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
|
|---|---|---|---|---|---|---|
CategoryTheory.Quotient.lift_spec
|
Mathlib/CategoryTheory/Quotient.lean
|
theorem lift_spec : functor r ⋙ lift r F H = F
|
case h_map
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
r : HomRel C
D : Type u_3
inst✝ : Category.{u_4, u_3} D
F : C ⥤ D
H : ∀ (x y : C) (f₁ f₂ : x ⟶ y), r f₁ f₂ → F.map f₁ = F.map f₂
⊢ autoParam (∀ (X Y : C) (f : X ⟶ Y), (functor r ⋙ lift r F H).map f = eqToHom ⋯ ≫ F.map f ≫ eqToHom ⋯) _auto✝
|
rintro X Y f
|
case h_map
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
r : HomRel C
D : Type u_3
inst✝ : Category.{u_4, u_3} D
F : C ⥤ D
H : ∀ (x y : C) (f₁ f₂ : x ⟶ y), r f₁ f₂ → F.map f₁ = F.map f₂
X Y : C
f : X ⟶ Y
⊢ (functor r ⋙ lift r F H).map f = eqToHom ⋯ ≫ F.map f ≫ eqToHom ⋯
|
9549d608fcca58b6
|
SymOptionSuccEquiv.encode_decode
|
Mathlib/Data/Sym/Basic.lean
|
theorem encode_decode [DecidableEq α] (s : Sym (Option α) n ⊕ Sym α n.succ) :
encode (decode s) = s
|
case inl
α : Type u_1
n : ℕ
inst✝ : DecidableEq α
s : Sym (Option α) n
⊢ encode (decode (Sum.inl s)) = Sum.inl s
|
simp
|
no goals
|
35ba1e8a60769782
|
ContinuousMap.compactOpen_eq_generateFrom
|
Mathlib/Topology/ContinuousMap/SecondCountableSpace.lean
|
theorem compactOpen_eq_generateFrom {S : Set (Set X)} {T : Set (Set Y)}
(hS₁ : ∀ K ∈ S, IsCompact K) (hT : IsTopologicalBasis T)
(hS₂ : ∀ f : C(X, Y), ∀ x, ∀ V ∈ T, f x ∈ V → ∃ K ∈ S, K ∈ 𝓝 x ∧ MapsTo f K V) :
compactOpen = .generateFrom (.image2 (fun K t ↦
{f : C(X, Y) | MapsTo f K (⋃₀ t)}) S {t : Set (Set Y) | t.Finite ∧ t ⊆ T})
|
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
S : Set (Set X)
T : Set (Set Y)
hS₁ : ∀ K ∈ S, IsCompact K
hT : IsTopologicalBasis T
hS₂ : ∀ (f : C(X, Y)) (x : X), ∀ V ∈ T, f x ∈ V → ∃ K ∈ S, K ∈ 𝓝 x ∧ MapsTo (⇑f) K V
f : C(X, Y)
K : Set X
hK : IsCompact K
U : Set Y
hU : IsOpen U
hfKU : MapsTo (⇑f) K U
t : Set (Set Y)
htT : t ⊆ T
htf : t.Finite
hTU : ∀ V ∈ t, V ⊆ U
hKT : K ⊆ ⇑f ⁻¹' ⋃₀ t
⊢ ∀ x ∈ K, ∃ L ∈ S, L ∈ 𝓝 x ∧ MapsTo (⇑f) L (⋃₀ t)
|
intro x hx
|
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
S : Set (Set X)
T : Set (Set Y)
hS₁ : ∀ K ∈ S, IsCompact K
hT : IsTopologicalBasis T
hS₂ : ∀ (f : C(X, Y)) (x : X), ∀ V ∈ T, f x ∈ V → ∃ K ∈ S, K ∈ 𝓝 x ∧ MapsTo (⇑f) K V
f : C(X, Y)
K : Set X
hK : IsCompact K
U : Set Y
hU : IsOpen U
hfKU : MapsTo (⇑f) K U
t : Set (Set Y)
htT : t ⊆ T
htf : t.Finite
hTU : ∀ V ∈ t, V ⊆ U
hKT : K ⊆ ⇑f ⁻¹' ⋃₀ t
x : X
hx : x ∈ K
⊢ ∃ L ∈ S, L ∈ 𝓝 x ∧ MapsTo (⇑f) L (⋃₀ t)
|
febdf58bf7ae25f1
|
map_inv_natCast_smul
|
Mathlib/Algebra/Module/Basic.lean
|
theorem map_inv_natCast_smul [AddCommMonoid M] [AddCommMonoid M₂] {F : Type*} [FunLike F M M₂]
[AddMonoidHomClass F M M₂] (f : F) (R S : Type*)
[DivisionSemiring R] [DivisionSemiring S] [Module R M]
[Module S M₂] (n : ℕ) (x : M) : f ((n⁻¹ : R) • x) = (n⁻¹ : S) • f x
|
case pos
M : Type u_3
M₂ : Type u_4
inst✝⁷ : AddCommMonoid M
inst✝⁶ : AddCommMonoid M₂
F : Type u_5
inst✝⁵ : FunLike F M M₂
inst✝⁴ : AddMonoidHomClass F M M₂
f : F
R : Type u_6
S : Type u_7
inst✝³ : DivisionSemiring R
inst✝² : DivisionSemiring S
inst✝¹ : Module R M
inst✝ : Module S M₂
n : ℕ
x : M
hR : ¬↑n = 0
hS : ↑n = 0
⊢ f ((↑n)⁻¹ • x) = (↑n)⁻¹ • f x
|
suffices ∀ y, f y = 0 by simp [this]
|
case pos
M : Type u_3
M₂ : Type u_4
inst✝⁷ : AddCommMonoid M
inst✝⁶ : AddCommMonoid M₂
F : Type u_5
inst✝⁵ : FunLike F M M₂
inst✝⁴ : AddMonoidHomClass F M M₂
f : F
R : Type u_6
S : Type u_7
inst✝³ : DivisionSemiring R
inst✝² : DivisionSemiring S
inst✝¹ : Module R M
inst✝ : Module S M₂
n : ℕ
x : M
hR : ¬↑n = 0
hS : ↑n = 0
⊢ ∀ (y : M), f y = 0
|
acfa63a0edc35b17
|
IsPurelyInseparable.trans
|
Mathlib/FieldTheory/PurelyInseparable/Basic.lean
|
theorem IsPurelyInseparable.trans [Algebra E K] [IsScalarTower F E K]
[h1 : IsPurelyInseparable F E] [h2 : IsPurelyInseparable E K] : IsPurelyInseparable F K
|
case intro.intro.intro
F : Type u
E : Type v
inst✝⁶ : Field F
inst✝⁵ : Field E
inst✝⁴ : Algebra F E
K : Type w
inst✝³ : Field K
inst✝² : Algebra F K
inst✝¹ : Algebra E K
inst✝ : IsScalarTower F E K
q : ℕ
h2✝ : ∀ (x : K), ∃ n, x ^ q ^ n ∈ (algebraMap E K).range
h1 : ∀ (x : E), ∃ n, x ^ q ^ n ∈ (algebraMap F E).range
h✝ : ExpChar F q
this : ExpChar E q
x : K
n : ℕ
y : E
h2 : (algebraMap E K) y = x ^ q ^ n
⊢ ∃ n, x ^ q ^ n ∈ (algebraMap F K).range
|
obtain ⟨m, z, h1⟩ := h1 y
|
case intro.intro.intro.intro.intro
F : Type u
E : Type v
inst✝⁶ : Field F
inst✝⁵ : Field E
inst✝⁴ : Algebra F E
K : Type w
inst✝³ : Field K
inst✝² : Algebra F K
inst✝¹ : Algebra E K
inst✝ : IsScalarTower F E K
q : ℕ
h2✝ : ∀ (x : K), ∃ n, x ^ q ^ n ∈ (algebraMap E K).range
h1✝ : ∀ (x : E), ∃ n, x ^ q ^ n ∈ (algebraMap F E).range
h✝ : ExpChar F q
this : ExpChar E q
x : K
n : ℕ
y : E
h2 : (algebraMap E K) y = x ^ q ^ n
m : ℕ
z : F
h1 : (algebraMap F E) z = y ^ q ^ m
⊢ ∃ n, x ^ q ^ n ∈ (algebraMap F K).range
|
db24f710628c1cfb
|
Finset.Nonempty.norm_prod_le_sup'_norm
|
Mathlib/Analysis/Normed/Group/Ultra.lean
|
/-- Nonarchimedean norm of a product is less than or equal the norm of any term in the product.
This version is phrased using `Finset.sup'` and `Finset.Nonempty` due to `Finset.sup`
operating over an `OrderBot`, which `ℝ` is not.
-/
@[to_additive "Nonarchimedean norm of a sum is less than or equal the norm of any term in the sum.
This version is phrased using `Finset.sup'` and `Finset.Nonempty` due to `Finset.sup`
operating over an `OrderBot`, which `ℝ` is not. "]
lemma _root_.Finset.Nonempty.norm_prod_le_sup'_norm {s : Finset ι} (hs : s.Nonempty) (f : ι → M) :
‖∏ i ∈ s, f i‖ ≤ s.sup' hs (‖f ·‖)
|
case cons.refine_2
M : Type u_1
ι : Type u_2
inst✝¹ : SeminormedCommGroup M
inst✝ : IsUltrametricDist M
s : Finset ι
f : ι → M
j : ι
t : Finset ι
hj : j ∉ t
hs✝ : t.Nonempty
IH : ∃ b ∈ t, ‖∏ i ∈ t, f i‖ ≤ ‖f b‖
h : ‖f j‖ ≤ ‖∏ i ∈ t, f i‖
⊢ ∃ a ∈ t, ‖f j * ∏ i ∈ t, f i‖ ≤ ‖f a‖
|
exact ⟨_, IH.choose_spec.left, (norm_mul_le_max _ _).trans <|
((max_eq_right h).le.trans IH.choose_spec.right)⟩
|
no goals
|
3c2c7b0860a7c76e
|
MeasureTheory.Lp.eLpNorm'_lim_eq_lintegral_liminf
|
Mathlib/MeasureTheory/Function/LpSpace/Basic.lean
|
theorem eLpNorm'_lim_eq_lintegral_liminf {ι} [Nonempty ι] [LinearOrder ι] {f : ι → α → G} {p : ℝ}
{f_lim : α → G} (h_lim : ∀ᵐ x : α ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))) :
eLpNorm' f_lim p μ = (∫⁻ a, atTop.liminf (‖f · a‖ₑ ^ p) ∂μ) ^ (1 / p)
|
α : Type u_1
G : Type u_6
m0 : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup G
ι : Type u_7
inst✝¹ : Nonempty ι
inst✝ : LinearOrder ι
f : ι → α → G
p : ℝ
f_lim : α → G
h_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))
⊢ eLpNorm' f_lim p μ = (∫⁻ (a : α), liminf (fun x => ‖f x a‖ₑ ^ p) atTop ∂μ) ^ (1 / p)
|
suffices h_no_pow : (∫⁻ a, ‖f_lim a‖ₑ ^ p ∂μ) = ∫⁻ a, atTop.liminf fun m => ‖f m a‖ₑ ^ p ∂μ by
rw [eLpNorm'_eq_lintegral_enorm, h_no_pow]
|
α : Type u_1
G : Type u_6
m0 : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup G
ι : Type u_7
inst✝¹ : Nonempty ι
inst✝ : LinearOrder ι
f : ι → α → G
p : ℝ
f_lim : α → G
h_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))
⊢ ∫⁻ (a : α), ‖f_lim a‖ₑ ^ p ∂μ = ∫⁻ (a : α), liminf (fun m => ‖f m a‖ₑ ^ p) atTop ∂μ
|
0eb42e6f44204d59
|
CategoryTheory.Limits.preservesPushout_symmetry
|
Mathlib/CategoryTheory/Limits/Preserves/Shapes/Pullbacks.lean
|
/-- If `F` preserves the pushout of `f, g`, it also preserves the pushout of `g, f`. -/
lemma preservesPushout_symmetry : PreservesColimit (span g f) G where
preserves {c} hc := ⟨by
apply (IsColimit.precomposeHomEquiv (diagramIsoSpan.{v₂} _).symm _).toFun
apply IsColimit.ofIsoColimit _ (PushoutCocone.isoMk _).symm
apply PushoutCocone.isColimitOfFlip
apply (isColimitMapCoconePushoutCoconeEquiv _ _).toFun
· refine @isColimitOfPreserves _ _ _ _ _ _ _ _ _ ?_ ?_ -- Porting note: more TC coddling
· exact PushoutCocone.flipIsColimit hc
· dsimp
infer_instance⟩
|
case ht.refine_2
C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
G : C ⥤ D
W X Y : C
f : W ⟶ X
g : W ⟶ Y
inst✝ : PreservesColimit (span f g) G
c : Cocone (span g f)
hc : IsColimit c
⊢ PreservesColimit (span f g) G
|
infer_instance
|
no goals
|
a31b72fa98bb4cd6
|
Matrix.inv_mulVec_eq_vec
|
Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean
|
lemma inv_mulVec_eq_vec {A : Matrix n n α} [Invertible A]
{u v : n → α} (hM : u = A.mulVec v) : A⁻¹.mulVec u = v
|
n : Type u'
α : Type v
inst✝³ : Fintype n
inst✝² : DecidableEq n
inst✝¹ : CommRing α
A : Matrix n n α
inst✝ : Invertible A
u v : n → α
hM : u = A *ᵥ v
⊢ A⁻¹ *ᵥ u = v
|
rw [hM, Matrix.mulVec_mulVec, Matrix.inv_mul_of_invertible, Matrix.one_mulVec]
|
no goals
|
973e1e1efd7de833
|
CategoryTheory.Functor.isContinuous_of_coverPreserving
|
Mathlib/CategoryTheory/Sites/CoverPreserving.lean
|
/-- If `F` is cover-preserving and compatible-preserving, then `F` is a continuous functor. -/
@[stacks 00WW "This is basically this Stacks entry."]
lemma Functor.isContinuous_of_coverPreserving (hF₁ : CompatiblePreserving.{w} K F)
(hF₂ : CoverPreserving J K F) : Functor.IsContinuous.{w} F J K where
op_comp_isSheaf_of_types G X S hS x hx
|
case hunique.intro.intro.intro.intro
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
F : C ⥤ D
J : GrothendieckTopology C
K : GrothendieckTopology D
hF₁ : CompatiblePreserving K F
hF₂ : CoverPreserving J K F
G : Sheaf K (Type w)
X : C
S : Sieve X
hS : S ∈ J X
x : FamilyOfElements (F.op ⋙ G.val) S.arrows
hx : x.Compatible
y₁ y₂ : (F.op ⋙ G.val).obj (op X)
hy₁ : x.IsAmalgamation y₁
hy₂ : x.IsAmalgamation y₂
Y : D
Z : C
g : Z ⟶ X
h : Y ⟶ F.obj Z
hg : S.arrows g
H : G.val.map (F.map g).op y₁ = G.val.map (F.map g).op y₂
⊢ G.val.map h.op (G.val.map (F.map g).op y₁) = G.val.map h.op (G.val.map (F.map g).op y₂)
|
rw [H]
|
no goals
|
343ebb90057ceb12
|
Polynomial.EisensteinCriterionAux.isUnit_of_natDegree_eq_zero_of_isPrimitive
|
Mathlib/RingTheory/EisensteinCriterion.lean
|
theorem isUnit_of_natDegree_eq_zero_of_isPrimitive {p q : R[X]}
-- Porting note: stated using `IsPrimitive` which is defeq to old statement.
