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
|
|---|---|---|---|---|---|---|
coprime_sq_sub_sq_add_of_odd_even
|
Mathlib/NumberTheory/PythagoreanTriples.lean
|
theorem coprime_sq_sub_sq_add_of_odd_even {m n : ℤ} (h : Int.gcd m n = 1) (hm : m % 2 = 1)
(hn : n % 2 = 0) : Int.gcd (m ^ 2 - n ^ 2) (m ^ 2 + n ^ 2) = 1
|
m n : ℤ
h : m.gcd n = 1
hm : m % 2 = 1
hn : n % 2 = 0
⊢ n.gcd m = 1
|
rwa [Int.gcd_comm]
|
no goals
|
c291954e54251a00
|
Module.Flat.trans
|
Mathlib/RingTheory/Flat/Stability.lean
|
theorem trans [Flat R S] [Flat S M] : Flat R M
|
case hf
R : Type u
S : Type v
M : Type w
inst✝¹⁰ : CommSemiring R
inst✝⁹ : CommSemiring S
inst✝⁸ : Algebra R S
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Module R M
inst✝⁵ : Module S M
inst✝⁴ : IsScalarTower R S M
inst✝³ : Flat R S
inst✝² : Flat S M
P : Type u
inst✝¹ : AddCommMonoid P
inst✝ : Module R P
N : Submodule R P
⊢ Function.Injective ⇑((AlgebraTensorModule.lTensor S S) N.subtype)
|
apply Flat.lTensor_preserves_injective_linearMap
|
case hf.hf
R : Type u
S : Type v
M : Type w
inst✝¹⁰ : CommSemiring R
inst✝⁹ : CommSemiring S
inst✝⁸ : Algebra R S
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Module R M
inst✝⁵ : Module S M
inst✝⁴ : IsScalarTower R S M
inst✝³ : Flat R S
inst✝² : Flat S M
P : Type u
inst✝¹ : AddCommMonoid P
inst✝ : Module R P
N : Submodule R P
⊢ Function.Injective ⇑N.subtype
|
3a1589fe28d19c18
|
SignType.map_cast
|
Mathlib/Data/Sign.lean
|
/-- Casting out of `SignType` respects composition with suitable bundled homomorphism types. -/
lemma map_cast {α β F : Type*} [AddGroupWithOne α] [One β] [SubtractionMonoid β]
[FunLike F α β] [AddMonoidHomClass F α β] [OneHomClass F α β] (f : F) (s : SignType) :
f s = s
|
α : Type u_2
β : Type u_3
F : Type u_4
inst✝⁵ : AddGroupWithOne α
inst✝⁴ : One β
inst✝³ : SubtractionMonoid β
inst✝² : FunLike F α β
inst✝¹ : AddMonoidHomClass F α β
inst✝ : OneHomClass F α β
f : F
s : SignType
⊢ f ↑s = ↑s
|
apply map_cast' <;> simp
|
no goals
|
1d59dc3f39335785
|
SimplexCategory.Truncated.morphismProperty_eq_top
|
Mathlib/AlgebraicTopology/SimplexCategory/MorphismProperty.lean
|
lemma Truncated.morphismProperty_eq_top
{d : ℕ} (W : MorphismProperty (Truncated d)) [W.IsMultiplicative]
(δ_mem : ∀ (n : ℕ) (hn : n < d) (i : Fin (n + 2)),
W (SimplexCategory.δ (n := n) i : ⟨.mk n, by dsimp; omega⟩ ⟶
⟨.mk (n + 1), by dsimp; omega⟩))
(σ_mem : ∀ (n : ℕ) (hn : n < d) (i : Fin (n + 1)),
W (SimplexCategory.σ (n := n) i : ⟨.mk (n + 1), by dsimp; omega⟩ ⟶
⟨.mk n, by dsimp; omega⟩)) :
W = ⊤
|
case pos
d : ℕ
W : MorphismProperty (Truncated d)
inst✝ : W.IsMultiplicative
δ_mem : ∀ (n : ℕ) (hn : n < d) (i : Fin (n + 2)), W (δ i)
σ_mem : ∀ (n : ℕ) (hn : n < d) (i : Fin (n + 1)), W (σ i)
c : ℕ
hc :
∀ (a : ℕ) (ha : a ≤ d) (b : ℕ) (hb : b ≤ d) (f : { obj := mk a, property := ha } ⟶ { obj := mk b, property := hb }),
a + b = c → W f
a : ℕ
ha : a ≤ d
b : ℕ
hb : b ≤ d
f : { obj := mk a, property := ha } ⟶ { obj := mk b, property := hb }
h : a + b = c + 1
f' : mk a ⟶ mk b := f
h₁ : Function.Surjective ⇑(Hom.toOrderHom f')
h₂ : Function.Injective ⇑(Hom.toOrderHom f')
⊢ W f
case neg
d : ℕ
W : MorphismProperty (Truncated d)
inst✝ : W.IsMultiplicative
δ_mem : ∀ (n : ℕ) (hn : n < d) (i : Fin (n + 2)), W (δ i)
σ_mem : ∀ (n : ℕ) (hn : n < d) (i : Fin (n + 1)), W (σ i)
c : ℕ
hc :
∀ (a : ℕ) (ha : a ≤ d) (b : ℕ) (hb : b ≤ d) (f : { obj := mk a, property := ha } ⟶ { obj := mk b, property := hb }),
a + b = c → W f
a : ℕ
ha : a ≤ d
b : ℕ
hb : b ≤ d
f : { obj := mk a, property := ha } ⟶ { obj := mk b, property := hb }
h : a + b = c + 1
f' : mk a ⟶ mk b := f
h₁ : Function.Surjective ⇑(Hom.toOrderHom f')
h₂ : ¬Function.Injective ⇑(Hom.toOrderHom f')
⊢ W f
|
swap
|
case neg
d : ℕ
W : MorphismProperty (Truncated d)
inst✝ : W.IsMultiplicative
δ_mem : ∀ (n : ℕ) (hn : n < d) (i : Fin (n + 2)), W (δ i)
σ_mem : ∀ (n : ℕ) (hn : n < d) (i : Fin (n + 1)), W (σ i)
c : ℕ
hc :
∀ (a : ℕ) (ha : a ≤ d) (b : ℕ) (hb : b ≤ d) (f : { obj := mk a, property := ha } ⟶ { obj := mk b, property := hb }),
a + b = c → W f
a : ℕ
ha : a ≤ d
b : ℕ
hb : b ≤ d
f : { obj := mk a, property := ha } ⟶ { obj := mk b, property := hb }
h : a + b = c + 1
f' : mk a ⟶ mk b := f
h₁ : Function.Surjective ⇑(Hom.toOrderHom f')
h₂ : ¬Function.Injective ⇑(Hom.toOrderHom f')
⊢ W f
case pos
d : ℕ
W : MorphismProperty (Truncated d)
inst✝ : W.IsMultiplicative
δ_mem : ∀ (n : ℕ) (hn : n < d) (i : Fin (n + 2)), W (δ i)
σ_mem : ∀ (n : ℕ) (hn : n < d) (i : Fin (n + 1)), W (σ i)
c : ℕ
hc :
∀ (a : ℕ) (ha : a ≤ d) (b : ℕ) (hb : b ≤ d) (f : { obj := mk a, property := ha } ⟶ { obj := mk b, property := hb }),
a + b = c → W f
a : ℕ
ha : a ≤ d
b : ℕ
hb : b ≤ d
f : { obj := mk a, property := ha } ⟶ { obj := mk b, property := hb }
h : a + b = c + 1
f' : mk a ⟶ mk b := f
h₁ : Function.Surjective ⇑(Hom.toOrderHom f')
h₂ : Function.Injective ⇑(Hom.toOrderHom f')
⊢ W f
|
ad7c40b6f9a9b699
|
List.takeWhile_replicate
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/TakeDrop.lean
|
theorem takeWhile_replicate (p : α → Bool) :
(replicate n a).takeWhile p = if p a then replicate n a else []
|
α : Type u_1
n : Nat
a : α
p : α → Bool
⊢ takeWhile p (replicate n a) = if p a = true then replicate n a else []
|
rw [takeWhile_replicate_eq_filter, filter_replicate]
|
no goals
|
5fd7fa1468541db7
|
CategoryTheory.Limits.Types.isColimit_iff_bijective_desc
|
Mathlib/CategoryTheory/Limits/Types.lean
|
theorem isColimit_iff_bijective_desc : Nonempty (IsColimit c) ↔ (Quot.desc c).Bijective
|
case refine_1
J : Type v
inst✝ : Category.{w, v} J
F : J ⥤ Type u
c : Cocone F
⊢ Nonempty (IsColimit c) → Function.Bijective (Quot.desc c)
|
refine fun ⟨hc⟩ => ⟨fun x y h => ?_, fun x => ?_⟩
|
case refine_1.refine_1
J : Type v
inst✝ : Category.{w, v} J
F : J ⥤ Type u
c : Cocone F
x✝ : Nonempty (IsColimit c)
hc : IsColimit c
x y : Quot F
h : Quot.desc c x = Quot.desc c y
⊢ x = y
case refine_1.refine_2
J : Type v
inst✝ : Category.{w, v} J
F : J ⥤ Type u
c : Cocone F
x✝ : Nonempty (IsColimit c)
hc : IsColimit c
x : c.pt
⊢ ∃ a, Quot.desc c a = x
|
53580cf8744b2af5
|
OrderEmbedding.minimal_apply_mem_inter_range_iff
|
Mathlib/Order/Minimal.lean
|
theorem minimal_apply_mem_inter_range_iff :
Minimal (· ∈ t ∩ range f) (f x) ↔ Minimal (fun x ↦ f x ∈ t) x
|
case refine_2
α : Type u_1
x : α
inst✝¹ : Preorder α
β : Type u_2
inst✝ : Preorder β
f : α ↪o β
t : Set β
h : Minimal (fun x => f x ∈ t) x
⊢ ∀ ⦃y : β⦄, (fun x => x ∈ t ∩ range ⇑f) y → y ≤ f x → f x ≤ y
|
rintro _ ⟨hyt, ⟨y, rfl⟩⟩
|
case refine_2.intro.intro
α : Type u_1
x : α
inst✝¹ : Preorder α
β : Type u_2
inst✝ : Preorder β
f : α ↪o β
t : Set β
h : Minimal (fun x => f x ∈ t) x
y : α
hyt : f y ∈ t
⊢ f y ≤ f x → f x ≤ f y
|
f37f6f9881265078
|
MeasureTheory.FiniteMeasure.tendsto_testAgainstNN_of_tendsto_normalize_testAgainstNN_of_tendsto_mass
|
Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean
|
theorem tendsto_testAgainstNN_of_tendsto_normalize_testAgainstNN_of_tendsto_mass {γ : Type*}
{F : Filter γ} {μs : γ → FiniteMeasure Ω}
(μs_lim : Tendsto (fun i ↦ (μs i).normalize) F (𝓝 μ.normalize))
(mass_lim : Tendsto (fun i ↦ (μs i).mass) F (𝓝 μ.mass)) (f : Ω →ᵇ ℝ≥0) :
Tendsto (fun i ↦ (μs i).testAgainstNN f) F (𝓝 (μ.testAgainstNN f))
|
case pos
Ω : Type u_1
inst✝² : Nonempty Ω
m0 : MeasurableSpace Ω
μ : FiniteMeasure Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
γ : Type u_2
F : Filter γ
μs : γ → FiniteMeasure Ω
μs_lim : Tendsto (fun i => (μs i).normalize) F (𝓝 μ.normalize)
f : Ω →ᵇ ℝ≥0
h_mass : μ.mass = 0
mass_lim : Tendsto (fun i => (μs i).mass) F (𝓝 0)
⊢ Tendsto (fun i => (μs i).testAgainstNN f) F (𝓝 0)
|
exact tendsto_zero_testAgainstNN_of_tendsto_zero_mass mass_lim f
|
no goals
|
9f1d40834d04327a
|
Complex.Gamma_eq_zero_iff
|
Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean
|
theorem Gamma_eq_zero_iff (s : ℂ) : Gamma s = 0 ↔ ∃ m : ℕ, s = -m
|
case mpr.intro
m : ℕ
⊢ Gamma (-↑m) = 0
|
exact Gamma_neg_nat_eq_zero m
|
no goals
|
9897780b3450ce6d
|
Configuration.HasLines.pointCount_le_lineCount
|
Mathlib/Combinatorics/Configuration.lean
|
theorem HasLines.pointCount_le_lineCount [HasLines P L] {p : P} {l : L} (h : p ∉ l)
[Finite { l : L // p ∈ l }] : pointCount P l ≤ lineCount L p
|
case pos
P : Type u_1
L : Type u_2
inst✝² : Membership P L
inst✝¹ : HasLines P L
p : P
l : L
h : p ∉ l
inst✝ : Finite { l // p ∈ l }
hf : Infinite { p // p ∈ l }
⊢ pointCount P l ≤ lineCount L p
|
exact (le_of_eq Nat.card_eq_zero_of_infinite).trans (zero_le (lineCount L p))
|
no goals
|
b4269a73233e853a
|
CompactT2.ExtremallyDisconnected.projective
|
Mathlib/Topology/ExtremallyDisconnected.lean
|
theorem CompactT2.ExtremallyDisconnected.projective [ExtremallyDisconnected A]
[CompactSpace A] [T2Space A] : CompactT2.Projective A
|
case intro.intro.intro
A : Type u
inst✝⁹ : TopologicalSpace A
inst✝⁸ : ExtremallyDisconnected A
inst✝⁷ : CompactSpace A
inst✝⁶ : T2Space A
B C : Type u
inst✝⁵ : TopologicalSpace B
inst✝⁴ : TopologicalSpace C
inst✝³ : CompactSpace B
inst✝² : T2Space B
inst✝¹ : CompactSpace C
inst✝ : T2Space C
φ : A → C
f : B → C
φ_cont : Continuous φ
f_cont : Continuous f
f_surj : Surjective f
D : Set (A × B) := {x | φ x.1 = f x.2}
D_comp : CompactSpace ↑D
π₁ : ↑D → A := Prod.fst ∘ Subtype.val
π₁_cont : Continuous π₁
π₁_surj : Surjective π₁
E : Set ↑D
left✝ : CompactSpace ↑E
E_onto : π₁ '' E = univ
E_min : ∀ (E₀ : Set ↑E), E₀ ≠ univ → IsClosed E₀ → E.restrict π₁ '' E₀ ≠ univ
ρ : ↑E → A := E.restrict π₁
ρ_cont : Continuous ρ
ρ_surj : Surjective ρ
ρ' : ↑E ≃ₜ A := ExtremallyDisconnected.homeoCompactToT2 ρ_cont ρ_surj E_min
π₂ : ↑D → B := Prod.snd ∘ Subtype.val
π₂_cont : Continuous π₂
⊢ f ∘ E.restrict π₂ ∘ ⇑ρ'.symm = φ
|
suffices f ∘ E.restrict π₂ = φ ∘ ρ' by
rw [← comp_assoc, this, comp_assoc, Homeomorph.self_comp_symm, comp_id]
|
case intro.intro.intro
A : Type u
inst✝⁹ : TopologicalSpace A
inst✝⁸ : ExtremallyDisconnected A
inst✝⁷ : CompactSpace A
inst✝⁶ : T2Space A
B C : Type u
inst✝⁵ : TopologicalSpace B
inst✝⁴ : TopologicalSpace C
inst✝³ : CompactSpace B
inst✝² : T2Space B
inst✝¹ : CompactSpace C
inst✝ : T2Space C
φ : A → C
f : B → C
φ_cont : Continuous φ
f_cont : Continuous f
f_surj : Surjective f
D : Set (A × B) := {x | φ x.1 = f x.2}
D_comp : CompactSpace ↑D
π₁ : ↑D → A := Prod.fst ∘ Subtype.val
π₁_cont : Continuous π₁
π₁_surj : Surjective π₁
E : Set ↑D
left✝ : CompactSpace ↑E
E_onto : π₁ '' E = univ
E_min : ∀ (E₀ : Set ↑E), E₀ ≠ univ → IsClosed E₀ → E.restrict π₁ '' E₀ ≠ univ
ρ : ↑E → A := E.restrict π₁
ρ_cont : Continuous ρ
ρ_surj : Surjective ρ
ρ' : ↑E ≃ₜ A := ExtremallyDisconnected.homeoCompactToT2 ρ_cont ρ_surj E_min
