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.FieldTheory.Separable
import Mathlib.RingTheory.IntegralDomain
import Mathlib.Algebra.CharP.Reduced
import Mathlib.Tactic.ApplyFun
#align_import field_theory.finite.basic from "leanprover-community/mathlib"@"12a85fac627bea918960da036049d611b1a3ee43"
variable {K : Type*} {R : Type*}
local notation ... | Mathlib/FieldTheory/Finite/Basic.lean | 179 | 210 | theorem sum_subgroup_pow_eq_zero [CommRing K] [NoZeroDivisors K]
{G : Subgroup Kˣ} [Fintype G] {k : ℕ} (k_pos : k ≠ 0) (k_lt_card_G : k < Fintype.card G) :
∑ x : G, ((x : Kˣ) : K) ^ k = 0 := by |
nontriviality K
have := NoZeroDivisors.to_isDomain K
rcases (exists_pow_ne_one_of_isCyclic k_pos k_lt_card_G) with ⟨a, ha⟩
rw [Finset.sum_eq_multiset_sum]
have h_multiset_map :
Finset.univ.val.map (fun x : G => ((x : Kˣ) : K) ^ k) =
Finset.univ.val.map (fun x : G => ((x : Kˣ) : K) ^ k * ((a : Kˣ) :... | 29 | 3,931,334,297,144.042 | 2 | 1.833333 | 6 | 1,921 |
import Mathlib.CategoryTheory.Functor.Hom
import Mathlib.CategoryTheory.Products.Basic
import Mathlib.Data.ULift
#align_import category_theory.yoneda from "leanprover-community/mathlib"@"369525b73f229ccd76a6ec0e0e0bf2be57599768"
namespace CategoryTheory
open Opposite
universe v₁ u₁ u₂
-- morphism levels before ... | Mathlib/CategoryTheory/Yoneda.lean | 255 | 258 | theorem coreprW_app_hom (X : C) (f : F.coreprX ⟶ X) :
(F.coreprW.app X).hom f = F.map f F.coreprx := by |
simp only [coyoneda_obj_obj, unop_op, Iso.app_hom, coreprx, ← FunctorToTypes.naturality,
coyoneda_obj_map, Category.id_comp]
| 2 | 7.389056 | 1 | 1 | 3 | 825 |
import Mathlib.Geometry.Manifold.Diffeomorph
import Mathlib.Geometry.Manifold.Instances.Real
import Mathlib.Geometry.Manifold.PartitionOfUnity
#align_import geometry.manifold.whitney_embedding from "leanprover-community/mathlib"@"86c29aefdba50b3f33e86e52e3b2f51a0d8f0282"
universe uι uE uH uM
variable {ι : Type u... | Mathlib/Geometry/Manifold/WhitneyEmbedding.lean | 118 | 133 | theorem exists_immersion_euclidean [Finite ι] (f : SmoothBumpCovering ι I M) :
∃ (n : ℕ) (e : M → EuclideanSpace ℝ (Fin n)),
Smooth I (𝓡 n) e ∧ Injective e ∧ ∀ x : M, Injective (mfderiv I (𝓡 n) e x) := by |
cases nonempty_fintype ι
set F := EuclideanSpace ℝ (Fin <| finrank ℝ (ι → E × ℝ))
letI : IsNoetherian ℝ (E × ℝ) := IsNoetherian.iff_fg.2 inferInstance
letI : FiniteDimensional ℝ (ι → E × ℝ) := IsNoetherian.iff_fg.1 inferInstance
set eEF : (ι → E × ℝ) ≃L[ℝ] F :=
ContinuousLinearEquiv.ofFinrankEq finrank_e... | 13 | 442,413.392009 | 2 | 2 | 4 | 2,356 |
import Mathlib.Algebra.Bounds
import Mathlib.Algebra.Order.Field.Basic -- Porting note: `LinearOrderedField`, etc
import Mathlib.Data.Set.Pointwise.SMul
#align_import algebra.order.pointwise from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function Set
open Pointwise
variable ... | Mathlib/Algebra/Order/Pointwise.lean | 183 | 194 | theorem smul_Ioo : r • Ioo a b = Ioo (r • a) (r • b) := by |
ext x
simp only [mem_smul_set, smul_eq_mul, mem_Ioo]
constructor
· rintro ⟨a, ⟨a_h_left_left, a_h_left_right⟩, rfl⟩
constructor
· exact (mul_lt_mul_left hr).mpr a_h_left_left
· exact (mul_lt_mul_left hr).mpr a_h_left_right
· rintro ⟨a_left, a_right⟩
use x / r
refine ⟨⟨(lt_div_iff' hr).mpr... | 11 | 59,874.141715 | 2 | 1.25 | 16 | 1,340 |
import Mathlib.Combinatorics.Hall.Finite
import Mathlib.CategoryTheory.CofilteredSystem
import Mathlib.Data.Rel
#align_import combinatorics.hall.basic from "leanprover-community/mathlib"@"8195826f5c428fc283510bc67303dd4472d78498"
open Finset CategoryTheory
universe u v
def hallMatchingsOn {ι : Type u} {α : Typ... | Mathlib/Combinatorics/Hall/Basic.lean | 77 | 86 | theorem hallMatchingsOn.nonempty {ι : Type u} {α : Type v} [DecidableEq α] (t : ι → Finset α)
(h : ∀ s : Finset ι, s.card ≤ (s.biUnion t).card) (ι' : Finset ι) :
Nonempty (hallMatchingsOn t ι') := by |
classical
refine ⟨Classical.indefiniteDescription _ ?_⟩
apply (all_card_le_biUnion_card_iff_existsInjective' fun i : ι' => t i).mp
intro s'
convert h (s'.image (↑)) using 1
· simp only [card_image_of_injective s' Subtype.coe_injective]
· rw [image_biUnion]
| 7 | 1,096.633158 | 2 | 2 | 2 | 2,193 |
import Mathlib.Combinatorics.SetFamily.Shadow
#align_import combinatorics.set_family.compression.uv from "leanprover-community/mathlib"@"6f8ab7de1c4b78a68ab8cf7dd83d549eb78a68a1"
open Finset
variable {α : Type*}
theorem sup_sdiff_injOn [GeneralizedBooleanAlgebra α] (u v : α) :
{ x | Disjoint u x ∧ v ≤ x }.... | Mathlib/Combinatorics/SetFamily/Compression/UV.lean | 98 | 102 | theorem compress_self (u a : α) : compress u u a = a := by |
unfold compress
split_ifs with h
· exact h.1.symm.sup_sdiff_cancel_right
· rfl
| 4 | 54.59815 | 2 | 1.6 | 10 | 1,727 |
import Mathlib.Algebra.Group.Equiv.Basic
import Mathlib.Algebra.Ring.Basic
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Order.Hom.Basic
#align_import algebra.order.sub.basic from "leanprover-community/mathlib"@"10b4e499f43088dd3bb7b5796184ad5216648ab1"
variable {α β : Type*}
section Add
variable [Preord... | Mathlib/Algebra/Order/Sub/Basic.lean | 25 | 28 | theorem AddHom.le_map_tsub [Preorder β] [Add β] [Sub β] [OrderedSub β] (f : AddHom α β)
(hf : Monotone f) (a b : α) : f a - f b ≤ f (a - b) := by |
rw [tsub_le_iff_right, ← f.map_add]
exact hf le_tsub_add
| 2 | 7.389056 | 1 | 0.5 | 2 | 422 |
import Mathlib.Analysis.Calculus.FDeriv.Basic
#align_import analysis.calculus.fderiv.restrict_scalars from "leanprover-community/mathlib"@"e3fb84046afd187b710170887195d50bada934ee"
open Filter Asymptotics ContinuousLinearMap Set Metric
open scoped Classical
open Topology NNReal Filter Asymptotics ENNReal
noncom... | Mathlib/Analysis/Calculus/FDeriv/RestrictScalars.lean | 92 | 95 | theorem HasFDerivWithinAt.of_restrictScalars {g' : E →L[𝕜] F} (h : HasFDerivWithinAt f g' s x)
(H : f'.restrictScalars 𝕜 = g') : HasFDerivWithinAt f f' s x := by |
rw [← H] at h
exact .of_isLittleO h.1
| 2 | 7.389056 | 1 | 1.25 | 4 | 1,301 |
import Mathlib.Geometry.Manifold.ContMDiff.Atlas
import Mathlib.Geometry.Manifold.VectorBundle.FiberwiseLinear
import Mathlib.Topology.VectorBundle.Constructions
#align_import geometry.manifold.vector_bundle.basic from "leanprover-community/mathlib"@"e473c3198bb41f68560cab68a0529c854b618833"
assert_not_exists mfde... | Mathlib/Geometry/Manifold/VectorBundle/Basic.lean | 178 | 196 | theorem contMDiffWithinAt_totalSpace (f : M → TotalSpace F E) {s : Set M} {x₀ : M} :
ContMDiffWithinAt IM (IB.prod 𝓘(𝕜, F)) n f s x₀ ↔
ContMDiffWithinAt IM IB n (fun x => (f x).proj) s x₀ ∧
ContMDiffWithinAt IM 𝓘(𝕜, F) n (fun x ↦ (trivializationAt F E (f x₀).proj (f x)).2) s x₀ := by |
simp (config := { singlePass := true }) only [contMDiffWithinAt_iff_target]
rw [and_and_and_comm, ← FiberBundle.continuousWithinAt_totalSpace, and_congr_right_iff]
intro hf
simp_rw [modelWithCornersSelf_prod, FiberBundle.extChartAt, Function.comp,
PartialEquiv.trans_apply, PartialEquiv.prod_coe, PartialEqu... | 15 | 3,269,017.372472 | 2 | 1.333333 | 3 | 1,437 |
import Mathlib.Algebra.ContinuedFractions.ContinuantsRecurrence
import Mathlib.Algebra.ContinuedFractions.TerminatedStable
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Ring
#align_import algebra.continued_fractions.convergents_equiv from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b348ce40a... | Mathlib/Algebra/ContinuedFractions/ConvergentsEquiv.lean | 155 | 181 | theorem succ_succ_nth_convergent'_aux_eq_succ_nth_convergent'_aux_squashSeq :
convergents'Aux s (n + 2) = convergents'Aux (squashSeq s n) (n + 1) := by |
cases s_succ_nth_eq : s.get? <| n + 1 with
| none =>
rw [squashSeq_eq_self_of_terminated s_succ_nth_eq,
convergents'Aux_stable_step_of_terminated s_succ_nth_eq]
| some gp_succ_n =>
induction n generalizing s gp_succ_n with
| zero =>
obtain ⟨gp_head, s_head_eq⟩ : ∃ gp_head, s.head = some g... | 25 | 72,004,899,337.38586 | 2 | 1.4 | 5 | 1,492 |
import Mathlib.LinearAlgebra.Dimension.LinearMap
import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition
