Context stringlengths 57 85k | file_name stringlengths 21 79 | start int64 14 2.42k | end int64 18 2.43k | theorem stringlengths 25 2.71k | proof stringlengths 5 10.6k |
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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 _)
|
import Mathlib.Algebra.BigOperators.Option
import Mathlib.Analysis.BoxIntegral.Box.Basic
import Mathlib.Data.Set.Pairwise.Lattice
#align_import analysis.box_integral.partition.basic from "leanprover-community/mathlib"@"84dc0bd6619acaea625086d6f53cb35cdd554219"
open Set Finset Function
open scoped Classical
open ... | Mathlib/Analysis/BoxIntegral/Partition/Basic.lean | 528 | 529 | theorem iUnion_restrict : (π.restrict J).iUnion = (J : Set (ι → ℝ)) ∩ (π.iUnion) := by |
simp [restrict, ← inter_iUnion, ← iUnion_def]
|
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
import Mathlib.Analysis.NormedSpace.AffineIsometry
#align_import geometry.euclidean.angle.unoriented.affine from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f... | Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean | 119 | 120 | theorem angle_const_sub (v : V) (v₁ v₂ v₃ : V) : ∠ (v - v₁) (v - v₂) (v - v₃) = ∠ v₁ v₂ v₃ := by |
simpa only [vsub_eq_sub] using angle_const_vsub v v₁ v₂ v₃
|
import Mathlib.Order.Interval.Multiset
#align_import data.nat.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
-- TODO
-- assert_not_exists Ring
open Finset Nat
variable (a b c : ℕ)
namespace Nat
instance instLocallyFiniteOrder : LocallyFiniteOrder ℕ where
finsetIcc a b... | Mathlib/Order/Interval/Finset/Nat.lean | 153 | 155 | theorem Ico_succ_right : Ico a b.succ = Icc a b := by |
ext x
rw [mem_Ico, mem_Icc, Nat.lt_succ_iff]
|
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.Probability.Independence.Basic
#align_import probability.integration from "leanprover-community/mathlib"@"2f8347015b12b0864dfaf366ec4909eb70c78740"
noncomputable section
open Set MeasureTheory
open scoped ENNReal MeasureTheory
variable {Ω : Type*... | Mathlib/Probability/Integration.lean | 138 | 153 | theorem IndepFun.integrable_mul {β : Type*} [MeasurableSpace β] {X Y : Ω → β}
[NormedDivisionRing β] [BorelSpace β] (hXY : IndepFun X Y μ) (hX : Integrable X μ)
(hY : Integrable Y μ) : Integrable (X * Y) μ := by |
let nX : Ω → ENNReal := fun a => ‖X a‖₊
let nY : Ω → ENNReal := fun a => ‖Y a‖₊
have hXY' : IndepFun (fun a => ‖X a‖₊) (fun a => ‖Y a‖₊) μ :=
hXY.comp measurable_nnnorm measurable_nnnorm
have hXY'' : IndepFun nX nY μ :=
hXY'.comp measurable_coe_nnreal_ennreal measurable_coe_nnreal_ennreal
have hnX : ... |
import Mathlib.Logic.Encodable.Lattice
import Mathlib.MeasureTheory.MeasurableSpace.Defs
#align_import measure_theory.pi_system from "leanprover-community/mathlib"@"98e83c3d541c77cdb7da20d79611a780ff8e7d90"
open MeasurableSpace Set
open scoped Classical
open MeasureTheory
def IsPiSystem {α} (C : Set (Set α)) :... | Mathlib/MeasureTheory/PiSystem.lean | 270 | 274 | theorem generateFrom_generatePiSystem_eq {α} {g : Set (Set α)} :
generateFrom (generatePiSystem g) = generateFrom g := by |
apply le_antisymm <;> apply generateFrom_le
· exact fun t h_t => generateFrom_measurableSet_of_generatePiSystem t h_t
· exact fun t h_t => measurableSet_generateFrom (generatePiSystem.base h_t)
|
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Tree.Basic
import Mathlib.Logic.Basic
import Mathlib.Tactic.NormNum.Core
import Mathlib.Util.SynthesizeUsing
import Mathlib.Util.Qq
open Lean Parser Tactic Mathlib Meta NormNum Qq
initialize registerTraceClass `CancelDen... | Mathlib/Tactic/CancelDenoms/Core.lean | 105 | 109 | theorem cancel_factors_ne {α} [Field α] {a b ad bd a' b' gcd : α} (ha : ad * a = a')
(hb : bd * b = b') (had : ad ≠ 0) (hbd : bd ≠ 0) (hgcd : gcd ≠ 0) :
(a ≠ b) = (1 / gcd * (bd * a') ≠ 1 / gcd * (ad * b')) := by |
classical
rw [eq_iff_iff, not_iff_not, cancel_factors_eq ha hb had hbd hgcd]
|
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 | 90 | 98 | theorem pi_univ_Ioc_update_left {x y : ∀ i, α i} {i₀ : ι} {m : α i₀} (hm : x i₀ ≤ m) :
(pi univ fun i ↦ Ioc (update x i₀ m i) (y i)) =
{ z | m < z i₀ } ∩ pi univ fun i ↦ Ioc (x i) (y i) := by |
have : Ioc m (y i₀) = Ioi m ∩ Ioc (x i₀) (y i₀) := by
rw [← Ioi_inter_Iic, ← Ioi_inter_Iic, ← inter_assoc,
inter_eq_self_of_subset_left (Ioi_subset_Ioi hm)]
simp_rw [univ_pi_update i₀ _ _ fun i z ↦ Ioc z (y i), ← pi_inter_compl ({i₀} : Set ι),
singleton_pi', ← inter_assoc, this]
rfl
|
import Mathlib.Data.Matrix.Basis
import Mathlib.Data.Matrix.DMatrix
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.Reindex
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.matrix.transvection from "leanprover-community/mathlib"@"0e2aab2b0d521f060f62a14d2cf2e2c54e... | Mathlib/LinearAlgebra/Matrix/Transvection.lean | 289 | 295 | theorem mul_sumInl_toMatrix_prod [Fintype n] [Fintype p] (M : Matrix n n R)
(L : List (TransvectionStruct n R)) (N : Matrix p p R) :
fromBlocks M 0 0 N * (L.map (toMatrix ∘ sumInl p)).prod =
fromBlocks (M * (L.map toMatrix).prod) 0 0 N := by |
induction' L with t L IH generalizing M N
· simp
· simp [IH, toMatrix_sumInl, fromBlocks_multiply]
|
import Mathlib.Data.Nat.Squarefree
import Mathlib.NumberTheory.Zsqrtd.QuadraticReciprocity
import Mathlib.Tactic.LinearCombination
#align_import number_theory.sum_two_squares from "leanprover-community/mathlib"@"5b2fe80501ff327b9109fb09b7cc8c325cd0d7d9"
section NegOneSquare
-- This could be formulated for ... | Mathlib/NumberTheory/SumTwoSquares.lean | 108 | 120 | theorem ZMod.isSquare_neg_one_iff {n : ℕ} (hn : Squarefree n) :
IsSquare (-1 : ZMod n) ↔ ∀ {q : ℕ}, q.Prime → q ∣ n → q % 4 ≠ 3 := by |
refine ⟨fun H q hqp hqd => hqp.mod_four_ne_three_of_dvd_isSquare_neg_one hqd H, fun H => ?_⟩
induction' n using induction_on_primes with p n hpp ih
· exact False.elim (hn.ne_zero rfl)
· exact ⟨0, by simp only [mul_zero, eq_iff_true_of_subsingleton]⟩
· haveI : Fact p.Prime := ⟨hpp⟩
have hcp : p.Coprime n ... |
import Mathlib.Algebra.Module.Zlattice.Basic
import Mathlib.NumberTheory.NumberField.Embeddings
import Mathlib.NumberTheory.NumberField.FractionalIdeal
#align_import number_theory.number_field.canonical_embedding from "leanprover-community/mathlib"@"60da01b41bbe4206f05d34fd70c8dd7498717a30"
variable (K : Type*) [F... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean | 281 | 284 | theorem normAtPlace_real (w : InfinitePlace K) (c : ℝ) :
normAtPlace w ((fun _ ↦ c, fun _ ↦ c) : (E K)) = |c| := by |
rw [show ((fun _ ↦ c, fun _ ↦ c) : (E K)) = c • 1 by ext <;> simp, normAtPlace_smul, map_one,
mul_one]
|
import Mathlib.Data.Set.Image
#align_import order.directed from "leanprover-community/mathlib"@"ffde2d8a6e689149e44fd95fa862c23a57f8c780"
open Function
universe u v w
variable {α : Type u} {β : Type v} {ι : Sort w} (r r' s : α → α → Prop)
local infixl:50 " ≼ " => r
def Directed (f : ι → α) :=
∀ x y, ∃ z, ... | Mathlib/Order/Directed.lean | 116 | 128 | theorem Directed.extend_bot [Preorder α] [OrderBot α] {e : ι → β} {f : ι → α}
(hf : Directed (· ≤ ·) f) (he : Function.Injective e) :
Directed (· ≤ ·) (Function.extend e f ⊥) := by |
intro a b
rcases (em (∃ i, e i = a)).symm with (ha | ⟨i, rfl⟩)
· use b
simp [Function.extend_apply' _ _ _ ha]
rcases (em (∃ i, e i = b)).symm with (hb | ⟨j, rfl⟩)
· use e i
simp [Function.extend_apply' _ _ _ hb]
rcases hf i j with ⟨k, hi, hj⟩
use e k
simp only [he.extend_apply, *, true_and_iff]... |
import Mathlib.MeasureTheory.Function.ConvergenceInMeasure
import Mathlib.MeasureTheory.Function.L1Space
