name
stringlengths 3
112
| file
stringlengths 21
116
| statement
stringlengths 17
8.64k
| state
stringlengths 7
205k
| tactic
stringlengths 3
4.55k
| result
stringlengths 7
205k
| id
stringlengths 16
16
|
|---|---|---|---|---|---|---|
Stirling.log_stirlingSeq_bounded_by_constant
|
Mathlib/Analysis/SpecialFunctions/Stirling.lean
|
theorem log_stirlingSeq_bounded_by_constant : ∃ c, ∀ n : ℕ, c ≤ log (stirlingSeq (n + 1))
|
⊢ ∃ c, ∀ (n : ℕ), c ≤ Real.log (stirlingSeq (n + 1))
|
obtain ⟨d, h⟩ := log_stirlingSeq_bounded_aux
|
case intro
d : ℝ
h : ∀ (n : ℕ), Real.log (stirlingSeq 1) - Real.log (stirlingSeq (n + 1)) ≤ d
⊢ ∃ c, ∀ (n : ℕ), c ≤ Real.log (stirlingSeq (n + 1))
|
c5d6b13878f90943
|
Ordnode.Valid'.balance'_lemma
|
Mathlib/Data/Ordmap/Ordset.lean
|
theorem Valid'.balance'_lemma {α l l' r r'} (H1 : BalancedSz l' r')
(H2 : Nat.dist (@size α l) l' ≤ 1 ∧ size r = r' ∨ Nat.dist (size r) r' ≤ 1 ∧ size l = l') :
2 * @size α r ≤ 9 * size l + 5 ∨ size r ≤ 3
|
α : Type u_2
l r : Ordnode α
r' : ℕ
hr : r.size.dist r' ≤ 1
left✝ : l.size ≤ delta * r'
h₂ : r' ≤ delta * l.size
⊢ 1 ≤ 3
|
decide
|
no goals
|
11dbedcfce2b4a42
|
gramSchmidt_orthogonal
|
Mathlib/Analysis/InnerProductSpace/GramSchmidtOrtho.lean
|
theorem gramSchmidt_orthogonal (f : ι → E) {a b : ι} (h₀ : a ≠ b) :
⟪gramSchmidt 𝕜 f a, gramSchmidt 𝕜 f b⟫ = 0
|
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : WellFoundedLT ι
f : ι → E
a b : ι
h₀ : a ≠ b
⊢ ∀ (a b : ι), a < b → inner (gramSchmidt 𝕜 f a) (gramSchmidt 𝕜 f b) = 0
|
clear h₀ a b
|
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : WellFoundedLT ι
f : ι → E
⊢ ∀ (a b : ι), a < b → inner (gramSchmidt 𝕜 f a) (gramSchmidt 𝕜 f b) = 0
|
a5e33da8a7158e66
|
RingHom.SurjectiveOnStalks.comp
|
Mathlib/RingTheory/SurjectiveOnStalks.lean
|
lemma SurjectiveOnStalks.comp (hg : SurjectiveOnStalks g) (hf : SurjectiveOnStalks f) :
SurjectiveOnStalks (g.comp f)
|
R : Type u_1
inst✝² : CommRing R
S : Type u_2
inst✝¹ : CommRing S
T : Type u_3
inst✝ : CommRing T
g : S →+* T
f : R →+* S
hg : g.SurjectiveOnStalks
hf : f.SurjectiveOnStalks
⊢ (g.comp f).SurjectiveOnStalks
|
intros I hI
|
R : Type u_1
inst✝² : CommRing R
S : Type u_2
inst✝¹ : CommRing S
T : Type u_3
inst✝ : CommRing T
g : S →+* T
f : R →+* S
hg : g.SurjectiveOnStalks
hf : f.SurjectiveOnStalks
I : Ideal T
hI : I.IsPrime
⊢ Function.Surjective ⇑(Localization.localRingHom (Ideal.comap (g.comp f) I) I (g.comp f) ⋯)
|
a57ce5a2a14f46e6
|
CategoryTheory.Pretriangulated.distinguished_cocone_triangle₂
|
Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean
|
/-- Any morphism `Z ⟶ X⟦1⟧` is part of a distinguished triangle `X ⟶ Y ⟶ Z ⟶ X⟦1⟧` -/
lemma distinguished_cocone_triangle₂ {Z X : C} (h : Z ⟶ X⟦(1 : ℤ)⟧) :
∃ (Y : C) (f : X ⟶ Y) (g : Y ⟶ Z), Triangle.mk f g h ∈ distTriang C
|
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
Z X : C
h : Z ⟶ (shiftFunctor C 1).obj X
Y' : C
f' : (shiftFunctor C 1).obj X ⟶ Y'
g' : Y' ⟶ (shiftFunctor C 1).obj Z
mem : Triangle.mk h f' g' ∈ distinguishedTriangles
T' : Triangle C := (Triangle.mk h f' g').invRotate.invRotate
⊢ (Triangle.mk (((shiftEquiv C 1).unitIso.app X).hom ≫ T'.mor₁) T'.mor₂ h).mor₁ ≫
(Iso.refl (Triangle.mk (((shiftEquiv C 1).unitIso.app X).hom ≫ T'.mor₁) T'.mor₂ h).obj₂).hom =
((shiftEquiv C 1).unitIso.app X).hom ≫ (Triangle.mk h f' g').invRotate.invRotate.mor₁
|
aesop_cat
|
no goals
|
05c412872a1053b2
|
gcd_eq_of_dvd_sub_right
|
Mathlib/Algebra/GCDMonoid/Basic.lean
|
theorem gcd_eq_of_dvd_sub_right {a b c : α} (h : a ∣ b - c) : gcd a b = gcd a c
|
case h
α : Type u_1
inst✝² : CommRing α
inst✝¹ : IsDomain α
inst✝ : NormalizedGCDMonoid α
a b c d : α
hd : b - c = a * d
e : α
he : b = gcd a b * e
f : α
hf : a = gcd a b * f
⊢ c = gcd a b * (e - f * d)
|
rw [mul_sub, ← he, ← mul_assoc, ← hf, ← hd, sub_sub_cancel]
|
no goals
|
43db9b878d1ce2f4
|
real_inner_add_sub_eq_zero_iff
|
Mathlib/Analysis/InnerProductSpace/Basic.lean
|
theorem real_inner_add_sub_eq_zero_iff (x y : F) : ⟪x + y, x - y⟫_ℝ = 0 ↔ ‖x‖ = ‖y‖
|
case mp
F : Type u_3
inst✝¹ : SeminormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
x y : F
h : ⟪y, x⟫_ℝ + ⟪x, x⟫_ℝ = ⟪y, x⟫_ℝ + ⟪y, y⟫_ℝ
⊢ ⟪x, x⟫_ℝ = ⟪y, y⟫_ℝ
|
linarith
|
no goals
|
f4e9bdcba8d76738
|
FirstOrder.Language.BoundedFormula.isPrenex_toPrenexImpRight
|
Mathlib/ModelTheory/Complexity.lean
|
theorem isPrenex_toPrenexImpRight {φ ψ : L.BoundedFormula α n} (hφ : IsQF φ) (hψ : IsPrenex ψ) :
IsPrenex (φ.toPrenexImpRight ψ)
|
case of_isQF
L : Language
α : Type u'
n : ℕ
ψ : L.BoundedFormula α n
n✝ : ℕ
φ✝ : L.BoundedFormula α n✝
hψ : φ✝.IsQF
φ : L.BoundedFormula α n✝
hφ : φ.IsQF
⊢ (φ.toPrenexImpRight φ✝).IsPrenex
|
rw [hψ.toPrenexImpRight]
|
case of_isQF
L : Language
α : Type u'
n : ℕ
ψ : L.BoundedFormula α n
n✝ : ℕ
φ✝ : L.BoundedFormula α n✝
hψ : φ✝.IsQF
φ : L.BoundedFormula α n✝
hφ : φ.IsQF
⊢ (φ ⟹ φ✝).IsPrenex
|
732ec074ac9fd833
|
Traversable.foldMap_map
|
Mathlib/Control/Fold.lean
|
theorem foldMap_map [Monoid γ] (f : α → β) (g : β → γ) (xs : t α) :
foldMap g (f <$> xs) = foldMap (g ∘ f) xs
|
α β γ : Type u
t : Type u → Type u
inst✝² : Traversable t
inst✝¹ : LawfulTraversable t
inst✝ : Monoid γ
f : α → β
g : β → γ
xs : t α
⊢ foldMap g (f <$> xs) = foldMap (g ∘ f) xs
|
simp only [foldMap, traverse_map, Function.comp_def]
|
no goals
|
82cf1f3bb7fef8e6
|
Finsupp.extendDomain_subtypeDomain
|
Mathlib/Data/Finsupp/Basic.lean
|
theorem extendDomain_subtypeDomain (f : α →₀ M) (hf : ∀ a ∈ f.support, P a) :
(subtypeDomain P f).extendDomain = f
|
α : Type u_1
M : Type u_13
inst✝¹ : Zero M
P : α → Prop
inst✝ : DecidablePred P
f : α →₀ M
hf : ∀ a ∈ f.support, P a
⊢ (subtypeDomain P f).extendDomain = f
|
ext a
|
case h
α : Type u_1
M : Type u_13
inst✝¹ : Zero M
P : α → Prop
inst✝ : DecidablePred P
f : α →₀ M
hf : ∀ a ∈ f.support, P a
a : α
⊢ (subtypeDomain P f).extendDomain a = f a
|
65d215dfe59b1c2f
|
Path.Homotopy.trans_assoc_reparam
|
Mathlib/AlgebraicTopology/FundamentalGroupoid/Basic.lean
|
theorem trans_assoc_reparam {x₀ x₁ x₂ x₃ : X} (p : Path x₀ x₁) (q : Path x₁ x₂) (r : Path x₂ x₃) :
(p.trans q).trans r =
(p.trans (q.trans r)).reparam
(fun t => ⟨transAssocReparamAux t, transAssocReparamAux_mem_I t⟩) (by continuity)
(Subtype.ext transAssocReparamAux_zero) (Subtype.ext transAssocReparamAux_one)
|
case neg
X : Type u
inst✝ : TopologicalSpace X
x₀ x₁ x₂ x₃ : X
p : Path x₀ x₁
q : Path x₁ x₂
r : Path x₂ x₃
x : ↑I
h₁ : ¬↑x ≤ 1 / 2
h✝² : ↑x ≤ 1 / 4
h✝¹ : ¬2 * ↑x ≤ 1 / 2
h✝ : ¬2 * (2 * ↑x) - 1 ≤ 1 / 2
⊢ r ⟨2 * ↑x - 1, ⋯⟩ = r ⟨2 * (2 * (2 * ↑x) - 1) - 1, ⋯⟩
|
exfalso
|
case neg
X : Type u
inst✝ : TopologicalSpace X
x₀ x₁ x₂ x₃ : X
p : Path x₀ x₁
q : Path x₁ x₂
r : Path x₂ x₃
x : ↑I
h₁ : ¬↑x ≤ 1 / 2
h✝² : ↑x ≤ 1 / 4
h✝¹ : ¬2 * ↑x ≤ 1 / 2
h✝ : ¬2 * (2 * ↑x) - 1 ≤ 1 / 2
⊢ False
|
fdaccc7c71c4d168
|
Polynomial.wronskian_self_eq_zero
|
Mathlib/RingTheory/Polynomial/Wronskian.lean
|
theorem wronskian_self_eq_zero (a : R[X]) : wronskian a a = 0
|
R : Type u_1
inst✝ : CommRing R
a : R[X]
⊢ a.wronskian a = 0
|
rw [wronskian, mul_comm, sub_self]
|
no goals
|
fb4d7ed5e30182c2
|
AlgebraicTopology.DoldKan.decomposition_Q
|
Mathlib/AlgebraicTopology/DoldKan/Decomposition.lean
|
theorem decomposition_Q (n q : ℕ) :
((Q q).f (n + 1) : X _⦋n + 1⦌ ⟶ X _⦋n + 1⦌) =
∑ i ∈ Finset.filter (fun i : Fin (n + 1) => (i : ℕ) < q) Finset.univ,
(P i).f (n + 1) ≫ X.δ i.rev.succ ≫ X.σ (Fin.rev i)
|
case zero
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
n : ℕ
⊢ (Q 0).f (n + 1) = ∑ i ∈ Finset.filter (fun i => ↑i < 0) Finset.univ, (P ↑i).f (n + 1) ≫ X.δ i.rev.succ ≫ X.σ i.rev
|
simp only [Q_zero, HomologicalComplex.zero_f_apply, Nat.not_lt_zero,
Finset.filter_False, Finset.sum_empty]
|
no goals
|
7a7706b8ab0d871c
|
HurwitzZeta.jacobiTheta₂'_functional_equation'
|
Mathlib/NumberTheory/LSeries/HurwitzZetaOdd.lean
|
lemma jacobiTheta₂'_functional_equation' (z τ : ℂ) :
jacobiTheta₂' z τ = (-2 * π) / (-I * τ) ^ (3 / 2 : ℂ) * jacobiTheta₂'' z (-1 / τ)
|
z τ : ℂ
⊢ jacobiTheta₂' z τ = -2 * ↑π / (-I * τ) ^ (3 / 2) * jacobiTheta₂'' z (-1 / τ)
|
rcases eq_or_ne τ 0 with rfl | hτ
|
case inl
z : ℂ
⊢ jacobiTheta₂' z 0 = -2 * ↑π / (-I * 0) ^ (3 / 2) * jacobiTheta₂'' z (-1 / 0)
case inr
z τ : ℂ
hτ : τ ≠ 0
⊢ jacobiTheta₂' z τ = -2 * ↑π / (-I * τ) ^ (3 / 2) * jacobiTheta₂'' z (-1 / τ)
|
dc79ad2579abd9e6
|
HomologicalComplex.extend.comp_d_eq_zero_iff
|
Mathlib/Algebra/Homology/Embedding/ExtendHomology.lean
|
lemma comp_d_eq_zero_iff ⦃W : C⦄ (φ : W ⟶ K.X j) :
φ ≫ K.d j k = 0 ↔ φ ≫ (K.extendXIso e hj').inv ≫ (K.extend e).d j' k' = 0
|
case neg
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝² : Category.{u_4, u_3} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasZeroObject C
K : HomologicalComplex C c
e : c.Embedding c'
j k : ι
j' k' : ι'
hj' : e.f j = j'
hk : c.next j = k
hk' : c'.next j' = k'
W : C
φ : W ⟶ K.X j
hjk : ¬c.Rel j k
⊢ φ ≫ K.d j k = 0 ↔ φ ≫ (K.extendXIso e hj').inv ≫ (K.extend e).d j' k' = 0
|
simp only [K.shape _ _ hjk, comp_zero, true_iff]
|
case neg
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝² : Category.{u_4, u_3} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasZeroObject C
K : HomologicalComplex C c
e : c.Embedding c'
j k : ι
j' k' : ι'
hj' : e.f j = j'
hk : c.next j = k
hk' : c'.next j' = k'
W : C
φ : W ⟶ K.X j
hjk : ¬c.Rel j k
⊢ φ ≫ (K.extendXIso e hj').inv ≫ (K.extend e).d j' k' = 0
|
fb2e617c809a5891
|
HomologicalComplex.extendMap_f
|
Mathlib/Algebra/Homology/Embedding/Extend.lean
|
lemma extendMap_f {i : ι} {i' : ι'} (h : e.f i = i') :
(extendMap φ e).f i' =
(extendXIso K e h).hom ≫ φ.f i ≫ (extendXIso L e h).inv
|
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝² : Category.{u_4, u_3} C
inst✝¹ : HasZeroObject C
inst✝ : HasZeroMorphisms C
K L : HomologicalComplex C c
φ : K ⟶ L
e : c.Embedding c'
i : ι
i' : ι'
h : e.f i = i'
⊢ (extend.XIso K ⋯).hom ≫ φ.f i ≫ (extend.XIso L ⋯).inv = (K.extendXIso e h).hom ≫ φ.f i ≫ (L.extendXIso e h).inv
|
rfl
|
no goals
|
37577a7201d0837b
|
ContinuousLinearMap.isInvertible_comp_equiv
|
Mathlib/Topology/Algebra/Module/Equiv.lean
|
@[simp] lemma isInvertible_comp_equiv {e : M₃ ≃L[R] M} {f : M →L[R] M₂} :
(f ∘L (e : M₃ →L[R] M)).IsInvertible ↔ f.IsInvertible
|
R : Type u_3
M : Type u_4
M₂ : Type u_5
M₃ : Type u_6
inst✝⁹ : TopologicalSpace M
inst✝⁸ : TopologicalSpace M₂
inst✝⁷ : TopologicalSpace M₃
inst✝⁶ : Semiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
inst✝³ : AddCommMonoid M₂
inst✝² : Module R M₂
inst✝¹ : AddCommMonoid M₃
inst✝ : Module R M₃
e : M₃ ≃L[R] M
f : M →L[R] M₂
⊢ (f.comp ↑e).IsInvertible ↔ f.IsInvertible
|
constructor
|
case mp
R : Type u_3
M : Type u_4
M₂ : Type u_5
M₃ : Type u_6
inst✝⁹ : TopologicalSpace M
inst✝⁸ : TopologicalSpace M₂
inst✝⁷ : TopologicalSpace M₃
inst✝⁶ : Semiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
inst✝³ : AddCommMonoid M₂
inst✝² : Module R M₂
inst✝¹ : AddCommMonoid M₃
inst✝ : Module R M₃
e : M₃ ≃L[R] M
f : M →L[R] M₂
⊢ (f.comp ↑e).IsInvertible → f.IsInvertible
case mpr
R : Type u_3
M : Type u_4
M₂ : Type u_5
M₃ : Type u_6
inst✝⁹ : TopologicalSpace M
inst✝⁸ : TopologicalSpace M₂
inst✝⁷ : TopologicalSpace M₃
inst✝⁶ : Semiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
inst✝³ : AddCommMonoid M₂
inst✝² : Module R M₂
inst✝¹ : AddCommMonoid M₃
inst✝ : Module R M₃
e : M₃ ≃L[R] M
f : M →L[R] M₂
⊢ f.IsInvertible → (f.comp ↑e).IsInvertible
|
58f6ac03b58e3c39
|
FreeRing.coe_eq
|
Mathlib/RingTheory/FreeCommRing.lean
|
theorem coe_eq : ((↑) : FreeRing α → FreeCommRing α) =
@Functor.map FreeAbelianGroup _ _ _ fun l : List α => (l : Multiset α)
|
α : Type u
⊢ castFreeCommRing = Functor.map fun l => ↑l
|
funext x
|
case h
α : Type u
x : FreeRing α
⊢ ↑x = (fun l => ↑l) <$> x
|
ef7da7a0cbb1d4a2
|
List.set_eq_take_append_cons_drop
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Nat/TakeDrop.lean
|
theorem set_eq_take_append_cons_drop (l : List α) (n : Nat) (a : α) :
l.set n a = if n < l.length then l.take n ++ a :: l.drop (n + 1) else l
|
case neg
α : Type u_1
l : List α
n : Nat
a : α
h : n < l.length
m : Nat
h' : ¬m < n
h'' : ¬m = n
⊢ (l.set n a)[m]? = (take n l ++ a :: drop (n + 1) l)[m]?
