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.MeasureTheory.Function.ConditionalExpectation.Indicator
import Mathlib.MeasureTheory.Function.UniformIntegrable
import Mathlib.MeasureTheory.Decomposition.RadonNikodym
#align_import measure_theory.function.conditional_expectation.real from "leanprover-community/mathlib"@"b2ff9a3d7a15fd5b0f060b135421d6a... | Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean | 59 | 89 | theorem snorm_one_condexp_le_snorm (f : α → ℝ) : snorm (μ[f|m]) 1 μ ≤ snorm f 1 μ := by |
by_cases hf : Integrable f μ
swap; · rw [condexp_undef hf, snorm_zero]; exact zero_le _
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm, snorm_zero]; exact zero_le _
by_cases hsig : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hsig, snorm_zero]; exact zero_le _
calc
snorm (... | 30 | 10,686,474,581,524.463 | 2 | 2 | 4 | 2,408 |
import Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral
#align_import analysis.special_functions.gamma.bohr_mollerup from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
set_option linter.uppercaseLean3 false
noncomputable section
open Filter Set MeasureTheory
open scoped Na... | Mathlib/Analysis/SpecialFunctions/Gamma/BohrMollerup.lean | 164 | 173 | theorem convexOn_log_Gamma : ConvexOn ℝ (Ioi 0) (log ∘ Gamma) := by |
refine convexOn_iff_forall_pos.mpr ⟨convex_Ioi _, fun x hx y hy a b ha hb hab => ?_⟩
have : b = 1 - a := by linarith
subst this
simp_rw [Function.comp_apply, smul_eq_mul]
simp only [mem_Ioi] at hx hy
rw [← log_rpow, ← log_rpow, ← log_mul]
· gcongr
exact Gamma_mul_add_mul_le_rpow_Gamma_mul_rpow_Gamma ... | 9 | 8,103.083928 | 2 | 2 | 2 | 2,350 |
import Mathlib.CategoryTheory.Abelian.Basic
#align_import category_theory.idempotents.basic from "leanprover-community/mathlib"@"3a061790136d13594ec10c7c90d202335ac5d854"
open CategoryTheory
open CategoryTheory.Category
open CategoryTheory.Limits
open CategoryTheory.Preadditive
open Opposite
namespace Catego... | Mathlib/CategoryTheory/Idempotents/Basic.lean | 143 | 154 | theorem split_iff_of_iso {X X' : C} (φ : X ≅ X') (p : X ⟶ X) (p' : X' ⟶ X')
(hpp' : p ≫ φ.hom = φ.hom ≫ p') :
(∃ (Y : C) (i : Y ⟶ X) (e : X ⟶ Y), i ≫ e = 𝟙 Y ∧ e ≫ i = p) ↔
∃ (Y' : C) (i' : Y' ⟶ X') (e' : X' ⟶ Y'), i' ≫ e' = 𝟙 Y' ∧ e' ≫ i' = p' := by |
constructor
· exact split_imp_of_iso φ p p' hpp'
· apply split_imp_of_iso φ.symm p' p
rw [← comp_id p, ← φ.hom_inv_id]
slice_rhs 2 3 => rw [hpp']
slice_rhs 1 2 => erw [φ.inv_hom_id]
simp only [id_comp]
rfl
| 8 | 2,980.957987 | 2 | 1.6 | 5 | 1,735 |
import Mathlib.Geometry.Manifold.ContMDiff.Basic
open Set Function Filter ChartedSpace SmoothManifoldWithCorners
open scoped Topology Manifold
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜]
-- declare a smooth manifold `M` over the pair `(E, H)`.
{E : Type*}
[NormedAddCommGroup E] [NormedSpace 𝕜 E] {H ... | Mathlib/Geometry/Manifold/ContMDiff/Product.lean | 149 | 162 | theorem contMDiffWithinAt_fst {s : Set (M × N)} {p : M × N} :
ContMDiffWithinAt (I.prod J) I n Prod.fst s p := by |
/- porting note: `simp` fails to apply lemmas to `ModelProd`. Was
rw [contMDiffWithinAt_iff']
refine' ⟨continuousWithinAt_fst, _⟩
refine' contDiffWithinAt_fst.congr (fun y hy => _) _
· simp only [mfld_simps] at hy
simp only [hy, mfld_simps]
· simp only [mfld_simps]
-/
rw [contMDiffWithinAt_iff']
... | 12 | 162,754.791419 | 2 | 1.5 | 4 | 1,657 |
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.MeasureTheory.Constructions.Prod.Integral
import Mathlib.MeasureTheory.Function.LocallyIntegrable
import Mathlib.MeasureTheory.Group.Integral
import Mathlib.MeasureTheory.Group.Prod
import Mathlib.Measure... | Mathlib/Analysis/Convolution.lean | 118 | 128 | theorem convolution_integrand_bound_right_of_le_of_subset {C : ℝ} (hC : ∀ i, ‖g i‖ ≤ C) {x t : G}
{s u : Set G} (hx : x ∈ s) (hu : -tsupport g + s ⊆ u) :
‖L (f t) (g (x - t))‖ ≤ u.indicator (fun t => ‖L‖ * ‖f t‖ * C) t := by |
-- Porting note: had to add `f := _`
refine le_indicator (f := fun t ↦ ‖L (f t) (g (x - t))‖) (fun t _ => ?_) (fun t ht => ?_) t
· apply_rules [L.le_of_opNorm₂_le_of_le, le_rfl]
· have : x - t ∉ support g := by
refine mt (fun hxt => hu ?_) ht
refine ⟨_, Set.neg_mem_neg.mpr (subset_closure hxt), _, ... | 8 | 2,980.957987 | 2 | 1.6 | 5 | 1,730 |
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 | 133 | 138 | theorem definable_finset_sup {ι : Type*} {f : ι → Set (α → M)} (hf : ∀ i, A.Definable L (f i))
(s : Finset ι) : A.Definable L (s.sup f) := by |
classical
refine Finset.induction definable_empty (fun i s _ h => ?_) s
rw [Finset.sup_insert]
exact (hf i).union h
| 4 | 54.59815 | 2 | 1.6 | 10 | 1,740 |
import Mathlib.CategoryTheory.Monoidal.Braided.Basic
import Mathlib.Algebra.Category.ModuleCat.Monoidal.Basic
#align_import algebra.category.Module.monoidal.symmetric from "leanprover-community/mathlib"@"74403a3b2551b0970855e14ef5e8fd0d6af1bfc2"
suppress_compilation
universe v w x u
open CategoryTheory MonoidalC... | Mathlib/Algebra/Category/ModuleCat/Monoidal/Symmetric.lean | 34 | 38 | theorem braiding_naturality {X₁ X₂ Y₁ Y₂ : ModuleCat.{u} R} (f : X₁ ⟶ Y₁) (g : X₂ ⟶ Y₂) :
(f ⊗ g) ≫ (Y₁.braiding Y₂).hom = (X₁.braiding X₂).hom ≫ (g ⊗ f) := by |
apply TensorProduct.ext'
intro x y
rfl
| 3 | 20.085537 | 1 | 1.2 | 5 | 1,276 |
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
#align_import linear_algebra.affine_space.restrict from "leanprover-community/mathlib"@"09258fb7f75d741b7eda9fa18d5c869e2135d9f1"
variable {k V₁ P₁ V₂ P₂ : Type*} [Ring k] [AddCommGroup V₁] [AddCommGroup V₂] [Module k V₁]
[Module k V₂] [AddTorsor V₁ P₁] [A... | Mathlib/LinearAlgebra/AffineSpace/Restrict.lean | 33 | 36 | theorem AffineSubspace.nonempty_map {E : AffineSubspace k P₁} [Ene : Nonempty E] {φ : P₁ →ᵃ[k] P₂} :
Nonempty (E.map φ) := by |
obtain ⟨x, hx⟩ := id Ene
exact ⟨⟨φ x, AffineSubspace.mem_map.mpr ⟨x, hx, rfl⟩⟩⟩
| 2 | 7.389056 | 1 | 1.25 | 4 | 1,334 |
import Mathlib.Algebra.Group.Prod
import Mathlib.Data.Set.Lattice
#align_import data.nat.pairing from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
assert_not_exists MonoidWithZero
open Prod Decidable Function
namespace Nat
-- Porting note: no pp_nodot
--@[pp_nodot]
def pair (a b : ... | Mathlib/Data/Nat/Pairing.lean | 93 | 100 | theorem unpair_lt {n : ℕ} (n1 : 1 ≤ n) : (unpair n).1 < n := by |
let s := sqrt n
simp only [unpair, ge_iff_le, Nat.sub_le_iff_le_add]
by_cases h : n - s * s < s <;> simp [h]
· exact lt_of_lt_of_le h (sqrt_le_self _)
· simp at h
have s0 : 0 < s := sqrt_pos.2 n1
exact lt_of_le_of_lt h (Nat.sub_lt n1 (Nat.mul_pos s0 s0))
| 7 | 1,096.633158 | 2 | 1.2 | 10 | 1,270 |
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Analysis.NormedSpace.OperatorNorm.Basic
import Mathlib.Analysis.NormedSpace.Pointwise
#align_import analysis.normed_space.is_R_or_C from "leanprover-community/mathlib"@"3f655f5297b030a87d641ad4e825af8d9679eb0b"
open Metric
variable {𝕜 : Type*} [RCLike 𝕜] {E :... | Mathlib/Analysis/NormedSpace/RCLike.lean | 55 | 75 | theorem LinearMap.bound_of_sphere_bound {r : ℝ} (r_pos : 0 < r) (c : ℝ) (f : E →ₗ[𝕜] 𝕜)
(h : ∀ z ∈ sphere (0 : E) r, ‖f z‖ ≤ c) (z : E) : ‖f z‖ ≤ c / r * ‖z‖ := by |
by_cases z_zero : z = 0
· rw [z_zero]
simp only [LinearMap.map_zero, norm_zero, mul_zero]
exact le_rfl
set z₁ := ((r : 𝕜) * (‖z‖ : 𝕜)⁻¹) • z with hz₁
have norm_f_z₁ : ‖f z₁‖ ≤ c := by
apply h
rw [mem_sphere_zero_iff_norm]
exact norm_smul_inv_norm' r_pos.le z_zero
have r_ne_zero : (r : �... | 19 | 178,482,300.963187 | 2 | 1.2 | 5 | 1,253 |
import Mathlib.Order.CompleteLattice
import Mathlib.Data.Finset.Lattice
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Category.Preorder
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.FiniteLimits
#align_import category_theory.limi... | Mathlib/CategoryTheory/Limits/Lattice.lean | 85 | 93 | theorem finite_product_eq_finset_inf [SemilatticeInf α] [OrderTop α] {ι : Type u} [Fintype ι]
(f : ι → α) : ∏ᶜ f = Fintype.elems.inf f := by |
trans
· exact
(IsLimit.conePointUniqueUpToIso (limit.isLimit _)
(finiteLimitCone (Discrete.functor f)).isLimit).to_eq
change Finset.univ.inf (f ∘ discreteEquiv.toEmbedding) = Fintype.elems.inf f
simp only [← Finset.inf_map, Finset.univ_map_equiv_to_embedding]
rfl
| 7 | 1,096.633158 | 2 | 2 | 3 | 2,234 |
import Mathlib.CategoryTheory.Idempotents.Karoubi
#align_import category_theory.idempotents.functor_extension from "leanprover-community/mathlib"@"5f68029a863bdf76029fa0f7a519e6163c14152e"
namespace CategoryTheory
namespace Idempotents
open Category Karoubi
variable {C D E : Type*} [Category C] [Category D] [Ca... | Mathlib/CategoryTheory/Idempotents/FunctorExtension.lean | 35 | 40 | theorem natTrans_eq {F G : Karoubi C ⥤ D} (φ : F ⟶ G) (P : Karoubi C) :
φ.app P = F.map (decompId_i P) ≫ φ.app P.X ≫ G.map (decompId_p P) := by |
rw [← φ.naturality, ← assoc, ← F.map_comp]
conv_lhs => rw [← id_comp (φ.app P), ← F.map_id]
congr
apply decompId
| 4 | 54.59815 | 2 | 2 | 1 | 2,111 |
