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
|
|---|---|---|---|---|---|---|
FiniteDimensional.of_locallyCompact_manifold
|
Mathlib/Geometry/Manifold/IsManifold/ExtChartAt.lean
|
theorem FiniteDimensional.of_locallyCompact_manifold
[CompleteSpace 𝕜] (I : ModelWithCorners 𝕜 E H) [Nonempty M] [LocallyCompactSpace M] :
FiniteDimensional 𝕜 E
|
E : Type u_8
𝕜 : Type u_9
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
H : Type u_10
inst✝⁵ : TopologicalSpace H
M : Type u_11
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : CompleteSpace 𝕜
I : ModelWithCorners 𝕜 E H
inst✝¹ : Nonempty M
inst✝ : LocallyCompactSpace M
this : LocallyCompactSpace E
⊢ FiniteDimensional 𝕜 E
|
exact FiniteDimensional.of_locallyCompactSpace 𝕜
|
no goals
|
2ad87ba12d29b468
|
MeasureTheory.setIntegral_tilted'
|
Mathlib/MeasureTheory/Measure/Tilted.lean
|
lemma setIntegral_tilted' (f : α → ℝ) (g : α → E) {s : Set α} (hs : MeasurableSet s) :
∫ x in s, g x ∂(μ.tilted f) = ∫ x in s, (exp (f x) / ∫ x, exp (f x) ∂μ) • (g x) ∂μ
|
case neg
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : α → ℝ
g : α → E
s : Set α
hs : MeasurableSet s
hf : ¬AEMeasurable f μ
hf' : ¬Integrable (fun x => rexp (f x)) μ
⊢ 0 = ∫ (x : α) in s, (rexp (f x) / 0) • g x ∂μ
|
simp
|
no goals
|
026f577fdf7186b7
|
ContinuousLinearMap.prod_ext_iff
|
Mathlib/Topology/Algebra/Module/LinearMapPiProd.lean
|
theorem prod_ext_iff {f g : M × M₂ →L[R] M₃} :
f = g ↔ f.comp (inl _ _ _) = g.comp (inl _ _ _) ∧ f.comp (inr _ _ _) = g.comp (inr _ _ _)
|
R : Type u_1
inst✝⁹ : Semiring R
M : Type u_2
inst✝⁸ : TopologicalSpace M
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Module R M
M₂ : Type u_3
inst✝⁵ : TopologicalSpace M₂
inst✝⁴ : AddCommMonoid M₂
inst✝³ : Module R M₂
M₃ : Type u_4
inst✝² : TopologicalSpace M₃
inst✝¹ : AddCommMonoid M₃
inst✝ : Module R M₃
f g : M × M₂ →L[R] M₃
⊢ ↑f ∘ₗ LinearMap.inl R M M₂ = ↑g ∘ₗ LinearMap.inl R M M₂ ∧ ↑f ∘ₗ LinearMap.inr R M M₂ = ↑g ∘ₗ LinearMap.inr R M M₂ ↔
↑(f.comp (inl R M M₂)) = ↑(g.comp (inl R M M₂)) ∧ ↑(f.comp (inr R M M₂)) = ↑(g.comp (inr R M M₂))
|
rfl
|
no goals
|
f0c9b8d2b77fd340
|
Set.Icc_add_bij
|
Mathlib/Algebra/Order/Interval/Set/Monoid.lean
|
theorem Icc_add_bij : BijOn (· + d) (Icc a b) (Icc (a + d) (b + d))
|
M : Type u_1
inst✝¹ : OrderedCancelAddCommMonoid M
inst✝ : ExistsAddOfLE M
a b d : M
⊢ BijOn (fun x => x + d) (Icc a b) (Icc (a + d) (b + d))
|
rw [← Ici_inter_Iic, ← Ici_inter_Iic]
|
M : Type u_1
inst✝¹ : OrderedCancelAddCommMonoid M
inst✝ : ExistsAddOfLE M
a b d : M
⊢ BijOn (fun x => x + d) (Ici a ∩ Iic b) (Ici (a + d) ∩ Iic (b + d))
|
69a7a3dbeaf26ca0
|
Relation.is_graph_iff
|
Mathlib/Data/Rel.lean
|
theorem Relation.is_graph_iff (r : Rel α β) : (∃! f, Function.graph f = r) ↔ ∀ x, ∃! y, r x y
|
case mp
α : Type u_1
β : Type u_2
r : Rel α β
⊢ (∃! f, (fun x y => f x = y) = r) → ∀ (x : α), ∃! y, r x y
|
rintro ⟨f, rfl, _⟩ x
|
case mp.intro.intro
α : Type u_1
β : Type u_2
f : α → β
right✝ : ∀ (y : α → β), (fun f_1 => (fun x y => f_1 x = y) = fun x y => f x = y) y → y = f
x : α
⊢ ∃! y, (fun x y => f x = y) x y
|
b9c04eef61e821ef
|
Polynomial.roots_pow
|
Mathlib/Algebra/Polynomial/Roots.lean
|
theorem roots_pow (p : R[X]) (n : ℕ) : (p ^ n).roots = n • p.roots
|
case zero
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R[X]
⊢ (p ^ 0).roots = 0 • p.roots
|
rw [pow_zero, roots_one, zero_smul, empty_eq_zero]
|
no goals
|
9515e924941f10c8
|
Complex.norm_exp_sub_sum_le_norm_mul_exp
|
Mathlib/Data/Complex/Exponential.lean
|
lemma norm_exp_sub_sum_le_norm_mul_exp (x : ℂ) (n : ℕ) :
‖exp x - ∑ m ∈ range n, x ^ m / m.factorial‖ ≤ ‖x‖ ^ n * Real.exp ‖x‖
|
x : ℂ
n j : ℕ
hj : j ≥ n
m : ℕ
hm : m < j ∧ n ≤ m
⊢ x ^ m / ↑m.factorial = x ^ n * (x ^ (m - n) / ↑m.factorial)
|
rw [← mul_div_assoc, ← pow_add, add_tsub_cancel_of_le hm.2]
|
no goals
|
0fd8b673128d7210
|
Pell.Solution₁.x_mul_pos
|
Mathlib/NumberTheory/Pell.lean
|
theorem x_mul_pos {a b : Solution₁ d} (ha : 0 < a.x) (hb : 0 < b.x) : 0 < (a * b).x
|
d : ℤ
a b : Solution₁ d
ha : 0 < a.x
hb : 0 < b.x
⊢ 0 < (a * b).x
|
simp only [x_mul]
|
d : ℤ
a b : Solution₁ d
ha : 0 < a.x
hb : 0 < b.x
⊢ 0 < a.x * b.x + d * (a.y * b.y)
|
5c93ee2e9cc7c71e
|
HurwitzZeta.hasSum_int_oddKernel
|
Mathlib/NumberTheory/LSeries/HurwitzZetaOdd.lean
|
lemma hasSum_int_oddKernel (a : ℝ) {x : ℝ} (hx : 0 < x) :
HasSum (fun n : ℤ ↦ (n + a) * rexp (-π * (n + a) ^ 2 * x)) (oddKernel ↑a x)
|
a x : ℝ
hx : 0 < x
⊢ 0 < im ?m.80841
|
rwa [I_mul_im, ofReal_re]
|
no goals
|
6db94613edf0e05c
|
ContDiffBump.ae_convolution_tendsto_right_of_locallyIntegrable
|
Mathlib/Analysis/Calculus/BumpFunction/Convolution.lean
|
theorem ae_convolution_tendsto_right_of_locallyIntegrable
{ι} {φ : ι → ContDiffBump (0 : G)} {l : Filter ι} {K : ℝ}
(hφ : Tendsto (fun i ↦ (φ i).rOut) l (𝓝 0))
(h'φ : ∀ᶠ i in l, (φ i).rOut ≤ K * (φ i).rIn) (hg : LocallyIntegrable g μ) : ∀ᵐ x₀ ∂μ,
Tendsto (fun i ↦ ((φ i).normed μ ⋆[lsmul ℝ ℝ, μ] g) x₀) l (𝓝 (g x₀))
|
case h
G : Type uG
E' : Type uE'
inst✝¹¹ : NormedAddCommGroup E'
g : G → E'
inst✝¹⁰ : MeasurableSpace G
μ : Measure G
inst✝⁹ : NormedSpace ℝ E'
inst✝⁸ : NormedAddCommGroup G
inst✝⁷ : NormedSpace ℝ G
inst✝⁶ : HasContDiffBump G
inst✝⁵ : CompleteSpace E'
inst✝⁴ : BorelSpace G
inst✝³ : IsLocallyFiniteMeasure μ
inst✝² : μ.IsOpenPosMeasure
inst✝¹ : FiniteDimensional ℝ G
inst✝ : μ.IsAddLeftInvariant
ι : Type u_1
φ : ι → ContDiffBump 0
l : Filter ι
K : ℝ
hφ : Tendsto (fun i => (φ i).rOut) l (𝓝 0)
h'φ : ∀ᶠ (i : ι) in l, (φ i).rOut ≤ K * (φ i).rIn
hg : LocallyIntegrable g μ
this✝ : μ.IsAddHaarMeasure
x₀ : G
h₀ : Tendsto (fun a => ⨍ (y : G) in a, ‖g y - g x₀‖ ∂μ) ((Besicovitch.vitaliFamily μ).filterAt x₀) (𝓝 0)
hφ' : Tendsto (fun i => (φ i).rOut) l (𝓝[>] 0)
this : Tendsto (((fun a => ⨍ (y : G) in a, ‖g y - g x₀‖ ∂μ) ∘ fun r => closedBall x₀ r) ∘ fun i => (φ i).rOut) l (𝓝 0)
⊢ Tendsto (fun i => ∫ (t : G), (φ i).normed μ (x₀ - t) • g t ∂μ) l (𝓝 (g x₀))
|
apply tendsto_integral_smul_of_tendsto_average_norm_sub (K ^ (Module.finrank ℝ G)) this
|
case h.f_int
G : Type uG
E' : Type uE'
inst✝¹¹ : NormedAddCommGroup E'
g : G → E'
inst✝¹⁰ : MeasurableSpace G
μ : Measure G
inst✝⁹ : NormedSpace ℝ E'
inst✝⁸ : NormedAddCommGroup G
inst✝⁷ : NormedSpace ℝ G
inst✝⁶ : HasContDiffBump G
inst✝⁵ : CompleteSpace E'
inst✝⁴ : BorelSpace G
inst✝³ : IsLocallyFiniteMeasure μ
inst✝² : μ.IsOpenPosMeasure
inst✝¹ : FiniteDimensional ℝ G
inst✝ : μ.IsAddLeftInvariant
ι : Type u_1
φ : ι → ContDiffBump 0
l : Filter ι
K : ℝ
hφ : Tendsto (fun i => (φ i).rOut) l (𝓝 0)
h'φ : ∀ᶠ (i : ι) in l, (φ i).rOut ≤ K * (φ i).rIn
hg : LocallyIntegrable g μ
this✝ : μ.IsAddHaarMeasure
x₀ : G
h₀ : Tendsto (fun a => ⨍ (y : G) in a, ‖g y - g x₀‖ ∂μ) ((Besicovitch.vitaliFamily μ).filterAt x₀) (𝓝 0)
hφ' : Tendsto (fun i => (φ i).rOut) l (𝓝[>] 0)
this : Tendsto (((fun a => ⨍ (y : G) in a, ‖g y - g x₀‖ ∂μ) ∘ fun r => closedBall x₀ r) ∘ fun i => (φ i).rOut) l (𝓝 0)
⊢ ∀ᶠ (i : ι) in l, IntegrableOn g ((fun r => closedBall x₀ r) ((fun i => (φ i).rOut) i)) μ
case h.hg
G : Type uG
E' : Type uE'
inst✝¹¹ : NormedAddCommGroup E'
g : G → E'
inst✝¹⁰ : MeasurableSpace G
μ : Measure G
inst✝⁹ : NormedSpace ℝ E'
inst✝⁸ : NormedAddCommGroup G
inst✝⁷ : NormedSpace ℝ G
inst✝⁶ : HasContDiffBump G
inst✝⁵ : CompleteSpace E'
inst✝⁴ : BorelSpace G
inst✝³ : IsLocallyFiniteMeasure μ
inst✝² : μ.IsOpenPosMeasure
inst✝¹ : FiniteDimensional ℝ G
inst✝ : μ.IsAddLeftInvariant
ι : Type u_1
φ : ι → ContDiffBump 0
l : Filter ι
K : ℝ
hφ : Tendsto (fun i => (φ i).rOut) l (𝓝 0)
h'φ : ∀ᶠ (i : ι) in l, (φ i).rOut ≤ K * (φ i).rIn
hg : LocallyIntegrable g μ
this✝ : μ.IsAddHaarMeasure
x₀ : G
h₀ : Tendsto (fun a => ⨍ (y : G) in a, ‖g y - g x₀‖ ∂μ) ((Besicovitch.vitaliFamily μ).filterAt x₀) (𝓝 0)
hφ' : Tendsto (fun i => (φ i).rOut) l (𝓝[>] 0)
this : Tendsto (((fun a => ⨍ (y : G) in a, ‖g y - g x₀‖ ∂μ) ∘ fun r => closedBall x₀ r) ∘ fun i => (φ i).rOut) l (𝓝 0)
⊢ Tendsto (fun i => ∫ (y : G), (φ i).normed μ (x₀ - y) ∂μ) l (𝓝 1)
case h.g_supp
G : Type uG
E' : Type uE'
inst✝¹¹ : NormedAddCommGroup E'
g : G → E'
inst✝¹⁰ : MeasurableSpace G
μ : Measure G
inst✝⁹ : NormedSpace ℝ E'
inst✝⁸ : NormedAddCommGroup G
inst✝⁷ : NormedSpace ℝ G
inst✝⁶ : HasContDiffBump G
inst✝⁵ : CompleteSpace E'
inst✝⁴ : BorelSpace G
inst✝³ : IsLocallyFiniteMeasure μ
inst✝² : μ.IsOpenPosMeasure
inst✝¹ : FiniteDimensional ℝ G
inst✝ : μ.IsAddLeftInvariant
ι : Type u_1
φ : ι → ContDiffBump 0
l : Filter ι
K : ℝ
hφ : Tendsto (fun i => (φ i).rOut) l (𝓝 0)
h'φ : ∀ᶠ (i : ι) in l, (φ i).rOut ≤ K * (φ i).rIn
hg : LocallyIntegrable g μ
this✝ : μ.IsAddHaarMeasure
x₀ : G
h₀ : Tendsto (fun a => ⨍ (y : G) in a, ‖g y - g x₀‖ ∂μ) ((Besicovitch.vitaliFamily μ).filterAt x₀) (𝓝 0)
hφ' : Tendsto (fun i => (φ i).rOut) l (𝓝[>] 0)
this : Tendsto (((fun a => ⨍ (y : G) in a, ‖g y - g x₀‖ ∂μ) ∘ fun r => closedBall x₀ r) ∘ fun i => (φ i).rOut) l (𝓝 0)
⊢ ∀ᶠ (i : ι) in l, (support fun y => (φ i).normed μ (x₀ - y)) ⊆ (fun r => closedBall x₀ r) ((fun i => (φ i).rOut) i)
case h.g_bound
G : Type uG
E' : Type uE'
inst✝¹¹ : NormedAddCommGroup E'
g : G → E'
inst✝¹⁰ : MeasurableSpace G
μ : Measure G
inst✝⁹ : NormedSpace ℝ E'
inst✝⁸ : NormedAddCommGroup G
inst✝⁷ : NormedSpace ℝ G
inst✝⁶ : HasContDiffBump G
inst✝⁵ : CompleteSpace E'
inst✝⁴ : BorelSpace G
inst✝³ : IsLocallyFiniteMeasure μ
inst✝² : μ.IsOpenPosMeasure
inst✝¹ : FiniteDimensional ℝ G
inst✝ : μ.IsAddLeftInvariant
ι : Type u_1
φ : ι → ContDiffBump 0
l : Filter ι
K : ℝ
hφ : Tendsto (fun i => (φ i).rOut) l (𝓝 0)
h'φ : ∀ᶠ (i : ι) in l, (φ i).rOut ≤ K * (φ i).rIn
hg : LocallyIntegrable g μ
this✝ : μ.IsAddHaarMeasure
x₀ : G
h₀ : Tendsto (fun a => ⨍ (y : G) in a, ‖g y - g x₀‖ ∂μ) ((Besicovitch.vitaliFamily μ).filterAt x₀) (𝓝 0)
hφ' : Tendsto (fun i => (φ i).rOut) l (𝓝[>] 0)
this : Tendsto (((fun a => ⨍ (y : G) in a, ‖g y - g x₀‖ ∂μ) ∘ fun r => closedBall x₀ r) ∘ fun i => (φ i).rOut) l (𝓝 0)
⊢ ∀ᶠ (i : ι) in l,
∀ (x : G),
|(φ i).normed μ (x₀ - x)| ≤
K ^ Module.finrank ℝ G / (μ ((fun r => closedBall x₀ r) ((fun i => (φ i).rOut) i))).toReal
|
22b99d3576c4b89d
|
InverseSystem.unique_pEquivOn
|
Mathlib/Order/DirectedInverseSystem.lean
|
theorem unique_pEquivOn (hs : IsLowerSet s) {e₁ e₂ : PEquivOn f equivSucc s} : e₁ = e₂
|
case mk.mk.equiv.h
ι : Type u_6
F : ι → Type u_7
X : ι → Type u_8
inst✝² : LinearOrder ι
f : ⦃i j : ι⦄ → i ≤ j → F j → F i
inst✝¹ : SuccOrder ι
equivSucc : ⦃i : ι⦄ → ¬IsMax i → F i⁺ ≃ F i × X i
s : Set ι
inst✝ : WellFoundedLT ι
hs : IsLowerSet s
e₁ : (i : ↑s) → F ↑i ≃ piLT X ↑i
nat₁ : IsNatEquiv f e₁
compat₁ : ∀ {i : ι} (hsi : i⁺ ∈ s) (hi : ¬IsMax i) (x : F ↑⟨i⁺, hsi⟩), (e₁ ⟨i⁺, hsi⟩) x ⟨i, ⋯⟩ = ((equivSucc hi) x).2
e₂ : (i : ↑s) → F ↑i ≃ piLT X ↑i
nat₂ : IsNatEquiv f e₂
compat₂ : ∀ {i : ι} (hsi : i⁺ ∈ s) (hi : ¬IsMax i) (x : F ↑⟨i⁺, hsi⟩), (e₂ ⟨i⁺, hsi⟩) x ⟨i, ⋯⟩ = ((equivSucc hi) x).2
i : ↑s
⊢ e₁ i = e₂ i
|
refine SuccOrder.prelimitRecOn i.1 (C := fun i ↦ ∀ h : i ∈ s, e₁ ⟨i, h⟩ = e₂ ⟨i, h⟩)
(fun i nmax ih hi ↦ ?_) (fun i lim ih hi ↦ ?_) i.2
|
case mk.mk.equiv.h.refine_1
ι : Type u_6
F : ι → Type u_7
X : ι → Type u_8
inst✝² : LinearOrder ι
f : ⦃i j : ι⦄ → i ≤ j → F j → F i
inst✝¹ : SuccOrder ι
equivSucc : ⦃i : ι⦄ → ¬IsMax i → F i⁺ ≃ F i × X i
s : Set ι
inst✝ : WellFoundedLT ι
hs : IsLowerSet s
e₁ : (i : ↑s) → F ↑i ≃ piLT X ↑i
nat₁ : IsNatEquiv f e₁
compat₁ : ∀ {i : ι} (hsi : i⁺ ∈ s) (hi : ¬IsMax i) (x : F ↑⟨i⁺, hsi⟩), (e₁ ⟨i⁺, hsi⟩) x ⟨i, ⋯⟩ = ((equivSucc hi) x).2
e₂ : (i : ↑s) → F ↑i ≃ piLT X ↑i
nat₂ : IsNatEquiv f e₂
compat₂ : ∀ {i : ι} (hsi : i⁺ ∈ s) (hi : ¬IsMax i) (x : F ↑⟨i⁺, hsi⟩), (e₂ ⟨i⁺, hsi⟩) x ⟨i, ⋯⟩ = ((equivSucc hi) x).2
i✝ : ↑s
i : ι
nmax : ¬IsMax i
ih : (fun i => ∀ (h : i ∈ s), e₁ ⟨i, h⟩ = e₂ ⟨i, h⟩) i
hi : i⁺ ∈ s
⊢ e₁ ⟨i⁺, hi⟩ = e₂ ⟨i⁺, hi⟩
case mk.mk.equiv.h.refine_2
ι : Type u_6
F : ι → Type u_7
X : ι → Type u_8
inst✝² : LinearOrder ι
f : ⦃i j : ι⦄ → i ≤ j → F j → F i
inst✝¹ : SuccOrder ι
equivSucc : ⦃i : ι⦄ → ¬IsMax i → F i⁺ ≃ F i × X i
s : Set ι
inst✝ : WellFoundedLT ι
hs : IsLowerSet s
e₁ : (i : ↑s) → F ↑i ≃ piLT X ↑i
nat₁ : IsNatEquiv f e₁
compat₁ : ∀ {i : ι} (hsi : i⁺ ∈ s) (hi : ¬IsMax i) (x : F ↑⟨i⁺, hsi⟩), (e₁ ⟨i⁺, hsi⟩) x ⟨i, ⋯⟩ = ((equivSucc hi) x).2
e₂ : (i : ↑s) → F ↑i ≃ piLT X ↑i
nat₂ : IsNatEquiv f e₂
compat₂ : ∀ {i : ι} (hsi : i⁺ ∈ s) (hi : ¬IsMax i) (x : F ↑⟨i⁺, hsi⟩), (e₂ ⟨i⁺, hsi⟩) x ⟨i, ⋯⟩ = ((equivSucc hi) x).2
i✝ : ↑s
i : ι
lim : IsSuccPrelimit i
ih : ∀ b < i, (fun i => ∀ (h : i ∈ s), e₁ ⟨i, h⟩ = e₂ ⟨i, h⟩) b
hi : i ∈ s
⊢ e₁ ⟨i, hi⟩ = e₂ ⟨i, hi⟩
|
dcc32ff31e12d17b
|
IsPrimitiveRoot.norm_pow_sub_one_of_prime_pow_ne_two
|
Mathlib/NumberTheory/Cyclotomic/PrimitiveRoots.lean
|
theorem norm_pow_sub_one_of_prime_pow_ne_two {k s : ℕ} (hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1)))
[hpri : Fact (p : ℕ).Prime] [IsCyclotomicExtension {p ^ (k + 1)} K L]
(hirr : Irreducible (cyclotomic (↑(p ^ (k + 1)) : ℕ) K)) (hs : s ≤ k)
(htwo : p ^ (k - s + 1) ≠ 2) : norm K (ζ ^ (p : ℕ) ^ s - 1) = (p : K) ^ (p : ℕ) ^ s
|
case refine_2.e_a
p : ℕ+
K : Type u
L : Type v
inst✝³ : Field L
ζ : L
inst✝² : Field K
inst✝¹ : Algebra K L
k s : ℕ
hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))
hpri : Fact (Nat.Prime ↑p)
inst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L
hirr : Irreducible (cyclotomic (↑(p ^ (k + 1))) K)
hs : s ≤ k
htwo : p ^ (k - s + 1) ≠ 2
hirr₁ : Irreducible (cyclotomic (↑(p ^ (k - s + 1))) K)
η : L := ζ ^ ↑p ^ s - 1
η₁ : ↥K⟮η⟯ := IntermediateField.AdjoinSimple.gen K η
this✝² : IsCyclotomicExtension {p ^ (k - s + 1)} K ↥K⟮η⟯
hη : IsPrimitiveRoot (η₁ + 1) ↑(p ^ (k - s + 1))
this✝¹ : FiniteDimensional K L
this✝ : IsGalois K L
H : (Algebra.norm K) η₁ = ↑(↑(p ^ (k - s + 1))).minFac
this : ↑p ^ ((k - s).succ - 1) * Module.finrank (↥K⟮η⟯) L = ↑p ^ (k.succ - 1)
Hex : k.succ - 1 = (k - s).succ - 1 + s
⊢ Module.finrank (↥K⟮η⟯) L = ↑p ^ s
|
rw [Hex, pow_add] at this
|
case refine_2.e_a
p : ℕ+
K : Type u
L : Type v
inst✝³ : Field L
ζ : L
inst✝² : Field K
inst✝¹ : Algebra K L
k s : ℕ
hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))
hpri : Fact (Nat.Prime ↑p)
inst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L
hirr : Irreducible (cyclotomic (↑(p ^ (k + 1))) K)
hs : s ≤ k
htwo : p ^ (k - s + 1) ≠ 2
hirr₁ : Irreducible (cyclotomic (↑(p ^ (k - s + 1))) K)
η : L := ζ ^ ↑p ^ s - 1
η₁ : ↥K⟮η⟯ := IntermediateField.AdjoinSimple.gen K η
this✝² : IsCyclotomicExtension {p ^ (k - s + 1)} K ↥K⟮η⟯
hη : IsPrimitiveRoot (η₁ + 1) ↑(p ^ (k - s + 1))
this✝¹ : FiniteDimensional K L
this✝ : IsGalois K L
H : (Algebra.norm K) η₁ = ↑(↑(p ^ (k - s + 1))).minFac
this : ↑p ^ ((k - s).succ - 1) * Module.finrank (↥K⟮η⟯) L = ↑p ^ ((k - s).succ - 1) * ↑p ^ s
Hex : k.succ - 1 = (k - s).succ - 1 + s
⊢ Module.finrank (↥K⟮η⟯) L = ↑p ^ s
|
460ee0dff43c909a
|
LieAlgebra.engel_isBot_of_isMin
|
Mathlib/Algebra/Lie/CartanExists.lean
|
/-- Let `L` be a Lie algebra of dimension `n` over a field `K` with at least `n` elements.