|
have h''' : n < m := by omega
|
case neg
α : Type u_1
l : List α
n : Nat
a : α
h : n < l.length
m : Nat
h' : ¬m < n
h'' : ¬m = n
h''' : n < m
⊢ (l.set n a)[m]? = (take n l ++ a :: drop (n + 1) l)[m]?
|
9fb4c99baced165c
|
FirstOrder.Language.Term.realize_relabel
|
Mathlib/ModelTheory/Semantics.lean
|
theorem realize_relabel {t : L.Term α} {g : α → β} {v : β → M} :
(t.relabel g).realize v = t.realize (v ∘ g)
|
case func
L : Language
M : Type w
inst✝ : L.Structure M
α : Type u'
β : Type v'
g : α → β
v : β → M
n : ℕ
f : L.Functions n
ts : Fin n → L.Term α
ih : ∀ (a : Fin n), realize v (relabel g (ts a)) = realize (v ∘ g) (ts a)
⊢ realize v (relabel g (func f ts)) = realize (v ∘ g) (func f ts)
|
simp [ih]
|
no goals
|
be0cdb57daa070d0
|
Grp.FilteredColimits.colimitInvAux_eq_of_rel
|
Mathlib/Algebra/Category/Grp/FilteredColimits.lean
|
theorem colimitInvAux_eq_of_rel (x y : Σ j, F.obj j)
(h : Types.FilteredColimit.Rel (F ⋙ forget Grp) x y) :
colimitInvAux.{v, u} F x = colimitInvAux F y
|
case h
J : Type v
inst✝¹ : SmallCategory J
inst✝ : IsFiltered J
F : J ⥤ Grp
x y : (j : J) × ↑(F.obj j)
h : Types.FilteredColimit.Rel (F ⋙ forget Grp) x y
⊢ ∃ k f g, (ConcreteCategory.hom (F.map f)) ⟨x.fst, x.snd⁻¹⟩.snd = (ConcreteCategory.hom (F.map g)) ⟨y.fst, y.snd⁻¹⟩.snd
|
obtain ⟨k, f, g, hfg⟩ := h
|
case h.intro.intro.intro
J : Type v
inst✝¹ : SmallCategory J
inst✝ : IsFiltered J
F : J ⥤ Grp
x y : (j : J) × ↑(F.obj j)
k : J
f : x.fst ⟶ k
g : y.fst ⟶ k
hfg : (F ⋙ forget Grp).map f x.snd = (F ⋙ forget Grp).map g y.snd
⊢ ∃ k f g, (ConcreteCategory.hom (F.map f)) ⟨x.fst, x.snd⁻¹⟩.snd = (ConcreteCategory.hom (F.map g)) ⟨y.fst, y.snd⁻¹⟩.snd
|
7ec8d6edded53dcd
|
FormalMultilinearSeries.ofScalars_radius_eq_top_of_tendsto
|
Mathlib/Analysis/Analytic/OfScalars.lean
|
theorem ofScalars_radius_eq_top_of_tendsto (hc : ∀ᶠ n in atTop, c n ≠ 0)
(hc' : Tendsto (fun n ↦ ‖c n.succ‖ / ‖c n‖) atTop (𝓝 0)) : (ofScalars E c).radius = ⊤
|
case h
𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedRing E
inst✝ : NormedAlgebra 𝕜 E
c : ℕ → 𝕜
hc : ∀ᶠ (n : ℕ) in atTop, c n ≠ 0
hc' : Tendsto (fun n => ‖c n.succ‖ / ‖c n‖) atTop (𝓝 0)
r' : ℝ≥0
hrz : ¬r' = 0
n : ℕ
hn : n ≠ 0
⊢ ‖ofScalars E c n‖ * ↑r' ^ n ≤ ‖c n‖ * ↑r' ^ n
|
gcongr
|
case h.h
𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedRing E
inst✝ : NormedAlgebra 𝕜 E
c : ℕ → 𝕜
hc : ∀ᶠ (n : ℕ) in atTop, c n ≠ 0
hc' : Tendsto (fun n => ‖c n.succ‖ / ‖c n‖) atTop (𝓝 0)
r' : ℝ≥0
hrz : ¬r' = 0
n : ℕ
hn : n ≠ 0
⊢ ‖ofScalars E c n‖ ≤ ‖c n‖
|
fed995db04eea233
|
List.find?_eq_some_iff_getElem
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Nat/Find.lean
|
theorem find?_eq_some_iff_getElem {xs : List α} {p : α → Bool} {b : α} :
xs.find? p = some b ↔ p b ∧ ∃ i h, xs[i] = b ∧ ∀ j : Nat, (hj : j < i) → !p xs[j]
|
α : Type u_1
xs : List α
p : α → Bool
b : α
⊢ (p b = true ∧ ∃ as bs, xs = as ++ b :: bs ∧ ∀ (a : α), a ∈ as → (!p a) = true) ↔
p b = true ∧ ∃ i h, xs[i] = b ∧ ∀ (j : Nat) (hj : j < i), (!p xs[j]) = true
|
simp only [Bool.not_eq_eq_eq_not, Bool.not_true, exists_and_right, and_congr_right_iff]
|
α : Type u_1
xs : List α
p : α → Bool
b : α
⊢ p b = true →
((∃ as, (∃ x, xs = as ++ b :: x) ∧ ∀ (a : α), a ∈ as → p a = false) ↔
∃ i h, xs[i] = b ∧ ∀ (j : Nat) (hj : j < i), p xs[j] = false)
|
61fa88fc2d6e44d7
|
AkraBazziRecurrence.rpow_p_mul_one_add_smoothingFn_ge
|
Mathlib/Computability/AkraBazzi/AkraBazzi.lean
|
lemma rpow_p_mul_one_add_smoothingFn_ge :
∀ᶠ (n : ℕ) in atTop, ∀ i, (b i) ^ (p a b) * n ^ (p a b) * (1 + ε n)
≤ (r i n) ^ (p a b) * (1 + ε (r i n))
|
α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
i : α
q : ℝ → ℝ := fun x => x ^ p a b * (1 + ε x)
h_diff_q : DifferentiableOn ℝ q (Set.Ioi 1)
h_deriv_q : deriv q =O[atTop] fun x => x ^ (p a b - 1)
h_main_norm :
(fun n => ‖q ↑(r i n) - q (b i * ↑n)‖) ≤ᶠ[atTop] fun n => ‖b i ^ p a b * ↑n ^ p a b * (ε (b i * ↑n) - ε ↑n)‖
n : ℕ
hn : ⌈(b i)⁻¹⌉₊ < n
hn' : 1 < n
h₁ : 0 < b i
this : b i < 1
⊢ b i * ↑n ≤ 1 * ↑n
|
gcongr
|
no goals
|
6b09e1ce84116b26
|
Ideal.quotientInfToPiQuotient_surj
|
Mathlib/RingTheory/Ideal/Quotient/Operations.lean
|
lemma quotientInfToPiQuotient_surj {I : ι → Ideal R}
(hI : Pairwise (IsCoprime on I)) : Surjective (quotientInfToPiQuotient I)
|
R : Type u_2
inst✝¹ : CommRing R
ι : Type u_3
inst✝ : Finite ι
I : ι → Ideal R
hI : Pairwise (IsCoprime on I)
val✝ : Fintype ι
g : (i : ι) → R ⧸ I i
f : ι → R
hf : ∀ (i : ι), (Quotient.mk (I i)) (f i) = g i
i : ι
hI' : ∀ j ∈ {i}ᶜ, IsCoprime (I i) (I j)
⊢ ∃ e, (Quotient.mk (I i)) e = 1 ∧ ∀ (j : ι), j ≠ i → (Quotient.mk (I j)) e = 0
|
rcases isCoprime_iff_exists.mp (isCoprime_biInf hI') with ⟨u, hu, e, he, hue⟩
|
case intro.intro.intro.intro
R : Type u_2
inst✝¹ : CommRing R
ι : Type u_3
inst✝ : Finite ι
I : ι → Ideal R
hI : Pairwise (IsCoprime on I)
val✝ : Fintype ι
g : (i : ι) → R ⧸ I i
f : ι → R
hf : ∀ (i : ι), (Quotient.mk (I i)) (f i) = g i
i : ι
hI' : ∀ j ∈ {i}ᶜ, IsCoprime (I i) (I j)
u : R
hu : u ∈ I i
e : R
he : e ∈ ⨅ j ∈ {i}ᶜ, I j
hue : u + e = 1
⊢ ∃ e, (Quotient.mk (I i)) e = 1 ∧ ∀ (j : ι), j ≠ i → (Quotient.mk (I j)) e = 0
|
df184cf9a63521b9
|
ArithmeticFunction.vonMangoldt.continuousOn_LFunctionResidueClassAux
|
Mathlib/NumberTheory/LSeries/PrimesInAP.lean
|
/-- The L-series of the von Mangoldt function restricted to the prime residue class `a` mod `q`
is continuous on `re s ≥ 1` except for a simple pole at `s = 1` with residue `(q.totient)⁻¹`.
The statement as given here in terms of `ArithmeticFunction.vonMangoldt.LFunctionResidueClassAux`
is equivalent. -/
lemma continuousOn_LFunctionResidueClassAux :
ContinuousOn (LFunctionResidueClassAux a) {s | 1 ≤ s.re}
|
q : ℕ
a : ZMod q
inst✝ : NeZero q
s : ℂ
hs : s ∈ {s | 1 ≤ s.re}
⊢ s ∈ {s | s = 1 ∨ ∀ (χ : DirichletCharacter ℂ q), LFunction χ s ≠ 0}
|
rcases eq_or_ne s 1 with rfl | hs₁
|
case inl
q : ℕ
a : ZMod q
inst✝ : NeZero q
hs : 1 ∈ {s | 1 ≤ s.re}
⊢ 1 ∈ {s | s = 1 ∨ ∀ (χ : DirichletCharacter ℂ q), LFunction χ s ≠ 0}
case inr
q : ℕ
a : ZMod q
inst✝ : NeZero q
s : ℂ
hs : s ∈ {s | 1 ≤ s.re}
hs₁ : s ≠ 1
⊢ s ∈ {s | s = 1 ∨ ∀ (χ : DirichletCharacter ℂ q), LFunction χ s ≠ 0}
|
9814c5299d106445
|
Matroid.Indep.fundCircuit_isCircuit
|
Mathlib/Data/Matroid/Circuit.lean
|
lemma Indep.fundCircuit_isCircuit (hI : M.Indep I) (hecl : e ∈ M.closure I) (heI : e ∉ I) :
M.IsCircuit (M.fundCircuit e I)
|
case refine_2
α : Type u_1
M : Matroid α
I : Set α
e : α
hI : M.Indep I
hecl : e ∈ M.closure I
heI : e ∉ I
aux : ⋂₀ {J | J ⊆ I ∧ e ∈ M.closure J} ⊆ I
⊢ e ∈ M.closure (⋂₀ {J | J ⊆ I ∧ e ∈ M.closure J})
|
rw [hI.closure_sInter_eq_biInter_closure_of_forall_subset ⟨I, by simpa⟩ (by simp +contextual)]
|
case refine_2
α : Type u_1
M : Matroid α
I : Set α
e : α
hI : M.Indep I
hecl : e ∈ M.closure I
heI : e ∉ I
aux : ⋂₀ {J | J ⊆ I ∧ e ∈ M.closure J} ⊆ I
⊢ e ∈ ⋂ J ∈ {J | J ⊆ I ∧ e ∈ M.closure J}, M.closure J
|
640cc0ceac07193c
|
Finset.small_alternating_pow_of_small_tripling
|
Mathlib/Combinatorics/Additive/SmallTripling.lean
|
/-- If `A` has small tripling, say with constant `K`, then `A` has small alternating powers, in the
sense that `|A^±1 * ... * A^±1|` is at most `|A|` times a constant exponential in the number of
terms in the product.
When `A` is symmetric (`A⁻¹ = A`), the base of the exponential can be lowered from `K ^ 3` to `K`,
where `K` is the tripling constant. See `Finset.small_pow_of_small_tripling`. -/
@[to_additive
"If `A` has small tripling, say with constant `K`, then `A` has small alternating powers, in the
sense that `|±A ± ... ± A|` is at most `|A|` times a constant exponential in the number of
terms in the product.
