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
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| id
stringlengths 16
16
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|---|---|---|---|---|---|---|
Finset.prod_ite_zero
|
Mathlib/Algebra/BigOperators/GroupWithZero/Finset.lean
|
lemma prod_ite_zero :
(∏ i ∈ s, if p i then f i else 0) = if ∀ i ∈ s, p i then ∏ i ∈ s, f i else 0
|
case neg
ι : Type u_1
M₀ : Type u_4
inst✝¹ : CommMonoidWithZero M₀
p : ι → Prop
inst✝ : DecidablePred p
f : ι → M₀
s : Finset ι
h : ∃ i ∈ s, ¬p i
⊢ (∏ i ∈ s, if p i then f i else 0) = 0
|
rcases h with ⟨i, hi, hq⟩
|
case neg.intro.intro
ι : Type u_1
M₀ : Type u_4
inst✝¹ : CommMonoidWithZero M₀
p : ι → Prop
inst✝ : DecidablePred p
f : ι → M₀
s : Finset ι
i : ι
hi : i ∈ s
hq : ¬p i
⊢ (∏ i ∈ s, if p i then f i else 0) = 0
|
78166aee35f52bfd
|
FirstOrder.Language.isExtensionPair_iff_exists_embedding_closure_singleton_sup
|
Mathlib/ModelTheory/PartialEquiv.lean
|
theorem isExtensionPair_iff_exists_embedding_closure_singleton_sup :
L.IsExtensionPair M N ↔
∀ (S : L.Substructure M) (_ : S.FG) (f : S ↪[L] N) (m : M),
∃ g : (closure L {m} ⊔ S : L.Substructure M) ↪[L] N, f =
g.comp (Substructure.inclusion le_sup_right)
|
case refine_1.intro.mk.intro.intro
L : Language
M : Type w
N : Type w'
inst✝¹ : L.Structure M
inst✝ : L.Structure N
h : L.IsExtensionPair M N
S : L.Substructure M
S_FG : S.FG
f : ↥S ↪[L] N
m : M
f' : M ≃ₚ[L] N
hf' : f'.dom.FG
mf' : m ∈ (↑⟨f', hf'⟩).dom
ff'1 : (↑⟨{ dom := S, cod := f.toHom.range, toEquiv := f.equivRange }, S_FG⟩).dom ≤ (↑⟨f', hf'⟩).dom
ff'2 :
(↑⟨f', hf'⟩).cod.subtype.comp ((↑⟨f', hf'⟩).toEquiv.toEmbedding.comp (Substructure.inclusion ff'1)) =
(↑⟨{ dom := S, cod := f.toHom.range, toEquiv := f.equivRange }, S_FG⟩).cod.subtype.comp
(↑⟨{ dom := S, cod := f.toHom.range, toEquiv := f.equivRange }, S_FG⟩).toEquiv.toEmbedding
⊢ ∃ g, f = g.comp (Substructure.inclusion ⋯)
|
refine ⟨f'.toEmbedding.comp (Substructure.inclusion ?_), ?_⟩
|
case refine_1.intro.mk.intro.intro.refine_1
L : Language
M : Type w
N : Type w'
inst✝¹ : L.Structure M
inst✝ : L.Structure N
h : L.IsExtensionPair M N
S : L.Substructure M
S_FG : S.FG
f : ↥S ↪[L] N
m : M
f' : M ≃ₚ[L] N
hf' : f'.dom.FG
mf' : m ∈ (↑⟨f', hf'⟩).dom
ff'1 : (↑⟨{ dom := S, cod := f.toHom.range, toEquiv := f.equivRange }, S_FG⟩).dom ≤ (↑⟨f', hf'⟩).dom
ff'2 :
(↑⟨f', hf'⟩).cod.subtype.comp ((↑⟨f', hf'⟩).toEquiv.toEmbedding.comp (Substructure.inclusion ff'1)) =
(↑⟨{ dom := S, cod := f.toHom.range, toEquiv := f.equivRange }, S_FG⟩).cod.subtype.comp
(↑⟨{ dom := S, cod := f.toHom.range, toEquiv := f.equivRange }, S_FG⟩).toEquiv.toEmbedding
⊢ (closure L).toFun {m} ⊔ S ≤ f'.dom
case refine_1.intro.mk.intro.intro.refine_2
L : Language
M : Type w
N : Type w'
inst✝¹ : L.Structure M
inst✝ : L.Structure N
h : L.IsExtensionPair M N
S : L.Substructure M
S_FG : S.FG
f : ↥S ↪[L] N
m : M
f' : M ≃ₚ[L] N
hf' : f'.dom.FG
mf' : m ∈ (↑⟨f', hf'⟩).dom
ff'1 : (↑⟨{ dom := S, cod := f.toHom.range, toEquiv := f.equivRange }, S_FG⟩).dom ≤ (↑⟨f', hf'⟩).dom
ff'2 :
(↑⟨f', hf'⟩).cod.subtype.comp ((↑⟨f', hf'⟩).toEquiv.toEmbedding.comp (Substructure.inclusion ff'1)) =
(↑⟨{ dom := S, cod := f.toHom.range, toEquiv := f.equivRange }, S_FG⟩).cod.subtype.comp
(↑⟨{ dom := S, cod := f.toHom.range, toEquiv := f.equivRange }, S_FG⟩).toEquiv.toEmbedding
⊢ f =
(f'.toEmbedding.comp (Substructure.inclusion ?refine_1.intro.mk.intro.intro.refine_1)).comp
(Substructure.inclusion ⋯)
|
5f72d9d9514858fa
|
le_limsup_mul
|
Mathlib/Topology/Algebra/Order/LiminfLimsup.lean
|
lemma le_limsup_mul (h₁ : 0 ≤ᶠ[f] u) (h₂ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u)
(h₃ : 0 ≤ᶠ[f] v) (h₄ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f v) :
(limsup u f) * liminf v f ≤ limsup (u * v) f
|
ι : Type u_1
f : Filter ι
inst✝ : f.NeBot
u v : ι → ℝ
h₁ : 0 ≤ᶠ[f] u
h₂ : IsBoundedUnder (fun x1 x2 => x1 ≤ x2) f u
h₃ : 0 ≤ᶠ[f] v
h₄ : IsBoundedUnder (fun x1 x2 => x1 ≤ x2) f v
h : IsCoboundedUnder (fun x1 x2 => x1 ≤ x2) f fun x => u x * v x
h' : IsBoundedUnder (fun x1 x2 => x1 ≤ x2) f (u * v)
u0 : 0 ≤ limsup u f
uv : 0 ≤ limsup (u * v) f
a : ℝ
x✝ : a ≥ 0
au : a < limsup u f
b : ℝ
b0 : b ≥ 0
bv : b < liminf v f
c : ℝ
c_ab : c < a * b
⊢ ∃ᶠ (a : ι) in f, c < u a * v a
|
refine ((frequently_lt_of_lt_limsup
(isBoundedUnder_of_eventually_ge h₁).isCoboundedUnder_le au).and_eventually
((eventually_lt_of_lt_liminf bv (isBoundedUnder_of_eventually_ge h₃)).and
(h₁.and h₃))).mono fun x ⟨xa, ⟨xb, u0, _⟩⟩ ↦ ?_
|
ι : Type u_1
f : Filter ι
inst✝ : f.NeBot
u v : ι → ℝ
h₁ : 0 ≤ᶠ[f] u
h₂ : IsBoundedUnder (fun x1 x2 => x1 ≤ x2) f u
h₃ : 0 ≤ᶠ[f] v
h₄ : IsBoundedUnder (fun x1 x2 => x1 ≤ x2) f v
h : IsCoboundedUnder (fun x1 x2 => x1 ≤ x2) f fun x => u x * v x
h' : IsBoundedUnder (fun x1 x2 => x1 ≤ x2) f (u * v)
u0✝ : 0 ≤ limsup u f
uv : 0 ≤ limsup (u * v) f
a : ℝ
x✝¹ : a ≥ 0
au : a < limsup u f
b : ℝ
b0 : b ≥ 0
bv : b < liminf v f
c : ℝ
c_ab : c < a * b
x : ι
x✝ : a < u x ∧ b < v x ∧ 0 x ≤ u x ∧ 0 x ≤ v x
xa : a < u x
xb : b < v x
u0 : 0 x ≤ u x
right✝ : 0 x ≤ v x
⊢ c < u x * v x
|
0975ba9cb811681a
|
MeasureTheory.Martingale.ae_not_tendsto_atTop_atBot
|
Mathlib/Probability/Martingale/BorelCantelli.lean
|
theorem Martingale.ae_not_tendsto_atTop_atBot [IsFiniteMeasure μ] (hf : Martingale f ℱ μ)
(hbdd : ∀ᵐ ω ∂μ, ∀ i, |f (i + 1) ω - f i ω| ≤ R) :
∀ᵐ ω ∂μ, ¬Tendsto (fun n => f n ω) atTop atBot
|
Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
ℱ : Filtration ℕ m0
f : ℕ → Ω → ℝ
R : ℝ≥0
inst✝ : IsFiniteMeasure μ
hf : Martingale f ℱ μ
hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R
⊢ ∀ᵐ (ω : Ω) ∂μ, ¬Tendsto (fun n => f n ω) atTop atBot
|
filter_upwards [hf.bddAbove_range_iff_bddBelow_range hbdd] with ω hω htop using
unbounded_of_tendsto_atBot htop (hω.1 <| bddAbove_range_of_tendsto_atTop_atBot htop)
|
no goals
|
e27ebab8a5fba485
|
Subspace.biUnion_ne_univ_of_top_nmem
|
Mathlib/GroupTheory/CosetCover.lean
|
theorem Subspace.biUnion_ne_univ_of_top_nmem (hs : ⊤ ∉ s) :
⋃ p ∈ s, (p : Set E) ≠ Set.univ
|
k : Type u_1
E : Type u_2
inst✝³ : DivisionRing k
inst✝² : Infinite k
inst✝¹ : AddCommGroup E
inst✝ : Module k E
s : Finset (Subspace k E)
hs : ⊤ ∉ s
hcovers : ⋃ p ∈ s, ↑p = Set.univ
p : Subspace k E
hp : p ∈ s
hfi : (Submodule.toAddSubgroup p).FiniteIndex
this : Finite (E ⧸ p)
⊢ False
|
have : Nontrivial (E ⧸ p) :=
Submodule.Quotient.nontrivial_of_lt_top p (ne_of_mem_of_not_mem hp hs).lt_top
|
k : Type u_1
E : Type u_2
inst✝³ : DivisionRing k
inst✝² : Infinite k
inst✝¹ : AddCommGroup E
inst✝ : Module k E
s : Finset (Subspace k E)
hs : ⊤ ∉ s
hcovers : ⋃ p ∈ s, ↑p = Set.univ
p : Subspace k E
hp : p ∈ s
hfi : (Submodule.toAddSubgroup p).FiniteIndex
this✝ : Finite (E ⧸ p)
this : Nontrivial (E ⧸ p)
⊢ False
|
dc216810ed8fea9b
|
Set.indicator_prod_one
|
Mathlib/Algebra/GroupWithZero/Indicator.lean
|
lemma indicator_prod_one {t : Set κ} {j : κ} :
(s ×ˢ t).indicator (1 : ι × κ → M₀) (i, j) = s.indicator 1 i * t.indicator 1 j
|
ι : Type u_1
κ : Type u_2
M₀ : Type u_4
inst✝ : MulZeroOneClass M₀
s : Set ι
i : ι
t : Set κ
j : κ
⊢ (if i ∈ s ∧ j ∈ t then 1 (i, j) else 0) = (if i ∈ s then 1 i else 0) * if j ∈ t then 1 j else 0
|
split_ifs with h₀ <;> simp only [Pi.one_apply, mul_one, mul_zero] <;> tauto
|
no goals
|
820eedd294002773
|
ENNReal.aemeasurable_of_tendsto'
|
Mathlib/MeasureTheory/Constructions/BorelSpace/Real.lean
|
/-- A limit (over a general filter) of a.e.-measurable `ℝ≥0∞` valued functions is
a.e.-measurable. -/
lemma aemeasurable_of_tendsto' {ι : Type*} {f : ι → α → ℝ≥0∞} {g : α → ℝ≥0∞}
{μ : Measure α} (u : Filter ι) [NeBot u] [IsCountablyGenerated u]
(hf : ∀ i, AEMeasurable (f i) μ) (hlim : ∀ᵐ a ∂μ, Tendsto (fun i ↦ f i a) u (𝓝 (g a))) :
AEMeasurable g μ
|
case intro
α : Type u_1
mα : MeasurableSpace α
ι : Type u_5
f : ι → α → ℝ≥0∞
g : α → ℝ≥0∞
μ : Measure α
u : Filter ι
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
hf : ∀ (i : ι), AEMeasurable (f i) μ
hlim : ∀ᵐ (a : α) ∂μ, Tendsto (fun i => f i a) u (𝓝 (g a))
v : ℕ → ι
hv : Tendsto v atTop u
⊢ AEMeasurable g μ
|
have h'f : ∀ n, AEMeasurable (f (v n)) μ := fun n ↦ hf (v n)
|
case intro
α : Type u_1
mα : MeasurableSpace α
ι : Type u_5
f : ι → α → ℝ≥0∞
g : α → ℝ≥0∞
μ : Measure α
u : Filter ι
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
hf : ∀ (i : ι), AEMeasurable (f i) μ
hlim : ∀ᵐ (a : α) ∂μ, Tendsto (fun i => f i a) u (𝓝 (g a))
v : ℕ → ι
hv : Tendsto v atTop u
h'f : ∀ (n : ℕ), AEMeasurable (f (v n)) μ
⊢ AEMeasurable g μ
|
6eeaf457edae86bb
|
CategoryTheory.Functor.eval_section_injective_of_eventually_injective
|
Mathlib/CategoryTheory/CofilteredSystem.lean
|
theorem eval_section_injective_of_eventually_injective {j}
(Finj : ∀ (i) (f : i ⟶ j), (F.map f).Injective) (i) (f : i ⟶ j) :
(fun s : F.sections => s.val j).Injective
|
case intro.intro.intro
J : Type u
inst✝¹ : Category.{u_1, u} J
F : J ⥤ Type v
inst✝ : IsCofilteredOrEmpty J
j : J
Finj : ∀ (i : J) (f : i ⟶ j), Function.Injective (F.map f)
i : J
f : i ⟶ j
s₀ s₁ : ↑F.sections
k m : J
mi : m ⟶ i
h : F.map (mi ≫ f) (↑s₀ m) = F.map (mi ≫ f) (↑s₁ m)
mk : m ⟶ k
h✝ : True
⊢ F.map mk (↑s₀ m) = F.map mk (↑s₁ m)
|
exact congr_arg _ (Finj m (mi ≫ f) h)
|
no goals
|
925c0614fa26e0a4
|
MvQPF.Cofix.bisim_aux
|
Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean
|
theorem Cofix.bisim_aux {α : TypeVec n} (r : Cofix F α → Cofix F α → Prop) (h' : ∀ x, r x x)
(h : ∀ x y, r x y →
appendFun id (Quot.mk r) <$$> Cofix.dest x = appendFun id (Quot.mk r) <$$> Cofix.dest y) :
∀ x y, r x y → x = y
|
case a
n : ℕ
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
r : Cofix F α → Cofix F α → Prop
h' : ∀ (x : Cofix F α), r x x
h : ∀ (x y : Cofix F α), r x y → (TypeVec.id ::: Quot.mk r) <$$> x.dest = (TypeVec.id ::: Quot.mk r) <$$> y.dest
x y : (P F).M α
rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)
r' : (P F).M α → (P F).M α → Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)
hr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)
a b : (P F).M α
r'ab : r' a b
h₀ :
(TypeVec.id ::: Quot.mk r ∘ Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =
(TypeVec.id ::: Quot.mk r ∘ Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b)
u v : (P F).M α
cuv : Mcongr u v
⊢ r' u v
|
dsimp [r', hr']
|
case a
n : ℕ
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
r : Cofix F α → Cofix F α → Prop
h' : ∀ (x : Cofix F α), r x x
h : ∀ (x y : Cofix F α), r x y → (TypeVec.id ::: Quot.mk r) <$$> x.dest = (TypeVec.id ::: Quot.mk r) <$$> y.dest
x y : (P F).M α
rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)
r' : (P F).M α → (P F).M α → Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)
hr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)
a b : (P F).M α
r'ab : r' a b
h₀ :
(TypeVec.id ::: Quot.mk r ∘ Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =
(TypeVec.id ::: Quot.mk r ∘ Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b)
u v : (P F).M α
cuv : Mcongr u v
⊢ r (Quot.mk Mcongr u) (Quot.mk Mcongr v)
|
0af76fe163f43a26
|
exists_sSupIndep_isCompl_sSup_atoms
|
Mathlib/Order/CompactlyGenerated/Basic.lean
|
theorem exists_sSupIndep_isCompl_sSup_atoms (h : sSup { a : α | IsAtom a } = ⊤) (b : α) :
∃ s : Set α, sSupIndep s ∧
IsCompl b (sSup s) ∧ ∀ ⦃a⦄, a ∈ s → IsAtom a
|
case inr
α : Type u_2
inst✝² : CompleteLattice α
inst✝¹ : IsModularLattice α
inst✝ : IsCompactlyGenerated α
h : sSup {a | IsAtom a} = ⊤
b : α
s : Set α
s_max : ∀ ⦃t : Set α⦄, t ∈ {s | sSupIndep s ∧ Disjoint b (sSup s) ∧ ∀ a ∈ s, IsAtom a} → s ⊆ t → s = t
s_ind : sSupIndep s
b_inf_Sup_s : Disjoint b (sSup s)
s_atoms : ∀ a ∈ s, IsAtom a
a : α
ha : a ∈ {a | IsAtom a}
con : Disjoint a (b ⊔ sSup s)
a_dis_Sup_s : Disjoint a (sSup s)
x : α
hx : x ∈ s ∨ x = a