Given a Lie subalgebra `U` of `L`, and an element `x ∈ U` such that `U ≤ engel K x`.
Suppose that `engel K x` is minimal amongst the Engel subalgebras `engel K y` for `y ∈ U`.
Then `engel K x ≤ engel K y` for all `y ∈ U`.
Lemma 2 in [barnes1967]. -/
lemma engel_isBot_of_isMin (hLK : finrank K L ≤ #K) (U : LieSubalgebra K L)
(E : {engel K x | x ∈ U}) (hUle : U ≤ E) (hmin : IsMin E) :
IsBot E
|
K : Type u_1
L : Type u_2
inst✝³ : Field K
inst✝² : LieRing L
inst✝¹ : LieAlgebra K L
inst✝ : Module.Finite K L
hLK : ↑(finrank K L) ≤ #K
U : LieSubalgebra K L
x : L
hxU : x ∈ U
y : L
hyU : y ∈ U
Ex : ↑{x | ∃ x_1 ∈ U, engel K x_1 = x} := ⟨engel K x, ⋯⟩
Ey : ↑{x | ∃ y ∈ U, engel K y = x} := ⟨engel K y, ⋯⟩
hUle : U ≤ ↑Ex
hmin : ∀ E ≤ Ex, Ex ≤ E
E : LieSubmodule K (↥U) L :=
let __src := engel K x;
{ toSubmodule := __src.toSubmodule, lie_mem := ⋯ }
hx₀ : x ≠ 0
Q : Type u_2 := L ⧸ E
r : ℕ := finrank K ↥E
hr : r < finrank K L
x' : ↥U := ⟨x, hxU⟩
y' : ↥U := ⟨y, hyU⟩
u : ↥U := y' - x'
χ : K[X][X] := LieAlgebra.engel_isBot_of_isMin.lieCharpoly K (↥E) x' u
ψ : K[X][X] := LieAlgebra.engel_isBot_of_isMin.lieCharpoly K Q x' u
hi : 0 < r
α : K
⊢ eval α (χ.coeff 0) = 0
|
rw [← coe_evalRingHom, ← coeff_map, lieCharpoly_map_eval,
← constantCoeff_apply, LinearMap.charpoly_constantCoeff_eq_zero_iff]
|
K : Type u_1
L : Type u_2
inst✝³ : Field K
inst✝² : LieRing L
inst✝¹ : LieAlgebra K L
inst✝ : Module.Finite K L
hLK : ↑(finrank K L) ≤ #K
U : LieSubalgebra K L
x : L
hxU : x ∈ U
y : L
hyU : y ∈ U
Ex : ↑{x | ∃ x_1 ∈ U, engel K x_1 = x} := ⟨engel K x, ⋯⟩
Ey : ↑{x | ∃ y ∈ U, engel K y = x} := ⟨engel K y, ⋯⟩
hUle : U ≤ ↑Ex
hmin : ∀ E ≤ Ex, Ex ≤ E
E : LieSubmodule K (↥U) L :=
let __src := engel K x;
{ toSubmodule := __src.toSubmodule, lie_mem := ⋯ }
hx₀ : x ≠ 0
Q : Type u_2 := L ⧸ E
r : ℕ := finrank K ↥E
hr : r < finrank K L
x' : ↥U := ⟨x, hxU⟩
y' : ↥U := ⟨y, hyU⟩
u : ↥U := y' - x'
χ : K[X][X] := LieAlgebra.engel_isBot_of_isMin.lieCharpoly K (↥E) x' u
ψ : K[X][X] := LieAlgebra.engel_isBot_of_isMin.lieCharpoly K Q x' u
hi : 0 < r
α : K
⊢ ∃ m, m ≠ 0 ∧ ((toEnd K ↥U ↥E) (α • u + x')) m = 0
|
53ed38f54130aaf6
|
List.zipWithAux_toArray_succ'
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/ToArray.lean
|
theorem zipWithAux_toArray_succ' (as : List α) (bs : List β) (f : α → β → γ) (i : Nat) (cs : Array γ) :
zipWithAux as.toArray bs.toArray f (i + 1) cs = zipWithAux (as.drop (i+1)).toArray (bs.drop (i+1)).toArray f 0 cs
|
α : Type u_1
β : Type u_2
γ : Type u_3
as : List α
bs : List β
f : α → β → γ
i : Nat
cs : Array γ
⊢ as.toArray.zipWithAux bs.toArray f (i + 1) cs = (drop (i + 1) as).toArray.zipWithAux (drop (i + 1) bs).toArray f 0 cs
|
induction i generalizing as bs cs with
| zero => simp [zipWithAux_toArray_succ]
| succ i ih =>
rw [zipWithAux_toArray_succ, ih]
simp
|
no goals
|
dbd8cc49bcc400e2
|
finrank_vectorSpan_insert_le
|
Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean
|
theorem finrank_vectorSpan_insert_le (s : AffineSubspace k P) (p : P) :
finrank k (vectorSpan k (insert p (s : Set P))) ≤ finrank k s.direction + 1
|
case neg
k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
s : AffineSubspace k P
p : P
hf : ¬FiniteDimensional k ↥s.direction
hf' : ¬FiniteDimensional k ↥(vectorSpan k (insert p ↑s))
⊢ 0 ≤ 1
|
exact zero_le_one
|
no goals
|
de5f4f906d2ae1ab
|
MeasureTheory.lintegral_withDensity_eq_lintegral_mul₀'
|
Mathlib/MeasureTheory/Measure/WithDensity.lean
|
theorem lintegral_withDensity_eq_lintegral_mul₀' {μ : Measure α} {f : α → ℝ≥0∞}
(hf : AEMeasurable f μ) {g : α → ℝ≥0∞} (hg : AEMeasurable g (μ.withDensity f)) :
∫⁻ a, g a ∂μ.withDensity f = ∫⁻ a, (f * g) a ∂μ
|
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : AEMeasurable f μ
g : α → ℝ≥0∞
f' : α → ℝ≥0∞ := AEMeasurable.mk f hf
hg : AEMeasurable g (μ.withDensity f')
this : μ.withDensity f = μ.withDensity f'
⊢ ∫⁻ (a : α), g a ∂μ.withDensity f' = ∫⁻ (a : α), (f * g) a ∂μ
|
let g' := hg.mk g
|
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : AEMeasurable f μ
g : α → ℝ≥0∞
f' : α → ℝ≥0∞ := AEMeasurable.mk f hf
hg : AEMeasurable g (μ.withDensity f')
this : μ.withDensity f = μ.withDensity f'
g' : α → ℝ≥0∞ := AEMeasurable.mk g hg
⊢ ∫⁻ (a : α), g a ∂μ.withDensity f' = ∫⁻ (a : α), (f * g) a ∂μ
|
8f99ec46bb14da3b
|
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')
|
case pos
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 ∈ ModelWithCorners.interior (M ⊕ M')
h : p.isLeft = true
⊢ p ∈ Sum.inl '' ModelWithCorners.interior M ∪ Sum.inr '' ModelWithCorners.interior M'
|
left
|
case pos.h
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 ∈ ModelWithCorners.interior (M ⊕ M')
h : p.isLeft = true
⊢ p ∈ Sum.inl '' ModelWithCorners.interior M
|
a37cb5a73686bbc7
|
IsUltrametricDist.isUltrametricDist_of_forall_norm_natCast_le_one
|
Mathlib/Analysis/Normed/Field/Ultra.lean
|
/-- To prove that a normed division ring is nonarchimedean, it suffices to prove that the norm
of the image of any natural is less than or equal to one. -/
lemma isUltrametricDist_of_forall_norm_natCast_le_one
(h : ∀ n : ℕ, ‖(n : R)‖ ≤ 1) : IsUltrametricDist R
|
R : Type u_1
inst✝ : NormedDivisionRing R
h : ∀ (n : ℕ), ‖↑n‖ ≤ 1
⊢ IsUltrametricDist R
|
refine isUltrametricDist_of_forall_pow_norm_le_nsmul_pow_max_one_norm (fun x m ↦ ?_)
|
R : Type u_1
inst✝ : NormedDivisionRing R
h : ∀ (n : ℕ), ‖↑n‖ ≤ 1
x : R
m : ℕ
⊢ ‖x + 1‖ ^ m ≤ (m + 1) • (1 ⊔ ‖x‖ ^ m)
|
065889d95ca2531c
|
Nat.multichoose_two
|
Mathlib/Data/Nat/Choose/Basic.lean
|
theorem multichoose_two (k : ℕ) : multichoose 2 k = k + 1
|
case succ
k : ℕ
IH : multichoose 2 k = k + 1
⊢ multichoose 2 (k + 1) = k + 1 + 1
|
rw [multichoose, IH]
|
case succ
k : ℕ
IH : multichoose 2 k = k + 1
⊢ multichoose 1 (k + 1) + (k + 1) = k + 1 + 1
|
07af28893cc1c434
|
Order.coheight_of_noMaxOrder
|
Mathlib/Order/KrullDimension.lean
|
@[simp] lemma coheight_of_noMaxOrder [NoMaxOrder α] (a : α) : coheight a = ⊤
|
α : Type u_1
inst✝¹ : Preorder α
inst✝ : NoMaxOrder α
a : α
f : ℕ → ↑(Set.Ioi a)
hstrictmono : StrictMono f
m : ℕ
⊢ { length := m, toFun := fun i => if i = 0 then a else ↑(f ↑i), step := ?step }.head = a ∧
{ length := m, toFun := fun i => if i = 0 then a else ↑(f ↑i), step := ?step }.length = m
|
simp [RelSeries.head]
|
no goals
|
f607a58b9d06f9d8
|
LipschitzWith.hasFDerivAt_of_hasLineDerivAt_of_closure
|
Mathlib/Analysis/Calculus/Rademacher.lean
|
theorem hasFDerivAt_of_hasLineDerivAt_of_closure
{f : E → F} (hf : LipschitzWith C f) {s : Set E} (hs : sphere 0 1 ⊆ closure s)
{L : E →L[ℝ] F} {x : E} (hL : ∀ v ∈ s, HasLineDerivAt ℝ f (L v) x v) :
HasFDerivAt f L x
|
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
F : Type u_2
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
C : ℝ≥0
inst✝ : FiniteDimensional ℝ E
f : E → F
hf : LipschitzWith C f
s : Set E
hs : sphere 0 1 ⊆ closure s
L : E →L[ℝ] F
x : E
hL : ∀ v ∈ s, HasLineDerivAt ℝ f (L v) x v
ε : ℝ
εpos : 0 < ε
δ : ℝ
δpos : 0 < δ
hδ : (↑C + ‖L‖ + 1) * δ = ε
q : Set E
hqs : q ⊆ s
q_fin : q.Finite
hq : sphere 0 1 ⊆ ⋃ y ∈ q, ball y δ
I : ∀ᶠ (t : ℝ) in 𝓝 0, ∀ v ∈ q, ‖f (x + t • v) - f x - t • L v‖ ≤ δ * ‖t‖
r : ℝ
r_pos : 0 < r
hr : ∀ (t : ℝ), ‖t‖ < r → ∀ v ∈ q, ‖f (x + t • v) - f x - t • L v‖ ≤ δ * ‖t‖
v : E
hv : v ∈ ball 0 r
v_ne : v ≠ 0
w : E
ρ : ℝ
w_mem : w ∈ sphere 0 1
hvw : v = ρ • w
hρ : ρ = ‖v‖
norm_rho : ‖ρ‖ = ρ
rho_pos : 0 ≤ ρ
y : E
yq : y ∈ q
hy : ‖w - y‖ < δ
⊢ ‖y - w‖ < δ
|
rwa [norm_sub_rev]
|
no goals
|
b201788e5c940afd
|
BitVec.bit_not_add_self
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Bitblast.lean
|
theorem bit_not_add_self (x : BitVec w) :
((iunfoldr (fun (i : Fin w) c => (c, !(x.getLsbD i)))) ()).snd + x = -1
|
w : Nat
x : BitVec w
⊢ ((iunfoldr (fun i c => (c, !x.getLsbD ↑i)) ()).snd.adc x false).snd = -1
|
apply iunfoldr_replace_snd (fun _ => false) (-1) false rfl
|
w : Nat
x : BitVec w
⊢ ∀ (i : Fin w),
adcb ((iunfoldr (fun i c => (c, !x.getLsbD ↑i)) ()).snd.getLsbD ↑i) (x.getLsbD ↑i) false = (false, (-1).getLsbD ↑i)
|
1929ded861ebdd30
|
RootPairing.Equiv.coweightHom_injective
|
Mathlib/LinearAlgebra/RootSystem/Hom.lean
|
lemma coweightHom_injective (P : RootPairing ι R M N) : Injective (Equiv.coweightHom P)
|
ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : RootPairing ι R M N
⊢ Injective ⇑(coweightHom P)
|
refine Injective.of_comp (f := fun a => MulOpposite.op a) fun g g' hgg' => ?_
|
ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : RootPairing ι R M N
g g' : P.Aut
hgg' : ((fun a => MulOpposite.op a) ∘ ⇑(coweightHom P)) g = ((fun a => MulOpposite.op a) ∘ ⇑(coweightHom P)) g'
⊢ g = g'
|
327d6663427025f5
|
Std.DHashMap.Internal.List.isEmpty_replaceEntry
|
Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/List/Associative.lean
|
theorem isEmpty_replaceEntry [BEq α] {l : List ((a : α) × β a)} {k : α} {v : β k} :
(replaceEntry k v l).isEmpty = l.isEmpty
|
α : Type u
β : α → Type v
inst✝ : BEq α
l : List ((a : α) × β a)
k : α
v : β k
⊢ (replaceEntry k v l).isEmpty = l.isEmpty
|
induction l using assoc_induction
|
case nil
α : Type u
β : α → Type v
inst✝ : BEq α
k : α
v : β k
⊢ (replaceEntry k v []).isEmpty = [].isEmpty
case cons
α : Type u
β : α → Type v
inst✝ : BEq α
k : α
v : β k
k✝ : α
v✝ : β k✝
tail✝ : List ((a : α) × β a)
a✝ : (replaceEntry k v tail✝).isEmpty = tail✝.isEmpty
⊢ (replaceEntry k v (⟨k✝, v✝⟩ :: tail✝)).isEmpty = (⟨k✝, v✝⟩ :: tail✝).isEmpty
|
f9d09d9271745f83
|
List.append_cancel_right_eq
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/BasicAux.lean
|
theorem append_cancel_right_eq (as bs cs : List α) : (as ++ bs = cs ++ bs) = (as = cs)
|
case a.mp
α : Type u_1
as bs cs : List α
⊢ as ++ bs = cs ++ bs → as = cs
case a.mpr
α : Type u_1
as bs cs : List α
⊢ as = cs → as ++ bs = cs ++ bs
|
next => apply append_cancel_right
|
case a.mpr
α : Type u_1
as bs cs : List α
⊢ as = cs → as ++ bs = cs ++ bs
|
6be9cff41bf607b4
|
Real.Angle.neg_pi_div_two_ne_zero
|
Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean
|
theorem neg_pi_div_two_ne_zero : ((-π / 2 : ℝ) : Angle) ≠ 0
|
⊢ -π / 2 ≠ 0
|
exact div_ne_zero (neg_ne_zero.2 Real.pi_ne_zero) two_ne_zero
|
no goals
|
76d7b0c884677ef3
|
IsDiscreteValuationRing.iff_pid_with_one_nonzero_prime
|
Mathlib/RingTheory/DiscreteValuationRing/Basic.lean
|
theorem iff_pid_with_one_nonzero_prime (R : Type u) [CommRing R] [IsDomain R] :
IsDiscreteValuationRing R ↔ IsPrincipalIdealRing R ∧ ∃! P : Ideal R, P ≠ ⊥ ∧ IsPrime P
|
case mpr.intro.intro.intro.intro
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
RPID : IsPrincipalIdealRing R
this : IsLocalRing R
P : Ideal R
right✝ : ∀ (y : Ideal R), (fun P => P ≠ ⊥ ∧ P.IsPrime) y → y = P
hP1 : P ≠ ⊥
hP2 : P.IsPrime
hPM : P = ⊥
h : maximalIdeal R = ⊥
⊢ False
|
exact hP1 hPM
|
no goals
|
f7521cb0caad85a8
|
Profinite.NobelingProof.projRestricts_eq_id
|
Mathlib/Topology/Category/Profinite/Nobeling.lean
|
theorem projRestricts_eq_id : ProjRestricts C (fun i (h : J i) ↦ h) = id
|
I : Type u
C : Set (I → Bool)
J : I → Prop
inst✝ : (i : I) → Decidable (J i)
⊢ ProjRestricts C ⋯ = id
|
ext ⟨x, y, hy, rfl⟩ i
|
case h.mk.intro.intro.a.h
I : Type u
C : Set (I → Bool)
J : I → Prop
inst✝ : (i : I) → Decidable (J i)
y : I → Bool
hy : y ∈ C
i : I
⊢ ↑(ProjRestricts C ⋯ ⟨Proj J y, ⋯⟩) i = ↑(id ⟨Proj J y, ⋯⟩) i
|
ba590648a7f71fc0
|
Batteries.AssocList.toList_eq_toListTR
|
Mathlib/.lake/packages/batteries/Batteries/Data/AssocList.lean
|
theorem toList_eq_toListTR : @toList = @toListTR
|
case h.h.h
α : Type u_2
β : Type u_1
as : AssocList α β
⊢ as.toList = (List.foldl (fun d x => d.push (x.fst, x.snd)) #[] as.toList).toList
|
exact .symm <| (Array.foldl_toList_eq_map (toList as) _ id).trans (List.map_id _)
|
no goals
|
8b1a4ef5b4991f78
|
CStarAlgebra.span_nonneg_inter_ball
|
Mathlib/Analysis/CStarAlgebra/SpecialFunctions/PosPart.lean
|
/-- A C⋆-algebra is spanned by nonnegative elements of norm less than `r`. -/
lemma span_nonneg_inter_ball {r : ℝ} (hr : 0 < r) :
span ℂ ({x : A | 0 ≤ x} ∩ Metric.ball 0 r) = ⊤
|
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
r : ℝ
hr : 0 < r
⊢ span ℂ ({x | 0 ≤ x} ∩ Metric.closedBall 0 (r / 2)) ≤ span ℂ ({x | 0 ≤ x} ∩ Metric.ball 0 r)
|
gcongr
|
case h.H
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
r : ℝ
hr : 0 < r
⊢ Metric.closedBall 0 (r / 2) ⊆ Metric.ball 0 r
|
eed7d32d179fb6f5
|
Fin.inv_partialProd_mul_eq_contractNth
|
Mathlib/Algebra/BigOperators/Fin.lean
|
theorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)
(j : Fin (n + 1)) (k : Fin n) :
(partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =
j.contractNth (· * ·) g k
|
case inr.inr
n : ℕ
G : Type u_3
inst✝ : Group G
g : Fin (n + 1) → G
