Context stringlengths 57 85k | file_name stringlengths 21 79 | start int64 14 2.42k | end int64 18 2.43k | theorem stringlengths 25 2.71k | proof stringlengths 5 10.6k |
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import Mathlib.Algebra.MvPolynomial.Degrees
#align_import data.mv_polynomial.variables from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Finsupp AddMonoidAlgebra
universe u v w
variable {R : Type u} {S : Type v}
namespace MvPolynomial
varia... | Mathlib/Algebra/MvPolynomial/Variables.lean | 226 | 228 | theorem vars_monomial_single (i : σ) {e : ℕ} {r : R} (he : e ≠ 0) (hr : r ≠ 0) :
(monomial (Finsupp.single i e) r).vars = {i} := by |
rw [vars_monomial hr, Finsupp.support_single_ne_zero _ he]
|
import Mathlib.FieldTheory.Perfect
#align_import field_theory.perfect_closure from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u v
open Function
section
variable (K : Type u) [CommRing K] (p : ℕ) [Fact p.Prime] [CharP K p]
@[mk_iff]
inductive PerfectClosure.R : ℕ × K → ℕ... | Mathlib/FieldTheory/PerfectClosure.lean | 328 | 350 | theorem mk_eq_iff (x y : ℕ × K) :
mk K p x = mk K p y ↔ ∃ z, (frobenius K p)^[y.1 + z] x.2 = (frobenius K p)^[x.1 + z] y.2 := by |
constructor
· intro H
replace H := Quot.exact _ H
induction H with
| rel x y H => cases' H with n x; exact ⟨0, rfl⟩
| refl H => exact ⟨0, rfl⟩
| symm x y H ih => cases' ih with w ih; exact ⟨w, ih.symm⟩
| trans x y z H1 H2 ih1 ih2 =>
cases' ih1 with z1 ih1
cases' ih2 with z2 ih2
... |
import Mathlib.Topology.Algebra.GroupWithZero
import Mathlib.Topology.Order.OrderClosed
#align_import topology.algebra.with_zero_topology from "leanprover-community/mathlib"@"3e0c4d76b6ebe9dfafb67d16f7286d2731ed6064"
open Topology Filter TopologicalSpace Filter Set Function
namespace WithZeroTopology
variable {α... | Mathlib/Topology/Algebra/WithZeroTopology.lean | 128 | 129 | theorem Iio_mem_nhds (h : γ₁ < γ₂) : Iio γ₂ ∈ 𝓝 γ₁ := by |
rcases eq_or_ne γ₁ 0 with (rfl | h₀) <;> simp [*, h.ne', Iio_mem_nhds_zero]
|
import Mathlib.Combinatorics.Quiver.Basic
import Mathlib.Combinatorics.Quiver.Path
#align_import combinatorics.quiver.cast from "leanprover-community/mathlib"@"fc2ed6f838ce7c9b7c7171e58d78eaf7b438fb0e"
universe v v₁ v₂ u u₁ u₂
variable {U : Type*} [Quiver.{u + 1} U]
namespace Quiver
def Hom.cast {u v u' v... | Mathlib/Combinatorics/Quiver/Cast.lean | 118 | 121 | theorem Path.cast_eq_iff_heq {u v u' v' : U} (hu : u = u') (hv : v = v') (p : Path u v)
(p' : Path u' v') : p.cast hu hv = p' ↔ HEq p p' := by |
rw [Path.cast_eq_cast]
exact _root_.cast_eq_iff_heq
|
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 | 91 | 93 | theorem dom_inv : r.inv.dom = r.codom := by |
ext x
rfl
|
import Mathlib.Algebra.Order.Monoid.Unbundled.MinMax
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Finset.Image
import Mathlib.Data.Multiset.Fold
#align_import data.finset.fold from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
-- TODO:
-- assert_not_exists OrderedComm... | Mathlib/Data/Finset/Fold.lean | 267 | 270 | theorem fold_max_add [Add β] [CovariantClass β β (Function.swap (· + ·)) (· ≤ ·)] (n : WithBot β)
(s : Finset α) : (s.fold max ⊥ fun x : α => ↑(f x) + n) = s.fold max ⊥ ((↑) ∘ f) + n := by |
classical
induction' s using Finset.induction_on with a s _ ih <;> simp [*, max_add_add_right]
|
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Group.Int
import Mathlib.Data.Nat.Dist
import Mathlib.Data.Ordmap.Ordnode
import Mathlib.Tactic.Abel
import Mathlib.Tactic.Linarith
#align_import data.ordmap.ordset from "leanprover-community/mathlib"@"47b51515e69f59bca5cf34ef456e6000fe205a69"
variable... | Mathlib/Data/Ordmap/Ordset.lean | 321 | 323 | theorem dual_node3R (l : Ordnode α) (x : α) (m : Ordnode α) (y : α) (r : Ordnode α) :
dual (node3R l x m y r) = node3L (dual r) y (dual m) x (dual l) := by |
simp [node3L, node3R, dual_node', add_comm]
|
import Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral
import Mathlib.Analysis.Complex.CauchyIntegral
import Mathlib.MeasureTheory.Integral.Pi
import Mathlib.Analysis.Fourier.FourierTransform
open Real Set MeasureTheory Filter Asymptotics intervalIntegral
open scoped Real Topology FourierTransform Re... | Mathlib/Analysis/SpecialFunctions/Gaussian/FourierTransform.lean | 51 | 55 | theorem norm_cexp_neg_mul_sq_add_mul_I (b : ℂ) (c T : ℝ) :
‖cexp (-b * (T + c * I) ^ 2)‖ = exp (-(b.re * T ^ 2 - 2 * b.im * c * T - b.re * c ^ 2)) := by |
rw [Complex.norm_eq_abs, Complex.abs_exp, neg_mul, neg_re, ← re_add_im b]
simp only [sq, re_add_im, mul_re, mul_im, add_re, add_im, ofReal_re, ofReal_im, I_re, I_im]
ring_nf
|
import Mathlib.Algebra.Homology.Homotopy
import Mathlib.AlgebraicTopology.DoldKan.Notations
#align_import algebraic_topology.dold_kan.homotopies from "leanprover-community/mathlib"@"b12099d3b7febf4209824444dd836ef5ad96db55"
open CategoryTheory CategoryTheory.Category CategoryTheory.Limits CategoryTheory.Preadditi... | Mathlib/AlgebraicTopology/DoldKan/Homotopies.lean | 104 | 108 | theorem hσ'_eq_zero {q n m : ℕ} (hnq : n < q) (hnm : c.Rel m n) :
(hσ' q n m hnm : X _[n] ⟶ X _[m]) = 0 := by |
simp only [hσ', hσ]
split_ifs
exact zero_comp
|
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Linear.Basic
#align_import category_theory.linear.linear_functor from "leanprover-community/mathlib"@"829895f162a1f29d0133f4b3538f4cd1fb5bffd3"
namespace CategoryTheory
variable (R : Type*) [Semiring R]
class Functor.Linear... | Mathlib/CategoryTheory/Linear/LinearFunctor.lean | 53 | 54 | theorem map_units_smul {X Y : C} (r : Rˣ) (f : X ⟶ Y) : F.map (r • f) = r • F.map f := by |
apply map_smul
|
import Mathlib.Data.List.Basic
#align_import data.list.infix from "leanprover-community/mathlib"@"26f081a2fb920140ed5bc5cc5344e84bcc7cb2b2"
open Nat
variable {α β : Type*}
namespace List
variable {l l₁ l₂ l₃ : List α} {a b : α} {m n : ℕ}
section Fix
#align list.prefix_append List.prefix_append
#align list.... | Mathlib/Data/List/Infix.lean | 178 | 178 | theorem tail_suffix (l : List α) : tail l <:+ l := by | rw [← drop_one]; apply drop_suffix
|
import Mathlib.Topology.EMetricSpace.Paracompact
import Mathlib.Topology.Instances.ENNReal
import Mathlib.Analysis.Convex.PartitionOfUnity
#align_import topology.metric_space.partition_of_unity from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Topology ENNReal NNReal Filter Set Fu... | Mathlib/Topology/MetricSpace/PartitionOfUnity.lean | 64 | 72 | theorem exists_forall_closedBall_subset_aux₁ (hK : ∀ i, IsClosed (K i)) (hU : ∀ i, IsOpen (U i))
(hKU : ∀ i, K i ⊆ U i) (hfin : LocallyFinite K) (x : X) :
∃ r : ℝ, ∀ᶠ y in 𝓝 x,
r ∈ Ioi (0 : ℝ) ∩ ENNReal.ofReal ⁻¹' ⋂ (i) (_ : y ∈ K i), { r | closedBall y r ⊆ U i } := by |
have := (ENNReal.continuous_ofReal.tendsto' 0 0 ENNReal.ofReal_zero).eventually
(eventually_nhds_zero_forall_closedBall_subset hK hU hKU hfin x).curry
rcases this.exists_gt with ⟨r, hr0, hr⟩
refine ⟨r, hr.mono fun y hy => ⟨hr0, ?_⟩⟩
rwa [mem_preimage, mem_iInter₂]
|
import Mathlib.Analysis.InnerProductSpace.TwoDim
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
#align_import geometry.euclidean.angle.oriented.basic from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
noncomputable section