When `A` is symmetric (`-A = A`), the base of the exponential can be lowered from `K ^ 3` to `K`,
where `K` is the tripling constant. See `Finset.small_nsmul_of_small_tripling`."]
lemma small_alternating_pow_of_small_tripling (hm : 3 ≤ m) (hA : #(A ^ 3) ≤ K * #A) (ε : Fin m → ℤ)
(hε : ∀ i, |ε i| = 1) :
#((finRange m).map fun i ↦ A ^ ε i).prod ≤ K ^ (3 * (m - 2)) * #A
|
case ha
G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : Group G
A : Finset G
K : ℝ
m : ℕ
hm : 3 ≤ m
hA : ↑(#(A ^ 3)) ≤ K * ↑(#A)
ε : Fin m → ℤ
hε : ∀ (i : Fin m), |ε i| = 1
hm₀ : m ≠ 0
hε₀ : ∀ (i : Fin m), ε i ≠ 0
hA₀ : A.Nonempty
hK₁ : 1 ≤ K
δ : Fin 3 → ℤ
hδ : (δ 0 = 1 ∨ δ 0 = -1) ∧ (δ 1 = 1 ∨ δ 1 = -1) ∧ (δ 2 = 1 ∨ δ 2 = -1)
this : K ≤ K ^ 3
⊢ 1 ≤ K
|
exact hK₁
|
no goals
|
647fdb3ae1591fd5
|
FreeSemigroup.length_mul
|
Mathlib/Algebra/Free.lean
|
theorem length_mul (x y : FreeSemigroup α) : (x * y).length = x.length + y.length
|
α : Type u
x y : FreeSemigroup α
⊢ (x * y).length = x.length + y.length
|
simp [length, Nat.add_right_comm, List.length, List.length_append]
|
no goals
|
2bf5ecbd31187c57
|
VitaliFamily.ae_tendsto_lintegral_enorm_sub_div_of_integrable
|
Mathlib/MeasureTheory/Covering/Differentiation.lean
|
theorem ae_tendsto_lintegral_enorm_sub_div_of_integrable {f : α → E} (hf : Integrable f μ) :
∀ᵐ x ∂μ, Tendsto (fun a => (∫⁻ y in a, ‖f y - f x‖ₑ ∂μ) / μ a) (v.filterAt x) (𝓝 0)
|
case e_a.h
α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : SecondCountableTopology α
inst✝¹ : BorelSpace α
inst✝ : IsLocallyFiniteMeasure μ
f : α → E
hf : Integrable f μ
I : Integrable (AEStronglyMeasurable.mk f ⋯) μ
x : α
hx :
Tendsto (fun a => (∫⁻ (y : α) in a, ‖AEStronglyMeasurable.mk f ⋯ y - AEStronglyMeasurable.mk f ⋯ x‖ₑ ∂μ) / μ a)
(v.filterAt x) (𝓝 0)
h'x : f x = AEStronglyMeasurable.mk f ⋯ x
a : Set α
⊢ (fun a => ‖AEStronglyMeasurable.mk f ⋯ a - AEStronglyMeasurable.mk f ⋯ x‖ₑ) =ᶠ[ae (μ.restrict a)] fun a =>
‖f a - f x‖ₑ
|
apply ae_restrict_of_ae
|
case e_a.h.h
α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : SecondCountableTopology α
inst✝¹ : BorelSpace α
inst✝ : IsLocallyFiniteMeasure μ
f : α → E
hf : Integrable f μ
I : Integrable (AEStronglyMeasurable.mk f ⋯) μ
x : α
hx :
Tendsto (fun a => (∫⁻ (y : α) in a, ‖AEStronglyMeasurable.mk f ⋯ y - AEStronglyMeasurable.mk f ⋯ x‖ₑ ∂μ) / μ a)
(v.filterAt x) (𝓝 0)
h'x : f x = AEStronglyMeasurable.mk f ⋯ x
a : Set α
⊢ ∀ᵐ (x_1 : α) ∂μ,
(fun a => ‖AEStronglyMeasurable.mk f ⋯ a - AEStronglyMeasurable.mk f ⋯ x‖ₑ) x_1 = (fun a => ‖f a - f x‖ₑ) x_1
|
2445f889d49b35fc
|
MeasureTheory.hausdorffMeasure_pi_real
|
Mathlib/MeasureTheory/Measure/Hausdorff.lean
|
theorem hausdorffMeasure_pi_real {ι : Type*} [Fintype ι] :
(μH[Fintype.card ι] : Measure (ι → ℝ)) = volume
|
ι : Type u_4
inst✝ : Fintype ι
a b : ι → ℚ
H : ∀ (i : ι), a i < b i
I : ∀ (i : ι), 0 ≤ ↑(b i) - ↑(a i)
γ : ℕ → Type u_4 := fun n => (i : ι) → Fin ⌈(↑(b i) - ↑(a i)) * ↑n⌉₊
t : (n : ℕ) → γ n → Set (ι → ℝ) := fun n f => univ.pi fun i => Icc (↑(a i) + ↑↑(f i) / ↑n) (↑(a i) + (↑↑(f i) + 1) / ↑n)
⊢ Tendsto (fun n => 1 / ↑n) atTop (𝓝 0)
|
simp only [one_div, ENNReal.tendsto_inv_nat_nhds_zero]
|
no goals
|
90212ad5e070d6f0
|
FDRep.forget₂_ρ
|
Mathlib/RepresentationTheory/FDRep.lean
|
theorem forget₂_ρ (V : FDRep R G) : ((forget₂ (FDRep R G) (Rep R G)).obj V).ρ = V.ρ
|
case h.h
R G : Type u
inst✝¹ : CommRing R
inst✝ : Monoid G
V : FDRep R G
g : G
v : ↑((forget₂ (FDRep R G) (Rep R G)).obj V).V
⊢ (((forget₂ (FDRep R G) (Rep R G)).obj V).ρ g) v = (V.ρ g) v
|
rfl
|
no goals
|
78e7451a209bf9e5
|
CategoryTheory.ShortComplex.LeftHomologyData.map_f'
|
Mathlib/Algebra/Homology/ShortComplex/PreservesHomology.lean
|
@[simp]
lemma map_f' : (h.map F).f' = F.map h.f'
|
C : Type u_1
D : Type u_2
inst✝⁵ : Category.{u_4, u_1} C
inst✝⁴ : Category.{u_3, u_2} D
inst✝³ : HasZeroMorphisms C
inst✝² : HasZeroMorphisms D
S : ShortComplex C
h : S.LeftHomologyData
F : C ⥤ D
inst✝¹ : F.PreservesZeroMorphisms
inst✝ : h.IsPreservedBy F
⊢ (h.map F).f' = F.map h.f'
|
rw [← cancel_mono (h.map F).i, f'_i, map_f, map_i, ← F.map_comp, f'_i]
|
no goals
|
7c179b1d073ec12c
|
Set.ncard_eq_three
|
Mathlib/Data/Set/Card.lean
|
theorem ncard_eq_three : s.ncard = 3 ↔ ∃ x y z, x ≠ y ∧ x ≠ z ∧ y ≠ z ∧ s = {x, y, z}
|
α : Type u_1
s : Set α
⊢ s.ncard = 3 ↔ ∃ x y z, x ≠ y ∧ x ≠ z ∧ y ≠ z ∧ s = {x, y, z}
|
rw [← encard_eq_three, ncard_def, ← Nat.cast_inj (R := ℕ∞), Nat.cast_ofNat]
|
α : Type u_1
s : Set α
⊢ ↑s.encard.toNat = 3 ↔ s.encard = 3
|
81e51a925750236b
|
Polynomial.isIntegral_isLocalization_polynomial_quotient
|
Mathlib/RingTheory/Jacobson/Ring.lean
|
theorem isIntegral_isLocalization_polynomial_quotient
(P : Ideal R[X]) (pX : R[X]) (hpX : pX ∈ P) [Algebra (R ⧸ P.comap (C : R →+* R[X])) Rₘ]
[IsLocalization.Away (pX.map (Ideal.Quotient.mk (P.comap (C : R →+* R[X])))).leadingCoeff Rₘ]
[Algebra (R[X] ⧸ P) Sₘ] [IsLocalization ((Submonoid.powers (pX.map (Ideal.Quotient.mk (P.comap
(C : R →+* R[X])))).leadingCoeff).map (quotientMap P C le_rfl) : Submonoid (R[X] ⧸ P)) Sₘ] :
(IsLocalization.map Sₘ (quotientMap P C le_rfl) (Submonoid.powers (pX.map (Ideal.Quotient.mk
(P.comap (C : R →+* R[X])))).leadingCoeff).le_comap_map : Rₘ →+* Sₘ).IsIntegral
|
R : Type u_1
inst✝⁶ : CommRing R
Rₘ : Type u_3
Sₘ : Type u_4
inst✝⁵ : CommRing Rₘ
inst✝⁴ : CommRing Sₘ
P : Ideal R[X]
pX : R[X]
hpX : pX ∈ P
inst✝³ : Algebra (R ⧸ comap C P) Rₘ
inst✝² : IsLocalization.Away (map (Ideal.Quotient.mk (comap C P)) pX).leadingCoeff Rₘ
inst✝¹ : Algebra (R[X] ⧸ P) Sₘ
inst✝ :
IsLocalization
(Submonoid.map (quotientMap P C ⋯) (Submonoid.powers (map (Ideal.Quotient.mk (comap C P)) pX).leadingCoeff)) Sₘ
P' : Ideal R := comap C P
M : Submonoid (R ⧸ P') := Submonoid.powers (map (Ideal.Quotient.mk (comap C P)) pX).leadingCoeff
⊢ (IsLocalization.map Sₘ (quotientMap P C ⋯) ⋯).IsIntegral
|
let M' : Submonoid (R[X] ⧸ P) :=
(Submonoid.powers (pX.map (Ideal.Quotient.mk (P.comap (C : R →+* R[X])))).leadingCoeff).map
(quotientMap P C le_rfl)
|
R : Type u_1
inst✝⁶ : CommRing R
Rₘ : Type u_3
Sₘ : Type u_4
inst✝⁵ : CommRing Rₘ
inst✝⁴ : CommRing Sₘ
P : Ideal R[X]
pX : R[X]
hpX : pX ∈ P
inst✝³ : Algebra (R ⧸ comap C P) Rₘ
inst✝² : IsLocalization.Away (map (Ideal.Quotient.mk (comap C P)) pX).leadingCoeff Rₘ
inst✝¹ : Algebra (R[X] ⧸ P) Sₘ
inst✝ :
IsLocalization
(Submonoid.map (quotientMap P C ⋯) (Submonoid.powers (map (Ideal.Quotient.mk (comap C P)) pX).leadingCoeff)) Sₘ
P' : Ideal R := comap C P
M : Submonoid (R ⧸ P') := Submonoid.powers (map (Ideal.Quotient.mk (comap C P)) pX).leadingCoeff
M' : Submonoid (R[X] ⧸ P) :=
Submonoid.map (quotientMap P C ⋯) (Submonoid.powers (map (Ideal.Quotient.mk (comap C P)) pX).leadingCoeff)
⊢ (IsLocalization.map Sₘ (quotientMap P C ⋯) ⋯).IsIntegral
|
4f255f198e9b3f92
|
le_nhds_iff
|
Mathlib/Topology/Basic.lean
|
theorem le_nhds_iff {f} : f ≤ 𝓝 x ↔ ∀ s : Set X, x ∈ s → IsOpen s → s ∈ f
|
X : Type u
x : X
inst✝ : TopologicalSpace X
f : Filter X
⊢ f ≤ 𝓝 x ↔ ∀ (s : Set X), x ∈ s → IsOpen s → s ∈ f
|
simp [nhds_def]
|
no goals
|
5415bef2b050be5e
|
AffineSubspace.direction_sup
|
Mathlib/LinearAlgebra/AffineSpace/AffineSubspace/Basic.lean
|
theorem direction_sup {s₁ s₂ : AffineSubspace k P} {p₁ p₂ : P} (hp₁ : p₁ ∈ s₁) (hp₂ : p₂ ∈ s₂) :
(s₁ ⊔ s₂).direction = s₁.direction ⊔ s₂.direction ⊔ k ∙ p₂ -ᵥ p₁
|
case refine_1.intro.intro.inr
k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : Ring k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
s₁ s₂ : AffineSubspace k P
p₁ p₂ : P
hp₁ : p₁ ∈ ↑s₁
hp₂ : p₂ ∈ s₂
p₃ : P
hp₃ : p₃ ∈ ↑s₂
⊢ ∃ y ∈ s₁.direction, ∃ z ∈ s₂.direction ⊔ Submodule.span k {p₂ -ᵥ p₁}, y + z = (fun x => x -ᵥ p₁) p₃
|
use 0, Submodule.zero_mem _, p₃ -ᵥ p₁
|
case h
k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : Ring k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
s₁ s₂ : AffineSubspace k P
p₁ p₂ : P
hp₁ : p₁ ∈ ↑s₁
hp₂ : p₂ ∈ s₂
p₃ : P
hp₃ : p₃ ∈ ↑s₂
⊢ p₃ -ᵥ p₁ ∈ s₂.direction ⊔ Submodule.span k {p₂ -ᵥ p₁} ∧ 0 + (p₃ -ᵥ p₁) = (fun x => x -ᵥ p₁) p₃
|
64c047987a3af743
|
MeasureTheory.OuterMeasure.ofFunction_eq_iInf_mem
|
Mathlib/MeasureTheory/OuterMeasure/OfFunction.lean
|
theorem ofFunction_eq_iInf_mem {P : Set α → Prop} (m_top : ∀ s, ¬ P s → m s = ∞) (s : Set α) :
OuterMeasure.ofFunction m m_empty s =
⨅ (t : ℕ → Set α) (_ : ∀ i, P (t i)) (_ : s ⊆ ⋃ i, t i), ∑' i, m (t i)
|
case neg
α : Type u_1
m : Set α → ℝ≥0∞
m_empty : m ∅ = 0
P : Set α → Prop
m_top : ∀ (s : Set α), ¬P s → m s = ⊤
s : Set α
t : ℕ → Set α
ht_subset : s ⊆ iUnion t
ht : ¬∀ (i : ℕ), P (t i)
⊢ ⨅ (_ : ∀ (i : ℕ), P (t i)), ⨅ (_ : s ⊆ ⋃ i, t i), ∑' (i : ℕ), m (t i) ≤ ∑' (n : ℕ), m (t n)
|
simp only [ht, not_false_eq_true, iInf_neg, top_le_iff]
|
case neg
α : Type u_1
m : Set α → ℝ≥0∞
m_empty : m ∅ = 0
P : Set α → Prop
m_top : ∀ (s : Set α), ¬P s → m s = ⊤
s : Set α
t : ℕ → Set α
ht_subset : s ⊆ iUnion t
ht : ¬∀ (i : ℕ), P (t i)
⊢ ∑' (i : ℕ), m (t i) = ⊤
|
6008c3e6e3e7ffd5
|
Finset.smul_univ₀'
|
Mathlib/Algebra/GroupWithZero/Pointwise/Finset.lean
|
lemma smul_univ₀' [Fintype β] {s : Finset α} (hs : s.Nontrivial) : s • (univ : Finset β) = univ :=
coe_injective <| by push_cast; exact Set.smul_univ₀' hs
|
α : Type u_1
β : Type u_2
inst✝³ : DecidableEq β
inst✝² : GroupWithZero α
inst✝¹ : MulAction α β
inst✝ : Fintype β
s : Finset α
hs : s.Nontrivial
⊢ ↑(s • univ) = ↑univ
|
push_cast
|
α : Type u_1
β : Type u_2
inst✝³ : DecidableEq β
inst✝² : GroupWithZero α
inst✝¹ : MulAction α β
inst✝ : Fintype β
s : Finset α
hs : s.Nontrivial
⊢ ↑s • Set.univ = Set.univ
|
361a54f04adc6b20
|
Rat.divInt_nonneg
|
Mathlib/Algebra/Order/Ring/Unbundled/Rat.lean
|
@[simp] lemma divInt_nonneg {a b : ℤ} (ha : 0 ≤ a) (hb : 0 ≤ b) : 0 ≤ a /. b
|
case inr
a b : ℤ
ha : 0 ≤ a
hb✝ : 0 ≤ b
hb : 0 < b
⊢ 0 ≤ a /. b
|
rwa [divInt_nonneg_iff_of_pos_right hb]
|
no goals
|
933133477f35d8d8
|
continuousAt_gauge