j : Fin (n + 1)
k : Fin n
h : ↑j < ↑k
⊢ (partialProd g (j.succ.succAbove k.castSucc))⁻¹ * partialProd g (j.succAbove k).succ =
j.contractNth (fun x1 x2 => x1 * x2) g k
|
rwa [succAbove_of_le_castSucc, succAbove_of_le_castSucc, partialProd_succ, partialProd_succ,
castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]
|
case inr.inr.h
n : ℕ
G : Type u_3
inst✝ : Group G
g : Fin (n + 1) → G
j : Fin (n + 1)
k : Fin n
h : ↑j < ↑k
⊢ j ≤ k.castSucc
case inr.inr.h
n : ℕ
G : Type u_3
inst✝ : Group G
g : Fin (n + 1) → G
j : Fin (n + 1)
k : Fin n
h : ↑j < ↑k
⊢ j.succ ≤ k.castSucc.castSucc
|
f84a17ef32860b19
|
ProbabilityTheory.Kernel.compProd_fst_condKernelReal
|
Mathlib/Probability/Kernel/Disintegration/StandardBorel.lean
|
lemma compProd_fst_condKernelReal (κ : Kernel α (γ × ℝ)) [IsFiniteKernel κ] :
fst κ ⊗ₖ condKernelReal κ = κ
|
α : Type u_1
γ : Type u_3
mα : MeasurableSpace α
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × ℝ)
inst✝ : IsFiniteKernel κ
⊢ κ.fst ⊗ₖ κ.condKernelReal = κ
|
rw [condKernelReal, compProd_toKernel]
|
no goals
|
22f942ab3ebf60df
|
GenContFract.compExactValue_correctness_of_stream_eq_some
|
Mathlib/Algebra/ContinuedFractions/Computation/CorrectnessTerminating.lean
|
theorem compExactValue_correctness_of_stream_eq_some :
∀ {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n →
v = compExactValue ((of v).contsAux n) ((of v).contsAux <| n + 1) ifp_n.fr
|
case succ.intro.intro.intro.inr
K : Type u_1
inst✝¹ : LinearOrderedField K
v : K
n✝ : ℕ
inst✝ : FloorRing K
g : GenContFract K := of v
n : ℕ
ifp_succ_n : IntFractPair K
succ_nth_stream_eq : IntFractPair.stream v (n + 1) = some ifp_succ_n
ifp_n : IntFractPair K
nth_stream_eq : IntFractPair.stream v n = some ifp_n
nth_fract_ne_zero : ifp_n.fr ≠ 0
conts : Pair K := g.contsAux (n + 2)
pconts : Pair K := g.contsAux (n + 1)
pconts_eq : pconts = g.contsAux (n + 1)
ppconts : Pair K := g.contsAux n
IH : ∀ {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n → v = compExactValue ppconts pconts ifp_n.fr
ppconts_eq : ppconts = g.contsAux n
ifp_succ_n_fr_ne_zero : ifp_succ_n.fr ≠ 0
⊢ compExactValue ppconts pconts ifp_n.fr = compExactValue pconts conts ifp_succ_n.fr
|
obtain ⟨ifp_n', nth_stream_eq', ifp_n_fract_ne_zero, ⟨refl⟩⟩ :
∃ ifp_n, IntFractPair.stream v n = some ifp_n ∧
ifp_n.fr ≠ 0 ∧ IntFractPair.of ifp_n.fr⁻¹ = ifp_succ_n :=
IntFractPair.succ_nth_stream_eq_some_iff.1 succ_nth_stream_eq
|
case succ.intro.intro.intro.inr.intro.intro.intro.refl
K : Type u_1
inst✝¹ : LinearOrderedField K
v : K
n✝ : ℕ
inst✝ : FloorRing K
g : GenContFract K := of v
n : ℕ
ifp_n : IntFractPair K
nth_stream_eq : IntFractPair.stream v n = some ifp_n
nth_fract_ne_zero : ifp_n.fr ≠ 0
conts : Pair K := g.contsAux (n + 2)
pconts : Pair K := g.contsAux (n + 1)
pconts_eq : pconts = g.contsAux (n + 1)
ppconts : Pair K := g.contsAux n
IH : ∀ {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n → v = compExactValue ppconts pconts ifp_n.fr
ppconts_eq : ppconts = g.contsAux n
ifp_n' : IntFractPair K
nth_stream_eq' : IntFractPair.stream v n = some ifp_n'
ifp_n_fract_ne_zero : ifp_n'.fr ≠ 0
succ_nth_stream_eq : IntFractPair.stream v (n + 1) = some (IntFractPair.of ifp_n'.fr⁻¹)
ifp_succ_n_fr_ne_zero : (IntFractPair.of ifp_n'.fr⁻¹).fr ≠ 0
⊢ compExactValue ppconts pconts ifp_n.fr = compExactValue pconts conts (IntFractPair.of ifp_n'.fr⁻¹).fr
|
434c0d1df2cfd41d
|
AnalyticAt.fderiv
|
Mathlib/Analysis/Calculus/FDeriv/Analytic.lean
|
theorem AnalyticAt.fderiv [CompleteSpace F] (h : AnalyticAt 𝕜 f x) :
AnalyticAt 𝕜 (fderiv 𝕜 f) x
|
case intro.intro
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
f : E → F
x : E
inst✝ : CompleteSpace F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
hp : HasFPowerSeriesOnBall f p x r
⊢ AnalyticAt 𝕜 (fderiv 𝕜 f) x
|
exact hp.fderiv.analyticAt
|
no goals
|
414431133816d537
|
HNNExtension.NormalWord.unitsSMul_one_group_smul
|
Mathlib/GroupTheory/HNNExtension.lean
|
theorem unitsSMul_one_group_smul (g : A) (w : NormalWord d) :
unitsSMul φ 1 ((g : G) • w) = (φ g : G) • (unitsSMul φ 1 w)
|
case pos.cons
G : Type u_1
inst✝ : Group G
A B : Subgroup G
φ : ↥A ≃* ↥B
d : TransversalPair G A B
g : ↥A
g✝ : G
u✝ : ℤˣ
w✝ : NormalWord d
h1✝ : w✝.head ∈ d.set u✝
h2✝ : ∀ u' ∈ Option.map Prod.fst w✝.toList.head?, w✝.head ∈ toSubgroup A B u✝ → u✝ = u'
this : Cancels 1 (↑g • cons g✝ u✝ w✝ h1✝ h2✝) ↔ Cancels 1 (cons g✝ u✝ w✝ h1✝ h2✝)
hcan : Cancels 1 (cons g✝ u✝ w✝ h1✝ h2✝)
a✝ :
cons g✝ u✝ w✝ h1✝ h2✝ = w✝ →
consRecOn (motive := fun x => Cancels 1 x → NormalWord d) (↑g • cons g✝ u✝ w✝ h1✝ h2✝)
(fun g_1 a => ↑g • cons g✝ u✝ w✝ h1✝ h2✝) (fun g x w x_1 x_2 x_3 can => ↑(φ ⟨g, ⋯⟩) • w) ⋯ =
↑(φ g) •
consRecOn (motive := fun x => Cancels 1 x → NormalWord d) (cons g✝ u✝ w✝ h1✝ h2✝)
(fun g a => cons g✝ u✝ w✝ h1✝ h2✝) (fun g x w x_1 x_2 x_3 can => ↑(φ ⟨g, ⋯⟩) • w) hcan
⊢ ↑(φ ⟨↑g * g✝, ⋯⟩) • w✝ = ↑(φ (g * ⟨g✝, ⋯⟩)) • w✝
|
rfl
|
no goals
|
17f5302f5114464d
|
ENNReal.Lp_add_le
|
Mathlib/Analysis/MeanInequalities.lean
|
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i ∈ s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i ∈ s, f i ^ p) ^ (1 / p) + (∑ i ∈ s, g i ^ p) ^ (1 / p)
|
case neg
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p : ℝ
hp : 1 ≤ p
pos : 0 < p
H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
this :
↑((∑ i ∈ s, ((fun i => (f i).toNNReal) i + (fun i => (g i).toNNReal) i) ^ p) ^ (1 / p)) ≤
↑((∑ i ∈ s, (fun i => (f i).toNNReal) i ^ p) ^ (1 / p) + (∑ i ∈ s, (fun i => (g i).toNNReal) i ^ p) ^ (1 / p))
⊢ (∑ i ∈ s, (f i + g i) ^ p) ^ (1 / p) ≤ (∑ i ∈ s, f i ^ p) ^ (1 / p) + (∑ i ∈ s, g i ^ p) ^ (1 / p)
|
push_cast [ENNReal.coe_rpow_of_nonneg, le_of_lt pos, le_of_lt (one_div_pos.2 pos)] at this
|
case neg
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p : ℝ
hp : 1 ≤ p
pos : 0 < p
H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
this :
(∑ x ∈ s, (↑(f x).toNNReal + ↑(g x).toNNReal) ^ p) ^ (1 / p) ≤
(∑ x ∈ s, ↑(f x).toNNReal ^ p) ^ (1 / p) + (∑ x ∈ s, ↑(g x).toNNReal ^ p) ^ (1 / p)
⊢ (∑ i ∈ s, (f i + g i) ^ p) ^ (1 / p) ≤ (∑ i ∈ s, f i ^ p) ^ (1 / p) + (∑ i ∈ s, g i ^ p) ^ (1 / p)
|
cfe53d149b9d93ea
|
Set.PartiallyWellOrderedOn.partiallyWellOrderedOn_sublistForall₂
|
Mathlib/Order/WellFoundedSet.lean
|
theorem partiallyWellOrderedOn_sublistForall₂ (r : α → α → Prop) [IsRefl α r] [IsTrans α r]
{s : Set α} (h : s.PartiallyWellOrderedOn r) :
{ l : List α | ∀ x, x ∈ l → x ∈ s }.PartiallyWellOrderedOn (List.SublistForall₂ r)
|
case inr
α : Type u_2
r : α → α → Prop
inst✝¹ : IsRefl α r
inst✝ : IsTrans α r
s : Set α
h : s.PartiallyWellOrderedOn r
h✝ : Nonempty α
⊢ {l | ∀ x ∈ l, x ∈ s}.PartiallyWellOrderedOn (List.SublistForall₂ r)
|
inhabit α
|
case inr
α : Type u_2
r : α → α → Prop
inst✝¹ : IsRefl α r
inst✝ : IsTrans α r
s : Set α
h : s.PartiallyWellOrderedOn r
h✝ : Nonempty α
inhabited_h : Inhabited α
⊢ {l | ∀ x ∈ l, x ∈ s}.PartiallyWellOrderedOn (List.SublistForall₂ r)
|
642b3f587e356afd
|
Ideal.IsHomogeneous.isPrime_of_homogeneous_mem_or_mem
|
Mathlib/RingTheory/GradedAlgebra/Radical.lean
|
theorem Ideal.IsHomogeneous.isPrime_of_homogeneous_mem_or_mem {I : Ideal A} (hI : I.IsHomogeneous 𝒜)
(I_ne_top : I ≠ ⊤)
(homogeneous_mem_or_mem :
∀ {x y : A}, IsHomogeneousElem 𝒜 x → IsHomogeneousElem 𝒜 y → x * y ∈ I → x ∈ I ∨ y ∈ I) :
Ideal.IsPrime I :=
⟨I_ne_top, by
intro x y hxy
by_contra! rid
obtain ⟨rid₁, rid₂⟩ := rid
classical
/-
The idea of the proof is the following :
since `x * y ∈ I` and `I` homogeneous, then `proj i (x * y) ∈ I` for any `i : ι`.
Then consider two sets `{i ∈ x.support | xᵢ ∉ I}` and `{j ∈ y.support | yⱼ ∉ J}`;
let `max₁, max₂` be the maximum of the two sets, then `proj (max₁ + max₂) (x * y) ∈ I`.
Then, `proj max₁ x ∉ I` and `proj max₂ j ∉ I`
but `proj i x ∈ I` for all `max₁ < i` and `proj j y ∈ I` for all `max₂ < j`.
` proj (max₁ + max₂) (x * y)`
`= ∑ {(i, j) ∈ supports | i + j = max₁ + max₂}, xᵢ * yⱼ`
`= proj max₁ x * proj max₂ y`
` + ∑ {(i, j) ∈ supports \ {(max₁, max₂)} | i + j = max₁ + max₂}, xᵢ * yⱼ`.
This is a contradiction, because both `proj (max₁ + max₂) (x * y) ∈ I` and the sum on the
right hand side is in `I` however `proj max₁ x * proj max₂ y` is not in `I`.
-/
set set₁ := {i ∈ (decompose 𝒜 x).support | proj 𝒜 i x ∉ I} with set₁_eq
set set₂ := {i ∈ (decompose 𝒜 y).support | proj 𝒜 i y ∉ I} with set₂_eq
have nonempty :
∀ x : A, x ∉ I → {i ∈ (decompose 𝒜 x).support | proj 𝒜 i x ∉ I}.Nonempty
|
ι : Type u_1
σ : Type u_2
A : Type u_3
inst✝⁴ : CommRing A
inst✝³ : LinearOrderedCancelAddCommMonoid ι
inst✝² : SetLike σ A
inst✝¹ : AddSubmonoidClass σ A
𝒜 : ι → σ
inst✝ : GradedRing 𝒜
I : Ideal A
hI : IsHomogeneous 𝒜 I
I_ne_top : I ≠ ⊤
homogeneous_mem_or_mem : ∀ {x y : A}, IsHomogeneousElem 𝒜 x → IsHomogeneousElem 𝒜 y → x * y ∈ I → x ∈ I ∨ y ∈ I
x y : A
rid₁ : x ∉ I
rid₂ : y ∉ I
set₁ : Finset ι := filter (fun i => (proj 𝒜 i) x ∉ I) (DFinsupp.support ((decompose 𝒜) x))
set₁_eq : set₁ = filter (fun i => (proj 𝒜 i) x ∉ I) (DFinsupp.support ((decompose 𝒜) x))
set₂ : Finset ι := filter (fun i => (proj 𝒜 i) y ∉ I) (DFinsupp.support ((decompose 𝒜) y))
set₂_eq : set₂ = filter (fun i => (proj 𝒜 i) y ∉ I) (DFinsupp.support ((decompose 𝒜) y))
nonempty : ∀ x ∉ I, (filter (fun i => (proj 𝒜 i) x ∉ I) (DFinsupp.support ((decompose 𝒜) x))).Nonempty
max₁ : ι := set₁.max' ⋯
max₂ : ι := set₂.max' ⋯
mem_max₁ : max₁ ∈ set₁
mem_max₂ : max₂ ∈ set₂
hxy : (proj 𝒜 (max₁ + max₂)) (x * y) ∈ I
mem_I : (proj 𝒜 max₁) x * (proj 𝒜 max₂) y ∈ I
⊢ (proj 𝒜 max₁) x ∉ I ∧ (proj 𝒜 max₂) y ∉ I
|
rw [mem_filter] at mem_max₁ mem_max₂
|
ι : Type u_1
σ : Type u_2
A : Type u_3
inst✝⁴ : CommRing A
inst✝³ : LinearOrderedCancelAddCommMonoid ι
inst✝² : SetLike σ A
inst✝¹ : AddSubmonoidClass σ A
𝒜 : ι → σ
inst✝ : GradedRing 𝒜
I : Ideal A
hI : IsHomogeneous 𝒜 I
I_ne_top : I ≠ ⊤
homogeneous_mem_or_mem : ∀ {x y : A}, IsHomogeneousElem 𝒜 x → IsHomogeneousElem 𝒜 y → x * y ∈ I → x ∈ I ∨ y ∈ I
x y : A
rid₁ : x ∉ I
rid₂ : y ∉ I
set₁ : Finset ι := filter (fun i => (proj 𝒜 i) x ∉ I) (DFinsupp.support ((decompose 𝒜) x))
set₁_eq : set₁ = filter (fun i => (proj 𝒜 i) x ∉ I) (DFinsupp.support ((decompose 𝒜) x))
set₂ : Finset ι := filter (fun i => (proj 𝒜 i) y ∉ I) (DFinsupp.support ((decompose 𝒜) y))
set₂_eq : set₂ = filter (fun i => (proj 𝒜 i) y ∉ I) (DFinsupp.support ((decompose 𝒜) y))
nonempty : ∀ x ∉ I, (filter (fun i => (proj 𝒜 i) x ∉ I) (DFinsupp.support ((decompose 𝒜) x))).Nonempty
max₁ : ι := set₁.max' ⋯
max₂ : ι := set₂.max' ⋯
mem_max₁ : max₁ ∈ DFinsupp.support ((decompose 𝒜) x) ∧ (proj 𝒜 max₁) x ∉ I
mem_max₂ : max₂ ∈ DFinsupp.support ((decompose 𝒜) y) ∧ (proj 𝒜 max₂) y ∉ I
hxy : (proj 𝒜 (max₁ + max₂)) (x * y) ∈ I
mem_I : (proj 𝒜 max₁) x * (proj 𝒜 max₂) y ∈ I
⊢ (proj 𝒜 max₁) x ∉ I ∧ (proj 𝒜 max₂) y ∉ I
|
7b2fd2fea167fa2f
|
MulAction.mem_subgroup_orbit_iff
|
Mathlib/GroupTheory/GroupAction/Defs.lean
|
@[to_additive]
lemma mem_subgroup_orbit_iff {H : Subgroup G} {x : α} {a b : orbit G x} :
a ∈ MulAction.orbit H b ↔ (a : α) ∈ MulAction.orbit H (b : α)
|
case refine_2.intro
G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
H : Subgroup G
x : α
a b : ↑(orbit G x)
g : ↥H
h : ↑g • b = a
⊢ a ∈ orbit (↥H) b
|
subst h
|
case refine_2.intro
G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
H : Subgroup G
x : α
b : ↑(orbit G x)
g : ↥H
⊢ ↑g • b ∈ orbit (↥H) b
|
455130db4c4a28b8
|
LinearMap.liftBaseChange_one_tmul
|
Mathlib/RingTheory/TensorProduct/Basic.lean
|
lemma liftBaseChange_one_tmul (l : M →ₗ[R] N) (y) : l.liftBaseChange A (1 ⊗ₜ y) = l y
|
R : Type u_1
M : Type u_2
N : Type u_3
A : Type u_4
inst✝⁸ : CommSemiring R
inst✝⁷ : CommSemiring A
inst✝⁶ : Algebra R A
inst✝⁵ : AddCommMonoid M
inst✝⁴ : AddCommMonoid N
inst✝³ : Module R M
inst✝² : Module R N
inst✝¹ : Module A N
inst✝ : IsScalarTower R A N
l : M →ₗ[R] N
y : M
⊢ (liftBaseChange A l) (1 ⊗ₜ[R] y) = l y
|
simp
|
no goals
|
ab42b9224ea6fc89
|
Ordnode.Valid'.node4L
|
Mathlib/Data/Ordmap/Ordset.lean
|
theorem Valid'.node4L {l} {x : α} {m} {y : α} {r o₁ o₂} (hl : Valid' o₁ l x) (hm : Valid' x m y)
(hr : Valid' (↑y) r o₂) (Hm : 0 < size m)
(H : size l = 0 ∧ size m = 1 ∧ size r ≤ 1 ∨
0 < size l ∧
ratio * size r ≤ size m ∧
delta * size l ≤ size m + size r ∧
3 * (size m + size r) ≤ 16 * size l + 9 ∧ size m ≤ delta * size r) :
Valid' o₁ (@node4L α l x m y r) o₂
|
α : Type u_1
inst✝ : Preorder α
l : Ordnode α
x y : α
r : Ordnode α
o₁ : WithBot α
o₂ : WithTop α
hl : Valid' o₁ l ↑x
hr : Valid' (↑y) r o₂
s : ℕ
ml : Ordnode α
z : α
mr : Ordnode α
hm : Valid' (↑x) (Ordnode.node s ml z mr) ↑y
Hm : 0 < (Ordnode.node s ml z mr).size
l0 : 0 < l.size
mr₁ : ratio * r.size ≤ ml.size + mr.size + 1