#align_import linear_algebra.free_module.finite.matrix from "leanprover-community/mathlib"@"b1c23399f01266afe392a0d8f71f599a0dad4f7b"
universe u u' v w
variable (R : Type u) (S : Type u') (M : Type v) (N ... | Mathlib/LinearAlgebra/FreeModule/Finite/Matrix.lean | 53 | 56 | theorem FiniteDimensional.rank_linearMap :
Module.rank S (M →ₗ[R] N) = lift.{w} (Module.rank R M) * lift.{v} (Module.rank S N) := by |
rw [(linearMapEquivFun R S M N).rank_eq, rank_fun_eq_lift_mul,
← finrank_eq_card_chooseBasisIndex, ← finrank_eq_rank R, lift_natCast]
| 2 | 7.389056 | 1 | 0.6 | 5 | 528 |
import Mathlib.Combinatorics.SimpleGraph.Dart
import Mathlib.Data.FunLike.Fintype
open Function
namespace SimpleGraph
variable {V W X : Type*} (G : SimpleGraph V) (G' : SimpleGraph W) {u v : V}
protected def map (f : V ↪ W) (G : SimpleGraph V) : SimpleGraph W where
Adj := Relation.Map G.Adj f f
symm a b... | Mathlib/Combinatorics/SimpleGraph/Maps.lean | 76 | 78 | theorem map_monotone (f : V ↪ W) : Monotone (SimpleGraph.map f) := by |
rintro G G' h _ _ ⟨u, v, ha, rfl, rfl⟩
exact ⟨_, _, h ha, rfl, rfl⟩
| 2 | 7.389056 | 1 | 0.75 | 4 | 677 |
import Mathlib.CategoryTheory.Abelian.Exact
import Mathlib.CategoryTheory.Preadditive.Injective
import Mathlib.CategoryTheory.Preadditive.Yoneda.Limits
import Mathlib.CategoryTheory.Preadditive.Yoneda.Injective
#align_import category_theory.abelian.injective from "leanprover-community/mathlib"@"f8d8465c3c392a93b9ed22... | Mathlib/CategoryTheory/Abelian/Injective.lean | 45 | 48 | theorem injective_of_preservesFiniteColimits_preadditiveYonedaObj (J : C)
[hP : PreservesFiniteColimits (preadditiveYonedaObj J)] : Injective J := by |
rw [injective_iff_preservesEpimorphisms_preadditive_yoneda_obj']
infer_instance
| 2 | 7.389056 | 1 | 1 | 1 | 1,107 |
import Mathlib.SetTheory.Game.Basic
import Mathlib.SetTheory.Ordinal.NaturalOps
#align_import set_theory.game.ordinal from "leanprover-community/mathlib"@"b90e72c7eebbe8de7c8293a80208ea2ba135c834"
universe u
open SetTheory PGame
open scoped NaturalOps PGame
namespace Ordinal
noncomputable def toPGame : Ordin... | Mathlib/SetTheory/Game/Ordinal.lean | 83 | 87 | theorem toPGame_moveLeft_hEq {o : Ordinal} :
have : IsWellOrder o.out.α (· < ·) := isWellOrder_out_lt o
HEq o.toPGame.moveLeft fun x : o.out.α => (typein (· < ·) x).toPGame := by |
rw [toPGame]
rfl
| 2 | 7.389056 | 1 | 0.222222 | 9 | 284 |
import Mathlib.Algebra.CharP.LocalRing
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.Tactic.FieldSimp
#align_import algebra.char_p.mixed_char_zero from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
variable (R : Type*) [CommRing R]
class MixedCharZero (p : ℕ) : Prop where
... | Mathlib/Algebra/CharP/MixedCharZero.lean | 264 | 270 | theorem to_not_mixedCharZero (h : ∀ I : Ideal R, I ≠ ⊤ → CharZero (R ⧸ I)) :
∀ p > 0, ¬MixedCharZero R p := by |
intro p p_pos
by_contra hp_mixedChar
rcases hp_mixedChar.charP_quotient with ⟨I, hI_ne_top, hI_p⟩
replace hI_zero : CharP (R ⧸ I) 0 := @CharP.ofCharZero _ _ (h I hI_ne_top)
exact absurd (CharP.eq (R ⧸ I) hI_p hI_zero) (ne_of_gt p_pos)
| 5 | 148.413159 | 2 | 1.875 | 8 | 1,930 |
import Mathlib.Analysis.Calculus.Deriv.Inv
import Mathlib.Analysis.NormedSpace.BallAction
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Analysis.InnerProductSpace.Calculus
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Geometry.Manifold.Algebra.LieGroup
import Mathlib.Geometry.Manifol... | Mathlib/Geometry/Manifold/Instances/Sphere.lean | 131 | 142 | theorem stereoInvFunAux_mem (hv : ‖v‖ = 1) {w : E} (hw : w ∈ (ℝ ∙ v)ᗮ) :
stereoInvFunAux v w ∈ sphere (0 : E) 1 := by |
have h₁ : (0 : ℝ) < ‖w‖ ^ 2 + 4 := by positivity
suffices ‖(4 : ℝ) • w + (‖w‖ ^ 2 - 4) • v‖ = ‖w‖ ^ 2 + 4 by
simp only [mem_sphere_zero_iff_norm, norm_smul, Real.norm_eq_abs, abs_inv, this,
abs_of_pos h₁, stereoInvFunAux_apply, inv_mul_cancel h₁.ne']
suffices ‖(4 : ℝ) • w + (‖w‖ ^ 2 - 4) • v‖ ^ 2 = (‖w... | 10 | 22,026.465795 | 2 | 1.5 | 6 | 1,585 |
import Mathlib.Data.Vector.Basic
import Mathlib.Data.Vector.Snoc
set_option autoImplicit true
namespace Vector
section Fold
section Bisim
variable {xs : Vector α n}
theorem mapAccumr_bisim {f₁ : α → σ₁ → σ₁ × β} {f₂ : α → σ₂ → σ₂ × β} {s₁ : σ₁} {s₂ : σ₂}
(R : σ₁ → σ₂ → Prop) (h₀ : R s₁ s₂)
(hR : ∀ {... | Mathlib/Data/Vector/MapLemmas.lean | 205 | 211 | theorem mapAccumr₂_bisim_tail {ys : Vector β n} {f₁ : α → β → σ₁ → σ₁ × γ}
{f₂ : α → β → σ₂ → σ₂ × γ} {s₁ : σ₁} {s₂ : σ₂}
(h : ∃ R : σ₁ → σ₂ → Prop, R s₁ s₂ ∧
∀ {s q} a b, R s q → R (f₁ a b s).1 (f₂ a b q).1 ∧ (f₁ a b s).2 = (f₂ a b q).2) :
(mapAccumr₂ f₁ xs ys s₁).2 = (mapAccumr₂ f₂ xs ys s₂).2 := by |
rcases h with ⟨R, h₀, hR⟩
exact (mapAccumr₂_bisim R h₀ hR).2
| 2 | 7.389056 | 1 | 0.333333 | 24 | 337 |
import Mathlib.CategoryTheory.Sites.Coherent.SheafComparison
import Mathlib.CategoryTheory.Sites.Equivalence
namespace CategoryTheory
variable {C : Type*} [Category C]
open GrothendieckTopology
namespace Equivalence
variable {D : Type*} [Category D]
variable (e : C ≌ D)
section Coherent
variable [Precoherent C... | Mathlib/CategoryTheory/Sites/Coherent/Equivalence.lean | 55 | 60 | theorem precoherent_isSheaf_iff (F : Cᵒᵖ ⥤ A) : haveI := e.precoherent
IsSheaf (coherentTopology C) F ↔ IsSheaf (coherentTopology D) (e.inverse.op ⋙ F) := by |
refine ⟨fun hF ↦ ((e.sheafCongrPrecoherent A).functor.obj ⟨F, hF⟩).cond, fun hF ↦ ?_⟩
rw [isSheaf_of_iso_iff (P' := e.functor.op ⋙ e.inverse.op ⋙ F)]
· exact (e.sheafCongrPrecoherent A).inverse.obj ⟨e.inverse.op ⋙ F, hF⟩ |>.cond
· exact isoWhiskerRight e.op.unitIso F
| 4 | 54.59815 | 2 | 2 | 2 | 2,253 |
import Mathlib.Topology.Compactness.SigmaCompact
import Mathlib.Topology.Connected.TotallyDisconnected
import Mathlib.Topology.Inseparable
#align_import topology.separation from "leanprover-community/mathlib"@"d91e7f7a7f1c7e9f0e18fdb6bde4f652004c735d"
open Function Set Filter Topology TopologicalSpace
open scoped... | Mathlib/Topology/Separation.lean | 125 | 127 | theorem separatedNhds_iff_disjoint {s t : Set X} : SeparatedNhds s t ↔ Disjoint (𝓝ˢ s) (𝓝ˢ t) := by |
simp only [(hasBasis_nhdsSet s).disjoint_iff (hasBasis_nhdsSet t), SeparatedNhds, exists_prop, ←
exists_and_left, and_assoc, and_comm, and_left_comm]
| 2 | 7.389056 | 1 | 0.5 | 4 | 446 |
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 | 390 | 394 | theorem det_fromBlocks₁₁ (A : Matrix m m α) (B : Matrix m n α) (C : Matrix n m α)
(D : Matrix n n α) [Invertible A] :
(Matrix.fromBlocks A B C D).det = det A * det (D - C * ⅟ A * B) := by |
rw [fromBlocks_eq_of_invertible₁₁ (A := A), det_mul, det_mul, det_fromBlocks_zero₂₁,
det_fromBlocks_zero₂₁, det_fromBlocks_zero₁₂, det_one, det_one, one_mul, one_mul, mul_one]
| 2 | 7.389056 | 1 | 1.1875 | 16 | 1,248 |
import Mathlib.RingTheory.MvPowerSeries.Basic
import Mathlib.RingTheory.Ideal.LocalRing
#align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60"
noncomputable section
open Finset (antidiagonal mem_antidiagonal)
namespace MvPowerSeries
open Fi... | Mathlib/RingTheory/MvPowerSeries/Inverse.lean | 107 | 137 | theorem mul_invOfUnit (φ : MvPowerSeries σ R) (u : Rˣ) (h : constantCoeff σ R φ = u) :
φ * invOfUnit φ u = 1 :=
ext fun n =>
letI := Classical.decEq (σ →₀ ℕ)
if H : n = 0 then by
rw [H]
simp [coeff_mul, support_single_ne_zero, h]
else by
classical
have : ((0 : σ →₀ ℕ), n) ∈ ant... | rw [mem_antidiagonal, zero_add]
rw [coeff_one, if_neg H, coeff_mul, ← Finset.insert_erase this,
Finset.sum_insert (Finset.not_mem_erase _ _), coeff_zero_eq_constantCoeff_apply, h,
coeff_invOfUnit, if_neg H, neg_mul, mul_neg, Units.mul_inv_cancel_left, ←
Finset.insert_erase this, Finset.su... | 22 | 3,584,912,846.131591 | 2 | 1 | 3 | 920 |
import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Subobject.Lattice
import Mathlib.Order.Atoms
#align_import category_theory.simple from "leanprover-community/mathlib"@"4ed0bcaef698011... | Mathlib/CategoryTheory/Simple.lean | 84 | 89 | theorem kernel_zero_of_nonzero_from_simple {X Y : C} [Simple X] {f : X ⟶ Y} [HasKernel f]
