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 | eval_complexity float64 0 1 |
|---|---|---|---|---|---|---|
import Mathlib.Order.Filter.AtTopBot
import Mathlib.Order.Filter.Pi
#align_import order.filter.cofinite from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
open Set Function
variable {ι α β : Type*} {l : Filter α}
namespace Filter
def cofinite : Filter α :=
comk Set.Finite finite_e... | Mathlib/Order/Filter/Cofinite.lean | 57 | 58 | theorem cofinite_eq_bot_iff : @cofinite α = ⊥ ↔ Finite α := by |
simp [← empty_mem_iff_bot, finite_univ_iff]
| 0.75 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Comp
import Mathlib.Analysis.Calculus.FDeriv.RestrictScalars
#align_import analysis.calculus.deriv.comp from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
open scoped Classical
open Top... | Mathlib/Analysis/Calculus/Deriv/Comp.lean | 357 | 361 | theorem HasFDerivWithinAt.comp_hasDerivWithinAt_of_eq {t : Set F}
(hl : HasFDerivWithinAt l l' t y)
(hf : HasDerivWithinAt f f' s x) (hst : MapsTo f s t) (hy : y = f x) :
HasDerivWithinAt (l ∘ f) (l' f') s x := by |
rw [hy] at hl; exact hl.comp_hasDerivWithinAt x hf hst
| 0.75 |
import Mathlib.Algebra.Polynomial.Expand
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.FieldTheory.Minpoly.Field
import Mathlib.RingTheory.PowerBasis
#align_import field_theory.separable from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
... | Mathlib/FieldTheory/Separable.lean | 87 | 89 | theorem Separable.of_mul_left {f g : R[X]} (h : (f * g).Separable) : f.Separable := by |
have := h.of_mul_left_left; rw [derivative_mul] at this
exact IsCoprime.of_mul_right_left (IsCoprime.of_add_mul_left_right this)
| 0.75 |
import Mathlib.Algebra.Ring.Prod
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.Tactic.FinCases
#align_import data.zmod.basic from "leanprover-community/mathlib"@"74ad1c88c77e799d2fea62801d1dbbd698cff1b7"
assert_not_exists Submodule
open Function
namespace ZMod
instance charZero : CharZero (ZMod 0) :=... | Mathlib/Data/ZMod/Basic.lean | 137 | 139 | theorem ringChar_zmod_n (n : ℕ) : ringChar (ZMod n) = n := by |
rw [ringChar.eq_iff]
exact ZMod.charP n
| 0.75 |
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 | 56 | 58 | theorem equalizerSubobject_arrow' :
(equalizerSubobjectIso f g).inv ≫ (equalizerSubobject f g).arrow = equalizer.ι f g := by |
simp [equalizerSubobjectIso]
| 0.75 |
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 | 248 | 251 | theorem HasDerivAt.mul_const (hc : HasDerivAt c c' x) (d : 𝔸) :
HasDerivAt (fun y => c y * d) (c' * d) x := by |
rw [← hasDerivWithinAt_univ] at *
exact hc.mul_const d
| 0.75 |
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 | 487 | 492 | theorem HasDerivWithinAt.clm_apply (hc : HasDerivWithinAt c c' s x)
(hu : HasDerivWithinAt u u' s x) :
HasDerivWithinAt (fun y => (c y) (u y)) (c' (u x) + c x u') s x := by |
have := (hc.hasFDerivWithinAt.clm_apply hu.hasFDerivWithinAt).hasDerivWithinAt
rwa [add_apply, comp_apply, flip_apply, smulRight_apply, smulRight_apply, one_apply, one_smul,
one_smul, add_comm] at this
| 0.75 |
import Mathlib.CategoryTheory.Preadditive.Basic
#align_import category_theory.preadditive.functor_category from "leanprover-community/mathlib"@"829895f162a1f29d0133f4b3538f4cd1fb5bffd3"
namespace CategoryTheory
open CategoryTheory.Limits Preadditive
variable {C D : Type*} [Category C] [Category D] [Preadditive D... | Mathlib/CategoryTheory/Preadditive/FunctorCategory.lean | 123 | 124 | theorem app_units_zsmul (X : C) (α : F ⟶ G) (n : ℤˣ) : (n • α).app X = n • α.app X := by |
apply app_zsmul
| 0.75 |
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 | 152 | 153 | theorem card_fintype_uIcc : Fintype.card (Set.uIcc a b) = (b - a : ℤ).natAbs + 1 := by |
rw [← card_uIcc, Fintype.card_ofFinset]
| 0.75 |
import Batteries.Data.List.Lemmas
import Batteries.Tactic.Classical
import Mathlib.Tactic.TypeStar
import Mathlib.Mathport.Rename
#align_import data.list.tfae from "leanprover-community/mathlib"@"5a3e819569b0f12cbec59d740a2613018e7b8eec"
namespace List
def TFAE (l : List Prop) : Prop :=
∀ x ∈ l, ∀ y ∈ l, x ↔ ... | Mathlib/Data/List/TFAE.lean | 110 | 115 | theorem exists_tfae {α : Type*} (l : List (α → Prop)) (H : ∀ a : α, (l.map (fun p ↦ p a)).TFAE) :
(l.map (fun p ↦ ∃ a, p a)).TFAE := by |
simp only [TFAE, List.forall_mem_map_iff]
intros p₁ hp₁ p₂ hp₂
exact exists_congr fun a ↦ H a (p₁ a) (mem_map_of_mem (fun p ↦ p a) hp₁)
(p₂ a) (mem_map_of_mem (fun p ↦ p a) hp₂)
| 0.75 |
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]
| 0.75 |
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Algebra.Module.BigOperators
import Mathlib.LinearAlgebra.Isomorphisms
import Mathlib.GroupTheory.Torsion
import Mathlib.RingTheory.Coprime.Ideal
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Set.Lattice
#align_import algebra.module.torsion from "leanpr... | Mathlib/Algebra/Module/Torsion.lean | 79 | 79 | theorem torsionOf_zero : torsionOf R M (0 : M) = ⊤ := by | simp [torsionOf]
| 0.75 |
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.Data.Nat.Lattice
#align_import combinatorics.simple_graph.metric from "leanprover-community/mathlib"@"352ecfe114946c903338006dd3287cb5a9955ff2"
namespace SimpleGraph
variable {V : Type*} (G : SimpleGraph V)
noncomputable def dist (u v : V)... | Mathlib/Combinatorics/SimpleGraph/Metric.lean | 95 | 96 | theorem dist_eq_zero_of_not_reachable {u v : V} (h : ¬G.Reachable u v) : G.dist u v = 0 := by |
simp [h]
| 0.75 |
import Mathlib.Data.Set.Subsingleton
import Mathlib.Order.WithBot
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
section Image
variable {f : α → β} {s t : Set... | Mathlib/Data/Set/Image.lean | 273 | 275 | theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by |
simp_rw [image_image, h_comm]
| 0.75 |