xa : x ≠ a
⊢ Disjoint x (sSup ((s ∪ {a}) \ {x}))
|
have h : (s ∪ {a}) \ {x} = s \ {x} ∪ {a} := by
simp only [Set.union_singleton]
rw [Set.insert_diff_of_not_mem]
rw [Set.mem_singleton_iff]
exact Ne.symm xa
|
case inr
α : Type u_2
inst✝² : CompleteLattice α
inst✝¹ : IsModularLattice α
inst✝ : IsCompactlyGenerated α
h✝ : sSup {a | IsAtom a} = ⊤
b : α
s : Set α
s_max : ∀ ⦃t : Set α⦄, t ∈ {s | sSupIndep s ∧ Disjoint b (sSup s) ∧ ∀ a ∈ s, IsAtom a} → s ⊆ t → s = t
s_ind : sSupIndep s
b_inf_Sup_s : Disjoint b (sSup s)
s_atoms : ∀ a ∈ s, IsAtom a
a : α
ha : a ∈ {a | IsAtom a}
con : Disjoint a (b ⊔ sSup s)
a_dis_Sup_s : Disjoint a (sSup s)
x : α
hx : x ∈ s ∨ x = a
xa : x ≠ a
h : (s ∪ {a}) \ {x} = s \ {x} ∪ {a}
⊢ Disjoint x (sSup ((s ∪ {a}) \ {x}))
|
04808f5eaf22927f
|
Dioph.inject_dummies_lem
|
Mathlib/NumberTheory/Dioph.lean
|
theorem inject_dummies_lem (f : β → γ) (g : γ → Option β) (inv : ∀ x, g (f x) = some x)
(p : Poly (α ⊕ β)) (v : α → ℕ) :
(∃ t, p (v ⊗ t) = 0) ↔ ∃ t, p.map (inl ⊗ inr ∘ f) (v ⊗ t) = 0
|
α β γ : Type u
f : β → γ
g : γ → Option β
inv : ∀ (x : β), g (f x) = some x
p : Poly (α ⊕ β)
v : α → ℕ
t : β → ℕ
ht : p (v ⊗ t) = 0
this : (v ⊗ (0 ::ₒ t) ∘ g) ∘ (inl ⊗ inr ∘ f) = v ⊗ t
⊢ p ((v ⊗ (0 ::ₒ t) ∘ g) ∘ (inl ⊗ inr ∘ f)) = 0
|
rwa [this]
|
no goals
|
92ce2273a39f5fe6
|
exists_seq_forall_proj_of_forall_finite
|
Mathlib/Order/KonigLemma.lean
|
theorem exists_seq_forall_proj_of_forall_finite {α : ℕ → Type*} [Finite (α 0)] [∀ i, Nonempty (α i)]
(π : {i j : ℕ} → (hij : i ≤ j) → α j → α i)
(π_refl : ∀ ⦃i⦄ (a : α i), π rfl.le a = a)
(π_trans : ∀ ⦃i j k⦄ (hij : i ≤ j) (hjk : j ≤ k) a, π hij (π hjk a) = π (hij.trans hjk) a)
(hfin : ∀ i a, {b : α (i+1) | π (Nat.le_add_right i 1) b = a}.Finite) :
∃ f : (i : ℕ) → α i, ∀ ⦃i j⦄ (hij : i ≤ j), π hij (f j) = f i
|
α : ℕ → Type u_1
inst✝¹ : Finite (α 0)
inst✝ : ∀ (i : ℕ), Nonempty (α i)
π : {i j : ℕ} → i ≤ j → α j → α i
π_refl : ∀ ⦃i : ℕ⦄ (a : α i), π ⋯ a = a
π_trans : ∀ ⦃i j k : ℕ⦄ (hij : i ≤ j) (hjk : j ≤ k) (a : α k), π hij (π hjk a) = π ⋯ a
hfin : ∀ (i : ℕ) (a : α i), {b | π ⋯ b = a}.Finite
αs : Type u_1 := (i : ℕ) × α i
x✝ : PartialOrder αs := PartialOrder.mk ⋯
hcovby : ∀ {a b : αs}, a ⋖ b ↔ a ≤ b ∧ a.fst + 1 = b.fst
i : ℕ
a b : α i
hne : ⟨i, a⟩ ≠ ⟨i, b⟩
hij : i ≤ i
h2 : π hij b = a
⊢ False
|
simp [← h2, π_refl] at hne
|
no goals
|
271fcb2e96bcf388
|
Polynomial.continuous_eval₂
|
Mathlib/Topology/Algebra/Polynomial.lean
|
theorem continuous_eval₂ [Semiring S] (p : S[X]) (f : S →+* R) :
Continuous fun x => p.eval₂ f x
|
R : Type u_1
S : Type u_2
inst✝³ : Semiring R
inst✝² : TopologicalSpace R
inst✝¹ : IsTopologicalSemiring R
inst✝ : Semiring S
p : S[X]
f : S →+* R
⊢ Continuous fun x => p.sum fun e a => f a * x ^ e
|
exact continuous_finset_sum _ fun c _ => continuous_const.mul (continuous_pow _)
|
no goals
|
d0ed11350349de73
|
CategoryTheory.Limits.prod.map_id_id
|
Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean
|
theorem prod.map_id_id {X Y : C} [HasBinaryProduct X Y] : prod.map (𝟙 X) (𝟙 Y) = 𝟙 _
|
C : Type u
inst✝¹ : Category.{v, u} C
X Y : C
inst✝ : HasBinaryProduct X Y
⊢ map (𝟙 X) (𝟙 Y) = 𝟙 (X ⨯ Y)
|
ext <;> simp
|
no goals
|
8a43ed8999e5542d
|
HasCompactSupport.enorm_le_lintegral_Ici_deriv
|
Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean
|
lemma _root_.HasCompactSupport.enorm_le_lintegral_Ici_deriv
{F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F]
{f : ℝ → F} (hf : ContDiff ℝ 1 f) (h'f : HasCompactSupport f) (x : ℝ) :
‖f x‖ₑ ≤ ∫⁻ y in Iic x, ‖deriv f y‖ₑ
|
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
hf : ContDiff ℝ 1 f
h'f : HasCompactSupport f
x : ℝ
I : F →L[ℝ] Completion F := Completion.toComplL
f' : ℝ → Completion F := ⇑I ∘ f
hf' : ContDiff ℝ 1 f'
⊢ ‖f x‖ₑ ≤ ∫⁻ (y : ℝ) in Iic x, ‖deriv f y‖ₑ
|
have h'f' : HasCompactSupport f' := h'f.comp_left rfl
|
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
hf : ContDiff ℝ 1 f
h'f : HasCompactSupport f
x : ℝ
I : F →L[ℝ] Completion F := Completion.toComplL
f' : ℝ → Completion F := ⇑I ∘ f
hf' : ContDiff ℝ 1 f'
h'f' : HasCompactSupport f'
⊢ ‖f x‖ₑ ≤ ∫⁻ (y : ℝ) in Iic x, ‖deriv f y‖ₑ
|
43d9795b86d3ea3e
|
Rat.finite_rat_abs_sub_lt_one_div_den_sq
|
Mathlib/NumberTheory/DiophantineApproximation/Basic.lean
|
theorem finite_rat_abs_sub_lt_one_div_den_sq (ξ : ℚ) :
{q : ℚ | |ξ - q| < 1 / (q.den : ℚ) ^ 2}.Finite
|
ξ : ℚ
f : ℚ → ℤ × ℕ := fun q => (q.num, q.den)
s : Set ℚ := {q | |ξ - q| < 1 / ↑q.den ^ 2}
hinj : Function.Injective f
H : f '' s ⊆ ⋃ y ∈ Ioc 0 ξ.den, Icc (⌈ξ * ↑y⌉ - 1) (⌊ξ * ↑y⌋ + 1) ×ˢ {y}
⊢ s.Finite
|
refine (Finite.subset ?_ H).of_finite_image hinj.injOn
|
ξ : ℚ
f : ℚ → ℤ × ℕ := fun q => (q.num, q.den)
s : Set ℚ := {q | |ξ - q| < 1 / ↑q.den ^ 2}
hinj : Function.Injective f
H : f '' s ⊆ ⋃ y ∈ Ioc 0 ξ.den, Icc (⌈ξ * ↑y⌉ - 1) (⌊ξ * ↑y⌋ + 1) ×ˢ {y}
⊢ (⋃ y ∈ Ioc 0 ξ.den, Icc (⌈ξ * ↑y⌉ - 1) (⌊ξ * ↑y⌋ + 1) ×ˢ {y}).Finite
|
dc62b04147c652be
|
Profinite.exists_isClopen_of_cofiltered
|
Mathlib/Topology/Category/Profinite/CofilteredLimit.lean
|
theorem exists_isClopen_of_cofiltered {U : Set C.pt} (hC : IsLimit C) (hU : IsClopen U) :
∃ (j : J) (V : Set (F.obj j)), IsClopen V ∧ U = C.π.app j ⁻¹' V
|
case refine_3.intro.intro.intro.intro.refine_2.h.mpr.intro.intro
J : Type v
inst✝¹ : SmallCategory J
inst✝ : IsCofiltered J
F : J ⥤ Profinite
C : Cone F
U : Set ↑C.pt.toTop
hC : IsLimit C
hU : IsClopen U
S : Set (Set ↑(toTopCat.mapCone C).pt)
hS :
S ⊆ {U | ∃ j, ∃ V ∈ (fun j => {W | IsClopen W}) j, U = ⇑(ConcreteCategory.hom ((toTopCat.mapCone C).π.app j)) ⁻¹' V}
h : U = ⋃₀ S
j : ↑S → J := fun s => Exists.choose ⋯
V : (s : ↑S) → Set ↑(F.obj (j s)).toTop := fun s => ⋯.choose
hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ⇑(ConcreteCategory.hom (C.π.app (j s))) ⁻¹' V s
hUo : ∀ (i : ↑S), IsOpen ((fun s => ⇑(ConcreteCategory.hom (C.π.app (j s))) ⁻¹' V s) i)
hsU : U ⊆ ⋃ i, (fun s => ⇑(ConcreteCategory.hom (C.π.app (j s))) ⁻¹' V s) i
G : Finset ↑S
hG : U ⊆ ⋃ i ∈ G, (fun s => ⇑(ConcreteCategory.hom (C.π.app (j s))) ⁻¹' V s) i
j0 : J
hj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X)
f : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => ⋯.some
W : ↑S → Set ↑(F.obj j0).toTop :=
fun s => if hs : s ∈ G then ⇑(ConcreteCategory.hom (F.map (f s hs))) ⁻¹' V s else Set.univ
x : ↑C.pt.toTop
s : ↑S
hs : s ∈ G
hx :
x ∈
⇑(ConcreteCategory.hom (C.π.app j0)) ⁻¹'
if h : s ∈ G then ⇑(ConcreteCategory.hom (F.map (f s ⋯))) ⁻¹' V s else Set.univ
⊢ x ∈ ↑s
|
rw [(hV s).2]
|
case refine_3.intro.intro.intro.intro.refine_2.h.mpr.intro.intro
J : Type v
inst✝¹ : SmallCategory J
inst✝ : IsCofiltered J
F : J ⥤ Profinite
C : Cone F
U : Set ↑C.pt.toTop
hC : IsLimit C
hU : IsClopen U
S : Set (Set ↑(toTopCat.mapCone C).pt)
hS :
S ⊆ {U | ∃ j, ∃ V ∈ (fun j => {W | IsClopen W}) j, U = ⇑(ConcreteCategory.hom ((toTopCat.mapCone C).π.app j)) ⁻¹' V}
h : U = ⋃₀ S
j : ↑S → J := fun s => Exists.choose ⋯
V : (s : ↑S) → Set ↑(F.obj (j s)).toTop := fun s => ⋯.choose
hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ⇑(ConcreteCategory.hom (C.π.app (j s))) ⁻¹' V s
hUo : ∀ (i : ↑S), IsOpen ((fun s => ⇑(ConcreteCategory.hom (C.π.app (j s))) ⁻¹' V s) i)
hsU : U ⊆ ⋃ i, (fun s => ⇑(ConcreteCategory.hom (C.π.app (j s))) ⁻¹' V s) i
G : Finset ↑S
hG : U ⊆ ⋃ i ∈ G, (fun s => ⇑(ConcreteCategory.hom (C.π.app (j s))) ⁻¹' V s) i
j0 : J
hj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X)
f : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => ⋯.some
W : ↑S → Set ↑(F.obj j0).toTop :=
fun s => if hs : s ∈ G then ⇑(ConcreteCategory.hom (F.map (f s hs))) ⁻¹' V s else Set.univ
x : ↑C.pt.toTop
s : ↑S
hs : s ∈ G
hx :
x ∈
⇑(ConcreteCategory.hom (C.π.app j0)) ⁻¹'
if h : s ∈ G then ⇑(ConcreteCategory.hom (F.map (f s ⋯))) ⁻¹' V s else Set.univ
⊢ x ∈ ⇑(ConcreteCategory.hom (C.π.app (j s))) ⁻¹' V s
|
0fb8cdb8a6309b67
|
Submodule.mem_annihilator_span
|
Mathlib/RingTheory/Ideal/Maps.lean
|
theorem mem_annihilator_span (s : Set M) (r : R) :
r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0
|
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
s : Set M
r : R
⊢ (∀ n ∈ span R s, r • n = 0) ↔ ∀ (n : ↑s), r • ↑n = 0
|
constructor
|
case mp
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
s : Set M
r : R
⊢ (∀ n ∈ span R s, r • n = 0) → ∀ (n : ↑s), r • ↑n = 0
case mpr
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
s : Set M
r : R
⊢ (∀ (n : ↑s), r • ↑n = 0) → ∀ n ∈ span R s, r • n = 0
|
c2ed1bab9d23d491
|
IsCompact.inf_nhdsSet_eq_biSup
|
Mathlib/Topology/Compactness/Compact.lean
|
theorem IsCompact.inf_nhdsSet_eq_biSup {K : Set X} (hK : IsCompact K) (l : Filter X) :
l ⊓ (𝓝ˢ K) = ⨆ x ∈ K, l ⊓ 𝓝 x
|
X : Type u
inst✝ : TopologicalSpace X
K : Set X
hK : IsCompact K
l : Filter X
⊢ l ⊓ 𝓝ˢ K = ⨆ x ∈ K, l ⊓ 𝓝 x
|
simp only [inf_comm l, hK.nhdsSet_inf_eq_biSup]
|
no goals
|
ecb4032335740849
|
Nat.mod_add_div'
|
Mathlib/Data/Nat/Init.lean
|
lemma mod_add_div' (a b : ℕ) : a % b + a / b * b = a
|
a b : ℕ
⊢ a % b + a / b * b = a
|
rw [Nat.mul_comm]
|
a b : ℕ
⊢ a % b + b * (a / b) = a
|
37db04288b7cd338
|
OrdinalApprox.lfpApprox_add_one
|
Mathlib/SetTheory/Ordinal/FixedPointApproximants.lean
|
theorem lfpApprox_add_one (h : x ≤ f x) (a : Ordinal) :
lfpApprox f x (a+1) = f (lfpApprox f x a)
|
case a
α : Type u
inst✝ : CompleteLattice α
f : α →o α
x : α
h : x ≤ f x
a : Ordinal.{u}
⊢ f (lfpApprox f x a) ≤ sSup ({x_1 | ∃ b, ∃ (_ : b < a + 1), f (lfpApprox f x b) = x_1} ∪ {x})
|
apply le_sSup
|
case a.a
α : Type u
inst✝ : CompleteLattice α
f : α →o α
x : α
h : x ≤ f x
a : Ordinal.{u}
⊢ f (lfpApprox f x a) ∈ {x_1 | ∃ b, ∃ (_ : b < a + 1), f (lfpApprox f x b) = x_1} ∪ {x}
|
27ca93c209470fa4
|
LSeries.mul_delta
|
Mathlib/NumberTheory/LSeries/Basic.lean
|
lemma mul_delta {f : ℕ → ℂ} (h : f 1 = 1) : f * δ = δ
|
f : ℕ → ℂ
h : f 1 = 1
⊢ f * δ = δ
|
rw [mul_delta_eq_smul_delta, h, one_smul]
|
no goals
|
258e3795b76ffed3
|
MeasureTheory.L2.norm_sq_eq_inner'
|
Mathlib/MeasureTheory/Function/L2Space.lean
|
theorem norm_sq_eq_inner' (f : α →₂[μ] E) : ‖f‖ ^ 2 = RCLike.re ⟪f, f⟫
|
α : Type u_1
E : Type u_2
𝕜 : Type u_4
inst✝³ : RCLike 𝕜
inst✝² : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
f : ↥(Lp E 2 μ)
h_two : ENNReal.toReal 2 = 2
⊢ ‖f‖ ^ 2 = RCLike.re (inner f f)
|
rw [inner_def, integral_inner_eq_sq_eLpNorm, norm_def, ← ENNReal.toReal_pow, RCLike.ofReal_re,
ENNReal.toReal_eq_toReal (ENNReal.pow_ne_top (Lp.eLpNorm_ne_top f)) _]
|
α : Type u_1
E : Type u_2
𝕜 : Type u_4
inst✝³ : RCLike 𝕜
inst✝² : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
f : ↥(Lp E 2 μ)
h_two : ENNReal.toReal 2 = 2
⊢ eLpNorm (↑↑f) 2 μ ^ 2 = ∫⁻ (a : α), ↑‖↑↑f a‖₊ ^ 2 ∂μ
α : Type u_1
E : Type u_2
𝕜 : Type u_4
inst✝³ : RCLike 𝕜
inst✝² : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
f : ↥(Lp E 2 μ)
h_two : ENNReal.toReal 2 = 2
⊢ ∫⁻ (a : α), ↑‖↑↑f a‖₊ ^ 2 ∂μ ≠ ⊤
|
629fb9c7e3401fd8
|
AlgebraicGeometry.sourceAffineLocally_respectsIso
|
Mathlib/AlgebraicGeometry/Morphisms/RingHomProperties.lean
|
theorem sourceAffineLocally_respectsIso (h₁ : RingHom.RespectsIso P) :
(sourceAffineLocally P).toProperty.RespectsIso
|
case h₁
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
h₁ : RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] => P
X Y Z : Scheme
e : X ≅ Y
f : Y ⟶ Z
inst✝ : IsAffine Z
H : sourceAffineLocally (fun {R S} [CommRing R] [CommRing S] => P) f
U : ↑X.affineOpens
this : IsIso (Scheme.Hom.appLE e.hom (e.hom ''ᵁ ↑U) ↑U ⋯)
⊢ P (CommRingCat.Hom.hom (Scheme.Hom.appLE (e.hom ≫ f) ⊤ ↑U ⋯))
|
rw [← Scheme.appLE_comp_appLE _ _ ⊤ (e.hom ''ᵁ U) U.1 le_top (e.hom.preimage_image_eq _).ge,
CommRingCat.hom_comp, h₁.cancel_right_isIso]
|
case h₁
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
h₁ : RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] => P