(w : f ≠ 0) : kernel.ι f = 0 := by |
classical
by_contra h
haveI := isIso_of_mono_of_nonzero h
exact w (eq_zero_of_epi_kernel f)
| 4 | 54.59815 | 2 | 1.5 | 8 | 1,594 |
import Mathlib.RepresentationTheory.Action.Limits
import Mathlib.RepresentationTheory.Action.Concrete
import Mathlib.CategoryTheory.Monoidal.FunctorCategory
import Mathlib.CategoryTheory.Monoidal.Transport
import Mathlib.CategoryTheory.Monoidal.Rigid.OfEquivalence
import Mathlib.CategoryTheory.Monoidal.Rigid.FunctorCa... | Mathlib/RepresentationTheory/Action/Monoidal.lean | 90 | 93 | theorem associator_inv_hom {X Y Z : Action V G} :
Hom.hom (α_ X Y Z).inv = (α_ X.V Y.V Z.V).inv := by |
dsimp
simp
| 2 | 7.389056 | 1 | 1 | 6 | 1,092 |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
#align_import ring_theory.coprime.lemmas from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
universe u v
section RelPrime
variable {α I} [Comm... | Mathlib/RingTheory/Coprime/Lemmas.lean | 299 | 301 | theorem pow_right (H : IsRelPrime x y) : IsRelPrime x (y ^ n) := by |
rw [← Finset.card_range n, ← Finset.prod_const]
exact IsRelPrime.prod_right fun _ _ ↦ H
| 2 | 7.389056 | 1 | 1.111111 | 18 | 1,195 |
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Convex.Normed
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.convex.side from "leanprover-community/mathlib"@"a63928c34ec358b5edcda2bf7513c50052a5230f"
variable {R V V' P P' : Type*}
open AffineEquiv AffineMap
namespace Affine... | Mathlib/Analysis/Convex/Side.lean | 70 | 80 | theorem _root_.Function.Injective.wSameSide_map_iff {s : AffineSubspace R P} {x y : P}
{f : P →ᵃ[R] P'} (hf : Function.Injective f) :
(s.map f).WSameSide (f x) (f y) ↔ s.WSameSide x y := by |
refine ⟨fun h => ?_, fun h => h.map _⟩
rcases h with ⟨fp₁, hfp₁, fp₂, hfp₂, h⟩
rw [mem_map] at hfp₁ hfp₂
rcases hfp₁ with ⟨p₁, hp₁, rfl⟩
rcases hfp₂ with ⟨p₂, hp₂, rfl⟩
refine ⟨p₁, hp₁, p₂, hp₂, ?_⟩
simp_rw [← linearMap_vsub, (f.linear_injective_iff.2 hf).sameRay_map_iff] at h
exact h
| 8 | 2,980.957987 | 2 | 1.6 | 5 | 1,714 |
import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Subobject.Lattice
import Mathlib.Order.Atoms
#align_import category_theory.simple from "leanprover-community/mathlib"@"4ed0bcaef698011... | Mathlib/CategoryTheory/Simple.lean | 96 | 100 | theorem epi_of_nonzero_to_simple [HasEqualizers C] {X Y : C} [Simple Y] {f : X ⟶ Y} [HasImage f]
(w : f ≠ 0) : Epi f := by |
rw [← image.fac f]
haveI : IsIso (image.ι f) := isIso_of_mono_of_nonzero fun h => w (eq_zero_of_image_eq_zero h)
apply epi_comp
| 3 | 20.085537 | 1 | 1.5 | 8 | 1,594 |
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 | 113 | 115 | theorem le_inv_mul_iff_mul_le : b ≤ a⁻¹ * c ↔ a * b ≤ c := by |
rw [← mul_le_mul_iff_left a]
simp
| 2 | 7.389056 | 1 | 0.4 | 25 | 400 |
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.LinearAlgebra.SesquilinearForm
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
variable {𝕜 E F : Type*} [RCLike 𝕜]
variable [NormedAddCommGroup E] [InnerProductSpace... | Mathlib/Analysis/InnerProductSpace/Orthogonal.lean | 93 | 97 | theorem sub_mem_orthogonal_of_inner_right {x y : E} (h : ∀ v : K, ⟪(v : E), x⟫ = ⟪(v : E), y⟫) :
x - y ∈ Kᗮ := by |
intro u hu
rw [inner_sub_right, sub_eq_zero]
exact h ⟨u, hu⟩
| 3 | 20.085537 | 1 | 1.1 | 10 | 1,190 |
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Data.Set.Lattice
#align_import data.set.intervals.pi from "leanprover-community/mathlib"@"e4bc74cbaf429d706cb9140902f7ca6c431e75a4"
-- Porting note: Added, since dot nota... | Mathlib/Order/Interval/Set/Pi.lean | 112 | 118 | theorem disjoint_pi_univ_Ioc_update_left_right {x y : ∀ i, α i} {i₀ : ι} {m : α i₀} :
Disjoint (pi univ fun i ↦ Ioc (x i) (update y i₀ m i))
(pi univ fun i ↦ Ioc (update x i₀ m i) (y i)) := by |
rw [disjoint_left]
rintro z h₁ h₂
refine (h₁ i₀ (mem_univ _)).2.not_lt ?_
simpa only [Function.update_same] using (h₂ i₀ (mem_univ _)).1
| 4 | 54.59815 | 2 | 2 | 3 | 1,935 |
import Mathlib.Analysis.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calculus.FDeriv.Comp
#align_import analysis.calculus.fderiv.prod from "leanprover-community/mathlib"@"e354e865255654389cc46e6032160238df2e0f40"
open Filter Asymptotics ContinuousLinearMap Set Metric
open scoped Classical
open Topology NNReal ... | Mathlib/Analysis/Calculus/FDeriv/Prod.lean | 400 | 403 | theorem hasStrictFDerivAt_pi' :
HasStrictFDerivAt Φ Φ' x ↔ ∀ i, HasStrictFDerivAt (fun x => Φ x i) ((proj i).comp Φ') x := by |
simp only [HasStrictFDerivAt, ContinuousLinearMap.coe_pi]
exact isLittleO_pi
| 2 | 7.389056 | 1 | 1.4 | 5 | 1,501 |
import Mathlib.Analysis.SpecialFunctions.Bernstein
import Mathlib.Topology.Algebra.Algebra
#align_import topology.continuous_function.weierstrass from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
open ContinuousMap Filter
open scoped unitInterval
theorem polynomialFunctions_closure... | Mathlib/Topology/ContinuousFunction/Weierstrass.lean | 114 | 122 | theorem exists_polynomial_near_of_continuousOn (a b : ℝ) (f : ℝ → ℝ)
(c : ContinuousOn f (Set.Icc a b)) (ε : ℝ) (pos : 0 < ε) :
∃ p : ℝ[X], ∀ x ∈ Set.Icc a b, |p.eval x - f x| < ε := by |
let f' : C(Set.Icc a b, ℝ) := ⟨fun x => f x, continuousOn_iff_continuous_restrict.mp c⟩
obtain ⟨p, b⟩ := exists_polynomial_near_continuousMap a b f' ε pos
use p
rw [norm_lt_iff _ pos] at b
intro x m
exact b ⟨x, m⟩
| 6 | 403.428793 | 2 | 1.8 | 5 | 1,892 |
import Mathlib.Data.List.Nodup
#align_import data.list.duplicate from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
variable {α : Type*}
namespace List
inductive Duplicate (x : α) : List α → Prop
| cons_mem {l : List α} : x ∈ l → Duplicate x (x :: l)
| cons_duplicate {y : α} {l ... | Mathlib/Data/List/Duplicate.lean | 52 | 55 | theorem Duplicate.mem_cons_self (h : x ∈+ x :: l) : x ∈ l := by |
cases' h with _ h _ _ h
· exact h
· exact h.mem
| 3 | 20.085537 | 1 | 0.818182 | 11 | 719 |
import Mathlib.CategoryTheory.Limits.Types
import Mathlib.CategoryTheory.Filtered.Basic
#align_import category_theory.limits.types from "leanprover-community/mathlib"@"4aa2a2e17940311e47007f087c9df229e7f12942"
open CategoryTheory CategoryTheory.Limits
universe v u w
namespace CategoryTheory.Limits.Types.Filtered... | Mathlib/CategoryTheory/Limits/TypesFiltered.lean | 112 | 117 | theorem colimit_eq_iff_aux {i j : J} {xi : F.obj i} {xj : F.obj j} :
(colimitCocone F).ι.app i xi = (colimitCocone F).ι.app j xj ↔
FilteredColimit.Rel.{v, u} F ⟨i, xi⟩ ⟨j, xj⟩ := by |
dsimp
rw [← (equivShrink _).symm.injective.eq_iff, Equiv.symm_apply_apply, Equiv.symm_apply_apply,
Quot.eq, FilteredColimit.rel_eq_eqvGen_quot_rel]
| 3 | 20.085537 | 1 | 1 | 1 | 1,129 |
import Mathlib.MeasureTheory.Constructions.Prod.Integral
import Mathlib.MeasureTheory.Integral.CircleIntegral
#align_import measure_theory.integral.torus_integral from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
variable {n : ℕ}
variable {E : Type*} [NormedAddCommGroup E]
noncomputa... | Mathlib/MeasureTheory/Integral/TorusIntegral.lean | 139 | 144 | theorem function_integrable [NormedSpace ℂ E] (hf : TorusIntegrable f c R) :
IntegrableOn (fun θ : ℝⁿ => (∏ i, R i * exp (θ i * I) * I : ℂ) • f (torusMap c R θ))
(Icc (0 : ℝⁿ) fun _ => 2 * π) volume := by |
refine (hf.norm.const_mul (∏ i, |R i|)).mono' ?_ ?_
· refine (Continuous.aestronglyMeasurable ?_).smul hf.1; continuity
simp [norm_smul, map_prod]
| 3 | 20.085537 | 1 | 0.428571 | 7 | 406 |
import Mathlib.FieldTheory.Galois
import Mathlib.Topology.Algebra.FilterBasis
import Mathlib.Topology.Algebra.OpenSubgroup
import Mathlib.Tactic.ByContra
#align_import field_theory.krull_topology from "leanprover-community/mathlib"@"039a089d2a4b93c761b234f3e5f5aeb752bac60f"
open scoped Classical Pointwise
theore... | Mathlib/FieldTheory/KrullTopology.lean | 93 | 100 | theorem IntermediateField.fixingSubgroup.bot {K L : Type*} [Field K] [Field L] [Algebra K L] :
IntermediateField.fixingSubgroup (⊥ : IntermediateField K L) = ⊤ := by |
ext f
refine ⟨fun _ => Subgroup.mem_top _, fun _ => ?_⟩