import Mathlib.Analysis.Convex.Basic
import Mathlib.Order.Closure
#align_import analysis.convex.hull from "leanprover-community/mathlib"@"92bd7b1ffeb306a89f450bee126ddd8a284c259d"
open Set
open Pointwise
variable {𝕜 E F : Type*}
section convexHull
section OrderedSemiring
variable [OrderedSemiring 𝕜]
secti... | Mathlib/Analysis/Convex/Hull.lean | 104 | 106 | theorem convexHull_nonempty_iff : (convexHull 𝕜 s).Nonempty ↔ s.Nonempty := by |
rw [nonempty_iff_ne_empty, nonempty_iff_ne_empty, Ne, Ne]
exact not_congr convexHull_empty_iff
| 0.75 |
import Mathlib.Data.Matrix.Basic
variable {l m n o : Type*}
universe u v w
variable {R : Type*} {α : Type v} {β : Type w}
namespace Matrix
def col (w : m → α) : Matrix m Unit α :=
of fun x _ => w x
#align matrix.col Matrix.col
-- TODO: set as an equation lemma for `col`, see mathlib4#3024
@[simp]
theorem col... | Mathlib/Data/Matrix/RowCol.lean | 148 | 151 | theorem diag_col_mul_row [Mul α] [AddCommMonoid α] (a b : n → α) :
diag (col a * row b) = a * b := by |
ext
simp [Matrix.mul_apply, col, row]
| 0.75 |
import Mathlib.MeasureTheory.Integral.Lebesgue
#align_import measure_theory.measure.giry_monad from "leanprover-community/mathlib"@"56f4cd1ef396e9fd389b5d8371ee9ad91d163625"
noncomputable section
open scoped Classical
open ENNReal
open scoped Classical
open Set Filter
variable {α β : Type*}
namespace MeasureT... | Mathlib/MeasureTheory/Measure/GiryMonad.lean | 85 | 88 | theorem measurable_dirac : Measurable (Measure.dirac : α → Measure α) := by |
refine measurable_of_measurable_coe _ fun s hs => ?_
simp_rw [dirac_apply' _ hs]
exact measurable_one.indicator hs
| 0.75 |
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 | 86 | 90 | theorem log_ofReal_mul {r : ℝ} (hr : 0 < r) {x : ℂ} (hx : x ≠ 0) :
log (r * x) = Real.log r + log x := by |
replace hx := Complex.abs.ne_zero_iff.mpr hx
simp_rw [log, map_mul, abs_ofReal, arg_real_mul _ hr, abs_of_pos hr, Real.log_mul hr.ne' hx,
ofReal_add, add_assoc]
| 0.75 |
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Matrix.RowCol
import Mathlib.GroupTheory.GroupAction.Ring
import Mathlib.GroupTheory.Perm.Fin
import Mathlib.LinearAlgebra.Alternating.Basic
#align_import linear_algebra.matrix.determinant from "leanprover-community/mathlib"@"c30... | Mathlib/LinearAlgebra/Matrix/Determinant/Basic.lean | 94 | 94 | theorem det_isEmpty [IsEmpty n] {A : Matrix n n R} : det A = 1 := by | simp [det_apply]
| 0.75 |
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.shapes.kernels from "leanprover-community/mathlib"@"956af7c76589f444f2e1313911bad16366ea476d"
noncomputable section
universe v v₂ u u' u₂
open CategoryTheory
open CategoryTheory.Limits.WalkingParallelPair
namespace... | Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean | 91 | 92 | theorem KernelFork.app_one (s : KernelFork f) : s.π.app one = 0 := by |
simp [Fork.app_one_eq_ι_comp_right]
| 0.75 |
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
import Mathlib.Logic.Basic
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort*} {f : α → β}
@[reducible, simp] de... | Mathlib/Logic/Function/Basic.lean | 89 | 91 | theorem Injective.beq_eq {α β : Type*} [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β] {f : α → β}
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by |
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
| 0.75 |
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 | 70 | 71 | theorem FiniteDimensional.finrank_linearMap_self : finrank S (M →ₗ[R] S) = finrank R M := by |
rw [finrank_linearMap, finrank_self, mul_one]
| 0.75 |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Finset.NatAntidiagonal
#align_import algebra.big_operators.nat_antidiagonal from "leanprover-community/mathlib"@"008205aa645b3f194c1da47025c5f110c8406eab"
variable {M N : Type*} [CommMonoid M] [AddCommMonoid N]
namespace Finset
namespace Nat
| Mathlib/Algebra/BigOperators/NatAntidiagonal.lean | 23 | 26 | theorem prod_antidiagonal_succ {n : ℕ} {f : ℕ × ℕ → M} :
(∏ p ∈ antidiagonal (n + 1), f p)
= f (0, n + 1) * ∏ p ∈ antidiagonal n, f (p.1 + 1, p.2) := by |
rw [antidiagonal_succ, prod_cons, prod_map]; rfl
| 0.75 |
import Mathlib.Algebra.Module.Submodule.EqLocus
import Mathlib.Algebra.Module.Submodule.RestrictScalars
import Mathlib.Algebra.Ring.Idempotents
import Mathlib.Data.Set.Pointwise.SMul
import Mathlib.LinearAlgebra.Basic
import Mathlib.Order.CompactlyGenerated.Basic
import Mathlib.Order.OmegaCompletePartialOrder
#align_... | Mathlib/LinearAlgebra/Span.lean | 147 | 150 | theorem span_insert_zero : span R (insert (0 : M) s) = span R s := by |
refine le_antisymm ?_ (Submodule.span_mono (Set.subset_insert 0 s))
rw [span_le, Set.insert_subset_iff]
exact ⟨by simp only [SetLike.mem_coe, Submodule.zero_mem], Submodule.subset_span⟩
| 0.75 |
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.GeomSum
import Mathlib.LinearAlgebra.Matrix.Block
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.Nondegenerate
#align_import linear_algebra.vandermonde from "leanprover-community/mathlib"@"70fd9563a21e7b963887c93... | Mathlib/LinearAlgebra/Vandermonde.lean | 72 | 74 | theorem vandermonde_transpose_mul_vandermonde {n : ℕ} (v : Fin n → R) (i j) :
((vandermonde v)ᵀ * vandermonde v) i j = ∑ k : Fin n, v k ^ (i + j : ℕ) := by |
simp only [vandermonde_apply, Matrix.mul_apply, Matrix.transpose_apply, pow_add]
| 0.75 |
import Mathlib.Analysis.InnerProductSpace.TwoDim
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
#align_import geometry.euclidean.angle.oriented.basic from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
noncomputable section
open FiniteDimensional Complex
open scoped Real Rea... | Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | 68 | 68 | theorem oangle_zero_left (x : V) : o.oangle 0 x = 0 := by | simp [oangle]
| 0.75 |
