mul_smul,
map_intCast_smul f R S, map_inv_natCast_smul f R S]
| 2 | 7.389056 | 1 | 1.666667 | 3 | 1,767 |
namespace Nat
@[reducible] def Coprime (m n : Nat) : Prop := gcd m n = 1
instance (m n : Nat) : Decidable (Coprime m n) := inferInstanceAs (Decidable (_ = 1))
theorem coprime_iff_gcd_eq_one : Coprime m n ↔ gcd m n = 1 := .rfl
theorem Coprime.gcd_eq_one : Coprime m n → gcd m n = 1 := id
theorem Coprime.symm ... | .lake/packages/batteries/Batteries/Data/Nat/Gcd.lean | 53 | 55 | theorem Coprime.gcd_mul_right_cancel_right (n : Nat)
(H : Coprime k m) : gcd m (n * k) = gcd m n := by |
rw [Nat.mul_comm n k, H.gcd_mul_left_cancel_right n]
| 1 | 2.718282 | 0 | 1 | 9 | 1,124 |
import Mathlib.Algebra.Algebra.Quasispectrum
import Mathlib.FieldTheory.IsAlgClosed.Spectrum
import Mathlib.Analysis.Complex.Liouville
import Mathlib.Analysis.Complex.Polynomial
import Mathlib.Analysis.Analytic.RadiusLiminf
import Mathlib.Topology.Algebra.Module.CharacterSpace
import Mathlib.Analysis.NormedSpace.Expon... | Mathlib/Analysis/NormedSpace/Spectrum.lean | 79 | 80 | theorem SpectralRadius.of_subsingleton [Subsingleton A] (a : A) : spectralRadius 𝕜 a = 0 := by |
simp [spectralRadius]
| 1 | 2.718282 | 0 | 1 | 4 | 1,004 |
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 | 112 | 113 | theorem le_dedup_self {s : Multiset α} : s ≤ dedup s ↔ Nodup s := by |
rw [le_dedup, and_iff_right le_rfl]
| 1 | 2.718282 | 0 | 0.25 | 4 | 296 |
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 | 107 | 109 | theorem binomial_eq_choose [DecidableEq α] (h : a ≠ b) :
multinomial {a, b} f = (f a + f b).choose (f a) := by |
simp [binomial_eq h, choose_eq_factorial_div_factorial (Nat.le_add_right _ _)]
| 1 | 2.718282 | 0 | 0.777778 | 9 | 694 |
import Mathlib.Probability.Kernel.Disintegration.Unique
import Mathlib.Probability.Notation
#align_import probability.kernel.cond_distrib from "leanprover-community/mathlib"@"00abe0695d8767201e6d008afa22393978bb324d"
open MeasureTheory Set Filter TopologicalSpace
open scoped ENNReal MeasureTheory ProbabilityTheo... | Mathlib/Probability/Kernel/CondDistrib.lean | 171 | 174 | theorem _root_.MeasureTheory.Integrable.norm_integral_condDistrib_map
(hY : AEMeasurable Y μ) (hf_int : Integrable f (μ.map fun a => (X a, Y a))) :
Integrable (fun x => ‖∫ y, f (x, y) ∂condDistrib Y X μ x‖) (μ.map X) := by |
rw [condDistrib, ← Measure.fst_map_prod_mk₀ (X := X) hY]; exact hf_int.norm_integral_condKernel
| 1 | 2.718282 | 0 | 0.777778 | 9 | 692 |
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 | 49 | 49 | theorem rotate'_nil (n : ℕ) : ([] : List α).rotate' n = [] := by | cases n <;> rfl
| 1 | 2.718282 | 0 | 0.153846 | 13 | 257 |
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 | 144 | 145 | theorem catalan_two : catalan 2 = 2 := by |
norm_num [catalan_eq_centralBinom_div, Nat.centralBinom, Nat.choose]
| 1 | 2.718282 | 0 | 0.428571 | 7 | 409 |
import Mathlib.Topology.Bases
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.Compactness.SigmaCompact
open Set Filter Topology TopologicalSpace
universe u v
variable {X : Type u} {Y : Type v} {ι : Type*}
variable [TopologicalSpace X] [TopologicalSpace Y] {s t : Set X}
section Lindelof
def I... | Mathlib/Topology/Compactness/Lindelof.lean | 69 | 75 | theorem IsLindelof.induction_on (hs : IsLindelof s) {p : Set X → Prop}
(hmono : ∀ ⦃s t⦄, s ⊆ t → p t → p s)
(hcountable_union : ∀ (S : Set (Set X)), S.Countable → (∀ s ∈ S, p s) → p (⋃₀ S))
(hnhds : ∀ x ∈ s, ∃ t ∈ 𝓝[s] x, p t) : p s := by |
let f : Filter X := ofCountableUnion p hcountable_union (fun t ht _ hsub ↦ hmono hsub ht)
have : sᶜ ∈ f := hs.compl_mem_sets_of_nhdsWithin (by simpa [f] using hnhds)
rwa [← compl_compl s]
| 3 | 20.085537 | 1 | 1.5 | 6 | 1,603 |
import Mathlib.MeasureTheory.Group.GeometryOfNumbers
import Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls
import Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic
#align_import number_theory.number_field.canonical_embedding from "leanprover-community/mathlib"@"60da01b41bbe4206f05d34fd70c8dd7498717a30"
... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean | 302 | 304 | theorem convexBodySumFun_neg (x : E K) :
convexBodySumFun (- x) = convexBodySumFun x := by |
simp_rw [convexBodySumFun, normAtPlace_neg]
| 1 | 2.718282 | 0 | 1.666667 | 15 | 1,803 |
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.special_functions.log.base from "leanprover-community/mathlib"@"f23a09ce6d3f367220dc3cecad6b7eb69eb01690"
open Set Filter Function
open Topology
noncomputable section
namespace Real
variable {b x y : ℝ}
-- @... | Mathlib/Analysis/SpecialFunctions/Log/Base.lean | 68 | 69 | theorem logb_neg_eq_logb (x : ℝ) : logb b (-x) = logb b x := by |
rw [← logb_abs x, ← logb_abs (-x), abs_neg]
| 1 | 2.718282 | 0 | 0.25 | 20 | 300 |
import Mathlib.Data.Set.Image
import Mathlib.Data.SProd
#align_import data.set.prod from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc9524185ae994553ebf4"
open Function
namespace Set
section Prod
