name
stringlengths 3
112
| file
stringlengths 21
116
| statement
stringlengths 17
8.64k
| state
stringlengths 7
205k
| tactic
stringlengths 3
4.55k
| result
stringlengths 7
205k
| id
stringlengths 16
16
|
|---|---|---|---|---|---|---|
MeasureTheory.Measure.haar.index_union_eq
|
Mathlib/MeasureTheory/Measure/Haar/Basic.lean
|
theorem index_union_eq (K₁ K₂ : Compacts G) {V : Set G} (hV : (interior V).Nonempty)
(h : Disjoint (K₁.1 * V⁻¹) (K₂.1 * V⁻¹)) :
index (K₁.1 ∪ K₂.1) V = index K₁.1 V + index K₂.1 V
|
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : (interior V).Nonempty
h : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)
s : Finset G
h1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' V
h2s : s.card = index (K₁.carrier ∪ K₂.carrier) V
⊢ ∀ K ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' V,
index K V ≤ (Finset.filter (fun g => ((fun h => g * h) ⁻¹' V ∩ K).Nonempty) s).card
|
intro K hK
|
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : (interior V).Nonempty
h : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)
s : Finset G
h1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' V
h2s : s.card = index (K₁.carrier ∪ K₂.carrier) V
K : Set G
hK : K ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' V
⊢ index K V ≤ (Finset.filter (fun g => ((fun h => g * h) ⁻¹' V ∩ K).Nonempty) s).card
|
ee267d5d5caa75d7
|
Commute.geom_sum₂_comm
|
Mathlib/Algebra/GeomSum.lean
|
theorem Commute.geom_sum₂_comm {α : Type u} [Semiring α] {x y : α} (n : ℕ)
(h : Commute x y) :
∑ i ∈ range n, x ^ i * y ^ (n - 1 - i) = ∑ i ∈ range n, y ^ i * x ^ (n - 1 - i)
|
case succ
α : Type u
inst✝ : Semiring α
x y : α
h : Commute x y
n✝ : ℕ
⊢ ∑ r ∈ range (n✝ + 1), x ^ (n✝ - r) * y ^ (n✝ - (n✝ - r)) = ∑ x_1 ∈ range (n✝ + 1), y ^ x_1 * x ^ (n✝ - x_1)
|
refine Finset.sum_congr rfl fun i hi => ?_
|
case succ
α : Type u
inst✝ : Semiring α
x y : α
h : Commute x y
n✝ i : ℕ
hi : i ∈ range (n✝ + 1)
⊢ x ^ (n✝ - i) * y ^ (n✝ - (n✝ - i)) = y ^ i * x ^ (n✝ - i)
|
d014dffa9922c8e6
|
LucasLehmer.order_ω
|
Mathlib/NumberTheory/LucasLehmer.lean
|
theorem order_ω (p' : ℕ) (h : lucasLehmerResidue (p' + 2) = 0) :
orderOf (ωUnit (p' + 2)) = 2 ^ (p' + 2)
|
case h₁
p' : ℕ
h✝ : lucasLehmerResidue (p' + 2) = 0
o : orderOf (ωUnit (p' + 2)) ∣ 2 ^ (p' + 1)
ω_pow : ω ^ 2 ^ (p' + 1) = 1
h : 1 = -1
this : Fact (2 < ↑(q (p' + 2)))
⊢ False
|
apply ZMod.neg_one_ne_one h.symm
|
no goals
|
9c324a727016e67e
|
IsPGroup.index
|
Mathlib/GroupTheory/PGroup.lean
|
theorem index (H : Subgroup G) [H.FiniteIndex] : ∃ n : ℕ, H.index = p ^ n
|
case intro
p : ℕ
G : Type u_1
inst✝¹ : Group G
hG : IsPGroup p G
hp : Fact (Nat.Prime p)
H : Subgroup G
inst✝ : H.FiniteIndex
n : ℕ
hn : Nat.card (G ⧸ H.normalCore) = p ^ n
⊢ ∃ n, H.index = p ^ n
|
obtain ⟨k, _, hk2⟩ :=
(Nat.dvd_prime_pow hp.out).mp
((congr_arg _ (H.normalCore.index_eq_card.trans hn)).mp
(Subgroup.index_dvd_of_le H.normalCore_le))
|
case intro.intro.intro
p : ℕ
G : Type u_1
inst✝¹ : Group G
hG : IsPGroup p G
hp : Fact (Nat.Prime p)
H : Subgroup G
inst✝ : H.FiniteIndex
n : ℕ
hn : Nat.card (G ⧸ H.normalCore) = p ^ n
k : ℕ
left✝ : k ≤ n
hk2 : H.index = p ^ k
⊢ ∃ n, H.index = p ^ n
|
9a50fce724440741
|
PMF.pure_bindOnSupport
|
Mathlib/Probability/ProbabilityMassFunction/Monad.lean
|
theorem pure_bindOnSupport (a : α) (f : ∀ (a' : α) (_ : a' ∈ (pure a).support), PMF β) :
(pure a).bindOnSupport f = f a ((mem_support_pure_iff a a).mpr rfl)
|
α : Type u_1
β : Type u_2
a : α
f : (a' : α) → a' ∈ (pure a).support → PMF β
b : β
⊢ ((pure a).bindOnSupport f) b = (f a ⋯) b
|
simp only [bindOnSupport_apply, pure_apply]
|
α : Type u_1
β : Type u_2
a : α
f : (a' : α) → a' ∈ (pure a).support → PMF β
b : β
⊢ (∑' (a_1 : α), (if a_1 = a then 1 else 0) * if h : (if a_1 = a then 1 else 0) = 0 then 0 else (f a_1 h) b) = (f a ⋯) b
|
da2d186076f1fc8c
|
List.one_lt_prod_of_one_lt
|
Mathlib/Algebra/Order/BigOperators/Group/List.lean
|
@[to_additive sum_pos]
lemma one_lt_prod_of_one_lt [OrderedCommMonoid M] :
∀ l : List M, (∀ x ∈ l, (1 : M) < x) → l ≠ [] → 1 < l.prod
| [], _, h => (h rfl).elim
| [b], h, _ => by simpa using h
| a :: b :: l, hl₁, _ => by
simp only [forall_eq_or_imp, List.mem_cons] at hl₁
rw [List.prod_cons]
apply one_lt_mul_of_lt_of_le' hl₁.1
apply le_of_lt ((b :: l).one_lt_prod_of_one_lt _ (l.cons_ne_nil b))
intro x hx; cases hx
· exact hl₁.2.1
· exact hl₁.2.2 _ ‹_›
|
M : Type u_3
inst✝ : OrderedCommMonoid M
a b : M
l : List M
hl₁ : ∀ (x : M), x ∈ a :: b :: l → 1 < x
x✝ : a :: b :: l ≠ []
⊢ 1 < (a :: b :: l).prod
|
simp only [forall_eq_or_imp, List.mem_cons] at hl₁
|
M : Type u_3
inst✝ : OrderedCommMonoid M
a b : M
l : List M
x✝ : a :: b :: l ≠ []
hl₁ : 1 < a ∧ 1 < b ∧ ∀ (a : M), a ∈ l → 1 < a
⊢ 1 < (a :: b :: l).prod
|
081aa2408de598f4
|
List.min?_mem
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/MinMax.lean
|
theorem min?_mem [Min α] (min_eq_or : ∀ a b : α, min a b = a ∨ min a b = b) :
{xs : List α} → xs.min? = some a → a ∈ xs
|
α : Type u_1
a : α
inst✝ : Min α
min_eq_or : ∀ (a b : α), min a b = a ∨ min a b = b
xs : List α
⊢ xs.min? = some a → a ∈ xs
|
match xs with
| nil => simp
| x :: xs =>
simp only [min?_cons', Option.some.injEq, List.mem_cons]
intro eq
induction xs generalizing x with
| nil =>
simp at eq
simp [eq]
| cons y xs ind =>
simp at eq
have p := ind _ eq
cases p with
| inl p =>
cases min_eq_or x y with | _ q => simp [p, q]
| inr p => simp [p, mem_cons]
|
no goals
|
f362dcf8097d474b
|
AlgebraicGeometry.StructureSheaf.locally_const_basicOpen
|
Mathlib/AlgebraicGeometry/StructureSheaf.lean
|
theorem locally_const_basicOpen (U : Opens (PrimeSpectrum.Top R))
(s : (structureSheaf R).1.obj (op U)) (x : U) :
∃ (f g : R) (i : PrimeSpectrum.basicOpen g ⟶ U), x.1 ∈ PrimeSpectrum.basicOpen g ∧
(const R f g (PrimeSpectrum.basicOpen g) fun _ hy => hy) =
(structureSheaf R).1.map i.op s
|
case right
R : Type u
inst✝ : CommRing R
U : Opens ↑(PrimeSpectrum.Top R)
s : ↑((structureSheaf R).val.obj (op U))
x : ↥U
V : Opens ↑(PrimeSpectrum.Top R)
hxV : ↑x ∈ V.carrier
iVU : V ⟶ U
f g : R
hVDg : V ≤ PrimeSpectrum.basicOpen g
s_eq : const R f g V hVDg = (ConcreteCategory.hom ((structureSheaf R).val.map iVU.op)) s
h : R
hxDh : ↑x ∈ (fun r => ↑(PrimeSpectrum.basicOpen r)) h
hDhV : PrimeSpectrum.basicOpen h ≤ V
n : ℕ
c : R
hc : c * g = h ^ (n + 1)
basic_opens_eq : PrimeSpectrum.basicOpen (h ^ (n + 1)) = PrimeSpectrum.basicOpen h
i_basic_open : PrimeSpectrum.basicOpen (h ^ (n + 1)) ⟶ V
⊢ const R (f * c) (h ^ (n + 1)) (PrimeSpectrum.basicOpen (h ^ (n + 1))) ⋯ =
const R f g (PrimeSpectrum.basicOpen (h ^ (n + 1))) ⋯
|
apply const_ext
|
case right.h
R : Type u
inst✝ : CommRing R
U : Opens ↑(PrimeSpectrum.Top R)
s : ↑((structureSheaf R).val.obj (op U))
x : ↥U
V : Opens ↑(PrimeSpectrum.Top R)
hxV : ↑x ∈ V.carrier
iVU : V ⟶ U
f g : R
hVDg : V ≤ PrimeSpectrum.basicOpen g
s_eq : const R f g V hVDg = (ConcreteCategory.hom ((structureSheaf R).val.map iVU.op)) s
h : R
hxDh : ↑x ∈ (fun r => ↑(PrimeSpectrum.basicOpen r)) h
hDhV : PrimeSpectrum.basicOpen h ≤ V
n : ℕ
c : R
hc : c * g = h ^ (n + 1)
basic_opens_eq : PrimeSpectrum.basicOpen (h ^ (n + 1)) = PrimeSpectrum.basicOpen h
i_basic_open : PrimeSpectrum.basicOpen (h ^ (n + 1)) ⟶ V
⊢ f * c * g = f * h ^ (n + 1)
|
2f7a4a8aeea28445
|
Nat.totient_dvd_of_dvd
|
Mathlib/Data/Nat/Totient.lean
|
theorem totient_dvd_of_dvd {a b : ℕ} (h : a ∣ b) : φ a ∣ φ b
|
case inr.inr
a b : ℕ
h : a ∣ b
ha0 : a ≠ 0
hb0 : b ≠ 0
hab' : a.primeFactors ⊆ b.primeFactors
⊢ (a.factorization.prod fun p k => p ^ (k - 1) * (p - 1)) ∣ b.factorization.prod fun p k => p ^ (k - 1) * (p - 1)
|
refine Finsupp.prod_dvd_prod_of_subset_of_dvd hab' fun p _ => mul_dvd_mul ?_ dvd_rfl
|
case inr.inr
a b : ℕ
h : a ∣ b
ha0 : a ≠ 0
hb0 : b ≠ 0
hab' : a.primeFactors ⊆ b.primeFactors
p : ℕ
x✝ : p ∈ a.factorization.support
⊢ p ^ (a.factorization p - 1) ∣ p ^ (b.factorization p - 1)
|
7f5cb7bbc1d70ca0
|
inf_eq_and_sup_eq_iff
|
Mathlib/Order/Lattice.lean
|
lemma inf_eq_and_sup_eq_iff : a ⊓ b = c ∧ a ⊔ b = c ↔ a = c ∧ b = c
|
case refine_2.intro
α : Type u
inst✝ : Lattice α
b : α
⊢ b ⊓ b = b ∧ b ⊔ b = b
|
exact ⟨inf_idem _, sup_idem _⟩
|
no goals
|
b5e04850b5913ca9
|
Cardinal.cast_toNat_eq_iff_lt_aleph0
|
Mathlib/SetTheory/Cardinal/ToNat.lean
|
theorem cast_toNat_eq_iff_lt_aleph0 {c : Cardinal} : (toNat c) = c ↔ c < ℵ₀
|
case mp
c : Cardinal.{u_1}
h : ↑0 = c
h' : ℵ₀ ≤ ↑0
⊢ False
|
absurd h'
|
case mp
c : Cardinal.{u_1}
h : ↑0 = c
h' : ℵ₀ ≤ ↑0
⊢ ¬ℵ₀ ≤ ↑0
|
253bbb93e898740d
|
LieSubmodule.inclusion_injective
|
Mathlib/Algebra/Lie/Submodule.lean
|
theorem inclusion_injective : Function.Injective (inclusion h) := fun x y ↦ by
simp only [inclusion_apply, imp_self, Subtype.mk_eq_mk, SetLike.coe_eq_coe]
|
R : Type u
L : Type v
M : Type w
inst✝⁴ : CommRing R
inst✝³ : LieRing L
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : LieRingModule L M
N N' : LieSubmodule R L M
h : N ≤ N'
x y : ↥N
⊢ (inclusion h) x = (inclusion h) y → x = y
|
simp only [inclusion_apply, imp_self, Subtype.mk_eq_mk, SetLike.coe_eq_coe]
|
no goals
|
65836dc5dc3baf09
|
LinearIndependent.lt_aleph0_of_finite
|
Mathlib/LinearAlgebra/Dimension/Finite.lean
|
theorem lt_aleph0_of_finite {ι : Type w}
[Module.Finite R M] {v : ι → M} (h : LinearIndependent R v) : #ι < ℵ₀
|
R : Type u
M : Type v
inst✝⁴ : Ring R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : StrongRankCondition R
ι : Type w
inst✝ : Module.Finite R M
v : ι → M
h : LinearIndependent R v
⊢ #ι < ℵ₀
|
apply Cardinal.lift_lt.1
|
R : Type u
M : Type v
inst✝⁴ : Ring R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : StrongRankCondition R
ι : Type w
inst✝ : Module.Finite R M
v : ι → M
h : LinearIndependent R v
⊢ lift.{?u.115417, w} #ι < lift.{?u.115417, w} ℵ₀
|
1685e2b4d3164a12
|
gal_C_isSolvable
|
Mathlib/FieldTheory/AbelRuffini.lean
|
theorem gal_C_isSolvable (x : F) : IsSolvable (C x).Gal
|
F : Type u_1
inst✝ : Field F
x : F
⊢ IsSolvable (C x).Gal
|
infer_instance
