Context stringlengths 57 6.04k | file_name stringlengths 21 79 | start int64 14 1.49k | end int64 18 1.5k | theorem stringlengths 25 1.55k | proof stringlengths 5 7.36k | num_lines int64 1 150 | complexity_score float64 2.72 139,370,958,066,637,970,000,000,000,000,000,000,000,000,000,000,000,000,000B | diff_level int64 0 2 | file_diff_level float64 0 2 | theorem_same_file int64 1 32 | rank_file int64 0 2.51k |
|---|---|---|---|---|---|---|---|---|---|---|---|
import Mathlib.Topology.Order.IsLUB
open Set Filter TopologicalSpace Topology Function
open OrderDual (toDual ofDual)
variable {α β γ : Type*}
section DenselyOrdered
variable [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a b : α}
{s : Set α}
theorem closure_Ioi' {a : α} (h : (Io... | Mathlib/Topology/Order/DenselyOrdered.lean | 111 | 112 | theorem interior_Ico [NoMinOrder α] {a b : α} : interior (Ico a b) = Ioo a b := by |
rw [← Ici_inter_Iio, interior_inter, interior_Ici, interior_Iio, Ioi_inter_Iio]
| 1 | 2.718282 | 0 | 0.769231 | 13 | 685 |
import Mathlib.AlgebraicGeometry.Morphisms.Basic
import Mathlib.RingTheory.LocalProperties
#align_import algebraic_geometry.morphisms.ring_hom_properties from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
-- Explicit universe annotations were used in this file to improve perfomance #127... | Mathlib/AlgebraicGeometry/Morphisms/RingHomProperties.lean | 86 | 102 | theorem RespectsIso.ofRestrict_morphismRestrict_iff (hP : RingHom.RespectsIso @P) {X Y : Scheme.{u}}
[IsAffine Y] (f : X ⟶ Y) (r : Y.presheaf.obj (Opposite.op ⊤)) (U : Opens X.carrier)
(hU : IsAffineOpen U) {V : Opens _}
(e : V = (Scheme.ιOpens <| f ⁻¹ᵁ Y.basicOpen r) ⁻¹ᵁ U) :
P (Scheme.Γ.map (Scheme.ιO... |
subst e
refine (hP.cancel_right_isIso _
(Scheme.Γ.mapIso (Scheme.restrictRestrictComm _ _ _).op).inv).symm.trans ?_
haveI : IsAffine _ := hU
rw [← hP.basicOpen_iff_localization, iff_iff_eq]
congr 1
simp only [Functor.mapIso_inv, Iso.op_inv, ← Functor.map_comp, ← op_comp, morphismRestrict_comp]
rw [← ... | 11 | 59,874.141715 | 2 | 2 | 6 | 1,934 |
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.Data.Nat.Lattice
#align_import combinatorics.simple_graph.metric from "leanprover-community/mathlib"@"352ecfe114946c903338006dd3287cb5a9955ff2"
namespace SimpleGraph
variable {V : Type*} (G : SimpleGraph V)
noncomputable def dist (u v : V)... | Mathlib/Combinatorics/SimpleGraph/Metric.lean | 99 | 102 | theorem nonempty_of_pos_dist {u v : V} (h : 0 < G.dist u v) :
(Set.univ : Set (G.Walk u v)).Nonempty := by |
simpa [Set.range_nonempty_iff_nonempty, Set.nonempty_iff_univ_nonempty] using
Nat.nonempty_of_pos_sInf h
| 2 | 7.389056 | 1 | 1 | 7 | 1,136 |
import Mathlib.Algebra.Divisibility.Basic
import Mathlib.Algebra.Group.Equiv.Basic
import Mathlib.Algebra.Ring.Defs
#align_import algebra.ring.divisibility from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
variable {α β : Type*}
section Semigroup
variable [Semigroup α] [Semigroup β]... | Mathlib/Algebra/Ring/Divisibility/Basic.lean | 31 | 38 | theorem MulEquiv.decompositionMonoid [DecompositionMonoid β] : DecompositionMonoid α where
primal a b c h := by |
rw [← map_dvd_iff f, map_mul] at h
obtain ⟨a₁, a₂, h⟩ := DecompositionMonoid.primal _ h
refine ⟨symm f a₁, symm f a₂, ?_⟩
simp_rw [← map_dvd_iff f, ← map_mul, eq_symm_apply]
iterate 2 erw [(f : α ≃* β).apply_symm_apply]
exact h
| 6 | 403.428793 | 2 | 2 | 1 | 2,244 |
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 | 114 | 115 | theorem sinh_log {x : ℝ} (hx : 0 < x) : sinh (log x) = (x - x⁻¹) / 2 := by |
rw [sinh_eq, exp_neg, exp_log hx]
| 1 | 2.718282 | 0 | 0.583333 | 12 | 525 |
import Mathlib.Analysis.SpecialFunctions.Complex.Arg
import Mathlib.Analysis.SpecialFunctions.Log.Basic
#align_import analysis.special_functions.complex.log from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
namespace Complex
open Set Filter Bornology
open scop... | Mathlib/Analysis/SpecialFunctions/Complex/Log.lean | 124 | 128 | theorem log_conj_eq_ite (x : ℂ) : log (conj x) = if x.arg = π then log x else conj (log x) := by |
simp_rw [log, abs_conj, arg_conj, map_add, map_mul, conj_ofReal]
split_ifs with hx
· rw [hx]
simp_rw [ofReal_neg, conj_I, mul_neg, neg_mul]
| 4 | 54.59815 | 2 | 0.375 | 16 | 378 |
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Analysis.NormedSpace.Dual
import Mathlib.Analysis.NormedSpace.Star.Basic
#align_import analysis.inner_product_space.dual from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped Classical
o... | Mathlib/Analysis/InnerProductSpace/Dual.lean | 157 | 159 | theorem toDual_symm_apply {x : E} {y : NormedSpace.Dual 𝕜 E} : ⟪(toDual 𝕜 E).symm y, x⟫ = y x := by |
rw [← toDual_apply]
simp only [LinearIsometryEquiv.apply_symm_apply]
| 2 | 7.389056 | 1 | 1.666667 | 3 | 1,798 |
import Mathlib.Algebra.MonoidAlgebra.Basic
import Mathlib.LinearAlgebra.Basis.VectorSpace
import Mathlib.RingTheory.SimpleModule
#align_import representation_theory.maschke from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u v w
noncomputable section
open Module MonoidAlgeb... | Mathlib/RepresentationTheory/Maschke.lean | 81 | 83 | theorem conjugate_i (g : G) (v : V) : (conjugate π g : W → V) (i v) = v := by |
rw [conjugate_apply, ← i.map_smul, h, ← mul_smul, single_mul_single, mul_one, mul_left_inv,
← one_def, one_smul]
| 2 | 7.389056 | 1 | 1 | 3 | 927 |
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 | 75 | 76 | theorem one_mul_hom {Z : C} (f : Z ⟶ M.X) : (M.one ⊗ f) ≫ M.mul = (λ_ Z).hom ≫ f := by |
rw [tensorHom_def'_assoc, M.one_mul, leftUnitor_naturality]
| 1 | 2.718282 | 0 | 0.5 | 4 | 503 |
import Mathlib.Data.Finset.Fold
import Mathlib.Algebra.GCDMonoid.Multiset
#align_import algebra.gcd_monoid.finset from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
#align_import algebra.gcd_monoid.div from "leanprover-community/mathlib"@"b537794f8409bc9598febb79cd510b1df5f4539d"
variab... | Mathlib/Algebra/GCDMonoid/Finset.lean | 203 | 205 | theorem gcd_image [DecidableEq β] {g : γ → β} (s : Finset γ) :
(s.image g).gcd f = s.gcd (f ∘ g) := by |
classical induction' s using Finset.induction with c s _ ih <;> simp [*]
| 1 | 2.718282 | 0 | 1 | 13 | 925 |
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.Prod
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Tactic.FinCases
import Mathlib.Tactic.LinearCombination
import Mathlib.Lean.Expr.ExtraRecognizers
import Mathlib.Data.Set.Subsingleton
#align_import lin... | Mathlib/LinearAlgebra/LinearIndependent.lean | 126 | 128 | theorem linearIndependent_iff :
LinearIndependent R v ↔ ∀ l, Finsupp.total ι M R v l = 0 → l = 0 := by |
simp [LinearIndependent, LinearMap.ker_eq_bot']
| 1 | 2.718282 | 0 | 1 | 7 | 908 |
import Mathlib.Data.Part
import Mathlib.Data.Nat.Upto
import Mathlib.Data.Stream.Defs
import Mathlib.Tactic.Common
#align_import control.fix from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
universe u v
open scoped Classical
variable {α : Type*} {β : α → Type*}
class Fix (α : Typ... | Mathlib/Control/Fix.lean | 111 | 113 | theorem fix_def' {x : α} (h' : ¬∃ i, (Fix.approx f i x).Dom) : Part.fix f x = none := by |
dsimp [Part.fix]
rw [assert_neg h']
| 2 | 7.389056 | 1 | 1 | 1 | 1,114 |
import Mathlib.CategoryTheory.Idempotents.Basic
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Equivalence
#align_import category_theory.idempotents.karoubi from "leanprover-community/mathlib"@"200eda15d8ff5669854ff6bcc10aaf37cb70498f"
noncomputable section
open CategoryT... | Mathlib/CategoryTheory/Idempotents/Karoubi.lean | 144 | 147 | theorem eqToHom_f {P Q : Karoubi C} (h : P = Q) :
Karoubi.Hom.f (eqToHom h) = P.p ≫ eqToHom (congr_arg Karoubi.X h) := by |
subst h
simp only [eqToHom_refl, Karoubi.id_eq, comp_id]
| 2 | 7.389056 | 1 | 0.625 | 8 | 543 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Div