variable {α β γ δ : Type*} {s s₁ s₂ : Set α} {t t₁ t₂ : Set β} {a : α} {b : β}
theorem Subsingleton.pro... | Mathlib/Data/Set/Prod.lean | 84 | 86 | theorem prod_empty : s ×ˢ (∅ : Set β) = ∅ := by |
ext
exact and_false_iff _
| 2 | 7.389056 | 1 | 0.692308 | 13 | 638 |
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev from "leanprover-community/mathlib"@"d774451114d6045faeb6751c396bea1eb9058946"
namespace Polynomial.Chebyshev
set_option linter.uppercaseLean3 false -- `T` `U` `X`
open Polynomial
v... | Mathlib/RingTheory/Polynomial/Chebyshev.lean | 156 | 157 | theorem U_sub_two (n : ℤ) : U R (n - 2) = 2 * X * U R (n - 1) - U R n := by |
linear_combination (norm := ring_nf) U_add_two R (n - 2)
| 1 | 2.718282 | 0 | 0.166667 | 12 | 265 |
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Analysis.NormedSpace.LinearIsometry
import Mathlib.Algebra.Star.SelfAdjoint
import Mathlib.Algebra.Star.Subalgebra
import Mathlib.Algebra.Star.Unitary
import Mathlib.Topology.Algebra.Module.Star
#align_import analysis.no... | Mathlib/Analysis/NormedSpace/Star/Basic.lean | 135 | 137 | theorem star_mul_self_eq_zero_iff (x : E) : x⋆ * x = 0 ↔ x = 0 := by |
rw [← norm_eq_zero, norm_star_mul_self]
exact mul_self_eq_zero.trans norm_eq_zero
| 2 | 7.389056 | 1 | 0.5 | 8 | 473 |
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Preserves.Basic
#align_import category_theory.limits.preserves.shapes.pullbacks from "leanprover-community/mathlib"@"f11e306adb9f2a393539d2bb4293bf1b42caa7ac"
noncomputable section
universe v₁ v₂ u₁ u₂
-- Porting note: ne... | Mathlib/CategoryTheory/Limits/Preserves/Shapes/Pullbacks.lean | 225 | 228 | theorem PreservesPushout.inl_iso_hom :
pushout.inl ≫ (PreservesPushout.iso G f g).hom = G.map pushout.inl := by |
delta PreservesPushout.iso
simp
| 2 | 7.389056 | 1 | 0.25 | 8 | 287 |
import Mathlib.Order.Filter.Basic
import Mathlib.Topology.Bases
import Mathlib.Data.Set.Accumulate
import Mathlib.Topology.Bornology.Basic
import Mathlib.Topology.LocallyFinite
open Set Filter Topology TopologicalSpace Classical Function
universe u v
variable {X : Type u} {Y : Type v} {ι : Type*}
variable [Topolog... | Mathlib/Topology/Compactness/Compact.lean | 48 | 52 | theorem IsCompact.compl_mem_sets (hs : IsCompact s) {f : Filter X} (hf : ∀ x ∈ s, sᶜ ∈ 𝓝 x ⊓ f) :
sᶜ ∈ f := by |
contrapose! hf
simp only [not_mem_iff_inf_principal_compl, compl_compl, inf_assoc] at hf ⊢
exact @hs _ hf inf_le_right
| 3 | 20.085537 | 1 | 1.6 | 5 | 1,733 |
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Algebra.Order.Ring.Int
import Mathlib.Data.Int.GCD
instance : GCDMonoid ℕ where
gcd := Nat.gcd
lcm := Nat.lcm
gcd_dvd_left := Nat.gcd_dvd_left
gcd_dvd_right := Nat.gcd_dvd_right
dvd_gcd := Nat.dvd_gcd
gcd_mul_lcm a b := by rw [Nat.gcd_mul_lcm]; rfl
... | Mathlib/Algebra/GCDMonoid/Nat.lean | 82 | 83 | theorem abs_eq_normalize (z : ℤ) : |z| = normalize z := by |
cases le_total 0 z <;> simp [-normalize_apply, normalize_of_nonneg, normalize_of_nonpos, *]
| 1 | 2.718282 | 0 | 1 | 4 | 1,077 |
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 | 97 | 99 | theorem reduceOption_get?_iff {l : List (Option α)} {x : α} :
(∃ i, l.get? i = some (some x)) ↔ ∃ i, l.reduceOption.get? i = some x := by |
rw [← mem_iff_get?, ← mem_iff_get?, reduceOption_mem_iff]
| 1 | 2.718282 | 0 | 0.615385 | 13 | 540 |
import Mathlib.Algebra.FreeNonUnitalNonAssocAlgebra
import Mathlib.Algebra.Lie.NonUnitalNonAssocAlgebra
import Mathlib.Algebra.Lie.UniversalEnveloping
import Mathlib.GroupTheory.GroupAction.Ring
#align_import algebra.lie.free from "leanprover-community/mathlib"@"841ac1a3d9162bf51c6327812ecb6e5e71883ac4"
universe ... | Mathlib/Algebra/Lie/Free.lean | 87 | 88 | theorem Rel.addLeft (a : lib R X) {b c : lib R X} (h : Rel R X b c) : Rel R X (a + b) (a + c) := by |
rw [add_comm _ b, add_comm _ c]; exact h.add_right _
| 1 | 2.718282 | 0 | 0.2 | 5 | 274 |
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.Rat.Denumerable
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.SetTheory.Cardinal.Continuum
#align_import data.real.cardinality from "leanprover-community/mathlib"@"7e7aaccf9b0182576cabdde36cf1b5ad3585b70d"
open Nat Set
open Cardinal
no... | Mathlib/Data/Real/Cardinality.lean | 82 | 83 | theorem cantorFunctionAux_zero (f : ℕ → Bool) : cantorFunctionAux c f 0 = cond (f 0) 1 0 := by |
cases h : f 0 <;> simp [h]
| 1 | 2.718282 | 0 | 0.909091 | 11 | 786 |
import Mathlib.Data.ENNReal.Real
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Topology.UniformSpace.Pi
import Mathlib.Topology.UniformSpace.UniformConvergence
import Mathlib.Topology.UniformSpace.UniformEmbedding
#align_import topology.metric_space.emetric_space from "leanprover-community/mathlib"@"c8f3055... | Mathlib/Topology/EMetricSpace/Basic.lean | 110 | 111 | theorem edist_triangle_left (x y z : α) : edist x y ≤ edist z x + edist z y := by |
rw [edist_comm z]; apply edist_triangle
| 1 | 2.718282 | 0 | 1 | 5 | 891 |
import Mathlib.CategoryTheory.Limits.Preserves.Finite
import Mathlib.CategoryTheory.Sites.Canonical
