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.Data.Finset.Image
import Mathlib.Data.List.FinRange
#align_import data.fintype.basic from "leanprover-community/mathlib"@"d78597269638367c3863d40d45108f52207e03cf"
assert_not_exists MonoidWithZero
assert_not_exists MulAction
open Function
open Nat
universe u v
variable {α β γ : Type*}
class Fi... | Mathlib/Data/Fintype/Basic.lean | 92 | 92 | theorem coe_univ : ↑(univ : Finset α) = (Set.univ : Set α) := by | ext; simp
| 1 | 2.718282 | 0 | 0.111111 | 9 | 248 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.iterated_deriv from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputable section
open scoped Classical Topology
open Filter Asymptotics Set
variable {𝕜... | Mathlib/Analysis/Calculus/IteratedDeriv/Defs.lean | 157 | 162 | theorem ContDiffWithinAt.differentiableWithinAt_iteratedDerivWithin {n : ℕ∞} {m : ℕ}
(h : ContDiffWithinAt 𝕜 n f s x) (hmn : (m : ℕ∞) < n) (hs : UniqueDiffOn 𝕜 (insert x s)) :
DifferentiableWithinAt 𝕜 (iteratedDerivWithin m f s) s x := by |
simpa only [iteratedDerivWithin_eq_equiv_comp,
LinearIsometryEquiv.comp_differentiableWithinAt_iff] using
h.differentiableWithinAt_iteratedFDerivWithin hmn hs
| 3 | 20.085537 | 1 | 0.727273 | 11 | 649 |
import Mathlib.Data.Set.Image
import Mathlib.Data.Set.Lattice
#align_import data.set.sigma from "leanprover-community/mathlib"@"2258b40dacd2942571c8ce136215350c702dc78f"
namespace Set
variable {ι ι' : Type*} {α β : ι → Type*} {s s₁ s₂ : Set ι} {t t₁ t₂ : ∀ i, Set (α i)}
{u : Set (Σ i, α i)} {x : Σ i, α i} {i j ... | Mathlib/Data/Set/Sigma.lean | 23 | 28 | theorem range_sigmaMk (i : ι) : range (Sigma.mk i : α i → Sigma α) = Sigma.fst ⁻¹' {i} := by |
apply Subset.antisymm
· rintro _ ⟨b, rfl⟩
simp
· rintro ⟨x, y⟩ (rfl | _)
exact mem_range_self y
| 5 | 148.413159 | 2 | 1.666667 | 3 | 1,824 |
import Mathlib.Algebra.CharP.ExpChar
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Algebra.MvPolynomial.Equiv
import Mathlib.RingTheory.Polynomial.Content
import Mathlib.RingTheory.UniqueFactorizationDomain
#align_import ring_theory.polynomial.basic from "leanprover-commun... | Mathlib/RingTheory/Polynomial/Basic.lean | 98 | 109 | theorem mem_degreeLT {n : ℕ} {f : R[X]} : f ∈ degreeLT R n ↔ degree f < n := by |
rw [degreeLT, Submodule.mem_iInf]
conv_lhs => intro i; rw [Submodule.mem_iInf]
rw [degree, Finset.max_eq_sup_coe]
rw [Finset.sup_lt_iff ?_]
rotate_left
· apply WithBot.bot_lt_coe
conv_rhs =>
simp only [mem_support_iff]
intro b
rw [Nat.cast_withBot, WithBot.coe_lt_coe, lt_iff_not_le, Ne, not_i... | 11 | 59,874.141715 | 2 | 1.5 | 4 | 1,612 |
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 | 72 | 74 | theorem mul_self : t * t = 1 := by |
rcases ht with ⟨w, i, rfl⟩
simp
| 2 | 7.389056 | 1 | 1.111111 | 9 | 1,198 |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
#align_import ring_theory.coprime.lemmas from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
universe u v
section RelPrime
variable {α I} [Comm... | Mathlib/RingTheory/Coprime/Lemmas.lean | 245 | 248 | theorem IsRelPrime.prod_left_iff : IsRelPrime (∏ i ∈ t, s i) x ↔ ∀ i ∈ t, IsRelPrime (s i) x := by |
classical
refine Finset.induction_on t (iff_of_true isRelPrime_one_left fun _ ↦ by simp) fun b t hbt ih ↦ ?_
rw [Finset.prod_insert hbt, IsRelPrime.mul_left_iff, ih, Finset.forall_mem_insert]
| 3 | 20.085537 | 1 | 1.111111 | 18 | 1,195 |
import Mathlib.RingTheory.DedekindDomain.Dvr
import Mathlib.RingTheory.DedekindDomain.Ideal
#align_import ring_theory.dedekind_domain.pid from "leanprover-community/mathlib"@"6010cf523816335f7bae7f8584cb2edaace73940"
variable {R : Type*} [CommRing R]
open Ideal
open UniqueFactorizationMonoid
open scoped nonZer... | Mathlib/RingTheory/DedekindDomain/PID.lean | 78 | 102 | theorem FractionalIdeal.isPrincipal_of_unit_of_comap_mul_span_singleton_eq_top {R A : Type*}
[CommRing R] [CommRing A] [Algebra R A] {S : Submonoid R} [IsLocalization S A]
(I : (FractionalIdeal S A)ˣ) {v : A} (hv : v ∈ (↑I⁻¹ : FractionalIdeal S A))
(h : Submodule.comap (Algebra.linearMap R A) ((I : Submodul... |
have hinv := I.mul_inv
set J := Submodule.comap (Algebra.linearMap R A) ((I : Submodule R A) * Submodule.span R {v})
have hJ : IsLocalization.coeSubmodule A J = ↑I * Submodule.span R {v} := by
-- Porting note: had to insert `val_eq_coe` into this rewrite.
-- Arguably this is because `Subtype.ext_iff` is ... | 20 | 485,165,195.40979 | 2 | 2 | 3 | 2,278 |
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 | 88 | 93 | theorem _root_.MeasureTheory.AEStronglyMeasurable.ae_integrable_condDistrib_map_iff
(hY : AEMeasurable Y μ) (hf : AEStronglyMeasurable f (μ.map fun a => (X a, Y a))) :
(∀ᵐ a ∂μ.map X, Integrable (fun ω => f (a, ω)) (condDistrib Y X μ a)) ∧
Integrable (fun a => ∫ ω, ‖f (a, ω)‖ ∂condDistrib Y X μ a) (μ.map ... |
rw [condDistrib, ← hf.ae_integrable_condKernel_iff, Measure.fst_map_prod_mk₀ hY]
| 1 | 2.718282 | 0 | 0.777778 | 9 | 692 |
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Data.Finsupp.Fin
import Mathlib.Data.Finsupp.Indicator
#align_import algebra.bi... | Mathlib/Algebra/BigOperators/Finsupp.lean | 101 | 104 | theorem prod_ite_eq [DecidableEq α] (f : α →₀ M) (a : α) (b : α → M → N) :
(f.prod fun x v => ite (a = x) (b x v) 1) = ite (a ∈ f.support) (b a (f a)) 1 := by |
dsimp [Finsupp.prod]
rw [f.support.prod_ite_eq]
| 2 | 7.389056 | 1 | 0.833333 | 6 | 729 |
import Mathlib.AlgebraicTopology.DoldKan.FunctorGamma
import Mathlib.AlgebraicTopology.DoldKan.SplitSimplicialObject
import Mathlib.CategoryTheory.Idempotents.HomologicalComplex
#align_import algebraic_topology.dold_kan.gamma_comp_n from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190f0b504"
no... | Mathlib/AlgebraicTopology/DoldKan/GammaCompN.lean | 95 | 100 | theorem N₁Γ₀_inv_app (K : ChainComplex C ℕ) :
N₁Γ₀.inv.app K = (toKaroubi _).map (Γ₀NondegComplexIso K).inv ≫
(Γ₀.splitting K).toKaroubiNondegComplexIsoN₁.hom := by |
change (N₁Γ₀.app K).inv = _
simp only [N₁Γ₀_app]
rfl
| 3 | 20.085537 | 1 | 1.2 | 5 | 1,264 |
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.TryThis
import Mathlib.Util.AtomM
set_option autoImplicit true
namespace Mathlib.Tactic.Abel
open Lean Elab Meta Tactic Qq
initialize registerTraceClass `abel
initialize registerTraceClass `abel.detail
structure Context where
α : Expr
univ :... | Mathlib/Tactic/Abel.lean | 148 | 152 | theorem term_add_termg {α} [AddCommGroup α] (n₁ x a₁ n₂ a₂ n' a')
(h₁ : n₁ + n₂ = n') (h₂ : a₁ + a₂ = a') :
@termg α _ n₁ x a₁ + @termg α _ n₂ x a₂ = termg n' x a' := by |
simp only [termg, h₁.symm, add_zsmul, h₂.symm]
exact add_add_add_comm (n₁ • x) a₁ (n₂ • x) a₂
| 2 | 7.389056 | 1 | 0.125 | 8 | 249 |
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.SesquilinearForm
import Mathlib.LinearAlgebra.Matrix.Symmetric
#align_import linear_algebra.quadratic_form.basic from "leanprover-community/mathlib"@"d11f435d4e34a6cea0a1797d6b625b0c170be845"
universe u v w
variable {S T : ... | Mathlib/LinearAlgebra/QuadraticForm/Basic.lean | 126 | 129 | theorem polar_comp {F : Type*} [CommRing S] [FunLike F R S] [AddMonoidHomClass F R S]
(f : M → R) (g : F) (x y : M) :
polar (g ∘ f) x y = g (polar f x y) := by |
simp only [polar, Pi.smul_apply, Function.comp_apply, map_sub]
| 1 | 2.718282 | 0 | 0.5 | 6 | 482 |
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Algebra.Group.ConjFinite
import Mathlib.Algebra.Group.Subgroup.Finite
import Mathlib.Data.Set.Card
import Mathlib.GroupTheory.Subgroup.Center
open MulAction ConjClasses
variable (G : Type*) [Group G]
| Mathlib/GroupTheory/ClassEquation.lean | 31 | 35 | theorem sum_conjClasses_card_eq_card [Fintype <| ConjClasses G] [Fintype G]
[∀ x : ConjClasses G, Fintype x.carrier] :
∑ x : ConjClasses G, x.carrier.toFinset.card = Fintype.card G := by |
suffices (Σ x : ConjClasses G, x.carrier) ≃ G by simpa using (Fintype.card_congr this)
simpa [carrier_eq_preimage_mk] using Equiv.sigmaFiberEquiv ConjClasses.mk
| 2 | 7.389056 | 1 | 1.75 | 4 | 1,858 |
import Mathlib.CategoryTheory.EqToHom
import Mathlib.CategoryTheory.Pi.Basic
import Mathlib.Data.ULift
#align_import category_theory.discrete_category from "leanprover-community/mathlib"@"369525b73f229ccd76a6ec0e0e0bf2be57599768"
