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.Probability.ProbabilityMassFunction.Constructions
import Mathlib.Tactic.FinCases
namespace PMF
open ENNReal
noncomputable
def binomial (p : ℝ≥0∞) (h : p ≤ 1) (n : ℕ) : PMF (Fin (n + 1)) :=
.ofFintype (fun i => p^(i : ℕ) * (1-p)^((Fin.last n - i) : ℕ) * (n.choose i : ℕ)) (by
convert (add_pow ... | Mathlib/Probability/ProbabilityMassFunction/Binomial.lean | 40 | 42 | theorem binomial_apply_zero (p : ℝ≥0∞) (h : p ≤ 1) (n : ℕ) :
binomial p h n 0 = (1-p)^n := by |
simp [binomial_apply]
| 1 | 2.718282 | 0 | 0 | 4 | 22 |
import Mathlib.Computability.Halting
import Mathlib.Computability.TuringMachine
import Mathlib.Data.Num.Lemmas
import Mathlib.Tactic.DeriveFintype
#align_import computability.tm_to_partrec from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
open Function (update)
open Relation
namespa... | Mathlib/Computability/TMToPartrec.lean | 183 | 183 | theorem id_eval (v) : id.eval v = pure v := by | simp [id]
| 1 | 2.718282 | 0 | 0.285714 | 14 | 314 |
import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Subobject.Lattice
import Mathlib.Order.Atoms
#align_import category_theory.simple from "leanprover-community/mathlib"@"4ed0bcaef698011... | Mathlib/CategoryTheory/Simple.lean | 193 | 201 | theorem Biprod.isIso_inl_iff_isZero (X Y : C) : IsIso (biprod.inl : X ⟶ X ⊞ Y) ↔ IsZero Y := by |
rw [biprod.isIso_inl_iff_id_eq_fst_comp_inl, ← biprod.total, add_right_eq_self]
constructor
· intro h
replace h := h =≫ biprod.snd
simpa [← IsZero.iff_isSplitEpi_eq_zero (biprod.snd : X ⊞ Y ⟶ Y)] using h
· intro h
rw [IsZero.iff_isSplitEpi_eq_zero (biprod.snd : X ⊞ Y ⟶ Y)] at h
rw [h, zero_comp... | 8 | 2,980.957987 | 2 | 1.5 | 8 | 1,594 |
import Mathlib.NumberTheory.Zsqrtd.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Real.Archimedean
#align_import number_theory.zsqrtd.gaussian_int from "leanprover-community/mathlib"@"5b2fe80501ff327b9109fb09b7cc8c325cd0d7d9"
open Zsqrtd Complex
open sc... | Mathlib/NumberTheory/Zsqrtd/GaussianInt.lean | 81 | 81 | theorem toComplex_def' (x y : ℤ) : ((⟨x, y⟩ : ℤ[i]) : ℂ) = x + y * I := by | simp [toComplex_def]
| 1 | 2.718282 | 0 | 0.090909 | 11 | 243 |
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.Perm
import Mathlib.GroupTheory.Perm.Finite
import Mathlib.GroupTheory.Perm.List
#align_import group_theory.perm.cycle.basic from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
open Equiv Function Finset
variable {... | Mathlib/GroupTheory/Perm/Cycle/Basic.lean | 157 | 158 | theorem sameCycle_pow_left {n : ℕ} : SameCycle f ((f ^ n) x) y ↔ SameCycle f x y := by |
rw [← zpow_natCast, sameCycle_zpow_left]
| 1 | 2.718282 | 0 | 0 | 7 | 112 |
import Mathlib.NumberTheory.FLT.Basic
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.Cyclotomic.Rat
section case1
open ZMod
private lemma cube_of_castHom_ne_zero {n : ZMod 9} :
castHom (show 3 ∣ 9 by norm_num) (ZMod 3) n ≠ 0 → n ^ 3 = 1 ∨ n ^ 3 = 8 := by
revert n; decide
private lemma cube_of_n... | Mathlib/NumberTheory/FLT/Three.lean | 36 | 44 | theorem fermatLastTheoremThree_case_1 {a b c : ℤ} (hdvd : ¬ 3 ∣ a * b * c) :
a ^ 3 + b ^ 3 ≠ c ^ 3 := by |
simp_rw [Int.prime_three.dvd_mul, not_or] at hdvd
apply mt (congrArg (Int.cast : ℤ → ZMod 9))
simp_rw [Int.cast_add, Int.cast_pow]
rcases cube_of_not_dvd hdvd.1.1 with ha | ha <;>
rcases cube_of_not_dvd hdvd.1.2 with hb | hb <;>
rcases cube_of_not_dvd hdvd.2 with hc | hc <;>
rw [ha, hb, hc] <;> decide
| 7 | 1,096.633158 | 2 | 2 | 1 | 2,102 |
import Mathlib.Logic.Function.Basic
import Mathlib.Logic.Relator
import Mathlib.Init.Data.Quot
import Mathlib.Tactic.Cases
import Mathlib.Tactic.Use
import Mathlib.Tactic.MkIffOfInductiveProp
import Mathlib.Tactic.SimpRw
#align_import logic.relation from "leanprover-community/mathlib"@"3365b20c2ffa7c35e47e5209b89ba9a... | Mathlib/Logic/Relation.lean | 312 | 316 | theorem symmetric (h : Symmetric r) : Symmetric (ReflTransGen r) := by |
intro x y h
induction' h with z w _ b c
· rfl
· apply Relation.ReflTransGen.head (h b) c
| 4 | 54.59815 | 2 | 1.6 | 15 | 1,743 |
import Mathlib.RingTheory.Polynomial.Hermite.Basic
import Mathlib.Analysis.Calculus.Deriv.Add
import Mathlib.Analysis.Calculus.Deriv.Polynomial
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
#align_import ring_theory.polynomial.hermite.gaussian from "leanprover-communit... | Mathlib/RingTheory/Polynomial/Hermite/Gaussian.lean | 67 | 70 | theorem hermite_eq_deriv_gaussian' (n : ℕ) (x : ℝ) : aeval x (hermite n) =
(-1 : ℝ) ^ n * deriv^[n] (fun y => Real.exp (-(y ^ 2 / 2))) x * Real.exp (x ^ 2 / 2) := by |
rw [hermite_eq_deriv_gaussian, Real.exp_neg]
field_simp [Real.exp_ne_zero]
| 2 | 7.389056 | 1 | 1.666667 | 3 | 1,780 |
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.MinMax
#align_import data.set.pointwise.interval from "leanprover-community/mathlib"@"2196ab363eb097c... | Mathlib/Data/Set/Pointwise/Interval.lean | 157 | 158 | theorem preimage_const_add_Ioc : (fun x => a + x) ⁻¹' Ioc b c = Ioc (b - a) (c - a) := by |
simp [← Ioi_inter_Iic]
| 1 | 2.718282 | 0 | 0.37931 | 29 | 381 |
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 | 77 | 78 | theorem vars_0 : (0 : MvPolynomial σ R).vars = ∅ := by |
classical rw [vars_def, degrees_zero, Multiset.toFinset_zero]
| 1 | 2.718282 | 0 | 0.9 | 20 | 778 |
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.LinearAlgebra.StdBasis
import Mathlib.RingTheory.AlgebraTower
import Mathlib.Algebra.Algebra.Subalgebra.Tower
#align_import linear_algebra.matrix.to_lin from "leanprover-community/mathlib"@"0e2aab2b0d521f060f62a14d2cf2e2c54e8491d6"
... | Mathlib/LinearAlgebra/Matrix/ToLin.lean | 112 | 123 | theorem Matrix.vecMul_injective_iff {R : Type*} [CommRing R] {M : Matrix m n R} :
Function.Injective M.vecMul ↔ LinearIndependent R (fun i ↦ M i) := by |
rw [← coe_vecMulLinear]
simp only [← LinearMap.ker_eq_bot, Fintype.linearIndependent_iff, Submodule.eq_bot_iff,
LinearMap.mem_ker, vecMulLinear_apply]
refine ⟨fun h c h0 ↦ congr_fun <| h c ?_, fun h c h0 ↦ funext <| h c ?_⟩
· rw [← h0]
ext i
simp [vecMul, dotProduct]
· rw [← h0]
ext j
sim... | 10 | 22,026.465795 | 2 | 1.75 | 4 | 1,846 |
import Mathlib.CategoryTheory.Iso
import Mathlib.CategoryTheory.Functor.Category
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.comma from "leanprover-community/mathlib"@"8a318021995877a44630c898d0b2bc376fceef3b"
namespace CategoryTheory
open Category
-- declare the `v`'s first; see `Catego... | Mathlib/CategoryTheory/Comma/Basic.lean | 173 | 176 | theorem eqToHom_right (X Y : Comma L R) (H : X = Y) :
CommaMorphism.right (eqToHom H) = eqToHom (by cases H; rfl) := by |
cases H
rfl
| 2 | 7.389056 | 1 | 1 | 2 | 987 |