X Y Z : Scheme
e : X ≅ Y
f : Y ⟶ Z
inst✝ : IsAffine Z
H : sourceAffineLocally (fun {R S} [CommRing R] [CommRing S] => P) f
U : ↑X.affineOpens
this : IsIso (Scheme.Hom.appLE e.hom (e.hom ''ᵁ ↑U) ↑U ⋯)
⊢ P (CommRingCat.Hom.hom (Scheme.Hom.appLE f ⊤ (e.hom ''ᵁ ↑U) ⋯))
|
37a0eb41f0553cd7
|
Std.DHashMap.Internal.List.distinctKeys_cons_iff
|
Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/List/Associative.lean
|
theorem distinctKeys_cons_iff [BEq α] [PartialEquivBEq α] {l : List ((a : α) × β a)} {k : α}
{v : β k} : DistinctKeys (⟨k, v⟩ :: l) ↔ DistinctKeys l ∧ (containsKey k l) = false
|
case refine_1
α : Type u
β : α → Type v
inst✝¹ : BEq α
inst✝ : PartialEquivBEq α
l : List ((a : α) × β a)
k : α
v : β k
x✝ : DistinctKeys (⟨k, v⟩ :: l)
h : List.Pairwise (fun a b => (a == b) = false) (keys (⟨k, v⟩ :: l))
⊢ DistinctKeys l ∧ containsKey k l = false
|
rw [keys_cons, pairwise_cons] at h
|
case refine_1
α : Type u
β : α → Type v
inst✝¹ : BEq α
inst✝ : PartialEquivBEq α
l : List ((a : α) × β a)
k : α
v : β k
x✝ : DistinctKeys (⟨k, v⟩ :: l)
h : (∀ (a' : α), a' ∈ keys l → (k == a') = false) ∧ List.Pairwise (fun a b => (a == b) = false) (keys l)
⊢ DistinctKeys l ∧ containsKey k l = false
|
62b2d49d16d3b84a
|
AlgebraicGeometry.Scheme.OpenCover.ext_elem
|
Mathlib/AlgebraicGeometry/Cover/Open.lean
|
/-- If two global sections agree after restriction to each member of an open cover, then
they agree globally. -/
lemma OpenCover.ext_elem {X : Scheme.{u}} {U : X.Opens} (f g : Γ(X, U)) (𝒰 : X.OpenCover)
(h : ∀ i : 𝒰.J, (𝒰.map i).app U f = (𝒰.map i).app U g) : f = g
|
case h
X : Scheme
U : X.Opens
f g : ↑Γ(X, U)
𝒰 : X.OpenCover
x : ↑↑X.toPresheafedSpace
h :
(ConcreteCategory.hom (X.presheaf.map (homOfLE ⋯).op ≫ (IsOpenImmersion.ΓIso (𝒰.map (𝒰.f x)) U).inv)) f =
(ConcreteCategory.hom (X.presheaf.map (homOfLE ⋯).op ≫ (IsOpenImmersion.ΓIso (𝒰.map (𝒰.f x)) U).inv)) g
⊢ (ConcreteCategory.hom (X.sheaf.val.map (homOfLE ⋯).op)) f = (ConcreteCategory.hom (X.sheaf.val.map (homOfLE ⋯).op)) g
|
exact (IsOpenImmersion.ΓIso (𝒰.map (𝒰.f x)) U).commRingCatIsoToRingEquiv.symm.injective h
|
no goals
|
dba4a0de0786d781
|
Polynomial.exists_partition_polynomial_aux
|
Mathlib/NumberTheory/ClassNumber/AdmissibleCardPowDegree.lean
|
theorem exists_partition_polynomial_aux (n : ℕ) {ε : ℝ} (hε : 0 < ε) {b : Fq[X]} (hb : b ≠ 0)
(A : Fin n → Fq[X]) : ∃ t : Fin n → Fin (Fintype.card Fq ^ ⌈-log ε / log (Fintype.card Fq)⌉₊),
∀ i₀ i₁ : Fin n, t i₀ = t i₁ ↔
(cardPowDegree (A i₁ % b - A i₀ % b) : ℝ) < cardPowDegree b • ε
|
Fq : Type u_1
inst✝¹ : Fintype Fq
inst✝ : Field Fq
n : ℕ
ε : ℝ
hε : 0 < ε
b : Fq[X]
hb : b ≠ 0
A : Fin n → Fq[X]
⊢ 0 < ↑(cardPowDegree b) * ε
|
exact mul_pos (Int.cast_pos.mpr (AbsoluteValue.pos _ hb)) hε
|
no goals
|
d529c9a5494d9218
|
CategoryTheory.Monoidal.Reflective.isIso_tfae
|
Mathlib/CategoryTheory/Monoidal/Braided/Reflection.lean
|
theorem isIso_tfae : List.TFAE
[ ∀ (c : C) (d : D), IsIso (adj.unit.app ((ihom d).obj (R.obj c)))
, ∀ (c : C) (d : D), IsIso ((pre (adj.unit.app d)).app (R.obj c))
, ∀ (d d' : D), IsIso (L.map ((adj.unit.app d) ▷ d'))
, ∀ (d d' : D), IsIso (L.map ((adj.unit.app d) ⊗ (adj.unit.app d')))]
|
C : Type u_1
D : Type u_2
inst✝⁶ : Category.{u_4, u_1} C
inst✝⁵ : Category.{u_3, u_2} D
inst✝⁴ : MonoidalCategory D
inst✝³ : SymmetricCategory D
inst✝² : MonoidalClosed D
R : C ⥤ D
inst✝¹ : R.Faithful
inst✝ : R.Full
L : D ⥤ C
adj : L ⊣ R
tfae_3_to_4 :
(∀ (d d' : D), IsIso (L.map (adj.unit.app d ▷ d'))) → ∀ (d d' : D), IsIso (L.map (adj.unit.app d ⊗ adj.unit.app d'))
tfae_4_to_1 :
(∀ (d d' : D), IsIso (L.map (adj.unit.app d ⊗ adj.unit.app d'))) →
∀ (c : C) (d : D), IsIso (adj.unit.app ((ihom d).obj (R.obj c)))
h : ∀ (c : C) (d : D), IsIso (adj.unit.app ((ihom d).obj (R.obj c)))
d d' : D
c : C
w₁ :
(coyoneda.map (L.map (adj.unit.app d ▷ d')).op).app c =
⇑(adj.homEquiv (Opposite.unop (Opposite.op ((𝟭 D).obj d ⊗ d'))) c).symm ∘
(coyoneda.map (adj.unit.app d ▷ d').op).app (R.obj c) ∘
⇑(adj.homEquiv (Opposite.unop (Opposite.op ((L ⋙ R).obj d ⊗ d'))) c)
w₂ :
(coyoneda.map (adj.unit.app d ▷ d').op).app (R.obj c) =
((yoneda.obj (R.obj c)).mapIso (β_ (Opposite.op d') (Opposite.op ((𝟭 D).toPrefunctor.1 d)))).hom ∘
(coyoneda.map (d' ◁ adj.unit.app d).op).app (R.obj c) ∘
((yoneda.obj (R.obj c)).mapIso (β_ (Opposite.op ((L ⋙ R).toPrefunctor.1 d)) (Opposite.op d'))).hom
w₃ :
(coyoneda.map (d' ◁ adj.unit.app d).op).app (R.obj c) =
⇑((ihom.adjunction d').homEquiv (Opposite.unop (Opposite.op ((𝟭 D).obj d))) (R.obj c)).symm ∘
(coyoneda.map (adj.unit.app d).op).app ((ihom d').obj (R.obj c)) ∘
⇑((ihom.adjunction d').homEquiv (Opposite.unop (Opposite.op ((L ⋙ R).obj d))) (R.obj c))
w₄ :
(coyoneda.map (adj.unit.app d).op).app ((ihom d').obj (R.obj c)) ≫
(coyoneda.obj (Opposite.op d)).map (adj.unit.app ((ihom d').obj (R.obj c))) =
(coyoneda.obj (Opposite.op ((L ⋙ R).obj d))).map (adj.unit.app ((ihom d').obj (R.obj c))) ≫
(coyoneda.map (adj.unit.app d).op).app ((L ⋙ R).obj ((ihom d').obj (R.obj c)))
⊢ IsIso
((coyoneda.map (adj.unit.app d).op).app ((ihom d').obj (R.obj c)) ≫
(coyoneda.obj (Opposite.op d)).map (adj.unit.app ((ihom d').obj (R.obj c))))
|
rw [w₄]
|
C : Type u_1
D : Type u_2
inst✝⁶ : Category.{u_4, u_1} C
inst✝⁵ : Category.{u_3, u_2} D
inst✝⁴ : MonoidalCategory D
inst✝³ : SymmetricCategory D
inst✝² : MonoidalClosed D
R : C ⥤ D
inst✝¹ : R.Faithful
inst✝ : R.Full
L : D ⥤ C
adj : L ⊣ R
tfae_3_to_4 :
(∀ (d d' : D), IsIso (L.map (adj.unit.app d ▷ d'))) → ∀ (d d' : D), IsIso (L.map (adj.unit.app d ⊗ adj.unit.app d'))
tfae_4_to_1 :
(∀ (d d' : D), IsIso (L.map (adj.unit.app d ⊗ adj.unit.app d'))) →
∀ (c : C) (d : D), IsIso (adj.unit.app ((ihom d).obj (R.obj c)))
h : ∀ (c : C) (d : D), IsIso (adj.unit.app ((ihom d).obj (R.obj c)))
d d' : D
c : C
w₁ :
(coyoneda.map (L.map (adj.unit.app d ▷ d')).op).app c =
⇑(adj.homEquiv (Opposite.unop (Opposite.op ((𝟭 D).obj d ⊗ d'))) c).symm ∘
(coyoneda.map (adj.unit.app d ▷ d').op).app (R.obj c) ∘
⇑(adj.homEquiv (Opposite.unop (Opposite.op ((L ⋙ R).obj d ⊗ d'))) c)
w₂ :
(coyoneda.map (adj.unit.app d ▷ d').op).app (R.obj c) =
((yoneda.obj (R.obj c)).mapIso (β_ (Opposite.op d') (Opposite.op ((𝟭 D).toPrefunctor.1 d)))).hom ∘
(coyoneda.map (d' ◁ adj.unit.app d).op).app (R.obj c) ∘
((yoneda.obj (R.obj c)).mapIso (β_ (Opposite.op ((L ⋙ R).toPrefunctor.1 d)) (Opposite.op d'))).hom
w₃ :
(coyoneda.map (d' ◁ adj.unit.app d).op).app (R.obj c) =
⇑((ihom.adjunction d').homEquiv (Opposite.unop (Opposite.op ((𝟭 D).obj d))) (R.obj c)).symm ∘
(coyoneda.map (adj.unit.app d).op).app ((ihom d').obj (R.obj c)) ∘
⇑((ihom.adjunction d').homEquiv (Opposite.unop (Opposite.op ((L ⋙ R).obj d))) (R.obj c))
w₄ :
(coyoneda.map (adj.unit.app d).op).app ((ihom d').obj (R.obj c)) ≫
(coyoneda.obj (Opposite.op d)).map (adj.unit.app ((ihom d').obj (R.obj c))) =
(coyoneda.obj (Opposite.op ((L ⋙ R).obj d))).map (adj.unit.app ((ihom d').obj (R.obj c))) ≫
(coyoneda.map (adj.unit.app d).op).app ((L ⋙ R).obj ((ihom d').obj (R.obj c)))
⊢ IsIso
((coyoneda.obj (Opposite.op ((L ⋙ R).obj d))).map (adj.unit.app ((ihom d').obj (R.obj c))) ≫
(coyoneda.map (adj.unit.app d).op).app ((L ⋙ R).obj ((ihom d').obj (R.obj c))))
|
8ccd82e9c49cc7ac
|
TopCat.GlueData.eqvGen_of_π_eq
|
Mathlib/Topology/Gluing.lean
|
theorem eqvGen_of_π_eq
-- Porting note: was `{x y : ∐ D.U} (h : 𝖣.π x = 𝖣.π y)`
{x y : sigmaObj (β := D.toGlueData.J) (C := TopCat) D.toGlueData.U}
(h : 𝖣.π x = 𝖣.π y) :
Relation.EqvGen
(Function.Coequalizer.Rel 𝖣.diagram.fstSigmaMap 𝖣.diagram.sndSigmaMap) x y
|
D : GlueData
x y : ↑(∐ D.U)
h :
(ConcreteCategory.hom
(coequalizer.π D.diagram.fstSigmaMap D.diagram.sndSigmaMap ≫ (Multicoequalizer.isoCoequalizer D.diagram).inv))
x =
(ConcreteCategory.hom
(coequalizer.π D.diagram.fstSigmaMap D.diagram.sndSigmaMap ≫ (Multicoequalizer.isoCoequalizer D.diagram).inv))
y
⊢ Relation.EqvGen
(Function.Coequalizer.Rel ⇑(ConcreteCategory.hom D.diagram.fstSigmaMap)
⇑(ConcreteCategory.hom D.diagram.sndSigmaMap))
x y
|
replace h : coequalizer.π D.diagram.fstSigmaMap D.diagram.sndSigmaMap x =
coequalizer.π D.diagram.fstSigmaMap D.diagram.sndSigmaMap y :=
(TopCat.mono_iff_injective (Multicoequalizer.isoCoequalizer 𝖣.diagram).inv).mp
inferInstance h
|
D : GlueData
x y : ↑(∐ D.U)
h :
(ConcreteCategory.hom (coequalizer.π D.diagram.fstSigmaMap D.diagram.sndSigmaMap)) x =
(ConcreteCategory.hom (coequalizer.π D.diagram.fstSigmaMap D.diagram.sndSigmaMap)) y
⊢ Relation.EqvGen
(Function.Coequalizer.Rel ⇑(ConcreteCategory.hom D.diagram.fstSigmaMap)
⇑(ConcreteCategory.hom D.diagram.sndSigmaMap))
x y
|
387fc6985553187a
|
Substring.ValidFor.dropWhile
|
Mathlib/.lake/packages/batteries/Batteries/Data/String/Lemmas.lean
|
theorem dropWhile (p : Char → Bool) : ∀ {s}, ValidFor l m r s →
ValidFor (l ++ m.takeWhile p) (m.dropWhile p) r (s.dropWhile p)
| _, ⟨⟩ => by
simp only [Substring.dropWhile, takeWhileAux_of_valid]
apply ValidFor.of_eq <;> simp
rw [Nat.add_assoc, ← utf8Len_append (m.takeWhile p), List.takeWhile_append_dropWhile]
|
l m r : List Char
p : Char → Bool
⊢ ValidFor (l ++ List.takeWhile p m) (List.dropWhile p m) r
{ str := { data := l ++ m ++ r }, startPos := { byteIdx := utf8Len l + utf8Len (List.takeWhile p m) },
stopPos := { byteIdx := utf8Len l + utf8Len m } }
|
apply ValidFor.of_eq <;> simp
|
case a
l m r : List Char
p : Char → Bool
⊢ utf8Len l + utf8Len m = utf8Len l + utf8Len (List.takeWhile p m) + utf8Len (List.dropWhile p m)
|
326f60c986ce8ca4
|
isNilpotent_of_pos_nilpotencyClass
|
Mathlib/RingTheory/Nilpotent/Defs.lean
|
lemma isNilpotent_of_pos_nilpotencyClass (hx : 0 < nilpotencyClass x) :
IsNilpotent x
|
R : Type u_1
x : R
inst✝¹ : Zero R
inst✝ : Pow R ℕ
hx : 0 < nilpotencyClass x
s : Set ℕ := {k | x ^ k = 0}
⊢ IsNilpotent x
|
change s.Nonempty
|
R : Type u_1
x : R
inst✝¹ : Zero R
inst✝ : Pow R ℕ
hx : 0 < nilpotencyClass x
s : Set ℕ := {k | x ^ k = 0}
⊢ s.Nonempty
|
47bfbe3b27fe953d
|
pow_add_pow_le
|
Mathlib/Algebra/Order/Ring/Basic.lean
|
theorem pow_add_pow_le (hx : 0 ≤ x) (hy : 0 ≤ y) (hn : n ≠ 0) : x ^ n + y ^ n ≤ (x + y) ^ n
|
case intro.succ
R : Type u_3
inst✝ : OrderedSemiring R
x y : R
hx : 0 ≤ x
hy : 0 ≤ y
k : ℕ
ih : k + 1 ≠ 0 → x ^ (k + 1) + y ^ (k + 1) ≤ (x + y) ^ (k + 1)
hn : k + 1 + 1 ≠ 0
n : ℕ := k.succ
h1 : 0 ≤ x * y ^ n + y * x ^ n
⊢ x ^ (k + 1 + 1) + y ^ (k + 1 + 1) ≤ (x + y) ^ (k + 1 + 1)
|
have h2 := add_nonneg hx hy
|
case intro.succ
R : Type u_3
inst✝ : OrderedSemiring R
x y : R
hx : 0 ≤ x
hy : 0 ≤ y
k : ℕ
ih : k + 1 ≠ 0 → x ^ (k + 1) + y ^ (k + 1) ≤ (x + y) ^ (k + 1)
hn : k + 1 + 1 ≠ 0
n : ℕ := k.succ
h1 : 0 ≤ x * y ^ n + y * x ^ n
h2 : 0 ≤ x + y
⊢ x ^ (k + 1 + 1) + y ^ (k + 1 + 1) ≤ (x + y) ^ (k + 1 + 1)
|
1b4313a1aa17f256
|
Ideal.FinrankQuotientMap.span_eq_top
|
Mathlib/NumberTheory/RamificationInertia/Basic.lean
|
theorem FinrankQuotientMap.span_eq_top [IsDomain R] [IsDomain S] [Algebra K L] [Module.Finite R S]
[Algebra R L] [IsScalarTower R S L] [IsScalarTower R K L] [Algebra.IsAlgebraic R S]
[NoZeroSMulDivisors R K] (hp : p ≠ ⊤) (b : Set S)
(hb' : Submodule.span R b ⊔ (p.map (algebraMap R S)).restrictScalars R = ⊤) :
Submodule.span K (algebraMap S L '' b) = ⊤
|
R : Type u
inst✝¹⁶ : CommRing R
S : Type v
inst✝¹⁵ : CommRing S
p : Ideal R
inst✝¹⁴ : Algebra R S
K : Type u_1
inst✝¹³ : Field K
inst✝¹² : Algebra R K
L : Type u_2
inst✝¹¹ : Field L
inst✝¹⁰ : Algebra S L
inst✝⁹ : IsFractionRing S L
inst✝⁸ : IsDomain R
inst✝⁷ : IsDomain S
inst✝⁶ : Algebra K L
inst✝⁵ : Module.Finite R S
inst✝⁴ : Algebra R L
inst✝³ : IsScalarTower R S L
inst✝² : IsScalarTower R K L
inst✝¹ : Algebra.IsAlgebraic R S
inst✝ : NoZeroSMulDivisors R K
hp : p ≠ ⊤
b : Set S
hb' : Submodule.span R b ⊔ Submodule.restrictScalars R (map (algebraMap R S) p) = ⊤