import Mathlib.Algebra.Regular.Basic
import Mathlib.LinearAlgebra.Matrix.MvPolynomial
import Mathlib.LinearAlgebra.Matrix.Polynomial
import Mathlib.RingTheory.Polynomial.Basic
#align_import linear_algebra.matrix.adjugate from "leanprover-community/mathlib"@"a99f85220eaf38f14f94e04699943e185a5e1d1a"
namespace Matr... | Mathlib/LinearAlgebra/Matrix/Adjugate.lean | 102 | 103 | theorem cramer_transpose_apply (i : n) : cramer Aᵀ b i = (A.updateRow i b).det := by |
rw [cramer_apply, updateColumn_transpose, det_transpose]
| 0.75 |
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 | 196 | 197 | theorem mul_mul_mul_comm (a b c d : G) : a * b * (c * d) = a * c * (b * d) := by |
simp only [mul_left_comm, mul_assoc]
| 0.75 |
import Mathlib.GroupTheory.Coxeter.Length
import Mathlib.Data.ZMod.Parity
namespace CoxeterSystem
open List Matrix Function
variable {B : Type*}
variable {W : Type*} [Group W]
variable {M : CoxeterMatrix B} (cs : CoxeterSystem M W)
local prefix:100 "s" => cs.simple
local prefix:100 "π" => cs.wordProd
local prefi... | Mathlib/GroupTheory/Coxeter/Inversion.lean | 95 | 100 | theorem length_mul_right_ne (w : W) : ℓ (t * w) ≠ ℓ w := by |
suffices cs.lengthParity (t * w) ≠ cs.lengthParity w by
contrapose! this
simp only [lengthParity_eq_ofAdd_length, this]
rcases ht with ⟨w, i, rfl⟩
simp [lengthParity_simple]
| 0.75 |
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 | 394 | 394 | theorem lt_inv' : a < b⁻¹ ↔ b < a⁻¹ := by | rw [← inv_lt_inv_iff, inv_inv]
| 0.75 |
import Mathlib.Algebra.MvPolynomial.Derivation
import Mathlib.Algebra.MvPolynomial.Variables
#align_import data.mv_polynomial.pderiv from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
universe u v
namespace MvPolynomial
open Set Function Finsupp
variable {R : ... | Mathlib/Algebra/MvPolynomial/PDeriv.lean | 96 | 96 | theorem pderiv_X_self (i : σ) : pderiv i (X i : MvPolynomial σ R) = 1 := by | classical simp
| 0.75 |
import Mathlib.Analysis.Calculus.FDeriv.Bilinear
#align_import analysis.calculus.fderiv.mul from "leanprover-community/mathlib"@"d608fc5d4e69d4cc21885913fb573a88b0deb521"
open scoped Classical
open Filter Asymptotics ContinuousLinearMap Set Metric Topology NNReal ENNReal
noncomputable section
section
variable ... | Mathlib/Analysis/Calculus/FDeriv/Mul.lean | 405 | 409 | theorem HasFDerivAt.mul (hc : HasFDerivAt c c' x) (hd : HasFDerivAt d d' x) :
HasFDerivAt (fun y => c y * d y) (c x • d' + d x • c') x := by |
convert hc.mul' hd
ext z
apply mul_comm
| 0.75 |
import Mathlib.CategoryTheory.NatTrans
import Mathlib.CategoryTheory.Iso
#align_import category_theory.functor.category from "leanprover-community/mathlib"@"63721b2c3eba6c325ecf8ae8cca27155a4f6306f"
namespace CategoryTheory
-- declare the `v`'s first; see note [CategoryTheory universes].
universe v₁ v₂ v₃ u₁ u₂ u... | Mathlib/CategoryTheory/Functor/Category.lean | 121 | 122 | theorem hcomp_id_app {H : D ⥤ E} (α : F ⟶ G) (X : C) : (α ◫ 𝟙 H).app X = H.map (α.app X) := by |
simp
| 0.75 |
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 | 365 | 366 | theorem setAverage_congr (h : s =ᵐ[μ] t) : ⨍ x in s, f x ∂μ = ⨍ x in t, f x ∂μ := by |
simp only [setAverage_eq, setIntegral_congr_set_ae h, measure_congr h]
| 0.75 |
import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
#align_import algebra.order.monoid.min_max from "leanprover-community/mathlib"@"de87d5053a9fe5cbde723172c0fb7e27e7436473"
open Function
variable {α β : Type*}
section CovariantClassMulLe
variable [LinearOrder α]
section Mul
variable [Mul α]
@[to_additive... | Mathlib/Algebra/Order/Monoid/Unbundled/MinMax.lean | 117 | 121 | theorem le_or_le_of_mul_le_mul [CovariantClass α α (· * ·) (· < ·)]
[CovariantClass α α (Function.swap (· * ·)) (· < ·)] {a₁ a₂ b₁ b₂ : α} :
a₁ * b₁ ≤ a₂ * b₂ → a₁ ≤ a₂ ∨ b₁ ≤ b₂ := by |
contrapose!
exact fun h => mul_lt_mul_of_lt_of_lt h.1 h.2
| 0.75 |
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
noncomputable section
universe u
namespace SetTheory
open scoped PGame
namespace PGame... | Mathlib/SetTheory/Game/Nim.lean | 73 | 75 | theorem moveLeft_nim_hEq (o : Ordinal) :
have : IsWellOrder (Quotient.out o).α (· < ·) := inferInstance
HEq (nim o).moveLeft fun i : o.out.α => nim (typein (· < ·) i) := by | rw [nim_def]; rfl
| 0.75 |
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 | 286 | 288 | theorem normAtPlace_apply_isReal {w : InfinitePlace K} (hw : IsReal w) (x : E K):
normAtPlace w x = ‖x.1 ⟨w, hw⟩‖ := by |
rw [normAtPlace, MonoidWithZeroHom.coe_mk, ZeroHom.coe_mk, dif_pos]
| 0.75 |
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 | 336 | 339 | theorem HasDerivAt.finset_prod (hf : ∀ i ∈ u, HasDerivAt (f i) (f' i) x) :
HasDerivAt (∏ i ∈ u, f i ·) (∑ i ∈ u, (∏ j ∈ u.erase i, f j x) • f' i) x := by |
simpa [ContinuousLinearMap.sum_apply, ContinuousLinearMap.smul_apply] using
(HasFDerivAt.finset_prod (fun i hi ↦ (hf i hi).hasFDerivAt)).hasDerivAt
| 0.75 |
import Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing
import Mathlib.AlgebraicGeometry.OpenImmersion
#align_import algebraic_geometry.gluing from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
set_option linter.uppercaseLean3 false
noncomputable section
universe u
open Topologica... | Mathlib/AlgebraicGeometry/Gluing.lean | 296 | 298 | theorem gluedCoverT'_fst_fst (x y z : 𝒰.J) :
𝒰.gluedCoverT' x y z ≫ pullback.fst ≫ pullback.fst = pullback.fst ≫ pullback.snd := by |
delta gluedCoverT'; simp
| 0.75 |
import Mathlib.Data.Set.Basic
#align_import data.bundle from "leanprover-community/mathlib"@"e473c3198bb41f68560cab68a0529c854b618833"
open Function Set
namespace Bundle
variable {B F : Type*} (E : B → Type*)
@[ext]
structure TotalSpace (F : Type*) (E : B → Type*) where
proj : B
snd : E proj
#align bund... | Mathlib/Data/Bundle.lean | 95 | 100 | theorem TotalSpace.range_mk (b : B) : range ((↑) : E b → TotalSpace F E) = π F E ⁻¹' {b} := by |