π₂ : ↑D → B := Prod.snd ∘ Subtype.val
π₂_cont : Continuous π₂
⊢ f ∘ E.restrict π₂ = φ ∘ ⇑ρ'
|
b8a9d37b6d16fdce
|
nhds_translation_mul_inv₀
|
Mathlib/Topology/Algebra/GroupWithZero.lean
|
theorem nhds_translation_mul_inv₀ (ha : a ≠ 0) : comap (· * a⁻¹) (𝓝 1) = 𝓝 a :=
((Homeomorph.mulRight₀ a ha).symm.comap_nhds_eq 1).trans <| by simp
|
G₀ : Type u_3
inst✝² : TopologicalSpace G₀
inst✝¹ : GroupWithZero G₀
inst✝ : ContinuousMul G₀
a : G₀
ha : a ≠ 0
⊢ 𝓝 ((Homeomorph.mulRight₀ a ha).symm.symm 1) = 𝓝 a
|
simp
|
no goals
|
52157c1cb0ba3371
|
EuclideanGeometry.cos_oangle_left_of_oangle_eq_pi_div_two
|
Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean
|
theorem cos_oangle_left_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ = ↑(π / 2)) :
Real.Angle.cos (∡ p₃ p₁ p₂) = dist p₁ p₂ / dist p₁ p₃
|
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
hd2 : Fact (finrank ℝ V = 2)
inst✝ : Oriented ℝ V (Fin 2)
p₁ p₂ p₃ : P
h : ∡ p₁ p₂ p₃ = ↑(π / 2)
hs : (∡ p₃ p₁ p₂).sign = 1
⊢ (∡ p₃ p₁ p₂).cos = dist p₁ p₂ / dist p₁ p₃
|
rw [oangle_eq_angle_of_sign_eq_one hs, angle_comm, Real.Angle.cos_coe,
cos_angle_of_angle_eq_pi_div_two (angle_rev_eq_pi_div_two_of_oangle_eq_pi_div_two h),
dist_comm p₁ p₃]
|
no goals
|
ba5dea16c292725b
|
Std.DHashMap.Raw.Const.size_insertMany_list
|
Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/RawLemmas.lean
|
theorem size_insertMany_list [EquivBEq α] [LawfulHashable α] (h : m.WF)
{l : List (α × β)}
(distinct : l.Pairwise (fun a b => (a.1 == b.1) = false)) :
(∀ (a : α), a ∈ m → (l.map Prod.fst).contains a = false) →
(insertMany m l).size = m.size + l.length
|
α : Type u
inst✝³ : BEq α
inst✝² : Hashable α
β : Type v
m : Raw α fun x => β
inst✝¹ : EquivBEq α
inst✝ : LawfulHashable α
h : m.WF
l : List (α × β)
distinct : List.Pairwise (fun a b => (a.fst == b.fst) = false) l
⊢ (∀ (a : α), a ∈ m → (List.map Prod.fst l).contains a = false) → (insertMany m l).size = m.size + l.length
|
simp [mem_iff_contains]
|
α : Type u
inst✝³ : BEq α
inst✝² : Hashable α
β : Type v
m : Raw α fun x => β
inst✝¹ : EquivBEq α
inst✝ : LawfulHashable α
h : m.WF
l : List (α × β)
distinct : List.Pairwise (fun a b => (a.fst == b.fst) = false) l
⊢ (∀ (a : α), m.contains a = true → (List.map Prod.fst l).contains a = false) →
(insertMany m l).size = m.size + l.length
|
a12d055fe38ab205
|
Set.mul_mem_center
|
Mathlib/Algebra/Group/Center.lean
|
theorem mul_mem_center {z₁ z₂ : M} (hz₁ : z₁ ∈ Set.center M) (hz₂ : z₂ ∈ Set.center M) :
z₁ * z₂ ∈ Set.center M where
comm a := calc
z₁ * z₂ * a = z₂ * z₁ * a
|
M : Type u_1
inst✝ : Mul M
z₁ z₂ : M
hz₁ : z₁ ∈ center M
hz₂ : z₂ ∈ center M
a : M
⊢ z₁ * z₂ * a = z₂ * z₁ * a
|
rw [hz₁.comm]
|
no goals
|
234c184ae8bc3c45
|
IncidenceAlgebra.mu_toDual
|
Mathlib/Combinatorics/Enumerative/IncidenceAlgebra.lean
|
@[simp]
lemma mu_toDual (a b : α) : mu 𝕜 (toDual a) (toDual b) = mu 𝕜 b a
|
𝕜 : Type u_2
α : Type u_5
inst✝³ : Ring 𝕜
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
this : DecidableRel fun x1 x2 => x1 ≤ x2 := Classical.decRel fun x1 x2 => x1 ≤ x2
mud : IncidenceAlgebra 𝕜 αᵒᵈ := { toFun := fun a b => (mu 𝕜) (ofDual b) (ofDual a), eq_zero_of_not_le' := ⋯ }
a b : αᵒᵈ
a✝ : a ≤ b
⊢ (∑ x ∈ Icc a b, if x ≤ b then mud a x else 0) = ∑ x ∈ Icc a b, mud a x
|
congr! with x hx
|
case a
𝕜 : Type u_2
α : Type u_5
inst✝³ : Ring 𝕜
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
this : DecidableRel fun x1 x2 => x1 ≤ x2 := Classical.decRel fun x1 x2 => x1 ≤ x2
mud : IncidenceAlgebra 𝕜 αᵒᵈ := { toFun := fun a b => (mu 𝕜) (ofDual b) (ofDual a), eq_zero_of_not_le' := ⋯ }
a b : αᵒᵈ
a✝ : a ≤ b
x : αᵒᵈ
hx : x ∈ Icc a b
⊢ (if x ≤ b then mud a x else 0) = mud a x
|
95f4eed14862eb7a
|
MeasureTheory.IsSetSemiring.exists_disjoint_finset_diff_eq
|
Mathlib/MeasureTheory/SetSemiring.lean
|
/-- In a semiring of sets `C`, for all set `s ∈ C` and finite set of sets `I ⊆ C`, there is a
finite set of sets in `C` whose union is `s \ ⋃₀ I`.
See `IsSetSemiring.disjointOfDiffUnion` for a definition that gives such a set. -/
lemma exists_disjoint_finset_diff_eq (hC : IsSetSemiring C) (hs : s ∈ C) (hI : ↑I ⊆ C) :
∃ J : Finset (Set α), ↑J ⊆ C ∧ PairwiseDisjoint (J : Set (Set α)) id ∧
s \ ⋃₀ I = ⋃₀ J
|
α : Type u_1
C : Set (Set α)
s : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
hI : ↑I ⊆ C
⊢ ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I = ⋃₀ ↑J
|
induction I using Finset.induction with
| empty =>
simp only [coe_empty, sUnion_empty, diff_empty, exists_prop]
refine ⟨{s}, singleton_subset_set_iff.mpr hs, ?_⟩
simp only [coe_singleton, pairwiseDisjoint_singleton, sUnion_singleton, eq_self_iff_true,
and_self_iff]
| @insert t I' _ h => ?_
|
case insert
α : Type u_1
C : Set (Set α)
s : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I' = ⋃₀ ↑J
hI : ↑(insert t I') ⊆ C
⊢ ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑(insert t I') = ⋃₀ ↑J
|
0bd1b80571fe1ad6
|
ContinuousMap.exists_extension_forall_mem
|
Mathlib/Topology/TietzeExtension.lean
|
theorem ContinuousMap.exists_extension_forall_mem (he : IsClosedEmbedding e)
{Y : Type v} [TopologicalSpace Y] (f : C(X₁, Y))
{t : Set Y} (hf : ∀ x, f x ∈ t) [ht : TietzeExtension.{u, v} t] :
∃ (g : C(X, Y)), (∀ x, g x ∈ t) ∧ g.comp ⟨e, he.continuous⟩ = f
|
X₁ : Type u₁
inst✝³ : TopologicalSpace X₁
X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : NormalSpace X
e : X₁ → X
he : IsClosedEmbedding e
Y : Type v
inst✝ : TopologicalSpace Y
f : C(X₁, Y)
t : Set Y
hf : ∀ (x : X₁), f x ∈ t
ht : TietzeExtension ↑t
g : C(X, ↑t)
hg : g.comp { toFun := e, continuous_toFun := ⋯ } = { toFun := Set.codRestrict (⇑f) t hf, continuous_toFun := ⋯ }
⊢ ∀ (x : X), ({ toFun := Subtype.val, continuous_toFun := ⋯ }.comp g) x ∈ t
|
simp
|
no goals
|
fe8dbdda96ca6b44
|
ONote.nf_repr_split'
|
Mathlib/SetTheory/Ordinal/Notation.lean
|
theorem nf_repr_split' : ∀ {o o' m} [NF o], split' o = (o', m) → NF o' ∧ repr o = ω * repr o' + m
| 0, o', m, _, p => by injection p; substs o' m; simp [NF.zero]
| oadd e n a, o', m, h, p => by
by_cases e0 : e = 0 <;> simp [e0, split, split'] at p ⊢
· rcases p with ⟨rfl, rfl⟩
simp [h.zero_of_zero e0, NF.zero]
· revert p
rcases h' : split' a with ⟨a', m'⟩
haveI := h.fst
haveI := h.snd
obtain ⟨IH₁, IH₂⟩ := nf_repr_split' h'
simp only [IH₂, and_imp]
intros
substs o' m
have : (ω : Ordinal.{0}) ^ repr e = ω ^ (1 : Ordinal.{0}) * ω ^ (repr e - 1)
|
case neg.mk.intro
e : ONote
n : ℕ+
a : ONote
h : (e.oadd n a).NF
e0 : ¬e = 0
a' : ONote
m' : ℕ
h' : a.split' = (a', m')
this✝ : e.NF
this : a.NF
IH₁ : a'.NF
IH₂ : a.repr = ω * a'.repr + ↑m'
⊢ ((e - 1).oadd n a').NF ∧ ω ^ e.repr * ↑↑n + (ω * a'.repr + ↑m') = ω * ((e - 1).oadd n a').repr + ↑m'
|
have : (ω : Ordinal.{0}) ^ repr e = ω ^ (1 : Ordinal.{0}) * ω ^ (repr e - 1) := by
have := mt repr_inj.1 e0
rw [← opow_add, Ordinal.add_sub_cancel_of_le (one_le_iff_ne_zero.2 this)]
|
case neg.mk.intro
e : ONote
n : ℕ+
a : ONote
h : (e.oadd n a).NF
e0 : ¬e = 0
a' : ONote
m' : ℕ
h' : a.split' = (a', m')
this✝¹ : e.NF
this✝ : a.NF
IH₁ : a'.NF
IH₂ : a.repr = ω * a'.repr + ↑m'
this : ω ^ e.repr = ω ^ 1 * ω ^ (e.repr - 1)
⊢ ((e - 1).oadd n a').NF ∧ ω ^ e.repr * ↑↑n + (ω * a'.repr + ↑m') = ω * ((e - 1).oadd n a').repr + ↑m'
|
d0a92888056b95d7
|
AlgebraicGeometry.Scheme.Hom.isAffineOpen_iff_of_isOpenImmersion
|
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
theorem _root_.AlgebraicGeometry.Scheme.Hom.isAffineOpen_iff_of_isOpenImmersion
(f : AlgebraicGeometry.Scheme.Hom X Y) [H : IsOpenImmersion f] {U : X.Opens} :
IsAffineOpen (f ''ᵁ U) ↔ IsAffineOpen U
|
case refine_1
X Y : Scheme
f : X.Hom Y
H : IsOpenImmersion f
U : X.Opens
hU : IsAffineOpen (f ''ᵁ U)
⊢ ⇑(ConcreteCategory.hom f.base) '' Set.range ⇑(ConcreteCategory.hom (X.ofRestrict ⋯).base) =
Set.range ⇑(ConcreteCategory.hom (Y.ofRestrict ⋯).base)
|
dsimp [Opens.coe_inclusion', Scheme.restrict]
|
case refine_1
X Y : Scheme
f : X.Hom Y
H : IsOpenImmersion f
U : X.Opens
hU : IsAffineOpen (f ''ᵁ U)
⊢ ⇑(ConcreteCategory.hom f.base) '' Set.range Subtype.val = Set.range Subtype.val
|
b53847fef8065733
|
Algebra.PowerBasis.norm_gen_eq_prod_roots
|
Mathlib/RingTheory/Norm/Basic.lean
|
theorem PowerBasis.norm_gen_eq_prod_roots [Algebra R F] (pb : PowerBasis R S)
(hf : (minpoly R pb.gen).Splits (algebraMap R F)) :
algebraMap R F (norm R pb.gen) = ((minpoly R pb.gen).aroots F).prod
|
R : Type u_1
S : Type u_2
inst✝⁴ : CommRing R
inst✝³ : Ring S
inst✝² : Algebra R S
F : Type u_6
inst✝¹ : Field F
inst✝ : Algebra R F
pb : PowerBasis R S
hf : Splits (algebraMap R F) (minpoly R pb.gen)
⊢ (algebraMap R F) ((norm R) pb.gen) = ((minpoly R pb.gen).aroots F).prod
|
haveI := Module.nontrivial R F
|
R : Type u_1
S : Type u_2
inst✝⁴ : CommRing R
inst✝³ : Ring S
inst✝² : Algebra R S
F : Type u_6
inst✝¹ : Field F
inst✝ : Algebra R F
pb : PowerBasis R S
hf : Splits (algebraMap R F) (minpoly R pb.gen)
this : Nontrivial R
⊢ (algebraMap R F) ((norm R) pb.gen) = ((minpoly R pb.gen).aroots F).prod
|
7493349ed609fb88
|
Submonoid.mem_iSup_of_directed
|
Mathlib/Algebra/Group/Submonoid/Membership.lean
|
theorem mem_iSup_of_directed {ι} [hι : Nonempty ι] {S : ι → Submonoid M} (hS : Directed (· ≤ ·) S)
{x : M} : (x ∈ ⨆ i, S i) ↔ ∃ i, x ∈ S i
|
case refine_2.intro.intro.intro.intro
M : Type u_1
inst✝ : MulOneClass M
ι : Sort u_4
hι : Nonempty ι
S : ι → Submonoid M
hS : Directed (fun x1 x2 => x1 ≤ x2) S
x✝ x y : M
i : ι
hi : x ∈ S i
j : ι
hj : y ∈ S j
k : ι
hki : S i ≤ S k
hkj : S j ≤ S k
⊢ ∃ i, x * y ∈ S i
|
exact ⟨k, (S k).mul_mem (hki hi) (hkj hj)⟩
|
no goals
|
69c68e2a28aaf244
|
Array.zip_map'
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Zip.lean
|
theorem zip_map' (f : α → β) (g : α → γ) (l : Array α) :
zip (l.map f) (l.map g) = l.map fun a => (f a, g a)
|
case mk
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → β
g : α → γ
toList✝ : List α
⊢ (map f { toList := toList✝ }).zip (map g { toList := toList✝ }) = map (fun a => (f a, g a)) { toList := toList✝ }
|
simp [List.zip_map']
|
no goals
|
0d061830ab950cab
|
Nat.choose_mul_add
|
Mathlib/Data/Nat/Choose/Mul.lean
|
theorem choose_mul_add {m n : ℕ} (hn : n ≠ 0) :
(m * n + n).choose n = (m + 1) * (m * n + n - 1).choose (n - 1)
|
m n : ℕ
hn : n ≠ 0
p : ℕ := n - 1
hp : n = p + 1
⊢ (m * n + n).choose n * ((m * n)! * n !) = (m + 1) * (m * n + n - 1).choose p * ((m * n)! * n !)
|
simp only [hp, add_succ_sub_one]
|
m n : ℕ
hn : n ≠ 0
p : ℕ := n - 1
hp : n = p + 1
⊢ (m * (p + 1) + (p + 1)).choose (p + 1) * ((m * (p + 1))! * (p + 1)!) =
(m + 1) * (m * (p + 1) + p).choose p * ((m * (p + 1))! * (p + 1)!)