#align_import data.polynomial.ring_division from "leanprover-community/mathlib"@"8efcf8022aac8e01df8d302dcebdbc25d6a886c8"
noncomputable ... | Mathlib/Algebra/Polynomial/RingDivision.lean | 368 | 396 | theorem eq_zero_of_mul_eq_zero_of_smul (P : R[X]) (h : ∀ r : R, r • P = 0 → r = 0) :
∀ (Q : R[X]), P * Q = 0 → Q = 0 := by |
intro Q hQ
suffices ∀ i, P.coeff i • Q = 0 by
rw [← leadingCoeff_eq_zero]
apply h
simpa [ext_iff, mul_comm Q.leadingCoeff] using fun i ↦ congr_arg (·.coeff Q.natDegree) (this i)
apply Nat.strong_decreasing_induction
· use P.natDegree
intro i hi
rw [coeff_eq_zero_of_natDegree_lt hi, zero_smu... | 27 | 532,048,240,601.79865 | 2 | 1.5 | 32 | 1,561 |
import Mathlib.Analysis.Convex.Basic
import Mathlib.Order.Closure
#align_import analysis.convex.hull from "leanprover-community/mathlib"@"92bd7b1ffeb306a89f450bee126ddd8a284c259d"
open Set
open Pointwise
variable {𝕜 E F : Type*}
section convexHull
section OrderedSemiring
variable [OrderedSemiring 𝕜]
secti... | Mathlib/Analysis/Convex/Hull.lean | 104 | 106 | theorem convexHull_nonempty_iff : (convexHull 𝕜 s).Nonempty ↔ s.Nonempty := by |
rw [nonempty_iff_ne_empty, nonempty_iff_ne_empty, Ne, Ne]
exact not_congr convexHull_empty_iff
| 2 | 7.389056 | 1 | 1.166667 | 6 | 1,236 |
import Mathlib.Topology.PartialHomeomorph
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Data.Real.Sqrt
#align_import analysis.normed_space.basic from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
open Set Metric Pointwise
var... | Mathlib/Analysis/NormedSpace/HomeomorphBall.lean | 140 | 141 | theorem univBall_apply_zero (c : P) (r : ℝ) : univBall c r 0 = c := by |
unfold univBall; split_ifs <;> simp
| 1 | 2.718282 | 0 | 0.375 | 8 | 380 |
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 | 81 | 85 | theorem hasDerivAt_iff_tendsto_slope_zero :
HasDerivAt f f' x ↔ Tendsto (fun t ↦ t⁻¹ • (f (x + t) - f x)) (𝓝[≠] 0) (𝓝 f') := by |
have : 𝓝[≠] x = Filter.map (fun t ↦ x + t) (𝓝[≠] 0) := by
simp [nhdsWithin, map_add_left_nhds_zero x, Filter.map_inf, add_right_injective x]
simp [hasDerivAt_iff_tendsto_slope, this, slope, Function.comp]
| 3 | 20.085537 | 1 | 1.2 | 5 | 1,257 |
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.Fin.Tuple.Reflection
#align_import data.matrix.reflection from "leanprover-community/mathlib"@"820b22968a2bc4a47ce5cf1d2f36a9ebe52510aa"
open Matrix
namespace Matrix
variable {l m n : ℕ} {α β : Type*}
def Forall : ∀ {m n}... | Mathlib/Data/Matrix/Reflection.lean | 132 | 135 | theorem dotProductᵣ_eq [Mul α] [AddCommMonoid α] {m} (a b : Fin m → α) :
dotProductᵣ a b = dotProduct a b := by |
simp_rw [dotProductᵣ, dotProduct, FinVec.sum_eq, FinVec.seq_eq, FinVec.map_eq,
Function.comp_apply]
| 2 | 7.389056 | 1 | 1 | 3 | 944 |
import Mathlib.Topology.Sets.Opens
#align_import topology.local_at_target from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open TopologicalSpace Set Filter
open Topology Filter
variable {α β : Type*} [TopologicalSpace α] [TopologicalSpace β] {f : α → β}
variable {s : Set β} {ι : Ty... | Mathlib/Topology/LocalAtTarget.lean | 116 | 126 | theorem isClosedMap_iff_isClosedMap_of_iSup_eq_top :
IsClosedMap f ↔ ∀ i, IsClosedMap ((U i).1.restrictPreimage f) := by |
refine ⟨fun h i => h.restrictPreimage _, ?_⟩
rintro H s hs
rw [isClosed_iff_coe_preimage_of_iSup_eq_top hU]
intro i
convert H i _ ⟨⟨_, hs.1, eq_compl_comm.mpr rfl⟩⟩
ext ⟨x, hx⟩
suffices (∃ y, y ∈ s ∧ f y = x) ↔ ∃ y, y ∈ s ∧ f y ∈ U i ∧ f y = x by
simpa [Set.restrictPreimage, ← Subtype.coe_inj]
exac... | 9 | 8,103.083928 | 2 | 1.714286 | 7 | 1,842 |
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Smooth.Basic
import Mathlib.RingTheory.Unramified.Basic
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
-- Porting note: added to make the syntax work below.
open scoped TensorProdu... | Mathlib/RingTheory/Etale/Basic.lean | 66 | 69 | theorem iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by |
rw [formallyUnramified_iff, formallySmooth_iff, formallyEtale_iff]
simp_rw [← forall_and, Function.Bijective]
| 2 | 7.389056 | 1 | 1 | 1 | 807 |
import Mathlib.Order.Ideal
#align_import order.pfilter from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
open OrderDual
namespace Order
structure PFilter (P : Type*) [Preorder P] where
dual : Ideal Pᵒᵈ
#align order.pfilter Order.PFilter
variable {P : Type*}
def IsPFilter [Preor... | Mathlib/Order/PFilter.lean | 120 | 120 | theorem principal_le_principal_iff {p q : P} : principal q ≤ principal p ↔ p ≤ q := by | simp
| 1 | 2.718282 | 0 | 0 | 1 | 123 |
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"
noncomputable section
section coevaluation
open TensorProduct FiniteDimensional
open TensorProduct
universe u v
variable (K : Type u) [Field K]
var... | Mathlib/LinearAlgebra/Coevaluation.lean | 81 | 95 | theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ∘ₗ
(TensorProduct.assoc K _ _ _).toLinearMap ∘ₗ (coevaluation K V).rTensor V =
(TensorProduct.rid K _).symm.toLinearMap ∘ₗ (TensorProduct.lid K _).toLinearMap := by |
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
apply LinearMap.ext_ring; apply (Basis.ofVectorSpace K V).ext; intro j
rw [LinearMap.compr₂_apply, LinearMap.compr₂_apply, TensorProduct.mk_apply]
simp only [LinearMap.coe_comp, Function.comp_apply, LinearEquiv.coe_toLinearMap]
... | 11 | 59,874.141715 | 2 | 2 | 3 | 2,162 |
import Mathlib.Order.Interval.Finset.Nat
#align_import data.fin.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
assert_not_exists MonoidWithZero
open Finset Fin Function
namespace Fin
variable (n : ℕ)
instance instLocallyFiniteOrder : LocallyFiniteOrder (Fin n) :=
Orde... | Mathlib/Order/Interval/Finset/Fin.lean | 161 | 163 | theorem Ioi_eq_finset_subtype : Ioi a = (Ioc (a : ℕ) n).fin n := by |
ext
simp
| 2 | 7.389056 | 1 | 0.125 | 16 | 251 |
import Mathlib.MeasureTheory.Integral.Periodic
import Mathlib.Data.ZMod.Quotient
#align_import measure_theory.group.add_circle from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Function Filter MeasureTheory MeasureTheory.Measure Metric
open scoped MeasureTheory Pointwise Top... | Mathlib/MeasureTheory/Group/AddCircle.lean | 95 | 104 | theorem volume_of_add_preimage_eq (s I : Set <| AddCircle T) (u x : AddCircle T)
(hu : IsOfFinAddOrder u) (hs : (u +ᵥ s : Set <| AddCircle T) =ᵐ[volume] s)
(hI : I =ᵐ[volume] ball x (T / (2 * addOrderOf u))) :
volume s = addOrderOf u • volume (s ∩ I) := by |
let G := AddSubgroup.zmultiples u
haveI : Fintype G := @Fintype.ofFinite _ hu.finite_zmultiples.to_subtype
have hsG : ∀ g : G, (g +ᵥ s : Set <| AddCircle T) =ᵐ[volume] s := by
rintro ⟨y, hy⟩; exact (vadd_ae_eq_self_of_mem_zmultiples hs hy : _)
rw [(isAddFundamentalDomain_of_ae_ball I u x hu hI).measure_eq_... | 6 | 403.428793 | 2 | 2 | 3 | 2,486 |
import Mathlib.Topology.Connected.Basic
open Set Topology
universe u v
variable {α : Type u} {β : Type v} {ι : Type*} {π : ι → Type*} [TopologicalSpace α]
{s t u v : Set α}
section LocallyConnectedSpace
class LocallyConnectedSpace (α : Type*) [TopologicalSpace α] : Prop where
open_connected_basis : ∀ x,... | Mathlib/Topology/Connected/LocallyConnected.lean | 78 | 81 | theorem isOpen_connectedComponent [LocallyConnectedSpace α] {x : α} :
IsOpen (connectedComponent x) := by |
rw [← connectedComponentIn_univ]
exact isOpen_univ.connectedComponentIn
| 2 | 7.389056 | 1 | 1.571429 | 7 | 1,702 |
import Mathlib.Order.CompleteLatticeIntervals
import Mathlib.Order.CompactlyGenerated.Basic
variable {ι α : Type*} [CompleteLattice α]
namespace Set.Iic
| Mathlib/Order/CompactlyGenerated/Intervals.lean | 18 | 24 | theorem isCompactElement {a : α} {b : Iic a} (h : CompleteLattice.IsCompactElement (b : α)) :
CompleteLattice.IsCompactElement b := by |