|
no goals
|
2a0583745baed498
|
MvQPF.liftP_iff_of_isUniform
|
Mathlib/Data/QPF/Multivariate/Basic.lean
|
theorem liftP_iff_of_isUniform (h : q.IsUniform) {α : TypeVec n} (x : F α) (p : ∀ i, α i → Prop) :
LiftP p x ↔ ∀ (i), ∀ u ∈ supp x i, p i u
|
case mk.mp
n : ℕ
F : TypeVec.{u} n → Type u_1
q : MvQPF F
h : IsUniform
α : TypeVec.{u} n
x : F α
p : (i : Fin2 n) → α i → Prop
a : (P F).A
f : (P F).B a ⟹ α
⊢ (∃ a_1 f_1, abs ⟨a, f⟩ = abs ⟨a_1, f_1⟩ ∧ ∀ (i : Fin2 n) (j : (P F).B a_1 i), p i (f_1 i j)) →
∀ (i : Fin2 n), ∀ u ∈ supp (abs ⟨a, f⟩) i, p i u
|
rintro ⟨a', f', abseq, hf⟩ u
|
case mk.mp.intro.intro.intro
n : ℕ
F : TypeVec.{u} n → Type u_1
q : MvQPF F
h : IsUniform
α : TypeVec.{u} n
x : F α
p : (i : Fin2 n) → α i → Prop
a : (P F).A
f : (P F).B a ⟹ α
a' : (P F).A
f' : (P F).B a' ⟹ α
abseq : abs ⟨a, f⟩ = abs ⟨a', f'⟩
hf : ∀ (i : Fin2 n) (j : (P F).B a' i), p i (f' i j)
u : Fin2 n
⊢ ∀ u_1 ∈ supp (abs ⟨a, f⟩) u, p u u_1
|
ce4f84e9bd6acfdc
|
LinearMap.IsSymmetric.inner_map_polarization
|
Mathlib/Analysis/InnerProductSpace/Symmetric.lean
|
theorem IsSymmetric.inner_map_polarization {T : E →ₗ[𝕜] E} (hT : T.IsSymmetric) (x y : E) :
⟪T x, y⟫ =
(⟪T (x + y), x + y⟫ - ⟪T (x - y), x - y⟫ - I * ⟪T (x + (I : 𝕜) • y), x + (I : 𝕜) • y⟫ +
I * ⟪T (x - (I : 𝕜) • y), x - (I : 𝕜) • y⟫) /
4
|
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
T : E →ₗ[𝕜] E
hT : T.IsSymmetric
x y : E
⊢ inner (T x) y =
(inner (T (x + y)) (x + y) - inner (T (x - y)) (x - y) - I * inner (T (x + I • y)) (x + I • y) +
I * inner (T (x - I • y)) (x - I • y)) /
4
|
rcases@I_mul_I_ax 𝕜 _ with (h | h)
|
case inl
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
T : E →ₗ[𝕜] E
hT : T.IsSymmetric
x y : E
h : I = 0
⊢ inner (T x) y =
(inner (T (x + y)) (x + y) - inner (T (x - y)) (x - y) - I * inner (T (x + I • y)) (x + I • y) +
I * inner (T (x - I • y)) (x - I • y)) /
4
case inr
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
T : E →ₗ[𝕜] E
hT : T.IsSymmetric
x y : E
h : I * I = -1
⊢ inner (T x) y =
(inner (T (x + y)) (x + y) - inner (T (x - y)) (x - y) - I * inner (T (x + I • y)) (x + I • y) +
I * inner (T (x - I • y)) (x - I • y)) /
4
|
fda02a1d3405d981
|
List.append_sublist_of_sublist_left
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Nat/Sublist.lean
|
theorem append_sublist_of_sublist_left {xs ys zs : List α} (h : zs <+ xs) :
xs ++ ys <+ zs ↔ ys = [] ∧ xs = zs
|
case mp
α : Type u_1
xs ys zs : List α
h : zs <+ xs
⊢ xs ++ ys <+ zs → ys = [] ∧ xs = zs
|
intro h'
|
case mp
α : Type u_1
xs ys zs : List α
h : zs <+ xs
h' : xs ++ ys <+ zs
⊢ ys = [] ∧ xs = zs
|
444ed8b01bf1eceb
|
Matrix.vecAlt0_vecAppend
|
Mathlib/Data/Fin/VecNotation.lean
|
theorem vecAlt0_vecAppend (v : Fin n → α) :
vecAlt0 rfl (vecAppend rfl v v) = v ∘ (fun n ↦ n + n)
|
case neg.e_a.e_val
α : Type u
n : ℕ
v : Fin n → α
i : Fin n
h : n ≤ ↑i + ↑i
⊢ ↑i + ↑i - n < n
|
omega
|
no goals
|
4c85d9edab2b8fd9
|
Hopf_.antipode_antipode
|
Mathlib/CategoryTheory/Monoidal/Hopf_.lean
|
theorem antipode_antipode (A : Hopf_ C) (comm : (β_ _ _).hom ≫ A.X.X.mul = A.X.X.mul) :
A.antipode ≫ A.antipode = 𝟙 A.X.X.X
|
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
A : Hopf_ C
comm : (β_ A.X.X.X A.X.X.X).hom ≫ A.X.X.mul = A.X.X.mul
⊢ A.antipode ≫ A.antipode = 𝟙 A.X.X.X
|
apply left_inv_eq_right_inv
(M := Conv ((Bimon_.toComon_ C).obj A.X) A.X.X)
(a := A.antipode)
|
case hba
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
A : Hopf_ C
comm : (β_ A.X.X.X A.X.X.X).hom ≫ A.X.X.mul = A.X.X.mul
⊢ A.antipode ≫ A.antipode * A.antipode = 1
case hac
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
A : Hopf_ C
comm : (β_ A.X.X.X A.X.X.X).hom ≫ A.X.X.mul = A.X.X.mul
⊢ A.antipode * 𝟙 A.X.X.X = 1
|
f8b404c4d62ad46c
|
Nat.any_eq_anyTR
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Nat/Fold.lean
|
theorem any_eq_anyTR : @any = @anyTR :=
funext fun n => funext fun f =>
let rec go : ∀ m n f, any (m + n) f = (any n (fun i h => f i (by omega)) || anyTR.loop (m + n) f m (by omega))
| 0, n, f => by
simp [anyTR.loop]
have t : 0 + n = n
|
n✝ : Nat
f✝ : (i : Nat) → i < n✝ → Bool
m n : Nat
f : (i : Nat) → i < m.succ + n → Bool
t : m + 1 + n = m + (n + 1)
⊢ (m + 1 + n).any f = ((n.any fun i h => f i ⋯) || (f (m + 1 + n - (m + 1)) ⋯ || anyTR.loop (m + 1 + n) f m ⋯))
|
rw [any_congr t, anyTR_loop_congr t, go, any, Bool.or_assoc]
|
n✝ : Nat
f✝ : (i : Nat) → i < n✝ → Bool
m n : Nat
f : (i : Nat) → i < m.succ + n → Bool
t : m + 1 + n = m + (n + 1)
⊢ ((n.any fun i h => f i ⋯) || (f n ⋯ || anyTR.loop (m + (n + 1)) (fun i h => f i ⋯) m ⋯)) =
((n.any fun i h => f i ⋯) || (f (m + 1 + n - (m + 1)) ⋯ || anyTR.loop (m + (n + 1)) (fun i h => f i ⋯) m ⋯))
|
323bfd38fa3324f5
|
LinearIndependent.disjoint_span_image
|
Mathlib/LinearAlgebra/LinearIndependent/Basic.lean
|
theorem LinearIndependent.disjoint_span_image (hv : LinearIndependent R v) {s t : Set ι}
(hs : Disjoint s t) : Disjoint (Submodule.span R <| v '' s) (Submodule.span R <| v '' t)
|
case intro.intro.intro.intro
ι : Type u'
R : Type u_2
M : Type u_4
v : ι → M
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hv : LinearIndependent R v
s t : Set ι
hs : Disjoint s t
l₁ : ι →₀ R
hl₁ : l₁ ∈ Finsupp.supported R R s
l₂ : ι →₀ R
hl₂ : l₂ ∈ Finsupp.supported R R t
H : l₂ = l₁
⊢ (Finsupp.linearCombination R v) l₁ = 0
|
subst l₂
|
case intro.intro.intro.intro
ι : Type u'
R : Type u_2
M : Type u_4
v : ι → M
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hv : LinearIndependent R v
s t : Set ι
hs : Disjoint s t
l₁ : ι →₀ R
hl₁ : l₁ ∈ Finsupp.supported R R s
hl₂ : l₁ ∈ Finsupp.supported R R t
⊢ (Finsupp.linearCombination R v) l₁ = 0
|
c57e657967127b39
|
CategoryTheory.ShortComplex.Splitting.ext_s
|
Mathlib/Algebra/Homology/ShortComplex/Exact.lean
|
lemma ext_s (s s' : S.Splitting) (h : s.s = s'.s) : s = s'
|
case mk.mk
C : Type u_1
inst✝¹ : Category.{u_3, u_1} C
inst✝ : Preadditive C
S : ShortComplex C
this : Mono S.f
r✝ : S.X₂ ⟶ S.X₁
s✝ : S.X₃ ⟶ S.X₂
f_r✝¹ : S.f ≫ r✝ = 𝟙 S.X₁
s_g✝¹ : s✝ ≫ S.g = 𝟙 S.X₃
id✝¹ : r✝ ≫ S.f + S.g ≫ s✝ = 𝟙 S.X₂
f_r✝ : S.f ≫ { r := r✝, s := s✝, f_r := f_r✝¹, s_g := s_g✝¹, id := id✝¹ }.r = 𝟙 S.X₁
s_g✝ : { r := r✝, s := s✝, f_r := f_r✝¹, s_g := s_g✝¹, id := id✝¹ }.s ≫ S.g = 𝟙 S.X₃
id✝ :
{ r := r✝, s := s✝, f_r := f_r✝¹, s_g := s_g✝¹, id := id✝¹ }.r ≫ S.f +
S.g ≫ { r := r✝, s := s✝, f_r := f_r✝¹, s_g := s_g✝¹, id := id✝¹ }.s =
𝟙 S.X₂
⊢ { r := r✝, s := s✝, f_r := f_r✝¹, s_g := s_g✝¹, id := id✝¹ } =
{ r := { r := r✝, s := s✝, f_r := f_r✝¹, s_g := s_g✝¹, id := id✝¹ }.r,
s := { r := r✝, s := s✝, f_r := f_r✝¹, s_g := s_g✝¹, id := id✝¹ }.s, f_r := f_r✝, s_g := s_g✝, id := id✝ }
|
rfl
|
no goals
|
500be6a8e246a01c
|
CategoryTheory.Functor.homologySequence_exact₁
|
Mathlib/CategoryTheory/Triangulated/HomologicalFunctor.lean
|
lemma homologySequence_exact₁ :
(ShortComplex.mk _ _ (F.homologySequenceδ_comp T hT _ _ h)).Exact
|
C : Type u_1
A : Type u_3
inst✝⁹ : Category.{u_5, u_1} C
inst✝⁸ : HasShift C ℤ
inst✝⁷ : Category.{u_4, u_3} A
F : C ⥤ A
inst✝⁶ : HasZeroObject C
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝³ : Pretriangulated C
inst✝² : Abelian A
inst✝¹ : F.IsHomological
inst✝ : F.ShiftSequence ℤ
T : Triangle C
hT : T ∈ distinguishedTriangles
n₀ n₁ : ℤ
h : n₀ + 1 = n₁
⊢ (-(F.shiftIso (-1) n₁ n₀ ⋯).hom.app T.obj₃) ≫ F.homologySequenceδ T n₀ n₁ h =
(F.shift n₁).map (-(shiftFunctor C (-1)).map T.mor₃ ≫ (shiftFunctorCompIsoId C 1 (-1) ⋯).hom.app T.obj₁) ≫
𝟙 ((F.shift n₁).obj T.obj₁)
|
simp only [homologySequenceδ, neg_comp, map_neg, comp_id,
F.shiftIso_hom_app_comp_shiftMap_of_add_eq_zero T.mor₃ (-1) (neg_add_cancel 1) n₀ n₁ (by omega)]
|
no goals
|
0cf33dbfc66ec25c
|
Profinite.NobelingProof.C1_projOrd
|
Mathlib/Topology/Category/Profinite/Nobeling.lean
|
theorem C1_projOrd {x : I → Bool} (hx : x ∈ C1 C ho) : SwapTrue o (Proj (ord I · < o) x) = x
|
case pos
I : Type u
C : Set (I → Bool)
inst✝¹ : LinearOrder I
inst✝ : WellFoundedLT I
o : Ordinal.{u}
hsC : contained C (Order.succ o)
ho : o < Ordinal.type fun x1 x2 => x1 < x2
x : I → Bool
hx : x ∈ C1 C ho
i : I
hi : ord I i = o
⊢ true = x i
|
rw [ord_term ho] at hi
|
case pos
I : Type u
C : Set (I → Bool)
inst✝¹ : LinearOrder I
inst✝ : WellFoundedLT I
o : Ordinal.{u}
hsC : contained C (Order.succ o)
ho : o < Ordinal.type fun x1 x2 => x1 < x2
x : I → Bool
hx : x ∈ C1 C ho
i : I
hi : term I ho = i
⊢ true = x i
|
3124430fb4afb78c
|
iSup_succ
|
Mathlib/Order/SuccPred/CompleteLinearOrder.lean
|
theorem iSup_succ [SuccOrder α] (x : α) : ⨆ a : Iio x, succ a.1 = x
|
α : Type u_2
inst✝¹ : ConditionallyCompleteLinearOrderBot α
inst✝ : SuccOrder α
x : α
H : BddAbove (range fun a => succ ↑a)
⊢ ∀ (i : ↑(Iio x)), succ ↑i ≤ x
|
exact fun a ↦ succ_le_of_lt a.2
|
no goals
|
1ac61462e616078c
|
TopCat.GlueData.ι_eq_iff_rel
|
Mathlib/Topology/Gluing.lean
|
theorem ι_eq_iff_rel (i j : D.J) (x : D.U i) (y : D.U j) :
𝖣.ι i x = 𝖣.ι j y ↔ D.Rel ⟨i, x⟩ ⟨j, y⟩
|
case mp
D : GlueData
i j : D.J
x : ↑(D.U i)
y : ↑(D.U j)
⊢ (ConcreteCategory.hom (D.ι i)) x = (ConcreteCategory.hom (D.ι j)) y → D.Rel ⟨i, x⟩ ⟨j, y⟩
|
delta GlueData.ι
|
case mp
D : GlueData
i j : D.J
x : ↑(D.U i)
y : ↑(D.U j)
⊢ (ConcreteCategory.hom (Multicoequalizer.π D.diagram i)) x =
(ConcreteCategory.hom (Multicoequalizer.π D.diagram j)) y →
D.Rel ⟨i, x⟩ ⟨j, y⟩
|
e0a110dda282edcf
|
Set.seq_seq
|
Mathlib/Data/Set/Lattice.lean
|
theorem seq_seq {s : Set (β → γ)} {t : Set (α → β)} {u : Set α} :
seq s (seq t u) = seq (seq ((· ∘ ·) '' s) t) u
|
α : Type u_1
β : Type u_2
γ : Type u_3
s : Set (β → γ)
t : Set (α → β)
u : Set α
⊢ image2 (fun f a => f a) s (image2 (fun f a => f a) t u) =
image2 (fun f a => f a) (image2 (fun x1 x2 => x1 ∘ x2) s t) u
|
exact .symm <| image2_assoc fun _ _ _ ↦ rfl
|
no goals
|
14b1c2e4f91292bb
|
IsPrime.to_maximal_ideal
|
Mathlib/RingTheory/PrincipalIdealDomain.lean
|
theorem to_maximal_ideal [CommRing R] [IsDomain R] [IsPrincipalIdealRing R] {S : Ideal R}
[hpi : IsPrime S] (hS : S ≠ ⊥) : IsMaximal S :=