(hu : IsPrimitive (p * q)) (hpm : p.natDegree = 0) : IsUnit p
|
R : Type u_1
inst✝ : CommRing R
p q : R[X]
hu : (p * q).IsPrimitive
hpm : p.natDegree = 0
⊢ IsUnit (p.coeff 0)
|
refine hu _ ?_
|
R : Type u_1
inst✝ : CommRing R
p q : R[X]
hu : (p * q).IsPrimitive
hpm : p.natDegree = 0
⊢ C (p.coeff 0) ∣ p * q
|
447194fa7b9cf99d
|
Associates.exists_prime_dvd_of_not_inf_one
|
Mathlib/RingTheory/UniqueFactorizationDomain/FactorSet.lean
|
theorem exists_prime_dvd_of_not_inf_one {a b : α} (ha : a ≠ 0) (hb : b ≠ 0)
(h : Associates.mk a ⊓ Associates.mk b ≠ 1) : ∃ p : α, Prime p ∧ p ∣ a ∧ p ∣ b
|
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : UniqueFactorizationMonoid α
a b : α
ha : a ≠ 0
hb : b ≠ 0
h : Associates.mk a ⊓ Associates.mk b ≠ 1
⊢ (Associates.mk a).factors ⊓ (Associates.mk b).factors ≠ 0
|
contrapose! h with hf
|
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : UniqueFactorizationMonoid α
a b : α
ha : a ≠ 0
hb : b ≠ 0
hf : (Associates.mk a).factors ⊓ (Associates.mk b).factors = 0
⊢ Associates.mk a ⊓ Associates.mk b = 1
|
4b4dfcc04fa14351
|
lt_mul_iff_one_lt_left'
|
Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean
|
theorem lt_mul_iff_one_lt_left' [MulRightStrictMono α]
[MulRightReflectLT α] (a : α) {b : α} : a < b * a ↔ 1 < b :=
Iff.trans (by rw [one_mul]) (mul_lt_mul_iff_right a)
|
α : Type u_1
inst✝³ : MulOneClass α
inst✝² : LT α
inst✝¹ : MulRightStrictMono α
inst✝ : MulRightReflectLT α
a b : α
⊢ a < b * a ↔ 1 * a < b * a
|
rw [one_mul]
|
no goals
|
f17213218151fc57
|
reesAlgebra.fg
|
Mathlib/RingTheory/ReesAlgebra.lean
|
theorem reesAlgebra.fg (hI : I.FG) : (reesAlgebra I).FG
|
case h
R : Type u
inst✝ : CommRing R
I : Ideal R
s : Finset R
hs : Ideal.span ↑s = I
⊢ Algebra.adjoin R (⇑(monomial 1) '' ↑s) = Algebra.adjoin R ↑(Submodule.map (monomial 1) (Ideal.span ↑s))
|
change
_ =
Algebra.adjoin R
(Submodule.map (monomial 1 : R →ₗ[R] R[X]) (Submodule.span R ↑s) : Set R[X])
|
case h
R : Type u
inst✝ : CommRing R
I : Ideal R
s : Finset R
hs : Ideal.span ↑s = I
⊢ Algebra.adjoin R (⇑(monomial 1) '' ↑s) = Algebra.adjoin R ↑(Submodule.map (monomial 1) (Submodule.span R ↑s))
|
69fde803654cdf7f
|
CompleteLattice.ωScottContinuous.sup
|
Mathlib/Order/OmegaCompletePartialOrder.lean
|
lemma ωScottContinuous.sup (hf : ωScottContinuous f) (hg : ωScottContinuous g) :
ωScottContinuous (f ⊔ g)
|
α : Type u_2
β : Type u_3
inst✝¹ : OmegaCompletePartialOrder α
inst✝ : CompleteLattice β
f g : α → β
hf : ωScottContinuous f
hg : ωScottContinuous g
⊢ ωScottContinuous (f ⊔ g)
|
rw [← sSup_pair]
|
α : Type u_2
β : Type u_3
inst✝¹ : OmegaCompletePartialOrder α
inst✝ : CompleteLattice β
f g : α → β
hf : ωScottContinuous f
hg : ωScottContinuous g
⊢ ωScottContinuous (SupSet.sSup {f, g})
|
43f8aa09f2e0bd75
|
Nat.Primes.summable_rpow
|
Mathlib/NumberTheory/SumPrimeReciprocals.lean
|
theorem Nat.Primes.summable_rpow {r : ℝ} :
Summable (fun p : Nat.Primes ↦ (p : ℝ) ^ r) ↔ r < -1
|
case pos
r : ℝ
h : r < -1
⊢ Summable fun p => ↑↑p ^ r
|
exact (Real.summable_nat_rpow.mpr h).subtype _
|
no goals
|
1efd7ce0900dcc13
|
MeasureTheory.AEMeasurable.ae_eq_of_forall_setLIntegral_eq
|
Mathlib/MeasureTheory/Function/AEEqOfLIntegral.lean
|
theorem AEMeasurable.ae_eq_of_forall_setLIntegral_eq {f g : α → ℝ≥0∞} (hf : AEMeasurable f μ)
(hg : AEMeasurable g μ) (hfi : ∫⁻ x, f x ∂μ ≠ ∞) (hgi : ∫⁻ x, g x ∂μ ≠ ∞)
(hfg : ∀ ⦃s⦄, MeasurableSet s → μ s < ∞ → ∫⁻ x in s, f x ∂μ = ∫⁻ x in s, g x ∂μ) :
f =ᵐ[μ] g
|
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
hfi : ∫⁻ (x : α), f x ∂μ ≠ ⊤
hgi : ∫⁻ (x : α), g x ∂μ ≠ ⊤
hfg : ∀ ⦃s : Set α⦄, MeasurableSet s → μ s < ⊤ → ∫⁻ (x : α) in s, f x ∂μ = ∫⁻ (x : α) in s, g x ∂μ
hf' : AEFinStronglyMeasurable f μ
hg' : AEFinStronglyMeasurable g μ
s : Set α := hf'.sigmaFiniteSet
t : Set α := hg'.sigmaFiniteSet
this : SigmaFinite (μ.restrict hf'.sigmaFiniteSet)
⊢ f =ᶠ[ae (μ.restrict (s ∪ t))] g
|
have := hg'.sigmaFinite_restrict
|
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
hfi : ∫⁻ (x : α), f x ∂μ ≠ ⊤
hgi : ∫⁻ (x : α), g x ∂μ ≠ ⊤
hfg : ∀ ⦃s : Set α⦄, MeasurableSet s → μ s < ⊤ → ∫⁻ (x : α) in s, f x ∂μ = ∫⁻ (x : α) in s, g x ∂μ
hf' : AEFinStronglyMeasurable f μ
hg' : AEFinStronglyMeasurable g μ
s : Set α := hf'.sigmaFiniteSet
t : Set α := hg'.sigmaFiniteSet
this✝ : SigmaFinite (μ.restrict hf'.sigmaFiniteSet)
this : SigmaFinite (μ.restrict hg'.sigmaFiniteSet)
⊢ f =ᶠ[ae (μ.restrict (s ∪ t))] g
|
928eae5949838171
|
numDerangements_tendsto_inv_e
|
Mathlib/Combinatorics/Derangements/Exponential.lean
|
theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1)))
|
case h
s : ℕ → ℝ := fun n => ∑ k ∈ Finset.range n, (-1) ^ k / ↑k.factorial
this : ∀ (n : ℕ), ↑(numDerangements n) / ↑n.factorial = s (n + 1)
⊢ HasSum (fun i => (-1) ^ i / ↑i.factorial) (exp ℝ (-1))
|
exact expSeries_div_hasSum_exp ℝ (-1 : ℝ)
|
no goals
|
368760d6fa6aef58
|
CategoryTheory.IsPushout.isVanKampen_iff
|
Mathlib/CategoryTheory/Adhesive.lean
|
theorem IsPushout.isVanKampen_iff (H : IsPushout f g h i) :
H.IsVanKampen ↔ IsVanKampenColimit (PushoutCocone.mk h i H.w)
|
case mp.refine_1
C : Type u
inst✝ : Category.{v, u} C
W X Y Z : C
f : W ⟶ X
g : W ⟶ Y
h : X ⟶ Z
i : Y ⟶ Z
H✝ : IsPushout f g h i
H : H✝.IsVanKampen
F' : WalkingSpan ⥤ C
c' : Cocone F'
α : F' ⟶ span f g
fα : c'.pt ⟶ (PushoutCocone.mk h i ⋯).pt
eα : α ≫ (PushoutCocone.mk h i ⋯).ι = c'.ι ≫ (Functor.const WalkingSpan).map fα
hα : NatTrans.Equifibered α
this : F'.map WalkingSpan.Hom.fst ≫ c'.ι.app WalkingSpan.left = F'.map WalkingSpan.Hom.snd ≫ c'.ι.app WalkingSpan.right
⊢ Nonempty (IsColimit c') ↔
IsPushout (F'.map WalkingSpan.Hom.fst) (F'.map WalkingSpan.Hom.snd) (c'.ι.app WalkingSpan.left)
(c'.ι.app WalkingSpan.right)
|
rw [(IsColimit.equivOfNatIsoOfIso (diagramIsoSpan F') c' (PushoutCocone.mk _ _ this)
_).nonempty_congr]
|
case mp.refine_1
C : Type u
inst✝ : Category.{v, u} C
W X Y Z : C
f : W ⟶ X
g : W ⟶ Y
h : X ⟶ Z
i : Y ⟶ Z
H✝ : IsPushout f g h i
H : H✝.IsVanKampen
F' : WalkingSpan ⥤ C
c' : Cocone F'
α : F' ⟶ span f g
fα : c'.pt ⟶ (PushoutCocone.mk h i ⋯).pt
eα : α ≫ (PushoutCocone.mk h i ⋯).ι = c'.ι ≫ (Functor.const WalkingSpan).map fα
hα : NatTrans.Equifibered α
this : F'.map WalkingSpan.Hom.fst ≫ c'.ι.app WalkingSpan.left = F'.map WalkingSpan.Hom.snd ≫ c'.ι.app WalkingSpan.right
⊢ Nonempty (IsColimit (PushoutCocone.mk (c'.ι.app WalkingSpan.left) (c'.ι.app WalkingSpan.right) this)) ↔
IsPushout (F'.map WalkingSpan.Hom.fst) (F'.map WalkingSpan.Hom.snd) (c'.ι.app WalkingSpan.left)
(c'.ι.app WalkingSpan.right)
C : Type u
inst✝ : Category.{v, u} C
W X Y Z : C
f : W ⟶ X
g : W ⟶ Y
h : X ⟶ Z
i : Y ⟶ Z
H✝ : IsPushout f g h i
H : H✝.IsVanKampen
F' : WalkingSpan ⥤ C
c' : Cocone F'
α : F' ⟶ span f g
fα : c'.pt ⟶ (PushoutCocone.mk h i ⋯).pt
eα : α ≫ (PushoutCocone.mk h i ⋯).ι = c'.ι ≫ (Functor.const WalkingSpan).map fα
hα : NatTrans.Equifibered α
this : F'.map WalkingSpan.Hom.fst ≫ c'.ι.app WalkingSpan.left = F'.map WalkingSpan.Hom.snd ≫ c'.ι.app WalkingSpan.right
⊢ (Cocones.precompose (diagramIsoSpan F').inv).obj c' ≅
PushoutCocone.mk (c'.ι.app WalkingSpan.left) (c'.ι.app WalkingSpan.right) this
|
03b3318bb2e454c2
|
Asymptotics.isTheta_bot
|
Mathlib/Analysis/Asymptotics/Theta.lean
|
theorem isTheta_bot : f =Θ[⊥] g
|
α : Type u_1
E : Type u_3
F : Type u_4
inst✝¹ : Norm E
inst✝ : Norm F
f : α → E
g : α → F
⊢ f =Θ[⊥] g
|
simp [IsTheta]
|
no goals
|
cdefb79688fb2d6c
|
norm_div_eq_norm_left
|
Mathlib/Analysis/Normed/Group/Basic.lean
|
@[to_additive]
lemma norm_div_eq_norm_left (x : E) {y : E} (h : ‖y‖ = 0) : ‖x / y‖ = ‖x‖
|
E : Type u_5
inst✝ : SeminormedGroup E
x y : E
h : ‖y‖ = 0
⊢ ‖x / y‖ ≤ ‖x‖
|
simpa [h] using norm_div_le x y
|
no goals
|
4d6cda813de16e1c
|
List.forIn'_pure_yield_eq_foldl
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Monadic.lean
|
theorem forIn'_pure_yield_eq_foldl [Monad m] [LawfulMonad m]
(l : List α) (f : (a : α) → a ∈ l → β → β) (init : β) :
forIn' l init (fun a m b => pure (.yield (f a m b))) =
pure (f := m) (l.attach.foldl (fun b ⟨a, h⟩ => f a h b) init)
|
m : Type u_1 → Type u_2
α : Type u_3
β : Type u_1
inst✝¹ : Monad m
inst✝ : LawfulMonad m
l : List α
f : (a : α) → a ∈ l → β → β
init : β
⊢ (forIn' l init fun a m_1 b => pure (ForInStep.yield (f a m_1 b))) =
pure
(foldl
(fun b x =>
match x with
| ⟨a, h⟩ => f a h b)
init l.attach)
|
simp only [forIn'_eq_foldlM]
|
m : Type u_1 → Type u_2
α : Type u_3
β : Type u_1
inst✝¹ : Monad m
inst✝ : LawfulMonad m
l : List α
f : (a : α) → a ∈ l → β → β
init : β
⊢ ForInStep.value <$>
List.foldlM
(fun b x =>
match b with
| ForInStep.yield b => pure (ForInStep.yield (f x.val ⋯ b))
| ForInStep.done b => pure (ForInStep.done b))
(ForInStep.yield init) l.attach =
pure (foldl (fun b x => f x.val ⋯ b) init l.attach)
|
242f86750c77fe00
|
BoxIntegral.unitPartition.mem_admissibleIndex_of_mem_box
|
Mathlib/Analysis/BoxIntegral/UnitPartition.lean
|
theorem mem_admissibleIndex_of_mem_box {B : Box ι} (hB : hasIntegralVertices B) {x : ι → ℝ}
(hx : x ∈ B) : index n x ∈ admissibleIndex n B
|
case intro.intro.intro.refine_2
ι : Type u_1
n : ℕ
inst✝¹ : NeZero n
inst✝ : Fintype ι
B : Box ι
x : ι → ℝ
hx : x ∈ B
l u : ι → ℤ
hl : ∀ (i : ι), B.lower i = ↑(l i)
hu : ∀ (i : ι), B.upper i = ↑(u i)
i : ι
⊢ (↑⌈↑n * x i⌉ - 1 + 1) / ↑n ≤ ↑(u i)
|
exact (mem_admissibleIndex_of_mem_box_aux₂ n (x i) (u i)).mp ((hu i) ▸ (hx i).2)
|
no goals
|
1dc74fce82988f92
|
Set.MapsTo.iterate_restrict
|
Mathlib/Data/Set/Function.lean
|
theorem MapsTo.iterate_restrict {f : α → α} {s : Set α} (h : MapsTo f s s) (n : ℕ) :
(h.restrict f s s)^[n] = (h.iterate n).restrict _ _ _
|
α : Type u_1
f : α → α
s : Set α
h : MapsTo f s s
n : ℕ
⊢ (restrict f s s h)^[n] = restrict f^[n] s s ⋯
|
funext x
|
case h
α : Type u_1
f : α → α
s : Set α
h : MapsTo f s s
n : ℕ
x : ↑s
⊢ (restrict f s s h)^[n] x = restrict f^[n] s s ⋯ x
|
46f3dbaf8f18289b
|
ContinuousMap.tendsto_of_tendstoLocallyUniformly
|
Mathlib/Topology/UniformSpace/CompactConvergence.lean
|
theorem tendsto_of_tendstoLocallyUniformly (h : TendstoLocallyUniformly (fun i a => F i a) f p) :
Tendsto F p (𝓝 f)
|
α : Type u₁
β : Type u₂
inst✝¹ : TopologicalSpace α
inst✝ : UniformSpace β
f : C(α, β)
ι : Type u₃
p : Filter ι
F : ι → C(α, β)
h : TendstoLocallyUniformly (fun i a => (F i) a) (⇑f) p
K : Set α
hK : IsCompact K
⊢ TendstoLocallyUniformlyOn (fun i a => (F i) a) (⇑f) p K
|
exact h.tendstoLocallyUniformlyOn
|
no goals
|
2a076f002ba6ff4c
|
norm_div_pos_iff
|
Mathlib/Analysis/Normed/Group/Basic.lean
|
theorem norm_div_pos_iff : 0 < ‖a / b‖ ↔ a ≠ b
|
E : Type u_5
inst✝ : NormedGroup E
a b : E
⊢ 0 < ‖a / b‖ ↔ a ≠ b
|
rw [(norm_nonneg' _).lt_iff_ne, ne_comm]
|
E : Type u_5
inst✝ : NormedGroup E
a b : E
⊢ ‖a / b‖ ≠ 0 ↔ a ≠ b
|
a5c810d9d84d6c6c
|
SSet.Quasicategory.hornFilling
|
Mathlib/AlgebraicTopology/Quasicategory/Basic.lean
|
lemma Quasicategory.hornFilling {S : SSet} [Quasicategory S] ⦃n : ℕ⦄ ⦃i : Fin (n+1)⦄
(h0 : 0 < i) (hn : i < Fin.last n)
(σ₀ : Λ[n, i] ⟶ S) : ∃ σ : Δ[n] ⟶ S, σ₀ = hornInclusion n i ≫ σ