import Mathlib.LinearAlgebra.Matrix.BilinearForm
import Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.Vandermonde
import Mathlib.LinearAlgebra.Trace
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosu... | Mathlib/RingTheory/Trace.lean | 134 | 146 | theorem trace_trace_of_basis [Algebra S T] [IsScalarTower R S T] {ι κ : Type*} [Finite ι]
[Finite κ] (b : Basis ι R S) (c : Basis κ S T) (x : T) :
trace R S (trace S T x) = trace R T x := by |
haveI := Classical.decEq ι
haveI := Classical.decEq κ
cases nonempty_fintype ι
cases nonempty_fintype κ
rw [trace_eq_matrix_trace (b.smul c), trace_eq_matrix_trace b, trace_eq_matrix_trace c,
Matrix.trace, Matrix.trace, Matrix.trace, ← Finset.univ_product_univ, Finset.sum_product]
refine Finset.sum_con... | 10 | 22,026.465795 | 2 | 1 | 8 | 843 |
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
#align_import measure_theory.function.ae_measurable_order from "leanprover-community/mathlib"@"bf6a01357ff5684b1ebcd0f1a13be314fc82c0bf"
open MeasureTheory Set TopologicalSpace
open scoped Classical
open ENNReal NNReal
| Mathlib/MeasureTheory/Function/AEMeasurableOrder.lean | 34 | 107 | theorem MeasureTheory.aemeasurable_of_exist_almost_disjoint_supersets {α : Type*}
{m : MeasurableSpace α} (μ : Measure α) {β : Type*} [CompleteLinearOrder β] [DenselyOrdered β]
[TopologicalSpace β] [OrderTopology β] [SecondCountableTopology β] [MeasurableSpace β]
[BorelSpace β] (s : Set β) (s_count : s.Coun... |
haveI : Encodable s := s_count.toEncodable
have h' : ∀ p q, ∃ u v, MeasurableSet u ∧ MeasurableSet v ∧
{ x | f x < p } ⊆ u ∧ { x | q < f x } ⊆ v ∧ (p ∈ s → q ∈ s → p < q → μ (u ∩ v) = 0) := by
intro p q
by_cases H : p ∈ s ∧ q ∈ s ∧ p < q
· rcases h p H.1 q H.2.1 H.2.2 with ⟨u, v, hu, hv, h'u, h'v... | 67 | 125,236,317,084,221,370,000,000,000,000 | 2 | 2 | 2 | 2,428 |
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 | 92 | 95 | theorem mem_perpBisector_iff_dist_eq : c ∈ perpBisector p₁ p₂ ↔ dist c p₁ = dist c p₂ := by |
rw [dist_eq_norm_vsub V, dist_eq_norm_vsub V, ← real_inner_add_sub_eq_zero_iff,
vsub_sub_vsub_cancel_left, inner_add_left, add_eq_zero_iff_eq_neg, ← inner_neg_right,
neg_vsub_eq_vsub_rev, mem_perpBisector_iff_inner_eq_inner]
| 3 | 20.085537 | 1 | 0.777778 | 9 | 693 |
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Logic.Equiv.Fin
import Mathlib.Topology.Algebra.InfiniteSum.Module
#align_import analysis.analytic.basic from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
noncomputable... | Mathlib/Analysis/Analytic/Basic.lean | 187 | 202 | theorem isLittleO_of_lt_radius (h : ↑r < p.radius) :
∃ a ∈ Ioo (0 : ℝ) 1, (fun n => ‖p n‖ * (r : ℝ) ^ n) =o[atTop] (a ^ ·) := by |
have := (TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 1 4
rw [this]
-- Porting note: was
-- rw [(TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 1 4]
simp only [radius, lt_iSup_iff] at h
rcases h with ⟨t, C, hC, rt⟩
rw [ENNReal.coe_lt_coe, ← NNReal.coe_lt_coe] at ... | 14 | 1,202,604.284165 | 2 | 1.5 | 2 | 1,600 |
import Mathlib.Data.Bracket
import Mathlib.LinearAlgebra.Basic
#align_import algebra.lie.basic from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
universe u v w w₁ w₂
open Function
class LieRing (L : Type v) extends AddCommGroup L, Bracket L L where
protected add_lie : ∀ x y z ... | Mathlib/Algebra/Lie/Basic.lean | 169 | 171 | theorem lie_neg : ⁅x, -m⁆ = -⁅x, m⁆ := by |
rw [← sub_eq_zero, sub_neg_eq_add, ← lie_add]
simp
| 2 | 7.389056 | 1 | 0.6 | 5 | 526 |
import Mathlib.Data.Int.Order.Units
import Mathlib.Data.ZMod.IntUnitsPower
import Mathlib.RingTheory.TensorProduct.Basic
import Mathlib.LinearAlgebra.DirectSum.TensorProduct
import Mathlib.Algebra.DirectSum.Algebra
suppress_compilation
open scoped TensorProduct DirectSum
variable {R ι A B : Type*}
namespace Tens... | Mathlib/LinearAlgebra/TensorProduct/Graded/External.lean | 85 | 90 | theorem gradedCommAux_lof_tmul (i j : ι) (a : 𝒜 i) (b : ℬ j) :
gradedCommAux R 𝒜 ℬ (lof R _ 𝒜ℬ (i, j) (a ⊗ₜ b)) =
(-1 : ℤˣ)^(j * i) • lof R _ ℬ𝒜 (j, i) (b ⊗ₜ a) := by |
rw [gradedCommAux]
dsimp
simp [mul_comm i j]
| 3 | 20.085537 | 1 | 1.666667 | 6 | 1,805 |
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Analytic.CPolynomial
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.fderiv_analytic from "leanprover-community/mathlib"@"3bce8d800a6f2... | Mathlib/Analysis/Calculus/FDeriv/Analytic.lean | 469 | 474 | theorem iteratedFDeriv_zero_apply_diag : iteratedFDeriv 𝕜 0 f x = p 0 := by |
ext
convert (h.hasSum <| EMetric.mem_ball_self h.r_pos).tsum_eq.symm
· rw [iteratedFDeriv_zero_apply, add_zero]
· rw [tsum_eq_single 0 fun n hn ↦ by haveI := NeZero.mk hn; exact (p n).map_zero]
exact congr(p 0 $(Subsingleton.elim _ _))
| 5 | 148.413159 | 2 | 1.857143 | 7 | 1,925 |
import Mathlib.Data.Matrix.Basic
variable {l m n o : Type*}
universe u v w
variable {R : Type*} {α : Type v} {β : Type w}
namespace Matrix
def col (w : m → α) : Matrix m Unit α :=
of fun x _ => w x
#align matrix.col Matrix.col
-- TODO: set as an equation lemma for `col`, see mathlib4#3024
@[simp]
theorem col... | Mathlib/Data/Matrix/RowCol.lean | 154 | 158 | theorem vecMulVec_eq [Mul α] [AddCommMonoid α] (w : m → α) (v : n → α) :
vecMulVec w v = col w * row v := by |
ext
simp only [vecMulVec, mul_apply, Fintype.univ_punit, Finset.sum_singleton]
rfl
| 3 | 20.085537 | 1 | 1 | 14 | 798 |
import Mathlib.Algebra.GradedMonoid
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Algebra.MvPolynomial.Basic
#align_import ring_theory.mv_polynomial.weighted_homogeneous from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Fins... | Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean | 168 | 173 | theorem weightedHomogeneousSubmodule_eq_finsupp_supported (w : σ → M) (m : M) :
weightedHomogeneousSubmodule R w m = Finsupp.supported R R { d | weightedDegree w d = m } := by |
ext x
rw [mem_supported, Set.subset_def]
simp only [Finsupp.mem_support_iff, mem_coe]
rfl
| 4 | 54.59815 | 2 | 1.125 | 8 | 1,208 |
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 | 143 | 152 | theorem ascPochhammer_succ_comp_X_add_one (n : ℕ) :
(ascPochhammer S (n + 1)).comp (X + 1) =
ascPochhammer S (n + 1) + (n + 1) • (ascPochhammer S n).comp (X + 1) := by |
suffices (ascPochhammer ℕ (n + 1)).comp (X + 1) =
ascPochhammer ℕ (n + 1) + (n + 1) * (ascPochhammer ℕ n).comp (X + 1)
by simpa [map_comp] using congr_arg (Polynomial.map (Nat.castRingHom S)) this
nth_rw 2 [ascPochhammer_succ_left]
rw [← add_mul, ascPochhammer_succ_right ℕ n, mul_comp, mul_comm, add_co... | 7 | 1,096.633158 | 2 | 0.96 | 25 | 796 |
import Mathlib.Topology.Algebra.Module.WeakDual
import Mathlib.MeasureTheory.Integral.BoundedContinuousFunction
import Mathlib.MeasureTheory.Measure.HasOuterApproxClosed
#align_import measure_theory.measure.finite_measure from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable... | Mathlib/MeasureTheory/Measure/FiniteMeasure.lean | 213 | 217 | theorem eq_of_forall_toMeasure_apply_eq (μ ν : FiniteMeasure Ω)
(h : ∀ s : Set Ω, MeasurableSet s → (μ : Measure Ω) s = (ν : Measure Ω) s) : μ = ν := by |
apply Subtype.ext
ext1 s s_mble
exact h s s_mble
| 3 | 20.085537 | 1 | 1.2 | 5 | 1,255 |
import Mathlib.Analysis.InnerProductSpace.Adjoint
#align_import analysis.inner_product_space.positive from "leanprover-community/mathlib"@"caa58cbf5bfb7f81ccbaca4e8b8ac4bc2b39cc1c"
open InnerProductSpace RCLike ContinuousLinearMap
open scoped InnerProduct ComplexConjugate
namespace ContinuousLinearMap
variable... | Mathlib/Analysis/InnerProductSpace/Positive.lean | 71 | 74 | theorem isPositive_zero : IsPositive (0 : E →L[𝕜] E) := by |
refine ⟨isSelfAdjoint_zero _, fun x => ?_⟩
change 0 ≤ re ⟪_, _⟫
rw [zero_apply, inner_zero_left, ZeroHomClass.map_zero]
| 3 | 20.085537 | 1 | 0.875 | 8 | 762 |
import Mathlib.Data.Int.Bitwise
import Mathlib.Data.Int.Order.Lemmas
import Mathlib.Data.Set.Function
import Mathlib.Order.Interval.Set.Basic
#align_import data.int.lemmas from "leanprover-community/mathlib"@"09597669f02422ed388036273d8848119699c22f"
open Nat
namespace Int
theorem le_natCast_sub (m n : ℕ) : (m ... | Mathlib/Data/Int/Lemmas.lean | 45 | 47 | theorem natAbs_eq_iff_sq_eq {a b : ℤ} : a.natAbs = b.natAbs ↔ a ^ 2 = b ^ 2 := by |
rw [sq, sq]
exact natAbs_eq_iff_mul_self_eq
| 2 | 7.389056 | 1 | 0.818182 | 11 | 720 |
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Maps
open Finset
namespace SimpleGraph
variable {V : Type*} [DecidableEq V] (G : SimpleGraph V) (s t : V)
section AddEdge
def edge : SimpleGraph V := fromEdgeSet {s(s, t)}
lemma edge_adj (v w : V) : (edge s t).Adj v w ↔ ... | Mathlib/Combinatorics/SimpleGraph/Operations.lean | 171 | 175 | theorem edgeFinset_sup_edge [Fintype (edgeSet (G ⊔ edge s t))] (hn : ¬G.Adj s t) (h : s ≠ t) :
(G ⊔ edge s t).edgeFinset = G.edgeFinset.cons s(s, t) (by simp_all) := by |
letI := Classical.decEq V
rw [edgeFinset_sup, cons_eq_insert, insert_eq, union_comm]
