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
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|---|---|---|---|---|---|---|
CategoryTheory.Subgroupoid.galoisConnection_map_comap
|
Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean
|
theorem galoisConnection_map_comap (hφ : Function.Injective φ.obj) :
GaloisConnection (map φ hφ) (comap φ)
|
case mpr.im
C : Type u
inst✝¹ : Groupoid C
D : Type u_1
inst✝ : Groupoid D
φ : C ⥤ D
hφ : Function.Injective φ.obj
S : Subgroupoid C
T : Subgroupoid D
h : ∀ {c d : C}, S.arrows c d ⊆ (comap φ T).arrows c d
c d : D
c✝ d✝ : C
a : c✝ ⟶ d✝
gφS : a ∈ S.arrows c✝ d✝
⊢ φ.map a ∈ T.arrows (φ.obj c✝) (φ.obj d✝)
|
exact h gφS
|
no goals
|
16163661f1ed9a8a
|
PFunctor.liftr_iff
|
Mathlib/Data/PFunctor/Univariate/Basic.lean
|
theorem liftr_iff {α : Type u} (r : α → α → Prop) (x y : P α) :
Liftr r x y ↔ ∃ a f₀ f₁, x = ⟨a, f₀⟩ ∧ y = ⟨a, f₁⟩ ∧ ∀ i, r (f₀ i) (f₁ i)
|
case h.left
P : PFunctor.{u}
α : Type u
r : α → α → Prop
x y : ↑P α
a : P.A
f₀ f₁ : P.B a → α
xeq : x = ⟨a, f₀⟩
yeq : y = ⟨a, f₁⟩
h : ∀ (i : P.B a), r (f₀ i) (f₁ i)
⊢ (fun t => (↑t).1) <$> ⟨a, fun i => ⟨(f₀ i, f₁ i), ⋯⟩⟩ = x
|
rw [xeq]
|
case h.left
P : PFunctor.{u}
α : Type u
r : α → α → Prop
x y : ↑P α
a : P.A
f₀ f₁ : P.B a → α
xeq : x = ⟨a, f₀⟩
yeq : y = ⟨a, f₁⟩
h : ∀ (i : P.B a), r (f₀ i) (f₁ i)
⊢ (fun t => (↑t).1) <$> ⟨a, fun i => ⟨(f₀ i, f₁ i), ⋯⟩⟩ = ⟨a, f₀⟩
|
50cdcbaa1a8a2d3e
|
StieltjesFunction.ae_hasDerivAt
|
Mathlib/Analysis/Calculus/Monotone.lean
|
theorem StieltjesFunction.ae_hasDerivAt (f : StieltjesFunction) :
∀ᵐ x, HasDerivAt f (rnDeriv f.measure volume x).toReal x
|
case h
f : StieltjesFunction
x : ℝ
hx : Tendsto (fun a => f.measure a / volume a) ((vitaliFamily volume 1).filterAt x) (𝓝 (f.measure.rnDeriv volume x))
h'x : f.measure.rnDeriv volume x < ⊤
h''x : ¬leftLim (↑f) x ≠ ↑f x
y : ℝ
hxy : x < y
⊢ (ENNReal.toReal ∘ (fun a => f.measure a / volume a) ∘ fun y => Icc x y) y = (↑f y - ↑f x) / (y - x)
|
simp only [comp_apply, StieltjesFunction.measure_Icc, Real.volume_Icc, Classical.not_not.1 h''x]
|
case h
f : StieltjesFunction
x : ℝ
hx : Tendsto (fun a => f.measure a / volume a) ((vitaliFamily volume 1).filterAt x) (𝓝 (f.measure.rnDeriv volume x))
h'x : f.measure.rnDeriv volume x < ⊤
h''x : ¬leftLim (↑f) x ≠ ↑f x
y : ℝ
hxy : x < y
⊢ (ENNReal.ofReal (↑f y - ↑f x) / ENNReal.ofReal (y - x)).toReal = (↑f y - ↑f x) / (y - x)
|
5dba92ed00d1be0b
|
LawfulFunctor.map_inj_right_of_nonempty
|
Mathlib/.lake/packages/batteries/Batteries/Control/Monad.lean
|
theorem _root_.LawfulFunctor.map_inj_right_of_nonempty [Functor f] [LawfulFunctor f] [Nonempty α]
{g : α → β} (h : ∀ {x y : α}, g x = g y → x = y) {x y : f α} :
g <$> x = g <$> y ↔ x = y
|
case mp
f : Type u_1 → Type u_2
α β : Type u_1
inst✝² : Functor f
inst✝¹ : LawfulFunctor f
inst✝ : Nonempty α
g : α → β
h : ∀ {x y : α}, g x = g y → x = y
x y : f α
⊢ g <$> x = g <$> y → x = y
|
let g' a := if h : ∃ b, g b = a then h.choose else Classical.ofNonempty
|
case mp
f : Type u_1 → Type u_2
α β : Type u_1
inst✝² : Functor f
inst✝¹ : LawfulFunctor f
inst✝ : Nonempty α
g : α → β
h : ∀ {x y : α}, g x = g y → x = y
x y : f α
g' : β → α := fun a => if h : ∃ b, g b = a then h.choose else Classical.ofNonempty
⊢ g <$> x = g <$> y → x = y
|
9ff9346398dc10b4
|
HasFPowerSeriesAt.locally_ne_zero
|
Mathlib/Analysis/Analytic/IsolatedZeros.lean
|
theorem locally_ne_zero (hp : HasFPowerSeriesAt f p z₀) (h : p ≠ 0) : ∀ᶠ z in 𝓝[≠] z₀, f z ≠ 0
|
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
p : FormalMultilinearSeries 𝕜 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
hp : HasFPowerSeriesAt f p z₀
h : p ≠ 0
h2 : ContinuousAt ((swap dslope z₀)^[p.order] f) z₀
⊢ ∀ᶠ (x : 𝕜) in 𝓝 z₀, x ∈ {z₀}ᶜ → f x ≠ 0
|
have h3 := h2.eventually_ne (iterate_dslope_fslope_ne_zero hp h)
|
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
p : FormalMultilinearSeries 𝕜 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
hp : HasFPowerSeriesAt f p z₀
h : p ≠ 0
h2 : ContinuousAt ((swap dslope z₀)^[p.order] f) z₀
h3 : ∀ᶠ (z : 𝕜) in 𝓝 z₀, (swap dslope z₀)^[p.order] f z ≠ 0
⊢ ∀ᶠ (x : 𝕜) in 𝓝 z₀, x ∈ {z₀}ᶜ → f x ≠ 0
|
225b3fe7c92728de
|
AlgebraicIndependent.aeval_comp_mvPolynomialOptionEquivPolynomialAdjoin
|
Mathlib/RingTheory/AlgebraicIndependent/Basic.lean
|
theorem AlgebraicIndependent.aeval_comp_mvPolynomialOptionEquivPolynomialAdjoin
(hx : AlgebraicIndependent R x) (a : A) :
RingHom.comp
(↑(Polynomial.aeval a : Polynomial (adjoin R (Set.range x)) →ₐ[_] A) :
Polynomial (adjoin R (Set.range x)) →+* A)
hx.mvPolynomialOptionEquivPolynomialAdjoin.toRingHom =
↑(MvPolynomial.aeval fun o : Option ι => o.elim a x : MvPolynomial (Option ι) R →ₐ[R] A)
|
case refine_1
ι : Type u_1
R : Type u_3
A : Type u_5
x : ι → A
inst✝² : CommRing R
inst✝¹ : CommRing A
inst✝ : Algebra R A
hx : AlgebraicIndependent R x
a : A
r : R
⊢ (Polynomial.aeval a) (hx.mvPolynomialOptionEquivPolynomialAdjoin (C r)) = (aeval fun o => o.elim a x) (C r)
|
rw [hx.mvPolynomialOptionEquivPolynomialAdjoin_C, aeval_C, Polynomial.aeval_C,
IsScalarTower.algebraMap_apply R (adjoin R (range x)) A]
|
no goals
|
ba3a3173da6809fd
|
MeasureTheory.AEStronglyMeasurable.integrable_truncation
|
Mathlib/Probability/StrongLaw.lean
|
theorem _root_.MeasureTheory.AEStronglyMeasurable.integrable_truncation [IsFiniteMeasure μ]
(hf : AEStronglyMeasurable f μ) {A : ℝ} : Integrable (truncation f A) μ
|
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
inst✝ : IsFiniteMeasure μ
hf : AEStronglyMeasurable f μ
A : ℝ
⊢ Integrable (truncation f A) μ
|
rw [← memLp_one_iff_integrable]
|
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
inst✝ : IsFiniteMeasure μ
hf : AEStronglyMeasurable f μ
A : ℝ
⊢ MemLp (truncation f A) 1 μ
|
cc0b527a01dab8e7
|
nhds_list
|
Mathlib/Topology/List.lean
|
theorem nhds_list (as : List α) : 𝓝 as = traverse 𝓝 as
|
case refine_2.intro.intro.intro.intro
α : Type u_1
inst✝ : TopologicalSpace α
l : List α
s : Set (List α)
u✝ : List (Set α)
hu✝ : List.Forall₂ (fun b s => s ∈ 𝓝 b) l u✝
hus : sequence u✝ ⊆ s
v : List (Set α)
hv : List.Forall₂ (fun a s => IsOpen s ∧ a ∈ s) l v
hvs : sequence v ⊆ s
this : sequence v ∈ traverse 𝓝 l
u : List α
hu : u ∈ sequence v
⊢ u ∈ {x | (fun y => s ∈ traverse 𝓝 y) x}
|
have hu := (List.mem_traverse _ _).1 hu
|
case refine_2.intro.intro.intro.intro
α : Type u_1
inst✝ : TopologicalSpace α
l : List α
s : Set (List α)
u✝ : List (Set α)
hu✝¹ : List.Forall₂ (fun b s => s ∈ 𝓝 b) l u✝
hus : sequence u✝ ⊆ s
v : List (Set α)
hv : List.Forall₂ (fun a s => IsOpen s ∧ a ∈ s) l v
hvs : sequence v ⊆ s
this : sequence v ∈ traverse 𝓝 l
u : List α
hu✝ : u ∈ sequence v
hu : List.Forall₂ (fun b a => b ∈ id a) u v
⊢ u ∈ {x | (fun y => s ∈ traverse 𝓝 y) x}
|
8c67666fd5977816
|
Primrec.vector_get
|
Mathlib/Computability/Primrec.lean
|
theorem vector_get {n} : Primrec₂ (@List.Vector.get α n) :=
option_some_iff.1 <|
(list_get?.comp (vector_toList.comp fst) (fin_val.comp snd)).of_eq fun a => by
rw [Vector.get_eq_get_toList, ← List.get?_eq_get]
rfl
|
α : Type u_1
inst✝ : Primcodable α
n : ℕ
a : List.Vector α n × Fin n
⊢ a.1.toList.get? ↑a.2 = some (a.1.get a.2)
|
rw [Vector.get_eq_get_toList, ← List.get?_eq_get]
|
α : Type u_1
inst✝ : Primcodable α
n : ℕ
a : List.Vector α n × Fin n
⊢ a.1.toList.get? ↑a.2 = a.1.toList.get? ↑(Fin.cast ⋯ a.2)
|
9cffdc77f2d561e0
|
circleIntegrable_sub_zpow_iff
|
Mathlib/MeasureTheory/Integral/CircleIntegral.lean
|
theorem circleIntegrable_sub_zpow_iff {c w : ℂ} {R : ℝ} {n : ℤ} :
CircleIntegrable (fun z => (z - w) ^ n) c R ↔ R = 0 ∨ 0 ≤ n ∨ w ∉ sphere c |R|
|
case mp.intro.intro.intro.intro
c : ℂ
R : ℝ
n : ℤ
hR : R ≠ 0
hn : n < 0
θ : ℝ
hθ : θ ∈ Ioc 0 (2 * π)
⊢ ¬IntervalIntegrable (fun θ_1 => (circleMap 0 R θ_1 * I) • (circleMap c R θ_1 - circleMap c R θ) ^ n) volume 0 (2 * π)
|
replace hθ : θ ∈ [[0, 2 * π]] := Icc_subset_uIcc (Ioc_subset_Icc_self hθ)
|
case mp.intro.intro.intro.intro
c : ℂ
R : ℝ
n : ℤ
hR : R ≠ 0
hn : n < 0
θ : ℝ
hθ : θ ∈ [[0, 2 * π]]
⊢ ¬IntervalIntegrable (fun θ_1 => (circleMap 0 R θ_1 * I) • (circleMap c R θ_1 - circleMap c R θ) ^ n) volume 0 (2 * π)
|
8d16963f19b22485
|
OreLocalization.smul_add
|
Mathlib/RingTheory/OreLocalization/Basic.lean
|
theorem smul_add (z : R[S⁻¹]) (x y : X[S⁻¹]) :
z • (x + y) = z • x + z • y
|
case c.c.c.mk.mk.intro
R : Type u_1
inst✝³ : Monoid R
S : Submonoid R
inst✝² : OreSet S
X : Type u_2
inst✝¹ : AddMonoid X
inst✝ : DistribMulAction R X
r₁ : X
s₁ : ↥S
r₂ : X
s₂ : ↥S
r₃ : R
s₃ : ↥S
ra : R
sa : ↥S
ha : ↑(sa * s₁) = ra * ↑s₂
⊢ oreNum r₃ (sa * s₁) • (sa • r₁ + ra • r₂) /ₒ (oreDenom r₃ (sa * s₁) * s₃) =
oreNum r₃ (sa * s₁) • sa • r₁ /ₒ (oreDenom r₃ (sa * s₁) * s₃) + (r₃ /ₒ s₃) • (ra • r₂ /ₒ (sa * s₁))
|
rw [oreDiv_smul_oreDiv]
|
case c.c.c.mk.mk.intro
R : Type u_1
inst✝³ : Monoid R
S : Submonoid R
inst✝² : OreSet S
X : Type u_2
inst✝¹ : AddMonoid X
inst✝ : DistribMulAction R X
r₁ : X
s₁ : ↥S
r₂ : X
s₂ : ↥S
r₃ : R
s₃ : ↥S
ra : R
sa : ↥S
ha : ↑(sa * s₁) = ra * ↑s₂
⊢ oreNum r₃ (sa * s₁) • (sa • r₁ + ra • r₂) /ₒ (oreDenom r₃ (sa * s₁) * s₃) =
oreNum r₃ (sa * s₁) • sa • r₁ /ₒ (oreDenom r₃ (sa * s₁) * s₃) +
oreNum r₃ (sa * s₁) • ra • r₂ /ₒ (oreDenom r₃ (sa * s₁) * s₃)
|
6684be5195a12015
|
HasProd.of_nat_of_neg_add_one
|
Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean
|
@[to_additive HasSum.of_nat_of_neg_add_one]
lemma HasProd.of_nat_of_neg_add_one {f : ℤ → M}
(hf₁ : HasProd (fun n : ℕ ↦ f n) m) (hf₂ : HasProd (fun n : ℕ ↦ f (-(n + 1))) m') :
HasProd f (m * m')
|
case disjoint
M : Type u_1
inst✝² : CommMonoid M
inst✝¹ : TopologicalSpace M
m m' : M
inst✝ : ContinuousMul M
f : ℤ → M
hf₁ : HasProd (fun n => f ↑n) m
hf₂ : HasProd (fun n => f (-(↑n + 1))) m'
hi₂ : Injective Int.negSucc
⊢ Disjoint (Set.range Nat.cast) (Set.range Int.negSucc)
|
rw [disjoint_iff_inf_le]
|
case disjoint
M : Type u_1
inst✝² : CommMonoid M
inst✝¹ : TopologicalSpace M
m m' : M
inst✝ : ContinuousMul M
f : ℤ → M
hf₁ : HasProd (fun n => f ↑n) m
hf₂ : HasProd (fun n => f (-(↑n + 1))) m'