|
case succ.succ
S : SSet
inst✝ : S.Quasicategory
n : ℕ
i : Fin (n + 1 + 1 + 1)
h0 : 0 < i
hn : i < Fin.last (n + 1 + 1)
σ₀ : Λ[n + 1 + 1, i] ⟶ S
⊢ ∃ σ, σ₀ = hornInclusion (n + 1 + 1) i ≫ σ
|
exact Quasicategory.hornFilling' σ₀ h0 hn
|
no goals
|
7effe73f43c70d45
|
WeierstrassCurve.Projective.negDblY_of_Z_eq_zero
|
Mathlib/AlgebraicGeometry/EllipticCurve/Projective.lean
|
lemma negDblY_of_Z_eq_zero [NoZeroDivisors R] {P : Fin 3 → R} (hP : W'.Equation P) (hPz : P z = 0) :
W'.negDblY P = -P y ^ 4
|
R : Type r
inst✝¹ : CommRing R
W' : Projective R
inst✝ : NoZeroDivisors R
P : Fin 3 → R
hP : W'.Equation P
hPz : P z = 0
⊢ ⋯ - ⋯ - ⋯ * 0 ^ 4 + ⋯ * P y ^ 2 * 0 + ⋯ * 0 * P y * 0 ^ 2 + 9 * W'.a₂ ^ 2 * 0 ^ 4 - 8 * W'.a₂ ^ 2 * 0 * P y ^ 2 * 0 -
9 * W'.a₂ * W'.a₃ * 0 ^ 2 *
P y *
0 +
9 * W'.a₂ * W'.a₄ * 0 ^ 3 *
0 -
4 * W'.a₂ * W'.a₄ * P y ^ 2 *
0 ^ 2 -
27 * W'.a₂ * W'.a₆ * 0 ^ 2 *
0 ^ 2 -
9 * W'.a₃ ^ 2 * 0 ^ 3 * 0 +
6 * W'.a₃ ^ 2 * P y ^ 2 * 0 ^ 2 -
12 * W'.a₃ * W'.a₄ * 0 * P y * 0 ^ 2 +
9 * W'.a₄ ^ 2 * 0 ^ 2 * 0 ^ 2 -
2 * W'.a₁ ^ 3 * 0 ^ 3 * P y +
W'.a₁ ^ 3 * P y ^ 3 * 0 +
3 * W'.a₁ ^ 2 * W'.a₂ * 0 ^ 4 +
2 * W'.a₁ ^ 2 * W'.a₂ * 0 * P y ^ 2 * 0 +
3 * W'.a₁ ^ 2 * W'.a₃ * 0 ^ 2 * P y * 0 +
3 * W'.a₁ ^ 2 * W'.a₄ * 0 ^ 3 * 0 -
W'.a₁ ^ 2 * W'.a₄ * P y ^ 2 * 0 ^ 2 -
12 * W'.a₁ * W'.a₂ ^ 2 * 0 ^ 2 * P y * 0 -
6 * W'.a₁ * W'.a₂ * W'.a₃ * 0 ^ 3 * 0 +
4 * W'.a₁ * W'.a₂ * W'.a₃ * P y ^ 2 * 0 ^ 2 -
8 * W'.a₁ * W'.a₂ * W'.a₄ * 0 * P y * 0 ^ 2 +
6 * W'.a₁ * W'.a₃ ^ 2 * 0 * P y * 0 ^ 2 -
W'.a₁ * W'.a₄ ^ 2 * P y * 0 ^ 3 +
8 * W'.a₂ ^ 3 * 0 ^ 3 * 0 -
8 * W'.a₂ ^ 2 * W'.a₃ * 0 * P y * 0 ^ 2 +
12 * W'.a₂ ^ 2 * W'.a₄ * 0 ^ 2 * 0 ^ 2 -
9 * W'.a₂ * W'.a₃ ^ 2 * 0 ^ 2 * 0 ^ 2 -
4 * W'.a₂ * W'.a₃ * W'.a₄ * P y * 0 ^ 3 +
6 * W'.a₂ * W'.a₄ ^ 2 * 0 * 0 ^ 3 +
W'.a₃ ^ 3 * P y * 0 ^ 3 -
3 * W'.a₃ ^ 2 * W'.a₄ * 0 * 0 ^ 3 +
W'.a₄ ^ 3 * 0 ^ 4 +
W'.a₁ ^ 4 * 0 * P y ^ 2 * 0 -
3 * W'.a₁ ^ 3 * W'.a₂ * 0 ^ 2 * P y * 0 +
W'.a₁ ^ 3 * W'.a₃ * P y ^ 2 * 0 ^ 2 -
2 * W'.a₁ ^ 3 * W'.a₄ * 0 * P y * 0 ^ 2 +
2 * W'.a₁ ^ 2 * W'.a₂ ^ 2 * 0 ^ 3 * 0 -
2 * W'.a₁ ^ 2 * W'.a₂ * W'.a₃ * 0 * P y * 0 ^ 2 +
3 * W'.a₁ ^ 2 * W'.a₂ * W'.a₄ * 0 ^ 2 * 0 ^ 2 -
2 * W'.a₁ ^ 2 * W'.a₃ * W'.a₄ * P y * 0 ^ 3 +
W'.a₁ ^ 2 * W'.a₄ ^ 2 * 0 * 0 ^ 3 +
W'.a₁ * W'.a₂ * W'.a₃ ^ 2 * P y * 0 ^ 3 +
2 * W'.a₁ * W'.a₂ * W'.a₃ * W'.a₄ * 0 * 0 ^ 3 +
W'.a₁ * W'.a₃ * W'.a₄ ^ 2 * 0 ^ 4 -
2 * W'.a₂ ^ 2 * W'.a₃ ^ 2 * 0 * 0 ^ 3 -
W'.a₂ * W'.a₃ ^ 2 * W'.a₄ * 0 ^ 4 =
-P y ^ 4
|
ring1
|
no goals
|
70a44ce1db5dd732
|
monotone_of_odd_of_monotoneOn_nonneg
|
Mathlib/Order/Monotone/Odd.lean
|
theorem monotone_of_odd_of_monotoneOn_nonneg {f : G → H} (h₁ : ∀ x, f (-x) = -f x)
(h₂ : MonotoneOn f (Ici 0)) : Monotone f
|
G : Type u_1
H : Type u_2
inst✝¹ : LinearOrderedAddCommGroup G
inst✝ : OrderedAddCommGroup H
f : G → H
h₁ : ∀ (x : G), f (-x) = -f x
h₂ : MonotoneOn f (Ici 0)
x : G
hx : x ∈ Iic 0
y : G
hy : y ∈ Iic 0
hxy : x ≤ y
⊢ f (-y) ≤ f (-x)
|
exact h₂ (neg_nonneg.2 hy) (neg_nonneg.2 hx) (neg_le_neg hxy)
|
no goals
|
fca0f3ebe0373e0a
|
Finset.UV.kruskal_katona_helper
|
Mathlib/Combinatorics/SetFamily/KruskalKatona.lean
|
/-- The main Kruskal-Katona helper: use induction with our measure to keep compressing until
we can't any more, which gives a set family which is fully compressed and has the nice properties we
want. -/
private lemma kruskal_katona_helper {r : ℕ} (𝒜 : Finset (Finset (Fin n)))
(h : (𝒜 : Set (Finset (Fin n))).Sized r) :
∃ ℬ : Finset (Finset (Fin n)), #(∂ ℬ) ≤ #(∂ 𝒜) ∧ #𝒜 = #ℬ ∧
(ℬ : Set (Finset (Fin n))).Sized r ∧ ∀ U V, UsefulCompression U V → IsCompressed U V ℬ
|
case inr.intro.mk.intro
n r : ℕ
𝒜 : Finset (Finset (Fin n))
h : Set.Sized r ↑𝒜
usable : Finset (Finset (Fin n) × Finset (Fin n)) :=
filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ∧ ¬IsCompressed t.1 t.2 𝒜) univ
husable : usable.Nonempty
U V : Finset (Fin n)
hUV : (U, V) ∈ usable
t : ∀ x' ∈ usable, #(U, V).1 ≤ #x'.1
⊢ ∃ ℬ,
#(∂ ℬ) ≤ #(∂ 𝒜) ∧
#𝒜 = #ℬ ∧ Set.Sized r ↑ℬ ∧ ∀ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V → IsCompressed U V ℬ
|
rw [mem_filter] at hUV
|
case inr.intro.mk.intro
n r : ℕ
𝒜 : Finset (Finset (Fin n))
h : Set.Sized r ↑𝒜
usable : Finset (Finset (Fin n) × Finset (Fin n)) :=
filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ∧ ¬IsCompressed t.1 t.2 𝒜) univ
husable : usable.Nonempty
U V : Finset (Fin n)
hUV : (U, V) ∈ univ ∧ Finset.UV.UsefulCompression (U, V).1 (U, V).2 ∧ ¬IsCompressed (U, V).1 (U, V).2 𝒜
t : ∀ x' ∈ usable, #(U, V).1 ≤ #x'.1
⊢ ∃ ℬ,
#(∂ ℬ) ≤ #(∂ 𝒜) ∧
#𝒜 = #ℬ ∧ Set.Sized r ↑ℬ ∧ ∀ (U V : Finset (Fin n)), Finset.UV.UsefulCompression U V → IsCompressed U V ℬ
|
ad1c271bd9c8d7ce
|
Nat.frequently_atTop_modEq_one
|
Mathlib/NumberTheory/PrimesCongruentOne.lean
|
theorem frequently_atTop_modEq_one {k : ℕ} (hk0 : k ≠ 0) :
∃ᶠ p in atTop, Nat.Prime p ∧ p ≡ 1 [MOD k]
|
k : ℕ
hk0 : k ≠ 0
n : ℕ
⊢ ∃ b ≥ n, Prime b ∧ b ≡ 1 [MOD k]
|
obtain ⟨p, hp⟩ := exists_prime_gt_modEq_one n hk0
|
case intro
k : ℕ
hk0 : k ≠ 0
n p : ℕ
hp : Prime p ∧ n < p ∧ p ≡ 1 [MOD k]
⊢ ∃ b ≥ n, Prime b ∧ b ≡ 1 [MOD k]
|
3062ae9187c2fafe
|
WittVector.ghostFun_natCast
|
Mathlib/RingTheory/WittVector/Basic.lean
|
theorem ghostFun_natCast (i : ℕ) : ghostFun (i : 𝕎 R) = i :=
show ghostFun i.unaryCast = _ by
induction i <;>
simp [*, Nat.unaryCast, ghostFun_zero, ghostFun_one, ghostFun_add, -Pi.natCast_def]
|
p : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : Fact (Nat.Prime p)
i : ℕ
⊢ WittVector.ghostFun i.unaryCast = ↑i
|
induction i <;>
simp [*, Nat.unaryCast, ghostFun_zero, ghostFun_one, ghostFun_add, -Pi.natCast_def]
|
no goals
|
22bebc9c8a72ffc9
|
PrimeSpectrum.zeroLocus_empty_iff_eq_top
|
Mathlib/RingTheory/Spectrum/Prime/Basic.lean
|
theorem zeroLocus_empty_iff_eq_top {I : Ideal R} : zeroLocus (I : Set R) = ∅ ↔ I = ⊤
|
case mpr
R : Type u
inst✝ : CommSemiring R
I : Ideal R
⊢ I = ⊤ → zeroLocus ↑I = ∅
|
rintro rfl
|
case mpr
R : Type u
inst✝ : CommSemiring R
⊢ zeroLocus ↑⊤ = ∅
|
3bed0d5e08a59c7f
|
CategoryTheory.MorphismProperty.IsStableUnderBaseChange.mk'
|
Mathlib/CategoryTheory/MorphismProperty/Limits.lean
|
theorem IsStableUnderBaseChange.mk' [RespectsIso P]
(hP₂ : ∀ (X Y S : C) (f : X ⟶ S) (g : Y ⟶ S) [HasPullback f g] (_ : P g),
P (pullback.fst f g)) :
IsStableUnderBaseChange P where
of_isPullback {X Y Y' S f g f' g'} sq hg
|
C : Type u
inst✝¹ : Category.{v, u} C
P : MorphismProperty C
inst✝ : P.RespectsIso
hP₂ : ∀ (X Y S : C) (f : X ⟶ S) (g : Y ⟶ S) [inst : HasPullback f g], P g → P (pullback.fst f g)
X Y Y' S : C
f : X ⟶ S
g : Y ⟶ S
f' : Y' ⟶ Y
g' : Y' ⟶ X
sq : IsPullback f' g' g f
hg : P g
this : HasPullback f g
e : Y' ≅ pullback f g := ⋯.isoPullback
⊢ P (pullback.fst f g)
|
exact hP₂ _ _ _ f g hg
|
no goals
|
a7faae7a4ba1aa38
|
ConvexOn.le_right_of_left_le'
|
Mathlib/Analysis/Convex/Function.lean
|
theorem ConvexOn.le_right_of_left_le' (hf : ConvexOn 𝕜 s f) {x y : E} {a b : 𝕜} (hx : x ∈ s)
(hy : y ∈ s) (ha : 0 ≤ a) (hb : 0 < b) (hab : a + b = 1) (hfx : f x ≤ f (a • x + b • y)) :
f (a • x + b • y) ≤ f y
|
𝕜 : Type u_1
E : Type u_2
β : Type u_5
inst✝⁵ : OrderedSemiring 𝕜
inst✝⁴ : AddCommMonoid E
inst✝³ : LinearOrderedCancelAddCommMonoid β
inst✝² : SMul 𝕜 E
inst✝¹ : Module 𝕜 β
inst✝ : OrderedSMul 𝕜 β
s : Set E
f : E → β
hf : ConvexOn 𝕜 s f
x y : E
a b : 𝕜
hx : x ∈ s
hy : y ∈ s
ha : 0 ≤ a
hb : 0 < b
hab : b + a = 1
hfx : f x ≤ f (b • y + a • x)
⊢ f (b • y + a • x) ≤ f y
|
exact hf.le_left_of_right_le' hy hx hb ha hab hfx
|
no goals
|
1b3839f8b3fe1c83
|
EulerProduct.one_sub_inv_eq_geometric_of_summable_norm
|
Mathlib/NumberTheory/EulerProduct/Basic.lean
|
lemma one_sub_inv_eq_geometric_of_summable_norm {f : ℕ →*₀ F} {p : ℕ} (hp : p.Prime)
(hsum : Summable fun x ↦ ‖f x‖) :
(1 - f p)⁻¹ = ∑' (e : ℕ), f (p ^ e)
|
F : Type u_1
inst✝¹ : NormedField F
inst✝ : CompleteSpace F
f : ℕ →*₀ F
p : ℕ
hp : Nat.Prime p
hsum : Summable fun x => ‖f x‖
⊢ Summable fun n => f p ^ n
|
refine Summable.of_norm ?_
|
F : Type u_1
inst✝¹ : NormedField F
inst✝ : CompleteSpace F
f : ℕ →*₀ F
p : ℕ
hp : Nat.Prime p
hsum : Summable fun x => ‖f x‖
⊢ Summable fun a => ‖f p ^ a‖
|
bde940928d778afb
|
List.cons_le_cons_iff
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Lex.lean
|
theorem cons_le_cons_iff [DecidableEq α] [LT α] [DecidableLT α]
[i₀ : Std.Irrefl (· < · : α → α → Prop)]
[i₁ : Std.Asymm (· < · : α → α → Prop)]
[i₂ : Std.Antisymm (¬ · < · : α → α → Prop)]
{a b} {l₁ l₂ : List α} :
(a :: l₁) ≤ (b :: l₂) ↔ a < b ∨ a = b ∧ l₁ ≤ l₂
|
case h
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : LT α
inst✝ : DecidableLT α
i₀ : Std.Irrefl fun x1 x2 => x1 < x2
i₁ : Std.Asymm fun x1 x2 => x1 < x2
i₂ : Std.Antisymm fun x1 x2 => ¬x1 < x2
a b : α
l₁ l₂ : List α
h₁ : ¬b < a
h₂ : ¬Lex (fun x1 x2 => x1 < x2) l₂ l₁
h₃ : ¬a < b
⊢ a = b ∧ ¬Lex (fun x1 x2 => x1 < x2) l₂ l₁
|
exact ⟨i₂.antisymm _ _ h₃ h₁, h₂⟩
|
no goals
|
75deba79ccf9ad38
|
Set.offDiag_union
|
Mathlib/Data/Set/Prod.lean
|
theorem offDiag_union (h : Disjoint s t) :
(s ∪ t).offDiag = s.offDiag ∪ t.offDiag ∪ s ×ˢ t ∪ t ×ˢ s
|
case h.mpr.inl.inr.intro
α : Type u_1
s t : Set α
h : Disjoint s t
x : α × α
h0 : x.2 ∈ s
h1 : x.2 ∈ t
h3 : x.1 = x.2
⊢ False
|
exact Set.disjoint_left.mp h h0 h1
|
no goals
|
e368a686903f78bc
|
cfcₙHom_eq_cfcₙ_extend
|
Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/NonUnital.lean
|
lemma cfcₙHom_eq_cfcₙ_extend {a : A} (g : R → R) (ha : p a) (f : C(σₙ R a, R)₀) :
cfcₙHom ha f = cfcₙ (Function.extend Subtype.val f g) a
|
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : Nontrivial R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R p
a : A
g : R → R
ha : p a
f : C(↑(σₙ R a), R)₀
⊢ ⇑f = (σₙ R a).restrict (Function.extend Subtype.val (⇑f) g)
|
ext
|
case h
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : Nontrivial R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R p
a : A
g : R → R
ha : p a
f : C(↑(σₙ R a), R)₀
x✝ : ↑(σₙ R a)
⊢ f x✝ = (σₙ R a).restrict (Function.extend Subtype.val (⇑f) g) x✝
|
21b58a7c62c97b99
|
TensorProduct.equivFinsuppOfBasisRight_apply_tmul_apply
|
Mathlib/LinearAlgebra/TensorProduct/Basis.lean
|
lemma TensorProduct.equivFinsuppOfBasisRight_apply_tmul_apply
(m : M) (n : N) (i : κ) :
(TensorProduct.equivFinsuppOfBasisRight 𝒞) (m ⊗ₜ n) i =
𝒞.repr n i • m
|
R : Type u_1
M : Type u_3
N : Type u_4
κ : Type u_6
inst✝⁵ : CommSemiring R
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
inst✝² : AddCommMonoid N
inst✝¹ : Module R N
inst✝ : DecidableEq κ
𝒞 : Basis κ R N
m : M
n : N
i : κ
⊢ ((equivFinsuppOfBasisRight 𝒞) (m ⊗ₜ[R] n)) i = (𝒞.repr n) i • m
|
simp only [equivFinsuppOfBasisRight_apply_tmul, Finsupp.mapRange_apply]
|
no goals
|
90ef4be1340cec36
|
AffineSubspace.isConnected_setOf_sOppSide
|
Mathlib/Analysis/Convex/Side.lean
|
theorem isConnected_setOf_sOppSide {s : AffineSubspace ℝ P} {x : P} (hx : x ∉ s)
(h : (s : Set P).Nonempty) : IsConnected { y | s.SOppSide x y }
|
case intro
V : Type u_2
P : Type u_4
inst✝³ : SeminormedAddCommGroup V
inst✝² : NormedSpace ℝ V
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor V P
