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.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 | 60 | 61 | theorem card_zmod : Fintype.card (TruncatedWittVector p n (ZMod p)) = p ^ n := by |
rw [card, ZMod.card]
| 1 | 2.718282 | 0 | 0.75 | 4 | 656 |
import Mathlib.Data.Finset.Lattice
import Mathlib.Data.Set.Sigma
#align_import data.finset.sigma from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function Multiset
variable {ι : Type*}
namespace Finset
section SigmaLift
variable {α β γ : ι → Type*} [DecidableEq ι]
def sigm... | Mathlib/Data/Finset/Sigma.lean | 184 | 187 | theorem not_mem_sigmaLift_of_ne_left (f : ∀ ⦃i⦄, α i → β i → Finset (γ i)) (a : Sigma α)
(b : Sigma β) (x : Sigma γ) (h : a.1 ≠ x.1) : x ∉ sigmaLift f a b := by |
rw [mem_sigmaLift]
exact fun H => h H.fst
| 2 | 7.389056 | 1 | 1.214286 | 14 | 1,292 |
import Mathlib.CategoryTheory.Limits.HasLimits
import Mathlib.CategoryTheory.Limits.Shapes.Equalizers
#align_import category_theory.limits.shapes.wide_equalizers from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
noncomputable section
namespace CategoryTheory.Limits
open CategoryTheo... | Mathlib/CategoryTheory/Limits/Shapes/WideEqualizers.lean | 218 | 219 | theorem Trident.app_zero (s : Trident f) (j : J) : s.π.app zero ≫ f j = s.π.app one := by |
rw [← s.w (line j), parallelFamily_map_left]
| 1 | 2.718282 | 0 | 0 | 2 | 10 |
import Mathlib.SetTheory.Cardinal.Finite
#align_import data.set.ncard from "leanprover-community/mathlib"@"74c2af38a828107941029b03839882c5c6f87a04"
namespace Set
variable {α β : Type*} {s t : Set α}
noncomputable def encard (s : Set α) : ℕ∞ := PartENat.withTopEquiv (PartENat.card s)
@[simp] theorem encard_uni... | Mathlib/Data/Set/Card.lean | 286 | 288 | theorem encard_pair {x y : α} (hne : x ≠ y) : ({x, y} : Set α).encard = 2 := by |
rw [encard_insert_of_not_mem (by simpa), ← one_add_one_eq_two,
WithTop.add_right_cancel_iff WithTop.one_ne_top, encard_singleton]
| 2 | 7.389056 | 1 | 0.5 | 14 | 454 |
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Data.Real.Sqrt
import Mathlib.RingTheory.Ideal.QuotientOperations
import Mathlib.Topology.MetricSpace.HausdorffDistance
#align_import analysis.normed.group.quotient from "leanprover-community/mathlib"@"2196ab363eb097c008... | Mathlib/Analysis/Normed/Group/Quotient.lean | 113 | 115 | theorem QuotientAddGroup.norm_eq_infDist {S : AddSubgroup M} (x : M ⧸ S) :
‖x‖ = infDist 0 { m : M | (m : M ⧸ S) = x } := by |
simp only [AddSubgroup.quotient_norm_eq, infDist_eq_iInf, sInf_image', dist_zero_left]
| 1 | 2.718282 | 0 | 1 | 8 | 984 |
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.special_functions.log.base from "leanprover-community/mathlib"@"f23a09ce6d3f367220dc3cecad6b7eb69eb01690"
open Set Filter Function
open Topology
noncomputable section
namespace Real
variable {b x y : ℝ}
-- @... | Mathlib/Analysis/SpecialFunctions/Log/Base.lean | 87 | 89 | theorem inv_logb_mul_base {a b : ℝ} (h₁ : a ≠ 0) (h₂ : b ≠ 0) (c : ℝ) :
(logb (a * b) c)⁻¹ = (logb a c)⁻¹ + (logb b c)⁻¹ := by |
simp_rw [inv_logb]; exact logb_mul h₁ h₂
| 1 | 2.718282 | 0 | 0.25 | 20 | 300 |
import Mathlib.Data.Vector.Basic
import Mathlib.Data.Vector.Snoc
set_option autoImplicit true
namespace Vector
section Fold
section Binary
variable (xs : Vector α n) (ys : Vector β n)
@[simp]
theorem mapAccumr₂_mapAccumr_left (f₁ : γ → β → σ₁ → σ₁ × ζ) (f₂ : α → σ₂ → σ₂ × γ) :
(mapAccumr₂ f₁ (mapAccumr f₂... | Mathlib/Data/Vector/MapLemmas.lean | 71 | 73 | theorem map₂_map_left (f₁ : γ → β → ζ) (f₂ : α → γ) :
map₂ f₁ (map f₂ xs) ys = map₂ (fun x y => f₁ (f₂ x) y) xs ys := by |
induction xs, ys using Vector.revInductionOn₂ <;> simp_all
| 1 | 2.718282 | 0 | 0.333333 | 24 | 337 |
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]
| 1 | 2.718282 | 0 | 0.1 | 10 | 245 |
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.List.Perm
import Mathlib.Data.List.Range
#align_import data.list.sublists from "leanprover-community/mathlib"@"ccad6d5093bd2f5c6ca621fc74674cce51355af6"
universe u v w
variable {α : Type u} {β : Type v} {γ : Type w}
open Nat
namespace List
@[simp]
theo... | Mathlib/Data/List/Sublists.lean | 120 | 129 | theorem sublistsAux_eq_array_foldl :
sublistsAux = fun (a : α) (r : List (List α)) =>
(r.toArray.foldl (init := #[])
fun r l => (r.push l).push (a :: l)).toList := by |
funext a r
simp only [sublistsAux, Array.foldl_eq_foldl_data, Array.mkEmpty]
have := foldl_hom Array.toList (fun r l => (r.push l).push (a :: l))
(fun (r : List (List α)) l => r ++ [l, a :: l]) r #[]
(by simp)
simpa using this
| 6 | 403.428793 | 2 | 1.666667 | 6 | 1,773 |
import Batteries.Data.List.Basic
import Batteries.Data.List.Lemmas
open Nat
namespace List
section countP
variable (p q : α → Bool)
@[simp] theorem countP_nil : countP p [] = 0 := rfl
protected theorem countP_go_eq_add (l) : countP.go p l n = n + countP.go p l 0 := by
induction l generalizing n with
| nil... | .lake/packages/batteries/Batteries/Data/List/Count.lean | 81 | 82 | theorem countP_eq_length : countP p l = l.length ↔ ∀ a ∈ l, p a := by |
rw [countP_eq_length_filter, filter_length_eq_length]
| 1 | 2.718282 | 0 | 0.8 | 10 | 703 |
import Mathlib.AlgebraicTopology.SimplexCategory
import Mathlib.CategoryTheory.Comma.Arrow
import Mathlib.CategoryTheory.Limits.FunctorCategory
import Mathlib.CategoryTheory.Opposites
#align_import algebraic_topology.simplicial_object from "leanprover-community/mathlib"@"5ed51dc37c6b891b79314ee11a50adc2b1df6fd6"
o... | Mathlib/AlgebraicTopology/SimplicialObject.lean | 131 | 134 | theorem δ_comp_δ_self {n} {i : Fin (n + 2)} :
X.δ (Fin.castSucc i) ≫ X.δ i = X.δ i.succ ≫ X.δ i := by |
dsimp [δ]
simp only [← X.map_comp, ← op_comp, SimplexCategory.δ_comp_δ_self]
| 2 | 7.389056 | 1 | 1 | 8 | 1,050 |
import Mathlib.Algebra.Homology.Homotopy
import Mathlib.Algebra.Homology.Linear
import Mathlib.CategoryTheory.MorphismProperty.IsInvertedBy
import Mathlib.CategoryTheory.Quotient.Linear
import Mathlib.CategoryTheory.Quotient.Preadditive
#align_import algebra.homology.homotopy_category from "leanprover-community/mathl... | Mathlib/Algebra/Homology/HomotopyCategory.lean | 138 | 139 | theorem quotient_map_out_comp_out {C D E : HomotopyCategory V c} (f : C ⟶ D) (g : D ⟶ E) :
(quotient V c).map (Quot.out f ≫ Quot.out g) = f ≫ g := by | simp
| 1 | 2.718282 | 0 | 0 | 1 | 84 |
import Mathlib.Data.Fin.VecNotation
import Mathlib.SetTheory.Cardinal.Basic
#align_import model_theory.basic from "leanprover-community/mathlib"@"369525b73f229ccd76a6ec0e0e0bf2be57599768"
set_option autoImplicit true
universe u v u' v' w w'
open Cardinal
open Cardinal
namespace FirstOrder
-- intended to b... | Mathlib/ModelTheory/Basic.lean | 95 | 100 | theorem lift_mk {i : ℕ} :
Cardinal.lift.{v,u} #(Sequence₂ a₀ a₁ a₂ i)
= #(Sequence₂ (ULift.{v,u} a₀) (ULift.{v,u} a₁) (ULift.{v,u} a₂) i) := by |
rcases i with (_ | _ | _ | i) <;>