rintro ⟨x, hx : x ∈ (⊥ : IntermediateField K L)⟩
rw [IntermediateField.mem_bot] at hx
rcases hx with ⟨y, rfl⟩
exact f.commutes y
| 6 | 403.428793 | 2 | 1.5 | 2 | 1,637 |
import Mathlib.Algebra.BigOperators.Finsupp
import Mathlib.Algebra.Module.Basic
import Mathlib.Algebra.Regular.SMul
import Mathlib.Data.Finset.Preimage
import Mathlib.Data.Rat.BigOperators
import Mathlib.GroupTheory.GroupAction.Hom
import Mathlib.Data.Set.Subsingleton
#align_import data.finsupp.basic from "leanprover... | Mathlib/Data/Finsupp/Basic.lean | 78 | 80 | theorem mem_graph_iff {c : α × M} {f : α →₀ M} : c ∈ f.graph ↔ f c.1 = c.2 ∧ c.2 ≠ 0 := by |
cases c
exact mk_mem_graph_iff
| 2 | 7.389056 | 1 | 1.25 | 4 | 1,325 |
import Mathlib.Algebra.Group.ConjFinite
import Mathlib.GroupTheory.Abelianization
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.GroupAction.Quotient
import Mathlib.GroupTheory.Index
import Mathlib.GroupTheory.SpecificGroups.Dihedral
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Line... | Mathlib/GroupTheory/CommutingProbability.lean | 119 | 128 | theorem Subgroup.commProb_quotient_le [H.Normal] : commProb (G ⧸ H) ≤ commProb G * Nat.card H := by |
/- After rewriting with `commProb_def'`, we reduce to showing that `G` has at least as many
conjugacy classes as `G ⧸ H`. -/
rw [commProb_def', commProb_def', div_le_iff, mul_assoc, ← Nat.cast_mul, ← Subgroup.index,
H.card_mul_index, div_mul_cancel₀, Nat.cast_le]
· apply Finite.card_le_of_surjective
... | 9 | 8,103.083928 | 2 | 1.625 | 8 | 1,749 |
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 | 50 | 62 | theorem card_functions_sum_skolem₁ :
#(Σ n, (L.sum L.skolem₁).Functions n) = #(Σ n, L.BoundedFormula Empty (n + 1)) := by |
simp only [card_functions_sum, skolem₁_Functions, mk_sigma, sum_add_distrib']
conv_lhs => enter [2, 1, i]; rw [lift_id'.{u, v}]
rw [add_comm, add_eq_max, max_eq_left]
· refine sum_le_sum _ _ fun n => ?_
rw [← lift_le.{_, max u v}, lift_lift, lift_mk_le.{v}]
refine ⟨⟨fun f => (func f default).bdEqual (f... | 11 | 59,874.141715 | 2 | 2 | 3 | 2,103 |
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Group.Measure
import Mathlib.Topology.Constructions
#align_import measure_theory.constructions.pi from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
noncomputable section
open Function Set MeasureTheory... | Mathlib/MeasureTheory/Constructions/Pi.lean | 166 | 174 | theorem piPremeasure_pi' {s : ∀ i, Set (α i)} : piPremeasure m (pi univ s) = ∏ i, m i (s i) := by |
cases isEmpty_or_nonempty ι
· simp [piPremeasure]
rcases (pi univ s).eq_empty_or_nonempty with h | h
· rcases univ_pi_eq_empty_iff.mp h with ⟨i, hi⟩
have : ∃ i, m i (s i) = 0 := ⟨i, by simp [hi]⟩
simpa [h, Finset.card_univ, zero_pow Fintype.card_ne_zero, @eq_comm _ (0 : ℝ≥0∞),
Finset.prod_eq_zero... | 8 | 2,980.957987 | 2 | 1.2 | 10 | 1,268 |
import Mathlib.Algebra.Field.Basic
import Mathlib.Deprecated.Subring
#align_import deprecated.subfield from "leanprover-community/mathlib"@"bd9851ca476957ea4549eb19b40e7b5ade9428cc"
variable {F : Type*} [Field F] (S : Set F)
structure IsSubfield extends IsSubring S : Prop where
inv_mem : ∀ {x : F}, x ∈ S → x⁻... | Mathlib/Deprecated/Subfield.lean | 75 | 77 | theorem Range.isSubfield {K : Type*} [Field K] (f : F →+* K) : IsSubfield (Set.range f) := by |
rw [← Set.image_univ]
apply Image.isSubfield _ Univ.isSubfield
| 2 | 7.389056 | 1 | 1.5 | 6 | 1,579 |
import Mathlib.Algebra.Polynomial.Div
import Mathlib.RingTheory.Polynomial.Basic
import Mathlib.RingTheory.Ideal.QuotientOperations
#align_import ring_theory.polynomial.quotient from "leanprover-community/mathlib"@"4f840b8d28320b20c87db17b3a6eef3d325fca87"
set_option linter.uppercaseLean3 false
open Polynomial
... | Mathlib/RingTheory/Polynomial/Quotient.lean | 205 | 209 | theorem quotient_map_C_eq_zero {I : Ideal R} {i : R} (hi : i ∈ I) :
(Ideal.Quotient.mk (Ideal.map (C : R →+* MvPolynomial σ R) I :
Ideal (MvPolynomial σ R))).comp C i = 0 := by |
simp only [Function.comp_apply, RingHom.coe_comp, Ideal.Quotient.eq_zero_iff_mem]
exact Ideal.mem_map_of_mem _ hi
| 2 | 7.389056 | 1 | 1.428571 | 7 | 1,526 |
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.RingTheory.DedekindDomain.Ideal
#align_import ring_theory.class_group from "leanprover-community/mathlib"@"565eb991e264d0db702722b4bde52ee5173c9950"
variable {R K L : Type*} [CommRing R]
variable [Field K] [Field L] [DecidableEq L]
variable [Algebra R K] [Is... | Mathlib/RingTheory/ClassGroup.lean | 72 | 79 | theorem mem_principal_ideals_iff {I : (FractionalIdeal R⁰ K)ˣ} :
I ∈ (toPrincipalIdeal R K).range ↔ ∃ x : K, spanSingleton R⁰ x = I := by |
simp only [MonoidHom.mem_range, toPrincipalIdeal_eq_iff]
constructor <;> rintro ⟨x, hx⟩
· exact ⟨x, hx⟩
· refine ⟨Units.mk0 x ?_, hx⟩
rintro rfl
simp [I.ne_zero.symm] at hx
| 6 | 403.428793 | 2 | 1.285714 | 7 | 1,351 |
import Mathlib.RingTheory.FiniteType
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Localization.Integer
import Mathlib.RingTheory.Localization.Submodule
import Mathlib.RingTheory.Nilpotent.Lemmas
import Mathlib.RingTheory.RingHomProperties
im... | Mathlib/RingTheory/LocalProperties.lean | 236 | 255 | theorem Ideal.le_of_localization_maximal {I J : Ideal R}
(h : ∀ (P : Ideal R) (hP : P.IsMaximal),
Ideal.map (algebraMap R (Localization.AtPrime P)) I ≤
Ideal.map (algebraMap R (Localization.AtPrime P)) J) :
I ≤ J := by |
intro x hx
suffices J.colon (Ideal.span {x}) = ⊤ by
simpa using Submodule.mem_colon.mp
(show (1 : R) ∈ J.colon (Ideal.span {x}) from this.symm ▸ Submodule.mem_top) x
(Ideal.mem_span_singleton_self x)
refine Not.imp_symm (J.colon (Ideal.span {x})).exists_le_maximal ?_
push_neg
intro P hP le
... | 15 | 3,269,017.372472 | 2 | 1.833333 | 6 | 1,920 |
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 | 331 | 339 | theorem descPochhammer_succ_comp_X_sub_one (n : ℕ) :
(descPochhammer R (n + 1)).comp (X - 1) =
descPochhammer R (n + 1) - (n + (1 : R[X])) • (descPochhammer R n).comp (X - 1) := by |
suffices (descPochhammer ℤ (n + 1)).comp (X - 1) =
descPochhammer ℤ (n + 1) - (n + 1) * (descPochhammer ℤ n).comp (X - 1)
by simpa [map_comp] using congr_arg (Polynomial.map (Int.castRingHom R)) this
nth_rw 2 [descPochhammer_succ_left]
rw [← sub_mul, descPochhammer_succ_right ℤ n, mul_comp, mul_comm, s... | 6 | 403.428793 | 2 | 0.96 | 25 | 796 |
import Mathlib.CategoryTheory.Subobject.Lattice
import Mathlib.CategoryTheory.EssentiallySmall
import Mathlib.CategoryTheory.Simple
#align_import category_theory.noetherian from "leanprover-community/mathlib"@"7c4c90f422a4a4477e4d8bc4dc9f1e634e6b2349"
namespace CategoryTheory
open CategoryTheory.Limits
variable... | Mathlib/CategoryTheory/Noetherian.lean | 82 | 87 | theorem exists_simple_subobject {X : C} [ArtinianObject X] (h : ¬IsZero X) :
∃ Y : Subobject X, Simple (Y : C) := by |
haveI : Nontrivial (Subobject X) := nontrivial_of_not_isZero h
haveI := isAtomic_of_orderBot_wellFounded_lt (ArtinianObject.subobject_lt_wellFounded X)
obtain ⟨Y, s⟩ := (IsAtomic.eq_bot_or_exists_atom_le (⊤ : Subobject X)).resolve_left top_ne_bot
exact ⟨Y, (subobject_simple_iff_isAtom _).mpr s.1⟩
| 4 | 54.59815 | 2 | 2 | 1 | 2,031 |
import Mathlib.Algebra.Group.Equiv.TypeTags
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Module.LinearMap.Basic
import Mathlib.Algebra.MonoidAlgebra.Basic
import Mathlib.LinearAlgebra.Dual
import Mathlib.LinearAlgebra.Contraction
import Mathlib.RingTheory.TensorProduct.Basic
#align_import representation_... | Mathlib/RepresentationTheory/Basic.lean | 166 | 169 | theorem asModuleEquiv_symm_map_rho (g : G) (x : V) :
ρ.asModuleEquiv.symm (ρ g x) = MonoidAlgebra.of k G g • ρ.asModuleEquiv.symm x := by |
apply_fun ρ.asModuleEquiv
simp
| 2 | 7.389056 | 1 | 0.75 | 8 | 651 |
import Mathlib.SetTheory.Game.State
#align_import set_theory.game.domineering from "leanprover-community/mathlib"@"b134b2f5cf6dd25d4bbfd3c498b6e36c11a17225"
namespace SetTheory
namespace PGame
namespace Domineering
open Function
@[simps!]
