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 |
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import Mathlib.Probability.Variance
#align_import probability.moments from "leanprover-community/mathlib"@"85453a2a14be8da64caf15ca50930cf4c6e5d8de"
open MeasureTheory Filter Finset Real
noncomputable section
open scoped MeasureTheory ProbabilityTheory ENNReal NNReal
namespace ProbabilityTheory
variable {Ω ι ... | Mathlib/Probability/Moments.lean | 62 | 64 | theorem moment_zero (hp : p ≠ 0) : moment 0 p μ = 0 := by |
simp only [moment, hp, zero_pow, Ne, not_false_iff, Pi.zero_apply, integral_const,
smul_eq_mul, mul_zero, integral_zero]
|
import Mathlib.Algebra.Group.NatPowAssoc
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Induction
import Mathlib.Algebra.Polynomial.Eval
namespace Polynomial
section MulActionWithZero
variable {R : Type*} [Semiring R] (r : R) (p : R[X]) {S : Type*} [AddCommMonoid S] [Pow S ℕ]
[Mu... | Mathlib/Algebra/Polynomial/Smeval.lean | 79 | 80 | theorem smeval_zero : (0 : R[X]).smeval x = 0 := by |
simp only [smeval_eq_sum, smul_pow, sum_zero_index]
|
import Mathlib.Data.SetLike.Basic
import Mathlib.Data.Finset.Preimage
import Mathlib.ModelTheory.Semantics
#align_import model_theory.definability from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u v w u₁
namespace Set
variable {M : Type w} (A : Set M) (L : FirstOrder.Lang... | Mathlib/ModelTheory/Definability.lean | 86 | 88 | theorem Definable.mono (hAs : A.Definable L s) (hAB : A ⊆ B) : B.Definable L s := by |
rw [definable_iff_empty_definable_with_params] at *
exact hAs.map_expansion (L.lhomWithConstantsMap (Set.inclusion hAB))
|
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.RingTheory.Ideal.Over
import Mathlib.RingTheory.Ideal.Prod
import Mathlib.RingTheory.Ideal.MinimalPrime
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Nilpotent.Lemmas
import Mathlib.Topology.Sets.Closeds
import Mathlib.Topology.Sober
#a... | Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean | 178 | 179 | theorem vanishingIdeal_singleton (x : PrimeSpectrum R) :
vanishingIdeal ({x} : Set (PrimeSpectrum R)) = x.asIdeal := by | simp [vanishingIdeal]
|
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 | 197 | 200 | theorem coe_stream'_rat_eq :
((IntFractPair.stream q).map (Option.map (mapFr (↑))) : Stream' <| Option <| IntFractPair K) =
IntFractPair.stream v := by |
funext n; exact IntFractPair.coe_stream_nth_rat_eq v_eq_q n
|
import Mathlib.Data.List.OfFn
import Mathlib.Data.List.Range
#align_import data.list.fin_range from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
universe u
namespace List
variable {α : Type u}
@[simp]
theorem map_coe_finRange (n : ℕ) : ((finRange n) : List (Fin n)).map (Fin.val) = ... | Mathlib/Data/List/FinRange.lean | 64 | 72 | theorem nodup_ofFn {n} {f : Fin n → α} : Nodup (ofFn f) ↔ Function.Injective f := by |
refine ⟨?_, nodup_ofFn_ofInjective⟩
refine Fin.consInduction ?_ (fun x₀ xs ih => ?_) f
· intro _
exact Function.injective_of_subsingleton _
· intro h
rw [Fin.cons_injective_iff]
simp_rw [ofFn_succ, Fin.cons_succ, nodup_cons, Fin.cons_zero, mem_ofFn] at h
exact h.imp_right ih
|
import Mathlib.Probability.Independence.Basic
import Mathlib.Probability.Independence.Conditional
#align_import probability.independence.zero_one from "leanprover-community/mathlib"@"2f8347015b12b0864dfaf366ec4909eb70c78740"
open MeasureTheory MeasurableSpace
open scoped MeasureTheory ENNReal
namespace Probabili... | Mathlib/Probability/Independence/ZeroOne.lean | 109 | 115 | theorem kernel.indep_biSup_limsup (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s κ μα)
(hf : ∀ t, p t → tᶜ ∈ f) {t : Set ι} (ht : p t) :
Indep (⨆ n ∈ t, s n) (limsup s f) κ μα := by |
refine indep_of_indep_of_le_right (indep_biSup_compl h_le h_indep t) ?_
refine limsSup_le_of_le (by isBoundedDefault) ?_
simp only [Set.mem_compl_iff, eventually_map]
exact eventually_of_mem (hf t ht) le_iSup₂
|
import Mathlib.Data.Nat.Defs
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Tactic.Monotonicity.Attr
#align_import data.nat.log from "leanprover-community/mathlib"@"3e00d81bdcbf77c8188bbd18f5524ddc3ed8cac6"
namespace Nat
--@[pp_nodot] porting note: unknown attribute
def log (b : ℕ) : ℕ → ℕ
| n => i... | Mathlib/Data/Nat/Log.lean | 42 | 44 | theorem log_eq_zero_iff {b n : ℕ} : log b n = 0 ↔ n < b ∨ b ≤ 1 := by |
rw [log, dite_eq_right_iff]