simp_rw [edgeFinset, edge_edgeSet_of_ne h]; rfl
| 3 | 20.085537 | 1 | 1.111111 | 9 | 1,199 |
import Mathlib.RingTheory.TensorProduct.Basic
import Mathlib.Algebra.Module.ULift
#align_import ring_theory.is_tensor_product from "leanprover-community/mathlib"@"c4926d76bb9c5a4a62ed2f03d998081786132105"
universe u v₁ v₂ v₃ v₄
open TensorProduct
section IsTensorProduct
variable {R : Type*} [CommSemiring R]
va... | Mathlib/RingTheory/IsTensorProduct.lean | 60 | 65 | theorem TensorProduct.isTensorProduct : IsTensorProduct (TensorProduct.mk R M N) := by |
delta IsTensorProduct
convert_to Function.Bijective (LinearMap.id : M ⊗[R] N →ₗ[R] M ⊗[R] N) using 2
· apply TensorProduct.ext'
simp
· exact Function.bijective_id
| 5 | 148.413159 | 2 | 1.4 | 5 | 1,506 |
import Mathlib.Combinatorics.SimpleGraph.Subgraph
import Mathlib.Data.List.Rotate
#align_import combinatorics.simple_graph.connectivity from "leanprover-community/mathlib"@"b99e2d58a5e6861833fa8de11e51a81144258db4"
open Function
universe u v w
namespace SimpleGraph
variable {V : Type u} {V' : Type v} {V'' : Typ... | Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | 146 | 149 | theorem copy_cons {u v w u' w'} (h : G.Adj u v) (p : G.Walk v w) (hu : u = u') (hw : w = w') :
(Walk.cons h p).copy hu hw = Walk.cons (hu ▸ h) (p.copy rfl hw) := by |
subst_vars
rfl
| 2 | 7.389056 | 1 | 1 | 6 | 918 |
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.Order.Invertible
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.LinearAlgebra.AffineSpace.Slope
import Mathlib.LinearAlgebra.AffineSpace.Midpoint
import Mathlib.Tactic.FieldSimp
#align_import li... | Mathlib/LinearAlgebra/AffineSpace/Ordered.lean | 57 | 59 | theorem lineMap_strict_mono_left (ha : a < a') (hr : r < 1) : lineMap a b r < lineMap a' b r := by |
simp only [lineMap_apply_module]
exact add_lt_add_right (smul_lt_smul_of_pos_left ha (sub_pos.2 hr)) _
| 2 | 7.389056 | 1 | 1.222222 | 9 | 1,293 |
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 | 284 | 291 | theorem Monic.irreducible_iff_lt_natDegree_lt {p : R[X]} (hp : p.Monic) (hp1 : p ≠ 1) :
Irreducible p ↔ ∀ q, Monic q → natDegree q ∈ Finset.Ioc 0 (natDegree p / 2) → ¬ q ∣ p := by |
rw [hp.irreducible_iff_natDegree', and_iff_right hp1]
constructor
· rintro h g hg hdg ⟨f, rfl⟩
exact h f g (hg.of_mul_monic_left hp) hg (mul_comm f g) hdg
· rintro h f g - hg rfl hdg
exact h g hg hdg (dvd_mul_left g f)
| 6 | 403.428793 | 2 | 1.5 | 32 | 1,561 |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.List.MinMax
import Mathlib.Algebra.Tropical.Basic
import Mathlib.Order.ConditionallyCompleteLattice.Finset
#align_import algebra.tropical.big_operators from "leanprover-community/mathlib"@"d6fad0e5bf2d6f48da9175d25c3dc5706b3834ce"
variable {R S :... | Mathlib/Algebra/Tropical/BigOperators.lean | 40 | 43 | theorem List.trop_sum [AddMonoid R] (l : List R) : trop l.sum = List.prod (l.map trop) := by |
induction' l with hd tl IH
· simp
· simp [← IH]
| 3 | 20.085537 | 1 | 0.928571 | 14 | 793 |
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Group.Commute.Defs
import Mathlib.Algebra.Ring.Defs
import Mathlib.Data.Subtype
import Mathlib.Order.Notation
#align_import algebra.ring.idempotents from "leanprover-community/mathlib"@"655994e298904d7e5bbd1e18c95defd7b543eb94"
variable {M N S M₀ M₁ R G G₀... | Mathlib/Algebra/Ring/Idempotents.lean | 93 | 97 | theorem iff_eq_zero_or_one {p : G₀} : IsIdempotentElem p ↔ p = 0 ∨ p = 1 := by |
refine
Iff.intro (fun h => or_iff_not_imp_left.mpr fun hp => ?_) fun h =>
h.elim (fun hp => hp.symm ▸ zero) fun hp => hp.symm ▸ one
exact mul_left_cancel₀ hp (h.trans (mul_one p).symm)
| 4 | 54.59815 | 2 | 0.666667 | 3 | 624 |
import Mathlib.Data.Multiset.Nodup
import Mathlib.Data.List.NatAntidiagonal
#align_import data.multiset.nat_antidiagonal from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace Multiset
namespace Nat
def antidiagonal (n : ℕ) : Multiset (ℕ × ℕ) :=
List.Nat.antidiagonal n
#align... | Mathlib/Data/Multiset/NatAntidiagonal.lean | 70 | 74 | theorem antidiagonal_succ_succ' {n : ℕ} :
antidiagonal (n + 2) =
(0, n + 2) ::ₘ (n + 2, 0) ::ₘ (antidiagonal n).map (Prod.map Nat.succ Nat.succ) := by |
rw [antidiagonal_succ, antidiagonal_succ', map_cons, map_map, Prod.map_apply]
rfl
| 2 | 7.389056 | 1 | 0.333333 | 6 | 362 |
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 | 62 | 65 | theorem lcm_dvd_iff {a : α} : s.lcm f ∣ a ↔ ∀ b ∈ s, f b ∣ a := by |
apply Iff.trans Multiset.lcm_dvd
simp only [Multiset.mem_map, and_imp, exists_imp]
exact ⟨fun k b hb ↦ k _ _ hb rfl, fun k a' b hb h ↦ h ▸ k _ hb⟩
| 3 | 20.085537 | 1 | 1 | 13 | 925 |
import Mathlib.CategoryTheory.Sites.DenseSubsite
#align_import category_theory.sites.induced_topology from "leanprover-community/mathlib"@"ba43124c37cfe0009bbfc57505f9503ae0e8c1af"
namespace CategoryTheory
universe v u
open Limits Opposite Presieve
section
variable {C : Type*} [Category C] {D : Type*} [Catego... | Mathlib/CategoryTheory/Sites/InducedTopology.lean | 112 | 121 | theorem Functor.locallyCoverDense_of_isCoverDense [Full G] [G.IsCoverDense K] :
LocallyCoverDense K G := by |
intro X T
refine K.superset_covering ?_ (K.bind_covering T.property
fun Y f _ => G.is_cover_of_isCoverDense _ Y)
rintro Y _ ⟨Z, _, f, hf, ⟨W, g, f', rfl : _ = _⟩, rfl⟩
use W; use G.preimage (f' ≫ f); use g
constructor
· simpa using T.val.downward_closed hf f'
· simp
| 8 | 2,980.957987 | 2 | 2 | 2 | 1,982 |
import Mathlib.FieldTheory.Finite.Basic
#align_import number_theory.wilson from "leanprover-community/mathlib"@"c471da714c044131b90c133701e51b877c246677"
open Finset Nat FiniteField ZMod
open scoped Nat
namespace ZMod
variable (p : ℕ) [Fact p.Prime]
@[simp]
theorem wilsons_lemma : ((p - 1)! : ZMod p) = -1 :=... | Mathlib/NumberTheory/Wilson.lean | 73 | 79 | theorem prod_Ico_one_prime : ∏ x ∈ Ico 1 p, (x : ZMod p) = -1 := by |
-- Porting note: was `conv in Ico 1 p =>`
conv =>
congr
congr
rw [← Nat.add_one_sub_one p, succ_sub (Fact.out (p := p.Prime)).pos]
rw [← prod_natCast, Finset.prod_Ico_id_eq_factorial, wilsons_lemma]
| 6 | 403.428793 | 2 | 2 | 3 | 2,454 |
import Mathlib.Topology.Order.Basic
open Set Filter OrderDual
open scoped Topology
section OrderClosedTopology
variable {α : Type*} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] {a b c d : α}
@[simp] theorem nhdsSet_Ioi : 𝓝ˢ (Ioi a) = 𝓟 (Ioi a) := isOpen_Ioi.nhdsSet_eq
@[simp] theorem nhdsSet... | Mathlib/Topology/Order/NhdsSet.lean | 47 | 50 | theorem nhdsSet_Icc (h : a ≤ b) : 𝓝ˢ (Icc a b) = 𝓝 a ⊔ 𝓝 b ⊔ 𝓟 (Ioo a b) := by |
rcases h.eq_or_lt with rfl | hlt
· simp
· rw [← Ioc_insert_left h, nhdsSet_insert, nhdsSet_Ioc hlt, sup_assoc]
| 3 | 20.085537 | 1 | 0.2 | 5 | 269 |
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 | 262 | 269 | theorem monic_descPochhammer (n : ℕ) [Nontrivial R] [NoZeroDivisors R] :
Monic <| descPochhammer R n := by |
induction' n with n hn
· simp
· have h : leadingCoeff (X - 1 : R[X]) = 1 := leadingCoeff_X_sub_C 1
have : natDegree (X - (1 : R[X])) ≠ 0 := ne_zero_of_eq_one <| natDegree_X_sub_C (1 : R)
rw [descPochhammer_succ_left, Monic.def, leadingCoeff_mul, leadingCoeff_comp this, hn, monic_X,
one_mul, one_m... | 6 | 403.428793 | 2 | 0.96 | 25 | 796 |
import Mathlib.Data.ENNReal.Inv
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
section Real
variable {a b c d : ℝ≥0∞} {r p q : ℝ≥0}
theorem toReal_add (ha : a ≠ ∞) (hb : b ≠ ∞) : (a + b).toReal = a.toReal ... | Mathlib/Data/ENNReal/Real.lean | 43 | 47 | theorem toReal_sub_of_le {a b : ℝ≥0∞} (h : b ≤ a) (ha : a ≠ ∞) :
(a - b).toReal = a.toReal - b.toReal := by |
lift b to ℝ≥0 using ne_top_of_le_ne_top ha h
lift a to ℝ≥0 using ha
simp only [← ENNReal.coe_sub, ENNReal.coe_toReal, NNReal.coe_sub (ENNReal.coe_le_coe.mp h)]
| 3 | 20.085537 | 1 | 0.857143 | 21 | 755 |
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.Analysis.Convex.Segment
import Mathlib.Tactic.GCongr
#align_import analysis.convex.star from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Set
open Convex Pointwise
variable {𝕜 E F : Type*}
section OrderedSemiring
va... | Mathlib/Analysis/Convex/Star.lean | 136 | 139 | theorem starConvex_sUnion {S : Set (Set E)} (hS : ∀ s ∈ S, StarConvex 𝕜 x s) :
StarConvex 𝕜 x (⋃₀ S) := by |
rw [sUnion_eq_iUnion]
exact starConvex_iUnion fun s => hS _ s.2
| 2 | 7.389056 | 1 | 1.6 | 5 | 1,732 |
import Mathlib.Algebra.Module.DedekindDomain
import Mathlib.LinearAlgebra.FreeModule.PID
import Mathlib.Algebra.Module.Projective
import Mathlib.Algebra.Category.ModuleCat.Biproducts
import Mathlib.RingTheory.SimpleModule
#align_import algebra.module.pid from "leanprover-community/mathlib"@"cdc34484a07418af43daf8198b... | Mathlib/Algebra/Module/PID.lean | 124 | 148 | theorem p_pow_smul_lift {x y : M} {k : ℕ} (hM' : Module.IsTorsionBy R M (p ^ pOrder hM y))
(h : p ^ k • x ∈ R ∙ y) : ∃ a : R, p ^ k • x = p ^ k • a • y := by |
-- Porting note: needed to make `smul_smul` work below.