lr₁ : delta * l.size ≤ ml.size + mr.size + 1 + r.size
lr₂ : 3 * (ml.size + mr.size + 1 + r.size) ≤ 16 * l.size + 9
mr₂ : ml.size + mr.size + 1 ≤ delta * r.size
r0 : r.size > 0
mm : ¬ml.size + mr.size ≤ 1
mm₁ : ml.size ≤ delta * mr.size
mm₂ : mr.size ≤ delta * ml.size
ml0 : ml.size > 0
this✝ : delta * (ratio * l.size) ≤ ratio * (ml.size + mr.size + 1) + ratio * r.size
this : delta * (ratio * l.size) ≤ ratio * (ml.size + mr.size + 1) + (ml.size + mr.size + 1)
⊢ 2 * l.size ≤ ml.size + mr.size + 1
|
rw [← Nat.succ_mul] at this
|
α : Type u_1
inst✝ : Preorder α
l : Ordnode α
x y : α
r : Ordnode α
o₁ : WithBot α
o₂ : WithTop α
hl : Valid' o₁ l ↑x
hr : Valid' (↑y) r o₂
s : ℕ
ml : Ordnode α
z : α
mr : Ordnode α
hm : Valid' (↑x) (Ordnode.node s ml z mr) ↑y
Hm : 0 < (Ordnode.node s ml z mr).size
l0 : 0 < l.size
mr₁ : ratio * r.size ≤ ml.size + mr.size + 1
lr₁ : delta * l.size ≤ ml.size + mr.size + 1 + r.size
lr₂ : 3 * (ml.size + mr.size + 1 + r.size) ≤ 16 * l.size + 9
mr₂ : ml.size + mr.size + 1 ≤ delta * r.size
r0 : r.size > 0
mm : ¬ml.size + mr.size ≤ 1
mm₁ : ml.size ≤ delta * mr.size
mm₂ : mr.size ≤ delta * ml.size
ml0 : ml.size > 0
this✝ : delta * (ratio * l.size) ≤ ratio * (ml.size + mr.size + 1) + ratio * r.size
this : delta * (ratio * l.size) ≤ ratio.succ * (ml.size + mr.size + 1)
⊢ 2 * l.size ≤ ml.size + mr.size + 1
|
4d0cc9fa23e9caa2
|
Vector.getElem_zero_flatten
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Vector/Find.lean
|
theorem getElem_zero_flatten {L : Vector (Vector α m) n} (h : 0 < n * m) :
(flatten L)[0] = (L.findSome? fun l => l[0]?).get (getElem_zero_flatten.proof h)
|
α : Type u_1
m n : Nat
L : Vector (Vector α m) n
h : 0 < n * m
t : L.flatten[0]? = findSome? (fun l => l[0]?) L
⊢ L.flatten[0] = (findSome? (fun l => l[0]?) L).get ⋯
|
simp [getElem?_eq_getElem, h] at t
|
α : Type u_1
m n : Nat
L : Vector (Vector α m) n
h : 0 < n * m
t : some L[0][0] = findSome? (fun l => l[0]?) L
⊢ L.flatten[0] = (findSome? (fun l => l[0]?) L).get ⋯
|
65e6e193ff3f3283
|
List.foldl_argAux_eq_none
|
Mathlib/Data/List/MinMax.lean
|
theorem foldl_argAux_eq_none : l.foldl (argAux r) o = none ↔ l = [] ∧ o = none :=
List.reverseRecOn l (by simp) fun tl hd => by
simp only [foldl_append, foldl_cons, argAux, foldl_nil, append_eq_nil_iff, and_false, false_and,
iff_false]
cases foldl (argAux r) o tl
· simp
· simp only [false_iff, not_and]
split_ifs <;> simp
|
α : Type u_1
r : α → α → Prop
inst✝ : DecidableRel r
l : List α
o : Option α
⊢ foldl (argAux r) o [] = none ↔ [] = [] ∧ o = none
|
simp
|
no goals
|
82e69bedad8e20e2
|
RelSeries.append_apply_right
|
Mathlib/Order/RelSeries.lean
|
lemma append_apply_right (p q : RelSeries r) (connect : r p.last q.head)
(i : Fin (q.length + 1)) :
p.append q connect (i.natAdd p.length + 1) = q i
|
case h.e'_2.h.e'_6.h.h
α : Type u_1
r : Rel α α
p q : RelSeries r
connect : r p.last q.head
i : Fin (q.length + 1)
⊢ p.length + ↑i + 1 < (p.length + q.length + 1).succ
|
omega
|
no goals
|
81a20ecaf5e7e778
|
SmoothPartitionOfUnity.exists_isSubordinate
|
Mathlib/Geometry/Manifold/PartitionOfUnity.lean
|
theorem exists_isSubordinate {s : Set M} (hs : IsClosed s) (U : ι → Set M) (ho : ∀ i, IsOpen (U i))
(hU : s ⊆ ⋃ i, U i) : ∃ f : SmoothPartitionOfUnity ι I M s, f.IsSubordinate U
|
case refine_1.intro
ι : Type uι
E : Type uE
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type uH
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : FiniteDimensional ℝ E
inst✝² : IsManifold I ∞ M
inst✝¹ : T2Space M
inst✝ : SigmaCompactSpace M
s✝ : Set M
hs✝ : IsClosed s✝
U : ι → Set M
ho : ∀ (i : ι), IsOpen (U i)
hU : s✝ ⊆ ⋃ i, U i
this✝ : LocallyCompactSpace H
this : LocallyCompactSpace M
s t : Set M
hs : IsClosed s
ht : IsClosed t
hd : Disjoint s t
f : C^∞⟮I, M; 𝓘(ℝ, ℝ), ℝ⟯
hf : EqOn (⇑f) 0 s ∧ EqOn (⇑f) 1 t ∧ ∀ (x : M), f x ∈ Icc 0 1
⊢ ∃ f, ContMDiff I 𝓘(ℝ, ℝ) ∞ ⇑f ∧ EqOn (⇑f) 0 s ∧ EqOn (⇑f) 1 t ∧ ∀ (x : M), f x ∈ Icc 0 1
|
exact ⟨f, f.contMDiff, hf⟩
|
no goals
|
bb02907964d5e497
|
AlgebraicGeometry.SurjectiveOnStalks.isEmbedding_pullback
|
Mathlib/AlgebraicGeometry/Morphisms/SurjectiveOnStalks.lean
|
/-- If `Y ⟶ S` is surjective on stalks, then for every `X ⟶ S`, `X ×ₛ Y` is a subset of
`X × Y` (cartesian product as topological spaces) with the induced topology. -/
lemma isEmbedding_pullback {X Y S : Scheme.{u}} (f : X ⟶ S) (g : Y ⟶ S) [SurjectiveOnStalks g] :
IsEmbedding (fun x ↦ ((pullback.fst f g).base x, (pullback.snd f g).base x))
|
case h.e'_5.h.h.intro.snd
X Y S : Scheme
f : X ⟶ S
g : Y ⟶ S
inst✝ : SurjectiveOnStalks g
L : ↑↑(pullback f g).toPresheafedSpace → ↑↑X.toPresheafedSpace × ↑↑Y.toPresheafedSpace :=
fun x => ((ConcreteCategory.hom (pullback.fst f g).base) x, (ConcreteCategory.hom (pullback.snd f g).base) x)
R A B : CommRingCat
iX : Spec A ⟶ X
iY : Spec B ⟶ Y
iS : Spec R ⟶ S
a✝² : IsOpenImmersion iX
a✝¹ : IsOpenImmersion iY
a✝ : IsOpenImmersion iS
φ : R ⟶ A
e₁ : Spec.map φ ≫ iS = iX ≫ f
ψ : R ⟶ B
e₂ : Spec.map ψ ≫ iS = iY ≫ g
H : (CommRingCat.Hom.hom ψ).SurjectiveOnStalks
algInst✝¹ : Algebra ↑R ↑A := (CommRingCat.Hom.hom φ).toAlgebra
algInst✝ : Algebra ↑R ↑B := (CommRingCat.Hom.hom ψ).toAlgebra
e_1✝ :
↑↑(pullback (Spec.map φ) (Spec.map ψ)).toPresheafedSpace =
↑↑(pullback (Spec.map (CommRingCat.ofHom (algebraMap ↑R ↑A)))
(Spec.map (CommRingCat.ofHom (algebraMap ↑R ↑B)))).toPresheafedSpace
x : ↑↑(Spec (CommRingCat.of (TensorProduct ↑R ↑A ↑B))).toPresheafedSpace
⊢ (L
((ConcreteCategory.hom (pullback.map (Spec.map φ) (Spec.map ψ) f g iX iY iS e₁ e₂).base)
((ConcreteCategory.hom (pullbackSpecIso ↑R ↑A ↑B).symm.hom.base) x))).2 =
(Prod.map (⇑(ConcreteCategory.hom iX.base)) (⇑(ConcreteCategory.hom iY.base))
(PrimeSpectrum.tensorProductTo (↑R) (↑A) (↑B)
((ConcreteCategory.hom (pullbackSpecIso ↑R ↑A ↑B).hom.base)
((ConcreteCategory.hom (pullbackSpecIso ↑R ↑A ↑B).symm.hom.base) x)))).2
|
simp only [L, ← Scheme.comp_base_apply, pullback.lift_snd, Iso.symm_hom,
Iso.inv_hom_id]
|
case h.e'_5.h.h.intro.snd
X Y S : Scheme
f : X ⟶ S
g : Y ⟶ S
inst✝ : SurjectiveOnStalks g
L : ↑↑(pullback f g).toPresheafedSpace → ↑↑X.toPresheafedSpace × ↑↑Y.toPresheafedSpace :=
fun x => ((ConcreteCategory.hom (pullback.fst f g).base) x, (ConcreteCategory.hom (pullback.snd f g).base) x)
R A B : CommRingCat
iX : Spec A ⟶ X
iY : Spec B ⟶ Y
iS : Spec R ⟶ S
a✝² : IsOpenImmersion iX
a✝¹ : IsOpenImmersion iY
a✝ : IsOpenImmersion iS
φ : R ⟶ A
e₁ : Spec.map φ ≫ iS = iX ≫ f
ψ : R ⟶ B
e₂ : Spec.map ψ ≫ iS = iY ≫ g
H : (CommRingCat.Hom.hom ψ).SurjectiveOnStalks
algInst✝¹ : Algebra ↑R ↑A := (CommRingCat.Hom.hom φ).toAlgebra
algInst✝ : Algebra ↑R ↑B := (CommRingCat.Hom.hom ψ).toAlgebra
e_1✝ :
↑↑(pullback (Spec.map φ) (Spec.map ψ)).toPresheafedSpace =
↑↑(pullback (Spec.map (CommRingCat.ofHom (algebraMap ↑R ↑A)))
(Spec.map (CommRingCat.ofHom (algebraMap ↑R ↑B)))).toPresheafedSpace
x : ↑↑(Spec (CommRingCat.of (TensorProduct ↑R ↑A ↑B))).toPresheafedSpace
⊢ (ConcreteCategory.hom (pullback.snd (Spec.map φ) (Spec.map ψ) ≫ iY).base)
((ConcreteCategory.hom (pullbackSpecIso ↑R ↑A ↑B).inv.base) x) =
(Prod.map (⇑(ConcreteCategory.hom iX.base)) (⇑(ConcreteCategory.hom iY.base))
(PrimeSpectrum.tensorProductTo (↑R) (↑A) (↑B)
((ConcreteCategory.hom (𝟙 (Spec (CommRingCat.of (TensorProduct ↑R ↑A ↑B)))).base) x))).2
|
9509c1734d72b9a6
|
exists_dist_le_le
|
Mathlib/Analysis/NormedSpace/Pointwise.lean
|
theorem exists_dist_le_le (hδ : 0 ≤ δ) (hε : 0 ≤ ε) (h : dist x z ≤ ε + δ) :
∃ y, dist x y ≤ δ ∧ dist y z ≤ ε
|
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x z : E
δ : ℝ
hδ : 0 ≤ δ
hε : 0 ≤ 0
h : dist x z ≤ 0 + δ
⊢ dist x z ≤ δ
|
rwa [zero_add] at h
|
no goals
|
59238fa715097bae
|
TopologicalSpace.IsOpenCover.quasiSober_iff_forall
|
Mathlib/Topology/Sober.lean
|
lemma TopologicalSpace.IsOpenCover.quasiSober_iff_forall {ι : Type*} {U : ι → Opens α}
(hU : TopologicalSpace.IsOpenCover U) : QuasiSober α ↔ ∀ i, QuasiSober (U i)
|
case h.a
α : Type u_1
inst✝ : TopologicalSpace α
ι : Type u_3
U : ι → Opens α
hU : IsOpenCover U
hU' : ∀ (i : ι), QuasiSober ↥(U i)
t : Set α
h : IsPreirreducible t
x : α
hx : x ∈ t
h' : IsClosed t
i : ι
hi : x ∈ U i
H : IsIrreducible (Subtype.val ⁻¹' t)
⊢ t ≤ closure (Subtype.val '' closure (Subtype.val ⁻¹' t))
|
refine (subset_closure_inter_of_isPreirreducible_of_isOpen h (U i).isOpen ⟨x, ⟨hx, hi⟩⟩).trans
(closure_mono ?_)
|
case h.a
α : Type u_1
inst✝ : TopologicalSpace α
ι : Type u_3
U : ι → Opens α
hU : IsOpenCover U
hU' : ∀ (i : ι), QuasiSober ↥(U i)
t : Set α
h : IsPreirreducible t
x : α
hx : x ∈ t
h' : IsClosed t
i : ι
hi : x ∈ U i
H : IsIrreducible (Subtype.val ⁻¹' t)
⊢ t ∩ ↑(U i) ⊆ Subtype.val '' closure (Subtype.val ⁻¹' t)
|
d49af6a604b6ce82
|
AlgebraicTopology.AlternatingFaceMapComplex.d_squared
|
Mathlib/AlgebraicTopology/AlternatingFaceMapComplex.lean
|
theorem d_squared (n : ℕ) : objD X (n + 1) ≫ objD X n = 0
|
case hi
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
n : ℕ
P : Type := Fin (n + 2) × Fin (n + 3)
S : Finset P := Finset.filter (fun ij => ↑ij.2 ≤ ↑ij.1) Finset.univ
φ : (ij : P) → ij ∈ S → P := fun ij hij => (ij.2.castLT ⋯, ij.1.succ)
ij : P
hij : ij ∈ S
⊢ φ ij hij ∈ Sᶜ
|
simp only [S, φ, Finset.mem_univ, Finset.compl_filter, Finset.mem_filter, true_and,
Fin.val_succ, Fin.coe_castLT] at hij ⊢
|
case hi
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
n : ℕ
P : Type := Fin (n + 2) × Fin (n + 3)
S : Finset P := Finset.filter (fun ij => ↑ij.2 ≤ ↑ij.1) Finset.univ
φ : (ij : P) → ij ∈ S → P := fun ij hij => (ij.2.castLT ⋯, ij.1.succ)
ij : P
hij : ↑ij.2 ≤ ↑ij.1
⊢ ¬↑ij.1 + 1 ≤ ↑ij.2
|
795ab48b2006c7af
|
ZMod.inv_coe_unit
|
Mathlib/Data/ZMod/Basic.lean
|
theorem inv_coe_unit {n : ℕ} (u : (ZMod n)ˣ) : (u : ZMod n)⁻¹ = (u⁻¹ : (ZMod n)ˣ)
|
n : ℕ
u : (ZMod n)ˣ
this : ↑((↑u).val.gcd n) = ↑1
⊢ (↑u)⁻¹ = ↑u⁻¹
|
rw [← mul_inv_eq_gcd, Nat.cast_one] at this
|
n : ℕ
u : (ZMod n)ˣ
this : ↑u * (↑u)⁻¹ = 1
⊢ (↑u)⁻¹ = ↑u⁻¹
|
18898df54cabae2d
|
List.dropInfix?_go_eq_some_iff
|
Mathlib/.lake/packages/batteries/Batteries/Data/List/Lemmas.lean
|
theorem dropInfix?_go_eq_some_iff [BEq α] {i l acc p s : List α} :
dropInfix?.go i l acc = some (p, s) ↔ ∃ p',
p = acc.reverse ++ p' ∧
-- `i` is an infix up to `==`
(∃ i', l = p' ++ i' ++ s ∧ i' == i) ∧
-- and there is no shorter prefix for which that is the case
(∀ p'' i'' s'', l = p'' ++ i'' ++ s'' → i'' == i → p''.length ≥ p'.length)
|
case h_2.h_1.mp.intro.intro.intro.intro.intro.inl.intro
α : Type u_1
inst✝ : BEq α
i acc s x✝² x✝¹ : List α
a : α
x✝ : Option (List α)
p' i' : List α
h₂✝ : (i' == i) = true
h : (a :: (p' ++ i' ++ s)).dropPrefix? i = none
w : ∀ (p'' i'' s'' : List α), p' ++ i' ++ s = p'' ++ i'' ++ s'' → (i'' == i) = true → p''.length ≥ p'.length
i'' s'' : List α
h₂ : (i'' == i) = true
h₁ : i'' ++ s'' = a :: (p' ++ (i' ++ s))
⊢ p'.length + 1 ≤ [].length
|
rw [append_assoc, ← h₁] at h
|
case h_2.h_1.mp.intro.intro.intro.intro.intro.inl.intro
α : Type u_1
inst✝ : BEq α
i acc s x✝² x✝¹ : List α
a : α
x✝ : Option (List α)
p' i' : List α
h₂✝ : (i' == i) = true
w : ∀ (p'' i'' s'' : List α), p' ++ i' ++ s = p'' ++ i'' ++ s'' → (i'' == i) = true → p''.length ≥ p'.length
i'' s'' : List α
h : (i'' ++ s'').dropPrefix? i = none
h₂ : (i'' == i) = true
h₁ : i'' ++ s'' = a :: (p' ++ (i' ++ s))
⊢ p'.length + 1 ≤ [].length
|
2a30116d5cc9db54
|
Submodule.mapQ_pow
|
Mathlib/LinearAlgebra/Quotient/Basic.lean
|
theorem mapQ_pow {f : M →ₗ[R] M} (h : p ≤ p.comap f) (k : ℕ)
(h' : p ≤ p.comap (f ^ k) := p.le_comap_pow_of_le_comap h k) :
p.mapQ p (f ^ k) h' = p.mapQ p f h ^ k
|
case zero
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
p : Submodule R M
f : M →ₗ[R] M
h : p ≤ comap f p
h' : optParam (p ≤ comap (f ^ 0) p) ⋯
⊢ p.mapQ p (f ^ 0) h' = p.mapQ p f h ^ 0
|
simp [LinearMap.one_eq_id]
|
no goals
|
f678a150cf10dce9
|
CoalgebraCat.MonoidalCategoryAux.comul_tensorObj_tensorObj_right
|
Mathlib/Algebra/Category/CoalgebraCat/ComonEquivalence.lean
|
theorem comul_tensorObj_tensorObj_right :
Coalgebra.comul (R := R) (A := (CoalgebraCat.of R M ⊗
(CoalgebraCat.of R N ⊗ CoalgebraCat.of R P) : CoalgebraCat R))
= Coalgebra.comul (A := M ⊗[R] N ⊗[R] P)
|
R : Type u
inst✝⁹ : CommRing R
M N P : Type u
inst✝⁸ : AddCommGroup M
inst✝⁷ : AddCommGroup N
inst✝⁶ : AddCommGroup P
inst✝⁵ : Module R M
inst✝⁴ : Module R N
inst✝³ : Module R P
inst✝² : Coalgebra R M
inst✝¹ : Coalgebra R N
inst✝ : Coalgebra R P
⊢ ModuleCat.Hom.hom
((comonEquivalence R).symm.inverse.obj (of R M) ⊗ (comonEquivalence R).symm.inverse.obj (of R N ⊗ of R P)).comul =
CoalgebraStruct.comul
|
dsimp only [Equivalence.symm_inverse, comonEquivalence_functor, toComon_obj,
instCoalgebraStruct_comul]
|
R : Type u