simp only [CompleteLattice.isCompactElement_iff, Finset.sup_eq_iSup] at h ⊢
intro ι s hb
replace hb : (b : α) ≤ iSup ((↑) ∘ s) := le_trans hb <| (coe_iSup s) ▸ le_refl _
obtain ⟨t, ht⟩ := h ι ((↑) ∘ s) hb
exact ⟨t, (by simpa using ht : (b : α) ≤ _)⟩
| 5 | 148.413159 | 2 | 2 | 1 | 2,315 |
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.Calculus.FDeriv.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
open Topology InnerProductSpace Set
noncomputable section
variable {𝕜 F : Type*} [RCLike 𝕜]
variable [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [CompleteSpace F]
variabl... | Mathlib/Analysis/Calculus/Gradient/Basic.lean | 127 | 131 | theorem DifferentiableWithinAt.hasGradientWithinAt (h : DifferentiableWithinAt 𝕜 f s x) :
HasGradientWithinAt f (gradientWithin f s x) s x := by |
rw [hasGradientWithinAt_iff_hasFDerivWithinAt, gradientWithin,
(toDual 𝕜 F).apply_symm_apply (fderivWithin 𝕜 f s x)]
exact h.hasFDerivWithinAt
| 3 | 20.085537 | 1 | 0.538462 | 13 | 511 |
import Mathlib.Algebra.GCDMonoid.Multiset
import Mathlib.Combinatorics.Enumerative.Partition
import Mathlib.Data.List.Rotate
import Mathlib.GroupTheory.Perm.Cycle.Factors
import Mathlib.GroupTheory.Perm.Closure
import Mathlib.Algebra.GCDMonoid.Nat
import Mathlib.Tactic.NormNum.GCD
#align_import group_theory.perm.cycl... | Mathlib/GroupTheory/Perm/Cycle/Type.lean | 94 | 98 | theorem two_le_of_mem_cycleType {σ : Perm α} {n : ℕ} (h : n ∈ σ.cycleType) : 2 ≤ n := by |
simp only [cycleType_def, ← Finset.mem_def, Function.comp_apply, Multiset.mem_map,
mem_cycleFactorsFinset_iff] at h
obtain ⟨_, ⟨hc, -⟩, rfl⟩ := h
exact hc.two_le_card_support
| 4 | 54.59815 | 2 | 1.375 | 8 | 1,473 |
import Mathlib.FieldTheory.SplittingField.IsSplittingField
import Mathlib.Algebra.CharP.Algebra
#align_import field_theory.splitting_field.construction from "leanprover-community/mathlib"@"e3f4be1fcb5376c4948d7f095bec45350bfb9d1a"
noncomputable section
open scoped Classical Polynomial
universe u v w
variable {... | Mathlib/FieldTheory/SplittingField/Construction.lean | 69 | 72 | theorem factor_dvd_of_not_isUnit {f : K[X]} (hf1 : ¬IsUnit f) : factor f ∣ f := by |
by_cases hf2 : f = 0; · rw [hf2]; exact dvd_zero _
rw [factor, dif_pos (WfDvdMonoid.exists_irreducible_factor hf1 hf2)]
exact (Classical.choose_spec <| WfDvdMonoid.exists_irreducible_factor hf1 hf2).2
| 3 | 20.085537 | 1 | 1.2 | 5 | 1,286 |
import Mathlib.Data.Fin.VecNotation
#align_import data.fin.tuple.monotone from "leanprover-community/mathlib"@"e3d9ab8faa9dea8f78155c6c27d62a621f4c152d"
open Set Fin Matrix Function
variable {α : Type*}
| Mathlib/Data/Fin/Tuple/Monotone.lean | 21 | 24 | theorem liftFun_vecCons {n : ℕ} (r : α → α → Prop) [IsTrans α r] {f : Fin (n + 1) → α} {a : α} :
((· < ·) ⇒ r) (vecCons a f) (vecCons a f) ↔ r a (f 0) ∧ ((· < ·) ⇒ r) f f := by |
simp only [liftFun_iff_succ r, forall_fin_succ, cons_val_succ, cons_val_zero, ← succ_castSucc,
castSucc_zero]
| 2 | 7.389056 | 1 | 1 | 1 | 1,097 |
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Int.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.Cases
import Mathlib.Tactic.SimpRw
import Mathlib.Tactic.SplitIfs
#align_import algebra.group.basic from "leanprover-community/mathlib"@"a07d750983b94c530a... | Mathlib/Algebra/Group/Basic.lean | 117 | 119 | theorem comp_mul_left (x y : α) : (x * ·) ∘ (y * ·) = (x * y * ·) := by |
ext z
simp [mul_assoc]
| 2 | 7.389056 | 1 | 0.333333 | 18 | 367 |
import Mathlib.Topology.Category.TopCat.Limits.Products
#align_import topology.category.Top.limits.pullbacks from "leanprover-community/mathlib"@"178a32653e369dce2da68dc6b2694e385d484ef1"
-- Porting note: every ML3 decl has an uppercase letter
set_option linter.uppercaseLean3 false
open TopologicalSpace
open Cat... | Mathlib/Topology/Category/TopCat/Limits/Pullbacks.lean | 447 | 452 | theorem coinduced_of_isColimit {F : J ⥤ TopCat.{max v u}} (c : Cocone F) (hc : IsColimit c) :
c.pt.str = ⨆ j, (F.obj j).str.coinduced (c.ι.app j) := by |
let homeo := homeoOfIso (hc.coconePointUniqueUpToIso (colimitCoconeIsColimit F))
ext
refine homeo.symm.isOpen_preimage.symm.trans (Iff.trans ?_ isOpen_iSup_iff.symm)
exact isOpen_iSup_iff
| 4 | 54.59815 | 2 | 0.714286 | 7 | 647 |
import Mathlib.Analysis.Quaternion
import Mathlib.Analysis.NormedSpace.Exponential
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Series
#align_import analysis.normed_space.quaternion_exponential from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open scoped Quaternion Nat
open... | Mathlib/Analysis/NormedSpace/QuaternionExponential.lean | 117 | 118 | theorem im_exp (q : ℍ[ℝ]) : (exp ℝ q).im = (exp ℝ q.re * (Real.sin ‖q.im‖ / ‖q.im‖)) • q.im := by |
simp [exp_eq, smul_smul]
| 1 | 2.718282 | 0 | 1.375 | 8 | 1,470 |
import Mathlib.Order.Filter.Basic
#align_import order.filter.prod from "leanprover-community/mathlib"@"d6fad0e5bf2d6f48da9175d25c3dc5706b3834ce"
open Set
open Filter
namespace Filter
variable {α β γ δ : Type*} {ι : Sort*}
section Prod
variable {s : Set α} {t : Set β} {f : Filter α} {g : Filter β}
protected ... | Mathlib/Order/Filter/Prod.lean | 126 | 128 | theorem prod_sup (f : Filter α) (g₁ g₂ : Filter β) : f ×ˢ (g₁ ⊔ g₂) = (f ×ˢ g₁) ⊔ (f ×ˢ g₂) := by |
dsimp only [SProd.sprod]
rw [Filter.prod, comap_sup, inf_sup_left, ← Filter.prod, ← Filter.prod]
| 2 | 7.389056 | 1 | 1 | 10 | 1,042 |
import Mathlib.RingTheory.WittVector.Frobenius
import Mathlib.RingTheory.WittVector.Verschiebung
import Mathlib.RingTheory.WittVector.MulP
#align_import ring_theory.witt_vector.identities from "leanprover-community/mathlib"@"0798037604b2d91748f9b43925fb7570a5f3256c"
namespace WittVector
variable {p : ℕ} {R : Typ... | Mathlib/RingTheory/WittVector/Identities.lean | 87 | 87 | theorem coeff_p_one [CharP R p] : (p : 𝕎 R).coeff 1 = 1 := by | rw [coeff_p, if_pos rfl]
| 1 | 2.718282 | 0 | 1 | 13 | 1,103 |
import Batteries.Data.List.Basic
namespace Batteries
inductive AssocList (α : Type u) (β : Type v) where
| nil
| cons (key : α) (value : β) (tail : AssocList α β)
deriving Inhabited
namespace AssocList
@[simp] def toList : AssocList α β → List (α × β)
| nil => []
| cons a b es => (a, b) :: es.toL... | .lake/packages/batteries/Batteries/Data/AssocList.lean | 55 | 56 | theorem length_toList (l : AssocList α β) : l.toList.length = l.length := by |
induction l <;> simp_all
| 1 | 2.718282 | 0 | 0 | 2 | 33 |
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.Algebra.GCDMonoid.Nat
#align_import ring_theory.int.basic from "leanprover-community/mathlib"@"e655e4ea5c6d02854696f97494997ba4c31be802"
theorem Int.Prime.dvd_mul {m n : ℤ} {p : ℕ} (hp : Nat.Prime p) (h : (p ... | Mathlib/RingTheory/Int/Basic.lean | 93 | 96 | theorem Int.Prime.dvd_mul' {m n : ℤ} {p : ℕ} (hp : Nat.Prime p) (h : (p : ℤ) ∣ m * n) :
(p : ℤ) ∣ m ∨ (p : ℤ) ∣ n := by |
rw [Int.natCast_dvd, Int.natCast_dvd]
exact Int.Prime.dvd_mul hp h
| 2 | 7.389056 | 1 | 1.153846 | 13 | 1,227 |
import Mathlib.Probability.ProbabilityMassFunction.Basic
#align_import probability.probability_mass_function.monad from "leanprover-community/mathlib"@"4ac69b290818724c159de091daa3acd31da0ee6d"
noncomputable section
variable {α β γ : Type*}
open scoped Classical
open NNReal ENNReal
open MeasureTheory
namespac... | Mathlib/Probability/ProbabilityMassFunction/Monad.lean | 170 | 182 | theorem toOuterMeasure_bind_apply :
(p.bind f).toOuterMeasure s = ∑' a, p a * (f a).toOuterMeasure s :=
calc
(p.bind f).toOuterMeasure s = ∑' b, if b ∈ s then ∑' a, p a * f a b else 0 := by |
simp [toOuterMeasure_apply, Set.indicator_apply]
_ = ∑' (b) (a), p a * if b ∈ s then f a b else 0 := tsum_congr fun b => by split_ifs <;> simp
_ = ∑' (a) (b), p a * if b ∈ s then f a b else 0 :=
(tsum_comm' ENNReal.summable (fun _ => ENNReal.summable) fun _ => ENNReal.summable)