open FiniteDimensional Complex
open scoped Real Rea... | Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | 337 | 341 | theorem oangle_smul_smul_self_of_nonneg (x : V) {r₁ r₂ : ℝ} (hr₁ : 0 ≤ r₁) (hr₂ : 0 ≤ r₂) :
o.oangle (r₁ • x) (r₂ • x) = 0 := by |
rcases hr₁.lt_or_eq with (h | h)
· simp [h, hr₂]
· simp [h.symm]
|
import Mathlib.Order.Interval.Multiset
#align_import data.nat.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
-- TODO
-- assert_not_exists Ring
open Finset Nat
variable (a b c : ℕ)
namespace Nat
instance instLocallyFiniteOrder : LocallyFiniteOrder ℕ where
finsetIcc a b... | Mathlib/Order/Interval/Finset/Nat.lean | 205 | 211 | theorem Ico_image_const_sub_eq_Ico (hac : a ≤ c) :
((Ico a b).image fun x => c - x) = Ico (c + 1 - b) (c + 1 - a) := by |
ext x
simp_rw [mem_image, mem_Ico]
refine ⟨?_, fun h ↦ ⟨c - x, by omega⟩⟩
rintro ⟨x, hx, rfl⟩
omega
|
import Mathlib.Algebra.GroupWithZero.Indicator
import Mathlib.Topology.ContinuousOn
import Mathlib.Topology.Instances.ENNReal
#align_import topology.semicontinuous from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Topology ENNReal
open Set Function Filter
variable {α : Type*} [... | Mathlib/Topology/Semicontinuous.lean | 604 | 613 | theorem lowerSemicontinuousWithinAt_sum {f : ι → α → γ} {a : Finset ι}
(ha : ∀ i ∈ a, LowerSemicontinuousWithinAt (f i) s x) :
LowerSemicontinuousWithinAt (fun z => ∑ i ∈ a, f i z) s x := by |
classical
induction' a using Finset.induction_on with i a ia IH
· exact lowerSemicontinuousWithinAt_const
· simp only [ia, Finset.sum_insert, not_false_iff]
exact
LowerSemicontinuousWithinAt.add (ha _ (Finset.mem_insert_self i a))
(IH fun j ja => ha j (Finset.mem_insert_of_mem ja)... |
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 | 509 | 510 | theorem _root_.Wbtw.oangle₃₁₂_eq_zero {p₁ p₂ p₃ : P} (h : Wbtw ℝ p₁ p₂ p₃) : ∡ p₃ p₁ p₂ = 0 := by |
rw [oangle_eq_zero_iff_oangle_rev_eq_zero, h.oangle₂₁₃_eq_zero]
|
import Mathlib.Topology.Algebra.UniformConvergence
#align_import topology.algebra.equicontinuity from "leanprover-community/mathlib"@"01ad394a11bf06b950232720cf7e8fc6b22f0d6a"
open Function
open UniformConvergence
@[to_additive]
| Mathlib/Topology/Algebra/Equicontinuity.lean | 20 | 31 | theorem equicontinuous_of_equicontinuousAt_one {ι G M hom : Type*} [TopologicalSpace G]
[UniformSpace M] [Group G] [Group M] [TopologicalGroup G] [UniformGroup M]
[FunLike hom G M] [MonoidHomClass hom G M] (F : ι → hom)
(hf : EquicontinuousAt ((↑) ∘ F) (1 : G)) :
Equicontinuous ((↑) ∘ F) := by |
rw [equicontinuous_iff_continuous]
rw [equicontinuousAt_iff_continuousAt] at hf
let φ : G →* (ι →ᵤ M) :=
{ toFun := swap ((↑) ∘ F)
map_one' := by dsimp [UniformFun]; ext; exact map_one _
map_mul' := fun a b => by dsimp [UniformFun]; ext; exact map_mul _ _ _ }
exact continuous_of_continuousAt_on... |
import Mathlib.Analysis.Calculus.FDeriv.Basic
#align_import analysis.calculus.fderiv.restrict_scalars from "leanprover-community/mathlib"@"e3fb84046afd187b710170887195d50bada934ee"
open Filter Asymptotics ContinuousLinearMap Set Metric
open scoped Classical
open Topology NNReal Filter Asymptotics ENNReal
noncom... | Mathlib/Analysis/Calculus/FDeriv/RestrictScalars.lean | 110 | 117 | theorem differentiableWithinAt_iff_restrictScalars (hf : DifferentiableWithinAt 𝕜 f s x)
(hs : UniqueDiffWithinAt 𝕜 s x) : DifferentiableWithinAt 𝕜' f s x ↔
∃ g' : E →L[𝕜'] F, g'.restrictScalars 𝕜 = fderivWithin 𝕜 f s x := by |
constructor
· rintro ⟨g', hg'⟩
exact ⟨g', hs.eq (hg'.restrictScalars 𝕜) hf.hasFDerivWithinAt⟩
· rintro ⟨f', hf'⟩
exact ⟨f', hf.hasFDerivWithinAt.of_restrictScalars 𝕜 hf'⟩
|
import Mathlib.Analysis.InnerProductSpace.TwoDim
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
#align_import geometry.euclidean.angle.oriented.basic from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
noncomputable section
open FiniteDimensional Complex
open scoped Real Rea... | Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | 897 | 926 | theorem oangle_sign_smul_add_right (x y : V) (r : ℝ) :
(o.oangle x (r • x + y)).sign = (o.oangle x y).sign := by |
by_cases h : o.oangle x y = 0 ∨ o.oangle x y = π
· rwa [Real.Angle.sign_eq_zero_iff.2 h, Real.Angle.sign_eq_zero_iff,
oangle_smul_add_right_eq_zero_or_eq_pi_iff]
have h' : ∀ r' : ℝ, o.oangle x (r' • x + y) ≠ 0 ∧ o.oangle x (r' • x + y) ≠ π := by
intro r'
rwa [← o.oangle_smul_add_right_eq_zero_or_eq... |
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.GroupTheory.MonoidLocalization
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.GroupTheory.GroupAction.Ring
#align_import ring_theory.localization.basic from "leanprover-community/mathlib"@"b69c9a770ecf37eb21... | Mathlib/RingTheory/Localization/Basic.lean | 348 | 349 | theorem mk'_eq_zero_iff (x : R) (s : M) : mk' S x s = 0 ↔ ∃ m : M, ↑m * x = 0 := by |
rw [← (map_units S s).mul_left_inj, mk'_spec, zero_mul, map_eq_zero_iff M]
|
import Mathlib.Init.Data.Nat.Notation
import Mathlib.Init.Order.Defs
set_option autoImplicit true
structure UFModel (n) where
parent : Fin n → Fin n
rank : Nat → Nat
rank_lt : ∀ i, (parent i).1 ≠ i → rank i < rank (parent i)
structure UFNode (α : Type*) where
parent : Nat
value : α
rank : Nat
inductive... | Mathlib/Data/UnionFind.lean | 73 | 77 | theorem mk' {arr : Array α} {f : α → β} {n} {g : Fin n → β} (e : n = arr.size)
(H : ∀ i h₁ h₂, f (arr.get ⟨i, h₁⟩) = g ⟨i, h₂⟩) : Agrees arr f g := by |
cases e
have : (fun i ↦ f (arr.get i)) = g := by funext ⟨i, h⟩; apply H
cases this; constructor
|
import Mathlib.AlgebraicGeometry.Restrict
import Mathlib.CategoryTheory.Adjunction.Limits
import Mathlib.CategoryTheory.Adjunction.Reflective
#align_import algebraic_geometry.Gamma_Spec_adjunction from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
-- Explicit universe annotations were u... | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | 128 | 134 | theorem isUnit_res_toΓSpecMapBasicOpen : IsUnit (X.toToΓSpecMapBasicOpen r r) := by |
convert
(X.presheaf.map <| (eqToHom <| X.toΓSpecMapBasicOpen_eq r).op).isUnit_map
(X.toRingedSpace.isUnit_res_basicOpen r)
-- Porting note: `rw [comp_apply]` to `erw [comp_apply]`
erw [← comp_apply, ← Functor.map_comp]
congr
|
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 | 187 | 190 | theorem restrict_restrict_of_subset (h : s ⊆ t) : (μ.restrict t).restrict s = μ.restrict s := by |
ext1 u hu
rw [restrict_apply hu, restrict_apply hu, restrict_eq_self]
exact inter_subset_right.trans h
|
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Topology.Algebra.Module.Basic
#align_import analysis.normed_space.basic from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
open Metric Set Function Filter
open scoped NNReal Topology
instance Real.punctured_nhds_module_neBot {E ... | Mathlib/Analysis/NormedSpace/Real.lean | 101 | 103 | theorem frontier_closedBall (x : E) {r : ℝ} (hr : r ≠ 0) :
frontier (closedBall x r) = sphere x r := by |
rw [frontier, closure_closedBall, interior_closedBall x hr, closedBall_diff_ball]
|
import Mathlib.Data.Fin.Fin2
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.Common
#align_import data.typevec from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc9524185ae994553ebf4"
universe u v w