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 | 48 | 50 | theorem Fintype.card_units [GroupWithZero α] [Fintype α] [DecidableEq α] :
Fintype.card αˣ = Fintype.card α - 1 := by |
rw [@Fintype.card_eq_card_units_add_one α, Nat.add_sub_cancel]
| 1 | 2.718282 | 0 | 0.666667 | 3 | 602 |
import Mathlib.Algebra.Field.Defs
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Ring.Commute
import Mathlib.Algebra.Ring.Invertible
import Mathlib.Order.Synonym
#align_import algebra.field.basic from "leanprover-community/mathlib"@"05101c3df9d9cfe9430edc205860c79b6d660102"
open Function ... | Mathlib/Algebra/Field/Basic.lean | 117 | 118 | theorem neg_div (a b : K) : -b / a = -(b / a) := by |
rw [neg_eq_neg_one_mul, mul_div_assoc, ← neg_eq_neg_one_mul]
| 1 | 2.718282 | 0 | 0.3125 | 16 | 321 |
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Deprecated.Submonoid
#align_import deprecated.subgroup from "leanprover-community/mathlib"@"f93c11933efbc3c2f0299e47b8ff83e9b539cbf6"
open Set Function
variable {G : Type*} {H : Type*} {A : Type*} {a a₁ a₂ b c : G}
section Group
variable [Group G] [Add... | Mathlib/Deprecated/Subgroup.lean | 57 | 58 | theorem IsSubgroup.div_mem {s : Set G} (hs : IsSubgroup s) {x y : G} (hx : x ∈ s) (hy : y ∈ s) :
x / y ∈ s := by | simpa only [div_eq_mul_inv] using hs.mul_mem hx (hs.inv_mem hy)
| 1 | 2.718282 | 0 | 0 | 1 | 180 |
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 | 114 | 115 | theorem edist_triangle_right (x y z : α) : edist x y ≤ edist x z + edist y z := by |
rw [edist_comm y]; apply edist_triangle
| 1 | 2.718282 | 0 | 1 | 5 | 891 |
import Mathlib.Order.Filter.AtTopBot
import Mathlib.Order.Filter.Subsingleton
open Set
variable {α β γ δ : Type*} {l : Filter α} {f : α → β}
namespace Filter
def EventuallyConst (f : α → β) (l : Filter α) : Prop := (map f l).Subsingleton
theorem HasBasis.eventuallyConst_iff {ι : Sort*} {p : ι → Prop} {s : ι → S... | Mathlib/Order/Filter/EventuallyConst.lean | 145 | 147 | theorem mulIndicator_const_iff :
EventuallyConst (s.mulIndicator fun _ ↦ c) l ↔ c = 1 ∨ EventuallyConst s l := by |
rcases eq_or_ne c 1 with rfl | hc <;> simp [mulIndicator_const_iff_of_ne, *]
| 1 | 2.718282 | 0 | 0 | 4 | 36 |
import Mathlib.LinearAlgebra.Basis
import Mathlib.Algebra.Module.LocalizedModule
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Localization.Integer
#align_import ring_theory.localization.module from "leanprover-community/mathlib"@"2e59a6de168f95d16b16d217b808a36290398c0a"
open nonZ... | Mathlib/RingTheory/Localization/Module.lean | 73 | 76 | theorem LinearIndependent.localization {ι : Type*} {b : ι → M} (hli : LinearIndependent R b) :
LinearIndependent Rₛ b := by |
have := isLocalizedModule_id S M Rₛ
exact hli.of_isLocalizedModule Rₛ S .id
| 2 | 7.389056 | 1 | 1.5 | 2 | 1,599 |
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 | 104 | 105 | theorem card_Icc : (Icc a b).card = b + 1 - a := by |
rw [← Nat.card_Icc, ← map_valEmbedding_Icc, card_map]
| 1 | 2.718282 | 0 | 0.125 | 16 | 251 |
import Mathlib.RepresentationTheory.Rep
import Mathlib.Algebra.Category.FGModuleCat.Limits
import Mathlib.CategoryTheory.Preadditive.Schur
import Mathlib.RepresentationTheory.Basic
#align_import representation_theory.fdRep from "leanprover-community/mathlib"@"19a70dceb9dff0994b92d2dd049de7d84d28112b"
suppress_comp... | Mathlib/RepresentationTheory/FdRep.lean | 113 | 114 | theorem forget₂_ρ (V : FdRep k G) : ((forget₂ (FdRep k G) (Rep k G)).obj V).ρ = V.ρ := by |
ext g v; rfl
| 1 | 2.718282 | 0 | 1 | 2 | 1,160 |
import Mathlib.Algebra.Module.Zlattice.Basic
import Mathlib.NumberTheory.NumberField.Embeddings
import Mathlib.NumberTheory.NumberField.FractionalIdeal
#align_import number_theory.number_field.canonical_embedding from "leanprover-community/mathlib"@"60da01b41bbe4206f05d34fd70c8dd7498717a30"
variable (K : Type*) [F... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean | 290 | 293 | theorem normAtPlace_apply_isComplex {w : InfinitePlace K} (hw : IsComplex w) (x : E K) :
normAtPlace w x = ‖x.2 ⟨w, hw⟩‖ := by |
rw [normAtPlace, MonoidWithZeroHom.coe_mk, ZeroHom.coe_mk,
dif_neg (not_isReal_iff_isComplex.mpr hw)]
| 2 | 7.389056 | 1 | 1.1875 | 16 | 1,249 |
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 | 445 | 445 | theorem inv_eq_one_div (x : G) : x⁻¹ = 1 / x := by | rw [div_eq_mul_inv, one_mul]
| 1 | 2.718282 | 0 | 0.333333 | 18 | 367 |
import Mathlib.CategoryTheory.Monoidal.Category
import Mathlib.CategoryTheory.Adjunction.FullyFaithful
import Mathlib.CategoryTheory.Products.Basic
#align_import category_theory.monoidal.functor from "leanprover-community/mathlib"@"3d7987cda72abc473c7cdbbb075170e9ac620042"
open CategoryTheory
universe v₁ v₂ v₃ u... | Mathlib/CategoryTheory/Monoidal/Functor.lean | 164 | 167 | theorem LaxMonoidalFunctor.left_unitality_inv (F : LaxMonoidalFunctor C D) (X : C) :
(λ_ (F.obj X)).inv ≫ F.ε ▷ F.obj X ≫ F.μ (𝟙_ C) X = F.map (λ_ X).inv := by |
rw [Iso.inv_comp_eq, F.left_unitality, Category.assoc, Category.assoc, ← F.toFunctor.map_comp,
Iso.hom_inv_id, F.toFunctor.map_id, comp_id]
| 2 | 7.389056 | 1 | 0.5 | 2 | 491 |
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 | 56 | 57 | theorem hammingDist_comm (x y : ∀ i, β i) : hammingDist x y = hammingDist y x := by |
simp_rw [hammingDist, ne_comm]
| 1 | 2.718282 | 0 | 0.7 | 10 | 642 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.Tactic.Abel
#align_import ring_theory.polynomial.pochhammer from "leanprover-community/mathlib"@"53b216bcc1146df1c4a0a868778... | Mathlib/RingTheory/Polynomial/Pochhammer.lean | 90 | 93 | theorem ascPochhammer_eval₂ (f : S →+* T) (n : ℕ) (t : T) :
(ascPochhammer T n).eval t = (ascPochhammer S n).eval₂ f t := by |
rw [← ascPochhammer_map f]
exact eval_map f t
| 2 | 7.389056 | 1 | 0.96 | 25 | 796 |
import Mathlib.Analysis.SpecialFunctions.Gamma.Basic
import Mathlib.Analysis.SpecialFunctions.PolarCoord
import Mathlib.Analysis.Convex.Complex
#align_import analysis.special_functions.gaussian from "leanprover-community/mathlib"@"7982767093ae38cba236487f9c9dd9cd99f63c16"
noncomputable section
open Real Set Measu... | Mathlib/Analysis/SpecialFunctions/Gaussian/GaussianIntegral.lean | 51 | 55 | theorem rpow_mul_exp_neg_mul_rpow_isLittleO_exp_neg (s : ℝ) {b p : ℝ} (hp : 1 < p) (hb : 0 < b) :
(fun x : ℝ => x ^ s * exp (- b * x ^ p)) =o[atTop] fun x : ℝ => exp (-(1 / 2) * x) := by |
apply ((isBigO_refl (fun x : ℝ => x ^ s) atTop).mul_isLittleO
(exp_neg_mul_rpow_isLittleO_exp_neg hb hp)).trans
simpa only [mul_comm] using Real.Gamma_integrand_isLittleO s
| 3 | 20.085537 | 1 | 1.5 | 6 | 1,554 |
import Mathlib.Init.Logic
import Mathlib.Init.Function
import Mathlib.Init.Algebra.Classes
import Batteries.Util.LibraryNote