s : AffineSubspace ℝ P
x : P
hx : x ∉ s
p : P
hp : p ∈ ↑s
this : Nonempty ↥s
⊢ IsConnected ((fun x_1 => x_1.1 • (x -ᵥ p) +ᵥ x_1.2) '' Set.Iio 0 ×ˢ ↑s)
|
refine (isConnected_Iio.prod (isConnected_iff_connectedSpace.2 ?_)).image _
((continuous_fst.smul continuous_const).vadd continuous_snd).continuousOn
|
case intro
V : Type u_2
P : Type u_4
inst✝³ : SeminormedAddCommGroup V
inst✝² : NormedSpace ℝ V
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor V P
s : AffineSubspace ℝ P
x : P
hx : x ∉ s
p : P
hp : p ∈ ↑s
this : Nonempty ↥s
⊢ ConnectedSpace ↑↑s
|
e51bcb5e18670498
|
ZMod.dft_even_iff
|
Mathlib/Analysis/Fourier/ZMod.lean
|
/-- The discrete Fourier transform of `Φ` is even if and only if `Φ` itself is. -/
lemma dft_even_iff {Φ : ZMod N → ℂ} : (𝓕 Φ).Even ↔ Φ.Even
|
N : ℕ
inst✝ : NeZero N
Φ : ZMod N → ℂ
h : ∀ {f : ZMod N → ℂ}, Function.Even f → Function.Even (𝓕 f)
⊢ Function.Even (𝓕 Φ) ↔ Function.Even Φ
|
refine ⟨fun hΦ x ↦ ?_, h⟩
|
N : ℕ
inst✝ : NeZero N
Φ : ZMod N → ℂ
h : ∀ {f : ZMod N → ℂ}, Function.Even f → Function.Even (𝓕 f)
hΦ : Function.Even (𝓕 Φ)
x : ZMod N
⊢ Φ (-x) = Φ x
|
16cf2a81b38e6f67
|
DividedPowers.ext
|
Mathlib/RingTheory/DividedPowers/Basic.lean
|
theorem DividedPowers.ext (hI : DividedPowers I) (hI' : DividedPowers I)
(h_eq : ∀ (n : ℕ) {x : A} (_ : x ∈ I), hI.dpow n x = hI'.dpow n x) :
hI = hI'
|
case mk
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI' : DividedPowers I
hI : ℕ → A → A
h₀ : ∀ {n : ℕ} {x : A}, x ∉ I → hI n x = 0
dpow_zero✝ : ∀ {x : A}, x ∈ I → hI 0 x = 1
dpow_one✝ : ∀ {x : A}, x ∈ I → hI 1 x = x
dpow_mem✝ : ∀ {n : ℕ} {x : A}, n ≠ 0 → x ∈ I → hI n x ∈ I
dpow_add✝ : ∀ {n : ℕ} {x y : A}, x ∈ I → y ∈ I → hI n (x + y) = ∑ k ∈ antidiagonal n, hI k.1 x * hI k.2 y
dpow_mul✝ : ∀ {n : ℕ} {a x : A}, x ∈ I → hI n (a * x) = a ^ n * hI n x
mul_dpow✝ : ∀ {m n : ℕ} {x : A}, x ∈ I → hI m x * hI n x = ↑((m + n).choose m) * hI (m + n) x
dpow_comp✝ : ∀ {m n : ℕ} {x : A}, n ≠ 0 → x ∈ I → hI m (hI n x) = ↑(m.uniformBell n) * hI (m * n) x
h_eq :
∀ (n : ℕ) {x : A},
x ∈ I →
{ dpow := hI, dpow_null := h₀, dpow_zero := dpow_zero✝, dpow_one := dpow_one✝, dpow_mem := dpow_mem✝,
dpow_add := dpow_add✝, dpow_mul := dpow_mul✝, mul_dpow := mul_dpow✝, dpow_comp := dpow_comp✝ }.dpow
n x =
hI'.dpow n x
⊢ { dpow := hI, dpow_null := h₀, dpow_zero := dpow_zero✝, dpow_one := dpow_one✝, dpow_mem := dpow_mem✝,
dpow_add := dpow_add✝, dpow_mul := dpow_mul✝, mul_dpow := mul_dpow✝, dpow_comp := dpow_comp✝ } =
hI'
|
obtain ⟨hI', h₀', _⟩ := hI'
|
case mk.mk
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : ℕ → A → A
h₀ : ∀ {n : ℕ} {x : A}, x ∉ I → hI n x = 0
dpow_zero✝¹ : ∀ {x : A}, x ∈ I → hI 0 x = 1
dpow_one✝¹ : ∀ {x : A}, x ∈ I → hI 1 x = x
dpow_mem✝¹ : ∀ {n : ℕ} {x : A}, n ≠ 0 → x ∈ I → hI n x ∈ I
dpow_add✝¹ : ∀ {n : ℕ} {x y : A}, x ∈ I → y ∈ I → hI n (x + y) = ∑ k ∈ antidiagonal n, hI k.1 x * hI k.2 y
dpow_mul✝¹ : ∀ {n : ℕ} {a x : A}, x ∈ I → hI n (a * x) = a ^ n * hI n x
mul_dpow✝¹ : ∀ {m n : ℕ} {x : A}, x ∈ I → hI m x * hI n x = ↑((m + n).choose m) * hI (m + n) x
dpow_comp✝¹ : ∀ {m n : ℕ} {x : A}, n ≠ 0 → x ∈ I → hI m (hI n x) = ↑(m.uniformBell n) * hI (m * n) x
hI' : ℕ → A → A
h₀' : ∀ {n : ℕ} {x : A}, x ∉ I → hI' n x = 0
dpow_zero✝ : ∀ {x : A}, x ∈ I → hI' 0 x = 1
dpow_one✝ : ∀ {x : A}, x ∈ I → hI' 1 x = x
dpow_mem✝ : ∀ {n : ℕ} {x : A}, n ≠ 0 → x ∈ I → hI' n x ∈ I
dpow_add✝ : ∀ {n : ℕ} {x y : A}, x ∈ I → y ∈ I → hI' n (x + y) = ∑ k ∈ antidiagonal n, hI' k.1 x * hI' k.2 y
dpow_mul✝ : ∀ {n : ℕ} {a x : A}, x ∈ I → hI' n (a * x) = a ^ n * hI' n x
mul_dpow✝ : ∀ {m n : ℕ} {x : A}, x ∈ I → hI' m x * hI' n x = ↑((m + n).choose m) * hI' (m + n) x
dpow_comp✝ : ∀ {m n : ℕ} {x : A}, n ≠ 0 → x ∈ I → hI' m (hI' n x) = ↑(m.uniformBell n) * hI' (m * n) x
h_eq :
∀ (n : ℕ) {x : A},
x ∈ I →
{ dpow := hI, dpow_null := h₀, dpow_zero := dpow_zero✝¹, dpow_one := dpow_one✝¹, dpow_mem := dpow_mem✝¹,
dpow_add := dpow_add✝¹, dpow_mul := dpow_mul✝¹, mul_dpow := mul_dpow✝¹, dpow_comp := dpow_comp✝¹ }.dpow
n x =
{ dpow := hI', dpow_null := h₀', dpow_zero := dpow_zero✝, dpow_one := dpow_one✝, dpow_mem := dpow_mem✝,
dpow_add := dpow_add✝, dpow_mul := dpow_mul✝, mul_dpow := mul_dpow✝, dpow_comp := dpow_comp✝ }.dpow
n x
⊢ { dpow := hI, dpow_null := h₀, dpow_zero := dpow_zero✝¹, dpow_one := dpow_one✝¹, dpow_mem := dpow_mem✝¹,
dpow_add := dpow_add✝¹, dpow_mul := dpow_mul✝¹, mul_dpow := mul_dpow✝¹, dpow_comp := dpow_comp✝¹ } =
{ dpow := hI', dpow_null := h₀', dpow_zero := dpow_zero✝, dpow_one := dpow_one✝, dpow_mem := dpow_mem✝,
dpow_add := dpow_add✝, dpow_mul := dpow_mul✝, mul_dpow := mul_dpow✝, dpow_comp := dpow_comp✝ }
|
3b635b1fbc30763f
|
Real.summable_exp_nat_mul_iff
|
Mathlib/Analysis/SpecialFunctions/Exp.lean
|
lemma summable_exp_nat_mul_iff {a : ℝ} :
Summable (fun n : ℕ ↦ exp (n * a)) ↔ a < 0
|
a : ℝ
⊢ (Summable fun n => rexp (↑n * a)) ↔ a < 0
|
simp only [exp_nat_mul, summable_geometric_iff_norm_lt_one, norm_of_nonneg (exp_nonneg _),
exp_lt_one_iff]
|
no goals
|
d02e87599149c19e
|
Path.trans_range
|
Mathlib/Topology/Path.lean
|
theorem trans_range {a b c : X} (γ₁ : Path a b) (γ₂ : Path b c) :
range (γ₁.trans γ₂) = range γ₁ ∪ range γ₂
|
case h
X : Type u_1
inst✝ : TopologicalSpace X
a b c : X
γ₁ : Path a b
γ₂ : Path b c
x : X
t : ℝ
ht0 : 0 ≤ t
ht1 : t ≤ 1
hxt : γ₂ ⟨t, ⋯⟩ = x
h : t = 0
⊢ { toFun := (fun t => if t ≤ 1 / 2 then γ₁.extend (2 * t) else γ₂.extend (2 * t - 1)) ∘ Subtype.val,
continuous_toFun := ⋯, source' := ⋯, target' := ⋯ }
⟨1 / 2, ⋯⟩ =
x
|
rw [coe_mk_mk, Function.comp_apply, if_pos le_rfl, Subtype.coe_mk,
mul_one_div_cancel (two_ne_zero' ℝ)]
|
case h
X : Type u_1
inst✝ : TopologicalSpace X
a b c : X
γ₁ : Path a b
γ₂ : Path b c
x : X
t : ℝ
ht0 : 0 ≤ t
ht1 : t ≤ 1
hxt : γ₂ ⟨t, ⋯⟩ = x
h : t = 0
⊢ γ₁.extend 1 = x
|
4045b3b0d5c91c52
|
Submodule.basis_of_pid_aux
|
Mathlib/LinearAlgebra/FreeModule/PID.lean
|
theorem Submodule.basis_of_pid_aux [Finite ι] {O : Type*} [AddCommGroup O] [Module R O]
(M N : Submodule R O) (b'M : Basis ι R M) (N_bot : N ≠ ⊥) (N_le_M : N ≤ M) :
∃ y ∈ M, ∃ a : R, a • y ∈ N ∧ ∃ M' ≤ M, ∃ N' ≤ N,
N' ≤ M' ∧ (∀ (c : R) (z : O), z ∈ M' → c • y + z = 0 → c = 0) ∧
(∀ (c : R) (z : O), z ∈ N' → c • a • y + z = 0 → c = 0) ∧
∀ (n') (bN' : Basis (Fin n') R N'),
∃ bN : Basis (Fin (n' + 1)) R N,
∀ (m') (hn'm' : n' ≤ m') (bM' : Basis (Fin m') R M'),
∃ (hnm : n' + 1 ≤ m' + 1) (bM : Basis (Fin (m' + 1)) R M),
∀ as : Fin n' → R,
(∀ i : Fin n', (bN' i : O) = as i • (bM' (Fin.castLE hn'm' i) : O)) →
∃ as' : Fin (n' + 1) → R,
∀ i : Fin (n' + 1), (bN i : O) = as' i • (bM (Fin.castLE hnm i) : O)
|
case neg.intro.intro.intro.refine_1.refine_2.intro
ι : Type u_1
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : IsDomain R
inst✝³ : IsPrincipalIdealRing R
inst✝² : Finite ι
O : Type u_4
inst✝¹ : AddCommGroup O
inst✝ : Module R O
M N : Submodule R O
b'M : Basis ι R ↥M
N_bot : N ≠ ⊥
N_le_M : N ≤ M
this : ∃ ϕ, ∀ (ψ : ↥M →ₗ[R] R), ¬ϕ.submoduleImage N < ψ.submoduleImage N
ϕ : ↥M →ₗ[R] R := this.choose
ϕ_max : ∀ (ψ : ↥M →ₗ[R] R), ¬this.choose.submoduleImage N < ψ.submoduleImage N
a : R := generator (ϕ.submoduleImage N)
a_mem : a ∈ ϕ.submoduleImage N
a_zero : ¬a = 0
y : O
yN : y ∈ N
ϕy_eq : ϕ ⟨y, ⋯⟩ = a
_ϕy_ne_zero : ϕ ⟨y, ⋯⟩ ≠ 0
c : ι → R
hc : ∀ (i : ι), (b'M.coord i) ⟨y, ⋯⟩ = a * c i
val✝ : Fintype ι
y' : O := ∑ i : ι, c i • ↑(b'M i)
y'M : y' ∈ M
mk_y' : ⟨y', y'M⟩ = ∑ i : ι, c i • b'M i
a_smul_y' : a • y' = y
ϕy'_eq : ϕ ⟨y', y'M⟩ = 1
ϕy'_ne_zero : ϕ ⟨y', y'M⟩ ≠ 0
M' : Submodule R O := map M.subtype (LinearMap.ker ϕ)
N' : Submodule R O := map N.subtype (LinearMap.ker (ϕ ∘ₗ inclusion N_le_M))
M'_le_M : M' ≤ M
N'_le_M' : N' ≤ M'
N'_le_N : N' ≤ N
y'_ortho_M' : ∀ (c : R), ∀ z ∈ M', c • y' + z = 0 → c = 0
ay'_ortho_N' : ∀ (c : R), ∀ z ∈ N', c • a • y' + z = 0 → c = 0
n' : ℕ
bN' : Basis (Fin n') R ↥N'
z : O
zN : z ∈ N
b : R
hb : ϕ ⟨z, ⋯⟩ = generator (ϕ.submoduleImage N) * b
⊢ ∃ c, z + c • y ∈ N'
|
refine ⟨-b, Submodule.mem_map.mpr ⟨⟨_, N.sub_mem zN (N.smul_mem b yN)⟩, ?_, ?_⟩⟩
|
case neg.intro.intro.intro.refine_1.refine_2.intro.refine_1
ι : Type u_1
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : IsDomain R
inst✝³ : IsPrincipalIdealRing R
inst✝² : Finite ι
O : Type u_4
inst✝¹ : AddCommGroup O
inst✝ : Module R O
M N : Submodule R O
b'M : Basis ι R ↥M
N_bot : N ≠ ⊥
N_le_M : N ≤ M
this : ∃ ϕ, ∀ (ψ : ↥M →ₗ[R] R), ¬ϕ.submoduleImage N < ψ.submoduleImage N
ϕ : ↥M →ₗ[R] R := this.choose
ϕ_max : ∀ (ψ : ↥M →ₗ[R] R), ¬this.choose.submoduleImage N < ψ.submoduleImage N
a : R := generator (ϕ.submoduleImage N)
a_mem : a ∈ ϕ.submoduleImage N
a_zero : ¬a = 0
y : O
yN : y ∈ N
ϕy_eq : ϕ ⟨y, ⋯⟩ = a
_ϕy_ne_zero : ϕ ⟨y, ⋯⟩ ≠ 0
c : ι → R
hc : ∀ (i : ι), (b'M.coord i) ⟨y, ⋯⟩ = a * c i
val✝ : Fintype ι
y' : O := ∑ i : ι, c i • ↑(b'M i)
y'M : y' ∈ M
mk_y' : ⟨y', y'M⟩ = ∑ i : ι, c i • b'M i
a_smul_y' : a • y' = y
ϕy'_eq : ϕ ⟨y', y'M⟩ = 1
ϕy'_ne_zero : ϕ ⟨y', y'M⟩ ≠ 0
M' : Submodule R O := map M.subtype (LinearMap.ker ϕ)
N' : Submodule R O := map N.subtype (LinearMap.ker (ϕ ∘ₗ inclusion N_le_M))
M'_le_M : M' ≤ M
N'_le_M' : N' ≤ M'
N'_le_N : N' ≤ N
y'_ortho_M' : ∀ (c : R), ∀ z ∈ M', c • y' + z = 0 → c = 0
ay'_ortho_N' : ∀ (c : R), ∀ z ∈ N', c • a • y' + z = 0 → c = 0
n' : ℕ
bN' : Basis (Fin n') R ↥N'
z : O
zN : z ∈ N
b : R
hb : ϕ ⟨z, ⋯⟩ = generator (ϕ.submoduleImage N) * b
⊢ ⟨z - b • y, ⋯⟩ ∈ LinearMap.ker (ϕ ∘ₗ inclusion N_le_M)
case neg.intro.intro.intro.refine_1.refine_2.intro.refine_2
ι : Type u_1
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : IsDomain R
inst✝³ : IsPrincipalIdealRing R
inst✝² : Finite ι
O : Type u_4
inst✝¹ : AddCommGroup O
inst✝ : Module R O
M N : Submodule R O
b'M : Basis ι R ↥M
N_bot : N ≠ ⊥
N_le_M : N ≤ M
this : ∃ ϕ, ∀ (ψ : ↥M →ₗ[R] R), ¬ϕ.submoduleImage N < ψ.submoduleImage N
ϕ : ↥M →ₗ[R] R := this.choose
ϕ_max : ∀ (ψ : ↥M →ₗ[R] R), ¬this.choose.submoduleImage N < ψ.submoduleImage N
a : R := generator (ϕ.submoduleImage N)
a_mem : a ∈ ϕ.submoduleImage N
a_zero : ¬a = 0
y : O
yN : y ∈ N
ϕy_eq : ϕ ⟨y, ⋯⟩ = a
_ϕy_ne_zero : ϕ ⟨y, ⋯⟩ ≠ 0
c : ι → R
hc : ∀ (i : ι), (b'M.coord i) ⟨y, ⋯⟩ = a * c i
val✝ : Fintype ι
y' : O := ∑ i : ι, c i • ↑(b'M i)
y'M : y' ∈ M
mk_y' : ⟨y', y'M⟩ = ∑ i : ι, c i • b'M i
a_smul_y' : a • y' = y
ϕy'_eq : ϕ ⟨y', y'M⟩ = 1
ϕy'_ne_zero : ϕ ⟨y', y'M⟩ ≠ 0
M' : Submodule R O := map M.subtype (LinearMap.ker ϕ)
N' : Submodule R O := map N.subtype (LinearMap.ker (ϕ ∘ₗ inclusion N_le_M))
M'_le_M : M' ≤ M
N'_le_M' : N' ≤ M'
N'_le_N : N' ≤ N
y'_ortho_M' : ∀ (c : R), ∀ z ∈ M', c • y' + z = 0 → c = 0
ay'_ortho_N' : ∀ (c : R), ∀ z ∈ N', c • a • y' + z = 0 → c = 0
n' : ℕ
bN' : Basis (Fin n') R ↥N'
z : O
zN : z ∈ N
b : R
hb : ϕ ⟨z, ⋯⟩ = generator (ϕ.submoduleImage N) * b
⊢ N.subtype ⟨z - b • y, ⋯⟩ = z + -b • y
|
48ee91e21f4317c3
|
NumberField.RingOfIntegers.HeightOneSpectrum.one_lt_absNorm
|
Mathlib/NumberTheory/NumberField/FinitePlaces.lean
|
/-- The norm of a maximal ideal is `> 1` -/
lemma one_lt_absNorm : 1 < absNorm v.asIdeal
|
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