isMaximal_iff.2
⟨(ne_top_iff_one S).1 hpi.1, by
intro T x hST hxS hxT
obtain ⟨z, hz⟩ := (mem_iff_generator_dvd _).1 (hST <| generator_mem S)
cases hpi.mem_or_mem (show generator T * z ∈ S from hz ▸ generator_mem S) with
| inl h =>
have hTS : T ≤ S
|
R : Type u
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : IsPrincipalIdealRing R
S : Ideal R
hpi : S.IsPrime
hS : S ≠ ⊥
T : Ideal R
x : R
hST : S ≤ T
hxS : x ∉ S
hxT : x ∈ T
z : R
hz : generator S = generator T * z
h : generator T ∈ S
⊢ T ≤ S
|
rwa [← T.span_singleton_generator, Ideal.span_le, singleton_subset_iff]
|
no goals
|
3054b0cfc56f1a76
|
MeasureTheory.integral_convolution
|
Mathlib/Analysis/Convolution.lean
|
theorem integral_convolution [MeasurableAdd₂ G] [MeasurableNeg G] [NormedSpace ℝ E]
[NormedSpace ℝ E'] [CompleteSpace E] [CompleteSpace E'] (hf : Integrable f ν)
(hg : Integrable g μ) : ∫ x, (f ⋆[L, ν] g) x ∂μ = L (∫ x, f x ∂ν) (∫ x, g x ∂μ)
|
𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
F : Type uF
inst✝¹⁹ : NormedAddCommGroup E
inst✝¹⁸ : NormedAddCommGroup E'
inst✝¹⁷ : NormedAddCommGroup F
f : G → E
g : G → E'
inst✝¹⁶ : RCLike 𝕜
inst✝¹⁵ : NormedSpace 𝕜 E
inst✝¹⁴ : NormedSpace 𝕜 E'
inst✝¹³ : NormedSpace ℝ F
inst✝¹² : NormedSpace 𝕜 F
inst✝¹¹ : MeasurableSpace G
μ ν : Measure G
L : E →L[𝕜] E' →L[𝕜] F
inst✝¹⁰ : CompleteSpace F
inst✝⁹ : AddGroup G
inst✝⁸ : SFinite μ
inst✝⁷ : SFinite ν
inst✝⁶ : μ.IsAddRightInvariant
inst✝⁵ : MeasurableAdd₂ G
inst✝⁴ : MeasurableNeg G
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace ℝ E'
inst✝¹ : CompleteSpace E
inst✝ : CompleteSpace E'
hf : Integrable f ν
hg : Integrable g μ
⊢ ∫ (y : G), (L (f y)) (∫ (x : G), g x ∂μ) ∂ν = (L (∫ (x : G), f x ∂ν)) (∫ (x : G), g x ∂μ)
|
exact (L.flip (∫ x, g x ∂μ)).integral_comp_comm hf
|
no goals
|
2dbbac4c3b182b0e
|
integrableOn_peak_smul_of_integrableOn_of_tendsto
|
Mathlib/MeasureTheory/Integral/PeakFunction.lean
|
theorem integrableOn_peak_smul_of_integrableOn_of_tendsto
(hs : MeasurableSet s) (h'st : t ∈ 𝓝[s] x₀)
(hlφ : ∀ u : Set α, IsOpen u → x₀ ∈ u → TendstoUniformlyOn φ 0 l (s \ u))
(hiφ : Tendsto (fun i ↦ ∫ x in t, φ i x ∂μ) l (𝓝 1))
(h'iφ : ∀ᶠ i in l, AEStronglyMeasurable (φ i) (μ.restrict s))
(hmg : IntegrableOn g s μ) (hcg : Tendsto g (𝓝[s] x₀) (𝓝 a)) :
∀ᶠ i in l, IntegrableOn (fun x => φ i x • g x) s μ
|
α : Type u_1
E : Type u_2
ι : Type u_3
hm : MeasurableSpace α
μ : Measure α
inst✝³ : TopologicalSpace α
inst✝² : BorelSpace α
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
g : α → E
l : Filter ι
x₀ : α
s t : Set α
φ : ι → α → ℝ
a : E
hs : MeasurableSet s
h'st : t ∈ 𝓝[s] x₀
hlφ : ∀ (u : Set α), IsOpen u → x₀ ∈ u → TendstoUniformlyOn φ 0 l (s \ u)
hiφ : Tendsto (fun e => ‖∫ (x : α) in t, φ e x ∂μ - 1‖) l (𝓝 0)
h'iφ : ∀ᶠ (i : ι) in l, AEStronglyMeasurable (φ i) (μ.restrict s)
hmg : IntegrableOn g s μ
hcg : Tendsto g (𝓝[s] x₀) (𝓝 a)
u : Set α
u_open : IsOpen u
x₀u : x₀ ∈ u
ut : s ∩ u ⊆ t
hu : ∀ x ∈ u ∩ s, g x ∈ ball a 1
i : ι
hi : ∀ x ∈ s \ u, dist (0 x) (φ i x) < 1
h'i : ‖∫ (x : α) in t, φ i x ∂μ - 1‖ < 1
h''i : AEStronglyMeasurable (φ i) (μ.restrict s)
I : IntegrableOn (φ i) t μ
⊢ MemLp (φ i) ⊤ (μ.restrict (s \ u))
|
apply memLp_top_of_bound (h''i.mono_set diff_subset) 1
|
α : Type u_1
E : Type u_2
ι : Type u_3
hm : MeasurableSpace α
μ : Measure α
inst✝³ : TopologicalSpace α
inst✝² : BorelSpace α
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
g : α → E
l : Filter ι
x₀ : α
s t : Set α
φ : ι → α → ℝ
a : E
hs : MeasurableSet s
h'st : t ∈ 𝓝[s] x₀
hlφ : ∀ (u : Set α), IsOpen u → x₀ ∈ u → TendstoUniformlyOn φ 0 l (s \ u)
hiφ : Tendsto (fun e => ‖∫ (x : α) in t, φ e x ∂μ - 1‖) l (𝓝 0)
h'iφ : ∀ᶠ (i : ι) in l, AEStronglyMeasurable (φ i) (μ.restrict s)
hmg : IntegrableOn g s μ
hcg : Tendsto g (𝓝[s] x₀) (𝓝 a)
u : Set α
u_open : IsOpen u
x₀u : x₀ ∈ u
ut : s ∩ u ⊆ t
hu : ∀ x ∈ u ∩ s, g x ∈ ball a 1
i : ι
hi : ∀ x ∈ s \ u, dist (0 x) (φ i x) < 1
h'i : ‖∫ (x : α) in t, φ i x ∂μ - 1‖ < 1
h''i : AEStronglyMeasurable (φ i) (μ.restrict s)
I : IntegrableOn (φ i) t μ
⊢ ∀ᵐ (x : α) ∂μ.restrict (s \ u), ‖φ i x‖ ≤ 1
|
4601b22b48a125d5
|
Std.Sat.AIG.denote_mkConst
|
Mathlib/.lake/packages/lean4/src/lean/Std/Sat/AIG/Lemmas.lean
|
theorem denote_mkConst {aig : AIG α} : ⟦(aig.mkConst val), assign⟧ = val
|
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
assign : α → Bool
val : Bool
aig : AIG α
b✝ : Bool
heq : val = b✝
⊢ b✝ = val
|
rw [heq]
|
no goals
|
d9e8c97541c0a83f
|
EuclideanGeometry.existsUnique_dist_eq_of_insert
|
Mathlib/Geometry/Euclidean/Circumcenter.lean
|
theorem existsUnique_dist_eq_of_insert {s : AffineSubspace ℝ P}
[HasOrthogonalProjection s.direction] {ps : Set P} (hnps : ps.Nonempty) {p : P} (hps : ps ⊆ s)
(hp : p ∉ s) (hu : ∃! cs : Sphere P, cs.center ∈ s ∧ ps ⊆ (cs : Set P)) :
∃! cs₂ : Sphere P,
cs₂.center ∈ affineSpan ℝ (insert p (s : Set P)) ∧ insert p ps ⊆ (cs₂ : Set P)
|
case h.right.mk.intro.intro.intro.intro.intro
V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
s : AffineSubspace ℝ P
inst✝ : HasOrthogonalProjection s.direction
ps : Set P
hnps : ps.Nonempty
p : P
hps : ps ⊆ ↑s
hp : p ∉ s
this : Nonempty ↥s
cc : P
cr : ℝ
hcccru : ∀ (y : Sphere P), y.center ∈ s ∧ ps ⊆ Metric.sphere y.center y.radius → y = { center := cc, radius := cr }
hcc : cc ∈ s
hcr : ps ⊆ Metric.sphere cc cr
x : ℝ := dist cc ↑((orthogonalProjection s) p)
y : ℝ := dist p ↑((orthogonalProjection s) p)
hy0 : y ≠ 0
ycc₂ : ℝ := (x * x + y * y - cr * cr) / (2 * y)
cc₂ : P := (ycc₂ / y) • (p -ᵥ ↑((orthogonalProjection s) p)) +ᵥ cc
cr₂ : ℝ := √(cr * cr + ycc₂ * ycc₂)
hpo : p = 1 • (p -ᵥ ↑((orthogonalProjection s) p)) +ᵥ ↑((orthogonalProjection s) p)
cc₃ : P
cr₃ : ℝ
hcc₃ : cc₃ ∈ affineSpan ℝ (insert p ↑s)
hcr₃ : insert p ps ⊆ Metric.sphere cc₃ cr₃
t₃ : ℝ
cc₃' : P
hcc₃' : cc₃' ∈ s
hcc₃'' : cc₃ = t₃ • (p -ᵥ ↑((orthogonalProjection s) p)) +ᵥ cc₃'
cr₃' : ℝ
hcr₃' : ∀ p₁ ∈ ps, dist p₁ ↑⟨cc₃', hcc₃'⟩ = cr₃'
⊢ { center := cc₃, radius := cr₃ } = { center := cc₂, radius := cr₂ }
|
have hu := hcccru ⟨cc₃', cr₃'⟩
|
case h.right.mk.intro.intro.intro.intro.intro
V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
s : AffineSubspace ℝ P
inst✝ : HasOrthogonalProjection s.direction
ps : Set P
hnps : ps.Nonempty
p : P
hps : ps ⊆ ↑s
hp : p ∉ s
this : Nonempty ↥s
cc : P
cr : ℝ
hcccru : ∀ (y : Sphere P), y.center ∈ s ∧ ps ⊆ Metric.sphere y.center y.radius → y = { center := cc, radius := cr }
hcc : cc ∈ s
hcr : ps ⊆ Metric.sphere cc cr
x : ℝ := dist cc ↑((orthogonalProjection s) p)
y : ℝ := dist p ↑((orthogonalProjection s) p)
hy0 : y ≠ 0
ycc₂ : ℝ := (x * x + y * y - cr * cr) / (2 * y)
cc₂ : P := (ycc₂ / y) • (p -ᵥ ↑((orthogonalProjection s) p)) +ᵥ cc
cr₂ : ℝ := √(cr * cr + ycc₂ * ycc₂)
hpo : p = 1 • (p -ᵥ ↑((orthogonalProjection s) p)) +ᵥ ↑((orthogonalProjection s) p)
cc₃ : P
cr₃ : ℝ
hcc₃ : cc₃ ∈ affineSpan ℝ (insert p ↑s)
hcr₃ : insert p ps ⊆ Metric.sphere cc₃ cr₃
t₃ : ℝ
cc₃' : P
hcc₃' : cc₃' ∈ s
hcc₃'' : cc₃ = t₃ • (p -ᵥ ↑((orthogonalProjection s) p)) +ᵥ cc₃'
cr₃' : ℝ
hcr₃' : ∀ p₁ ∈ ps, dist p₁ ↑⟨cc₃', hcc₃'⟩ = cr₃'
hu :
{ center := cc₃', radius := cr₃' }.center ∈ s ∧
ps ⊆ Metric.sphere { center := cc₃', radius := cr₃' }.center { center := cc₃', radius := cr₃' }.radius →
{ center := cc₃', radius := cr₃' } = { center := cc, radius := cr }
⊢ { center := cc₃, radius := cr₃ } = { center := cc₂, radius := cr₂ }
|
6d31314b3548c9e7
|
TopologicalSpace.exists_isInducing_l_infty
|
Mathlib/Topology/Metrizable/Urysohn.lean
|
theorem exists_isInducing_l_infty : ∃ f : X → ℕ →ᵇ ℝ, IsInducing f
|
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : RegularSpace X
inst✝ : SecondCountableTopology X
B : Set (Set X)
hBc : B.Countable
hB : IsTopologicalBasis B
s : Set (Set X × Set X) := {UV | UV ∈ B ×ˢ B ∧ closure UV.1 ⊆ UV.2}
this✝¹ : Encodable ↑s
this✝ : TopologicalSpace ↑s := ⊥
this : DiscreteTopology ↑s
hd : ∀ (UV : ↑s), Disjoint (closure (↑UV).1) (↑UV).2ᶜ
ε : ↑s → ℝ
ε01 : ∀ (UV : ↑s), ε UV ∈ Ioc 0 1
hε : Tendsto ε cofinite (𝓝 0)
UV : ↑s
⊢ ∃ f, EqOn (⇑f) 0 (↑UV).1 ∧ EqOn (⇑f) (fun x => ε UV) (↑UV).2ᶜ ∧ ∀ (x : X), f x ∈ Icc 0 (ε UV)
|
rcases exists_continuous_zero_one_of_isClosed isClosed_closure
(hB.isOpen UV.2.1.2).isClosed_compl (hd UV) with
⟨f, hf₀, hf₁, hf01⟩
|
case intro.intro.intro
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : RegularSpace X
inst✝ : SecondCountableTopology X
B : Set (Set X)
hBc : B.Countable
hB : IsTopologicalBasis B
s : Set (Set X × Set X) := {UV | UV ∈ B ×ˢ B ∧ closure UV.1 ⊆ UV.2}
this✝¹ : Encodable ↑s
this✝ : TopologicalSpace ↑s := ⊥
this : DiscreteTopology ↑s
hd : ∀ (UV : ↑s), Disjoint (closure (↑UV).1) (↑UV).2ᶜ
ε : ↑s → ℝ
ε01 : ∀ (UV : ↑s), ε UV ∈ Ioc 0 1
hε : Tendsto ε cofinite (𝓝 0)
UV : ↑s
f : C(X, ℝ)
hf₀ : EqOn (⇑f) 0 (closure (↑UV).1)
hf₁ : EqOn (⇑f) 1 (↑UV).2ᶜ
hf01 : ∀ (x : X), f x ∈ Icc 0 1
⊢ ∃ f, EqOn (⇑f) 0 (↑UV).1 ∧ EqOn (⇑f) (fun x => ε UV) (↑UV).2ᶜ ∧ ∀ (x : X), f x ∈ Icc 0 (ε UV)
|
7bd5745f003136ce
|
Complex.sum_div_factorial_le
|
Mathlib/Data/Complex/Exponential.lean
|
theorem sum_div_factorial_le {α : Type*} [LinearOrderedField α] (n j : ℕ) (hn : 0 < n) :
(∑ m ∈ range j with n ≤ m, (1 / m.factorial : α)) ≤ n.succ / (n.factorial * n) :=
calc
(∑ m ∈ range j with n ≤ m, (1 / m.factorial : α)) =
∑ m ∈ range (j - n), (1 / ((m + n).factorial : α))
|
α : Type u_1
inst✝ : LinearOrderedField α
n j : ℕ
hn : 0 < n
h₁ : ↑n.succ ≠ 1
h₂ : ↑n.succ ≠ 0
h₃ : ↑n.factorial * ↑n ≠ 0
⊢ (↑n.factorial)⁻¹ * ∑ m ∈ range (j - n), (↑n.succ)⁻¹ ^ m =
(↑n.succ - ↑n.succ * (↑n.succ)⁻¹ ^ (j - n)) / (↑n.factorial * ↑n)
|
have h₄ : (n.succ - 1 : α) = n := by simp
|
α : Type u_1
inst✝ : LinearOrderedField α
n j : ℕ
hn : 0 < n
h₁ : ↑n.succ ≠ 1
h₂ : ↑n.succ ≠ 0
h₃ : ↑n.factorial * ↑n ≠ 0
h₄ : ↑n.succ - 1 = ↑n
⊢ (↑n.factorial)⁻¹ * ∑ m ∈ range (j - n), (↑n.succ)⁻¹ ^ m =
(↑n.succ - ↑n.succ * (↑n.succ)⁻¹ ^ (j - n)) / (↑n.factorial * ↑n)
|
e59fd4ab785bbc73
|
CategoryTheory.ShortComplex.exact_iff_mono_cokernel_desc
|