def shiftUp : ℤ × ℤ ≃ ℤ × ℤ :=
(Equiv.refl ℤ).prodCongr (Equiv.addRig... | Mathlib/SetTheory/Game/Domineering.lean | 86 | 90 | theorem snd_pred_mem_erase_of_mem_left {b : Board} {m : ℤ × ℤ} (h : m ∈ left b) :
(m.1, m.2 - 1) ∈ b.erase m := by |
rw [mem_left] at h
apply Finset.mem_erase_of_ne_of_mem _ h.2
exact ne_of_apply_ne Prod.snd (pred_ne_self m.2)
| 3 | 20.085537 | 1 | 1.428571 | 7 | 1,521 |
import Mathlib.Data.Countable.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Order.Disjointed
import Mathlib.MeasureTheory.OuterMeasure.Defs
#align_import measure_theory.measure.outer_measure from "leanprover-community/mathlib"@"343e80208d29d2d15f8050b929aa50fe4ce71b55"
noncomputable section
open Set F... | Mathlib/MeasureTheory/OuterMeasure/Basic.lean | 72 | 76 | theorem measure_biUnion_le {I : Set ι} (μ : F) (hI : I.Countable) (s : ι → Set α) :
μ (⋃ i ∈ I, s i) ≤ ∑' i : I, μ (s i) := by |
have := hI.to_subtype
rw [biUnion_eq_iUnion]
apply measure_iUnion_le
| 3 | 20.085537 | 1 | 0.5 | 10 | 470 |
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.LinearAlgebra.StdBasis
import Mathlib.RingTheory.AlgebraTower
import Mathlib.Algebra.Algebra.Subalgebra.Tower
#align_import linear_algebra.matrix.to_lin from "leanprover-community/mathlib"@"0e2aab2b0d521f060f62a14d2cf2e2c54e8491d6"
... | Mathlib/LinearAlgebra/Matrix/ToLin.lean | 91 | 99 | theorem Matrix.vecMul_stdBasis (M : Matrix m n R) (i j) :
(LinearMap.stdBasis R (fun _ ↦ R) i 1 ᵥ* M) j = M i j := by |
have : (∑ i', (if i = i' then 1 else 0) * M i' j) = M i j := by
simp_rw [boole_mul, Finset.sum_ite_eq, Finset.mem_univ, if_true]
simp only [vecMul, dotProduct]
convert this
split_ifs with h <;> simp only [stdBasis_apply]
· rw [h, Function.update_same]
· rw [Function.update_noteq (Ne.symm h), Pi.zero_ap... | 7 | 1,096.633158 | 2 | 1.75 | 4 | 1,846 |
import Mathlib.Topology.MetricSpace.Basic
#align_import topology.metric_space.infsep from "leanprover-community/mathlib"@"5316314b553dcf8c6716541851517c1a9715e22b"
variable {α β : Type*}
namespace Set
section Einfsep
open ENNReal
open Function
noncomputable def einfsep [EDist α] (s : Set α) : ℝ≥0∞ :=
⨅ (x... | Mathlib/Topology/MetricSpace/Infsep.lean | 93 | 95 | theorem Subsingleton.einfsep (hs : s.Subsingleton) : s.einfsep = ∞ := by |
rw [einfsep_top]
exact fun _ hx _ hy hxy => (hxy <| hs hx hy).elim
| 2 | 7.389056 | 1 | 0.25 | 12 | 302 |
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.Nat.Digits
import Mathlib.Data.Nat.MaxPowDiv
import Mathlib.Data.Nat.Multiplicity
import Mathlib.Tactic.IntervalCases
#align_import number_theory.padics.padic_val from "leanprover-community/mathlib"@"60fa54e778c9e85d930efae172435f42fb0d71f7"
universe u
ope... | Mathlib/NumberTheory/Padics/PadicVal.lean | 162 | 169 | theorem of_ne_one_ne_zero {z : ℤ} (hp : p ≠ 1) (hz : z ≠ 0) :
padicValInt p z =
(multiplicity (p : ℤ) z).get
(by
apply multiplicity.finite_int_iff.2
simp [hp, hz]) := by |
rw [padicValInt, padicValNat, dif_pos (And.intro hp (Int.natAbs_pos.mpr hz))]
simp only [multiplicity.Int.natAbs p z]
| 2 | 7.389056 | 1 | 1.333333 | 6 | 1,377 |
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks
#align_import category_theory.limits.constructions.epi_mono from "leanprover-community/mathlib"@"f7baecbb54bd0f24f228576f97b1752fc3c9b318"
... | Mathlib/CategoryTheory/Limits/Constructions/EpiMono.lean | 32 | 36 | theorem preserves_mono_of_preservesLimit {X Y : C} (f : X ⟶ Y) [PreservesLimit (cospan f f) F]
[Mono f] : Mono (F.map f) := by |
have := isLimitPullbackConeMapOfIsLimit F _ (PullbackCone.isLimitMkIdId f)
simp_rw [F.map_id] at this
apply PullbackCone.mono_of_isLimitMkIdId _ this
| 3 | 20.085537 | 1 | 1.25 | 4 | 1,309 |
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 | 171 | 176 | theorem discr_powerBasis_eq_prod' [IsSeparable K L] (e : Fin pb.dim ≃ (L →ₐ[K] E)) :
algebraMap K E (discr K pb.basis) =
∏ i : Fin pb.dim, ∏ j ∈ Ioi i, -((e j pb.gen - e i pb.gen) * (e i pb.gen - e j pb.gen)) := by |
rw [discr_powerBasis_eq_prod _ _ _ e]
congr; ext i; congr; ext j
ring
| 3 | 20.085537 | 1 | 1.2 | 10 | 1,266 |
import Mathlib.Analysis.Complex.CauchyIntegral
import Mathlib.Analysis.NormedSpace.Completion
import Mathlib.Analysis.NormedSpace.Extr
import Mathlib.Topology.Order.ExtrClosure
#align_import analysis.complex.abs_max from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open TopologicalSpa... | Mathlib/Analysis/Complex/AbsMax.lean | 181 | 196 | theorem norm_eqOn_closedBall_of_isMaxOn {f : E → F} {z : E} {r : ℝ}
(hd : DiffContOnCl ℂ f (ball z r)) (hz : IsMaxOn (norm ∘ f) (ball z r) z) :
EqOn (norm ∘ f) (const E ‖f z‖) (closedBall z r) := by |
intro w hw
rw [mem_closedBall, dist_comm] at hw
rcases eq_or_ne z w with (rfl | hne); · rfl
set e := (lineMap z w : ℂ → E)
have hde : Differentiable ℂ e := (differentiable_id.smul_const (w - z)).add_const z
suffices ‖(f ∘ e) (1 : ℂ)‖ = ‖(f ∘ e) (0 : ℂ)‖ by simpa [e]
have hr : dist (1 : ℂ) 0 = 1 := by sim... | 13 | 442,413.392009 | 2 | 2 | 4 | 2,117 |
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from "leanprover-community/mathlib"@"07992a1d1f7a4176c6d3f160209608be4e198566"
open scoped ComplexOrder
namespace Matrix
variable {m n R 𝕜 : Type*}
variable [Fintype m] [Fint... | Mathlib/LinearAlgebra/Matrix/PosDef.lean | 81 | 87 | theorem submatrix {M : Matrix n n R} (hM : M.PosSemidef) (e : m → n) :
(M.submatrix e e).PosSemidef := by |
classical
rw [(by simp : M = 1 * M * 1), submatrix_mul (he₂ := Function.bijective_id),
submatrix_mul (he₂ := Function.bijective_id), submatrix_id_id]
simpa only [conjTranspose_submatrix, conjTranspose_one] using
conjTranspose_mul_mul_same hM (Matrix.submatrix 1 id e)
| 5 | 148.413159 | 2 | 1.5 | 2 | 1,567 |
import Mathlib.Combinatorics.SetFamily.Shadow
#align_import combinatorics.set_family.compression.uv from "leanprover-community/mathlib"@"6f8ab7de1c4b78a68ab8cf7dd83d549eb78a68a1"
open Finset
variable {α : Type*}
theorem sup_sdiff_injOn [GeneralizedBooleanAlgebra α] (u v : α) :
{ x | Disjoint u x ∧ v ≤ x }.... | Mathlib/Combinatorics/SetFamily/Compression/UV.lean | 107 | 110 | theorem compress_sdiff_sdiff (a b : α) : compress (a \ b) (b \ a) b = a := by |
refine (compress_of_disjoint_of_le disjoint_sdiff_self_left sdiff_le).trans ?_
rw [sup_sdiff_self_right, sup_sdiff, disjoint_sdiff_self_right.sdiff_eq_left, sup_eq_right]
exact sdiff_sdiff_le
| 3 | 20.085537 | 1 | 1.6 | 10 | 1,727 |
import Mathlib.LinearAlgebra.CliffordAlgebra.Fold
import Mathlib.LinearAlgebra.ExteriorAlgebra.Basic
#align_import linear_algebra.exterior_algebra.of_alternating from "leanprover-community/mathlib"@"ce11c3c2a285bbe6937e26d9792fda4e51f3fe1a"
variable {R M N N' : Type*}
variable [CommRing R] [AddCommGroup M] [AddCo... | Mathlib/LinearAlgebra/ExteriorAlgebra/OfAlternating.lean | 79 | 85 | theorem liftAlternating_ι_mul (f : ∀ i, M [⋀^Fin i]→ₗ[R] N) (m : M)
(x : ExteriorAlgebra R M) :
liftAlternating (R := R) (M := M) (N := N) f (ι R m * x) =
liftAlternating (R := R) (M := M) (N := N) (fun i => (f i.succ).curryLeft m) x := by |
dsimp [liftAlternating]
rw [foldl_mul, foldl_ι]
rfl
| 3 | 20.085537 | 1 | 1.333333 | 6 | 1,397 |
import Mathlib.Data.Finsupp.Basic
import Mathlib.Data.List.AList
#align_import data.finsupp.alist from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
namespace AList
variable {α M : Type*} [Zero M]
open List
noncomputable def lookupFinsupp (l : AList fun _x : α => M) : α →₀ M where
... | Mathlib/Data/Finsupp/AList.lean | 116 | 120 | theorem singleton_lookupFinsupp (a : α) (m : M) :
(singleton a m).lookupFinsupp = Finsupp.single a m := by |
classical
-- porting note (#10745): was `simp [← AList.insert_empty]` but timeout issues
simp only [← AList.insert_empty, insert_lookupFinsupp, empty_lookupFinsupp, Finsupp.zero_update]
| 3 | 20.085537 | 1 | 0.777778 | 9 | 691 |
import Mathlib.Data.List.Basic
#align_import data.list.infix from "leanprover-community/mathlib"@"26f081a2fb920140ed5bc5cc5344e84bcc7cb2b2"
open Nat
variable {α β : Type*}
namespace List
variable {l l₁ l₂ l₃ : List α} {a b : α} {m n : ℕ}
section Fix
#align list.prefix_append List.prefix_append
#align list.... | Mathlib/Data/List/Infix.lean | 73 | 76 | theorem prefix_concat_iff {l₁ l₂ : List α} {a : α} :
l₁ <+: l₂ ++ [a] ↔ l₁ = l₂ ++ [a] ∨ l₁ <+: l₂ := by |
simpa only [← reverse_concat', reverse_inj, reverse_suffix] using
suffix_cons_iff (l₁ := l₁.reverse) (l₂ := l₂.reverse)
| 2 | 7.389056 | 1 | 0.5 | 2 | 457 |
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Set.Finite
#align_import data.finset.n_ary from "leanprover-community/mathlib"@"eba7871095e834365616b5e43c8c7bb0b37058d0"
open Function Set
variable {α α' β β' γ γ' δ δ' ε ε' ζ ζ' ν : Type*}
namespace Finset
variable [DecidableEq α'] [DecidableEq β'] [Decidabl... | Mathlib/Data/Finset/NAry.lean | 77 | 79 | theorem image₂_subset (hs : s ⊆ s') (ht : t ⊆ t') : image₂ f s t ⊆ image₂ f s' t' := by |
rw [← coe_subset, coe_image₂, coe_image₂]
exact image2_subset hs ht
| 2 | 7.389056 | 1 | 0.375 | 8 | 379 |
import Mathlib.Logic.Function.CompTypeclasses
import Mathlib.Algebra.Group.Hom.Defs
section MonoidHomCompTriple
namespace MonoidHom