simp only [Sequence₂, mk_uLift, Nat.succ_ne_zero, IsEmpty.forall_iff, Nat.succ.injEq,
add_eq_zero, OfNat.ofNat_ne_zero, and_false, one_ne_zero, mk_eq_zero, lift_zero]
| 3 | 20.085537 | 1 | 0.666667 | 3 | 555 |
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Set.Card
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.Trace
open BigOperators Matrix Equiv
variable {n R : Type*} [DecidableEq n] [Fintype n] (σ : Perm n)
variable (R) in
abbrev Equiv.Perm.permMatrix [Zero R] [One... | Mathlib/LinearAlgebra/Matrix/Permutation.lean | 41 | 43 | theorem det_permutation [CommRing R] : det (σ.permMatrix R) = Perm.sign σ := by |
rw [← Matrix.mul_one (σ.permMatrix R), PEquiv.toPEquiv_mul_matrix,
det_permute, det_one, mul_one]
| 2 | 7.389056 | 1 | 1 | 2 | 1,007 |
import Mathlib.RingTheory.Nilpotent.Basic
import Mathlib.RingTheory.UniqueFactorizationDomain
#align_import algebra.squarefree from "leanprover-community/mathlib"@"00d163e35035c3577c1c79fa53b68de17781ffc1"
variable {R : Type*}
def Squarefree [Monoid R] (r : R) : Prop :=
∀ x : R, x * x ∣ r → IsUnit x
#align sq... | Mathlib/Algebra/Squarefree/Basic.lean | 154 | 163 | theorem irreducible_sq_not_dvd_iff_eq_zero_and_no_irreducibles_or_squarefree (r : R) :
(∀ x : R, Irreducible x → ¬x * x ∣ r) ↔ (r = 0 ∧ ∀ x : R, ¬Irreducible x) ∨ Squarefree r := by |
refine ⟨fun h ↦ ?_, ?_⟩
· rcases eq_or_ne r 0 with (rfl | hr)
· exact .inl (by simpa using h)
· exact .inr ((squarefree_iff_no_irreducibles hr).mpr h)
· rintro (⟨rfl, h⟩ | h)
· simpa using h
intro x hx t
exact hx.not_unit (h x t)
| 8 | 2,980.957987 | 2 | 1.714286 | 7 | 1,837 |
import Mathlib.Analysis.Convex.Between
import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.Topology.MetricSpace.Holder
import Mathlib.Topology.MetricSpace.MetricSeparated
#align_import measure_theory.measure.hausdorff from "leanprover-communit... | Mathlib/MeasureTheory/Measure/Hausdorff.lean | 309 | 315 | theorem trim_pre [MeasurableSpace X] [OpensMeasurableSpace X] (m : Set X → ℝ≥0∞)
(hcl : ∀ s, m (closure s) = m s) (r : ℝ≥0∞) : (pre m r).trim = pre m r := by |
refine le_antisymm (le_pre.2 fun s hs => ?_) (le_trim _)
rw [trim_eq_iInf]
refine iInf_le_of_le (closure s) <| iInf_le_of_le subset_closure <|
iInf_le_of_le measurableSet_closure ((pre_le ?_).trans_eq (hcl _))
rwa [diam_closure]
| 5 | 148.413159 | 2 | 1.428571 | 7 | 1,520 |
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 | 125 | 126 | theorem pderiv_C_mul {f : MvPolynomial σ R} {i : σ} : pderiv i (C a * f) = C a * pderiv i f := by |
rw [C_mul', Derivation.map_smul, C_mul']
| 1 | 2.718282 | 0 | 0.222222 | 9 | 283 |
import Mathlib.Geometry.Manifold.ContMDiff.Atlas
import Mathlib.Geometry.Manifold.VectorBundle.FiberwiseLinear
import Mathlib.Topology.VectorBundle.Constructions
#align_import geometry.manifold.vector_bundle.basic from "leanprover-community/mathlib"@"e473c3198bb41f68560cab68a0529c854b618833"
assert_not_exists mfde... | Mathlib/Geometry/Manifold/VectorBundle/Basic.lean | 108 | 114 | theorem FiberBundle.chartedSpace_chartAt (x : TotalSpace F E) :
chartAt (ModelProd HB F) x =
(trivializationAt F E x.proj).toPartialHomeomorph ≫ₕ
(chartAt HB x.proj).prod (PartialHomeomorph.refl F) := by |
dsimp only [chartAt_comp, prodChartedSpace_chartAt, FiberBundle.chartedSpace'_chartAt,
chartAt_self_eq]
rw [Trivialization.coe_coe, Trivialization.coe_fst' _ (mem_baseSet_trivializationAt F E x.proj)]
| 3 | 20.085537 | 1 | 1.333333 | 3 | 1,437 |
import Mathlib.Algebra.Field.Subfield
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Topology.Algebra.GroupWithZero
import Mathlib.Topology.Algebra.Ring.Basic
import Mathlib.Topology.Order.LocalExtr
#align_import topology.algebra.field from "leanprover-community/mathlib"@"c10e724be91096453ee3db13862... | Mathlib/Topology/Algebra/Field.lean | 112 | 114 | theorem IsLocalMin.inv {f : α → β} {a : α} (h1 : IsLocalMin f a) (h2 : ∀ᶠ z in 𝓝 a, 0 < f z) :
IsLocalMax f⁻¹ a := by |
filter_upwards [h1, h2] with z h3 h4 using(inv_le_inv h4 h2.self_of_nhds).mpr h3
| 1 | 2.718282 | 0 | 1.333333 | 3 | 1,409 |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.LinearAlgebra.Vandermonde
import Mathlib.RingTheory.Polynomial.Basic
#align_import linear_algebra.lagrange from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open Polynomial
section PolynomialDetermination
namespace Poly... | Mathlib/LinearAlgebra/Lagrange.lean | 63 | 67 | theorem eq_of_degrees_lt_of_eval_finset_eq (degree_f_lt : f.degree < s.card)
(degree_g_lt : g.degree < s.card) (eval_fg : ∀ x ∈ s, f.eval x = g.eval x) : f = g := by |
rw [← mem_degreeLT] at degree_f_lt degree_g_lt
refine eq_of_degree_sub_lt_of_eval_finset_eq _ ?_ eval_fg
rw [← mem_degreeLT]; exact Submodule.sub_mem _ degree_f_lt degree_g_lt
| 3 | 20.085537 | 1 | 1.75 | 4 | 1,852 |
import Mathlib.Algebra.Module.DedekindDomain
import Mathlib.LinearAlgebra.FreeModule.PID
import Mathlib.Algebra.Module.Projective
import Mathlib.Algebra.Category.ModuleCat.Biproducts
import Mathlib.RingTheory.SimpleModule
#align_import algebra.module.pid from "leanprover-community/mathlib"@"cdc34484a07418af43daf8198b... | Mathlib/Algebra/Module/PID.lean | 89 | 98 | theorem Submodule.exists_isInternal_prime_power_torsion_of_pid [Module.Finite R M]
(hM : Module.IsTorsion R M) :
∃ (ι : Type u) (_ : Fintype ι) (_ : DecidableEq ι) (p : ι → R) (_ : ∀ i, Irreducible <| p i)
(e : ι → ℕ), DirectSum.IsInternal fun i => torsionBy R M <| p i ^ e i := by |
refine ⟨_, ?_, _, _, ?_, _, Submodule.isInternal_prime_power_torsion_of_pid hM⟩
· exact Finset.fintypeCoeSort _
· rintro ⟨p, hp⟩
have hP := prime_of_factor p (Multiset.mem_toFinset.mp hp)
haveI := Ideal.isPrime_of_prime hP
exact (IsPrincipal.prime_generator_of_isPrime p hP.ne_zero).irreducible
| 6 | 403.428793 | 2 | 2 | 5 | 2,030 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Data.Matrix.Basis
import Mathlib.Data.Matrix.DMatrix
import Mathlib.RingTheory.MatrixAlgebra
#align_import ring_theory.polynomial_algebra from "leanprover-community/mathlib"@"565eb991e264d0db702722b4bde52ee5173c9950"
universe u v w
open Polynomial Tensor... | Mathlib/RingTheory/PolynomialAlgebra.lean | 56 | 61 | theorem toFunBilinear_apply_eq_sum (a : A) (p : R[X]) :
toFunBilinear R A a p = p.sum fun n r => monomial n (a * algebraMap R A r) := by |
simp only [toFunBilinear_apply_apply, aeval_def, eval₂_eq_sum, Polynomial.sum, Finset.smul_sum]
congr with i : 1
rw [← Algebra.smul_def, ← C_mul', mul_smul_comm, C_mul_X_pow_eq_monomial, ← Algebra.commutes,
← Algebra.smul_def, smul_monomial]
| 4 | 54.59815 | 2 | 0.833333 | 6 | 728 |
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import analysis.inner_product_space.adjoint from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open RCLike
open scoped ComplexConjugate
variable {𝕜 E F G : Type... | Mathlib/Analysis/InnerProductSpace/Adjoint.lean | 155 | 158 | theorem apply_norm_sq_eq_inner_adjoint_right (A : E →L[𝕜] F) (x : E) :