apply Subset.antisymm
· rintro _ ⟨x, rfl⟩
rfl
· rintro ⟨_, x⟩ rfl
exact ⟨x, rfl⟩
| 0.75 |
import Mathlib.SetTheory.Ordinal.Arithmetic
#align_import set_theory.ordinal.exponential from "leanprover-community/mathlib"@"b67044ba53af18680e1dd246861d9584e968495d"
noncomputable section
open Function Cardinal Set Equiv Order
open scoped Classical
open Cardinal Ordinal
universe u v w
namespace Ordinal
in... | Mathlib/SetTheory/Ordinal/Exponential.lean | 68 | 69 | theorem opow_le_of_limit {a b c : Ordinal} (a0 : a ≠ 0) (h : IsLimit b) :
a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ c := by | rw [opow_limit a0 h, bsup_le_iff]
| 0.75 |
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 | 95 | 97 | theorem trinomial_leadingCoeff (hkm : k < m) (hmn : m < n) (hw : w ≠ 0) :
(trinomial k m n u v w).leadingCoeff = w := by |
rw [leadingCoeff, trinomial_natDegree hkm hmn hw, trinomial_leading_coeff' hkm hmn]
| 0.75 |
import Mathlib.Order.Filter.Cofinite
import Mathlib.Order.Filter.CountableInter
import Mathlib.Order.Filter.CardinalInter
import Mathlib.SetTheory.Cardinal.Ordinal
import Mathlib.SetTheory.Cardinal.Cofinality
import Mathlib.Order.Filter.Bases
open Set Filter Cardinal
universe u
variable {ι : Type u} {α β : Type u}... | Mathlib/Order/Filter/Cocardinal.lean | 70 | 72 | theorem frequently_cocardinal {p : α → Prop} :
(∃ᶠ x in cocardinal α hreg, p x) ↔ c ≤ # { x | p x } := by |
simp only [Filter.Frequently, eventually_cocardinal, not_not,coe_setOf, not_lt]
| 0.75 |
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]
| 0.75 |
import Mathlib.CategoryTheory.Monoidal.Braided.Basic
import Mathlib.Algebra.Category.ModuleCat.Monoidal.Basic
#align_import algebra.category.Module.monoidal.symmetric from "leanprover-community/mathlib"@"74403a3b2551b0970855e14ef5e8fd0d6af1bfc2"
suppress_compilation
universe v w x u
open CategoryTheory MonoidalC... | Mathlib/Algebra/Category/ModuleCat/Monoidal/Symmetric.lean | 43 | 46 | theorem braiding_naturality_left {X Y : ModuleCat R} (f : X ⟶ Y) (Z : ModuleCat R) :
f ▷ Z ≫ (braiding Y Z).hom = (braiding X Z).hom ≫ Z ◁ f := by |
simp_rw [← id_tensorHom]
apply braiding_naturality
| 0.75 |
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
open Filter Metric Set
open scoped ComplexConjugate Real To... | Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean | 87 | 89 | theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by |
ext x
simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range]
| 0.75 |
import Mathlib.Algebra.Associated
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Group.Abs
import Mathlib.Algebra.Ring.Divisibility.Basic
#align_import ring_theory.prime from "leanprover-community/mathlib"@"008205aa645b3f194c1da47025c5f110c8406eab"
section CommRing
variable {α : Ty... | Mathlib/RingTheory/Prime.lean | 70 | 73 | theorem Prime.abs [LinearOrder α] {p : α} (hp : Prime p) : Prime (abs p) := by |
obtain h | h := abs_choice p <;> rw [h]
· exact hp
· exact hp.neg
| 0.75 |
import Mathlib.Data.Nat.Cast.Basic
import Mathlib.Algebra.CharZero.Defs
import Mathlib.Algebra.Order.Group.Abs
import Mathlib.Data.Nat.Cast.NeZero
import Mathlib.Algebra.Order.Ring.Nat
#align_import data.nat.cast.basic from "leanprover-community/mathlib"@"acebd8d49928f6ed8920e502a6c90674e75bd441"
variable {α β : T... | Mathlib/Data/Nat/Cast/Order.lean | 142 | 143 | theorem cast_lt_one : (n : α) < 1 ↔ n = 0 := by |
rw [← cast_one, cast_lt, Nat.lt_succ_iff, ← bot_eq_zero, le_bot_iff]
| 0.75 |
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
| 0.75 |
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Range
#align_import data.list.nat_antidiagonal from "leanprover-community/mathlib"@"7b78d1776212a91ecc94cf601f83bdcc46b04213"
open List Function Nat
namespace List
namespace Nat
def antidiagonal (n : ℕ) : List (ℕ × ℕ) :=
(range (n + 1)).map fun i ↦ (i,... | Mathlib/Data/List/NatAntidiagonal.lean | 52 | 53 | theorem length_antidiagonal (n : ℕ) : (antidiagonal n).length = n + 1 := by |
rw [antidiagonal, length_map, length_range]
| 0.75 |
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
namespace Con... | Mathlib/Analysis/NormedSpace/ContinuousAffineMap.lean | 66 | 67 | theorem coe_contLinear_eq_linear (f : P →ᴬ[R] Q) :
(f.contLinear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear := by | ext; rfl
| 0.75 |
import Mathlib.Data.Finset.Lattice
import Mathlib.Data.Finset.NAry
import Mathlib.Data.Multiset.Functor
#align_import data.finset.functor from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78ad6527b7e4"
universe u
open Function
namespace Finset
protected instance pure : Pure Finset :=
⟨fun... | Mathlib/Data/Finset/Functor.lean | 200 | 202 | theorem id_traverse [DecidableEq α] (s : Finset α) : traverse (pure : α → Id α) s = s := by |
rw [traverse, Multiset.id_traverse]
exact s.val_toFinset
| 0.75 |
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Fintype.BigOperators
#align_import data.sign from "leanprover-community/mathlib"@"2445c98ae4b87eabebdde552593519b9b6dc350c"
-- Porting note (#11081): cannot automatically derive Fintype, adde... | Mathlib/Data/Sign.lean | 162 | 162 | theorem nonneg_iff {a : SignType} : 0 ≤ a ↔ a = 0 ∨ a = 1 := by | cases a <;> decide
| 0.75 |
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 | 140 | 140 | theorem zero'_eval : zero'.eval = fun v => pure (0 :: v) := by | simp [eval]
| 0.75 |
import Mathlib.Analysis.Convex.Normed
import Mathlib.Analysis.NormedSpace.Connected
import Mathlib.LinearAlgebra.AffineSpace.ContinuousAffineEquiv
open Set
variable {F : Type*} [AddCommGroup F] [Module ℝ F] [TopologicalSpace F]
def AmpleSet (s : Set F) : Prop :=
∀ x ∈ s, convexHull ℝ (connectedComponentIn s ... | Mathlib/Analysis/Convex/AmpleSet.lean | 94 | 96 | theorem preimage {s : Set F} (h : AmpleSet s) (L : E ≃ᵃL[ℝ] F) : AmpleSet (L ⁻¹' s) := by |