simp only [Nat.add_eq_zero_iff, Nat.one_ne_zero, and_false, imp_false, not_and_or, not_le, not_lt]
|
import Mathlib.Algebra.EuclideanDomain.Defs
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Regular
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Ring.Basic
#align_import algebra.euclidean_domain.basic from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d0... | Mathlib/Algebra/EuclideanDomain/Basic.lean | 123 | 128 | theorem dvd_div_of_mul_dvd {a b c : R} (h : a * b ∣ c) : b ∣ c / a := by |
rcases eq_or_ne a 0 with (rfl | ha)
· simp only [div_zero, dvd_zero]
rcases h with ⟨d, rfl⟩
refine ⟨d, ?_⟩
rw [mul_assoc, mul_div_cancel_left₀ _ ha]
|
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.Calculus.BumpFunction.Normed
import Mathlib.MeasureTheory.Integral.Average
import Mathlib.MeasureTheory.Covering.Differentiation
import Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace
import Mathlib.MeasureTheory.Measure.Haar.Unique
#align_import analy... | Mathlib/Analysis/Calculus/BumpFunction/Convolution.lean | 65 | 68 | theorem normed_convolution_eq_right {x₀ : G} (hg : ∀ x ∈ ball x₀ φ.rOut, g x = g x₀) :
(φ.normed μ ⋆[lsmul ℝ ℝ, μ] g : G → E') x₀ = g x₀ := by |
rw [convolution_eq_right' _ φ.support_normed_eq.subset hg]
exact integral_normed_smul φ μ (g x₀)
|
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.Deriv.Linear
import Mathlib.Analysis.Complex.Conformal
import Mathlib.Analysis.Calculus.Conformal.NormedSpace
#align_import analysis.complex.real_deriv from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
se... | Mathlib/Analysis/Complex/RealDeriv.lean | 106 | 108 | theorem HasDerivAt.complexToReal_fderiv' {f : ℂ → E} {x : ℂ} {f' : E} (h : HasDerivAt f f' x) :
HasFDerivAt f (reCLM.smulRight f' + I • imCLM.smulRight f') x := by |
simpa only [Complex.restrictScalars_one_smulRight'] using h.hasFDerivAt.restrictScalars ℝ
|
import Mathlib.Data.Finsupp.Lex
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.GameAdd
#align_import logic.hydra from "leanprover-community/mathlib"@"48085f140e684306f9e7da907cd5932056d1aded"
namespace Relation
open Multiset Prod
variable {α : Type*}
def CutExpand (r : α → α → Prop) (s' s : Multise... | Mathlib/Logic/Hydra.lean | 138 | 146 | theorem _root_.Acc.cutExpand [IsIrrefl α r] {a : α} (hacc : Acc r a) : Acc (CutExpand r) {a} := by |
induction' hacc with a h ih
refine Acc.intro _ fun s ↦ ?_
classical
simp only [cutExpand_iff, mem_singleton]
rintro ⟨t, a, hr, rfl, rfl⟩
refine acc_of_singleton fun a' ↦ ?_
rw [erase_singleton, zero_add]
exact ih a' ∘ hr a'
|
import Mathlib.Probability.Independence.Basic
import Mathlib.Probability.Independence.Conditional
#align_import probability.independence.zero_one from "leanprover-community/mathlib"@"2f8347015b12b0864dfaf366ec4909eb70c78740"
open MeasureTheory MeasurableSpace
open scoped MeasureTheory ENNReal
namespace Probabili... | Mathlib/Probability/Independence/ZeroOne.lean | 46 | 49 | theorem measure_eq_zero_or_one_or_top_of_indepSet_self {t : Set Ω}
(h_indep : IndepSet t t μ) : μ t = 0 ∨ μ t = 1 ∨ μ t = ∞ := by |
simpa only [ae_dirac_eq, Filter.eventually_pure]
using kernel.measure_eq_zero_or_one_or_top_of_indepSet_self h_indep
|
import Mathlib.Init.Function
import Mathlib.Init.Order.Defs
#align_import data.bool.basic from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
namespace Bool
@[deprecated (since := "2024-06-07")] alias decide_True := decide_true_eq_true
#align bool.to_bool_true decide_true_eq_true
@[dep... | Mathlib/Data/Bool/Basic.lean | 112 | 112 | theorem and_intro : ∀ {a b : Bool}, a → b → a && b := by | decide
|
import Mathlib.Algebra.Homology.ComplexShape
import Mathlib.CategoryTheory.Subobject.Limits
import Mathlib.CategoryTheory.GradedObject
import Mathlib.Algebra.Homology.ShortComplex.Basic
#align_import algebra.homology.homological_complex from "leanprover-community/mathlib"@"88bca0ce5d22ebfd9e73e682e51d60ea13b48347"
... | Mathlib/Algebra/Homology/HomologicalComplex.lean | 177 | 182 | theorem next_nat_zero : (ComplexShape.down ℕ).next 0 = 0 := by |
classical
refine dif_neg ?_
push_neg
intro
apply Nat.noConfusion
|