hi₂ : Injective Int.negSucc
⊢ Set.range Nat.cast ⊓ Set.range Int.negSucc ≤ ⊥
|
edc6df3233be9c8f
|
MeasureTheory.tendstoInMeasure_of_tendsto_ae_of_stronglyMeasurable
|
Mathlib/MeasureTheory/Function/ConvergenceInMeasure.lean
|
theorem tendstoInMeasure_of_tendsto_ae_of_stronglyMeasurable [IsFiniteMeasure μ]
(hf : ∀ n, StronglyMeasurable (f n)) (hg : StronglyMeasurable g)
(hfg : ∀ᵐ x ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))) : TendstoInMeasure μ f atTop g
|
case neg.intro.intro.intro.intro
α : Type u_1
E : Type u_4
m : MeasurableSpace α
μ : Measure α
inst✝¹ : MetricSpace E
f : ℕ → α → E
g : α → E
inst✝ : IsFiniteMeasure μ
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ
hε : 0 < ε
δ : ℝ≥0
hδ : 0 < ↑δ
t : Set α
left✝ : MeasurableSet t
ht : μ t ≤ ↑δ
hunif : ∀ ε > 0, ∀ᶠ (n : ℕ) in atTop, ∀ x ∈ tᶜ, dist (g x) (f n x) < ε
⊢ ∃ N, ∀ n ≥ N, μ {x | ε ≤ dist (f n x) (g x)} ≤ ↑δ
|
obtain ⟨N, hN⟩ := eventually_atTop.1 (hunif ε hε)
|
case neg.intro.intro.intro.intro.intro
α : Type u_1
E : Type u_4
m : MeasurableSpace α
μ : Measure α
inst✝¹ : MetricSpace E
f : ℕ → α → E
g : α → E
inst✝ : IsFiniteMeasure μ
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ
hε : 0 < ε
δ : ℝ≥0
hδ : 0 < ↑δ
t : Set α
left✝ : MeasurableSet t
ht : μ t ≤ ↑δ
hunif : ∀ ε > 0, ∀ᶠ (n : ℕ) in atTop, ∀ x ∈ tᶜ, dist (g x) (f n x) < ε
N : ℕ
hN : ∀ b ≥ N, ∀ x ∈ tᶜ, dist (g x) (f b x) < ε
⊢ ∃ N, ∀ n ≥ N, μ {x | ε ≤ dist (f n x) (g x)} ≤ ↑δ
|
6928bb0aed59df15
|
Cardinal.add_lt_of_lt
|
Mathlib/SetTheory/Cardinal/Arithmetic.lean
|
theorem add_lt_of_lt {a b c : Cardinal} (hc : ℵ₀ ≤ c) (h1 : a < c) (h2 : b < c) : a + b < c :=
(add_le_add (le_max_left a b) (le_max_right a b)).trans_lt <|
(lt_or_le (max a b) ℵ₀).elim (fun h => (add_lt_aleph0 h h).trans_le hc) fun h => by
rw [add_eq_self h]; exact max_lt h1 h2
|
a b c : Cardinal.{u_1}
hc : ℵ₀ ≤ c
h1 : a < c
h2 : b < c
h : ℵ₀ ≤ a ⊔ b
⊢ a ⊔ b + a ⊔ b < c
|
rw [add_eq_self h]
|
a b c : Cardinal.{u_1}
hc : ℵ₀ ≤ c
h1 : a < c
h2 : b < c
h : ℵ₀ ≤ a ⊔ b
⊢ a ⊔ b < c
|
9be722ee8004ffbc
|
MeasureTheory.tendsto_of_integral_tendsto_of_monotone
|
Mathlib/MeasureTheory/Integral/Bochner.lean
|
/-- If a monotone sequence of functions has an upper bound and the sequence of integrals of these
functions tends to the integral of the upper bound, then the sequence of functions converges
almost everywhere to the upper bound. -/
lemma tendsto_of_integral_tendsto_of_monotone {μ : Measure α} {f : ℕ → α → ℝ} {F : α → ℝ}
(hf_int : ∀ n, Integrable (f n) μ) (hF_int : Integrable F μ)
(hf_tendsto : Tendsto (fun i ↦ ∫ a, f i a ∂μ) atTop (𝓝 (∫ a, F a ∂μ)))
(hf_mono : ∀ᵐ a ∂μ, Monotone (fun i ↦ f i a))
(hf_bound : ∀ᵐ a ∂μ, ∀ i, f i a ≤ F a) :
∀ᵐ a ∂μ, Tendsto (fun i ↦ f i a) atTop (𝓝 (F a))
|
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ
F : α → ℝ
hf_int : ∀ (n : ℕ), Integrable (f n) μ
hF_int : Integrable F μ
hf_tendsto : Tendsto (fun i => ∫ (a : α), f i a ∂μ) atTop (𝓝 (∫ (a : α), F a ∂μ))
hf_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i => f i a
hf_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤ F a
f' : ℕ → α → ℝ≥0∞ := fun n a => ENNReal.ofReal (f n a - f 0 a)
F' : α → ℝ≥0∞ := fun a => ENNReal.ofReal (F a - f 0 a)
hf'_int_eq : ∀ (i : ℕ), ∫⁻ (a : α), f' i a ∂μ = ENNReal.ofReal (∫ (a : α), f i a ∂μ - ∫ (a : α), f 0 a ∂μ)
⊢ ∫⁻ (a : α), F' a ∂μ = ENNReal.ofReal (∫ (a : α), F a ∂μ - ∫ (a : α), f 0 a ∂μ)
|
unfold F'
|
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ
F : α → ℝ
hf_int : ∀ (n : ℕ), Integrable (f n) μ
hF_int : Integrable F μ
hf_tendsto : Tendsto (fun i => ∫ (a : α), f i a ∂μ) atTop (𝓝 (∫ (a : α), F a ∂μ))
hf_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i => f i a
hf_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤ F a
f' : ℕ → α → ℝ≥0∞ := fun n a => ENNReal.ofReal (f n a - f 0 a)
F' : α → ℝ≥0∞ := fun a => ENNReal.ofReal (F a - f 0 a)
hf'_int_eq : ∀ (i : ℕ), ∫⁻ (a : α), f' i a ∂μ = ENNReal.ofReal (∫ (a : α), f i a ∂μ - ∫ (a : α), f 0 a ∂μ)
⊢ ∫⁻ (a : α), ENNReal.ofReal (F a - f 0 a) ∂μ = ENNReal.ofReal (∫ (a : α), F a ∂μ - ∫ (a : α), f 0 a ∂μ)
|
a54f15f3a6cf0f39
|
Equiv.Perm.alternatingGroup_le_of_index_le_two
|
Mathlib/GroupTheory/SpecificGroups/Alternating.lean
|
theorem alternatingGroup_le_of_index_le_two
{G : Subgroup (Equiv.Perm α)} (hG : G.index ≤ 2) :
alternatingGroup α ≤ G
|
case inr.inl
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Perm α)
hG : G.index ≤ 2
h✝ : G.index > 0
h : G.index = Nat.succ 0
⊢ alternatingGroup α ≤ G
|
exact index_eq_one.mp h ▸ le_top
|
no goals
|
c86d5a49b767e282
|
Nat.preimage_Ioc
|
Mathlib/Algebra/Order/Floor.lean
|
theorem preimage_Ioc {a b : α} (ha : 0 ≤ a) (hb : 0 ≤ b) :
(Nat.cast : ℕ → α) ⁻¹' Set.Ioc a b = Set.Ioc ⌊a⌋₊ ⌊b⌋₊
|
α : Type u_2
inst✝¹ : LinearOrderedSemiring α
inst✝ : FloorSemiring α
a b : α
ha : 0 ≤ a
hb : 0 ≤ b
⊢ Nat.cast ⁻¹' Ioc a b = Ioc ⌊a⌋₊ ⌊b⌋₊
|
ext
|
case h
α : Type u_2
inst✝¹ : LinearOrderedSemiring α
inst✝ : FloorSemiring α
a b : α
ha : 0 ≤ a
hb : 0 ≤ b
x✝ : ℕ
⊢ x✝ ∈ Nat.cast ⁻¹' Ioc a b ↔ x✝ ∈ Ioc ⌊a⌋₊ ⌊b⌋₊
|
d1f291a40be12756
|
EMetric.hausdorffEdist_triangle
|
Mathlib/Topology/MetricSpace/HausdorffDistance.lean
|
theorem hausdorffEdist_triangle : hausdorffEdist s u ≤ hausdorffEdist s t + hausdorffEdist t u
|
α : Type u
inst✝ : PseudoEMetricSpace α
s t u : Set α
⊢ (⨆ x ∈ s, infEdist x u) ⊔ ⨆ y ∈ u, infEdist y s ≤ hausdorffEdist s t + hausdorffEdist t u
|
simp only [sup_le_iff, iSup_le_iff]
|
α : Type u
inst✝ : PseudoEMetricSpace α
s t u : Set α
⊢ (∀ i ∈ s, infEdist i u ≤ hausdorffEdist s t + hausdorffEdist t u) ∧
∀ i ∈ u, infEdist i s ≤ hausdorffEdist s t + hausdorffEdist t u
|
91d661feb399f899
|
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
|
case inr.intro.intro.intro.intro.intro.intro.left.a.a
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
j : ↥J
⊢ ⁅↑j, z⁆ ∈ ⁅J, sSup ↑s'⁆
|
apply LieSubmodule.lie_mem_lie j.2
|
case inr.intro.intro.intro.intro.intro.intro.left.a.a
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
j : ↥J
⊢ z ∈ sSup ↑s'
|
9feaa310a91f7482
|
LaurentPolynomial.smeval_zero
|
Mathlib/Algebra/Polynomial/Laurent.lean
|
theorem smeval_zero : (0 : R[T;T⁻¹]).smeval x = (0 : S)
|
R : Type u_1
S : Type u_2
inst✝³ : Semiring R
inst✝² : AddCommMonoid S
inst✝¹ : SMulWithZero R S
inst✝ : Monoid S
x : Sˣ
⊢ smeval 0 x = 0
|
simp only [smeval_eq_sum, Finsupp.sum_zero_index]
|
no goals
|
117a0a2d2c710f52
|
FormalMultilinearSeries.ofScalars_series_injective
|
Mathlib/Analysis/Analytic/OfScalars.lean
|
theorem ofScalars_series_injective [Nontrivial E] : Function.Injective (ofScalars E (𝕜 := 𝕜))
|
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : Field 𝕜
inst✝⁴ : Ring E
inst✝³ : Algebra 𝕜 E
inst✝² : TopologicalSpace E
inst✝¹ : IsTopologicalRing E
inst✝ : Nontrivial E
a₁✝ a₂✝ : ℕ → 𝕜
h : ¬a₁✝ = a₂✝
⊢ ¬ofScalars E a₁✝ = ofScalars E a₂✝
|
simp_rw [FormalMultilinearSeries.ext_iff, ofScalars, ContinuousMultilinearMap.ext_iff,
ContinuousMultilinearMap.smul_apply]
|
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : Field 𝕜
inst✝⁴ : Ring E
inst✝³ : Algebra 𝕜 E
inst✝² : TopologicalSpace E
inst✝¹ : IsTopologicalRing E
inst✝ : Nontrivial E
a₁✝ a₂✝ : ℕ → 𝕜
h : ¬a₁✝ = a₂✝
⊢ ¬∀ (n : ℕ) (x : Fin n → E),
a₁✝ n • (ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 n E) x =
a₂✝ n • (ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 n E) x
|
006f2d7f0b972d3b
|
DirichletCharacter.summable_neg_log_one_sub_mul_prime_cpow
|
Mathlib/NumberTheory/LSeries/Nonvanishing.lean
|
/-- The logarithms of the Euler factors of a Dirichlet L-series form a summable sequence. -/
lemma summable_neg_log_one_sub_mul_prime_cpow {s : ℂ} (hs : 1 < s.re) :
Summable fun p : Nat.Primes ↦ -log (1 - χ p * (p : ℂ) ^ (-s))
|
N : ℕ
χ : DirichletCharacter ℂ N
s : ℂ
hs : 1 < s.re
p : Nat.Primes
⊢ ‖χ ↑↑p * ↑↑p ^ (-s)‖ ≤ ↑↑p ^ (-s).re
|
simpa only [norm_mul, norm_natCast_cpow_of_re_ne_zero _ <| re_neg_ne_zero_of_one_lt_re hs]
using mul_le_of_le_one_left (by positivity) (χ.norm_le_one _)
|
no goals
|
c65e3b63625e4ce9
|
Polynomial.natDegree_add_le_iff_left
|
Mathlib/Algebra/Polynomial/Degree/Lemmas.lean
|
theorem natDegree_add_le_iff_left {n : ℕ} (p q : R[X]) (qn : q.natDegree ≤ n) :
(p + q).natDegree ≤ n ↔ p.natDegree ≤ n
|
R : Type u
inst✝ : Semiring R
n : ℕ
p q : R[X]
qn : q.natDegree ≤ n
⊢ (p + q).natDegree ≤ n ↔ p.natDegree ≤ n
|
refine ⟨fun h => ?_, fun h => natDegree_add_le_of_degree_le h qn⟩
|
R : Type u
inst✝ : Semiring R
n : ℕ
p q : R[X]
qn : q.natDegree ≤ n
h : (p + q).natDegree ≤ n
⊢ p.natDegree ≤ n
|
581fd3e77b8887b0
|
AlgebraicGeometry.stalkwise_isLocalAtSource_of_respectsIso
|
Mathlib/AlgebraicGeometry/Morphisms/Constructors.lean
|
/-- If `P` respects isos, then `stalkwise P` is local at the source. -/
lemma stalkwise_isLocalAtSource_of_respectsIso (hP : RingHom.RespectsIso P) :
IsLocalAtSource (stalkwise P)
|
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hP : RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] => P
this : (stalkwise fun {R S} [CommRing R] [CommRing S] => P).RespectsIso := stalkwise_respectsIso hP
X Y : Scheme
f : X ⟶ Y
ι : Type u
U : ι → X.Opens
hU : iSup U = ⊤
hf : ∀ (i : ι), stalkwise (fun {R S} [CommRing R] [CommRing S] => P) ((U i).ι ≫ f)
x : ↑↑X.toPresheafedSpace
⊢ x ∈ iSup U
|
rw [hU]
|
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hP : RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] => P
this : (stalkwise fun {R S} [CommRing R] [CommRing S] => P).RespectsIso := stalkwise_respectsIso hP
X Y : Scheme
f : X ⟶ Y
ι : Type u
U : ι → X.Opens
hU : iSup U = ⊤
hf : ∀ (i : ι), stalkwise (fun {R S} [CommRing R] [CommRing S] => P) ((U i).ι ≫ f)
x : ↑↑X.toPresheafedSpace
⊢ x ∈ ⊤
|
6976c8cc8afa0527
|
exists_isIntegralCurve_of_isIntegralCurveOn
|
Mathlib/Geometry/Manifold/IntegralCurve/UniformTime.lean
|
/-- If there exists `ε > 0` such that the local integral curve at each point `x : M` is defined at
least on an open interval `Ioo (-ε) ε`, then every point on `M` has a global integral curve
passing through it.