import Batteries.Tactic.Lint.Basic
#align_import logic.basic from "leanprover-community/mathlib"@"3365b20c2ffa7c35e47e5209b89ba9abdddf3ffe"
#align_import init.ite_simp from "leanprover-communit... | Mathlib/Logic/Basic.lean | 611 | 612 | theorem heq_rec_iff_heq {C : α → Sort*} {x : β} {y : C a} {e : a = b} :
HEq x (e ▸ y) ↔ HEq x y := by | subst e; rfl
| 1 | 2.718282 | 0 | 0 | 8 | 105 |
import Mathlib.Analysis.BoxIntegral.Box.Basic
import Mathlib.Analysis.SpecificLimits.Basic
#align_import analysis.box_integral.box.subbox_induction from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Finset Function Filter Metric Classical Topology Filter ENNReal
noncomputable... | Mathlib/Analysis/BoxIntegral/Box/SubboxInduction.lean | 101 | 103 | theorem upper_sub_lower_splitCenterBox (I : Box ι) (s : Set ι) (i : ι) :
(I.splitCenterBox s).upper i - (I.splitCenterBox s).lower i = (I.upper i - I.lower i) / 2 := by |
by_cases i ∈ s <;> field_simp [splitCenterBox] <;> field_simp [mul_two, two_mul]
| 1 | 2.718282 | 0 | 1.4 | 5 | 1,487 |
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 | 80 | 80 | theorem expand_zero (f : R[X]) : expand R 0 f = C (eval 1 f) := by | simp [expand]
| 1 | 2.718282 | 0 | 0.285714 | 7 | 311 |
import Mathlib.Order.CompleteLattice
import Mathlib.Order.Cover
import Mathlib.Order.Iterate
import Mathlib.Order.WellFounded
#align_import order.succ_pred.basic from "leanprover-community/mathlib"@"0111834459f5d7400215223ea95ae38a1265a907"
open Function OrderDual Set
variable {α β : Type*}
@[ext]
class SuccOr... | Mathlib/Order/SuccPred/Basic.lean | 284 | 286 | theorem succ_le_succ_iff_of_not_isMax (ha : ¬IsMax a) (hb : ¬IsMax b) :
succ a ≤ succ b ↔ a ≤ b := by |
rw [succ_le_iff_of_not_isMax ha, lt_succ_iff_of_not_isMax hb]
| 1 | 2.718282 | 0 | 0.666667 | 3 | 560 |
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
open Set Filter Topology
variable {X : Type*}
def OnePoint (X : Type*) :=
Option X
#ali... | Mathlib/Topology/Compactification/OnePoint.lean | 165 | 167 | theorem coe_preimage_infty : ((↑) : X → OnePoint X) ⁻¹' {∞} = ∅ := by |
ext
simp
| 2 | 7.389056 | 1 | 0.25 | 4 | 289 |
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.RingTheory.Localization.Basic
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Surreal.Basic
#align_import set_theory.surreal.dyadic from "leanprover-community/mathlib"@"9... | Mathlib/SetTheory/Surreal/Dyadic.lean | 64 | 64 | theorem powHalf_moveLeft (n i) : (powHalf n).moveLeft i = 0 := by | cases n <;> cases i <;> rfl
| 1 | 2.718282 | 0 | 0.714286 | 7 | 646 |
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.MinMax
#align_import data.set.pointwise.interval from "leanprover-community/mathlib"@"2196ab363eb097c... | Mathlib/Data/Set/Pointwise/Interval.lean | 110 | 113 | theorem Ici_mul_Ioi_subset' (a b : α) : Ici a * Ioi b ⊆ Ioi (a * b) := by |
haveI := covariantClass_le_of_lt
rintro x ⟨y, hya, z, hzb, rfl⟩
exact mul_lt_mul_of_le_of_lt hya hzb
| 3 | 20.085537 | 1 | 0.37931 | 29 | 381 |
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 | 104 | 107 | theorem IsDiag.smul [Monoid R] [AddMonoid α] [DistribMulAction R α] (k : R) {A : Matrix n n α}
(ha : A.IsDiag) : (k • A).IsDiag := by |
intro i j h
simp [ha h]
| 2 | 7.389056 | 1 | 1.25 | 8 | 1,302 |
import Mathlib.Probability.Kernel.Composition
#align_import probability.kernel.invariance from "leanprover-community/mathlib"@"3b92d54a05ee592aa2c6181a4e76b1bb7cc45d0b"
open MeasureTheory
open scoped MeasureTheory ENNReal ProbabilityTheory
namespace ProbabilityTheory
variable {α β γ : Type*} {mα : MeasurableSp... | Mathlib/Probability/Kernel/Invariance.lean | 83 | 84 | theorem Invariant.comp_const (hκ : Invariant κ μ) : κ ∘ₖ const α μ = const α μ := by |
rw [← const_bind_eq_comp_const κ μ, hκ.def]
| 1 | 2.718282 | 0 | 1 | 6 | 1,165 |
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
import Mathlib.Analysis.NormedSpace.AffineIsometry
#align_import geometry.euclidean.angle.unoriented.affine from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f... | Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean | 61 | 64 | theorem _root_.AffineIsometry.angle_map {V₂ P₂ : Type*} [NormedAddCommGroup V₂]
[InnerProductSpace ℝ V₂] [MetricSpace P₂] [NormedAddTorsor V₂ P₂]
(f : P →ᵃⁱ[ℝ] P₂) (p₁ p₂ p₃ : P) : ∠ (f p₁) (f p₂) (f p₃) = ∠ p₁ p₂ p₃ := by |
simp_rw [angle, ← AffineIsometry.map_vsub, LinearIsometry.angle_map]
| 1 | 2.718282 | 0 | 0.4 | 5 | 403 |
import Mathlib.CategoryTheory.Elements
import Mathlib.CategoryTheory.IsConnected
import Mathlib.CategoryTheory.SingleObj
import Mathlib.GroupTheory.GroupAction.Quotient
import Mathlib.GroupTheory.SemidirectProduct
#align_import category_theory.action from "leanprover-community/mathlib"@"aa812bd12a4dbbd2c129b38205f222... | Mathlib/CategoryTheory/Action.lean | 89 | 89 | theorem back_coe (x : ActionCategory M X) : ↑x.back = x := by | cases x; rfl
| 1 | 2.718282 | 0 | 0 | 1 | 31 |
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 | 97 | 98 | theorem sum_smul_dirac [Countable α] [MeasurableSingletonClass α] (μ : Measure α) :
(sum fun a => μ {a} • dirac a) = μ := by | simpa using (map_eq_sum μ id measurable_id).symm
| 1 | 2.718282 | 0 | 1.5 | 6 | 1,557 |
import Mathlib.Data.Real.Sqrt
import Mathlib.Analysis.NormedSpace.Star.Basic
import Mathlib.Analysis.NormedSpace.ContinuousLinearMap
import Mathlib.Analysis.NormedSpace.Basic
#align_import data.is_R_or_C.basic from "leanprover-community/mathlib"@"baa88307f3e699fa7054ef04ec79fa4f056169cb"
section
local notation "�... | Mathlib/Analysis/RCLike/Basic.lean | 162 | 162 | theorem one_re : re (1 : K) = 1 := by | rw [← ofReal_one, ofReal_re]
| 1 | 2.718282 | 0 | 0 | 3 | 103 |
import Mathlib.Init.Function
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
universe u
open Function
namespace Option
variable {α β γ δ : Type*} {f : α → β → γ} {a : Option α} {b : Option β} {c : Option γ}
def map₂ (f : α → β → γ) (a : Option α) ... | Mathlib/Data/Option/NAry.lean | 99 | 100 | theorem map₂_map_right (f : α → γ → δ) (g : β → γ) :
map₂ f a (b.map g) = map₂ (fun a b => f a (g b)) a b := by | cases b <;> rfl
| 1 | 2.718282 | 0 | 0 | 14 | 191 |
import Mathlib.Algebra.Field.Defs
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Ring.Commute
import Mathlib.Algebra.Ring.Invertible
import Mathlib.Order.Synonym
#align_import algebra.field.basic from "leanprover-community/mathlib"@"05101c3df9d9cfe9430edc205860c79b6d660102"
open Function ... | Mathlib/Algebra/Field/Basic.lean | 96 | 98 | theorem one_div_neg_one_eq_neg_one : (1 : K) / -1 = -1 :=
have : -1 * -1 = (1 : K) := by | rw [neg_mul_neg, one_mul]
Eq.symm (eq_one_div_of_mul_eq_one_right this)
| 2 | 7.389056 | 1 | 0.3125 | 16 | 321 |