|
Mathlib/Analysis/Convex/Gauge.lean
|
theorem continuousAt_gauge (hc : Convex ℝ s) (hs₀ : s ∈ 𝓝 0) : ContinuousAt (gauge s) x
|
case h.left
E : Type u_2
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
s : Set E
x : E
inst✝² : TopologicalSpace E
inst✝¹ : IsTopologicalAddGroup E
inst✝ : ContinuousSMul ℝ E
hc : Convex ℝ s
hs₀ : s ∈ 𝓝 0
ha : Absorbent ℝ s
ε : ℝ
hε₀ : 0 < ε
this : ε • s ∩ -(ε • s) ∈ 𝓝 0
y : E
hy : y ∈ ε • s ∩ -(ε • s)
⊢ gauge s x ≤ gauge s (x + y) + ε
|
calc
gauge s x = gauge s (x + y + (-y)) := by simp
_ ≤ gauge s (x + y) + gauge s (-y) := gauge_add_le hc ha _ _
_ ≤ gauge s (x + y) + ε := add_le_add_left (gauge_le_of_mem hε₀.le (mem_neg.1 hy.2)) _
|
no goals
|
7f81bfcfb910ec17
|
Matrix.pow_inv_comm'
|
Mathlib/LinearAlgebra/Matrix/ZPow.lean
|
theorem pow_inv_comm' (A : M) (m n : ℕ) : A⁻¹ ^ m * A ^ n = A ^ n * A⁻¹ ^ m
|
case succ.zero
n' : Type u_1
inst✝² : DecidableEq n'
inst✝¹ : Fintype n'
R : Type u_2
inst✝ : CommRing R
A : M
n : ℕ
IH : ∀ (m : ℕ), A⁻¹ ^ m * A ^ n = A ^ n * A⁻¹ ^ m
⊢ A⁻¹ ^ 0 * A ^ (n + 1) = A ^ (n + 1) * A⁻¹ ^ 0
|
simp
|
no goals
|
1a086ad17decf402
|
Complex.differentiableAt_GammaAux
|
Mathlib/Analysis/SpecialFunctions/Gamma/Deriv.lean
|
theorem differentiableAt_GammaAux (s : ℂ) (n : ℕ) (h1 : 1 - s.re < n) (h2 : ∀ m : ℕ, s ≠ -m) :
DifferentiableAt ℂ (GammaAux n) s
|
n : ℕ
s : ℂ
h1 : 1 - s.re < ↑n + 1
h2 : ∀ (m : ℕ), s ≠ -↑m
hn : 1 - (s + 1).re < ↑n → (∀ (m : ℕ), s + 1 ≠ -↑m) → DifferentiableAt ℂ (GammaAux n) (s + 1)
⊢ 1 - (s.re + 1) < ↑n
|
linarith
|
no goals
|
943fb7e67daeabdc
|
FractionalIdeal.eq_zero_or_one
|
Mathlib/RingTheory/FractionalIdeal/Operations.lean
|
theorem eq_zero_or_one (I : FractionalIdeal K⁰ L) : I = 0 ∨ I = 1
|
case mp
K : Type u_4
L : Type u_5
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsFractionRing K L
I : FractionalIdeal K⁰ L
hI : ¬I = 0
x : L
x_mem : x ∈ I
⊢ ∃ x', (algebraMap K L) x' = x
|
obtain ⟨n, d, rfl⟩ := IsLocalization.mk'_surjective K⁰ x
|
case mp.intro.intro
K : Type u_4
L : Type u_5
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsFractionRing K L
I : FractionalIdeal K⁰ L
hI : ¬I = 0
n : K
d : ↥K⁰
x_mem : mk' L n d ∈ I
⊢ ∃ x', (algebraMap K L) x' = mk' L n d
|
a46b0893f2e92a27
|
MvPolynomial.IsHomogeneous.coeff_isHomogeneous_of_optionEquivLeft_symm
|
Mathlib/RingTheory/MvPolynomial/Homogeneous.lean
|
lemma coeff_isHomogeneous_of_optionEquivLeft_symm
[hσ : Finite σ] {p : Polynomial (MvPolynomial σ R)}
(hp : ((optionEquivLeft R σ).symm p).IsHomogeneous n) (i j : ℕ) (h : i + j = n) :
(p.coeff i).IsHomogeneous j
|
case h.e'_4
σ : Type u_1
R : Type u_3
inst✝ : CommSemiring R
n : ℕ
hσ : Finite σ
p : Polynomial (MvPolynomial σ R)
hp : ((optionEquivLeft R σ).symm p).IsHomogeneous n
i j : ℕ
h : i + j = n
k : ℕ
e : σ ≃ Fin k
e' : Option σ ≃ Fin (k + 1) := e.optionCongr.trans (_root_.finSuccEquiv k).symm
F : MvPolynomial σ R ≃ₐ[R] MvPolynomial (Fin k) R := renameEquiv R e
F' : MvPolynomial (Option σ) R ≃ₐ[R] MvPolynomial (Fin (k + 1)) R := renameEquiv R e'
φ : MvPolynomial (Fin (k + 1)) R := F' ((optionEquivLeft R σ).symm p)
hφ : φ.IsHomogeneous n
⊢ (rename ⇑e) (p.coeff i) = ((finSuccEquiv R k) ((rename ⇑e') ((optionEquivLeft R σ).symm p))).coeff i
|
rw [finSuccEquiv_rename_finSuccEquiv, AlgEquiv.apply_symm_apply]
|
case h.e'_4
σ : Type u_1
R : Type u_3
inst✝ : CommSemiring R
n : ℕ
hσ : Finite σ
p : Polynomial (MvPolynomial σ R)
hp : ((optionEquivLeft R σ).symm p).IsHomogeneous n
i j : ℕ
h : i + j = n
k : ℕ
e : σ ≃ Fin k
e' : Option σ ≃ Fin (k + 1) := e.optionCongr.trans (_root_.finSuccEquiv k).symm
F : MvPolynomial σ R ≃ₐ[R] MvPolynomial (Fin k) R := renameEquiv R e
F' : MvPolynomial (Option σ) R ≃ₐ[R] MvPolynomial (Fin (k + 1)) R := renameEquiv R e'
φ : MvPolynomial (Fin (k + 1)) R := F' ((optionEquivLeft R σ).symm p)
hφ : φ.IsHomogeneous n
⊢ (rename ⇑e) (p.coeff i) = (Polynomial.map (rename ⇑e).toRingHom p).coeff i
|
5e0fbe3d40322b86
|
IsSeparable.of_algebra_isSeparable_of_isSeparable
|
Mathlib/FieldTheory/SeparableDegree.lean
|
theorem IsSeparable.of_algebra_isSeparable_of_isSeparable [Algebra E K] [IsScalarTower F E K]
[Algebra.IsSeparable F E] {x : K} (hsep : IsSeparable E x) : IsSeparable F x
|
F : Type u
E : Type v
inst✝⁷ : Field F
inst✝⁶ : Field E
inst✝⁵ : Algebra F E
K : Type w
inst✝⁴ : Field K
inst✝³ : Algebra F K
inst✝² : Algebra E K
inst✝¹ : IsScalarTower F E K
inst✝ : Algebra.IsSeparable F E
x : K
hsep : IsSeparable E x
f : E[X] := minpoly E x
hf : f = minpoly E x
E' : IntermediateField F E := adjoin F ↑f.coeffs
this : FiniteDimensional F ↥E'
⊢ IsSeparable F x
|
let g : E'[X] := f.toSubring E'.toSubring (subset_adjoin F _)
|
F : Type u
E : Type v
inst✝⁷ : Field F
inst✝⁶ : Field E
inst✝⁵ : Algebra F E
K : Type w
inst✝⁴ : Field K
inst✝³ : Algebra F K
inst✝² : Algebra E K
inst✝¹ : IsScalarTower F E K
inst✝ : Algebra.IsSeparable F E
x : K
hsep : IsSeparable E x
f : E[X] := minpoly E x
hf : f = minpoly E x
E' : IntermediateField F E := adjoin F ↑f.coeffs
this : FiniteDimensional F ↥E'
g : (↥E')[X] := f.toSubring E'.toSubring ⋯
⊢ IsSeparable F x
|
6901ddfd7b743413
|
MvPFunctor.comp_wPathCasesOn
|
Mathlib/Data/PFunctor/Multivariate/W.lean
|
theorem comp_wPathCasesOn {α β : TypeVec n} (h : α ⟹ β) {a : P.A} {f : P.last.B a → P.last.W}
(g' : P.drop.B a ⟹ α) (g : ∀ j : P.last.B a, P.WPath (f j) ⟹ α) :
h ⊚ P.wPathCasesOn g' g = P.wPathCasesOn (h ⊚ g') fun i => h ⊚ g i
|
case a.h
n : ℕ
P : MvPFunctor.{u} (n + 1)
α : TypeVec.{u_1} n
β : TypeVec.{u_2} n
h : α ⟹ β
a : P.A
f : P.last.B a → P.last.W
g' : P.drop.B a ⟹ α
g : (j : P.last.B a) → P.WPath (f j) ⟹ α
i : Fin2 n
x : P.WPath (WType.mk a f) i
⊢ (h ⊚ P.wPathCasesOn g' g) i x = P.wPathCasesOn (h ⊚ g') (fun i => h ⊚ g i) i x
|
cases x <;> rfl
|
no goals
|
c6ed33f023d791f0
|
List.unzip_toArray
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean
|
theorem unzip_toArray (as : List (α × β)) :
as.toArray.unzip = Prod.map List.toArray List.toArray as.unzip
|
α : Type u_1
β : Type u_2
as : List (α × β)
⊢ as.toArray.unzip = Prod.map toArray toArray as.unzip
|
ext1 <;> simp
|
no goals
|
8bd3794a6a3f4550
|
SimpleGraph.iUnion_connectedComponentSupp
|
Mathlib/Combinatorics/SimpleGraph/Path.lean
|
lemma iUnion_connectedComponentSupp (G : SimpleGraph V) :
⋃ c : G.ConnectedComponent, c.supp = Set.univ
|
V : Type u
G : SimpleGraph V
v : V
⊢ ∃ y, y.supp = ↑(G.connectedComponentMk v)
|
use G.connectedComponentMk v
|
case h
V : Type u
G : SimpleGraph V
v : V
⊢ (G.connectedComponentMk v).supp = ↑(G.connectedComponentMk v)
|
3a9ae24cb75c2aa7
|
Array.setIfInBounds_setIfInBounds
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean
|
theorem setIfInBounds_setIfInBounds (a b : α) (as : Array α) (i : Nat) :
(as.setIfInBounds i a).setIfInBounds i b = as.setIfInBounds i b
|
case mk
α : Type u_1
a b : α
i : Nat
toList✝ : List α
⊢ ({ toList := toList✝ }.setIfInBounds i a).setIfInBounds i b = { toList := toList✝ }.setIfInBounds i b
|
simp
|
no goals
|
ec9965e281a40a9f
|
Real.two_mul_arctan_add_pi
|
Mathlib/Analysis/SpecialFunctions/Trigonometric/Arctan.lean
|
theorem two_mul_arctan_add_pi {x : ℝ} (h : 1 < x) :
2 * arctan x = arctan (2 * x / (1 - x ^ 2)) + π
|
case e_a.e_x
x : ℝ
h : 1 < x
⊢ (x + x) / (1 - x * x) = 2 * x / (1 - x ^ 2)
|
ring
|
no goals
|
d1c8265a24878c41
|
Group.card_center_add_sum_card_noncenter_eq_card
|
Mathlib/GroupTheory/ClassEquation.lean
|
theorem Group.card_center_add_sum_card_noncenter_eq_card (G) [Group G]
[∀ x : ConjClasses G, Fintype x.carrier] [Fintype G] [Fintype <| Subgroup.center G]
[Fintype <| noncenter G] : Fintype.card (Subgroup.center G) +
∑ x ∈ (noncenter G).toFinset, x.carrier.toFinset.card = Fintype.card G
|
case h.e'_3
G : Type u_2
inst✝⁴ : Group G
inst✝³ : (x : ConjClasses G) → Fintype ↑x.carrier
inst✝² : Fintype G
inst✝¹ : Fintype ↥(Subgroup.center G)
inst✝ : Fintype ↑(noncenter G)
⊢ Fintype.card G = Nat.card G
|
simp
|
no goals
|
e8d5dab4b658ff9d
|
blimsup_cthickening_mul_ae_eq
|
Mathlib/MeasureTheory/Covering/LiminfLimsup.lean
|
theorem blimsup_cthickening_mul_ae_eq (p : ℕ → Prop) (s : ℕ → Set α) {M : ℝ} (hM : 0 < M)
(r : ℕ → ℝ) (hr : Tendsto r atTop (𝓝 0)) :
(blimsup (fun i => cthickening (M * r i) (s i)) atTop p : Set α) =ᵐ[μ]
(blimsup (fun i => cthickening (r i) (s i)) atTop p : Set α)
|
α : Type u_1
inst✝⁵ : PseudoMetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
s : ℕ → Set α
M : ℝ
hM : 0 < M
p : ℕ → Prop
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝[>] 0)
⊢ Tendsto (fun i => M * r i) atTop (𝓝[>] 0)
|
convert TendstoNhdsWithinIoi.const_mul hM hr <;> simp only [mul_zero]
|
no goals
|
8749d72cf710712d
|
Stream'.Seq.map_cons
|
Mathlib/Data/Seq/Seq.lean
|
theorem map_cons (f : α → β) (a) : ∀ s, map f (cons a s) = cons (f a) (map f s)
| ⟨s, al⟩ => by apply Subtype.eq; dsimp [cons, map]; rw [Stream'.map_cons]; rfl
|
case a
α : Type u
β : Type v
f : α → β
a : α
s : Stream' (Option α)
al : s.IsSeq
⊢ Option.map f (some a) :: Stream'.map (Option.map f) s = some (f a) :: Stream'.map (Option.map f) s
|
rfl
|
no goals
|
53104f98914cff40
|
ENNReal.toReal_eq_toReal
|
Mathlib/Data/ENNReal/Real.lean
|
theorem toReal_eq_toReal (ha : a ≠ ∞) (hb : b ≠ ∞) : a.toReal = b.toReal ↔ a = b
|
case intro.intro
a b : ℝ≥0
⊢ (↑a).toReal = (↑b).toReal ↔ ↑a = ↑b
|
simp only [coe_inj, NNReal.coe_inj, coe_toReal]
|
no goals
|
30f5aebaa93cabcd
|
MeasureTheory.aemeasurable_of_exist_almost_disjoint_supersets
|
Mathlib/MeasureTheory/Function/AEMeasurableOrder.lean
|
theorem MeasureTheory.aemeasurable_of_exist_almost_disjoint_supersets {α : Type*}
{m : MeasurableSpace α} (μ : Measure α) {β : Type*} [CompleteLinearOrder β] [DenselyOrdered β]
[TopologicalSpace β] [OrderTopology β] [SecondCountableTopology β] [MeasurableSpace β]
[BorelSpace β] (s : Set β) (s_count : s.Countable) (s_dense : Dense s) (f : α → β)