namespace CategoryTheory
-- morphism levels before object levels. See note [Category... | Mathlib/CategoryTheory/DiscreteCategory.lean | 56 | 57 | theorem Discrete.mk_as {α : Type u₁} (X : Discrete α) : Discrete.mk X.as = X := by |
rfl
| 1 | 2.718282 | 0 | 0 | 2 | 70 |
import Mathlib.Algebra.Group.Hom.Defs
import Mathlib.Algebra.Group.Units
#align_import algebra.hom.units from "leanprover-community/mathlib"@"a07d750983b94c530ab69a726862c2ab6802b38c"
assert_not_exists MonoidWithZero
assert_not_exists DenselyOrdered
open Function
universe u v w
namespace Units
variable {α : Ty... | Mathlib/Algebra/Group/Units/Hom.lean | 94 | 94 | theorem map_id : map (MonoidHom.id M) = MonoidHom.id Mˣ := by | ext; rfl
| 1 | 2.718282 | 0 | 0 | 4 | 43 |
import Mathlib.Tactic.CategoryTheory.Coherence
import Mathlib.CategoryTheory.Bicategory.Coherence
namespace CategoryTheory
namespace Bicategory
open Category
open scoped Bicategory
open Mathlib.Tactic.BicategoryCoherence (bicategoricalComp bicategoricalIsoComp)
universe w v u
variable {B : Type u} [Bicategory... | Mathlib/CategoryTheory/Bicategory/Adjunction.lean | 136 | 149 | theorem comp_left_triangle_aux (adj₁ : f₁ ⊣ g₁) (adj₂ : f₂ ⊣ g₂) :
leftZigzag (compUnit adj₁ adj₂) (compCounit adj₁ adj₂) = (λ_ _).hom ≫ (ρ_ _).inv := by |
calc
_ = 𝟙 _ ⊗≫
adj₁.unit ▷ (f₁ ≫ f₂) ⊗≫
f₁ ◁ (adj₂.unit ▷ (g₁ ≫ f₁) ≫ (f₂ ≫ g₂) ◁ adj₁.counit) ▷ f₂ ⊗≫
(f₁ ≫ f₂) ◁ adj₂.counit ⊗≫ 𝟙 _ := by
simp [bicategoricalComp]; coherence
_ = 𝟙 _ ⊗≫
(leftZigzag adj₁.unit adj₁.counit) ▷ f₂ ⊗≫
f₁ ◁ (leftZ... | 12 | 162,754.791419 | 2 | 1.428571 | 7 | 1,519 |
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Localization.Ideal
import Mathlib.RingTheory.Noetherian
#align_import ring_theory.localization.submodule from "leanprover-community/mathlib"@"1ebb20602a8caef435ce47f6373e1aa40851a177"
variable {R : Type*} [CommRing R] (M : Submonoid R) ... | Mathlib/RingTheory/Localization/Submodule.lean | 75 | 78 | theorem coeSubmodule_span (s : Set R) :
coeSubmodule S (Ideal.span s) = Submodule.span R (algebraMap R S '' s) := by |
rw [IsLocalization.coeSubmodule, Ideal.span, Submodule.map_span]
rfl
| 2 | 7.389056 | 1 | 0.857143 | 7 | 753 |
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 | 117 | 117 | theorem lift_one : q.lift (1 : ℍ[R,c₁,c₂]) = 1 := by | simp [lift]
| 1 | 2.718282 | 0 | 0.4 | 10 | 394 |
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.Data.Finset.NoncommProd
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Order.SupIndep
#align_import group_theory.noncomm_pi_coprod from "leanprover-community/mathlib"@"6f9f36364eae3f42368b04858fd66d6d9ae730d8"
... | Mathlib/GroupTheory/NoncommPiCoprod.lean | 282 | 285 | theorem commute_subtype_of_commute (i j : ι) (hne : i ≠ j) :
∀ (x : H i) (y : H j), Commute ((H i).subtype x) ((H j).subtype y) := by |
rintro ⟨x, hx⟩ ⟨y, hy⟩
exact hcomm hne x y hx hy
| 2 | 7.389056 | 1 | 1.75 | 4 | 1,878 |
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
t... | Mathlib/Algebra/BigOperators/NatAntidiagonal.lean | 42 | 45 | theorem prod_antidiagonal_succ' {n : ℕ} {f : ℕ × ℕ → M} : (∏ p ∈ antidiagonal (n + 1), f p) =
f (n + 1, 0) * ∏ p ∈ antidiagonal n, f (p.1, p.2 + 1) := by |
rw [← prod_antidiagonal_swap, prod_antidiagonal_succ, ← prod_antidiagonal_swap]
rfl
| 2 | 7.389056 | 1 | 0.666667 | 3 | 586 |
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 | 120 | 121 | theorem invOf_mul_self_assoc [Monoid α] (a b : α) [Invertible a] : ⅟ a * (a * b) = b := by |
rw [← mul_assoc, invOf_mul_self, one_mul]
| 1 | 2.718282 | 0 | 0.1 | 10 | 245 |
import Mathlib.CategoryTheory.Opposites
#align_import category_theory.eq_to_hom from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
universe v₁ v₂ v₃ u₁ u₂ u₃
-- morphism levels before object levels. See note [CategoryTheory universes].
namespace CategoryTheory
open Opposite
variable ... | Mathlib/CategoryTheory/EqToHom.lean | 116 | 119 | theorem congrArg_mpr_hom_left {X Y Z : C} (p : X = Y) (q : Y ⟶ Z) :
(congrArg (fun W : C => W ⟶ Z) p).mpr q = eqToHom p ≫ q := by |
cases p
simp
| 2 | 7.389056 | 1 | 0.9 | 10 | 777 |
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Topology.MetricSpace.HausdorffDistance
import Mathlib.Topology.Sets.Compacts
#align_import topology.metric_space.closeds from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open scoped Classical
open Topo... | Mathlib/Topology/MetricSpace/Closeds.lean | 56 | 69 | theorem continuous_infEdist_hausdorffEdist :
Continuous fun p : α × Closeds α => infEdist p.1 p.2 := by |
refine continuous_of_le_add_edist 2 (by simp) ?_
rintro ⟨x, s⟩ ⟨y, t⟩
calc
infEdist x s ≤ infEdist x t + hausdorffEdist (t : Set α) s :=
infEdist_le_infEdist_add_hausdorffEdist
_ ≤ infEdist y t + edist x y + hausdorffEdist (t : Set α) s :=
(add_le_add_right infEdist_le_infEdist_add_edist _)
... | 12 | 162,754.791419 | 2 | 2 | 2 | 2,497 |
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Shift
import Mathlib.Analysis.Calculus.IteratedDeriv.Defs
variable
{𝕜 : Type*} [NontriviallyNormedField 𝕜]
{F : Type*} [NormedAddCommGroup F] [NormedSpace 𝕜 F]
{R : Type*} [Semi... | Mathlib/Analysis/Calculus/IteratedDeriv/Lemmas.lean | 102 | 104 | theorem iteratedDeriv_const_mul {n : ℕ} {f : 𝕜 → 𝕜} (h : ContDiff 𝕜 n f) (c : 𝕜) :
iteratedDeriv n (fun x => f (c * x)) = fun x => c ^ n * iteratedDeriv n f (c * x) := by |
simpa only [smul_eq_mul] using iteratedDeriv_const_smul h c
| 1 | 2.718282 | 0 | 1.2 | 10 | 1,290 |
import Mathlib.Data.Int.Bitwise
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.Symmetric
#align_import linear_algebra.matrix.zpow from "leanprover-community/mathlib"@"03fda9112aa6708947da13944a19310684bfdfcb"
open Matrix
namespace Matrix
variable {n' : Type*} [Decidab... | Mathlib/LinearAlgebra/Matrix/ZPow.lean | 44 | 47 | theorem inv_pow' (A : M) (n : ℕ) : A⁻¹ ^ n = (A ^ n)⁻¹ := by |
induction' n with n ih
· simp
· rw [pow_succ A, mul_inv_rev, ← ih, ← pow_succ']
| 3 | 20.085537 | 1 | 1.25 | 4 | 1,330 |
import Mathlib.MeasureTheory.Measure.MeasureSpace
open scoped ENNReal NNReal Topology
open Set MeasureTheory Measure Filter MeasurableSpace ENNReal Function
variable {R α β δ γ ι : Type*}
namespace MeasureTheory
variable {m0 : MeasurableSpace α} [MeasurableSpace β] [MeasurableSpace γ]
variable {μ μ₁ μ₂ μ₃ ν ν' ν... | Mathlib/MeasureTheory/Measure/Restrict.lean | 140 | 141 | theorem restrict_apply_univ (s : Set α) : μ.restrict s univ = μ s := by |
rw [restrict_apply MeasurableSet.univ, Set.univ_inter]
| 1 | 2.718282 | 0 | 0.666667 | 6 | 583 |
import Mathlib.Probability.ProbabilityMassFunction.Monad
#align_import probability.probability_mass_function.constructions from "leanprover-community/mathlib"@"4ac69b290818724c159de091daa3acd31da0ee6d"
universe u
namespace PMF
noncomputable section
variable {α β γ : Type*}
open scoped Classical
open NNReal ENN... | Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | 259 | 259 | theorem mem_support_normalize_iff (a : α) : a ∈ (normalize f hf0 hf).support ↔ f a ≠ 0 := by | simp
| 1 | 2.718282 | 0 | 0.2 | 10 | 279 |
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 | 576 | 578 | theorem toReal_sInf (s : Set ℝ≥0∞) (hf : ∀ r ∈ s, r ≠ ∞) :
(sInf s).toReal = sInf (ENNReal.toReal '' s) := by |
simp only [ENNReal.toReal, toNNReal_sInf s hf, NNReal.coe_sInf, Set.image_image]
| 1 | 2.718282 | 0 | 0.857143 | 21 | 755 |
import Mathlib.Algebra.ContinuedFractions.Computation.Basic
import Mathlib.Algebra.ContinuedFractions.Translations
#align_import algebra.continued_fractions.computation.translations from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b348ce40ad"
namespace GeneralizedContinuedFraction
open Generali... | Mathlib/Algebra/ContinuedFractions/Computation/Translations.lean | 170 | 171 | theorem of_h_eq_floor : (of v).h = ⌊v⌋ := by |
simp [of_h_eq_intFractPair_seq1_fst_b, IntFractPair.of]
| 1 | 2.718282 | 0 | 1.307692 | 13 | 1,366 |
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.Tactic.PPWithUniv
import Mathlib.Data.Set.Defs
#align_import category_theory.types from "leanprover-community/mathlib"@"48085f140e684306f9e7da907cd5932056d1aded"