import Mathlib.Algebra.Polynomial.Module.Basic
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv.Defs
import Mathlib.Analysis.Calculus.MeanValue
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
open scoped... | Mathlib/Analysis/Calculus/Taylor.lean | 107 | 111 | theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) :
taylorWithinEval f n s x₀ x₀ = f x₀ := by |
induction' n with k hk
· exact taylor_within_zero_eval _ _ _ _
simp [hk]
| 3 | 20.085537 | 1 | 1.714286 | 7 | 1,841 |
import Mathlib.Tactic.NormNum.Core
import Mathlib.Tactic.HaveI
import Mathlib.Data.Nat.Cast.Commute
import Mathlib.Algebra.Ring.Int
import Mathlib.Algebra.GroupWithZero.Invertible
import Mathlib.Tactic.ClearExclamation
import Mathlib.Data.Nat.Cast.Basic
set_option autoImplicit true
namespace Mathlib
open Lean hidi... | Mathlib/Tactic/NormNum/Basic.lean | 119 | 120 | theorem isNat_intCast {R} [Ring R] (n : ℤ) (m : ℕ) :
IsNat n m → IsNat (n : R) m := by | rintro ⟨⟨⟩⟩; exact ⟨by simp⟩
| 1 | 2.718282 | 0 | 0 | 3 | 207 |
import Mathlib.MeasureTheory.Measure.MeasureSpace
import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
#align_import measure_theory.measure.open_pos from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Topology ENNReal MeasureTheory
open Set Function Filter
namespace Measur... | Mathlib/MeasureTheory/Measure/OpenPos.lean | 107 | 110 | theorem _root_.IsClosed.measure_eq_one_iff_eq_univ [OpensMeasurableSpace X] [IsProbabilityMeasure μ]
(hF : IsClosed F) :
μ F = 1 ↔ F = univ := by |
rw [← measure_univ (μ := μ), hF.measure_eq_univ_iff_eq]
| 1 | 2.718282 | 0 | 0.666667 | 6 | 594 |
import Mathlib.Algebra.Module.Zlattice.Basic
import Mathlib.NumberTheory.NumberField.Embeddings
import Mathlib.NumberTheory.NumberField.FractionalIdeal
#align_import number_theory.number_field.canonical_embedding from "leanprover-community/mathlib"@"60da01b41bbe4206f05d34fd70c8dd7498717a30"
variable (K : Type*) [F... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean | 259 | 262 | theorem normAtPlace_nonneg (w : InfinitePlace K) (x : E K) :
0 ≤ normAtPlace w x := by |
rw [normAtPlace, MonoidWithZeroHom.coe_mk, ZeroHom.coe_mk]
split_ifs <;> exact norm_nonneg _
| 2 | 7.389056 | 1 | 1.1875 | 16 | 1,249 |
import Mathlib.Algebra.Order.Floor
import Mathlib.Topology.Algebra.Order.Group
import Mathlib.Topology.Order.Basic
#align_import topology.algebra.order.floor from "leanprover-community/mathlib"@"84dc0bd6619acaea625086d6f53cb35cdd554219"
open Filter Function Int Set Topology
variable {α β γ : Type*} [LinearOrdere... | Mathlib/Topology/Algebra/Order/Floor.lean | 101 | 105 | theorem tendsto_ceil_right_pure_floor_add_one (x : α) :
Tendsto (ceil : α → ℤ) (𝓝[>] x) (pure (⌊x⌋ + 1)) :=
have : ↑(⌊x⌋ + 1) - 1 ≤ x := by | rw [cast_add, cast_one, add_sub_cancel_right]; exact floor_le _
tendsto_pure.2 <| mem_of_superset (Ioc_mem_nhdsWithin_Ioi' <| lt_succ_floor _) fun _y hy =>
ceil_eq_on_Ioc _ _ ⟨this.trans_lt hy.1, hy.2⟩
| 3 | 20.085537 | 1 | 0.5 | 6 | 492 |
import Mathlib.Algebra.Algebra.Spectrum
import Mathlib.FieldTheory.IsAlgClosed.Basic
#align_import field_theory.is_alg_closed.spectrum from "leanprover-community/mathlib"@"58a272265b5e05f258161260dd2c5d247213cbd3"
namespace spectrum
open Set Polynomial
open scoped Pointwise Polynomial
universe u v
section Scal... | Mathlib/FieldTheory/IsAlgClosed/Spectrum.lean | 55 | 63 | theorem exists_mem_of_not_isUnit_aeval_prod [IsDomain R] {p : R[X]} {a : A}
(h : ¬IsUnit (aeval a (Multiset.map (fun x : R => X - C x) p.roots).prod)) :
∃ k : R, k ∈ σ a ∧ eval k p = 0 := by |
rw [← Multiset.prod_toList, AlgHom.map_list_prod] at h
replace h := mt List.prod_isUnit h
simp only [not_forall, exists_prop, aeval_C, Multiset.mem_toList, List.mem_map, aeval_X,
exists_exists_and_eq_and, Multiset.mem_map, AlgHom.map_sub] at h
rcases h with ⟨r, r_mem, r_nu⟩
exact ⟨r, by rwa [mem_iff, ← I... | 6 | 403.428793 | 2 | 2 | 2 | 2,470 |
import Mathlib.RingTheory.WittVector.Truncated
import Mathlib.RingTheory.WittVector.Identities
import Mathlib.NumberTheory.Padics.RingHoms
#align_import ring_theory.witt_vector.compare from "leanprover-community/mathlib"@"168ad7fc5d8173ad38be9767a22d50b8ecf1cd00"
noncomputable section
variable {p : ℕ} [hp : Fact... | Mathlib/RingTheory/WittVector/Compare.lean | 127 | 130 | theorem commutes_symm {m : ℕ} (hm : n ≤ m) :
(zmodEquivTrunc p n).symm.toRingHom.comp (truncate hm) =
(ZMod.castHom (pow_dvd_pow p hm) _).comp (zmodEquivTrunc p m).symm.toRingHom := by |
ext; apply commutes_symm'
| 1 | 2.718282 | 0 | 0.75 | 4 | 656 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.deriv.mul from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
noncomputable section
open scoped Classical... | Mathlib/Analysis/Calculus/Deriv/Mul.lean | 114 | 117 | theorem HasStrictDerivAt.smul_const (hc : HasStrictDerivAt c c' x) (f : F) :
HasStrictDerivAt (fun y => c y • f) (c' • f) x := by |
have := hc.smul (hasStrictDerivAt_const x f)
rwa [smul_zero, zero_add] at this
| 2 | 7.389056 | 1 | 1 | 25 | 997 |
import Mathlib.Data.Set.Image
import Mathlib.Order.SuccPred.Relation
import Mathlib.Topology.Clopen
import Mathlib.Topology.Irreducible
#align_import topology.connected from "leanprover-community/mathlib"@"d101e93197bb5f6ea89bd7ba386b7f7dff1f3903"
open Set Function Topology TopologicalSpace Relation
open scoped C... | Mathlib/Topology/Connected/Basic.lean | 142 | 145 | theorem IsPreconnected.union' {s t : Set α} (H : (s ∩ t).Nonempty) (hs : IsPreconnected s)
(ht : IsPreconnected t) : IsPreconnected (s ∪ t) := by |
rcases H with ⟨x, hxs, hxt⟩
exact hs.union x hxs hxt ht
| 2 | 7.389056 | 1 | 1.4 | 5 | 1,507 |
import Mathlib.Analysis.SpecialFunctions.Complex.Arg
import Mathlib.Analysis.SpecialFunctions.Log.Basic
#align_import analysis.special_functions.complex.log from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
namespace Complex
open Set Filter Bornology
open scop... | Mathlib/Analysis/SpecialFunctions/Complex/Log.lean | 39 | 39 | theorem neg_pi_lt_log_im (x : ℂ) : -π < (log x).im := by | simp only [log_im, neg_pi_lt_arg]
| 1 | 2.718282 | 0 | 0.375 | 16 | 378 |
import Mathlib.Analysis.InnerProductSpace.Adjoint
import Mathlib.Analysis.Matrix
import Mathlib.Analysis.RCLike.Basic
import Mathlib.LinearAlgebra.UnitaryGroup
import Mathlib.Topology.UniformSpace.Matrix
#align_import analysis.normed_space.star.matrix from "leanprover-community/mathlib"@"468b141b14016d54b479eb7a0fff1... | Mathlib/Analysis/NormedSpace/Star/Matrix.lean | 49 | 77 | theorem entry_norm_bound_of_unitary {U : Matrix n n 𝕜} (hU : U ∈ Matrix.unitaryGroup n 𝕜)
(i j : n) : ‖U i j‖ ≤ 1 := by |
-- The norm squared of an entry is at most the L2 norm of its row.