Mathlib/Algebra/Homology/ShortComplex/Exact.lean
|
lemma exact_iff_mono_cokernel_desc [S.HasHomology] [HasCokernel S.f] :
S.Exact ↔ Mono (cokernel.desc S.f S.g S.zero)
|
C : Type u_1
inst✝³ : Category.{u_3, u_1} C
inst✝² : Preadditive C
S : ShortComplex C
inst✝¹ : S.HasHomology
inst✝ : HasCokernel S.f
⊢ S.opcyclesIsoCokernel.symm.hom ≫ (Arrow.mk S.fromOpcycles).hom =
(Arrow.mk (cokernel.desc S.f S.g ⋯)).hom ≫ (Iso.refl (Arrow.mk (cokernel.desc S.f S.g ⋯)).right).hom
|
aesop_cat
|
no goals
|
180cebf98bee6054
|
Plausible.InjectiveFunction.applyId_injective
|
Mathlib/Testing/Plausible/Functions.lean
|
theorem applyId_injective [DecidableEq α] {xs ys : List α} (h₀ : List.Nodup xs) (h₁ : xs ~ ys) :
Injective.{u + 1, u + 1} (List.applyId (xs.zip ys))
|
case pos
α : Type u
inst✝ : DecidableEq α
xs ys : List α
h₀ : xs.Nodup
h₁ : xs ~ ys
x y : α
h : applyId (xs.zip ys) x = applyId (xs.zip ys) y
hx : applyId (xs.zip ys) x ∉ ys
hy : applyId (xs.zip ys) y ∈ ys
⊢ x = y
|
rw [h] at hx
|
case pos
α : Type u
inst✝ : DecidableEq α
xs ys : List α
h₀ : xs.Nodup
h₁ : xs ~ ys
x y : α
h : applyId (xs.zip ys) x = applyId (xs.zip ys) y
hx : applyId (xs.zip ys) y ∉ ys
hy : applyId (xs.zip ys) y ∈ ys
⊢ x = y
|
159dee04a35ae3d2
|
orthogonalProjectionFn_norm_sq
|
Mathlib/Analysis/InnerProductSpace/Projection.lean
|
theorem orthogonalProjectionFn_norm_sq (v : E) :
‖v‖ * ‖v‖ =
‖v - orthogonalProjectionFn K v‖ * ‖v - orthogonalProjectionFn K v‖ +
‖orthogonalProjectionFn K v‖ * ‖orthogonalProjectionFn K v‖
|
𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
inst✝ : HasOrthogonalProjection K
v : E
p : E := orthogonalProjectionFn K v
h' : ⟪v - p, p⟫_𝕜 = 0
⊢ ‖v‖ * ‖v‖ = ‖v - p‖ * ‖v - p‖ + ‖p‖ * ‖p‖
|
convert norm_add_sq_eq_norm_sq_add_norm_sq_of_inner_eq_zero (v - p) p h' using 2 <;> simp
|
no goals
|
40e939907977a6d5
|
MeasureTheory.FiniteMeasure.tendsto_of_forall_integral_tendsto
|
Mathlib/MeasureTheory/Measure/FiniteMeasure.lean
|
theorem tendsto_of_forall_integral_tendsto {γ : Type*} {F : Filter γ} {μs : γ → FiniteMeasure Ω}
{μ : FiniteMeasure Ω}
(h : ∀ f : Ω →ᵇ ℝ,
Tendsto (fun i ↦ ∫ x, f x ∂(μs i : Measure Ω)) F (𝓝 (∫ x, f x ∂(μ : Measure Ω)))) :
Tendsto μs F (𝓝 μ)
|
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
γ : Type u_2
F : Filter γ
μs : γ → FiniteMeasure Ω
μ : FiniteMeasure Ω
h : ∀ (f : Ω →ᵇ ℝ), Tendsto (fun i => ∫ (x : Ω), f x ∂↑(μs i)) F (𝓝 (∫ (x : Ω), f x ∂↑μ))
f : Ω →ᵇ ℝ≥0
lip : LipschitzWith 1 NNReal.toReal
f₀ : Ω →ᵇ ℝ := comp NNReal.toReal lip f
_def_f₀ : f₀ = comp NNReal.toReal lip f
f₀_eq : ⇑f₀ = NNReal.toReal ∘ ⇑f
f₀_nn : 0 ≤ ⇑f₀
f₀_ae_nn : 0 ≤ᶠ[ae ↑μ] ⇑f₀
f₀_ae_nns : ∀ (i : γ), 0 ≤ᶠ[ae ↑(μs i)] ⇑f₀
⊢ Tendsto (fun n => (∫⁻ (x : Ω), ↑(f x) ∂↑(μs n)).toReal) F (𝓝 (∫⁻ (x : Ω), ↑(f x) ∂↑μ).toReal)
|
have aux :=
integral_eq_lintegral_of_nonneg_ae f₀_ae_nn f₀.continuous.measurable.aestronglyMeasurable
|
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
γ : Type u_2
F : Filter γ
μs : γ → FiniteMeasure Ω
μ : FiniteMeasure Ω
h : ∀ (f : Ω →ᵇ ℝ), Tendsto (fun i => ∫ (x : Ω), f x ∂↑(μs i)) F (𝓝 (∫ (x : Ω), f x ∂↑μ))
f : Ω →ᵇ ℝ≥0
lip : LipschitzWith 1 NNReal.toReal
f₀ : Ω →ᵇ ℝ := comp NNReal.toReal lip f
_def_f₀ : f₀ = comp NNReal.toReal lip f
f₀_eq : ⇑f₀ = NNReal.toReal ∘ ⇑f
f₀_nn : 0 ≤ ⇑f₀
f₀_ae_nn : 0 ≤ᶠ[ae ↑μ] ⇑f₀
f₀_ae_nns : ∀ (i : γ), 0 ≤ᶠ[ae ↑(μs i)] ⇑f₀
aux : ∫ (a : Ω), f₀ a ∂↑μ = (∫⁻ (a : Ω), ENNReal.ofReal (f₀ a) ∂↑μ).toReal
⊢ Tendsto (fun n => (∫⁻ (x : Ω), ↑(f x) ∂↑(μs n)).toReal) F (𝓝 (∫⁻ (x : Ω), ↑(f x) ∂↑μ).toReal)
|
921a8ca53571d4e5
|
continuousOn_extendFrom
|
Mathlib/Topology/ExtendFrom.lean
|
theorem continuousOn_extendFrom [RegularSpace Y] {f : X → Y} {A B : Set X} (hB : B ⊆ closure A)
(hf : ∀ x ∈ B, ∃ y, Tendsto f (𝓝[A] x) (𝓝 y)) : ContinuousOn (extendFrom A f) B
|
case intro.intro.intro.intro
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : RegularSpace Y
f : X → Y
A B : Set X
hB : B ⊆ closure A
hf : ∀ x ∈ B, ∃ y, Tendsto f (𝓝[A] x) (𝓝 y)
φ : X → Y := extendFrom A f
x : X
x_in : x ∈ B
V' : Set Y
V'_in : V' ∈ 𝓝 (φ x)
V'_closed : IsClosed V'
V : Set X
V_in : V ∈ 𝓝 x
V_op : IsOpen V
hV : V ∩ A ⊆ f ⁻¹' V'
y : X
hyV : y ∈ V
hyB : y ∈ B
this : (𝓝[A] y).NeBot
limy : Tendsto f (𝓝[A] y) (𝓝 (φ y))
⊢ φ y ∈ V'
|
have hVy : V ∈ 𝓝 y := IsOpen.mem_nhds V_op hyV
|
case intro.intro.intro.intro
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : RegularSpace Y
f : X → Y
A B : Set X
hB : B ⊆ closure A
hf : ∀ x ∈ B, ∃ y, Tendsto f (𝓝[A] x) (𝓝 y)
φ : X → Y := extendFrom A f
x : X
x_in : x ∈ B
V' : Set Y
V'_in : V' ∈ 𝓝 (φ x)
V'_closed : IsClosed V'
V : Set X
V_in : V ∈ 𝓝 x
V_op : IsOpen V
hV : V ∩ A ⊆ f ⁻¹' V'
y : X
hyV : y ∈ V
hyB : y ∈ B
this : (𝓝[A] y).NeBot
limy : Tendsto f (𝓝[A] y) (𝓝 (φ y))
hVy : V ∈ 𝓝 y
⊢ φ y ∈ V'
|
a5bc0b035e892139
|
ModularGroup.three_lt_four_mul_im_sq_of_mem_fdo
|
Mathlib/NumberTheory/Modular.lean
|
theorem three_lt_four_mul_im_sq_of_mem_fdo (h : z ∈ 𝒟ᵒ) : 3 < 4 * z.im ^ 2
|
z : ℍ
h : z ∈ 𝒟ᵒ
⊢ 3 < 4 * z.im ^ 2
|
have : 1 < z.re * z.re + z.im * z.im := by simpa [Complex.normSq_apply] using h.1
|
z : ℍ
h : z ∈ 𝒟ᵒ
this : 1 < z.re * z.re + z.im * z.im
⊢ 3 < 4 * z.im ^ 2
|
7788c720e6fb7c46
|
Semifield.isCoprime_iff
|
Mathlib/RingTheory/Coprime/Basic.lean
|
/-- `IsCoprime` is not a useful definition if an inverse is available. -/
@[simp]
lemma Semifield.isCoprime_iff {R : Type*} [Semifield R] {m n : R} :
IsCoprime m n ↔ m ≠ 0 ∨ n ≠ 0
|
case inl
R : Type u_1
inst✝ : Semifield R
m : R
⊢ IsCoprime m 0 ↔ m ≠ 0 ∨ 0 ≠ 0
|
simp [isCoprime_zero_right]
|
no goals
|
746c9755a6938d1d
|
CategoryTheory.Pretriangulated.productTriangle_distinguished
|
Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean
|
/-- A product of distinguished triangles is distinguished -/
lemma productTriangle_distinguished {J : Type*} (T : J → Triangle C)
(hT : ∀ j, T j ∈ distTriang C)
[HasProduct (fun j => (T j).obj₁)] [HasProduct (fun j => (T j).obj₂)]
[HasProduct (fun j => (T j).obj₃)] [HasProduct (fun j => (T j).obj₁⟦(1 : ℤ)⟧)] :
productTriangle T ∈ distTriang C
|
C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : HasZeroObject C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
J : Type u_1
T : J → Triangle C
hT : ∀ (j : J), T j ∈ distinguishedTriangles
inst✝³ : HasProduct fun j => (T j).obj₁
inst✝² : HasProduct fun j => (T j).obj₂
inst✝¹ : HasProduct fun j => (T j).obj₃
inst✝ : HasProduct fun j => (shiftFunctor C 1).obj (T j).obj₁
f₁ : (∏ᶜ fun j => (T j).obj₁) ⟶ ∏ᶜ fun j => (T j).obj₂ := Limits.Pi.map fun j => (T j).mor₁
Z : C
f₂ : (∏ᶜ fun j => (T j).obj₂) ⟶ Z
f₃ : Z ⟶ (shiftFunctor C 1).obj (∏ᶜ fun j => (T j).obj₁)
T' : Triangle C := Triangle.mk f₁ f₂ f₃
hT' : T' ∈ distinguishedTriangles
φ : (j : J) → T' ⟶ T j :=
fun j =>
completeDistinguishedTriangleMorphism T' (T j) hT' ⋯ (Pi.π (fun j => (T j).obj₁) j) (Pi.π (fun j => (T j).obj₂) j) ⋯
φ' : T' ⟶ productTriangle T := productTriangle.lift T φ
h₁ : φ'.hom₁ = 𝟙 T'.obj₁
h₂ : φ'.hom₂ = 𝟙 T'.obj₂
⊢ IsIso φ'.hom₁
|
rw [h₁]
|
C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : HasZeroObject C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
J : Type u_1
T : J → Triangle C
hT : ∀ (j : J), T j ∈ distinguishedTriangles
inst✝³ : HasProduct fun j => (T j).obj₁
inst✝² : HasProduct fun j => (T j).obj₂
inst✝¹ : HasProduct fun j => (T j).obj₃
inst✝ : HasProduct fun j => (shiftFunctor C 1).obj (T j).obj₁
f₁ : (∏ᶜ fun j => (T j).obj₁) ⟶ ∏ᶜ fun j => (T j).obj₂ := Limits.Pi.map fun j => (T j).mor₁
Z : C
f₂ : (∏ᶜ fun j => (T j).obj₂) ⟶ Z
f₃ : Z ⟶ (shiftFunctor C 1).obj (∏ᶜ fun j => (T j).obj₁)
T' : Triangle C := Triangle.mk f₁ f₂ f₃
hT' : T' ∈ distinguishedTriangles
φ : (j : J) → T' ⟶ T j :=
fun j =>
completeDistinguishedTriangleMorphism T' (T j) hT' ⋯ (Pi.π (fun j => (T j).obj₁) j) (Pi.π (fun j => (T j).obj₂) j) ⋯
φ' : T' ⟶ productTriangle T := productTriangle.lift T φ
h₁ : φ'.hom₁ = 𝟙 T'.obj₁
h₂ : φ'.hom₂ = 𝟙 T'.obj₂
⊢ IsIso (𝟙 T'.obj₁)
|
ce9c86c420201fe7
|
Set.Countable.isPathConnected_compl_of_one_lt_rank
|
Mathlib/Analysis/NormedSpace/Connected.lean
|
theorem Set.Countable.isPathConnected_compl_of_one_lt_rank
(h : 1 < Module.rank ℝ E) {s : Set E} (hs : s.Countable) :
IsPathConnected sᶜ
|
case h
E : Type u_1
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : TopologicalSpace E
inst✝¹ : ContinuousAdd E
inst✝ : ContinuousSMul ℝ E
h : 1 < Module.rank ℝ E
s : Set E
hs : s.Countable
this : Nontrivial E
a : E
ha : a ∈ sᶜ
b : E
hb : b ∈ sᶜ
hab : a ≠ b
c : E := 2⁻¹ • (a + b)
x : E := 2⁻¹ • (b - a)
Ia : c - x = a
Ib : c + x = b
x_ne_zero : x ≠ 0
y : E
hy : LinearIndependent ℝ ![x, y]
A : {t | ([c + x-[ℝ]c + t • y] ∩ s).Nonempty}.Countable
B : {t | ([c - x-[ℝ]c + t • y] ∩ s).Nonempty}.Countable
t : ℝ
z : E := c + t • y
ht : [c + x-[ℝ]c + t • y] ∩ s = ∅ ∧ [c - x-[ℝ]c + t • y] ∩ s = ∅
⊢ [a-[ℝ]z] ⊆ sᶜ
|
rw [subset_compl_iff_disjoint_right, disjoint_iff_inter_eq_empty]
|
case h
E : Type u_1
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : TopologicalSpace E
inst✝¹ : ContinuousAdd E
inst✝ : ContinuousSMul ℝ E
h : 1 < Module.rank ℝ E
s : Set E
hs : s.Countable
this : Nontrivial E
a : E
ha : a ∈ sᶜ
b : E
hb : b ∈ sᶜ
hab : a ≠ b
c : E := 2⁻¹ • (a + b)
x : E := 2⁻¹ • (b - a)
Ia : c - x = a
Ib : c + x = b
x_ne_zero : x ≠ 0
y : E
hy : LinearIndependent ℝ ![x, y]
A : {t | ([c + x-[ℝ]c + t • y] ∩ s).Nonempty}.Countable
B : {t | ([c - x-[ℝ]c + t • y] ∩ s).Nonempty}.Countable
t : ℝ
z : E := c + t • y
ht : [c + x-[ℝ]c + t • y] ∩ s = ∅ ∧ [c - x-[ℝ]c + t • y] ∩ s = ∅
⊢ [a-[ℝ]z] ∩ s = ∅
|
a2405a5e761a2c4d
|
Std.DHashMap.Raw.get!_modify_self
|
Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/RawLemmas.lean
|
theorem get!_modify_self [LawfulBEq α] {k : α} [Inhabited (β k)] {f : β k → β k} (h : m.WF) :
(m.modify k f).get! k = ((m.get? k).map f).get!
|
α : Type u
β : α → Type v
inst✝³ : BEq α
inst✝² : Hashable α
m : Raw α β
inst✝¹ : LawfulBEq α
k : α
inst✝ : Inhabited (β k)
f : β k → β k
h : m.WF
⊢ (m.modify k f).get! k = (Option.map f (m.get? k)).get!