_ = ∑' a, p a * ... | 9 | 8,103.083928 | 2 | 1 | 6 | 1,147 |
import Mathlib.Analysis.InnerProductSpace.Orthogonal
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import geometry.euclidean.basic from "leanprover-community/mathlib"@"2de9c37fa71dde2f1c6feff19876dd6a7b1519f0"
open Set
open scoped RealInnerProductSpace
variable {V P : Type*} [NormedAddCommGroup V] [InnerP... | Mathlib/Geometry/Euclidean/PerpBisector.lean | 86 | 90 | theorem mem_perpBisector_iff_inner_eq :
c ∈ perpBisector p₁ p₂ ↔ ⟪c -ᵥ p₁, p₂ -ᵥ p₁⟫ = (dist p₁ p₂) ^ 2 / 2 := by |
rw [mem_perpBisector_iff_inner_eq_zero, ← vsub_sub_vsub_cancel_right _ _ p₁, inner_sub_left,
sub_eq_zero, midpoint_vsub_left, invOf_eq_inv, real_inner_smul_left, real_inner_self_eq_norm_sq,
dist_eq_norm_vsub' V, div_eq_inv_mul]
| 3 | 20.085537 | 1 | 0.777778 | 9 | 693 |
import Mathlib.Data.Fin.VecNotation
import Mathlib.Logic.Embedding.Set
#align_import logic.equiv.fin from "leanprover-community/mathlib"@"bd835ef554f37ef9b804f0903089211f89cb370b"
assert_not_exists MonoidWithZero
universe u
variable {m n : ℕ}
def finZeroEquiv : Fin 0 ≃ Empty :=
Equiv.equivEmpty _
#align fin_... | Mathlib/Logic/Equiv/Fin.lean | 56 | 60 | theorem Fin.preimage_apply_01_prod {α : Fin 2 → Type u} (s : Set (α 0)) (t : Set (α 1)) :
(fun f : ∀ i, α i => (f 0, f 1)) ⁻¹' s ×ˢ t =
Set.pi Set.univ (Fin.cons s <| Fin.cons t finZeroElim) := by |
ext f
simp [Fin.forall_fin_two]
| 2 | 7.389056 | 1 | 0.25 | 4 | 298 |
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 | 99 | 106 | theorem coe_vanishingIdeal (t : Set (ProjectiveSpectrum 𝒜)) :
(vanishingIdeal t : Set A) =
{ f | ∀ x : ProjectiveSpectrum 𝒜, x ∈ t → f ∈ x.asHomogeneousIdeal } := by |
ext f
rw [vanishingIdeal, SetLike.mem_coe, ← HomogeneousIdeal.mem_iff, HomogeneousIdeal.toIdeal_iInf,
Submodule.mem_iInf]
refine forall_congr' fun x => ?_
rw [HomogeneousIdeal.toIdeal_iInf, Submodule.mem_iInf, HomogeneousIdeal.mem_iff]
| 5 | 148.413159 | 2 | 0.8 | 5 | 697 |
import Mathlib.Algebra.Group.Prod
import Mathlib.Order.Cover
#align_import algebra.support from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
assert_not_exists MonoidWithZero
open Set
namespace Function
variable {α β A B M N P G : Type*}
section One
variable [One M] [One N] [One P]
... | Mathlib/Algebra/Group/Support.lean | 88 | 90 | theorem mulSupport_update_of_ne_one [DecidableEq α] (f : α → M) (x : α) {y : M} (hy : y ≠ 1) :
mulSupport (update f x y) = insert x (mulSupport f) := by |
ext a; rcases eq_or_ne a x with rfl | hne <;> simp [*]
| 1 | 2.718282 | 0 | 0.333333 | 6 | 332 |
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 Inf
-- can be defined with just `[Top α]` where some lemmas hold with... | Mathlib/Data/Multiset/Lattice.lean | 163 | 164 | theorem inf_ndunion (s₁ s₂ : Multiset α) : (ndunion s₁ s₂).inf = s₁.inf ⊓ s₂.inf := by |
rw [← inf_dedup, dedup_ext.2, inf_dedup, inf_add]; simp
| 1 | 2.718282 | 0 | 0.285714 | 7 | 313 |
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Monoidal.Functor
#align_import category_theory.monoidal.preadditive from "leanprover-community/mathlib"@"986c4d5761f938b2e1c43c01f001b6d9d88c2055"
noncomputable section
open scoped Classical
namespace CategoryTheory
open Cat... | Mathlib/CategoryTheory/Monoidal/Preadditive.lean | 151 | 158 | theorem leftDistributor_hom {J : Type} [Fintype J] (X : C) (f : J → C) :
(leftDistributor X f).hom =
∑ j : J, (X ◁ biproduct.π f j) ≫ biproduct.ι (fun j => X ⊗ f j) j := by |
ext
dsimp [leftDistributor, Functor.mapBiproduct, Functor.mapBicone]
erw [biproduct.lift_π]
simp only [Preadditive.sum_comp, Category.assoc, biproduct.ι_π, comp_dite, comp_zero,
Finset.sum_dite_eq', Finset.mem_univ, ite_true, eqToHom_refl, Category.comp_id]
| 5 | 148.413159 | 2 | 0.5 | 8 | 481 |
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#align_import algebra.order.group.defs from "leanprover-community/mathlib"@"b599f4e4e5cf1fbcb4194503671d3d9e569c1fce"
open Function
universe u
variable {α : Type u}
class OrderedAddCommGroup (α : Ty... | Mathlib/Algebra/Order/Group/Defs.lean | 215 | 217 | theorem Right.inv_le_one_iff : a⁻¹ ≤ 1 ↔ 1 ≤ a := by |
rw [← mul_le_mul_iff_right a]
simp
| 2 | 7.389056 | 1 | 0.4 | 25 | 400 |
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.Topology.Algebra.Nonarchimedean.Bases
import Mathlib.Topology.Algebra.UniformRing
#align_import topology.algebra.nonarchimedean.adic_topology from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
variable {R : Type*} [CommRing R]
open S... | Mathlib/Topology/Algebra/Nonarchimedean/AdicTopology.lean | 116 | 126 | theorem adic_module_basis :
I.ringFilterBasis.SubmodulesBasis fun n : ℕ => I ^ n • (⊤ : Submodule R M) :=
{ inter := fun i j =>
⟨max i j,
le_inf_iff.mpr
⟨smul_mono_left <| pow_le_pow_right (le_max_left i j),
smul_mono_left <| pow_le_pow_right (le_max_right i j)⟩⟩
smul := fu... | simpa [(I ^ i).mul_top] using a_in
exact smul_mem_smul a_in mem_top⟩ }
| 2 | 7.389056 | 1 | 1.5 | 4 | 1,581 |
import Mathlib.Order.PartialSups
#align_import order.disjointed from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
variable {α β : Type*}
section GeneralizedBooleanAlgebra
variable [GeneralizedBooleanAlgebra α]
def disjointed (f : ℕ → α) : ℕ → α
| 0 => f 0
| n + 1 => f (n + 1) ... | Mathlib/Order/Disjointed.lean | 114 | 118 | theorem partialSups_disjointed (f : ℕ → α) : partialSups (disjointed f) = partialSups f := by |
ext n
induction' n with k ih
· rw [partialSups_zero, partialSups_zero, disjointed_zero]
· rw [partialSups_succ, partialSups_succ, disjointed_succ, ih, sup_sdiff_self_right]
| 4 | 54.59815 | 2 | 2 | 4 | 2,431 |
import Mathlib.Algebra.Group.Hom.Defs
#align_import algebra.group.ext from "leanprover-community/mathlib"@"e574b1a4e891376b0ef974b926da39e05da12a06"
assert_not_exists MonoidWithZero
assert_not_exists DenselyOrdered
open Function
universe u
@[to_additive (attr := ext)]
theorem Monoid.ext {M : Type u} ⦃m₁ m₂ : Mo... | Mathlib/Algebra/Group/Ext.lean | 103 | 106 | theorem CancelMonoid.toLeftCancelMonoid_injective {M : Type u} :
Function.Injective (@CancelMonoid.toLeftCancelMonoid M) := by |
rintro ⟨⟩ ⟨⟩ h
congr
| 2 | 7.389056 | 1 | 1.333333 | 6 | 1,372 |
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Ring.Invertible
import Mathlib.Data.Nat.Cast.Order
#align_import algebra.order.invertible from "leanprover-community/mathlib"@"ee0c179cd3c8a45aa5bffbf1b41d8dbede452865"
variable {α : Type*} [LinearOrderedSemiring α] {a : α}
@[simp]
theorem invOf_pos [I... | Mathlib/Algebra/Order/Invertible.lean | 29 | 31 | theorem invOf_nonneg [Invertible a] : 0 ≤ ⅟ a ↔ 0 ≤ a :=
haveI : 0 < a * ⅟ a := by | simp only [mul_invOf_self, zero_lt_one]
⟨fun h => (pos_of_mul_pos_left this h).le, fun h => (pos_of_mul_pos_right this h).le⟩
| 2 | 7.389056 | 1 | 0.5 | 4 | 441 |
import Mathlib.Data.Finset.Fold
import Mathlib.Algebra.GCDMonoid.Multiset
#align_import algebra.gcd_monoid.finset from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
#align_import algebra.gcd_monoid.div from "leanprover-community/mathlib"@"b537794f8409bc9598febb79cd510b1df5f4539d"
variab... | Mathlib/Algebra/GCDMonoid/Finset.lean | 92 | 92 | theorem normalize_lcm : normalize (s.lcm f) = s.lcm f := by | simp [lcm_def]
| 1 | 2.718282 | 0 | 1 | 13 | 925 |
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 | 42 | 44 | theorem Ico_filter_dvd_card : ((Ico a b).filter (r ∣ ·)).card =
max (⌈b / (r : ℚ)⌉ - ⌈a / (r : ℚ)⌉) 0 := by |
rw [Ico_filter_dvd_eq _ _ hr, card_map, card_Ico, toNat_eq_max]
| 1 | 2.718282 | 0 | 0 | 4 | 158 |
import Mathlib.Analysis.Complex.AbsMax
import Mathlib.Analysis.Asymptotics.SuperpolynomialDecay
#align_import analysis.complex.phragmen_lindelof from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Function Filter Asymptotics Metric Complex Bornology