have norm_sum : ‖U i j‖ ^ 2 ≤ ∑ x, ‖U i x‖ ^ 2 := by
apply Multiset.single_le_sum
· intro x h_x
rw [Multiset.mem_map] at h_x
cases' h_x with a h_a
rw [← h_a.2]
apply sq_nonneg
· rw [Multiset.mem_map]
use j
... | 27 | 532,048,240,601.79865 | 2 | 2 | 2 | 2,259 |
import Mathlib.LinearAlgebra.Quotient
#align_import linear_algebra.isomorphisms from "leanprover-community/mathlib"@"2738d2ca56cbc63be80c3bd48e9ed90ad94e947d"
universe u v
variable {R M M₂ M₃ : Type*}
variable [Ring R] [AddCommGroup M] [AddCommGroup M₂] [AddCommGroup M₃]
variable [Module R M] [Module R M₂] [Modul... | Mathlib/LinearAlgebra/Isomorphisms.lean | 88 | 93 | theorem quotientInfEquivSupQuotient_surjective (p p' : Submodule R M) :
Function.Surjective (quotientInfToSupQuotient p p') := by |
rw [← range_eq_top, quotientInfToSupQuotient, range_liftQ, eq_top_iff']
rintro ⟨x, hx⟩; rcases mem_sup.1 hx with ⟨y, hy, z, hz, rfl⟩
use ⟨y, hy⟩; apply (Submodule.Quotient.eq _).2
simp only [mem_comap, map_sub, coeSubtype, coe_inclusion, sub_add_cancel_left, neg_mem_iff, hz]
| 4 | 54.59815 | 2 | 1.333333 | 3 | 1,385 |
import Mathlib.Topology.MetricSpace.PiNat
import Mathlib.Topology.MetricSpace.Isometry
import Mathlib.Topology.MetricSpace.Gluing
import Mathlib.Topology.Sets.Opens
import Mathlib.Analysis.Normed.Field.Basic
#align_import topology.metric_space.polish from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78... | Mathlib/Topology/MetricSpace/Polish.lean | 91 | 94 | theorem complete_polishSpaceMetric (α : Type*) [ht : TopologicalSpace α] [h : PolishSpace α] :
@CompleteSpace α (polishSpaceMetric α).toUniformSpace := by |
convert h.complete.choose_spec.2
exact MetricSpace.replaceTopology_eq _ _
| 2 | 7.389056 | 1 | 1.5 | 2 | 1,651 |
import Mathlib.Algebra.Group.ConjFinite
import Mathlib.GroupTheory.Abelianization
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.GroupAction.Quotient
import Mathlib.GroupTheory.Index
import Mathlib.GroupTheory.SpecificGroups.Dihedral
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Line... | Mathlib/GroupTheory/CommutingProbability.lean | 108 | 116 | theorem Subgroup.commProb_subgroup_le : commProb H ≤ commProb G * (H.index : ℚ) ^ 2 := by |
/- After rewriting with `commProb_def`, we reduce to showing that `G` has at least as many
commuting pairs as `H`. -/
rw [commProb_def, commProb_def, div_le_iff, mul_assoc, ← mul_pow, ← Nat.cast_mul,
mul_comm H.index, H.card_mul_index, div_mul_cancel₀, Nat.cast_le]
· refine Finite.card_le_of_injective ... | 8 | 2,980.957987 | 2 | 1.625 | 8 | 1,749 |
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 | 99 | 101 | theorem exists_eq_mul_self [IsAlgClosed k] (x : k) : ∃ z, x = z * z := by |
rcases exists_pow_nat_eq x zero_lt_two with ⟨z, rfl⟩
exact ⟨z, sq z⟩
| 2 | 7.389056 | 1 | 1.5 | 6 | 1,683 |
import Mathlib.Data.Fintype.Card
import Mathlib.Order.UpperLower.Basic
#align_import combinatorics.set_family.intersecting from "leanprover-community/mathlib"@"d90e4e186f1d18e375dcd4e5b5f6364b01cb3e46"
open Finset
variable {α : Type*}
namespace Set
section SemilatticeInf
variable [SemilatticeInf α] [OrderBot ... | Mathlib/Combinatorics/SetFamily/Intersecting.lean | 122 | 130 | theorem Intersecting.isUpperSet' {s : Finset α} (hs : (s : Set α).Intersecting)
(h : ∀ t : Finset α, (t : Set α).Intersecting → s ⊆ t → s = t) : IsUpperSet (s : Set α) := by |
classical
rintro a b hab ha
rw [h (Insert.insert b s) _ (Finset.subset_insert _ _)]
· exact mem_insert_self _ _
rw [coe_insert]
exact
hs.insert (mt (eq_bot_mono hab) <| hs.ne_bot ha) fun c hc hbc => hs ha hc <| hbc.mono_left hab
| 7 | 1,096.633158 | 2 | 1.5 | 4 | 1,627 |
import Mathlib.CategoryTheory.Subobject.Limits
#align_import algebra.homology.image_to_kernel from "leanprover-community/mathlib"@"618ea3d5c99240cd7000d8376924906a148bf9ff"
universe v u w
open CategoryTheory CategoryTheory.Limits
variable {ι : Type*}
variable {V : Type u} [Category.{v} V] [HasZeroMorphisms V]
o... | Mathlib/Algebra/Homology/ImageToKernel.lean | 95 | 98 | theorem imageToKernel_zero_left [HasKernels V] [HasZeroObject V] {w} :
imageToKernel (0 : A ⟶ B) g w = 0 := by |
ext
simp
| 2 | 7.389056 | 1 | 0.888889 | 9 | 768 |
import Mathlib.Topology.MetricSpace.ProperSpace
import Mathlib.Topology.MetricSpace.Cauchy
open Set Filter Bornology
open scoped ENNReal Uniformity Topology Pointwise
universe u v w
variable {α : Type u} {β : Type v} {X ι : Type*}
variable [PseudoMetricSpace α]
namespace Metric
#align metric.bounded Bornology.I... | Mathlib/Topology/MetricSpace/Bounded.lean | 133 | 134 | theorem comap_dist_left_atTop (c : α) : comap (dist c) atTop = cobounded α := by |
simpa only [dist_comm _ c] using comap_dist_right_atTop c
| 1 | 2.718282 | 0 | 0 | 3 | 140 |
import Mathlib.Order.Atoms
import Mathlib.Order.OrderIsoNat
import Mathlib.Order.RelIso.Set
import Mathlib.Order.SupClosed
import Mathlib.Order.SupIndep
import Mathlib.Order.Zorn
import Mathlib.Data.Finset.Order
import Mathlib.Order.Interval.Set.OrderIso
import Mathlib.Data.Finite.Set
import Mathlib.Tactic.TFAE
#alig... | Mathlib/Order/CompactlyGenerated/Basic.lean | 110 | 149 | theorem isCompactElement_iff_le_of_directed_sSup_le (k : α) :
IsCompactElement k ↔
∀ s : Set α, s.Nonempty → DirectedOn (· ≤ ·) s → k ≤ sSup s → ∃ x : α, x ∈ s ∧ k ≤ x := by |
classical
constructor
· intro hk s hne hdir hsup
obtain ⟨t, ht⟩ := hk s hsup
-- certainly every element of t is below something in s, since ↑t ⊆ s.
have t_below_s : ∀ x ∈ t, ∃ y ∈ s, x ≤ y := fun x hxt => ⟨x, ht.left hxt, le_rfl⟩
obtain ⟨x, ⟨hxs, hsupx⟩⟩ := Finset.sup_le_of_le_directe... | 37 | 11,719,142,372,802,612 | 2 | 2 | 3 | 2,488 |
import Mathlib.Order.Filter.Prod
#align_import order.filter.n_ary from "leanprover-community/mathlib"@"78f647f8517f021d839a7553d5dc97e79b508dea"
open Function Set
open Filter
namespace Filter
variable {α α' β β' γ γ' δ δ' ε ε' : Type*} {m : α → β → γ} {f f₁ f₂ : Filter α}
{g g₁ g₂ : Filter β} {h h₁ h₂ : Filt... | Mathlib/Order/Filter/NAry.lean | 91 | 91 | theorem map₂_eq_bot_iff : map₂ m f g = ⊥ ↔ f = ⊥ ∨ g = ⊥ := by | simp [← map_prod_eq_map₂]
| 1 | 2.718282 | 0 | 0 | 5 | 17 |
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 | 373 | 380 | theorem average_add_measure [IsFiniteMeasure μ] {ν : Measure α} [IsFiniteMeasure ν] {f : α → E}
(hμ : Integrable f μ) (hν : Integrable f ν) :
⨍ x, f x ∂(μ + ν) =
((μ univ).toReal / ((μ univ).toReal + (ν univ).toReal)) • ⨍ x, f x ∂μ +
((ν univ).toReal / ((μ univ).toReal + (ν univ).toReal)) • ⨍ x, f... |
simp only [div_eq_inv_mul, mul_smul, measure_smul_average, ← smul_add,
← integral_add_measure hμ hν, ← ENNReal.toReal_add (measure_ne_top μ _) (measure_ne_top ν _)]
rw [average_eq, Measure.add_apply]
| 3 | 20.085537 | 1 | 0.347826 | 23 | 374 |
import Mathlib.CategoryTheory.Opposites
#align_import category_theory.eq_to_hom from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
universe v₁ v₂ v₃ u₁ u₂ u₃
-- morphism levels before object levels. See note [CategoryTheory universes].