@[pp_with_univ]
def TypeVec (n : ℕ) :=
Fin2 n → Type*
#align typevec TypeVec
instance {n} : Inh... | Mathlib/Data/TypeVec.lean | 437 | 438 | theorem id_eq_nilFun {α : TypeVec 0} : @id _ α = nilFun := by |
ext x; cases x
|
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Maps
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b"
universe u v
namespace SimpleGraph
@[ext]
structure Subgraph {V : Type u} (G : SimpleGra... | Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | 177 | 178 | theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by |
simp [Subgraph.spanningCoe]
|
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.Algebra.Polynomial.Lifts
import Mathlib.Data.List.Prime
#align_import data.polynomial.splits from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
noncomputable section
open Polynomial
universe u v w
variable {R : Type*} {F... | Mathlib/Algebra/Polynomial/Splits.lean | 164 | 165 | theorem splits_id_iff_splits {f : K[X]} : (f.map i).Splits (RingHom.id L) ↔ f.Splits i := by |
rw [splits_map_iff, RingHom.id_comp]
|
import Mathlib.Data.Finset.Lattice
#align_import combinatorics.set_family.compression.down from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
variable {α : Type*} [DecidableEq α] {𝒜 ℬ : Finset (Finset α)} {s : Finset α} {a : α}
namespace Finset
def nonMemberSubfamily (a : α) (𝒜 : ... | Mathlib/Combinatorics/SetFamily/Compression/Down.lean | 133 | 136 | theorem nonMemberSubfamily_nonMemberSubfamily :
(𝒜.nonMemberSubfamily a).nonMemberSubfamily a = 𝒜.nonMemberSubfamily a := by |
ext
simp
|
import Mathlib.CategoryTheory.Category.Grpd
import Mathlib.CategoryTheory.Groupoid
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.Topology.Homotopy.Path
import Mathlib.Data.Set.Subsingleton
#align_import algebraic_topology.fundamental_groupoid.basic from "leanprover-community/mathlib"@"3d7987cda72abc473... | Mathlib/AlgebraicTopology/FundamentalGroupoid/Basic.lean | 152 | 163 | theorem trans_refl_reparam (p : Path x₀ x₁) :
p.trans (Path.refl x₁) =
p.reparam (fun t => ⟨transReflReparamAux t, transReflReparamAux_mem_I t⟩) (by continuity)
(Subtype.ext transReflReparamAux_zero) (Subtype.ext transReflReparamAux_one) := by |
ext
unfold transReflReparamAux
simp only [Path.trans_apply, not_le, coe_reparam, Function.comp_apply, one_div, Path.refl_apply]
split_ifs
· rfl
· rfl
· simp
· simp
|
import Mathlib.MeasureTheory.Decomposition.RadonNikodym
import Mathlib.Probability.Kernel.Disintegration.CdfToKernel
#align_import probability.kernel.cond_cdf from "leanprover-community/mathlib"@"3b88f4005dc2e28d42f974cc1ce838f0dafb39b8"
open MeasureTheory Set Filter TopologicalSpace
open scoped NNReal ENNReal Me... | Mathlib/Probability/Kernel/Disintegration/CondCdf.lean | 87 | 89 | theorem iInf_IicSnd_gt (t : ℚ) {s : Set α} (hs : MeasurableSet s) [IsFiniteMeasure ρ] :
⨅ r : { r' : ℚ // t < r' }, ρ.IicSnd r s = ρ.IicSnd t s := by |
simp_rw [ρ.IicSnd_apply _ hs, Measure.iInf_rat_gt_prod_Iic hs]
|
import Mathlib.Data.Set.Function
import Mathlib.Order.Interval.Set.OrdConnected
#align_import data.set.intervals.proj_Icc from "leanprover-community/mathlib"@"4e24c4bfcff371c71f7ba22050308aa17815626c"
variable {α β : Type*} [LinearOrder α]
open Function
namespace Set
def projIci (a x : α) : Ici a := ⟨max a x,... | Mathlib/Order/Interval/Set/ProjIcc.lean | 219 | 220 | theorem range_IciExtend (f : Ici a → β) : range (IciExtend f) = range f := by |
simp only [IciExtend, range_comp f, range_projIci, range_id', image_univ]
|
import Mathlib.Mathport.Rename
import Mathlib.Tactic.Lemma
import Mathlib.Tactic.TypeStar
#align_import data.option.defs from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
namespace Option
#align option.lift_or_get Option.liftOrGet
protected def traverse.{u, v}
{F : Type u → Type... | Mathlib/Data/Option/Defs.lean | 96 | 97 | theorem mem_toList {a : α} {o : Option α} : a ∈ toList o ↔ a ∈ o := by |
cases o <;> simp [toList, eq_comm]
|
import Mathlib.Topology.Algebra.GroupWithZero
import Mathlib.Topology.Order.OrderClosed
#align_import topology.algebra.with_zero_topology from "leanprover-community/mathlib"@"3e0c4d76b6ebe9dfafb67d16f7286d2731ed6064"
open Topology Filter TopologicalSpace Filter Set Function
namespace WithZeroTopology
variable {α... | Mathlib/Topology/Algebra/WithZeroTopology.lean | 47 | 49 | theorem nhds_eq_update : (𝓝 : Γ₀ → Filter Γ₀) = update pure 0 (⨅ γ ≠ 0, 𝓟 (Iio γ)) := by |
rw [nhds_nhdsAdjoint, sup_of_le_right]
exact le_iInf₂ fun γ hγ ↦ le_principal_iff.2 <| zero_lt_iff.2 hγ
|
import Mathlib.Data.List.Nodup
#align_import data.list.duplicate from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
variable {α : Type*}
namespace List
inductive Duplicate (x : α) : List α → Prop
| cons_mem {l : List α} : x ∈ l → Duplicate x (x :: l)
| cons_duplicate {y : α} {l ... | Mathlib/Data/List/Duplicate.lean | 46 | 49 | theorem Duplicate.mem (h : x ∈+ l) : x ∈ l := by |
induction' h with l' _ y l' _ hm
· exact mem_cons_self _ _
· exact mem_cons_of_mem _ hm
|
import Mathlib.Order.Bounds.Basic
import Mathlib.Order.WellFounded
import Mathlib.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Data.Set.Lattice
#align_import order.conditionally_complete_lattice.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
open Func... | Mathlib/Order/ConditionallyCompleteLattice/Basic.lean | 116 | 121 | theorem WithTop.coe_sSup' [SupSet α] {s : Set α} (hs : BddAbove s) :
↑(sSup s) = (sSup ((fun (a : α) ↦ ↑a) '' s) : WithTop α) := by |
change _ = ite _ _ _
rw [if_neg, preimage_image_eq, if_pos hs]
· exact Option.some_injective _
· rintro ⟨x, _, ⟨⟩⟩
|
import Mathlib.Algebra.Order.Field.Power
import Mathlib.NumberTheory.Padics.PadicVal
#align_import number_theory.padics.padic_norm from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
def padicNorm (p : ℕ) (q : ℚ) : ℚ :=
if q = 0 then 0 else (p : ℚ) ^ (-padicValRat p q)
#align padic_n... | Mathlib/NumberTheory/Padics/PadicNorm.lean | 327 | 338 | theorem sum_le {α : Type*} {F : α → ℚ} {t : ℚ} {s : Finset α} :
s.Nonempty → (∀ i ∈ s, padicNorm p (F i) ≤ t) → padicNorm p (∑ i ∈ s, F i) ≤ t := by |
classical
refine s.induction_on (by rintro ⟨-, ⟨⟩⟩) ?_
rintro a S haS IH - ht
by_cases hs : S.Nonempty
· rw [Finset.sum_insert haS]
exact
padicNorm.nonarchimedean.trans
(max_le (ht a (Finset.mem_insert_self a S))
(IH hs fun b hb ↦ ht b (Finset.mem_insert_of_mem hb)... |
import Mathlib.Algebra.MvPolynomial.Supported
import Mathlib.RingTheory.WittVector.Truncated
#align_import ring_theory.witt_vector.mul_coeff from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
namespace WittVector
variable (p : ℕ) [hp : Fact p.Prime]
variable {k ... | Mathlib/RingTheory/WittVector/MulCoeff.lean | 138 | 142 | theorem mul_polyOfInterest_aux2 (n : ℕ) :
(p : 𝕄) ^ n * wittMul p n + wittPolyProdRemainder p n = wittPolyProd p n := by |
convert mul_polyOfInterest_aux1 p n
rw [sum_range_succ, add_comm, Nat.sub_self, pow_zero, pow_one]
rfl
|
import Mathlib.Algebra.BigOperators.WithTop
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.ENNReal.Basic
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
variable {a b c d : ℝ≥0∞} {r p q... | Mathlib/Data/ENNReal/Operations.lean | 212 | 212 | theorem top_mul' : ∞ * a = if a = 0 then 0 else ∞ := by | convert WithTop.top_mul' a