import Mathlib.CategoryTheory.Sites.Coherent.Basic
import Mathlib.CategoryTheory.Sites.Preserves
universe v u w
namespace CategoryTheory
open Limits
variable {C : Type u} [Category.{v} C]
variable [FinitaryPreExten... | Mathlib/CategoryTheory/Sites/Coherent/ExtensiveSheaves.lean | 115 | 132 | theorem Presheaf.isSheaf_iff_preservesFiniteProducts {D : Type*} [Category D]
[FinitaryExtensive C] (F : Cᵒᵖ ⥤ D) :
IsSheaf (extensiveTopology C) F ↔ Nonempty (PreservesFiniteProducts F) := by |
constructor
· intro h
rw [IsSheaf] at h
refine ⟨⟨fun J _ ↦ ⟨fun {K} ↦ ⟨fun {c} hc ↦ ?_⟩⟩⟩⟩
apply coyonedaJointlyReflectsLimits
intro ⟨E⟩
specialize h E
rw [Presieve.isSheaf_iff_preservesFiniteProducts] at h
have : PreservesLimit K (F.comp (coyoneda.obj ⟨E⟩)) := (h.some.preserves J).pres... | 15 | 3,269,017.372472 | 2 | 2 | 4 | 2,378 |
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Int.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.Cases
import Mathlib.Tactic.SimpRw
import Mathlib.Tactic.SplitIfs
#align_import algebra.group.basic from "leanprover-community/mathlib"@"a07d750983b94c530a... | Mathlib/Algebra/Group/Basic.lean | 245 | 245 | theorem bit1_zero [One M] : bit1 (0 : M) = 1 := by | rw [bit1, bit0_zero, zero_add]
| 1 | 2.718282 | 0 | 0.333333 | 18 | 367 |
import Mathlib.AlgebraicTopology.SimplexCategory
import Mathlib.CategoryTheory.Comma.Arrow
import Mathlib.CategoryTheory.Limits.FunctorCategory
import Mathlib.CategoryTheory.Opposites
#align_import algebraic_topology.simplicial_object from "leanprover-community/mathlib"@"5ed51dc37c6b891b79314ee11a50adc2b1df6fd6"
o... | Mathlib/AlgebraicTopology/SimplicialObject.lean | 146 | 149 | theorem δ_comp_σ_of_le {n} {i : Fin (n + 2)} {j : Fin (n + 1)} (H : i ≤ Fin.castSucc j) :
X.σ j.succ ≫ X.δ (Fin.castSucc i) = X.δ i ≫ X.σ j := by |
dsimp [δ, σ]
simp only [← X.map_comp, ← op_comp, SimplexCategory.δ_comp_σ_of_le H]
| 2 | 7.389056 | 1 | 1 | 8 | 1,050 |
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
import Qq
#align_import analysis.special_functions.pow.real from "leanprover-community/mathlib"@"4fa54b337f7d52805480306db1b1439c741848c8"
noncomputable section
open scoped Classical
open Real ComplexConjugate
open Finset Set
namespace Real
variable {x y z... | Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 120 | 121 | theorem rpow_pos_of_pos {x : ℝ} (hx : 0 < x) (y : ℝ) : 0 < x ^ y := by |
rw [rpow_def_of_pos hx]; apply exp_pos
| 1 | 2.718282 | 0 | 0.384615 | 13 | 383 |
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.MeanValue
#align_import analysis.calculus.fderiv_symmetric from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
open Asymptotics Set
open scoped Topology
variable {E F : Type*} [NormedAddCommGroup E] [NormedSpa... | Mathlib/Analysis/Calculus/FDeriv/Symmetric.lean | 68 | 172 | theorem Convex.taylor_approx_two_segment {v w : E} (hv : x + v ∈ interior s)
(hw : x + v + w ∈ interior s) :
(fun h : ℝ => f (x + h • v + h • w)
- f (x + h • v) - h • f' x w - h ^ 2 • f'' v w - (h ^ 2 / 2) • f'' w w) =o[𝓝[>] 0]
fun h => h ^ 2 := by |
-- it suffices to check that the expression is bounded by `ε * ((‖v‖ + ‖w‖) * ‖w‖) * h^2` for
-- small enough `h`, for any positive `ε`.
refine IsLittleO.trans_isBigO
(isLittleO_iff.2 fun ε εpos => ?_) (isBigO_const_mul_self ((‖v‖ + ‖w‖) * ‖w‖) _ _)
-- consider a ball of radius `δ` around `x` in which the ... | 100 | 26,881,171,418,161,356,000,000,000,000,000,000,000,000,000 | 2 | 2 | 1 | 2,461 |
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Order.Interval.Set.OrdConnected
#align_import order.complete_lattice_intervals from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
open scoped Classical
open Set
variable {ι : ... | Mathlib/Order/CompleteLatticeIntervals.lean | 265 | 265 | theorem coe_biSup : (↑(⨆ i, ⨆ (_ : p i), f i) : α) = ⨆ i, ⨆ (_ : p i), (f i : α) := by | simp
| 1 | 2.718282 | 0 | 0.125 | 8 | 250 |
import Mathlib.Data.Finset.Card
#align_import data.finset.prod from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
assert_not_exists MonoidWithZero
open Multiset
variable {α β γ : Type*}
namespace Finset
section Prod
variable {s s' : Finset α} {t t' : Finset β} {a : α} {b : β}
... | Mathlib/Data/Finset/Prod.lean | 137 | 139 | theorem product_biUnion [DecidableEq γ] (s : Finset α) (t : Finset β) (f : α × β → Finset γ) :
(s ×ˢ t).biUnion f = s.biUnion fun a => t.biUnion fun b => f (a, b) := by |
classical simp_rw [product_eq_biUnion, biUnion_biUnion, image_biUnion]
| 1 | 2.718282 | 0 | 0.666667 | 3 | 616 |
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.Algebra.Module.Torsion
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v v' u₁' w w'
variable {R S : Type u} {M : Type v} {M' : Type v'} {M₁ : Type v}... | Mathlib/LinearAlgebra/Dimension/Constructions.lean | 235 | 236 | theorem finrank_finsupp_self {ι : Type v} [Fintype ι] : finrank R (ι →₀ R) = card ι := by |
rw [finrank, rank_finsupp_self, ← mk_toNat_eq_card, toNat_lift]
| 1 | 2.718282 | 0 | 0.75 | 24 | 667 |
