Context stringlengths 57 85k | file_name stringlengths 21 79 | start int64 14 2.42k | end int64 18 2.43k | theorem stringlengths 25 2.71k | proof stringlengths 5 10.6k |
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import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Monoidal.Functor
#align_import category_theory.monoidal.preadditive from "leanprover-community/mathlib"@"986c4d5761f938b2e1c43c01f001b6d9d88c2055"
noncomputable section
open scoped Classical
namespace CategoryTheory
open Cat... | Mathlib/CategoryTheory/Monoidal/Preadditive.lean | 221 | 226 | theorem rightDistributor_inv {J : Type} [Fintype J] (f : J → C) (X : C) :
(rightDistributor f X).inv = ∑ j : J, biproduct.π _ j ≫ (biproduct.ι f j ▷ X) := by |
ext
dsimp [rightDistributor, Functor.mapBiproduct, Functor.mapBicone]
simp only [biproduct.ι_desc, Preadditive.comp_sum, ne_eq, biproduct.ι_π_assoc, dite_comp,
zero_comp, Finset.sum_dite_eq, Finset.mem_univ, eqToHom_refl, Category.id_comp, ite_true]
|
import Mathlib.Topology.Separation
import Mathlib.Topology.NoetherianSpace
#align_import topology.quasi_separated from "leanprover-community/mathlib"@"5dc6092d09e5e489106865241986f7f2ad28d4c8"
open TopologicalSpace
variable {α β : Type*} [TopologicalSpace α] [TopologicalSpace β] {f : α → β}
def IsQuasiSeparate... | Mathlib/Topology/QuasiSeparated.lean | 89 | 96 | theorem OpenEmbedding.isQuasiSeparated_iff (h : OpenEmbedding f) {s : Set α} :
IsQuasiSeparated s ↔ IsQuasiSeparated (f '' s) := by |
refine ⟨fun hs => hs.image_of_embedding h.toEmbedding, ?_⟩
intro H U V hU hU' hU'' hV hV' hV''
rw [h.toEmbedding.isCompact_iff, Set.image_inter h.inj]
exact
H (f '' U) (f '' V) (Set.image_subset _ hU) (h.isOpenMap _ hU') (hU''.image h.continuous)
(Set.image_subset _ hV) (h.isOpenMap _ hV') (hV''.imag... |
import Mathlib.Data.Matrix.Basic
import Mathlib.LinearAlgebra.Matrix.Trace
#align_import data.matrix.basis from "leanprover-community/mathlib"@"320df450e9abeb5fc6417971e75acb6ae8bc3794"
variable {l m n : Type*}
variable {R α : Type*}
namespace Matrix
open Matrix
variable [DecidableEq l] [DecidableEq m] [Decida... | Mathlib/Data/Matrix/Basis.lean | 174 | 176 | theorem trace_eq : trace (stdBasisMatrix i i c) = c := by |
-- Porting note: added `-diag_apply`
simp [trace, -diag_apply]
|
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle
#align_import geometry.euclidean.angle.oriented.right_angle from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped EuclideanGeometry
ope... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 155 | 159 | theorem cos_oangle_add_left_mul_norm_of_oangle_eq_pi_div_two {x y : V}
(h : o.oangle x y = ↑(π / 2)) : Real.Angle.cos (o.oangle (x + y) y) * ‖x + y‖ = ‖y‖ := by |
rw [← neg_inj, oangle_rev, ← oangle_neg_orientation_eq_neg, neg_inj] at h ⊢
rw [add_comm]
exact (-o).cos_oangle_add_right_mul_norm_of_oangle_eq_pi_div_two h
|
import Mathlib.Algebra.BigOperators.Option
import Mathlib.Analysis.BoxIntegral.Box.Basic
import Mathlib.Data.Set.Pairwise.Lattice
#align_import analysis.box_integral.partition.basic from "leanprover-community/mathlib"@"84dc0bd6619acaea625086d6f53cb35cdd554219"
open Set Finset Function
open scoped Classical
open ... | Mathlib/Analysis/BoxIntegral/Partition/Basic.lean | 365 | 367 | theorem biUnionIndex_mem (hJ : J ∈ π.biUnion πi) : π.biUnionIndex πi J ∈ π := by |
rw [biUnionIndex, dif_pos hJ]
exact (π.mem_biUnion.1 hJ).choose_spec.1
|
import Mathlib.Geometry.Manifold.MFDeriv.FDeriv
noncomputable section
open scoped Manifold
open Bundle Set Topology
section SpecificFunctions
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜] {E : Type*} [NormedAddCommGroup E]
[NormedSpace 𝕜 E] {H : Type*} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H)... | Mathlib/Geometry/Manifold/MFDeriv/SpecificFunctions.lean | 157 | 160 | theorem mfderivWithin_id (hxs : UniqueMDiffWithinAt I s x) :
mfderivWithin I I (@id M) s x = ContinuousLinearMap.id 𝕜 (TangentSpace I x) := by |
rw [MDifferentiable.mfderivWithin (mdifferentiableAt_id I) hxs]
exact mfderiv_id I
|
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#align_import algebra.order.group.defs from "leanprover-community/mathlib"@"b599f4e4e5cf1fbcb4194503671d3d9e569c1fce"
open Function
universe u
variable {α : Type u}
class OrderedAddCommGroup (α : Ty... | Mathlib/Algebra/Order/Group/Defs.lean | 343 | 345 | theorem inv_le_inv_iff : a⁻¹ ≤ b⁻¹ ↔ b ≤ a := by |
rw [← mul_le_mul_iff_left a, ← mul_le_mul_iff_right b]
simp
|
import Mathlib.Init.ZeroOne
import Mathlib.Data.Set.Defs
import Mathlib.Order.Basic
import Mathlib.Order.SymmDiff
import Mathlib.Tactic.Tauto
import Mathlib.Tactic.ByContra
import Mathlib.Util.Delaborators
#align_import data.set.basic from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
... | Mathlib/Data/Set/Basic.lean | 2,271 | 2,272 | theorem ite_compl (t s s' : Set α) : tᶜ.ite s s' = t.ite s' s := by |
rw [Set.ite, Set.ite, diff_compl, union_comm, diff_eq]
|
import Mathlib.Order.CompleteLattice
import Mathlib.Order.GaloisConnection
import Mathlib.Data.Set.Lattice
import Mathlib.Tactic.AdaptationNote
#align_import data.rel from "leanprover-community/mathlib"@"706d88f2b8fdfeb0b22796433d7a6c1a010af9f2"
variable {α β γ : Type*}
def Rel (α β : Type*) :=
α → β → Prop --... | Mathlib/Data/Rel.lean | 375 | 380 | theorem graph_injective : Injective (graph : (α → β) → Rel α β) := by |
intro _ g h
ext x
have h2 := congr_fun₂ h x (g x)
simp only [graph_def, eq_iff_iff, iff_true] at h2
exact h2
|
import Mathlib.AlgebraicTopology.SimplexCategory
import Mathlib.CategoryTheory.Comma.Arrow
import Mathlib.CategoryTheory.Limits.FunctorCategory
import Mathlib.CategoryTheory.Opposites
#align_import algebraic_topology.simplicial_object from "leanprover-community/mathlib"@"5ed51dc37c6b891b79314ee11a50adc2b1df6fd6"
o... | Mathlib/AlgebraicTopology/SimplicialObject.lean | 174 | 177 | theorem δ_comp_σ_succ' {n} {j : Fin (n + 2)} {i : Fin (n + 1)} (H : j = i.succ) :
X.σ i ≫ X.δ j = 𝟙 _ := by |
subst H
rw [δ_comp_σ_succ]
|
import Mathlib.RingTheory.AdjoinRoot
import Mathlib.FieldTheory.Minpoly.Field
import Mathlib.RingTheory.Polynomial.GaussLemma
#align_import field_theory.minpoly.is_integrally_closed from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open scoped Classical Polynomial
open Polynomial Set... | Mathlib/FieldTheory/Minpoly/IsIntegrallyClosed.lean | 202 | 205 | theorem _root_.PowerBasis.ofGenMemAdjoin'_gen (B : PowerBasis R S) (hint : IsIntegral R x)
(hx : B.gen ∈ adjoin R ({x} : Set S)) :
(B.ofGenMemAdjoin' hint hx).gen = x := by |
simp [PowerBasis.ofGenMemAdjoin']
|
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.FractionalIdeal.Basic
#align_import ring_theory.fractional_ideal from "leanprover-community/mathlib"@"ed90a7d327c3a5caf65a6faf7e8a0d63c4605df7"
open IsLocalization Pointwise nonZeroDivisors
namespace FractionalIdeal
open Set Submodule
variable... | Mathlib/RingTheory/FractionalIdeal/Operations.lean | 603 | 606 | theorem spanFinset_eq_zero {ι : Type*} {s : Finset ι} {f : ι → K} :
spanFinset R₁ s f = 0 ↔ ∀ j ∈ s, f j = 0 := by |
simp only [← coeToSubmodule_inj, spanFinset_coe, coe_zero, Submodule.span_eq_bot,
Set.mem_image, Finset.mem_coe, forall_exists_index, and_imp, forall_apply_eq_imp_iff₂]
|
import Mathlib.AlgebraicGeometry.Morphisms.QuasiCompact
import Mathlib.Topology.QuasiSeparated
#align_import algebraic_geometry.morphisms.quasi_separated from "leanprover-community/mathlib"@"1a51edf13debfcbe223fa06b1cb353b9ed9751cc"
noncomputable section
open CategoryTheory CategoryTheory.Limits Opposite Topolog... | Mathlib/AlgebraicGeometry/Morphisms/QuasiSeparated.lean | 271 | 274 | theorem IsAffineOpen.isQuasiSeparated {X : Scheme} {U : Opens X.carrier} (hU : IsAffineOpen U) :
IsQuasiSeparated (U : Set X.carrier) := by |
rw [isQuasiSeparated_iff_quasiSeparatedSpace]
exacts [@AlgebraicGeometry.quasiSeparatedSpace_of_isAffine _ hU, U.isOpen]
|
import Mathlib.Data.Nat.Count