#align_import measure_theory.function.uniform_integrable from "leanprover-community/mathlib"@"57ac39bd365c2f80589a700f9fbb664d3a1a30c2"
noncomputable section
open scoped Classical MeasureTheory NNReal ENNReal... | Mathlib/MeasureTheory/Function/UniformIntegrable.lean | 891 | 904 | theorem UniformIntegrable.spec (hp : p ≠ 0) (hp' : p ≠ ∞) (hfu : UniformIntegrable f p μ) {ε : ℝ}
(hε : 0 < ε) :
∃ C : ℝ≥0, ∀ i, snorm ({ x | C ≤ ‖f i x‖₊ }.indicator (f i)) p μ ≤ ENNReal.ofReal ε := by |
set g : ι → α → β := fun i => (hfu.1 i).choose
have hgmeas : ∀ i, StronglyMeasurable (g i) := fun i => (Exists.choose_spec <| hfu.1 i).1
have hgunif : UniformIntegrable g p μ := hfu.ae_eq fun i => (Exists.choose_spec <| hfu.1 i).2
obtain ⟨C, hC⟩ := hgunif.spec' hp hp' hgmeas hε
refine ⟨C, fun i => le_trans (... |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Finset.NatAntidiagonal
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Logic.Function.Iterate
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Zify
#align_import data.nat.fib from "leanprover-community/mathlib"@"... | Mathlib/Data/Nat/Fib/Basic.lean | 87 | 88 | theorem fib_add_two {n : ℕ} : fib (n + 2) = fib n + fib (n + 1) := by |
simp [fib, Function.iterate_succ_apply']
|
import Mathlib.Order.Interval.Set.OrdConnected
import Mathlib.Data.Set.Lattice
#align_import data.set.intervals.ord_connected_component from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
open Interval Function OrderDual
namespace Set
variable {α : Type*} [LinearOrder α] {s t : Set α}... | Mathlib/Order/Interval/Set/OrdConnectedComponent.lean | 127 | 133 | theorem ordConnectedProj_eq {x y : s} :
ordConnectedProj s x = ordConnectedProj s y ↔ [[(x : α), y]] ⊆ s := by |
constructor <;> intro h
· rw [← mem_ordConnectedComponent, ← ordConnectedComponent_ordConnectedProj, h,
ordConnectedComponent_ordConnectedProj, self_mem_ordConnectedComponent]
exact y.2
· simp only [ordConnectedProj, ordConnectedComponent_eq h]
|
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Data.Finite.Card
import Mathlib.GroupTheory.Finiteness
import Mathlib.GroupTheory.GroupAction.Quotient
#align_import group_theory.index from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
namespace Subgroup
open Ca... | Mathlib/GroupTheory/Index.lean | 435 | 440 | theorem relindex_inf_le : (H ⊓ K).relindex L ≤ H.relindex L * K.relindex L := by |
by_cases h : H.relindex L = 0
· exact (le_of_eq (relindex_eq_zero_of_le_left inf_le_left h)).trans (zero_le _)
rw [← inf_relindex_right, inf_assoc, ← relindex_mul_relindex _ _ L inf_le_right inf_le_right,
inf_relindex_right, inf_relindex_right]
exact mul_le_mul_right' (relindex_le_of_le_right inf_le_right ... |
import Mathlib.Algebra.Module.Equiv
import Mathlib.Algebra.Module.Hom
import Mathlib.Algebra.Module.Prod
import Mathlib.Algebra.Module.Submodule.Range
import Mathlib.Data.Set.Finite
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Tactic.Abel
#align_import linear_algebra.basic from "leanprover-c... | Mathlib/LinearAlgebra/Basic.lean | 295 | 297 | theorem ofBijective_symm_apply_apply [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {h} (x : M) :
(ofBijective f h).symm (f x) = x := by |
simp [LinearEquiv.symm_apply_eq]
|
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Group.Int
import Mathlib.Data.Nat.Dist
import Mathlib.Data.Ordmap.Ordnode
import Mathlib.Tactic.Abel
import Mathlib.Tactic.Linarith
#align_import data.ordmap.ordset from "leanprover-community/mathlib"@"47b51515e69f59bca5cf34ef456e6000fe205a69"
variable... | Mathlib/Data/Ordmap/Ordset.lean | 1,404 | 1,409 | theorem Valid'.eraseMin_aux {s l} {x : α} {r o₁ o₂} (H : Valid' o₁ (.node s l x r) o₂) :
Valid' ↑(findMin' l x) (@eraseMin α (.node' l x r)) o₂ ∧
size (.node' l x r) = size (eraseMin (.node' l x r)) + 1 := by |
have := H.dual.eraseMax_aux
rwa [← dual_node', size_dual, ← dual_eraseMin, size_dual, ← Valid'.dual_iff, findMax'_dual]
at this
|
import Mathlib.MeasureTheory.Group.Measure
import Mathlib.MeasureTheory.Integral.IntegrableOn
import Mathlib.MeasureTheory.Function.LocallyIntegrable
open Asymptotics MeasureTheory Set Filter
variable {α E F : Type*} [MeasurableSpace α] [NormedAddCommGroup E] [NormedAddCommGroup F]
{f : α → E} {g : α → F} {a b :... | Mathlib/MeasureTheory/Integral/Asymptotics.lean | 89 | 93 | theorem LocallyIntegrableOn.integrableOn_of_isBigO_atTop [IsMeasurablyGenerated (atTop (α := α))]
(hf : LocallyIntegrableOn f (Ici a) μ) (ho : f =O[atTop] g)
(hg : IntegrableAtFilter g atTop μ) : IntegrableOn f (Ici a) μ := by |
refine integrableOn_Ici_iff_integrableAtFilter_atTop.mpr ⟨ho.integrableAtFilter ?_ hg, hf⟩
exact ⟨Ici a, Ici_mem_atTop a, hf.aestronglyMeasurable⟩
|
import Mathlib.Topology.Algebra.Module.WeakDual
import Mathlib.MeasureTheory.Integral.BoundedContinuousFunction
import Mathlib.MeasureTheory.Measure.HasOuterApproxClosed
#align_import measure_theory.measure.finite_measure from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable... | Mathlib/MeasureTheory/Measure/FiniteMeasure.lean | 389 | 393 | theorem testAgainstNN_add (μ : FiniteMeasure Ω) (f₁ f₂ : Ω →ᵇ ℝ≥0) :
μ.testAgainstNN (f₁ + f₂) = μ.testAgainstNN f₁ + μ.testAgainstNN f₂ := by |
simp only [← ENNReal.coe_inj, BoundedContinuousFunction.coe_add, ENNReal.coe_add, Pi.add_apply,
testAgainstNN_coe_eq]
exact lintegral_add_left (BoundedContinuousFunction.measurable_coe_ennreal_comp _) _
|
import Mathlib.Order.Interval.Multiset
#align_import data.nat.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
-- TODO
-- assert_not_exists Ring
open Finset Nat
variable (a b c : ℕ)
namespace Nat
instance instLocallyFiniteOrder : LocallyFiniteOrder ℕ where
finsetIcc a b... | Mathlib/Order/Interval/Finset/Nat.lean | 138 | 139 | theorem card_fintypeIic : Fintype.card (Set.Iic b) = b + 1 := by |
rw [Fintype.card_ofFinset, card_Iic]
|
import Mathlib.MeasureTheory.Function.LpOrder
#align_import measure_theory.function.l1_space from "leanprover-community/mathlib"@"ccdbfb6e5614667af5aa3ab2d50885e0ef44a46f"
noncomputable section
open scoped Classical
open Topology ENNReal MeasureTheory NNReal
open Set Filter TopologicalSpace ENNReal EMetric Meas... | Mathlib/MeasureTheory/Function/L1Space.lean | 128 | 130 | theorem hasFiniteIntegral_iff_ofNNReal {f : α → ℝ≥0} :
HasFiniteIntegral (fun x => (f x : ℝ)) μ ↔ (∫⁻ a, f a ∂μ) < ∞ := by |
simp [hasFiniteIntegral_iff_norm]
|
import Mathlib.Data.Finset.Lattice
import Mathlib.Data.Set.Sigma
#align_import data.finset.sigma from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function Multiset
variable {ι : Type*}
namespace Finset
section SigmaLift
variable {α β γ : ι → Type*} [DecidableEq ι]
def sigm... | Mathlib/Data/Finset/Sigma.lean | 184 | 187 | theorem not_mem_sigmaLift_of_ne_left (f : ∀ ⦃i⦄, α i → β i → Finset (γ i)) (a : Sigma α)
(b : Sigma β) (x : Sigma γ) (h : a.1 ≠ x.1) : x ∉ sigmaLift f a b := by |
rw [mem_sigmaLift]
exact fun H => h H.fst
|
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.DominatedConvergence
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
noncomputable s... | Mathlib/MeasureTheory/Constructions/Prod/Integral.lean | 158 | 167 | theorem integrable_measure_prod_mk_left {s : Set (α × β)} (hs : MeasurableSet s)
(h2s : (μ.prod ν) s ≠ ∞) : Integrable (fun x => (ν (Prod.mk x ⁻¹' s)).toReal) μ := by |
refine ⟨(measurable_measure_prod_mk_left hs).ennreal_toReal.aemeasurable.aestronglyMeasurable, ?_⟩
simp_rw [HasFiniteIntegral, ennnorm_eq_ofReal toReal_nonneg]
convert h2s.lt_top using 1
-- Porting note: was `simp_rw`
rw [prod_apply hs]
apply lintegral_congr_ae
filter_upwards [ae_measure_lt_top hs h2s] w... |