open scoped Topology Filter R... | Mathlib/Analysis/Complex/PhragmenLindelof.lean | 80 | 94 | theorem isBigO_sub_exp_rpow {a : ℝ} {f g : ℂ → E} {l : Filter ℂ}
(hBf : ∃ c < a, ∃ B, f =O[cobounded ℂ ⊓ l] fun z => expR (B * abs z ^ c))
(hBg : ∃ c < a, ∃ B, g =O[cobounded ℂ ⊓ l] fun z => expR (B * abs z ^ c)) :
∃ c < a, ∃ B, (f - g) =O[cobounded ℂ ⊓ l] fun z => expR (B * abs z ^ c) := by |
have : ∀ {c₁ c₂ B₁ B₂ : ℝ}, c₁ ≤ c₂ → 0 ≤ B₂ → B₁ ≤ B₂ →
(fun z : ℂ => expR (B₁ * abs z ^ c₁)) =O[cobounded ℂ ⊓ l]
fun z => expR (B₂ * abs z ^ c₂) := fun hc hB₀ hB ↦ .of_bound 1 <| by
filter_upwards [(eventually_cobounded_le_norm 1).filter_mono inf_le_left] with z hz
simp only [one_mul, Real.no... | 11 | 59,874.141715 | 2 | 2 | 3 | 2,267 |
import Mathlib.MeasureTheory.Group.Measure
assert_not_exists NormedSpace
namespace MeasureTheory
open Measure TopologicalSpace
open scoped ENNReal
variable {G : Type*} [MeasurableSpace G] {μ : Measure G} {g : G}
section TopologicalGroup
variable [TopologicalSpace G] [Group G] [TopologicalGroup G] [BorelSpace ... | Mathlib/MeasureTheory/Group/LIntegral.lean | 71 | 73 | theorem lintegral_eq_zero_of_isMulLeftInvariant [Regular μ] [NeZero μ] {f : G → ℝ≥0∞}
(hf : Continuous f) : ∫⁻ x, f x ∂μ = 0 ↔ f = 0 := by |
rw [lintegral_eq_zero_iff hf.measurable, hf.ae_eq_iff_eq μ continuous_zero]
| 1 | 2.718282 | 0 | 0.5 | 4 | 488 |
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
#align_import linear_algebra.affine_space.affine_subspace from "leanprover-community/mathlib"@"e96bdfbd1e8c98a09ff75f7ac6204d142debc840"
noncomputable section
open Affine
open Set
section
variable (k : Type*) {V : Type*} {P : Type*} [Ring k] [AddCommGroup V]... | Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean | 136 | 143 | theorem vsub_mem_vectorSpan_of_mem_spanPoints_of_mem_spanPoints {s : Set P} {p1 p2 : P}
(hp1 : p1 ∈ spanPoints k s) (hp2 : p2 ∈ spanPoints k s) : p1 -ᵥ p2 ∈ vectorSpan k s := by |
rcases hp1 with ⟨p1a, ⟨hp1a, ⟨v1, ⟨hv1, hv1p⟩⟩⟩⟩
rcases hp2 with ⟨p2a, ⟨hp2a, ⟨v2, ⟨hv2, hv2p⟩⟩⟩⟩
rw [hv1p, hv2p, vsub_vadd_eq_vsub_sub (v1 +ᵥ p1a), vadd_vsub_assoc, add_comm, add_sub_assoc]
have hv1v2 : v1 - v2 ∈ vectorSpan k s := (vectorSpan k s).sub_mem hv1 hv2
refine (vectorSpan k s).add_mem ?_ hv1v2
e... | 6 | 403.428793 | 2 | 1 | 5 | 815 |
import Mathlib.LinearAlgebra.AffineSpace.Independent
import Mathlib.LinearAlgebra.Basis
#align_import linear_algebra.affine_space.basis from "leanprover-community/mathlib"@"2de9c37fa71dde2f1c6feff19876dd6a7b1519f0"
open Affine
open Set
universe u₁ u₂ u₃ u₄
structure AffineBasis (ι : Type u₁) (k : Type u₂) {V ... | Mathlib/LinearAlgebra/AffineSpace/Basis.lean | 168 | 170 | theorem coord_apply_eq (i : ι) : b.coord i (b i) = 1 := by |
simp only [coord, Basis.coe_sumCoords, LinearEquiv.map_zero, LinearEquiv.coe_coe, sub_zero,
AffineMap.coe_mk, Finsupp.sum_zero_index, vsub_self]
| 2 | 7.389056 | 1 | 0.833333 | 6 | 735 |
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds
#align_import data.real.pi.bounds from "leanprover-community/mathlib"@"402f8982dddc1864bd703da2d6e2ee304a866973"
-- Porting note: needed to add a lot of type ascriptions for lean to interpret numbers as reals.
open scoped Real
namespace Real
theorem ... | Mathlib/Data/Real/Pi/Bounds.lean | 85 | 93 | theorem sqrtTwoAddSeries_step_up (c d : ℕ) {a b n : ℕ} {z : ℝ} (hz : sqrtTwoAddSeries (c / d) n ≤ z)
(hb : 0 < b) (hd : 0 < d) (h : (2 * b + a) * d ^ 2 ≤ c ^ 2 * b) :
sqrtTwoAddSeries (a / b) (n + 1) ≤ z := by |
refine le_trans ?_ hz; rw [sqrtTwoAddSeries_succ]; apply sqrtTwoAddSeries_monotone_left
have hb' : 0 < (b : ℝ) := Nat.cast_pos.2 hb
have hd' : 0 < (d : ℝ) := Nat.cast_pos.2 hd
rw [sqrt_le_left (div_nonneg c.cast_nonneg d.cast_nonneg), div_pow,
add_div_eq_mul_add_div _ _ (ne_of_gt hb'), div_le_div_iff hb' (... | 6 | 403.428793 | 2 | 1.833333 | 6 | 1,911 |
import Mathlib.MeasureTheory.Measure.Dirac
set_option autoImplicit true
open Set
open scoped ENNReal Classical
variable [MeasurableSpace α] [MeasurableSpace β] {s : Set α}
noncomputable section
namespace MeasureTheory.Measure
def count : Measure α :=
sum dirac
#align measure_theory.measure.count MeasureTheo... | Mathlib/MeasureTheory/Measure/Count.lean | 92 | 96 | theorem count_apply_eq_top [MeasurableSingletonClass α] : count s = ∞ ↔ s.Infinite := by |
by_cases hs : s.Finite
· exact count_apply_eq_top' hs.measurableSet
· change s.Infinite at hs
simp [hs, count_apply_infinite]
| 4 | 54.59815 | 2 | 1.1 | 10 | 1,189 |
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 | 585 | 588 | theorem map_comp (f : X ⟶ Y) (g : Y ⟶ Z) [Mono f] [Mono g] (x : Subobject X) :
(map (f ≫ g)).obj x = (map g).obj ((map f).obj x) := by |
induction' x using Quotient.inductionOn' with t
exact Quotient.sound ⟨(MonoOver.mapComp _ _).app t⟩
| 2 | 7.389056 | 1 | 1 | 5 | 901 |
import Mathlib.LinearAlgebra.CliffordAlgebra.Contraction
variable {R M : Type*}
variable [CommRing R] [AddCommGroup M] [Module R M] {Q : QuadraticForm R M}
namespace CliffordAlgebra
variable (Q)
def invertibleιOfInvertible (m : M) [Invertible (Q m)] : Invertible (ι Q m) where
invOf := ι Q (⅟ (Q m) • m)
invO... | Mathlib/LinearAlgebra/CliffordAlgebra/Inversion.lean | 44 | 47 | theorem ι_mul_ι_mul_invOf_ι (a b : M) [Invertible (ι Q a)] [Invertible (Q a)] :
ι Q a * ι Q b * ⅟ (ι Q a) = ι Q ((⅟ (Q a) * QuadraticForm.polar Q a b) • a - b) := by |
rw [invOf_ι, map_smul, mul_smul_comm, ι_mul_ι_mul_ι, ← map_smul, smul_sub, smul_smul, smul_smul,
invOf_mul_self, one_smul]
| 2 | 7.389056 | 1 | 1 | 5 | 870 |
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.Calculus.BumpFunction.Normed
import Mathlib.MeasureTheory.Integral.Average
import Mathlib.MeasureTheory.Covering.Differentiation
import Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace
import Mathlib.MeasureTheory.Measure.Haar.Unique
#align_import analy... | Mathlib/Analysis/Calculus/BumpFunction/Convolution.lean | 54 | 56 | theorem convolution_eq_right {x₀ : G} (hg : ∀ x ∈ ball x₀ φ.rOut, g x = g x₀) :
(φ ⋆[lsmul ℝ ℝ, μ] g : G → E') x₀ = integral μ φ • g x₀ := by |
simp_rw [convolution_eq_right' _ φ.support_eq.subset hg, lsmul_apply, integral_smul_const]
| 1 | 2.718282 | 0 | 1 | 3 | 1,037 |
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 | 89 | 92 | theorem agree_children {n : ℕ} (x : CofixA F (succ n)) (y : CofixA F (succ n + 1)) {i j}
(h₀ : HEq i j) (h₁ : Agree x y) : Agree (children' x i) (children' y j) := by |
cases' h₁ with _ _ _ _ _ _ hagree; cases h₀
apply hagree
| 2 | 7.389056 | 1 | 1.166667 | 6 | 1,239 |
import Mathlib.Data.Opposite
import Mathlib.Data.Set.Defs
#align_import data.set.opposite from "leanprover-community/mathlib"@"fc2ed6f838ce7c9b7c7171e58d78eaf7b438fb0e"
variable {α : Type*}
open Opposite
namespace Set
protected def op (s : Set α) : Set αᵒᵖ :=
unop ⁻¹' s
#align set.op Set.op
protected def u... | Mathlib/Data/Set/Opposite.lean | 48 | 48 | theorem unop_mem_unop {s : Set αᵒᵖ} {a : αᵒᵖ} : unop a ∈ s.unop ↔ a ∈ s := by | rfl
| 1 | 2.718282 | 0 | 1.333333 | 6 | 1,444 |
import Mathlib.Data.Nat.Defs
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Tactic.Monotonicity.Attr
#align_import data.nat.log from "leanprover-community/mathlib"@"3e00d81bdcbf77c8188bbd18f5524ddc3ed8cac6"
namespace Nat
--@[pp_nodot] porting note: unknown attribute
def log (b : ℕ) : ℕ → ℕ
| n => i... | Mathlib/Data/Nat/Log.lean | 108 | 111 | theorem pow_le_of_le_log {b x y : ℕ} (hy : y ≠ 0) (h : x ≤ log b y) : b ^ x ≤ y := by |
refine (le_or_lt b 1).elim (fun hb => ?_) fun hb => (pow_le_iff_le_log hb hy).2 h
rw [log_of_left_le_one hb, Nat.le_zero] at h
rwa [h, Nat.pow_zero, one_le_iff_ne_zero]
| 3 | 20.085537 | 1 | 1 | 6 | 1,106 |
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 | 74 | 84 | theorem isClosed_subsets_of_isClosed (hs : IsClosed s) :
IsClosed { t : Closeds α | (t : Set α) ⊆ s } := by |