import Mathlib.Data.Nat.SuccPred
import Mathlib.Order.Interval.Set.Monotone
import Mathlib.Order.OrderIsoNat
#align_import data.nat.nth from "leanprover-community/mathlib"@"7fdd4f3746cb059edfdb5d52cba98f66fce418c0"
open Finset
namespace Nat
variable (p : ℕ → Prop)
noncomputable d... | Mathlib/Data/Nat/Nth.lean | 76 | 80 | theorem nth_strictMonoOn (hf : (setOf p).Finite) :
StrictMonoOn (nth p) (Set.Iio hf.toFinset.card) := by |
rintro m (hm : m < _) n (hn : n < _) h
simp only [nth_eq_orderEmbOfFin, *]
exact OrderEmbedding.strictMono _ h
|
import Mathlib.Order.Filter.FilterProduct
import Mathlib.Analysis.SpecificLimits.Basic
#align_import data.real.hyperreal from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open scoped Classical
open Filter Germ Topology
def Hyperreal : Type :=
Germ (hyperfilter ℕ : Filter ℕ) ℝ deri... | Mathlib/Data/Real/Hyperreal.lean | 751 | 753 | theorem infiniteNeg_iff_infinitesimal_inv_neg {x : ℝ*} :
InfiniteNeg x ↔ Infinitesimal x⁻¹ ∧ x⁻¹ < 0 := by |
rw [← infinitePos_neg, infinitePos_iff_infinitesimal_inv_pos, inv_neg, neg_pos, infinitesimal_neg]
|
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#align_import number_theory.modular_forms.congruence_subgroups from "leanprover-community/mathlib"@"ae690b0c236e488a0043f6faa8ce3546e7f2f... | Mathlib/NumberTheory/ModularForms/CongruenceSubgroups.lean | 78 | 88 | theorem Gamma_zero_bot : Gamma 0 = ⊥ := by |
ext
simp only [Gamma_mem, coe_matrix_coe, Int.coe_castRingHom, map_apply, Int.cast_id,
Subgroup.mem_bot]
constructor
· intro h
ext i j
fin_cases i <;> fin_cases j <;> simp only [h]
exacts [h.1, h.2.1, h.2.2.1, h.2.2.2]
· intro h
simp [h]
|
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.Order.Fin
import Mathlib.Order.PiLex
import Mathlib.Order.Interval.Set.Basic
#align_import data.fin.tuple.basic from "leanprover-community/mathlib"@"ef997baa41b5c428be3fb50089a7139bf4ee886b"
assert_not_exists MonoidWithZero
un... | Mathlib/Data/Fin/Tuple/Basic.lean | 1,074 | 1,083 | theorem nat_find_mem_find {p : Fin n → Prop} [DecidablePred p]
(h : ∃ i, ∃ hin : i < n, p ⟨i, hin⟩) :
(⟨Nat.find h, (Nat.find_spec h).fst⟩ : Fin n) ∈ find p := by |
let ⟨i, hin, hi⟩ := h
cases' hf : find p with f
· rw [find_eq_none_iff] at hf
exact (hf ⟨i, hin⟩ hi).elim
· refine Option.some_inj.2 (le_antisymm ?_ ?_)
· exact find_min' hf (Nat.find_spec h).snd
· exact Nat.find_min' _ ⟨f.2, by convert find_spec p hf⟩
|
import Mathlib.Analysis.Convex.Normed
import Mathlib.Analysis.NormedSpace.Connected
import Mathlib.LinearAlgebra.AffineSpace.ContinuousAffineEquiv
open Set
variable {F : Type*} [AddCommGroup F] [Module ℝ F] [TopologicalSpace F]
def AmpleSet (s : Set F) : Prop :=
∀ x ∈ s, convexHull ℝ (connectedComponentIn s ... | Mathlib/Analysis/Convex/AmpleSet.lean | 53 | 56 | theorem ampleSet_univ {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] :
AmpleSet (univ : Set F) := by |
intro x _
rw [connectedComponentIn_univ, PreconnectedSpace.connectedComponent_eq_univ, convexHull_univ]
|
import Mathlib.Analysis.BoxIntegral.Partition.Filter
import Mathlib.Analysis.BoxIntegral.Partition.Measure
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Init.Data.Bool.Lemmas
#align_import analysis.box_integral.basic from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open... | Mathlib/Analysis/BoxIntegral/Basic.lean | 682 | 711 | theorem integrable_of_continuousOn [CompleteSpace E] {I : Box ι} {f : ℝⁿ → E}
(hc : ContinuousOn f (Box.Icc I)) (μ : Measure ℝⁿ) [IsLocallyFiniteMeasure μ] :
Integrable.{u, v, v} I l f μ.toBoxAdditive.toSMul := by |
have huc := I.isCompact_Icc.uniformContinuousOn_of_continuous hc
rw [Metric.uniformContinuousOn_iff_le] at huc
refine integrable_iff_cauchy_basis.2 fun ε ε0 => ?_
rcases exists_pos_mul_lt ε0 (μ.toBoxAdditive I) with ⟨ε', ε0', hε⟩
rcases huc ε' ε0' with ⟨δ, δ0 : 0 < δ, Hδ⟩
refine ⟨fun _ _ => ⟨δ / 2, half_po... |
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 | 107 | 111 | theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) :
taylorWithinEval f n s x₀ x₀ = f x₀ := by |
induction' n with k hk
· exact taylor_within_zero_eval _ _ _ _
simp [hk]
|
import Mathlib.Data.Finsupp.Defs
#align_import data.finsupp.ne_locus from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
variable {α M N P : Type*}
namespace Finsupp
variable [DecidableEq α]
section NHasZero
variable [DecidableEq N] [Zero N] (f g : α →₀ N)
def neLocus (f g : α →₀ ... | Mathlib/Data/Finsupp/NeLocus.lean | 74 | 76 | theorem neLocus_zero_right : f.neLocus 0 = f.support := by |
ext
rw [mem_neLocus, mem_support_iff, coe_zero, Pi.zero_apply]
|
import Mathlib.MeasureTheory.Group.Action
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Group.Pointwise
#align_import measure_theory.group.fundamental_domain from "leanprover-community/mathlib"@"3b52265189f3fb43aa631edffce5d060fafaf82f"
open scoped ENNReal Pointwise Topology NNRea... | Mathlib/MeasureTheory/Group/FundamentalDomain.lean | 426 | 430 | theorem integral_eq_tsum_of_ac (h : IsFundamentalDomain G s μ) (hν : ν ≪ μ) (f : α → E)
(hf : Integrable f ν) : ∫ x, f x ∂ν = ∑' g : G, ∫ x in g • s, f x ∂ν := by |
rw [← MeasureTheory.integral_sum_measure, h.sum_restrict_of_ac hν]
rw [h.sum_restrict_of_ac hν]
exact hf
|
import Mathlib.Data.List.OfFn
import Mathlib.Data.List.Range
#align_import data.list.indexes from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
assert_not_exists MonoidWithZero
universe u v
open Function
namespace List
variable {α : Type u} {β : Type v}
section MapIdx
-- Porting n... | Mathlib/Data/List/Indexes.lean | 198 | 202 | theorem mapIdx_append (K L : List α) (f : ℕ → α → β) :
(K ++ L).mapIdx f = K.mapIdx f ++ L.mapIdx fun i a ↦ f (i + K.length) a := by |
induction' K with a J IH generalizing f
· rfl
· simp [IH fun i ↦ f (i + 1), Nat.add_assoc, Nat.succ_eq_add_one]
|
import Mathlib.Probability.ProbabilityMassFunction.Monad
#align_import probability.probability_mass_function.constructions from "leanprover-community/mathlib"@"4ac69b290818724c159de091daa3acd31da0ee6d"
universe u
namespace PMF
noncomputable section
variable {α β γ : Type*}
open scoped Classical
open NNReal ENN... | Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | 96 | 97 | theorem toOuterMeasure_map_apply : (p.map f).toOuterMeasure s = p.toOuterMeasure (f ⁻¹' s) := by |
simp [map, Set.indicator, toOuterMeasure_apply p (f ⁻¹' s)]
|
import Mathlib.Algebra.BigOperators.Group.List
import Mathlib.Data.Vector.Defs
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.OfFn
import Mathlib.Data.List.InsertNth
import Mathlib.Control.Applicative
import Mathlib.Control.Traversable.Basic
#align_import data.vector.basic from "leanprover-community/mathlib"... | Mathlib/Data/Vector/Basic.lean | 383 | 394 | theorem scanl_get (i : Fin n) :
(scanl f b v).get i.succ = f ((scanl f b v).get (Fin.castSucc i)) (v.get i) := by |
cases' n with n
· exact i.elim0
induction' n with n hn generalizing b
· have i0 : i = 0 := Fin.eq_zero _
simp [scanl_singleton, i0, get_zero]; simp [get_eq_get, List.get]
· rw [← cons_head_tail v, scanl_cons, get_cons_succ]
refine Fin.cases ?_ ?_ i
· simp only [get_zero, scanl_head, Fin.castSucc_... |
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Int.Defs
import Mathlib.Data.Rat.Init
import Mathlib.Order.Basic
import Mathlib.Tactic.Common
#align_import data.rat.defs from "leanprover-community/mathlib"@"18a5306c091183ac90884daa9373fa3b178e8607"
-- TODO: If `Inv` was defined earlier than `Algebra.Group.De... | Mathlib/Data/Rat/Defs.lean | 127 | 128 | theorem divInt_eq_zero {a b : ℤ} (b0 : b ≠ 0) : a /. b = 0 ↔ a = 0 := by |
rw [← zero_divInt b, divInt_eq_iff b0 b0, Int.zero_mul, Int.mul_eq_zero, or_iff_left b0]
|
import Mathlib.MeasureTheory.Integral.Lebesgue
open Set hiding restrict restrict_apply
open Filter ENNReal NNReal MeasureTheory.Measure
namespace MeasureTheory
variable {α : Type*} {m0 : MeasurableSpace α} {μ : Measure α}
noncomputable