import Mathlib.Data.List.Basic
#align_import data.list.join from "leanprover-community/mathlib"@"18a5306c091183ac90884daa9373fa3b178e8607"
-- Make sure we don't import algebra
assert_not_exists Monoid
variable {α β : Type*}
namespace List
attribute [simp] join
-- Porting note (#10618): simp can prove this
-- @... | Mathlib/Data/List/Join.lean | 204 | 206 | theorem join_reverse (L : List (List α)) :
L.reverse.join = (L.map reverse).join.reverse := by |
simpa [reverse_reverse, map_reverse] using congr_arg List.reverse (reverse_join L.reverse)
|
import Mathlib.Algebra.ContinuedFractions.Computation.Approximations
import Mathlib.Algebra.ContinuedFractions.Computation.CorrectnessTerminating
import Mathlib.Data.Rat.Floor
#align_import algebra.continued_fractions.computation.terminates_iff_rat from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b3... | Mathlib/Algebra/ContinuedFractions/Computation/TerminatesIffRat.lean | 119 | 123 | theorem exists_rat_eq_nth_convergent : ∃ q : ℚ, (of v).convergents n = (q : K) := by |
rcases exists_rat_eq_nth_numerator v n with ⟨Aₙ, nth_num_eq⟩
rcases exists_rat_eq_nth_denominator v n with ⟨Bₙ, nth_denom_eq⟩
use Aₙ / Bₙ
simp [nth_num_eq, nth_denom_eq, convergent_eq_num_div_denom]
|
import Mathlib.Algebra.IsPrimePow
import Mathlib.NumberTheory.ArithmeticFunction
import Mathlib.Analysis.SpecialFunctions.Log.Basic
#align_import number_theory.von_mangoldt from "leanprover-community/mathlib"@"c946d6097a6925ad16d7ec55677bbc977f9846de"
namespace ArithmeticFunction
open Finset Nat
open scoped Arit... | Mathlib/NumberTheory/VonMangoldt.lean | 83 | 87 | theorem vonMangoldt_nonneg {n : ℕ} : 0 ≤ Λ n := by |
rw [vonMangoldt_apply]
split_ifs
· exact Real.log_nonneg (one_le_cast.2 (Nat.minFac_pos n))
rfl
|
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Data.Real.Sqrt
import Mathlib.RingTheory.Ideal.QuotientOperations
import Mathlib.Topology.MetricSpace.HausdorffDistance
#align_import analysis.normed.group.quotient from "leanprover-community/mathlib"@"2196ab363eb097c008... | Mathlib/Analysis/Normed/Group/Quotient.lean | 200 | 206 | theorem norm_mk_lt' (S : AddSubgroup M) (m : M) {ε : ℝ} (hε : 0 < ε) :
∃ s ∈ S, ‖m + s‖ < ‖mk' S m‖ + ε := by |
obtain ⟨n : M, hn : mk' S n = mk' S m, hn' : ‖n‖ < ‖mk' S m‖ + ε⟩ :=
norm_mk_lt (QuotientAddGroup.mk' S m) hε
erw [eq_comm, QuotientAddGroup.eq] at hn
use -m + n, hn
rwa [add_neg_cancel_left]
|
import Mathlib.Algebra.MonoidAlgebra.Division
import Mathlib.Algebra.MvPolynomial.Basic
#align_import data.mv_polynomial.division from "leanprover-community/mathlib"@"72c366d0475675f1309d3027d3d7d47ee4423951"
variable {σ R : Type*} [CommSemiring R]
namespace MvPolynomial
theorem monomial_dvd_monomial {r s : ... | Mathlib/Algebra/MvPolynomial/Division.lean | 244 | 247 | theorem monomial_one_dvd_monomial_one [Nontrivial R] {i j : σ →₀ ℕ} :
monomial i (1 : R) ∣ monomial j 1 ↔ i ≤ j := by |
rw [monomial_dvd_monomial]
simp_rw [one_ne_zero, false_or_iff, dvd_rfl, and_true_iff]
|
import Mathlib.Algebra.Group.Equiv.Basic
import Mathlib.Data.ENat.Lattice
import Mathlib.Data.Part
import Mathlib.Tactic.NormNum
#align_import data.nat.part_enat from "leanprover-community/mathlib"@"3ff3f2d6a3118b8711063de7111a0d77a53219a8"
open Part hiding some
def PartENat : Type :=
Part ℕ
#align part_enat ... | Mathlib/Data/Nat/PartENat.lean | 405 | 406 | theorem ne_top_iff_dom {x : PartENat} : x ≠ ⊤ ↔ x.Dom := by |
classical exact not_iff_comm.1 Part.eq_none_iff'.symm
|
import Mathlib.MeasureTheory.Integral.SetToL1
#align_import measure_theory.integral.bochner from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc9524185ae994553ebf4"
assert_not_exists Differentiable
noncomputable section
open scoped Topology NNReal ENNReal MeasureTheory
open Set Filter TopologicalSpace EN... | Mathlib/MeasureTheory/Integral/Bochner.lean | 249 | 251 | theorem weightedSMul_nonneg (s : Set α) (x : ℝ) (hx : 0 ≤ x) : 0 ≤ weightedSMul μ s x := by |
simp only [weightedSMul, Algebra.id.smul_eq_mul, coe_smul', _root_.id, coe_id', Pi.smul_apply]
exact mul_nonneg toReal_nonneg hx
|
import Mathlib.Data.Finset.Card
#align_import data.finset.prod from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
assert_not_exists MonoidWithZero
open Multiset
variable {α β γ : Type*}
namespace Finset
section Prod
variable {s s' : Finset α} {t t' : Finset β} {a : α} {b : β}
... | Mathlib/Data/Finset/Prod.lean | 169 | 185 | theorem filter_product_card (s : Finset α) (t : Finset β) (p : α → Prop) (q : β → Prop)
[DecidablePred p] [DecidablePred q] :
((s ×ˢ t).filter fun x : α × β => (p x.1) = (q x.2)).card =
(s.filter p).card * (t.filter q).card +
(s.filter (¬ p ·)).card * (t.filter (¬ q ·)).card := by |
classical
rw [← card_product, ← card_product, ← filter_product, ← filter_product, ← card_union_of_disjoint]
· apply congr_arg
ext ⟨a, b⟩
simp only [filter_union_right, mem_filter, mem_product]
constructor <;> intro h <;> use h.1
· simp only [h.2, Function.comp_apply, Decidable.em, and_self]
·... |
import Mathlib.Algebra.Associated
import Mathlib.Algebra.Star.Unitary
import Mathlib.RingTheory.Int.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.Tactic.Ring
#align_import number_theory.zsqrtd.basic from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
@[ext]
struct... | Mathlib/NumberTheory/Zsqrtd/Basic.lean | 315 | 315 | theorem muld_val (x y : ℤ) : sqrtd (d := d) * ⟨x, y⟩ = ⟨d * y, x⟩ := by | ext <;> simp
|
import Mathlib.Algebra.Polynomial.Module.Basic
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv.Defs
import Mathlib.Analysis.Calculus.MeanValue
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
open scoped... | Mathlib/Analysis/Calculus/Taylor.lean | 97 | 102 | theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) :
taylorWithinEval f 0 s x₀ x = f x₀ := by |
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
|
import Mathlib.MeasureTheory.Measure.Regular
import Mathlib.MeasureTheory.Function.SimpleFuncDenseLp
import Mathlib.Topology.UrysohnsLemma
import Mathlib.MeasureTheory.Integral.Bochner
#align_import measure_theory.function.continuous_map_dense from "leanprover-community/mathlib"@"e0736bb5b48bdadbca19dbd857e12bee38ccf... | Mathlib/MeasureTheory/Function/ContinuousMapDense.lean | 226 | 233 | theorem Integrable.exists_hasCompactSupport_integral_sub_le
[WeaklyLocallyCompactSpace α] [μ.Regular]
{f : α → E} (hf : Integrable f μ) {ε : ℝ} (hε : 0 < ε) :
∃ g : α → E, HasCompactSupport g ∧ (∫ x, ‖f x - g x‖ ∂μ) ≤ ε ∧
Continuous g ∧ Integrable g μ := by |
simp only [← memℒp_one_iff_integrable, ← snorm_one_eq_lintegral_nnnorm, ← ENNReal.ofReal_one]
at hf ⊢
simpa using hf.exists_hasCompactSupport_integral_rpow_sub_le zero_lt_one hε
|
import Mathlib.Algebra.Homology.HomologicalComplex
import Mathlib.CategoryTheory.DifferentialObject
#align_import algebra.homology.differential_object from "leanprover-community/mathlib"@"b535c2d5d996acd9b0554b76395d9c920e186f4f"
open CategoryTheory CategoryTheory.Limits
open scoped Classical
noncomputable secti... | Mathlib/Algebra/Homology/DifferentialObject.lean | 53 | 54 | theorem objEqToHom_d {x y : β} (h : x = y) :
X.objEqToHom h ≫ X.d y = X.d x ≫ X.objEqToHom (by cases h; rfl) := by | cases h; dsimp; simp
|
import Mathlib.Data.Real.Sqrt
import Mathlib.Analysis.NormedSpace.Star.Basic
import Mathlib.Analysis.NormedSpace.ContinuousLinearMap
import Mathlib.Analysis.NormedSpace.Basic
#align_import data.is_R_or_C.basic from "leanprover-community/mathlib"@"baa88307f3e699fa7054ef04ec79fa4f056169cb"
section