v : HeightOneSpectrum (𝓞 K)
h : absNorm v.asIdeal ≤ 1
⊢ Finite (𝓞 K ⧸ v.asIdeal)
|
exact (v.asIdeal.fintypeQuotientOfFreeOfNeBot v.ne_bot).finite
|
no goals
|
849af9e987d28ab6
|
List.mem_range'
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Range.lean
|
theorem mem_range' : ∀{n}, m ∈ range' s n step ↔ ∃ i < n, m = s + step * i
| 0 => by simp [range', Nat.not_lt_zero]
| n + 1 => by
have h (i) : i ≤ n ↔ i = 0 ∨ ∃ j, i = succ j ∧ j < n
|
s step m : Nat
⊢ m ∈ range' s 0 step ↔ ∃ i, i < 0 ∧ m = s + step * i
|
simp [range', Nat.not_lt_zero]
|
no goals
|
1c34e10b4040f331
|
ContractingWith.isFixedPt_fixedPoint_iterate
|
Mathlib/Topology/MetricSpace/Contracting.lean
|
theorem isFixedPt_fixedPoint_iterate {n : ℕ} (hf : ContractingWith K f^[n]) :
IsFixedPt f (hf.fixedPoint f^[n])
|
α : Type u_1
inst✝² : MetricSpace α
K : ℝ≥0
f : α → α
inst✝¹ : Nonempty α
inst✝ : CompleteSpace α
n : ℕ
hf : ContractingWith K f^[n]
x : α := fixedPoint f^[n] hf
hx : f^[n] x = x
this✝ : ¬IsFixedPt f x
this : 0 < dist x (f x)
⊢ ↑K * dist x (f x) < dist x (f x)
|
simpa only [NNReal.coe_one, one_mul, NNReal.val_eq_coe] using (mul_lt_mul_right this).mpr hf.left
|
no goals
|
c937df74ec08dd68
|
FormalMultilinearSeries.changeOrigin_eval
|
Mathlib/Analysis/Analytic/ChangeOrigin.lean
|
theorem changeOrigin_eval (h : (‖x‖₊ + ‖y‖₊ : ℝ≥0∞) < p.radius) :
(p.changeOrigin x).sum y = p.sum (x + y)
|
case mk.mk.mk
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
p : FormalMultilinearSeries 𝕜 E F
x y : E
h : ↑‖x‖₊ + ↑‖y‖₊ < p.radius
radius_pos : 0 < p.radius
x_mem_ball : x ∈ EMetric.ball 0 p.radius
y_mem_ball : y ∈ EMetric.ball 0 (p.changeOrigin x).radius
x_add_y_mem_ball : x + y ∈ EMetric.ball 0 p.radius
f : (k : ℕ) × (l : ℕ) × { s // s.card = l } → F :=
fun s => ((p.changeOriginSeriesTerm s.fst s.snd.fst ↑s.snd.snd ⋯) fun x_1 => x) fun x => y
k l : ℕ
s : Finset (Fin (k + l))
hs : s.card = l
⊢ ‖f ⟨k, ⟨l, ⟨s, hs⟩⟩⟩‖₊ ≤ ‖p (k + l)‖₊ * ‖x‖₊ ^ l * ‖y‖₊ ^ k
|
exact p.nnnorm_changeOriginSeriesTerm_apply_le _ _ _ _ _ _
|
no goals
|
ccc324cf6a7a6ae5
|
SimpleGraph.Subgraph.comap_monotone
|
Mathlib/Combinatorics/SimpleGraph/Subgraph.lean
|
theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f)
|
case right
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
f : G →g G'
H H' : G'.Subgraph
h : H ≤ H'
v w : V
⊢ G.Adj v w → H.Adj (f v) (f w) → H'.Adj (f v) (f w)
|
intro
|
case right
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
f : G →g G'
H H' : G'.Subgraph
h : H ≤ H'
v w : V
a✝ : G.Adj v w
⊢ H.Adj (f v) (f w) → H'.Adj (f v) (f w)
|
b65a5ecf003263d1
|
Stonean.epi_iff_surjective
|
Mathlib/Topology/Category/Stonean/Basic.lean
|
/--
A morphism in `Stonean` is an epi iff it is surjective.
-/
lemma epi_iff_surjective {X Y : Stonean} (f : X ⟶ Y) :
Epi f ↔ Function.Surjective f
|
X Y : Stonean
f : X ⟶ Y
h : Epi f
y : ↑Y.toTop
hy : ∀ (a : ↑X.toTop), (ConcreteCategory.hom f) a ≠ y
⊢ False
|
let C := Set.range f
|
X Y : Stonean
f : X ⟶ Y
h : Epi f
y : ↑Y.toTop
hy : ∀ (a : ↑X.toTop), (ConcreteCategory.hom f) a ≠ y
C : Set ((fun X => ↑X.toTop) Y) := Set.range ⇑(ConcreteCategory.hom f)
⊢ False
|
333cfe532dfd4152
|
AkraBazziRecurrence.GrowsPolynomially.mul
|
Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean
|
protected lemma GrowsPolynomially.mul {f g : ℝ → ℝ} (hf : GrowsPolynomially f)
(hg : GrowsPolynomially g) : GrowsPolynomially fun x => f x * g x
|
f g : ℝ → ℝ
hf✝¹ : GrowsPolynomially f
hg✝¹ : GrowsPolynomially g
b : ℝ
hb : b ∈ Set.Ioo 0 1
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf✝ :
∀ᶠ (x : ℝ) in atTop,
∀ u ∈ Set.Icc (b * x) x, (fun x => |f x|) u ∈ Set.Icc (c₁ * (fun x => |f x|) x) (c₂ * (fun x => |f x|) x)
c₃ : ℝ
hc₃_mem : c₃ > 0
c₄ : ℝ
hc₄_mem : c₄ > 0
hg✝ :
∀ᶠ (x : ℝ) in atTop,
∀ u ∈ Set.Icc (b * x) x, (fun x => |g x|) u ∈ Set.Icc (c₃ * (fun x => |g x|) x) (c₄ * (fun x => |g x|) x)
x : ℝ
hf : ∀ u ∈ Set.Icc (b * x) x, |f u| ∈ Set.Icc (c₁ * |f x|) (c₂ * |f x|)
hg : ∀ u ∈ Set.Icc (b * x) x, |g u| ∈ Set.Icc (c₃ * |g x|) (c₄ * |g x|)
u : ℝ
hu : u ∈ Set.Icc (b * x) x
⊢ c₂ * |f x| * (c₄ * |g x|) = c₂ * c₄ * (|f x| * |g x|)
|
ring
|
no goals
|
c399539f7cb9b401
|
MeasureTheory.Measure.eq_withDensity_rnDeriv
|
Mathlib/MeasureTheory/Decomposition/Lebesgue.lean
|
theorem eq_withDensity_rnDeriv {s : Measure α} {f : α → ℝ≥0∞} (hf : Measurable f) (hs : s ⟂ₘ ν)
(hadd : μ = s + ν.withDensity f) : ν.withDensity f = ν.withDensity (μ.rnDeriv ν)
|
case intro.intro
α : Type u_1
m : MeasurableSpace α
μ ν s : Measure α
f : α → ℝ≥0∞
hf : Measurable f
hs : s ⟂ₘ ν
hadd : μ = s + ν.withDensity f
this : μ.HaveLebesgueDecomposition ν
hmeas : Measurable (μ.rnDeriv ν)
hsing : μ.singularPart ν ⟂ₘ ν
hadd' : μ = μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)
⊢ ν.withDensity f = ν.withDensity (μ.rnDeriv ν)
|
obtain ⟨⟨S, hS₁, hS₂, hS₃⟩, ⟨T, hT₁, hT₂, hT₃⟩⟩ := hs, hsing
|
case intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
m : MeasurableSpace α
μ ν s : Measure α
f : α → ℝ≥0∞
hf : Measurable f
hadd : μ = s + ν.withDensity f
this : μ.HaveLebesgueDecomposition ν
hmeas : Measurable (μ.rnDeriv ν)
hadd' : μ = μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)
S : Set α
hS₁ : MeasurableSet S
hS₂ : s S = 0
hS₃ : ν Sᶜ = 0
T : Set α
hT₁ : MeasurableSet T
hT₂ : (μ.singularPart ν) T = 0
hT₃ : ν Tᶜ = 0
⊢ ν.withDensity f = ν.withDensity (μ.rnDeriv ν)
|
fc3f06ec8038d87c
|
Cycle.Chain.imp
|
Mathlib/Data/List/Cycle.lean
|
theorem Chain.imp {r₁ r₂ : α → α → Prop} (H : ∀ a b, r₁ a b → r₂ a b) (p : Chain r₁ s) :
Chain r₂ s
|
case HI
α : Type u_1
s : Cycle α
r₁ r₂ : α → α → Prop
H : ∀ (a b : α), r₁ a b → r₂ a b
a✝¹ : α
l✝ : List α
a✝ : Chain r₁ ↑l✝ → Chain r₂ ↑l✝
p : List.Chain r₁ a✝¹ (l✝ ++ [a✝¹])
⊢ List.Chain r₂ a✝¹ (l✝ ++ [a✝¹])
|
exact p.imp H
|
no goals
|
f7c8f7679a6fc82c
|
Std.DHashMap.Raw.Const.getKeyD_ofList_of_contains_eq_false
|
Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/RawLemmas.lean
|
theorem getKeyD_ofList_of_contains_eq_false [EquivBEq α] [LawfulHashable α]
{l : List (α × β)} {k fallback : α}
(contains_eq_false : (l.map Prod.fst).contains k = false) :
(ofList l).getKeyD k fallback = fallback
|
α : Type u
inst✝³ : BEq α
inst✝² : Hashable α
β : Type v
inst✝¹ : EquivBEq α
inst✝ : LawfulHashable α
l : List (α × β)
k fallback : α
contains_eq_false : (List.map Prod.fst l).contains k = false
⊢ (ofList l).getKeyD k fallback = fallback
|
simp_to_raw using Raw₀.Const.getKeyD_insertMany_empty_list_of_contains_eq_false
|
no goals
|
d55c5bbc0972f7ce
|
Ideal.mem_pointwise_smul_iff_inv_smul_mem
|
Mathlib/RingTheory/Ideal/Pointwise.lean
|
theorem mem_pointwise_smul_iff_inv_smul_mem {a : M} {S : Ideal R} {x : R} :
x ∈ a • S ↔ a⁻¹ • x ∈ S :=
⟨fun h => by simpa using smul_mem_pointwise_smul a⁻¹ _ _ h,
fun h => by simpa using smul_mem_pointwise_smul a _ _ h⟩
|
M : Type u_1
R : Type u_2
inst✝² : Group M
inst✝¹ : Semiring R
inst✝ : MulSemiringAction M R
a : M
S : Ideal R
x : R
h : x ∈ a • S
⊢ a⁻¹ • x ∈ S
|
simpa using smul_mem_pointwise_smul a⁻¹ _ _ h
|
no goals
|
02576b44d4ec2894
|
MeasureTheory.Measure.OuterRegular.of_restrict
|
Mathlib/MeasureTheory/Measure/Regular.lean
|
/-- If the restrictions of a measure to countably many open sets covering the space are
outer regular, then the measure itself is outer regular. -/
lemma of_restrict [OpensMeasurableSpace α] {μ : Measure α} {s : ℕ → Set α}
(h : ∀ n, OuterRegular (μ.restrict (s n))) (h' : ∀ n, IsOpen (s n)) (h'' : univ ⊆ ⋃ n, s n) :
OuterRegular μ
|
α : Type u_1
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
inst✝ : OpensMeasurableSpace α
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).OuterRegular
h' : ∀ (n : ℕ), IsOpen (s n)
h'' : univ ⊆ ⋃ n, s n
r : ℝ≥0∞
hm : ∀ (n : ℕ), MeasurableSet (s n)
A : ℕ → Set α
hAm : ∀ (n : ℕ), MeasurableSet (A n)
hAs : ∀ (n : ℕ), A n ⊆ s n
hAd : Pairwise (Disjoint on A)
hA : MeasurableSet (⋃ n, A n)
hr : r > μ (⋃ n, A n)
HA : μ (⋃ n, A n) < ⊤
δ : ℕ → ℝ≥0∞
δ0 : ∀ (i : ℕ), 0 < δ i
hδε : μ (⋃ n, A n) + ∑' (i : ℕ), δ i < r
⊢ ∀ (n : ℕ), ∃ U ⊇ A n, IsOpen U ∧ μ U < μ (A n) + δ n
|
intro n
|
α : Type u_1
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
inst✝ : OpensMeasurableSpace α
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).OuterRegular
h' : ∀ (n : ℕ), IsOpen (s n)
h'' : univ ⊆ ⋃ n, s n
r : ℝ≥0∞
hm : ∀ (n : ℕ), MeasurableSet (s n)
A : ℕ → Set α
hAm : ∀ (n : ℕ), MeasurableSet (A n)
hAs : ∀ (n : ℕ), A n ⊆ s n
hAd : Pairwise (Disjoint on A)
hA : MeasurableSet (⋃ n, A n)
hr : r > μ (⋃ n, A n)
HA : μ (⋃ n, A n) < ⊤
δ : ℕ → ℝ≥0∞
δ0 : ∀ (i : ℕ), 0 < δ i
hδε : μ (⋃ n, A n) + ∑' (i : ℕ), δ i < r
n : ℕ
⊢ ∃ U ⊇ A n, IsOpen U ∧ μ U < μ (A n) + δ n
|
aae30475f9af97a9
|
Real.fourierCoeff_tsum_comp_add
|
Mathlib/Analysis/Fourier/PoissonSummation.lean
|
theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)}
(hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖)
(m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 f m
|
f : C(ℝ, ℂ)
hf : ∀ (K : Compacts ℝ), Summable fun n => ‖ContinuousMap.restrict (↑K) (f.comp (ContinuousMap.addRight ↑n))‖
m : ℤ
e : C(ℝ, ℂ) := (fourier (-m)).comp { toFun := QuotientAddGroup.mk, continuous_toFun := ⋯ }
neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖ContinuousMap.restrict (↑K) (e * g)‖ = ‖ContinuousMap.restrict (↑K) g‖
eadd : ∀ (n : ℤ), e.comp (ContinuousMap.addRight ↑n) = e
⊢ ∑' (n : ℤ), ∫ (x : ℝ) in 0 ..1, (e * f.comp (ContinuousMap.addRight ↑n)) x =
∑' (n : ℤ), ∫ (x : ℝ) in 0 ..1, (e.comp (ContinuousMap.addRight ↑n) * f.comp (ContinuousMap.addRight ↑n)) x
|
simp_rw [eadd]
|
no goals
|
ec2dd51bd198a0fb
|
AkraBazziRecurrence.GrowsPolynomially.eventually_atTop_nonneg_or_nonpos
|
Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean
|
lemma eventually_atTop_nonneg_or_nonpos (hf : GrowsPolynomially f) :
(∀ᶠ x in atTop, 0 ≤ f x) ∨ (∀ᶠ x in atTop, f x ≤ 0)
|
f : ℝ → ℝ
hf : GrowsPolynomially f
c₁ : ℝ
left✝¹ : c₁ > 0
c₂ : ℝ
left✝ : c₂ > 0
h : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (1 / 2 * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
heq : c₁ = c₂
c : ℝ
hc✝ : ∀ᶠ (x : ℝ) in atTop, f x = c
hneg : c < 0
x : ℝ
hc : f x = c
⊢ f x < 0
|
simpa only [hc]
|
no goals
|
69674f7264744f71
|
Std.DHashMap.Internal.List.mem_alterKey_of_key_ne
|
Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/List/Associative.lean
|
theorem mem_alterKey_of_key_ne {a : α} {f : Option (β a) → Option (β a)}
{l : List ((a : α) × β a)} (p : (a : α) × β a) (hne : p.1 ≠ a) :
p ∈ alterKey a f l ↔ p ∈ l
|
α : Type u
β : α → Type v
inst✝¹ : BEq α
inst✝ : LawfulBEq α
a : α
f : Option (β a) → Option (β a)
l : List ((a : α) × β a)
p : (a : α) × β a
hne : p.fst ≠ a
⊢ p ∈ alterKey a f l ↔ p ∈ l
|
rw [alterKey]
|
α : Type u
β : α → Type v
inst✝¹ : BEq α
inst✝ : LawfulBEq α
a : α
f : Option (β a) → Option (β a)
l : List ((a : α) × β a)
p : (a : α) × β a
hne : p.fst ≠ a
⊢ (p ∈
match f (getValueCast? a l) with
| none => eraseKey a l
| some v => insertEntry a v l) ↔
p ∈ l
|
c3b5d8c9bced3123
|
Set.range_list_getElem?
