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.LinearAlgebra.QuadraticForm.IsometryEquiv
#align_import linear_algebra.quadratic_form.prod from "leanprover-community/mathlib"@"9b2755b951bc323c962bd072cd447b375cf58101"
universe u v w
variable {ι : Type*} {R : Type*} {M₁ M₂ N₁ N₂ : Type*} {Mᵢ Nᵢ : ι → Type*}
namespace QuadraticForm
section Pro... | Mathlib/LinearAlgebra/QuadraticForm/Prod.lean | 257 | 261 | theorem pi_apply_single [Fintype ι] [DecidableEq ι]
(Q : ∀ i, QuadraticForm R (Mᵢ i)) (i : ι) (m : Mᵢ i) :
pi Q (Pi.single i m) = Q i m := by |
rw [pi_apply, Fintype.sum_eq_single i fun j hj => ?_, Pi.single_eq_same]
rw [Pi.single_eq_of_ne hj, map_zero]
| 2 | 7.389056 | 1 | 1.833333 | 6 | 1,914 |
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Fintype.BigOperators
import Mathlib.RingTheory.PowerSeries.Inverse
import Mathlib.RingTheory.PowerSeries.WellKnown
import Mathlib.Tactic.FieldSimp
#align_import number_theory.bernoulli from "leanprover-community/mat... | Mathlib/NumberTheory/Bernoulli.lean | 128 | 131 | theorem bernoulli'_four : bernoulli' 4 = -1 / 30 := by |
have : Nat.choose 4 2 = 6 := by decide -- shrug
rw [bernoulli'_def]
norm_num [sum_range_succ, sum_range_succ, sum_range_zero, this]
| 3 | 20.085537 | 1 | 1.272727 | 11 | 1,346 |
import Mathlib.Analysis.Complex.Basic
import Mathlib.FieldTheory.IntermediateField
import Mathlib.Topology.Algebra.Field
import Mathlib.Topology.Algebra.UniformRing
#align_import topology.instances.complex from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
section ComplexSubfield
open... | Mathlib/Topology/Instances/Complex.lean | 50 | 116 | theorem Complex.uniformContinuous_ringHom_eq_id_or_conj (K : Subfield ℂ) {ψ : K →+* ℂ}
(hc : UniformContinuous ψ) : ψ.toFun = K.subtype ∨ ψ.toFun = conj ∘ K.subtype := by |
letI : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk
letI : TopologicalRing K.topologicalClosure :=
Subring.instTopologicalRing K.topologicalClosure.toSubring
set ι : K → K.topologicalClosure := ⇑(Subfield.inclusion K.le_topologicalClosure)
have ui : UniformInducing ι :=
⟨by
erw [unifor... | 65 | 16,948,892,444,103,336,000,000,000,000 | 2 | 2 | 2 | 2,026 |
import Mathlib.Topology.UniformSpace.UniformConvergence
import Mathlib.Topology.UniformSpace.Equicontinuity
import Mathlib.Topology.Separation
import Mathlib.Topology.Support
#align_import topology.uniform_space.compact from "leanprover-community/mathlib"@"735b22f8f9ff9792cf4212d7cb051c4c994bc685"
open scoped Cla... | Mathlib/Topology/UniformSpace/Compact.lean | 69 | 75 | theorem unique_uniformity_of_compact [t : TopologicalSpace γ] [CompactSpace γ]
{u u' : UniformSpace γ} (h : u.toTopologicalSpace = t) (h' : u'.toTopologicalSpace = t) :
u = u' := by |
refine UniformSpace.ext ?_
have : @CompactSpace γ u.toTopologicalSpace := by rwa [h]
have : @CompactSpace γ u'.toTopologicalSpace := by rwa [h']
rw [@compactSpace_uniformity _ u, compactSpace_uniformity, h, h']
| 4 | 54.59815 | 2 | 2 | 2 | 2,312 |
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.Setoid.Basic
import Mathlib.Dynamics.FixedPoints.Topology
import Mathlib.Topology.MetricSpace.Lipschitz
#align_import topology.metric_space.contracting from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open scoped Classi... | Mathlib/Topology/MetricSpace/Contracting.lean | 62 | 64 | theorem one_sub_K_ne_top : (1 : ℝ≥0∞) - K ≠ ∞ := by |
norm_cast
exact ENNReal.coe_ne_top
| 2 | 7.389056 | 1 | 0.6 | 5 | 531 |
import Mathlib.Data.Set.Image
import Mathlib.Data.SProd
#align_import data.set.prod from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc9524185ae994553ebf4"
open Function
namespace Set
section Prod
variable {α β γ δ : Type*} {s s₁ s₂ : Set α} {t t₁ t₂ : Set β} {a : α} {b : β}
theorem Subsingleton.pro... | Mathlib/Data/Set/Prod.lean | 142 | 144 | theorem prod_inter : s ×ˢ (t₁ ∩ t₂) = s ×ˢ t₁ ∩ s ×ˢ t₂ := by |
ext ⟨x, y⟩
simp only [← and_and_left, mem_inter_iff, mem_prod]
| 2 | 7.389056 | 1 | 0.692308 | 13 | 638 |
import Mathlib.AlgebraicGeometry.Morphisms.QuasiCompact
import Mathlib.Topology.QuasiSeparated
#align_import algebraic_geometry.morphisms.quasi_separated from "leanprover-community/mathlib"@"1a51edf13debfcbe223fa06b1cb353b9ed9751cc"
noncomputable section
open CategoryTheory CategoryTheory.Limits Opposite Topolog... | Mathlib/AlgebraicGeometry/Morphisms/QuasiSeparated.lean | 57 | 83 | theorem quasiSeparatedSpace_iff_affine (X : Scheme) :
QuasiSeparatedSpace X.carrier ↔ ∀ U V : X.affineOpens, IsCompact (U ∩ V : Set X.carrier) := by |
rw [quasiSeparatedSpace_iff]
constructor
· intro H U V; exact H U V U.1.2 U.2.isCompact V.1.2 V.2.isCompact
· intro H
suffices
∀ (U : Opens X.carrier) (_ : IsCompact U.1) (V : Opens X.carrier) (_ : IsCompact V.1),
IsCompact (U ⊓ V).1
by intro U V hU hU' hV hV'; exact this ⟨U, hU⟩ hU' ⟨V... | 25 | 72,004,899,337.38586 | 2 | 0.833333 | 6 | 726 |
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathlib.CategoryTheory.FullSubcategory
import Mathlib.CategoryTheory.Skeletal
import Mathlib.Data.Fintype.Card
#align_import category_theory.Fintype from "leanprover-community/mathlib"@"c3019c79074b0619edb4b27553a91b2e82242395"
open scoped Classical
ope... | Mathlib/CategoryTheory/FintypeCat.lean | 211 | 213 | theorem incl_mk_nat_card (n : ℕ) : Fintype.card (incl.obj (mk n)) = n := by |
convert Finset.card_fin n
apply Fintype.ofEquiv_card
| 2 | 7.389056 | 1 | 1.5 | 2 | 1,559 |
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 | 204 | 209 | theorem pnatCast_one [Fact (∀ I : Ideal R, I ≠ ⊤ → CharZero (R ⧸ I))] : ((1 : ℕ+) : Rˣ) = 1 := by |
apply Units.ext
rw [Units.val_one]
change ((PNat.isUnit_natCast (R := R) 1).unit : R) = 1
rw [IsUnit.unit_spec (PNat.isUnit_natCast 1)]
rw [PNat.one_coe, Nat.cast_one]
| 5 | 148.413159 | 2 | 1.875 | 8 | 1,930 |
import Mathlib.NumberTheory.ModularForms.SlashInvariantForms
import Mathlib.NumberTheory.ModularForms.CongruenceSubgroups
noncomputable section
open ModularForm UpperHalfPlane Matrix
namespace SlashInvariantForm
| Mathlib/NumberTheory/ModularForms/Identities.lean | 22 | 32 | theorem vAdd_width_periodic (N : ℕ) (k n : ℤ) (f : SlashInvariantForm (Gamma N) k) (z : ℍ) :
f (((N * n) : ℝ) +ᵥ z) = f z := by |
norm_cast
rw [← modular_T_zpow_smul z (N * n)]
have Hn := (ModularGroup_T_pow_mem_Gamma N (N * n) (by simp))
simp only [zpow_natCast, Int.natAbs_ofNat] at Hn
convert (SlashInvariantForm.slash_action_eqn' k (Gamma N) f ⟨((ModularGroup.T ^ (N * n))), Hn⟩ z)
unfold SpecialLinearGroup.coeToGL
simp only [Fin.... | 9 | 8,103.083928 | 2 | 1.5 | 2 | 1,536 |
import Mathlib.Analysis.Calculus.LineDeriv.Measurable
import Mathlib.Analysis.NormedSpace.FiniteDimension
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.Analysis.BoundedVariation
import Mathlib.MeasureTheory.Group.Integral
import Mathlib.Analysis.Distribution.AEEqOfIntegralContDiff
import Mathlib.... | Mathlib/Analysis/Calculus/Rademacher.lean | 97 | 117 | theorem integral_inv_smul_sub_mul_tendsto_integral_lineDeriv_mul
(hf : LipschitzWith C f) (hg : Integrable g μ) (v : E) :
Tendsto (fun (t : ℝ) ↦ ∫ x, (t⁻¹ • (f (x + t • v) - f x)) * g x ∂μ) (𝓝[>] 0)
(𝓝 (∫ x, lineDeriv ℝ f x v * g x ∂μ)) := by |
apply tendsto_integral_filter_of_dominated_convergence (fun x ↦ (C * ‖v‖) * ‖g x‖)
· filter_upwards with t
apply AEStronglyMeasurable.mul ?_ hg.aestronglyMeasurable
apply aestronglyMeasurable_const.smul
apply AEStronglyMeasurable.sub _ hf.continuous.measurable.aestronglyMeasurable
apply AEMeasurabl... | 17 | 24,154,952.753575 | 2 | 2 | 3 | 2,148 |
import Mathlib.CategoryTheory.Category.Grpd
import Mathlib.CategoryTheory.Groupoid
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.Topology.Homotopy.Path