|
import Mathlib.Init.Order.Defs
import Mathlib.Logic.Nontrivial.Defs
import Mathlib.Tactic.Attr.Register
import Mathlib.Data.Prod.Basic
import Mathlib.Data.Subtype
import Mathlib.Logic.Function.Basic
import Mathlib.Logic.Unique
#align_import logic.nontrivial from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc95... | Mathlib/Logic/Nontrivial/Basic.lean | 90 | 93 | theorem nontrivial_at (i' : I) [inst : ∀ i, Nonempty (f i)] [Nontrivial (f i')] :
Nontrivial (∀ i : I, f i) := by |
letI := Classical.decEq (∀ i : I, f i)
exact (Function.update_injective (fun i ↦ Classical.choice (inst i)) i').nontrivial
|
import Mathlib.MeasureTheory.Function.LpOrder
#align_import measure_theory.function.l1_space from "leanprover-community/mathlib"@"ccdbfb6e5614667af5aa3ab2d50885e0ef44a46f"
noncomputable section
open scoped Classical
open Topology ENNReal MeasureTheory NNReal
open Set Filter TopologicalSpace ENNReal EMetric Meas... | Mathlib/MeasureTheory/Function/L1Space.lean | 859 | 869 | theorem Integrable.measure_norm_ge_lt_top {f : α → β} (hf : Integrable f μ) {ε : ℝ} (hε : 0 < ε) :
μ { x | ε ≤ ‖f x‖ } < ∞ := by |
rw [show { x | ε ≤ ‖f x‖ } = { x | ENNReal.ofReal ε ≤ ‖f x‖₊ } by
simp only [ENNReal.ofReal, Real.toNNReal_le_iff_le_coe, ENNReal.coe_le_coe, coe_nnnorm]]
refine (meas_ge_le_mul_pow_snorm μ one_ne_zero ENNReal.one_ne_top hf.1 ?_).trans_lt ?_
· simpa only [Ne, ENNReal.ofReal_eq_zero, not_le] using hε
appl... |
import Mathlib.GroupTheory.Archimedean
import Mathlib.Topology.Order.Basic
#align_import topology.algebra.order.archimedean from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
open Set
theorem Rat.denseRange_cast {𝕜} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜]
... | Mathlib/Topology/Algebra/Order/Archimedean.lean | 67 | 71 | theorem dense_or_cyclic (S : AddSubgroup G) : Dense (S : Set G) ∨ ∃ a : G, S = closure {a} := by |
refine (em _).imp (dense_of_not_isolated_zero S) fun h => ?_
push_neg at h
rcases h with ⟨ε, ε0, hε⟩
exact cyclic_of_isolated_zero ε0 (disjoint_left.2 hε)
|
import Mathlib.Analysis.Convex.Cone.Extension
import Mathlib.Analysis.Convex.Gauge
import Mathlib.Topology.Algebra.Module.FiniteDimension
import Mathlib.Topology.Algebra.Module.LocallyConvex
#align_import analysis.normed_space.hahn_banach.separation from "leanprover-community/mathlib"@"915591b2bb3ea303648db07284a161a... | Mathlib/Analysis/NormedSpace/HahnBanach/Separation.lean | 84 | 112 | theorem geometric_hahn_banach_open (hs₁ : Convex ℝ s) (hs₂ : IsOpen s) (ht : Convex ℝ t)
(disj : Disjoint s t) : ∃ (f : E →L[ℝ] ℝ) (u : ℝ), (∀ a ∈ s, f a < u) ∧ ∀ b ∈ t, u ≤ f b := by |
obtain rfl | ⟨a₀, ha₀⟩ := s.eq_empty_or_nonempty
· exact ⟨0, 0, by simp, fun b _hb => le_rfl⟩
obtain rfl | ⟨b₀, hb₀⟩ := t.eq_empty_or_nonempty
· exact ⟨0, 1, fun a _ha => zero_lt_one, by simp⟩
let x₀ := b₀ - a₀
let C := x₀ +ᵥ (s - t)
have : (0 : E) ∈ C :=
⟨a₀ - b₀, sub_mem_sub ha₀ hb₀, by simp_rw [x₀... |
import Mathlib.Logic.Encodable.Lattice
import Mathlib.MeasureTheory.MeasurableSpace.Defs
#align_import measure_theory.pi_system from "leanprover-community/mathlib"@"98e83c3d541c77cdb7da20d79611a780ff8e7d90"
open MeasurableSpace Set
open scoped Classical
open MeasureTheory
def IsPiSystem {α} (C : Set (Set α)) :... | Mathlib/MeasureTheory/PiSystem.lean | 85 | 92 | theorem IsPiSystem.insert_empty {α} {S : Set (Set α)} (h_pi : IsPiSystem S) :
IsPiSystem (insert ∅ S) := by |
intro s hs t ht hst
cases' hs with hs hs
· simp [hs]
· cases' ht with ht ht
· simp [ht]
· exact Set.mem_insert_of_mem _ (h_pi s hs t ht hst)
|
import Mathlib.Algebra.Order.Ring.Abs
#align_import data.int.order.lemmas from "leanprover-community/mathlib"@"fc2ed6f838ce7c9b7c7171e58d78eaf7b438fb0e"
open Function Nat
namespace Int
variable {a b : ℤ} {n : ℕ}
theorem natAbs_eq_iff_mul_self_eq {a b : ℤ} : a.natAbs = b.natAbs ↔ a * a = b * b := by
rw [← a... | Mathlib/Data/Int/Order/Lemmas.lean | 62 | 68 | theorem eq_zero_of_abs_lt_dvd {m x : ℤ} (h1 : m ∣ x) (h2 : |x| < m) : x = 0 := by |
obtain rfl | hm := eq_or_ne m 0
· exact Int.zero_dvd.1 h1
rcases h1 with ⟨d, rfl⟩
apply mul_eq_zero_of_right
rw [← abs_lt_one_iff, ← mul_lt_iff_lt_one_right (abs_pos.mpr hm), ← abs_mul]
exact lt_of_lt_of_le h2 (le_abs_self m)
|
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Iterate
import Mathlib.Order.SemiconjSup
import Mathlib.Tactic.Monotonicity
import Mathlib.Topology.Order.MonotoneContinuity
#align_import dynamics.circle.rotation_number.translation_number from "leanprover-... | Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean | 793 | 797 | theorem tendsto_translationNumber (x : ℝ) :
Tendsto (fun n : ℕ => ((f ^ n) x - x) / n) atTop (𝓝 <| τ f) := by |
rw [← translationNumber_conj_eq' (translate <| Multiplicative.ofAdd x)]
refine (tendsto_translation_number₀ _).congr fun n ↦ ?_
simp [sub_eq_neg_add, Units.conj_pow']
|
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 | 135 | 137 | theorem coe_smul (r : ℝ≥0) : (r : ℝ) • j = r • j := by |
-- Porting note: replaced `show`
rw [real_smul_def, if_pos (NNReal.coe_nonneg r), Real.toNNReal_coe]
|
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 | 1,259 | 1,284 | theorem lintegral_iSup_directed [Countable β] {f : β → α → ℝ≥0∞} (hf : ∀ b, AEMeasurable (f b) μ)
(h_directed : Directed (· ≤ ·) f) : ∫⁻ a, ⨆ b, f b a ∂μ = ⨆ b, ∫⁻ a, f b a ∂μ := by |
simp_rw [← iSup_apply]
let p : α → (β → ENNReal) → Prop := fun x f' => Directed LE.le f'
have hp : ∀ᵐ x ∂μ, p x fun i => f i x := by
filter_upwards [] with x i j
obtain ⟨z, hz₁, hz₂⟩ := h_directed i j
exact ⟨z, hz₁ x, hz₂ x⟩
have h_ae_seq_directed : Directed LE.le (aeSeq hf p) := by
intro b₁ b₂... |
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.GroupAction.Quotient
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.Topology.Algebra.Monoid
import Mathlib.Topology.Algebra.Constructions
#align_import topology.algebra.group.basic from "leanprover-community/mathlib"@"3b1890e71632be9e3... | Mathlib/Topology/Algebra/Group/Basic.lean | 623 | 624 | theorem tendsto_inv_nhdsWithin_Ici_inv {a : H} : Tendsto Inv.inv (𝓝[≥] a⁻¹) (𝓝[≤] a) := by |
simpa only [inv_inv] using @tendsto_inv_nhdsWithin_Ici _ _ _ _ a⁻¹
|
import Mathlib.Topology.Maps
import Mathlib.Topology.NhdsSet
#align_import topology.constructions from "leanprover-community/mathlib"@"f7ebde7ee0d1505dfccac8644ae12371aa3c1c9f"
noncomputable section
open scoped Classical
open Topology TopologicalSpace Set Filter Function
universe u v
variable {X : Type u} {Y :... | Mathlib/Topology/Constructions.lean | 1,042 | 1,044 | theorem isOpenMap_sum {f : X ⊕ Y → Z} :
IsOpenMap f ↔ (IsOpenMap fun a => f (inl a)) ∧ IsOpenMap fun b => f (inr b) := by |
simp only [isOpenMap_iff_nhds_le, Sum.forall, nhds_inl, nhds_inr, Filter.map_map, comp]
|
import Mathlib.LinearAlgebra.ExteriorAlgebra.Basic
import Mathlib.LinearAlgebra.CliffordAlgebra.Fold
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.Dual
#align_import linear_algebra.clifford_algebra.contraction from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2... | Mathlib/LinearAlgebra/CliffordAlgebra/Contraction.lean | 159 | 161 | theorem contractRight_mul_algebraMap (a : CliffordAlgebra Q) (r : R) :