refine isClosed_of_closure_subset fun
(t : Closeds α) (ht : t ∈ closure {t : Closeds α | (t : Set α) ⊆ s}) (x : α) (hx : x ∈ t) => ?_
have : x ∈ closure s := by
refine mem_closure_iff.2 fun ε εpos => ?_
obtain ⟨u : Closeds α, hu : u ∈ {t : Closeds α | (t : Set α) ⊆ s}, Dtu : edist t u < ε⟩ :=
mem... | 9 | 8,103.083928 | 2 | 2 | 2 | 2,497 |
import Mathlib.Algebra.Group.Units
import Mathlib.Algebra.GroupWithZero.Basic
import Mathlib.Logic.Equiv.Defs
import Mathlib.Tactic.Contrapose
import Mathlib.Tactic.Nontriviality
import Mathlib.Tactic.Spread
import Mathlib.Util.AssertExists
#align_import algebra.group_with_zero.units.basic from "leanprover-community/... | Mathlib/Algebra/GroupWithZero/Units/Basic.lean | 152 | 154 | theorem inverse_zero : inverse (0 : M₀) = 0 := by |
nontriviality
exact inverse_non_unit _ not_isUnit_zero
| 2 | 7.389056 | 1 | 0.375 | 8 | 377 |
import Mathlib.Data.Fintype.Card
import Mathlib.Order.UpperLower.Basic
#align_import combinatorics.set_family.intersecting from "leanprover-community/mathlib"@"d90e4e186f1d18e375dcd4e5b5f6364b01cb3e46"
open Finset
variable {α : Type*}
namespace Set
section SemilatticeInf
variable [SemilatticeInf α] [OrderBot ... | Mathlib/Combinatorics/SetFamily/Intersecting.lean | 81 | 92 | theorem intersecting_iff_pairwise_not_disjoint :
s.Intersecting ↔ (s.Pairwise fun a b => ¬Disjoint a b) ∧ s ≠ {⊥} := by |
refine ⟨fun h => ⟨fun a ha b hb _ => h ha hb, ?_⟩, fun h a ha b hb hab => ?_⟩
· rintro rfl
exact intersecting_singleton.1 h rfl
have := h.1.eq ha hb (Classical.not_not.2 hab)
rw [this, disjoint_self] at hab
rw [hab] at hb
exact
h.2
(eq_singleton_iff_unique_mem.2
⟨hb, fun c hc => not_n... | 10 | 22,026.465795 | 2 | 1.5 | 4 | 1,627 |
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Analysis.Normed.Group.Basic
import Mathlib.Topology.Instances.NNReal
#align_import analysis.normed.group.infinite_sum from "leanprover-community/mathlib"@"9a59dcb7a2d06bf55da57b9030169219980660cd"
open Topology NNReal
open Finset Filter Metric
variabl... | Mathlib/Analysis/Normed/Group/InfiniteSum.lean | 78 | 89 | theorem cauchySeq_range_of_norm_bounded {f : ℕ → E} (g : ℕ → ℝ)
(hg : CauchySeq fun n => ∑ i ∈ range n, g i) (hf : ∀ i, ‖f i‖ ≤ g i) :
CauchySeq fun n => ∑ i ∈ range n, f i := by |
refine Metric.cauchySeq_iff'.2 fun ε hε => ?_
refine (Metric.cauchySeq_iff'.1 hg ε hε).imp fun N hg n hn => ?_
specialize hg n hn
rw [dist_eq_norm, ← sum_Ico_eq_sub _ hn] at hg ⊢
calc
‖∑ k ∈ Ico N n, f k‖ ≤ ∑ k ∈ _, ‖f k‖ := norm_sum_le _ _
_ ≤ ∑ k ∈ _, g k := sum_le_sum fun x _ => hf x
_ ≤ ‖∑ k ... | 9 | 8,103.083928 | 2 | 1.4 | 5 | 1,502 |
import Mathlib.CategoryTheory.Limits.Shapes.Terminal
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
#align_import category_theory.limits.shapes.strict_initial from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe v u
namespace CategoryTheory
namespace Limits
open C... | Mathlib/CategoryTheory/Limits/Shapes/StrictInitial.lean | 74 | 77 | theorem IsInitial.strict_hom_ext (hI : IsInitial I) {A : C} (f g : A ⟶ I) : f = g := by |
haveI := hI.isIso_to f
haveI := hI.isIso_to g
exact eq_of_inv_eq_inv (hI.hom_ext (inv f) (inv g))
| 3 | 20.085537 | 1 | 1.333333 | 3 | 1,432 |
import Mathlib.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.Content
#align_import field_theory.ratfunc from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
universe u v
noncompu... | Mathlib/FieldTheory/RatFunc/Basic.lean | 117 | 118 | theorem ofFractionRing_neg (p : FractionRing K[X]) :
ofFractionRing (-p) = -ofFractionRing p := by | simp only [Neg.neg, RatFunc.neg]
| 1 | 2.718282 | 0 | 0.416667 | 12 | 404 |
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.MvPolynomial.Rename
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
#align_import data.mv_polynomial.variables from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Finsupp Ad... | Mathlib/Algebra/MvPolynomial/Degrees.lean | 358 | 362 | theorem totalDegree_eq (p : MvPolynomial σ R) :
p.totalDegree = p.support.sup fun m => Multiset.card (toMultiset m) := by |
rw [totalDegree]
congr; funext m
exact (Finsupp.card_toMultiset _).symm
| 3 | 20.085537 | 1 | 0.846154 | 13 | 743 |
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Ideal.Over
import Mathlib.RingTheory.JacobsonIdeal
#align_import ring_theory.jacobson from "leanprover-community/mathlib"@"a7c017d750512a352b623b1824d75da5998457d0"
set_option autoImplicit true
universe u
namespace Ideal
open Polynomial
... | Mathlib/RingTheory/Jacobson.lean | 303 | 355 | theorem isIntegral_isLocalization_polynomial_quotient
(P : Ideal R[X]) (pX : R[X]) (hpX : pX ∈ P) [Algebra (R ⧸ P.comap (C : R →+* R[X])) Rₘ]
[IsLocalization.Away (pX.map (Quotient.mk (P.comap (C : R →+* R[X])))).leadingCoeff Rₘ]
[Algebra (R[X] ⧸ P) Sₘ] [IsLocalization ((Submonoid.powers (pX.map (Quotient.m... |
let P' : Ideal R := P.comap C
let M : Submonoid (R ⧸ P') :=
Submonoid.powers (pX.map (Quotient.mk (P.comap (C : R →+* R[X])))).leadingCoeff
let M' : Submonoid (R[X] ⧸ P) :=
(Submonoid.powers (pX.map (Quotient.mk (P.comap (C : R →+* R[X])))).leadingCoeff).map
(quotientMap P C le_rfl)
let φ : R ⧸ P... | 46 | 94,961,194,206,024,480,000 | 2 | 2 | 4 | 2,189 |
import Mathlib.Analysis.BoxIntegral.Partition.Filter
import Mathlib.Analysis.BoxIntegral.Partition.Measure
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Init.Data.Bool.Lemmas
#align_import analysis.box_integral.basic from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open... | Mathlib/Analysis/BoxIntegral/Basic.lean | 143 | 145 | theorem integralSum_neg (f : ℝⁿ → E) (vol : ι →ᵇᵃ E →L[ℝ] F) (π : TaggedPrepartition I) :
integralSum (-f) vol π = -integralSum f vol π := by |
simp only [integralSum, Pi.neg_apply, (vol _).map_neg, Finset.sum_neg_distrib]
| 1 | 2.718282 | 0 | 1 | 6 | 835 |
import Mathlib.Algebra.Module.Equiv
import Mathlib.Data.DFinsupp.Basic
import Mathlib.Data.Finsupp.Basic
#align_import data.finsupp.to_dfinsupp from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
variable {ι : Type*} {R : Type*} {M : Type*}
section Defs
def Finsupp.toDFinsupp [Zer... | Mathlib/Data/Finsupp/ToDFinsupp.lean | 88 | 91 | theorem Finsupp.toDFinsupp_single (i : ι) (m : M) :
(Finsupp.single i m).toDFinsupp = DFinsupp.single i m := by |
ext
simp [Finsupp.single_apply, DFinsupp.single_apply]
| 2 | 7.389056 | 1 | 1 | 4 | 1,019 |
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Polynomial.IntegralNormalization
#align_import ring_theory.algebraic from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
universe u v w
open scoped Classical
open Polynomi... | Mathlib/RingTheory/Algebraic.lean | 113 | 115 | theorem isAlgebraic_one [Nontrivial R] : IsAlgebraic R (1 : A) := by |
rw [← _root_.map_one (algebraMap R A)]
exact isAlgebraic_algebraMap 1
| 2 | 7.389056 | 1 | 1.125 | 8 | 1,205 |
import Mathlib.MeasureTheory.Integral.Lebesgue
open Set hiding restrict restrict_apply
open Filter ENNReal NNReal MeasureTheory.Measure
namespace MeasureTheory
variable {α : Type*} {m0 : MeasurableSpace α} {μ : Measure α}
noncomputable
def Measure.withDensity {m : MeasurableSpace α} (μ : Measure α) (f : α → ℝ≥... | Mathlib/MeasureTheory/Measure/WithDensity.lean | 116 | 119 | theorem withDensity_sum {ι : Type*} {m : MeasurableSpace α} (μ : ι → Measure α) (f : α → ℝ≥0∞) :
(sum μ).withDensity f = sum fun n => (μ n).withDensity f := by |
ext1 s hs
simp_rw [sum_apply _ hs, withDensity_apply f hs, restrict_sum μ hs, lintegral_sum_measure]
| 2 | 7.389056 | 1 | 1.272727 | 11 | 1,348 |
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 | 40 | 43 | theorem inv_norm_smul_mem_closed_unit_ball (x : E) :
‖x‖⁻¹ • x ∈ closedBall (0 : E) 1 := by |
simp only [mem_closedBall_zero_iff, norm_smul, norm_inv, norm_norm, ← div_eq_inv_mul,
div_self_le_one]