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.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.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.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.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.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.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.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.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.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.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.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.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.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.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.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.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 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.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.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.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.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.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.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.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]
|
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 | 214 | 217 | theorem pnatCast_eq_natCast [Fact (∀ I : Ideal R, I ≠ ⊤ → CharZero (R ⧸ I))] (n : ℕ+) :
((n : Rˣ) : R) = ↑n := by |
change ((PNat.isUnit_natCast (R := R) n).unit : R) = ↑n
simp only [IsUnit.unit_spec]
|
import Mathlib.Algebra.Group.WithOne.Defs
import Mathlib.Algebra.GroupWithZero.InjSurj
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.GroupWithZero.WithZero
import Mathlib.Algebra.Order.Group.Units
import Mathlib.Algebra.Order.GroupWithZero.Synonym
import Mathlib.Algebra.Order.Monoid.Basic
imp... | Mathlib/Algebra/Order/GroupWithZero/Canonical.lean | 128 | 129 | theorem le_of_le_mul_right (h : c ≠ 0) (hab : a * c ≤ b * c) : a ≤ b := by |
simpa only [mul_inv_cancel_right₀ h] using mul_le_mul_right' hab c⁻¹
|
import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.MeasureTheory.Integral.PeakFunction
#align_import analysis.special_functions.trigonometric.euler_sine_prod from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
open scoped Real Topology
open Real Set Filter intervalIntegra... | Mathlib/Analysis/SpecialFunctions/Trigonometric/EulerSineProd.lean | 59 | 85 | theorem integral_cos_mul_cos_pow_aux (hn : 2 ≤ n) (hz : z ≠ 0) :
(∫ x in (0 : ℝ)..π / 2, Complex.cos (2 * z * x) * (cos x : ℂ) ^ n) =
n / (2 * z) *
∫ x in (0 : ℝ)..π / 2, Complex.sin (2 * z * x) * sin x * (cos x : ℂ) ^ (n - 1) := by |
have der1 :
∀ x : ℝ,
x ∈ uIcc 0 (π / 2) →
HasDerivAt (fun y : ℝ => (cos y : ℂ) ^ n) (-n * sin x * (cos x : ℂ) ^ (n - 1)) x := by
intro x _
have b : HasDerivAt (fun y : ℝ => (cos y : ℂ)) (-sin x) x := by
simpa using (hasDerivAt_cos x).ofReal_comp
convert HasDerivAt.comp x (hasDeriv... |
import Mathlib.Order.Filter.Lift
import Mathlib.Topology.Separation
import Mathlib.Order.Interval.Set.Monotone
#align_import topology.filter from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
open Set Filter TopologicalSpace
open Filter Topology
variable {ι : Sort*} {α β X Y : Type*}... | Mathlib/Topology/Filter.lean | 125 | 125 | theorem nhds_top : 𝓝 (⊤ : Filter α) = ⊤ := by | simp [nhds_eq]
|
import Mathlib.Data.ZMod.Basic
import Mathlib.RingTheory.Int.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
#align_import data.zmod.coprime from "leanprover-community/mathlib"@"4b4975cf92a1ffe2ddfeff6ff91b0c46a9162bf5"
namespace ZMod
| Mathlib/Data/ZMod/Coprime.lean | 24 | 28 | theorem eq_zero_iff_gcd_ne_one {a : ℤ} {p : ℕ} [pp : Fact p.Prime] :
(a : ZMod p) = 0 ↔ a.gcd p ≠ 1 := by |
rw [Ne, Int.gcd_comm, Int.gcd_eq_one_iff_coprime,
(Nat.prime_iff_prime_int.1 pp.1).coprime_iff_not_dvd, Classical.not_not,
intCast_zmod_eq_zero_iff_dvd]
|
import Mathlib.GroupTheory.Archimedean
import Mathlib.Topology.Order.Basic
#align_import topology.algebra.order.archimedean from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
open Set
theorem Rat.denseRange_cast {𝕜} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜]
... | Mathlib/Topology/Algebra/Order/Archimedean.lean | 58 | 62 | theorem dense_of_no_min (S : AddSubgroup G) (hbot : S ≠ ⊥)
(H : ¬∃ a : G, IsLeast { g : G | g ∈ S ∧ 0 < g } a) : Dense (S : Set G) := by |
refine S.dense_of_not_isolated_zero fun ε ε0 => ?_
contrapose! H
exact exists_isLeast_pos hbot ε0 (disjoint_left.2 H)
|
import Mathlib.Combinatorics.SimpleGraph.Clique
open Finset