a * algebraMap _ _ r⌊d = a⌊d * algebraMap _ _ r := by |
rw [← Algebra.commutes, contractRight_algebraMap_mul, Algebra.commutes]
|
import Mathlib.Probability.Notation
import Mathlib.Probability.Independence.Basic
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic
#align_import probability.conditional_expectation from "leanprover-community/mathlib"@"2f8347015b12b0864dfaf366ec4909eb70c78740"
open TopologicalSpace Filter
open s... | Mathlib/Probability/ConditionalExpectation.lean | 40 | 77 | theorem condexp_indep_eq (hle₁ : m₁ ≤ m) (hle₂ : m₂ ≤ m) [SigmaFinite (μ.trim hle₂)]
(hf : StronglyMeasurable[m₁] f) (hindp : Indep m₁ m₂ μ) : μ[f|m₂] =ᵐ[μ] fun _ => μ[f] := by |
by_cases hfint : Integrable f μ
swap; · rw [condexp_undef hfint, integral_undef hfint]; rfl
refine (ae_eq_condexp_of_forall_setIntegral_eq hle₂ hfint
(fun s _ hs => integrableOn_const.2 (Or.inr hs)) (fun s hms hs => ?_)
stronglyMeasurable_const.aeStronglyMeasurable').symm
rw [setIntegral_const]
rw ... |
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.AddTorsor
#align_import analysis.convex.normed from "leanprover-community/mathlib"@"a63928c34ec358b5edcda2bf7513c50052... | Mathlib/Analysis/Convex/Normed.lean | 144 | 147 | theorem isConnected_setOf_sameRay (x : E) : IsConnected { y | SameRay ℝ x y } := by |
by_cases hx : x = 0; · simpa [hx] using isConnected_univ (α := E)
simp_rw [← exists_nonneg_left_iff_sameRay hx]
exact isConnected_Ici.image _ (continuous_id.smul continuous_const).continuousOn
|
import Mathlib.CategoryTheory.Linear.LinearFunctor
import Mathlib.CategoryTheory.Monoidal.Preadditive
#align_import category_theory.monoidal.linear from "leanprover-community/mathlib"@"986c4d5761f938b2e1c43c01f001b6d9d88c2055"
namespace CategoryTheory
open CategoryTheory.Limits
open CategoryTheory.MonoidalCateg... | Mathlib/CategoryTheory/Monoidal/Linear.lean | 58 | 70 | theorem monoidalLinearOfFaithful {D : Type*} [Category D] [Preadditive D] [Linear R D]
[MonoidalCategory D] [MonoidalPreadditive D] (F : MonoidalFunctor D C) [F.Faithful]
[F.toFunctor.Additive] [F.toFunctor.Linear R] : MonoidalLinear R D :=
{ whiskerLeft_smul := by |
intros X Y Z r f
apply F.toFunctor.map_injective
rw [F.map_whiskerLeft]
simp
smul_whiskerRight := by
intros r X Y f Z
apply F.toFunctor.map_injective
rw [F.map_whiskerRight]
simp }
|
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Preserves.Basic
#align_import category_theory.limits.preserves.shapes.pullbacks from "leanprover-community/mathlib"@"f11e306adb9f2a393539d2bb4293bf1b42caa7ac"
noncomputable section
universe v₁ v₂ u₁ u₂
-- Porting note: ne... | Mathlib/CategoryTheory/Limits/Preserves/Shapes/Pullbacks.lean | 239 | 241 | theorem PreservesPushout.inl_iso_inv :
G.map pushout.inl ≫ (PreservesPushout.iso G f g).inv = pushout.inl := by |
simp [PreservesPushout.iso, Iso.comp_inv_eq]
|
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Data.Nat.SuccPred
#align_import set_theory.ordinal.arithmetic from "leanprover-community/mathlib"@"31b269b60935483943542d547a6dd83a66b37dc7"
assert_not_exists Field
assert_not_exists Module
noncomputable section
open Function Cardinal Set Equiv Order
open sc... | Mathlib/SetTheory/Ordinal/Arithmetic.lean | 744 | 746 | theorem le_mul_right (a : Ordinal) {b : Ordinal} (hb : 0 < b) : a ≤ b * a := by |
convert mul_le_mul_right' (one_le_iff_pos.2 hb) a
rw [one_mul a]
|
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.centering from "leanprover-community/mathlib"@"bea6c853b6edbd15e9d0941825abd04d77933ed0"
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory
namespace MeasureTheory
variable {Ω E : Type*} {m0 : ... | Mathlib/Probability/Martingale/Centering.lean | 86 | 90 | theorem integrable_martingalePart (hf_int : ∀ n, Integrable (f n) μ) (n : ℕ) :
Integrable (martingalePart f ℱ μ n) μ := by |
rw [martingalePart_eq_sum]
exact (hf_int 0).add
(integrable_finset_sum' _ fun i _ => ((hf_int _).sub (hf_int _)).sub integrable_condexp)
|
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.GroupTheory.Perm.Basic
#align_import group_theory.perm.support from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Equiv Finset
namespace Equiv.Perm
variable {α : Type*}
section support
v... | Mathlib/GroupTheory/Perm/Support.lean | 297 | 298 | theorem mem_support {x : α} : x ∈ f.support ↔ f x ≠ x := by |
rw [support, mem_filter, and_iff_right (mem_univ x)]
|
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 | 126 | 127 | theorem toLp_congr {f g : α → E} (hf : Memℒp f p μ) (hg : Memℒp g p μ) (hfg : f =ᵐ[μ] g) :
hf.toLp f = hg.toLp g := by | simp [toLp, hfg]
|
import Mathlib.Analysis.Convex.Hull
#align_import analysis.convex.join from "leanprover-community/mathlib"@"951bf1d9e98a2042979ced62c0620bcfb3587cf8"
open Set
variable {ι : Sort*} {𝕜 E : Type*}
section OrderedSemiring
variable (𝕜) [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] {s t s₁ s₂ t₁ t₂ u : Set ... | Mathlib/Analysis/Convex/Join.lean | 98 | 100 | theorem convexJoin_iUnion_right (s : Set E) (t : ι → Set E) :
convexJoin 𝕜 s (⋃ i, t i) = ⋃ i, convexJoin 𝕜 s (t i) := by |
simp_rw [convexJoin_comm s, convexJoin_iUnion_left]
|
import Mathlib.Data.Set.Image
import Mathlib.Data.SProd
#align_import data.set.prod from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc9524185ae994553ebf4"
open Function
namespace Set
section Prod
variable {α β γ δ : Type*} {s s₁ s₂ : Set α} {t t₁ t₂ : Set β} {a : α} {b : β}
theorem Subsingleton.pro... | Mathlib/Data/Set/Prod.lean | 231 | 233 | theorem mk_preimage_prod_right (ha : a ∈ s) : Prod.mk a ⁻¹' s ×ˢ t = t := by |
ext b
simp [ha]
|
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.Ring.Action.Basic
import Mathlib.GroupTheory.GroupAction.Hom
import Mathlib.GroupTheory.GroupAction.Quotient
#align_import algebra.polynomial.group_ring_action from "leanprover-community/mathlib"@"afad8e438d03f... | Mathlib/Algebra/Polynomial/GroupRingAction.lean | 76 | 78 | theorem smul_eval [MulSemiringAction G S] (g : G) (f : S[X]) (x : S) :
(g • f).eval x = g • f.eval (g⁻¹ • x) := by |
rw [← smul_eval_smul, smul_inv_smul]
|
import Mathlib.NumberTheory.Cyclotomic.Discriminant
import Mathlib.RingTheory.Polynomial.Eisenstein.IsIntegral
import Mathlib.RingTheory.Ideal.Norm
#align_import number_theory.cyclotomic.rat from "leanprover-community/mathlib"@"b353176c24d96c23f0ce1cc63efc3f55019702d9"
universe u
open Algebra IsCyclotomicExtensio... | Mathlib/NumberTheory/Cyclotomic/Rat.lean | 74 | 119 | theorem isIntegralClosure_adjoin_singleton_of_prime_pow [hcycl : IsCyclotomicExtension {p ^ k} ℚ K]
(hζ : IsPrimitiveRoot ζ ↑(p ^ k)) : IsIntegralClosure (adjoin ℤ ({ζ} : Set K)) ℤ K := by |
refine ⟨Subtype.val_injective, @fun x => ⟨fun h => ⟨⟨x, ?_⟩, rfl⟩, ?_⟩⟩
swap
· rintro ⟨y, rfl⟩
exact
IsIntegral.algebraMap
((le_integralClosure_iff_isIntegral.1
(adjoin_le_integralClosure (hζ.isIntegral (p ^ k).pos))).isIntegral _)
let B := hζ.subOnePowerBasis ℚ
have hint : IsInte... |
import Mathlib.Algebra.Associated
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
variable {α β... | Mathlib/RingTheory/Multiplicity.lean | 123 | 126 | theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) :
0 < (multiplicity a b).get hfin := by |