def Measure.withDensity {m : MeasurableSpace α} (μ : Measure α) (f : α → ℝ≥... | Mathlib/MeasureTheory/Measure/WithDensity.lean | 447 | 453 | theorem lintegral_withDensity_le_lintegral_mul (μ : Measure α) {f : α → ℝ≥0∞}
(f_meas : Measurable f) (g : α → ℝ≥0∞) : (∫⁻ a, g a ∂μ.withDensity f) ≤ ∫⁻ a, (f * g) a ∂μ := by |
rw [← iSup_lintegral_measurable_le_eq_lintegral, ← iSup_lintegral_measurable_le_eq_lintegral]
refine iSup₂_le fun i i_meas => iSup_le fun hi => ?_
have A : f * i ≤ f * g := fun x => mul_le_mul_left' (hi x) _
refine le_iSup₂_of_le (f * i) (f_meas.mul i_meas) ?_
exact le_iSup_of_le A (le_of_eq (lintegral_withD... |
import Mathlib.Algebra.Field.Subfield
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Topology.Algebra.GroupWithZero
import Mathlib.Topology.Algebra.Ring.Basic
import Mathlib.Topology.Order.LocalExtr
#align_import topology.algebra.field from "leanprover-community/mathlib"@"c10e724be91096453ee3db13862... | Mathlib/Topology/Algebra/Field.lean | 142 | 149 | theorem IsPreconnected.eq_or_eq_neg_of_sq_eq [Field 𝕜] [HasContinuousInv₀ 𝕜] [ContinuousMul 𝕜]
(hS : IsPreconnected S) (hf : ContinuousOn f S) (hg : ContinuousOn g S)
(hsq : EqOn (f ^ 2) (g ^ 2) S) (hg_ne : ∀ {x : α}, x ∈ S → g x ≠ 0) :
EqOn f g S ∨ EqOn f (-g) S := by |
have hsq : EqOn ((f / g) ^ 2) 1 S := fun x hx => by
simpa [div_eq_one_iff_eq (pow_ne_zero _ (hg_ne hx))] using hsq hx
simpa (config := { contextual := true }) [EqOn, div_eq_iff (hg_ne _)]
using hS.eq_one_or_eq_neg_one_of_sq_eq (hf.div hg fun z => hg_ne) hsq
|
import Mathlib.Algebra.Order.Ring.Nat
#align_import data.nat.dist from "leanprover-community/mathlib"@"d50b12ae8e2bd910d08a94823976adae9825718b"
namespace Nat
def dist (n m : ℕ) :=
n - m + (m - n)
#align nat.dist Nat.dist
-- Should be aligned to `Nat.dist.eq_def`, but that is generated on demand and isn't pr... | Mathlib/Data/Nat/Dist.lean | 31 | 31 | theorem dist_self (n : ℕ) : dist n n = 0 := by | simp [dist, tsub_self]
|
import Mathlib.MeasureTheory.Integral.IntegrableOn
#align_import measure_theory.function.locally_integrable from "leanprover-community/mathlib"@"08a4542bec7242a5c60f179e4e49de8c0d677b1b"
open MeasureTheory MeasureTheory.Measure Set Function TopologicalSpace Bornology
open scoped Topology Interval ENNReal
variabl... | Mathlib/MeasureTheory/Function/LocallyIntegrable.lean | 141 | 156 | theorem locallyIntegrableOn_iff [LocallyCompactSpace X] [T2Space X] (hs : IsClosed s ∨ IsOpen s) :
LocallyIntegrableOn f s μ ↔ ∀ (k : Set X), k ⊆ s → (IsCompact k → IntegrableOn f k μ) := by |
-- The correct condition is that `s` be *locally closed*, i.e. for every `x ∈ s` there is some
-- `U ∈ 𝓝 x` such that `U ∩ s` is closed. But mathlib doesn't have locally closed sets yet.
refine ⟨fun hf k hk => hf.integrableOn_compact_subset hk, fun hf x hx => ?_⟩
cases hs with
| inl hs =>
exact
le... |
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 | 161 | 164 | theorem set_lintegral_const_lt_top [IsFiniteMeasure μ] (s : Set α) {c : ℝ≥0∞} (hc : c ≠ ∞) :
∫⁻ _ in s, c ∂μ < ∞ := by |
rw [lintegral_const]
exact ENNReal.mul_lt_top hc (measure_ne_top (μ.restrict s) univ)
|
import Mathlib.AlgebraicGeometry.Gluing
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.AlgebraicGeometry.AffineScheme
import Mathlib.CategoryTheory.Limits.Shapes.Diagonal
#align_import algebraic_geometry.pullbacks from "leanprover-community/mathlib"@"7316286ff2942aa14e540add9058c6b0aa1c8070"
set_opt... | Mathlib/AlgebraicGeometry/Pullbacks.lean | 407 | 409 | theorem pullbackP1Iso_hom_snd (i : 𝒰.J) :
(pullbackP1Iso 𝒰 f g i).hom ≫ pullback.snd = pullback.fst ≫ p2 𝒰 f g := by |
simp_rw [pullbackP1Iso, pullback.lift_snd]
|
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 | 311 | 313 | theorem degree_C_mul_X_pow_le (n : ℕ) (a : R) : degree (C a * X ^ n) ≤ n := by |
rw [C_mul_X_pow_eq_monomial]
apply degree_monomial_le
|
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology.Instances.Sign
#align_import analysis.special_functions.trigonometric.angle from "leanprover-community/mathlib"@"213b0cff7bc5ab6696ee07cceec80829... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 737 | 749 | theorem cos_nonneg_iff_abs_toReal_le_pi_div_two {θ : Angle} : 0 ≤ cos θ ↔ |θ.toReal| ≤ π / 2 := by |
nth_rw 1 [← coe_toReal θ]
rw [abs_le, cos_coe]
refine ⟨fun h => ?_, cos_nonneg_of_mem_Icc⟩
by_contra hn
rw [not_and_or, not_le, not_le] at hn
refine (not_lt.2 h) ?_
rcases hn with (hn | hn)
· rw [← Real.cos_neg]
refine cos_neg_of_pi_div_two_lt_of_lt (by linarith) ?_
linarith [neg_pi_lt_toReal θ... |
import Mathlib.Algebra.Quaternion
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Topology.Algebra.Algebra
#align_import analysis.quaternion from "leanprover-community/mathlib"@"07992a1d1f7a4176c6d3f160209608be4e198566"
@[inherit_doc] scoped[Quaternion... | Mathlib/Analysis/Quaternion.lean | 244 | 247 | theorem summable_coe {f : α → ℝ} : (Summable fun a => (f a : ℍ)) ↔ Summable f := by |
simpa only using
Summable.map_iff_of_leftInverse (algebraMap ℝ ℍ) (show ℍ →ₗ[ℝ] ℝ from QuaternionAlgebra.reₗ _ _)
(continuous_algebraMap _ _) continuous_re coe_re
|
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.MinMax
#align_import data.set.pointwise.interval from "leanprover-community/mathlib"@"2196ab363eb097c... | Mathlib/Data/Set/Pointwise/Interval.lean | 162 | 163 | theorem preimage_const_add_Ioo : (fun x => a + x) ⁻¹' Ioo b c = Ioo (b - a) (c - a) := by |
simp [← Ioi_inter_Iio]
|
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Prod.Basic
import Mathlib.Data.Sum.Basic
import Mathlib.Logic.Unique
import Mathlib.Tactic.Spread
#align_import data.pi.algebra from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
-- We enforce to only import `Algebra.Group.Defs` and ... | Mathlib/Algebra/Group/Pi/Basic.lean | 390 | 392 | theorem mulSingle_comm [One β] (i : I) (x : β) (i' : I) :
(mulSingle i x : I → β) i' = (mulSingle i' x : I → β) i := by |
simp [mulSingle_apply, eq_comm]
|
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan
import Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv
#align_import analysis.special_functions.trigonometric.arctan_deriv from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
namespace Real... | Mathlib/Analysis/SpecialFunctions/Trigonometric/ArctanDeriv.lean | 48 | 51 | theorem continuousAt_tan {x : ℝ} : ContinuousAt tan x ↔ cos x ≠ 0 := by |
refine ⟨fun hc h₀ => ?_, fun h => (hasDerivAt_tan h).continuousAt⟩
exact not_tendsto_nhds_of_tendsto_atTop (tendsto_abs_tan_of_cos_eq_zero h₀) _
(hc.norm.tendsto.mono_left inf_le_left)
|
import Mathlib.CategoryTheory.Bicategory.Functor.Oplax
#align_import category_theory.bicategory.natural_transformation from "leanprover-community/mathlib"@"4ff75f5b8502275a4c2eb2d2f02bdf84d7fb8993"
namespace CategoryTheory
open Category Bicategory
open scoped Bicategory
universe w₁ w₂ v₁ v₂ u₁ u₂
variable {B :... | Mathlib/CategoryTheory/Bicategory/NaturalTransformation.lean | 128 | 137 | theorem whiskerRight_naturality_comp (f : a ⟶ b) (g : b ⟶ c) (h : G.obj c ⟶ a') :
η.naturality (f ≫ g) ▷ h ≫ (α_ _ _ _).hom ≫ η.app a ◁ G.mapComp f g ▷ h =
F.mapComp f g ▷ η.app c ▷ h ≫
(α_ _ _ _).hom ▷ h ≫
(α_ _ _ _).hom ≫
F.map f ◁ η.naturality g ▷ h ≫
(α_ _ _ _).... |
rw [← associator_naturality_middle, ← comp_whiskerRight_assoc, naturality_comp]; simp
|
import Mathlib.Geometry.Euclidean.Circumcenter
#align_import geometry.euclidean.monge_point from "leanprover-community/mathlib"@"1a4df69ca1a9a0e5e26bfe12e2b92814216016d0"
noncomputable section
open scoped Classical
open scoped RealInnerProductSpace
namespace Affine
namespace Simplex
open Finset AffineSubspac... | Mathlib/Geometry/Euclidean/MongePoint.lean | 362 | 365 | theorem vectorSpan_isOrtho_altitude_direction {n : ℕ} (s : Simplex ℝ P (n + 1)) (i : Fin (n + 2)) :
vectorSpan ℝ (s.points '' ↑(Finset.univ.erase i)) ⟂ (s.altitude i).direction := by |
rw [direction_altitude]
exact (Submodule.isOrtho_orthogonal_right _).mono_right inf_le_left
|