rw [← L.image_symm_eq_preimage]
exact h.image L.symm
| 0.75 |
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.FieldTheory.Minpoly.Basic
import Mathlib.RingTheory.Adjoin.Basic
import Mathlib.RingTheory.FinitePresentation
import Mathlib.RingTheory.FiniteType
import Mathlib.RingTheory.PowerBasis
import Mathlib.RingTheory.PrincipalI... | Mathlib/RingTheory/AdjoinRoot.lean | 120 | 121 | theorem smul_of [DistribSMul S R] [IsScalarTower S R R] (a : S) (x : R) :
a • of f x = of f (a • x) := by | rw [of, RingHom.comp_apply, RingHom.comp_apply, smul_mk, smul_C]
| 0.75 |
import Mathlib.Algebra.Group.Semiconj.Defs
import Mathlib.Init.Algebra.Classes
#align_import algebra.group.commute from "leanprover-community/mathlib"@"05101c3df9d9cfe9430edc205860c79b6d660102"
assert_not_exists MonoidWithZero
assert_not_exists DenselyOrdered
variable {G M S : Type*}
@[to_additive "Two elements... | Mathlib/Algebra/Group/Commute/Defs.lean | 262 | 263 | theorem mul_inv_cancel_assoc (h : Commute a b) : a * (b * a⁻¹) = b := by |
rw [← mul_assoc, h.mul_inv_cancel]
| 0.75 |
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 | 65 | 67 | theorem exp_inj_of_neg_pi_lt_of_le_pi {x y : ℂ} (hx₁ : -π < x.im) (hx₂ : x.im ≤ π) (hy₁ : -π < y.im)
(hy₂ : y.im ≤ π) (hxy : exp x = exp y) : x = y := by |
rw [← log_exp hx₁ hx₂, ← log_exp hy₁ hy₂, hxy]
| 0.75 |
import Mathlib.Data.Set.Lattice
import Mathlib.Order.Hom.Lattice
#align_import order.hom.complete_lattice from "leanprover-community/mathlib"@"9d684a893c52e1d6692a504a118bfccbae04feeb"
open Function OrderDual Set
variable {F α β γ δ : Type*} {ι : Sort*} {κ : ι → Sort*}
-- Porting note: mathport made this & sInf... | Mathlib/Order/Hom/CompleteLattice.lean | 134 | 135 | theorem map_iSup₂ [SupSet α] [SupSet β] [sSupHomClass F α β] (f : F) (g : ∀ i, κ i → α) :
f (⨆ (i) (j), g i j) = ⨆ (i) (j), f (g i j) := by | simp_rw [map_iSup]
| 0.75 |
import Mathlib.Data.Finsupp.Defs
#align_import data.finsupp.ne_locus from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
variable {α M N P : Type*}
namespace Finsupp
variable [DecidableEq α]
section NHasZero
variable [DecidableEq N] [Zero N] (f g : α →₀ N)
def neLocus (f g : α →₀ ... | Mathlib/Data/Finsupp/NeLocus.lean | 42 | 44 | theorem mem_neLocus {f g : α →₀ N} {a : α} : a ∈ f.neLocus g ↔ f a ≠ g a := by |
simpa only [neLocus, Finset.mem_filter, Finset.mem_union, mem_support_iff,
and_iff_right_iff_imp] using Ne.ne_or_ne _
| 0.75 |
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing
import Mathlib.CategoryTheory.Adjunction.Evaluation
import Mathlib.CategoryTheory.Limits.Types
import Mathlib.CategoryTheory.Limits.Preserves.Filtered
import Mathlib.Cat... | Mathlib/Topology/Sheaves/Stalks.lean | 113 | 114 | theorem germ_res_apply (F : X.Presheaf C) {U V : Opens X} (i : U ⟶ V) (x : U) [ConcreteCategory C]
(s) : germ F x (F.map i.op s) = germ F (i x) s := by | rw [← comp_apply, germ_res]
| 0.75 |
import Mathlib.Algebra.Group.Units.Equiv
import Mathlib.CategoryTheory.Endomorphism
#align_import category_theory.conj from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
universe v u
namespace CategoryTheory
namespace Iso
variable {C : Type u} [Category.{v} C]
def homCongr {X Y X₁... | Mathlib/CategoryTheory/Conj.lean | 64 | 66 | theorem homCongr_trans {X₁ Y₁ X₂ Y₂ X₃ Y₃ : C} (α₁ : X₁ ≅ X₂) (β₁ : Y₁ ≅ Y₂) (α₂ : X₂ ≅ X₃)
(β₂ : Y₂ ≅ Y₃) (f : X₁ ⟶ Y₁) :
(α₁ ≪≫ α₂).homCongr (β₁ ≪≫ β₂) f = (α₁.homCongr β₁).trans (α₂.homCongr β₂) f := by | simp
| 0.75 |
import Mathlib.Analysis.Calculus.FDeriv.Basic
import Mathlib.Analysis.NormedSpace.BoundedLinearMaps
#align_import analysis.calculus.fderiv.linear from "leanprover-community/mathlib"@"e3fb84046afd187b710170887195d50bada934ee"
open Filter Asymptotics ContinuousLinearMap Set Metric
open scoped Classical
open Topolo... | Mathlib/Analysis/Calculus/FDeriv/Linear.lean | 136 | 139 | theorem IsBoundedLinearMap.fderivWithin (h : IsBoundedLinearMap 𝕜 f)
(hxs : UniqueDiffWithinAt 𝕜 s x) : fderivWithin 𝕜 f s x = h.toContinuousLinearMap := by |
rw [DifferentiableAt.fderivWithin h.differentiableAt hxs]
exact h.fderiv
| 0.75 |
import Mathlib.Analysis.NormedSpace.Basic
#align_import analysis.normed_space.enorm from "leanprover-community/mathlib"@"57ac39bd365c2f80589a700f9fbb664d3a1a30c2"
noncomputable section
attribute [local instance] Classical.propDecidable
open ENNReal
structure ENorm (𝕜 : Type*) (V : Type*) [NormedField 𝕜] [Ad... | Mathlib/Analysis/NormedSpace/ENorm.lean | 107 | 110 | theorem map_neg (x : V) : e (-x) = e x :=
calc
e (-x) = ‖(-1 : 𝕜)‖₊ * e x := by | rw [← map_smul, neg_one_smul]
_ = e x := by simp
| 0.75 |
import Mathlib.Computability.DFA
import Mathlib.Data.Fintype.Powerset
#align_import computability.NFA from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
open Set
open Computability
universe u v
-- Porting note: Required as `NFA` is used in mathlib3
set_option linter.uppercaseLean3 fa... | Mathlib/Computability/NFA.lean | 78 | 80 | theorem evalFrom_append_singleton (S : Set σ) (x : List α) (a : α) :
M.evalFrom S (x ++ [a]) = M.stepSet (M.evalFrom S x) a := by |
simp only [evalFrom, List.foldl_append, List.foldl_cons, List.foldl_nil]
| 0.75 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Data.Complex.Exponential
import Mathlib.Data.Complex.Module
import Mathlib.RingTheory.Polynomial.Chebyshev
#align_import analysis.special_functions.trigonometric.chebyshev from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
set_... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Chebyshev.lean | 29 | 30 | theorem aeval_T (x : A) (n : ℤ) : aeval x (T R n) = (T A n).eval x := by |
rw [aeval_def, eval₂_eq_eval_map, map_T]
| 0.75 |
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.GroupTheory.Perm.Basic