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 | 92 | 95 | theorem IsDiag.add [AddZeroClass α] {A B : Matrix n n α} (ha : A.IsDiag) (hb : B.IsDiag) :
(A + B).IsDiag := by |
intro i j h
simp [ha h, hb h]
| 2 | 7.389056 | 1 | 1.25 | 8 | 1,302 |
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathlib.Util.AddRelatedDecl
import Batteries.Tactic.Lint
set_option autoImplicit true
open Lean Meta Elab Tactic
open Mathlib.Tactic
namespace Tactic.Elementwise
open CategoryTheory
section theorems
theorem forall_congr_forget_Type (α : Type u) (p : α... | Mathlib/Tactic/CategoryTheory/Elementwise.lean | 52 | 53 | theorem hom_elementwise [Category C] [ConcreteCategory C]
{X Y : C} {f g : X ⟶ Y} (h : f = g) (x : X) : f x = g x := by | rw [h]
| 1 | 2.718282 | 0 | 0 | 1 | 108 |
import Mathlib.LinearAlgebra.Dimension.Constructions
import Mathlib.LinearAlgebra.Dimension.Finite
universe u v
open Function Set Cardinal
variable {R} {M M₁ M₂ M₃ : Type u} {M' : Type v} [Ring R]
variable [AddCommGroup M] [AddCommGroup M₁] [AddCommGroup M₂] [AddCommGroup M₃] [AddCommGroup M']
variable [Module R M... | Mathlib/LinearAlgebra/Dimension/RankNullity.lean | 81 | 84 | theorem lift_rank_eq_of_surjective {f : M →ₗ[R] M'} (h : Surjective f) :
lift.{v} (Module.rank R M) =
lift.{u} (Module.rank R M') + lift.{v} (Module.rank R (LinearMap.ker f)) := by |
rw [← lift_rank_range_add_rank_ker f, ← rank_range_of_surjective f h]
| 1 | 2.718282 | 0 | 1 | 7 | 820 |
import Mathlib.Probability.Notation
import Mathlib.Probability.Density
import Mathlib.Probability.ConditionalProbability
import Mathlib.Probability.ProbabilityMassFunction.Constructions
open scoped Classical MeasureTheory NNReal ENNReal
-- TODO: We can't `open ProbabilityTheory` without opening the `ProbabilityThe... | Mathlib/Probability/Distributions/Uniform.lean | 77 | 78 | theorem absolutelyContinuous {X : Ω → E} {s : Set E} (hu : IsUniform X s ℙ μ) : map X ℙ ≪ μ := by |
rw [hu]; exact ProbabilityTheory.cond_absolutelyContinuous
| 1 | 2.718282 | 0 | 1.428571 | 7 | 1,510 |
import Mathlib.LinearAlgebra.Dimension.LinearMap
import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition
#align_import linear_algebra.free_module.finite.matrix from "leanprover-community/mathlib"@"b1c23399f01266afe392a0d8f71f599a0dad4f7b"
universe u u' v w
variable (R : Type u) (S : Type u') (M : Type v) (N ... | Mathlib/LinearAlgebra/FreeModule/Finite/Matrix.lean | 59 | 61 | theorem FiniteDimensional.finrank_linearMap :
finrank S (M →ₗ[R] N) = finrank R M * finrank S N := by |
simp_rw [finrank, rank_linearMap, toNat_mul, toNat_lift]
| 1 | 2.718282 | 0 | 0.6 | 5 | 528 |
import Mathlib.Order.CompleteLattice
import Mathlib.Order.Atoms
def Order.radical (α : Type*) [Preorder α] [OrderTop α] [InfSet α] : α :=
⨅ a ∈ {H | IsCoatom H}, a
variable {α : Type*} [CompleteLattice α]
lemma Order.radical_le_coatom {a : α} (h : IsCoatom a) : radical α ≤ a := biInf_le _ h
variable {β : Typ... | Mathlib/Order/Radical.lean | 38 | 48 | theorem Order.radical_nongenerating [IsCoatomic α] {a : α} (h : a ⊔ radical α = ⊤) : a = ⊤ := by |
-- Since the lattice is coatomic, either `a` is already the top element,
-- or there is a coatom above it.
obtain (rfl | w) := eq_top_or_exists_le_coatom a
· -- In the first case, we're done, this was already the goal.
rfl
· obtain ⟨m, c, le⟩ := w
have q : a ⊔ radical α ≤ m := sup_le le (radical_le_c... | 10 | 22,026.465795 | 2 | 2 | 2 | 2,290 |
import Mathlib.Algebra.Group.Equiv.TypeTags
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Module.LinearMap.Basic
import Mathlib.Algebra.MonoidAlgebra.Basic
import Mathlib.LinearAlgebra.Dual
import Mathlib.LinearAlgebra.Contraction
import Mathlib.RingTheory.TensorProduct.Basic
#align_import representation_... | Mathlib/RepresentationTheory/Basic.lean | 110 | 110 | theorem asAlgebraHom_single_one (g : G) : asAlgebraHom ρ (Finsupp.single g 1) = ρ g := by | simp
| 1 | 2.718282 | 0 | 0.75 | 8 | 651 |
import Batteries.Tactic.Init
import Batteries.Tactic.Alias
import Batteries.Tactic.Lint.Misc
instance {f : α → β} [DecidablePred p] : DecidablePred (p ∘ f) :=
inferInstanceAs <| DecidablePred fun x => p (f x)
@[deprecated] alias proofIrrel := proof_irrel
theorem Function.id_def : @id α = fun x => x := rfl
al... | .lake/packages/batteries/Batteries/Logic.lean | 142 | 143 | theorem congr_eqRec {β : α → Sort _} (f : (x : α) → β x → γ) (h : x = x') (y : β x) :
f x' (Eq.rec y h) = f x y := by | cases h; rfl
| 1 | 2.718282 | 0 | 0 | 8 | 67 |
import Mathlib.MeasureTheory.Decomposition.RadonNikodym
import Mathlib.Probability.Kernel.Disintegration.CdfToKernel
#align_import probability.kernel.cond_cdf from "leanprover-community/mathlib"@"3b88f4005dc2e28d42f974cc1ce838f0dafb39b8"
open MeasureTheory Set Filter TopologicalSpace
open scoped NNReal ENNReal Me... | Mathlib/Probability/Kernel/Disintegration/CondCdf.lean | 72 | 75 | theorem IicSnd_le_fst (r : ℝ) : ρ.IicSnd r ≤ ρ.fst := by |
refine Measure.le_iff.2 fun s hs ↦ ?_
simp_rw [fst_apply hs, IicSnd_apply ρ r hs]