local notation "�... | Mathlib/Analysis/RCLike/Basic.lean | 369 | 371 | theorem conj_smul (r : ℝ) (z : K) : conj (r • z) = r • conj z := by |
rw [conj_eq_re_sub_im, conj_eq_re_sub_im, smul_re, smul_im, ofReal_mul, ofReal_mul,
real_smul_eq_coe_mul r (_ - _), mul_sub, mul_assoc]
|
import Mathlib.Topology.Separation
import Mathlib.Topology.UniformSpace.Basic
import Mathlib.Topology.UniformSpace.Cauchy
#align_import topology.uniform_space.uniform_convergence from "leanprover-community/mathlib"@"2705404e701abc6b3127da906f40bae062a169c9"
noncomputable section
open Topology Uniformity Filter S... | Mathlib/Topology/UniformSpace/UniformConvergence.lean | 349 | 352 | theorem tendstoUniformlyOn_singleton_iff_tendsto :
TendstoUniformlyOn F f p {x} ↔ Tendsto (fun n : ι => F n x) p (𝓝 (f x)) := by |
simp_rw [tendstoUniformlyOn_iff_tendsto, Uniform.tendsto_nhds_right, tendsto_def]
exact forall₂_congr fun u _ => by simp [mem_prod_principal, preimage]
|
import Mathlib.Topology.Maps
import Mathlib.Topology.NhdsSet
#align_import topology.constructions from "leanprover-community/mathlib"@"f7ebde7ee0d1505dfccac8644ae12371aa3c1c9f"
noncomputable section
open scoped Classical
open Topology TopologicalSpace Set Filter Function
universe u v
variable {X : Type u} {Y :... | Mathlib/Topology/Constructions.lean | 1,727 | 1,729 | theorem ULift.isOpen_iff [TopologicalSpace X] {s : Set (ULift.{v} X)} :
IsOpen s ↔ IsOpen (ULift.up ⁻¹' s) := by |
rw [ULift.topologicalSpace, ← Equiv.ulift_apply, ← Equiv.ulift.coinduced_symm, ← isOpen_coinduced]
|
import Mathlib.MeasureTheory.Covering.VitaliFamily
import Mathlib.MeasureTheory.Measure.Regular
import Mathlib.MeasureTheory.Function.AEMeasurableOrder
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.MeasureTheory.Integral.Average
import Mathlib.MeasureTheory.Decomposition.Lebesgue
#align_import measure... | Mathlib/MeasureTheory/Covering/Differentiation.lean | 211 | 228 | theorem null_of_frequently_le_of_frequently_ge {c d : ℝ≥0} (hcd : c < d) (s : Set α)
(hc : ∀ x ∈ s, ∃ᶠ a in v.filterAt x, ρ a ≤ c * μ a)
(hd : ∀ x ∈ s, ∃ᶠ a in v.filterAt x, (d : ℝ≥0∞) * μ a ≤ ρ a) : μ s = 0 := by |
apply measure_null_of_locally_null s fun x _ => ?_
obtain ⟨o, xo, o_open, μo⟩ : ∃ o : Set α, x ∈ o ∧ IsOpen o ∧ μ o < ∞ :=
Measure.exists_isOpen_measure_lt_top μ x
refine ⟨s ∩ o, inter_mem_nhdsWithin _ (o_open.mem_nhds xo), ?_⟩
let s' := s ∩ o
by_contra h
apply lt_irrefl (ρ s')
calc
ρ s' ≤ c * μ ... |
import Batteries.Control.ForInStep.Lemmas
import Batteries.Data.List.Basic
import Batteries.Tactic.Init
import Batteries.Tactic.Alias
namespace List
open Nat
@[simp] theorem mem_toArray {a : α} {l : List α} : a ∈ l.toArray ↔ a ∈ l := by
simp [Array.mem_def]
@[simp]
theorem drop_one : ∀ l : List α, drop 1 l =... | .lake/packages/batteries/Batteries/Data/List/Lemmas.lean | 539 | 544 | theorem erase_eq_eraseP' (a : α) (l : List α) : l.erase a = l.eraseP (· == a) := by |
induction l
· simp
· next b t ih =>
rw [erase_cons, eraseP_cons, ih]
if h : b == a then simp [h] else simp [h]
|
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Localization.Ideal
import Mathlib.RingTheory.Noetherian
#align_import ring_theory.localization.submodule from "leanprover-community/mathlib"@"1ebb20602a8caef435ce47f6373e1aa40851a177"
variable {R : Type*} [CommRing R] (M : Submonoid R) ... | Mathlib/RingTheory/Localization/Submodule.lean | 165 | 175 | theorem mem_span_map {x : S} {a : Set R} :
x ∈ Ideal.span (algebraMap R S '' a) ↔ ∃ y ∈ Ideal.span a, ∃ z : M, x = mk' S y z := by |
refine (mem_span_iff M).trans ?_
constructor
· rw [← coeSubmodule_span]
rintro ⟨_, ⟨y, hy, rfl⟩, z, hz⟩
refine ⟨y, hy, z, ?_⟩
rw [hz, Algebra.linearMap_apply, smul_eq_mul, mul_comm, mul_mk'_eq_mk'_of_mul, mul_one]
· rintro ⟨y, hy, z, hz⟩
refine ⟨algebraMap R S y, Submodule.map_mem_span_algebraM... |
import Mathlib.Topology.ContinuousOn
import Mathlib.Order.Minimal
open Set Classical
variable {X : Type*} {Y : Type*} [TopologicalSpace X] [TopologicalSpace Y] {s t : Set X}
section Preirreducible
def IsPreirreducible (s : Set X) : Prop :=
∀ u v : Set X, IsOpen u → IsOpen v → (s ∩ u).Nonempty → (s ∩ v).Nonempt... | Mathlib/Topology/Irreducible.lean | 226 | 230 | theorem Subtype.preirreducibleSpace (h : IsPreirreducible s) : PreirreducibleSpace s where
isPreirreducible_univ := by |
rintro _ _ ⟨u, hu, rfl⟩ ⟨v, hv, rfl⟩ ⟨⟨x, hxs⟩, -, hxu⟩ ⟨⟨y, hys⟩, -, hyv⟩
rcases h u v hu hv ⟨x, hxs, hxu⟩ ⟨y, hys, hyv⟩ with ⟨x, hxs, ⟨hxu, hxv⟩⟩
exact ⟨⟨x, hxs⟩, ⟨Set.mem_univ _, ⟨hxu, hxv⟩⟩⟩
|
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Analysis.Convex.Hull
import Mathlib.LinearAlgebra.AffineSpace.Basis
#align_import analysis.convex.combination from "leanprover-community/mathlib"@"92bd7b1ffeb306a89f450bee126ddd8a284c259d"
open Set Function
open scoped Classical
open Pointwise
... | Mathlib/Analysis/Convex/Combination.lean | 216 | 232 | theorem convex_iff_sum_mem : Convex R s ↔ ∀ (t : Finset E) (w : E → R),
(∀ i ∈ t, 0 ≤ w i) → ∑ i ∈ t, w i = 1 → (∀ x ∈ t, x ∈ s) → (∑ x ∈ t, w x • x) ∈ s := by |
refine ⟨fun hs t w hw₀ hw₁ hts => hs.sum_mem hw₀ hw₁ hts, ?_⟩
intro h x hx y hy a b ha hb hab
by_cases h_cases : x = y
· rw [h_cases, ← add_smul, hab, one_smul]
exact hy
· convert h {x, y} (fun z => if z = y then b else a) _ _ _
-- Porting note: Original proof had 2 `simp_intro i hi`
· simp only ... |
import Mathlib.Algebra.Star.Basic
import Mathlib.Algebra.FreeAlgebra
#align_import algebra.star.free from "leanprover-community/mathlib"@"07c3cf2d851866ff7198219ed3fedf42e901f25c"
namespace FreeAlgebra
variable {R : Type*} [CommSemiring R] {X : Type*}
instance : StarRing (FreeAlgebra R X) where
star := MulOp... | Mathlib/Algebra/Star/Free.lean | 68 | 68 | theorem star_ι (x : X) : star (ι R x) = ι R x := by | simp [star, Star.star]
|
import Mathlib.MeasureTheory.Integral.Lebesgue
#align_import measure_theory.measure.giry_monad from "leanprover-community/mathlib"@"56f4cd1ef396e9fd389b5d8371ee9ad91d163625"
noncomputable section
open scoped Classical
open ENNReal
open scoped Classical
open Set Filter
variable {α β : Type*}
namespace MeasureT... | Mathlib/MeasureTheory/Measure/GiryMonad.lean | 91 | 96 | theorem measurable_lintegral {f : α → ℝ≥0∞} (hf : Measurable f) :
Measurable fun μ : Measure α => ∫⁻ x, f x ∂μ := by |
simp only [lintegral_eq_iSup_eapprox_lintegral, hf, SimpleFunc.lintegral]
refine measurable_iSup fun n => Finset.measurable_sum _ fun i _ => ?_
refine Measurable.const_mul ?_ _
exact measurable_coe ((SimpleFunc.eapprox f n).measurableSet_preimage _)
|
import Batteries.Tactic.Lint.Basic
import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Order.ZeroLEOne
import Mathlib.Data.Nat.Cast.Order
import Mathlib.Init.Data.Int.Order
set_option autoImplicit true
namespace Linarith
theorem lt_irrefl {α : Type u} ... | Mathlib/Tactic/Linarith/Lemmas.lean | 36 | 37 | theorem le_of_le_of_eq {α} [OrderedSemiring α] {a b : α} (ha : a ≤ 0) (hb : b = 0) : a + b ≤ 0 := by |
simp [*]
|
import Mathlib.RingTheory.Polynomial.Basic
import Mathlib.RingTheory.Ideal.LocalRing
#align_import data.polynomial.expand from "leanprover-community/mathlib"@"bbeb185db4ccee8ed07dc48449414ebfa39cb821"
universe u v w
open Polynomial
open Finset
namespace Polynomial
section CommSemiring
variable (R : Type u) [... | Mathlib/Algebra/Polynomial/Expand.lean | 153 | 172 | theorem natDegree_expand (p : ℕ) (f : R[X]) : (expand R p f).natDegree = f.natDegree * p := by |
rcases p.eq_zero_or_pos with hp | hp
· rw [hp, coe_expand, pow_zero, mul_zero, ← C_1, eval₂_hom, natDegree_C]
by_cases hf : f = 0
· rw [hf, AlgHom.map_zero, natDegree_zero, zero_mul]