See Lemma 9.15, [J.M. Lee (2012)][lee2012]. -/
lemma exists_isIntegralCurve_of_isIntegralCurveOn [BoundarylessManifold I M]
{v : (x : M) → TangentSpace I x}
(hv : ContMDiff I I.tangent 1 (fun x ↦ (⟨x, v x⟩ : TangentBundle I M)))
{ε : ℝ} (hε : 0 < ε) (h : ∀ x : M, ∃ γ : ℝ → M, γ 0 = x ∧ IsIntegralCurveOn γ v (Ioo (-ε) ε))
(x : M) : ∃ γ : ℝ → M, γ 0 = x ∧ IsIntegralCurve γ v
|
case intro.intro.intro.intro
E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
H : Type u_2
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : IsManifold I 1 M
inst✝¹ : T2Space M
inst✝ : BoundarylessManifold I M
v : (x : M) → TangentSpace I x
hv : ContMDiff I I.tangent 1 fun x => { proj := x, snd := v x }
ε : ℝ
hε : 0 < ε
h : ∀ (x : M), ∃ γ, γ 0 = x ∧ IsIntegralCurveOn γ v (Ioo (-ε) ε)
x : M
s : Set ℝ := {a | ∃ γ, γ 0 = x ∧ IsIntegralCurveOn γ v (Ioo (-a) a)}
hbdd : ∀ (x : ℝ), ∃ y ∈ s, x < y
a y : ℝ
hlt : a < y
γ : ℝ → M
hγ1 : γ 0 = x
hγ2 : IsIntegralCurveOn γ v (Ioo (-y) y)
⊢ ∃ γ, γ 0 = x ∧ IsIntegralCurveOn γ v (Ioo (-a) a)
|
exact ⟨γ, hγ1, hγ2.mono <| Ioo_subset_Ioo (neg_le_neg hlt.le) hlt.le⟩
|
no goals
|
bd02ea3fe1e0f263
|
CanonicallyOrderedAdd.list_prod_pos
|
Mathlib/Algebra/Order/BigOperators/Ring/List.lean
|
/-- A variant of `List.prod_pos` for `CanonicallyOrderedAdd`. -/
@[simp] lemma CanonicallyOrderedAdd.list_prod_pos {α : Type*}
[CommSemiring α] [PartialOrder α] [CanonicallyOrderedAdd α] [NoZeroDivisors α] [Nontrivial α] :
∀ {l : List α}, 0 < l.prod ↔ (∀ x ∈ l, (0 : α) < x)
| [] => by simp
| (x :: xs) => by simp_rw [List.prod_cons, List.forall_mem_cons, CanonicallyOrderedAdd.mul_pos,
list_prod_pos]
|
α : Type u_2
inst✝⁴ : CommSemiring α
inst✝³ : PartialOrder α
inst✝² : CanonicallyOrderedAdd α
inst✝¹ : NoZeroDivisors α
inst✝ : Nontrivial α
x : α
xs : List α
⊢ 0 < (x :: xs).prod ↔ ∀ (x_1 : α), x_1 ∈ x :: xs → 0 < x_1
|
simp_rw [List.prod_cons, List.forall_mem_cons, CanonicallyOrderedAdd.mul_pos,
list_prod_pos]
|
no goals
|
fd38ef0dd441caee
|
HomologicalComplex₂.D₁_D₂
|
Mathlib/Algebra/Homology/TotalComplex.lean
|
@[reassoc]
lemma D₁_D₂ (i₁₂ i₁₂' i₁₂'' : I₁₂) :
K.D₁ c₁₂ i₁₂ i₁₂' ≫ K.D₂ c₁₂ i₁₂' i₁₂'' = - K.D₂ c₁₂ i₁₂ i₁₂' ≫ K.D₁ c₁₂ i₁₂' i₁₂''
|
C : Type u_1
inst✝⁴ : Category.{u_5, u_1} C
inst✝³ : Preadditive C
I₁ : Type u_2
I₂ : Type u_3
I₁₂ : Type u_4
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K : HomologicalComplex₂ C c₁ c₂
c₁₂ : ComplexShape I₁₂
inst✝² : TotalComplexShape c₁ c₂ c₁₂
inst✝¹ : DecidableEq I₁₂
inst✝ : K.HasTotal c₁₂
i₁₂ i₁₂' i₁₂'' : I₁₂
⊢ K.D₁ c₁₂ i₁₂ i₁₂' ≫ K.D₂ c₁₂ i₁₂' i₁₂'' = -K.D₂ c₁₂ i₁₂ i₁₂' ≫ K.D₁ c₁₂ i₁₂' i₁₂''
|
simp
|
no goals
|
a22ae8c4124b91f0
|
CategoryTheory.Functor.pointwiseLeftKanExtension_desc_app
|
Mathlib/CategoryTheory/Functor/KanExtension/Pointwise.lean
|
@[simp]
lemma pointwiseLeftKanExtension_desc_app (G : D ⥤ H) (α : F ⟶ L ⋙ G) (Y : D) :
((pointwiseLeftKanExtension L F).descOfIsLeftKanExtension (pointwiseLeftKanExtensionUnit L F)
G α |>.app Y) = colimit.desc _ (costructuredArrowMapCocone L F G α Y)
|
C : Type u_1
D : Type u_2
H : Type u_3
inst✝³ : Category.{u_6, u_1} C
inst✝² : Category.{u_4, u_2} D
inst✝¹ : Category.{u_5, u_3} H
L : C ⥤ D
F : C ⥤ H
inst✝ : L.HasPointwiseLeftKanExtension F
G : D ⥤ H
α : F ⟶ L ⋙ G
Y : D
β : L.pointwiseLeftKanExtension F ⟶ G :=
{ app := fun Y => colimit.desc (CostructuredArrow.proj L Y ⋙ F) (L.costructuredArrowMapCocone F G α Y),
naturality := ⋯ }
⊢ ((L.pointwiseLeftKanExtension F).descOfIsLeftKanExtension (L.pointwiseLeftKanExtensionUnit F) G α).app Y =
colimit.desc (CostructuredArrow.proj L Y ⋙ F) (L.costructuredArrowMapCocone F G α Y)
|
have h : (pointwiseLeftKanExtension L F).descOfIsLeftKanExtension
(pointwiseLeftKanExtensionUnit L F) G α = β := by
apply hom_ext_of_isLeftKanExtension (α := pointwiseLeftKanExtensionUnit L F)
aesop
|
C : Type u_1
D : Type u_2
H : Type u_3
inst✝³ : Category.{u_6, u_1} C
inst✝² : Category.{u_4, u_2} D
inst✝¹ : Category.{u_5, u_3} H
L : C ⥤ D
F : C ⥤ H
inst✝ : L.HasPointwiseLeftKanExtension F
G : D ⥤ H
α : F ⟶ L ⋙ G
Y : D
β : L.pointwiseLeftKanExtension F ⟶ G :=
{ app := fun Y => colimit.desc (CostructuredArrow.proj L Y ⋙ F) (L.costructuredArrowMapCocone F G α Y),
naturality := ⋯ }
h : (L.pointwiseLeftKanExtension F).descOfIsLeftKanExtension (L.pointwiseLeftKanExtensionUnit F) G α = β
⊢ ((L.pointwiseLeftKanExtension F).descOfIsLeftKanExtension (L.pointwiseLeftKanExtensionUnit F) G α).app Y =
colimit.desc (CostructuredArrow.proj L Y ⋙ F) (L.costructuredArrowMapCocone F G α Y)
|
a3d14f6988b1bd2f
|
HasFPowerSeriesOnBall.fderiv
|
Mathlib/Analysis/Calculus/FDeriv/Analytic.lean
|
theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F]
(h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (fderiv 𝕜 f) p.derivSeries x r
|
case refine_2
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
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
f : E → F
x : E
inst✝ : CompleteSpace F
h : HasFPowerSeriesOnBall f p x r
z : E
hz : z ∈ EMetric.ball x r
⊢ (fun z => (continuousMultilinearCurryFin1 𝕜 E F) (p.changeOrigin (z - x) 1)) z = fderiv 𝕜 f z
|
dsimp only
|
case refine_2
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
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
f : E → F
x : E
inst✝ : CompleteSpace F
h : HasFPowerSeriesOnBall f p x r
z : E
hz : z ∈ EMetric.ball x r
⊢ (continuousMultilinearCurryFin1 𝕜 E F) (p.changeOrigin (z - x) 1) = fderiv 𝕜 f z
|
fd4e9439d16c4d4e
|
ProbabilityTheory.iCondIndepSets_iff
|
Mathlib/Probability/Independence/Conditional.lean
|
lemma iCondIndepSets_iff (π : ι → Set (Set Ω)) (hπ : ∀ i s (_hs : s ∈ π i), MeasurableSet s)
(μ : Measure Ω) [IsFiniteMeasure μ] :
iCondIndepSets m' hm' π μ ↔ ∀ (s : Finset ι) {f : ι → Set Ω} (_H : ∀ i, i ∈ s → f i ∈ π i),
μ⟦⋂ i ∈ s, f i | m'⟧ =ᵐ[μ] ∏ i ∈ s, (μ⟦f i | m'⟧)
|
Ω : Type u_1
ι : Type u_2
m' mΩ : MeasurableSpace Ω
inst✝¹ : StandardBorelSpace Ω
hm' : m' ≤ mΩ
π : ι → Set (Set Ω)
hπ : ∀ (i : ι), ∀ s ∈ π i, MeasurableSet s
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
h_eq' :
∀ (s : Finset ι) (f : ι → Set Ω),
(∀ i ∈ s, f i ∈ π i) →
∀ i ∈ s, (fun ω => (((condExpKernel μ m') ω) (f i)).toReal) =ᶠ[ae μ] μ[(f i).indicator fun ω => 1|m']
⊢ (∀ (s : Finset ι) {f : ι → Set Ω},
(∀ i ∈ s, f i ∈ π i) →
∀ᵐ (a : Ω) ∂μ.trim hm', ((condExpKernel μ m') a) (⋂ i ∈ s, f i) = ∏ i ∈ s, ((condExpKernel μ m') a) (f i)) ↔
∀ (s : Finset ι) {f : ι → Set Ω},
(∀ i ∈ s, f i ∈ π i) →
μ[(⋂ i ∈ s, f i).indicator fun ω => 1|m'] =ᶠ[ae μ] ∏ i ∈ s, μ[(f i).indicator fun ω => 1|m']
|
have h_eq : ∀ (s : Finset ι) (f : ι → Set Ω) (_H : ∀ i, i ∈ s → f i ∈ π i), ∀ᵐ ω ∂μ,
∀ i ∈ s, ENNReal.toReal (condExpKernel μ m' ω (f i)) = (μ⟦f i | m'⟧) ω := by
intros s f H
simp_rw [← Finset.mem_coe]
rw [ae_ball_iff (Finset.countable_toSet s)]
exact h_eq' s f H
|
Ω : Type u_1
ι : Type u_2
m' mΩ : MeasurableSpace Ω
inst✝¹ : StandardBorelSpace Ω
hm' : m' ≤ mΩ
π : ι → Set (Set Ω)
hπ : ∀ (i : ι), ∀ s ∈ π i, MeasurableSet s
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
h_eq' :
∀ (s : Finset ι) (f : ι → Set Ω),
(∀ i ∈ s, f i ∈ π i) →
∀ i ∈ s, (fun ω => (((condExpKernel μ m') ω) (f i)).toReal) =ᶠ[ae μ] μ[(f i).indicator fun ω => 1|m']
h_eq :
∀ (s : Finset ι) (f : ι → Set Ω),
(∀ i ∈ s, f i ∈ π i) →
∀ᵐ (ω : Ω) ∂μ, ∀ i ∈ s, (((condExpKernel μ m') ω) (f i)).toReal = (μ[(f i).indicator fun ω => 1|m']) ω
⊢ (∀ (s : Finset ι) {f : ι → Set Ω},
(∀ i ∈ s, f i ∈ π i) →
∀ᵐ (a : Ω) ∂μ.trim hm', ((condExpKernel μ m') a) (⋂ i ∈ s, f i) = ∏ i ∈ s, ((condExpKernel μ m') a) (f i)) ↔
∀ (s : Finset ι) {f : ι → Set Ω},
(∀ i ∈ s, f i ∈ π i) →
μ[(⋂ i ∈ s, f i).indicator fun ω => 1|m'] =ᶠ[ae μ] ∏ i ∈ s, μ[(f i).indicator fun ω => 1|m']
|
7769d795b81eae16
|
Std.DHashMap.Internal.Raw₀.toListModel_containsThenInsertIfNew
|
Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/WF.lean
|
theorem toListModel_containsThenInsertIfNew [BEq α] [Hashable α] [EquivBEq α] [LawfulHashable α]
{m : Raw₀ α β} (h : Raw.WFImp m.1) {a : α} {b : β a} :
Perm (toListModel (m.containsThenInsertIfNew a b).2.1.2)
(insertEntryIfNew a b (toListModel m.1.2))
|
α : Type u
β : α → Type v
inst✝³ : BEq α
inst✝² : Hashable α
inst✝¹ : EquivBEq α
inst✝ : LawfulHashable α
m : Raw₀ α β
h : Raw.WFImp m.val
a : α
b : β a
⊢ toListModel (m.containsThenInsertIfNew a b).snd.val.buckets ~ insertEntryIfNew a b (toListModel m.val.buckets)
|
rw [containsThenInsertIfNew_eq_insertIfNewₘ]
|
α : Type u
β : α → Type v
inst✝³ : BEq α
inst✝² : Hashable α
inst✝¹ : EquivBEq α
inst✝ : LawfulHashable α
m : Raw₀ α β
h : Raw.WFImp m.val
a : α
b : β a
⊢ toListModel (m.insertIfNewₘ a b).val.buckets ~ insertEntryIfNew a b (toListModel m.val.buckets)
|
93b483d3edaad7a9
|
measurableSet_eq_fun'
|
Mathlib/MeasureTheory/Group/Arithmetic.lean
|
@[measurability]
lemma measurableSet_eq_fun' {β : Type*} [AddCommMonoid β] [PartialOrder β]
[CanonicallyOrderedAdd β] [Sub β] [OrderedSub β]
{_ : MeasurableSpace β} [MeasurableSub₂ β] [MeasurableSingletonClass β]
{f g : α → β} (hf : Measurable f) (hg : Measurable g) :
MeasurableSet {x | f x = g x}
|
case h
α : Type u_3
m : MeasurableSpace α
β : Type u_5
inst✝⁶ : AddCommMonoid β
inst✝⁵ : PartialOrder β
inst✝⁴ : CanonicallyOrderedAdd β
inst✝³ : Sub β
inst✝² : OrderedSub β
x✝¹ : MeasurableSpace β
inst✝¹ : MeasurableSub₂ β
inst✝ : MeasurableSingletonClass β
f g : α → β
hf : Measurable f
hg : Measurable g
x✝ : α
⊢ f x✝ = g x✝ ↔ f x✝ ≤ g x✝ ∧ g x✝ ≤ f x✝
|
exact ⟨fun h ↦ ⟨h.le, h.symm.le⟩, fun h ↦ le_antisymm h.1 h.2⟩
|
no goals
|
c0e843245b8a108c
|
Nat.Prime.deficient_pow
|
Mathlib/NumberTheory/FactorisationProperties.lean
|
theorem Prime.deficient_pow (h : Prime n) : Deficient (n ^ m)
|
case inr
n m : ℕ
h : Prime n
h✝ : m > 0
h1 : (n ^ m).properDivisors = image (fun x => n ^ x) (range m)
⊢ (n ^ m).Deficient
|
have h2 : ∑ i ∈ image (fun x => n ^ x) (range m), i = ∑ i ∈ range m, n^i := by
rw [Finset.sum_image]
rintro x _ y _
apply pow_injective_of_not_isUnit h.not_unit <| Prime.ne_zero h
|
case inr
n m : ℕ
h : Prime n
h✝ : m > 0
h1 : (n ^ m).properDivisors = image (fun x => n ^ x) (range m)
h2 : ∑ i ∈ image (fun x => n ^ x) (range m), i = ∑ i ∈ range m, n ^ i
⊢ (n ^ m).Deficient
|
244ad8ad5eb9d7f5
|
Polynomial.Sequence.linearIndependent
|
Mathlib/Algebra/Polynomial/Sequence.lean
|
/-- Polynomials in a polynomial sequence are linearly independent. -/
lemma linearIndependent :
LinearIndependent R S := linearIndependent_iff'.mpr <| fun s g eqzero i hi ↦ by
by_cases hsupzero : s.sup (fun i ↦ (g i • S i).degree) = ⊥
· have le_sup := Finset.le_sup hi (f := fun i ↦ (g i • S i).degree)
exact (smul_eq_zero_iff_left (S.ne_zero i)).mp <| degree_eq_bot.mp (eq_bot_mono le_sup hsupzero)
have hpairwise : {i | i ∈ s ∧ g i • S i ≠ 0}.Pairwise (Ne on fun i ↦ (g i • S i).degree)
|
case pos
R : Type u_1
inst✝¹ : Ring R
S : Sequence R
inst✝ : NoZeroDivisors R
s : Finset ℕ
g : ℕ → R
eqzero : ∑ i ∈ s, g i • ↑S i = 0
i : ℕ
hi : i ∈ s
hsupzero : (s.sup fun i => (g i • ↑S i).degree) = ⊥
le_sup : (fun i => (g i • ↑S i).degree) i ≤ s.sup fun i => (g i • ↑S i).degree
⊢ g i = 0
|
exact (smul_eq_zero_iff_left (S.ne_zero i)).mp <| degree_eq_bot.mp (eq_bot_mono le_sup hsupzero)
|
no goals
|
760e019bda62fdf2
|
ContinuousSMul.of_basis_zero
|
Mathlib/Topology/Algebra/FilterBasis.lean
|
theorem _root_.ContinuousSMul.of_basis_zero {ι : Type*} [IsTopologicalRing R] [TopologicalSpace M]
[IsTopologicalAddGroup M] {p : ι → Prop} {b : ι → Set M} (h : HasBasis (𝓝 0) p b)
(hsmul : ∀ {i}, p i → ∃ V ∈ 𝓝 (0 : R), ∃ j, p j ∧ V • b j ⊆ b i)
(hsmul_left : ∀ (x₀ : R) {i}, p i → ∃ j, p j ∧ MapsTo (x₀ • ·) (b j) (b i))