#align_import group_theory.perm.support from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Equiv Finset
namespace Equiv.Perm
variable {α : Type*}
section support
s... | Mathlib/GroupTheory/Perm/Support.lean | 264 | 267 | theorem set_support_inv_eq : { x | p⁻¹ x ≠ x } = { x | p x ≠ x } := by |
ext x
simp only [Set.mem_setOf_eq, Ne]
rw [inv_def, symm_apply_eq, eq_comm]
| 0.75 |
import Mathlib.Data.ENNReal.Basic
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.MetricSpace.Thickening
#align_import topology.metric_space.thickened_indicator from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open scoped Classical
open NNReal ENNReal Topol... | Mathlib/Topology/MetricSpace/ThickenedIndicator.lean | 79 | 81 | theorem thickenedIndicatorAux_closure_eq (δ : ℝ) (E : Set α) :
thickenedIndicatorAux δ (closure E) = thickenedIndicatorAux δ E := by |
simp (config := { unfoldPartialApp := true }) only [thickenedIndicatorAux, infEdist_closure]
| 0.75 |
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Data.Set.Function
import Mathlib.Order.Interval.Set.Basic
#align_import data.set.intervals.monoid from "leanprover-community/mathlib"@"aba57d4d3dae35460225919dcd82fe91355162f9"
namespace Set
variable {M : Type*} ... | Mathlib/Algebra/Order/Interval/Set/Monoid.lean | 128 | 129 | theorem image_const_add_Ico : (fun x => a + x) '' Ico b c = Ico (a + b) (a + c) := by |
simp only [add_comm a, image_add_const_Ico]
| 0.75 |
import Mathlib.Algebra.Group.Int
import Mathlib.GroupTheory.GroupAction.Opposite
import Mathlib.Logic.Function.Iterate
#align_import algebra.hom.iterate from "leanprover-community/mathlib"@"792a2a264169d64986541c6f8f7e3bbb6acb6295"
assert_not_exists DenselyOrdered
assert_not_exists Ring
open Function
variable {M... | Mathlib/Algebra/GroupPower/IterateHom.lean | 111 | 111 | theorem mul_right_iterate_apply_one : (· * a)^[n] 1 = a ^ n := by | simp [mul_right_iterate]
| 0.75 |
import Mathlib.Data.List.Basic
#align_import data.bool.all_any from "leanprover-community/mathlib"@"5a3e819569b0f12cbec59d740a2613018e7b8eec"
variable {α : Type*} {p : α → Prop} [DecidablePred p] {l : List α} {a : α}
namespace List
-- Porting note: in Batteries
#align list.all_nil List.all_nil
#align list.all_... | Mathlib/Data/Bool/AllAny.lean | 48 | 48 | theorem any_iff_exists_prop : (any l fun a => p a) ↔ ∃ a ∈ l, p a := by | simp [any_iff_exists]
| 0.75 |
import Mathlib.Topology.ContinuousOn
#align_import topology.algebra.order.left_right from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78ad6527b7e4"
open Set Filter Topology
section TopologicalSpace
variable {α β : Type*} [TopologicalSpace α] [LinearOrder α] [TopologicalSpace β]
| Mathlib/Topology/Order/LeftRight.lean | 111 | 112 | theorem nhds_left_sup_nhds_right (a : α) : 𝓝[≤] a ⊔ 𝓝[≥] a = 𝓝 a := by |
rw [← nhdsWithin_union, Iic_union_Ici, nhdsWithin_univ]
| 0.75 |
import Mathlib.Algebra.Group.Subsemigroup.Basic
#align_import group_theory.subsemigroup.membership from "leanprover-community/mathlib"@"6cb77a8eaff0ddd100e87b1591c6d3ad319514ff"
assert_not_exists MonoidWithZero
variable {ι : Sort*} {M A B : Type*}
section NonAssoc
variable [Mul M]
open Set
namespace Subsemigr... | Mathlib/Algebra/Group/Subsemigroup/Membership.lean | 89 | 91 | theorem mem_sup_right {S T : Subsemigroup M} : ∀ {x : M}, x ∈ T → x ∈ S ⊔ T := by |
have : T ≤ S ⊔ T := le_sup_right
tauto
| 0.75 |
import Mathlib.Algebra.Group.Prod
import Mathlib.Order.Cover
#align_import algebra.support from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
assert_not_exists MonoidWithZero
open Set
namespace Function
variable {α β A B M N P G : Type*}
section One
variable [One M] [One N] [One P]
... | Mathlib/Algebra/Group/Support.lean | 98 | 100 | theorem mulSupport_update_eq_ite [DecidableEq α] [DecidableEq M] (f : α → M) (x : α) (y : M) :
mulSupport (update f x y) = if y = 1 then mulSupport f \ {x} else insert x (mulSupport f) := by |
rcases eq_or_ne y 1 with rfl | hy <;> simp [mulSupport_update_one, mulSupport_update_of_ne_one, *]
| 0.75 |
import Mathlib.Algebra.Group.Defs
import Mathlib.Algebra.GroupWithZero.Defs
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Tactic.Spread
import Mathlib.Util.AssertExists
#align_import algebra.ring.defs from "leanprover-community/mathlib"@"76de8ae01554c3b37d66544866659ff174e66e1f"
universe u v w x
variable {α : ... | Mathlib/Algebra/Ring/Defs.lean | 164 | 165 | theorem one_add_mul [RightDistribClass α] (a b : α) : (1 + a) * b = b + a * b := by |
rw [add_mul, one_mul]
| 0.75 |
import Mathlib.Algebra.MvPolynomial.Variables
#align_import data.mv_polynomial.supported from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
universe u v w
namespace MvPolynomial
variable {σ τ : Type*} {R : Type u} {S : Type v} {r : R} {e : ℕ} {n m : σ}
section CommSemiring
variable... | Mathlib/Algebra/MvPolynomial/Supported.lean | 91 | 92 | theorem mem_supported_vars (p : MvPolynomial σ R) : p ∈ supported R (↑p.vars : Set σ) := by |
rw [mem_supported]
| 0.75 |
import Mathlib.Data.Complex.Module
import Mathlib.Data.Complex.Order
import Mathlib.Data.Complex.Exponential
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Topology.Algebra.InfiniteSum.Module
import Mathlib.Topology.Instances.RealVectorSpace
#align_import analysis.complex.basic from "leanprover-community/mathlib... | Mathlib/Analysis/Complex/Basic.lean | 133 | 134 | theorem edist_of_im_eq {z w : ℂ} (h : z.im = w.im) : edist z w = edist z.re w.re := by |
rw [edist_nndist, edist_nndist, nndist_of_im_eq h]
| 0.75 |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Combinatorics.SimpleGraph.Dart
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Data.ZMod.Parity
#align_import combinatorics.simple_graph.degree_sum from "leanprover-community/mathlib"@"90659cbe25e59ec302e2fb92b00e9732160cc620"