|
a6f9d9f9cd7f0083
|
LinearMap.eventually_iSup_ker_pow_eq
|
Mathlib/RingTheory/Noetherian/Defs.lean
|
lemma LinearMap.eventually_iSup_ker_pow_eq (f : M →ₗ[R] M) :
∀ᶠ n in atTop, ⨆ m, LinearMap.ker (f ^ m) = LinearMap.ker (f ^ n)
|
case intro
R : Type u_1
M : Type u_2
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : IsNoetherian R M
f : M →ₗ[R] M
n : ℕ
hn : ∀ (m : ℕ), n ≤ m → ker (f ^ n) = ker (f ^ m)
m : ℕ
hm : m ≥ n
l : ℕ
⊢ ker (f ^ l) ≤ ker (f ^ m)
|
rcases le_or_lt m l with h | h
|
case intro.inl
R : Type u_1
M : Type u_2
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : IsNoetherian R M
f : M →ₗ[R] M
n : ℕ
hn : ∀ (m : ℕ), n ≤ m → ker (f ^ n) = ker (f ^ m)
m : ℕ
hm : m ≥ n
l : ℕ
h : m ≤ l
⊢ ker (f ^ l) ≤ ker (f ^ m)
case intro.inr
R : Type u_1
M : Type u_2
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : IsNoetherian R M
f : M →ₗ[R] M
n : ℕ
hn : ∀ (m : ℕ), n ≤ m → ker (f ^ n) = ker (f ^ m)
m : ℕ
hm : m ≥ n
l : ℕ
h : l < m
⊢ ker (f ^ l) ≤ ker (f ^ m)
|
f69959d360e4ff43
|
Std.Tactic.BVDecide.Normalize.BitVec.ofNatLt_reduce
|
Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Normalize/BitVec.lean
|
theorem BitVec.ofNatLt_reduce (n : Nat) (h) : BitVec.ofNatLt n h = BitVec.ofNat w n
|
w n : Nat
h : n < 2 ^ w
⊢ n#'h = BitVec.ofNat w n
|
simp [BitVec.ofNatLt, BitVec.ofNat, Fin.ofNat', Nat.mod_eq_of_lt h]
|
no goals
|
cf3efc0727a0b39d
|
CauSeq.trichotomy
|
Mathlib/Algebra/Order/CauSeq/Basic.lean
|
theorem trichotomy (f : CauSeq α abs) : Pos f ∨ LimZero f ∨ Pos (-f)
|
case inr.intro.intro.intro.refine_2.intro
α : Type u_1
inst✝ : LinearOrderedField α
f : CauSeq α abs
h✝ : ¬f.LimZero
K : α
K0 : K > 0
hK : ∃ i, ∀ j ≥ i, K ≤ |↑f j|
i : ℕ
hi : ∀ j ≥ i, K ≤ |↑f j| ∧ ∀ k ≥ j, |↑f k - ↑f j| < K
h : ↑f i ≤ 0
j : ℕ
ij : j ≥ i
this : K ≤ |↑f j|
h₁ : K ≤ |↑f i|
h₂ : ∀ k ≥ i, |↑f k - ↑f i| < K
⊢ ↑f j ≤ 0
|
rw [abs_of_nonpos h] at h₁
|
case inr.intro.intro.intro.refine_2.intro
α : Type u_1
inst✝ : LinearOrderedField α
f : CauSeq α abs
h✝ : ¬f.LimZero
K : α
K0 : K > 0
hK : ∃ i, ∀ j ≥ i, K ≤ |↑f j|
i : ℕ
hi : ∀ j ≥ i, K ≤ |↑f j| ∧ ∀ k ≥ j, |↑f k - ↑f j| < K
h : ↑f i ≤ 0
j : ℕ
ij : j ≥ i
this : K ≤ |↑f j|
h₁ : K ≤ -↑f i
h₂ : ∀ k ≥ i, |↑f k - ↑f i| < K
⊢ ↑f j ≤ 0
|
b071776ee71c7fea
|
Matrix.det_mul
|
Mathlib/LinearAlgebra/Matrix/Determinant/Basic.lean
|
theorem det_mul (M N : Matrix n n R) : det (M * N) = det M * det N :=
calc
det (M * N) = ∑ p : n → n, ∑ σ : Perm n, ε σ * ∏ i, M (σ i) (p i) * N (p i) i
|
n : Type u_2
inst✝² : DecidableEq n
inst✝¹ : Fintype n
R : Type v
inst✝ : CommRing R
M N : Matrix n n R
σ : Perm n
x✝ : σ ∈ univ
τ : Perm n
⊢ (∏ i : n, N (σ i) i) * ↑↑(sign τ) * ∏ j : n, M (τ j) (σ j) =
(∏ i : n, N (σ i) i) * (↑↑(sign σ) * ↑↑(sign ((Equiv.mulRight σ⁻¹) τ))) * ∏ i : n, M (((Equiv.mulRight σ⁻¹) τ) i) i
|
have : (∏ j, M (τ j) (σ j)) = ∏ j, M ((τ * σ⁻¹) j) j := by
rw [← (σ⁻¹ : _ ≃ _).prod_comp]
simp only [Equiv.Perm.coe_mul, apply_inv_self, Function.comp_apply]
|
n : Type u_2
inst✝² : DecidableEq n
inst✝¹ : Fintype n
R : Type v
inst✝ : CommRing R
M N : Matrix n n R
σ : Perm n
x✝ : σ ∈ univ
τ : Perm n
this : ∏ j : n, M (τ j) (σ j) = ∏ j : n, M ((τ * σ⁻¹) j) j
⊢ (∏ i : n, N (σ i) i) * ↑↑(sign τ) * ∏ j : n, M (τ j) (σ j) =
(∏ i : n, N (σ i) i) * (↑↑(sign σ) * ↑↑(sign ((Equiv.mulRight σ⁻¹) τ))) * ∏ i : n, M (((Equiv.mulRight σ⁻¹) τ) i) i
|
396c3d4a14f910cc
|
Polynomial.X_pow_sub_one_eq_prod
|
Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean
|
theorem X_pow_sub_one_eq_prod {ζ : R} {n : ℕ} (hpos : 0 < n) (h : IsPrimitiveRoot ζ n) :
X ^ n - 1 = ∏ ζ ∈ nthRootsFinset n R, (X - C ζ)
|
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
ζ : R
n : ℕ
hpos : 0 < n
h : IsPrimitiveRoot ζ n
⊢ X ^ n - 1 = (Multiset.map (fun ζ => X - C ζ) (nthRoots n 1)).prod
|
rw [nthRoots]
|
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
ζ : R
n : ℕ
hpos : 0 < n
h : IsPrimitiveRoot ζ n
⊢ X ^ n - 1 = (Multiset.map (fun ζ => X - C ζ) (X ^ n - C 1).roots).prod
|
7a9fe34619bf1b6b
|
measurable_of_isOpen
|
Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean
|
theorem measurable_of_isOpen {f : δ → γ} (hf : ∀ s, IsOpen s → MeasurableSet (f ⁻¹' s)) :
Measurable f
|
γ : Type u_3
δ : Type u_5
inst✝³ : TopologicalSpace γ
inst✝² : MeasurableSpace γ
inst✝¹ : BorelSpace γ
inst✝ : MeasurableSpace δ
f : δ → γ
hf : ∀ (s : Set γ), IsOpen s → MeasurableSet (f ⁻¹' s)
⊢ Measurable f
|
exact measurable_generateFrom hf
|
no goals
|
dde88b7086bdeb51
|
Multiset.le_inter
|
Mathlib/Data/Multiset/UnionInter.lean
|
lemma le_inter (h₁ : s ≤ t) (h₂ : s ≤ u) : s ≤ t ∩ u
|
α : Type u_1
inst✝ : DecidableEq α
t : Multiset α
⊢ ∀ {s u : Multiset α}, s ≤ t → s ≤ u → s ≤ t ∩ u
|
refine @(Multiset.induction_on t ?_ fun a t IH => ?_) <;> intros s u h₁ h₂
|
case refine_1
α : Type u_1
inst✝ : DecidableEq α
t s u : Multiset α
h₁ : s ≤ 0
h₂ : s ≤ u
⊢ s ≤ 0 ∩ u
case refine_2
α : Type u_1
inst✝ : DecidableEq α
t✝ : Multiset α
a : α
t : Multiset α
IH : ∀ {s u : Multiset α}, s ≤ t → s ≤ u → s ≤ t ∩ u
s u : Multiset α
h₁ : s ≤ a ::ₘ t
h₂ : s ≤ u
⊢ s ≤ (a ::ₘ t) ∩ u
|
eac334b2668878dc
|
exists_norm_eq_iInf_of_complete_convex
|
Mathlib/Analysis/InnerProductSpace/Projection.lean
|
theorem exists_norm_eq_iInf_of_complete_convex {K : Set F} (ne : K.Nonempty) (h₁ : IsComplete K)
(h₂ : Convex ℝ K) : ∀ u : F, ∃ v ∈ K, ‖u - v‖ = ⨅ w : K, ‖u - w‖ := fun u => by
let δ := ⨅ w : K, ‖u - w‖
letI : Nonempty K := ne.to_subtype
have zero_le_δ : 0 ≤ δ := le_ciInf fun _ => norm_nonneg _
have δ_le : ∀ w : K, δ ≤ ‖u - w‖ := ciInf_le ⟨0, Set.forall_mem_range.2 fun _ => norm_nonneg _⟩
have δ_le' : ∀ w ∈ K, δ ≤ ‖u - w‖ := fun w hw => δ_le ⟨w, hw⟩
-- Step 1: since `δ` is the infimum, can find a sequence `w : ℕ → K` in `K`
-- such that `‖u - w n‖ < δ + 1 / (n + 1)` (which implies `‖u - w n‖ --> δ`);
-- maybe this should be a separate lemma
have exists_seq : ∃ w : ℕ → K, ∀ n, ‖u - w n‖ < δ + 1 / (n + 1)
|
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
K : Set F
ne : K.Nonempty
h₁ : IsComplete K
h₂ : Convex ℝ K
u : F
δ : ℝ := ⨅ w, ‖u - ↑w‖
this✝ : Nonempty ↑K := Set.Nonempty.to_subtype ne
zero_le_δ : 0 ≤ δ
δ_le : ∀ (w : ↑K), δ ≤ ‖u - ↑w‖
δ_le' : ∀ w ∈ K, δ ≤ ‖u - w‖
w : ℕ → ↑K
hw : ∀ (n : ℕ), ‖u - ↑(w n)‖ < δ + 1 / (↑n + 1)
norm_tendsto : Tendsto (fun n => ‖u - ↑(w n)‖) atTop (𝓝 δ)
b✝ : ℕ → ℝ := fun n => 8 * δ * (1 / (↑n + 1)) + 4 * (1 / (↑n + 1)) * (1 / (↑n + 1))
p q N : ℕ
hp : N ≤ p
hq : N ≤ q
wp : F := ↑(w p)
wq : F := ↑(w q)
a : F := u - wq
b : F := u - wp
half : ℝ := 1 / 2
div : ℝ := 1 / (↑N + 1)
this : 4 * ‖u - half • (wq + wp)‖ * ‖u - half • (wq + wp)‖ + ‖wp - wq‖ * ‖wp - wq‖ = 2 * (‖a‖ * ‖a‖ + ‖b‖ * ‖b‖)
eq : δ ≤ ‖u - half • (wq + wp)‖
eq₁ : 4 * δ * δ ≤ 4 * ‖u - half • (wq + wp)‖ * ‖u - half • (wq + wp)‖
eq₂ : ‖a‖ ≤ δ + div
eq₂' : ‖b‖ ≤ δ + div
⊢ 2 * ((δ + div) * (δ + div) + (δ + div) * (δ + div)) - 4 * δ * δ = 8 * δ * div + 4 * div * div
|
ring
|
no goals
|
1aed30586c16c1ed
|
MeasureTheory.lintegral_trim
|
Mathlib/MeasureTheory/Integral/Lebesgue.lean
|
theorem lintegral_trim {μ : Measure α} (hm : m ≤ m0) {f : α → ℝ≥0∞} (hf : Measurable[m] f) :
∫⁻ a, f a ∂μ.trim hm = ∫⁻ a, f a ∂μ
|
case refine_2
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
f✝ : α → ℝ≥0∞
hf✝ : Measurable f✝
f g : α → ℝ≥0∞
a✝¹ : Disjoint (support f) (support g)
hf : Measurable f
a✝ : Measurable g
hf_prop : ∫⁻ (a : α), f a ∂μ.trim hm = ∫⁻ (a : α), f a ∂μ
hg_prop : ∫⁻ (a : α), g a ∂μ.trim hm = ∫⁻ (a : α), g a ∂μ
⊢ ∫⁻ (a : α), (f + g) a ∂μ.trim hm = ∫⁻ (a : α), (f + g) a ∂μ
|
have h_m := lintegral_add_left (μ := Measure.trim μ hm) hf g
|
case refine_2
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
f✝ : α → ℝ≥0∞
hf✝ : Measurable f✝
f g : α → ℝ≥0∞
a✝¹ : Disjoint (support f) (support g)
hf : Measurable f
a✝ : Measurable g
hf_prop : ∫⁻ (a : α), f a ∂μ.trim hm = ∫⁻ (a : α), f a ∂μ
hg_prop : ∫⁻ (a : α), g a ∂μ.trim hm = ∫⁻ (a : α), g a ∂μ
h_m : ∫⁻ (a : α), f a + g a ∂μ.trim hm = ∫⁻ (a : α), f a ∂μ.trim hm + ∫⁻ (a : α), g a ∂μ.trim hm
⊢ ∫⁻ (a : α), (f + g) a ∂μ.trim hm = ∫⁻ (a : α), (f + g) a ∂μ
|
ba450faf648821cd
|
FractionalIdeal.spanSingleton_one
|
Mathlib/RingTheory/FractionalIdeal/Operations.lean
|
theorem spanSingleton_one : spanSingleton S (1 : P) = 1
|
case a
R : Type u_1
inst✝³ : CommRing R
S : Submonoid R
P : Type u_2
inst✝² : CommRing P
inst✝¹ : Algebra R P
inst✝ : IsLocalization S P
x✝ : P
⊢ ∀ (a : R), a • 1 = x✝ ↔ (algebraMap R P) a = x✝
|
intro x'
|
case a
R : Type u_1
inst✝³ : CommRing R
S : Submonoid R
P : Type u_2
inst✝² : CommRing P
inst✝¹ : Algebra R P
inst✝ : IsLocalization S P
x✝ : P
x' : R
⊢ x' • 1 = x✝ ↔ (algebraMap R P) x' = x✝
|
dd02abd64f210f57
|
List.findIdx_cons
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Find.lean
|
theorem findIdx_cons (p : α → Bool) (b : α) (l : List α) :
(b :: l).findIdx p = bif p b then 0 else (l.findIdx p) + 1
|
case nil
α : Type u_1
p✝ : α → Bool
b : α
l : List α
p : α → Bool
n : Nat
⊢ findIdx.go p [] (n + 1) = findIdx.go p [] n + 1
|
unfold findIdx.go
|
case nil
α : Type u_1
p✝ : α → Bool
b : α
l : List α
p : α → Bool
n : Nat
⊢ n + 1 = n + 1
|
5f956aebe5d1eb40
|
PosNum.divMod_to_nat
|
Mathlib/Data/Num/Lemmas.lean
|
theorem divMod_to_nat (d n : PosNum) :
(n / d : ℕ) = (divMod d n).1 ∧ (n % d : ℕ) = (divMod d n).2
|
case bit1.mk.h₁
d n : PosNum
q r : Num
IH : ↑r + ↑d * ↑q = ↑n ∧ ↑r < ↑d
⊢ 2 * ↑r + 1 + ↑d * (↑q + ↑q) = ↑n + ↑n + 1
|
rw [← two_mul, ← two_mul, add_right_comm, mul_left_comm, ← mul_add, IH.1]
|
no goals
|
16ea79c4525f4751
|
Set.image2_eq_seq
|
Mathlib/Data/Set/Lattice.lean
|
theorem image2_eq_seq (f : α → β → γ) (s : Set α) (t : Set β) : image2 f s t = seq (f '' s) t
|
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → β → γ
s : Set α
t : Set β
⊢ image2 f s t = (f '' s).seq t
|
rw [seq_eq_image2, image2_image_left]
|
no goals
|
c71db0f6c3851dfd
|
PiLp.nndist_eq_of_L1
|
Mathlib/Analysis/Normed/Lp/PiLp.lean
|
theorem nndist_eq_of_L1 (x y : PiLp 1 β) : nndist x y = ∑ i, nndist (x i) (y i) :=
NNReal.eq <| by push_cast; exact dist_eq_of_L1 _ _
|
ι : Type u_2
β : ι → Type u_4
inst✝¹ : Fintype ι
inst✝ : (i : ι) → SeminormedAddCommGroup (β i)
x y : PiLp 1 β
⊢ ↑(nndist x y) = ↑(∑ i : ι, nndist (x i) (y i))
|
push_cast
|
ι : Type u_2
β : ι → Type u_4
inst✝¹ : Fintype ι
inst✝ : (i : ι) → SeminormedAddCommGroup (β i)
x y : PiLp 1 β
⊢ dist x y = ∑ x_1 : ι, dist (x x_1) (y x_1)
|
abdf87638648d230
|
ContMDiffWithinAt.mfderivWithin
|
Mathlib/Geometry/Manifold/ContMDiffMFDeriv.lean
|
theorem ContMDiffWithinAt.mfderivWithin {x₀ : N} {f : N → M → M'} {g : N → M}
{t : Set N} {u : Set M}
(hf : ContMDiffWithinAt (J.prod I) I' n (Function.uncurry f) (t ×ˢ u) (x₀, g x₀))
(hg : ContMDiffWithinAt J I m g t x₀) (hx₀ : x₀ ∈ t)
(hu : MapsTo g t u) (hmn : m + 1 ≤ n) (h'u : UniqueMDiffOn I u) :
haveI : IsManifold I 1 M := .of_le (le_trans le_add_self hmn)
haveI : IsManifold I' 1 M' := .of_le (le_trans le_add_self hmn)