namespace CategoryTheory
-- morphism levels be... | Mathlib/CategoryTheory/Types.lean | 170 | 172 | theorem eqToHom_map_comp_apply (p : X = Y) (q : Y = Z) (x : F.obj X) :
F.map (eqToHom q) (F.map (eqToHom p) x) = F.map (eqToHom <| p.trans q) x := by |
aesop_cat
| 1 | 2.718282 | 0 | 0.8 | 5 | 702 |
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.Data.Set.Basic
import Mathlib.Data.Setoid.Basic
import Mathlib.GroupTheory.Coset
#align_import group_theory.double_coset from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
-- Porting note: removed import
-- import Mathlib.Tac... | Mathlib/GroupTheory/DoubleCoset.lean | 44 | 45 | theorem mem_doset {s t : Set α} {a b : α} : b ∈ doset a s t ↔ ∃ x ∈ s, ∃ y ∈ t, b = x * a * y := by |
simp only [doset_eq_image2, Set.mem_image2, eq_comm]
| 1 | 2.718282 | 0 | 1.428571 | 7 | 1,515 |
import Mathlib.Algebra.Module.Defs
import Mathlib.Data.Fintype.BigOperators
import Mathlib.GroupTheory.GroupAction.BigOperators
#align_import algebra.module.big_operators from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
variable {ι κ α β R M : Type*}
section AddCommMonoid
variable [... | Mathlib/Algebra/Module/BigOperators.lean | 41 | 45 | theorem Finset.sum_smul_sum {f : α → R} {g : β → M} {s : Finset α} {t : Finset β} :
((∑ i ∈ s, f i) • ∑ i ∈ t, g i) = ∑ p ∈ s ×ˢ t, f p.fst • g p.snd := by |
rw [Finset.sum_product, Finset.sum_smul, Finset.sum_congr rfl]
intros
rw [Finset.smul_sum]
| 3 | 20.085537 | 1 | 0.666667 | 3 | 600 |
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 | 83 | 87 | theorem vonMangoldt_nonneg {n : ℕ} : 0 ≤ Λ n := by |
rw [vonMangoldt_apply]
split_ifs
· exact Real.log_nonneg (one_le_cast.2 (Nat.minFac_pos n))
rfl
| 4 | 54.59815 | 2 | 0.636364 | 11 | 552 |
import Mathlib.Order.Interval.Set.OrdConnected
import Mathlib.Order.Antisymmetrization
#align_import order.cover from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
open Set OrderDual
variable {α β : Type*}
section WeaklyCovers
section LT
variable [LT α] {a b : α}
def CovBy (a b :... | Mathlib/Order/Cover.lean | 233 | 234 | theorem not_covBy_iff (h : a < b) : ¬a ⋖ b ↔ ∃ c, a < c ∧ c < b := by |
simp_rw [CovBy, h, true_and_iff, not_forall, exists_prop, not_not]
| 1 | 2.718282 | 0 | 0.666667 | 3 | 588 |
import Mathlib.CategoryTheory.Preadditive.Yoneda.Basic
import Mathlib.CategoryTheory.Preadditive.Injective
import Mathlib.Algebra.Category.GroupCat.EpiMono
import Mathlib.Algebra.Category.ModuleCat.EpiMono
#align_import category_theory.preadditive.yoneda.injective from "leanprover-community/mathlib"@"f8d8465c3c392a93... | Mathlib/CategoryTheory/Preadditive/Yoneda/Injective.lean | 32 | 40 | theorem injective_iff_preservesEpimorphisms_preadditiveYoneda_obj (J : C) :
Injective J ↔ (preadditiveYoneda.obj J).PreservesEpimorphisms := by |
rw [injective_iff_preservesEpimorphisms_yoneda_obj]
refine
⟨fun h : (preadditiveYoneda.obj J ⋙ (forget AddCommGroupCat)).PreservesEpimorphisms => ?_, ?_⟩
· exact
Functor.preservesEpimorphisms_of_preserves_of_reflects (preadditiveYoneda.obj J) (forget _)
· intro
exact (inferInstance : (preadditive... | 7 | 1,096.633158 | 2 | 2 | 2 | 2,336 |
import Mathlib.Data.Complex.Basic
import Mathlib.MeasureTheory.Integral.CircleIntegral
#align_import measure_theory.integral.circle_transform from "leanprover-community/mathlib"@"d11893b411025250c8e61ff2f12ccbd7ee35ab15"
open Set MeasureTheory Metric Filter Function
open scoped Interval Real
noncomputable secti... | Mathlib/MeasureTheory/Integral/CircleTransform.lean | 48 | 55 | theorem circleTransformDeriv_periodic (f : ℂ → E) :
Periodic (circleTransformDeriv R z w f) (2 * π) := by |
have := periodic_circleMap
simp_rw [Periodic] at *
intro x
simp_rw [circleTransformDeriv, this]
congr 2
simp [this]
| 6 | 403.428793 | 2 | 1.777778 | 9 | 1,880 |
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 | 79 | 80 | theorem one_sub_gold : 1 - ψ = φ := by |
linarith [gold_add_goldConj]
| 1 | 2.718282 | 0 | 0.894737 | 19 | 776 |
import Mathlib.CategoryTheory.SingleObj
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Pi.Basic
import Mathlib.CategoryTheory.Limits.IsLimit
#align_import category_theory.category.Groupoid from "leanprover-community/mathlib"@"c9c9fa15fec7ca18e9ec97306fb8764bfe988a7e"
universe v... | Mathlib/CategoryTheory/Category/Grpd.lean | 152 | 155 | theorem piIsoPi_hom_π (J : Type u) (f : J → Grpd.{u, u}) (j : J) :
(piIsoPi J f).hom ≫ Limits.Pi.π f j = CategoryTheory.Pi.eval _ j := by |
simp [piIsoPi]
rfl
| 2 | 7.389056 | 1 | 1 | 1 | 821 |
import Mathlib.Algebra.Order.Monoid.Unbundled.Pow
import Mathlib.Data.Finset.Fold
import Mathlib.Data.Finset.Option
import Mathlib.Data.Finset.Pi
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Multiset.Lattice
import Mathlib.Data.Set.Lattice
import Mathlib.Order.Hom.Lattice
import Mathlib.Order.Nat
#align_import... | Mathlib/Data/Finset/Lattice.lean | 140 | 143 | theorem sup_bot (s : Finset β) : (s.sup fun _ => ⊥) = (⊥ : α) := by |
obtain rfl | hs := s.eq_empty_or_nonempty
· exact sup_empty
· exact sup_const hs _
| 3 | 20.085537 | 1 | 1.333333 | 3 | 1,459 |
import Mathlib.MeasureTheory.MeasurableSpace.Basic
import Mathlib.MeasureTheory.Measure.MeasureSpaceDef
#align_import measure_theory.function.ae_measurable_sequence from "leanprover-community/mathlib"@"d003c55042c3cd08aefd1ae9a42ef89441cdaaf3"
open MeasureTheory
open scoped Classical
variable {ι : Sort*} {α β γ... | Mathlib/MeasureTheory/Function/AEMeasurableSequence.lean | 50 | 56 | theorem mk_eq_fun_of_mem_aeSeqSet (hf : ∀ i, AEMeasurable (f i) μ) {x : α} (hx : x ∈ aeSeqSet hf p)
(i : ι) : (hf i).mk (f i) x = f i x :=
haveI h_ss : aeSeqSet hf p ⊆ { x | ∀ i, f i x = (hf i).mk (f i) x } := by |
rw [aeSeqSet, ← compl_compl { x | ∀ i, f i x = (hf i).mk (f i) x }, Set.compl_subset_compl]
refine Set.Subset.trans (Set.compl_subset_compl.mpr fun x h => ?_) (subset_toMeasurable _ _)
exact h.1
(h_ss hx i).symm
| 4 | 54.59815 | 2 | 1.333333 | 6 | 1,404 |
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 | 117 | 118 | theorem hammingDist_pos {x y : ∀ i, β i} : 0 < hammingDist x y ↔ x ≠ y := by |
rw [← hammingDist_ne_zero, iff_not_comm, not_lt, Nat.le_zero]
| 1 | 2.718282 | 0 | 0.7 | 10 | 642 |
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.Algebra.Order.Group.OrderIso
import Mathlib.Data.Set.Pointwise.SMul
import Mathlib.Order.UpperLower.Basic
#align_import algebra.order.upper_lower from "leanprover-community/mathlib"@"c0c52abb75074ed8b73a948341f50521fbf43b4c"
open Function Set
open Pointw... | Mathlib/Algebra/Order/UpperLower.lean | 63 | 65 | theorem IsUpperSet.mul_left (ht : IsUpperSet t) : IsUpperSet (s * t) := by |
rw [← smul_eq_mul, ← Set.iUnion_smul_set]
exact isUpperSet_iUnion₂ fun x _ ↦ ht.smul
| 2 | 7.389056 | 1 | 1 | 5 | 961 |
import Mathlib.Algebra.Group.ConjFinite
import Mathlib.GroupTheory.Abelianization
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.GroupAction.Quotient
import Mathlib.GroupTheory.Index
import Mathlib.GroupTheory.SpecificGroups.Dihedral
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Line... | Mathlib/GroupTheory/CommutingProbability.lean | 98 | 102 | theorem commProb_def' : commProb G = Nat.card (ConjClasses G) / Nat.card G := by |
rw [commProb, card_comm_eq_card_conjClasses_mul_card, Nat.cast_mul, sq]
by_cases h : (Nat.card G : ℚ) = 0
· rw [h, zero_mul, div_zero, div_zero]
· exact mul_div_mul_right _ _ h
| 4 | 54.59815 | 2 | 1.625 | 8 | 1,749 |
import Mathlib.Topology.UniformSpace.CompactConvergence
import Mathlib.Topology.UniformSpace.Equicontinuity
import Mathlib.Topology.UniformSpace.Equiv
open Set Filter Uniformity Topology Function UniformConvergence
variable {ι X Y α β : Type*} [TopologicalSpace X] [UniformSpace α] [UniformSpace β]
variable {F : ι ... | Mathlib/Topology/UniformSpace/Ascoli.lean | 85 | 125 | theorem Equicontinuous.comap_uniformFun_eq [CompactSpace X] (F_eqcont : Equicontinuous F) :
(UniformFun.uniformSpace X α).comap F =
(Pi.uniformSpace _).comap F := by |
-- The `≤` inequality is trivial
refine le_antisymm (UniformSpace.comap_mono UniformFun.uniformContinuous_toFun) ?_
-- A bit of rewriting to get a nice intermediate statement.