‖A x‖ ^ 2 = re ⟪x, (A† ∘L A) x⟫ := by |
have h : ⟪x, (A† ∘L A) x⟫ = ⟪A x, A x⟫ := by rw [← adjoint_inner_right]; rfl
rw [h, ← inner_self_eq_norm_sq (𝕜 := 𝕜) _]
| 2 | 7.389056 | 1 | 0.875 | 8 | 763 |
import Mathlib.Algebra.Group.Even
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Algebra.Order.Sub.Defs
#align_import algebra.order.sub.canonical from "leanprover-community/mathlib"@"62a5626868683c104774de8d85b9855234ac807c"
variable {α : Type*}
section ExistsAddOfLE
variable [AddCommSemigrou... | Mathlib/Algebra/Order/Sub/Canonical.lean | 44 | 45 | theorem tsub_le_tsub_iff_right (h : c ≤ b) : a - c ≤ b - c ↔ a ≤ b := by |
rw [tsub_le_iff_right, tsub_add_cancel_of_le h]
| 1 | 2.718282 | 0 | 0.571429 | 7 | 515 |
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.Adjugate
import Mathlib.LinearAlgebra.Matrix.Transvection
import Mathlib.RingTheory.RootsOfUnity.Basic
#align_import linear_algebra.matrix.special_linear_group from "leanprover-community/mathlib"@"f06058e64b7e8397234455038f3f8aec83aab... | Mathlib/LinearAlgebra/Matrix/SpecialLinearGroup.lean | 181 | 183 | theorem det_ne_zero [Nontrivial R] (g : SpecialLinearGroup n R) : det ↑ₘg ≠ 0 := by |
rw [g.det_coe]
norm_num
| 2 | 7.389056 | 1 | 1 | 1 | 960 |
import Mathlib.LinearAlgebra.TensorProduct.Graded.External
import Mathlib.RingTheory.GradedAlgebra.Basic
import Mathlib.GroupTheory.GroupAction.Ring
suppress_compilation
open scoped TensorProduct
variable {R ι A B : Type*}
variable [CommSemiring ι] [Module ι (Additive ℤˣ)] [DecidableEq ι]
variable [CommRing R] [R... | Mathlib/LinearAlgebra/TensorProduct/Graded/Internal.lean | 133 | 135 | theorem auxEquiv_one : auxEquiv R 𝒜 ℬ 1 = 1 := by |
rw [← of_one, Algebra.TensorProduct.one_def, auxEquiv_tmul 𝒜 ℬ, DirectSum.decompose_one,
DirectSum.decompose_one, Algebra.TensorProduct.one_def]
| 2 | 7.389056 | 1 | 1 | 1 | 873 |
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import analysis.inner_product_space.adjoint from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open RCLike
open scoped ComplexConjugate
variable {𝕜 E F G : Type... | Mathlib/Analysis/InnerProductSpace/Adjoint.lean | 138 | 141 | theorem adjoint_comp (A : F →L[𝕜] G) (B : E →L[𝕜] F) : (A ∘L B)† = B† ∘L A† := by |
ext v
refine ext_inner_left 𝕜 fun w => ?_
simp only [adjoint_inner_right, ContinuousLinearMap.coe_comp', Function.comp_apply]
| 3 | 20.085537 | 1 | 0.875 | 8 | 763 |
import Mathlib.LinearAlgebra.Basis
import Mathlib.LinearAlgebra.Multilinear.Basic
#align_import linear_algebra.multilinear.basis from "leanprover-community/mathlib"@"ce11c3c2a285bbe6937e26d9792fda4e51f3fe1a"
open MultilinearMap
variable {R : Type*} {ι : Type*} {n : ℕ} {M : Fin n → Type*} {M₂ : Type*} {M₃ : Type*... | Mathlib/LinearAlgebra/Multilinear/Basis.lean | 56 | 61 | theorem Basis.ext_multilinear [Finite ι] {f g : MultilinearMap R (fun _ : ι => M₂) M₃} {ι₁ : Type*}
(e : Basis ι₁ R M₂) (h : ∀ v : ι → ι₁, (f fun i => e (v i)) = g fun i => e (v i)) : f = g := by |
cases nonempty_fintype ι
exact
(domDomCongr_eq_iff (Fintype.equivFin ι) f g).mp
(Basis.ext_multilinear_fin (fun _ => e) fun i => h (i ∘ _))
| 4 | 54.59815 | 2 | 2 | 2 | 2,194 |
import Mathlib.Data.Real.Irrational
import Mathlib.Data.Nat.Fib.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Algebra.LinearRecurrence
import Mathlib.Tactic.NormNum.NatFib
import Mathlib.Tactic.NormNum.Prime
#align_import data.real.golden_ratio from "leanprover-community/mathlib"@"2196ab363eb097c008d449712... | Mathlib/Data/Real/GoldenRatio.lean | 44 | 47 | theorem inv_gold : φ⁻¹ = -ψ := by |
have : 1 + √5 ≠ 0 := ne_of_gt (add_pos (by norm_num) <| Real.sqrt_pos.mpr (by norm_num))
field_simp [sub_mul, mul_add]
norm_num
| 3 | 20.085537 | 1 | 0.894737 | 19 | 776 |
import Mathlib.Data.Finsupp.ToDFinsupp
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.LinearIndependent
#align_import linear_algebra.dfinsupp from "leanprover-community/mathlib"@"a148d797a1094ab554ad4183a4ad6f130358ef64"
variable {ι : Type*} {R : Type*} {S : Type*} {M : ι → Type*} {N : Type*}
n... | Mathlib/LinearAlgebra/DFinsupp.lean | 170 | 172 | theorem lsum_single [Semiring S] [Module S N] [SMulCommClass R S N] (F : ∀ i, M i →ₗ[R] N) (i)
(x : M i) : lsum S (M := M) F (single i x) = F i x := by |
simp
| 1 | 2.718282 | 0 | 0.666667 | 3 | 596 |
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Coxeter.Basic
namespace CoxeterSystem
open List Matrix Function Classical
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
... | Mathlib/GroupTheory/Coxeter/Length.lean | 71 | 73 | theorem exists_reduced_word (w : W) : ∃ ω, ω.length = ℓ w ∧ w = π ω := by |
have := Nat.find_spec (cs.exists_word_with_prod w)
tauto
| 2 | 7.389056 | 1 | 1.363636 | 11 | 1,468 |
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 | 104 | 106 | theorem kernelSubobject_arrow' :
(kernelSubobjectIso f).inv ≫ (kernelSubobject f).arrow = kernel.ι f := by |
simp [kernelSubobjectIso]
| 1 | 2.718282 | 0 | 0.263158 | 19 | 308 |
import Mathlib.Data.Set.Pointwise.SMul
import Mathlib.Topology.MetricSpace.Isometry
import Mathlib.Topology.MetricSpace.Lipschitz
#align_import topology.metric_space.isometric_smul from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
open Set
open ENNReal Pointwise
universe u v w
vari... | Mathlib/Topology/MetricSpace/IsometricSMul.lean | 143 | 144 | theorem edist_inv [PseudoEMetricSpace G] [IsometricSMul G G] [IsometricSMul Gᵐᵒᵖ G]
(x y : G) : edist x⁻¹ y = edist x y⁻¹ := by | rw [← edist_inv_inv, inv_inv]
| 1 | 2.718282 | 0 | 0.25 | 4 | 304 |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Combinatorics.SimpleGraph.Density
import Mathlib.Data.Nat.Cast.Field
import Mathlib.Order.Partition.Equipartition
import Mathlib.SetTheory.Ordinal.Basic
#align_import combinatorics.simple_graph.regularity.uniform from "leanprover-community/mathlib"@"bf7ef0e83e5b... | Mathlib/Combinatorics/SimpleGraph/Regularity/Uniform.lean | 154 | 157 | theorem right_nonuniformWitnesses_card (h : ¬G.IsUniform ε s t) :
(t.card : 𝕜) * ε ≤ (G.nonuniformWitnesses ε s t).2.card := by |
rw [nonuniformWitnesses, dif_pos h]
exact (not_isUniform_iff.1 h).choose_spec.2.choose_spec.2.2.1
| 2 | 7.389056 | 1 | 1 | 5 | 922 |
import Mathlib.Probability.Kernel.Basic
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.DominatedConvergence
#align_import probability.kernel.measurable_integral from "leanprover-community/mathlib"@"28b2a92f2996d28e580450863c130955de0ed398"
open MeasureTheory Probabilit... | Mathlib/Probability/Kernel/MeasurableIntegral.lean | 42 | 99 | theorem measurable_kernel_prod_mk_left_of_finite {t : Set (α × β)} (ht : MeasurableSet t)
(hκs : ∀ a, IsFiniteMeasure (κ a)) : Measurable fun a => κ a (Prod.mk a ⁻¹' t) := by |
-- `t` is a measurable set in the product `α × β`: we use that the product σ-algebra is generated
-- by boxes to prove the result by induction.