open Finset
nam... | Mathlib/Combinatorics/SimpleGraph/DegreeSum.lean | 56 | 64 | theorem dart_fst_fiber [DecidableEq V] (v : V) :
(univ.filter fun d : G.Dart => d.fst = v) = univ.image (G.dartOfNeighborSet v) := by |
ext d
simp only [mem_image, true_and_iff, mem_filter, SetCoe.exists, mem_univ, exists_prop_of_true]
constructor
· rintro rfl
exact ⟨_, d.adj, by ext <;> rfl⟩
· rintro ⟨e, he, rfl⟩
rfl
| 0.75 |
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.NormedSpace.FiniteDimension
#align_import analysis.calculus.cont_diff from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputable section
universe uD uE uF uG
variable {𝕜 : Type*} [NontriviallyNormedField ... | Mathlib/Analysis/Calculus/ContDiff/FiniteDimension.lean | 71 | 75 | theorem contDiffOn_succ_iff_fderiv_apply [FiniteDimensional 𝕜 E] {n : ℕ} {f : E → F} {s : Set E}
(hs : UniqueDiffOn 𝕜 s) :
ContDiffOn 𝕜 (n + 1 : ℕ) f s ↔
DifferentiableOn 𝕜 f s ∧ ∀ y, ContDiffOn 𝕜 n (fun x => fderivWithin 𝕜 f s x y) s := by |
rw [contDiffOn_succ_iff_fderivWithin hs, contDiffOn_clm_apply]
| 0.75 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.HasseDeriv
#align_import data.polynomial.taylor from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
noncomputable section
namespace Polynomial
open Polynomial... | Mathlib/Algebra/Polynomial/Taylor.lean | 106 | 107 | theorem taylor_mul {R} [CommSemiring R] (r : R) (p q : R[X]) :
taylor r (p * q) = taylor r p * taylor r q := by | simp only [taylor_apply, mul_comp]
| 0.75 |
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.Calculus.FDeriv.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
open Topology InnerProductSpace Set
noncomputable section
variable {𝕜 F : Type*} [RCLike 𝕜]
variable [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [CompleteSpace F]
variabl... | Mathlib/Analysis/Calculus/Gradient/Basic.lean | 118 | 121 | theorem DifferentiableAt.hasGradientAt (h : DifferentiableAt 𝕜 f x) :
HasGradientAt f (∇ f x) x := by |
rw [hasGradientAt_iff_hasFDerivAt, gradient, (toDual 𝕜 F).apply_symm_apply (fderiv 𝕜 f x)]
exact h.hasFDerivAt
| 0.75 |
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
#align_import analysis.special_functions.pow.asymptotics from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8"
set_option linter.uppercaseLean3 false
noncomputable section
open scoped Classical
open Real Topology NNReal ENNReal Filter C... | Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean | 126 | 128 | theorem tendsto_rpow_neg_div : Tendsto (fun x => x ^ (-(1 : ℝ) / x)) atTop (𝓝 1) := by |
convert tendsto_rpow_div_mul_add (-(1 : ℝ)) _ (0 : ℝ) zero_ne_one
ring
| 0.71875 |
import Mathlib.Order.BooleanAlgebra
import Mathlib.Logic.Equiv.Basic
#align_import order.symm_diff from "leanprover-community/mathlib"@"6eb334bd8f3433d5b08ba156b8ec3e6af47e1904"
open Function OrderDual
variable {ι α β : Type*} {π : ι → Type*}
def symmDiff [Sup α] [SDiff α] (a b : α) : α :=
a \ b ⊔ b \ a
#ali... | Mathlib/Order/SymmDiff.lean | 125 | 125 | theorem symmDiff_bot : a ∆ ⊥ = a := by | rw [symmDiff, sdiff_bot, bot_sdiff, sup_bot_eq]
| 0.71875 |
import Mathlib.Analysis.Convex.Gauge
import Mathlib.Analysis.Convex.Normed
open Metric Bornology Filter Set
open scoped NNReal Topology Pointwise
noncomputable section
section Module
variable {E : Type*} [AddCommGroup E] [Module ℝ E]
def gaugeRescale (s t : Set E) (x : E) : E := (gauge s x / gauge t x) • x
the... | Mathlib/Analysis/Convex/GaugeRescale.lean | 48 | 52 | theorem gaugeRescale_self_apply {s : Set E} (hsa : Absorbent ℝ s) (hsb : IsVonNBounded ℝ s)
(x : E) : gaugeRescale s s x = x := by |
rcases eq_or_ne x 0 with rfl | hx; · simp
rw [gaugeRescale, div_self, one_smul]
exact ((gauge_pos hsa hsb).2 hx).ne'
| 0.71875 |
import Mathlib.Data.ENNReal.Inv
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
section iInf
variable {ι : Sort*} {f g : ι → ℝ≥0∞}
variable {a b c d : ℝ≥0∞} {r p q : ℝ≥0}
theorem toNNReal_iInf (hf : ∀ i, f ... | Mathlib/Data/ENNReal/Real.lean | 556 | 561 | theorem toNNReal_iSup (hf : ∀ i, f i ≠ ∞) : (iSup f).toNNReal = ⨆ i, (f i).toNNReal := by |
lift f to ι → ℝ≥0 using hf
simp_rw [toNNReal_coe]
by_cases h : BddAbove (range f)
· rw [← coe_iSup h, toNNReal_coe]
· rw [NNReal.iSup_of_not_bddAbove h, iSup_coe_eq_top.2 h, top_toNNReal]
| 0.71875 |
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Data.Nat.Totient
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.GroupTheory.Subgroup.Simple
import Mathlib.Tactic.Group
import Mathlib.GroupTheory.Exponent
#align_import group_theory.specific_groups.cyclic from "leanprover-community/mathli... | Mathlib/GroupTheory/SpecificGroups/Cyclic.lean | 152 | 154 | theorem mem_zpowers_of_prime_card {G : Type*} [Group G] {_ : Fintype G} {p : ℕ} [hp : Fact p.Prime]
(h : Fintype.card G = p) {g g' : G} (hg : g ≠ 1) : g' ∈ zpowers g := by |
simp_rw [zpowers_eq_top_of_prime_card h hg, Subgroup.mem_top]
| 0.71875 |
import Mathlib.Algebra.Divisibility.Basic
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Ring.Defs
#align_import algebra.euclidean_domain.defs from "leanprover-community/mathlib"@"ee7b9f9a9ac2a8d9f04ea39bbfe6b1a3be053b38"
universe u
class EuclideanDomain (R : Type u) extends CommRing R, Nontrivial R ... | Mathlib/Algebra/EuclideanDomain/Defs.lean | 136 | 138 | theorem div_add_mod' (m k : R) : m / k * k + m % k = m := by |
rw [mul_comm]
exact div_add_mod _ _
| 0.71875 |
import Mathlib.CategoryTheory.Preadditive.Injective
import Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex
import Mathlib.Algebra.Homology.QuasiIso
#align_import category_theory.preadditive.injective_resolution from "leanprover-community/mathlib"@"14b69e9f3c16630440a2cbd46f1ddad0d561dee7"
noncomputable s... | Mathlib/CategoryTheory/Preadditive/InjectiveResolution.lean | 111 | 113 | theorem complex_d_comp (n : ℕ) :