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Connected.PathConnected
import Mathlib.Topology.Homotopy.Basic
#align_import topology.homotopy.H_spaces from "leanprover-community/mathlib"@"729d23f9e1640e1687141be89b106d3c8f9d10c0"
-- Porting note: `HSpace` already contains an upper case letter
set_optio... | Mathlib/Topology/Homotopy/HSpaces.lean | 193 | 202 | theorem qRight_zero_right (t : I) :
(qRight (t, 0) : ℝ) = if (t : ℝ) ≤ 1 / 2 then (2 : ℝ) * t else 1 := by |
simp only [qRight, coe_zero, add_zero, div_one]
split_ifs
· rw [Set.projIcc_of_mem _ ((mul_pos_mem_iff zero_lt_two).2 _)]
refine ⟨t.2.1, ?_⟩
tauto
· rw [(Set.projIcc_eq_right _).2]
· linarith
· exact zero_lt_one
|
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 | 83 | 85 | theorem hasseDeriv_zero' : hasseDeriv 0 f = f := by |
simp only [hasseDeriv_apply, tsub_zero, Nat.choose_zero_right, Nat.cast_one, one_mul,
sum_monomial_eq]
|
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
#align_import linear_algebra.symplectic_group from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open Matrix
variable {l R : Type*}
namespace Matrix
variable (l) [DecidableEq l] (R) [CommRing R]
section JMatrixLemmas
def J : ... | Mathlib/LinearAlgebra/SymplecticGroup.lean | 59 | 62 | theorem J_inv : (J l R)⁻¹ = -J l R := by |
refine Matrix.inv_eq_right_inv ?_
rw [Matrix.mul_neg, J_squared]
exact neg_neg 1
|
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.RingTheory.Localization.FractionRing
#align_import data.polynomial.ring_division from "leanprover-community/mathlib"@"8efcf8022aac8e01df8d302dcebdbc25d6a886c8"
noncomputable section
namespace Polynomial
universe u v w z
variable {R : Type u} {S : Ty... | Mathlib/Algebra/Polynomial/Roots.lean | 452 | 453 | theorem aroots_zero (S) [CommRing S] [IsDomain S] [Algebra T S] : (0 : T[X]).aroots S = 0 := by |
rw [← C_0, aroots_C]
|
import Mathlib.Order.Interval.Set.OrdConnected
import Mathlib.Data.Set.Lattice
#align_import data.set.intervals.ord_connected_component from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
open Interval Function OrderDual
namespace Set
variable {α : Type*} [LinearOrder α] {s t : Set α}... | Mathlib/Order/Interval/Set/OrdConnectedComponent.lean | 53 | 54 | theorem self_mem_ordConnectedComponent : x ∈ ordConnectedComponent s x ↔ x ∈ s := by |
rw [mem_ordConnectedComponent, uIcc_self, singleton_subset_iff]
|
import Mathlib.MeasureTheory.Measure.Sub
import Mathlib.MeasureTheory.Decomposition.SignedHahn
import Mathlib.MeasureTheory.Function.AEEqOfIntegral
#align_import measure_theory.decomposition.lebesgue from "leanprover-community/mathlib"@"b2ff9a3d7a15fd5b0f060b135421d6a89a999c2f"
open scoped MeasureTheory NNReal ENN... | Mathlib/MeasureTheory/Decomposition/Lebesgue.lean | 102 | 106 | theorem measurable_rnDeriv (μ ν : Measure α) : Measurable <| μ.rnDeriv ν := by |
by_cases h : HaveLebesgueDecomposition μ ν
· exact (haveLebesgueDecomposition_spec μ ν).1
· rw [rnDeriv_of_not_haveLebesgueDecomposition h]
exact measurable_zero
|
import Mathlib.Analysis.Analytic.Composition
#align_import analysis.analytic.inverse from "leanprover-community/mathlib"@"284fdd2962e67d2932fa3a79ce19fcf92d38e228"
open scoped Classical Topology
open Finset Filter
namespace FormalMultilinearSeries
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜] {E : Type*} ... | Mathlib/Analysis/Analytic/Inverse.lean | 248 | 262 | theorem comp_rightInv (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F)
(h : p 1 = (continuousMultilinearCurryFin1 𝕜 E F).symm i) (h0 : p 0 = 0) :
p.comp (rightInv p i) = id 𝕜 F := by |
ext (n v)
match n with
| 0 =>
simp only [h0, ContinuousMultilinearMap.zero_apply, id_apply_ne_one, Ne, not_false_iff,
zero_ne_one, comp_coeff_zero']
| 1 =>
simp only [comp_coeff_one, h, rightInv_coeff_one, ContinuousLinearEquiv.apply_symm_apply,
id_apply_one, ContinuousLinearEquiv.coe_apply... |
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
#align_import measure_theory.constructions.borel_space.basic from "leanprover-community/mathlib"@"9f55d0d4363ae59948c33864cbc52e0b12e0e8ce"
open Set Filter MeasureTheory MeasurableSpace
open scoped Classical Topology NNReal ENNReal MeasureTheory
univers... | Mathlib/MeasureTheory/Constructions/BorelSpace/Real.lean | 44 | 54 | theorem borel_eq_generateFrom_Iio_rat : borel ℝ = .generateFrom (⋃ a : ℚ, {Iio (a : ℝ)}) := by |
rw [borel_eq_generateFrom_Iio]
refine le_antisymm
(generateFrom_le ?_)
(generateFrom_mono <| iUnion_subset fun q ↦ singleton_subset_iff.mpr <| mem_range_self _)
rintro _ ⟨a, rfl⟩
have : IsLUB (range ((↑) : ℚ → ℝ) ∩ Iio a) a := by
simp [isLUB_iff_le_iff, mem_upperBounds, ← le_iff_forall_rat_lt_imp_l... |
import Mathlib.Order.CompleteLattice
import Mathlib.Order.Cover
import Mathlib.Order.Iterate
import Mathlib.Order.WellFounded
#align_import order.succ_pred.basic from "leanprover-community/mathlib"@"0111834459f5d7400215223ea95ae38a1265a907"
open Function OrderDual Set
variable {α β : Type*}
@[ext]
class SuccOr... | Mathlib/Order/SuccPred/Basic.lean | 343 | 344 | theorem Ico_succ_left_of_not_isMax (ha : ¬IsMax a) : Ico (succ a) b = Ioo a b := by |
rw [← Ici_inter_Iio, Ici_succ_of_not_isMax ha, Ioi_inter_Iio]
|
import Mathlib.Combinatorics.SimpleGraph.Subgraph
import Mathlib.Data.List.Rotate
#align_import combinatorics.simple_graph.connectivity from "leanprover-community/mathlib"@"b99e2d58a5e6861833fa8de11e51a81144258db4"
open Function
universe u v w
namespace SimpleGraph
variable {V : Type u} {V' : Type v} {V'' : Typ... | Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | 979 | 982 | theorem isTrail_copy {u v u' v'} (p : G.Walk u v) (hu : u = u') (hv : v = v') :
(p.copy hu hv).IsTrail ↔ p.IsTrail := by |
subst_vars
rfl
|
import Mathlib.Algebra.CharZero.Lemmas
import Mathlib.Algebra.Order.Interval.Set.Group
import Mathlib.Algebra.Group.Int
import Mathlib.Data.Int.Lemmas
import Mathlib.Data.Set.Subsingleton
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Order.GaloisConnection
import Mathlib.Tactic.Abel
import Mathlib.Tactic.Linarith... | Mathlib/Algebra/Order/Floor.lean | 577 | 589 | theorem subsingleton_floorSemiring {α} [LinearOrderedSemiring α] :
Subsingleton (FloorSemiring α) := by |
refine ⟨fun H₁ H₂ => ?_⟩
have : H₁.ceil = H₂.ceil := funext fun a => (H₁.gc_ceil.l_unique H₂.gc_ceil) fun n => rfl
have : H₁.floor = H₂.floor := by
ext a
cases' lt_or_le a 0 with h h
· rw [H₁.floor_of_neg, H₂.floor_of_neg] <;> exact h
· refine eq_of_forall_le_iff fun n => ?_
rw [H₁.gc_floor... |
import Mathlib.Analysis.Complex.AbsMax
import Mathlib.Analysis.Complex.RemovableSingularity
#align_import analysis.complex.schwarz from "leanprover-community/mathlib"@"3f655f5297b030a87d641ad4e825af8d9679eb0b"
open Metric Set Function Filter TopologicalSpace
open scoped Topology
namespace Complex
section Space... | Mathlib/Analysis/Complex/Schwarz.lean | 154 | 160 | theorem dist_le_div_mul_dist_of_mapsTo_ball (hd : DifferentiableOn ℂ f (ball c R₁))
(h_maps : MapsTo f (ball c R₁) (ball (f c) R₂)) (hz : z ∈ ball c R₁) :
dist (f z) (f c) ≤ R₂ / R₁ * dist z c := by |
rcases eq_or_ne z c with (rfl | hne);
· simp only [dist_self, mul_zero, le_rfl]
simpa only [dslope_of_ne _ hne, slope_def_module, norm_smul, norm_inv, ← div_eq_inv_mul, ←
dist_eq_norm, div_le_iff (dist_pos.2 hne)] using norm_dslope_le_div_of_mapsTo_ball hd h_maps hz
|
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 | 242 | 272 | theorem FinrankQuotientMap.linearIndependent_of_nontrivial [IsDedekindDomain R]
(hRS : RingHom.ker (algebraMap R S) ≠ ⊤) (f : V'' →ₗ[R] V) (hf : Function.Injective f)
(f' : V'' →ₗ[R] V') {ι : Type*} {b : ι → V''} (hb' : LinearIndependent S (f' ∘ b)) :
LinearIndependent K (f ∘ b) := by |
contrapose! hb' with hb
-- Informally, if we have a nontrivial linear dependence with coefficients `g` in `K`,
-- then we can find a linear dependence with coefficients `I.Quotient.mk g'` in `R/I`,
-- where `I = ker (algebraMap R S)`.