|
simp_to_raw using Raw₀.get!_modify_self
|
no goals
|
7dca714267dac1d9
|
Real.Gamma_ne_zero
|
Mathlib/Analysis/SpecialFunctions/Gamma/Basic.lean
|
theorem Gamma_ne_zero {s : ℝ} (hs : ∀ m : ℕ, s ≠ -m) : Gamma s ≠ 0
|
case hs
n✝ : ℕ
n_ih : ∀ {s : ℝ}, (∀ (m : ℕ), s ≠ -↑m) → -↑n✝ < s → Gamma s ≠ 0
s : ℝ
hs' : -↑(n✝ + 1) < s
m : ℕ
hs : s + 1 = -↑m
⊢ s = -↑(1 + m)
|
rw [← eq_sub_iff_add_eq] at hs
|
case hs
n✝ : ℕ
n_ih : ∀ {s : ℝ}, (∀ (m : ℕ), s ≠ -↑m) → -↑n✝ < s → Gamma s ≠ 0
s : ℝ
hs' : -↑(n✝ + 1) < s
m : ℕ
hs : s = -↑m - 1
⊢ s = -↑(1 + m)
|
b27fef48da598597
|
List.isSome_isPrefixOf?_eq_isPrefixOf
|
Mathlib/.lake/packages/batteries/Batteries/Data/List/Lemmas.lean
|
theorem isSome_isPrefixOf?_eq_isPrefixOf [BEq α] (xs ys : List α) :
(xs.isPrefixOf? ys).isSome = xs.isPrefixOf ys
|
α : Type u_1
inst✝ : BEq α
xs ys : List α
head✝¹ : α
tail✝¹ : List α
head✝ : α
tail✝ : List α
⊢ ((head✝¹ :: tail✝¹).isPrefixOf? (head✝ :: tail✝)).isSome = (head✝¹ :: tail✝¹).isPrefixOf (head✝ :: tail✝)
|
simp only [List.isPrefixOf?, List.isPrefixOf]
|
α : Type u_1
inst✝ : BEq α
xs ys : List α
head✝¹ : α
tail✝¹ : List α
head✝ : α
tail✝ : List α
⊢ (if (head✝¹ == head✝) = true then tail✝¹.isPrefixOf? tail✝ else none).isSome =
(head✝¹ == head✝ && tail✝¹.isPrefixOf tail✝)
|
fcd4d9aaeffcffb4
|
MeasureTheory.integral_le_measure
|
Mathlib/MeasureTheory/Integral/SetIntegral.lean
|
lemma integral_le_measure {f : X → ℝ} {s : Set X}
(hs : ∀ x ∈ s, f x ≤ 1) (h's : ∀ x ∈ sᶜ, f x ≤ 0) :
ENNReal.ofReal (∫ x, f x ∂μ) ≤ μ s
|
case pos.hf
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
f : X → ℝ
s : Set X
hs : ∀ x ∈ s, f x ≤ 1
h's : ∀ x ∈ sᶜ, f x ≤ 0
H : Integrable f μ
g : X → ℝ := fun x => f x ⊔ 0
g_int : Integrable g μ
this : ENNReal.ofReal (∫ (x : X), f x ∂μ) ≤ ENNReal.ofReal (∫ (x : X), g x ∂μ)
⊢ ∀ (a : X), ENNReal.ofReal (g a) ≤ 1
|
intro x
|
case pos.hf
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
f : X → ℝ
s : Set X
hs : ∀ x ∈ s, f x ≤ 1
h's : ∀ x ∈ sᶜ, f x ≤ 0
H : Integrable f μ
g : X → ℝ := fun x => f x ⊔ 0
g_int : Integrable g μ
this : ENNReal.ofReal (∫ (x : X), f x ∂μ) ≤ ENNReal.ofReal (∫ (x : X), g x ∂μ)
x : X
⊢ ENNReal.ofReal (g x) ≤ 1
|
3c1b6cc50ba267ee
|
HahnSeries.map_single
|
Mathlib/RingTheory/HahnSeries/Basic.lean
|
@[simp]
protected lemma map_single [Zero S] (f : ZeroHom R S) : (single a r).map f = single a (f r)
|
Γ : Type u_1
R : Type u_3
S : Type u_4
inst✝² : PartialOrder Γ
inst✝¹ : Zero R
a : Γ
r : R
inst✝ : Zero S
f : ZeroHom R S
⊢ ((single a) r).map f = (single a) (f r)
|
ext g
|
case coeff.h
Γ : Type u_1
R : Type u_3
S : Type u_4
inst✝² : PartialOrder Γ
inst✝¹ : Zero R
a : Γ
r : R
inst✝ : Zero S
f : ZeroHom R S
g : Γ
⊢ (((single a) r).map f).coeff g = ((single a) (f r)).coeff g
|
7a9473dc78274d3e
|
EuclideanGeometry.collinear_iff_eq_or_eq_or_angle_eq_zero_or_angle_eq_pi
|
Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean
|
theorem collinear_iff_eq_or_eq_or_angle_eq_zero_or_angle_eq_pi {p₁ p₂ p₃ : P} :
Collinear ℝ ({p₁, p₂, p₃} : Set P) ↔ p₁ = p₂ ∨ p₃ = p₂ ∨ ∠ p₁ p₂ p₃ = 0 ∨ ∠ p₁ p₂ p₃ = π
|
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
p₁ p₂ p₃ : P
⊢ Collinear ℝ {p₁, p₂, p₃} ↔ p₁ = p₂ ∨ p₃ = p₂ ∨ ∠ p₁ p₂ p₃ = 0 ∨ ∠ p₁ p₂ p₃ = π
|
refine ⟨fun h => ?_, fun h => ?_⟩
|
case refine_1
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
p₁ p₂ p₃ : P
h : Collinear ℝ {p₁, p₂, p₃}
⊢ p₁ = p₂ ∨ p₃ = p₂ ∨ ∠ p₁ p₂ p₃ = 0 ∨ ∠ p₁ p₂ p₃ = π
case refine_2
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
p₁ p₂ p₃ : P
h : p₁ = p₂ ∨ p₃ = p₂ ∨ ∠ p₁ p₂ p₃ = 0 ∨ ∠ p₁ p₂ p₃ = π
⊢ Collinear ℝ {p₁, p₂, p₃}
|
6d4fda70b0d8e02f
|
LieIdeal.coe_killingCompl_top
|
Mathlib/Algebra/Lie/TraceForm.lean
|
lemma coe_killingCompl_top :
killingCompl R L ⊤ = LinearMap.ker (killingForm R L)
|
case h
R : Type u_1
L : Type u_3
inst✝² : CommRing R
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
x : L
⊢ x ∈ (toLieSubalgebra R L (killingCompl R L ⊤)).toSubmodule ↔ x ∈ LinearMap.ker (killingForm R L)
|
simp [LinearMap.ext_iff, LinearMap.BilinForm.IsOrtho, LieModule.traceForm_comm R L L x]
|
no goals
|
3f3bf71632928c7f
|
Real.sin_pi_div_six
|
Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean
|
theorem sin_pi_div_six : sin (π / 6) = 1 / 2
|
⊢ cos (π / 2 - π / 6) = cos (π / 3)
|
congr
|
case e_x
⊢ π / 2 - π / 6 = π / 3
|
bd3bee9dce573540
|
Ideal.ideal_prod_eq
|
Mathlib/RingTheory/Ideal/Prod.lean
|
theorem ideal_prod_eq (I : Ideal (R × S)) :
I = Ideal.prod (map (RingHom.fst R S) I : Ideal R) (map (RingHom.snd R S) I)
|
case h.mk
R : Type u
S : Type v
inst✝¹ : Semiring R
inst✝ : Semiring S
I : Ideal (R × S)
r : R
s : S
⊢ (r, s) ∈ I ↔ (r, s) ∈ (map (RingHom.fst R S) I).prod (map (RingHom.snd R S) I)
|
rw [mem_prod, mem_map_iff_of_surjective (RingHom.fst R S) Prod.fst_surjective,
mem_map_iff_of_surjective (RingHom.snd R S) Prod.snd_surjective]
|
case h.mk
R : Type u
S : Type v
inst✝¹ : Semiring R
inst✝ : Semiring S
I : Ideal (R × S)
r : R
s : S
⊢ (r, s) ∈ I ↔ (∃ x ∈ I, (RingHom.fst R S) x = r) ∧ ∃ x ∈ I, (RingHom.snd R S) x = s
|
01de65c949d940b4
|
Real.Angle.abs_cos_eq_of_two_zsmul_eq
|
Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean
|
theorem abs_cos_eq_of_two_zsmul_eq {θ ψ : Angle} (h : (2 : ℤ) • θ = (2 : ℤ) • ψ) :
|cos θ| = |cos ψ|
|
θ ψ : Angle
h : 2 • θ = 2 • ψ
⊢ |θ.cos| = |ψ.cos|
|
exact abs_cos_eq_of_two_nsmul_eq h
|
no goals
|
b40ff988772aa5a4
|
norm_image_sub_le_of_norm_deriv_right_le_segment
|
Mathlib/Analysis/Calculus/MeanValue.lean
|
theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ}
(hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x)
(bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a)
|
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b : ℝ
f' : ℝ → E
C : ℝ
hf : ContinuousOn f (Icc a b)
hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x
bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C
g : ℝ → E := fun x => f x - f a
hg : ContinuousOn g (Icc a b)
hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x
B : ℝ → ℝ := fun x => C * (x - a)
hB : ∀ (x : ℝ), HasDerivAt B C x
⊢ ‖g a‖ ≤ B a
|
simp only [g, B]
|
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b : ℝ
f' : ℝ → E
C : ℝ
hf : ContinuousOn f (Icc a b)
hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x
bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C
g : ℝ → E := fun x => f x - f a
hg : ContinuousOn g (Icc a b)
hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x
B : ℝ → ℝ := fun x => C * (x - a)
hB : ∀ (x : ℝ), HasDerivAt B C x
⊢ ‖f a - f a‖ ≤ C * (a - a)
|
2ebbef8de7fd661c
|
IsDiscreteValuationRing.aux_pid_of_ufd_of_unique_irreducible
|
Mathlib/RingTheory/DiscreteValuationRing/Basic.lean
|
theorem aux_pid_of_ufd_of_unique_irreducible (R : Type u) [CommRing R] [IsDomain R]
[UniqueFactorizationMonoid R] (h₁ : ∃ p : R, Irreducible p)
(h₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q) :
IsPrincipalIdealRing R
|
case h
R : Type u
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : UniqueFactorizationMonoid R
h₁ : ∃ p, Irreducible p
h₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q
I : Ideal R
I0 : I = ⊥
⊢ ⊥ = Submodule.span R {0}
|
simp only [Set.singleton_zero, Submodule.span_zero]
|
no goals
|
378c4d5b6b1ec199
|
Set.smul_Icc
|
Mathlib/Algebra/Order/Group/Pointwise/Interval.lean
|
@[to_additive (attr := simp)]
lemma smul_Icc (a b c : α) : a • Icc b c = Icc (a * b) (a * c)
|
case h.mp
α : Type u_1
inst✝² : LinearOrderedCommMonoid α
inst✝¹ : MulLeftReflectLE α
inst✝ : ExistsMulOfLE α
a b c x : α
⊢ x ∈ a • Icc b c → x ∈ Icc (a * b) (a * c)
|
rintro ⟨y, ⟨hby, hyc⟩, rfl⟩
|
case h.mp.intro.intro.intro
α : Type u_1
inst✝² : LinearOrderedCommMonoid α
inst✝¹ : MulLeftReflectLE α
inst✝ : ExistsMulOfLE α
a b c y : α
hby : b ≤ y
hyc : y ≤ c
⊢ (fun x => a • x) y ∈ Icc (a * b) (a * c)
|
2d2bbce6d2f58d9e
|
BitVec.setWidth_setWidth
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean
|
theorem setWidth_setWidth {x : BitVec u} {w v : Nat} (h : ¬ (v < u ∧ v < w)) :
setWidth w (setWidth v x) = setWidth w x
|
case pred
u : Nat
x : BitVec u
w v : Nat
h : ¬(v < u ∧ v < w)
i✝ : Nat
a✝ : i✝ < w
⊢ x.getLsbD i✝ = true → i✝ < v
|
intro h
|
case pred
u : Nat
x : BitVec u
w v : Nat
h✝ : ¬(v < u ∧ v < w)
i✝ : Nat
a✝ : i✝ < w
h : x.getLsbD i✝ = true
⊢ i✝ < v
|
46fa8a6e672650cc
|
Real.pi_lt_sqrtTwoAddSeries
|
Mathlib/Data/Real/Pi/Bounds.lean
|
theorem pi_lt_sqrtTwoAddSeries (n : ℕ) :
π < 2 ^ (n + 1) * √(2 - sqrtTwoAddSeries 0 n) + 1 / 4 ^ n
|
n : ℕ
⊢ 0 < π / 2 ^ (n + 2)
|
positivity
|
no goals
|
02d82b689d50a592
|
Array.getElem_extract_loop_ge
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean
|
theorem getElem_extract_loop_ge (as bs : Array α) (size start : Nat) (hge : i ≥ bs.size)
(h : i < (extract.loop as size start bs).size)
(h' := getElem_extract_loop_ge_aux as bs size start hge h) :
(extract.loop as size start bs)[i] = as[start + i - bs.size]
|
α : Type u_1
as : Array α
size : Nat
bs : Array α
start : Nat
this✝ : start < as.size
ih :
∀ (bs_1 : Array α) (start : Nat) (hge : bs.size ≥ bs_1.size) (h : bs.size < (extract.loop as size start bs_1).size)
(h' : optParam (start + bs.size - bs_1.size < as.size) ⋯),
(extract.loop as size start bs_1)[bs.size] = as[start + bs.size - bs_1.size]
hge : bs.size ≥ bs.size
h : bs.size < (extract.loop as (size + 1) start bs).size
h' : optParam (start + bs.size - bs.size < as.size) ⋯
this : bs.size < (extract.loop as size (start + 1) (bs.push as[start])).size
heq : (extract.loop as (size + 1) start bs)[bs.size] = (extract.loop as size (start + 1) (bs.push as[start]))[bs.size]
h₁ : bs.size < (bs.push as[start]).size
h₂ : bs.size < (extract.loop as size (start + 1) (bs.push as[start])).size
⊢ (extract.loop as size (start + 1) (bs.push as[start]))[bs.size] = as[start]
|
rw [getElem_extract_loop_lt as (bs.push as[start]) size (start+1) h₁ h₂, getElem_push_eq]
|
no goals
|
89ddb2da716cb704
|
Ideal.mem_iInf_smul_pow_eq_bot_iff
|
Mathlib/RingTheory/Filtration.lean
|
theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) :
x ∈ (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) ↔ ∃ r : I, (r : R) • x = x
|
case mp
R : Type u_1
M : Type u_2
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
I : Ideal R
inst✝¹ : IsNoetherianRing R
inst✝ : Module.Finite R M
x : M
N : Submodule R M := ⨅ i, I ^ i • ⊤
hN : ∀ (k : ℕ), (I.stableFiltration ⊤ ⊓ I.trivialFiltration N).N k = N
⊢ x ∈ ⨅ i, I ^ i • ⊤ → ∃ r, ↑r • x = x
|
obtain ⟨r, hr₁, hr₂⟩ :=
Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by
obtain ⟨k, hk⟩ := (I.stableFiltration_stable ⊤).inter_right (I.trivialFiltration N)
have := hk k (le_refl _)
rw [hN, hN] at this
exact le_of_eq this.symm)
|
case mp.intro.intro
R : Type u_1
M : Type u_2
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
I : Ideal R
inst✝¹ : IsNoetherianRing R
inst✝ : Module.Finite R M
x : M
N : Submodule R M := ⨅ i, I ^ i • ⊤
hN : ∀ (k : ℕ), (I.stableFiltration ⊤ ⊓ I.trivialFiltration N).N k = N
r : R
hr₁ : r ∈ I
hr₂ : ∀ n ∈ N, r • n = n
⊢ x ∈ ⨅ i, I ^ i • ⊤ → ∃ r, ↑r • x = x