-- Porting note: added motive
refine MeasurableSpace.induction_on_inter
(C := fun t => Measurable fun a => κ a (Prod.mk a ⁻¹' t))
generateFrom_prod.symm isPiSy... | 56 | 2,091,659,496,012,996,000,000,000 | 2 | 2 | 3 | 2,256 |
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calculus.FDeriv.Comp
#align_import analysis.calculus.fderiv.equiv from "leanprover-community/mathlib"@"e3fb84046afd187b710170887195d50bada934ee"
open Filter Asymptotics ContinuousLinearMa... | Mathlib/Analysis/Calculus/FDeriv/Equiv.lean | 116 | 118 | theorem comp_differentiable_iff {f : G → E} : Differentiable 𝕜 (iso ∘ f) ↔ Differentiable 𝕜 f := by |
rw [← differentiableOn_univ, ← differentiableOn_univ]
exact iso.comp_differentiableOn_iff
| 2 | 7.389056 | 1 | 1.555556 | 9 | 1,700 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.LinearAlgebra.AffineSpace.Slope
#align_import analysis.calculus.deriv.slope from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
noncomputable section
open Topology Filter TopologicalSpace
open Filter Set
secti... | Mathlib/Analysis/Calculus/Deriv/Slope.lean | 72 | 74 | theorem hasDerivWithinAt_iff_tendsto_slope' (hs : x ∉ s) :
HasDerivWithinAt f f' s x ↔ Tendsto (slope f x) (𝓝[s] x) (𝓝 f') := by |
rw [hasDerivWithinAt_iff_tendsto_slope, diff_singleton_eq_self hs]
| 1 | 2.718282 | 0 | 1.2 | 5 | 1,257 |
import Mathlib.LinearAlgebra.DirectSum.Finsupp
import Mathlib.LinearAlgebra.FinsuppVectorSpace
#align_import linear_algebra.tensor_product_basis from "leanprover-community/mathlib"@"f784cc6142443d9ee623a20788c282112c322081"
noncomputable section
open Set LinearMap Submodule
section CommSemiring
variable {R : T... | Mathlib/LinearAlgebra/TensorProduct/Basis.lean | 44 | 46 | theorem Basis.tensorProduct_apply' (b : Basis ι R M) (c : Basis κ R N) (i : ι × κ) :
Basis.tensorProduct b c i = b i.1 ⊗ₜ c i.2 := by |
simp [Basis.tensorProduct]
| 1 | 2.718282 | 0 | 0 | 3 | 34 |
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 | 244 | 245 | theorem mul_boole {α} [MulZeroOneClass α] (P : Prop) [Decidable P] (a : α) :
(a * if P then 1 else 0) = if P then a else 0 := by | simp
| 1 | 2.718282 | 0 | 0.181818 | 11 | 267 |
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Algebra.Lie.OfAssociative
import Mathlib.Algebra.Lie.Submodule
import Mathlib.Algebra.Lie.Basic
#align_import algebra.lie.direct_sum from "leanprover-community/mathlib"@"c0cc689babd41c0e9d5f02429211ffbe2403472a"
universe u v w w₁
namespace DirectSum
open DF... | Mathlib/Algebra/Lie/DirectSum.lean | 130 | 136 | theorem lie_of_of_ne [DecidableEq ι] {i j : ι} (hij : i ≠ j) (x : L i) (y : L j) :
⁅of L i x, of L j y⁆ = 0 := by |
refine DFinsupp.ext fun k => ?_
rw [bracket_apply]
obtain rfl | hik := Decidable.eq_or_ne i k
· rw [of_eq_of_ne _ _ _ _ hij.symm, lie_zero, zero_apply]
· rw [of_eq_of_ne _ _ _ _ hik, zero_lie, zero_apply]
| 5 | 148.413159 | 2 | 1.5 | 2 | 1,635 |
import Mathlib.Tactic.Qify
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.DiophantineApproximation
import Mathlib.NumberTheory.Zsqrtd.Basic
#align_import number_theory.pell from "leanprover-community/mathlib"@"7ad820c4997738e2f542f8a20f32911f52020e26"
namespace Pell
open Zsqrtd
theorem is_pell_s... | Mathlib/NumberTheory/Pell.lean | 133 | 133 | theorem prop_x (a : Solution₁ d) : a.x ^ 2 = 1 + d * a.y ^ 2 := by | rw [← a.prop]; ring
| 1 | 2.718282 | 0 | 1 | 7 | 981 |
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Analysis.Calculus.Deriv.Inv
#align_import analysis.calculus.lhopital from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
open Filter Set
open scoped Filter Topology Pointwise
variable {a b : ℝ} (hab : a < b) {l : Filter ℝ} {f f... | Mathlib/Analysis/Calculus/LHopital.lean | 107 | 129 | theorem lhopital_zero_left_on_Ioo (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x)
(hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x) (hg' : ∀ x ∈ Ioo a b, g' x ≠ 0)
(hfb : Tendsto f (𝓝[<] b) (𝓝 0)) (hgb : Tendsto g (𝓝[<] b) (𝓝 0))
(hdiv : Tendsto (fun x => f' x / g' x) (𝓝[<] b) l) :
Tendsto (fun x => f x / ... |
-- Here, we essentially compose by `Neg.neg`. The following is mostly technical details.
have hdnf : ∀ x ∈ -Ioo a b, HasDerivAt (f ∘ Neg.neg) (f' (-x) * -1) x := fun x hx =>
comp x (hff' (-x) hx) (hasDerivAt_neg x)
have hdng : ∀ x ∈ -Ioo a b, HasDerivAt (g ∘ Neg.neg) (g' (-x) * -1) x := fun x hx =>
comp ... | 18 | 65,659,969.137331 | 2 | 2 | 3 | 2,002 |
import Mathlib.Algebra.Module.Torsion
import Mathlib.RingTheory.DedekindDomain.Ideal
#align_import algebra.module.dedekind_domain from "leanprover-community/mathlib"@"cdc34484a07418af43daf8198beaf5c00324bca8"
universe u v
variable {R : Type u} [CommRing R] [IsDomain R] {M : Type v} [AddCommGroup M] [Module R M]
... | Mathlib/Algebra/Module/DedekindDomain.lean | 65 | 72 | theorem isInternal_prime_power_torsion [Module.Finite R M] (hM : Module.IsTorsion R M) :
DirectSum.IsInternal fun p : (factors (⊤ : Submodule R M).annihilator).toFinset =>
torsionBySet R M (p ^ (factors (⊤ : Submodule R M).annihilator).count ↑p : Ideal R) := by |
have hM' := Module.isTorsionBySet_annihilator_top R M
have hI := Submodule.annihilator_top_inter_nonZeroDivisors hM
refine isInternal_prime_power_torsion_of_is_torsion_by_ideal ?_ hM'
rw [← Set.nonempty_iff_ne_empty] at hI; rw [Submodule.ne_bot_iff]
obtain ⟨x, H, hx⟩ := hI; exact ⟨x, H, nonZeroDivisors.ne_ze... | 5 | 148.413159 | 2 | 2 | 2 | 2,196 |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Fin
import Mathlib.GroupTheory.GroupAction.Pi
import Mathlib.Logic.Equiv.Fin
#align_import algebra.big_operators.fin from "leanprover-community/mathlib"@"cc5dd6244981976cc9da7afc4eee5682b037a013"
open Fins... | Mathlib/Algebra/BigOperators/Fin.lean | 113 | 113 | theorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by | simp
| 1 | 2.718282 | 0 | 0.333333 | 12 | 353 |
import Mathlib.CategoryTheory.Sites.CompatiblePlus
import Mathlib.CategoryTheory.Sites.ConcreteSheafification
#align_import category_theory.sites.compatible_sheafification from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
namespace CategoryTheory.GrothendieckTopology
open CategoryThe... | Mathlib/CategoryTheory/Sites/CompatibleSheafification.lean | 110 | 114 | theorem sheafificationWhiskerRightIso_inv_app :
(J.sheafificationWhiskerRightIso F).inv.app P = (J.sheafifyCompIso F P).inv := by |
dsimp [sheafificationWhiskerRightIso, sheafifyCompIso]
simp only [Category.id_comp, Category.comp_id]
erw [Category.id_comp]
| 3 | 20.085537 | 1 | 1.2 | 5 | 1,281 |
import Mathlib.NumberTheory.Zsqrtd.GaussianInt
import Mathlib.NumberTheory.LegendreSymbol.Basic
import Mathlib.Analysis.Normed.Field.Basic
#align_import number_theory.zsqrtd.quadratic_reciprocity from "leanprover-community/mathlib"@"5b2fe80501ff327b9109fb09b7cc8c325cd0d7d9"
open Zsqrtd Complex
open scoped Comple... | Mathlib/NumberTheory/Zsqrtd/QuadraticReciprocity.lean | 86 | 93 | theorem prime_of_nat_prime_of_mod_four_eq_three (p : ℕ) [hp : Fact p.Prime] (hp3 : p % 4 = 3) :
Prime (p : ℤ[i]) :=
irreducible_iff_prime.1 <|
by_contradiction fun hpi =>
let ⟨a, b, hab⟩ := sq_add_sq_of_nat_prime_of_not_irreducible p hpi
have : ∀ a b : ZMod 4, a ^ 2 + b ^ 2 ≠ (p : ZMod 4) := by |
erw [← ZMod.natCast_mod p 4, hp3]; decide
this a b (hab ▸ by simp)
| 2 | 7.389056 | 1 | 1.5 | 2 | 1,553 |
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Polynomial.AlgebraMap
#align_import ring_theory.polynomial.tower from "leanprover-community/mathlib"@"bb168510ef455e9280a152e7f31673cabd3d7496"
open Polynomial
variable (R A B : Type*)
namespace Polynomial
section CommSemiring
variable [CommSemiring ... | Mathlib/RingTheory/Polynomial/Tower.lean | 60 | 63 | theorem aeval_algebraMap_eq_zero_iff [NoZeroSMulDivisors A B] [Nontrivial B] (x : A) (p : R[X]) :
aeval (algebraMap A B x) p = 0 ↔ aeval x p = 0 := by |
rw [aeval_algebraMap_apply, Algebra.algebraMap_eq_smul_one, smul_eq_zero,