refine zero_lt_iff.2 fun h => ?_
simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h)
|
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.MvPolynomial.Variables
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Algebra.MvPolynomial.Expand
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.ZMod.Basic
#align_import ring_theory.witt_vector.witt_polynomial from "leanprover-c... | Mathlib/RingTheory/WittVector/WittPolynomial.lean | 154 | 163 | theorem wittPolynomial_zmod_self (n : ℕ) :
W_ (ZMod (p ^ (n + 1))) (n + 1) = expand p (W_ (ZMod (p ^ (n + 1))) n) := by |
simp only [wittPolynomial_eq_sum_C_mul_X_pow]
rw [sum_range_succ, ← Nat.cast_pow, CharP.cast_eq_zero (ZMod (p ^ (n + 1))) (p ^ (n + 1)), C_0,
zero_mul, add_zero, AlgHom.map_sum, sum_congr rfl]
intro k hk
rw [AlgHom.map_mul, AlgHom.map_pow, expand_X, algHom_C, ← pow_mul, ← pow_succ']
congr
rw [mem_range... |
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.Order.Filter.IndicatorFunction
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner
import Mathlib.MeasureTheory.Function.LpSeminorm.Trim
#align_import measure_theory.function.conditional... | Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean | 102 | 110 | theorem const_inner {𝕜 β} [RCLike 𝕜] [NormedAddCommGroup β] [InnerProductSpace 𝕜 β] {f : α → β}
(hfm : AEStronglyMeasurable' m f μ) (c : β) :
AEStronglyMeasurable' m (fun x => (inner c (f x) : 𝕜)) μ := by |
rcases hfm with ⟨f', hf'_meas, hf_ae⟩
refine
⟨fun x => (inner c (f' x) : 𝕜), (@stronglyMeasurable_const _ _ m _ c).inner hf'_meas,
hf_ae.mono fun x hx => ?_⟩
dsimp only
rw [hx]
|
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 | 847 | 850 | theorem mul_meas_ge_le_lintegral₀ {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) (ε : ℝ≥0∞) :
ε * μ { x | ε ≤ f x } ≤ ∫⁻ a, f a ∂μ := by |
simpa only [lintegral_zero, zero_add] using
lintegral_add_mul_meas_add_le_le_lintegral (ae_of_all _ fun x => zero_le (f x)) hf ε
|
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.GroupAction.Units
import Mathlib.Logic.Basic
import Mathlib.Tactic.Ring
#align_import ring_theory.coprime.basic from "leanprover-community/mathlib"@"a95b16cb... | Mathlib/RingTheory/Coprime/Basic.lean | 114 | 121 | theorem IsCoprime.mul_left (H1 : IsCoprime x z) (H2 : IsCoprime y z) : IsCoprime (x * y) z :=
let ⟨a, b, h1⟩ := H1
let ⟨c, d, h2⟩ := H2
⟨a * c, a * x * d + b * c * y + b * d * z,
calc a * c * (x * y) + (a * x * d + b * c * y + b * d * z) * z
_ = (a * x + b * z) * (c * y + d * z) := by | ring
_ = 1 := by rw [h1, h2, mul_one]
⟩
|
import Mathlib.Analysis.Complex.Circle
import Mathlib.MeasureTheory.Group.Integral
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Haar.OfBasis
import Mathlib.MeasureTheory.Constructions.Prod.Integral
import Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace
import Mathlib.Alge... | Mathlib/Analysis/Fourier/FourierTransform.lean | 96 | 100 | theorem norm_fourierIntegral_le_integral_norm (e : AddChar 𝕜 𝕊) (μ : Measure V)
(L : V →ₗ[𝕜] W →ₗ[𝕜] 𝕜) (f : V → E) (w : W) :
‖fourierIntegral e μ L f w‖ ≤ ∫ v : V, ‖f v‖ ∂μ := by |
refine (norm_integral_le_integral_norm _).trans (le_of_eq ?_)
simp_rw [norm_circle_smul]
|
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.Seminorm
import Mathlib.Analysis.LocallyConvex.Bounded
import Mathlib.Analysis.RCLike.Basic
#align_import analysis.convex.gauge from "leanprover-community/mathlib"@"373b03b5b9d0486534edbe94747f23cb3712f93d"
... | Mathlib/Analysis/Convex/Gauge.lean | 459 | 475 | theorem continuousAt_gauge (hc : Convex ℝ s) (hs₀ : s ∈ 𝓝 0) : ContinuousAt (gauge s) x := by |
have ha : Absorbent ℝ s := absorbent_nhds_zero hs₀
refine (nhds_basis_Icc_pos _).tendsto_right_iff.2 fun ε hε₀ ↦ ?_
rw [← map_add_left_nhds_zero, eventually_map]
have : ε • s ∩ -(ε • s) ∈ 𝓝 0 :=
inter_mem ((set_smul_mem_nhds_zero_iff hε₀.ne').2 hs₀)
(neg_mem_nhds_zero _ ((set_smul_mem_nhds_zero_iff ... |
import Mathlib.Algebra.Field.Opposite
import Mathlib.Algebra.Group.Subgroup.ZPowers
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Ring.NegOnePow
import Mathlib.Algebra.Order.Archimedean
import Mathlib.GroupTheory.Coset
#align_import algebra.periodic from "leanprover-community/mathlib"@"3041... | Mathlib/Algebra/Periodic.lean | 480 | 482 | theorem Antiperiodic.int_mul_sub_eq [Ring α] [Ring β] (h : Antiperiodic f c) (n : ℤ) :
f (n * c - x) = (n.negOnePow : ℤ) * f (-x) := by |
simpa only [zsmul_eq_mul] using h.zsmul_sub_eq n
|
import Mathlib.CategoryTheory.Limits.Shapes.CommSq
import Mathlib.CategoryTheory.Limits.Shapes.Diagonal
import Mathlib.CategoryTheory.MorphismProperty.Composition
universe v u
namespace CategoryTheory
open Limits
namespace MorphismProperty
variable {C : Type u} [Category.{v} C]
def StableUnderBaseChange (P : ... | Mathlib/CategoryTheory/MorphismProperty/Limits.lean | 219 | 228 | theorem RespectsIso.diagonal (hP : P.RespectsIso) : P.diagonal.RespectsIso := by |
constructor
· introv H
rwa [diagonal_iff, pullback.diagonal_comp, hP.cancel_left_isIso, hP.cancel_left_isIso,
← hP.cancel_right_isIso _
(pullback.map (e.hom ≫ f) (e.hom ≫ f) f f e.hom e.hom (𝟙 Z) (by simp) (by simp)),
← pullback.condition, hP.cancel_left_isIso]
· introv H
delta diago... |
import Mathlib.Algebra.BigOperators.Group.Multiset
import Mathlib.Data.PNat.Prime
import Mathlib.Data.Nat.Factors
import Mathlib.Data.Multiset.Sort
#align_import data.pnat.factors from "leanprover-community/mathlib"@"e3d9ab8faa9dea8f78155c6c27d62a621f4c152d"
-- Porting note: `deriving` contained Inhabited, Canonic... | Mathlib/Data/PNat/Factors.lean | 89 | 91 | theorem coeNat_prime (v : PrimeMultiset) (p : ℕ) (h : p ∈ (v : Multiset ℕ)) : p.Prime := by |
rcases Multiset.mem_map.mp h with ⟨⟨_, hp'⟩, ⟨_, h_eq⟩⟩
exact h_eq ▸ hp'
|
import Mathlib.Init.Control.Combinators
import Mathlib.Data.Option.Defs
import Mathlib.Logic.IsEmpty
import Mathlib.Logic.Relator
import Mathlib.Util.CompileInductive
import Aesop
#align_import data.option.basic from "leanprover-community/mathlib"@"f340f229b1f461aa1c8ee11e0a172d0a3b301a4a"
universe u
namespace Op... | Mathlib/Data/Option/Basic.lean | 260 | 270 | theorem pmap_eq_some_iff {hf} {y : β} :
pmap f x hf = some y ↔ ∃ (a : α) (H : x = some a), f a (hf a H) = y := by |
rcases x with (_|x)
· simp only [not_mem_none, exists_false, pmap, not_false_iff, exists_prop_of_false]
· constructor
· intro h
simp only [pmap, Option.some_inj] at h
exact ⟨x, rfl, h⟩
· rintro ⟨a, H, rfl⟩
simp only [mem_def, Option.some_inj] at H
simp only [H, pmap]
|
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 | 1,119 | 1,128 | theorem continuous_setIntegral [NormedSpace ℝ E] (s : Set X) :
Continuous fun f : X →₁[μ] E => ∫ x in s, f x ∂μ := by |
haveI : Fact ((1 : ℝ≥0∞) ≤ 1) := ⟨le_rfl⟩
have h_comp :
(fun f : X →₁[μ] E => ∫ x in s, f x ∂μ) =
integral (μ.restrict s) ∘ fun f => LpToLpRestrictCLM X E ℝ μ 1 s f := by
ext1 f
rw [Function.comp_apply, integral_congr_ae (LpToLpRestrictCLM_coeFn ℝ s f)]
rw [h_comp]
exact continuous_integral.c... |
import Mathlib.Analysis.SpecialFunctions.Complex.Circle
import Mathlib.Geometry.Euclidean.Angle.Oriented.Basic
#align_import geometry.euclidean.angle.oriented.rotation from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