| 2 | 7.389056 | 1 | 0.9 | 10 | 783 |
import Mathlib.Analysis.Calculus.TangentCone
import Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics
#align_import analysis.calculus.fderiv.basic from "leanprover-community/mathlib"@"41bef4ae1254365bc190aee63b947674d2977f01"
open Filter Asymptotics ContinuousLinearMap Set Metric
open scoped Classical
open To... | Mathlib/Analysis/Calculus/FDeriv/Basic.lean | 246 | 276 | theorem HasFDerivWithinAt.lim (h : HasFDerivWithinAt f f' s x) {α : Type*} (l : Filter α)
{c : α → 𝕜} {d : α → E} {v : E} (dtop : ∀ᶠ n in l, x + d n ∈ s)
(clim : Tendsto (fun n => ‖c n‖) l atTop) (cdlim : Tendsto (fun n => c n • d n) l (𝓝 v)) :
Tendsto (fun n => c n • (f (x + d n) - f x)) l (𝓝 (f' v)) :=... |
have tendsto_arg : Tendsto (fun n => x + d n) l (𝓝[s] x) := by
conv in 𝓝[s] x => rw [← add_zero x]
rw [nhdsWithin, tendsto_inf]
constructor
· apply tendsto_const_nhds.add (tangentConeAt.lim_zero l clim cdlim)
· rwa [tendsto_principal]
have : (fun y => f y - f x - f' (y - x)) =o[𝓝[s] x] fun y... | 27 | 532,048,240,601.79865 | 2 | 1.333333 | 6 | 1,387 |
import Mathlib.NumberTheory.LegendreSymbol.Basic
import Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum
#align_import number_theory.legendre_symbol.quadratic_reciprocity from "leanprover-community/mathlib"@"5b2fe80501ff327b9109fb09b7cc8c325cd0d7d9"
open Nat
section Values
variable {p : ℕ} [Fact p.Pri... | Mathlib/NumberTheory/LegendreSymbol/QuadraticReciprocity.lean | 158 | 162 | theorem quadratic_reciprocity_three_mod_four (hp : p % 4 = 3) (hq : q % 4 = 3) :
legendreSym q p = -legendreSym p q := by |
let nop := @neg_one_pow_div_two_of_three_mod_four
rw [quadratic_reciprocity', pow_mul, nop hp, nop hq, neg_one_mul] <;>
rwa [← Prime.mod_two_eq_one_iff_ne_two, odd_of_mod_four_eq_three]
| 3 | 20.085537 | 1 | 1.5 | 8 | 1,563 |
import Mathlib.Data.SetLike.Basic
import Mathlib.Data.Finset.Preimage
import Mathlib.ModelTheory.Semantics
#align_import model_theory.definability from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u v w u₁
namespace Set
variable {M : Type w} (A : Set M) (L : FirstOrder.Lang... | Mathlib/ModelTheory/Definability.lean | 116 | 122 | theorem Definable.union {f g : Set (α → M)} (hf : A.Definable L f) (hg : A.Definable L g) :
A.Definable L (f ∪ g) := by |
rcases hf with ⟨φ, hφ⟩
rcases hg with ⟨θ, hθ⟩
refine ⟨φ ⊔ θ, ?_⟩
ext
rw [hφ, hθ, mem_setOf_eq, Formula.realize_sup, mem_union, mem_setOf_eq, mem_setOf_eq]
| 5 | 148.413159 | 2 | 1.6 | 10 | 1,740 |
import Mathlib.Order.Interval.Multiset
#align_import data.nat.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
-- TODO
-- assert_not_exists Ring
open Finset Nat
variable (a b c : ℕ)
namespace Nat
instance instLocallyFiniteOrder : LocallyFiniteOrder ℕ where
finsetIcc a b... | Mathlib/Order/Interval/Finset/Nat.lean | 67 | 67 | theorem Ico_zero_eq_range : Ico 0 = range := by | rw [← Nat.bot_eq_zero, ← Iio_eq_Ico, Iio_eq_range]
| 1 | 2.718282 | 0 | 0.125 | 8 | 253 |
import Mathlib.Data.Finsupp.Encodable
import Mathlib.LinearAlgebra.Pi
import Mathlib.LinearAlgebra.Span
import Mathlib.Data.Set.Countable
#align_import linear_algebra.finsupp from "leanprover-community/mathlib"@"9d684a893c52e1d6692a504a118bfccbae04feeb"
noncomputable section
open Set LinearMap Submodule
namespa... | Mathlib/LinearAlgebra/Finsupp.lean | 266 | 284 | theorem disjoint_lsingle_lsingle (s t : Set α) (hs : Disjoint s t) :
Disjoint (⨆ a ∈ s, LinearMap.range (lsingle a : M →ₗ[R] α →₀ M))
(⨆ a ∈ t, LinearMap.range (lsingle a : M →ₗ[R] α →₀ M)) := by |
-- Porting note: 2 placeholders are added to prevent timeout.
refine
(Disjoint.mono
(lsingle_range_le_ker_lapply s sᶜ ?_)
(lsingle_range_le_ker_lapply t tᶜ ?_))
?_
· apply disjoint_compl_right
· apply disjoint_compl_right
rw [disjoint_iff_inf_le]
refine le_trans (le_iInf fun i => ?_) ... | 16 | 8,886,110.520508 | 2 | 1 | 9 | 1,040 |
import Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots
import Mathlib.FieldTheory.Finite.Trace
import Mathlib.Algebra.Group.AddChar
import Mathlib.Data.ZMod.Units
import Mathlib.Analysis.Complex.Polynomial
#align_import number_theory.legendre_symbol.add_character from "leanprover-community/mathlib"@"0723536a0522d24fc2... | Mathlib/NumberTheory/LegendreSymbol/AddCharacter.lean | 76 | 83 | theorem to_mulShift_inj_of_isPrimitive {ψ : AddChar R R'} (hψ : IsPrimitive ψ) :
Function.Injective ψ.mulShift := by |
intro a b h
apply_fun fun x => x * mulShift ψ (-b) at h
simp only [mulShift_mul, mulShift_zero, add_right_neg] at h
have h₂ := hψ (a + -b)
rw [h, isNontrivial_iff_ne_trivial, ← sub_eq_add_neg, sub_ne_zero] at h₂
exact not_not.mp fun h => h₂ h rfl
| 6 | 403.428793 | 2 | 1.5 | 6 | 1,658 |
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 | 116 | 117 | theorem logb_rpow_eq_mul_logb_of_pos (hx : 0 < x) : logb b (x ^ y) = y * logb b x := by |
rw [logb, log_rpow hx, logb, mul_div_assoc]
| 1 | 2.718282 | 0 | 0.25 | 20 | 300 |
import Mathlib.Data.Finset.Card
#align_import data.finset.prod from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
assert_not_exists MonoidWithZero
open Multiset
variable {α β γ : Type*}
namespace Finset
section Prod
variable {s s' : Finset α} {t t' : Finset β} {a : α} {b : β}
... | Mathlib/Data/Finset/Prod.lean | 81 | 83 | theorem product_image_snd [DecidableEq β] (ht : s.Nonempty) : (s ×ˢ t).image Prod.snd = t := by |
ext i
simp [mem_image, ht.exists_mem]
| 2 | 7.389056 | 1 | 0.666667 | 3 | 616 |
import Mathlib.AlgebraicTopology.SplitSimplicialObject
import Mathlib.AlgebraicTopology.DoldKan.PInfty
#align_import algebraic_topology.dold_kan.functor_gamma from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190f0b504"
noncomputable section
open CategoryTheory CategoryTheory.Category CategoryT... | Mathlib/AlgebraicTopology/DoldKan/FunctorGamma.lean | 105 | 107 | theorem mapMono_id : mapMono K (𝟙 Δ) = 𝟙 _ := by |
unfold mapMono
simp only [eq_self_iff_true, eqToHom_refl, dite_eq_ite, if_true]
| 2 | 7.389056 | 1 | 1.5 | 4 | 1,648 |
import Mathlib.Algebra.Exact
import Mathlib.RingTheory.TensorProduct.Basic
section Modules
open TensorProduct LinearMap
section Semiring
variable {R : Type*} [CommSemiring R] {M N P Q: Type*}
[AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [AddCommMonoid Q]
[Module R M] [Module R N] [Module R P] [... | Mathlib/LinearAlgebra/TensorProduct/RightExactness.lean | 149 | 158 | theorem LinearMap.rTensor_range :
range (rTensor Q g) =
range (rTensor Q (Submodule.subtype (range g))) := by |
have : g = (Submodule.subtype _).comp g.rangeRestrict := rfl
nth_rewrite 1 [this]
rw [rTensor_comp]
apply range_comp_of_range_eq_top
rw [range_eq_top]
apply rTensor_surjective
rw [← range_eq_top, range_rangeRestrict]
| 7 | 1,096.633158 | 2 | 2 | 4 | 2,132 |
import Mathlib.CategoryTheory.Category.Grpd
import Mathlib.CategoryTheory.Groupoid
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.Topology.Homotopy.Path
import Mathlib.Data.Set.Subsingleton
#align_import algebraic_topology.fundamental_groupoid.basic from "leanprover-community/mathlib"@"3d7987cda72abc473... | Mathlib/AlgebraicTopology/FundamentalGroupoid/Basic.lean | 200 | 202 | theorem transAssocReparamAux_mem_I (t : I) : transAssocReparamAux t ∈ I := by |
unfold transAssocReparamAux
split_ifs <;> constructor <;> linarith [unitInterval.le_one t, unitInterval.nonneg t]
| 2 | 7.389056 | 1 | 1.166667 | 12 | 1,229 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.deriv.mul from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
noncomputable section
open scoped Classical... | Mathlib/Analysis/Calculus/Deriv/Mul.lean | 487 | 492 | theorem HasDerivWithinAt.clm_apply (hc : HasDerivWithinAt c c' s x)
(hu : HasDerivWithinAt u u' s x) :
HasDerivWithinAt (fun y => (c y) (u y)) (c' (u x) + c x u') s x := by |