iff_false_intro (one_ne_zero' B), or_false_iff]
| 2 | 7.389056 | 1 | 0.25 | 4 | 294 |
import Mathlib.CategoryTheory.Monoidal.Braided.Basic
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.CategoryTheory.Limits.Shapes.Terminal
import Mathlib.Algebra.PUnitInstances
#align_import category_theory.monoidal.Mon_ from "leanprover-community/... | Mathlib/CategoryTheory/Monoidal/Mon_.lean | 80 | 81 | theorem mul_one_hom {Z : C} (f : Z ⟶ M.X) : (f ⊗ M.one) ≫ M.mul = (ρ_ Z).hom ≫ f := by |
rw [tensorHom_def_assoc, M.mul_one, rightUnitor_naturality]
| 1 | 2.718282 | 0 | 0.5 | 4 | 503 |
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Data.Int.LeastGreatest
#align_import data.int.conditionally_complete_order from "leanprover-community/mathlib"@"1e05171a5e8cf18d98d9cf7b207540acb044acae"
open Int
noncomputable section
open scoped Classical
instance instConditionallyComplet... | Mathlib/Data/Int/ConditionallyCompleteOrder.lean | 78 | 82 | theorem csInf_eq_least_of_bdd {s : Set ℤ} [DecidablePred (· ∈ s)] (b : ℤ) (Hb : ∀ z ∈ s, b ≤ z)
(Hinh : ∃ z : ℤ, z ∈ s) : sInf s = leastOfBdd b Hb Hinh := by |
have : s.Nonempty ∧ BddBelow s := ⟨Hinh, b, Hb⟩
simp only [sInf, this, and_self, dite_true]
convert (coe_leastOfBdd_eq Hb (Classical.choose_spec (⟨b, Hb⟩ : BddBelow s)) Hinh).symm
| 3 | 20.085537 | 1 | 1 | 4 | 857 |
import Mathlib.Algebra.Module.LinearMap.Basic
import Mathlib.LinearAlgebra.Basic
import Mathlib.LinearAlgebra.Basis
import Mathlib.LinearAlgebra.BilinearMap
#align_import linear_algebra.sesquilinear_form from "leanprover-community/mathlib"@"87c54600fe3cdc7d32ff5b50873ac724d86aef8d"
variable {R R₁ R₂ R₃ M M₁ M₂ M₃... | Mathlib/LinearAlgebra/SesquilinearForm.lean | 246 | 252 | theorem isSymm_iff_eq_flip {B : LinearMap.BilinForm R M} : B.IsSymm ↔ B = B.flip := by |
constructor <;> intro h
· ext
rw [← h, flip_apply, RingHom.id_apply]
intro x y
conv_lhs => rw [h]
rfl
| 6 | 403.428793 | 2 | 1.25 | 4 | 1,319 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Algebra.Polynomial.Module.AEval
import Mathlib.RingTheory.Derivation.Basic
noncomputable section
namespace Polynomial
section CommSemiring
variable {R A : Type*} [CommSemiring R]
@[simps]
def derivative' : D... | Mathlib/Algebra/Polynomial/Derivation.lean | 67 | 67 | theorem mkDerivation_X (a : A) : mkDerivation R a X = a := by | simp [mkDerivation_apply]
| 1 | 2.718282 | 0 | 0.5 | 2 | 479 |
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology MeasureTheory
names... | Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean | 106 | 106 | theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by | rw [condexp, dif_neg hm_not]
| 1 | 2.718282 | 0 | 1.222222 | 9 | 1,296 |
import Mathlib.SetTheory.Cardinal.Finite
#align_import data.set.ncard from "leanprover-community/mathlib"@"74c2af38a828107941029b03839882c5c6f87a04"
namespace Set
variable {α β : Type*} {s t : Set α}
noncomputable def encard (s : Set α) : ℕ∞ := PartENat.withTopEquiv (PartENat.card s)
@[simp] theorem encard_uni... | Mathlib/Data/Set/Card.lean | 82 | 83 | theorem encard_coe_eq_coe_finsetCard (s : Finset α) : encard (s : Set α) = s.card := by |
rw [Finite.encard_eq_coe_toFinset_card (Finset.finite_toSet s)]; simp
| 1 | 2.718282 | 0 | 0.5 | 14 | 454 |
import Mathlib.Algebra.Ring.Semiconj
import Mathlib.Algebra.Ring.Units
import Mathlib.Algebra.Group.Commute.Defs
import Mathlib.Data.Bracket
#align_import algebra.ring.commute from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
universe u v w x
variable {α : Type u} {β : Type v} {γ : T... | Mathlib/Algebra/Ring/Commute.lean | 82 | 85 | theorem mul_self_eq_mul_self_iff [NonUnitalNonAssocRing R] [NoZeroDivisors R] {a b : R}
(h : Commute a b) : a * a = b * b ↔ a = b ∨ a = -b := by |
rw [← sub_eq_zero, h.mul_self_sub_mul_self_eq, mul_eq_zero, or_comm, sub_eq_zero,
add_eq_zero_iff_eq_neg]
| 2 | 7.389056 | 1 | 0.333333 | 3 | 342 |
import Mathlib.Topology.Sheaves.Forget
import Mathlib.Topology.Sheaves.SheafCondition.PairwiseIntersections
import Mathlib.CategoryTheory.Limits.Shapes.Types
#align_import topology.sheaves.sheaf_condition.unique_gluing from "leanprover-community/mathlib"@"5dc6092d09e5e489106865241986f7f2ad28d4c8"
noncomputable sec... | Mathlib/Topology/Sheaves/SheafCondition/UniqueGluing.lean | 125 | 134 | theorem isSheaf_iff_isSheafUniqueGluing_types : F.IsSheaf ↔ F.IsSheafUniqueGluing := by |
simp_rw [isSheaf_iff_isSheafPairwiseIntersections, IsSheafPairwiseIntersections,
Types.isLimit_iff, IsSheafUniqueGluing, isGluing_iff_pairwise]
refine forall₂_congr fun ι U ↦ ⟨fun h sf cpt ↦ ?_, fun h s hs ↦ ?_⟩
· exact h _ cpt.sectionPairwise.prop
· specialize h (fun i ↦ s <| op <| Pairwise.single i) fun ... | 9 | 8,103.083928 | 2 | 2 | 2 | 1,992 |
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 | 83 | 105 | theorem isCompactElement_iff.{u} {α : Type u} [CompleteLattice α] (k : α) :
CompleteLattice.IsCompactElement k ↔
∀ (ι : Type u) (s : ι → α), k ≤ iSup s → ∃ t : Finset ι, k ≤ t.sup s := by |
classical
constructor
· intro H ι s hs
obtain ⟨t, ht, ht'⟩ := H (Set.range s) hs
have : ∀ x : t, ∃ i, s i = x := fun x => ht x.prop
choose f hf using this
refine ⟨Finset.univ.image f, ht'.trans ?_⟩
rw [Finset.sup_le_iff]
intro b hb
rw [← show s (f ⟨b, hb⟩) = id b fro... | 20 | 485,165,195.40979 | 2 | 2 | 3 | 2,488 |
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analysis.inner_product_space.calculus from "leanprover-community/mathlib"@"f9dd3204df14a0749cd456fac1e6849dfe7d2b88"
noncomputable section
open RCLike Real ... | Mathlib/Analysis/InnerProductSpace/Calculus.lean | 109 | 112 | theorem HasDerivWithinAt.inner {f g : ℝ → E} {f' g' : E} {s : Set ℝ} {x : ℝ}
(hf : HasDerivWithinAt f f' s x) (hg : HasDerivWithinAt g g' s x) :
HasDerivWithinAt (fun t => ⟪f t, g t⟫) (⟪f x, g'⟫ + ⟪f', g x⟫) s x := by |
simpa using (hf.hasFDerivWithinAt.inner 𝕜 hg.hasFDerivWithinAt).hasDerivWithinAt
| 1 | 2.718282 | 0 | 0.833333 | 12 | 734 |
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 | 64 | 66 | theorem ascPochhammer_succ_left (n : ℕ) :
ascPochhammer S (n + 1) = X * (ascPochhammer S n).comp (X + 1) := by |
rw [ascPochhammer]
| 1 | 2.718282 | 0 | 0.96 | 25 | 796 |
import Mathlib.Algebra.Group.Ext
import Mathlib.CategoryTheory.Simple
import Mathlib.CategoryTheory.Linear.Basic
import Mathlib.CategoryTheory.Endomorphism
import Mathlib.FieldTheory.IsAlgClosed.Spectrum
#align_import category_theory.preadditive.schur from "leanprover-community/mathlib"@"58a272265b5e05f258161260dd2c5... | Mathlib/CategoryTheory/Preadditive/Schur.lean | 114 | 125 | theorem finrank_endomorphism_eq_one {X : C} (isIso_iff_nonzero : ∀ f : X ⟶ X, IsIso f ↔ f ≠ 0)
[I : FiniteDimensional 𝕜 (X ⟶ X)] : finrank 𝕜 (X ⟶ X) = 1 := by |
have id_nonzero := (isIso_iff_nonzero (𝟙 X)).mp (by infer_instance)
refine finrank_eq_one (𝟙 X) id_nonzero ?_
intro f
have : Nontrivial (End X) := nontrivial_of_ne _ _ id_nonzero
have : FiniteDimensional 𝕜 (End X) := I
obtain ⟨c, nu⟩ := spectrum.nonempty_of_isAlgClosed_of_finiteDimensional 𝕜 (End.of f)... | 10 | 22,026.465795 | 2 | 2 | 1 | 2,367 |
import Mathlib.Algebra.Module.Submodule.Map
#align_import linear_algebra.basic from "leanprover-community/mathlib"@"9d684a893c52e1d6692a504a118bfccbae04feeb"
open Function
open Pointwise
variable {R : Type*} {R₁ : Type*} {R₂ : Type*} {R₃ : Type*}
variable {K : Type*}
variable {M : Type*} {M₁ : Type*} {M₂ : Type*... | Mathlib/Algebra/Module/Submodule/Ker.lean | 112 | 113 | theorem ker_eq_bot' {f : F} : ker f = ⊥ ↔ ∀ m, f m = 0 → m = 0 := by |
simpa [disjoint_iff_inf_le] using disjoint_ker (f := f) (p := ⊤)
| 1 | 2.718282 | 0 | 0.142857 | 7 | 255 |
import Mathlib.Algebra.IsPrimePow
import Mathlib.NumberTheory.ArithmeticFunction
import Mathlib.Analysis.SpecialFunctions.Log.Basic
#align_import number_theory.von_mangoldt from "leanprover-community/mathlib"@"c946d6097a6925ad16d7ec55677bbc977f9846de"