-- We make use of the same principle but stay in `R` everywhere.
simp on... |
import Mathlib.Geometry.Manifold.PartitionOfUnity
import Mathlib.Geometry.Manifold.Metrizable
import Mathlib.MeasureTheory.Function.AEEqOfIntegral
open MeasureTheory Filter Metric Function Set TopologicalSpace
open scoped Topology Manifold
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimen... | Mathlib/Analysis/Distribution/AEEqOfIntegralContDiff.lean | 156 | 169 | theorem ae_eq_of_integral_smooth_smul_eq
(hf : LocallyIntegrable f μ) (hf' : LocallyIntegrable f' μ) (h : ∀ (g : M → ℝ),
Smooth I 𝓘(ℝ) g → HasCompactSupport g → ∫ x, g x • f x ∂μ = ∫ x, g x • f' x ∂μ) :
∀ᵐ x ∂μ, f x = f' x := by |
have : ∀ᵐ x ∂μ, (f - f') x = 0 := by
apply ae_eq_zero_of_integral_smooth_smul_eq_zero I (hf.sub hf')
intro g g_diff g_supp
simp only [Pi.sub_apply, smul_sub]
rw [integral_sub, sub_eq_zero]
· exact h g g_diff g_supp
· exact hf.integrable_smul_left_of_hasCompactSupport g_diff.continuous g_supp
... |
import Mathlib.Logic.Function.Basic
import Mathlib.Logic.Relator
import Mathlib.Init.Data.Quot
import Mathlib.Tactic.Cases
import Mathlib.Tactic.Use
import Mathlib.Tactic.MkIffOfInductiveProp
import Mathlib.Tactic.SimpRw
#align_import logic.relation from "leanprover-community/mathlib"@"3365b20c2ffa7c35e47e5209b89ba9a... | Mathlib/Logic/Relation.lean | 159 | 164 | theorem comp_assoc : (r ∘r p) ∘r q = r ∘r p ∘r q := by |
funext a d
apply propext
constructor
· exact fun ⟨c, ⟨b, hab, hbc⟩, hcd⟩ ↦ ⟨b, hab, c, hbc, hcd⟩
· exact fun ⟨b, hab, c, hbc, hcd⟩ ↦ ⟨c, ⟨b, hab, hbc⟩, hcd⟩
|
import Mathlib.Algebra.MvPolynomial.Funext
import Mathlib.Algebra.Ring.ULift
import Mathlib.RingTheory.WittVector.Basic
#align_import ring_theory.witt_vector.is_poly from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc9524185ae994553ebf4"
namespace WittVector
universe u
variable {p : ℕ} {R S : Type u} {σ id... | Mathlib/RingTheory/WittVector/IsPoly.lean | 349 | 359 | theorem IsPoly.map [Fact p.Prime] {f} (hf : IsPoly p f) (g : R →+* S) (x : 𝕎 R) :
map g (f x) = f (map g x) := by |
-- this could be turned into a tactic “macro” (taking `hf` as parameter)
-- so that applications do not have to worry about the universe issue
-- see `IsPoly₂.map` for a slightly more general proof strategy
obtain ⟨φ, hf⟩ := hf
ext n
simp only [map_coeff, hf, map_aeval]
apply eval₂Hom_congr (RingHom.ext_... |
import Mathlib.Probability.Notation
import Mathlib.Probability.Integration
import Mathlib.MeasureTheory.Function.L2Space
#align_import probability.variance from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open MeasureTheory Filter Finset
noncomputable section
open scoped MeasureThe... | Mathlib/Probability/Variance.lean | 128 | 139 | theorem _root_.MeasureTheory.Memℒp.variance_eq [IsFiniteMeasure μ] (hX : Memℒp X 2 μ) :
variance X μ = μ[(X - fun _ => μ[X] :) ^ (2 : Nat)] := by |
rw [variance, evariance_eq_lintegral_ofReal, ← ofReal_integral_eq_lintegral_ofReal,
ENNReal.toReal_ofReal (by positivity)]
· rfl
· -- Porting note: `μ[X]` without whitespace is ambiguous as it could be GetElem,
-- and `convert` cannot disambiguate based on typeclass inference failure.
convert (hX.sub... |
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.BigOperators
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.Tactic.FinCases
#align_import linear_algebra.affine_space.combination from ... | Mathlib/LinearAlgebra/AffineSpace/Combination.lean | 722 | 724 | theorem affineCombinationLineMapWeights_apply_of_ne [DecidableEq ι] {i j t : ι} (hi : t ≠ i)
(hj : t ≠ j) (c : k) : affineCombinationLineMapWeights i j c t = 0 := by |
simp [affineCombinationLineMapWeights, hi, hj]
|
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 | 159 | 169 | theorem ramificationIdx_ne_zero (hp0 : map f p ≠ ⊥) (hP : P.IsPrime) (le : map f p ≤ P) :
ramificationIdx f p P ≠ 0 := by |
have hP0 : P ≠ ⊥ := by
rintro rfl
have := le_bot_iff.mp le
contradiction
have hPirr := (Ideal.prime_of_isPrime hP0 hP).irreducible
rw [IsDedekindDomain.ramificationIdx_eq_normalizedFactors_count hp0 hP hP0]
obtain ⟨P', hP', P'_eq⟩ :=
exists_mem_normalizedFactors_of_dvd hp0 hPirr (Ideal.dvd_iff_... |
import Mathlib.Algebra.Homology.ShortComplex.Basic
import Mathlib.CategoryTheory.Limits.Constructions.FiniteProductsOfBinaryProducts
import Mathlib.CategoryTheory.Triangulated.TriangleShift
#align_import category_theory.triangulated.pretriangulated from "leanprover-community/mathlib"@"6876fa15e3158ff3e4a4e2af1fb6e194... | Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean | 126 | 130 | theorem comp_distTriang_mor_zero₁₂ (T) (H : T ∈ (distTriang C)) : T.mor₁ ≫ T.mor₂ = 0 := by |
obtain ⟨c, hc⟩ :=
complete_distinguished_triangle_morphism _ _ (contractible_distinguished T.obj₁) H (𝟙 T.obj₁)
T.mor₁ rfl
simpa only [contractibleTriangle_mor₂, zero_comp] using hc.left.symm
|
import Mathlib.Data.DFinsupp.Lex
import Mathlib.Order.GameAdd
import Mathlib.Order.Antisymmetrization
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Tactic.AdaptationNote
#align_import data.dfinsupp.well_founded from "leanprover-community/mathlib"@"e9b8651eb1ad354f4de6be35a38ef31efcd2cfaa"
variable {ι : Ty... | Mathlib/Data/DFinsupp/WellFounded.lean | 118 | 129 | theorem Lex.acc_of_single [DecidableEq ι] [∀ (i) (x : α i), Decidable (x ≠ 0)] (x : Π₀ i, α i) :
(∀ i ∈ x.support, Acc (DFinsupp.Lex r s) <| single i (x i)) → Acc (DFinsupp.Lex r s) x := by |
generalize ht : x.support = t; revert x
classical
induction' t using Finset.induction with b t hb ih
· intro x ht
rw [support_eq_empty.1 ht]
exact fun _ => Lex.acc_zero hbot
refine fun x ht h => Lex.acc_of_single_erase b (h b <| t.mem_insert_self b) ?_
refine ih _ (by rw [support_erase,... |
import Mathlib.Analysis.Normed.Group.InfiniteSum
import Mathlib.Analysis.Normed.MulAction
import Mathlib.Topology.Algebra.Order.LiminfLimsup
import Mathlib.Topology.PartialHomeomorph
#align_import analysis.asymptotics.asymptotics from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open ... | Mathlib/Analysis/Asymptotics/Asymptotics.lean | 113 | 114 | theorem isBigO_iff : f =O[l] g ↔ ∃ c : ℝ, ∀ᶠ x in l, ‖f x‖ ≤ c * ‖g x‖ := by |
simp only [IsBigO_def, IsBigOWith_def]
|
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator
import Mathlib.MeasureTheory.Function.UniformIntegrable
import Mathlib.MeasureTheory.Decomposition.RadonNikodym
#align_import measure_theory.function.conditional_expectation.real from "leanprover-community/mathlib"@"b2ff9a3d7a15fd5b0f060b135421d6a... | Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean | 184 | 223 | theorem Integrable.uniformIntegrable_condexp {ι : Type*} [IsFiniteMeasure μ] {g : α → ℝ}
(hint : Integrable g μ) {ℱ : ι → MeasurableSpace α} (hℱ : ∀ i, ℱ i ≤ m0) :
UniformIntegrable (fun i => μ[g|ℱ i]) 1 μ := by |
let A : MeasurableSpace α := m0
have hmeas : ∀ n, ∀ C, MeasurableSet {x | C ≤ ‖(μ[g|ℱ n]) x‖₊} := fun n C =>
measurableSet_le measurable_const (stronglyMeasurable_condexp.mono (hℱ n)).measurable.nnnorm
have hg : Memℒp g 1 μ := memℒp_one_iff_integrable.2 hint
refine uniformIntegrable_of le_rfl ENNReal.one_n... |
import Mathlib.CategoryTheory.Subobject.Lattice
import Mathlib.CategoryTheory.EssentiallySmall
import Mathlib.CategoryTheory.Simple
#align_import category_theory.noetherian from "leanprover-community/mathlib"@"7c4c90f422a4a4477e4d8bc4dc9f1e634e6b2349"
namespace CategoryTheory
open CategoryTheory.Limits
variable... | Mathlib/CategoryTheory/Noetherian.lean | 82 | 87 | theorem exists_simple_subobject {X : C} [ArtinianObject X] (h : ¬IsZero X) :