import Mathlib.Data.Set.Subsingleton
#align_import algebraic_topology.fundamental_groupoid.basic from "leanprover-community/mathlib"@"3d7987cda72abc473... | Mathlib/AlgebraicTopology/FundamentalGroupoid/Basic.lean | 131 | 135 | theorem continuous_transReflReparamAux : Continuous transReflReparamAux := by |
refine continuous_if_le ?_ ?_ (Continuous.continuousOn ?_) (Continuous.continuousOn ?_) ?_ <;>
[continuity; continuity; continuity; continuity; skip]
intro x hx
simp [hx]
| 4 | 54.59815 | 2 | 1.166667 | 12 | 1,229 |
import Mathlib.Logic.UnivLE
import Mathlib.SetTheory.Ordinal.Basic
set_option autoImplicit true
noncomputable section
open Cardinal
| Mathlib/SetTheory/Cardinal/UnivLE.lean | 19 | 27 | theorem univLE_iff_cardinal_le : UnivLE.{u, v} ↔ univ.{u, v+1} ≤ univ.{v, u+1} := by |
rw [← not_iff_not, UnivLE]; simp_rw [small_iff_lift_mk_lt_univ]; push_neg
-- strange: simp_rw [univ_umax.{v,u}] doesn't work
refine ⟨fun ⟨α, le⟩ ↦ ?_, fun h ↦ ?_⟩
· rw [univ_umax.{v,u}, ← lift_le.{u+1}, lift_univ, lift_lift] at le
exact le.trans_lt (lift_lt_univ'.{u,v+1} #α)
· obtain ⟨⟨α⟩, h⟩ := lt_univ'... | 8 | 2,980.957987 | 2 | 1 | 2 | 808 |
import Mathlib.Data.List.Defs
import Mathlib.Data.Option.Basic
import Mathlib.Data.Nat.Defs
import Mathlib.Init.Data.List.Basic
import Mathlib.Util.AssertExists
-- Make sure we haven't imported `Data.Nat.Order.Basic`
assert_not_exists OrderedSub
namespace List
universe u v
variable {α : Type u} {β : Type v} (l :... | Mathlib/Data/List/GetD.lean | 77 | 80 | theorem getD_replicate_default_eq (r n : ℕ) : (replicate r d).getD n d = d := by |
induction r generalizing n with
| zero => simp
| succ n ih => cases n <;> simp [ih]
| 3 | 20.085537 | 1 | 1.375 | 8 | 1,471 |
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.CategoryTheory.Bicategory.LocallyDiscrete
#align_import category_theory.bicategory.coherence from "leanprover-community/mathlib"@"f187f1074fa1857c94589cc653c786ca... | Mathlib/CategoryTheory/Bicategory/Coherence.lean | 94 | 98 | theorem preinclusion_map₂ {a b : B} (f g : Discrete (Path.{v + 1} a b)) (η : f ⟶ g) :
(preinclusion B).map₂ η = eqToHom (congr_arg _ (Discrete.ext _ _ (Discrete.eq_of_hom η))) := by |
rcases η with ⟨⟨⟩⟩
cases Discrete.ext _ _ (by assumption)
convert (inclusionPath a b).map_id _
| 3 | 20.085537 | 1 | 1.75 | 4 | 1,860 |
import Mathlib.Analysis.NormedSpace.Exponential
import Mathlib.Analysis.NormedSpace.ProdLp
import Mathlib.Topology.Instances.TrivSqZeroExt
#align_import analysis.normed_space.triv_sq_zero_ext from "leanprover-community/mathlib"@"88a563b158f59f2983cfad685664da95502e8cdd"
variable (𝕜 : Type*) {S R M : Type*}
loca... | Mathlib/Analysis/NormedSpace/TrivSqZeroExt.lean | 83 | 88 | theorem snd_expSeries_of_smul_comm
(x : tsze R M) (hx : MulOpposite.op x.fst • x.snd = x.fst • x.snd) (n : ℕ) :
snd (expSeries 𝕜 (tsze R M) (n + 1) fun _ => x) = (expSeries 𝕜 R n fun _ => x.fst) • x.snd := by |
simp_rw [expSeries_apply_eq, snd_smul, snd_pow_of_smul_comm _ _ hx, nsmul_eq_smul_cast 𝕜 (n + 1),
smul_smul, smul_assoc, Nat.factorial_succ, Nat.pred_succ, Nat.cast_mul, mul_inv_rev,
inv_mul_cancel_right₀ ((Nat.cast_ne_zero (R := 𝕜)).mpr <| Nat.succ_ne_zero n)]
| 3 | 20.085537 | 1 | 1 | 4 | 872 |
import Mathlib.GroupTheory.QuotientGroup
#align_import algebra.char_zero.quotient from "leanprover-community/mathlib"@"d90e4e186f1d18e375dcd4e5b5f6364b01cb3e46"
variable {R : Type*} [DivisionRing R] [CharZero R] {p : R}
namespace AddSubgroup
theorem zsmul_mem_zmultiples_iff_exists_sub_div {r : R} {z : ℤ} (hz :... | Mathlib/Algebra/CharZero/Quotient.lean | 42 | 47 | theorem nsmul_mem_zmultiples_iff_exists_sub_div {r : R} {n : ℕ} (hn : n ≠ 0) :
n • r ∈ AddSubgroup.zmultiples p ↔
∃ k : Fin n, r - (k : ℕ) • (p / n : R) ∈ AddSubgroup.zmultiples p := by |
rw [← natCast_zsmul r, zsmul_mem_zmultiples_iff_exists_sub_div (Int.natCast_ne_zero.mpr hn),
Int.cast_natCast]
rfl
| 3 | 20.085537 | 1 | 1.5 | 2 | 1,570 |
import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.MeasureTheory.Integral.Layercake
#align_import analysis.special_functions.japanese_bracket from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
noncomputable section
op... | Mathlib/Analysis/SpecialFunctions/JapaneseBracket.lean | 41 | 46 | theorem one_add_norm_le_sqrt_two_mul_sqrt (x : E) :
(1 : ℝ) + ‖x‖ ≤ √2 * √(1 + ‖x‖ ^ 2) := by |
rw [← sqrt_mul zero_le_two]
have := sq_nonneg (‖x‖ - 1)
apply le_sqrt_of_sq_le
linarith
| 4 | 54.59815 | 2 | 1.571429 | 7 | 1,710 |
import Mathlib.Analysis.SpecialFunctions.JapaneseBracket
import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.MeasureTheory.Group.Integral
import Mathlib.MeasureTheory.Integral.IntegralEqImproper
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.special_functions.improper_inte... | Mathlib/Analysis/SpecialFunctions/ImproperIntegrals.lean | 93 | 101 | theorem not_integrableOn_Ioi_rpow (s : ℝ) : ¬ IntegrableOn (fun x ↦ x ^ s) (Ioi (0 : ℝ)) := by |
intro h
rcases le_or_lt s (-1) with hs|hs
· have : IntegrableOn (fun x ↦ x ^ s) (Ioo (0 : ℝ) 1) := h.mono Ioo_subset_Ioi_self le_rfl
rw [integrableOn_Ioo_rpow_iff zero_lt_one] at this
exact hs.not_lt this
· have : IntegrableOn (fun x ↦ x ^ s) (Ioi (1 : ℝ)) := h.mono (Ioi_subset_Ioi zero_le_one) le_rfl
... | 8 | 2,980.957987 | 2 | 1.5 | 8 | 1,667 |
import Mathlib.Combinatorics.SimpleGraph.Connectivity
#align_import combinatorics.simple_graph.prod from "leanprover-community/mathlib"@"2985fa3c31a27274aed06c433510bc14b73d6488"
variable {α β γ : Type*}
namespace SimpleGraph
-- Porting note: pruned variables to keep things out of local contexts, which
-- can im... | Mathlib/Combinatorics/SimpleGraph/Prod.lean | 69 | 73 | theorem boxProd_neighborSet (x : α × β) :
(G □ H).neighborSet x = G.neighborSet x.1 ×ˢ {x.2} ∪ {x.1} ×ˢ H.neighborSet x.2 := by |
ext ⟨a', b'⟩
simp only [mem_neighborSet, Set.mem_union, boxProd_adj, Set.mem_prod, Set.mem_singleton_iff]
simp only [eq_comm, and_comm]
| 3 | 20.085537 | 1 | 0.333333 | 3 | 334 |
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 | 117 | 119 | theorem image₂_nonempty_iff : (image₂ f s t).Nonempty ↔ s.Nonempty ∧ t.Nonempty := by |
rw [← coe_nonempty, coe_image₂]
exact image2_nonempty_iff
| 2 | 7.389056 | 1 | 0.375 | 8 | 379 |
import Mathlib.Data.Finsupp.Defs
#align_import data.finsupp.ne_locus from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
variable {α M N P : Type*}
namespace Finsupp
variable [DecidableEq α]
section NHasZero
variable [DecidableEq N] [Zero N] (f g : α →₀ N)
def neLocus (f g : α →₀ ... | Mathlib/Data/Finsupp/NeLocus.lean | 74 | 76 | theorem neLocus_zero_right : f.neLocus 0 = f.support := by |
ext
rw [mem_neLocus, mem_support_iff, coe_zero, Pi.zero_apply]
| 2 | 7.389056 | 1 | 0.75 | 4 | 669 |
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Algebra.Order.Monoid.WithTop
#align_import data.nat.with_bot from "leanprover-community/mathlib"@"966e0cf0685c9cedf8a3283ac69eef4d5f2eaca2"
namespace Nat
namespace WithBot
instance : WellFoundedRelation (WithBot ℕ) where
rel := (· < ·)
wf := IsWellFounde... | Mathlib/Data/Nat/WithBot.lean | 27 | 32 | theorem add_eq_zero_iff {n m : WithBot ℕ} : n + m = 0 ↔ n = 0 ∧ m = 0 := by |
rcases n, m with ⟨_ | _, _ | _⟩
repeat (· exact ⟨fun h => Option.noConfusion h, fun h => Option.noConfusion h.1⟩)
· exact ⟨fun h => Option.noConfusion h, fun h => Option.noConfusion h.2⟩
repeat erw [WithBot.coe_eq_coe]
exact add_eq_zero_iff' (zero_le _) (zero_le _)
| 5 | 148.413159 | 2 | 1.857143 | 7 | 1,928 |
import Mathlib.Mathport.Rename
set_option autoImplicit true
namespace Thunk
#align thunk.mk Thunk.mk
-- Porting note: Added `Thunk.ext` to get `ext` tactic to work.