|
Mathlib/Data/Set/List.lean
|
theorem range_list_getElem? :
range (l[·]? : ℕ → Option α) = insert none (some '' { x | x ∈ l })
|
case refine_2
α : Type u_1
l : List α
⊢ none ∈ range fun x => l[x]?
|
exact ⟨_, getElem?_eq_none_iff.mpr le_rfl⟩
|
no goals
|
ac656803bb38f6d4
|
doublyStochastic_sum_perm_aux
|
Mathlib/Analysis/Convex/Birkhoff.lean
|
/--
If M is a scalar multiple of a doubly stochastic matrix, then it is a conical combination of
permutation matrices. This is most useful when M is a doubly stochastic matrix, in which case
the combination is convex.
This particular formulation is chosen to make the inductive step easier: we no longer need to
rescale each time a permutation matrix is subtracted.
-/
private lemma doublyStochastic_sum_perm_aux (M : Matrix n n R)
(s : R) (hs : 0 ≤ s)
(hM : ∃ M' ∈ doublyStochastic R n, M = s • M') :
∃ w : Equiv.Perm n → R, (∀ σ, 0 ≤ w σ) ∧ ∑ σ, w σ • σ.permMatrix R = M
|
R : Type u_1
n : Type u_2
inst✝² : Fintype n
inst✝¹ : DecidableEq n
inst✝ : LinearOrderedField R
h✝ : Nonempty n
d : ℕ
ih :
∀ m < d,
∀ (M : Matrix n n R) (s : R),
0 ≤ s →
(∃ M' ∈ doublyStochastic R n, M = s • M') →
#(filter (fun i => M i.1 i.2 ≠ 0) univ) = m →
∃ w, (∀ (σ : Equiv.Perm n), 0 ≤ w σ) ∧ ∑ σ : Equiv.Perm n, w σ • Equiv.Perm.permMatrix R σ = M
M : Matrix n n R
s : R
hs : 0 ≤ s
hM : (∀ (i j : n), 0 ≤ M i j) ∧ (∀ (i : n), ∑ j : n, M i j = s) ∧ ∀ (j : n), ∑ i : n, M i j = s
hd : #(filter (fun i => M i.1 i.2 ≠ 0) univ) = d
hs' : 0 < s
σ : Equiv.Perm n
hσ : ∀ (i j : n), M i j = 0 → Equiv.Perm.permMatrix R σ i j = 0
i : n
hi : i ∈ univ
hi' : ∀ x' ∈ univ, M i (σ i) ≤ M x' (σ x')
N : Matrix n n R := M - M i (σ i) • Equiv.Perm.permMatrix R σ
hMi' : 0 < M i (σ i)
s' : R := s - M i (σ i)
⊢ M i (σ i) ≤ ∑ j : n, M i j
|
exact single_le_sum (fun j _ => hM.1 i j) (by simp)
|
no goals
|
fe82d3c50ee8248f
|
MeasureTheory.aestronglyMeasurable_condExpL1CLM
|
Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean
|
theorem aestronglyMeasurable_condExpL1CLM (f : α →₁[μ] F') :
AEStronglyMeasurable[m] (condExpL1CLM F' hm μ f) μ
|
case refine_1
α : Type u_1
F' : Type u_3
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
f : ↥(Lp F' 1 μ)
c : F'
s : Set α
hs : MeasurableSet s
hμs : μ s < ⊤
⊢ AEStronglyMeasurable (↑↑((condExpL1CLM F' hm μ) ↑(simpleFunc.indicatorConst 1 hs ⋯ c))) μ
|
rw [condExpL1CLM_indicatorConst hs hμs.ne c]
|
case refine_1
α : Type u_1
F' : Type u_3
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
f : ↥(Lp F' 1 μ)
c : F'
s : Set α
hs : MeasurableSet s
hμs : μ s < ⊤
⊢ AEStronglyMeasurable (↑↑((condExpInd F' hm μ s) c)) μ
|
3f7d93520dda47e8
|
IntermediateField.Lifts.union_isExtendible
|
Mathlib/FieldTheory/Extension.lean
|
theorem union_isExtendible [alg : Algebra.IsAlgebraic F E]
[Nonempty c] (hext : ∀ σ ∈ c, σ.IsExtendible) :
(union c hc).IsExtendible := fun S ↦ by
let Ω := adjoin F (S : Set E) →ₐ[F] K
have ⟨ω, hω⟩ : ∃ ω : Ω, ∀ π : c, ∃ θ ≥ π.1, ⟨_, ω⟩ ≤ θ ∧ θ.carrier = π.1.1 ⊔ adjoin F S
|
F : Type u_1
E : Type u_2
K : Type u_3
inst✝⁵ : Field F
inst✝⁴ : Field E
inst✝³ : Field K
inst✝² : Algebra F E
inst✝¹ : Algebra F K
c : Set (Lifts F E K)
hc : IsChain (fun x1 x2 => x1 ≤ x2) c
alg : Algebra.IsAlgebraic F E
inst✝ : Nonempty ↑c
hext : ∀ σ ∈ c, σ.IsExtendible
S : Finset E
Ω : Type (max u_2 u_3) := ↥(adjoin F ↑S) →ₐ[F] K
ω : Ω
θ : ↑c → Lifts F E K
ge : ∀ (π : ↑c), θ π ≥ ↑π
hθ : ∀ (π : ↑c), { carrier := adjoin F ↑S, emb := ω } ≤ θ π
eq : ∀ (π : ↑c), (θ π).carrier = (↑π).carrier ⊔ adjoin F ↑S
this : IsChain (fun x1 x2 => x1 ≤ x2) (Set.range θ)
⊢ ↑S ⊆ ↑(union (Set.range θ) this).carrier
|
simp_rw [carrier_union, iSup_range', eq]
|
F : Type u_1
E : Type u_2
K : Type u_3
inst✝⁵ : Field F
inst✝⁴ : Field E
inst✝³ : Field K
inst✝² : Algebra F E
inst✝¹ : Algebra F K
c : Set (Lifts F E K)
hc : IsChain (fun x1 x2 => x1 ≤ x2) c
alg : Algebra.IsAlgebraic F E
inst✝ : Nonempty ↑c
hext : ∀ σ ∈ c, σ.IsExtendible
S : Finset E
Ω : Type (max u_2 u_3) := ↥(adjoin F ↑S) →ₐ[F] K
ω : Ω
θ : ↑c → Lifts F E K
ge : ∀ (π : ↑c), θ π ≥ ↑π
hθ : ∀ (π : ↑c), { carrier := adjoin F ↑S, emb := ω } ≤ θ π
eq : ∀ (π : ↑c), (θ π).carrier = (↑π).carrier ⊔ adjoin F ↑S
this : IsChain (fun x1 x2 => x1 ≤ x2) (Set.range θ)
⊢ ↑S ⊆ ↑(⨆ i, (↑i).carrier ⊔ adjoin F ↑S)
|
f447a1810182010c
|
frontier_univ_prod_eq
|
Mathlib/Topology/Constructions.lean
|
theorem frontier_univ_prod_eq (s : Set Y) :
frontier ((univ : Set X) ×ˢ s) = univ ×ˢ frontier s
|
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set Y
⊢ frontier (univ ×ˢ s) = univ ×ˢ frontier s
|
simp [frontier_prod_eq]
|
no goals
|
b3fc2d5c16ff578b
|
List.fst_lt_add_of_mem_enumFrom
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Nat/Range.lean
|
theorem fst_lt_add_of_mem_enumFrom {x : Nat × α} {n : Nat} {l : List α} (h : x ∈ enumFrom n l) :
x.1 < n + length l
|
case intro
α : Type u_1
n : Nat
l : List α
i : Fin (enumFrom n l).length
h : (enumFrom n l).get i ∈ enumFrom n l
⊢ ((enumFrom n l).get i).fst < n + l.length
|
simpa using i.isLt
|
no goals
|
c1ce1a46fa439db5
|
MeasureTheory.IsSetSemiring.disjointOfUnion_props
|
Mathlib/MeasureTheory/SetSemiring.lean
|
theorem disjointOfUnion_props (hC : IsSetSemiring C) (h1 : ↑J ⊆ C) :
∃ K : Set α → Finset (Set α),
PairwiseDisjoint J K
∧ (∀ i ∈ J, ↑(K i) ⊆ C)
∧ PairwiseDisjoint (⋃ x ∈ J, (K x : Set (Set α))) id
∧ (∀ j ∈ J, ⋃₀ K j ⊆ j)
∧ (∀ j ∈ J, ∅ ∉ K j)
∧ ⋃₀ J = ⋃₀ (⋃ x ∈ J, (K x : Set (Set α)))
|
case h.refine_6
α : Type u_1
C : Set (Set α)
J✝ : Finset (Set α)
hC : IsSetSemiring C
s : Set α
J : Finset (Set α)
hJ : s ∉ J
hind :
↑J ⊆ C →
∃ K,
(↑J).PairwiseDisjoint K ∧
(∀ i ∈ J, ↑(K i) ⊆ C) ∧
(⋃ x ∈ J, ↑(K x)).PairwiseDisjoint id ∧
(∀ j ∈ J, ⋃₀ ↑(K j) ⊆ j) ∧ (∀ j ∈ J, ∅ ∉ K j) ∧ ⋃₀ ↑J = ⋃₀ ⋃ x ∈ J, ↑(K x)
h1 : s ∈ C ∧ ↑J ⊆ C
K : Set α → Finset (Set α)
hK0 : (↑J).PairwiseDisjoint K
hK1 : ∀ i ∈ J, ↑(K i) ⊆ C
hK2 : (⋃ x ∈ J, ↑(K x)).PairwiseDisjoint id
hK3 : ∀ j ∈ J, ⋃₀ ↑(K j) ⊆ j
hK4 : ∀ j ∈ J, ∅ ∉ K j
hK5 : ⋃₀ ↑J = ⋃₀ ⋃ x ∈ J, ↑(K x)
K1 : Set α → Finset (Set α) := fun t => if t = s then hC.disjointOfDiffUnion ⋯ ⋯ else K t
hK1s : K1 s = hC.disjointOfDiffUnion ⋯ ⋯
hK1_of_ne : ∀ (t : Set α), t ≠ s → K1 t = K t
ht1' : ∀ x ∈ J, K1 x = K x
ht2 : ⋃ x ∈ J, ↑(K1 x) = ⋃ x ∈ J, ↑(K x)
⊢ s ∪ ⋃₀ ↑J = ⋃₀ ⋃ x, ⋃ (_ : x = s ∨ x ∈ J), ↑(K1 x)
|
simp only [iUnion_iUnion_eq_or_left, ht2, sUnion_union, ht2, K1]
|
case h.refine_6
α : Type u_1
C : Set (Set α)
J✝ : Finset (Set α)
hC : IsSetSemiring C
s : Set α
J : Finset (Set α)
hJ : s ∉ J
hind :
↑J ⊆ C →
∃ K,
(↑J).PairwiseDisjoint K ∧
(∀ i ∈ J, ↑(K i) ⊆ C) ∧
(⋃ x ∈ J, ↑(K x)).PairwiseDisjoint id ∧
(∀ j ∈ J, ⋃₀ ↑(K j) ⊆ j) ∧ (∀ j ∈ J, ∅ ∉ K j) ∧ ⋃₀ ↑J = ⋃₀ ⋃ x ∈ J, ↑(K x)
h1 : s ∈ C ∧ ↑J ⊆ C
K : Set α → Finset (Set α)
hK0 : (↑J).PairwiseDisjoint K
hK1 : ∀ i ∈ J, ↑(K i) ⊆ C
hK2 : (⋃ x ∈ J, ↑(K x)).PairwiseDisjoint id
hK3 : ∀ j ∈ J, ⋃₀ ↑(K j) ⊆ j
hK4 : ∀ j ∈ J, ∅ ∉ K j
hK5 : ⋃₀ ↑J = ⋃₀ ⋃ x ∈ J, ↑(K x)
K1 : Set α → Finset (Set α) := fun t => if t = s then hC.disjointOfDiffUnion ⋯ ⋯ else K t
hK1s : K1 s = hC.disjointOfDiffUnion ⋯ ⋯
hK1_of_ne : ∀ (t : Set α), t ≠ s → K1 t = K t
ht1' : ∀ x ∈ J, K1 x = K x
ht2 : ⋃ x ∈ J, ↑(K1 x) = ⋃ x ∈ J, ↑(K x)
⊢ s ∪ ⋃₀ ↑J = ⋃₀ ↑(if True then hC.disjointOfDiffUnion ⋯ ⋯ else K s) ∪ ⋃₀ ⋃ x ∈ J, ↑(K x)
|
defbf789029f4cb3
|
FractionalIdeal.isFractional_span_iff
|
Mathlib/RingTheory/FractionalIdeal/Operations.lean
|
theorem isFractional_span_iff {s : Set P} :
IsFractional S (span R s) ↔ ∃ a ∈ S, ∀ b : P, b ∈ s → IsInteger R (a • b) :=
⟨fun ⟨a, a_mem, h⟩ => ⟨a, a_mem, fun b hb => h b (subset_span hb)⟩, fun ⟨a, a_mem, h⟩ =>
⟨a, a_mem, fun _ hb =>
span_induction (hx := hb) h
(by
rw [smul_zero]
exact isInteger_zero)
(fun x y _ _ hx hy => by
rw [smul_add]
exact isInteger_add hx hy)
fun s x _ hx => by
rw [smul_comm]
exact isInteger_smul hx⟩⟩
|
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
s : Set P
x✝³ : ∃ a ∈ S, ∀ b ∈ s, IsInteger R (a • b)
a : R
a_mem : a ∈ S
h : ∀ b ∈ s, IsInteger R (a • b)
x✝² : P
hb : x✝² ∈ span R s
x y : P
x✝¹ : x ∈ span R s
x✝ : y ∈ span R s
hx : IsInteger R (a • x)
hy : IsInteger R (a • y)
⊢ IsInteger R (a • (x + y))
|
rw [smul_add]
|
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
s : Set P
x✝³ : ∃ a ∈ S, ∀ b ∈ s, IsInteger R (a • b)
a : R
a_mem : a ∈ S
h : ∀ b ∈ s, IsInteger R (a • b)
x✝² : P
hb : x✝² ∈ span R s
x y : P
x✝¹ : x ∈ span R s
x✝ : y ∈ span R s
hx : IsInteger R (a • x)
hy : IsInteger R (a • y)
⊢ IsInteger R (a • x + a • y)
|
6aeb6df4d04dfba9
|
MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos'
|
Mathlib/MeasureTheory/Decomposition/SignedHahn.lean
|
theorem exists_subset_restrict_nonpos' (hi₁ : MeasurableSet i) (hi₂ : s i < 0)
(hn : ¬∀ n : ℕ, ¬s ≤[i \ ⋃ l < n, restrictNonposSeq s i l] 0) :
∃ j : Set α, MeasurableSet j ∧ j ⊆ i ∧ s ≤[j] 0 ∧ s j < 0
|
case neg.h
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi₁ : MeasurableSet i
hi₂ : ↑s i < 0
h : ¬s ≤[i] 0
hn : ∃ n, s ≤[i \ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l] 0
k : ℕ := Nat.find hn
hk₂ : s ≤[i \ ⋃ l, ⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l] 0
hmeas : MeasurableSet (⋃ l, ⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
a : ℕ
x : α
a✝ : a < k
hx : x ∈ MeasureTheory.SignedMeasure.restrictNonposSeq s i a
⊢ x ∈ i
|
exact restrictNonposSeq_subset _ hx
|
no goals
|
37ca223bdee11c33
|
Mathlib.Tactic.LinearCombination'.eq_of_add
|
Mathlib/Tactic/LinearCombination'.lean
|
theorem eq_of_add [AddGroup α] (p : (a:α) = b) (H : (a' - b') - (a - b) = 0) : a' = b'
|
α : Type u_1
a a' b b' : α
inst✝ : AddGroup α
p : a - b = 0
H : a' - b' - (a - b) = 0
⊢ a' - b' = 0
|
rwa [sub_eq_zero, p] at H
|
no goals
|
cd0168b15ad88650
|
CategoryTheory.ChosenFiniteProducts.prodComparisonBifunctorNatTrans_comp
|
Mathlib/CategoryTheory/ChosenFiniteProducts.lean
|
theorem prodComparisonBifunctorNatTrans_comp {E : Type u₂} [Category.{v₂} E]