ContMDiffWithinAt J 𝓘(𝕜, E →L[𝕜] E') m
(inTangentCoordinates I I' g (fun x => f x (g x))
(fun x => mfderivWithin I I' (f x) u (g x)) x₀) t x₀
|
𝕜 : Type u_1
inst✝¹⁵ : NontriviallyNormedField 𝕜
m n : WithTop ℕ∞
E : Type u_2
inst✝¹⁴ : NormedAddCommGroup E
inst✝¹³ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹¹ : TopologicalSpace M
inst✝¹⁰ : ChartedSpace H M
E' : Type u_5
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁷ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁶ : TopologicalSpace M'
inst✝⁵ : ChartedSpace H' M'
F : Type u_8
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
G : Type u_9
inst✝² : TopologicalSpace G
J : ModelWithCorners 𝕜 F G
N : Type u_10
inst✝¹ : TopologicalSpace N
inst✝ : ChartedSpace G N
Js : IsManifold J n N
Is : IsManifold I n M
I's : IsManifold I' n M'
x₀ : N
f : N → M → M'
g : N → M
t : Set N
u : Set M
hf : ContMDiffWithinAt (J.prod I) I' n (uncurry f) (t ×ˢ u) (x₀, g x₀)
hg : ContMDiffWithinAt J I m g t x₀
hx₀ : x₀ ∈ t
hu : MapsTo g t u
hmn : m + 1 ≤ n
h'u : UniqueMDiffOn I u
this✝² : IsManifold I 1 M
this✝¹ : IsManifold I' 1 M'
this✝ : IsManifold J 1 N
this : IsManifold J m N
t' : Set N := t ∩ g ⁻¹' (extChartAt I (g x₀)).source
ht't : t' ⊆ t
hx₀gx₀ : (x₀, g x₀) ∈ t ×ˢ u
h4f✝ : ContinuousWithinAt (fun x => f x (g x)) t x₀
h4f : (fun x => f x (g x)) ⁻¹' (extChartAt I' (f x₀ (g x₀))).source ∈ 𝓝[t] x₀
⊢ 1 ≠ ↑⊤
|
simp
|
no goals
|
b011e1f26afdb375
|
LinearMap.finrank_maxGenEigenspace
|
Mathlib/LinearAlgebra/Eigenspace/Zero.lean
|
lemma finrank_maxGenEigenspace (φ : Module.End K M) :
finrank K (φ.maxGenEigenspace 0) = natTrailingDegree (φ.charpoly)
|
K : Type u_2
M : Type u_3
inst✝³ : Field K
inst✝² : AddCommGroup M
inst✝¹ : Module K M
inst✝ : Module.Finite K M
φ : End K M
V : Submodule K M := φ.maxGenEigenspace 0
hV : V = ⨆ n, ker (φ ^ n)
W : Submodule K M := ⨅ n, range (φ ^ n)
hVW : IsCompl V W
⊢ ∀ (x : M) (x_1 : ℕ), (φ ^ x_1) x = 0 → ∃ k, (φ ^ k) (φ x) = 0
|
intro x n hx
|
K : Type u_2
M : Type u_3
inst✝³ : Field K
inst✝² : AddCommGroup M
inst✝¹ : Module K M
inst✝ : Module.Finite K M
φ : End K M
V : Submodule K M := φ.maxGenEigenspace 0
hV : V = ⨆ n, ker (φ ^ n)
W : Submodule K M := ⨅ n, range (φ ^ n)
hVW : IsCompl V W
x : M
n : ℕ
hx : (φ ^ n) x = 0
⊢ ∃ k, (φ ^ k) (φ x) = 0
|
b669b1274d0f8ced
|
IsLocalizedModule.lift_rank_eq
|
Mathlib/LinearAlgebra/Dimension/Localization.lean
|
lemma IsLocalizedModule.lift_rank_eq :
Cardinal.lift.{v} (Module.rank S N) = Cardinal.lift.{v'} (Module.rank R M)
|
R : Type u
S : Type u'
M : Type v
N : Type v'
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : AddCommGroup M
inst✝⁷ : AddCommGroup N
inst✝⁶ : Module R M
inst✝⁵ : Module R N
inst✝⁴ : Algebra R S
inst✝³ : Module S N
inst✝² : IsScalarTower R S N
p : Submonoid R
inst✝¹ : IsLocalization p S
f : M →ₗ[R] N
inst✝ : IsLocalizedModule p f
hp : p ≤ R⁰
⊢ Cardinal.lift.{v, v'} (Module.rank S N) = Cardinal.lift.{v', v} (Module.rank R M)
|
cases subsingleton_or_nontrivial R
|
case inl
R : Type u
S : Type u'
M : Type v
N : Type v'
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : AddCommGroup M
inst✝⁷ : AddCommGroup N
inst✝⁶ : Module R M
inst✝⁵ : Module R N
inst✝⁴ : Algebra R S
inst✝³ : Module S N
inst✝² : IsScalarTower R S N
p : Submonoid R
inst✝¹ : IsLocalization p S
f : M →ₗ[R] N
inst✝ : IsLocalizedModule p f
hp : p ≤ R⁰
h✝ : Subsingleton R
⊢ Cardinal.lift.{v, v'} (Module.rank S N) = Cardinal.lift.{v', v} (Module.rank R M)
case inr
R : Type u
S : Type u'
M : Type v
N : Type v'
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : AddCommGroup M
inst✝⁷ : AddCommGroup N
inst✝⁶ : Module R M
inst✝⁵ : Module R N
inst✝⁴ : Algebra R S
inst✝³ : Module S N
inst✝² : IsScalarTower R S N
p : Submonoid R
inst✝¹ : IsLocalization p S
f : M →ₗ[R] N
inst✝ : IsLocalizedModule p f
hp : p ≤ R⁰
h✝ : Nontrivial R
⊢ Cardinal.lift.{v, v'} (Module.rank S N) = Cardinal.lift.{v', v} (Module.rank R M)
|
85b7978350d6d455
|
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.deleteOne_preserves_strongAssignmentsInvariant
|
Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/Lemmas.lean
|
theorem deleteOne_preserves_strongAssignmentsInvariant {n : Nat} (f : DefaultFormula n) (id : Nat) :
StrongAssignmentsInvariant f → StrongAssignmentsInvariant (deleteOne f id)
|
n : Nat
f : DefaultFormula n
id : Nat
hsize : f.assignments.size = n
hf : ∀ (i : PosFin n) (b : Bool), hasAssignment b f.assignments[i.val] = true → unit (i, b) ∈ f.toList
hsize' : (f.deleteOne id).assignments.size = n
i : PosFin n
b : Bool
hb : hasAssignment b (f.deleteOne id).assignments[i.val] = true
i_in_bounds : i.val < f.assignments.size
c : DefaultClause n
heq : f.clauses[id]! = some c
hl :
∀ (x : PosFin n),
¬c = { clause := [(x, false)], nodupkey := ⋯, nodup := ⋯ } ∧
¬c = { clause := [(x, true)], nodupkey := ⋯, nodup := ⋯ }
x✝² : Option (DefaultClause n)
val✝ : DefaultClause n
x✝¹ :
∀ (l : Literal (PosFin n)) (nodupkey : ∀ (l_1 : PosFin n), ¬(l_1, true) ∈ [l] ∨ ¬(l_1, false) ∈ [l])
(nodup : [l].Nodup), val✝ = { clause := [l], nodupkey := nodupkey, nodup := nodup } → False
heq✝ : some c = some val✝
x✝ : Option (DefaultClause n)
heq2 : f.clauses[id]! = none
⊢ { clauses := f.clauses, rupUnits := f.rupUnits, ratUnits := f.ratUnits, assignments := f.assignments } =
{ clauses := f.clauses.set! id none, rupUnits := f.rupUnits, ratUnits := f.ratUnits, assignments := f.assignments }
|
simp [heq] at heq2
|
no goals
|
0ab20a88767ce69e
|
CategoryTheory.PreGaloisCategory.toAut_continuous
|
Mathlib/CategoryTheory/Galois/IsFundamentalgroup.lean
|
lemma toAut_continuous [TopologicalSpace G] [IsTopologicalGroup G]
[∀ (X : C), ContinuousSMul G (F.obj X)] :
Continuous (toAut F G)
|
case hf
C : Type u₁
inst✝⁸ : Category.{u₂, u₁} C
F : C ⥤ FintypeCat
G : Type u_1
inst✝⁷ : Group G
inst✝⁶ : (X : C) → MulAction G (F.obj X).carrier
inst✝⁵ : IsNaturalSMul F G
inst✝⁴ : GaloisCategory C
inst✝³ : FiberFunctor F
inst✝² : TopologicalSpace G
inst✝¹ : IsTopologicalGroup G
inst✝ : ∀ (X : C), ContinuousSMul G (F.obj X).carrier
⊢ ∀ A ∈ nhds 1, ⇑(toAut F G) ⁻¹' A ∈ nhds 1
|
intro A hA
|
case hf
C : Type u₁
inst✝⁸ : Category.{u₂, u₁} C
F : C ⥤ FintypeCat
G : Type u_1
inst✝⁷ : Group G
inst✝⁶ : (X : C) → MulAction G (F.obj X).carrier
inst✝⁵ : IsNaturalSMul F G
inst✝⁴ : GaloisCategory C
inst✝³ : FiberFunctor F
inst✝² : TopologicalSpace G
inst✝¹ : IsTopologicalGroup G
inst✝ : ∀ (X : C), ContinuousSMul G (F.obj X).carrier
A : Set (Aut F)
hA : A ∈ nhds 1
⊢ ⇑(toAut F G) ⁻¹' A ∈ nhds 1
|
6f49297602372823
|
finprod_eq_dif
|
Mathlib/Algebra/BigOperators/Finprod.lean
|
theorem finprod_eq_dif {p : Prop} [Decidable p] (f : p → M) :
∏ᶠ i, f i = if h : p then f h else 1
|
case neg
M : Type u_2
inst✝¹ : CommMonoid M
p : Prop
inst✝ : Decidable p
f : p → M
h : ¬p
⊢ ∏ᶠ (i : p), f i = 1
|
haveI : IsEmpty p := ⟨h⟩
|
case neg
M : Type u_2
inst✝¹ : CommMonoid M
p : Prop
inst✝ : Decidable p
f : p → M
h : ¬p
this : IsEmpty p
⊢ ∏ᶠ (i : p), f i = 1
|
79648c5d95d89f12
|
nhds_hasBasis_absConvex_open
|
Mathlib/Analysis/LocallyConvex/AbsConvex.lean
|
theorem nhds_hasBasis_absConvex_open :
(𝓝 (0 : E)).HasBasis (fun s => (0 : E) ∈ s ∧ IsOpen s ∧ AbsConvex 𝕜 s) id
|
case refine_2
𝕜 : Type u_1
E : Type u_2
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : AddCommGroup E
inst✝⁷ : Module 𝕜 E
inst✝⁶ : Module ℝ E
inst✝⁵ : SMulCommClass ℝ 𝕜 E
inst✝⁴ : TopologicalSpace E
inst✝³ : LocallyConvexSpace ℝ E
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : ContinuousSMul ℝ E
inst✝ : IsTopologicalAddGroup E
⊢ ∀ (i' : Set E), 0 ∈ i' ∧ IsOpen i' ∧ AbsConvex 𝕜 i' → ∃ i, (i ∈ 𝓝 0 ∧ AbsConvex 𝕜 i) ∧ id i ⊆ id i'
|
rintro s ⟨hs_zero, hs_open, hs_balanced, hs_convex⟩
|
case refine_2.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : AddCommGroup E
inst✝⁷ : Module 𝕜 E
inst✝⁶ : Module ℝ E
inst✝⁵ : SMulCommClass ℝ 𝕜 E
inst✝⁴ : TopologicalSpace E
inst✝³ : LocallyConvexSpace ℝ E
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : ContinuousSMul ℝ E
inst✝ : IsTopologicalAddGroup E
s : Set E
hs_zero : 0 ∈ s
hs_open : IsOpen s
hs_balanced : Balanced 𝕜 s
hs_convex : Convex ℝ s
⊢ ∃ i, (i ∈ 𝓝 0 ∧ AbsConvex 𝕜 i) ∧ id i ⊆ id s
|
168de266ed5a05f9
|
AddGrp.epi_iff_surjective
|
Mathlib/Algebra/Category/Grp/EpiMono.lean
|
theorem epi_iff_surjective : Epi f ↔ Function.Surjective f
|
A B : AddGrp
f : A ⟶ B
e' : Epi f
⊢ Epi (groupAddGroupEquivalence.inverse.map f)
|
apply groupAddGroupEquivalence.inverse.map_epi
|
no goals
|
faf0569f4a3fa45b
|
CategoryTheory.toNerve₂.mk_naturality_σ1i
|
Mathlib/AlgebraicTopology/SimplicialSet/NerveAdjunction.lean
|
lemma toNerve₂.mk_naturality_σ1i (i : Fin 2) : toNerve₂.mk.naturalityProperty F (σ₂ i)
|
case e_f.«1»
C : Type u
inst✝ : SmallCategory C
X : SSet.Truncated 2
F : oneTruncation₂.obj X ⟶ ReflQuiv.of C
hyp : ∀ (φ : X.obj (op { obj := [2], property := ⋯ })), F.map (ev02₂ φ) = F.map (ev01₂ φ) ≫ F.map (ev12₂ φ)
⊢ mk.naturalityProperty F (δ₂ 2 ⋯ ⋯ ≫ σ₂ ((fun i => i) ⟨1, ⋯⟩) ⋯ ⋯)
|
dsimp only [Fin.mk_one]
|
case e_f.«1»
C : Type u
inst✝ : SmallCategory C
X : SSet.Truncated 2
F : oneTruncation₂.obj X ⟶ ReflQuiv.of C
hyp : ∀ (φ : X.obj (op { obj := [2], property := ⋯ })), F.map (ev02₂ φ) = F.map (ev01₂ φ) ≫ F.map (ev12₂ φ)
⊢ mk.naturalityProperty F (δ₂ 2 ⋯ ⋯ ≫ σ₂ 1 ⋯ ⋯)
|
6f808b30451e4c20
|
MeasureTheory.Submartingale.exists_ae_tendsto_of_bdd
|
Mathlib/Probability/Martingale/Convergence.lean
|
theorem Submartingale.exists_ae_tendsto_of_bdd [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(hbdd : ∀ n, eLpNorm (f n) 1 μ ≤ R) : ∀ᵐ ω ∂μ, ∃ c, Tendsto (fun n => f n ω) atTop (𝓝 c)
|
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 : Submartingale f ℱ μ
hbdd : ∀ (n : ℕ), eLpNorm (f n) 1 μ ≤ ↑R
⊢ ∀ᵐ (ω : Ω) ∂μ, ∃ c, Tendsto (fun n => f n ω) atTop (𝓝 c)
|
filter_upwards [hf.upcrossings_ae_lt_top hbdd, ae_bdd_liminf_atTop_of_eLpNorm_bdd one_ne_zero
(fun n => (hf.stronglyMeasurable n).measurable.mono (ℱ.le n) le_rfl) hbdd] with ω h₁ h₂
|
case h
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 : Submartingale f ℱ μ
hbdd : ∀ (n : ℕ), eLpNorm (f n) 1 μ ≤ ↑R
ω : Ω
h₁ : ∀ (a b : ℚ), a < b → upcrossings (↑a) (↑b) f ω < ⊤
h₂ : liminf (fun n => ‖f n ω‖ₑ) atTop < ⊤
⊢ ∃ c, Tendsto (fun n => f n ω) atTop (𝓝 c)
|
9cb271f428a74c4f
|
Action.SingleObj.preservesColimit
|
Mathlib/CategoryTheory/Action/Limits.lean
|
/-- `F : C ⥤ SingleObj G ⥤ V` preserves the colimit of some `K : J ⥤ C` if it does
evaluated at `SingleObj.star G`. -/
private lemma SingleObj.preservesColimit (F : C ⥤ SingleObj G ⥤ V)
{J : Type w₁} [Category.{w₂} J] (K : J ⥤ C)
(h : PreservesColimit K (F ⋙ (evaluation (SingleObj G) V).obj (SingleObj.star G))) :
PreservesColimit K F
|
case H
V : Type (u + 1)
inst✝³ : LargeCategory V
G : Type u
inst✝² : Monoid G
C : Type t₁
inst✝¹ : Category.{t₂, t₁} C
F : C ⥤ SingleObj G ⥤ V
J : Type w₁
inst✝ : Category.{w₂, w₁} J