(hsmul_right : ∀ (m₀ : M) {i}, p i → ∀ᶠ x in 𝓝 (0 : R), x • m₀ ∈ b i) : ContinuousSMul R M
|
case hmulleft
R : Type u_1
M : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : TopologicalSpace R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
ι : Type u_3
inst✝² : IsTopologicalRing R
inst✝¹ : TopologicalSpace M
inst✝ : IsTopologicalAddGroup M
p : ι → Prop
b : ι → Set M
h : (𝓝 0).HasBasis p b
hsmul : ∀ {i : ι}, p i → ∃ V ∈ 𝓝 0, ∃ j, p j ∧ V • b j ⊆ b i
hsmul_left : ∀ (x₀ : R) {i : ι}, p i → ∃ j, p j ∧ MapsTo (fun x => x₀ • x) (b j) (b i)
hsmul_right : ∀ (m₀ : M) {i : ι}, p i → ∀ᶠ (x : R) in 𝓝 0, x • m₀ ∈ b i
m₀ : M
i : ι
hi : p i
⊢ ∀ᶠ (x : R) in 𝓝 0, x • m₀ ∈ b i
|
exact hsmul_right m₀ hi
|
no goals
|
4f90af5932a68739
|
LinearIndependent.map_of_isPurelyInseparable_of_isSeparable
|
Mathlib/FieldTheory/PurelyInseparable/Tower.lean
|
theorem LinearIndependent.map_of_isPurelyInseparable_of_isSeparable [IsPurelyInseparable F E]
{ι : Type*} {v : ι → K} (hsep : ∀ i : ι, IsSeparable F (v i))
(h : LinearIndependent F v) : LinearIndependent E v
|
case neg
F : Type u
E : Type v
inst✝⁷ : Field F
inst✝⁶ : Field E
inst✝⁵ : Algebra F E
K : Type w
inst✝⁴ : Field K
inst✝³ : Algebra F K
inst✝² : Algebra E K
inst✝¹ : IsScalarTower F E K
inst✝ : IsPurelyInseparable F E
ι : Type u_1
v : ι → K
hsep : ∀ (i : ι), IsSeparable F (v i)
h : LinearIndependent F v
q : ℕ
h✝ : ExpChar F q
this✝ : ExpChar K q
l : ι →₀ E
hl : (Finsupp.linearCombination E v) l = 0
i✝ : ι
f : ι → ℕ
hf : ∀ (i : ι), l i ^ q ^ f i ∈ (algebraMap F E).range
n : ℕ := l.support.sup f
this : q ^ n ≠ 0
i : ι
hs : i ∉ l.support
⊢ l i ^ q ^ n ∈ (algebraMap F E).range
|
exact ⟨0, by rw [map_zero, Finsupp.not_mem_support_iff.1 hs, zero_pow this]⟩
|
no goals
|
e47247a3e0a1480f
|
AlgebraicGeometry.isLocallyNoetherian_of_isOpenImmersion
|
Mathlib/AlgebraicGeometry/Noetherian.lean
|
lemma isLocallyNoetherian_of_isOpenImmersion {Y : Scheme} (f : X ⟶ Y) [IsOpenImmersion f]
[IsLocallyNoetherian Y] : IsLocallyNoetherian X
|
X Y : Scheme
f : X ⟶ Y
inst✝¹ : IsOpenImmersion f
inst✝ : IsLocallyNoetherian Y
U : ↑X.affineOpens
V : ↑Y.affineOpens := ⟨f ''ᵁ ↑U, ⋯⟩
this : Scheme.Hom.opensRange f ⊓ ↑V = ↑V
⊢ Γ(Y, Scheme.Hom.opensRange f ⊓ f ''ᵁ ↑U) ≅ Γ(Y, ↑V)
|
rw [this]
|
no goals
|
2ec02523b8c91c80
|
IsPrimal.mul
|
Mathlib/Algebra/GroupWithZero/Divisibility.lean
|
theorem IsPrimal.mul {α} [CancelCommMonoidWithZero α] {m n : α}
(hm : IsPrimal m) (hn : IsPrimal n) : IsPrimal (m * n)
|
case inr.intro.intro.intro.intro.intro.intro
α : Type u_2
inst✝ : CancelCommMonoidWithZero α
n : α
hn : IsPrimal n
a₁ a₂ b c : α
hm : IsPrimal (a₁ * a₂)
h0 : a₁ * a₂ ≠ 0
h : a₁ * a₂ * n ∣ a₁ * b * (a₂ * c)
⊢ ∃ a₁_1 a₂_1, a₁_1 ∣ a₁ * b ∧ a₂_1 ∣ a₂ * c ∧ a₁ * a₂ * n = a₁_1 * a₂_1
|
rw [mul_mul_mul_comm, mul_dvd_mul_iff_left h0] at h
|
case inr.intro.intro.intro.intro.intro.intro
α : Type u_2
inst✝ : CancelCommMonoidWithZero α
n : α
hn : IsPrimal n
a₁ a₂ b c : α
hm : IsPrimal (a₁ * a₂)
h0 : a₁ * a₂ ≠ 0
h : n ∣ b * c
⊢ ∃ a₁_1 a₂_1, a₁_1 ∣ a₁ * b ∧ a₂_1 ∣ a₂ * c ∧ a₁ * a₂ * n = a₁_1 * a₂_1
|
ab5031a4a2a30aa7
|
Complex.exp_ofReal_mul_I_im
|
Mathlib/Data/Complex/Trigonometric.lean
|
theorem exp_ofReal_mul_I_im (x : ℝ) : (exp (x * I)).im = Real.sin x
|
x : ℝ
⊢ (cexp (↑x * I)).im = Real.sin x
|
simp [exp_mul_I, sin_ofReal_re]
|
no goals
|
b2d7c31527804ab0
|
IsLocalization.Away.mul
|
Mathlib/RingTheory/Localization/Away/Basic.lean
|
/-- Localizing the localization of `R` at `x` at the image of `y` is the same as localizing
`R` at `y * x`. See `IsLocalization.Away.mul'` for the `x * y` version. -/
lemma mul (T : Type*) [CommSemiring T] [Algebra S T]
[Algebra R T] [IsScalarTower R S T] (x y : R)
[IsLocalization.Away x S] [IsLocalization.Away (algebraMap R S y) T] :
IsLocalization.Away (y * x) T
|
case refine_3
R : Type u_1
inst✝⁸ : CommSemiring R
S : Type u_2
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
T : Type u_5
inst✝⁵ : CommSemiring T
inst✝⁴ : Algebra S T
inst✝³ : Algebra R T
inst✝² : IsScalarTower R S T
x y : R
inst✝¹ : Away x S
inst✝ : Away ((algebraMap R S) y) T
a b : R
h : (algebraMap S T) ((algebraMap R S) a) = (algebraMap R T) b
⊢ ∃ n, (y * x) ^ n * a = (y * x) ^ n * b
|
rw [IsScalarTower.algebraMap_apply R S T] at h
|
case refine_3
R : Type u_1
inst✝⁸ : CommSemiring R
S : Type u_2
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
T : Type u_5
inst✝⁵ : CommSemiring T
inst✝⁴ : Algebra S T
inst✝³ : Algebra R T
inst✝² : IsScalarTower R S T
x y : R
inst✝¹ : Away x S
inst✝ : Away ((algebraMap R S) y) T
a b : R
h : (algebraMap S T) ((algebraMap R S) a) = (algebraMap S T) ((algebraMap R S) b)
⊢ ∃ n, (y * x) ^ n * a = (y * x) ^ n * b
|
f779f7b45ac969ec
|
List.mergeSort_cons
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Sort/Lemmas.lean
|
theorem mergeSort_cons {le : α → α → Bool}
(trans : ∀ (a b c : α), le a b → le b c → le a c)
(total : ∀ (a b : α), le a b || le b a)
(a : α) (l : List α) :
∃ l₁ l₂, mergeSort (a :: l) le = l₁ ++ a :: l₂ ∧ mergeSort l le = l₁ ++ l₂ ∧
∀ b, b ∈ l₁ → !le a b
|
case intro.intro
α : Type u_1
le : α → α → Bool
trans : ∀ (a b c : α), le a b = true → le b c = true → le a c = true
total : ∀ (a b : α), (le a b || le b a) = true
a : α
l : List α
nd : (map (fun x => x.snd) (a :: l).zipIdx).Nodup
m₁ : (a, 0) ∈ (a :: l).zipIdx.mergeSort (zipIdxLE le)
l₁ l₂ : List (α × Nat)
h : (a :: l).zipIdx.mergeSort (zipIdxLE le) = l₁ ++ (a, 0) :: l₂
⊢ ∃ l₁ l₂,
map (fun x => x.fst) (((a, 0) :: l.zipIdx (0 + 1)).mergeSort (zipIdxLE le)) = l₁ ++ a :: l₂ ∧
l.mergeSort le = l₁ ++ l₂ ∧ ∀ (b : α), b ∈ l₁ → (!le a b) = true
|
have s := sorted_mergeSort (zipIdxLE_trans trans) (zipIdxLE_total total) ((a :: l).zipIdx)
|
case intro.intro
α : Type u_1
le : α → α → Bool
trans : ∀ (a b c : α), le a b = true → le b c = true → le a c = true
total : ∀ (a b : α), (le a b || le b a) = true
a : α
l : List α
nd : (map (fun x => x.snd) (a :: l).zipIdx).Nodup
m₁ : (a, 0) ∈ (a :: l).zipIdx.mergeSort (zipIdxLE le)
l₁ l₂ : List (α × Nat)
h : (a :: l).zipIdx.mergeSort (zipIdxLE le) = l₁ ++ (a, 0) :: l₂
s : Pairwise (fun a b => zipIdxLE le a b = true) ((a :: l).zipIdx.mergeSort (zipIdxLE le))
⊢ ∃ l₁ l₂,
map (fun x => x.fst) (((a, 0) :: l.zipIdx (0 + 1)).mergeSort (zipIdxLE le)) = l₁ ++ a :: l₂ ∧
l.mergeSort le = l₁ ++ l₂ ∧ ∀ (b : α), b ∈ l₁ → (!le a b) = true
|
eac9e2d7b83f8a69
|
Matrix.adjugate_diagonal
|
Mathlib/LinearAlgebra/Matrix/Adjugate.lean
|
theorem adjugate_diagonal (v : n → α) :
adjugate (diagonal v) = diagonal fun i => ∏ j ∈ Finset.univ.erase i, v j
|
case a.inr
n : Type v
α : Type w
inst✝² : DecidableEq n
inst✝¹ : Fintype n
inst✝ : CommRing α
v : n → α
i j : n
hij : i ≠ j
⊢ ((diagonal v).updateCol j (Pi.single i 1)).det = diagonal (fun i => ∏ j ∈ univ.erase i, v j) i j
|
rw [diagonal_apply_ne _ hij]
|
case a.inr
n : Type v
α : Type w
inst✝² : DecidableEq n
inst✝¹ : Fintype n
inst✝ : CommRing α
v : n → α
i j : n
hij : i ≠ j
⊢ ((diagonal v).updateCol j (Pi.single i 1)).det = 0
|
d170811fd1d837fa
|
Matrix.eval_matrixOfPolynomials_eq_vandermonde_mul_matrixOfPolynomials
|
Mathlib/LinearAlgebra/Vandermonde.lean
|
theorem eval_matrixOfPolynomials_eq_vandermonde_mul_matrixOfPolynomials {n : ℕ}
(v : Fin n → R) (p : Fin n → R[X]) (h_deg : ∀ i, (p i).natDegree ≤ i) :
Matrix.of (fun i j => ((p j).eval (v i))) =
(Matrix.vandermonde v) * (Matrix.of (fun (i j : Fin n) => (p j).coeff i))
|
case a
R : Type u_1
inst✝ : CommRing R
n : ℕ
v : Fin n → R
p : Fin n → R[X]
h_deg : ∀ (i : Fin n), (p i).natDegree ≤ ↑i
i j : Fin n
this : (p j).support ⊆ range n
⊢ ∑ i_1 : Fin n, (RingHom.id R) ((p j).coeff ↑i_1) * v i ^ ↑i_1 =
∑ x : Fin n, of (fun i j => v i ^ ↑j) i x * of (fun i j => (p j).coeff ↑i) x j
|
congr
|
case a.e_f
R : Type u_1
inst✝ : CommRing R
n : ℕ
v : Fin n → R
p : Fin n → R[X]
h_deg : ∀ (i : Fin n), (p i).natDegree ≤ ↑i
i j : Fin n
this : (p j).support ⊆ range n
⊢ (fun i_1 => (RingHom.id R) ((p j).coeff ↑i_1) * v i ^ ↑i_1) = fun x =>
of (fun i j => v i ^ ↑j) i x * of (fun i j => (p j).coeff ↑i) x j
|
9ae3c6287bae2d19
|
Topology.IsConstructible.image_of_isClosedEmbedding
|
Mathlib/Topology/Constructible.lean
|
@[stacks 09YG]
lemma IsConstructible.image_of_isClosedEmbedding (hf : IsClosedEmbedding f)
(hfcomp : IsRetrocompact (range f)ᶜ) (hs : IsConstructible s) : IsConstructible (f '' s)
|
X : Type u_2
Y : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
s : Set X
hf : IsClosedEmbedding f
hfcomp : IsRetrocompact (range f)ᶜ
U : Set X
hUopen : IsOpen U
hUcomp : IsRetrocompact U
hfU : IsOpen (f '' U ∪ (range f)ᶜ)
h : IsRetrocompact (f '' U ∪ (range f)ᶜ)
⊢ IsConstructible (f '' U)
|
simpa [union_inter_distrib_right, inter_eq_left.2 (image_subset_range ..)]
using (h.isConstructible hfU).sdiff (hfcomp.isConstructible hf.isClosed_range.isOpen_compl)
|
no goals
|
98bccc08ab186c11
|
LinearMap.det_pi
|
Mathlib/LinearAlgebra/Determinant.lean
|
theorem det_pi [Module.Free R M] [Module.Finite R M] (f : ι → M →ₗ[R] M) :
(LinearMap.pi (fun i ↦ (f i).comp (LinearMap.proj i))).det = ∏ i, (f i).det
|
case a.mk.mk
R : Type u_1
inst✝⁵ : CommRing R
M : Type u_2
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
ι : Type u_4
inst✝² : Fintype ι
inst✝¹ : Module.Free R M
inst✝ : Module.Finite R M
f : ι → M →ₗ[R] M
b : Basis (Module.Free.ChooseBasisIndex R M) R M := Module.Free.chooseBasis R M
B : Basis (Module.Free.ChooseBasisIndex R M × ι) R (ι → M) :=
(Pi.basis fun x => b).reindex
((Equiv.sigmaEquivProd ι (Module.Free.ChooseBasisIndex R M)).trans
(Equiv.prodComm ι (Module.Free.ChooseBasisIndex R M)))
i₁ : Module.Free.ChooseBasisIndex R M
i₂ : ι
j₁ : Module.Free.ChooseBasisIndex R M
j₂ : ι
⊢ (b.repr ((f i₂) (if i₂ = j₂ then b j₁ else 0))) i₁ = if i₂ = j₂ then (b.repr ((f i₂) (b j₁))) i₁ else 0
|
split_ifs with h
|
case pos
R : Type u_1
inst✝⁵ : CommRing R
M : Type u_2
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
ι : Type u_4
inst✝² : Fintype ι
inst✝¹ : Module.Free R M
inst✝ : Module.Finite R M
f : ι → M →ₗ[R] M
b : Basis (Module.Free.ChooseBasisIndex R M) R M := Module.Free.chooseBasis R M
B : Basis (Module.Free.ChooseBasisIndex R M × ι) R (ι → M) :=
(Pi.basis fun x => b).reindex
((Equiv.sigmaEquivProd ι (Module.Free.ChooseBasisIndex R M)).trans
(Equiv.prodComm ι (Module.Free.ChooseBasisIndex R M)))
i₁ : Module.Free.ChooseBasisIndex R M
i₂ : ι
j₁ : Module.Free.ChooseBasisIndex R M
j₂ : ι
h : i₂ = j₂
⊢ (b.repr ((f i₂) (b j₁))) i₁ = (b.repr ((f i₂) (b j₁))) i₁
case neg
R : Type u_1
inst✝⁵ : CommRing R
M : Type u_2
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
ι : Type u_4
inst✝² : Fintype ι
inst✝¹ : Module.Free R M
inst✝ : Module.Finite R M
f : ι → M →ₗ[R] M
b : Basis (Module.Free.ChooseBasisIndex R M) R M := Module.Free.chooseBasis R M
B : Basis (Module.Free.ChooseBasisIndex R M × ι) R (ι → M) :=
(Pi.basis fun x => b).reindex
((Equiv.sigmaEquivProd ι (Module.Free.ChooseBasisIndex R M)).trans
(Equiv.prodComm ι (Module.Free.ChooseBasisIndex R M)))
i₁ : Module.Free.ChooseBasisIndex R M
i₂ : ι
j₁ : Module.Free.ChooseBasisIndex R M
j₂ : ι
h : ¬i₂ = j₂
⊢ (b.repr ((f i₂) 0)) i₁ = 0
|
7d793b62306d1e2f
|
Mon_.whiskerLeft_hom
|
Mathlib/CategoryTheory/Monoidal/Mon_.lean
|
theorem whiskerLeft_hom {X Y : Mon_ C} (f : X ⟶ Y) (Z : Mon_ C) :
(f ▷ Z).hom = f.hom ▷ Z.X
|
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
X Y : Mon_ C
f : X ⟶ Y
Z : Mon_ C
⊢ (f ▷ Z).hom = f.hom ▷ Z.X
|
rw [← tensorHom_id]
|
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
X Y : Mon_ C
f : X ⟶ Y
Z : Mon_ C
⊢ (f ▷ Z).hom = f.hom ⊗ 𝟙 Z.X
|
e932038810eafaac
|
cross_anticomm'
|
Mathlib/LinearAlgebra/CrossProduct.lean
|
theorem cross_anticomm' (v w : Fin 3 → R) : v ×₃ w + w ×₃ v = 0