letI : MulAction R M := MulActionWithZero.toMulAction
by_cases hk : k ≤ pOrder hM y
· let f :=
((R ∙ p ^ (pOrder hM y - k) * p ^ k).quotEquivOfEq _ ?_).trans
(quotTorsionOfEquivSpanSingleton R M y)
· have : f.symm ⟨p ^ k • x, h⟩ ∈
... | 23 | 9,744,803,446.248903 | 2 | 2 | 5 | 2,030 |
import Mathlib.Data.Real.Basic
#align_import data.real.sign from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace Real
noncomputable def sign (r : ℝ) : ℝ :=
if r < 0 then -1 else if 0 < r then 1 else 0
#align real.sign Real.sign
theorem sign_of_neg {r : ℝ} (hr : r < 0) : si... | Mathlib/Data/Real/Sign.lean | 85 | 89 | theorem sign_neg {r : ℝ} : sign (-r) = -sign r := by |
obtain hn | rfl | hp := lt_trichotomy r (0 : ℝ)
· rw [sign_of_neg hn, sign_of_pos (neg_pos.mpr hn), neg_neg]
· rw [sign_zero, neg_zero, sign_zero]
· rw [sign_of_pos hp, sign_of_neg (neg_lt_zero.mpr hp)]
| 4 | 54.59815 | 2 | 1.363636 | 11 | 1,467 |
import Mathlib.Topology.UniformSpace.CompleteSeparated
import Mathlib.Topology.EMetricSpace.Lipschitz
import Mathlib.Topology.MetricSpace.Basic
import Mathlib.Topology.MetricSpace.Bounded
#align_import topology.metric_space.antilipschitz from "leanprover-community/mathlib"@"c8f305514e0d47dfaa710f5a52f0d21b588e6328"
... | Mathlib/Topology/MetricSpace/Antilipschitz.lean | 64 | 67 | theorem antilipschitzWith_iff_le_mul_dist :
AntilipschitzWith K f ↔ ∀ x y, dist x y ≤ K * dist (f x) (f y) := by |
simp only [antilipschitzWith_iff_le_mul_nndist, dist_nndist]
norm_cast
| 2 | 7.389056 | 1 | 0.6 | 5 | 537 |
import Mathlib.Algebra.CharP.ExpChar
import Mathlib.GroupTheory.OrderOfElement
#align_import algebra.char_p.two from "leanprover-community/mathlib"@"7f1ba1a333d66eed531ecb4092493cd1b6715450"
variable {R ι : Type*}
namespace CharTwo
section Semiring
variable [Semiring R] [CharP R 2]
theorem two_eq_zero : (2 : ... | Mathlib/Algebra/CharP/Two.lean | 49 | 51 | theorem bit1_eq_one : (bit1 : R → R) = 1 := by |
funext
simp [bit1]
| 2 | 7.389056 | 1 | 0.2 | 10 | 273 |
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.PosDef
#align_import linear_algebra.matrix.schur_complement from "leanprover-community/mathlib"@"a176cb1219e300e85793d44583dede42377b51af"
variable {l m n α : Type*}
namespace Matrix
... | Mathlib/LinearAlgebra/Matrix/SchurComplement.lean | 100 | 104 | theorem invOf_fromBlocks_zero₂₁_eq (A : Matrix m m α) (B : Matrix m n α) (D : Matrix n n α)
[Invertible A] [Invertible D] [Invertible (fromBlocks A B 0 D)] :
⅟ (fromBlocks A B 0 D) = fromBlocks (⅟ A) (-(⅟ A * B * ⅟ D)) 0 (⅟ D) := by |
letI := fromBlocksZero₂₁Invertible A B D
convert (rfl : ⅟ (fromBlocks A B 0 D) = _)
| 2 | 7.389056 | 1 | 1.1875 | 16 | 1,248 |
import Mathlib.RingTheory.IntegrallyClosed
import Mathlib.RingTheory.Trace
import Mathlib.RingTheory.Norm
#align_import ring_theory.discriminant from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa1"
universe u v w z
open scoped Matrix
open Matrix FiniteDimensional Fintype Polynomial Fin... | Mathlib/RingTheory/Discriminant.lean | 113 | 116 | theorem discr_of_matrix_vecMul (b : ι → B) (P : Matrix ι ι A) :
discr A (b ᵥ* P.map (algebraMap A B)) = P.det ^ 2 * discr A b := by |
rw [discr_def, traceMatrix_of_matrix_vecMul, det_mul, det_mul, det_transpose, mul_comm, ←
mul_assoc, discr_def, pow_two]
| 2 | 7.389056 | 1 | 1.2 | 10 | 1,266 |
import Mathlib.Combinatorics.SimpleGraph.Regularity.Increment
#align_import combinatorics.simple_graph.regularity.lemma from "leanprover-community/mathlib"@"1d4d3ca5ec44693640c4f5e407a6b611f77accc8"
open Finpartition Finset Fintype Function SzemerediRegularity
variable {α : Type*} [DecidableEq α] [Fintype α] (G ... | Mathlib/Combinatorics/SimpleGraph/Regularity/Lemma.lean | 74 | 151 | theorem szemeredi_regularity (hε : 0 < ε) (hl : l ≤ card α) :
∃ P : Finpartition univ,
P.IsEquipartition ∧ l ≤ P.parts.card ∧ P.parts.card ≤ bound ε l ∧ P.IsUniform G ε := by |
obtain hα | hα := le_total (card α) (bound ε l)
-- If `card α ≤ bound ε l`, then the partition into singletons is acceptable.
· refine ⟨⊥, bot_isEquipartition _, ?_⟩
rw [card_bot, card_univ]
exact ⟨hl, hα, bot_isUniform _ hε⟩
-- Else, let's start from a dummy equipartition of size `initialBound ε l`.