namespace ArithmeticFunction
open Finset Nat
open scoped Arit... | Mathlib/NumberTheory/VonMangoldt.lean | 135 | 136 | theorem log_mul_moebius_eq_vonMangoldt : log * μ = Λ := by |
rw [← vonMangoldt_mul_zeta, mul_assoc, coe_zeta_mul_coe_moebius, mul_one]
| 1 | 2.718282 | 0 | 0.636364 | 11 | 552 |
import Mathlib.NumberTheory.ZetaValues
import Mathlib.NumberTheory.LSeries.RiemannZeta
open Complex Real Set
open scoped Nat
open HurwitzZeta
theorem riemannZeta_two_mul_nat {k : ℕ} (hk : k ≠ 0) :
riemannZeta (2 * k) = (-1) ^ (k + 1) * (2 : ℂ) ^ (2 * k - 1)
* (π : ℂ) ^ (2 * k) * bernoulli (2 * k) / (... | Mathlib/NumberTheory/LSeries/HurwitzZetaValues.lean | 220 | 224 | theorem riemannZeta_two : riemannZeta 2 = (π : ℂ) ^ 2 / 6 := by |
convert congr_arg ((↑) : ℝ → ℂ) hasSum_zeta_two.tsum_eq
· rw [← Nat.cast_two, zeta_nat_eq_tsum_of_gt_one one_lt_two]
simp only [push_cast]
· norm_cast
| 4 | 54.59815 | 2 | 2 | 8 | 2,019 |
import Mathlib.Analysis.Calculus.Deriv.Inv
import Mathlib.Analysis.NormedSpace.BallAction
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Analysis.InnerProductSpace.Calculus
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Geometry.Manifold.Algebra.LieGroup
import Mathlib.Geometry.Manifol... | Mathlib/Geometry/Manifold/Instances/Sphere.lean | 145 | 160 | theorem hasFDerivAt_stereoInvFunAux (v : E) :
HasFDerivAt (stereoInvFunAux v) (ContinuousLinearMap.id ℝ E) 0 := by |
have h₀ : HasFDerivAt (fun w : E => ‖w‖ ^ 2) (0 : E →L[ℝ] ℝ) 0 := by
convert (hasStrictFDerivAt_norm_sq (0 : E)).hasFDerivAt
simp
have h₁ : HasFDerivAt (fun w : E => (‖w‖ ^ 2 + 4)⁻¹) (0 : E →L[ℝ] ℝ) 0 := by
convert (hasFDerivAt_inv _).comp _ (h₀.add (hasFDerivAt_const 4 0)) <;> simp
have h₂ : HasFDer... | 14 | 1,202,604.284165 | 2 | 1.5 | 6 | 1,585 |
import Mathlib.LinearAlgebra.QuadraticForm.TensorProduct
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.TensorProduct.Opposite
import Mathlib.RingTheory.TensorProduct.Basic
variable {R A V : Type*}
variable [CommRing R] [CommRing A] [AddCommGroup V]
variable [Algebra R A] [Mod... | Mathlib/LinearAlgebra/CliffordAlgebra/BaseChange.lean | 124 | 137 | theorem toBaseChange_comp_reverseOp (Q : QuadraticForm R V) :
(toBaseChange A Q).op.comp reverseOp =
((Algebra.TensorProduct.opAlgEquiv R A A (CliffordAlgebra Q)).toAlgHom.comp <|
(Algebra.TensorProduct.map
(AlgEquiv.toOpposite A A).toAlgHom (reverseOp (Q := Q))).comp
(toBaseChange A... |
ext v
show op (toBaseChange A Q (reverse (ι (Q.baseChange A) (1 ⊗ₜ[R] v)))) =
Algebra.TensorProduct.opAlgEquiv R A A (CliffordAlgebra Q)
(Algebra.TensorProduct.map (AlgEquiv.toOpposite A A).toAlgHom (reverseOp (Q := Q))
(toBaseChange A Q (ι (Q.baseChange A) (1 ⊗ₜ[R] v))))
rw [toBaseChange_ι, re... | 8 | 2,980.957987 | 2 | 2 | 2 | 2,353 |
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 | 134 | 134 | theorem T_neg_two : T R (-2) = 2 * X ^ 2 - 1 := by | simp [T_two]
| 1 | 2.718282 | 0 | 0.166667 | 12 | 265 |
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.Order.Filter.IndicatorFunction
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner
import Mathlib.MeasureTheory.Function.LpSeminorm.Trim
#align_import measure_theory.function.conditional... | Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean | 95 | 99 | theorem const_smul [SMul 𝕜 β] [ContinuousConstSMul 𝕜 β] (c : 𝕜) (hf : AEStronglyMeasurable' m f μ) :
AEStronglyMeasurable' m (c • f) μ := by |
rcases hf with ⟨f', h_f'_meas, hff'⟩
refine ⟨c • f', h_f'_meas.const_smul c, ?_⟩
exact EventuallyEq.fun_comp hff' fun x => c • x
| 3 | 20.085537 | 1 | 1.142857 | 7 | 1,222 |
import Mathlib.Algebra.MonoidAlgebra.Support
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.Algebra.Regular.Basic
import Mathlib.Data.Nat.Choose.Sum
#align_import data.polynomial.coeff from "leanprover-community/mathlib"@"2651125b48fc5c170ab1111afd0817c903b1fc6c"
set_option linter.uppercaseLean3 false
no... | Mathlib/Algebra/Polynomial/Coeff.lean | 40 | 44 | theorem coeff_add (p q : R[X]) (n : ℕ) : coeff (p + q) n = coeff p n + coeff q n := by |
rcases p with ⟨⟩
rcases q with ⟨⟩
simp_rw [← ofFinsupp_add, coeff]
exact Finsupp.add_apply _ _ _
| 4 | 54.59815 | 2 | 1.125 | 8 | 1,203 |
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.MeasureTheory.Group.Measure
#align_import measure_theory.group.integration from "leanprover-community/mathlib"@"ec247d43814751ffceb33b758e8820df2372bf6f"
namespace MeasureTheory
open Measure TopologicalSpace
open scoped ENNReal
variable {𝕜 M α G E F ... | Mathlib/MeasureTheory/Group/Integral.lean | 40 | 43 | theorem integral_inv_eq_self (f : G → E) (μ : Measure G) [IsInvInvariant μ] :
∫ x, f x⁻¹ ∂μ = ∫ x, f x ∂μ := by |
have h : MeasurableEmbedding fun x : G => x⁻¹ := (MeasurableEquiv.inv G).measurableEmbedding
rw [← h.integral_map, map_inv_eq_self]
| 2 | 7.389056 | 1 | 0.75 | 8 | 660 |
import Mathlib.Data.List.Sigma
#align_import data.list.alist from "leanprover-community/mathlib"@"f808feb6c18afddb25e66a71d317643cf7fb5fbb"
universe u v w
open List
variable {α : Type u} {β : α → Type v}
structure AList (β : α → Type v) : Type max u v where
entries : List (Sigma β)
nodupKeys : entri... | Mathlib/Data/List/AList.lean | 183 | 190 | theorem keys_subset_keys_of_entries_subset_entries
{s₁ s₂ : AList β} (h : s₁.entries ⊆ s₂.entries) : s₁.keys ⊆ s₂.keys := by |
intro k hk
letI : DecidableEq α := Classical.decEq α
have := h (mem_lookup_iff.1 (Option.get_mem (lookup_isSome.2 hk)))
rw [← mem_lookup_iff, Option.mem_def] at this
rw [← mem_keys, ← lookup_isSome, this]
exact Option.isSome_some
| 6 | 403.428793 | 2 | 1 | 2 | 848 |
import Mathlib.Analysis.SpecialFunctions.JapaneseBracket
import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.MeasureTheory.Group.Integral
import Mathlib.MeasureTheory.Integral.IntegralEqImproper
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.special_functions.improper_inte... | Mathlib/Analysis/SpecialFunctions/ImproperIntegrals.lean | 106 | 116 | theorem integral_Ioi_rpow_of_lt {a : ℝ} (ha : a < -1) {c : ℝ} (hc : 0 < c) :
∫ t : ℝ in Ioi c, t ^ a = -c ^ (a + 1) / (a + 1) := by |
have hd : ∀ x ∈ Ici c, HasDerivAt (fun t => t ^ (a + 1) / (a + 1)) (x ^ a) x := by
intro x hx
convert (hasDerivAt_rpow_const (p := a + 1) (Or.inl (hc.trans_le hx).ne')).div_const _ using 1
field_simp [show a + 1 ≠ 0 from ne_of_lt (by linarith), mul_comm]
have ht : Tendsto (fun t => t ^ (a + 1) / (a + 1... | 9 | 8,103.083928 | 2 | 1.5 | 8 | 1,667 |
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 | 135 | 137 | theorem toComplex_star (x : ℤ[i]) : ((star x : ℤ[i]) : ℂ) = conj (x : ℂ) := by |
rw [toComplex_def₂, toComplex_def₂]
exact congr_arg₂ _ rfl (Int.cast_neg _)
| 2 | 7.389056 | 1 | 0.090909 | 11 | 243 |
import Mathlib.Algebra.Quaternion
import Mathlib.Tactic.Ring
#align_import algebra.quaternion_basis from "leanprover-community/mathlib"@"3aa5b8a9ed7a7cabd36e6e1d022c9858ab8a8c2d"
open Quaternion
namespace QuaternionAlgebra
structure Basis {R : Type*} (A : Type*) [CommRing R] [Ring A] [Algebra R A] (c₁ c₂ : R) ... | Mathlib/Algebra/QuaternionBasis.lean | 138 | 139 | theorem lift_smul (r : R) (x : ℍ[R,c₁,c₂]) : q.lift (r • x) = r • q.lift x := by |
simp [lift, mul_smul, ← Algebra.smul_def]
| 1 | 2.718282 | 0 | 0.4 | 10 | 394 |
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.RingTheory.Localization.FractionRing
#align_import data.polynomial.ring_division from "leanprover-community/mathlib"@"8efcf8022aac8e01df8d302dcebdbc25d6a886c8"
noncomputable section
namespace Polynomial
universe u v w z
variable {R : Type u} {S : Ty... | Mathlib/Algebra/Polynomial/Roots.lean | 82 | 87 | theorem card_roots_sub_C {p : R[X]} {a : R} (hp0 : 0 < degree p) :
(Multiset.card (p - C a).roots : WithBot ℕ) ≤ degree p :=
calc
(Multiset.card (p - C a).roots : WithBot ℕ) ≤ degree (p - C a) :=