class CompTriple {M N P : Type*} [Monoid M] [Monoid N] [Monoid P]
(φ : M →* N) (ψ : N →* P) (χ : outParam (M →* P)) : Prop where
comp_eq : ψ.comp φ = χ
attribute [simp] C... | Mathlib/Algebra/Group/Hom/CompTypeclasses.lean | 98 | 106 | theorem comp_assoc {Q : Type*} [Monoid Q]
{φ₁ : M →* N} {φ₂ : N →* P} {φ₁₂ : M →* P}
(κ : CompTriple φ₁ φ₂ φ₁₂)
{φ₃ : P →* Q} {φ₂₃ : N →* Q} (κ' : CompTriple φ₂ φ₃ φ₂₃)
{φ₁₂₃ : M →* Q} :
CompTriple φ₁ φ₂₃ φ₁₂₃ ↔ CompTriple φ₁₂ φ₃ φ₁₂₃ := by |
constructor <;>
· rintro ⟨h⟩
exact ⟨by simp only [← κ.comp_eq, ← h, ← κ'.comp_eq, MonoidHom.comp_assoc]⟩
| 3 | 20.085537 | 1 | 1 | 1 | 822 |
import Mathlib.Analysis.RCLike.Lemmas
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.function.l2_space from "leanprover-community/mathlib"@"83a66c8775fa14ee5180c85cab98e970956401ad"
set_option linter.uppercaseLean3 false... | Mathlib/MeasureTheory/Function/L2Space.lean | 54 | 57 | theorem memℒp_two_iff_integrable_sq {f : α → ℝ} (hf : AEStronglyMeasurable f μ) :
Memℒp f 2 μ ↔ Integrable (fun x => f x ^ 2) μ := by |
convert memℒp_two_iff_integrable_sq_norm hf using 3
simp
| 2 | 7.389056 | 1 | 0.888889 | 9 | 770 |
import Mathlib.CategoryTheory.Limits.Shapes.Equalizers
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Shapes.StrongEpi
import Mathlib.CategoryTheory.MorphismProperty.Factorization
#align_import category_theory.limits.shapes.images from "leanprover-community/mathlib"@"563aed... | Mathlib/CategoryTheory/Limits/Shapes/Images.lean | 108 | 115 | theorem ext {F F' : MonoFactorisation f} (hI : F.I = F'.I)
(hm : F.m = eqToHom hI ≫ F'.m) : F = F' := by |
cases' F with _ Fm _ _ Ffac; cases' F' with _ Fm' _ _ Ffac'
cases' hI
simp? at hm says simp only [eqToHom_refl, Category.id_comp] at hm
congr
apply (cancel_mono Fm).1
rw [Ffac, hm, Ffac']
| 6 | 403.428793 | 2 | 2 | 1 | 2,177 |
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 | 117 | 123 | theorem spanPoints_nonempty (s : Set P) : (spanPoints k s).Nonempty ↔ s.Nonempty := by |
constructor
· contrapose
rw [Set.not_nonempty_iff_eq_empty, Set.not_nonempty_iff_eq_empty]
intro h
simp [h, spanPoints]
· exact fun h => h.mono (subset_spanPoints _ _)
| 6 | 403.428793 | 2 | 1 | 5 | 815 |
import Mathlib.GroupTheory.CoprodI
import Mathlib.GroupTheory.Coprod.Basic
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.GroupTheory.Complement
namespace Monoid
open CoprodI Subgroup Coprod Function List
variable {ι : Type*} {G : ι → Type*} {H : Type*} {K : Type*} [Monoid K]
def PushoutI.con [∀ i, Mo... | Mathlib/GroupTheory/PushoutI.lean | 167 | 184 | theorem induction_on {motive : PushoutI φ → Prop}
(x : PushoutI φ)
(of : ∀ (i : ι) (g : G i), motive (of i g))
(base : ∀ h, motive (base φ h))
(mul : ∀ x y, motive x → motive y → motive (x * y)) : motive x := by |
delta PushoutI PushoutI.of PushoutI.base at *
induction x using Con.induction_on with
| H x =>
induction x using Coprod.induction_on with
| inl g =>
induction g using CoprodI.induction_on with
| h_of i g => exact of i g
| h_mul x y ihx ihy =>
rw [map_mul]
exact mul _ _ i... | 13 | 442,413.392009 | 2 | 1 | 6 | 991 |
import Mathlib.Data.Set.Basic
#align_import data.set.bool_indicator from "leanprover-community/mathlib"@"fc2ed6f838ce7c9b7c7171e58d78eaf7b438fb0e"
open Bool
namespace Set
variable {α : Type*} (s : Set α)
noncomputable def boolIndicator (x : α) :=
@ite _ (x ∈ s) (Classical.propDecidable _) true false
#align s... | Mathlib/Data/Set/BoolIndicator.lean | 27 | 29 | theorem mem_iff_boolIndicator (x : α) : x ∈ s ↔ s.boolIndicator x = true := by |
unfold boolIndicator
split_ifs with h <;> simp [h]
| 2 | 7.389056 | 1 | 1 | 4 | 906 |
import Mathlib.LinearAlgebra.TensorProduct.RightExactness
import Mathlib.LinearAlgebra.TensorProduct.Finiteness
universe u
variable (R : Type u) [CommRing R]
variable {M : Type u} [AddCommGroup M] [Module R M]
variable {N : Type u} [AddCommGroup N] [Module R N]
open Classical DirectSum LinearMap Function Submodul... | Mathlib/LinearAlgebra/TensorProduct/Vanishing.lean | 175 | 192 | theorem vanishesTrivially_of_sum_tmul_eq_zero_of_rTensor_injective
(hm : Injective (rTensor N (span R (Set.range m)).subtype))
(hmn : ∑ i, m i ⊗ₜ n i = (0 : M ⊗[R] N)) : VanishesTrivially R m n := by |
-- Restrict `m` on the codomain to $M'$, then apply `vanishesTrivially_of_sum_tmul_eq_zero`.
have mem_M' i : m i ∈ span R (Set.range m) := subset_span ⟨i, rfl⟩
set m' : ι → span R (Set.range m) := Subtype.coind m mem_M' with m'_eq
have hm' : span R (Set.range m') = ⊤ := by
apply map_injective_of_injective ... | 15 | 3,269,017.372472 | 2 | 2 | 3 | 2,251 |
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Data.Nat.Totient
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.GroupTheory.Subgroup.Simple
import Mathlib.Tactic.Group
import Mathlib.GroupTheory.Exponent
#align_import group_theory.specific_groups.cyclic from "leanprover-community/mathli... | Mathlib/GroupTheory/SpecificGroups/Cyclic.lean | 110 | 116 | theorem MonoidHom.map_cyclic {G : Type*} [Group G] [h : IsCyclic G] (σ : G →* G) :
∃ m : ℤ, ∀ g : G, σ g = g ^ m := by |
obtain ⟨h, hG⟩ := IsCyclic.exists_generator (α := G)
obtain ⟨m, hm⟩ := hG (σ h)
refine ⟨m, fun g => ?_⟩
obtain ⟨n, rfl⟩ := hG g
rw [MonoidHom.map_zpow, ← hm, ← zpow_mul, ← zpow_mul']
| 5 | 148.413159 | 2 | 1.333333 | 6 | 1,386 |
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 | 81 | 83 | theorem coeff_p_zero [CharP R p] : (p : 𝕎 R).coeff 0 = 0 := by |
rw [coeff_p, if_neg]
exact zero_ne_one
| 2 | 7.389056 | 1 | 1 | 13 | 1,103 |
import Mathlib.Geometry.Manifold.MFDeriv.Defs
#align_import geometry.manifold.mfderiv from "leanprover-community/mathlib"@"e473c3198bb41f68560cab68a0529c854b618833"
noncomputable section
open scoped Topology Manifold
open Set Bundle
section DerivativesProperties
variable
{𝕜 : Type*} [NontriviallyNormedFiel... | Mathlib/Geometry/Manifold/MFDeriv/Basic.lean | 54 | 59 | theorem uniqueMDiffWithinAt_iff {s : Set M} {x : M} :
UniqueMDiffWithinAt I s x ↔
UniqueDiffWithinAt 𝕜 ((extChartAt I x).symm ⁻¹' s ∩ (extChartAt I x).target)
((extChartAt I x) x) := by |
apply uniqueDiffWithinAt_congr
rw [nhdsWithin_inter, nhdsWithin_inter, nhdsWithin_extChartAt_target_eq]
| 2 | 7.389056 | 1 | 1 | 2 | 890 |
import Mathlib.Analysis.NormedSpace.ConformalLinearMap
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.conformal.normed_space from "leanprover-community/mathlib"@"e3fb84046afd187b710170887195d50bada934ee"
noncomputable section
variable {X Y Z : Type*} [NormedAddCommGroup X] [NormedAdd... | Mathlib/Analysis/Calculus/Conformal/NormedSpace.lean | 98 | 102 | theorem comp {f : X → Y} {g : Y → Z} (x : X) (hg : ConformalAt g (f x)) (hf : ConformalAt f x) :
ConformalAt (g ∘ f) x := by |
rcases hf with ⟨f', hf₁, cf⟩
rcases hg with ⟨g', hg₁, cg⟩
exact ⟨g'.comp f', hg₁.comp x hf₁, cg.comp cf⟩
| 3 | 20.085537 | 1 | 1.5 | 2 | 1,611 |
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Analysis.SpecialFunctions.Complex.Circle
import Mathlib.Analysis.InnerProductSpace.l2Space
import Mathlib.MeasureTheory.Function.ContinuousMapDense
import Mathlib.MeasureTheory.Function.L2Space
import Mathlib.MeasureTheory.Group.Integral
import Mathlib.M... | Mathlib/Analysis/Fourier/AddCircle.lean | 154 | 159 | theorem fourier_neg {n : ℤ} {x : AddCircle T} : fourier (-n) x = conj (fourier n x) := by |
induction x using QuotientAddGroup.induction_on'
simp_rw [fourier_apply, toCircle]
rw [← QuotientAddGroup.mk_zsmul, ← QuotientAddGroup.mk_zsmul]
simp_rw [Function.Periodic.lift_coe, ← coe_inv_circle_eq_conj, ← expMapCircle_neg,
neg_smul, mul_neg]
| 5 | 148.413159 | 2 | 0.916667 | 12 | 791 |
import Mathlib.ModelTheory.Satisfiability
#align_import model_theory.types from "leanprover-community/mathlib"@"98bd247d933fb581ff37244a5998bd33d81dd46d"
set_option linter.uppercaseLean3 false
universe u v w w'
open Cardinal Set
open scoped Classical
open Cardinal FirstOrder
namespace FirstOrder
namespace La... | Mathlib/ModelTheory/Types.lean | 115 | 126 | theorem setOf_subset_eq_empty_iff (S : L[[α]].Theory) :
{ p : T.CompleteType α | S ⊆ ↑p } = ∅ ↔
¬((L.lhomWithConstants α).onTheory T ∪ S).IsSatisfiable := by |
rw [iff_not_comm, ← not_nonempty_iff_eq_empty, Classical.not_not, Set.Nonempty]
refine
⟨fun h =>
⟨⟨L[[α]].completeTheory h.some, (subset_union_left (t := S)).trans completeTheory.subset,
completeTheory.isMaximal (L[[α]]) h.some⟩,
(((L.lhomWithConstants α).onTheory T).subset_union_right)... | 9 | 8,103.083928 | 2 | 1.8 | 5 | 1,884 |
import Mathlib.Geometry.Euclidean.Circumcenter
#align_import geometry.euclidean.monge_point from "leanprover-community/mathlib"@"1a4df69ca1a9a0e5e26bfe12e2b92814216016d0"
noncomputable section
open scoped Classical
open scoped RealInnerProductSpace
namespace Affine
namespace Simplex
open Finset AffineSubspac... | Mathlib/Geometry/Euclidean/MongePoint.lean | 130 | 154 | theorem mongePoint_eq_affineCombination_of_pointsWithCircumcenter {n : ℕ}
(s : Simplex ℝ P (n + 2)) :
s.mongePoint =
(univ : Finset (PointsWithCircumcenterIndex (n + 2))).affineCombination ℝ
s.pointsWithCircumcenter (mongePointWeightsWithCircumcenter n) := by |
rw [mongePoint_eq_smul_vsub_vadd_circumcenter,
centroid_eq_affineCombination_of_pointsWithCircumcenter,