change comap _ _ ≤ comap _ _
simp_rw [Pi.uniformity, Filter.comap_iInf, comap_comap, Function.comp]
refine ((UniformFun.hasBasis_un... | 38 | 31,855,931,757,113,756 | 2 | 2 | 2 | 2,106 |
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.Algebra.MonoidAlgebra.Support
import Mathlib.Algebra.DirectSum.Internal
import Mathlib.RingTheory.GradedAlgebra.Basic
#align_import algebra.monoid_algebra.grading from "leanprover-community/mathlib"@"feb99064803fd3108e37c18b0f77d0a8344677a3"
noncomputable sectio... | Mathlib/Algebra/MonoidAlgebra/Grading.lean | 63 | 64 | theorem mem_gradeBy_iff (f : M → ι) (i : ι) (a : R[M]) :
a ∈ gradeBy R f i ↔ (a.support : Set M) ⊆ f ⁻¹' {i} := by | rfl
| 1 | 2.718282 | 0 | 1.2 | 5 | 1,256 |
import Mathlib.RingTheory.FractionalIdeal.Basic
import Mathlib.RingTheory.Ideal.Norm
namespace FractionalIdeal
open scoped Pointwise nonZeroDivisors
variable {R : Type*} [CommRing R] [IsDedekindDomain R] [Module.Free ℤ R] [Module.Finite ℤ R]
variable {K : Type*} [CommRing K] [Algebra R K] [IsFractionRing R K]
th... | Mathlib/RingTheory/FractionalIdeal/Norm.lean | 84 | 84 | theorem absNorm_nonneg (I : FractionalIdeal R⁰ K) : 0 ≤ absNorm I := by | dsimp [absNorm]; positivity
| 1 | 2.718282 | 0 | 1.142857 | 7 | 1,212 |
import Mathlib.Analysis.LocallyConvex.Bounded
import Mathlib.Topology.Algebra.Module.StrongTopology
#align_import analysis.normed_space.compact_operator from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open Function Set Filter Bornology Metric Pointwise Topology
def IsCompactOperat... | Mathlib/Analysis/NormedSpace/CompactOperator.lean | 260 | 265 | theorem IsCompactOperator.continuous_comp {f : M₁ → M₂} (hf : IsCompactOperator f) {g : M₂ → M₃}
(hg : Continuous g) : IsCompactOperator (g ∘ f) := by |
rcases hf with ⟨K, hK, hKf⟩
refine ⟨g '' K, hK.image hg, mem_of_superset hKf ?_⟩
rw [preimage_comp]
exact preimage_mono (subset_preimage_image _ _)
| 4 | 54.59815 | 2 | 1.6 | 5 | 1,713 |
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Order.Interval.Set.OrdConnected
#align_import order.complete_lattice_intervals from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
open scoped Classical
open Set
variable {ι : ... | Mathlib/Order/CompleteLatticeIntervals.lean | 102 | 104 | theorem subset_sInf_emptyset [Inhabited s] :
sInf (∅ : Set s) = default := by |
simp [sInf]
| 1 | 2.718282 | 0 | 0.125 | 8 | 250 |
import Mathlib.Topology.Defs.Sequences
import Mathlib.Topology.UniformSpace.Cauchy
#align_import topology.sequences from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Function Filter TopologicalSpace Bornology
open scoped Topology Uniformity
variable {X Y : Type*}
section ... | Mathlib/Topology/Sequences.lean | 113 | 116 | theorem mem_closure_iff_seq_limit [FrechetUrysohnSpace X] {s : Set X} {a : X} :
a ∈ closure s ↔ ∃ x : ℕ → X, (∀ n : ℕ, x n ∈ s) ∧ Tendsto x atTop (𝓝 a) := by |
rw [← seqClosure_eq_closure]
rfl
| 2 | 7.389056 | 1 | 1.666667 | 3 | 1,823 |
import Mathlib.LinearAlgebra.Basis
import Mathlib.LinearAlgebra.Dual
import Mathlib.Data.Fin.FlagRange
open Set Submodule
namespace Basis
section Semiring
variable {R M : Type*} [Semiring R] [AddCommMonoid M] [Module R M] {n : ℕ}
def flag (b : Basis (Fin n) R M) (k : Fin (n + 1)) : Submodule R M :=
.span R <... | Mathlib/LinearAlgebra/Basis/Flag.lean | 32 | 32 | theorem flag_zero (b : Basis (Fin n) R M) : b.flag 0 = ⊥ := by | simp [flag]
| 1 | 2.718282 | 0 | 0.333333 | 3 | 335 |
import Mathlib.Data.Set.Lattice
import Mathlib.Data.Set.Pairwise.Basic
#align_import data.set.pairwise.lattice from "leanprover-community/mathlib"@"c4c2ed622f43768eff32608d4a0f8a6cec1c047d"
open Function Set Order
variable {α β γ ι ι' : Type*} {κ : Sort*} {r p q : α → α → Prop}
section Pairwise
variable {f g : ... | Mathlib/Data/Set/Pairwise/Lattice.lean | 147 | 153 | theorem Set.PairwiseDisjoint.subset_of_biUnion_subset_biUnion (h₀ : (s ∪ t).PairwiseDisjoint f)
(h₁ : ∀ i ∈ s, (f i).Nonempty) (h : ⋃ i ∈ s, f i ⊆ ⋃ i ∈ t, f i) : s ⊆ t := by |
rintro i hi
obtain ⟨a, hai⟩ := h₁ i hi
obtain ⟨j, hj, haj⟩ := mem_iUnion₂.1 (h <| mem_iUnion₂_of_mem hi hai)
rwa [h₀.eq (subset_union_left hi) (subset_union_right hj)
(not_disjoint_iff.2 ⟨a, hai, haj⟩)]
| 5 | 148.413159 | 2 | 1.666667 | 6 | 1,759 |
import Mathlib.Init.Function
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
universe u
open Function
namespace Option
variable {α β γ δ : Type*} {f : α → β → γ} {a : Option α} {b : Option β} {c : Option γ}
def map₂ (f : α → β → γ) (a : Option α) ... | Mathlib/Data/Option/NAry.lean | 134 | 137 | theorem map₂_left_comm {f : α → δ → ε} {g : β → γ → δ} {f' : α → γ → δ'} {g' : β → δ' → ε}
(h_left_comm : ∀ a b c, f a (g b c) = g' b (f' a c)) :
map₂ f a (map₂ g b c) = map₂ g' b (map₂ f' a c) := by |
cases a <;> cases b <;> cases c <;> simp [h_left_comm]
| 1 | 2.718282 | 0 | 0 | 14 | 191 |
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Tree.Basic
import Mathlib.Logic.Basic
import Mathlib.Tactic.NormNum.Core
import Mathlib.Util.SynthesizeUsing
import Mathlib.Util.Qq
open Lean Parser Tactic Mathlib Meta NormNum Qq
initialize registerTraceClass `CancelDen... | Mathlib/Tactic/CancelDenoms/Core.lean | 66 | 68 | theorem pow_subst {α} [CommRing α] {n e1 t1 k l : α} {e2 : ℕ}
(h1 : n * e1 = t1) (h2 : l * n ^ e2 = k) : k * (e1 ^ e2) = l * t1 ^ e2 := by |
rw [← h2, ← h1, mul_pow, mul_assoc]
| 1 | 2.718282 | 0 | 0.545455 | 11 | 512 |
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.Analysis.MeanInequalities
import Mathlib.Analysis.MeanInequalitiesPow
import Mathlib.MeasureTheory.Function.SpecialFunctions.Basic
#align_import measure_theory.integral.mean_inequalities from "leanprover-community/mathlib"@"13bf7613c96a9fd66a81b9020a82cad9... | Mathlib/MeasureTheory/Integral/MeanInequalities.lean | 87 | 90 | theorem fun_eq_funMulInvSnorm_mul_snorm {p : ℝ} (f : α → ℝ≥0∞) (hf_nonzero : (∫⁻ a, f a ^ p ∂μ) ≠ 0)