exact measure_mono (prod_subset_preimage_fst _ _)
| 3 | 20.085537 | 1 | 1.333333 | 6 | 1,412 |
import Mathlib.Data.Finset.Lattice
import Mathlib.Data.Fintype.Vector
import Mathlib.Data.Multiset.Sym
#align_import data.finset.sym from "leanprover-community/mathlib"@"02ba8949f486ebecf93fe7460f1ed0564b5e442c"
namespace Finset
variable {α : Type*}
@[simps]
protected def sym2 (s : Finset α) : Finset (Sym2 α) :... | Mathlib/Data/Finset/Sym.lean | 156 | 156 | theorem diag_mem_sym2_iff : Sym2.diag a ∈ s.sym2 ↔ a ∈ s := by | simp [diag_mem_sym2_mem_iff]
| 1 | 2.718282 | 0 | 0.769231 | 13 | 684 |
import Mathlib.Algebra.Quotient
import Mathlib.Algebra.Group.Subgroup.Actions
import Mathlib.Algebra.Group.Subgroup.MulOpposite
import Mathlib.GroupTheory.GroupAction.Basic
import Mathlib.SetTheory.Cardinal.Finite
#align_import group_theory.coset from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce4... | Mathlib/GroupTheory/Coset.lean | 117 | 118 | theorem leftCoset_rightCoset (s : Set α) (a b : α) : op b • a • s = a • (op b • s) := by |
simp [← image_smul, (image_comp _ _ _).symm, Function.comp, mul_assoc]
| 1 | 2.718282 | 0 | 0.25 | 4 | 295 |
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 | 120 | 122 | theorem dedup_map_dedup_eq [DecidableEq β] (f : α → β) (s : Multiset α) :
dedup (map f (dedup s)) = dedup (map f s) := by |
simp [dedup_ext]
| 1 | 2.718282 | 0 | 0.25 | 4 | 296 |
import Mathlib.Combinatorics.Quiver.Cast
import Mathlib.Combinatorics.Quiver.Symmetric
import Mathlib.Data.Sigma.Basic
import Mathlib.Logic.Equiv.Basic
import Mathlib.Tactic.Common
#align_import combinatorics.quiver.covering from "leanprover-community/mathlib"@"188a411e916e1119e502dbe35b8b475716362401"
open Funct... | Mathlib/Combinatorics/Quiver/Covering.lean | 114 | 118 | theorem Prefunctor.IsCovering.map_injective (hφ : φ.IsCovering) {u v : U} :
Injective fun f : u ⟶ v => φ.map f := by |
rintro f g he
have : φ.star u (Quiver.Star.mk f) = φ.star u (Quiver.Star.mk g) := by simpa using he
simpa using (hφ.star_bijective u).left this
| 3 | 20.085537 | 1 | 1.333333 | 3 | 1,426 |
import Mathlib.Topology.MetricSpace.PseudoMetric
#align_import topology.metric_space.basic from "leanprover-community/mathlib"@"c8f305514e0d47dfaa710f5a52f0d21b588e6328"
open Set Filter Bornology
open scoped NNReal Uniformity
universe u v w
variable {α : Type u} {β : Type v} {X ι : Type*}
variable [PseudoMetricS... | Mathlib/Topology/MetricSpace/Basic.lean | 82 | 83 | theorem dist_le_zero {x y : γ} : dist x y ≤ 0 ↔ x = y := by |
simpa [le_antisymm_iff, dist_nonneg] using @dist_eq_zero _ _ x y
| 1 | 2.718282 | 0 | 0.166667 | 12 | 258 |
import Mathlib.Analysis.Convex.Between
import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.Topology.MetricSpace.Holder
import Mathlib.Topology.MetricSpace.MetricSeparated
#align_import measure_theory.measure.hausdorff from "leanprover-communit... | Mathlib/MeasureTheory/Measure/Hausdorff.lean | 293 | 297 | theorem tendsto_pre_nat (m : Set X → ℝ≥0∞) (s : Set X) :
Tendsto (fun n : ℕ => pre m n⁻¹ s) atTop (𝓝 <| mkMetric' m s) := by |
refine (tendsto_pre m s).comp (tendsto_inf.2 ⟨ENNReal.tendsto_inv_nat_nhds_zero, ?_⟩)
refine tendsto_principal.2 (eventually_of_forall fun n => ?_)
simp
| 3 | 20.085537 | 1 | 1.428571 | 7 | 1,520 |
import Mathlib.RingTheory.HahnSeries.Addition
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Data.Finset.MulAntidiagonal
#align_import ring_theory.hahn_series from "leanprover-community/mathlib"@"a484a7d0eade4e1268f4fb402859b6686037f965"
set_option linter.uppercaseLean3 false
open Finset Function
... | Mathlib/RingTheory/HahnSeries/Multiplication.lean | 152 | 161 | theorem smul_coeff_right [SMulZeroClass R W] {x : HahnSeries Γ R}
{y : HahnModule Γ R W} {a : Γ} {s : Set Γ} (hs : s.IsPWO) (hys : ((of R).symm y).support ⊆ s) :
((of R).symm <| x • y).coeff a =
∑ ij ∈ addAntidiagonal x.isPWO_support hs a,
x.coeff ij.fst • ((of R).symm y).coeff ij.snd := by |
rw [smul_coeff]
apply sum_subset_zero_on_sdiff (addAntidiagonal_mono_right hys) _ fun _ _ => rfl
intro b hb
simp only [not_and, mem_sdiff, mem_addAntidiagonal, HahnSeries.mem_support, not_imp_not] at hb
rw [hb.2 hb.1.1 hb.1.2.2, smul_zero]
| 5 | 148.413159 | 2 | 1.666667 | 3 | 1,815 |
import Mathlib.Data.List.Basic
#align_import data.list.lattice from "leanprover-community/mathlib"@"dd71334db81d0bd444af1ee339a29298bef40734"
open Nat
namespace List
variable {α : Type*} {l l₁ l₂ : List α} {p : α → Prop} {a : α}
variable [DecidableEq α]
section BagInter
@[simp]
theorem nil_bagInt... | Mathlib/Data/List/Lattice.lean | 203 | 207 | theorem cons_bagInter_of_pos (l₁ : List α) (h : a ∈ l₂) :
(a :: l₁).bagInter l₂ = a :: l₁.bagInter (l₂.erase a) := by |
cases l₂
· exact if_pos h
· simp only [List.bagInter, if_pos (elem_eq_true_of_mem h)]
| 3 | 20.085537 | 1 | 0.3 | 10 | 318 |
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Nat.Factors
import Mathlib.Order.Interval.Finset.Nat
#align_import number_theory.divisors from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