∃ Y : Subobject X, Simple (Y : C) := by |
haveI : Nontrivial (Subobject X) := nontrivial_of_not_isZero h
haveI := isAtomic_of_orderBot_wellFounded_lt (ArtinianObject.subobject_lt_wellFounded X)
obtain ⟨Y, s⟩ := (IsAtomic.eq_bot_or_exists_atom_le (⊤ : Subobject X)).resolve_left top_ne_bot
exact ⟨Y, (subobject_simple_iff_isAtom _).mpr s.1⟩
|
import Mathlib.Init.ZeroOne
import Mathlib.Data.Set.Defs
import Mathlib.Order.Basic
import Mathlib.Order.SymmDiff
import Mathlib.Tactic.Tauto
import Mathlib.Tactic.ByContra
import Mathlib.Util.Delaborators
#align_import data.set.basic from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
... | Mathlib/Data/Set/Basic.lean | 1,325 | 1,326 | theorem inter_singleton_eq_empty : s ∩ {a} = ∅ ↔ a ∉ s := by |
rw [inter_comm, singleton_inter_eq_empty]
|
import Mathlib.LinearAlgebra.Matrix.Reindex
import Mathlib.LinearAlgebra.Matrix.ToLin
#align_import linear_algebra.matrix.basis from "leanprover-community/mathlib"@"6c263e4bfc2e6714de30f22178b4d0ca4d149a76"
noncomputable section
open LinearMap Matrix Set Submodule
open Matrix
section BasisToMatrix
variable {ι... | Mathlib/LinearAlgebra/Matrix/Basis.lean | 97 | 102 | theorem toMatrix_unitsSMul [DecidableEq ι] (e : Basis ι R₂ M₂) (w : ι → R₂ˣ) :
e.toMatrix (e.unitsSMul w) = diagonal ((↑) ∘ w) := by |
ext i j
by_cases h : i = j
· simp [h, toMatrix_apply, unitsSMul_apply, Units.smul_def]
· simp [h, toMatrix_apply, unitsSMul_apply, Units.smul_def, Ne.symm h]
|
import Mathlib.Data.List.Range
import Mathlib.Data.List.Perm
#align_import data.list.sigma from "leanprover-community/mathlib"@"f808feb6c18afddb25e66a71d317643cf7fb5fbb"
universe u v
namespace List
variable {α : Type u} {β : α → Type v} {l l₁ l₂ : List (Sigma β)}
def keys : List (Sigma β) → List α :=
map ... | Mathlib/Data/List/Sigma.lean | 510 | 523 | theorem dlookup_kerase_ne {a a'} {l : List (Sigma β)} (h : a ≠ a') :
dlookup a (kerase a' l) = dlookup a l := by |
induction l with
| nil => rfl
| cons hd tl ih =>
cases' hd with ah bh
by_cases h₁ : a = ah <;> by_cases h₂ : a' = ah
· substs h₁ h₂
cases Ne.irrefl h
· subst h₁
simp [h₂]
· subst h₂
simp [h]
· simp [h₁, h₂, ih]
|
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
theor... | Mathlib/Topology/Order/ExtendFrom.lean | 68 | 74 | theorem continuousOn_Ioc_extendFrom_Ioo [TopologicalSpace α] [LinearOrder α] [DenselyOrdered α]
[OrderTopology α] [TopologicalSpace β] [RegularSpace β] {f : α → β} {a b : α} {lb : β}
(hab : a < b) (hf : ContinuousOn f (Ioo a b)) (hb : Tendsto f (𝓝[<] b) (𝓝 lb)) :
ContinuousOn (extendFrom (Ioo a b) f) (Ioc... |
have := @continuousOn_Ico_extendFrom_Ioo αᵒᵈ _ _ _ _ _ _ _ f _ _ lb hab
erw [dual_Ico, dual_Ioi, dual_Ioo] at this
exact this hf hb
|
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology.Instances.Sign
#align_import analysis.special_functions.trigonometric.angle from "leanprover-community/mathlib"@"213b0cff7bc5ab6696ee07cceec80829... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 169 | 169 | theorem two_zsmul_coe_pi : (2 : ℤ) • (π : Angle) = 0 := by | simp [← intCast_mul_eq_zsmul]
|
import Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction
import Mathlib.Analysis.BoxIntegral.Partition.Split
#align_import analysis.box_integral.partition.filter from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
open Set Function Filter Metric Finset Bool
open scoped Classical
o... | Mathlib/Analysis/BoxIntegral/Partition/Filter.lean | 372 | 383 | theorem MemBaseSet.exists_common_compl (h₁ : l.MemBaseSet I c₁ r₁ π₁) (h₂ : l.MemBaseSet I c₂ r₂ π₂)
(hU : π₁.iUnion = π₂.iUnion) :
∃ π : Prepartition I, π.iUnion = ↑I \ π₁.iUnion ∧
(l.bDistortion → π.distortion ≤ c₁) ∧ (l.bDistortion → π.distortion ≤ c₂) := by |
wlog hc : c₁ ≤ c₂ with H
· simpa [hU, _root_.and_comm] using
@H _ _ I J c c₂ c₁ r r₂ r₁ π π₂ π₁ _ l₂ l₁ h₂ h₁ hU.symm (le_of_not_le hc)
by_cases hD : (l.bDistortion : Prop)
· rcases h₁.4 hD with ⟨π, hπU, hπc⟩
exact ⟨π, hπU, fun _ => hπc, fun _ => hπc.trans hc⟩
· exact ⟨π₁.toPrepartition.compl, π₁.t... |
import Mathlib.Combinatorics.SimpleGraph.Subgraph
import Mathlib.Data.List.Rotate
#align_import combinatorics.simple_graph.connectivity from "leanprover-community/mathlib"@"b99e2d58a5e6861833fa8de11e51a81144258db4"
open Function
universe u v w
namespace SimpleGraph
variable {V : Type u} {V' : Type v} {V'' : Typ... | Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | 1,194 | 1,203 | theorem take_spec {u v w : V} (p : G.Walk v w) (h : u ∈ p.support) :
(p.takeUntil u h).append (p.dropUntil u h) = p := by |
induction p
· rw [mem_support_nil_iff] at h
subst u
rfl
· cases h
· simp!
· simp! only
split_ifs with h' <;> subst_vars <;> simp [*]
|
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Convex.Normed
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.convex.side from "leanprover-community/mathlib"@"a63928c34ec358b5edcda2bf7513c50052a5230f"
variable {R V V' P P' : Type*}
open AffineEquiv AffineMap
namespace Affine... | Mathlib/Analysis/Convex/Side.lean | 321 | 323 | theorem sOppSide_vadd_right_iff {s : AffineSubspace R P} {x y : P} {v : V} (hv : v ∈ s.direction) :
s.SOppSide x (v +ᵥ y) ↔ s.SOppSide x y := by |
rw [sOppSide_comm, sOppSide_vadd_left_iff hv, sOppSide_comm]
|
import Mathlib.Analysis.ODE.Gronwall
import Mathlib.Analysis.ODE.PicardLindelof
import Mathlib.Geometry.Manifold.InteriorBoundary
import Mathlib.Geometry.Manifold.MFDeriv.Atlas
open scoped Manifold Topology
open Function Set
variable
{E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
{H : ... | Mathlib/Geometry/Manifold/IntegralCurve.lean | 433 | 469 | theorem isIntegralCurveOn_Ioo_eqOn_of_contMDiff (ht₀ : t₀ ∈ Ioo a b)
(hγt : ∀ t ∈ Ioo a b, I.IsInteriorPoint (γ t))
(hv : ContMDiff I I.tangent 1 (fun x ↦ (⟨x, v x⟩ : TangentBundle I M)))
(hγ : IsIntegralCurveOn γ v (Ioo a b)) (hγ' : IsIntegralCurveOn γ' v (Ioo a b))
(h : γ t₀ = γ' t₀) : EqOn γ γ' (Ioo ... |
set s := {t | γ t = γ' t} ∩ Ioo a b with hs
-- since `Ioo a b` is connected, we get `s = Ioo a b` by showing that `s` is clopen in `Ioo a b`
-- in the subtype topology (`s` is also non-empty by assumption)
-- here we use a slightly weaker alternative theorem
suffices hsub : Ioo a b ⊆ s from fun t ht ↦ mem_se... |
import Mathlib.Data.Set.Image
import Mathlib.Order.SuccPred.Relation
import Mathlib.Topology.Clopen
import Mathlib.Topology.Irreducible
#align_import topology.connected from "leanprover-community/mathlib"@"d101e93197bb5f6ea89bd7ba386b7f7dff1f3903"
open Set Function Topology TopologicalSpace Relation
open scoped C... | Mathlib/Topology/Connected/Basic.lean | 866 | 868 | theorem nonempty_inter [PreconnectedSpace α] {s t : Set α} :
IsOpen s → IsOpen t → s ∪ t = univ → s.Nonempty → t.Nonempty → (s ∩ t).Nonempty := by |
simpa only [univ_inter, univ_subset_iff] using @PreconnectedSpace.isPreconnected_univ α _ _ s t
|
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Group.Measure
#align_import measure_theory.group.prod from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
noncomputable section
open Set hiding prod_eq
open Function MeasureTheory
open Filter hiding ma... | Mathlib/MeasureTheory/Group/Prod.lean | 247 | 262 | theorem measure_mul_lintegral_eq [IsMulLeftInvariant ν] (sm : MeasurableSet s) (f : G → ℝ≥0∞)
(hf : Measurable f) : (μ s * ∫⁻ y, f y ∂ν) = ∫⁻ x, ν ((fun z => z * x) ⁻¹' s) * f x⁻¹ ∂μ := by |
rw [← set_lintegral_one, ← lintegral_indicator _ sm,
← lintegral_lintegral_mul (measurable_const.indicator sm).aemeasurable hf.aemeasurable,