(hf_top : (∫⁻ a, f a ^ p ∂μ) ≠ ⊤) {a : α} :
f a = funMulInvSnorm f p μ a * (∫⁻ c, f c ^ p ∂μ) ^ (1 / p) := by |
simp [funMulInvSnorm, mul_assoc, ENNReal.inv_mul_cancel, hf_nonzero, hf_top]
| 1 | 2.718282 | 0 | 1.4 | 5 | 1,491 |
import Mathlib.CategoryTheory.Functor.ReflectsIso
import Mathlib.CategoryTheory.MorphismProperty.Basic
universe w v v' u u'
namespace CategoryTheory
namespace MorphismProperty
variable {C : Type u} [Category.{v} C] {D : Type u'} [Category.{v'} D]
def IsInvertedBy (P : MorphismProperty C) (F : C ⥤ D) : Prop :=
... | Mathlib/CategoryTheory/MorphismProperty/IsInvertedBy.lean | 128 | 131 | theorem IsInvertedBy.iff_of_iso (W : MorphismProperty C) {F₁ F₂ : C ⥤ D} (e : F₁ ≅ F₂) :
W.IsInvertedBy F₁ ↔ W.IsInvertedBy F₂ := by |
dsimp [IsInvertedBy]
simp only [NatIso.isIso_map_iff e]
| 2 | 7.389056 | 1 | 1 | 1 | 1,118 |
import Mathlib.Data.PFunctor.Univariate.Basic
#align_import data.pfunctor.univariate.M from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
universe u v w
open Nat Function
open List
variable (F : PFunctor.{u})
-- Porting note: the ♯ tactic is never used
-- local prefix:0 "♯" => cast... | Mathlib/Data/PFunctor/Univariate/M.lean | 66 | 67 | theorem approx_eta {n : ℕ} (x : CofixA F (n + 1)) : x = CofixA.intro (head' x) (children' x) := by |
cases x; rfl
| 1 | 2.718282 | 0 | 1.166667 | 6 | 1,239 |
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.Algebra.Module.Torsion
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v v' u₁' w w'
variable {R S : Type u} {M : Type v} {M' : Type v'} {M₁ : Type v}... | Mathlib/LinearAlgebra/Dimension/Constructions.lean | 544 | 547 | theorem subalgebra_top_finrank_eq_submodule_top_finrank :
finrank F (⊤ : Subalgebra F E) = finrank F (⊤ : Submodule F E) := by |
rw [← Algebra.top_toSubmodule]
rfl
| 2 | 7.389056 | 1 | 0.75 | 24 | 667 |
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Choose.Cast
import Mathlib.Data.Nat.Choose.Vandermonde
import Mathlib.Tactic.FieldSimp
#align_import data.polynomial.hasse_deriv from "leanprover-community/mathlib"@"a148d797a1094ab554ad4183a4ad6f130358... | Mathlib/Algebra/Polynomial/HasseDeriv.lean | 133 | 134 | theorem hasseDeriv_apply_one (hk : 0 < k) : hasseDeriv k (1 : R[X]) = 0 := by |
rw [← C_1, hasseDeriv_C k _ hk]
| 1 | 2.718282 | 0 | 1.2 | 10 | 1,278 |
import Mathlib.Data.Vector.Basic
import Mathlib.Data.Vector.Snoc
set_option autoImplicit true
namespace Vector
section Fold
section Unary
variable (xs : Vector α n) (f₁ : β → σ₁ → σ₁ × γ) (f₂ : α → σ₂ → σ₂ × β)
@[simp]
theorem mapAccumr_mapAccumr :
mapAccumr f₁ (mapAccumr f₂ xs s₂).snd s₁
= let m := (... | Mathlib/Data/Vector/MapLemmas.lean | 38 | 40 | theorem mapAccumr_map (f₂ : α → β) :
(mapAccumr f₁ (map f₂ xs) s) = (mapAccumr (fun x s => f₁ (f₂ x) s) xs s) := by |
induction xs using Vector.revInductionOn generalizing s <;> simp_all
| 1 | 2.718282 | 0 | 0.333333 | 24 | 337 |
import Mathlib.Data.Option.NAry
import Mathlib.Data.Seq.Computation
#align_import data.seq.seq from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b348ce40ad"
namespace Stream'
universe u v w
def IsSeq {α : Type u} (s : Stream' (Option α)) : Prop :=
∀ {n : ℕ}, s n = none → s (n + 1) = none
#al... | Mathlib/Data/Seq/Seq.lean | 174 | 178 | theorem ge_stable (s : Seq α) {aₙ : α} {n m : ℕ} (m_le_n : m ≤ n)
(s_nth_eq_some : s.get? n = some aₙ) : ∃ aₘ : α, s.get? m = some aₘ :=
have : s.get? n ≠ none := by | simp [s_nth_eq_some]
have : s.get? m ≠ none := mt (s.le_stable m_le_n) this
Option.ne_none_iff_exists'.mp this
| 3 | 20.085537 | 1 | 0.666667 | 3 | 601 |
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.Sym.Sym2
namespace List
variable {α : Type*}
section Sym2
protected def sym2 : List α → List (Sym2 α)
| [] => []
| x :: xs => (x :: xs).map (fun y => s(x, y)) ++ xs.sym2
| Mathlib/Data/List/Sym.lean | 40 | 43 | theorem mem_sym2_cons_iff {x : α} {xs : List α} {z : Sym2 α} :
z ∈ (x :: xs).sym2 ↔ z = s(x, x) ∨ (∃ y, y ∈ xs ∧ z = s(x, y)) ∨ z ∈ xs.sym2 := by |
simp only [List.sym2, map_cons, cons_append, mem_cons, mem_append, mem_map]
simp only [eq_comm]
| 2 | 7.389056 | 1 | 1.444444 | 9 | 1,529 |
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
#align_import linear_algebra.affine_space.midpoint from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
open AffineMap AffineEquiv
section
variable (R : Type*) {V V' P P' : Type*} [Ring R] [Invertible (2 : R)] [AddCommGroup V]
[Modu... | Mathlib/LinearAlgebra/AffineSpace/Midpoint.lean | 129 | 130 | theorem right_vsub_midpoint (p₁ p₂ : P) : p₂ -ᵥ midpoint R p₁ p₂ = (⅟ 2 : R) • (p₂ -ᵥ p₁) := by |
rw [midpoint_comm, left_vsub_midpoint]
| 1 | 2.718282 | 0 | 0.444444 | 9 | 412 |
import Mathlib.Geometry.Euclidean.Sphere.Basic
#align_import geometry.euclidean.sphere.second_inter from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open RealInnerProductSpace
namespace EuclideanGeometry
variable {V : Type*} {P : Type*} [NormedAddCommGroup V]... | Mathlib/Geometry/Euclidean/Sphere/SecondInter.lean | 44 | 49 | theorem Sphere.secondInter_dist (s : Sphere P) (p : P) (v : V) :
dist (s.secondInter p v) s.center = dist p s.center := by |
rw [Sphere.secondInter]
by_cases hv : v = 0; · simp [hv]
rw [dist_smul_vadd_eq_dist _ _ hv]
exact Or.inr rfl
| 4 | 54.59815 | 2 | 1.25 | 8 | 1,314 |
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Polynomial.Degree.Lemmas
#align_import data.polynomial.erase_lead from "leanprover-community/mathlib"@"fa256f00ce018e7b40e1dc756e403c86680bf448"
noncomputable section
open Polynomial
open Polynomial Finset
namespace Polynomial
variable {R : Type*}... | Mathlib/Algebra/Polynomial/EraseLead.lean | 60 | 60 | theorem eraseLead_zero : eraseLead (0 : R[X]) = 0 := by | simp only [eraseLead, erase_zero]
| 1 | 2.718282 | 0 | 0.5 | 14 | 465 |
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 | 276 | 280 | theorem descPochhammer_map (f : R →+* T) (n : ℕ) :