(h : ∀ p ∈ s, ∀ q ∈ s, p < q → ∃ u v, MeasurableSet u ∧ MeasurableSet v ∧
{ x | f x < p } ⊆ u ∧ { x | q < f x } ⊆ v ∧ μ (u ∩ v) = 0) :
AEMeasurable f μ
|
α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_2
inst✝⁶ : CompleteLinearOrder β
inst✝⁵ : DenselyOrdered β
inst✝⁴ : TopologicalSpace β
inst✝³ : OrderTopology β
inst✝² : SecondCountableTopology β
inst✝¹ : MeasurableSpace β
inst✝ : BorelSpace β
s : Set β
s_count : s.Countable
s_dense : Dense s
f : α → β
h :
∀ p ∈ s,
∀ q ∈ s, p < q → ∃ u v, MeasurableSet u ∧ MeasurableSet v ∧ {x | f x < p} ⊆ u ∧ {x | q < f x} ⊆ v ∧ μ (u ∩ v) = 0
this : Encodable ↑s
u v : β → β → Set α
huv :
∀ (p q : β),
MeasurableSet (u p q) ∧
MeasurableSet (v p q) ∧
{x | f x < p} ⊆ u p q ∧ {x | q < f x} ⊆ v p q ∧ (p ∈ s → q ∈ s → p < q → μ (u p q ∩ v p q) = 0)
u' : β → Set α := fun p => ⋂ q ∈ s ∩ Ioi p, u p q
u'_meas : ∀ (i : β), MeasurableSet (u' i)
f' : α → β := fun x => ⨅ i, (u' ↑i).piecewise (fun x => ↑i) (fun x => ⊤) x
⊢ AEMeasurable f μ
|
have f'_meas : Measurable f' := by fun_prop (disch := aesop)
|
α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_2
inst✝⁶ : CompleteLinearOrder β
inst✝⁵ : DenselyOrdered β
inst✝⁴ : TopologicalSpace β
inst✝³ : OrderTopology β
inst✝² : SecondCountableTopology β
inst✝¹ : MeasurableSpace β
inst✝ : BorelSpace β
s : Set β
s_count : s.Countable
s_dense : Dense s
f : α → β
h :
∀ p ∈ s,
∀ q ∈ s, p < q → ∃ u v, MeasurableSet u ∧ MeasurableSet v ∧ {x | f x < p} ⊆ u ∧ {x | q < f x} ⊆ v ∧ μ (u ∩ v) = 0
this : Encodable ↑s
u v : β → β → Set α
huv :
∀ (p q : β),
MeasurableSet (u p q) ∧
MeasurableSet (v p q) ∧
{x | f x < p} ⊆ u p q ∧ {x | q < f x} ⊆ v p q ∧ (p ∈ s → q ∈ s → p < q → μ (u p q ∩ v p q) = 0)
u' : β → Set α := fun p => ⋂ q ∈ s ∩ Ioi p, u p q
u'_meas : ∀ (i : β), MeasurableSet (u' i)
f' : α → β := fun x => ⨅ i, (u' ↑i).piecewise (fun x => ↑i) (fun x => ⊤) x
f'_meas : Measurable f'
⊢ AEMeasurable f μ
|
6b0c36117865acf2
|
isClopen_iff_frontier_eq_empty
|
Mathlib/Topology/Clopen.lean
|
theorem isClopen_iff_frontier_eq_empty : IsClopen s ↔ frontier s = ∅
|
X : Type u
inst✝ : TopologicalSpace X
s : Set X
h : closure s ⊆ interior s
⊢ closure s = s ∧ interior s = s
|
exact ⟨(h.trans interior_subset).antisymm subset_closure,
interior_subset.antisymm (subset_closure.trans h)⟩
|
no goals
|
fd204de967800dba
|
Polynomial.mul_scaleRoots
|
Mathlib/RingTheory/Polynomial/ScaleRoots.lean
|
/-- Multiplication and `scaleRoots` commute up to a power of `r`. The factor disappears if we
assume that the product of the leading coeffs does not vanish. See `Polynomial.mul_scaleRoots'`. -/
lemma mul_scaleRoots (p q : R[X]) (r : R) :
r ^ (natDegree p + natDegree q - natDegree (p * q)) • (p * q).scaleRoots r =
p.scaleRoots r * q.scaleRoots r
|
case inr.inr
R : Type u_1
inst✝ : CommSemiring R
p q : R[X]
r : R
n a b : ℕ
e : a + b = n
ha : a ≤ p.natDegree
hb : b ≤ q.natDegree
⊢ p.coeff a * q.coeff b * r ^ (p.natDegree + q.natDegree - n) =
p.coeff a * r ^ (p.natDegree - a) * (q.coeff b * r ^ (q.natDegree - b))
|
simp only [← e, mul_assoc, mul_comm (r ^ (_ - a)), ← pow_add]
|
case inr.inr
R : Type u_1
inst✝ : CommSemiring R
p q : R[X]
r : R
n a b : ℕ
e : a + b = n
ha : a ≤ p.natDegree
hb : b ≤ q.natDegree
⊢ p.coeff a * (q.coeff b * r ^ (p.natDegree + q.natDegree - (a + b))) =
p.coeff a * (q.coeff b * r ^ (q.natDegree - b + (p.natDegree - a)))
|
ac865a3551bb16a7
|
CochainComplex.HomComplex.Cochain.rightShift_leftShift
|
Mathlib/Algebra/Homology/HomotopyCategory/HomComplexShift.lean
|
lemma rightShift_leftShift (a n' : ℤ) (hn' : n + a = n') :
(γ.leftShift a n' hn').rightShift a n hn' =
(a * n' + (a * (a - 1)) / 2).negOnePow • γ.shift a
|
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
K L : CochainComplex C ℤ
n : ℤ
γ : Cochain K L n
a n' : ℤ
hn' : n + a = n'
⊢ (γ.leftShift a n' hn').rightShift a n hn' = (a * n' + a * (a - 1) / 2).negOnePow • γ.shift a
|
ext p q hpq
|
case h
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
K L : CochainComplex C ℤ
n : ℤ
γ : Cochain K L n
a n' : ℤ
hn' : n + a = n'
p q : ℤ
hpq : p + n = q
⊢ ((γ.leftShift a n' hn').rightShift a n hn').v p q hpq = ((a * n' + a * (a - 1) / 2).negOnePow • γ.shift a).v p q hpq
|
97919e60b1c652b4
|
Std.Tactic.BVDecide.BVPred.denote_bitblast
|
Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Lemmas/Pred.lean
|
theorem denote_bitblast (aig : AIG BVBit) (pred : BVPred) (assign : BVExpr.Assignment) :
⟦bitblast aig pred, assign.toAIGAssignment⟧ = pred.eval assign
|
case bin.eq.hleft
aig : AIG BVBit
assign : BVExpr.Assignment
w✝ : Nat
lhs rhs : BVExpr w✝
idx✝ : Nat
hidx✝ : idx✝ < w✝
⊢ ({ lhs := (BVExpr.bitblast aig lhs).vec.cast ⋯,
rhs := (BVExpr.bitblast (BVExpr.bitblast aig lhs).aig rhs).vec }.lhs.get
idx✝ hidx✝).gate <
(BVExpr.bitblast aig lhs).aig.decls.size
|
simp [Ref.hgate]
|
no goals
|
e5d3f10f6565948a
|
Std.Tactic.BVDecide.Normalize.BitVec.add_const_right
|
Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Normalize/BitVec.lean
|
theorem BitVec.add_const_right (a b c : BitVec w) : a + (b + c) = (a + c) + b
|
w : Nat
a b c : BitVec w
⊢ a + (b + c) = a + c + b
|
ac_rfl
|
no goals
|
8a66f00a112967f4
|
Nat.testBit_two_pow_add_gt
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Nat/Bitwise/Lemmas.lean
|
theorem testBit_two_pow_add_gt {i j : Nat} (j_lt_i : j < i) (x : Nat) :
testBit (2^i + x) j = testBit x j
|
i j : Nat
j_lt_i : j < i
x : Nat
i_def : i = j + (i - j)
i_sub_j_eq : i - j = 0
⊢ decide ((2 ^ 0 + x / 2 ^ j) % 2 = 1) = decide (x / 2 ^ j % 2 = 1)
|
exfalso
|
i j : Nat
j_lt_i : j < i
x : Nat
i_def : i = j + (i - j)
i_sub_j_eq : i - j = 0
⊢ False
|
e0a351492794664a
|
Polynomial.natTrailingDegree_le_natDegree
|
Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean
|
theorem natTrailingDegree_le_natDegree (p : R[X]) : p.natTrailingDegree ≤ p.natDegree
|
R : Type u
inst✝ : Semiring R
p : R[X]
⊢ p.natTrailingDegree ≤ p.natDegree
|
by_cases hp : p = 0
|
case pos
R : Type u
inst✝ : Semiring R
p : R[X]
hp : p = 0
⊢ p.natTrailingDegree ≤ p.natDegree
case neg
R : Type u
inst✝ : Semiring R
p : R[X]
hp : ¬p = 0
⊢ p.natTrailingDegree ≤ p.natDegree
|
4e2cdbf8a94a668c
|
Metric.hausdorffDist_image
|
Mathlib/Topology/MetricSpace/HausdorffDistance.lean
|
theorem hausdorffDist_image (h : Isometry Φ) :
hausdorffDist (Φ '' s) (Φ '' t) = hausdorffDist s t
|
α : Type u
β : Type v
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
s t : Set α
Φ : α → β
h : Isometry Φ
⊢ hausdorffDist (Φ '' s) (Φ '' t) = hausdorffDist s t
|
simp [hausdorffDist, hausdorffEdist_image h]
|
no goals
|
80d7381c2de4781b
|
AffineBasis.coord_apply_combination_of_mem
|
Mathlib/LinearAlgebra/AffineSpace/Basis.lean
|
theorem coord_apply_combination_of_mem (hi : i ∈ s) {w : ι → k} (hw : s.sum w = 1) :
b.coord i (s.affineCombination k b w) = w i
|
ι : Type u_1
k : Type u_5
V : Type u_6
P : Type u_7
inst✝³ : AddCommGroup V
inst✝² : AffineSpace V P
inst✝¹ : Ring k
inst✝ : Module k V
b : AffineBasis ι k P
s : Finset ι
i : ι
hi : i ∈ s
w : ι → k
hw : s.sum w = 1
⊢ (b.coord i) ((Finset.affineCombination k s ⇑b) w) = w i
|
simp only [coord_apply, hi, Finset.affineCombination_eq_linear_combination, if_true,
mul_boole, hw, Function.comp_apply, smul_eq_mul, s.sum_ite_eq,
s.map_affineCombination b w hw]
|
no goals
|
3f38e0252350b813
|
nhds_basis_clopen
|
Mathlib/Topology/Separation/Profinite.lean
|
theorem nhds_basis_clopen (x : X) : (𝓝 x).HasBasis (fun s : Set X => x ∈ s ∧ IsClopen s) id :=
⟨fun U => by
constructor
· have hx : connectedComponent x = {x} :=
totallyDisconnectedSpace_iff_connectedComponent_singleton.mp ‹_› x
rw [connectedComponent_eq_iInter_isClopen] at hx
intro hU
let N := { s // IsClopen s ∧ x ∈ s }
rsuffices ⟨⟨s, hs, hs'⟩, hs''⟩ : ∃ s : N, s.val ⊆ U
· exact ⟨s, ⟨hs', hs⟩, hs''⟩
haveI : Nonempty N := ⟨⟨univ, isClopen_univ, mem_univ x⟩⟩
have hNcl : ∀ s : N, IsClosed s.val := fun s => s.property.1.1
have hdir : Directed Superset fun s : N => s.val
|
X : Type u_1
inst✝³ : TopologicalSpace X
inst✝² : T2Space X
inst✝¹ : CompactSpace X
inst✝ : TotallyDisconnectedSpace X
x : X
U : Set X
hx : ⋂ s, ↑s = {x}
hU : U ∈ 𝓝 x
N : Type (max 0 u_1) := { s // IsClopen s ∧ x ∈ s }
this : Nonempty N
hNcl : ∀ (s : N), IsClosed ↑s
hdir : Directed Superset fun s => ↑s
⊢ ∃ s, ↑s ⊆ U
|
have h_nhd : ∀ y ∈ ⋂ s : N, s.val, U ∈ 𝓝 y := fun y y_in => by
rw [hx, mem_singleton_iff] at y_in
rwa [y_in]
|
X : Type u_1
inst✝³ : TopologicalSpace X
inst✝² : T2Space X
inst✝¹ : CompactSpace X
inst✝ : TotallyDisconnectedSpace X
x : X
U : Set X
hx : ⋂ s, ↑s = {x}
hU : U ∈ 𝓝 x
N : Type (max 0 u_1) := { s // IsClopen s ∧ x ∈ s }
this : Nonempty N
hNcl : ∀ (s : N), IsClosed ↑s
hdir : Directed Superset fun s => ↑s
h_nhd : ∀ y ∈ ⋂ s, ↑s, U ∈ 𝓝 y
⊢ ∃ s, ↑s ⊆ U
|
9480ae471dafed2e
|
affineCombination_mem_affineSpan
|
Mathlib/LinearAlgebra/AffineSpace/Combination.lean
|
theorem affineCombination_mem_affineSpan [Nontrivial k] {s : Finset ι} {w : ι → k}
(h : ∑ i ∈ s, w i = 1) (p : ι → P) :
s.affineCombination k p w ∈ affineSpan k (Set.range p)
|
ι : Type u_1
k : Type u_2
V : Type u_3
P : Type u_4
inst✝⁴ : Ring k
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AffineSpace V P
inst✝ : Nontrivial k
s : Finset ι
w : ι → k
h : ∑ i ∈ s, w i = 1
p : ι → P
⊢ (Finset.affineCombination k s p) w ∈ affineSpan k (Set.range p)
|
classical
have hnz : ∑ i ∈ s, w i ≠ 0 := h.symm ▸ one_ne_zero
have hn : s.Nonempty := Finset.nonempty_of_sum_ne_zero hnz
obtain ⟨i1, hi1⟩ := hn
let w1 : ι → k := Function.update (Function.const ι 0) i1 1
have hw1 : ∑ i ∈ s, w1 i = 1 := by
simp only [w1, Function.const_zero, Finset.sum_update_of_mem hi1, Pi.zero_apply,
Finset.sum_const_zero, add_zero]
have hw1s : s.affineCombination k p w1 = p i1 :=
s.affineCombination_of_eq_one_of_eq_zero w1 p hi1 (Function.update_self ..) fun _ _ hne =>
Function.update_of_ne hne ..