K : J ⥤ C
h : PreservesColimit K (F ⋙ (evaluation (SingleObj G) V).obj (SingleObj.star G))
⊢ ∀ (k : SingleObj G), PreservesColimit K (F ⋙ (evaluation (SingleObj G) V).obj k)
|
intro _
|
case H
V : Type (u + 1)
inst✝³ : LargeCategory V
G : Type u
inst✝² : Monoid G
C : Type t₁
inst✝¹ : Category.{t₂, t₁} C
F : C ⥤ SingleObj G ⥤ V
J : Type w₁
inst✝ : Category.{w₂, w₁} J
K : J ⥤ C
h : PreservesColimit K (F ⋙ (evaluation (SingleObj G) V).obj (SingleObj.star G))
k✝ : SingleObj G
⊢ PreservesColimit K (F ⋙ (evaluation (SingleObj G) V).obj k✝)
|
ab7f5b714fcc49f6
|
CategoryTheory.ShortComplex.SnakeInput.δ_apply
|
Mathlib/Algebra/Homology/ShortComplex/ConcreteCategory.lean
|
/-- This lemma allows the computation of the connecting homomorphism
`D.δ` when `D : SnakeInput C` and `C` is a concrete category. -/
lemma δ_apply (x₃ : ToType (D.L₀.X₃)) (x₂ : ToType (D.L₁.X₂)) (x₁ : ToType (D.L₂.X₁))
(h₂ : D.L₁.g x₂ = D.v₀₁.τ₃ x₃) (h₁ : D.L₂.f x₁ = D.v₁₂.τ₂ x₂) :
D.δ x₃ = D.v₂₃.τ₁ x₁
|
case a
C : Type u
inst✝⁶ : Category.{v, u} C
FC : C → C → Type u_1
CC : C → Type v
inst✝⁵ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
inst✝⁴ : ConcreteCategory C FC
inst✝³ : HasForget₂ C Ab
inst✝² : Abelian C
inst✝¹ : (forget₂ C Ab).Additive
inst✝ : (forget₂ C Ab).PreservesHomology
D : SnakeInput C
x₃ : ToType D.L₀.X₃
x₂ : ToType D.L₁.X₂
x₁ : ToType D.L₂.X₁
h₂ : (ConcreteCategory.hom D.L₁.g) x₂ = (ConcreteCategory.hom D.v₀₁.τ₃) x₃
h₁ : (ConcreteCategory.hom D.L₂.f) x₁ = (ConcreteCategory.hom D.v₁₂.τ₂) x₂
this✝ : PreservesFiniteLimits (forget₂ C Ab)
this : PreservesFiniteLimits (forget C)
eq :
(ConcreteCategory.hom D.δ) x₃ =
(ConcreteCategory.hom D.v₂₃.τ₁) ((ConcreteCategory.hom D.φ₁) (Concrete.pullbackMk D.L₁.g D.v₀₁.τ₃ x₂ x₃ h₂))
eq₁ : (ConcreteCategory.hom (pullback.fst D.L₁.g D.v₀₁.τ₃)) (Concrete.pullbackMk D.L₁.g D.v₀₁.τ₃ x₂ x₃ h₂) = x₂
eq₂ : (ConcreteCategory.hom (pullback.snd D.L₁.g D.v₀₁.τ₃)) (Concrete.pullbackMk D.L₁.g D.v₀₁.τ₃ x₂ x₃ h₂) = x₃
⊢ (ConcreteCategory.hom D.L₂.f) ((ConcreteCategory.hom D.φ₁) (Concrete.pullbackMk D.L₁.g D.v₀₁.τ₃ x₂ x₃ h₂)) =
(ConcreteCategory.hom D.L₂.f) x₁
|
rw [← ConcreteCategory.comp_apply, φ₁_L₂_f]
|
case a
C : Type u
inst✝⁶ : Category.{v, u} C
FC : C → C → Type u_1
CC : C → Type v
inst✝⁵ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
inst✝⁴ : ConcreteCategory C FC
inst✝³ : HasForget₂ C Ab
inst✝² : Abelian C
inst✝¹ : (forget₂ C Ab).Additive
inst✝ : (forget₂ C Ab).PreservesHomology
D : SnakeInput C
x₃ : ToType D.L₀.X₃
x₂ : ToType D.L₁.X₂
x₁ : ToType D.L₂.X₁
h₂ : (ConcreteCategory.hom D.L₁.g) x₂ = (ConcreteCategory.hom D.v₀₁.τ₃) x₃
h₁ : (ConcreteCategory.hom D.L₂.f) x₁ = (ConcreteCategory.hom D.v₁₂.τ₂) x₂
this✝ : PreservesFiniteLimits (forget₂ C Ab)
this : PreservesFiniteLimits (forget C)
eq :
(ConcreteCategory.hom D.δ) x₃ =
(ConcreteCategory.hom D.v₂₃.τ₁) ((ConcreteCategory.hom D.φ₁) (Concrete.pullbackMk D.L₁.g D.v₀₁.τ₃ x₂ x₃ h₂))
eq₁ : (ConcreteCategory.hom (pullback.fst D.L₁.g D.v₀₁.τ₃)) (Concrete.pullbackMk D.L₁.g D.v₀₁.τ₃ x₂ x₃ h₂) = x₂
eq₂ : (ConcreteCategory.hom (pullback.snd D.L₁.g D.v₀₁.τ₃)) (Concrete.pullbackMk D.L₁.g D.v₀₁.τ₃ x₂ x₃ h₂) = x₃
⊢ (ConcreteCategory.hom D.φ₂) (Concrete.pullbackMk D.L₁.g D.v₀₁.τ₃ x₂ x₃ h₂) = (ConcreteCategory.hom D.L₂.f) x₁
|
ed36ceab2c4785e2
|
List.mk_mem_sym2
|
Mathlib/Data/List/Sym.lean
|
theorem mk_mem_sym2 {xs : List α} {a b : α} (ha : a ∈ xs) (hb : b ∈ xs) :
s(a, b) ∈ xs.sym2
|
case cons.inl.inr.h
α : Type u_1
a b : α
xs : List α
ih : a ∈ xs → b ∈ xs → s(a, b) ∈ xs.sym2
hb : b ∈ xs
⊢ (∃ y ∈ xs, s(a, b) = s(a, y)) ∨ s(a, b) ∈ xs.sym2
|
left
|
case cons.inl.inr.h.h
α : Type u_1
a b : α
xs : List α
ih : a ∈ xs → b ∈ xs → s(a, b) ∈ xs.sym2
hb : b ∈ xs
⊢ ∃ y ∈ xs, s(a, b) = s(a, y)
|
2e917f6e418f8710
|
MeasureTheory.mul_le_integral_rnDeriv_of_ac
|
Mathlib/MeasureTheory/Decomposition/IntegralRNDeriv.lean
|
/-- For a convex continuous function `f` on `[0, ∞)`, if `μ` is absolutely continuous
with respect to `ν`, then
`(ν univ).toReal * f ((μ univ).toReal / (ν univ).toReal) ≤ ∫ x, f (μ.rnDeriv ν x).toReal ∂ν`. -/
lemma mul_le_integral_rnDeriv_of_ac [IsFiniteMeasure μ] [IsFiniteMeasure ν]
(hf_cvx : ConvexOn ℝ (Ici 0) f) (hf_cont : ContinuousWithinAt f (Ici 0) 0)
(hf_int : Integrable (fun x ↦ f (μ.rnDeriv ν x).toReal) ν) (hμν : μ ≪ ν) :
(ν univ).toReal * f ((μ univ).toReal / (ν univ).toReal)
≤ ∫ x, f (μ.rnDeriv ν x).toReal ∂ν
|
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
f : ℝ → ℝ
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
hf_cvx : ConvexOn ℝ (Ici 0) f
hf_cont : ContinuousWithinAt f (Ici 0) 0
hf_int : Integrable (fun x => f (μ.rnDeriv ν x).toReal) ν
hμν : μ ≪ ν
hν : ¬ν = 0
this✝ : NeZero ν
μ' : Measure α := (ν univ)⁻¹ • μ
ν' : Measure α := (ν univ)⁻¹ • ν
this : IsFiniteMeasure μ'
hμν' : μ' ≪ ν'
h1' : μ'.rnDeriv ν' =ᶠ[ae ν'] (ν univ)⁻¹ • μ.rnDeriv ν'
⊢ μ'.rnDeriv ν' =ᶠ[ae ν] (ν univ)⁻¹ • μ.rnDeriv ν'
|
rwa [Measure.ae_smul_measure_eq] at h1'
|
case hc
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
f : ℝ → ℝ
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
hf_cvx : ConvexOn ℝ (Ici 0) f
hf_cont : ContinuousWithinAt f (Ici 0) 0
hf_int : Integrable (fun x => f (μ.rnDeriv ν x).toReal) ν
hμν : μ ≪ ν
hν : ¬ν = 0
this✝ : NeZero ν
μ' : Measure α := (ν univ)⁻¹ • μ
ν' : Measure α := (ν univ)⁻¹ • ν
this : IsFiniteMeasure μ'
hμν' : μ' ≪ ν'
h1' : μ'.rnDeriv ν' =ᶠ[ae ν'] (ν univ)⁻¹ • μ.rnDeriv ν'
⊢ (ν univ)⁻¹ ≠ 0
|
0004fdd48300255a
|
Stream'.Seq.of_mem_append
|
Mathlib/Data/Seq/Seq.lean
|
theorem of_mem_append {s₁ s₂ : Seq α} {a : α} (h : a ∈ append s₁ s₂) : a ∈ s₁ ∨ a ∈ s₂
|
case cons.inr.intro
α : Type u
s₂ : Seq α
a : α
ss : Seq α
h : a ∈ ss
b : α
s' : Seq α
o : a = b ∨ ∀ {s₁ : Seq α}, a ∈ s₁.append s₂ → s₁.append s₂ = s' → a ∈ s₁ ∨ a ∈ s₂
c : α
t₁ : Seq α
m✝ : a ∈ (cons c t₁).append s₂
e : (cons c t₁).append s₂ = cons b s'
this : ((cons c t₁).append s₂).destruct = (cons b s').destruct
m : a ∈ t₁.append s₂
i1 : c = b
i2 : t₁.append s₂ = s'
⊢ a ∈ cons c t₁ ∨ a ∈ s₂
|
rcases o with e' | IH
|
case cons.inr.intro.inl
α : Type u
s₂ : Seq α
a : α
ss : Seq α
h : a ∈ ss
b : α
s' : Seq α
c : α
t₁ : Seq α
m✝ : a ∈ (cons c t₁).append s₂
e : (cons c t₁).append s₂ = cons b s'
this : ((cons c t₁).append s₂).destruct = (cons b s').destruct
m : a ∈ t₁.append s₂
i1 : c = b
i2 : t₁.append s₂ = s'
e' : a = b
⊢ a ∈ cons c t₁ ∨ a ∈ s₂
case cons.inr.intro.inr
α : Type u
s₂ : Seq α
a : α
ss : Seq α
h : a ∈ ss
b : α
s' : Seq α
c : α
t₁ : Seq α
m✝ : a ∈ (cons c t₁).append s₂
e : (cons c t₁).append s₂ = cons b s'
this : ((cons c t₁).append s₂).destruct = (cons b s').destruct
m : a ∈ t₁.append s₂
i1 : c = b
i2 : t₁.append s₂ = s'
IH : ∀ {s₁ : Seq α}, a ∈ s₁.append s₂ → s₁.append s₂ = s' → a ∈ s₁ ∨ a ∈ s₂
⊢ a ∈ cons c t₁ ∨ a ∈ s₂
|
616ea8064ee30260
|
List.beq_nil_iff
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean
|
theorem beq_nil_iff [BEq α] {l : List α} : (l == []) = l.isEmpty
|
α : Type u_1
inst✝ : BEq α
l : List α
⊢ (l == []) = l.isEmpty
|
cases l <;> rfl
|
no goals
|
05ad7d584fefaf2c
|
Set.uIcc_self
|
Mathlib/Order/Interval/Set/UnorderedInterval.lean
|
lemma uIcc_self : [[a, a]] = {a}
|
α : Type u_1
inst✝ : Lattice α
a : α
⊢ [[a, a]] = {a}
|
simp [uIcc]
|
no goals
|
0006f3407891f91d
|
countable_right_of_prod_of_nonempty
|
Mathlib/Data/Countable/Basic.lean
|
lemma countable_right_of_prod_of_nonempty [Nonempty α] (h : Countable (α × β)) : Countable β
|
α : Type u
β : Type v
inst✝ : Nonempty α
h : ¬Countable β
⊢ ¬Countable (α × β)
|
rw [not_countable_iff] at *
|
α : Type u
β : Type v
inst✝ : Nonempty α
h : Uncountable β
⊢ Uncountable (α × β)
|
92df1e3fe766c302
|
IsLocalization.Away.commutes
|
Mathlib/RingTheory/Localization/Away/Basic.lean
|
/-- If `S₁` is the localization of `R` away from `f` and `S₂` is the localization away from `g`,
then any localization `T` of `S₂` away from `f` is also a localization of `S₁` away from `g`. -/
lemma commutes {R : Type*} [CommSemiring R] (S₁ S₂ T : Type*) [CommSemiring S₁]
[CommSemiring S₂] [CommSemiring T] [Algebra R S₁] [Algebra R S₂] [Algebra R T] [Algebra S₁ T]
[Algebra S₂ T] [IsScalarTower R S₁ T] [IsScalarTower R S₂ T] (x y : R)
[IsLocalization.Away x S₁] [IsLocalization.Away y S₂]
[IsLocalization.Away (algebraMap R S₂ x) T] :
IsLocalization.Away (algebraMap R S₁ y) T
|
case h.e.h.e'_3.h
R : Type u_5
inst✝¹³ : CommSemiring R
S₁ : Type u_6
S₂ : Type u_7
T : Type u_8
inst✝¹² : CommSemiring S₁
inst✝¹¹ : CommSemiring S₂
inst✝¹⁰ : CommSemiring T
inst✝⁹ : Algebra R S₁
inst✝⁸ : Algebra R S₂
inst✝⁷ : Algebra R T
inst✝⁶ : Algebra S₁ T
inst✝⁵ : Algebra S₂ T
inst✝⁴ : IsScalarTower R S₁ T
inst✝³ : IsScalarTower R S₂ T
x y : R
inst✝² : Away x S₁
inst✝¹ : Away y S₂
inst✝ : Away ((algebraMap R S₂) x) T
this : IsLocalization (Algebra.algebraMapSubmonoid S₂ (Submonoid.powers x)) T
⊢ Submonoid.powers ((algebraMap R S₁) y) = Algebra.algebraMapSubmonoid S₁ (Submonoid.powers y)
|
ext x
|
case h.e.h.e'_3.h.h
R : Type u_5
inst✝¹³ : CommSemiring R
S₁ : Type u_6
S₂ : Type u_7
T : Type u_8
inst✝¹² : CommSemiring S₁
inst✝¹¹ : CommSemiring S₂
inst✝¹⁰ : CommSemiring T
inst✝⁹ : Algebra R S₁
inst✝⁸ : Algebra R S₂
inst✝⁷ : Algebra R T
inst✝⁶ : Algebra S₁ T
inst✝⁵ : Algebra S₂ T
inst✝⁴ : IsScalarTower R S₁ T
inst✝³ : IsScalarTower R S₂ T
x✝ y : R
inst✝² : Away x✝ S₁
inst✝¹ : Away y S₂
inst✝ : Away ((algebraMap R S₂) x✝) T
this : IsLocalization (Algebra.algebraMapSubmonoid S₂ (Submonoid.powers x✝)) T
x : S₁
⊢ x ∈ Submonoid.powers ((algebraMap R S₁) y) ↔ x ∈ Algebra.algebraMapSubmonoid S₁ (Submonoid.powers y)
|
ce56f98c4df0dd7c
|
Cardinal.mk_subset_mk_lt_cof
|
Mathlib/SetTheory/Cardinal/Cofinality.lean
|
theorem mk_subset_mk_lt_cof {α : Type*} (h : ∀ x < #α, 2 ^ x < #α) :
#{ s : Set α // #s < cof (#α).ord } = #α
|
case inl
α : Type u_1
h : ∀ x < #α, 2 ^ x < #α
ha : #α = 0
⊢ #{ s // #↑s < (#α).ord.cof } = #α
|
simp [ha]
|
no goals
|
6c44834aaa390522
|
IsProperMap.prodMap
|
Mathlib/Topology/Maps/Proper/Basic.lean
|
/-- A binary product of proper maps is proper. -/
lemma IsProperMap.prodMap {g : Z → W} (hf : IsProperMap f) (hg : IsProperMap g) :
IsProperMap (Prod.map f g)
|
X : Type u_1
Y : Type u_2
Z : Type u_3
W : Type u_4
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : TopologicalSpace W
f : X → Y
g : Z → W
hf : Continuous f ∧ ∀ ⦃𝒰 : Ultrafilter X⦄ ⦃y : Y⦄, Tendsto f (↑𝒰) (𝓝 y) → ∃ x, f x = y ∧ ↑𝒰 ≤ 𝓝 x
hg : Continuous g ∧ ∀ ⦃𝒰 : Ultrafilter Z⦄ ⦃y : W⦄, Tendsto g (↑𝒰) (𝓝 y) → ∃ x, g x = y ∧ ↑𝒰 ≤ 𝓝 x
𝒰 : Ultrafilter (X × Z)
y : Y
w : W
hyw : Tendsto (fun n => (Prod.map f g n).1) (↑𝒰) (𝓝 y) ∧ Tendsto (fun n => (Prod.map f g n).2) (↑𝒰) (𝓝 w)
x : X
hxy : f x = y
hx : ↑(Ultrafilter.map fst 𝒰) ≤ 𝓝 x
⊢ Tendsto g (↑(Ultrafilter.map snd 𝒰)) (𝓝 w)
|
simpa using hyw.2
|
no goals
|
137a7c14fc720f6e
|
Equiv.Perm.Disjoint.disjoint_cycleFactorsFinset
|
Mathlib/GroupTheory/Perm/Cycle/Factors.lean