(descPochhammer R n).map f = descPochhammer T n := by |
induction' n with n ih
· simp
· simp [ih, descPochhammer_succ_left, map_comp]
| 3 | 20.085537 | 1 | 0.96 | 25 | 796 |
import Mathlib.Algebra.MvPolynomial.Supported
import Mathlib.RingTheory.Derivation.Basic
#align_import data.mv_polynomial.derivation from "leanprover-community/mathlib"@"b608348ffaeb7f557f2fd46876037abafd326ff3"
namespace MvPolynomial
noncomputable section
variable {σ R A : Type*} [CommSemiring R] [AddCommMonoi... | Mathlib/Algebra/MvPolynomial/Derivation.lean | 96 | 114 | theorem leibniz_iff_X (D : MvPolynomial σ R →ₗ[R] A) (h₁ : D 1 = 0) :
(∀ p q, D (p * q) = p • D q + q • D p) ↔ ∀ s i, D (monomial s 1 * X i) =
(monomial s 1 : MvPolynomial σ R) • D (X i) + (X i : MvPolynomial σ R) • D (monomial s 1) := by |
refine ⟨fun H p i => H _ _, fun H => ?_⟩
have hC : ∀ r, D (C r) = 0 := by intro r; rw [C_eq_smul_one, D.map_smul, h₁, smul_zero]
have : ∀ p i, D (p * X i) = p • D (X i) + (X i : MvPolynomial σ R) • D p := by
intro p i
induction' p using MvPolynomial.induction_on' with s r p q hp hq
· rw [← mul_one r,... | 16 | 8,886,110.520508 | 2 | 1.5 | 2 | 1,575 |
import Mathlib.RingTheory.PowerSeries.Trunc
import Mathlib.RingTheory.PowerSeries.Inverse
import Mathlib.RingTheory.Derivation.Basic
namespace PowerSeries
open Polynomial Derivation Nat
section CommutativeSemiring
variable {R} [CommSemiring R]
noncomputable def derivativeFun (f : R⟦X⟧) : R⟦X⟧ := mk fun n ↦ coef... | Mathlib/RingTheory/PowerSeries/Derivative.lean | 87 | 88 | theorem derivativeFun_one : derivativeFun (1 : R⟦X⟧) = 0 := by |
rw [← map_one (C R), derivativeFun_C (1 : R)]
| 1 | 2.718282 | 0 | 1.111111 | 9 | 1,200 |
import Mathlib.Analysis.Calculus.LineDeriv.Basic
import Mathlib.MeasureTheory.Integral.IntegralEqImproper
open MeasureTheory Measure FiniteDimensional
variable {E F G W : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F]
[NormedSpace ℝ F] [NormedAddCommGroup G] [NormedSpace ℝ G] [NormedAddCo... | Mathlib/Analysis/Calculus/LineDeriv/IntegrationByParts.lean | 101 | 151 | theorem integral_bilinear_hasLineDerivAt_right_eq_neg_left_of_integrable
{f f' : E → F} {g g' : E → G} {v : E} {B : F →L[ℝ] G →L[ℝ] W}
(hf'g : Integrable (fun x ↦ B (f' x) (g x)) μ) (hfg' : Integrable (fun x ↦ B (f x) (g' x)) μ)
(hfg : Integrable (fun x ↦ B (f x) (g x)) μ)
(hf : ∀ x, HasLineDerivAt ℝ f ... |
by_cases hW : CompleteSpace W; swap
· simp [integral, hW]
rcases eq_or_ne v 0 with rfl|hv
· have Hf' x : f' x = 0 := by
simpa [(hasLineDerivAt_zero (f := f) (x := x)).lineDeriv] using (hf x).lineDeriv.symm
have Hg' x : g' x = 0 := by
simpa [(hasLineDerivAt_zero (f := g) (x := x)).lineDeriv] usi... | 45 | 34,934,271,057,485,095,000 | 2 | 2 | 1 | 2,175 |
import Mathlib.Order.Bounds.Basic
import Mathlib.Order.Hom.Set
#align_import order.bounds.order_iso from "leanprover-community/mathlib"@"a59dad53320b73ef180174aae867addd707ef00e"
set_option autoImplicit true
open Set
namespace OrderIso
variable [Preorder α] [Preorder β] (f : α ≃o β)
theorem upperBounds_image {... | Mathlib/Order/Bounds/OrderIso.lean | 55 | 56 | theorem isLUB_preimage {s : Set β} {x : α} : IsLUB (f ⁻¹' s) x ↔ IsLUB s (f x) := by |
rw [← f.symm_symm, ← image_eq_preimage, isLUB_image]
| 1 | 2.718282 | 0 | 0 | 3 | 77 |
import Mathlib.Algebra.Algebra.Equiv
import Mathlib.LinearAlgebra.Span
#align_import algebra.algebra.tower from "leanprover-community/mathlib"@"71150516f28d9826c7341f8815b31f7d8770c212"
open Pointwise
universe u v w u₁ v₁
variable (R : Type u) (S : Type v) (A : Type w) (B : Type u₁) (M : Type v₁)
namespace IsS... | Mathlib/Algebra/Algebra/Tower.lean | 162 | 164 | theorem _root_.AlgHom.map_algebraMap (f : A →ₐ[S] B) (r : R) :
f (algebraMap R A r) = algebraMap R B r := by |
rw [algebraMap_apply R S A r, f.commutes, ← algebraMap_apply R S B]
| 1 | 2.718282 | 0 | 0 | 4 | 181 |
import Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear
import Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm
import Mathlib.Analysis.NormedSpace.Span
suppress_compilation
open Bornology
open Filter hiding map_smul
open scoped Classical NNReal Topology Uniformity
-- the `ₗ` subscript variables are for special... | Mathlib/Analysis/NormedSpace/OperatorNorm/NormedSpace.lean | 140 | 146 | theorem homothety_norm [RingHomIsometric σ₁₂] [Nontrivial E] (f : E →SL[σ₁₂] F) {a : ℝ}
(hf : ∀ x, ‖f x‖ = a * ‖x‖) : ‖f‖ = a := by |
obtain ⟨x, hx⟩ : ∃ x : E, x ≠ 0 := exists_ne 0
rw [← norm_pos_iff] at hx
have ha : 0 ≤ a := by simpa only [hf, hx, mul_nonneg_iff_of_pos_right] using norm_nonneg (f x)
apply le_antisymm (f.opNorm_le_bound ha fun y => le_of_eq (hf y))
simpa only [hf, hx, mul_le_mul_right] using f.le_opNorm x
| 5 | 148.413159 | 2 | 1.666667 | 6 | 1,809 |
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.MinMax
#align_import data.set.pointwise.interval from "leanprover-community/mathlib"@"2196ab363eb097c... | Mathlib/Data/Set/Pointwise/Interval.lean | 634 | 635 | theorem preimage_mul_const_Icc (a b : α) {c : α} (h : 0 < c) :
(fun x => x * c) ⁻¹' Icc a b = Icc (a / c) (b / c) := by | simp [← Ici_inter_Iic, h]
| 1 | 2.718282 | 0 | 0.37931 | 29 | 381 |
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.GroupAction.Units
import Mathlib.Logic.Basic
import Mathlib.Tactic.Ring
#align_import ring_theory.coprime.basic from "leanprover-community/mathlib"@"a95b16cb... | Mathlib/RingTheory/Coprime/Basic.lean | 84 | 86 | theorem IsCoprime.ne_zero [Nontrivial R] {p : Fin 2 → R} (h : IsCoprime (p 0) (p 1)) : p ≠ 0 := by |
rintro rfl
exact not_isCoprime_zero_zero h
| 2 | 7.389056 | 1 | 1.142857 | 7 | 1,219 |
import Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity
#align_import number_theory.legendre_symbol.jacobi_symbol from "leanprover-community/mathlib"@"74a27133cf29446a0983779e37c8f829a85368f3"
section Jacobi
open Nat ZMod
-- Since we need the fact that the factors are prime, we use `List.pmap`.