namespace CategoryTheory
open Opposite
variable ... | Mathlib/CategoryTheory/EqToHom.lean | 86 | 89 | theorem eqToHom_iso_hom_naturality {f g : β → C} (z : ∀ b, f b ≅ g b) {j j' : β} (w : j = j') :
(z j).hom ≫ eqToHom (by simp [w]) = eqToHom (by simp [w]) ≫ (z j').hom := by |
cases w
simp
| 2 | 7.389056 | 1 | 0.9 | 10 | 777 |
import Mathlib.Topology.Algebra.InfiniteSum.Group
import Mathlib.Topology.Algebra.Star
noncomputable section
open Filter Finset Function
open scoped Topology
variable {α β γ δ : Type*}
section ProdCodomain
variable [CommMonoid α] [TopologicalSpace α] [CommMonoid γ] [TopologicalSpace γ]
@[to_additive HasSum... | Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean | 68 | 70 | theorem HasProd.prod_mk {f : β → α} {g : β → γ} {a : α} {b : γ}
(hf : HasProd f a) (hg : HasProd g b) : HasProd (fun x ↦ (⟨f x, g x⟩ : α × γ)) ⟨a, b⟩ := by |
simp [HasProd, ← prod_mk_prod, Filter.Tendsto.prod_mk_nhds hf hg]
| 1 | 2.718282 | 0 | 1 | 4 | 817 |
import Batteries.Data.DList
import Mathlib.Mathport.Rename
import Mathlib.Tactic.Cases
#align_import data.dlist from "leanprover-community/lean"@"855e5b74e3a52a40552e8f067169d747d48743fd"
universe u
#align dlist Batteries.DList
namespace Batteries.DList
open Function
variable {α : Type u}
#align dlist.of_list... | Mathlib/Data/DList/Defs.lean | 84 | 85 | theorem toList_push (x : α) (l : DList α) : toList (push l x) = toList l ++ [x] := by |
cases' l with _ l_invariant; simp; rw [l_invariant]
| 1 | 2.718282 | 0 | 0.333333 | 6 | 333 |
import Mathlib.Algebra.DirectSum.Finsupp
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.DirectSum.TensorProduct
#align_import linear_algebra.direct_sum.finsupp from "leanprover-community/mathlib"@"9b9d125b7be0930f564a68f1d73ace10cf46064d"
noncomputable section
open DirectSum TensorProduct
ope... | Mathlib/LinearAlgebra/DirectSum/Finsupp.lean | 293 | 295 | theorem finsuppTensorFinsuppLid_apply_apply (f : ι →₀ R) (g : κ →₀ N) (a : ι) (b : κ) :
finsuppTensorFinsuppLid R N ι κ (f ⊗ₜ[R] g) (a, b) = f a • g b := by |
simp [finsuppTensorFinsuppLid]
| 1 | 2.718282 | 0 | 0.75 | 8 | 652 |
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 | 96 | 97 | theorem sym2_eq_empty : s.sym2 = ∅ ↔ s = ∅ := by |
rw [← val_eq_zero, sym2_val, Multiset.sym2_eq_zero_iff, val_eq_zero]
| 1 | 2.718282 | 0 | 0.769231 | 13 | 684 |
import Mathlib.CategoryTheory.Filtered.Connected
import Mathlib.CategoryTheory.Limits.TypesFiltered
import Mathlib.CategoryTheory.Limits.Final
universe v₁ v₂ u₁ u₂
namespace CategoryTheory
open CategoryTheory.Limits CategoryTheory.Functor Opposite
section ArbitraryUniverses
variable {C : Type u₁} [Category.{v₁}... | Mathlib/CategoryTheory/Filtered/Final.lean | 91 | 95 | theorem Functor.final_of_exists_of_isFiltered [IsFilteredOrEmpty C]
(h₁ : ∀ d, ∃ c, Nonempty (d ⟶ F.obj c)) (h₂ : ∀ {d : D} {c : C} (s s' : d ⟶ F.obj c),
∃ (c' : C) (t : c ⟶ c'), s ≫ F.map t = s' ≫ F.map t) : Functor.Final F := by |
suffices ∀ d, IsFiltered (StructuredArrow d F) from final_of_isFiltered_structuredArrow F
exact isFiltered_structuredArrow_of_isFiltered_of_exists F h₁ h₂
| 2 | 7.389056 | 1 | 1.666667 | 6 | 1,795 |
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Ext
local macro:max "local_hAdd[" type:term ", " inst:term "]" : term =>
`(term| (letI := $inst; HAdd.hAdd : $type → $type → $type))
local macro:max "local_hMul[" type:term ", " inst:term "]" : term =>
`(term| (letI := $inst; HMul.hMul : $type → $typ... | Mathlib/Algebra/Ring/Ext.lean | 73 | 77 | theorem toDistrib_injective : Function.Injective (@toDistrib R) := by |
intro _ _ h
ext x y
· exact congrArg (·.toAdd.add x y) h
· exact congrArg (·.toMul.mul x y) h
| 4 | 54.59815 | 2 | 0.4 | 10 | 397 |
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#align_import algebra.order.group.defs from "leanprover-community/mathlib"@"b599f4e4e5cf1fbcb4194503671d3d9e569c1fce"
open Function
universe u
variable {α : Type u}
class OrderedAddCommGroup (α : Ty... | Mathlib/Algebra/Order/Group/Defs.lean | 196 | 197 | theorem lt_inv_mul_iff_lt : 1 < b⁻¹ * a ↔ b < a := by |
rw [← mul_lt_mul_iff_left b, mul_one, mul_inv_cancel_left]
| 1 | 2.718282 | 0 | 0.4 | 25 | 400 |
import Mathlib.Data.Multiset.Dedup
#align_import data.multiset.finset_ops from "leanprover-community/mathlib"@"c227d107bbada5d0d9d20287e3282c0a7f1651a0"
namespace Multiset
open List
variable {α : Type*} [DecidableEq α] {s : Multiset α}
def ndinsert (a : α) (s : Multiset α) : Multiset α :=
Quot.liftOn s (... | Mathlib/Data/Multiset/FinsetOps.lean | 83 | 84 | theorem dedup_cons {a : α} {s : Multiset α} : dedup (a ::ₘ s) = ndinsert a (dedup s) := by |
by_cases h : a ∈ s <;> simp [h]
| 1 | 2.718282 | 0 | 0.4 | 5 | 392 |
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Data.List.Rotate
import Mathlib.GroupTheory.Perm.Support
#align_import group_theory.perm.list from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace List
variable {α β : Type*}
section FormPerm
variable [DecidableEq α] (l :... | Mathlib/GroupTheory/Perm/List.lean | 100 | 103 | theorem support_formPerm_le [Fintype α] : support (formPerm l) ≤ l.toFinset := by |
intro x hx
have hx' : x ∈ { x | formPerm l x ≠ x } := by simpa using hx
simpa using support_formPerm_le' _ hx'
| 3 | 20.085537 | 1 | 0.7 | 10 | 641 |
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Sym.Basic
import Mathlib.Data.Sym.Sym2.Init
import Mathlib.Data.SetLike.Basic
#align_import data.sym.sym2 from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
assert_not_exists MonoidWithZero
open Finset Function Sym
universe u
variab... | Mathlib/Data/Sym/Sym2.lean | 69 | 69 | theorem Rel.symm {x y : α × α} : Rel α x y → Rel α y x := by | aesop (rule_sets := [Sym2])
| 1 | 2.718282 | 0 | 0 | 4 | 173 |
import Mathlib.AlgebraicTopology.SimplicialObject
import Mathlib.CategoryTheory.Limits.Shapes.Products
#align_import algebraic_topology.split_simplicial_object from "leanprover-community/mathlib"@"dd1f8496baa505636a82748e6b652165ea888733"
noncomputable section
open CategoryTheory CategoryTheory.Category Category... | Mathlib/AlgebraicTopology/SplitSimplicialObject.lean | 143 | 151 | theorem eqId_iff_len_eq : A.EqId ↔ A.1.unop.len = Δ.unop.len := by |
rw [eqId_iff_eq]
constructor
· intro h
rw [h]
· intro h
rw [← unop_inj_iff]
ext
exact h
| 8 | 2,980.957987 | 2 | 2 | 5 | 2,188 |
import Mathlib.Algebra.Order.Ring.Nat
#align_import data.nat.dist from "leanprover-community/mathlib"@"d50b12ae8e2bd910d08a94823976adae9825718b"
namespace Nat
def dist (n m : ℕ) :=
n - m + (m - n)
#align nat.dist Nat.dist
-- Should be aligned to `Nat.dist.eq_def`, but that is generated on demand and isn't pr... | Mathlib/Data/Nat/Dist.lean | 92 | 96 | theorem dist.triangle_inequality (n m k : ℕ) : dist n k ≤ dist n m + dist m k := by |
have : dist n m + dist m k = n - m + (m - k) + (k - m + (m - n)) := by