have hf1 : expand R p f ≠ 0 := mt (expand_eq_zero hp).1 hf
rw [← WithBot.coe_eq_coe]
convert (degree_eq_natDegree hf1).sym... |
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 | 237 | 244 | theorem ContinuousOn.tendstoUniformly [LocallyCompactSpace α] [CompactSpace β] [UniformSpace γ]
{f : α → β → γ} {x : α} {U : Set α} (hxU : U ∈ 𝓝 x) (h : ContinuousOn (↿f) (U ×ˢ univ)) :
TendstoUniformly f (f x) (𝓝 x) := by |
rcases LocallyCompactSpace.local_compact_nhds _ _ hxU with ⟨K, hxK, hKU, hK⟩
have : UniformContinuousOn (↿f) (K ×ˢ univ) :=
IsCompact.uniformContinuousOn_of_continuous (hK.prod isCompact_univ)
(h.mono <| prod_mono hKU Subset.rfl)
exact this.tendstoUniformly hxK
|
import Mathlib.Algebra.BigOperators.Group.List
import Mathlib.Algebra.Group.Prod
import Mathlib.Data.Multiset.Basic
#align_import algebra.big_operators.multiset.basic from "leanprover-community/mathlib"@"6c5f73fd6f6cc83122788a80a27cdd54663609f4"
assert_not_exists MonoidWithZero
variable {F ι α β γ : Type*}
names... | Mathlib/Algebra/BigOperators/Group/Multiset.lean | 105 | 106 | theorem prod_pair (a b : α) : ({a, b} : Multiset α).prod = a * b := by |
rw [insert_eq_cons, prod_cons, prod_singleton]
|
import Mathlib.Topology.Bases
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.Compactness.SigmaCompact
open Set Filter Topology TopologicalSpace
universe u v
variable {X : Type u} {Y : Type v} {ι : Type*}
variable [TopologicalSpace X] [TopologicalSpace Y] {s t : Set X}
section Lindelof
def I... | Mathlib/Topology/Compactness/Lindelof.lean | 98 | 110 | theorem IsLindelof.image_of_continuousOn {f : X → Y} (hs : IsLindelof s) (hf : ContinuousOn f s) :
IsLindelof (f '' s) := by |
intro l lne _ ls
have : NeBot (l.comap f ⊓ 𝓟 s) :=
comap_inf_principal_neBot_of_image_mem lne (le_principal_iff.1 ls)
obtain ⟨x, hxs, hx⟩ : ∃ x ∈ s, ClusterPt x (l.comap f ⊓ 𝓟 s) := @hs _ this _ inf_le_right
haveI := hx.neBot
use f x, mem_image_of_mem f hxs
have : Tendsto f (𝓝 x ⊓ (comap f l ⊓ 𝓟 s)... |
import Mathlib.CategoryTheory.Adjunction.Unique
import Mathlib.CategoryTheory.Adjunction.FullyFaithful
import Mathlib.CategoryTheory.Sites.Sheaf
import Mathlib.CategoryTheory.Limits.Preserves.Finite
universe v₁ v₂ u₁ u₂
namespace CategoryTheory
open Limits
variable {C : Type u₁} [Category.{v₁} C] (J : Grothendiec... | Mathlib/CategoryTheory/Sites/Sheafification.lean | 100 | 102 | theorem sheafifyMap_comp {P Q R : Cᵒᵖ ⥤ D} (η : P ⟶ Q) (γ : Q ⟶ R) :
sheafifyMap J (η ≫ γ) = sheafifyMap J η ≫ sheafifyMap J γ := by |
simp [sheafifyMap, sheafify]
|
import Mathlib.Analysis.InnerProductSpace.TwoDim
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
#align_import geometry.euclidean.angle.oriented.basic from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
noncomputable section
open FiniteDimensional Complex
open scoped Real Rea... | Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | 1,039 | 1,042 | theorem oangle_sign_smul_add_smul_left (x y : V) (r₁ r₂ : ℝ) :
(o.oangle (r₁ • x + r₂ • y) y).sign = SignType.sign r₁ * (o.oangle x y).sign := by |
simp_rw [o.oangle_rev y, Real.Angle.sign_neg, add_comm (r₁ • x), oangle_sign_smul_add_smul_right,
mul_neg]
|
import Mathlib.Algebra.Group.Defs
#align_import algebra.invertible from "leanprover-community/mathlib"@"722b3b152ddd5e0cf21c0a29787c76596cb6b422"
assert_not_exists MonoidWithZero
assert_not_exists DenselyOrdered
universe u
variable {α : Type u}
class Invertible [Mul α] [One α] (a : α) : Type u where
invOf... | Mathlib/Algebra/Group/Invertible/Defs.lean | 259 | 260 | theorem mul_invOf_eq_iff_eq_mul_right : a * ⅟c = b ↔ a = b * c := by |
rw [← mul_right_inj_of_invertible (c := c), mul_invOf_mul_self_cancel]
|
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 | 47 | 53 | theorem getD_map {n : ℕ} (f : α → β) : (map f l).getD n (f d) = f (l.getD n d) := by |
induction l generalizing n with
| nil => rfl
| cons head tail ih =>
cases n
· rfl
· simp [ih]
|
import Mathlib.Data.Finset.Lattice
import Mathlib.Data.Set.Sigma
#align_import data.finset.sigma from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function Multiset
variable {ι : Type*}
namespace Finset
section Sigma
variable {α : ι → Type*} {β : Type*} (s s₁ s₂ : Finset ι) (... | Mathlib/Data/Finset/Sigma.lean | 99 | 104 | theorem sup_sigma [SemilatticeSup β] [OrderBot β] :
(s.sigma t).sup f = s.sup fun i => (t i).sup fun b => f ⟨i, b⟩ := by |
simp only [le_antisymm_iff, Finset.sup_le_iff, mem_sigma, and_imp, Sigma.forall]
exact
⟨fun i a hi ha => (le_sup hi).trans' <| le_sup (f := fun a => f ⟨i, a⟩) ha, fun i hi a ha =>
le_sup <| mem_sigma.2 ⟨hi, ha⟩⟩
|
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Data.List.Rotate
import Mathlib.GroupTheory.Perm.Support
#align_import group_theory.perm.list from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace List
variable {α β : Type*}
section FormPerm
variable [DecidableEq α] (l :... | Mathlib/GroupTheory/Perm/List.lean | 100 | 103 | theorem support_formPerm_le [Fintype α] : support (formPerm l) ≤ l.toFinset := by |
intro x hx
have hx' : x ∈ { x | formPerm l x ≠ x } := by simpa using hx
simpa using support_formPerm_le' _ hx'
|
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 | 159 | 167 | theorem zmod_congr_of_sub_mem_max_ideal (x : ℤ_[p]) (m n : ℕ) (hm : x - m ∈ maximalIdeal ℤ_[p])
(hn : x - n ∈ maximalIdeal ℤ_[p]) : (m : ZMod p) = n := by |
rw [maximalIdeal_eq_span_p] at hm hn
have := zmod_congr_of_sub_mem_span_aux 1 x m n
simp only [pow_one] at this
specialize this hm hn
apply_fun ZMod.castHom (show p ∣ p ^ 1 by rw [pow_one]) (ZMod p) at this
simp only [map_intCast] at this
simpa only [Int.cast_natCast] using this
|
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.CategoryTheory.Adjunction.FullyFaithful
#align_import algebraic_geometry.presheafed_space from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
open Opposite CategoryTheory CategoryTheory.Category CategoryTheory.Functor TopCat Topolo... | Mathlib/Geometry/RingedSpace/PresheafedSpace.lean | 112 | 121 | theorem Hom.ext {X Y : PresheafedSpace C} (α β : Hom X Y) (w : α.base = β.base)
(h : α.c ≫ whiskerRight (eqToHom (by rw [w])) _ = β.c) : α = β := by |
rcases α with ⟨base, c⟩
rcases β with ⟨base', c'⟩
dsimp at w
subst w
dsimp at h
erw [whiskerRight_id', comp_id] at h
subst h
rfl
|
import Mathlib.Topology.Constructions
import Mathlib.Topology.ContinuousOn
#align_import topology.bases from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78ad6527b7e4"
open Set Filter Function Topology
noncomputable section
namespace TopologicalSpace
universe u
variable {α : Type u} {β : Type*} ... | Mathlib/Topology/Bases.lean | 539 | 542 | theorem _root_.Dense.isSeparable_iff (hs : Dense s) :
IsSeparable s ↔ SeparableSpace α := by |
simp_rw [IsSeparable, separableSpace_iff, dense_iff_closure_eq, ← univ_subset_iff,
← hs.closure_eq, isClosed_closure.closure_subset_iff]
|
import Mathlib.MeasureTheory.Integral.IntervalIntegral
import Mathlib.Analysis.Calculus.Deriv.ZPow
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.SpecialFunctions.NonIntegrable
import Mathlib.Analysis.Analytic.Basic
#align_import measure_theory.integral.circle_integral from "leanprover-communit... | Mathlib/MeasureTheory/Integral/CircleIntegral.lean | 105 | 106 | theorem circleMap_sub_center (c : ℂ) (R : ℝ) (θ : ℝ) : circleMap c R θ - c = circleMap 0 R θ := by |
simp [circleMap]
|
import Mathlib.NumberTheory.FLT.Basic
import Mathlib.NumberTheory.PythagoreanTriples
import Mathlib.RingTheory.Coprime.Lemmas
import Mathlib.Tactic.LinearCombination