|
12c98f9d25221f64
|
DihedralGroup.exponent
|
Mathlib/GroupTheory/SpecificGroups/Dihedral.lean
|
theorem exponent : Monoid.exponent (DihedralGroup n) = lcm n 2
|
case inr.a.a
n : ℕ
hn : NeZero n
⊢ ∀ (g : DihedralGroup n), g ^ lcm n 2 = 1
|
rintro (m | m)
|
case inr.a.a.r
n : ℕ
hn : NeZero n
m : ZMod n
⊢ r m ^ lcm n 2 = 1
case inr.a.a.sr
n : ℕ
hn : NeZero n
m : ZMod n
⊢ sr m ^ lcm n 2 = 1
|
5619adb1153445d8
|
OreLocalization.add_smul
|
Mathlib/RingTheory/OreLocalization/Ring.lean
|
theorem add_smul (y z : R[S⁻¹]) (x : X[S⁻¹]) :
(y + z) • x = y • x + z • x
|
R : Type u_1
inst✝³ : Semiring R
S : Submonoid R
inst✝² : OreSet S
X : Type u_2
inst✝¹ : AddCommMonoid X
inst✝ : Module R X
r₁ : X
s₁ : ↥S
r₂ : R
s₂ : ↥S
r₃ : R
s₃ : ↥S
ra : R
sa : ↥S
ha : ↑sa * ↑s₂ = ra * ↑s₃
rb : R
sb : ↥S
hb : ↑sb * sa • r₂ = rb * ↑s₁
⊢ ↑sb * ra * ↑s₃ ∈ S
|
rw [mul_assoc, ← ha]
|
R : Type u_1
inst✝³ : Semiring R
S : Submonoid R
inst✝² : OreSet S
X : Type u_2
inst✝¹ : AddCommMonoid X
inst✝ : Module R X
r₁ : X
s₁ : ↥S
r₂ : R
s₂ : ↥S
r₃ : R
s₃ : ↥S
ra : R
sa : ↥S
ha : ↑sa * ↑s₂ = ra * ↑s₃
rb : R
sb : ↥S
hb : ↑sb * sa • r₂ = rb * ↑s₁
⊢ ↑sb * (↑sa * ↑s₂) ∈ S
|
79dc4eb2119c8abc
|
MeasureTheory.Measure.haarScalarFactor_smul
|
Mathlib/MeasureTheory/Measure/Haar/Unique.lean
|
@[to_additive (attr := simp) addHaarScalarFactor_smul]
lemma haarScalarFactor_smul [LocallyCompactSpace G] (μ' μ : Measure G) [IsHaarMeasure μ]
[IsFiniteMeasureOnCompacts μ'] [IsMulLeftInvariant μ'] {c : ℝ≥0} :
haarScalarFactor (c • μ') μ = c • haarScalarFactor μ' μ
|
G : Type u_1
inst✝⁸ : TopologicalSpace G
inst✝⁷ : Group G
inst✝⁶ : IsTopologicalGroup G
inst✝⁵ : MeasurableSpace G
inst✝⁴ : BorelSpace G
inst✝³ : LocallyCompactSpace G
μ' μ : Measure G
inst✝² : μ.IsHaarMeasure
inst✝¹ : IsFiniteMeasureOnCompacts μ'
inst✝ : μ'.IsMulLeftInvariant
c : ℝ≥0
g : G → ℝ
g_cont : Continuous g
g_comp : HasCompactSupport ⇑{ toFun := g, continuous_toFun := g_cont }
g_nonneg : 0 ≤ { toFun := g, continuous_toFun := g_cont }
g_one : { toFun := g, continuous_toFun := g_cont } 1 ≠ 0
int_g_ne_zero : ∫ (x : G), g x ∂μ ≠ 0
⊢ (∫ (x : G), g x ∂c • μ') / ∫ (x : G), g x ∂μ = (c • ∫ (x : G), g x ∂μ') / ∫ (x : G), g x ∂μ
|
simp
|
no goals
|
25416b28ad97352c
|
CategoryTheory.Limits.Types.isIso_colimitPointwiseProductToProductColimit
|
Mathlib/CategoryTheory/Limits/FilteredColimitCommutesProduct.lean
|
theorem Types.isIso_colimitPointwiseProductToProductColimit (F : ∀ i, I i ⥤ Type u) :
IsIso (colimitPointwiseProductToProductColimit F)
|
α : Type u
I : α → Type u
inst✝¹ : (i : α) → SmallCategory (I i)
inst✝ : ∀ (i : α), IsFiltered (I i)
F : (i : α) → I i ⥤ Type u
ky : (i : α) → I i
yk₀ : (pointwiseProduct F).obj ky
ky' : (i : α) → I i
yk₀' : (pointwiseProduct F).obj ky'
k : (i : α) → I i := IsFiltered.max ky ky'
yk : ∏ᶜ fun s => (F s).obj (k s) := (pointwiseProduct F).map (IsFiltered.leftToMax ky ky') yk₀
yk' : ∏ᶜ fun s => (F s).obj (k s) := (pointwiseProduct F).map (IsFiltered.rightToMax ky ky') yk₀'
hyk₀ : colimit.ι (pointwiseProduct F) ky yk₀ = colimit.ι (pointwiseProduct F) k yk
hyk₀' : colimit.ι (pointwiseProduct F) ky' yk₀' = colimit.ι (pointwiseProduct F) k yk'
hy :
colimitPointwiseProductToProductColimit F (colimit.ι (pointwiseProduct F) k yk) =
colimitPointwiseProductToProductColimit F (colimit.ι (pointwiseProduct F) k yk')
k' : (s : α) → I s
f : (s : α) → k s ⟶ k' s
hk' :
∀ (s : α),
(F s).map (f s) (Pi.π (fun s => (F s).obj (k s)) s yk) = (F s).map (f s) (Pi.π (fun s => (F s).obj (k s)) s yk')
x✝ : Discrete α
s : α
⊢ limit.π (Discrete.functor fun s => (F s).obj (k' s)) { as := s } ((pointwiseProduct F).map f yk) =
limit.π (Discrete.functor fun s => (F s).obj (k' s)) { as := s } ((pointwiseProduct F).map f yk')
|
simpa using hk' _
|
no goals
|
fc31391d5c273e31
|
TwoSidedIdeal.mem_span_iff
|
Mathlib/RingTheory/TwoSidedIdeal/Operations.lean
|
lemma mem_span_iff {s : Set R} {x} :
x ∈ span s ↔ ∀ (I : TwoSidedIdeal R), s ⊆ I → x ∈ I
|
R : Type u_1
inst✝ : NonUnitalNonAssocRing R
s : Set R
x : R
h : x ∈ { ringCon := sInf {s_1 | ∀ (x y : R), x - y ∈ s → s_1 x y} }
I : TwoSidedIdeal R
hI : s ⊆ ↑I
⊢ x ∈ I
|
refine sInf_le (α := RingCon R) ?_ h
|
R : Type u_1
inst✝ : NonUnitalNonAssocRing R
s : Set R
x : R
h : x ∈ { ringCon := sInf {s_1 | ∀ (x y : R), x - y ∈ s → s_1 x y} }
I : TwoSidedIdeal R
hI : s ⊆ ↑I
⊢ I.ringCon ∈ {s_1 | ∀ (x y : R), x - y ∈ s → s_1 x y}
|
3e62e9a67bfb15af
|
DyckWord.infix_of_le
|
Mathlib/Combinatorics/Enumerative/DyckWord.lean
|
lemma infix_of_le (h : p ≤ q) : p.toList <:+: q.toList
|
case tail.inr
p q m r : DyckWord
_pm : Relation.ReflTransGen (fun p q => p = q.insidePart ∨ p = q.outsidePart) p m
mq : m = r.insidePart ∨ m = r.outsidePart
ih : ↑p <:+: ↑m
hr : r ≠ 0
this : [U] ++ ↑r.insidePart ++ [D] ++ ↑r.outsidePart = ↑r
⊢ ↑p <:+: ↑r
|
rcases mq with hm | hm
|
case tail.inr.inl
p q m r : DyckWord
_pm : Relation.ReflTransGen (fun p q => p = q.insidePart ∨ p = q.outsidePart) p m
ih : ↑p <:+: ↑m
hr : r ≠ 0
this : [U] ++ ↑r.insidePart ++ [D] ++ ↑r.outsidePart = ↑r
hm : m = r.insidePart
⊢ ↑p <:+: ↑r
case tail.inr.inr
p q m r : DyckWord
_pm : Relation.ReflTransGen (fun p q => p = q.insidePart ∨ p = q.outsidePart) p m
ih : ↑p <:+: ↑m
hr : r ≠ 0
this : [U] ++ ↑r.insidePart ++ [D] ++ ↑r.outsidePart = ↑r
hm : m = r.outsidePart
⊢ ↑p <:+: ↑r
|
b036e2b83cf27c51
|
FiniteField.isSquare_odd_prime_iff
|
Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/GaussSum.lean
|
theorem FiniteField.isSquare_odd_prime_iff (hF : ringChar F ≠ 2) {p : ℕ} [Fact p.Prime]
(hp : p ≠ 2) :
IsSquare (p : F) ↔ quadraticChar (ZMod p) (χ₄ (Fintype.card F) * Fintype.card F) ≠ -1
|
case neg
F : Type u_1
inst✝² : Field F
inst✝¹ : Fintype F
hF : ringChar F ≠ 2
p : ℕ
inst✝ : Fact (Nat.Prime p)
hp : p ≠ 2
hFp : ¬ringChar F = p
⊢ IsSquare ↑p ↔ (quadraticChar (ZMod p)) (↑(χ₄ ↑(Fintype.card F)) * ↑(Fintype.card F)) ≠ -1
|
rw [← Iff.not_left (@quadraticChar_neg_one_iff_not_isSquare F _ _ _ _),
quadraticChar_odd_prime hF hp]
|
case neg
F : Type u_1
inst✝² : Field F
inst✝¹ : Fintype F
hF : ringChar F ≠ 2
p : ℕ
inst✝ : Fact (Nat.Prime p)
hp : p ≠ 2
hFp : ¬ringChar F = p
⊢ ringChar F ≠ p
|
355421dc3af479ff
|
Prop.forall_iff
|
Mathlib/Logic/Basic.lean
|
theorem Prop.forall_iff {p : Prop → Prop} : (∀ h, p h) ↔ p False ∧ p True :=
⟨fun H ↦ ⟨H _, H _⟩, fun ⟨h₁, h₂⟩ h ↦ by by_cases H : h <;> simpa only [H]⟩
|
p : Prop → Prop
x✝ : p False ∧ p True
h : Prop
h₁ : p False
h₂ : p True
⊢ p h
|
by_cases H : h <;> simpa only [H]
|
no goals
|
908b1823c34063a7
|
MvPolynomial.coeff_X_pow
|
Mathlib/Algebra/MvPolynomial/Basic.lean
|
theorem coeff_X_pow [DecidableEq σ] (i : σ) (m) (k : ℕ) :
coeff m (X i ^ k : MvPolynomial σ R) = if Finsupp.single i k = m then 1 else 0
|
R : Type u
σ : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq σ
i : σ
m : σ →₀ ℕ
k : ℕ
⊢ coeff m (X i ^ k) = if Finsupp.single i k = m then 1 else 0
|
have := coeff_monomial m (Finsupp.single i k) (1 : R)
|
R : Type u
σ : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq σ
i : σ
m : σ →₀ ℕ
k : ℕ
this : coeff m ((monomial (Finsupp.single i k)) 1) = if Finsupp.single i k = m then 1 else 0
⊢ coeff m (X i ^ k) = if Finsupp.single i k = m then 1 else 0
|
969e05a9f8c23cb9
|
Matrix.det_succ_row
|
Mathlib/LinearAlgebra/Matrix/Determinant/Basic.lean
|
theorem det_succ_row {n : ℕ} (A : Matrix (Fin n.succ) (Fin n.succ) R) (i : Fin n.succ) :
det A =
∑ j : Fin n.succ, (-1) ^ (i + j : ℕ) * A i j * det (A.submatrix i.succAbove j.succAbove)
|
case e_a
R : Type v
inst✝ : CommRing R
n : ℕ
A : Matrix (Fin n.succ) (Fin n.succ) R
i : Fin n.succ
this : A.det = (-1) ^ ↑i * ↑↑(sign i.cycleRange⁻¹) * A.det
⊢ ∑ j : Fin n.succ,
(-1) ^ ↑j * A.submatrix (⇑i.cycleRange⁻¹) id 0 j *
((A.submatrix (⇑i.cycleRange⁻¹) id).submatrix Fin.succ j.succAbove).det =
∑ i_1 : Fin n.succ, (-1) ^ ↑i_1 * (A i i_1 * (A.submatrix i.succAbove i_1.succAbove).det)
|
refine Finset.sum_congr rfl fun j _ => ?_
|
case e_a
R : Type v
inst✝ : CommRing R
n : ℕ
A : Matrix (Fin n.succ) (Fin n.succ) R
i : Fin n.succ
this : A.det = (-1) ^ ↑i * ↑↑(sign i.cycleRange⁻¹) * A.det
j : Fin n.succ
x✝ : j ∈ univ
⊢ (-1) ^ ↑j * A.submatrix (⇑i.cycleRange⁻¹) id 0 j *
((A.submatrix (⇑i.cycleRange⁻¹) id).submatrix Fin.succ j.succAbove).det =
(-1) ^ ↑j * (A i j * (A.submatrix i.succAbove j.succAbove).det)
|
d17ec63e8ba7211a
|
Subgroup.exists_finiteIndex_of_leftCoset_cover_aux
|
Mathlib/GroupTheory/CosetCover.lean
|
theorem exists_finiteIndex_of_leftCoset_cover_aux [DecidableEq (Subgroup G)]
(j : ι) (hj : j ∈ s) (hcovers' : ⋃ i ∈ s.filter (H · = H j), g i • (H i : Set G) ≠ Set.univ) :
∃ i ∈ s, H i ≠ H j ∧ (H i).FiniteIndex
|
case refine_2
G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq (Subgroup G)
n : ℕ
ih :
∀ m < n,
∀ {ι : Type u_2} {H : ι → Subgroup G} {g : ι → G} {s : Finset ι},
⋃ i ∈ s, g i • ↑(H i) = Set.univ →
∀ j ∈ s,
⋃ i ∈ Finset.filter (fun x => H x = H j) s, g i • ↑(H i) ≠ Set.univ →
m = (Finset.image H s).card → ∃ i ∈ s, H i ≠ H j ∧ (H i).FiniteIndex
ι : Type u_2
H : ι → Subgroup G
g : ι → G
s : Finset ι
hcovers : ⋃ i ∈ s, g i • ↑(H i) = Set.univ
j : ι
hj : j ∈ s
hcovers' : ⋃ i ∈ Finset.filter (fun x => H x = H j) s, g i • ↑(H i) ≠ Set.univ
hn : n = (Finset.image H s).card
x : G
hx : ∀ i ∈ s, H i = H j → ↑(g i) ≠ ↑x
y z : G
hz : z ∈ y • ↑(H j)
i : ι
hi : i ∈ s
hmem : ↑(g i) = ↑(x * (y⁻¹ * z))
⊢ z ∈ (y * x⁻¹ * g i) • ↑(H i)
|
simpa [mem_leftCoset_iff, SetLike.mem_coe, QuotientGroup.eq, mul_assoc] using hmem
|
no goals
|
d6e7c1f1fddf36c9
|
Matrix.zpow_add
|
Mathlib/LinearAlgebra/Matrix/ZPow.lean
|
theorem zpow_add {A : M} (ha : IsUnit A.det) (m n : ℤ) : A ^ (m + n) = A ^ m * A ^ n
|
case hp
n' : Type u_1
inst✝² : DecidableEq n'
inst✝¹ : Fintype n'
R : Type u_2
inst✝ : CommRing R
A : M
ha : IsUnit A.det
m : ℤ
n : ℕ
ihn : A ^ (m + ↑n) = A ^ m * A ^ ↑n
⊢ A ^ (m + (↑n + 1)) = A ^ m * A ^ (↑n + 1)
|
simp only [← add_assoc, zpow_add_one ha, ihn, mul_assoc]
|
no goals
|
825209b07812538f
|
Real.norm_deriv_mulExpNegMulSq_le_one
|
Mathlib/Analysis/SpecialFunctions/MulExpNegMulSq.lean
|
theorem norm_deriv_mulExpNegMulSq_le_one (hε : 0 < ε) (x : ℝ) :
‖deriv (mulExpNegMulSq ε) x‖ ≤ 1
|
ε : ℝ
hε : 0 < ε
x : ℝ
y : ℝ := ε * x * x
heq : rexp (-y) + x * (rexp (-y) * (-2 * ε * x)) = rexp (-y) * (1 - 2 * y)
hy : y = ε * x * x
hynonneg : 0 ≤ y
⊢ 2 * y ≤ 1 + rexp y ∧ 1 ≤ rexp y + 2 * y
|
refine ⟨le_trans two_mul_le_exp ((le_add_iff_nonneg_left (exp y)).mpr zero_le_one), ?_⟩
|
ε : ℝ
hε : 0 < ε
x : ℝ
y : ℝ := ε * x * x
heq : rexp (-y) + x * (rexp (-y) * (-2 * ε * x)) = rexp (-y) * (1 - 2 * y)
hy : y = ε * x * x
hynonneg : 0 ≤ y
⊢ 1 ≤ rexp y + 2 * y
|
fd284f88458aa015
|
CategoryTheory.RelCat.rel_iso_iff
|
Mathlib/CategoryTheory/Category/RelCat.lean
|
theorem rel_iso_iff {X Y : RelCat} (r : X ⟶ Y) :
IsIso (C := RelCat) r ↔ ∃ f : (Iso (C := Type u) X Y), graphFunctor.map f.hom = r
|
case mp
X Y : RelCat
r : X ⟶ Y
h : IsIso r
h1 : ∀ (a b : X), (r ≫ inv r) a b = 𝟙 X a b
h2 : ∀ (a b : Y), (inv r ≫ r) a b = 𝟙 Y a b
⊢ ∃ f, graphFunctor.map f.hom = r
|