open scoped Classical
open Finset
namespace Nat
variable (n : ℕ)
d... | Mathlib/NumberTheory/Divisors.lean | 75 | 75 | theorem properDivisors.not_self_mem : ¬n ∈ properDivisors n := by | simp [properDivisors]
| 1 | 2.718282 | 0 | 1 | 11 | 1,070 |
import Mathlib.Data.W.Basic
#align_import data.pfunctor.univariate.basic from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
-- "W", "Idx"
set_option linter.uppercaseLean3 false
universe u v v₁ v₂ v₃
@[pp_with_univ]
structure PFunctor where
A : Type u
B : A → Type u
#align p... | Mathlib/Data/PFunctor/Univariate/Basic.lean | 154 | 154 | theorem fst_map (x : P α) (f : α → β) : (P.map f x).1 = x.1 := by | cases x; rfl
| 1 | 2.718282 | 0 | 0.25 | 4 | 305 |
import Mathlib.Algebra.FreeNonUnitalNonAssocAlgebra
import Mathlib.Algebra.Lie.NonUnitalNonAssocAlgebra
import Mathlib.Algebra.Lie.UniversalEnveloping
import Mathlib.GroupTheory.GroupAction.Ring
#align_import algebra.lie.free from "leanprover-community/mathlib"@"841ac1a3d9162bf51c6327812ecb6e5e71883ac4"
universe ... | Mathlib/Algebra/Lie/Free.lean | 103 | 106 | theorem Rel.smulOfTower {S : Type*} [Monoid S] [DistribMulAction S R] [IsScalarTower S R R] (t : S)
(a b : lib R X) (h : Rel R X a b) : Rel R X (t • a) (t • b) := by |
rw [← smul_one_smul R t a, ← smul_one_smul R t b]
exact h.smul _
| 2 | 7.389056 | 1 | 0.2 | 5 | 274 |
import Mathlib.Algebra.Algebra.Spectrum
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.RingTheory.Nilpotent.Basic
#align_import linear_algebra.eigenspace.basic from "leanprover-community/mathlib"@"6b0169218d01f2837d79ea2784882009a0da1aa1"
universe u v... | Mathlib/LinearAlgebra/Eigenspace/Basic.lean | 104 | 105 | theorem mem_eigenspace_iff {f : End R M} {μ : R} {x : M} : x ∈ eigenspace f μ ↔ f x = μ • x := by |
rw [eigenspace, LinearMap.mem_ker, LinearMap.sub_apply, algebraMap_end_apply, sub_eq_zero]
| 1 | 2.718282 | 0 | 0.714286 | 7 | 644 |
import Mathlib.Topology.Order.IsLUB
open Set Filter TopologicalSpace Topology Function
open OrderDual (toDual ofDual)
variable {α β γ : Type*}
section DenselyOrdered
variable [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a b : α}
{s : Set α}
theorem closure_Ioi' {a : α} (h : (Io... | Mathlib/Topology/Order/DenselyOrdered.lean | 116 | 117 | theorem Ico_mem_nhds_iff [NoMinOrder α] {a b x : α} : Ico a b ∈ 𝓝 x ↔ x ∈ Ioo a b := by |
rw [← interior_Ico, mem_interior_iff_mem_nhds]
| 1 | 2.718282 | 0 | 0.769231 | 13 | 685 |
import Mathlib.Algebra.MonoidAlgebra.Division
import Mathlib.Algebra.MvPolynomial.Basic
#align_import data.mv_polynomial.division from "leanprover-community/mathlib"@"72c366d0475675f1309d3027d3d7d47ee4423951"
variable {σ R : Type*} [CommSemiring R]
namespace MvPolynomial
theorem monomial_dvd_monomial {r s : ... | Mathlib/Algebra/MvPolynomial/Division.lean | 260 | 263 | theorem X_dvd_monomial {i : σ} {j : σ →₀ ℕ} {r : R} :
(X i : MvPolynomial σ R) ∣ monomial j r ↔ r = 0 ∨ j i ≠ 0 := by |
refine monomial_dvd_monomial.trans ?_
simp_rw [one_dvd, and_true_iff, Finsupp.single_le_iff, Nat.one_le_iff_ne_zero]
| 2 | 7.389056 | 1 | 1.25 | 4 | 1,339 |
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open ENNReal MeasureTheory MeasureTheory.Measure Metric Set Filter TopologicalSpace Function
open scoped Topology ENNReal Convex
variable... | Mathlib/MeasureTheory/Integral/Average.lean | 332 | 333 | theorem average_eq (f : α → E) : ⨍ x, f x ∂μ = (μ univ).toReal⁻¹ • ∫ x, f x ∂μ := by |
rw [average_eq', integral_smul_measure, ENNReal.toReal_inv]
| 1 | 2.718282 | 0 | 0.347826 | 23 | 374 |
import Mathlib.Algebra.Algebra.Spectrum
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.RingTheory.Nilpotent.Basic
#align_import linear_algebra.eigenspace.basic from "leanprover-community/mathlib"@"6b0169218d01f2837d79ea2784882009a0da1aa1"
universe u v... | Mathlib/LinearAlgebra/Eigenspace/Basic.lean | 113 | 115 | theorem HasEigenvector.pow_apply {f : End R M} {μ : R} {v : M} (hv : f.HasEigenvector μ v) (n : ℕ) :
(f ^ n) v = μ ^ n • v := by |
induction n <;> simp [*, pow_succ f, hv.apply_eq_smul, smul_smul, pow_succ' μ]
| 1 | 2.718282 | 0 | 0.714286 | 7 | 644 |
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.Order.Interval.Set.Group
import Mathlib.Analysis.Convex.Segment
import Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional
import Mathlib.Tactic.FieldSimp
#align_import analysis.convex.between from "leanprover-community/mathlib"@"571e13cacbed7bf042fd3058c... | Mathlib/Analysis/Convex/Between.lean | 49 | 55 | theorem affineSegment_comm (x y : P) : affineSegment R x y = affineSegment R y x := by |
refine Set.ext fun z => ?_
constructor <;>
· rintro ⟨t, ht, hxy⟩
refine ⟨1 - t, ?_, ?_⟩
· rwa [Set.sub_mem_Icc_iff_right, sub_self, sub_zero]
· rwa [lineMap_apply_one_sub]
| 6 | 403.428793 | 2 | 0.625 | 8 | 545 |
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 | 122 | 123 | theorem hammingDist_lt_one {x y : ∀ i, β i} : hammingDist x y < 1 ↔ x = y := by |
rw [Nat.lt_one_iff, hammingDist_eq_zero]
| 1 | 2.718282 | 0 | 0.7 | 10 | 642 |