@[ext]
| Mathlib/Lean/Thunk.lean | 20 | 24 | theorem ext {α : Type u} {a b : Thunk α} (eq : a.get = b.get) : a = b := by |
have ⟨_⟩ := a
have ⟨_⟩ := b
congr
exact funext fun _ ↦ eq
| 4 | 54.59815 | 2 | 2 | 1 | 1,998 |
import Mathlib.Algebra.CharP.Defs
import Mathlib.Data.Nat.Prime
import Mathlib.ModelTheory.Algebra.Ring.FreeCommRing
import Mathlib.ModelTheory.Algebra.Field.Basic
variable {p : ℕ} {K : Type*}
namespace FirstOrder
namespace Field
open Language Ring
noncomputable def eqZero (n : ℕ) : Language.ring.Sentence :=
... | Mathlib/ModelTheory/Algebra/Field/CharP.lean | 63 | 78 | theorem charP_iff_model_fieldOfChar [Field K] [CompatibleRing K] :
(Theory.fieldOfChar p).Model K ↔ CharP K p := by |
simp only [Theory.fieldOfChar, Theory.model_union_iff,
(show (Theory.field.Model K) by infer_instance), true_and]
split_ifs with hp0 hp
· subst hp0
simp only [Theory.model_iff, Set.mem_image, Set.mem_setOf_eq, Sentence.Realize,
forall_exists_index, and_imp, forall_apply_eq_imp_iff₂, Formula.realize... | 14 | 1,202,604.284165 | 2 | 2 | 1 | 2,207 |
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Tactic.AdaptationNote
#align_import analysis.special_functions.log.deriv from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
ope... | Mathlib/Analysis/SpecialFunctions/Log/Deriv.lean | 78 | 81 | theorem contDiffOn_log {n : ℕ∞} : ContDiffOn ℝ n log {0}ᶜ := by |
suffices ContDiffOn ℝ ⊤ log {0}ᶜ from this.of_le le_top
refine (contDiffOn_top_iff_deriv_of_isOpen isOpen_compl_singleton).2 ?_
simp [differentiableOn_log, contDiffOn_inv]
| 3 | 20.085537 | 1 | 1.666667 | 3 | 1,796 |
import Batteries.Data.HashMap.Basic
import Batteries.Data.Array.Lemmas
import Batteries.Data.Nat.Lemmas
namespace Batteries.HashMap
namespace Imp
attribute [-simp] Bool.not_eq_true
namespace Buckets
@[ext] protected theorem ext : ∀ {b₁ b₂ : Buckets α β}, b₁.1.data = b₂.1.data → b₁ = b₂
| ⟨⟨_⟩, _⟩, ⟨⟨_⟩, _⟩, rfl ... | .lake/packages/batteries/Batteries/Data/HashMap/WF.lean | 38 | 40 | theorem mk_size (h) : (mk n h : Buckets α β).size = 0 := by |
simp only [mk, mkArray, size_eq]; clear h
induction n <;> simp [*]
| 2 | 7.389056 | 1 | 1.4 | 5 | 1,476 |
import Mathlib.Analysis.Analytic.Constructions
import Mathlib.Analysis.Calculus.Dslope
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Analysis.Analytic.Uniqueness
#align_import analysis.analytic.isolated_zeros from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
open sco... | Mathlib/Analysis/Analytic/IsolatedZeros.lean | 48 | 62 | theorem exists_hasSum_smul_of_apply_eq_zero (hs : HasSum (fun m => z ^ m • a m) s)
(ha : ∀ k < n, a k = 0) : ∃ t : E, z ^ n • t = s ∧ HasSum (fun m => z ^ m • a (m + n)) t := by |
obtain rfl | hn := n.eq_zero_or_pos
· simpa
by_cases h : z = 0
· have : s = 0 := hs.unique (by simpa [ha 0 hn, h] using hasSum_at_zero a)
exact ⟨a n, by simp [h, hn.ne', this], by simpa [h] using hasSum_at_zero fun m => a (m + n)⟩
· refine ⟨(z ^ n)⁻¹ • s, by field_simp [smul_smul], ?_⟩
have h1 : ∑ i ... | 13 | 442,413.392009 | 2 | 1.2 | 5 | 1,274 |
import Mathlib.Topology.Order.IsLUB
open Set Filter TopologicalSpace Topology Function
open OrderDual (toDual ofDual)
variable {α β γ : Type*}
section DenselyOrdered
variable [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a b : α}
{s : Set α}
theorem closure_Ioi' {a : α} (h : (Io... | Mathlib/Topology/Order/DenselyOrdered.lean | 52 | 61 | theorem closure_Ioo {a b : α} (hab : a ≠ b) : closure (Ioo a b) = Icc a b := by |
apply Subset.antisymm
· exact closure_minimal Ioo_subset_Icc_self isClosed_Icc
· cases' hab.lt_or_lt with hab hab
· rw [← diff_subset_closure_iff, Icc_diff_Ioo_same hab.le]
have hab' : (Ioo a b).Nonempty := nonempty_Ioo.2 hab
simp only [insert_subset_iff, singleton_subset_iff]
exact ⟨(isGLB... | 9 | 8,103.083928 | 2 | 0.769231 | 13 | 685 |
import Mathlib.Algebra.Group.Even
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Algebra.Order.Sub.Defs
#align_import algebra.order.sub.canonical from "leanprover-community/mathlib"@"62a5626868683c104774de8d85b9855234ac807c"
variable {α : Type*}
section ExistsAddOfLE
variable [AddCommSemigrou... | Mathlib/Algebra/Order/Sub/Canonical.lean | 25 | 28 | theorem add_tsub_cancel_of_le (h : a ≤ b) : a + (b - a) = b := by |
refine le_antisymm ?_ le_add_tsub
obtain ⟨c, rfl⟩ := exists_add_of_le h
exact add_le_add_left add_tsub_le_left a
| 3 | 20.085537 | 1 | 0.571429 | 7 | 515 |
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 | 103 | 115 | theorem liftAlternating_apply_ιMulti {n : ℕ} (f : ∀ i, M [⋀^Fin i]→ₗ[R] N)
(v : Fin n → M) : liftAlternating (R := R) (M := M) (N := N) f (ιMulti R n v) = f n v := by |
rw [ιMulti_apply]
-- Porting note: `v` is generalized automatically so it was removed from the next line
induction' n with n ih generalizing f
· -- Porting note: Lean does not automatically synthesize the instance
-- `[Subsingleton (Fin 0 → M)]` which is needed for `Subsingleton.elim 0 v` on line 114.
... | 11 | 59,874.141715 | 2 | 1.333333 | 6 | 1,397 |
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 | 129 | 132 | theorem setOf_mem_eq_univ_iff (φ : L[[α]].Sentence) :
{ p : T.CompleteType α | φ ∈ p } = Set.univ ↔ (L.lhomWithConstants α).onTheory T ⊨ᵇ φ := by |
rw [models_iff_not_satisfiable, ← compl_empty_iff, compl_setOf_mem, ← setOf_subset_eq_empty_iff]
simp
| 2 | 7.389056 | 1 | 1.8 | 5 | 1,884 |
import Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear
import Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm
import Mathlib.Analysis.NormedSpace.Span
suppress_compilation
open Bornology
open Filter hiding map_smul
open scoped Classical NNReal Topology Uniformity
-- the `ₗ` subscript variables are for special... | Mathlib/Analysis/NormedSpace/OperatorNorm/NormedSpace.lean | 42 | 46 | theorem bound_of_shell [RingHomIsometric σ₁₂] (f : E →ₛₗ[σ₁₂] F) {ε C : ℝ} (ε_pos : 0 < ε) {c : 𝕜}
(hc : 1 < ‖c‖) (hf : ∀ x, ε / ‖c‖ ≤ ‖x‖ → ‖x‖ < ε → ‖f x‖ ≤ C * ‖x‖) (x : E) :
‖f x‖ ≤ C * ‖x‖ := by |
by_cases hx : x = 0; · simp [hx]
exact SemilinearMapClass.bound_of_shell_semi_normed f ε_pos hc hf (norm_ne_zero_iff.2 hx)
| 2 | 7.389056 | 1 | 1.666667 | 6 | 1,809 |
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.MinMax
#align_import data.set.pointwise.interval from "leanprover-community/mathlib"@"2196ab363eb097c... | Mathlib/Data/Set/Pointwise/Interval.lean | 92 | 95 | theorem Iic_mul_Iio_subset' (a b : α) : Iic a * Iio b ⊆ Iio (a * b) := by |
haveI := covariantClass_le_of_lt
rintro x ⟨y, hya, z, hzb, rfl⟩
exact mul_lt_mul_of_le_of_lt hya hzb
| 3 | 20.085537 | 1 | 0.37931 | 29 | 381 |
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 | 156 | 158 | theorem natDegree_le_of_dvd {p q : R[X]} (h1 : p ∣ q) (h2 : q ≠ 0) : p.natDegree ≤ q.natDegree := by |
rcases h1 with ⟨q, rfl⟩; rw [mul_ne_zero_iff] at h2
rw [natDegree_mul h2.1 h2.2]; exact Nat.le_add_right _ _
| 2 | 7.389056 | 1 | 1.5 | 32 | 1,561 |
import Mathlib.Algebra.Module.Card
import Mathlib.SetTheory.Cardinal.CountableCover
import Mathlib.SetTheory.Cardinal.Continuum
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Topology.MetricSpace.Perfect
universe u v
open Filter Pointwise Set Function Cardinal
open scoped Cardinal Topology
theorem c... | Mathlib/Topology/Algebra/Module/Cardinality.lean | 49 | 54 | theorem continuum_le_cardinal_of_module
(𝕜 : Type u) (E : Type v) [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜]
[AddCommGroup E] [Module 𝕜 E] [Nontrivial E] : 𝔠 ≤ #E := by |
have A : lift.{v} (𝔠 : Cardinal.{u}) ≤ lift.{v} (#𝕜) := by
simpa using continuum_le_cardinal_of_nontriviallyNormedField 𝕜
simpa using A.trans (Cardinal.mk_le_of_module 𝕜 E)
| 3 | 20.085537 | 1 | 1.2 | 5 | 1,284 |
import Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing
import Mathlib.AlgebraicGeometry.OpenImmersion
#align_import algebraic_geometry.gluing from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
set_option linter.uppercaseLean3 false
noncomputable section
universe u
open Topologica... | Mathlib/AlgebraicGeometry/Gluing.lean | 331 | 335 | theorem glued_cover_cocycle (x y z : 𝒰.J) :
gluedCoverT' 𝒰 x y z ≫ gluedCoverT' 𝒰 y z x ≫ gluedCoverT' 𝒰 z x y = 𝟙 _ := by |
apply pullback.hom_ext <;> simp_rw [Category.id_comp, Category.assoc]
· apply glued_cover_cocycle_fst
· apply glued_cover_cocycle_snd
| 3 | 20.085537 | 1 | 0.142857 | 7 | 256 |
import Mathlib.Analysis.SpecialFunctions.Pow.Real
#align_import analysis.special_functions.log.monotone from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8"
open Set Filter Function
open Topology
noncomputable section
namespace Real
variable {x y : ℝ}
theorem log_mul_self_monotoneOn... | Mathlib/Analysis/SpecialFunctions/Log/Monotone.lean | 41 | 53 | theorem log_div_self_antitoneOn : AntitoneOn (fun x : ℝ => log x / x) { x | exp 1 ≤ x } := by |
simp only [AntitoneOn, mem_setOf_eq]
intro x hex y hey hxy
have x_pos : 0 < x := (exp_pos 1).trans_le hex
have y_pos : 0 < y := (exp_pos 1).trans_le hey
have hlogx : 1 ≤ log x := by rwa [le_log_iff_exp_le x_pos]
have hyx : 0 ≤ y / x - 1 := by rwa [le_sub_iff_add_le, le_div_iff x_pos, zero_add, one_mul]
r... | 12 | 162,754.791419 | 2 | 1.75 | 4 | 1,850 |
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 | 84 | 86 | theorem nontrivial_of_einfsep_lt_top (hs : s.einfsep < ∞) : s.Nontrivial := by |
rcases einfsep_lt_top.1 hs with ⟨_, hx, _, hy, hxy, _⟩
exact ⟨_, hx, _, hy, hxy⟩
| 2 | 7.389056 | 1 | 0.25 | 12 | 302 |
import Mathlib.CategoryTheory.Limits.ColimitLimit
import Mathlib.CategoryTheory.Limits.Preserves.FunctorCategory
import Mathlib.CategoryTheory.Limits.Preserves.Finite
import Mathlib.CategoryTheory.Limits.Shapes.FiniteLimits
import Mathlib.CategoryTheory.Limits.TypesFiltered
import Mathlib.CategoryTheory.ConcreteCatego... | Mathlib/CategoryTheory/Limits/FilteredColimitCommutesFiniteLimit.lean | 72 | 142 | theorem colimitLimitToLimitColimit_injective :
Function.Injective (colimitLimitToLimitColimit F) := by |
classical
cases nonempty_fintype J
-- Suppose we have two terms `x y` in the colimit (over `K`) of the limits (over `J`),
-- and that these have the same image under `colimitLimitToLimitColimit F`.