simp only [RelCat.Hom.rel_comp_apply₂, RelCat.Hom.rel_id_apply₂, eq_iff_iff] at h1 h2
|
case mp
X Y : RelCat
r : X ⟶ Y
h : IsIso r
h1 : ∀ (a b : X), (∃ y, r a y ∧ inv r y b) ↔ a = b
h2 : ∀ (a b : Y), (∃ y, inv r a y ∧ r y b) ↔ a = b
⊢ ∃ f, graphFunctor.map f.hom = r
|
a2d7764e00cdb24d
|
monovary_inv_right₀
|
Mathlib/Algebra/Order/Monovary.lean
|
@[simp] lemma monovary_inv_right₀ (hg : StrongLT 0 g) : Monovary f g⁻¹ ↔ Antivary f g :=
forall_swap.trans <| forall₂_congr fun i j ↦ by simp [inv_lt_inv₀ (hg _) (hg _)]
|
ι : Type u_1
α : Type u_2
β : Type u_3
inst✝¹ : LinearOrderedSemifield α
inst✝ : LinearOrderedSemifield β
f : ι → α
g : ι → β
hg : StrongLT 0 g
i j : ι
⊢ g⁻¹ j < g⁻¹ i → f j ≤ f i ↔ g i < g j → f j ≤ f i
|
simp [inv_lt_inv₀ (hg _) (hg _)]
|
no goals
|
2f45565a54105e52
|
Nat.Partrec.Code.hG
|
Mathlib/Computability/PartrecCode.lean
|
theorem hG : Primrec G
|
case hpc
a : Primrec fun a => ofNat (ℕ × Code) a.length
k✝¹ : Primrec fun a => (ofNat (ℕ × Code) a.1.length).1
n✝¹ : Primrec Prod.snd
k✝ : Primrec fun a => (ofNat (ℕ × Code) a.1.1.length).1
n✝ : Primrec fun a => a.1.2
k' : Primrec Prod.snd
c : Primrec fun a => (ofNat (ℕ × Code) a.1.1.length).2
L : Primrec fun a => a.1.1.1
k : Primrec fun a => (ofNat (ℕ × Code) a.1.1.1.length).1
n : Primrec fun a => a.1.1.2
⊢ Primrec fun a =>
let z := (unpair a.1.1.2).1;
Nat.casesOn (unpair a.1.1.2).2 (Nat.Partrec.Code.lup a.1.1.1 ((ofNat (ℕ × Code) a.1.1.1.length).1, a.2.1) z)
fun y => do
let i ← Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (ℕ × Code) a.1.1.1.length).2) (Nat.pair z y)
Nat.Partrec.Code.lup a.1.1.1 ((ofNat (ℕ × Code) a.1.1.1.length).1, a.2.2.1) (Nat.pair z (Nat.pair y i))
|
have cf := Primrec.fst.comp (Primrec.snd (α := (List (List (Option ℕ)) × ℕ) × ℕ)
(β := Code × Code × Option ℕ × Option ℕ))
|
case hpc
a : Primrec fun a => ofNat (ℕ × Code) a.length
k✝¹ : Primrec fun a => (ofNat (ℕ × Code) a.1.length).1
n✝¹ : Primrec Prod.snd
k✝ : Primrec fun a => (ofNat (ℕ × Code) a.1.1.length).1
n✝ : Primrec fun a => a.1.2
k' : Primrec Prod.snd
c : Primrec fun a => (ofNat (ℕ × Code) a.1.1.length).2
L : Primrec fun a => a.1.1.1
k : Primrec fun a => (ofNat (ℕ × Code) a.1.1.1.length).1
n : Primrec fun a => a.1.1.2
cf : Primrec fun a => a.2.1
⊢ Primrec fun a =>
let z := (unpair a.1.1.2).1;
Nat.casesOn (unpair a.1.1.2).2 (Nat.Partrec.Code.lup a.1.1.1 ((ofNat (ℕ × Code) a.1.1.1.length).1, a.2.1) z)
fun y => do
let i ← Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (ℕ × Code) a.1.1.1.length).2) (Nat.pair z y)
Nat.Partrec.Code.lup a.1.1.1 ((ofNat (ℕ × Code) a.1.1.1.length).1, a.2.2.1) (Nat.pair z (Nat.pair y i))
|
4867ddf124efd4e0
|
MeasureTheory.Lp.simpleFunc.dense
|
Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean
|
theorem dense (hp_ne_top : p ≠ ∞) : Dense (Lp.simpleFunc E p μ : Set (Lp E p μ))
|
α : Type u_1
E : Type u_4
inst✝² : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
inst✝ : Fact (1 ≤ p)
hp_ne_top : p ≠ ⊤
⊢ Dense ↑(simpleFunc E p μ)
|
simpa only [denseRange_subtype_val] using simpleFunc.denseRange (E := E) (μ := μ) hp_ne_top
|
no goals
|
74254ecfcc7f02f0
|
QuadraticMap.polar_add
|
Mathlib/LinearAlgebra/QuadraticForm/Basic.lean
|
theorem polar_add (f g : M → N) (x y : M) : polar (f + g) x y = polar f x y + polar g x y
|
M : Type u_4
N : Type u_5
inst✝¹ : AddCommGroup M
inst✝ : AddCommGroup N
f g : M → N
x y : M
⊢ polar (f + g) x y = polar f x y + polar g x y
|
simp only [polar, Pi.add_apply]
|
M : Type u_4
N : Type u_5
inst✝¹ : AddCommGroup M
inst✝ : AddCommGroup N
f g : M → N
x y : M
⊢ f (x + y) + g (x + y) - (f x + g x) - (f y + g y) = f (x + y) - f x - f y + (g (x + y) - g x - g y)
|
bc9c98d787f29584
|
exists_lt_ack_of_nat_primrec
|
Mathlib/Computability/Ackermann.lean
|
theorem exists_lt_ack_of_nat_primrec {f : ℕ → ℕ} (hf : Nat.Primrec f) :
∃ m, ∀ n, f n < ack m n
|
case pair
f✝ f g : ℕ → ℕ
hf : Nat.Primrec f
hg : Nat.Primrec g
IHf : ∃ m, ∀ (n : ℕ), f n < ack m n
IHg : ∃ m, ∀ (n : ℕ), g n < ack m n
⊢ ∃ m, ∀ (n : ℕ), (fun n => pair (f n) (g n)) n < ack m n
case comp
f✝ f g : ℕ → ℕ
hf : Nat.Primrec f
hg : Nat.Primrec g
IHf : ∃ m, ∀ (n : ℕ), f n < ack m n
IHg : ∃ m, ∀ (n : ℕ), g n < ack m n
⊢ ∃ m, ∀ (n : ℕ), (fun n => f (g n)) n < ack m n
case prec
f✝ f g : ℕ → ℕ
hf : Nat.Primrec f
hg : Nat.Primrec g
IHf : ∃ m, ∀ (n : ℕ), f n < ack m n
IHg : ∃ m, ∀ (n : ℕ), g n < ack m n
⊢ ∃ m, ∀ (n : ℕ), unpaired (fun z n => rec (f z) (fun y IH => g (pair z (pair y IH))) n) n < ack m n
|
all_goals obtain ⟨a, ha⟩ := IHf; obtain ⟨b, hb⟩ := IHg
|
case pair.intro.intro
f✝ f g : ℕ → ℕ
hf : Nat.Primrec f
hg : Nat.Primrec g
a : ℕ
ha : ∀ (n : ℕ), f n < ack a n
b : ℕ
hb : ∀ (n : ℕ), g n < ack b n
⊢ ∃ m, ∀ (n : ℕ), (fun n => pair (f n) (g n)) n < ack m n
case comp.intro.intro
f✝ f g : ℕ → ℕ
hf : Nat.Primrec f
hg : Nat.Primrec g
a : ℕ
ha : ∀ (n : ℕ), f n < ack a n
b : ℕ
hb : ∀ (n : ℕ), g n < ack b n
⊢ ∃ m, ∀ (n : ℕ), (fun n => f (g n)) n < ack m n
case prec.intro.intro
f✝ f g : ℕ → ℕ
hf : Nat.Primrec f
hg : Nat.Primrec g
a : ℕ
ha : ∀ (n : ℕ), f n < ack a n
b : ℕ
hb : ∀ (n : ℕ), g n < ack b n
⊢ ∃ m, ∀ (n : ℕ), unpaired (fun z n => rec (f z) (fun y IH => g (pair z (pair y IH))) n) n < ack m n
|
e97338bb483aa942
|
Matrix.PosSemidef.inv
|
Mathlib/LinearAlgebra/Matrix/PosDef.lean
|
protected lemma inv [DecidableEq n] {M : Matrix n n R} (hM : M.PosSemidef) : M⁻¹.PosSemidef
|
case neg
n : Type u_2
R : Type u_3
inst✝⁴ : Fintype n
inst✝³ : CommRing R
inst✝² : PartialOrder R
inst✝¹ : StarRing R
inst✝ : DecidableEq n
M : Matrix n n R
hM : M.PosSemidef
h : ¬IsUnit M.det
⊢ PosSemidef 0
|
exact .zero
|
no goals
|
665a36634fb462d8
|
MeasureTheory.integral_llr_tilted_right
|
Mathlib/MeasureTheory/Measure/LogLikelihoodRatio.lean
|
lemma integral_llr_tilted_right [IsProbabilityMeasure μ] [SigmaFinite ν]
(hμν : μ ≪ ν) (hfμ : Integrable f μ) (hfν : Integrable (fun x ↦ exp (f x)) ν)
(h_int : Integrable (llr μ ν) μ) :
∫ x, llr μ (ν.tilted f) x ∂μ = ∫ x, llr μ ν x ∂μ - ∫ x, f x ∂μ + log (∫ x, exp (f x) ∂ν)
|
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
f : α → ℝ
inst✝¹ : IsProbabilityMeasure μ
inst✝ : SigmaFinite ν
hμν : μ ≪ ν
hfμ : Integrable f μ
hfν : Integrable (fun x => rexp (f x)) ν
h_int : Integrable (llr μ ν) μ
⊢ ∫ (a : α), -f a ∂μ + ∫ (a : α), log (∫ (x : α), rexp (f x) ∂ν) ∂μ + ∫ (a : α), llr μ ν a ∂μ =
∫ (a : α), -f a ∂μ + log (∫ (x : α), rexp (f x) ∂ν) + ∫ (x : α), llr μ ν x ∂μ
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
f : α → ℝ
inst✝¹ : IsProbabilityMeasure μ
inst✝ : SigmaFinite ν
hμν : μ ≪ ν
hfμ : Integrable f μ
hfν : Integrable (fun x => rexp (f x)) ν
h_int : Integrable (llr μ ν) μ
⊢ Integrable (fun a => -f a) μ
|
swap
|
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
f : α → ℝ
inst✝¹ : IsProbabilityMeasure μ
inst✝ : SigmaFinite ν
hμν : μ ≪ ν
hfμ : Integrable f μ
hfν : Integrable (fun x => rexp (f x)) ν
h_int : Integrable (llr μ ν) μ
⊢ Integrable (fun a => -f a) μ
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
f : α → ℝ
inst✝¹ : IsProbabilityMeasure μ
inst✝ : SigmaFinite ν
hμν : μ ≪ ν
hfμ : Integrable f μ
hfν : Integrable (fun x => rexp (f x)) ν
h_int : Integrable (llr μ ν) μ
⊢ ∫ (a : α), -f a ∂μ + ∫ (a : α), log (∫ (x : α), rexp (f x) ∂ν) ∂μ + ∫ (a : α), llr μ ν a ∂μ =
∫ (a : α), -f a ∂μ + log (∫ (x : α), rexp (f x) ∂ν) + ∫ (x : α), llr μ ν x ∂μ
|
618388524e9eb506
|
InnerProductGeometry.cos_eq_zero_iff_angle_eq_pi_div_two
|
Mathlib/Geometry/Euclidean/Angle/Unoriented/Basic.lean
|
theorem cos_eq_zero_iff_angle_eq_pi_div_two : cos (angle x y) = 0 ↔ angle x y = π / 2
|
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
⊢ π / 2 ∈ Icc 0 π
|
constructor <;> linarith [pi_pos]
|
no goals
|
5da138e792432e8b
|
Commute.isNilpotent_sum
|
Mathlib/RingTheory/Nilpotent/Basic.lean
|
protected lemma isNilpotent_sum {ι : Type*} {s : Finset ι} {f : ι → R}
(hnp : ∀ i ∈ s, IsNilpotent (f i)) (h_comm : ∀ i j, i ∈ s → j ∈ s → Commute (f i) (f j)) :
IsNilpotent (∑ i ∈ s, f i)
|
case insert.hx
R : Type u_1
inst✝ : Semiring R
ι : Type u_3
f : ι → R
j : ι
s : Finset ι
hj : j ∉ s
ih : (∀ i ∈ s, IsNilpotent (f i)) → (∀ (i j : ι), i ∈ s → j ∈ s → Commute (f i) (f j)) → IsNilpotent (∑ i ∈ s, f i)
hnp : ∀ i ∈ insert j s, IsNilpotent (f i)
h_comm : ∀ (i j_1 : ι), i ∈ insert j s → j_1 ∈ insert j s → Commute (f i) (f j_1)
⊢ IsNilpotent (f j)
|
apply hnp
|
case insert.hx.a
R : Type u_1
inst✝ : Semiring R
ι : Type u_3
f : ι → R
j : ι
s : Finset ι
hj : j ∉ s
ih : (∀ i ∈ s, IsNilpotent (f i)) → (∀ (i j : ι), i ∈ s → j ∈ s → Commute (f i) (f j)) → IsNilpotent (∑ i ∈ s, f i)
hnp : ∀ i ∈ insert j s, IsNilpotent (f i)
h_comm : ∀ (i j_1 : ι), i ∈ insert j s → j_1 ∈ insert j s → Commute (f i) (f j_1)
⊢ j ∈ insert j s
|
d22c464defd4e821
|
mem_closure_of_gauge_le_one
|
Mathlib/Analysis/Convex/Gauge.lean
|
theorem mem_closure_of_gauge_le_one (hc : Convex ℝ s) (hs₀ : 0 ∈ s) (ha : Absorbent ℝ s)
(h : gauge s x ≤ 1) : x ∈ closure s
|
E : Type u_2
inst✝³ : AddCommGroup E
inst✝² : Module ℝ E
s : Set E
x : E
inst✝¹ : TopologicalSpace E
inst✝ : ContinuousSMul ℝ E
hc : Convex ℝ s
hs₀ : 0 ∈ s
ha : Absorbent ℝ s
h : gauge s x ≤ 1
this : ∀ᶠ (r : ℝ) in 𝓝[<] 1, r • x ∈ s
⊢ Tendsto (fun x_1 => x_1 • x) (𝓝[<] 1) (𝓝 x)
|
exact Filter.Tendsto.mono_left (Continuous.tendsto' (by fun_prop) _ _ (one_smul _ _))
inf_le_left
|
no goals
|
1bb11cde2da67157
|
MeasureTheory.L2.inner_indicatorConstLp_eq_setIntegral_inner
|
Mathlib/MeasureTheory/Function/L2Space.lean
|
theorem inner_indicatorConstLp_eq_setIntegral_inner (f : Lp E 2 μ) (hs : MeasurableSet s) (c : E)
(hμs : μ s ≠ ∞) : (⟪indicatorConstLp 2 hs hμs c, f⟫ : 𝕜) = ∫ x in s, ⟪c, f x⟫ ∂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
s : Set α
f : ↥(Lp E 2 μ)
hs : MeasurableSet s
c : E
hμs : μ s ≠ ⊤
⊢ ∫ (x : α) in s, inner (↑↑(indicatorConstLp 2 hs hμs c) x) (↑↑f x) ∂μ +
∫ (x : α) in sᶜ, inner (↑↑(indicatorConstLp 2 hs hμs c) x) (↑↑f x) ∂μ =
∫ (x : α) in s, inner c (↑↑f x) ∂μ
|
have h_left : (∫ x in s, ⟪(indicatorConstLp 2 hs hμs c) x, f x⟫ ∂μ) = ∫ x in s, ⟪c, f x⟫ ∂μ := by
suffices h_ae_eq : ∀ᵐ x ∂μ, x ∈ s → ⟪indicatorConstLp 2 hs hμs c x, f x⟫ = ⟪c, f x⟫ from
setIntegral_congr_ae hs h_ae_eq
have h_indicator : ∀ᵐ x : α ∂μ, x ∈ s → indicatorConstLp 2 hs hμs c x = c :=
indicatorConstLp_coeFn_mem
refine h_indicator.mono fun x hx hxs => ?_
congr
exact hx hxs
|
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
s : Set α
f : ↥(Lp E 2 μ)
hs : MeasurableSet s
c : E
hμs : μ s ≠ ⊤
h_left : ∫ (x : α) in s, inner (↑↑(indicatorConstLp 2 hs hμs c) x) (↑↑f x) ∂μ = ∫ (x : α) in s, inner c (↑↑f x) ∂μ
⊢ ∫ (x : α) in s, inner (↑↑(indicatorConstLp 2 hs hμs c) x) (↑↑f x) ∂μ +
∫ (x : α) in sᶜ, inner (↑↑(indicatorConstLp 2 hs hμs c) x) (↑↑f x) ∂μ =
∫ (x : α) in s, inner c (↑↑f x) ∂μ
|
550df0062a1bcf16
|
SetTheory.PGame.insertRight_equiv_of_lf
|
Mathlib/SetTheory/Game/PGame.lean
|
/-- Adding a gift horse right option does not change the value of `x`. A gift horse right option is
a game `x'` with `x ⧏ x'`. It is called "gift horse" because it seems like Right has gotten the