card_roots <| mt sub_eq_zero.1 fun h => not_le_of_gt hp0 <| h.symm ▸ degree_C_le
_ = degree p := by | rw [sub_eq_add_neg, ← C_neg]; exact degree_add_C hp0
| 1 | 2.718282 | 0 | 1.285714 | 7 | 1,352 |
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Topology.Algebra.Module.Basic
#align_import analysis.normed_space.basic from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
open Metric Set Function Filter
open scoped NNReal Topology
instance Real.punctured_nhds_module_neBot {E ... | Mathlib/Analysis/NormedSpace/Real.lean | 50 | 59 | theorem dist_smul_add_one_sub_smul_le {r : ℝ} {x y : E} (h : r ∈ Icc 0 1) :
dist (r • x + (1 - r) • y) x ≤ dist y x :=
calc
dist (r • x + (1 - r) • y) x = ‖1 - r‖ * ‖x - y‖ := by |
simp_rw [dist_eq_norm', ← norm_smul, sub_smul, one_smul, smul_sub, ← sub_sub, ← sub_add,
sub_right_comm]
_ = (1 - r) * dist y x := by
rw [Real.norm_eq_abs, abs_eq_self.mpr (sub_nonneg.mpr h.2), dist_eq_norm']
_ ≤ (1 - 0) * dist y x := by gcongr; exact h.1
_ = dist y x := by rw [sub_zero... | 6 | 403.428793 | 2 | 0.9 | 10 | 783 |
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Analysis.NormedSpace.Real
#align_import analysis.special_functions.log.basic from "leanprover-community/mathlib"@"f23a09ce6d3f367220dc3cecad6b7eb69eb01690"
open Set Filter Function
open Topology
noncomputable ... | Mathlib/Analysis/SpecialFunctions/Log/Basic.lean | 69 | 74 | theorem le_exp_log (x : ℝ) : x ≤ exp (log x) := by |
by_cases h_zero : x = 0
· rw [h_zero, log, dif_pos rfl, exp_zero]
exact zero_le_one
· rw [exp_log_eq_abs h_zero]
exact le_abs_self _
| 5 | 148.413159 | 2 | 0.583333 | 12 | 525 |
import Mathlib.Analysis.SpecialFunctions.Complex.Circle
import Mathlib.Geometry.Euclidean.Angle.Oriented.Basic
#align_import geometry.euclidean.angle.oriented.rotation from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
noncomputable section
open FiniteDimensional Complex
open scoped ... | Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean | 155 | 157 | theorem rotation_pi_div_two : o.rotation (π / 2 : ℝ) = J := by |
ext x
simp [rotation]
| 2 | 7.389056 | 1 | 0.857143 | 7 | 744 |
import Mathlib.CategoryTheory.Preadditive.Yoneda.Projective
import Mathlib.CategoryTheory.Preadditive.Yoneda.Limits
import Mathlib.Algebra.Category.ModuleCat.EpiMono
universe v u
namespace CategoryTheory
open Limits Projective Opposite
variable {C : Type u} [Category.{v} C] [Abelian C]
noncomputable def preser... | Mathlib/CategoryTheory/Abelian/Projective.lean | 37 | 42 | theorem projective_of_preservesFiniteColimits_preadditiveCoyonedaObj (P : C)
[hP : PreservesFiniteColimits (preadditiveCoyonedaObj (op P))] : Projective P := by |
rw [projective_iff_preservesEpimorphisms_preadditiveCoyoneda_obj']
-- Porting note: this next line wasn't necessary in Lean 3
dsimp only [preadditiveCoyoneda]
infer_instance
| 4 | 54.59815 | 2 | 2 | 1 | 2,309 |
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
| 2 | 7.389056 | 1 | 1.2 | 5 | 1,252 |
import Mathlib.Algebra.BigOperators.Finsupp
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Data.Fintype.BigOperators
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.SetTheory.Cardinal.Cofinality
#align_import linear_algebra.basis from "leanprover-communit... | Mathlib/LinearAlgebra/Basis.lean | 137 | 141 | theorem repr_symm_single : b.repr.symm (Finsupp.single i c) = c • b i :=
calc
b.repr.symm (Finsupp.single i c) = b.repr.symm (c • Finsupp.single i (1 : R)) := by |
{ rw [Finsupp.smul_single', mul_one] }
_ = c • b i := by rw [LinearEquiv.map_smul, repr_symm_single_one]
| 2 | 7.389056 | 1 | 0.9 | 10 | 779 |
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 | 256 | 259 | theorem finsuppTensorFinsupp_single (i : ι) (m : M) (k : κ) (n : N) :
finsuppTensorFinsupp R S M N ι κ (Finsupp.single i m ⊗ₜ Finsupp.single k n) =
Finsupp.single (i, k) (m ⊗ₜ n) := by |
simp [finsuppTensorFinsupp]
| 1 | 2.718282 | 0 | 0.75 | 8 | 652 |
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
| 2 | 7.389056 | 1 | 1 | 1 | 940 |
import Mathlib.Data.Int.Interval
import Mathlib.Data.Int.ModEq
import Mathlib.Data.Nat.Count
import Mathlib.Data.Rat.Floor
import Mathlib.Order.Interval.Finset.Nat
open Finset Int
namespace Int
variable (a b : ℤ) {r : ℤ} (hr : 0 < r)
lemma Ico_filter_dvd_eq : (Ico a b).filter (r ∣ ·) =
(Ico ⌈a / (r : ℚ)⌉ ⌈b... | Mathlib/Data/Int/CardIntervalMod.lean | 71 | 73 | theorem Ioc_filter_modEq_card (v : ℤ) : ((Ioc a b).filter (· ≡ v [ZMOD r])).card =
max (⌊(b - v) / (r : ℚ)⌋ - ⌊(a - v) / (r : ℚ)⌋) 0 := by |
simp [Ioc_filter_modEq_eq, Ioc_filter_dvd_eq, toNat_eq_max, hr]
| 1 | 2.718282 | 0 | 0 | 4 | 158 |
import Mathlib.Algebra.Order.Pointwise
import Mathlib.Analysis.NormedSpace.SphereNormEquiv
import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.MeasureTheory.Constructions.Prod.Integral
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
open Set Function Metric MeasurableSpace intervalIntegral
open s... | Mathlib/MeasureTheory/Constructions/HaarToSphere.lean | 49 | 53 | theorem toSphere_apply_aux (s : Set (sphere (0 : E) 1)) (r : Ioi (0 : ℝ)) :
μ ((↑) '' (homeomorphUnitSphereProd E ⁻¹' s ×ˢ Iio r)) = μ (Ioo (0 : ℝ) r • ((↑) '' s)) := by |
rw [← image2_smul, image2_image_right, ← Homeomorph.image_symm, image_image,
← image_subtype_val_Ioi_Iio, image2_image_left, image2_swap, ← image_prod]
rfl
| 3 | 20.085537 | 1 | 1.2 | 5 | 1,289 |
import Mathlib.Analysis.Seminorm
import Mathlib.Topology.Algebra.Equicontinuity
import Mathlib.Topology.MetricSpace.Equicontinuity
import Mathlib.Topology.Algebra.FilterBasis
import Mathlib.Topology.Algebra.Module.LocallyConvex
#align_import analysis.locally_convex.with_seminorms from "leanprover-community/mathlib"@"... | Mathlib/Analysis/LocallyConvex/WithSeminorms.lean | 115 | 118 | theorem basisSets_zero (U) (hU : U ∈ p.basisSets) : (0 : E) ∈ U := by |
rcases p.basisSets_iff.mp hU with ⟨ι', r, hr, hU⟩
rw [hU, mem_ball_zero, map_zero]
exact hr
| 3 | 20.085537 | 1 | 1.272727 | 11 | 1,349 |
import Mathlib.Analysis.Normed.Order.Basic
import Mathlib.Analysis.Asymptotics.Asymptotics
import Mathlib.Analysis.NormedSpace.Basic
#align_import analysis.asymptotics.specific_asymptotics from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Filter Asymptotics
open Topology
sectio... | Mathlib/Analysis/Asymptotics/SpecificAsymptotics.lean | 63 | 67 | theorem tendsto_pow_div_pow_atTop_zero [TopologicalSpace 𝕜] [OrderTopology 𝕜] {p q : ℕ}
(hpq : p < q) : Tendsto (fun x : 𝕜 => x ^ p / x ^ q) atTop (𝓝 0) := by |
rw [tendsto_congr' pow_div_pow_eventuallyEq_atTop]
apply tendsto_zpow_atTop_zero
omega
| 3 | 20.085537 | 1 | 1.25 | 8 | 1,299 |
import Mathlib.Analysis.Normed.Group.InfiniteSum
import Mathlib.Analysis.Normed.MulAction
import Mathlib.Topology.Algebra.Order.LiminfLimsup
import Mathlib.Topology.PartialHomeomorph
#align_import analysis.asymptotics.asymptotics from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open ... | Mathlib/Analysis/Asymptotics/Asymptotics.lean | 118 | 132 | theorem isBigO_iff' {g : α → E'''} :
f =O[l] g ↔ ∃ c > 0, ∀ᶠ x in l, ‖f x‖ ≤ c * ‖g x‖ := by |
refine ⟨fun h => ?mp, fun h => ?mpr⟩
case mp =>
rw [isBigO_iff] at h
obtain ⟨c, hc⟩ := h
refine ⟨max c 1, zero_lt_one.trans_le (le_max_right _ _), ?_⟩
filter_upwards [hc] with x hx
apply hx.trans
gcongr
exact le_max_left _ _
case mpr =>
rw [isBigO_iff]
obtain ⟨c, ⟨_, hc⟩⟩ := h... | 13 | 442,413.392009 | 2 | 0.8 | 5 | 709 |
import Mathlib.Topology.Bases
import Mathlib.Topology.DenseEmbedding
#align_import topology.stone_cech from "leanprover-community/mathlib"@"0a0ec35061ed9960bf0e7ffb0335f44447b58977"
noncomputable section
open Filter Set
open Topology
universe u v
section Ultrafilter
def ultrafilterBasis (α : Type u) : Set ... | Mathlib/Topology/StoneCech.lean | 138 | 143 | theorem induced_topology_pure :
TopologicalSpace.induced (pure : α → Ultrafilter α) Ultrafilter.topologicalSpace = ⊥ := by |