intro x y h
-- These elements of the colimit have representatives somewhere:
obtain ⟨kx, x, rfl... | 69 | 925,378,172,558,778,900,000,000,000,000 | 2 | 2 | 1 | 2,082 |
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 | 494 | 503 | theorem schur_complement_eq₂₂ [Fintype m] [Fintype n] [DecidableEq n] (A : Matrix m m 𝕜)
(B : Matrix m n 𝕜) {D : Matrix n n 𝕜} (x : m → 𝕜) (y : n → 𝕜) [Invertible D]
(hD : D.IsHermitian) :
(star (x ⊕ᵥ y)) ᵥ* (fromBlocks A B Bᴴ D) ⬝ᵥ (x ⊕ᵥ y) =
(star ((D⁻¹ * Bᴴ) *ᵥ x + y)) ᵥ* D ⬝ᵥ ((D⁻¹ * Bᴴ) *ᵥ x... |
simp [Function.star_sum_elim, fromBlocks_mulVec, vecMul_fromBlocks, add_vecMul,
dotProduct_mulVec, vecMul_sub, Matrix.mul_assoc, vecMul_mulVec, hD.eq,
conjTranspose_nonsing_inv, star_mulVec]
abel
| 4 | 54.59815 | 2 | 1.1875 | 16 | 1,248 |
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Convex.Uniform
import Mathlib.Analysis.NormedSpace.Completion
import Mathlib.Analysis.NormedSpace.BoundedLinearMaps
#align_import analysis.inner_product_space.basic from "leanprover-community/mathlib"@"3f655f5297b030... | Mathlib/Analysis/InnerProductSpace/Basic.lean | 239 | 241 | theorem inner_smul_right (x y : F) {r : 𝕜} : ⟪x, r • y⟫ = r * ⟪x, y⟫ := by |
rw [← inner_conj_symm, inner_smul_left];
simp only [conj_conj, inner_conj_symm, RingHom.map_mul]
| 2 | 7.389056 | 1 | 0.5 | 6 | 449 |
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Module.LinearMap.Basic
import Mathlib.Data.Finset.Preimage
import Mathlib.Data.Set.Finite
import Mathlib.GroupTheory.GroupAction.BigOperators
#align_import data.dfinsupp.basic from "leanpr... | Mathlib/Data/DFinsupp/Basic.lean | 144 | 147 | theorem mapRange_id (h : ∀ i, id (0 : β₁ i) = 0 := fun i => rfl) (g : Π₀ i : ι, β₁ i) :
mapRange (fun i => (id : β₁ i → β₁ i)) h g = g := by |
ext
rfl
| 2 | 7.389056 | 1 | 1 | 3 | 915 |
import Mathlib.MeasureTheory.SetSemiring
open MeasurableSpace Set
namespace MeasureTheory
variable {α : Type*} {𝒜 : Set (Set α)} {s t : Set α}
structure IsSetAlgebra (𝒜 : Set (Set α)) : Prop where
empty_mem : ∅ ∈ 𝒜
compl_mem : ∀ ⦃s⦄, s ∈ 𝒜 → sᶜ ∈ 𝒜
union_mem : ∀ ⦃s t⦄, s ∈ 𝒜 → t ∈ 𝒜 → s ∪ t ∈ 𝒜
... | Mathlib/MeasureTheory/SetAlgebra.lean | 151 | 158 | theorem generateSetAlgebra_subset {ℬ : Set (Set α)} (h : 𝒜 ⊆ ℬ)
(hℬ : IsSetAlgebra ℬ) : generateSetAlgebra 𝒜 ⊆ ℬ := by |
intro s hs
induction hs with
| base t t_mem => exact h t_mem
| empty => exact hℬ.empty_mem
| compl t _ t_mem => exact hℬ.compl_mem t_mem
| union t u _ _ t_mem u_mem => exact hℬ.union_mem t_mem u_mem
| 6 | 403.428793 | 2 | 2 | 4 | 1,933 |
import Mathlib.Control.Bitraversable.Basic
#align_import control.bitraversable.lemmas from "leanprover-community/mathlib"@"58581d0fe523063f5651df0619be2bf65012a94a"
universe u
variable {t : Type u → Type u → Type u} [Bitraversable t]
variable {β : Type u}
namespace Bitraversable
open Functor LawfulApplicative
... | Mathlib/Control/Bitraversable/Lemmas.lean | 79 | 83 | theorem tfst_tsnd {α₀ α₁ β₀ β₁} (f : α₀ → F α₁) (f' : β₀ → G β₁) (x : t α₀ β₀) :
Comp.mk (tfst f <$> tsnd f' x)
= bitraverse (Comp.mk ∘ pure ∘ f) (Comp.mk ∘ map pure ∘ f') x := by |
rw [← comp_bitraverse]
simp only [Function.comp, map_pure]
| 2 | 7.389056 | 1 | 0.666667 | 6 | 606 |
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Algebra.Ring.Regular
import Mathlib.Order.Interval.Set.Basic
#align_import data.set.intervals.instances from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
variable {α : Type*}
section OrderedSemiring
variable [OrderedSe... | Mathlib/Algebra/Order/Interval/Set/Instances.lean | 89 | 91 | theorem coe_eq_one {x : Icc (0 : α) 1} : (x : α) = 1 ↔ x = 1 := by |
symm
exact Subtype.ext_iff
| 2 | 7.389056 | 1 | 1 | 3 | 892 |
import Mathlib.Order.Filter.Bases
import Mathlib.Order.ConditionallyCompleteLattice.Basic
#align_import order.filter.lift from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
open Set Classical Filter Function
namespace Filter
variable {α β γ : Type*} {ι : Sort*}
section lift
protect... | Mathlib/Order/Filter/Lift.lean | 45 | 53 | theorem HasBasis.mem_lift_iff {ι} {p : ι → Prop} {s : ι → Set α} {f : Filter α}
(hf : f.HasBasis p s) {β : ι → Type*} {pg : ∀ i, β i → Prop} {sg : ∀ i, β i → Set γ}
{g : Set α → Filter γ} (hg : ∀ i, (g <| s i).HasBasis (pg i) (sg i)) (gm : Monotone g)
{s : Set γ} : s ∈ f.lift g ↔ ∃ i, p i ∧ ∃ x, pg i x ∧ sg... |
refine (mem_biInf_of_directed ?_ ⟨univ, univ_sets _⟩).trans ?_
· intro t₁ ht₁ t₂ ht₂
exact ⟨t₁ ∩ t₂, inter_mem ht₁ ht₂, gm inter_subset_left, gm inter_subset_right⟩
· simp only [← (hg _).mem_iff]
exact hf.exists_iff fun t₁ t₂ ht H => gm ht H
| 5 | 148.413159 | 2 | 0.666667 | 6 | 581 |
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.affine_space.slope from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open AffineMap
variable {k E PE : Type*} [Field k] [AddCommGroup E] [Module k E] [AddTorsor E PE]
def ... | Mathlib/LinearAlgebra/AffineSpace/Slope.lean | 102 | 116 | theorem sub_div_sub_smul_slope_add_sub_div_sub_smul_slope (f : k → PE) (a b c : k) :
((b - a) / (c - a)) • slope f a b + ((c - b) / (c - a)) • slope f b c = slope f a c := by |
by_cases hab : a = b
· subst hab
rw [sub_self, zero_div, zero_smul, zero_add]
by_cases hac : a = c
· simp [hac]
· rw [div_self (sub_ne_zero.2 <| Ne.symm hac), one_smul]
by_cases hbc : b = c;
· subst hbc
simp [sub_ne_zero.2 (Ne.symm hab)]
rw [add_comm]
simp_rw [slope, div_eq_inv_mul, mul... | 13 | 442,413.392009 | 2 | 0.7 | 10 | 639 |
import Mathlib.Algebra.Homology.ShortComplex.ModuleCat
import Mathlib.RepresentationTheory.GroupCohomology.Basic
import Mathlib.RepresentationTheory.Invariants
universe v u
noncomputable section
open CategoryTheory Limits Representation
variable {k G : Type u} [CommRing k] [Group G] (A : Rep k G)
namespace grou... | Mathlib/RepresentationTheory/GroupCohomology/LowDegree.lean | 423 | 427 | theorem smul_map_inv_sub_map_inv_of_isTwoCocycle {f : G × G → A} (hf : IsTwoCocycle f) (g : G) :
g • f (g⁻¹, g) - f (g, g⁻¹) = f (1, 1) - f (g, 1) := by |
have := hf g g⁻¹ g
simp only [mul_right_inv, mul_left_inv, map_one_fst_of_isTwoCocycle hf g] at this
exact sub_eq_sub_iff_add_eq_add.2 this.symm
| 3 | 20.085537 | 1 | 0.333333 | 9 | 355 |
import Mathlib.Data.Set.Pointwise.SMul
#align_import algebra.add_torsor from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
class AddTorsor (G : outParam Type*) (P : Type*) [AddGroup G] extends AddAction G P,
VSub G P where
[nonempty : Nonempty P]
vsub_vadd' : ∀ p₁ p₂ : P, (p₁ ... | Mathlib/Algebra/AddTorsor.lean | 146 | 148 | theorem vsub_add_vsub_cancel (p₁ p₂ p₃ : P) : p₁ -ᵥ p₂ + (p₂ -ᵥ p₃) = p₁ -ᵥ p₃ := by |
apply vadd_right_cancel p₃
rw [add_vadd, vsub_vadd, vsub_vadd, vsub_vadd]
| 2 | 7.389056 | 1 | 0.555556 | 9 | 513 |
import Mathlib.Analysis.SpecialFunctions.Complex.Log
#align_import analysis.special_functions.pow.complex from "leanprover-community/mathlib"@"4fa54b337f7d52805480306db1b1439c741848c8"
open scoped Classical
open Real Topology Filter ComplexConjugate Finset Set
namespace Complex
noncomputable def cpow (x y : ℂ) ... | Mathlib/Analysis/SpecialFunctions/Pow/Complex.lean | 96 | 99 | theorem cpow_mul {x y : ℂ} (z : ℂ) (h₁ : -π < (log x * y).im) (h₂ : (log x * y).im ≤ π) :
x ^ (y * z) = (x ^ y) ^ z := by |
simp only [cpow_def]
split_ifs <;> simp_all [exp_ne_zero, log_exp h₁ h₂, mul_assoc]
| 2 | 7.389056 | 1 | 0.636364 | 11 | 551 |
import Mathlib.Topology.Order.MonotoneContinuity
import Mathlib.Topology.Algebra.Order.LiminfLimsup
import Mathlib.Topology.Instances.NNReal
import Mathlib.Topology.EMetricSpace.Lipschitz
import Mathlib.Topology.Metrizable.Basic
import Mathlib.Topology.Order.T5
#align_import topology.instances.ennreal from "leanprove... | Mathlib/Topology/Instances/ENNReal.lean | 60 | 62 | theorem isOpen_Ico_zero : IsOpen (Ico 0 b) := by |
rw [ENNReal.Ico_eq_Iio]
exact isOpen_Iio
| 2 | 7.389056 | 1 | 1.285714 | 7 | 1,362 |
import Mathlib.AlgebraicTopology.DoldKan.Projections
import Mathlib.CategoryTheory.Idempotents.FunctorCategories
import Mathlib.CategoryTheory.Idempotents.FunctorExtension
#align_import algebraic_topology.dold_kan.p_infty from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190f0b504"
open Category... | Mathlib/AlgebraicTopology/DoldKan/PInfty.lean | 36 | 42 | theorem P_is_eventually_constant {q n : ℕ} (hqn : n ≤ q) :
((P (q + 1)).f n : X _[n] ⟶ _) = (P q).f n := by |
rcases n with (_|n)
· simp only [Nat.zero_eq, P_f_0_eq]
· simp only [P_succ, add_right_eq_self, comp_add, HomologicalComplex.comp_f,
