Context stringlengths 57 92.3k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 18 3.69k | theorem stringlengths 25 2.71k | proof stringlengths 5 10.6k |
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import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Order.Bounds.OrderIso
import Mathlib.Tactic.Positivity.Core
#align_import algebra.order.field.basic from "leanprover-community/mathlib"@"8477... | Mathlib/Algebra/Order/Field/Basic.lean | 382 | 382 | theorem le_one_div (ha : 0 < a) (hb : 0 < b) : a ≤ 1 / b ↔ b ≤ 1 / a := by | simpa using le_inv ha hb
|
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Function.SimpleFunc
import Mathlib.MeasureTheory.Measure.MutuallySingular
import Mathlib.MeasureTheory.Measure.Count
import Mathlib.Topology.IndicatorConstPointwise
import Mathlib.MeasureTheory.Constructions.BorelSpace.Real
#align_import m... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 689 | 694 | theorem lintegral_const_mul'' (r : ℝ≥0∞) {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) :
∫⁻ a, r * f a ∂μ = r * ∫⁻ a, f a ∂μ := by |
have A : ∫⁻ a, f a ∂μ = ∫⁻ a, hf.mk f a ∂μ := lintegral_congr_ae hf.ae_eq_mk
have B : ∫⁻ a, r * f a ∂μ = ∫⁻ a, r * hf.mk f a ∂μ :=
lintegral_congr_ae (EventuallyEq.fun_comp hf.ae_eq_mk _)
rw [A, B, lintegral_const_mul _ hf.measurable_mk]
|
import Mathlib.Analysis.Normed.Group.Basic
import Mathlib.Topology.MetricSpace.Thickening
import Mathlib.Topology.MetricSpace.IsometricSMul
#align_import analysis.normed.group.pointwise from "leanprover-community/mathlib"@"c8f305514e0d47dfaa710f5a52f0d21b588e6328"
open Metric Set Pointwise Topology
variable {E :... | Mathlib/Analysis/Normed/Group/Pointwise.lean | 183 | 184 | theorem closedBall_mul_singleton : closedBall x δ * {y} = closedBall (x * y) δ := by |
simp [mul_comm _ {y}, mul_comm y]
|
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Algebra.Pi
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Algebra.BigOperators.RingEquiv
import Mathlib.Algebra.Module.LinearMap.Basic
import Mathlib.Algebra.Module.Pi
import Mathlib.Algebra.Star.BigOperators... | Mathlib/Data/Matrix/Basic.lean | 1,148 | 1,151 | theorem map_mul [Fintype n] {L : Matrix m n α} {M : Matrix n o α} [NonAssocSemiring β]
{f : α →+* β} : (L * M).map f = L.map f * M.map f := by |
ext
simp [mul_apply, map_sum]
|
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Fintype.BigOperators
#align_import data.sign from "leanprover-community/mathlib"@"2445c98ae4b87eabebdde552593519b9b6dc350c"
-- Porting note (#11081): cannot automatically derive Fintype, adde... | Mathlib/Data/Sign.lean | 525 | 526 | theorem sign_eq_sign (n : ℤ) : Int.sign n = SignType.sign n := by |
obtain (n | _) | _ := n <;> simp [sign, Int.sign_neg, negSucc_lt_zero]
|
import Mathlib.RingTheory.Flat.Basic
import Mathlib.LinearAlgebra.TensorProduct.Vanishing
import Mathlib.Algebra.Module.FinitePresentation
universe u
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M]
open Classical DirectSum LinearMap TensorProduct Finsupp
open scoped BigOperators
namespace Modu... | Mathlib/RingTheory/Flat/EquationalCriterion.lean | 248 | 252 | theorem exists_factorization_of_apply_eq_zero_of_free [Flat R M] {N : Type u}
[AddCommGroup N] [Module R N] [Free R N] [Finite R N] {f : N} {x : N →ₗ[R] M} (h : x f = 0) :
∃ (κ : Type u) (_ : Fintype κ) (a : N →ₗ[R] (κ →₀ R)) (y : (κ →₀ R) →ₗ[R] M),
x = y ∘ₗ a ∧ a f = 0 := by |
exact ((tfae_equational_criterion R M).out 0 5 rfl rfl).mp ‹Flat R M› h
|
import Mathlib.Algebra.Ring.Divisibility.Lemmas
import Mathlib.Algebra.Lie.Nilpotent
import Mathlib.Algebra.Lie.Engel
import Mathlib.LinearAlgebra.Eigenspace.Triangularizable
import Mathlib.RingTheory.Artinian
import Mathlib.LinearAlgebra.Trace
import Mathlib.LinearAlgebra.FreeModule.PID
#align_import algebra.lie.wei... | Mathlib/Algebra/Lie/Weights/Basic.lean | 334 | 336 | theorem isNilpotent_toEnd_weightSpace_zero [IsNoetherian R M] (x : L) :
_root_.IsNilpotent <| toEnd R L (weightSpace M (0 : L → R)) x := by |
simpa using isNilpotent_toEnd_sub_algebraMap M (0 : L → R) x
|
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.CharZero.Lemmas
import Mathlib.Data.Finset.NatAntidiagonal
import Mathlib.Data.Nat.Choose.Central
import Mathlib.Data.Tree.Basic
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.GCongr
import Mathlib... | Mathlib/Combinatorics/Enumerative/Catalan.lean | 79 | 79 | theorem catalan_one : catalan 1 = 1 := by | simp [catalan_succ]
|
import Mathlib.Algebra.Module.Submodule.EqLocus
import Mathlib.Algebra.Module.Submodule.RestrictScalars
import Mathlib.Algebra.Ring.Idempotents
import Mathlib.Data.Set.Pointwise.SMul
import Mathlib.LinearAlgebra.Basic
import Mathlib.Order.CompactlyGenerated.Basic
import Mathlib.Order.OmegaCompletePartialOrder
#align_... | Mathlib/LinearAlgebra/Span.lean | 276 | 277 | theorem span_nat_eq (s : AddSubmonoid M) : (span ℕ (s : Set M)).toAddSubmonoid = s := by |
rw [span_nat_eq_addSubmonoid_closure, s.closure_eq]
|
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.Polynomial.Coeff
import Mathlib.Algebra.Polynomial.Monomial
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Nat.WithBot
import Mathlib.Data.Nat.Cast.WithTop
import Mathlib.Data.Nat.SuccPred
#align_import data.polynomial.degree.definitions... | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 1,575 | 1,577 | theorem leadingCoeff_pow_X_add_C (r : R) (i : ℕ) : leadingCoeff ((X + C r) ^ i) = 1 := by |
nontriviality
rw [leadingCoeff_pow'] <;> simp
|
import Mathlib.Algebra.Lie.OfAssociative
import Mathlib.Algebra.Lie.IdealOperations
#align_import algebra.lie.abelian from "leanprover-community/mathlib"@"8983bec7cdf6cb2dd1f21315c8a34ab00d7b2f6d"
universe u v w w₁ w₂
class LieModule.IsTrivial (L : Type v) (M : Type w) [Bracket L M] [Zero M] : Prop where
triv... | Mathlib/Algebra/Lie/Abelian.lean | 201 | 203 | theorem maxTrivEquiv_of_refl_eq_refl :
maxTrivEquiv (LieModuleEquiv.refl : M ≃ₗ⁅R,L⁆ M) = LieModuleEquiv.refl := by |
ext; simp only [coe_maxTrivEquiv_apply, LieModuleEquiv.refl_apply]
|
import Mathlib.CategoryTheory.NatIso
#align_import category_theory.bicategory.basic from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
namespace CategoryTheory
universe w v u
open Category Iso
-- intended to be used with explicit universe parameters
@[nolint checkUnivs]
class Bicate... | Mathlib/CategoryTheory/Bicategory/Basic.lean | 327 | 329 | theorem triangle_assoc_comp_right_inv (f : a ⟶ b) (g : b ⟶ c) :
(ρ_ f).inv ▷ g ≫ (α_ f (𝟙 b) g).hom = f ◁ (λ_ g).inv := by |
simp [← cancel_mono (f ◁ (λ_ g).hom)]
|
import Mathlib.Data.Finset.NAry
import Mathlib.Data.Finset.Preimage
import Mathlib.Data.Set.Pointwise.Finite
import Mathlib.Data.Set.Pointwise.SMul
import Mathlib.Data.Set.Pointwise.ListOfFn
import Mathlib.GroupTheory.GroupAction.Pi
import Mathlib.SetTheory.Cardinal.Finite
#align_import data.finset.pointwise from "le... | Mathlib/Data/Finset/Pointwise.lean | 2,213 | 2,213 | theorem smul_zero_subset (s : Finset α) : s • (0 : Finset β) ⊆ 0 := by | simp [subset_iff, mem_smul]
|
import Mathlib.Analysis.BoxIntegral.Partition.Additive
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
#align_import analysis.box_integral.partition.measure from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
open Set
noncomputable section
open scoped ENNReal Classical BoxIntegral... | Mathlib/Analysis/BoxIntegral/Partition/Measure.lean | 85 | 89 | theorem Prepartition.measure_iUnion_toReal [Finite ι] {I : Box ι} (π : Prepartition I)
(μ : Measure (ι → ℝ)) [IsLocallyFiniteMeasure μ] :
(μ π.iUnion).toReal = ∑ J ∈ π.boxes, (μ J).toReal := by |
erw [← ENNReal.toReal_sum, π.iUnion_def, measure_biUnion_finset π.pairwiseDisjoint]
exacts [fun J _ => J.measurableSet_coe, fun J _ => (J.measure_coe_lt_top μ).ne]
|
import Mathlib.Algebra.Order.Monoid.Unbundled.Pow
import Mathlib.Data.Finset.Fold
import Mathlib.Data.Finset.Option
import Mathlib.Data.Finset.Pi
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Multiset.Lattice
import Mathlib.Data.Set.Lattice
import Mathlib.Order.Hom.Lattice
import Mathlib.Order.Nat