|
R : Type u_1
inst✝ : CommRing R
v w : Fin 3 → R
⊢ (crossProduct v) w + (crossProduct w) v = 0
|
rw [add_eq_zero_iff_eq_neg, cross_anticomm]
|
no goals
|
e8c4b0a2397903c5
|
Finset.card_div_choose_le_card_shadow_div_choose
|
Mathlib/Combinatorics/SetFamily/LYM.lean
|
theorem card_div_choose_le_card_shadow_div_choose (hr : r ≠ 0)
(h𝒜 : (𝒜 : Set (Finset α)).Sized r) : (#𝒜 : 𝕜) / (Fintype.card α).choose r
≤ #(∂ 𝒜) / (Fintype.card α).choose (r - 1)
|
case inr
𝕜 : Type u_1
α : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 : Finset (Finset α)
r : ℕ
hr : r ≠ 0
hr' : r ≤ Fintype.card α
h𝒜 : #𝒜 * r ≤ #(∂ 𝒜) * (Fintype.card α - r + 1)
⊢ ↑(#𝒜) / ↑((Fintype.card α).choose r) ≤ ↑(#(∂ 𝒜)) / ↑((Fintype.card α).choose (r - 1))
|
rw [div_le_div_iff₀] <;> norm_cast
|
case inr
𝕜 : Type u_1
α : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 : Finset (Finset α)
r : ℕ
hr : r ≠ 0
hr' : r ≤ Fintype.card α
h𝒜 : #𝒜 * r ≤ #(∂ 𝒜) * (Fintype.card α - r + 1)
⊢ #𝒜 * (Fintype.card α).choose (r - 1) ≤ #(∂ 𝒜) * (Fintype.card α).choose r
case inr.hb
𝕜 : Type u_1
α : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 : Finset (Finset α)
r : ℕ
hr : r ≠ 0
hr' : r ≤ Fintype.card α
h𝒜 : #𝒜 * r ≤ #(∂ 𝒜) * (Fintype.card α - r + 1)
⊢ 0 < (Fintype.card α).choose r
case inr.hd
𝕜 : Type u_1
α : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 : Finset (Finset α)
r : ℕ
hr : r ≠ 0
hr' : r ≤ Fintype.card α
h𝒜 : #𝒜 * r ≤ #(∂ 𝒜) * (Fintype.card α - r + 1)
⊢ 0 < (Fintype.card α).choose (r - 1)
|
39fd5f73c0c4b048
|
DFinsupp.comul_comp_lapply
|
Mathlib/RingTheory/Coalgebra/Basic.lean
|
theorem comul_comp_lapply (i : ι) :
comul ∘ₗ (lapply i : _ →ₗ[R] A i) = TensorProduct.map (lapply i) (lapply i) ∘ₗ comul
|
case h.inr
R : Type u
ι : Type v
A : ι → Type w
inst✝⁴ : DecidableEq ι
inst✝³ : CommSemiring R
inst✝² : (i : ι) → AddCommMonoid (A i)
inst✝¹ : (i : ι) → Module R (A i)
inst✝ : (i : ι) → Coalgebra R (A i)
i j : ι
hij : i ≠ j
⊢ (comul ∘ₗ lapply i) ∘ₗ lsingle j = TensorProduct.map (lapply i ∘ₗ lsingle j) (lapply i ∘ₗ lsingle j) ∘ₗ comul
|
rw [comp_assoc, lapply_comp_lsingle_of_ne _ _ hij, comp_zero, TensorProduct.map_zero_left,
zero_comp]
|
no goals
|
65bcafb1b82fb31c
|
Real.rpow_add_rpow_le
|
Mathlib/Analysis/MeanInequalitiesPow.lean
|
lemma rpow_add_rpow_le {p q : ℝ} {a b : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b) (hp_pos : 0 < p)
(hpq : p ≤ q) :
(a ^ q + b ^ q) ^ (1 / q) ≤ (a ^ p + b ^ p) ^ (1 / p)
|
case intro.intro
p q : ℝ
hp_pos : 0 < p
hpq : p ≤ q
a b : ℝ≥0
⊢ (↑a ^ q + ↑b ^ q) ^ (1 / q) ≤ (↑a ^ p + ↑b ^ p) ^ (1 / p)
|
exact_mod_cast NNReal.rpow_add_rpow_le a b hp_pos hpq
|
no goals
|
078c357d2c7ecf53
|
Matrix.coeff_charpolyRev_eq_neg_trace
|
Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean
|
@[simp] lemma coeff_charpolyRev_eq_neg_trace (M : Matrix n n R) :
coeff M.charpolyRev 1 = - trace M
|
case inl
R : Type u
inst✝² : CommRing R
n : Type v
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
a✝ : Nontrivial R
h✝ : IsEmpty n
⊢ M.charpolyRev.coeff 1 = -M.trace
|
simp [charpolyRev, coeff_one]
|
no goals
|
d1ce8fd4828bc34f
|
infinite_sum
|
Mathlib/Data/Fintype/Sum.lean
|
theorem infinite_sum : Infinite (α ⊕ β) ↔ Infinite α ∨ Infinite β
|
α : Type u_1
β : Type u_2
H : ¬Infinite α ∧ ¬Infinite β
this✝ : Fintype α
this : Fintype β
⊢ ¬Infinite (α ⊕ β)
|
exact Infinite.false
|
no goals
|
f9ba300294526652
|
AddCircle.continuousAt_equivIoc
|
Mathlib/Topology/Instances/AddCircle.lean
|
theorem continuousAt_equivIoc (hx : x ≠ a) : ContinuousAt (equivIoc p a) x
|
case H
𝕜 : Type u_1
inst✝³ : LinearOrderedAddCommGroup 𝕜
p : 𝕜
hp : Fact (0 < p)
a : 𝕜
inst✝² : Archimedean 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
x : AddCircle p
z✝ : 𝕜
hx : ↑z✝ ≠ ↑a
⊢ Filter.map (⇑(equivIoc p a) ∘ QuotientAddGroup.mk) (𝓝 z✝) ≤ 𝓝 ((equivIoc p a) ↑z✝)
|
exact (continuousAt_toIocMod hp.out a hx).codRestrict _
|
no goals
|
d8a2df19a895e89d
|
AddCircle.ae_empty_or_univ_of_forall_vadd_ae_eq_self
|
Mathlib/Dynamics/Ergodic/AddCircle.lean
|
theorem ae_empty_or_univ_of_forall_vadd_ae_eq_self {s : Set <| AddCircle T}
(hs : NullMeasurableSet s volume) {ι : Type*} {l : Filter ι} [l.NeBot] {u : ι → AddCircle T}
(hu₁ : ∀ i, (u i +ᵥ s : Set _) =ᵐ[volume] s) (hu₂ : Tendsto (addOrderOf ∘ u) l atTop) :
s =ᵐ[volume] (∅ : Set <| AddCircle T) ∨ s =ᵐ[volume] univ
|
case inr.h
T : ℝ
hT : Fact (0 < T)
s : Set (AddCircle T)
ι : Type u_1
l : Filter ι
inst✝ : l.NeBot
u : ι → AddCircle T
μ : Measure (AddCircle T) := volume
hs : NullMeasurableSet s μ
hu₁ : ∀ (i : ι), u i +ᵥ s =ᶠ[ae μ] s
n : ι → ℕ := addOrderOf ∘ u
hu₂ : Tendsto n l atTop
hT₀ : 0 < T
hT₁ : ENNReal.ofReal T ≠ 0
h : μ s ≠ 0
⊢ μ s = ENNReal.ofReal T
|
obtain ⟨d, -, hd⟩ : ∃ d, d ∈ s ∧ ∀ {ι'} {l : Filter ι'} (w : ι' → AddCircle T) (δ : ι' → ℝ),
Tendsto δ l (𝓝[>] 0) → (∀ᶠ j in l, d ∈ closedBall (w j) (1 * δ j)) →
Tendsto (fun j => μ (s ∩ closedBall (w j) (δ j)) / μ (closedBall (w j) (δ j))) l (𝓝 1) :=
exists_mem_of_measure_ne_zero_of_ae h
(IsUnifLocDoublingMeasure.ae_tendsto_measure_inter_div μ s 1)
|
case inr.h.intro.intro
T : ℝ
hT : Fact (0 < T)
s : Set (AddCircle T)
ι : Type u_1
l : Filter ι
inst✝ : l.NeBot
u : ι → AddCircle T
μ : Measure (AddCircle T) := volume
hs : NullMeasurableSet s μ
hu₁ : ∀ (i : ι), u i +ᵥ s =ᶠ[ae μ] s
n : ι → ℕ := addOrderOf ∘ u
hu₂ : Tendsto n l atTop
hT₀ : 0 < T
hT₁ : ENNReal.ofReal T ≠ 0
h : μ s ≠ 0
d : AddCircle T
hd :
∀ {ι' : Type ?u.6517} {l : Filter ι'} (w : ι' → AddCircle T) (δ : ι' → ℝ),
Tendsto δ l (𝓝[>] 0) →
(∀ᶠ (j : ι') in l, d ∈ closedBall (w j) (1 * δ j)) →
Tendsto (fun j => μ (s ∩ closedBall (w j) (δ j)) / μ (closedBall (w j) (δ j))) l (𝓝 1)
⊢ μ s = ENNReal.ofReal T
|
caa1d183a972f27c
|
Submodule.LinearDisjoint.of_basis_right'
|
Mathlib/LinearAlgebra/LinearDisjoint.lean
|
theorem of_basis_right' {ι : Type*} (n : Basis ι R N)
(H : Function.Injective (mulRightMap M n)) : M.LinearDisjoint N
|
R : Type u
S : Type v
inst✝² : CommSemiring R
inst✝¹ : Semiring S
inst✝ : Algebra R S
M N : Submodule R S
ι : Type u_1
n : Basis ι R ↥N
H : Function.Injective ⇑(M.mulRightMap ⇑n)
⊢ M.LinearDisjoint N
|
simp_rw [mulRightMap_eq_mulMap_comp, ← Basis.coe_repr_symm,
← LinearEquiv.coe_lTensor, LinearEquiv.comp_coe, LinearMap.coe_comp,
LinearEquiv.coe_coe, EquivLike.injective_comp] at H
|
R : Type u
S : Type v
inst✝² : CommSemiring R
inst✝¹ : Semiring S
inst✝ : Algebra R S
M N : Submodule R S
ι : Type u_1
n : Basis ι R ↥N
H : Function.Injective ⇑(M.mulMap N)
⊢ M.LinearDisjoint N
|
515c1bdb72ddad0c
|
balancedCoreAux_balanced
|
Mathlib/Analysis/LocallyConvex/BalancedCoreHull.lean
|
theorem balancedCoreAux_balanced (h0 : (0 : E) ∈ balancedCoreAux 𝕜 s) :
Balanced 𝕜 (balancedCoreAux 𝕜 s)
|
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedDivisionRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s : Set E
h0 : 0 ∈ balancedCoreAux 𝕜 s
a : 𝕜
ha : ‖a‖ ≤ 1
y : E
hy : ∀ (r : 𝕜), 1 ≤ ‖r‖ → y ∈ r • s
h : a ≠ 0
r : 𝕜
hr : 1 ≤ ‖r‖
⊢ 1 ≤ ‖a‖⁻¹ * ‖r‖
|
exact one_le_mul_of_one_le_of_one_le ((one_le_inv₀ (norm_pos_iff.mpr h)).2 ha) hr
|
no goals
|
d8dfb328ccf0ac15
|
Equiv.Perm.zpow_apply_eq_of_apply_apply_eq_self
|
Mathlib/GroupTheory/Perm/Support.lean
|
theorem zpow_apply_eq_of_apply_apply_eq_self {x : α} (hffx : f (f x) = x) :
∀ i : ℤ, (f ^ i) x = x ∨ (f ^ i) x = f x
| (n : ℕ) => pow_apply_eq_of_apply_apply_eq_self hffx n
| Int.negSucc n => by
rw [zpow_negSucc, inv_eq_iff_eq, ← f.injective.eq_iff, ← mul_apply, ← pow_succ', eq_comm,
inv_eq_iff_eq, ← mul_apply, ← pow_succ, @eq_comm _ x, or_comm]
exact pow_apply_eq_of_apply_apply_eq_self hffx _
|
α : Type u_1
f : Perm α
x : α
hffx : f (f x) = x
n : ℕ
⊢ (f ^ Int.negSucc n) x = x ∨ (f ^ Int.negSucc n) x = f x
|
rw [zpow_negSucc, inv_eq_iff_eq, ← f.injective.eq_iff, ← mul_apply, ← pow_succ', eq_comm,
inv_eq_iff_eq, ← mul_apply, ← pow_succ, @eq_comm _ x, or_comm]
|
α : Type u_1
f : Perm α
x : α
hffx : f (f x) = x
n : ℕ
⊢ (f ^ (n + 1 + 1)) x = x ∨ (f ^ (n + 1 + 1)) x = f x
|
3a8c355114bd8eaa
|
Batteries.RBNode.balance1_toList
|
Mathlib/.lake/packages/batteries/Batteries/Data/RBMap/Lemmas.lean
|
theorem balance1_toList {l : RBNode α} {v r} :
(l.balance1 v r).toList = l.toList ++ v :: r.toList
|
α : Type u_1
l : RBNode α
v : α
r : RBNode α
⊢ (l.balance1 v r).toList = l.toList ++ v :: r.toList
|
unfold balance1
|
α : Type u_1
l : RBNode α
v : α
r : RBNode α
⊢ (match l, v, r with
| node red (node red a x b) y c, z, d => node red (node black a x b) y (node black c z d)
| node red a x (node red b y c), z, d => node red (node black a x b) y (node black c z d)
| a, x, b => node black a x b).toList =
l.toList ++ v :: r.toList
|
a86912f9c8139c07
|
Profinite.exists_locallyConstant_finite_aux
|
Mathlib/Topology/Category/Profinite/CofilteredLimit.lean
|
theorem exists_locallyConstant_finite_aux {α : Type*} [Finite α] (hC : IsLimit C)
(f : LocallyConstant C.pt α) : ∃ (j : J) (g : LocallyConstant (F.obj j) (α → Fin 2)),
(f.map fun a b => if a = b then (0 : Fin 2) else 1) = g.comap (C.π.app _).hom
|
case intro.intro
J : Type v
inst✝² : SmallCategory J
inst✝¹ : IsCofiltered J
F : J ⥤ Profinite
C : Cone F
α : Type u_1
inst✝ : Finite α
hC : IsLimit C
f : LocallyConstant (↑C.pt.toTop) α
val✝ : Fintype α
ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1
ff : α → LocallyConstant (↑C.pt.toTop) (Fin 2) := (LocallyConstant.map ι f).flip
j : α → J
g : (a : α) → LocallyConstant (↑(F.obj (j a)).toTop) (Fin 2)
h : ∀ (a : α), ff a = LocallyConstant.comap (TopCat.Hom.hom (C.π.app (j a))) (g a)
G : Finset J := Finset.image j Finset.univ
j0 : J
hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X)
⊢ ∃ j g,
LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (TopCat.Hom.hom (C.π.app j)) g
|
have hj : ∀ a, j a ∈ (Finset.univ.image j : Finset J) := by
intro a
simp only [Finset.mem_image, Finset.mem_univ, true_and, exists_apply_eq_apply]
|
case intro.intro
J : Type v
inst✝² : SmallCategory J
inst✝¹ : IsCofiltered J
F : J ⥤ Profinite
C : Cone F
α : Type u_1
inst✝ : Finite α
hC : IsLimit C
f : LocallyConstant (↑C.pt.toTop) α
val✝ : Fintype α
ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1
ff : α → LocallyConstant (↑C.pt.toTop) (Fin 2) := (LocallyConstant.map ι f).flip
j : α → J
g : (a : α) → LocallyConstant (↑(F.obj (j a)).toTop) (Fin 2)
h : ∀ (a : α), ff a = LocallyConstant.comap (TopCat.Hom.hom (C.π.app (j a))) (g a)
G : Finset J := Finset.image j Finset.univ
j0 : J
hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X)
hj : ∀ (a : α), j a ∈ Finset.image j Finset.univ
⊢ ∃ j g,
LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (TopCat.Hom.hom (C.π.app j)) g
|
ad55d9835f0fad2e
|
MeasureTheory.maximal_ineq
|
Mathlib/Probability/Martingale/OptionalStopping.lean
|
theorem maximal_ineq [IsFiniteMeasure μ] (hsub : Submartingale f 𝒢 μ) (hnonneg : 0 ≤ f) {ε : ℝ≥0}
(n : ℕ) : ε • μ {ω | (ε : ℝ) ≤ (range (n + 1)).sup' nonempty_range_succ fun k => f k ω} ≤
ENNReal.ofReal (∫ ω in {ω | (ε : ℝ) ≤ (range (n + 1)).sup' nonempty_range_succ fun k => f k ω},
f n ω ∂μ)
|
Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
𝒢 : Filtration ℕ m0
f : ℕ → Ω → ℝ
inst✝ : IsFiniteMeasure μ
hsub : Submartingale f 𝒢 μ
hnonneg : 0 ≤ f
ε : ℝ≥0
n : ℕ
this :
ε • μ {ω | ↑ε ≤ (range (n + 1)).sup' ⋯ fun k => f k ω} +
ENNReal.ofReal (∫ (ω : Ω) in {ω | ((range (n + 1)).sup' ⋯ fun k => f k ω) < ↑ε}, f n ω ∂μ) ≤
ENNReal.ofReal (∫ (x : Ω), f n x ∂μ)
⊢ ENNReal.ofReal (∫ (ω : Ω), f n ω ∂μ) =
ENNReal.ofReal
(∫ (x : Ω) in
{ω | ↑ε ≤ (range (n + 1)).sup' ⋯ fun k => f k ω} ∪ {ω | ((range (n + 1)).sup' ⋯ fun k => f k ω) < ↑ε}, f n x ∂μ)
|