... | 75 | 373,324,199,679,900,150,000,000,000,000,000 | 2 | 2 | 1 | 2,465 |
import Mathlib.Algebra.Group.Center
import Mathlib.Data.Int.Cast.Lemmas
#align_import group_theory.subsemigroup.center from "leanprover-community/mathlib"@"1ac8d4304efba9d03fa720d06516fac845aa5353"
variable {M : Type*}
namespace Set
variable (M)
@[simp]
theorem natCast_mem_center [NonAssocSemiring M] (n : ℕ) :... | Mathlib/Algebra/Ring/Center.lean | 81 | 86 | theorem neg_mem_center [NonUnitalNonAssocRing M] {a : M} (ha : a ∈ Set.center M) :
-a ∈ Set.center M where
comm _ := by | rw [← neg_mul_comm, ← ha.comm, neg_mul_comm]
left_assoc _ _ := by rw [neg_mul, ha.left_assoc, neg_mul, neg_mul]
mid_assoc _ _ := by rw [← neg_mul_comm, ha.mid_assoc, neg_mul_comm, neg_mul]
right_assoc _ _ := by rw [mul_neg, ha.right_assoc, mul_neg, mul_neg]
| 4 | 54.59815 | 2 | 2 | 4 | 2,386 |
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Analysis.SpecialFunctions.Complex.Arg
#align_import analysis.complex.arg from "leanprover-community/mathlib"@"45a46f4f03f8ae41491bf3605e8e0e363ba192fd"
variable {x y : ℂ}
namespace Complex
| Mathlib/Analysis/Complex/Arg.lean | 31 | 38 | theorem sameRay_iff : SameRay ℝ x y ↔ x = 0 ∨ y = 0 ∨ x.arg = y.arg := by |
rcases eq_or_ne x 0 with (rfl | hx)
· simp
rcases eq_or_ne y 0 with (rfl | hy)
· simp
simp only [hx, hy, false_or_iff, sameRay_iff_norm_smul_eq, arg_eq_arg_iff hx hy]
field_simp [hx, hy]
rw [mul_comm, eq_comm]
| 7 | 1,096.633158 | 2 | 2 | 2 | 1,972 |
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.GroupTheory.Solvable
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.Sylow
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Tactic.TFAE
#align_import group_theory.nilpotent from "leanprover-community/mathlib"@"2bbc7e3884ba234309d2a43b19144... | Mathlib/GroupTheory/Nilpotent.lean | 151 | 155 | theorem upperCentralSeries_one : upperCentralSeries G 1 = center G := by |
ext
simp only [upperCentralSeries, upperCentralSeriesAux, upperCentralSeriesStep,
Subgroup.mem_center_iff, mem_mk, mem_bot, Set.mem_setOf_eq]
exact forall_congr' fun y => by rw [mul_inv_eq_one, mul_inv_eq_iff_eq_mul, eq_comm]
| 4 | 54.59815 | 2 | 2 | 4 | 2,285 |
import Mathlib.Data.Nat.Choose.Central
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.Nat.Multiplicity
#align_import data.nat.choose.factorization from "leanprover-community/mathlib"@"dc9db541168768af03fe228703e758e649afdbfc"
namespace Nat
variable {p n k : ℕ}
theorem factorization_choose_le_l... | Mathlib/Data/Nat/Choose/Factorization.lean | 61 | 88 | theorem factorization_choose_of_lt_three_mul (hp' : p ≠ 2) (hk : p ≤ k) (hk' : p ≤ n - k)
(hn : n < 3 * p) : (choose n k).factorization p = 0 := by |
cases' em' p.Prime with hp hp
· exact factorization_eq_zero_of_non_prime (choose n k) hp
cases' lt_or_le n k with hnk hkn
· simp [choose_eq_zero_of_lt hnk]
rw [factorization_def _ hp, @padicValNat_def _ ⟨hp⟩ _ (choose_pos hkn)]
simp only [hp.multiplicity_choose hkn (lt_add_one _), PartENat.get_natCast, Fin... | 26 | 195,729,609,428.83878 | 2 | 1.5 | 6 | 1,538 |
import Mathlib.Algebra.DualNumber
import Mathlib.Algebra.QuaternionBasis
import Mathlib.Data.Complex.Module
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Star
import Mathlib.LinearAlgebra.QuadraticForm.Prod
#align_import linear_algebra.clifford_algebra.equivs fr... | Mathlib/LinearAlgebra/CliffordAlgebra/Equivs.lean | 194 | 198 | theorem ofComplex_comp_toComplex : ofComplex.comp toComplex = AlgHom.id ℝ (CliffordAlgebra Q) := by |
ext
dsimp only [LinearMap.comp_apply, Subtype.coe_mk, AlgHom.id_apply, AlgHom.toLinearMap_apply,
AlgHom.comp_apply]
rw [toComplex_ι, one_smul, ofComplex_I]
| 4 | 54.59815 | 2 | 1.4 | 10 | 1,508 |
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 | 139 | 147 | theorem sqrtTwoAddSeries_step_down (a b : ℕ) {c d n : ℕ} {z : ℝ}
(hz : z ≤ sqrtTwoAddSeries (a / b) n) (hb : 0 < b) (hd : 0 < d)
(h : a ^ 2 * d ≤ (2 * d + c) * b ^ 2) : z ≤ sqrtTwoAddSeries (c / d) (n + 1) := by |
apply le_trans hz; rw [sqrtTwoAddSeries_succ]; apply sqrtTwoAddSeries_monotone_left
apply le_sqrt_of_sq_le
have hb' : 0 < (b : ℝ) := Nat.cast_pos.2 hb
have hd' : 0 < (d : ℝ) := Nat.cast_pos.2 hd
rw [div_pow, add_div_eq_mul_add_div _ _ (ne_of_gt hd'), div_le_div_iff (pow_pos hb' _) hd']
exact mod_cast h
| 6 | 403.428793 | 2 | 1.833333 | 6 | 1,911 |
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Interval.Set.IsoIoo
import Mathlib.Topology.Order.MonotoneContinuity
import Mathlib.Topology.UrysohnsBounded
#align_import topology.tietze_extension from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
section TietzeExten... | Mathlib/Topology/TietzeExtension.lean | 134 | 143 | theorem TietzeExtension.of_retract {Y : Type v} {Z : Type w} [TopologicalSpace Y]
[TopologicalSpace Z] [TietzeExtension.{u, w} Z] (ι : C(Y, Z)) (r : C(Z, Y))
(h : r.comp ι = .id Y) : TietzeExtension.{u, v} Y where
exists_restrict_eq' s hs f := by |
obtain ⟨g, hg⟩ := (ι.comp f).exists_restrict_eq hs
use r.comp g
ext1 x
have := congr(r.comp $(hg))
rw [← r.comp_assoc ι, h, f.id_comp] at this
congrm($this x)
| 6 | 403.428793 | 2 | 1.5 | 6 | 1,691 |
import Mathlib.Data.Nat.Prime
import Mathlib.Tactic.NormNum.Basic
#align_import data.nat.prime_norm_num from "leanprover-community/mathlib"@"10b4e499f43088dd3bb7b5796184ad5216648ab1"
open Nat Qq Lean Meta
namespace Mathlib.Meta.NormNum
theorem not_prime_mul_of_ble (a b n : ℕ) (h : a * b = n) (h₁ : a.ble 1 = fals... | Mathlib/Tactic/NormNum/Prime.lean | 84 | 88 | theorem minFacHelper_2 {n k k' : ℕ} (e : k + 2 = k') (nk : ¬ Nat.Prime k)
(h : MinFacHelper n k) : MinFacHelper n k' := by |
refine minFacHelper_1 e h λ h2 ↦ ?_
rw [← h2] at nk
exact nk <| minFac_prime h.one_lt.ne'
| 3 | 20.085537 | 1 | 1.8 | 5 | 1,898 |
import Mathlib.Data.List.Lattice
import Mathlib.Data.List.Range
import Mathlib.Data.Bool.Basic
#align_import data.list.intervals from "leanprover-community/mathlib"@"7b78d1776212a91ecc94cf601f83bdcc46b04213"
open Nat
namespace List
def Ico (n m : ℕ) : List ℕ :=
range' n (m - n)
#align list.Ico List.Ico
names... | Mathlib/Data/List/Intervals.lean | 136 | 139 | theorem pred_singleton {m : ℕ} (h : 0 < m) : Ico (m - 1) m = [m - 1] := by |
dsimp [Ico]
rw [Nat.sub_sub_self (succ_le_of_lt h)]
simp [← Nat.one_eq_succ_zero]
| 3 | 20.085537 | 1 | 0.9375 | 16 | 794 |
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Index
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.GroupAction.Quotient
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.SpecificGroups.Cyclic
import Mathlib.Tactic.IntervalCases
#align_import group_theory.p_gr... | Mathlib/GroupTheory/PGroup.lean | 74 | 77 | theorem of_injective {H : Type*} [Group H] (ϕ : H →* G) (hϕ : Function.Injective ϕ) :
IsPGroup p H := by |
simp_rw [IsPGroup, ← hϕ.eq_iff, ϕ.map_pow, ϕ.map_one]
exact fun h => hG (ϕ h)
| 2 | 7.389056 | 1 | 1 | 4 | 836 |
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv
#align_import analysis.special_functions.trigonometric.inverse_deriv from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open scoped Classic... | Mathlib/Analysis/SpecialFunctions/Trigonometric/InverseDeriv.lean | 74 | 79 | theorem hasDerivWithinAt_arcsin_Iic {x : ℝ} (h : x ≠ 1) :
HasDerivWithinAt arcsin (1 / √(1 - x ^ 2)) (Iic x) x := by |
rcases em (x = -1) with (rfl | h')
· convert (hasDerivWithinAt_const (-1 : ℝ) _ (-(π / 2))).congr _ _ <;>
simp (config := { contextual := true }) [arcsin_of_le_neg_one]
· exact (hasDerivAt_arcsin h' h).hasDerivWithinAt
| 4 | 54.59815 | 2 | 2 | 6 | 2,119 |
import Mathlib.Algebra.Homology.ComplexShape
import Mathlib.CategoryTheory.Subobject.Limits
import Mathlib.CategoryTheory.GradedObject
import Mathlib.Algebra.Homology.ShortComplex.Basic
#align_import algebra.homology.homological_complex from "leanprover-community/mathlib"@"88bca0ce5d22ebfd9e73e682e51d60ea13b48347"
... | Mathlib/Algebra/Homology/HomologicalComplex.lean | 236 | 240 | theorem Hom.comm {A B : HomologicalComplex V c} (f : A.Hom B) (i j : ι) :
f.f i ≫ B.d i j = A.d i j ≫ f.f j := by |
by_cases hij : c.Rel i j
· exact f.comm' i j hij
· rw [A.shape i j hij, B.shape i j hij, comp_zero, zero_comp]
| 3 | 20.085537 | 1 | 1.4 | 10 | 1,494 |
import Mathlib.Analysis.RCLike.Lemmas
import Mathlib.MeasureTheory.Constructions.BorelSpace.Complex
#align_import measure_theory.function.special_functions.is_R_or_C from "leanprover-community/mathlib"@"83a66c8775fa14ee5180c85cab98e970956401ad"
noncomputable section
open NNReal ENNReal
namespace RCLike
variabl... | Mathlib/MeasureTheory/Function/SpecialFunctions/RCLike.lean | 80 | 84 | theorem aemeasurable_of_re_im (hre : AEMeasurable (fun x => RCLike.re (f x)) μ)
(him : AEMeasurable (fun x => RCLike.im (f x)) μ) : AEMeasurable f μ := by |
convert AEMeasurable.add (M := 𝕜) (RCLike.measurable_ofReal.comp_aemeasurable hre)
((RCLike.measurable_ofReal.comp_aemeasurable him).mul_const RCLike.I)
exact (RCLike.re_add_im _).symm