#align_import... | Mathlib/Data/Finset/Lattice.lean | 196 | 200 | theorem sup_erase_bot [DecidableEq α] (s : Finset α) : (s.erase ⊥).sup id = s.sup id := by |
refine (sup_mono (s.erase_subset _)).antisymm (Finset.sup_le_iff.2 fun a ha => ?_)
obtain rfl | ha' := eq_or_ne a ⊥
· exact bot_le
· exact le_sup (mem_erase.2 ⟨ha', ha⟩)
|
import Mathlib.Algebra.CharP.Quotient
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Data.Finsupp.Fintype
import Mathlib.Data.Int.AbsoluteValue
import Mathlib.Data.Int.Associated
import Mathlib.LinearAlgebra.FreeModule.Determinant
import Mathlib.LinearAlgebra.FreeModule.IdealQuotient
import Mathli... | Mathlib/RingTheory/Ideal/Norm.lean | 273 | 273 | theorem absNorm_top : absNorm (⊤ : Ideal S) = 1 := by | rw [← Ideal.one_eq_top, _root_.map_one]
|
import Mathlib.Algebra.Star.Subalgebra
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.Tactic.NoncommRing
#align_import algebra.algebra.spectrum from "leanprover-community/mathlib"@"58a272265b5e05f258161260dd2c5d247213cbd3"
open Set
open scoped Pointwise
universe u v
namespace spectrum
section ScalarSemir... | Mathlib/Algebra/Algebra/Spectrum.lean | 162 | 163 | theorem of_subsingleton [Subsingleton A] (a : A) : spectrum R a = ∅ := by |
rw [spectrum, resolventSet_of_subsingleton, Set.compl_univ]
|
import Mathlib.Topology.MetricSpace.Basic
#align_import topology.metric_space.infsep from "leanprover-community/mathlib"@"5316314b553dcf8c6716541851517c1a9715e22b"
variable {α β : Type*}
namespace Set
section Einfsep
open ENNReal
open Function
noncomputable def einfsep [EDist α] (s : Set α) : ℝ≥0∞ :=
⨅ (x... | Mathlib/Topology/MetricSpace/Infsep.lean | 155 | 156 | theorem einfsep_pair_le_right (hxy : x ≠ y) : ({x, y} : Set α).einfsep ≤ edist y x := by |
rw [pair_comm]; exact einfsep_pair_le_left hxy.symm
|
import Mathlib.RingTheory.DedekindDomain.Ideal
#align_import number_theory.ramification_inertia from "leanprover-community/mathlib"@"039a089d2a4b93c761b234f3e5f5aeb752bac60f"
namespace Ideal
universe u v
variable {R : Type u} [CommRing R]
variable {S : Type v} [CommRing S] (f : R →+* S)
variable (p : Ideal R) (... | Mathlib/NumberTheory/RamificationInertia.lean | 808 | 839 | theorem sum_ramification_inertia (K L : Type*) [Field K] [Field L] [IsDedekindDomain R]
[Algebra R K] [IsFractionRing R K] [Algebra S L] [IsFractionRing S L] [Algebra K L]
[Algebra R L] [IsScalarTower R S L] [IsScalarTower R K L] [IsNoetherian R S]
[IsIntegralClosure S R L] [p.IsMaximal] (hp0 : p ≠ ⊥) :
... |
set e := ramificationIdx (algebraMap R S) p
set f := inertiaDeg (algebraMap R S) p
have inj_RL : Function.Injective (algebraMap R L) := by
rw [IsScalarTower.algebraMap_eq R K L, RingHom.coe_comp]
exact (RingHom.injective _).comp (IsFractionRing.injective R K)
have inj_RS : Function.Injective (algebraMa... |
import Mathlib.Data.Set.Image
import Mathlib.Data.List.InsertNth
import Mathlib.Init.Data.List.Lemmas
#align_import data.list.lemmas from "leanprover-community/mathlib"@"2ec920d35348cb2d13ac0e1a2ad9df0fdf1a76b4"
open List
variable {α β γ : Type*}
namespace List
theorem injOn_insertNth_index_of_not_mem (l : List... | Mathlib/Data/List/Lemmas.lean | 44 | 52 | theorem foldr_range_subset_of_range_subset {f : β → α → α} {g : γ → α → α}
(hfg : Set.range f ⊆ Set.range g) (a : α) : Set.range (foldr f a) ⊆ Set.range (foldr g a) := by |
rintro _ ⟨l, rfl⟩
induction' l with b l H
· exact ⟨[], rfl⟩
· cases' hfg (Set.mem_range_self b) with c hgf
cases' H with m hgf'
rw [foldr_cons, ← hgf, ← hgf']
exact ⟨c :: m, rfl⟩
|
import Mathlib.Algebra.Order.Monoid.Unbundled.Pow
import Mathlib.Data.Finset.Fold
import Mathlib.Data.Finset.Option
import Mathlib.Data.Finset.Pi
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Multiset.Lattice
import Mathlib.Data.Set.Lattice
import Mathlib.Order.Hom.Lattice
import Mathlib.Order.Nat
#align_import... | Mathlib/Data/Finset/Lattice.lean | 247 | 253 | theorem sup_induction {p : α → Prop} (hb : p ⊥) (hp : ∀ a₁, p a₁ → ∀ a₂, p a₂ → p (a₁ ⊔ a₂))
(hs : ∀ b ∈ s, p (f b)) : p (s.sup f) := by |
induction s using Finset.cons_induction with
| empty => exact hb
| cons _ _ _ ih =>
simp only [sup_cons, forall_mem_cons] at hs ⊢
exact hp _ hs.1 _ (ih hs.2)
|
import Mathlib.CategoryTheory.Monoidal.Category
import Mathlib.CategoryTheory.Adjunction.FullyFaithful
import Mathlib.CategoryTheory.Products.Basic
#align_import category_theory.monoidal.functor from "leanprover-community/mathlib"@"3d7987cda72abc473c7cdbbb075170e9ac620042"
open CategoryTheory
universe v₁ v₂ v₃ u... | Mathlib/CategoryTheory/Monoidal/Functor.lean | 164 | 167 | theorem LaxMonoidalFunctor.left_unitality_inv (F : LaxMonoidalFunctor C D) (X : C) :
(λ_ (F.obj X)).inv ≫ F.ε ▷ F.obj X ≫ F.μ (𝟙_ C) X = F.map (λ_ X).inv := by |
rw [Iso.inv_comp_eq, F.left_unitality, Category.assoc, Category.assoc, ← F.toFunctor.map_comp,
Iso.hom_inv_id, F.toFunctor.map_id, comp_id]
|
import Mathlib.Data.Num.Lemmas
import Mathlib.Data.Nat.Prime
import Mathlib.Tactic.Ring
#align_import data.num.prime from "leanprover-community/mathlib"@"58581d0fe523063f5651df0619be2bf65012a94a"
namespace PosNum
def minFacAux (n : PosNum) : ℕ → PosNum → PosNum
| 0, _ => n
| fuel + 1, k =>
if n < k.bit1... | Mathlib/Data/Num/Prime.lean | 44 | 54 | theorem minFacAux_to_nat {fuel : ℕ} {n k : PosNum} (h : Nat.sqrt n < fuel + k.bit1) :
(minFacAux n fuel k : ℕ) = Nat.minFacAux n k.bit1 := by |
induction' fuel with fuel ih generalizing k <;> rw [minFacAux, Nat.minFacAux]
· rw [Nat.zero_add, Nat.sqrt_lt] at h
simp only [h, ite_true]
simp_rw [← mul_to_nat]
simp only [cast_lt, dvd_to_nat]
split_ifs <;> try rfl
rw [ih] <;> [congr; convert Nat.lt_succ_of_lt h using 1] <;>
simp only [_root_.bit... |
import Mathlib.Analysis.Convex.Side
import Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
#align_import geometry.euclidean.angle.oriented.affine from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open ... | Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean | 85 | 86 | theorem left_ne_right_of_oangle_ne_zero {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ ≠ 0) : p₁ ≠ p₃ := by |
rw [← (vsub_left_injective p₂).ne_iff]; exact o.ne_of_oangle_ne_zero h
|
import Mathlib.Algebra.Star.Order
import Mathlib.Topology.Instances.NNReal
import Mathlib.Topology.Order.MonotoneContinuity
#align_import data.real.sqrt from "leanprover-community/mathlib"@"31c24aa72e7b3e5ed97a8412470e904f82b81004"
open Set Filter
open scoped Filter NNReal Topology
namespace Real
noncomputable ... | Mathlib/Data/Real/Sqrt.lean | 252 | 254 | theorem sqrt_le_left (hy : 0 ≤ y) : √x ≤ y ↔ x ≤ y ^ 2 := by |
rw [sqrt, ← Real.le_toNNReal_iff_coe_le hy, NNReal.sqrt_le_iff_le_sq, sq, ← Real.toNNReal_mul hy,
Real.toNNReal_le_toNNReal_iff (mul_self_nonneg y), sq]
|
import Mathlib.Topology.Algebra.Module.WeakDual
import Mathlib.MeasureTheory.Integral.BoundedContinuousFunction
import Mathlib.MeasureTheory.Measure.HasOuterApproxClosed
#align_import measure_theory.measure.finite_measure from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable... | Mathlib/MeasureTheory/Measure/FiniteMeasure.lean | 207 | 209 | theorem mass_nonzero_iff (μ : FiniteMeasure Ω) : μ.mass ≠ 0 ↔ μ ≠ 0 := by |
rw [not_iff_not]
exact FiniteMeasure.mass_zero_iff μ
|
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 | 1,714 | 1,719 | theorem _root_.MeasurableEmbedding.integral_map {β} {_ : MeasurableSpace β} {f : α → β}
(hf : MeasurableEmbedding f) (g : β → G) : ∫ y, g y ∂Measure.map f μ = ∫ x, g (f x) ∂μ := by |
by_cases hgm : AEStronglyMeasurable g (Measure.map f μ)
· exact MeasureTheory.integral_map hf.measurable.aemeasurable hgm
· rw [integral_non_aestronglyMeasurable hgm, integral_non_aestronglyMeasurable]
exact fun hgf => hgm (hf.aestronglyMeasurable_map_iff.2 hgf)
|
import Mathlib.Algebra.Lie.Semisimple.Defs
import Mathlib.Order.BooleanGenerators
#align_import algebra.lie.semisimple from "leanprover-community/mathlib"@"356447fe00e75e54777321045cdff7c9ea212e60"
namespace LieAlgebra
variable (R L : Type*) [CommRing R] [LieRing L] [LieAlgebra R L]
variable {R L} in
theorem Has... | Mathlib/Algebra/Lie/Semisimple/Basic.lean | 318 | 324 | theorem abelian_radical_iff_solvable_is_abelian [IsNoetherian R L] :