rw [← setIntegral_univ]
|
Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
𝒢 : Filtration ℕ m0
f : ℕ → Ω → ℝ
inst✝ : IsFiniteMeasure μ
hsub : Submartingale f 𝒢 μ
hnonneg : 0 ≤ f
ε : ℝ≥0
n : ℕ
this :
ε • μ {ω | ↑ε ≤ (range (n + 1)).sup' ⋯ fun k => f k ω} +
ENNReal.ofReal (∫ (ω : Ω) in {ω | ((range (n + 1)).sup' ⋯ fun k => f k ω) < ↑ε}, f n ω ∂μ) ≤
ENNReal.ofReal (∫ (x : Ω), f n x ∂μ)
⊢ ENNReal.ofReal (∫ (x : Ω) in Set.univ, f n x ∂μ) =
ENNReal.ofReal
(∫ (x : Ω) in
{ω | ↑ε ≤ (range (n + 1)).sup' ⋯ fun k => f k ω} ∪ {ω | ((range (n + 1)).sup' ⋯ fun k => f k ω) < ↑ε}, f n x ∂μ)
|
7389c69afd10c321
|
DiscreteQuotient.map_ofLE
|
Mathlib/Topology/DiscreteQuotient.lean
|
theorem map_ofLE (cond : LEComap f A B) (h : A' ≤ A) (c : A') :
map f cond (ofLE h c) = map f (cond.mono h le_rfl) c
|
X : Type u_2
Y : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : C(X, Y)
A A' : DiscreteQuotient X
B : DiscreteQuotient Y
cond : LEComap f A B
h : A' ≤ A
c : Quotient A'.toSetoid
⊢ map f cond (ofLE h c) = map f ⋯ c
|
rcases c with ⟨⟩
|
case mk
X : Type u_2
Y : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : C(X, Y)
A A' : DiscreteQuotient X
B : DiscreteQuotient Y
cond : LEComap f A B
h : A' ≤ A
c : Quotient A'.toSetoid
a✝ : X
⊢ map f cond (ofLE h (Quot.mk (⇑A'.toSetoid) a✝)) = map f ⋯ (Quot.mk (⇑A'.toSetoid) a✝)
|
639aa028a2f2cc81
|
Profinite.NobelingProof.GoodProducts.maxTail_isGood
|
Mathlib/Topology/Category/Profinite/Nobeling.lean
|
theorem maxTail_isGood (l : MaxProducts C ho)
(h₁ : ⊤ ≤ Submodule.span ℤ (Set.range (eval (π C (ord I · < o))))) :
l.val.Tail.isGood (C' C ho)
|
case h
I : Type u
C : Set (I → Bool)
inst✝¹ : LinearOrder I
inst✝ : WellFoundedLT I
o : Ordinal.{u}
hC : IsClosed C
hsC : contained C (Order.succ o)
ho : o < Ordinal.type fun x1 x2 => x1 < x2
l : ↑(MaxProducts C ho)
h₁ : ⊤ ≤ Submodule.span ℤ (Set.range (eval (π C fun x => ord I x < o)))
this : Inhabited I
m : Products I →₀ ℤ
hmmem : ↑m.support ⊆ {m | m < (↑l).Tail}
hmsum : (m.sum fun i a => a • Products.eval (C' C ho) i) = (Linear_CC' C hsC ho) (Products.eval C ↑l)
q : Products I
hq : q ∈ m.support
⊢ m q • Products.eval (C' C ho) q = m q • (Linear_CC' C hsC ho) (List.map (e C) (term I ho :: ↑q)).prod
|
have hx'' : q < l.val.Tail := hmmem hq
|
case h
I : Type u
C : Set (I → Bool)
inst✝¹ : LinearOrder I
inst✝ : WellFoundedLT I
o : Ordinal.{u}
hC : IsClosed C
hsC : contained C (Order.succ o)
ho : o < Ordinal.type fun x1 x2 => x1 < x2
l : ↑(MaxProducts C ho)
h₁ : ⊤ ≤ Submodule.span ℤ (Set.range (eval (π C fun x => ord I x < o)))
this : Inhabited I
m : Products I →₀ ℤ
hmmem : ↑m.support ⊆ {m | m < (↑l).Tail}
hmsum : (m.sum fun i a => a • Products.eval (C' C ho) i) = (Linear_CC' C hsC ho) (Products.eval C ↑l)
q : Products I
hq : q ∈ m.support
hx'' : q < (↑l).Tail
⊢ m q • Products.eval (C' C ho) q = m q • (Linear_CC' C hsC ho) (List.map (e C) (term I ho :: ↑q)).prod
|
71fdaf74452759e3
|
ENNReal.exists_mem_Ico_zpow
|
Mathlib/Data/ENNReal/Inv.lean
|
theorem exists_mem_Ico_zpow {x y : ℝ≥0∞} (hx : x ≠ 0) (h'x : x ≠ ∞) (hy : 1 < y) (h'y : y ≠ ⊤) :
∃ n : ℤ, x ∈ Ico (y ^ n) (y ^ (n + 1))
|
x : ℝ≥0
hx : ↑x ≠ 0
y : ℝ≥0
hy : 1 < ↑y
⊢ y ≠ 0
|
simpa only [Ne, coe_eq_zero] using (zero_lt_one.trans hy).ne'
|
no goals
|
cffdb470f1d2f180
|
Asymptotics.IsEquivalent.smul
|
Mathlib/Analysis/Asymptotics/AsymptoticEquivalent.lean
|
theorem IsEquivalent.smul {α E 𝕜 : Type*} [NormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E]
{a b : α → 𝕜} {u v : α → E} {l : Filter α} (hab : a ~[l] b) (huv : u ~[l] v) :
(fun x ↦ a x • u x) ~[l] fun x ↦ b x • v x
|
α : Type u_1
E : Type u_2
𝕜 : Type u_3
inst✝² : NormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
a b : α → 𝕜
u v : α → E
l : Filter α
hab : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x : α) in l, ‖(a - b) x‖ ≤ c * ‖b x‖
φ : α → 𝕜
habφ : a =ᶠ[l] φ * b
this : ((fun x => a x • u x) - fun x => b x • v x) =ᶠ[l] fun x => b x • (φ x • u x - v x)
C : ℝ
hC : C > 0
hCuv : ∀ᶠ (x : α) in l, ‖u x‖ ≤ C * ‖v x‖
c : ℝ
hc : 0 < c
hφ : ∀ᶠ (x : α) in l, ‖φ x - 1‖ < c / 2 / C
huv : ∀ᶠ (x : α) in l, ‖(u - v) x‖ ≤ c / 2 * ‖v x‖
x : α
hCuvx : ‖u x‖ ≤ C * ‖v x‖
huvx : ‖(u - v) x‖ ≤ c / 2 * ‖v x‖
hφx : ‖φ x - 1‖ < c / 2 / C
⊢ ‖φ x - 1‖ * ‖u x‖ ≤ c / 2 / C * ‖u x‖
|
gcongr
|
no goals
|
1fc300a902bb7f39
|
Int.eq_succ_of_zero_lt
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Int/Order.lean
|
theorem eq_succ_of_zero_lt {a : Int} (h : 0 < a) : ∃ n : Nat, a = n.succ :=
let ⟨n, (h : ↑(1 + n) = a)⟩ := le.dest h
⟨n, by rw [Nat.add_comm] at h; exact h.symm⟩
|
a : Int
h✝ : 0 < a
n : Nat
h : ↑(n + 1) = a
⊢ a = ↑n.succ
|
exact h.symm
|
no goals
|
fe9a752e18ea3ab6
|
IsLindelof.countable_of_discrete
|
Mathlib/Topology/Compactness/Lindelof.lean
|
theorem IsLindelof.countable_of_discrete [DiscreteTopology X] (hs : IsLindelof s) :
s.Countable
|
X : Type u
inst✝¹ : TopologicalSpace X
s : Set X
inst✝ : DiscreteTopology X
hs : IsLindelof s
⊢ ∀ (x : X), {x} ∈ 𝓝 x
|
simp [nhds_discrete]
|
no goals
|
1f0daf8fd68879c4
|
ite_iff_ite
|
Mathlib/.lake/packages/lean4/src/lean/Init/PropLemmas.lean
|
theorem ite_iff_ite (p : Prop) {h h' : Decidable p} (x y : Prop) :
(@ite _ p h x y ↔ @ite _ p h' x y) ↔ True
|
p : Prop
h h' : Decidable p
x y : Prop
⊢ ((if p then x else y) ↔ if p then x else y) ↔ True
|
rw [iff_true]
|
p : Prop
h h' : Decidable p
x y : Prop
⊢ (if p then x else y) ↔ if p then x else y
|
328ab7a364efd5f6
|
Filter.EventuallyEq.of_mulIndicator
|
Mathlib/Order/Filter/IndicatorFunction.lean
|
theorem Filter.EventuallyEq.of_mulIndicator [One β] {l : Filter α} {f : α → β}
(hf : ∀ᶠ x in l, f x ≠ 1) {s t : Set α} (h : s.mulIndicator f =ᶠ[l] t.mulIndicator f) :
s =ᶠ[l] t
|
α : Type u_1
β : Type u_2
inst✝ : One β
l : Filter α
f : α → β
hf : ∀ᶠ (x : α) in l, f x ≠ 1
s t : Set α
h : s.mulIndicator f =ᶠ[l] t.mulIndicator f
⊢ s =ᶠ[l] t
|
have : ∀ {s : Set α}, Function.mulSupport (s.mulIndicator f) =ᶠ[l] s := fun {s} ↦ by
rw [mulSupport_mulIndicator]
exact (hf.mono fun x hx ↦ and_iff_left hx).set_eq
|
α : Type u_1
β : Type u_2
inst✝ : One β
l : Filter α
f : α → β
hf : ∀ᶠ (x : α) in l, f x ≠ 1
s t : Set α
h : s.mulIndicator f =ᶠ[l] t.mulIndicator f
this : ∀ {s : Set α}, Function.mulSupport (s.mulIndicator f) =ᶠ[l] s
⊢ s =ᶠ[l] t
|
2dada3411376bc4f
|
CategoryTheory.Localization.Construction.morphismProperty_is_top
|
Mathlib/CategoryTheory/Localization/Construction.lean
|
theorem morphismProperty_is_top (P : MorphismProperty W.Localization)
[P.IsStableUnderComposition] (hP₁ : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), P (W.Q.map f))
(hP₂ : ∀ ⦃X Y : C⦄ (w : X ⟶ Y) (hw : W w), P (wInv w hw)) :
P = ⊤
|
case h.h.h.a.mpr.cons
C : Type uC
inst✝¹ : Category.{uC', uC} C
W : MorphismProperty C
P : MorphismProperty W.Localization
inst✝ : P.IsStableUnderComposition
hP₁ : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), P (W.Q.map f)
hP₂ : ∀ ⦃X Y : C⦄ (w : X ⟶ Y) (hw : W w), P (wInv w hw)
X Y : W.Localization
f : X ⟶ Y
a✝ : ⊤ f
G : Paths (LocQuiver W) ⥤ W.Localization := Quotient.functor (relations W)
this : G.Full
X₁ X₂✝ X₂ X₃ : Paths (LocQuiver W)
p : Quiver.Path X₁ X₂
g : X₂ ⟶ X₃
hp : P (G.map p)
p' : X₁ ⟶ X₂ := p
⊢ P (G.map p' ≫ G.map g.toPath)
|
refine P.comp_mem _ _ hp ?_
|
case h.h.h.a.mpr.cons
C : Type uC
inst✝¹ : Category.{uC', uC} C
W : MorphismProperty C
P : MorphismProperty W.Localization
inst✝ : P.IsStableUnderComposition
hP₁ : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), P (W.Q.map f)
hP₂ : ∀ ⦃X Y : C⦄ (w : X ⟶ Y) (hw : W w), P (wInv w hw)
X Y : W.Localization
f : X ⟶ Y
a✝ : ⊤ f
G : Paths (LocQuiver W) ⥤ W.Localization := Quotient.functor (relations W)
this : G.Full
X₁ X₂✝ X₂ X₃ : Paths (LocQuiver W)
p : Quiver.Path X₁ X₂
g : X₂ ⟶ X₃
hp : P (G.map p)
p' : X₁ ⟶ X₂ := p
⊢ P (G.map g.toPath)
|
bfd1f03091fbe383
|
Ordinal.bsup_succ_le_blsub
|
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
theorem bsup_succ_le_blsub {o : Ordinal.{u}} (f : ∀ a < o, Ordinal.{max u v}) :
succ (bsup.{_, v} o f) ≤ blsub.{_, v} o f ↔ ∃ i hi, f i hi = bsup.{_, v} o f
|
case refine_1
o : Ordinal.{u}
f : (a : Ordinal.{u}) → a < o → Ordinal.{max u v}
h : succ (o.bsup f) ≤ o.blsub f
⊢ ∃ i, ∃ (hi : i < o), f i hi = o.bsup f
|
by_contra! hf
|
case refine_1
o : Ordinal.{u}
f : (a : Ordinal.{u}) → a < o → Ordinal.{max u v}
h : succ (o.bsup f) ≤ o.blsub f
hf : ∀ (i : Ordinal.{u}) (hi : i < o), f i hi ≠ o.bsup f
⊢ False
|
a42b5f68d8abe58d
|
ContinuousOn.if
|
Mathlib/Topology/ContinuousOn.lean
|
theorem ContinuousOn.if {p : α → Prop} [∀ a, Decidable (p a)]
(hp : ∀ a ∈ s ∩ frontier { a | p a }, f a = g a)
(hf : ContinuousOn f <| s ∩ closure { a | p a })
(hg : ContinuousOn g <| s ∩ closure { a | ¬p a }) :
ContinuousOn (fun a => if p a then f a else g a) s
|
case hpg.intro.intro
α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
f g : α → β
s : Set α
p : α → Prop
inst✝ : (a : α) → Decidable (p a)
hp : ∀ a ∈ s ∩ frontier {a | p a}, f a = g a
hf : ContinuousOn f (s ∩ closure {a | p a})
hg : ContinuousOn g (s ∩ closure {a | ¬p a})
a : α
has : a ∈ s
left✝ : a ∈ closure {a | p a}
ha : a ∉ interior {a | p a}
⊢ Tendsto g (𝓝[s ∩ closure {a | ¬p a}] a) (𝓝 (g a))
|
rw [← mem_compl_iff, ← closure_compl] at ha
|
case hpg.intro.intro
α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
f g : α → β
s : Set α
p : α → Prop
inst✝ : (a : α) → Decidable (p a)
hp : ∀ a ∈ s ∩ frontier {a | p a}, f a = g a
hf : ContinuousOn f (s ∩ closure {a | p a})
hg : ContinuousOn g (s ∩ closure {a | ¬p a})
a : α
has : a ∈ s
left✝ : a ∈ closure {a | p a}
ha : a ∈ closure {a | p a}ᶜ
⊢ Tendsto g (𝓝[s ∩ closure {a | ¬p a}] a) (𝓝 (g a))
|
342fb5057472ed38
|
Submodule.one_le_finrank_iff
|
Mathlib/LinearAlgebra/Dimension/Finite.lean
|
@[simp]
lemma Submodule.one_le_finrank_iff [StrongRankCondition R] [NoZeroSMulDivisors R M]
{S : Submodule R M} [Module.Finite R S] :
1 ≤ finrank R S ↔ S ≠ ⊥
|
R : Type u
M : Type v
inst✝⁵ : Ring R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : StrongRankCondition R
inst✝¹ : NoZeroSMulDivisors R M
S : Submodule R M
inst✝ : Module.Finite R ↥S
⊢ 1 ≤ finrank R ↥S ↔ S ≠ ⊥
|
simp [← not_iff_not]
|
no goals
|
88f93c598a8c1a70
|
RingHom.locally_iff_isLocalization
|
Mathlib/RingTheory/RingHom/Locally.lean
|
/-- In the definition of `Locally` we may replace `Localization.Away` with an arbitrary
algebra satisfying `IsLocalization.Away`. -/
lemma locally_iff_isLocalization (hP : RespectsIso P) (f : R →+* S) :
Locally P f ↔ ∃ (s : Finset S) (_ : Ideal.span (s : Set S) = ⊤),
∀ t ∈ s, ∀ (Sₜ : Type u) [CommRing Sₜ] [Algebra S Sₜ] [IsLocalization.Away t Sₜ],
P ((algebraMap S Sₜ).comp f)
|
case refine_1
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
R S : Type u
inst✝¹ : CommRing R
inst✝ : CommRing S
hP : RespectsIso fun {R S} [CommRing R] [CommRing S] => P
f : R →+* S
x✝³ : ∃ s, ∃ (_ : Ideal.span ↑s = ⊤), ∀ t ∈ s, P ((algebraMap S (Localization.Away t)).comp f)
s : Finset S
hsone : Ideal.span ↑s = ⊤
hs : ∀ t ∈ s, P ((algebraMap S (Localization.Away t)).comp f)
t : S
ht : t ∈ s
Sₜ : Type u
x✝² : CommRing Sₜ
x✝¹ : Algebra S Sₜ
x✝ : IsLocalization.Away t Sₜ
e : Localization.Away t ≃+* Sₜ := (IsLocalization.algEquiv (Submonoid.powers t) (Localization.Away t) Sₜ).toRingEquiv
this : algebraMap S Sₜ = e.toRingHom.comp (algebraMap S (Localization.Away t))
⊢ P (e.toRingHom.comp ((algebraMap S (Localization.Away t)).comp f))
|
exact hP.left _ _ (hs t ht)
|
no goals
|
48f562d83319536b
|
isProperMap_iff_tendsto_cocompact
|
Mathlib/Topology/Maps/Proper/CompactlyGenerated.lean
|
/-- Version of `isProperMap_iff_isCompact_preimage` in terms of `cocompact`.