|
theorem Disjoint.disjoint_cycleFactorsFinset {f g : Perm α} (h : Disjoint f g) :
_root_.Disjoint (cycleFactorsFinset f) (cycleFactorsFinset g)
|
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f g : Perm α
h : _root_.Disjoint f.support g.support
x : Perm α
hf : ∀ (a : α), x a ≠ a → x a = f a
a : α
ha : x a ≠ a
hg : ∀ (a : α), x a ≠ a → x a = g a
⊢ a ∈ f.support ∩ g.support
|
simp [ha, ← hf a ha, ← hg a ha]
|
no goals
|
f04b07da2504b5bf
|
Multiset.Subset.ndunion_eq_right
|
Mathlib/Data/Multiset/FinsetOps.lean
|
theorem Subset.ndunion_eq_right {s t : Multiset α} (h : s ⊆ t) : s.ndunion t = t
|
α : Type u_1
inst✝ : DecidableEq α
s t : Multiset α
h : s ⊆ t
⊢ s.ndunion t = t
|
induction s, t using Quot.induction_on₂
|
case h
α : Type u_1
inst✝ : DecidableEq α
a✝ b✝ : List α
h : Quot.mk (⇑(isSetoid α)) a✝ ⊆ Quot.mk (⇑(isSetoid α)) b✝
⊢ ndunion (Quot.mk (⇑(isSetoid α)) a✝) (Quot.mk (⇑(isSetoid α)) b✝) = Quot.mk (⇑(isSetoid α)) b✝
|
a3900f9c2876fc14
|
DirichletCharacter.convolution_mul_moebius
|
Mathlib/NumberTheory/LSeries/Dirichlet.lean
|
/-- The convolution of a Dirichlet character `χ` with the twist `χ * μ` is `δ`,
the indicator function of `{1}`. -/
lemma convolution_mul_moebius {n : ℕ} (χ : DirichletCharacter ℂ n) : ↗χ ⍟ (↗χ * ↗μ) = δ
|
n : ℕ
χ : DirichletCharacter ℂ n
this : (1 ⍟ fun x => ↑(μ x)) = δ
⊢ (fun n_1 => χ ↑n_1) * 1 ⍟ ((fun n_1 => χ ↑n_1) * fun n => ↑(μ n)) = δ
|
simpa only [mul_convolution_distrib χ 1 ↗μ, this] using mul_delta _
|
no goals
|
1c03f0abaacee504
|
NNReal.concaveOn_rpow
|
Mathlib/Analysis/Convex/SpecificFunctions/Pow.lean
|
lemma concaveOn_rpow {p : ℝ} (hp₀ : 0 ≤ p) (hp₁ : p ≤ 1) :
ConcaveOn ℝ≥0 univ fun x : ℝ≥0 ↦ x ^ p
|
p : ℝ
hp₀ : 0 ≤ p
hp₁ : p ≤ 1
⊢ ConcaveOn ℝ≥0 univ fun x => x ^ p
|
rcases eq_or_lt_of_le hp₀ with (rfl | hp₀)
|
case inl
hp₀ : 0 ≤ 0
hp₁ : 0 ≤ 1
⊢ ConcaveOn ℝ≥0 univ fun x => x ^ 0
case inr
p : ℝ
hp₀✝ : 0 ≤ p
hp₁ : p ≤ 1
hp₀ : 0 < p
⊢ ConcaveOn ℝ≥0 univ fun x => x ^ p
|
825e34473ac8096f
|
List.lex_eq_decide_lex
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Lex.lean
|
theorem lex_eq_decide_lex [DecidableEq α] (lt : α → α → Bool) :
lex l₁ l₂ lt = decide (Lex (fun x y => lt x y) l₁ l₂)
|
case cons.nil
α : Type u_1
inst✝ : DecidableEq α
lt : α → α → Bool
a : α
l₁ : List α
ih : ∀ {l₂ : List α}, l₁.lex l₂ lt = decide (Lex (fun x y => lt x y = true) l₁ l₂)
⊢ (a :: l₁).lex [] lt = decide (Lex (fun x y => lt x y = true) (a :: l₁) [])
|
simp [lex]
|
no goals
|
9226dc6b1512e752
|
Basis.prod_apply_inl_snd
|
Mathlib/LinearAlgebra/Basis/Basic.lean
|
theorem prod_apply_inl_snd (i) : (b.prod b' (Sum.inl i)).2 = 0 :=
b'.repr.injective <| by
ext j
simp only [Basis.prod, Basis.coe_ofRepr, LinearEquiv.symm_trans_apply, LinearEquiv.prod_symm,
LinearEquiv.prod_apply, b'.repr.apply_symm_apply, LinearEquiv.symm_symm, repr_self,
Equiv.toFun_as_coe, Finsupp.snd_sumFinsuppLEquivProdFinsupp, LinearEquiv.map_zero,
Finsupp.zero_apply]
apply Finsupp.single_eq_of_ne Sum.inl_ne_inr
|
case h
ι : Type u_1
ι' : Type u_2
R : Type u_3
M : Type u_5
M' : Type u_6
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
b : Basis ι R M
b' : Basis ι' R M'
i : ι
j : ι'
⊢ (single (Sum.inl i) 1) (Sum.inr j) = 0
|
apply Finsupp.single_eq_of_ne Sum.inl_ne_inr
|
no goals
|
30421bfd5860cfaf
|
AlgebraicIndependent.isTranscendenceBasis_iff
|
Mathlib/RingTheory/AlgebraicIndependent/TranscendenceBasis.lean
|
theorem AlgebraicIndependent.isTranscendenceBasis_iff {ι : Type w} {R : Type u} [CommRing R]
[Nontrivial R] {A : Type v} [CommRing A] [Algebra R A] {x : ι → A}
(i : AlgebraicIndependent R x) :
IsTranscendenceBasis R x ↔
∀ (κ : Type v) (w : κ → A) (_ : AlgebraicIndependent R w) (j : ι → κ) (_ : w ∘ j = x),
Surjective j
|
case right
ι : Type w
R : Type u
inst✝³ : CommRing R
inst✝² : Nontrivial R
A : Type v
inst✝¹ : CommRing A
inst✝ : Algebra R A
x : ι → A
i : AlgebraicIndependent R x
w : Set A
i' : AlgebraicIndependent R Subtype.val
h : range x ≤ w
p : Surjective fun i => ⟨x i, ⋯⟩
q : (fun s => Subtype.val '' s) (range fun i => ⟨x i, ⋯⟩) = (fun s => Subtype.val '' s) univ
⊢ range x = w
|
dsimp at q
|
case right
ι : Type w
R : Type u
inst✝³ : CommRing R
inst✝² : Nontrivial R
A : Type v
inst✝¹ : CommRing A
inst✝ : Algebra R A
x : ι → A
i : AlgebraicIndependent R x
w : Set A
i' : AlgebraicIndependent R Subtype.val
h : range x ≤ w
p : Surjective fun i => ⟨x i, ⋯⟩
q : (Subtype.val '' range fun i => ⟨x i, ⋯⟩) = Subtype.val '' univ
⊢ range x = w
|
5618ce02d41569c5
|
Matrix.Pivot.exists_isTwoBlockDiagonal_list_transvec_mul_mul_list_transvec
|
Mathlib/LinearAlgebra/Matrix/Transvection.lean
|
theorem exists_isTwoBlockDiagonal_list_transvec_mul_mul_list_transvec
(M : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜) :
∃ L L' : List (TransvectionStruct (Fin r ⊕ Unit) 𝕜),
IsTwoBlockDiagonal ((L.map toMatrix).prod * M * (L'.map toMatrix).prod)
|
case neg.intro.inl
𝕜 : Type u_3
inst✝ : Field 𝕜
r : ℕ
M : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜
hM : M (inr ()) (inr ()) = 0
H :
(¬∀ (i : Fin r) (j : Unit), of (fun i j => M (inl i) (inr j)) i j = 0 i j) ∨
¬∀ (i : Unit) (j : Fin r), of (fun i j => M (inr i) (inl j)) i j = 0 i j
i : Fin r
h : M (inl i) (inr ()) ≠ 0
M' : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜 := transvection (inr ()) (inl i) 1 * M
⊢ ∃ L L', ((List.map toMatrix L).prod * M * (List.map toMatrix L').prod).IsTwoBlockDiagonal
|
have hM' : M' (inr unit) (inr unit) ≠ 0 := by simpa [M', hM]
|
case neg.intro.inl
𝕜 : Type u_3
inst✝ : Field 𝕜
r : ℕ
M : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜
hM : M (inr ()) (inr ()) = 0
H :
(¬∀ (i : Fin r) (j : Unit), of (fun i j => M (inl i) (inr j)) i j = 0 i j) ∨
¬∀ (i : Unit) (j : Fin r), of (fun i j => M (inr i) (inl j)) i j = 0 i j
i : Fin r
h : M (inl i) (inr ()) ≠ 0
M' : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜 := transvection (inr ()) (inl i) 1 * M
hM' : M' (inr ()) (inr ()) ≠ 0
⊢ ∃ L L', ((List.map toMatrix L).prod * M * (List.map toMatrix L').prod).IsTwoBlockDiagonal
|
ef47f5e7783abe91
|
exists_pos_right_iff_sameRay_and_ne_zero
|
Mathlib/LinearAlgebra/Ray.lean
|
theorem exists_pos_right_iff_sameRay_and_ne_zero (hy : y ≠ 0) :
(∃ r : R, 0 < r ∧ x = r • y) ↔ SameRay R x y ∧ x ≠ 0
|
R : Type u_1
inst✝² : LinearOrderedField R
M : Type u_2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
x y : M
hy : y ≠ 0
⊢ (∃ r, 0 < r ∧ x = r • y) ↔ SameRay R y x ∧ x ≠ 0
|
simp_rw [eq_comm (a := x)]
|
R : Type u_1
inst✝² : LinearOrderedField R
M : Type u_2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
x y : M
hy : y ≠ 0
⊢ (∃ r, 0 < r ∧ r • y = x) ↔ SameRay R y x ∧ x ≠ 0
|
37f9b198aa5d64db
|
Array.foldr_reverse'
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean
|
theorem foldr_reverse' (l : Array α) (f : α → β → β) (b) (w : start = l.size) :
l.reverse.foldr f b start 0 = l.foldl (fun x y => f y x) b
|
α : Type u_1
β : Type u_2
start : Nat
l : Array α
f : α → β → β
b : β
w : start = l.size
⊢ foldr f b l.reverse start = foldl (fun x y => f y x) b l
|
simp [w, foldl_eq_foldlM, foldr_eq_foldrM]
|
no goals
|
cf5c1b5bec5a35f5
|
ModelWithCorners.interior_disjointUnion
|
Mathlib/Geometry/Manifold/IsManifold/InteriorBoundary.lean
|
lemma interior_disjointUnion :
ModelWithCorners.interior (I := I) (M ⊕ M') =
Sum.inl '' (ModelWithCorners.interior (I := I) M)
∪ Sum.inr '' (ModelWithCorners.interior (I := I) M')
|
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
M' : Type u_5
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H M'
p : M ⊕ M'
hp : p ∈ Sum.inl '' ModelWithCorners.interior M ∪ Sum.inr '' ModelWithCorners.interior M'
h : p.isLeft = true
x : M := p.getLeft h
x_eq : x = p.getLeft h
⊢ p ∈ Sum.inl '' ModelWithCorners.interior M
|
obtain (good | ⟨y, hy, hxy⟩) := hp
|
case inl
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
M' : Type u_5
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H M'
p : M ⊕ M'
h : p.isLeft = true
x : M := p.getLeft h
x_eq : x = p.getLeft h
good : p ∈ Sum.inl '' ModelWithCorners.interior M
⊢ p ∈ Sum.inl '' ModelWithCorners.interior M
case inr.intro.intro
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
M' : Type u_5
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H M'
p : M ⊕ M'
h : p.isLeft = true
x : M := p.getLeft h
x_eq : x = p.getLeft h
y : M'
hy : y ∈ ModelWithCorners.interior M'
hxy : Sum.inr y = p
⊢ p ∈ Sum.inl '' ModelWithCorners.interior M
|
a37cb5a73686bbc7
|
EisensteinSeries.div_max_sq_ge_one
|
Mathlib/NumberTheory/ModularForms/EisensteinSeries/UniformConvergence.lean
|
lemma div_max_sq_ge_one (x : Fin 2 → ℤ) (hx : x ≠ 0) :
1 ≤ (x 0 / ‖x‖) ^ 2 ∨ 1 ≤ (x 1 / ‖x‖) ^ 2
|
x : Fin 2 → ℤ
hx : x ≠ 0
⊢ 1 ≤ (↑(x 0) / ‖x‖) ^ 2 ∨ 1 ≤ (↑(x 1) / ‖x‖) ^ 2
|
refine (max_choice (x 0).natAbs (x 1).natAbs).imp (fun H0 ↦ ?_) (fun H1 ↦ ?_)
|
case refine_1
x : Fin 2 → ℤ
hx : x ≠ 0
H0 : (x 0).natAbs ⊔ (x 1).natAbs = (x 0).natAbs
⊢ 1 ≤ (↑(x 0) / ‖x‖) ^ 2
case refine_2
x : Fin 2 → ℤ
hx : x ≠ 0
H1 : (x 0).natAbs ⊔ (x 1).natAbs = (x 1).natAbs
⊢ 1 ≤ (↑(x 1) / ‖x‖) ^ 2
|
897aecfdc409c454
|
VectorFourier.norm_fourierPowSMulRight_iteratedFDeriv_fourierIntegral_le
|
Mathlib/Analysis/Fourier/FourierTransformDeriv.lean
|
theorem norm_fourierPowSMulRight_iteratedFDeriv_fourierIntegral_le [FiniteDimensional ℝ V]
{μ : Measure V} [Measure.IsAddHaarMeasure μ] {K N : ℕ∞} (hf : ContDiff ℝ N f)
(h'f : ∀ (k n : ℕ), k ≤ K → n ≤ N → Integrable (fun v ↦ ‖v‖^k * ‖iteratedFDeriv ℝ n f v‖) μ)
{k n : ℕ} (hk : k ≤ K) (hn : n ≤ N) {w : W} :
‖fourierPowSMulRight (-L.flip)
(iteratedFDeriv ℝ k (fourierIntegral 𝐞 μ L.toLinearMap₂ f)) w n‖ ≤
(2 * π) ^ k * (2 * k + 2) ^ n * ‖L‖ ^ k * ∑ p ∈ Finset.range (k + 1) ×ˢ Finset.range (n + 1),
∫ v, ‖v‖ ^ p.1 * ‖iteratedFDeriv ℝ p.2 f v‖ ∂μ
|
E : Type u_1
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℂ E
V : Type u_2
W : Type u_3
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : NormedSpace ℝ V
inst✝⁵ : NormedAddCommGroup W
inst✝⁴ : NormedSpace ℝ W
L : V →L[ℝ] W →L[ℝ] ℝ
f : V → E
inst✝³ : MeasurableSpace V
inst✝² : BorelSpace V
inst✝¹ : FiniteDimensional ℝ V
μ : Measure V
inst✝ : μ.IsAddHaarMeasure
K N : ℕ∞
hf : ContDiff ℝ (↑N) f
h'f : ∀ (k n : ℕ), ↑k ≤ K → ↑n ≤ N → Integrable (fun v => ‖v‖ ^ k * ‖iteratedFDeriv ℝ n f v‖) μ
k n : ℕ
hk : ↑k ≤ K
hn : ↑n ≤ N
w : W
⊢ ∫ (v : V), ‖iteratedFDeriv ℝ n (fun v => fourierPowSMulRight L f v k) v‖ ∂μ ≤
(2 * π) ^ k * (2 * ↑k + 2) ^ n * ‖L‖ ^ k *
∑ p ∈ Finset.range (k + 1) ×ˢ Finset.range (n + 1), ∫ (v : V), ‖v‖ ^ p.1 * ‖iteratedFDeriv ℝ p.2 f v‖ ∂μ
|
have I p (hp : p ∈ Finset.range (k + 1) ×ˢ Finset.range (n + 1)) :
Integrable (fun v ↦ ‖v‖ ^ p.1 * ‖iteratedFDeriv ℝ p.2 f v‖) μ := by
simp only [Finset.mem_product, Finset.mem_range_succ_iff] at hp
exact h'f _ _ (le_trans (by simpa using hp.1) hk) (le_trans (by simpa using hp.2) hn)
|
E : Type u_1