IsLieAbelian (radical R L) ↔ ∀ I : LieIdeal R L, IsSolvable R I → IsLieAbelian I := by |
constructor
· rintro h₁ I h₂
rw [LieIdeal.solvable_iff_le_radical] at h₂
exact (LieIdeal.inclusion_injective h₂).isLieAbelian h₁
· intro h; apply h; infer_instance
|
import Mathlib.Order.Filter.SmallSets
import Mathlib.Tactic.Monotonicity
import Mathlib.Topology.Compactness.Compact
import Mathlib.Topology.NhdsSet
import Mathlib.Algebra.Group.Defs
#align_import topology.uniform_space.basic from "leanprover-community/mathlib"@"195fcd60ff2bfe392543bceb0ec2adcdb472db4c"
open Set F... | Mathlib/Topology/UniformSpace/Basic.lean | 1,655 | 1,665 | theorem uniformContinuous_inf_dom_right₂ {α β γ} {f : α → β → γ} {ua1 ua2 : UniformSpace α}
{ub1 ub2 : UniformSpace β} {uc1 : UniformSpace γ}
(h : by haveI := ua2; haveI := ub2; exact UniformContinuous fun p : α × β => f p.1 p.2) : by
haveI := ua1 ⊓ ua2; haveI := ub1 ⊓ ub2;
exact UniformContinuous... |
-- proof essentially copied from `continuous_inf_dom_right₂`
have ha := @UniformContinuous.inf_dom_right _ _ id ua1 ua2 ua2 (@uniformContinuous_id _ (id _))
have hb := @UniformContinuous.inf_dom_right _ _ id ub1 ub2 ub2 (@uniformContinuous_id _ (id _))
have h_unif_cont_id :=
@UniformContinuous.prod_map _ _... |
import Batteries.Classes.Order
namespace Batteries.PairingHeapImp
inductive Heap (α : Type u) where
| nil : Heap α
| node (a : α) (child sibling : Heap α) : Heap α
deriving Repr
def Heap.size : Heap α → Nat
| .nil => 0
| .node _ c s => c.size + 1 + s.size
def Heap.singleton (a : α) : Heap α := .... | .lake/packages/batteries/Batteries/Data/PairingHeap.lean | 95 | 101 | theorem Heap.noSibling_combine (le) (s : Heap α) :
(s.combine le).NoSibling := by |
unfold combine; split
· exact noSibling_merge _ _ _
· match s with
| nil | node _ _ nil => constructor
| node _ _ (node _ _ s) => rename_i h; exact (h _ _ _ _ _ rfl).elim
|
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Data.Finsupp.Fin
import Mathlib.Data.Finsupp.Indicator
#align_import algebra.bi... | Mathlib/Algebra/BigOperators/Finsupp.lean | 581 | 591 | theorem prod_add_index_of_disjoint [AddCommMonoid M] {f1 f2 : α →₀ M}
(hd : Disjoint f1.support f2.support) {β : Type*} [CommMonoid β] (g : α → M → β) :
(f1 + f2).prod g = f1.prod g * f2.prod g := by |
have :
∀ {f1 f2 : α →₀ M},
Disjoint f1.support f2.support → (∏ x ∈ f1.support, g x (f1 x + f2 x)) = f1.prod g :=
fun hd =>
Finset.prod_congr rfl fun x hx => by
simp only [not_mem_support_iff.mp (disjoint_left.mp hd hx), add_zero]
classical simp_rw [← this hd, ← this hd.symm, add_comm (f2 _)... |
import Mathlib.MeasureTheory.Decomposition.SignedHahn
import Mathlib.MeasureTheory.Measure.MutuallySingular
#align_import measure_theory.decomposition.jordan from "leanprover-community/mathlib"@"70a4f2197832bceab57d7f41379b2592d1110570"
noncomputable section
open scoped Classical MeasureTheory ENNReal NNReal
va... | Mathlib/MeasureTheory/Decomposition/Jordan.lean | 375 | 425 | theorem toSignedMeasure_injective : Injective <| @JordanDecomposition.toSignedMeasure α _ := by |
/- The main idea is that two Jordan decompositions of a signed measure provide two
Hahn decompositions for that measure. Then, from `of_symmDiff_compl_positive_negative`,
the symmetric difference of the two Hahn decompositions has measure zero, thus, allowing us to
show the equality of the underlying mea... |
import Mathlib.MeasureTheory.Constructions.Pi
import Mathlib.MeasureTheory.Integral.Lebesgue
open scoped Classical ENNReal
open Set Function Equiv Finset
noncomputable section
namespace MeasureTheory
section LMarginal
variable {δ δ' : Type*} {π : δ → Type*} [∀ x, MeasurableSpace (π x)]
variable {μ : ∀ i, Measu... | Mathlib/MeasureTheory/Integral/Marginal.lean | 137 | 139 | theorem lmarginal_union' (f : (∀ i, π i) → ℝ≥0∞) (hf : Measurable f) {s t : Finset δ}
(hst : Disjoint s t) : ∫⋯∫⁻_s ∪ t, f ∂μ = ∫⋯∫⁻_t, ∫⋯∫⁻_s, f ∂μ ∂μ := by |
rw [Finset.union_comm, lmarginal_union μ f hf hst.symm]
|
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Tactic.NthRewrite
#align_import data.nat.gcd.basic from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
namespace Nat
theorem gcd_greatest {a b d : ℕ} (hda : d ∣ a) (hdb : d ∣ b) (hd ... | Mathlib/Data/Nat/GCD/Basic.lean | 308 | 317 | theorem pow_dvd_pow_iff {a b n : ℕ} (n0 : n ≠ 0) : a ^ n ∣ b ^ n ↔ a ∣ b := by |
refine ⟨fun h => ?_, fun h => pow_dvd_pow_of_dvd h _⟩
rcases Nat.eq_zero_or_pos (gcd a b) with g0 | g0
· simp [eq_zero_of_gcd_eq_zero_right g0]
rcases exists_coprime' g0 with ⟨g, a', b', g0', co, rfl, rfl⟩
rw [mul_pow, mul_pow] at h
replace h := Nat.dvd_of_mul_dvd_mul_right (pow_pos g0' _) h
have := pow_... |
import Mathlib.Topology.Separation
open Topology Filter Set TopologicalSpace
section Basic
variable {α : Type*} [TopologicalSpace α] {C : Set α}
| Mathlib/Topology/Perfect.lean | 62 | 68 | theorem AccPt.nhds_inter {x : α} {U : Set α} (h_acc : AccPt x (𝓟 C)) (hU : U ∈ 𝓝 x) :
AccPt x (𝓟 (U ∩ C)) := by |
have : 𝓝[≠] x ≤ 𝓟 U := by
rw [le_principal_iff]
exact mem_nhdsWithin_of_mem_nhds hU
rw [AccPt, ← inf_principal, ← inf_assoc, inf_of_le_left this]
exact h_acc
|
import Mathlib.FieldTheory.Galois
#align_import field_theory.polynomial_galois_group from "leanprover-community/mathlib"@"e3f4be1fcb5376c4948d7f095bec45350bfb9d1a"
noncomputable section
open scoped Polynomial
open FiniteDimensional
namespace Polynomial
variable {F : Type*} [Field F] (p q : F[X]) (E : Type*) [... | Mathlib/FieldTheory/PolynomialGaloisGroup.lean | 155 | 168 | theorem mapRoots_bijective [h : Fact (p.Splits (algebraMap F E))] :
Function.Bijective (mapRoots p E) := by |
constructor
· exact fun _ _ h => Subtype.ext (RingHom.injective _ (Subtype.ext_iff.mp h))
· intro y
-- this is just an equality of two different ways to write the roots of `p` as an `E`-polynomial
have key :=
roots_map (IsScalarTower.toAlgHom F p.SplittingField E : p.SplittingField →+* E)
(... |
import Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots
import Mathlib.NumberTheory.NumberField.Discriminant
#align_import number_theory.cyclotomic.discriminant from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa1"
universe u v
open Algebra Polynomial Nat IsPrimitiveRoot PowerBasis
open s... | Mathlib/NumberTheory/Cyclotomic/Discriminant.lean | 37 | 48 | theorem discr_zeta_eq_discr_zeta_sub_one (hζ : IsPrimitiveRoot ζ n) :
discr ℚ (hζ.powerBasis ℚ).basis = discr ℚ (hζ.subOnePowerBasis ℚ).basis := by |
haveI : NumberField K := @NumberField.mk _ _ _ (IsCyclotomicExtension.finiteDimensional {n} ℚ K)
have H₁ : (aeval (hζ.powerBasis ℚ).gen) (X - 1 : ℤ[X]) = (hζ.subOnePowerBasis ℚ).gen := by simp
have H₂ : (aeval (hζ.subOnePowerBasis ℚ).gen) (X + 1 : ℤ[X]) = (hζ.powerBasis ℚ).gen := by simp
refine discr_eq_discr_... |
import Mathlib.Analysis.Convex.Basic
import Mathlib.Order.Filter.Extr
import Mathlib.Tactic.GCongr
#align_import analysis.convex.function from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
open scoped Classical
open LinearMap Set Convex Pointwise
variable {𝕜 E F α β ι : Type*}
secti... | Mathlib/Analysis/Convex/Function.lean | 805 | 808 | theorem ConvexOn.lt_left_of_right_lt (hf : ConvexOn 𝕜 s f) {x y z : E} (hx : x ∈ s) (hy : y ∈ s)
(hz : z ∈ openSegment 𝕜 x y) (hyz : f y < f z) : f z < f x := by |
obtain ⟨a, b, ha, hb, hab, rfl⟩ := hz
exact hf.lt_left_of_right_lt' hx hy ha hb hab hyz
|
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 | 145 | 150 | theorem EuclideanSpace.sphere_zero_eq {n : Type*} [Fintype n] (r : ℝ) (hr : 0 ≤ r) :
Metric.sphere (0 : EuclideanSpace ℝ n) r = {x | ∑ i, x i ^ 2 = r ^ 2} := by |
ext x
have : (0 : ℝ) ≤ ∑ i, x i ^ 2 := Finset.sum_nonneg fun _ _ => sq_nonneg _
simp_rw [mem_setOf, mem_sphere_zero_iff_norm, norm_eq, norm_eq_abs, sq_abs,
Real.sqrt_eq_iff_sq_eq this hr, eq_comm]
|
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 | 708 | 710 | theorem integral_sub (f g : α →₁[μ] E) : integral (f - g) = integral f - integral g := by |
simp only [integral]
exact map_sub integralCLM f g
|
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 Complex
theorem ... | Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 280 | 287 | theorem ofReal_cpow_of_nonpos {x : ℝ} (hx : x ≤ 0) (y : ℂ) :