have hv : s.affineCombination k p w -ᵥ p i1 ∈ (affineSpan k (Set.range p)).direction := by
rw [direction_affineSpan, ← hw1s, Finset.affineCombination_vsub]
apply weightedVSub_mem_vectorSpan
simp [Pi.sub_apply, h, hw1]
rw [← vsub_vadd (s.affineCombination k p w) (p i1)]
exact AffineSubspace.vadd_mem_of_mem_direction hv (mem_affineSpan k (Set.mem_range_self _))
|
no goals
|
b0078337a6cd7107
|
ZMod.cast_one
|
Mathlib/Data/ZMod/Basic.lean
|
theorem cast_one (h : m ∣ n) : (cast (1 : ZMod n) : R) = 1
|
case succ.zero
R : Type u_1
inst✝¹ : Ring R
m : ℕ
inst✝ : CharP R m
h : m = 1
⊢ ↑(1 % (0 + 1)) = 1
|
subst m
|
case succ.zero
R : Type u_1
inst✝¹ : Ring R
inst✝ : CharP R 1
⊢ ↑(1 % (0 + 1)) = 1
|
e6a9c00445bb61a1
|
DoubleQuot.quotQuotEquivComm_symmₐ
|
Mathlib/RingTheory/Ideal/Quotient/Operations.lean
|
theorem quotQuotEquivComm_symmₐ : (quotQuotEquivCommₐ R I J).symm = quotQuotEquivCommₐ R J I
|
case h
R : Type u
A : Type u_1
inst✝² : CommSemiring R
inst✝¹ : CommRing A
inst✝ : Algebra R A
I J : Ideal A
a✝ : (A ⧸ J) ⧸ map (Quotient.mkₐ R J) I
⊢ (quotQuotEquivCommₐ R I J).symm a✝ = (quotQuotEquivCommₐ R J I) a✝
|
unfold quotQuotEquivCommₐ
|
case h
R : Type u
A : Type u_1
inst✝² : CommSemiring R
inst✝¹ : CommRing A
inst✝ : Algebra R A
I J : Ideal A
a✝ : (A ⧸ J) ⧸ map (Quotient.mkₐ R J) I
⊢ (AlgEquiv.ofRingEquiv ⋯).symm a✝ = (AlgEquiv.ofRingEquiv ⋯) a✝
|
2d73434f9d628cbe
|
PhragmenLindelof.right_half_plane_of_tendsto_zero_on_real
|
Mathlib/Analysis/Complex/PhragmenLindelof.lean
|
theorem right_half_plane_of_tendsto_zero_on_real (hd : DiffContOnCl ℂ f {z | 0 < z.re})
(hexp : ∃ c < (2 : ℝ), ∃ B,
f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z => expR (B * ‖z‖ ^ c))
(hre : Tendsto (fun x : ℝ => f x) atTop (𝓝 0)) (him : ∀ x : ℝ, ‖f (x * I)‖ ≤ C)
(hz : 0 ≤ z.re) : ‖f z‖ ≤ C
|
case intro.intro.intro
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
C : ℝ
f : ℂ → E
hd : DiffContOnCl ℂ f {z | 0 < z.re}
hre : Tendsto (fun x => f ↑x) atTop (𝓝 0)
him : ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C
C' : ℝ
hC' : ∀ (x : ℝ), 0 ≤ x → ‖f ↑x‖ ≤ C'
z : ℂ
hz : 0 ≤ z.re
c : ℝ
hc : c < 2
B : ℝ
hO : f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z => expR (B * ‖z‖ ^ c)
⊢ ‖f z‖ ≤ C ⊔ C'
|
rcases le_total z.im 0 with h | h
|
case intro.intro.intro.inl
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
C : ℝ
f : ℂ → E
hd : DiffContOnCl ℂ f {z | 0 < z.re}
hre : Tendsto (fun x => f ↑x) atTop (𝓝 0)
him : ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C
C' : ℝ
hC' : ∀ (x : ℝ), 0 ≤ x → ‖f ↑x‖ ≤ C'
z : ℂ
hz : 0 ≤ z.re
c : ℝ
hc : c < 2
B : ℝ
hO : f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z => expR (B * ‖z‖ ^ c)
h : z.im ≤ 0
⊢ ‖f z‖ ≤ C ⊔ C'
case intro.intro.intro.inr
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
C : ℝ
f : ℂ → E
hd : DiffContOnCl ℂ f {z | 0 < z.re}
hre : Tendsto (fun x => f ↑x) atTop (𝓝 0)
him : ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C
C' : ℝ
hC' : ∀ (x : ℝ), 0 ≤ x → ‖f ↑x‖ ≤ C'
z : ℂ
hz : 0 ≤ z.re
c : ℝ
hc : c < 2
B : ℝ
hO : f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z => expR (B * ‖z‖ ^ c)
h : 0 ≤ z.im
⊢ ‖f z‖ ≤ C ⊔ C'
|
8c2477e37fa78449
|
cardinal_eq_of_mem_nhds
|
Mathlib/Topology/Algebra/Module/Cardinality.lean
|
theorem cardinal_eq_of_mem_nhds
{E : Type*} (𝕜 : Type*) [NontriviallyNormedField 𝕜] [AddCommGroup E] [Module 𝕜 E]
[TopologicalSpace E] [ContinuousAdd E] [ContinuousSMul 𝕜 E]
{s : Set E} {x : E} (hs : s ∈ 𝓝 x) : #s = #E
|
E : Type u_1
𝕜 : Type u_2
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : Module 𝕜 E
inst✝² : TopologicalSpace E
inst✝¹ : ContinuousAdd E
inst✝ : ContinuousSMul 𝕜 E
s : Set E
x : E
hs : s ∈ 𝓝 x
g : E ≃ₜ E := Homeomorph.addLeft x
t : Set E := ⇑g ⁻¹' s
⊢ #↑s = #E
|
have : t ∈ 𝓝 0 := g.continuous.continuousAt.preimage_mem_nhds (by simpa [g] using hs)
|
E : Type u_1
𝕜 : Type u_2
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : Module 𝕜 E
inst✝² : TopologicalSpace E
inst✝¹ : ContinuousAdd E
inst✝ : ContinuousSMul 𝕜 E
s : Set E
x : E
hs : s ∈ 𝓝 x
g : E ≃ₜ E := Homeomorph.addLeft x
t : Set E := ⇑g ⁻¹' s
this : t ∈ 𝓝 0
⊢ #↑s = #E
|
fd663fd33aaad9d7
|
AkraBazziRecurrence.asympBound_pos
|
Mathlib/Computability/AkraBazzi/AkraBazzi.lean
|
lemma asympBound_pos (n : ℕ) (hn : 0 < n) : 0 < asympBound g a b n
|
α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
n : ℕ
hn : 0 < n
⊢ ↑n ^ p a b * (1 + 0) ≤ ↑n ^ p a b * (1 + ∑ u ∈ range n, g ↑u / ↑u ^ (p a b + 1))
|
gcongr n^p a b * (1 + ?_)
|
case h.bc
α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
n : ℕ
hn : 0 < n
⊢ 0 ≤ ∑ u ∈ range n, g ↑u / ↑u ^ (p a b + 1)
|
a0987b7c03f9cda1
|
Vector.zipIdx_eq_append_iff
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Vector/Range.lean
|
theorem zipIdx_eq_append_iff {l : Vector α (n + m)} {k : Nat} :
zipIdx l k = l₁ ++ l₂ ↔
∃ (l₁' : Vector α n) (l₂' : Vector α m),
l = l₁' ++ l₂' ∧ l₁ = zipIdx l₁' k ∧ l₂ = zipIdx l₂' (k + n)
|
case mk.mk.mk
α : Type u_1
k : Nat
l : Array α
l₁ l₂ : Array (α × Nat)
h : l.size = l₁.size + l₂.size
⊢ { toArray := l, size_toArray := h }.zipIdx k =
{ toArray := l₁, size_toArray := ⋯ } ++ { toArray := l₂, size_toArray := ⋯ } ↔
∃ l₁' l₂',
{ toArray := l, size_toArray := h } = l₁' ++ l₂' ∧
{ toArray := l₁, size_toArray := ⋯ } = l₁'.zipIdx k ∧
{ toArray := l₂, size_toArray := ⋯ } = l₂'.zipIdx (k + l₁.size)
|
simp only [zipIdx_mk, mk_append_mk, eq_mk, Array.zipIdx_eq_append_iff, mk_eq, toArray_append,
toArray_zipIdx]
|
case mk.mk.mk
α : Type u_1
k : Nat
l : Array α
l₁ l₂ : Array (α × Nat)
h : l.size = l₁.size + l₂.size
⊢ (∃ l₁' l₂', l = l₁' ++ l₂' ∧ l₁ = l₁'.zipIdx k ∧ l₂ = l₂'.zipIdx (k + l₁'.size)) ↔
∃ l₁' l₂', l = l₁'.toArray ++ l₂'.toArray ∧ l₁ = l₁'.zipIdx k ∧ l₂ = l₂'.zipIdx (k + l₁.size)
|
e21d92e5105fe40c
|
TendstoLocallyUniformlyOn.differentiableOn
|
Mathlib/Analysis/Complex/LocallyUniformLimit.lean
|
theorem _root_.TendstoLocallyUniformlyOn.differentiableOn [φ.NeBot]
(hf : TendstoLocallyUniformlyOn F f φ U) (hF : ∀ᶠ n in φ, DifferentiableOn ℂ (F n) U)
(hU : IsOpen U) : DifferentiableOn ℂ f U
|
E : Type u_1
ι : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
U : Set ℂ
φ : Filter ι
F : ι → ℂ → E
f : ℂ → E
inst✝¹ : CompleteSpace E
inst✝ : φ.NeBot
hf : TendstoLocallyUniformlyOn F f φ U
hF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U
hU : IsOpen U
⊢ DifferentiableOn ℂ f U
|
rintro x hx
|
E : Type u_1
ι : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
U : Set ℂ
φ : Filter ι
F : ι → ℂ → E
f : ℂ → E
inst✝¹ : CompleteSpace E
inst✝ : φ.NeBot
hf : TendstoLocallyUniformlyOn F f φ U
hF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U
hU : IsOpen U
x : ℂ
hx : x ∈ U
⊢ DifferentiableWithinAt ℂ f U x
|
4179b164980029db
|
MonoidAlgebra.of_mem_span_of_iff
|
Mathlib/RingTheory/FiniteType.lean
|
theorem of_mem_span_of_iff [Nontrivial R] {m : M} {S : Set M} :
of R M m ∈ span R (of R M '' S) ↔ m ∈ S
|
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : Monoid M
inst✝ : Nontrivial R
m : M
S : Set M
h : ↑{m} ⊆ S
⊢ m ∈ S
|
simpa using h
|
no goals
|
e166d2db530130a9
|
CategoryTheory.GrothendieckTopology.yonedaEquiv_symm_naturality_right
|
Mathlib/CategoryTheory/Sites/Subcanonical.lean
|
lemma yonedaEquiv_symm_naturality_right (X : C) {F F' : Sheaf J (Type v)} (f : F ⟶ F')
(x : F.val.obj ⟨X⟩) : J.yonedaEquiv.symm x ≫ f = J.yonedaEquiv.symm (f.val.app ⟨X⟩ x)
|
C : Type u
inst✝¹ : Category.{v, u} C
J : GrothendieckTopology C
inst✝ : J.Subcanonical
X : C
F F' : Sheaf J (Type v)
f : F ⟶ F'
x : F.val.obj (op X)
⊢ J.yonedaEquiv.symm x ≫ f = J.yonedaEquiv.symm (f.val.app (op X) x)
|
apply J.yonedaEquiv.injective
|
case a
C : Type u
inst✝¹ : Category.{v, u} C
J : GrothendieckTopology C
inst✝ : J.Subcanonical
X : C
F F' : Sheaf J (Type v)
f : F ⟶ F'
x : F.val.obj (op X)
⊢ J.yonedaEquiv (J.yonedaEquiv.symm x ≫ f) = J.yonedaEquiv (J.yonedaEquiv.symm (f.val.app (op X) x))
|
41b8ba8609f5fbc6
|
HallMarriageTheorem.hall_hard_inductive_step_B
|
Mathlib/Combinatorics/Hall/Finite.lean
|
theorem hall_hard_inductive_step_B {n : ℕ} (hn : Fintype.card ι = n + 1)
(ht : ∀ s : Finset ι, #s ≤ #(s.biUnion t))
(ih :
∀ {ι' : Type u} [Fintype ι'] (t' : ι' → Finset α),
Fintype.card ι' ≤ n →
(∀ s' : Finset ι', #s' ≤ #(s'.biUnion t')) →
∃ f : ι' → α, Function.Injective f ∧ ∀ x, f x ∈ t' x)
(s : Finset ι) (hs : s.Nonempty) (hns : s ≠ univ) (hus : #s = #(s.biUnion t)) :
∃ f : ι → α, Function.Injective f ∧ ∀ x, f x ∈ t x
|
case intro.intro.intro.intro
ι : Type u
α : Type v
inst✝¹ : DecidableEq α
t : ι → Finset α
inst✝ : Fintype ι
n : ℕ
hn : Fintype.card ι = n.succ
ht : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)
ih :
∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),
Fintype.card ι' ≤ n →
(∀ (s' : Finset ι'), #s' ≤ #(s'.biUnion t')) → ∃ f, Function.Injective f ∧ ∀ (x : ι'), f x ∈ t' x
s : Finset ι
hs : s.Nonempty
hns : s ≠ univ
hus : #s = #(s.biUnion t)
this : DecidableEq ι
card_ι'_le : Fintype.card { x // x ∈ s } ≤ n
t' : { x // x ∈ s } → Finset α := fun x' => t ↑x'
f' : { x // x ∈ s } → α
hf' : Function.Injective f'
hsf' : ∀ (x : { x // x ∈ s }), f' x ∈ t' x
ι'' : Set ι := (↑s)ᶜ
t'' : ↑ι'' → Finset α := fun a'' => t ↑a'' \ s.biUnion t
card_ι''_le : Fintype.card ↑ι'' ≤ n
f'' : ↑ι'' → α
hf'' : Function.Injective f''
hsf'' : ∀ (x : ↑ι''), f'' x ∈ t'' x
f'_mem_biUnion : ∀ (x' : ι) (hx' : x' ∈ s), f' ⟨x', hx'⟩ ∈ s.biUnion t
⊢ ∃ f, Function.Injective f ∧ ∀ (x : ι), f x ∈ t x
|
have f''_not_mem_biUnion : ∀ (x'') (hx'' : ¬x'' ∈ s), ¬f'' ⟨x'', hx''⟩ ∈ s.biUnion t := by
intro x'' hx''
have h := hsf'' ⟨x'', hx''⟩
rw [mem_sdiff] at h
exact h.2
|
case intro.intro.intro.intro
ι : Type u
α : Type v
inst✝¹ : DecidableEq α
t : ι → Finset α
inst✝ : Fintype ι
n : ℕ
hn : Fintype.card ι = n.succ
ht : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)
ih :
∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),
Fintype.card ι' ≤ n →
(∀ (s' : Finset ι'), #s' ≤ #(s'.biUnion t')) → ∃ f, Function.Injective f ∧ ∀ (x : ι'), f x ∈ t' x
s : Finset ι
hs : s.Nonempty
hns : s ≠ univ
hus : #s = #(s.biUnion t)
this : DecidableEq ι
card_ι'_le : Fintype.card { x // x ∈ s } ≤ n
t' : { x // x ∈ s } → Finset α := fun x' => t ↑x'
f' : { x // x ∈ s } → α
hf' : Function.Injective f'
hsf' : ∀ (x : { x // x ∈ s }), f' x ∈ t' x
ι'' : Set ι := (↑s)ᶜ
t'' : ↑ι'' → Finset α := fun a'' => t ↑a'' \ s.biUnion t
card_ι''_le : Fintype.card ↑ι'' ≤ n
f'' : ↑ι'' → α
hf'' : Function.Injective f''
hsf'' : ∀ (x : ↑ι''), f'' x ∈ t'' x
f'_mem_biUnion : ∀ (x' : ι) (hx' : x' ∈ s), f' ⟨x', hx'⟩ ∈ s.biUnion t
f''_not_mem_biUnion : ∀ (x'' : ι) (hx'' : x'' ∉ s), f'' ⟨x'', hx''⟩ ∉ s.biUnion t
⊢ ∃ f, Function.Injective f ∧ ∀ (x : ι), f x ∈ t x
|
de2e47a85ff6c58c