← lintegral_lintegral_mul_inv μ ν]
swap
· exact (((measurable_const.indicator sm).comp measurable_fst).mul
(hf.comp measurable_snd)).aemeasurable
have ms :
... |
import Mathlib.Logic.Relation
import Mathlib.Data.Option.Basic
import Mathlib.Data.Seq.Seq
#align_import data.seq.wseq from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b348ce40ad"
namespace Stream'
open Function
universe u v w
def WSeq (α) :=
Seq (Option α)
#align stream.wseq Stream'.WSeq
... | Mathlib/Data/Seq/WSeq.lean | 1,694 | 1,709 | theorem join_map_ret (s : WSeq α) : join (map ret s) ~ʷ s := by |
refine ⟨fun s1 s2 => join (map ret s2) = s1, rfl, ?_⟩
intro s' s h; rw [← h]
apply liftRel_rec fun c1 c2 => ∃ s, c1 = destruct (join (map ret s)) ∧ c2 = destruct s
· exact fun {c1 c2} h =>
match c1, c2, h with
| _, _, ⟨s, rfl, rfl⟩ => by
clear h
-- Porting note: `ret` is simplified ... |
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 | 185 | 193 | theorem memℒp_finset_sum {ι} (s : Finset ι) {f : ι → α → E} (hf : ∀ i ∈ s, Memℒp (f i) p μ) :
Memℒp (fun a => ∑ i ∈ s, f i a) p μ := by |
haveI : DecidableEq ι := Classical.decEq _
revert hf
refine Finset.induction_on s ?_ ?_
· simp only [zero_mem_ℒp', Finset.sum_empty, imp_true_iff]
· intro i s his ih hf
simp only [his, Finset.sum_insert, not_false_iff]
exact (hf i (s.mem_insert_self i)).add (ih fun j hj => hf j (Finset.mem_insert_of_... |
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Nat.ModEq
import Mathlib.Data.Set.Finite
#align_import combinatorics.pigeonhole from "leanprover-community/mathlib"@"d6fad0e5bf2d6f48da9175d25c3dc5706b3834ce"
un... | Mathlib/Combinatorics/Pigeonhole.lean | 291 | 296 | theorem exists_card_fiber_le_of_card_le_nsmul (ht : t.Nonempty) (hb : ↑s.card ≤ t.card • b) :
∃ y ∈ t, ↑(s.filter fun x => f x = y).card ≤ b := by |
simp_rw [cast_card] at hb ⊢
refine
exists_sum_fiber_le_of_sum_fiber_nonneg_of_sum_le_nsmul
(fun _ _ => sum_nonneg fun _ _ => zero_le_one) ht hb
|
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 | 207 | 209 | theorem decode₂_encode [Encodable α] (a : α) : decode₂ α (encode a) = some a := by |
ext
simp [mem_decode₂, eq_comm, decode₂_eq_some]
|
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap
import Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.MeasureTheory.Constructio... | Mathlib/MeasureTheory/Function/Jacobian.lean | 1,207 | 1,215 | theorem det_one_smulRight {𝕜 : Type*} [NormedField 𝕜] (v : 𝕜) :
((1 : 𝕜 →L[𝕜] 𝕜).smulRight v).det = v := by |
have : (1 : 𝕜 →L[𝕜] 𝕜).smulRight v = v • (1 : 𝕜 →L[𝕜] 𝕜) := by
ext1
simp only [ContinuousLinearMap.smulRight_apply, ContinuousLinearMap.one_apply,
Algebra.id.smul_eq_mul, one_mul, ContinuousLinearMap.coe_smul', Pi.smul_apply, mul_one]
rw [this, ContinuousLinearMap.det, ContinuousLinearMap.coe_s... |
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 | 631 | 642 | theorem norm_setIntegral_le_of_norm_le_const_ae' {C : ℝ} (hs : μ s < ∞)
(hC : ∀ᵐ x ∂μ, x ∈ s → ‖f x‖ ≤ C) (hfm : AEStronglyMeasurable f (μ.restrict s)) :
‖∫ x in s, f x ∂μ‖ ≤ C * (μ s).toReal := by |
apply norm_setIntegral_le_of_norm_le_const_ae hs
have A : ∀ᵐ x : X ∂μ, x ∈ s → ‖AEStronglyMeasurable.mk f hfm x‖ ≤ C := by
filter_upwards [hC, hfm.ae_mem_imp_eq_mk] with _ h1 h2 h3
rw [← h2 h3]
exact h1 h3
have B : MeasurableSet {x | ‖hfm.mk f x‖ ≤ C} :=
hfm.stronglyMeasurable_mk.norm.measurable ... |
import Mathlib.AlgebraicGeometry.GammaSpecAdjunction
import Mathlib.AlgebraicGeometry.Restrict
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.RingTheory.Localization.InvSubmonoid
#align_import algebraic_geometry.AffineScheme from "leanprover-community/mathlib"@"88474d1b5af6d37c2ab728b757771bced7f5194c"... | Mathlib/AlgebraicGeometry/AffineScheme.lean | 295 | 300 | theorem imageIsOpenImmersion (f : X ⟶ Y) [H : IsOpenImmersion f] :
IsAffineOpen (f.opensFunctor.obj U) := by |
have : IsAffine _ := hU
convert rangeIsAffineOpenOfOpenImmersion (X.ofRestrict U.openEmbedding ≫ f)
ext1
exact Set.image_eq_range _ _
|
import Mathlib.RingTheory.LocalProperties
#align_import ring_theory.ring_hom.surjective from "leanprover-community/mathlib"@"831c494092374cfe9f50591ed0ac81a25efc5b86"
namespace RingHom
open scoped TensorProduct
open TensorProduct Algebra.TensorProduct
local notation "surjective" => fun {X Y : Type _} [CommRing... | Mathlib/RingTheory/RingHom/Surjective.lean | 26 | 27 | theorem surjective_stableUnderComposition : StableUnderComposition surjective := by |
introv R hf hg; exact hg.comp hf
|
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Analysis.Convex.Slope
open Metric Set Asymptotics ContinuousLinearMap Filter
open scoped Classical Topology NNReal
theorem MonotoneOn.convexOn_of_deriv {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ}
(hf : ContinuousOn f D) (hf' : DifferentiableOn ℝ f (in... | Mathlib/Analysis/Convex/Deriv.lean | 65 | 75 | theorem StrictMonoOn.exists_slope_lt_deriv_aux {x y : ℝ} {f : ℝ → ℝ} (hf : ContinuousOn f (Icc x y))
(hxy : x < y) (hf'_mono : StrictMonoOn (deriv f) (Ioo x y)) (h : ∀ w ∈ Ioo x y, deriv f w ≠ 0) :
∃ a ∈ Ioo x y, (f y - f x) / (y - x) < deriv f a := by |
have A : DifferentiableOn ℝ f (Ioo x y) := fun w wmem =>
(differentiableAt_of_deriv_ne_zero (h w wmem)).differentiableWithinAt
obtain ⟨a, ⟨hxa, hay⟩, ha⟩ : ∃ a ∈ Ioo x y, deriv f a = (f y - f x) / (y - x) :=
exists_deriv_eq_slope f hxy hf A
rcases nonempty_Ioo.2 hay with ⟨b, ⟨hab, hby⟩⟩
refine ⟨b, ⟨hxa... |
import Mathlib.Algebra.Algebra.RestrictScalars
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.LinearAlgebra.Quotient
import Mathlib.LinearAlgebra.StdBasis
import Mathlib.GroupTheory.Finiteness
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.RingTheory.Nilpotent.Defs
#align_import ring_theory.finit... | Mathlib/RingTheory/Finiteness.lean | 335 | 344 | theorem fg_induction (R M : Type*) [Semiring R] [AddCommMonoid M] [Module R M]
(P : Submodule R M → Prop) (h₁ : ∀ x, P (Submodule.span R {x}))
(h₂ : ∀ M₁ M₂, P M₁ → P M₂ → P (M₁ ⊔ M₂)) (N : Submodule R M) (hN : N.FG) : P N := by |
classical
obtain ⟨s, rfl⟩ := hN
induction s using Finset.induction
· rw [Finset.coe_empty, Submodule.span_empty, ← Submodule.span_zero_singleton]
apply h₁
· rw [Finset.coe_insert, Submodule.span_insert]
apply h₂ <;> apply_assumption
|
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.BigOperators
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.Tactic.FinCases
#align_import linear_algebra.affine_space.combination from ... | Mathlib/LinearAlgebra/AffineSpace/Combination.lean | 680 | 681 | theorem weightedVSubVSubWeights_apply_right [DecidableEq ι] {i j : ι} (h : i ≠ j) :
weightedVSubVSubWeights k i j j = -1 := by | simp [weightedVSubVSubWeights, h.symm]
|
import Mathlib.Algebra.Group.Commute.Defs
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Set.Function
import Mathlib.Logic.Pairwise
#align_import algebra.group.pi from "leanprover-community/mathlib"@"e4bc74cbaf429d706cb9140902f7ca6c431e75a4"
assert_not_exists AddMonoidWithOne
assert_not_exists Mono... | Mathlib/Algebra/Group/Pi/Lemmas.lean | 546 | 549 | theorem curry_mulSingle [DecidableEq α] [∀ a, DecidableEq (β a)] [∀ a b, One (γ a b)]
(i : Σ a, β a) (x : γ i.1 i.2) :
Sigma.curry (Pi.mulSingle i x) = Pi.mulSingle i.1 (Pi.mulSingle i.2 x) := by |
simp only [Pi.mulSingle, Sigma.curry_update, Sigma.curry_one, Pi.one_apply]
|
import Mathlib.Algebra.Order.Floor
import Mathlib.Algebra.Order.Field.Power
import Mathlib.Data.Nat.Log