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℂ E
V : Type u_2
W : Type u_3
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : NormedSpace ℝ V
inst✝⁵ : NormedAddCommGroup W
inst✝⁴ : NormedSpace ℝ W
L : V →L[ℝ] W →L[ℝ] ℝ
f : V → E
inst✝³ : MeasurableSpace V
inst✝² : BorelSpace V
inst✝¹ : FiniteDimensional ℝ V
μ : Measure V
inst✝ : μ.IsAddHaarMeasure
K N : ℕ∞
hf : ContDiff ℝ (↑N) f
h'f : ∀ (k n : ℕ), ↑k ≤ K → ↑n ≤ N → Integrable (fun v => ‖v‖ ^ k * ‖iteratedFDeriv ℝ n f v‖) μ
k n : ℕ
hk : ↑k ≤ K
hn : ↑n ≤ N
w : W
I : ∀ p ∈ Finset.range (k + 1) ×ˢ Finset.range (n + 1), Integrable (fun v => ‖v‖ ^ p.1 * ‖iteratedFDeriv ℝ p.2 f v‖) μ
⊢ ∫ (v : V), ‖iteratedFDeriv ℝ n (fun v => fourierPowSMulRight L f v k) v‖ ∂μ ≤
(2 * π) ^ k * (2 * ↑k + 2) ^ n * ‖L‖ ^ k *
∑ p ∈ Finset.range (k + 1) ×ˢ Finset.range (n + 1), ∫ (v : V), ‖v‖ ^ p.1 * ‖iteratedFDeriv ℝ p.2 f v‖ ∂μ
|
8318f0ae43be69ad
|
BoundedContinuousFunction.dist_extend_extend
|
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
theorem dist_extend_extend (f : α ↪ δ) (g₁ g₂ : α →ᵇ β) (h₁ h₂ : δ →ᵇ β) :
dist (g₁.extend f h₁) (g₂.extend f h₂) =
max (dist g₁ g₂) (dist (h₁.restrict (range f)ᶜ) (h₂.restrict (range f)ᶜ))
|
case refine_3
α : Type u
β : Type v
inst✝³ : TopologicalSpace α
inst✝² : PseudoMetricSpace β
δ : Type u_2
inst✝¹ : TopologicalSpace δ
inst✝ : DiscreteTopology δ
f : α ↪ δ
g₁ g₂ : α →ᵇ β
h₁ h₂ : δ →ᵇ β
x : ↑(range ⇑f)ᶜ
⊢ dist ((h₁.restrict (range ⇑f)ᶜ) x) ((h₂.restrict (range ⇑f)ᶜ) x) ≤ dist (extend f g₁ h₁) (extend f g₂ h₂)
|
calc
dist (h₁ x) (h₂ x) = dist (extend f g₁ h₁ x) (extend f g₂ h₂ x) := by
rw [extend_apply' x.coe_prop, extend_apply' x.coe_prop]
_ ≤ _ := dist_coe_le_dist _
|
no goals
|
45cb1096070ba961
|
Int.not_even_iff
|
Mathlib/Algebra/Group/Int/Even.lean
|
lemma not_even_iff : ¬Even n ↔ n % 2 = 1
|
n : ℤ
⊢ ¬Even n ↔ n % 2 = 1
|
rw [even_iff, emod_two_ne_zero]
|
no goals
|
2a3dce372a304bf1
|
Submodule.exists_fg_le_eq_rTensor_subtype
|
Mathlib/RingTheory/Finiteness/TensorProduct.lean
|
theorem exists_fg_le_eq_rTensor_subtype (x : N ⊗ M) :
∃ (J : Submodule R N) (_ : J.FG) (y : J ⊗ M), x = rTensor M J.subtype y
|
case add.intro.intro.intro.intro.intro.intro
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid N
inst✝¹ : Module R M
inst✝ : Module R N
J₁ : Submodule R N
fg₁ : J₁.FG
y₁ : ↥J₁ ⊗[R] M
J₂ : Submodule R N
fg₂ : J₂.FG
y₂ : ↥J₂ ⊗[R] M
⊢ ∃ J, ∃ (_ : J.FG), ∃ y, (rTensor M J₁.subtype) y₁ + (rTensor M J₂.subtype) y₂ = (rTensor M J.subtype) y
|
refine ⟨J₁ ⊔ J₂, fg₁.sup fg₂,
rTensor M (J₁.inclusion le_sup_left) y₁ + rTensor M (J₂.inclusion le_sup_right) y₂, ?_⟩
|
case add.intro.intro.intro.intro.intro.intro
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid N
inst✝¹ : Module R M
inst✝ : Module R N
J₁ : Submodule R N
fg₁ : J₁.FG
y₁ : ↥J₁ ⊗[R] M
J₂ : Submodule R N
fg₂ : J₂.FG
y₂ : ↥J₂ ⊗[R] M
⊢ (rTensor M J₁.subtype) y₁ + (rTensor M J₂.subtype) y₂ =
(rTensor M (J₁ ⊔ J₂).subtype) ((rTensor M (inclusion ⋯)) y₁ + (rTensor M (inclusion ⋯)) y₂)
|
24b6c2f045083d77
|
List.prev_reverse_eq_next
|
Mathlib/Data/List/Cycle.lean
|
theorem prev_reverse_eq_next (l : List α) (h : Nodup l) (x : α) (hx : x ∈ l) :
prev l.reverse x (mem_reverse.mpr hx) = next l x hx
|
α : Type u_1
inst✝ : DecidableEq α
l : List α
h : l.Nodup
x : α
hx : x ∈ l
⊢ l.reverse.prev x ⋯ = l.next x hx
|
obtain ⟨k, hk, rfl⟩ := getElem_of_mem hx
|
case intro.intro
α : Type u_1
inst✝ : DecidableEq α
l : List α
h : l.Nodup
k : ℕ
hk : k < l.length
hx : l[k] ∈ l
⊢ l.reverse.prev l[k] ⋯ = l.next l[k] hx
|
223a9f0719f3b161
|
LinearMap.mem_submoduleImage
|
Mathlib/Algebra/Module/Submodule/Range.lean
|
theorem mem_submoduleImage {M' : Type*} [AddCommMonoid M'] [Module R M'] {O : Submodule R M}
{ϕ : O →ₗ[R] M'} {N : Submodule R M} {x : M'} :
x ∈ ϕ.submoduleImage N ↔ ∃ (y : _) (yO : y ∈ O), y ∈ N ∧ ϕ ⟨y, yO⟩ = x
|
case refine_1
R : Type u_1
M : Type u_5
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
M' : Type u_10
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
O : Submodule R M
ϕ : ↥O →ₗ[R] M'
N : Submodule R M
x : M'
⊢ (∃ y, O.subtype y ∈ N ∧ ϕ y = x) → ∃ y, ∃ (yO : y ∈ O), y ∈ N ∧ ϕ ⟨y, yO⟩ = x
|
rintro ⟨⟨y, yO⟩, yN : y ∈ N, h⟩
|
case refine_1.intro.mk.intro
R : Type u_1
M : Type u_5
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
M' : Type u_10
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
O : Submodule R M
ϕ : ↥O →ₗ[R] M'
N : Submodule R M
x : M'
y : M
yO : y ∈ O
yN : y ∈ N
h : ϕ ⟨y, yO⟩ = x
⊢ ∃ y, ∃ (yO : y ∈ O), y ∈ N ∧ ϕ ⟨y, yO⟩ = x
|
956940ccbba13812
|
PiTensorProduct.algebraMap_apply
|
Mathlib/RingTheory/PiTensorProduct.lean
|
lemma algebraMap_apply (r : R') (i : ι) [DecidableEq ι] :
algebraMap R' (⨂[R] i, A i) r = tprod R (Pi.mulSingle i (algebraMap R' (A i) r))
|
ι : Type u_1
R' : Type u_2
R : Type u_3
A : ι → Type u_4
inst✝⁷ : CommSemiring R'
inst✝⁶ : CommSemiring R
inst✝⁵ : (i : ι) → Semiring (A i)
inst✝⁴ : Algebra R' R
inst✝³ : (i : ι) → Algebra R (A i)
inst✝² : (i : ι) → Algebra R' (A i)
inst✝¹ : ∀ (i : ι), IsScalarTower R' R (A i)
r : R'
i : ι
inst✝ : DecidableEq ι
⊢ Pi.mulSingle i (r • 1) = update (fun i => 1) i (r • 1)
|
rfl
|
no goals
|
182d0b1194f24ced
|
SzemerediRegularity.card_chunk
|
Mathlib/Combinatorics/SimpleGraph/Regularity/Chunk.lean
|
theorem card_chunk (hm : m ≠ 0) : #(chunk hP G ε hU).parts = 4 ^ #P.parts
|
case neg
α : Type u_1
inst✝² : Fintype α
inst✝¹ : DecidableEq α
P : Finpartition univ
hP : P.IsEquipartition
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
ε : ℝ
U : Finset α
hU : U ∈ P.parts
hm : m ≠ 0
h✝ : ¬#U = m * 4 ^ #P.parts + (Fintype.card α / #P.parts - m * 4 ^ #P.parts)
⊢ #(equitabilise ⋯).parts = 4 ^ #P.parts
|
rw [card_parts_equitabilise _ _ hm, tsub_add_cancel_of_le a_add_one_le_four_pow_parts_card]
|
no goals
|
ed00f8abb571263d
|
Subgroup.exists_pow_mem_of_index_ne_zero
|
Mathlib/GroupTheory/Index.lean
|
@[to_additive]
lemma exists_pow_mem_of_index_ne_zero (h : H.index ≠ 0) (a : G) :
∃ n, 0 < n ∧ n ≤ H.index ∧ a ^ n ∈ H
|
G : Type u_1
inst✝ : Group G
H : Subgroup G
h : H.index ≠ 0
a : G
n₁ n₂ : ℕ
hlt : n₁ < n₂
hle : n₂ ≤ H.index
he : ↑(a ^ n₂) = ↑(a ^ n₁)
⊢ n₂ - n₁ ≤ H.index
|
omega
|
no goals
|
e678927c1f4a5913
|
minpolyDiv_monic
|
Mathlib/FieldTheory/Minpoly/MinpolyDiv.lean
|
lemma minpolyDiv_monic : Monic (minpolyDiv R x)
|
R : Type u_2
S : Type u_1
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S
hx : IsIntegral R x
a✝ : Nontrivial S
this : (minpolyDiv R x).leadingCoeff * (X - C x).leadingCoeff = 1
⊢ (minpolyDiv R x).Monic
|
simpa using this
|
no goals
|
2b365b24f6e61a1f
|
Polynomial.eq_of_degree_sub_lt_of_eval_finset_eq
|
Mathlib/LinearAlgebra/Lagrange.lean
|
theorem eq_of_degree_sub_lt_of_eval_finset_eq (degree_fg_lt : (f - g).degree < #s)
(eval_fg : ∀ x ∈ s, f.eval x = g.eval x) : f = g
|
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
f g : R[X]
s : Finset R
degree_fg_lt : (f - g).degree < ↑(#s)
eval_fg : ∀ x ∈ s, eval x f = eval x g
⊢ ∀ x ∈ s, eval x f = eval x g
|
exact eval_fg
|
no goals
|
a45b7536783fc75f
|
Vector.eq_push_append_of_mem
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Vector/Lemmas.lean
|
theorem eq_push_append_of_mem {xs : Vector α n} {x : α} (h : x ∈ xs) :
∃ (n₁ n₂ : Nat) (as : Vector α n₁) (bs : Vector α n₂) (h : n₁ + 1 + n₂ = n),
xs = (as.push x ++ bs).cast h ∧ x ∉ as
|
case mk.intro.intro.intro
α : Type u_1
x : α
xs : Array α
h✝ : x ∈ { toArray := xs, size_toArray := ⋯ }
as bs : Array α
h : xs = as.push x ++ bs
w : ¬x ∈ as
⊢ ∃ n₁ n₂ as bs h, { toArray := xs, size_toArray := ⋯ } = Vector.cast h (as.push x ++ bs) ∧ ¬x ∈ as
|
obtain rfl := h
|
case mk.intro.intro.intro
α : Type u_1
x : α
as bs : Array α
w : ¬x ∈ as
h : x ∈ { toArray := as.push x ++ bs, size_toArray := ⋯ }
⊢ ∃ n₁ n₂ as_1 bs_1 h,
{ toArray := as.push x ++ bs, size_toArray := ⋯ } = Vector.cast h (as_1.push x ++ bs_1) ∧ ¬x ∈ as_1
|
10d47003372b5a57
|
Polynomial.map_mod_divByMonic
|
Mathlib/Algebra/Polynomial/Div.lean
|
theorem map_mod_divByMonic [Ring S] (f : R →+* S) (hq : Monic q) :
(p /ₘ q).map f = p.map f /ₘ q.map f ∧ (p %ₘ q).map f = p.map f %ₘ q.map f
|
R : Type u
S : Type v
inst✝¹ : Ring R
p q : R[X]
inst✝ : Ring S
f : R →+* S
hq : q.Monic
⊢ map f (p /ₘ q) = map f p /ₘ map f q ∧ map f (p %ₘ q) = map f p %ₘ map f q
|
nontriviality S
|
R : Type u
S : Type v
inst✝¹ : Ring R
p q : R[X]
inst✝ : Ring S
f : R →+* S
hq : q.Monic
a✝ : Nontrivial S
⊢ map f (p /ₘ q) = map f p /ₘ map f q ∧ map f (p %ₘ q) = map f p %ₘ map f q
|
bad20178d8d590e7
|
AnalyticOnNhd.isClopen_setOf_order_eq_top
|
Mathlib/Analysis/Analytic/Order.lean
|
theorem isClopen_setOf_order_eq_top (h₁f : AnalyticOnNhd 𝕜 f U) :
IsClopen { u : U | (h₁f u.1 u.2).order = ⊤ }
|
case right
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
U : Set 𝕜
h₁f : AnalyticOnNhd 𝕜 f U
z : ↑U
hz : ∃ t, (∀ y ∈ t, f y = 0) ∧ IsOpen t ∧ ↑z ∈ t
⊢ ∃ x ⊆ {x | ∃ t, (∀ y ∈ t, f y = 0) ∧ IsOpen t ∧ ↑x ∈ t}, IsOpen x ∧ z ∈ x
|
obtain ⟨t', h₁t', h₂t', h₃t'⟩ := hz
|
case right.intro.intro.intro
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
U : Set 𝕜
h₁f : AnalyticOnNhd 𝕜 f U
z : ↑U
t' : Set 𝕜
h₁t' : ∀ y ∈ t', f y = 0
h₂t' : IsOpen t'
h₃t' : ↑z ∈ t'
⊢ ∃ x ⊆ {x | ∃ t, (∀ y ∈ t, f y = 0) ∧ IsOpen t ∧ ↑x ∈ t}, IsOpen x ∧ z ∈ x
|
d5d16b4618a2912b
|
CoxeterSystem.length_eq_one_iff
|
Mathlib/GroupTheory/Coxeter/Length.lean
|
theorem length_eq_one_iff {w : W} : ℓ w = 1 ↔ ∃ i : B, w = s i
|
case mpr
B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
w : W
⊢ (∃ i, w = cs.simple i) → cs.length w = 1
|
rintro ⟨i, rfl⟩
|
case mpr.intro
B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
i : B
⊢ cs.length (cs.simple i) = 1
|
cc0b68a3032cf331
|
HolderOnWith.hausdorffMeasure_image_le
|
Mathlib/MeasureTheory/Measure/Hausdorff.lean
|
theorem hausdorffMeasure_image_le (h : HolderOnWith C r f s) (hr : 0 < r) {d : ℝ} (hd : 0 ≤ d) :
μH[d] (f '' s) ≤ (C : ℝ≥0∞) ^ d * μH[r * d] s
|
case inl.inr.intro
X : Type u_2
Y : Type u_3
inst✝⁵ : EMetricSpace X
inst✝⁴ : EMetricSpace Y
inst✝³ : MeasurableSpace X
inst✝² : BorelSpace X
inst✝¹ : MeasurableSpace Y
inst✝ : BorelSpace Y
r : ℝ≥0
f : X → Y
s : Set X
hr : 0 < r
d : ℝ
hd : 0 ≤ d
h : HolderOnWith 0 r f s
x : X
hx : x ∈ s
this : f '' s = {f x}
⊢ μH[d] (f '' s) ≤ ↑0 ^ d * μH[↑r * d] s
|
rw [this]
|
case inl.inr.intro
X : Type u_2
Y : Type u_3
inst✝⁵ : EMetricSpace X
inst✝⁴ : EMetricSpace Y
inst✝³ : MeasurableSpace X
inst✝² : BorelSpace X
inst✝¹ : MeasurableSpace Y
inst✝ : BorelSpace Y
r : ℝ≥0
f : X → Y
s : Set X
hr : 0 < r
d : ℝ
hd : 0 ≤ d
h : HolderOnWith 0 r f s
x : X
hx : x ∈ s
this : f '' s = {f x}
⊢ μH[d] {f x} ≤ ↑0 ^ d * μH[↑r * d] s
|
cf145f55a786d8cb
|
TopologicalSpace.isTopologicalBasis_of_subbasis
|
Mathlib/Topology/Bases.lean
|
theorem isTopologicalBasis_of_subbasis {s : Set (Set α)} (hs : t = generateFrom s) :
IsTopologicalBasis ((fun f => ⋂₀ f) '' { f : Set (Set α) | f.Finite ∧ f ⊆ s })