| 3 | 20.085537 | 1 | 1 | 2 | 862 |
import Mathlib.Algebra.GroupWithZero.Hom
import Mathlib.Algebra.GroupWithZero.Units.Basic
import Mathlib.Algebra.Ring.Defs
import Mathlib.Data.Nat.Lattice
#align_import ring_theory.nilpotent from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
universe u v
open Function Set
variable {R ... | Mathlib/RingTheory/Nilpotent/Defs.lean | 81 | 85 | theorem IsNilpotent.map [MonoidWithZero R] [MonoidWithZero S] {r : R} {F : Type*}
[FunLike F R S] [MonoidWithZeroHomClass F R S] (hr : IsNilpotent r) (f : F) :
IsNilpotent (f r) := by |
use hr.choose
rw [← map_pow, hr.choose_spec, map_zero]
| 2 | 7.389056 | 1 | 1.333333 | 3 | 1,458 |
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Tactic.Ring
#align_import data.nat.count from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
open Finset
namespace Nat
variable (p : ℕ → Prop)
section Count
variable [DecidablePred p]
def count (n : ℕ) : ℕ :=
(List.range n).... | Mathlib/Data/Nat/Count.lean | 86 | 88 | theorem count_add' (a b : ℕ) : count p (a + b) = count (fun k ↦ p (k + b)) a + count p b := by |
rw [add_comm, count_add, add_comm]
simp_rw [add_comm b]
| 2 | 7.389056 | 1 | 0.642857 | 14 | 554 |
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 | 83 | 85 | theorem hasseDeriv_zero' : hasseDeriv 0 f = f := by |
simp only [hasseDeriv_apply, tsub_zero, Nat.choose_zero_right, Nat.cast_one, one_mul,
sum_monomial_eq]
| 2 | 7.389056 | 1 | 1.2 | 10 | 1,278 |
import Mathlib.Algebra.CharP.Defs
#align_import algebra.char_p.invertible from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
variable {K : Type*}
section Field
variable [Field K]
def invertibleOfRingCharNotDvd {t : ℕ} (not_dvd : ¬ringChar K ∣ t) : Invertible (t : K) :=
invertible... | Mathlib/Algebra/CharP/Invertible.lean | 32 | 34 | theorem not_ringChar_dvd_of_invertible {t : ℕ} [Invertible (t : K)] : ¬ringChar K ∣ t := by |
rw [← ringChar.spec, ← Ne]
exact nonzero_of_invertible (t : K)
| 2 | 7.389056 | 1 | 1 | 1 | 967 |
import Mathlib.Topology.Compactness.Compact
open Set Filter Topology TopologicalSpace Classical
variable {X : Type*} {Y : Type*} {ι : Type*}
variable [TopologicalSpace X] [TopologicalSpace Y] {s t : Set X}
instance [WeaklyLocallyCompactSpace X] [WeaklyLocallyCompactSpace Y] :
WeaklyLocallyCompactSpace (X × Y) ... | Mathlib/Topology/Compactness/LocallyCompact.lean | 141 | 144 | theorem exists_compact_subset [LocallyCompactSpace X] {x : X} {U : Set X} (hU : IsOpen U)
(hx : x ∈ U) : ∃ K : Set X, IsCompact K ∧ x ∈ interior K ∧ K ⊆ U := by |
rcases LocallyCompactSpace.local_compact_nhds x U (hU.mem_nhds hx) with ⟨K, h1K, h2K, h3K⟩
exact ⟨K, h3K, mem_interior_iff_mem_nhds.2 h1K, h2K⟩
| 2 | 7.389056 | 1 | 1.5 | 2 | 1,574 |
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 | 64 | 68 | theorem eq_δ₀ {n : ℕ} {i : ([n] : SimplexCategory) ⟶ [n + 1]} [Mono i] (hi : Isδ₀ i) :
i = SimplexCategory.δ 0 := by |
obtain ⟨j, rfl⟩ := SimplexCategory.eq_δ_of_mono i
rw [iff] at hi
rw [hi]
| 3 | 20.085537 | 1 | 1.5 | 4 | 1,648 |
import Mathlib.Data.Fintype.Order
import Mathlib.Data.Set.Finite
import Mathlib.Order.Category.FinPartOrd
import Mathlib.Order.Category.LinOrd
import Mathlib.CategoryTheory.Limits.Shapes.Images
import Mathlib.CategoryTheory.Limits.Shapes.RegularMono
import Mathlib.Data.Set.Subsingleton
#align_import order.category.No... | Mathlib/Order/Category/NonemptyFinLinOrd.lean | 171 | 209 | theorem epi_iff_surjective {A B : NonemptyFinLinOrd.{u}} (f : A ⟶ B) :
Epi f ↔ Function.Surjective f := by |
constructor
· intro
dsimp only [Function.Surjective]
by_contra! hf'
rcases hf' with ⟨m, hm⟩
let Y := NonemptyFinLinOrd.of (ULift (Fin 2))
let p₁ : B ⟶ Y :=
⟨fun b => if b < m then ULift.up 0 else ULift.up 1, fun x₁ x₂ h => by
simp only
split_ifs with h₁ h₂ h₂
any_g... | 37 | 11,719,142,372,802,612 | 2 | 2 | 2 | 2,035 |
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Data.Finset.Antidiagonal
import Mathlib.Data.Finset.Card
import Mathlib.Data.Multiset.NatAntidiagonal
#align_import data.finset.nat_antidiagonal from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function
namespace Finset
name... | Mathlib/Data/Finset/NatAntidiagonal.lean | 67 | 75 | theorem antidiagonal_succ (n : ℕ) :
antidiagonal (n + 1) =
cons (0, n + 1)
((antidiagonal n).map
(Embedding.prodMap ⟨Nat.succ, Nat.succ_injective⟩ (Embedding.refl _)))
(by simp) := by |
apply eq_of_veq
rw [cons_val, map_val]
apply Multiset.Nat.antidiagonal_succ
| 3 | 20.085537 | 1 | 0.75 | 4 | 673 |
import Mathlib.Data.Set.Subsingleton
import Mathlib.Order.WithBot
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
section Image
variable {f : α → β} {s t : Set... | Mathlib/Data/Set/Image.lean | 291 | 293 | theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by |
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
| 2 | 7.389056 | 1 | 0.666667 | 15 | 590 |
import Mathlib.Data.Rat.Cast.Defs
import Mathlib.Algebra.Field.Basic
#align_import data.rat.cast from "leanprover-community/mathlib"@"acebd8d49928f6ed8920e502a6c90674e75bd441"
namespace NNRat
@[simp, norm_cast]
| Mathlib/Data/Rat/Cast/Lemmas.lean | 64 | 67 | theorem cast_pow {K} [DivisionSemiring K] (q : ℚ≥0) (n : ℕ) :
NNRat.cast (q ^ n) = (NNRat.cast q : K) ^ n := by |
rw [cast_def, cast_def, den_pow, num_pow, Nat.cast_pow, Nat.cast_pow, div_eq_mul_inv, ← inv_pow,
← (Nat.cast_commute _ _).mul_pow, ← div_eq_mul_inv]
| 2 | 7.389056 | 1 | 1.333333 | 6 | 1,436 |
import Mathlib.AlgebraicGeometry.Morphisms.Basic
import Mathlib.Topology.Spectral.Hom
import Mathlib.AlgebraicGeometry.Limits
#align_import algebraic_geometry.morphisms.quasi_compact from "leanprover-community/mathlib"@"5dc6092d09e5e489106865241986f7f2ad28d4c8"
noncomputable section
open CategoryTheory CategoryT... | Mathlib/AlgebraicGeometry/Morphisms/QuasiCompact.lean | 129 | 158 | theorem isCompact_basicOpen (X : Scheme) {U : Opens X.carrier} (hU : IsCompact (U : Set X.carrier))
(f : X.presheaf.obj (op U)) : IsCompact (X.basicOpen f : Set X.carrier) := by |
classical
refine ((isCompact_open_iff_eq_finset_affine_union _).mpr ?_).1
obtain ⟨s, hs, e⟩ := (isCompact_open_iff_eq_finset_affine_union _).mp ⟨hU, U.isOpen⟩
let g : s → X.affineOpens := by
intro V
use V.1 ⊓ X.basicOpen f
have : V.1.1 ⟶ U := by
apply homOfLE; change _ ⊆ (U : Set X.carrier); ... | 28 | 1,446,257,064,291.475 | 2 | 1.5 | 6 | 1,632 |
import Mathlib.Computability.Halting
import Mathlib.Computability.TuringMachine
import Mathlib.Data.Num.Lemmas
import Mathlib.Tactic.DeriveFintype
#align_import computability.tm_to_partrec from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
open Function (update)
open Relation
namespa... | Mathlib/Computability/TMToPartrec.lean | 550 | 554 | theorem Cont.then_eval {k k' : Cont} {v} : (k.then k').eval v = k.eval v >>= k'.eval := by |
induction' k with _ _ _ _ _ _ _ _ _ k_ih _ _ k_ih generalizing v <;>
simp only [Cont.eval, Cont.then, bind_assoc, pure_bind, *]
· simp only [← k_ih]
· split_ifs <;> [rfl; simp only [← k_ih, bind_assoc]]
| 4 | 54.59815 | 2 | 0.285714 | 14 | 314 |
import Mathlib.NumberTheory.FLT.Basic
import Mathlib.NumberTheory.PythagoreanTriples
import Mathlib.RingTheory.Coprime.Lemmas
import Mathlib.Tactic.LinearCombination
#align_import number_theory.fermat4 from "leanprover-community/mathlib"@"10b4e499f43088dd3bb7b5796184ad5216648ab1"
noncomputable section
open scope... | Mathlib/NumberTheory/FLT/Four.lean | 159 | 162 | theorem Int.coprime_of_sq_sum' {r s : ℤ} (h : IsCoprime r s) :
IsCoprime (r ^ 2 + s ^ 2) (r * s) := by |
apply IsCoprime.mul_right (Int.coprime_of_sq_sum (isCoprime_comm.mp h))
rw [add_comm]; apply Int.coprime_of_sq_sum h
| 2 | 7.389056 | 1 | 1.666667 | 9 | 1,779 |
import Mathlib.Topology.EMetricSpace.Paracompact
import Mathlib.Topology.Instances.ENNReal
import Mathlib.Analysis.Convex.PartitionOfUnity
#align_import topology.metric_space.partition_of_unity from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Topology ENNReal NNReal Filter Set Fu... | Mathlib/Topology/MetricSpace/PartitionOfUnity.lean | 100 | 106 | theorem exists_continuous_nnreal_forall_closedBall_subset (hK : ∀ i, IsClosed (K i))
(hU : ∀ i, IsOpen (U i)) (hKU : ∀ i, K i ⊆ U i) (hfin : LocallyFinite K) :
∃ δ : C(X, ℝ≥0), (∀ x, 0 < δ x) ∧ ∀ (i), ∀ x ∈ K i, closedBall x (δ x) ⊆ U i := by |
rcases exists_continuous_real_forall_closedBall_subset hK hU hKU hfin with ⟨δ, hδ₀, hδ⟩
lift δ to C(X, ℝ≥0) using fun x => (hδ₀ x).le
refine ⟨δ, hδ₀, fun i x hi => ?_⟩
simpa only [← ENNReal.ofReal_coe_nnreal] using hδ i x hi
| 4 | 54.59815 | 2 | 1.75 | 4 | 1,871 |
import Mathlib.Order.Disjoint
#align_import order.prop_instances from "leanprover-community/mathlib"@"6623e6af705e97002a9054c1c05a980180276fc1"
instance Prop.instDistribLattice : DistribLattice Prop where
sup := Or
le_sup_left := @Or.inl
le_sup_right := @Or.inr
sup_le := fun _ _ _ => Or.rec
inf := And