noncomputable section
open FiniteDimensional Complex
open scoped ... | Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean | 307 | 308 | theorem eq_rotation_self_iff (x : V) (θ : Real.Angle) : x = o.rotation θ x ↔ x = 0 ∨ θ = 0 := by |
rw [← rotation_eq_self_iff, eq_comm]
|
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
import Mathlib.MeasureTheory.Decomposition.Jordan
import Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure
noncomputable section
open scoped Classical MeasureTheory NNReal ENNReal
open Set
variable {α β : Type*... | Mathlib/MeasureTheory/Decomposition/SignedLebesgue.lean | 395 | 398 | theorem singularPart_sub (s t : SignedMeasure α) (μ : Measure α) [s.HaveLebesgueDecomposition μ]
[t.HaveLebesgueDecomposition μ] :
(s - t).singularPart μ = s.singularPart μ - t.singularPart μ := by |
rw [sub_eq_add_neg, sub_eq_add_neg, singularPart_add, singularPart_neg]
|
import Mathlib.CategoryTheory.Balanced
import Mathlib.CategoryTheory.Limits.EssentiallySmall
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
import Mathlib.CategoryTheory.Subobject.Lattice
import Mathlib.CategoryTheory.Subobject.WellPowered
import Mathlib.Data.S... | Mathlib/CategoryTheory/Generator.lean | 113 | 114 | theorem isSeparating_unop_iff (𝒢 : Set Cᵒᵖ) : IsSeparating 𝒢.unop ↔ IsCoseparating 𝒢 := by |
rw [← isCoseparating_op_iff, Set.unop_op]
|
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Data.Set.Lattice
#align_import data.set.intervals.pi from "leanprover-community/mathlib"@"e4bc74cbaf429d706cb9140902f7ca6c431e75a4"
-- Porting note: Added, since dot nota... | Mathlib/Order/Interval/Set/Pi.lean | 101 | 109 | theorem pi_univ_Ioc_update_right {x y : ∀ i, α i} {i₀ : ι} {m : α i₀} (hm : m ≤ y i₀) :
(pi univ fun i ↦ Ioc (x i) (update y i₀ m i)) =
{ z | z i₀ ≤ m } ∩ pi univ fun i ↦ Ioc (x i) (y i) := by |
have : Ioc (x i₀) m = Iic m ∩ Ioc (x i₀) (y i₀) := by
rw [← Ioi_inter_Iic, ← Ioi_inter_Iic, inter_left_comm,
inter_eq_self_of_subset_left (Iic_subset_Iic.2 hm)]
simp_rw [univ_pi_update i₀ y m fun i z ↦ Ioc (x i) z, ← pi_inter_compl ({i₀} : Set ι),
singleton_pi', ← inter_assoc, this]
rfl
|
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.Group.FiniteSupport
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Set.Subsingleton
#align_import algebra.big_operators.finprod from "leanprover-community/mathlib"@"d6fad0e5bf... | Mathlib/Algebra/BigOperators/Finprod.lean | 258 | 261 | theorem finprod_congr_Prop {p q : Prop} {f : p → M} {g : q → M} (hpq : p = q)
(hfg : ∀ h : q, f (hpq.mpr h) = g h) : finprod f = finprod g := by |
subst q
exact finprod_congr hfg
|
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 | 146 | 154 | theorem evariance_eq_zero_iff (hX : AEMeasurable X μ) :
evariance X μ = 0 ↔ X =ᵐ[μ] fun _ => μ[X] := by |
rw [evariance, lintegral_eq_zero_iff']
constructor <;> intro hX <;> filter_upwards [hX] with ω hω
· simpa only [Pi.zero_apply, sq_eq_zero_iff, ENNReal.coe_eq_zero, nnnorm_eq_zero, sub_eq_zero]
using hω
· rw [hω]
simp
· exact (hX.sub_const _).ennnorm.pow_const _ -- TODO `measurability` and `fun_prop... |
import Mathlib.Order.Hom.CompleteLattice
import Mathlib.Topology.Bases
import Mathlib.Topology.Homeomorph
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Order.CompactlyGenerated.Basic
import Mathlib.Order.Copy
#align_import topology.sets.opens from "leanprover-community/mathlib"@"dc6c365e751e34d100e8... | Mathlib/Topology/Sets/Opens.lean | 247 | 249 | theorem mem_iSup {ι} {x : α} {s : ι → Opens α} : x ∈ iSup s ↔ ∃ i, x ∈ s i := by |
rw [← SetLike.mem_coe]
simp
|
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 | 447 | 449 | theorem ext_iff_of_biUnion_eq_univ {S : Set ι} {s : ι → Set α} (hc : S.Countable)
(hs : ⋃ i ∈ S, s i = univ) : μ = ν ↔ ∀ i ∈ S, μ.restrict (s i) = ν.restrict (s i) := by |
rw [← restrict_biUnion_congr hc, hs, restrict_univ, restrict_univ]
|
import Mathlib.Algebra.Order.Kleene
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.Data.List.Join
import Mathlib.Data.Set.Lattice
import Mathlib.Tactic.DeriveFintype
#align_import computability.language from "leanprover-community/mathlib"@"a239cd3e7ac2c7cde36c913808f9d40c411344f6"
open List Set Computability... | Mathlib/Computability/Language.lean | 200 | 203 | theorem le_mul_congr {l₁ l₂ m₁ m₂ : Language α} : l₁ ≤ m₁ → l₂ ≤ m₂ → l₁ * l₂ ≤ m₁ * m₂ := by |
intro h₁ h₂ x hx
simp only [mul_def, exists_and_left, mem_image2, image_prod] at hx ⊢
tauto
|
import Mathlib.CategoryTheory.Sites.Spaces
import Mathlib.Topology.Sheaves.Sheaf
import Mathlib.CategoryTheory.Sites.DenseSubsite
#align_import topology.sheaves.sheaf_condition.sites from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
noncomputable section
set_option linter.uppercaseLe... | Mathlib/Topology/Sheaves/SheafCondition/Sites.lean | 137 | 144 | theorem coverDense_iff_isBasis [Category ι] (B : ι ⥤ Opens X) :
B.IsCoverDense (Opens.grothendieckTopology X) ↔ Opens.IsBasis (Set.range B.obj) := by |
rw [Opens.isBasis_iff_nbhd]
constructor
· intro hd U x hx; rcases hd.1 U x hx with ⟨V, f, ⟨i, f₁, f₂, _⟩, hV⟩
exact ⟨B.obj i, ⟨i, rfl⟩, f₁.le hV, f₂.le⟩
intro hb; constructor; intro U x hx; rcases hb hx with ⟨_, ⟨i, rfl⟩, hx, hi⟩
exact ⟨B.obj i, ⟨⟨hi⟩⟩, ⟨⟨i, 𝟙 _, ⟨⟨hi⟩⟩, rfl⟩⟩, hx⟩
|
import Mathlib.Analysis.Convex.Combination
import Mathlib.Analysis.Convex.Strict
import Mathlib.Topology.Connected.PathConnected
import Mathlib.Topology.Algebra.Affine
import Mathlib.Topology.Algebra.Module.Basic
#align_import analysis.convex.topology from "leanprover-community/mathlib"@"0e3aacdc98d25e0afe035c452d876... | Mathlib/Analysis/Convex/Topology.lean | 95 | 97 | theorem segment_subset_closure_openSegment : [x -[𝕜] y] ⊆ closure (openSegment 𝕜 x y) := by |
rw [segment_eq_image, openSegment_eq_image, ← closure_Ioo (zero_ne_one' 𝕜)]
exact image_closure_subset_closure_image (by continuity)
|
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.Sym.Sym2
namespace List
variable {α : Type*}
section Sym2
protected def sym2 : List α → List (Sym2 α)
| [] => []
| x :: xs => (x :: xs).map (fun y => s(x, y)) ++ xs.sym2
theorem mem_sym2_cons_iff {x : α} {xs : List α} {z : Sym2 α} :
z ∈ (x :: xs)... | Mathlib/Data/List/Sym.lean | 68 | 79 | theorem mk_mem_sym2 {xs : List α} {a b : α} (ha : a ∈ xs) (hb : b ∈ xs) :
s(a, b) ∈ xs.sym2 := by |
induction xs with
| nil => simp at ha
| cons x xs ih =>
rw [mem_sym2_cons_iff]
rw [mem_cons] at ha hb
obtain (rfl | ha) := ha <;> obtain (rfl | hb) := hb
· left; rfl
· right; left; use b
· right; left; rw [Sym2.eq_swap]; use a
· right; right; exact ih ha hb
|
import Batteries.Data.RBMap.Basic
import Mathlib.Init.Data.Nat.Notation
import Mathlib.Mathport.Rename
import Mathlib.Tactic.TypeStar
import Mathlib.Util.CompileInductive
#align_import data.tree from "leanprover-community/mathlib"@"ed989ff568099019c6533a4d94b27d852a5710d8"
inductive Tree.{u} (α : Type u) : Type ... | Mathlib/Data/Tree/Basic.lean | 90 | 91 | theorem numLeaves_eq_numNodes_succ (x : Tree α) : x.numLeaves = x.numNodes + 1 := by |
induction x <;> simp [*, Nat.add_comm, Nat.add_assoc, Nat.add_left_comm]
|
import Mathlib.LinearAlgebra.AffineSpace.Basis
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