simp [dist, add_comm, add_left_comm, add_assoc]
rw [this, dist]
exact add_le_add tsub_le_tsub_add_tsub tsub_le_tsub_add_tsub
| 4 | 54.59815 | 2 | 0.266667 | 15 | 309 |
import Mathlib.Data.Matroid.Restrict
variable {α : Type*} {M : Matroid α} {E B I X R J : Set α}
namespace Matroid
open Set
section EmptyOn
def emptyOn (α : Type*) : Matroid α where
E := ∅
Base := (· = ∅)
Indep := (· = ∅)
indep_iff' := by simp [subset_empty_iff]
exists_base := ⟨∅, rfl⟩
base_exchange... | Mathlib/Data/Matroid/Constructions.lean | 71 | 73 | theorem eq_emptyOn [IsEmpty α] (M : Matroid α) : M = emptyOn α := by |
rw [← ground_eq_empty_iff]
exact M.E.eq_empty_of_isEmpty
| 2 | 7.389056 | 1 | 1 | 3 | 1,145 |
import Mathlib.GroupTheory.GroupAction.BigOperators
import Mathlib.Logic.Equiv.Fin
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.Algebra.Module.Prod
import Mathlib.Algebra.Module.Submodule.Ker
#align_import linear_algebra.pi from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
un... | Mathlib/LinearAlgebra/Pi.lean | 64 | 66 | theorem pi_eq_zero (f : (i : ι) → M₂ →ₗ[R] φ i) : pi f = 0 ↔ ∀ i, f i = 0 := by |
simp only [LinearMap.ext_iff, pi_apply, funext_iff];
exact ⟨fun h a b => h b a, fun h a b => h b a⟩
| 2 | 7.389056 | 1 | 0.333333 | 3 | 346 |
import Mathlib.Data.Set.Prod
import Mathlib.Logic.Equiv.Fin
import Mathlib.ModelTheory.LanguageMap
#align_import model_theory.syntax from "leanprover-community/mathlib"@"d565b3df44619c1498326936be16f1a935df0728"
universe u v w u' v'
namespace FirstOrder
namespace Language
variable (L : Language.{u, v}) {L' : L... | Mathlib/ModelTheory/Syntax.lean | 284 | 290 | theorem comp_onTerm {L'' : Language} (φ : L' →ᴸ L'') (ψ : L →ᴸ L') :
((φ.comp ψ).onTerm : L.Term α → L''.Term α) = φ.onTerm ∘ ψ.onTerm := by |
ext t
induction' t with _ _ _ _ ih
· rfl
· simp_rw [onTerm, ih]
rfl
| 5 | 148.413159 | 2 | 1.5 | 4 | 1,562 |
import Mathlib.RingTheory.FiniteType
#align_import ring_theory.rees_algebra from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open Polynomial
def reesAlgebra : Subalgebra... | Mathlib/RingTheory/ReesAlgebra.lean | 68 | 73 | theorem mem_reesAlgebra_iff_support (f : R[X]) :
f ∈ reesAlgebra I ↔ ∀ i ∈ f.support, f.coeff i ∈ I ^ i := by |
apply forall_congr'
intro a
rw [mem_support_iff, Iff.comm, Classical.imp_iff_right_iff, Ne, ← imp_iff_not_or]
exact fun e => e.symm ▸ (I ^ a).zero_mem
| 4 | 54.59815 | 2 | 1.8 | 5 | 1,902 |
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 | 38 | 40 | theorem bit0_eq_zero : (bit0 : R → R) = 0 := by |
funext
exact add_self_eq_zero _
| 2 | 7.389056 | 1 | 0.2 | 10 | 273 |
import Mathlib.Data.Set.Function
import Mathlib.Logic.Relation
import Mathlib.Logic.Pairwise
#align_import data.set.pairwise.basic from "leanprover-community/mathlib"@"c4c2ed622f43768eff32608d4a0f8a6cec1c047d"
open Function Order Set
variable {α β γ ι ι' : Type*} {r p q : α → α → Prop}
section Pairwise
variabl... | Mathlib/Data/Set/Pairwise/Basic.lean | 121 | 125 | theorem pairwise_iff_exists_forall [Nonempty ι] (s : Set α) (f : α → ι) {r : ι → ι → Prop}
[IsEquiv ι r] : s.Pairwise (r on f) ↔ ∃ z, ∀ x ∈ s, r (f x) z := by |
rcases s.eq_empty_or_nonempty with (rfl | hne)
· simp
· exact hne.pairwise_iff_exists_forall
| 3 | 20.085537 | 1 | 1 | 5 | 1,138 |
import Mathlib.Data.Option.Basic
import Mathlib.Data.Set.Basic
#align_import data.pequiv from "leanprover-community/mathlib"@"7c3269ca3fa4c0c19e4d127cd7151edbdbf99ed4"
universe u v w x
structure PEquiv (α : Type u) (β : Type v) where
toFun : α → Option β
invFun : β → Option α
inv : ∀ (a : α) (b :... | Mathlib/Data/PEquiv.lean | 136 | 136 | theorem symm_symm (f : α ≃. β) : f.symm.symm = f := by | cases f; rfl
| 1 | 2.718282 | 0 | 0.25 | 4 | 290 |
import Mathlib.Analysis.NormedSpace.BoundedLinearMaps
import Mathlib.Topology.FiberBundle.Basic
#align_import topology.vector_bundle.basic from "leanprover-community/mathlib"@"e473c3198bb41f68560cab68a0529c854b618833"
noncomputable section
open scoped Classical
open Bundle Set
open scoped Topology
variable (R : ... | Mathlib/Topology/VectorBundle/Basic.lean | 157 | 160 | theorem linearMapAt_symmₗ (e : Pretrivialization F (π F E)) [e.IsLinear R] {b : B}
(hb : b ∈ e.baseSet) (y : F) : e.linearMapAt R b (e.symmₗ R b y) = y := by |
rw [e.linearMapAt_def_of_mem hb]
exact (e.linearEquivAt R b hb).right_inv y
| 2 | 7.389056 | 1 | 0.6 | 5 | 539 |
import Mathlib.Data.SetLike.Fintype
import Mathlib.Algebra.Divisibility.Prod
import Mathlib.RingTheory.Nakayama
import Mathlib.RingTheory.SimpleModule
import Mathlib.Tactic.RSuffices
#align_import ring_theory.artinian from "leanprover-community/mathlib"@"210657c4ea4a4a7b234392f70a3a2a83346dfa90"
open Set Filter Po... | Mathlib/RingTheory/Artinian.lean | 175 | 195 | theorem IsArtinian.finite_of_linearIndependent [Nontrivial R] [IsArtinian R M] {s : Set M}
(hs : LinearIndependent R ((↑) : s → M)) : s.Finite := by |
refine by_contradiction fun hf => (RelEmbedding.wellFounded_iff_no_descending_seq.1
(wellFounded_submodule_lt (R := R) (M := M))).elim' ?_
have f : ℕ ↪ s := Set.Infinite.natEmbedding s hf
have : ∀ n, (↑) ∘ f '' { m | n ≤ m } ⊆ s := by
rintro n x ⟨y, _, rfl⟩
exact (f y).2
have : ∀ a b : ℕ, a ≤ b ↔
... | 19 | 178,482,300.963187 | 2 | 2 | 1 | 1,948 |
import Mathlib.Probability.ProbabilityMassFunction.Constructions
import Mathlib.Tactic.FinCases
namespace PMF
open ENNReal
noncomputable
def binomial (p : ℝ≥0∞) (h : p ≤ 1) (n : ℕ) : PMF (Fin (n + 1)) :=
.ofFintype (fun i => p^(i : ℕ) * (1-p)^((Fin.last n - i) : ℕ) * (n.choose i : ℕ)) (by
convert (add_pow ... | Mathlib/Probability/ProbabilityMassFunction/Binomial.lean | 49 | 50 | theorem binomial_apply_self (p : ℝ≥0∞) (h : p ≤ 1) (n : ℕ) :
binomial p h n n = p^n := by | simp
| 1 | 2.718282 | 0 | 0 | 4 | 22 |
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 | 162 | 163 | theorem U_eq (n : ℤ) : U R n = 2 * X * U R (n - 1) - U R (n - 2) := by |
linear_combination (norm := ring_nf) U_add_two R (n - 2)
| 1 | 2.718282 | 0 | 0.166667 | 12 | 265 |
import Mathlib.Topology.MetricSpace.HausdorffDistance
#align_import topology.metric_space.pi_nat from "leanprover-community/mathlib"@"49b7f94aab3a3bdca1f9f34c5d818afb253b3993"
noncomputable section
open scoped Classical
open Topology Filter
open TopologicalSpace Set Metric Filter Function
attribute [local simp... | Mathlib/Topology/MetricSpace/PiNat.lean | 134 | 147 | theorem mem_cylinder_iff_eq {x y : ∀ n, E n} {n : ℕ} :
y ∈ cylinder x n ↔ cylinder y n = cylinder x n := by |
constructor
· intro hy
apply Subset.antisymm
· intro z hz i hi
rw [← hy i hi]
exact hz i hi
· intro z hz i hi
rw [hy i hi]
exact hz i hi
· intro h
rw [← h]
exact self_mem_cylinder _ _
| 12 | 162,754.791419 | 2 | 1.083333 | 12 | 1,184 |
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.LinearAlgebra.SesquilinearForm