namespace SimpleGraph
variable {V : Type*} [Fintype V] [DecidableEq V] (G H : SimpleGraph V) [DecidableRel G.Adj]
{n r : ℕ}
def IsTuranMaximal (r : ℕ) : Prop :=
G.CliqueFree (r + 1) ∧ ∀ (H : SimpleGraph V) [DecidableRel H.Adj],
H.CliqueFree (r +... | Mathlib/Combinatorics/SimpleGraph/Turan.lean | 84 | 92 | theorem not_cliqueFree_of_isTuranMaximal (hn : r ≤ Fintype.card V) (hG : G.IsTuranMaximal r) :
¬G.CliqueFree r := by |
rintro h
obtain ⟨K, _, rfl⟩ := exists_smaller_set (univ : Finset V) r hn
obtain ⟨a, -, b, -, hab, hGab⟩ : ∃ a ∈ K, ∃ b ∈ K, a ≠ b ∧ ¬ G.Adj a b := by
simpa only [isNClique_iff, IsClique, Set.Pairwise, mem_coe, ne_eq, and_true, not_forall,
exists_prop, exists_and_right] using h K
exact hGab <| le_sup_... |
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Comp
#align_import analysis.calculus.deriv.inv from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
open scoped Classical
open Topology Filter ENNReal
open Filter Asymptotics Set
open Cont... | Mathlib/Analysis/Calculus/Deriv/Inv.lean | 98 | 101 | theorem derivWithin_inv (x_ne_zero : x ≠ 0) (hxs : UniqueDiffWithinAt 𝕜 s x) :
derivWithin (fun x => x⁻¹) s x = -(x ^ 2)⁻¹ := by |
rw [DifferentiableAt.derivWithin (differentiableAt_inv.2 x_ne_zero) hxs]
exact deriv_inv
|
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanprover-community/mathlib"@"00f91228655eecdcd3ac97a7fd8dbcb139fe990a"
universe u v w
open scoped Classical Polynomial
open Polynomial
vari... | Mathlib/FieldTheory/IsAlgClosed/Basic.lean | 68 | 69 | theorem IsAlgClosed.splits_codomain {k K : Type*} [Field k] [IsAlgClosed k] [Field K] {f : K →+* k}
(p : K[X]) : p.Splits f := by | convert IsAlgClosed.splits (p.map f); simp [splits_map_iff]
|
import Mathlib.Data.Finset.Sort
import Mathlib.Data.List.FinRange
import Mathlib.Data.Prod.Lex
import Mathlib.GroupTheory.Perm.Basic
import Mathlib.Order.Interval.Finset.Fin
#align_import data.fin.tuple.sort from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
namespace Tuple
variable {... | Mathlib/Data/Fin/Tuple/Sort.lean | 50 | 57 | theorem graph.card (f : Fin n → α) : (graph f).card = n := by |
rw [graph, Finset.card_image_of_injective]
· exact Finset.card_fin _
· intro _ _
-- porting note (#10745): was `simp`
dsimp only
rw [Prod.ext_iff]
simp
|
import Mathlib.Analysis.NormedSpace.Multilinear.Basic
import Mathlib.Analysis.NormedSpace.Units
import Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness
import Mathlib.Analysis.NormedSpace.OperatorNorm.Mul
#align_import analysis.normed_space.bounded_linear_maps from "leanprover-community/mathlib"@"ce11c3c2a285b... | Mathlib/Analysis/NormedSpace/BoundedLinearMaps.lean | 121 | 124 | theorem snd : IsBoundedLinearMap 𝕜 fun x : E × F => x.2 := by |
refine (LinearMap.snd 𝕜 E F).isLinear.with_bound 1 fun x => ?_
rw [one_mul]
exact le_max_right _ _
|
import Mathlib.Algebra.Category.ModuleCat.Abelian
import Mathlib.CategoryTheory.Limits.Shapes.Images
#align_import algebra.category.Module.images from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open CategoryTheory
open CategoryTheory.Limits
universe u v
namespace ModuleCat
set_op... | Mathlib/Algebra/Category/ModuleCat/Images.lean | 117 | 119 | theorem imageIsoRange_hom_subtype {G H : ModuleCat.{v} R} (f : G ⟶ H) :
(imageIsoRange f).hom ≫ ModuleCat.ofHom f.range.subtype = Limits.image.ι f := by |
erw [← imageIsoRange_inv_image_ι f, Iso.hom_inv_id_assoc]
|
import Mathlib.CategoryTheory.Galois.Basic
import Mathlib.RepresentationTheory.Action.Basic
import Mathlib.RepresentationTheory.Action.Concrete
import Mathlib.RepresentationTheory.Action.Limits
import Mathlib.CategoryTheory.Limits.FintypeCat
import Mathlib.CategoryTheory.Limits.Shapes.Types
import Mathlib.Logic.Equiv.... | Mathlib/CategoryTheory/Galois/Examples.lean | 127 | 145 | theorem Action.isConnected_of_transitive (X : FintypeCat) [MulAction G X]
[MulAction.IsPretransitive G X] [h : Nonempty X] :
IsConnected (Action.FintypeCat.ofMulAction G X) where
notInitial := not_initial_of_inhabited (Action.forget _ _) h.some
noTrivialComponent Y i hm hni := by |
/- We show that the induced inclusion `i.hom` of finite sets is surjective, using the
transitivity of the `G`-action. -/
obtain ⟨(y : Y.V)⟩ := (not_initial_iff_fiber_nonempty (Action.forget _ _) Y).mp hni
have : IsIso i.hom := by
refine (ConcreteCategory.isIso_iff_bijective i.hom).mpr ⟨?_, fun x'... |