#align_import linear_algebra.affine_space.matrix from "leanprover-community/mathlib"@"2de9c37fa71dde2f1c6feff19876dd6a7b1519f0"
open Affine Matrix
open Set
universe u₁ u₂ u₃ u₄
variable {ι : Type u₁} {k : Type... | Mathlib/LinearAlgebra/AffineSpace/Matrix.lean | 124 | 127 | theorem toMatrix_mul_toMatrix : b.toMatrix b₂ * b₂.toMatrix b = 1 := by |
ext l m
change (b.coords (b₂ l) ᵥ* b₂.toMatrix b) m = _
rw [toMatrix_vecMul_coords, coords_apply, ← toMatrix_apply, toMatrix_self]
|
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Data.Nat.SuccPred
#align_import set_theory.ordinal.arithmetic from "leanprover-community/mathlib"@"31b269b60935483943542d547a6dd83a66b37dc7"
assert_not_exists Field
assert_not_exists Module
noncomputable section
open Function Cardinal Set Equiv Order
open sc... | Mathlib/SetTheory/Ordinal/Arithmetic.lean | 1,781 | 1,784 | theorem blsub_eq_blsub {ι : Type u} (r r' : ι → ι → Prop) [IsWellOrder ι r] [IsWellOrder ι r']
(f : ι → Ordinal.{max u v}) :
blsub.{_, v} _ (bfamilyOfFamily' r f) = blsub.{_, v} _ (bfamilyOfFamily' r' f) := by |
rw [blsub_eq_lsub', blsub_eq_lsub']
|
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.Algebra.Module.Torsion
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v v' u₁' w w'
variable {R S : Type u} {M : Type v} {M' : Type v'} {M₁ : Type v}... | Mathlib/LinearAlgebra/Dimension/Constructions.lean | 58 | 64 | theorem LinearIndependent.union_of_quotient
{M' : Submodule R M} {s : Set M} (hs : s ⊆ M') (hs' : LinearIndependent (ι := s) R Subtype.val)
{t : Set M} (ht : LinearIndependent (ι := t) R (Submodule.Quotient.mk (p := M') ∘ Subtype.val)) :
LinearIndependent (ι := (s ∪ t : _)) R Subtype.val := by |
refine (LinearIndependent.sum_elim_of_quotient (f := Set.embeddingOfSubset s M' hs)
(of_comp M'.subtype (by simpa using hs')) Subtype.val ht).to_subtype_range' ?_
simp only [embeddingOfSubset_apply_coe, Sum.elim_range, Subtype.range_val]
|
import Mathlib.Algebra.Quaternion
import Mathlib.Tactic.Ring
#align_import algebra.quaternion_basis from "leanprover-community/mathlib"@"3aa5b8a9ed7a7cabd36e6e1d022c9858ab8a8c2d"
open Quaternion
namespace QuaternionAlgebra
structure Basis {R : Type*} (A : Type*) [CommRing R] [Ring A] [Algebra R A] (c₁ c₂ : R) ... | Mathlib/Algebra/QuaternionBasis.lean | 138 | 139 | theorem lift_smul (r : R) (x : ℍ[R,c₁,c₂]) : q.lift (r • x) = r • q.lift x := by |
simp [lift, mul_smul, ← Algebra.smul_def]
|
import Mathlib.Data.Finset.Image
import Mathlib.Data.List.FinRange
#align_import data.fintype.basic from "leanprover-community/mathlib"@"d78597269638367c3863d40d45108f52207e03cf"
assert_not_exists MonoidWithZero
assert_not_exists MulAction
open Function
open Nat
universe u v
variable {α β γ : Type*}
class Fi... | Mathlib/Data/Fintype/Basic.lean | 241 | 243 | theorem compl_insert : (insert a s)ᶜ = sᶜ.erase a := by |
ext
simp only [not_or, mem_insert, iff_self_iff, mem_compl, mem_erase]
|
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.RingTheory.Polynomial.Bernstein
import Mathlib.Topology.ContinuousFunction.Polynomial
import Mathlib.Topology.ContinuousFunction.Compact
#align_import analysis.special_functions.bernstein from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba14... | Mathlib/Analysis/SpecialFunctions/Bernstein.lean | 67 | 71 | theorem bernstein_nonneg {n ν : ℕ} {x : I} : 0 ≤ bernstein n ν x := by |
simp only [bernstein_apply]
have h₁ : (0:ℝ) ≤ x := by unit_interval
have h₂ : (0:ℝ) ≤ 1 - x := by unit_interval
positivity
|
import Mathlib.Algebra.Algebra.Subalgebra.Unitization
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Topology.Algebra.StarSubalgebra
import Mathlib.Topology.ContinuousFunction.ContinuousMapZero
import Mathlib.Topology.ContinuousFunction.Weierstrass
#align_import topology.continuous_function.stone_weierstrass fro... | Mathlib/Topology/ContinuousFunction/StoneWeierstrass.lean | 297 | 300 | theorem continuousMap_mem_subalgebra_closure_of_separatesPoints (A : Subalgebra ℝ C(X, ℝ))
(w : A.SeparatesPoints) (f : C(X, ℝ)) : f ∈ A.topologicalClosure := by |
rw [subalgebra_topologicalClosure_eq_top_of_separatesPoints A w]
simp
|
import Mathlib.Algebra.Group.Equiv.TypeTags
import Mathlib.GroupTheory.FreeAbelianGroup
import Mathlib.GroupTheory.FreeGroup.IsFreeGroup
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
#align_import group_theory.free_abelian_group_finsupp from "leanprover-community/mathlib"@"47b51515e69f59bca5cf34ef456e600... | Mathlib/GroupTheory/FreeAbelianGroupFinsupp.lean | 72 | 74 | theorem Finsupp.toFreeAbelianGroup_toFinsupp {X} (x : FreeAbelianGroup X) :
Finsupp.toFreeAbelianGroup (FreeAbelianGroup.toFinsupp x) = x := by |
rw [← AddMonoidHom.comp_apply, Finsupp.toFreeAbelianGroup_comp_toFinsupp, AddMonoidHom.id_apply]
|
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.PosDef
#align_import linear_algebra.matrix.schur_complement from "leanprover-community/mathlib"@"a176cb1219e300e85793d44583dede42377b51af"
variable {l m n α : Type*}
namespace Matrix
... | Mathlib/LinearAlgebra/Matrix/SchurComplement.lean | 398 | 401 | theorem det_fromBlocks_one₁₁ (B : Matrix m n α) (C : Matrix n m α) (D : Matrix n n α) :
(Matrix.fromBlocks 1 B C D).det = det (D - C * B) := by |
haveI : Invertible (1 : Matrix m m α) := invertibleOne
rw [det_fromBlocks₁₁, invOf_one, Matrix.mul_one, det_one, one_mul]
|
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 | 699 | 705 | theorem two_nsmul_toReal_eq_two_mul_sub_two_pi {θ : Angle} :
((2 : ℕ) • θ).toReal = 2 * θ.toReal - 2 * π ↔ π / 2 < θ.toReal := by |
nth_rw 1 [← coe_toReal θ]
rw [← coe_nsmul, two_nsmul, ← two_mul, toReal_coe_eq_self_sub_two_pi_iff, Set.mem_Ioc]
exact
⟨fun h => by linarith, fun h =>
⟨(div_lt_iff' (zero_lt_two' ℝ)).1 h, by linarith [pi_pos, toReal_le_pi θ]⟩⟩
|
import Mathlib.NumberTheory.Zsqrtd.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Real.Archimedean
#align_import number_theory.zsqrtd.gaussian_int from "leanprover-community/mathlib"@"5b2fe80501ff327b9109fb09b7cc8c325cd0d7d9"
open Zsqrtd Complex
open sc... | Mathlib/NumberTheory/Zsqrtd/GaussianInt.lean | 154 | 155 | theorem intCast_real_norm (x : ℤ[i]) : (x.norm : ℝ) = Complex.normSq (x : ℂ) := by |
rw [Zsqrtd.norm, normSq]; simp
|
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Iterate
import Mathlib.Order.SemiconjSup
import Mathlib.Tactic.Monotonicity
import Mathlib.Topology.Order.MonotoneContinuity
#align_import dynamics.circle.rotation_number.translation_number from "leanprover-... | Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean | 867 | 872 | theorem map_lt_add_floor_translationNumber_add_one (x : ℝ) : f x < x + ⌊τ f⌋ + 1 := by |
rw [add_assoc]
norm_cast
refine map_lt_of_translationNumber_lt_int _ ?_ _
push_cast
exact lt_floor_add_one _
|
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Data.Finset.Preimage
import Mathlib.Order.Interval.Set.Disjoint
import Mathlib.Order.Int... | Mathlib/Order/Filter/AtTopBot.lean | 323 | 326 | theorem Subsingleton.atTop_eq (α) [Subsingleton α] [Preorder α] : (atTop : Filter α) = ⊤ := by |
refine top_unique fun s hs x => ?_
rw [atTop, ciInf_subsingleton x, mem_principal] at hs
exact hs left_mem_Ici
|
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