#align_import number_theory.fermat4 from "leanprover-community/mathlib"@"10b4e499f43088dd3bb7b5796184ad5216648ab1"
noncomputable section
open scope... | Mathlib/NumberTheory/FLT/Four.lean | 154 | 156 | theorem Int.coprime_of_sq_sum {r s : ℤ} (h2 : IsCoprime s r) : IsCoprime (r ^ 2 + s ^ 2) r := by |
rw [sq, sq]
exact (IsCoprime.mul_left h2 h2).mul_add_left_left r
|
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 | 110 | 110 | theorem asAlgebraHom_single_one (g : G) : asAlgebraHom ρ (Finsupp.single g 1) = ρ g := by | simp
|
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.Order.Invertible
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.LinearAlgebra.AffineSpace.Slope
import Mathlib.LinearAlgebra.AffineSpace.Midpoint
import Mathlib.Tactic.FieldSimp
#align_import li... | Mathlib/LinearAlgebra/AffineSpace/Ordered.lean | 83 | 86 | theorem lineMap_lt_lineMap_iff_of_lt (h : r < r') : lineMap a b r < lineMap a b r' ↔ a < b := by |
simp only [lineMap_apply_module]
rw [← lt_sub_iff_add_lt, add_sub_assoc, ← sub_lt_iff_lt_add', ← sub_smul, ← sub_smul,
sub_sub_sub_cancel_left, smul_lt_smul_iff_of_pos_left (sub_pos.2 h)]
|
import Mathlib.MeasureTheory.Measure.GiryMonad
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.MeasureTheory.Measure.OpenPos
#align_import measure_theory.constructions.prod.basic from "leanprover-community/mathlib"@"00abe0695d8767201e6d008afa22393978bb32... | Mathlib/MeasureTheory/Constructions/Prod/Basic.lean | 191 | 195 | theorem Measurable.map_prod_mk_left [SFinite ν] :
Measurable fun x : α => map (Prod.mk x) ν := by |
apply measurable_of_measurable_coe; intro s hs
simp_rw [map_apply measurable_prod_mk_left hs]
exact measurable_measure_prod_mk_left hs
|
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 | 159 | 161 | theorem image_diag_union_image_offDiag :
s.diag.image Sym2.mk ∪ s.offDiag.image Sym2.mk = s.sym2 := by |
rw [← image_union, diag_union_offDiag, sym2_eq_image]
|
import Mathlib.Logic.Relation
import Mathlib.Data.Option.Basic
import Mathlib.Data.Seq.Seq
#align_import data.seq.wseq from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b348ce40ad"
namespace Stream'
open Function
universe u v w
def WSeq (α) :=
Seq (Option α)
#align stream.wseq Stream'.WSeq
... | Mathlib/Data/Seq/WSeq.lean | 1,106 | 1,107 | theorem liftRel_think_left (R : α → β → Prop) (s t) : LiftRel R (think s) t ↔ LiftRel R s t := by |
rw [liftRel_destruct_iff, liftRel_destruct_iff]; simp
|
import Mathlib.MeasureTheory.Measure.MeasureSpace
import Mathlib.MeasureTheory.Measure.Regular
import Mathlib.Topology.Sets.Compacts
#align_import measure_theory.measure.content from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
universe u v w
noncomputable section
open Set Topologic... | Mathlib/MeasureTheory/Measure/Content.lean | 331 | 337 | theorem outerMeasure_caratheodory (A : Set G) :
MeasurableSet[μ.outerMeasure.caratheodory] A ↔
∀ U : Opens G, μ.outerMeasure (U ∩ A) + μ.outerMeasure (U \ A) ≤ μ.outerMeasure U := by |
rw [Opens.forall]
apply inducedOuterMeasure_caratheodory
· apply innerContent_iUnion_nat
· apply innerContent_mono'
|
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.Matrix.RowCol
import Mathlib.Data.Fin.VecNotation
import Mathlib.Tactic.FinCases
#align_import data.matrix.notation from "leanprover-community/mathlib"@"a99f85220eaf38f14f94e04699943e185a5e1d1a"
namespace Matrix
universe u uₘ uₙ uₒ
variable {α : Type u} {o n m... | Mathlib/Data/Matrix/Notation.lean | 174 | 175 | theorem cons_dotProduct_cons (x : α) (v : Fin n → α) (y : α) (w : Fin n → α) :
dotProduct (vecCons x v) (vecCons y w) = x * y + dotProduct v w := by | simp
|
import Mathlib.Topology.UniformSpace.Cauchy
import Mathlib.Topology.UniformSpace.Separation
import Mathlib.Topology.DenseEmbedding
#align_import topology.uniform_space.uniform_embedding from "leanprover-community/mathlib"@"195fcd60ff2bfe392543bceb0ec2adcdb472db4c"
open Filter Function Set Uniformity Topology
sec... | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | 116 | 118 | theorem UniformInducing.inducing {f : α → β} (h : UniformInducing f) : Inducing f := by |
obtain rfl := h.comap_uniformSpace
exact inducing_induced f
|
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.Adjugate
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import linear_algebra.matrix.nonsingular_inverse from "leanprover-community/mathlib"@"722b3b152ddd5e0cf21c0a29787c76596cb6b422"
namespace Matrix
universe u u' v
variable {l : ... | Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean | 79 | 81 | theorem invOf_eq [Invertible A.det] [Invertible A] : ⅟ A = ⅟ A.det • A.adjugate := by |
letI := invertibleOfDetInvertible A
convert (rfl : ⅟ A = _)
|
import Batteries.Data.List.Lemmas
import Batteries.Data.Array.Basic
import Batteries.Tactic.SeqFocus
import Batteries.Util.ProofWanted
namespace Array
theorem forIn_eq_data_forIn [Monad m]
(as : Array α) (b : β) (f : α → β → m (ForInStep β)) :
forIn as b f = forIn as.data b f := by
let rec loop : ∀ {i h b ... | .lake/packages/batteries/Batteries/Data/Array/Lemmas.lean | 127 | 129 | theorem size_shrink (a : Array α) (n) : (a.shrink n).size = min a.size n := by |
simp [shrink, size_shrink_loop]
omega
|
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 | 101 | 103 | theorem sym2_nonempty : s.sym2.Nonempty ↔ s.Nonempty := by |
rw [← not_iff_not]
simp_rw [not_nonempty_iff_eq_empty, sym2_eq_empty]
|
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.Order.Filter.IndicatorFunction
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner
import Mathlib.MeasureTheory.Function.LpSeminorm.Trim
#align_import measure_theory.function.conditional... | Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean | 78 | 83 | theorem neg [AddGroup β] [TopologicalAddGroup β] {f : α → β} (hfm : AEStronglyMeasurable' m f μ) :
AEStronglyMeasurable' m (-f) μ := by |
rcases hfm with ⟨f', hf'_meas, hf_ae⟩
refine ⟨-f', hf'_meas.neg, hf_ae.mono fun x hx => ?_⟩
simp_rw [Pi.neg_apply]
rw [hx]
|
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.Orientation
import Mathlib.Data.Complex.Orientation
import Mathlib.Tactic.LinearCombination
#align_import analysis.inner_product_space.two_dim from "leanprover-community/mathlib"@"cd8fafa2fac98e1a67097e8a91ad9901cfde48af"
non... | Mathlib/Analysis/InnerProductSpace/TwoDim.lean | 559 | 565 | theorem eq_zero_or_eq_zero_of_kahler_eq_zero {x y : E} (hx : o.kahler x y = 0) : x = 0 ∨ y = 0 := by |
have : ‖x‖ * ‖y‖ = 0 := by simpa [hx] using (o.norm_kahler x y).symm
cases' eq_zero_or_eq_zero_of_mul_eq_zero this with h h
· left
simpa using h
· right
simpa using h
|
import Mathlib.CategoryTheory.Subobject.Lattice
#align_import category_theory.subobject.limits from "leanprover-community/mathlib"@"956af7c76589f444f2e1313911bad16366ea476d"
universe v u
noncomputable section
open CategoryTheory CategoryTheory.Category CategoryTheory.Limits CategoryTheory.Subobject Opposite
var... | Mathlib/CategoryTheory/Subobject/Limits.lean | 419 | 422 | theorem imageSubobjectCompIso_inv_arrow (f : X ⟶ Y) [HasImage f] {Y' : C} (h : Y ⟶ Y') [IsIso h] :
(imageSubobjectCompIso f h).inv ≫ (imageSubobject (f ≫ h)).arrow =
(imageSubobject f).arrow ≫ h := by |
simp [imageSubobjectCompIso]
|
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap
import Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.MeasureTheory.Constructio... | Mathlib/MeasureTheory/Function/Jacobian.lean | 760 | 766 | theorem aemeasurable_toNNReal_abs_det_fderivWithin (hs : MeasurableSet s)
(hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) :