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 | 95 | 97 | theorem support_formPerm_le' : { x | formPerm l x ≠ x } ≤ l.toFinset := by |
refine (zipWith_swap_prod_support' l l.tail).trans ?_
simpa [Finset.subset_iff] using tail_subset l
|
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Real.Cardinality
#align_import data.complex.cardinality from "leanprover-community/mathlib"@"1c4e18434eeb5546b212e830b2b39de6a83c473c"
-- Porting note: the lemmas `mk_complex` and `mk_univ_complex` should be in the namespace `Cardinal`
-- like their real counter... | Mathlib/Data/Complex/Cardinality.lean | 31 | 31 | theorem mk_univ_complex : #(Set.univ : Set ℂ) = 𝔠 := by | rw [mk_univ, mk_complex]
|
import Mathlib.CategoryTheory.Adjunction.Basic
open CategoryTheory
variable {C D : Type*} [Category C] [Category D]
namespace CategoryTheory.Adjunction
@[simps]
def natTransEquiv {F F' : C ⥤ D} {G G' : D ⥤ C} (adj1 : F ⊣ G) (adj2 : F' ⊣ G') :
(G ⟶ G') ≃ (F' ⟶ F) where
toFun f := {
app := fun X ↦ F'.map... | Mathlib/CategoryTheory/Adjunction/Unique.lean | 123 | 127 | theorem unit_leftAdjointUniq_hom {F F' : C ⥤ D} {G : D ⥤ C} (adj1 : F ⊣ G) (adj2 : F' ⊣ G) :
adj1.unit ≫ whiskerRight (leftAdjointUniq adj1 adj2).hom G = adj2.unit := by |
ext x
rw [NatTrans.comp_app, ← homEquiv_leftAdjointUniq_hom_app adj1 adj2]
simp [← G.map_comp]
|
import Mathlib.LinearAlgebra.Dimension.Basic
import Mathlib.SetTheory.Cardinal.ToNat
#align_import linear_algebra.finrank from "leanprover-community/mathlib"@"347636a7a80595d55bedf6e6fbd996a3c39da69a"
universe u v w
open Cardinal Submodule Module Function
variable {R : Type u} {M : Type v} {N : Type w}
variable... | Mathlib/LinearAlgebra/Dimension/Finrank.lean | 72 | 75 | theorem finrank_le_of_rank_le {n : ℕ} (h : Module.rank R M ≤ ↑n) : finrank R M ≤ n := by |
rwa [← Cardinal.toNat_le_iff_le_of_lt_aleph0, toNat_natCast] at h
· exact h.trans_lt (nat_lt_aleph0 n)
· exact nat_lt_aleph0 n
|
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.RingTheory.MvPowerSeries.Basic
#align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60"
noncomputable section
... | Mathlib/RingTheory/PowerSeries/Basic.lean | 150 | 151 | theorem coeff_def {s : Unit →₀ ℕ} {n : ℕ} (h : s () = n) : coeff R n = MvPowerSeries.coeff R s := by |
erw [coeff, ← h, ← Finsupp.unique_single s]
|
import Mathlib.Algebra.Ring.Int
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.Size
#align_import data.int.bitwise from "leanprover-community/mathlib"@"0743cc5d9d86bcd1bba10f480e948a257d65056f"
#align_import init.data.int.bitwise from "leanprover-community/lean"@"855e5b74e3a52a40552e8f067169d747d48743fd"
... | Mathlib/Data/Int/Bitwise.lean | 159 | 167 | theorem bodd_neg (n : ℤ) : bodd (-n) = bodd n := by |
cases n with
| ofNat =>
rw [← negOfNat_eq, bodd_negOfNat]
simp
| negSucc n =>
rw [neg_negSucc, bodd_coe, Nat.bodd_succ]
change (!Nat.bodd n) = !(bodd n)
rw [bodd_coe]
|
import Mathlib.FieldTheory.SeparableClosure
import Mathlib.Algebra.CharP.IntermediateField
open FiniteDimensional Polynomial IntermediateField Field
noncomputable section
universe u v w
variable (F : Type u) (E : Type v) [Field F] [Field E] [Algebra F E]
variable (K : Type w) [Field K] [Algebra F K]
section IsP... | Mathlib/FieldTheory/PurelyInseparable.lean | 230 | 243 | theorem isPurelyInseparable_iff_pow_mem (q : ℕ) [ExpChar F q] :
IsPurelyInseparable F E ↔ ∀ x : E, ∃ n : ℕ, x ^ q ^ n ∈ (algebraMap F E).range := by |
rw [isPurelyInseparable_iff]
refine ⟨fun h x ↦ ?_, fun h x ↦ ?_⟩
· obtain ⟨g, h1, n, h2⟩ := (minpoly.irreducible (h x).1).hasSeparableContraction q
exact ⟨n, (h _).2 <| h1.of_dvd <| minpoly.dvd F _ <| by
simpa only [expand_aeval, minpoly.aeval] using congr_arg (aeval x) h2⟩
have hdeg := (minpoly.natS... |
import Mathlib.Algebra.Ring.Pi
import Mathlib.Algebra.Ring.Prod
import Mathlib.Algebra.Ring.InjSurj
import Mathlib.Tactic.Monotonicity.Attr
#align_import algebra.order.kleene from "leanprover-community/mathlib"@"98e83c3d541c77cdb7da20d79611a780ff8e7d90"
open Function
universe u
variable {α β ι : Type*} {π : ι →... | Mathlib/Algebra/Order/Kleene.lean | 154 | 154 | theorem add_eq_right_iff_le : a + b = b ↔ a ≤ b := by | simp
|
import Mathlib.Combinatorics.Quiver.Cast
import Mathlib.Combinatorics.Quiver.Symmetric
#align_import combinatorics.quiver.single_obj from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
namespace Quiver
-- Porting note: Removed `deriving Unique`.