... | Mathlib/Order/PropInstances.lean | 106 | 108 | theorem Prop.isCompl_iff {P Q : Prop} : IsCompl P Q ↔ ¬(P ↔ Q) := by |
rw [_root_.isCompl_iff, Prop.disjoint_iff, Prop.codisjoint_iff, not_iff]
by_cases P <;> by_cases Q <;> simp [*]
| 2 | 7.389056 | 1 | 1 | 3 | 855 |
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
import Mathlib.MeasureTheory.Integral.FundThmCalculus
#align_import analysis.special_functions.non_integrable from "leanprover-community/mathlib"@"55ec6e9af7d3e0043f57e394cb06a72f6275273e"
open scoped MeasureTheory Topology Interval NNReal ENNReal
open MeasureTh... | Mathlib/Analysis/SpecialFunctions/NonIntegrable.lean | 98 | 121 | theorem not_integrableOn_of_tendsto_norm_atTop_of_deriv_isBigO_filter
{f : ℝ → E} {g : ℝ → F}
{k : Set ℝ} (l : Filter ℝ) [NeBot l] [TendstoIxxClass Icc l l]
(hl : k ∈ l) (hd : ∀ᶠ x in l, DifferentiableAt ℝ f x) (hf : Tendsto (fun x => ‖f x‖) l atTop)
(hfg : deriv f =O[l] g) : ¬IntegrableOn g k := by |
let a : E →ₗᵢ[ℝ] UniformSpace.Completion E := UniformSpace.Completion.toComplₗᵢ
let f' := a ∘ f
have h'd : ∀ᶠ x in l, DifferentiableAt ℝ f' x := by
filter_upwards [hd] with x hx using a.toContinuousLinearMap.differentiableAt.comp x hx
have h'f : Tendsto (fun x => ‖f' x‖) l atTop := hf.congr (fun x ↦ by sim... | 19 | 178,482,300.963187 | 2 | 2 | 2 | 1,967 |
import Mathlib.Topology.Constructions
#align_import topology.continuous_on from "leanprover-community/mathlib"@"d4f691b9e5f94cfc64639973f3544c95f8d5d494"
open Set Filter Function Topology Filter
variable {α : Type*} {β : Type*} {γ : Type*} {δ : Type*}
variable [TopologicalSpace α]
@[simp]
theorem nhds_bind_nhdsW... | Mathlib/Topology/ContinuousOn.lean | 63 | 67 | theorem eventually_nhdsWithin_nhdsWithin {a : α} {s : Set α} {p : α → Prop} :
(∀ᶠ y in 𝓝[s] a, ∀ᶠ x in 𝓝[s] y, p x) ↔ ∀ᶠ x in 𝓝[s] a, p x := by |
refine ⟨fun h => ?_, fun h => (eventually_nhds_nhdsWithin.2 h).filter_mono inf_le_left⟩
simp only [eventually_nhdsWithin_iff] at h ⊢
exact h.mono fun x hx hxs => (hx hxs).self_of_nhds hxs
| 3 | 20.085537 | 1 | 0.5 | 6 | 483 |
import Mathlib.Combinatorics.SetFamily.HarrisKleitman
import Mathlib.Combinatorics.SetFamily.Intersecting
#align_import combinatorics.set_family.kleitman from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
open Finset
open Fintype (card)
variable {ι α : Type*} [Fintype α] [DecidableEq... | Mathlib/Combinatorics/SetFamily/Kleitman.lean | 37 | 85 | theorem Finset.card_biUnion_le_of_intersecting (s : Finset ι) (f : ι → Finset (Finset α))
(hf : ∀ i ∈ s, (f i : Set (Finset α)).Intersecting) :
(s.biUnion f).card ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - s.card) := by |
have : DecidableEq ι := by
classical
infer_instance
obtain hs | hs := le_total (Fintype.card α) s.card
· rw [tsub_eq_zero_of_le hs, pow_zero]
refine (card_le_card <| biUnion_subset.2 fun i hi a ha ↦
mem_compl.2 <| not_mem_singleton.2 <| (hf _ hi).ne_bot ha).trans_eq ?_
rw [card_compl, Finty... | 46 | 94,961,194,206,024,480,000 | 2 | 2 | 1 | 2,222 |
import Mathlib.Combinatorics.Quiver.Cast
import Mathlib.Combinatorics.Quiver.Symmetric
import Mathlib.Data.Sigma.Basic
import Mathlib.Logic.Equiv.Basic
import Mathlib.Tactic.Common
#align_import combinatorics.quiver.covering from "leanprover-community/mathlib"@"188a411e916e1119e502dbe35b8b475716362401"
open Funct... | Mathlib/Combinatorics/Quiver/Covering.lean | 132 | 136 | theorem Prefunctor.IsCovering.of_comp_left (hφ : φ.IsCovering) (hφψ : (φ ⋙q ψ).IsCovering)
(φsur : Surjective φ.obj) : ψ.IsCovering := by |
refine ⟨fun v => ?_, fun v => ?_⟩ <;> obtain ⟨u, rfl⟩ := φsur v
exacts [(Bijective.of_comp_iff _ (hφ.star_bijective u)).mp (hφψ.star_bijective u),
(Bijective.of_comp_iff _ (hφ.costar_bijective u)).mp (hφψ.costar_bijective u)]
| 3 | 20.085537 | 1 | 1.333333 | 3 | 1,426 |
import Mathlib.CategoryTheory.Sites.Spaces
import Mathlib.Topology.Sheaves.Sheaf
import Mathlib.CategoryTheory.Sites.DenseSubsite
#align_import topology.sheaves.sheaf_condition.sites from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
noncomputable section
set_option linter.uppercaseLe... | Mathlib/Topology/Sheaves/SheafCondition/Sites.lean | 103 | 107 | theorem mem_grothendieckTopology :
Sieve.generate (presieveOfCovering U) ∈ Opens.grothendieckTopology X (iSup U) := by |
intro x hx
obtain ⟨i, hxi⟩ := Opens.mem_iSup.mp hx
exact ⟨U i, Opens.leSupr U i, ⟨U i, 𝟙 _, Opens.leSupr U i, ⟨i, rfl⟩, Category.id_comp _⟩, hxi⟩
| 3 | 20.085537 | 1 | 1.6 | 5 | 1,729 |
import Mathlib.Analysis.Calculus.Deriv.ZPow
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv
import Mathlib.Analysis.Convex.Deriv
#align_import analysis.convex.specific_functions.deriv from "leanprover-communi... | Mathlib/Analysis/Convex/SpecificFunctions/Deriv.lean | 57 | 69 | theorem Finset.prod_nonneg_of_card_nonpos_even {α β : Type*} [LinearOrderedCommRing β] {f : α → β}
[DecidablePred fun x => f x ≤ 0] {s : Finset α} (h0 : Even (s.filter fun x => f x ≤ 0).card) :
0 ≤ ∏ x ∈ s, f x :=
calc
0 ≤ ∏ x ∈ s, (if f x ≤ 0 then (-1 : β) else 1) * f x :=
Finset.prod_nonneg fun x ... |
rw [Finset.prod_mul_distrib, Finset.prod_ite, Finset.prod_const_one, mul_one,
Finset.prod_const, neg_one_pow_eq_pow_mod_two, Nat.even_iff.1 h0, pow_zero, one_mul]
| 2 | 7.389056 | 1 | 1.6 | 10 | 1,744 |
import Mathlib.Algebra.MonoidAlgebra.Division
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Induction
import Mathlib.Algebra.Polynomial.EraseLead
import Mathlib.Order.Interval.Finset.Nat
#align_import data.polynomial.inductions from "leanprover-community/mathlib"@"57e09a1296bf... | Mathlib/Algebra/Polynomial/Inductions.lean | 119 | 143 | theorem degree_divX_lt (hp0 : p ≠ 0) : (divX p).degree < p.degree := by |
haveI := Nontrivial.of_polynomial_ne hp0
calc
degree (divX p) < (divX p * X + C (p.coeff 0)).degree :=
if h : degree p ≤ 0 then by
have h' : C (p.coeff 0) ≠ 0 := by rwa [← eq_C_of_degree_le_zero h]
rw [eq_C_of_degree_le_zero h, divX_C, degree_zero, zero_mul, zero_add]
exact lt_of_... | 24 | 26,489,122,129.84347 | 2 | 1.142857 | 7 | 1,211 |
import Mathlib.RingTheory.Localization.Basic
#align_import ring_theory.localization.integer from "leanprover-community/mathlib"@"9556784a5b84697562e9c6acb40500d4a82e675a"
variable {R : Type*} [CommSemiring R] {M : Submonoid R} {S : Type*} [CommSemiring S]
variable [Algebra R S] {P : Type*} [CommSemiring P]
open ... | Mathlib/RingTheory/Localization/Integer.lean | 149 | 159 | theorem finsetIntegerMultiple_image [DecidableEq R] (s : Finset S) :
algebraMap R S '' finsetIntegerMultiple M s = commonDenomOfFinset M s • (s : Set S) := by |
delta finsetIntegerMultiple commonDenom
rw [Finset.coe_image]
ext
constructor
· rintro ⟨_, ⟨x, -, rfl⟩, rfl⟩
rw [map_integerMultiple]
exact Set.mem_image_of_mem _ x.prop
· rintro ⟨x, hx, rfl⟩
exact ⟨_, ⟨⟨x, hx⟩, s.mem_attach _, rfl⟩, map_integerMultiple M s id _⟩
| 9 | 8,103.083928 | 2 | 1.4 | 5 | 1,499 |
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.LinearAlgebra.Matrix.ZPow
#align_import linear_algebra.matrix.hermitian from "leanprover-community/mathlib"@"caa58cbf5bfb7f81ccbaca4e8b8ac4bc2b39cc1c"
namespace Matrix
variable {α β : Type*} {m n : Type*} {A : Matrix n n α}
open scoped Matrix
local ... | Mathlib/LinearAlgebra/Matrix/Hermitian.lean | 74 | 76 | theorem IsHermitian.transpose {A : Matrix n n α} (h : A.IsHermitian) : Aᵀ.IsHermitian := by |
rw [IsHermitian, conjTranspose, transpose_map]
exact congr_arg Matrix.transpose h
| 2 | 7.389056 | 1 | 0.5 | 2 | 440 |
import Mathlib.MeasureTheory.Measure.Typeclasses
import Mathlib.Analysis.Complex.Basic
#align_import measure_theory.measure.vector_measure from "leanprover-community/mathlib"@"70a4f2197832bceab57d7f41379b2592d1110570"
noncomputable section
open scoped Classical
open NNReal ENNReal MeasureTheory
namespace Measur... | Mathlib/MeasureTheory/Measure/VectorMeasure.lean | 128 | 136 | theorem ext_iff (v w : VectorMeasure α M) : v = w ↔ ∀ i : Set α, MeasurableSet i → v i = w i := by |
constructor
· rintro rfl _ _
rfl
· rw [ext_iff']
intro h i
by_cases hi : MeasurableSet i
· exact h i hi
· simp_rw [not_measurable _ hi]
| 8 | 2,980.957987 | 2 | 1.333333 | 3 | 1,442 |
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Data.Real.ConjExponents
#align_import analysis.mean_inequalities from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
universe u... | Mathlib/Analysis/MeanInequalities.lean | 138 | 148 | theorem geom_mean_le_arith_mean {ι : Type*} (s : Finset ι) (w : ι → ℝ) (z : ι → ℝ)
(hw : ∀ i ∈ s, 0 ≤ w i) (hw' : 0 < ∑ i ∈ s, w i) (hz : ∀ i ∈ s, 0 ≤ z i) :
(∏ i ∈ s, z i ^ w i) ^ (∑ i ∈ s, w i)⁻¹ ≤ (∑ i ∈ s, w i * z i) / (∑ i ∈ s, w i) := by |
convert geom_mean_le_arith_mean_weighted s (fun i => (w i) / ∑ i ∈ s, w i) z ?_ ?_ hz using 2
· rw [← finset_prod_rpow _ _ (fun i hi => rpow_nonneg (hz _ hi) _) _]
refine Finset.prod_congr rfl (fun _ ih => ?_)
rw [div_eq_mul_inv, rpow_mul (hz _ ih)]