have := (hc.hasFDerivWithinAt.clm_apply hu.hasFDerivWithinAt).hasDerivWithinAt
rwa [add_apply, comp_apply, flip_apply, smulRight_apply, smulRight_apply, one_apply, one_smul,
one_smul, add_comm] at this
| 3 | 20.085537 | 1 | 1 | 25 | 997 |
import Mathlib.Data.Set.Lattice
#align_import data.set.intervals.disjoint from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
universe u v w
variable {ι : Sort u} {α : Type v} {β : Type w}
open Set
open OrderDual (toDual)
namespace Set
section Preorder
variable [Preorder α] {a b c... | Mathlib/Order/Interval/Set/Disjoint.lean | 97 | 98 | theorem iUnion_Icc_left (b : α) : ⋃ a, Icc a b = Iic b := by |
simp only [← Ici_inter_Iic, ← iUnion_inter, iUnion_Ici, univ_inter]
| 1 | 2.718282 | 0 | 0.333333 | 18 | 364 |
import Mathlib.Data.Set.Basic
#align_import data.bundle from "leanprover-community/mathlib"@"e473c3198bb41f68560cab68a0529c854b618833"
open Function Set
namespace Bundle
variable {B F : Type*} (E : B → Type*)
@[ext]
structure TotalSpace (F : Type*) (E : B → Type*) where
proj : B
snd : E proj
#align bund... | Mathlib/Data/Bundle.lean | 69 | 70 | theorem TotalSpace.mk_cast {x x' : B} (h : x = x') (b : E x) :
.mk' F x' (cast (congr_arg E h) b) = TotalSpace.mk x b := by | subst h; rfl
| 1 | 2.718282 | 0 | 0.666667 | 3 | 578 |
import Mathlib.Algebra.Algebra.Unitization
import Mathlib.Algebra.Star.NonUnitalSubalgebra
import Mathlib.Algebra.Star.Subalgebra
import Mathlib.GroupTheory.GroupAction.Ring
section Subalgebra
variable {R A : Type*} [CommSemiring R] [Semiring A] [Algebra R A]
def Subalgebra.toNonUnitalSubalgebra (S : Subalgebr... | Mathlib/Algebra/Algebra/Subalgebra/Unitization.lean | 70 | 71 | theorem Subalgebra.toNonUnitalSubalgebra_toSubalgebra (S : Subalgebra R A) :
S.toNonUnitalSubalgebra.toSubalgebra S.one_mem = S := by | cases S; rfl
| 1 | 2.718282 | 0 | 0.75 | 4 | 668 |
import Mathlib.MeasureTheory.OuterMeasure.OfFunction
import Mathlib.MeasureTheory.PiSystem
#align_import measure_theory.measure.outer_measure from "leanprover-community/mathlib"@"343e80208d29d2d15f8050b929aa50fe4ce71b55"
noncomputable section
open Set Function Filter
open scoped Classical NNReal Topology ENNReal
... | Mathlib/MeasureTheory/OuterMeasure/Caratheodory.lean | 97 | 100 | theorem isCaratheodory_inter (h₁ : IsCaratheodory m s₁) (h₂ : IsCaratheodory m s₂) :
IsCaratheodory m (s₁ ∩ s₂) := by |
rw [← isCaratheodory_compl_iff, Set.compl_inter]
exact isCaratheodory_union _ (isCaratheodory_compl _ h₁) (isCaratheodory_compl _ h₂)
| 2 | 7.389056 | 1 | 0.6 | 5 | 530 |
import Mathlib.Analysis.Calculus.BumpFunction.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
#align_import analysis.calculus.bump_function_inner from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputable section
open F... | Mathlib/Analysis/Calculus/BumpFunction/Normed.lean | 117 | 123 | theorem normed_le_div_measure_closedBall_rIn (x : E) :
f.normed μ x ≤ 1 / (μ (closedBall c f.rIn)).toReal := by |
rw [normed_def]
gcongr
· exact ENNReal.toReal_pos (measure_closedBall_pos _ _ f.rIn_pos).ne' measure_closedBall_lt_top.ne
· exact f.le_one
· exact f.measure_closedBall_le_integral μ
| 5 | 148.413159 | 2 | 0.818182 | 11 | 722 |
import Mathlib.Data.Fin.Tuple.Basic
import Mathlib.Data.List.Join
#align_import data.list.of_fn from "leanprover-community/mathlib"@"bf27744463e9620ca4e4ebe951fe83530ae6949b"
universe u
variable {α : Type u}
open Nat
namespace List
#noalign list.length_of_fn_aux
@[simp]
theorem length_ofFn_go {n} (f : Fin n ... | Mathlib/Data/List/OfFn.lean | 75 | 77 | theorem nthLe_ofFn {n} (f : Fin n → α) (i : Fin n) :
nthLe (ofFn f) i ((length_ofFn f).symm ▸ i.2) = f i := by |
simp [nthLe]
| 1 | 2.718282 | 0 | 0.6 | 10 | 535 |
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2
#align_import measure_theory.function.conditional_expectation.condexp_L1 from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
noncomputable section
open TopologicalSpace MeasureTheory.Lp Filter ContinuousLinearMap
o... | Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean | 116 | 125 | theorem condexpIndL1Fin_smul' [NormedSpace ℝ F] [SMulCommClass ℝ 𝕜 F] (hs : MeasurableSet s)
(hμs : μ s ≠ ∞) (c : 𝕜) (x : F) :
condexpIndL1Fin hm hs hμs (c • x) = c • condexpIndL1Fin hm hs hμs x := by |
ext1
refine (Memℒp.coeFn_toLp q).trans ?_
refine EventuallyEq.trans ?_ (Lp.coeFn_smul _ _).symm
rw [condexpIndSMul_smul' hs hμs c x]
refine (Lp.coeFn_smul _ _).trans ?_
refine (condexpIndL1Fin_ae_eq_condexpIndSMul hm hs hμs x).mono fun y hy => ?_
simp only [Pi.smul_apply, hy]
| 7 | 1,096.633158 | 2 | 2 | 4 | 2,398 |
import Mathlib.MeasureTheory.Measure.Typeclasses
#align_import measure_theory.measure.sub from "leanprover-community/mathlib"@"562bbf524c595c153470e53d36c57b6f891cc480"
open Set
namespace MeasureTheory
namespace Measure
noncomputable instance instSub {α : Type*} [MeasurableSpace α] : Sub (Measure α) :=
⟨fun ... | Mathlib/MeasureTheory/Measure/Sub.lean | 100 | 102 | theorem sub_add_cancel_of_le [IsFiniteMeasure ν] (h₁ : ν ≤ μ) : μ - ν + ν = μ := by |
ext1 s h_s_meas
rw [add_apply, sub_apply h_s_meas h₁, tsub_add_cancel_of_le (h₁ s)]
| 2 | 7.389056 | 1 | 1.25 | 4 | 1,324 |
import Mathlib.Init.Function
import Mathlib.Logic.Function.Basic
#align_import data.sigma.basic from "leanprover-community/mathlib"@"a148d797a1094ab554ad4183a4ad6f130358ef64"
open Function
section Sigma
variable {α α₁ α₂ : Type*} {β : α → Type*} {β₁ : α₁ → Type*} {β₂ : α₂ → Type*}
namespace Sigma
instance inst... | Mathlib/Data/Sigma/Basic.lean | 70 | 71 | theorem ext_iff {x₀ x₁ : Sigma β} : x₀ = x₁ ↔ x₀.1 = x₁.1 ∧ HEq x₀.2 x₁.2 := by |
cases x₀; cases x₁; exact Sigma.mk.inj_iff
| 1 | 2.718282 | 0 | 0.5 | 4 | 421 |
import Mathlib.Analysis.Normed.Order.Lattice
import Mathlib.MeasureTheory.Function.LpSpace
#align_import measure_theory.function.lp_order from "leanprover-community/mathlib"@"5dc275ec639221ca4d5f56938eb966f6ad9bc89f"
set_option linter.uppercaseLean3 false
open TopologicalSpace MeasureTheory
open scoped ENNReal
... | Mathlib/MeasureTheory/Function/LpOrder.lean | 41 | 42 | theorem coeFn_le (f g : Lp E p μ) : f ≤ᵐ[μ] g ↔ f ≤ g := by |
rw [← Subtype.coe_le_coe, ← AEEqFun.coeFn_le]
| 1 | 2.718282 | 0 | 1 | 2 | 974 |
import Mathlib.AlgebraicTopology.SplitSimplicialObject
import Mathlib.AlgebraicTopology.DoldKan.Degeneracies
import Mathlib.AlgebraicTopology.DoldKan.FunctorN
#align_import algebraic_topology.dold_kan.split_simplicial_object from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190f0b504"
open Categ... | Mathlib/AlgebraicTopology/DoldKan/SplitSimplicialObject.lean | 73 | 85 | theorem σ_comp_πSummand_id_eq_zero {n : ℕ} (i : Fin (n + 1)) :
X.σ i ≫ s.πSummand (IndexSet.id (op [n + 1])) = 0 := by |
apply s.hom_ext'
intro A
dsimp only [SimplicialObject.σ]
rw [comp_zero, s.cofan_inj_epi_naturality_assoc A (SimplexCategory.σ i).op,
cofan_inj_πSummand_eq_zero]
rw [ne_comm]
change ¬(A.epiComp (SimplexCategory.σ i).op).EqId
rw [IndexSet.eqId_iff_len_eq]
have h := SimplexCategory.len_le_of_epi (infe... | 11 | 59,874.141715 | 2 | 1.428571 | 7 | 1,516 |
import Mathlib.Data.List.Join
#align_import data.list.permutation from "leanprover-community/mathlib"@"dd71334db81d0bd444af1ee339a29298bef40734"
-- Make sure we don't import algebra
assert_not_exists Monoid
open Nat
variable {α β : Type*}
namespace List
theorem permutationsAux2_fst (t : α) (ts : List α) (r : L... | Mathlib/Data/List/Permutation.lean | 133 | 137 | theorem map_map_permutations'Aux (f : α → β) (t : α) (ts : List α) :
map (map f) (permutations'Aux t ts) = permutations'Aux (f t) (map f ts) := by |
induction' ts with a ts ih
· rfl
· simp only [permutations'Aux, map_cons, map_map, ← ih, cons.injEq, true_and, Function.comp_def]
| 3 | 20.085537 | 1 | 1 | 9 | 903 |
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