|
Basis.eq_bot_of_rank_eq_zero
|
Mathlib/LinearAlgebra/Basis/Basic.lean
|
theorem Basis.eq_bot_of_rank_eq_zero [NoZeroDivisors R] (b : Basis ι R M) (N : Submodule R M)
(rank_eq : ∀ {m : ℕ} (v : Fin m → N), LinearIndependent R ((↑) ∘ v : Fin m → M) → m = 0) :
N = ⊥
|
case mk
ι : Type u_1
R : Type u_3
M : Type u_5
inst✝³ : Ring R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : NoZeroDivisors R
b : Basis ι R M
N : Submodule R M
x : M
hx : x ∈ N
x_ne : x ≠ 0
g : Fin 1 → R
val✝ : ℕ
hi : val✝ < 1
sum_eq : g 0 • (Subtype.val ∘ fun x_1 => ⟨x, hx⟩) 0 = 0
⊢ g ⟨val✝, hi⟩ = 0
|
convert (b.smul_eq_zero.mp sum_eq).resolve_right x_ne
|
no goals
|
2aad7cb72b7cf502
|
MeasureTheory.Filtration.filtrationOfSet_eq_natural
|
Mathlib/Probability/Process/Filtration.lean
|
theorem filtrationOfSet_eq_natural [MulZeroOneClass β] [Nontrivial β] {s : ι → Set Ω}
(hsm : ∀ i, MeasurableSet[m] (s i)) :
filtrationOfSet hsm = natural (fun i => (s i).indicator (fun _ => 1 : Ω → β)) fun i =>
stronglyMeasurable_one.indicator (hsm i)
|
case h.refine_2.intro
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁵ : TopologicalSpace β
inst✝⁴ : MetrizableSpace β
mβ : MeasurableSpace β
inst✝³ : BorelSpace β
inst✝² : Preorder ι
inst✝¹ : MulZeroOneClass β
inst✝ : Nontrivial β
s : ι → Set Ω
hsm : ∀ (i : ι), MeasurableSet (s i)
i : ι
t : Set Ω
n : ι
ht : MeasurableSet t
⊢ MeasurableSet t
|
suffices MeasurableSpace.generateFrom {t | n ≤ i ∧
MeasurableSet[MeasurableSpace.comap ((s n).indicator (fun _ => 1 : Ω → β)) mβ] t} ≤
MeasurableSpace.generateFrom {t | ∃ (j : ι), j ≤ i ∧ s j = t} by
exact this _ ht
|
case h.refine_2.intro
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁵ : TopologicalSpace β
inst✝⁴ : MetrizableSpace β
mβ : MeasurableSpace β
inst✝³ : BorelSpace β
inst✝² : Preorder ι
inst✝¹ : MulZeroOneClass β
inst✝ : Nontrivial β
s : ι → Set Ω
hsm : ∀ (i : ι), MeasurableSet (s i)
i : ι
t : Set Ω
n : ι
ht : MeasurableSet t
⊢ MeasurableSpace.generateFrom {t | n ≤ i ∧ MeasurableSet t} ≤ MeasurableSpace.generateFrom {t | ∃ j ≤ i, s j = t}
|
7ffbace60117795e
|
totallyBounded_interUnionBalls
|
Mathlib/Topology/UniformSpace/Cauchy.lean
|
lemma totallyBounded_interUnionBalls {p : ℕ → Prop} {U : ℕ → Set (α × α)}
(H : (uniformity α).HasBasis p U) (xs : ℕ → α) (u : ℕ → ℕ) :
TotallyBounded (interUnionBalls xs u U)
|
α : Type u
uniformSpace : UniformSpace α
p : ℕ → Prop
U : ℕ → Set (α × α)
H : (𝓤 α).HasBasis p U
xs : ℕ → α
u : ℕ → ℕ
i : ℕ
a✝ : p i
h_subset : interUnionBalls xs u U ⊆ ⋃ m, ⋃ (_ : m ≤ u i), ball (xs m) (Prod.swap ⁻¹' U i)
x : α
hx : x ∈ interUnionBalls xs u U
⊢ x ∈ ⋃ y ∈ ↑(Finset.image xs (Finset.range (u i + 1))), {x | (x, y) ∈ U i}
|
simp only [Finset.coe_image, Finset.coe_range, mem_image, mem_Iio, iUnion_exists, biUnion_and',
iUnion_iUnion_eq_right, Nat.lt_succ_iff]
|
α : Type u
uniformSpace : UniformSpace α
p : ℕ → Prop
U : ℕ → Set (α × α)
H : (𝓤 α).HasBasis p U
xs : ℕ → α
u : ℕ → ℕ
i : ℕ
a✝ : p i
h_subset : interUnionBalls xs u U ⊆ ⋃ m, ⋃ (_ : m ≤ u i), ball (xs m) (Prod.swap ⁻¹' U i)
x : α
hx : x ∈ interUnionBalls xs u U
⊢ x ∈ ⋃ y, ⋃ (_ : y ≤ u i), {x | (x, xs y) ∈ U i}
|
a7e97fbc5a5e4030
|
StieltjesFunction.measure_const
|
Mathlib/MeasureTheory/Measure/Stieltjes.lean
|
@[simp]
lemma measure_const (c : ℝ) : (StieltjesFunction.const c).measure = 0 :=
Measure.ext_of_Ioc _ _ (by simp)
|
c : ℝ
⊢ ∀ ⦃a b : ℝ⦄, a < b → (StieltjesFunction.const c).measure (Ioc a b) = 0 (Ioc a b)
|
simp
|
no goals
|
ddf913c1bd3042b4
|
Derivation.tensorProductTo_mul
|
Mathlib/RingTheory/Kaehler/Basic.lean
|
theorem Derivation.tensorProductTo_mul (D : Derivation R S M) (x y : S ⊗[R] S) :
D.tensorProductTo (x * y) =
TensorProduct.lmul' (S := S) R x • D.tensorProductTo y +
TensorProduct.lmul' (S := S) R y • D.tensorProductTo x
|
case refine_2.refine_1
R : Type u
S : Type v
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module S M
inst✝ : IsScalarTower R S M
D : Derivation R S M
x y : S ⊗[R] S
x₁ x₂ : S
⊢ D.tensorProductTo (x₁ ⊗ₜ[R] x₂ * 0) =
(TensorProduct.lmul' R) (x₁ ⊗ₜ[R] x₂) • D.tensorProductTo 0 +
(TensorProduct.lmul' R) 0 • D.tensorProductTo (x₁ ⊗ₜ[R] x₂)
|
rw [mul_zero, map_zero, map_zero, zero_smul, smul_zero, add_zero]
|
no goals
|
5acdf1d1a845072a
|
Real.pow_mul_norm_iteratedFDeriv_fourierIntegral_le
|
Mathlib/Analysis/Fourier/FourierTransformDeriv.lean
|
/-- One can bound `‖w‖^n * ‖D^k (𝓕 f) w‖` in terms of integrals of the derivatives of `f` (or order
at most `n`) multiplied by powers of `v` (of order at most `k`). -/
lemma pow_mul_norm_iteratedFDeriv_fourierIntegral_le
{K N : ℕ∞} (hf : ContDiff ℝ N f)
(h'f : ∀ (k n : ℕ), k ≤ K → n ≤ N → Integrable (fun v ↦ ‖v‖^k * ‖iteratedFDeriv ℝ n f v‖))
{k n : ℕ} (hk : k ≤ K) (hn : n ≤ N) (w : V) :
‖w‖ ^ n * ‖iteratedFDeriv ℝ k (𝓕 f) w‖ ≤ (2 * π) ^ k * (2 * k + 2) ^ n *
∑ p ∈ Finset.range (k + 1) ×ˢ Finset.range (n + 1),
∫ v, ‖v‖ ^ p.1 * ‖iteratedFDeriv ℝ p.2 f v‖
|
case inr.inr
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℂ E
V : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : FiniteDimensional ℝ V
inst✝¹ : MeasurableSpace V
inst✝ : BorelSpace V
f : V → E
K N : ℕ∞
hf : ContDiff ℝ (↑N) f
h'f : ∀ (k n : ℕ), ↑k ≤ K → ↑n ≤ N → Integrable (fun v => ‖v‖ ^ k * ‖iteratedFDeriv ℝ n f v‖) volume
k n : ℕ
hk : ↑k ≤ K
hn✝ : ↑n ≤ N
w : V
Z :
‖w‖ ^ n * ‖iteratedFDeriv ℝ k (𝓕 f) w‖ ≤
(2 * (π * ‖innerSL ℝ‖)) ^ k *
((2 * ↑k + 2) ^ n *
∑ p ∈ Finset.range (k + 1) ×ˢ Finset.range (n + 1), ∫ (v : V), ‖v‖ ^ p.1 * ‖iteratedFDeriv ℝ p.2 f v‖)
hn : n ≠ 0
hw : w ≠ 0
⊢ (2 * (π * ‖innerSL ℝ‖)) ^ k *
((2 * ↑k + 2) ^ n *
∑ p ∈ Finset.range (k + 1) ×ˢ Finset.range (n + 1), ∫ (v : V), ‖v‖ ^ p.1 * ‖iteratedFDeriv ℝ p.2 f v‖) ≤
(2 * (π * 1)) ^ k *
((2 * ↑k + 2) ^ n *
∑ p ∈ Finset.range (k + 1) ×ˢ Finset.range (n + 1), ∫ (v : V), ‖v‖ ^ p.1 * ‖iteratedFDeriv ℝ p.2 f v‖)
|
gcongr
|
case inr.inr.h.hab.h.h
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℂ E
V : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : FiniteDimensional ℝ V
inst✝¹ : MeasurableSpace V
inst✝ : BorelSpace V
f : V → E
K N : ℕ∞
hf : ContDiff ℝ (↑N) f
h'f : ∀ (k n : ℕ), ↑k ≤ K → ↑n ≤ N → Integrable (fun v => ‖v‖ ^ k * ‖iteratedFDeriv ℝ n f v‖) volume
k n : ℕ
hk : ↑k ≤ K
hn✝ : ↑n ≤ N
w : V
Z :
‖w‖ ^ n * ‖iteratedFDeriv ℝ k (𝓕 f) w‖ ≤
(2 * (π * ‖innerSL ℝ‖)) ^ k *
((2 * ↑k + 2) ^ n *
∑ p ∈ Finset.range (k + 1) ×ˢ Finset.range (n + 1), ∫ (v : V), ‖v‖ ^ p.1 * ‖iteratedFDeriv ℝ p.2 f v‖)
hn : n ≠ 0
hw : w ≠ 0
⊢ ‖innerSL ℝ‖ ≤ 1
|
d6081cebd8a737ff
|
HomologicalComplex.singleMapHomologicalComplex_inv_app_self
|
Mathlib/Algebra/Homology/Additive.lean
|
theorem singleMapHomologicalComplex_inv_app_self (j : ι) (X : W₁) :
((singleMapHomologicalComplex F c j).inv.app X).f j =
(singleObjXSelf c j (F.obj X)).hom ≫ F.map (singleObjXSelf c j X).inv
|
ι : Type u_1
W₁ : Type u_3
W₂ : Type u_4
inst✝⁷ : Category.{u_6, u_3} W₁
inst✝⁶ : Category.{u_5, u_4} W₂
inst✝⁵ : HasZeroMorphisms W₁
inst✝⁴ : HasZeroMorphisms W₂
inst✝³ : HasZeroObject W₁
inst✝² : HasZeroObject W₂
F : W₁ ⥤ W₂
inst✝¹ : F.PreservesZeroMorphisms
c : ComplexShape ι
inst✝ : DecidableEq ι
j : ι
X : W₁
⊢ ((singleMapHomologicalComplex F c j).inv.app X).f j =
(singleObjXSelf c j (F.obj X)).hom ≫ F.map (singleObjXSelf c j X).inv
|
simp [singleMapHomologicalComplex, singleObjXSelf, singleObjXIsoOfEq, eqToHom_map]
|
no goals
|
d9b018e698e4cb14
|
Nat.add_factorial_succ_le_factorial_add_succ
|
Mathlib/Data/Nat/Factorial/Basic.lean
|
theorem add_factorial_succ_le_factorial_add_succ (i : ℕ) (n : ℕ) :
i + (n + 1)! ≤ (i + (n + 1))!
|
case inl
i n : ℕ
h✝ : 2 ≤ i
⊢ i + (n + 1)! ≤ (i + (n + 1))!
|
rw [← Nat.add_assoc]
|
case inl
i n : ℕ
h✝ : 2 ≤ i
⊢ i + (n + 1)! ≤ (i + n + 1)!
|
26356308d2e4db6a
|
Submodule.spanRank_span_le_card
|
Mathlib/Algebra/Module/SpanRank.lean
|
lemma spanRank_span_le_card (s : Set M) : (Submodule.span R s).spanRank ≤ #s
|
R : Type u_1
M : Type u
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
s : Set M
⊢ ⨅ s_1, #↑↑s_1 ≤ #↑s
|
let s' : {s1 : Set M // span R s1 = span R s} := ⟨s, rfl⟩
|
R : Type u_1
M : Type u
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
s : Set M
s' : { s1 // span R s1 = span R s } := ⟨s, ⋯⟩
⊢ ⨅ s_1, #↑↑s_1 ≤ #↑s
|
0eb082337e985dd3
|
ProbabilityTheory.IsRatStieltjesPoint.ite
|
Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean
|
lemma IsRatStieltjesPoint.ite {f g : α → ℚ → ℝ} {a : α} (p : α → Prop) [DecidablePred p]
(hf : p a → IsRatStieltjesPoint f a) (hg : ¬ p a → IsRatStieltjesPoint g a) :
IsRatStieltjesPoint (fun a ↦ if p a then f a else g a) a where
mono
|
α : Type u_1
f g : α → ℚ → ℝ
a : α
p : α → Prop
inst✝ : DecidablePred p
hf : p a → IsRatStieltjesPoint f a
hg : ¬p a → IsRatStieltjesPoint g a
⊢ Monotone (if p a then f a else g a)
|
split_ifs with h
|
case pos
α : Type u_1
f g : α → ℚ → ℝ
a : α
p : α → Prop
inst✝ : DecidablePred p
hf : p a → IsRatStieltjesPoint f a
hg : ¬p a → IsRatStieltjesPoint g a
h : p a
⊢ Monotone (f a)
case neg
α : Type u_1
f g : α → ℚ → ℝ
a : α
p : α → Prop
inst✝ : DecidablePred p
hf : p a → IsRatStieltjesPoint f a
hg : ¬p a → IsRatStieltjesPoint g a
h : ¬p a
⊢ Monotone (g a)
|
07e5c46b6daf969c
|
ProbabilityTheory.Kernel.iIndepFun.cond_iInter
|
Mathlib/Probability/Independence/Kernel.lean
|
/-- The probability of an intersection of preimages conditioning on another intersection factors
into a product. -/
lemma iIndepFun.cond_iInter [Finite ι] (hY : ∀ i, Measurable (Y i))
(hindep : iIndepFun (fun _ ↦ mα.prod mβ) (fun i ω ↦ (X i ω, Y i ω)) κ μ)
(hf : ∀ i ∈ s, MeasurableSet[mα.comap (X i)] (f i))
(hy : ∀ᵐ a ∂μ, ∀ i ∉ s, κ a (Y i ⁻¹' t i) ≠ 0) (ht : ∀ i, MeasurableSet (t i)) :
∀ᵐ a ∂μ, (κ a)[⋂ i ∈ s, f i | ⋂ i, Y i ⁻¹' t i] = ∏ i ∈ s, (κ a)[f i | Y i in t i]
|
case intro
ι : Type u_4
Ω : Type u_5
α : Type u_6
β : Type u_7
mΩ : MeasurableSpace Ω
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α Ω
μ : Measure α
X : ι → Ω → α
Y : ι → Ω → β
f : ι → Set Ω
t : ι → Set β
s : Finset ι
inst✝ : Finite ι
hY : ∀ (i : ι), Measurable (Y i)
hindep : iIndepFun (fun x => mα.prod mβ) (fun i ω => (X i ω, Y i ω)) κ μ
hf : ∀ i ∈ s, MeasurableSet (f i)
hy : ∀ᵐ (a : α) ∂μ, ∀ i ∉ s, (κ a) (Y i ⁻¹' t i) ≠ 0
ht : ∀ (i : ι), MeasurableSet (t i)
val✝ : Fintype ι
⊢ ∀ᵐ (a : α) ∂μ, (κ a)[⋂ i ∈ s, f i | ⋂ i, Y i ⁻¹' t i] = ∏ i ∈ s, (κ a)[f i | Y i ⁻¹' t i]
|
let g (i' : ι) := if i' ∈ s then Y i' ⁻¹' t i' ∩ f i' else Y i' ⁻¹' t i'
|
case intro
ι : Type u_4
Ω : Type u_5
α : Type u_6
β : Type u_7
mΩ : MeasurableSpace Ω
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α Ω
μ : Measure α
X : ι → Ω → α
Y : ι → Ω → β
f : ι → Set Ω
t : ι → Set β
s : Finset ι
inst✝ : Finite ι
hY : ∀ (i : ι), Measurable (Y i)