inst✝⁹ : CommRing R
M N P : Type u
inst✝⁸ : AddCommGroup M
inst✝⁷ : AddCommGroup N
inst✝⁶ : AddCommGroup P
inst✝⁵ : Module R M
inst✝⁴ : Module R N
inst✝³ : Module R P
inst✝² : Coalgebra R M
inst✝¹ : Coalgebra R N
inst✝ : Coalgebra R P
⊢ ModuleCat.Hom.hom ((of R M).toComonObj ⊗ (of R N ⊗ of R P).toComonObj).comul =
↑(tensorTensorTensorComm R M M (N ⊗[R] P) (N ⊗[R] P)) ∘ₗ
map CoalgebraStruct.comul (↑(tensorTensorTensorComm R N N P P) ∘ₗ map CoalgebraStruct.comul CoalgebraStruct.comul)
|
f4373b2177e1dea4
|
MeasureTheory.exists_upperSemicontinuous_le_integral_le
|
Mathlib/MeasureTheory/Integral/VitaliCaratheodory.lean
|
theorem exists_upperSemicontinuous_le_integral_le (f : α → ℝ≥0)
(fint : Integrable (fun x => (f x : ℝ)) μ) {ε : ℝ} (εpos : 0 < ε) :
∃ g : α → ℝ≥0,
(∀ x, g x ≤ f x) ∧
UpperSemicontinuous g ∧
Integrable (fun x => (g x : ℝ)) μ ∧ (∫ x, (f x : ℝ) ∂μ) - ε ≤ ∫ x, ↑(g x) ∂μ
|
case intro.intro.intro.intro.refine_2.hfm
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : μ.WeaklyRegular
f : α → ℝ≥0
fint : Integrable (fun x => ↑(f x)) μ
ε : ℝ≥0
εpos : 0 < ↑ε
If : ∫⁻ (x : α), ↑(f x) ∂μ < ⊤
g : α → ℝ≥0
gf : ∀ (x : α), g x ≤ f x
gcont : UpperSemicontinuous g
gint : ∫⁻ (x : α), ↑(f x) ∂μ ≤ ∫⁻ (x : α), ↑(g x) ∂μ + ↑ε
Ig : ∫⁻ (x : α), ↑(g x) ∂μ < ⊤
⊢ AEStronglyMeasurable (fun x => ↑(f x)) μ
|
exact fint.aestronglyMeasurable
|
no goals
|
e7f3a2333862318c
|
ProbabilityTheory.Kernel.measure_eq_zero_or_one_or_top_of_indepSet_self
|
Mathlib/Probability/Independence/ZeroOne.lean
|
theorem Kernel.measure_eq_zero_or_one_or_top_of_indepSet_self {t : Set Ω}
(h_indep : Kernel.IndepSet t t κ μα) :
∀ᵐ a ∂μα, κ a t = 0 ∨ κ a t = 1 ∨ κ a t = ∞
|
case neg
α : Type u_1
Ω : Type u_2
_mα : MeasurableSpace α
m0 : MeasurableSpace Ω
κ : Kernel α Ω
μα : Measure α
t : Set Ω
h_indep : ∀ᵐ (a : α) ∂μα, (κ a) (t ∩ t) = (κ a) t * (κ a) t
a : α
ha : 1 = (κ a) t
h0 : ¬(κ a) t = 0
h_top : ¬(κ a) t = ⊤
⊢ (κ a) t = 0 ∨ (κ a) t = 1 ∨ (κ a) t = ⊤
|
exact Or.inr (Or.inl ha.symm)
|
no goals
|
8b17ec0015226de1
|
QPF.Cofix.bisim_aux
|
Mathlib/Data/QPF/Univariate/Basic.lean
|
theorem Cofix.bisim_aux (r : Cofix F → Cofix F → Prop) (h' : ∀ x, r x x)
(h : ∀ x y, r x y → Quot.mk r <$> Cofix.dest x = Quot.mk r <$> Cofix.dest y) :
∀ x y, r x y → x = y
|
F : Type u → Type u
q : QPF F
r : Cofix F → Cofix F → Prop
h' : ∀ (x : Cofix F), r x x
h : ∀ (x y : Cofix F), r x y → Quot.mk r <$> x.dest = Quot.mk r <$> y.dest
x✝ : Cofix F
x : (P F).M
y✝ : Cofix F
y : (P F).M
rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)
r' : (P F).M → (P F).M → Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)
a b : (P F).M
r'ab : r' a b
⊢ abs ((P F).map (Quot.mk r') a.dest) = abs ((P F).map (Quot.mk r') b.dest)
|
have h₀ :
Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) =
Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b) :=
h _ _ r'ab
|
F : Type u → Type u
q : QPF F
r : Cofix F → Cofix F → Prop
h' : ∀ (x : Cofix F), r x x
h : ∀ (x y : Cofix F), r x y → Quot.mk r <$> x.dest = Quot.mk r <$> y.dest
x✝ : Cofix F
x : (P F).M
y✝ : Cofix F
y : (P F).M
rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)
r' : (P F).M → (P F).M → Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)
a b : (P F).M
r'ab : r' a b
h₀ : Quot.mk r <$> Quot.mk Mcongr <$> abs a.dest = Quot.mk r <$> Quot.mk Mcongr <$> abs b.dest
⊢ abs ((P F).map (Quot.mk r') a.dest) = abs ((P F).map (Quot.mk r') b.dest)
|
ee671efbc83b977d
|
Finset.weightedVSubOfPoint_eq_of_weights_eq
|
Mathlib/LinearAlgebra/AffineSpace/Combination.lean
|
theorem weightedVSubOfPoint_eq_of_weights_eq (p : ι → P) (j : ι) (w₁ w₂ : ι → k)
(hw : ∀ i, i ≠ j → w₁ i = w₂ i) :
s.weightedVSubOfPoint p (p j) w₁ = s.weightedVSubOfPoint p (p j) w₂
|
case e_f.h
k : Type u_1
V : Type u_2
P : Type u_3
inst✝² : Ring k
inst✝¹ : AddCommGroup V
inst✝ : Module k V
S : AffineSpace V P
ι : Type u_4
s : Finset ι
p : ι → P
j : ι
w₁ w₂ : ι → k
hw : ∀ (i : ι), i ≠ j → w₁ i = w₂ i
i : ι
⊢ w₁ i • (p i -ᵥ p j) = w₂ i • (p i -ᵥ p j)
|
rcases eq_or_ne i j with h | h
|
case e_f.h.inl
k : Type u_1
V : Type u_2
P : Type u_3
inst✝² : Ring k
inst✝¹ : AddCommGroup V
inst✝ : Module k V
S : AffineSpace V P
ι : Type u_4
s : Finset ι
p : ι → P
j : ι
w₁ w₂ : ι → k
hw : ∀ (i : ι), i ≠ j → w₁ i = w₂ i
i : ι
h : i = j
⊢ w₁ i • (p i -ᵥ p j) = w₂ i • (p i -ᵥ p j)
case e_f.h.inr
k : Type u_1
V : Type u_2
P : Type u_3
inst✝² : Ring k
inst✝¹ : AddCommGroup V
inst✝ : Module k V
S : AffineSpace V P
ι : Type u_4
s : Finset ι
p : ι → P
j : ι
w₁ w₂ : ι → k
hw : ∀ (i : ι), i ≠ j → w₁ i = w₂ i
i : ι
h : i ≠ j
⊢ w₁ i • (p i -ᵥ p j) = w₂ i • (p i -ᵥ p j)
|
8a064655adca0142
|
Real.hasDerivAt_fourierChar
|
Mathlib/Analysis/Fourier/FourierTransformDeriv.lean
|
lemma hasDerivAt_fourierChar (x : ℝ) : HasDerivAt (𝐞 · : ℝ → ℂ) (2 * π * I * 𝐞 x) x
|
x y : ℝ
⊢ cexp (↑(2 * π * y) * I) = cexp (2 * ↑π * I * ↑1 * ↑y / ↑1)
|
push_cast
|
x y : ℝ
⊢ cexp (2 * ↑π * ↑y * I) = cexp (2 * ↑π * I * 1 * ↑y / 1)
|
09c33dc599e467bd
|
Turing.TM1to0.tr_supports
|
Mathlib/Computability/PostTuringMachine.lean
|
theorem tr_supports {S : Finset Λ} (ss : TM1.Supports M S) :
TM0.Supports (tr M) ↑(trStmts M S)
|
case right.mk.some.mk.some
Γ : Type u_1
Λ : Type u_2
inst✝² : Inhabited Λ
σ : Type u_3
inst✝¹ : Inhabited σ
M : Λ → TM1.Stmt Γ Λ σ
inst✝ : Fintype σ
S : Finset Λ
ss : TM1.Supports M S
a : Γ
s : TM0.Stmt Γ
v : σ
q : TM1.Stmt Γ Λ σ
v' : σ
h₂ : some q ∈ TM1.stmts M S
val✝ : TM1.Stmt Γ Λ σ
h₁ : ((some val✝, v'), s) ∈ tr M (some q, v) a
⊢ some val✝ ∈ TM1.stmts M S
|
simp only [tr, Option.mem_def] at h₁
|
case right.mk.some.mk.some
Γ : Type u_1
Λ : Type u_2
inst✝² : Inhabited Λ
σ : Type u_3
inst✝¹ : Inhabited σ
M : Λ → TM1.Stmt Γ Λ σ
inst✝ : Fintype σ
S : Finset Λ
ss : TM1.Supports M S
a : Γ
s : TM0.Stmt Γ
v : σ
q : TM1.Stmt Γ Λ σ
v' : σ
h₂ : some q ∈ TM1.stmts M S
val✝ : TM1.Stmt Γ Λ σ
h₁ : some (trAux M a q v) = some ((some val✝, v'), s)
⊢ some val✝ ∈ TM1.stmts M S
|
c5e09604fd205550
|
Equiv.Perm.cycle_induction_on
|
Mathlib/GroupTheory/Perm/Cycle/Factors.lean
|
theorem cycle_induction_on [Finite β] (P : Perm β → Prop) (σ : Perm β) (base_one : P 1)
(base_cycles : ∀ σ : Perm β, σ.IsCycle → P σ)
(induction_disjoint : ∀ σ τ : Perm β,
Disjoint σ τ → IsCycle σ → P σ → P τ → P (σ * τ)) : P σ
|
β : Type u_3
inst✝ : Finite β
P : Perm β → Prop
σ : Perm β
base_one : P 1
base_cycles : ∀ (σ : Perm β), σ.IsCycle → P σ
induction_disjoint : ∀ (σ τ : Perm β), σ.Disjoint τ → σ.IsCycle → P σ → P τ → P (σ * τ)
⊢ P σ
|
cases nonempty_fintype β
|
case intro
β : Type u_3
inst✝ : Finite β
P : Perm β → Prop
σ : Perm β
base_one : P 1
base_cycles : ∀ (σ : Perm β), σ.IsCycle → P σ
induction_disjoint : ∀ (σ τ : Perm β), σ.Disjoint τ → σ.IsCycle → P σ → P τ → P (σ * τ)
val✝ : Fintype β
⊢ P σ
|
a7d4f63bf9ff66c2
|
Matroid.mapEmbedding_isBasis_iff
|
Mathlib/Data/Matroid/Map.lean
|
@[simp] lemma mapEmbedding_isBasis_iff {f : α ↪ β} {I X : Set β} :
(M.mapEmbedding f).IsBasis I X ↔ M.IsBasis (f ⁻¹' I) (f ⁻¹' X) ∧ I ⊆ X ∧ X ⊆ range f
|
case refine_2.intro
α : Type u_1
β : Type u_2
M : Matroid α
f : α ↪ β
I : Set β
X : Set α
x✝ : M.IsBasis (⇑f ⁻¹' I) (⇑f ⁻¹' (⇑f '' X)) ∧ I ⊆ ⇑f '' X ∧ ⇑f '' X ⊆ range ⇑f
hb : M.IsBasis (⇑f ⁻¹' I) (⇑f ⁻¹' (⇑f '' X))
hIX : I ⊆ ⇑f '' X
hX : ⇑f '' X ⊆ range ⇑f
⊢ ∃ I₀ X₀, M.IsBasis I₀ X₀ ∧ I = ⇑f '' I₀ ∧ ⇑f '' X = ⇑f '' X₀
|
obtain ⟨I, -, rfl⟩ := subset_image_iff.1 hIX
|
case refine_2.intro.intro.intro
α : Type u_1
β : Type u_2
M : Matroid α
f : α ↪ β
X : Set α
hX : ⇑f '' X ⊆ range ⇑f
I : Set α
x✝ : M.IsBasis (⇑f ⁻¹' (⇑f '' I)) (⇑f ⁻¹' (⇑f '' X)) ∧ ⇑f '' I ⊆ ⇑f '' X ∧ ⇑f '' X ⊆ range ⇑f
hb : M.IsBasis (⇑f ⁻¹' (⇑f '' I)) (⇑f ⁻¹' (⇑f '' X))
hIX : ⇑f '' I ⊆ ⇑f '' X
⊢ ∃ I₀ X₀, M.IsBasis I₀ X₀ ∧ ⇑f '' I = ⇑f '' I₀ ∧ ⇑f '' X = ⇑f '' X₀
|
aceb5823b23814d0
|
GromovHausdorff.totallyBounded
|
Mathlib/Topology/MetricSpace/GromovHausdorff.lean
|
theorem totallyBounded {t : Set GHSpace} {C : ℝ} {u : ℕ → ℝ} {K : ℕ → ℕ}
(ulim : Tendsto u atTop (𝓝 0)) (hdiam : ∀ p ∈ t, diam (univ : Set (GHSpace.Rep p)) ≤ C)
(hcov : ∀ p ∈ t, ∀ n : ℕ, ∃ s : Set (GHSpace.Rep p),
(#s) ≤ K n ∧ univ ⊆ ⋃ x ∈ s, ball x (u n)) :
TotallyBounded t
|
t : Set GHSpace
C : ℝ
u : ℕ → ℝ
K : ℕ → ℕ
ulim : Tendsto u atTop (𝓝 0)
hdiam : ∀ p ∈ t, diam univ ≤ C
hcov : ∀ p ∈ t, ∀ (n : ℕ), ∃ s, #↑s ≤ ↑(K n) ∧ univ ⊆ ⋃ x ∈ s, ball x (u n)
δ : ℝ
δpos : δ > 0
ε : ℝ := 1 / 5 * δ
εpos : 0 < ε
n : ℕ
hn : ∀ n_1 ≥ n, dist (u n_1) 0 < ε
u_le_ε : u n ≤ ε
s : (p : GHSpace) → Set p.Rep
N : GHSpace → ℕ
hN : ∀ (p : GHSpace), N p ≤ K n
E : (p : GHSpace) → ↑(s p) ≃ Fin (N p)
hs : ∀ p ∈ t, univ ⊆ ⋃ x ∈ s p, ball x (u n)
M : ℕ := ⌊ε⁻¹ * (C ⊔ 0)⌋₊
F : GHSpace → (k : Fin (K n).succ) × (Fin ↑k → Fin ↑k → Fin M.succ) :=
fun p => ⟨⟨N p, ⋯⟩, fun a b => ⟨M ⊓ ⌊ε⁻¹ * dist ((E p).symm a) ((E p).symm b)⌋₊, ⋯⟩⟩
p : GHSpace
pt : p ∈ t
q : GHSpace
qt : q ∈ t
hpq : (fun p => F ↑p) ⟨p, pt⟩ = (fun p => F ↑p) ⟨q, qt⟩
Npq : N p = N q
Ψ : ↑(s p) → ↑(s q) := fun x => (E q).symm (Fin.cast Npq ((E p) x))
Φ : ↑(s p) → q.Rep := fun x => ↑(Ψ x)
x y : ↑(s p)
this : dist (Φ x) (Φ y) = dist (Ψ x) (Ψ y)
i : ℕ := ↑((E p) x)
hip : i < N p
hiq : i < N q
i' : i = ↑((E q) (Ψ x))
j : ℕ := ↑((E p) y)
hjp : j < N p
hjq : j < N q
j' : j = ↑((E q) (Ψ y))
Ap : ↑((F p).snd ⟨i, hip⟩ ⟨j, hjp⟩) = ⌊ε⁻¹ * dist x y⌋₊
⊢ ↑((F q).snd ((E q) (Ψ x)) ((E q) (Ψ y))) = M ⊓ ⌊ε⁻¹ * dist (Ψ x) (Ψ y)⌋₊
|
simp only [F, (E q).symm_apply_apply]
|
no goals
|
c10025d34debaff9
|
DividedPowers.coincide_on_smul
|
Mathlib/RingTheory/DividedPowers/Basic.lean
|
theorem coincide_on_smul {J : Ideal A} (hJ : DividedPowers J) {n : ℕ} (ha : a ∈ I • J) :
hI.dpow n a = hJ.dpow n a
|
case add
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
a : A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
x : A
hx : x ∈ I • J
y : A
hy : y ∈ I • J
hx' : ∀ {n : ℕ}, hI.dpow n x = hJ.dpow n x
hy' : ∀ {n : ℕ}, hI.dpow n y = hJ.dpow n y
n : ℕ
k : ℕ × ℕ
a✝ : k ∈ antidiagonal n
⊢ hI.dpow k.1 x * hI.dpow k.2 y = hJ.dpow k.1 x * hJ.dpow k.2 y
|
rw [hx', hy']
|
no goals
|
404c3d86efc5a9c7
|
Real.geom_mean_weighted_of_constant
|
Mathlib/Analysis/MeanInequalities.lean
|
theorem geom_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i ∈ s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i ∈ s, z i ^ w i = x :=
calc
∏ i ∈ s, z i ^ w i = ∏ i ∈ s, x ^ w i
|
case intro.intro
ι : Type u
s : Finset ι
w z : ι → ℝ
x : ℝ
hw : ∀ i ∈ s, 0 ≤ w i
hw' : ∑ i ∈ s, w i = 1
hz : ∀ i ∈ s, 0 ≤ z i
hx : ∀ i ∈ s, w i ≠ 0 → z i = x
this : ∑ i ∈ s, w i ≠ 0
i : ι
his : i ∈ s
hi : w i ≠ 0
⊢ 0 ≤ z i
|
exact hz i his
|
no goals
|
332f81523d8d5681
|
AlgebraicClosure.spanCoeffs_ne_top
|
Mathlib/FieldTheory/IsAlgClosed/AlgebraicClosure.lean
|
theorem spanCoeffs_ne_top : spanCoeffs k ≠ ⊤
|
case intro.intro
k : Type u
inst✝ : Field k
v : Monics k × ℕ →₀ MvPolynomial (Vars k) k
left✝ : v ∈ Finsupp.supported (MvPolynomial (Vars k) k) (MvPolynomial (Vars k) k) Set.univ
hv : ∑ x ∈ v.support, (toSplittingField (Finset.image Prod.fst v.support)) (v x • (subProdXSubC x.1).coeff x.2) = 1
j : Monics k × ℕ
hj : j ∈ v.support
⊢ (toSplittingField (Finset.image Prod.fst v.support)) (v j • (subProdXSubC j.1).coeff j.2) = 0
|
rw [smul_eq_mul, map_mul, toSplittingField_coeff (Finset.mem_image_of_mem _ hj), mul_zero]
|
no goals
|
7ddf9f30cf89e546
|
Polynomial.monic_restriction
|
Mathlib/RingTheory/Polynomial/Basic.lean
|
theorem monic_restriction {p : R[X]} : Monic (restriction p) ↔ Monic p
|
R : Type u
inst✝ : Ring R
p : R[X]
⊢ p.restriction.coeff p.natDegree = 1 ↔ ↑(p.restriction.coeff p.natDegree) = 1
|
exact ⟨fun H => by rw [H, OneMemClass.coe_one], fun H => Subtype.coe_injective H⟩
|
no goals
|
7b55e8e0a768e58b
|
MeasureTheory.volume_sum_rpow_lt_one
|
Mathlib/MeasureTheory/Measure/Lebesgue/VolumeOfBalls.lean
|
theorem MeasureTheory.volume_sum_rpow_lt_one (hp : 1 ≤ p) :
volume {x : ι → ℝ | ∑ i, |x i| ^ p < 1} =
.ofReal ((2 * Gamma (1 / p + 1)) ^ card ι / Gamma (card ι / p + 1))
|
ι : Type u_1
inst✝ : Fintype ι
p : ℝ
hp : 1 ≤ p
h₁ : 0 < p
h₂ : ∀ (x : ι → ℝ), 0 ≤ ∑ i : ι, |x i| ^ p
eq_norm : ∀ (x : ι → ℝ), ‖x‖ = (∑ x_1 : ι, |x x_1| ^ p) ^ (1 / p)
this : Fact (1 ≤ ENNReal.ofReal p)
nm_zero : ‖0‖ = 0
eq_zero : ∀ (x : ι → ℝ), ‖x‖ = 0 ↔ x = 0
nm_neg : ∀ (x : ι → ℝ), ‖-x‖ = ‖x‖
nm_add : ∀ (x y : ι → ℝ), ‖x + y‖ ≤ ‖x‖ + ‖y‖
⊢ volume {x | ∑ i : ι, |x i| ^ p < 1} = ENNReal.ofReal ((2 * Gamma (1 / p + 1)) ^ card ι / Gamma (↑(card ι) / p + 1))
|
simp_rw [eq_norm] at eq_zero nm_zero nm_neg nm_add
|
ι : Type u_1
inst✝ : Fintype ι
p : ℝ
hp : 1 ≤ p
h₁ : 0 < p
h₂ : ∀ (x : ι → ℝ), 0 ≤ ∑ i : ι, |x i| ^ p
eq_norm : ∀ (x : ι → ℝ), ‖x‖ = (∑ x_1 : ι, |x x_1| ^ p) ^ (1 / p)
this : Fact (1 ≤ ENNReal.ofReal p)