hRL : Function.Injective ⇑(algebraMap R L)
M : Submodule R S := Submodule.span R b
n : ℕ
a : Fin n → S ⧸ M
ha : Submodule.span R (Set.range a) = ⊤
smul_top_eq : p • ⊤ = ⊤
exists_sum : ∀ (x : S ⧸ M), ∃ a', (∀ (i : Fin n), a' i ∈ p) ∧ ∑ i : Fin n, a' i • a i = x
A' : Fin n → Fin n → R
hA'p : ∀ (i i_1 : Fin n), A' i i_1 ∈ p
hA' : ∀ (i : Fin n), ∑ i_1 : Fin n, A' i i_1 • a i_1 = a i
A : Matrix (Fin n) (Fin n) R := Matrix.of A' - 1
B : Matrix (Fin n) (Fin n) R := A.adjugate
A_smul : ∀ (i : Fin n), ∑ j : Fin n, A i j • a j = 0
d_smul : ∀ (i : Fin n), A.det • a i = 0
span_d : Submodule.restrictScalars R (Submodule.span S {(algebraMap R S) A.det}) ≤ M
this : Nontrivial (R ⧸ p)
⊢ (Quotient.mk p) A.det = ((Quotient.mk p).mapMatrix A).det
|
rw [RingHom.map_det]
|
no goals
|
53ab591d29bce2fe
|
Batteries.HashMap.Imp.expand_size
|
Mathlib/.lake/packages/batteries/Batteries/Data/HashMap/WF.lean
|
theorem expand_size [Hashable α] {buckets : Buckets α β} :
(expand sz buckets).buckets.size = buckets.size
|
case b
α : Type u_1
β : Type u_2
sz : Nat
inst✝ : Hashable α
buckets : Buckets α β
i : Nat
source : Array (AssocList α β)
target : Buckets α β
hs : ∀ (j : Nat), j < i → source.toList[j]?.getD AssocList.nil = AssocList.nil
H : i < source.size
⊢ (List.map (fun x => x.toList.length) (source.set i AssocList.nil H).toList).sum +
(AssocList.foldl reinsertAux target source[i]).size =
(List.map (fun x => x.toList.length) source.toList).sum + target.size
|
case b =>
simp only [Array.length_toList, Array.toList_set, Array.get_eq_getElem, AssocList.foldl_eq]
refine have ⟨l₁, l₂, h₁, _, eq⟩ := List.exists_of_set H; eq ▸ ?_
rw [h₁]
simp only [Buckets.size_eq, List.map_append, List.map_cons, AssocList.toList,
List.length_nil, Nat.sum_append, List.sum_cons, Nat.zero_add, Array.length_toList]
rw [Nat.add_assoc, Nat.add_assoc, Nat.add_assoc]; congr 1
(conv => rhs; rw [Nat.add_left_comm]); congr 1
rw [Array.getElem_toList]
have := @reinsertAux_size α β _; simp [Buckets.size] at this
induction source[i].toList generalizing target <;> simp [*, Nat.succ_add]; rfl
|
no goals
|
5e56d234c0d941c0
|
singleton_mem_nhdsWithin_of_mem_discrete
|
Mathlib/Topology/Separation/Basic.lean
|
theorem singleton_mem_nhdsWithin_of_mem_discrete {s : Set X} [DiscreteTopology s] {x : X}
(hx : x ∈ s) : {x} ∈ 𝓝[s] x
|
X : Type u_1
inst✝¹ : TopologicalSpace X
s : Set X
inst✝ : DiscreteTopology ↑s
x : X
hx : x ∈ s
this : {⟨x, hx⟩} ∈ 𝓝 ⟨x, hx⟩
⊢ {x} ∈ 𝓝[s] x
|
simpa only [nhdsWithin_eq_map_subtype_coe hx, image_singleton] using
@image_mem_map _ _ _ ((↑) : s → X) _ this
|
no goals
|
235f75a071720217
|
transcendental_aeval_iff
|
Mathlib/RingTheory/Algebraic/Integral.lean
|
theorem transcendental_aeval_iff {r : A} {f : K[X]} :
Transcendental K (Polynomial.aeval r f) ↔ Transcendental K r ∧ Transcendental K f
|
K : Type u_1
A : Type u_4
inst✝² : Field K
inst✝¹ : Ring A
inst✝ : Algebra K A
r : A
f : K[X]
h : IsIntegral K r
⊢ IsIntegral K ((aeval r) f)
|
exact .of_mem_of_fg _ h.fg_adjoin_singleton _ (aeval_mem_adjoin_singleton _ _)
|
no goals
|
71dfaf2de2b26676
|
PythagoreanTriple.coprime_classification'
|
Mathlib/NumberTheory/PythagoreanTriples.lean
|
theorem coprime_classification' {x y z : ℤ} (h : PythagoreanTriple x y z)
(h_coprime : Int.gcd x y = 1) (h_parity : x % 2 = 1) (h_pos : 0 < z) :
∃ m n,
x = m ^ 2 - n ^ 2 ∧
y = 2 * m * n ∧
z = m ^ 2 + n ^ 2 ∧
Int.gcd m n = 1 ∧ (m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0) ∧ 0 ≤ m
|
case h.inl.right.inr
x y z : ℤ
h : PythagoreanTriple x y z
h_coprime : x.gcd y = 1
h_parity : x % 2 = 1
h_pos : 0 < z
m n : ℤ
ht3 : m.gcd n = 1
ht4 : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
hm : m < 0
h_odd : x = m ^ 2 - n ^ 2 ∧ y = 2 * m * n
h_neg : z = -(m ^ 2 + n ^ 2)
⊢ False
|
revert h_pos
|
case h.inl.right.inr
x y z : ℤ
h : PythagoreanTriple x y z
h_coprime : x.gcd y = 1
h_parity : x % 2 = 1
m n : ℤ
ht3 : m.gcd n = 1
ht4 : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
hm : m < 0
h_odd : x = m ^ 2 - n ^ 2 ∧ y = 2 * m * n
h_neg : z = -(m ^ 2 + n ^ 2)
⊢ 0 < z → False
|
36cd8cbffd99f6d1
|
essSup_comp_quotientGroup_mk
|
Mathlib/MeasureTheory/Measure/Haar/Quotient.lean
|
/-- The `essSup` of a function `g` on the quotient space `G ⧸ Γ` with respect to the pushforward
of the restriction, `μ_𝓕`, of a right-invariant measure `μ` to a fundamental domain `𝓕`, is the
same as the `essSup` of `g`'s lift to the universal cover `G` with respect to `μ`. -/
@[to_additive "The `essSup` of a function `g` on the additive quotient space `G ⧸ Γ` with respect
to the pushforward of the restriction, `μ_𝓕`, of a right-invariant measure `μ` to a fundamental
domain `𝓕`, is the same as the `essSup` of `g`'s lift to the universal cover `G` with respect
to `μ`."]
lemma essSup_comp_quotientGroup_mk [μ.IsMulRightInvariant] {g : G ⧸ Γ → ℝ≥0∞}
(g_ae_measurable : AEMeasurable g μ_𝓕) : essSup g μ_𝓕 = essSup (fun (x : G) ↦ g x) μ
|
G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : MeasurableSpace G
inst✝⁶ : TopologicalSpace G
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : BorelSpace G
μ : Measure G
Γ : Subgroup G
𝓕 : Set G
h𝓕 : IsFundamentalDomain (↥Γ.op) 𝓕 μ
inst✝³ : Countable ↥Γ
inst✝² : MeasurableSpace (G ⧸ Γ)
inst✝¹ : BorelSpace (G ⧸ Γ)
inst✝ : μ.IsMulRightInvariant
g : G ⧸ Γ → ℝ≥0∞
g_ae_measurable : AEMeasurable g (map QuotientGroup.mk (μ.restrict 𝓕))
⊢ essSup g (map QuotientGroup.mk (μ.restrict 𝓕)) = essSup (fun x => g ↑x) μ
|
have hπ : Measurable (QuotientGroup.mk : G → G ⧸ Γ) := continuous_quotient_mk'.measurable
|
G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : MeasurableSpace G
inst✝⁶ : TopologicalSpace G
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : BorelSpace G
μ : Measure G
Γ : Subgroup G
𝓕 : Set G
h𝓕 : IsFundamentalDomain (↥Γ.op) 𝓕 μ
inst✝³ : Countable ↥Γ
inst✝² : MeasurableSpace (G ⧸ Γ)
inst✝¹ : BorelSpace (G ⧸ Γ)
inst✝ : μ.IsMulRightInvariant
g : G ⧸ Γ → ℝ≥0∞
g_ae_measurable : AEMeasurable g (map QuotientGroup.mk (μ.restrict 𝓕))
hπ : Measurable QuotientGroup.mk
⊢ essSup g (map QuotientGroup.mk (μ.restrict 𝓕)) = essSup (fun x => g ↑x) μ
|
88454486251e53cb
|
fourierCoeffOn_of_hasDeriv_right
|
Mathlib/Analysis/Fourier/AddCircle.lean
|
theorem fourierCoeffOn_of_hasDeriv_right {a b : ℝ} (hab : a < b) {f f' : ℝ → ℂ}
{n : ℤ} (hn : n ≠ 0)
(hf : ContinuousOn f [[a, b]])
(hff' : ∀ x, x ∈ Ioo (min a b) (max a b) → HasDerivWithinAt f (f' x) (Ioi x) x)
(hf' : IntervalIntegrable f' volume a b) :
fourierCoeffOn hab f n = 1 / (-2 * π * I * n) *
(fourier (-n) (a : AddCircle (b - a)) * (f b - f a) - (b - a) * fourierCoeffOn hab f' n)
|
a b : ℝ
hab : a < b
f f' : ℝ → ℂ
n : ℤ
hn : n ≠ 0
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (a ⊓ b) (a ⊔ b), HasDerivWithinAt f (f' x) (Ioi x) x
hf' : IntervalIntegrable f' volume a b
hT : Fact (0 < b - a)
this : ∀ (u v w : ℂ), u * (↑(b - a) / v * w) = ↑(b - a) / v * (u * w)
⊢ ↑(b - a) / (-2 * ↑π * I * ↑n) = ↑(b - a) * (1 / (-2 * ↑π * I * ↑n))
|
ring
|
no goals
|
0be290c2ece914fe
|
CochainComplex.HomComplex.δ_neg_one_cochain
|
Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean
|
lemma δ_neg_one_cochain (z : Cochain F G (-1)) :
δ (-1) 0 z = Cochain.ofHom (Homotopy.nullHomotopicMap'
(fun i j hij => z.v i j (by dsimp at hij; rw [← hij, add_neg_cancel_right])))
|
case h
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
F G : CochainComplex C ℤ
z : Cochain F G (-1)
p : ℤ
⊢ (δ (-1) 0 z).v p p ⋯ = (Cochain.ofHom (Homotopy.nullHomotopicMap' fun i j hij => z.v i j ⋯)).v p p ⋯
|
rw [δ_v (-1) 0 (neg_add_cancel 1) _ p p (add_zero p) (p-1) (p+1) rfl rfl]
|
case h
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
F G : CochainComplex C ℤ
z : Cochain F G (-1)
p : ℤ
⊢ z.v p (p - 1) ⋯ ≫ G.d (p - 1) p + Int.negOnePow 0 • F.d p (p + 1) ≫ z.v (p + 1) p ⋯ =
(Cochain.ofHom (Homotopy.nullHomotopicMap' fun i j hij => z.v i j ⋯)).v p p ⋯
|
f568289a63d09e4d
|
WeierstrassCurve.b₄_of_isCharTwoJEqZeroNF
|
Mathlib/AlgebraicGeometry/EllipticCurve/NormalForms.lean
|
theorem b₄_of_isCharTwoJEqZeroNF : W.b₄ = 2 * W.a₄
|
R : Type u_1
inst✝¹ : CommRing R
W : WeierstrassCurve R
inst✝ : W.IsCharTwoJEqZeroNF
⊢ 2 * W.a₄ + 0 * W.a₃ = 2 * W.a₄
|
ring1
|
no goals
|
8570716c4ac7b64b
|
Complex.norm_log_one_add_half_le_self
|
Mathlib/Analysis/SpecialFunctions/Complex/LogBounds.lean
|
/-- For `‖z‖ ≤ 1/2`, the complex logarithm is bounded by `(3/2) * ‖z‖`. -/
lemma norm_log_one_add_half_le_self {z : ℂ} (hz : ‖z‖ ≤ 1/2) : ‖(log (1 + z))‖ ≤ (3/2) * ‖z‖
|
case hab.h₁
z : ℂ
hz : ‖z‖ ≤ 1 / 2
hz3 : (1 - ‖z‖)⁻¹ ≤ 2
⊢ ‖z‖ * ‖z‖ ≤ ‖z‖ * (1 / 2)
|
apply mul_le_mul (by simp only [mul_one, le_refl])
(by simpa only [one_div] using hz) (norm_nonneg z) (by simp only [mul_one, norm_nonneg])
|
no goals
|
f3a688a3bed721f7
|
EReal.left_distrib_of_nonneg
|
Mathlib/Data/Real/EReal.lean
|
lemma left_distrib_of_nonneg {a b c : EReal} (ha : 0 ≤ a) (hb : 0 ≤ b) :
c * (a + b) = c * a + c * b
|
a b c : EReal
ha : 0 ≤ a
hb : 0 ≤ b
⊢ (a + b) * c = a * c + b * c
|
exact right_distrib_of_nonneg ha hb
|
no goals
|
bba678c65cf093fc
|
Convex.helly_theorem'
|
Mathlib/Analysis/Convex/Radon.lean
|
theorem helly_theorem' {F : ι → Set E} {s : Finset ι}
(h_convex : ∀ i ∈ s, Convex 𝕜 (F i))
(h_inter : ∀ I ⊆ s, #I ≤ finrank 𝕜 E + 1 → (⋂ i ∈ I, F i).Nonempty) :
(⋂ i ∈ s, F i).Nonempty
|
case h.h
𝕜 : Type u_2
E : Type u_3
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
n k : ℕ
h_card : finrank 𝕜 E + 1 ≤ k
hk :
∀ {ι : Type u_1} {F : ι → Set E} {s : Finset ι},
(∀ i ∈ s, Convex 𝕜 (F i)) →
(∀ I ⊆ s, #I ≤ finrank 𝕜 E + 1 → (⋂ i ∈ I, F i).Nonempty) → #s = k → (⋂ i ∈ s, F i).Nonempty
ι : Type u_1
F : ι → Set E
s : Finset ι
h_convex : ∀ i ∈ s, Convex 𝕜 (F i)
h_inter : ∀ I ⊆ s, #I ≤ finrank 𝕜 E + 1 → (⋂ i ∈ I, F i).Nonempty
hn : #s = k + 1
a : { x // x ∈ s } → E := fun i => ⋯.some
h_ind : ¬AffineIndependent 𝕜 a
I : Set { x // x ∈ s }
p : E
hp_I : p ∈ (convexHull 𝕜) (a '' I)
hp_Ic : p ∈ (convexHull 𝕜) (a '' Iᶜ)
i✝ : ι
hi : i✝ ∈ Membership.mem s.val
i : { x // x ∈ s } := ⟨i✝, hi⟩
⊢ ∀ (J : Set { x // x ∈ s }), i ∈ J → (convexHull 𝕜) (a '' Jᶜ) ⊆ F ↑i
|
intro J hi
|
case h.h
𝕜 : Type u_2
E : Type u_3
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
n k : ℕ
h_card : finrank 𝕜 E + 1 ≤ k
hk :
∀ {ι : Type u_1} {F : ι → Set E} {s : Finset ι},
(∀ i ∈ s, Convex 𝕜 (F i)) →
(∀ I ⊆ s, #I ≤ finrank 𝕜 E + 1 → (⋂ i ∈ I, F i).Nonempty) → #s = k → (⋂ i ∈ s, F i).Nonempty
ι : Type u_1
F : ι → Set E
s : Finset ι
h_convex : ∀ i ∈ s, Convex 𝕜 (F i)
h_inter : ∀ I ⊆ s, #I ≤ finrank 𝕜 E + 1 → (⋂ i ∈ I, F i).Nonempty
hn : #s = k + 1
a : { x // x ∈ s } → E := fun i => ⋯.some
h_ind : ¬AffineIndependent 𝕜 a
I : Set { x // x ∈ s }
p : E
hp_I : p ∈ (convexHull 𝕜) (a '' I)
hp_Ic : p ∈ (convexHull 𝕜) (a '' Iᶜ)
i✝ : ι
hi✝ : i✝ ∈ Membership.mem s.val
i : { x // x ∈ s } := ⟨i✝, hi✝⟩
J : Set { x // x ∈ s }
hi : i ∈ J
⊢ (convexHull 𝕜) (a '' Jᶜ) ⊆ F ↑i
|
d589b7c5c4686ede
|
CStarAlgebra.directedOn_nonneg_ball
|
Mathlib/Analysis/CStarAlgebra/ApproximateUnit.lean
|
lemma CStarAlgebra.directedOn_nonneg_ball :
DirectedOn (· ≤ ·) ({x : A | 0 ≤ x} ∩ Metric.ball 0 1)
|
case intro.intro.refine_1
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
f : ℝ≥0 → ℝ≥0 := fun x => 1 - (1 + x)⁻¹
g : ℝ≥0 → ℝ≥0 := fun x => x * (1 - x)⁻¹
this : ∀ (a b : A), 0 ≤ a → 0 ≤ b → ‖a‖ < 1 → ‖b‖ < 1 → a ≤ cfcₙ f (cfcₙ g a + cfcₙ g b)
a : A
ha₁ : 0 ≤ a
b : A
hb₁ : 0 ≤ b
ha₂ : ‖a‖ < 1
hb₂ : ‖b‖ < 1
⊢ cfcₙ f (cfcₙ g a + cfcₙ g b) ∈ Metric.ball 0 1
|
simpa only [Metric.mem_ball, dist_zero_right] using norm_cfcₙ_one_sub_one_add_inv_lt_one _
|
no goals
|
dd87a5ac42b34be2
|
ProbabilityTheory.Kernel.setIntegral_density_of_measurableSet
|
Mathlib/Probability/Kernel/Disintegration/Density.lean