HomologicalComplex.add_f_apply, comp_id]
exact (HigherFacesVanish.of_P q n).comp_Hσ_eq_zero (Nat.succ_le_iff.mp hqn)
| 5 | 148.413159 | 2 | 0.833333 | 6 | 736 |
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.Galois
universe u v w
open scoped Classical Polynomial
open Polynomial
variable (k : Type u) [Field k] (K : Type v) [Field K]
class IsSepClosed : Prop where
splits_of_separable : ∀ p : k[X], p.Separable → (p.Splits <| RingHom.... | Mathlib/FieldTheory/IsSepClosed.lean | 168 | 179 | theorem algebraMap_surjective
[IsSepClosed k] [Algebra k K] [IsSeparable k K] :
Function.Surjective (algebraMap k K) := by |
refine fun x => ⟨-(minpoly k x).coeff 0, ?_⟩
have hq : (minpoly k x).leadingCoeff = 1 := minpoly.monic (IsSeparable.isIntegral k x)
have hsep : (minpoly k x).Separable := IsSeparable.separable k x
have h : (minpoly k x).degree = 1 :=
degree_eq_one_of_irreducible k (minpoly.irreducible (IsSeparable.isIntegr... | 9 | 8,103.083928 | 2 | 1.5 | 6 | 1,639 |
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.Algebra.MonoidAlgebra.Support
import Mathlib.Algebra.DirectSum.Internal
import Mathlib.RingTheory.GradedAlgebra.Basic
#align_import algebra.monoid_algebra.grading from "leanprover-community/mathlib"@"feb99064803fd3108e37c18b0f77d0a8344677a3"
noncomputable sectio... | Mathlib/Algebra/MonoidAlgebra/Grading.lean | 86 | 89 | theorem single_mem_gradeBy {R} [CommSemiring R] (f : M → ι) (m : M) (r : R) :
Finsupp.single m r ∈ gradeBy R f (f m) := by |
intro x hx
rw [Finset.mem_singleton.mp (Finsupp.support_single_subset hx)]
| 2 | 7.389056 | 1 | 1.2 | 5 | 1,256 |
import Mathlib.Algebra.MvPolynomial.Equiv
import Mathlib.Algebra.MvPolynomial.Supported
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.RingTheory.Adjoin.Basic
import Mathlib.RingTheory.Algebraic
import Mathlib.RingTheory.MvPolynomial.Basic
#align_import ring_theory.algebraic_independent from "leanprove... | Mathlib/RingTheory/AlgebraicIndependent.lean | 129 | 131 | theorem comp (f : ι' → ι) (hf : Function.Injective f) : AlgebraicIndependent R (x ∘ f) := by |
intro p q
simpa [aeval_rename, (rename_injective f hf).eq_iff] using @hx (rename f p) (rename f q)
| 2 | 7.389056 | 1 | 1.285714 | 7 | 1,361 |
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.ContDiff.RCLike
import Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv
noncomputable section
namespace ContDiffAt
variable {𝕂 : Type*} [RCLike 𝕂]
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕂 E]
variable {F : ... | Mathlib/Analysis/Calculus/InverseFunctionTheorem/ContDiff.lean | 68 | 75 | theorem to_localInverse {n : ℕ∞} (hf : ContDiffAt 𝕂 n f a)
(hf' : HasFDerivAt f (f' : E →L[𝕂] F) a) (hn : 1 ≤ n) :
ContDiffAt 𝕂 n (hf.localInverse hf' hn) (f a) := by |
have := hf.localInverse_apply_image hf' hn
apply (hf.toPartialHomeomorph f hf' hn).contDiffAt_symm
(image_mem_toPartialHomeomorph_target hf hf' hn)
· convert hf'
· convert hf
| 5 | 148.413159 | 2 | 2 | 1 | 2,476 |
import Mathlib.Data.Fintype.Basic
import Mathlib.GroupTheory.Perm.Sign
import Mathlib.Logic.Equiv.Defs
#align_import logic.equiv.fintype from "leanprover-community/mathlib"@"9407b03373c8cd201df99d6bc5514fc2db44054f"
section Fintype
variable {α β : Type*} [Fintype α] [DecidableEq β] (e : Equiv.Perm α) (f : α ↪ β)
... | Mathlib/Logic/Equiv/Fintype.lean | 78 | 82 | theorem Equiv.Perm.viaFintypeEmbedding_apply_mem_range {b : β} (h : b ∈ Set.range f) :
e.viaFintypeEmbedding f b = f (e (f.invOfMemRange ⟨b, h⟩)) := by |
simp only [viaFintypeEmbedding, Function.Embedding.invOfMemRange]
rw [Equiv.Perm.extendDomain_apply_subtype]
congr
| 3 | 20.085537 | 1 | 0.7 | 10 | 640 |
import Mathlib.Data.Set.Image
#align_import data.nat.set from "leanprover-community/mathlib"@"cf9386b56953fb40904843af98b7a80757bbe7f9"
namespace Nat
section Set
open Set
theorem zero_union_range_succ : {0} ∪ range succ = univ := by
ext n
cases n <;> simp
#align nat.zero_union_range_succ Nat.zero_union_ran... | Mathlib/Data/Nat/Set.lean | 37 | 46 | theorem range_rec {α : Type*} (x : α) (f : ℕ → α → α) :
(Set.range fun n => Nat.rec x f n : Set α) =
{x} ∪ Set.range fun n => Nat.rec (f 0 x) (f ∘ succ) n := by |
convert (range_of_succ (fun n => Nat.rec x f n : ℕ → α)).symm using 4
dsimp
rename_i n
induction' n with n ihn
· rfl
· dsimp at ihn ⊢
rw [ihn]
| 7 | 1,096.633158 | 2 | 1 | 3 | 1,045 |
import Mathlib.RingTheory.Valuation.Basic
import Mathlib.NumberTheory.Padics.PadicNorm
import Mathlib.Analysis.Normed.Field.Basic
#align_import number_theory.padics.padic_numbers from "leanprover-community/mathlib"@"b9b2114f7711fec1c1e055d507f082f8ceb2c3b7"
noncomputable section
open scoped Classical
open Nat m... | Mathlib/NumberTheory/Padics/PadicNumbers.lean | 121 | 135 | theorem norm_zero_iff (f : PadicSeq p) : f.norm = 0 ↔ f ≈ 0 := by |
constructor
· intro h
by_contra hf
unfold norm at h
split_ifs at h
· contradiction
apply hf
intro ε hε
exists stationaryPoint hf
intro j hj
have heq := stationaryPoint_spec hf le_rfl hj
simpa [h, heq]
· intro h
simp [norm, h]
| 14 | 1,202,604.284165 | 2 | 1.4 | 5 | 1,505 |
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Group.Int
import Mathlib.Algebra.GroupWithZero.Semiconj
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Order.Bounds.Basic
#align_import data.int.gcd from "leanprover-community/mathlib"@"47a1a73351de8dd6c8d3d32b569c8e434b03ca47"
namespace Nat
... | Mathlib/Data/Int/GCD.lean | 146 | 154 | theorem exists_mul_emod_eq_gcd {k n : ℕ} (hk : gcd n k < k) : ∃ m, n * m % k = gcd n k := by |
have hk' := Int.ofNat_ne_zero.2 (ne_of_gt (lt_of_le_of_lt (zero_le (gcd n k)) hk))
have key := congr_arg (fun (m : ℤ) => (m % k).toNat) (gcd_eq_gcd_ab n k)
simp only at key
rw [Int.add_mul_emod_self_left, ← Int.natCast_mod, Int.toNat_natCast, mod_eq_of_lt hk] at key
refine ⟨(n.gcdA k % k).toNat, Eq.trans (In... | 8 | 2,980.957987 | 2 | 1.090909 | 11 | 1,188 |
import Mathlib.Algebra.MvPolynomial.Monad
#align_import data.mv_polynomial.expand from "leanprover-community/mathlib"@"5da451b4c96b4c2e122c0325a7fce17d62ee46c6"
namespace MvPolynomial
variable {σ τ R S : Type*} [CommSemiring R] [CommSemiring S]
noncomputable def expand (p : ℕ) : MvPolynomial σ R →ₐ[R] MvPolyno... | Mathlib/Algebra/MvPolynomial/Expand.lean | 59 | 61 | theorem expand_one : expand 1 = AlgHom.id R (MvPolynomial σ R) := by |
ext1 f
rw [expand_one_apply, AlgHom.id_apply]
| 2 | 7.389056 | 1 | 0.571429 | 7 | 523 |
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Data.Finsupp.Fin
import Mathlib.Data.Finsupp.Indicator
#align_import algebra.bi... | Mathlib/Algebra/BigOperators/Finsupp.lean | 131 | 134 | theorem prod_ite_eq' [DecidableEq α] (f : α →₀ M) (a : α) (b : α → M → N) :
(f.prod fun x v => ite (x = a) (b x v) 1) = ite (a ∈ f.support) (b a (f a)) 1 := by |
dsimp [Finsupp.prod]
rw [f.support.prod_ite_eq']
| 2 | 7.389056 | 1 | 0.833333 | 6 | 729 |
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Ring.Pi
import Mathlib.GroupTheory.GroupAction.Pi
import Mathlib.GroupTheory.GroupAction.Ring
import Mathlib.Init.Align
import Mathlib.Tactic.GCongr
import Mathlib.Tactic... | Mathlib/Algebra/Order/CauSeq/Basic.lean | 58 | 71 | theorem rat_mul_continuous_lemma {ε K₁ K₂ : α} (ε0 : 0 < ε) :
∃ δ > 0, ∀ {a₁ a₂ b₁ b₂ : β}, abv a₁ < K₁ → abv b₂ < K₂ → abv (a₁ - b₁) < δ →
abv (a₂ - b₂) < δ → abv (a₁ * a₂ - b₁ * b₂) < ε := by |
have K0 : (0 : α) < max 1 (max K₁ K₂) := lt_of_lt_of_le zero_lt_one (le_max_left _ _)
have εK := div_pos (half_pos ε0) K0
refine ⟨_, εK, fun {a₁ a₂ b₁ b₂} ha₁ hb₂ h₁ h₂ => ?_⟩
replace ha₁ := lt_of_lt_of_le ha₁ (le_trans (le_max_left _ K₂) (le_max_right 1 _))
replace hb₂ := lt_of_lt_of_le hb₂ (le_trans (le_ma... | 11 | 59,874.141715 | 2 | 2 | 3 | 2,249 |
import Mathlib.Algebra.IsPrimePow
import Mathlib.SetTheory.Cardinal.Ordinal
import Mathlib.Tactic.WLOG
#align_import set_theory.cardinal.divisibility from "leanprover-community/mathlib"@"ea050b44c0f9aba9d16a948c7cc7d2e7c8493567"
namespace Cardinal
open Cardinal
universe u
variable {a b : Cardinal.{u}} {n m : ℕ... | Mathlib/SetTheory/Cardinal/Divisibility.lean | 76 | 89 | theorem prime_of_aleph0_le (ha : ℵ₀ ≤ a) : Prime a := by |
refine ⟨(aleph0_pos.trans_le ha).ne', ?_, fun b c hbc => ?_⟩
· rw [isUnit_iff]
exact (one_lt_aleph0.trans_le ha).ne'
rcases eq_or_ne (b * c) 0 with hz | hz
· rcases mul_eq_zero.mp hz with (rfl | rfl) <;> simp
wlog h : c ≤ b
· cases le_total c b <;> [solve_by_elim; rw [or_comm]]