@[nolint unusedArguments]
def SingleObj ... | Mathlib/Combinatorics/Quiver/SingleObj.lean | 110 | 112 | theorem toPrefunctor_symm_comp (f : SingleObj α ⥤q SingleObj β) (g : SingleObj β ⥤q SingleObj γ) :
toPrefunctor.symm (f ⋙q g) = toPrefunctor.symm g ∘ toPrefunctor.symm f := by |
simp only [Equiv.symm_apply_eq, toPrefunctor_comp, Equiv.apply_symm_apply]
|
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 | 140 | 145 | theorem natDegree_pow (p : R[X]) (n : ℕ) : natDegree (p ^ n) = n * natDegree p := by |
classical
obtain rfl | hp := eq_or_ne p 0
· obtain rfl | hn := eq_or_ne n 0 <;> simp [*]
exact natDegree_pow' $ by
rw [← leadingCoeff_pow, Ne, leadingCoeff_eq_zero]; exact pow_ne_zero _ hp
|
import Mathlib.Data.List.Join
#align_import data.list.permutation from "leanprover-community/mathlib"@"dd71334db81d0bd444af1ee339a29298bef40734"
-- Make sure we don't import algebra
assert_not_exists Monoid
open Nat
variable {α β : Type*}
namespace List
theorem permutationsAux2_fst (t : α) (ts : List α) (r : L... | Mathlib/Data/List/Permutation.lean | 77 | 79 | theorem permutationsAux2_append (t : α) (ts : List α) (r : List β) (ys : List α) (f : List α → β) :
(permutationsAux2 t ts nil ys f).2 ++ r = (permutationsAux2 t ts r ys f).2 := by |
induction ys generalizing f <;> simp [*]
|
import Mathlib.Geometry.Manifold.ContMDiff.Defs
open Set Filter Function
open scoped Topology Manifold
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜]
-- declare a smooth manifold `M` over the pair `(E, H)`.
{E : Type*}
[NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type*} [TopologicalSpace H]
(I : Mode... | Mathlib/Geometry/Manifold/ContMDiff/Basic.lean | 262 | 262 | theorem smooth_one [One M'] : Smooth I I' (1 : M → M') := by | simp only [Pi.one_def, smooth_const]
|
import Mathlib.Analysis.InnerProductSpace.Adjoint
#align_import analysis.inner_product_space.positive from "leanprover-community/mathlib"@"caa58cbf5bfb7f81ccbaca4e8b8ac4bc2b39cc1c"
open InnerProductSpace RCLike ContinuousLinearMap
open scoped InnerProduct ComplexConjugate
namespace ContinuousLinearMap
variable... | Mathlib/Analysis/InnerProductSpace/Positive.lean | 67 | 68 | theorem IsPositive.inner_nonneg_right {T : E →L[𝕜] E} (hT : IsPositive T) (x : E) :
0 ≤ re ⟪x, T x⟫ := by | rw [inner_re_symm]; exact hT.inner_nonneg_left x
|
import Mathlib.MeasureTheory.Group.GeometryOfNumbers
import Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls
import Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic
#align_import number_theory.number_field.canonical_embedding from "leanprover-community/mathlib"@"60da01b41bbe4206f05d34fd70c8dd7498717a30"
... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean | 316 | 324 | theorem convexBodySumFun_eq_zero_iff (x : E K) :
convexBodySumFun x = 0 ↔ x = 0 := by |
rw [← normAtPlace_eq_zero, convexBodySumFun, Finset.sum_eq_zero_iff_of_nonneg fun _ _ =>
mul_nonneg (Nat.cast_pos.mpr mult_pos).le (normAtPlace_nonneg _ _)]
conv =>
enter [1, w, hw]
rw [mul_left_mem_nonZeroDivisors_eq_zero_iff
(mem_nonZeroDivisors_iff_ne_zero.mpr <| Nat.cast_ne_zero.mpr mult_ne_z... |
import Mathlib.LinearAlgebra.Isomorphisms
import Mathlib.Algebra.Category.ModuleCat.Kernels
import Mathlib.Algebra.Category.ModuleCat.Limits
import Mathlib.CategoryTheory.Abelian.Exact
#align_import algebra.category.Module.abelian from "leanprover-community/mathlib"@"09f981f72d43749f1fa072deade828d9c1e185bb"
open... | Mathlib/Algebra/Category/ModuleCat/Abelian.lean | 123 | 127 | theorem exact_iff : Exact f g ↔ LinearMap.range f = LinearMap.ker g := by |
rw [abelian.exact_iff' f g (kernelIsLimit _) (cokernelIsColimit _)]
exact
⟨fun h => le_antisymm (range_le_ker_iff.2 h.1) (ker_le_range_iff.2 h.2), fun h =>