|
/-- Auxiliary lemma for `setIntegral_density`. -/
lemma setIntegral_density_of_measurableSet (hκν : fst κ ≤ ν)
[IsFiniteKernel ν] (n : ℕ) (a : α) {s : Set β} (hs : MeasurableSet s) {A : Set γ}
(hA : MeasurableSet[countableFiltration γ n] A) :
∫ x in A, density κ ν a x s ∂(ν a) = (κ a (A ×ˢ s)).toReal
|
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
inst✝ : IsFiniteKernel ν
n : ℕ
a : α
s : Set β
hs : MeasurableSet s
A : Set γ
hA : MeasurableSet A
h :
Tendsto (fun i => ∫ (x : γ) in A, κ.densityProcess ν i a x s ∂ν a) atTop (𝓝 (∫ (x : γ) in A, κ.density ν a x s ∂ν a))
⊢ ∫ (x : γ) in A, κ.density ν a x s ∂ν a = limsup (fun i => ∫ (x : γ) in A, κ.densityProcess ν i a x s ∂ν a) atTop
|
rw [h.limsup_eq]
|
no goals
|
973b1e917f8f26ce
|
SemilatticeInf.ext
|
Mathlib/Order/Lattice.lean
|
theorem SemilatticeInf.ext {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B
|
case mk.mk.refl.e_inf.h.h
α : Type u_1
toPartialOrder✝ : PartialOrder α
inf✝¹ : α → α → α
inf_le_left✝¹ : ∀ (a b : α), inf✝¹ a b ≤ a
inf_le_right✝¹ : ∀ (a b : α), inf✝¹ a b ≤ b
le_inf✝¹ : ∀ (a b c : α), a ≤ b → a ≤ c → a ≤ inf✝¹ b c
inf✝ : α → α → α
inf_le_left✝ : ∀ (a b : α), inf✝ a b ≤ a
inf_le_right✝ : ∀ (a b : α), inf✝ a b ≤ b
le_inf✝ : ∀ (a b c : α), a ≤ b → a ≤ c → a ≤ inf✝ b c
H : ∀ (x y : α), x ≤ y ↔ x ≤ y
x✝¹ x✝ : α
⊢ inf✝¹ x✝¹ x✝ = inf✝ x✝¹ x✝
|
apply SemilatticeInf.ext_inf H
|
no goals
|
e11959820adb9936
|
isClosed_of_spaced_out
|
Mathlib/Topology/UniformSpace/Separation.lean
|
theorem isClosed_of_spaced_out [T0Space α] {V₀ : Set (α × α)} (V₀_in : V₀ ∈ 𝓤 α) {s : Set α}
(hs : s.Pairwise fun x y => (x, y) ∉ V₀) : IsClosed s
|
case intro.intro.intro.h.intro.intro.h.intro.intro
α : Type u
inst✝¹ : UniformSpace α
inst✝ : T0Space α
V₀ : Set (α × α)
V₀_in : V₀ ∈ 𝓤 α
s : Set α
hs : s.Pairwise fun x y => (x, y) ∉ V₀
V₁ : Set (α × α)
V₁_in : V₁ ∈ 𝓤 α
V₁_symm : SymmetricRel V₁
h_comp : V₁ ○ V₁ ⊆ V₀
x : α
hx : ∀ {V : Set (α × α)}, V ∈ 𝓤 α → (ball x V ∩ s).Nonempty
V : Set (α × α)
V_in : V ∈ 𝓤 α
a✝ : SymmetricRel V
z : α
hz : z ∈ ball x (V₁ ∩ V)
hz' : z ∈ s
hy : z ∈ ball x V₁
hy' : z ∈ s
⊢ (x, z) ∈ V
|
exact ball_inter_right x _ _ hz
|
no goals
|
3bca2cffbaff4806
|
Bool.lt_trans
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Bool.lean
|
theorem lt_trans : ∀ {x y z : Bool}, x < y → y < z → x < z
|
⊢ ∀ {x y z : Bool}, x < y → y < z → x < z
|
decide
|
no goals
|
8582e02182357a8e
|
mulLeftLinearMap_eq_zero_iff
|
Mathlib/Data/Matrix/Bilinear.lean
|
theorem mulLeftLinearMap_eq_zero_iff [Nonempty n] (a : Matrix l m A) :
mulLeftLinearMap n R a = 0 ↔ a = 0
|
case mpr
l : Type u_1
m : Type u_2
n : Type u_3
R : Type u_5
A : Type u_6
inst✝⁶ : Fintype m
inst✝⁵ : DecidableEq m
inst✝⁴ : Semiring R
inst✝³ : Semiring A
inst✝² : Module R A
inst✝¹ : SMulCommClass R A A
inst✝ : Nonempty n
a : Matrix l m A
h : a = 0
⊢ mulLeftLinearMap n R a = 0
|
rw [h]
|
case mpr
l : Type u_1
m : Type u_2
n : Type u_3
R : Type u_5
A : Type u_6
inst✝⁶ : Fintype m
inst✝⁵ : DecidableEq m
inst✝⁴ : Semiring R
inst✝³ : Semiring A
inst✝² : Module R A
inst✝¹ : SMulCommClass R A A
inst✝ : Nonempty n
a : Matrix l m A
h : a = 0
⊢ mulLeftLinearMap n R 0 = 0
|
4d8d625502868ba9
|
MvQPF.suppPreservation_iff_isUniform
|
Mathlib/Data/QPF/Multivariate/Basic.lean
|
theorem suppPreservation_iff_isUniform : q.SuppPreservation ↔ q.IsUniform
|
case mpr
n : ℕ
F : TypeVec.{u} n → Type u_1
q : MvQPF F
⊢ IsUniform → SuppPreservation
|
rintro h α ⟨a, f⟩
|
case mpr.mk
n : ℕ
F : TypeVec.{u} n → Type u_1
q : MvQPF F
h : IsUniform
α : TypeVec.{u} n
a : (P F).A
f : (P F).B a ⟹ α
⊢ supp (abs ⟨a, f⟩) = supp ⟨a, f⟩
|
08d317fabec2531c
|
quotient_norm_add_le
|
Mathlib/Analysis/Normed/Group/Quotient.lean
|
theorem quotient_norm_add_le (S : AddSubgroup M) (x y : M ⧸ S) : ‖x + y‖ ≤ ‖x‖ + ‖y‖
|
case intro.intro.intro
M : Type u_1
inst✝ : SeminormedAddCommGroup M
S : AddSubgroup M
x y : M
⊢ ‖↑x + ↑y‖ ≤ ‖↑x‖ + ‖↑y‖
|
simp only [← mk'_apply, ← map_add, quotient_norm_mk_eq, sInf_image']
|
case intro.intro.intro
M : Type u_1
inst✝ : SeminormedAddCommGroup M
S : AddSubgroup M
x y : M
⊢ ⨅ a, ‖x + y + ↑a‖ ≤ (⨅ a, ‖x + ↑a‖) + ⨅ a, ‖y + ↑a‖
|
410baaa27ccbacc2
|
List.cons_subperm_of_not_mem_of_mem
|
Mathlib/.lake/packages/batteries/Batteries/Data/List/Perm.lean
|
theorem cons_subperm_of_not_mem_of_mem {a : α} {l₁ l₂ : List α} (h₁ : a ∉ l₁) (h₂ : a ∈ l₂)
(s : l₁ <+~ l₂) : a :: l₁ <+~ l₂
|
α : Type u_1
a : α
l₂ l r₁ l₂✝ : List α
b : α
s'✝ : r₁ <+ l₂✝
ih : ∀ {l₁ : List α}, ¬a ∈ l₁ → a ∈ l₂✝ → r₁ ~ l₁ → a :: l₁ <+~ l₂✝
l₁ : List α
h₁ : ¬a ∈ l₁
p : r₁ ~ l₁
h₂ : a = b ∨ a ∈ l₂✝
m : a ∈ l₂✝
t : List α
p' : t ~ a :: l₁
s' : t <+ l₂✝
⊢ a :: l₁ <+~ b :: l₂✝
|
exact ⟨t, p', s'.cons _⟩
|
no goals
|
5432dae05155b132
|
GaussianInt.mod_four_eq_three_of_nat_prime_of_prime
|
Mathlib/NumberTheory/Zsqrtd/QuadraticReciprocity.lean
|
theorem mod_four_eq_three_of_nat_prime_of_prime (p : ℕ) [hp : Fact p.Prime]
(hpi : Prime (p : ℤ[i])) : p % 4 = 3 :=
hp.1.eq_two_or_odd.elim
(fun hp2 => by
have := hpi.irreducible.isUnit_or_isUnit (a := ⟨1, 1⟩) (b := ⟨1, -1⟩)
simp [hp2, Zsqrtd.ext_iff, ← norm_eq_one_iff, norm_def] at this)
fun hp1 =>
by_contradiction fun hp3 : p % 4 ≠ 3 => by
have hp41 : p % 4 = 1
|
p : ℕ
hp : Fact (Nat.Prime p)
hpi : Prime ↑p
hp1 : p % 2 = 1
hp3 : p % 4 ≠ 3
hp41 : p % 4 = 1
k : ℕ
k_lt_p : k < p
hk : -1 = ↑k * ↑k
hpk : p ∣ k ^ 2 + 1
⊢ ↑k ^ 2 + 1 = { re := ↑k, im := 1 } * { re := ↑k, im := -1 }
|
ext <;> simp [sq]
|
no goals
|
9ab49eaa8ecd58e2
|
Polynomial.coeff_hermite_explicit
|
Mathlib/RingTheory/Polynomial/Hermite/Basic.lean
|
theorem coeff_hermite_explicit :
∀ n k : ℕ, coeff (hermite (2 * n + k)) k = (-1) ^ n * (2 * n - 1)‼ * Nat.choose (2 * n + k) k
| 0, _ => by simp
| n + 1, 0 => by
convert coeff_hermite_succ_zero (2 * n + 1) using 1
-- Porting note: ring_nf did not solve the goal on line 165
rw [coeff_hermite_explicit n 1, (by rw [Nat.left_distrib, mul_one, Nat.add_one_sub_one] :
2 * (n + 1) - 1 = 2 * n + 1), Nat.doubleFactorial_add_one, Nat.choose_zero_right,
Nat.choose_one_right, pow_succ]
push_cast
ring
| n + 1, k + 1 => by
let hermite_explicit : ℕ → ℕ → ℤ := fun n k =>
(-1) ^ n * (2 * n - 1)‼ * Nat.choose (2 * n + k) k
have hermite_explicit_recur :
∀ n k : ℕ,
hermite_explicit (n + 1) (k + 1) =
hermite_explicit (n + 1) k - (k + 2) * hermite_explicit n (k + 2)
|
n : ℕ
⊢ 2 * (n + 1) - 1 = 2 * n + 1
|
rw [Nat.left_distrib, mul_one, Nat.add_one_sub_one]
|
no goals
|
76b4bea6433d05b5
|
Std.DHashMap.Internal.List.getValueCast_insertEntry_self
|
Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/List/Associative.lean
|
theorem getValueCast_insertEntry_self [BEq α] [LawfulBEq α] {l : List ((a : α) × β a)} {k : α}
{v : β k} : getValueCast k (insertEntry k v l) containsKey_insertEntry_self = v
|
α : Type u
β : α → Type v
inst✝¹ : BEq α
inst✝ : LawfulBEq α
l : List ((a : α) × β a)
k : α
v : β k
⊢ getValueCast k (insertEntry k v l) ⋯ = v
|
simp [getValueCast_insertEntry]
|
no goals
|
a32e05cc0c740ceb
|
MeasureTheory.lintegral_pow_le_pow_lintegral_fderiv
|
Mathlib/Analysis/FunctionalSpaces/SobolevInequality.lean
|
theorem lintegral_pow_le_pow_lintegral_fderiv {u : E → F}
(hu : ContDiff ℝ 1 u) (h2u : HasCompactSupport u)
{p : ℝ} (hp : Real.IsConjExponent (finrank ℝ E) p) :
∫⁻ x, ‖u x‖ₑ ^ p ∂μ ≤
lintegralPowLePowLIntegralFDerivConst μ p * (∫⁻ x, ‖fderiv ℝ u x‖ₑ ∂μ) ^ p
|
case h.e'_5.h.e'_4.h.h.e'_3
F : Type u_3
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
E : Type u_4
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
u : E → F
hu : ContDiff ℝ 1 u
h2u : HasCompactSupport u
p : ℝ
hp✝ : (↑(finrank ℝ E)).IsConjExponent p
C : ℝ≥0 := lintegralPowLePowLIntegralFDerivConst μ p
ι : Type := Fin (finrank ℝ E)
hιcard : #ι = finrank ℝ E
this✝ : finrank ℝ E = finrank ℝ (ι → ℝ)
e : E ≃L[ℝ] ι → ℝ := ContinuousLinearEquiv.ofFinrankEq this✝
this : (Measure.map (⇑e.symm) volume).IsAddHaarMeasure
hp : (↑#ι).IsConjExponent p
h0p : 0 ≤ p
c : ℝ≥0 := μ.addHaarScalarFactor (Measure.map (⇑e.symm) volume)
hc : 0 < c
h2c : μ = c • Measure.map (⇑e.symm) volume
h3c : ↑c ≠ 0
h0C : C = c * ‖↑e.symm‖₊ ^ p * (c ^ p)⁻¹
hC : C * c ^ p = c * ‖↑e.symm‖₊ ^ p
v : (ι → ℝ) → F := u ∘ ⇑e.symm
hv : ContDiff ℝ 1 v
h2v : HasCompactSupport v
y : ι → ℝ
⊢ DifferentiableAt ℝ (⇑e.symm) y
|
exact e.symm.differentiableAt
|
no goals
|
7e7c86c82b1c72d8
|
tendsto_sum_mul_atTop_nhds_one_sub_integral
|
Mathlib/NumberTheory/AbelSummation.lean
|
theorem tendsto_sum_mul_atTop_nhds_one_sub_integral
(hf_diff : ∀ t ∈ Set.Ici 0, DifferentiableAt ℝ f t)
(hf_int : LocallyIntegrableOn (deriv f) (Set.Ici 0)) {l : 𝕜}
(h_lim : Tendsto (fun n : ℕ ↦ f n * ∑ k ∈ Icc 0 n, c k) atTop (𝓝 l))
{g : ℝ → 𝕜} (hg_dom : (fun t ↦ deriv f t * ∑ k ∈ Icc 0 ⌊t⌋₊, c k) =O[atTop] g)
(hg_int : IntegrableAtFilter g atTop) :
Tendsto (fun n : ℕ ↦ ∑ k ∈ Icc 0 n, f k * c k) atTop
(𝓝 (l - ∫ t in Set.Ioi 0, deriv f t * ∑ k ∈ Icc 0 ⌊t⌋₊, c k))
|
𝕜 : Type u_1
inst✝ : RCLike 𝕜
c : ℕ → 𝕜
f : ℝ → 𝕜
hf_diff : ∀ t ∈ Set.Ici 0, DifferentiableAt ℝ f t
hf_int : LocallyIntegrableOn (deriv f) (Set.Ici 0) volume
l : 𝕜
h_lim : Tendsto (fun n => f ↑n * ∑ k ∈ Icc 0 n, c k) atTop (𝓝 l)
g : ℝ → 𝕜
hg_dom : (fun t => deriv f t * ∑ k ∈ Icc 0 ⌊t⌋₊, c k) =O[atTop] g
hg_int : IntegrableAtFilter g atTop volume
h_lim' :
Tendsto (fun n => ∫ (t : ℝ) in Set.Ioc 0 ↑n, deriv f t * ∑ k ∈ Icc 0 ⌊t⌋₊, c k) atTop
(𝓝 (∫ (t : ℝ) in Set.Ioi 0, deriv f t * ∑ k ∈ Icc 0 ⌊t⌋₊, c k))
⊢ Tendsto (fun n => ∑ k ∈ Icc 0 n, f ↑k * c k) atTop (𝓝 (l - ∫ (t : ℝ) in Set.Ioi 0, deriv f t * ∑ k ∈ Icc 0 ⌊t⌋₊, c k))
|
refine (h_lim.sub h_lim').congr (fun _ ↦ ?_)
|
𝕜 : Type u_1
inst✝ : RCLike 𝕜
c : ℕ → 𝕜
f : ℝ → 𝕜
hf_diff : ∀ t ∈ Set.Ici 0, DifferentiableAt ℝ f t
hf_int : LocallyIntegrableOn (deriv f) (Set.Ici 0) volume
l : 𝕜
h_lim : Tendsto (fun n => f ↑n * ∑ k ∈ Icc 0 n, c k) atTop (𝓝 l)
g : ℝ → 𝕜
hg_dom : (fun t => deriv f t * ∑ k ∈ Icc 0 ⌊t⌋₊, c k) =O[atTop] g
hg_int : IntegrableAtFilter g atTop volume
h_lim' :
Tendsto (fun n => ∫ (t : ℝ) in Set.Ioc 0 ↑n, deriv f t * ∑ k ∈ Icc 0 ⌊t⌋₊, c k) atTop
(𝓝 (∫ (t : ℝ) in Set.Ioi 0, deriv f t * ∑ k ∈ Icc 0 ⌊t⌋₊, c k))
x✝ : ℕ
⊢ f ↑x✝ * ∑ k ∈ Icc 0 x✝, c k - ∫ (t : ℝ) in Set.Ioc 0 ↑x✝, deriv f t * ∑ k ∈ Icc 0 ⌊t⌋₊, c k =
∑ k ∈ Icc 0 x✝, f ↑k * c k
|
b58ce824ebf3d435
|
HasFDerivAtFilter.iterate
|
Mathlib/Analysis/Calculus/FDeriv/Comp.lean
|
theorem HasFDerivAtFilter.iterate {f : E → E} {f' : E →L[𝕜] E}
(hf : HasFDerivAtFilter f f' x L) (hL : Tendsto f L L) (hx : f x = x) (n : ℕ) :
HasFDerivAtFilter f^[n] (f' ^ n) x L
|
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : E
L : Filter E
f : E → E
f' : E →L[𝕜] E
hf : HasFDerivAtFilter f f' x L
hL : Tendsto f L L
hx : f x = x
n : ℕ
⊢ HasFDerivAtFilter f^[n] (f' ^ n) x L
|
induction n with
| zero => exact hasFDerivAtFilter_id x L
| succ n ihn =>
rw [Function.iterate_succ, pow_succ]
rw [← hx] at ihn
exact ihn.comp x hf hL
|
no goals
|
1115050efc37856a
|
MeasureTheory.withDensityᵥ_smul_eq_withDensityᵥ_withDensity
|
Mathlib/MeasureTheory/VectorMeasure/WithDensity.lean
|
theorem withDensityᵥ_smul_eq_withDensityᵥ_withDensity {f : α → ℝ≥0} {g : α → E}
(hf : AEMeasurable f μ) (hfg : Integrable (f • g) μ) :
μ.withDensityᵥ (f • g) = (μ.withDensity (fun x ↦ f x)).withDensityᵥ g
|
case h
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : α → ℝ≥0
g : α → E
hf : AEMeasurable f μ
hfg : Integrable (f • g) μ
s : Set α
hs : MeasurableSet s
⊢ ∫ (x : α) in s, (f • g) x ∂μ = ∫ (x : α) in s, f x • g x ∂μ
|
simp only [Pi.smul_apply']
|
no goals
|
d5e2cb953e5e689a
|
ProbabilityTheory.integral_compProd
|
Mathlib/Probability/Kernel/Composition/IntegralCompProd.lean
|
theorem integral_compProd :
∀ {f : β × γ → E} (_ : Integrable f ((κ ⊗ₖ η) a)),
∫ z, f z ∂(κ ⊗ₖ η) a = ∫ x, ∫ y, f (x, y) ∂η (a, x) ∂κ a