There was an older version of this theorem which was changed to this one to make use
of the `CompactlyGeneratedSpace` typeclass. (since 2024-11-10) -/
lemma isProperMap_iff_tendsto_cocompact :
IsProperMap f ↔ Continuous f ∧ Tendsto f (cocompact X) (cocompact Y)
|
X : Type u_1
Y : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : T2Space Y
inst✝ : CompactlyGeneratedSpace Y
f : X → Y
f_cont : Continuous f
H : ∀ (i : Set Y), IsCompact i → (f ⁻¹' i)ᶜ ∈ cocompact X
K : Set Y
hK : IsCompact K
⊢ IsCompact (f ⁻¹' K)
|
rcases mem_cocompact.mp (H K hK) with ⟨K', hK', hK'y⟩
|
case intro.intro
X : Type u_1
Y : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : T2Space Y
inst✝ : CompactlyGeneratedSpace Y
f : X → Y
f_cont : Continuous f
H : ∀ (i : Set Y), IsCompact i → (f ⁻¹' i)ᶜ ∈ cocompact X
K : Set Y
hK : IsCompact K
K' : Set X
hK' : IsCompact K'
hK'y : K'ᶜ ⊆ (f ⁻¹' K)ᶜ
⊢ IsCompact (f ⁻¹' K)
|
695fdbdec5ba9365
|
measurableSet_bddAbove_range
|
Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean
|
lemma measurableSet_bddAbove_range {ι} [Countable ι] {f : ι → δ → α} (hf : ∀ i, Measurable (f i)) :
MeasurableSet {b | BddAbove (range (fun i ↦ f i b))}
|
case inr
α : Type u_1
δ : Type u_4
inst✝⁵ : TopologicalSpace α
mα : MeasurableSpace α
inst✝⁴ : BorelSpace α
mδ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_5
inst✝ : Countable ι
f : ι → δ → α
hf : ∀ (i : ι), Measurable (f i)
hα : Nonempty α
A : ∀ (i : ι) (c : α), MeasurableSet {x | f i x ≤ c}
B : ∀ (c : α), MeasurableSet {x | ∀ (i : ι), f i x ≤ c}
⊢ MeasurableSet {b | BddAbove (range fun i => f i b)}
|
obtain ⟨u, hu⟩ : ∃ (u : ℕ → α), Tendsto u atTop atTop := exists_seq_tendsto (atTop : Filter α)
|
case inr.intro
α : Type u_1
δ : Type u_4
inst✝⁵ : TopologicalSpace α
mα : MeasurableSpace α
inst✝⁴ : BorelSpace α
mδ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_5
inst✝ : Countable ι
f : ι → δ → α
hf : ∀ (i : ι), Measurable (f i)
hα : Nonempty α
A : ∀ (i : ι) (c : α), MeasurableSet {x | f i x ≤ c}
B : ∀ (c : α), MeasurableSet {x | ∀ (i : ι), f i x ≤ c}
u : ℕ → α
hu : Tendsto u atTop atTop
⊢ MeasurableSet {b | BddAbove (range fun i => f i b)}
|
2ea9408e1624118d
|
BitVec.append_zero_width
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean
|
theorem append_zero_width (x : BitVec w) (y : BitVec 0) : x ++ y = x
|
case pred
w : Nat
x : BitVec w
y : BitVec 0
i✝ : Nat
a✝ : i✝ < w + 0
⊢ (if i✝ < 0 then y.getLsbD i✝ else x.getLsbD (i✝ - 0)) = x.getLsbD i✝
|
simp
|
no goals
|
f9b160d56cb6e16e
|
CategoryTheory.SingleFunctors.shiftIso_zero_hom_app
|
Mathlib/CategoryTheory/Shift/SingleFunctors.lean
|
@[simp]
lemma shiftIso_zero_hom_app (a : A) (X : C) :
(F.shiftIso 0 a a (zero_add a)).hom.app X = (shiftFunctorZero D A).hom.app _
|
C : Type u_1
D : Type u_2
inst✝³ : Category.{u_7, u_1} C
inst✝² : Category.{u_6, u_2} D
A : Type u_5
inst✝¹ : AddMonoid A
inst✝ : HasShift D A
F : SingleFunctors C D A
a : A
X : C
⊢ (F.shiftIso 0 a a ⋯).hom.app X = (shiftFunctorZero D A).hom.app ((F.functor a).obj X)
|
rw [shiftIso_zero]
|
C : Type u_1
D : Type u_2
inst✝³ : Category.{u_7, u_1} C
inst✝² : Category.{u_6, u_2} D
A : Type u_5
inst✝¹ : AddMonoid A
inst✝ : HasShift D A
F : SingleFunctors C D A
a : A
X : C
⊢ (isoWhiskerLeft (F.functor a) (shiftFunctorZero D A)).hom.app X = (shiftFunctorZero D A).hom.app ((F.functor a).obj X)
|
904568f0628d1381
|
MeasureTheory.Martingale.condExp_stopping_time_ae_eq_restrict_eq_const_of_le_const
|
Mathlib/Probability/Martingale/OptionalSampling.lean
|
theorem condExp_stopping_time_ae_eq_restrict_eq_const_of_le_const (h : Martingale f ℱ μ)
(hτ : IsStoppingTime ℱ τ) (hτ_le : ∀ x, τ x ≤ n)
[SigmaFinite (μ.trim (hτ.measurableSpace_le_of_le hτ_le))] (i : ι) :
μ[f n|hτ.measurableSpace] =ᵐ[μ.restrict {x | τ x = i}] f i
|
case neg.h
Ω : Type u_1
E : Type u_2
m : MeasurableSpace Ω
μ : Measure Ω
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : CompleteSpace E
ι : Type u_3
inst✝⁵ : LinearOrder ι
inst✝⁴ : TopologicalSpace ι
inst✝³ : OrderTopology ι
inst✝² : FirstCountableTopology ι
ℱ : Filtration ι m
inst✝¹ : SigmaFiniteFiltration μ ℱ
τ : Ω → ι
f : ι → Ω → E
n : ι
h : Martingale f ℱ μ
hτ : IsStoppingTime ℱ τ
hτ_le : ∀ (x : Ω), τ x ≤ n
inst✝ : SigmaFinite (μ.trim ⋯)
i : ι
hin : ¬i ≤ n
x : Ω
⊢ x ∈ {x | τ x = i} ↔ x ∈ ∅
|
simp only [Set.mem_setOf_eq, Set.mem_empty_iff_false, iff_false]
|
case neg.h
Ω : Type u_1
E : Type u_2
m : MeasurableSpace Ω
μ : Measure Ω
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : CompleteSpace E
ι : Type u_3
inst✝⁵ : LinearOrder ι
inst✝⁴ : TopologicalSpace ι
inst✝³ : OrderTopology ι
inst✝² : FirstCountableTopology ι
ℱ : Filtration ι m
inst✝¹ : SigmaFiniteFiltration μ ℱ
τ : Ω → ι
f : ι → Ω → E
n : ι
h : Martingale f ℱ μ
hτ : IsStoppingTime ℱ τ
hτ_le : ∀ (x : Ω), τ x ≤ n
inst✝ : SigmaFinite (μ.trim ⋯)
i : ι
hin : ¬i ≤ n
x : Ω
⊢ ¬τ x = i
|
f56f2f7145da6813
|
Int.lt_zpow_succ_log_self
|
Mathlib/Data/Int/Log.lean
|
theorem lt_zpow_succ_log_self {b : ℕ} (hb : 1 < b) (r : R) : r < (b : R) ^ (log b r + 1)
|
case inr.inl
R : Type u_1
inst✝¹ : LinearOrderedSemifield R
inst✝ : FloorSemiring R
b : ℕ
hb : 1 < b
r : R
hr : 0 < r
hr1 : 1 ≤ r
⊢ r < ↑b ^ (↑(Nat.log b ⌊r⌋₊) + 1)
|
rw [Int.ofNat_add_one_out, zpow_natCast, ← Nat.cast_pow]
|
case inr.inl
R : Type u_1
inst✝¹ : LinearOrderedSemifield R
inst✝ : FloorSemiring R
b : ℕ
hb : 1 < b
r : R
hr : 0 < r
hr1 : 1 ≤ r
⊢ r < ↑(b ^ (Nat.log b ⌊r⌋₊).succ)
|
d923e983e06cf99e
|
MeasureTheory.Submartingale.neg
|
Mathlib/Probability/Martingale/Basic.lean
|
theorem neg [Preorder E] [AddLeftMono E] (hf : Submartingale f ℱ μ) :
Supermartingale (-f) ℱ μ
|
case h
Ω : Type u_1
E : Type u_2
ι : Type u_3
inst✝⁵ : Preorder ι
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : CompleteSpace E
f : ι → Ω → E
ℱ : Filtration ι m0
inst✝¹ : Preorder E
inst✝ : AddLeftMono E
hf : Submartingale f ℱ μ
i j : ι
hij : i ≤ j
a✝¹ : Ω
a✝ : f i a✝¹ ≤ (μ[f j|↑ℱ i]) a✝¹
⊢ (-μ[f j|↑ℱ i]) a✝¹ ≤ (-f) i a✝¹
|
simpa
|
no goals
|
f0607f80a235e382
|
SimpleGraph.Walk.not_nil_of_isCycle_cons
|
Mathlib/Combinatorics/SimpleGraph/Path.lean
|
lemma not_nil_of_isCycle_cons {p : G.Walk u v} {h : G.Adj v u} (hc : (Walk.cons h p).IsCycle) :
¬ p.Nil
|
V : Type u
G : SimpleGraph V
u v : V
p : G.Walk u v
h : G.Adj v u
hc : (cons h p).IsCycle
this : 3 ≤ p.length + 1
⊢ 0 < p.length
|
omega
|
no goals
|
74acab2b384981d7
|
List.eraseIdx_insertIdx
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Nat/InsertIdx.lean
|
theorem eraseIdx_insertIdx (n : Nat) (l : List α) : (l.insertIdx n a).eraseIdx n = l
|
α : Type u
a : α
n : Nat
l : List α
⊢ (insertIdx n a l).eraseIdx n = l
|
rw [eraseIdx_eq_modifyTailIdx, insertIdx, modifyTailIdx_modifyTailIdx_self]
|
α : Type u
a : α
n : Nat
l : List α
⊢ modifyTailIdx (tail ∘ cons a) n l = l
|
b0bfb81c607a14dd
|
exists_vector_succ
|
Mathlib/Data/Vector3.lean
|
theorem exists_vector_succ (f : Vector3 α (succ n) → Prop) : Exists f ↔ ∃ x v, f (x :: v) :=
⟨fun ⟨v, fv⟩ => ⟨_, _, by rw [cons_head_tail v]; exact fv⟩, fun ⟨_, _, fxv⟩ => ⟨_, fxv⟩⟩
|
α : Type u_1
n : ℕ
f : Vector3 α n.succ → Prop
x✝ : Exists f
v : Vector3 α n.succ
fv : f v
⊢ f (?m.22125 x✝ v fv :: ?m.22126 x✝ v fv)
|
rw [cons_head_tail v]
|
α : Type u_1
n : ℕ
f : Vector3 α n.succ → Prop
x✝ : Exists f
v : Vector3 α n.succ
fv : f v
⊢ f v
|
25a8351d10864f6e
|
ENNReal.biSup_add'
|
Mathlib/Data/ENNReal/Inv.lean
|
lemma biSup_add' {p : ι → Prop} (h : ∃ i, p i) (f : ι → ℝ≥0∞) :
(⨆ i, ⨆ _ : p i, f i) + a = ⨆ i, ⨆ _ : p i, f i + a
|
ι : Sort u_1
a : ℝ≥0∞
p : ι → Prop
h : ∃ i, p i
f : ι → ℝ≥0∞
⊢ (⨆ i, ⨆ (_ : p i), f i) + a = ⨆ i, ⨆ (_ : p i), f i + a
|
simp only [add_comm, add_biSup' h]
|
no goals
|
1898be706f93d49b
|
SetTheory.PGame.Impartial.forall_leftMoves_fuzzy_iff_equiv_zero
|
Mathlib/SetTheory/Game/Impartial.lean
|
theorem forall_leftMoves_fuzzy_iff_equiv_zero : (∀ i, G.moveLeft i ‖ 0) ↔ G ≈ 0
|
case refine_2
G : PGame
inst✝ : G.Impartial
hp : G ≈ 0
i : G.LeftMoves
⊢ G.moveLeft i ‖ 0
|
rw [fuzzy_zero_iff_lf]
|
case refine_2
G : PGame
inst✝ : G.Impartial
hp : G ≈ 0
i : G.LeftMoves
⊢ G.moveLeft i ⧏ 0
|
fc4280f0775ba6f7
|
ProbabilityTheory.condExp_eq_zero_or_one_of_condIndepSet_self
|
Mathlib/Probability/Independence/ZeroOne.lean
|
theorem condExp_eq_zero_or_one_of_condIndepSet_self
[StandardBorelSpace Ω]
(hm : m ≤ m0) [hμ : IsFiniteMeasure μ] {t : Set Ω} (ht : MeasurableSet t)
(h_indep : CondIndepSet m hm t t μ) :
∀ᵐ ω ∂μ, (μ⟦t | m⟧) ω = 0 ∨ (μ⟦t | m⟧) ω = 1
|
case h.inl
Ω : Type u_2
m m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝ : StandardBorelSpace Ω
hm : m ≤ m0
hμ : IsFiniteMeasure μ
t : Set Ω
ht : MeasurableSet t
h_indep : CondIndepSet m hm t t μ
this : ∀ (a : Ω), IsFiniteMeasure ((condExpKernel μ m) a)
h✝ : ∀ᵐ (x : Ω) ∂μ, ((condExpKernel μ m) x) t = 0 ∨ ((condExpKernel μ m) x) t = 1
ω : Ω
hω_eq : (((condExpKernel μ m) ω) t).toReal = (μ[t.indicator fun ω => 1|m]) ω
h : ((condExpKernel μ m) ω) t = 0
⊢ (((condExpKernel μ m) ω) t = 0 ∨ ((condExpKernel μ m) ω) t = ⊤) ∨ ((condExpKernel μ m) ω) t = 1
|
exact Or.inl (Or.inl h)
|
no goals
|
aecb617e5da492aa
|
AddMonoidAlgebra.apply_add_of_supDegree_le
|
Mathlib/Algebra/MonoidAlgebra/Degree.lean
|
theorem apply_add_of_supDegree_le (hadd : ∀ a1 a2, D (a1 + a2) = D a1 + D a2)
[AddLeftStrictMono B] [AddRightStrictMono B]
(hD : D.Injective) {ap aq : A} (hp : p.supDegree D ≤ D ap) (hq : q.supDegree D ≤ D aq) :
(p * q) (ap + aq) = p ap * q aq
|
case h₀
R : Type u_1
A : Type u_3
B : Type u_5
inst✝⁶ : Semiring R
inst✝⁵ : SemilatticeSup B
inst✝⁴ : OrderBot B
D : A → B
inst✝³ : AddZeroClass A
p q : R[A]
inst✝² : Add B
hadd : ∀ (a1 a2 : A), D (a1 + a2) = D a1 + D a2
inst✝¹ : AddLeftStrictMono B
inst✝ : AddRightStrictMono B
hD : Function.Injective D
ap aq : A
hp : supDegree D p ≤ D ap
hq : supDegree D q ≤ D aq
a : A
ha : a ∈ q.support
hne : a ≠ aq
he : D (ap + a) = D (ap + aq)
⊢ False
|
simp_rw [hadd] at he
|
case h₀
R : Type u_1
A : Type u_3
B : Type u_5
inst✝⁶ : Semiring R
inst✝⁵ : SemilatticeSup B
inst✝⁴ : OrderBot B
D : A → B
inst✝³ : AddZeroClass A
p q : R[A]
inst✝² : Add B
hadd : ∀ (a1 a2 : A), D (a1 + a2) = D a1 + D a2
inst✝¹ : AddLeftStrictMono B
inst✝ : AddRightStrictMono B
hD : Function.Injective D
ap aq : A
hp : supDegree D p ≤ D ap
hq : supDegree D q ≤ D aq
a : A
ha : a ∈ q.support
hne : a ≠ aq
he : D ap + D a = D ap + D aq
⊢ False
|
dca14f257dfac70d
|
LinearMap.iterateMapComap_eq_succ
|
Mathlib/Algebra/Module/Submodule/IterateMapComap.lean
|
theorem iterateMapComap_eq_succ (K : Submodule R N)
(m : ℕ) (heq : f.iterateMapComap i m K = f.iterateMapComap i (m + 1) K)
(hf : Surjective f) (hi : Injective i) (n : ℕ) :
f.iterateMapComap i n K = f.iterateMapComap i (n + 1) K
|
case zero.succ
R : Type u_1
N : Type u_2
M : Type u_3
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid N
inst✝² : Module R N
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
f i : N →ₗ[R] M
K : Submodule R N
hf : Surjective ⇑f
hi : Injective ⇑i
heq : f.iterateMapComap i 0 K ≠ f.iterateMapComap i (0 + 1) K
m : ℕ
ih : f.iterateMapComap i m K ≠ f.iterateMapComap i (m + 1) K
⊢ ((fun K => comap f (map i K)) ∘ (fun K => comap f (map i K))^[m]) K ≠
((fun K => comap f (map i K)) ∘ (fun K => comap f (map i K))^[m + 1]) K
|
exact fun H ↦ ih (map_injective_of_injective hi (comap_injective_of_surjective hf H))
|
no goals
|
aefaed022eb94959
|
coe_setBasisOfLinearIndependentOfCardEqFinrank
|
Mathlib/LinearAlgebra/FiniteDimensional.lean
|