(x : ℂ) ^ y = (-x : ℂ) ^ y * exp (π * I * y) := by |
rcases hx.eq_or_lt with (rfl | hlt)
· rcases eq_or_ne y 0 with (rfl | hy) <;> simp [*]
have hne : (x : ℂ) ≠ 0 := ofReal_ne_zero.mpr hlt.ne
rw [cpow_def_of_ne_zero hne, cpow_def_of_ne_zero (neg_ne_zero.2 hne), ← exp_add, ← add_mul, log,
log, abs.map_neg, arg_ofReal_of_neg hlt, ← ofReal_neg,
arg_ofReal_o... |
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.MeasureTheory.Function.Egorov
import Mathlib.MeasureTheory.Function.LpSpace
#align_import measure_theory.function.convergence_in_measure from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8"
open TopologicalSpace Filter
ope... | Mathlib/MeasureTheory/Function/ConvergenceInMeasure.lean | 337 | 343 | theorem tendstoInMeasure_of_tendsto_snorm {l : Filter ι} (hp_ne_zero : p ≠ 0)
(hf : ∀ n, AEStronglyMeasurable (f n) μ) (hg : AEStronglyMeasurable g μ)
(hfg : Tendsto (fun n => snorm (f n - g) p μ) l (𝓝 0)) : TendstoInMeasure μ f l g := by |
by_cases hp_ne_top : p = ∞
· subst hp_ne_top
exact tendstoInMeasure_of_tendsto_snorm_top hfg
· exact tendstoInMeasure_of_tendsto_snorm_of_ne_top hp_ne_zero hp_ne_top hf hg hfg
|
import Mathlib.Algebra.Algebra.Spectrum
import Mathlib.FieldTheory.IsAlgClosed.Basic
#align_import field_theory.is_alg_closed.spectrum from "leanprover-community/mathlib"@"58a272265b5e05f258161260dd2c5d247213cbd3"
namespace spectrum
open Set Polynomial
open scoped Pointwise Polynomial
universe u v
section Scal... | Mathlib/FieldTheory/IsAlgClosed/Spectrum.lean | 153 | 159 | theorem nonempty_of_isAlgClosed_of_finiteDimensional [IsAlgClosed 𝕜] [Nontrivial A]
[I : FiniteDimensional 𝕜 A] (a : A) : (σ a).Nonempty := by |
obtain ⟨p, ⟨h_mon, h_eval_p⟩⟩ := isIntegral_of_noetherian (IsNoetherian.iff_fg.2 I) a
have nu : ¬IsUnit (aeval a p) := by rw [← aeval_def] at h_eval_p; rw [h_eval_p]; simp
rw [eq_prod_roots_of_monic_of_splits_id h_mon (IsAlgClosed.splits p)] at nu
obtain ⟨k, hk, _⟩ := exists_mem_of_not_isUnit_aeval_prod nu
e... |
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.Nat.Digits
import Mathlib.Data.Nat.MaxPowDiv
import Mathlib.Data.Nat.Multiplicity
import Mathlib.Tactic.IntervalCases
#align_import number_theory.padics.padic_val from "leanprover-community/mathlib"@"60fa54e778c9e85d930efae172435f42fb0d71f7"
universe u
ope... | Mathlib/NumberTheory/Padics/PadicVal.lean | 715 | 720 | theorem sub_one_mul_padicValNat_factorial [hp : Fact p.Prime] (n : ℕ):
(p - 1) * padicValNat p (n !) = n - (p.digits n).sum := by |
rw [padicValNat_factorial <| lt_succ_of_lt <| lt.base (log p n)]
nth_rw 2 [← zero_add 1]
rw [Nat.succ_eq_add_one, ← Finset.sum_Ico_add' _ 0 _ 1,
Ico_zero_eq_range, ← sub_one_mul_sum_log_div_pow_eq_sub_sum_digits, Nat.succ_eq_add_one]
|
import Mathlib.Probability.Process.Adapted
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
#align_import probability.process.stopping from "leanprover-community/mathlib"@"ba074af83b6cf54c3104e59402b39410ddbd6dca"
open Filter Order TopologicalSpace
open scoped Classical MeasureTheory NNReal ENNReal Top... | Mathlib/Probability/Process/Stopping.lean | 1,085 | 1,089 | theorem stoppedProcess_eq (n : ℕ) : stoppedProcess u τ n = Set.indicator {a | n ≤ τ a} (u n) +
∑ i ∈ Finset.range n, Set.indicator {ω | τ ω = i} (u i) := by |
rw [stoppedProcess_eq'' n]
congr with i
rw [Finset.mem_Iio, Finset.mem_range]
|
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align_import order.lattice from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
@[gcongr_forward] def exactSubsetOfSSubset : Mat... | Mathlib/Order/Lattice.lean | 808 | 813 | theorem lt_sup_iff : a < b ⊔ c ↔ a < b ∨ a < c := by |
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩
|
import Mathlib.Order.Filter.Lift
import Mathlib.Topology.Defs.Filter
#align_import topology.basic from "leanprover-community/mathlib"@"e354e865255654389cc46e6032160238df2e0f40"
noncomputable section
open Set Filter
universe u v w x
def TopologicalSpace.ofClosed {X : Type u} (T : Set (Set X)) (empty_mem : ∅ ∈... | Mathlib/Topology/Basic.lean | 527 | 528 | theorem interior_compl : interior sᶜ = (closure s)ᶜ := by |
simp [closure_eq_compl_interior_compl]
|
import Mathlib.Algebra.Group.Indicator
import Mathlib.Data.Finset.Piecewise
import Mathlib.Data.Finset.Preimage
#align_import algebra.big_operators.basic from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
-- TODO
-- assert_not_exists AddCommMonoidWithOne
assert_not_exists MonoidWithZero... | Mathlib/Algebra/BigOperators/Group/Finset.lean | 1,850 | 1,853 | theorem prod_inter_mul_prod_diff [DecidableEq α] (s t : Finset α) (f : α → β) :
(∏ x ∈ s ∩ t, f x) * ∏ x ∈ s \ t, f x = ∏ x ∈ s, f x := by |
convert (s.prod_piecewise t f f).symm
simp (config := { unfoldPartialApp := true }) [Finset.piecewise]
|
import Mathlib.AlgebraicGeometry.OpenImmersion
-- Explicit universe annotations were used in this file to improve perfomance #12737
set_option linter.uppercaseLean3 false
noncomputable section
open TopologicalSpace CategoryTheory Opposite
open CategoryTheory.Limits
namespace AlgebraicGeometry
universe v v₁ v₂... | Mathlib/AlgebraicGeometry/Restrict.lean | 138 | 144 | theorem Scheme.restrictFunctor_map_app_aux {U V : Opens X} (i : U ⟶ V) (W : Opens V) :
U.openEmbedding.isOpenMap.functor.obj ((X.restrictFunctor.map i).1 ⁻¹ᵁ W) ≤
V.openEmbedding.isOpenMap.functor.obj W := by |
simp only [← SetLike.coe_subset_coe, IsOpenMap.functor_obj_coe, Set.image_subset_iff,
Scheme.restrictFunctor_map_base, Opens.map_coe, Opens.inclusion_apply]
rintro _ h
exact ⟨_, h, rfl⟩
|
import Mathlib.Probability.ProbabilityMassFunction.Basic
import Mathlib.Probability.ProbabilityMassFunction.Constructions
import Mathlib.MeasureTheory.Integral.Bochner
namespace PMF
open MeasureTheory ENNReal TopologicalSpace
section General
variable {α : Type*} [MeasurableSpace α] [MeasurableSingletonClass α]
v... | Mathlib/Probability/ProbabilityMassFunction/Integrals.lean | 28 | 41 | theorem integral_eq_tsum (p : PMF α) (f : α → E) (hf : Integrable f p.toMeasure) :
∫ a, f a ∂(p.toMeasure) = ∑' a, (p a).toReal • f a := calc
_ = ∫ a in p.support, f a ∂(p.toMeasure) := by | rw [restrict_toMeasure_support p]
_ = ∑' (a : support p), (p.toMeasure {a.val}).toReal • f a := by
apply integral_countable f p.support_countable
rwa [restrict_toMeasure_support p]
_ = ∑' (a : support p), (p a).toReal • f a := by
congr with x; congr 2
apply PMF.toMeasure_apply_singleton p x (Measur... |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Fintype.Card
#align_import data.multiset.fintype from "leanprover-community/mathlib"@"e3d9ab8faa9dea8f78155c6c27d62a621f4c152d"
variable {α : Type*} [DecidableEq α] {m : Multiset α}
def Multiset.ToType (m : Multiset α) : Type _ := (x : α) × Fi... | Mathlib/Data/Multiset/Fintype.lean | 219 | 224 | theorem Multiset.map_univ_coe (m : Multiset α) :
(Finset.univ : Finset m).val.map (fun x : m ↦ (x : α)) = m := by |
have := m.map_toEnumFinset_fst
rw [← m.map_univ_coeEmbedding] at this
simpa only [Finset.map_val, Multiset.coeEmbedding_apply, Multiset.map_map,
Function.comp_apply] using this
|
import Mathlib.Data.List.Lattice
import Mathlib.Data.List.Range
import Mathlib.Data.Bool.Basic
#align_import data.list.intervals from "leanprover-community/mathlib"@"7b78d1776212a91ecc94cf601f83bdcc46b04213"
open Nat
namespace List
def Ico (n m : ℕ) : List ℕ :=
range' n (m - n)
#align list.Ico List.Ico
names... | Mathlib/Data/List/Intervals.lean | 95 | 100 | theorem append_consecutive {n m l : ℕ} (hnm : n ≤ m) (hml : m ≤ l) :
Ico n m ++ Ico m l = Ico n l := by |
dsimp only [Ico]
convert range'_append n (m-n) (l-m) 1 using 2
· rw [Nat.one_mul, Nat.add_sub_cancel' hnm]
· rw [Nat.sub_add_sub_cancel hml hnm]
|
import Mathlib.Algebra.Order.Module.Defs
import Mathlib.Data.Finsupp.Basic
#align_import data.finsupp.order from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
-- Porting note: removed from module documentation because it moved to `Data.Finsupp.Multiset`
-- TODO: move to `Data.Finsupp.Mu... | Mathlib/Data/Finsupp/Order.lean | 265 | 267 | theorem subset_support_tsub [DecidableEq ι] {f1 f2 : ι →₀ α} :
f1.support \ f2.support ⊆ (f1 - f2).support := by |