#align_import data.int.log from "leanprover-community/mathlib"@"1f0096e6caa61e9c849ec2adbd227e960e9dff58"
variable {R : Type*} [LinearOrderedSemifield R] [FloorSemiring R]
namespace Int
def log (b : ℕ) (r : ... | Mathlib/Data/Int/Log.lean | 260 | 264 | theorem zpow_pred_clog_lt_self {b : ℕ} {r : R} (hb : 1 < b) (hr : 0 < r) :
(b : R) ^ (clog b r - 1) < r := by |
rw [← neg_log_inv_eq_clog, ← neg_add', zpow_neg, inv_lt _ hr]
· exact lt_zpow_succ_log_self hb _
· exact zpow_pos_of_pos (Nat.cast_pos.mpr <| zero_le_one.trans_lt hb) _
|
import Mathlib.Algebra.GeomSum
import Mathlib.Order.Filter.Archimedean
import Mathlib.Order.Iterate
import Mathlib.Topology.Algebra.Algebra
import Mathlib.Topology.Algebra.InfiniteSum.Real
#align_import analysis.specific_limits.basic from "leanprover-community/mathlib"@"57ac39bd365c2f80589a700f9fbb664d3a1a30c2"
n... | Mathlib/Analysis/SpecificLimits/Basic.lean | 195 | 198 | theorem geom_le {u : ℕ → ℝ} {c : ℝ} (hc : 0 ≤ c) (n : ℕ) (h : ∀ k < n, c * u k ≤ u (k + 1)) :
c ^ n * u 0 ≤ u n := by |
apply (monotone_mul_left_of_nonneg hc).seq_le_seq n _ _ h <;>
simp [_root_.pow_succ', mul_assoc, le_refl]
|
import Mathlib.Data.Set.Image
import Mathlib.Data.Set.Lattice
#align_import data.set.sigma from "leanprover-community/mathlib"@"2258b40dacd2942571c8ce136215350c702dc78f"
namespace Set
variable {ι ι' : Type*} {α β : ι → Type*} {s s₁ s₂ : Set ι} {t t₁ t₂ : ∀ i, Set (α i)}
{u : Set (Σ i, α i)} {x : Σ i, α i} {i j ... | Mathlib/Data/Set/Sigma.lean | 131 | 133 | theorem sigma_inter_sigma : s₁.sigma t₁ ∩ s₂.sigma t₂ = (s₁ ∩ s₂).sigma fun i ↦ t₁ i ∩ t₂ i := by |
ext ⟨x, y⟩
simp [and_assoc, and_left_comm]
|
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.SesquilinearForm
import Mathlib.LinearAlgebra.Matrix.Symmetric
#align_import linear_algebra.quadratic_form.basic from "leanprover-community/mathlib"@"d11f435d4e34a6cea0a1797d6b625b0c170be845"
universe u v w
variable {S T : ... | Mathlib/LinearAlgebra/QuadraticForm/Basic.lean | 103 | 104 | theorem polar_neg (f : M → R) (x y : M) : polar (-f) x y = -polar f x y := by |
simp only [polar, Pi.neg_apply, sub_eq_add_neg, neg_add]
|
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.Matrix.Block
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.TensorProduct.Basic
import Mathlib.RingTheory.TensorProduct.Basic
#align_import data.matrix.kronecker from "leanpr... | Mathlib/Data/Matrix/Kronecker.lean | 132 | 136 | theorem kroneckerMap_diagonal_right [Zero β] [Zero γ] [DecidableEq n] (f : α → β → γ)
(hf : ∀ a, f a 0 = 0) (A : Matrix l m α) (b : n → β) :
kroneckerMap f A (diagonal b) = blockDiagonal fun i => A.map fun a => f a (b i) := by |
ext ⟨i₁, i₂⟩ ⟨j₁, j₂⟩
simp [diagonal, blockDiagonal, apply_ite (f (A i₁ j₁)), hf]
|
import Mathlib.SetTheory.Ordinal.Arithmetic
import Mathlib.Tactic.Abel
#align_import set_theory.ordinal.natural_ops from "leanprover-community/mathlib"@"31b269b60935483943542d547a6dd83a66b37dc7"
set_option autoImplicit true
universe u v
open Function Order
noncomputable section
def NatOrdinal : Type _ :=
... | Mathlib/SetTheory/Ordinal/NaturalOps.lean | 237 | 239 | theorem lt_nadd_iff : a < b ♯ c ↔ (∃ b' < b, a ≤ b' ♯ c) ∨ ∃ c' < c, a ≤ b ♯ c' := by |
rw [nadd_def]
simp [lt_blsub_iff]
|
import Batteries.Data.Sum.Basic
import Batteries.Logic
open Function
namespace Sum
@[simp] protected theorem «forall» {p : α ⊕ β → Prop} :
(∀ x, p x) ↔ (∀ a, p (inl a)) ∧ ∀ b, p (inr b) :=
⟨fun h => ⟨fun _ => h _, fun _ => h _⟩, fun ⟨h₁, h₂⟩ => Sum.rec h₁ h₂⟩
@[simp] protected theorem «exists» {p : α ⊕ β ... | .lake/packages/batteries/Batteries/Data/Sum/Lemmas.lean | 75 | 75 | theorem not_isLeft {x : α ⊕ β} : ¬x.isLeft ↔ x.isRight := by | simp
|
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 | 361 | 363 | theorem norm_real (c : ℝ) :
mixedEmbedding.norm ((fun _ ↦ c, fun _ ↦ c) : (E K)) = |c| ^ finrank ℚ K := by |
rw [show ((fun _ ↦ c, fun _ ↦ c) : (E K)) = c • 1 by ext <;> simp, norm_smul, map_one, mul_one]
|
import Mathlib.Topology.UniformSpace.UniformConvergenceTopology
#align_import topology.uniform_space.equicontinuity from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
section
open UniformSpace Filter Set Uniformity Topology UniformConvergence Function
variable {ι κ X X' Y Z α α' β β'... | Mathlib/Topology/UniformSpace/Equicontinuity.lean | 648 | 653 | theorem Filter.HasBasis.equicontinuousAt_iff_right {p : κ → Prop} {s : κ → Set (α × α)}
{F : ι → X → α} {x₀ : X} (hα : (𝓤 α).HasBasis p s) :
EquicontinuousAt F x₀ ↔ ∀ k, p k → ∀ᶠ x in 𝓝 x₀, ∀ i, (F i x₀, F i x) ∈ s k := by |
rw [equicontinuousAt_iff_continuousAt, ContinuousAt,
(UniformFun.hasBasis_nhds_of_basis ι α _ hα).tendsto_right_iff]
rfl
|
import Mathlib.NumberTheory.ZetaValues
import Mathlib.NumberTheory.LSeries.RiemannZeta
open Complex Real Set
open scoped Nat
open HurwitzZeta
theorem riemannZeta_two_mul_nat {k : ℕ} (hk : k ≠ 0) :
riemannZeta (2 * k) = (-1) ^ (k + 1) * (2 : ℂ) ^ (2 * k - 1)
* (π : ℂ) ^ (2 * k) * bernoulli (2 * k) / (... | Mathlib/NumberTheory/LSeries/HurwitzZetaValues.lean | 227 | 232 | theorem riemannZeta_four : riemannZeta 4 = π ^ 4 / 90 := by |
convert congr_arg ((↑) : ℝ → ℂ) hasSum_zeta_four.tsum_eq
· rw [← Nat.cast_one, show (4 : ℂ) = (4 : ℕ) by norm_num,
zeta_nat_eq_tsum_of_gt_one (by norm_num : 1 < 4)]
simp only [push_cast]
· norm_cast
|
import Mathlib.CategoryTheory.Sites.Sieves
#align_import category_theory.sites.sheaf_of_types from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe w v₁ v₂ u₁ u₂
namespace CategoryTheory
open Opposite CategoryTheory Category Limits Sieve
namespace Presieve
variable {C : Type ... | Mathlib/CategoryTheory/Sites/IsSheafFor.lean | 386 | 391 | theorem is_compatible_of_exists_amalgamation (x : FamilyOfElements P R)
(h : ∃ t, x.IsAmalgamation t) : x.Compatible := by |
cases' h with t ht
intro Y₁ Y₂ Z g₁ g₂ f₁ f₂ h₁ h₂ comm
rw [← ht _ h₁, ← ht _ h₂, ← FunctorToTypes.map_comp_apply, ← op_comp, comm]
simp
|
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Algebra.Star.Unitary
import Mathlib.Data.Nat.ModEq
import Mathlib.NumberTheory.Zsqrtd.Basic
import Mathlib.Tactic.Monotonicity
#align_import number_theory.pell_matiyasevic from "leanprover-community/mathlib"@"795b501869b9f... | Mathlib/NumberTheory/PellMatiyasevic.lean | 520 | 526 | theorem xy_succ_succ (n) :
xn a1 (n + 2) + xn a1 n =
2 * a * xn a1 (n + 1) ∧ yn a1 (n + 2) + yn a1 n = 2 * a * yn a1 (n + 1) := by |
have := pellZd_succ_succ a1 n; unfold pellZd at this
erw [Zsqrtd.smul_val (2 * a : ℕ)] at this
injection this with h₁ h₂
constructor <;> apply Int.ofNat.inj <;> [simpa using h₁; simpa using h₂]
|
import Mathlib.Data.Real.Pi.Bounds
import Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody
-- TODO. Rewrite some of the FLT results on the disciminant using the definitions and results of
-- this file
namespace NumberField
open FiniteDimensional NumberField NumberField.InfinitePlace Matrix
open sco... | Mathlib/NumberTheory/NumberField/Discriminant.lean | 71 | 103 | theorem _root_.NumberField.mixedEmbedding.volume_fundamentalDomain_latticeBasis :
volume (fundamentalDomain (latticeBasis K)) =
(2 : ℝ≥0∞)⁻¹ ^ NrComplexPlaces K * sqrt ‖discr K‖₊ := by |
let f : Module.Free.ChooseBasisIndex ℤ (𝓞 K) ≃ (K →+* ℂ) :=
(canonicalEmbedding.latticeBasis K).indexEquiv (Pi.basisFun ℂ _)
let e : (index K) ≃ Module.Free.ChooseBasisIndex ℤ (𝓞 K) := (indexEquiv K).trans f.symm
let M := (mixedEmbedding.stdBasis K).toMatrix ((latticeBasis K).reindex e.symm)
let N := Alg... |
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