theorem coe_setBasisOfLinearIndependentOfCardEqFinrank {s : Set V} [Nonempty s] [Fintype s]
(lin_ind : LinearIndependent K ((↑) : s → V)) (card_eq : s.toFinset.card = finrank K V) :
⇑(setBasisOfLinearIndependentOfCardEqFinrank lin_ind card_eq) = ((↑) : s → V)
|
K : Type u
V : Type v
inst✝⁴ : DivisionRing K
inst✝³ : AddCommGroup V
inst✝² : Module K V
s : Set V
inst✝¹ : Nonempty ↑s
inst✝ : Fintype ↑s
lin_ind : LinearIndependent K Subtype.val
card_eq : s.toFinset.card = finrank K V
⊢ ⇑(basisOfLinearIndependentOfCardEqFinrank lin_ind ⋯) = Subtype.val
|
exact Basis.coe_mk _ _
|
no goals
|
c8ff7cf4a9065a35
|
frontier_Ioo
|
Mathlib/Topology/Order/DenselyOrdered.lean
|
theorem frontier_Ioo {a b : α} (h : a < b) : frontier (Ioo a b) = {a, b}
|
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a b : α
h : a < b
⊢ frontier (Ioo a b) = {a, b}
|
rw [frontier, closure_Ioo h.ne, interior_Ioo, Icc_diff_Ioo_same h.le]
|
no goals
|
d4762eec44141e63
|
Int.lt_of_add_lt_add_left
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Int/Order.lean
|
theorem lt_of_add_lt_add_left {a b c : Int} (h : a + b < a + c) : b < c
|
a b c : Int
h : a + b < a + c
⊢ b < c
|
have : -a + (a + b) < -a + (a + c) := Int.add_lt_add_left h _
|
a b c : Int
h : a + b < a + c
this : -a + (a + b) < -a + (a + c)
⊢ b < c
|
a415f2d257d47c22
|
Complex.hasSum_taylorSeries_neg_log
|
Mathlib/Analysis/SpecialFunctions/Complex/LogBounds.lean
|
/-- The series `∑ z^n/n` converges to `-log (1-z)` on the open unit disk. -/
lemma hasSum_taylorSeries_neg_log {z : ℂ} (hz : ‖z‖ < 1) :
HasSum (fun n : ℕ ↦ z ^ n / n) (-log (1 - z))
|
case h.e'_5.h.inr.hb
z : ℂ
hz : ‖z‖ < 1
n : ℕ
hn : n > 0
⊢ ↑n ≠ 0
case h.e'_5.h.inr.hd
z : ℂ
hz : ‖z‖ < 1
n : ℕ
hn : n > 0
⊢ ↑n ≠ 0
|
all_goals {norm_cast; exact hn.ne'}
|
no goals
|
e0a75ab7c4c4ee6b
|
MvPowerSeries.constantCoeff_invOfUnit
|
Mathlib/RingTheory/MvPowerSeries/Inverse.lean
|
theorem constantCoeff_invOfUnit (φ : MvPowerSeries σ R) (u : Rˣ) :
constantCoeff σ R (invOfUnit φ u) = ↑u⁻¹
|
σ : Type u_1
R : Type u_2
inst✝ : Ring R
φ : MvPowerSeries σ R
u : Rˣ
⊢ (constantCoeff σ R) (φ.invOfUnit u) = ↑u⁻¹
|
rw [← coeff_zero_eq_constantCoeff_apply, coeff_invOfUnit, if_pos rfl]
|
no goals
|
c9d098c27bb1f219
|
CategoryTheory.InjectiveResolution.toRightDerivedZero_eq
|
Mathlib/CategoryTheory/Abelian/RightDerived.lean
|
lemma InjectiveResolution.toRightDerivedZero_eq
{X : C} (I : InjectiveResolution X) (F : C ⥤ D) [F.Additive] :
F.toRightDerivedZero.app X = I.toRightDerivedZero' F ≫
(CochainComplex.isoHomologyπ₀ _).hom ≫ (I.isoRightDerivedObj F 0).inv
|
C : Type u
inst✝⁵ : Category.{v, u} C
D : Type u_1
inst✝⁴ : Category.{u_2, u_1} D
inst✝³ : Abelian C
inst✝² : HasInjectiveResolutions C
inst✝¹ : Abelian D
X : C
I : InjectiveResolution X
F : C ⥤ D
inst✝ : F.Additive
⊢ F.toRightDerivedZero.app X =
I.toRightDerivedZero' F ≫
(CochainComplex.isoHomologyπ₀ ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj I.cocomplex)).hom ≫
(I.isoRightDerivedObj F 0).inv
|
dsimp [Functor.toRightDerivedZero, isoRightDerivedObj]
|
C : Type u
inst✝⁵ : Category.{v, u} C
D : Type u_1
inst✝⁴ : Category.{u_2, u_1} D
inst✝³ : Abelian C
inst✝² : HasInjectiveResolutions C
inst✝¹ : Abelian D
X : C
I : InjectiveResolution X
F : C ⥤ D
inst✝ : F.Additive
⊢ (injectiveResolution X).toRightDerivedZero' F ≫
((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj (injectiveResolution X).cocomplex).homologyπ 0 ≫
(HomotopyCategory.homologyFunctorFactors D (ComplexShape.up ℕ) 0).inv.app
((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj (injectiveResolution X).cocomplex) =
I.toRightDerivedZero' F ≫
((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj I.cocomplex).homologyπ 0 ≫
(HomotopyCategory.homologyFunctorFactors D (ComplexShape.up ℕ) 0).inv.app
((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj I.cocomplex) ≫
(HomotopyCategory.homologyFunctor D (ComplexShape.up ℕ) 0).map (I.isoRightDerivedToHomotopyCategoryObj F).inv
|
5b33edafb83a6e1a
|
MvPolynomial.IsHomogeneous.C_mul
|
Mathlib/RingTheory/MvPolynomial/Homogeneous.lean
|
lemma C_mul (hφ : φ.IsHomogeneous m) (r : R) :
(C r * φ).IsHomogeneous m
|
σ : Type u_1
R : Type u_3
inst✝ : CommSemiring R
φ : MvPolynomial σ R
m : ℕ
hφ : φ.IsHomogeneous m
r : R
⊢ (C r * φ).IsHomogeneous m
|
simpa only [zero_add] using (isHomogeneous_C _ _).mul hφ
|
no goals
|
fab9c664e294e76f
|
Ordnode.balance_eq_balance'
|
Mathlib/Data/Ordmap/Ordset.lean
|
theorem balance_eq_balance' {l x r} (hl : Balanced l) (hr : Balanced r) (sl : Sized l)
(sr : Sized r) : @balance α l x r = balance' l x r
|
case pos.node.nil
α : Type u_1
x : α
ls : ℕ
lx : α
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
hr : (node rs rl rx rr).Balanced
sr : (node rs rl rx rr).Sized
h : ¬delta * ls < rs
h_1 : delta * rs < ls
lls : ℕ
lll : Ordnode α
llx : α
llr : Ordnode α
hl : (node ls (node lls lll llx llr) lx nil).Balanced
sl : (node ls (node lls lll llx llr) lx nil).Sized
ld : delta ≤ (node lls lll llx llr).size + nil.size
⊢ (node ls (node lls lll llx llr) lx nil).rotateR x (node rs rl rx rr) =
rec nil
(fun size l x_1 r l_ih r_ih =>
rec nil
(fun size_1 l x_2 r l_ih r_ih =>
if size_1 < ratio * size then
node (ls + rs + 1) (node lls lll llx llr) lx (node (size_1 + rs + 1) nil x (node rs rl rx rr))
else
node (ls + rs + 1) (node (size + l.size + 1) (node lls lll llx llr) lx l) x_2
(node (r.size + rs + 1) r x (node rs rl rx rr)))
nil)
(node lls lll llx llr)
|
exact absurd (le_trans ld (balancedSz_zero.1 hl.1)) (by decide)
|
no goals
|
23a52200299a6569
|
StieltjesFunction.measure_singleton
|
Mathlib/MeasureTheory/Measure/Stieltjes.lean
|
theorem measure_singleton (a : ℝ) : f.measure {a} = ofReal (f a - leftLim f a)
|
f : StieltjesFunction
a : ℝ
u : ℕ → ℝ
u_mono : StrictMono u
u_lt_a : ∀ (n : ℕ), u n < a
u_lim : Tendsto u atTop (𝓝 a)
⊢ {a} = ⋂ n, Ioc (u n) a
|
refine Subset.antisymm (fun x hx => by simp [mem_singleton_iff.1 hx, u_lt_a]) fun x hx => ?_
|
f : StieltjesFunction
a : ℝ
u : ℕ → ℝ
u_mono : StrictMono u
u_lt_a : ∀ (n : ℕ), u n < a
u_lim : Tendsto u atTop (𝓝 a)
x : ℝ
hx : x ∈ ⋂ n, Ioc (u n) a
⊢ x ∈ {a}
|
852c2535f58a09d3
|
CategoryTheory.Limits.colimitLimitToLimitColimit_injective
|
Mathlib/CategoryTheory/Limits/FilteredColimitCommutesFiniteLimit.lean
|
theorem colimitLimitToLimitColimit_injective :
Function.Injective (colimitLimitToLimitColimit F)
|
case intro.intro.intro.intro.intro.intro.intro.w
J : Type u₁
K : Type u₂
inst✝⁴ : Category.{v₁, u₁} J
inst✝³ : Category.{v₂, u₂} K
inst✝² : Small.{v, u₂} K
F : J × K ⥤ Type v
inst✝¹ : IsFiltered K
inst✝ : Finite J
val✝ : Fintype J
kx : K
x : limit ((curry.obj (swap K J ⋙ F)).obj kx)
ky : K
y : limit ((curry.obj (swap K J ⋙ F)).obj ky)
h :
∀ (j : J),
∃ k f g,
F.map (𝟙 j, f) (limit.π ((curry.obj (swap K J ⋙ F)).obj kx) j x) =
F.map (𝟙 j, g) (limit.π ((curry.obj (swap K J ⋙ F)).obj ky) j y)
k : J → K := fun j => ⋯.choose
f : (j : J) → kx ⟶ k j := fun j => ⋯.choose
g : (j : J) → ky ⟶ k j := fun j => ⋯.choose
w :
∀ (j : J),
F.map (𝟙 j, f j) (limit.π ((curry.obj (swap K J ⋙ F)).obj kx) j x) =
F.map (𝟙 j, g j) (limit.π ((curry.obj (swap K J ⋙ F)).obj ky) j y)
O : Finset K := Finset.image k Finset.univ ∪ {kx, ky}
kxO : kx ∈ O
kyO : ky ∈ O
kjO : ∀ (j : J), k j ∈ O
H : Finset ((X : K) ×' (Y : K) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y)) :=
Finset.image (fun j => ⟨kx, ⟨k j, ⟨kxO, ⟨⋯, f j⟩⟩⟩⟩) Finset.univ ∪
Finset.image (fun j => ⟨ky, ⟨k j, ⟨kyO, ⟨⋯, g j⟩⟩⟩⟩) Finset.univ
S : K
T : {X : K} → X ∈ O → (X ⟶ S)
W : ∀ {X Y : K} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y}, ⟨X, ⟨Y, ⟨mX, ⟨mY, f⟩⟩⟩⟩ ∈ H → f ≫ T mY = T mX
fH : ∀ (j : J), ⟨kx, ⟨k j, ⟨kxO, ⟨⋯, f j⟩⟩⟩⟩ ∈ H
gH : ∀ (j : J), ⟨ky, ⟨k j, ⟨kyO, ⟨⋯, g j⟩⟩⟩⟩ ∈ H
j : J
⊢ limit.π ((curry.obj (swap K J ⋙ F)).obj S) j ((curry.obj (swap K J ⋙ F) ⋙ lim).map (T kxO) x) =
limit.π ((curry.obj (swap K J ⋙ F)).obj S) j ((curry.obj (swap K J ⋙ F) ⋙ lim).map (T kyO) y)
|
simp only [Functor.comp_map, Limit.map_π_apply, curry_obj_map_app, swap_map]
|
case intro.intro.intro.intro.intro.intro.intro.w
J : Type u₁
K : Type u₂
inst✝⁴ : Category.{v₁, u₁} J
inst✝³ : Category.{v₂, u₂} K
inst✝² : Small.{v, u₂} K
F : J × K ⥤ Type v
inst✝¹ : IsFiltered K
inst✝ : Finite J
val✝ : Fintype J
kx : K
x : limit ((curry.obj (swap K J ⋙ F)).obj kx)
ky : K
y : limit ((curry.obj (swap K J ⋙ F)).obj ky)
h :
∀ (j : J),
∃ k f g,
F.map (𝟙 j, f) (limit.π ((curry.obj (swap K J ⋙ F)).obj kx) j x) =
F.map (𝟙 j, g) (limit.π ((curry.obj (swap K J ⋙ F)).obj ky) j y)
k : J → K := fun j => ⋯.choose
f : (j : J) → kx ⟶ k j := fun j => ⋯.choose
g : (j : J) → ky ⟶ k j := fun j => ⋯.choose
w :
∀ (j : J),
F.map (𝟙 j, f j) (limit.π ((curry.obj (swap K J ⋙ F)).obj kx) j x) =
F.map (𝟙 j, g j) (limit.π ((curry.obj (swap K J ⋙ F)).obj ky) j y)
O : Finset K := Finset.image k Finset.univ ∪ {kx, ky}
kxO : kx ∈ O
kyO : ky ∈ O
kjO : ∀ (j : J), k j ∈ O
H : Finset ((X : K) ×' (Y : K) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y)) :=
Finset.image (fun j => ⟨kx, ⟨k j, ⟨kxO, ⟨⋯, f j⟩⟩⟩⟩) Finset.univ ∪
Finset.image (fun j => ⟨ky, ⟨k j, ⟨kyO, ⟨⋯, g j⟩⟩⟩⟩) Finset.univ
S : K
T : {X : K} → X ∈ O → (X ⟶ S)
W : ∀ {X Y : K} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y}, ⟨X, ⟨Y, ⟨mX, ⟨mY, f⟩⟩⟩⟩ ∈ H → f ≫ T mY = T mX
fH : ∀ (j : J), ⟨kx, ⟨k j, ⟨kxO, ⟨⋯, f j⟩⟩⟩⟩ ∈ H
gH : ∀ (j : J), ⟨ky, ⟨k j, ⟨kyO, ⟨⋯, g j⟩⟩⟩⟩ ∈ H
j : J
⊢ limit.π ((curry.obj (swap K J ⋙ F)).obj S) j (lim.map ((curry.obj (swap K J ⋙ F)).map (T kxO)) x) =
limit.π ((curry.obj (swap K J ⋙ F)).obj S) j (lim.map ((curry.obj (swap K J ⋙ F)).map (T kyO)) y)
|
dd8ec51392784757
|
FirstOrder.Language.BoundedFormula.relabel_sumInl
|
Mathlib/ModelTheory/Syntax.lean
|
theorem relabel_sumInl (φ : L.BoundedFormula α n) :
(φ.relabel Sum.inl : L.BoundedFormula α (0 + n)) = φ.castLE (ge_of_eq (zero_add n))
|
case imp
L : Language
α : Type u'
n n✝ : ℕ
f₁✝ f₂✝ : L.BoundedFormula α n✝
ih1 :
mapTermRel (fun x t => Term.relabel (Sum.map id (natAdd 0)) t) (fun x => id) (fun x => castLE ⋯) f₁✝ = castLE ⋯ f₁✝
ih2 :
mapTermRel (fun x t => Term.relabel (Sum.map id (natAdd 0)) t) (fun x => id) (fun x => castLE ⋯) f₂✝ = castLE ⋯ f₂✝
⊢ mapTermRel (fun x t => Term.relabel (Sum.map id (natAdd 0)) t) (fun x => id) (fun x => castLE ⋯) (f₁✝.imp f₂✝) =
castLE ⋯ (f₁✝.imp f₂✝)
|
simp_all [mapTermRel]
|
no goals
|
958a4bc95e5e8f6e
|
cbiSup_eq_of_not_forall
|
Mathlib/Order/ConditionallyCompleteLattice/Indexed.lean
|
theorem cbiSup_eq_of_not_forall {p : ι → Prop} {f : Subtype p → α} (hp : ¬ (∀ i, p i)) :
⨆ (i) (h : p i), f ⟨i, h⟩ = iSup f ⊔ sSup ∅
|
case intro.intro
α : Type u_1
ι : Sort u_4
inst✝ : ConditionallyCompleteLinearOrder α
p : ι → Prop
f : Subtype p → α
hp : ¬∀ (i : ι), p i
i₀ : ι
hi₀ : ¬p i₀
this : Nonempty ι
c : α
hc : c ∈ upperBounds (range f)
i : ι
⊢ (fun i => if h : p i then f ⟨i, h⟩ else sSup ∅) i ≤ c ⊔ sSup ∅
|
by_cases hi : p i
|
case pos
α : Type u_1
ι : Sort u_4
inst✝ : ConditionallyCompleteLinearOrder α
p : ι → Prop
f : Subtype p → α
hp : ¬∀ (i : ι), p i
i₀ : ι
hi₀ : ¬p i₀
this : Nonempty ι
c : α
hc : c ∈ upperBounds (range f)
i : ι
hi : p i
⊢ (fun i => if h : p i then f ⟨i, h⟩ else sSup ∅) i ≤ c ⊔ sSup ∅
case neg
α : Type u_1
ι : Sort u_4
inst✝ : ConditionallyCompleteLinearOrder α
p : ι → Prop
f : Subtype p → α
hp : ¬∀ (i : ι), p i
i₀ : ι
hi₀ : ¬p i₀
this : Nonempty ι
c : α
hc : c ∈ upperBounds (range f)
i : ι
hi : ¬p i
⊢ (fun i => if h : p i then f ⟨i, h⟩ else sSup ∅) i ≤ c ⊔ sSup ∅
|
f8e2dff97f22af64
|
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