circumcenter_eq_affineCombination_of_pointsWithCircumcenter, affineCombination_vsub,
← LinearMap.map_smul, weightedVSub_vadd_affineCombination]
congr with i
rw [Pi.add_apply, Pi.smul_apply, smul_eq... | 20 | 485,165,195.40979 | 2 | 1.75 | 4 | 1,853 |
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 | 67 | 82 | theorem minFacHelper_1 {n k k' : ℕ} (e : k + 2 = k') (h : MinFacHelper n k)
(np : minFac n ≠ k) : MinFacHelper n k' := by |
rw [← e]
refine ⟨Nat.lt_add_right _ h.1, ?_, ?_⟩
· rw [add_mod, mod_self, add_zero, mod_mod]
exact h.2.1
rcases h.2.2.eq_or_lt with rfl|h2
· exact (np rfl).elim
rcases (succ_le_of_lt h2).eq_or_lt with h2|h2
· refine ((h.1.trans_le h.2.2).ne ?_).elim
have h3 : 2 ∣ minFac n := by
rw [Nat.dvd_... | 14 | 1,202,604.284165 | 2 | 1.8 | 5 | 1,898 |
import Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho
import Mathlib.LinearAlgebra.Orientation
#align_import analysis.inner_product_space.orientation from "leanprover-community/mathlib"@"bd65478311e4dfd41f48bf38c7e3b02fb75d0163"
noncomputable section
variable {E : Type*} [NormedAddCommGroup E] [InnerProduct... | Mathlib/Analysis/InnerProductSpace/Orientation.lean | 103 | 105 | theorem orthonormal_adjustToOrientation : Orthonormal ℝ (e.toBasis.adjustToOrientation x) := by |
apply e.orthonormal.orthonormal_of_forall_eq_or_eq_neg
simpa using e.toBasis.adjustToOrientation_apply_eq_or_eq_neg x
| 2 | 7.389056 | 1 | 1.111111 | 9 | 1,197 |
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Algebra.Order.Ring.Int
import Mathlib.Data.Int.GCD
instance : GCDMonoid ℕ where
gcd := Nat.gcd
lcm := Nat.lcm
gcd_dvd_left := Nat.gcd_dvd_left
gcd_dvd_right := Nat.gcd_dvd_right
dvd_gcd := Nat.dvd_gcd
gcd_mul_lcm a b := by rw [Nat.gcd_mul_lcm]; rfl
... | Mathlib/Algebra/GCDMonoid/Nat.lean | 139 | 145 | theorem exists_unit_of_abs (a : ℤ) : ∃ (u : ℤ) (_ : IsUnit u), (Int.natAbs a : ℤ) = u * a := by |
cases' natAbs_eq a with h h
· use 1, isUnit_one
rw [← h, one_mul]
· use -1, isUnit_one.neg
rw [← neg_eq_iff_eq_neg.mpr h]
simp only [neg_mul, one_mul]
| 6 | 403.428793 | 2 | 1 | 4 | 1,077 |
import Mathlib.RingTheory.OrzechProperty
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.PrincipalIdealDomain
#align_import linear_algebra.invariant_basis_number from "leanprover-community/mathlib"@"5fd3186f1ec30a75d5f65732e3ce5e623382556f"
noncomputable section
open Function
universe u v w
... | Mathlib/LinearAlgebra/InvariantBasisNumber.lean | 158 | 164 | theorem card_le_of_injective [StrongRankCondition R] {α β : Type*} [Fintype α] [Fintype β]
(f : (α → R) →ₗ[R] β → R) (i : Injective f) : Fintype.card α ≤ Fintype.card β := by |
let P := LinearEquiv.funCongrLeft R R (Fintype.equivFin α)
let Q := LinearEquiv.funCongrLeft R R (Fintype.equivFin β)
exact
le_of_fin_injective R ((Q.symm.toLinearMap.comp f).comp P.toLinearMap)
(((LinearEquiv.symm Q).injective.comp i).comp (LinearEquiv.injective P))
| 5 | 148.413159 | 2 | 2 | 5 | 2,178 |
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.Algebra.Module.Torsion
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v v' u₁' w w'
variable {R S : Type u} {M : Type v} {M' : Type v'} {M₁ : Type v}... | Mathlib/LinearAlgebra/Dimension/Constructions.lean | 550 | 552 | theorem Subalgebra.rank_top : Module.rank F (⊤ : Subalgebra F E) = Module.rank F E := by |
rw [subalgebra_top_rank_eq_submodule_top_rank]
exact _root_.rank_top F E
| 2 | 7.389056 | 1 | 0.75 | 24 | 667 |
import Mathlib.Algebra.Polynomial.Smeval
import Mathlib.GroupTheory.GroupAction.Ring
import Mathlib.RingTheory.Polynomial.Pochhammer
section Multichoose
open Function Polynomial
class BinomialRing (R : Type*) [AddCommMonoid R] [Pow R ℕ] where
nsmul_right_injective (n : ℕ) (h : n ≠ 0) : Injective (n • · : R →... | Mathlib/RingTheory/Binomial.lean | 107 | 115 | theorem descPochhammer_smeval_eq_ascPochhammer (r : R) (n : ℕ) :
(descPochhammer ℤ n).smeval r = (ascPochhammer ℕ n).smeval (r - n + 1) := by |
induction n with
| zero => simp only [descPochhammer_zero, ascPochhammer_zero, smeval_one, npow_zero]
| succ n ih =>
rw [Nat.cast_succ, sub_add, add_sub_cancel_right, descPochhammer_succ_right, smeval_mul, ih,
ascPochhammer_succ_left, X_mul, smeval_mul_X, smeval_comp, smeval_sub, ← C_eq_natCast,
... | 7 | 1,096.633158 | 2 | 1.6 | 5 | 1,723 |
import Mathlib.Data.List.OfFn
import Mathlib.Data.List.Range
#align_import data.list.indexes from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
assert_not_exists MonoidWithZero
universe u v
open Function
namespace List
variable {α : Type u} {β : Type v}
section MapIdx
-- Porting n... | Mathlib/Data/List/Indexes.lean | 132 | 138 | theorem mapIdxGo_length : ∀ (f : ℕ → α → β) (l : List α) (arr : Array β),
length (mapIdx.go f l arr) = length l + arr.size := by |
intro f l
induction' l with head tail ih
· intro; simp only [mapIdx.go, Array.toList_eq, length_nil, Nat.zero_add]
· intro; simp only [mapIdx.go]; rw [ih]; simp only [Array.size_push, length_cons];
simp only [Nat.add_succ, add_zero, Nat.add_comm]
| 5 | 148.413159 | 2 | 1.5 | 6 | 1,681 |
import Mathlib.LinearAlgebra.Eigenspace.Basic
import Mathlib.FieldTheory.Minpoly.Field
#align_import linear_algebra.eigenspace.minpoly from "leanprover-community/mathlib"@"c3216069e5f9369e6be586ccbfcde2592b3cec92"
universe u v w
namespace Module
namespace End
open Polynomial FiniteDimensional
open scoped Poly... | Mathlib/LinearAlgebra/Eigenspace/Minpoly.lean | 75 | 91 | theorem hasEigenvalue_of_isRoot (h : (minpoly K f).IsRoot μ) : f.HasEigenvalue μ := by |
cases' dvd_iff_isRoot.2 h with p hp
rw [HasEigenvalue, eigenspace]
intro con
cases' (LinearMap.isUnit_iff_ker_eq_bot _).2 con with u hu
have p_ne_0 : p ≠ 0 := by
intro con
apply minpoly.ne_zero (Algebra.IsIntegral.isIntegral (R := K) f)
rw [hp, con, mul_zero]
have : (aeval f) p = 0 := by
ha... | 16 | 8,886,110.520508 | 2 | 1.8 | 5 | 1,893 |
import Mathlib.Data.Int.Interval
import Mathlib.Data.Int.SuccPred
import Mathlib.Data.Int.ConditionallyCompleteOrder
import Mathlib.Topology.Instances.Discrete
import Mathlib.Topology.MetricSpace.Bounded
import Mathlib.Order.Filter.Archimedean
#align_import topology.instances.int from "leanprover-community/mathlib"@"... | Mathlib/Topology/Instances/Int.lean | 41 | 43 | theorem pairwise_one_le_dist : Pairwise fun m n : ℤ => 1 ≤ dist m n := by |
intro m n hne
rw [dist_eq]; norm_cast; rwa [← zero_add (1 : ℤ), Int.add_one_le_iff, abs_pos, sub_ne_zero]
| 2 | 7.389056 | 1 | 0.333333 | 6 | 327 |
import Mathlib.Probability.Martingale.BorelCantelli
import Mathlib.Probability.ConditionalExpectation
import Mathlib.Probability.Independence.Basic
#align_import probability.borel_cantelli from "leanprover-community/mathlib"@"2f8347015b12b0864dfaf366ec4909eb70c78740"
open scoped MeasureTheory ProbabilityTheory EN... | Mathlib/Probability/BorelCantelli.lean | 43 | 48 | theorem iIndepFun.indep_comap_natural_of_lt (hf : ∀ i, StronglyMeasurable (f i))
(hfi : iIndepFun (fun _ => mβ) f μ) (hij : i < j) :
Indep (MeasurableSpace.comap (f j) mβ) (Filtration.natural f hf i) μ := by |
suffices Indep (⨆ k ∈ ({j} : Set ι), MeasurableSpace.comap (f k) mβ)
(⨆ k ∈ {k | k ≤ i}, MeasurableSpace.comap (f k) mβ) μ by rwa [iSup_singleton] at this
exact indep_iSup_of_disjoint (fun k => (hf k).measurable.comap_le) hfi (by simpa)
| 3 | 20.085537 | 1 | 1.333333 | 3 | 1,369 |
import Mathlib.Dynamics.PeriodicPts
import Mathlib.GroupTheory.Exponent
import Mathlib.GroupTheory.GroupAction.Basic
namespace MulAction
universe u v
variable {α : Type v}
variable {G : Type u} [Group G] [MulAction G α]
variable {M : Type u} [Monoid M] [MulAction M α]
@[to_additive "If the action is periodic, t... | Mathlib/GroupTheory/GroupAction/Period.lean | 117 | 120 | theorem period_bounded_of_exponent_pos (exp_pos : 0 < Monoid.exponent M) (m : M) :
BddAbove (Set.range (fun a : α => period m a)) := by |
use Monoid.exponent M
simpa [upperBounds] using period_le_exponent exp_pos _
| 2 | 7.389056 | 1 | 0.75 | 4 | 663 |
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.MeasureTheory.Measure.Haar.Quotient
import Mathlib.MeasureTheory.Constructions.Polish
import Mathlib.MeasureTheory.Integral.IntervalIntegral
import Mathlib.Topology.Algebra.Order.Floor
#align_import measure_theory.integral.periodic from "leanprover-c... | Mathlib/MeasureTheory/Integral/Periodic.lean | 256 | 262 | theorem intervalIntegral_add_eq_of_pos (hf : Periodic f T) (hT : 0 < T) (t s : ℝ) :
∫ x in t..t + T, f x = ∫ x in s..s + T, f x := by |
simp only [integral_of_le, hT.le, le_add_iff_nonneg_right]
haveI : VAddInvariantMeasure (AddSubgroup.zmultiples T) ℝ volume :=
⟨fun c s _ => measure_preimage_add _ _ _⟩
apply IsAddFundamentalDomain.setIntegral_eq (G := AddSubgroup.zmultiples T)
exacts [isAddFundamentalDomain_Ioc hT t, isAddFundamentalDomai... | 5 | 148.413159 | 2 | 1.666667 | 6 | 1,765 |
import Mathlib.Algebra.MonoidAlgebra.Basic
import Mathlib.Algebra.Group.UniqueProds
#align_import algebra.monoid_algebra.no_zero_divisors from "leanprover-community/mathlib"@"3e067975886cf5801e597925328c335609511b1a"
open Finsupp
variable {R A : Type*} [Semiring R]
namespace MonoidAlgebra