eq_zero : ∀ (x : ι → ℝ), (∑ x_1 : ι, |x x_1| ^ p) ^ (1 / p) = 0 ↔ x = 0
nm_zero : (∑ x : ι, |0 x| ^ p) ^ (1 / p) = 0
nm_neg : ∀ (x : ι → ℝ), (∑ x_1 : ι, |(-x) x_1| ^ p) ^ (1 / p) = (∑ x_1 : ι, |x x_1| ^ p) ^ (1 / p)
nm_add :
∀ (x y : ι → ℝ),
(∑ x_1 : ι, |(x + y) x_1| ^ p) ^ (1 / p) ≤ (∑ x_1 : ι, |x x_1| ^ p) ^ (1 / p) + (∑ x : ι, |y x| ^ p) ^ (1 / p)
⊢ volume {x | ∑ i : ι, |x i| ^ p < 1} = ENNReal.ofReal ((2 * Gamma (1 / p + 1)) ^ card ι / Gamma (↑(card ι) / p + 1))
|
412f34b3848867a2
|
IsClosed.ae_eq_univ_iff_eq
|
Mathlib/MeasureTheory/Measure/OpenPos.lean
|
theorem _root_.IsClosed.ae_eq_univ_iff_eq (hF : IsClosed F) :
F =ᵐ[μ] univ ↔ F = univ
|
X : Type u_1
inst✝¹ : TopologicalSpace X
m : MeasurableSpace X
μ : Measure X
inst✝ : μ.IsOpenPosMeasure
F : Set X
hF : IsClosed F
h : F =ᶠ[ae μ] univ
⊢ F = univ
|
rwa [ae_eq_univ, hF.isOpen_compl.measure_eq_zero_iff μ, compl_empty_iff] at h
|
no goals
|
bd0f56f21b466be3
|
Complex.integral_cpow_mul_exp_neg_mul_Ioi
|
Mathlib/Analysis/SpecialFunctions/Gamma/Basic.lean
|
/-- Expresses the integral over `Ioi 0` of `t ^ (a - 1) * exp (-(r * t))` in terms of the Gamma
function, for complex `a`. -/
lemma integral_cpow_mul_exp_neg_mul_Ioi {a : ℂ} {r : ℝ} (ha : 0 < a.re) (hr : 0 < r) :
∫ (t : ℝ) in Ioi 0, t ^ (a - 1) * exp (-(r * t)) = (1 / r) ^ a * Gamma a
|
a : ℂ
r : ℝ
ha : 0 < a.re
hr : 0 < r
⊢ (1 / ↑r) ^ a = 1 / ↑r * (1 / ↑r) ^ (a - 1)
|
nth_rewrite 2 [← cpow_one (1 / r : ℂ)]
|
a : ℂ
r : ℝ
ha : 0 < a.re
hr : 0 < r
⊢ (1 / ↑r) ^ a = (1 / ↑r) ^ 1 * (1 / ↑r) ^ (a - 1)
|
5d070e1c4207a464
|
Lean.Order.List.monotone_forIn'_loop
|
Mathlib/.lake/packages/lean4/src/lean/Init/Internal/Order/Lemmas.lean
|
theorem monotone_forIn'_loop {α : Type uu}
(as : List α) (f : γ → (a : α) → a ∈ as → β → m (ForInStep β)) (as' : List α) (b : β)
(p : Exists (fun bs => bs ++ as' = as)) (hmono : monotone f) :
monotone (fun x => List.forIn'.loop as (f x) as' b p)
|
case nil
m : Type u → Type v
inst✝³ : Monad m
inst✝² : (α : Type u) → PartialOrder (m α)
inst✝¹ : MonoBind m
β : Type u
γ : Type w
inst✝ : PartialOrder γ
α : Type uu
as : List α
f : γ → (a : α) → a ∈ as → β → m (ForInStep β)
hmono : monotone f
b : β
p : ∃ bs, bs ++ [] = as
⊢ monotone fun x => List.forIn'.loop as (f x) [] b p
|
apply monotone_const
|
no goals
|
54f63850339d8283
|
VectorField.leibniz_identity_lieBracketWithin_of_isSymmSndFDerivWithinAt
|
Mathlib/Analysis/Calculus/VectorField.lean
|
/-- The Lie bracket of vector fields in vector spaces satisfies the Leibniz identity
`[U, [V, W]] = [[U, V], W] + [V, [U, W]]`. -/
lemma leibniz_identity_lieBracketWithin_of_isSymmSndFDerivWithinAt
{U V W : E → E} {s : Set E} {x : E} (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s)
(hU : ContDiffWithinAt 𝕜 2 U s x) (hV : ContDiffWithinAt 𝕜 2 V s x)
(hW : ContDiffWithinAt 𝕜 2 W s x)
(h'U : IsSymmSndFDerivWithinAt 𝕜 U s x) (h'V : IsSymmSndFDerivWithinAt 𝕜 V s x)
(h'W : IsSymmSndFDerivWithinAt 𝕜 W s x) :
lieBracketWithin 𝕜 U (lieBracketWithin 𝕜 V W s) s x =
lieBracketWithin 𝕜 (lieBracketWithin 𝕜 U V s) W s x
+ lieBracketWithin 𝕜 V (lieBracketWithin 𝕜 U W s) s x
|
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
U V W : E → E
s : Set E
x : E
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
hU : ContDiffWithinAt 𝕜 2 U s x
hV : ContDiffWithinAt 𝕜 2 V s x
hW : ContDiffWithinAt 𝕜 2 W s x
h'U : IsSymmSndFDerivWithinAt 𝕜 U s x
h'V : IsSymmSndFDerivWithinAt 𝕜 V s x
h'W : IsSymmSndFDerivWithinAt 𝕜 W s x
⊢ lieBracketWithin 𝕜 U (lieBracketWithin 𝕜 V W s) s x =
lieBracketWithin 𝕜 (lieBracketWithin 𝕜 U V s) W s x + lieBracketWithin 𝕜 V (lieBracketWithin 𝕜 U W s) s x
|
simp only [lieBracketWithin_eq, map_sub]
|
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
U V W : E → E
s : Set E
x : E
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
hU : ContDiffWithinAt 𝕜 2 U s x
hV : ContDiffWithinAt 𝕜 2 V s x
hW : ContDiffWithinAt 𝕜 2 W s x
h'U : IsSymmSndFDerivWithinAt 𝕜 U s x
h'V : IsSymmSndFDerivWithinAt 𝕜 V s x
h'W : IsSymmSndFDerivWithinAt 𝕜 W s x
⊢ (fderivWithin 𝕜 (fun x => (fderivWithin 𝕜 W s x) (V x) - (fderivWithin 𝕜 V s x) (W x)) s x) (U x) -
((fderivWithin 𝕜 U s x) ((fderivWithin 𝕜 W s x) (V x)) - (fderivWithin 𝕜 U s x) ((fderivWithin 𝕜 V s x) (W x))) =
(fderivWithin 𝕜 W s x) ((fderivWithin 𝕜 V s x) (U x)) - (fderivWithin 𝕜 W s x) ((fderivWithin 𝕜 U s x) (V x)) -
(fderivWithin 𝕜 (fun x => (fderivWithin 𝕜 V s x) (U x) - (fderivWithin 𝕜 U s x) (V x)) s x) (W x) +
((fderivWithin 𝕜 (fun x => (fderivWithin 𝕜 W s x) (U x) - (fderivWithin 𝕜 U s x) (W x)) s x) (V x) -
((fderivWithin 𝕜 V s x) ((fderivWithin 𝕜 W s x) (U x)) - (fderivWithin 𝕜 V s x) ((fderivWithin 𝕜 U s x) (W x))))
|
a8ebc5b229cab643
|
deriv_const_smul'
|
Mathlib/Analysis/Calculus/Deriv/Mul.lean
|
/-- A variant of `deriv_const_smul` without differentiability assumption when the scalar
multiplication is by field elements. -/
lemma deriv_const_smul' {f : 𝕜 → F} {x : 𝕜} {R : Type*} [Field R] [Module R F] [SMulCommClass 𝕜 R F]
[ContinuousConstSMul R F] (c : R) :
deriv (fun y ↦ c • f y) x = c • deriv f x
|
𝕜 : Type u
inst✝⁶ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
f : 𝕜 → F
x : 𝕜
R : Type u_3
inst✝³ : Field R
inst✝² : Module R F
inst✝¹ : SMulCommClass 𝕜 R F
inst✝ : ContinuousConstSMul R F
c : R
hc : c ≠ 0
hf : DifferentiableAt 𝕜 (fun y => c • f y) x
⊢ DifferentiableAt 𝕜 (fun y => c⁻¹ • c • f y) x
|
exact DifferentiableAt.const_smul hf c⁻¹
|
no goals
|
6906aeb992f7f336
|
LieAlgebra.IsSemisimple.finitelyAtomistic
|
Mathlib/Algebra/Lie/Semisimple/Basic.lean
|
/--
In a semisimple Lie algebra,
Lie ideals that are contained in the supremum of a finite collection of atoms
are themselves the supremum of a finite subcollection of those atoms.
By a compactness argument, this statement can be extended to arbitrary sets of atoms.
See `atomistic`.
The proof is by induction on the finite set of atoms.
-/
private
lemma finitelyAtomistic : ∀ s : Finset (LieIdeal R L), ↑s ⊆ {I : LieIdeal R L | IsAtom I} →
∀ I : LieIdeal R L, I ≤ s.sup id → ∃ t ⊆ s, I = t.sup id
|
R : Type u_1
L : Type u_2
inst✝³ : CommRing R
inst✝² : LieRing L
inst✝¹ : LieAlgebra R L
inst✝ : IsSemisimple R L
s : Finset (LieIdeal R L)
hs : ↑s ⊆ {I | IsAtom I}
I : LieIdeal R L
hI✝ : I ≤ s.sup id
S : Set (LieIdeal R L) := {I | IsAtom I}
hI : I < s.sup id
J : LieIdeal R L
hJs : J ∈ s
hJI : ¬J ≤ I
s' : Finset (LieIdeal R L) := s.erase J
hs' : s' ⊂ s
hs'S : ↑s' ⊆ S
K : LieIdeal R L := s'.sup id
y : L
hy : y ∈ id J
z : L
hz : z ∈ K
hx : y + z ∈ I
this : ⟨y, hy⟩ ∈ center R ↥J
⊢ y + z ∈ K
|
have _inst := isSimple_of_isAtom J (hs hJs)
|
R : Type u_1
L : Type u_2
inst✝³ : CommRing R
inst✝² : LieRing L
inst✝¹ : LieAlgebra R L
inst✝ : IsSemisimple R L
s : Finset (LieIdeal R L)
hs : ↑s ⊆ {I | IsAtom I}
I : LieIdeal R L
hI✝ : I ≤ s.sup id
S : Set (LieIdeal R L) := {I | IsAtom I}
hI : I < s.sup id
J : LieIdeal R L
hJs : J ∈ s
hJI : ¬J ≤ I
s' : Finset (LieIdeal R L) := s.erase J
hs' : s' ⊂ s
hs'S : ↑s' ⊆ S
K : LieIdeal R L := s'.sup id
y : L
hy : y ∈ id J
z : L
hz : z ∈ K
hx : y + z ∈ I
this : ⟨y, hy⟩ ∈ center R ↥J
_inst : IsSimple R ↥J
⊢ y + z ∈ K
|
9feaa310a91f7482
|
LaurentSeries.valuation_le_iff_coeff_lt_eq_zero
|
Mathlib/RingTheory/LaurentSeries.lean
|
theorem valuation_le_iff_coeff_lt_eq_zero {D : ℤ} {f : K⸨X⸩} :
Valued.v f ≤ ↑(Multiplicative.ofAdd (-D : ℤ)) ↔ ∀ n : ℤ, n < D → f.coeff n = 0
|
case neg
K : Type u_2
inst✝ : Field K
D : ℤ
f : K⸨X⸩
h_val_f : ∀ n < D, f.coeff n = 0
F : K⟦X⟧ := f.powerSeriesPart
ord_nonpos : HahnSeries.order f ≤ 0
s : ℕ
hs : HahnSeries.order f = -↑s
hDs : ¬D + ↑s ≤ 0
⊢ Valued.v ((ofPowerSeries ℤ K) f.powerSeriesPart) ≤ ↑(Multiplicative.ofAdd (-(D + ↑s)))
|
obtain ⟨d, hd⟩ := Int.eq_ofNat_of_zero_le (le_of_lt <| not_le.mp hDs)
|
case neg.intro
K : Type u_2
inst✝ : Field K
D : ℤ
f : K⸨X⸩
h_val_f : ∀ n < D, f.coeff n = 0
F : K⟦X⟧ := f.powerSeriesPart
ord_nonpos : HahnSeries.order f ≤ 0
s : ℕ
hs : HahnSeries.order f = -↑s
hDs : ¬D + ↑s ≤ 0
d : ℕ
hd : D + ↑s = ↑d
⊢ Valued.v ((ofPowerSeries ℤ K) f.powerSeriesPart) ≤ ↑(Multiplicative.ofAdd (-(D + ↑s)))
|
34ebed25af8874e3
|
edist_lt_coe
|
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
theorem edist_lt_coe {x y : α} {c : ℝ≥0} : edist x y < c ↔ nndist x y < c
|
α : Type u
inst✝ : PseudoMetricSpace α
x y : α
c : ℝ≥0
⊢ edist x y < ↑c ↔ nndist x y < c
|
rw [edist_nndist, ENNReal.coe_lt_coe]
|
no goals
|
b3c5e1c1c01dd681
|
εNFA.isPath_append
|
Mathlib/Computability/EpsilonNFA.lean
|
theorem isPath_append {x y : List (Option α)} :
M.IsPath s u (x ++ y) ↔ ∃ t, M.IsPath s t x ∧ M.IsPath t u y where
mp
|
α : Type u
σ : Type v
M : εNFA α σ
s u : σ
x y : List (Option α)
⊢ M.IsPath s u (x ++ y) → ∃ t, M.IsPath s t x ∧ M.IsPath t u y
|
induction' x with x a ih generalizing s
|
case nil
α : Type u
σ : Type v
M : εNFA α σ
u : σ
y : List (Option α)
s : σ
⊢ M.IsPath s u ([] ++ y) → ∃ t, M.IsPath s t [] ∧ M.IsPath t u y
case cons
α : Type u
σ : Type v
M : εNFA α σ
u : σ
y : List (Option α)
x : Option α
a : List (Option α)
ih : ∀ {s : σ}, M.IsPath s u (a ++ y) → ∃ t, M.IsPath s t a ∧ M.IsPath t u y
s : σ
⊢ M.IsPath s u (x :: a ++ y) → ∃ t, M.IsPath s t (x :: a) ∧ M.IsPath t u y
|
e5aadd4797a0d54e
|
HomologicalComplex.mapBifunctorAssociatorX_hom_D₂
|
Mathlib/Algebra/Homology/BifunctorAssociator.lean
|
@[reassoc]
lemma mapBifunctorAssociatorX_hom_D₂ (j j' : ι₄) :
(mapBifunctorAssociatorX associator K₁ K₂ K₃ c₁₂ c₂₃ c₄ j).hom ≫
mapBifunctor₂₃.D₂ F G₂₃ K₁ K₂ K₃ c₁₂ c₂₃ c₄ j j' =
mapBifunctor₁₂.D₂ F₁₂ G K₁ K₂ K₃ c₁₂ c₄ j j' ≫
(mapBifunctorAssociatorX associator K₁ K₂ K₃ c₁₂ c₂₃ c₄ j').hom
|
case hfg
C₁ : Type u_1
C₂ : Type u_2
C₁₂ : Type u_3
C₂₃ : Type u_4
C₃ : Type u_5
C₄ : Type u_6
inst✝³³ : Category.{u_16, u_1} C₁
inst✝³² : Category.{u_17, u_2} C₂
inst✝³¹ : Category.{u_15, u_5} C₃
inst✝³⁰ : Category.{u_13, u_6} C₄
inst✝²⁹ : Category.{u_14, u_3} C₁₂
inst✝²⁸ : Category.{u_18, u_4} C₂₃
inst✝²⁷ : HasZeroMorphisms C₁
inst✝²⁶ : HasZeroMorphisms C₂
inst✝²⁵ : HasZeroMorphisms C₃
inst✝²⁴ : Preadditive C₁₂
inst✝²³ : Preadditive C₂₃
inst✝²² : Preadditive C₄
F₁₂ : C₁ ⥤ C₂ ⥤ C₁₂
G : C₁₂ ⥤ C₃ ⥤ C₄
F : C₁ ⥤ C₂₃ ⥤ C₄
G₂₃ : C₂ ⥤ C₃ ⥤ C₂₃
inst✝²¹ : F₁₂.PreservesZeroMorphisms
inst✝²⁰ : ∀ (X₁ : C₁), (F₁₂.obj X₁).PreservesZeroMorphisms
inst✝¹⁹ : G.Additive
inst✝¹⁸ : ∀ (X₁₂ : C₁₂), (G.obj X₁₂).PreservesZeroMorphisms
inst✝¹⁷ : G₂₃.PreservesZeroMorphisms
inst✝¹⁶ : ∀ (X₂ : C₂), (G₂₃.obj X₂).PreservesZeroMorphisms
inst✝¹⁵ : F.PreservesZeroMorphisms
inst✝¹⁴ : ∀ (X₁ : C₁), (F.obj X₁).Additive
associator : bifunctorComp₁₂ F₁₂ G ≅ bifunctorComp₂₃ F G₂₃
ι₁ : Type u_7
ι₂ : Type u_8
ι₃ : Type u_9
ι₁₂ : Type u_10
ι₂₃ : Type u_11
ι₄ : Type u_12
inst✝¹³ : DecidableEq ι₄
c₁ : ComplexShape ι₁
c₂ : ComplexShape ι₂
c₃ : ComplexShape ι₃
K₁ : HomologicalComplex C₁ c₁
K₂ : HomologicalComplex C₂ c₂
K₃ : HomologicalComplex C₃ c₃
c₁₂ : ComplexShape ι₁₂
c₂₃ : ComplexShape ι₂₃
c₄ : ComplexShape ι₄
inst✝¹² : TotalComplexShape c₁ c₂ c₁₂
inst✝¹¹ : TotalComplexShape c₁₂ c₃ c₄
inst✝¹⁰ : TotalComplexShape c₂ c₃ c₂₃
inst✝⁹ : TotalComplexShape c₁ c₂₃ c₄
inst✝⁸ : K₁.HasMapBifunctor K₂ F₁₂ c₁₂
inst✝⁷ : K₂.HasMapBifunctor K₃ G₂₃ c₂₃
inst✝⁶ : c₁.Associative c₂ c₃ c₁₂ c₂₃ c₄
inst✝⁵ : DecidableEq ι₁₂
inst✝⁴ : DecidableEq ι₂₃
inst✝³ : (K₁.mapBifunctor K₂ F₁₂ c₁₂).HasMapBifunctor K₃ G c₄
inst✝² : K₁.HasMapBifunctor (K₂.mapBifunctor K₃ G₂₃ c₂₃) F c₄
inst✝¹ : HasGoodTrifunctor₁₂Obj F₁₂ G K₁ K₂ K₃ c₁₂ c₄
inst✝ : HasGoodTrifunctor₂₃Obj F G₂₃ K₁ K₂ K₃ c₁₂ c₂₃ c₄
j j' : ι₄
i₁ : ι₁
i₂ : ι₂
i₃ : ι₃
h : c₁.r c₂ c₃ c₁₂ c₄ (i₁, i₂, i₃) = j
⊢ ((associator.hom.app (K₁.X i₁)).app (K₂.X i₂)).app (K₃.X i₃) ≫
mapBifunctor₂₃.d₂ F G₂₃ K₁ K₂ K₃ c₁₂ c₂₃ c₄ i₁ i₂ i₃ j' =
mapBifunctor₁₂.d₂ F₁₂ G K₁ K₂ K₃ c₁₂ c₄ i₁ i₂ i₃ j' ≫
(mapBifunctorAssociatorX associator K₁ K₂ K₃ c₁₂ c₂₃ c₄ j').hom
|
by_cases h₁ : c₂.Rel i₂ (c₂.next i₂)
|
case pos
C₁ : Type u_1
C₂ : Type u_2
C₁₂ : Type u_3
C₂₃ : Type u_4
C₃ : Type u_5
C₄ : Type u_6
inst✝³³ : Category.{u_16, u_1} C₁
inst✝³² : Category.{u_17, u_2} C₂
inst✝³¹ : Category.{u_15, u_5} C₃
inst✝³⁰ : Category.{u_13, u_6} C₄
inst✝²⁹ : Category.{u_14, u_3} C₁₂
inst✝²⁸ : Category.{u_18, u_4} C₂₃
inst✝²⁷ : HasZeroMorphisms C₁
inst✝²⁶ : HasZeroMorphisms C₂
inst✝²⁵ : HasZeroMorphisms C₃
inst✝²⁴ : Preadditive C₁₂
inst✝²³ : Preadditive C₂₃
inst✝²² : Preadditive C₄
F₁₂ : C₁ ⥤ C₂ ⥤ C₁₂
G : C₁₂ ⥤ C₃ ⥤ C₄
F : C₁ ⥤ C₂₃ ⥤ C₄
G₂₃ : C₂ ⥤ C₃ ⥤ C₂₃
inst✝²¹ : F₁₂.PreservesZeroMorphisms
inst✝²⁰ : ∀ (X₁ : C₁), (F₁₂.obj X₁).PreservesZeroMorphisms
inst✝¹⁹ : G.Additive
inst✝¹⁸ : ∀ (X₁₂ : C₁₂), (G.obj X₁₂).PreservesZeroMorphisms
inst✝¹⁷ : G₂₃.PreservesZeroMorphisms
inst✝¹⁶ : ∀ (X₂ : C₂), (G₂₃.obj X₂).PreservesZeroMorphisms