⟨range_le_ker_iff.1 <| le_of_eq h, ker_le_range_iff.1 <| le_of_eq h.symm⟩⟩
|
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 | 62 | 65 | theorem le_rpow_one_add_norm_iff_norm_le {r t : ℝ} (hr : 0 < r) (ht : 0 < t) (x : E) :
t ≤ (1 + ‖x‖) ^ (-r) ↔ ‖x‖ ≤ t ^ (-r⁻¹) - 1 := by |
rw [le_sub_iff_add_le', neg_inv]
exact (Real.le_rpow_inv_iff_of_neg (by positivity) ht (neg_lt_zero.mpr hr)).symm
|
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.Tactic.ByContra
import Mathlib.Topology.Algebra.Polynomial
import Mathlib.NumberTheory.Padics.PadicVal
import Mathlib.Analysis.Complex.Arg
#align_import ring_theory.polynomial.cyclotomic.eval from "leanprover-community/mathlib"@"5bfbcca0a7ffdd21cf16... | Mathlib/RingTheory/Polynomial/Cyclotomic/Eval.lean | 36 | 37 | theorem eval₂_one_cyclotomic_prime {R S : Type*} [CommRing R] [Semiring S] (f : R →+* S) {p : ℕ}
[Fact p.Prime] : eval₂ f 1 (cyclotomic p R) = p := by | simp
|
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 | 125 | 125 | theorem rpow_zero (x : ℝ) : x ^ (0 : ℝ) = 1 := by | simp [rpow_def]
|
import Mathlib.Dynamics.Ergodic.AddCircle
import Mathlib.MeasureTheory.Covering.LiminfLimsup
#align_import number_theory.well_approximable from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open Set Filter Function Metric MeasureTheory
open scoped MeasureTheory Topology Pointwise
@[... | Mathlib/NumberTheory/WellApproximable.lean | 147 | 166 | theorem smul_eq_of_mul_dvd (hn : 0 < n) (han : orderOf a ^ 2 ∣ n) :
a • approxOrderOf A n δ = approxOrderOf A n δ := by |
simp_rw [approxOrderOf, thickening_eq_biUnion_ball, ← image_smul, image_iUnion₂, image_smul,
smul_ball'', smul_eq_mul, mem_setOf_eq]
replace han : ∀ {b : A}, orderOf b = n → orderOf (a * b) = n := by
intro b hb
rw [← hb] at han hn
rw [sq] at han
rwa [(Commute.all a b).orderOf_mul_eq_right_of_fo... |
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Data.Rat.Cast.Defs
#align_import data.rat.cast from "leanprover-community/mathlib"@"acebd8d49928f6ed8920e502a6c90674e75bd441"
variable {F ι α β : Type*}
namespace Rat
open Rat
section WithDivRing
variable [DivisionRing α]
@[simp, norm_cast]
th... | Mathlib/Data/Rat/Cast/CharZero.lean | 46 | 46 | theorem cast_eq_zero [CharZero α] {n : ℚ} : (n : α) = 0 ↔ n = 0 := by | rw [← cast_zero, cast_inj]
|
import Mathlib.Analysis.Calculus.FDeriv.Bilinear
#align_import analysis.calculus.fderiv.mul from "leanprover-community/mathlib"@"d608fc5d4e69d4cc21885913fb573a88b0deb521"
open scoped Classical
open Filter Asymptotics ContinuousLinearMap Set Metric Topology NNReal ENNReal
noncomputable section
section
variable ... | Mathlib/Analysis/Calculus/FDeriv/Mul.lean | 230 | 233 | theorem fderiv_continuousMultilinear_apply_const_apply (hc : DifferentiableAt 𝕜 c x)
(u : ∀ i, M i) (m : E) :
(fderiv 𝕜 (fun y ↦ (c y) u) x) m = (fderiv 𝕜 c x) m u := by |
simp [fderiv_continuousMultilinear_apply_const hc]
|
import Mathlib.Topology.Category.TopCat.EpiMono
import Mathlib.Topology.Category.TopCat.Limits.Basic
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.ConcreteCategory
import Mathlib.Data.Set.Subsingleton
import Mathlib.Tactic.CategoryTheory.Elementwise
#align_import topology.c... | Mathlib/Topology/Category/TopCat/Limits/Products.lean | 82 | 86 | theorem piIsoPi_hom_apply {ι : Type v} (α : ι → TopCat.{max v u}) (i : ι)
(x : (∏ᶜ α : TopCat.{max v u})) : (piIsoPi α).hom x i = (Pi.π α i : _) x := by |
have := piIsoPi_inv_π α i
rw [Iso.inv_comp_eq] at this
exact ConcreteCategory.congr_hom this x
|
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Analysis.Convex.Star
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