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Analysis.Normed.Field.UnitBall
#align_import analysis.complex.circle from "leanprover-community/mathlib"@"ad3dfaca9ea2465198bcf58aa114401c324e29d1"
noncomputable section
open Complex Metric
open ComplexC... | Mathlib/Analysis/Complex/Circle.lean | 81 | 82 | theorem coe_inv_circle_eq_conj (z : circle) : ↑z⁻¹ = conj (z : ℂ) := by |
rw [coe_inv_circle, inv_def, normSq_eq_of_mem_circle, inv_one, ofReal_one, mul_one]
| 1 | 2.718282 | 0 | 0 | 3 | 129 |
import Mathlib.Data.List.Cycle
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.Perm.List
#align_import group_theory.perm.cycle.concrete from "leanprover-community/mathlib"@"00638177efd1b2534fc5269363ebf42a7871df9a"
open Equiv Equiv.Perm List
variable {α : Type*}
namespace Equiv.Perm
secti... | Mathlib/GroupTheory/Perm/Cycle/Concrete.lean | 253 | 254 | theorem nthLe_toList (n : ℕ) (hn : n < length (toList p x)) :
(toList p x).nthLe n hn = (p ^ n) x := by | simp [toList]
| 1 | 2.718282 | 0 | 1 | 18 | 1,030 |
import Mathlib.Analysis.Complex.Basic
import Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle
#align_import analysis.complex.re_im_topology from "leanprover-community/mathlib"@"468b141b14016d54b479eb7a0fff1e360b7e3cf6"
open Set
noncomputable section
namespace Complex
theorem isHomeomorphicTrivialFiber... | Mathlib/Analysis/Complex/ReImTopology.lean | 134 | 135 | theorem frontier_setOf_re_le (a : ℝ) : frontier { z : ℂ | z.re ≤ a } = { z | z.re = a } := by |
simpa only [frontier_Iic] using frontier_preimage_re (Iic a)
| 1 | 2.718282 | 0 | 0 | 10 | 135 |
import Mathlib.SetTheory.Cardinal.Finite
#align_import data.set.ncard from "leanprover-community/mathlib"@"74c2af38a828107941029b03839882c5c6f87a04"
namespace Set
variable {α β : Type*} {s t : Set α}
noncomputable def encard (s : Set α) : ℕ∞ := PartENat.withTopEquiv (PartENat.card s)
@[simp] theorem encard_uni... | Mathlib/Data/Set/Card.lean | 116 | 117 | theorem encard_insert_of_not_mem {a : α} (has : a ∉ s) : (insert a s).encard = s.encard + 1 := by |
rw [← union_singleton, encard_union_eq (by simpa), encard_singleton]
| 1 | 2.718282 | 0 | 0.5 | 14 | 454 |
import Mathlib.Algebra.QuadraticDiscriminant
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
#align_import analysis.special_functions.trigonometric.complex from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
noncomputable secti... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Complex.lean | 60 | 61 | theorem sin_ne_zero_iff {θ : ℂ} : sin θ ≠ 0 ↔ ∀ k : ℤ, θ ≠ k * π := by |
rw [← not_exists, not_iff_not, sin_eq_zero_iff]
| 1 | 2.718282 | 0 | 0.909091 | 11 | 787 |
import Mathlib.Algebra.MvPolynomial.Degrees
#align_import data.mv_polynomial.variables from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Finsupp AddMonoidAlgebra
universe u v w
variable {R : Type u} {S : Type v}
namespace MvPolynomial
varia... | Mathlib/Algebra/MvPolynomial/Variables.lean | 222 | 223 | theorem vars_map_of_injective (hf : Injective f) : (map f p).vars = p.vars := by |
simp [vars, degrees_map_of_injective _ hf]
| 1 | 2.718282 | 0 | 0.9 | 20 | 778 |
import Mathlib.Algebra.MvPolynomial.Variables
#align_import data.mv_polynomial.comm_ring from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Finsupp AddMonoidAlgebra
universe u v
variable {R : Type u} {S : Type v}
namespace MvPolynomial
varia... | Mathlib/Algebra/MvPolynomial/CommRing.lean | 203 | 204 | theorem totalDegree_neg (a : MvPolynomial σ R) : (-a).totalDegree = a.totalDegree := by |
simp only [totalDegree, support_neg]
| 1 | 2.718282 | 0 | 0.833333 | 6 | 723 |
import Mathlib.Algebra.Module.Defs
import Mathlib.Data.Fintype.BigOperators
import Mathlib.GroupTheory.GroupAction.BigOperators
#align_import algebra.module.big_operators from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
variable {ι κ α β R M : Type*}
| Mathlib/Algebra/Module/BigOperators.lean | 50 | 51 | theorem Finset.cast_card [CommSemiring R] (s : Finset α) : (s.card : R) = ∑ a ∈ s, 1 := by |
rw [Finset.sum_const, Nat.smul_one_eq_cast]
| 1 | 2.718282 | 0 | 0.666667 | 3 | 600 |
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanprover-community/mathlib"@"00f91228655eecdcd3ac97a7fd8dbcb139fe990a"
universe u v w
open scoped Classical Polynomial
open Polynomial
vari... | Mathlib/FieldTheory/IsAlgClosed/Basic.lean | 68 | 69 | theorem IsAlgClosed.splits_codomain {k K : Type*} [Field k] [IsAlgClosed k] [Field K] {f : K →+* k}
(p : K[X]) : p.Splits f := by | convert IsAlgClosed.splits (p.map f); simp [splits_map_iff]
| 1 | 2.718282 | 0 | 1.5 | 6 | 1,683 |
import Mathlib.MeasureTheory.Measure.VectorMeasure
import Mathlib.MeasureTheory.Function.AEEqOfIntegral
#align_import measure_theory.measure.with_density_vector_measure from "leanprover-community/mathlib"@"d1bd9c5df2867c1cb463bc6364446d57bdd9f7f1"
noncomputable section
open scoped Classical MeasureTheory NNReal ... | Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean | 64 | 65 | theorem withDensityᵥ_zero : μ.withDensityᵥ (0 : α → E) = 0 := by |