AEMeasurable (fun x => |(f' x).det|.toNNReal) (μ.restrict s) := by |
apply measurable_real_toNNReal.comp_aemeasurable
refine continuous_abs.measurable.comp_aemeasurable ?_
refine ContinuousLinearMap.continuous_det.measurable.comp_aemeasurable ?_
exact aemeasurable_fderivWithin μ hs hf'
|
import Mathlib.MeasureTheory.Function.LpSeminorm.Basic
import Mathlib.MeasureTheory.Integral.MeanInequalities
#align_import measure_theory.function.lp_seminorm from "leanprover-community/mathlib"@"c4015acc0a223449d44061e27ddac1835a3852b9"
open Filter
open scoped ENNReal Topology
namespace MeasureTheory
variable ... | Mathlib/MeasureTheory/Function/LpSeminorm/TriangleInequality.lean | 36 | 44 | theorem snorm'_add_le_of_le_one {f g : α → E} (hf : AEStronglyMeasurable f μ) (hq0 : 0 ≤ q)
(hq1 : q ≤ 1) : snorm' (f + g) q μ ≤ (2 : ℝ≥0∞) ^ (1 / q - 1) * (snorm' f q μ + snorm' g q μ) :=
calc
(∫⁻ a, (‖(f + g) a‖₊ : ℝ≥0∞) ^ q ∂μ) ^ (1 / q) ≤
(∫⁻ a, ((fun a => (‖f a‖₊ : ℝ≥0∞)) + fun a => (‖g a‖₊ : ℝ≥0... |
gcongr with a
simp only [Pi.add_apply, ← ENNReal.coe_add, ENNReal.coe_le_coe, nnnorm_add_le]
_ ≤ (2 : ℝ≥0∞) ^ (1 / q - 1) * (snorm' f q μ + snorm' g q μ) :=
ENNReal.lintegral_Lp_add_le_of_le_one hf.ennnorm hq0 hq1
|
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.GroupTheory.GroupAction.Ring
#align_import data.polynomial.derivative from "leanprover-community/mathlib"@"bbeb185db4ccee8ed07dc48449414ebfa39cb821"
noncomputable section
open Finset
open Polynomial
namespace Pol... | Mathlib/Algebra/Polynomial/Derivative.lean | 395 | 435 | theorem iterate_derivative_mul {n} (p q : R[X]) :
derivative^[n] (p * q) =
∑ k ∈ range n.succ, (n.choose k • (derivative^[n - k] p * derivative^[k] q)) := by |
induction' n with n IH
· simp [Finset.range]
calc
derivative^[n + 1] (p * q) =
derivative (∑ k ∈ range n.succ,
n.choose k • (derivative^[n - k] p * derivative^[k] q)) := by
rw [Function.iterate_succ_apply', IH]
_ = (∑ k ∈ range n.succ,
n.choose k • (derivative^[n - k +... |
import Mathlib.Init.ZeroOne
import Mathlib.Data.Set.Defs
import Mathlib.Order.Basic
import Mathlib.Order.SymmDiff
import Mathlib.Tactic.Tauto
import Mathlib.Tactic.ByContra
import Mathlib.Util.Delaborators
#align_import data.set.basic from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
... | Mathlib/Data/Set/Basic.lean | 2,232 | 2,235 | theorem mem_dite_empty_right (p : Prop) [Decidable p] (t : p → Set α) (x : α) :
(x ∈ if h : p then t h else ∅) ↔ ∃ h : p, x ∈ t h := by |
simp only [mem_dite, mem_empty_iff_false, imp_false, not_not]
exact ⟨fun h => ⟨h.2, h.1 h.2⟩, fun ⟨h₁, h₂⟩ => ⟨fun _ => h₂, h₁⟩⟩
|
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.EuclideanDist
import Mathlib.MeasureTheory.Function.ContinuousMapDense
import Mathlib.MeasureTheory.Group.Integral
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.M... | Mathlib/Analysis/Fourier/RiemannLebesgueLemma.lean | 243 | 256 | theorem tendsto_integral_exp_smul_cocompact_of_inner_product (μ : Measure V) [μ.IsAddHaarMeasure] :
Tendsto (fun w : V →L[ℝ] ℝ => ∫ v, 𝐞 (-w v) • f v ∂μ) (cocompact (V →L[ℝ] ℝ)) (𝓝 0) := by |
rw [μ.isAddLeftInvariant_eq_smul volume]
simp_rw [integral_smul_nnreal_measure]
rw [← (smul_zero _ : Measure.addHaarScalarFactor μ volume • (0 : E) = 0)]
apply Tendsto.const_smul
let A := (InnerProductSpace.toDual ℝ V).symm
have : (fun w : V →L[ℝ] ℝ ↦ ∫ v, 𝐞 (-w v) • f v) = (fun w : V ↦ ∫ v, 𝐞 (-⟪v, w⟫) ... |
import Mathlib.Data.List.Chain
import Mathlib.Data.List.Enum
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Pairwise
import Mathlib.Data.List.Zip
#align_import data.list.range from "leanprover-community/mathlib"@"7b78d1776212a91ecc94cf601f83bdcc46b04213"
set_option autoImplicit true
universe u
open Nat... | Mathlib/Data/List/Range.lean | 191 | 193 | theorem unzip_enumFrom_eq_prod (l : List α) {n : ℕ} :
(l.enumFrom n).unzip = (range' n l.length, l) := by |
simp only [enumFrom_eq_zip_range', unzip_zip, length_range']
|
import Mathlib.Data.Fin.Fin2
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.Common
#align_import data.typevec from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc9524185ae994553ebf4"
universe u v w
@[pp_with_univ]
def TypeVec (n : ℕ) :=
Fin2 n → Type*
#align typevec TypeVec
instance {n} : Inh... | Mathlib/Data/TypeVec.lean | 441 | 442 | theorem const_nil {β} (x : β) (α : TypeVec 0) : TypeVec.const x α = nilFun := by |
ext i : 1; cases i
|
import Mathlib.LinearAlgebra.Matrix.Basis
import Mathlib.LinearAlgebra.Matrix.Nondegenerate
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.ToLinearEquiv
import Mathlib.LinearAlgebra.BilinearForm.Properties
import Mathlib.LinearAlgebra.Matrix.SesquilinearForm
#align_import l... | Mathlib/LinearAlgebra/Matrix/BilinearForm.lean | 337 | 342 | theorem Matrix.toBilin_comp (M : Matrix n n R₂) (P Q : Matrix n o R₂) :
(Matrix.toBilin b M).comp (toLin c b P) (toLin c b Q) = Matrix.toBilin c (Pᵀ * M * Q) := by |
ext x y
rw [Matrix.toBilin, BilinForm.toMatrix, Matrix.toBilin, BilinForm.toMatrix, toMatrix₂_symm,
toMatrix₂_symm, ← Matrix.toLinearMap₂_compl₁₂ b b c c]
simp
|
import Mathlib.Data.Finset.Attr
import Mathlib.Data.Multiset.FinsetOps
import Mathlib.Logic.Equiv.Set
import Mathlib.Order.Directed
import Mathlib.Order.Interval.Set.Basic
#align_import data.finset.basic from "leanprover-community/mathlib"@"442a83d738cb208d3600056c489be16900ba701d"
-- Assert that we define `Finset... | Mathlib/Data/Finset/Basic.lean | 389 | 389 | theorem not_subset : ¬s ⊆ t ↔ ∃ x ∈ s, x ∉ t := by | simp only [← coe_subset, Set.not_subset, mem_coe]
|
import Mathlib.Order.Filter.Lift
import Mathlib.Order.Filter.AtTopBot
#align_import order.filter.small_sets from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
open Filter
open Filter Set
variable {α β : Type*} {ι : Sort*}
namespace Filter
variable {l l' la : Filter α} {lb : Filter ... | Mathlib/Order/Filter/SmallSets.lean | 125 | 128 | theorem smallSets_comap_eq_comap_image (l : Filter β) (f : α → β) :
(comap f l).smallSets = comap (image f) l.smallSets := by |
refine (gc_map_comap _).u_comm_of_l_comm (gc_map_comap _) bind_smallSets_gc bind_smallSets_gc ?_
simp [Function.comp, map_bind, bind_map]
|
import Mathlib.Algebra.Module.Torsion
import Mathlib.SetTheory.Cardinal.Cofinality
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
... | Mathlib/LinearAlgebra/Dimension/Finite.lean | 34 | 40 | theorem rank_le {n : ℕ}
(H : ∀ s : Finset M, (LinearIndependent R fun i : s => (i : M)) → s.card ≤ n) :
Module.rank R M ≤ n := by |
rw [Module.rank_def]
apply ciSup_le'
rintro ⟨s, li⟩
exact linearIndependent_bounded_of_finset_linearIndependent_bounded H _ li
|
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Topology.Algebra.Field
import Mathlib.Topology.Algebra.Order.Group
#align_import topology.algebra.order.field from "leanprover-community/mathlib"@"9a59dcb7a2d06bf55da57b9030169219980660cd"
open Set Filter TopologicalSpace Function
open scoped Pointwise Top... | Mathlib/Topology/Algebra/Order/Field.lean | 30 | 51 | theorem TopologicalRing.of_norm {R 𝕜 : Type*} [NonUnitalNonAssocRing R] [LinearOrderedField 𝕜]
[TopologicalSpace R] [TopologicalAddGroup R] (norm : R → 𝕜)
(norm_nonneg : ∀ x, 0 ≤ norm x) (norm_mul_le : ∀ x y, norm (x * y) ≤ norm x * norm y)
(nhds_basis : (𝓝 (0 : R)).HasBasis ((0 : 𝕜) < ·) (fun ε ↦ { x ... |
have h0 : ∀ f : R → R, ∀ c ≥ (0 : 𝕜), (∀ x, norm (f x) ≤ c * norm x) →
Tendsto f (𝓝 0) (𝓝 0) := by
refine fun f c c0 hf ↦ (nhds_basis.tendsto_iff nhds_basis).2 fun ε ε0 ↦ ?_
rcases exists_pos_mul_lt ε0 c with ⟨δ, δ0, hδ⟩
refine ⟨δ, δ0, fun x hx ↦ (hf _).trans_lt ?_⟩