inst✝¹⁵ : F.PreservesZeroMorphisms
inst✝¹⁴ : ∀ (X₁ : C₁), (F.obj X₁).Additive
associator : bifunctorComp₁₂ F₁₂ G ≅ bifunctorComp₂₃ F G₂₃
ι₁ : Type u_7
ι₂ : Type u_8
ι₃ : Type u_9
ι₁₂ : Type u_10
ι₂₃ : Type u_11
ι₄ : Type u_12
inst✝¹³ : DecidableEq ι₄
c₁ : ComplexShape ι₁
c₂ : ComplexShape ι₂
c₃ : ComplexShape ι₃
K₁ : HomologicalComplex C₁ c₁
K₂ : HomologicalComplex C₂ c₂
K₃ : HomologicalComplex C₃ c₃
c₁₂ : ComplexShape ι₁₂
c₂₃ : ComplexShape ι₂₃
c₄ : ComplexShape ι₄
inst✝¹² : TotalComplexShape c₁ c₂ c₁₂
inst✝¹¹ : TotalComplexShape c₁₂ c₃ c₄
inst✝¹⁰ : TotalComplexShape c₂ c₃ c₂₃
inst✝⁹ : TotalComplexShape c₁ c₂₃ c₄
inst✝⁸ : K₁.HasMapBifunctor K₂ F₁₂ c₁₂
inst✝⁷ : K₂.HasMapBifunctor K₃ G₂₃ c₂₃
inst✝⁶ : c₁.Associative c₂ c₃ c₁₂ c₂₃ c₄
inst✝⁵ : DecidableEq ι₁₂
inst✝⁴ : DecidableEq ι₂₃
inst✝³ : (K₁.mapBifunctor K₂ F₁₂ c₁₂).HasMapBifunctor K₃ G c₄
inst✝² : K₁.HasMapBifunctor (K₂.mapBifunctor K₃ G₂₃ c₂₃) F c₄
inst✝¹ : HasGoodTrifunctor₁₂Obj F₁₂ G K₁ K₂ K₃ c₁₂ c₄
inst✝ : HasGoodTrifunctor₂₃Obj F G₂₃ K₁ K₂ K₃ c₁₂ c₂₃ c₄
j j' : ι₄
i₁ : ι₁
i₂ : ι₂
i₃ : ι₃
h : c₁.r c₂ c₃ c₁₂ c₄ (i₁, i₂, i₃) = j
h₁ : c₂.Rel i₂ (c₂.next i₂)
⊢ ((associator.hom.app (K₁.X i₁)).app (K₂.X i₂)).app (K₃.X i₃) ≫
mapBifunctor₂₃.d₂ F G₂₃ K₁ K₂ K₃ c₁₂ c₂₃ c₄ i₁ i₂ i₃ j' =
mapBifunctor₁₂.d₂ F₁₂ G K₁ K₂ K₃ c₁₂ c₄ i₁ i₂ i₃ j' ≫
(mapBifunctorAssociatorX associator K₁ K₂ K₃ c₁₂ c₂₃ c₄ j').hom
case neg
C₁ : Type u_1
C₂ : Type u_2
C₁₂ : Type u_3
C₂₃ : Type u_4
C₃ : Type u_5
C₄ : Type u_6
inst✝³³ : Category.{u_16, u_1} C₁
inst✝³² : Category.{u_17, u_2} C₂
inst✝³¹ : Category.{u_15, u_5} C₃
inst✝³⁰ : Category.{u_13, u_6} C₄
inst✝²⁹ : Category.{u_14, u_3} C₁₂
inst✝²⁸ : Category.{u_18, u_4} C₂₃
inst✝²⁷ : HasZeroMorphisms C₁
inst✝²⁶ : HasZeroMorphisms C₂
inst✝²⁵ : HasZeroMorphisms C₃
inst✝²⁴ : Preadditive C₁₂
inst✝²³ : Preadditive C₂₃
inst✝²² : Preadditive C₄
F₁₂ : C₁ ⥤ C₂ ⥤ C₁₂
G : C₁₂ ⥤ C₃ ⥤ C₄
F : C₁ ⥤ C₂₃ ⥤ C₄
G₂₃ : C₂ ⥤ C₃ ⥤ C₂₃
inst✝²¹ : F₁₂.PreservesZeroMorphisms
inst✝²⁰ : ∀ (X₁ : C₁), (F₁₂.obj X₁).PreservesZeroMorphisms
inst✝¹⁹ : G.Additive
inst✝¹⁸ : ∀ (X₁₂ : C₁₂), (G.obj X₁₂).PreservesZeroMorphisms
inst✝¹⁷ : G₂₃.PreservesZeroMorphisms
inst✝¹⁶ : ∀ (X₂ : C₂), (G₂₃.obj X₂).PreservesZeroMorphisms
inst✝¹⁵ : F.PreservesZeroMorphisms
inst✝¹⁴ : ∀ (X₁ : C₁), (F.obj X₁).Additive
associator : bifunctorComp₁₂ F₁₂ G ≅ bifunctorComp₂₃ F G₂₃
ι₁ : Type u_7
ι₂ : Type u_8
ι₃ : Type u_9
ι₁₂ : Type u_10
ι₂₃ : Type u_11
ι₄ : Type u_12
inst✝¹³ : DecidableEq ι₄
c₁ : ComplexShape ι₁
c₂ : ComplexShape ι₂
c₃ : ComplexShape ι₃
K₁ : HomologicalComplex C₁ c₁
K₂ : HomologicalComplex C₂ c₂
K₃ : HomologicalComplex C₃ c₃
c₁₂ : ComplexShape ι₁₂
c₂₃ : ComplexShape ι₂₃
c₄ : ComplexShape ι₄
inst✝¹² : TotalComplexShape c₁ c₂ c₁₂
inst✝¹¹ : TotalComplexShape c₁₂ c₃ c₄
inst✝¹⁰ : TotalComplexShape c₂ c₃ c₂₃
inst✝⁹ : TotalComplexShape c₁ c₂₃ c₄
inst✝⁸ : K₁.HasMapBifunctor K₂ F₁₂ c₁₂
inst✝⁷ : K₂.HasMapBifunctor K₃ G₂₃ c₂₃
inst✝⁶ : c₁.Associative c₂ c₃ c₁₂ c₂₃ c₄
inst✝⁵ : DecidableEq ι₁₂
inst✝⁴ : DecidableEq ι₂₃
inst✝³ : (K₁.mapBifunctor K₂ F₁₂ c₁₂).HasMapBifunctor K₃ G c₄
inst✝² : K₁.HasMapBifunctor (K₂.mapBifunctor K₃ G₂₃ c₂₃) F c₄
inst✝¹ : HasGoodTrifunctor₁₂Obj F₁₂ G K₁ K₂ K₃ c₁₂ c₄
inst✝ : HasGoodTrifunctor₂₃Obj F G₂₃ K₁ K₂ K₃ c₁₂ c₂₃ c₄
j j' : ι₄
i₁ : ι₁
i₂ : ι₂
i₃ : ι₃
h : c₁.r c₂ c₃ c₁₂ c₄ (i₁, i₂, i₃) = j
h₁ : ¬c₂.Rel i₂ (c₂.next i₂)
⊢ ((associator.hom.app (K₁.X i₁)).app (K₂.X i₂)).app (K₃.X i₃) ≫
mapBifunctor₂₃.d₂ F G₂₃ K₁ K₂ K₃ c₁₂ c₂₃ c₄ i₁ i₂ i₃ j' =
mapBifunctor₁₂.d₂ F₁₂ G K₁ K₂ K₃ c₁₂ c₄ i₁ i₂ i₃ j' ≫
(mapBifunctorAssociatorX associator K₁ K₂ K₃ c₁₂ c₂₃ c₄ j').hom
|
9e9b646514c4ff68
|
Ring.DirectLimit.lift_injective
|
Mathlib/Algebra/Colimit/Ring.lean
|
lemma lift_injective [Nonempty ι] [IsDirected ι (· ≤ ·)]
(injective : ∀ i, Function.Injective <| g i) :
Function.Injective (lift G f P g Hg)
|
ι : Type u_1
inst✝⁴ : Preorder ι
G : ι → Type u_2
inst✝³ : (i : ι) → CommRing (G i)
f : (i j : ι) → i ≤ j → G i → G j
P : Type u_3
inst✝² : CommRing P
g : (i : ι) → G i →+* P
Hg : ∀ (i j : ι) (hij : i ≤ j) (x : G i), (g j) (f i j hij x) = (g i) x
inst✝¹ : Nonempty ι
inst✝ : IsDirected ι fun x1 x2 => x1 ≤ x2
injective : ∀ (i : ι) (a : G i), (g i) a = 0 → a = 0
z : DirectLimit G f
hz : (lift G f P g Hg) z = 0
⊢ z = 0
|
induction z using DirectLimit.induction_on with
| ih _ g => rw [lift_of] at hz; rw [injective _ g hz, _root_.map_zero]
|
no goals
|
ec9feee4de21e3d5
|
Nat.sub_le_of_le_add
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean
|
theorem sub_le_of_le_add {a b c : Nat} (h : a ≤ c + b) : a - b ≤ c
|
a b c : Nat
h : a ≤ c + b
d : Nat
hd : a + d = c + b
hge : b ≤ a
⊢ a - b ≤ c
|
apply @le.intro _ _ d
|
a b c : Nat
h : a ≤ c + b
d : Nat
hd : a + d = c + b
hge : b ≤ a
⊢ a - b + d = c
|
738662ec716bb284
|
AlgebraicGeometry.HasRingHomProperty.isStableUnderBaseChange
|
Mathlib/AlgebraicGeometry/Morphisms/RingHomProperties.lean
|
lemma isStableUnderBaseChange (hP : RingHom.IsStableUnderBaseChange Q) :
P.IsStableUnderBaseChange
|
case hP'.H.inr
P : MorphismProperty Scheme
Q : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
inst✝² : HasRingHomProperty P Q
hP : RingHom.IsStableUnderBaseChange fun {R S} [CommRing R] [CommRing S] => Q
this✝ : (sourceAffineLocally fun {R S} [CommRing R] [CommRing S] => Q).IsLocal :=
HasAffineProperty.isLocal_affineProperty P
X Y S : Scheme
inst✝¹ : IsAffine S
inst✝ : IsAffine X
f : X ⟶ S
g : Y ⟶ S
H : P g
this : ∀ ⦃Y : Scheme⦄ (g : Y ⟶ S), P g → IsAffine Y → P (pullback.fst f g)
hX : ¬IsAffine Y
i : (Scheme.Pullback.openCoverOfRight Y.affineCover f g).J
⊢ P (pullback.fst f (Y.affineCover.map i ≫ g))
|
apply this _ (comp_of_isOpenImmersion _ _ _ H) inferInstance
|
no goals
|
12b3b3ff9d099e59
|
Real.sSup_smul_of_nonneg
|
Mathlib/Data/Real/Pointwise.lean
|
theorem Real.sSup_smul_of_nonneg (ha : 0 ≤ a) (s : Set ℝ) : sSup (a • s) = a • sSup s
|
case inr.inr
α : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : MulActionWithZero α ℝ
inst✝ : OrderedSMul α ℝ
a : α
ha : 0 ≤ a
s : Set ℝ
hs : s.Nonempty
ha' : 0 < a
⊢ sSup (a • s) = a • sSup s
|
by_cases h : BddAbove s
|
case pos
α : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : MulActionWithZero α ℝ
inst✝ : OrderedSMul α ℝ
a : α
ha : 0 ≤ a
s : Set ℝ
hs : s.Nonempty
ha' : 0 < a
h : BddAbove s
⊢ sSup (a • s) = a • sSup s
case neg
α : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : MulActionWithZero α ℝ
inst✝ : OrderedSMul α ℝ
a : α
ha : 0 ≤ a
s : Set ℝ
hs : s.Nonempty
ha' : 0 < a
h : ¬BddAbove s
⊢ sSup (a • s) = a • sSup s
|
87e20576dbe946d5
|
Subgroup.fg_iff_add_fg
|
Mathlib/GroupTheory/Finiteness.lean
|
theorem Subgroup.fg_iff_add_fg (P : Subgroup G) : P.FG ↔ P.toAddSubgroup.FG
|
G : Type u_3
inst✝ : Group G
P : Subgroup G
⊢ P.FG ↔ (toAddSubgroup P).FG
|
exact (Subgroup.toSubmonoid P).fg_iff_add_fg
|
no goals
|
df9b2c8f40405e93
|
Multiset.replicate_le_replicate
|
Mathlib/Data/Multiset/Replicate.lean
|
theorem replicate_le_replicate (a : α) {k n : ℕ} : replicate k a ≤ replicate n a ↔ k ≤ n :=
_root_.trans (by rw [← replicate_le_coe, coe_replicate]) (List.replicate_sublist_replicate a)
|
α : Type u_1
a : α
k n : ℕ
⊢ replicate k a ≤ replicate n a ↔ List.replicate k a <+ List.replicate n a
|
rw [← replicate_le_coe, coe_replicate]
|
no goals
|
f94b5364c732b628
|
TopologicalSpace.ext_iff_isClosed
|
Mathlib/Topology/Basic.lean
|
theorem TopologicalSpace.ext_iff_isClosed {X} {t₁ t₂ : TopologicalSpace X} :
t₁ = t₂ ↔ ∀ s, IsClosed[t₁] s ↔ IsClosed[t₂] s
|
X : Type u_3
t₁ t₂ : TopologicalSpace X
⊢ t₁ = t₂ ↔ ∀ (s : Set X), IsClosed s ↔ IsClosed s
|
rw [TopologicalSpace.ext_iff, compl_surjective.forall]
|
X : Type u_3
t₁ t₂ : TopologicalSpace X
⊢ (∀ (x : Set X), IsOpen xᶜ ↔ IsOpen xᶜ) ↔ ∀ (s : Set X), IsClosed s ↔ IsClosed s
|
9a9b5b388220d5cd
|
BitVec.setWidth_succ
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean
|
theorem setWidth_succ (x : BitVec w) :
setWidth (i+1) x = cons (getLsbD x i) (setWidth i x)
|
case pred
w i : Nat
x : BitVec w
j : Nat
h : j < i + 1
⊢ x.getLsbD j = if j = i then x.getLsbD i else decide (j < i) && x.getLsbD j
|
if j_eq : j = i then
simp [j_eq]
else
have j_lt : j < i := Nat.lt_of_le_of_ne (Nat.le_of_succ_le_succ h) j_eq
simp [j_eq, j_lt]
|
no goals
|
127c2c3d6ebd3d56
|
polynomial_expand_eq
|
Mathlib/FieldTheory/Perfect.lean
|
lemma polynomial_expand_eq (f : R[X]) :
expand R p f = (f.map (frobeniusEquiv R p).symm) ^ p
|
R : Type u_1
p : ℕ
inst✝² : CommSemiring R
inst✝¹ : ExpChar R p
inst✝ : PerfectRing R p
f : R[X]
⊢ (expand R p) f = map (↑(frobeniusEquiv R p).symm) f ^ p
|
rw [← (f.map (S := R) (frobeniusEquiv R p).symm).expand_char p, map_expand, map_map,
frobenius_comp_frobeniusEquiv_symm, map_id]
|
no goals
|
2cb846c588a95df2
|
Equiv.refl_trans
|
Mathlib/Logic/Equiv/Defs.lean
|
theorem refl_trans (e : α ≃ β) : (Equiv.refl α).trans e = e
|
case mk
α : Sort u
β : Sort v
toFun✝ : α → β
invFun✝ : β → α
left_inv✝ : LeftInverse invFun✝ toFun✝
right_inv✝ : RightInverse invFun✝ toFun✝
⊢ (Equiv.refl α).trans { toFun := toFun✝, invFun := invFun✝, left_inv := left_inv✝, right_inv := right_inv✝ } =
{ toFun := toFun✝, invFun := invFun✝, left_inv := left_inv✝, right_inv := right_inv✝ }
|
rfl
|
no goals
|
9d117295fb2de6db
|
Ideal.spanIntNorm_localization
|
Mathlib/RingTheory/Ideal/Norm/RelNorm.lean
|
theorem spanIntNorm_localization (I : Ideal S) (M : Submonoid R) (hM : M ≤ R⁰)
{Rₘ : Type*} (Sₘ : Type*) [CommRing Rₘ] [Algebra R Rₘ] [CommRing Sₘ] [Algebra S Sₘ]
[Algebra Rₘ Sₘ] [Algebra R Sₘ] [IsScalarTower R Rₘ Sₘ] [IsScalarTower R S Sₘ]
[IsLocalization M Rₘ] [IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ]
[IsIntegrallyClosed Rₘ] [IsDomain Rₘ] [IsDomain Sₘ] [NoZeroSMulDivisors Rₘ Sₘ]
[Module.Finite Rₘ Sₘ] [IsIntegrallyClosed Sₘ]
[Algebra.IsSeparable (FractionRing Rₘ) (FractionRing Sₘ)] :
spanNorm Rₘ (I.map (algebraMap S Sₘ)) = (spanNorm R I).map (algebraMap R Rₘ)
|
R : Type u_1
inst✝²⁶ : CommRing R
inst✝²⁵ : IsDomain R
S : Type u_3
inst✝²⁴ : CommRing S
inst✝²³ : IsDomain S
inst✝²² : IsIntegrallyClosed R
inst✝²¹ : IsIntegrallyClosed S
inst✝²⁰ : Algebra R S
inst✝¹⁹ : Module.Finite R S
inst✝¹⁸ : NoZeroSMulDivisors R S
inst✝¹⁷ : Algebra.IsSeparable (FractionRing R) (FractionRing S)
I : Ideal S
M : Submonoid R
hM : M ≤ R⁰
Rₘ : Type u_4
Sₘ : Type u_5
inst✝¹⁶ : CommRing Rₘ
inst✝¹⁵ : Algebra R Rₘ
inst✝¹⁴ : CommRing Sₘ
inst✝¹³ : Algebra S Sₘ
inst✝¹² : Algebra Rₘ Sₘ
inst✝¹¹ : Algebra R Sₘ
inst✝¹⁰ : IsScalarTower R Rₘ Sₘ
inst✝⁹ : IsScalarTower R S Sₘ
inst✝⁸ : IsLocalization M Rₘ
inst✝⁷ : IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ
inst✝⁶ : IsIntegrallyClosed Rₘ
inst✝⁵ : IsDomain Rₘ
inst✝⁴ : IsDomain Sₘ
inst✝³ : NoZeroSMulDivisors Rₘ Sₘ
inst✝² : Module.Finite Rₘ Sₘ
inst✝¹ : IsIntegrallyClosed Sₘ
inst✝ : Algebra.IsSeparable (FractionRing Rₘ) (FractionRing Sₘ)
K : Type u_1 := FractionRing R
f : Rₘ →+* K := IsLocalization.map K (RingHom.id R) hM
L : Type u_3 := FractionRing S
g : Sₘ →+* L := IsLocalization.map L (RingHom.id S) ⋯
algInst✝² : Algebra Rₘ K := f.toAlgebra
algInst✝¹ : Algebra Sₘ L := g.toAlgebra
algInst✝ : Algebra Rₘ L := ((algebraMap K L).comp f).toAlgebra
scalarTowerInst✝ : IsScalarTower Rₘ K L := IsScalarTower.of_algebraMap_eq' (Eq.refl (algebraMap Rₘ L))
this✝ : IsScalarTower R Rₘ K
x✝ : IsFractionRing Rₘ K := IsFractionRing.isFractionRing_of_isDomain_of_isLocalization M Rₘ K
this : IsScalarTower S Sₘ L
⊢ spanNorm Rₘ (map (algebraMap S Sₘ) I) = map (algebraMap R Rₘ) (spanNorm R I)
|
have : IsScalarTower Rₘ Sₘ L := by
apply IsScalarTower.of_algebraMap_eq'
apply IsLocalization.ringHom_ext M
rw [RingHom.algebraMap_toAlgebra, RingHom.algebraMap_toAlgebra (R := Sₘ), RingHom.comp_assoc,
RingHom.comp_assoc, ← IsScalarTower.algebraMap_eq, IsScalarTower.algebraMap_eq R S Sₘ,
IsLocalization.map_comp, RingHom.comp_id, ← RingHom.comp_assoc, IsLocalization.map_comp,
RingHom.comp_id, ← IsScalarTower.algebraMap_eq, ← IsScalarTower.algebraMap_eq]
|
R : Type u_1
inst✝²⁶ : CommRing R
inst✝²⁵ : IsDomain R
S : Type u_3
inst✝²⁴ : CommRing S
inst✝²³ : IsDomain S
inst✝²² : IsIntegrallyClosed R
inst✝²¹ : IsIntegrallyClosed S
inst✝²⁰ : Algebra R S
inst✝¹⁹ : Module.Finite R S
inst✝¹⁸ : NoZeroSMulDivisors R S
inst✝¹⁷ : Algebra.IsSeparable (FractionRing R) (FractionRing S)
I : Ideal S
M : Submonoid R
hM : M ≤ R⁰
Rₘ : Type u_4
Sₘ : Type u_5
inst✝¹⁶ : CommRing Rₘ
inst✝¹⁵ : Algebra R Rₘ
inst✝¹⁴ : CommRing Sₘ
inst✝¹³ : Algebra S Sₘ
inst✝¹² : Algebra Rₘ Sₘ
inst✝¹¹ : Algebra R Sₘ
inst✝¹⁰ : IsScalarTower R Rₘ Sₘ
inst✝⁹ : IsScalarTower R S Sₘ
inst✝⁸ : IsLocalization M Rₘ
inst✝⁷ : IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ
inst✝⁶ : IsIntegrallyClosed Rₘ