def ... | Mathlib/NumberTheory/LegendreSymbol/JacobiSymbol.lean | 110 | 111 | theorem one_right (a : ℤ) : J(a | 1) = 1 := by |
simp only [jacobiSym, factors_one, List.prod_nil, List.pmap]
| 1 | 2.718282 | 0 | 0.833333 | 6 | 730 |
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv
import Mathlib.Analysis.SpecialFunctions.Log.Basic
#align_import analysis.special_functions.arsinh from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open Function Filter Set
open scoped Topology
name... | Mathlib/Analysis/SpecialFunctions/Arsinh.lean | 168 | 168 | theorem arsinh_nonpos_iff : arsinh x ≤ 0 ↔ x ≤ 0 := by | rw [← sinh_le_sinh, sinh_zero, sinh_arsinh]
| 1 | 2.718282 | 0 | 0.625 | 8 | 547 |
import Mathlib.Topology.Algebra.Constructions
import Mathlib.Topology.Bases
import Mathlib.Topology.UniformSpace.Basic
#align_import topology.uniform_space.cauchy from "leanprover-community/mathlib"@"22131150f88a2d125713ffa0f4693e3355b1eb49"
universe u v
open scoped Classical
open Filter TopologicalSpace Set Uni... | Mathlib/Topology/UniformSpace/Cauchy.lean | 70 | 72 | theorem cauchy_map_iff {l : Filter β} {f : β → α} :
Cauchy (l.map f) ↔ NeBot l ∧ Tendsto (fun p : β × β => (f p.1, f p.2)) (l ×ˢ l) (𝓤 α) := by |
rw [Cauchy, map_neBot_iff, prod_map_map_eq, Tendsto]
| 1 | 2.718282 | 0 | 0.5 | 2 | 430 |
import Mathlib.Algebra.QuadraticDiscriminant
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
#align_import analysis.special_functions.trigonometric.complex from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
noncomputable secti... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Complex.lean | 69 | 72 | theorem tan_eq_zero_iff {θ : ℂ} : tan θ = 0 ↔ ∃ k : ℤ, k * π / 2 = θ := by |
rw [tan, div_eq_zero_iff, ← mul_eq_zero, ← mul_right_inj' two_ne_zero, mul_zero,
← mul_assoc, ← sin_two_mul, sin_eq_zero_iff]
field_simp [mul_comm, eq_comm]
| 3 | 20.085537 | 1 | 0.909091 | 11 | 787 |
import Mathlib.CategoryTheory.Functor.Hom
import Mathlib.CategoryTheory.Products.Basic
import Mathlib.Data.ULift
#align_import category_theory.yoneda from "leanprover-community/mathlib"@"369525b73f229ccd76a6ec0e0e0bf2be57599768"
namespace CategoryTheory
open Opposite
universe v₁ u₁ u₂
-- morphism levels before ... | Mathlib/CategoryTheory/Yoneda.lean | 221 | 224 | theorem reprW_app_hom (X : Cᵒᵖ) (f : unop X ⟶ F.reprX) :
(F.reprW.app X).hom f = F.map f.op F.reprx := by |
simp only [yoneda_obj_obj, Iso.app_hom, op_unop, reprx, ← FunctorToTypes.naturality,
yoneda_obj_map, unop_op, Quiver.Hom.unop_op, Category.comp_id]
| 2 | 7.389056 | 1 | 1 | 3 | 825 |
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Algebra.Order.Monoid.WithTop
#align_import data.nat.with_bot from "leanprover-community/mathlib"@"966e0cf0685c9cedf8a3283ac69eef4d5f2eaca2"
namespace Nat
namespace WithBot
instance : WellFoundedRelation (WithBot ℕ) where
rel := (· < ·)
wf := IsWellFounde... | Mathlib/Data/Nat/WithBot.lean | 81 | 85 | theorem add_one_le_of_lt {n m : WithBot ℕ} (h : n < m) : n + 1 ≤ m := by |
cases n
· exact bot_le
cases m
exacts [(not_lt_bot h).elim, WithBot.coe_le_coe.2 (WithBot.coe_lt_coe.1 h)]
| 4 | 54.59815 | 2 | 1.857143 | 7 | 1,928 |
import Mathlib.MeasureTheory.OuterMeasure.Operations
import Mathlib.Analysis.SpecificLimits.Basic
#align_import measure_theory.measure.outer_measure from "leanprover-community/mathlib"@"343e80208d29d2d15f8050b929aa50fe4ce71b55"
noncomputable section
open Set Function Filter
open scoped Classical NNReal Topology E... | Mathlib/MeasureTheory/OuterMeasure/OfFunction.lean | 139 | 169 | theorem ofFunction_union_of_top_of_nonempty_inter {s t : Set α}
(h : ∀ u, (s ∩ u).Nonempty → (t ∩ u).Nonempty → m u = ∞) :
OuterMeasure.ofFunction m m_empty (s ∪ t) =
OuterMeasure.ofFunction m m_empty s + OuterMeasure.ofFunction m m_empty t := by |
refine le_antisymm (measure_union_le _ _) (le_iInf₂ fun f hf ↦ ?_)
set μ := OuterMeasure.ofFunction m m_empty
rcases Classical.em (∃ i, (s ∩ f i).Nonempty ∧ (t ∩ f i).Nonempty) with (⟨i, hs, ht⟩ | he)
· calc
μ s + μ t ≤ ∞ := le_top
_ = m (f i) := (h (f i) hs ht).symm
_ ≤ ∑' i, m (f i) := ENNR... | 25 | 72,004,899,337.38586 | 2 | 2 | 1 | 1,956 |
set_option autoImplicit true
namespace Array
@[simp]
theorem extract_eq_nil_of_start_eq_end {a : Array α} :
a.extract i i = #[] := by
refine extract_empty_of_stop_le_start a ?h
exact Nat.le_refl i
theorem extract_append_left {a b : Array α} {i j : Nat} (h : j ≤ a.size) :
(a ++ b).extract i j = a.extrac... | Mathlib/Data/Array/ExtractLemmas.lean | 44 | 50 | theorem extract_extract {a : Array α} (h : s1 + e2 ≤ e1) :
(a.extract s1 e1).extract s2 e2 = a.extract (s1 + s2) (s1 + e2) := by |
apply ext
· simp only [size_extract]
omega
· intro i h1 h2
simp only [get_extract, Nat.add_assoc]
| 5 | 148.413159 | 2 | 1.4 | 5 | 1,495 |
import Mathlib.Data.Set.Pairwise.Basic
import Mathlib.Data.Set.Lattice
import Mathlib.Data.SetLike.Basic
#align_import order.chain from "leanprover-community/mathlib"@"c227d107bbada5d0d9d20287e3282c0a7f1651a0"
open scoped Classical
open Set
variable {α β : Type*}
section Chain
variable (r : α → α → Prop)
... | Mathlib/Order/Chain.lean | 184 | 188 | theorem IsChain.superChain_succChain (hs₁ : IsChain r s) (hs₂ : ¬IsMaxChain r s) :
SuperChain r s (SuccChain r s) := by |
simp only [IsMaxChain, _root_.not_and, not_forall, exists_prop, exists_and_left] at hs₂
obtain ⟨t, ht, hst⟩ := hs₂ hs₁
exact succChain_spec ⟨t, hs₁, ht, ssubset_iff_subset_ne.2 hst⟩
| 3 | 20.085537 | 1 | 1.2 | 5 | 1,280 |
import Mathlib.CategoryTheory.Subobject.MonoOver
import Mathlib.CategoryTheory.Skeletal
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathlib.Tactic.ApplyFun
import Mathlib.Tactic.CategoryTheory.Elementwise
#align_import category_theory.subobject.basic from "leanprover-community/mathlib"@"70fd9563a21e7b... | Mathlib/CategoryTheory/Subobject/Basic.lean | 556 | 558 | theorem pullback_id (x : Subobject X) : (pullback (𝟙 X)).obj x = x := by |
induction' x using Quotient.inductionOn' with f
exact Quotient.sound ⟨MonoOver.pullbackId.app f⟩
| 2 | 7.389056 | 1 | 1 | 5 | 901 |
import Mathlib.RingTheory.GradedAlgebra.HomogeneousIdeal
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.Topology.Sets.Opens
import Mathlib.Data.Set.Subsingleton
#align_import algebraic_geometry.projective_spectrum.topology from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
... | Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Topology.lean | 137 | 141 | theorem gc_set :
@GaloisConnection (Set A) (Set (ProjectiveSpectrum 𝒜))ᵒᵈ _ _
(fun s => zeroLocus 𝒜 s) fun t => vanishingIdeal t := by |
have ideal_gc : GaloisConnection Ideal.span _ := (Submodule.gi A _).gc
simpa [zeroLocus_span, Function.comp] using GaloisConnection.compose ideal_gc (gc_ideal 𝒜)
| 2 | 7.389056 | 1 | 0.8 | 5 | 697 |
import Mathlib.Analysis.Convex.Cone.Basic
import Mathlib.Analysis.InnerProductSpace.Projection
#align_import analysis.convex.cone.dual from "leanprover-community/mathlib"@"915591b2bb3ea303648db07284a161a7f2a9e3d4"
open Set LinearMap
open scoped Classical
open Pointwise
variable {𝕜 E F G : Type*}
section Dua... | Mathlib/Analysis/Convex/Cone/InnerDual.lean | 105 | 107 | theorem innerDualCone_insert (x : H) (s : Set H) :
(insert x s).innerDualCone = Set.innerDualCone {x} ⊓ s.innerDualCone := by |
rw [insert_eq, innerDualCone_union]
| 1 | 2.718282 | 0 | 1.142857 | 7 | 1,218 |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Fintype.Option
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.Fintype.Sum
#align_import combinatorics.hales_jewett from "leanprover-community/mathlib"@"1126441d6bccf98c81214a0780c73d499f6721fe"
open scoped Classical
universe u v
namespace ... | Mathlib/Combinatorics/HalesJewett.lean | 204 | 207 | theorem prod_apply {α ι ι'} (l : Line α ι) (l' : Line α ι') (x : α) :
l.prod l' x = Sum.elim (l x) (l' x) := by |
funext i
cases i <;> rfl
| 2 | 7.389056 | 1 | 0.571429 | 7 | 522 |
import Mathlib.Algebra.MvPolynomial.Degrees
#align_import data.mv_polynomial.variables from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Finsupp AddMonoidAlgebra
universe u v w
variable {R : Type u} {S : Type v}
namespace MvPolynomial
varia... | Mathlib/Algebra/MvPolynomial/Variables.lean | 98 | 99 | theorem mem_vars (i : σ) : i ∈ p.vars ↔ ∃ d ∈ p.support, i ∈ d.support := by |
classical simp only [vars_def, Multiset.mem_toFinset, mem_degrees, mem_support_iff, exists_prop]