simp (config := { contextual := true }) [subset_iff]
|
import Mathlib.Algebra.Module.Submodule.Lattice
import Mathlib.Algebra.Module.Submodule.LinearMap
open Function Pointwise Set
variable {R : Type*} {R₁ : Type*} {R₂ : Type*} {R₃ : Type*}
variable {M : Type*} {M₁ : Type*} {M₂ : Type*} {M₃ : Type*}
namespace Submodule
section AddCommMonoid
variable [Semiring R] [... | Mathlib/Algebra/Module/Submodule/Map.lean | 564 | 572 | theorem map_symm_eq_iff (e : M ≃ₛₗ[τ₁₂] M₂) {K : Submodule R₂ M₂} :
K.map e.symm = p ↔ p.map e = K := by |
constructor <;> rintro rfl
· calc
map e (map e.symm K) = comap e.symm (map e.symm K) := map_equiv_eq_comap_symm _ _
_ = K := comap_map_eq_of_injective e.symm.injective _
· calc
map e.symm (map e p) = comap e (map e p) := (comap_equiv_eq_map_symm _ _).symm
_ = p := comap_map_eq_of_injectiv... |
import Mathlib.Data.Set.Card
import Mathlib.Order.Minimal
import Mathlib.Data.Matroid.Init
set_option autoImplicit true
open Set
def Matroid.ExchangeProperty {α : Type _} (P : Set α → Prop) : Prop :=
∀ X Y, P X → P Y → ∀ a ∈ X \ Y, ∃ b ∈ Y \ X, P (insert b (X \ {a}))
def Matroid.ExistsMaximalSubsetProperty {... | Mathlib/Data/Matroid/Basic.lean | 852 | 856 | theorem exists_basis_union_inter_basis (M : Matroid α) (X Y : Set α) (hX : X ⊆ M.E := by | aesop_mat)
(hY : Y ⊆ M.E := by aesop_mat) : ∃ I, M.Basis I (X ∪ Y) ∧ M.Basis (I ∩ Y) Y :=
let ⟨J, hJ⟩ := M.exists_basis Y
(hJ.exists_basis_inter_eq_of_superset subset_union_right).imp
(fun I hI ↦ ⟨hI.1, by rwa [hI.2]⟩)
|
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Choose.Cast
import Mathlib.Data.Nat.Choose.Vandermonde
import Mathlib.Tactic.FieldSimp
#align_import data.polynomial.hasse_deriv from "leanprover-community/mathlib"@"a148d797a1094ab554ad4183a4ad6f130358... | Mathlib/Algebra/Polynomial/HasseDeriv.lean | 93 | 97 | theorem hasseDeriv_eq_zero_of_lt_natDegree (p : R[X]) (n : ℕ) (h : p.natDegree < n) :
hasseDeriv n p = 0 := by |
rw [hasseDeriv_apply, sum_def]
refine Finset.sum_eq_zero fun x hx => ?_
simp [Nat.choose_eq_zero_of_lt ((le_natDegree_of_mem_supp _ hx).trans_lt h)]
|
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Inverse
#align_import analysis.calculus.cont_diff from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputab... | Mathlib/Analysis/Calculus/ContDiff/Basic.lean | 2,058 | 2,061 | theorem contDiffOn_top_iff_deriv_of_isOpen (hs : IsOpen s₂) :
ContDiffOn 𝕜 ∞ f₂ s₂ ↔ DifferentiableOn 𝕜 f₂ s₂ ∧ ContDiffOn 𝕜 ∞ (deriv f₂) s₂ := by |
rw [contDiffOn_top_iff_derivWithin hs.uniqueDiffOn]
exact Iff.rfl.and <| contDiffOn_congr fun _ => derivWithin_of_isOpen hs
|
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
#align_import ring_theory.coprime.lemmas from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
universe u v
section IsCoprime
variable {R : Type ... | Mathlib/RingTheory/Coprime/Lemmas.lean | 69 | 70 | theorem IsCoprime.prod_right : (∀ i ∈ t, IsCoprime x (s i)) → IsCoprime x (∏ i ∈ t, s i) := by |
simpa only [isCoprime_comm] using IsCoprime.prod_left (R := R)
|
import Mathlib.Algebra.Module.DedekindDomain
import Mathlib.LinearAlgebra.FreeModule.PID
import Mathlib.Algebra.Module.Projective
import Mathlib.Algebra.Category.ModuleCat.Biproducts
import Mathlib.RingTheory.SimpleModule
#align_import algebra.module.pid from "leanprover-community/mathlib"@"cdc34484a07418af43daf8198b... | Mathlib/Algebra/Module/PID.lean | 238 | 260 | theorem equiv_directSum_of_isTorsion [h' : Module.Finite R N] (hN : Module.IsTorsion R N) :
∃ (ι : Type u) (_ : Fintype ι) (p : ι → R) (_ : ∀ i, Irreducible <| p i) (e : ι → ℕ),
Nonempty <| N ≃ₗ[R] ⨁ i : ι, R ⧸ R ∙ p i ^ e i := by |
obtain ⟨I, fI, _, p, hp, e, h⟩ := Submodule.exists_isInternal_prime_power_torsion_of_pid hN
haveI := fI
have :
∀ i,
∃ (d : ℕ) (k : Fin d → ℕ),
Nonempty <| torsionBy R N (p i ^ e i) ≃ₗ[R] ⨁ j, R ⧸ R ∙ p i ^ k j := by
haveI := fun i => isNoetherian_submodule' (torsionBy R N <| p i ^ e i)
... |
import Mathlib.Logic.Equiv.Nat
import Mathlib.Data.PNat.Basic
import Mathlib.Order.Directed
import Mathlib.Data.Countable.Defs
import Mathlib.Order.RelIso.Basic
import Mathlib.Data.Fin.Basic
#align_import logic.encodable.basic from "leanprover-community/mathlib"@"7c523cb78f4153682c2929e3006c863bfef463d0"
open Opt... | Mathlib/Logic/Encodable/Basic.lean | 385 | 390 | theorem decode_prod_val [i : Encodable α] (n : ℕ) :
(@decode (α × β) _ n : Option (α × β))
= (decode n.unpair.1).bind fun a => (decode n.unpair.2).map <| Prod.mk a := by |
simp only [decode_ofEquiv, Equiv.symm_symm, decode_sigma_val]
cases (decode n.unpair.1 : Option α) <;> cases (decode n.unpair.2 : Option β)
<;> rfl
|
import Mathlib.Analysis.NormedSpace.Multilinear.Basic
#align_import analysis.normed_space.multilinear from "leanprover-community/mathlib"@"f40476639bac089693a489c9e354ebd75dc0f886"
suppress_compilation
noncomputable section
open NNReal Finset Metric ContinuousMultilinearMap Fin Function
universe u v v' wE wE... | Mathlib/Analysis/NormedSpace/Multilinear/Curry.lean | 86 | 92 | theorem ContinuousMultilinearMap.norm_map_cons_le (f : ContinuousMultilinearMap 𝕜 Ei G) (x : Ei 0)
(m : ∀ i : Fin n, Ei i.succ) : ‖f (cons x m)‖ ≤ ‖f‖ * ‖x‖ * ∏ i, ‖m i‖ :=
calc
‖f (cons x m)‖ ≤ ‖f‖ * ∏ i, ‖cons x m i‖ := f.le_opNorm _
_ = ‖f‖ * ‖x‖ * ∏ i, ‖m i‖ := by |
rw [prod_univ_succ]
simp [mul_assoc]
|
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Data.List.Cycle
import Mathlib.Data.Nat.Prime
import Mathlib.Data.PNat.Basic
import Mathlib.Dynamics.FixedPoints.Basic
import Mathlib.GroupTheory.GroupAction.Group
#align_import dynamics.periodic_pts from "leanp... | Mathlib/Dynamics/PeriodicPts.lean | 156 | 159 | theorem left_of_comp {g : α → α} (hco : Commute f g) (hfg : IsPeriodicPt (f ∘ g) n x)
(hg : IsPeriodicPt g n x) : IsPeriodicPt f n x := by |
rw [IsPeriodicPt, hco.comp_iterate] at hfg
exact hfg.left_of_comp hg
|
import Mathlib.Algebra.Order.ToIntervalMod
import Mathlib.Algebra.Ring.AddAut
import Mathlib.Data.Nat.Totient
import Mathlib.GroupTheory.Divisible
import Mathlib.Topology.Connected.PathConnected
import Mathlib.Topology.IsLocalHomeomorph
#align_import topology.instances.add_circle from "leanprover-community/mathlib"@"... | Mathlib/Topology/Instances/AddCircle.lean | 82 | 89 | theorem continuous_left_toIocMod : ContinuousWithinAt (toIocMod hp a) (Iic x) x := by |
rw [(funext fun y => Eq.trans (by rw [neg_neg]) <| toIocMod_neg _ _ _ :
toIocMod hp a = (fun x => p - x) ∘ toIcoMod hp (-a) ∘ Neg.neg)]
-- Porting note: added
have : ContinuousNeg 𝕜 := TopologicalAddGroup.toContinuousNeg
exact
(continuous_sub_left _).continuousAt.comp_continuousWithinAt <|
(co... |
import Mathlib.Algebra.CharP.ExpChar
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Algebra.MvPolynomial.Equiv
import Mathlib.RingTheory.Polynomial.Content
import Mathlib.RingTheory.UniqueFactorizationDomain
#align_import ring_theory.polynomial.basic from "leanprover-commun... | Mathlib/RingTheory/Polynomial/Basic.lean | 1,061 | 1,068 | theorem disjoint_ker_aeval_of_coprime (f : M →ₗ[R] M) {p q : R[X]} (hpq : IsCoprime p q) :
Disjoint (LinearMap.ker (aeval f p)) (LinearMap.ker (aeval f q)) := by |
rw [disjoint_iff_inf_le]
intro v hv
rcases hpq with ⟨p', q', hpq'⟩
simpa [LinearMap.mem_ker.1 (Submodule.mem_inf.1 hv).1,
LinearMap.mem_ker.1 (Submodule.mem_inf.1 hv).2] using
congr_arg (fun p : R[X] => aeval f p v) hpq'.symm
|
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calculus.LocalExtr.Rolle
import Mathlib.Analysis.Convex.Normed
import Mathlib.Analysis.RCLike.Basic
#align_import... | Mathlib/Analysis/Calculus/MeanValue.lean | 92 | 124 | theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ}
(hf : ContinuousOn f (Icc a b))
-- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x`
(hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r)
{B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Ic... |
change Icc a b ⊆ { x | f x ≤ B x }
set s := { x | f x ≤ B x } ∩ Icc a b
have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB
have : IsClosed s := by
simp only [s, inter_comm]
exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le'
apply this.Icc_subset_of_fo... |