inst✝⁵ : IsDomain Rₘ
inst✝⁴ : IsDomain Sₘ
inst✝³ : NoZeroSMulDivisors Rₘ Sₘ
inst✝² : Module.Finite Rₘ Sₘ
inst✝¹ : IsIntegrallyClosed Sₘ
inst✝ : Algebra.IsSeparable (FractionRing Rₘ) (FractionRing Sₘ)
K : Type u_1 := FractionRing R
f : Rₘ →+* K := IsLocalization.map K (RingHom.id R) hM
L : Type u_3 := FractionRing S
g : Sₘ →+* L := IsLocalization.map L (RingHom.id S) ⋯
algInst✝² : Algebra Rₘ K := f.toAlgebra
algInst✝¹ : Algebra Sₘ L := g.toAlgebra
algInst✝ : Algebra Rₘ L := ((algebraMap K L).comp f).toAlgebra
scalarTowerInst✝ : IsScalarTower Rₘ K L := IsScalarTower.of_algebraMap_eq' (Eq.refl (algebraMap Rₘ L))
this✝¹ : IsScalarTower R Rₘ K
x✝ : IsFractionRing Rₘ K := IsFractionRing.isFractionRing_of_isDomain_of_isLocalization M Rₘ K
this✝ : IsScalarTower S Sₘ L
this : IsScalarTower Rₘ Sₘ L
⊢ spanNorm Rₘ (map (algebraMap S Sₘ) I) = map (algebraMap R Rₘ) (spanNorm R I)
|
c51e5741dd2367a5
|
List.mem_mapFinIdx
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/MapIdx.lean
|
theorem mem_mapFinIdx {b : β} {l : List α} {f : (i : Nat) → α → (h : i < l.length) → β} :
b ∈ l.mapFinIdx f ↔ ∃ (i : Nat) (h : i < l.length), f i l[i] h = b
|
case mpr
β : Type u_1
α : Type u_2
b : β
l : List α
f : (i : Nat) → α → i < l.length → β
⊢ (∃ i h, f i l[i] h = b) → b ∈ l.mapFinIdx f
|
rintro ⟨i, h, rfl⟩
|
case mpr.intro.intro
β : Type u_1
α : Type u_2
l : List α
f : (i : Nat) → α → i < l.length → β
i : Nat
h : i < l.length
⊢ f i l[i] h ∈ l.mapFinIdx f
|
00d1fdd38a729f1a
|
Polynomial.sum_fin
|
Mathlib/Algebra/Polynomial/Degree/Support.lean
|
theorem sum_fin [AddCommMonoid S] (f : ℕ → R → S) (hf : ∀ i, f i 0 = 0) {n : ℕ} {p : R[X]}
(hn : p.degree < n) : (∑ i : Fin n, f i (p.coeff i)) = p.sum f
|
case pos
R : Type u
S : Type v
inst✝¹ : Semiring R
inst✝ : AddCommMonoid S
f : ℕ → R → S
hf : ∀ (i : ℕ), f i 0 = 0
n : ℕ
p : R[X]
hn : p.degree < ↑n
hp : p = 0
⊢ ∀ x ∈ univ, f (↑x) (coeff 0 ↑x) = 0
|
intro i _
|
case pos
R : Type u
S : Type v
inst✝¹ : Semiring R
inst✝ : AddCommMonoid S
f : ℕ → R → S
hf : ∀ (i : ℕ), f i 0 = 0
n : ℕ
p : R[X]
hn : p.degree < ↑n
hp : p = 0
i : Fin n
a✝ : i ∈ univ
⊢ f (↑i) (coeff 0 ↑i) = 0
|
15253ffb89262ddf
|
Polynomial.isIntegral_isLocalization_polynomial_quotient
|
Mathlib/RingTheory/Jacobson/Ring.lean
|
theorem isIntegral_isLocalization_polynomial_quotient
(P : Ideal R[X]) (pX : R[X]) (hpX : pX ∈ P) [Algebra (R ⧸ P.comap (C : R →+* R[X])) Rₘ]
[IsLocalization.Away (pX.map (Ideal.Quotient.mk (P.comap (C : R →+* R[X])))).leadingCoeff Rₘ]
[Algebra (R[X] ⧸ P) Sₘ] [IsLocalization ((Submonoid.powers (pX.map (Ideal.Quotient.mk (P.comap
(C : R →+* R[X])))).leadingCoeff).map (quotientMap P C le_rfl) : Submonoid (R[X] ⧸ P)) Sₘ] :
(IsLocalization.map Sₘ (quotientMap P C le_rfl) (Submonoid.powers (pX.map (Ideal.Quotient.mk
(P.comap (C : R →+* R[X])))).leadingCoeff).le_comap_map : Rₘ →+* Sₘ).IsIntegral
|
case h.left
R : Type u_1
inst✝⁶ : CommRing R
Rₘ : Type u_3
Sₘ : Type u_4
inst✝⁵ : CommRing Rₘ
inst✝⁴ : CommRing Sₘ
P : Ideal R[X]
pX : R[X]
hpX : pX ∈ P
inst✝³ : Algebra (R ⧸ comap C P) Rₘ
inst✝² : IsLocalization.Away (map (Ideal.Quotient.mk (comap C P)) pX).leadingCoeff Rₘ
inst✝¹ : Algebra (R[X] ⧸ P) Sₘ
inst✝ :
IsLocalization
(Submonoid.map (quotientMap P C ⋯) (Submonoid.powers (map (Ideal.Quotient.mk (comap C P)) pX).leadingCoeff)) Sₘ
P' : Ideal R := comap C P
M : Submonoid (R ⧸ P') := Submonoid.powers (map (Ideal.Quotient.mk (comap C P)) pX).leadingCoeff
M' : Submonoid (R[X] ⧸ P) :=
Submonoid.map (quotientMap P C ⋯) (Submonoid.powers (map (Ideal.Quotient.mk (comap C P)) pX).leadingCoeff)
φ : R ⧸ P' →+* R[X] ⧸ P := quotientMap P C ⋯
φ' : Rₘ →+* Sₘ := IsLocalization.map Sₘ φ ⋯
hφ' : φ.comp (Ideal.Quotient.mk P') = (Ideal.Quotient.mk P).comp C
p✝ : Sₘ
p' q : R[X] ⧸ P
hq : q ∈ M'
hp : p✝ * (algebraMap (R[X] ⧸ P) Sₘ) q = (algebraMap (R[X] ⧸ P) Sₘ) p'
p : R[X]
hy : p = C (p.coeff 0)
⊢ (X - C ((algebraMap (R ⧸ P') Rₘ) ((Ideal.Quotient.mk P') (p.coeff 0)))).Monic
|
apply monic_X_sub_C
|
no goals
|
4f255f198e9b3f92
|
eq_pos_convex_span_of_mem_convexHull
|
Mathlib/Analysis/Convex/Caratheodory.lean
|
theorem eq_pos_convex_span_of_mem_convexHull {x : E} (hx : x ∈ convexHull 𝕜 s) :
∃ (ι : Sort (u + 1)) (_ : Fintype ι),
∃ (z : ι → E) (w : ι → 𝕜), Set.range z ⊆ s ∧ AffineIndependent 𝕜 z ∧ (∀ i, 0 < w i) ∧
∑ i, w i = 1 ∧ ∑ i, w i • z i = x
|
case intro.intro.intro.intro.intro.intro.refine_5
𝕜 : Type u_1
E : Type u
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s : Set E
x : E
t : Finset E
ht₁ : ↑t ⊆ s
ht₂ : AffineIndependent 𝕜 Subtype.val
w : E → 𝕜
hw₁ : ∀ y ∈ t, 0 ≤ w y
hw₂ : ∑ y ∈ t, w y = 1
hw₃ : t.centerMass w id = x
t' : Finset E := filter (fun i => w i ≠ 0) t
⊢ ∑ i ∈ t'.attach, (fun e => w e • e) ↑i = x
|
rw [Finset.sum_attach (f := fun e => w e • e), Finset.sum_filter_of_ne]
|
case intro.intro.intro.intro.intro.intro.refine_5
𝕜 : Type u_1
E : Type u
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s : Set E
x : E
t : Finset E
ht₁ : ↑t ⊆ s
ht₂ : AffineIndependent 𝕜 Subtype.val
w : E → 𝕜
hw₁ : ∀ y ∈ t, 0 ≤ w y
hw₂ : ∑ y ∈ t, w y = 1
hw₃ : t.centerMass w id = x
t' : Finset E := filter (fun i => w i ≠ 0) t
⊢ ∑ x ∈ t, w x • x = x
case intro.intro.intro.intro.intro.intro.refine_5
𝕜 : Type u_1
E : Type u
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s : Set E
x : E
t : Finset E
ht₁ : ↑t ⊆ s
ht₂ : AffineIndependent 𝕜 Subtype.val
w : E → 𝕜
hw₁ : ∀ y ∈ t, 0 ≤ w y
hw₂ : ∑ y ∈ t, w y = 1
hw₃ : t.centerMass w id = x
t' : Finset E := filter (fun i => w i ≠ 0) t
⊢ ∀ x ∈ t, w x • x ≠ 0 → w x ≠ 0
|
c36d0bcdbb3360db
|
IsCauSeq.bounded
|
Mathlib/Algebra/Order/CauSeq/Basic.lean
|
lemma bounded (hf : IsCauSeq abv f) : ∃ r, ∀ i, abv (f i) < r
|
case intro
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
hf : IsCauSeq abv f
i : ℕ
h : ∀ j ≥ i, abv (f j - f i) < 1
⊢ ∃ r, ∀ (i : ℕ), abv (f i) < r
|
set R : ℕ → α := @Nat.rec (fun _ => α) (abv (f 0)) fun i c => max c (abv (f i.succ)) with hR
|
case intro
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
hf : IsCauSeq abv f
i : ℕ
h : ∀ j ≥ i, abv (f j - f i) < 1
R : ℕ → α := Nat.rec (abv (f 0)) fun i c => c ⊔ abv (f i.succ)
hR : R = Nat.rec (abv (f 0)) fun i c => c ⊔ abv (f i.succ)
⊢ ∃ r, ∀ (i : ℕ), abv (f i) < r
|
9032991fc685129f
|
EulerSine.tendsto_integral_cos_pow_mul_div
|
Mathlib/Analysis/SpecialFunctions/Trigonometric/EulerSineProd.lean
|
theorem tendsto_integral_cos_pow_mul_div {f : ℝ → ℂ} (hf : ContinuousOn f (Icc 0 (π / 2))) :
Tendsto
(fun n : ℕ => (∫ x in (0 : ℝ)..π / 2, (cos x : ℂ) ^ n * f x) /
(∫ x in (0 : ℝ)..π / 2, cos x ^ n : ℝ))
atTop (𝓝 <| f 0)
|
f : ℝ → ℂ
hf : ContinuousOn f (Icc 0 (π / 2))
c_lt : ∀ y ∈ Icc 0 (π / 2), y ≠ 0 → cos y < cos 0
c_nonneg : ∀ x ∈ Icc 0 (π / 2), 0 ≤ cos x
⊢ 0 < 1
|
exact zero_lt_one
|
no goals
|
6e4d26a12747c680
|
Array.findIdx_subtype
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Find.lean
|
theorem findIdx_subtype {p : α → Prop} {l : Array { x // p x }}
{f : { x // p x } → Bool} {g : α → Bool} (hf : ∀ x h, f ⟨x, h⟩ = g x) :
l.findIdx f = l.unattach.findIdx g
|
case mk
α : Type u_1
p : α → Prop
f : { x // p x } → Bool
g : α → Bool
hf : ∀ (x : α) (h : p x), f ⟨x, h⟩ = g x
toList✝ : List { x // p x }
⊢ findIdx f { toList := toList✝ } = findIdx g { toList := toList✝ }.unattach
|
simp [hf]
|
no goals
|
ea378e9c0e6d7dec
|
SimpleGraph.FarFromTriangleFree.le_card_cliqueFinset
|
Mathlib/Combinatorics/SimpleGraph/Triangle/Removal.lean
|
/-- **Triangle Removal Lemma**. If not all triangles can be removed by removing few edges (on the
order of `(card α)^2`), then there were many triangles to start with (on the order of
`(card α)^3`). -/
lemma FarFromTriangleFree.le_card_cliqueFinset (hG : G.FarFromTriangleFree ε) :
triangleRemovalBound ε * card α ^ 3 ≤ #(G.cliqueFinset 3)
|
case inr.inr.inr.intro.intro.intro.intro
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
ε : ℝ
hG : G.FarFromTriangleFree ε
h✝ : Nonempty α
hε : 0 < ε
l : ℕ := ⌈4 / ε⌉₊
hl : 4 / ε ≤ ↑l
hl' : l ≤ Fintype.card α
P : Finpartition univ
hP₁ : P.IsEquipartition
hP₂ : l ≤ #P.parts
hP₃ : #P.parts ≤ bound (ε / 8) l
hP₄ : P.IsUniform G (ε / 8)
this : 4 / ε ≤ ↑(#P.parts)
k : ↑(#G.edgeFinset) - ↑(#(regularityReduced P G (ε / 8) (ε / 4)).edgeFinset) < ε * ↑(Fintype.card α ^ 2)
⊢ triangleRemovalBound ε * ↑(Fintype.card α) ^ 3 ≤ ↑(#(G.cliqueFinset 3))
|
obtain ⟨t, ht⟩ := hG.cliqueFinset_nonempty' regularityReduced_le k
|
case inr.inr.inr.intro.intro.intro.intro.intro
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
ε : ℝ
hG : G.FarFromTriangleFree ε
h✝ : Nonempty α
hε : 0 < ε
l : ℕ := ⌈4 / ε⌉₊
hl : 4 / ε ≤ ↑l
hl' : l ≤ Fintype.card α
P : Finpartition univ
hP₁ : P.IsEquipartition
hP₂ : l ≤ #P.parts
hP₃ : #P.parts ≤ bound (ε / 8) l
hP₄ : P.IsUniform G (ε / 8)
this : 4 / ε ≤ ↑(#P.parts)
k : ↑(#G.edgeFinset) - ↑(#(regularityReduced P G (ε / 8) (ε / 4)).edgeFinset) < ε * ↑(Fintype.card α ^ 2)
t : Finset α
ht : t ∈ (regularityReduced P G (ε / 8) (ε / 4)).cliqueFinset 3
⊢ triangleRemovalBound ε * ↑(Fintype.card α) ^ 3 ≤ ↑(#(G.cliqueFinset 3))
|
68b85a2891fa9866
|
preNormEDS_odd
|
Mathlib/NumberTheory/EllipticDivisibilitySequence.lean
|
lemma preNormEDS_odd (m : ℤ) : preNormEDS b c d (2 * m + 1) =
preNormEDS b c d (m + 2) * preNormEDS b c d m ^ 3 * (if Even m then b else 1) -
preNormEDS b c d (m - 1) * preNormEDS b c d (m + 1) ^ 3 * (if Even m then 1 else b)
|
case nat.succ.zero
R : Type u
inst✝ : CommRing R
b c d : R
⊢ preNormEDS b c d (2 * ↑(0 + 1) + 1) =
(preNormEDS b c d (↑(0 + 1) + 2) * preNormEDS b c d ↑(0 + 1) ^ 3 * if Even ↑(0 + 1) then b else 1) -
preNormEDS b c d (↑(0 + 1) - 1) * preNormEDS b c d (↑(0 + 1) + 1) ^ 3 * if Even ↑(0 + 1) then 1 else b
|
simp
|
no goals
|
a81418ac2a260128
|
Complex.HadamardThreeLines.norm_le_interp_of_mem_verticalClosedStrip₀₁'
|
Mathlib/Analysis/Complex/Hadamard.lean
|
/-- **Hadamard three-line theorem** on `re ⁻¹' [0, 1]` (Variant in simpler terms): Let `f` be a
bounded function, continuous on the closed strip `re ⁻¹' [0, 1]` and differentiable on open strip
`re ⁻¹' (0, 1)`. If, for all `z.re = 0`, `‖f z‖ ≤ a` for some `a ∈ ℝ` and, similarly, for all
`z.re = 1`, `‖f z‖ ≤ b` for some `b ∈ ℝ` then for all `z` in the closed strip
`re ⁻¹' [0, 1]` the inequality `‖f(z)‖ ≤ a ^ (1 - z.re) * b ^ z.re` holds. -/
lemma norm_le_interp_of_mem_verticalClosedStrip₀₁' (f : ℂ → E) {z : ℂ} {a b : ℝ}
(hz : z ∈ verticalClosedStrip 0 1) (hd : DiffContOnCl ℂ f (verticalStrip 0 1))
(hB : BddAbove ((norm ∘ f) '' verticalClosedStrip 0 1))
(ha : ∀ z ∈ re ⁻¹' {0}, ‖f z‖ ≤ a) (hb : ∀ z ∈ re ⁻¹' {1}, ‖f z‖ ≤ b) :
‖f z‖ ≤ a ^ (1 - z.re) * b ^ z.re
|
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
a b : ℝ
hz : z ∈ verticalClosedStrip 0 1
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
ha : ∀ z ∈ re ⁻¹' {0}, ‖f z‖ ≤ a
hb : ∀ z ∈ re ⁻¹' {1}, ‖f z‖ ≤ b
this : ‖interpStrip f z‖ ≤ sSupNormIm f 0 ^ (1 - z.re) * sSupNormIm f 1 ^ z.re
⊢ (norm ∘ f '' (re ⁻¹' {1})).Nonempty
|
use ‖(f 1)‖, 1
|
case h
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
a b : ℝ
hz : z ∈ verticalClosedStrip 0 1
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
ha : ∀ z ∈ re ⁻¹' {0}, ‖f z‖ ≤ a
hb : ∀ z ∈ re ⁻¹' {1}, ‖f z‖ ≤ b
this : ‖interpStrip f z‖ ≤ sSupNormIm f 0 ^ (1 - z.re) * sSupNormIm f 1 ^ z.re
⊢ 1 ∈ re ⁻¹' {1} ∧ (norm ∘ f) 1 = ‖f 1‖
|
0fab8ad239caab3f
|
Affine.Simplex.inner_mongePoint_vsub_face_centroid_vsub
|
Mathlib/Geometry/Euclidean/MongePoint.lean
|
theorem inner_mongePoint_vsub_face_centroid_vsub {n : ℕ} (s : Simplex ℝ P (n + 2))
{i₁ i₂ : Fin (n + 3)} :
⟪s.mongePoint -ᵥ ({i₁, i₂}ᶜ : Finset (Fin (n + 3))).centroid ℝ s.points,
s.points i₁ -ᵥ s.points i₂⟫ =
0
|
case right
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
n : ℕ
s : Simplex ℝ P (n + 2)
i₁ i₂ : Fin (n + 3)
h : ¬i₁ = i₂
hs :
∑ i : PointsWithCircumcenterIndex (n + 2), (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) i = 0
fs : Finset (Fin (n + 3)) := {i₁, i₂}
i : Fin (n + 3)
hi : i ∉ fs
hj : i = i₂
⊢ False
|
simp [fs, ← hj] at hi
|
no goals
|
4dd62aff1c506de0
|
Complex.HadamardThreeLines.scale_diffContOnCl
|
Mathlib/Analysis/Complex/Hadamard.lean
|
/-- The function `scale f l u` is `diffContOnCl`. -/
lemma scale_diffContOnCl {f : ℂ → E} {l u : ℝ} (hul : l < u)
(hd : DiffContOnCl ℂ f (verticalStrip l u)) :
DiffContOnCl ℂ (scale f l u) (verticalStrip 0 1)
|
case hg.hf.hc
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
l u : ℝ
hul : l < u
hd : DiffContOnCl ℂ f (verticalStrip l u)
⊢ DiffContOnCl ℂ (fun x => x) (verticalStrip 0 1)
|
exact Differentiable.diffContOnCl differentiable_id'
|
no goals
|
3fd41ceba2ac6041
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.