apply_assumption
... | 13 | 442,413.392009 | 2 | 2 | 7 | 2,430 |
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
#align_import analysis.special_functions.pow.asymptotics from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8"
set_option linter.uppercaseLean3 false
noncomputable section
open scoped Classical
open Real Topology NNReal ENNReal Filter C... | Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean | 279 | 283 | theorem IsLittleO.rpow (hr : 0 < r) (hg : 0 ≤ᶠ[l] g) (h : f =o[l] g) :
(fun x => f x ^ r) =o[l] fun x => g x ^ r := by |
refine .of_isBigOWith fun c hc ↦ ?_
rw [← rpow_inv_rpow hc.le hr.ne']
refine (h.forall_isBigOWith ?_).rpow ?_ ?_ hg <;> positivity
| 3 | 20.085537 | 1 | 1.5 | 10 | 1,605 |
import Mathlib.Topology.MetricSpace.PiNat
#align_import topology.metric_space.cantor_scheme from "leanprover-community/mathlib"@"49b7f94aab3a3bdca1f9f34c5d818afb253b3993"
namespace CantorScheme
open List Function Filter Set PiNat
open scoped Classical
open Topology
variable {β α : Type*} (A : List β → Set α)
... | Mathlib/Topology/MetricSpace/CantorScheme.lean | 99 | 115 | theorem Disjoint.map_injective (hA : CantorScheme.Disjoint A) : Injective (inducedMap A).2 := by |
rintro ⟨x, hx⟩ ⟨y, hy⟩ hxy
refine Subtype.coe_injective (res_injective ?_)
dsimp
ext n : 1
induction' n with n ih; · simp
simp only [res_succ, cons.injEq]
refine ⟨?_, ih⟩
contrapose hA
simp only [CantorScheme.Disjoint, _root_.Pairwise, Ne, not_forall, exists_prop]
refine ⟨res x n, _, _, hA, ?_⟩
r... | 16 | 8,886,110.520508 | 2 | 1.666667 | 3 | 1,827 |
import Mathlib.Data.Set.Equitable
import Mathlib.Logic.Equiv.Fin
import Mathlib.Order.Partition.Finpartition
#align_import order.partition.equipartition from "leanprover-community/mathlib"@"b363547b3113d350d053abdf2884e9850a56b205"
open Finset Fintype
namespace Finpartition
variable {α : Type*} [DecidableEq α] ... | Mathlib/Order/Partition/Equipartition.lean | 114 | 134 | theorem IsEquipartition.exists_partsEquiv (hP : P.IsEquipartition) :
∃ f : P.parts ≃ Fin P.parts.card,
∀ t, t.1.card = s.card / P.parts.card + 1 ↔ f t < s.card % P.parts.card := by |
let el := (P.parts.filter fun p ↦ p.card = s.card / P.parts.card + 1).equivFin
let es := (P.parts.filter fun p ↦ p.card = s.card / P.parts.card).equivFin
simp_rw [mem_filter, hP.card_large_parts_eq_mod] at el
simp_rw [mem_filter, hP.card_small_parts_eq_mod] at es
let sneg : { x // x ∈ P.parts ∧ ¬x.card = s.c... | 18 | 65,659,969.137331 | 2 | 1.375 | 8 | 1,469 |
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition
import Mathlib.LinearAlgebra.Projection
import Mathlib.LinearAlgebra.SesquilinearForm
import Mathlib.RingTheory.TensorProduct.Basic
import Mathlib.RingTheory.I... | Mathlib/LinearAlgebra/Dual.lean | 388 | 392 | theorem sum_dual_apply_smul_coord (f : Module.Dual R M) :
(∑ x, f (b x) • b.coord x) = f := by |
ext m
simp_rw [LinearMap.sum_apply, LinearMap.smul_apply, smul_eq_mul, mul_comm (f _), ← smul_eq_mul, ←
f.map_smul, ← _root_.map_sum, Basis.coord_apply, Basis.sum_repr]
| 3 | 20.085537 | 1 | 1.111111 | 9 | 1,196 |
import Mathlib.Algebra.Group.Equiv.TypeTags
import Mathlib.GroupTheory.FreeAbelianGroup
import Mathlib.GroupTheory.FreeGroup.IsFreeGroup
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
#align_import group_theory.free_abelian_group_finsupp from "leanprover-community/mathlib"@"47b51515e69f59bca5cf34ef456e600... | Mathlib/GroupTheory/FreeAbelianGroupFinsupp.lean | 45 | 50 | theorem Finsupp.toFreeAbelianGroup_comp_singleAddHom (x : X) :
Finsupp.toFreeAbelianGroup.comp (Finsupp.singleAddHom x) =
(smulAddHom ℤ (FreeAbelianGroup X)).flip (of x) := by |
ext
simp only [AddMonoidHom.coe_comp, Finsupp.singleAddHom_apply, Function.comp_apply, one_smul,
toFreeAbelianGroup, Finsupp.liftAddHom_apply_single]
| 3 | 20.085537 | 1 | 0.857143 | 7 | 747 |
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
import Qq
#align_import analysis.special_functions.pow.real from "leanprover-community/mathlib"@"4fa54b337f7d52805480306db1b1439c741848c8"
noncomputable section
open scoped Classical
open Real ComplexConjugate
open Finset Set
namespace Real
variable {x y z... | Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 85 | 87 | theorem rpow_eq_zero_iff_of_nonneg (hx : 0 ≤ x) : x ^ y = 0 ↔ x = 0 ∧ y ≠ 0 := by |
simp only [rpow_def_of_nonneg hx]
split_ifs <;> simp [*, exp_ne_zero]
| 2 | 7.389056 | 1 | 0.384615 | 13 | 383 |
import Mathlib.Data.Finset.Pi
import Mathlib.Data.Fintype.Basic
#align_import data.fintype.pi from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
variable {α : Type*}
open Finset
namespace Fintype
variable [DecidableEq α] [Fintype α] {γ δ : α → Type*} {s : ∀ a, Finset (γ a)}
def pi... | Mathlib/Data/Fintype/Pi.lean | 34 | 42 | theorem mem_piFinset {t : ∀ a, Finset (δ a)} {f : ∀ a, δ a} : f ∈ piFinset t ↔ ∀ a, f a ∈ t a := by |
constructor
· simp only [piFinset, mem_map, and_imp, forall_prop_of_true, exists_prop, mem_univ, exists_imp,
mem_pi]
rintro g hg hgf a
rw [← hgf]
exact hg a
· simp only [piFinset, mem_map, forall_prop_of_true, exists_prop, mem_univ, mem_pi]
exact fun hf => ⟨fun a _ => f a, hf, rfl⟩
| 8 | 2,980.957987 | 2 | 2 | 1 | 1,946 |
import Mathlib.LinearAlgebra.Matrix.Gershgorin
import Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody
import Mathlib.NumberTheory.NumberField.Units.Basic
import Mathlib.RingTheory.RootsOfUnity.Basic
#align_import number_theory.number_field.units from "leanprover-community/mathlib"@"00f91228655eecdcd3ac... | Mathlib/NumberTheory/NumberField/Units/DirichletTheorem.lean | 128 | 151 | theorem log_le_of_logEmbedding_le {r : ℝ} {x : (𝓞 K)ˣ} (hr : 0 ≤ r) (h : ‖logEmbedding K x‖ ≤ r)
(w : InfinitePlace K) : |Real.log (w x)| ≤ (Fintype.card (InfinitePlace K)) * r := by |
have tool : ∀ x : ℝ, 0 ≤ x → x ≤ mult w * x := fun x hx => by
nth_rw 1 [← one_mul x]
refine mul_le_mul ?_ le_rfl hx ?_
all_goals { rw [mult]; split_ifs <;> norm_num }
by_cases hw : w = w₀
· have hyp := congr_arg (‖·‖) (sum_logEmbedding_component x).symm
replace hyp := (le_of_eq hyp).trans (norm_s... | 22 | 3,584,912,846.131591 | 2 | 1.833333 | 6 | 1,909 |
import Mathlib.Order.RelIso.Set
import Mathlib.Data.Multiset.Sort
import Mathlib.Data.List.NodupEquivFin
import Mathlib.Data.Finset.Lattice
import Mathlib.Data.Fintype.Card
#align_import data.finset.sort from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
namespace Finset
open Multiset... | Mathlib/Data/Finset/Sort.lean | 79 | 81 | theorem sort_perm_toList (s : Finset α) : sort r s ~ s.toList := by |
rw [← Multiset.coe_eq_coe]
simp only [coe_toList, sort_eq]
| 2 | 7.389056 | 1 | 1 | 1 | 1,137 |
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.RingTheory.Adjoin.Basic
import Mathlib.RingTheory.AdjoinRoot
#align_import ring_theory.adjoin.field from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
noncomputable section
open Polynomial
variable {R K L M : Type*} [CommRing R]... | Mathlib/RingTheory/Adjoin/Field.lean | 106 | 110 | theorem IsIntegral.minpoly_splits_tower_top [Algebra K L] [IsScalarTower R K L]
(h : Splits (algebraMap R L) (minpoly R x)) :
Splits (algebraMap K L) (minpoly K x) := by |
rw [IsScalarTower.algebraMap_eq R K L] at h
exact int.minpoly_splits_tower_top' h
| 2 | 7.389056 | 1 | 1.5 | 2 | 1,558 |
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Set.Finite
import Mathlib.Data.Set.Pointwise.SMul
import Mathlib.Data.Setoid.Basic
import Mathlib.GroupTheory.GroupAction.Defs
import Mathlib.GroupTheory.GroupAction.Group
#align_import group_theory.group_action.basic fro... | Mathlib/GroupTheory/GroupAction/Basic.lean | 312 | 317 | theorem smul_cancel_of_non_zero_divisor {M R : Type*} [Monoid M] [NonUnitalNonAssocRing R]
[DistribMulAction M R] (k : M) (h : ∀ x : R, k • x = 0 → x = 0) {a b : R} (h' : k • a = k • b) :
a = b := by |
rw [← sub_eq_zero]
refine h _ ?_
rw [smul_sub, h', sub_self]
| 3 | 20.085537 | 1 | 1 | 1 | 1,054 |
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 | 82 | 86 | theorem natAbs_coe_sub_coe_le_of_le {a b n : ℕ} (a_le_n : a ≤ n) (b_le_n : b ≤ n) :
natAbs (a - b : ℤ) ≤ n := by |
rw [← Nat.cast_le (α := ℤ), natCast_natAbs]
exact abs_sub_le_of_nonneg_of_le (ofNat_nonneg a) (ofNat_le.mpr a_le_n)
(ofNat_nonneg b) (ofNat_le.mpr b_le_n)
| 3 | 20.085537 | 1 | 0.818182 | 11 | 720 |
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Measure.Haar.Unique