[ChosenFiniteProducts E] (G : D ⥤ E) : prodComparisonBifunctorNatTrans (F ⋙ G) =
whiskerRight (prodComparisonBifunctorNatTrans F) ((whiskeringRight _ _ _).obj G) ≫
whiskerLeft F (whiskerRight (prodComparisonBifunctorNatTrans G)
((whiskeringLeft _ _ _).obj F))
|
case w.h.w.h
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : ChosenFiniteProducts C
D : Type u₁
inst✝³ : Category.{v₁, u₁} D
inst✝² : ChosenFiniteProducts D
F : C ⥤ D
E : Type u₂
inst✝¹ : Category.{v₂, u₂} E
inst✝ : ChosenFiniteProducts E
G : D ⥤ E
x✝¹ x✝ : C
⊢ ((prodComparisonBifunctorNatTrans (F ⋙ G)).app x✝¹).app x✝ =
((whiskerRight (prodComparisonBifunctorNatTrans F) ((whiskeringRight C D E).obj G) ≫
whiskerLeft F (whiskerRight (prodComparisonBifunctorNatTrans G) ((whiskeringLeft C D E).obj F))).app
x✝¹).app
x✝
|
simp [prodComparison_comp]
|
no goals
|
f893aa99a63240f8
|
List.Duplicate.mono_sublist
|
Mathlib/Data/List/Duplicate.lean
|
theorem Duplicate.mono_sublist {l' : List α} (hx : x ∈+ l) (h : l <+ l') : x ∈+ l'
|
case cons₂
α : Type u_1
l : List α
x : α
l' l₁ l₂ : List α
y : α
h : l₁ <+ l₂
IH : x ∈+ l₁ → x ∈+ l₂
hx : x ∈+ y :: l₁
⊢ x ∈+ y :: l₂
|
rw [duplicate_cons_iff] at hx ⊢
|
case cons₂
α : Type u_1
l : List α
x : α
l' l₁ l₂ : List α
y : α
h : l₁ <+ l₂
IH : x ∈+ l₁ → x ∈+ l₂
hx : y = x ∧ x ∈ l₁ ∨ x ∈+ l₁
⊢ y = x ∧ x ∈ l₂ ∨ x ∈+ l₂
|
81309db93c0eb7b6
|
MeasureTheory.Content.innerContent_pos_of_is_mul_left_invariant
|
Mathlib/MeasureTheory/Measure/Content.lean
|
theorem innerContent_pos_of_is_mul_left_invariant [Group G] [IsTopologicalGroup G]
(h3 : ∀ (g : G) {K : Compacts G}, μ (K.map _ <| continuous_mul_left g) = μ K) (K : Compacts G)
(hK : μ K ≠ 0) (U : Opens G) (hU : (U : Set G).Nonempty) : 0 < μ.innerContent U
|
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : Group G
inst✝ : IsTopologicalGroup G
h3 : ∀ (g : G) {K : Compacts G}, μ (Compacts.map (fun b => g * b) ⋯ K) = μ K
K : Compacts G
hK : μ K ≠ 0
U : Opens G
hU : (↑U).Nonempty
this : (interior ↑U).Nonempty
s : Finset G
hs : K.carrier ⊆ ⋃ g ∈ s, (fun x => g * x) ⁻¹' ↑U
⊢ ↑K ⊆ ↑(⨆ g ∈ s, (Opens.comap ↑(Homeomorph.mulLeft g)) U)
|
simpa only [Opens.iSup_def, Opens.coe_comap, Subtype.coe_mk]
|
no goals
|
08eb0f8d74959da1
|
SmoothBumpCovering.embeddingPiTangent_ker_mfderiv
|
Mathlib/Geometry/Manifold/WhitneyEmbedding.lean
|
theorem embeddingPiTangent_ker_mfderiv (x : M) (hx : x ∈ s) :
LinearMap.ker (mfderiv I 𝓘(ℝ, ι → E × ℝ) f.embeddingPiTangent x) = ⊥
|
ι : Type uι
E : Type uE
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : FiniteDimensional ℝ E
H : Type uH
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : IsManifold I ∞ M
inst✝¹ : T2Space M
inst✝ : Fintype ι
s : Set M
f : SmoothBumpCovering ι I M s
x : M
hx : x ∈ s
⊢ LinearMap.ker (mfderiv I 𝓘(ℝ, ι → E × ℝ) (⇑f.embeddingPiTangent) x) = ⊥
|
apply bot_unique
|
case h
ι : Type uι
E : Type uE
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : FiniteDimensional ℝ E
H : Type uH
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : IsManifold I ∞ M
inst✝¹ : T2Space M
inst✝ : Fintype ι
s : Set M
f : SmoothBumpCovering ι I M s
x : M
hx : x ∈ s
⊢ LinearMap.ker (mfderiv I 𝓘(ℝ, ι → E × ℝ) (⇑f.embeddingPiTangent) x) ≤ ⊥
|
a0f8631339390c09
|
Set.preimage_iUnionLift
|
Mathlib/Data/Set/UnionLift.lean
|
theorem preimage_iUnionLift (t : Set β) :
iUnionLift S f hf T hT ⁻¹' t =
inclusion hT ⁻¹' (⋃ i, inclusion (subset_iUnion S i) '' (f i ⁻¹' t))
|
case h.mp
α : Type u_1
ι : Sort u_3
β : Type u_2
S : ι → Set α
f : (i : ι) → ↑(S i) → β
hf : ∀ (i j : ι) (x : α) (hxi : x ∈ S i) (hxj : x ∈ S j), f i ⟨x, hxi⟩ = f j ⟨x, hxj⟩
T : Set α
hT : T ⊆ iUnion S
t : Set β
x : ↑T
⊢ iUnionLift S f hf T hT x ∈ t → ∃ i x_1, f i x_1 ∈ t ∧ inclusion ⋯ x_1 = inclusion hT x
|
rcases mem_iUnion.1 (hT x.prop) with ⟨i, hi⟩
|
case h.mp.intro
α : Type u_1
ι : Sort u_3
β : Type u_2
S : ι → Set α
f : (i : ι) → ↑(S i) → β
hf : ∀ (i j : ι) (x : α) (hxi : x ∈ S i) (hxj : x ∈ S j), f i ⟨x, hxi⟩ = f j ⟨x, hxj⟩
T : Set α
hT : T ⊆ iUnion S
t : Set β
x : ↑T
i : ι
hi : ↑x ∈ S i
⊢ iUnionLift S f hf T hT x ∈ t → ∃ i x_1, f i x_1 ∈ t ∧ inclusion ⋯ x_1 = inclusion hT x
|
553a23558fd28573
|
Ideal.Filtration.submodule_eq_span_le_iff_stable_ge
|
Mathlib/RingTheory/Filtration.lean
|
theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1)
|
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F : I.Filtration M
n₀ : ℕ
F' : Submodule (↥(reesAlgebra I)) (PolynomialModule R M) :=
Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n₀), ⇑(single R i) '' ↑(F.N i))
hF : ∀ n ≥ n₀, I • F.N n = F.N (n + 1)
i✝ i : ℕ
hi : i ≤ n₀
⊢ ⇑(single R i) '' ↑(F.N i) ⊆ ↑F'
|
refine Set.Subset.trans ?_ Submodule.subset_span
|
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F : I.Filtration M
n₀ : ℕ
F' : Submodule (↥(reesAlgebra I)) (PolynomialModule R M) :=
Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n₀), ⇑(single R i) '' ↑(F.N i))
hF : ∀ n ≥ n₀, I • F.N n = F.N (n + 1)
i✝ i : ℕ
hi : i ≤ n₀
⊢ ⇑(single R i) '' ↑(F.N i) ⊆ ⋃ i, ⋃ (_ : i ≤ n₀), ⇑(single R i) '' ↑(F.N i)
|
fba02d553ffa8877
|
AntilipschitzWith.hausdorffMeasure_preimage_le
|
Mathlib/MeasureTheory/Measure/Hausdorff.lean
|
theorem hausdorffMeasure_preimage_le (hf : AntilipschitzWith K f) (hd : 0 ≤ d) (s : Set Y) :
μH[d] (f ⁻¹' s) ≤ (K : ℝ≥0∞) ^ d * μH[d] s
|
case inl.inr.intro.inl
X : Type u_2
Y : Type u_3
inst✝⁵ : EMetricSpace X
inst✝⁴ : EMetricSpace Y
inst✝³ : MeasurableSpace X
inst✝² : BorelSpace X
inst✝¹ : MeasurableSpace Y
inst✝ : BorelSpace Y
f : X → Y
s : Set Y
hf : AntilipschitzWith 0 f
x : X
hx : x ∈ f ⁻¹' s
this : f ⁻¹' s = {x}
hd : 0 ≤ 0
⊢ 1 ≤ μH[0] s
|
exact one_le_hausdorffMeasure_zero_of_nonempty ⟨f x, hx⟩
|
no goals
|
d3852939648a0879
|
CategoryTheory.Functor.mem_mapTriangle_essImage_of_distinguished
|
Mathlib/CategoryTheory/Triangulated/Functor.lean
|
lemma mem_mapTriangle_essImage_of_distinguished
[F.IsTriangulated] [F.mapArrow.EssSurj] (T : Triangle D) (hT : T ∈ distTriang D) :
∃ (T' : Triangle C) (_ : T' ∈ distTriang C), Nonempty (F.mapTriangle.obj T' ≅ T)
|
case intro.intro.intro.intro.intro.intro.intro.intro
C : Type u_1
D : Type u_2
inst✝¹⁴ : Category.{u_4, u_1} C
inst✝¹³ : Category.{u_5, u_2} D
inst✝¹² : HasShift C ℤ
inst✝¹¹ : HasShift D ℤ
F : C ⥤ D
inst✝¹⁰ : F.CommShift ℤ
inst✝⁹ : HasZeroObject C
inst✝⁸ : HasZeroObject D
inst✝⁷ : Preadditive C
inst✝⁶ : Preadditive D
inst✝⁵ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝³ : Pretriangulated C
inst✝² : Pretriangulated D
inst✝¹ : F.IsTriangulated
inst✝ : F.mapArrow.EssSurj
T : Triangle D
hT : T ∈ distinguishedTriangles
X Y : C
f : X ⟶ Y
e₁ : F.obj X ≅ T.obj₁
e₂ : F.obj Y ≅ T.obj₂
w : F.map f ≫ e₂.hom = e₁.hom ≫ T.mor₁
W : C
g : Y ⟶ W
h : W ⟶ (shiftFunctor C 1).obj X
H : Triangle.mk f g h ∈ distinguishedTriangles
⊢ ∃ T', ∃ (_ : T' ∈ distinguishedTriangles), Nonempty (F.mapTriangle.obj T' ≅ T)
|
exact ⟨_, H, ⟨isoTriangleOfIso₁₂ _ _ (F.map_distinguished _ H) hT e₁ e₂ w⟩⟩
|
no goals
|
5c95eca80a040aef
|
toIcoDiv_zsmul_add
|
Mathlib/Algebra/Order/ToIntervalMod.lean
|
theorem toIcoDiv_zsmul_add (a b : α) (m : ℤ) : toIcoDiv hp a (m • p + b) = m + toIcoDiv hp a b
|
α : Type u_1
inst✝ : LinearOrderedAddCommGroup α
hα : Archimedean α
p : α
hp : 0 < p
a b : α
m : ℤ
⊢ toIcoDiv hp a (m • p + b) = m + toIcoDiv hp a b
|
rw [add_comm, toIcoDiv_add_zsmul, add_comm]
|
no goals
|
df5425d9871ec943
|
MeasureTheory.OuterMeasure.IsMetric.borel_le_caratheodory
|
Mathlib/MeasureTheory/Measure/Hausdorff.lean
|
theorem borel_le_caratheodory (hm : IsMetric μ) : borel X ≤ μ.caratheodory
|
X : Type u_2
inst✝ : EMetricSpace X
μ : OuterMeasure X
hm : μ.IsMetric
⊢ MeasurableSpace.generateFrom {s | IsClosed s} ≤ μ.caratheodory
|
refine MeasurableSpace.generateFrom_le fun t ht => μ.isCaratheodory_iff_le.2 fun s => ?_
|
X : Type u_2
inst✝ : EMetricSpace X
μ : OuterMeasure X
hm : μ.IsMetric
t : Set X
ht : t ∈ {s | IsClosed s}
s : Set X
⊢ μ (s ∩ t) + μ (s \ t) ≤ μ s
|
603b48ab3190787f
|
Matrix.PosSemidef.fromBlocks₁₁
|
Mathlib/LinearAlgebra/Matrix/SchurComplement.lean
|
theorem PosSemidef.fromBlocks₁₁ [Fintype m] [DecidableEq m] [Fintype n] {A : Matrix m m 𝕜}
(B : Matrix m n 𝕜) (D : Matrix n n 𝕜) (hA : A.PosDef) [Invertible A] :
(fromBlocks A B Bᴴ D).PosSemidef ↔ (D - Bᴴ * A⁻¹ * B).PosSemidef
|
case mpr
m : Type u_2
n : Type u_3
𝕜 : Type u_5
inst✝⁷ : CommRing 𝕜
inst✝⁶ : StarRing 𝕜
inst✝⁵ : PartialOrder 𝕜
inst✝⁴ : StarOrderedRing 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
⊢ (D - Bᴴ * A⁻¹ * B).PosSemidef →
(D - Bᴴ * A⁻¹ * B).IsHermitian ∧ ∀ (x : m ⊕ n → 𝕜), 0 ≤ star x ⬝ᵥ fromBlocks A B Bᴴ D *ᵥ x
|
refine fun h => ⟨h.1, fun x => ?_⟩
|
case mpr
m : Type u_2
n : Type u_3
𝕜 : Type u_5
inst✝⁷ : CommRing 𝕜
inst✝⁶ : StarRing 𝕜
inst✝⁵ : PartialOrder 𝕜
inst✝⁴ : StarOrderedRing 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
h : (D - Bᴴ * A⁻¹ * B).PosSemidef
x : m ⊕ n → 𝕜
⊢ 0 ≤ star x ⬝ᵥ fromBlocks A B Bᴴ D *ᵥ x
|
80a4e91088bca52a
|
mul_inv_mul_cancel
|
Mathlib/Algebra/GroupWithZero/Basic.lean
|
theorem mul_inv_mul_cancel (a : G₀) : a * a⁻¹ * a = a
|
case pos
G₀ : Type u_2
inst✝ : GroupWithZero G₀
a : G₀
h : a = 0
⊢ a * a⁻¹ * a = a
|
rw [h, inv_zero, mul_zero]
|
no goals
|
c905b8d43fbe3da1
|
BitVec.le_zero_iff
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean
|
theorem le_zero_iff {x : BitVec w} : x ≤ 0#w ↔ x = 0#w
|
case mp
w : Nat
x : BitVec w
h : x ≤ 0#w
⊢ x = 0#w
|
have : x ≥ 0 := not_lt_iff_le.mp not_lt_zero
|
case mp
w : Nat
x : BitVec w
h : x ≤ 0#w
this : x ≥ 0
⊢ x = 0#w
|
d0bc1f43385390a3
|
Topology.IsClosedEmbedding.preimage_closedPoints
|
Mathlib/Topology/JacobsonSpace.lean
|
lemma Topology.IsClosedEmbedding.preimage_closedPoints (hf : IsClosedEmbedding f) :
f ⁻¹' closedPoints Y = closedPoints X
|
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
hf : IsClosedEmbedding f
⊢ f ⁻¹' closedPoints Y = closedPoints X
|
ext x
|
case h
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