I.cocomplex.d n (n + 1) ≫ I.cocomplex.d (n + 1) (n + 2) = 0 := by |
simp
| 0.71875 |
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Ring.Invertible
import Mathlib.Data.Nat.Cast.Order
#align_import algebra.order.invertible from "leanprover-community/mathlib"@"ee0c179cd3c8a45aa5bffbf1b41d8dbede452865"
variable {α : Type*} [LinearOrderedSemiring α] {a : α}
@[simp]
| Mathlib/Algebra/Order/Invertible.lean | 19 | 21 | theorem invOf_pos [Invertible a] : 0 < ⅟ a ↔ 0 < a :=
haveI : 0 < a * ⅟ a := by | simp only [mul_invOf_self, zero_lt_one]
⟨fun h => pos_of_mul_pos_left this h.le, fun h => pos_of_mul_pos_right this h.le⟩
| 0.71875 |
import Mathlib.RingTheory.Localization.Basic
#align_import ring_theory.localization.integer from "leanprover-community/mathlib"@"9556784a5b84697562e9c6acb40500d4a82e675a"
variable {R : Type*} [CommSemiring R] {M : Submonoid R} {S : Type*} [CommSemiring S]
variable [Algebra R S] {P : Type*} [CommSemiring P]
open ... | Mathlib/RingTheory/Localization/Integer.lean | 107 | 111 | theorem exist_integer_multiples_of_finite {ι : Type*} [Finite ι] (f : ι → S) :
∃ b : M, ∀ i, IsLocalization.IsInteger R ((b : R) • f i) := by |
cases nonempty_fintype ι
obtain ⟨b, hb⟩ := exist_integer_multiples M Finset.univ f
exact ⟨b, fun i => hb i (Finset.mem_univ _)⟩
| 0.71875 |
import Mathlib.Algebra.Homology.ShortComplex.ModuleCat
import Mathlib.RepresentationTheory.GroupCohomology.Basic
import Mathlib.RepresentationTheory.Invariants
universe v u
noncomputable section
open CategoryTheory Limits Representation
variable {k G : Type u} [CommRing k] [Group G] (A : Rep k G)
namespace grou... | Mathlib/RepresentationTheory/GroupCohomology/LowDegree.lean | 524 | 526 | theorem map_one_of_isMulOneCocycle {f : G → M} (hf : IsMulOneCocycle f) :
f 1 = 1 := by |
simpa only [mul_one, one_smul, self_eq_mul_right] using hf 1 1
| 0.71875 |
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 | 93 | 93 | theorem to_real_im (x : ℤ[i]) : ((x.im : ℤ) : ℝ) = (x : ℂ).im := by | simp [toComplex_def]
| 0.71875 |
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]
| Mathlib/Data/List/Lattice.lean | 195 | 195 | theorem nil_bagInter (l : List α) : [].bagInter l = [] := by | cases l <;> rfl
| 0.71875 |
import Mathlib.Combinatorics.Quiver.Basic
import Mathlib.Combinatorics.Quiver.Path
#align_import combinatorics.quiver.cast from "leanprover-community/mathlib"@"fc2ed6f838ce7c9b7c7171e58d78eaf7b438fb0e"
universe v v₁ v₂ u u₁ u₂
variable {U : Type*} [Quiver.{u + 1} U]
namespace Quiver
def Hom.cast {u v u' v... | Mathlib/Combinatorics/Quiver/Cast.lean | 118 | 121 | theorem Path.cast_eq_iff_heq {u v u' v' : U} (hu : u = u') (hv : v = v') (p : Path u v)
(p' : Path u' v') : p.cast hu hv = p' ↔ HEq p p' := by |
rw [Path.cast_eq_cast]
exact _root_.cast_eq_iff_heq
| 0.71875 |
import Mathlib.Mathport.Rename
import Mathlib.Tactic.Lemma
import Mathlib.Tactic.TypeStar
#align_import data.option.defs from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
namespace Option
#align option.lift_or_get Option.liftOrGet
protected def traverse.{u, v}
{F : Type u → Type... | Mathlib/Data/Option/Defs.lean | 96 | 97 | theorem mem_toList {a : α} {o : Option α} : a ∈ toList o ↔ a ∈ o := by |
cases o <;> simp [toList, eq_comm]
| 0.71875 |
import Mathlib.Data.Finset.Fold
import Mathlib.Algebra.GCDMonoid.Multiset
#align_import algebra.gcd_monoid.finset from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
#align_import algebra.gcd_monoid.div from "leanprover-community/mathlib"@"b537794f8409bc9598febb79cd510b1df5f4539d"
variab... | Mathlib/Algebra/GCDMonoid/Finset.lean | 100 | 103 | theorem lcm_congr {f g : β → α} (hs : s₁ = s₂) (hfg : ∀ a ∈ s₂, f a = g a) :
s₁.lcm f = s₂.lcm g := by |
subst hs
exact Finset.fold_congr hfg
| 0.71875 |
import Mathlib.Topology.Sets.Opens
#align_import topology.local_at_target from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open TopologicalSpace Set Filter
open Topology Filter
variable {α β : Type*} [TopologicalSpace α] [TopologicalSpace β] {f : α → β}
variable {s : Set β} {ι : Ty... | Mathlib/Topology/LocalAtTarget.lean | 78 | 84 | theorem IsOpenMap.restrictPreimage (H : IsOpenMap f) (s : Set β) :
IsOpenMap (s.restrictPreimage f) := by |
intro t
suffices ∀ u, IsOpen u → Subtype.val ⁻¹' u = t →
∃ v, IsOpen v ∧ Subtype.val ⁻¹' v = s.restrictPreimage f '' t by
simpa [isOpen_induced_iff]
exact fun u hu e => ⟨f '' u, H u hu, by simp [← e, image_restrictPreimage]⟩
| 0.71875 |
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 | 69 | 72 | theorem sym2_mono (h : s ⊆ t) : s.sym2 ⊆ t.sym2 := by |
rw [← val_le_iff, sym2_val, sym2_val]
apply Multiset.sym2_mono
rwa [val_le_iff]
| 0.71875 |
import Mathlib.Algebra.MvPolynomial.Counit
import Mathlib.Algebra.MvPolynomial.Invertible
import Mathlib.RingTheory.WittVector.Defs
#align_import ring_theory.witt_vector.basic from "leanprover-community/mathlib"@"9556784a5b84697562e9c6acb40500d4a82e675a"
noncomputable section
open MvPolynomial Function
variable... | Mathlib/RingTheory/WittVector/Basic.lean | 183 | 185 | theorem matrix_vecEmpty_coeff {R} (i j) :
@coeff p R (Matrix.vecEmpty i) j = (Matrix.vecEmpty i : ℕ → R) j := by |
rcases i with ⟨_ | _ | _ | _ | i_val, ⟨⟩⟩
| 0.71875 |
import Mathlib.Algebra.Group.Defs
#align_import algebra.invertible from "leanprover-community/mathlib"@"722b3b152ddd5e0cf21c0a29787c76596cb6b422"
assert_not_exists MonoidWithZero
assert_not_exists DenselyOrdered
universe u
variable {α : Type u}
class Invertible [Mul α] [One α] (a : α) : Type u where
invOf... | Mathlib/Algebra/Group/Invertible/Defs.lean | 128 | 129 | theorem mul_invOf_self_assoc [Monoid α] (a b : α) [Invertible a] : a * (⅟ a * b) = b := by |
rw [← mul_assoc, mul_invOf_self, one_mul]
| 0.71875 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.