import Mathlib.Data.Set.Function
import Mathlib.Logic.Function.Iterate
import Mathlib.GroupTheory.Perm.Basic
#align_import dynamics.fixed_points.basic from "leanprover-community/mathlib"@"b86832321b586c6ac23ef8cdef6a7a27e42b13bd"
open Equiv
universe u v
variable {α : Type u} {β : Type v} {f fa g : α → α} {x y :... | Mathlib/Dynamics/FixedPoints/Basic.lean | 192 | 194 | theorem Commute.left_bijOn_fixedPoints_comp (h : Commute f g) :
Set.BijOn f (fixedPoints <| f ∘ g) (fixedPoints <| f ∘ g) := by |
simpa only [h.comp_eq] using bijOn_fixedPoints_comp g f
|
import Mathlib.Algebra.BigOperators.Associated
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Data.Nat.Factors
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.Multiplicity
#align_import ring_theory.unique_factorization_domain from "leanprover-community/mathli... | Mathlib/RingTheory/UniqueFactorizationDomain.lean | 594 | 599 | theorem factors_eq_normalizedFactors {M : Type*} [CancelCommMonoidWithZero M]
[UniqueFactorizationMonoid M] [Unique Mˣ] (x : M) : factors x = normalizedFactors x := by |
unfold normalizedFactors
convert (Multiset.map_id (factors x)).symm
ext p
exact normalize_eq p
|
import Mathlib.MeasureTheory.Measure.MeasureSpace
open scoped ENNReal NNReal Topology
open Set MeasureTheory Measure Filter MeasurableSpace ENNReal Function
variable {R α β δ γ ι : Type*}
namespace MeasureTheory
variable {m0 : MeasurableSpace α} [MeasurableSpace β] [MeasurableSpace γ]
variable {μ μ₁ μ₂ μ₃ ν ν' ν... | Mathlib/MeasureTheory/Measure/Restrict.lean | 221 | 222 | theorem restrict_eq_zero : μ.restrict s = 0 ↔ μ s = 0 := by |
rw [← measure_univ_eq_zero, restrict_apply_univ]
|
import Mathlib.Topology.Category.Profinite.Basic
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.Topology.Category.CompHaus.Limits
namespace Profinite
universe u w
attribute [local instance] CategoryTheory.ConcreteCategory.instFunLike
open CategoryTheory Limits
section Pullbacks
variable ... | Mathlib/Topology/Category/Profinite/Limits.lean | 123 | 126 | theorem pullback_fst_eq :
Profinite.pullback.fst f g = (pullbackIsoPullback f g).hom ≫ Limits.pullback.fst := by |
dsimp [pullbackIsoPullback]
simp only [Limits.limit.conePointUniqueUpToIso_hom_comp, pullback.cone_pt, pullback.cone_π]
|
import Mathlib.Topology.Separation
import Mathlib.Algebra.Group.Defs
#align_import topology.algebra.semigroup from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
@[to_additive
"Any nonempty compact Hausdorff additive semigroup where right-addition is continuous
contains an ... | Mathlib/Topology/Algebra/Semigroup.lean | 27 | 72 | theorem exists_idempotent_of_compact_t2_of_continuous_mul_left {M} [Nonempty M] [Semigroup M]
[TopologicalSpace M] [CompactSpace M] [T2Space M]
(continuous_mul_left : ∀ r : M, Continuous (· * r)) : ∃ m : M, m * m = m := by |
/- We apply Zorn's lemma to the poset of nonempty closed subsemigroups of `M`.
It will turn out that any minimal element is `{m}` for an idempotent `m : M`. -/
let S : Set (Set M) :=
{ N | IsClosed N ∧ N.Nonempty ∧ ∀ (m) (_ : m ∈ N) (m') (_ : m' ∈ N), m * m' ∈ N }
rsuffices ⟨N, ⟨N_closed, ⟨m, hm⟩, N_mul... |
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 | 374 | 378 | theorem toLp_denseRange [CompactSpace α] [μ.WeaklyRegular] [IsFiniteMeasure μ] :
DenseRange (toLp p μ 𝕜 : C(α, E) →L[𝕜] Lp E p μ) := by |
refine (BoundedContinuousFunction.toLp_denseRange _ _ hp 𝕜).mono ?_
refine range_subset_iff.2 fun f ↦ ?_
exact ⟨f.toContinuousMap, rfl⟩
|
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Combinatorics.Derangements.Basic
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Tactic.Ring
#align_import combinatorics.derangements.finite from "leanprover-community/mathlib"@"c3019c79074b0619edb4b27553a91b2e82242395"
open derangements Equiv Fintyp... | Mathlib/Combinatorics/Derangements/Finite.lean | 107 | 110 | theorem card_derangements_eq_numDerangements (α : Type*) [Fintype α] [DecidableEq α] :
card (derangements α) = numDerangements (card α) := by |
rw [← card_derangements_invariant (card_fin _)]
exact card_derangements_fin_eq_numDerangements
|
import Mathlib.MeasureTheory.Measure.Dirac
set_option autoImplicit true
open Set
open scoped ENNReal Classical
variable [MeasurableSpace α] [MeasurableSpace β] {s : Set α}
noncomputable section
namespace MeasureTheory.Measure
def count : Measure α :=
sum dirac
#align measure_theory.measure.count MeasureTheo... | Mathlib/MeasureTheory/Measure/Count.lean | 39 | 40 | theorem count_apply (hs : MeasurableSet s) : count s = ∑' i : s, 1 := by |
simp only [count, sum_apply, hs, dirac_apply', ← tsum_subtype s (1 : α → ℝ≥0∞), Pi.one_apply]
|
import Mathlib.MeasureTheory.Integral.IntegrableOn
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.MeasureTheory.Function.LocallyIntegrable
import Mathlib.Topology.MetricSpace.ThickenedIndicator
import Mathlib.Topology.ContinuousFunction.Compact
import Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDua... | Mathlib/MeasureTheory/Integral/SetIntegral.lean | 498 | 520 | theorem integral_norm_eq_pos_sub_neg {f : X → ℝ} (hfi : Integrable f μ) :
∫ x, ‖f x‖ ∂μ = ∫ x in {x | 0 ≤ f x}, f x ∂μ - ∫ x in {x | f x ≤ 0}, f x ∂μ :=
have h_meas : NullMeasurableSet {x | 0 ≤ f x} μ :=
aestronglyMeasurable_const.nullMeasurableSet_le hfi.1
calc
∫ x, ‖f x‖ ∂μ = ∫ x in {x | 0 ≤ f x}, ‖f ... |
rw [← integral_add_compl₀ h_meas hfi.norm]
_ = ∫ x in {x | 0 ≤ f x}, f x ∂μ + ∫ x in {x | 0 ≤ f x}ᶜ, ‖f x‖ ∂μ := by
congr 1
refine setIntegral_congr₀ h_meas fun x hx => ?_
dsimp only
rw [Real.norm_eq_abs, abs_eq_self.mpr _]
exact hx
_ = ∫ x in {x | 0 ≤ f x}, f x ∂μ - ∫ x in ... |
import Mathlib.Algebra.CharP.Two
import Mathlib.Algebra.CharP.Reduced
import Mathlib.Algebra.NeZero
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.GroupTheory.SpecificGroups.Cyclic
import Mathlib.NumberTheory.Divisors
import Mathlib.RingTheory.IntegralDomain
import Mathlib.Tactic.Zify
#align_import rin... | Mathlib/RingTheory/RootsOfUnity/Basic.lean | 645 | 651 | theorem primitiveRoots_one : primitiveRoots 1 R = {(1 : R)} := by |
apply Finset.eq_singleton_iff_unique_mem.2
constructor
· simp only [IsPrimitiveRoot.one_right_iff, mem_primitiveRoots zero_lt_one]
· intro x hx
rw [mem_primitiveRoots zero_lt_one, IsPrimitiveRoot.one_right_iff] at hx
exact hx
|
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Set.Image
import Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov
import Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp
import Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality... | Mathlib/MeasureTheory/Function/LpSpace.lean | 338 | 339 | theorem norm_measure_zero (f : Lp E p (0 : MeasureTheory.Measure α)) : ‖f‖ = 0 := by |
simp [norm_def]
|
import Mathlib.Probability.Kernel.Basic
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.DominatedConvergence
#align_import probability.kernel.measurable_integral from "leanprover-community/mathlib"@"28b2a92f2996d28e580450863c130955de0ed398"
open MeasureTheory Probabilit... | Mathlib/Probability/Kernel/MeasurableIntegral.lean | 305 | 308 | theorem StronglyMeasurable.integral_kernel_prod_right' ⦃f : α × β → E⦄ (hf : StronglyMeasurable f) :
StronglyMeasurable fun x => ∫ y, f (x, y) ∂κ x := by |
rw [← uncurry_curry f] at hf
exact hf.integral_kernel_prod_right
|
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Ideal.Over
import Mathlib.RingTheory.JacobsonIdeal
#align_import ring_theory.jacobson from "leanprover-community/mathlib"@"a7c017d750512a352b623b1824d75da5998457d0"
set_option autoImplicit true
universe u
namespace Ideal
open Polynomial
... | Mathlib/RingTheory/Jacobson.lean | 579 | 583 | theorem isMaximal_comap_C_of_isJacobson : (P.comap (C : R →+* R[X])).IsMaximal := by |
rw [← @mk_ker _ _ P, RingHom.ker_eq_comap_bot, comap_comap]
have := (bot_quotient_isMaximal_iff _).mpr hP
exact isMaximal_comap_of_isIntegral_of_isMaximal' _ (quotient_mk_comp_C_isIntegral_of_jacobson P)
⊥
|
import Mathlib.SetTheory.Game.State
#align_import set_theory.game.domineering from "leanprover-community/mathlib"@"b134b2f5cf6dd25d4bbfd3c498b6e36c11a17225"
namespace SetTheory
namespace PGame
namespace Domineering
open Function
@[simps!]