#align_import analysis.convex.basic from "leanprover-community/mathlib"@"92bd7b1ffeb306a89f450bee126ddd8a284c259d"
variable {𝕜 E F β : Type*}
open LinearMap Set
open scope... | Mathlib/Analysis/Convex/Basic.lean | 121 | 128 | theorem Directed.convex_iUnion {ι : Sort*} {s : ι → Set E} (hdir : Directed (· ⊆ ·) s)
(hc : ∀ ⦃i : ι⦄, Convex 𝕜 (s i)) : Convex 𝕜 (⋃ i, s i) := by |
rintro x hx y hy a b ha hb hab
rw [mem_iUnion] at hx hy ⊢
obtain ⟨i, hx⟩ := hx
obtain ⟨j, hy⟩ := hy
obtain ⟨k, hik, hjk⟩ := hdir i j
exact ⟨k, hc (hik hx) (hjk hy) ha hb hab⟩
|
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.SchwartzSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.fourier.poisson_summation from "leanprover-community/mathlib"@"fd5... | Mathlib/Analysis/Fourier/PoissonSummation.lean | 56 | 103 | theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)}
(hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖)
(m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 f m := by |
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩
have neK : ∀ (K : Comp... |
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
#align_import linear_algebra.clifford_algebra.star from "leanprover-community/mathlib"@"4d66277cfec381260ba05c68f9ae6ce2a118031d"
variable {R : Type*} [CommRing R]
variable {M : Type*} [AddCommGroup M] [Module R M]
variable {Q : QuadraticForm R M}
namespac... | Mathlib/LinearAlgebra/CliffordAlgebra/Star.lean | 57 | 58 | theorem star_smul (r : R) (x : CliffordAlgebra Q) : star (r • x) = r • star x := by |
rw [star_def, star_def, map_smul, map_smul]
|
import Mathlib.Data.Countable.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Order.Disjointed
import Mathlib.MeasureTheory.OuterMeasure.Defs
#align_import measure_theory.measure.outer_measure from "leanprover-community/mathlib"@"343e80208d29d2d15f8050b929aa50fe4ce71b55"
noncomputable section
open Set F... | Mathlib/MeasureTheory/OuterMeasure/Basic.lean | 84 | 86 | theorem measure_iUnion_fintype_le [Fintype ι] (μ : F) (s : ι → Set α) :
μ (⋃ i, s i) ≤ ∑ i, μ (s i) := by |
simpa using measure_biUnion_finset_le Finset.univ s
|
import Mathlib.Topology.ExtendFrom
import Mathlib.Topology.Order.DenselyOrdered
#align_import topology.algebra.order.extend_from from "leanprover-community/mathlib"@"0a0ec35061ed9960bf0e7ffb0335f44447b58977"
set_option autoImplicit true
open Filter Set TopologicalSpace
open scoped Classical
open Topology
| Mathlib/Topology/Order/ExtendFrom.lean | 23 | 33 | theorem continuousOn_Icc_extendFrom_Ioo [TopologicalSpace α] [LinearOrder α] [DenselyOrdered α]
[OrderTopology α] [TopologicalSpace β] [RegularSpace β] {f : α → β} {a b : α} {la lb : β}
(hab : a ≠ b) (hf : ContinuousOn f (Ioo a b)) (ha : Tendsto f (𝓝[>] a) (𝓝 la))
(hb : Tendsto f (𝓝[<] b) (𝓝 lb)) : Cont... |
apply continuousOn_extendFrom
· rw [closure_Ioo hab]
· intro x x_in
rcases eq_endpoints_or_mem_Ioo_of_mem_Icc x_in with (rfl | rfl | h)
· exact ⟨la, ha.mono_left <| nhdsWithin_mono _ Ioo_subset_Ioi_self⟩
· exact ⟨lb, hb.mono_left <| nhdsWithin_mono _ Ioo_subset_Iio_self⟩
· exact ⟨f x, hf x h⟩
|
import Mathlib.Algebra.Order.Pointwise
import Mathlib.Analysis.NormedSpace.SphereNormEquiv
import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.MeasureTheory.Constructions.Prod.Integral
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
open Set Function Metric MeasurableSpace intervalIntegral
open s... | Mathlib/MeasureTheory/Constructions/HaarToSphere.lean | 62 | 63 | theorem toSphere_apply_univ' : μ.toSphere univ = dim E * μ (ball 0 1 \ {0}) := by |
rw [μ.toSphere_apply' .univ, image_univ, Subtype.range_coe, Ioo_smul_sphere_zero] <;> simp
|
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