|
case h.e'_2
α : Type u_1
β : Type u_2
γ : Type u_3
E : Type u_4
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝³ : NormedAddCommGroup E
a : α
κ : Kernel α β
inst✝² : IsSFiniteKernel κ
η : Kernel (α × β) γ
inst✝¹ : IsSFiniteKernel η
inst✝ : NormedSpace ℝ E
f✝ : β × γ → E
hE : CompleteSpace E
f g : β × γ → E
hfg : f =ᶠ[ae ((κ ⊗ₖ η) a)] g
a✝ : Integrable f ((κ ⊗ₖ η) a)
hf : ∫ (z : β × γ), f z ∂(κ ⊗ₖ η) a = ∫ (x : β), ∫ (y : γ), f (x, y) ∂η (a, x) ∂κ a
⊢ ∫ (z : β × γ), g z ∂(κ ⊗ₖ η) a = ∫ (z : β × γ), f z ∂(κ ⊗ₖ η) a
|
exact integral_congr_ae hfg.symm
|
no goals
|
0414f2f4340089bd
|
Monoid.CoprodI.Word.mem_smul_iff
|
Mathlib/GroupTheory/CoprodI.lean
|
theorem mem_smul_iff {i j : ι} {m₁ : M i} {m₂ : M j} {w : Word M} :
⟨_, m₁⟩ ∈ (of m₂ • w).toList ↔
(¬i = j ∧ ⟨i, m₁⟩ ∈ w.toList)
∨ (m₁ ≠ 1 ∧ ∃ (hij : i = j),(⟨i, m₁⟩ ∈ w.toList.tail) ∨
(∃ m', ⟨j, m'⟩ ∈ w.toList.head? ∧ m₁ = hij ▸ (m₂ * m')) ∨
(w.fstIdx ≠ some j ∧ m₁ = hij ▸ m₂))
|
case neg
ι : Type u_1
M : ι → Type u_2
inst✝² : (i : ι) → Monoid (M i)
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → DecidableEq (M i)
j : ι
m₂ : M j
w : Word M
m₁ : M j
hw : ⟨j, m₁⟩ ∉ w.toList.tail
⊢ (⟨j, m₁⟩ ∈ w.toList.tail ∨
¬m₁ = 1 ∧
m₁ = m₂ * if h : ∃ (h : ¬w.toList = []), (w.toList.head h).fst = j then ⋯ ▸ (w.toList.head ⋯).snd else 1) ↔
¬m₁ = 1 ∧ (⟨j, m₁⟩ ∈ w.toList.tail ∨ (∃ m', ⟨j, m'⟩ ∈ w.toList.head? ∧ m₁ = m₂ * m') ∨ ¬w.fstIdx = some j ∧ m₁ = m₂)
|
simp only [hw, false_or, Option.mem_def, ne_eq, and_congr_right_iff]
|
case neg
ι : Type u_1
M : ι → Type u_2
inst✝² : (i : ι) → Monoid (M i)
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → DecidableEq (M i)
j : ι
m₂ : M j
w : Word M
m₁ : M j
hw : ⟨j, m₁⟩ ∉ w.toList.tail
⊢ ¬m₁ = 1 →
((m₁ = m₂ * if h : ∃ (h : ¬w.toList = []), (w.toList.head h).fst = j then ⋯ ▸ (w.toList.head ⋯).snd else 1) ↔
(∃ m', w.toList.head? = some ⟨j, m'⟩ ∧ m₁ = m₂ * m') ∨ ¬w.fstIdx = some j ∧ m₁ = m₂)
|
1bd393897e7ca3ac
|
EReal.tendsto_nhds_bot_iff_real
|
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
theorem tendsto_nhds_bot_iff_real {α : Type*} {m : α → EReal} {f : Filter α} :
Tendsto m f (𝓝 ⊥) ↔ ∀ x : ℝ, ∀ᶠ a in f, m a < x :=
nhds_bot_basis.tendsto_right_iff.trans <| by simp only [true_implies, mem_Iio]
|
α : Type u_2
m : α → EReal
f : Filter α
⊢ (∀ (i : ℝ), True → ∀ᶠ (x : α) in f, m x ∈ Iio ↑i) ↔ ∀ (x : ℝ), ∀ᶠ (a : α) in f, m a < ↑x
|
simp only [true_implies, mem_Iio]
|
no goals
|
4549be55b30cd7f3
|
legendreSym.quadratic_reciprocity
|
Mathlib/NumberTheory/LegendreSymbol/QuadraticReciprocity.lean
|
theorem quadratic_reciprocity (hp : p ≠ 2) (hq : q ≠ 2) (hpq : p ≠ q) :
legendreSym q p * legendreSym p q = (-1) ^ (p / 2 * (q / 2))
|
p q : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Fact (Nat.Prime q)
hp : p ≠ 2
hq : q ≠ 2
hpq : p ≠ q
hp₁ : p % 2 = 1
hq₁ : q % 2 = 1
hq₂ : ringChar (ZMod q) ≠ 2
h : (quadraticChar (ZMod p)) ↑q = (quadraticChar (ZMod q)) (↑(χ₄ ↑p) * ↑p)
⊢ legendreSym q ↑p * legendreSym p ↑q = (-1) ^ (p / 2 * (q / 2))
|
have nc : ∀ n r : ℕ, ((n : ℤ) : ZMod r) = n := fun n r => by norm_cast
|
p q : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Fact (Nat.Prime q)
hp : p ≠ 2
hq : q ≠ 2
hpq : p ≠ q
hp₁ : p % 2 = 1
hq₁ : q % 2 = 1
hq₂ : ringChar (ZMod q) ≠ 2
h : (quadraticChar (ZMod p)) ↑q = (quadraticChar (ZMod q)) (↑(χ₄ ↑p) * ↑p)
nc : ∀ (n r : ℕ), ↑↑n = ↑n
⊢ legendreSym q ↑p * legendreSym p ↑q = (-1) ^ (p / 2 * (q / 2))
|
1533f920a3f1cd59
|
Std.Tactic.BVDecide.LRAT.Internal.CNF.unsat_of_convertLRAT_unsat
|
Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Convert.lean
|
theorem CNF.unsat_of_convertLRAT_unsat (cnf : CNF Nat) :
Unsatisfiable (PosFin (cnf.numLiterals + 1)) (CNF.convertLRAT cnf)
→
cnf.Unsat
|
case a
cnf : CNF Nat
assignment : PosFin (cnf.numLiterals + 1) → Bool
h1 : ¬(assignment ⊨ DefaultFormula.ofArray (convertLRAT' (lift cnf)).toArray)
⊢ CNF.eval assignment (lift cnf) = false
|
apply eq_false_of_ne_true
|
case a.a
cnf : CNF Nat
assignment : PosFin (cnf.numLiterals + 1) → Bool
h1 : ¬(assignment ⊨ DefaultFormula.ofArray (convertLRAT' (lift cnf)).toArray)
⊢ ¬CNF.eval assignment (lift cnf) = true
|
c8ce4969f2703677
|
List.find?_flatten_eq_some_iff
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Find.lean
|
theorem find?_flatten_eq_some_iff {xs : List (List α)} {p : α → Bool} {a : α} :
xs.flatten.find? p = some a ↔
p a ∧ ∃ as ys zs bs, xs = as ++ (ys ++ a :: zs) :: bs ∧
(∀ a ∈ as, ∀ x ∈ a, !p x) ∧ (∀ x ∈ ys, !p x)
|
case mpr.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
p : α → Bool
a : α
h : p a = true
as : List (List α)
ys zs : List α
bs : List (List α)
h₁ : ∀ (a : List α), a ∈ as → ∀ (x : α), x ∈ a → (!p x) = true
h₂ : ∀ (x : α), x ∈ ys → (!p x) = true
⊢ p a = true ∧
∃ as_1 bs_1, (as ++ (ys ++ a :: zs) :: bs).flatten = as_1 ++ a :: bs_1 ∧ ∀ (a : α), a ∈ as_1 → (!p a) = true
|
refine ⟨h, as.flatten ++ ys, zs ++ bs.flatten, by simp, ?_⟩
|
case mpr.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
p : α → Bool
a : α
h : p a = true
as : List (List α)
ys zs : List α
bs : List (List α)
h₁ : ∀ (a : List α), a ∈ as → ∀ (x : α), x ∈ a → (!p x) = true
h₂ : ∀ (x : α), x ∈ ys → (!p x) = true
⊢ ∀ (a : α), a ∈ as.flatten ++ ys → (!p a) = true
|
b3081438100b7db8
|
Wbtw.sameRay_vsub
|
Mathlib/Analysis/Convex/Between.lean
|
theorem Wbtw.sameRay_vsub {x y z : P} (h : Wbtw R x y z) : SameRay R (y -ᵥ x) (z -ᵥ y)
|
case intro.intro.intro.inr
R : Type u_1
V : Type u_2
P : Type u_4
inst✝³ : StrictOrderedCommRing R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
x z : P
ht0 : 0 ≤ 0
ht1 : 0 ≤ 1
⊢ SameRay R ((0 • (z -ᵥ x) +ᵥ x) -ᵥ x) (z -ᵥ (0 • (z -ᵥ x) +ᵥ x))
|
simp
|
no goals
|
420bf32587db852e
|
IsometryEquiv.midpoint_fixed
|
Mathlib/Analysis/Normed/Affine/MazurUlam.lean
|
theorem midpoint_fixed {x y : PE} :
∀ e : PE ≃ᵢ PE, e x = x → e y = y → e (midpoint ℝ x y) = midpoint ℝ x y
|
E : Type u_1
PE : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : MetricSpace PE
inst✝ : NormedAddTorsor E PE
x y : PE
z : PE := midpoint ℝ x y
s : Set (PE ≃ᵢ PE) := {e | e x = x ∧ e y = y}
this : Nonempty ↑s
h_bdd : BddAbove (range fun e => dist (↑e z) z)
R : PE ≃ᵢ PE := (pointReflection ℝ z).toIsometryEquiv
f : PE ≃ᵢ PE → PE ≃ᵢ PE := fun e => ((e.trans R).trans e.symm).trans R
hf_dist : ∀ (e : PE ≃ᵢ PE), dist ((f e) z) z = 2 * dist (e z) z
⊢ ∀ (e : PE ≃ᵢ PE), e x = x → e y = y → e z = z
|
have hf_maps_to : MapsTo f s s := by
rintro e ⟨hx, hy⟩
constructor <;> simp [f, R, z, hx, hy, e.symm_apply_eq.2 hx.symm, e.symm_apply_eq.2 hy.symm]
|
E : Type u_1
PE : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : MetricSpace PE
inst✝ : NormedAddTorsor E PE
x y : PE
z : PE := midpoint ℝ x y
s : Set (PE ≃ᵢ PE) := {e | e x = x ∧ e y = y}
this : Nonempty ↑s
h_bdd : BddAbove (range fun e => dist (↑e z) z)
R : PE ≃ᵢ PE := (pointReflection ℝ z).toIsometryEquiv
f : PE ≃ᵢ PE → PE ≃ᵢ PE := fun e => ((e.trans R).trans e.symm).trans R
hf_dist : ∀ (e : PE ≃ᵢ PE), dist ((f e) z) z = 2 * dist (e z) z
hf_maps_to : MapsTo f s s
⊢ ∀ (e : PE ≃ᵢ PE), e x = x → e y = y → e z = z
|
b718555705b87dec
|
sSup_sUnion
|
Mathlib/Data/Set/Lattice.lean
|
theorem sSup_sUnion (s : Set (Set β)) : sSup (⋃₀ s) = ⨆ t ∈ s, sSup t
|
β : Type u_2
inst✝ : CompleteLattice β
s : Set (Set β)
⊢ sSup (⋃₀ s) = ⨆ t ∈ s, sSup t
|
simp only [sUnion_eq_biUnion, sSup_eq_iSup, iSup_iUnion]
|
no goals
|
b34aa5abf2637a7b
|
Filter.HasBasis.le_basis_iff
|
Mathlib/Order/Filter/Bases.lean
|
theorem HasBasis.le_basis_iff (hl : l.HasBasis p s) (hl' : l'.HasBasis p' s') :
l ≤ l' ↔ ∀ i', p' i' → ∃ i, p i ∧ s i ⊆ s' i'
|
α : Type u_1
ι : Sort u_4
ι' : Sort u_5
l l' : Filter α
p : ι → Prop
s : ι → Set α
p' : ι' → Prop
s' : ι' → Set α
hl : l.HasBasis p s
hl' : l'.HasBasis p' s'
⊢ l ≤ l' ↔ ∀ (i' : ι'), p' i' → ∃ i, p i ∧ s i ⊆ s' i'
|
simp only [hl'.ge_iff, hl.mem_iff]
|
no goals
|
407295017a0d1906
|
SetTheory.PGame.grundyValue_nim_add_nim
|
Mathlib/SetTheory/Game/Nim.lean
|
theorem grundyValue_nim_add_nim (x y : Ordinal) : grundyValue (nim x + nim y) = ∗x + ∗y
|
x y : Ordinal.{u_1}
⊢ (nim x + nim y).grundyValue = toNimber x + toNimber y
|
apply (grundyValue_le_of_forall_moveLeft _).antisymm (le_grundyValue_of_Iio_subset_moveLeft _)
|
x y : Ordinal.{u_1}
⊢ ∀ (i : (nim x + nim y).LeftMoves), ((nim x + nim y).moveLeft i).grundyValue ≠ toNimber x + toNimber y
x y : Ordinal.{u_1}
⊢ Set.Iio (toNimber x + toNimber y) ⊆ Set.range (grundyValue ∘ (nim x + nim y).moveLeft)
|
1ed3a29368c61fe4
|
List.sublist_append_iff
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Sublist.lean
|
theorem sublist_append_iff {l : List α} :
l <+ r₁ ++ r₂ ↔ ∃ l₁ l₂, l = l₁ ++ l₂ ∧ l₁ <+ r₁ ∧ l₂ <+ r₂
|
case cons.mpr.intro.intro.intro.intro.cons₂
α : Type u_1
r₂ : List α
r : α
r₁ : List α
ih : ∀ {l : List α}, l <+ r₁ ++ r₂ ↔ ∃ l₁ l₂, l = l₁ ++ l₂ ∧ l₁ <+ r₁ ∧ l₂ <+ r₂
l₂ : List α
w₂ : l₂ <+ r₂
l₁✝ : List α
w₁ : l₁✝ <+ r₁
⊢ r :: l₁✝ ++ l₂ <+ r :: r₁ ++ r₂
|
rename_i l
|
case cons.mpr.intro.intro.intro.intro.cons₂
α : Type u_1
r₂ : List α
r : α
r₁ : List α
ih : ∀ {l : List α}, l <+ r₁ ++ r₂ ↔ ∃ l₁ l₂, l = l₁ ++ l₂ ∧ l₁ <+ r₁ ∧ l₂ <+ r₂
l₂ : List α
w₂ : l₂ <+ r₂
l : List α
w₁ : l <+ r₁
⊢ r :: l ++ l₂ <+ r :: r₁ ++ r₂
|
1d98f3fb3ce2dcde
|
Polynomial.hasStrictDerivAt
|
Mathlib/Analysis/Calculus/Deriv/Polynomial.lean
|
theorem hasStrictDerivAt (x : 𝕜) :
HasStrictDerivAt (fun x => p.eval x) (p.derivative.eval x) x
|
case h_add
𝕜 : Type u
inst✝ : NontriviallyNormedField 𝕜
p✝ : 𝕜[X]
x : 𝕜
p q : 𝕜[X]
hp : HasStrictDerivAt (fun x => eval x p) (eval x (derivative p)) x
hq : HasStrictDerivAt (fun x => eval x q) (eval x (derivative q)) x
⊢ HasStrictDerivAt (fun x => eval x (p + q)) (eval x (derivative (p + q))) x
|
simpa using hp.add hq
|
no goals
|
a78192851f8d6b6d
|
ZMod.isSquare_neg_one_of_eq_sq_add_sq_of_isCoprime
|
Mathlib/NumberTheory/SumTwoSquares.lean
|
theorem ZMod.isSquare_neg_one_of_eq_sq_add_sq_of_isCoprime {n x y : ℤ} (h : n = x ^ 2 + y ^ 2)
(hc : IsCoprime x y) : IsSquare (-1 : ZMod n.natAbs)
|
n x y : ℤ
h : n = x ^ 2 + y ^ 2
hc : IsCoprime x y
⊢ IsCoprime x n
|
have hc2 : IsCoprime (x ^ 2) (y ^ 2) := hc.pow
|
n x y : ℤ
h : n = x ^ 2 + y ^ 2
hc : IsCoprime x y
hc2 : IsCoprime (x ^ 2) (y ^ 2)
⊢ IsCoprime x n
|
1f943d046c7310cd
|
PowerSeries.trunc_one_X
|
Mathlib/RingTheory/PowerSeries/Trunc.lean
|
lemma trunc_one_X : trunc (R := R) 1 X = 0
|
R : Type u_1
inst✝ : Semiring R
⊢ trunc 1 X = 0
|
simp
|
no goals
|
800ca48eab182a1c
|
Rat.numberField_discr
|
Mathlib/NumberTheory/NumberField/Discriminant/Defs.lean
|
theorem numberField_discr : discr ℚ = 1
|
b : Basis (Fin 1) ℤ (𝓞 ℚ) := (Basis.singleton (Fin 1) ℤ).map ringOfIntegersEquiv.toAddEquiv.toIntLinearEquiv.symm
⊢ discr ℚ = 1
|
calc NumberField.discr ℚ
_ = Algebra.discr ℤ b := by convert (discr_eq_discr ℚ b).symm
_ = Algebra.trace ℤ (𝓞 ℚ) (b default * b default) := by
rw [Algebra.discr_def, Matrix.det_unique, Algebra.traceMatrix_apply, Algebra.traceForm_apply]
_ = Algebra.trace ℤ (𝓞 ℚ) 1 := by
rw [Basis.map_apply, RingEquiv.toAddEquiv_eq_coe, AddEquiv.toIntLinearEquiv_symm,
AddEquiv.coe_toIntLinearEquiv, Basis.singleton_apply,
show (AddEquiv.symm ↑ringOfIntegersEquiv) (1 : ℤ) = ringOfIntegersEquiv.symm 1 by rfl,
map_one, mul_one]
_ = 1 := by rw [Algebra.trace_eq_matrix_trace b]; norm_num
|
no goals
|
d094a928336499b7
|
WeierstrassCurve.Projective.nonsingularLift_addMap
|
Mathlib/AlgebraicGeometry/EllipticCurve/Projective.lean
|
lemma nonsingularLift_addMap {P Q : PointClass F} (hP : W.NonsingularLift P)
(hQ : W.NonsingularLift Q) : W.NonsingularLift <| W.addMap P Q
|
case mk
F : Type u
inst✝ : Field F
W : Projective F
P Q : PointClass F
hQ : W.NonsingularLift Q
a✝ : Fin 3 → F