def shiftUp : ℤ × ℤ ≃ ℤ × ℤ :=
(Equiv.refl ℤ).prodCongr (Equiv.addRig... | Mathlib/SetTheory/Game/Domineering.lean | 117 | 122 | theorem moveRight_card {b : Board} {m : ℤ × ℤ} (h : m ∈ right b) :
Finset.card (moveRight b m) + 2 = Finset.card b := by |
dsimp [moveRight]
rw [Finset.card_erase_of_mem (fst_pred_mem_erase_of_mem_right h)]
rw [Finset.card_erase_of_mem (Finset.mem_of_mem_inter_left h)]
exact tsub_add_cancel_of_le (card_of_mem_right h)
|
import Mathlib.Analysis.Convex.Basic
import Mathlib.Topology.Algebra.Group.Basic
import Mathlib.Topology.Order.Basic
#align_import analysis.convex.strict from "leanprover-community/mathlib"@"84dc0bd6619acaea625086d6f53cb35cdd554219"
open Set
open Convex Pointwise
variable {𝕜 𝕝 E F β : Type*}
open Function Se... | Mathlib/Analysis/Convex/Strict.lean | 231 | 233 | theorem StrictConvex.preimage_add_left (hs : StrictConvex 𝕜 s) (z : E) :
StrictConvex 𝕜 ((fun x => x + z) ⁻¹' s) := by |
simpa only [add_comm] using hs.preimage_add_right z
|
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Option.Basic
import Mathlib.Data.List.Defs
import Mathlib.Init.Data.List.Basic
import Mathlib.Init.Data.List.Instances
import Mathlib.Init.Data.List.Lemmas
import Mathlib.Logic.Unique
import Mathlib.Order.Basic
import Mathlib.Tactic.Common
#align_import data.list.basic... | Mathlib/Data/List/Basic.lean | 3,125 | 3,128 | theorem map_erase [DecidableEq β] {f : α → β} (finj : Injective f) {a : α} (l : List α) :
map f (l.erase a) = (map f l).erase (f a) := by |
have this : (a == ·) = (f a == f ·) := by ext b; simp [beq_eq_decide, finj.eq_iff]
rw [erase_eq_eraseP, erase_eq_eraseP, eraseP_map, this]; rfl
|
import Mathlib.Algebra.Star.Subalgebra
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.Tactic.NoncommRing
#align_import algebra.algebra.spectrum from "leanprover-community/mathlib"@"58a272265b5e05f258161260dd2c5d247213cbd3"
open Set
open scoped Pointwise
universe u v
namespace spectrum
section ScalarSemir... | Mathlib/Algebra/Algebra/Spectrum.lean | 226 | 227 | theorem add_mem_add_iff {a : A} {r s : R} : r + s ∈ σ (↑ₐ s + a) ↔ r ∈ σ a := by |
rw [add_mem_iff, neg_add_cancel_left]
|
import Mathlib.Data.Complex.Basic
import Mathlib.MeasureTheory.Integral.CircleIntegral
#align_import measure_theory.integral.circle_transform from "leanprover-community/mathlib"@"d11893b411025250c8e61ff2f12ccbd7ee35ab15"
open Set MeasureTheory Metric Filter Function
open scoped Interval Real
noncomputable secti... | Mathlib/MeasureTheory/Integral/CircleTransform.lean | 133 | 152 | theorem circleTransformDeriv_bound {R : ℝ} (hR : 0 < R) {z x : ℂ} {f : ℂ → ℂ} (hx : x ∈ ball z R)
(hf : ContinuousOn f (sphere z R)) : ∃ B ε : ℝ, 0 < ε ∧
ball x ε ⊆ ball z R ∧ ∀ (t : ℝ), ∀ y ∈ ball x ε, ‖circleTransformDeriv R z y f t‖ ≤ B := by |
obtain ⟨r, hr, hrx⟩ := exists_lt_mem_ball_of_mem_ball hx
obtain ⟨ε', hε', H⟩ := exists_ball_subset_ball hrx
obtain ⟨⟨⟨a, b⟩, ⟨ha, hb⟩⟩, hab⟩ :=
abs_circleTransformBoundingFunction_le hr (pos_of_mem_ball hrx).le z
let V : ℝ → ℂ → ℂ := fun θ w => circleTransformDeriv R z w (fun _ => 1) θ
obtain ⟨X, -, HX2⟩... |
import Mathlib.Data.Real.Irrational
import Mathlib.Data.Nat.Fib.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Algebra.LinearRecurrence
import Mathlib.Tactic.NormNum.NatFib
import Mathlib.Tactic.NormNum.Prime
#align_import data.real.golden_ratio from "leanprover-community/mathlib"@"2196ab363eb097c008d449712... | Mathlib/Data/Real/GoldenRatio.lean | 70 | 72 | theorem gold_add_goldConj : φ + ψ = 1 := by |
rw [goldenRatio, goldenConj]
ring
|
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
import Mathlib.Algebra.Category.Ring.Colimits
import Mathlib.Algebra.Category.Ring.Limits
import Mathlib.Topology.Sheaves.LocalPredicate
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.Algebra.Ring.Subring.Basic
#align_import algebraic_geometry.struct... | Mathlib/AlgebraicGeometry/StructureSheaf.lean | 1,080 | 1,097 | theorem comapFunIsLocallyFraction (f : R →+* S) (U : Opens (PrimeSpectrum.Top R))
(V : Opens (PrimeSpectrum.Top S)) (hUV : V.1 ⊆ PrimeSpectrum.comap f ⁻¹' U.1)
(s : ∀ x : U, Localizations R x) (hs : (isLocallyFraction R).toPrelocalPredicate.pred s) :
(isLocallyFraction S).toPrelocalPredicate.pred (comapFun ... |
rintro ⟨p, hpV⟩
-- Since `s` is locally fraction, we can find a neighborhood `W` of `PrimeSpectrum.comap f p`
-- in `U`, such that `s = a / b` on `W`, for some ring elements `a, b : R`.