apply eq_bot_of_singletons_open
intro x
use { u : Ultrafilter α | {x} ∈ u }, ultrafilter_isOpen_basic _
simp
| 4 | 54.59815 | 2 | 2 | 5 | 2,361 |
import Mathlib.SetTheory.Ordinal.Arithmetic
import Mathlib.SetTheory.Ordinal.Exponential
#align_import set_theory.ordinal.fixed_point from "leanprover-community/mathlib"@"0dd4319a17376eda5763cd0a7e0d35bbaaa50e83"
noncomputable section
universe u v
open Function Order
namespace Ordinal
section
variable {ι ... | Mathlib/SetTheory/Ordinal/FixedPoint.lean | 102 | 106 | theorem nfpFamily_le_apply [Nonempty ι] (H : ∀ i, IsNormal (f i)) {a b} :
(∃ i, nfpFamily.{u, v} f a ≤ f i b) ↔ nfpFamily.{u, v} f a ≤ b := by |
rw [← not_iff_not]
push_neg
exact apply_lt_nfpFamily_iff H
| 3 | 20.085537 | 1 | 1.666667 | 3 | 1,808 |
import Mathlib.Order.Filter.Basic
import Mathlib.Data.Set.Countable
#align_import order.filter.countable_Inter from "leanprover-community/mathlib"@"b9e46fe101fc897fb2e7edaf0bf1f09ea49eb81a"
open Set Filter
open Filter
variable {ι : Sort*} {α β : Type*}
class CountableInterFilter (l : Filter α) : Prop where
... | Mathlib/Order/Filter/CountableInter.lean | 65 | 68 | theorem eventually_countable_forall [Countable ι] {p : α → ι → Prop} :
(∀ᶠ x in l, ∀ i, p x i) ↔ ∀ i, ∀ᶠ x in l, p x i := by |
simpa only [Filter.Eventually, setOf_forall] using
@countable_iInter_mem _ _ l _ _ fun i => { x | p x i }
| 2 | 7.389056 | 1 | 1 | 5 | 868 |
import Mathlib.Data.Set.Image
#align_import order.directed from "leanprover-community/mathlib"@"ffde2d8a6e689149e44fd95fa862c23a57f8c780"
open Function
universe u v w
variable {α : Type u} {β : Type v} {ι : Sort w} (r r' s : α → α → Prop)
local infixl:50 " ≼ " => r
def Directed (f : ι → α) :=
∀ x y, ∃ z, ... | Mathlib/Order/Directed.lean | 58 | 60 | theorem directedOn_iff_directed {s} : @DirectedOn α r s ↔ Directed r (Subtype.val : s → α) := by |
simp only [DirectedOn, Directed, Subtype.exists, exists_and_left, exists_prop, Subtype.forall]
exact forall₂_congr fun x _ => by simp [And.comm, and_assoc]
| 2 | 7.389056 | 1 | 1 | 4 | 976 |
import Mathlib.Algebra.Module.Torsion
import Mathlib.RingTheory.DedekindDomain.Ideal
#align_import algebra.module.dedekind_domain from "leanprover-community/mathlib"@"cdc34484a07418af43daf8198beaf5c00324bca8"
universe u v
variable {R : Type u} [CommRing R] [IsDomain R] {M : Type v} [AddCommGroup M] [Module R M]
... | Mathlib/Algebra/Module/DedekindDomain.lean | 37 | 59 | theorem isInternal_prime_power_torsion_of_is_torsion_by_ideal {I : Ideal R} (hI : I ≠ ⊥)
(hM : Module.IsTorsionBySet R M I) :
DirectSum.IsInternal fun p : (factors I).toFinset =>
torsionBySet R M (p ^ (factors I).count ↑p : Ideal R) := by |
let P := factors I
have prime_of_mem := fun p (hp : p ∈ P.toFinset) =>
prime_of_factor p (Multiset.mem_toFinset.mp hp)
apply torsionBySet_isInternal (p := fun p => p ^ P.count p) _
· convert hM
rw [← Finset.inf_eq_iInf, IsDedekindDomain.inf_prime_pow_eq_prod, ← Finset.prod_multiset_count,
← assoc... | 19 | 178,482,300.963187 | 2 | 2 | 2 | 2,196 |
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Data.Finset.Basic
import Mathlib.Order.Interval.Finset.Defs
open Function
namespace Finset
class HasAntidiagonal (A : Type*) [AddMonoid A] where
antidiagonal : A → Finset (A × A)
mem_antidiagonal {n} {a} : a ∈ antidiagonal n ↔ a.fst + a.snd = n
exp... | Mathlib/Data/Finset/Antidiagonal.lean | 100 | 104 | theorem antidiagonal_congr (hp : p ∈ antidiagonal n) (hq : q ∈ antidiagonal n) :
p = q ↔ p.1 = q.1 := by |
refine ⟨congr_arg Prod.fst, fun h ↦ Prod.ext h ((add_right_inj q.fst).mp ?_)⟩
rw [mem_antidiagonal] at hp hq
rw [hq, ← h, hp]
| 3 | 20.085537 | 1 | 1.142857 | 7 | 1,213 |
import Mathlib.Combinatorics.SimpleGraph.Basic
namespace SimpleGraph
variable {V : Type*} (G : SimpleGraph V)
structure Dart extends V × V where
adj : G.Adj fst snd
deriving DecidableEq
#align simple_graph.dart SimpleGraph.Dart
initialize_simps_projections Dart (+toProd, -fst, -snd)
attribute [simp] Dart.a... | Mathlib/Combinatorics/SimpleGraph/Dart.lean | 33 | 34 | theorem Dart.ext_iff (d₁ d₂ : G.Dart) : d₁ = d₂ ↔ d₁.toProd = d₂.toProd := by |
cases d₁; cases d₂; simp
| 1 | 2.718282 | 0 | 0.75 | 4 | 655 |
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 | 94 | 96 | theorem transpose_col (v : m → α) : (Matrix.col v)ᵀ = Matrix.row v := by |
ext
rfl
| 2 | 7.389056 | 1 | 1 | 14 | 798 |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Fin
import Mathlib.GroupTheory.GroupAction.Pi
import Mathlib.Logic.Equiv.Fin
#align_import algebra.big_operators.fin from "leanprover-community/mathlib"@"cc5dd6244981976cc9da7afc4eee5682b037a013"
open Fins... | Mathlib/Algebra/BigOperators/Fin.lean | 143 | 146 | theorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :
∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by |
rw [prod_univ_castSucc, prod_univ_four]
rfl
| 2 | 7.389056 | 1 | 0.333333 | 12 | 353 |
import Mathlib.Data.Multiset.FinsetOps
import Mathlib.Data.Multiset.Fold
#align_import data.multiset.lattice from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
namespace Multiset
variable {α : Type*}
section Sup
-- can be defined with just `[Bot α]` where some lemmas hold without... | Mathlib/Data/Multiset/Lattice.lean | 93 | 99 | theorem nodup_sup_iff {α : Type*} [DecidableEq α] {m : Multiset (Multiset α)} :
m.sup.Nodup ↔ ∀ a : Multiset α, a ∈ m → a.Nodup := by |
-- Porting note: this was originally `apply m.induction_on`, which failed due to
-- `failed to elaborate eliminator, expected type is not available`
induction' m using Multiset.induction_on with _ _ h
· simp
· simp [h]
| 5 | 148.413159 | 2 | 0.285714 | 7 | 313 |
import Mathlib.Dynamics.BirkhoffSum.Basic
import Mathlib.Algebra.Module.Basic
open Finset
section birkhoffAverage
variable (R : Type*) {α M : Type*} [DivisionSemiring R] [AddCommMonoid M] [Module R M]
def birkhoffAverage (f : α → α) (g : α → M) (n : ℕ) (x : α) : M := (n : R)⁻¹ • birkhoffSum f g n x
| Mathlib/Dynamics/BirkhoffSum/Average.lean | 44 | 45 | theorem birkhoffAverage_zero (f : α → α) (g : α → M) (x : α) :
birkhoffAverage R f g 0 x = 0 := by | simp [birkhoffAverage]
| 1 | 2.718282 | 0 | 0.2 | 5 | 276 |
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 | 84 | 85 | theorem tendsto_ceil_left_pure (n : ℤ) : Tendsto (ceil : α → ℤ) (𝓝[≤] n) (pure n) := by |
simpa only [ceil_intCast] using tendsto_ceil_left_pure_ceil (n : α)
| 1 | 2.718282 | 0 | 0.5 | 6 | 492 |
import Mathlib.LinearAlgebra.Span
import Mathlib.LinearAlgebra.BilinearMap
#align_import algebra.module.submodule.bilinear from "leanprover-community/mathlib"@"6010cf523816335f7bae7f8584cb2edaace73940"
universe uι u v
open Set
open Pointwise
namespace Submodule
variable {ι : Sort uι} {R M N P : Type*}
variabl... | Mathlib/Algebra/Module/Submodule/Bilinear.lean | 59 | 73 | theorem map₂_span_span (f : M →ₗ[R] N →ₗ[R] P) (s : Set M) (t : Set N) :
map₂ f (span R s) (span R t) = span R (Set.image2 (fun m n => f m n) s t) := by |
apply le_antisymm
· rw [map₂_le]
apply @span_induction' R M _ _ _ s
intro a ha
apply @span_induction' R N _ _ _ t
intro b hb
exact subset_span ⟨_, ‹_›, _, ‹_›, rfl⟩
all_goals intros; simp only [*, add_mem, smul_mem, zero_mem, _root_.map_zero, map_add,
Linear... | 13 | 442,413.392009 | 2 | 2 | 1 | 2,469 |
import Mathlib.Data.List.Basic
#align_import data.list.count from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
assert_not_exists Set.range
assert_not_exists GroupWithZero
assert_not_exists Ring
open Nat
variable {α : Type*} {l : List α}
namespace List
section Count
variable [Dec... | Mathlib/Data/List/Count.lean | 90 | 93 | theorem count_cons' (a b : α) (l : List α) :
count a (b :: l) = count a l + if a = b then 1 else 0 := by |
simp only [count, beq_iff_eq, countP_cons, Nat.add_right_inj]
simp only [eq_comm]
| 2 | 7.389056 | 1 | 1 | 2 | 818 |
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