hP : W.NonsingularLift (Quot.mk (⇑(MulAction.orbitRel Fˣ (Fin 3 → F))) a✝)
⊢ W.NonsingularLift (W.addMap (Quot.mk (⇑(MulAction.orbitRel Fˣ (Fin 3 → F))) a✝) Q)
|
rcases Q
|
case mk.mk
F : Type u
inst✝ : Field F
W : Projective F
P Q : PointClass F
a✝¹ : Fin 3 → F
hP : W.NonsingularLift (Quot.mk (⇑(MulAction.orbitRel Fˣ (Fin 3 → F))) a✝¹)
a✝ : Fin 3 → F
hQ : W.NonsingularLift (Quot.mk (⇑(MulAction.orbitRel Fˣ (Fin 3 → F))) a✝)
⊢ W.NonsingularLift
(W.addMap (Quot.mk (⇑(MulAction.orbitRel Fˣ (Fin 3 → F))) a✝¹) (Quot.mk (⇑(MulAction.orbitRel Fˣ (Fin 3 → F))) a✝))
|
3e0b375aec5be89b
|
SimpleGraph.Walk.mem_support_iff_exists_getVert
|
Mathlib/Combinatorics/SimpleGraph/Connectivity/WalkDecomp.lean
|
theorem mem_support_iff_exists_getVert {u v w : V} {p : G.Walk v w} :
u ∈ p.support ↔ ∃ n, p.getVert n = u ∧ n ≤ p.length
|
case h
V : Type u
G : SimpleGraph V
u v w : V
p : G.Walk v w
n : ℕ
hn : p.getVert (n + 1) = u ∧ n + 1 ≤ p.length
hnp : ¬p.Nil
⊢ p.tail.getVert n = u ∧ n ≤ p.tail.length
|
rwa [getVert_tail, ← Nat.add_one_le_add_one_iff, length_tail_add_one hnp]
|
no goals
|
2d1fd9978307d9f8
|
Equiv.Perm.cycleFactorsFinset_mul_inv_mem_eq_sdiff
|
Mathlib/GroupTheory/Perm/Cycle/Factors.lean
|
theorem cycleFactorsFinset_mul_inv_mem_eq_sdiff [DecidableEq α] [Fintype α] {f g : Perm α}
(h : f ∈ cycleFactorsFinset g) : cycleFactorsFinset (g * f⁻¹) = cycleFactorsFinset g \ {f}
|
case refine_3.inl
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g f✝ σ τ : Perm α
hd : σ.Disjoint τ
a✝ : σ.IsCycle
hσ : ∀ {f : Perm α}, f ∈ σ.cycleFactorsFinset → (σ * f⁻¹).cycleFactorsFinset = σ.cycleFactorsFinset \ {f}
hτ : ∀ {f : Perm α}, f ∈ τ.cycleFactorsFinset → (τ * f⁻¹).cycleFactorsFinset = τ.cycleFactorsFinset \ {f}
f : Perm α
hf : f ∈ σ.cycleFactorsFinset
⊢ (σ * τ * f⁻¹).cycleFactorsFinset = (σ.cycleFactorsFinset ∪ τ.cycleFactorsFinset) \ {f}
|
rw [hd.commute.eq, union_comm, union_sdiff_distrib, sdiff_singleton_eq_erase,
erase_eq_of_not_mem, mul_assoc, Disjoint.cycleFactorsFinset_mul_eq_union, hσ hf]
|
case refine_3.inl
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g f✝ σ τ : Perm α
hd : σ.Disjoint τ
a✝ : σ.IsCycle
hσ : ∀ {f : Perm α}, f ∈ σ.cycleFactorsFinset → (σ * f⁻¹).cycleFactorsFinset = σ.cycleFactorsFinset \ {f}
hτ : ∀ {f : Perm α}, f ∈ τ.cycleFactorsFinset → (τ * f⁻¹).cycleFactorsFinset = τ.cycleFactorsFinset \ {f}
f : Perm α
hf : f ∈ σ.cycleFactorsFinset
⊢ τ.Disjoint (σ * f⁻¹)
case refine_3.inl
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g f✝ σ τ : Perm α
hd : σ.Disjoint τ
a✝ : σ.IsCycle
hσ : ∀ {f : Perm α}, f ∈ σ.cycleFactorsFinset → (σ * f⁻¹).cycleFactorsFinset = σ.cycleFactorsFinset \ {f}
hτ : ∀ {f : Perm α}, f ∈ τ.cycleFactorsFinset → (τ * f⁻¹).cycleFactorsFinset = τ.cycleFactorsFinset \ {f}
f : Perm α
hf : f ∈ σ.cycleFactorsFinset
⊢ f ∉ τ.cycleFactorsFinset
|
3d77fd0de031f082
|
Stream.fst_take_succ
|
Mathlib/.lake/packages/batteries/Batteries/Data/Stream.lean
|
theorem fst_take_succ [Stream σ α] (s : σ) :
(take s (n+1)).fst = match next? s with
| none => []
| some (a, s) => a :: (take s n).fst
|
σ : Type u_1
α : Type u_2
n : Nat
inst✝ : Stream σ α
s : σ
⊢ (match next? s with
| none => ([], s)
| some (a, s) => (a :: (take s n).fst, (take s n).snd)).fst =
match next? s with
| none => []
| some (a, s) => a :: (take s n).fst
|
split <;> rfl
|
no goals
|
ae89cbd25284fdc5
|
VectorFourier.fourierIntegral_continuous
|
Mathlib/Analysis/Fourier/FourierTransform.lean
|
theorem fourierIntegral_continuous [FirstCountableTopology W] (he : Continuous e)
(hL : Continuous fun p : V × W ↦ L p.1 p.2) {f : V → E} (hf : Integrable f μ) :
Continuous (fourierIntegral e μ L f)
|
case h_cont
𝕜 : Type u_1
inst✝¹³ : CommRing 𝕜
V : Type u_2
inst✝¹² : AddCommGroup V
inst✝¹¹ : Module 𝕜 V
inst✝¹⁰ : MeasurableSpace V
W : Type u_3
inst✝⁹ : AddCommGroup W
inst✝⁸ : Module 𝕜 W
E : Type u_4
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℂ E
inst✝⁵ : TopologicalSpace 𝕜
inst✝⁴ : IsTopologicalRing 𝕜
inst✝³ : TopologicalSpace V
inst✝² : BorelSpace V
inst✝¹ : TopologicalSpace W
e : AddChar 𝕜 𝕊
μ : Measure V
L : V →ₗ[𝕜] W →ₗ[𝕜] 𝕜
inst✝ : FirstCountableTopology W
he : Continuous ⇑e
hL : Continuous fun p => (L p.1) p.2
f : V → E
hf : Integrable f μ
v : V
⊢ Continuous fun x => -(L v) x
|
exact (hL.comp (continuous_prod_mk.mpr ⟨continuous_const, continuous_id⟩)).neg
|
no goals
|
eee0139c18fc1a81
|
PresheafOfModules.Sheafify.smul_add
|
Mathlib/Algebra/Category/ModuleCat/Presheaf/Sheafify.lean
|
protected lemma smul_add : smul α φ r (m + m') = smul α φ r m + smul α φ r m'
|
case a.intro.intro.intro.intro.intro
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
J : GrothendieckTopology C
R₀ : Cᵒᵖ ⥤ RingCat
R : Sheaf J RingCat
α : R₀ ⟶ R.val
inst✝³ : Presheaf.IsLocallyInjective J α
inst✝² : Presheaf.IsLocallySurjective J α
M₀ : PresheafOfModules R₀
A : Sheaf J AddCommGrp
φ : M₀.presheaf ⟶ A.val
inst✝¹ : Presheaf.IsLocallyInjective J φ
inst✝ : Presheaf.IsLocallySurjective J φ
X : Cᵒᵖ
r : ↑(R.val.obj X)
m m' : ↑(A.val.obj X)
S : Sieve (Opposite.unop X) := Presheaf.imageSieve α r ⊓ Presheaf.imageSieve φ m ⊓ Presheaf.imageSieve φ m'
hS : S ∈ J (Opposite.unop X)
Y : C
f : Y ⟶ Opposite.unop X
r₀ : ToType (R₀.obj (Opposite.op Y))
hr₀ : (ConcreteCategory.hom (α.app (Opposite.op Y))) r₀ = (ConcreteCategory.hom (R.val.map f.op)) r
m₀ : ↑(M₀.obj (Opposite.op Y))
hm₀ : (ConcreteCategory.hom (φ.app (Opposite.op Y))) m₀ = (ConcreteCategory.hom (A.val.map f.op)) m
m₀' : ↑(M₀.obj (Opposite.op Y))
hm₀' : (ConcreteCategory.hom (φ.app (Opposite.op Y))) m₀' = (ConcreteCategory.hom (A.val.map f.op)) m'
⊢ (ConcreteCategory.hom (A.val.map f.op)) (smul α φ r (m + m')) =
(ConcreteCategory.hom (A.val.map f.op)) (smul α φ r m + smul α φ r m')
|
rw [(A.val.map f.op).hom.map_add, map_smul_eq α φ r m f.op r₀ hr₀ m₀ hm₀,
map_smul_eq α φ r m' f.op r₀ hr₀ m₀' hm₀',
map_smul_eq α φ r (m + m') f.op r₀ hr₀ (m₀ + m₀')
(by rw [map_add, map_add, hm₀, hm₀']),
smul_add, map_add]
|
no goals
|
e45a9bac6c91f436
|
Fin.accumulate_injective
|
Mathlib/RingTheory/MvPolynomial/Symmetric/FundamentalTheorem.lean
|
lemma accumulate_injective {n m} (hnm : n ≤ m) : Function.Injective (accumulate n m)
|
n m : ℕ
hnm : n ≤ m
⊢ Function.Injective ⇑(accumulate n m)
|
refine fun t s he ↦ funext fun i ↦ ?_
|
n m : ℕ
hnm : n ≤ m
t s : Fin n → ℕ
he : (accumulate n m) t = (accumulate n m) s
i : Fin n
⊢ t i = s i
|
48331ce5ab536779
|
SchwartzMap.pow_mul_le_of_le_of_pow_mul_le
|
Mathlib/Analysis/Distribution/SchwartzSpace.lean
|
/-- Pointwise inequality to control `x ^ k * f` in terms of `1 / (1 + x) ^ l` if one controls both
`f` (with a bound `C₁`) and `x ^ (k + l) * f` (with a bound `C₂`). This will be used to check
integrability of `x ^ k * f x` when `f` is a Schwartz function, and to control explicitly its
integral in terms of suitable seminorms of `f`. -/
lemma pow_mul_le_of_le_of_pow_mul_le {C₁ C₂ : ℝ} {k l : ℕ} {x f : ℝ} (hx : 0 ≤ x) (hf : 0 ≤ f)
(h₁ : f ≤ C₁) (h₂ : x ^ (k + l) * f ≤ C₂) :
x ^ k * f ≤ 2 ^ l * (C₁ + C₂) * (1 + x) ^ (- (l : ℝ))
|
C₁ C₂ : ℝ
k l : ℕ
x f : ℝ
hx : 0 ≤ x
hf : 0 ≤ f
h₁ : f ≤ C₁
h₂ : x ^ (k + l) * f ≤ C₂
this✝ : 0 ≤ C₂
this : 2 ^ l * (C₁ + C₂) * (1 + x) ^ (-↑l) = ((1 + x) / 2) ^ (-↑l) * (C₁ + C₂)
h'x : 1 ≤ x
⊢ 0 ≤ ((1 + x) / 2) ^ (-↑l)
|
positivity
|
no goals
|
53024ad59cf95cae
|
div_eq_quo_add_rem_div_add_rem_div
|
Mathlib/Algebra/Polynomial/PartialFractions.lean
|
theorem div_eq_quo_add_rem_div_add_rem_div (f : R[X]) {g₁ g₂ : R[X]} (hg₁ : g₁.Monic)
(hg₂ : g₂.Monic) (hcoprime : IsCoprime g₁ g₂) :
∃ q r₁ r₂ : R[X],
r₁.degree < g₁.degree ∧
r₂.degree < g₂.degree ∧ (f : K) / (↑g₁ * ↑g₂) = ↑q + ↑r₁ / ↑g₁ + ↑r₂ / ↑g₂
|
case intro.intro
R : Type
inst✝⁴ : CommRing R
inst✝³ : IsDomain R
K : Type
inst✝² : Field K
inst✝¹ : Algebra R[X] K
inst✝ : IsFractionRing R[X] K
f g₁ g₂ : R[X]
hg₁ : g₁.Monic
hg₂ : g₂.Monic
c d : R[X]
hcd : c * g₁ + d * g₂ = 1
hg₁' : ↑g₁ ≠ 0
hg₂' : ↑g₂ ≠ 0
hfc : f * c %ₘ g₂ + g₂ * (f * c /ₘ g₂) = f * c
⊢ ↑f / (↑g₁ * ↑g₂) = ↑(f * d /ₘ g₁ + f * c /ₘ g₂) + ↑(f * d %ₘ g₁) / ↑g₁ + ↑(f * c %ₘ g₂) / ↑g₂
|
have hfd := modByMonic_add_div (f * d) hg₁
|
case intro.intro
R : Type
inst✝⁴ : CommRing R
inst✝³ : IsDomain R
K : Type
inst✝² : Field K
inst✝¹ : Algebra R[X] K
inst✝ : IsFractionRing R[X] K
f g₁ g₂ : R[X]
hg₁ : g₁.Monic
hg₂ : g₂.Monic
c d : R[X]
hcd : c * g₁ + d * g₂ = 1
hg₁' : ↑g₁ ≠ 0
hg₂' : ↑g₂ ≠ 0
hfc : f * c %ₘ g₂ + g₂ * (f * c /ₘ g₂) = f * c
hfd : f * d %ₘ g₁ + g₁ * (f * d /ₘ g₁) = f * d
⊢ ↑f / (↑g₁ * ↑g₂) = ↑(f * d /ₘ g₁ + f * c /ₘ g₂) + ↑(f * d %ₘ g₁) / ↑g₁ + ↑(f * c %ₘ g₂) / ↑g₂
|
950f38e99f21aa33
|
linearIndependent_le_span_aux'
|
Mathlib/LinearAlgebra/Dimension/StrongRankCondition.lean
|
theorem linearIndependent_le_span_aux' {ι : Type*} [Fintype ι] (v : ι → M)
(i : LinearIndependent R v) (w : Set M) [Fintype w] (s : range v ≤ span R w) :
Fintype.card ι ≤ Fintype.card w
|
case i
R : Type u
M : Type v
inst✝⁵ : Semiring R
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
inst✝² : StrongRankCondition R
ι : Type u_1
inst✝¹ : Fintype ι
v : ι → M
i : LinearIndependent R v
w : Set M
inst✝ : Fintype ↑w
s : range v ≤ ↑(span R w)
f g : ι →₀ R
h :
(linearCombination R Subtype.val) ((linearCombination R fun i => Span.repr R w ⟨v i, ⋯⟩) f) =
(linearCombination R Subtype.val) ((linearCombination R fun i => Span.repr R w ⟨v i, ⋯⟩) g)
⊢ f = g
|
simp only [linearCombination_linearCombination, Submodule.coe_mk,
Span.finsupp_linearCombination_repr] at h
|
case i
R : Type u
M : Type v
inst✝⁵ : Semiring R
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
inst✝² : StrongRankCondition R
ι : Type u_1
inst✝¹ : Fintype ι
v : ι → M
i : LinearIndependent R v
w : Set M
inst✝ : Fintype ↑w
s : range v ≤ ↑(span R w)
f g : ι →₀ R
h : (linearCombination R fun b => v b) f = (linearCombination R fun b => v b) g
⊢ f = g
|
675e1368e54dbc1d
|
continuousOn_list_prod
|
Mathlib/Topology/Algebra/Monoid.lean
|
theorem continuousOn_list_prod {f : ι → X → M} (l : List ι) {t : Set X}
(h : ∀ i ∈ l, ContinuousOn (f i) t) :
ContinuousOn (fun a => (l.map fun i => f i a).prod) t
|
ι : Type u_1
M : Type u_3
X : Type u_5
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace M
inst✝¹ : Monoid M
inst✝ : ContinuousMul M
f : ι → X → M
l : List ι
t : Set X
h : ∀ i ∈ l, ContinuousOn (f i) t
x : X
hx : x ∈ t
⊢ ContinuousWithinAt (fun a => (List.map (fun i => f i a) l).prod) t x
|
rw [continuousWithinAt_iff_continuousAt_restrict _ hx]
|
ι : Type u_1
M : Type u_3
X : Type u_5
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace M
inst✝¹ : Monoid M
inst✝ : ContinuousMul M
f : ι → X → M
l : List ι
t : Set X
h : ∀ i ∈ l, ContinuousOn (f i) t
x : X
hx : x ∈ t
⊢ ContinuousAt (t.restrict fun a => (List.map (fun i => f i a) l).prod) ⟨x, hx⟩
|
df53e6c7d2182796
|
Polynomial.sumIDeriv_C
|
Mathlib/Algebra/Polynomial/SumIteratedDerivative.lean
|
theorem sumIDeriv_C (a : R) : sumIDeriv (C a) = C a
|
R : Type u_1
inst✝ : Semiring R
a : R
⊢ sumIDeriv (C a) = C a
|
rw [sumIDeriv_apply, natDegree_C, zero_add, sum_range_one, Function.iterate_zero_apply]
|
no goals
|
bb038e3d9f21b78a
|
IsModuleTopology.continuous_bilinear_of_pi_fintype
|
Mathlib/Topology/Algebra/Module/ModuleTopology.lean
|
theorem continuous_bilinear_of_pi_fintype (ι : Type*) [Finite ι]
(bil : (ι → R) →ₗ[R] B →ₗ[R] C) : Continuous (fun ab ↦ bil ab.1 ab.2 : ((ι → R) × B → C))
|
R : Type u_1
inst✝¹⁰ : TopologicalSpace R
inst✝⁹ : CommSemiring R
B : Type u_2
inst✝⁸ : AddCommMonoid B
inst✝⁷ : Module R B
inst✝⁶ : TopologicalSpace B
inst✝⁵ : IsModuleTopology R B
C : Type u_3
inst✝⁴ : AddCommMonoid C
inst✝³ : Module R C
inst✝² : TopologicalSpace C
inst✝¹ : IsModuleTopology R C
ι : Type u_4
inst✝ : Finite ι
bil : (ι → R) →ₗ[R] B →ₗ[R] C
val✝ : Fintype ι