#align_import measure_theory.measure.lebesgue.integral from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
open Set Filter MeasureTheory... | Mathlib/MeasureTheory/Measure/Lebesgue/Integral.lean | 55 | 69 | theorem Real.integrable_of_summable_norm_Icc {E : Type*} [NormedAddCommGroup E] {f : C(ℝ, E)}
(hf : Summable fun n : ℤ => ‖(f.comp <| ContinuousMap.addRight n).restrict (Icc 0 1)‖) :
Integrable f := by |
refine integrable_of_summable_norm_restrict (.of_nonneg_of_le
(fun n : ℤ => mul_nonneg (norm_nonneg
(f.restrict (⟨Icc (n : ℝ) ((n : ℝ) + 1), isCompact_Icc⟩ : Compacts ℝ)))
ENNReal.toReal_nonneg) (fun n => ?_) hf) ?_
· simp only [Compacts.coe_mk, Real.volume_Icc, add_sub_cancel_left,
ENNReal... | 12 | 162,754.791419 | 2 | 1.8 | 5 | 1,894 |
import Mathlib.MeasureTheory.Covering.Differentiation
import Mathlib.MeasureTheory.Covering.VitaliFamily
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.MeasureTheory.Measure.Regular
import Mathlib.SetTheory.Ordinal.Arithmetic
import Mathlib.Topology.MetricSpace.Basic
import Mathlib.Data.Set.Pairwise.Lat... | Mathlib/MeasureTheory/Covering/Besicovitch.lean | 278 | 281 | theorem monotone_iUnionUpTo : Monotone p.iUnionUpTo := by |
intro i j hij
simp only [iUnionUpTo]
exact iUnion_mono' fun r => ⟨⟨r, r.2.trans_le hij⟩, Subset.rfl⟩
| 3 | 20.085537 | 1 | 1.666667 | 3 | 1,792 |
import Mathlib.Order.Filter.Germ
import Mathlib.Topology.NhdsSet
import Mathlib.Topology.LocallyConstant.Basic
import Mathlib.Analysis.NormedSpace.Basic
variable {F G : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F]
[NormedAddCommGroup G] [NormedSpace ℝ G]
open scoped Topology
open Filter Set
variable {X Y Z ... | Mathlib/Topology/Germ.lean | 104 | 110 | theorem restrictGermPredicate_congr {P : ∀ x : X, Germ (𝓝 x) Y → Prop}
(hf : RestrictGermPredicate P A x f) (h : ∀ᶠ z in 𝓝ˢ A, g z = f z) :
RestrictGermPredicate P A x g := by |
intro hx
apply ((hf hx).and <| (eventually_nhdsSet_iff_forall.mp h x hx).eventually_nhds).mono
rintro y ⟨hy, h'y⟩
rwa [Germ.coe_eq.mpr h'y]
| 4 | 54.59815 | 2 | 1.666667 | 3 | 1,810 |
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 | 74 | 105 | theorem measure_limsup_eq_one {s : ℕ → Set Ω} (hsm : ∀ n, MeasurableSet (s n)) (hs : iIndepSet s μ)
(hs' : (∑' n, μ (s n)) = ∞) : μ (limsup s atTop) = 1 := by |
rw [measure_congr (eventuallyEq_set.2 (ae_mem_limsup_atTop_iff μ <|
measurableSet_filtrationOfSet' hsm) : (limsup s atTop : Set Ω) =ᵐ[μ]
{ω | Tendsto (fun n => ∑ k ∈ Finset.range n,
(μ[(s (k + 1)).indicator (1 : Ω → ℝ)|filtrationOfSet hsm k]) ω) atTop atTop})]
suffices {ω | Tendsto (fun n => ∑ k ... | 30 | 10,686,474,581,524.463 | 2 | 1.333333 | 3 | 1,369 |
import Mathlib.Data.List.Sym
namespace Multiset
variable {α : Type*}
section Sym2
protected def sym2 (m : Multiset α) : Multiset (Sym2 α) :=
m.liftOn (fun xs => xs.sym2) fun _ _ h => by rw [coe_eq_coe]; exact h.sym2
@[simp] theorem sym2_coe (xs : List α) : (xs : Multiset α).sym2 = xs.sym2 := rfl
@[simp]
the... | Mathlib/Data/Multiset/Sym.lean | 63 | 66 | theorem sym2_mono {m m' : Multiset α} (h : m ≤ m') : m.sym2 ≤ m'.sym2 := by |
refine Quotient.inductionOn₂ m m' (fun xs ys h => ?_) h
suffices xs <+~ ys from this.sym2
simpa only [quot_mk_to_coe, coe_le, sym2_coe] using h
| 3 | 20.085537 | 1 | 1 | 2 | 878 |
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#align_import number_theory.modular_forms.congruence_subgroups from "leanprover-community/mathlib"@"ae690b0c236e488a0043f6faa8ce3546e7f2f... | Mathlib/NumberTheory/ModularForms/CongruenceSubgroups.lean | 78 | 88 | theorem Gamma_zero_bot : Gamma 0 = ⊥ := by |
ext
simp only [Gamma_mem, coe_matrix_coe, Int.coe_castRingHom, map_apply, Int.cast_id,
Subgroup.mem_bot]
constructor
· intro h
ext i j
fin_cases i <;> fin_cases j <;> simp only [h]
exacts [h.1, h.2.1, h.2.2.1, h.2.2.2]
· intro h
simp [h]
| 10 | 22,026.465795 | 2 | 1.25 | 4 | 1,329 |
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 | 87 | 91 | theorem quotient_map_C_eq_zero {I : Ideal R} :
∀ a ∈ I, ((Quotient.mk (map (C : R →+* R[X]) I : Ideal R[X])).comp C) a = 0 := by |
intro a ha
rw [RingHom.comp_apply, Quotient.eq_zero_iff_mem]
exact mem_map_of_mem _ ha
| 3 | 20.085537 | 1 | 1.428571 | 7 | 1,526 |
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 | 98 | 100 | theorem leftUnitor_hom_hom {X : Action V G} : Hom.hom (λ_ X).hom = (λ_ X.V).hom := by |
dsimp
simp
| 2 | 7.389056 | 1 | 1 | 6 | 1,092 |
import Mathlib.Algebra.Homology.Homotopy
import Mathlib.AlgebraicTopology.DoldKan.Notations
#align_import algebraic_topology.dold_kan.homotopies from "leanprover-community/mathlib"@"b12099d3b7febf4209824444dd836ef5ad96db55"
open CategoryTheory CategoryTheory.Category CategoryTheory.Limits CategoryTheory.Preadditi... | Mathlib/AlgebraicTopology/DoldKan/Homotopies.lean | 111 | 119 | theorem hσ'_eq {q n a m : ℕ} (ha : n = a + q) (hnm : c.Rel m n) :
(hσ' q n m hnm : X _[n] ⟶ X _[m]) =
((-1 : ℤ) ^ a • X.σ ⟨a, Nat.lt_succ_iff.mpr (Nat.le.intro (Eq.symm ha))⟩) ≫
eqToHom (by congr) := by |
simp only [hσ', hσ]
split_ifs
· omega
· have h' := tsub_eq_of_eq_add ha
congr
| 5 | 148.413159 | 2 | 1.5 | 6 | 1,625 |
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 | 80 | 86 | theorem isCompact_open_iff_eq_finset_affine_union {X : Scheme} (U : Set X.carrier) :
IsCompact U ∧ IsOpen U ↔
∃ s : Set X.affineOpens, s.Finite ∧ U = ⋃ (i : X.affineOpens) (_ : i ∈ s), i := by |
apply Opens.IsBasis.isCompact_open_iff_eq_finite_iUnion
(fun (U : X.affineOpens) => (U : Opens X.carrier))
· rw [Subtype.range_coe]; exact isBasis_affine_open X
· exact fun i => i.2.isCompact
| 4 | 54.59815 | 2 | 1.5 | 6 | 1,632 |
import Mathlib.Analysis.SpecialFunctions.ImproperIntegrals
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.MeasureTheory.Measure.Haar.NormedSpace
#align_import analysis.mellin_transform from "leanprover-community/mathlib"@"917c3c072e487b3cccdbfeff17e75b40e45f66cb"
open MeasureTheory Set Filter A... | Mathlib/Analysis/MellinTransform.lean | 53 | 56 | theorem MellinConvergent.cpow_smul {f : ℝ → E} {s a : ℂ} :
MellinConvergent (fun t => (t : ℂ) ^ a • f t) s ↔ MellinConvergent f (s + a) := by |
refine integrableOn_congr_fun (fun t ht => ?_) measurableSet_Ioi
simp_rw [← sub_add_eq_add_sub, cpow_add _ _ (ofReal_ne_zero.2 <| ne_of_gt ht), mul_smul]
| 2 | 7.389056 | 1 | 1.333333 | 12 | 1,415 |
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.LinearAlgebra.Matrix.ToLin
#align_import linear_algebra.matrix.charpoly.linear_map from "leanprover-community/mathlib"@"62c0a4ef1441edb463095ea02a06e87f3dfe135c"
variable {ι : Type*} [Fintype ι]
variable {M : Type*} [AddCommGroup M] (R : Type*) [Co... | Mathlib/LinearAlgebra/Matrix/Charpoly/LinearMap.lean | 43 | 46 | theorem PiToModule.fromMatrix_apply_single_one [DecidableEq ι] (A : Matrix ι ι R) (j : ι) :
PiToModule.fromMatrix R b A (Pi.single j 1) = ∑ i : ι, A i j • b i := by |
rw [PiToModule.fromMatrix_apply, Fintype.total_apply, Matrix.mulVec_single]
simp_rw [mul_one]
| 2 | 7.389056 | 1 | 1.444444 | 9 | 1,527 |
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.BigOperators
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.Tactic.FinCases
#align_import linear_algebra.affine_space.combination from ... | Mathlib/LinearAlgebra/AffineSpace/Combination.lean | 96 | 104 | theorem weightedVSubOfPoint_eq_of_weights_eq (p : ι → P) (j : ι) (w₁ w₂ : ι → k)
(hw : ∀ i, i ≠ j → w₁ i = w₂ i) :
s.weightedVSubOfPoint p (p j) w₁ = s.weightedVSubOfPoint p (p j) w₂ := by |
simp only [Finset.weightedVSubOfPoint_apply]
congr
ext i
rcases eq_or_ne i j with h | h
· simp [h]
· simp [hw i h]
| 6 | 403.428793 | 2 | 1.083333 | 12 | 1,183 |
import Mathlib.Data.Finset.Lattice
import Mathlib.Data.Fintype.Vector
import Mathlib.Data.Multiset.Sym
#align_import data.finset.sym from "leanprover-community/mathlib"@"02ba8949f486ebecf93fe7460f1ed0564b5e442c"
namespace Finset
variable {α : Type*}
@[simps]
protected def sym2 (s : Finset α) : Finset (Sym2 α) :... | Mathlib/Data/Finset/Sym.lean | 77 | 80 | theorem injective_sym2 : Function.Injective (Finset.sym2 : Finset α → _) := by |
intro s t h
ext x
simpa using congr(s(x, x) ∈ $h)
| 3 | 20.085537 | 1 | 0.769231 | 13 | 684 |
import Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity
#align_import number_theory.legendre_symbol.jacobi_symbol from "leanprover-community/mathlib"@"74a27133cf29446a0983779e37c8f829a85368f3"
section Jacobi
open Nat ZMod
-- Since we need the fact that the factors are prime, we use `List.pmap`.