exact (mul_le_mul_of_nonn... |
import Mathlib.RingTheory.Polynomial.Basic
import Mathlib.RingTheory.Ideal.LocalRing
#align_import data.polynomial.expand from "leanprover-community/mathlib"@"bbeb185db4ccee8ed07dc48449414ebfa39cb821"
universe u v w
open Polynomial
open Finset
namespace Polynomial
section CommSemiring
variable (R : Type u) [... | Mathlib/Algebra/Polynomial/Expand.lean | 288 | 295 | theorem rootMultiplicity_expand_pow :
(expand R (p ^ n) f).rootMultiplicity r = p ^ n * f.rootMultiplicity (r ^ p ^ n) := by |
obtain rfl | h0 := eq_or_ne f 0; · simp
obtain ⟨g, hg, ndvd⟩ := f.exists_eq_pow_rootMultiplicity_mul_and_not_dvd h0 (r ^ p ^ n)
rw [dvd_iff_isRoot, ← eval_X (x := r), ← eval_pow, ← isRoot_comp, ← expand_eq_comp_X_pow] at ndvd
conv_lhs => rw [hg, map_mul, map_pow, map_sub, expand_X, expand_C, map_pow, ← sub_pow... |
import Mathlib.Analysis.Convex.Cone.Basic
import Mathlib.Data.Real.Archimedean
import Mathlib.LinearAlgebra.LinearPMap
#align_import analysis.convex.cone.basic from "leanprover-community/mathlib"@"915591b2bb3ea303648db07284a161a7f2a9e3d4"
open Set LinearMap
variable {𝕜 E F G : Type*}
variable [AddCommGroup E... | Mathlib/Analysis/Convex/Cone/Extension.lean | 115 | 139 | theorem exists_top (p : E →ₗ.[ℝ] ℝ) (hp_nonneg : ∀ x : p.domain, (x : E) ∈ s → 0 ≤ p x)
(hp_dense : ∀ y, ∃ x : p.domain, (x : E) + y ∈ s) :
∃ q ≥ p, q.domain = ⊤ ∧ ∀ x : q.domain, (x : E) ∈ s → 0 ≤ q x := by |
set S := { p : E →ₗ.[ℝ] ℝ | ∀ x : p.domain, (x : E) ∈ s → 0 ≤ p x }
have hSc : ∀ c, c ⊆ S → IsChain (· ≤ ·) c → ∀ y ∈ c, ∃ ub ∈ S, ∀ z ∈ c, z ≤ ub := by
intro c hcs c_chain y hy
clear hp_nonneg hp_dense p
have cne : c.Nonempty := ⟨y, hy⟩
have hcd : DirectedOn (· ≤ ·) c := c_chain.directedOn
ref... |
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 | 427 | 429 | theorem Inducing.r0Space [TopologicalSpace Y] {f : Y → X} (hf : Inducing f) : R0Space Y where
specializes_symmetric a b := by |
simpa only [← hf.specializes_iff] using Specializes.symm
|
import Mathlib.Dynamics.FixedPoints.Basic
import Mathlib.GroupTheory.Perm.Option
import Mathlib.Logic.Equiv.Defs
import Mathlib.Logic.Equiv.Option
#align_import combinatorics.derangements.basic from "leanprover-community/mathlib"@"9407b03373c8cd201df99d6bc5514fc2db44054f"
open Equiv Function
def derangements (α... | Mathlib/Combinatorics/Derangements/Basic.lean | 123 | 126 | theorem RemoveNone.mem_fiber (a : Option α) (f : Perm α) :
f ∈ RemoveNone.fiber a ↔
∃ F : Perm (Option α), F ∈ derangements (Option α) ∧ F none = a ∧ removeNone F = f := by |
simp [RemoveNone.fiber, derangements]
|
import Mathlib.Data.Finset.Image
import Mathlib.Data.List.FinRange
#align_import data.fintype.basic from "leanprover-community/mathlib"@"d78597269638367c3863d40d45108f52207e03cf"
assert_not_exists MonoidWithZero
assert_not_exists MulAction
open Function
open Nat
universe u v
variable {α β γ : Type*}
class Fi... | Mathlib/Data/Fintype/Basic.lean | 113 | 114 | theorem univ_eq_empty_iff : (univ : Finset α) = ∅ ↔ IsEmpty α := by |
rw [← not_nonempty_iff, ← univ_nonempty_iff, not_nonempty_iff_eq_empty]
|
import Mathlib.Topology.UniformSpace.Cauchy
import Mathlib.Topology.UniformSpace.Separation
import Mathlib.Topology.DenseEmbedding
#align_import topology.uniform_space.uniform_embedding from "leanprover-community/mathlib"@"195fcd60ff2bfe392543bceb0ec2adcdb472db4c"
open Filter Function Set Uniformity Topology
sec... | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | 93 | 97 | theorem uniformInducing_of_compose {f : α → β} {g : β → γ} (hf : UniformContinuous f)
(hg : UniformContinuous g) (hgf : UniformInducing (g ∘ f)) : UniformInducing f := by |
refine ⟨le_antisymm ?_ hf.le_comap⟩
rw [← hgf.1, ← Prod.map_def, ← Prod.map_def, ← Prod.map_comp_map f f g g, ← comap_comap]
exact comap_mono hg.le_comap
|
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Data.Nat.PrimeFin
import Mathlib.NumberTheory.Padics.PadicVal
import Mathlib.Order.Interval.Finset.Nat
#align_import data.nat.factorization.basic from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
... | Mathlib/Data/Nat/Factorization/Basic.lean | 198 | 202 | theorem factorization_mul {a b : ℕ} (ha : a ≠ 0) (hb : b ≠ 0) :
(a * b).factorization = a.factorization + b.factorization := by |
ext p
simp only [add_apply, ← factors_count_eq, perm_iff_count.mp (perm_factors_mul ha hb) p,
count_append]
|
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.LinearAlgebra.FreeModule.PID
import Mathlib.LinearAlgebra.Matrix.AbsoluteValue
import Mathlib.NumberTheory.ClassNumber.AdmissibleAbsoluteValue
import Mathlib.RingTheory.ClassGroup
import Mathlib.RingTheory.DedekindDomain.IntegralClosure
import Mathlib.Ri... | Mathlib/NumberTheory/ClassNumber/Finite.lean | 197 | 247 | theorem exists_mem_finsetApprox (a : S) {b} (hb : b ≠ (0 : R)) :
∃ q : S,
∃ r ∈ finsetApprox bS adm, abv (Algebra.norm R (r • a - b • q)) <
abv (Algebra.norm R (algebraMap R S b)) := by |
have dim_pos := Fintype.card_pos_iff.mpr bS.index_nonempty
set ε : ℝ := normBound abv bS ^ (-1 / Fintype.card ι : ℝ) with ε_eq
have hε : 0 < ε := Real.rpow_pos_of_pos (Int.cast_pos.mpr (normBound_pos abv bS)) _
have ε_le : (normBound abv bS : ℝ) * (abv b • ε) ^ (Fintype.card ι : ℝ)
≤ abv b ^ (F... |
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Analysis.NormedSpace.PiLp
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.UnitaryGroup
#align_import analysis.inner_product_space.pi_L2 from "leanprover-community/mathlib"@"13bce9a6b6c44f6b4c91ac1c1d2a816e2533d395"
set_... | Mathlib/Analysis/InnerProductSpace/PiL2.lean | 114 | 116 | theorem EuclideanSpace.norm_eq {𝕜 : Type*} [RCLike 𝕜] {n : Type*} [Fintype n]
(x : EuclideanSpace 𝕜 n) : ‖x‖ = √(∑ i, ‖x i‖ ^ 2) := by |
simpa only [Real.coe_sqrt, NNReal.coe_sum] using congr_arg ((↑) : ℝ≥0 → ℝ) x.nnnorm_eq
|
import Mathlib.Control.Functor.Multivariate
import Mathlib.Data.PFunctor.Univariate.Basic
#align_import data.pfunctor.multivariate.basic from "leanprover-community/mathlib"@"e3d9ab8faa9dea8f78155c6c27d62a621f4c152d"
universe u v
open MvFunctor
@[pp_with_univ]
structure MvPFunctor (n : ℕ) where
A : Type u
... | Mathlib/Data/PFunctor/Multivariate/Basic.lean | 206 | 217 | theorem supp_eq {α : TypeVec n} (a : P.A) (f : P.B a ⟹ α) (i) :
@supp.{u} _ P.Obj _ α (⟨a, f⟩ : P α) i = f i '' univ := by |
ext x; simp only [supp, image_univ, mem_range, mem_setOf_eq]
constructor <;> intro h
· apply @h fun i x => ∃ y : P.B a i, f i y = x
rw [liftP_iff']
intros
exact ⟨_, rfl⟩
· simp only [liftP_iff']
cases h
subst x
tauto
|
import Mathlib.Order.Cover
import Mathlib.Order.Interval.Finset.Defs
#align_import data.finset.locally_finite from "leanprover-community/mathlib"@"442a83d738cb208d3600056c489be16900ba701d"
assert_not_exists MonoidWithZero
assert_not_exists Finset.sum
open Function OrderDual
open FinsetInterval
variable {ι α : T... | Mathlib/Order/Interval/Finset/Basic.lean | 235 | 237 | theorem Ioo_subset_Ico_self : Ioo a b ⊆ Ico a b := by |
rw [← coe_subset, coe_Ioo, coe_Ico]
exact Set.Ioo_subset_Ico_self
|
import Mathlib.Algebra.MvPolynomial.Variables
#align_import data.mv_polynomial.supported from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
universe u v w
namespace MvPolynomial
variable {σ τ : Type*} {R : Type u} {S : Type v} {r : R} {e : ℕ} {n m : σ}
section CommSemiring
variable... | Mathlib/Algebra/MvPolynomial/Supported.lean | 91 | 92 | theorem mem_supported_vars (p : MvPolynomial σ R) : p ∈ supported R (↑p.vars : Set σ) := by |
rw [mem_supported]
|
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