⊢ (fun fb => (bil fb.1) fb.2) = fun fb => ∑ i : ι, fb.1 i • (bil ⇑(Finsupp.single i 1)) fb.2
|
ext ⟨f, b⟩
|
case h.mk
R : Type u_1
inst✝¹⁰ : TopologicalSpace R
inst✝⁹ : CommSemiring R
B : Type u_2
inst✝⁸ : AddCommMonoid B
inst✝⁷ : Module R B
inst✝⁶ : TopologicalSpace B
inst✝⁵ : IsModuleTopology R B
C : Type u_3
inst✝⁴ : AddCommMonoid C
inst✝³ : Module R C
inst✝² : TopologicalSpace C
inst✝¹ : IsModuleTopology R C
ι : Type u_4
inst✝ : Finite ι
bil : (ι → R) →ₗ[R] B →ₗ[R] C
val✝ : Fintype ι
f : ι → R
b : B
⊢ (bil (f, b).1) (f, b).2 = ∑ i : ι, (f, b).1 i • (bil ⇑(Finsupp.single i 1)) (f, b).2
|
81eedbff87e7f06b
|
InnerProductSpace.Core.inner_smul_right
|
Mathlib/Analysis/InnerProductSpace/Defs.lean
|
theorem inner_smul_right (x y : F) {r : 𝕜} : ⟪x, r • y⟫ = r * ⟪x, y⟫
|
𝕜 : Type u_1
F : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
c : PreInnerProductSpace.Core 𝕜 F
x y : F
r : 𝕜
⊢ ⟪x, r • y⟫_𝕜 = r * ⟪x, y⟫_𝕜
|
rw [← inner_conj_symm, inner_smul_left]
|
𝕜 : Type u_1
F : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
c : PreInnerProductSpace.Core 𝕜 F
x y : F
r : 𝕜
⊢ (starRingEnd 𝕜) ((starRingEnd 𝕜) r * ⟪y, x⟫_𝕜) = r * ⟪x, y⟫_𝕜
|
e16f7c85017c8904
|
AlgebraicGeometry.HasAffineProperty.diagonal_of_openCover
|
Mathlib/AlgebraicGeometry/Morphisms/Constructors.lean
|
theorem HasAffineProperty.diagonal_of_openCover (P) {Q} [HasAffineProperty P Q]
{X Y : Scheme.{u}} (f : X ⟶ Y) (𝒰 : Scheme.OpenCover.{u} Y) [∀ i, IsAffine (𝒰.obj i)]
(𝒰' : ∀ i, Scheme.OpenCover.{u} (pullback f (𝒰.map i))) [∀ i j, IsAffine ((𝒰' i).obj j)]
(h𝒰' : ∀ i j k,
Q (pullback.mapDesc ((𝒰' i).map j) ((𝒰' i).map k) (𝒰.pullbackHom f i))) :
P.diagonal f
|
case mk.mk.convert_2
P : MorphismProperty Scheme
Q : AffineTargetMorphismProperty
inst✝² : HasAffineProperty P Q
X Y : Scheme
f : X ⟶ Y
𝒰 : Y.OpenCover
inst✝¹ : ∀ (i : 𝒰.J), IsAffine (𝒰.obj i)
𝒰' : (i : 𝒰.J) → (pullback f (𝒰.map i)).OpenCover
inst✝ : ∀ (i : 𝒰.J) (j : (𝒰' i).J), IsAffine ((𝒰' i).obj j)
h𝒰' : ∀ (i : 𝒰.J) (j k : (𝒰' i).J), Q (pullback.mapDesc ((𝒰' i).map j) ((𝒰' i).map k) (Scheme.Cover.pullbackHom 𝒰 f i))
this : Q.IsLocal := isLocal_affineProperty P
𝒱 : Scheme.Cover (@IsOpenImmersion) (pullback f f) :=
Scheme.Cover.bind (Scheme.Pullback.openCoverOfBase 𝒰 f f) fun i =>
Scheme.Pullback.openCoverOfLeftRight (𝒰' i) (𝒰' i) (pullback.snd f (𝒰.map i)) (pullback.snd f (𝒰.map i))
i1 : ∀ (i : 𝒱.J), IsAffine (𝒱.obj i)
i : (Scheme.Pullback.openCoverOfBase 𝒰 f f).J
j k : (𝒰' i).J
⊢ pullback.map ((𝒰' i).map j ≫ pullback.snd f (𝒰.map i)) ((𝒰' i).map k ≫ pullback.snd f (𝒰.map i)) f f
((𝒰' i).map j ≫ pullback.fst f (𝒰.map i)) ((𝒰' i).map k ≫ pullback.fst f (𝒰.map i)) (𝒰.map i) ⋯ ⋯ ≫
𝟙 (pullback.diagonalObj f) =
𝟙 (pullback ((𝒰' i).map j ≫ pullback.snd f (𝒰.map i)) ((𝒰' i).map k ≫ pullback.snd f (𝒰.map i))) ≫
pullback.map ((𝒰' i).map j ≫ pullback.snd f (𝒰.map i)) ((𝒰' i).map k ≫ pullback.snd f (𝒰.map i))
(pullback.snd f (𝒰.map i)) (pullback.snd f (𝒰.map i)) ((𝒰' i).map j) ((𝒰' i).map k) (𝟙 (𝒰.obj i)) ⋯ ⋯ ≫
pullback.map (pullback.snd f (𝒰.map i)) (pullback.snd f (𝒰.map i)) f f (pullback.fst f (𝒰.map i))
(pullback.fst f (𝒰.map i)) (𝒰.map i) ⋯ ⋯
|
ext1 <;> simp
|
no goals
|
69fa16ea5623a629
|
ENNReal.zero_rpow_def
|
Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean
|
theorem zero_rpow_def (y : ℝ) : (0 : ℝ≥0∞) ^ y = if 0 < y then 0 else if y = 0 then 1 else ⊤
|
case inl
y : ℝ
H : 0 < y
⊢ 0 ^ y = if 0 < y then 0 else if y = 0 then 1 else ⊤
|
simp [H, ne_of_gt, zero_rpow_of_pos, lt_irrefl]
|
no goals
|
ccd7a3d0a0150b8a
|
intervalIntegral.integrableOn_deriv_right_of_nonneg
|
Mathlib/MeasureTheory/Integral/FundThmCalculus.lean
|
theorem integrableOn_deriv_right_of_nonneg (hcont : ContinuousOn g (Icc a b))
(hderiv : ∀ x ∈ Ioo a b, HasDerivWithinAt g (g' x) (Ioi x) x)
(g'pos : ∀ x ∈ Ioo a b, 0 ≤ g' x) : IntegrableOn g' (Ioc a b)
|
case pos
g' g : ℝ → ℝ
a b : ℝ
hcont : ContinuousOn g (Icc a b)
hderiv : ∀ x ∈ Ioo a b, HasDerivWithinAt g (g' x) (Ioi x) x
g'pos : ∀ x ∈ Ioo a b, 0 ≤ g' x
hab : a < b
⊢ IntegrableOn g' (Ioc a b) volume
|
rw [integrableOn_Ioc_iff_integrableOn_Ioo]
|
case pos
g' g : ℝ → ℝ
a b : ℝ
hcont : ContinuousOn g (Icc a b)
hderiv : ∀ x ∈ Ioo a b, HasDerivWithinAt g (g' x) (Ioi x) x
g'pos : ∀ x ∈ Ioo a b, 0 ≤ g' x
hab : a < b
⊢ IntegrableOn g' (Ioo a b) volume
|
87311e9302ce7d99
|
List.zipLeft_eq_zipLeft'
|
Mathlib/Data/List/Map2.lean
|
theorem zipLeft_eq_zipLeft' (as : List α) (bs : List β) : zipLeft as bs = (zipLeft' as bs).fst
|
case cons
α : Type u
β : Type v
bs : List β
head✝ : α
atl : List α
⊢ zipWithLeft Prod.mk (head✝ :: atl) bs = (zipWithLeft' Prod.mk (head✝ :: atl) bs).fst
|
cases bs with
| nil => rfl
| cons _ btl =>
rw [zipWithLeft, zipWithLeft', cons_inj_right]
exact @zipLeft_eq_zipLeft' atl btl
|
no goals
|
9c125819a53618fa
|
List.head_filterMap_of_eq_some
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean
|
theorem head_filterMap_of_eq_some {f : α → Option β} {l : List α} (w : l ≠ []) {b : β} (h : f (l.head w) = some b) :
(filterMap f l).head ((ne_nil_of_mem (mem_filterMap.2 ⟨_, head_mem w, h⟩))) =
b
|
case cons
α : Type u_1
β : Type u_2
f : α → Option β
b : β
a : α
l : List α
w : a :: l ≠ []
h : f ((a :: l).head w) = some b
⊢ (filterMap f (a :: l)).head ⋯ = b
|
simp only [head_cons] at h
|
case cons
α : Type u_1
β : Type u_2
f : α → Option β
b : β
a : α
l : List α
w : a :: l ≠ []
h : f a = some b
⊢ (filterMap f (a :: l)).head ⋯ = b
|
b256857532480341
|
Int.bmod_zero
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Int/DivModLemmas.lean
|
theorem bmod_zero : Int.bmod 0 m = 0
|
m✝ m : Nat
h : ↑m / 2 + 2 / 2 ≤ 0
⊢ 2 ≠ 0
m✝ m : Nat
h : (↑m + (1 + 1)) / 2 ≤ 0
⊢ 2 ∣ 1 + 1
|
all_goals decide
|
no goals
|
aafdd0ab2162fca6
|
CoxeterSystem.getD_rightInvSeq_mul_self
|
Mathlib/GroupTheory/Coxeter/Inversion.lean
|
theorem getD_rightInvSeq_mul_self (ω : List B) (j : ℕ) :
((ris ω).getD j 1) * ((ris ω).getD j 1) = 1
|
B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
ω : List B
j : ℕ
⊢ (cs.rightInvSeq ω).getD j 1 * (cs.rightInvSeq ω).getD j 1 = 1
|
simp_rw [getD_rightInvSeq, mul_assoc]
|
B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
ω : List B
j : ℕ
⊢ (cs.wordProd (drop (j + 1) ω))⁻¹ *
((Option.map cs.simple ω[j]?).getD 1 *
(cs.wordProd (drop (j + 1) ω) *
((cs.wordProd (drop (j + 1) ω))⁻¹ * ((Option.map cs.simple ω[j]?).getD 1 * cs.wordProd (drop (j + 1) ω))))) =
1
|
6874aa1b2b35417c
|
MeasureTheory.LevyProkhorov.continuous_equiv_symm_probabilityMeasure
|
Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.lean
|
lemma LevyProkhorov.continuous_equiv_symm_probabilityMeasure :
Continuous (LevyProkhorov.equiv (α := ProbabilityMeasure Ω)).symm
|
Ω : Type u_1
inst✝³ : PseudoMetricSpace Ω
inst✝² : MeasurableSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
inst✝ : SeparableSpace Ω
P : ProbabilityMeasure Ω
ε : ℝ
ε_pos : ε > 0
third_ε_pos : 0 < ε / 3
third_ε_pos' : 0 < ENNReal.ofReal (ε / 3)
Es : ℕ → Set Ω
Es_mble : ∀ (n : ℕ), MeasurableSet (Es n)
Es_bdd : ∀ (n : ℕ), Bornology.IsBounded (Es n)
Es_diam : ∀ (n : ℕ), diam (Es n) ≤ ε / 3
Es_cover : ⋃ n, Es n = univ
Es_disjoint : Pairwise fun n m => Disjoint (Es n) (Es m)
N : ℕ
hN : ↑P (⋃ j ∈ Iio N, Es j)ᶜ < ENNReal.ofReal (ε / 3)
Js_finite : {J | J ⊆ Iio N}.Finite
Gs : Set (Set Ω) := (fun J => thickening (ε / 3) (⋃ j ∈ J, Es j)) '' {J | J ⊆ Iio N}
Gs_open : ∀ (J : Set ℕ), IsOpen (thickening (ε / 3) (⋃ j ∈ J, Es j))
mem_nhds_P : ∀ (G : Set Ω), IsOpen G → {Q | ↑P G < ↑Q G + ENNReal.ofReal (ε / 3)} ∈ 𝓝 P
Q : ProbabilityMeasure Ω
hQ :
∀ i ⊆ Iio N,
↑P (⋃ i_2 ∈ i, thickening (ε / 3) (Es i_2)) < ↑Q (⋃ i_2 ∈ i, thickening (ε / 3) (Es i_2)) + ENNReal.ofReal (ε / 3)
δ : ℝ
B : Set Ω
δ_gt : 2 * (ε / 3) < δ
x✝ : MeasurableSet B
JB : Set ℕ := {i | B ∩ Es i ≠ ∅ ∧ i ∈ Iio N}
B_subset : B ⊆ (⋃ i ∈ JB, thickening (ε / 3) (Es i)) ∪ (⋃ j ∈ Iio N, Es j)ᶜ
subset_thickB : ⋃ i ∈ JB, thickening (ε / 3) (Es i) ⊆ thickening δ B
⊢ ↑((equiv (ProbabilityMeasure Ω)).symm P) B ≤
↑((equiv (ProbabilityMeasure Ω)).symm Q) (thickening δ B) + ENNReal.ofReal δ
|
apply (measure_mono B_subset).trans ((measure_union_le _ _).trans ?_)
|
Ω : Type u_1
inst✝³ : PseudoMetricSpace Ω
inst✝² : MeasurableSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
inst✝ : SeparableSpace Ω
P : ProbabilityMeasure Ω
ε : ℝ
ε_pos : ε > 0
third_ε_pos : 0 < ε / 3
third_ε_pos' : 0 < ENNReal.ofReal (ε / 3)
Es : ℕ → Set Ω
Es_mble : ∀ (n : ℕ), MeasurableSet (Es n)
Es_bdd : ∀ (n : ℕ), Bornology.IsBounded (Es n)
Es_diam : ∀ (n : ℕ), diam (Es n) ≤ ε / 3
Es_cover : ⋃ n, Es n = univ
Es_disjoint : Pairwise fun n m => Disjoint (Es n) (Es m)
N : ℕ
hN : ↑P (⋃ j ∈ Iio N, Es j)ᶜ < ENNReal.ofReal (ε / 3)
Js_finite : {J | J ⊆ Iio N}.Finite
Gs : Set (Set Ω) := (fun J => thickening (ε / 3) (⋃ j ∈ J, Es j)) '' {J | J ⊆ Iio N}
Gs_open : ∀ (J : Set ℕ), IsOpen (thickening (ε / 3) (⋃ j ∈ J, Es j))
mem_nhds_P : ∀ (G : Set Ω), IsOpen G → {Q | ↑P G < ↑Q G + ENNReal.ofReal (ε / 3)} ∈ 𝓝 P
Q : ProbabilityMeasure Ω
hQ :
∀ i ⊆ Iio N,
↑P (⋃ i_2 ∈ i, thickening (ε / 3) (Es i_2)) < ↑Q (⋃ i_2 ∈ i, thickening (ε / 3) (Es i_2)) + ENNReal.ofReal (ε / 3)
δ : ℝ
B : Set Ω
δ_gt : 2 * (ε / 3) < δ
x✝ : MeasurableSet B
JB : Set ℕ := {i | B ∩ Es i ≠ ∅ ∧ i ∈ Iio N}
B_subset : B ⊆ (⋃ i ∈ JB, thickening (ε / 3) (Es i)) ∪ (⋃ j ∈ Iio N, Es j)ᶜ
subset_thickB : ⋃ i ∈ JB, thickening (ε / 3) (Es i) ⊆ thickening δ B
⊢ ↑((equiv (ProbabilityMeasure Ω)).symm P) (⋃ i ∈ JB, thickening (ε / 3) (Es i)) +
↑((equiv (ProbabilityMeasure Ω)).symm P) (⋃ j ∈ Iio N, Es j)ᶜ ≤
↑((equiv (ProbabilityMeasure Ω)).symm Q) (thickening δ B) + ENNReal.ofReal δ
|
1196bd5a9632bf03
|
CategoryTheory.FinitaryPreExtensive.sigma_desc_iso
|
Mathlib/CategoryTheory/Extensive.lean
|
lemma FinitaryPreExtensive.sigma_desc_iso [FinitaryPreExtensive C] {α : Type} [Finite α] {X : C}
{Z : α → C} (π : (a : α) → Z a ⟶ X) {Y : C} (f : Y ⟶ X) (hπ : IsIso (Sigma.desc π)) :
IsIso (Sigma.desc ((fun _ ↦ pullback.fst _ _) : (a : α) → pullback f (π a) ⟶ _))
|
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : FinitaryPreExtensive C
α : Type
inst✝ : Finite α
X : C
Z : α → C
π : (a : α) → Z a ⟶ X
Y : C
f : Y ⟶ X
hπ : IsIso (Sigma.desc π)
this : IsColimit (Cofan.mk X π) := (coproductIsCoproduct Z).ofPointIso
⊢ IsColimit (Cofan.mk Y fun x => pullback.fst f (π x))
|
refine (FinitaryPreExtensive.isUniversal_finiteCoproducts this
(Cofan.mk _ ((fun _ ↦ pullback.fst _ _) : (a : α) → pullback f (π a) ⟶ _))
(Discrete.natTrans fun i ↦ pullback.snd _ _) f ?_
(NatTrans.equifibered_of_discrete _) ?_).some
|
case refine_1
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : FinitaryPreExtensive C
α : Type
inst✝ : Finite α
X : C
Z : α → C
π : (a : α) → Z a ⟶ X
Y : C
f : Y ⟶ X
hπ : IsIso (Sigma.desc π)
this : IsColimit (Cofan.mk X π) := (coproductIsCoproduct Z).ofPointIso
⊢ (Discrete.natTrans fun i => pullback.snd f (π i.as)) ≫ (Cofan.mk X π).ι =
(Cofan.mk Y fun x => pullback.fst f (π x)).ι ≫ (Functor.const (Discrete α)).map f
case refine_2
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : FinitaryPreExtensive C
α : Type
inst✝ : Finite α
X : C
Z : α → C
π : (a : α) → Z a ⟶ X
Y : C
f : Y ⟶ X
hπ : IsIso (Sigma.desc π)
this : IsColimit (Cofan.mk X π) := (coproductIsCoproduct Z).ofPointIso
⊢ ∀ (j : Discrete α),
IsPullback ((Cofan.mk Y fun x => pullback.fst f (π x)).ι.app j)
((Discrete.natTrans fun i => pullback.snd f (π i.as)).app j) f ((Cofan.mk X π).ι.app j)
|
800d0cda5060197f
|
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_1
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 ∉ ⋂₀ {J | J ⊆ I ∧ e ∈ M.closure J}
|
simp [show ∃ x ⊆ I, e ∈ M.closure x ∧ e ∉ x from ⟨I, by simp [hecl, heI]⟩]
|
no goals
|
640cc0ceac07193c
|
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