def ... | Mathlib/NumberTheory/LegendreSymbol/JacobiSymbol.lean | 159 | 163 | theorem mul_left (a₁ a₂ : ℤ) (b : ℕ) : J(a₁ * a₂ | b) = J(a₁ | b) * J(a₂ | b) := by |
simp_rw [jacobiSym, List.pmap_eq_map_attach, legendreSym.mul _ _ _];
exact List.prod_map_mul (α := ℤ) (l := (factors b).attach)
(f := fun x ↦ @legendreSym x {out := prime_of_mem_factors x.2} a₁)
(g := fun x ↦ @legendreSym x {out := prime_of_mem_factors x.2} a₂)
| 4 | 54.59815 | 2 | 0.833333 | 6 | 730 |
import Mathlib.MeasureTheory.Measure.VectorMeasure
import Mathlib.MeasureTheory.Function.AEEqOfIntegral
#align_import measure_theory.measure.with_density_vector_measure from "leanprover-community/mathlib"@"d1bd9c5df2867c1cb463bc6364446d57bdd9f7f1"
noncomputable section
open scoped Classical MeasureTheory NNReal ... | Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean | 69 | 76 | theorem withDensityᵥ_neg : μ.withDensityᵥ (-f) = -μ.withDensityᵥ f := by |
by_cases hf : Integrable f μ
· ext1 i hi
rw [VectorMeasure.neg_apply, withDensityᵥ_apply hf hi, ← integral_neg,
withDensityᵥ_apply hf.neg hi]
rfl
· rw [withDensityᵥ, withDensityᵥ, dif_neg hf, dif_neg, neg_zero]
rwa [integrable_neg_iff]
| 7 | 1,096.633158 | 2 | 1.142857 | 7 | 1,217 |
import Mathlib.Algebra.Homology.Homotopy
import Mathlib.Algebra.Category.ModuleCat.Abelian
import Mathlib.Algebra.Category.ModuleCat.Subobject
import Mathlib.CategoryTheory.Limits.Shapes.ConcreteCategory
#align_import algebra.homology.Module from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225... | Mathlib/Algebra/Homology/ModuleCat.lean | 37 | 49 | theorem homology'_ext {L M N K : ModuleCat.{u} R} {f : L ⟶ M} {g : M ⟶ N} (w : f ≫ g = 0)
{h k : homology' f g w ⟶ K}
(w :
∀ x : LinearMap.ker g,
h (cokernel.π (imageToKernel _ _ w) (toKernelSubobject x)) =
k (cokernel.π (imageToKernel _ _ w) (toKernelSubobject x))) :
h = k := by |
refine Concrete.cokernel_funext fun n => ?_
-- Porting note: as `equiv_rw` was not ported, it was replaced by `Equiv.surjective`
-- Gosh it would be nice if `equiv_rw` could directly use an isomorphism, or an enriched `≃`.
obtain ⟨n, rfl⟩ := (kernelSubobjectIso g ≪≫
ModuleCat.kernelIsoKer g).toLinearEquiv.... | 6 | 403.428793 | 2 | 1.25 | 4 | 1,337 |
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Data.List.Infix
import Mathlib.Data.List.MinMax
import Mathlib.Data.List.EditDistance.Defs
set_option autoImplicit true
variable {C : Levenshtein.Cost α β δ} [CanonicallyLinearOrderedAddCommMonoid δ]
theorem suffixLevenshtein_minimum_le_levenshtein... | Mathlib/Data/List/EditDistance/Bounds.lean | 89 | 92 | theorem le_levenshtein_cons (xs : List α) (y ys) :
∃ xs', xs' <:+ xs ∧ levenshtein C xs' ys ≤ levenshtein C xs (y :: ys) := by |
simpa [suffixLevenshtein_eq_tails_map, List.minimum_le_coe_iff] using
suffixLevenshtein_minimum_le_levenshtein_cons (δ := δ) xs y ys
| 2 | 7.389056 | 1 | 1.5 | 6 | 1,674 |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Algebra.Field.Rat
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Field.Rat
import Mathlib.Combinatorics.Enumerative.DoubleCounting
import Mathlib.Combinatorics.SetFamily.Shadow
#align_import combinatorics.set_family.lym from "leanprover-co... | Mathlib/Combinatorics/SetFamily/LYM.lean | 131 | 133 | theorem mem_falling : s ∈ falling k 𝒜 ↔ (∃ t ∈ 𝒜, s ⊆ t) ∧ s.card = k := by |
simp_rw [falling, mem_sup, mem_powersetCard]
aesop
| 2 | 7.389056 | 1 | 1.75 | 4 | 1,868 |
import Mathlib.Order.BoundedOrder
import Mathlib.Order.MinMax
import Mathlib.Algebra.NeZero
import Mathlib.Algebra.Order.Monoid.Defs
#align_import algebra.order.monoid.canonical.defs from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
universe u
variable {α : Type u}
class ExistsMulOf... | Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean | 56 | 58 | theorem exists_one_lt_mul_of_lt' (h : a < b) : ∃ c, 1 < c ∧ a * c = b := by |
obtain ⟨c, rfl⟩ := exists_mul_of_le h.le
exact ⟨c, one_lt_of_lt_mul_right h, rfl⟩
| 2 | 7.389056 | 1 | 0.666667 | 3 | 622 |
import Mathlib.Probability.ProbabilityMassFunction.Basic
import Mathlib.Probability.ProbabilityMassFunction.Constructions
import Mathlib.MeasureTheory.Integral.Bochner
namespace PMF
open MeasureTheory ENNReal TopologicalSpace
section General
variable {α : Type*} [MeasurableSpace α] [MeasurableSingletonClass α]
v... | Mathlib/Probability/ProbabilityMassFunction/Integrals.lean | 43 | 47 | theorem integral_eq_sum [Fintype α] (p : PMF α) (f : α → E) :
∫ a, f a ∂(p.toMeasure) = ∑ a, (p a).toReal • f a := by |
rw [integral_fintype _ (.of_finite _ f)]
congr with x; congr 2
exact PMF.toMeasure_apply_singleton p x (MeasurableSet.singleton _)
| 3 | 20.085537 | 1 | 1.5 | 2 | 1,573 |
import Mathlib.Topology.MetricSpace.HausdorffDistance
#align_import topology.metric_space.hausdorff_distance from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
noncomputable section
open NNReal ENNReal Topology Set Filter Bornology
universe u v w
variable {ι : Sort*} {α : Type u} {β :... | Mathlib/Topology/MetricSpace/Thickening.lean | 242 | 244 | theorem cthickening_of_nonpos {δ : ℝ} (hδ : δ ≤ 0) (E : Set α) : cthickening δ E = closure E := by |
ext x
simp [mem_closure_iff_infEdist_zero, cthickening, ENNReal.ofReal_eq_zero.2 hδ]
| 2 | 7.389056 | 1 | 0.777778 | 9 | 686 |
import Mathlib.Algebra.Module.Submodule.Ker
#align_import linear_algebra.basic from "leanprover-community/mathlib"@"9d684a893c52e1d6692a504a118bfccbae04feeb"
variable {R : Type*} {R₂ : Type*}
variable {M : Type*} {M₂ : Type*}
namespace LinearMap
section AddCommMonoid
variable [Semiring R] [Semiring R₂]
varia... | Mathlib/Algebra/Module/Submodule/EqLocus.lean | 73 | 76 | theorem eqOn_sup {f g : F} {S T : Submodule R M} (hS : Set.EqOn f g S) (hT : Set.EqOn f g T) :
Set.EqOn f g ↑(S ⊔ T) := by |
rw [← le_eqLocus] at hS hT ⊢
exact sup_le hS hT
| 2 | 7.389056 | 1 | 0.5 | 2 | 433 |
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