rcases hs ⟨PrimeSpectrum.comap f p, hUV hpV⟩ with ⟨W, m, iWU, a, b, h_frac⟩
-- We claim that we can write our new section a... |
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
import Mathlib.Logic.Embedding.Basic
import Mathlib.Data.Fintype.CardEmbedding
import Mathlib.Topology.Algebra.Module.Multilinear.Topology
#align_import analysis.normed_space.multilinear from "leanprover-community/mathlib"@"f40476639bac089693a489c9e354ebd75... | Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean | 1,428 | 1,429 | theorem norm_ofSubsingleton_id [Subsingleton ι] [Nontrivial G] (i : ι) :
‖ofSubsingleton 𝕜 G G i (.id _ _)‖ = 1 := by | simp
|
import Mathlib.Topology.Algebra.Constructions
import Mathlib.Topology.Bases
import Mathlib.Topology.UniformSpace.Basic
#align_import topology.uniform_space.cauchy from "leanprover-community/mathlib"@"22131150f88a2d125713ffa0f4693e3355b1eb49"
universe u v
open scoped Classical
open Filter TopologicalSpace Set Uni... | Mathlib/Topology/UniformSpace/Cauchy.lean | 131 | 134 | theorem Cauchy.prod [UniformSpace β] {f : Filter α} {g : Filter β} (hf : Cauchy f) (hg : Cauchy g) :
Cauchy (f ×ˢ g) := by |
have := hf.1; have := hg.1
simpa [cauchy_prod_iff, hf.1] using ⟨hf, hg⟩
|
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Field.Defs
import Mathlib.RingTheory.OreLocalization.Basic
#align_import ring_theory.ore_localization.basic from "leanprover-community/mathlib"@"861a26926586cd46ff80264d121cdb6fa0e35cc1"
universe u
name... | Mathlib/RingTheory/OreLocalization/Ring.lean | 182 | 186 | theorem subsingleton_iff :
Subsingleton R[S⁻¹] ↔ 0 ∈ S := by |
rw [← subsingleton_iff_zero_eq_one, OreLocalization.one_def,
OreLocalization.zero_def, oreDiv_eq_iff]
simp
|
import Mathlib.Algebra.Polynomial.Mirror
import Mathlib.Analysis.Complex.Polynomial
#align_import data.polynomial.unit_trinomial from "leanprover-community/mathlib"@"302eab4f46abb63de520828de78c04cb0f9b5836"
namespace Polynomial
open scoped Polynomial
open Finset
section Semiring
variable {R : Type*} [Semirin... | Mathlib/Algebra/Polynomial/UnitTrinomial.lean | 49 | 52 | theorem trinomial_leading_coeff' (hkm : k < m) (hmn : m < n) :
(trinomial k m n u v w).coeff n = w := by |
rw [trinomial_def, coeff_add, coeff_add, coeff_C_mul_X_pow, coeff_C_mul_X_pow, coeff_C_mul_X_pow,
if_neg (hkm.trans hmn).ne', if_neg hmn.ne', if_pos rfl, zero_add, zero_add]
|
import Mathlib.Order.Lattice
import Mathlib.Data.List.Sort
import Mathlib.Logic.Equiv.Fin
import Mathlib.Logic.Equiv.Functor
import Mathlib.Data.Fintype.Card
import Mathlib.Order.RelSeries
#align_import order.jordan_holder from "leanprover-community/mathlib"@"91288e351d51b3f0748f0a38faa7613fb0ae2ada"
universe u
... | Mathlib/Order/JordanHolder.lean | 116 | 117 | theorem second_iso_of_eq {x y a b : X} (hm : IsMaximal x a) (ha : x ⊔ y = a) (hb : x ⊓ y = b) :
Iso (x, a) (b, y) := by | substs a b; exact second_iso hm
|
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Topology.MetricSpace.HausdorffDistance
import Mathlib.Topology.Sets.Compacts
#align_import topology.metric_space.closeds from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open scoped Classical
open Topo... | Mathlib/Topology/MetricSpace/Closeds.lean | 256 | 291 | theorem NonemptyCompacts.isClosed_in_closeds [CompleteSpace α] :
IsClosed (range <| @NonemptyCompacts.toCloseds α _ _) := by |
have :
range NonemptyCompacts.toCloseds =
{ s : Closeds α | (s : Set α).Nonempty ∧ IsCompact (s : Set α) } := by
ext s
refine ⟨?_, fun h => ⟨⟨⟨s, h.2⟩, h.1⟩, Closeds.ext rfl⟩⟩
rintro ⟨s, hs, rfl⟩
exact ⟨s.nonempty, s.isCompact⟩
rw [this]
refine isClosed_of_closure_subset fun s hs => ⟨?_... |
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Tactic.NthRewrite
#align_import data.nat.gcd.basic from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
namespace Nat
theorem gcd_greatest {a b d : ℕ} (hda : d ∣ a) (hdb : d ∣ b) (hd ... | Mathlib/Data/Nat/GCD/Basic.lean | 264 | 269 | theorem gcd_mul_of_coprime_of_dvd {a b c : ℕ} (hac : Coprime a c) (b_dvd_c : b ∣ c) :
gcd (a * b) c = b := by |
rcases exists_eq_mul_left_of_dvd b_dvd_c with ⟨d, rfl⟩
rw [gcd_mul_right]
convert one_mul b
exact Coprime.coprime_mul_right_right hac
|
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Algebra.Pi
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Algebra.BigOperators.RingEquiv
import Mathlib.Algebra.Module.LinearMap.Basic
import Mathlib.Algebra.Module.Pi
import Mathlib.Algebra.Star.BigOperators... | Mathlib/Data/Matrix/Basic.lean | 519 | 522 | theorem diagonal_conjTranspose [AddMonoid α] [StarAddMonoid α] (v : n → α) :
(diagonal v)ᴴ = diagonal (star v) := by |
rw [conjTranspose, diagonal_transpose, diagonal_map (star_zero _)]
rfl
|
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 | 231 | 238 | theorem IsLindelof.elim_countable_subcover_image {b : Set ι} {c : ι → Set X} (hs : IsLindelof s)
(hc₁ : ∀ i ∈ b, IsOpen (c i)) (hc₂ : s ⊆ ⋃ i ∈ b, c i) :
∃ b', b' ⊆ b ∧ Set.Countable b' ∧ s ⊆ ⋃ i ∈ b', c i := by |
simp only [Subtype.forall', biUnion_eq_iUnion] at hc₁ hc₂
rcases hs.elim_countable_subcover (fun i ↦ c i : b → Set X) hc₁ hc₂ with ⟨d, hd⟩
refine ⟨Subtype.val '' d, by simp, Countable.image hd.1 Subtype.val, ?_⟩
rw [biUnion_image]
exact hd.2
|
import Mathlib.RepresentationTheory.Rep
import Mathlib.Algebra.Category.FGModuleCat.Limits
import Mathlib.CategoryTheory.Preadditive.Schur
import Mathlib.RepresentationTheory.Basic
#align_import representation_theory.fdRep from "leanprover-community/mathlib"@"19a70dceb9dff0994b92d2dd049de7d84d28112b"
suppress_comp... | Mathlib/RepresentationTheory/FdRep.lean | 95 | 100 | theorem Iso.conj_ρ {V W : FdRep k G} (i : V ≅ W) (g : G) :
W.ρ g = (FdRep.isoToLinearEquiv i).conj (V.ρ g) := by |
-- Porting note: Changed `rw` to `erw`
erw [FdRep.isoToLinearEquiv, ← FGModuleCat.Iso.conj_eq_conj, Iso.conj_apply]
rw [Iso.eq_inv_comp ((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i)]
exact (i.hom.comm g).symm
|
import Mathlib.Algebra.Polynomial.Degree.TrailingDegree
import Mathlib.Algebra.Polynomial.EraseLead
import Mathlib.Algebra.Polynomial.Eval
#align_import data.polynomial.reverse from "leanprover-community/mathlib"@"44de64f183393284a16016dfb2a48ac97382f2bd"
namespace Polynomial
open Polynomial Finsupp Finset
open... | Mathlib/Algebra/Polynomial/Reverse.lean | 398 | 399 | theorem reflect_neg (f : R[X]) (N : ℕ) : reflect N (-f) = -reflect N f := by |
rw [neg_eq_neg_one_mul, ← C_1, ← C_neg, reflect_C_mul, C_neg, C_1, ← neg_eq_neg_one_mul]
|
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.CharP.Algebra
import Mathlib.Data.Nat.Prime
#align_import algebra.char_p.exp_char from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u
variable (R : Type u)
section Semiring
variable [Semiring R]
class inductive Ex... | Mathlib/Algebra/CharP/ExpChar.lean | 120 | 125 | theorem expChar_one_iff_char_zero (p q : ℕ) [CharP R p] [ExpChar R q] : q = 1 ↔ p = 0 := by |
constructor
· rintro rfl
exact char_zero_of_expChar_one R p
· rintro rfl
exact expChar_one_of_char_zero R q
|
import Mathlib.Algebra.GCDMonoid.Multiset
import Mathlib.Combinatorics.Enumerative.Partition
import Mathlib.Data.List.Rotate
import Mathlib.GroupTheory.Perm.Cycle.Factors
import Mathlib.GroupTheory.Perm.Closure
import Mathlib.Algebra.GCDMonoid.Nat
import Mathlib.Tactic.NormNum.GCD
#align_import group_theory.perm.cycl... | Mathlib/GroupTheory/Perm/Cycle/Type.lean | 184 | 192 | theorem orderOf_cycleOf_dvd_orderOf (f : Perm α) (x : α) : orderOf (cycleOf f x) ∣ orderOf f := by |
by_cases hx : f x = x
· rw [← cycleOf_eq_one_iff] at hx
simp [hx]
· refine dvd_of_mem_cycleType ?_
rw [cycleType, Multiset.mem_map]
refine ⟨f.cycleOf x, ?_, ?_⟩
· rwa [← Finset.mem_def, cycleOf_mem_cycleFactorsFinset_iff, mem_support]
· simp [(isCycle_cycleOf _ hx).orderOf]
|
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 | 854 | 861 | theorem lt_find (n : ℕ) (h : ∀ m ≤ n, ¬P m) : (n : PartENat) < find P := by |
rw [coe_lt_iff]
intro h₁
rw [find_get]
have h₂ := @Nat.find_spec P _ h₁
revert h₂
contrapose!
exact h _
|
import Mathlib.AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf
import Mathlib.AlgebraicGeometry.GammaSpecAdjunction
import Mathlib.RingTheory.GradedAlgebra.Radical
#align_import algebraic_geometry.projective_spectrum.scheme from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
noncomp... | Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.lean | 294 | 305 | theorem mem_carrier_iff_of_mem (hm : 0 < m) (q : Spec.T A⁰_ f) (a : A) {n} (hn : a ∈ 𝒜 n) :
a ∈ carrier f_deg q ↔
(HomogeneousLocalization.mk ⟨m * n, ⟨a ^ m, pow_mem_graded m hn⟩,
⟨f ^ n, by rw [mul_comm]; mem_tac⟩, ⟨_, rfl⟩⟩ : A⁰_ f) ∈ q.asIdeal := by |
trans (HomogeneousLocalization.mk ⟨m * n, ⟨proj 𝒜 n a ^ m, by mem_tac⟩,
⟨f ^ n, by rw [mul_comm]; mem_tac⟩, ⟨_, rfl⟩⟩ : A⁰_ f) ∈ q.asIdeal
· refine ⟨fun h ↦ h n, fun h i ↦ if hi : i = n then hi ▸ h else ?_⟩
convert zero_mem q.asIdeal
apply HomogeneousLocalization.val_injective
simp only [proj_appl... |
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