· simp_rw [div_eq_mul_inv, mul_assoc, mul_comm, ← mul_a... | 8 | 2,980.957987 | 2 | 1.6 | 5 | 1,716 |
import Mathlib.RingTheory.Ideal.QuotientOperations
import Mathlib.RingTheory.Localization.Basic
#align_import ring_theory.localization.ideal from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74"
namespace IsLocalization
section CommSemiring
variable {R : Type*} [CommSemiring R] (M : Sub... | Mathlib/RingTheory/Localization/Ideal.lean | 53 | 64 | theorem mem_map_algebraMap_iff {I : Ideal R} {z} : z ∈ Ideal.map (algebraMap R S) I ↔
∃ x : I × M, z * algebraMap R S x.2 = algebraMap R S x.1 := by |
constructor
· change _ → z ∈ map_ideal M S I
refine fun h => Ideal.mem_sInf.1 h fun z hz => ?_
obtain ⟨y, hy⟩ := hz
let Z : { x // x ∈ I } := ⟨y, hy.left⟩
use ⟨Z, 1⟩
simp [hy.right]
· rintro ⟨⟨a, s⟩, h⟩
rw [← Ideal.unit_mul_mem_iff_mem _ (map_units S s), mul_comm]
exact h.symm ▸ Ideal... | 10 | 22,026.465795 | 2 | 2 | 4 | 1,952 |
import Mathlib.Analysis.Normed.Order.Basic
import Mathlib.Analysis.Asymptotics.Asymptotics
import Mathlib.Analysis.NormedSpace.Basic
#align_import analysis.asymptotics.specific_asymptotics from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Filter Asymptotics
open Topology
sectio... | Mathlib/Analysis/Asymptotics/SpecificAsymptotics.lean | 131 | 136 | theorem Asymptotics.isLittleO_sum_range_of_tendsto_zero {α : Type*} [NormedAddCommGroup α]
{f : ℕ → α} (h : Tendsto f atTop (𝓝 0)) :
(fun n => ∑ i ∈ range n, f i) =o[atTop] fun n => (n : ℝ) := by |
have := ((isLittleO_one_iff ℝ).2 h).sum_range fun i => zero_le_one
simp only [sum_const, card_range, Nat.smul_one_eq_cast] at this
exact this tendsto_natCast_atTop_atTop
| 3 | 20.085537 | 1 | 1.25 | 8 | 1,299 |
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Order.Group.Int
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Algebra.Ring.Rat
import Mathlib.Data.PNat.Defs
#align_import data.rat.lemmas from "leanprover-community/mathlib"@"550b58538991c8977703fdeb7c9d51a5aa27df11"
namespace Rat
o... | Mathlib/Data/Rat/Lemmas.lean | 87 | 90 | theorem mul_den_dvd (q₁ q₂ : ℚ) : (q₁ * q₂).den ∣ q₁.den * q₂.den := by |
rw [mul_def, normalize_eq]
apply Nat.div_dvd_of_dvd
apply Nat.gcd_dvd_right
| 3 | 20.085537 | 1 | 1.333333 | 12 | 1,389 |
import Mathlib.Analysis.InnerProductSpace.Calculus
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import analysis.inner_product_space.euclidean_dist from "leanprover-community/mathlib"@"9425b6f8220e53b059f5a4904786c3c4b50fc057"
open scoped Topology
open Set
variable {E : Type*} [AddCommGroup E] [Topologi... | Mathlib/Analysis/InnerProductSpace/EuclideanDist.lean | 117 | 119 | theorem nhds_basis_ball {x : E} : (𝓝 x).HasBasis (fun r : ℝ => 0 < r) (ball x) := by |
rw [toEuclidean.toHomeomorph.nhds_eq_comap x]
exact Metric.nhds_basis_ball.comap _
| 2 | 7.389056 | 1 | 0.666667 | 3 | 613 |
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.CategoryTheory.Adjunction.FullyFaithful
#align_import algebraic_geometry.presheafed_space from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
open Opposite CategoryTheory CategoryTheory.Category CategoryTheory.Functor TopCat Topolo... | Mathlib/Geometry/RingedSpace/PresheafedSpace.lean | 126 | 130 | theorem hext {X Y : PresheafedSpace C} (α β : Hom X Y) (w : α.base = β.base) (h : HEq α.c β.c) :
α = β := by |
cases α
cases β
congr
| 3 | 20.085537 | 1 | 1.5 | 2 | 1,622 |
import Mathlib.Algebra.BigOperators.Finsupp
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Data.Fintype.BigOperators
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.SetTheory.Cardinal.Cofinality
#align_import linear_algebra.basis from "leanprover-communit... | Mathlib/LinearAlgebra/Basis.lean | 231 | 236 | theorem coe_sumCoords_eq_finsum : (b.sumCoords : M → R) = fun m => ∑ᶠ i, b.coord i m := by |
ext m
simp only [Basis.sumCoords, Basis.coord, Finsupp.lapply_apply, LinearMap.id_coe,
LinearEquiv.coe_coe, Function.comp_apply, Finsupp.coe_lsum, LinearMap.coe_comp,
finsum_eq_sum _ (b.repr m).finite_support, Finsupp.sum, Finset.finite_toSet_toFinset, id,
Finsupp.fun_support_eq]
| 5 | 148.413159 | 2 | 0.9 | 10 | 779 |
import Mathlib.MeasureTheory.Measure.Haar.Basic
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import measure_theory.measure.haar.of_basis from "leanprover-community/mathlib"@"92bd7b1ffeb306a89f450bee126ddd8a284c259d"
open Set TopologicalSpace MeasureTheory MeasureTheory.Measure FiniteDimensional
open sco... | Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean | 67 | 71 | theorem image_parallelepiped (f : E →ₗ[ℝ] F) (v : ι → E) :
f '' parallelepiped v = parallelepiped (f ∘ v) := by |
simp only [parallelepiped, ← image_comp]
congr 1 with t
simp only [Function.comp_apply, _root_.map_sum, LinearMap.map_smulₛₗ, RingHom.id_apply]
| 3 | 20.085537 | 1 | 1.5 | 6 | 1,672 |
import Mathlib.Topology.Algebra.Module.WeakDual
import Mathlib.MeasureTheory.Integral.BoundedContinuousFunction
import Mathlib.MeasureTheory.Measure.HasOuterApproxClosed
#align_import measure_theory.measure.finite_measure from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable... | Mathlib/MeasureTheory/Measure/FiniteMeasure.lean | 207 | 209 | theorem mass_nonzero_iff (μ : FiniteMeasure Ω) : μ.mass ≠ 0 ↔ μ ≠ 0 := by |
rw [not_iff_not]
exact FiniteMeasure.mass_zero_iff μ
| 2 | 7.389056 | 1 | 1.2 | 5 | 1,255 |
import Mathlib.Algebra.BigOperators.Associated
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.Data.Nat.Choose.Dvd
import Mathlib.Data.Nat.Prime
#align_import number_theory.primorial from "leanprover-community/mathlib"@"0a0ec35061ed9960bf0e7ffb0335f44447b58977"
open Finset
... | Mathlib/NumberTheory/Primorial.lean | 73 | 91 | theorem primorial_le_4_pow (n : ℕ) : n# ≤ 4 ^ n := by |
induction' n using Nat.strong_induction_on with n ihn
cases' n with n; · rfl
rcases n.even_or_odd with (⟨m, rfl⟩ | ho)
· rcases m.eq_zero_or_pos with (rfl | hm)
· decide
calc
(m + m + 1)# = (m + 1 + m)# := by rw [add_right_comm]
_ ≤ (m + 1)# * choose (m + 1 + m) (m + 1) := primorial_add_le ... | 18 | 65,659,969.137331 | 2 | 1.333333 | 3 | 1,394 |
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analysis.inner_product_space.calculus from "leanprover-community/mathlib"@"f9dd3204df14a0749cd456fac1e6849dfe7d2b88"
noncomputable section
open RCLike Real ... | Mathlib/Analysis/InnerProductSpace/Calculus.lean | 365 | 367 | theorem contDiff_euclidean {n : ℕ∞} : ContDiff 𝕜 n f ↔ ∀ i, ContDiff 𝕜 n fun x => f x i := by |
rw [← (EuclideanSpace.equiv ι 𝕜).comp_contDiff_iff, contDiff_pi]
rfl
| 2 | 7.389056 | 1 | 0.833333 | 12 | 734 |
import Mathlib.RingTheory.Ideal.Operations
import Mathlib.Algebra.Module.Torsion
import Mathlib.Algebra.Ring.Idempotents
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Filtration
import Mathlib.RingTheory.Nakayama
#align_import ring_theory.ideal.cota... | Mathlib/RingTheory/Ideal/Cotangent.lean | 74 | 76 | theorem toCotangent_eq {x y : I} : I.toCotangent x = I.toCotangent y ↔ (x - y : R) ∈ I ^ 2 := by |
rw [← sub_eq_zero]
exact I.mem_toCotangent_ker
| 2 | 7.389056 | 1 | 1.333333 | 6 | 1,400 |
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.LinearAlgebra.Matrix.Trace
import Mathlib.LinearAlgebra.Matrix.Symmetric
#align_import combinatorics.simple_graph.adj_matrix from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa1... | Mathlib/Combinatorics/SimpleGraph/AdjMatrix.lean | 109 | 111 | theorem compl_apply [Zero α] [One α] (i j : V) : A.compl i j = 0 ∨ A.compl i j = 1 := by |
unfold compl
split_ifs <;> simp
| 2 | 7.389056 | 1 | 0.285714 | 7 | 315 |
import Mathlib.ModelTheory.ElementarySubstructures
#align_import model_theory.skolem from "leanprover-community/mathlib"@"3d7987cda72abc473c7cdbbb075170e9ac620042"
universe u v w w'
namespace FirstOrder
namespace Language
open Structure Cardinal
open Cardinal
variable (L : Language.{u, v}) {M : Type w} [None... | Mathlib/ModelTheory/Skolem.lean | 65 | 73 | theorem card_functions_sum_skolem₁_le : #(Σ n, (L.sum L.skolem₁).Functions n) ≤ max ℵ₀ L.card := by |
rw [card_functions_sum_skolem₁]
trans #(Σ n, L.BoundedFormula Empty n)
· exact
⟨⟨Sigma.map Nat.succ fun _ => id,
Nat.succ_injective.sigma_map fun _ => Function.injective_id⟩⟩
· refine _root_.trans BoundedFormula.card_le (lift_le.{max u v}.1 ?_)
simp only [mk_empty, lift_zero, lift_uzero, ze... | 8 | 2,980.957987 | 2 | 2 | 3 | 2,103 |
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.Adjugate
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import linear_algebra.matrix.nonsingular_inverse from "leanprover-community/mathlib"@"722b3b152ddd5e0cf21c0a29787c76596cb6b422"
namespace Matrix
universe u u' v
variable {l : ... | Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean | 79 | 81 | theorem invOf_eq [Invertible A.det] [Invertible A] : ⅟ A = ⅟ A.det • A.adjugate := by |
letI := invertibleOfDetInvertible A
convert (rfl : ⅟ A = _)
| 2 | 7.389056 | 1 | 1 | 9 | 1,111 |
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