hindep : iIndepFun (fun x => mα.prod mβ) (fun i ω => (X i ω, Y i ω)) κ μ
hf : ∀ i ∈ s, MeasurableSet (f i)
hy : ∀ᵐ (a : α) ∂μ, ∀ i ∉ s, (κ a) (Y i ⁻¹' t i) ≠ 0
ht : ∀ (i : ι), MeasurableSet (t i)
val✝ : Fintype ι
g : ι → Set Ω := fun i' => if i' ∈ s then Y i' ⁻¹' t i' ∩ f i' else Y i' ⁻¹' t i'
⊢ ∀ᵐ (a : α) ∂μ, (κ a)[⋂ i ∈ s, f i | ⋂ i, Y i ⁻¹' t i] = ∏ i ∈ s, (κ a)[f i | Y i ⁻¹' t i]
|
31a9bf09c9f649df
|
Multiset.rel_cons_right
|
Mathlib/Data/Multiset/ZeroCons.lean
|
theorem rel_cons_right {as b bs} :
Rel r as (b ::ₘ bs) ↔ ∃ a as', r a b ∧ Rel r as' bs ∧ as = a ::ₘ as'
|
α : Type u_1
β : Type v
r : α → β → Prop
as : Multiset α
b : β
bs : Multiset β
a : α
as' : Multiset α
⊢ flip r b a ∧ Rel (flip r) bs as' ∧ as = a ::ₘ as' ↔ r a b ∧ Rel r as' bs ∧ as = a ::ₘ as'
|
rw [rel_flip, flip]
|
no goals
|
173c90a4cd6cfbdd
|
Linarith.without_one_mul
|
Mathlib/Tactic/Linarith/Preprocessing.lean
|
theorem without_one_mul {M : Type*} [MulOneClass M] {a b : M} (h : 1 * a = b) : a = b
|
M : Type u_1
inst✝ : MulOneClass M
a b : M
h : 1 * a = b
⊢ a = b
|
rwa [one_mul] at h
|
no goals
|
6b0b47cad70d6773
|
GromovHausdorff.ghDist_le_of_approx_subsets
|
Mathlib/Topology/MetricSpace/GromovHausdorff.lean
|
theorem ghDist_le_of_approx_subsets {s : Set X} (Φ : s → Y) {ε₁ ε₂ ε₃ : ℝ}
(hs : ∀ x : X, ∃ y ∈ s, dist x y ≤ ε₁) (hs' : ∀ x : Y, ∃ y : s, dist x (Φ y) ≤ ε₃)
(H : ∀ x y : s, |dist x y - dist (Φ x) (Φ y)| ≤ ε₂) : ghDist X Y ≤ ε₁ + ε₂ / 2 + ε₃
|
case intro
X : Type u
inst✝⁵ : MetricSpace X
inst✝⁴ : CompactSpace X
inst✝³ : Nonempty X
Y : Type v
inst✝² : MetricSpace Y
inst✝¹ : CompactSpace Y
inst✝ : Nonempty Y
s : Set X
Φ : ↑s → Y
ε₁ ε₂ ε₃ : ℝ
hs : ∀ (x : X), ∃ y ∈ s, dist x y ≤ ε₁
hs' : ∀ (x : Y), ∃ y, dist x (Φ y) ≤ ε₃
H : ∀ (x y : ↑s), |dist x y - dist (Φ x) (Φ y)| ≤ ε₂
δ : ℝ
δ0 : 0 < δ
xX : X
h✝ : xX ∈ univ
⊢ ghDist X Y ≤ ε₁ + ε₂ / 2 + ε₃ + δ
|
rcases hs xX with ⟨xs, hxs, Dxs⟩
|
case intro.intro.intro
X : Type u
inst✝⁵ : MetricSpace X
inst✝⁴ : CompactSpace X
inst✝³ : Nonempty X
Y : Type v
inst✝² : MetricSpace Y
inst✝¹ : CompactSpace Y
inst✝ : Nonempty Y
s : Set X
Φ : ↑s → Y
ε₁ ε₂ ε₃ : ℝ
hs : ∀ (x : X), ∃ y ∈ s, dist x y ≤ ε₁
hs' : ∀ (x : Y), ∃ y, dist x (Φ y) ≤ ε₃
H : ∀ (x y : ↑s), |dist x y - dist (Φ x) (Φ y)| ≤ ε₂
δ : ℝ
δ0 : 0 < δ
xX : X
h✝ : xX ∈ univ
xs : X
hxs : xs ∈ s
Dxs : dist xX xs ≤ ε₁
⊢ ghDist X Y ≤ ε₁ + ε₂ / 2 + ε₃ + δ
|
ca73aa8d17706ff3
|
IsUnifLocDoublingMeasure.ae_tendsto_average_norm_sub
|
Mathlib/MeasureTheory/Covering/DensityTheorem.lean
|
theorem ae_tendsto_average_norm_sub {f : α → E} (hf : LocallyIntegrable f μ) (K : ℝ) : ∀ᵐ x ∂μ,
∀ {ι : Type*} {l : Filter ι} (w : ι → α) (δ : ι → ℝ) (_ : Tendsto δ l (𝓝[>] 0))
(_ : ∀ᶠ j in l, x ∈ closedBall (w j) (K * δ j)),
Tendsto (fun j => ⨍ y in closedBall (w j) (δ j), ‖f y - f x‖ ∂μ) l (𝓝 0)
|
α : Type u_1
inst✝⁶ : PseudoMetricSpace α
inst✝⁵ : MeasurableSpace α
μ : Measure α
inst✝⁴ : IsUnifLocDoublingMeasure μ
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
E : Type u_2
inst✝ : NormedAddCommGroup E
f : α → E
hf : LocallyIntegrable f μ
K : ℝ
⊢ ∀ᵐ (x : α) ∂μ,
∀ {ι : Type u_3} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
Tendsto δ l (𝓝[>] 0) →
(∀ᶠ (j : ι) in l, x ∈ closedBall (w j) (K * δ j)) →
Tendsto (fun j => ⨍ (y : α) in closedBall (w j) (δ j), ‖f y - f x‖ ∂μ) l (𝓝 0)
|
filter_upwards [(vitaliFamily μ K).ae_tendsto_average_norm_sub hf] with x hx ι l w δ δlim
xmem using hx.comp (tendsto_closedBall_filterAt μ _ _ δlim xmem)
|
no goals
|
86187b49abc7e85e
|
OrderIso.isMin_apply
|
Mathlib/Order/Hom/Basic.lean
|
theorem OrderIso.isMin_apply {α β : Type*} [Preorder α] [Preorder β] (f : α ≃o β) {x : α} :
IsMin (f x) ↔ IsMin x
|
α : Type u_6
β : Type u_7
inst✝¹ : Preorder α
inst✝ : Preorder β
f : α ≃o β
x : α
⊢ IsMin (f x) ↔ IsMin x
|
refine ⟨f.strictMono.isMin_of_apply, ?_⟩
|
α : Type u_6
β : Type u_7
inst✝¹ : Preorder α
inst✝ : Preorder β
f : α ≃o β
x : α
⊢ IsMin x → IsMin (f x)
|
6e3aa051fe221b6a
|
Polynomial.ite_le_natDegree_coeff
|
Mathlib/Algebra/Polynomial/Degree/Operations.lean
|
theorem ite_le_natDegree_coeff (p : R[X]) (n : ℕ) (I : Decidable (n < 1 + natDegree p)) :
@ite _ (n < 1 + natDegree p) I (coeff p n) 0 = coeff p n
|
case neg
R : Type u
inst✝ : Semiring R
p : R[X]
n : ℕ
I : Decidable (n < 1 + p.natDegree)
h : ¬n < 1 + p.natDegree
⊢ 0 = p.coeff n
|
exact (coeff_eq_zero_of_natDegree_lt (not_le.1 fun w => h (Nat.lt_one_add_iff.2 w))).symm
|
no goals
|
00b03bbf28d36178
|
Polynomial.quotient_mk_comp_C_isIntegral_of_jacobson'
|
Mathlib/RingTheory/Jacobson/Ring.lean
|
theorem quotient_mk_comp_C_isIntegral_of_jacobson' [Nontrivial R] (hR : IsJacobsonRing R)
(hP' : ∀ x : R, C x ∈ P → x = 0) :
((Ideal.Quotient.mk P).comp C : R →+* R[X] ⧸ P).IsIntegral
|
case intro.intro
R : Type u_1
inst✝¹ : CommRing R
P : Ideal R[X]
hP : P.IsMaximal
inst✝ : Nontrivial R
hR : IsJacobsonRing R
hP' : ∀ (x : R), C x ∈ P → x = 0
P' : Ideal R := comap C P
pX : R[X]
hpX : pX ∈ P
hp0 : map (Ideal.Quotient.mk (comap C P)) pX ≠ 0
⊢ (quotientMap P C ⋯).IsIntegral
|
let a : R ⧸ P' := (pX.map (Ideal.Quotient.mk P')).leadingCoeff
|
case intro.intro
R : Type u_1
inst✝¹ : CommRing R
P : Ideal R[X]
hP : P.IsMaximal
inst✝ : Nontrivial R
hR : IsJacobsonRing R
hP' : ∀ (x : R), C x ∈ P → x = 0
P' : Ideal R := comap C P
pX : R[X]
hpX : pX ∈ P
hp0 : map (Ideal.Quotient.mk (comap C P)) pX ≠ 0
a : R ⧸ P' := (map (Ideal.Quotient.mk P') pX).leadingCoeff
⊢ (quotientMap P C ⋯).IsIntegral
|
fe75ad9300033bac
|
trace_eq_trace_adjoin
|
Mathlib/RingTheory/Trace/Basic.lean
|
theorem trace_eq_trace_adjoin [FiniteDimensional K L] (x : L) :
trace K L x = finrank K⟮x⟯ L • trace K K⟮x⟯ (AdjoinSimple.gen K x)
|
K : Type u_4
L : Type u_5
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : FiniteDimensional K L
x : L
⊢ (Algebra.trace K L) x = finrank (↥K⟮x⟯) L • (Algebra.trace K ↥K⟮x⟯) (AdjoinSimple.gen K x)
|
rw [← trace_trace (S := K⟮x⟯)]
|
K : Type u_4
L : Type u_5
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : FiniteDimensional K L
x : L
⊢ (Algebra.trace K ↥K⟮x⟯) ((Algebra.trace (↥K⟮x⟯) L) x) =
finrank (↥K⟮x⟯) L • (Algebra.trace K ↥K⟮x⟯) (AdjoinSimple.gen K x)
|
aaab260abefe4404
|
Vitali.exists_disjoint_covering_ae
|
Mathlib/MeasureTheory/Covering/Vitali.lean
|
theorem exists_disjoint_covering_ae
[PseudoMetricSpace α] [MeasurableSpace α] [OpensMeasurableSpace α]
[SecondCountableTopology α] (μ : Measure α) [IsLocallyFiniteMeasure μ] (s : Set α) (t : Set ι)
(C : ℝ≥0) (r : ι → ℝ) (c : ι → α) (B : ι → Set α) (hB : ∀ a ∈ t, B a ⊆ closedBall (c a) (r a))
(μB : ∀ a ∈ t, μ (closedBall (c a) (3 * r a)) ≤ C * μ (B a))
(ht : ∀ a ∈ t, (interior (B a)).Nonempty) (h't : ∀ a ∈ t, IsClosed (B a))
(hf : ∀ x ∈ s, ∀ ε > (0 : ℝ), ∃ a ∈ t, r a ≤ ε ∧ c a = x) :
∃ u ⊆ t, u.Countable ∧ u.PairwiseDisjoint B ∧ μ (s \ ⋃ a ∈ u, B a) = 0
|
case inr.intro.intro.refine_2.h
α : Type u_1
ι : Type u_2
inst✝⁴ : PseudoMetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ a ∈ t, B a ⊆ closedBall (c a) (r a)
μB : ∀ a ∈ t, μ (closedBall (c a) (3 * r a)) ≤ ↑C * μ (B a)
ht : ∀ a ∈ t, (interior (B a)).Nonempty
h't : ∀ a ∈ t, IsClosed (B a)
hf : ∀ x ∈ s, ∀ ε > 0, ∃ a ∈ t, r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : u.PairwiseDisjoint B
hu : ∀ a ∈ t', ∃ b ∈ u, (B a ∩ B b).Nonempty ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : u.Countable
x : α
x✝ : x ∈ s \ ⋃ a ∈ u, B a
v : Set ι := {a | a ∈ u ∧ (B a ∩ ball x (R x)).Nonempty}
vu : v ⊆ u
Idist_v : ∀ a ∈ v, dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
R0_bdd : BddAbove (r '' v)
H : R x ≤ R0
R0pos : 0 < R0
vnonempty : v.Nonempty
a : ι
hav : a ∈ v
R0a : R0 / 2 < r a
b : ι
bu : b ∈ u
hbx : (B b ∩ ball x (R x)).Nonempty
this : r b ≤ R0
⊢ r b + dist (c b) x ≤ 8 * R0
|
linarith [Idist_v b ⟨bu, hbx⟩]
|
no goals
|
4466d2fc3e80cb3f
|
Std.Tactic.BVDecide.BVExpr.bitblast.blastConst.go_denote_eq
|
Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Lemmas/Const.lean
|
theorem go_denote_eq (aig : AIG α) (c : BitVec w) (assign : α → Bool)
(curr : Nat) (hcurr : curr ≤ w) (s : AIG.RefVec aig curr) :
∀ (idx : Nat) (hidx1 : idx < w),
curr ≤ idx
→
⟦
(go aig c curr s hcurr).aig,
(go aig c curr s hcurr).vec.get idx hidx1,
assign
⟧
=
c.getLsbD idx
|
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
c : BitVec w
assign : α → Bool
curr : Nat
hcurr : curr ≤ w
s : aig.RefVec curr
idx : Nat
hidx1 : idx < w
hidx2 : curr ≤ idx
res : RefVecEntry α w
hgo :
(if hcurr : curr < w then
let res := aig.mkConstCached (c.getLsbD curr);
let aig_1 := res.aig;
let bitRef := res.ref;
let s := s.cast ⋯;
let s := s.push bitRef;
go aig_1 c (curr + 1) s ⋯
else
let_fun hcurr := ⋯;
{ aig := aig, vec := hcurr ▸ s }) =
res
⊢ ⟦assign, { aig := res.aig, ref := res.vec.get idx hidx1 }⟧ = c.getLsbD idx
|
split at hgo
|
case isTrue
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
c : BitVec w
assign : α → Bool
curr : Nat
hcurr : curr ≤ w
s : aig.RefVec curr
idx : Nat
hidx1 : idx < w
hidx2 : curr ≤ idx
res : RefVecEntry α w
h✝ : curr < w
hgo :
(let res := aig.mkConstCached (c.getLsbD curr);
let aig_1 := res.aig;
let bitRef := res.ref;
let s := s.cast ⋯;
let s := s.push bitRef;
go aig_1 c (curr + 1) s ⋯) =
res
⊢ ⟦assign, { aig := res.aig, ref := res.vec.get idx hidx1 }⟧ = c.getLsbD idx
case isFalse
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
c : BitVec w
assign : α → Bool
curr : Nat
hcurr : curr ≤ w
s : aig.RefVec curr
idx : Nat
hidx1 : idx < w
hidx2 : curr ≤ idx
res : RefVecEntry α w
h✝ : ¬curr < w
hgo :
(let_fun hcurr := ⋯;
{ aig := aig, vec := hcurr ▸ s }) =
res
⊢ ⟦assign, { aig := res.aig, ref := res.vec.get idx hidx1 }⟧ = c.getLsbD idx
|
bf1ceda9c97ad4d3
|
smul_eq_of_le_smul
|
Mathlib/GroupTheory/OrderOfElement.lean
|
lemma smul_eq_of_le_smul
{G : Type*} [Group G] [Finite G] {α : Type*} [PartialOrder α] {g : G} {a : α}
[MulAction G α] [CovariantClass G α HSMul.hSMul LE.le] (h : a ≤ g • a) : g • a = a
|
G : Type u_6
inst✝⁴ : Group G
inst✝³ : Finite G
α : Type u_7
inst✝² : PartialOrder α
g : G
a : α
inst✝¹ : MulAction G α
inst✝ : CovariantClass G α HSMul.hSMul LE.le
h : a ≤ g • a
⊢ g • a = a
|
have key := smul_mono_right g (le_pow_smul h (Nat.card G - 1))
|
G : Type u_6
inst✝⁴ : Group G
inst✝³ : Finite G
α : Type u_7
inst✝² : PartialOrder α
g : G
a : α
inst✝¹ : MulAction G α
inst✝ : CovariantClass G α HSMul.hSMul LE.le
h : a ≤ g • a
key : g • a ≤ g • g ^ (Nat.card G - 1) • a
⊢ g • a = a
|
ddaf2e7f08767f26
|
List.zip_nil_right
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Basic.lean
|
theorem zip_nil_right : zip (l : List α) ([] : List β) = []
|
α : Type u
β : Type v
l : List α
⊢ l.zip nil = nil
|
simp [zip, zipWith]
|
no goals
|
0bf5b75b986e3f15
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.