|
case refine_3.intro.intro.intro
α : Type u
s : Set (Set α)
this : TopologicalSpace α := generateFrom s
t : Set (Set α)
hft : t.Finite
htb : t ⊆ s
⊢ IsOpen ((fun f => ⋂₀ f) t)
|
exact hft.isOpen_sInter fun s hs ↦ GenerateOpen.basic _ <| htb hs
|
no goals
|
0ac20a1a81c48653
|
LinearRecurrence.eq_mk_of_is_sol_of_eq_init
|
Mathlib/Algebra/LinearRecurrence.lean
|
theorem eq_mk_of_is_sol_of_eq_init {u : ℕ → α} {init : Fin E.order → α} (h : E.IsSolution u)
(heq : ∀ n : Fin E.order, u n = init n) : ∀ n, u n = E.mkSol init n
|
case neg.e_f.h
α : Type u_1
inst✝ : CommSemiring α
E : LinearRecurrence α
u : ℕ → α
init : Fin E.order → α
h : E.IsSolution u
heq : ∀ (n : Fin E.order), u ↑n = init n
n : ℕ
h' : ¬n < E.order
k : Fin E.order
this : n - E.order + ↑k < n
⊢ E.coeffs k * E.mkSol init (n - E.order + ↑k) = E.coeffs k * E.mkSol init (n - E.order + E.order - E.order + ↑k)
|
simp
|
no goals
|
c844dd4d1e0a7cc3
|
AddAction.automorphize_smul_left
|
Mathlib/Topology/Algebra/InfiniteSum/Module.lean
|
/-- Automorphization of a function into an `R`-`Module` distributes, that is, commutes with the
`R`-scalar multiplication. -/
lemma AddAction.automorphize_smul_left [AddGroup α] [AddAction α β] (f : β → M)
(g : Quotient (AddAction.orbitRel α β) → R) :
AddAction.automorphize ((g ∘ (@Quotient.mk' _ (_))) • f)
= g • (AddAction.automorphize f : Quotient (AddAction.orbitRel α β) → M)
|
α : Type u_1
β : Type u_2
M : Type u_11
inst✝⁷ : TopologicalSpace M
inst✝⁶ : AddCommMonoid M
inst✝⁵ : T2Space M
R : Type u_12
inst✝⁴ : DivisionRing R
inst✝³ : Module R M
inst✝² : ContinuousConstSMul R M
inst✝¹ : AddGroup α
inst✝ : AddAction α β
f : β → M
g : Quotient (orbitRel α β) → R
x : Quotient (orbitRel α β)
b : β
π : β → Quotient (orbitRel α β) := Quotient.mk (orbitRel α β)
a : α
⊢ (orbitRel α β) (a +ᵥ b) b
|
use a
|
no goals
|
f7dea7bb2091769a
|
TopologicalSpace.Opens.isCompactElement_iff
|
Mathlib/Topology/Sets/Opens.lean
|
theorem isCompactElement_iff (s : Opens α) :
CompleteLattice.IsCompactElement s ↔ IsCompact (s : Set α)
|
case refine_1.intro
α : Type u_2
inst✝ : TopologicalSpace α
s : Opens α
H : ∀ (ι : Type u_2) (s_1 : ι → Opens α), s ≤ iSup s_1 → ∃ t, s ≤ t.sup s_1
ι : Type u_2
U : ι → Set α
hU : ∀ (i : ι), IsOpen (U i)
hU' : ↑s ⊆ ⋃ i, U i
t : Finset ι
ht : s ≤ t.sup fun i => { carrier := U i, is_open' := ⋯ }
⊢ ∃ t, ↑s ⊆ ⋃ i ∈ t, U i
|
refine ⟨t, Set.Subset.trans ht ?_⟩
|
case refine_1.intro
α : Type u_2
inst✝ : TopologicalSpace α
s : Opens α
H : ∀ (ι : Type u_2) (s_1 : ι → Opens α), s ≤ iSup s_1 → ∃ t, s ≤ t.sup s_1
ι : Type u_2
U : ι → Set α
hU : ∀ (i : ι), IsOpen (U i)
hU' : ↑s ⊆ ⋃ i, U i
t : Finset ι
ht : s ≤ t.sup fun i => { carrier := U i, is_open' := ⋯ }
⊢ ↑(t.sup fun i => { carrier := U i, is_open' := ⋯ }) ⊆ ⋃ i ∈ t, U i
|
1b1f1c194abbad2e
|
IsFractionRing.stabilizerHom_surjective
|
Mathlib/RingTheory/Invariant.lean
|
theorem IsFractionRing.stabilizerHom_surjective :
Function.Surjective (stabilizerHom G P Q K L)
|
A : Type u_1
B : Type u_2
inst✝¹⁹ : CommRing A
inst✝¹⁸ : CommRing B
inst✝¹⁷ : Algebra A B
G : Type u_3
inst✝¹⁶ : Group G
inst✝¹⁵ : Finite G
inst✝¹⁴ : MulSemiringAction G B
inst✝¹³ : SMulCommClass G A B
P : Ideal A
Q : Ideal B
inst✝¹² : Q.IsPrime
inst✝¹¹ : Q.LiesOver P
K : Type u_4
L : Type u_5
inst✝¹⁰ : Field K
inst✝⁹ : Field L
inst✝⁸ : Algebra (A ⧸ P) K
inst✝⁷ : Algebra (B ⧸ Q) L
inst✝⁶ : Algebra (A ⧸ P) L
inst✝⁵ : IsScalarTower (A ⧸ P) (B ⧸ Q) L
inst✝⁴ : Algebra K L
inst✝³ : IsScalarTower (A ⧸ P) K L
inst✝² : Algebra.IsInvariant A B G
inst✝¹ : IsFractionRing (A ⧸ P) K
inst✝ : IsFractionRing (B ⧸ Q) L
x✝ : MulSemiringAction (↥(MulAction.stabilizer G Q)) L := MulSemiringAction.compHom L (stabilizerHom G P Q K L)
f : L ≃ₐ[K] L
⊢ ∃ a, (stabilizerHom G P Q K L) a = f
|
obtain ⟨g, hg⟩ := FixedPoints.toAlgAut_surjective (MulAction.stabilizer G Q) L
(AlgEquiv.ofRingEquiv (f := f) (fun x ↦ fixed_of_fixed2 G P Q K L f x x.2))
|
case intro
A : Type u_1
B : Type u_2
inst✝¹⁹ : CommRing A
inst✝¹⁸ : CommRing B
inst✝¹⁷ : Algebra A B
G : Type u_3
inst✝¹⁶ : Group G
inst✝¹⁵ : Finite G
inst✝¹⁴ : MulSemiringAction G B
inst✝¹³ : SMulCommClass G A B
P : Ideal A
Q : Ideal B
inst✝¹² : Q.IsPrime
inst✝¹¹ : Q.LiesOver P
K : Type u_4
L : Type u_5
inst✝¹⁰ : Field K
inst✝⁹ : Field L
inst✝⁸ : Algebra (A ⧸ P) K
inst✝⁷ : Algebra (B ⧸ Q) L
inst✝⁶ : Algebra (A ⧸ P) L
inst✝⁵ : IsScalarTower (A ⧸ P) (B ⧸ Q) L
inst✝⁴ : Algebra K L
inst✝³ : IsScalarTower (A ⧸ P) K L
inst✝² : Algebra.IsInvariant A B G
inst✝¹ : IsFractionRing (A ⧸ P) K
inst✝ : IsFractionRing (B ⧸ Q) L
x✝ : MulSemiringAction (↥(MulAction.stabilizer G Q)) L := MulSemiringAction.compHom L (stabilizerHom G P Q K L)
f : L ≃ₐ[K] L
g : ↥(MulAction.stabilizer G Q)
hg :
(MulSemiringAction.toAlgAut (↥(MulAction.stabilizer G Q)) (↥(FixedPoints.subfield (↥(MulAction.stabilizer G Q)) L)) L)
g =
AlgEquiv.ofRingEquiv ⋯
⊢ ∃ a, (stabilizerHom G P Q K L) a = f
|
1aebdd9199a9aae9
|
Polynomial.mem_closure_X_union_C
|
Mathlib/RingTheory/Jacobson/Ring.lean
|
lemma mem_closure_X_union_C {R : Type*} [Ring R] (p : R[X]) :
p ∈ Subring.closure (insert X {f | f.degree ≤ 0} : Set R[X])
|
case refine_1.a.a
R : Type u_1
inst✝ : Ring R
p : R[X]
r : R
⊢ C r ∈ {f | f.degree ≤ 0}
|
exact degree_C_le
|
no goals
|
9a8a4355ff754a01
|
iSupIndep.linearIndependent
|
Mathlib/LinearAlgebra/DFinsupp.lean
|
theorem iSupIndep.linearIndependent [NoZeroSMulDivisors R N] {ι} (p : ι → Submodule R N)
(hp : iSupIndep p) {v : ι → N} (hv : ∀ i, v i ∈ p i) (hv' : ∀ i, v i ≠ 0) :
LinearIndependent R v
|
R : Type u_2
N : Type u_5
inst✝³ : Ring R
inst✝² : AddCommGroup N
inst✝¹ : Module R N
inst✝ : NoZeroSMulDivisors R N
ι : Type u_6
p : ι → Submodule R N
hp : iSupIndep p
v : ι → N
hv : ∀ (i : ι), v i ∈ p i
hv' : ∀ (i : ι), v i ≠ 0
x✝¹ : DecidableEq ι := Classical.decEq ι
x✝ : DecidableEq R := Classical.decEq R
⊢ LinearIndependent R v
|
rw [linearIndependent_iff]
|
R : Type u_2
N : Type u_5
inst✝³ : Ring R
inst✝² : AddCommGroup N
inst✝¹ : Module R N
inst✝ : NoZeroSMulDivisors R N
ι : Type u_6
p : ι → Submodule R N
hp : iSupIndep p
v : ι → N
hv : ∀ (i : ι), v i ∈ p i
hv' : ∀ (i : ι), v i ≠ 0
x✝¹ : DecidableEq ι := Classical.decEq ι
x✝ : DecidableEq R := Classical.decEq R
⊢ ∀ (l : ι →₀ R), (Finsupp.linearCombination R v) l = 0 → l = 0
|
5ab7ea3a741397a7
|
Besicovitch.TauPackage.color_lt
|
Mathlib/MeasureTheory/Covering/Besicovitch.lean
|
theorem color_lt {i : Ordinal.{u}} (hi : i < p.lastStep) {N : ℕ}
(hN : IsEmpty (SatelliteConfig α N p.τ)) : p.color i < N
|
case h
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
⋃ j,
⋃ (_ :
(closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
{p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
⊢ sInf (univ \ A) ≠ N
|
intro Inf_eq_N
|
case h
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
⋃ j,
⋃ (_ :
(closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
{p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
⊢ False
|
70d4d5d8aa8a7de2
|
Std.Tactic.BVDecide.BVExpr.bitblast.go_decl_eq
|
Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Impl/Expr.lean
|
theorem bitblast.go_decl_eq (aig : AIG BVBit) (expr : BVExpr w) :
∀ (idx : Nat) (h1) (h2), (go aig expr).val.aig.decls[idx]'h2 = aig.decls[idx]'h1
|
case h2
w idx w✝ : Nat
op : BVUnOp
expr : BVExpr w✝
ih :
∀ (aig : AIG BVBit) (h1 : idx < aig.decls.size) (h2 : idx < (go aig expr).val.aig.decls.size),
(go aig expr).val.aig.decls[idx] = aig.decls[idx]
aig : AIG BVBit
h1 : idx < aig.decls.size
n✝ : Nat
h2 : idx < (go aig (un (BVUnOp.arithShiftRightConst n✝) expr)).val.aig.decls.size
this : aig.decls.size ≤ (go aig expr).val.aig.decls.size
⊢ idx < (go aig expr).val.aig.decls.size
|
omega
|
no goals
|
f92988d5a1595b39
|
ZMod.eq_one_or_isUnit_sub_one
|
Mathlib/FieldTheory/Finite/Basic.lean
|
theorem ZMod.eq_one_or_isUnit_sub_one {n p k : ℕ} [Fact p.Prime] (hn : n = p ^ k) (a : ZMod n)
(ha : (orderOf a).Coprime n) : a = 1 ∨ IsUnit (a - 1)
|
n p k : ℕ
inst✝ : Fact (Nat.Prime p)
hn : n = p ^ k
a : ZMod n
ha : (orderOf a).Coprime n
⊢ a = 1 ∨ IsUnit (a - 1)
|
rcases eq_or_ne n 0 with rfl | hn0
|
case inl
p k : ℕ
inst✝ : Fact (Nat.Prime p)
hn : 0 = p ^ k
a : ZMod 0
ha : (orderOf a).Coprime 0
⊢ a = 1 ∨ IsUnit (a - 1)
case inr
n p k : ℕ
inst✝ : Fact (Nat.Prime p)
hn : n = p ^ k
a : ZMod n
ha : (orderOf a).Coprime n
hn0 : n ≠ 0
⊢ a = 1 ∨ IsUnit (a - 1)
|
e140f007bb03a360
|
CategoryTheory.Sheaf.isPullback_square_op_map_yoneda_presheafToSheaf_yoneda_iff
|
Mathlib/CategoryTheory/Sites/MayerVietorisSquare.lean
|
lemma Sheaf.isPullback_square_op_map_yoneda_presheafToSheaf_yoneda_iff
[HasWeakSheafify J (Type v)]
(F : Sheaf J (Type v)) (sq : Square C) :
(sq.op.map ((yoneda ⋙ presheafToSheaf J _).op ⋙ yoneda.obj F)).IsPullback ↔
(sq.op.map F.val).IsPullback
|
case refine_4.h
C : Type u
inst✝¹ : Category.{v, u} C
J : GrothendieckTopology C
inst✝ : HasWeakSheafify J (Type v)
F : Sheaf J (Type v)
sq : Square C
x : (sq.op.map ((yoneda ⋙ presheafToSheaf J (Type v)).op ⋙ yoneda.obj F)).X₃
⊢ (⇑(((sheafificationAdjunction J (Type v)).homEquiv (yoneda.obj (unop sq.op.X₄)) F).trans yonedaEquiv) ∘
(sq.op.map ((yoneda ⋙ presheafToSheaf J (Type v)).op ⋙ yoneda.obj F)).f₃₄)
x =
((sq.op.map F.val).f₃₄ ∘
⇑(((sheafificationAdjunction J (Type v)).homEquiv (yoneda.obj (unop sq.op.X₃)) F).trans yonedaEquiv))
x
|
dsimp
|
case refine_4.h
C : Type u
inst✝¹ : Category.{v, u} C
J : GrothendieckTopology C
inst✝ : HasWeakSheafify J (Type v)
F : Sheaf J (Type v)
sq : Square C
x : (sq.op.map ((yoneda ⋙ presheafToSheaf J (Type v)).op ⋙ yoneda.obj F)).X₃
⊢ yonedaEquiv
(((sheafificationAdjunction J (Type v)).homEquiv (yoneda.obj sq.X₁) F)
((presheafToSheaf J (Type v)).map (yoneda.map sq.f₁₃) ≫ x)) =
F.val.map sq.f₁₃.op (yonedaEquiv (((sheafificationAdjunction J (Type v)).homEquiv (yoneda.obj sq.X₃) F) x))
|
bfe4e0c25299bb4f
|
AkraBazziRecurrence.exists_eventually_const_mul_le_r
|
Mathlib/Computability/AkraBazzi/AkraBazzi.lean
|
lemma exists_eventually_const_mul_le_r :
∃ c ∈ Set.Ioo (0 : ℝ) 1, ∀ᶠ (n : ℕ) in atTop, ∀ i, c * n ≤ r i n
|
α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
⊢ ∃ c ∈ Set.Ioo 0 1, ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), c * ↑n ≤ ↑(r i n)
|
have gt_zero : 0 < b (min_bi b) := R.b_pos (min_bi b)
|
α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
gt_zero : 0 < b (min_bi b)
⊢ ∃ c ∈ Set.Ioo 0 1, ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), c * ↑n ≤ ↑(r i n)
|
bc15a40b01d9e13e
|
MeasureTheory.Measure.withDensity_rnDeriv_le
|
Mathlib/MeasureTheory/Decomposition/Lebesgue.lean
|
theorem withDensity_rnDeriv_le (μ ν : Measure α) : ν.withDensity (μ.rnDeriv ν) ≤ μ
|
case neg
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
hl : ¬μ.HaveLebesgueDecomposition ν
⊢ 0 ≤ μ
|
exact Measure.zero_le μ
|
no goals
|
2275dc5f7cdbb54c
|
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