"gift" of a new option, but actually the value of the game did not change. -/
lemma insertRight_equiv_of_lf {x x' : PGame} (h : x ⧏ x') : insertRight x x' ≈ x
|
x x' : PGame
h : x ⧏ x'
⊢ x.insertRight x' ≈ x
|
rw [← neg_equiv_neg_iff, ← neg_insertLeft_neg]
|
x x' : PGame
h : x ⧏ x'
⊢ (-x).insertLeft (-x') ≈ -x
|
50f28abf4680854a
|
AffineSubspace.coe_direction_eq_vsub_set_right
|
Mathlib/LinearAlgebra/AffineSpace/AffineSubspace/Defs.lean
|
theorem coe_direction_eq_vsub_set_right {s : AffineSubspace k P} {p : P} (hp : p ∈ s) :
(s.direction : Set V) = (· -ᵥ p) '' s
|
case refine_1
k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : Ring k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
s : AffineSubspace k P
p : P
hp : p ∈ s
⊢ ↑s -ᵥ ↑s ≤ (fun x => x -ᵥ p) '' ↑s
|
rintro v ⟨p₁, hp₁, p₂, hp₂, rfl⟩
|
case refine_1.intro.intro.intro.intro
k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : Ring k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
s : AffineSubspace k P
p : P
hp : p ∈ s
p₁ : P
hp₁ : p₁ ∈ ↑s
p₂ : P
hp₂ : p₂ ∈ ↑s
⊢ (fun x1 x2 => x1 -ᵥ x2) p₁ p₂ ∈ (fun x => x -ᵥ p) '' ↑s
|
d2d0dab6124cb47b
|
Nat.mem_factoredNumbers'
|
Mathlib/NumberTheory/SmoothNumbers.lean
|
/-- `m` is `s`-factored if and only if all prime divisors of `m` are in `s`. -/
lemma mem_factoredNumbers' {s : Finset ℕ} {m : ℕ} :
m ∈ factoredNumbers s ↔ ∀ p, p.Prime → p ∣ m → p ∈ s
|
case intro.intro
s : Finset ℕ
m p : ℕ
hp₁ : 1 + s.sup id ≤ p
hp₂ : Prime p
⊢ (m ≠ 0 ∧ ∀ p ≤ m, Prime p → p ∣ m → p ∈ s) ↔ ∀ (p : ℕ), Prime p → p ∣ m → p ∈ s
|
refine ⟨fun ⟨H₀, H₁⟩ ↦ fun p hp₁ hp₂ ↦ H₁ p (le_of_dvd (Nat.pos_of_ne_zero H₀) hp₂) hp₁ hp₂,
fun H ↦ ⟨fun h ↦ lt_irrefl p ?_, fun p _ ↦ H p⟩⟩
|
case intro.intro
s : Finset ℕ
m p : ℕ
hp₁ : 1 + s.sup id ≤ p
hp₂ : Prime p
H : ∀ (p : ℕ), Prime p → p ∣ m → p ∈ s
h : m = 0
⊢ p < p
|
6c2d6475c8785202
|
Real.aux₂
|
Mathlib/NumberTheory/DiophantineApproximation/Basic.lean
|
theorem aux₂ : 0 < u - ⌊ξ⌋ * v ∧ u - ⌊ξ⌋ * v < v
|
case intro.intro.intro.refine_2
ξ : ℝ
u v : ℤ
hv : 2 ≤ v
hcop : IsCoprime u v
left✝ : v = 1 → -(1 / 2) < ξ - ↑u
h : -1 + ξ * (↑v * (2 * ↑v - 1)) < ↑u * (2 * ↑v - 1) ∧ ↑u * (2 * ↑v - 1) < 1 + ξ * (↑v * (2 * ↑v - 1))
hv₀ : 0 < ↑v
hv₀' : 0 < 2 * ↑v - 1
hv₁ : 0 < 2 * v - 1
hu₀ : 0 ≤ u - ⌊ξ⌋ * v
hu₁ : u - ⌊ξ⌋ * v ≤ v
hf : u - ⌊ξ⌋ * v = v
huv_cop : v = 1 ∨ v = -1
⊢ False
|
rcases huv_cop with huv_cop | huv_cop <;> linarith only [hv, huv_cop]
|
no goals
|
b733bf52b4cb0f24
|
Subsemigroup.le_prod_iff
|
Mathlib/Algebra/Group/Subsemigroup/Operations.lean
|
theorem le_prod_iff {s : Subsemigroup M} {t : Subsemigroup N} {u : Subsemigroup (M × N)} :
u ≤ s.prod t ↔ u.map (fst M N) ≤ s ∧ u.map (snd M N) ≤ t
|
case mp
M : Type u_1
N : Type u_2
inst✝¹ : Mul M
inst✝ : Mul N
s : Subsemigroup M
t : Subsemigroup N
u : Subsemigroup (M × N)
h : u ≤ s.prod t
⊢ map (fst M N) u ≤ s ∧ map (snd M N) u ≤ t
|
constructor
|
case mp.left
M : Type u_1
N : Type u_2
inst✝¹ : Mul M
inst✝ : Mul N
s : Subsemigroup M
t : Subsemigroup N
u : Subsemigroup (M × N)
h : u ≤ s.prod t
⊢ map (fst M N) u ≤ s
case mp.right
M : Type u_1
N : Type u_2
inst✝¹ : Mul M
inst✝ : Mul N
s : Subsemigroup M
t : Subsemigroup N
u : Subsemigroup (M × N)
h : u ≤ s.prod t
⊢ map (snd M N) u ≤ t
|
b59429101fcfc181
|
Ideal.quotientInfToPiQuotient_surj
|
Mathlib/RingTheory/Ideal/Quotient/Operations.lean
|
lemma quotientInfToPiQuotient_surj {I : ι → Ideal R}
(hI : Pairwise (IsCoprime on I)) : Surjective (quotientInfToPiQuotient I)
|
case intro
R : Type u_2
inst✝¹ : CommRing R
ι : Type u_3
inst✝ : Finite ι
I : ι → Ideal R
hI : Pairwise (IsCoprime on I)
val✝ : Fintype ι
g : (i : ι) → R ⧸ I i
f : ι → R
hf : ∀ (i : ι), (Quotient.mk (I i)) (f i) = g i
e : ι → R
he : ∀ (i : ι), (Quotient.mk (I i)) (e i) = 1 ∧ ∀ (j : ι), j ≠ i → (Quotient.mk (I j)) (e i) = 0
⊢ ∃ a, (quotientInfToPiQuotient I) a = g
|
use mk _ (∑ i, f i*e i)
|
case h
R : Type u_2
inst✝¹ : CommRing R
ι : Type u_3
inst✝ : Finite ι
I : ι → Ideal R
hI : Pairwise (IsCoprime on I)
val✝ : Fintype ι
g : (i : ι) → R ⧸ I i
f : ι → R
hf : ∀ (i : ι), (Quotient.mk (I i)) (f i) = g i
e : ι → R
he : ∀ (i : ι), (Quotient.mk (I i)) (e i) = 1 ∧ ∀ (j : ι), j ≠ i → (Quotient.mk (I j)) (e i) = 0
⊢ (quotientInfToPiQuotient I) ((Quotient.mk (⨅ i, I i)) (∑ i : ι, f i * e i)) = g
|
df184cf9a63521b9
|
MulOpposite.unop_list_prod
|
Mathlib/Algebra/BigOperators/Group/List/Lemmas.lean
|
lemma unop_list_prod (l : List Mᵐᵒᵖ) : l.prod.unop = (l.map unop).reverse.prod
|
M : Type u_4
inst✝ : Monoid M
l : List Mᵐᵒᵖ
⊢ unop l.prod = (map unop l).reverse.prod
|
rw [← op_inj, op_unop, MulOpposite.op_list_prod, map_reverse, map_map, reverse_reverse,
op_comp_unop, map_id]
|
no goals
|
f143ab63d179e213
|
Measurable.ennreal_tsum'
|
Mathlib/MeasureTheory/Constructions/BorelSpace/Real.lean
|
theorem Measurable.ennreal_tsum' {ι} [Countable ι] {f : ι → α → ℝ≥0∞} (h : ∀ i, Measurable (f i)) :
Measurable (∑' i, f i)
|
case h.e'_5.h
α : Type u_1
mα : MeasurableSpace α
ι : Type u_5
inst✝ : Countable ι
f : ι → α → ℝ≥0∞
h : ∀ (i : ι), Measurable (f i)
x : α
⊢ (∑' (i : ι), f i) x = ∑' (i : ι), f i x
|
exact tsum_apply (Pi.summable.2 fun _ => ENNReal.summable)
|
no goals
|
d25ac079611c84ba
|
MeasureTheory.BoundedContinuousFunction.integral_le_of_levyProkhorovEDist_lt
|
Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.lean
|
/-- Assuming `levyProkhorovEDist μ ν < ε`, we can bound `∫ f ∂μ` in terms of
`∫ t in (0, ‖f‖], ν (thickening ε {x | f(x) ≥ t}) dt` and `‖f‖`. -/
lemma BoundedContinuousFunction.integral_le_of_levyProkhorovEDist_lt (μ ν : Measure Ω)
[IsFiniteMeasure μ] [IsFiniteMeasure ν] {ε : ℝ} (ε_pos : 0 < ε)
(hμν : levyProkhorovEDist μ ν < ENNReal.ofReal ε) (f : Ω →ᵇ ℝ) (f_nn : 0 ≤ᵐ[μ] f) :
∫ ω, f ω ∂μ
≤ (∫ t in Ioc 0 ‖f‖, ENNReal.toReal (ν (thickening ε {a | t ≤ f a}))) + ε * ‖f‖
|
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : PseudoMetricSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ ν : Measure Ω
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
ε : ℝ
ε_pos : 0 < ε
hμν : levyProkhorovEDist μ ν < ENNReal.ofReal ε
f : Ω →ᵇ ℝ
f_nn : 0 ≤ᶠ[ae μ] ⇑f
key : (fun t => (μ {a | t ≤ f a}).toReal) ≤ fun t => (ν (thickening ε {a | t ≤ f a})).toReal + ε
intble₁ : IntegrableOn (fun t => (μ {a | t ≤ f a}).toReal) (Ioc 0 ‖f‖) volume
t : ℝ
⊢ (ν (thickening ε {a | t ≤ f a})).toReal ≤ (ν univ).toReal
|
exact ENNReal.toReal_mono (measure_ne_top _ _) <| measure_mono (subset_univ _)
|
no goals
|
abad1a50e7cd6063
|
FormalMultilinearSeries.apply_eq_prod_smul_coeff
|
Mathlib/Analysis/Calculus/FormalMultilinearSeries.lean
|
theorem apply_eq_prod_smul_coeff : p n y = (∏ i, y i) • p.coeff n
|
𝕜 : Type u
E : Type v
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
p : FormalMultilinearSeries 𝕜 𝕜 E
n : ℕ
y : Fin n → 𝕜
⊢ (p n) y = (∏ i : Fin n, y i) • p.coeff n
|
convert (p n).toMultilinearMap.map_smul_univ y 1
|
case h.e'_2.h.e'_1.h
𝕜 : Type u
E : Type v
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
p : FormalMultilinearSeries 𝕜 𝕜 E
n : ℕ
y : Fin n → 𝕜
x✝ : Fin n
⊢ y x✝ = y x✝ • 1 x✝
|
f6c046f6e19a80f3
|
isLowerSet_iff_Iio_subset
|
Mathlib/Order/UpperLower/Basic.lean
|
theorem isLowerSet_iff_Iio_subset : IsLowerSet s ↔ ∀ ⦃a⦄, a ∈ s → Iio a ⊆ s
|
α : Type u_1
inst✝ : PartialOrder α
s : Set α
⊢ IsLowerSet s ↔ ∀ ⦃a : α⦄, a ∈ s → Iio a ⊆ s
|
simp [isLowerSet_iff_forall_lt, subset_def, @forall_swap (_ ∈ s)]
|
no goals
|
49a750b788a9af3f
|
Complex.analyticAt_iff_eventually_differentiableAt
|
Mathlib/Analysis/Complex/CauchyIntegral.lean
|
theorem analyticAt_iff_eventually_differentiableAt {f : ℂ → E} {c : ℂ} :
AnalyticAt ℂ f c ↔ ∀ᶠ z in 𝓝 c, DifferentiableAt ℂ f z
|
case h
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
fa : AnalyticAt ℂ f c
⊢ ∀ (a : ℂ), AnalyticAt ℂ f a → DifferentiableAt ℂ f a
|
apply AnalyticAt.differentiableAt
|
no goals
|
6373504a7ca915e7
|
bernoulli'PowerSeries_mul_exp_sub_one
|
Mathlib/NumberTheory/Bernoulli.lean
|
theorem bernoulli'PowerSeries_mul_exp_sub_one :
bernoulli'PowerSeries A * (exp A - 1) = X * exp A
|
case h.e'_2
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n : ℕ
⊢ ∑ i ∈ antidiagonal n, bernoulli' i.1 / ↑i.1! * ((↑i.2 + 1) * ↑i.2!)⁻¹ * ↑n ! =
∑ k ∈ antidiagonal n, ↑((k.1 + k.2).choose k.2) / (↑k.2 + 1) * bernoulli' k.1
|
apply sum_congr rfl
|
case h.e'_2
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n : ℕ
⊢ ∀ x ∈ antidiagonal n,
bernoulli' x.1 / ↑x.1! * ((↑x.2 + 1) * ↑x.2!)⁻¹ * ↑n ! = ↑((x.1 + x.2).choose x.2) / (↑x.2 + 1) * bernoulli' x.1
|
42e6710e61d9a85a
|
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.reduce_postcondition
|
Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RupAddSound.lean
|
theorem reduce_postcondition {n : Nat} (c : DefaultClause n) (assignment : Array Assignment) :
(reduce c assignment = reducedToEmpty → Incompatible (PosFin n) c assignment) ∧
(∀ l : Literal (PosFin n), reduce c assignment = reducedToUnit l → ∀ (p : (PosFin n) → Bool), p ⊨ assignment → p ⊨ c → p ⊨ l)
|
n : Nat
c : DefaultClause n
assignment : Array Assignment
c_arr : Array (Literal (PosFin n)) := List.toArray c.clause
c_clause_rw : c.clause = c_arr.toList
motive : Nat → ReduceResult (PosFin n) → Prop := ReducePostconditionInductionMotive c_arr assignment
⊢ ∀ (a : PosFin n) (b : Bool), (reducedToEmpty = reducedToUnit (a, b)) = False
|
intros
|
n : Nat
c : DefaultClause n
assignment : Array Assignment
c_arr : Array (Literal (PosFin n)) := List.toArray c.clause
c_clause_rw : c.clause = c_arr.toList
motive : Nat → ReduceResult (PosFin n) → Prop := ReducePostconditionInductionMotive c_arr assignment
a✝ : PosFin n
b✝ : Bool
⊢ (reducedToEmpty = reducedToUnit (a✝, b✝)) = False
|
bc01533c0ab5cdb8
|
IsDenseInducing.extend_Z_bilin_aux
|
Mathlib/Topology/Algebra/UniformGroup/Defs.lean
|
theorem extend_Z_bilin_aux (x₀ : α) (y₁ : δ) : ∃ U₂ ∈ comap e (𝓝 x₀), ∀ x ∈ U₂, ∀ x' ∈ U₂,
(fun p : β × δ => φ p.1 p.2) (x' - x, y₁) ∈ W'
|
α : Type u_1
β : Type u_2
δ : Type u_4
G : Type u_5
inst✝⁸ : TopologicalSpace α
inst✝⁷ : AddCommGroup α
inst✝⁶ : IsTopologicalAddGroup α
inst✝⁵ : TopologicalSpace β
inst✝⁴ : AddCommGroup β
inst✝³ : TopologicalSpace δ
inst✝² : AddCommGroup δ
inst✝¹ : UniformSpace G
inst✝ : AddCommGroup G
e : β →+ α
de : IsDenseInducing ⇑e
φ : β →+ δ →+ G
hφ : Continuous fun p => (φ p.1) p.2
W' : Set G
W'_nhd : W' ∈ 𝓝 0
x₀ : α
y₁ : δ
⊢ ∃ U₂ ∈ comap (⇑e) (𝓝 x₀), ∀ x ∈ U₂, ∀ x' ∈ U₂, (fun p => (φ p.1) p.2) (x' - x, y₁) ∈ W'
|
let Nx := 𝓝 x₀
|
α : Type u_1
β : Type u_2
δ : Type u_4
G : Type u_5
inst✝⁸ : TopologicalSpace α
inst✝⁷ : AddCommGroup α
inst✝⁶ : IsTopologicalAddGroup α
inst✝⁵ : TopologicalSpace β
inst✝⁴ : AddCommGroup β
inst✝³ : TopologicalSpace δ
inst✝² : AddCommGroup δ
inst✝¹ : UniformSpace G
inst✝ : AddCommGroup G
e : β →+ α
de : IsDenseInducing ⇑e
φ : β →+ δ →+ G
hφ : Continuous fun p => (φ p.1) p.2
W' : Set G
W'_nhd : W' ∈ 𝓝 0
x₀ : α
y₁ : δ
Nx : Filter α := 𝓝 x₀
⊢ ∃ U₂ ∈ comap (⇑e) (𝓝 x₀), ∀ x ∈ U₂, ∀ x' ∈ U₂, (fun p => (φ p.1) p.2) (x' - x, y₁) ∈ W'
|
6766ff934c6fdbf1
|
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