hf : IsClosedEmbedding f
x : X
⊢ x ∈ f ⁻¹' closedPoints Y ↔ x ∈ closedPoints X
|
7d8dbc1c3380d18e
|
SimpleGraph.Walk.IsEulerian.even_degree_iff
|
Mathlib/Combinatorics/SimpleGraph/Trails.lean
|
theorem IsEulerian.even_degree_iff {x u v : V} {p : G.Walk u v} (ht : p.IsEulerian) [Fintype V]
[DecidableRel G.Adj] : Even (G.degree x) ↔ u ≠ v → x ≠ u ∧ x ≠ v
|
case h.e'_1.h.e'_3
V : Type u_1
G : SimpleGraph V
inst✝² : DecidableEq V
x u v : V
p : G.Walk u v
ht : p.IsEulerian
inst✝¹ : Fintype V
inst✝ : DecidableRel G.Adj
⊢ (G.incidenceFinset x).val.card = (Multiset.filter (fun e => x ∈ e) ↑p.edges).card
|
congr 1
|
case h.e'_1.h.e'_3.e_a
V : Type u_1
G : SimpleGraph V
inst✝² : DecidableEq V
x u v : V
p : G.Walk u v
ht : p.IsEulerian
inst✝¹ : Fintype V
inst✝ : DecidableRel G.Adj
⊢ (G.incidenceFinset x).val = Multiset.filter (fun e => x ∈ e) ↑p.edges
|
5fed760b6a554578
|
Finset.Colex.erase_le_erase_min'
|
Mathlib/Combinatorics/Colex.lean
|
/-- If `s ≤ t` in colex and `#s ≤ #t`, then `s \ {a} ≤ t \ {min t}` for any `a ∈ s`. -/
lemma erase_le_erase_min' (hst : toColex s ≤ toColex t) (hcard : #s ≤ #t) (ha : a ∈ s) :
toColex (s.erase a) ≤
toColex (t.erase <| min' t <| card_pos.1 <| (card_pos.2 ⟨a, ha⟩).trans_le hcard)
|
case inr.intro.intro.intro.inr.inl
α : Type u_1
inst✝ : LinearOrder α
s t : Finset α
a : α
hcard : #s ≤ #t
ha : a ∈ s
ht : t.Nonempty
m : α := t.min' ht
h' : s ≠ t
hwt : m ∈ t
hws : m ∉ s
hw : ∀ ⦃a : α⦄, m < a → (a ∈ s ↔ a ∈ t)
haw : a < m
⊢ { ofColex := s.erase a } ≤ { ofColex := t.erase m }
|
have : erase t m ⊆ erase s a := by
rintro b hb
rw [mem_erase] at hb ⊢
exact ⟨(haw.trans_le <| min'_le _ _ hb.2).ne',
(hw <| hb.1.lt_of_le' <| min'_le _ _ hb.2).2 hb.2⟩
|
case inr.intro.intro.intro.inr.inl
α : Type u_1
inst✝ : LinearOrder α
s t : Finset α
a : α
hcard : #s ≤ #t
ha : a ∈ s
ht : t.Nonempty
m : α := t.min' ht
h' : s ≠ t
hwt : m ∈ t
hws : m ∉ s
hw : ∀ ⦃a : α⦄, m < a → (a ∈ s ↔ a ∈ t)
haw : a < m
this : t.erase m ⊆ s.erase a
⊢ { ofColex := s.erase a } ≤ { ofColex := t.erase m }
|
94805d4e030ffc34
|
isLUB_Ioo
|
Mathlib/Order/Bounds/Basic.lean
|
theorem isLUB_Ioo {a b : γ} (hab : a < b) : IsLUB (Ioo a b) b
|
γ : Type v
inst✝¹ : SemilatticeInf γ
inst✝ : DenselyOrdered γ
a b : γ
hab : a < b
⊢ IsLUB (Ioo a b) b
|
simpa only [dual_Ioo] using isGLB_Ioo hab.dual
|
no goals
|
7075546f82949525
|
Estimator.improveUntilAux_spec
|
Mathlib/Order/Estimator.lean
|
theorem Estimator.improveUntilAux_spec (a : Thunk α) (p : α → Bool)
[Estimator a ε] [WellFoundedGT (range (bound a : ε → α))] (e : ε) (r : Bool) :
match Estimator.improveUntilAux a p e r with
| .error _ => ¬ p a.get
| .ok e' => p (bound a e')
|
case pos
α : Type u_1
ε : Type u_2
inst✝² : Preorder α
a : Thunk α
p : α → Bool
inst✝¹ : Estimator a ε
inst✝ : WellFoundedGT ↑(range (bound a))
e : ε
r : Bool
h : p (bound a e) = true
⊢ match
if True then pure e
else
match improve a e, ⋯ with
| none, x => Except.error (if r = true then none else some e)
| some e', x => improveUntilAux a p e' true with
| Except.error a_1 => ¬p a.get = true
| Except.ok e' => p (bound a e') = true
|
exact h
|
no goals
|
66dc4c70c5091567
|
Dynamics.coverMincard_univ
|
Mathlib/Dynamics/TopologicalEntropy/CoverEntropy.lean
|
lemma coverMincard_univ (T : X → X) {F : Set X} (h : F.Nonempty) (n : ℕ) :
coverMincard T F univ n = 1
|
X : Type u_1
T : X → X
F : Set X
h : F.Nonempty
n : ℕ
⊢ coverMincard T F univ n = 1
|
apply le_antisymm _ ((one_le_coverMincard_iff T F univ n).2 h)
|
X : Type u_1
T : X → X
F : Set X
h : F.Nonempty
n : ℕ
⊢ coverMincard T F univ n ≤ 1
|
1832f6ac09cf7d05
|
KaehlerDifferential.span_range_derivation
|
Mathlib/RingTheory/Kaehler/Basic.lean
|
theorem KaehlerDifferential.span_range_derivation :
Submodule.span S (Set.range <| KaehlerDifferential.D R S) = ⊤
|
case intro.mk.refine_4.intro
R : Type u
S : Type v
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x✝ : S ⊗[R] S
hx : x✝ ∈ ideal R S
this : x✝ ∈ Submodule.span S (Set.range fun s => 1 ⊗ₜ[R] s - s ⊗ₜ[R] 1)
r : S
x : S ⊗[R] S
hx₁ : x ∈ ideal R S
hx₂ : (ideal R S).toCotangent ⟨x, hx₁⟩ ∈ Submodule.span S (Set.range ⇑(D R S))
⊢ ∃ (hx : r • x ∈ ideal R S), (ideal R S).toCotangent ⟨r • x, hx⟩ ∈ Submodule.span S (Set.range ⇑(D R S))
|
exact ⟨((KaehlerDifferential.ideal R S).restrictScalars S).smul_mem r hx₁,
Submodule.smul_mem _ r hx₂⟩
|
no goals
|
c1fdba8f7521637c
|
schnirelmannDensity_finset
|
Mathlib/Combinatorics/Schnirelmann.lean
|
/-- The Schnirelmann density of any finset is `0`. -/
lemma schnirelmannDensity_finset (A : Finset ℕ) : schnirelmannDensity A = 0
|
A : Finset ℕ
ε : ℝ
hε : 0 < ε
hε₁ : ε ≤ 1
⊢ ∃ n, 0 < n ∧ ↑(#(filter (fun a => a ∈ ↑A) (Ioc 0 n))) / ↑n < ε
|
let n : ℕ := ⌊#A / ε⌋₊ + 1
|
A : Finset ℕ
ε : ℝ
hε : 0 < ε
hε₁ : ε ≤ 1
n : ℕ := ⌊↑(#A) / ε⌋₊ + 1
⊢ ∃ n, 0 < n ∧ ↑(#(filter (fun a => a ∈ ↑A) (Ioc 0 n))) / ↑n < ε
|
8e68ea4bfdcfb76e
|
Polynomial.iterate_derivative_C_mul
|
Mathlib/Algebra/Polynomial/Derivative.lean
|
theorem iterate_derivative_C_mul (a : R) (p : R[X]) (k : ℕ) :
derivative^[k] (C a * p) = C a * derivative^[k] p
|
R : Type u
inst✝ : Semiring R
a : R
p : R[X]
k : ℕ
⊢ (⇑derivative)^[k] (C a * p) = C a * (⇑derivative)^[k] p
|
simp_rw [← smul_eq_C_mul, iterate_derivative_smul]
|
no goals
|
e6a4491999f3fb6b
|
CategoryTheory.Limits.Cofork.app_zero_eq_comp_π_left
|
Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean
|
theorem Cofork.app_zero_eq_comp_π_left (s : Cofork f g) : s.ι.app zero = f ≫ s.π
|
C : Type u
inst✝ : Category.{v, u} C
X Y : C
f g : X ⟶ Y
s : Cofork f g
⊢ s.ι.app zero = f ≫ s.π
|
rw [← s.app_one_eq_π, ← s.w left, parallelPair_map_left]
|
no goals
|
d64991ddd51fc3ff
|
LaurentSeries.single_order_mul_powerSeriesPart
|
Mathlib/RingTheory/LaurentSeries.lean
|
theorem single_order_mul_powerSeriesPart (x : R⸨X⸩) :
(single x.order 1 : R⸨X⸩) * x.powerSeriesPart = x
|
case neg
R : Type u_1
inst✝ : Semiring R
x : R⸨X⸩
n : ℤ
h : 0 ≠ x.coeff n
⊢ order x ≤ n
|
exact order_le_of_coeff_ne_zero h.symm
|
no goals
|
29116b940ffcf48a
|
Rat.substr_num_den'
|
Mathlib/Data/Rat/Lemmas.lean
|
theorem substr_num_den' (q r : ℚ) :
(q - r).num * q.den * r.den = (q.num * r.den - r.num * q.den) * (q - r).den
|
q r : ℚ
⊢ (q - r).num * ↑q.den * ↑r.den = (q.num * ↑r.den - r.num * ↑q.den) * ↑(q - r).den
|
rw [sub_eq_add_neg, sub_eq_add_neg, ← neg_mul, ← num_neg_eq_neg_num, ← den_neg_eq_den r,
add_num_den' q (-r)]
|
no goals
|
e22aecd803aed83c
|
map_le_nonZeroDivisors_of_injective
|
Mathlib/Algebra/GroupWithZero/NonZeroDivisors.lean
|
theorem map_le_nonZeroDivisors_of_injective [NoZeroDivisors M₀'] [MonoidWithZeroHomClass F M₀ M₀']
(f : F) (hf : Injective f) {S : Submonoid M₀} (hS : S ≤ M₀⁰) : S.map f ≤ M₀'⁰
|
case inr.intro.intro
F : Type u_1
M₀ : Type u_2
M₀' : Type u_3
inst✝⁴ : MonoidWithZero M₀
inst✝³ : MonoidWithZero M₀'
inst✝² : FunLike F M₀ M₀'
inst✝¹ : NoZeroDivisors M₀'
inst✝ : MonoidWithZeroHomClass F M₀ M₀'
f : F
hf : Injective ⇑f
S : Submonoid M₀
hS : S ≤ M₀⁰
h✝ : Nontrivial M₀
x : M₀
hx : x ∈ ↑S
hx0 : f x = 0
⊢ False
|
exact zero_not_mem_nonZeroDivisors <| hS <| map_eq_zero_iff f hf |>.mp hx0 ▸ hx
|
no goals
|
c454728866afa604
|
iUnion_Iic_eq_Iio_of_lt_of_tendsto
|
Mathlib/Topology/Order/OrderClosed.lean
|
theorem iUnion_Iic_eq_Iio_of_lt_of_tendsto {ι : Type*} {F : Filter ι} [F.NeBot]
[ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [ClosedIicTopology α]
{a : α} {f : ι → α} (hlt : ∀ i, f i < a) (hlim : Tendsto f F (𝓝 a)) :
⋃ i : ι, Iic (f i) = Iio a
|
case intro
α : Type u
ι : Type u_1
F : Filter ι
inst✝³ : F.NeBot
inst✝² : ConditionallyCompleteLinearOrder α
inst✝¹ : TopologicalSpace α
inst✝ : ClosedIicTopology α
f : ι → α
i : ι
hlt : ∀ (i_1 : ι), f i_1 < f i
hlim : Tendsto f F (𝓝 (f i))
⊢ False
|
exact (hlt i).false
|
no goals
|
01d4e3e5ecc81e53
|
Pell.n_lt_a_pow
|
Mathlib/NumberTheory/PellMatiyasevic.lean
|
theorem n_lt_a_pow : ∀ n : ℕ, n < a ^ n
| 0 => Nat.le_refl 1
| n + 1 => by
have IH := n_lt_a_pow n
have : a ^ n + a ^ n ≤ a ^ n * a
|
a : ℕ
a1 : 1 < a
n : ℕ
IH : n < a ^ n
this : a ^ n + a ^ n ≤ a ^ n * a
⊢ n + 1 < a ^ n + a ^ n
|
exact add_lt_add_of_lt_of_le IH (lt_of_le_of_lt (Nat.zero_le _) IH)
|
no goals
|
e47a66206aecfba3
|
solvableByRad.induction
|
Mathlib/FieldTheory/AbelRuffini.lean
|
theorem induction (P : solvableByRad F E → Prop)
(base : ∀ α : F, P (algebraMap F (solvableByRad F E) α))
(add : ∀ α β : solvableByRad F E, P α → P β → P (α + β))
(neg : ∀ α : solvableByRad F E, P α → P (-α))
(mul : ∀ α β : solvableByRad F E, P α → P β → P (α * β))
(inv : ∀ α : solvableByRad F E, P α → P α⁻¹)
(rad : ∀ α : solvableByRad F E, ∀ n : ℕ, n ≠ 0 → P (α ^ n) → P α) (α : solvableByRad F E) :
P α
|
F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
P : ↥(solvableByRad F E) → Prop
base : ∀ (α : F), P ((algebraMap F ↥(solvableByRad F E)) α)
add : ∀ (α β : ↥(solvableByRad F E)), P α → P β → P (α + β)
neg : ∀ (α : ↥(solvableByRad F E)), P α → P (-α)
mul : ∀ (α β : ↥(solvableByRad F E)), P α → P β → P (α * β)
inv : ∀ (α : ↥(solvableByRad F E)), P α → P α⁻¹
rad : ∀ (α : ↥(solvableByRad F E)) (n : ℕ), n ≠ 0 → P (α ^ n) → P α
⊢ ∀ (α : ↥(solvableByRad F E)), P α
|
suffices ∀ α : E, IsSolvableByRad F α → ∃ β : solvableByRad F E, ↑β = α ∧ P β by
intro α
obtain ⟨α₀, hα₀, Pα⟩ := this α (Subtype.mem α)
convert Pα
exact Subtype.ext hα₀.symm
|
F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
P : ↥(solvableByRad F E) → Prop
base : ∀ (α : F), P ((algebraMap F ↥(solvableByRad F E)) α)
add : ∀ (α β : ↥(solvableByRad F E)), P α → P β → P (α + β)
neg : ∀ (α : ↥(solvableByRad F E)), P α → P (-α)
mul : ∀ (α β : ↥(solvableByRad F E)), P α → P β → P (α * β)
inv : ∀ (α : ↥(solvableByRad F E)), P α → P α⁻¹
rad : ∀ (α : ↥(solvableByRad F E)) (n : ℕ), n ≠ 0 → P (α ^ n) → P α
⊢ ∀ (α : E), IsSolvableByRad F α → ∃ β, ↑β = α ∧ P β
|
856cf0182c98d000
|
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