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
variable {𝕜 E F : Type*} [RCLike 𝕜]
variable [NormedAddCommGroup E] [InnerProductSpace... | Mathlib/Analysis/InnerProductSpace/Orthogonal.lean | 68 | 69 | theorem inner_left_of_mem_orthogonal {u v : E} (hu : u ∈ K) (hv : v ∈ Kᗮ) : ⟪v, u⟫ = 0 := by |
rw [inner_eq_zero_symm]; exact inner_right_of_mem_orthogonal hu hv
| 1 | 2.718282 | 0 | 1.1 | 10 | 1,190 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Div
#align_import data.polynomial.ring_division from "leanprover-community/mathlib"@"8efcf8022aac8e01df8d302dcebdbc25d6a886c8"
noncomputable ... | Mathlib/Algebra/Polynomial/RingDivision.lean | 178 | 180 | theorem not_dvd_of_degree_lt {p q : R[X]} (h0 : q ≠ 0) (hl : q.degree < p.degree) : ¬p ∣ q := by |
by_contra hcontra
exact h0 (eq_zero_of_dvd_of_degree_lt hcontra hl)
| 2 | 7.389056 | 1 | 1.5 | 32 | 1,561 |
import Mathlib.Algebra.ContinuedFractions.Basic
import Mathlib.Algebra.GroupWithZero.Basic
#align_import algebra.continued_fractions.translations from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b348ce40ad"
namespace GeneralizedContinuedFraction
section WithDivisionRing
variable {K : Type*}... | Mathlib/Algebra/ContinuedFractions/Translations.lean | 150 | 152 | theorem second_continuant_aux_eq {gp : Pair K} (zeroth_s_eq : g.s.get? 0 = some gp) :
g.continuantsAux 2 = ⟨gp.b * g.h + gp.a, gp.b⟩ := by |
simp [zeroth_s_eq, continuantsAux, nextContinuants, nextDenominator, nextNumerator]
| 1 | 2.718282 | 0 | 0.052632 | 19 | 240 |
import Mathlib.Algebra.Group.Hom.Defs
import Mathlib.Algebra.Group.Units
#align_import algebra.hom.units from "leanprover-community/mathlib"@"a07d750983b94c530ab69a726862c2ab6802b38c"
assert_not_exists MonoidWithZero
assert_not_exists DenselyOrdered
open Function
universe u v w
namespace Units
variable {α : Ty... | Mathlib/Algebra/Group/Units/Hom.lean | 198 | 199 | theorem map [MonoidHomClass F M N] (f : F) {x : M} (h : IsUnit x) : IsUnit (f x) := by |
rcases h with ⟨y, rfl⟩; exact (Units.map (f : M →* N) y).isUnit
| 1 | 2.718282 | 0 | 0 | 4 | 43 |
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 | 69 | 69 | theorem eigenspace_zero (f : End R M) : f.eigenspace 0 = LinearMap.ker f := by | simp [eigenspace]
| 1 | 2.718282 | 0 | 0.714286 | 7 | 644 |
import Mathlib.FieldTheory.RatFunc.AsPolynomial
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.Content
noncomputable section
universe u
variable {K : Type u}
namespace RatFunc
section IntDegree
open Polynomial
variable [Field... | Mathlib/FieldTheory/RatFunc/Degree.lean | 102 | 107 | theorem natDegree_num_mul_right_sub_natDegree_denom_mul_left_eq_intDegree {x : RatFunc K}
(hx : x ≠ 0) {s : K[X]} (hs : s ≠ 0) :
((x.num * s).natDegree : ℤ) - (s * x.denom).natDegree = x.intDegree := by |
apply natDegree_sub_eq_of_prod_eq (mul_ne_zero (num_ne_zero hx) hs)
(mul_ne_zero hs x.denom_ne_zero) (num_ne_zero hx) x.denom_ne_zero
rw [mul_assoc]
| 3 | 20.085537 | 1 | 1 | 9 | 886 |
import Mathlib.Analysis.Convex.Cone.InnerDual
import Mathlib.Algebra.Order.Nonneg.Module
import Mathlib.Algebra.Module.Submodule.Basic
variable {𝕜 E F G : Type*}
local notation3 "𝕜≥0" => {c : 𝕜 // 0 ≤ c}
abbrev PointedCone (𝕜 E) [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] :=
Submodule {c : 𝕜 // 0... | Mathlib/Analysis/Convex/Cone/Pointed.lean | 51 | 52 | theorem toConvexCone_pointed (S : PointedCone 𝕜 E) : (S : ConvexCone 𝕜 E).Pointed := by |
simp [toConvexCone, ConvexCone.Pointed]
| 1 | 2.718282 | 0 | 0 | 1 | 137 |
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.affine_space.slope from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open AffineMap
variable {k E PE : Type*} [Field k] [AddCommGroup E] [Module k E] [AddTorsor E PE]
def ... | Mathlib/LinearAlgebra/AffineSpace/Slope.lean | 92 | 93 | theorem slope_comm (f : k → PE) (a b : k) : slope f a b = slope f b a := by |
rw [slope, slope, ← neg_vsub_eq_vsub_rev, smul_neg, ← neg_smul, neg_inv, neg_sub]
| 1 | 2.718282 | 0 | 0.7 | 10 | 639 |
import Mathlib.Tactic.CategoryTheory.Elementwise
import Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer
import Mathlib.CategoryTheory.Limits.Constructions.EpiMono
import Mathlib.CategoryTheory.Limits.Preserves.Limits
import Mathlib.CategoryTheory.Limits.Shapes.Types
#align_import category_theory.glue_data from "l... | Mathlib/CategoryTheory/GlueData.lean | 108 | 111 | theorem t'_inv (i j k : D.J) :
D.t' i j k ≫ (pullbackSymmetry _ _).hom ≫ D.t' j i k ≫ (pullbackSymmetry _ _).hom = 𝟙 _ := by |
rw [← cancel_mono (pullback.fst : pullback (D.f i j) (D.f i k) ⟶ _)]
simp [t_fac, t_fac_assoc]
| 2 | 7.389056 | 1 | 1.5 | 6 | 1,591 |
import Mathlib.Data.Set.Lattice
import Mathlib.Init.Set
import Mathlib.Control.Basic
import Mathlib.Lean.Expr.ExtraRecognizers
#align_import data.set.functor from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
universe u
open Function
namespace Set
variable {α β : Type u} {s : Set α} ... | Mathlib/Data/Set/Functor.lean | 93 | 94 | theorem coe_subset : (γ : Set α) ⊆ β := by |
intro _ ⟨_, ⟨⟨⟨_, ha⟩, rfl⟩, _, ⟨_, rfl⟩, _⟩⟩; convert ha
| 1 | 2.718282 | 0 | 0.285714 | 7 | 312 |
import Mathlib.Geometry.Manifold.MFDeriv.Basic
noncomputable section
open scoped Manifold
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜] {E : Type*} [NormedAddCommGroup E]
[NormedSpace 𝕜 E] {E' : Type*} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {f : E → E'}
{s : Set E} {x : E}
section MFDerivFderiv
t... | Mathlib/Geometry/Manifold/MFDeriv/FDeriv.lean | 108 | 110 | theorem mdifferentiable_iff_differentiable :
MDifferentiable 𝓘(𝕜, E) 𝓘(𝕜, E') f ↔ Differentiable 𝕜 f := by |
simp only [MDifferentiable, Differentiable, mdifferentiableAt_iff_differentiableAt]
| 1 | 2.718282 | 0 | 0.5 | 8 | 455 |
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 | 130 | 131 | theorem card_fintypeIcc : Fintype.card (Set.Icc a b) = b + 1 - a := by |
rw [← card_Icc, Fintype.card_ofFinset]
| 1 | 2.718282 | 0 | 0.125 | 16 | 251 |
import Mathlib.MeasureTheory.Measure.VectorMeasure
import Mathlib.Order.SymmDiff
#align_import measure_theory.decomposition.signed_hahn from "leanprover-community/mathlib"@"bc7d81beddb3d6c66f71449c5bc76c38cb77cf9e"
noncomputable section
open scoped Classical NNReal ENNReal MeasureTheory
variable {α β : Type*} [... | Mathlib/MeasureTheory/Decomposition/SignedHahn.lean | 342 | 364 | theorem bddBelow_measureOfNegatives : BddBelow s.measureOfNegatives := by |
simp_rw [BddBelow, Set.Nonempty, mem_lowerBounds]
by_contra! h
have h' : ∀ n : ℕ, ∃ y : ℝ, y ∈ s.measureOfNegatives ∧ y < -n := fun n => h (-n)
choose f hf using h'
have hf' : ∀ n : ℕ, ∃ B, MeasurableSet B ∧ s ≤[B] 0 ∧ s B < -n := by
intro n
rcases hf n with ⟨⟨B, ⟨hB₁, hBr⟩, hB₂⟩, hlt⟩
exact ⟨B, ... | 22 | 3,584,912,846.131591 | 2 | 2 | 1 | 1,986 |
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