| Mathlib/Algebra/MonoidAlgebra/NoZeroDivisors.lean | 68 | 79 | theorem mul_apply_mul_eq_mul_of_uniqueMul [Mul A] {f g : MonoidAlgebra R A} {a0 b0 : A}
(h : UniqueMul f.support g.support a0 b0) :
(f * g) (a0 * b0) = f a0 * g b0 := by |
classical
simp_rw [mul_apply, sum, ← Finset.sum_product']
refine (Finset.sum_eq_single (a0, b0) ?_ ?_).trans (if_pos rfl) <;> simp_rw [Finset.mem_product]
· refine fun ab hab hne => if_neg (fun he => hne <| Prod.ext ?_ ?_)
exacts [(h hab.1 hab.2 he).1, (h hab.1 hab.2 he).2]
· refine fun hnmem => ite_eq_r... | 9 | 8,103.083928 | 2 | 2 | 1 | 2,114 |
import Mathlib.CategoryTheory.Balanced
import Mathlib.CategoryTheory.LiftingProperties.Basic
#align_import category_theory.limits.shapes.strong_epi from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
universe v u
namespace CategoryTheory
variable {C : Type u} [Category.{v} C]
variable... | Mathlib/CategoryTheory/Limits/Shapes/StrongEpi.lean | 114 | 123 | theorem strongEpi_of_strongEpi [StrongEpi (f ≫ g)] : StrongEpi g :=
{ epi := epi_of_epi f g
llp := fun {X Y} z _ => by
constructor
intro u v sq
have h₀ : (f ≫ u) ≫ z = (f ≫ g) ≫ v := by | simp only [Category.assoc, sq.w]
exact
CommSq.HasLift.mk'
⟨(CommSq.mk h₀).lift, by
simp only [← cancel_mono z, Category.assoc, CommSq.fac_right, sq.w], by simp⟩ }
| 5 | 148.413159 | 2 | 1.428571 | 7 | 1,511 |
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 | 427 | 436 | theorem le_rootMultiplicity_iff {p : R[X]} (p0 : p ≠ 0) {a : R} {n : ℕ} :
n ≤ rootMultiplicity a p ↔ (X - C a) ^ n ∣ p := by |
classical
rw [rootMultiplicity_eq_nat_find_of_nonzero p0, @Nat.le_find_iff _ (_)]
simp_rw [Classical.not_not]
refine ⟨fun h => ?_, fun h m hm => (pow_dvd_pow _ hm).trans h⟩
cases' n with n;
· rw [pow_zero]
apply one_dvd;
· exact h n n.lt_succ_self
| 8 | 2,980.957987 | 2 | 1.5 | 32 | 1,561 |
import Mathlib.GroupTheory.FreeGroup.Basic
import Mathlib.GroupTheory.QuotientGroup
#align_import group_theory.presented_group from "leanprover-community/mathlib"@"d90e4e186f1d18e375dcd4e5b5f6364b01cb3e46"
variable {α : Type*}
def PresentedGroup (rels : Set (FreeGroup α)) :=
FreeGroup α ⧸ Subgroup.normalClosu... | Mathlib/GroupTheory/PresentedGroup.lean | 53 | 58 | theorem closure_range_of (rels : Set (FreeGroup α)) :
Subgroup.closure (Set.range (PresentedGroup.of : α → PresentedGroup rels)) = ⊤ := by |
have : (PresentedGroup.of : α → PresentedGroup rels) = QuotientGroup.mk' _ ∘ FreeGroup.of := rfl
rw [this, Set.range_comp, ← MonoidHom.map_closure (QuotientGroup.mk' _),
FreeGroup.closure_range_of, ← MonoidHom.range_eq_map]
exact MonoidHom.range_top_of_surjective _ (QuotientGroup.mk'_surjective _)
| 4 | 54.59815 | 2 | 1.333333 | 3 | 1,392 |
import Mathlib.Data.Set.Prod
import Mathlib.Logic.Function.Conjugate
#align_import data.set.function from "leanprover-community/mathlib"@"996b0ff959da753a555053a480f36e5f264d4207"
variable {α β γ : Type*} {ι : Sort*} {π : α → Type*}
open Equiv Equiv.Perm Function
namespace Set
section restrict
def restrict (... | Mathlib/Data/Set/Function.lean | 124 | 127 | theorem restrict_extend_compl_range (f : α → β) (g : α → γ) (g' : β → γ) :
(range f)ᶜ.restrict (extend f g g') = g' ∘ Subtype.val := by |
classical
exact restrict_dite_compl _ _
| 2 | 7.389056 | 1 | 0.8 | 10 | 704 |
import Mathlib.Probability.IdentDistrib
import Mathlib.MeasureTheory.Integral.DominatedConvergence
import Mathlib.Analysis.SpecificLimits.FloorPow
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Asymptotics.SpecificAsymptotics
#align_import probability.strong_law from "leanprover-community/mathlib"@"f2ce60867... | Mathlib/Probability/StrongLaw.lean | 99 | 103 | theorem abs_truncation_le_abs_self (f : α → ℝ) (A : ℝ) (x : α) : |truncation f A x| ≤ |f x| := by |
simp only [truncation, indicator, Set.mem_Icc, id, Function.comp_apply]
split_ifs
· exact le_rfl
· simp [abs_nonneg]
| 4 | 54.59815 | 2 | 1.444444 | 9 | 1,531 |
import Mathlib.Data.Set.Prod
import Mathlib.Logic.Function.Conjugate
#align_import data.set.function from "leanprover-community/mathlib"@"996b0ff959da753a555053a480f36e5f264d4207"
variable {α β γ : Type*} {ι : Sort*} {π : α → Type*}
open Equiv Equiv.Perm Function
namespace Set
section restrict
def restrict (... | Mathlib/Data/Set/Function.lean | 130 | 136 | theorem range_extend_subset (f : α → β) (g : α → γ) (g' : β → γ) :
range (extend f g g') ⊆ range g ∪ g' '' (range f)ᶜ := by |
classical
rintro _ ⟨y, rfl⟩
rw [extend_def]
split_ifs with h
exacts [Or.inl (mem_range_self _), Or.inr (mem_image_of_mem _ h)]
| 5 | 148.413159 | 2 | 0.8 | 10 | 704 |
import Mathlib.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
#align_import data.set.intervals.with_bot_top from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
variable {α : Type*}
namespace WithTop
@[simp]
theorem preimage_coe_top : (some : α → WithTop α) ⁻¹' {⊤} =... | Mathlib/Order/Interval/Set/WithBotTop.lean | 102 | 104 | theorem image_coe_Iic : (some : α → WithTop α) '' Iic a = Iic (a : WithTop α) := by |
rw [← preimage_coe_Iic, image_preimage_eq_inter_range, range_coe,
inter_eq_self_of_subset_left (Iic_subset_Iio.2 <| coe_lt_top a)]
| 2 | 7.389056 | 1 | 0.4 | 15 | 401 |
import Mathlib.RingTheory.AdjoinRoot
import Mathlib.FieldTheory.Minpoly.Field
import Mathlib.RingTheory.Polynomial.GaussLemma
#align_import field_theory.minpoly.is_integrally_closed from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open scoped Classical Polynomial
open Polynomial Set... | Mathlib/FieldTheory/Minpoly/IsIntegrallyClosed.lean | 103 | 108 | theorem ker_eval {s : S} (hs : IsIntegral R s) :
RingHom.ker ((Polynomial.aeval s).toRingHom : R[X] →+* S) =
Ideal.span ({minpoly R s} : Set R[X]) := by |
ext p
simp_rw [RingHom.mem_ker, AlgHom.toRingHom_eq_coe, AlgHom.coe_toRingHom,
isIntegrallyClosed_dvd_iff hs, ← Ideal.mem_span_singleton]
| 3 | 20.085537 | 1 | 1.571429 | 7 | 1,708 |
import Mathlib.Data.Nat.Lattice
import Mathlib.Logic.Denumerable
import Mathlib.Logic.Function.Iterate
import Mathlib.Order.Hom.Basic
import Mathlib.Data.Set.Subsingleton
#align_import order.order_iso_nat from "leanprover-community/mathlib"@"210657c4ea4a4a7b234392f70a3a2a83346dfa90"
variable {α : Type*}
namespa... | Mathlib/Order/OrderIsoNat.lean | 99 | 101 | theorem not_wellFounded_of_decreasing_seq (f : ((· > ·) : ℕ → ℕ → Prop) ↪r r) : ¬WellFounded r := by |
rw [wellFounded_iff_no_descending_seq, not_isEmpty_iff]
exact ⟨f⟩
| 2 | 7.389056 | 1 | 1.6 | 5 | 1,718 |
import Mathlib.Dynamics.Ergodic.Ergodic
import Mathlib.MeasureTheory.Function.AEEqFun
open Function Set Filter MeasureTheory Topology TopologicalSpace
variable {α X : Type*} [MeasurableSpace α] {μ : MeasureTheory.Measure α}
theorem QuasiErgodic.ae_eq_const_of_ae_eq_comp_of_ae_range₀ [Nonempty X] [MeasurableSpace... | Mathlib/Dynamics/Ergodic/Function.lean | 77 | 82 | theorem ae_eq_const_of_ae_eq_comp_ae {g : α → X} (h : QuasiErgodic f μ)
(hgm : AEStronglyMeasurable g μ) (hg_eq : g ∘ f =ᵐ[μ] g) : ∃ c, g =ᵐ[μ] const α c := by |
borelize X
rcases hgm.isSeparable_ae_range with ⟨t, ht, hgt⟩
haveI := ht.secondCountableTopology
exact h.ae_eq_const_of_ae_eq_comp_of_ae_range₀ hgt hgm.aemeasurable.nullMeasurable hg_eq
| 4 | 54.59815 | 2 | 2 | 2 | 2,080 |
import Mathlib.Data.Finite.Card
import Mathlib.GroupTheory.Commutator
import Mathlib.GroupTheory.Finiteness
#align_import group_theory.abelianization from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
universe u v w
-- Let G be a group.
variable (G : Type u) [Group G]
open Subgroup (... | Mathlib/GroupTheory/Abelianization.lean | 65 | 68 | theorem rank_commutator_le_card [Finite (commutatorSet G)] :
Group.rank (commutator G) ≤ Nat.card (commutatorSet G) := by |
rw [Subgroup.rank_congr (commutator_eq_closure G)]
apply Subgroup.rank_closure_finite_le_nat_card
| 2 | 7.389056 | 1 | 0.8 | 5 | 705 |
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 | 85 | 92 | theorem mem_prod_principal {s : Set (α × β)} :
s ∈ f ×ˢ 𝓟 t ↔ { a | ∀ b ∈ t, (a, b) ∈ s } ∈ f := by |
rw [← @exists_mem_subset_iff _ f, mem_prod_iff]
refine exists_congr fun u => Iff.rfl.and ⟨?_, fun h => ⟨t, mem_principal_self t, ?_⟩⟩
· rintro ⟨v, v_in, hv⟩ a a_in b b_in
exact hv (mk_mem_prod a_in <| v_in b_in)
· rintro ⟨x, y⟩ ⟨hx, hy⟩
exact h hx y hy
| 6 | 403.428793 | 2 | 1 | 10 | 1,042 |
import Mathlib.NumberTheory.Padics.PadicIntegers
import Mathlib.RingTheory.ZMod
#align_import number_theory.padics.ring_homs from "leanprover-community/mathlib"@"565eb991e264d0db702722b4bde52ee5173c9950"
noncomputable section
open scoped Classical
open Nat LocalRing Padic
namespace PadicInt
variable {p : ℕ} [h... | Mathlib/NumberTheory/Padics/RingHoms.lean | 563 | 575 | theorem nthHomSeq_mul (r s : R) :
nthHomSeq f_compat (r * s) ≈ nthHomSeq f_compat r * nthHomSeq f_compat s := by |
intro ε hε
obtain ⟨n, hn⟩ := exists_pow_neg_lt_rat p hε
use n
intro j hj
dsimp [nthHomSeq]
apply lt_of_le_of_lt _ hn
rw [← Int.cast_mul, ← Int.cast_sub, ← padicNorm.dvd_iff_norm_le, ←
ZMod.intCast_zmod_eq_zero_iff_dvd]
dsimp [nthHom]
simp only [ZMod.natCast_val, RingHom.map_mul, Int.cast_sub, ZMo... | 11 | 59,874.141715 | 2 | 1.833333 | 12 | 1,916 |
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