| 1 | 2.718282 | 0 | 0.9 | 20 | 778 |
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 | 110 | 115 | theorem indicator_le_thickenedIndicatorAux (δ : ℝ) (E : Set α) :
(E.indicator fun _ => (1 : ℝ≥0∞)) ≤ thickenedIndicatorAux δ E := by |
intro a
by_cases h : a ∈ E
· simp only [h, indicator_of_mem, thickenedIndicatorAux_one δ E h, le_refl]
· simp only [h, indicator_of_not_mem, not_false_iff, zero_le]
| 4 | 54.59815 | 2 | 1 | 8 | 1,081 |
import Mathlib.Data.Finset.Image
import Mathlib.Data.List.FinRange
#align_import data.fintype.basic from "leanprover-community/mathlib"@"d78597269638367c3863d40d45108f52207e03cf"
assert_not_exists MonoidWithZero
assert_not_exists MulAction
open Function
open Nat
universe u v
variable {α β γ : Type*}
class Fi... | Mathlib/Data/Fintype/Basic.lean | 84 | 84 | theorem eq_univ_iff_forall : s = univ ↔ ∀ x, x ∈ s := by | simp [ext_iff]
| 1 | 2.718282 | 0 | 0.111111 | 9 | 248 |
import Mathlib.MeasureTheory.Function.LpSeminorm.Basic
import Mathlib.MeasureTheory.Integral.MeanInequalities
#align_import measure_theory.function.lp_seminorm from "leanprover-community/mathlib"@"c4015acc0a223449d44061e27ddac1835a3852b9"
open Filter
open scoped ENNReal Topology
namespace MeasureTheory
section S... | Mathlib/MeasureTheory/Function/LpSeminorm/CompareExp.lean | 26 | 45 | theorem snorm'_le_snorm'_mul_rpow_measure_univ {p q : ℝ} (hp0_lt : 0 < p) (hpq : p ≤ q)
(hf : AEStronglyMeasurable f μ) :
snorm' f p μ ≤ snorm' f q μ * μ Set.univ ^ (1 / p - 1 / q) := by |
have hq0_lt : 0 < q := lt_of_lt_of_le hp0_lt hpq
by_cases hpq_eq : p = q
· rw [hpq_eq, sub_self, ENNReal.rpow_zero, mul_one]
have hpq : p < q := lt_of_le_of_ne hpq hpq_eq
let g := fun _ : α => (1 : ℝ≥0∞)
have h_rw : (∫⁻ a, (‖f a‖₊ : ℝ≥0∞) ^ p ∂μ) = ∫⁻ a, ((‖f a‖₊ : ℝ≥0∞) * g a) ^ p ∂μ :=
lintegral_cong... | 17 | 24,154,952.753575 | 2 | 1.833333 | 6 | 1,918 |
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.Algebra.Module.Basic
#align_import topology.algebra.continuous_affine_map from "leanprover-community/mathlib"@"bd1fc183335ea95a9519a1630bcf901fe9326d83"
structure ContinuousAffineMap (R : T... | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | 108 | 111 | theorem to_continuousMap_injective {f g : P →ᴬ[R] Q} (h : (f : C(P, Q)) = (g : C(P, Q))) :
f = g := by |
ext a
exact ContinuousMap.congr_fun h a
| 2 | 7.389056 | 1 | 1 | 3 | 1,146 |
def SatisfiesM {m : Type u → Type v} [Functor m] (p : α → Prop) (x : m α) : Prop :=
∃ x' : m {a // p a}, Subtype.val <$> x' = x
@[simp] theorem SatisfiesM_Id_eq : SatisfiesM (m := Id) p x ↔ p x :=
⟨fun ⟨y, eq⟩ => eq ▸ y.2, fun h => ⟨⟨_, h⟩, rfl⟩⟩
@[simp] theorem SatisfiesM_Option_eq : SatisfiesM (m := Option... | .lake/packages/batteries/Batteries/Classes/SatisfiesM.lean | 165 | 166 | theorem SatisfiesM_StateRefT_eq [Monad m] :
SatisfiesM (m := StateRefT' ω σ m) p x ↔ ∀ s, SatisfiesM p (x s) := by | simp
| 1 | 2.718282 | 0 | 0 | 1 | 75 |
import Mathlib.Order.Filter.Bases
#align_import order.filter.pi from "leanprover-community/mathlib"@"ce64cd319bb6b3e82f31c2d38e79080d377be451"
open Set Function
open scoped Classical
open Filter
namespace Filter
variable {ι : Type*} {α : ι → Type*} {f f₁ f₂ : (i : ι) → Filter (α i)} {s : (i : ι) → Set (α i)}
... | Mathlib/Order/Filter/Pi.lean | 248 | 250 | theorem coprodᵢ_eq_bot_iff' : Filter.coprodᵢ f = ⊥ ↔ (∃ i, IsEmpty (α i)) ∨ f = ⊥ := by |
simpa only [not_neBot, not_and_or, funext_iff, not_forall, not_exists, not_nonempty_iff]
using coprodᵢ_neBot_iff'.not
| 2 | 7.389056 | 1 | 0.666667 | 12 | 565 |
import Mathlib.Order.Lattice
import Mathlib.Data.List.Sort
import Mathlib.Logic.Equiv.Fin
import Mathlib.Logic.Equiv.Functor
import Mathlib.Data.Fintype.Card
import Mathlib.Order.RelSeries
#align_import order.jordan_holder from "leanprover-community/mathlib"@"91288e351d51b3f0748f0a38faa7613fb0ae2ada"
universe u
... | Mathlib/Order/JordanHolder.lean | 173 | 177 | theorem total {s : CompositionSeries X} {x y : X} (hx : x ∈ s) (hy : y ∈ s) : x ≤ y ∨ y ≤ x := by |
rcases Set.mem_range.1 hx with ⟨i, rfl⟩
rcases Set.mem_range.1 hy with ⟨j, rfl⟩
rw [s.strictMono.le_iff_le, s.strictMono.le_iff_le]
exact le_total i j
| 4 | 54.59815 | 2 | 1 | 4 | 952 |
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.NormedSpace.FiniteDimension
#align_import analysis.calculus.bump_function_inner from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputable section
open Function Set Filter
open scoped Topology Filter
variable... | Mathlib/Analysis/Calculus/BumpFunction/Basic.lean | 154 | 157 | theorem one_of_mem_closedBall (hx : x ∈ closedBall c f.rIn) : f x = 1 := by |
apply ContDiffBumpBase.eq_one _ _ f.one_lt_rOut_div_rIn
simpa only [norm_smul, Real.norm_eq_abs, abs_inv, abs_of_nonneg f.rIn_pos.le, ← div_eq_inv_mul,
div_le_one f.rIn_pos] using mem_closedBall_iff_norm.1 hx
| 3 | 20.085537 | 1 | 1 | 4 | 1,090 |
import Mathlib.Init.Control.Combinators
import Mathlib.Data.Option.Defs
import Mathlib.Logic.IsEmpty
import Mathlib.Logic.Relator
import Mathlib.Util.CompileInductive
import Aesop
#align_import data.option.basic from "leanprover-community/mathlib"@"f340f229b1f461aa1c8ee11e0a172d0a3b301a4a"
universe u
namespace Op... | Mathlib/Data/Option/Basic.lean | 151 | 153 | theorem map_comm {f₁ : α → β} {f₂ : α → γ} {g₁ : β → δ} {g₂ : γ → δ} (h : g₁ ∘ f₁ = g₂ ∘ f₂)
(a : α) :
(Option.map f₁ a).map g₁ = (Option.map f₂ a).map g₂ := by | rw [map_map, h, ← map_map]
| 1 | 2.718282 | 0 | 0 | 8 | 110 |
import Mathlib.CategoryTheory.Galois.GaloisObjects
import Mathlib.CategoryTheory.Limits.Shapes.CombinedProducts
universe u₁ u₂ w
namespace CategoryTheory
open Limits Functor
variable {C : Type u₁} [Category.{u₂} C]
namespace PreGaloisCategory
variable [GaloisCategory C]
section Decomposition
private lemma... | Mathlib/CategoryTheory/Galois/Decomposition.lean | 118 | 121 | theorem has_decomp_connected_components' (X : C) :
∃ (ι : Type) (_ : Finite ι) (f : ι → C) (_ : ∐ f ≅ X), ∀ i, IsConnected (f i) := by |
obtain ⟨ι, f, g, hl, hc, hf⟩ := has_decomp_connected_components X
exact ⟨ι, hf, f, colimit.isoColimitCocone ⟨Cofan.mk X g, hl⟩, hc⟩
| 2 | 7.389056 | 1 | 1 | 2 | 1,102 |
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.Seminorm
import Mathlib.Analysis.LocallyConvex.Bounded
import Mathlib.Analysis.RCLike.Basic
#align_import analysis.convex.gauge from "leanprover-community/mathlib"@"373b03b5b9d0486534edbe94747f23cb3712f93d"
... | Mathlib/Analysis/Convex/Gauge.lean | 148 | 163 | theorem gauge_le_eq (hs₁ : Convex ℝ s) (hs₀ : (0 : E) ∈ s) (hs₂ : Absorbent ℝ s) (ha : 0 ≤ a) :
{ x | gauge s x ≤ a } = ⋂ (r : ℝ) (_ : a < r), r • s := by |
ext x
simp_rw [Set.mem_iInter, Set.mem_setOf_eq]
refine ⟨fun h r hr => ?_, fun h => le_of_forall_pos_lt_add fun ε hε => ?_⟩
· have hr' := ha.trans_lt hr
rw [mem_smul_set_iff_inv_smul_mem₀ hr'.ne']
obtain ⟨δ, δ_pos, hδr, hδ⟩ := exists_lt_of_gauge_lt hs₂ (h.trans_lt hr)
suffices (r⁻¹ * δ) • δ⁻¹ • x ∈... | 14 | 1,202,604.284165 | 2 | 1.090909 | 11 | 1,185 |
import Mathlib.Order.Filter.Bases
import Mathlib.Order.ConditionallyCompleteLattice.Basic
#align_import order.filter.lift from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
open Set Classical Filter Function
namespace Filter
variable {α β γ : Type*} {ι : Sort*}
section lift
protect... | Mathlib/Order/Filter/Lift.lean | 65 | 70 | theorem HasBasis.lift {ι} {p : ι → Prop} {s : ι → Set α} {f : Filter α} (hf : f.HasBasis p s)
{β : ι → Type*} {pg : ∀ i, β i → Prop} {sg : ∀ i, β i → Set γ} {g : Set α → Filter γ}
(hg : ∀ i, (g (s i)).HasBasis (pg i) (sg i)) (gm : Monotone g) :
(f.lift g).HasBasis (fun i : Σi, β i => p i.1 ∧ pg i.1 i.2) fun... |
refine ⟨fun t => (hf.mem_lift_iff hg gm).trans ?_⟩
simp [Sigma.exists, and_assoc, exists_and_left]
| 2 | 7.389056 | 1 | 0.666667 | 6 | 581 |
import Mathlib.Data.List.Chain
import Mathlib.Data.List.Enum
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Pairwise
import Mathlib.Data.List.Zip
#align_import data.list.range from "leanprover-community/mathlib"@"7b78d1776212a91ecc94cf601f83bdcc46b04213"
set_option autoImplicit true
universe u
open Nat... | Mathlib/Data/List/Range.lean | 66 | 67 | theorem nthLe_range'_1 {n m} (i) (H : i < (range' n m).length) :
nthLe (range' n m) i H = n + i := by | simp
| 1 | 2.718282 | 0 | 0.571429 | 7 | 516 |
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