ext1 s hs; erw [withDensityᵥ_apply (integrable_zero α E μ) hs]; simp
| 1 | 2.718282 | 0 | 1.142857 | 7 | 1,217 |
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 | 59 | 62 | theorem hermite_eq_iterate (n : ℕ) : hermite n = (fun p => X * p - derivative p)^[n] 1 := by |
induction' n with n ih
· rfl
· rw [Function.iterate_succ_apply', ← ih, hermite_succ]
| 3 | 20.085537 | 1 | 1.1 | 10 | 1,191 |
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Analysis.Normed.Field.UnitBall
#align_import analysis.complex.circle from "leanprover-community/mathlib"@"ad3dfaca9ea2465198bcf58aa114401c324e29d1"
noncomputable section
open Complex Metric
open ComplexC... | Mathlib/Analysis/Complex/Circle.lean | 62 | 62 | theorem mem_circle_iff_normSq {z : ℂ} : z ∈ circle ↔ normSq z = 1 := by | simp [Complex.abs]
| 1 | 2.718282 | 0 | 0 | 3 | 129 |
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.LinearAlgebra.Matrix.Block
#align_import analysis.inner_product_space.gram_schmidt_ortho from "leanprover-community/mathlib"@"1a4df69ca1a9a0e5e26bfe12e2b92814216016d0"
open Finset Submodule FiniteDimensional
variable (𝕜 : Type*) {E : Type*} [RCLike �... | Mathlib/Analysis/InnerProductSpace/GramSchmidtOrtho.lean | 58 | 60 | theorem gramSchmidt_def (f : ι → E) (n : ι) :
gramSchmidt 𝕜 f n = f n - ∑ i ∈ Iio n, orthogonalProjection (𝕜 ∙ gramSchmidt 𝕜 f i) (f n) := by |
rw [← sum_attach, attach_eq_univ, gramSchmidt]
| 1 | 2.718282 | 0 | 1.125 | 8 | 1,201 |
import Mathlib.Algebra.Homology.Homotopy
import Mathlib.Algebra.Category.ModuleCat.Abelian
import Mathlib.Algebra.Category.ModuleCat.Subobject
import Mathlib.CategoryTheory.Limits.Shapes.ConcreteCategory
#align_import algebra.homology.Module from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225... | Mathlib/Algebra/Homology/ModuleCat.lean | 91 | 93 | theorem homology'_ext' {M : ModuleCat R} (i : ι) {h k : C.homology' i ⟶ M}
(w : ∀ x : LinearMap.ker (C.dFrom i), h (toHomology' x) = k (toHomology' x)) : h = k := by |
apply homology'_ext _ w
| 1 | 2.718282 | 0 | 1.25 | 4 | 1,337 |
import Mathlib.SetTheory.Cardinal.ENat
#align_import set_theory.cardinal.basic from "leanprover-community/mathlib"@"3ff3f2d6a3118b8711063de7111a0d77a53219a8"
universe u v
open Function Set
namespace Cardinal
variable {α : Type u} {c d : Cardinal.{u}}
noncomputable def toNat : Cardinal →*₀ ℕ :=
ENat.toNat.com... | Mathlib/SetTheory/Cardinal/ToNat.lean | 47 | 49 | theorem cast_toNat_of_lt_aleph0 {c : Cardinal} (h : c < ℵ₀) : ↑(toNat c) = c := by |
lift c to ℕ using h
rw [toNat_natCast]
| 2 | 7.389056 | 1 | 0.4 | 5 | 391 |
import Mathlib.Data.Finsupp.Basic
import Mathlib.Data.Finsupp.Order
#align_import data.finsupp.multiset from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
open Finset
variable {α β ι : Type*}
namespace Finsupp
def toMultiset : (α →₀ ℕ) →+ Multiset α where
toFun f := Finsupp.sum f... | Mathlib/Data/Finsupp/Multiset.lean | 67 | 68 | theorem card_toMultiset (f : α →₀ ℕ) : Multiset.card (toMultiset f) = f.sum fun _ => id := by |
simp [toMultiset_apply, map_finsupp_sum, Function.id_def]
| 1 | 2.718282 | 0 | 1.111111 | 9 | 1,194 |
import Mathlib.Init.Data.Sigma.Lex
import Mathlib.Data.Prod.Lex
import Mathlib.Data.Sigma.Lex
import Mathlib.Order.Antichain
import Mathlib.Order.OrderIsoNat
import Mathlib.Order.WellFounded
import Mathlib.Tactic.TFAE
#align_import order.well_founded_set from "leanprover-community/mathlib"@"2c84c2c5496117349007d97104... | Mathlib/Order/WellFoundedSet.lean | 112 | 113 | theorem wellFoundedOn_image {s : Set β} : (f '' s).WellFoundedOn r ↔ s.WellFoundedOn (r on f) := by |
rw [image_eq_range]; exact wellFoundedOn_range
| 1 | 2.718282 | 0 | 1.5 | 10 | 1,649 |
import Mathlib.Init.Algebra.Classes
import Mathlib.Data.FunLike.Basic
import Mathlib.Logic.Embedding.Basic
import Mathlib.Order.RelClasses
#align_import order.rel_iso.basic from "leanprover-community/mathlib"@"f29120f82f6e24a6f6579896dfa2de6769fec962"
set_option autoImplicit true
open Function
universe u v w
v... | Mathlib/Order/RelIso/Basic.lean | 168 | 180 | theorem injective_of_increasing (r : α → α → Prop) (s : β → β → Prop) [IsTrichotomous α r]
[IsIrrefl β s] (f : α → β) (hf : ∀ {x y}, r x y → s (f x) (f y)) : Injective f := by |
intro x y hxy
rcases trichotomous_of r x y with (h | h | h)
· have := hf h
rw [hxy] at this
exfalso
exact irrefl_of s (f y) this
· exact h
· have := hf h
rw [hxy] at this
exfalso
exact irrefl_of s (f y) this
| 11 | 59,874.141715 | 2 | 2 | 1 | 2,146 |
import Mathlib.Data.List.Infix
#align_import data.list.rdrop from "leanprover-community/mathlib"@"26f081a2fb920140ed5bc5cc5344e84bcc7cb2b2"
-- Make sure we don't import algebra
assert_not_exists Monoid
variable {α : Type*} (p : α → Bool) (l : List α) (n : ℕ)
namespace List
def rdrop : List α :=
l.take (l.leng... | Mathlib/Data/List/DropRight.lean | 51 | 51 | theorem rdrop_zero : rdrop l 0 = l := by | simp [rdrop]
| 1 | 2.718282 | 0 | 0.631579 | 19 | 550 |
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